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0704.1633

Hilbert Spaces with Generic Predicates

HILBERT SPACES WITH GENERIC PREDICATES ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES Abstract. We study themodel theory of expansions of Hilbert spaces by generic predicates. We first prove the existence of model companions for generic ex- pansions of Hilbert spaces in the form first of a distance function to a random substructure, then a distance to a random subset. The theory obtained with the random substructure isω-stable, while the one obtained with the distance to a random subset is TP2 and NSOP1. That example is the first continuous struc- ture in that class. 1. Introduction This paper deals with Hilbert spaces expanded with random predicates in the framework of continuous logic as developed in [2]. The model theory of Hilbert spaces is very well understood, see [2, Chapter 15] or [5]. However, we briefly review some of its properties at the end of this section. In this paper we build several new expansions, by various kinds of random predicates (random substructure and the distance to a random subset) of Hilbert spaces, and study them within the framework of continuous logic. While our constructions are not exactlymetric Fraïssé (failing the hereditaryproperty), some of them are indeed amalgamation classes and we study the model theory of their limits. Several papers deal with generic expansions of Hilbert spaces. Ben Yaacov, Usvyatsov and Zadka [3] studied the expansion of a Hilbert space with a generic This workwas partially supported by Colciencias grantMétodos de Estabilidad en Clases No Esta- bles. The second and third author were also sponsored by Catalonia’s Centre de Recerca Matemàtica (Intensive Research Program in Strong Logics) and by the University of Helsinki in 2016 for part of this work. The third author was also partially sponsored by Colciencias (Proy. 1101-05-13605 CT-210-2003). http://arxiv.org/abs/0704.1633v2 2 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES automorphism. The models of this theory are expansions of Hilbert spaces with a unitary map whose spectrum is S1. A model of this theory can be constructed by amalgamating together the collection ofn-dimensional Hilbert spaces with a uni- tary map whose eigenvalues are the n-th roots of unity as n varies in the positive integers. More work on generic automorphisms can be found in [4], where the first author of this paper studied Hilbert spaces expanded with a random group of automorphisms G. There are also several papers about expansions of Hilbert spaces with random subspaces. In [5] the first author and Buechler identified the saturated models of the theory of beautiful pairs of a Hilbert space. An analysis of lovely pairs (the generalization of beautiful pairs (belles paires) to simple theories) in the setting of compact abstract theories is carried out in [1]. In the (very short) second section of this paper we build the beautiful pairs of Hilbert spaces as the model companion of the theory of Hilbert spaces with an orthonormal projection. We provide an axiomatization for this class and we show that the projection operator into the subspace is interdefinable with a predicate for the distance to the subspace. We also prove that the theory of beautiful pairs of Hilbert spaces is ω-stable. Many of the properties of beautiful pairs of Hilbert spaces are known from the literature or folklore, so this section is mostly a compilation of results. In the third section we add a predicate for the distance to a random subset. This construction was inspired by the idea of finding an analogue to the first order generic predicates studied by Chatzidakis and Pillay in [7]. The axiomatization we found for the model companion was inspired in the ideas of [7] together with the following observation: in Hilbert spaces there is a definable function that measures the distance between a point and a model. We prove that the theory of Hilbert spaces with a generic predicate is unstable. We also study a natural notion of independence in a monster model of this theory and prove some of its properties. Several natural independence notions have various good properties, but the theory fails to be simple and even fails to be NTP2. 1.1. Model theory of Hilbert spaces (quick review). HILBERT SPACES WITH GENERIC PREDICATES 3 1.1.1. Hilbert spaces. We follow [2] in our study of the model theory of a real Hilbert space and its expansions. We assume the reader is familiar with the basic concepts of continuous logic as presented in [2]. A Hilbert spaceH can be seen as amulti-sorted structure (Bn(H), 0,+, 〈, 〉, {λr : r ∈ R})0<n<ω, whereBn(H) is the ball of radiusn,+ stands for addition of vectors (defined fromBn(H)×Bn(H) into B2n(H)), 〈, 〉 : Bn(H) × Bn(H) → [−n2, n2] is the inner product, 0 is a constant for the zero vector and λr : Bn(H) → Bn(⌈|r|⌉)H is the multiplication by r ∈ R. We denote by L the language of Hilbert spaces and by T the theory of Hilbert spaces. By a universal domainH of T we mean a Hilbert spaceH which is κ-saturated and κ-strongly homogeneous with respect to types in the language L, where κ is a cardinal larger than 2ℵ0 . Constructing such a structure is straightforward —just consider a Hilbert space with an orthonormal basis of cardinality at least κ. We will assume that the reader is familiar with the metric versions of definable closure and non-dividing. The reader can check [2, 5] for the definitions. 1.1. Notation. Let dcl stand for the definable closure and acl stand for the alge- braic closure in the language L. 1.2. Fact. Let A ⊂ H be small. Then dcl(A) = acl(A) = the smallest Hilbert subspace ofH containing A. Proof. See Lemma 3 in [5, p. 80] � Recall a characterization of non-dividing in pure Hilbert spaces (that will be useful in the more sophisticated constructions in forthcoming sections): 1.3. Proposition. Let B,C ⊂ H be small, let (a1, . . . , an) ∈ Hn and assume that C = dcl(C), soC is a Hilbert subspace ofH. Denote by PC the projection onC. Then tp(a1, . . . , an/C ∪ B) does not divide over C if and only if for all i ≤ n and all b ∈ B, ai − PC(ai) ⊥ b− PC(b). Proof. Proved as Corollary 2 and Lemma 8 of [5, pp. 81–82]. � 4 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES For A,B,C ⊂ H small, we say that A is independent from B over C if for all n ≥ 1 and ā ∈ An, tp(ā/C ∪ B) does not divide over C. Under non-dividing independence, types over sets are stationary. In particular, the independence theorem holds over sets, and we may refer to this property as 3-existence. It is also important to point out that non-dividing is trivial, that is, for all sets B,C and tuples (a1, . . . , an) from H, tp(a1, . . . , an/C ∪ B) does not divide over C if and only if tp(ai/B ∪ C) does not divide over C for i ≤ n. 2. Random subspaces and beautiful pairs We now deal with the easiest situation: a Hilbert space with an orthonormal projection operator onto a subspace. Let Lp = L ∪ {P} where P is a new unary function andwe consider structures of the form (H, P), whereP : H→ H is a pro- jection into a subspace. Note that P : Bn(H) → Bn(H) and that P is determined by its action onB1(H). Recall that projections are bounded linear operators, char- acterized by two properties: (1) P2 = P (2) P∗ = P The second condition means that for any u, v ∈ H, 〈P(u), v〉 = 〈u, P(v)〉. A projection also satisfies, for any u, v ∈ H, ‖P(u) − P(v)‖ ≤ ‖u − v‖. In particular, it is a uniformly continuous map and its modulus of uniform continuity is ∆P(ǫ) = ǫ. We start by showing that the class of Hilbert spaces with projections has the free amalgamation property: 2.1. Lemma. Let (H0, P0) ⊂ (Hi, Pi) where i = 1, 2 and H1 | H2 be (possibly finite dimensional) Hilbert spaces with projections. Then H3 = span{H1, H2} is a Hilbert space and P3(v3) = P1(v1) + P2(v2) is a well defined projection, where v3 = v1 + v2 and v1 ∈ H1, v2 ∈ H2. HILBERT SPACES WITH GENERIC PREDICATES 5 Proof. It is clear that H3 = span{H1 ∪ H2} is a Hilbert space containing H1 and H2. It remains to prove that P3 is a projection map and that it is well defined. We denote byQ0,Q1,Q2 the projections onto the spacesH0,H1 andH2 respectively. Since H0 ⊂ H1, we can write H1 = H0 ⊕ (H1 ∩ H⊥0 ). Similarly H2 = H0 ⊕ (H2 ∩H⊥0 ). Finally, since H1 |⌣H0 H2, H3 = H0 ⊕ (H1 ∩H⊥0 )⊕ (H2 ∩H⊥0 ) Let v3 ∈ H3. Let u0 = Q0(v3), u1 = PH⊥ ∩H1(v3) = Q1(v3) − u0, u2 = ∩H2(v3) = Q2(v3) − u0. Then v3 = u0 + u1 + u2. AsH1∩H2 = H0, we can write v3 in many different ways as a sum of elements inH1 andH2. Given one such expression, v3 = v1+v2, with v1 ∈ H1 and v2 ∈ H2, it is easy to see that P1(v1)+P2(v2) = P0(u0)+P1(u1)+P2(u2), and thus prove that P3 is well defined. Let TP be the theory of Hilbert spaces with a projection. It is axiomatized by the theory of Hilbert spaces together with the axioms (1) and (2) that say that P is a projection. Consider first the finite dimensional models. Given an n-dimensional Hilbert space Hn, there are only n + 1 many pairs (Hn, P), where P is a projection, modulo isomorphism. They are classified by the dimension of P(H), which ranges from 0 to n. In order to characterize the existentially closedmodels of TP, note the following facts: (1) Let (H, P) be existentially closed, and (Hn, Pn) be ann-dimensional Hilbert space with an orthonormal projection with the property that Pn(Hn) = Hn. Then (H, P) ⊕ (Hn, Pn) is an extension of (H, P) with dim([P ⊕ Pn](H ⊕Hn)) ≥ n. Since n can be chosen as big as we want and (H, P) is existentially closed, dim(P(H)) = ∞. 6 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES (2) Let (H, P) be existentially closed, and (Hn, P0) be ann-dimensionalHilbert space with an orthonormal projection such that P0(Hn) = {0}. Then (H, P)⊕ (Hn, P0) is an extension of (H,P) such that dim(([P⊕ P0](H⊕ ⊥) ≥ n. Since n can be chosen as big as we want and (H, P) is existentially closed, dim(P(H)⊥) = ∞. The theory TPω extending T P, stating that Pω is a projection and that there are infinitely many pairwise orthonormal vectors v satisfying Pω(v) = v and also infinitely many pairwise orthonormal vectors u satisfying Pω(u) = 0 gives an axiomatization for the the model companion of TP, which corresponds to the theory of beautiful pairs of Hilbert spaces. We will now study some properties of Let (H, P) |= TPω and for any v ∈ H let dP(v) = ‖v − P(v)‖. Then dP(v) measures the distance between v and the subspace P(H). The distance function dP(x) is definable in (H, P). We will now prove the converse, that is, that we can definably recover P from dP. 2.2. Lemma. Let (H, P) |= TPω. For any v ∈ Hω let dP(v) = ‖v − P(v)‖. Then P(v) ∈ dcl(v) in the structure (H, dP). Proof. Note that P(v) is the unique element x in P(H) satisfying ‖v−x‖ = dP(v). Thus P(v) is the unique realization of the conditionϕ(x) = max{dP(x), |‖v−x‖− dP(v)|} = 0. � 2.3. Proposition. Let (H, P) |= TPω. For any v ∈ Hω let dP(v) = ‖v − P(v)‖. Then the projection function P(x) is definable in the structure (H, dP) Proof. Let (H, P) |= TPω be κ-saturated for κ > ℵ0 and let dP(v) = ‖v − P(v)‖. Since dP is definable in the structure (H, P), the new structure (H, dP) is still κ-saturated. Let GP be the graph of the function P. Then by the previous lemma GP is type-definable in (H, dP) and thus by [2, Proposition 9.24] P is definable in the structure (H, dP). � HILBERT SPACES WITH GENERIC PREDICATES 7 2.4. Notation. We write tp for L-types, tpP for LP-types and q�pP for quantifier free LP-types. We write aclP for the algebraic closure in the language LP. We follow a similar convention for dclP . 2.5. Lemma. TPω has quantifier elimination. Proof. It suffices to show that quantifier free LP-types determine the LP-types. Let (H, P) |= TPω and let ā = (a1, . . . , an), b̄ = (b1, . . . , bn) ∈ Hn. Assume that q�pP(ā) = q�pP(b̄). Then tp(P(a1), . . . , P(an)) = tp(P(b1), . . . , P(bn)) tp(a1 − P(a1), . . . , an − P(an)) = tp(b1 − P(b1), . . . , bn − P(bn)). Let H0 = P(H) and let H1 = H 0 , both are then infinite dimensional Hilbert spaces and H = H0 ⊕ H1. Let f0 ∈ Aut(H0) satisfy f0(P(a1), . . . , P(an)) = (P(b1), . . . , P(bn)) and let f1 ∈ Aut(H1) be such that f1(a1 − P(a1), . . . , an − P(an)) = (b1−P(b1), . . . , bn−P(bn)). Let f be the automorphism ofH induced by by f0 and f1, that is, f = f0 ⊕ f1. Then f ∈ Aut(H, P) and f(a1, . . . , an) = (b1, . . . , bn), so tpP(ā) = tpP(b̄). � Characterization of types: By the previous lemma, the LP-type of an n-tuple ā = (a1, . . . , an) inside a structure (H, P) |= T ω is determined by the type of its projections tp(P(a1), . . . , P(an), a1 − P(a1), . . . , an − P(an)). In particular, we may regard (H, P) as the direct sum of the two independent pure Hilbert spaces (P(H),+, 0, 〈, 〉) and (P(H)⊥,+, 0, 〈, 〉). We may therefore characterize definable and algebraic closure, as follows. 2.6. Proposition. Let (H, P) |= TPω and let A ⊂ H. Then dclP(A) = aclP(A) = dcl(A ∪ P(A)). We leave the proof to the reader. Another consequence of the description of types is: 8 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES 2.7. Proposition. The theory TPω isω-stable. Proof. Let (H, P) |= TPω be separable and let A ⊂ H be countable. Replacing (H, P) for (H, P) ⊕ (H, P) if necessary, we may assume that P(H)∩dclP(A)⊥ is infinite dimensional and that P(H)⊥∩dclP(A)⊥ is infinite dimensional. Thus every Lp-type over A is realized in the structure (H, P) and (S1(A), d) is separable. � 2.8. Proposition. Let (H, P) |= TPω be a κ-saturated domain and let A,B,C ⊂ H be small. Then tpP(A/B ∪C) does not fork over C if and only if tp(A ∪ P(A)/B ∪ P(B) ∪ C ∪ P(C)) does not fork over C ∪ P(C). Again the proof is straightforward. 3. Continuous random predicates We now come to our main theory and to our first set of results. We study the expansion of a Hilbert space with a distance function to a subset of H. Let dN be a new unary predicate and let LN be the language of Hilbert spaces together with dN. We denote the LN structures by (H, dN), where dN : H → [0, 1] and we want to consider the structures where dN is a distance to a subset ofH. Instead of measuring the actual distance to the subset, we truncate the distance at one. We start by characterizing the functions dN corresponding to distances. 3.1. The basic theory T0. We denote by T0 the theory of Hilbert spaces together with the next two axioms (compare with Theorem 9.11 in [2]): (1) supxmin{1− · dN(x), infymax{|dN(x) − ‖x− y‖|, dN(y)}} = 0 (2) supx supy[dN(y) − ‖x − y‖− dN(x)] ≤ 0 We say a point is black if dN(x) = 0 and white if dN(x) = 1. All other points are gray, darker if d(x) is close to zero and whiter if dN(x) is close to one. This terminology follows [10]. From the second axiom we get that dN is uniformly continuous (with modulus of uniform continuity ∆(ǫ) = ǫ). Thus we can apply the tools of continuous model theory to analyze these structures. HILBERT SPACES WITH GENERIC PREDICATES 9 3.1. Lemma. Let (H, d) |= T0 be ℵ0-saturated and let N = {x ∈ H : dN(x) = 0}. Then for any x ∈ H, dN(x) = dist(x,N). Proof. Let v ∈ H and let w ∈ N. Then by the second axiom dN(v) ≤ ‖v − w‖ and thus dN(v) ≤ dist(v,N). Now let v ∈ H. If dN(v) = 1 there is nothing to prove, so we may assume that dN(v) < 1. Consider now the set of statements p(x) given by dN(x) = 0, ‖x− v‖ = dN(v). Claim The type p(x) is approximately satisfiable. Let ε > 0. We want to show that there is a realization of the statements dN(x) ≤ ε, dN(v) ≤ ‖x − v‖ + ε. By the first axiom there is w such that dN(w) ≤ ε and dN(v) ≤ ‖v −w‖+ ε. Since (H, d) is ℵ0-saturated, there is w ∈ N such that ‖v − w‖ = dN(v) as we wanted. � There are several ages that need to be considered. We fix r ∈ [0, 1) and we consider the class Kr of all models of T0 such that dN(0) = r. Note that in all finite dimensional spaces in Kr we have at least a point v with dN(v) = 0. 3.2. Notation. If (Hi, d N) |= T0 for i ∈ {0, 1}, we write (H0, d0N) ⊂ (H1, d1N) if H0 ⊂ H1 and d0N = d1N ↾H0 (for each sort). We will work in Kr. We start with constructing free amalgamations: 3.3. Lemma. Let (H0, d N) ⊂ (Hi, diN) where i = 1, 2 and H1 |⌣H0 H2 be Hilbert spaces with distance functions, all of them in Kr. Let H3 = span{H1, H2} and let d3N(v) = min (PH1(v)) 2 + ‖PH2∩H⊥0 (v)‖ (PH2(v)) 2 + ‖PH1∩H⊥0 (v)‖ Then (Hi, d N) ⊂ (H3, d3N) for i = 1, 2, and (H3, d3N) ∈ Kr. Proof. For arbitrary v ∈ H1, (PH1(v)) 2 + ‖PH2∩H⊥0 (v)‖ 2 = d1N(v). Since (H0, d N) ⊂ (Hi, diN) we also have 10 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES (PH2(v)) 2 + ‖PH1∩H⊥0 (v)‖ (PH0(v)) 2 + ‖PH1∩H⊥0 (v)‖ 2 ≥ d1N(v). Similarly, for any v ∈ H2, d3N(v) = d2N(v). Therefore (H3, d N) ⊃ (Hi, diN) for i ∈ {1, 2}. Now we have to prove that the function d3N that we defined is indeed a distance function. Geometrically, d3N(v) takes the minimum of the distances of v to the selected black subsets of H1 and H2. That is, the random subset of the amalgamation of (H1, d N) and (H2, d N) is the union of the two random subsets. It is easy to check that (H3, d N) |= T0. Since each of (H1, d N), (H2, d N) belongs to Kr, we have d1N(0) = r = d N(0) and thus d N(0) = r. � The classK0 also has the JEP: let (H1, d N), (H2, d N) belong toK0 and assume that H1 ⊥ H2. Let N1 = {v ∈ H1 : d1N(v) = 0} and let N2 = {v ∈ H2 : d2N(v) = 0}. Let H3 = span(H1 ∪H2) and let N3 = N1 ∪N2 ⊂ H3 and finally, let d3N(v) = dist(v,N3). Then (H3, d N) is a witnesses of the JEP in K0. 3.4. Lemma. There is a model (H, dN) |= T0 such that H is a 2n-dimensional Hilbert space and there are orthonormal vectors v1, . . . , vn ∈ H, u1, . . . , un ∈ H such that dN((ui+ vj)/2) = 2/2 for i ≤ j, dN(0) = 0 and dN((ui+ vj)/2) = 0 for i > j. Proof. LetH be a Hilbert space of dimension 2n, and fix some orthonormal basis 〈v1, . . . , vn, u1, . . . , un〉 for H. Let N = {(ui + vj)/2 : i > j} ∪ {0} and let dN(x) = dist(x,N). Then dN(0) = 0 and dN((ui + vj)/2) = 0 for i > j. Since ‖(ui + vj)/2− (uk + vj)/2‖ = 2/2 for i 6= k and ‖(ui + vj)/2− 0‖ = we get that dN(ui + vj) = 2/2 for i ≤ j � A similar construction can be made in order to get the Lemma with dN(0) = r for any r ∈ [0, 1]. In particular, if we fix an infinite cardinal κ and we amalga- mate all possible pairs (H,d) in Kr for dim(H) ≤ κ, the theory of the resulting structure will be unstable. 3.2. The model companion. HILBERT SPACES WITH GENERIC PREDICATES 11 3.2.1. Basic notations. We now provide the axioms of the model companion of T0 ∪ {dN(0) = 0}. Call Td0 the theory of the structure built out of amalgamating all separable Hilbert spaces together with a distance function belonging to the age K0. Infor- mally speaking, Td0 = Th(lim−→(K0)). We show how to axiomatize Td0. The idea for the axiomatization of this part (unlike our third example, in next section) follows the lines of Theorem 2.4 of [7]. There are however important differences in the behavior of algebraic closures and independence, due to the metric character of our examples. Let (M,dN) inK0 be an existentially closed structure and take some extension (M1, dN) ⊃ (M,dN). Let x̄ = (x1, . . . , xn+k) be elements in M1 \M and let z1, . . . , zn+k be their projections on M. Assume that for i ≤ n there are ȳ = (y1, . . . , yn) inM1 \M that satisfy dN(xi) = ‖xi − yi‖ and dN(yi) = 0. Also assume that for i > n, the witnesses for the distances to the black points belong toM, that is, d2N(xi) = ‖xi − zi‖2 + d2N(zi) for i > n. Also, let us assume that all points in x̄, ȳ live in a ball of radius L around the origin. Let ū = (u1, . . . , un) be the projection of ȳ = (y1, . . . , yn) overM. Let ϕ(x̄, ȳ, z̄, ū) be a formula such that ϕ(x̄, ȳ, z̄, ū) = 0 describes the val- ues of the inner products between all the elements of the tuples, that is, it de- termines the (Hilbert space) geometric locus of the tuple (x̄, ȳ, z̄, ū). The state- ment ϕ(x̄, ȳ, z̄, ū) = 0 expresses the position of the potentially new points x̄, ȳ with respect to their projections into a model. Since dN(xi) = ‖xi − yi‖ and dN(yi) = 0, we have ‖xi − yj‖ ≥ ‖xi − yi‖ for j ≤ n, i ≤ n. Also, for i > n, d2N(xi) = ‖xi − zi‖2 + d2N(zi), and get ‖xi − yj‖2 ≥ ‖xi − zi‖2 + d2N(zi) for j ≤ n. Note that as (M1, dN) ⊃ (M,dN), for all z ∈ M, d2N(z) ≤ ‖z − yi‖2 = ‖z − ui‖2 + ‖yi − ui‖2 for i ≤ n. We may also assume that there is a positive real ηϕ such that ‖xi − zi‖ ≥ ηϕ for i ≤ n + k and ‖yi − ui‖ ≥ ηϕ for i ≤ n. 3.2.2. An informal description of the axioms. We want to express that for any pa- rameters z̄, ū in the structure 12 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES if we can find realizations x̄, ȳ of ϕ(x̄, ȳ, z̄, ū) = 0 such that for allw and i ≤ n, d2N(w) ≤ ‖w− ui‖2 + ‖ui − yi‖2, ‖xi − yi‖2 ≤ ‖xi − zi‖2 + d2N(zi) for i ≤ n, ‖xi − yj‖2 ≥ ‖xi − zj‖2 + d2N(zj) for i > n and j ≤ n, then there are tuples x̄ ′, ȳ ′ such that ϕ(x̄ ′, ȳ ′, z̄, ū) = 0, dN(y i) = 0, dN(x ‖x ′i − y ′i‖ for i ≤ n and d2N(xj) = ‖xj − zj‖2 + d2N(zj) for j > n. That is, for any z̄, ū in the structure, if we can find realizations x̄, ȳ of the Hilbert space locus given byϕ, and we prescribe “distances” dN that do not clash with the dN information we already had, in such a way that for i ≤ n, the yi’s are black and are witnesses for the distance to the black set for the xi’s, and for i > n the xi’s do not require new witnesses, then we can actually find arbitrarily close realizations, with the prescribed distances. The only problemwith this idea is that we do not have an implication in contin- uous logic. We replace the expression “p → q” by a sequence of approximations indexed by ε. 3.2.3. The axioms of TN. 3.5. Notation. Let z̄, ū be tuples inM and let x ∈M1. By Pspan(z̄ū)(x) we mean the projection of x in the space spanned by (z̄, ū). For fixed ε ∈ (0, 1), let f : [0, 1] → [0, 1] be a continuous function such that whenever ϕ(t̄) < f(ε) and ϕ(t̄ ′) = 0, then (a): ‖Pspan(z̄ū)(xi) − zi‖ < ε. (b): ‖Pspan(z̄ū)(yi) − ui‖ < ε. (c): |‖ti − tj‖− ‖t ′i − t ′j‖| < ε where t̄ is the concatenation of x̄, ȳ, z̄, ū. Choosing ε small enough, we may assume that (d): ‖xi − Pspan(z̄ū)(xi)‖ ≥ ηϕ/2 for i ≤ n + k. (e): ‖yi − Pspan(z̄ū)(yi)‖ ≥ ηϕ/2 for i ≤ n. Let δ = 2 ε(L+ 2) and consider the following axiom ψϕ,ε (which we write as a positive bounded formula for clarity) where the quantifiers range over a ball of radius L+ 1: HILBERT SPACES WITH GENERIC PREDICATES 13 ∀z̄∀ū ∀x̄∀ȳϕ(x̄, ȳ, z̄, ū) ≥ f(ε)∨∃w i≤n(d N(w) ≥ ‖w−ui‖2+‖yi−ui‖2+ i>n,j≤n(‖xi − yj‖2 ≤ ‖xi − zi‖2 + d2N(zi) + ε2)∨ i,j≤n,j6=i(‖xi−yj‖ ≤ ‖xi−yi‖−ε)∨ i≤n(‖xi−zi‖2+d2N(zi) ≤ ‖xi−yi‖2−ε2) ∨∃x̄∃ȳ (ϕ(x̄, ȳ, z̄, ū) ≤ f(ε)∧ i≤n dN(yi) ≤ δ)∧ i≤n |dN(xi)−‖xi−yi‖| ≤ i>n |d N(xi) − ‖xi − zi‖2 − d2N(zi)| ≤ 4δL Let TN be the theory T0 together with this scheme of axiomsψϕ,ε indexed by all Hilbert space geometric locus formulas ϕ(x̄, ȳ, z̄, ū) = 0 and ε ∈ (0, 1) ∩ Q. The radius of the ball that contains all elements, L, as well as n and k are determined from the configuration of points described by the formula ϕ(x̄, ȳ, z̄, ū) = 0. 3.2.4. Existentially closed models of T0. 3.6. Theorem. Assume that (M,dN) |= T0 is existentially closed. Then (M,dN) |= Proof. Fix ε > 0 and ϕ as above. Let z̄ ∈Mn+k, ū ∈Mn and assume that there are x̄, ȳwithϕ(x̄, ȳ, z̄, ū) < f(ε) and d2N(w) < ‖w−ui‖2+‖yi−ui‖2+ε2 for all w ∈M, ‖xi−yi‖2 < ‖xi−zi‖2+d2N(zi)+ε2 for i ≤ n, ‖xi−yj‖ > ‖xi−yi‖−ε for i, j ≤ n, i 6= j, ‖xi − yj‖2 > ‖xi − zi‖2 + d2N(zj) + ε2 for i > n, j ≤ n. Let ε ′ < ε be such that ϕ(x̄, ȳ, z̄, ū) < f(ε ′) and (f): d2N(w) < ‖w − ui‖2 + ‖yi − ui‖2 + ε ′2 for allw ∈M. (g1): ‖xi − yi‖2 > ‖xi − zi‖2 + dN(zi) + ε ′2 for i ≤ n. (g2): ‖xi − yj‖ > ‖xi − yi‖− ε ′ for i, j ≤ n, i 6= j (h): ‖xi − yj‖2 ≥ ‖xi − zi‖2 + d2N(zi) + ε ′2 for i > n, j ≤ n. We construct an extension (H,dN) ⊃ (M,dN) where the conclusion of the axiom indexed by ε ′ holds. Since (M,dN) is existentially closed and the conclu- sion of the axiom is true for (H,dN) replacing ε for ε ′ < ε, then the conclusion of the axiom indexed by ε will hold for (M,dN). So let H ⊃ M be such that dim(H ∩ M⊥) = ∞. Let a1, . . . , an+k and c1, . . . , cn ∈ H be such that tp(ā, c̄/z̄ū) = tp(x̄, ȳ/z̄ū) and āc̄ | ⌣z̄ū M. We 14 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES can write ai = a i + z i and ci = c i + u i for some z i ∈ M and a ′i, c ′i ∈M⊥. By (d) and (e) ‖a ′i‖ ≥ η/2 for i ≤ n + k and ‖c ′i‖ ≥ η/2 for i ≤ n. Now let ĉi = c i + u i + δ ′c ′i/‖c ′i‖, where δ ′ = 2ε ′(L+ 2). Let the black points in H be the ones fromM plus the points ĉ1, . . . , ĉn. Now we check that the conclusion of the axiom ψϕ,ε ′ holds. (1) ϕ(ā, c̄, z̄, ū) ≤ f(ε ′) since tp(ā, c̄/z̄ū) = tp(x̄, ȳ/z̄ū). (2) Since ‖ci − ĉi‖ ≤ δ ′ and ĉi is black we have dN(ci) ≤ δ ′. (3) We check that the distance from ai to the black set is as prescribed for i ≤ n. dN(ai) ≤ ‖ai − ĉi‖ ≤ ‖ai − ci‖+ δ ′ for i ≤ n. Also, for i 6= j, i, j ≤ n, using (g2)we prove ‖ai− ĉj‖ ≥ ‖ai−cj‖−δ ′ ≥ ‖ai − ci‖ − ε ′ − δ ′ ≥ ‖ai − ci‖− 2δ ′. Finally by (a) ‖ai − PM(ai)‖2 + d2N(PM(ai)) ≥ (‖ai − zi‖− ε ′)2+(dN(zi)− ε ′)2 ≥ ‖ai− zi‖2− 2Lε ′ + ε ′2 + d2N(zi) − 2ε ′ + ε ′2 and by (g1), we get ‖ai − zi‖2 − 2Lε ′ + ε ′2 + d2N(zi) − 2ε ′ + ε ′2 ≥ ‖ai − ci‖2 − 2Lε ′ − 2ε ′ ≥ ‖ai − ci‖2 − 4δ ′2. (4) We check that dN(ai) is as desired for i > n. Clearly ‖aj − ĉi‖ ≥ ‖aj − ci‖ − δ ′, so ‖aj − ĉi‖2 ≥ ‖aj − ci‖2 + δ ′2 − 2δ ′2L and by (h) we get ‖aj − ci‖2 + δ ′2 − 4δ ′L ≥ ‖aj − zj‖2 + d2N(zj) − 4δ ′L − ε ′2 + δ ′2 ≥ ‖aj − zj‖2 + d2N(zj) − 4δ ′L. It remains to show that (M,dN) ⊂ (H,dN), i.e., the function dN onH extends the function dN on M . Since we added the black points in the ball of radius L + 1, we only have to check that for any w ∈ M in the ball of radius L + 2, d2N(w) ≤ ‖w − ĉi‖2 = ‖w − u ′i‖2 + ‖c ′i + δ ′(c ′i/‖c ′i‖)‖2. But by (f) d2N(w) ≤ ‖w − ui‖2 + ‖ci − ui‖2 + ε ′2, so it suffices to show that ‖w− ui‖2 + ‖ci − ui‖2 + ε ′2 ≤ ‖w− u ′i‖2 + ‖c ′i‖2 + 2δ ′‖c ′i‖+ δ ′2 By (a) ‖w− u ′i‖2 ≥ (‖w − ui‖− ε ′)2 and is enough to prove that ‖w− ui‖2 + ‖ci − ui‖2 + ε ′2 ≤ (‖w − ui‖− ε ′)2 + ‖c ′i‖2 + 2δ ′‖c ′i‖+ δ ′2 But (‖w−ui‖−ε ′)2+‖c ′i‖2+2δ ′‖c ′i‖+δ ′2 = ‖w−ui‖2−2ε ′‖w−ui‖+ε ′2+ ‖c ′i‖2+2δ ′‖c ′i‖+δ ′2 and ‖ci−ui‖2 ≤ ‖ci−u ′i‖2+2ε ′‖ci−u ′i‖+ε ′2 = ‖c ′i‖2+ HILBERT SPACES WITH GENERIC PREDICATES 15 2ε ′‖c ′i‖+ε ′2. Thus, after simplifying, we only need to check 2ε ′‖w−ui‖+ε ′2 ≤ δ ′2 which is true since 2ε ′‖w−ui‖+ ε ′2 ≤ 2ε ′(2L+ 2) + ε ′2 ≤ 4ε ′(L+ 2). � 3.7. Theorem. Assume that (M,dN) |= TN. Then (M,dN) is existentially closed. Proof. Let (H,dN) ⊃ (M,dN) and assume that (H,dN) is ℵ0-saturated. Let ψ(x̄, v̄) be a quantifier free LN-formula, where x̄ = (x1, . . . xn+k) and v̄ = (v1, . . . vl). Suppose that there are a1, . . . , an+k ∈ H \ M and e1, . . . el ∈ M such that (H,dN) |= ψ(ā, ē) = 0. After enlarging the formula ψ if necessary, we may assume that ψ(x̄, v̄) = 0 describes the values of dN(xi) for i ≤ n+ k, the values of dN(vj) for j ≤ l and the inner products between those elements. We may as- sume that for i ≤ n there is ρ > 0 such that dN(ai)−d(ai, z) ≥ 2ρ for all z ∈M with dN(z) ≤ ρ. Since (H,dN) isℵ0-saturated, there are c1, . . . cn ∈ H such that dN(ai) = ‖ai− ci‖ and dN(ci) = 0. Then d(ci,M) ≥ ρ. Fix ε > 0, ε < ρ, 1. We may also assume that for i > n, |d2N(ai)−‖ai−PM(ai)‖2−d2N(PM(ai))| ≤ ε/2. Also, assume that all points mentioned so far live in a ball of radius L around the origin. Let b1, . . . , bn+k ∈M be the projections of a1, . . . , an+k ontoM and let d1, . . . , dn ∈ M be the projections of c1, . . . , cn ontoM. Let ϕ(x̄, ȳ, z̄, ū) = 0 be an L-statement that describes the inner products between the elements listed and such that ϕ(ā, c̄, b̄, d̄) = 0. Using the axioms we can find ā ′, c̄ ′ inM such that ϕ(ā ′, c̄ ′, b̄, d̄) ≤ f(ε), dN(c ′i) ≤ δ for i ≤ n, |dN(a ′i) − ‖a ′i − c ′i‖| ≤ δ for i ≤ n and |d2N(ai) − ‖ai − bi‖2 −d2N(bi)| ≤ 4Lδ, where δ = 2ε(L + 2). Since ε > 0 was arbitrary we get (M,dn) |= infx1 . . . infxn+k ψ(x̄, v̄) = 0. � 4. Model theoretic analysis of TN We prove three theorems in this section about the theory TN: • TN is not simple, • TN is not even NTP2! (Of course, this implies the previous, but we will provide the proof of non-simplicity as well.) 16 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES • TN is NSOP1. Therefore, in spite of having a tree property, our theory is still “close to being simple” in the precise sense of not having the SOP1 tree property. These results place TN in a very interesting situation in the stability hierarchy for continuous logic. 4.1. Notation. We write tp for types of elements in the language L and tpN for types of elements in the language LN. Similarly we denote by aclN the algebraic closure in the language LN and by acl the algebraic closure for pure Hilbert spaces. Recall that for a set A, acl(A) = dcl(A), and this corresponds to the closure of the space spanned by A (Fact 1.2). 4.2. Observation. The theory TN does not have elimination of quantifiers. We use the characterization of quantifier elimination given in Theorem 8.4.1 from [9]. Let H1 be a two dimensional Hilbert space, let {u1, u2} be an orthonormal basis for H1 and let N1 = {0, u0 + u1} and let d N(x) = min{1, dist(x,N1)}. Then (H1, d N) |= T0. Let a = u0, b = u0 − u1 and c = u0 + u1. Note that d1N(b) = . Let (H ′1, d N) ⊃ (H1, d1N) be existentially closed. Now let H2 be an infinite dimensional separable Hilbert space and let {vi : i ∈ ω} be an orthonormal basis. LetN2 = {x ∈ H : ‖x− v1‖ = 14 , Pspan(v1)(x) = v1} ∪ {0} and let d2N(x) = min{1, dist(x,N2)}. Let (H N) ⊃ (H2, d2N) be existentially closed. Then (span(a), d1N ↾span(a)) ∼= (span(v1), d N ↾span(v1)) and they can be identified say by a function F. But (H ′1, d N) and (H N) cannot be amalgamated over this common substructure: If they could, then we would have dist(F(b), v1 + vi) = dist(b, v1 + vi) < for some i > 1 and thus d1N(b) < , a contradiction. In this case, the main reason for this failure of amalgamation resides in the fact that (span(a), d1N ↾span(a)) ∼= (span(v1), d N ↾span(v1)) is not a model of T0: informally, the distance values around v1 are determinedby an “external attractor” (the black point u0 + u1 or the black ring orthogonal to v1 at distance ) that the subspace (span(a), d1N ↾span(a)) simply cannot see. This violates Axiom (1) in HILBERT SPACES WITH GENERIC PREDICATES 17 the description of T0. This “noise external to the substructure” accounts for the failure of amalgamation, and ultimately for the lack of quantifier elimination. In [7, Corollary 2.6], the authors show that the algebraic closure of the expan- sion of a simple structure with a generic subset corresponds to the algebraic in the original language. However, in our setting, the new algebraic closure aclN(X) does not agree with the old algebraic closure acl(X): 4.3. Observation. The previous construction shows that aclN does not coincide with acl. Indeed, c ∈ aclN(a) \ acl(a) - the set of solutions of the type tpN(c/a) is {c}, but c /∈ dcl(a) as c /∈ span(a). However, models of the basic theory T0 are LN-algebraically closed. The proof is similar to [7, Proposition 2.6(3)]: 4.4. Lemma. Let (M,dN) |= TN and let A ⊂ M be such that A = dcl(A) and (A,dN ↾A) |= T0. Let a ∈M. Then a ∈ aclN(A) if and only if a ∈ A. Proof. Assume a /∈ A. We will show that a /∈ aclN(A). Let a ′ |= tp(a/A) be such that a ′ | M. Let (M ′, dN) be an isomorphic copy of (M,dN) over A through f : M →A M ′ such that f(a) = a ′. We may assume thatM ′ | Since (A,dN ↾A) is an amalgamation base, (N,dN) = (M ⊕A M ′, dN) |= T0. Let (N ′, dN) ⊃ (N,dN) be an existentially closed structure. Then tpN(a/A) = tpN(a ′/A) and therefore a /∈ aclN(A). � As TN is model complete, the types in the extended language are determined by the existential formulas within them, i.e. formulas of the form inf ȳϕ(ȳ, x̄) = 0 Another difference with the work of Chatzidakis and Pillay is that the analogue to [7, Proposition 2.5] no longer holds. Let a, b, c be as in Observation 4.3; notice that (span(a), dN ↾span(a)) ∼= (span(v1), dN ↾span(v1)). However, (H 1, dN, a) 6≡ (H ′2, dN, v1). Instead, we can show the following weaker version of the Proposi- tion. 18 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES 4.5.Proposition. Let (M,dN) and (N,dN) bemodels of TN and letA be a common subset ofM and N such that (span(A), dN ↾span(A)) |= T0. Then (M,dN) ≡A (N,dN). Proof. Assume thatM∩N = span(A). Since (span(A), dN ↾span(A)) |= T0, it is an amalgamation base and therefore we may consider the free amalgam (M⊕span(A) N,dN) of (M,dN) and (N,dN) over (span(A), dN ↾span(A)). Let now (E, dN) be a model of TN extending (M⊕span(A) N,dN). By the model completeness of TN, we have that (M,dN) ≺ (E, dN) and (N,dN) ≺ (E, dN) and thus (M,dN) ≡A (N,dN). � 4.1. Generic independence. In this section we define an abstract notion of in- dependence and study its properties. Fix (U, dN) |= TN be a κ-universal domain. 4.6. Definition. Let A,B,C ⊂ U be small sets. We say that A is ∗-independent from B over C and write A |∗ B if aclN(A ∪ C) is independent (in the sense of Hilbert spaces) from aclN(C ∪ B) over aclN(C). That is, A |∗ B if for all a ∈ aclN(A ∪ C), PB∪C(a) = PC(a), where B ∪ C = aclN(C ∪ B) and C = aclN(C). 4.7. Proposition. The relation |∗ satisfies the following properties (here A, B, etc., are any small subsets of U): (1) Invariance under automorphisms of U. (2) Symmetry: A |∗ B⇐⇒ B |∗ (3) Transitivity: A |∗ BD if and only if A |∗ B and A |∗ (4) Finite Character: A |∗ B if and only ā |∗ B for all ā ∈ A finite. (5) Local Character: If ā is any finite tuple, then there is countable B0 ⊆ B such that ā |∗ (6) Extension property over models of T0. If (C,dN ↾C) |= T0, then we can find A ′ such that tpN(A/C) = tpN(A ′/C) and A ′ |∗ (7) Existence over models: ā |∗ M for any ā. (8) Monotonicity: āā ′ |∗ b̄b̄ ′ implies ā | HILBERT SPACES WITH GENERIC PREDICATES 19 Proof. (1) Is clear. (2) It follows from the fact that independence in Hilbert spaces satisfies Sym- metry (see Proposition 1.3). (3) It follows from the fact that independence in Hilbert spaces satisfies Tran- sitivity (see Proposition 1.3). (4) Clearly A |∗ B implies that ā |∗ B for all ā ∈ A finite. On the other hand if ā |∗ B for all ā ∈ A finite, then for a dense subset A0 of A, B and thus A |∗ (5) Local Character: let ā be a finite tuple. Since independence in Hilbert spaces satisfies local character, there is B1 ⊆ aclN(B) countable such that ā |∗ B. Now let B0 ⊆ B be countable such that aclN(B0) ⊃ B1. Then ā |∗ (6) LetC be such that (C,dN ↾C) |= T0. LetD ⊃ A∪C be such that (D,dN ↾D ) |= T0 and let E ⊃ B ∪ C be such that (E, dN ↾E) |= T0. Changing D for another set D ′ with tpN(D ′/C) = tpN(D/C), we may assume that the space generated by D ′ ∪ E is the free amalgamation of D ′ and E over C. By lemma 4.4 D ′, E are algebraically closed andD ′ |∗ (7) It follows from the definition of ∗-independence. (8) It follows from the definition of ∗-independence and transitivity. Therefore we have a natural independence notion that satisfies many good properties, but not enough to guarantee the simplicity of TN. We will show below that the theory TN has both TP2 and NSOP1. This places it in an interesting area of the stability hierarchy for continuous model theory: while having the tree property TP2 and therefore lacking the good properties of NTP2 theories, it still has a quite well-behaved independence notion | , good enough to guarantee that it does not have the SOP1 tree property. Therefore, although the theory is not simple, it is reasonably close to this family of theories. 4.2. The failure of simplicity. 20 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES 4.8. Theorem. The theory TN is not simple. The proof’s idea uses a characterization of simplicity in terms of the number of partial types due to Shelah (see [11]; see also Casanovas [6] for further analysis): T is simple iff for all κ, λ such thatNT(κ, λ) < 2κ+λω, whereNT(κ, λ) counts the supremum of the cardinalities of families of pairwise incompatible partial types of size ≤ κ over a set of cardinality≤ λ. This holds for continuous logic as well. We show that TN fails this criterion. Proof. Fix κ an infinite cardinal and λ ≥ κ. We will find a complete submodel Mf of the monster model, of density character λ, and λ κ many types over sub- sets of Mf of power κ in such a way that we guarantee that they are pairwise incompatible in a uniform way. Now also fix some orthonormal basis ofMf, listed as {bi|i < κ} ∪ {aj|j < λ} ∪ {cX|X ∈ Pκ(λ)}. Also fix, for every X ∈ Pκ(λ), a bijection fX : {bi|i < κ} → {aj|j ∈ X}. Let the “black points” ofMf consist of the set N = {cX + bi + (1/2)fX(bi) | i < κ,X ∈ Pκ(λ)} ∪ {0} and as usual define dN(x) as the distance from x to N. This is a submodel of the monster. Let AX := {bi|i < κ} ∪ {aj|j ∈ X} for each X ∈ Pκ(λ). The crux of the proof is to notice that if X 6= Y then the types tp(cX/AX) and tp(cY/AY) are incompatible, thereby witnessing that there are λ κ many incom- patible types: Suppose there is some c such that tp(c/AX) = tp(cX/AX) and tp(c/AY) = tp(cY/AY). Take (wlog) j ∈ Y \ X. Pick ℓ < κ such that fY(bℓ) = aj. Let k ∈ X be such that fX(bℓ) = ak. HILBERT SPACES WITH GENERIC PREDICATES 21 InMf, the distance to black of cX+bℓ− ak is 1: by definition, cX+bℓ+ cX + bℓ + fX(bℓ) ∈ N and the only difference between cX + bℓ − 12ak and cX + bℓ + ak is the sign in front of an element of an orthonormal basis. Therefore the distance to black of d = c+ bℓ − ak is also 1 (in the monster). However, e = c + bℓ + aj must be a black point, since e ′ = cY + bℓ + aj is black (by definition of N and since aj = fY(bℓ) and tp(c/AY) = tp(cY/AY)). On the other hand, the distance from e to d is < 1. This contradicts that the color of d is 1. � This stands in sharp contrast with respect to the result by Chatzidakis and Pillay in the (discrete) first order case. The existence of these incompatible types is rendered possible here by the presence of “euclidean” interactions between the elements of the basis chosen. So far we have two kinds of expansions of Hilbert spaces by predicates: either they remain stable (as in the case of the distance to a Hilbert subspace as in the previous section) or they are not even simple. 4.3. TN has the tree property TP2. 4.9. Theorem. The theory TN has the tree property TP2. Proof. We will construct a complete submodel M |= T0 of the monster model, of density character 2ℵ0 , and a quantifier free formula ϕ(x;y, z) that witnesses TP2 inside M. Since this model can be embedded in the monster model of TN preserving the distance to black points, this will show that TN has TP2. We fix some orthonormal basis ofM, listed as {bi|i < ω} ∪ {cn,i|n, i < ω} ∪ {af|f : ω→ ω}. Also let the “black points” ofM consist of the set N = {af + bn + (1/2)cn,f(n) | n < ω, f : ω→ ω} ∪ {0} and as usual define dN(x) as the distance from x to N. This is a model of T0 and thus a submodel of the monster. 22 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES Let ϕ(x, y, z) = max{1− dN(x+ y− (1/2)z), dN(x+ y − (1/2)z)}. Claim1 For each i, the conditions {ϕ(x, bi, ci,j) = 0 : j ∈ ω} are 2-inconsistent. Assume otherwise, so we can find a (in an extension ofM) such that dN(a + bi + (1/2)ci,j) = 0 and dN(a + bi − (1/2)ci,l) = 1 for some j < l. But then d(a+bi+(1/2)ci,j, a+bi−(1/2)ci,l) = d((1/2)ci,j,−(1/2)ci,l) = 2/2 < 1. Sincea+bi+(1/2)ci,j is a black point, we get thatdN(a+bi−(1/2)ci,l) ≤ a contradiction. Claim 2 For each f the conditions {ϕ(x, bi, ci,f(i)) = 0 : i ∈ ω} are consistent. Indeed fix f and consider af, then by construction dN(af+bn+(1/2)cn,f(n)) = 0 and d(af + bn − (1/2)cn,f(n), af + bn + (1/2)cn,f(n)) = 1, so dN(af + bn − (1/2)cn,f(n)) ≤ 1. Now we check the distance to the other points in N. It is easy to see that d(af + bn − (1/2)cn,f(n), af + bm + (1/2)cm,f(m)) > 1 form 6= n, d(af + bn − (1/2)cn,f(n), ag + bk + (1/2)ck,g(k)) > 1 for g 6= f and all indexes k. Finally, d(af + bn − (1/2)cn,f(n), 0) > 1. This shows that af is a witness for the claim. 4.4. TN and the property NSOP1. Chernikov and Ramsey have proved that whenever a first order discrete theory satisfies the following properties (for ar- bitrary models and tuples), then the theory satisfies theNSOP1 property (see [8, Prop. 5.3]). • Strong finite character: whenever ā depends on b̄ overM, there is a for- mula ϕ(x, b̄, m̄) ∈ tp(ā/b̄M) such that every ā ′ |= ϕ(x̄, b̄, m̄) depends on b̄ overM. • Existence over models: ā | M for any ā. • Monotonicity: āā ′ | b̄b̄ ′ implies ā | • Symmetry: ā | b̄ ⇐⇒ b̄ | • Independent amalgamation: c̄0 | c̄1, b̄0 | c̄0, b̄1 | c̄1, b̄0 ≡M b̄1 implies there exists b̄ with b̄ ≡c̄0M b̄0, b̄ ≡c̄1M b̄1. HILBERT SPACES WITH GENERIC PREDICATES 23 We prove next that in TN, | ∗ satisfies analogues of these five properties - we may thereby conclude that TN can be regarded (following the analogy) as a NSOP1 continuous theory. In what remains of the paper, we prove that TN satisfies these properties. We focus our efforts in strong finite character and independent amalgamation, the other properties were proved in Proposition 4.7. We need the following setting: Let M be the monster model of TN andA ⊂M. FixAwithA ⊂ A ⊂ M be such that A |= T0 and let ā = (a0, . . . , an) ∈ M. We say that (ā, A,B) is minimal if tp(B/A) = tp(A/A) and for all b̄ ∈ M, if tp(b̄/A) = tp(ā/A) then ‖ prB(b0)‖+ · · · + ‖ prB(bn)‖ ≥ ‖ prB(a0)‖+ · · · + ‖ prB(an)‖. By compactness, for all p ∈ S(A) there is a minimal (ā, A,B) such that ā |= p. Now let cl0(A) be the set of all x such that for some minimal (ā, A,B), x = prB(a0) (the first coordinate of ā). 4.10. Lemma. If tp(B/A) = tp(A/A) and x ∈ cl0(A) then x ∈ B. Proof. Suppose not. Let B witness this and let C and ā be such that (ā, A,C) is minimal and x = prC(a0). Since C |= T0, wlog B |⌣C a (independence in the sense of Hilbert spaces). But then prB(ai) = prB(prC(ai)) for each i and thus ‖ prB(a0)‖+ · · ·+‖ prB(an)‖ < ‖ prC(a0)‖+ · · ·+‖ prC(an)‖. This contradicts minimality. � A direct consequence of the previous lemma is that cl0(A) ⊂ bclN(A) = ∩A⊂B|=TNB, as cl0(A) belongs to every model of the theory TN. We now define the essential closure ecl. Let clα+1(A) = cl0(clα(A)) for all ordinals α, clδ(A) = α<δ(clα(A)), and ecl(A) = α∈On clα(A). 4.11. Lemma. For all ā, B,A, if ecl(A) = A then there is b̄ such that tp(b̄/A) = tp(ā/A) and b̄ | Proof. Choose A |= T0 such that A ⊂ A and c̄ such that tp(c̄/A) = tp(ā/A) and (c̄, A,A) is minimal. Since cl0(A) = A, prA(ci) ∈ A for all i ≤ n (c̄ = 24 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES (c0, . . . , cn)), i.e. c̄ | A. Now choose b̄ such that tp(b̄/A) = tp(c̄/A) and B. Then b̄ is as needed. � 4.12. Corollary. ecl(A) = aclN(A). Proof. Clearly aclN(A) ⊂ bclN(A). On the other hand, assume that x /∈ aclN(A). Let B be a model of TN such that A ⊂ B. By Lemma 4.11, we may assume that B. Then x /∈ B, so x /∈ bcl(A), so x /∈ ecl(A). � 4.13. Theorem. Suppose ecl(A) = A, A ⊂ B,C, B |∗ C (i.e. ecl(B) | ecl(C)), ā |∗ B, b̄ |∗ C and tp(ā/A) = tp(b̄/A). Then there is c̄ such that tp(c̄/B) = tp(ā/B), tp(c̄/C) = tp(b̄/C) and c̄ |∗ Proof. Wlog ecl(B) = B and ecl(C) = C. By Lemma 4.11 we can find modelsA0, A1, B ∗ and C∗ of T0 such that Aā ⊂ A0, Ab̄ ⊂ A1, B ⊂ B∗ and C ⊂ C∗, such that B∗ |∗ C∗,A0 | B∗ andA1 | C∗. We can also find models of T0,A and D∗ such that A0B ∗ ⊂ A∗0, A1C∗ ⊂ A∗1 and B∗C∗ ⊂ D∗ and wlog we may assume that ā and b̄ are chosen so thatA∗0 |⌣B∗ D ∗,A∗1 |⌣C∗ D ∗, and that there is an automorphism F of the monster model fixingA pointwise such that F(ā) = b̄, F(A0) = A1 and F(A 0) |⌣A1 A∗1 . Notice that now D∗ and A1 | We can now find Hilbert spaces A∗, A∗∗0 , A 1 and E such that (i) E is generated byD∗A∗∗0 A (ii) A ⊂ A∗ ⊂ A∗∗0 ∩A∗∗1 , B∗ ⊂ A∗∗0 , C∗ ⊂ A∗∗1 , (iii) There are Hilbert space isomorphisms G : A∗∗0 → A 0 and H : A 1 → A such that a) F ◦G ↾ A∗ = H ↾ A∗, b) G ↾ B∗ = idB∗ , H ↾ C ∗ = idC∗ , c) G ∪ idD∗ generate an isomorphism 〈A∗∗0 D∗〉 → 〈A∗0D∗〉 HILBERT SPACES WITH GENERIC PREDICATES 25 d) H ∪ idD∗ generate an isomorphism 〈A∗∗1 D∗〉 → 〈A∗1D∗〉 e) F ∪G ∪H generate an isomorphism 〈A∗∗0 A∗∗1 〉 → 〈F(A∗0)A∗1〉. We can find these because non-dividing independence in Hilbert spaces has 3- existence (the independence theorem holds for types over sets). Now we choose the “black points” of our model: a ∈ E is black if one of the following holds: (i) a ∈ A∗∗0 and G(a) is black, (ii) a ∈ A∗∗0 and H(a) is black, (iii) a ∈ D∗ and is black. Then in E we define the “distance to black” function simply as the real distance. Then in D∗ there is no change and G and H remain isomorphisms after adding this structure;D∗, A∗0, A 1 and F(A 0) witness this. So we can assume that E is a submodel of the monster, and letting c̄ = G−1(a), G witnesses that tp(c̄/B) = tp(ā/B) and H witnesses that tp(c̄/C) = tp(b̄/C). We have already seen that A∗ | D∗ and thus c̄ |∗ BC. � 4.14. Proposition. Suppose b̄ 6 |∗ C, A ⊂ B ∩ C and (wlog) C = bcl(C). Then there exists a formula χ ∈ tp(b̄/C) such that for all ā |= χ, ā 6 |∗ Proof. By compactness, we can find ε > 0 such that (letting b̄ = (b0, . . . , bn), (ā = (a0, . . . , an)), ∀ā |= tp(b̄/B), ‖ prC(a0)‖+· · ·+‖ prC(an)‖ ≥ ε+‖ prbcl(A)(a0)‖+· · ·+‖ prbcl(A)(an)‖. Again by compactness we can find χ ∈ tp(b̄/B) such that (1) holds when we replace tp(b̄/B) by χ and ε by ε/2, that is: 26 ALEXANDER BERENSTEIN, TAPANI HYTTINEN, AND ANDRÉS VILLAVECES ∀ā |= χ, ‖ prC(a0)‖+· · ·+‖ prC(an)‖ ≥ ε/2+‖ prbcl(A)(a0)‖+· · ·+‖ prbcl(A)(an)‖. and in particular ā 6 |∗ C, as we wanted � References [1] Itaï Ben Yaacov, Lovely pairs of models: the non first order case, Journal of Symbolic Logic, volume 69 (2004), 641-662. [2] Itaï Ben Yaacov, Alexander Berenstein, Ward C. Henson and Alexander Usvyatsov, Model The- ory for metric structures in Model Theory with Applications to Algebra and Analysis, Volume 2, Cambridge University Press, 2008. [3] Itaï Ben Yaacov, Alexander Usvyatsov and Moshe Zadka, Generic automorphism of a Hilbert space, preprint avaliable at http://ptmat.fc.ul.pt/∼alexus/papers.html. [4] Alexander Berenstein, Hilbert Spaces with Generic Groups of Automorphisms, Arch. Math. Logic, vol 46 (2007) no. 3, 289–299. [5] Alexander Berenstein and Steven Buechler, Simple stable homogeneous expansions of Hilbert spaces. Ann. Pure Appl. Logic 128 (2004), no. 1-3, 75–101. [6] Enrique Casanovas, The number of types in simple theories. Ann. Pure Appl. Logic 98 (1999), 69–86. [7] Zoé Chatzidakis and Anand Pillay, Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998), no. 1-3, 71–92. [8] Artem Chernikov and Nicholas Ramsey, On model-theoretic tree properties, Journal of Mathe- matical Logic, 16 (2), 2016. [9] Wilfrid Hodges,Model Theory, Cambridge University Press 1993. [10] Bruno Poizat, Le carré de l’égalité, Journal of Symbolic Logic 64 (1999), 1339–1355. [11] Saharon Shelah, Simple Unstable Theories, Ann. Math Logic 19 (1980), 177–203. http://ptmat.fc.ul.pt/~alexus/papers.html HILBERT SPACES WITH GENERIC PREDICATES 27 Alexander Berenstein, Universidad de los Andes, Departamento de Matemáticas, Cra 1 # 18A-10, Bogotá, Colombia. URL: www.matematicas.uniandes.edu.co/~aberenst Tapani Hyttinen, University of Helsinki, Department of Mathematics and Statistics, Gustaf Hällströminkatu 2b. Helsinki 00014, Finland. E-mail address: [email protected] Andrés Villaveces, Universidad Nacional de Colombia, Departamento de Matemáticas, Av. Cra 30 # 45-03, Bogotá 111321, Colombia. E-mail address: [email protected] 1. Introduction 1.1. Model theory of Hilbert spaces (quick review) 2. Random subspaces and beautiful pairs 3. Continuous random predicates 3.1. The basic theory T0 3.2. The model companion 4. Model theoretic analysis of TN 4.1. Generic independence 4.2. The failure of simplicity 4.3. TN has the tree property TP2 4.4. TN and the property NSOP1 References

0704.1634

Riggings of locally compact abelian groups

RIGGINGS OF LOCALLY COMPACT ABELIAN GROUPS. M. Gadella, F. Gómez, S. Wickramasekara. Abstract We obtain a set of generalized eigenvectors that provides a gen- eralized spectral decomposition for a given unitary representation of a commutative, locally compact topological group. These generalized eigenvectors are functionals belonging to the dual space of a rigging on the space of square integrable functions on the character group. These riggings are obtained through suitable spectral measure spaces. 1 Introduction The purpose of the present paper is to take a first step towards a general formalism of unitary representations of groups and semigroups on rigged Hilbert spaces. To begin with, we want to introduce the theory correspond- ing to Abelian locally compact groups, leaving the more general nonabelian case as well as semigroups for a later work. We recall that a rigged Hilbert space or a rigging of a Hilbert space H is a triplet of the form Φ ⊂ H ⊂ Φ× , (1) where Φ is a locally convex space dense in H with a topology stronger than that inherited from H and Φ× is the dual space of Φ. In this paper, we shall always assume that H is separable. To each self adjoint operator A on H, the von Neumann theorem [1] associates a spectral measure space. This is the quadruple (Λ,A,H, P ), whereH is the Hilbert space on which A acts, Λ = σ(A) is the spectrum of A, A is the family of Borel sets in Λ, and P is the projection valued measure on A determined by A through the von Neumann theorem. Obviously Λ ⊂ R. A complete discussion on the relation between these concepts can be found in [2]. We say that the topological vector space (Φ, τΦ) (vector space Φ with http://arxiv.org/abs/0704.1634v1 the locally convex topology given by τΦ) equips or rigs the spectral measure (Λ,A,H, P ) if the following conditions hold: i. There exists a one-to-one linear mapping I : Φ 7−→ H with range dense in H. We can assume that Φ ⊂ H is a dense subspace of H and I, the canonical injection from Φ into H. ii. There exists a σ-finite measure µ on (Λ,A), a set Λ0 ⊂ Λ with zero µ measure and a family of vectors in Φ× of the form {|λk×〉 ∈ Φ× : λ ∈ Λ\Λ0, k ∈ {1, 2, . . . ,m}}, (2) where m ∈ {∞, 1, 2, . . .}, such that (φ, P (E)ϕ)H = 〈φ|λk×〉 〈ϕ|λk×〉∗ dµ(λ), ∀φ,ϕ ∈ Φ, ∀E ∈ A. Each family of the form (2) satisfying (3) is called a complete system of Dirac kets of the spectral measure (Λ,A,H, P ) in (Φ, τΦ). In this case, the triplet Φ ⊂ H ⊂ Φ× is a rigged Hilbert space, which is called a rigging of (Λ,A,H, P ). Conversely, the von Neumann theorem asserts that a projection valued measure defined on the σ-algebra of Borel sets on a subset of the real line determines a self adjoint operator A. If (Λ,A,H, P ), where Λ ⊂ R, is such a measure space, then for ϕ and φ on a suitable dense domain, the self-adjoint operator A such that Λ = σ(A) is defined by (φ,Aϕ)H = λ 〈φ|λk×〉 〈ϕ|λk×〉∗ dµ(λ) (4) where µ, |λk×〉 and m are as defined in (3). Further, if f(λ) is a measurable complex valued function on Λ, then, for φ,ϕ on a suitable dense domain, which is the whole of H if f(λ) is bounded, the operator valued function f(A) is defined by (φ, f(A)ϕ)H = f(λ) 〈φ|λk×〉 〈ϕ|λk×〉∗ dµ(λ) . (5) The functionals |λk×〉 ∈ Φ× and the complex numbers f(λ) are the gen- eralized eigenvectors and respective generalized eigenvalues of f(A) [3]. In particular, if f(λ) = eitλ, where t ∈ R, the set of operators eitA forms a one parameter commutative group of unitary operators and Φ ⊂ H ⊂ Φ× as defined above is a rigging for this group. One can expect that similar riggings exist for unitary representations of arbitrary groups and semigroups and that the operators of the representa- tions can be expanded in terms of generalized eigenvectors and eigenvalues as in (5). Riggings that make use of Hardy functions on a half plane exist for one parameter dynamical semigroups e−itH , t ≤ 0 and e−itH , t ≥ 0, where H is the Hamiltonian [3]. In the present paper, we show that riggings along the above lines al- ways exist for unitary representations of Abelian locally compact groups. In particular, let G be an Abelian locally compact group and π, a unitary representation of G on a separable Hilbert space H. We will see that the Fourier transform on G, or equivalently, the Gelfand transformation on the C∗-algebra L1(G) allows us to represent π in terms of generalized eigenfunc- tions and riggings of H in a manner similar to the description given in [2] for the action of a spectral measure. 2 Characters of Abelian Locally Compact Groups. Let G be a locally compact abelian group with Haar measure µ. A character χ of G is any continuous mapping from G into the set of complex numbers C such that χ(g1g2) = χ(g1)χ(g2) for all g1, g2 ∈ G and |χ(g)| = 1 for all g ∈ G, i.e., a character of G is a continuous homomorphism from G into the unit circle T. The set of all the characters of G forms a group, Ĝ, which is often called the dual group of G. We shall use the notation χ(g) := 〈g|χ〉. Let L1(G) be the space of complex valued functions, integrable in the modulus with respect to the Haar measure µ on G. L1(G) is an abelian ∗- algebra, with the convolution product. The dual group Ĝ can be identified with the set of maximal ideals of L1(G) [4]. When endowed with the Gelfand topology, Ĝ is a compact Hausdorf space (see [5] page 268). For any χ ∈ Ĝ, we may define a linear functional Λχ on L 1(G) by Λχ(f) = 〈g|χ〉∗f(g)dµ(g) . (6) Let C(Ĝ) be the space of complex continuous functions on Ĝ with the supremun norm topology. The Gelfand-Fourier transform is the mapping F : L1(G) 7−→ C(Ĝ) defined by: [Ff ](χ) = f̂(χ) = Λχ(f) = 〈g|χ〉∗ f(g) dµ(g) . (7) Let (π,H) be a unitary representation of G. Then (see [6] page 105), there is a unique spectral measure (Ĝ,B,H, P ), where B is the σ-algebra of Borel sets on Ĝ, such that for all g ∈ G and all f ∈ L1(G), we have π(g) = 〈g|χ〉 dP (χ) ; π(f) = Λχ(f) dP (χ) . (8) There is a one to one correspondence between unitary representations of G and non degenerate ∗-representations1 of L1(G) as given by (7) and (8). 2.1 Riggings of functions of characters. Let us consider the spectral measure space (Ĝ,B,H, P ) introduced in the previous section. For simplicity in the discussion, we assume the existence of a cyclic vector u ∈ H. This means that the subspace spanned by the vectors of the form P (E)u with E ∈ A is dense in H. The general case can be easily obtained as a finite or countable direct sum of cyclic subspaces of Then, the von Neumann decomposition theorem [1] establishes that be- ing given the spectral measure space (Ĝ,B,H, P ) and a positive measure ν on (Ĝ,B) with maximal spectral type2 [P ] (ν ∈ [P ]), there exists a uni- tary mapping U : H 7−→ L2(Ĝ, dµ) , such that πν(g) := Uπ(g)U −1 is the multiplication by 〈g|χ〉 on L2(Ĝ, dν): πν(g)φ(χ) = Uπ(g)U −1φ(χ) = 〈g|χ〉φ(χ) , ∀φ(χ) ∈ L2(Ĝ, dν) . (9) Since πν(g) is a multiplication operator, it is easy to see that the Dirac delta type Radon measures λ(χ)δχ form a complete system of Dirac kets for the spectral measure space (Ĝ,B,H, P ) in the sense given by (3). For any f(χ) ∈ Φ these deltas satisfy f(χ) δχ′ dν = f(χ ′) . (10) Thus, a possible choice for Φ is C(Ĝ), the space of continuous functions on Ĝ endowed with a topology τΦ stronger than both the topologies of the supremun and the || · || L2( bG,dν) norm. In this case, the dual Φ× of Φ includes the space of all Radon measures on (Ĝ,B). We have the rigged Hilbert space Φ ⊂ L2(Ĝ) ⊂ Φ×. 1Non degenerate means that π(f)v = 0 for every f implies v = 0. The representation has also the property that π(f∗) = π†(f), where f 7→ f∗ is the involution on L1(G), see 2For a definition and properties of the spectral type, see [1, 2]. 3 Positive Type Functions and Riggings. Next, we shall introduce another representation πφ of G linked to a function of positive type, that can be defined as follows: Let φ(g) ∈ L∞(G). We say that φ(g) is a function of positive type if for any f(g) ∈ L1(G), we have that f∗(g)f(gg′)φ(g′) dµ(g) dµ(g′) ≥ 0 , (11) where the star ∗ denotes complex conjugation. If φ(g) is a function of positive type, then, the following positive Hermi- tian form on L1(G) 〈h|f〉φ := h∗(g′)f(g)φ(g−1g′) dµ(g′) dµ(g) (12) is semi-definite in the sense that it may exist non-zero functions f ∈ L1(G) such that 〈f |f〉φ = 0. These functions form a subspace of L 1(G) that we denote by N . Consider the factor space L1(G)/N and again denote by 〈·|·〉φ the scalar product induced on L1(G)/N by the Hermitian form (12). The completion of L1(G)/N by 〈·|·〉φ gives a Hilbert space usually denoted as Hφ. Then, for any g ∈ G and f(g) ∈ L 1(G), we define: (Lg)f(g ′) := f(g−1g′) . (13) Note that Lg preserves the scalar product 〈·|·〉φ: 〈Lgh|Lgf〉φ = h∗(g−1g′)f(g−1g′′)φ(g −1g′′) dµ(g′) dµ(g′′) h∗(g′)f(g′′)φ((gg′)−1(gg′′)) dµ(g′) dµ(g′′) = 〈h|f〉φ , (14) for all f(g) ∈ L1(G). This also shows that LgN ⊂ N and therefore Lg induces a transformation on the factor space L1(G)/N , that we also denote as Lg, defined as Lg(f(g ′) +N ) := f(g−1g′) +N = Lg(f(g ′)) +N . (15) By (14), we easily see that Lg preserves the scalar product on L 1(G)/N . It is obviously invertible. Therefore, it can be uniquely extended into a unitary operator on Hφ. Then, if for each g ∈ G we write πφ(g)f := Lgf , ∀ f ∈ Hφ , (16) then, πφ(g) determines a unitary representation of G on Hφ. The proof of this statement is straightforward. The representation πφ(g) of G on Hφ can be lifted to a unitary repre- sentation of the group algebra L1(G) on Hφ that we shall also denote as πφ. In this case, for all f ∈ L1(G), we have πφ(f)h := f ∗ h. Here, ∗ denotes convolution. The existence of a cyclic vector η ∈ Hφ for the representation πφ is proven in [6]. Recall that η is cyclic vector if the subspace {πφ(f)η, ∃f ∈ L 1(G)} is dense in Hφ. In addition, this result also gives the following formula that allows to find the function φ(g) in terms of η and the unitary representation πφ of G on Hφ: φ(g) = 〈η|πφ(g)η〉 . (17) Now, let us consider the unitary πν representation of G given by (9) with cyclic vector ξ and define the following complex valued function on G: φ(g−1g′) := 〈ξ|πν(g −1g′)ξ〉 L2( bG,dν) = 〈πν(g)ξ|πν(g L2( bG,dν) . (18) Then, as shown in [6], Chapter 3, i.) the function φ is of positive type in the sense of (11), and ii.) the representation ofG onHφ given by πφ, where φ is as (18) is equivalent to πν . Note that this result implies in particular that for this φ as in (18) φ(g) = 〈η|πφ(g)η〉φ = 〈ξ|πν(g)ξ〉L2( bG,dν) , ∀ g ∈ G . (19) According to (9) and (18), we have that φ(g−1g′) = 〈πν(g)ξ|πν(g L2( bG,dν) [〈g|χ〉 ξ(χ)]∗ 〈g′|χ〉 ξ(χ) dν(χ) . If we carry this formula into (12) and apply the Fubini theorem of the change of the order of integration, we have for all f, h ∈ L1(G): 〈f |h〉φ = [f(g)〈g|χ〉]∗ dµ(g) h(g′)〈g′|χ〉 dµ(g′) |ξ(χ)|2 dν(χ) [f̂(χ)]∗ ĥ(χ) |ξ(χ)|2 dν(χ) = 〈f̂ |χ〉〈χ|ĥ〉 |ξ(χ)|2 dν(χ) . (21) This latter formula shows that the generalized eigenvalues Fχ of πφ(g) are the following: if f ∈ Φ := L1(G) ∩ L2(G) |Fχ〉 ≡ Fχ : f 7−→ |η(χ)|f̂ ∗(χ) = |η(χ)| 〈g|χ〉f∗(g) dµ(g) . (22) We endow Φ with any topology stronger than the topologies L1(G) and L2(G). For instance, we can choose a locally convex topology with the semi- norms p1(f) := ||f ||L1(G) and p2(f) := ||f ||L2(G), for all f ∈ Φ. With this topology or another stronger one, the antilinear functional Fχ is continuous. Then, if we use (8) in the scalar product on Hφ, we have: 〈f |πφ(g)h〉φ = 〈g|χ〉 d〈f |P (χ)h〉φ = 〈g|χ〉 |η(χ)|2 f̂∗(χ)ĝ(χ) dν(χ) 〈g|χ〉 〈f |Fχ〉〈Fχ|h〉 dν(χ) . (23) If we omit the arbitrary f, h ∈ Φ in (23), we have the following spectral decomposition for πφ(g) for all g ∈ G: πφ(g) = 〈χ|g〉 |Fχ〉〈Fχ| dν(χ) . (24) Note that in the antidual space Φ×, the generalized eigenvalue equation πφ(g)|Fχ〉 = 〈χ|g〉|Fχ〉 is valid, where we use the same notation πφ(g) for the extensions of these unitary operators into Φ×. In conclusion, for each unitary representation of a locally compact Abelian topological group, we have found an equivalent representation and a rigged Hilbert space such that each of the unitary operators of the representation admits a generalized spectral decomposition in terms of generalized eigen- vectors of them. The eigenvectors of the decomposition are labeled by the group characters only and their respective eigenvalues, complex numbers with modulus one, depend on both the corresponding character and the group element. The spectral decomposition and the corresponding rigging comes after the existence of a spectral measure space. Note that the Abelian property is crucial in our derivation and in partic- ular in the existence of the spectral measure space (Ĝ,B,H, P ), since then, the group algebra is also Abelian and the Gelfand theory applies. An ex- tension of the present formalism to nonabelian locally compact groups will require an extension of the Gelfand formalism that at least allows for a new and consistent definition of the Gelfand Fourier transform (7), an essential feature of our construction. Acknowledgements We acknowledge the financial support from the Junta de Castilla y León Project VA013C05 and the Ministry of Education and Science of Spain, projects MTM2005-09183 and FIS2005-03988. S.W. acknowledges the fi- nancial support from the University of Valladolid where he was a visitor while this work was done and additional financial support from Grinnell College. References [1] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, N.J., 1955). [2] M. Gadella, F. Gómez, Foundations of Physics, 32, 815 (2002); M. Gadella, F. Gómez, International Journal of Theoretical Physiscs, 42, 2225-2254 (2003); M. Gadella, F Gómez, Bulletin des Sciences Mathèmatiques, 129, 567 (2005); M. Gadella, F. Gómez, Reports on Mathematical Physics, 59, 127 (2007). [3] A. Bohm, M. Gadella, Dirac kets, Gamow vectors and Gelfand triplets, Springer Lecture Notes in Physics, 348 (Springer, Berlin 1989). [4] M.A. Naimark, Normed Rings (Wolters-Noordhoff, Groningen, The Netherlands, 1970). [5] W. Rudin, Functional Analysis (McGraw-Hill, New York 1973). [6] G. B. Folland, A Course in Abstract Harmonic Analysis (CRC, Boca Raton, London, 1995). M. Gadella Departamento de F́ısica Teórica Facultad de Ciencias c. Real de Burgos, s.n. 47011 Valladolid, Spain E-mail address: [email protected] F. Gómez Departamento de Análisis Matemático Facultad de Ciencias c. Real de Burgos, s.n. 47011 Valladolid, Spain E-mail address: [email protected] S. Wickramasekara Department of Physics, Grinnell College, Grinnell, IA 50112, USA E-mail address: [email protected] Characters of Abelian Locally Compact Groups. Riggings of functions of characters. Positive Type Functions and Riggings.

0704.1635

Weak Amenability of Hyperbolic Groups

WEAK AMENABILITY OF HYPERBOLIC GROUPS NARUTAKA OZAWA Abstract. We prove that hyperbolic groups are weakly amenable. This par- tially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combina- torial approach in the spirit of Haagerup and prove that for the word length distance d of a hyperbolic group, the Schur multipliers associated with the kernel rd have uniformly bounded norms for 0 < r < 1. We then combine this with a Bożejko-Picardello type inequality to obtain weak amenability. 1. Introduction The notion of weak amenability for groups was introduced by Cowling and Haagerup [CH]. (It has almost nothing to do with the notion of weak amenability for Banach algebras.) We use the following equivalent form of the definition. See Section 2 and [BO, CH, Pi] for more information. Definition. A countable discrete group Γ is said to be weakly amenable with constant C if there exists a sequence of finitely supported functions ϕn on Γ such that ϕn → 1 pointwise and supn ‖ϕn‖cb ≤ C, where ‖ϕ‖cb denotes the (completely bounded) norm of the Schur multiplier on B(ℓ2Γ) associated with (x, y) 7→ ϕ(x−1y). In the pioneering paper [Ha], Haagerup proved that the group C∗-algebra of a free group has a very interesting approximation property. Among other things, he proved that the graph distance d on a tree Γ is conditionally negatively definite; in particular, the Schur multiplier on B(ℓ2Γ) associated with the kernel r d has (completely bounded) norm one for every 0 < r < 1. For information of Schur multipliers and completely bounded maps, see Section 2 and [BO, CH, Pi]. Bożejko and Picardello [BP] proved that the Schur multiplier associated with the charac- teristic function of the subset {(x, y) : d(x, y) = n} has (completely bounded) norm at most 2(n + 1). These two results together imply that a group acting properly on a tree is weakly amenable with constant one. Recently, this result was extended to the case of finite-dimensional CAT(0) cube complexes by Guentner and Higson [GH]. See also [Mi]. Cowling and Haagerup [dCH, Co, CH] proved that lattices in simple Lie groups of real rank one are weakly amenable and computed explicitly the associated constants. It is then natural to explore this property for hyperbolic groups in the sense of Gromov [GdH, Gr]. We prove that hyperbolic 2000 Mathematics Subject Classification. Primary 20F67; Secondary 43A65, 46L07. Key words and phrases. hyperbolic groups, weak amenability, Schur multipliers. Supported by Sloan Foundation and NSF grant. http://arxiv.org/abs/0704.1635v3 2 NARUTAKA OZAWA groups are weakly amenable, without giving estimates of the associated constants. The results and proofs are inspired by and partially generalize those of Haagerup [Ha], Pytlik-Szwarc [PS] and Bożejko-Picardello [BP]. We denote by N0 the set of non-negative integers, and by D the unit disk {z ∈ C : |z| < 1}. Theorem 1. Let Γ be a hyperbolic graph with bounded degree and d be the graph distance on Γ. Then, there exists a constant C such that the following are true. (1) For every z ∈ D, the Schur multiplier θz on B(ℓ2Γ) associated with the kernel Γ× Γ ∋ (x, y) 7→ zd(x,y) ∈ C has (completely bounded) norm at most C|1−z|/(1−|z|). Moreover, z 7→ θz is a holomorphic map from D into the space V2(Γ) of Schur multipliers. (2) For every n ∈ N0, the Schur multiplier on B(ℓ2Γ) associated with the characteristic function of the subset {(x, y) ∈ Γ× Γ : d(x, y) = n} has (completely bounded) norm at most C(n+ 1). (3) There exists a sequence of finitely supported functions fn : N0 → [0, 1] such that fn → 1 pointwise and that the Schur multiplier on B(ℓ2Γ) associated with the kernel Γ× Γ ∋ (x, y) 7→ fn(d(x, y)) ∈ [0, 1] has (completely bounded) norm at most C for every n. Let Γ be a hyperbolic group and d be the word length distance associated with a fixed finite generating subset of Γ. Then, for the sequence fn as above, the sequence of functions ϕn(x) = fn(d(e, x)) satisfy the properties required for weak amenability. Thus we obtain the following as a corollary. Theorem 2. Every hyperbolic group is weakly amenable. This solves affirmatively a problem raised by Roe at the end of [Ro]. We close the introduction with a few problems and remarks. Is it possible to construct a family of uniformly bounded representations as it is done in [Do, PS]? Is it true that a group which is hyperbolic relative to weakly amenable groups is again weakly amenable? There is no serious difficulty in extending Theorem 1 to (uniformly) fine hyperbolic graphs in the sense of Bowditch [Bo]. Ricard and Xu [RX] proved that weak amenability with constant one is closed under free products with finite amalgamation. The author is grateful to Professor Masaki Izumi for conversations and encouragement. 2. Preliminary on Schur multipliers Let Γ be a set and denote by B(ℓ2Γ) the Banach space of bounded linear oper- ators on ℓ2Γ. We view an element A ∈ B(ℓ2Γ) as a Γ× Γ-matrix: A = [Ax,y]x,y∈Γ with Ax,y = 〈Aδy, δx〉. For a kernel k : Γ×Γ → C, the Schur multiplier associated with k is the map mk on B(ℓ2Γ) defined by mk(A) = [k(x, y)Ax,y]. We recall the necessary and sufficient condition for mk to be bounded (and everywhere-defined). See [BO, Pi] for more information of completely bounded maps and the proof of the following theorem. WEAK AMENABILITY OF HYPERBOLIC GROUPS 3 Theorem 3. Let a kernel k : Γ × Γ → C and a constant C ≥ 0 be given. Then the following are equivalent. (1) The Schur multiplier mk is bounded and ‖mk‖ ≤ C. (2) The Schur multiplier mk is completely bounded and ‖mk‖cb ≤ C. (3) There exist a Hilbert space H and vectors ζ+(x), ζ−(y) in H with norms at most C such that 〈ζ−(y), ζ+(x)〉 = k(x, y) for every x, y ∈ Γ. We denote by V2(Γ) = {mk : ‖mk‖ < ∞} the Banach space of Schur multipliers. The above theorem says that the sesquilinear form ℓ∞(Γ,H)× ℓ∞(Γ,H) ∋ (ζ−, ζ+) 7→ mk ∈ V2(Γ), where k(x, y) = 〈ζ−(y), ζ+(x)〉, is contractive for any Hilbert space H. Let Pf(Γ) be the set of finite subsets of Γ. We note that the empty set ∅ belongs to Pf (Γ). For S ∈ Pf(Γ), we define ξ̃+S and ξ̃ ∈ ℓ2(Pf (Γ)) by ξ̃+S (ω) = 1 if ω ⊂ S 0 otherwise and ξ̃−S (ω) = (−1)|ω| if ω ⊂ S 0 otherwise We also set ξ+ = ξ̃+ − δ∅ and ξ−S = −(ξ̃ − δ∅). Note that ξ±S ⊥ ξ if S ∩ T = ∅. The following lemma is a trivial consequence of the binomial theorem. Lemma 4. One has ‖ξ± ‖2 + 1 = ‖ξ̃± ‖2 = 2|S| and 〉 = 1− 〈ξ̃− , ξ̃+ 1 if S ∩ T 6= ∅ 0 otherwise for every S, T ∈ Pf(Γ). 3. Preliminary on hyperbolic graphs We recall and prove some facts of hyperbolic graphs. We identify a graph Γ with its vertex set and equip it with the graph distance: d(x, y) = min{n : ∃x = x0, x1, . . . , xn = y such that xi and xi+1 are adjacent}. We assume the graph Γ to be connected so that d is well-defined. For a subset E ⊂ Γ and R > 0, we define the R-neighborhood of E by NR(E) = {x ∈ Γ : d(x,E) < R}, where d(x,E) = inf{d(x, y) : y ∈ E}. We write BR(x) = NR({x}) for the ball with center x and radius R. A geodesic path p is a finite or infinite sequence of points in Γ such that d(p(m), p(n)) = |m − n| for every m,n. Most of the time, we view a geodesic path p as a subset of Γ. We note the following fact (see e.g., Lemma E.8 in [BO]). Lemma 5. Let Γ be a connected graph. Then, for any infinite geodesic path p : N0 → Γ and any x ∈ Γ, there exists an infinite geodesic path px which starts at x and eventually flows into p (i.e., the symmetric difference p△ px is finite). Definition. We say a graph Γ is hyperbolic if there exists a constant δ > 0 such that for every geodesic triangle each edge is contained in the δ-neighborhood of the union of the other two. We say a finitely generated group Γ is hyperbolic if its 4 NARUTAKA OZAWA Cayley graph is hyperbolic. Hyperbolicity is a property of Γ which is independent of the choice of the finite generating subset [GdH, Gr]. From now on, we consider a hyperbolic graph Γ which has bounded degree: supx |BR(x)| < ∞ for every R > 0. We fix δ > 1 satisfying the above definition. We fix once for all an infinite geodesic path p : N0 → Γ and, for every x ∈ Γ, choose an infinite geodesic path px which starts at x and eventually flows into p. For x, y, w ∈ Γ, the Gromov product is defined by 〈x, y〉w = (d(x,w) + d(y, w)− d(x, y)) ≥ 0. See [BO, GdH, Gr] for more information on hyperbolic spaces and the proof of the following lemma which says every geodesic triangle is “thin”. Lemma 6 (Proposition 2.21 in [GdH]). Let x, y, w ∈ Γ be arbitrary. Then, for any geodesic path [x, y] connecting x to y, one has d(w, [x, y]) ≤ 〈x, y〉w + 10δ. Lemma 7. For x ∈ Γ and k ∈ Z, we set T (x, k) = {w ∈ N100δ(px) : d(w, x) ∈ {k − 1, k} }, where T (x, k) = ∅ if k < 0. Then, there exists a constant R0 satisfying the following: For every x ∈ Γ and k ∈ N0, if we denote by v the point on px such that d(v, x) = k, then T (x, k) ⊂ BR0(v). Proof. Let w ∈ T (x, k) and choose a point w′ on px such that d(w,w′) < 100δ. Then, one has |d(w′, x)− d(w, x)| < 100δ and d(w, v) ≤ d(w,w′) + d(w′, v) ≤ 100δ + |d(w′, x)− k| < 200δ + 1. Thus the assertion holds for R0 = 200δ + 1. � Lemma 8. For k, l ∈ Z, we set W (k, l) = {(x, y) ∈ Γ× Γ : T (x, k) ∩ T (y, l) 6= ∅}. Then, for every n ∈ N0, one has E(n) := {(x, y) ∈ Γ× Γ : d(x, y) ≤ n} = W (k, n− k). Moreover, there exists a constant R1 such that W (k, l) ∩W (k + j, l − j) = ∅ for all j > R1. Proof. First, if (x, y) ∈ W (k, n−k), then one can find w ∈ T (x, k)∩T (y, n−k) and d(x, y) ≤ d(x,w) + d(w, y) ≤ n. This proves that the right hand side is contained in the left hand side. To prove the other inclusion, let (x, y) and n ≥ d(x, y) be given. Choose a point p on px ∩ py such that d(p, x) + d(p, y) ≥ n, and a geodesic path [x, y] connecting x to y. By Lemma 6, there is a point a on [x, y] such that d(a, p) ≤ 〈x, y〉p + 10δ. It follows that 〈x, p〉a + 〈y, p〉a = d(a, p)− 〈x, y〉p ≤ 10δ. WEAK AMENABILITY OF HYPERBOLIC GROUPS 5 We choose a geodesic path [a, p] connecting a to p and denote by w(m) the point on [a, p] such that d(w(m), a) = m. Consider the function f(m) = d(w(m), x) + d(w(m), y). Then, one has that f(0) = d(x, y) ≤ n ≤ d(p, x) + d(p, y) = f(d(a, p)) and that f(m + 1) ≤ f(m) + 2 for every m. Therefore, there is m0 ∈ N0 such that f(m0) ∈ {n− 1, n}. We claim that w := w(m0) ∈ T (x, k) ∩ T (y, n− k) for k = d(w, x). First, note that d(w, y) = f(m0)− k ∈ {n− k − 1, n− k}. Since 〈x, p〉w ≤ (d(x, a) + d(a, w) + d(p, w) − d(x, p)) (d(x, a) + d(p, a)− d(x, p)) = 〈x, p〉a ≤ 10δ, one has that d(w, px) ≤ 20δ by Lemma 6. This proves that w ∈ T (x, k). One proves likewise that w ∈ T (y, n − k). Therefore, T (x, k) ∩ T (y, n − k) 6= ∅ and (x, y) ∈ W (k, n− k). Suppose now that (x, y) ∈ W (k, l) ∩ W (k + j, l − j) exists. We choose v ∈ T (x, k) ∩ T (y, l) and w ∈ T (x, k + j) ∩ T (y, l− j). Let vx (resp. wx) be the point on px such that d(vx, x) = k (resp. d(wx, x) = k + j). Then, by Lemma 7, one has d(v, vx) ≤ R0 and d(w,wx) ≤ R0. We choose vy, wy on py likewise for y. It follows that d(vx, vy) ≤ 2R0 and d(wx, wy) ≤ 2R0. Choose a point p on px ∩ py. Then, one has |d(vx, p)− d(vy , p)| ≤ 2R0 and |d(wx, p)− d(wy , p)| ≤ 2R0. On the other hand, one has d(vx, p) = d(wx, p) + j and d(vy , p) = d(wy , p)− j. It follows 2j = d(vx, p)− d(wx, p)− d(vy, p) + d(wy , p) ≤ 4R0. This proves the second assertion for R1 = 2R0. � Lemma 9. We set Z(k, l) = W (k, l) ∩ W (k + j, l − j)c. Then, for every n ∈ N0, one has χE(n) = χZ(k,n−k). Proof. We first note that Lemma 8 implies Z(k, l) = W (k, l)∩ j=1 W (k+j, l−j)c k=0 Z(k, n−k) ⊂ k=0 W (k, n−k) = E(n). It is left to show that for every (x, y) and n ≥ d(x, y), there exists one and only one k such that (x, y) ∈ Z(k, n−k). For this, we observe that (x, y) ∈ Z(k, n− k) if and only if k is the largest integer that satisfies (x, y) ∈ W (k, n− k). � 4. Proof of Theorem Proposition 10. Let Γ be a hyperbolic graph with bounded degree and define E(n) = {(x, y) : d(x, y) ≤ n}. Then, there exist a constant C0 > 0, subsets Z(k, l) ⊂ Γ, a Hilbert space H and vectors η+ (x) and η− (y) in H which satisfy the following properties: 6 NARUTAKA OZAWA (1) η±m(w) ⊥ η±m′(w) for every w ∈ Γ and m,m′ ∈ N0 with |m−m′| ≥ 2. (2) ‖η±m(w)‖ ≤ C0 for every w ∈ Γ and m ∈ N0. (3) 〈η− (y), η+ (x)〉 = χZ(k,l)(x, y) for every x, y ∈ Γ and k, l ∈ N0. (4) χE(n) = k=0 χZ(k,n−k) for every n ∈ N0. Proof. We use the same notations as in the previous sections. Let H = ℓ2(Pf (Γ))⊗(1+R1) and define η+k (x) and η (y) in H by (x) = ξ+ T (x,k) ⊗ ξ̃+ T (x,k+1) ⊗ · · · ⊗ ξ̃+ T (x,k+R1) (y) = ξ− T (y,l) ⊗ ξ̃− T (y,l−1) ⊗ · · · ⊗ ξ̃− T (y,l−R1) If |m−m′| ≥ 2, then T (w,m)∩T (w,m′) = ∅ and ξ± T (w,m) T (w,m′) . This implies the first assertion. By Lemma 7 and the assumption that Γ has bounded degree, one has C1 := supw,m |T (w,m)| ≤ supv |BR0(v)| < ∞. Now the second assertion follows from Lemma 4 with C0 = 2 C1(1+R1). Finally, by Lemma 4, one has (y), η+ (x)〉 = χW (k,l)(x, y) χW (k+j,l−j)c (x, y) = χZ(k,l)(x, y). This proves the third assertion. The fourth is nothing but Lemma 9. � Proof of Theorem 1. Take η±m ∈ ℓ∞(Γ,H) as in Proposition 10 and set C = 2C0. For every z ∈ D, we define ζ±z ∈ ℓ∞(Γ,H) by the absolutely convergent series ζ+z (x) = ζ−z (y) = where 1− z denotes the principal branch of the square root. The construction of ζ±z draws upon [PS]. We note that the map D ∋ z 7→ (ζ±z (w))w ∈ ℓ∞(Γ,H) is (anti-)holomorphic. By Proposition 10, one has 〈ζ−z (y), ζ+z (x)〉 = (1 − z) zk+lχZ(k,l)(x, y) = (1 − z) znχE(n)(x, y) = (1 − z) n=d(x,y) = zd(x,y) WEAK AMENABILITY OF HYPERBOLIC GROUPS 7 for all x, y ∈ Γ, and ‖ζ±z (w)‖2 ≤ 2|1− z| j=0,1 (z±)2m+jη±2m+j(w)‖ = 2|1− z| j=0,1 |z|4m+2j‖η±2m+j(w)‖ ≤ 2|1− z| 1 1− |z|2C0 |1− z| 1− |z| for all w ∈ Γ. Therefore the Schur multiplier θz associated with the kernel zd has (completely bounded) norm at most C|1 − z|/(1− |z|) by Theorem 3. Moreover, the map D ∋ z 7→ θz ∈ V2(Γ) is holomorphic. For the second assertion, we simply write ‖Z‖ for the (completely bounded) norm of the Schur multiplier associated with the characteristic function χZ of a subset Z ⊂ Γ× Γ. By Proposition 10 and Theorem 3, one has ‖E(n)‖ ≤ ‖Z(k, n− k)‖ ≤ C0(n+ 1). and ‖{(x, y) : d(x, y) = n}‖ = ‖E(n) \ E(n − 1)‖ ≤ C(n + 1). This proves the second assertion. The third assertion follows from the previous two, by choosing fn(d) = χE(Kn)(d)r n for suitable 0 < rn < 1 and Kn ∈ N0 with rn → 1 and Kn → ∞. We refer to [BP, Ha] for the proof of this fact. � References [Bo] B.H. Bowditch, Relatively hyperbolic groups. Preprint. 1999. [BP] M. Bożejko and M.A. Picardello, Weakly amenable groups and amalgamated products. Proc. Amer. Math. Soc. 117 (1993), 1039–1046. [BO] N. Brown and N. Ozawa, C∗-algebras and Finite-Dimensional Approximations. Graduate Studies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008. [dCH] J. de Cannière and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107 (1985), 455–500. [Co] M. Cowling, Harmonic analysis on some nilpotent Lie groups (with application to the representation theory of some semisimple Lie groups). Topics in modern harmonic anal- ysis, Vol. I, II (Turin/Milan, 1982), 81–123, Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983. [CH] M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96 (1989), 507–549. [Do] A.H. Dooley, Heisenberg-type groups and intertwining operators. J. Funct. Anal. 212 (2004), 261–286. [GdH] E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’aprés Mikhael Gromov. Progress in Math., 83, Birkaüser, 1990. [Gr] M. Gromov, Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987. [GH] E. Guentner and N. Higson, Weak amenability of CAT(0) cubical groups. Preprint arXiv:math/0702568. [Ha] U. Haagerup, An example of a nonnuclear C∗-algebra, which has the metric approximation property. Invent. Math. 50 (1978/79), 279–293. http://arxiv.org/abs/math/0702568 8 NARUTAKA OZAWA [Mi] N. Mizuta, A Bożejko-Picardello type inequality for finite dimensional CAT(0) cube com- plexes. J. Funct. Anal., in press. [Pi] G. Pisier, Similarity problems and completely bounded maps. Second, expanded edition. Includes the solution to ”The Halmos problem”. Lecture Notes in Mathematics, 1618. Springer-Verlag, Berlin, 2001. [PS] T. Pytlik and R. Szwarc, An analytic family of uniformly bounded representations of free groups. Acta Math. 157 (1986), 287–309. [RX] É. Ricard and Q. Xu, Khintchine type inequalities for reduced free products and appli- cations. J. Reine Angew. Math. 599 (2006), 27–59. [Ro] J. Roe, Lectures on coarse geometry, University Lecture Series, 31. American Mathemat- ical Society, Providence, RI, 2003. Department of Mathematical Sciences, University of Tokyo, Komaba, 153-8914, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555 E-mail address: [email protected] 1. Introduction 2. Preliminary on Schur multipliers 3. Preliminary on hyperbolic graphs 4. Proof of Theorem References

0704.1636

Light Curves of Dwarf Plutonian Planets and other Large Kuiper Belt Objects: Their Rotations, Phase Functions and Absolute Magnitudes

arXiv:0704.1636v2 [astro-ph] 13 Apr 2007 Light Curves of Dwarf Plutonian Planets and other Large Kuiper Belt Objects: Their Rotations, Phase Functions and Absolute Magnitudes Scott S. Sheppard Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Rd. NW, Washington, DC 20015 [email protected] ABSTRACT I report new time-resolved light curves and determine the rotations and phase functions of several large Kuiper Belt objects, which includes the dwarf planet Eris (2003 UB313). Three of the new sample of ten Trans-Neptunian objects display obvious short-term periodic light curves. (120348) 2004 TY364 shows a light curve which if double-peaked has a period of 11.70±0.01 hours and a peak- to-peak amplitude of 0.22±0.02 magnitudes. (84922) 2003 VS2 has a well defined double-peaked light curve of 7.41±0.02 hours with a 0.21±0.02 magnitude range. (126154) 2001 YH140 shows variability of 0.21± 0.04 magnitudes with a possible 13.25±0.2 hour single-peaked period. The seven new KBOs in the sample which show no discernible variations within the uncertainties on short rotational time scales are 2001 UQ18, (55565) 2002 AW197, (119979) 2002 WC19, (120132) 2003 FY128, (136108) Eris 2003 UB313, (90482) Orcus 2004 DW, and (90568) 2004 GV9. Four of the ten newly sampled Kuiper Belt objects were observed over a significant range of phase angles to determine their phase functions and absolute magnitudes. The three medium to large sized Kuiper Belt objects 2004 TY364, Orcus and 2004 GV9 show fairly steep linear phase curves (∼ 0.18 to 0.26 mags per degree) between phase angles of 0.1 and 1.5 degrees. This is consistent with previous measurements obtained for moderately sized Kuiper Belt objects. The extremely large dwarf planet Eris (2003 UB313) shows a shallower phase curve (0.09 ± 0.03 mags per degree) which is more similar to the other known dwarf planet Pluto. It appears the surface properties of the largest dwarf planets in the Kuiper Belt maybe different than the smaller Kuiper Belt objects. This may have to do with the larger objects ability to hold more volatile ices as well as sustain atmospheres. Finally, it is found that the absolute magnitudes obtained using the phase slopes found for individual objects are several tenths of magnitudes different than that given by the Minor Planet Center. http://arxiv.org/abs/0704.1636v2 – 2 – Subject headings: Kuiper Belt — Oort Cloud — minor planets, asteroids — solar system: general — planets and satellites: individual (2001 UQ18, (126154) 2001 YH140, (55565) 2002 AW197, (119979) 2002 WC19, (120132) 2003 FY128, (136199) Eris 2003 UB313, (84922) 2003 VS2, (90482) Orcus 2004 DW, (90568) 2004 GV9, and (120348) 2004 TY364) 1. Introduction To date only about 1% of the Trans-Neptunian objects (TNOs) are known of the nearly one hundred thousand expected larger than about 50 km in radius just beyond Neptune’s orbit (Trujillo et al. 2001). The majority of the largest Kuiper Belt objects (KBOs) now being called dwarf Plutonian planets (radii > 400 km) have only recently been discovered in the last few years (Brown et al. 2005). The large self gravity of the dwarf planets will allow them to be near Hydrostatic equilibrium, have possible tenuous atmospheres, retain extremely volatile ices such as Methane and are likely to be differentiated. Thus the surfaces as well as the interior physical characteristics of the largest TNOs may be significantly different than the smaller TNOs. The largest TNOs have not been observed to have any remarkable differences from the smaller TNOs in optical and near infrared broad band color measurements (Doressoundiram et al. 2005; Barucci et al. 2005). But near infrared spectra has shown that only the three largest TNOs (Pluto, Eris (2003 UB313) and (136472) 2005 FY9) have obvious Methane on their surfaces while slightly smaller objects are either spectrally featureless or have strong water ice signatures (Brown et al. 2005; Licandro et al. 2006). In addition to the Near infrared spectra differences, the albedos of the larger objects appear to be predominately higher than those for the smaller objects (Cruikshank et al. 2005; Bertoldi et al. 2006; Brown et al. 2006). A final indication that the larger objects are indeed different is that the shapes of the largest KBOs seem to signify they are more likely to be in hydrostatic equilibrium than that for the smaller KBOs (Sheppard and Jewitt 2002; Trilling and Bernstein 2006; Lacerda and Luu 2006). The Kuiper Belt has been dynamically and collisionally altered throughout the age of the solar system. The largest KBOs should have rotations that have been little influenced since the sculpting of the primordial Kuiper Belt. This is not the case for the smaller KBOs where recent collisions and fragmentation processes will have highly modified their spins throughout the age of the solar system (Davis and Farinella 1997). The large volatile rich KBOs show significantly different median period and possible amplitude rotational differ- ences when compared to the rocky large main belt asteroids which is expected because of – 3 – their differing compositions and collisional histories (Sheppard and Jewitt 2002; Lacerda and Luu 2006). I have furthered the photometric monitoring of large KBOs (absolute magnitudes H < 5.5 or radii greater than about 100 km assuming moderate albedos) in order to determine their short term rotational and long term phase related light curves to better understand their rotations, shapes and possible surface characteristics. This is a continuation of previous works (Jewitt and Sheppard 2002; Sheppard and Jewitt 2002; Sheppard and Jewitt 2003; Sheppard and Jewitt 2004). 2. Observations The data for this work were obtained at the Dupont 2.5 meter telescope at Las Campanas in Chile and the University of Hawaii 2.2 meter telescope atop Mauna Kea in Hawaii. Observations at the Dupont 2.5 meter telescope were performed on the nights of Febru- ary 14, 15 and 16, March 9 and 10, October 25, 26, and 27, November 28, 29, and 30 and December 1, 2005 UT. The instrument used was the Tek5 with a 2048 × 2048 pixel CCD with 24 µm pixels giving a scale of 0.′′259 pixel−1 at the f/7.5 Cassegrain focus for a field of view of about 8′.85 × 8′.85. Images were acquired through a Harris R-band filter while the telescope was autoguided on nearby bright stars at sidereal rates (Table 1). Seeing was generally good and ranged from 0.′′6 to 1.′′5 FWHM. Observations at the University of Hawaii 2.2 meter telescope were obtained on the nights of December 19, 21, 23 and 24, 2003 UT and used the Tektronix 2048 × 2048 pixel CCD. The pixels were 24 µm in size giving 0.′′219 pixel−1 scale at the f/10 Cassegrain focus for a field of view of about 7′.5 × 7′.5. Images were obtained in the R-band filter based on the Johnson-Kron-Cousins system with the telescope auto-guiding at sidereal rates using nearby bright stars. Seeing was very good over the several nights ranging from 0.′′6 to 1.′′2 FWHM. For all observations the images were first bias subtracted and then flat-fielded using the median of a set of dithered images of the twilight sky. The photometry for the KBOs was done in two ways in order to optimize the signal-to-noise ratio. First, aperture correction photometry was performed by using a small aperture on the KBOs (0.′′65 to 1.′′04 in radius) and both the same small aperture and a large aperture (2.′′63 to 3.′′63 in radius) on several nearby unsaturated bright field stars. The magnitude within the small aperture used for the KBOs was corrected by determining the correction from the small to the large aperture using the PSF of the field stars. Second, I performed photometry on the KBOs using the same field stars but only using the large aperture on the KBOs. The smaller apertures allow better – 4 – photometry for the fainter objects since it uses only the high signal-to-noise central pixels. The range of radii varies because the actual radii used depends on the seeing. The worse the seeing the larger the radius of the aperture needed in order to optimize the photometry. Both techniques found similar results, though as expected, the smaller aperture gives less scatter for the fainter objects while the larger aperture is superior for the brighter objects. Photometric standard stars from Landolt (1992) were used for calibration. Each in- dividual object was observed at all times in the same filter and with the same telescope setup. Relative photometric calibration from night to night was very stable since the same fields stars were observed. The few observations that were taken in mildly non-photometric conditions (i.e. thin cirrus) were easily calibrated to observations of the same field stars on the photometric nights. Thus, the data points on these mildly non-photometric nights are almost as good as the other data with perhaps a slightly larger error bar. The dominate source of error in the photometry comes from simple root N noise. 3. Light Curve Causes The apparent magnitude or brightness of an atmospherless inert body in our solar system is mainly from reflected sunlight and can be calculated as mR = m⊙ − 2.5log 2φ(α)/(2.25× 1016R2∆2) in which r [km] is the radius of the KBO, R [AU] is the heliocentric distance, ∆ [AU] is the geocentric distance, m⊙ is the apparent red magnitude of the sun (−27.1), mR is the apparent red magnitude, pR is the red geometric albedo, and φ(α) is the phase function in which the phase angle α = 0 deg at opposition and φ(0) = 1. The apparent magnitude of the TNO may vary for the main following reasons: 1) The geometry in which R,∆ and/or α changes for the TNO. Geometrical consider- ations at the distances of the TNOs are usually only noticeable over a few weeks or longer and thus are considered long-term variations. These are further discussed in section 5. 2) The TNOs albedo, pR, may not be uniform on its surface causing the apparent magnitude to vary as the different albedo markings on the TNOs surface rotate in and out of our line of sight. Albedo or surface variations on an object usually cause less than a 30% difference from maximum to minimum brightness of an object. (134340) Pluto, because of its atmosphere (Spencer et al. 1997), has one of the highest known amplitudes from albedo variations (∼ 0.3 magnitudes; Buie et al. 1997). 3) Shape variations or elongation of an object will cause the effective radius of an object – 5 – to our line of sight to change as the TNO rotates. A double peaked periodic light curve is expected to be seen in this case since the projected cross section would go between two minima (short axis) and two maxima (long axis) during one complete rotation of the TNO. Elongation from material strength is likely for small TNOs (r < 100 km) but for the larger TNOs observed in this paper no significant elongation is expected from material strength because their large self gravity. A large TNO (r > 100 km) may be significantly elongated if it has a large amount of rotational angular momentum. An object will be near breakup if it has a rotation period near the critical rotation period (Pcrit) at which centripetal acceleration equals gravitational acceleration towards the center of a rotating spherical object, Pcrit = where G is the gravitational constant and ρ is the density of the object. With ρ = 103 kg m−3 the critical period is about 3.3 hours. At periods just below the critical period the object will likely break apart. For objects with rotations significantly above the critical period the shapes will be bimodal Maclaurin spheroids which do not shown any significant rotational light curves produced by shape (Jewitt and Sheppard 2002). For periods just above the critical period the equilibrium figures are triaxial ellipsoids which are elongated from the large centripetal force and usually show prominent rotational light curves (Weidenschilling 1981; Holsapple 2001; Jewitt and Sheppard 2002). For an object that is triaxially elongated the peak-to-peak amplitude of the rotational light curve allows for the determination of the projection of the body shape into the plane of the sky by (Binzel et al. 1989) ∆m = 2.5log − 1.25log a2cos2θ + c2sin2θ b2cos2θ + c2sin2θ where a ≥ b ≥ c are the semiaxes with the object in rotation about the c axis, ∆m is expressed in magnitudes, and θ is the angle at which the rotation (c) axis is inclined to the line of sight (an object with θ = 90 deg. is viewed equatorially). The amplitudes of the light curves produced from rotational elongation can range up to about 0.9 magnitudes (Leone et al. 1984). Assuming θ = 90 degrees gives a/b = 100.4∆m. Thus the easily measured quantities of the rotation period and amplitude can be used to determine a minimum density for an object if it is assumed to be rotational elongated and strengthless (i.e. the bodies structure behaves like a fluid, Chandrasekhar 1969). The two best cases of this high angular momentum – 6 – elongation in the Kuiper Belt are (20000) Varuna (Jewitt and Sheppard 2002) and (136108) 2003 EL61 (Rabinowitz et al. 2006). 4) Periodic light curves may be produced if a TNO is an eclipsing or contact binary. A double-peaked light curve would be expected with a possible characteristic notch shape near the minimum of the light curve. Because the two objects may be tidally elongated the light curves can range up to about 1.2 magnitudes (Leone et al. 1984). The best example of such an object in the Kuiper Belt is 2001 QG298 (Sheppard and Jewitt 2004). 5) A non-periodic short-term light curve may occur from a complex rotational state, a recent collision, a binary with each component having a large light curve amplitude and a different rotation period or outgassing/cometary activity. These types of short term vari- ability are expected to be extremely rare and none have yet been reliably detected in the Kuiper Belt (Sheppard and Jewitt 2003; Belskaya et al. 2006) 4. Light Curve Results and Analysis The photometric measurements for the 10 newly observed KBOs are listed in Table 1, where the columns include the start time of each integration, the corresponding Julian date, and the magnitude. No correction for light travel time has been made. Results of the light curve analysis for all the KBOs newly observed are summarized in Table 2. The phase dispersion minimization (PDM) method (Stellingwerf 1978) was used to search for periodicity in the individual light curves. In PDM, the metric is the so-called Theta parameter, which is essentially the variance of the unphased data divided by the variance of the data when phased by a given period. The best fit period should have a very small dispersion compared to the unphased data and thus Theta << 1 indicates that a good fit has been found. In practice, a Theta less than 0.4 indicates a possible periodic signature. 4.1. (120348) 2004 TY364 Through the PDM analysis I found a strong Theta minima for 2004 TY364 near a period of P = 5.85 hours with weaker alias periods flanking this (Figure 1). Phasing the data to all possible periods in the PDM plot with Theta < 0.4 found that only the single-peaked period near 5.85 hours and the double-peaked period near 11.70 hours fits all the data obtained from October, November and December 2005. Both periods have an equally low Theta parameter of about 0.15 and either could be the true rotation period (Figures 2 and 3). The peak-to-peak amplitude is 0.22± 0.02 magnitudes. – 7 – If 2004 TY364 has a double-peaked period it may be elongated from its high angular momentum. If the TNO is assumed to be observed equator on then from Equation 3 the a : b axis ratio is about 1.2. Following Jewitt and Sheppard (2002) I assume the TNO is a rotationally elongated strengthless rubble pile. Using the spin period of 11.7 hours, the 1.2 a : b axis ratio found above and the Jacobi ellipsoid tables produced by Chandrasekhar (1969) I find the minimum density of 2004 TY364 is about 290 kg m −3 with an a : c axis ratio of about 1.9. This density is quite low which leads one to believe either the TNO is not being viewed equator on or the relatively long double-peaked period is not created from high angular momentum of the object. 4.2. (84922) 2003 VS2 The KBO 2003 VS2 has a very low Theta of less than 0.1 near 7.41 hours in the PDM plot (Figure 4). Phasing the December 2003 data to this period shows a well defined double- peaked period (Figure 5). The single peaked period for this result would be near 3.71 hours which was a possible period determined for this object by Ortiz et al. (2006). The 3.71 hour single-peaked period does not look as convincing (Figure 6) which confirms the PDM result that the single-peaked period has about three times more dispersion than the double-peaked period. This is likely because one of the peaks is taller in amplitude (∼ 0.05 mags) and a little wider. The other single-peaked period of 4.39 hours (Figure 7) and the double-peaked period of 8.77 hours (Figure 8) mentioned by Oritz et al. (2006) do not show a low Theta in the PDM and also do not look convincing when examining the phased data. The peak- to-peak amplitude is 0.21 ± 0.02 magnitudes, which is similar to that detected by Ortiz et al. (2006). The fast rotation of 7.41 hours and double-peaked nature suggests that 2003 VS2 may be elongated from its high angular momentum. Using Equation 3 and assuming the TNO is observed equator on the a : b axis ratio is about 1.2. Using the spin period of 7.41 hours, the 1.2 a : b axis ratio and the Jacobi ellipsoid tables produced by Chandrasekhar (1969) I find the minimum density of 2003 VS2 is about 720 kg m −3 with an a : c axis ratio of about 1.9. This result is similar to other TNO densities found through the Jacobian Ellipsoid assumption (Jewitt and Sheppard 2002; Sheppard and Jewitt 2002; Rabinowitz et al. 2006) as well as recent thermal results from the Spitzer space telescope (Stansberry et al. 2006). – 8 – 4.3. (126154) 2001 YH140 (126154) 2001 YH140 shows variability of 0.21 ± 0.04 magnitudes. The PDM for this TNO shows possible periods near 8.5, 9.15, 10.25 and 13.25 hours though only the 13.25 hour period has a Theta less than 0.4 (Figure 9). Visibly examining the phased data finds only the 13.25 hour period is viable (Figure 10). This is consistent with the observation that one minimum and one maximum were shown on December 23, 2003 in about six and a half hours, which would give a single-peaked light curve of twice this time or about 13.25 hours. Ortiz et al. (2006) found this object to have a similar variability but with very limited data could not obtain a reliable period. Ortiz et al. did have one period of 12.99 hours which may be consistent with our result. 4.4. Flat Rotation Curves Seven of the ten newly observed KBOs; 2001 UQ18, (55565) 2002 AW197, (119979) 2002 WC19, (120132) 2003 FY128, (136199) Eris 2003 UB313, (90482) Orcus 2004 DW, and (90568) 2004 GV9 showed no variability within the photometric uncertainties of the observations (Table 2; Figures 11 to 21). These KBOs thus either have extremely long rotational periods, are viewed nearly pole-on or most likely have small peak-to-peak rotational amplitudes. The upper limits for the objects short-term rotational variability as shown in Table 2 were determined through a monte carlo simulation. The monte carlo simulation determined the lowest possible amplitude that would be seen in the data from the time sampling and variance of the photometry as well as the errors on the individual points. Ortiz et al. (2006) reported a possible 0.04± 0.02 photometric range for (90482) Orcus 2004 DW and a period near 10 hours. I do not confirm this result here. Ortiz et al. (2006) also reported a marginal 0.08± 0.03 photometric range for (55565) 2002 AW197 with no one clear best period. I can not confirm this result and find that for 2002 AW197 the rotational variability appears significantly less than 0.08 magnitudes. Some of the KBOs in this sample appear to have variability which is just below the threshold of the data detection and thus no significant period could be obtained with the current data. In particular 2001 UQ18 appears to have a light curve with a significant amplitude above 0.1 magnitudes but the data is sparser for this object than most the others and thus no significant period is found. Followup observations will be required in order to determine if most of these flat light curve objects do have any significant short-term variability. – 9 – 4.5. Comparisons with Size, Amplitude, Period, and MBAs In Figures 22 and 23 are plotted the diameters of the largest TNOs and Main Belt Asteroids (MBAs) versus rotational amplitude and period, respectively. Most outliers on Figure 22 can easily be explained from the discussion in section 3. Varuna, 2003 EL61 and the other unmarked TNOs with photometric ranges above about 0.4 magnitudes are all spinning faster than about 8 hours. They are thus likely hydrostatic equilibrium triaxial Jacobian ellipsoids which are elongated from their rotational angular momentum (Jewitt and Sheppard 2002; Sheppard and Jewitt 2002; Rabinowitz et al. 2006). 2001 QG298’s large photometric range is probably because this object is a contact binary indicative of its longer period and notched shaped light curve (Sheppard and Jewitt 2004). Pluto’s relatively large amplitude light curve is best explained through its active atmosphere (Spencer et al. 1997). Like the MBAs, the photometric amplitudes of the TNOs start to increase significantly at sizes less than about 300 km in diameter. The likely reason is this size range is where the objects are still large enough to be dominated by self-gravity and are not easily disrupted through collisions but can still have their angular momentum highly altered from the collisional process (Farinella et al. 1982; Davis and Farinella 1997). Thus this is the region most likely to be populated by high angular momentum triaxial Jacobian ellipsoids (Farinella et al. 1992). From this work Eris (2003 UB313) has one of the highest signal-to-noise time-resolved photometry measurements of any TNO searched for a rotational period. There is no obvi- ous rotational light curve larger than about 0.01 magnitudes in our extensive data which indicates a very uniform surface, a rotation period of over a few days or a pole-on view- ing geometry. Carraro et al. (2006) suggest a possible 0.05 magnitude variability for Eris between nights but this is not obvious in this data set. The similar inferred composition and size of Eris to Pluto suggests these objects should behave very similar (Brown et al. 2005,2006). Since Pluto has a relatively substantial atmosphere at its current position of about 30 AU (Elliot et al. 2003; Sicardy et al. 2003) it is very likely that Eris has an active atmosphere when near its perihelion of 38 AU. At Eris’ current distance of 97 AU its surface thermal temperature should be over 20 degrees colder than when at perihelion. Like Pluto, Eris’ putative atmosphere near perihelion would likely be composed of N2, CH4 or CO which would mostly condense when near aphelion (Spencer et al. 1997; Hubbard 2003), effectively resurfacing the TNO every few hundred years. This is the most likely explanation as to why the surface of Eris appears so uniform. This may also be true for 2005 FY9 which appears compositionally similar to Pluto (Licandro et al. 2006) and at 52 AU is about 15 degrees colder than Pluto. Figure 23 shows that the median rotation period distribution for TNOs is about 9.5 ± – 10 – 1 hours which is marginally larger than for similarly sized main belt asteroids (7.0 ± 1 hours)(Sheppard and Jewitt 2002; and Lacerda and Luu 2006). If confirmed, the likely reason for this difference are the collisional histories of each reservoir as well as the objects compositions. 5. Phase Curve Results The phase function of an objects surface mostly depends on the albedo, texture and particle structure of the regolith. Four of the newly imaged TNOs (Eris 2003 UB313, (120348) 2004 TY364, Orcus 2004 DW, and (90568) 2004 GV9) were viewed on two separate telescope observing runs occurring at significantly different phase angles (Figures 24 to 27). This allowed their linear phase functions, φ(α) = 10−0.4βα (4) to be estimated where α is the phase angle in degrees and β is the linear phase coefficient in magnitudes per degree (Table 3). The phase angles for TNOs are always less than about 2 degrees as seen from the Earth. Most atmosphereless bodies show opposition effects at such small phase angles (Muinonen et al. 2002). The TNOs appear to have mostly linear phase curves between phase angles of about 2 and 0.1 degrees (Sheppard and Jewitt 2002,2003; Rabinowitz et al. 2007). For phase angles smaller than about 0.1 degrees TNOs may display an opposition spike (Hicks et al. 2005; Belskaya et al. 2006). The moderate to large KBOs Orcus, 2004 TY364, and 2004 GV9 show steep linear R-band phase slopes (0.18 to 0.26 mags per degree) similar to previous measurements of similarly sized moderate to large TNOs (Sheppard and Jewitt 2002,2003; Rabinowitz et al. 2007). In contrast the extremely large dwarf planet Eris (2003 UB313) has a shallower phase slope (0.09 mags per degree) more similar to Charon (∼ 0.09 mags/deg; Buie et al. (1997)) and possibly Pluto (∼ 0.03 mags/deg; Buratti et al. (2003)). Empirically lower phase coefficients between 0.5 and 2 degrees may correspond to bright icy objects whose surfaces have probably been recently resurfaced such as Triton, Pluto and Europa (Buie et al. 1997; Buratti et al. 2003; Rabinowitz et al. 2007). Thus Eris’ low β is consistent with it having an icy surface that has recently been resurfaced. In Figures 28 to 32 are plotted the linear phase coefficients found for several TNOs versus several different parameters (reduced magnitude, albedo, rotational photometric amplitude and B − I broad band color). Table 4 shows the significance of any correlations. Based on only a few large objects it appears that the larger TNOs may have lower β values. This is true for the R-band and V-band data at the 97% confidence level but interestingly using – 11 – data from Rabinowitz et al. (2007) no correlation is seen in the I-band (Table 4). Thus further measurements are needed to determine if there is a significantly strong correlation between the size and phase function of TNOs. Further, it may be that the albedos are anti- correlated with β, but since we have such a small number of albedos known the statistics don’t give a good confidence in this correlation. If confirmed with additional observations, these correlations may be an indication that larger TNOs surfaces are less susceptible to phase angle opposition effects at optical wavelengths. This could be because the larger TNOs have different surface properties from smaller TNOs due to active atmospheres, stronger self-gravity or different surface layers from possible differentiation. 5.1. Absolute Magnitudes From the linear phase coefficient the reduced magnitude, mR(1, 1, 0) = mR−5log(R∆) or absolute magnitude H (Bowell et al. 1989), which is the magnitude of an object if it could be observed at heliocentric and geocentric distances of 1 AU and a phase angle of 0 degrees, can be estimated (see Sheppard and Jewitt 2002 for further details). The results for mR(1, 1, 0) and H are found to be consistent to within a couple hundreths of a magnitude (Table 3 and Figures 24 to 27). It is found that the R-band empirically determined absolute magnitudes of individual TNOs appears to be several tenths of a magnitude different than what is given by the Minor Planet Center (Table 3). This is likely because the MPC assumes a generic phase function and color for all TNOs while these two physical properties appear to be significantly different for individual KBOs (Jewitt and Luu 1998). The work by Romanishin and Tegler (2005) attempts to determine various absolute magnitudes of TNOs by using main belt asteroid type phase curves which are not appropriate for TNOs (Sheppard and Jewitt 2002). 6. Summary Ten large trans-Neptunian objects were observed in the R-band to determine photomet- ric variability on times scales of hours, days and months. 1) Three of the TNOs show obvious short-term photometric variability which is taken to correspond to their rotational states. • (120348) 2004 TY364 shows a double-peaked period of 11.7 hours and if single-peaked is 5.85 hours. The peak-to-peak amplitude of the light curve is 0.22± 0.02 mags. – 12 – • (84922) 2003 VS2 has a well defined double-peaked period of 7.41 hours with a peak- to-peak amplitude of 0.21± 0.02 mags. If the light curve is from elongation than 2003 VS2’s a/b axis ratio is at least 1.2 and the a/c axis ratio is about 1.9. Assuming 2003 VS2 is elongated from its high angular momentum and is a strengthless rubble pile it would have a minimum density of about 720 kg m−3. • (126154) 2001 YH140 has a single-peaked period of about 13.25 hours with a photo- metric range of 0.21± 0.04 mags. 2) Seven of the TNOs show no short-term photometric variability within the measure- ment uncertainties. • Photometric measurements of the large TNOs (90482) Orcus and (55565) 2002 AW197 showed no variability within or uncertainties. Thus these measurements do not confirm possible small photometric variability found for these TNOs by Ortiz et al. (2006). • No short-term photometric variability was found for (136199) Eris 2003 UB313 to about the 0.01 magnitude level. This high signal to noise photometry suggests Eris is nearly spherical with a very uniform surface. Such a nearly uniform surface may be explained by an atmosphere which is frozen onto the surface of Eris when near aphelion. The atmosphere, like Pluto’s, may become active when near perihelion effectively resurfac- ing Eris every few hundred years. The Methane rich TNO 2005 FY9 may also be in a similar situation. 3) Four of the TNOs were observed over significantly different phase angles allowing their long term photometric variability to be measured between phase angles of 0.1 and 1.5 degrees. • TNOs Orcus, 2004 TY364 and 2004 GV9 show steep linear R-band phase slopes between 0.18 and 0.26 mags/degree. • Eris 2003 UB313 shows a shallower R-band phase slope of 0.09 mags/degree. This is consistent with Eris having a high albedo, icy surface which may have recently been resurfaced. • At the 97% confidence level the largest TNOs have shallower R-band linear phase slopes compared to smaller TNOs. The largest TNOs surfaces may differ from the smaller TNOs because of their more volatile ice inventory, increased self-gravity, active atmospheres, differentiation process or collisional history. – 13 – 3) The absolute magnitudes determined for several TNOs through measuring their phase curves show a difference of several tenths of a magnitude from the Minor Planet Center values. • The values found for the reduced magnitude, mR(1, 1, 0), and absolute magnitude, H , are similar to within a few hundreths of a magnitude for most TNOs. 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Observations of Kuiper Belt Objects Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) 2001 UQ18 uq1223n3025 1.17 320 2003 12 23.22561 22.20 0.04 uq1223n3026 1.15 320 2003 12 23.23060 22.30 0.04 uq1223n3038 1.02 320 2003 12 23.28384 22.38 0.04 uq1223n3039 1.01 320 2003 12 23.28882 22.56 0.04 uq1223n3051 1.01 350 2003 12 23.34333 22.40 0.04 uq1223n3052 1.01 350 2003 12 23.35123 22.48 0.04 uq1223n3070 1.15 350 2003 12 23.41007 22.37 0.04 uq1223n3071 1.17 350 2003 12 23.41540 22.28 0.04 uq1224n4024 1.21 350 2003 12 24.21433 22.30 0.03 uq1224n4025 1.19 350 2003 12 24.21969 22.15 0.03 uq1224n4033 1.03 350 2003 12 24.27591 22.07 0.03 uq1224n4034 1.02 350 2003 12 24.28125 22.04 0.03 uq1224n4041 1.00 350 2003 12 24.31300 22.11 0.03 uq1224n4042 1.00 350 2003 12 24.31834 22.14 0.03 uq1224n4051 1.04 350 2003 12 24.36433 22.22 0.03 uq1224n4052 1.05 350 2003 12 24.36967 22.18 0.03 uq1224n4061 1.17 350 2003 12 24.41216 22.27 0.03 uq1224n4062 1.20 350 2003 12 24.41750 22.22 0.03 uq1224n4072 1.50 350 2003 12 24.46253 22.14 0.03 uq1224n4073 1.56 350 2003 12 24.46781 22.09 0.03 (126154) 2001 YH140 yh1219n1073 1.10 300 2003 12 19.42900 20.85 0.02 yh1219n1074 1.08 300 2003 12 19.43381 20.82 0.02 yh1219n1084 1.01 300 2003 12 19.47450 20.81 0.02 yh1219n1085 1.01 300 2003 12 19.47935 20.79 0.02 yh1219n1092 1.00 300 2003 12 19.51172 20.77 0.02 yh1219n1093 1.00 300 2003 12 19.51657 20.80 0.02 yh1219n1112 1.06 300 2003 12 19.56332 20.86 0.02 yh1219n1113 1.08 300 2003 12 19.56815 20.80 0.02 yh1219n1116 1.15 350 2003 12 19.59215 20.81 0.02 yh1219n1117 1.18 350 2003 12 19.59764 20.79 0.02 – 17 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) yh1219n1122 1.36 350 2003 12 19.63042 20.87 0.02 yh1219n1123 1.41 350 2003 12 19.63587 20.85 0.02 yh1219n1125 1.50 350 2003 12 19.64669 20.95 0.03 yh1221n2067 1.46 300 2003 12 21.35652 20.98 0.02 yh1221n2068 1.42 300 2003 12 21.36124 20.92 0.02 yh1223n3059 1.24 300 2003 12 23.38285 20.86 0.02 yh1223n3060 1.21 300 2003 12 23.38762 20.89 0.02 yh1223n3078 1.03 300 2003 12 23.44626 20.88 0.02 yh1223n3079 1.02 300 2003 12 23.45102 20.87 0.02 yh1223n3086 1.00 300 2003 12 23.48192 20.92 0.02 yh1223n3087 1.00 300 2003 12 23.48668 20.91 0.02 yh1223n3091 1.00 300 2003 12 23.51268 20.94 0.02 yh1223n3092 1.01 300 2003 12 23.51744 20.95 0.02 yh1223n3101 1.08 300 2003 12 23.55695 20.92 0.02 yh1223n3102 1.09 300 2003 12 23.56169 20.96 0.03 yh1223n3106 1.18 300 2003 12 23.58754 20.98 0.03 yh1223n3107 1.20 300 2003 12 23.59231 21.01 0.03 yh1223n3114 1.31 300 2003 12 23.61182 21.03 0.03 yh1223n3115 1.34 300 2003 12 23.61659 20.99 0.03 yh1223n3119 1.56 300 2003 12 23.64084 20.99 0.03 yh1224n4047 1.49 300 2003 12 24.34589 20.90 0.02 yh1224n4048 1.44 300 2003 12 24.35066 20.91 0.02 yh1224n4057 1.18 300 2003 12 24.39217 20.85 0.02 yh1224n4058 1.16 300 2003 12 24.39693 20.85 0.02 yh1224n4068 1.03 300 2003 12 24.44421 20.87 0.02 yh1224n4069 1.02 300 2003 12 24.44898 20.87 0.02 yh1224n4080 1.00 300 2003 12 24.49899 20.84 0.02 yh1224n4081 1.00 300 2003 12 24.50375 20.86 0.02 yh1224n4088 1.02 300 2003 12 24.52567 20.82 0.02 yh1224n4089 1.03 300 2003 12 24.53043 20.83 0.02 – 18 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) yh1224n4093 1.08 300 2003 12 24.55588 20.81 0.02 yh1224n4094 1.09 300 2003 12 24.56065 20.82 0.02 yh1224n4102 1.23 300 2003 12 24.59447 20.87 0.02 yh1224n4103 1.25 300 2003 12 24.59926 20.82 0.02 yh1224n4107 1.44 300 2003 12 24.62661 20.87 0.02 (55565) 2002 AW197 aw1223n3088 1.09 220 2003 12 23.49223 19.89 0.01 aw1223n3089 1.08 220 2003 12 23.49606 19.89 0.01 aw1223n3093 1.03 220 2003 12 23.52320 19.87 0.01 aw1223n3094 1.03 220 2003 12 23.52704 19.88 0.01 aw1223n3103 1.02 220 2003 12 23.56663 19.89 0.01 aw1223n3104 1.02 220 2003 12 23.57046 19.89 0.01 aw1223n3108 1.05 220 2003 12 23.59807 19.89 0.01 aw1223n3109 1.06 220 2003 12 23.60190 19.89 0.01 aw1223n3116 1.11 220 2003 12 23.62171 19.87 0.01 aw1223n3117 1.12 220 2003 12 23.62556 19.89 0.01 aw1223n3122 1.26 220 2003 12 23.65822 19.87 0.01 aw1223n3123 1.28 220 2003 12 23.66201 19.89 0.01 aw1224n4066 1.34 220 2003 12 24.43521 19.87 0.01 aw1224n4067 1.32 220 2003 12 24.43903 19.86 0.01 aw1224n4078 1.09 220 2003 12 24.48975 19.89 0.01 aw1224n4079 1.08 220 2003 12 24.49358 19.89 0.01 aw1224n4086 1.04 220 2003 12 24.51683 19.86 0.01 aw1224n4087 1.03 220 2003 12 24.52066 19.89 0.01 aw1224n4091 1.01 220 2003 12 24.54768 19.90 0.01 aw1224n4092 1.01 220 2003 12 24.55158 19.90 0.01 aw1224n4100 1.04 220 2003 12 24.58659 19.86 0.01 aw1224n4101 1.04 220 2003 12 24.59042 19.86 0.01 aw1224n4105 1.10 220 2003 12 24.61789 19.86 0.01 aw1224n4106 1.12 220 2003 12 24.62172 19.87 0.01 aw1224n4111 1.25 220 2003 12 24.65382 19.86 0.01 – 19 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) aw1224n4112 1.27 220 2003 12 24.65766 19.87 0.01 aw1224n4113 1.30 220 2003 12 24.66150 19.88 0.01 aw1224n4114 1.32 220 2003 12 24.66534 19.88 0.01 (119979) 2002 WC19 wc1219n1033 1.19 350 2003 12 19.27766 20.56 0.02 wc1219n1045 1.05 300 2003 12 19.32341 20.61 0.02 wc1219n1046 1.04 300 2003 12 19.32826 20.59 0.02 wc1219n1057 1.00 300 2003 12 19.36042 20.60 0.02 wc1219n1058 1.00 300 2003 12 19.36529 20.57 0.02 wc1219n1066 1.01 300 2003 12 19.40263 20.56 0.02 wc1219n1067 1.02 300 2003 12 19.40748 20.57 0.02 wc1219n1077 1.11 300 2003 12 19.44804 20.61 0.02 wc1219n1078 1.12 300 2003 12 19.45289 20.58 0.02 wc1219n1088 1.33 300 2003 12 19.49419 20.59 0.02 wc1219n1089 1.37 300 2003 12 19.49909 20.55 0.02 wc1219n1094 1.58 300 2003 12 19.52222 20.57 0.02 wc1219n1095 1.64 300 2003 12 19.52704 20.58 0.02 wc1221n2026 1.64 300 2003 12 21.21505 20.56 0.02 wc1221n2027 1.59 300 2003 12 21.21980 20.57 0.02 wc1221n2042 1.26 300 2003 12 21.25881 20.55 0.02 wc1221n2043 1.24 300 2003 12 21.26356 20.53 0.02 wc1221n2065 1.01 300 2003 12 21.33897 20.58 0.02 wc1221n2066 1.01 300 2003 12 21.34373 20.63 0.02 wc1223n3027 1.38 300 2003 12 23.23616 20.57 0.02 wc1223n3028 1.34 300 2003 12 23.24092 20.60 0.02 wc1223n3044 1.05 300 2003 12 23.30891 20.57 0.02 wc1223n3045 1.04 300 2003 12 23.31367 20.57 0.02 wc1223n3057 1.00 300 2003 12 23.37221 20.58 0.02 wc1223n3058 1.00 300 2003 12 23.37696 20.56 0.02 wc1223n3076 1.10 320 2003 12 23.43506 20.57 0.02 wc1223n3077 1.12 320 2003 12 23.44005 20.60 0.02 – 20 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) wc1223n3084 1.25 320 2003 12 23.47067 20.61 0.02 wc1223n3085 1.28 320 2003 12 23.47566 20.60 0.02 wc1224n4026 1.44 300 2003 12 24.22597 20.55 0.02 wc1224n4027 1.40 300 2003 12 24.23073 20.56 0.02 wc1224n4035 1.10 300 2003 12 24.28804 20.58 0.02 wc1224n4036 1.09 300 2003 12 24.29281 20.61 0.02 wc1224n4043 1.02 300 2003 12 24.32492 20.58 0.02 wc1224n4044 1.02 300 2003 12 24.32969 20.58 0.02 wc1224n4053 1.00 300 2003 12 24.37574 20.54 0.02 wc1224n4054 1.01 300 2003 12 24.38049 20.56 0.02 wc1224n4063 1.07 350 2003 12 24.42313 20.57 0.02 wc1224n4076 1.32 300 2003 12 24.47888 20.58 0.02 wc1224n4077 1.35 300 2003 12 24.48365 20.61 0.02 (120132) 2003 FY128 fy0309n037 1.16 350 2005 03 09.30416 20.29 0.02 fy0309n038 1.17 350 2005 03 09.30906 20.31 0.02 fy0309n045 1.32 350 2005 03 09.34449 20.29 0.02 fy0309n046 1.35 350 2005 03 09.34942 20.28 0.02 fy0309n051 1.55 350 2005 03 09.37484 20.28 0.02 fy0309n052 1.60 350 2005 03 09.37975 20.30 0.02 fy0310n113 1.37 300 2005 03 10.13114 20.33 0.02 fy0310n114 1.34 300 2005 03 10.13543 20.31 0.02 fy0310n121 1.15 300 2005 03 10.18141 20.27 0.02 fy0310n122 1.14 300 2005 03 10.18572 20.29 0.02 fy0310n131 1.08 250 2005 03 10.23854 20.28 0.02 fy0310n132 1.08 250 2005 03 10.24229 20.27 0.02 fy0310n142 1.17 300 2005 03 10.30636 20.29 0.02 fy0310n146 1.27 300 2005 03 10.33107 20.27 0.02 fy0310n147 1.29 300 2005 03 10.33564 20.27 0.02 fy0310n152 1.51 300 2005 03 10.36726 20.25 0.02 fy0310n153 1.55 300 2005 03 10.37157 20.22 0.02 – 21 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) (136199) Eris 2003 UB313 ub1026c142 1.71 350 2005 10 25.02653 18.372 0.006 ub1026c143 1.65 350 2005 10 25.03144 18.374 0.005 ub1026c150 1.37 300 2005 10 25.06369 18.370 0.005 ub1026c156 1.35 300 2005 10 25.06800 18.361 0.005 ub1026c162 1.11 250 2005 10 25.19559 18.361 0.005 ub1026c170 1.11 250 2005 10 25.19931 18.364 0.005 ub1026c171 1.16 250 2005 10 25.22449 18.360 0.005 ub1026c174 1.16 250 2005 10 25.22825 18.370 0.005 ub1026c175 1.25 300 2005 10 25.25305 18.327 0.005 ub1026c178 1.27 300 2005 10 25.25694 18.365 0.005 ub1026c179 1.38 300 2005 10 25.27710 18.350 0.005 ub1026c183 1.41 300 2005 10 25.28146 18.369 0.005 ub1026c184 1.54 300 2005 10 25.29766 18.365 0.005 ub1026c187 1.58 300 2005 10 25.30202 18.351 0.005 ub1026c188 1.78 350 2005 10 25.31871 18.364 0.006 ub1026c189 1.85 350 2005 10 25.32363 18.364 0.006 ub1027c043 1.83 250 2005 10 26.01513 18.356 0.006 ub1027c044 1.78 250 2005 10 26.01890 18.362 0.006 ub1027c049 1.34 200 2005 10 26.06597 18.360 0.005 ub1027c050 1.18 300 2005 10 26.10460 18.352 0.005 ub1027c069 1.10 300 2005 10 26.14440 18.348 0.005 ub1027c070 1.11 250 2005 10 26.19650 18.352 0.005 ub1027c074 1.11 250 2005 10 26.20049 18.365 0.005 ub1027c075 1.15 300 2005 10 26.21742 18.348 0.005 ub1027c084 1.16 300 2005 10 26.22174 18.359 0.005 ub1027c085 1.20 300 2005 10 26.23858 18.351 0.005 ub1027c088 1.22 300 2005 10 26.24295 18.331 0.005 ub1027c089 1.36 300 2005 10 26.27118 18.352 0.005 ub1027c092 1.39 300 2005 10 26.27555 18.345 0.005 ub1027c093 1.51 350 2005 10 26.29210 18.344 0.005 – 22 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ub1027c096 1.56 350 2005 10 26.29702 18.345 0.005 ub1027c097 1.61 350 2005 10 26.30193 18.357 0.005 ub1028c240 1.65 300 2005 10 27.02670 18.350 0.005 ub1028c246 1.33 300 2005 10 27.06572 18.362 0.005 ub1028c258 1.10 250 2005 10 27.14482 18.357 0.005 ub1028c266 1.12 250 2005 10 27.19952 18.359 0.005 ub1028c267 1.12 250 2005 10 27.20321 18.367 0.005 ub1028c271 1.18 300 2005 10 27.22789 18.370 0.005 ub1028c272 1.19 300 2005 10 27.23216 18.366 0.005 ub1028c276 1.29 300 2005 10 27.25643 18.272 0.005 ub1028c277 1.31 300 2005 10 27.26070 18.376 0.005 ub1028c280 1.42 300 2005 10 27.27739 18.375 0.005 ub1028c281 1.45 300 2005 10 27.28171 18.369 0.005 ub1028c282 1.48 300 2005 10 27.28598 18.371 0.005 ub1028c283 1.52 300 2005 10 27.29033 18.372 0.005 ub1028c284 1.56 300 2005 10 27.29462 18.371 0.005 ub1028c285 1.61 300 2005 10 27.29900 18.381 0.005 ub1028c286 1.65 300 2005 10 27.30333 18.392 0.005 ub1028c287 1.70 300 2005 10 27.30761 18.393 0.006 ub1028c288 1.76 300 2005 10 27.31197 18.383 0.006 ub1028c289 1.82 300 2005 10 27.31625 18.369 0.006 ub1028c290 1.89 300 2005 10 27.32052 18.388 0.006 ub1028c291 1.96 300 2005 10 27.32484 18.367 0.006 ub1028c292 2.04 300 2005 10 27.32916 18.405 0.007 ub1028c293 2.13 300 2005 10 27.33347 18.378 0.007 ub1128n027 1.11 250 2005 11 28.10847 18.389 0.005 ub1128n028 1.12 250 2005 11 28.11219 18.382 0.005 ub1128n029 1.12 250 2005 11 28.11593 18.401 0.005 ub1128n032 1.16 250 2005 11 28.13378 18.391 0.005 ub1128n033 1.17 250 2005 11 28.13749 18.383 0.005 – 23 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ub1128n034 1.18 250 2005 11 28.14118 18.391 0.005 ub1128n035 1.19 250 2005 11 28.14496 18.389 0.005 ub1128n036 1.21 250 2005 11 28.14866 18.386 0.005 ub1128n037 1.22 250 2005 11 28.15236 18.380 0.005 ub1128n038 1.23 250 2005 11 28.15614 18.376 0.005 ub1128n039 1.25 250 2005 11 28.15983 18.397 0.005 ub1128n040 1.27 250 2005 11 28.16353 18.393 0.005 ub1128n041 1.28 250 2005 11 28.16732 18.379 0.005 ub1128n042 1.30 250 2005 11 28.17102 18.378 0.005 ub1128n043 1.32 250 2005 11 28.17472 18.402 0.005 ub1128n044 1.34 250 2005 11 28.17850 18.379 0.005 ub1128n045 1.37 250 2005 11 28.18220 18.385 0.005 ub1128n046 1.39 250 2005 11 28.18589 18.380 0.005 ub1128n047 1.42 250 2005 11 28.18969 18.375 0.005 ub1128n048 1.44 250 2005 11 28.19339 18.387 0.005 ub1128n049 1.47 250 2005 11 28.19708 18.396 0.005 ub1128n050 1.51 250 2005 11 28.20084 18.393 0.005 ub1128n051 1.54 250 2005 11 28.20453 18.390 0.005 ub1128n052 1.58 250 2005 11 28.20822 18.391 0.005 ub1128n053 1.61 250 2005 11 28.21195 18.391 0.005 ub1128n054 1.66 250 2005 11 28.21565 18.376 0.005 ub1128n055 1.70 250 2005 11 28.21935 18.386 0.005 ub1128n056 1.75 250 2005 11 28.22311 18.377 0.006 ub1128n057 1.80 250 2005 11 28.22680 18.379 0.006 ub1128n058 1.85 250 2005 11 28.23050 18.382 0.006 ub1128n059 1.91 250 2005 11 28.23437 18.393 0.006 ub1128n060 1.98 250 2005 11 28.23800 18.386 0.006 ub1128n061 2.05 250 2005 11 28.24173 18.376 0.007 ub1128n062 2.13 250 2005 11 28.24546 18.383 0.007 ub1129n112 1.15 250 2005 11 29.02268 18.424 0.005 – 24 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ub1129n119 1.09 250 2005 11 29.08304 18.432 0.005 ub1129n120 1.09 250 2005 11 29.08673 18.429 0.005 ub1129n121 1.10 250 2005 11 29.09043 18.421 0.005 ub1129n122 1.10 250 2005 11 29.09412 18.430 0.005 ub1129n123 1.10 250 2005 11 29.09782 18.426 0.005 ub1129n124 1.11 250 2005 11 29.10161 18.426 0.005 ub1129n125 1.11 250 2005 11 29.10530 18.431 0.005 ub1129n126 1.12 250 2005 11 29.10900 18.435 0.005 ub1129n127 1.12 250 2005 11 29.11269 18.418 0.005 ub1129n128 1.13 250 2005 11 29.11639 18.422 0.005 ub1129n129 1.14 250 2005 11 29.12018 18.435 0.005 ub1129n130 1.14 250 2005 11 29.12387 18.425 0.005 ub1129n131 1.15 250 2005 11 29.12757 18.418 0.005 ub1129n132 1.16 250 2005 11 29.13136 18.421 0.005 ub1129n133 1.17 250 2005 11 29.13506 18.421 0.005 ub1129n134 1.18 250 2005 11 29.13876 18.420 0.005 ub1129n135 1.20 250 2005 11 29.14254 18.415 0.005 ub1129n136 1.21 250 2005 11 29.14624 18.419 0.005 ub1129n137 1.22 250 2005 11 29.14993 18.424 0.005 ub1129n138 1.24 250 2005 11 29.15373 18.426 0.005 ub1129n139 1.25 250 2005 11 29.15742 18.422 0.005 ub1129n142 1.35 250 2005 11 29.17679 18.418 0.005 ub1129n143 1.37 250 2005 11 29.18049 18.421 0.005 ub1129n144 1.40 250 2005 11 29.18418 18.408 0.005 ub1129n145 1.43 250 2005 11 29.18788 18.422 0.005 ub1129n146 1.45 250 2005 11 29.19158 18.397 0.005 ub1129n147 1.48 250 2005 11 29.19531 18.412 0.005 ub1129n148 1.52 250 2005 11 29.19901 18.403 0.005 ub1129n149 1.55 250 2005 11 29.20270 18.394 0.005 ub1129n150 1.59 250 2005 11 29.20640 18.401 0.005 – 25 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ub1129n151 1.63 250 2005 11 29.21010 18.400 0.005 ub1129n152 1.67 250 2005 11 29.21388 18.405 0.005 ub1129n153 1.71 250 2005 11 29.21758 18.401 0.005 ub1129n154 1.76 250 2005 11 29.22127 18.391 0.005 ub1129n155 1.81 250 2005 11 29.22496 18.397 0.006 ub1129n156 1.87 250 2005 11 29.22866 18.396 0.006 ub1129n157 1.93 250 2005 11 29.23238 18.415 0.006 ub1129n158 2.00 250 2005 11 29.23607 18.399 0.006 ub1130n226 1.13 250 2005 11 30.11178 18.386 0.005 ub1130n227 1.13 250 2005 11 30.11548 18.386 0.005 ub1130n228 1.14 250 2005 11 30.11918 18.394 0.005 ub1130n229 1.15 250 2005 11 30.12288 18.394 0.005 ub1130n230 1.16 250 2005 11 30.12657 18.390 0.005 ub1130n231 1.17 250 2005 11 30.13027 18.383 0.005 ub1130n232 1.18 250 2005 11 30.13397 18.398 0.005 ub1130n233 1.19 250 2005 11 30.13766 18.394 0.005 ub1130n234 1.20 250 2005 11 30.14136 18.392 0.005 ub1130n235 1.21 250 2005 11 30.14515 18.384 0.005 ub1130n236 1.23 250 2005 11 30.14884 18.391 0.005 ub1130n237 1.24 250 2005 11 30.15254 18.387 0.005 ub1130n238 1.26 250 2005 11 30.15624 18.391 0.005 ub1130n239 1.28 250 2005 11 30.15993 18.397 0.005 ub1130n240 1.29 250 2005 11 30.16370 18.388 0.005 ub1130n241 1.31 250 2005 11 30.16740 18.405 0.005 ub1130n242 1.33 250 2005 11 30.17110 18.379 0.005 ub1130n243 1.36 250 2005 11 30.17480 18.388 0.005 ub1130n244 1.38 250 2005 11 30.17849 18.383 0.005 ub1130n247 1.50 250 2005 11 30.19434 18.386 0.005 ub1130n248 1.53 250 2005 11 30.19804 18.394 0.005 ub1130n249 1.57 250 2005 11 30.20173 18.393 0.005 – 26 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ub1130n250 1.60 250 2005 11 30.20543 18.400 0.005 ub1130n251 1.64 250 2005 11 30.20912 18.397 0.005 ub1130n252 1.69 250 2005 11 30.21281 18.390 0.005 ub1130n253 1.73 250 2005 11 30.21651 18.387 0.005 ub1130n254 1.78 250 2005 11 30.22020 18.403 0.006 ub1130n255 1.84 250 2005 11 30.22389 18.399 0.006 ub1130n256 1.90 250 2005 11 30.22768 18.379 0.006 ub1130n257 1.96 250 2005 11 30.23138 18.394 0.006 ub1130n258 2.03 250 2005 11 30.23514 18.393 0.007 ub1201n327 1.10 300 2005 12 01.04542 18.378 0.005 ub1201n333 1.10 250 2005 12 01.08574 18.376 0.005 ub1201n334 1.10 250 2005 12 01.08943 18.397 0.005 ub1201n335 1.10 250 2005 12 01.09313 18.386 0.005 ub1201n338 1.12 250 2005 12 01.10868 18.391 0.005 ub1201n339 1.13 250 2005 12 01.11237 18.381 0.005 ub1201n340 1.14 250 2005 12 01.11606 18.398 0.005 ub1201n341 1.15 250 2005 12 01.11976 18.382 0.005 ub1201n342 1.16 250 2005 12 01.12347 18.385 0.005 ub1201n343 1.17 250 2005 12 01.12725 18.389 0.005 ub1201n344 1.18 250 2005 12 01.13095 18.388 0.005 ub1201n345 1.19 250 2005 12 01.13465 18.386 0.005 ub1201n346 1.20 250 2005 12 01.13843 18.384 0.005 ub1201n347 1.21 250 2005 12 01.14212 18.381 0.005 ub1201n348 1.23 250 2005 12 01.14581 18.381 0.005 ub1201n351 1.30 250 2005 12 01.16207 18.379 0.005 ub1201n352 1.32 250 2005 12 01.16577 18.394 0.005 ub1201n353 1.34 250 2005 12 01.16946 18.394 0.005 ub1201n354 1.36 250 2005 12 01.17316 18.385 0.005 ub1201n355 1.39 250 2005 12 01.17685 18.383 0.005 ub1201n356 1.41 250 2005 12 01.18055 18.391 0.005 – 27 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ub1201n357 1.44 250 2005 12 01.18424 18.379 0.005 ub1201n358 1.47 250 2005 12 01.18793 18.377 0.005 ub1201n359 1.50 250 2005 12 01.19163 18.381 0.005 ub1201n360 1.53 250 2005 12 01.19548 18.394 0.005 ub1201n361 1.57 250 2005 12 01.19918 18.389 0.005 ub1201n362 1.61 250 2005 12 01.20287 18.388 0.005 ub1201n363 1.65 250 2005 12 01.20688 18.388 0.005 ub1201n364 1.69 250 2005 12 01.21058 18.377 0.005 ub1201n365 1.74 250 2005 12 01.21427 18.390 0.006 ub1201n366 1.79 250 2005 12 01.21797 18.396 0.006 ub1201n367 1.85 250 2005 12 01.22166 18.396 0.006 ub1201n368 1.96 250 2005 12 01.22846 18.390 0.007 ub1201n369 2.03 250 2005 12 01.23228 18.382 0.007 (84922) 2003 VS2 vs1219n1031 1.07 250 2003 12 19.26838 19.39 0.01 vs1219n1032 1.06 250 2003 12 19.27266 19.36 0.01 vs1219n1043 1.02 250 2003 12 19.31376 19.37 0.01 vs1219n1044 1.02 250 2003 12 19.31810 19.41 0.01 vs1219n1055 1.03 220 2003 12 19.35203 19.53 0.01 vs1219n1056 1.04 220 2003 12 19.35594 19.52 0.01 vs1219n1064 1.11 220 2003 12 19.39432 19.53 0.01 vs1219n1065 1.12 220 2003 12 19.39821 19.52 0.01 vs1219n1075 1.29 220 2003 12 19.43959 19.39 0.01 vs1219n1076 1.31 220 2003 12 19.44346 19.38 0.01 vs1219n1086 1.67 230 2003 12 19.48544 19.34 0.01 vs1219n1087 1.72 230 2003 12 19.48944 19.38 0.01 vs1221n2024 1.26 220 2003 12 21.20617 19.52 0.01 vs1221n2025 1.24 220 2003 12 21.21014 19.52 0.01 vs1221n2040 1.10 220 2003 12 21.24958 19.53 0.01 vs1221n2041 1.09 220 2003 12 21.25340 19.52 0.01 vs1221n2046 1.05 220 2003 12 21.27486 19.46 0.01 – 28 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) vs1221n2047 1.05 220 2003 12 21.27870 19.45 0.01 vs1223n3022 1.21 200 2003 12 23.21214 19.41 0.01 vs1223n3023 1.19 200 2003 12 23.21579 19.44 0.01 vs1223n3024 1.17 250 2003 12 23.22026 19.46 0.01 vs1223n3042 1.02 220 2003 12 23.30008 19.34 0.01 vs1223n3043 1.02 220 2003 12 23.30391 19.33 0.01 vs1223n3055 1.07 220 2003 12 23.36359 19.46 0.01 vs1223n3056 1.07 220 2003 12 23.36743 19.50 0.01 vs1223n3074 1.28 220 2003 12 23.42704 19.47 0.01 vs1223n3075 1.31 220 2003 12 23.43087 19.46 0.01 vs1223n3082 1.55 200 2003 12 23.46289 19.36 0.01 vs1223n3083 1.59 200 2003 12 23.46644 19.35 0.01 vs1224n4022 1.23 250 2003 12 24.20522 19.35 0.01 vs1224n4023 1.21 250 2003 12 24.20940 19.38 0.01 vs1224n4030 1.05 220 2003 12 24.26469 19.37 0.01 vs1224n4031 1.05 220 2003 12 24.26848 19.38 0.01 vs1224n4039 1.02 220 2003 12 24.30385 19.51 0.01 vs1224n4040 1.02 220 2003 12 24.30871 19.52 0.01 vs1224n4049 1.06 220 2003 12 24.35630 19.45 0.01 vs1224n4050 1.06 220 2003 12 24.36014 19.44 0.01 vs1224n4059 1.19 220 2003 12 24.40395 19.34 0.01 vs1224n4060 1.20 220 2003 12 24.40778 19.32 0.01 vs1224n4070 1.50 220 2003 12 24.45457 19.43 0.01 vs1224n4071 1.53 220 2003 12 24.45839 19.48 0.01 (90482) Orcus 2004 DW dw0214n028 1.22 200 2005 02 14.11873 18.63 0.01 dw0214n029 1.21 200 2005 02 14.12189 18.65 0.01 dw0215n106 1.84 250 2005 02 15.03735 18.64 0.01 dw0215n107 1.78 250 2005 02 15.04156 18.66 0.01 dw0215n108 1.73 250 2005 02 15.04534 18.65 0.01 dw0215n109 1.69 250 2005 02 15.04911 18.64 0.01 – 29 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) dw0215n113 1.42 250 2005 02 15.07806 18.65 0.01 dw0215n114 1.39 250 2005 02 15.08183 18.65 0.01 dw0215n118 1.26 220 2005 02 15.10658 18.65 0.01 dw0215n119 1.25 220 2005 02 15.10998 18.64 0.01 dw0215n128 1.11 220 2005 02 15.17545 18.65 0.01 dw0215n129 1.11 220 2005 02 15.17885 18.65 0.01 dw0215n140 1.18 220 2005 02 15.24664 18.65 0.01 dw0215n141 1.19 220 2005 02 15.25007 18.65 0.01 dw0215n147 1.33 220 2005 02 15.28450 18.63 0.01 dw0215n148 1.35 220 2005 02 15.28789 18.66 0.01 dw0215n155 1.68 230 2005 02 15.32663 18.65 0.01 dw0215n156 1.73 230 2005 02 15.33014 18.64 0.01 dw0216n199 1.76 250 2005 02 16.04005 18.65 0.01 dw0216n200 1.72 250 2005 02 16.04379 18.67 0.01 dw0216n205 1.51 250 2005 02 16.06390 18.66 0.01 dw0216n206 1.47 250 2005 02 16.06767 18.67 0.01 dw0216n209 1.37 250 2005 02 16.08251 18.65 0.01 dw0216n210 1.35 250 2005 02 16.08625 18.66 0.01 dw0216n217 1.17 250 2005 02 16.13055 18.66 0.01 dw0216n218 1.16 250 2005 02 16.13437 18.66 0.01 dw0216n235 1.21 250 2005 02 16.25223 18.64 0.01 dw0216n247 1.81 300 2005 02 16.33285 18.66 0.01 dw0309n014 1.21 250 2005 03 09.05919 18.71 0.01 dw0309n015 1.20 250 2005 03 09.06295 18.70 0.01 dw0309n022 1.11 300 2005 03 09.11334 18.72 0.01 dw0309n023 1.11 300 2005 03 09.11762 18.71 0.01 dw0309n027 1.11 300 2005 03 09.13928 18.69 0.01 dw0309n028 1.11 300 2005 03 09.14363 18.70 0.01 dw0310n091 1.43 250 2005 03 10.01315 18.71 0.01 dw0310n092 1.40 250 2005 03 10.01688 18.71 0.01 – 30 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) dw0310n097 1.27 250 2005 03 10.04113 18.72 0.01 dw0310n098 1.25 250 2005 03 10.04487 18.72 0.01 dw0310n107 1.12 250 2005 03 10.09569 18.71 0.01 dw0310n108 1.12 250 2005 03 10.09945 18.72 0.01 dw0310n125 1.26 250 2005 03 10.20552 18.70 0.01 dw0310n126 1.28 250 2005 03 10.20927 18.71 0.01 dw0310n135 1.67 300 2005 03 10.26076 18.72 0.01 (90568) 2004 GV9 gv0215n130 1.75 250 2005 02 15.18402 19.75 0.03 gv0215n131 1.70 250 2005 02 15.18792 19.81 0.03 gv0215n142 1.19 250 2005 02 15.25502 19.77 0.03 gv0215n143 1.17 250 2005 02 15.25891 19.73 0.03 gv0215n153 1.03 250 2005 02 15.31558 19.74 0.03 gv0215n154 1.03 250 2005 02 15.31931 19.79 0.03 gv0215n159 1.00 250 2005 02 15.34893 19.77 0.03 gv0215n160 1.00 250 2005 02 15.35268 19.80 0.03 gv0215n165 1.01 250 2005 02 15.38618 19.79 0.03 gv0215n166 1.02 250 2005 02 15.38992 19.83 0.03 gv0216n229 1.41 250 2005 02 16.21394 19.74 0.03 gv0216n230 1.38 250 2005 02 16.21768 19.76 0.03 gv0216n242 1.05 300 2005 02 16.29937 19.76 0.03 gv0216n254 1.01 300 2005 02 16.37745 19.75 0.03 gv0309n029 1.46 300 2005 03 09.14960 19.64 0.02 gv0309n030 1.42 300 2005 03 09.15392 19.66 0.02 gv0309n033 1.01 300 2005 03 09.27972 19.75 0.02 gv0309n034 1.00 300 2005 03 09.28405 19.73 0.02 gv0309n039 1.01 300 2005 03 09.31533 19.70 0.02 gv0309n040 1.01 300 2005 03 09.31965 19.67 0.02 gv0309n047 1.06 300 2005 03 09.35654 19.70 0.02 gv0309n048 1.07 300 2005 03 09.36090 19.68 0.02 gv0309n054 1.18 300 2005 03 09.39525 19.68 0.02 – 31 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) gv0309n055 1.20 300 2005 03 09.39959 19.69 0.02 gv0310n115 1.51 300 2005 03 10.14103 19.64 0.02 gv0310n116 1.44 300 2005 03 10.14905 19.66 0.02 gv0310n127 1.11 300 2005 03 10.21489 19.64 0.02 gv0310n128 1.09 300 2005 03 10.21918 19.68 0.02 gv0310n143 1.01 300 2005 03 10.31197 19.75 0.02 gv0310n148 1.04 300 2005 03 10.34106 19.72 0.02 gv0310n149 1.05 300 2005 03 10.34539 19.77 0.02 gv0310n155 1.14 300 2005 03 10.38172 19.76 0.02 gv0310n158 1.20 300 2005 03 10.39711 19.72 0.02 gv0310n159 1.22 300 2005 03 10.40147 19.73 0.02 (120348) 2004 TY364 ty1025n041 1.89 400 2005 10 25.01425 19.89 0.01 ty1025n042 1.81 400 2005 10 25.01944 19.86 0.01 ty1025n047 1.45 400 2005 10 25.05162 19.87 0.01 ty1025n048 1.41 400 2005 10 25.05711 19.92 0.01 ty1025n067 1.04 350 2005 10 25.18450 19.98 0.01 ty1025n068 1.04 350 2005 10 25.18939 19.99 0.01 ty1025n072 1.06 350 2005 10 25.21357 19.95 0.01 ty1025n073 1.07 350 2005 10 25.21847 19.92 0.01 ty1025n082 1.12 350 2005 10 25.24193 19.89 0.01 ty1025n083 1.13 350 2005 10 25.24683 19.90 0.01 ty1025n086 1.19 350 2005 10 25.26632 19.90 0.01 ty1025n087 1.21 350 2005 10 25.27123 19.90 0.01 ty1025n090 1.29 350 2005 10 25.28657 19.89 0.01 ty1025n091 1.32 350 2005 10 25.29147 19.95 0.01 ty1025n094 1.42 350 2005 10 25.30718 19.95 0.01 ty1025n095 1.46 350 2005 10 25.31207 20.00 0.01 ty1025n098 1.63 350 2005 10 25.32928 19.98 0.01 ty1025n099 1.69 350 2005 10 25.33418 19.97 0.01 ty1025n100 1.76 350 2005 10 25.33909 20.02 0.01 – 32 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ty1025n101 1.83 350 2005 10 25.34410 20.03 0.01 ty1026n144 1.71 400 2005 10 26.02367 19.85 0.01 ty1026n145 1.64 400 2005 10 26.02917 19.86 0.01 ty1026n151 1.30 350 2005 10 26.07121 20.02 0.01 ty1026n157 1.13 400 2005 10 26.11043 20.07 0.01 ty1026n163 1.06 400 2005 10 26.14943 20.03 0.01 ty1026n172 1.06 400 2005 10 26.20516 19.92 0.01 ty1026n176 1.09 400 2005 10 26.22689 19.88 0.01 ty1026n177 1.10 400 2005 10 26.23234 19.90 0.01 ty1026n181 1.16 450 2005 10 26.25425 19.94 0.01 ty1026n182 1.19 450 2005 10 26.26393 19.92 0.01 ty1026n185 1.27 400 2005 10 26.28048 19.95 0.01 ty1026n186 1.30 400 2005 10 26.28599 19.98 0.01 ty1026n190 1.45 450 2005 10 26.30748 20.03 0.01 ty1026n191 1.50 450 2005 10 26.31356 20.09 0.01 ty1026n192 1.56 450 2005 10 26.31963 20.09 0.01 ty1026n193 1.62 450 2005 10 26.32568 20.07 0.01 ty1026n194 1.70 450 2005 10 26.33173 20.09 0.01 ty1026n195 1.78 450 2005 10 26.33774 20.10 0.01 ty1026n196 1.87 450 2005 10 26.34378 20.08 0.01 ty1027n241 1.58 400 2005 10 27.03210 20.01 0.01 ty1027n247 1.28 400 2005 10 27.07100 20.04 0.01 ty1027n264 1.05 400 2005 10 27.18749 19.89 0.01 ty1027n265 1.05 400 2005 10 27.19292 19.87 0.01 ty1027n269 1.07 400 2005 10 27.21551 19.91 0.01 ty1027n270 1.08 400 2005 10 27.22094 19.90 0.01 ty1027n274 1.14 400 2005 10 27.24440 19.90 0.01 ty1027n275 1.15 400 2005 10 27.24988 19.89 0.01 ty1027n278 1.22 350 2005 10 27.26622 19.98 0.01 ty1027n279 1.24 350 2005 10 27.27115 19.99 0.01 – 33 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ty1027n294 1.83 400 2005 10 27.33881 20.09 0.01 ty1027n295 1.92 400 2005 10 27.34429 20.09 0.01 ty1027n296 2.02 400 2005 10 27.34976 20.10 0.01 ty1128n030 1.07 350 2005 11 28.12175 19.99 0.01 ty1128n031 1.07 350 2005 11 28.12665 19.99 0.01 ty1128n063 1.87 400 2005 11 28.25204 20.17 0.01 ty1128n064 1.96 400 2005 11 28.25747 20.15 0.01 ty1129n117 1.05 350 2005 11 29.07101 20.00 0.01 ty1129n118 1.04 350 2005 11 29.07586 20.00 0.01 ty1129n140 1.17 350 2005 11 29.16375 20.11 0.01 ty1129n141 1.19 350 2005 11 29.16860 20.12 0.01 ty1129n159 1.76 350 2005 11 29.24196 20.17 0.01 ty1129n160 1.83 350 2005 11 29.24682 20.13 0.01 ty1129n161 1.91 350 2005 11 29.25167 20.15 0.01 ty1129n162 1.99 350 2005 11 29.25657 20.08 0.01 ty1130n224 1.05 350 2005 11 30.10029 19.98 0.01 ty1130n225 1.05 350 2005 11 30.10514 20.00 0.01 ty1130n245 1.27 350 2005 11 30.18325 20.18 0.01 ty1130n246 1.30 350 2005 11 30.18810 20.15 0.01 ty1130n259 1.77 350 2005 11 30.23992 20.12 0.01 ty1130n260 1.84 350 2005 11 30.24478 20.10 0.01 ty1130n261 1.92 350 2005 11 30.24963 20.13 0.01 ty1130n262 2.01 350 2005 11 30.25454 20.11 0.01 ty1201n328 1.06 400 2005 12 01.05104 19.99 0.01 ty1201n336 1.05 350 2005 12 01.09816 20.08 0.01 ty1201n337 1.05 350 2005 12 01.10301 20.07 0.01 ty1201n349 1.14 350 2005 12 01.15006 20.18 0.01 ty1201n350 1.16 350 2005 12 01.15492 20.13 0.01 ty1201n370 1.77 350 2005 12 01.23777 20.05 0.01 ty1201n371 1.85 350 2005 12 01.24262 20.07 0.01 – 34 – Table 1—Continued Name Imagea Airmass Expb UT Datec Mag.d Err (sec) yyyy mm dd.ddddd (mR) (mR) ty1201n372 1.93 350 2005 12 01.24748 20.08 0.01 ty1201n373 2.02 350 2005 12 01.25239 20.04 0.01 aImage number. bExposure time for the image. cDecimal Universal Date at the start of the integration. dApparent red magnitude. eUncertainties on the individual photometric measurements. – 35 – Table 2. Properties of Observed KBOs Name Ha mR b Nightsc ∆mR d Singlee Doublef (mag) (mag) (#) (mag) (hrs) (hrs) 2001 UQ18 5.4 22.3 2 < 0.3 - - (126154) 2001 YH140 5.4 20.85 4 0.21± 0.04 13.25± 0.2 - (55565) 2002 AW197 3.3 19.88 2 < 0.03 - - (119979) 2002 WC19 5.1 20.58 4 < 0.05 - - (120132) 2003 FY128 5.0 20.28 2 < 0.08 - - (136199) Eris 2003 UB313 -1.2 18.36 7 < 0.01 - - (84922) 2003 VS2 4.2 19.45 4 0.21± 0.02 - 7.41± 0.02 (90482) Orcus 2004 DW 2.3 18.65 5 < 0.03 - - (90568) 2004 GV9 4.0 19.68 4 < 0.08 - - (120348) 2004 TY364 4.5 19.98 7 0.22± 0.02 5.85± 0.01 11.70± 0.01 aThe visible absolute magnitude of the object from the Minor Planet Center. The values from the MPC differ than the R-band absolute magnitudes found for the few objects in which we have actual phase curves as shown in Table 3. bMean red magnitude of the object. For the four objects observed at significantly different phase angles the data near the lowest phase angle is used: Eris in Oct. 2005, Orcus in Feb. 2005, 90568 in Mar. 2005, and 120348 in Oct. 2005. cNumber of nights data were taken to determine the lightcurve. dThe peak to peak range of the lightcurve. eThe lightcurve period if there is one maximum per period. fThe lightcurve period if there are two maximum per period. – 36 – Table 3. Phase Function Data for KBOs Name mR(1, 1, 0) a Hb MPCc β(α < 2◦)d (mag) (mag) (mag) (mag/deg) (136199) Eris 2003 UB313 −1.50± 0.02 −1.50± 0.02 −1.65 0.09± 0.03 (90482) Orcus 2004 DW 1.81± 0.05 1.81± 0.05 1.93 0.26± 0.05 (90568) 2004 GV9 3.64± 0.06 3.62± 0.06 3.5 0.18± 0.06 (120348) 2004 TY364 3.91± 0.03 3.90± 0.03 4.0 0.19± 0.03 aThe R-band reduced magnitude determined from the linear phase coefficient found in this work. bThe R-band absolute magnitude determined as described in Bowell et al. (1989). cThe R-band absolute magnitude from the Minor Planet Center converted from the V-band as is shown in Table 2 to the R-band using the known colors of the objects: V-R= 0.45 for Eris (Brown et al. 2005), V-R= 0.37 for Orcus (de Bergh et al. 2005), and a nominal value of V-R= 0.5 for 90568 and 120348 since these objects don’t have known V-R colors. dβ(α < 2◦) is the phase coefficient in magnitudes per degree at phase angles < 2◦. – 37 – Table 4. Phase Function Correlations β vs.a rcorr b Nc Sigd mR(1, 1, 0) 0.50 19 97% mV (1, 1, 0) 0.54 16 97% mI(1, 1, 0) 0.12 14 < 60% pR -0.51 5 65% pV -0.38 9 70% pI -0.27 10 < 60% ∆m -0.21 19 < 60% B − I -0.20 11 < 60% aβ is the linear phase coefficient in magnitudes per degree at phase angles < 2◦. In the column are what β is compared to in order to see if there is any correlation; mR(1, 1, 0), mV (1, 1, 0) and mI(1, 1, 0) are the reduced mangitudes in the R, V and I-band respectively and are compared to the value of β determined at the same wavelength; pR, pV and pI are the geometric albedos compared to β in the R, V and I-band respectively; ∆m is the peak-to-peak amplitude of the rotational light curve; and B − I is the color. The phase curves in the R-band are from this work and Sheppard and Jewitt (2002;2003) while the V and I-band data are from Buie et al. (1997) and Rabinowitz et al. (2007). The albedo information is from Cruikshank et al. (2006) and the colors from Barucci et al. (2005). brcorr is the Pearson correlation coefficient. cN is the number of TNOs used for the correlation. dSig is the confidence of significance of the correlation. – 38 – Fig. 1.— The Phase Dispersion Minimization (PDM) plot for (120348) 2004 TY364. The best fit single-peaked period is near 5.85 hours. – 39 – Fig. 2.— The phased best fit single-peaked period for (120348) 2004 TY364 of 5.85 hours. The peak-to-peak amplitude is about 0.22 magnitudes. The data from November and December has been vertically shifted to correspond to the same phase angle as the data from October using the phase function found for this object in this work. Individual error bars for the measurements are not shown for clarity but are generally ±0.01 mags as seen in Table 1. – 40 – Fig. 3.— The phased double-peaked period for (120348) 2004 TY364 of 11.70 hours. The data from November and December has been vertically shifted to correspond to the same phase angle as the data from October using the phase function found for this object in this work. Individual error bars for the measurements are not shown for clarity but are generally ±0.01 mags as seen in Table 1. – 41 – Fig. 4.— The Phase Dispersion Minimization (PDM) plot for (84922) 2003 VS2. The best fit is the double-peaked period near 7.41 hours. – 42 – Fig. 5.— The phased best fit double-peaked period for (84922) 2003 VS2 of 7.41 hours. The peak-to-peak amplitude is about 0.21 magnitudes. The two peaks have differences since one is slightly wider while the other is slightly shorter in amplitude. This is the best fit period for (84922) 2003 VS2. Individual error bars for the measurements are not shown for clarity but are generally ±0.01 mags as seen in Table 1. – 43 – Fig. 6.— The phased single-peaked period for (84922) 2003 VS2 of 3.70 hours. The single peaked period for 2003 VS2 does not look well matched and has a larger scatter about the solution compared to the double-peaked period shown in Figure 5. Individual error bars for the measurements are not shown for clarity but are generally ±0.01 mags as seen in Table – 44 – Fig. 7.— The phased single-peaked period for (84922) 2003 VS2 of 4.39 hours. Again, the single peaked period for 2003 VS2 does not look well matched and has a larger scatter about the solution compared to the double-peaked period shown in Figure 5. Individual error bars for the measurements are not shown for clarity but are generally ±0.01 mags as seen in Table – 45 – Fig. 8.— The phased double-peaked period for (84922) 2003 VS2 of 8.77 hours. This double- peaked period for 2003 VS2 does not look well matched and has a larger scatter about the solution compared to the 7.41 hour double-peaked period shown in Figure 5. Individual error bars for the measurements are not shown for clarity but are generally ±0.01 mags as seen in Table 1. – 46 – Fig. 9.— The Phase Dispersion Minimization (PDM) plot for 2001 YH140. The best fit is the single-peaked period near 13.25 hours. The other possible fits near 8.5, 9.15 and 10.25 hours don’t look good when phasing the data and viewing the result by eye. – 47 – Fig. 10.— The phased best fit single-peaked period for 2001 YH140 of 13.25 hours. The peak- to-peak amplitude is about 0.21 magnitudes. Individual error bars for the measurements are not shown for clarity but are generally ±0.02 mags as seen in Table 1. – 48 – Fig. 11.— The flat light curve of 2001 UQ18. The KBO may have a significant amplitude light curve but further observations are needed to confirm. – 49 – Fig. 12.— The flat light curve of (55565) 2002 AW197. The KBO has no significant short-term variations larger than 0.03 magnitudes over two days. – 50 – Fig. 13.— The flat light curve of (119979) 2002 WC19. The KBO has no significant short- term variations larger than 0.03 magnitudes over four days. – 51 – Fig. 14.— The flat light curve of (119979) 2002 WC19. The KBO has no significant short- term variations larger than 0.03 magnitudes over four days. – 52 – Fig. 15.— The flat light curve of (120132) 2003 FY128. The KBO has no significant short- term variations larger than 0.08 magnitudes over two days. – 53 – Fig. 16.— The flat light curve of Eris (2003 UB313) in October 2005. The KBO has no significant short-term variations larger than 0.01 magnitudes over several days. – 54 – Fig. 17.— The flat light curve of Eris (2003 UB313) in November and December 2005. The KBO has no significant short-term variations larger than 0.01 magnitudes over several days. – 55 – Fig. 18.— The flat light curve of (90482) Orcus 2004 DW in February 2005. The KBO has no significant short-term variations larger than 0.03 magnitudes over several days. – 56 – Fig. 19.— The flat light curve of (90482) Orcus 2004 DW in March 2005. The KBO has no significant short-term variations larger than 0.03 magnitudes over several days. – 57 – Fig. 20.— The flat light curve of (90568) 2004 GV9 in February 2005. The KBO has no significant short-term variations larger than 0.1 magnitudes over several days. – 58 – Fig. 21.— The flat light curve of (90568) 2004 GV9 in March 2005. The KBO has no significant short-term variations larger than 0.1 magnitudes over several days. – 59 – Fig. 22.— This plot shows the diameter of asteroids and TNOs versus their light curve amplitudes. The TNOs sizes if unknown assume they have moderate albedos of about 10 percent. For objects with flat light curves they are plotted at the variation limit found by observations. – 60 – Fig. 23.— Same as the previous figure except the diameter versus the light curve period is plotted. The dashed line is the median of known TNOs rotation periods (9.5 ± 1 hours) which is significantly above the median large MBAs rotation periods (7.0± 1 hours). Pluto falls off the graph in the upper right corner because of its slow rotation created by the tidal locking to its satellite Charon. – 61 – Fig. 24.— The phase curve for Eris (2003 UB313). The dashed line is the linear fit to the data while the solid line uses the Bowell et al. (1989) H-G scattering formalism. In order to create only a few points with small error bars, the data has been averaged for each observing night. – 62 – Fig. 25.— The phase curve for (90482) Orcus 2004 DW. The dashed line is the linear fit to the data while the solid line uses the Bowell et al. (1989) H-G scattering formalism. In order to create only a few points with small error bars, the data has been averaged for each observing night. – 63 – Fig. 26.— The phase curve for (120348) 2004 TY364. The dashed line is the linear fit to the data while the solid line uses the Bowell et al. (1989) H-G scattering formalism. In order to create only a few points with small error bars, the data has been averaged for each observing night. – 64 – Fig. 27.— The phase curve for (90568) 2004 GV9. The dashed line is the linear fit to the data while the solid line uses the Bowell et al. (1989) H-G scattering formalism. In order to create only a few points with small error bars, the data has been averaged for each observing night. – 65 – Fig. 28.— The R-band reduced magnitude versus the R-band linear phase coefficient β(α < 2 degrees) for TNOs. R-band data is from this work and Sheppard and Jewitt (2002),(2003) as well as Sedna from Rabinowitz et al. (2007) and Pluto from Buratti et al. (2003). A linear fit is shown by the dahsed line. Larger objects (smaller reduced magnitudes) may have smaller β at the 97% confidence level using the Pearson correlation coefficient. – 66 – Fig. 29.— Same as Figure 28 except for the V-band (squares) and I-band (diamonds). Pluto and Charon data are from Buie et al. (1997) and the other data are from Rabinowitz et al. (2007). Error bars are usually less than 0.04 mags/deg. The V-band data shows a similar correlation (97% confidence, dashed line) as found for the R-band data in Figure 28, that is larger objects may have smaller β. There is no correlation found using the I-band data (dotted line). – 67 – Fig. 30.— Same as Figures 28 and 29 except is the albedo versus linear phase coefficient for TNOs. Filled circles are R-band data, squares are V-band and diamonds are I-band data. Albedos are from Cruikshank et al. (2006). – 68 – Fig. 31.— Same as Figure 28 except is the light curve amplitude versus the linear phase coefficient for TNOs. TNOs with no measured rotational variability are plotted with their possible amplitude upper limits. No significant correlation is found. – 69 – Fig. 32.— Same as Figure 28 except is the B-I broad band colors versus the linear phase coefficient for TNOs. Colors are from Barucci et al. (2005). No significant correlation is found.

0704.1637

Fischler-Susskind holographic cosmology revisited

Fischler-Susskind holographic cosmology revisited Pablo Diaz∗ M. A. Per†, Antonio Segui‡ Departamento de Fisica Teorica Universidad de Zaragoza. 50009-Zaragoza. Spain Abstract When Fischler and Susskind proposed a holographic prescription based on the Particle Horizon, they found that spatially closed cosmological models do not verify it due to the apparently unavoidable recontraction of the Particle Horizon area. In this article, after a short review of their original work, we expose graphically and analytically that spatially closed cosmological models can avoid this problem if they expand fast enough. It has been also shown that the Holographic Principle is saturated for a codimension one brane dominated Universe. The Fischler-Susskind prescription is used to obtain the maximum number of degrees of freedom per Planck volume at the Planck era compatible with the Holographic Principle. ∗e-mail: [email protected] †e-mail: [email protected] ‡e-mail: [email protected] http://arxiv.org/abs/0704.1637v2 1 Introduction One of the most promising ideas that emerged in theoretical physics during the last decade was the Holographic Principle according to which a physical system can be described uniquely by degrees of freedom living on its boundary [1, 2]. If the Holographic Principle is indeed a primary principle of fundamental physics it should be verified when the entire universe is considered as a physical system. That is, the physical information inside any cosmological domain should be holographically codified on its boundary area. But obviously, if an unlimited region of scale L is considered, its entropy content will scale like volume L3 and its boundary area like L2; so inevitably the former will grow quicker than the second and the holographic codification will be impossible for big size cosmological domains. The origin of the Holographic Principle is related to black hole horizons; so, it seems natural to relate it now to any kind of cosmological horizon. It is at this stage when the causal relationship that gives rise to cosmological horizons should be taken into account. William Fischler and Leonard Susskind proposed a cosmological holographic prescription based on the particle horizon [3] SPH ≤ . (1) The entropy content inside the particle horizon of a cosmological observer cannot be greater than one quarter of the horizon area in Planck units. Enforcing this condition for the future of any cosmological model with constant ω = p/ρ (Friedmann-Robertson- Walker models, FRW) spatially flat, Fischler and Susskind found the limit ω < 1. The compatibility of this limit with the dominant energy condition seems to support the Fischler-Susskind (FS) holographic prescription. In section 2, a detailed deduction of this limit is shown. Moreover, the verification of the FS prescription in the past is enforced, finding a limit for the entropy density in the Planck era. On the other hand, in spatially closed cosmological models, the FS holographic pre- scription yields to apparently unavoidable problems. Indeed, if the model has compact homogeneous spatial sections, all of them of finite volume, then a physical system cannot have an arbitrary big size at a given time. But for this kind of cosmological models the boundary area does not grow uniformly when the size of a cosmological domain increases. Graphically, it is shown that when the domain crosses the equator the boundary area begins to decrease, going to zero when the domain reaches the antipodes and covers the entire universe [3, 4]. Figure 1 show this behavior for spatial dimension n = 2. Raphael Bousso proposed a different holographic prescription [4, 5] based on the evalua- tion of the entropy content over certain null sections named light-sheets. This prescription solves the problems associated to spatially closed cosmological models, but it also lacks the simplicity of the FS prescription. The Bousso prescription will not be used here but it can be shown that both prescriptions are closely related: Two of the light-sheets de- fined by Bousso give rise to the past light cone of a cosmological observer1. According to our previous work [6], the entropy content over the past light cone is proportional to the entropy content over the particle horizon (defined over the homogeneous spatial section of the observer), and for adiabatic expansion both will be exactly the same. In fact, the 1According to the Bousso’s nomenclature, every past light cone can be built with the light sheets (+-) and (-+) associated to the maximum of that cosmological light cone, also called apparent horizon [4, 5]. Figure 1: Decrease of the area of a domain defined in a compact spatial section when its volume increases and goes beyond one half of the total volume (further than the equator). original FS prescription applies to the entropy content over the ingoing past directed null section associated to a given spherical boundary; the key is that the verification for the particle horizon (1) guarantees the verification for every spherical boundary. In conclu- sion, the FS holographic prescription (1) also imposes a limit on the entropy content over the past light cone, and then it may also be regarded covariant as well as the Bousso prescription. In section 3 of this paper general explicit solutions for the area and the volume of spherical cosmological domains are obtained in spatially closed (n+1)-dimensional FRW models. It is shown that, in fact, the boundary area of the particle horizon defined in recontract- ing models (dominated by conventional matter) tends to zero; so, the FS holographic prescription will be violated for this kind of models. But it is also shown that non- recontracting models, that is, spatially closed (n+1)-dimensional FRW models dominated by quintessence matter (bouncing models), do not necessarily present this problematical behavior. These models present accelerated expansion, and particularly only the most accelerated models avoid the collapse of the particle horizon. So, it is deduced that a rapid enough cosmological expansion does not allow the particle horizon to evolve enough over the hyperspheric spatial section to reach the antipodes, so the boundary area never decreases. It will be shown that the sufficiently accelerated FRW model corresponds to universes dominated by a codimension one brane gas; thus, such a fluid could saturate the Holographic Principle. Section 3 concludes with a discussion of our results in contrast with other related works. Especially interesting are the recent works about holographic dark energy. The simplified argument is that a holographic limit on the entropy of a cosmological domain could also imply a limit of its energy content; thus, the Holographic Principle applied to cosmology might illuminate the dark energy problem [7, 8]. It is argued how our results could improve the compatibility between the particle horizon and the holographic dark energy. Finally, section 4 exposes the basic conclusions of our work. 2 Fischler-Susskind holography in flat universes We will consider (n+1)-dimensional cosmological models with constant parameter ω = p/ρ (FRW models). Here we study the spatially flat case k = 0; the scale factor grows according to the potential function R(t) = R0 n(1+ω) ∝ t1− α (2) where subscript 0 refers to the value of a magnitude in an arbitrary reference time t0. For later convenience we have defined n(1 + ω) n(1 + ω)− 2 n being the spatial dimension of the model. In this section, only conventional matter dominated models –which are decelerated and verify α > 1– will be considered, and quintessence dominated models –which are accelerated and verify α < 0– are left for the next section. Table 1 summarizes these cases and gives the specific limiting values acceleration ω-range α-range denomination R̈ < 0 − 1 < ω ≤ +1 α ≥ > 0 conventional matter R̈ = 0 ω = − 1 α = ∞ curvature dominated R̈ > 0 −1 ≤ ω < − 1 α ≤ 0 quintessence matter Table 1: Relation among the cosmological acceleration, the dynamically dominant matter and the parameters of its equation of state ω and α. The ranges can be obtained from the spatially flat case (2) but they are also valid for the positively (18) and negatively curved case. The dominant energy condition |ω| ≤ 1 and the value ω = −1 related with a cosmological constant (de Sitter universe) has been also included. Given the scale factor, the particle horizon (named in [9] like future event horizon) for decelerated FRW models can be obtained as [10, 11, 12] DPH(t) = R(t) R(t′) = αt . (4) Assuming adiabatic expansion, the entropy in a comoving volume must be constant; so, the spatial entropy density scales like s(t)R(t)n = s0R 0 = constant ⇒ s(t) = s0R 0 R(t) −n. (5) Now the entropy content inside the particle horizon can be computed SPH(t) = s(t)VPH(t) = s0R 0 R(t) −n ωn−1 DPH(t) n , (6) where ωn−1 is the area of the unit sphere. The FS holographic prescription [3] demands that the above entropy content must not be greater than one quarter of the particle horizon area (1). Then SPH(t) = s(t) DPH(t) APH(t) = ωn−1DPH(t) n−1 , (7) performing some cancelations and introducing (5) we arrive at DPH(t) ≤ 4s(t) R(t)n . (8) This inequality is the simplified form of the FS holographic prescription for spatially flat cosmological models. Now, according to the FS work the inequality should be imposed in the future of any FRW model. For this purpose, comparing the exponents of temporal evolution is sufficient: the particle horizon evolves linearly (4) and the scale factor evolves according to (2). Thus, we obtain a family of cosmological models which will verify the FS holographic prescription in the future n(1 + ω) ⇒ ω < 1 . (9) This bound on the parameter of the equation of state coincides with the limit of Special Relativity; the sound speed in a fluid given by v2 = δp/δρ must not be greater than the speed of light. When ω = 1, the entropic limit could be also verified depending on the numerical prefactors (see condition (11) below). So, according to this, the dominant energy condition enables the verification of the FS holographic prescription2 in the future. But the previous FS argument presents an objection that we will not obviate. If we enforce that in the future the particle horizon area dominates over its entropy content, being potential functions, it is unavoidable that in the past the entropy content dominates over the horizon area. In other words, these mathematical functions intersects in a given time, so that at any previous time the holographic codification will be impossible. This intersection time depends on the numeric prefactors that we have previously left out. Our proposal is the enforcement of the intersection time near the Planck time; thus, the apparent violation of the holographic prescription will be restricted to the Planck era. Imposing this limit we will obtain an interesting relation involving the numeric prefactors; so, we have to enforce the simplified holographic relation (8) at the Planck time (tP l = 1). Using (4) and (3) we reach SPH(tP l) ≤ APH(tP l) ⇒ α < 4 sP l ⇒ sP l < 1 + ω . (10) The first idea about this result is that the verification of the Holographic Principle needs, in general, not too high an entropy density; concretely, the FS prescription gives us a limit 2The reverse implication is not valid: the FS prescription allows temporal violations of the dominant energy condition [13]. on the entropy density at the Planck time. This fact is usually skipped in the literature. Perhaps it is assumed that an entropy density at the Planck time sP l of the same order as one is not problematic. A second view at the previous result may take one to interpret it as a restriction the Holographic Principle imposes on the complexity of our world: the number of degrees of freedom per Planck volume at the Planck era must not be greater than the previous value. Thus, taking n = 3 and assuming a radiation dominated universe (ω = 1/3) at early times, we get sP l < 3/8. Note also that this result does not depend on the final behavior of the model, in a way that is also valid for our universe which is supposed to be dominated now by some kind of dark energy. Restriction (10) is not trivial. If we consider a cosmological model dynamically dominated by a fluid with ω very near to the limit ωlim = − 1 (α = ∞ ) , (11) then, the entropy density required at Planck time (10) will be absurdly small. This is because the models with fluid of matter driven by (11) do not present particle horizon (R(t) ∝ t); near this limit the particle horizon becomes arbitrarily big, so the entropy content –scaled with the volume– can hardly be codified on the horizon area. Moreover, according to [14] the observational data are compatible with a universe very near the linear evolution; so this case cannot be discarded. Bousso [4], Kaloper and Linde [15] proposed an ad hoc solution based on a redefinition of the particle horizon. They took integral (4) from the Planck time t = 1 instead of t = 0 as the starting point. However, it is not a valid solution for accelerated models (ω < ωlim ∼ α < 0); let us see the reason. According to the new prescription, the redefined particle horizon D̃PH grows as the scale factor (2) D̃PH(t) = R(t) R(t′) = α(t− t1−1/α) ∼ −α t1−1/α . (12) So, computing the associated entropy content S̃PH –with the entropy density (5)– leads to a function that approaches a constant value; it can be simplified taking the Planck time as reference time S̃PH(t) = s0R 0 R(t) −n ωn−1 D̃PH(t) n ⇒ lim S̃PH(t) = sP l|α| n . (13) This limit for the entropy content seems fairly unnatural because it is of the same order as one. 3 Fischler-Susskind holography in closed universes Let us focus on Robertson-Walker metrics with closed spatial sections (curvature param- eter k = +1). The line element in conformal coordinates (η, χ) reads ds2 = R2(η) − dη2 + dχ2 + sin2(χ)dΩ2n−1 , (14) where dΩn−1 is the metric of the (n-1)-dimensional unit sphere. The inner volume and area of a spherical domain of coordinate radius χ can be obtained by integrating this metric at a given cosmological time A(η, χ) = ωn−1R(η) n−1 sinn−1(χ) (15) V (η, χ) = R(η)nωn−1 sinn−1(χ′) dχ′ . (16) The entropy content inside this volume is obtained using the entropy density (5) S(χ) = s0R 0 ωn−1 sinn−1(χ′) dχ′ , (17) where scale factors R(t) have been cancelled; thus, the entropy content inside a comoving volume is constant (adiabatic expansion). Note that S(χ) strictly grows with the confor- mal size χ of the spherical domain; however boundary area A(η, χ) reaches a maximum near the equator : for χ > π/2 the boundary area decreases, going to zero at the antipodes, where χ → π (see Fig. 1). Similar problems appear when the cosmological model recon- tracts to a Big Crunch, because every boundary area will shrink to zero. In both cases holographic codification will be impossible. This problem will be reviewed in detail and a solution based on the cosmological acceleration will be proposed in the next section. 3.1 Conventional matter dominated cosmological models Fischler and Susskind applied the previous ideas to a FRW (3+1)-dimensional spatially closed cosmological model, dynamically dominated by conventional matter [3]; the explicit solution for the scale factor is R(η) = Rm . (18) Here Rm is the maximum value of the scale factor on decelerated models (α > 1 for conventional matter, see Table 1); it depends on the relation Ω between the energy density of the model and the critical density Rm ≡ R0 1− Ω−10 . (19) Introducing this scale factor on (15), and computing (17) for the usual case n = 3, the relation between the entropy content and the boundary area of a spherical domain of coordinate size χ at the conformal time η is obtained (η, χ) = 2χ− sin 2χ (sin η )2(α−1) sin2 χ . (20) It should also be kept in mind that the maximum domain accessible at a given time η is the particle horizon; so this relation must be evaluated for χPH(η), the value that locates the particle horizon for each η [10, 12] χPH(η) = η − ηBB, (21) where ηBB is the value of the conformal time assigned to the beginning of the universe (usually the Big Bang). A quick observation of relation (20) shows that the denominator goes to zero at χPH = π (antipodes) and also when the scale factor collapses in a Big Crunch; for both cases the ratio SPH/APH diverges and so the holographic codification (1) is impossible. All FRW spatially closed dynamically dominated by conventional matter models (that is −1/3 < ω ≤ 1 for n = 3) will finally recollapse; so, these models will violate the FS holographic prescription. 3.2 Quintessence dominated cosmological models As seen in the last section, some scenarios can become problematic for the holographic prescription. This section aims to expose an alternative solution for some of those trou- bling cosmological models. The key point in what follows lies in the fact that not all spatially closed cosmological models do recollapse; for example a positive cosmological constant could avoid the recontraction and finally provide an accelerated expansion. The same can be said for different mechanisms which drive acceleration. The present study provides an example where the final accelerated expansion is driven by a negative pres- sure fluid; this means considering FRW spatially closed (curvature parameter k = +1) cosmological models dynamically dominated by quintessence matter, that is α < 0 (see Table 1). The explicit solution for this kind of models is (18) as well, but its behavior is very differ- ent: a negative exponent for the scale factor prevents it from reaching the problematic zero value and so these models are safe from recollapsing in a Big-Crunch and from presenting a singular Big-Bang. Now, the scale factor take a minimum value at same η; firstly the universe contracts, but after this minimum it undergoes an accelerated expansion for ever; these are called bouncing models [16]. Bouncing models present the obvious advantage of being free of singularities [17], and they also enjoy a renewed interest [18] due to the observed cosmological acceleration [21] and especially in relation with brane-cosmology [16]3. On the other hand bouncing cosmologies meets with many problems when trying to reproduce the universe we observe; so the solution (18) must be only considered like a toy model to study the final behavior of an spatially closed and finally accelerated cos- mological model. Now, formula (19) gives the minimum value of scale factor Rm, and according to it Rm tends to zero when the energy density tends to the critical density (Ω → 1). For an almost flat bouncing cosmology, near the minimum on the scale fac- tor Rm quantum gravity effects could dominate erasing every correlation coming from the previous era4. So, in following calculations the beginning of the cosmological time is going to be taken at the minimum on the scale factor (like a no-singular Big-Bang); according to (18), this corresponds to a conformal time ηBB = π(1−α)/2. The coordinate distance to the particle horizon (21) is then χPH(η) = η − ηBB = η − (1− α) . (22) 3However, our simplest bouncing models associated to the general solution (18) usually are not con- sidered in the literature. 4George Gamow words refering to bouncing models: “from the physical point of view we must forget entirely about the precollapse period” [19]. It was also obtained from (18) that the scale factor diverges for η∞ = π(1 − α). This bounded value of the conformal time implies a bounded value for the coordinate size of the particle horizon χPH(η∞) too. As argued before, problems for the FS holographic prescription arise at χPH = π, i. e. the value at which a refocusing of the particle horizon on the antipodes of the observer takes place (the horizon area goes to zero). However, this scenario can be avoided by preventing the conformal time from reaching the problematic value (see Fig. 2); such FRW spatially closed models will never present any particle horizon recontraction χPH∞ < π ⇔ η∞ − ηBB = (1− α) < π ⇔ α > −1 . (23) Quintessence models also verify α < 0; then the allowed range becomes 0 > α > −1 which corresponds to very accelerated cosmological models. This result can be physically interpreted as follows: For very accelerated spatially closed cosmological models the growing rate of the scale factor is so high that it does not permit null geodesics to develop even half a rotation over the spatial sections (see Fig. 3). So the particle horizon, far from reaching the antipodal point, presents an eternally increasing area. It also happens for the limiting case α = −1 (ω = −2/3 if n = 3) due to the diver- gence of the scale factor. This can be summarized in the next statement: every spatially closed quintessence model with α ≥ −1 has an eternally increasing particle horizon area. The volume of the spatial sections for spatially closed cosmological models is always finite, and so the entropy content will be; moreover the entropy content of the universe for adiabatic expansion is constant. Then, in accordance with the previous result, the relation SPH/APH remains finite and goes to zero (see Fig. 4); now, using (3) leads to the conclusion that the FS holographic limit is also compatible with FRW spatially closed models verifying − 1 (n = 3, ω ≤ − ). (24) D. Youm [22] applies the same argument to brane universes and arrives to similar con- clusions. Note that the limiting value ω = 1 − 1 corresponds to a gas of co-dimension one branes [23]; with this kind of matter the FS holographic limit could be saturated depending on the numerical prefactors (like the value of the entropy density s0). The FS prescription is neither violated in the past since entropy content SPH goes to zero quicker than the particle horizon area APH as the beginning is approached, in a way that the relation SPH/APH also goes to zero. This behavior may be checked by introducing (22) in the general equation (20) (χPH) = sm χPH − sinχPH cosχPH sin2 χPH )2(1−α) χPH ≪ π : ≃ sm χPH , (26) where sm is the spatial entropy density at the beginning of the universe, which is chosen as reference time (so s0 = sm and R0 = Rm). Fig. 4 shows function (25) for different values of α(ω); there, the behavior that has been analytically deduced may be graphically Figure 2: Penrose diagrams for spatially closed FRW universes dominated by quintessence (spatial dimension n = 3); at the “Big-Bounce” the scale factor reaches a minimum but at the “future infinite” diverges. Depending on the particle horizon behavior two very different cases are shown: • On the left the particle horizon reaches the antipodes χ = π; in this case the particle horizon area firstly grows but later it surpasses the equator of the hyperspherical spatial section and finally decreases and shrinks to zero (see Fig. 1) in a finite time. In this case the holographic codification will be impossible. • But on the right the model is more accelerated and so the scale factor diverges for a lower value of the conformal time; so the diagram height is shorter and the particle horizon cannot reach the antipodes. In this case the particle horizon area diverges (due to the divergence of the scale factor at the future infinite) and the holographic codification is always possible. The height of diagram ∆η discriminates both behaviors; so, the limit case is obviously ∆η = π; then the limit value ω = −2/3 is obtained. For this limiting case the particle horizon reaches the antipodes at the future infinite; the scale factor diverges, the particle horizon area also diverges and, as a consequence, the holographic codification is allowed. So, the ω-range compatible to the holographic codification on the particle horizon is −1 ≤ ω ≤ −2/3 which corresponds to very accelerated spatially closed cosmological models. In general, a sufficient cosmological acceleration do not permit the recontraction of the particle horizon at the antipodes and enables the Fischler-Susskind holographic prescription. Different Particle Horizon Behavior accelerated FRW models k=+1 Hα<0: quintessence L observer at the Big-Bounce -expanding particle horizon Figure 3: Polar representation of particle horizons for quintessence dominated (α < 0) spatially closed FRW models. Future light cones are represented from the beginning η = ηBB (Big- Bounce) for an observer at χ = 0. For α < −1 the particle horizon reconverges in the antipodes (it reaches and surpasses value χ = π), so the particle horizon area shrinks to zero; this shrinkage for a particular future light cone is also shown in the figure. However, for α ≥ −1 the particle horizon does not reconverge since the cosmological acceleration does not allow it. The FS holographic prescription would be verified in this case. A thick line has been used to show the limit case α = −1 (ω = −2/3 if n = 3). The accelerated growth of the closed spatial sections (3-spheres) is shown by concentric circles; the smallest of them is considered the beginning of the universe, so all the particle horizons (future light cones) arise from it. In this kind of representations the radial distance coincides with the physical radius of the spatially closed model. So, in the figure, light cones do not show the usual 45 degrees evolution. In fact, at the beginning, the future light cones are very flattened since the scale factor of bouncing models evolves very slowly near the minimum which is considered the beginning of time. 0.5 π π Evolution of the Entropy - Area relation α=-0.2 ω=-8 9 α=-1.6 ω=-3 5 ω=-2 3 Figure 4: Evolution of quotient SPH/APH depending on the coordinate distance χPH as the particle horizon evolves and assuming sm = 1. Functions for different values of the parameter α(ω) are shown. A thick line represents the limit case α = −1. For α < −1 (ω > −2/3 if n = 3) the quotient diverges as the particle horizon reaches χPH = π (the particle horizon area shrinks to zero at the antipodes of a fiducial observer). But for very accelerated models, α ≥ −1 (ω ≤ −2/3 if n = 3), the quotient is always finite which is a necessary condition for the FS holographic prescription to be verified. verified. Looking at maxima of the SPH/APH functions proves that, for non-problematic cases (α ≥ −1), value 0.5 is an upper bound, so that α ≥ −1 (n = 3, ω ≤ −2/3) ⇒ (η) < 0.5 sm . (27) The maximum initial entropy density compatible with the FS entropic limit depends on this bound and this turns out to be sm ≤ 1/2 ⇒ SPH ≤ . (28) This means that to impose not to have more than one degree of freedom for each two Planck volumes is enough to ensure the verification of the FS prescription for spatially closed and accelerated FRW models with α > −1. 3.3 A more realistic cosmological model The previous results are based on a simple explicit solution for the scale factor (18) but its beginning (the bounce) probably is far from the real evolution of our universe. Here the opposite point of view is exposed: a two-fluid explicit, but not simple, solution mimics a spatially closed cosmological model according to the observed behavior. The Friedmann equations with curvature parameter k = +1 can be solved exactly for a universe initially dominated by radiation plus a positive cosmological constant Λ that finally provides the desired final acceleration5. The scale factor then evolves as R(t) = 2− 2 cosh , (29) where Cγ is a constant related to the radiation density ργ 0 measured in an arbitrary reference time: ργ 0R 0 . (30) Due to the initial deceleration (radiation dominated era) this model presents a genuine particle horizon defined by the future light-cone from the Big-Bang. The evolution of this light-front over the compact spatial sections is better described by the conformal angle χPH(t) = . (31) Like in the previous section if this conformal angle reaches the value π for a finite time this means that the particle horizon has covered all the spatial section, that is, it has reached the antipodes. There the particle horizon area is zero and the FS holographic prescription is not verified. But the proposed model is finally dominated by a positive Λ that provides an extreme (exponential) cosmological acceleration that could prevent the refocusing of the particle horizon. It can be checked that the conformal angle never reaches the problematic value π when the parameters verify CγΛ > 1.2482 (in Planck units). Experimental measurements suggest that our universe is flat or almost flat; here the second case is assumed, based on the value Ω = 1.02±0.02 from the combination of SDSS and WMAP data [20]. The best fit of the scale factor (29) to the standard cosmological parameters H0, t0 and ΩΛ takes place for CγΛ ∼ 700. Thus, the final acceleration of our universe seems to be enough to avoid the refocusing of the particle horizon; particularly it will tend to the asymptotic value χPH∞ ∼ 0.5 rad. The conclusion is that if our universe is positively curved and its evolution is similar to (29) then it could verify the FS holographic prescription far from saturation due to the ever increasing character of the particle horizon area. 3.4 Discussion and related works After the Fischler and Susskind exposition of the problematic application of the holo- graphic principle for spatially closed models [3] and R. Easther and D. Lowe confirmed these difficulties [24], several authors proposed feasible solutions. Kalyana Rama [25] 5For a small enough Λ the attractive character of the radiation always dominates and the universe recollapses in a Big-Crunch. Like in the classical Lemâıtre’s model (initially dominated by pressureless matter) there exists a critical value Λ which provides a static but inestable model. proposed a two-fluid cosmological model, and found that when one was of quintessence type, the FS prescription would be verified under some additional conditions. N. Cruz and S. Lepe [26] studied cosmological models with spatial dimension n = 2, and found also that models with negative pressure could verify the FS prescription. There are some alternative ways such as [13] which are worth quoting. All these authors analyzed math- ematically the functional behavior of relation S/A; our work however claims to endorse the mathematical work with a simple picture: ever expanding spatially closed cosmolog- ical models could verify the FS holographic prescription, since, due to the cosmological acceleration, future light cones could not reconverge into focal points and, so, the particle horizon area would never shrink to zero. As one can imagine, by virtue of the previous argument there are many spatially closed cosmological models which fulfill the FS holographic prescription; ensuring a sufficiently accelerated final era is enough. Examples other than quintessence concern spatially closed models with conventional matter and a positive cosmological constant, the so-called oscil- lating models of the second kind [27]. In fact, the late evolution of this family of models is dominated by the cosmological constant which is compatible with ω = −1, and this value verifies (24). Roughly speaking, an asymptotically exponential expansion will provide acceleration enough to avoid the reconvergence of future light cones. One more remark about observational result comes to support the study of quintessence models. If the fundamental character of the Holographic Principle as a primary princi- ple guiding the behavior of our universe is assumed, it looks reasonable to suppose the saturation of the holographic limit. This is one of the arguments used by T. Banks and W. Fischler [28, 29] to propose a holographic cosmology based on a an early universe, spatially flat, dominated by a fluid with ω = 16. According to (9) this value saturates the FS prescription for spatially flat FRW models, but it seems fairly incompatible with observational results. However, for spatially closed FRW cosmological models, it has been found that the saturation of the Holographic Principle is related to the value ω = −2/3 which is compatible with current observations (according to [30], ω < −0.76 at the 95% confidence level). It is likely that the simplest bouncing model (18) does not describe our universe correctly; however, as shown in this paper, the initial behavior of the universe can enforce the evolution of the particle horizon (future light cone from the beginning) to a saturated scenario compatible with the observed cosmological acceleration7. Thus, the dark energy computation based on the Holographic Principle [7, 8] seems much more plausible ρDE ∼ s T ∼ SPH/VPH APH/VPH ∼ D−2PH . (32) Taking DPH ∼ 10Gy gives ρDE ∼ 10 −10 eV4 in agreement the measured value [31]. Finally, two recent conjectures concerning holography in spatially closed universes deserve some comments. W. Zimdahl and D. Pavon [32] claim that dynamics of the holographic dark energy in a spatially closed universe could solve the coincidence problem; however the cosmological scale necessary for the definition of the holographic dark energy seems to be incompatible with the particle horizon [7, 8, 33]. In a more recent paper F. Simpson 6Banks and Fischler propose a scenario where black holes of the maximum possible size –the size of the particle horizon– coalesce saturating the holographic limit; this “fluid” evolves according to ω = 1. 7Work in progress. [34] proposed an imaginative mechanism in which the non-monotonic evolution of the particle horizon over a spatially closed universe controls the equation of state of the dark energy. The abundant work in that line is still inconclusive but it seems to be a fairly promising line of work. 4 Conclusions It is usually believed that we live in a very complex and chaotic universe. The Holographic Principle puts a bound for the complexity on our world arguing that a more complex universe would undergo a gravitational collapse. So, one dare say that gravitational interaction is responsible for the simplicity of our world. In this paper a measure of the maximum complexity of the universe compatible with the FS prescription of the Holographic Principle has been deduced. The maximum entropy density at the Planck era under the assumption of a flat FRW universe (10) and a quintessence dominated spatially closed FRW universe (28) has been computed as well. One of the main points of this paper is to get over an extended prejudice which states that the FS holographic prescription is, in general, incompatible with spatially closed cosmo- logical models. Only two very particular solutions –[25] and [26]– solved the problem but no physical arguments were given. It has been shown along this paper that cosmological acceleration actually allows the verification of the FS prescription for a wide range of spatially closed cosmological models. Finally, let us take a further step, a step to a more clear suggestion. First let us assume that the FS prescription is a correct method for the application of the Holographic Prin- ciple in Cosmology, then if our universe is spatially closed (although almost flat) it should be accelerated by virtue of the FS prescription. In this sense, the observed acceleration [30] enforces the previous assumption. In fact, the experimental results are compatible with k = 0 [31], but a very small positive curvature cannot be discarded [20, 30, 35, 36]. This reductionist use of the Holographic Principle is not usual in the literature. The most common way is to search a valid prescription for every cosmological model and every sce- nario (like the Bousso solution [4, 5]). However, the only possible world we have evidence of is the one which is observed, and maybe it is so because the Holographic Principle does not permit a different one. Acknowledgements We acknowledge R. Bousso criticism and suggestions. This work has been supported by MCYT (Spain) under grant FPA 2003-02948. References [1] G. ’t Hooft: Dimensional reduction in quantum gravity ; in Salanfestschrift pp. 284- 296, ed. A. Alo, J. Ellis, S. Randjbar-Daemi, World Scientific Co, Singapore (1993) [gr-qc/9310026]. http://arxiv.org/abs/gr-qc/9310026 [2] L. Susskind: The world as a hologram; J. Math. Phys. 36, 6377 (1995) [hep-th/9409089]. [3] W. Fischler, L. Susskind: Holography and Cosmology ; [hep-th/9806039]. [4] R. Bousso: The Holographic Principle; Rev. Mod. Phys. 74, 825 (2002) [hep-th/0203101]. [5] R. Bousso: Holography in general space-times ; JHEP 9906, 028 (1999) [hep-th/9906022]. [6] M. A. Per, A. J. Segui: Encoding the scaling of the cosmological variables with the Euler Beta function; Int. J. Mod. Phys. A20, 4917 (2005) [hep-th/0210266]. [7] S. D. H. Hsu: Entropy bounds and dark energy ; Phys. Lett. B594, 13 (2004) [hep-th/0403052]. [8] M. Li: A model of holographic dark energy ; Phys. Lett. B603, 1 (2004) [hep-th/0403127]. [9] S. W. Hawking, G. F. R. Ellis: The Large Scale Structure of Spacetime; Cambridge University Press, Cambridge (1973). [10] M. Trodden, S. M. Carroll: TASI Lectures: Introduction to Cosmology ; [astro-ph/0401547]. [11] G. F. R. Ellis, T. Rothman: Lost horizons ; Am. J. Phys. 61, 10 (1993). [12] W. Rindler: Visual horizons and world models ; Mon. Not. Roy. Astr. Soc. 116, 662 (1956). [13] D. N. Vollick: Holography in closed universes ; [hep-th/0306149]. [14] M. Kaplinghat; G. Steigman; I. Tkachev; T. P. Walker: Observational Constraints On Power-Law Cosmologies ; Phys. Rev. D59, 043514 (1999) [astro-ph/9805114]. [15] N. Kaloper, A. Linde: Cosmology vs. holography ; Phys. Rev. D60, 103509 (1999) [hep-th/9904120]. [16] C. P. Burgess, F. Quevedo, R. Rabadan, G. Tasinato, I. Zavala: On bouncing brane- worlds, S-branes and branonium cosmology ; JCAP 0402 008 (2004) [hep-th/0310122]. [17] J. D. Bekenstein: Nonsingular General Relativistic Cosmologies ; Phys. Rev. D 11, 2072 (1975). [18] C. Molina-Paris, M. Visser: Minimal conditions for the creation of a Friedmann- Robertson-Walker universe from a ‘bounce’ ; Phys. Lett. B455, 90 (1999) [gr-qc/9810023]. [19] H. Kragh: George Gamow and the “factual approach” to relativistic cosmology ; In The universe of general relativity, A. J. Kox, Jean Eisenstaedt (eds.), Einstein Stud- ies, Vol. 11 p. 175, Boston (2005). http://arxiv.org/abs/hep-th/9409089 http://arxiv.org/abs/hep-th/9806039 http://arxiv.org/abs/hep-th/0203101 http://arxiv.org/abs/hep-th/9906022 http://arxiv.org/abs/hep-th/0210266 http://arxiv.org/abs/hep-th/0403052 http://arxiv.org/abs/hep-th/0403127 http://arxiv.org/abs/astro-ph/0401547 http://arxiv.org/abs/hep-th/0306149 http://arxiv.org/abs/astro-ph/9805114 http://arxiv.org/abs/hep-th/9904120 http://arxiv.org/abs/hep-th/0310122 http://arxiv.org/abs/gr-qc/9810023 [20] M. Texmark, M. A. Strauss, M. R. Blanton et al.: Cosmological parameters from SDSS and WMAP ; Phys. Rev. D69, 103501 (2004) [astro-ph/0310723]. [21] Benjamin K. Tippet, Kayll Lake: Energy conditions and a bounce in FLRW cosmolo- gies ; [gr-qc/9810023]. [22] D. Youm: A Note on Thermodynamics and Holography of Moving Giant Gravitons ; Phys. Rev. D64, 065014 (2001) [hep-th/0104011]. [23] A. Karch, L. Randall: Relaxing to Three Dimensions ; Phys. Rev. Lett. 95, 161601 (2005) [hep-th/0506053]. [24] R. Easther, D. A. Lowe: Holography, cosmology and the second law of thermodynam- ics ; Phys. Rev. Lett. 82 4967 (1999) [hep-th/9902088]. [25] S. Kalyana Rama: Holographic Principle in the closed universe: a resolution with negative pressure matter ; Phys. Lett. B457, 268 [hep-th/9904110]. [26] N. Cruz, S. Lepe: Closed universes can satisfy the holographic principle in three dimensions ; Phys. Lett. B521, 343 (2002) [hep-th/0110175]. [27] J. V. Narlikar: Introduction to Cosmology p.136; Cambridge University Press, Cam- bridge (1993). [28] T. Banks, W. Fischler: Holographic Cosmology 3.0 ; Phys. Scripta T 117, 56-63 (2005) [hep-th/0310288]. [29] T. Banks, W. Fischler, L. Mannelli: Microscopic Quantum Mechanics of the p = ρ Universe; [hep-th/0408076]. [30] A. G. Riess, L. G. Strolger, J. Tonry et al.: Type Ia Supernova Discoveries at z > 1 From the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution; Astrophys. J. 607, 665 (2004) [astro-ph/0402512]. [31] D. N. Spergel, R. Bean, O. Dore et al. : Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology ; [astro-ph/0603449]. [32] W. Zimdahl, D. Pavon: Spatial curvature and holographic dark energy ; [hep-th/0606555]. [33] M. R. Setare: Interacting holographic dark energy model in non-flat universe; Phys. Lett. B 642, 1-4 (2006) [hep-th/0609069]. [34] F. Simpson: An alternative approach to holographic dark energy ; [astro-ph/0609755]. [35] M. Tegmark: Measuring Spacetime: from Big Bang to Black Holes ; Lect. Notes Phys. 646, 169 (2004) [astro-ph/0207199]. [36] K. Ichikawa, M. Kawasaki, T. Sekiguchi et al.: Implication of dark en- ergy parametrizations on the determination of the curvature of the universe; [astro-ph/0605481]. http://arxiv.org/abs/astro-ph/0310723 http://arxiv.org/abs/gr-qc/9810023 http://arxiv.org/abs/hep-th/0104011 http://arxiv.org/abs/hep-th/0506053 http://arxiv.org/abs/hep-th/9902088 http://arxiv.org/abs/hep-th/9904110 http://arxiv.org/abs/hep-th/0110175 http://arxiv.org/abs/hep-th/0310288 http://arxiv.org/abs/hep-th/0408076 http://arxiv.org/abs/astro-ph/0402512 http://arxiv.org/abs/astro-ph/0603449 http://arxiv.org/abs/hep-th/0606555 http://arxiv.org/abs/hep-th/0609069 http://arxiv.org/abs/astro-ph/0609755 http://arxiv.org/abs/astro-ph/0207199 http://arxiv.org/abs/astro-ph/0605481 Introduction Fischler-Susskind holography in flat universes Fischler-Susskind holography in closed universes Conventional matter dominated cosmological models Quintessence dominated cosmological models A more realistic cosmological model Discussion and related works Conclusions

0704.1638

Accelerated expansion of the Universe filled up with the scalar gravitons

Accelerated expansion of the Universe filled up with the scalar gravitons Yu. F. Pirogov Theory Division, Institute for High Energy Physics, Protvino, RU-142281 Moscow Region, Russia Abstract The concept of the scalar graviton as the source of the dark matter and dark energy of the gravitaional origin is applied to study the evolution of the isotropic homo- geneous Universe. A realistic self-consistent solution to the modified pure gravity equations, which correctly describes the accelerated expansion of the spatially flat Universe, is found and investigated. It is argued that the scenario with the scalar gravitons filling up the Universe may emulate the LCDM model, reducing thus the true dark matter to an artefact. 1 Introduction According to the present-day cosmological paradigm our Universe is fairly isotropic, homogeneous, spatially flat and experiences presently the accelerated expansion. The conventional description of the latter phenomenon is given by the model with the Λ-term and the cold dark matter (CDM).1 Nevertheless, such a description may be just a phenomenological reflection of a more fundamental mechanism. A realistic candidate on such a role is presented in the given paper. In a preceding paper [2], we proposed a modification of the General Relativity (GR), with the massive scalar graviton in addition to the massless tensor one.2 The scalar graviton was put forward as a source of the dark matter (DM) and the dark energy (DE) of the gravitational origin. In ref. [4], this concept was applied to study the evolution of the isotropic homogeneous Universe. The evolution equations were derived and the plausible arguments in favour of the reality of the evolution scenario with the scalar gravitons were presented. In the present paper, we expose an explicit solution to the evolution equations in the vacuum, which gives the correct description of the accelerated expansion of the spatially flat Universe. It is shown that the emulation of the LCDM model can indeed be reached as it was anticipated earlier [4]. In Section 2, we first briefly remind the evolution equations in the vacuum filled up only with the scalar gravi- tons. Then the master equation for the Hubble parameter is presented. Finally, a self-consistent solution of the latter equation, possessing the desired properties, is found and investigated. In the Conclusion, the proposed solution to the DM and DE problems is recapitulated. 1Hereof, the LCDM model. For a review on cosmology, see, e.g., ref. [1]. 2For a brief exposition of such a modified GR, see ref. [3]. http://arxiv.org/abs/0704.1638v1 2 Accelerated expansion Evolution equations We consider the isotropic homogeneous Universe without the true DM. Besides, we neglect by the luminous matter missing thus the initial period of the Universe evolution. Then, the vacuum evolution equations look like3 (ρs + ρΛ), (ps + pΛ), (1) with a(t) being the dynamical scale factor of the Universe, t being the comoving time and ȧ = da/dt, etc. In the above, κ2 is proportional to the spatial curvature, with κ2 = 0 for the spatially flat Universe. The parameter mP is the Planck mass. On the r.h.s. of eq. (1), ρΛ and pΛ are the energy density and the pressure corresponding to the cosmological constant Λ: ρΛ = −ρΛ = m2PΛ ≥ 0. Likewise, ρs and ps are, respectively, the energy density and pressure of the scalar gravitons: ρs = f σ̇2 + 3 σ̇ + σ̈ +m2PΛs(σ), ps = f σ̇2 − 3 σ̇ − σ̈ −m2PΛs(σ). (2) Here, fs = O(mP) is a constant with the dimension of mass entering the kinetic term of the scalar graviton field σ. The latter in the given context looks like σ = 3 ln , (3) with ã(t) being a nondynamical scale factor given a priori. The σ-field is defined up to an additive constant. Without any loose of generality, we can fix the constant by the asymptotic condition: σ(t) → 0 at t → ∞. In eq. (2), we put Vs + ∂Vs/∂σ ≡ m2P Λs(σ) + Λ , (4) where Vs(σ) is the scalar graviton potential. More particularly, we put Vs = V0 + s (σ − σ0)2 +O (σ − σ0)3 , (5) with σ0 being a constant, fs(σ−σ0) the physical field of the scalar graviton and ms the mass of the latter. By their nature, Λs and Λ are quite similar. To make the division onto these two parts unambiguous we normalize Λs by an additive constant so that Λs(0) = 0. Clearly, we get from eq. (2) that ρs + ps = f 2. Here, the contribution of Λs exactly cancels what is quite similar to the relation ρΛ + pΛ = 0. So, the contribution of Λs is a kind of the dark energy. In what follows, we put σ0 = 0 and V0 = m PΛ, with σ → 0 at t → ∞ becoming the ground state. The nondynamical functions Vs and ã being the two characteristics of the vacuum are not quite independent. More particularly, adopting the isotropic homogeneous ansatz for the solution of the gravity equations, with only one dynamical variable a, we tacitly put a consistency relation between ã and Vs. As a result, only one combination of the two lines of eq. (1) is the true equation of evolution, with the second independent combination giving just the required consistency condition. 3We refer the reader to ref. [4] for more details. Master equation In what follows, we restrict ourselves by the case of the spa- tially flat Universe, κ = 0. Subtracting the first line of eq. (1) from the second one and accounting for eq. (2) we get the relation Ḣ = − ασ̇2, (6) where H ≡ ȧ/a is the Hubble parameter and α = 2 . (7) We assume that α = O(1). Substituting σ̇ given by eq. (6) into the first line of eq. (1) we get the integro-differential master equation for the Hubble parameter: H2 = −1 Ḧ + 6HḢ Λs(σ) + Λ . (8) where it is to be understood −Ḣ(τ) dτ. (9) Remind that we assume σ(t) → 0 at t → ∞. Equations (8) and (6) supersede the pair of the original evolution equations (1). Self-consistent solution Let us put in what follows Λs ≡ 0. This will be justified afterwards. Iterating eq. (8), with Λ considered as a perturbation, we can get the solution with any desired accuracy. In particular, substituting into the r.h.s. of eq. (8) the solution H = α/t from the zeroth approximation (Λ = 0) we get the first approximation as follows O(1) at t → 0 , O(1/t3) at t → ∞, (10) or otherwise (tΛ/t) 2 + 1 α2/t2 at t/tΛ < 1, α2/t2Λ at t/tΛ > 1, being the characteristic time of the evolution of the Universe. Numerically, tΛ ∼ 1010yr is of order the age of the Universe. Equation (11) is the basis for the quali- tative discussion in what follows. Integrating eq. (11) we get the scale factor as follows: + 1− ln α ln(t/tΛ) at t/tΛ < 1, αt/tΛ at t/tΛ > 1, where a0 is an integration constant. 4 Explicitly, the scale factor looks like (t/tΛ) α at t/tΛ < 1, exp(αt/tΛ) at t/tΛ > 1, 4To phenomenologically account for the effect of the initial inflation we could formally shift the origin of time: t → t+ t0, with t0 > 0. with tΛ bordering thus the epoch of the the power law expansion from the epoch of the exponential expansion. Equation (13) gives the two-parametric representation for the scale factor of the acceleratedly expanding Universe after the initial period. With account for eq. (9) the σ-field behaves as σ = − (t/tΛ) ξ(1 + ξ)1/4 2 ln(t/tΛ) at t/tΛ < 1, tΛ/t at t/tΛ > 1. Note that at tΛ → ∞ or, equivalently, Λ → 0 the integral above diverges and the σ-field can not be normalized properly. Λ 6= 0 is thus necessary as a regulator in the theory. Now, the consistency condition looks like ã = a exp (−σ/3) ∼ (t/tΛ) α−2/3 at t/tΛ < 1, exp (αt/tΛ) at t/tΛ > 1. Clearly, Λ should be already presupposed in ã. Note that in the case α = 2/3, the parameter ã is approximately constant at t/tΛ < 1. Substituting equations (11) and (6) into the first line of eq. (2) we can explicitly verify that . (17) This is to be anticipated already from the relation ρΛ/m P = Λ, as well as eq. (10) and the first line of eq. (1). Clearly, ρs is positive. At t/tΛ < 1 we get from equations (14) and (17) that ρsa 3 ∼ t3α−2. In the case α = 2/3, we have ρs ∼ 1/a3 as it should be for the true CDM. On the other hand, the pressure of the scalar gravitons is as follows ps = − (tΛ/t) (tΛ/t)2 + 1 3α(2/3 − α)/t2 − α/t2Λ at t/tΛ < 1, (2α/t2Λ)(tΛ/t) 3 at t/tΛ > 1. At the same conditions as before, the pressure is ps/m P = −Λ/2, being near constant though not zero as it should be anticipated for the true CDM. Nevertheless, we see that the value α = 2/3 is exceptional in many respects. Conceivably, such a value is distinguished by a more fundamental theory. Introducing the critical energy density ρc = 3m 2, we get for the partial energy densities Ωs = ρs/ρc and ΩΛ = ρΛ/ρc, respectively, of the scalar gravitons and the Λ-term the following: 1 + (t/tΛ)2 1− (t/tΛ)2 at t/tΛ < 1, (tΛ/t) 2 at t/tΛ > 1, with ΩΛ = 1 − Ωs. Note that Ωs = ΩΛ = 1/2 at t/tΛ = 1. Presently, we have Ωs/ΩΛ ≃ 1/3 and thus the respective time t, in the neglect by the effect of the initial inflation, is somewhat larger tΛ. Finally, the condition Λs = 0 adopted earlier can be justified as follows. First of all, Λs is indeed negligible at t → ∞ due to σ → 0 and Λs(0) = 0. On the other hand, at t ∼ tΛ we have |σ| ∼ 1 and hence Λs ∼ m2s . For Λs to be negligible in this region, too, we should require ms ≤ Λ ∼ 1/tΛ. Nevertheless, in the early period of evolution when |σ| > 1 the contribution of Λs may be significant. The parameter α = 2/3 being fixed the theory may be terminated just by two mass parameters: the ultraviolet mP and the infrared t Λ or, otherwise, ms. 3 Conclusion To conclude, let us recapitulate the proposed solution to the DM and DE problems in the context of the evolution of the Universe. According to the viewpoint adopted, there is neither true DM nor DE in the Universe (at least, in a sizable amount). Instead, the field σ of the scalar graviton serves as a common source of both the DM and DE of the gravitational origin. DM is represented by the derivative contribution of σ, with DE being reflected by the derivativeless contribution. In this, the constant part of the latter contribution corresponds to the conventional Λ-term, while the σ-dependent part corresponds to DE. The latter is less important than the Λ-term at present, becoming conceivably more crucial at the early time. The self-consistent evolution of the Universe may be considered as the transition of the “sea” of the scalar gravitons, produced in the early period, from the excited state with |σ| > 1 to the ground state with σ = 0. The ground state is characterized by the cosmological constant Λ which, in turn, predetermines the characteristic evolution time of the Universe, tΛ ∼ 1/ Λ. The scenario is in the possession to naturally describe the accelerated expansion of the spatially flat Universe, correctly emulating thus the conventional LCDM model. The more complete study of the scenario, the initial period of the evolution including, is in order. The author is grateful to O. V. Zenin for the useful discussions. References [1] M. Trodden and S. M. Carroll, astro-ph/0401547. [2] Yu. F. Pirogov, Phys. Atom. Nucl. 69, 1338 (2006) [Yad. Fiz. 69, 1374 (2006)]; gr-qc/0505031. [3] Yu. F. Pirogov, gr-qc/0609103. [4] Yu. F. Pirogov, gr-qc/0612053. http://arxiv.org/abs/astro-ph/0401547 http://arxiv.org/abs/gr-qc/0505031 http://arxiv.org/abs/gr-qc/0609103 http://arxiv.org/abs/gr-qc/0612053 Introduction Accelerated expansion Conclusion

0704.1641

U Geminorum: a test case for orbital parameters determination

U Geminorum: a test case for orbital parameters determination Juan Echevarŕıa1, Eduardo de la Fuente2 and Rafael Costero3 Instituto de Astronomı́a, Universidad Nacional Autónoma de México, Apartado Postal 70-264, México, D.F., México ABSTRACT High-resolution spectroscopy of U Gem was obtained during quiescence. We did not find a hot spot or gas stream around the outer boundaries of the accretion disk. Instead, we detected a strong narrow emission near the location of the secondary star. We measured the radial velocity curve from the wings of the double-peaked Hα emission line, and obtained a semi-amplitude value that is in excellent agreement with the obtained from observations in the ultraviolet spectral region by Sion et al. (1998). We present also a new method to obtain K2, which enhances the detection of absorption or emission features arising in the late-type companion. Our results are compared with published values derived from the near-infrared NaI line doublet. From a comparison of the TiO band with those of late type M stars, we find that a best fit is obtained for a M6V star, contributing 5 percent of the total light at that spectral region. Assuming that the radial velocity semi-amplitudes reflect accurately the motion of the binary components, then from our results: Kem = 107 ± 2 km s −1; Kabs = 310 ± 5 km s−1, and using the inclination angle given by Zhang & Robinson (1987); i = 69.7◦ ± 0.7, the system parameters become: MWD = 1.20 ± 0.05M⊙; MRD = 0.42 ± 0.04M⊙; and a = 1.55± 0.02R⊙. Based on the separation of the double emission peaks, we calculate an outer disk radius of Rout/a ∼ 0.61, close to the distance of the inner Lagrangian point L1/a ∼ 0.63. Therefore we suggest that, at the time of observations, the accretion disk was filling the Roche-Lobe of the primary, and that the matter leaving the L1 point was colliding with the disc directly, producing the hot spot at this location. Subject headings: binaries: close — novae, cataclysmic variables — stars: indi- vidual (U Geminorum) 1email: [email protected] 2present address: Departamento de F́ısica, CUCEI, Universidad de Guadalajara. Av. Revolución 1500 S/R Guadalajara, Jalisco, Mexico. email: [email protected] 3email: [email protected] http://arxiv.org/abs/0704.1641v1 – 2 – 1. Introduction Discovered by Hind (1856), U Geminorum is the prototype of a subclass of dwarf novae, a descriptive term suggested by Payne-Gaposchkin & Gaposchkin (1938) due to the small scale similarity of the outbursts in these objects to those of Novae. After the work by Kraft (1962), who found U Gem to be a single-lined spectroscopic binary with an orbital period around 4.25 hr, and from the studies by Kreminski (1965), who establish the eclipsing nature of this binary, Warner & Nather (1971) and Smak (1971), established the classical model for Cataclysmic Variable stars. The model includes a white dwarf primary surrounded by a disc accreted from a Roche-Lobe filling late-type secondary star. The stream of material, coming through the L1 point intersects the edge of the disc producing a bright spot, which can contribute a large fraction of the visual flux. The bright spot is observed as a strong hump in the light curves of U Gem and precedes a partial eclipse of the accretion disk and bright spot themselves (the white dwarf is not eclipsed in this object). A mean recurrence time for U Gem outbursts of ≈ 118 days, with ∆mV =5 and out- burst width of 12 d, was first found by Szkody & Mattei (1984). However, recent analysis shows that the object has a complex outburst behavior (Cook 1987; Mattei et al. 1987; Cannizo, Gehrels & Mattei 2002). Smak (2004), using the AAVSO data on the 1985 out- burst, has discovered the presence of super-humps, a fact that challenges the current theories of super-outbursts and super-humps for long period system with mass ratios above 1/3. The latter author also points out the fact that calculations of the radius of the disc – obtained from the separation of the emission peaks (Kraft 1975) in quiescence – are in disagreement with the calculations of the disc radii obtained from the photometric eclipse data (Smak 2001). Several radial velocity studies have been conducted since the first results published by Kraft (1962). In the visible spectral range, where the secondary star has not been detected, their results are mainly based on spectroscopic radial velocity analysis of the emission lines arising from the accretion disc (Kraft 1962; Smak 1976; Stover 1981; Unda-Sanzana et al. 2006). In other wavelengths, works are based on absorption lines: in the near-infrared, on the Na I doublet from the secondary star (Wade 1981; Friend et al. 1990; Naylor et al. 2005) and in the ultraviolet, on lines coming from the white dwarf itself (Sion et al. 1998; Long & Gilliland 1999). Although the research work on U Gem has been of paramount importance in our under- standing of cataclysmic variables, the fact that it is a partially-eclipsed and – in the visual range – a single-lined spectroscopic binary, make the determination of its physical param- – 3 – eters difficult to achieve through precise measurements of the semi-amplitudes K1,2 and of the inclination angle i of the orbit. Spectroscopic results of K1,2 differ in the ultraviolet, visual and infrared ranges. Therefore, auxiliary assumptions have been used to derive its more fundamental parameters (Smak 2001). In this paper we present a value of K1, obtained from our high-dispersion Echelle spectra, which is in agreement with the ultraviolet results, and of K2 from a new method applicable to optical spectroscopy. By chance, the system was observed at a peculiar low state, when the classical hot spot was absent. 2. Observations U Geminorum was observed in 1999, January 15 with the Echelle spectrograph at the f/7.5 Cassegrain focus of the 2.1 m telescope of the Observatorio Astrónomico Nacional at San Pedro Mártir, B.C., México. A Thomson 2048×2048 CCD was used to cover the spectral range between λ5200 and λ9100 Å, with spectral resolution of R=18,000. An echellette grating of 150 l/mm, with Blaze around 7000 Å , was used. The observations were obtained at quiescence (V ≈ 14), about 20 d after a broad outburst (data provided by the AAVSO: www.aavso.org). The spectra show a strong Hα emission line. No absorption features were detected from the secondary star. A first complete orbital cycle was covered through twenty- one spectra, each with 10 min exposure time. Thirteen further spectra were subsequently acquired with an exposure of 5 min each. The latter cover an additional half orbital period. The heliocentric mid-time of each observation is shown in column one in Table 1. The flux standard HR17520 and the late spectral M star HR3950 were also observed on the same night. Data reduction was carried out with the IRAF package1. The spectra were wavelength calibrated using a Th-Ar lamp and the standard star was also used to properly subtract the telluric absorption lines using the IRAF routine telluric. 3. Radial Velocities In this section we derive radial velocities from the prominent Hα emission line observed in U Gem, first by measuring the peaks, secondly by using a method based on a cross- correlating technique, and thirdly by using the standard double-Gaussian technique designed to measure only the wings of the line. In the case of the secondary star, we were unable to detect any single absorption line in the individual spectra; therefore it was not possible to 1IRAF is distributed by the National Optical Observatories, operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. – 4 – use any standard method. However, here we propose and use a new method, based on a co- adding technique, to derive the semi-amplitude of the orbital radial velocity of the companion star. In this section, we compare our results with published values for both components in the binary. We first discuss the basic mathematical method used here to derive the orbital parameters and its limitation in the context of Cataclysmic Variables; then we present our results for the orbital parameters – calculated from the different methods – and finally discuss an improved ephemeris for U Gem. 3.1. Orbital Parameters Calculations To find the orbital parameters of the components in a cataclysmic variable – in which no eccentricity is expected (Zahn 1966; Warner 1995) – we use an equation of the form V(t)(em,abs) = γ +K(em,abs)sin[(2π(t− HJD⊙)/Porb)], where V(t)(em,abs) are the observed radial velocities as measured from the emission lines in the accretion disc or from the absorption lines of the red star; γ is the systemic velocity; K(em,abs) are the corresponding semi-amplitudes derived from the radial velocity curve; HJD⊙ is the heliocentric Julian time of the inferior conjunction of the companion; and Porb is the orbital period of the binary. A minimum least-squares sinusoidal fit is run, which uses initial values for the four (Porb, γ, Kem,abs, and HJD⊙) orbital parameters. The program allows for one or more of these variables to be fixed, i.e. they can be set to constant values in the initial parameters file. If the orbital period is not previously known, a frequency search – using a variety of methods for evenly- or unevenly-sampled time series data (Schwarzenberg-Czerny 1999) – may be applied to the measured radial velocities in order to obtain an initial value for Porb, which is then used in the minimum least-squares sinusoidal fit. If the time coverage of the observations is not sufficient or is uneven, period aliases may appear and their values have to be considered in the least-squares fits. A tentative orbital period is selected by comparing the quality of each result. In these cases, additional radial velocity observations should be sought, until the true orbital period is found unequivocally. Time series photometric observations are usually helpful to find orbital modulations and are definitely important in establishing the orbital period of eclipsing binaries. In the case of U Gem, the presence of eclipses and the ample photometric coverage since the early work of Kreminski (1965), has permitted to establish its orbital period with a high degree of accuracy (Marsh et al. 1990). Although in eclipsing binaries a zero phase is also usually determined, in the case of U Gem the variable – 5 – positions of the hot spot and stream, causes the zero point to oscillate, as mentioned by the latter authors. Accurate spectroscopic observations are necessary to correctly establish the time when the secondary star is closest to Earth, i.e in inferior conjunction. Further discussion on this subject is given in section 4. To obtain the real semi-amplitudes of the binary, i.e K(em,abs)=K(1,2), some reasonable auxiliary assumptions are made. First, that the measurements of the emission lines, produced in the accretion disc, are free from distortions and accurately follow the orbital motion of the unseen white dwarf. Second, that the profiles of the measured absorption lines are symmetric, which implies that the brightness at the surface of the secondary star is the same for all its longitudes and latitudes. Certainly, a hot spot in the disc or irradiation in the secondary from the energy sources related to the primary will invalidate either the first, the second, or both assumptions. Corrections may be introduced if these effects are present. In the case of U Gem, a three-body correction was introduced by Smak (1976) in order to account for the radial velocity distortion produced by the hot spot, and a correction to heating effects on the face of the secondary star facing the primary was applied by Friend et al. (1990) before equating Kabs = K2. As initial values in our least-squared sinusoidal fits, we use Porb = 0.1769061911 d and HJD⊙ = 2, 437, 638.82325 d from Marsh et al. (1990), a systemic velocity of 42 km s −1 from Smak (2001), and K1 = 107 km s −1 and K2 = 295 km s −1 from Long & Gilliland (1999) and Friend et al. (1990), respectively. In our calculations, the orbital period was set fixed at the above mentioned value, since our observations have a very limited time coverage. This allow us to increase the precision for the other three parameters. 3.2. The Primary Star In this section we compare three methods for determining the radial velocity of the primary star, based on measurements of the Hα emission line. Although, as we will see in the next subsections, the last method results in far better accuracy and agrees with the ultraviolet results, we have included all of them here because the first method essentially provides an accurate way to determine the separations of the blue and red peaks, which is an indicator of the outer radius of the disc (Smak 2001), and the second yields a Kem value much closer to that obtained from UV results than any other published method. This cross- correlation method might be worthwhile to consider for its use in other objects. Furthermore, as we will see in the discussion, all three methods yield a consistent value of the systemic velocity, which is essential to the understanding of other parameters in the binary system. – 6 – Table 1: Measured Hα Radial Velocities. HJD φ∗ Peaksa Fxcb Wingsc (240000+) (km s−1) 51193.67651 0.68 166.1 139.1 121.1 51193.68697 0.75 183.4 130.0 133.8 51193.69679 0.80 181.9 125.0 126.9 51193.70723 0.86 167.9 102.0 101.1 51193.71744 0.92 137.1 81.7 90.9 51193.72726 0.97 90.0 46.8 41.7 51193.73581 0.02 14.0 -17.9 6.9 51193.74700 0.09 -47.9 -48.1 -27.1 51193.75691 0.14 -67.1 -66.7 -48.2 51193.76743 0.20 -99.6 -84.6 -79.3 51193.77738 0.26 -132.3 -86.1 -75.7 51193.78900 0.32 -152.6 -60.2 -48.8 51193.80174 0.39 -77.9 -32.9 -33.6 51193.81211 0.45 9.0 10.9 14.5 51193.82196 0.51 104.3 79.2 65.1 51193.83176 0.56 134.6 113.7 107.0 51193.84175 0.62 141.0 142.8 124.9 51193.85156 0.67 159.3 158.6 147.6 51193.86133 0.73 165.6 148.0 131.7 51193.87101 0.79 192.9 142.8 130.3 51193.88116 0.84 175.0 120.7 110.6 51193.88306 0.91 154.6 106.5 91.1 51193.90530 0.98 90.6 32.3 31.9 51193.91751 0.05 -70.5 8.0 -23.1 51193.93029 0.12 -88.5 -71.8 -51.6 51193.94259 0.19 -97.1 -79.0 -66.7 51193.95483 0.26 -114.4 -88.8 -75.6 51193.95955 0.29 -142.2 -70.9 -67.9 ∗Orbital phases derived from the ephemeris given in section 4 aVelocities derived as described in section 3.2.1 bVelocities derived as described in section 3.2.2 cVelocities derived as described in section 3.2.3 – 7 – To match the signal to noise ratio of the first twenty-one spectra, we have co-added, in pairs, the thirteen 5-minute exposures. The last three spectra were added to form two different spectra, in order to avoid losing the last single spectrum. A handicap to this approach is that, due to the large read-out time of the Thomson CCD, we are effectively smearing the phase coverage of the co-added spectra to nearly 900 s. However, the mean heliocentric time was accordingly corrected for each sum. This adds to a total sample of twenty-eight 600 s spectra. 3.2.1. Measurements from the double-peaks We have measured the position of the peaks using a double-gaussian fit, with their sepa- ration, width and position as free parameters. The results yield a mean half-peak separation Vout of about 460 km s −1. The average value of the velocities of the red and blue peaks, for each spectrum, is shown in column 3 of Table 1. We then applied our nonlinear least-squares fit to these radial velocities. The obtained orbital parameters are shown in column 2 of Table 2. The numbers in parentheses after the zero point results are the evaluated errors of the last digit. We will use this notation for large numbers throughout the paper. The radial velocities are also shown in Figure 1, folded with the orbital period and the time of inferior conjunction adopted in the section 4. The solid lines in this figure correspond to sinusoidal fits using the derived parameters in our program. Although we have not independently tab- ulated the measured velocities of the blue and red peaks, they are shown in Figure 1 together with their average. The semi-amplitudes of the plotted curves are 154 km s−1 and 167 km s−1 for the blue and red peaks, respectively. 3.2.2. Cross Correlation using a Template We have also cross-correlated the Hα line in our spectra with a template constructed as follows: First, we selected a spectrum from the first observed orbital cycle close to phase 0.02 when, in the case of our observations, we should expect a minimum distortion in the double-peaked line due to asymmetric components (see section 5). The blue peak in this spectrum is slightly stronger than the red one. This is probably caused by the hot spot near the L1 point (see section 3.3.2), which might be visible at this phase due to the fact that the binary has an inclination angle smaller than 70 degrees. The half-separation of the peaks is 470 km s−1, a value similar to that measured in a spectrum taken during the same orbital phase in next cycle. The chosen spectrum was then highly smoothed to minimize high-frequency correlations. The resulting template is shown in Figure 2. A radial velocity – 8 – Fig. 1.— Radial velocity curve of the double peaks. The half-separation of the peaks, shown at the top of the diagram has a mean value of about 460 km s−1. The curve at the middle is the mean from blue (bottom curve) and red (top curve). – 9 – Fig. 2.— Hα template near phase 0.02. The half-separation of the peaks has a value of 470 km s−1. – 10 – for the template was derived from the wavelength measured at the dip between the two peaks and corrected to give an heliocentric velocity. The IRAF fxc task was then used to derive the radial velocities, which are shown in column 4 of Table 1. As in the previous section, we have fitted the radial velocities with our nonlinear least-squares fit algorithm. The resulting orbital parameters are given in column 3 of Table 2. In Figure 3 the obtained velocities and the corresponding sinusoidal fit (solid line) are plotted. 3.2.3. Measurements from the wings and Diagnostic Diagrams The Hα emission line was additionally measured using the standard double Gaussian technique and its diagnostic diagrams, as described in Shafter, Szkody and Thorstensen (1986). We refer to this paper for the details on the interpretation of our results. We have used the convolve routine from the IRAF rvsao package, kindly made available to us by Thorstensen (private communication). The double peaked Hα emission line – with a sepa- ration of about 20 Å – shows broad wings reaching up to 40 Å from the line center. Unlike the case of low resolution spectra – where for over-sampled data the fitting is made with individual Gaussians having a FWHM of about one resolution element – in our spectra, with resolution ≈ 0.34 Å, such Gaussians would be inadequately narrow, as they will cover only a very small region in the wings. To measure the wings appropriately and, at the same time, avoid possible low velocity asymmetric features, we must select a σ value which fits the line regions corresponding to disc velocities from about 700 to 1000 kms−1. As a first step, we evaluated the width of the Gaussians by setting this as a free pa- rameter from 10 to 40 pixels and for a wide range of Gaussian separations (between 180 and 280 pixels). For each run, we applied a nonlinear least-squares fit of the computed radial velocities to sinusoids of the form described in section 3.1. The results are shown in Figure 4, in particular for three different Gaussian separations: a = 180, 230 and 280 pixels. These correspond to the low and upper limits as well as to the value for a preferred solution, all of which are self-consistent with the second step (see below). In the bottom panel of the figure we have plotted the overall rms value for each least-squares fit, as this parameter is very sen- sitive to the selected Gaussian separations. As expected at this high spectral resolution, the parameters in the diagram change rapidly for low values of σ, and there are even cases when no solution was found. At low values of a (e.g. crosses) there are no solutions for widths narrower than 20 pixels. The rms values increase rapidly with width, while the σ(K)/K, γ and phase shift values differ strongly from the other cases. For higher values of a (open circles) we obtain lower values for σ(K)/K, but the rms results are still large, in particular for intermediate values of the width of the Gaussians. For the middle solution (dots) the – 11 – Fig. 3.— Radial velocities obtained from cross correlation using the template. The solid line correspond to the solution from column 3 in Table 2. – 12 – results are comparable with those for large a values, but the rms is much lower. Similar results were found for other intermediate values of a, and they all converge to a minimum rms for a width of 26 pixels at a = 230 pixels. For the second step we have fixed the width to a value of 26 Å and ran the double- Gaussian program for a range of a separations, from about 60 to 120 Å. The results obtained are shown in Figure 5. If only an asymmetric low velocity component is present, the semi-amplitude should decrease asymptotically as a increases, until K1 reaches the correct value. Here we observe such behavior, although for larger values of a, there is a K1 increase for values of a up to 40 Å, before it decreases strongly with high values of a. This behavior might be due to the fact that we are observing a narrow hot-spot near the L1 point (see section 5). On the other hand, as expected, the σ(K)/K vs a curve has a change in slope, at a value of a for which the individual Gaussians have reached the velocity width of the line at the continuum. For larger values of a the velocity measurements become dominated by noise. For low values of a, the phase shift usually gives spurious results, although in our case it approaches a stable value around 0.015. We believe this value reflects the difference between the eclipse ephemeris, which is based mainly on the eclipse of the hot spot, and the true inferior conjunction of the secondary star. This problem is further discussed in section 5. Finally, we must point out that the systemic velocity smoothly increases up to a maximum of about 40 km s−1 at Gaussian separation of nearly 42 Å, while the best results, as seen from the Figure, are obtained for a = 31Å. This discrepancy may be also be related to the narrow hot-spot near the L1 point and might be due to the phase-shift between the hot-spot eclipse and the true inferior conjunction. This problem will also be address in section 4. The radial velocities, corresponding to the adopted solution, are shown in column 5 of Table 1 and plotted in Figure 6, while the corresponding orbital parameters – obtained from the nonlinear least-squares fit – are given in column 4 of Table 2. 3.3. The Secondary Star We were unable to detect single features from the secondary star in any individual spectra, after careful correction for telluric lines. In particular we found no radial velocity results using a standard cross-correlation technique near the NaI λλ8183.3, 8194.8 Å doublet. As we will see below, this doublet was very weak compared with previous observations (Wade 1981; Friend et al. 1990; Naylor et al. 2005). We have been able, however, to detect the NaI doublet and the TiO Head band around λ7050 Å with a new technique, which enables us to derive the semi-amplitude Kabs of the secondary star velocity curve. We first present here – 13 – the general method for deriving the semi-amplitude and then apply it to U Gem, using not only the absorption features but the Hα emission as well. 3.3.1. A new method to determine K2 In many cataclysmic variables the secondary star is poorly visible, or even absent, in the optical spectral range. Consequently, no V (t) measurements are feasible for this component. Among these systems are dwarf novae with orbital periods under 0.25 days, for which it is thought that the disc luminosity dominates over the luminosity of the Roche-Lobe filling secondary, whose brightness depends on the orbital period of the binary (Echevarŕıa & Jones 1984). For such binaries, the orbital parameters have been derived only for the white-dwarf- accretion disc system, in a way similar to that described in section 3.1. In order to determine a value of Kabs from a set of spectra of a cataclysmic variable, for which the orbital period and time of inferior conjunction have been already determined from the emission lines, we propose to reverse the process: derive V (t)abs using Kpr as the initial value for the semi-amplitude, and set the values of Porb and HJD⊙, derived from the emission lines, as constants. The initial value for the systemic velocity is set to zero, and its final value may be calculated later (see below). The individual spectra are then co-added in the frame of reference of the secondary star, i.e. by Doppler-shifting the spectra using the calculated V (t)calc from the equation given in section 3.1, and then add them together. Hereinafter we will refer to this procedure as the co-phasing process. Ideally, as the proposed Kpr is changed through a range of possible values, there will be a one for which the co-phased spectral features associated with the absorption spectrum will have an optimal signal-to-noise ratio. In fact, this will also be the case for any emission line features associated with the red star, if present. In a way, this process works in a similar fashion as the double Gaussian fitting used in the previous section, provided that adequate criteria are set in order to select the best value for Kabs. We propose three criteria or tests that, for late type stars, may be used with this method: The first one consists in analyzing the behavior of the measured depths or widths of a well identified absorption line in the co-phased spectra, as a function of the proposed Kpr; one would expect that the width of the line will show a minimum and its depth a maximum value at the optimal solution. This method could be particularly useful for K-type stars which have strong single metallic lines like Ca I and Fe I. The second criterion is based upon measurements of the slope of head-bands, like that of TiO at λ7050 Å. It should be relevant to short period systems, with low mass M-type secondaries with spectra featuring strong molecular bands. In this case one could expect that the slope of the head-band will be a function of Kpr, and will have a maximum negative value at the best solution. A third test – 14 – is to measure the strength of a narrow emission arising from the secondary. This emission, if present, would be particularly visible in the co-phased spectrum and will have minimum width and maximum height at the best selected semi-amplitude Kpr. We have tested these three methods by means of an artificial spectrum with simulated narrow absorption lines, a TiO-like head band and a narrow emission line. The spectrum with these artificial features was then Doppler shifted using pre-established inferior conjunction phase and orbital period, to produce a series of test spectra. An amount of random Gaussian noise was added to each Doppler shifted spectrum, sufficient to mask the artificial features. We then proceeded to apply the co-phasing process to recover our pre-determined orbital values. All three criteria reproduced back the original set of values, as long as the random noise amplitude was of the same order of magnitude as the strength of the clean artificial features. 3.3.2. Determination of K2 for U Gem We have applied the above-mentioned criteria to U Gem. The time of the inferior conjunction of the secondary and the orbital period were taken from section 4. To attain the best signal to noise ratio we have used all the 28 observed spectra. Although they span over slightly more than 1.5 orbital periods, any departure from a real K2 value will not depend on selecting data in exact multiples of the orbital period, as any possible deviation from the real semi-amplitude will already be present in one complete orbital period and will depend mainly on the intrinsic intensity distribution of the selected feature around the secondary itself (also see below the results for γ). Figure 7 shows the application of the first test to the NaI doublet λλ 8183,8195 Å. The spectra were co-phased varying Kpr between 250 to 450 km s −1. The line depth of the blue and red components of the doublet (stars and open circles, respectively), as well as their mean value (dots) are shown in the diagram. We find a best solution for K2 = 310 ± 5 km s The error has been estimated from the intrinsic modulation of the solution curve. As it approaches its maximum value, the line depth value oscillates slightly, but in the same way for both lines. A similar behavior was present when low signal to noise features were used on the artificial spectra process described above. Figure 8 shows the co-phased spectrum of the NaI doublet of our best solution for K2. These lines appear very weak as compared with those reported by Friend et al. (1990) and Naylor et al. (2005). We have also measured the gamma velocity from the co-phased spectrum by fitting a double-gaussian to the Na I doublet (dotted line in Figure 7) and find a mean value γ = 69± 10 km s−1 (corrected to the heliocentric standard of motion). We did a similar calculation for γ by co-phasing the – 15 – selected spectra used in section 5, covering a full cycle only. The results were very similar to those obtained by using all spectra. The second test, to measure the slope of the TiO band has at λ7050 Å was not suc- cessful. The solution curve oscillates strongly near values between 250 and 350 km s−1. We believe that the signal to noise ratio in our spectra is too poor for this test and that more ob- servations, accumulated during several orbital cycles, have to be obtained in order to attain a reliable result using this method. However, we have co-phased our spectra for K2 = 310 km s −1, with the results shown in Figure 9. The TiO band is clearly seen while the noise is prominent, particulary along the slope of the head-band. We have used this co-added spectrum to compare it with several late-type M stars extracted from the published data by Montes et al. (1997) fitted to our co-phased spectrum. A gray continuum has been added to the comparison spectra in order to compensate for the fill-in effect arising from the other light sources in the system, so as to obtain the best fit. In particular, we show in the same figure the fits when two close candidates – GJ406 (M6 V, upper panel) and GJ402 (M4-5 V, lower panel) – are used. The best fit is obtained for the M6 V star, to which we have added a 95 percent continuum. For the M4-5 V star the fit is poor, as we observe a flux excess around 7000 Å and a stronger TiO head-band. Increasing the grey flux contribution will fit the TiO head band, but will result in a larger excess at the 7000 Å region. On the other hand, the fit with the M6 V star is much better all along the spectral interval. There are a number of publications which assign to U Gem spectral types M4 (Harrison et al. 2000), M5 (Wade 1981) and possibly as far as M5.5 (Berriman et al. 1983). Even in the case that the spectral type of the secondary star were variable, its spectral classification is still incompatible with its mass determination (Echevarŕıa 1983). For the third test, we have selected the region around Hα, as in the individual spectra we see evidence of a narrow spot, which is very well defined in our spectrum near orbital phase 0.5. In this test we have co-phased the spectra as before, and have adopted as the test parameter the peak intensity around the emission line. The results are shown in Figure 10. A clear and smooth maximum is obtained for Kpr = 310 ± 3 km s −1. The co-phased spectrum obtained from this solution is shown in Figure 11. The double-peak structure has been completely smeared – as expected when co-adding in the reference frame of the secondary star, as opposed to that of the primary star- and instead we observe a narrow and strong peak at the center of the line. We have also fitted the peak to find the radial velocity of the spot. We find γ = 33± 10 km s−1, compatible with the gamma velocity derived from the radial velocity analysis of the emission line, γ = 34± 2 km s−1 (see section 3.2.3). This is a key result for the determination of the true systemic velocity and can be compared with the – 16 – values derived from the secondary star (see section 7). 4. Improved Ephemeris of U Gem As mentioned in section 3.1, the presence of eclipses in U Gem and an ample photo- metric coverage during 30 years has permitted to establish, with a high degree of accuracy, the value of orbital period. This has been discussed in detail by Marsh et al. (1990). How- ever, as pointed by these authors, this object shows erratic variations in the timing of the photometric mid-eclipse that may be caused either by orbital period changes, variations in the position of the hot spot, or they may even be the consequence of the different methods of measuring of the eclipse phases. A variation in position and intensity of the gas stream will also contribute to such changes. A date for the zero phase determined independently from spectroscopic measurements would evidently be desirable. Marsh et al. (1990) discuss two spectroscopic measurements by Marsh & Horne (1988) and Wade (1981), and conclude that the spectroscopic inferior conjunction of the secondary star occurs about 0.016 in phase prior to the mean photometric zero phase. There are two published spectroscopic studies (Honeycutt et al et al. 1987; Stover 1981), as well as one in this paper, that could be used to confirm this result. Unfortunately there is no radial velocity analysis in the former paper, nor in the excellent Doppler Imaging paper by Marsh et al. (1990) based on their original observations. However, the results by Stover (1981) are of particular interest since he finds the spectroscopic conjunction in agreement with the time of the eclipse when using the pho- tometric ephemerides by Wade (1981), taken from Arnold et al. (1976). The latter authors introduce a small quadratic term which is consistent with the O-C oscillations shown in Marsh et al. (1990). It is difficult to compare results derived from emission lines to those obtained from absorption lines, especially if they are based on different ephemerides. Furthermore, the contamination on the timing of the spectroscopic conjunction – either caused by a hot spot, by gas stream or by irradiation on the secondary – has not been properly evaluated. However, since our observations were made at a time when the hot spot in absent (or, at least, is along the line between the two components in the binary) and the disc was very symmetric (see section 5), we can safely assume that in our case, the photometric and spectroscopic phases must coincide. If we then take the orbital period derived by Marsh et al. (1990) and use the zero point value derived from our measurements of the Hα wings, (section 3.2.3), we can improve the ephemeris: HJD = 2, 437, 638.82566(4) + 0.1769061911(28) E , – 17 – for the inferior conjunction of the secondary star. These ephemeris are used throughout this paper for all our phase folded diagrams and Doppler Tomography. 5. Doppler Tomography Doppler Tomography is a useful and powerful tool to study the material orbiting the white dwarf, including the gas stream coming from the secondary star as well as emission regions arising from the companion itself. It uses the emission line profiles observed as a function of the orbital phase to reconstruct a two-dimensional velocity map of the emitting material. A detailed formulation of this technique can be found in Marsh & Horne (1988). A careful interpretation of these velocity maps has to be made, as the main assumption invoked by tomography is that all the observed material is in the orbital plane and is visible at all times. The Doppler Tomography, derived here from the Hα emission line in U Gem, was constructed using the code developed by Spruit (1998). Our observations of the object cover 1.5 orbital cycles. Consequently – to avoid disparities on the intensity of the trailed and reconstructed spectra, as well as on the tomographic map – we have carefully selected spectra covering a full cycle only. For this purpose we discarded the first 3 spectra (which have the largest airmass) and used only 18 spectra out of the first 21, 600 s exposures, starting with the spectrum at orbital phase 0.88 and ending with the one at phase 0.86 (see Table 1). In addition, in generating the Tomography map we have excluded the spectra taken during the partial eclipse of the accretion disc (phases between 0.95 and 0.05). The original and reconstructed trailed spectra are shown in Figure 12. They show the sinusoidal variation of the blue and read peaks, which are strong at all phases. The typical S-wave is also seen showing the same simple sinusoidal variation, but shifted by 0.5 in orbital phase with respect to the double-peaks. The Doppler tomogram is shown in Figure 13; as customary, the oval represents the Roche-Lobe of the secondary and the solid lines the Keplerian (upper) and ballistic (lower) trajectories. The Tomogram reveals a disc reaching to the distance of to the inner Lagrangian point in most phases. A compact and strong emission is seen close to the center of velocities of the secondary star. A blow-up of this region is shown in Figure 14. Both maps have been constructed using the parameters shown at the top of the diagrams and a γ velocity of 34 km s−1. The velocity resolution of the map near the secondary star is about 10 km s−1. The V (x, y) position of the hot-spot (in km s−1) is (-50,305), within the uncertainties. The tomography shown in Figure 13 is very different from what we expected to find and from what has been observed by other authors. We find a very symmetric full disc, – 18 – reaching close to the inner Lagrangian point and a compact bright spot also close to the L1 point, instead of a complex system like that observed by Unda-Sanzana et al. (2006), who find U Gem at a stage when the Doppler Tomographs show: emission at low velocity close to the center of mass; a transient narrow absorption in the Balmer lines; as well as two distinct spots, one very narrow and close in velocity to the accretion disc near the impact region and another much broader, located between the ballistic and Keplerian trajectories. They present also tentative evidence of a weak spiral structure, which have been seen as strong spiral shocks during an outburst observed by Groot (1991). Our results also differ from those of Marsh et al. (1990), who also find that the bulk of the bright spot arising from the Balmer, He I and He II emission come from a region between the ballistic and Keplerian trajectories. We interpret the difference between our results and previous studies simply by the fact that we have observed the system at a peculiar low state not detected before (see sections 1 and 7) . This should not be at all surprising because, although U Gem is a well observed object, it is also a very unusual and variable system. Figure 14 shows a blow-up of the region around the secondary star. The bright spot is shown close to the center of mass of the late-type star, slightly located towards the leading hemisphere. Since this is a velocity map and not a geometrical one, there are at two possible interpretations of the position in space of the bright spot (assuming the observed material is in the orbital plane). The first one is that the emission is been produced at the surface of the secondary, i.e. still attached to its gravitational field. The second is that the emission is the result of a direct shock front with the accretion disc and that the compact spot is starting to gain velocity towards the Keplerian trajectory. We believe that the second explanation is more plausible, as it is consistent with the well accepted mechanism to produce a bright spot. On the other hand, at this peculiar low state it is difficult to invoke an external source strong enough to produce a back-illuminated secondary and especially a bright and compact spot on its leading hemisphere. 6. Basic system parameters Assuming that the radial velocity semi-amplitudes reflect accurately the motion of the binary components, then from our results –Kem = K1 = 107±2 km s −1; Kabs = K2 = 310±5 km s−1 – and adopting P = 0.1769061911 we obtain: = 0.35± 0.05, – 19 – M1 sin 3 i = PK2(K1 +K2) = 0.99± 0.03M⊙, M2 sin 3 i = PK1(K1 +K2) = 0.35± 0.02M⊙, a sin i = P (K1 +K2) = 1.46± 0.02R⊙. Using the inclination angle derived by Zhang & Robinson (1987), i = 69.7◦ ± 0.7, the system parameters become: MWD = 1.20 ± 0.05M⊙; MRD = 0.42 ± 0.04M⊙; and a = 1.55± 0.02R⊙. 6.1. The inner and outer size of the disc A first order estimate of the dimensions of the disc – the inner and outer radius – can be made from the observed Balmer emission line. Its peak-to peak velocity separation is related to the outer radius of the accreted material, while the wings of the line, coming from the high velocity regions of the disc, can give an estimate of the inner radius (Smak 2001). The peak-to-peak velocity separation of the 31 individual spectra were measured (see section 3.2.1), as well as the velocity of the blue and red wings of Hα at ten percent level of the continuum level. ¿From these measurements we derive mean values of Vout = 460 km s and Vin = 1200 km s These velocities can be related to the disc radii from numerical disc simulations, tidal limitations and analytical approximations (see Warner (1995) and references therein). If we assume the material in the disc at radius r is moving with Keplerian rotational velocity V (r), then the radius in units of the binary separation is given by (Horne, Wade & Szkody 1986): r/a = (Kem +Kabs)Kabs/V (r) The observed maximum intensity of the double-peak emission in Keplerian discs occurs close to the velocity of its outer radius (Smak 1981). From the observed Vout and Vin values we obtain an outer radius of Rout/a = 0.61 and an inner radius of Rin/a = 0.09. If we take a = 1.55±0.02R⊙ from the last section we obtain an inner radius of the disc Rin = 0.1395R⊙ – 20 – equivalent to about 97 000 km. This is about 25 times larger than the expected radius of the white dwarf (see section 7). On the other hand, the distance from the center of the primary to the inner Lagrangian point, RL1/a, is RL1/a = 1− w + 1/3w 2 + 1/9W 3, where w3 = q/(3(1 + q) ((Kopal 1959)). Using q = 0.35 we obtain RL1/a = 0.63. The disc, therefore, appears to be large, almost filling the Roche-Lobe of the primary, with the matter leaving the secondary component through the L1 point colliding with the disc directly and producing the hot spot near this location. 7. Discussion For the first time, a radial velocity semi-amplitude of the primary component of U Gem has been obtained in the visual spectral region, which agrees with the value obtained from ultraviolet observations by Sion et al. (1998) and Long & Gilliland (1999). In a recent paper, Unda-Sanzana et al. (2006) present high-resolution spectroscopy around Hα and Hβ and conclude that they cannot recover the ultraviolet value for K1 to better than about 20 percent by any method. Although the spectral resolution at Hα of the instrument they used is only a factor of two smaller than that of the one we used, the diagnostic diagrams they obtain show a completely different behavior as compared to those we present here, with best values for K1 of about 95 km s −1 from Hα and 150 km s−1 from Hβ (see their Figures 13 and 14, respectively). We believe that the disagreement with our result lies not in the quality of the data or the measuring method, but in the distortion of the emission lines due to the presence of a complex accretion disc at the time of their observations, as the authors themselves suggest. Their Doppler tomograms show emission at low velocity, close to the center of mass, two distinct spots, a narrow component close to the L1 point, and a broader and larger one between the Keplerian and the ballistic trajectories. There is even evidence of a weak spiral structure. In contrast, we have observed U Gem during a favorable stage, one in which the disc was fully symmetric, and the hot-spot was narrow and near the inner Lagrangian point. This allowed us to measure the real motion of the white dwarf by means of the time-resolved behavior of the Hα emission line. Our highly consistent results for the systemic velocity derived from the Hα spot (γ = 33± 10 km s−1 and those found from the different methods used for the radial velocity analysis of the emission arising from the accretion disk (see section 3.2 and Table 2), give strong support to our adopting a true systemic velocity value of γ = 34± 2 km s−1. If we are – 21 – indeed detecting the true motion of the white dwarf, we can use this adopted value, to make an independent check on the mass of the primary: The observed total redshift of the white dwarf (gravitational plus systemic)– found by Long & Gilliland (1999) – is 172 km s−1, from which, after subtraction of the adopted systemic velocity, we derive a gravitational shift of the white dwarf of 138 km s−1. From the mass-radius relationship for white dwarfs (Anderson 1988), we obtain consistent results for Mwd = 1.23M⊙ and Rwd = 3900 km (see Figure 7 in (Long & Gilliland 1999)). This mass is in excellent agreement with that obtained in this paper from the radial velocity analysis. ¿From our new method to determine the radial velocity curve of the secondary (sec- tion 3.3.2), we obtain a value for the semi-amplitude close to 310 km s−1. Three previous papers have determinations of the radial velocity curves from the observed Na I doublet in the near-infrared. In order to evaluate if our method is valid, we here compare our re- sult with these direct determinations. The published values are: Krd = 283 km s ((Wade 1981)); Krd = 309 km s ±3, (before correction for irradiation effects, (Friend et al. 1990)); and Krd = 300 km s −1 (Naylor et al. 2005). Wade (1981) notes that an elliptical orbital (e = 0.086) may better fit his data, as the velocity extremum near phase 0.25 ap- pears somewhat sharper than that near phase 0.75 (see his Figure 3). However, he also finds a very large systemic velocity, γ = 85 km s−1, much larger than the values found by Kraft (1962) (γ = 42 km s−1) and Smak (1976) (γ = 40± 6 km s−1), both obtained from the emission lines. Since the discrepancy with the results of these two authors was large, Wade (1981) defers this discussion to further confirmation of his results. Instead, and more important, this author discusses two scenarios that may significantly alter the real value of K2: the non-sphericity and the back-illumination of the secondary. In the latter effect, each particular absorption line may move further away from, or closer to the center of mass of the binary. He estimates the magnitude of this effect and concludes that the deviation of the photocenter would probably be much less than 0.1 radii. Friend et al. (1990) further discusses the circumstances that might cause the photocenter to deviate, and concludes that their observed value for the semi-amplitude should be corrected down by 3.5 percent, to yield K2 = 298 km s ± 9. Although they discuss the results by Martin (1988) – which indicate that the relatively small heating effects in quiescent dwarf novae always lead to a decrease in the measured Krd for the Na I lines – they argue that line quenching, produced by ionization of the same lines, may also be important, and result in an increased Krd. Another disturbing effect, considered by the same authors, is line contamination by the presence of weak disc features, like the Paschen lines. In this respect we point out here that a poor correction for telluric lines will function as an anchor, reducing also the amplitude of the radial velocity measurements. Friend et al. (1990) also find an observed systemic velocity of γ = 43± 6 km s−1 and a small eccentricity of e = 0.027. Naylor et al. (2005) also discuss the distortion – 22 – effects on the Na I lines and, based on their fit residuals, argue in favor of a depletion of the doublet in the leading hemisphere of the secondary, around phases 0.4 and 0.6, as removing flux from the blueward wing of the lines results in an apparent redshift, which would explain the observed residuals. However, they additionally find that fitting the data to an eccentric orbit, with e = 0.024, results in a significant decrease in the residuals caused by this deple- tion, and conclude that it may be unnecessary to further correct the radial velocity curve. We must point out that a depletion of the blueward wing of the Na I lines will results in a contraction of the observed radial velocity curves, as the measured velocities – especially around phases 0.25 and 0.75 – will be pulled towards the systemic velocity. Naylor et al. (2005) present their results derived from the Na I doublet and the K I/TiO region (around 7550-7750 Å), compared with several spectral standards, all giving values between 289 and 305 km s−1 (no errors are quoted). Based on the radial velocity measurements for Na I, obtained by these authors in 2001 January (115 spectra), and using GJ213 as template (see their Table 1), we have recalculated the circular orbital parameters through our nonlinear least-squares fit. We find K2 = 300 km s ± 1, in close agreement with their published value. It would be advisable to establish a link between the observed gamma velocity of the secondary and the semi-amplitude K2, under the assumption that its value may be distorted by heating effects. We take as a reference our results from the radial velocity analysis of the broad Hα line and the hot-spot from the secondary, which support a true systemic velocity of 34 km s−1. However, we find no positive correlation in the available results derived from the Na I lines, either between different authors or even among one data set. In the case of Naylor et al. (2005), the gamma values show a range between 11 and 43 km s−1, depending on the standard star used as a template, for K2 velocities in the range 289 to 305 km s −1. Wade (1981) finds γ = 85± 10 km s−1 for a low K2) value of 283 km s −1, while Friend et al. (1990) finds γ = 43± 6 km s−1 for K2 about 309 km s −1, and we obtain a large gamma velocity of about 69 km s−1 for a K2 value of 310 km s −1. We believe that further and more specific spectroscopic observations of the secondary star should be conducted in order to understand the possible distortion effects on lines like the Na I doublet, and their implications on the derived semi-amplitude and systemic velocity values. Acknowledgments E. de la F wishes to thank Andrés Rodriguez J. for his useful computer help. The Thomson detector, used in our observations, was obtained through PACIME-CONACYT project F325-E9211. – 23 – REFERENCES Anderson, N., 1988, ApJ., 326, 266 Arnold, S., Berg, R.A. & Duthie, J.G., 1976, ApJ., 206, 790 Berriman, G., Beattie, I.A., Lee, T.J., Mochnacki, S.W. & Szkody, P., 1983, MNRAS, 204, Cannizzo, J., K., Gehrels, N., & Mattei, J.A., 2002, ApJ, 579, 760 Cook, L.M., 1987, JAVSO, 16, 83 Echevarŕıa, J, 1983, RMAA, 8, 109 Echevarŕıa, J & Jones, D.H.P., 1984, MNRAS, 206, 919 Friend, M. T., Martin, J. S., Connon Smith, R., & Jones, D. H. P., 1990, MNRAS, 246, 637 Groot, P.J., 1991, ApJ, , 2649 Harrison, T.E., McNamara, B.J., Szkody, P. & Gilliland, R.L., 2000, ApJ, 120, 2649 Hind, J. R., 1856, MNRAS, 16, 56 Honeycutt, R. K., Kaitchuck, R. H., & Schlegel, E. M., 1987, ApJS, 65, 451 Horne, K., Wade, R.A. & Szkody, P., 1986, MNRAS, 219, 791 Kopal, Z., Close Binary Systems, Champan & Hall, London. Kraft, R. P., 1962, ApJ, 135, 408 Kraft, R. 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Orbital Peaks (a) Fxc (b) Wings (c) Parameters γ (km s−1) 38 ± 5 35 ± 3 34 ± 2 K (km s−1) 162 ± 7 119 ± 3 107 ± 2 HJD⊙ 0.8259(2) 0.82462(6) 0.82152(9) (+2437638 days) Porb (days) (d) (d) (d) σ 25.2 12.2 9.1 aDerived from measurements of the double-peaks bDerived from cross correlation methods cResults from the fitting of fixed double gaussians to the wings dPeriod fixed, P=0.1769061911 d – 26 – Fig. 4.— Diagnostic Diagram One. Orbital Parameters as a function of width of individual Gaussians for several separations. Crosses correspond to a = 180 pixels; Dots to a = 230 pixels (≈ 34 Å) and Open circles to a = 280 pixels; – 27 – Fig. 5.— Diagnostic Diagram Two. The best estimate of the semi-amplitude of the white dwarf is 107 km s−1, corresponding to a ≈ 34 Å. – 28 – 0 0.5 1 1.5 2 Fig. 6.— Radial velocities for U Gem. The open circles correspond to the measurements of the first 21 spectra single spectra, while the dots correspond to those of the co-added spectra (see section 3.2). The solid line, close to the points, correspond to the solution with Kem = 107 km s −1 (see text), while the large amplitude line correspond to the solution found for K2 (see section 3.3.1). – 29 – Fig. 7.— Maximum flux depth of the individual NaI lines λ8183.3 Å (top), λ8194.8 Å (bot- tom) and mean (middle) as a function of Kpr. – 30 – Fig. 8.— Co-phased spectrum around the NaI doublet. – 31 – Fig. 9.— U Gem TiO Head Band near 7050 Å compared with GJ406, an M6V star (upper diagram), and GJ402, an M4 V star (lower diagram) (see text). – 32 – Fig. 10.— Maximum peak flux of the co-added Hα spectra as a function of Kpr – 33 – Fig. 11.— Shape of the co-added Hα spectrum for K2 = 310 km s – 34 – Fig. 12.— Trailed spectra of the Hα emission line. Original (left) and reconstructed data (right). – 35 – Fig. 13.— Doppler Tomography of U Gem. The various features are discussed in the text. The vx and vy axes are in km s −1. A compact hot spot, close to the inner Lagrangian point is detected instead of the usual bright spot and/or broad stream, where the material, following a Keplerian or ballistic trajectory strikes the disc. The Tomogram reveals a full disc whose outer edge is very close to the L1 point (see text). – 36 – Fig. 14.— Blow-up of the region around the hot spot. Note that this feature is slightly ahead of the center of mass of the secondary star. Since this is a velocity map and not a geometrical one, its physical position in the binary is carefully discussed in the text. Introduction Observations Radial Velocities Orbital Parameters Calculations The Primary Star Measurements from the double-peaks Cross Correlation using a Template Measurements from the wings and Diagnostic Diagrams A new method to determine K2 Determination of K2 for U Gem Improved Ephemeris of U Gem Doppler Tomography Discussion

0704.1642

Collective excitations of hard-core Bosons at half filling on square and triangular lattices: Development of roton minima and collapse of roton gap

Collective excitations of hard-core Bosons at half filling on square and triangular lattices: Development of roton minima and collapse of roton gap Tyler Bryant and Rajiv R. P. Singh Department of Physics, University of California, Davis, CA 95616, USA (Dated: November 28, 2018) We study ground state properties and excitation spectra for hard-core Bosons on square and tri- angular lattices, at half filling, using series expansion methods. Nearest-neighbor repulsion between the Bosons leads to the development of short-range density order at the antiferromagnetic wavevec- tor, and simultaneously a roton minima in the density excitation spectra. On the square-lattice, the model maps on to the well studied XXZ model, and the roton gap collapses to zero precisely at the Heisenberg symmetry point, leading to the well known spectra for the Heisenberg antiferromagnet. On the triangular-lattice, the collapse of the roton gap signals the onset of the supersolid phase. Our results suggest that the transition from the superfluid to the supersolid phase maybe weakly first order. We also find several features in the density of states, including two-peaks and a sharp discontinuity, which maybe observable in experimental realization of such systems. PACS numbers: I. INTRODUCTION A microscopic theory for rotons in the excitation spec- tra of superfluids was first developed by Feynman, where he showed that the roton minima was related to a peak in the static structure factor.1 This study has had broad impact in condensed matter physics ranging from Quantum Hall Effect2 to frustrated antiferromagnets.3,4,5 In recent years considerable interest has also centered on Supersolid phases of matter.6 While the existence of such homogeneous bulk phases in Helium remains controversial,7,8 in case of lattice models such phases have been clearly established. One such example is that of hard-core Bosons hopping on a triangular-lattice, where a large enough nearest-neighbor repulsion leads to su- persolid order.9,10,11,12,13,14 The nature of the excitation spectra in the superfluid phase and on approach to the supersolid transition has not been addressed for the spin- half model. Here we use series expansion methods to study the ground state properties and excitation spectra of hard- core Bosons, at half filling, on square and triangular lat- tices, with nearest neighbor repulsion. On the square- lattice, the model is equivalent to the antiferromagnetic XXZ model, and we present the elementary excitation spectra for the XXZ model with XY type anisotropy. To our knowledge this calculation has not been done before. It should be useful for experimental studies of antiferro- magnetic materials with XY anisotropy. We set the XY coupling to unity and study the spectra as a function of the Ising coupling Jz. For the XY model, the spectra is gapless at q = 0 (the Goldstone mode of the superfluid) and has a maximum at the antiferromagnetic wavevector (π, π). As the Ising coupling is increased a roton minima develops at the antiferromagnetic wavevector, which goes to zero at the point of Heisenberg symmetry (Jz = 1), as expected for the system with doubled unit cell. For the triangular-lattice, the hard-core Boson model maps onto a ferromagnetic XY model, which is unfrus- trated. The nearest-neighbor repulsion, on the other hand corresponds to an antiferromagnetic Ising coupling, which is frustrated. This model cannot be mapped onto an antiferromagnetic XXZ model on the triangular lat- tice. For this model, we calculate the equal-time struc- ture factor S(q) as well as the excitation spectra, ω(q). Once again, we find that in the absence of nearest- neighbor repulsion, the excitation spectra is gapless at q = 0 and has a maximum at the antiferromagnetic wavevector ((4π/3, 0) and equivalent points). As the re- pulsion is increased, a pronounced peak develops in S(q) at these wavevectors and simultaneously a sharp roton minima develops in the spectra. Series extrapolations suggest that the roton gap vanishes when the repulsion term (Jz) reaches a value of ≈ 4.5. However, we are un- able to estimate any critical exponents for the vanishing of the gap or for the divergence of the structure factor. A comparison of our structure factor data with the Quan- tum Monte Carlo data of Wessel and Troyer, leads us to suggest that the transition to the supersolid phase maybe weakly first order and occurs for a value of Jz slightly less than 4.5. Our calculations also show a near minimum and flat re- gions in the spectra at the wavevectors (π, π/ 3), which correspond to the midpoint of the faces of the Bril- louin zone. These are points where the antiferromagnetic Heisenberg model has a well defined minima.4 In our case the dispersion is very flat along some directions and a minimum along others. There are several distinguishing features in the density of states (DOS) of the excitation spectra. The largest maximum in the DOS is close to the maximum excitation energy and is not unlike many other antiferromagnets. But, here, in addition, we get a second maximum in the DOS from the flat regions in the spectra at the midpoint of the faces of the Brillouin zone and a sharp drop in the DOS at the roton energy. It maybe possible to engineer such hard-core Boson systems on a triangular-lattice in cold atomic gases. It should, then, be possible to excite these collective exciations either op- http://arxiv.org/abs/0704.1642v2 tically or by driving the system out of equilibrium. A measurement of the energies associated with the charac- teristic features in the density of states can be used to accurately determine the microscopic parameters of the system. II. METHOD The linked-cluster series expansions performed here in- volve writing the Hamiltonian of interest as H = H0 + λH1 (1) where the eigenstates of H0 define the basis to be used and H1 is the perturbation to be applied in a linked clus- ter expansion. Ground state properties are then obtained as a power series in λ using Raleigh-Schrodinger pertur- bation theory. Excited state properties are obtained following the pro- cedure outlined in22, in which a similarity transformation is obtained in order to block diagonalize the Hamiltonian where the ground state sits in a block by itself and the one-particle states form another block. Heff = S−1HS (2) where Heff is an effective Hamiltonian for the states which are the perturbatively constructed extensions of the single spin-flip states. The effective Hamiltonian is then used to obtain a set of transition amplitudes r=0 λ rcr,m,n that describe propagation of the excita- tion through a distance (mx̂+ nŷ) for the square lattice and (1 nŷ) with m and n both even or both odd for the triangular lattice. These transition amplitudes are used to obtain the transition amplitudes for the bulk lattice by summing over clusters. Fourier transformation of the bulk transi- tion amplitudes then gives the excitation energy in mo- mentum space. ∆(qx, qy) = cr,m,nfm,n(qx, qy) (3) where fm,n(qx, qy) is given by the symmetry of the lattice, f sqrm,n(qx, qy) = [cos(mqx + nqy) + cos(mqx − nqy) +cos(nqx +mqy) + cos(nqx −mqy)] /4 (4) for the square lattice and f trim,n(qx, qy) = qx)cos( + cos( 3(m+ n) qy)cos( m− 3n qx) (5) + cos( 3(m− n) qy)cos( m+ 3n for the triangular lattice. In order to access values of the expansion parameter λ up to and including λ = 1, we use standard first order integrated differential approximants18 (IDAs) of the form QL(x) +RM (x)f + ST (x) = 0 (6) where QL,RM ,ST are polynomials of degree L,M,and T determined uniquely from the expansion coefficients. When gapless modes are present, estimates of the spin- wave velocity are made using the technique of Singh and Gelfand15. For small q = |q| the spectrum is assumed to have the form ∆(q) ∼ [A(λ) + B(λ)q2]1/2. To calcu- late the spin-wave velocity, we expand ∆(q) in powers of q, ∆(q) = C(λ) +D(λ)q2 + ... and identify C = A1/2 and D = B/2A1/2. Thus the series 2C(λ)D(λ) provides an estimate for B, which is the square of the spin-wave velocity. III. SQUARE LATTICE On the square lattice we perform two distinct types of expansions. For Jz ≥ J⊥, one can expand directly in J⊥/Jz by choosing <i,j> Szi S <i,j> (Sxi S j + S j ) (7) In this case λ in (1) is J⊥ (setting Jz = 1). Since H1 conserves the total Sz, one can perform the computation to high order by restricting the full Hilbert space to the total Sz sector of interest, which in this paper will be restricted to total Sz = 0 (half filling). Series expansion studies of the excitation spectra by Singh et al.15 and subsequently extended by Zheng et al.16 have been performed for Jz ≥ J⊥, with expansions involving linked clusters of up to 11 sites (λ10) and 15 sites (λ14) respectively. Fig. 1 shows the results of the spin-wave disper- sion analysis for J⊥ from the dispersionless Ising model J⊥ = 0 to the Heisenberg model J⊥ = 1. One can see the development of minima at (0, 0) and (π, π) with increas- ing J⊥, with the gap completely closing at J⊥ = 1. Since IDAs are not accurate near the gapless points, the dot- ted line shows the estimated spin-wave velocity v = 1.666 when J⊥ = 1. To obtain the spectra with XY anisotropy (Jz ≤ J⊥), we need to develop a different type of expansion. We consider the following break up of the Hamiltonian: (for Jz ≤ J⊥) <i,j> Sxi S <i,j> j + JzS j ) (8) (π/2,π/2)(π,0)(0,0)(π,π)(π,0) FIG. 1: (Color online) The spin-wave dispersion of the XXZ model on the square lattice for various values of J⊥ (Jz = 1). The error bars give an indication of the spread of various IDAs. The lines around the gapless points for J⊥ = 1 show the calculated spin-wave velocity. where J⊥ = 1. Now, a new series is obtained for each value of Jz, and the XXZ model is only obtained upon extrapolation to λ = 1. In contrast to the first type of expansion, H1 does not conserve total Sz, and so the entire Hilbert space must be used, limiting the order of computation of the series to λ10 (11 sites). Fig. 2 shows the results of the spin-wave dispersion analysis for several values of Jz from the XY model (Jz = 0) to the Heisenberg model (J⊥ = 1). We find that for the pure XY model, there is gapless excitations at q = 0 (Goldstone modes of the superfluid phase), but there is no roton minima at the antiferromegnetic wavevector. As Jz is increased, the spin-wave veloc- ity increases and a clear roton-minima develops at the antiferromagnetic wavevector. This minima collapses to zero as the Heisenberg point is approached. In fact, the doubling of the unit cell implies that for the Heisenberg limit, the spectra at q and at q+(π, π) become identi- cal. Another point of interest is that along the direction (π, 0) to (π/2, π/2), which corresponds to the antiferro- magnetic zone boundary, the dispersion is very flat for the pure XY model. A weak minimum develops at (π, 0) as the Heisenberg symmetry point is reached. These re- sults should be useful in comparing with spectra of two- dimensional antiferromagnets, where there is significant exchange anisotropy. IV. TRIANGULAR LATTICE There has been much recent interest in the XXZ model on the triangular lattice. The spin- 1 XXZ model with ferromagnetic in-plane coupling J⊥ < 0 and antiferro- magnetic coupling in the z direction Jz > 0 can be mapped to a hard-core boson model with nearest neigh- (π/2,π/2)(π,0)(0,0)(π,π)(π,0) FIG. 2: (Color online) The spin-wave dispersion of the XXZ model on the square lattice for various values of Jz (J⊥ = 1). The error bars give an indication of the spread of various IDAs. The lines around the gapless points show the calculated spin-wave velocities. TABLE I: Series coefficients for the ground state energy per site E0/N and M n E0/N for Jz=0 M for Jz=0 0 -5.000000e-01 1.250000e-01 2 -4.166667e-02 -6.944444e-03 4 -4.282407e-03 -2.267072e-03 6 -1.251190e-03 -1.141688e-03 8 -5.538567e-04 -7.184375e-04 10 -2.990401e-04 -5.039687e-04 12 -1.823004e-04 -3.784068e-04 14 -1.015895e-04 -2.494459e-04 bor repulsion. Hb = −t <i,j> ibj + bib j) + V <i,j> ninj (9) where b i is the bosonic creation operator, ni = b ibi. The parameters are related by t = −J⊥/2 and V = Jz . For the rest of this section, we let J⊥ = −1, and so V/t = −2Jz/J⊥ = 2Jz. We will continue to use the spin language as it is natu- ral for our study. For Jz = 0, the ferromagnetic in-plane coupling is unfrustrated. As Jz is increased, the com- peting interaction leads to an emergence of a supersolid order. We have performed expansions for the triangular lat- tice XXZ model of the form H0 = − <i,j> Sxi S <i,j> (−Syi S j + JzS j ) (10) where J⊥ = −1. Series are obtained for each value of Jz, and the XXZ model is obtained upon extrapolation to λ = 1. The static structure factor S(k) = eik·r〈Sz0Szr 〉 (11) is shown in Fig. 3 along contours shown in Fig. 4. As Jz increases, a peak forms at wavevector q=(4π/3, 0). A plot of this point is shown in fig. 5 along with QMC data from Wessel and Troyer.10 EBQPCOBA FIG. 3: (Color online) The static structure factor of the XXZ model on the triangular lattice for various values of Jz (J⊥=−1). The error bars give an indication of the spread of IDAs. 2π/31/2 -2π/31/2 4π/32π/30-2π/3-4π/3 O P Q FIG. 4: The hexagonal first Brillouin zone of the triangu- lar lattice and the path ABOCPQBE along which the static structure factor and spin-wave dispersion have been plotted in Figs. 3 and 6. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Series 12x12 24x24 36x36 FIG. 5: (Color online) The static structure factor at q=(4π/3, 0) of the XXZ model on the triangular lattice ver- sus Jz (J⊥=−1). The error bars for the series data give an indication of the spread of IDAs. Also shown are QMC data for 12x12,24x24,and 36x36 site clusters from Wessel and Troyer.10 Fig. 6 shows the results of the spin-wave dispersion analysis for various values of Jz (J⊥ = −1). The error bars give an indication of the spread of IDAs. The lines around the gapless points show the calculated spin-wave velocities. One can see the development of minima at Q with increasing Jz, with the gap completely closing at Jz ∼ 4.5. Since IDAs are not accurate near the gapless point (q = 0), the dotted line shows the estimated spin- wave velocities. We have been unable to get any consistent estimates for the critical exponents characterizing the divergence of the antiferromagnetic structure factor and the van- ishing of the roton gap as the supersolid phase is ap- proached. Furthermore, the comparison with the QMC data of Wessel and Troyer show that the QMC data begin to show deviations from our series expansion results be- fore Jz = 4.5. We believe, this implies that the superfluid to supersolid transition is weakly first order. Wessel and Troyer estimate the transition to be at Jz ≈ 4.3 ± 0.2 (|t/V | = 0.115 ± 0.005). Note that the spin-wave the- ory gives the transition point to be at Jz = 2, 9 so that quantum fluctuations play a substantial role here. Addi- tional QMC studies, should provide further insight into the nature of the transition.24 The calculations also show that near the midpoint of the faces of the Brillouin Zone (point B in Fig. 4), the dis- persion is a minima in the direction perpendicular to the zone face QB and is very flat in other directions. This be- havior is reminiscent of the dispersion in the Heisenberg antiferromagnet on the traingular lattice where there is a true minimum at this point.4,5 Note that this behav- ior is unrelated to any peak in the static structure factor and thus, as in case of the Heisenberg model, is more quantum mechanical in nature. In Fig. 7, we show the density of states for the spectra for Jz = 2. There are several distinguishing features in the density of states. First the largest peak in the density of states occurs close to the highest excitation energies. This is not unlike what is found in many other antiferromegnets. However, here, there is a second peak that corresponds to the flat regions in the spectra near the point B. Finally, at the roton energy there is a sharp drop in the density of states. The only contributions to the density of states below the roton gap comes from the Goldstone modes near q = 0. Since the latter have very small density of states, there is a discontinuity in the density of states at the roton energy. EBQPCOBA FIG. 6: (Color online) The spin-wave dispersion of the XXZ model on the triangular lattice for various values of Jz (J⊥ = −1). The error bars give an indication of the spread of IDAs. The lines around the gapless points show the calculated spin- wave velocities. 0 0.2 0.4 0.6 0.8 1 FIG. 7: The density of states for the XXZ model on the tri- angular lattice for Jz = 2 (J⊥ = −1). V. SUMMARY AND CONCLUSIONS In this paper, we have studied the excitation spectra of hard-core Boson models at half-filling on square and tri- angular lattices. The calculations show the development of the roton minima at the antiferromagnetic wavevector, due to nearest-neighbor repulsion. In accord with Feyn- man’s ideas, the development of the minima is correlated with the emergence of a sharp peak in the static structure factor. The case of triangular-lattice is clearly more in- teresting as one has a phase transition from a superfluid to a supersolid phase, where the roton gap goes to zero. Our series results suggest that the roton-gap vanishes at Jz ≈ 4.5. However, there maybe a weakly first order transition slightly before this Jz value. A more careful finite-size scaling analysis of the QMC data should pro- vide further insight into this issue. Our results of the spectra suggest two peaks in the density of states and a sharp drop in the density of states at the energy of the roton minima. If such a hard-core Boson system on a triangular-lattice is realized in cold- atom experiments, a measurement of the two peaks in the density of states and the roton minima can be used to determine independently the hopping parameter t and the nearest-neighbor repulsion V . Acknowledgments This research is supported in part by the National Sci- ence Foundation Grant Number DMR-0240918. We are greatful to Stefan Wessel for providing us with the QMC data for the structure factors and to Marcos Rigol and Stefan Wessel for discussions. 1 R. P. Feynman, Phys. Rev. 94, 262 (1954). 2 S. M. Girvin, A. H. MacDonald, and P. M. Platzman Phys. Rev. Lett. 54, 581-583 (1985). 3 P.Chandra, P. Coleman and A.I. Larkin, J. Phys. Cond. Matter 2, 7933 (1990). 4 W. Zheng, et al, Phys. Rev. Lett. 96, 057201 (2006); Phys. Rev. B 74, 224420 (2006). 5 O. A. Starykh, A. V. Chubukov and A. G. Abanov, Phys Rev. B74, 180403 (2006); A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. Lett. 97, 207202 (2006). 6 E. Kim and M. H. W. Chan, Nature 427, 225 (2004). 7 See for example M. Boninsegni et al, Phys. Rev. Lett. 97, 080401 (2006). 8 P. W. Anderson, W. F. Brinkman, D. A. Huse, Science 310, 1164 (2005). 9 G. Murthy, D. Arovas, and A. Auerbach, Phys. Rev. B, 55, 3104 (1997). 10 S. Wessel and M. Troyer, Phys. Rev. Lett. 95, 127205 (2005). 11 D. Heidarian and K. Damle, Phys. Rev. Lett. 95, 127206 (2005). 12 R. G. Melko, A. Paramekanti, A. A. Burkov, A. Vish- wanath, D.N. Sheng, and L. Balents Phys. Rev. Lett. 95, 127207 (2005). 13 M. Boninsegni and N. Prokof’ev Phys. Rev. Lett. 95, 237204 (2005). 14 E. Zhao and Arun Paramekanti Phys. Rev. Lett. 96, 105303 (2006). 15 Rajiv R. P. Singh, Martin P. Gelfand, Phys. Rev. B, 52, R15 695 (1995) 16 W. Zheng, J. Oitmaa, and C.J. Hamer, Phys. Rev. B, 71, 184440 (2005). 17 W. Zheng, J. Oitmaa, and C.J. Hamer, Phys. Rev. B, 43, 8321 (1991). 18 J. Oitmaa, C. Hamer, and W. Zheng, Series Expansion Methods for Strongly Interacting Lattice Models (Cam- bridge: Cambridge University Press) (2006). 19 A. W. Sandvik and R. R. P. Singh, Phys. Rev. Lett., 86, 528 (2001). 20 W. Zheng, C.J. Hamer, R. R. P. Singh, S. Trebst and H. Monien, Phys. Rev. B, 63, 144410 (2001). 21 H.-Q. Lin, J. S. Flynn and D. D. Betts, Phys. Rev. B, 64, 214411 (2001). 22 M. P. Gelfand and R. R. P. Singh, Adv. Phys. 49, 93 (2000). 23 T. Bryant, Ph.D. Dissertation, University of California, Davis, to be submitted. 24 S. Wessel, to be published. TABLE II: Series coefficients for the magnon dispersion on the square lattice for Jz = 0 (XY model), nonzero coefficients up to r=9 are listed for compactness (the complete series can be found in Ref. 23) (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (0,0,0) 2.000000e+00 (3,2,1) -7.291667e-02 (8,3,3) -1.715283e-03 (9,5,2) -1.334928e-03 (2,0,0) -4.166667e-02 (5,2,1) -1.968093e-02 (4,4,0) -2.712674e-03 (8,5,3) -7.225832e-04 (4,0,0) -1.023582e-02 (7,2,1) -8.897152e-03 (6,4,0) -2.980614e-03 (9,5,4) -3.993759e-04 (6,0,0) -5.390283e-03 (9,2,1) -4.845897e-03 (8,4,0) -1.932294e-03 (6,6,0) -1.156309e-04 (8,0,0) -2.781363e-03 (4,2,2) -1.627604e-02 (5,4,1) -5.303277e-03 (8,6,0) -3.216877e-04 (1,1,0) -1.000000e+00 (6,2,2) -5.758412e-03 (7,4,1) -4.614353e-03 (7,6,1) -3.803429e-04 (3,1,0) 4.340278e-02 (8,2,2) -2.558526e-03 (9,4,1) -3.055177e-03 (9,6,1) -6.801940e-04 (5,1,0) 1.811921e-02 (3,3,0) -1.215278e-02 (6,4,2) -3.468926e-03 (8,6,2) -3.612916e-04 (7,1,0) 7.679634e-03 (5,3,0) -7.265535e-03 (8,4,2) -2.946432e-03 (9,6,3) -2.662506e-04 (9,1,0) 4.056254e-03 (7,3,0) -3.368594e-03 (7,4,3) -1.901715e-03 (7,7,0) -2.716735e-05 (2,1,1) -2.500000e-01 (9,3,0) -1.895272e-03 (9,4,3) -1.807997e-03 (9,7,0) -1.043517e-04 (4,1,1) -2.314815e-02 (4,3,1) -2.170139e-02 (8,4,4) -4.516145e-04 (8,7,1) -1.032262e-04 (6,1,1) -5.841368e-03 (6,3,1) -1.033207e-02 (5,5,0) -5.303277e-04 (9,7,2) -1.141074e-04 (8,1,1) -1.566143e-03 (8,3,1) -4.852573e-03 (7,5,0) -1.004891e-03 (8,8,0) -6.451636e-06 (2,2,0) -1.250000e-01 (5,3,2) -1.060655e-02 (9,5,0) -9.122910e-04 (9,8,1) -2.852685e-05 (4,2,0) -3.067130e-02 (7,3,2) -6.456544e-03 (6,5,1) -1.387570e-03 (9,9,0) -1.584825e-06 (6,2,0) -9.598676e-03 (9,3,2) -3.716341e-03 (8,5,1) -1.771298e-03 (8,2,0) -3.989385e-03 (6,3,3) -2.312617e-03 (7,5,2) -1.141029e-03 TABLE III: Series coefficients for the ground state energy per site E0/N and M n E0/N for Jz=0 M for Jz=0 E0/N for Jz=1 M for Jz=1 E0/N for Jz=2 M for Jz=2 0 -7.500000e-01 -2.500000e-01 -7.500000e-01 -2.500000e-01 -7.500000e-01 -2.500000e-01 1 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2 -3.750000e-02 7.500000e-03 -1.500000e-01 3.000000e-02 -3.375000e-01 6.750000e-02 3 -7.500000e-03 3.000000e-03 6.750000e-02 -2.700000e-02 4 -3.102679e-03 1.989902e-03 -6.428572e-04 3.271769e-03 -3.081696e-02 3.263208e-02 5 -1.557668e-03 1.356060e-03 4.457109e-02 -4.706087e-02 6 -9.211778e-04 1.018008e-03 -1.686432e-03 2.253907e-03 -5.708928e-02 7.245005e-02 7 -5.949646e-04 7.975125e-04 7.181401e-02 -1.117528e-01 8 -4.102048e-04 6.468307e-04 -7.027097e-04 1.498661e-03 -1.001587e-01 1.839365e-01 9 -2.965850e-04 5.380214e-04 1.440002e-01 -3.025679e-01 10 -2.225228e-04 4.565960e-04 -3.752974e-04 1.029273e-03 -2.164303e-01 5.141882e-01 11 -1.719314e-04 3.937734e-04 3.343757e-01 -8.849178e-01 12 -1.360614e-04 3.441169e-04 -2.322484e-04 7.779323e-04 -5.294397e-01 1.545967e+00 TABLE IV: Series coefficients for the magnon dispersion on the triangular lattice Jz = 0, J⊥ = −1, nonzero coefficients up to r=9 are listed for compactness (the complete series can be found in Ref. 23) (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (0,0,0) 3.000000e+00 (6,5,1) -6.186994e-03 (5,8,0) -2.318836e-03 (9,10,6) -3.135408e-05 (2,0,0) -3.125000e-02 (7,5,1) -4.204807e-03 (6,8,0) -2.496675e-03 (9,10,8) -4.493212e-07 (3,0,0) 6.927083e-03 (8,5,1) -3.031857e-03 (7,8,0) -2.214348e-03 (6,11,1) -9.073618e-05 (4,0,0) -5.786823e-03 (9,5,1) -2.274746e-03 (8,8,0) -1.849982e-03 (7,11,1) -2.637766e-04 (5,0,0) -2.071746e-03 (4,5,3) -3.466797e-03 (9,8,0) -1.528161e-03 (8,11,1) -3.828143e-04 (6,0,0) -2.263644e-03 (5,5,3) -4.350420e-03 (5,8,2) -1.098718e-03 (9,11,1) -4.387479e-04 (7,0,0) -1.418733e-03 (6,5,3) -3.723124e-03 (6,8,2) -1.554950e-03 (7,11,3) -7.642054e-05 (8,0,0) -1.158308e-03 (7,5,3) -2.959033e-03 (7,8,2) -1.579331e-03 (8,11,3) -1.651312e-04 (9,0,0) -8.857566e-04 (8,5,3) -2.305863e-03 (8,8,2) -1.427534e-03 (9,11,3) -2.316277e-04 (1,2,0) -1.500000e+00 (9,5,3) -1.817374e-03 (9,8,2) -1.241335e-03 (8,11,5) -1.824976e-05 (2,2,0) -2.500000e-01 (3,6,0) -7.265625e-03 (6,8,4) -2.268405e-04 (9,11,5) -4.925111e-05 (3,2,0) 2.236979e-02 (4,6,0) -1.113487e-02 (7,8,4) -4.458355e-04 (9,11,7) -1.797285e-06 (4,2,0) -3.470238e-03 (5,6,0) -7.839022e-03 (8,8,4) -5.558869e-04 (6,12,0) -1.512270e-05 (5,2,0) 6.223272e-03 (6,6,0) -5.440344e-03 (9,8,4) -5.861054e-04 (7,12,0) -9.486853e-05 (6,2,0) 2.258837e-03 (7,6,0) -3.834575e-03 (7,8,6) -1.528411e-05 (8,12,0) -1.917140e-04 (7,2,0) 2.729869e-03 (8,6,0) -2.794337e-03 (8,8,6) -6.113630e-05 (9,12,0) -2.589374e-04 (8,2,0) 1.793769e-03 (9,6,0) -2.110041e-03 (9,8,6) -1.121505e-04 (7,12,2) -4.585233e-05 (9,2,0) 1.431100e-03 (4,6,2) -5.200195e-03 (5,9,1) -5.493588e-04 (8,12,2) -1.209317e-04 (2,3,1) -1.875000e-01 (5,6,2) -5.161886e-03 (6,9,1) -1.094328e-03 (9,12,2) -1.836224e-04 (3,3,1) -5.677083e-02 (6,6,2) -4.187961e-03 (7,9,1) -1.233659e-03 (8,12,4) -2.281220e-05 (4,3,1) -2.743217e-02 (7,6,2) -3.218811e-03 (8,9,1) -1.182811e-03 (9,12,4) -5.685542e-05 (5,3,1) -1.275959e-02 (8,6,2) -2.454043e-03 (9,9,1) -1.066119e-03 (9,12,6) -4.193665e-06 (6,3,1) -8.214468e-03 (9,6,2) -1.905072e-03 (6,9,3) -3.024539e-04 (7,13,1) -1.528411e-05 (7,3,1) -4.959005e-03 (5,6,4) -5.493588e-04 (7,9,3) -5.232866e-04 (8,13,1) -6.113630e-05 (8,3,1) -3.456161e-03 (6,6,4) -1.094328e-03 (8,9,3) -6.255503e-04 (9,13,1) -1.121505e-04 (9,3,1) -2.443109e-03 (7,6,4) -1.233659e-03 (9,9,3) -6.426095e-04 (8,13,3) -1.824976e-05 (2,4,0) -9.375000e-02 (8,6,4) -1.182811e-03 (7,9,5) -4.585233e-05 (9,13,3) -4.925111e-05 (3,4,0) -4.916667e-02 (9,6,4) -1.066119e-03 (8,9,5) -1.209317e-04 (9,13,5) -6.290497e-06 (4,4,0) -2.605934e-02 (4,7,1) -3.466797e-03 (9,9,5) -1.836224e-04 (7,14,0) -2.183444e-06 (5,4,0) -1.289340e-02 (5,7,1) -4.350420e-03 (8,9,7) -2.607109e-06 (8,14,0) -1.878307e-05 (6,4,0) -8.269582e-03 (6,7,1) -3.723124e-03 (9,9,7) -1.376709e-05 (9,14,0) -4.937808e-05 (7,4,0) -5.280757e-03 (7,7,1) -2.959033e-03 (5,10,0) -1.098718e-04 (8,14,2) -9.124881e-06 (8,4,0) -3.753955e-03 (8,7,1) -2.305863e-03 (6,10,0) -4.726374e-04 (9,14,2) -3.135408e-05 (9,4,0) -2.735539e-03 (9,7,1) -1.817374e-03 (7,10,0) -7.110879e-04 (9,14,4) -6.290497e-06 (3,4,2) -2.179687e-02 (5,7,3) -1.098718e-03 (8,10,0) -7.825389e-04 (8,15,1) -2.607109e-06 (4,4,2) -1.596136e-02 (6,7,3) -1.554950e-03 (9,10,0) -7.669912e-04 (9,15,1) -1.376709e-05 (5,4,2) -9.470639e-03 (7,7,3) -1.579331e-03 (6,10,2) -2.268405e-04 (9,15,3) -4.193665e-06 (6,4,2) -6.186994e-03 (8,7,3) -1.427534e-03 (7,10,2) -4.458355e-04 (8,16,0) -3.258886e-07 (7,4,2) -4.204807e-03 (9,7,3) -1.241335e-03 (8,10,2) -5.558869e-04 (9,16,0) -3.685726e-06 (8,4,2) -3.031857e-03 (6,7,5) -9.073618e-05 (9,10,2) -5.861054e-04 (9,16,2) -1.797285e-06 (9,4,2) -2.274746e-03 (7,7,5) -2.637766e-04 (7,10,4) -7.642054e-05 (9,17,1) -4.493212e-07 (3,5,1) -2.179687e-02 (8,7,5) -3.828143e-04 (8,10,4) -1.651312e-04 (9,18,0) -4.992458e-08 (4,5,1) -1.596136e-02 (9,7,5) -4.387479e-04 (9,10,4) -2.316277e-04 (5,5,1) -9.470639e-03 (4,8,0) -8.666992e-04 (8,10,6) -9.124881e-06 TABLE V: Series coefficients for the magnon dispersion on the triangular lattice Jz = 1, J⊥ = −1, nonzero coefficients up to r=9 are listed for compactness (the complete series can be found in Ref. 23) (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (0,0,0) 3.000000e+00 (8,4,2) 1.641818e-02 (8,7,3) -3.898852e-03 (6,10,2) -1.174079e-03 (2,0,0) -1.250000e-01 (4,5,1) -7.072545e-02 (6,7,5) -4.696316e-04 (8,10,2) -2.957178e-03 (4,0,0) 8.773202e-02 (6,5,1) -2.266297e-02 (8,7,5) -2.220857e-03 (8,10,4) -1.051498e-03 (6,0,0) -3.747008e-02 (8,5,1) 1.641818e-02 (4,8,0) -3.632812e-03 (8,10,6) -5.518909e-05 (8,0,0) 2.658737e-02 (4,5,3) -1.453125e-02 (6,8,0) -1.296471e-02 (6,11,1) -4.696316e-04 (2,2,0) -1.000000e-00 (6,5,3) -1.709052e-02 (8,8,0) -2.634458e-03 (8,11,1) -2.220857e-03 (4,2,0) 3.067262e-01 (8,5,3) -1.306500e-05 (6,8,2) -8.548979e-03 (8,11,3) -1.051498e-03 (6,2,0) -1.478629e-01 (4,6,0) -4.674479e-02 (8,8,2) -3.898852e-03 (8,11,5) -1.103782e-04 (8,2,0) 1.192248e-01 (6,6,0) -2.107435e-02 (6,8,4) -1.174079e-03 (6,12,0) -7.827194e-05 (2,3,1) -7.500000e-01 (8,6,0) 8.083746e-03 (8,8,4) -2.957178e-03 (8,12,0) -1.178397e-03 (4,3,1) 8.683532e-02 (4,6,2) -2.179687e-02 (8,8,6) -3.787294e-04 (8,12,2) -7.614846e-04 (6,3,1) -7.198726e-02 (6,6,2) -1.775987e-02 (6,9,1) -5.919357e-03 (8,12,4) -1.379727e-04 (8,3,1) 6.383529e-02 (8,6,2) 1.136255e-03 (8,9,1) -4.075885e-03 (8,13,1) -3.787294e-04 (2,4,0) -3.750000e-01 (6,6,4) -5.919357e-03 (6,9,3) -1.565439e-03 (8,13,3) -1.103782e-04 (4,4,0) -2.357440e-02 (8,6,4) -4.075885e-03 (8,9,3) -3.182356e-03 (8,14,0) -1.146580e-04 (6,4,0) -4.667627e-02 (4,7,1) -1.453125e-02 (8,9,5) -7.614846e-04 (8,14,2) -5.518909e-05 (8,4,0) 4.476721e-02 (6,7,1) -1.709052e-02 (8,9,7) -1.576831e-05 (8,15,1) -1.576831e-05 (4,4,2) -7.072545e-02 (8,7,1) -1.306500e-05 (6,10,0) -2.499303e-03 (8,16,0) -1.971039e-06 (6,4,2) -2.266297e-02 (6,7,3) -8.548979e-03 (8,10,0) -3.646508e-03 TABLE VI: Series coefficients for the magnon dispersion on the triangular lattice Jz = 2, J⊥ = −1, nonzero coefficients up to r=9 are listed for compactness (the complete series can be found in Ref. 23) (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (r,m,n) cr,m,n (0,0,0) 3.000000e+00 (6,5,1) -5.113269e-01 (5,8,0) 1.114660e-01 (9,10,6) 3.094665e-02 (2,0,0) -2.812500e-01 (7,5,1) 3.541550e-01 (6,8,0) -2.789923e-01 (9,10,8) 4.332352e-04 (3,0,0) -6.234375e-02 (8,5,1) -5.927875e-01 (7,8,0) 3.889189e-01 (6,11,1) -1.020493e-02 (4,0,0) 3.427127e-01 (9,5,1) 1.818953e+00 (8,8,0) -5.937472e-01 (7,11,1) 5.778696e-02 (5,0,0) 1.248868e-01 (4,5,3) -9.659180e-02 (9,8,0) 1.024914e+00 (8,11,1) -1.791293e-01 (6,0,0) -3.138557e-01 (5,5,3) 2.102338e-01 (5,8,2) 5.215530e-02 (9,11,1) 3.824414e-01 (7,0,0) -2.065582e-01 (6,5,3) -3.793451e-01 (6,8,2) -1.818856e-01 (7,11,3) 1.646341e-02 (8,0,0) 2.679692e-01 (7,5,3) 4.434905e-01 (7,8,2) 3.088576e-01 (8,11,3) -8.048695e-02 (9,0,0) 6.130976e-01 (8,5,3) -6.470161e-01 (8,8,2) -5.178979e-01 (9,11,3) 2.165711e-01 (1,2,0) 1.500000e+00 (9,5,3) 1.205017e+00 (9,8,2) 8.731811e-01 (8,11,5) -8.618625e-03 (2,2,0) -2.250000e+00 (3,6,0) 6.539063e-02 (6,8,4) -2.551233e-02 (9,11,5) 4.893766e-02 (3,2,0) -2.013281e-01 (4,6,0) -3.105654e-01 (7,8,4) 9.867811e-02 (9,11,7) 1.732941e-03 (4,2,0) 1.349036e+00 (5,6,0) 3.819521e-01 (8,8,4) -2.507655e-01 (6,12,0) -1.700822e-03 (5,2,0) 3.245835e-01 (6,6,0) -4.794666e-01 (9,8,4) 4.867085e-01 (7,12,0) 2.057252e-02 (6,2,0) -9.772865e-01 (7,6,0) 4.227344e-01 (7,8,6) 3.292682e-03 (8,12,0) -9.194406e-02 (7,2,0) -1.324972e+00 (8,6,0) -6.511353e-01 (8,8,6) -2.934241e-02 (9,12,0) 2.400559e-01 (8,2,0) 1.736288e+00 (9,6,0) 1.561357e+00 (9,8,6) 1.091019e-01 (7,12,2) 9.878045e-03 (9,2,0) 1.998149e+00 (4,6,2) -1.448877e-01 (5,9,1) 2.607765e-02 (8,12,2) -5.858223e-02 (2,3,1) -1.687500e+00 (5,6,2) 2.496909e-01 (6,9,1) -1.264872e-01 (9,12,2) 1.748291e-01 (3,3,1) 5.109375e-01 (6,6,2) -4.028414e-01 (7,9,1) 2.546730e-01 (8,12,4) -1.077328e-02 (4,3,1) 1.438694e-01 (7,6,2) 4.542205e-01 (8,9,1) -4.585226e-01 (9,12,4) 5.664036e-02 (5,3,1) 4.631412e-01 (8,6,2) -6.635572e-01 (9,9,1) 7.791699e-01 (9,12,6) 4.043528e-03 (6,3,1) -7.686440e-01 (9,6,2) 1.278006e+00 (6,9,3) -3.401644e-02 (7,13,1) 3.292682e-03 (7,3,1) -2.673657e-01 (5,6,4) 2.607765e-02 (7,9,3) 1.163420e-01 (8,13,1) -2.934241e-02 (8,3,1) 2.424913e-01 (6,6,4) -1.264872e-01 (8,9,3) -2.770943e-01 (9,13,1) 1.091019e-01 (9,3,1) 2.292894e+00 (7,6,4) 2.546730e-01 (9,9,3) 5.242482e-01 (8,13,3) -8.618625e-03 (2,4,0) -8.437500e-01 (8,6,4) -4.585226e-01 (7,9,5) 9.878045e-03 (9,13,3) 4.893766e-02 (3,4,0) 4.425000e-01 (9,6,4) 7.791699e-01 (8,9,5) -5.858223e-02 (9,13,5) 6.065292e-03 (4,4,0) -3.406189e-01 (4,7,1) -9.659180e-02 (9,9,5) 1.748291e-01 (7,14,0) 4.703831e-04 (5,4,0) 5.419257e-01 (5,7,1) 2.102338e-01 (8,9,7) -1.231232e-03 (8,14,0) -8.933341e-03 (6,4,0) -6.661367e-01 (6,7,1) -3.793451e-01 (9,9,7) 1.347384e-02 (9,14,0) 4.854225e-02 (7,4,0) 3.675761e-02 (7,7,1) 4.434905e-01 (5,10,0) 5.215530e-03 (8,14,2) -4.309313e-03 (8,4,0) -2.016564e-01 (8,7,1) -6.470161e-01 (6,10,0) -5.380211e-02 (9,14,2) 3.094665e-02 (9,4,0) 2.293150e+00 (9,7,1) 1.205017e+00 (7,10,0) 1.541458e-01 (9,14,4) 6.065292e-03 (3,4,2) 1.961719e-01 (5,7,3) 5.215530e-02 (8,10,0) -3.358010e-01 (8,15,1) -1.231232e-03 (4,4,2) -4.619167e-01 (6,7,3) -1.818856e-01 (9,10,0) 6.055844e-01 (9,15,1) 1.347384e-02 (5,4,2) 4.570889e-01 (7,7,3) 3.088576e-01 (6,10,2) -2.551233e-02 (9,15,3) 4.043528e-03 (6,4,2) -5.113269e-01 (8,7,3) -5.178979e-01 (7,10,2) 9.867811e-02 (8,16,0) -1.539040e-04 (7,4,2) 3.541550e-01 (9,7,3) 8.731811e-01 (8,10,2) -2.507655e-01 (9,16,0) 3.577909e-03 (8,4,2) -5.927875e-01 (6,7,5) -1.020493e-02 (9,10,2) 4.867085e-01 (9,16,2) 1.732941e-03 (9,4,2) 1.818953e+00 (7,7,5) 5.778696e-02 (7,10,4) 1.646341e-02 (9,17,1) 4.332352e-04 (3,5,1) 1.961719e-01 (8,7,5) -1.791293e-01 (8,10,4) -8.048695e-02 (9,18,0) 4.813724e-05 (4,5,1) -4.619167e-01 (9,7,5) 3.824414e-01 (9,10,4) 2.165711e-01 (5,5,1) 4.570889e-01 (4,8,0) -2.414795e-02 (8,10,6) -4.309313e-03

0704.1643

The LIL for $U$-statistics in Hilbert spaces

The LIL for U -statistics in Hilbert spaces Rados law Adamczak∗ Rafa l Lata la† November 13, 2018 Abstract We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U -statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U -statistics of arbitrary order, which are of independent inter- Keywords: U -statistics, law of the iterated logarithm, tail and moment estimates. AMS 2000 Subject Classification: Primary: 60F15, Secondary: 60E15 1 Introduction In the last two decades we have witnessed a rapid development in the asymp- totic theory of U -statistics, boosted by the introduction of the so called ’decoupling’ techniques (see [5, 6, 7]), which allow to treat U -statistics con- ditionally as sums of independent random variables. This approach yielded better understanding of U -statistics versions of the classical limit theorems of probability. Necessary and sufficient conditions were found for the strong law of large numbers [17], the central limit theorem [19, 10] and the law of the iterated logarithm [11, 2]. Also some sharp exponential inequalities for canonical U -statistics have been found [8, 1, 14]. Analysis of the afore- mentioned results shows an interesting phenomenon. Namely, the natural counterparts of the necessary and sufficient conditions for sums of i.i.d. ran- dom variables (U -statistics of degree 1), remain sufficient for U -statistics Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland. Email: [email protected]. Research partially supported by MEiN Grant 2 PO3A 019 †Institute of Mathematics, Warsaw University, Warsaw, Poland. Email: [email protected]. Research partially supported by MEiN Grant 1 PO3A 012 29. http://arxiv.org/abs/0704.1643v1 of arbitrary degree, but with an exception for the CLT, they cease to be necessary. The correct conditions turn out to be much more involved and are expressed for instance in terms of convergence of some series (LLN) or as growth conditions for some functions (LIL). A natural problem is an extension of the above results to the infinite- dimensional setting. There has been some progress in this direction, and partial answers have been found, usually under the assumption on the geo- metrical structure of the space in which the values of a U -statistic are taken. In general however the picture is far from being complete and the necessary and sufficient conditions are known only in the case of the CLT for Hilbert space valued U -statistics (see [5, 10] for the proof of sufficiency in type 2 spaces and necessity in cotype 2 spaces respectively). In this article we generalize to separable Hilbert spaces the results from [2] on necessary and sufficient conditions for the LIL for real valued U - statistics. The conditions are expressed only in terms of the U -statistic kernel and the distribution of the underlying i.i.d. sequence and can be also considered a generalization of results from [13], where the LIL for i.i.d. sums in Hilbert spaces was characterized. We consider only the bounded version of the LIL and do not give the exact value of the lim sup nor determine the limiting set. Except for the classical case of sums of i.i.d. random variables, the problem of finding the lim sup is at the moment open even in the one dimensional case (see [3, 5, 15] for some partial results) and the problem of the geometry of the limiting set and the compact LIL is solved only under suboptimal integrability conditions [3]. The organization of the paper is as follows. First, in Section 3 we prove sharp exponential inequalities for canonical U -statistics, which generalize the results of [1, 8] for the real-valued case. Then, after recalling some basic facts about the LIL we give necessary and sufficient condition for the LIL for decoupled, canonical U -statistics (Theorem 2). The quite involved proof is given in the two subsequent sections. Finally we conclude with our main result (Theorem 4), which gives a characterization of the LIL for undecoupled U -statistics and follows quite easily from Theorem 2 and the one dimensional result. 2 Notation For an integer d, let (Xi)i∈N, (X i )i∈N,1≤k≤d be independent random vari- ables with values in a Polish space Σ, equipped with the Borel σ-field F . Let also (εi)i∈N, (ε i )i∈N,1≤k≤d be independent Rademacher variables, in- dependent of (Xi)i∈N, (X i )i∈N,1≤k≤d. Consider moreover measurable functions hi : Σ d → H, where (H, | · |) is a separable Hilbert space (we will denote both the norm in H and the absolute value of a real number by | · |, the context will however prevent ambiguity). To shorten the notation, we will use the following convention. For i = (i1, . . . , id) ∈ {1, . . . , n}d we will write Xi (resp. Xdeci ) for (Xi1 , . . . ,Xid), (resp. (X , . . . ,X )) and ǫi (resp. ǫ i ) for the product εi1 · . . . · εid (resp. · . . . · ε(d)id ), the notation being thus slightly inconsistent, which however should not lead to a misunderstanding. The U -statistics will therefore be denoted i∈Idn hi(Xi) (an undecoupled U -statistic) |i|≤n i ) (a decoupled U -statistic) i∈Idn ǫihi(Xi) (an undecoupled randomized U -statistic) |i|≤n ǫdeci hi(X i ) (a decoupled randomized U -statistic), where |i| = max k=1,...,d Idn = {i : |i| ≤ n, ij 6= ik for j 6= k}. Since in this notation {1, . . . , d} = I1d we will write Id = {1, 2, . . . , d}. Throughout the article we will write Ld, L to denote constants depending only on d and universal constants respectively. In all those cases the values of a constant may differ at each occurrence. For I ⊆ Id, we will write EI to denote integration with respect to vari- ables (X i )i∈N,j∈I . We will consider mainly canonical (or completely de- generated) kernels, i.e. kernels hi, such that for all j ∈ Id, Ejhi(Xdeci ) = 0 3 Moment inequalities for U-statistics in Hilbert space In this section we will present sharp moment and tail inequalities for Hilbert space valued U -statistics, which in the sequel will constitute an important ingredient in the analysis of the LIL. These estimates are a natural general- ization of inequalities for real valued U-statistics presented in [1]. Let us first introduce some definitions. Definition 1. For a nonempty, finite set I let PI be the family consisting of all partitions J = {J1, . . . , Jk} of I into nonempty, pairwise disjoint subsets. Let us also define for J as above deg(J ) = k. Additionally let P∅ = {∅} with deg(∅) = 0. Definition 2. For a nonempty set I ⊆ Id consider J = {J1, . . . , Jk} ∈ PI . For an array (hi)i∈Idn of H-valued kernels and fixed value of iI c, define ‖(hi)iI‖J = sup EI [hi(X deg(J ) (XdeciJj : ΣJj → R |f (j)iJj (X )|2 ≤ 1 for j = 1, . . . ,deg(J ) Let moreover ‖(hi)i∅‖∅ = |hi|. Remark It is worth mentioning that for I = Id, ‖ · ‖J is a deterministic norm, whereas for I ( Id it is a random variable, depending on X Quantities given by the above definition suffice to obtain precise moment estimates for real valued U -statistics. However, to bound the moments of U -statistics with values in general Hilbert spaces, we will need to introduce one more definition. Definition 3. For nonempty sets K ⊆ I ⊆ Id consider J = {J1, . . . , Jk} ∈ PI\K . For an array (hi)i∈Idn of H-valued kernels and fixed value of iIc, define ‖(hi)iI‖K,J = sup EI [〈hi(Xdeci ), giK (XdeciK )〉 deg(J ) (XdeciJj )]| : : ΣJj → R, giK : ΣK → H ,E |giK (XdeciK )| 2 ≤ 1 |f (j)iJj (X )|2 ≤ 1 for j = 1, . . . ,deg(J ) Remark One can see that the only difference between the above definition and Definition 2 is that the latter distinguishes one set of coordinates and allows functions corresponding to this set to take values in H. Moreover, since the norm in H satisfies | · | = sup|φ|≤1〈φ, ·〉, we can treat Definition 2 as a counterpart of Definition 3 for K = ∅. We will use this convention to simplify the statements of the subsequent theorems. Thus, from now on, we will write ‖ · ‖∅,J := ‖ · ‖J . Example For d = 2 and I = {1, 2}, the above definition gives ‖(hij(Xi, Yj))i,j‖∅,{{1,2}} = sup hij(Xi, Yj)fij(Xi, Yj) f(Xi, Yj) 2 ≤ 1 = sup φ∈H,|φ|≤1 〈φ, hij(Xi, Yj)〉2, ‖(hij(Xi, Yj))i,j‖∅,{{1}{2}} = sup hij(Xi, Yj)fi(Xi)gj(Yj) Ef(Xi) Eg(Yj) 2 ≤ 1 ‖(hij(Xi, Yj))i,j‖{1},{{2}} = sup 〈fi(Xi), hij(Xi, Yj)〉gj(Yj) : |f(Xi)|2,E g(Yj) 2 ≤ 1 ‖(hij(Xi, Yj))i,j‖{1,2},∅ = sup 〈fij(Xi, Yj), hij(Xi, Yj)〉 : |f(Xi, Yj)|2 ≤ 1 E|hij(Xi, Yj)|2. We can now present the main result of this section. Theorem 1. For any array of H-valued, completely degenerate kernels (hi)i and any p ≥ 2, we have h(Xdeci ) p ≤ Lpd K⊆I⊆Id J∈PI\K pp(#I c+degJ /2) EIc max ‖(hi)iI‖ The proof of the above theorem proceeds along the lines of arguments presented in [1, 8]. In particular we will need the following moment estimates for suprema of empirical processes [8]. Lemma 1 ([8, Proposition 3.1], see also [4, Theorem 12]). Let X1, . . . ,Xn be independent random variables with values in (Σ,F) and T be a countable class of measurable real functions on Σ, such that for all f ∈ T and i ∈ In, Ef(Xi) = 0 and Ef(Xi) 2 < ∞. Consider the random variable S := supf∈T | i f(Xi)|. Then for all p ≥ 1, ESp ≤ Lp (ES)p + pp/2σp + ppEmax |f(Xi)|p where σ2 = sup Ef(Xi) We will also need the following technical lemma. Lemma 2 (Lemma 5 in [1]). For α > 0 and arbitrary nonnegative kernels gi : Σ d → R+ and p > 1 we have i ≤ L pαpEmax I({1,...,d} p#IpEI max EIcgi) Before stating the next lemma, let us introduce some more definitions, concerning J –norms of deterministic matrices Definition 4. Let (ai)i∈Idn be a d-indexed array of real numbers. For J = {J1, . . . , Jk} ∈ PId define ‖(ai)i‖J = sup · · · x(k)iJk : )2 ≤ 1, . . . , )2 ≤ 1 We will also need Definition 5. For i ∈ Nd−1 × In let ai : Σ → R be measurable functions and Z1, . . . , Zn be independent random variables with values in Σ. For a partition J = {J1, . . . , Jk} ∈ PId (d ∈ J1), let us define ‖(ai(Zid))i‖J = sup iId\J1 ai(Zid)x · · · x(k)iJk )2 ≤ 1, . . . , )2 ≤ 1 Remark All the definitions of norms presented so far, seem quite similar and indeed they can be all interpreted as injective tensor-product norms on proper spaces. We have decided to introduce them separately by explicit formulas, because this form appears in our applications. The next lemma is crucial for obtaining moment inequalities for canon- ical real-valued U -statistics of order greater than 2. In the context of U - statistics in Hilbert spaces we will need it already for d = 2. Lemma 3 (Theorem 5 in [1]). Let Z1, . . . , Zn be independent random vari- ables with values in (Σ,F). For i ∈ Nd−1 × In let ai : Σ → R be measurable functions, such that EZai(Zid) = 0. Then, for all p ≥ 2 we have ai(Zid))iId−1 ‖ ≤ Ld J∈PId p(1+deg (J )−d)/2‖(ai(Zid))i‖J J∈PId−1 p1+(1+deg(J )−d)/2 ‖(ai(Zid))iId−1‖ where ‖·‖ denotes the norm of a (d−1)-indexed matrix, regarded as a (d−1)- linear operator on (l2) d−1 (thus the ‖ · ‖{1}...{d−1}–norm in our notation). To prove Theorem 1, we will need to adapt the above lemma to be able to bound the (K,J )-norms of sums of independent kernels. Definition 6. We define a partial order ≺ on PI as I ≺ J if and only if for all I ∈ I, there exists J ∈ J , such that I ⊆ J . Lemma 4. Assume that i E|hi(Xdeci )|2 < ∞. Then for any K ⊆ Id−1 and J = {J1, . . . , Jk} ∈ PId−1\K and all p ≥ 2, i ))iId−1 ‖K,J (1) K⊆L⊆Id, K∈PId\L J∪{K,{d}}≺K∪{L} p(degK−degJ )/2‖(hi)iId‖L,K K⊆L⊆Id−1, K∈PId−1\L J∪{K}≺K∪{L} p1+(degK−degJ )/2 Edmax ‖(hi)iId−1‖ Remark In the above lemma we slightly abuse the notation, by identifying for K = ∅ the partition {∅} ∪ J with J . Given Lemma 3, the proof of Lemma 4 is not complicated, the main idea is just a change of basis, however due to complicated notation it is quite difficult to write it directly. We find it more convenient to write the proof in terms of tensor products of Hilbert spaces. Let us begin with a classical fact. Lemma 5. Let H be a separable Hilber space and X a Σ-valued random variable. Then H ⊗ L2(X) ≃ L2(X,H), where L2(X,H) is the space of square integrable random variables of the form f(X), f : Σ → H-measurable. With the above identification, for h ∈ H, f(X) ∈ L2(X), we have h⊗f(X) = hf(X) ∈ L2(X,H). Proof of Lemma 4. To avoid problems with notation, which would lengthen an intuitively easy proof, we will omit some technical details, related to obvious identification of some tensor product of Hilbert spaces (in the spirit of Lemma 5). Similarly, when considering linear functionals on a space, which can be written as a tensor product in several ways, we will switch to the most convenient notation, without further explanations. H0 = H ⊗ ⊗l∈K (⊕ni=1L2(X i )] ≃ ⊕|iK |≤nL 2(XdeciK ,H) and, for j = 1, . . . , k, Hi = ⊗l∈Jj(⊕ni=1L2(X i )) ≃ ⊕|iJj |≤nL 2(XdeciJj In the case K = ∅, we have (using the common convention for empty products) H0 ≃ H. For id = 1, . . . , n and fixed value of X , let Aid be a linear functional on H̃ = ⊕|iId−1 |≤nL 2(XdeciId−1 ,H) ≃ ⊗kj=0Hk, given by (hi(Xdeci ))|iId−1 |≤n ∈ H̃, with the formula Aid((giId−1 (XdeciId−1 ))iId−1 ) = 〈(giId−1 (X iId−1 ))iId−1 , (hi(X i ))iId−1 |iId−1 |≤n E{1,...,d−1}〈giId−1 (X iId−1 ), hi(X i )〉H . As functions of X , Aid = Aid(X ) are independent random linear func- tionals. Thus they determine also random (k + 1)-linear functionals on ⊕kj=0Hk, given by (h0, h1, . . . , hk) 7→ Aid(h0 ⊗ h1 ⊗ . . .⊗ hk). If we denote by ‖·‖ the norm of a (k+1)-linear functional, the left hand-side of (1), can be written as Aid(X Moreover, denoting by ‖Aid‖HS the norm of Aid seen as a linear operator on ⊗kj=0Hj (by analogy with the Hilbert-Schmidt norm of a matrix), we have E‖Aid(X )‖2HS = ‖(hi)i‖2Id,∅ < ∞, so the sequence Aid(X ), determines a linear functional A on H̃⊗[⊕nid=1L ⊕|i|≤nL2(Xdeci ,H) ≃ ⊕nid=1L , H̃), given by the formula A(g1(X 1 ), . . . , gn(X n )) = E[Aid(X )(gid(X It is easily seen, that if we interpret the domain of this functional as⊕|i|≤nL2(Xdeci ,H), then it corresponds to the multimatrix (hi(X i ))i. Let us now introduce the following notation, consistent with the defini- tion of ‖ · ‖J . If T is a linear functional on ⊗mj=0Ej for some Hilbert spaces Ej , and I = {L1, . . . , Lr} ∈ PIm∪{0}, then let ‖T‖I denote the norm of T as a r-linear functional on ⊕ri=1[⊗j∈LiEj ], given by (e1, . . . , er) 7→ T (e1 ⊗ . . .⊗ er). Now, denotingHk+1 = ⊕nid=1L ), we can apply the above definition to H̃ ⊗ [⊕nid=1L )] ≃ ⊗k+1j=0Hj and use Lemma 3 to obtain Aid(X ∥ ≤Ld I∈PIk+1∪{0} p(1+deg (I)−(k+2))/2‖A‖I I∈PIk∪{0} p1+(1+deg(I)−(k+2))/2 ‖Aid(X )‖2I . This inequality is just the statement of the Lemma, which follows from ,,associativity” of the tensor product and its ,,distributivity” with respect to the simple sum of Hilbert spaces. Indeed, denoting Jk+1 = {d}, we have for 0 /∈ Li and U = ⊗j∈LiHj ≃ ⊗j∈Li ⊗l∈Jj (⊕ns=1L2(X(l)s )) ≃ ⊗l∈U (⊕ns=1L2(X(l)s )) ≃ ⊕|iU |≤nL 2(XdeciU ). Similarly, if 0 ∈ Li, ⊗j∈LiHj ≃ [⊕|iK |≤nL 2(XdeciK ,H)]× [⊗06=j∈Li ⊗l∈Jj (⊕ 2(X(l)s ))] ≃ ⊕|iU |≤nL 2(XdeciU ,H), where U = ( 06=j∈Li Jj) ∪ K. Using the fact that for fixed X(d)id , Aid corresponds to the multimatrix (hi(X i ))|iId−1 |≤n , and A corresponds to (hi(X i ))|i|≤n, we can see, that each summand ‖ · ‖I on the right hand side of (2) is equal to some summand ‖ · ‖L,K on the right hand side of (1). In- formally speaking and abusing slightly the notation (in the case K = ∅), we ,,merge” the elements of the partition {{d}, J1, . . . , Jk,K} or {J1, . . . , Jk,K} in a way described by the partition I, thus obtaining the partition {L}∪K, where L is the set corresponding in the new partition to the set Li ∈ I, containing 0 (in particular, if K = ∅ and {0} ∈ I, then L = ∅). Let us also notice, that deg(I) = deg(K) + 1, hence 1 + deg(I)− (k + 2) = deg(K)− deg(J ), which shows, that also the powers of p on the right hand sides of (1) and (2) are the same, completing the proof. Proof of Theorem 1. For d = 1, the theorem is an obvious consequence of Lemma 1. Indeed, since | · | = sup|φ|≤1 |φ(·)|, and we can restrict the supre- mum to a countable set of functionals, we have hi(Xi)|p ≤ Lp hi(Xi)|)p + pp/2 sup |φ|≤1 E〈φ, hi(Xi)〉2)p/2 + ppEmax |hi(Xi)|p But E| i hi(Xi)| ≤ i hi(Xi)|2 = i E|hi(Xi)|2 = ‖(hi)i‖{1},∅ and we also have sup|φ|≤1( i E〈φ, hi(Xi)〉2)1/2 = ‖(hi)i‖∅,{1} and maxi |hi(Xi)| = maxi ‖hi‖∅,∅. We will now proceed by induction with respect to d. Assume that the theorem is true for all integers smaller than d ≥ 2 and denote Ĩc = Ic\{d} for I ⊆ Id. Then, applying it for fixed X(d)id to the array of functions hi(x1, . . . , xd−1,X )iId−1 , we get by the Fubini theorem i )|p K⊆I⊆Id−1 J∈PI\K pp(#Ĩ c+degJ /2) EIc‖( hi)iI‖ where we have replaced the maxima in iIc by sums (we can afford this apparent loss, since we will be able to fix it with Lemma 2). Now, from Lemma 1 (applied to Ed) it follows that hi)iI‖ K,J ≤ L (Ed‖( hi)iI‖K,J )p + pp/2‖(hi)iI∪{d}‖ K,J∪{{d}} Ed‖(hi)iI‖ Since Ĩc = (I ∪ {d})c, degJ ∪ {{d}} = degJ + 1 and #Ic = #Ĩc + 1, combining the above inequalities gives i )|p ≤ L K⊆I⊆Id J∈PI\K pp(#I c+degJ /2) ‖(hi)iI‖ K⊆I⊆Id−1 J∈PI\K pp(#Ĩ c+degJ /2) (Ed‖( hi)iI‖K,J )p By applying Lemma 4 to the second sum on the right hand side, we get i )|p ≤ L K⊆I⊆Id J∈PI\K pp(#I c+degJ /2) ‖(hi)iI‖ We can now finish the proof using Lemma 2. We apply it to EIc for I 6= Id, with #Ic instead of d and p/2 instead of p (for p = 2 the theorem is trivial, so we can assume that p > 2) and α = 2#Ic + degJ +#Ic. Using the fact that (p/2)α#I c ≤ Lpd and E‖(hi)iI‖2K,J ≤ EI |hi|2, we get ‖(hi)iI‖ K,J ≤ p −αp/2L̃ pαp/2EIc max ‖(hi)iI‖ p#Jp/2EJ max iIc\J EIc\J‖(hi)iI‖2K,J ≤ L̄pd EIc max ‖(hi)iI‖ + p−(#I c+degJ /2)p max EJ max EJc |h(Xdeci )|2)p/2 EIc max ‖(hi)iI‖ K,J + p −(#Ic+degJ /2)p max EJ max ‖(hi)iJc‖ which allows us to replace the sums in iIc on the right-hand side of (3) by the corresponding maxima, proving the inequality in question. Theorem 1 gives a precise estimate for moments of canonical Hilbert space valued U -statistics. In the sequel however we will need a weaker estimate, using the ‖·‖K,J norms only for I = Id and specialized to the case hi = h. Before we formulate a proper corollary, let us introduce Definition 7. Let h : Σd → H be a canonical kernel. Let moreover X1,X2, . . . ,Xd be i.i.d random variables with values in Σ. Denote X = (X1, . . . ,Xd) and for J ⊆ Id, XJ = (Xj)j∈J . For K ⊆ I ⊆ Id and J = {J1, . . . , Jk} ∈ PI\K , we define ‖h‖K,J = sup EI〈h(X), g(XK )〉 fj(XJj ) : g : Σ #K → H, E|g(XK)|2 ≤ 1, fj : Σ #Jj → R, Efj(XJj ))2 ≤ 1, j = 1, . . . , k In other words ‖h‖K,J is the ‖ · ‖K,J of an array (hi)|i|=1, with h(1,...,1) = h. Remark For I = Id, ‖h‖K,J is a norm, whereas for I ( Id, it is a random variable, depending on XIc . It is also easy to see that if all the variables X i are i.i.d. and for all |i| ≤ n we have hi = h, then for any fixed value of iIc , ‖(hi)|iI |≤n‖K,J = ‖h‖K,Jn #I/2, where ‖h‖K,J is defined with respect to any i.i.d. sequence X1, . . . ,Xd of the form Xj = X for j ∈ Ic. We also have ‖h‖K,J ≤ EI |h(X)|2, which together with the above observations allows us to derive the following Corollary 1. For all p ≥ 2, we have h(Xdeci )|p ≤L J∈PId\K ppdegJ /2ndp/2‖h‖pK,J pp(d+#I c)/2n#Ip/2EIc max (EI |h(Xdeci )|2)p/2 The Chebyshev inequality gives the following corollary for bounded ker- Corollary 2. If h is bounded, then for all t ≥ 0, h(Xdeci )| ≥ Ld(nd/2(E|h|2)1/2 + t) Ld exp K(Id,J∈PId\K nd/2‖h‖K,J )2/deg(J )) n#I/2‖(EI |h|2)1/2‖∞ )2/(d+#Ic))] Before we formulate the version of exponential inequalities that will be useful for the analysis of the LIL, let us recall the classical definition of Hoeffding projections. Definition 8. For an integrable kernel h : Σd → H, define πdh : Σk → R with the formula πdh(x1, . . . , xk) = (δx1 −P)× (δx2 −P)× . . .× (δxd −P)h, where P is the law of X1. Remark It is easy to see that πkh is canonical. Moreover πdh = h iff h is canonical. The following Lemma was proven for H = R in [2] (Lemma 1). The proof given there works for an arbitrary Banach space. Lemma 6. Consider an arbitrary family of integrable kernels hi : Σ d → H, |i| ≤ n. For any p ≥ 1 we have |i|≤n πdhi(X |i|≤n ǫdeci hi(X In the sequel we will use exponential inequalities to U -statistics gener- ated by πdh, where h will be a non-necessarily canonical kernel of order d. Since the kernel h̃((ε1,X1), . . . , (εd,Xd)) = ε1 · · · εdh(X1, . . . ,Xd), where εi’s are i.i.d. Rademacher variables independent of Xi’s is always canoni- cal, Corollary 1, Lemma 6 and the Chebyshev inequality give us also the following corollary (note that ‖h̃‖K,J = ‖h‖K,J ) Corollary 3. If h is bounded, then for all p ≥ 0, πdh(X ≥ Ld(nd/2(E|h|2)1/2 + t) Ld exp K(Id,J∈PId\K nd/2‖h‖K,J )2/ deg(J )) n#I/2‖(EI |h|2)1/2‖∞ )2/(d+#Ic))] 4 The equivalence of several LIL statements In this section we will recall general results on the correspondence of various statements of the LIL. We will state them without proofs, since all of them have been proven in [9] and [2] in the real case and the proofs can be directly transferred to the Hilbert space case, with some simple modifications that we will indicate. Before we proceed, let us introduce the assumptions and notation com- mon for the remaining part of the article. • We assume that (Xi)i∈N, (X(k)i )i∈N,1≤k≤d are i.i.d. and h : Σd → H is a measurable function. • Recall that (εi)i∈N, (ε(k)i )i∈N,1≤k≤d are independent Rademacher vari- ables, independent of (Xi)i∈N, (X i )i∈N,1≤k≤d. • To avoid technical problems with small values of h let us also define LLx = loglog (x ∨ ee). • We will also occasionally write X for (X1, . . . ,Xd) and for I ⊆ Id, XI = (Xi)i∈I . Sometimes we will write simply h instead of h(X). • We will use the letter K to denote constants depending only on the function h. We will need the following simple fact Lemma 7. If E|h|2/(LL|h|)d = K < ∞ then E(|h|2∧u) ≤ L(loglog u)d with L depending only on K and d. The next lemma comes from [9]. It is proven there for H = R but the argument is valid also for general Banach spaces. Lemma 8. Let h : Σd → H be a symmetric function. There exist constants Ld, such that if lim sup (nloglog n)d/2 i∈Idn h(Xi) ∣ < C a.s., (4) |i|≤2n ǫdeci h(X ∣ ≥ D2nd/2 logd/2 n < ∞ (5) for D = LdC. Lemma 9. For a symmetric function h : Σd → H, the LIL (4) is equivalent to the decoupled LIL lim sup (nloglog n)d/2 i∈Idn h(Xdeci ) ∣ < D a.s., (6) meaning that (4) implies (6) with D = LdC, and conversely (6) implies (4) with C = LdD. Proof. This is Lemma 8 in [2]. The proof is the same as there, one needs only to replace l∞ with l∞(H) – the space of bounded H-valued sequences. The next lemma also comes from [2] (Lemma 9). Although stated for real kernels, its proof relies on an inductive argument with a stronger, Banach- valued hypothesis. Lemma 10. There exists a universal constant L < ∞, such that for any kernel h : Σd → H we have |j|≤n i : ik≤jk,k=1...d h(Xdeci ) ∣ ≥ t ≤ LdP |i|≤n h(Xdeci ) ∣ ≥ t/Ld Corollary 4. Consider a kernel h : Σd → H and α > 0. If |i|≤2n h(Xdeci )| ≥ C2nα logα n) < ∞, lim sup (nloglog n)α |i|≤2n h(Xdeci ) ∣ ≤ Ld,αC a.s. Proof. Given Lemma 10, the proof is the same as the one for real kernels, presented in [2] (Corollary 1 therein). The next lemma shows that the contribution to a decoupled U-statistic from the ’diagonal’, i.e. from the sum over multiindices i /∈ Idn is negligible. The proof given in [2] (Lemma 10) is still valid, since the only part which cannot be directly transferred to the Banach space setting is the estimate of variance of canonical U-statistics, which is the same in the real and general Hilbert space case. Lemma 11. If h : Σd → H is canonical and satisfies E(|h|2 ∧ u) = O((loglog u)β), for some β, then lim sup (nloglog n)d/2 |i|≤n ∃j 6=kij=ik h(Xdeci ) ∣ = 0 a.s. (7) Corollary 5. The randomized decoupled LIL lim sup (nloglog n)d/2 |i|≤n ǫdeci h(X ∣ < C (8) is equivalent to (5), meaning then if (8) holds then so does (5) with D = LdC and (5) implies (8) with C = LdD. The proof is the same as for the real-valued case, given in [2] (Corollary 2), one only needs to replace h2 by |h|2 and use the formula for the second moments in Hilbert spaces. Corollary 6. For a symmetric, canonical kernel h : Σd → H, the LIL (4) is equivalent to the decoupled LIL ’with diagonal’ lim sup (nloglog n)d/2 |i|≤n h(Xdeci ) ∣ < D (9) again meaning that there are constants Ld such that if (4) holds for some D then so does (9) for D = LdC, and conversely, (9) implies (4) for C = LdD. Proof. The proof is the same as in the real case (see [2], Corollary 3). Al- though the integrability of the kernel guaranteed by the LIL is worse in the Hilbert space case, it still allows one to use Lemma 11. 5 The canonical decoupled case Before we formulate the necessary and sufficient conditions for the bounded LIL in Hilbert spaces, we need Definition 9. For a canonical kernel h : Σd → H, K ⊆ Id, J = {J1, . . . , Jk} ∈ PId\K and u > 0 we define ‖h‖K,J ,u = sup{E〈h(X), g(XK )〉 fi(XJi) : g : Σ K → H, fi : Σ Ji → R, ‖g‖2, ‖fi‖2 ≤ 1, ‖g‖∞, ‖fi‖∞ ≤ u}, where for K = ∅ by g(XK) we mean an element g ∈ H, and ‖g‖2 denotes just the norm of g in H (alternatively we may think of g as of a random variable measurable with respect to σ((Xi)i∈∅), hence constant). Thus the condition on g becomes in this case just |g| ≤ 1. Example For d = 2, the above definition reads as ‖h(X1,X2)‖∅,{{1,2}},u = sup{|Eh(X1,X2)f(X1,X2)| : Ef(X1,X2) 2 ≤ 1, ‖f‖∞ ≤ u}, ‖h(X1,X2)‖∅,{{1}{2}},u = sup{|Eh(X1,X2)f(X1)g(X2)| : Ef(X1) 2,Eg(X2) 2 ≤ 1 ‖f‖∞, ‖g‖∞ ≤ u}, ‖h(X1,X2)‖{1},{{2}},u = sup{E〈f(X1), h(X1,X2)〉g(X2) : E|f(X1)|2,Eg(X2)2 ≤ 1 ‖f‖∞, ‖g‖∞,≤ u} ‖h(X1,X2)‖{1,2},∅,u = sup{E〈f(X1,X2), h(X1,X2)〉 : E|f(X1,X2)|2 ≤ 1, ‖f‖∞ ≤ u}. Theorem 2. Let h be a canonical H-valued symmetric kernel in d variables. Then the decoupled LIL lim sup nd/2(loglog n)d/2 |i|≤n h(Xdeci )| < C (10) holds if and only if (LL|h|)d < ∞ (11) and for all K ⊆ Id,J ∈ PId\K lim sup (loglog u)(d−deg J )/2 ‖h‖K,J ,u < D. (12) More precisely, if (10) holds for some C then (12) is satisfied for D = LdC and conversely, (11) and (12) implies (10) with C = LdD. Remark Using Lemma 7 one can easily check that the condition (12) with D < ∞ for I = Id is implied by (11). 6 Necessity The proof is a refinement of ideas from [16], used to study random matrix approximations of the operator norm of kernel integral operators. Lemma 12. If a, t > 0 and h is a nonnegative d-dimensional kernel such that NdEh(X) ≥ ta and ‖EIh(X)‖∞ ≤ N−#Ia for all ∅ ⊆ I ( {1, . . . , d}, ∀λ∈(0,1) P( |i|≤N h(Xdeci ) ≥ λta) ≥ (1−λ)2 t+ 2d − 1 ≥ (1−λ) 22−d min(1, t). Proof. We have |i|≤N h(Xdeci ) |i|≤N |j|≤N Eh(Xdeci )h(X |i|≤N |j|≤N : {k : ik=jk}=I Eh(Xdeci )h(X |i|≤N |j|≤N : {k : ik=jk}=I E[h(Xdeci )EIch(X ≤ N2d(Eh(X))2 + ∅6=I⊆Id Nd+#I Eh(X)‖EIch(X)‖∞ ≤ N2d(Eh(X))2 + (2d − 1)NdaEh(X) ≤ N2d(Eh(X))2 + (2d − 1)t−1N2d(Eh(X))2 ≤ t+ 2 d − 1 |i|≤n h(Xdeci ) The lemma follows now from the Paley-Zygmund inequality (see e.g. [5], Corollary 3.3.2.), which says that for an arbitrary nonnegative random vari- able S, P(S ≥ λS) ≥ (1− λ)2 (ES) Corollary 7. Let A ⊆ Σd be a measurable set, such that ∀∅(I({1,...,d}∀xIc∈ΣIc PI((xIc ,XI) ∈ A) ≤ N −#I . P(∃|i|≤N Xdeci ∈ A) ≥ 2−d min(NdP(X ∈ A), 1). Proof. We apply Lemma 12 with h = IA, a = 1, t = N dP(X ∈ A) and λ → 0+. Lemma 13. Suppose that Zj are nonnegative r.v.’s, p > 0 and aj ∈ R are such that P(Zj ≥ aj) ≥ p for all j. Then Zj ≥ p aj/2) ≥ p/2. Proof. Let α := P( j Zj ≥ p j aj/2), then aj ≤ E( min(Zj , aj)) ≤ α aj + p aj/2. Theorem 3. Let Y be a r.v. independent of X i . Suppose that for each n, an ∈ R, hn is a d+ 1-dimensional nonnegative kernel such that |i|≤2n i , Y ) ≥ an Let p > 0, then there exists a constant Cd(p) depending only on p and d such that the sets An := x ∈ Sd : ∀n≤m≤2d−1n PY (hm(x, Y ) ≥ Cd(p)2d(n−m)am) ≥ p satisfy 2dnP(X ∈ An) < ∞. Proof. We will show by induction on d, that the assertion holds with C1(p) := 1, C2(p) := 12/p and Cd(p) := 12p −1 max 1≤l≤d−1 Cd−l(2 −l−4p/3) for d = 3, 4, . . . . For d = 1 we have min(2nP(X ∈ An), 1) ≤ P(∃|i|≤2n Xdeci ∈ An) = P(∃|i|≤2n PY (hn(Xdeci , Y ) ≥ an) ≥ p) ≤ P(PY ( |i|≤2n i , Y ) ≥ an) ≥ p) ≤ p−1P( |i|≤2n i , Y ) ≥ an). Before investigating the case d > 1 let us define Ãn := An \ The sets Ãn are pairwise disjoint and obviously Ãn ⊂ An. Notice that since Cd(p) ≥ 1, P(X ∈ An) ≤ P(PY (hn(X,Y ) ≥ an)) ≤ p−1P( |i|≤2n i , Y ) ≥ an). Hence n P(X ∈ An) < ∞, so P(X ∈ lim supAn) = 0. But if x /∈ lim supAn, then ndIAn(x) ≤ nd+1I (x). So it is enough to show 2dnP(X ∈ Ãn) < ∞. Induction step Suppose that the statement holds for all d′ < d, we will show it for d. First we will inductively construct sets Ãn = A n ⊃ A1n ⊃ . . . ⊃ Ad−1n such that for 1 ≤ l ≤ d− 1, ∀∅(I({1,...,d−1}, #I≤l ∀xIc PI((xIc ,XI) ∈ A n) ≤ 2−n#I (13) 2ndP(X ∈ Al−1n \Aln) < ∞. (14) Suppose that 1 ≤ l ≤ d − 1 and the set Al−1n was already defined. Let I ⊂ {1, . . . , d} be such that #I = l and let j ∈ I. Notice that PI((xIc ,XI) ∈ Al−1n ) = EjPI\{j}((xIc ,Xj ,XI\{j}) ∈ Al−1n ) ≤ 2−n(l−1) by the property (13) of the set Al−1n . Let us define for n(l− 1)+1 ≤ k ≤ nl, BIn,k := {xIc : PI((xIc ,XI) ∈ Al−1n ) ∈ (2−k, 2−k+1]} BIn := k=n(l−1)+1 BIn,k = {xIc : PI((xIc ,XI) ∈ Al−1n ) > 2−nl}. We have 2dnP(X ∈ Al−1n ,XIc ∈ BIn) ≤ 2 k=n(l−1)+1 2dn−kP(XIc ∈ BIn) = 2EkI1(XIc), where kI1(xIc) := k=n(l−1)+1 2dn−kIBI (xIc). Let m ≥ 1 and CIm := {xIc : 2(m+1)(d−l) > k1(xIc) ≥ 2m(d−l)}. Notice that for n > m and k ≤ nl, 2dn−k ≥ 2(d−l)(m+1), moreover n<m/2 k=n(l−1)+1 2dn−k ≤ n<m/2 2(d−l+1)n ≤ 4 2(d−l+1)(m−1)/2 ≤ 2 2(d−l)m. Hence xIc ∈ CIm ⇒ m/2≤n≤m k=n(l−1)+1 2dn−kIBI (xIc) ≥ 2(d−l)m. (15) Let m ≤ r ≤ 2d−2m, if m/2 ≤ n ≤ m, then since Al−1n ⊂ An we have for all x ∈ Sd, PY (hr(x, Y ) ≥ Cd(p)2d(n−r)arIAl−1n (x)) ≥ p, therefore, since Al−1n ⊂ Ãn are pairwise disjoint, hr(x, Y ) ≥ Cd(p)2−drar m/2≤n≤m Al−1n Hence, by Lemma 13, |iI |≤2r hr(xIc ,X , Y ) ≥ p Cd(p)2 −drar |iI |≤2r k2,xIc (X , (16) where k2,xIc (xI) := m/2≤n≤m (xIc , xI). We have ‖k2,xIc‖∞ ≤ 2dm and for ∅ 6= J ( I, by the property (13) of Al−1n , EJk2,xIc (xI\J ,XJ ) = m/2≤n≤m 2dnPJ (xIc , xI\J ,XJ) ∈ Al−1n m/2≤n≤m 2(d−#J)n ≤ 2(d−#J)m+1. Moreover for xIc ∈ CIm, by the definition of BIn,k and (15), Ek2,xIc (XI) ≥ m/2≤n≤m k=n(l−1)+1 2dnPI((xIc ,XI) ∈ Al−1n )IBI (xIc) m/2≤n≤m k=n(l−1)+1 2dn−kIBI (xIc) ≥ 2(d−l)m. Therefore by Lemma 12 (with l instead of d and a = 2(d−l)m+rl+1, t = 1/6, N = 2r, λ = 1/2), for m ≤ r ≤ 2d−2m, |iI |≤2r k2,xIc (X ) ≥ 1 2(d−l)m+rl 2−l−3. Combining the above estimate with (16) we get (for xIc ∈ CIm and m ≤ r ≤ 2d−2m), |iI |≤2r hr(xIc ,X , Y ) ≥ p Cd(p)2 (d−l)(m−r)ar 2−l−4p. Let us define Ỹ := ((X i )j∈I , Y ) and h̃n(xIc , Ỹ ) := |iI |≤2n hn(xIc ,X , Y ). |iIc |≤2 h̃n(X , Ỹ ) ≥ an) = |i|≤2n i , Y ) ≥ an) < ∞. Moreover (since Cd(p) ≥ 12p−1Cd−l(2−l−4p/3)), CIm ⊆ ∀m≤r≤2d−2m PỸ (h̃r(xIc , Ỹ ) ≥ Cd−l(2 −l−4p/3)2(d−l)(m−r)ar) ≥ 2−l−4p/3 Hence by the induction assumption, 2(d−l)mP(XIc ∈ CIm) < ∞, so EkI1(XIc) < ∞ and thus ∀#I=l 2dnP(X ∈ Al−1n ,XIc ∈ BIn) < ∞. (17) We set Aln := {x ∈ Al−1n : xIc /∈ BIn for all I ⊂ {1, . . . , d},#I = l}. The set Aln satisfies the condition (13) by the definition of B n and the property (13) for Al−1n . The condition (14) follows by (17). Notice that the set Ad−1n satisfies the assumptions of Corollary 7 with N = 2n, therefore if Cd(p) ≥ 1, 2−d min(1, 2ndP(X ∈ Ad−1n )) ≤ P(∃|i|≤2n Xdeci ∈ Ad−1n ) ≤ P(∃|i|≤2n Xdeci ∈ Ãn) ≤ P(∃|i|≤2n PY (hn(Xdeci , Y ) ≥ Cd(p)an) ≥ p) ≤ P(PY ( |i|≤2n i , Y ) ≥ an) ≥ p) ≤ p−1P( |i|≤2n i , Y ) ≥ an). Therefore ndP(X ∈ Ad−1n ) < ∞, so by (14) we get X ∈ Ãn) = P(X ∈ Al−1n \ Aln) + P(X ∈ Ad−1n ) Corollary 8. If |i|≤2n h2(Xdeci ) ≥ ε2nd(log n)α for some ε > 0 and α ∈ R, then E h (LL|h|)α Proof. We apply Theorem 3 with hn = h 2 and an = ε2 nd logd n in the degenerate case when Y is deterministic. It is easy to notice that h2 ≥ C̃d(p, ε)2 dn logd n implies that ∀n≤m≤2d−1n h2 ≥ Cd(p)2d(n−m)am. To prove the necessity part of Theorem 2 we will also need the following Lemmas Lemma 14 ([2], Lemma 12). Let g : Σd → R be a square integrable function. |i|≤n g(Xdeci )) ≤ (2d − 1)n2d−1Eg(X)2. Lemma 15 ([2], Lemma 5). If E(|h|2 ∧ u) = O((loglog u)β) then E|h|1{|h|≥s} = O( (loglog s)β Lemma 16. Let (ai)i∈Idn be a d–indexed array of vectors from a Hilbert space H. Consider a random variable |i|≤n |i|≤n For any set K ⊆ Id and a partition J = {J1, . . . , Jm} ∈ PId\K let us define ‖(ai)‖∗K,J ,p := sup |i|≤n 〈ai, α(0)iK 〉 |α(0)iK | 2 ≤ 1, )2 ≤ p, ∀imaxJk∈In )2 ≤ 1, k = 1, . . . ,m where ⋄J = J\{max J} (here ai = ai). Then, for all p ≥ 1, ‖S‖p ≥ K⊆Id,J∈PId\K ‖(ai)‖∗K,J ,p. In particular for some constant cd P(S ≥ cd K⊆Id,J∈PId\K ‖(ai)‖∗K,J ,p) ≥ cd ∧ e−p. Remark For K = ∅, we define ‖(ai)‖∗∅,J ,p := sup |i|≤n ∈ R(I )2 ≤ p, ∀imaxJk∈In )2 ≤ 1, k = 1, . . . ,m It is also easy to see that for a d-indexed matrix, ‖(ai)i‖Id,{∅},p = i |ai|2 = ‖S‖2 and thus does not depend on p. Since it will not be important in the applications, we keep a uniform notation with the subscript p. Examples For d = 1, we have ‖(ai)i≤n‖∗∅,{{1}},p = sup α2i ≤ p, |αi| ≤ 1, i = 1, . . . , n ‖(ai)i≤n‖∗{1},∅,p = sup 〈ai, αi〉 : |αi|2 ≤ 1 |ai|2, whereas for d = 2, we get ‖(aij)i,j≤n‖∗∅,{{1},{2}},p = sup i,j=1 aijαiβj α2i ≤ p, β2j ≤ p, ∀i∈In |αi| ≤ 1,∀j∈In |βj | ≤ 1 ‖(aij)i,j≤n‖∗∅,{I2},p = sup i,j=1 aijαij i,j=1 α2ij ≤ p,∀j∈In α2ij ≤ 1 ‖(aij)i,j≤n‖∗{1},{{2}},p = sup i,j=1 〈aij , αi〉βj |αi|2 ≤ 1, β2j ≤ p,∀j∈In|βj | ≤ 1 ‖(aij)i,j≤n‖∗I2,∅,p = sup i,j=1 〈aij , αij〉 i,j=1 α2ij ≤ 1 |aij |2. Proof of Lemma 16. We will combine the classical hypercontractivity prop- erty of Rademacher chaoses (see e.g. [5], p. 110-116) with Lemma 3 in [2], which says that for H = R we have ‖S‖p ≥ J∈PId ‖(ai)‖∅,J ,p. (18) Since ‖(ai)‖Id,{∅},p = i |ai|2 = ‖S‖2, the inequality ‖S‖p ≥ L−1‖(ai)‖Id,{∅},p is just Jensen’s inequality (p ≥ 2) or the aforesaid hypercontractivity of Rademacher chaos (p ∈ (1, 2)). On the other hand, for K 6= Id and J ∈ PId\K , we have ‖S‖p = EId\KEK iId\K p)1/p EId\K iId\K 2)p/2)1/p EId\K sup iId\K 〈α(0)iK , ai〉 p)1/p EId\K iId\K 〈α(0)iK , ai〉 p)1/p L#KLd−#K 〈α(0)iK , ai〉)iId\K ∅,J ,p L#KLd−#K ‖(ai)‖K,J ,p, where the first inequality follows from hypercontractivity applied condition- ally on (ε i )k/∈K,i∈In, the second is Jensen’s inequality and the third is (18) applied for a chaos of order d−#K. The tail estimate follows from moment estimates by the Paley-Zygmund inequality and the inequality ‖(ai)‖K,J ,tp ≤ tdegJ ‖(ai)‖K,J ,p for t ≥ 1 just like in [12, 18]. Proof of necessity. First we will prove the integrability condition (11). Let us notice that by classical hypercontractive estimates for Rademacher chaoses and the Paley-Zygmund inequality (or by Lemma 16), we have |i|≤2n ǫdeci h(X |i|≤2n h(Xdeci ) for some constant cd > 0. By the Fubini theorem it gives |i|≤2n ǫdeci h(X ≥ D2nd/2 logd/2 n ≥ cdP |i|≤2n h(Xdeci ) 2 ≥ D2c−2d 2 nd logd n which together with Lemma 8 yields |i|≤2n h(Xdeci ) 2 ≥ D2c−2 2nd logd n The integrability condition (11) follows now from Corollary 8. Before we proceed to the proof of (12), let us notice that (11) and Lemma 7 imply that E(|h|2 ∧ u) ≤ K(loglog u)d (19) for n large enough. The proof of (12) can be now obtained by adapting the argument for the real valued case. Since limn→∞ = log 2, (5) implies that there exists N0, such that for all N > N0, there exists N ≤ n ≤ 2N , satisfying |i|≤2n ǫdeci h(X > LdC2 nd/2 logd/2 n . (20) Let us thus fix N > N0 and consider n as above. Let K ⊆ Id, J = {J1, . . . , Jk} ∈ PId\K . Let us also fix functions g : Σ#K → H, fj : Σ#Jj → R, j = 1, . . . , k, such that ‖g(Xk)‖2 ≤ 1, ‖g(XK )‖∞ ≤ 2n/(2k+3), ‖fj(XJj )‖2 ≤ 1, ‖fj(XJj)‖∞ ≤ 2n/(2k+3). The Chebyshev inequality gives |iJj |≤2 )2 log n ≤ 10 · 2d2#Jjn log n) ≥ 1− 1 10 · 2d . (21) Similarly, if K 6= ∅, |iK |≤2n |g(XdeciK )| 2 ≤ 10 · 2d2#Kn) ≥ 1− 1 10 · 2d (22) and for K = ∅, |g| ≤ 1 (recall that for K = ∅, the function g is constant). Moreover for j = 1, . . . , k and sufficiently large N , |i⋄Jj |≤2 2n#Jj )2 · log n ≤ 2 n#⋄Jj22n/(2k+3) log n 2n#Jj 2n/(2k+3) log n Without loss of generality we may assume that the sequences (X i )i,j and (ε i )i,j are defined as coordinates of a product probability space. If for each j = 1, . . . , k we denote the set from (21) by Ak, and the set from (22) by A0, we have P( j=0Ak) ≥ 0.9. Recall now Lemma 16. On j=0Ak we can estimate the ‖ · ‖∗K,J ,logn norms of the matrix (h(Xdeci ))|i|≤2n by using the test sequences log n 101/22d/22n#Jj/2 for j = 1, . . . , k and g(XdeciK ) 101/22d/22n#K/2 Therefore with probability at least 0.9 we have ‖(h(Xdeci ))|i|≤2n‖∗K,J ,logn (23) ≥ (log n) 2d(k+1)/210(k+1)/22 j #Jj)n/2 |i|≤2n 〈g(XdeciK ), h(X (log n)k/2 2d(k+1)/210(k+1)/22dn/2 |i|≤2n 〈g(XdeciK ), h(X Our aim is now to further bound from below the right hand side of the above inequality, to have, via Lemma 16, control from below on the conditional tail probability of |i|≤2n ǫ i h(X i ), given the sample (X From now on let us assume that |E〈g(XK), h(X)〉 fj(XJj )| > 1. (24) The Markov inequality, (19) and Lemma 15 give |i|≤2n 〈g(XK), h(XdeciK )〉1{|h(Xdeci )|>2n} 2nd|E〈g, h〉 j=1 fj| 2nd(‖g‖∞ j=1 ‖fj‖∞) · E|h|1{|h|>2n} 2nd|E〈g, h〉 j=1 fj| ≤ 42n(k+1)/(2k+3)E|h|1{|h|>2n} ≤ 4K (log n) n(k+2) . (25) Let now hn = h1{|h|≤2n}. By the Chebyshev inequality, Lemma 14 and (19) |i|≤2n 〈g(XdeciK ),hn(X )− 2ndE〈g, hn〉 |E〈g, hn〉 |i|≤2n〈g(XdeciK ), hn(X j=1 fj(X 22nd|E〈g, hn〉 j=1 fj|2 ≤ 25 (2 d − 1)2n(2d−1) 22nd|E〈g, hn〉 j=1 fj|2 E|〈g, hn〉 ≤ 25(2d − 1) 2 2n(k+1)/(2k+3)E|hn|2 2n|E〈g, hn〉 j=1 fj|2 ≤ 25K(2d − 1) log 2n/(2k+3)|E〈g, hn〉 j=1 fj|2 . (26) Let us also notice that for large n, by (19), Lemma 15 and (24) |E〈g, hn〉 fj| ≥ |E〈g, h〉 fj| − |E〈g, h〉1{|h|>2n} ≥ |E〈g, h〉 fj| − 2n(k+1)/(2k+3)K (log n)d |E〈g, h〉 fj| ≥ Inequalities (25), (26) and (27) imply, that for large n with probability at least 0.9 we have |i|≤2n 〈g(XdeciK ), h(X |i|≤2n 〈g(XdeciK ), hn(X |i|≤2n 〈g(XdeciK ), h(X i )〉1{|h(Xdec )|>2n} ≥ 2nd |E〈g, hn〉 fj| − |E〈g, h〉 ≥ 2nd |E〈g, h〉 fj| − |E〈g, h〉 |E〈g, h〉 fj |. Together with (23) this yields that for large n with probability at least ‖(hi)|i|≤2n‖∗K,J ,logn ≥ 2nd/2 logk/2 n 4 · 2d(k+1)/210(k+1)/2 |E〈g, h〉 Thus, by Lemma 16, for large n |i|≤2n ǫdeci h(X ∣ ≥ cd 2nd/2 logk/2 n 4 · 2d(k+1)/210(k+1)/2 |E〈g, h〉 which together with (20) gives |E〈g, h〉 fj| ≤ LdC 4 · 2d(k+1)/210(k+1)/2 log(d−k)/2 n. In particular for sufficiently large N , for arbitrary functions g : Σ#K → H, fj : Σ #Jj → R, j = 1, . . . , k, such that ‖g(XK)‖∞, ‖fj(XJj )‖2 ≤ 1, ‖g(XK)‖2, ‖fj(XJj )‖∞ ≤ 2N/(2k+3) we have |E〈g, h〉 fj| ≤ LdC 4 · 2d(k+1)/210(k+1)/2 log(d−k)/2 n ≤ L̃dC log(d−k)/2 N, which clearly implies (12). 7 Sufficiency Lemma 17. Let H = H(X1, . . . ,Xd) be a nonnegative random variable, such that EH2 < ∞. Then for I ⊆ Id, I 6= ∅,Id, 2l+#I PIc(EIH 2 ≥ 22l+#Icn) < ∞. Proof. 2l+#I PIc(EIH 2 ≥ 22l+#Icn) = 2lEIc 1{EI |H|2≥22l+#I 21−lEIcEIH 2 ≤ 4EH2 < ∞. Lemma 18. Let X = (X1, . . . ,Xd) and X̃(I) = ((Xi)i∈I , (X i )i∈Ic). De- note H = |h|/(LL|h|)d/2. If E|H|2 < ∞ and hn = h1An , where x : |h(x)|2 ≤ 2nd logd n and ∀I 6=∅,Id EIH 2 ≤ 2#Icn then for I ⊆ Id, I 6= ∅, we have 2−n#I log2d n E[|hn(X)|2|hn(X̃(I))|2] < ∞. Proof. a) I = Id E|hn|4 2nd log2d n ≤ E|h|4 2nd log2d n 1{|h|2≤2nd logd n} ≤ LdE|h|4 |h|2(LL|h|)d < ∞. b) I 6= Id, ∅. Let us denote by EI ,EIc , ẼIc respectively, the expectation with respect to (Xi)i∈I , (Xi)i∈Ic and (X i )i∈Ic . Let also h̃, h̃n stand for h(X̃(I)), hn(X̃(I)) respectively. Then E(|hn|2 · |h̃n|2) 2n#I log2d n E(|hn|2 · |h̃n|21{|h|≤|h̃|}) 2n#I log2d n |h|2|h̃|21 {|h|≤|h̃|} 2n#I log2d n {EIc |h| {|h|2≤22nd} #In logd n, |h̃|2≤22nd} |h|2|h̃|21 {|h|≤|h̃|} 2n#I log2d n 1{EIc |h|21{|h|2≤|h̃|2}≤Ld2 #In logd n, |h̃|2≤22nd} ≤ L̃dE |h|2|h̃|21 {|h|≤|h̃|} (EIc |h|21{|h|2≤|h̃|2})(LL|h̃|)d = L̃dEI ẼIc |h̃|2EIc |h|21 {|h|≤|h̃|} (EIc |h|21{|h|2≤|h̃|2})(LL|h̃|)d ≤ L̃dE |h̃|2 (LL|h̃|)d where to obtain the second inequality, we used the fact that EIc |h|21{|h|2≤22nd,EIcH2≤2#In} ≤ EIc (LL|h|)d (loglog 2 nd)d1{EIcH2≤2#In} ≤ LdEIcH21{EIcH2≤2#In} log d n ≤ Ld2#In logd n. Lemma 19. Consider a square integrable, nonnegative random variable Y . Let Yn = Y 1Bn , with Bn = k∈K(n)Ck, where C0, C1, C2, . . . are pairwise disjoint subsets of Ω and K(n) = {k ≤ n : E(Y 21Ck) ≤ 2 k−n}. (EY 2n ) 2 < ∞ Proof. Let us first notice that by the Schwarz inequality, we have k∈K(n) E(Y 21Ck) k∈K(n) 2(n−k)/22(k−n)/2E(Y 21Ck) k∈K(n) [2n−k(E(Y 21Ck)) k∈K(n) [2n−k(E(Y 21Ck)) (EY 2n ) k∈K(n) [2n−k(E(Y 21Ck)) k : E(Y 21Ck )>0 (E(Y 21Ck)) n : k∈K(n) k : E(Y 21Ck )>0 (E(Y 21Ck)) 2 max n : k∈K(n) k : E(Y 21Ck )>0 (E(Y 21Ck)) E(Y 21Ck) E(Y 21Ck) = 4EY 2 < ∞. Proof of sufficiency. The proof consists of several truncation arguments. The first part of it follows the proofs presented in [11] and [2] for the real- valued case. Then some modifications are required, reflecting the diminished integrability condition in the Hilbert space case. At each step we will show |i|≤2n πdhn(X ∣ ≥ C2nd/2 logd/2 n < ∞, (28) with hn = h1An for some sequence of sets An. In the whole proof we keep the notation H = |h|/(LL|h|)d/2. Let us also fix ηd ∈ (0, 1), such that the following implication holds ∀n=1,2,... |h|2 ≤ η2d2nd logd n =⇒ H2 ≤ 2nd. (29) Step 1 Inequality (28) holds for any C > 0 if x : |h(x)|2 ≥ η2d2nd logd n We have, by the Chebyshev inequality and the inequality E|πdhn| ≤ 2dE|hn| (which follows directly from the definition of πd or may be considered a trivial case of Lemma 6), |i|≤2n πdhn(X ∣ ≥ C2nd/2 logd/2 n |i|≤2n πdhn(X C2nd/2 logd/2 n 2ndE|h|1 {|h|>ηd2 nd/2 logd/2 n} C2nd/2 logd/2 n = 2dC−1E 2nd/2 logd/2 n {|h|>ηd2 nd/2 logd/2 n} ≤ LdC−1E (LL|h|)d < ∞. Step 2 Inequality (28) holds for any C > 0 if x : |h(x)|2 ≤ η2d2nd logd n, ∃I 6=∅,Id EIH 2 ≥ 2#Icn As in the previous step, it is enough to prove that |i|≤2n ǫ i hn(X 2nd/2 logd/2 n The set An can be written as I⊆Id,I 6=Id,∅ An(I), where the sets An(I) are pairwise disjoint and An(I) ⊆ {x : |h(x)|2 ≤ 22nd, EIH2 ≥ 2#I Therefore it suffices to prove that |i|≤2n ǫ i h(X i )1An(I)(X 2nd/2 logd/2 n < ∞. (30) Let for l ∈ N, An,l(I) := {x : |h(x)|2 ≤ 22nd, 22l+2+#I cn > EIH 2 ≥ 22l+#Icn} ∩An(I). Then hn1An(I) = l=0 hn,l, where hn,l := hn1An,l(I) (notice that the sum is actually finite in each point x ∈ Σd as for large l, x /∈ An,l(I)). We have |i|≤2n ǫdeci hn,l(X i )| ≤ |iIc |≤2 EIcEI | |iI |≤2n ǫdeciI hn,l(X |iIc |≤2 EIc(EI | |iI |≤2n ǫdeciI hn,l(X i )|2)1/2 ≤ 2(#Ic+#I/2)nEIc(EI |hn,l|2)1/2 ≤ Ld[2(#I c+d/2)n+l+1 logd/2 n]PIc(EIH 2 ≥ 22l+#Icn), where in the last inequality we used the estimate n,l ≤LdEI [(log n)dH21{22l+2+#Icn>EIH2≥22l+#Icn}] ≤Ld22l+2+#I cn(log n)d1{EIH2≥22l+#I Therefore to get (30) it is enough to show that 2l+#I PIc(EIH 2 ≥ 22l+#Icn) < ∞. But this is just the statement of Lemma 17. Step 3 Inequality (28) holds for any C > 0 if x : |h(x)|2 ≤ η2d2nd logd n, ∀I 6=∅,Id EIH 2 ≤ 2#Icn} ∩ with BIn = k∈K(I,n)C k and C 0 = {x : EIH2 ≤ 1}, CIk = {x : 2#I c(k−1) < 2 ≤ 2#Ick}, k ≥ 1, K(I, n) = {k ≤ n : E(H21CI ) ≤ 2k−n}. By Lemma 6 and the Chebyshev inequality, it is enough to show that |i|≤2n ǫ i hn(X i )|4 22nd log2d n The Khintchine inequality for Rademacher chaoses gives |i|≤2n ǫdeci hn(X i )|4 ≤ E( |i|≤2n |hn(Xdeci )|2)2 |i|≤2n |j|≤2n : {k : ik=jk}=I E|[hn(Xdeci )|2|hn(Xdecj )|2] 2nd2n(d−#I)E[|hn(X)|2 · |hn(X̃(I))|2], where X = (X1, . . . ,Xd) and X̃(I) = ((Xi)i∈I , (X i )i∈Ic). To prove the statement of this step it thus suffices to show that for all I ⊆ Id, S(I) := 2−n#I log2d n E[|hn(X)|2|hn(X̃(I))|2] < ∞. (31) The case of nonempty I follows from Lemma 18. It thus remains to consider the case I = ∅. Set H2I = EIH2. We have S(∅) = (E|hn|2)2 log2d n logd n 1An)) 2 ≤ Ld (E(H21An)) (E(H2 ))2 ≤ L̃d (E(H21BIn)) = L̃d (E(H2I1BIn)) 2 < ∞ by Lemma 19, applied for Y 2 = EIH 2, since EH2I = EH 2 < ∞. Step 4 Inequality (28) holds for some C ≤ LdD if x : |h(x)|2 ≤ η2d2nd logd n, ∀I 6=∅,IdEIH 2 ≤ 2#Icn} ∩ (BIn) where BIn is defined as in the previous step. Let us first estimate ‖(EI |hn|2)1/2‖∞ for I ( Id. We have EI |hn|2 ≤ EI |h|21{|h|2≤ηd2nd logd n} k≤n,k/∈K(I,n) ≤ Ld logd n k≤n,k/∈K(I,n) The fact that we can restrict the summation to k ≤ n follows directly from the definition of An for I 6= ∅ and for I = ∅ from (29). The sets CIk are pairwise disjoint and thus ‖EI |hn|2‖∞ ≤ (Ld logd n) max k≤n,k/∈K(I,n) ck = Ld2 #IckI(n) logd n, (32) where kI(n) = max{k ≤ n : k /∈ K(I, n)}. Therefore for C > 0, ( C2nd/2 logd/2 n 2#In/2‖(EI |hn|2)1/2‖∞ )2/(d+#Ic)] k≤n,k/∈K(I,n) ( C2nd/2 logd/2 n 2#In/22#I ck/2 logd/2 n )2/(d+#Ic)] n≥k, k /∈K(I,n) c(n−k)/2 )2/(d+#Ic)] Notice that for each k the inner series is bounded by a geometric series with the ratio smaller than some qd,C < 1 (qd,C depending only on d and C). Therefore the right hand side of the above inequality is bounded by n≥k, k /∈K(I,n) c(n−k)/2 )2/(d+#Ic)] with the convention sup ∅ = 0. But k /∈ K(I, n) implies that 2#Ic(n−k)/2 ≥ (E(H21CI ))−#I c/2. Therefore the above quantity is further bounded by C−2/#I E(H21CI )−#Ic/(d+#Ic)] ≤ L̄dC−2/#I E(H21CI = L̄dC −2/#IcEH2 < ∞, where we used the inequality ex ≥ cdxα for all x ≥ 0 and 0 ≤ α ≤ 2d. We have thus proven that for all I ( Id and C,Ld > 0, n : An 6=∅ ( C2nd/2 logd/2 n 2#In/2‖(EI |hn|2)1/2‖∞ )2/(d+#Ic)] < ∞. (33) Now we will turn to the estimation of ‖hn‖J0,J . Let us consider J0 ⊆ Id, J = {J1, . . . , Jl} ∈ PId\J0 and denote as before X = (X1, . . . ,Xd), XI = (Xi)i∈I . Recall that ‖hn‖J0,J = sup E〈hn(X), f0(XJ0)〉 fi(XJi) : E|f0(XJ0)|2 ≤ 1, Ef2i (XJi) ≤ 1, i ≥ 1 In what follows, to simplify the already quite complicated notation, let us suppress the arguments of all the functions and write just h instead of h(X) and fi instead of fi(XJi). Let us also remark that although f0 plays special role in the definition of ‖ · ‖J0,J , in what follows the same arguments will apply to all fi’s with the obvious use of Schwarz inequality for the scalar product in H. We will therefore not distinguish the case i = 0 and f2i will denote either the usual power or 〈f0, f0〉, whereas ‖fi‖2 for i = 0 will be the norm in L2(H,XJ0), which may happen to be equal just H if J0 = ∅. Since E|fi|2 ≤ 1, i = 0, . . . , l, then for each j = 0, . . . , l and J ( Jj by the Schwarz inequality applied conditionally to XJj\J E|〈hn, f0〉 fi1{EJf2j >a ≤ EJj\J (E(Jj\J)c f2i ) 1{EJf2j ≥a 2}(E(Jj\J)c |hn| 2)1/2 ≤ EJj\J 1{EJf2j ≥a 2}(E(Jj\J)c |hn| 2)1/2 ≤ Ld2 k(Jj\J) c(n)#(Jj\J)/2 logd/2 nEJj\J [(EJf 1{EJf2j ≥a ≤ Ld[2 k(Jj\J) c(n)#(Jj\J)/2 logd/2 n]a−1, where the third inequality follows from (32) and the last one from the ele- mentary fact E|X|1{|X|≥a} ≤ a−1E|X|2. This way we obtain ‖hn‖J0,J (34) ≤ sup{E[〈hn, f0〉 fi] : ‖fi‖2 ≤ 1,∀J(Ji ‖(EJf2i )1/2‖∞ ≤ 2n#(Ji\J)/2} 2(k(Ji\J)c(n)−n)#(Ji\J)/2 logd/2 n ≤ sup{E[〈hn, f0〉 fi] : ‖fi‖2 ≤ 1,∀J(Ji ‖(EJf2i )1/2‖∞ ≤ 2n#(Ji\J)/2} 2(kI (n)−n)#I c/2 logd/2 n. Let us thus consider arbitrary fi, i = 0, . . . , k such that ‖fi‖2 ≤ 1, ‖(EJf2i )1/2‖∞ ≤ 2n#(Ji\J)/2 for all J ( Ji (note that the latter condition means in particular that ‖fi‖∞ ≤ 2n#Ji/2). We have by assumption (12) for sufficiently large n, |E[〈h, f0〉 fi]| ≤ ‖h‖K,J ,2nd/2 ≤ LdD log(d−degJ )/2 n. We have also E|〈h, f0〉1{|h|2≥ηd2nd logd n} fi| ≤ E[|h|1{|h|2≥ηd2nd logd n}] ‖fi‖∞ ≤ 2nd/2E[|h|1{|h|2≥ηd2nd logd n}] =: αn. Also for I ⊆ Id, I 6= ∅, Id, denoting h̃n = h1{|h|2≤ηd2nd logd n}, we get E|〈h̃n,f0〉 fi1{EIH2≥2n#I ≤ EIc (EI |h̃n|2)1/21{EIH2≥2n#Ic} (EJi∩I |fi|2)1/2 2n#(Ji∩I c)/2]EIc [(EI |h̃n|2)1/21{EIH2≥2n#Ic}] ≤ Ld2n#I EIc[(EIH 2 logd n)1/21{EIH2≥2n#I ≤ Ld[2n#I c/2 logd/2 n]EIc[(EIH 2)1/21{EIH2≥2n#I }] =: β Let us denote h̄n = h̃n ∅6=I(Id 1{EIH2≤2#I cn} and γ n = E|h̄n1BIn | Combining the three last inequalities we obtain |E〈hn, f0〉 fi| ≤|E〈h, f0〉 fi|+ |E〈hn1Acn , f0〉 ≤LdD log(d−deg J )/2 n+ E|〈h1{|h|2≥2nd logd n}, f0〉 ∅6=I(Id E|〈h̃n1{EIH2≥2n#Ic}, f0〉 E|〈h̄n1BIn , f0〉 ≤LdD log(d−deg J )/2 n+ αn + ∅6=I(Id βIn + Now, combining the above estimate with (34), we obtain ‖hn‖J0,J ≤ Ld 2(kI (n)−n)#I c/2 logd/2 n+ LdD log (d−degJ )/2 n (35) + αn + ∅6=I(Id βIn + Let us notice that logd/2 n ∀I 6=∅,Id logd/2 n < ∞, (36) ∀I 6=∅,Id (γIn) log2d n The first inequality was proved in Step 1. The proof of the second one is straightforward. Indeed, we have logd/2 n = LdEIc[(EIH 2)1/2 1{EIH2≥2n#I ≤ L̃dEIcEIH2 = L̃dEH2 < ∞. The third inequality is implicitly proved in Step 3. Let us however present an explicit argument. (γIn) log2d n |h|21 {|h|≤ηd2 nd/2 logd/2 n} logd n (EIcEI(H ))2 < ∞ by Lemma 19 applied to the random variable EIH2. We are now in position to finish the proof. Let us notice that we have either E(|h|21{|h|2≤22nd}) ≤ 1, or we can use the function h1{‖h|2≤22nd} (E(|h|21{|h|2≤22nd}))1/2 as a test function in the definition of ‖h‖Id,∅,2nd , obtaining (E(|h|21{|h|2≤22nd}))1/2 = E〈h, g〉 ≤ ‖h‖Id,∅,2nd < D log for large n. Combining this estimate with Corollary 3, we can now write |i|≤2n πdhn(X i )| ≥ L̃d(D + C)2nd/2 logd/2 n ≤ L̃d J0(Id J∈PId\J0 (C2nd/2 logd/2 n 2nd/2‖hn‖Jo,J )2/degJ ] + L̃d ( C2nd/2 logd/2 n 2n#I/2‖(EI |hn|2)1/2‖∞ )2/(d+#Ic)] The second series is convergent by (33). Thus it remains to prove the convergence of the first series. By (35), we have for all J0,J (C logd/2 n ‖hn‖Jo,J )2/degJ ] ( C logd/2 n 2(kI (n)−n)#I c/2 logd/2 n )2/degJ ] + exp ( C logd/2 n D log(d−deg J )/2 n )2/degJ ] + exp (C logd/2 n )2/degJ ] + exp ( C logd/2 n ∅6=I(Id )2/degJ ] + exp ( C logd/2 n )2/ degJ ] (under our permanent convention that the values of Ld in different equations need not be the same). The series determined by the three last components at the right-hand side are convergent by (36) since e−x ≤ Lrx−r for r > 0. The series corresponding to the second component is convergent for C large enough and we can take C = LdD. As for the series corresponding to the first term, we have, just as in the proof of (33) for any I ( Id, ( C logd/2 n (kI (n)−n)#Ic/2 logd/2 n )2/degJ ] n≥k,k /∈K(I,n) C2(n−k)#I )2/degJ ] n≥k,k /∈K(I,n) C2(n−k)#I )2/degJ ] E(H21CI ) = K̄EH2 < ∞. We have thus proven the convergence of the series at the left-hand side of (37) with C ≤ LdD, which ends Step 5. Now to finish the proof, we just split Σd for each n into four sets, described by steps 1–4 and use the triangle inequality, to show that |i|≤2n h(Xdeci )| ≥ LdD2nd/2 logd/2 n which proves the sufficiency part of the theorem by Corollary 4. 8 The undecoupled case Theorem 4. For any function h : Σd → H and a sequence X1,X2, . . . of i.i.d., Σ-valued random variables, the LIL (4) holds if and only if h (LL|h|)d < ∞, h is completely degenerate for the law of X1 and the growth conditions (12) are satisfied. More precisely, if (4) holds, then (12) is satisfied with D = LdC and conversely, (12) together with complete degeneration and the integrability condition imply (4) with C = LdD. Proof. Sufficiency follows from Corollary 6 and Theorem 2. To prove the necessity assume that (4) holds and observe that from Lemma 8 and Corol- lary 5, h satisfies the randomized decoupled LIL (8) and thus, by Theo- rem 2, (11) holds and the growth conditions (12) on functions ‖h‖K,J ,u are satisfied (note that the ‖ · ‖J ,u norms of the kernel h(X1, . . . ,Xd) and ε1 · · · εdh(X1, . . . ,Xd) are equal). The complete degeneracy of 〈ϕ, h〉 for any ϕ ∈ H follows from the necessary conditions for real-valued kernels. Since by (11), Eih is well defined in the Bochner sense, we must have Eih = 0. References [1] Adamczak, R. (2005). Moment Inequalities for U -statistics. Ann. Probab. 34, No. 6, 2288-2314. [2] Adamczak, R., Lata la, R. (2006) LIL for canonical U-statistics. Sub- mitted [3] Arcones, M. and Giné, E. (1995). On the law of the iterated logarithm for canonical U -statistics and processes. Stochastic Process. Appl. 58 217–245. MR1348376. [4] Bousquet O., Boucheron S., Lugosi G., Massart P.(2005). Mo- ment inequalities for functions of independent random variables. Ann. Probab. 33, no. 2, 514-560. MR2123200. [5] de la Peña, V.H. and Giné, E. (1999). Decoupling. From Dependence to Independence. Springer, New York. MR1666908. [6] de la Peña, V.H. and Montgomery-Smith, S. (1994). Bounds for the tail probabilities of U -statistics and quadratic forms. Bull. Amer. Math. Soc. 31 223–227. MR1261237. [7] de la Peña, V. H., Montgomery-Smith, S. J. (1995) Decoupling inequalities for the tail probabilities of multivariate U -statistics. Ann. Probab. 23, no. 2, 806–816. MR1334173. [8] Giné E., Lata la, R. and Zinn, J. (2000). Exponential and moment in- equalities for U -statistics. High Dimensional Probability II, 13–38. Progr. Probab., 47, Birkhauser, Boston. MR1857312. [9] Giné, E. and Zhang, C.-H. (1996). On integrability in the LIL for degenerate U -statistics. J. Theoret. Probab. 9 385–412. MR1385404. [10] Giné, E., Zinn, J. (1994). A remark on convergence in distribution of U -statistics. Ann. Probab. 22 117–125. MR1258868. [11] Giné E., Kwapień, S., Lata la, R. and Zinn, J. (2001). The LIL for canonical U -statistics of order 2, Ann. Probab. 29 520–557. MR1825163. [12] Gluskin, E.D., Kwapień, S. (1995). Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math. 114 , no. 3, 303–309. MR1338834. [13] Goodman, V., Kuelbs, J., Zinn, J. (1981). Some results on the LIL in Banach space with applications to weighted empirical processes. Ann. Probab. 9 , no. 5, 713–752. MR0628870. [14] Houdré, C. and Reynaud-Bouret, P. (2003). Exponential Inequal- ities, with Constants, for U -statistics of Order Two. Stochastic inequal- ities and applications, 55–69. Progr. Probab., 56, Birkhauser, Basel. MR2073426. [15] Kwapień, S., Lata la, R., Oleszkiewicz, K. and Zinn, J. (2003). On the limit set in the law of the iterated logarithm for U -statistics of order two. High dimensional probability, III (Sandjberg), 111–126. Progr. Probab., 55, Birkhauser, Basel. MR2033884. [16] Lata la, R. (1998) On the almost sure boundedness of norms of some empirical operators. Statist. Probab. Lett. 38 , no. 2, 177–182. MR1627869. [17] Lata la, R. and Zinn, J. (2000). Necessary and sufficient conditions for the strong law of large numbers for U -statistics. Ann. Probab. 28 1908–1924. MR1813848. [18] Lata la, R. (2006). Estimation of moments and tails of Gaussian chaoses. Ann. Probab. 34, No. 6. 2061-2440. [19] Rubin, H. and Vitale, R.A. (1980). Asymptotic distribution of sym- metric statistics. Ann. Statist. 8 165–170. MR0557561. Introduction Notation Moment inequalities for U-statistics in Hilbert space The equivalence of several LIL statements The canonical decoupled case Necessity Sufficiency The undecoupled case

0704.1644

Circulating Current States in Bilayer Fermionic and Bosonic Systems

Circulating Current States in Bilayer Fermionic and Bosonic Systems A. K. Kolezhuk∗ Institut für Theoretische Physik C, RWTH Aachen, 52056 Aachen, Germany and Institut für Theoretische Physik, Universität Hannover, 30167 Hannover, Germany It is shown that fermionic polar molecules or atoms in a bilayer optical lattice can undergo the transition to a state with circulating currents, which spontaneously breaks the time reversal symmetry. Estimates of relevant temperature scales are given and experimental signatures of the circulating current phase are identified. Related phenomena in bosonic and spin systems with ring exchange are discussed. PACS numbers: 05.30.Fk, 05.30.Jp, 42.50.Fx, 75.10.Jm Introduction.– The technique of ultracold gases loaded into optical lattices [1, 2] allows a direct experimental study of paradigmatic models of strongly correlated systems. The pos- sibility of unprecedented control over the model parameters has opened wide perspectives for the study of quantum phase transitions. Detection of the Mott insulator to superfluid tran- sition in bosonic atomic gases [3, 4, 5], of superfluidity [6, 7] and Fermi liquid [8] in cold Fermi gases, realization of Fermi systems with low dimensionality [9, 10] mark some of the re- cent achievements in this rapidly developing field [11]. While the atomic interactions can be treated as contact ones for most purposes, polar molecules [12, 13, 14] could provide further opportunities of controlling longer-range interactions. In this Letter, I propose several models on a bilayer op- tical lattice which exhibit a phase transition into an exotic circulating current state with spontaneously broken time re- versal symmetry. Those states are closely related to the “or- bital antiferromagnetic states” proposed first by Halperin and Rice nearly 40 years ago [15], rediscovered two decades later [16, 17, 18] and recently found in numerical studies in ex- tended t-J model on a ladder [19] and on a two-dimensional bilayer [20]. Our goal is to show how such states can be real- ized and detected in a relatively simple optical lattice setup. Model of fermions on a bilayer optical lattice.– Consider spin-polarized fermions in a bilayer optical lattice shown in Fig. 1. The system is described by the Hamiltonian H = V n1,rn2,r + 〈rr′〉 V ′σσ′nσ,rnσ′,r′ (1) 1,ra2,r + h.c.)− t 〈rr′〉 (a†σ,raσ,r′ + h.c.) where r labels the vertical dimers arranged in a two- dimensional (2d) square lattice, σ = 1, 2 labels two layers, and 〈rr′〉 denotes a sum over nearest neighbors. Amplitudes t and t′ describe hopping between the layers and within a layer, respectively. A strong “on-dimer” nearest-neighbor repulsion V ≫ t, t′ > 0 is assumed, and there is an interaction between the nearest-neighbor dimers V ′σσ′ which can be of either sign. This seemingly exotic setup can be realized by using po- lar molecules [13, 14], or atoms with a large dipolar magnetic moment such as 53Cr [12], and adjusting the direction of the dipoles with respect to the bilayer plane. Let θ, ϕ be the polar and azimuthal angles of the dipolar moment (the coordinate axes are along the basis vectors of the lattice, z axis is perpen- dicular to the bilayer plane). Setting ϕ = ±π ,± 3π ensures the dipole-dipole interaction is the same along the x and y di- rections. The nearest neighbor interaction parameters in (1) take the following values: V = (d20/ℓ ⊥)(1 − 3 cos 2 θ), and V ′12 = V 21 = (d 3){1−3R−2(ℓ‖ cos θ+ℓ⊥ sin θ cosϕ) V ′11 = V 22 = V 12(ℓ‖ = 0), where d0 is the dipole moment of the particle, ℓ⊥ and ℓ‖ are the lattice spacings in the direc- tions perpendicular and parallel to the layers, respectively, and R2 = ℓ2 + ℓ2⊥. The strength and the sign of interactions V , Ṽ ′ can be controlled by tuning the angles θ, ϕ and the lattice constants ℓ⊥, ℓ‖. Below we will see that the physics of the problem depends on the difference Ṽ ′ = V ′11 − V 12, (2) with the most interesting regime corresponding to Ṽ ′ < 0. Consider the model at half-filling. Since V ≫ t, t′, we may restrict ourselves to the reduced Hilbert space containing only states with one fermion per dimer. Two states of each dimer can be identified with pseudospin- 1 states |↑〉 and |↓〉. Second- order perturbation theory in t′ yields the effective Hamiltonian 〈rr′〉 J(SxrS r′ + S ) + JzS Sxr , J = 4(t′)2/V, Jz ≡ J∆ = J + Ṽ ′, H = 2t, (3) describing a 2d anisotropic Heisenberg antiferromagnet in a magnetic field perpendicular to the anisotropy axis. The twofold degenerate ground state has the Néel antiferromag- netic (AF) order transverse to the field, with spins canted to- wards the field direction. The AF order is along the y axis for ∆ < 1 (i.e., Ṽ ′ < 0), and along the z axis for∆ > 1 (Ṽ ′ > 0). t , V2 FIG. 1: Bilayer lattice model described by the Hamiltonian (1). The arrows denote particle flow in the circulating current phase. http://arxiv.org/abs/0704.1644v3 The angle α between the spins and the field is classically given by cosα = H/(2ZJS), where S is the spin value and Z = 4 is the lattice coordination number. This classical ground state is exact at the special point H = 2SJ 2(1 + ∆) [21]. The transversal AF order vanishes above a certain critical field Hc; classically Hc = 2ZJS, and the same result follows from the spin-wave analysis of (3) (one starts with the fully polarized spin state at large H and looks when the magnon gap van- ishes). This expression becomes exact at the isotropic point ∆ = 1 and is a good approximation for ∆ close to 1. The long-range AF order along the y direction translates in the original fermionic language into the staggered arrange- ment of currents flowing from one layer to the other: Ny = (−) r〈Syr 〉 7→ (−) 1,ra2,r − a 2,ra1,r)〉. (4) In terms of the original model (1), the condition H < Hc for the existence of such a staggered current order becomes t < 8(t′)2/V. (5) The continuity equation for the current and the lattice symme- try dictate the current pattern shown in Fig. 1. This circulating current (CC) state has a spontaneously broken time reversal symmetry, and is realized only for attractive inter-dimer inter- action Ṽ ′ < 0 (i.e., the easy-plane anisotropy ∆ < 1) [22]. If ∆ = 1, the direction of the AF order in the xy plane is arbitrary, so there is no long-range order at any finite temper- ature. For ∆ > 1 (i.e., Ṽ ′ > 0) the AF order along the z axis corresponds to the density wave (DW) phase with in-layer oc- cupation numbers having a finite staggered component. The phase diagram in the temperature-anisotropy plane is sketched in Fig. 2. At the critical temperature T = Tc the dis- crete Z2 symmetry gets spontaneously broken, so the corre- sponding thermal phase transition belongs to the 2d Ising uni- versality class (except the two lines ∆ = 1 and H = 0 where the symmetry is enlarged to U(1) and the transition becomes the Kosterlitz-Thouless one). Away from the phase bound- aries the critical temperature Tc ∼ J , but at the isotropic point ∆ = 0, H = 0 it vanishes due to divergent thermal fluctua- tions: for 1−∆ ≪ 1 and H ≪ J , it can be estimated as Tc ∼ J/ ln[min(|1−∆| −1, J2/H2)]. (6) <sz><s >y current circulating density wave FIG. 2: Schematic phase diagram of the model (1), (3) at fixed H = 2t. The line ∆ = 1 corresponds to the Kosterlitz-Thouless phase, with the transition temperature TKT ∝ J/ ln(J/H) at small H . FIG. 3: Noise correlation function G(r, r′) from time-of-flight im- ages in the circulating current (CC) phase, shown as the function of the relative distance Q(r) − Q(r′), with Q(r) = mr/(~t) ex- pressed in 1/ℓ‖ units: (a) Q(r) = (0, 0); (b) Q(r) = (1, 1). Changing the initial point Q(r) leads to the change of relative weight of the two systems of dips, which is the fingerprint of the CC phase. The quantum phase transition at T = 0, H = Hc is of the 3d Ising type (except at the U(1)-symmetric point ∆ = 1 where the universality class is that of the 2d dilute Bose gas [23]), so in its vicinity the CC order parameter Ny ∝ (Hc −H) β with β ≃ 0.313 [24], and Tc ∝ JN y ∝ J(Hc −H) 2β . At T > Tc or H > Hc the only order parameter is 〈S x〉, corresponding to the Mott phase with one particle per dimer. Bilayer lattice design and hierarchy of scales.– The bi- layer can be realized, e.g., by employing three pairs of mutu- ally perpendicular counter-propagating laser beams with the same polarization and adding another pair of beams with an orthogonal polarization and additional phase shift δ, so that the resulting field intensity has the form E⊥(cos kx + cos ky)+Ez cos kz +Ẽ2z cos 2(kz+δ). Taking δ = π (1+ζ) and Ẽ2z > Ez(2E⊥+ ζEz), with ζ = ±1 for blue and red de- tuning, respectively, one obtains a three-dimensional stack of bilayers, separated by large potential barriers U3d. Eq. (5) im- plies V ≫ t′ ≫ t, |Ṽ ′|, which can be achieved by making the z-direction potential barrier U⊥ inside the bilayer sufficiently larger than the in-plane barrier U‖, so that the condition t ≪ t will be met; e.g., Ẽz/E⊥ ≈ 20, Ez/E⊥ ≈ 15 yields the bar- rier ratio U3d : U⊥ : U‖ of approximately 16 : 8 : 1, and the lattice constants ℓ⊥ ≈ 0.45λ, ℓ‖ = λ, where λ = 2π/k is the laser wave length. The parameter Ṽ ′ has a zero as a function of the angle θ, so it can be made as small as needed. Taking λ = 400 nm, one obtains an estimate of Tc = (0.1÷ 0.3) µK for cyanide molecules ClCN and HCN with the dipolar mo- ment d0 ≈ 3 Debye, while the Fermi temperature for the same parameters is Tf ≈ (0.6÷1.3) µK. This estimate corresponds to the maximum value of Tc ∼ J reached when Ṽ ′ ∼ −J and t . J . The hopping t′ was estimated assuming the in- plane potential barrier U‖ is roughly equal to the recoil energy Er = (~k) 2/2m, where m is the particle mass. Experimental signatures. – Signatures of the ordered phases can be observed [25, 26] in time-of-flight experi- ments by measuring the density noise correlator G(r, r′) = 〈n(r)n(r′)〉 − 〈n(r)〉〈n(r′)〉. If the imaging axis is perpen- dicular to the bilayer, n(r) = σ,raσ,r〉 is the local net density of two layers. For large flight times t it is proportional to the momentum distribution nQ(r), where Q(r) = mr/~t. In the Mott phase the response shows fermionic “Bragg dips” at reciprocal lattice vectors g = (2πh/ℓ‖, 2πk/ℓ‖), GM(r, r ′) ∝ f0(r, r ′) = −2〈Sx〉2 Q(r)−Q(r′)−g In the CC and DW phases the noise correlator contains an additional system of dips shifted by QB = (π/ℓ‖, π/ℓ‖): GCC,DW(r, r ′) ∝ f0(r, r ′)− 2 〈Sz〉2 + 〈Sy〉2 Qx(r)ℓ‖ + cos Qy(r)ℓ‖ ))2]} Q(r)−Q(r′)−QB − g In the DW phase 〈Sz〉 6= 0, 〈Sy〉 = 0, and so the density cor- relator depends only on r − r′. In the CC phase 〈Sz〉 = 0, 〈Sy〉 6= 0, and the relative strength of the two systems of dips varies periodically when one changes the initial point r, see Fig. 3. This Q-dependent contribution stems from the intra- layer currents 〈a†σ,raσ,r′〉 = (−) σ(−)rδ〈rr′〉i〈S y〉/4, where comes from the fact that the inter-layer current splits into four equivalent intra-layer ones (see Fig. 1), δ〈rr′〉 means r and r′ must be nearest neighbors, and (−)r ≡ eiQB ·r de- notes an oscillating factor. If the correlator is averaged over the particle positions, the CC and DW phases become indis- tinguishable. A direct way to observe the CC phase could be to use the laser-induced fluorescence spectroscopy to detect the Doppler line splitting proportional to the current. Bosonic models.– Consider the bosonic version of the model (1), with the additional on-site repulsion U . The ef- fective Hamiltonian has the form (3) with J = −4(t′)2/V and Jz = Ṽ ′ + 4(t′)2(1/V − 1/U). Due to ferromag- netic (FM) transverse exchange, instead of spontaneous cur- rent one obtains the usual Mott phase. CC states can be in- duced by artificial gauge fields [27]: The vector potential A(x) = π (x + 1/2) makes hopping along the x axis imagi- nary, t′ 7→ it′. The unitary transformation Sx,yr 7→ (−) rSx,yr maps the system onto a set of FM chains along the x axis, AF-coupled in the y direction and subject to a staggered field H = 2t along the x axis in the easy (xy) plane. In the ground state net chain moments are arranged in a staggered way along the y axis, so a current pattern similar to that of Fig. 1 emerges, now staggered along only one of the two in-plane directions. A different type of CC states, with orbital currents localized at lattice sites, can be achieved with p-band bosons [28]. Yet another way to create a CC state in a bosonic bilayer is to introduce the ring exchange on vertical plaquettes: Hring = 〈rr′〉 2,r′b2,rb1,r′ + h.c.). (8) In pseudospin language, the ring interaction modifies the transverse exchange, J 7→ J + K , so for K > 0 one can achieve the conditions J > 0, J > |Jz| necessary for the CC phase to exist. However, engineering a sizeable ring exchange in bosonic systems is difficult (see [29] for recent proposals). Spin- 1 bilayer with four-spin ring exchange.– Consider the Hubbard model for spinful fermions on a bilayer shown in Fig. 1, with the on-site repulsion U and inter- and intra- layer hoppings t and t′, respectively. At half filling (i.e., two fermions per dimer), one can effectively describe the system in terms of spin degrees of freedom represented by the opera- tors S = 1 a†ασαβaβ . The leading term in t/U yields the AF Heisenberg model with the nearest-neighbor exchange con- stants J⊥ = 4t 2/U (inter-layer) and J‖ = 4(t ′)2/U (intra- layer), while the next term, with the interaction strength J4 ≃ 10t2(t′)2/U3, corresponds to the ring exchange [30, 31]: H4 = 2J4 (S1 · S2)(S1′ · S2′) + (S1 · S1′)(S2 · S2′)− (S1 · S2′)(S2 · S1′) , (9) where the sum is over vertical plaquettes only (the interaction for intra-layer plaquettes is of the order of (t′)4/U3 and is ne- glected), and the sites (1, 2, 2′, 1′) form a plaquette (traversed counter-clockwise). In the same order of the perturbation the- ory, the nearest-neighbor exchange constants get corrections, J⊥ 7→ JR = J⊥ + J4, J‖ 7→ JL = J‖ + J4/2, and the interaction JD = J4 along the diagonals of verti- cal plaquettes is generated. Generalization for any 2d bipar- tite lattice built of vertically arranged dimers is trivial. Since J⊥ ≫ J‖, J4, we can treat the system as a set of weakly cou- pled spin dimers. The dynamics can be described with the help of the effective field theory [32] which is a continuum version of the bond boson approach [33] and is based on dimer coherent states |u,v〉 = (1−u2−v2)|s〉+ j(uj + ivj)|tj〉. Here |s〉 and |tj〉, j = (x, y, z) are the singlet and triplet states, and u, v are real vectors related to the staggered mag- netization 〈S1 − S2〉 = 2u(1 − u 2 − v2)1/2 and vector chi- rality 〈S1×S2〉 = v(1−u 2−v2)1/2 of the dimer. Using the ansatz u(r) = (−)rϕ(r), v(r) = (−)rχ(r), passing to the continuum in the coherent states path integral, and retaining up to quartic terms in u, v, one obtains the Euclidean action dτd2r ~(ϕ · ∂τχ− χ · ∂τϕ) (10) + (ϕ2 + χ2)(JR − 3ZJ4/2)− Z[J‖ϕ 2 + J4χ + (Z/2)[J‖(∂kϕ) 2 + J4(∂kχ) 2] + ZU4(ϕ,χ) where the quartic potential U4 has the form U4 = (ϕ 2 + χ2)[J‖ϕ 2 + J4χ + J4(ϕ 2 + χ2)2 + (J‖ + J4)(ϕ× χ) 2. (11) Interdimer interactions J‖ and J4 favor two competing types of order: while J‖ tends to establish the AF order (ϕ 6= 0, χ = 0), strong ring exchange J4 favors another solution with ϕ = 0, χ 6= 0, describing the state with a staggered vector chirality. It wins over the AF one for J4 > J‖, J4 > which for the square lattice (Z = 4) translates into J4 > max(J‖, J⊥/9). (12) On the line J4 = J‖ the symmetry is enhanced from SU(2) to SU(2)× U(1), and the AF and chiral orders can coexist: a rotation (ϕ+ iχ) 7→ (ϕ+ iχ)eiα leaves the action invariant. The chiral state may be viewed as an analog of the cir- culating current state considered above: in terms of the original fermions of the Hubbard model, the z-component of the chirality (S1 × S2)z = 1↓a2↓)(a 2↑a1↑) − 2↓a1↓)(a 1↑a2↑) corresponds to the spin current (particles with up and down spins moving in opposite directions). Summary.– I have considered fermionic and bosonic models on a bilayer optical lattice which exhibit a phase tran- sition into a circulating current state with spontaneously bro- ken time reversal symmetry. The simplest of those models includes just nearest-neighbor interactions and hoppings, and can possibly be realized with the help of polar molecules. Acknowledgments.– I sincerely thank U. Schollwöck, T. Vekua, and S. Wessel for fruitful discussions. Support by Deutsche Forschungsgemeinschaft (the Heisenberg Program, KO 2335/1-1) is gratefully acknowledged. ∗ On leave from: Institute of Magnetism, National Academy of Sciences and Ministry of Education, 03142 Kiev, Ukraine. [1] A. Kastberg et al., Phys. Rev. Lett. 74, 1542 (1995). [2] G. Raithel, W. D. Phillips, and S. L. Rolston, Phys. Rev. Lett. 81, 3615 (1998). [3] M. Greiner et al., Nature (London) 415, 39 (2002). [4] T. Stöferle et al., Phys. Rev. Lett. 92, 130403 (2004). [5] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998). 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Thomas, and G. Müller, Physica 112A, 235 (1982). [22] Note that this regime cannot be reached within the model with controlled hopping discussed in L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003). [23] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, 1999). [24] J. Garcı́a and J. A. Gonzalo, Physica 326A, 464 (2003). [25] E. Altman, E. Demler, M. D. Lukin, Phys. Rev. A 70, 013603 (2004). [26] S. Fölling et al., Nature 434, 481 (2005). [27] D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003); G. Juzeli- unas and P. Ohberg, Phys. Rev. Lett. 93, 033602 (2004); E. J. Mueller, Phys. Rev. A 70, 041603(R) (2004); A. S. Sorensen, E. Demler, M. D. Lukin, Phys. Rev. Lett. 94, 086803 (2005). [28] W. V. Liu and C. Wu, Phys. Rev. A 74, 013607 (2006); C. Wu et al., Phys. Rev. Lett. 97, 190406 (2006). [29] H. P. Büchler et al., Phys. Rev. Lett. 95, 040402 (2005). [30] M. Takahashi, J. Phys. C 10, 1289 (1977). [31] A. H. MacDonald, S. M. Girvin, and D. 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0704.1645

Non-Relativistic Propagators via Schwinger's Method

Non-Relativistic Propagators via Schwinger’s Method A. Aragão,∗ H. Boschi-Filho,† and C. Farina‡ Instituto de F́ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68.528, 21941-972 Rio de Janeiro, RJ, Brazil. F. A. Barone§ Universidade Federal de Itajubá, Av. BPS 1303 Caixa Postal 50 - 37500-903, Itajubá, MG, Brazil. In order to popularize the so called Schwinger’s method we reconsider the Feynman propagator of two non-relativistic systems: a charged particle in a uniform magnetic field and a charged harmonic oscillator in a uniform magnetic field. Instead of solving the Heisenberg equations for the position and the canonical momentum operators, R and P, we apply this method by solving the Heisenberg equations for the gauge invariant operators R and π = P − eA, the latter being the mechanical momentum operator. In our procedure we avoid fixing the gauge from the beginning and the result thus obtained shows explicitly the gauge dependence of the Feynman propagator. PACS numbers: 42.50.Dv Keywords: Schwinger’s method, Feynman Propagator, Magnetic Field, Harmonic Oscillator. I. INTRODUCTION In a recent paper published in this journal [1], three methods were used to compute the Feynman propaga- tors of a one-dimensional harmonic oscillator, with the purpose of allowing a student to compare the advan- tadges and disadvantadges of each method. The above mentioned methods were the following: the so called Schwinger’s method (SM), the algebraic method and the path integral one. Though extremely powerful and el- egant, Schwinger’s method is by far the less popular among them. The main purpose of the present paper is to popularize Schwinger’s method providing the reader with two examples slightly more difficult than the har- monic oscillator case and whose solutions may serve as a preparation for attacking relativistic problems. In some sense, this paper is complementary to reference [1]. The method we shall be concerned with was introduced by Schwinger in 1951 [2] in a paper about QED enti- tled “Gauge invariance and vacuum polarization”. After introducing the proper time representation for comput- ing effetive actions in QED, Schwinger was faced with a kind of non-relativistic propagator in one extra dimen- sion. The way he solved this problem is what we mean by Schwinger’s method for computing quantum propa- gators. For relativistic Green functions of charged parti- cles under external electromagnetic fields, the main steps of this method are summarized in Itzykson and Zuber’s textbook [3] (apart, of course, from Schwinger’s work [2]). Since then, this method has been used mainly in relativis- tic quantum theory [4–16]. ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] §Electronic address: [email protected] However, as mentioned before, Schwinger’s method is also well suited for computing non-relativistic propaga- tors, though it has rarely been used in this context. As far as we know, this method was used for the first time in non-relativistic quantum mechanics by Urrutia and Hernandez [17]. These authors used Schwinger’s action principle to obtain the Feynman propagator for a damped harmonic oscillator with a time-dependent frequency un- der a time-dependent external force. Up to our knowl- edge, since then only a few papers have been written with this method, namely: in 1986, Urrutia and Manterola [18] used it in the problem of an anharmonic charged oscillator under a magnetic field; in the same year, Hor- ing, Cui, and Fiorenza [19] applied Schwinger’s method to obtain the Green function for crossed time-dependent electric and magnetic fields; the method was later applied in a rederivation of the Feynman propagator for a har- monic oscillator with a time-dependent frequency [20]; a connection with the mid-point-rule for path integrals involving electromagnetic interactions was discussed in [21]. Finally, pedagogical presentations of this method can be found in the recent publication [1] as well as in Schwinger’s original lecture notes recently published [22], which includes a discussion of the quantum action prin- ciple and a derivation of the method to calculate propa- gators with some examples. It is worth mentioning that this same method was inde- pendently developed by M. Goldberger and M. GellMann in the autumn of 1951 in connection with an unpublished paper about density matrix in statistical mechanics [23]. Our purpose in this paper is to provide the reader with two other examples of non-relativistic quantum propa- gators that can be computed in a straightforward way by Schwinger’s method, namely: the propagator for a charged particle in a uniformmagnetic field and this same problem with an additional harmonic oscillator potential. Though these problems have already been treated in the context of the quantum action principle [18], we decided http://arxiv.org/abs/0704.1645v2 mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] to reconsider them for the following reasons: instead of solving the Heisenberg equations for the position and the canonical momentum operators, R and P, as is done in [18], we apply Schwinger’s method by solving the Heisen- berg equations for the gauge invariant operators R and π = P − eA, the latter being the mechanical momen- tum operator. This is precisely the procedure followed by Schwinger in his seminal paper of gauge invariance and vacuum polarization [2]. This procedures has some nice properties. For instance, we are not obligued to choose a particular gauge at the beginning of calcula- tions. As a consequence, we end up with an expression for the propagator written in an arbitrary gauge. As a bonus, the transformation law for the propagator under gauge transformations can be readly obtained. In order to prepare the students to attack more com- plex problems, we solve the Heisenberg equations in ma- trix form, which is well suited for generalizations involv- ing Green functions of relativistic charged particles under the influence of electromagnetic fields (constant Fµν , a plane wave field or even combinations of both). For ped- agogical reasons, at the end of each calculation, we show how to extract the corresponding energy spectrum from the Feynman propagator. Although the way Schwniger’s method must be applied to non-relativistic problems has already been explained in the literature [1, 18, 22], it is not of common knowledge so that we start this paper by summarizing its main steps. The paper is organized as follows: in the next section we review Schwinger’s method, in section III we present our examples and sec- tion IV is left for the final remarks. II. MAIN STEPS OF SCHWINGER’S METHOD For simplicity, consider a one-dimensional time- independent Hamiltonian H and the corresponding non- relativistic Feynman propagator defined as K(x, x′; τ) = θ(τ)〈x| exp [−iHτ |x′〉, (1) where θ(τ) is the Heaviside step function and |x〉, |x′〉 are the eingenkets of the position operator X (in the Schrödinger picture) with eingenvalues x and x′, respec- tively. The extension for 3D systems is straightforward and will be done in the next section. For τ > 0 we have, from equation (1), that K(x, x′; τ) = 〈x|H exp [−iHτ |x′〉. (2) Inserting the unity 1l = exp [−(i/~)Hτ ] exp [(i/~)Hτ ] in the r.h.s. of the above expression and using the well known relation between operators in the Heisenberg and Schrödinger pictures, we get the equation for the Feyn- man propagator in the Heisenberg picture, K(x, x′; τ) = 〈x, τ |H(X(0), P (0))|x′, 0〉, (3) where |x, τ〉 and |x′, 0〉 are the eingenvectors of opera- tors X(τ) and X(0), respectively, with the correspond- ing eingenvalues x and x′: X(τ)|x, τ〉 = x|x, τ〉 and X(0)|x′, 0〉 = x′|x′, 0〉, with K(x, x′; τ) = 〈x, τ |x′, 0〉. Be- sides, X(τ) and P (τ) satisfy the Heisenberg equations, (τ) = [X(τ),H] ; i~ (τ) = [P (τ),H]. (4) Schwinger’s method consists in the following steps: (i) we solve the Heisenberg equations forX(τ) and P (τ), and write the solution for P (0) only in terms of the operators X(τ) and X(0); (ii) then, we substitute the results obtained in (i) into the expression for H(X(0), P (0)) in (3) and using the commutator [X(0), X(τ)] we rewrite each term ofH in a time ordered form with all operatorsX(τ) to the left and all operators X(0) to the right; (iii) with such an ordered hamiltonian, equation (3) can be readly cast into the form K(x, x′; τ) = F (x, x′; τ)K(x, x′; τ), (5) with F (x, x′; τ) being an ordinary function defined F (x, x′; τ) = 〈x, τ |Hord(X(τ), X(0))|x ′, 0〉 〈x, τ |x′, 0〉 . (6) Integrating in τ , the Feynman propagator takes the K(x, x′; τ) = C(x, x′) exp F (x, x′; τ ′)dτ ′ , (7) where C(x, x′) is an integration constant indepen- dent of τ and means an indefinite integral; (iv) last step is concerned with the evaluation of C(x, x′). This is done after imposing the follow- ing conditions 〈x, τ |x′, 0〉 = 〈x, τ |P (τ)|x′, 0〉 , (8) 〈x, τ |x′, 0〉 = 〈x, τ |P (0)|x′, 0〉 , (9) as well as the initial condition K(x, x′; τ) = δ(x− x′) . (10) Imposing conditions (8) and (9) means to substitute in their left hand sides the expression for 〈x, τ |x′, 0〉 given by (7), while in their right hand sides the operators P (τ) and P (0), respectively, written in terms of the operators X(τ) and X(0) with the appropriate time ordering. III. EXAMPLES A. Charged particle in an uniform magnetic field As our first example, we consider the propagator of a non-relativistic particle with electric charge e and mass m, submitted to a constant and uniform magnetic fieldB. Even though this is a genuine three-dimensional problem, the extension of the results reviewed in the last section to this case is straightforward. Since there is no electric field present, the hamiltonian can be written as (P− eA) , (11) where P is the canonical momentum operator, A is the vector potential and π = P − eA is the gauge invariant mechanical momentum operation. We choose the axis such that the magnetic field is given by B = Be3. Hence, the hamiltonian can be decomposed as π21 + π = H⊥ + , (12) with an obvious definition for H⊥. Since the motion along the OX 3 direction is free, the three-dimensional propagator K(x,x′; τ) can be written as a product of a two-dimensional propagator, K⊥(r, r ′; τ), related to the magnetic field and a one- dimensional free propagator, K 3 (x3, x 3; τ): K(x,x′; τ) = K⊥(r, r ′; τ)K 3 (x3, x 3; τ), (τ > 0) where r = x1e1 + x2e2 and K 3 (x3, x 3; τ) is the well known propagator of the free particle [24], 3 (x3, x 3; τ) = 2πi~τ (x3 − x . (14) In order to use Schwinger’s method to compute the two- dimensional propagator K⊥(r, r ′; τ) = 〈r, τ |r′, 0〉, we start by writing the differential equation 〈r, τ |r′, 0〉 = 〈r, τ |H⊥(R⊥(0),π⊥(0))|r ′, 0〉 , (15) where R⊥(τ) = X1(τ)e1 + X2(τ)e2 and π⊥(τ) = π1(τ)e1+π2(τ)e2. In (15) |r, τ〉 and |r ′, 0〉 are the eigen- vectors of position operators R(τ) = X1(τ)e1 +X2(τ)e2 and R(0) = X1(0)e1 + X2(0)e2, respectively. More especifically, operators X1(0), X1(τ), X2(0) and X2(τ) have the eigenvalues x′1, x1, x 2 and x2, respectively. In order to solve the Heisenberg equations for operators R⊥(τ) and π⊥(τ), we need the commutators Xi(τ), π j (τ) = 2i~πi(τ) , πi(τ), π j (τ) = 2i~eBǫij3πj(τ), (16) where ǫij3 is the usual Levi-Civita symbol. Introducing the matrix notation R(τ) = X1(τ) X2(τ) ; Π(τ) = π1(τ) π2(τ) , (17) and using the previous commutators the Heisenberg equations of motion can be cast into the form dR(τ) , (18) dΠ(τ) = 2ωCΠ(τ) , (19) where 2ω = eB/m is the cyclotron frequency and we defined the anti-diagonal matrix . (20) Integrating equation (19) we find Π(τ) = e2ωCτΠ(0) . (21) Substituting this solution in equation (18) and integrat- ing once more, we get R(τ) −R(0) = sin (ωτ) eωCτΠ(0) , (22) where we used the following properties of C matrix: 2 = −1l; C−1 = −C = CT , eαC = cos (α)1l + sin (α)C with CT being the transpose of C. Combining equations (22) and (21) we can write Π(0) in terms of the operators R(τ) and R(0) as Π(0) = sin (ωτ) e−ωCτ R(τ)−R(0) . (23) In order to express H⊥ = (π 2)/2m in terms of R(τ) and R(0), we use (23). In matrix notation, we have ΠT (0)Π(0) 2 sin2 (ωτ) RT (τ)R(τ) +RT (0)R(0) + −RT (τ)R(0) −RT (0)R(τ) . (24) Last term on the r.h.s. of (24) is not ordered appropri- ately as required in the step (ii). The correct ordering may be obtained as follows: first, we write R(0)TR(τ) = R(τ)TR(0) + [Xi(0), Xi(τ)] . (25) Using equation (22), the usual commutator [Xi(0), πj(0)] = i~δij1l and the properties of matrix C it is easy to show that [Xi(0), Xi(τ)] = 2i~ sin(ωτ) cos(ωτ) , (26) so that hamiltonian H⊥ with the appropriate time order- ing takes the form 2 sin2 (ωτ) R2(τ) +R2(0)− 2RT (τ)R(0) − i~ω cot(ωτ). (27) Substituting this hamiltonian into equation (15) and in- tegrating in τ , we obtain 〈r, τ |r′, 0〉 = C(r, r′) sin (ωτ) cot(ωτ)(r− r′)2 , (28) where C(r, r ′) is an integration constant to be deter- mined by conditions (8), (9) and (10), which for the case of hand read 〈r, τ |πj(τ)|r ′, 0〉 = − eAj(r) 〈r, τ |r′, 0〉 (29) 〈r, τ |πj(0)|r ′, 0〉 = − eAj(r 〈r, τ |r′, 0〉 , (30) 〈r, τ |r′, 0〉 = δ(2)(r− r′). (31) In order to compute the matrix element on the l.h.s. of (29), we need to express Π(τ) in terms of R(τ) and R(0). From equaitons (21) and (23), we have Π(τ) = sin (ωτ) R(τ)−R(0) , (32) which leads to the matrix element 〈r, τ |πj(τ)|r ′, 0〉 = mω[cot(ωτ) xj − x + ǫjk3 (xk − x k)]〈r, τ |r ′, 0〉 , (33) where we used the properties of matrix C and Einstein convention for repeated indices is summed. Analogously, the l.h.s. of equation (30) can be computed from (23), 〈r, τ |πj(0)|r ′, 0〉 = mω[cot(ωτ) xj − x − ǫjk3 (xk − x k)]〈r, τ |r ′, 0〉 . (34) Substituting equations (33) and (34) into (29) and (30), respectively, and using (28), we have + eAj(r)+ eFjk(xk−x C(r, r ′)= 0,(35) − eAj(r eFjk(xk−x C(r, r ′)= 0,(36) where we defined Fjk = ǫjk3 B. Our strategy to solve the above system of differential equations is the following: we first equation (35) assum- ing in this equation variables r ′ as constants. Then, we impose that the result thus obtained is a solution of equa- tion (36). With this goal, we multiply both sides of (35) by dxj and sum over j, to obtain Aj(r)+ Fjk (xk − x dxj . (37) Integration of the previous equation leads to C(r, r ′) = C(r ′, r ′) e [Aj(ξ)+ Fjk(ξk−x k)] dξj} , where the line integral is assumed to be along curve Γ, to be specified in a moment. As we shall see, this line inte- gral does not depend on the curve Γ joining r ′ and r, as expected, since the l.h.s. of (37) is an exact differencial. In order to determine the differential equation for C(r ′, r ′) we must substitue expression (38) into equa- tion (36). Doing that and using carefully the fundamen- tal theorem of differential calculus, it is straightforward to show that (r ′, r ′) = 0 , (39) which means that C(r ′, r ′) is a constant, C0, indepen- dent of r ′. Noting that [B× (ξ − r ′)]j = −Fjk (ξk − x k) , (40) equation (38) can be written as C(r, r ′)= C0 exp A(ξ)− B× (ξ − r ′) Observe, now, that the integrand in the previous equa- tion has a vanishing curl, A(ξ)− B× (ξ − r ′) = B−B = 0 , which means that the line integral in (42) is path in- dependent. Choosing, for convenience, the straightline from r ′ to r, it can be readly shown that [B× (ξ − r ′)] · dξ = 0 , where Γsl means a straightline from r ′ to r. With this simplification, the C(r ′, r) takes the form C(r, r ′)= C0 exp ~ Γsl A(ξ) · dξ . (42) Substituting last equation into (28) and using the initial condition (10), we readly obtain C0 = . Therefore the complete Feynman propagator for a charged particle under the influence of a constant and uniform magnetic field takes the form K(x,x′; τ) 2πi~ sin (ωτ) 2πi~τ A(ξ) · dξ cot(ωτ)(r − r ′)2 (x3 − x ,(43) where in the above equation we omitted the symbol Γsl but, of course, it is implicit that the line integral must be done along a straightline, and we brought back the free propagation along the OX 3 direction. A few comments about the above result are in order. 1. Firstly, we should emphasize that the line integral which appears in the first exponencial on the r.h.s. of (43) must be evaluated along a straight line be- tween r′ and r. If for some reason we want to choose another path, instead of integral A(ξ) · dξ, we must evaluate [A(ξ)− (1/2)B× (ξ − r ′)] · dξ. 2. Since we solved the Heisenberg equations for the gauge invariant operators R⊥ and π⊥, our final result is written for a generic gauge. Note that the gauge-independent and gauge-dependent parts of the propagator are clearly separated. The gauge fixing corresponds to choose a particular expression for A(ξ). Besides, from (43) we imediately obtain the transformation law for the propagator under a gauge transformation A → A+∇Λ, namely, K(r, r ′; τ) 7−→ e Λ(r)K(r, r ′; τ) e− Λ(r ′) . Although this transformation law was obtained in a particular case, it can be shown that it is quite general. 3. It is interesting to show how the energy spectrum (Landau levels), with the corresponding degener- acy per unit area, can be extracted from prop- agator (43). With this purpose, we recall that the partition function can be obtained from the Feynman propagator by taking τ = −i~β, with β = 1/(KBT ), and taking the spatial trace, Z(β) = dx2 K(r, r;−i~β) . Substituting (43) into last expression, we get Z(β) = 2π~ senh(~βω) where we used the fact that sin(−iθ) = −i sinh θ. Observe that the above result is divergent, since the area of the OX 1X2 plane is infinite. This is a consequence of the fact that each Landau level is infinitely degenerated, though the degeneracy per unit area is finite. In order to proceed, let us as- sume an area as big as we want, but finite. Adopt- ing this kind or regularization, we write ∫ L/2 ∫ L/2 dx2 K (r, r;−i~β) ≈ 2π~ senh(~βω) L2 eB ~βω − e−~βω L2 eB 1− e−~βωc L2 eB −β(n+ 1 )~ωc , where we denoted by ωc = eB/2m the ciclotron frequency. Comparing this result with that of a partition function whose energy level En has de- generacy gn, given by Z(β) = −βEn , we imediately identify the so called Landau leves and the corresponding degeneracy per unit area, ~ωc ; (n = 0, 1, ...) . B. Charged harmonic oscillator in a uniform magnetic field In this section we consider a particle with mass m and charge e in the presence of a constant and uniform mag- netic field B = Be3 and submitted to a 2-dimensional isotropic harmonic oscillator potential in the OX 1X2 plane, with natural frequency ω0. Using the same no- tation as before, we can write the hamiltonian of the system in the form H = H⊥ + , (44) where π21 + π X21 +X . (45) As before, the Feynman propagator for this problem takes the form K(x,x′; τ) = K⊥(r, r ′; τ)K 3 (x3, x 3; τ), with K 3 (x3, x 3; τ) given by equation (14). The propa- gator in the OX 1X2-plane satisfies the differential equa- tion (15) and will be determined by the same used in the previous example. Using hamiltonian (45) and the usual commutation re- lations the Heisenberg equations are given by dR(τ) , (46) dΠ(τ) = 2ωCΠ(τ) −mω20R(τ) , (47) where we have used the matrix notation introduced in (17) and (20). Equation (46) is the same as (18), but equation (47) contains an extra term when compared to (19). In order to decouple equations (46) and (47), we differentiate (46) with respect to τ and then use (47). This procedure leads to the following uncoupled equation d2R(τ) − 2ωC dR(τ) + ω20R(τ) = 0 (48) After solving this equation, R(τ) and Π(τ) are con- strained to satisfy equations (46) and (47), respectively. A straightforward algebra yields the solution R(τ) = M−R(0) + NΠ(0) (49) Π(τ) = M+Π(0)−m2ω20NR(0) , (50) where we defined the matrices sin (Ωτ) eωτC (51) ± = eωτC cos (Ωτ)1l± sin (Ωτ)C , (52) and frequency Ω = ω2 + ω20 . Using (49) and (50), we write Π(0) and Π(τ) in terms of R(τ) and R(0), Π(0) = N−1R(τ)− N−1M−R(0) , (53) Π(τ)= M+N−1R(τ)− −+m2ω20N R(0). (54) Now, we must order appropriately the hamiltonian operator H⊥ = Π T (0)Π(0)/(2m) + mω20R T (0)R(0)/2, which, with the aid of equation (53), can be written as RT (τ)(N−1)T −RT (0)(M−)T (N−1)T −1R(τ)− N−1M−R(0) +mω20R T (0)R(0) 2 sin2 (Ωτ) RT (τ) −RT (0)(M−)T R(τ)−M−R(0) +mω20R T (0)R(0) 2 sin2 (Ωτ) RT (τ)R(τ)−RT (τ)M−R(0) −RT (0)(M−)TRT (τ) +RT (0)(M−)TM−R(0) +mω20R T (0)R(0) 2 sin2 (Ωτ) R2(τ) −RT (τ)M−R(0) −RT (0)(M−)TR(τ) +R2(0) , (55) where superscript T means transpose and we have used the properties of the matrices N and M− given by (51) and (52). In order to get the right time ordering, observe first that RT (0)(M−)TR(τ) = RT (τ)M−R(0)+ −R(0) ,Xi(τ) where −R(0) ,Xi(τ) = i~Tr N(M−)T sin (2Ωτ) . Using the last two equations into (55) we rewrite the hamiltonian in the desired ordered form, namely, 2 sin2 (Ωτ) R2(τ) +R2(0)− 2RT (τ)M−R(0) sin (2Ωτ) . (56) For future convenience, let us define U(τ) = cos (ωτ) cos (Ωτ) + sin (ωτ) sin (Ωτ) ,(57) V (τ) = sin (ωτ) cos (Ωτ) − cos (ωτ) sin (Ωτ) (58) and write matrix M−, defined in (52), in the form − = U(τ)1l + V(τ)C. (59) Substituting (59) in (56) we have 2 sin2 (Ωτ) R2(τ) +R2(0)− 2U(τ)RT (τ)R(0) − 2V (τ)RT (τ)CR(0)− sin (2Ωτ) . (60) The next step is to compute the classical function F (r, r′; τ). Using the following identities ΩU(τ) sin2 (Ωτ) [cos (ωτ) sin (Ωτ) , (61) ΩV (τ) sin2 (Ωτ) [ sin (ωτ) sin (Ωτ) , (62) into (60), we write F (r, r′; τ) in the convenient form F (r, r′; τ) = (r2 + r′ )csc(Ωτ)2+mΩr · r′ [cos (ωτ) sin (Ωτ) + mΩr · Cr′ [ sin (ωτ) sin (Ωτ) − i~Ω cos (Ωτ) sin (Ωτ) . (63) Inserting this result into the differential equation 〈r, τ |r′, 0〉 = F (r, r′; τ)〈r, τ |r′, 0〉 , and integrating in τ , we obtain 〈r, τ |r′,0〉 = C(r, r′) sin (Ωτ) (r2 + r′ ) cot (Ωτ) r · r′ cos (ωτ) sin (Ωτ) + r · Cr′ sin (ωτ) sin (Ωτ) . (64) where C(r, r ′) is an arbitrary integration constantto be determined by conditions (29), (30) and (31). Using (54) we can calculate the l.h.s. of condition (29), 〈r,τ |πj(τ)|r ′, 0〉 = sin (Ωτ) cos (Ωτ)xj − cos (ωτ)x sin (Ωτ)xk − sin (ωτ)x 〈r, τ |r′, 0〉, (65) and using (53) we get the l.h.s. of condition (30), 〈r,τ |πj(0)|r ′, 0〉 = sin (Ωτ) cos (ωτ)xj − cos (Ωτ)x sin (Ωτ)x′k − sin (ωτ)xk 〈r, τ |r′, 0〉. (66) With the help of the simple identities (r2 + r′ ) = 2xj ; (r2 + r′ ) = 2x′j r · r′ = x′j ; r · r′ = xj r · Cr′ = ǫjk3x r · Cr′ = −ǫjk3xk. and also using equation (64), we are able to compute the right hand sides of conditions (29) and (30), which are given, respectively, by C(r, r′) ∂C(r, r′) cos (Ωτ) sin (Ωτ) xj −mΩ cos (ωτ) sin (Ωτ) sin (ωτ) sin (Ωτ) ǫjk3x k − eAj(r) 〈r, τ |r′, 0〉 (67) C(r, r′) ∂C(r, r′) cos (Ωτ) sin (Ωτ) x′j +mΩ cos (ωτ) sin (Ωτ) sin (ωτ) sin (Ωτ) ǫjk3xk − eAj(r 〈r, τ |r′, 0〉. (68) Equating (65) and (67), and also (66) and (68)), we get the system of differential equations for C(r, r′) ∂C(r, r′) Aj(r) + C(r, r′) = 0 , (69) ∂C(r, r′) C(r, r′) = 0. (70) Proceeding as in the previous example, we first integrate (69). With this goal, we multiply it by dxj , sum in j and integrate it to obtain C(r, r ′) = C(r ′, r ′) exp Aj(ξ) + where the path of integration Γ will be specified in a moment. Inserting expression (71) into the second differ- ential equation (70), we get C(r ′, r ′) = 0 =⇒ C(r ′, r ′) = C0 , where C0 is a constant independent of r ′, so that equa- tion (71) can be cast, after some convenient rearrange- ments, into the form C(r, r′) = C0 exp A(ξ)− Note that the integrand has a vanishing curl so that we can choose the path of integration Γ at our will. Choos- ing, as before, the straight line between r′ and r, it can be shown that A(ξ)− ·dξ = A(ξ)·dξ+ Br ·Cr′ , (73) where, for simplicity of notation, we omitted the symbol Γsl indicating that the line integral must be done along a straight line. From equations (71), (72) e (73), we get C(r, r′) = C0 exp A(ξ) · dξ r · Cr′ which substituted back into equation (64) yields 〈r, τ |r′, 0〉 = sin (Ωτ) A(ξ) · dξ { imΩ 2~ sin (Ωτ) (r2 + r′ ) cos (Ωτ) −2r · r′ cos (ωτ)− 2 sin (ωτ) − sinΩτ The initial condition implies C0 = mΩ/(2πi~). Hence, the desired Feynman propagator is finally given by K(x,x′; τ) = K⊥(r, r ′; τ)K 3 (x3, x 3; τ) 2 π i ~ sin (Ωτ) 2πi~τ A(ξ) · dξ 2~ sin (Ωτ) cos (Ωτ)(r2 + r′ − 2 cos (ωτ)r · r′ − 2 sin (ωτ)− sin (Ωτ) r · Cr′ (x3 − x , (76) where we brought back the free part of the propagator corresponding to the movement along the OX 3 direction. Of course, for ω0 = 0 we reobtain the propagator found in our first example and for B = 0 we reobtain the prop- agator for a bidimensional oscillator in the OX 1X2 plane multiplied by a free propagator in the OX 3 direction, as can be easily checked. Regarding the gauge dependence of the propagator, the same comments done before are still valid here, namely, the above expression is written for a generic gauge, the transformation law for the propagator under a gauge transformation is the same as before, etc. We fin- ish this section, extracting from the previous propagator, the corresponding energy spectrum. With this purpose, we first compute the trace of the propagator, dx2 K ⊥(x1, x1, x2, x2; τ) = 2πi~ sin(Ωτ) dx2 exp 2~ sin(Ωτ) cos(Ωτ) − cos(ωτ) (x21 + x 2[cos(Ωτ) − cos(ωτ)] , (77) where we used the well known result for the Fresnel in- tegral. Using now the identity cos(Ωτ)−cos(ωτ) = −2 sin[(Ω+ω)τ/2] sin[(Ω−ω)τ/2)] , we get for the corresponding energy Green function G (E) =−i dx2 K ⊥(x1, x1, x2, x2; τ) sen(Ωτ τ) sen(Ω−ω e−(l+ )(Ω+ω)τ e−i(n+ (Ω−ω)τ where is tacitly assumed that E → E − iε and we also used that (with the assumption ν → ν − iǫ) sen(ν e−i(n+ )ντ . Changing the order of integration and summations, and integrating in τ , we finally obtain G(E) = l,n=0 E − Enl , (78) where the poles of G(E), which give the desired energy levels, are identified as Enl = (l+n+1)~Ω+(l−n)~ω , (l, n = 0, 1, ...) . (79) The Landau levels can be reobtained from the previous result by simply taking the limit ω0 → 0: Enl −→ (2l + 1)~ω = (l + )~ωc , (80) with l = 0, 1, ... and ωc = eB/m, in agreement to the result we had already obtained before. IV. FINAL REMARKS In this paper we reconsidered, in the context of Schwinger’s method, the Feynman propagators of two well known problems, namely, a charged particle under the influence of a constant and uniform magnetic field (Landau problem) and the same problem in which we added a bidimensional harmonic oscillator potential. Al- though these problems had already been treated from the point of view of Schwinger’s action principle, the novelty of our work relies on the fact that we solved the Heisenberg equations for gauge invariant operators. This procedure has some nice properties, as for instance: (i) the Feynman propagator is obtained in a generic gauge; (ii) the gauge-dependent and gauge-independent parts of the propagator appear clearly separated and (iii) the transformation law for the propagator under gauge transformation can be readly obtained. Besides, we adopted a matrix notation which can be straightfor- wardly generalized to cases of relativistic charged parti- cles in the presence of constant electromagnetic fields and a plane wave electromagnetic field, treated by Schwinger [2]. For completeness, we showed explicitly how one can obtain the energy spectrum directly from que Feynman propagator. In the Landau problem, we obtained the (in- finitely degenerated) Landau levels with the correspond- ing degeneracy per unit area. For the case where we included the bidimensional harmonic potential, we ob- tained the energy spectrum after identifying the poles of the corresponding energy Green function. We hope that this pedagogical paper may be useful for undergraduate as well as graduate students and that these two simple examples may enlarge the (up to now) small list of non- relativistic problems that have been treated by such a powerful and elegant method. Acknowledgments F.A. Barone, H. Boschi-Filho and C. Farina would like to thank Professor Marvin Goldberger for a private com- munication and for kindly sending his lecture notes on quantum mechanics where this method was explicitly used. We would like to thank CNPq and Fapesp (brazil- ian agencies) for partial financial support. [1] F.A. Barone, H. Boschi-Filho and C. Farina, Three meth- ods for calculating the Feynman propagator, Am. J. Phys. 71, 483-491 (2003). [2] J. Schwinger, On Gauge Invariance And Vacuum Polar- ization, Phys. Rev. 82, 664 (1951). [3] Claude Itzykson and Jean-Bernard Zuber, Quantum Field Theory, (McGraw-Hill Inc., NY, 1980), pg 100. [4] E.S Fradkin, D.M Gitman and S.M. Shvartsman, Quan- tum Eletrodinamics with Unstable Vacuum (Springer, Berlim, 1991). [5] V. V. Dodonov, I. A. Malkin, and V. I. Manko, “In- variants and Green’s functions of a relativistic charged particle in electromagnetic fields,” Lett. Nuovo Cimento Soc. Ital. Fis. 14, 241-244 (1975). [6] V. V. Dodonov, I. A. Malkin, and V. I. Manko, “Green’s functions for relativistic particles in non-uniform external fields,” J. Phys. A 9, 1791- 1796 (1976). [7] J. D. Likken, J. Sonnenschein, and N. Weiss, “The theory of anyonic superconductivity: A review,” Int. J. Mod. Phys. A 6, 5155-5214 (1991). [8] A. Ferrando and V. Vento, “Hadron correlators and the structure of the quark propagator,” Z. Phys. C63, 485 (1994). [9] H. Boschi-Filho, C. Farina and A. N. Vaidya, “Schwinger’s method for the electron propagator in a plane wave field revisited,” Phys. Lett. A 215, 109-112 (1996). [10] S. P. Gavrilov, D. M. Gitman and A. E. Goncalves, “QED in external field with space-time uniform invariants: Ex- act solutions,” J. Math. Phys. 39, 3547 (1998). [11] D. G. C. McKeon, I. Sachs and I. A. Shovkovy, “SU(2) Yang-Mills theory with extended supersymmetry in a background magnetic field,” Phys. Rev. D59, 105010 (1999). [12] T. K. Chyi, C. W. Hwang, W. F. Kao, G. L. Lin, K. W. Ng and J. J. Tseng, “The weak-field expansion for processes in a homogeneous background magnetic field,” Phys. Rev. D 62, 105014 (2000). [13] N. C. Tsamis and R. P. Woodard, “Schwinger’s propaga- tor is only a Green’s function,” Class. Quant. Grav. 18, 83 (2001). [14] M. Chaichian, W. F. Chen and R. Gonzalez Felipe, “Radiatively induced Lorentz and CPT violation in Schwinger constant field approximation,” Phys. Lett. B 503, 215 (2001). [15] J. M. Chung and B. K. Chung, “Induced Lorentz- and CPT-violating Chern-Simons term in QED: Fock- Schwinger proper time method,” Phys. Rev. D 63, 105015 (2001). [16] H. Boschi-Filho, C. Farina e A.N. Vaidya, Schwinger’s method for the electron propagator in a plane wave field revisited, Phys. Let. A215 (1996) 109-112. [17] Luis F. Urrutia and Eduardo Hernandez, Calculation of the Propagator for a Time-Dependent Damped, Forced Harmonic Oscillator Using the Schwinger Action Princi- ple, Int. J. Theor. Phys. 23, 1105-1127 (1984) [18] L.F. Urrutia and C. Manterola, Propagator for the Anisotropic Three-Dimensional Charged Harmonic Os- cillator in a Constant Magnetic Field Using the Schwinger Action Principle, Int. J. Theor. Phys. 25, 75- 88 (1986). [19] N. J. M. Horing, H. L. Cui, and G. Fiorenza, Nonrel- ativistic Schrodinger Greens function for crossed time- dependent electric and magnetic fields, Phys. Rev. A34, 612615 (1986). [20] C. Farina and Antonio Segui-Santonja, Schwinger’s method for a harmonic oscillator with a time-dependent frequency, Phys. Lett. A184, 23-28 (1993). [21] S.J. Rabello and C. Farina, Gauge invariance and the path integral, Phys. Rev. A51, 2614-2615 (1995). [22] J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, edited by B.G. Englert (Springer, 2001). [23] Private communication with Professor M. Goldberger. We thank him for kindly sending us a copy of his notes on quantum mechanics given at Princeton for more than ten years. [24] R.P. Feynman e A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

0704.1646

A linear RFQ ion trap for the Enriched Xenon Observatory

A linear RFQ ion trap for the Enriched Xenon Observatory B. Flatt a,∗, M. Green a, J. Wodin a, R. DeVoe a, P. Fierlinger a, G. Gratta a, F. LePort a, M. Montero Dı́ez a, R. Neilson a, K. O’Sullivan a, A. Pocar a, S. Waldman a,1, E. Baussan b, M. Breidenbach c, R. Conley c, W. Fairbank Jr. d, J. Farine e C. Hall c,2, K. Hall d, D. Hallman e, C. Hargrove f , M. Hauger b, J. Hodgson c, F. Juget b, D.S. Leonard g, D. Mackay c, Y. Martin b, B. Mong d, A. Odian c, L. Ounalli b, A. Piepke g, C.Y. Prescott c, P.C. Rowson c, K. Skarpaas c, D. Schenker b, D. Sinclair f , V. Strickland f , C. Virtue e, J.-L. Vuilleuimier b, J.-M. Vuilleuimier b, K. Wamba c, P. Weber b aPhysics Department, Stanford University, Stanford CA, USA bInstitut de Physique, Université de Neuchatel, Neuchatel, Switzerland cStanford Linear Accelerator Center, Menlo Park CA, USA dPhysics Department, Colorado State University, Fort Collins CO, USA ePhysics Department, Laurentian University, Sudbury ON, Canada fPhysics Department, Carleton Univerisity, Ottawa ON, Canada gDept. of Physics and Astronomy, University of Alabama, Tuscaloosa AL, USA Abstract The design, construction, and performance of a linear radio-frequency ion trap (RFQ) intended for use in the Enriched Xenon Observatory (EXO) are described. EXO aims to detect the neutrinoless double-beta decay of 136Xe to 136Ba. To sup- press possible backgrounds EXO will complement the measurement of decay energy and, to some extent, topology of candidate events in a Xe filled detector with the identification of the daughter nucleus (136Ba). The ion trap described here is capa- ble of accepting, cooling, and confining individual Ba ions extracted from the site of the candidate double-beta decay event. A single trapped ion can then be identified, with a large signal-to-noise ratio, via laser spectroscopy. Key words: RFQ trap, EXO, fluorescence spectroscopy PACS: 34.10.+x, 42.62.Fi, 14.60.Pq Preprint submitted to Elsevier 30 October 2018 1 Introduction In the last decade, compelling evidence for flavor mixing in the neutrino sec- tor has clearly shown that neutrinos have finite masses [1]. These experiments reveal mass differences between single mass eigenstates, but not their absolute values. The measurement of such masses has become arguably the most im- portant frontier in neutrino physics, with implications in astrophysics, particle physics, and cosmology. β-decay endpoint spectroscopy measurements provide an increasingly sensitive probe of neutrino mass [2]. However, a less direct but potentially more sensitive technique is the observation and measurement of the rate of neutrinoless double-beta (0νββ) decay [3]. The discovery of this exotic nuclear decay mode would provide an absolute scale for neutrino masses and establish the existence of two-component Majorana particles [4]. Sensitivity to Majorana neutrino masses in the interesting 10 - 100 meV re- gion is achievable by experiments utilizing a ton-scale 0νββ isotope source [3]. This assumes that backgrounds from natural radioactivity, cosmic rays, and the standard-model two-neutrino double-beta (2νββ) decay can be sufficiently reduced and understood. Several proposals exist to perform this daunting task [3]. The Enriched Xenon Observatory (EXO) is designed to identify the atomic species (136Ba) produced in the decay process, using high resolution atomic spectroscopy [5]. This isotope specific “Ba tagging,” working in con- junction with more conventional measurements of decay energy and crude event topology, will potentially provide a clean signature of 0νββ decay. The EXO collaboration is currently pursuing a 0νββ detector R&D program, focusing on a time projection chamber (TPC) filled with xenon enriched to 80% 136Xe in liquid (LXe) or gaseous (GXe) phase. Many of the detector parameters and, in particular, the details of the Ba tagging technique would be different in LXe and GXe. The ion trap described here is designed to accept, trap, and cool individual Ba ions extracted from a 0νββ detector. While the technique to efficiently transport ions from their production site is still under investigation (and is beyond the scope of this article), the ion trap discussed here is optimized to operate with a LXe detector and a mechanical system to retrieve and inject the ions. This ion trap is capable of confining ions for extended periods of time (∼ min) to a small volume (∼ (500 µm)3), essential for observing single ions via laser spectroscopy with a high signal-to-noise ratio. These properties are required to drastically suppress candidate decays ∗ Corresponding author. Address: Physics Department, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA. Tel: +1-650-723-2946; fax: +1-650- 725-6544. Email address: [email protected] (B. Flatt). 1 Now at Caltech, Pasadena CA, USA 2 Now at University of Maryland, College Park MD, USA that do not create Ba ions in the TPC, while maintaining a high detection efficiency for Ba-ion-producing events. In addition, this trap can operate in the presence of some Xe contamination, which is likely in any Ba tagging system coupled to a Xe filled detector. The ion trap system described here is designed to be appropriately flexible as an R&D device. Simplifications and modifications of this system can be adopted for the actual trap to be used in 2 Linear RFQ traps and buffer gas cooling RF Paul traps confine charged particles in a quadrupole RF field [7]. Spherical traps have a closed geometry consisting of a ring and two endcap electrodes, providing trapping in three spatial dimensions. Linear Paul traps generally consist of four parallel cylindrical electrodes, placed symmetrically about a central (longitudinal) axis, as shown in fig. 1. An RF field applied across diagonally opposing electrodes provides transverse (x− y plane in fig. 1) con- finement of the ion. ...UDC 0V S1 S4 S5 S16S14S13 S15 -1.2 V -1.2 V Ba oven e--gun -0.1 V 10V-0.2 V Spectro- scopy lasers Trapped Fig. 1. Schematic of the linear RF trap. Ions are loaded in S3 and stored in S14. The lower part of the figure shows the DC potential distribution. Appropriately chosen DC voltages, applied to longitudinally (z axis in fig. 1) segmented electrodes, provide longitudinal confinement. A single group of four symmetrically placed electrodes is referred to as a “segment”. Electrodes are constructed and positioned such that their radius, re, is related to the char- acteristic radial trap size, r0, by re = 1.148r0 (1) where r0 is the distance from the axis of the trap to the innermost edge of an electrode. This configuration creates the closest approximation to a hyperbolic potential at the trap center for cylindrically shaped electrodes [6]. The ion’s orbit in the transverse plane is described by the Mathieu equation [9]. Analysis of the solutions to this equation reveal stability criteria for the ion’s motion in the trap. The dimensionless Mathieu stability parameters, a and q, are defined q = 2 mr20ω a = 4 mr20ω where where e and m are the ion’s charge and mass, ωRF is the angular RF frequency, URF is the RF voltage (amplitude), and UDC is the DC voltage. Values of a and q between 0 and 0.91, falling within a region defined by the characteristic numbers of the Mathieu equation, correspond to stable ion orbits [8]. Transverse confinement is attributed to a pseudopotential VRF (r) = r2. (4) quadratically dependent on the radial distance, r, from the longitudinal axis of the trap. The DC voltages applied to the longitudinally segmented electrodes are chosen to create a trapping potential VDC(r, z) = where z and z0 are the longitudinal coordinate and length of the segment in which the ion is trapped (the “trapping segment”). The radial dependence of the longitudinal potential well arises from Laplace’s equation applied to the interior region of the ion trap. This radial defocusing reduces the depth of the transverse pseudopotential well, created by the RF field. The total potential well used to trap ions is the sum of eqns. 4 and 5. The open geometry of this type of trap allows for an unobstructed view of the trapping region, with large optical angular acceptance. In addition, the longitudinal electrode segmentation allows for multiple configurations of the longitudinal potentials as required for the injection, trapping, and ejection of a single ion. In order to confine an energetic ion of mass m injected from outside the trap, a mechanism of energy loss must be provided in order to dissipate the ion’s kinetic energy to below both eVRF and eVDC . Collisions with a ”buffer” gas of mass mB can provide such an energy-loss mechanism. The phenomenology of ion-neutral interactions in an RF Paul trap can be divided into three cases. If mB � m, the ion is cooled via a large number of ion-neutral collisions, each exchanging a small amount of energy and momentum. In this case, the cooling process is adiabatic compared to the period of the ion’s motion in the RF field, and the pseudopotential formulation is valid during the cooling process. If mB � m, each collision can add or remove substantial momentum and energy from a trapped ion. This large instantaneous momentum transfer can alter the ion’s trajectory appreciably, which may result in energy transfer from the RF-field to the ion (”RF heating”). Under these conditions, the ion is unstable in the trap, and is rapidly ejected. In the intermediate regime, mB ≈ m, a form of RF heating also occurs and the amount of time that a single ion is trapped depends on trap parameters. 3 Simulation of ion cooling and trapping The DC and RF voltage amplitudes, the longitudinal dimensions of the indi- vidual segments, and the buffer gas pressure and type are optimized using the SIMION 7.0 simulation package 3 for single ion stability. Ion-neutral collisions are implemented using a hard-sphere model with a variable radius, depending on the buffer gas and trapped ion species. This model, applicable in the case of a single atomic ion interacting with a noble buffer gas [12], uses a cross sec- tion dependent on the ion’s velocity, v, and buffer gas polarizability to account for the dipole moment of the neutral atom induced by the ion. Collisions are implemented by specifying a mean free path λ, the buffer gas mass mB, and a buffer gas temperature TB. The probability that a trapped ion collides with a buffer gas atom in a time interval ∆t is given by P (∆t) = 1− e−v∆t/λ (6) Before each time-step of the ion’s trajectory, a random number is chosen. This number is used to decide if a collision occurs during that time-step. If a collision occurs, the kinematics of the collision are calculated assuming that the velocity distribution of the neutral buffer gas atoms follows Maxwell- Boltzmann statistics with a temperature TB. The total longitudinal trap length, 604 mm, is chosen as a result of cooling simulations of an ion at various initial kinetic energies, interacting with a 3 http://www.sisweb.com/simion.htm range of buffer gases (He, Ar, Kr, and Xe). The trap is split into 16 segments, to provide sufficient versatility in shaping the longitudinal field for different phases of the R&D program. The segments are labeled Si, where i runs from 1 to 16 as shown in fig. 1. The segments are chosen to be 40 mm long, except for a single short 4 mm segment (S14, the “trapping segment”), used to tightly confine the ion longitudinally, optimizing the single ion fluorescence signal-to- noise ratio. The DC potential profile chosen for most operations is UDC = {0V, -0.1V, - 0.2V, ..., -1.2V, -8.0V, -1.2V, +10V}, with the minimum of -8 V at S14. Because of a limitation in the number of input parameters of SIMION, this profile is approximated as UDC = {+10V, +10V, +10V, -0.4V, -0.5V, ..., -1.3V, -1.2V, -8.0V, -1.2V, +10V}, in the simulation. This does not appreciably affect the ion cooling at the potential minimum. The values of the trap radius, r0, and electrode diameter, re are chosen to optimize the external optical access to a trapped ion, as well as the shape of the RF field. The electrode radius is re = 3 mm, resulting in r0 = 2.61 mm (see eqn. 1). The RF frequency is ωRF/2π = 1.2 MHz, with an amplitude of 150 V. These parameters correspond to q = 0.52 and a = 0.05 (see eqn. 3), well within the region of stability of the Mathieu equation. An example of the simulated cooling process is shown in fig. 2. In this simula- tion, a single Ba ion is created in S3, with an energy of 10 eV in 1× 10−2 torr He. The ion’s kinetic energy and z-trajectory are plotted during the initial cooling (panels a and b), and after the ion is confined in the potential well at S14 (panels c and d). During the initial cooling, the ion is reflected back and forth longitudinally in the trap. On average, the ion loses energy with each col- lision with a He atom. Once the ion is confined to S14, the ion continues to cool to the minimum, until it comes into thermal equilibrium with the buffer gas. The same processes are shown in fig. 3 in the case of Ar as a buffer gas. The ion cools much faster in the presence of Ar; however, the frequency and ampli- tude of RF heating collisions increase as well. Ar is therefore a more efficient cooling gas for Ba ions, though the higher rate of RF heating likely decreases the ion’s stability in the trap. Whereas SIMION is useful for studying these cooling and heating processes, reliable trajectory simulation is limited to the timescale of a few seconds. This is due to finite computational resources, as well as error buildup during trajectory integration. For this reason, single ion storage times longer than a few seconds, relevant for the study of RF heating and ion deconfinement, cannot be simulated. Fig. 2. Simulation of a single 136Ba+ in He (1 · 10−2torr) using SIMION. The ion in the simulation was started in the center of segment S3 of the trap. Panel (a) shows the collisional cooling during the first few hundred µs after the start of the simulation. Panel (b) shows the respective trajectory. Panels (c) and (d) show the evolution of the same quantities on a longer time scale. The ion is confined to the trapping segment and it cools further down to the buffer gas temperature. Fig. 3. The same simulation as in fig.2 of a single 136Ba+ in Ar at a pressure of 3.7 · 10−3torr. Faster cooling and larger momentum transfer collisions are evident. 4 Trap construction A single trap segment is made of four stainless steel tubes threaded onto a center stainless steel rod, as shown in an exploded view in fig. 4. Vespel 4 tube spacers insulate each segment from its neighbors, and from the center 4 Vespel is a trademark of DuPont de Nemours rod. Special care is taken to insure that all vespel parts are recessed behind conductors, in order to avoid any insulator charging that could affect the DC field inside the trap. Details of the RF and DC feed circuitry for two segments are shown in fig. 5. A DC voltage is applied to all four electrodes in a segment, using a 16-bit computer controlled DAC. The RF is applied to one diagonal pair of electrodes in a segment, while the other pair is RF grounded through a capacitor. The RF signal is supplied by a function generator 5 , which is amplified by a broadband 50 dB amplifier 6 , internally back-terminated with 50 Ω. The system can deliver the RF voltage required without the use of a tuned circuit. Each segment has a capacitance of ∼ 18 pF, however the total capacitance of the trap is closer to 600 pF due to contributions from cables and vacuum feedthroughs. Fig. 4. Exploded view of electrodes 13-16, showing the internal support and electrical insulation. The whole trap is housed in a custom-made, electropolished stainless steel UHV tank pumped by a turbomolecular pump 7 backed by a dry scroll pump 8 (fig. 6). A septum inside the vacuum tank allows for the installation of an aperture, to be used in a differentially pumped scheme (not used for the work described here), to maintain different buffer gas pressures in the injection and trapping regions of the system. The pressure in the tank is read out in the upper (injec- tion) and lower (trapping) regions of the vacuum system by vacuum gauges 9 . A gas handling manifold, connected to the trap by a computer-controlled leak 5 HP 8656B 6 ENI A150 7 Pfeiffer TMU521P 8 BOC Edwards XDS5 9 Pfeiffer PKR251 valve 10 , allows for the introduction of individual buffer gas species and binary mixtures (fig. 7). The leak valve keeps the buffer gas pressure in the vacuum chamber constant by regulating gas flow into the chamber, based on the vac- uum gauge measurement closest to S14. The turbo pump runs continuously, so that the gas pressure can be either increased or decreased at any time. Using this method, the gas pressure in the vacuum system can be regulated between 3× 10−9 and 1× 10−2 torr, with a stability of ≤ 1 %. The lower range is the limit of the vacuum gauges, while the upper limit is the maximum allowable pressure in front of the turbo-pump running at full speed. The upper pressure limit can be extended if required, with simple modifications to the vacuum system. Before any ion trapping operations begin, the entire vacuum system is baked for two days at 135 ◦C in an oven that completely encloses the tank. After bakeout, the system reaches a base pressure of < 3× 10−9 torr. Vacuum system A1 RF Monitor 1 nF 5 pF 100 nF 100 nF 1 MΩ 100 nF Fig. 5. Electrical schematics of the ion trap. Only two of 16 identical segments are shown. Fluorescence from a trapped Ba ion is induced, following the classic “shelving” scheme [13], by resonant lasers cycling the ion between the 6S1/2 (ground), 6P1/2 (excited), and 5D3/2 (metastable) states. The ion undergoes sponta- neous emission when in the 6P1/2 state, emitting either a 493 nm or 650 nm photon. The 493 nm fluorescence photons are the signal collected for this ex- periment. The 6S1/2 ↔ 6P1/2 transition at 493 nm is excited by a frequency- 10 Pfeiffer EVR116 Fig. 6. Cutaway view of the vacuum system with the linear trap mounted inside. doubled external-cavity diode laser (ECDL) 11 . The 6P1/2 ↔ 5D3/2 transition at 650 nm is excited by a ECDL 12 . 20 mW (10 mW) of 493 nm (650 nm) light is available for spectroscopy, far in excess of that required to observe a single ion. The blue laser is frequency stabilized to ∼ 20 MHz (relative) using an 11 TOPTICA SHG 100 12 TOPTICA DL 100 V6 V8 Ar He Xe V3 V2 V1 Getter P1 (70 l/s turbo) (400 l/s turbo) Linear trap vacuum system Pressure control Pressure feedback loop P3 (Scroll) V12 V13 Fig. 7. Schematic view of the gas handling system. Invar Fabry-Perot reference cavity. Long term absolute frequency stabilization is achieved by locking both lasers to a hollow-cathode Ba lamp 13 . A schematic of the laser setup is shown in fig. 8. The laser systems reside on a vibration isolated, optics table in a dust con- trolled environment, completely separated from the linear ion trap vacuum system. Both lasers are fed into one single-mode fiber, which is routed over an arbitrary distance to the ion trap system. The output beams are coupled into the ion trap via injection optics, consisting of an aspheric focusing lens, an iris to reduce beam halo, and a beam-steering mirror. The beams are directed into the vacuum system through an anti-reflection (AR) coated window 14 along the longitudinal axis of the trap, from the end of the trap closest to S14. An additional aperture inside the vacuum system aids in beam alignment and further halo reduction. Due to the chromaticity of the aspheric lens, the beam waists are separated by 260 mm, which is on the order of the beams Rayleigh length. The 493 nm and 650 nm waists at the trapping segment (S14) are 370 µm and 570 µ m, respectively. The laser powers at the injection region are sampled in real-time by photodiodes. These powers are fed-back to acousto- optic modulators before the fiber input on the laser table, and used to keep the injected beam powers stable to ∼ 1 % indefinitely. This configuration is found to suppress background laser light levels to the level required for observing a single ion. 13 Perkin-Elmer Ba Lumina HCL model N305-0109 14 “Super-V” AR coating (99.98 % transmission at 493 nm) by OptoSigma Chopper wheel Ba hollow cathode lamp Lock-in amplifier Laser Controller 650 nm ECDL 493 nm ECDL (986 SHG) Isolator Isolator Lock-in amplifier Laser Controller Chopper wheel Ba hollow cathode lamp Wavemeter Scanning Fabry-Perot AOM 3 EOM 1 Cavity controller Optogalvanic lock power control Pound-Drever-Hall lock Optogalvanic lock Optogalvanic lock Optogalvanic lock power control AOM 1 AOM 4 Fiber launch to Burleigh confocal cavity Scope Red power feedback Blue power feedback Fig. 8. Schematic view of the laser setup. The fluorescence from a trapped ion is detected by an electron-multiplied CCD camera (EMCCD), sensitive to single photons 15 . The fluorescence is imaged onto the EMCCD by a 64 mm working distance microscope 16 , with a numerical aperture of 0.195. The outer lens of the microscope objective is placed close to S14, outside a re-entrant vacuum window (see fig. 6). A spherical mirror inside the vacuum tank, directly behind S14, reflects fluorescence light back through the trap that would otherwise be lost. This roughly doubles the fluorescence light collection efficiency. A filter in front of the EMCCD attenuates the 650 nm light by more than 99 % while transmitting ∼ 85% of the 493 nm light. The light collection efficiency of the system is estimated to be 10−2, including the 90 % quantum efficiency of the EMCCD. 15 Andor iXon EM+ 16 Infinity InFocus KC with IF4 objective 5 Trap operation and results After the initial pump-down and bakeout, the trap is loaded with ions by ionizing neutral Ba in the central region of S3 (see fig. 1). Barium is chemically produced, after the system has been pumped to good vacuum, by heating a “barium dispenser” 17 loaded with BaAl4-Ni, and depositing Ba on a Ta foil. The foil can be resistively heated repeatedly, producing a Ba vapor in S3, which is ionized by a 500 eV electron beam from an electron gun 18 . A 32- gauge thermocouple on the Ta foil is used to control the temperature of the foil, regulating the amount of Ba emitted. Before loading ions into the trap, a buffer gas is introduced into the system. To load small numbers of ions (< 10), the oven is operated at 100 ◦C and the e-gun pulsed for 10 s. Ions are cooled by the buffer gas, and trapped at S14 as discussed earlier. The trap can also be loaded by turning on one of the cold cathode vacuum gauges. This effect is explained by the possible emission of electrons and ions from the gauge, which is presumably coated with Ba from the initial Ba source creation process. This should not be considered a background for Ba tagging in EXO, as the trap used in EXO will not be heavily contaminated with Ba, nor will vacuum gauges be operated during the tagging process. Fig. 9 shows a grayscale image of the fluorescence from three ions in the trap, imaged by the EMCCD over 20 s. The DC potentials are set as shown in fig. 1. The brightness of each pixel is proportional to the number of collected photons. The cloud of white pixels (encompassed by the black square) is flu- orescence from three ions trapped in S14. The cloud has two lobes, due to a slight misalignment of the spherical mirror behind S14. The white vertical bands on either side of the cloud are due to laser light scattering off the elec- trodes, which is at the same wavelength as the fluorescence photons. Precise alignment of the laser beams is required to minimize scattered laser light and optimize the signal to noise in the region of interest, in order to observe the fluorescence from a single ion in the trap. A time-series of the Ba ion fluorescence signal is shown in fig. 10 [10], starting with four ions loaded into the trap at 4.4 × 10−3 torr He. The x-axis is in seconds, and the y-axis pepresents the fluorescence in arbitrary units of EM- CCD counts. Each data point in this series is the sum of the pixels the the square box in fig. 9, after 5 s of integration by the EMCCD. The y-axis is zero- suppressed due to the large EMCCD pedestal. Over time, ions spontaneously eject from the trap due to RF heating, and possibly ion-ion Coulomb interac- tions. As individual ions eject, the fluorescence signal decreases in quantized steps. The difference in the fluorescence signals for the single steps is constant 17 SAES: http://www.saesgetters.com/default.aspx?idpage=460 18 Kimball Physics FRA-2X1-2016 http://www.saesgetters.com/default.aspx?idpage=460 r (mm) -1.5 -1 -0.5 0 0.5 1 1.5 Fig. 9. Greyscale picture of three ions contained in segment 14 of the trap. The image was taken at 5 · 10−4 torr He. The signal in the region of interest (black box) was integrated over 20 seconds with the EMCCD. within 4%, clearly establishing the capability of the system in detecting single ions. A high signal-to-noise ratio of the fluorescence from a single trapped ion will be required to confirm a 0νββ decay. The signal-to-noise ratio for this purpose is defined as S/N = 〈RI〉 − 〈RB〉√ where 〈RI〉 and 〈RS〉 are the average single-ion fluorescence and background rates, σI and σB are the Gaussian widths of the single-ion fluorescence and background rates, tI and tB are the total single-ion fluorescence and back- ground rate observation times, and ∆t is the integration time of single mea- surement (hence tI/∆t and tB/∆t are the number of measurements for the signal and the background, respectively). This metric assumes that both the single ion fluorescence and background rates follow Gaussian statistics. If the Time [s] 0 200 400 600 800 1000 1200 1400 1600 1800 4 ions 3 ions 2 ions 1 ion Background level Fig. 10. Time series of the 493 nm ion fluorescence rate in the trap at 4.4·10−3 torr He. Ions unload, causing clear quantized drops in the fluorescence rate. Each point represents 5 s of integration with the EMCCD [10]. two integration times are equal (tI = tB = t), eqn. 7 becomes S/N = 〈RI〉 − 〈RB〉√ σ2I + σ t (8) where the signal and background rates have been absorbed into the constant k, in units of Hz−1/2. The signal-to-noise ratio of the fluorescence from an individ- ual ion increases with the square root of the measurement time, or equivalently number of measurements. For the single ion fluorescence and background rates in fig. 10, k = 2.75 Hz−1/2, so that S/N = 18 for a 60s measurement time. Similar values are found in the cases of Ar, and He/Xe mixtures as buffer gases. Drifts in the laser beam position at S14 may lead to fluctuations in the background rate, RB, that are not accounted for by the assumptions of Gaus- sian statistics made here. Such drifts are caused primarily by temperature variations in the lab, affecting the alignment of the trap injection optics. In the setup used for the data presented here, no provision is made for the tem- perature stabilization of such optics that drift by as much as ±2◦C on a daily cycle. Even under these non-ideal conditions, the system is stable enough to apply the S/N description of eqn. 7 for periods of minutes, much longer than required for non-ambiguous single Ba ion identification. Temperature stabi- lization of the injection optics capable of ±0.1◦C would be straightforward to implement and would reduce non-Gaussian background fluctuations to a negligible level. The lifetimes of single Ba ions in the trap at different He pressures have been measured and reported elsewhere [10]. In the same paper, the destabilizing effects of Xe contaminations are also studied and modeled. It is found that a sufficient partial pressure of He in the trap can counter the effects of Xe, and provide lifetimes that are sufficient for single-ion detection with very high significance. 6 Summary A linear RFQ ion trap, designed and built within the R&D program towards the EXO experiment, is described. The trap is capable of confining individual Ba ions for observation by laser spectroscopy, in the presence of light buffer gases and low Xe concentrations. Single trapped Ba ions are observed, with a high signal-to-noise ratio. A similar trap will be used to identify single Ba ions produced in the 0νββ decay of 136Xe in the EXO experiment. The successful operation of this trap, as described here, is one of the cornerstones of a full Ba tagging system for EXO, which will lead to a new method of background suppression in low-background experiments. In parallel, several systems to capture single Ba ions in LXe, and transfer them into the ion trap are under development. Acknowledgements This work was supported, in part, by DoE grant FG03-90ER40569-A019 and by private funding from Stanford University. We also gratefully acknowledge a substantial equipment donation from the IBM corporation, as well as very informative discussions with Guy Savard. References [1] Y. Fukuda, T. Hayakawa, E. Ichihara, K. Inoue,et al., Phys. Rev. Lett. 81 1562 (1998). M.H. Ahn, E. Aliu, S. Andringa, S. Aoki et al., Phys. Rev. D 74, 072003 (2006). Q.R. Ahmad, R.C. Allen, T.C. Andersen, J.D Anglin et al., Phys. Rev. Lett 89 011301 (2002). T. Araki, K. Eguchi, S. Enomoto, K. Furuno, Phys. Rev. Lett. 94 081801 (2005). B.T. Cleveland,T. Daily, R. Davis, Jr., J.R. Distel et al., Astrophys. J. 496, 505 (1998). J.N. Abdurashitov, V.N. Gavrin, S.V. Girin, V.V. Gorbachev et al., Phys. Rev. C 60, 055801 (1999). W. Hampel, J. Handt, G. Heusser, J. Kiko et al., Phys. Lett. B 447, 127 (1999). D.G. Michael, P. Adamson, T. Alexopoulos, W.W.M. Allison et al., Phys. Rev. Lett. 97, 191801 (2006). [2] A. Osipowicz, H. Blumer, G. Drexlin, K. Eitel et al., arxiv hep-ex/0109033. A. Minfardini, C. Arnaboldi, C. Brofferio, S. Capelli et al., Nucl. Instr. Meth. A 559 346 (2006). [3] S. Elliott, P. Vogel, Ann. Rev. Nucl. Part. Sci. 52, 11551 (2002). [4] E. Majorana, Nuovo Cim. 14 (1937) 171. [5] M. Danilov, R. DeVoe, A. Dolgolenko, G. Giannini et al., Phys. Lett. B 480 (2000) 12. M. Breidenbach, M Danilov, J. Detwiler et al., R&D proposal, Feb 2000, Unpublished. [6] D. Denison, J. Vac. Sci. Tech. 8, 266 (1971). [7] W. Paul, Rev. Mod. Phys. 62, 531 (1990). [8] R. Marchetal., Quadrupole Ion Trap Mass Spectrometry, Wiley-Interscience, (2005). [9] N. McLachlan, Theory and Application of Mathieu Functions (Dover, 1964). [10] M. Green, J. Wodin, R. deVoe, P. Fierlinger et al., arXiv:physics/0702122 (2007), submitted to PRL. [11] S. Waldman, PhD Thesis, Stanford University 2005. J. Wodin, PhD Thesis, Stanford University 2007. [12] K. Taeman, PhD Thesis, McGill University 1997. [13] W. Neuhauser, M. Hohenstatt, P. Toscheck, H. Dehmelt, Phys. Rev. Lett. 41 (1978) 233. http://arxiv.org/abs/hep-ex/0109033 http://arxiv.org/abs/physics/0702122 Introduction Linear RFQ traps and buffer gas cooling Simulation of ion cooling and trapping Trap construction Trap operation and results Summary References

0704.1647

How much entropy is produced in strongly coupled Quark-Gluon Plasma (sQGP) by dissipative effects?

How much entropy is produced in strongly coupled Quark-Gluon Plasma (sQGP) by dissipative effects? M.Lublinsky and E.Shuryak Department of Physics and Astronomy, State University of New York, Stony Brook NY 11794-3800, USA (Dated: November 4, 2018) We argue that estimates of dissipative effects based on the first-order hydrodynamics with shear viscosity are potentially misleading because higher order terms in the gradient expansion of the dissipative part of the stress tensor tend to reduce them. Using recently obtained sound dispersion relation in thermal N=4 supersymmetric plasma, we calculate the resummed effect of these high order terms for Bjorken expansion appropriate to RHIC/LHC collisions. A reduction of entropy production is found to be substantial, up to an order of magnitude. PACS numbers: Hydrodynamical description of matter created in high energy collisions have been proposed by Landau [1] more than 50 years ago, motivated by large coupling at small distance, as followed from the beta functions of QED and scalar theories known at the time. Hadronic matter is of course described by QCD, in which the coupling runs in the opposite way. And yet, recent RHIC experiments have shown spectacular collective flows, well described by relativistic hydrodynamics. More specifically, one ob- served three types of flow: (i) outward expansion in trans- verse plane, or radial flow, (ii) azimuthal asymmetry or “elliptic flow” [2, 3], as well as recently proposed (iii) “conical flow” from quenched jets [4]. These observation lead to conclusion that QGP at RHIC is a near-perfect liquid, in a strongly coupled regime [5]. The issue we discuss below is at what “initial time” τ0 one is able to start hydrodynamical description of heavy ion collisions, without phenomenological/theoretical contradictions. Phenomenologically, it was argued in [2, 3] that elliptic flow is especially sensitive to τ0. Indeed, ballistic motion of partons may quickly erase the initial spatial anisotropy on which this effect is based. In practice, hydrodynamics at RHIC is usually used starting from time τ0 ∼ 1/2fm, otherwise the observed ellipticity is not reproduced. Can one actually use hydrodynamics reliably at such short time? How large is τ0 compared to a relevant “mi- croscopic scales” of sQGP? How much dissipation occurs in the system at this time? As a measure of that, we will calculate below the ratio of the amount of entropy pro- duced at τ > τ0 to its “primordial” value at τ0, ∆S/S0. To set up the problem, let us start with a very crude dimensional estimate. If we think that the QCD effective coupling is large αs ∼ 1 and the only reasonable micro- scopic length is given by temperature [14], then the rele- vant micro-to-macro ratio of scales is simply T0τ0. With T0 ∼ 400MeV at RHIC, one finds this ratio to be close to one. We are then lead to a pessimistic conclusion: at such time application of any macroscopic theory, thermo- or hydro-dynamics, seems to be impossible, since order one corrections are expected. Let us then do the first approximation, including the explicit viscosity term to the first order. Zeroth order (in mean free path) stress tensor used in the ideal hydrody- namics has the form T (0)µν = (ǫ+ p)uµuν + p gµν (1) while dissipative corrections are induced by gradients of the velocity field. The well known first order corrections are due to shear (η) and bulk (ξ) viscosities δT (1)µν = η(∇µuν +∇νuµ − ∆µν∇ρuρ) + ξ(∆µν∇ρuρ)(2) In this equation the following projection operator onto the matter rest frame was used: ∇µ ≡ ∆µν∂ν , ∆µν ≡ gµν − uµuν (3) The energy-momentum conservation ∂µ Tµν at this order corresponds to Navier-Stokes equation. Because colliding nuclei are Lorentz-compressed, the largest gradients at early time are longitudinal, along the beam direction. The expansion at this time can be ap- proximated by well known Bjorken rapidity-independent setup [6], in which hydrodynamical equations depend on only one coordinate – proper time τ = t2 − x2. ǫ + p = − 1 1− (4/3)η + ξ (ǫ+ p)τ where we have introduced the entropy density s = (ǫ + p)/T . Note that for traceless Tµν (conformally invariant plasma), the bulk viscosity ξ = 0. For reasons which will become clear soon, let us com- pare this eqn to another problem, in which large longi- tudinal gradients appear as well, namely sound wave in the medium. The dispersion relation (the pole position) for a sound wave with frequency ω and wave vector q is, at small q ω = csq − q2Γs, Γs ≡ Notice that the right hand side of (4) contains precisely the same combination of viscosity and thermodynamical http://arxiv.org/abs/0704.1647v1 parameters as appears in the sound attenuation problem: the length Γs, which measures directly the magnitude of the dissipative corrections. At proper times τ ∼ Γs one has to abandon the hydrodynamics altogether, as the dissipative corrections cannot be ignored. For the entropy production (4) the first correction to the ideal case is (1−Γs/τ). Since the correction to one is negative, it reduces the rate of the entropy decrease with time. Equivalently statement is that the total positive sign shows that some amount of entropy is generated by the dissipative term. Danielewicz and Gyulassy [7] have analyzed eq. (4) in great details considering vari- ous values of η. Their results indicate that the entropy production can be substantial. Our present study is motivated by the following ar- gument. If the hydrodynamical description is forced to begin at early time τ0 which is not large compared to the intrinsic micro scale 1/T , then limiting dissipative effects to the first gradient only (δT µν ) is parametrically not justified and higher order terms have to be accounted for. Ideally those effects need to be resummed. As a first step, however, we may attempt to guess their sign and estimate the magnitude. Formally one can think of the dissipative part of the stress tensor δTµν as expended in a series containing all derivatives of the velocity field u, δT 1µν being the first term in the expansion. In general 3+1 dimensional case there are many structures, each entering with a new and independent viscosity coefficient. We call them “higher order viscosities” and the expansion is somewhat similar to twist expansion. For 1+1 Bjorken problem, the ap- pearance of the extra terms modifies eq. (4), which can be written as a series in inverse proper time ∂τ (sτ) s (τ T ) (τ T )2 (Tτ)2n We have put T here simply for dimensional reasons: clearly Tτ is a micro-to-macro scale ratio which deter- mines convergence of these series and the total amount of produced entropy. Similarly, the sound wave dispersion relation becomes nonlinear as we go beyond the lowest order: ω = ℜ[ω(q)] + iℑ[ω(q)] ; (7) 2 π T 2 π T 2 π T )2n+1 = − 4πη Based on T-parity arguments we keep only odd (even) powers of q for the real (imaginary) parts of ω. The co- efficients cn, rn and ηn are related since they originate from the very same gradient expansion of Tµν . Although both the entropy production series above and sound ab- sorption should converge to sign-definite answer, the co- efficients of the series may well be of alternating sign (as we will see shortly). Clearly, keeping these next order terms can be use- ful only provided there is some microscopic theory which would make it possible to determine the values of the high order viscosities. For strongly coupled QCD plasma this information is at the moment beyond current theo- retical reach, and we have to rely on models. A particu- larly useful and widely studied model of QCD plasma is N = 4 supersymmetric plasma, which is also conformal (CFT). The AdS/CFT correspondence [8] (see [9] for re- view) relates the strongly coupled gauge theory descrip- tion to weakly coupled gravity problem in the background of AdS5 black hole metric. Remarkably, certain informa- tion on higher order viscosities in the CFT plasma can be read of from the literature and we exploit this possibility below. The viscosity-to-entropy ratio (η/s = 1/4π) deduced from AdS [10] turns out to be quite a reasonable ap- proximation to the values appropriate for the RHIC data description. Thus one may hope that the information on the higher viscosities gained from the very same model can be well trusted as a model for QCD. Admittedly hav- ing no convincing argument in favor, we simply assume that the viscosity expansion of the QCD plasma displays very similar behavior, both qualitative and quantitative, as its CFT sister. Our estimates are based on the analysis of the quasi- normal modes in the AdS black hole background due to Kovtun and Starinets [11]. The dispersion relation for the sound mode, calculated in ref. [11], is shown in Fig.1. The real and imaginary parts of ω correspond to the ex- pressions given in (7). At q → 0 they agree with the leading order hydrodynamical dispersion relation (5). The first important observation is that the next order coefficient η2 is negative, reducing the effect of the first one when gradients are large. The second is that |ℑ[ω]| has maximum at q/2πT ∼ 1, and at large q the imaginary part starts to decrease. This means that the expansion (7) has a radius of convergence q/2πT ∼ 1. In order to estimate the effect of higher viscosities on the entropy production in the Bjorken setup we first iden- tify τ in (6) with 2π/q in (7). Second we identify the coef- ficients cn with ηn. Both sound attenuation and entropy production in question are one dimensional problems as- sociated with the same longitudinal gradients and pre- sumably the same physics. In practice we use the curve for the imaginary part of ω (Fig. 1) as an input for the right hand side of (6). The numerical results are shown in Figs. 2 and 3 in which we compare our estimates with the “conventional” shear viscosity results from (4). To be fully consistent with the model we set η/s = 1/4π. We also set the initial temperature T0 = 300MeV while the standard equation of state s = 4 kSB T 3. For the coefficient kSB we use the “QCD” value kSB = 2(Nc − 1)2 + ; nf = 3; Nc = 3 Fig. 2 presents the results for entropy production as a 1 2 3 4 5 w q qq 1 2 3 4 5 Imw q FIG. 1: Sound dispersion (real and imaginary parts) obtained from the analysis of quasinormal modes in the AdS black hole background. The result and figure are taken from Ref.[11]. function of proper time for two initial times τ0 = 0.2 fm and τ0 = 0.5 fm. The dashed lines correspond to the first order result (4) while the solid curves include the higher order viscosity corrections. Noticeably there is a dra- matic effect toward reduction of the entropy production as we start the hydro evolution at earlier times (the ef- fect is almost invisible on the temperature profile). This is the central message of the present paper. Fig. 3 illustrates the relative amount of entropy pro- duced during the hydro phase as a function of initial time. If the fist order hydrodynamics is launched at very early times, the hydro phase produces too large amount of en- tropy, up to 250%. (Such a large discrepancy is not seen in the RHIC data.) In sharp contrast, the results from the resummed viscous hydrodynamics is very stable, and does not produce more than some 25% of initial entropy, even if pushed to start from extremely early times. The right figure displays the absence of any pathological ex- plosion at small τ0. It is worth commenting that we carried the analysis using the minimal value for the ratio η/s = 1/4π. We expect that if this ratio is taken larger, the discrepancy between the first order dissipative hydro and all orders will be even stronger. Before concluding this paper we note that a practi- cal implementation of relativistic viscous hydrodynamics had followed Israel-Stewart second order formalism (for recent publications see [12]) in which one introduces ad- 0 2 4 6 8 10 0 2 4 6 8 10 (fm)τ τ (fm) = 0.2 (fm)0τ = 0.5 (fm)0τ FIG. 2: Entropy production as a function of proper time for initial time τ0 = 0.2 fm (left) and τ0 = 0.5 fm (right). The initial temperature T0 = 300MeV. The dashed (blue) curves correspond to the first order (shear) viscosity approximation Eq.(4). The solid curve (red) is the all order dissipative re- summation Eq.(6). 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 1.25T = 300 (MeV)0 T = 300 (MeV)0 τ0 (fm) τ0 (fm) τs 0 0 ∆ ( ) FIG. 3: Fraction of entropy produced during the hydro phase as a function of initial proper time. The initial temperature T0 = 300MeV. The left (blue) points correspond to the first order (shear) viscosity approximation. The right (red) points are for the all order resummation. ditional parameter , the relaxation time for the system. Then the dissipative part of the stress tensor is found as a solution of an evolution equation, with the relaxation time being its parameter. For the Bjorken setup, the dis- sipative tensor thus obtained has all powers in 1/τ and might resemble the expansion in (6) and (7). The use of AdS/CFT may shed light on the interrelation between the two approaches: the first step in this direction has been made recently [13], resulting in numerically very small relaxation time. Finally, why can it be that macroscopic approaches like hydrodynamics can be rather accurate at such a short time scale? Trying to answer this central question one should keep in mind that 1/T is not the shortest micro- scopic scale. The inter-parton distance is much smaller, ∼ 1/(T ∗N1/3dof ) where the number of effective degrees of freedom Ndof ∼ 40 in QCD while Ndof ∼ N2c → ∞ in the AdS/CFT approach. In summary, we have argued that the higher order dis- sipative terms strongly reduce the effect of the usual vis- cosity. Therefore an “effective” viscosity-to-entropy ratio found from comparison Navier-Stokes results to experi- ment, can even be below the (proposed) lower bound of 1/4π. We conclude that it is not impossible to use a hy- drodynamic description of RHIC collision starting from very early times. In particular, our study suggests that the final entropy observed and its “primordial” value ob- tained right after collision should indeed match, with an accuracy of 10-20 percent. Acknowledgment We are thankful to Adrian Dumitru whose results (pre- sented in his talk at Stony Brook) inspired us to think about the issue of entropy production during the hydro phase. He emphasized to us the important problem of matching the final entropy measured after late hydro stage with the early-time partonic predictions, based on approaches such as color glass condensate. This work is supported by the US-DOE grants DE-FG02-88ER40388 and DE-FG03-97ER4014. [1] L. D. Landau, Izv. Akad Nauk SSSR, ser. fiz. 17 (1953) 51. Reprinted in Collected works by L.D.Landau. [2] D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev. Lett. 86, 4783 (2001) [arXiv:nucl-th/0011058]. “A hydrody- namic description of heavy ion collisions at the SPS and RHIC,” arXiv:nucl-th/0110037. [3] P. F. Kolb and U. W. Heinz, “Hydrodynamic description of ultrarelativistic heavy-ion collisions,” arXiv:nucl-th/0305084. [4] J. Casalderrey-Solana, E. V. Shuryak and D. Teaney, J. Phys. Conf. Ser. 27, 22 (2005) [Nucl. Phys. A 774, 577 (2006)] [arXiv:hep-ph/0411315]. [5] E.V.Shuryak, Prog. Part. Nucl. Phys. 53, 273 (2004) [ hep-ph/0312227]. [6] J. Bjorken, Phys. Rev. D27(1983)140 [7] P. Danielewicz and M. Gyulassy, Phys. Rev. D 31 (1985) [8] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. [9] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000) [arXiv:hep-th/9905111]. [10] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005) [arXiv:hep-th/0405231]. [11] P. K. Kovtun and A. O. Starinets, Phys. Rev. D 72, 086009 (2005) [arXiv:hep-th/0506184]. [12] U. W. Heinz, arXiv:nucl-th/0512049. R. Baier and P. Romatschke, arXiv:nucl-th/0610108. R. Baier, P. Ro- matschke and U. A. Wiedemann, Nucl. Phys. A 782, 313 (2007) [arXiv:nucl-th/0604006]. [13] M. P. Heller and R. A. Janik, arXiv:hep-th/0703243. [14] Note we have ignored e.g. ΛQCD . http://arxiv.org/abs/nucl-th/0011058 http://arxiv.org/abs/nucl-th/0110037 http://arxiv.org/abs/nucl-th/0305084 http://arxiv.org/abs/hep-ph/0411315 http://arxiv.org/abs/hep-ph/0312227 http://arxiv.org/abs/hep-th/9711200 http://arxiv.org/abs/hep-th/9905111 http://arxiv.org/abs/hep-th/0405231 http://arxiv.org/abs/hep-th/0506184 http://arxiv.org/abs/nucl-th/0512049 http://arxiv.org/abs/nucl-th/0610108 http://arxiv.org/abs/nucl-th/0604006 http://arxiv.org/abs/hep-th/0703243

0704.1648

Spectral Analysis of the Chandra Comet Survey

Astronomy & Astrophysics manuscript no. ms c© ESO 2021 August 26, 2021 Spectral Analysis of the Chandra Comet Survey D. Bodewits1, D. J. Christian2, M. Torney3, M. Dryer4, C. M. Lisse5, K. Dennerl6, T. H. Zurbuchen7, S. J. Wolk8, A. G. G. M. Tielens9, and R. Hoekstra1 1 kvi atomic physics, University of Groningen, Zernikelaan 25, NL-9747 AA Groningen, The Netherlands e-mail: [email protected], [email protected] 2 Queen’s University Belfast, Department of Physics and Astronomy, Belfast, BT7 1NN, UK e-mail: [email protected] 3 Atoms Beams and Plasma Group, University of Strathclyde, Glasgow, G4 0NG, UK e-mail: [email protected] 4 noaa Space Environment Center, 325 Broadway, Boulder, CO 80305, USA e-mail: [email protected] 5 Planetary Exploration Group, Space Department, Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Rd, Laurel, MD 20723, USA e-mail: [email protected] 6 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse, 85748 Garching, Germany e-mail: [email protected] 7 The University of Michigan, Department of Atmospheric, Oceanic and Space Sciences, Space Research Building, Ann Arbor, MI 48109-2143, USA e-mail: [email protected] 8 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA e-mail: [email protected] 9 nasa Ames Research Center, MS 245-3, Moffett Field, CA 9435-1000, USA e-mail: [email protected] Received August 26, 2021 ABSTRACT Aims. We present results of the analysis of cometary X-ray spectra with an extended version of our charge exchange emission model (Bodewits et al. 2006). We have applied this model to the sample of 8 comets thus far observed with the Chandra X-ray observatory and acis spectrometer in the 300–1000 eV range. The surveyed comets are C/1999 S4 (linear), C/1999 T1 (McNaught– Hartley), C/2000 WM1 (linear), 153P/2002 (Ikeya–Zhang), 2P/2003 (Encke), C/2001 Q4 (neat), 9P/2005 (Tempel 1) and 73P/2006- B (Schwassmann–Wachmann 3) and the observations include a broad variety of comets, solar wind environments and observational conditions. Methods. The interaction model is based on state selective, velocity dependent charge exchange cross sections and is used to explore how cometary X-ray emission depend on cometary, observational and solar wind characteristics. It is further demonstrated that cometary X-ray spectra mainly reflect the state of the local solar wind. The current sample of Chandra observations was fit using the constrains of the charge exchange model, and relative solar wind abundances were derived from the X-ray spectra. Results. Our analysis showed that spectral differences can be ascribed to different solar wind states, as such identifying comets interacting with (I) fast, cold wind, (II), slow, warm wind and (III) disturbed, fast, hot winds associated with interplanetary coronal mass ejections. We furthermore predict the existence of a fourth spectral class, associated with the cool, fast high latitude wind. Key words. Surveys, atomic processes, molecular processes, Sun: solar wind, coronal mass ejections (cmes), X-rays: solar system, Comets: general Comets: individual: C/1999 S4 (linear), C/1999 T1 (McNaught–Hartley), C/2000 WM1, 153P/2002 (Ikeya–Zhang), 2P/2003 (Encke), C/2001 Q4 (neat), 9P/2005 (Tempel 1) and 73/P-B 2006 (Schwassmann–Wachmann 3B) 1. Introduction When highly charged ions from the solar wind collide on a neutral gas, the ions get partially neutralized by capturing elec- trons into an excited state. These ions subsequently decay to the ground state by the emission of one or more photons. This pho- ton emission is called charge exchange emission (cxe) and it has been observed from comets, planets and the interstellar medium in X-rays and the Far-UV Lisse et al. (1996); Krasnopolsky (1997); Snowden et al. (2004); Dennerl (2002). The spec- tral shape of the cxe depends on properties of both the neutral Send offprint requests to: D. Bodewits gas and the solar wind and the subsequent emission can there- fore be regarded as a fingerprint of the underlying interactions Cravens et al. (1997); Kharchenko and Dalgarno (2000, 2001); Beiersdorfer et al. (2003); Bodewits et al. (2004a, 2006). Since the first observations of cometary X-ray emission, more than 20 comets have been observed with various X-ray and Far-UV observatories Lisse et al. (2004); Krasnopolsky et al. (2004). This observational sample contains a broad variety of comets, solar wind environments and observational conditions. The observations clearly demonstrate the diagnostics available from cometary charge exchange emission. First of all, the emission morphology is a tomography of the distribution of neutral gas around the nucleus Wegmann et 2 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey al. (2004). Gaseous structures in the collisionally thin parts of the coma brighten, such as the jets in 2P/Encke Lisse et al. (2005), the Deep Impact triggered plume in 9P/Tempel 1 Lisse et al. (2007) and the unusual morphology of comet 6P/d’Arrest Mumma et al (1997). In other comets, the X-ray emission clearly mapped a spherical gas distribution. This resulted in a characteristic crescent shape for larger and hence collisionally thick comets observed at phase angles of roughly 90 degrees (e.g. Hyakutake - Lisse et al. (1996), linear S4 - Lisse et al. (2001)). Macroscopic features of the plasma interaction such as the bowshock are observable, too Wegmann & Dennerl (2005). Secondly, by observing the temporal behavior of the comets X-ray emission, the activity of the solar wind and comet can be monitored. This was first shown for comet C/1996 B2 (Hyakutake) Neugebauer et al. (2000) and re- cently in great detail by long term observations of comet 9P/2005 (Tempel 1) Willingale et al. (2006); Lisse et al. (2007) and 73P/2006 (Schwassmann–Wachmann 3C) Brown et al. (2007), where cometary X-ray flares could be assigned to either cometary outbursts and/or solar wind enhancements. Thirdly, cometary spectra reflect the physical characteristics of the solar wind; e.g. spectra resulting from either fast, cold (polar) wind and slow, warm equatorial solar wind should be clearly different Schwadron and Cravens (2000); Kharchenko and Dalgarno (2001); Bodewits et al. (2004a). Several attempts were made to extract ionic abundances from the X-ray spectra. The first generation spectral models have all made strong assumptions when modelling the X-ray spectra Haeberli et al (1997); Wegmann et al. (1998); Kharchenko and Dalgarno (2000); Schwadron and Cravens (2000); Lisse et al. (2001); Kharchenko and Dalgarno (2001); Krasnopolsky et al. (2002); Beiersdorfer et al. (2003); Wegmann et al. (2004); Bodewits et al. (2004a); Krasnopolsky (2004); Lisse et al. (2005). Here, we present a more elaborate and sophisticated procedure to an- alyze cometary X-ray spectra based on atomic physics input, which for the first time allows for a comparative study of all existing cometary X-ray spectra. In Section 2, our comet-wind interaction model is briefly introduced. In Section 3, it is demon- strated how cometary spectra are affected by the velocity and target dependencies of charge exchange reactions. In Section 4, the various existing observations performed with the Chandra X-ray Observatory, as well as the solar wind data available are introduced. Based upon our modelling, we construct an analyt- ical method of which the details and results are presented in Section 5. In Section 6, we discuss our results in terms of comet and solar wind characteristics. Lastly, in Section 7 we summa- rize our findings. Details of the individual Chandra comet ob- servations are given in Appendix A. 2. Charge Exchange Model 2.1. Atomic structure of He-like ions Electron capture by highly charged ions populates highly excited states, which subsequently decay to the ground state. These cas- cading pathways follow ionic branching ratio statistics. Because decay schemes work as a funnel, the lowest transitions (n = 2→ 1) are the strongest emission lines in cxe spectra. For helium-like ions, these are the forbidden line (z: 1s2 1S0–1s2s 3S1), the inter- combination lines (y, x: 1s2 1S0–1s2p 3P1,2), and the resonance line (w: 1s2 1S0–1s2p 1P1), see Figure 1. The apparent branching ratio, Beff , for the intercombination transitions is determined by weighting branching ratios (B j) de- rived from theoretical transition rates compiled by Porquet et al. Fig. 1. Part of the decay scheme of a helium–like ion. The 1S0 decays to the ground state via two-photon processes (not indicated). Table 1. Apparent effective branching ratios (Beff) for the relaxation of the 23P-state of He-like carbon, nitrogen, oxygen and neon. transition C v N vi O vii Ne ix 1s2 (1S0)–1s2p (3P1,2) 0.11 0.22 0.30 0.34 1s2s (3S1)–1s2p (3P0,1,2) 0.89 0.78 0.70 0.66 (2000, 2001), by an assumed statistical population of the triplet P-term: Beff = (2 j + 1) (2L + 1)(2S + 1) · B j (1) The resulting effective branching ratios are given in Table 1. These ratios can only be observed at conditions where the metastable state is not destroyed (e.g. by UV flux or collisions) before it decays. In contrast to many other astrophysical X- ray sources, this condition is fulfilled in cometary atmospheres, making the forbidden lines strong markers of cxe emission. 2.2. Emission Cross Sections To obtain line emission cross sections we start with an initial state population based on state selective electron capture cross sections and then track the relaxation pathways defined by the ion’s branching ratios. Electron capture reactions can be strongly dependent on target effects. An important difference between reactions with atomic hydrogen and the other species is the presence of multi- ple electrons, hence allowing for multiple (mostly double) elec- tron transfer. It has been demonstrated both experimentally and theoretically that double electron capture can be an important reaction channel in multi-electron targets and that after autoion- ization to an excited state it may contribute to the X-ray emis- sion Ali et al. (2005); Hoekstra et al. (1989); Beiersdorfer et al. (2003); Otranto et al (2006); Bodewits et al. (2006). Unfortunately, experimental data on reactions with species typ- ical for cometary atmospheres, such as H2O, atomic O and CO are at best scarcely available. Because the first ionization po- tentials of these species are all close to that of atomic H, using state selective one electron capture cross sections for bare ions charge exchanging with atomic hydrogen from theory is a rea- sonable assumption, which is also confirmed by experimental studies Greenwood et al. (2000, 2001); Bodewits et al. (2006). Here, we will use the working hypothesis that effective one elec- tron cross sections for multi-electron targets present in cometary atmospheres are at least roughly comparable to cross sections for D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey 3 Table 2. Compilation of theoretical, velocity dependent emission cross sections for collisions between bare- and H-like solar wind ions and atomic hydrogen, in units of 10−16 cm2. See text for details. We estimate uncertainties to be ca. 20%. The ion column contains the resulting ion, not the original solar wind ion. Line energies compiled from Garcia & Mack (1965); Vainshtein & Safronova (1985); Drake (1988); Savukov et al. (2003) and the chianti database Dere et al. (1997); Landi et al. (2006). E (eV) Ion Transition 200 km s−1 400 km s−1 600 km s−1 800 km s−1 1000 km s−1 299.0 C v z 8.7 12 16 18 20 304.4 C v x,y 0.65 1.0 1.5 1.7 1.8 307.9 C v w 1.8 3.0 4.1 4.8 5.2 354.5 C v 1s3p-1s2 0.55 0.71 0.81 1.0 1.3 367.5 C v 1s4p-1s2 0.70 0.66 0.76 0.74 0.72 367.5 C vi 2p-1s 15 26 30 33 34 378.9 C v 1s5p-1s2 0.00 0.02 0.05 0.04 0.04 419.8 N vi z 13 23 28 29 29 426.3 N vi x,y 2.7 4.3 5.3 5.7 6.0 430.7 N vi w 3.8 6.0 7.4 8.1 8.5 435.5 C vi 3p-1s 1.6 4.0 4.7 4.7 4.8 459.4 C vi 4p-1s 2.9 5.9 7.0 6.4 6.0 471.4 C vi 5p-1s 0.55 1.0 1.3 0.85 0.54 497.9 N vi 1s3p-1s2 0.43 0.99 1.3 1.3 1.3 500.3 N vii 2p-1s 40 45 44 42 42 523.0 N vi 1s4p-1s2 0.81 1.6 1.9 1.8 1.7 534.1 N vi 1s5p-1s2 0.14 0.31 0.33 0.21 0.14 561.1 O vii z 37 34 33 32 31 568.6 O vii x,y 10 10 10 9.9 9.7 574.0 O vii w 9.9 11 11 11 10 592.9 N vii 3p-1s 6.3 4.9 4.8 4.5 4.3 625.3 N vii 4p-1s 2.9 2.9 3.7 4.3 4.6 640.4 N vii 5p-1s 11 5.2 3.7 2.7 2.2 650.2 N vii 6p-1s 0.00 0.21 0.13 0.09 0.08 653.5 O viii 2p-1s 27 40 48 51 53 665.6 O vii 1s3p-1s2 1.7 1.3 1.3 1.2 1.2 697.8 O vii 1s4p-1s2 0.81 0.79 1.0 1.2 1.3 712.8 O vii 1s5p-1s2 2.8 1.3 0.92 0.68 0.54 722.7 O vii 1s6p-1s2 0.00 0.06 0.04 0.02 0.02 774.6 O viii 3p-1s 2.6 4.7 5.6 5.3 5.0 817.0 O viii 4p-1s 1.0 1.6 2.0 2.2 2.3 836.5 O viii 5p-1s 2.4 4.0 4.6 4.1 3.7 849.1 O viii 6p-1s 1.6 1.6 1.5 1.1 0.67 one electron capture from H. Based on this hypothesis, we will use our comet-wind interaction model to evaluate the contribu- tion of the different species. For our calculations, we use a compilation of theoretical state selective, velocity dependent cross sections for collisions with atomic hydrogen Errea et al. (2004); Fritsch and Lin (1984); Green et al. (1982); Shipsey et al. (1983). We furthermore assume that capture by H-like ions leads to a statistical triplet to singlet ratio of 3:1, based on measurements by Suraud et al. (1991); Bliek et al. (1998). We will first focus on the strongest emission features, which are the n = 2 → 1 transitions, i.e., the Ly-α transition (H-like ions) or the forbidden, resonance and intercombination lines (He-like ions). In Fig. 2, the emission cross sections of the Ly-α or the sum of the emission cross sections of the forbidden, resonance and intercombination lines of different ions (C, N, O) are shown as a function of collision velocity, for one electron capture reac- tions with atomic hydrogen. This figure sets the stage for solar wind velocity induced effects in cometary X-ray spectra. Most important is the effect of the velocity on the two carbon emis- sion features; their prime emission features increase by a factor of almost two when going from typical ‘slow’ to typical ‘fast’ solar wind velocities. The O viii Ly-α emission cross section can be seen to drop steeply below ca. 300 km s−1. The N vi K-α dis- plays a similar, though somewhat less strong behavior. Fig. 2. Velocity dependence of Ly-α or the sum of the forbid- den/resonance/intercombination emission cross sections of different so- lar wind ions: O viii (dashed, grey line), O vii (solid, black line), N vii (dotted, black line), N vi (solid, grey line), C vi (dashed, black line) and C v (dash-dotted, black line). The relative intensity of the emission lines (per species) is governed by the state selective electron capture cross sections 4 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey Fig. 3. Velocity dependence of the hardness ratio of different solar wind ions: O viii (solid line), O vii (dashed line) N vii (dashed line) and C vi (dash-dotted line). Also shown are two experimentally obtained hard- ness ratios by Beiersdorfer et al. (2001) and Greenwood et al. (2000) for O8+ colliding on CO2 and H2O, respectively (see text). of the charge exchange reaction and the branching ratios of the resulting ion. A measure of these intensities is the hardness ra- tio (Beiersdorfer et al. 2001), which is defined as the ratio be- tween the emission cross sections of the higher order terms of the Lyman-series and Ly-α (or between the higher order K-series and K-α in case of He-like ions):∑ n>2 σem(Ly−n) σem(Ly−α) For electron capture by H-like ions, we will use the ratio be- tween the sum of the resonance-, intercombination and forbid- den emission lines and the rest of the K-series as the hardness ratio. Fig. 3 shows the hardness ratios of cxe from abundant so- lar wind ions. The figure shows that most hardness ratios are constant at typical solar wind velocities (above 300 km s−1) but it also clearly demonstrates the suggestion made by Beiersdorfer et al. (2001) that hardness ratios are good candidates for studies of velocimetry deep within the coma when the solar wind has slowed down by mass loading. 2.3. Interaction Model Cometary high-energy emission depends upon certain properties of both the comet (gas production rate, composition, distance to the Sun) and the solar wind (speed, composition). Recently, we developed a model that takes each of these effects into account Bodewits et al. (2006), which we will briefly describe here. The neutral gas model is based on the Haser-equation, which assumes that a comet has a spherically expanding neutral coma Haser (1957); Festou (1981). The lifetime of neutrals in the solar radiation field varies greatly amongst species typical for cometary atmospheres Huebner et al. (1992). The dissociation and ionization scale lengths also depend on absolute UV fluxes, and therefore on the distance to the Sun. The coma interacts with solar wind ions, penetrating from the sunward side follow- ing straight line trajectories. The charge exchange processes be- tween solar wind ions and coma neutrals are explicitly followed both in the change of the ionization state of the solar wind ions Fig. 4. Modeled charge state distribution along the comet-Sun line, as- suming an equatorial 300 km s−1 wind interacting with a comet with outgassing rate Q=1029 molecules s−1 at 1 AU from the Sun. A compo- sition typical for the slow, equatorial wind was assumed. and in the relaxation cascade of the excited ions (as discussed above). Due to its interaction with the cometary atmosphere, the so- lar wind is both decelerated and heated in the bow shock. This bow shock does not affect the ionic charge state distribution. The bow shock lowers the drift velocity of the wind but at the same time increases its temperature and the net collision velocity of the ions is ca. 77% of the initial velocity v(∞) throughout the interaction zone. We use a rule of thumb derived by Wegmann et al. (2004) to estimate the stand-off distance Rbs of the bow shock. Deep within the coma, the solar wind finally cools down as the hot wind ions, neutralized by charge exchange, are replaced by cooler cometary ions. For simplicity however, we shall as- sume that the wind keeps a constant velocity and temperature after crossing the bow shock. Initially, the charge state distribution depends on the solar wind state. For most simulation purposes, we will assume the ‘average’ ionic composition for the slow, equatorial solar wind as given by Schwadron and Cravens (2000). Using our compi- lation of charge changing cross sections, we can solve the differ- ential equations that describe the charge state distribution in the coma in the 2D-geometry fixed by the comet-Sun axis. Figure 4 shows the charge state distribution for a 300 km s−1 equato- rial wind interacting with a comet with an outgassing rate Q of = 1029 molecules s−1 comet. From this charge state distribution, it can be seen that along the comet-Sun axis, the comet becomes collisionally thick between 3500 km (O8+) to 2000 km (C6+), depending on the cross section of the ions. A maximum in the C5+ abundance can be seen around 2,000 km, which is due to the relatively large initial C6+ population and the small cross section of C5+ charge exchange. A 3D integration assuming cylindrical symmetry around the comet-Sun axis finally yields the absolute intensity of the emis- sion lines. Effects due to the observational geometry (i.e. field of view and phase angle) are included at this step in the model. D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey 5 Fig. 5. Relative contribution of target species to the total intensity of O vii 570 eV emission complex with increasing field of view, for an active Q= 1029 molecules s−1 comet, interacting with a 300 km s−1 solar wind at 1 AU from the Sun. The shaded area indicates the range of apertures used to obtain spectra discussed within this survey. 3. Model Results 3.1. Relative Contribution of Target Species Figure 5 shows the dominant collisions which underly the X-ray emission of comets. Shown is the total intensity projected on the sky, with increasing field of view. Within 104 km around the nu- cleus, water is the dominant collision partner. Farther outward (≥ 2 × 105 km), the atomic dissociation products of water take over, and atomic oxygen becomes the most important collision partner. When the field of view exceeds 107 km, atomic hydro- gen becomes the sole collision partner. Note that collisions with water never account for 100% of the emission, even with very small apertures, due to the contribution of collisions with atomic hydrogen, OH and oxygen in the line of sight towards the nu- cleus. The comets observed with Chandra are all observed with an aperture of ca. 7.5′ centered on the nucleus. This corresponds to a range of 1.6−22×104 km (as indicated in Figure 5). Our model predicts that the emission from nearby comets will be dominated by cxe from water, but that for comets observed with a larger field of view, up to 60% of the emission can come from cxe interactions with the water dissociation products atomic oxygen and OH, and 10% from interactions with atomic hydrogen. 3.2. Solar Wind Velocity To illustrate solar wind velocity induced variations in charge ex- change spectra, we simulated charge exchange spectra follow- ing solar wind interactions between an equatorial wind and a Q = 1029 molecules s−1 comet, and assumed the same solar wind composition in all cases. In Fig. 6, spectra resulting from collisional velocities of 300 km s−1 and 700 km s−1 are shown. In the spectrum from the faster wind, the C vi 367 eV and O vii 570 eV emission features are roughly equally strong, whereas at 300 km s−1, the oxygen feature is clearly stronger. Assuming the wind’s composition remains the same, within the range of typ- ical solar wind velocities (300–700 km s−1), the cross sectional dependence on solar wind velocity does not affect cometary X- ray spectra by more than a factor 1.5. In practice, the composi- tional differences between slow and fast wind will induce much stronger spectral changes. Fig. 6. Simulated X-ray spectra for a 1029 molecules s−1 comet interact- ing with an equatorial wind with velocities of 300 km s−1 (solid grey line) and 700 km s−1 (dashed black line). The spectra are convolved with Gaussians with a width of σ = 50 eV to simulate the Chandra spectral resolution. To indicate the different lines, also the 700 km s−1 σ = 1 eV spectrum is indicated (not to scale). A field of view of 105 km and ‘typical’ slow wind composition were used. 3.3. Collisional Opacity Many of the 20+ comets that have been observed in X-ray dis- play a typical crescent shape as the solar wind ion content is depleted via charge exchange. Comets with low outgassing rates around 1028 molecules s−1, such as 2P/2002 (Encke) and 9P/2005 (Tempel 1), did not display this emission morphology Lisse et al. (2005, 2007). Whether or not the crescent shape can be resolved depends mainly on properties of the comet (out- gassing rate), but, to a minor extent, also on the solar wind (velocity dependence of cross sections). Other parameters (sec- ondary, but important), are the spatial resolution of the instru- ment and the distance of the comet to the observer. In a collisionally thin environment, the ratio between emis- sion features is the product of the ion abundance ratios and the ratio between the relevant emission cross sections: rthin = n(Aq+) n(Bq+) em (v) em (v) The flux ratio for a collisionally thick system depends on the charge states considered. In case of a bare ion A and a hydro- genic ion B, the ratio between the photon fluxes from A and B is given by the abundance ratio weighted by efficiency factors µ and η: rthick = n(Aq+) n(B(r−1)+) + µ(Br+)n(Br+) η(Aq+) η(B(r−1)+) The efficiency factor µ is a measure of how much B(r−1)+ is pro- duced by charge exchange reactions by Bq+: σr,r−1(v) σr(v) where σr is the total charge exchange cross section and σr,r−1 the one electron charge changing cross section. The efficiency factor η describes the emission yield per reaction and is given by the ratio between the relevant emission cross section σem and the total charge changing cross section σr: σem(v) σr(v) 6 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey Fig. 7. Collisional opacity effects on flux ratios within the field of view. The outer bounds of the fields of view within this survey were between 104 − 105 km, as indicated by the shaded area. We considered a 500 km s−1 equatorial wind interacting with comets with different activities: Q = 1028 molecules s−1 (dashed lines) and Q = 1029 molecules s−1 (solid lines). All flux ratios are normalized to 1 at infinity. To explore the effect of collisional opacity on spectra, we simulated two comets at 1 AU from the Sun, with gas pro- duction rates of 1028 and 1029 molecules s−1, interacting with a solar wind with a velocity of 500 km s−1 and an averaged slow wind composition Schwadron and Cravens (2000). The results are summarized in Figure 7 where different flux ratios are shown. The behavior of these ratios as a function of aper- ture is important because they can be used to derive relative ionic abundances. All ratios are normalized to 1 at infinite dis- Fig. 8. Simulated X-ray spectra for a 1029 molecules s−1 comet inter- acting with an equatorial wind with a velocity of 300 km s−1 for fields of view decreasing from 105 km (solid line), 104 km (dashed line) and 103 km (dotted line). tance from the comet’s nucleus. For low activity comets with Q ≤ 1028 molecules s−1, the collisional opacity does not affect the comet’s X-ray spectrum. Within typical field of views all line flux ratios are close to the collisionally thin value. For more ac- tive comets (Q = 1029 molecules s−1), collisional opacity can become important within the field of view. Observed flux ratios involving C v should be treated with care, see e.g. C v/O vii and C vi/C v, because the flux ratios within the field of view can be affected by almost 50% and 35%, respectively. The effect is the strongest in these cases because of the large relative abundance of C6+, that contributes to the C v emission via sequential elec- tron capture reactions in the collisionally thick zones. For N vii and O viii, a small field of view of 104 km could affect the ob- served ionic ratios by some 20%. To further illustrate these results, we show the result- ing X-ray spectra in Fig. 8. There, we consider a Q = 1029 molecules s−1 comet interacting with a 300 km s−1 wind and show the effect of slowly zooming from the collisionally thin to the collisionally thick zone around the nucleus. The field of view decreases from 105 to 103 km. At 105 km, the spectrum is not affected by collisionally thick emission, whereas the emis- sion within an aperture of 1000 km is almost purely from the interactions within the collisionally thick zones of the comet, which can be most clearly seen by the strong enhancement of the C v emission around 300 eV. The results of our model efforts demonstrate that cometary X-ray spectra reflect characteristics of the comet, the solar wind and the observational conditions. Firstly, charge exchange cross sections depend on the velocity of the solar wind, but its effects are the strongest at velocities below regular solar wind velocities. Secondly, collisional opacity can affect cometary X-ray spectra but mainly when an active comet (Q = 1029 molecules s−1) is observed with a small field of view (≤ 5×104 km). The dominant factor however to explain differences in cometary CXE spectra is therefore the state and hence composition of the solar wind. This implies that the spectral analysis of cometary X-ray spectra can be used as a direct, remote quantitative and qualitative probe of the solar wind. D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey 7 8 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey Fig. 9. Chandra comet observations during the descending phase of so- lar cycle # 23. Monthly sunspot numbers (grey line) and smoothed monthly sunspot number (black lines) from the Solar Influences Data Analysis Center of the Department of Solar Physics, Royal Observatory of Belgium (http://sidc.oma.be/). Letters refer to the chronological or- der of observation. 4. Observations In this section, we will briefly introduce the different comet ob- servations performed with Chandra. A summary of comet and solar wind parameters is given in Table 3. More observational details on the comet and a summary of the state of the solar wind at the location of the comet during the X-ray observations can be found in Appendix A. 4.1. Solar Wind Data Our survey spans the whole period between solar maximum (mid 2000) and solar minimum (mid 2006), see Fig. 9. During solar minimum, the solar wind can be classified in polar- and equa- torial streams, where the polar can be found at latitudes larger than 30◦ and the equatorial wind within 15◦ of the helioequator. Polar streams are fast (ca. 700 km s−1) and show only small vari- ations in time, in contrast to the irregular equatorial wind. Cold, fast wind is also ejected from coronal holes around the equa- tor, and when these streams interact with the slower background wind corotating interaction regions (cirs) are formed. As was illustrated by Schwadron and Cravens (2000), different wind types vary greatly in their compositions, with the cooler, fast wind consisting of on average lower charged ions than the hot- ter equatorial wind. This clear distinction disappears during solar maximum, when at all latitudes the equatorial type of wind dom- inates. In addition, coronal mass ejections are far more common around solar maximum. There is a strong variability of heavy ion densities due to variations in the solar source regions and dynamic changes in the solar wind itself Zurbuchen & Richardson (2006). The vari- ations mainly concern the charge state of the wind as elemental variations are only on the order of a factor of 2 (Von Steiger et al. (2000), and references therein). We obtained solar wind data from the online data archives of ace (proton velocities and densities from the swepam instru- ment, heavy ion fluxes from the swics and swims instruments1) and soho (proton fluxes from the Proton Monitor Instrument2). Both ace and soho are located near Earth, at its Lagrangian 1 http://www.srl.caltech.edu/ace/ASC/level2/index.html 2 http://umtof.umd.edu/pm/crn/ point L1. In order to map the solar wind from L1 to the posi- tion of the comets, we used the time shift procedure described by Neugebauer et al. (2000). The calculations are based on the comet ephemeris, the location of L1 and the measured wind speed. With this procedure, the time delay between an element of the corotating solar wind arriving at L1 and the comet can be predicted. A disadvantage of this procedure is that it cannot ac- count for latitudinal structures in the wind or the magnetohydro- dynamical behavior of the wind (i.e., the propagation of shocks and cmes). These shortcomings imply that especially for comets that have large longitudinal, latitudinal and/or radial separations from Earth, the solar wind data is at best an estimate of the local wind conditions. The resulting proton velocities at the comets near the time of the Chandra observations are shown in Fig. 10. Parallel to this helioradial and heliolongitudinal mapping, we compared our comet survey to a 3D mhd time–dependent so- lar wind model that was employed during most of Solar Cycle 23 (1997 - 2006) on a continuous basis when significant solar flares were observed. The model (reported by Fry et al. (2003); McKenna-Lawlor et al. (2006) and Z.K. Smith, private com- munication, for, respectively, the ascending, maximum, and de- scending phases) treats solar flare observations and maps the progress of interplanetary shocks and cmes. The papers men- tioned above provide an rms error for ”hits” of ±11 hours Smith et al. (2000); McKenna-Lawlor et al. (2006). cir fast forward shocks were also taken into account in order to differentiate be- tween the co-rotating ”quiet” and transient structures. It was im- portant, in this differentiating analysis, to examine (as we have done here) the ecliptic plane plots of both of these structures as simulated by the deforming interplanetary magnetic field lines (see, for example, Lisse et al. (2005, 2007) for several of the comets discussed here.) Therefore, the various comet locations (Table 3) were used to estimate the probability of their X-ray emission during the observations being influenced by either of these heliospheric situations. 4.2. X-ray Observations After its launch in 1999, 8 comets have been observed with the Chandra X-ray Observatory and Advanced ccd Imaging Spectrometer (acis). Here, we have mainly considered obser- vations made with the acis-S3 chip, which has the most sensi- tive low energy response and for which the majority of comets were centered. The Chandra’s acis-S instrument provides mod- erate energy resolution (σ ≈ 50 eV) in the 300 to 1500 eV en- ergy range, the primary range for the relatively soft cometary emission. All comets in our sample were re-mapped into comet- centered coordinates using the standard Chandra Interactive Analysis of Observations (ciao v3.4) software ‘sso freeze’ al- gorithm. Comet source spectra were extracted from the S3 chip with a circular aperture with a diameter of 7.5′, centered on the cometary emission. The exception was comet C/2001 Q4, which filled the chip and a 50% larger aperture was used. acis’ re- sponse matrices were used to model the instrument’s effective area and energy dependent sensitivity matrices were created for each comet separately using the standard ciao tools. Due to the large extent of cometary X-ray emission, and Chandra’s relatively narrow field of view, it is not trivial to ob- tain a background uncontaminated by the comet and sufficiently close in time and viewing direction. We extracted background spectra using several techniques: spectra from the S3 chip in an outer region generally > 8′, an available acis S3 blank sky observation, and backgrounds extracted from the S1 ccd. For D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey 9 Fig. 10. Solar wind proton velocities estimated from ace and soho data. For all comets, the time of the observations is indicated with a dotted line. Letters refer to the chronological order of observation. several comets there are still a significant number of cometary counts in the outer region of the S3 ccd. Background spec- tra taken from the S1 chip have the advantage of having been taken simultaneous with the S3 observation and thus having the same space environment as the S3 observation. In general the background spectra were extracted with the same 7.5′ aper- ture as the source spectra but centered on the S1 chip. For comet Encke, where the S1 chip was off during the observa- tion the background from the outer region of the S3 chip was used. Comet C/2000 WM1 (linear) was observed with the Low- Energy Transmission Grating (letg) and acis-S array. For the latter, we analyzed the zero-th order spectrum, and used a back- ground extracted from the outer region of the S3 chip. It is possi- ble that the proportion of incident X-rays diffracted onto the S3 chip will vary with photon energy. Background-subtracted spec- tra generally have a signal-to-noise at 561 eV of at least 10, and over 50 for 153P/2002 C1 (Ikeya–Zhang). 5. Spectroscopy The observed spectra are shown in Figure 11. The spectra suggest a classification based upon three competing emission features, i.e. the combined carbon and nitrogen emission (be- low 500 eV), O vii emission around 565 eV and O viii emis- sion at 654 eV. Firstly, the C+N emission (<500 eV) seems to be anti-correlated with the oxygen emission. This clearly sets the spectra of 73P/2006 S.–W.3B and 2P/2003 (Encke) apart, as for those two comets the C+N features are roughly as strong as the O vii emission. In the spectra of the remaining five comets, oxygen emission dominates over the carbon and nitrogen emis- sion below 500 eV. The O viii/O vii ratio can be seen to increase continuously, culminating in the spectrum of 153P/2002 (Ikeya– Zhang) where the spectrum is completely dominated by oxygen emission with almost comparable O viii and O vii emission fea- tures. From our modelling, we expect that the separate classes reflect different states of the solar wind, which imply different ionic abundances. To explore the obtained spectra more quanti- tatively, we will use a spectral fitting technique based on our cxe model to extract X-ray line fluxes. 5.1. Spectral Fitting The charge exchange mechanism implies that cometary X-ray spectra result from a set of solar wind ions, which produce at least 35 emission lines in the regime visible with Chandra. As comets are extended sources, these lines cannot all be resolved. All spectra were therefore fit using the 6 groups of fixed lines of our cxe model (see Table 2) and spectral parameters were de- rived using the least squares fitting procedure with the xspec package. The relative strengths from all lines were fixed per ionic species, according to their velocity dependent emission cross sections. Thus, the free parameters were the relative fluxes of the C, N and O ions contained in our model. Two additional Ne lines at 907 eV (Ne ix) and 1024 eV (Ne x) were also included, giving a total of 8 free parameters. All line widths were fixed at the acis-S3 instrument resolution. The spectra were fit in the 300 to 1000 eV range. This pro- vided 49 spectral bins, and thus 41 degrees of freedom. acis spectra below 300 eV are discarded because of the rising back- ground contributions, calibration problems and a decreased ef- fective area near the instrument’s carbon edge. As a more detailed example of the cxe model and compari- son to the data, we show in Fig 12 the acis-S3 data for C/1999 S4 (linear). The figure shows the background subtracted source spectrum over-plotted with the background spectrum, the differ- ence between the model and data, and the model spectrum and 10 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey Fig. 11. Observed spectrum and fit of all 8 comets observed with Chandra, grouped by their spectral shape (see text). The histogram lines indicate the CXE model fit. data to indicate to contribution of the different ions. Only the emission lines with >3% strength of the strongest line in their species are shown for ease of presentation. The fluxes obtained by our fitting are converted into relative ionic abundances by weighting them by their velocity dependent emission cross sections. For comets observed near the ecliptic plane (< 15◦), solar wind conditions mapped to the comet were used (Section 4.1). For comets observed at higher latitudes, these data are most likely not applicable and a solar wind velocity of 500 km s−1 was assumed. Fig. 12. Details of the cxe fit for the spectrum of comet 1999/S4 (linear). Top panel: Comet (filled triangles) and background (open squares) spectrum. Middle panel: Residuals of cxe fit Bottom panel: cxemodel and observed spectrum indicating the different lines and their strengths. Carbon - red; nitrogen - orange; oxygen - blue; neon - green. The unfolded model is scaled above the emission lines for the ease of presentation. Fig. 13. Parameter sensitivity for the major emission features in the fit of C/1999 S4 (linear), with respect to the O vii 561 eV feature. All units are 10−4 photons cm−2 s−1. The contours indicate a χ2R of 9.2 (or 99% confidence, largest, green contour), a χ2R of 4.6 (90%, red contour) and a χ2R of 2.3 (68%, smallest, blue contour). 5.2. Spectroscopic Results The fits to all cometary spectra are shown in Fig. 11 and the results of the fits are given in Table 4. For the majority of the comets, the model is a good fit to the data within a 95% con- fidence limit (χ2R ≈ 1.4). Results for comet 153P/2002 (Ikeya– Zhang) are presented in Table 5 with an additional systematic error to account for its brightness and any uncertainties in the response. The spectra for all comets are well reproduced in the 300 to 1000 eV range. The nitrogen contribution is statistically signifi- cant for all comets except the fainter ones, 2P/2003 (Encke) and D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey 11 12 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey Table 6. Solar wind abundance relative to O7+, obtained for comet linear S4. References: Bei ’03 – Beiersdorfer et al. (2003), Kra ’04 – Krasnopolsky (2004), Kra ’06 – Krasnopolsky (2006), Otr ’06 – Otranto et al (2006) and S&C ’00 – Schwadron and Cravens (2000). Dots indicate that an ion was included in the fitting, but no abundances were derived; dash means that an ion was not included in the fitting. Otranto et al (2006) did not fit the observed spectrum, but used a combination of ace-data and solar wind averages from Schwadron and Cravens (2000) to compute a syntectic spectrum of the comet. Solar wind averages are given for comparison Schwadron and Cravens (2000) Ion this work Bei 03 Kra 04 Kra 06 Otr 06 O8+ 0.32 ± 0.03 0.13 ± 0.03 0.13 ± 0.05 0.15 ± 0.03 0.35 0.35 C6+ 1.4 ± 0.4 0.9 ± 0.3 0.7 ± 0.2 0.7 ± 0.2 1.02 1.59 C5+ 12 ± 4.0 11 ± 9 . . . 1.7 ± 0.7 1.05 1.05 N7+ 0.07 ± 0.06 0.06 ± 0.02 – – 0.03 0.03 N6+ 0.63 ± 0.21 0.5 ± 0.3 – – 0.29 0.29 Ne10+ 0.02 ± 0.01 – – – – – Ne9+ . . . – (15 ± 6) × 10−3 (20 ± 7) × 10−3 – – 73P/2006 (S.-W.3B). For example, removing the nitrogen com- ponents from linear S4’s cxe model and re-fitting, increases χ2R to over 7. χ2 contours for C/1999 S4 (linear) are presented in Fig 13. The line strengths for each ionic species are generally well con- strained, except where spectral features overlap. This can be readily seen when comparing the contours for the N vii 500 eV and O vii 561 eV features where a strong anti-correlation exists (Figure 12). Due to the limited resolution of acis an increase in the N vii feature will decrease the O vii strength. Similar anti- correlations exist between the nitrogen N vi or N vii and C v 299 eV lines. Since the line strength for the main line in each ionic species is linked to weaker lines, a range of energies can contribute and better constrain its strength. However with O vii as the strongest spectral feature the nitrogen and carbon compo- nents may be artificially lower as a result of the aforementioned anti-correlations. The lack of effective area due to the carbon edge in the acis response also may over-estimate the C v line flux. The neon features were well constrained for the brighter comets, but this is a region of lower signal and some caution must be taken when treating the neon line strengths and they are included here largely for completeness. In the case of 153P/Ikeya–Zhang, the χ2R > 1.4. The main discrepancy is that the model produces not enough flux in the 700 to 850 eV range compared to the observed spectrum. This may reflect an underestimation of higher O viii transitions or the presence of species not (yet) included in the model, such as Fe. This will be discussed further in the last section of this paper and in a separate paper dedicated to the observations of this comet (K. Dennerl, private communication). One of the best studied comets is C/1999 S4 (linear), be- cause of its good signal-to-noise ratio. To discuss our results, we will compare our findings with earlier studies of this comet. In general, the spectra analyzed here have more counts than ear- lier analyzes, because of improvements in the Chandra process- ing software and because we took special care to use a back- ground that is as comet-free as possible. Previous studies appear to have removed true comet signal when the background subtrac- tion was performed. In particular, both the Krasnopolsky (2004) and Lisse et al. (2001) studies used background regions from the outer part of the S3 chip and this may have still had true cometary emission. Krasnopolsky (2004) subtracted over 70% of the total signal as background. We find that using the S1-chip, the background contributes only 20% of the total counts. Different attempts to derive relative ionic abundances from C1999/S4’s X-ray spectrum are compared in Table 6. Our atomic physics based spectral analysis combines the benefits of ear- lier analytical approaches by Kharchenko and Dalgarno (2000, 2001); Beiersdorfer et al. (2003). These methods were all ap- plied to just one or two comets. Beiersdorfer et al. (2003) inter- pret C1999/S4’s X-ray spectrum by fitting it with 6 experimental spectra obtained with their ebit setup. The resulting abundances are very similar to ours. The advantage of their method is that it includes multiple electron capture, but in order to observe the forbidden line emission, the spectra were obtained with trapped ions colliding at CO2, at collision energies of 200 to 300 eV or ca. 30 km s−1. As was shown in Fig. 3, the cxe hardness ratio may change rapidly below 300 km s−1, implying an overesti- mation of the higher order lines compared to the n = 2 → 1 transition, which for O vii overlap with the O viii emission. We therefore find higher abundances of O8+. Krasnopolsky (2004, 2006) obtained fluxes and ionic abun- dances by fitting the spectrum with 10 lines of which the energies were semi-free. Their analysis thus does not take the contamina- tion of unresolved emission into account, and N vi and N vii are not included in the fit. The line energies were attributed to cxe lines of mainly solar wind C and O but also to ions of Mg and Ne. The inclusion of the resulting low energy emission (near 300 eV) results in lower C5+ fluxes (see also Otranto et al (2006)). There are several factors that may contribute to the unexpect- edly low C vi/C v ratios: 1) There may be a small contribution to the C v line from other ions in the 250-300 eV range (e.g. Si, Mg, Ne) that are currently not included in the model. Including these species in the model would lower the C v flux, but proba- bly only with a small amount. 2) The low acis effective area in the 250-300 eV region allows the C v flux to be unconstrained, and this increases the uncertainty in the C v flux. We estimate that the uncertainty in the effective area, introduced by the car- bon edge, can account for an uncertainty as large as a factor of 10 in the observed C v/C vi ratios. We will not compare our results with measured ace/swics ionic data. As discussed in section 4, the solar wind is highly variable in time and its composition can change dramatically over the course of less than a day. Variations in the solar wind’s ionic composition are often more than 50% during the course of an observation. Data on N, Ne, and O8+ ions have not been well documented as the errors of these abundances are dominated by counting statistics. As discussed above, latitudinal and corota- tional separations imply large inaccuracies in any solar wind mapping procedure. These conditions clearly disfavor modelling based on either average solar wind data or ace/swics data. 6. Comparative Results As noted in Section 5, spectral differences show up in the be- havior of the low energy C+N emission (< 500 eV), the O vii D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey 13 Table 7. Correlation between classification according to spectral shape and comet/solar wind characteristics during the observations. Comet families from Marsden & Williams (2005). Phase refers to where in the solar cycle the comet was observed, where 1 is the solar maximum and 0 the solar minimum of cycle #23’s descending phase. For other references, see Table 3. Class # Comet Comet Q Latitude Wind Type Family (1028 mol. s−1) cold H 73P/2006 (S.-W.3B) Jupiter 2 0.5 cir E 2P/2003 (Encke) Jupiter 0.7 11.4 Flare/PS warm F C/2001 Q4 (neat) unknown 10 -3 Quiet G 9P/2005 (Tempel 1) Jupiter 0.9 0.8 Quiet hot C C/2000 WM1 (linear) unknown 3-9 -34 PS A C/1999 S4 (linear) unknown 3 24 icme B C/1999 T1 (McNaught–Hartley) unknown 6-20 15 Flare/cir D C/2002 C1 (Ikeya–Zhang) Oort 20 26 icme Fig. 14. Flux ratios of all observed comets. The low energy C+N feature is anti-correlated to the oxygen ionic ratio. Letters refer to the chrono- logical order of observation. Fig. 15. Ion ratios of all observed comets. The C+N ionic abundantie is anti-correlated to the oxygen ionic ratio. Letters refer to the chronolog- ical order of observation. emission at 561 eV and the O viii emission at 653 eV. Figure 14 shows a color plot of the fluxes of these three emission features, and Figure 15 the corresponding ionic abundances. There is a clear separation between the two comets with a large C+N con- tribution and the other ‘oxygen-dominated’ comets, which on their turn show a gradual increase in the oxygen ionic ratio. This sample of comet observations suggest that we can distinguish two or three spectral classes. Table 7 surveys the comet parameters for the different spec- tral classes. The outgassing rate, heliocentric- or geocentric dis- tance and comet family do not correlate to the different classes, in accordance with our model findings. The data does suggest a correlation between latitude and wind conditions during the observations. At first sight, the apparent correlation between lat- itude and oxygen ratio seems paradoxical. According to the bi- modal structure of the solar wind the fast, cold wind dominates at latitudes > 15◦, implying less O viii emission. In Figure 9, the comet observations are shown with respect to the phase of the last solar cycle. Interestingly, we note that all comets that were observed at higher latitudes were observed around solar maximum. The solar wind is highly chaotic during solar maxi- mum and the frequency of impulsive events like CMEs is much higher than during the descending and minimum phase of the cycle. This explains both why the comets observed in the period 2000–2002 encountered a disturbed solar wind and why our sur- vey does not contain a sample of the cool fast wind from polar coronal holes. The observed classification can therefore be fully ascribed to solar wind states. The first class is associated with cold, fast winds with lower average ionization. These winds are found in cirs and behind flare related shocks. The spectra due to these winds are dominated by the low energy x-rays, because of the low abundances of highly charged oxygen. At the relevant tem- peratures, most of the solar wind oxygen is He-like O6+, which does not produce any emission visible in the 300–1000 eV regime accessible with Chandra. Secondly, there is an interme- diate class with two comets that were all observed during periods of quiet solar wind. These comets interacted with the equatorial, warm slow wind. The third class then comprises comets that in- teracted with a fast, hot, disturbed wind associated with icmes or flares. From the solar wind data, Ikeya–Zhang was proba- bly the most extreme example of this case. This comet had 10 times more signal than any other comet in our sample and small discrepancies in the response may be important at this level. Extending into the 1-2 keV regime, a preliminary analysis indi- cates the presence of bare and H-like Si, Mg and Fe xv-xx ions, in accordance with ace measurements of icme compositions Lepri & Zurbuchen (2004). The variability and complex nature of the solar wind allows for many intermediate states in between these three categories Zurbuchen et al. (2002), which explain the gradual increase of the O viii/O vii ratio that we observed in the cometary spec- tra. As the solar wind is a collisionless plasma, the charge state 14 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey Fig. 16. Spectrum derived ionic oxygen ratios and corresponding freezing-in temperatures from Mazotta et al. (1998). The shaded area indicates the typical range of slow wind associated with streamers. Letters refer to the chronological order of observation. distribution in the solar wind is linked to the temperature in its source region. Ionic temperatures are therefore a good indicator of the state of the wind encountered by a comet. The ratio be- tween O7+ and O6+ ionic abundances has been demonstrated to be a good probe of solar wind states. Zurbuchen et al. (2002) observed that slow, warm wind associated with streamers typi- cally lies within 0.1 < O7+/O6+ < 1.0, corresponding to freez- ing in temperatures of 1.3–2.1 MK. The corresponding temper- ature range is indicated in the Figure 16. In the figure, we show the observed O8+ to O7+ ratios and the corresponding freezing- in temperatures from the ionizational/recombination equilibrium model by Mazotta et al. (1998). Most observations are within or near to the streamer-associated range of oxygen freezing in tem- peratures. Four comets interacted with a wind significantly hot- ter than typical streamer winds, and in all four cases we found evidence in solar wind archives that the comets most likely en- countered a disturbed wind. 7. Conclusions Cometary X-ray emission arises from collisions between bare- and H-like ions (such as C, N, O, Ne, . . . ) with mainly water and its dissociation products OH, O and H. The manifold of depen- dencies of the cxe mechanism on characteristics of both comet and wind offers many diagnostic opportunities, which are ex- plored in the first part of this paper. Charge exchange cross sec- tions are strongly dependent on the velocity of the solar wind, and these effects are strongest at velocities below the regular wind conditions. This dependency might be used as a remote plasma diagnostics in future observations. Ruling out collisional opacity effects, we used our model to demonstrate that the spec- tral shape of cometary cxe emission is in the first place deter- mined by local solar wind conditions. Cometary X-ray spectra hence reflect the state of the solar wind. Based on atomic physic modelling of cometary charge ex- change emission, we developed an analytical method to study cometary X-ray spectra. First, the data of 8 comets observed with Chandra were carefully reprocessed to avoid the subtrac- tion of cometary signal as background. The spectra were then fit using an extensive data set of velocity dependent emission cross sections for eight different solar wind species. Although the limited observational resolution currently available hampers the interpretation of cometary X-ray spectra to some degree, our spectral analysis allows for the unravelling of cometary X-ray spectra and allowed us to derive relative solar wind abundances from the spectra. Because the solar wind is a collisionless plasma, local ionic charge states reflect conditions of its source regions. Comparing the fluxes of the C+N emission below 500 eV, the O vii emission and the O viii emission yields a quantitative probe of the state of the wind. In accordance with our modelling, we found that spectral differences amongst the comets in our survey could be very well understood in terms of solar wind conditions. We are able to distinguish interactions with three different wind types, being the cold, fast wind (I), the warm, slow wind (II); and the hot, fast, disturbed wind (III). Based on our findings, we pre- dict the existence of even cooler cometary X-ray spectra when a comet interacts with the fast, cool high latitude wind from polar coronal holes. The upcoming solar minimum offers the perfect opportunity for such an observation. Acknowledgements. DB and RH acknowledge support within the framework of the fom–euratom association agreement and by the Netherlands Organization for Scientific Research (nwo). MD thanks the noaa Space Environment Center for its post-retirement hospitality. We are grateful for the cometary ephemerides of D. K. Yeomans published at the jpl/horizons website. Proton velocities used here are courtesy of the soho/celias/pm team. soho is a mission of international cooperation between esa and nasa. 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Due to the low signal-to-noise ratio of the second detection, only the July 14th 2000 pre-breakup observation is discussed here. Summing the 8 pointings of the satellite gave a total time interval of 9390 s. In this period, the acis-S3 ccd collected a total of 11 710 photons were detected in the range 300–1000 eV. Detections out side this range or on other acis-ccds were not attributed to the comet. As a result, data from the S1-ccd (which is configured identically to S3) may be used as an indicator of the local X-ray background. The morphology can be described by a crescent shape, with the maximum brightness point 24 000 km from the nucleus on the Sun-facing side. The brightness dims to 10% of the maximum level at 110 000 km from the nucleus. Solar wind. A large velocity jump can be seen around DoY 199, which was due to the famous ”Bastille Day” flare on 14 July (FF#153, Dryer et al (2001); Fry et al. (2003)). This flare reached the comet only after the first observation. At July 12, 2017UT a solar flare started at N17W65 (FF#152), which was nicely placed to hit this comet with a very high probability dur- ing the first observations Fry et al. (2003). As for the second observation, there was another flare on July 28, S17E24, at 1713 UT (FF#164) and there was a high probability that its shock’s weaker flank hit the comet. A.2. C/1999 T1 (McNaught–Hartley) X-rays. The allocated observing time of comet McNaught– Hartley was partitioned into 5 one-hour-slots between January 8th and January 15th, 2001 Krasnopolsky et al. (2002). The strongest observing period was on January 8th, when ∆ = 1.37 AU and rh = 1.26 AU. There were 15 000 total counts observed by the acis-S3 ccd between 300 and 1000 eV. The emission region can be described by a crescent, with the peak brightness is at 29 000 km from the nucleus. The brightness dims to 10% of the maximum at a cometocentric distance of 260 000 km. Again, the acis-S1 ccd may be used to indicate the local background signal. Solar wind. The comet was not within the heliospheric cur- rent/plasma sheet (HCS/HPS). Two corotating cirs are probably associated with the first two observations. Two flares (FF#233 and #234) took place; however, another corotating cir more likely arrived before the flare’s transient shock’s effects did McKenna-Lawlor et al. (2006). A.3. C/2000 WM1 (linear) X-rays. The only attempt to use the high-resolution grat- ing capability of the acis-S array was made with comet C/2000 WM1 (linear). Here, the Low-Energy Transmission Grating (letg) was used. The dimness of the observed X-rays, and the extended nature of the emitting atmosphere meant that the grated spectra did not yield significant results. It is still 16 D. Bodewits et al.: Spectral Analysis of the Chandra Comet Survey possible to extract a spectrum based on the pulse-heights gener- ated by each X-ray detection on the acis-S3 chip, although the morphology is not recorded. 6300 total counts were recorded for the pulse-height spectrum of the S3 chip in the 300 to 1000 eV range. Solar wind. Comet WM1 was observed at the highest latitude available within this survey, and at a latitude of 34 degrees, it was far outside the hcs. During the observations, this comet might have experienced the southerly flank of the shock of a strong X3.4 flare at S20E97 and its icme and shock on December 28, 2001 (FF#359) McKenna-Lawlor et al. (2006). A.4. 153P/2002 (Ikeya–Zhang) X-rays. The brightest X-ray comet in the Chandra archive is 153P/2002 (Ikeya–Zhang). The heliographic latitude, geocentric distance and heliocentric distance were comparable to those for comet C/1999 S4 (linear), with a latitude of 26◦, ∆ = 0.457 AU and rh = 0.8 AU. Rather than periodically re-point the detector to track the comet, the pointing direction was fixed and the comet was monitored as it passed through the field of view, thus increasing the effective FoV. There were two observing periods on April 15th 2002, each lasting for approximately 3 hours and 15 minutes. In both periods, a strong cometary signal is detected on all of the activated acis-ccds. Consequently, a background signal cannot be taken from the observation. A crescent shape on the Sun side of the comet is observed over all of the ccd array. Over 200 000 total counts were observed from the S3 chip in the 300 to 1000 eV range. The time intervals for each observing period are 11 570 and 11 813 seconds. Solar wind. Like C/2000 WM1, this comet was observed at a relatively high heliographic latitude. Solar wind data obtained in the ecliptic plane can therefore not be used to determine the wind state at the comet. 153P/2002 (Ikeya–Zhang) was well- positioned during the first observation on 15 April 2002 for a flare at N16E05 (FF#388) on 12 April 2002. During the second observation on 16 April, there was an earlier flare on 14 April at N14W57, but this flare was probably too far to the west to be ef- fective McKenna-Lawlor et al. (2006). The comet was observed at a high latitude, and hence ace solar wind data is most likely not applicable. A.5. 2P/2003 (Encke) X-rays. The Chandra observation of Encke took place on the 24th of November 2003 Lisse et al. (2005), when the comet had a heliocentric distance of rh = 0.891 AU and a geocentric distance of ∆ = 0.275 AU and a heliographic latitude of 11.4 degrees. The comet was continuously tracked for over 15 hours, resulting in a useful exposure of 44 000 seconds. The acis-S3 ccd counted 6140 X-rays in the range 300–1000 eV. The brightest point was offset from the nucleus by 11 000 km, dimming to 10% of this value at a distance of 60 000 km. The acis-S1 ccd was not activated in this observation. The low quantum efficiency of the other activated ccds below 0.5 keV makes them unsuitable as background references. Solar wind. The proton velocity decreased during observations from 600 km s−1 to 500 km s−1. A flare on 20 November 2003, at N01W08 (FF#525), was well-positioned to affect the observations on 23 November (data from work in progress by Z.K. Smith et al.). The comet most likely interacted with the overexpanded, rarified plasma flow that followed the earlier hot shocked and compressed flow behind the flare’s shock. A.6. C/2001 Q4 (neat) X-rays. A short observation of comet C/2001 Q4 was made on May 12 2004, when the geocentric and heliocentric distances were ∆ = 0.362 AU and rh = 0.964 AU respectively. With a heliographic latitude of 3 degrees, the comet was almost in the ecliptic plane. From 3 pointings, the useful exposure was 10 328 seconds. The acis-S3 chip detected 6540 X-rays in be- tween 300 and 1000 eV. The acis-S1 was used as a background signal. Solar wind. There was no significant solar activity during the observations (Z.K. Smith et al., ibid.). From solar wind data, the comet interacted with a quiet, slow 352 km s−1 wind. A.7. 9P/2005 (Tempel 1) X-rays. The observation of comet 9P/2005 (Tempel 1) was de- signed to coincide with the Deep Impact mission Lisse et al. (2007). The allocated observation time of 291.6 ks was split into 7 periods, starting on June 30th, July 4th (encompassing the Deep Impact collision), July 5th, July 8th, July 10th, July 13th and July 24th. The brightest observing periods were June 30th and July 8th. The focus here is on the June 30th observation. On this date, rh = 1.507 AU and ∆ = 0.872 AU. The useful exposure was 50 059 seconds, with a total of 7300 counts, 4000 from the June 30th flare alone, were detected in the energy range of 300–1000 eV. The brightest point for the June 30th observation was located 11 000 km from the nucleus. The morphology appears to be more spherical than in other comet observations. Solar wind. Observations were taken over a long time span cov- ering different solar wind environments. There was no signifi- cant solar activity during the 30 June 2005 observations (Z.K. Smith et al., ibid. Lisse et al. (2007)). From the ace data, it can be seen that at June 30, the comet most likely interacted with a quiet, slow solar wind. A.8. 73P/2006 (Schwassmann–Wachmann 3B) X-rays. The close approach of comet 73P/2006 (Schwassmann– Wachmann 3B) in May 2005 (∆ = 0.106 AU, rh = 0.965 AU) provided an opportunity to examine cometary X-rays in high spatial resolution. Chandra was one of several X-ray missions to focus on one of the large fragments of the comet. Between 300 and 1000 eV, 6285 counts were obtained in a useful exposure of 20 600 seconds. Solar wind. There was a weak flare on 22 May 2006 (FF#655, Z.K. Smith, priv. comm.). A sequence of three high speed coro- nal hole streams passed the comet in the period around the ob- servations and a corotating cir might have reached the comet in association with the observations on 23 May, which is confirmed by the mapped solar wind data.

0704.1649

Hamiltonian formalism in a problem of 3-th waves hierarchy

arXiv:0704.1649v1 [hep-lat] 12 Apr 2007 Hamiltonian formalism in a problem of 3-th waves hierarchy A. N. Leznov Abstract By the method of discrete transformation equations of 3-th wave hier- archy are constructed. We present in explicit form two Poisson structures, which allow to construct Hamiltonian operator consequent application of which leads to all equations of this hierarchy. For calculations it will be necessary results of previous paper [1], which for convenience of the reader we present in corresponding place of the text. The obtained formulae are checked by independent calculations. 1 Introduction All system equations of 3-th wave hierarchy are invariant with respect to two mutually commutative discrete transformation of this problem [1],[3],[5],[6][1]. In this introduction we present the solution of the same problem in the case A1 algebra follow to the paper [2]. We repeat here briefly the most important punks of general construction from [2]. The discrete invertible substitution (mapping) defined as ũ = T (u, u′, ..., ur) ≡ T (u) (1) u is s dimensional vector function; ur its derivatives of corresponding order with respect to ”space” coordinates. The property of invertibility means that (1) can be resolved and ”old” func- tion u may expressed in terms of new one ũ and its derivatives. Freshet derivative T ′(u) of (1) is s× s matrix operator defined as T ′(u) = Tu + Tu′D + Tu′′D 2 + ... (2) where Dm is operator of m-times differentiation with respect to space coordi- nates. ∗Universidad Autonoma del Estado de Morelos, CCICAp,Cuernavaca, Mexico http://arxiv.org/abs/0704.1649v1 Let us consider equation Fn(T (u)) = T ′(u)Fn(u) (3) where Fn(u) is s-component unknown vector function, each component of which depend on u and its derivatives not more than n order. It is not difficult to understand that evolution type equation ut = Fn(u) is invariant with respect substitution (1). Two other equations and its solutions are important in what follows T ′(u)J(u)(T ′(u))T = J(T (u)), T ′(u)H(u)(T ′(u))−1 = H(T (u)) (4) where (T ′(u))T = T Tu −DT u′ +D 2T Tu′′ + ... and J(u), H(u) are unknown s× s matrix operators, the matrix elements of which are polynomial of some finite order with respect to operator of differentiation (of its positive and negative degrees). JT (u) = −J(u) may be connected with the Poisson structure and equation (4) means its invariance with respect to discrete transformation T . The second equation (4) determine operator H(u), which after application to arbitrary solution of (3) F (u) leads to new solution of the same system F̃ (u) = H(u)F (u) And thus we obtain reccurent procedure to construct solutions of (3) from few simple ones. If it is possible to find two different J1, J2 (Hamiltonian operators,Poisson structures) then H(u) = J2J 1 (5) satisfy second equation (4). In [2] are presented arguments that Hamiltonian operator it is the sense find in a form J(u) = Fn(u)D −1Fn(u) i (6) where Fn some solution of (3) and Ai some s × s matrices constructed from u and its derivatives. The direct generalization of (7) which will be used below is the following J(u) = Fi(u)D −1Fi(u) i (7) where first term in (6) is changed on sum of some number of different solutions of (3). 2 Necessary facts from [1] In [1] was constructed equations of 3-th waves hierarchy of zero, first and second order for six unknown functions f±1.0, f 0.1, f 1.1. The form of these equations will be essentially used in what follows. 2.1 Equations of the zero order 1.0 = ±bf 0.1 = ±cf 1.1 = ±af 1.1 (8) where (b, c) arbitrary numerical parameters a = b+ c. 2.2 Equations of the first order 1.1 = θ1.1(f ′ + σ1.1f 1.0 = ν1.0(f ′ + σ1.0f 0.1 = ν0.1(f ′ + σ0.1f 0.1 = ν0.1(f ′ + σ0.1f 1.1 (9) 1.0 = ν1.0(f ′ + σ1.0f 0.1(f 1.1 = θ1.1(f ′ + σ1.1f where νi,j , σij , θ11 are numerical parameters connected by condition ν01 − ν10 = 2σ10, ν01 + ν10 = 2θ11, −2σ11 = σ10 = σ01 (10) Thus solution is defined by two independent parameters ν01, ν10 as in the case of zero degree solution of the previous subsection. 2.3 Equations of the second order 1.1 = ν1.1(f ′′ + γ1.1f 1.0(f ′ + δ1.1f 0.1(f ′ + f+1.1R11, where a ≡ (b+ c), Rij ≡ 2aijf 1.1 + bijf 1.0 + cijf 1.0 = ν1.0(f ′′ + γ1.0f 0.1(f ′ + δ1.0f 1.1(f ′ + f+1.0R10 0.1 = ν0.1(f ′′ + γ0.1f 1.0(f ′ + δ0.1f 1.1(f ′ + f+0.1R01, − ˙f−0.1 = ν0.1(f ′′ + γ0.1f 1.0(f ′ + δ0.1f 1.1(f ′ + f−0.1R01, − ˙f−1.0 = ν1.0(f ′′ + γ1.0f 0.1(f ′ + δ1.0f 1.1(f ′ + f−1.0R10, − ˙f−1.1 = ν1.1(f ′′ + γ1.1f 1.0(f ′ + δ1.1f 0.1(f ′) + f−1.1R11. All numerical parameters in (11) may be expressed in terms of only two ones and are connected by relations ν11 = a11, γ1.1 + δ1.1 = (b11 − c11), ν10 = 2b10 γ1.0 + δ1.0 = 2(c10 − b10), ν01 = 2c01, γ0.1 + δ0.1 = 2(c01 − b01) a11 = −2c10 b11 = a10 c11 = −3c10 − b10 a10 = c10 + b10 b10 = b10 c10 = c10 a01 = −3c10 − b10 b01 = c10 c01 = −b10 − 4c10  (12) δ10 = 4c10, γ10 = −2(c10 + b10), 2γ11 = δ10 − γ10, 2δ11 = −γ10, δ01 = −δ10, γ01 = γ10 − δ10 2.4 Hamiltonian form of equations As it was shown in (1) equations of the previous subsections may be considered as Hamiltonian ones with following non zero Poisson breakets {f+1.1, f 1.1} = , {f+1.0, f 1.0} = 1, {f 0.1, f 0.1} = 1 (13) This fact leads to existence of the first Poisson structure, inverse to which is the following 0 0 0 0 0 −2 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 3 Hamiltonian operator of 3-th waves problem Let us seek in connection the proposition (7) of introduction the second Poisson structure in a form J2 = J 2 + J 2 , where J 2 contain terms with D −1 and J terms with non negative degree of D 2 = −F −1(F 10 ) T − F 20D −1(F 20 ) where F 0 are two different solutions of equations of the zero order (first sub- section of the previous section). 0 0 0 − 1 0 0 f+1.1 0 D − 0 −f+1.1 0 D 0 1.0 0 D 0 −f 1.1 0 0.1 D 0 f 1.1 0 0 0.1 − 1.0 0 0 0 In the last expression we present finally result. Really it is necessary to write anti symmetrical matrix with arbitrary coefficients, which will be found after calculations described below. Now let us consider how reccurent operator H (5) acts on some solution of (3) F . At first F it is necessary multiply on J−11 with the result 1 F = −2F−1.1 −F−1.0 −F−0.1 2F+1.1 and this column vector multiply on J2 = J 2 from (15). In two terms of J it is necessary multiply vector line (F i0) T on the last vector column with scalar result (F i0) 1 F = −2a i(f+1.1F 1.1+f 1.1)−b i(f+1.0F 1.0+f 1.0)−c i(f+0.1F 0.1+f Thus input of two first terms of Jn2 into ”new solution” will be 1.1 = −f −1(2a2(f+1.1F 1.1+f 1.1)+(ab)(f 1.0+f 1.0)+(ac)(f 0.1+f 0.1)) 1.0 = −f −1(2(ba)(f+1.1F 1.1+f 1.1)+b 2(f+1.0F 1.0+f 1.0)+(bc)(f 0.1+f 0.1)) 0.1 = −f −1(2(ca)(f+1.1F 1.1+f 1.1)+(cb)(f 1.0+f 1.0)+c 2(f+0.1F 0.1+f 0.1)) and the same expressions with opposite sign for components with negative upper indexes. a2 = aiai, (ab) = aibi and so on. The result of multiplication J 1 F is determined by usual rules of multi- plication matrix on vector 1 F = (F+1.1) ′ + 1 (f+0.1F 1.0 − f (F+1.0) ′ − f+1.1F 0.1 − (F+0.1) ′ + f+1.1F 1.0 + −(F−0.1) ′ − f−1.1F 1.0 − −(F+1.0) ′ + f−1.1F 0.1 + (F−1.1) ′ − 1 (f−0.1F 1.0 − f Now let us take for F right hand side zero degree equations (subsection 1 from previous section) F±1.1 = ±(ν1.0 + ν0.1)f 1.1, F 1.0 = ±ν1.0f 1.0, F 0.1 = ±ν0.1f (b = ν1.0, c = ν0.1). In this case input from J 2 terms (16) equal to zero and input from J 2 terms exactly coincides with right hand side of equations of the first order (subsection 2 from previous section). Really calculations must be done in a back direction: in definition of J 2 (15) it is necessary to use arbitrary skew symmetrical matrix and after comparison result of calculations above with first order equations obtain finally form J 2 (15). Now we repeat the same trick with equations of the first degree. In this case 1 F = −(ν1.0 + ν0.1)(f ′ − ν1.0−ν0.1 −ν1.0(f ′ + ν1.0−ν0.1 −ν0.1(f ′ + ν1.0−ν0.1 ν0.1(f ′ − ν1.0−ν0.1 ν1.0(f ′ − ν1.0−ν0.1 (ν1.0 + ν0.1)(f ′ + ν1.0−ν0.1 We present result of action J 1 F below ν1.0 + ν0.1 (f+1.1) 3ν1.0 − ν0.1 (f+1.0) 0.1 + ν1.0 − 3ν0.1 1.0)(f ν1.0 − ν0.1 1.1(f 1.0 − f 0.1) (18) The terms in the first line exactly coincide with terms with derivatives in equa- tion of the second order for f+1.1 component. Indeed (ν1.0+ν0.1) = 2b10+2c01 = −8c10 = 4a11 = 4ν11, b10 = , c10 = − ν1.0+ν0.1 , δ11 = b10 + c10 = 3ν1.0−ν0.1 and so on. The same situation takes place with respect all other components: terms with derivatives coincide with calculated from J 1 F . Terms without derivatives arise from terms of the second line of (18) and from (16). In the last equations after substitution F from equation of the first order (and in all other cases) under the sign D−1 arises sign D which lead to unity and (16) plus terms of second line of (18) look as 1.1(2θ11a 1.1+[(ab)ν10+ ν1.0 − ν0.1 ]f+1.0f 1.0+[(ac)ν01− ν1.0 − ν0.1 ]f+0.1f 1.0([2(ba)θ11+ ν1.0 − ν0.1 1.1+b 2ν10f 1.0+[(bc)ν01− ν1.0 − ν0.1 ]f+0.1f 0.1([2(ca)θ11− ν1.0 − ν0.1 ]f+1.1f 1.1+[(cb)ν10+ ν1.0 − ν0.1 ]f+1.0f 1.0+c All terms above with scalar products arise from (16). All others from the ”second line” of (18). Now it is necessary compleat these expressions with terms f+ijRij of the right hand side of equation of the second order (subsection 3 of the previous section). This comparison leads to the following conclusion: a2 = b2 = c2 = , (ab) = (ac) = −(bc) = And now recurrent operator H is defined uniquely. From this result it is clear that all attempts to construct H using only one solution in the anzats for J2 lead to contradiction and it was the main difficult to the author. The following observation take place. Let us consider three elements of Cartan subalgebra R3 = h1+h2 , R2 = h2, R1 = h1 and the positive root system of A2 algebra system X 3 = X 12, X 2 , X 1 . Let us define 3× 3 ”Cartan matrix” KW by the condition [Ri, X j ] = K a2 (ab) (ac) (ba) b2 (bc) (ca) (cb) c2 4 Hamiltonian formalism II In calculations of the previous section results of [1] were used in the whole measure. But the corresponding calculations were not simple and not strait forward. Now having the explicit expression for J2 we are able to check that equation it defined (4) is satisfied. Let us rewrite it notations of the previous section −T ′(f)F 10D −1(F 10 ) T (T ′(f))T − T ′(f)F 20D −1(F 20 ) T (T ′(f))T+ T ′(f)J ′(f))T = (Jn2 + J 2 )(T (f)) (20) ( we think that the same sign T for substitution and transposition will not lead to mixing). We will do all calculations below with respect to T3 transformation (explicit formulae for it and F ′3(f) reader can find in Appendix). We remind the rule of multiplication of quadratical in derivatives operator on scalar function (A+DB+DC2)R = AR+BRA′+CRT ′′+(BRA+2CRT ′)D+CDR2 (21) Matrix elements of 3 first lines of F ′3(f) do not contain operator of differen- tiation. Its fourth and fifth lines linear in D and its sixth one quadratical in D. By definition (3) all terms without operator D lead to F i0 . All others can be simple calculated using (21) with the result T ′3(f)F F i0 + 1.0D − a F i0 + Γi (22) where ∆i = ci − bi, ai = ci + bi, 03- three dimensional zero vector After substi- tution (22) into (20) and cancelation equivalent terms in both sides we come to the following equality have to be checked Γi D−1 Γi D−1 Γi T )+ T ′3(f)J 3(f)) T = J 2 (T3(f)) (23) The matrix of the first sum in the first line has different from zero only elements of its last three columns. The second sum - only elements of its three last line. And the last sum only elements of 3× 3 matrix in its left down corner. At first let us calculate the first sum. Result is the following (we present below only two first lines of this matrix operator) F i0 D 03 0 0 D + 1 0.1D −  (24) (where 03 is three dimensional row vector) (T ′3(f)J 3(f)) T )1l = (T ′3(f))1m(J 2 )mk((T 3(f)) T )kl = (T ′3(f))16(J 2 )6k(T t)lk = (because only one elements of the first line ((T ′3(f))16) of Frechet derivative matrix and three elements of sixth line matrix J 2 are different from zero). (f−1.1) D(T ′3(f)) 0.1(T 3(f)) 1.0(T 3(f)) where (kD)t ≡ −Dkt. or the first line of the matrix T ′3(f)J 3(f)) T looks as 0.1 − D − 1 Now let us calculate second line (T ′3(f)J 3(f)) T )2l = (T ′3(f))2m(J 2 )mk((T 3(f)) T )kl = (T ′3(f))26(J 2 )6k(T t)lk + (T ′3(f))24(J 2 )4k(T t)lk = (because only two elements (T ′3(f))24, (T 3(f))26 of second line of Frechet deriva- tive matrix are different from zero ) (f−1.1) D(T ′3) 0.1(T 1.0(T 1.0(T l1+D(T 1.1(T Result of simple algebraical calculation lead to explicit form of second row 1.1 − D + 3 0.1D + 1.0 + [(f−0.1) After summation (25) and (25) with corresponding lines of (24) we obtain exactly first two lines of left hand side of (23). By similar calculation it is possible to check that (23) is satisfied. Finally expression for Jn2 27) and (15) solve the problem of the construction of the second Poisson struc- ture for the equations of 3-th waves hierarchy. 5 Comments about multi-soliton solutions of the systems of 3-th waves hierarchy Let us find solution of the system equations of the second order (subsection 3 of section 2) under additional condition f+i.j = 0. The system under consideration looks as − ˙f−0.1 = ν0.1(f ′′, − ˙f−1.0 = ν1.0(f − ˙f−1.1 = ν1.1(f ′′ + γ1.1f 1.0(f ′ + δ1.1f 0.1(f where ν11 = ν10+ν01 , δ1.1 = 3ν10−ν01 , γ1.1 = ν10−3ν01 Solution of two first linear equation are obvious 0.1 = dµe−ν01tµ 2+µxq(µ), f−1.0 = dλe−ν10tλ 2+λxp(λ) Let us find partial solution on nonhomogineous third equation in a form 1.1 = −(ν01µ 2+ν10λ 2)t+(µ+λ)x r(µ, λ) After substitution this anzats for f−1.1 and obtained above solutions for f 1.0, f into third equation we come to an equality [ν01µ 2 + ν10λ ν10 + ν01 (µ+ λ)2]r(µ, λ) = (γ1.1µ+ δ1.1λ)q(µ)p(λ) Quadratical multiplier of the left side is equal to 2(λ − µ)(γ1.1µ + δ1.1λ) and finally we obtain r(µ, λ) = q(µ)p(λ) 1.1 = dλe−(ν01µ 2+ν10λ 2)t+(µ+λ)x 1 q(µ)p(λ) The last expression coincides (up to nonessential multiplier 1 ) with solution in the case of 3-th problem. But in resolving of equations of discrete transfor- mation only differentiation with respect to space coordinate take place. Thus all equations of discrete transformation in the case of 3-th wave problem and in the case under consideration will be have the same solution except of time dependent multiplier. And the form of multi soliton solution (up to this factor) will be the same for all systems of 3-th waves hierarchy. 6 Outlook The main result of the present paper the explicit expression for second Pois- son structure (15) and (27), which allow to construct Hamiltonian reccurent operator and obtain all equations of 3-th wave hierarchy in explicit form. From the physical point of view these systems may be considered as three interacting fields of nonlinear Schredinger hierarchy connected with A1 algebra. All equa- tions depended on one arbitrary numerical parameter, which can be connected with parameters of the particles of the fields describing by nonlinear Schredinger hierarchy. The discrete transformation for n-wave problem in the case of arbitrary semisimple algebra was presented in [4] form and author have no doubts that the problem of equations of n-wave hierarchy and its multi-soliton solution may be resolved in explicit form. The most riddle to the author remain question about the nature of the group of discrete transformation. As it follows from its introduction [4] it has some connection with the Weil group of the root space of semisimple algebra. Weil group is discrete one but non commutative. Discrete transformation in the case of the present paper is some reduction from group of discrete transformation of four dimensional self-dual Yang-Mills equations [9]. And thus understanding this situation in the case n-waves interaction will give possible guess to solution of this problem in Yang-Mills case. Aknowledgements The author thanks CONACYT for financial support. 7 Appendix We present here different from zero matrix elements of Frechet derivative for T3 discrete transformation - 6× 6 matrix operator (see Introduction). All calcula- tions are done in connection with its definition (2) and explicit formulae for T3 discrete transformation presented below. 7.1 Discrete transformation T3 1.0= − 0.1= −2(f ′ − f−1.1f 1.0 − (f−0.1) (f−1.1) 1.0= −2(f ′ + f−1.1f 0.1 + (f−1.0) (f−1.1) 1.1= f 1.1+(ln f (f−0.1) 0.1 − (f (f+1.0f 1.0+f 0.1)+ 7.2 Frechet derivative We present below different from zero matrix elements of Frechet operator F ′16= − (f−1.1) F ′24= − F ′26= (f−1.1) F ′35= F ′36= − (f−1.1) F ′42= −f F ′44= −2D− Z + (f−1.1) F ′45= − (f−0.1) 2f−1.1 F ′46= −f 1.0 + 2f−1.1 0.1(f (f−1.1) 53= f (f−1.0) 2f−1.1 55= −2D+ Z + (f−1.1) F ′56= f 0.1 − 2f−1.1 1.0(f (f−1.1) F ′61= (f F ′62= F ′63= F ′64= (f−1.0D−(f 1.0Z+ F ′65= − (f−0.1D − (f 0.1Z + F ′66= D 2 − 2 (f−1.1) (f−1.1) ′(f−1.1) + 2f−1.1f 1.1 + (f−1.0f 1.0 + f 0.1)− where Z = References [1] A.N.Leznov , Equations of 3-th wave hierarchy arXiv:math-ph/0703063 [2] Derjagin V.B. and A.N.Leznov Preprint MPI 96-3 Discrete Symmetries and and multi-Poisson structures of 1+1 integrable systems [3] A.N.Leznov and R.Torres-CordobaJ.Math.Phys 44(5):2342-2352(2003) A.N.Leznov, J.Escobedo-Alatrore and R.Torres-Cordoba J.Nonlinear Math.Phys 10(2):243-251(2003) [4] A.N.Leznov, Discrete symmetries of the n-wave problem, Theoretical and mathematical physics, 132(1): 955-969 (2002) [5] A.N.Leznov, G.R.Toker and R.Torres-Cordoba, Multisoliton solution of 3-th wave problemhep-th/060500906 [6] A.N.Leznov, G.R.Toker and R.Torres-Cordoba, Resolving of discrete trans- formation and multisoliton solution of 3-th wave problem Non.Lin.Math- Phys. v 14 N2 238-249 (2007) [7] A.N.Leznov and M.V.Saveliev Group theoretical methods for integration of nonlinear dynamic systems. Birchoiser, 1992 [8] A.N.Leznov Theoretical and mathematical physics, +122(2): 211-228 (1998) [9] A.N. Leznov Discrete and backlund (!) transformations of SDYM system. math-ph/0504004

0704.1650

Correlations, fluctuations and stability of a finite-size network of coupled oscillators

Correlations, fluctuations and stability of a finite-size network of coupled oscillators Michael A. Buice and Carson C. Chow Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD (Dated: November 19, 2018) The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e. when the population is not synchronous. This demonstration is facilitated by the construction of a non-equilibrium statistical field theoretic formulation of a generic model of coupled oscillators. This theory is consistent with previous results. In the all-to-all case, the fluctuations in this theory are due completely to finite size corrections, which can be calculated in an expansion in 1/N , where N is the number of oscillators. The N → ∞ limit of this theory is what is traditionally called mean field theory for the Kuramoto model. I. INTRODUCTION Systems of coupled oscillators have been used to de- scribe the dynamics of an extraordinary range of phe- nomena [1], including networks of neurons [2, 3], syn- chronization of blinking fireflies [4, 5], chorusing of chirp- ing crickets [6], neutrino flavor oscillations [7], arrays of lasers [8], and coupled Josephson junctions [9]. A com- mon model of coupled oscillators is the Kuramoto model [10], which describes the evolution of N coupled oscilla- tors. A generalized form is given by θ̇i = ωi + f(θj − θi) (1) where i labels the oscillators, θi is the phase of oscillator i, f(θ) is the phase dependent coupling, and the intrin- sic driving frequencies ωi are distributed according to some distribution g(ω). In the original Kuramoto model, f(θ) = sin(θ). Here, we consider f to be any smooth odd function. The system can be characterized by the complex order parameter Z(t) = eiθj(t) ≡ r(t)eiΨ(t) (2) where the magnitude r gives a measure of synchrony in the system. In the limit of an infinite oscillator system, Kuramoto showed that there is a bifurcation or continuous phase transition as the coupling K is increased beyond some critical value, Kc [10]. Below the critical point the steady state solution has r = 0 (the “incoherent” state). Be- yond the critical point, a new steady state solution with r > 0 emerges. Strogatz and Mirollo analyzed the lin- ear stability of the incoherent state of this system us- ing a Fokker-Planck formalism [11]. In the absence of external noise, the system displays marginal modes as- sociated with the driving frequencies of the oscillators. However, numerical simulations of the Kuramoto model for a large but finite number of oscillators show that the oscillators quickly settle into the incoherent state below the critical point. The paradox of why the marginally stable incoherent state seemed to be an attractor in sim- ulations was partially resolved by Strogatz, Mirollo and Matthews [12] who demonstrated (within the context of the N → ∞ limit) that there was a dephasing effect akin to Landau damping in plasma physics which brought r to zero with a time constant that is inversely proportional to the width of the frequency distribution. Recently, Stro- gatz and Mirollo have shown that the fully locked state r = 1 is stable [13] but the partially locked state is again marginally stable [14]. Although dephasing can explain how the order parameter can go to zero, the question of whether the incoherent state is truly stable for a finite number of oscillators remains unknown. Even with de- phasing, in the infinite oscillator limit the system still has an infinite memory of the initial state so there may be classes of initial conditions for which the order parameter or the density exhibits oscillations. The applicability of the results for the infinite size Kuramoto model to a finite-size network of oscillators is largely unknown. The intractability of the finite size case suggests a statistical approach to understanding the dynamics. Accordingly, the infinite oscillator theories should be the limits of some averaging process for a finite system. While the behavior of a finite system is expected to converge to the “infinite” oscillator behavior, for a fi- nite number of oscillators the dynamics of the system will exhibit fluctuations. For example, Daido [15, 16] consid- ered his analytical treatments of the Kuramoto model using time averages and he was able to compute an ana- lytical estimate of the variance. In contrast, we will pur- sue ensemble averages over oscillator phases and driving frequencies. As the Kuramoto dynamics are determin- istic, this is equivalent to an average over initial phases and driving frequencies. Furthermore, the averaging pro- cess imparts a distinction between the order parameter Z and its magnitude r. Namely, do we consider 〈Z〉 or 〈r〉 = 〈|Z|〉 to be the order parameter? This is important as the two are not equal. In keeping with the density as the proper degree of freedom for the system (as in the in- finite oscillator theories mentioned above), we assert that 〈Z〉 is the natural order parameter, as it is obtained via a linear transformation applied to the density. Recently, Hildebrand et al. [17] produced a kinetic theory inspired by plasma physics to describe the fluctu- ations within the system. They produced a Bogoliubov- Born-Green-Kirkwood-Yvon (BBGKY) moment hierar- http://arxiv.org/abs/0704.1650v3 chy and truncation at second order in the hierarchy yielded analytical results for the two point correlation function from which the fluctuations in the order param- eter could be computed. At this order, the system still manifested marginal modes. Going beyond second or- der was impractical within the kinetic theory formalism. Thus, it remained an open question as to whether going to higher order would show that finite size fluctuations could stabilize the marginal modes. Here, we introduce a statistical field theory approach to calculate the moments of the distribution function gov- erning the Kuramoto model. The formalism is equivalent to the Doi-Peliti path integral method used to derive sta- tistical field theories for Markov processes, even though our model is fully deterministic [18, 19, 20, 21]. The field theoretic action we derive produces exactly the same BBGKY hierarchy of the kinetic theory approach [17]. The advantages of the field theory approach are that 1) difficult calculations are easily visualized and performed through Feynman graph analysis, 2) the theory is eas- ily extendable and generalizable (e.g. to local coupling), 3) the field theoretic formalism permits the use of the renormalization group (which will be necessary near the critical point), and 4) in the case of the all-to-all ho- mogeneous coupling of the Kuramoto model proper, the formalism results in an expansion in 1/N and verifies that mean field theory is exact in the N → ∞ limit. We will demonstrate that this theory predicts that finite size corrections will stabilize the marginal modes of the infi- nite oscillator theory. Readers unfamiliar with the tools of field theory are directed to one of the standard texts [22]. A review of field theory for non-equilibrium dynam- ics is [23]. In section II, we present the derivation of the the- ory and elaborate on this theory’s relationship to the BBGKY hierarchy. In section III, we describe the com- putation of correlation functions in this theory and, in particular, describe the tree level linear response. This will connect the present work directly with what was computed using the kinetic theory approach [17]. In sec- tion IV, after describing two example perturbations, we calculate the one loop correction to the linear response and demonstrate that the modes which are marginal at mean field level are rendered stable by finite size effects. In addition, we demonstrate how generalized Landau damping arises quite naturally within our formalism. We compare these results to simulations in section V. II. FIELD THEORY FOR THE KURAMOTO MODEL The Kuramoto model (1) can be described in terms of a density of oscillators in θ, ω space η(θ, ω, t) = δ(θ − θi(t))δ(ω − ωi) (3) that obeys the continuity equation C(η) ≡ f(θ′ − θ)η(θ′, ω′, t)η(θ, ω, t)dθ′dω′ = 0 (4) Equation (4) remains an exact description of the dynam- ics of the Kuramoto model (1) [17]. Although equa- tion (4) has the same form as that used by Strogatz and Mirollo [24], it is fundamentally different because solutions need not be smooth. Rather, the solutions of Eq. (4) are treated in the sense of distributions as defined by Eq. (3). As we will show, imposing smooth solutions is equivalent to mean field theory (the infinite oscilla- tor limit). Drawing an analogy to the kinetic theory of plasmas, Eq. (4) is equivalent to the Klimontovich equa- tion while the mean field equation used by Strogatz and Mirollo [24] is equivalent to the Vlasov equation [25, 26]. Our goal is to construct a field theory to calculate the response functions and moments of the density η. Even- tually we will construct a theory akin to a Doi-Peliti field theory [18, 19, 20, 21], a standard approach for reaction- diffusion systems. We will arrive at this through a con- struction using a Martin-Siggia-Rose response field [27]. Since the model is deterministic, the time evolution of η(θ, ω, t) serves to map the initial distribution forward in time. We can therefore represent the functional proba- bility measure P [η(θ, ω, t)] for the density η(θ, ω, t) as a delta functional which enforces the deterministic evolu- tion from equation (4) along with an expectation taken over the distribution P0[η0] of the initial configuration η0(θ, ω) = η(θ, ω, t0). We emphasize that no external noise is added to our system. Any statistical uncertainty completely derives from the distribution of the initial state. Hence we arrive at the following path integral, P [η(θ, ω, t)] Dη0P0[η0]δ [N {C(η(θ, ω, t))− δ(t− t0)η0(θ, ω)}] The definition of the delta functional contains an ar- bitrary scaling factor, which we have taken to be N . We will show later that this choice is necessary for the field theory to correctly describe the statistics of η. The probability measure obeys the normalization condition DηP [η]. We first write the generalized Fourier decomposition of the delta functional. P [η(θ, ω, t)] Dη̃Dη0P0[η0] × exp dθdωdt η̃ [C(η)− δ(t− t0)η0(θ, ω)] where η̃(θ, ω, t) is usually called the “response field” after Martin-Siggia-Rose [27] and the integral is taken along the imaginary axis. It is more convenient to work with the generalized probability including the response field: P̃[η, η̃] = Dη0P0[η0] exp dθdωdt η̃C(η) dθdωη̃(θ, ω, t0)η0(θ, ω) which obeys the normalization DηDη̃P̃[η, η̃; η0] (8) We now compute the integral over η0 in equation (7), which is an ensemble average over initial phases and driv- ing frequencies. We assume that the initial phases and driving frequencies for each of the N oscillators are inde- pendent and obey the distribution ρ0(θ, ω). P [η0] repre- sents the functional probability distribution for the ini- tial number density of oscillators. Noting that η0(θ, ω) is given by η0(θ, ω) = δ(θ − θi(0))δ(ω − ωi) (9) Dη0P0[η0] = dθidωiρ0(θi, ωi), one can show that the distribution from equation (7) is given by P̃ [η, η̃] = exp dθdωdt η̃C(η) + N ln eη̃(θ,ω,t0) − 1 ρ0(θ, ω) In deriving Eq. (10), we have used the fact that Dη0P0[η0] exp dθdωη̃(θ, ω, t0)η0(θ, ω) dθidωiρ0(θi, ωi) exp η̃(θi, ωi, t0) dθdωρ0(θ, ω)e η̃(θ,ω,t0) = exp dθdωρ0(θ, ω) eη̃(θ,ω,t0) − 1 We see that the fluctuations (i.e. terms non-linear in η̃) appear only in the initial condition of (10), which is to be expected since the Kuramoto system is deterministic. In this form the continuity equation (4) appears as a Langevin equation sourced by the noise from the initial state. Although the noise is contained entirely within the ini- tial conditions, it is still relatively complicated. We can simplify the structure of the noise in (10) by performing the following transformation [21]: ϕ(θ, ω, t) = η exp(−η̃) ϕ̃(θ, ω, t) + 1 = exp(η̃) (12) Under the transformation (12), P̃ [n, ñ] becomes P̃[ϕ, ϕ̃], which is given by P̃ [ϕ, ϕ̃] = exp (−NS[ϕ, ϕ̃]) (13) where the action S[ϕ, ϕ̃] is S[ϕ, ϕ̃] = dωdθdt dω′dθ′ (ϕ̃′ϕ̃+ ϕ̃) {f(θ′ − θ)ϕ′ϕ} dθdωϕ̃(θ, ω, t0)ρ0(θ, ω) The form (14) is obtained from the transformation (12) only after several integrations by parts. In most cases, these integrations do not yield boundary terms because of the periodic domain (i.e. those in θ). In the case of the ∂t operator, however, we are left with boundary terms of the form [ln(ϕ̃ + 1) − 1]ϕ̃ϕ. These terms will not affect computations of the moments because of the causality of the propagator (see section III A). We are interested in fluctuations around a smooth so- lution ρ(θ, ω, t) of the continuity equation (4) with initial condition ρ(θ, ω, t0) = ρ0(θ, ω). We transform the field variables via ψ = ϕ−ρ and ψ̃ = ϕ̃ in (14) and obtain the following action: S[ψ, ψ̃] = dωdθdtψ̃ dω′dθ′ {f(θ′ − θ) (ψ′ρ+ ρ′ψ + ψ′ψ)} dω′dθ′ψ̃′ {f(θ′ − θ) (ψ′ + ρ′) (ψ + ρ)} (−1)k+1 dθdωψ̃(θ, ω, t0)ρ0(θ, ω) For fluctuations about the incoherent state: ρ(θ, ω, t) = ρ0(θ, ω) = g(ω)/2π, where g(ω) is a fixed frequency dis- tribution. The incoherent state is an exact solution of the continuity equation (4). Due to the homogeneity in θ and the derivative couplings, there are no corrections to it at any order in 1/N . The action (15) with ρ = g(ω)/2π therefore describes fluctuations about the true mean dis- tribution of the theory, i.e. 〈η(θ, ω, t)〉 = g(ω)/2π. We can evaluate the moments of the probability distribution (13) with (15) using the method of steepest descents, which treats 1/N as an expansion parameter. This is a standard method in field theory which produces the loop expansion [22]. We first separate the action (15) into “free” and “interacting” terms. S[ψ, ψ̃] = SF [ψ, ψ̃] + SI [ψ, ψ̃] (16) where SF [ψ, ψ̃] dωdθdtψ̃ dω′dθ′ {f(θ′ − θ) (ψ′ρ+ ρ′ψ)} dxdtdx′dt′ψ̃(x′, t′)Γ0(x, t;x ′, t′)ψ(x, t) SI [ψ, ψ̃] dωdθdtψ̃ dω′dθ′ {f(θ′ − θ) (ψ′ψ)} dω′dθ′ψ̃′ {f(θ′ − θ) (ψ′ + ρ′) (ψ + ρ)} (−1)k+1 dθdωψ̃(θ, ω, t0)ρ0(θ, ω) In deriving the loop expansion, the action is expanded around a saddle point, resulting in an asymptotic series whose terms consist of moments of the Gaussian func- tional defined by the terms in the action (15) which are bilinear in ψ and ψ̃, i.e. SF [ψ, ψ̃]. Hence, the loop expan- sion terms consist of various combinations of the inverse of the operator Γ0, defined by SF , called the bare prop- agator, with the higher order terms in the action, called the vertices. Vertices are given by the terms in SI . The terms in the loop expansion are conveniently rep- resented by diagrams. The bare propagator is repre- sented diagrammatically by a line, and should be com- pared to the variance of a gaussian distribution. Each term in the action (other than the bilinear term) with n powers of ψ and m powers of ψ̃ is represented by a vertex with n incoming lines and m outgoing lines. The initial state vertices produce only outgoing lines and, like the non-initial state or “bulk” vertices, are integrated over θ, ω, and t for each point at which the operators are de- fined. The bulk vertices are represented by a solid black dot (or square, see Figure 1) and initial state vertices by an open circle. The bare propagator and vertices are shown in Figure 1. Unlike conventional Feynman dia- grams used in field theory, the vertices in Figure 1 rep- resent nonlocal operators defined at multiple points. In particular, the initial state terms involve operators at a different point for each outgoing line. Although uncon- ventional, this is the natural way of characterizing the 1/N expansion. Adopting the shorthand notation of x ≡ {θ, ω}, each arm of a vertex must be connected to a line (propagator) at either x or x′ and lines connect outgoing arms in one vertex to incoming arms in another. The moment with n powers of ψ and m powers of ψ̃ is calculated by sum- ming all diagrams with n outgoing lines and m incoming { f (θ′−θ) . . .} { f (θ′−θ) . . .} g(ω′) { f (θ′−θ) . . .} { f (θ′−θ)} (2π)2 g(ω)g(ω′) { f (θ′−θ)} P0(x, t|x ′)x x (−1)k+1 ρ0(θ j,ω j) FIG. 1: Diagrammatic (Feynman) rules for the fluctuations about the mean. Time moves from right to left, as indicated by the arrow. The bare propagator P0(x, t|x ′, t′) (see Eq. (31)) connects points at x′ to x, where x ≡ {θ, ω}. Each branch of a vertex is labeled by x and x′ and is connected to a factor of the propagator at x or x′. Each vertex represents an operator given to the right of that vertex. The “. . . ” on which the derivatives act only include the incoming propagators, but not the outgoing ones. There are integrations over θ, θ′, ω, ω′ and t at each vertex. lines. This means that each diagram will stand for sev- eral terms which are equivalent by permutations of the vertices or the edges in the graph, equivalently permuta- tions of the factors of ψ̃ and ψ in the terms in the series expansion. In a typical field theory, this results in combi- natoric factors. In the present case, diagrams which are not topologically distinct can produce different contribu- tions to a given moment. Nonetheless, we will designate the sum of these terms with a single graph. Generically, the combinatoric factors we expect are due to the ex- change of equivalent vertices, which typically cancel the factorial in the series expansion. Additionally, each line in a diagram contributes a factor of 1/N and each vertex contributes a factor of N . Hence each loop in a diagram carries a factor of 1/N . The terms in the expansion with- out loops are called “tree level”. The bare propagator is the tree level expansion of 〈ψ(θ, ω, t)ψ̃(θ′, ω′, t′)〉. The tree level expansion of each moment beyond the first car- ries an additional factor of 1/N , i.e. the propagator and two-point correlators are each O(1/N). Mean field theory is defined as the N → ∞ limit of this field theory. In the infinite size limit, all moments higher than the first are zero (provided the terms in the series are not at a singular point, i.e. the onset of synchrony). Hence, the only surviving terms in the action (15) are those which contribute to the mean of the field at tree level. These terms sum to give solutions to the continuity equation (4). If the initial conditions are smooth, then mean field theory is given by the relevant smooth solution of (4). In most of the previous work (e.g. [12, 24]), smooth solutions to (4) were taken as the starting point and hence automatically assumed mean field theory. We can now validate our choice of N as the correct scaling factor in the delta functional of (5) by considering the equal time two-point correlator. Using the definition of η from (3) we get 〈η(x, t)η(x′, t)〉 = C(x, x′, t) + ρ(x, t)ρ(x′, t) ρ(x, t)ρ(x′, t) + δ(x − x′)ρ(x′, t) where ρ(x, t) = 〈η(x, t)〉 = 〈ϕ(x, t)〉. Using the fields ϕ and ϕ̃ (defined in (12)) and taking η at different times gives 〈η(x, t)η(x′, t′)〉 = 〈[ϕ̃(x, t)ϕ(x, t) + ϕ(x, t)] [ϕ̃(x′, t′)ϕ(x′, t′) + ϕ(x′, t′)]〉 for t > t′. The response field has the property that ex- pectation values containing ϕ̃(x, t) are zero unless an- other field insertion of ϕ(x, t) is also present but at a later time (this is because the propagator is causal; it is zero for t− t′ ≤ 0). Therefore 〈η(x, t)η(x′, t′)〉 = 〈ϕ(x, t)ϕ(x′, t′)〉 + 〈ϕ(x, t)ϕ̃(x′, t′)〉〈ϕ(x′, t′)〉 (21) As we will show later when we discuss the propagator in more detail, we have 〈ϕ(x, t)ϕ̃(x′, t′)〉 = δ(x − x′) (22) Comparing (21) in the limit t → t′ with (19) allows the immediate identification of 〈ϕ(x, t)ϕ(x′, t)〉 = C(x, x′, t) + ρ(x, t)ρ(x′, t) ρ(x, t)ρ(x′, t) (23) C(x, x′, t) is the two oscillator “connected” correlation or moment function. This is consistent with (15) which gives 〈ϕ(x, t0)ϕ(x ′, t0)〉 = ρ(x, t0)ρ(x ′, t0)− ρ(x, t0)ρ(x ′, t0) as the initial condition. Thus, comparing the second mo- ment of η using the Doi-Peliti fields (20) with the expres- sion from the direct computation given by (19) shows that the factor of N in the delta functional of (5) was necessary to obtain the correct scaling for the moments. The Doi-Peliti action (15) can also be derived by con- sidering an effective Markov process on a circular lattice representing the angle θ where the probability of an oscil- lator moving to a new point on the lattice is determined by its native driving frequency ω and the relative phases of the other oscillators (see Appendix C). This is the primary reason we refer to the theory as being of Doi- Peliti type. The continuum limit of this process yields a theory described by the action (15). The Markov picture provides an intuitive description and underscores the fun- damental idea that we have produced a statistical theory obeyed by a deterministic process. Although our formalism is statistical, we emphasize that no approximations have been introduced. The sta- tistical uncertainty is inherited from averaging over the initial phases and driving frequencies. This formalism could be applied to a wide variety of deterministic dy- namical systems that can be represented by a distribu- tional continuity equation like Eq. (4). In general, a solu- tion for the moment generating functional for our action (15) is as difficult to obtain as solving the original sys- tem. The advantage of formulating the system as a field theory is that a controlled perturbation expansion with the inverse system size as the small parameter is possible. A. Relation to Kinetic Theory and Moment Hierarchies The theory defined by the action (15) is equivalently expressed as a Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) moment hierarchy starting with the continuity equation (4). To construct the moment hierarchy, one takes expectation values of the continuity equation with products of the density, η(θ, ω, t). This results in coupled equations of motion for the various moments of η(θ, ω, t), where each equation depends upon one higher moment in the hierarchy. The Dyson-Schwinger equations derivable from (15) with 〈ϕ̃〉 = 0 are exactly the BBGKY hierarchy derived in Ref. [17]. Thus the kinetic theory and field theory approaches are entirely equivalent. The moments of η can be computed in the BBGKY hi- erarchy by truncating at some level. In Ref. [17], this was done at Gaussian order by assuming that the connected three-point function was zero. The first two equations of the hierarchy then form a closed system which can by solved. The first two equations of the BBGKY hierar- chy [17] are f(θ′ − θ)ρ(x, t)ρ(x′, t)dθ′dω′ f(θ′ − θ)C(x, x′, t)dθ′dω′ (25) where ρ(x, t) = 〈η(x, t)〉, and the connected (equal-time) correlation function C(x, x′, t) = 〈η(x, t)η(x′, t)〉 − ρ(x, t)ρ(x′, t) ρ(x, t)ρ(x′, t)− δ(x− x′)ρ(x′, t) obeys f(θ3 − θ1) + f(θ3 − θ2)]ρ(x3, t)dθ3dω3 C(x1, x2, t) f(θ3 − θ1)ρ(x1, t)C(x2, x3, t)dθ3dω3 +K f(θ3 − θ2)ρ(x2, t)C(x3, x1, t)dθ3dω3} f(θ2 − θ1) + f(θ1 − θ2)]ρ(x1, t)ρ(x2, t). In the field theoretic approach, instead of truncating the BBGKY hierarchy, one instead truncates the loop expansion. Truncating the moment hierarchy at the mth order is equivalent to truncating the loop expansion for the lth moment at the (m − l)th order. Thus the solu- tion to the moment equations (25) and (27) is the one loop expression for the first moment and the tree level expression for the second moment. The advantage of us- ing the action (15) is that the terms in the perturbation expansion are given automatically by the relevant dia- grams at any level of the hierarchy. Ref. [17] suggested that a higher order in the hierarchy would be necessary to check whether the mean field marginal modes are stable for finite N . We demonstrate below that the field the- ory facilitates the calculation of the linearization of equa- tion (25) to higher order in 1/N and show that marginal modes are stabilized by finite size fluctuations. One can compare this approach to the maximum en- tropy approach of Rangan and Cai [28] for developing consistent moment closures for such hierarchies. In the moment hierarchy approach of Ref. [17], moment closure is obtained via the somewhat ad hoc approach of setting the nth cumulant to zero. In contrast, Rangan and Cai maximize the entropy of the distribution subject to cer- tain normalization constraints. The moment closure is facilitated by constraining higher moments from the hi- erarchy. However, one still must solve the resulting equa- tions. In the loop expansion, moment closure is obtained implicitly via truncating the loop expansion. The loop expansion approach offers the advantage of providing a natural means for determining when the approximation, thus the implicit closure, breaks down and avoids deal- ing with the moment hierarchy explicitly. In fact, Ran- gan and Cai’s procedure has a natural interpretation in field theory, namely the minimization of a generalized effective action in terms of various moments. The sim- plest and most common is the effective action in terms of the mean field, which is the generating functional of one partical irreducible (1PI) graphs [22]. The next level of approximation is a generalized effective action (the “effective action for composite operators” [29]) in terms of the mean and the two-point function (or functions), which is the generating functional of two particle irre- ducible (2PI) graphs. One can continue in this way. The equations of motion of these effective actions will produce a closure of the moment hierarchy implicit in the action for the theory. At tree level these equations will be equiv- alent to those produced by Rangan and Cai’s maximum entropy approach. The loop expansion allows for sys- tematic corrections to these equations without explicitly invoking higher equations in the hierarchy. III. TREE LEVEL LINEAR RESPONSE, CORRELATIONS AND FLUCTUATIONS As a first example, we reproduce the calculation of the variation of the order parameter Z, which was calculated previously using the BBGKY moment hierarchy [17]. To do so requires the calculation of the tree level linear re- sponse or bare propagator and the tree level connected two-oscillator correlation function. A. The Propagator The propagator P (θ, ω, t|θ′, ω′, t′) is given by the ex- pectation value P (θ, ω, t|θ′, ω′, t′) = 〈ϕ(θ, ω, t)ϕ̃(θ′, ω′, t′)〉 (28) It is the linear response of (4). This can be shown by considering a small perturbation δρ0 to the initial state ρ in the action (15). Expanding to first order then yields: δρ(θ, ω, t) = N dθ′dω′ 〈ϕ(θ, ω, t)ϕ̃(θ′, ω′, t′)〉 δρ0(θ ′, ω′) The tree level linear response or bare propagator P0(θ, ω, t|θ ′, ω′, t′) ≡ P0(x, t;x ′, t′) is the functional in- verse of the operator Γ0 defined by the free part of the action (17). The bare propagator is therefore given by Γ0 · P0 ≡ dx′′dt′′Γ0(x, t;x ′′, t′′)P0(x ′′, t′′;x′, t′) δ(x− x′)δ(t− t′) (30) Using the action (15) with Eq. (30) gives Γ0 · P0 ≡ f(θ1 − θ)ρ(x1, t)dθ1dω1 P0(x, x ′, t− t′) f(θ1 − θ)ρ(x, t)P0(x1, x ′, t− t′)dθ1dω1 δ(θ − θ′)δ(ω − ω′)δ(t− t′) (31) Due to the rotational invariance in θ of f(θ), P (x, t;x′, t′) ≡ P (θ − θ′, ω, ω′, t− t′). In the incoherent state, ρ(θ, ω, t) = g(ω)/2π. Thus, for f(θ) odd, Eq. (31) becomes P0(x, x ′, t− t′) f(θ1 − θ)P0(x1, x ′, t− t′)dθ1dω1 δ(θ − θ′)δ(ω − ω′)δ(t− t′) (32) We can invert this equation using Fourier and Laplace transforms. Taking the Fourier transform of Eq. (32) (with respect to θ and θ′) yields + inω P0(n, ω;m,ω ′, t− t′) + inKg(ω)f(−n) P0(n, ω1;m,ω ′, t− t′)dω1 δ(t− t′)δ(ω − ω′)δn+m (33) where we use the following convention for the Fourier transform f(n) = f(θ)e−inθdθ f(θ) = f(n)einθ (34) Hereon, we will suppress the index m since the propaga- tor must be diagonal (i.e. P0(n,m)) ∝ δm+n). We Laplace transform in τ = t− t′ to get [s+ inω] P̃0(n, ω, ω ′, s) + inKg(ω)f(−n) P̃0(n, ω1, ω ′, s)dω1 δ(ω − ω′) (35) using the convention f̃(s) = f(t)e−sτdτ f(τ) = f̃(s)esτds (36) where the contour L is to the right of all poles in f̃(s). We can solve for P̃0(n, ω, ω ′, s) using a self-consistency condition. Integrate (35) over ω after dividing by s+ inω to get dωP̃0(n, ω, ω ′, s) inKg(ω)f(−n) s+ inω dω1P̃0(n, ω1, ω ′, s) s+ inω′ which we can solve to obtain dωP̃0(n, ω, ω ′, s) = s+ inω′ Λn(s) where Λn(s) = 1 + inKf(−n) s+ inω Λn(s) is defined for Re(s) ≤ 0 via analytic continuation. In the kinetic theory context of an oscillator density obey- ing the continuity equation (4), Λn(s) is analogous to a plasma dielectric function [17]. If we assume that g(ω) is even and f(θ) is odd, then there is a single real number sn such that Λn(sn) = 0. (Mirollo and Strogatz proved that there is at most one single, real root of (39) and that it must satisfy Re(s) ≥ 0 [11, 30]. In our case, Λn(s) is defined for Re(s) < 0 not by (39), but rather via ana- lytic continuation.) Using (39) in (35) and solving for P̃0(n, ω, ω ′, s) gives P̃0(n, ω, ω ′, s) = δ(ω − ω′) s+ inω inKg(ω)f(−n) (s+ inω) (s+ inω′) Λn(s) Here we identify the spectrum with the zeroes of Γ0 or, equivalently, the poles of the propagator, as these will de- termine the time evolution of perturbations. Analogous to the analysis of Strogatz and Mirollo [24] we define the operator O by O[bn(ω, t)] ≡ inωbn(ω, t) + inKf−n bn(ω1, t)g(ω1)dω1 (41) FIG. 2: Diagrams for the connected two-point function at tree level a) and to one loop b). (cf. equation (62) below). The continuous spectrum of O consists of the frequencies inω whereas the discrete spectrum (according to Ref. [24]) only exists for K > Kc. Consistent with that approach (i.e. linear operator theory), we identify the poles in P due to s+ inω as the continuous spectrum and those due to the zeroes of Λn(s) as the discrete spectrum. If Λn(s) is not analytically continued for Re(s) < 0, it will not have zeroes for that domain, as in Ref. [24]. However, zeros can exist for Re(s) < 0 when Λn(s) is analytically continued and this is why analytic continuation is of such crucial importance to the conclusions of Ref. [12]. B. Correlation Function The connected correlation function (cumulant func- tion) is given by C(x1, t1;x2, t2) = 〈ψ(x1, t1)ψ(x2, t2)〉 (42) This is equivalent to (26), which was computed in Ref. [17] when t1 = t2. If the initial phases are un- correlated (i.e. C(x1, 0;x2, 0) = 0) then at tree level C(x1, t1;x2, t2) is given by the diagram shown in Fig- ure 2 a). It is comprised of vertex III (see Figure 1) com- bined with a bare propagator on each arm. For general (i.e. not odd) f(θ), the diagram in Figure 2 a) actually corresponds to two different terms because the arms of vertex III can be interchanged, giving two terms in equa- tion (43) below. Unlike, conventional field theory, these interchanges are not symmetric. These two terms are equal when f(θ) is odd. More generally other vertices do not exhibit the symmetries typical of Feynman dia- grams even for odd f(θ). Applying the Feynman rules then gives at tree level C(x1, t1;x2, t2) (2π)2 dω′1dω dθ′1dθ × P0(x1, x 1, t1 − t ′)P0(x2, x 2, t2 − t f(θ′2 − θ f(θ′1 − θ 2)]g(ω 1)g(ω 2)(43) where t = min(t1, t2). This is essentially identical to the ansatz used in Ref. [17] for the solution of the second moment equation in the BBGKY hierarchy (27). For f(θ) = sin θ and t1 > t2, Fourier transforming Eq. (43) and inserting the bare propagator from Eq. (40) (after an inverse Laplace transformation) gives C1(ω1, t1, ω2, t2) = g(ω1)g(ω2) (iω1 + )(−iω2 + (iω1 + )(−iω2 + e(i(ω1−ω2)t2) i(ω1 − ω2) i(ω1 − ω2) eiω1(t1−t2) (iω2 − − iω1)(( )2 + (ω2)2) +iω2)(t2) − 1 )(t1−t2) (−iω1 − + iω2)(( )2 + (ω1)2) −iω1)t2 − 1 e−iω1(t1−t2) 4(K − 2Kc − iω1)( + iω2) e−(Kc−K)t2 − 1 e−(Kc−K))(t1−t2) The equal time correlator is given by (t1 = t2 = τ): C1(ω1, ω2, τ) = g(ω1)g(ω2) (iω1 + )(−iω2 + (iω1 + )(−iω2 + e(i(ω1−ω2)τ) i(ω1 − ω2) i(ω1 − ω2) (iω2 − − iω1)(( )2 + (ω2)2) +iω2)τ − 1 (−iω1 − + iω2)(( )2 + (ω1)2) −iω1)τ − 1 4(K − 2Kc − iω1)( + iω2) e−(Kc−K)τ − 1 C−1 = C 1 and the other modes vanish. Note that the initial condition C1(ω, ω ′, 0) = 0 is satisfied and that the time constants and frequencies which appear are every possible way of pairing those from the tree level propa- gator. For illustrative purposes, Figure 2 b) shows the one loop diagrams which contribute to C; these diagrams are O(1/N2). We should note here the special role played by the diagrams with initial state terms, in particular the vertex proportional to ψ̃(θ, ω, t0) 2. This diagram evalu- ates to exactly the same result as (44), with an additional factor of −1/N . It serves to provide the proper normal- ization for the two point function, which should go as (N − 1)/N since the self-interaction (diagonal) terms are not included. The other diagrams diverge faster as one approaches criticality (K = Kc). They are of negligible importance at small coupling but become increasingly important near the onset of synchrony. C. Order Parameter Fluctuations We now compute the fluctuations in the order parame- ter Z given in Eq. (2). The variance of Z (second moment 〈ZZ̄〉) is given by 〈ZZ̄〉 = 〈r2(t)〉 dωdω′dθdθ′〈η(ω, θ, t)η(ω′, θ′, t)〉ei(θ−θ Using equation (19) in equation (46) gives 〈r2(t)〉 = dωdω′dθdθ′C(x, x′, t)ei(θ−θ since in the incoherent state ρ(x, t) = g(ω)/2π is inde- pendent of θ, so that 〈Z〉 = 0. Hence 〈r2(τ)〉 = 4π2 dωdω′C−1(ω, ω ′, τ) + which evaluates to [17] 〈r2(τ)〉 = Λ1(s− s0)− 1 Λ1(s− s0) × Res Λ1(s) esτ + The time evolution of 〈r2〉 is then determined by the poles of Λ1(s). As an example, consider f(θ) = sin θ and g(ω) a Lorentz distribution (see Appendix B); we have the result 〈r2(τ)〉 = Kc −K Kc −K e−(Kc−K)τ (50) where Kc = 2γ. Note that this diverges as τ → ∞ for K = Kc. In the mean field limit N → ∞, 〈r 2〉 = 0 as expected. As was shown in Ref [17], the tree level calcu- lation adequately captures the fluctuations except near the onset of synchrony (K = Kc). An advantage of the field theoretic formalism is that it allows us to approach even higher moments without needing to worry about the moment hierarchy. In particular, for f(θ) = sin θ it is straightforward to show that higher cumulants, such as 〈(ZZ̄)2〉 − 〈ZZ̄〉2, must be zero at tree level because of rotational invariance (more precisely, any cumulant of Z higher than quadratic). The cumulants are given by graphs which are connected. Vertex III produces two lines with wave numbers n = ±1. Additionally, the IV and V vertices impose a shift in wave number, whereas Z and Z̄ project onto ±1. In order to calculate these higher fluctuations it is necessary to go to the one loop level. Note that this does not imply that the higher cu- mulants of η are zero. Figure 2 b) gives the diagrams for the correlation function at one loop. The one loop calculation would also give a better estimate for 〈r2〉, es- pecially nearer to criticality. The non-interacting distri- bution is Gaussian, with non-Gaussian behavior growing as one approaches criticality. IV. LINEAR STABILITY AND MARGINAL MODES We analyze linear stability by convolving the linear response or propagator with an initial perturbation. Al- though we are free in our formalism to consider arbitrary perturbations, we will consider two specific kinds for il- lustrative purposes. In the first case we perturb only the angular distribution. In the second case we consider a perturbation which fixes one oscillator to be at a given angle θ and frequency ω at a given time t. We first calcu- late the results at tree level which reproduces the mean field theory results of Ref. [24]. In particular, we arrive at the same spectrum and Landau damping results of Refs. [24] and [12]. We then define and calculate the op- erator Γ (an extension of Γ0) to one loop order which ultimately allows us to calculate the corrections to the spectrum to order 1/N . A. Mean field theory The bare propagator which is the full linear response for mean field theory is given by Eq. (40). The zeroes of the operator Γ with respect to s specify the spectrum of the linear response. There is a set of marginal modes (continuous spectrum) along the imaginary axis spanning an interval given by the support of g(ω). There is also a set of discrete modes given by the zeros of the dielectric function Λn(s) as was found in Ref. [24], aside from the issue of the analytic continuation of Λn(s). However, even though there are marginally stable modes, the order parameter Z can still decay to zero due to a generalized Landau damping effect as was shown in Ref. [12]. Consider a generalization of the order param- Zn(t) = einθj , (51) which represents Fourier modes of the density integrated over all frequencies: Zn(t) = dθdωη(θ, ω, t)einθ (52) and hence 〈Zn(t)〉 = dθdωρ(θ, ω, t)einθ dωρ(−n, ω, t). (53) In the incoherent state, ρ(θ, ω, t) is independent of θ so 〈Zn〉 is zero for n > 0. The density response δρ(θ, ω, t) to an initial perturbation δρ(θ, ω, 0) is given by δρ(θ, ω, t) = N P0(x, x ′, t)δρ(θ′, ω′, 0)dθ′dω′ Recall from the definition of the action (15), the prop- agator operates on an initial condition defined by Nρ0. The perturbed order parameter thus obeys δ〈Zn(t)〉 = dθdωδρ(θ, ω, t)einθ. (55) We will show that for any initial condition involving a smooth distribution in frequency and angle, δ〈Zn(t)〉 will decay to zero. However, for non-smooth initial perturba- tions, δ〈Zn(t)〉 will not decay to zero but will oscillate. We first consider an initial perturbation of the form δρ(θ, ω, 0) = g(ω)c(θ) (56) where c(θ)dθ = 0 (57) Inserting into (54) yields δρ(θ, ω, t) = N P0(x, x ′, t)c(θ′)g(ω′)dθ′dω′ which is consistent with the perturbation considered in Ref. [24]. Taking the Laplace transform of (58) gives δρ̃n(ω, s) = Ncn P̃0(n, ω, ω ′, s)g(ω′)dω′ (59) Using the tree level propagator (40), we can show that P̃0(n, ω, ω ′, s)g(ω′)dω′ = s+ inω Λn(s) where Λn(s) is given in (39). Hence δρ̃(n, ω, s) = s+ inω Λn(s) From Eq. (61), we see that the continuous spectrum is given by inω and the discrete spectrum by the zeros of Λn(s). If we define bn(ω, t) = δρ(n, ω, t)/(Ncng(ω)) then using (58) and (40) we can show + inω bn(ω, t) + inKf(−n) bn(ω1, t)g(ω1)dω1 δ(t) (62) which is equivalent to the linearized perturbation equa- tion derived by Strogatz and Mirollo [24], with the ex- ception that (62) includes the effects of the initial config- uration through the source term proportional to δ(t). Inserting (61) into the the Laplace transform of (55) yields δ〈Z̃n(s)〉 = c−n s− inω Λ−n(s) = c−n Λ−n(s)− 1 −inKf(n) Λ−n(s) . (63) For f(θ) = sin θ, f(±1) = ∓i/2 which leads to δ〈Z̃1(s)〉 = c−1 Λ−1(s)− 1 Λ−1(s) . (64) We note that δZ1(s) is identical to what was calculated in Ref. [12] in which it was shown that δZ1(t) → 0 as t → ∞. Even in the presence of marginal modes, the order parameter decays to zero through dephasing of the oscillators. This dephasing effect is similar to Landau damping in plasma physics. We can see this explicitly for the case of the Lorentz distribution g(ω) = γ2 + ω2 From (39) we can calculate Λ±1(s) = s+ γ − K and Λn(s) = 1 for n 6= ±1. The zero of Λ±1(s) is at s±1 = −(γ −K/2), which provides a critical coupling Kc = 2γ (67) above which the system begins to synchronize. The in- coherent state is reached when K < Kc, which gives s±1 < 0. Thus δ〈Z±1〉 = c∓1e −(γ−K δ〈Zn6=±1〉 = c−ne −|n|γτ (68) Hence angular perturbations decay away in the order pa- rameter. Landau damping due to dephasing is sufficient to de- scribe the relaxation of Zn(t) to zero for a smooth per- turbation. However, for non-smooth perturbations, this may not be true. Consider the linear response to a stim- ulus consisting of perturbing a single oscillator to have initial position θ0 and frequency ω0: ρ0(θ, ω) = (N − 1) + δ(θ − θ0)δ(ω − ω0) and so the initial perturbation is δρ0(θ, ω) = + δ(θ − θ0)δ(ω − ω0) Inserting into (54) gives the time evolution of this initial perturbation δρ(θ, ω, t) = − + P (θ, ω, t; θ0, ω0, t ′) (71) Substituting into (55) and taking the Lapace transform gives δ〈Z̃n(s)〉 = 2π dωδρ̃−n(ω, s) dωP̃ (−n, ω, ω0, s) s− inω0 Λ−n(s) There are therefore two modes in δZn(t), one which de- cays due to dephasing (determined by the zero of Λ−n(s)) and one which oscillates at frequency ω0. Thus, for this perturbation, the tree level prediction is that δ〈Z〉 is not zero but oscillates. Inverse Laplace transforming (72), gives the time dependence of the order parameter δ〈Z1(t)〉 = ω20 + γ − K + ω20 − e−iω0t −iω0 + γ − e−(γ− The perturbed oscillator has phase θ0 − ω0t. It can al- ways be located; no information is lost as the time evo- lution progresses. Hence, for a single oscillator pertur- bation, the tree level calculation predicts that the order parameter will not decay to zero. In the next section, we show that to the next order in the loop expansion, which accounts for finite size effects, the marginal modes are moved off of the imaginary axis stabilizing the incoher- ent state, and the order parameter for a single oscillator perturbation decays to zero. B. Finite size effects Let us define a generalization of the operator Γ0. Γ · P = δ(x− x′)δ(t− t′) (74) where P without a subscript denotes the full propagator and the operator Γ is the functional inverse of P . We can estimate the effect of finite size on the stability of the incoherent state by calculating the one loop correction to the operator Γ. We will see that the one loop correction FIG. 3: The diagrams contributing to the propagator at one loop order, organized by topology. We consider d) to be of different topology than c) because it is equivalent to a tree level diagram with an additional factor of 1/N , due to the initial state vertex. FIG. 4: Diagrammatic equation for the propagator. The dou- ble lines represent the summation of the entire series in 1/N for the propagator. produces the effect, among others, of adding a diffusion operator to Γ, which is enough to stabilize the continuum of marginal modes because the continuous spectrum is pushed off the imaginary axis by an amount proportional to the diffusion coefficient. We calculate the correction to Γ to one loop order. The propagator is represented by diagrams with one incom- ing line and one outgoing line. There are four groups of diagrams which contribute to the propagator at one loop order. They are shown in Figure 3 and are labeled by a), b), c), and d). Using these graphs to calculate the propagator to order 1/N is not sufficient to demonstrate the behavior of the spectrum to order 1/N . However, we can use these graphs to construct an approximation of Γ to order 1/N and derive the spectrum from this. If we denote the full propagator (i.e. the entire series in 1/N for P ) by a double line, we can approximate the full propagator recursively by the diagrammatic equation shown in Figure 4. The only terms which are neglected in this relation are those which are from two loop and higher graphs and therefore would contribute O(1/N2) to Γ. Readers familiar with field theory will note that we are simply calculating the two point proper vertex, which is the inverse of the full propagator, to one loop order. If we act on both sides of this equation with Γ0 (the operator whose inverse is the tree level propagator), we arrive at an equation of the form: Γ0 · P = δ(x− x′)δ(t− t′)− Γ1 · P (75) where we have implicitly defined the one loop correction to Γ, which we label Γ1. The action of Γ0 converts the leftmost propagator in each diagram into a delta func- tion, so that the delta function term in (75) arises from the tree level propagator line and Γ1 is then comprised of the loop portions of the remaining diagrams (the “ampu- tated” graphs). We denote the contribution to Γ1 from each group of one loop diagrams by Γ1r(θ, ω;φ, η; t− t where r represents a, b, c, or d indicating the group of diagrams in question. Γ1b = 0 because the derivative coupling acts on the incoherent state, which is homoge- neous in θ. The equation of motion for the one loop propagator P1(x, x ′, t) then has the form Γ0 · P1 + Γ1 · P1 = δ(θ − θ′)δ(ω − ω′)δ(t− t′) (76) where Γ0 · P1 is given by Γ0 · P1 = f(θ1 − θ)ρ(x1, t)dθ1dω1 P1(x, x ′, t− t′) f(θ1 − θ)ρ(x, t)P1(x1, x ′, t− t′)dθ1dω1 (77) Γ1 · P1 = dt′′Γ1a(θ, ω;φ, η; t− t ′′)P1(φ, η, t ′′; θ′, ω′; t′) dt′Γ1c(θ, ω;φ, η; t− t ′′)P1(φ, η, t ′′; θ′, ω′; t′) dt′Γ1d(θ, ω;φ, η; t− t ′′)P1(φ, η, t ′′; θ′, ω′; t′) (78) is the one loop contribution. The kernels Γ1a, Γ1c, and Γ1d are explicitly computed in Appendix A. The expressions for the one loop contribution to Γ are rather complicated but several key features can be ex- tracted, namely 1) the introduction of a diffusion opera- tor, 2) a shift in the driving frequency, and 3) the addi- tion of higher order harmonics to the coupling function f . The diffusion operator has the effect of shifting the marginal spectrum from the imaginary axis into the left hand plane. The effect is that the finite size fluctuations to order 1/N stabilize the incoherent state. We can see these effects more easily by considering the special case of f(θ) = sin θ and g(ω) being a Lorentz dis- tribution (see Appendix B). The Fourier-Laplace trans- formed equation of motion for the one loop propagator has the form α±1(s;ω)P̃1(n, ω, ω ′, s) g(ω)(1 + β1(s;ω)) dνP̃1(n, ν, ω ′, s) δ(ω − ω′), n = ±1, (79) α±2(s;ω)P̃1(n, ω, ω ′, s) dνβ±2(s;ω, ν)P̃1(n, ν, ω ′, s) δ(ω − ω′), n = ±2, (80) αn(s;ω)P̃1(n, ω, ω ′, s) = δ(ω − ω′), |n| > 2 where αn(s;ω) = s+ inω + s+ γ − K + i(m+ n)ω n(2m+ n) + n(m+ n) 2γ −K β±1(s;ω) = − s+ γ − K ± 2iω γ − K 2γ − K 2γ −K s+ γ − K β±2(s;ω, η) = − s∓ iω s∓ iω + γ − K s± i(η − ω) ±iω + γ γ − K s+ γ − K s+ 2γ −K s+ γ − K 2γ −K −γ + K 2γ −K s+ γ − K s+ γ − K s+ 2γ − K s+ 2γ −K . (82) We can solve for P̃1 using the same kind of self- consistency computation that we used for P̃0. This pro- duces an analogously defined dialectric function, Λ1n(s). Λ1n(s) = 1− g(ω)(1 + βn(s;ω)) αn(s;ω) Stability of the incoherent state is determined by the spectrum of the operator Γ. Analogous to tree level, the continuous spectrum is given by the zeros of αn(s;ω) and the discrete spectrum by the zeros of Λ1n(s) given by (83). However, at one loop order, the expressions for P̃1 rep- resent solutions to a coupled system of equations. Thus, the poles of tree level are shifted, and there are also new poles reflecting the interaction of the mean density with the two-point correlation function. These poles will have residues of O(1/N2) owing to their higher order nature. For n = ±1,±2, we cannot solve for the poles exactly but we can approximate the shift in the tree level spec- trum by evaluating the loop correction at the value of the tree level pole, s = ∓inω, which is equivalent to using the “on-shell” condition in field theory. Since the higher order modes will decay faster than the tree level modes, this essentially amounts to ignoring short time scales and is similar to the Bogoliubov approximation [25, 26]. The remaining effective equation for P̃1 is now first order and, consequently, we can consider the spectrum of the implic- itly defined operator analogous to (41). The continuous spectrum consists of all the zeros of the function αn. We expect a term of the form inω+O(1/N) because of the tree level continuous spectrum. This will govern the behavior at large times. In this case, the “on- shell” condition is equivalent to Taylor expanding the loop correction via s = inω + O(1/N) and keeping only terms which are O(1/N). This yields: αn(s;ω) = s+ in(ω + δω) + n 2D (84) where δω = − γ − K 4γ −K 2γ −K is a frequency shift and γ − K is a diffusion coefficient. The frequency shift, which is negative, serves to tighten the distribution around the average frequency. The diffusion operator serves to damp the modes which are marginal at tree level. The discrete spectrum arises from the zeroes of Λ1n(s). We can again approximate the shift in the tree level zero by using the on-shell condition. This gives Λ1n(s) = 1− s+ in(ω + δω) + n2D 2γ − K 2γ −K γ − K + inω γ − K − inω We assume the shift will be small which allows us to write the zero of Λ1n(s) as s = − δΛ1n(s0) ) (88) where s0 = − γ − K and δΛ1n(s) is the O(1/N) correc- tion to Λ1n(s). This results in sn = − 2γ −K γ − K 6γ −K 2γ −K Away from criticality (K ≪ 2γ) or for large N , this cor- rection is small. We conclude this section by writing down an effective equation of motion for the density function which incor- porates the effect of fluctuations. Recall the first equa- tion of the BBGKY hierarchy is f(θ′ − θ)ρ(x, t)ρ(x′, t)dθ′dω′ f(θ′ − θ)C(x;x′, t)dθ′dω′(90) This equation has a term (the “collision” integral) involv- ing the 2-point correlation function, C, on the right hand side. The equation determining the tree level propaga- tor (31) is the linearization of the first BBGKY equation with C considered to be zero. The one loop correction to this equation (78) incorporates the effect of the cor- relations on the linearization. The diagrams in Figure 3 provide the linearization of the collision term, where C is considered as a functional of ρ. Using our one loop calculation, we can propose an effective density equation at one loop order sin(θ′ − θ)ρ(θ′,Ω′, t)ρ(θ,Ω, t)dθ′dω′ = −K2(ω) sin(2θ′ − 2θ)ρ(θ′,Ω′, t)ρ(θ,Ω, t)dθ′dω′ (91) where Ω = ω + δω and D is given by Eq. (86). The field ρ(θ,Ω, t) is now defined in terms of the shifted frequency distribution G(Ω) ≈ g(ω)(1− ) (92) The new coupling constant K2(ω) is O(1/N) and is due solely to the fluctuations. It arises from the term β±2 in the equation for P̃1. In fact, given the structure of the diagrams, it is clear that for O(1/Nn) there will be a new coupling, Kn+1, which corresponds to a sin[(n+1)θ] term. In the language of field theory, all odd couplings are generated under renormalization. The generation of higher order couplings is especially interesting in light of the results of Crawford and Davies concerning the scaling of the density η beyond the onset of synchronization, i.e. η − ρ0 ∼ (K − Kc) β [31, 32]. Although our calculations pertain to the incoherent state, the fact that the loop corrections generate higher order couplings is a general feature of the bulk theory defined by the action of Eq. (14) as well. Thus, we expect a crossover from β = 1/2 to β = 1 behavior to occur as N gets smaller. This is consistent with [31], wherein a crossover manifested as the rate constant became smaller than the externally applied diffusion. In our case, the magnitude of the diffusion is governed by the distance to criticality and the number of oscillators. Our proposed effective equation (91) is not self- consistent because we use the propagator to infer the form of the mean field equation. Thus, we neglect non- linear terms which may arise due to the loop corrections. In addition, our calculation applies specifically to per- turbations in the incoherent state. There are likely other terms we are neglecting for both of these reasons. The consistent approach would be to calculate the effective ac- tion to one loop order and derive the equation for ρ from that. This would involve essentially the same calculation we have performed here, but for arbitrary ρ(θ, ω, t) (i.e. we would need to solve (31) for the propagator in the presence of an arbitrary mean). V. NUMERICAL SIMULATIONS We compare our analytical results to simulations of sin- gle oscillator perturbations, since this provides a direct measurement of the propagator per equation (71). We perform simulations of N oscillators with f(θ) = sin θ. We fix 2% of the oscillators at a specific angle (θ0 = 0) and driving frequency (unless N = 10, in which case we fix a single oscillator; the plots with N = 10 have been rescaled to match the other data). The remaining oscilla- tors are initially uniformly distributed over angle θ with driving frequencies drawn from a Lorentz distribution. We measure the real part of Z1(t). This measurement allows us to observe the behavior of the modes which are marginal at tree level. Equation (73) gives the behavior of δZ1(t) with a single oscillator fixed at θ0 and ω0 at time t = 0. Recall that Z1 = 0 in the incoherent state, so that we expect δZ1 ≈ Z1. To tree level Z1(t) = ω20 + γ − K (γ(γ − ) + ω20 − iω0)e −iω0t − (−iω0 + γ − e−(γ− In other words, the initially fixed oscillator has phase θ0−ω0t; no information is lost as the time evolution progresses. Incorporating the one loop computation gives Z1(t) = ω20 + γ − K (γ(γ − ) + ω20 − iω0)e i(ω+δω)t−Dt − (−iω0 + γ − where we have ignored a term of amplitude O(1/N2); we are only considering the contributions coming from the poles described in the previous section. With the one loop corrections taken into account, we see that Z1(t) relaxes back to zero as t→ ∞. In the simulations, we compute the real part of Z1(t) with θ0 = 0. This gives Re(Z1(t)) = ω20 + γ − K γ(γ − ) + ω20 cos ((ω0 + δω)t) e sin ((ω0 + δω)t) e −Dt − )es1t The special case of ω0 = 0 and θ0 = 0 gives: Z1(t) = γ − K γe−Dt − The imaginary part vanishes so that Re(Z1(t)) = Z1(t). We first compare our estimate of the diffusion coef- ficient D given by (86) with the simulations. We plot the measured decay constant D of Z1 compared to the theoretical estimate of (96) for the long time behavior in Figure 5. These data only include values of K = 0.3Kc and K = 0.5Kc (Kc = 2γ = 0.1). Higher values of K did not yield good fits due to the neglected contributions to Z1. These decay constants are obtained via fitting the time evolution of Z1 to an exponential for t > 200s. In both cases they behave as 1/N for large N as predicted. There is a consistent discrepancy likely due to round- ing error after simulating for such a long period of time (≈ 30000 time steps). This error appears as a small de- gree of noise which further damps the response, hence the decay constants appear slightly larger in Figure 5. This effect can be seen in Figures 6 and 7 as well. For large times, the simulation data consistently fall slightly under 10 100 0.00001 0.00010 0.00100 0.01000 K = 0.03 K = 0.05 FIG. 5: The large time (> 200s) decay constants with zero driving frequency for the perturbed oscillator. Lines are the predictions given by Eq. 86). Solid line and circles represent K = 0.03. Dashed line and boxes represent K = 0.05. the analytic prediction. Similarly, the data is noisier at large times. Figures 6 and 7 show the evolution of Z1(t) over time along with the analytical predictions and the tree level result for K = 0.3Kc and K = 0.5Kc respectively. For K = 0.3Kc, the prediction works quite well, with perhaps the beginning of a systematic deviation appearing atN = 10 and N = 50 (there is a slight initial overshoot followed by an undershoot at larger times). This same deviation is more pronounced for K = 0.5Kc, although the data follow the prediction quite well nonetheless. Consistent with our expectations from the loop expan- sion, as we move closer to criticality, i.e. the onset of syn- chronization, the results for K = 0.7Kc and K = 0.9Kc do not fare as well. Figure 8 demonstrates a marked devi- ation from the prediction. We have not shown analytical results for the lower values of N because the deviation is so severe. The same holds true for all the results for K = 0.9Kc, so that we have just plotted the simula- tion data in Figure 9. The general trend of approaching the mean field result still holds. The primary feature to take from these plots is that the fluctuations increase the decay constant. The closer to criticality, the more impor- tant the fluctuations and the faster the decay, hence the systematic undershoot which grows as one nears critical- ity. The fastest relaxation to the incoherent state appears at high K for a given N and at low N for a given K, ei- ther limit results in increased effects from fluctuations. It would be necessary to carry the loop expansion to two or more loops in order to obtain good matches with these data. In Figures 10 and 11, we plot the time evolution of Z1(t) given that the favored oscillator has a driving fre- quency of ω0 = 0.05. Note first that Z1(t) approaches the tree level calculation as N → ∞. The amplitude of the oscillation also shows the same deviation as the ω0 = 0 data, namely that of a slight initial overshoot of the one loop prediction followed by an undershoot. In 0 500 1000 1500 Tree Level N = 10 N = 50 N=100 N = 500 N = 1000 FIG. 6: Z1(t) vs. t for various values of N and K = 0.3Kc. Each graph shows N = {10, 50, 100, 500, 1000}. Note that as N → ∞ the curve approaches the tree level value. From top to bottom: Black line represents tree level. X’s and violet line represent N = 1000. Triangles and blue line represent N = 500. Diamonds and green line represent N = 100. Boxes and red line represent N = 50. Circles and purple line represent N = 10. 0 500 1000 1500 Tree Level N = 10 N = 50 N = 100 N = 500 N = 1000 FIG. 7: Z1(t) vs. t for various values of N and K = 0.5Kc. Each graph shows N = {10, 50, 100, 500, 1000}. Note that as N → ∞ the curve approaches the tree level value. Symbols as in Figure 6. addition to this, we can see an increasing frequency shift as N → 0. The data, prediction, and mean field results eventually become out of phase. For intermediate values of N , one can see that the one loop correction follows this shift, while for N = 10, the mean field, data, and one loop results each have a different phase. In the case of n > 2 it is easier to write down a complete analytical solution for the time evolution. With ω0 = 0, θ0 = 0 we have, Zn(t) = γ − K γ − K )e−(γ− )t+Dn2t 0 500 1000 1500 Tree Level N = 50 N = 100 N = 500 N = 1000 0 500 1000 1500 Tree Level N = 10 N = 50 N = 100 N = 500 N = 1000 FIG. 8: Z1(t) vs. t for various values of N and K = 0.7Kc. Each graph shows N = {10, 50, 100, 500, 1000}. Note that as N → ∞ the curve approaches the tree level value. Symbols as in Figure 6. This is compared with a simulation result in Figure 12. We see the same general trends as the previous graphs. At large N , the simulation follows the prediction quite well. For small N , the simulation seems consistently higher than the one loop prediction with K = 0.3Kc. For K = 0.5Kc, the prediction is again sufficiently sin- gular that we have not plotted N = 10. The deviation is already apparent for N = 50. VI. DISCUSSION Using techniques from field theory, we have produced a theory which captures the fluctuations and correlations of the Kuramoto model of coupled oscillators. Although we have used the Kuramoto model as an example system, the methodology is readily extendible to other systems of coupled oscillators, even those which are not interacting via all-to-all couplings. Moreover, the methodology can be readily applied to any system which obeys a conti- nuity equation. We derive an action that describes the dynamics of the Kuramoto model. The path integral de- fined by this action constitutes an ensemble average over the configurations of the system, i.e. the phases and driv- ing frequencies of the oscillators. Because the dynamics 0 500 1000 1500 Tree Level N = 10 N = 50 N = 100 N = 500 N = 1000 FIG. 9: Z1(t) vs. t for various values of N and K = 0.9Kc. Each graph shows N = {10, 50, 100, 500, 1000}. Note that as N → ∞ the curve approaches the tree level value. Symbols as in Figure 6. of the model are deterministic, this is equivalent to an ensemble average over initial phases and driving frequen- cies. Using the loop expansion, we can compute moments of the oscillator density function perturbatively with the inverse system size as an expansion parameter. However, it is important to point out that the loop expansion is equivalent to an expansion in 1/N only because of the all-to-all coupling. A local coupling will produce fluctu- ations which do not vanish in the thermodynamic limit. Our previous work in this direction developed a mo- ment hierarchy analogous to the BBGKY hierarchy in plasma physics. This paper fully encompasses that earlier work. The equations of motion for the multi-oscillator density functions derivable from the action are in fact the equations of that BBGKY hierarchy. The calculation in Ref. [17] is in the present context the tree level calculation of the 2-point correlation function, given by the Feyn- man graph in Figure 2a. With the BBGKY hierarchy, the calculational approach involves arbitrary truncation at some order, with no a priori knowledge of how this approximation is related to the system size, N . Here we show that this approximation is entirely equivalent to the loop expansion approximation. Truncating the hierarchy at the nth moment is equivalent to truncating the loop expansion at the (n− l)th loop for the lth moment. The one loop calculation is performed in the BBGKY context by considering the linear response in the presence of the 2-point correlation function. This would produce a more roundabout manner of arriving at our one loop linear response. One should also compare our one loop calcula- tion with the Direct-Interaction-Approximation of fluid dynamics; the path integral approach in that context is the Martin-Siggia-Rose formalism [27]. Another possible equivalent means of approaching this problem is through the Ito Calculus, treating the density as a stochastic vari- able and developing a stochastic differential equation for An important aspect of our theory is that it is di- 0 500 1000 1500 -0.03 -0.02 -0.01 Tree Level One Loop N = 10 0 500 1000 1500 -0.03 -0.02 -0.01 Tree Level One Loop N = 50 0 500 1000 1500 -0.03 -0.02 -0.01 Tree Level One Loop N = 100 0 500 1000 1500 -0.03 -0.02 -0.01 N = 500 One Loop Tree Level FIG. 10: As the previous figures, but with K = 0.3Kc and ω0 = 0.05. Solid line represents tree level. Dashed blue line represents the one loop calculation. Circles represent the sim- ulation data. 0 500 1000 1500 Tree Level One Loop N = 10 0 500 1000 1500 -0.03 -0.02 -0.01 Tree Level One Loop N = 50 0 500 1000 1500 -0.03 -0.02 -0.01 Tree Level One Loop N = 100 0 500 1000 1500 -0.03 -0.02 -0.01 Tree Level One Loop N = 500 FIG. 11: As the previous figures, but with K = 0.5Kc and ω0 = 0.05. Symbols as in Figure 10. 0 500 Tree Level N = 10 N = 50 N = 100 N = 500 N = 1000 0 100 200 300 400 500 0.005 0.015 Tree Level N = 50 N = 100 N = 500 N = 1000 FIG. 12: δZ3(t) vs. t for K = 0.3Kc (top) and K = 0.5Kc (bottom). Symbols as in Figure 6. rectly related to a Markov process derivable from the Kuramoto equation. One can employ the standard Doi- Peliti method for deriving an action from a Markov pro- cess to arrive at the same theory. Although the Ku- ramoto model is deterministic, the probability distribu- tion evolves in a manner indistinguishable from a funda- mentally random process. The stochasticity of the effec- tive Markov process is due to the distribution of phases and driving frequencies. In other words, it is a state- ment about information available to us about the state of the system. The incoherent state is a state of high en- tropy. The single oscillator perturbation is one in which we have gained a small amount of information about the system and we ask a question concerning our knowledge about future states. In the mean field limit for the sin- gle oscillator perturbation, we always know where to find the perturbed oscillator given a prescription of its initial state. In the finite case, our ignorance of the positions and driving frequencies of the other oscillators makes a determination of its future location difficult. Eventually, we lose all ability to locate the perturbed oscillator as it interacts with the “heat bath” of the population. Fur- thermore, this result should be time reversal invariant. Just as we have no way of determining with accuracy where to find the oscillator in the future, likewise we have no means of determining where it has been at some time in the past. To prove this statement in the con- text of our theory would require an analysis of the “time reversed” theory, obtained essentially by switching the roles of ϕ̃ and ϕ. The relevant propagator for this time reversed theory will be the solution of the linearization of the adjoint of the mean field equation. Accordingly, this adjoint theory will have loop corrections which will damp the time reversed propagator as well. It is important to point out that our formulation ac- counts for the local stability of the incoherent state ρ(θ, ω, t) = g(ω)/2π to linear perturbations along with demonstrating the order parameters Zn approach zero. In mean field theory, there is the possibility of quasiperi- odic oscillations so that Zn = 0, i.e. the modes de-phase, while the incoherent state is marginally stable and infor- mation of the initial state is retained. Our work shows that in a finite size system, this is does not happen; the incoherent state is linearly stable. We have considered exclusively the case of fluctua- tions about the incoherent state below the critical point. Above criticality, a fraction of the population synchro- nizes. In this case, to analyze the fluctuations one may need to employ a “low temperature” expansion in con- trast to our “high temperature” treatment. In essence, one separates the populations into locked and unlocked oscillators and derives a perturbation expansion from the locked action. At criticality, each term in the loop expan- sion diverges. This is an indication that fluctuations at all scales become relevant near the transition and thus a renormalization group approach is suggested. Our for- malism provides a natural basis for this approach. In summary, we have provided a method for deriving the statistics of theories defined via a Klimontovich, or continuity, equation for a number density. This method produces a consistent means for approximating arbitrary multi-point functions. In the case of all-to-all coupling, this approximation becomes a system size expansion. We have demonstrated further that the system size correc- tions are sufficient to render the incoherent state of the Kuramoto model stable to perturbations. ACKNOWLEDGMENTS This research was supported by the Intramural Re- search Program of NIH/NIDDK. APPENDIX A: ONE LOOP CALCULATION OF THE PROPAGATOR The loop correction Γ1a applied to P is given by dt′Γ1a(θ, ω;φ, ν; t− t ′′)P (φ, ν, t′′; θ′, ω′; t′) = (A1) dθ1dω1dθ [f(θ′2 − θ) {P0(θ 2, t; θ 1, t1)P0(θ, ω, t; θ1, ω1, t1) + P0(θ 2, t; θ1, ω1, t1)P0(θ, ω, t; θ 1, t1)}] [f(θ′1 − θ1) {ρ(θ 1, t1)P (θ1, ω1, t1; θ ′, ω′, t′) + ρ(θ1, ω1, t1)P (θ 1, t1; θ ′, ω′, t′)}] This term arises from one vertex with a single incoming line and two outgoing lines and one vertex with two incoming lines and a single outgoing line, hence the product of two tree level propagators, P0. We can represent Γ1a in Fourier/Laplace space as Γ1a(n, ω, ν, t− t ′) = (A2) dω′1dω 2(2π) n(m+ n)f(−m)g(ω′1)P 0(−m,ω′2, t;ω ′)P 0(m+ n, ω, t; ν, t′) (−nm)f(m)g(ω′1)P 0(−m,ω′2, t;ω ′)P 0(m+ n, ω, t; ν, t′) (−mn)f(m+ n)g(ω′1)P 0(−m,ω′2, t; ν, t ′)P 0(m+ n, ω, t;ω′1, t n(m+ n)f(−n−m)g(ω′1)P 0(−m,ω′2, t; ν, t ′)P 0(m+ n, ω, t;ω′1, t If f(θ) is odd then f(m) = −f(−m). Therefore, Γ1a(n, ω, ν, t− t ′) = (A3) dω′1dω 2(2π) n(2m+ n)f(−m)g(ω′1)P 0(−m,ω′2, t;ω ′)P 0(m+ n, ω, t; ν, t′) n(2m+ n)f(−n−m)g(ω′1)P 0(−m,ω′2, t; ν, t ′)P 0(m+ n, ω, t;ω′1, t In this form it is easy to see the different channels which appear in the correction. Evaluating this expression using the tree level propagator (40) gives us Γ̃1a(n, ω, ν, s) = (A4) (2π)2 n(2m+ n)f(−m) imKf(m) Λ−m(s1) s1=sn NP̃ 0(m+ n, ω; ν, s− sn) n(2m+ n)f(−m− n) Λ−m(s1) s1=sn (sn − imν) (s− sn + i(m+ n)ω) Λm+n(s− sn) n(2m+ n)f(−m− n) Λ−m(imν) (s− imν + i(m+ n)ω) Λm+n(s− imν) The diagram Γ1c is given by dt′Γ1c(n, θ, ω;φ, ν; t− t ′)P (φ, ν; θ′, ω′; t′) = (A6) (2π)2 dθ′3dω dθ′idω idθidωidti f(θ′3 − θ) f(θ′1 − θ1)g(ω1)g(ω {P0(θ 3, t; θ2, ω2, t2)P0(θ, ω, t; θ1, ω1, t1) + P0(θ 3, t; θ1, ω1, t1)P0(θ, ω, t; θ2, ω2, t2)} f(θ′2 − θ2) {P0(θ 2, t2; θ 1, t1)P (θ2, ω2, t2; θ ′, ω′, t′) + P (θ′2, ω 2, t2; θ ′, ω′, t′)P0(θ2, ω2, t2; θ 1, t1)}] (A7) In Fourier space, we can write: Γ1c(n, ω, ν, t− t ′) = 2N2K3(2π)3 dω′3dω1dω 1dt1g(ω1)g(ω dω′2(imn)(m+ n)f(−m− n)f(m)f(−m)P (n+m,ω 3, t; ν, t ′)P (−m,ω, t;ω1, t1)P (m,ω ′;ω′1, t1) dω′2inm(m+ n)f(m)f(m)f(−m)P (−m,ω 3, t;ω1, t1)P (n+m,ω, t; ν, t ′)P (m,ω′2, t ′;ω′1, t1) dω2inm(m+ n)f(−n−m)f(m)f(−n)P (n+m,ω 3, t;ω2, t ′)P (−m,ω, t;ω1, t1)P (m,ω2, t ′;ω′1, t1) dω2inm(m+ n)f(m)f(m)f(−n)P (−m,ω 3, t;ω1, t1)P (n+m,ω, t;ω2, t ′)P (m,ω2, t ′;ω′1, t1) and taking the Laplace transform: Γ̃1c(n, ω, ν, s) = 2N2K3(2π)3 dω′3dω1dω 1ds1g(ω1)g(ω dω′2(imn)(m+ n)f(−m− n)f(m)f(−m)P̃ (n+m,ω 3; ν, s− s1)P (−m,ω;ω1, s1)P (m)ω 1,−s1) dω′2inm(m+ n)f(m)f(m)f(−m)P (−m,ω 3;ω1, s1)P (n+m,ω; ν, s− s1)P (m,ω 1,−s1) dω2inm(m+ n)f(−n−m)f(m)f(−n)P (n+m,ω 3;ω2, s− s1)P (−m,ω;ω1, s1)P (m,ω2;ω 1,−s1) dω2inm(m+ n)f(m)f(m)f(−n)P (−m,ω 3;ω1, s1)P (n+m,ω;ω2, s− s1)P (m,ω2;ω 1,−s1) where the contour for s1 lies between 0 and 0 < s < −Re(sn). Performing the integrals, we have: Γ̃1c(n, ω, ν, s) = (imn)(m+ n) f(−m− n)f(m)f(−m) Λn+m(s− s1) s− s1 + i(m+ n)ν s1 − imω Λ−m(s1) imKf(−m) Λm(−s1) + f(m)f(m)f(−m) imKf(m) Λ−m(s1) (2πN)P (n+m,ω; ν, s− s1) imKf(−m) Λm(−s1) dω2f(−n−m)f(m)f(−n) Λn+m(s− s1) s− s1 + i(m+ n)ω2 s1 − imω Λ−m(s1) g(ω2) −s1 + imω2 Λm(−s1) dω2f(m)f(m)f(−n) imKf(m) Λ−m(s1) (2πN)P (n+m,ω;ω2, s− s1) g(ω2) −s1 + imω2 Λm(−s1) (A10) Finally, the diagram Γ1d is given by dt′Γ1d(θ, ω;φ, ν; t− t ′)P (φ, ν; θ′, ω′; t′) = (A11) (2π)2 dθ′3dω dθ′idω idθidωi [f(θ′3 − θ)g(ω1)g(ω {P0(θ 3, t; θ2, ω2, t2)P0(θ, ω, t; θ1, ω1, t0) + P0(θ 3, t; θ1, ω1, t0)P0(θ, ω, t; θ2, ω2, t2)} f(θ′2 − θ2) {P0(θ 2, t2; θ 1, t0)P (θ2, ω2, t2; θ ′, ω′, t′) + P (θ′2, ω 2, t2; θ ′, ω′, t′)P0(θ2, ω2, t2; θ 1, t0)}] (A12) Using the fact that dω′dθ′P (θ, ω, t; θ′, ω′, t′)g(ω′) = g(ω)/N and dθf(θ) = 0 we have dt′Γ1d(θ, ω;φ, ν; t− t ′)P (φ, ν; θ′, ω′; t′) = (A13) (2π)2 dθ′3dω dθ′idω idθidωi [f(θ′3 − θ)g(ω1)g(ω 1)P0(θ 3, t; θ2, ω2, t2) f(θ′2 − θ2) P (θ 2, t2; θ ′, ω′, t′)] (A14) In Fourier-Laplace we have Γ̃1d(n;ω, ν; s) = inKf(−n)g(ω) Λn(s) (A15) APPENDIX B: f(θ) = sin θ AND g(ω) LORENTZ In order to both simplify the correction and to provide a concrete example, we will specialize to the case that g(ω) is a Lorentz distribution and f(θ) = sin θ. f(θ) = sin θ is the traditional coupling for the Kuramoto model (and has the advantage of being bounded in Fourier space so that we avoid “ultraviolet” singularities) and using a Lorentz frequency distribution yields analytical results. The Lorentz frequency distribution is given by g(ω) = γ2 + ω2 From this we can calculate Λ±1(s) = s+ γ − K and Λn(s) = 1 for n 6= ±1. The residue of the function Λ±1(s) is Λ−m(s1) s1=sn and the pole is at s±1 = −(γ − K/2). This provides a critical coupling Kc = 2γ (B4) above which the system begins to synchronize. The in- coherent state is reached when K < Kc, which gives s± < 0. From f(θ) = sin θ we also have f(±1) = ∓i/2. In this case, diagram a evaluates to Γ̃1a(n, ω, ν, s) = (B5) (2π)2 n(2m+ n)f(−m) imKf(m) NP̃ 0(m+ n, ω; ν, s+ γ − n(2m+ n)f(−m− n) − γ − imν s+ γ − K + i(m+ n)ω s+ 2γ − K s+ 2γ −K n(2m+ n)f(−m− n) Λ−m(imν) (s− imν + i(m+ n)ω) s+ γ − imν s+ γ − K − imν There is no contribution for n = 0 which we expect from probability conservation. The terms proportional to n(2m + n)f(−n − m) will always evaluate to 0, be- cause f(n 6= ±1) = 0. Since the tree level propagator contains such a term as well, we have the simplification Γ̃1a(n, ω, ν, s) = (B7) n(2m+ n) δ(ω − ν) s+ γ − K + i(m+ n)ω For diagram c, we see that we can immediately ignore the third term because nmf(−n)f(−m− n) is always 0. We also see that the first term is only non-zero for n = ±2, and the last term only for n = ±1. After performing the s1 integration, this leaves us with Γ̃1c(n, ω, ν, s) = n(m+ n) δ(ω − ν) s+ γ − K + i(n+m)ω 2γ −K + δn±1 s+ γ − K + 2inω γ − K + inω 2γ − K 2γ −K + δn±2(−g(ω)) iω + γ − K i(ν − ω) iω + γ γ − K s+ γ − K s+ 2γ −K s+ γ − K 2γ −K −γ + K 2γ −K s+ γ − K s+ γ − K s+ 2γ − K s+ 2γ −K Γ1d is given simply by Γ̃1d(±1;ω, ν; s) = − s+ γ − K Γ̃1d(n 6= ±1;ω, ν; s) = 0 (B9) APPENDIX C: EQUIVALENT MARKOV PROCESS The action (15) can be derived by applying the Doi- Peliti method to a Markov process equivalent to the Ku- ramoto dynamics. Consider a two dimensional lattice L, periodic in one dimension, with lattice constants aθ in the periodic direction and aω in the other. (The radius of the eventual cylinder is R.) The indices i and j will be used for the frequency and periodic domains, respec- tively. The oscillators obey an equation of the form: θ̇i = v ~θ, ~ω The indices on ~θ and ~ω run over the lattice points of the periodic and frequency variables. The state of the system is described by the number of oscillators ni,j at each site. Given this, the fraction of oscillators found on the lattice sites is governed by the following Master equation: dP (~n, t) ni,jP (~n, t) vi,j−1 (ni,j−1 + 1)P ~nj , t where the indices of the vector ~n run over the lattice points, L, and ~nj (note the superscript) is equal to ~n except for the jth and j − 1st components. At those points we have n i,j = ni,j − 1 and n i,j−1 = ni,j−1 + 1. The first term on the RHS represents the outward flux of oscillators from the state with nj oscillators at each periodic lattice point while the second term is the inward flux due to oscillators “hopping” from j − 1 to j. There is no flux in the other direction (ω); this lattice variable simply serves to label each oscillator by its fundamental frequency. Consider a generalization of the Kuramoto model of the form: θ̇i = ωi + f (θj − θi) (C3) N is the total number of oscillators and we impose f(0) = 0. The velocity in equation (C1) now has the form: vij = iaω + i′,j′ f ([j′ − j]aθ)ni′,j′ (C4) In the limit aω → 0 we have iaω = ω. Similarly, iaθ → θ. The factor of ni′,j has been added because the sum must cover all oscillators, and this factor describes the number at each site. We also sum over all frequency sites i′. The master equation (C2) now takes the form: dP (~n, t) i′,j′ f ([j′ − j]aθ)ni′,j′ ×ni,jP (~n, t) i′,j′ f ([j′ − j + 1]aθ)ni′,j′ ×(ni,j−1 + 1)P ~nj , t = −HP (~n, t) (C5) The matrix H is the Hamiltonian. From this point, one can develop an operator representation as in Doi-Peliti. Using coherent states and taking the continuum and ther- modynamic limits results in the action (15), after “shift- ing” the field ϕ̃. [1] A. T. Winfree, Journal of Theoretical Biology 16, 15 (1967). [2] C. Liu, D. Weaver, S. Strogatz, and S. Reppert, Cell 91, 855 (1987). [3] D. Golomb and D. Hansel, Neural Computation 12, 1095 (2000). [4] G. B. Ermentrout and J. Rinzel, Am. J. Physiol 246, R102 (1984). [5] G. B. Ermentrout, Journal of Mathematical Biology 29, 571 (1991). [6] T. J. Walker, Science 166, 891 (1969). [7] J. Pantaleone, Physical Review D 58 (1998). [8] S. Y. Kourtchatov, V. V. Likhanskii, and A. P. Npar- tovich, Physical Review A 52 (1995). [9] K. Wiesenfeld, P. Colet, and S. H. Strogatz, Physical Review Letters 76 (1996). [10] Y. Kuramoto, Chemical Oscillations, Waves, and Turbu- lence (Springer-Verlag, 1984). [11] S. H. Strogatz, Physica D 143, 1 (2000). [12] S. H. Strogatz, R. E. Mirollo, and P. C. Matthews, Phys- ical Review Letters 68, 2730 (1992). [13] R. E. Mirollo and S. H. Strogatz, Physica D 205, 249 (2005). [14] R. E. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the kuramoto model, eprint arXiv:nlin/0702043. [15] H. Daido, Journal of Statistical Physics 60, 753 (1990). [16] H. Daido, Progress of Theoretical Physics 75, 1460 (1986). [17] E. J. Hildebrand, M. A. Buice, and C. C. Chow, Physical Review Letters 98 (2007). [18] L. Peliti, Journal de Physique 46, 1469 (1985). [19] M. Doi, Journal of Physics A: Mathematical and General 9, 1465 (1976). [20] M. Doi, Journal of Physics A: Mathematical and General 9, 1479 (1976). [21] H.-K. Janssen and U. C. Tauber, Annals of Physics 315, 147 (2005). [22] J. Zinn-Justin, Quantum Field Theory and Critical Phe- nomena (Oxford Science Publications, 2002), 4th ed. [23] U. C. Tauber, Lecture Notes in Physics 716, 295 (2006). [24] S. H. Strogatz and R. E. Mirollo, Journal of Statistical Physics 63, 613 (1991). [25] S. Ichimaru, Basic principles of Plasma Physics: A Sta- tistical Approach (W.A. Benjamin Advanced Book Pro- gram, 1973). [26] D. R. Nicholson, Introduction to Plasma Theory (Krieger Publishing Company, 1992). [27] P. C. Martin, E. D. Siggia, and H. A. Rose, Physical Review A 8, 423 (1973). [28] A. V. Rangan and D. Cai, Physical Review Letters 96 (2006). [29] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Physical Review D 10, 2428 (1974). [30] R. E. Mirollo and S. H. Strogatz, Journal of Statistical Physics 60, 245 (1990). [31] J. D. Crawford, Physical Review Letters 74 (1995). [32] J. D. Crawford and K. Davies, Physica D 125, 1 (1999).

0704.1651

Route to Lambda in conformally coupled phantom cosmology

Route to Lambda in conformally coupled phantom cosmology Orest Hrycyna∗ Department of Theoretical Physics, Faculty of Philosophy, The John Paul II Catholic University of Lublin, Al. Rac lawickie 14, 20-950 Lublin, Poland and Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland Marek Szyd lowski† Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland and Mark Kac Complex Systems Research Centre, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland In this letter we investigate acceleration in the flat cosmological model with a conformally coupled phantom field and we show that acceleration is its generic feature. We reduce the dynamics of the model to a 3-dimensional dynamical system and analyze it on a invariant 2-dimensional submanifold. Then the concordance FRW model with the cosmological constant Λ is a global attractor situated on a 2-dimensional invariant space. We also study the behaviour near this attractor, which can be approximated by the dynamics of the linearized part of the system. We demonstrate that trajectories of the conformally coupled phantom scalar field with a simple quadratic potential crosses the cosmological constant barrier infinitely many times in the phase space. The universal behaviour of the scalar field and its potential is also calculated. We conclude that the phantom scalar field conformally coupled to gravity gives a natural dynamical mechanism of concentration of the equation of state coefficient around the magical value weff = −1. We demonstrate route to Lambda through the infinite times crossing the weff = −1 phantom divide. PACS numbers: 98.80.Bp, 98.80.Cq, 11.15.Ex At present the scalar fields play a crucial role in modern cosmology. In an inflationary scenario they generate an exponential rate of evolution of the universe as well as density fluctuations due to vacuum energy. The Lagrangian for a phantom scalar field on the background of the Friedmann-Robertson-Walker (FRW) universe is assumed in the [gµν∂µψ∂νψ + ξRψ 2 − 2U(ψ)], (1) where gµν is the metric of the spacetime manifold, ψ = ψ(t), t is the cosmological time, R = R(g) is the Ricci scalar for the spacetime metricg, ξ is a coupling constant which assumes zero for a scalar field minimally coupled to gravity and 1/6 for a conformally coupled scalar field, U(ψ) is a potential of the scalar field. The minimally coupled slowly evolving scalar fields with a potential function U(ψ) are good candidates for a description of dark energy. In this model, called quintessence [1, 2], the energy density and pressure from the scalar field are ρψ = −1/2ψ̇2 +U(ψ), pψ = −1/2ψ̇2−U(ψ). From recent studies of observational constraints we obtain that wψ ≡ pψ/ρψ < −0.55 [3]. This model has been also extended to the case of a complex scalar field [4, 5]. Observations of distant supernovae support the cosmological constant term which corresponds to the case ψ̇ ≃ 0. Then we obtain that wψ = −1. But there emerge two problems in this context. Namely, the fine tuning and the cosmic coincidence problems. The first problem comes from the quantum field theory where the vacuum expectation value is of 123 orders of magnitude larger than the observed value of 10−47GeV4. The lack of a fundamental mechanism which sets the cosmological constant almost zero is called the cosmological constant problem. The second problem called “cosmic conundrum” is a question why the energy densities of both dark energy and dark matter are nearly equal at the present epoch. One of the solutions to this problem offers the idea of quintessence, which is a version of the time varying cosmological constant conception. Quintessence solves the first problem through the decaying Λ term from the beginning of the Universe to a small value observed at the present epoch. Also the ratio of energy density of this field to the matter density increases slowly during the expansion of the Universe because the specific feature of this model is the variation of the coefficient of the equation of state with respect to time. The quintessence models [2, 6] describe the dark energy with the time varying equation of state for which wX > −1, but recently quintessence models have been extended to the phantom quintessence models with wX < −1. In this class of models the weak energy condition is violated and ∗Electronic address: [email protected] †Electronic address: [email protected] http://arxiv.org/abs/0704.1651v3 mailto:[email protected] mailto:[email protected] such a theoretical possibility is realized by a scalar field with a switched sign in the kinetic term ψ̇2 → −ψ̇2 [7, 8, 9]. From theoretical point of view it is necessary to explore different evolutional scenarios for dark energy which provide a simple and natural transition to wX = −1. The methods of dynamical systems with notion of attractor (a limit set with an open inset) offers the possibility of description of transition trajectories to the regime with wX = −1. Moreover they demonstrate whether this mechanism is generic. Inflation and quintessence with non-minimal coupling constant are studied in the context of formulation of necessary conditions for the acceleration of the universe [10] (see also [11, 12]). We can find two important arguments which favour the choice of conformal coupling over ξ 6= 1/6. The first, equation for the massless scalar field is conformally invariant [13, 14]. The second argument is that if the scalar field satisfy Klein-Gordon equation in the curved space then ψ does not violate the equivalence principle, and ξ is forced to assume the value 1/6 [15]. While recent astronomical observations give support that the equation of state parameter for dark energy is close to constant value −1 they do not give a “corridor” around this value. Moreover, Alam et al. [16] pointed out that evolving state parameter is favoured over constant wX = −1. The first step in the direction of description of the dynamics of the dark energy seems to be investigation of the system with evolving dark energy in the close neighbourhood of the value wX = −1. For this aim we linearize dynamical system at this critical point and then describe the system in a good approximation (following the Hartman-Grobman theorem [17]) by its linearized part. Other dark energy models like the Chaplygin gas model [18, 19, 20], [9, and references therein] and the model with tachyonic matter can also be interpreted in terms of a scalar field with some form of a potential function. Recent applications of the Bayesian framework [21, 22, 23, 24, 25] of model selection to the broad class of cosmo- logical models with acceleration indicate that a posteriori probability for the ΛCDM model is 96%. Therefore the explanation why the current universe is such close to the ΛCDM model seems to be a major challenge for modern theoretical cosmology. In this letter we present the simplest mechanism of concentration around wX = −1 basing on the influence of a single scalar field conformally coupled to gravity acting in the radiation epoch. Phantom cosmology non-minimally coupled to the Ricci scalar was explored in the context of superquintesence (wX < −1) by Faraoni [26, 27] and there was pointed out that the superacceleration regime can be achieved by the conformally coupled scalar field in contrast to the minimally coupled scalar field. Let us consider the flat FRW model which contains a negative kinetic scalar field conformally coupled to gravity (ξ = 1/6) (phantom) with the potential function U(ψ). For the simplicity of presentation we assume U(ψ) ∝ ψ2. In this model the phantom scalar field is coupled to gravity via the term ξRψ2. We consider massive scalar fields (for recent discussion of cosmological implications of massive and massless scalar fields see [28]). The dynamics of a non-minimally coupled scalar field for some self-interacting potential U(ψ) and for an arbitrary ξ is equivalent to the action of the phantom scalar field (which behaves like a perfect fluid) with energy density ρψ and pressure pψ [29] ρψ = − ψ̇2 + U(ψ) − 3ξH2ψ2 − 3ξH(ψ2)̇, (2) pψ = − ψ̇2 − U(ψ) + ξ 2H(ψ2)̇ + (ψ2)̈ 2Ḣ + 3H2 ψ2, (3) where the conservation condition ρ̇ψ = −3H(ρψ + pψ) gives rise to the equation of motion for the field ψ̈ + 3Hψ̇ + ξRψ2 − U ′(ψ) = 0, (4) where R = 6 Ḣ + 2H2 is the Ricci scalar. Let us assume that both the homogeneous scalar field ψ(t) and the potential U(ψ) depend on time through the scale factor, i.e. ψ(t) = ψ(a(t)), U(ψ) = U(ψ(a)); (5) then due to this simplified assumption the coefficient of the equation of state wψ is parameterized by the scale factor wψ = wψ(a), pψ = wψ(a)ρψ(a), (6) ψ′2H2a2 − U(ψ) + ξ 2(ψ2)′H2a+ (ψ2 )̈ Ḣ + 3H2 ψ′2H2a2 + U(ψ) − 3ξH2ψ2 − 3ξ(ψ2)′H2a where prime denotes the differentiation with respect to the scale factor. We assume the flat model with the FRW geometry, i.e., the line element has the form ds2 = −dt2 + a2(t)[dr2 + r2(dθ2 + sin2 θdϕ2)], (8) where 0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π and 0 ≤ r ≤ ∞ are comoving coordinates, t stands for the cosmological time. It is also assumed that a source of gravity is the phantom scalar field ψ with the conformal coupling to gravity ξ = 1/6. The dynamics is governed by the action m2pR+ (g µνψµψν + Rψ2 − 2U(ψ)) where m2p = (8πG) −1; for simplicity and without lost of generality we assume 4πG/3 = 1 and U(ψ) is the scalar field potential U(ψ) = m2ψ2. (10) After dropping the full derivatives with respect to time, rescaling phantom field ψ → φ = ψa and the time variable to the conformal time dt = adη we obtain the energy conservation condition a′2 + φ′2 − m2a2φ2 = ρr,0 (11) where ρr,0 is constant corresponding to the radiation in the model. The equations of motion are a′′ = m2aφ2, φ′′ = m2a2φ where a prime denotes the differentiation with respect to the conformal time dt = adη and m2 > 0. From the energy conservation condition we have a′2 = ρr,0 + m2a2φ2 1 + φ̇2 and now from the equations of motion (12) we receive (ρr,0 + m2a2φ2)φ̈+ m2aφ(1 + φ̇2)(φφ̇ − a) = 0. (14) The effective equation of state parameter is weff = ρφ + ρr , (15) for our model this parameter reduces to weff = − φ′2 + 1 m2a2φ2 − ρr,0 φ′2 + 1 m2a2φ2 + ρr,0 where a prime denotes the differentiation with respect to the conformal time and finally taking into account equa- tion (13) we have weff = − φ̇2 + m2a2φ2 − ρr,0 m2a2φ2 + ρr,0 (1 + φ̇2) . (17) For a, φ≫ ρr,0 this equation reduces to weff = − (2φ̇2 + 1), (18) and it is clear that for any value of φ̇ weff is always negative. To analyze equation (14) we reintroduce the original phantom field variable ψ = φ and da/a = d ln a. Now equation (14) reads (ψ′′ + ψ′) + m2ψ(1 + (ψ′ + ψ)2)(ψ(ψ′ + ψ) − 1) = 0 (19) where a prime now denotes the differentiation with respect to a natural logarithm of the scale factor. Introducing new variables y = ψ′ and ρr = ρr,0a −4 we can represent this equation as an autonomous dynamical system ψ′ = y y′ = −y − (ψ(y + ψ) − 1)(1 + (y + ψ)2) (20) ρ′r = −4ρr. There are the two critical points in the phase space (ψ, y, ρr), namely ψ = ±1, y = 0, ρr = 0. The linearization matrix reads 0 1 0 −2(1 + ψ2) −1 − (1 + ψ2) 2 (ψ2 − 1)(1 + ψ2) 0 0 −4 y=0,ρr=0 0 1 0 −4 −3 0 0 0 −4 y=0,ρr=0,ψ=±1 . (21) The eigenvalues for this matrix are λ1,2 = (−3 ± i 7) and λ3 = −4. To find a global phase portrait it is necessary to study the system in the neighbourhood of the critical points which correspond, from the physical point of view, stationary states (or asymptotic solutions). Then the Hartman-Grobman theorem guaranties us that the linearized system at this point is a well approximation of the nonlinear system. First, we must note that ρr = 0 is in the invariant submanifold of the 3-dimensional nonlinear system. It is also useful to calculate the eigenvectors for any eigenvalue. We obtain following eigenvectors v1,2 =  , v3 =  . (22) They are helpful in construction of the exact solution of the linearized system ~x(t) = ~x(0) exp t 0 1 0 −4 −3 0 0 0 −4 0 − 3 0 1 0 1 0 0 e−4t 0 0 t cos t −e− 32 t sin t sin t cos 0 0 1 0 1 0 where x = ψ − ψ0, y = ψ′ − ψ′0, z = ρr − ρ0 and x0, y0, z0 are initial conditions and we have substituted ln a = t. If we consider linearized system on the invariant stable submanifold z = 0, it is easy to find the exact solution. If we return to the original variables ψ, ψ′, then ψ(ln a) is the solution of the linear equation (ψ − ψ0)′′ + 3(ψ − ψ0)′ + 4(ψ − ψ0) = 0, (24) i. e., (ψ − ψ0) = C1 exp + C2 exp (φ− φ0) = C1a− 2 cos + C2a 2 sin . (26) Because of the lack of alternatives to the mysterious cosmological constant [21, 22] we allow that energy might vary in time following assumed a priori parameterization of w(z). In the popular parameterization [30, 31, 32, 33] appears a free function in most scenarios which is a source of difficulties in constraining parameters by observations. However most parameterizations of the dark energy equation of state cannot reflect real dynamics of cosmological models with dark energy. The assumed form of w(z) can be incompatible with the w(z) obtained from the underlying dynamics of the cosmological model. For example some of parameters can be determined from the dynamics which can be crucial in testing and selection of cosmological models [21]. Our point of view is to obtain the form of w(z) specific for given class of cosmological models from dynamics of this models and apply it in further analysis both theoretical and empirical. In practice we put the cosmological model in the form of the dynamical system and linearize it around the neighbourhood of the present epoch to find the exact formula of w(z). For the phantom scalar field model this incompability manifests by the presence of a focus type critical point (therefore damping oscillations) in the phase space rather than a stable node (Fig. 1 and its 3D version Fig. 2). The properties of the minimally coupled phantom field in the FRW cosmology using the phase portrait have been investigated by Singh et al. [34] (see also [35] for more recent studies). Authors showed the existence of the deSitter attractor and damped oscillations (the ratio of the kinetic to the potential energy |T/U | to oscillate to zero). We can also express weff in these new variables weff = − (ψ + ψ′)2 − ρr − 12m 1 + (ψ + ψ′)2 w′eff = dweff d ln a (ψ + ψ′)(ψ′ + ψ′′) + ρrm 2 + ψ (ρr + m2ψ2)2 1 + (ψ + ψ′)2 . (28) Recently Caldwell and Linder [36] have discussed dynamics of quintessence models of dark energy in terms of w−w′ phase variables, where w′ was the differentiation with respect to the logarithm of the scale factor. These methods were extended to the phantom and quintom models of dark energy [37, 38]. Guo et al. [38] examined the two-field quintom models as the illustration of the simplest model of transition across the wX = −1 barrier. The interesting mechanism of acceleration with a periodic crossing of the w = −1 barrier have been recently discussed in the context of the cubic superstring field theory [39]. In the model under consideration we obtain this effect but trajectories cross the barrier infinitely many times. The main advantage of the discovered road to Λ is that it takes place in the simple flat FRW model with the quadratic potential of the scalar field. It is easy to check that at the critical points weff = −1 and dweffd lna = 0. Since these points are sinks there is infinite many crossings of weff = −1 during the evolution. The methods of the Lyapunov function are useful in discussion of stability of the critical point of the non-linear system. The stability of any hyperbolic critical point of dynamical system is determined by the signs of the real parts of the eigenvalues λi of the Jacobi matrix. A hyperbolic critical point is asymptotically stable iff real λi < 0 ∀i, if x0 is a sink. The hyperbolic critical point is unstable iff it is either a source or a saddle. The method of the Lyapunov function is especially useful in deciding the stability of a non-hyperbolic critical points [17, p.129]. The construction of the Lyapunov function was used by [40] for demonstration that periodic behaviour of a single scalar field is not possible for minimally coupled phantom scalar field (see also [41]). The quantity w′eff in terms of weff and (lnψ) ′ reads w′eff = −(1 − 3weff)(1 + weff + (lnψ)′). (29) It is interesting that equation (29) can be solved in terms of w̄(a) – the mean of the equation of the state parameter in the logarithmic scale defined by Rahvar and Movahed [42] as w̄(a) = w(a′)d(ln a′) d(ln a′) , (30) namely: w(a) = a3(1+w̄(a))ψ2. (31) They argued that this phenomenological parameterization removes the fine tuning of dark energy and ρX/ρm ∝ a−3w̄(a) approaches a unity at the early universe. Note that in w̄(a) = −1 that w(z) + 1 = (1 − ψ2), (32) FIG. 1: The phase portrait (weff, w eff) of the investigated model on the submanifold ρr = 0. This figure illustrates the evolution of the dark energy equation of the state parameter as a function of redshift for different initial conditions. In all cases trajectories cross the boundary line weff = −1 infinite many times but this state also represents the global attractor. where ψ = ψ0 + (1 + z) C1 cos( ln(1 + z)) − C2 sin( ln(1 + z)) . (33) In Fig. 3 we present the relation w(z) for different values of parameters ψ0 = ±1, C1 and C2. In this letter we regarded the phantom scalar field conformally coupled to gravity in the context of the problem of acceleration of the Universe. We applied the methods of dynamical systems and the Hartman-Grobman theorem to find universal behaviour at the late times – damping oscillations around weff = −1. We argued that most parameterizations of the dark energy, such as linear evolution of w(z) in redshift or the scale factor, cannot reflect realistic physical models because of the presence of non-hyperbolic critical point of a focus type on the phase plane (w,w′). We suggested a parameterization of a type wX(z) = −1 + (1 + z)3 C1 cos(ln(1 + z)) + C2 sin(ln(1 + z)) which parameterizes damping oscillations around wX = −1 “phantom divide”, and finally, with the help of this formula one can simply calculate energy density for dark energy ρX ρX = ρX,0 exp (1 + z)3 A sin(ln(1 + z)) +B cos(ln(1 + z)) . (35) Acknowledgments The work of M.S. has been supported by the Marie Curie Actions Transfer of Knowledge project COCOS (contract MTKD-CT-2004-517186). [1] C. Wetterich, Nucl. Phys. B 302, 668 (1988). [2] B. Ratra and P. J. E. Peebles, Phys. Rev. D37, 3406 (1988). [3] K.-H. Chae, A. Biggs, R. Blandford, I. Browne, A. de Bruyn, C. Fassnacht, P. Helbig, N. Jackson, L. King, L. Koopmans, et al. 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0704.1652

Interaction of Supernova Ejecta with Nearby Protoplanetary Disks

Interaction of Supernova Ejecta with Nearby Protoplanetary Disks N. Ouellette Department of Physics, Arizona State University, PO Box 871504, Tempe, AZ 85287-1504 S. J. Desch and J. J. Hester School of Earth and Space exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404 Received ; accepted http://arxiv.org/abs/0704.1652v1 – 2 – ABSTRACT The early Solar System contained short-lived radionuclides such as 60Fe (t1/2 = 1.5 Myr) whose most likely source was a nearby supernova. Previous models of Solar System formation considered a supernova shock that triggered the collapse of the Sun’s nascent molecular cloud. We advocate an alternative hy- pothesis, that the Solar System’s protoplanetary disk had already formed when a very close (< 1 pc) supernova injected radioactive material directly into the disk. We conduct the first numerical simulations designed to answer two questions re- lated to this hypothesis: will the disk be destroyed by such a close supernova; and will any of the ejecta be mixed into the disk? Our simulations demonstrate that the disk does not absorb enough momentum from the shock to escape the protostar to which it is bound. Only low amounts (< 1%) of mass loss occur, due to stripping by Kelvin-Helmholtz instabilities across the top of the disk, which also mix into the disk about 1% of the intercepted ejecta. These low efficiencies of destruction and injectation are due to the fact that the high disk pressures prevent the ejecta from penetrating far into the disk before stalling. Injection of gas-phase ejecta is too inefficient to be consistent with the abundances of ra- dionuclides inferred from meteorites. On the other hand, the radionuclides found in meteorites would have condensed into dust grains in the supernova ejecta, and we argue that such grains will be injected directly into the disk with nearly 100% efficiency. The meteoritic abundances of the short-lived radionuclides such as 60Fe therefore are consistent with injection of grains condensed from the ejecta of a nearby (< 1 pc) supernova, into an already-formed protoplanetary disk. Subject headings: methods: numerical—shock waves—solar system: formation—stars: formation—supernovae: general – 3 – 1. Introduction Many aspects of the formation of the Solar System are fundamentally affected by the Sun’s stellar birth environment, but to this day the type of environment has not been well constrained. Did the Sun form in a quiescent molecular cloud like the Taurus molecular cloud in which many T Tauri stars are observed today? Or did the Sun form in the vicinity of massive O stars that ionized surrounding gas, creating an H ii region before exploding as core-collapse supernovae? Recent isotopic analyses of meteorites reveal that the early Solar System held live 60Fe at moderately high abundances, 60Fe/56Fe ∼ 3− 7× 10−7 (Tachibana & Huss 2003; Huss & Tachibana 2004; Mostefaoui et al. 2004, 2005; Quitte et al. 2005; Tachibana et al. 2006). Given these high initial abundances, the origin of this short-lived radionuclide (SLR), with a half-life of 1.5 Myr, is almost certainly a nearby supernova, and these meteoritic isotopic measurements severely constrain the Sun’s birth environment. Since its discovery, the high initial abundance of 60Fe in the early Solar System has been recognized as demanding an origin in a nearby stellar nucleosynthetic source, almost certainly a supernova (Jacobsen 2005; Goswami et al. 2005; Ouellette et al. 2005; Tachibana et al. 2006, Looney et al. 2006). Inheritance from the interstellar medium (ISM) can be ruled out: the average abundance of 60Fe maintained by ongoing Galactic nucleosynthesis in supernovae and asymptotic-giant-branch (AGB) stars is estimated at 60Fe/56Fe = 3× 10−8 (Wasserburg et al. 1998) to 3 × 10−7 (Harper 1996), lower than the meteoritic ratio. Moreover, this 60Fe is injected into the hot phase of the ISM (Meyer & Clayton 2000), and incorporation into molecular clouds and solar systems takes ∼ 107 years or more (Meyer & Clayton 2000; Jacobsen 2005), by which time the 60Fe has decayed. A late source is argued for (Jacobsen 2005; see also Harper 1996, Meyer & Clayton 2000). Production within the Solar System itself by irradiation of rocky material by solar energetic particles has been proposed for the origin of other SLRs (e.g., Lee et al. 1998; Gounelle et al. 2001), but – 4 – neutron-rich 60Fe is produced in very low yields by this process. Predicted abundances are 60Fe/56Fe ∼ 10−11, too low by orders of magnitude to explain the meteoritic abundance (Lee et al. 1998; Leya et al. 2003; Gounelle et al. 2006). The late source is therefore a stellar nucleosynthetic source, either a supernova or an AGB star. AGB stars are not associated with star-forming regions: Kastner & Myers (1994) used astronomical observations to estimate a firm upper limit of ≈ 3× 10−6 per Myr to the probability that our Solar System was contaminated by material from an AGB star. The yields of 60Fe from an AGB star also may not be sufficient to explain the meteoritic ratio (Tachibana et al. 2006). Supernovae, on the other hand, are commonly associated with star-forming regions, and a core-collapse supernova is by far the most plausible source of the Solar System’s 60Fe. Supernovae are naturally associated with star-forming regions because the typical lifetimes of the stars massive enough to explode as supernovae (>∼ 8M⊙) are 7 yr, too short a time for them to disperse away from the star-forming region they were born in. Low-mass (∼ 1M⊙) stars are also born in such regions. In fact, astronomical observations indicate that the majority of low-mass stars form in association with massive stars. Lada & Lada (2003) conducted a census of protostars in deeply embedded clusters complete to 2 kpc and found that 70-90% of stars form in clusters with > 100 stars. Integration of the cluster initial mass function indicates that of all stars born in clusters of at least 100 members, about 70% will form in clusters with at least one star massive enough to supernova (Adams & Laughlin 2001; Hester & Desch 2005). Thus at least 50% of all low-mass stars form in association with a supernova, and it is reasonable to assume the Sun was one such star. Astronomical observations are consistent with, and the presence of 60Fe demands, formation of the Sun in association with at least one massive star that went supernova. While the case for a supernova is strong, constraining the proximity and the timing of the supernova is more difficult. The SLRs in meteorites provide some constraints on – 5 – the timing. The SLR 60Fe must have made its way from the supernova to the Solar System in only a few half-lives; models in which multiple SLRs are injected by a single supernova provide a good match to meteoritic data only if the meteoritic components containing the SLRs formed <∼ 1 Myr after the supernova (e.g., Meyer 2005, Looney et al. 2007). The significance of this tight timing constraint is that the formation of the Solar System was somehow associated with the supernova. Cameron & Truran (1977) suggested that the formation of the Solar System was triggered by the shock wave from the same supernova that injected the SLRs, and subseqeuent numerical simulations show this is a viable mechanism, provided several parsecs of molecular gas lies between the supernova and the Solar System’s cloud core, or else the supernova shock will shred the molecular cloud (Vanhala & Boss 2000, 2002). The likelihood of this initial condition has not yet been established by astronomical observations. Also in 1977, T. Gold proposed that the Solar System acquired its radionuclides from a nearby supernova, after its protoplanetary disk had already formed (Clayton 1977). Astronomical observations strongly support this scenario, especially since protoplanetary disks were directly imaged ∼ 0.2 pc from the massive star θ1 Ori C in the Orion Nebula (McCaughrean & O’Dell 1996). Further imaging has revealed protostars with disks near (≤ 1 pc) massive stars in the Carina Nebula (Smith et al. 2003), NGC 6611 (Oliveira et al. 2005), and M17 and Pismis 24 (de Marco et al. 2006). This hypothesis, that the Solar System acquired SLRs from a supernova that occurred < 1 pc away, after the Sun’s protoplanetary disk had formed, is the focus of this paper. In this paper we address two main questions pertinent to this model. First, are protoplanetary disks destroyed by the explosion of a supernova a fraction of a parsec away? Second, can supernova ejecta containing SLRs be mixed into the disk? These questions were analytically examined in a limited manner by Chevalier (2000). Here we present the first multidimensional numerical simulations of the interaction of supernova ejecta with protoplanetary disks. In §2 we describe the numerical code, Perseus, we have written to – 6 – study this problem. In §3 we discuss the results of one canonical case in particular, run at moderate spatial resolution. We examine closely the effects of our limited numerical resolution in §4, and show that we have achieved sufficient convergence to draw conclusions about the survivability of protoplanetary disks hit by supernova shocks. We conduct a parameter study, investigating the effects of supernova energy and distance and disk mass, as described in §5. Finally, we summarize our results in §6, in which we conclude that disks are not destroyed by a nearby supernova, that gaseous ejecta is not effectively mixed into the disks, but that solid grains from the supernova likely are, thereby explaining the presence of SLRs like 60Fe in the early Solar System. 2. Perseus We have written a 2-D (cylindrical) hydrodynamics code we call Perseus. Perseus (son of Zeus) is based heavily on the Zeus algorithms (Stone & Norman 1992). The code evolves the system while obeying the equations of conservation of mass, momentum and energy: + ρ∇ · ~v = 0 (1) = −∇p− ρ∇Φ (2) = −p∇ · ~v, (3) where ρ is the mass density, ~v is the velocity, p is the pressure, e is the internal energy density and Φ is the gravitational potential (externally imposed). The Lagrangean, or comoving derivative D/Dt is defined as + ~v · ∇. (4) The pressure and energy are related by the simple equation of state appropriate for the ideal gas law, p = e(γ − 1), where γ is the adiabatic index. The term p∇ · ~v represents mechanical work. – 7 – Currently, the only gravitational potential Φ used is a simple point source, representing a star at the center of a disk. This point mass is constrained to remain at the origin. Technically this violates conservation of momentum by a minute amount by excluding the gravitational force of the disk on the central star. As discussed in §4, the star should acquire a velocity ∼ 102 cm s−1 at the end of our simulations. In future simulations we will include this effect, but for the problem explored here this is completely negligible. The variables evolved by Perseus are set on a cylindrical grid. The program is separated in two steps: the source and the transport step. The source step calculates the changes in velocity and energy due to sources and sinks. Using finite difference approximations, it evolves ~v and e according to = −∇p− ρ∇Φ−∇ ·Q (5) = −p∇ · ~v −Q : ∇~v, (6) where Q is the tensor artificial viscosity. Detailed expressions for the artificial viscosity can be found in Stone & Norman (1992). The transport step evolves the variables according to the velocities present on the grid. For a given variable A, the conservation equation is solved, using finite difference approximations: AdV = − A~v · d~S. (7) The variables A advected in this way are density ρ, linear and angular momentum ρ~v and Rρvφ, and energy density e. As in the Zeus code, A on each surface element is found with an upwind interpolation scheme; we use second-order van Leer interpolation. Perseus is an explicit code and must satisfy the Courant-Friedrichs-Lewis (CFL) stability criterion. The amount of time advanced per timestep, essentially, must not exceed – 8 – the time it could take for information to cross a grid zone in the physical system. In every grid zone, the thermal time step δtcs = ∆x/(cs) is computed, where ∆x is the size of the zone (smallest of the r and z dimension) and cs is the sound speed. Also computed are δtr = δr/(|vr|) and δtz = δz/(|vz|), where ∆r and ∆z are the sizes of the zone in the r and z directions respectively. Because of artificial viscosity, a viscous time step must also be added for stability. For a given grid zone, the viscous time step δtvisc = max(|(l ∇ · ~v/δr |(l ∇ · ~v/δz2)|) is computed, where l is a length chosen to be a 3 zone widths. The final ∆t is taken to be ∆t = C0 (δt + δt−2r + δt z + δt visc) −1/2, (8) where C0 is the Courant number, a safety factor, taken to be C0=0.5. To insure stability, ∆t is computed over all zones, and the smallest value is kept for the next step of the simulation. Boundary conditions were implemented using ghost zones as in the Zeus code. To allow for supernova ejecta to flow past the disk, inflow boundary conditions were used at the upper boundary (z = zmax), and outflow boundary conditions were used at the lower boundary (z = zmin) and outer boundary (r = rmax). Reflecting boundary conditions, were used on the inner boundary (r = rmin 6= 0) to best model the symmetry about the protoplanetary disk’s axis. The density and velocity of gas flowing into the upper boundary were varied with time to match the ejecta properties (see §3). A more detailed description of the algorithms used in Perseus can be found in Stone & Norman (1992). – 9 – 2.1. Additions to Zeus To consider the particular problem of high-velocity ejecta hitting a protoplanetary disk, we wrote Perseus with the following additions to the Zeus code. One minor change is the use of a non-uniform grid. In all of our simulations we used an orthogonal grid with uniform spacing in r but non-uniform spacing in the z direction. For example, in the canonical simulation (§3), the computational domain extends from r = 4 to 80 AU, with spacing ∆r = 1AU, for a total of 76 zones in r. The computational domain extends from z = −50AU to +90AU, but zone spacings vary with z, from ∆z = 0.2AU at z = 0, to ∆z ≈ 3AU at the upper boundary. Grid spacings increased geometrically by 5% per zone, for a total of 120 zones in z. Another addition was the use of a radiative cooling term. The simulations bear out the expectation that almost all of the shocked supernova ejecta flow past the disk before they have time to cool significantly. Cooling is significant only where the ejecta collide with the dense gas of the disk itself, but there the cooling is sensitive to many unconstrained physical properties to do with the the chemical state of the gas, properties of dust, etc. To capture the gross effects of cooling (especially compression of gas near the dense disk gas) in a computationally simple way, we have adopted the following additional term in the energy equation, implemented in the source step: = −nenpΛ, (9) where ne and np are the number of protons and electrons in the gas, and Λ is the cooling function. The densities ne and np are obtained simply by assuming the hydrogen gas is fully ionized, so ne = np = ρ/1.4mH. For gas temperatures above 10 4K, we take Λ of a solar-metallicity gas from Sutherland and Dopita (1993); Λ typically ranges between 10−24 erg cm3 s−1 (at T = 104K) and Λ = 10−21 erg cm3 s−1 (at T = 105K). Below 104K we adopted a flat cooling function of Λ = 10−24 erg cm3 s−1. At very low temperatures it – 10 – is necessary to include heating processes as well as cooling, or else the gas rapidly cools to unreasonable temperatures. Rather than handle transfer of radiation from the central star, we defined a minimum temperature below which the gas is not allowed to cool: Tmin = 300 (r/1AU) −3/4 K. Perseus uses a simple first-order, finite-difference equation to handle cooling. Although this method is not as precise as a predictor-corrector method, in §2.4 we show that it is sufficiently accurate for our purposes. Because Perseus is an explicit code, the implementation of a cooling term demands the introduction of a cooling time step to insure that the gas doesn’t cool too rapidly during one time step, resulting in negative temperatures or other instabilities. For a radiating gas, the cooling timescale can be approximated by tcool ≈ kBT/nΛ, where kB is the Boltzmann constant, T is the temperature of the gas, n is the number density and Λ is the appropriate cooling function. This cooling timescale is calculated on all the grid zones where the temperature exceeds 103K, and the cooling time step δtcool is defined to be 0.025 times the shortest cooling timescale on the grid. If the smallest cooling time step is shorter that the previously calculated ∆t as defined by eq. [8], then it becomes the new time step. We ignore zones where the temperature is below 103K because heating and cooling are not fully calculated anyway, and because these zones are always associated with very high densities and cool extremely rapidly, on timescales as short as hours, too rapidly to reasonably follow anyway. Finally, to follow the evolution of the ejecta gas with respect to the disk gas, a tracer “color density” was added. By defining a different density, the color density ρc, it is possible to follow the mixing of a two specific parts of a system, in this case the ejecta and the disk. By comparing ρc to ρ, it is possible to know how much of the ejecta is present in a given zone relative to the original material. It is important to note that ρc is a tracer and does not affect the simulation in any way. – 11 – 2.2. Sod Shock-Tube We have benchmarked the Perseus code against a well-known analytic solution, the Sod shock tube (Sod 1978). Tests were performed to verify the validity of Perseus’s results. It is a 1-D test, and hence was only done in the z direction, as curvature effects would render this test invalid in the r direction. Therefore, the gas was initially set spatially uniform in r. 120 zones were used in the z direction. The other initial conditions of the Sod shock-tube are as follows: the simulation domain is split in half and filled with a γ=1.4 gas; in one half (z < 0.5 cm), the gas has a pressure of 1.0 dyne cm−2 and a density of 1.0 g cm−3, while in the other half (z > 0.5 cm) the gas has a pressure of 0.1 dyne cm−2 and a density of 0.125 g cm−3. The results of the simulation and the analytical solution at t = 0.245 s are shown in Figure 1. The slight discrepancies between the analytic and numerical results are attributable to numerical diffusion associated with the upwind interpolation (see Stone & Norman 1992), match the results of Stone & Norman (1992) almost exactly, and are entirely acceptable. 2.3. Gravitational Collapse As a test problem involving curvature terms, we also simulated the pressure-free gravitational collapse of a spherical clump of gas. A uniform density gas (ρ = 10−14 g cm−3) was imposed everywhere within 30 AU of the star. As stated above, the only source of gravitational acceleration in our simulations is the central protostar, with mass M = 1M⊙. The grid on which this simulation takes place has 120 zones in the z direction and 80 in the r direction The free-fall timescale under the gravitational potential of a 1 M⊙ star is 29.0 yrs. The results of the simulation can be seen in Figure 2. After 28 years, the 30AU clump has contracted to the edge of the computational volume. Spherical symmetry is maintained throughout as the gas is advected despite the presence of the inner boundary condition. – 12 – 2.4. Cooling To test the accuracy of the cooling algorithm, a simple 2D grid of 64 zones by 64 zones was set up. The simulation starts with gas at T = 1010K. The temperature of the gas is followed until it reaches T = 104K. Simulations were run varying the cooling time step δtcool. As the cooling subroutine does not use a predictor-corrector method, decreasing the time step increases the precision. A range of cooling time steps, varying from 10 times what is used in the code to 0.1 times what is used in the code, were tested. Since in the range of T = 104K − 1010K, the cooling rate varies with temperature (according to Sutherland & Dopita 1993), the size of the time step should affect the time evolution of the temperature. This evolution is depicted in Figure 3, from which one can see that δtcool used in the code is sufficient, as using smaller time steps gives the same result. In addition, we can see that even the lesser precision runs give comparably good results, as the thermal time step of the CFL condition prevents a catastrophically rapid cooling. The precision of the cooling is limited by the accuracy of the cooling factors used, not the algorithm. 2.5. “Relaxed Disk” Finally, we have modeled the long-term evolution of an isolated protoplanetary disk. To begin, a minimum-mass solar nebula disk (Hayashi et al. 1985) in Keplerian rotation is truncated at 30AU. The code then runs for 2000 years, allowing the disk to find its equilibrium configuration under gravity from the central star (1M⊙), pressure and angular momentum. We call this the “relaxed disk”, and use it as the initial state for the runs that follow. To check the long term stability of the system, we allow the relaxed disk to evolve an extra 2000 years. This test verifies the stability of the simulated disk against numerical effects. In addition, using a color density, we can assess how much numerical diffusion occurs in the code. – 13 – After the extra 2000 years, the disk maintains its shape, and is deformed only at its lowest isodensity contour, because of the gravitational infall of the surrounding gas (Figure 4). Comparing this deformation to the results from the canonical run (§3), this is a negligible effect. Some of the surrounding gas has accreted on the disk due to the gravitational potential of the central star. The color density allows us to follow the location of the accreted gas. After 2000 years, roughly 20% of the accreted mass has found its way to the midplane of the disk due to the effects of numerical diffusion. Hence some numerical diffusion exists and must be considered in what follows. 3. Canonical Case In this section, we adopt a particular set of parameters pertinent to the disk and the supernova, and follow the evolution of the disk and ejecta in some detail. The simulation begins with our relaxed disk (§2.5), seen in Figure 5. Its mass is about 0.00838 M⊙, and it extends from 4AU to 40AU, the inner parts of the disk being removed to improve code performance. The gas density around the disk is taken to be a uniform 10 cm−3, which is a typical density for an H ii region. This disk has similar characteristics to those found in the Orion nebula, which have been photoevaporated down to tens of AU by the radiation of nearby massive O stars (Johnstone, Hollenbach & Bally 1998). In setting up our disk, we have ignored the effects of the UV flash that accompanies the supernova, in which approximately 3 × 1047 erg of high-energy ultraviolet photons are emitted over several days (Hamuy et al. 1988). The typical UV opacities of protoplanetary disk dust are κ ∼ 102 cm2 g−1 (D’Allesio et al. 2006), so this UV energy does not penetrate below a column density ∼ κ−1 ∼ 10−2 g cm−2. The gas density at the base of this layer is typically ρ ∼ 10−15 g cm−3; if the gas reaches temperatures < 105K, tcool will not exceed a few hours (§2.1). The upper layer of the disk absorbing the UV is not heated above a – 14 – temperature T ∼ (EUV/4πd 2)mHκ/kB ∼ 10 5K. Because the gas in the disk absorbs and then reradiates the energy it absorbs from the UV flash, we have ignored it. We have also neglected low-density gas structures that are likely to have surrounded the disk, including photoevaporative flows and bow shocks from stellar winds, as these are beyond the scope of this paper. It is likely that the UV flash would greatly heat this low-density gas and cause it to rapidly escape the disk anyway. Our “relaxed disk” initial state is a reasonable, simplified model of the disks seen in H ii regions before they are struck by supernova shocks. After a stable disk is obtained, supernova ejecta are added to the system. The canonical simulation assumes Mej = 20M⊙ of material was ejected isotropically by a supernova d = 0.3 pc away, with an explosion kinetic energy Eej = 10 51 erg, (1 f.o.e.). This is typical of the mass ejected by a 25M⊙ progenitor star, as considered by Woosley & Weaver (1995), and although more recent models show that progenitor winds are likely to reduce the ejecta mass to < 10M⊙ (Woosley, Heger & Weaver 2002), we retain the larger ejecta mass as a worst-case scenario for disk survivability. The ejecta are assumed to explode isotropically, but with density and velocity decreasing with time. The time dependence is taken from the scaling solutions of Matzner & McKee (1999); in analogy to their eq. [1], we define the following quantities: where R∗ is the radius of the exploding star, taken to be 50R⊙. The travel time from the supernova to the disk is computed as ttrav = d/v∗, and is typically ∼ 100 years. Finally, expressions for the time dependence of velocity, density and pressure of the ejecta, are – 15 – obtained for any given time t after the shock strikes the disk: vej(t) = v∗ ttrav t+ ttrav ρej(t) = ρ∗ ttrav ttrav t + ttrav pej(t) = p∗ ttrav ttrav t+ ttrav We acknowledge that supernova ejecta are not distributed homogeneously within the progenitor (Matzner & McKee 1999), nor are they ejected isotropically (Woosley, Heger & Weaver 2002), but more detailed modeling lies beyond the scope of this paper. Our assumption of homologous expansion is in any case a worst-case scenario for disk survivability in that the ejecta are front-loaded in a way that overestimates the ram pressure (C. Matzner, private communication). As our parameter study (§5) shows, density and velocity variations have little influence on the results. The incoming ejecta and the shock they create while propagating through the low-density gas of the H ii region can be seen in Figure 6. When the shock reaches the disk, the lower-density outer edges are swept away, as the ram pressure of the ejecta is much higher than the gas pressure in those areas. However, the shock stalls at the higher density areas of the disk, as the gas pressure is higher there. A snapshot of the stalling shock can be seen in Figure 7. As the ejecta hit the disk, they shock and thermalize, heating the gas on the upper layers of the disk. This increases the pressure in that area, causing a reverse shock to propagate into the incoming ejecta. The reverse shock will eventually stall, forming a bow shock around the disk (Figures 8 and 9). Roughly 4 months have passed between the initial contact and the formation of the bow shock. Some stripping of the low density gas at the disk’s edge (> 30 AU) may occur as the supernova ejecta is deflected around it, due primarily to the ram pressure of the ejecta. As the stripped gas is removed from the top and the sides of the disk, it either is snowplowed – 16 – away from the disk if enough momentum has been imparted to it, or it is pushed behind the disk, where it can fall back onto it (Figure 10). In addition to stripping the outer layers of the disk, the pressure of the thermalized shocked gas will compress the disk to a smaller size; although they do not destroy the disk, the ejecta do temporarily deform the disk considerably. Figure 11 shows the effect of the pressure on the disk, which has been reduced in thickness and has shrunk to a radius of 30 AU. The extra external pressure effectively aids gravity and allows the gas to orbit at a smaller radius with the same angular momentum. As the ejecta is deflected across the top edge of the disk, some mixing between the disk gas and the ejecta may occur through Kelvin-Helmholtz instabilities. Figure 12 shows a close up of the disk where a Kelvin-Helmholtz roll is occurring at the boundary between the disk and the flowing ejecta. In addition, some ejecta mixed in with the stripped material under the disk might also accrete onto the disk. As time goes by and slower ejecta hit the disk, the ram pressure affecting the disk diminishes, and the disk slowly returns to its original state, recovering almost completely after 2000 years (Figure 13). The exchange of material between the disk and the ejecta is mediated through the ejecta-disk interface, which in our simulations is only moderately well resolved. As discussed in §4, the numerical resolution will affect how well we quantify both the destruction of the disk and the mixing of ejecta into the disk. In the canonical run, at least, disk destruction and gas mixing are minimal. Although some stripping has occurred while the disk was being hit by the ejecta, it has lost less than 0.1% of its mass. The final disk mass, computed from the zones where the density is greater than 100 cm−3, remains roughly at 0.00838 M⊙. Some of the ejecta have also been mixed into the disk, but only with very low efficiency. A 30AU disk sitting 0.3 pc from the supernova intercepts roughly one part in 1.7 × 107 of the total ejecta from the supernova, assuming isotropic ejecta distribution. For 20 M⊙ of ejecta, this corresponds to roughly 1.18 × 10−6M⊙ intercepted. At the end of the simulation, we find only 1.48× 10−8M⊙ of supernova ejecta was injected in the disk, for an injection efficiency – 17 – of about 1.3%. Some of the injected material could be attributed to numerical diffusion between the outer parts of the disk and the inner layers: as seen in §2.5, Perseus is diffusive over long periods of time. However, the distribution of the colored mass is qualitatively different from that obtained from a simple numerical diffusion process. Figure 14 compares the percentage of colored mass within a given isodensity contour for the canonical case and the relaxed disk simulation of §2.5, at a time 500 years after the beginning of each of these simulations. From this graph, it is clear that the process that injects the supernova ejecta is not simply numerical diffusion, as it is much more efficient at injecting material deep within the disk. The post-shock pressure of the ejecta gas, 100 years after initial contact, when its forward progession in the disk has stalled is ∼ 2ρejv ej/(γ + 1) = 2.8 × 10 −5 dyne cm−2. (After 100 years, ρej = 2.2 × 10 −21 g cm−3 and vej = 1300 km s −1.) The shock stalls where the post-shock pressure is comparable to the disk pressure ∼ ρkBT/m̄. Hence at 20AU, where the temperature of the disk is T ≈ 30K, the shock stalls at the isodensity contour ∼ 1.5 × 10−14 g cm−3. As about half of the color mass is mixed to just this depth, this is further evidence that the color field in the disk represents a real physical mixing. 4. Numerical Resolution The results of canonical run show many similarities to related problems that have been studied extensively in the literature. The interaction of a supernova shock with a protoplanetary disk resembles the interaction of a shock with a molecular cloud, as modeled by Nittmann et al. (1982), Bedogni & Woodward (1990), Klein, McKee & Colella (1994; hereafter KMC), Mac Low et al. (1994), Xu & Stone (1995), Orlando et al. (2005) and Nakamura et al. (2006). Especially in Nakamura et al. (2006), the numerical resolutions achieved in these simulations are state-of-the-art, reaching several ×103 zones per axis. In those simulations, as in our canonical run, the evolution is dominated by two physical – 18 – effects: the transfer of momentum to the cloud or disk; and the onset of Kelvin-Helmholtz (KH) instabilities that fragment and strip gas from the cloud or disk. KH instabilities are the most difficult aspect of either simulation to model, because there is no practical lower limit to the lengthscales on which KH instabilities operate (they are only suppressed at scales smaller than the sheared surface). Increasing the numerical resolution generally reveals increasingly small-scale structure at the interface between the shock and the cloud or disk (see Figure 1 of Mac Low et al. 1994). The numerical resolution in our canonical run is about 100 zones per axis; more specifically, there are about 26 zones in one disk radius (of 30 AU), and about 20 zones across two scale heights of the disk (one scale-height being about 2 AU at 20 AU). Our highest-resolution run used about 50 zones along the radius of the disk, and placed about 30 zones across the disk vertically. In the notation of KMC, then, our simulations employ about 20-30 zones per cloud radius, a factor of 3 lower than the resolutions of 100 zones per cloud radius argued by Nakamura et al. (2006) to be necessary to resolve the hydrodynamics of a shock hitting a molecular cloud. Higher numerical resolutions are difficult to achieve; unlike the case of a supernova shock with speed ∼ 2000 km s−1 striking a molecular cloud with radius of 1 pc, our simulations deal with a shock with the same speed striking an object whose intrinsic lengthscale is ∼ 0.1AU. Satisfying our CFL condition requires us to use timesteps that are only ∼ 103 s, four orders of magnitude smaller than the timesteps needed for the case of a molecular cloud. This and other factors conspire to make simulations of a shock striking a protoplanetary disk about 100 times more computationally intensive than the case of a shock striking a molecular cloud. Due to the numerous lengthscales in the problem imposed by the star’s gravity and the rotation of the disk, it is not possible to run the simulations at low Mach numbers and then scale the results to higher Mach numbers. We intend to create a parallelized version of Perseus to run on a computer cluster in the near future, but until then, our numerical resolution cannot match that of simulations of shocks interacting – 19 – with molecular clouds. This begs the question, if our resolution is not as good as has been achieved by others, is it good enough? To quantify what numerical resolutions are sufficient, we examine the physics of a shock interacting with a molecular cloud, and review the convergence studies of the same undertaken by previous authors. In the most well-known simulations (Nittmann et al. 1982; KMC; Mac Low et al. 1994; Nakamura et al. 2006), it is assumed that a low-density molecular cloud with no gravity or magnetic fields is exposed to a steady shock. The shock collides with the cloud, producing a reverse shock that develops into a bow shock; a shock propagates through the cloud, passing through it in a “cloud-crushing” time tcc. The cloud is accelerated, but as long as a velocity difference between the high-velocity gas and the cloud exists, KH instabilities grow that create fragments with significant velocity dispersions, ∼ 10% of the shock speed (Nakamura et al. 2006). Cloud destruction takes place before the cloud is fully accelerated, and the cloud is effectively fragmented in a few × tcc before the velocity difference diminishes. These fragments are not gravitationally bound to the cloud and easily escape. As long as the shock remains steady for a few × tcc, it is inevitable that the cloud is destroyed. As KH instabilities are what fragment the cloud and accelerate the fragments, it is important to model them carefully, with numerical resolution as high as can be achieved. KMC stated in their abstract and throughout their paper that 100 zones per cloud radius were required for “accurate results”; however, all definitions of what was meant by “accurate”, or what were the physically relevant “results” were deferred to a future “Paper II”. A companion paper by Mac Low et al. (1994) referred to the same Paper II and repeated the claim that 100 zones per axis were required. Nakamura et al. (2006), published this year, appears to be the Paper II that reports the relevant convergence study and quantifies what is meant by accurate results. Global quantities, including the – 20 – morphology of the cloud, its forward mass and momentum, and the velocity dispersions of cloud fragments, were defined and calculated at various levels of numerical resolution. These were then compared to the same quantities calculated using the highest achievable resolutions, about 500 zones per cloud radius (over 1000 zones per axis). The quantities slowest to converge with higher numerical resolution were the velocity dispersions, probably, they claim, because these quantities are so sensitive to the hydrodynamics at shocks and contact discontinuities where the code becomes first-order accurate only. The velocity dispersions converged to within 10% of the highest-resolution values only when at least 100 zones per cloud radius were used. For this single arbitrary reason, Nakamura et al. (2006) claimed numerical resolutions of 100 zones per cloud radius were necessary. We note, however, that the other quantities to do with cloud morphology and momentum were found to converge much more readily; according to Figure 1 of Nakamura et al. (2006), numerical resolutions of only 30 zones per cloud radius are sufficient to yield values within 10% of the values found in the highest-resolution simulations. And although the velocity dispersions are not so well converged at 30 zones per cloud radius, even then the errors do not exceed a factor of 2. Assuming that the problem we have investigated is similar enough to that investigated by Nakamura et al. (2006) so that their convergence study could be applied to our problem, we would conclude that even our canonical run is sufficiently resolving relevant physical quantities, the one possible exception being the velocities of fragments generated by KH instabilities, where the errors could be a factor of 2. Of course, the problem we have investigated, a supernova shock striking a protoplanetary disk, is different in four very important ways from the cases considered by KMC, Mac Low et al. (1994) and Nakamura et al. (2006). The most important fundamental difference is that the disk is gravitationally bound to the central protostar. Thus, even if gas is accelerated to supersonic speeds ∼ 10 km s−1, it is not guaranteed to escape the star. Second, the densities of gas in the disk, ρdisk, are significantly higher than the density – 21 – in the gas colliding with the disk, ρej. In the notation of KMC, χ = ρdisk/ρej. Because the disk density is not uniform, no single value of χ applies, but if χ is understood to refer to different parcels of disk gas, χ would vary from 104 to over 108. This affects the magnitudes of certain variables (see, e.g., Figure 17 of KMC regarding mix fractions), but also qualitatively alters the problem: the densities and pressures in the disk are so high that the supernova shock cannot cross through the disk, instead stalling at several scale heights above the disk. Unlike the case of a shock shredding a molecular cloud, the cloud-crushing timescale tcc is not even a relevant quantity for our calculations. The third difference is that shocks cannot remain non-radiative when gas is as dense as it is near the disk. Using ρ = 10−14 g cm−3 and Λ = 10−24 erg cm3 s−1, tcool is only a few hours, and shocks in the disk are effectively isothermal. Shocks propagating into the disk therefore stall at somewhat higher locations above the disk than they would have if they were adiabatic. Finally, the fourth fundamental difference between our simulations and those investigated in KMC, Mac Low et al. (1994) and Nakamura et al. (2006) is that we do not assume steady shocks. For supernova shocks striking protoplanetary disks about 0.3 pc away, the most intense effects are felt only for a time ∼ 102 years, and after only 2000 years the shock has for all purposes passed. There are limits, therefore, to the energy and momentum that can be delivered to the disk. Very much unlike the case of a steady, non-radiative shock striking a low-density, gravitationally unbound molecular cloud, where ultimately destruction of the cloud is inevitable, many factors contribute to the survivability of protoplanetary disks struck by supernova shocks. This conclusion is borne out by a resolution study we have conducted that shows that the vertical momentum delivered to the disk is certainly too small to destroy it, and that we are not significantly underresolving the KH instabilities at the top of the disk. Using the parameters of our canonical case, we have conducted 6 simulations with different numerical resolutions. The resolutions range from truly awful, with only 8 zones in the – 22 – radial direction (∆r = 10AU) and 18 zones in the vertical direction (with ∆z = 1AU at the midplane, barely sufficient to resolve a scale height), to our canonical run (76 x 120), to one high-resolution run with 152 radial zones (∆r = 0.5AU) and 240 vertical zones (∆z = 0.13AU at the midplane). On an Apple G5 desktop with two 2.0-GHz processors, these simulations took from less than a day to 80 days to run. To test for convergence, we calculated several global quanities Q, including: the density-weighted cloud radius, a; the density-weighted cloud thickness, c; the density-weighted vertical velocity, 〈vz〉; the density-weighted velocity dispersion in r, δvr; the density-weighted velocity dispersion in z, 〈vz〉; as well as the mass of ejecta injected into the disk, Minj. Except for the last quantity, these are defined exactly as in Nakamura et al. (2006), but using a density threshold corresponding to 100 cm−3. Each global quantity was measured at a time 500 years into each simulation. We define each global quantity Q as a function of numerical resolution n, where n is the geometric mean of the number of zones along each axis, which ranges from 12 to 191. To compare to the resolutions of KMC, one must divide this number by about 3 to get the number of zones per “cloud radius” (two scale heights at 20 AU) in the vertical direction, and divide by about 2 to get the number of zones per cloud radius in the radial direction. The convergence is measured by computing |Q(n)−Q(nmax)| /Q(nmax), where nmax = 191 corresponds to our highest resolution case. In Figure 15 we plot each quantity Q(n) as a function of resolution n (except 〈vz〉). All of the quantities have converged to within 10%, the criterion imposed by Nakamura et al. (2006) as signifying adequate convergence. It is significant that δvr has converged to within 10%, because this is the quantity relevant to disk destruction by KH instabilities. Material is stripped from the disk only if supersonic gas streaming radially above the top of the disk can generate KH instabilities and fragments of gas that can then be accelerated radially to escape velocities. If we were underresolving this layer significantly, one would expect large differences in δvr as the resolution was increased, but instead this quantity has converged. Higher-resolution – 23 – simulations are likely to reveal smaller-scale KH instabilities and perhaps more stripping of the top of the disk, but not an order of mangitude more. The convergence of 〈vz〉 with resolution is handled differently because unlike the other quantities, 〈vz〉 can vanish at certain times. The disk absorbs the momentum of the ejecta and is pushed downward, but unlike the case of an isolated molecular cloud, the disk feels a restoring force from the gravity of the central star. The disk then undergoes damped vertical oscillations about the origin as it collides with incoming ejecta at lower and lower speeds. This behavior is illustrated by the time-dependence of 〈vz〉, shown in Figure 16 for two numerical resolutions, our canonical run (n = 95) and our highest-resolution run (n = 191). Figure 16 shows that the vertical velocity of the disk oscillates about zero, but with an amplitude ∼ 0.1 km s−1. The time-average of this amplitude can be quantified by −< vz >2 , where the bar represents an average over time; the result is 825 cm s−1 for the highest-resolution run and is only 2% smaller for the canonical resolution. The difference between the two runs is generally much smaller than this; except for a few times around t = 150 yr, and t = 300 yr, when the discrepancies approach 30%, the agreement between the two resolutions is within 10%. The time-averaged dispersion of the amplitude of the difference (defined as above for 〈vz〉 itself) is only 12.0 cm s −1, which is only 1.5% of the value for 〈vz〉 itself. Taking a time average of |〈vz〉95 − 〈vz〉191| / |〈vz〉191| yields 8.7%. We therefore claim convergence at about the 10% level for 〈vz〉 as well. Using these velocities, we also note here that the neglect of the star’s motion is entirely justified. The amplitude of 〈vz〉 is entirely understandable as reflecting the momentum delivered to the disk by the supernova ejecta, which is ∼ 20M⊙ (πR disk/4πd 2) Vej ∼ 10−3M⊙ km s −1, and which should yield a disk velocity ∼ 0.1 km s−1. The period of oscillation is about 150 years, which is consistent with most of this momentum being delivered to the outer reaches of the disk from 25 to 30 AU where the orbital periods are – 24 – 125 to 165 years. These velocities are completely unaffected by the neglected velocity of the central star, whose mass is 120 times greater than the disk’s mass. If the central star, with mass ∼ 1M⊙, had been allowed to absorb the ejecta’s momentum, it would only move at ∼ 100 cm s−1 and be displaced at most 0.4 AU after 2000 years. This neglected velocity, is much smaller than all other relevant velocities in the problem, including |〈vz〉| ∼ 800 cm s as well as the escape velocities (∼ 10 km s−1), the velocities of gas flowing over the disk (∼ 102 km s−1), and of course the shock speeds (∼ 103 km s−1). Our analysis shows that we have reached adequate convergence with our canonical numerical resolution (n = 95). We observe KH instabilities in all of our simulations (except n = 12), and we see the role they play in stripping the disk and mixing ejecta gas into it. We are therefore confident that we are adequately resolving these hydrodynamic features; nevertheless, we now consider a worst-case scenario in which we KH instabilities can strip the disk with 100% efficiency where they act, and ask how much mass the disk could possibly lose under such conditions. Supernova ejecta that has passed through the bow shock and strikes the disk necessarily stalls where the gas pressure in the disk exceeds the ram pressure of the ejecta. Below this level, the momentum of the ejecta is transferred not as a shock but as a pressure (sound) wave. Gas motions below this level are subsonic. Note that this is drastically different from the case of an isolated molecular cloud as studied by KMC and others; the high pressure in the disk is maintained only because of the gravitational pull of the central star. The location where the incoming ejecta stall is easily found. Assuming the vertical isothermal minimum-mass solar nebula disk of Hayashi et al. (1985), the gas density varies as ρ(r, z) = 1.4×10−9 (r/1AU)−21/8 exp(−z2/2H2) g cm−3, where H = cs/Ω, cs is the sound speed and Ω is the Keplerian orbital frequency. Using the maximum density and velocity of the incoming ejecta (ρej = 1.2 × 10 −20 g cm−3 and Vej = 2200 km s −1), the ram pressure – 25 – of the shock striking the disk does not exceed pram = ρejV ej/4 = 1.5 × 10 −4 dyne cm−2 (the factor of 1/4 arises because the gas must pass through the bow shock before it strikes the disk). At 10 AU the pressure in the disk, ρc2s , exceeds the ram pressure at z = 2.7H , and at 20 AU the ejecta stall at z = 1.7H ; the gas densities at these locations are ≈ 10−13 g cm−3. At later times, the ejecta stall even higher above the disk, because pram ∝ t −5 (cf. eq. [11]). For example, at t = 100 yr, the ram pressure drops below 1×10−5 dyne cm−2, and the ejecta stall above z = 3.6H (10 AU) and z = 2.9H (20 AU). The column density above a height z in a vertically isothermal disk is easily found to be Σ(> z) ≈ ρ(z)H2/z = p(z)/(Ω2z). Integrating over radius, the total amount of disk gas that ever comes into contact with ejecta is (approximating z = 2H): Mss = pramr πpramR . (12) Using a disk radius Rd = 30AU, the maximum amount of disk gas that is actually exposed to a shock at any time is only 1.5 × 10−5M⊙, or 0.2% of the disk mass. This fraction decreases with time as pram ∝ t −5 (eq. [11]); the integral over time of pram is pram(t = 0) × ttrav/4. The ram pressure drops so quickly, that effectively ejecta interact with this uppermost 0.2% of the disk mass only for about 30 years. This is equivalent to one orbital timescale at 10 AU, so the amount of disk gas that is able to mix or otherwise interact with the ejecta hitting the upper layers of the disk is very small, probably a few percent at most. As for KH instabilities, they are initiated when the Richardson number drops below a critical value, when (∂U/∂z)2 , (13) where g = −Ω2z is the vertical gravitational acceleration, Ω is the Keplerian orbital frequency, and (∂U/∂z) is the velocity gradient at the top of the disk. Below the stall point, all gas motions are subsonic and the velocity gradient would have to be execptionally – 26 – steep, with an unreasonably thin shear layer thickness, <∼H/10, to initiate KH instabilities. Mixing of ejecta into the disk is quite effective above where the shock stalls, as illustrated by Figure 14; it is in these same layers (experiencing supersonic velocities) that we expect that KH instabilities to occur, but again <∼ 1% of the disk mass can be expected to interact with these layers. To summarize, our numerical simulations are run at a lower simulation (by a factor of about 3) than has been claimed necessary to study the interaction of steady shocks with gravitationally unbound molecular clouds, but the drastically different physics of the problem studied here as allowed us to achieve numerical convergence and allowed us to reach meaningful conclusions. Our global quantities have converged to within 10%, the same criterion used by Nakamura et al. (2006) to claim convergence. The problem is so different because the disk is tightly gravitationally bound to the star and the supernova shock is of finite duration. The high pressure in the disk makes the concept of a cloud-crushing time meaningless, because the ejecta stall before they drive through even 1% of the disk gas. Rather than a sharp interface between the ejecta and the disk, the two interact via sound waves within the disk, which entails smoother gradients. While we do resolve KH instabilities in this interface, we allow that we may be underresolving this layer; but even if we are, this will not affect our conclusions regarding the disk survival or the amount of gas mixed into the disk. This is because we already find that mass is stripped from the disk and ejecta are mixed into the disk very effectively (see Figure 14) above the layer where the ejecta stall, and below this layer mixing is much less efficient and all the gas is subsonic and bound to the star. It is inevitable that mass loss and mixing of ejecta should be only at the ∼ 1% level. Similar studies using higher numerical resolutions are likely to reveal more detailed structures at the disk-ejecta interface, but it is doubtful that more than a few percent of the disk mass can be mixed-in ejecta, and it is even more doubtful that even 1% of the disk mass can be lost. We therefore have sufficient confidence in our – 27 – canonical resolution to use it to test the effects of varying parameters on gas mixing and disk destruction. 5. Parameter Study 5.1. Distance Various parameters were changed from the canonical case to study their effect on the survival of the disk and the injection efficiency of ejecta, including: the distance between the supernova and the disk, d; the explosion energy of the supernova, Eej; and the mass of gas in the disk, Mdisk. In all these scenarios, the resolution stayed the same as in the canonical case. The first parameter studied was the distance between the supernova and the disk. From the canonical distance of 0.3 pc, the disk was moved to 0.5 pc and 0.1 pc. The main effect of this change is to vary the density of the ejecta hitting the disk (see eq. [11]). If the disk is closer, the gaseous ejecta is less diluted as it hits the disk. Hence these simulations are essentially equivalent to simulating a denser or a more tenuous clump of gas hitting the disk in an non-homogeneous supernova explosion. The results of these simulations can be seen in Table 2. The “% injected” column gives the percentage of the ejecta intercepted by the disk [with an assumed cross-section of π(30 AU)2] that was actually mixed into the disk. The third column gives the estimated 26Al/27Al ratio that one would expect in the disk if the SLRs were delivered in the gas phase. This quantity was calculated using a disk chemical composition taken from Lodders (2003), and the ejecta isotopic composition from a 25 M⊙ supernova taken from Woosley & Weaver (1995), which ejects M = 1.27× 10 of 26Al. Although the injection efficiency increases for denser ejecta, and the geometric dilution decreases for a closer supernova, gas-phase injection of ejecta into a disk at 0.1 pc cannot explain the SLR ratios in meteorites. The 26Al/27Al ratio is off by roughly an order of magnitude from the measured value of 5 × 10−5 (e.g., MacPherson et al. 1995). Stripping – 28 – was more important with denser ejecta (d = 0.1 pc), although still negligible compared to the mass of the disk; only 0.7% of the disk mass was lost. 5.2. Explosion Energy We next varied the explosion energy, which defines the velocity at which the ejecta travel. The explosion energy was changed from 1 f.o.e. to 0.25 and 4 f.o.e., effectively modifying the ejecta velocity from 2200 km/s to 1100 km/s and 4400 km/s, respectively. The results of the simulations can be seen in Table 3. Slower ejecta thermalizes to a lower temperature, and does not form such a strong reverse shock. Therefore, slower ejecta is injected at a slightly higher efficiency into a disk. Primarily, though, the results are insensitive to the velocity of the incoming supernova ejecta. 5.3. Disk Mass The final parameter varied was the mass of the disk. From these simulations, the mass of the the minimum mass disk used in the canonical simulation was increased by a factor of 10, and decreased by a factor of 10. The results of the simulations can be seen in Table 4. Increasing the mass by a factor of 10 slightly increases, but this could be due to the fact that the disk does not get compressed as much as the canonical disk (it has a higher density and pressure at each radius). Hence the disk has a larger surface to intercept the ejecta (the calculation for injection efficiency assumes a radius of 30 AU). Reducing the mass by a factor of 10 increases the efficiency. As the gas density in the disk is less, the pressure is less, and hence the ejecta is able to get closer to the midplane, increasing the amount injected. – 29 – 6. Conclusions In this paper, we have described a 2-D cylindrical hydrodynamics code we wrote, Perseus, and the results from the application of this code to the problem of the interaction of supernova shocks with protoplanetary disks. A main conclusion of this paper is that disks are not destroyed by a nearby supernova, even one as close as 0.1 pc. The robustness of the disks is a fundamentally new result that differs from previous 1-D analytical estimates (Chevalier 2000) and numerical simulations (Ouellette et al. 2005). In those simulations, in which gas could not be deflected around the disk, the full momentum of the supernova ejecta was transferred directly to each annulus of gas in the disk. Chevalier (2000) had estimated that disk annuli would be stripped away from the disk wherever MejVej/4πd 2 > ΣdVesc, where Σd is the surface density of the disk [Σd = 1700 (r/1AU) −3/2 g cm−2 for a minimum mass disk; Hayashi et al. 1985), and Vesc is the escape velocity at the radius of the annulus. In the geometry considered here, the momentum is applied at right angles to the disk rotation, so vesc can be replaced with the Keplerian orbital velocity, as the total kinetic energy would then be sufficient for escape. Also, integrating the momentum transfer over time (eq. [11]), we find Vej = 3v⋆/4. Therefore, using the criterion of Chevalier (2000), and considering the parameters of the canonical case but with d = 0.1 pc, the disk should have been destroyed everywhere outside of 30.2AU, representing a loss of 13% of the mass of a 40 AU radius disk. Comparable conclusions were reached by Ouellette et al. (2005). In contrast, as these 2-D simulations show, the disk becomes surrounded by high- pressure shocked gas that cushions the disk and deflects ejecta around the disk. This high-pressure gas has many effects. First, the bow shock deviates the gas, making part of the ejecta that would have normally hit the disk flow around it. From Figure 11, by following the velocity vectors, it is possible to estimate that the gas initially on trajectories withr > 20AU will be deflected by > 14◦ after passing through the bow showk, and will – 30 – miss the disk. For a disk 30 AU in size, this represents a reduction in the mass flux hitting by ≈ 45%; more thorough calculations give a reduction of ≈ 50%. Second, the bow shock reduces the forward velocity of the gas that does hit the disk. Gas deviated sideways about 14◦, will have lost more than 10% of its forward velocity upon reaching the disk. These two effects combined conspire to reduce the amount of momentum hitting the disk by 55% overall. By virtue of the smaller escape velocity and the lower disk surface density, gas at the disk edges is most vulnerable to loss by the momentum of the shock, but it at the disk edges that the momentum of the supernova shock is most sharply reduced. Because of the loss of momentum, the disk in the previous paragraph could survive out to a radius of about 45AU. A third, significant effect of the surrounding high-pressure shocked gas, though, is its ability to shrink the disk to a smaller radius. The pressure in the post-shock gas is ∼ 2ρejv ej/(γ + 1) = 4.4 × 10 −4 dyne cm−2, so the average pressure gradient in the disk between about 30 and 35 AU is ≈ 1.9 × 10−18 dyne cm−3. This is to be compared to the gravitational force per volume at 35AU, ρg = 4.8 × 10−19 dyne cm−3 (at 35 AU, ρ ∼ 1.0× 10−15 in the canonical disk.) The pressure of the shocked gas enhances the inward gravitational force by a significant amount, causing gas of a given angular momentum to orbit at a smaller radius than it would if in pure Keplerian rotation. When this high pressure is relieved after the supernova shock has passed, the disk is restored to Keplerian rotation and expands to its original size. While the shock is strongest, the high-pressure gas forces a protoplanetary disk to orbit at a reduced size, ≈ 30AU, where it is invulnerable to being stripped by direct transfers of momentum. Because of these combined effects of the cushion of high-pressure shocked gas surrounding the disk—reduction in ejecta momentum and squeezing of the disk—protoplanetary disks even 0.1 pc from the supernova lose < 1% of their mass. – 31 – Destruction of the disk, if it occurs at all, is due to stripping of the low-density upper layers of the disk by Kelvin-Helmholtz (KH) instabilities. We observed KH instabilities in all of our simulations (except n = 12), and we observe their role in stripping gas from the disk and mixing supernova ejecta into the disk (e.g., Figure 12). Our canonical numerical resolution (n = 95) and our highest-resolution simulation (n = 191), corresponding to effectively 20-30 zones per cloud radius in the terminology of KMC, are just adequate to provide convergence at the 10% level, as described in §4. We are confident we are capturing the relevant physics in our simulations, but we have shown that even if KH instabilities are considerably more effective than we are modeling, that no more than about 1% of the disk mass could ever be affected by KH instabilities. This is because the supernova shock stalls where the ram pressure is balanced by the pressure in the disk, and for typical protoplanetary disk conditions, this occurs several scale heights above the midplane. It is unlikely that higher-resolution simulations would observe loss of more than ∼ 1% of the disk mass. We observed that the ratio of injected mass to disk mass was typically ∼ 1% as well, for similar reasons. We stipulate that mixing of ejecta into the disk is more subtle than stripping of disk mass, but given the limited ability of the supernova ejecta to enter the disk, we find it doubtful that higher-resolution simulations would increase by more than a few the amount of gas-phase radionuclides injected into the disk. Therefore, while disks like those observed in the Orion Nebula (McCaughrean & O’Dell 1995) should survive the explosions of the massive stars in their vicinity, and while these disks would then contain some fraction of the supernova’s gas-phase ejecta, they would not retain more than a small fraction (∼ 1%) of the gaseous ejecta actually intercepted by the disk. If SLRs like 26Al are in the gas phase of the supernova (as modeled here), they will not be injected into the disk in quantities large enough to explain the observed meteoritic ratios, failing by 1-2 orders of magnitude. Of course, the SLRs inferred from meteorites, e.g., 60Fe and 26Al, would not be detected – 32 – if they were not refractory elements. These elements should condense out of the supernova ejecta as dust grains before colliding with a disk (Ebel & Grossman 2001). Colgan et al. (1994) observed the production of dust at the temperature at which FeS condenses, 640 days after SN 1987A, suggesting that the Fe and other refractory elements should condense out of the cooling supernova ejecta in less than a few years. (The supernova ejecta is actually quite cool because of the adiabatic expansion of the gas.) As the travel times from the supernova to the disks in our simulations are typically 20 − 500 years, SLRs can be expected to be sequestered into dust grains condensed from the supernova before striking a disk. Dust grains will be injected into the disk much more effectively than gas-phase ejecta. When the ejecta gas and dust, moving together at the same speed, encounter the bow shock, the gas is almost instantaneously deflected around the disk, but the dust grains will continue forward by virtue of their momentum. The dust grains will be slowed only as fast as drag forces can act. The drag force F on a highly supersonic particle is F ≈ πa2 ρg ∆v where a is the dust radius, ρg is the gas density, and ∆v is the velocity difference between the gas and the dust. Assuming the dust grains are spherical with internal density ρs, the resultant acceleration is dv/dt = −(3ρg∆v 2)/(4ρsa). Immediately after passage through the bow shock, the gas velocity has dropped to 1/4 of the ejecta velocity, so ∆v ≈ (3/4)vej. Integrating the acceleration, we find the time t1/2 for the dust to lose half its initial velocity: t1/2 = 16ρsa 9ρgvej . (14) Measurements of SiC grains with isotopic ratios indicative of formation in supernova ejecta reveal typical radii of a ∼ 0.5µm (Amari et al. 1994; Hoppe et al. 2000). Assuming similar values for all supernova grains, and an internal density ρs = 2.5 g cm −3, and using the maximum typical gas density in the region between the bow shock and the disk, ρg ≈ 5 × 10 −20 g cm−3 , we find a minimum dust stopping time t1/2 ≈ 2 × 10 7 s. In that – 33 – time, the dust will have travelled about 300AU. As the bow shock lies about 20 AU from the disk, the dust will encounter the protoplanetary disk well before travelling this distance, and we conclude that the dust the size of typical supernova SiC grains is not deflected around the disk. We estimate that nearly all the dust in the ejecta intercepted by the disk will be injected into the disk. With nearly 100% injection efficiency, the abundances of 26Al and 60Fe in a disk 0.15 pc from a supernova would be 26Al/27Al = 6.8 × 10−5 and 60Fe/56Fe = 4.8 × 10−7 (using the yields from Woosley & Weaver 1995). These values compare quite favorably to the meteoritic ratios (26Al/27Al = 5.0 × 10−5 and 60Fe/56Fe = 3−7×10−7; MacPherson et al. 1995, Tachibana & Huss 2003), and we conclude that injection of SLRs into an already formed protoplanetary disk by a nearby supernova is a viable mechanism for delivering radionuclides to the early Solar System, provided the SLRs have condensed into dust. In future work we will present numerical simulations of this process (Ouellette et al. 2007, in preparation). We thank an anonymous referee for two very thorough reviews that significantly improved the manuscript. We also thank Chris Matzner for helpful discussions. – 34 – REFERENCES Adams, F. C. & Laughlin, G., 2001, Icarus, 150, 151 Amari, S., Lewis, R. 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P., 2000 ApJ538, 911 Vanhala H. A. T. & Boss A. P., 2002 ApJ575, 1144 Wasserburg, G. J., Gallino, R. & Busso, M.,1998, ApJ, 500, 189 Woosley, S. E.,Heger, A. & Weaver, T. A., 2002, RvMP, 74,1015 Woosley, S. E. & Weaver, T., 1995. ApJS, 101, 181 Xu, J., & Stone, J. M. 1995, ApJ, 454, 172 This manuscript was prepared with the AAS LATEX macros v5.2. – 38 – Table 1 Mass injected # of zones (r × z) % injected 8×18 0.83 16×30 0.77 30×54 0.96 39×74 1.28 60×88 1.31 76 ×120 1.26 152×240 1.25 – 39 – Table 2 Effect of Distance d % injected 26Al/27Al 0.1 pc 4.3 6.4 × 10−6 0.3 pc 1.3 2.2 × 10−7 0.5 pc 1.0 5.6 × 10−8 – 40 – Table 3 Effect of Explosion Energy Eej % injected 26Al/27Al 4.0 f.o.e. 1.0 1.7 × 10−7 1.0 f.o.e. 1.3 2.2 × 10−7 0.25 f.o.e. 1.7 2.8 × 10−7 – 41 – Table 4 Effect of Disk Mass Mdisk % injected 26Al/27Al 0.084 M⊙ 1.4 2.3 × 10 0.0084 M⊙ 1.3 2.2 × 10 0.00084 M⊙ 2.2 3.6 × 10 – 42 – Fig. 1.— Sod shock-tube problem benchmark. The squares are the results of the simulation using Perseus, and the solid line is the analytical solution from Hawley et al. (1984). – 43 – Fig. 2.— Pressure-free collapse of a 30 AU, uniform-density clump of gas, as simulated by Perseus. Spherical symmetry is maintained despite the cylindrical geometry and the inner boundary condition at r = 2 AU. – 44 – Fig. 3.— (a) Gas temperature evolution using various cooling time steps (tc is the time step normally used in the code) (b) Close-up of the time interval 31-33 years. Convergence is achieved using tc and higher resolution timesteps. – 45 – Fig. 4.— (a) Isodensity contours of an equilibrium (“relaxed”) protoplanetary disk. Con- tours are spaced a factor of 10 apart, with the outermost contour representing a density 10−20 g cm−3. The rotating disk has already evolved for 2000 years and is stable; this is the configuration used as the initial state for our subsequent runs. (b) The “relaxed” disk, after an additional 2000 years of evolution. While some slight deformation of the lowest-density contours is seen, attributable to gravitational infall of surrounding gas, the disk is stable and non-evolving over the spans of time relevant to supernova shock passage, ≈ 2000 years. – 46 – Fig. 5.— Isodensity contours of the relaxed disk, just before impact of the supernova ejecta. Contours are spaced a factor of 10 apart, with the outermost contour representing a density 10−21 g cm−3. – 47 – Fig. 6.— Protoplanetary disk immediately prior to impact by the supernova shock. Isoden- sity contours are as in Figure 4. Arrows represent gas velocities. The supernova ejecta are travelling through the H ii region toward the disk at about 2200 km s−1. – 48 – Fig. 7.— Protoplanetary disk 0.05 years after being first hit by supernova ejecta. As the supernova shock sweeps around the disk edge, it snowplows the low-density disk material with it, but the shock stalls in the high-density gas in the disk proper. Isodensity contours and velocity vectors as in Figure 5. – 49 – Fig. 8.— Protoplanetary disk 0.1 years after first being hit by supernova ejecta. As the pressure increases on the side of the disk facing the ejecta, a reverse shock forms, visible as the outermost (dashed) isodensity contour. Isodensity contours and velocity vectors as in Figure 5. – 50 – Fig. 9.— Protoplanetary disk 0.3 years after first being hit by supernova ejecta. The reverse shock, visible as the outermost (dashed) contour, has stalled and formed a bow shock. The bow shock deflects incoming gas around the disk, which is effectively protected in a high- pressure “bubble” of gas. Isodensity contours and velocity vectors as in Figure 5. – 51 – Fig. 10.— Protoplanetary disk 4 years after first being hit by supernova ejecta. Gas is being stripped from the disk (e.g., the clump between R = 35 and 45 AU). Gas stripped from the top of the disk either is entrained in the flow of ejecta and escapes the simulation domain, or flows under the disk and falls back onto it. Isodensity contours and velocity vectors as in Figure 5. – 52 – Fig. 11.— Protoplanetary disk 50 years after first being hit by supernova ejecta. The disk is substantially deformed by the high pressures in the surrounding shocked gas. The pressures compress the disk in the z direction, and also effectively aid gravity in the r direction, allowing the gas to orbit at smaller radii with the same angular momentum. Isodensity contours and velocity vectors as in Figure 5. – 53 – Fig. 12.— (a) Protoplanetary disk 400 years after first being hit by supernova ejecta. At this instant mass is being stripped off the top of the disk by a Kelvin-Helmholtz instability, seen in detail in (b). Isodensity contours and velocity vectors as in Figure 5. – 54 – Fig. 13.— (a) Protoplanetary disk prior to impact by supernova ejecta (same as the relaxed disk of Figure 4), and (b) 2000 years after first being struck by supernova ejecta. This “before and after” picture of the disk illustrates how the disk recovers almost completely from the shock of a nearby supernova. Isodensity contours and velocity vectors as in Figure 5. – 55 – Fig. 14.— Ratio of color mass to total mass within a given isodensity contour (abscissa). The dashed line represents the mass ratio after allowing the relaxed disk to evolve for 2000 years in the absence of a supernova; the solid line represents the mass ratio after 2000 years of interaction with the supernova shock (our canonical simulation). Supernova ejecta is injected very effectively up to densities where the shock would stall (∼ 10−14 g cm−3), much more effectively than can be accounted for by numerical diffusion alone. – 56 – Fig. 15.— Convergence properties of selected global variables. The global variables are: the mass-weighted radius of the cloud, a; the mass-weighted cloud thickness, c; the dispersions in the mass-weighted radial (δvr) and vertical (〈vz〉) velocities; and the mass of ejecta gas injected into the disk, Minj. The quantities are calculated at a time t = 500 years, but using 6 different numerical resolutions, n = 12, 22, 40, 54, 98 and 191. The deviation of each global quantity Q from the highest-resolution value Q191 is plotted against numerical resolution n. For our canonical simulation (n = 98), all quantities have converged at about the 10% level. – 57 – Fig. 16.— Density-weighted velocity along the z axis using the highest numerical simulation n = 191 (solid line) and the canonical resolution n = 98 (dotted line). The difference between them is plotted as the dashed line. After absorbing the initial impulse of downward momentum from the supernova ejecta, the disk oscillates vertically about the position of the central protostar with a period ∼ 150 years, characteristic of the most affected gas at about 30 AU. Introduction Perseus Additions to Zeus Sod Shock-Tube Gravitational Collapse Cooling ``Relaxed Disk'' Canonical Case Numerical Resolution Parameter Study Distance Explosion Energy Disk Mass Conclusions

0704.1653

Scaling cosmologies, geodesic motion and pseudo-susy

UG-07-01 Scaling cosmologies, geodesic motion and pseudo-susy Wissam Chemissany, André Ploegh and Thomas Van Riet Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands w.chemissany, a.r.ploegh, [email protected] Abstract One-parameter solutions in supergravity carried by scalars and a metric trace out curves on the scalar manifold. In ungauged supergravity these curves describe a geodesic mo- tion. It is known that a geodesic motion sometimes occurs in the presence of a scalar potential and for time-dependent solutions this can happen for scaling cosmologies. This note contains a further study of such solutions in the context of pseudo-supersymmetry for multi-field systems whose first-order equations we derive using a Bogomol’nyi-like method. In particular we show that scaling solutions that are pseudo-BPS must describe geodesic curves. Furthermore, we clarify how to solve the geodesic equations of motion when the scalar manifold is a maximally non-compact coset such as occurs in maximal supergravity. This relies upon a parametrization of the coset in the Borel gauge. We then illustrate this with the cosmological solutions of higher-dimensional gravity compactified on a n-torus. http://arxiv.org/abs/0704.1653v3 Contents 1 Preliminaries 2 2 (Pseudo-) supersymmetry 3 3 Multi-field scaling cosmologies 4 3.1 Pure kinetic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Potential-kinetic scaling solutions . . . . . . . . . . . . . . . . . . . . . . . 5 4 Geodesic curves and the Borel gauge 8 4.1 A solution-generating technique . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 An illustration from dimensional reduction . . . . . . . . . . . . . . . . . . 9 5 Discussion 10 A Curvatures 11 B The coset SL(N, IR)/ SO(N) 11 1 Preliminaries We consider scalar fields Φi that parametrize a Riemannian manifold with metric Gij coupled to gravity through the standard action µν∂µΦ j − V (Φ) . (1) We restrict to solutions with the following D-dimensional space-time metric ds2D = g(y) 2ds2D−1 + ǫf(y) 2dy2 , ds2D−1 = (ηǫ)abdx adxb , (2) where ǫ = ±1 and ηǫ = diag(−ǫ, 1, . . . , 1). The case ǫ = −1 describes a flat FLRW-space- time and ǫ = +1 a Minkowski-sliced domain wall (DW) space-time. The scalar fields that source these space-times can only depend on the y-coordinate Φi = Φi(y). The function f corresponds to the gauge freedom of reparameterizing the y-coordinate. Of particular interest in this note are scaling comologies, which have received a great deal of attention in the dark-energy literature, see [1] for a review and references. One def- inition (amongst many) of scaling cosmologies is that they are solutions for which all terms in the Friedmann equation have the same time dependence. For pure scalar cosmologies this implies that H2 ∼ V ∼ T ∼ τ−2 , (3) where τ denotes cosmic time, H the Hubble parameter and T is the kinetic energy T = GijΦ̇ iΦ̇j . These relations imply that the scale factor is power-law a(τ) ∼ τ p. In the case of curved FLRW-universes we also demand that H ∼ k/a2, which is only possible for p = 1. Interestingly, scaling solutions correspond to the FLRW-geometries that possess a time-like conformal vectorfield ξ coming from the transformation τ → eλτ , xa → e(1−p)λxa , (4) where xa are the space-like cartesian coordinates1. In the forthcoming we reserve the indices a, b, . . . to denote space-like coordinates when we consider cosmological space-times. Apart from the intriguing cosmological properties of scaling solutions they are also interesting for understanding the dynamics of a general cosmological solution since scaling solutions are often critical points of an autonomous system of differential equations and therefore correspond to attractors, repellors or saddle points [2]. Scaling cosmologies often appear in supergravity theories (see for instance [3,4]) but, remarkably, they also appear by spatially averaging inhomogeneous cosmologies in classical general relativity [5]. We will use two coordinate frames to describe scaling comologies τ − frame : ds2 = −dτ 2 + τ 2p ds2D−1 , (5) t− frame : ds2 = −e2t dt2 + e2ptds2D−1 . (6) The first is the usual FLRW-coordinate system and the second can be obtained by the substitution t = ln τ . 2 (Pseudo-) supersymmetry If the scalar potential V (Φ) can be written in terms of another function W (Φ) as follows V = ǫ Gij∂iW∂jW − D−14(D−2)W , (7) then the action can be written as “a sum of squares” plus a boundary term when reduced to one dimension: dy fgD−1 (D−1) 4(D−2) W − 2(D − 2) ġ +Gij∂jW ||2 gD−1W − 2(D − 1)ġgD−2f−1 , (8) where a dot denotes a derivative w.r.t. y. The term ||Φ̇i/f + Gij∂jW ||2 is a shorthand notation and the square involves a contraction with the field metric Gij. It is clear that the action is stationary under variations if the terms within brackets are zero2, leading to the following first-order equations of motion W = 2(D − 2) +Gij∂jW = 0 . (9) 1For curved FLRW-space-times the space-like coordinates are invariant. 2For completeness we should have added the Gibbons-Hawking term [6] in the action which deletes that part of the above boundary term that contains ġ. For ǫ = +1 these equations are the standard Bogomol’nyi-Prasad-Sommerfield (BPS) equations for domain walls that arise from demanding the susy-variation of the fermions to vanish, which guarantees that the DW preserves a fraction of the total supersymmetry of the theory. The function W is then the superpotential that appears in the susy-variation rules and equation (7) with ǫ = +1 is natural for supergravity theories. It is clear that for every W that obeys (7) we can find a corresponding DW-solution, and if W is not related to the susy-variations we call the solutions fake supersymmetric [7]. For ǫ = −1 these equations are the generalization to arbitrary space-time dimension D and field metric Gij of the framework found in references [8–11]. So here we generalized and derived in a different way (some of) the results of [8–11] by showing that analogously to DW’s we can write the Lagrangian as a sum of squares. We refer to these first-order equations as pseudo-BPS equations and W is named the pseudo-superpotential because of the immediate analogy with BPS domain walls in supergravity [10, 11]. For the case of cosmologies there is no natural choice for W as cosmologies cannot be found by demanding vanishing susy-variations of the fermions3. In [11] it is proven that for all single-scalar cosmologies (and domain walls) a pseudo- superpotential W exists such that the cosmology is pseudo-BPS and that one can give a fermionic interpretation of the pseudo-BPS flow in terms of so-called pseudo-Killing spinors. This does not necessarily carry over to multi-scalar solutions as was shown in [14]. Nonetheless, a multi-field solution can locally be seen as a single-field solution [15] because locally we can redefine the scalar coordinates such that the curve Φ(y) is aligned with a scalar axis and all other scalars are constant on this solution. A necessary condition for the single-field pseudo-BPS flow to carry over (locally) to the multi-field system is that the truncation down to a single scalar is consistent (this means that apart from the solution one can put the other scalars always to zero) [14]. 3 Multi-field scaling cosmologies Let us turn to scaling solutions in the framework of pseudo-supersymmetry and see how geodesic motion arises. First we consider the rather trivial case with vanishing scalar potential V and then in section 3.2 we add a scalar potential V . Pseudo-supersymmetry is only discussed in the case of non-vanishing V . 3.1 Pure kinetic solutions If there is no scalar potential the solutions trace out geodesics since after a change of coordinates y → ỹ(y) via dỹ = fg1−Ddy, the scalar field action becomes ′iΦ′jdỹ, where a prime means a derivative w.r.t. ỹ. This new action describes geodesic curves with affine parameter ỹ. The affine velocity is constant by definition and positive since the metric is positive definite ′iΦ′j = ||v||2 . (10) 3Star supergravity is an exception [12] and that seems related to pseudo-supersymmetry [13]. The Einstein equation is Ryy = 12GijΦ̇ iΦ̇j = ||v||2 g2−2Df 2 , Rab = 0 . (11) In the gauge f = 1 the solution is given by g = eC2(y +C1) D−1 , with C1 and C2 arbitrary integration constants, but with a shift of y we can always put C1 = 0 and C2 can always be put to zero by re-scaling the space-like coordinates. In the case of a four-dimensional cosmology the geometry is a power-law FLRW-solution with p = 1/3. 3.2 Potential-kinetic scaling solutions In a recent paper of Tolley and Wesley an interesting interpretation was given to scaling solutions [16], which we repeat here. The finite transformation (4) leaves the equations of motion invariant if the action S scales with a constant factor, which is exactly what happens for scaling solutions since all terms in the Lagrangian scale like τ−2. Under (4) the metric scales like e2λgµν and in order for the action to scale as a whole we must have V → e−2λV , T = 1 gττGijΦ̇ iΦ̇j → e−2λT . (12) Equations (12) imply that GijΦ̇ iΦ̇j remains invariant from which one deduces that dΦ must be a Killing vector. The curve that describes a scaling solution follows an isometry of the scalar manifold. It depends on the parametrization whether the tangent vector Φ̇ itself is Killing. This happens for the parametrization in terms of t = ln τ since = limλ→0 Φi(eλτ)− Φi(τ) d ln τ . (13) Thus a scaling solution is associated with an invariance of the equations of motion for a rescaling of cosmic time and is therefore associated with a conformal Killing vector on space-time and a Killing vector on the scalar manifold. Pseudo-supersymmetry comes into play when we check the geodesic equation of motion ∇Φ̇Φ̇i = Φ̇ j∇jΦ̇i = Φ̇j ∇(jΦ̇i) +∇[jΦ̇i] , (14) where we denote Φ̇i = GikΦ̇ k. Now we have that the symmetric part is zero if we parametrize the curve with t = ln τ since scaling makes Φ̇ a Killing vector. We also have that ∇[jΦ̇i] = 0 since the pseudo-BPS condition makes Φ̇ a curl-free flow Φ̇i = −f∂iW . To check that the curl is indeed zero (when f 6= 1) one has to notice that in the parametriza- tion of the curve in terms of t = ln τ the gauge is such that ġ/g is constant and that f ∼ W−1. Since the curl is also zero we notice that the curve is a geodesic with ln τ as affine parametrization4 ∇Φ̇Φ̇ i = 0 = Φ̈i + ΓijkΦ̇ jΦ̇k . (15) 4 One could wonder whether the results works in two ways. Imagine that a scaling solution is a geodesic. This then implies that ∇[jΦ̇i] = 0 and therefore the flow is locally a gradient flow Φ̇i = ∂i lnW ∼ f∂iW . The link between scaling and geodesics was discovered by Karthauser and Saffin in [17], but no conditions on the Lagrangian were given in [17] such that the relation scaling- geodesic holds. An example of a scaling solution that is not a geodesic was given by Sonner and Townsend in [18]. A more intuitive understanding of the origin of the geodesic motion for some scaling cosmologies comes from the on-shell substitution V = (3p− 1) T in the Lagrangian to get a new Lagrangian describing seemingly massless fields. Although this is rarely a consistent procedure we believe that this is nonetheless related to the existence of geodesic scaling solutions. Single field For single-field models the potential must be exponential V = Λeαφ in order to have scaling solutions. The simplest pseudo-superpotential belonging to an exponential potential is itself exponential W = ± 2 . (16) If we choose the plus sign the solution to the pseudo-BPS equation is φ(τ) = − 2 ln τ + 1 ln[6−2α ] , g(τ) ∼ τ α2 . (17) The minus sign corresponds to the time reversed solution. Multiple fields For a general multi-field model a scaling solution with power-law scale factor τ p obeys V = (3p− 1)T from which we derive the on-shell relation Gij∂iW∂jW = ⇒ W = ± 8 p V 3p− 1 . (18) In general the above expression for the superpotential W ∼ V does not hold off-shell, unless the potential is a function of a specific kind: Gij∂iV ∂jV . (19) Scalar potentials that obey (19) with the extra condition that p ≷ 1 ↔ V ≷ 0 allow for multi-field scaling solutions. For a given scalar potential that obeys (19) there probably exist many pseudo-superpotentials W compatible with V but if we make the specific choice 8 p V/(3p− 1) then all pseudo-BPS solutions must be scaling and hence geodesic. As a consistency check we substitute the first-order pseudo-BPS equations into the right- hand-side of the following second-order equations of motion Φ̈i + ΓijkΦ̇ kΦ̇j = −f 2Gij∂jV − 3 ˙(ln g)− ˙(ln f) Φ̇i , (20) and choose a gauge for which W , (21) then we indeed find an affine geodesic motion since the right hand side of (20) vanishes. For some systems one first needs to perform a truncation in order to find the above relation (19). A good example is the multi-field potential appearing in Assisted Inflation V (Φ1, . . . ,Φn) = , Gij = δij . (22) The scaling solution of this system was proven to be the same as the single-exponential scaling [20]. The reason is that one can perform an orthogonal transformation in field space such that the form of the kinetic term is preserved but the scalar potential is given V = eαϕ U(Φ1, . . . ,Φn−1) , . (23) The scaling solution is such that Φ1, . . . ,Φn−1 are frozen in a stationary point of U and therefore the system is truncated to a single-field system that obeys (19). The same was proven for Generalized Assisted Inflation [21] in reference [22]. The scaling solution in the original field coordinates reads Φi = Ai ln τ + Bi, which is clearly a straight line and thus a geodesic. The scaling solutions of [14, 18] were constructed for an axion-dilaton system with an exponential potential for the dilaton (∂φ)2 − 1 eµφ(∂χ)2 − Λeαφ . (24) Clearly this two-field system obeys (19) and (one of) the pseudo-superpotential(s) is given by (16). The pseudo-BPS scaling solution therefore has constant axion and is effectively described by the dilaton in an exponential potential. Note that this solution indeed de- scribes a geodesic on SL(2, IR)/ SO(2) with ln τ as affine parameter. All examples of scaling solutions in the literature seem to occur for exponential potentials, however by performing a SL(2, IR)-transformation on the Lagrangian (24) the kinetic term is unchanged and the potential becomes a more complicated function of the axion and the dilaton. The same scaling solution then trivially still exists (and (19) still holds) but the axion is not con- stant in the new frame and instead the solution follows a more complicated geodesic on SL(2, IR)/ SO(2). However another scaling solution is given in [18] that is not geodesic and with varying axion in the frame of the above action (24). This is an illustration of the above, since the solution is not geodesic we know that there does not exists any other pseudo-superpotential for which the varying axion solution is pseudo-BPS, consistent with what is shown in [14] for that particular solution. 4 Geodesic curves and the Borel gauge For the last example of the previous section the pseudo-BPS scaling solutions described geodesics on the symmetric space SL(2, IR)/ SO(2). In this section we consider a general class of symmetric spaces of which SL(2, IR)/ SO(2) is an example and they are known as maximally non-compact cosets U/K. It seems that for this class of spaces the geodesic equations of motion can be solved easily. The symmetry of the geodesic equations is the symmetry of the scalar coset U/K. In the case of maximal supergravity the symmetry U is a U-duality and is a maximal non-compact real slice of a complex semisimple group. The isotropy group K is the maximal compact subgroup of U . 4.1 A solution-generating technique In the Borel gauge the scalar fields are divided into r dilatons φI and (n − r) axions χα, with r the rank of U and n the dimension of U/K (see for instance [23]). The dilatons are related to the generators HI of the Cartan sub-algebra (CSA) and the axions to the positive root generators Eα through the following expression for the coset representative L in the Borel gauge L = Παexp[χ αEα]ΠIexp[−12φ IHI ] . (25) In this language the geodesic equation is φ̈I + ΓIJKφ̇ J φ̇K + ΓIαJ χ̇ αφ̇J + ΓIαβχ̇ αχ̇β = 0 , (26) χ̈α + ΓαJKφ̇ J φ̇K + ΓαβJ χ̇ βφ̇J + Γαβγχ̇ βχ̇γ = 0 . (27) Since ΓIJK = 0 and Γ JK = 0 at points for which χ α = 0 a trivial solution is given by φI = vI y , χα = 0 . (28) How many other solutions are there? A first thing we notice is that every global U - transformation Φ → Φ̃ brings us from one solution to another solution. Since U generically mixes dilatons and axions we can construct solutions with non-trivial axions in this way. We now prove that in this way all geodesics are obtained and this depends on the fact that U is maximally non-compact with K the maximal compact subgroup of U . Consider an arbitrary geodesic curve Φ(t) on U/K. The point Φ(0) can be mapped to the origin L = 1 using a U -transformation, since we can identify Φ(0) with an element of U and then we multiply the geodesic curve Φ(t) with Φ(0)−1, generating a new geodesic curve Φ2(t) = Φ(0) −1Φ(t) that goes through the origin. The origin is invariant under K-rotations but the tangent space at the origin transforms under the adjoint of K. One can prove that there always exists an element k ∈ K, such that AdjkΦ̇2(0) ∈ CSA [24]. Therefore χ̇α2 = 0 and this solution must be a straight line. So we started out with a general curve Φ(t) and proved that the curve Φ3(t) = kΦ(0) −1Φ(t) is a straight line. 4.2 An illustration from dimensional reduction The metric Ansatz for the dimensional reduction of (4 + n)-dimensional Einstein-gravity on the n-torus (Tn) is ds24+n = e 2αϕds24 + e 2βϕMabdza ⊗ dzb , (29) where 4(n+ 2) , β = − . (30) The matrix M is a positive-definite symmetric n×n matrix with unit determinant, which depends on the 4-dimensional coordinates, describing the moduli of Tn. The modulus ϕ controls the overall volume and is named the breathing mode or radion field. Notice that we already truncated the Kaluza–Klein vectors in the Ansatz. The reduction of the Einstein–Hilbert term gives −g{R− 1 (∂ϕ)2 + 1 Tr∂M∂M−1} . (31) The scalars parametrize IR × SL(n, IR)/ SO(n) where ϕ belongs to the decoupled IR-part and M is the SL(n, IR)/ SO(n) part. If we take the four-dimensional part of space-time to be a flat FLRW-space then that part of the metric will be power-law with p = 1/3 and the scalars follow a geodesic with ln τ as an affine parameter. According to the solution-generating technique, the Ansatz for the scalars is ϕ = v0 ln τ + c0 , M = ΩDΩT , D = diag(e− ~βa·~φ) , (32) with ~φ = ~v ln τ and ~β the weights of SL(n, IR) in the fundamental representation (see appendix B for some explanations on the SL(n, IR)/ SO(n)-coset in this representation). The diagonal matrix D represents the straight-line solution and Ω is an arbitrary SL(n, IR)- matrix in the fundamental representation. Therefore M = ΩDΩT is the most general coset matrix describing a geodesic curve. The Friedmann equation implies that the affine velocity is restricted to be v20 + ||v||2 = 43 , (33) which is the only constraint coming from the 4-dimensional Einstein equation. If we sub- stitute this solution in (29) and define new coordinates ~y = ~zΩ we find ds24+n = −τ 2α v0dτ 2 + τ +2αv0d~x23 + ~βa·~v+2β v0dy2a . (34) This is similar to what is called a Kasner solution in general relativity (see for instance [25]). Kasner solutions are a general class of time-dependent geometries that look like ds2 = −τ 2p0dτ 2 + τ 2padx2a . (35) Kasner solutions solve the Einstein equations in vacuum if the following two conditions are satisfied p0 + 1 = pa , (p0 + 1) p2a . (36) For the metric (34) these conditions are satisfied if the lower-dimensional Friedmann equa- tion is satisfied. For this calculation one needs the properties of the weight-vectors ~βa (given in appendix B) and the relation between α and β (30). We therefore conclude that the general spatially flat FLRW-solution lifts up to the most general Kasner solution with SO(3)-symmetry in 4 + n dimensions. 5 Discussion In this note we have studied multi-field scaling solutions using a first-order formalism for scalar cosmologies a.k.a. pseudo-supersymmetry. We derived these first-order equations via a Bogomol’nyi-like method that was known to work for domain wall solutions as was first shown in [26, 27]5 and we showed that it trivially extends to cosmological solutions. This first-order formalism allows a better understanding of the geodesic motion that comes with a specific class of scaling solutions. One of the main results of this note is a proof that shows that all pseudo-BPS cosmologies that are scaling solutions must be geodesic. This complements to the discussion in [14] where the first example of a non-geodesic scaling cosmology was shown to be non-pseudo-BPS. Moreover we gave constraints on multi-field Lagrangians for which the pseudo-BPS cosmologies are geodesic scaling solutions. Having illustrated the importance of geodesic motion in scalar cosmology, we tackled the problem of solving the geodesic equations in the second part of this note. We showed that the most general geodesic curve can be written down for maximally non-compact coset spaces U/K. These coset spaces appear in all maximal and some less-extended supergravities [29]. We used a solution-generating technique based on the symmetries of the coset. We were able to prove that the most general solution is given by a U- transformation on the “straight line”, (φI(t) = vIt, χα = 0) in the Borel gauge. We illustrated this technique for the coset SL(n, IR)/ SO(n). Since SL(n, IR)/ SO(n) is also the moduli space of the n-torus we applied it to find the cosmological solutions of higher- dimensional gravity compactified on a n-torus. This exercise nicely illustrates why the straight line is the generating solution since, from a higher-dimensional point of view, all solutions that correspond to the non-straight line geodesics can be seen as coordinate transformations of the solutions associated with the straight line. The oxidation of the straight line solutions corresponds to the most general SO(3)-invariant Kasner solution of (4 + n)-dimensional vacuum GR. The same technique was used in [3] to find all geodesic scaling cosmologies of the CSO-gaugings in maximal supergravity. The solution-generating technique presented here should be considered complementary to the “compensator method” developed by Fré et al in [30]. There the straight line 5See also [28]. also serves as a generating solution but instead of rigid U -transformations one uses local K transformations that preserve the solvable gauge to generate new non-trivial solutions. This technique is a nice illustration of the integrability of the second–order geodesic equations of motion [31]. Acknowledgments We are grateful to Dennis Westra for useful discussions and comments on the manuscript and to Jan Rosseel for many useful discussions. This work is supported in part by the Eu- ropean Communitys Human Potential Programme under contract MRTN-CT-2004-005104 in which the authors are associated to Utrecht University. The work of AP and TVR is part of the research programme of the Stichting voor Fundamenteel Onderzoek der Materie (FOM). A Curvatures For the metric Ansatz (2) the Ricci tensor is given by Rab = −ǫ(ηǫ)ab gġḟ + (D − 3) ġ , Ryy = (D − 1) − ( g̈ . (37) B The coset SL(N, IR)/ SO(N) Consider a general coset U/K. It is not difficult to construct a coset representative using the Lie algebras U and K of U and K respectively. Since K is a subgroup of U we have the decomposition U = K ⊕ F, with F the complement of K in U. For a given representation of the algebra U we define a coset representative via L(y) = exp(yifi) where the fi form a basis of F in some representation of U. To derive the metric we define a Lie algebra valued one-form from the coset represen- tative L(y) via L−1dL ≡ E + Ω , (38) where E takes values in F and Ω in K. We notice that L−1dL is invariant under left multiplication with a y-independent element g ∈ U . Multiplying L from the right with local elements k ∈ K results in E → k−1E k , Ω → k−1Ω k + k−1dk . (39) In supergravity the parameters yi are scalar fields that depend on the space-time coordi- nates yi = φi(x). The one-form L−1dL can be written out in terms of coset-coordinate one-forms dφi which themselves can be pulled back to space-time coordinate one-forms dφi = ∂µφ idxµ. Now we can write L−1dL = Eµdx µ + Ωµdx µ . (40) Under the φ-dependent K-transformations k(φ(x)) we have that Ωµ → k−1Ωµk + k−1∂µk and Eµ → k−1Eµk. It is clear that Eµ is covariant under local K-transformations and Ωµ transforms like a connection. Using this connection Ωµ we can make the following K-covariant derivative on L and L−1 DµL = ∂µL− LΩµ , DµL−1 = ∂µL−1 + ΩµL−1 . (41) To find a kinetic term for the scalars we notice that the object Tr[DµLD µL−1] = −Tr[EµEµ] , (42) has all the right properties as it contains single derivatives on the scalars, it is a space-time scalar, it is invariant under rigid U transformations and under local K-transformations. Thus, e−1Lscalar = −Tr[EµEµ] ≡ −12g(φ)ij∂µφ i∂µφj . (43) If SO(N) is the maximal compact subgroup of U and we work in the fundamental representation, then the Lie algebra of SO(N) is the vector space of antisymmetric matrices, L−1dL+ (L−1dL)T , Ω = L−1dL− (L−1dL)T , (44) and a calculation shows that e−1Lscalar = −Tr[E2] = +14Tr[∂M∂M −1] , (45) where M is the SO(N)-invariant matrix M = LLT . No we specify to U = SL(N, IR). 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0704.1654

The Peculiar Velocities of Local Type Ia Supernovae and their Impact on Cosmology

Draft version September 15, 2021 Preprint typeset using LATEX style emulateapj v. 08/22/09 THE PECULIAR VELOCITIES OF LOCAL TYPE Ia SUPERNOVAE AND THEIR IMPACT ON COSMOLOGY James D. Neill California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125 Michael J. Hudson University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, CANADA Alex Conley University of Toronto, 60 Saint George Street, Toronto, ON M5S 3H8, CANADA Draft version September 15, 2021 ABSTRACT We quantify the effect of supernova Type Ia peculiar velocities on the derivation of cosmological parameters. The published distant and local Ia SNe used for the Supernova Legacy Survey first-year cosmology report form the sample for this study. While previous work has assumed that the local SNe are at rest in the CMB frame (the No Flow assumption), we test this assumption by applying peculiar velocity corrections to the local SNe using three different flow models. The models are based on the IRAS PSCz galaxy redshift survey, have varying β = Ω0.6m /b, and reproduce the Local Group motion in the CMB frame. These datasets are then fit for w, Ωm, and ΩΛ using flatness or ΛCDM and a BAO prior. The χ2 statistic is used to examine the effect of the velocity corrections on the quality of the fits. The most favored model is the β = 0.5 model, which produces a fit significantly better than the No Flow assumption, consistent with previous peculiar velocity studies. By comparing the No Flow assumption with the favored models we derive the largest potential systematic error in w caused by ignoring peculiar velocities to be ∆w = +0.04. For ΩΛ, the potential error is ∆ΩΛ = −0.04 and for Ωm, the potential error is ∆Ωm < +0.01. The favored flow model (β = 0.5) produces the following cosmological parameters: w = −1.08+0.09 −0.08, Ωm = 0.27 +0.02 −0.02 assuming a flat cosmology, and ΩΛ = 0.80 +0.08 −0.07 and Ωm = 0.27 +0.02 −0.02 for a w = −1 (ΛCDM) cosmology. Subject headings: cosmology: large-scale structure of the universe – galaxies: distances and redshifts – supernovae: general 1. INTRODUCTION Dark Energy has challenged our knowledge of funda- mental physics since the direct evidence for its existence was discovered using Type Ia supernovae (Riess et al. 1998; Perlmutter et al. 1999). Because there are cur- rently no compelling theoretical explanations for Dark Energy, the correct emphasis, as pointed out by the Dark Energy Task Force (DETF, Albrecht et al. 2006), is on refining our observations of the accelerated expansion of the universe. Recommendation V from the DETF Re- port (Albrecht et al. 2006) calls for an exploration of the systematic effects that could impair the needed observa- tional refinements. A couple of recent studies (Hui & Greene 2006; Cooray & Caldwell 2006) point out that the redshift lever arm needed to accurately measure the universal expansion requires the use of a local sample, but that coherent large-scale local (z < 0.2) peculiar velocities add additional uncertainty to the Hubble diagram and hence to the derived cosmological parameters. Current analyses (e.g., Astier et al. 2006; Riess et al. 2007; Wood-Vasey et al. 2007) of the cosmological pa- rameters do not attempt to correct for the effect of local peculiar velocities. As briefly noted by Hui & Greene Electronic address: [email protected] Electronic address: [email protected] Electronic address: [email protected] (2006) and Cooray & Caldwell (2006), it is possible to use local data to measure the local velocity field and hence limit the impact on the derived cosmo- logical parameters. Measurements of the local ve- locity field have improved to the point where there is consistency among surveys and methods (Hudson 2003; Hudson et al. 2004; Radburn-Smith et al. 2004; Pike & Hudson 2005; Sarkar et al. 2006). Type Ia supernova peculiar velocities have been studied re- cently by Radburn-Smith et al. (2004); Pike & Hudson (2005); Jha et al. (2006); Haugboelle et al. (2006); Watkins & Feldman (2007) and others. Their results demonstrate that the local flows derived from SNe are in agreement with those derived from other distance in- dicators, such as the Tully-Fisher relation and the Funda- mental Plane. Our aim is to use the current knowledge of the local peculiar motions to correct local SNe and, together with a homogeneous set of distant SNe, fit for cosmological parameters and measure the effect of the corrections on the cosmological fits. To produce this measurement, we analyze the lo- cal and distant SN Ia sample used in the first-year cosmology results from the Supernova Legacy Survey (SNLS, Astier et al. 2006, hence A06). This sam- ple is composed of 44 local SNe (A06, Table 8: Hamuy et al. 1996; Riess et al. 1999; Krisciunas et al. 2001; Jha 2002; Strolger et al. 2002; Altavilla et al. http://arxiv.org/abs/0704.1654v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] 2 Neill, Hudson, & Conley 2004; Krisciunas et al. 2004a,b) and 71 distant SNe (A06, Table 9). The distant SNe are the largest homo- geneous set currently in the literature. The local sample span the redshift range 0.015 < z < 0.125 and were selected to have good lightcurve sampling (A06, § 5.2). Using three different models encompassing the range of plausible local large-scale flow, we assign and correct for the peculiar velocity of each local SN. We then re-fit the entire sample for w, Ωm, and ΩΛ to assess the system- atics due to the peculiar velocity field, and to asses the change in the quality of the resulting fits. 2. PECULIAR VELOCITY MODELS Peculiar velocities, v, arise due to inhomogeneities in the mass density and hence in the expansion. Their ef- fect is to perturb the observed redshifts from their cos- mological values: czCMB = cz + v · r̂, where cz is the cosmological redshift the SN would have in the absence of peculiar velocities. With the advent of all-sky galaxy redshift surveys, it is possible to predict peculiar veloc- ities from the galaxy distribution provided one knows β = f(Ω)/b, where b is a linear biasing parameter relating fluctuations in the galaxy density, δ, to fluctuations in the mass density. The peculiar velocity in the CMB frame is then given by linear perturbation theory (Peebles 1980) applied to the density field (see, e.g. Yahil et al. 1991; Hudson 1993): ∫ Rmax δ(r′) (r′ − r) |r′ − r|3 d3r′ +V. (1) In this Letter, we use the density field of IRAS PSCz galaxies (Branchini et al. 1999), which extends to a depth Rmax = 20000 km s −1. Contributions to the pe- culiar velocity arising from masses on scales larger than Rmax are modeled by a simple residual dipole, V. Thus, given a density field, the parameters β and V describe the velocity field within Rmax. For galaxies with dis- tances greater than Rmax, the first term above is set to zero. The predicted peculiar velocities from the PSCz den- sity field are subject to two sources of uncertainty: the noisiness of the predictions due to the sparsely-sampled density field, and the inapplicability of linear perturba- tion theory on small scales. Typically these uncertainties are accounted for by adding an additional “thermal” dis- persion, which is assumed to be Gaussian. From a care- ful analysis of predicted and observed peculiar velocities, Willick & Strauss (1998) estimated these uncertainties to be ∼ 100 km s−1, albeit with a dependence on den- sity. Radburn-Smith et al. (2004) found reasonable χ2 values if 150 km s−1 was assumed in the field, with an extra contribution to the small-scale dispersion added in quadrature for SNe in clusters. Here we adopt a thermal dispersion of 150 km s−1. For this study, we explore the results of three different models of large-scale flows and compare them to a case where no flow model is used. These models have been chosen to span the range of flow models permitted by peculiar velocity data, and all of these models reproduce the observed ∼ 600 km s−1 motion of the Local Group with respect to the CMB. The first model assumes a pure bulk flow (model PBF, hence β = 0) with V having vector components (57,−540, 314) km s−1 in Galactic Cartesian coordinates. The second model assumes β = 0.5 (model B05), with a dipole vector of (70,−194, 0) km s−1. The third model adopts β = 0.7 (model B07) which requires no residual dipole. We compare these models to the no-correction scenario adopted by A06 and others with β = 0, V = 0 which we call the “No Flow” or NF scenario. Note that a recent comparison (Pike & Hudson 2005) of results from IRAS predictions versus peculiar velocity data yields a mean value fit with β = 0.50±0.02 (stat), so the B05 model is strongly favored over the NF scenario by independent peculiar velocity analyses. 3. COSMOLOGICAL FITS Prior to the fitting procedure, the peculiar veloci- ties for each model are used to correct the local SNe (using a variation of Hui & Greene 2006, equations 11 and 13). We then fit our corrected SN data in two ways using a χ2-gridding cosmology fitter1 (also used by Wood-Vasey et al. 2007). The first fit uses a flat cos- mology (Ω = 1) with the equation of state parameter w and Ωm as free parameters. The second fit assumes a ΛCDM (w = −1) cosmology with ΩΛ and Ωm as free parameters. We used the same intrinsic SN photometric scatter (σint = 0.13 mag, A06) for every fit. The result- ing χ2 probability surfaces for both fits are then further constrained using the BAO result from Eisenstein et al. (2005). The final derived cosmological parameters are then used to calculate the χ2 for each fit (see A06, § 5.4). The fitting procedure employed here differs in imple- mentation from that used in A06. Three additional pa- rameters, often called nuisance parameters, must be fit along with the two cosmological parameters. These pa- rameters are the constant of proportionality for the SN lightcurve shape, αs, the correction for the SN observed color, βc, and a SN brightness normalization,M. We dis- tinguish βc from the β used to describe the flow models above. A06 used analytic marginalization of the nuisance parameters αs and βc in their fits. Here these parame- ters are fully gridded like the cosmological parameters. This avoids a bias in the nuisance parameters that re- sults because, in the analytic method, their values must be held fixed to compute the errors. The result is that our fits using the NF scenario produces slightly different cosmological parameters than quoted in A06. 4. RESULTS The results of the cosmological fits for each model are listed in Table 1 and plotted in Figure 1 and Figure 2. They demonstrate two effects of the peculiar velocity corrections: a change in the values of the cosmological parameters, and a change in the quality of the fits as measured by the χ2 statistic. We expect, if a given model is correct, to improve the fitting since our corrected data should more closely re- semble the homogeneous universe described by a few cos- mological parameters. The χ2 of the fits for each flow model can be compared to the χ2 for the NF scenario (shown by the dashed line in the figures) as a test of this hypothesis. Using ∆χ2 = −2 lnL/LNF , where L is the likelihood, we find that the pure bulk flow is over 103 times less likely than the NF scenario, while the B05 and 1 http://qold.astro.utoronto.ca/conley/simple cosfitter/ http://qold.astro.utoronto.ca/conley/simple_cosfitter/ Peculiar Velocities of SNe 3 TABLE 1 Peculiar Velocity Model Parameters and Results Ω = 1 + BAO prior w = −1 + BAO prior Model β V (km s−1) w Ωm χ ΩΛ Ωm χ ΩΛ,Ωm A06a 0.0 · · · −1.023± 0.090 0.271± 0.021 · · · 0.751± 0.082 0.271 ± 0.020 · · · NF 0.0 · · · −1.054 +0.086 −0.084 0.270 +0.024 −0.018 115.5 0.770 +0.083 −0.071 0.269 +0.033 −0.017 115.4 PBF 0.0 57,-540,314 −1.026+0.085 −0.083 0.273+0.024 −0.019 129.4 0.741+0.084 −0.073 0.273+0.034 −0.017 129.2 B05 0.5b 70,-194,0 −1.081 +0.087 −0.085 0.268 +0.024 −0.018 110.3 0.796 +0.081 −0.070 0.267 +0.032 −0.017 110.1 B07 0.7 · · · −1.094 +0.087 −0.085 0.267 +0.024 −0.018 111.2 0.809 +0.082 −0.069 0.265 +0.032 −0.017 111.1 a results quoted in A06 marginalizing analytically over αs and βc (see § 3) b best fit value from Pike & Hudson (2005) Fig. 1.— Parameter values for the w, Ωm fit (Ω = 1 + BAO prior) for each of the four peculiar velocity models in Table 1. The values for the NF scenario are indicted by the dashed lines. The largest systematic error in w compared with the NF fit is +0.040 for the B07 model, which demonstrates the amplitude of the systematic error if peculiar velocity is not accounted for. The offsets for Ωm are all within ±0.003 showing that this parameter is not sensitive to the peculiar velocity corrections due to the BAO prior. The χ2 of the fits improve when using the two β models (B05, B07), while the PBF model provides a significantly worse fit. B07 models are 13.5 and 8.6 times more likely, respec- tively. We also use these data to assess the systematic er- rors made in the parameters if no peculiar velocities are accounted for. The largest of these are obtained by com- paring the B07 model with the NF scenario. This com- parison yields ∆wB07 = +0.040 and ∆ΩΛ,B07 = −0.039. The same comparison for the B05 model, which is only slightly preferred by the χ2 statistic over model B07, pro- duces ∆wB05 = +0.027 and ∆ΩΛ,B05 = −0.026. The systematic offsets for Ωm are all 0.004 or less, demon- strating the insensitivity of this parameter to peculiar velocities. This is due to the BAO prior which is insensi- tive to local flow and provides a much stronger constraint for Ωm than for w or ΩΛ (see A06, Figures 5 and 6). 5. DISCUSSION AND SUMMARY The systematic effect of different flow models is at the level of ±0.04 in w. This is smaller than the present level of random error in w, which is largely due to the small numbers of high- and low-redshift SNe. However, com- pared to other systematics discussed in A06, which total Fig. 2.— Parameter values for the ΩΛ, Ωm fit (w = −1 + BAO prior) for each of the four peculiar velocity models as in Figure 1. Again, comparing the NF fits to the B07 model produces the largest systematic in ΩΛ of −0.039. We also find Ωm insensitive to the corrections, having all offsets within ±0.004. The χ2 values show the same pattern as in Figure 1, favoring the β models over no correction (NF), and over pure bulk flow. ∆w = ±0.054, the systematic effect of large-scale flows is important. Wood-Vasey et al. (2007, Table 5) list 16 sources of systematic error which total ∆w = ±0.13. Aside from three method-dependent systematics and the photometric zero-point error, they are all smaller than the flow systematic. As the number of SNe continues to increase, and understanding of other systematics (e.g. photometric zero-points) improves, it is possible that large-scale flows will become one of the dominant sources of systematic uncertainty. The peculiar velocities of SN host galaxies arise from large-scale structures over a range of scales. The compo- nent arising from small-scale, local structure is the least important: it is essentially a random variable which is reduced by N . More problematic is the large-scale co- herent component. Such a large-scale component can take several forms: an overdensity or underdensity; a large-scale dipole, or “bulk” flow. The existence of a large-scale, but local (< 7400 km s−1) underdensity, or “Hubble Bubble” was first discussed by Zehavi et al. (1998). Recently Jha et al. (2006) have re-enforced this claim with a larger SN data set: they find that the difference in the Hubble constant inside the Bubble and outside is ∆H/H = 6.5 ± 1.8%. 4 Neill, Hudson, & Conley If correct, this could have a dramatic effect on the de- rived cosmological parameters (Jha et al. 2006, Fig 17), especially for those studies that extend their local sample down below z < 0.015. However, the “Hubble Bubble” was not confirmed by Giovanelli et al. (1999) who found ∆H/H = 1.0 ± 2.2% using the Tully-Fisher (TF) pe- culiar velocities, nor by Hudson et al. (2004) who found ∆H/H = 2.3± 1.9% using the Fundamental Plane (FP) distances. According to equation 1, a mean underdensity of IRAS galaxies of order ∼ 40% within 7400 km s−1 would be needed to generate the “Hubble Bubble” quoted by Jha et al. (2006). However, we find that the IRAS PSCz density field of Branchini et al. (1999) is not un- derdense in this distance range; instead it is mildly over- dense (by a few percent) within 7400 km s−1 (see also Branchini et al. 1999, Figure 2). As a further cross- check, when we refit the Jha et al. (2006) data after hav- ing subtracted the predictions of the B05 flow model, the “Bubble” remains in the Jha et al. (2006) data. Thus, the Jha et al “Bubble” cannot be explained by local structure, unless that structure is not traced by IRAS galaxies. Moreover, when we analyze the 99 SNe within 15000 km s−1 from Tonry et al. (2003) in the same way, we find no evidence of a significant “Hubble Bubble” (∆H/H = 1.5 ± 2.0%), in agreement with the results from TF and FP surveys. The Tonry et al. (2003) sam- ple and that of Jha et al. (2006) have 67 SNe in common. The high degree of overlap suggests that the difference lies in the different methods for converting the photom- etry into SN distance moduli. A local large-scale flow can also introduce systematic errors if the low-z sample is biased in its sky coverage: in this case, an uncorrected dipole term can corrupt the monopole term, which then biases the cosmological pa- rameters. For the large-scale flow directions considered here, this does not appear to affect the A06 sample: we note that the PBF-corrected case has similar cosmolog- ical parameters to the “No Flow” case. However, if co- herent flows exist on large scales, this may affect surveys with unbalanced sky coverage, such as the SN Factory (Aldering et al. 2002) or the SDSS SN survey2. 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W., Jha, S., Riess, A. G., Garnavich, P. M., Kirshner, R. P., Aguilera, C., et al. 2007, preprint (astro-ph/0701041) Yahil, A., Strauss, M. A., Davis, M., & Huchra, J. P. 1991, ApJ, 372, 380 Zehavi, I., Riess, A. G., Kirshner, R. P., & Dekel, A. 1998, ApJ, 503, 483 http://arxiv.org/abs/astro-ph/0702751 http://arxiv.org/abs/astro-ph/0701041

0704.1655

Creation of Quark-gluon Plasma in Celestial Laboratories

Creation of Quark-gluon Plasma in Celestial Laboratories R. K. Thakur *Retired Professor of Physics, School of Studies in Physics Pt.Ravishakar Shukla University, Raipur, India 21 College Road, Choube Colony, Raipur-492001, India Abstract It is shown that a gravitationally collapsing black hole acts as an ultrahigh energy particle accelerator that can accelerate particles to energies inconceivable in any ter- restrial particle accelerator, and that when the energy E of the particles comprising the matter in the black hole is ∼ 102 GeV or more,or equivalently the temperature T is ∼ 1015 K or more, the entire matter in the black hole will be in the form of quark-gluon plasma permeated by leptons. Key words: Quark-gluon plasma, black holes, particle accelerators PACS: 12.38 Mh, 25.75 Nq, 97.60 Lf, 04.70−s 1 Introduction Efforts are being made to create quark-gluon plasma (QGP)in terrestrial labo- ratories. A report released by CERN, the European Organization for Nuclear Research, at Geneva, on February 10, 2000 said, “A series of experiments using CERN’s lead beam have presented compelling evidence for the exis- tence of a new state of matter 20 times denser than nuclear matter, in which quarks instead of being bound up into more complex particles such as pro- tons and neutrons, are liberated to roam freely“. By smashing together lead ions at CERN’s accelerator at temperatures 100,000 times as hot as sun’s cen- tre, i.e. at temperatures T ∼ 1.5 × 1012 K, and energy densities never before reached in laboratory experiments, a team of 350 scientists from institutes in 20 countries succeeded in isolating quarks from more complex particles, e.g. protons and neutrons. However, the evidence of creation QGP at CERN Email address: [email protected] (R. K. Thakur). Preprint submitted to Elsevier 10 August 2021 http://arxiv.org/abs/0704.1655v1 is indirect,involving detection of particles produced when QGP changes back to hadrons. The production of these particles can be explained alternatively without having to have QGP. Therefore, the evidence of the creation of QGP at CERN is not enough and conclusive. In view of this CERN will start a new experiment, ALICE (A Large Ion Collider Experiment), at much higher energies available at its LHC (Large Hadron Collider). First collisions in the LHC will occur in November 2007. A two months run in 2007, with beams colliding at an energy of 0.9 TeV, will give the accelerator and detector teams the opportunity to run-in their equipment, ready for a run at the full collision energy of 14 Tev to start in spring 2008. In the meantime, the focus of research on QGP has shifted to the Relativistic Heavy Ion Collider (RHIC), the world’s newest and largest particle accelerator for nuclear research, at Brookhaven National Laboratory (BNL) in Upton, New York. RHIC’s goal is to create and study QGP by head-on collisions of two beams of gold ions at energies 10 times those of CERN’s programme, which ought to produce QGP with higher temperature and longer life time thereby allowing much clear and direct observation. The programme at RHIC started in June 2000.Researchers at RHIC generated thousands of head-on collisions between gold ions at energies of 130 GeV creating fireballs of matter having density hundred times greater than that of the nuclear matter and temperature ∼ 2 × 1012 K (175 MeV in the energy scale). Fireballs were of size ∼ 5 femtometre which lasted a few times 10−24 second. All the four detector systems, viz., STAR, PHENIX, BRAHMS, PHOBOS, detected “jet quenching“ and suppression of “leading particles“, highly energetic individual particles that emerge from the nuclear fireballs in gold-gold collisions. Jet quenching and suppression of leading particles are signs of QGP formation. Eventually, with plenty of data in hand, all the four detector collaborations - STAR, PHENIX, BRAHMS, PHOBOS - operating at the BNL have converged on a consensus opinion that the fireball is a liquid of strongly interacting quarks and gluons rather than a gas of weakly interacting quarks and gluons. More- over, this liquid is almost a “perfect“ liquid with very low viscosity. The RHIC findings were reported at the meeting of the American Physical Society (APS) held during April 16-19, 2005 in Tampa, Florida in a talk delivered by Gary Westfall. Thus, it is obvious that the existence of QGP theoretically predicted by Quantum Chromodynamics (QCD) has been experimentally validated at RHIC. But the QGP created hitherto in terrestrial laboratories is ephemeral, its life- time is, as mentioned earlier, a few times 10−24 second, presumably because its temperature is not well above the transition temperature for transition from the hadronic phase to the QGP phase. In addition to this, it is difficult to maintain it even at that temperature for long enough time. However, as shown in the sequel, in nature we have celestial laboratories in the form of gravitationally collapsing black holes wherein QGP is created naturally; this QGP is at much higher temperature than the transition temperature, and pre- sumably therefore it is not ephemeral. More so, because the temperature of the QGP created in black holes continually increases and as such it is always above the transition temperature. 2 Gravitationally collapsing black hole as a particle accelerator We consider a gravitationally collapsing black hole (BH). In the simplest treat- ment (1) a BH is considered to be a spherically symmetric ball of dust with negligible pressure, uniform density ρ = ρ(t), and at rest at t = 0. These assumptions lead to the unique solution of the Einstein field equations, and in the comoving co-ordinate system the metric inside the BH is given by ds2 = dt2 − R2(t) 1− k r2 + r2dθ2 + r2 sin2 θ dφ2 in units in which the speed of light in vacuum, c = 1, and where k = 8πGρ(0)/3 is a constant. On neglecting mutual interactions the energy E of any one of the particles comprising the matter in the BH is given by E2 = p2 + m2 > p2, in units in which again c = 1, and where p is the magnitude of the 3-momentum of the particle and m its rest mass. But p = h , where λ is the de Broglie wavelength of the particle and h Planck’s constant of action. Since all length in the collapsing BH scale down in proportion to the scale factor R(t) in equation (1), it is obvious that λ ∝ R(t). Therefore it follows that p ∝ R−1(t), and hence p = aR−1(t), where a is the constant of proportionality. From this it follows that E > a/R(t). Consequently, E as well as p increases continually as R decreases. It is also obvious that E and p → ∞ as R → 0. Thus, in effect, we have an ultrahigh energy particle accelerator, so far inconceivable in any terrestrial laboratory, in the form of a gravitationally collapsing BH, which can, in the absence of any physical process inhibiting the collapse, accelerate particles to an arbitrarily high energy and momentum without any limit. What has been concluded above can also be demonstrated alternatively, with- out resorting to the general theory of relativity, as follows. As an object col- lapses under self-gravitation, the inter-particle distance s between any pair of particles in the object decreases. Obviously, the de Broglie wavelength λ of any particle in the object is less than or equal to s, a simple consequence of Heisenberg’s uncertainty principle. Therefore, s ≥ h . Consequently, p ≥ h hence E ≥ h . Since during the gravitational collapse of an object s decreases continually, the energy E as well as p, the magnitude of the 3-momentum of each of the particles is the object increases continually. Moreover, from E ≥ h and p ≥ h it follows that E and p → ∞ as s → 0. Thus, any gravitation- ally collapsing object in general, and a BH in particular, acts as an ultrahigh energy particle accelerator. It is also obvious that ρ, the density of matter in the BH, continually increases as the BH collapses. In fact, ρ ∝ R−3, and hence ρ → ∞ as R → 0. 3 Creation of quark-gluon plasma inside gravitationally collapsing black holes It has been shown theoretically that when the energy E of the particles in mat- ter is ∼ 102 GeV (s ∼ 10−16 cm ) corresponding to a temperature T ∼ 1015 K, all interactions are of the Yang-Mills type with SUc(3)×SUIW (2)×UYW (1) gauge symmetry, where c stands for colour, IW for weak isospin, and YW for weak hypercharge; and at this stage quark deconfinement occurs as a result of which the matter now consists of its fundamental constituents: spin 1/2 leptons, namely, the electrons, the muons, the tau leptons, and their neu- tirnos, which interact only through the electroweak interaction; and the spin 1/2 quarks, u(up), d(down), s(strange), c(charm), b(bottom), t(top), which interact electroweakly as well as through the colour force generated by gluons (2). In this context it may be noted that, as shown in section 2, the energy E of each of the particles comprising the matter in a gravitationally collaps- ing BH continually increases, and so does the density ρ of the matter in the BH. During the continual collapse of a BH a stage will be reached when E and ρ will be so large and s so small that the quarks confined in the hadrons will be liberated from the infrared slavery and acquire asymptotic freedom, i.e., the quark deconfinement will occur.This will happen when E ∼ 102 GeV (s ∼ 10−16 cm) corresponding to T ∼ 1015 K. Consequently, during the con- tinual gravitational collapse of a BH, when E ≥ 102 GeV (s ≤ 10−16 cm) corresponding to T ≥ 1015 K, the entire matter in the BH will be in the form QGP permeated by leptons. One may understand what happens eventually to the matter in a gravitation- ally collapsing BH in another way as follows. As a BH collapses continually, gravitational energy is released continually. Since, inter alia, gravitational en- ergy so released cannot escape the BH, it will continually heat the matter comprising the BH. Consequently, the temperature of the matter in the BH will increase continually. When the temperature reaches the transition tem- perature for transition from the hadronic phase to the QGP phase, which is predicted to be ∼ 170 MeV (∼ 1012 K) by the Lattice Gauge Theory, the entire matter in the BH will be converted into QGP permeated by leptons. It may be noted that in a BH the QGP will not be ephemeral like what it has hitherto been in the case of the QGP created in terrestrial laboratories, it will not go back to the hadronic phase, because the temperature of the matter in the BH continually increases and, after crossing the transition temperature for the transition from the hadronic phase to the QGP phase, it will be more and more above the transition temperature. Consequently, once the transition from the hadronic phase to the QGP phase occurs in a BH, there is no going back; the entire matter in the BH will remain in the form of QGP permeated by leptons. 4 Conclusion From the foregoing it is obvious that a BH acts as an ultrahigh energy par- ticle accelerator that can accelerate particles to energies inconceivable in any terrestrial particle accelerator, and that the matter in any gravitationally col- lapsing BH is eventually converted into QGP permeated by leptons. However, the snag is that it is not possible to probe and study the properties of the QGP in a BH because nothing can escape outside the event horizon of a BH. 5 Acknowledgment The author thanks Professor S. K. Pandey, the Co-ordinator of the Refer- ence Centre at Pt. Ravishankar Shukla University, Raipur of the University Grants Commission’s Inter-university Centre for Astronomy and Astrophysics at Pune. He also thanks Mr. Laxmikant Chaware and Miss Leena Madharia for typing the manuscript. References [1] S. Weinberg, Gravitation and Cosmology (John Wiley & Sons, New York, 1972), 342. [2] P. Ramond, Ann. Rev. Nucl. Part.Sc., 33 (1983) 31. Introduction Gravitationally collapsing black hole as a particle accelerator Creation of quark-gluon plasma inside gravitationally collapsing black holes Conclusion Acknowledgment

0704.1656

Temperature-driven transition from the Wigner Crystal to the Bond-Charge-Density Wave in the Quasi-One-Dimensional Quarter-Filled band

Temperature-driven transition from the Wigner Crystal to the Bond-Charge-Density Wave in the Quasi-One-Dimensional Quarter-Filled band R.T. Clay,1 R.P. Hardikar,1 and S. Mazumdar2 1Department of Physics and Astronomy and HPC2 Center for Computational Sciences, Mississippi State University, Mississippi State MS 39762 2 Department of Physics, University of Arizona Tucson, AZ 85721 (Dated: August 21, 2021) It is known that within the interacting electron model Hamiltonian for the one-dimensional 1 filled band, the singlet ground state is a Wigner crystal only if the nearest neighbor electron-electron repulsion is larger than a critical value. We show that this critical nearest neighbor Coulomb interaction is different for each spin subspace, with the critical value decreasing with increasing spin. As a consequence, with the lowering of temperature, there can occur a transition from a Wigner crystal charge-ordered state to a spin-Peierls state that is a Bond-Charge-Density Wave with charge occupancies different from the Wigner crystal. This transition is possible because spin excitations from the spin-Peierls state in the 1 -filled band are necessarily accompanied by changes in site charge densities. We apply our theory to the 1 -filled band quasi-one-dimensional organic charge-transfer solids in general and to 2:1 tetramethyltetrathiafulvalene (TMTTF) and tetramethyltetraselenafulvalene (TMTSF) cationic salts in particular. We believe that many recent experiments strongly indicate the Wigner crystal to Bond-Charge-Density Wave transition in several members of the TMTTF family. We explain the occurrence of two different antiferromagnetic phases but a single spin-Peierls state in the generic phase diagram for the 2:1 cationic solids. The antiferromagnetic phases can have either the Wigner crystal or the Bond-Charge-Spin-Density Wave charge occupancies. The spin-Peierls state is always a Bond-Charge-Density Wave. PACS numbers: 71.30.+h, 71.45.Lr, 74.70.Kn I. INTRODUCTION Spatial broken symmetries in the quasi-one- dimensional (quasi-1D) 1 -filled organic charge- transfer solids (CTS) have been of strong experimental1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 and theoretical18,19,20,21,22,23,24,25,26 interest. The bro- ken symmetry states include charge order (hereafter CO, this is usually accompanied by intramolecular distortions), intermolecular lattice distortions (hereafter bond order wave or BOW), antiferromagnetism (AFM) and spin-Peierls (SP) order. Multiple orderings may compete or even coexist simultaneously. Interestingly, these unconventional insulating states in the CTS are often proximate to superconductivity27, the mecha- nism of which has remained perplexing after intensive investigations over several decades. Unconventional behavior at or near 1 -filling has also been observed in the quasi-two-dimensional organic CTS with higher superconducting critical temperatures11,28,29,30,31, sodium cobaltate32,33,34 and oxides of titanium35 and vanadium36,37. In spite of extensive research on 1D instabilities in the CTS, detailed comparisons of theory and experiments re- main difficult. Strong electron-electron (e-e) Coulomb interactions in these systems make calculations particu- larly challenging, and with few exceptions38,39 existing theoretical discussions of broken symmetries in the in- teracting 1 -filled band have been limited to the ground state18,19,20,21,22,23,24,25,26. This leaves a number of im- portant questions unresolved, as we point out below. 4k 1010F AsF6 PF6SbF6 PF6 Pressure FIG. 1: Schematic of the proposed T-vs.-P phase diagram for (TMTCF)2X, where C=T or S, along with the charge occu- pancies of the sites in the low T phases as determined in this work. The P-axis reflects the extent of interchain coupling. Open (filled) arrows indicate the ambient pressure locations of TMTTF (TMTSF) salts. Quasi-1D CTS undergo two distinct phase transitions as the temperature (hereafter T) is reduced40. The 4kF transition at higher T involves charge degrees of free- dom (T4kF ∼ 100 K), with the semiconducting state be- low T4kF exhibiting either a dimerized CO or a dimer- ized BOW. Charges alternate as 0.5 + ǫ and 0.5 – ǫ on the molecules along the stack in the dimerized CO. The site charge occupancy in the dimerized CO state is commonly written as · · · 1010· · · (with ‘1’ and ‘0’ denot- ing charge-rich and charge-poor sites), and this state is http://arxiv.org/abs/0704.1656v2 also referred to as the Wigner crystal. The 4kF BOW has alternating intermolecular bond strengths, but site charges are uniformly 0.5. It is generally accepted that the dimer CO (dimer BOW) is obtained for strong (weak) intermolecular Coulomb interactions (the intramolecular Coulomb interaction can be large in either case, see Sec- tion III). At T < T2kF ∼ 10 – 20 K, there occurs a sec- ond transition in the CTS, involving spin degrees of free- dom, to either an SP or an AFM state. Importantly, the SP and the AFM states both can be derived from the dimer CO or the dimer BOW. Coexistence of the SP state with the Wigner crystal would require that the SP state has the structure 1 = 0 = 1 · · · 0 · · · 1, with singlet “bonds” alternating in strength between the charge-rich sites of the Wigner crystal. Similarly, coexisting AFM and Wigner crystal would imply site charge-spin occu- pancies ↑ 0 ↓ 0. The occurrence of both have been sug- gested in the literature20,26,38,39. The SP state can also be obtained from dimerization of the dimer 4kF BOW, in which case there occurs a spontaneous transition from the uniform charge-density state to a coexisting bond- charge-density wave (BCDW)18,22,25 1− 1 = 0 · · · 0. The unit cells here are dimers with site occupancies ‘1’ and ‘0’ or ‘0’ and ‘1’, with strong and weak interunit 1–1 and 0· · · 0 bonds, respectively. The AFM state with the same site occupancies · · · 1100· · · is referred to as the bond- charge-spin density wave (BCSDW)21 and is denoted as ↑↓ 00. The above characterizations of the different states and the related bond, charge and spin patterns are largely based on ground state calculations. The mechanism of the transition from the 4kF state to the 2kF state lies outside the scope of such theories. Questions that remain unresolved as a consequence include: (i) Does the nature of the 4kF state (charge versus bond dimerized) predeter- mine the site charge occupancies of the 2kF state? This is assumed in many of the published works. (ii) What de- termines the nature of the 2kF state (AFM versus SP)? (iii) How does one understand the occurrence of two dif- ferent AFM phases (hereafter AFM1 and AFM2) strad- dling a single SP phase in the proposed T vs. P (where P is pressure) phase diagram (see Fig. 1)9 for the cationic 1 filled band quasi-1D CTS? (iv) What is the nature of the intermediate T states between the 4kF dimerized state and the 2kF tetramerized state? As we point out in the next section, where we present a brief review of the ex- perimental results, answers to these questions are crucial for a deeper understanding of the underlying physics of the 1 -filled band CTS. In the present paper we report the results of our calcu- lations of T-dependent behavior within theoretical mod- els incorporating both e-e and electron-phonon (e-p) in- teractions to answer precisely the above questions. One key result of our work is as follows: in between the strong and weak intermolecular Coulomb interaction parame- ter regimes, there exists a third parameter regime within which the intermolecular Coulomb interactions are in- termediate, and within which there can occur a novel transition from a Wigner crystal CO state to a BCDW SP state as the T is lowered. Thus the charge or bond ordering in the 4kF phase does not necessarily decide the same in the 2kF state. For realistic intramolecular Coulomb interactions (Hubbard U), we show that the width of this intermediate parameter regime is compara- ble to the strong and weak intersite interaction regimes. We believe that our results are directly applicable to the family of the cationic CTS (TMTTF)2X, where a redis- tribution of charge upon entering the SP state from the CO state has been observed15,16 in X=AsF6 and PF6. A natural explanation of this redistribution emerges within our theory. The SP state within our theory is unique and has the BCDW charge occupancy, while the two AFM re- gions in the phase diagram of Fig. 1 have different site occupancies. Our theory therefore provides a simple di- agnostic to determine the pattern of CO coexisting with low-temperature magnetic states in the 1 -filled CTS. In addition to the above theoretical results directly pertaining to the quasi-1D CTS, our work gives new in- sight to excitations from a SP ground state in a non- half-filled band. In the case of the usual SP transition within the 1 -filled band, the SP state is bond-dimerized at T = 0 and has uniform bonds at T > T2kF . The site charges are uniform at all T. This is in contrast to the -filled band, where the SP state at T = 0 is bond and charge-tetramerized and the T > T2kF state is dimerized as opposed to being uniform. Furthermore, we show that the high T phase here can be either charge- or bond- dimerized, starting from the same low T state. This clearly requires two different kinds of spin excitations in the 1 -filled band. We demonstrate that spin excitations from the SP state in the 1 -filled band can lead to two different kinds of defects in the background BCDW. In the next section we present a brief yet detailed sum- mary of relevant experimental results in the quasi-1D CTS. The scope of this summary makes the need for hav- ing T-dependent theory clear. Following this, in Section III we present our theoretical model along with conjec- tures based on physical intuitive pictures. In Section IV we substantiate these conjectures with accurate quantum Monte Carlo (QMC) and exact diagonalization (ED) nu- merical calculations. Finally in Section V we compare our theoretical results and experiments, and present our conclusions. II. REVIEW OF EXPERIMENTAL RESULTS Examples of both CO and BOW broken symmetry at T < T4kF are found in the -filled CTS. The 4kF phase in the anionic 1:2 CTS is commonly bond-dimerized. The most well known example is MEM(TCNQ)2, which un- dergoes a metal-insulator transition accompanied with bond-dimerization at 335 K41. Site charges are uniform in this 4kF phase. The 2kF phase in the TCNQ-based systems is universally SP and not AFM. The SP transi- tion in MEM(TCNQ)2 occurs below T2kF = 19 K, and low T neutron diffraction measurements of deuterated samples41 have established that the bond tetramerization is accompanied by 2kF CO · · · 1100· · · . X-ray 42,43 and neutron diffraction44 experiments have confirmed a sim- ilar low T phase in TEA(TCNQ)2. We will not discuss these further in the present paper, as they are well de- scribed within our previous work18,25. We will, however, argue that the SP ground state in (DMe-DCNQI)2Ag (as opposed to AFM) indicates · · · 1100· · · CO in this. The cationic (TMTCF)2X, C= S and Se, exhibit more variety, presumably because the counterions affect pack- ing as well as site energies in the cation stack. Differ- ences between systems with centrosymmetric and non- centrosymmetric anions are also observed. Their overall behavior is summarized in Fig. 1, where as is custom- ary pressure P can also imply larger interchain coupling. We have indicated schematically the possible locations of different materials on the phase diagram. The most sig- nificant aspect of the phase diagram is the occurrence of two distinct antiferromagnetic phases, AFM1 and AFM2, straddling a single SP phase. Most TMTTF lie near the low P region of the phase diagram and are insulating already at or near room tem- perature because of charge localization, which is due to the intrinsic dimerization along the cationic stacks1,4,11. CO at intermediate temperatures TCO has been found in dielectric permittivity4, NMR7, and ESR46 experi- ments on materials near the low and intermediate P end. Although the pattern of the CO has not been de- termined directly, the observation of ferroelectric behav- ior below TCO is consistent with · · · 1010· · · type CO in this region5,23. With further lowering of T, most (TMTTF)2X undergo transitions to the AFM1 or SP phase (with X = Br a possible exception, see below). X = SbF6 at low T lies in the AFM1 region 9, with a very high TCO and relatively low Neel temperature TN = 8 K. As the schematic phase diagram indicates, pressure suppresses both TCO and TN in this region. For P > 0.5 GPa, (TMTTF)2SbF6 undergoes a transition from the AFM1 to the SP phase9, the details of which are not com- pletely understood; any charge disproportionation in the SP phase is small9. (TMTTF)2ReO4 also has a relatively high TCO = 225 K, but the low T phase here, reached following an anion-ordering transition is spin singlet14. Nakamura et al. have suggested, based on NMR exper- iments, that the CO involves the Wigner crystal state, but the low T state is the · · · 1100· · · BCDW14. Further along the P axis lie X = AsF6 and PF6, where TCO are re- duced to 100 K and 65 K, respectively7. The low T phase in both cases is now SP. Neutron scattering experiments on (TMTTF)2PF6 have found that the lattice distortion in the SP state is the expected 2kF BOW distortion, but that the amplitude of the lattice distortion is much smaller10 than that found in other organic SP materials such as MEM(TCNQ)2. The exact pattern of the BOW has not been determined yet. Experimental evidence ex- ists that some form of CO persists in the magnetic phases. For example, the splitting in vibronic modes below TCO in (TMTTF)2PF6 and (TMTTF)2AsF6, a signature of charge disproportionation, persists into the SP phase17, indicating coexistence of CO and SP. At the same time, the high T CO is in competition with the SP ground state8, as is inferred from the different effects of pressure on TCO and TSP : while pressure reduces TCO, it in- creases TSP . This is in clear contrast to the effect of pres- sure on TN in X = SbF6. Similarly, deuteration of the hy- drogen atoms of TMTTF increases TCO but decreases TSP . That higher TCO is accompanied by lower TSP for centrosymmetric X (TSP= 16.4 K in X = PF6 and 11.1 K in X = AsF6) has also been noted 48. This trend is in ob- vious agreement with the occurrence of AFM instead of SP state under ambient pressure in X = SbF6. Most in- terestingly, Nakamura et al. have very recently observed redistribution of the charges on the TMTTF molecules in (TMTTF)2AsF6 and (TMTTF)2PF6 as these systems enter the SP phase from CO states15,16. Charge dispro- portionation, if any, in the SP phase is much smaller than in the CO phase15,16, which is in apparent agree- ment with the above observations9,14 in X = ReO4 and SbF6. The bulk of the (TMTTF)2X therefore lie in the AFM1 and SP regions of Fig. 1. (TMTSF)2X, in contrast, occupy the AFM2 region. Coexisting 2kF CDW and spin-density wave, SDW, with the same 2kF periodicity 49,50 here is explained naturally as the · · · 1100· · · BCSDW21,22,51. In contrast to the TMTTF salts discussed above, charge and magnetic ordering in (TMTTF)2Br occur almost simultaneously 46,52. X-ray studies of lattice distortions point to similarities with (TMTSF)2PF6 49, indicating that (TMTTF)2Br is also a · · · 1100· · · BCSDW21. We do not discuss AFM2 re- gion in the present paper, as this can be found in our earlier work21,25. III. THEORETICAL MODEL AND CONJECTURES The 1D Hamiltonian we investigate is written as H = HSSH +HHol +Hee (1a) HSSH = t [1 + α(a i + ai)](c i,σci+1,σ + h.c.) + ~ωS iai (1b) HHol = g i + bi)ni + ~ωH ibi (1c) Hee = U ni,↑ni,↓ + V nini+1 (1d) In the above, c i,σ creates an electron with spin σ (↑,↓) on molecular site i, ni,σ = c i,σci,σ is the number of electrons with spin σ on site i, and ni = ni,σ. U and V are the on-site and intersite Coulomb repulsions, and a i and b create (dispersionless) Su-Schrieffer-Heeger (SSH)53 and Holstein (Hol)54 phonons on the ith bond and site respec- tively, with frequencies ωS and ωH . Because the Peierls instability involves only phonon modes near q = π, keep- ing single dispersionless phonon modes is sufficient for the Peierls transitions to occur55,56. Although purely 1D calculations cannot yield a finite temperature phase tran- sition, as in all low dimensional theories57,58 we antici- pate that the 3D ordering in the real system is principally determined by the dominant 1D instability. The above Hamiltonian includes the most important terms necessary to describe the family of quasi-1D CTS, but ignores nonessential terms that may be necessary for understanding the detailed behavior of individual sys- tems. Such nonessential terms include (i) the intrinsic dimerization that characterizes many (TMTTF)2X, (ii) interaction between counterions and the carriers on the quasi-1D cations stacks, (iii) interchain Coulomb inter- action, and (iv) interchain hopping. Inclusion of the in- trinsic dimerization will make the Wigner crystal ground state even less likely24, and this is the reason for exclud- ing it. We have verified the conclusions of reference 24 from exact diagonalization calculations. The inclusion of interactions with counterions may enhance the Wigner crystal ordering5,23 for some (TMTTF)2X. We will dis- cuss this point further below, and argue that it is im- portant in the AFM1 region of the phase diagram. The effects of intrinsic dimerization and counterion interac- tions can be reproduced by modifying the V/|t| in our Hamiltonian, and thus these are not included explicitly. Rather the V in Eq. (1a) should be considered as the effective V for the quasi-1D CTS. Interchain hopping, at least in the TMTTF (though not in the TMTSF), are in- deed negligible. The interchain Coulomb interaction can promote the Wigner crystal within a rectangular lattice framework, but for the realistic nearly triangular lattice will cause frustration, thereby making the BCDW state of interest here even more likely. We therefore believe that Hamiltonian (1a) captures the essential physics of the quasi-1D CTS. For applications to the quasi-1D CTS we will be inter- ested in the parameter regime18,24,25,26 |t| = 0.1 − 0.25 eV, U/|t| = 6 − 8. The exact value of V is less known, but since the same cationic molecules of interest as well as other related molecules (for e.g., HMTTF, HMTSF, etc.) also form quasi-1D 1 -filled band Mott-Hubbard semiconductors with short-range antiferromagnetic spin correlations59,60, it must be true that V < 1 U (since for V > 1 U the 1D 1 -filled band is a CDW61,62). Two other known theoretical results now fix the range of V that will be of interest. First, the ground state of the 1 -filled band within Hamiltonian (1a) in the limit of zero e-p coupling is the Wigner crystal · · · 1010· · · only for suffi- ciently large V > Vc(U). Second, Vc(U → ∞) = 2|t|, and is larger for finite U19,24,25. With the known U/|t| and the above restrictions in place, it is now easily concluded that (i) the Wigner crystal is obtained in the CTS for a relatively narrow range of realistic parameters, and (ii) even in such cases the material’s V is barely larger than Vc(U) 24,25. We now go beyond the above ground state theories of spatial broken symmetries to make the following cru- cial observation: each different spin subspace of Hamil- tonian (1a) must have its own Vc at which the Wigner crystal is formed. This conclusion of a spin-dependent Vc = Vc(U, S) follows from the comparison of the fer- romagnetic subspace with total spin S = Smax and the S = 0 subspace. The ferromagnetic subspace is equiva- lent to the 1 -filled spinless fermion band, and therefore Vc(U, Smax) is independent of U and exactly 2|t|. The increase of Vc(U, S = 0) with decreasing U 19,25 is then clearly related to the occurrence of doubly occupied and vacant sites at finite U in the S = 0 subspace. Since the probability of double occupancy (for fixed U and V ) decreases monotonically with increasing S, we can in- terpolate between the two extreme cases of S = 0 and S = Smax to conclude Vc(U, S) > Vc(U, S+1). We prove this explicitly from numerical calculations in the next section. Our conjecture regarding spin-dependent Vc(U) in turn implies that there exist three distinct parameter regimes for realistic U and V : (i) V ≤ V (Smax) = 2|t|, in which case the ground state is the BCDW with 2kF CO and the high temperature 4kF state is a BOW 18,25; (ii) V > Vc(U, S = 0), in which case both the 4kF and the 2kF phases have Wigner crystal CO; (iii) and the intermediate regime 2|t| ≤ V ≤ Vc(U, S = 0). Pa- rameter regime (i) has been discussed in our previous work18,21,25,63. This is the case where the description of the 1 -filled band as an effective 1 -filled band is appropri- ate: the unit cell in the 4kF state is a dimer of two sites, and the 2kF transition can be described as the usual SP dimerization of the dimer lattice. We will not discuss this parameter region further. Given the U/|t| values for the CTS, parameter regime (ii) can have rather narrow width. For U/|t| = 6, Vc(U, S = 0) = 3.3|t|, and there is no value of realistic V for which the ground state is a Wigner crystal. For U/|t| = 8, Vc(U, S = 0) = 3.0|t|, and the widths of parameter regime (iii), 2|t| ≤ V ≤ 3|t| and of parameter regime (ii), 3|t| ≤ V ≤ 4|t|, are comparable. We will discuss parameter regime (ii) only briefly, as the physics of this region has been discussed by other authors20,26. We investigated thoroughly the intermedi- ate parameter regime (iii), which has not been studied previously. Within the intermediate parameter regime we expect a novel transition from the BCDW in the 2kF state at low T to a · · · 1010· · · CO in the 4kF phase at high T that has not been discussed in the theoretical lit- erature. Our observation regarding the T-dependent behavior of these CTS follows from the more general observation that thermodynamic behavior depends on the free en- ergy and the partition function. For the strong e-e inter- actions of interest here, thermodynamics of 1D systems at temperatures of interest is determined almost entirely by spin excitations. Since multiplicities of spin states increase with the total spin S, the partition function is dominated by high (low) spin states with large (small) multiplicities at high (low) temperatures. While at T=0 such a system must be a BCDW for V < Vc(U, S = 0), as the temperature is raised, higher and higher spin states begin to dominate the free energy, until V for the ma- terial in question exceeds Vc(U, S), at which point the charge occupancy reverts to · · · 1010· · · We demonstrate this explicitly in the next section. A charge redistribu- tion is expected in such a system at T2kF , as the devi- ation from the average charge of 0.5 is much smaller in the BCDW than in the Wigner crystal25. The above conjecture leads to yet another novel impli- cation. The ground state for both weak and intermediate intersite Coulomb interaction parameters are the BCDW, even as the 4kF phases are different in the two cases: BOW in the former and CO in the latter. This necessarily requires the existence of two different kinds of spin exci- tations from the BCDW. Recall that within the standard theory of the SP transition in the 1 -filled band57,64,65, thermal excitations from the T = 0 ground state gener- ates spin excitations with bond-alternation domain walls (solitons), with the phase of bond alternation in between the solitons being opposite to that outside the solitons. Progressive increase in T generates more and more soli- tons with reversed bond alternation phases, until overlaps between the two phases of bond alternations lead ulti- mately to the uniform state. A key difference between the 1 -filled and 1 -filled SP states is that site charge oc- cupancies in spin excitations in the former continue to be uniform, while they are necessarily nonuniform in the lat- ter case, as we show in the next section. We demonstrate that defect centers with two distinct charge occupancies (that we will term as type I and II), depending upon the actual value of V , are possible in the 1 -filled band. Pre- ponderance of one or another type of defects generates the distinct 4kF states. IV. RESULTS We present in this section the results of QMC investi- gations of Eq. (1a), for both zero and nonzero e-p cou- plings, and of ED studies of the adiabatic (semiclassical) limit of Eq. (1a). Using QMC techniques, we demon- strate explicitly the spin-dependence of Vc(U), as well as the transition from the Wigner crystal to the BCDW for the intermediate parameter regime (iii). The ED studies demonstrate the exotic nature of spin excitations from the BCDW ground state. In what follows, all quantities are expressed in units of |t| (|t|=1). The QMC method we use is the Stochastic Series Ex- pansion (SSE) method using the directed loop update for the electron degrees of freedom66. For 1D fermions with nearest-neighbor hopping SSE provides statistically exact results with no systematic errors. While the SSE directed-loop method is grand canonical (with fluctuat- ing particle density), we restrict measurements to only the 1 -filled density sector to obtain results in the canoni- cal ensemble. Quantum phonons are treated within SSE by directly adding the bosonic phonon creation and an- nihilation operators in Eq. (1a) to the series expansion67. An upper limit in the phonon spectrum must be imposed, but can be set arbitrarily large to avoid any systematic errors67. For the results shown below we used a cutoff of 100 SSH phonons per bond and either 30 (for g = 0.5) or 50 (for g = 0.75) Holstein phonons per site. The observables we calculate within SSE are the standard wave-vector dependent charge structure factor Sρ(q), defined as, Sρ(q) = eiq(j−k)〈O 〉 (2) and charge and bond-order susceptibilities χρ(q) and χB(q), defined as, χx(q) = eiq(j−k) dτ〈Oxj (τ)O k (0)〉 (3) In Eqs. (2) and (3) N is the number of lattice sites, O nj,↑ + nj,↓, O j+1,σcj,σ + h.c.), and β is the inverse temperature in units of t. The presence of CO or BOW can be detected by the divergence of the 2kF or 4kF charge or bond-order sus- ceptibility as a function of increasing system size. Strictly speaking, in a purely 1D model these functions diverge only at T = 0; as already explained above, we make the reasonable assumption58 that in the presence of realis- tic inter-chain couplings transitions involving charge or bond-order instabilities, as determined by the dominant susceptibility, occur at finite T. A. Spin-dependent Vc(U) We first present computational results within Hamil- tonian (1a) in the absence of e-p coupling to demon- strate that the Vc(U) at which the Wigner crystal order is established in the lowest state of a given spin sub- space decreases with increasing spin S. Our computa- tional approach conserves the total z-component of spin Sz and not the total S. Since the Lieb-Mattis theorem E(S) < E(S+1), where E(S) is the energy of the lowest state in the spin subspace S, applies to the 1D Hamilto- nian (1a), and since in the absence of a magnetic field all Sz states for a given S are degenerate, our results for the lowest state within each different Sz must pertain to S = Sz. To determine Vc(U, S) we use the fact that the purely electronic model is a Luttinger liquid (LL) for V < Vc with correlation functions determined by a single expo- nent Kρ (see Reference 69 for a review). The Wigner crystal state is reached when Kρ = . The exponent Kρ may be calculated from the long-wavelength limit of Sρ(q) 70. In Fig. 2(a) we have plotted our calculated Kρ for U = 8 as a function of V for different Sz sectors. The 2.2 2.4 2.6 2.8 3 3.2 0 2 4 6 8 FIG. 2: Vc for Eq. (1a) in the limit of zero e-p interactions (α = g = 0) as a function of Sz. Results are for a N=32 site periodic ring with U = 8. For N = 32 Sz=8 corresponds to the fully polarized (spinless fermion) limit. (a) Luttinger Liquid exponent Kρ as a function of V . Kρ = determines the boundary for the · · · 1010· · · CO phase. (b) Vc plotted vs. temperature chosen is small enough (β = 2N) that in all cases the results correspond to the lowest state within a given Sz. In Fig. 2(b) we have plotted our calculated Vc(U = 8), as obtained from Fig. 2(a), as a function of Sz. Vc is largest for Sz = 0 and decreases with increas- ing Sz, in agreement with the conjecture of Section III. Importantly, the calculated Vc for Sz = 8 is close to the correct limiting value of 2, indicating the validity of our approach. We have not performed any finite-size scaling in Fig. 2, which accounts for the the slight deviation from the exact value of 2. B. T-dependent susceptibilities We next present the results of QMC calculations within the full Hamiltonian (1a). To reproduce correct relative energy scales of intra- and intermolecular phonon modes in the CTS, we choose ωH > ωS , specifically ωH = 0.5 and ωS = 0.1 in our calculations. Small deviations from these values do not make any significant difference. In all cases we have chosen the electron-molecular vibration coupling g larger than the coupling between electrons and the SSH phonons α, thereby deliberately enhancing the T=0.25t T=0.125t T=0.042t 0 0.2 0.4 0.6 0.8 1 T=0.25t T=0.125t T=0.042t 0 0.05 0.1 0.15 0.2 0.25 FIG. 3: (color online) QMC results for the temperature- dependent charge susceptibilities for a N=64 site periodic ring with U = 8, V = 2.75, α = 0.15, ωS = 0.1, g = 0.5, and ωH = 0.5. (a) and (b) Wavevector-dependent charge and bond-order susceptibilities. (c) 2kF and 4kF charge sus- ceptibilities as a function of temperature. (d) 2kF and 4kF bond-order susceptibilities as a function of temperature. If error bars are not shown, statistical error bars are smaller than the symbol sizes. Lines are guides to the eye. likelihood of the Wigner crystal CO. We report results only for intermediate and strong weak intersite Coulomb interactions; the weak interaction regime V < 2 has been discussed extensively in our previous work25. 1. 2 ≤ V ≤ Vc(U, S = 0) Our results are summarized in Figs. 3–5, where we re- port results for two different V of intermediate strength and several different e-p couplings. Fig. 3 first shows re- sults for relatively weak SSH coupling α. The charge sus- ceptibility is dominated by a peak at 4kF = π (Fig. 3(a)). Figs. 3(c) and (d) show the T-dependence of the 2kF as well as 4kF charge and bond susceptibility. For V < Vc(U, S = 0) the 4kF charge susceptibility does not diverge with system size, and the purely 1D system remains a LL with no long range CO at zero tempera- ture. The dominance of χρ(4kF ) over χρ(2kF ) suggests, however, that · · · 1010· · · CO will likely occur in the 3D T=0.25t T=0.125t T=0.042t 0 0.2 0.4 0.6 0.8 1 T=0.25t T=0.125t T=0.042t g=0.50 g=0.50 g=0.75 g=0.75 0 0.05 0.1 0.15 0.2 0.25 g=0.50 g=0.50 g=0.75 g=0.75 FIG. 4: (color online) Same as Fig. 3, but with parameters U = 8, V = 2.25, α = 0.27, ωS = 0.1, and ωH = 0.5. In panels (a) and (b), data are for g = 0.50 only. In panels (c) and (d), data for both g = 0.50 and g = 0.75 are shown. Arrows indicate temperature where χρ(2kF ) = χρ(4kF ) (solid and broken arrows correspond to g = 0.50 and 0.75, respectively.) system, especially if this order is further enhanced due to interactions with the counterions9. We have plotted the bond susceptibilities χB(2kF ) and χB(4kF ) in Fig. 3(d). A SP transition requires that χB(2kF ) diverges as T → 0. χB(2kF ) weaker than χB(4kF ) at low T indicates that the SP order is absent in the present case with weak SSH e-p coupling. This result is in agreement with earlier result55 that the SP order is obtained only above a critical αc (αc may be smaller for the infinite system than found in our calculation). The most likely scenario with the present parameters is the persistence of the · · · 1010· · · CO to the lowest T with no SP transition. These pa- rameters, along with counterion interactions, could then describe the (TMTTF)2Xmaterials with an AFM ground state9. In Figs. 4 and Fig. 5 we show our results for larger SSH e-p coupling α and two different intersite Coulomb interaction V=2.25 and 2.75. The calculations of Fig. 4 were done for two different Holstein e-p couplings g. For both the V parameters, χρ(4kF ) dominates at high T but χρ(2kF ) is stronger at low T in both Fig. 4(c) and Fig. 5(c). The crossing between the two susceptibilities T=0.25t T=0.125t T=0.042t 0 0.2 0.4 0.6 0.8 1 T=0.25t T=0.125t T=0.042t g=0.50 g=0.50 0 0.05 0.1 0.15 0.2 0.25 g=0.50 g=0.50 FIG. 5: (color online) Same as Fig. 3, but with parameters U = 8, V = 2.75, α = 0.27, ωS = 0.1, g = 0.5, and ωH = 0.5. Arrow indicates temperature where χρ(2kF ) = χρ(4kF ). is clear indication that in the intermediate parameter regime, as T is lowered the 2kF CDW instability domi- nates over the 4kF CO. The rise in χρ(2kF ) at low T is accompanied by a steep rise in χB(2kF ) in both cases (see Fig. 4(d) and Fig. 5(d)). Importantly, unlike in Fig. 3(d), χB(2kF ) in these cases clearly dominates over χB(4kF ) by an order of magnitude at low temperatures. There is thus a clear signature of the SP instability for these parameters. The simultaneous rise in χB(2kF ) and χρ(2kF ) indicates that the SP state is the · · · 1100· · · BCDW. Comparison of Figs. 4 and 5 indicates that the effect of larger V is to decrease the T where the 2kF and 4kF susceptibilities cross. Since larger V would imply larger TCO, this result implies that larger TCO is accompanied by lower TSP . Our calculations are for relatively modest α < g. Larger α (not shown) further strengthens the divergence of 2kF susceptibilities. The motivation for performing the calculations of Fig. 4 with multiple Holstein couplings was to deter- mine whether it is possible to have a Wigner crystal at low T even for V < V (U, S = 0), by simply increas- ing g. The argument for this would be that in strong coupling, increasing g amounts to an effective increase in V 71. The Holstein coupling cannot be increased arbi- T=0.25t T=0.125t T=0.042t 0 0.2 0.4 0.6 0.8 1 T=0.25t T=0.125t T=0.042t 0 0.05 0.1 0.15 0.2 0.25 FIG. 6: (color online) Same as Fig. 3, but with parameters U = 8, V = 3.5, α = 0.24, ωS = 0.1, g = 0.5, and ωH = 0.5. trarily, however, as beyond a certain point, g promotes formation of on-site bipolarons67. Importantly, the co- operative interaction between the 2kF BOW and CDW in the BCDW21,25 implies that in the V < V (U, S = 0) region, larger g not only promotes the 4kF CO but also enhances the BCDW. In our calculations in Fig. 4(b) and (c) we we find both these effects: a weak increase in χρ(4kF ) at intermediate T, and an even stronger increase in the T → 0 values of χρ(2kF ) and χB(2kF ). Actually, this result is in qualitative agreement with our obser- vation for the ground state within the adiabatic limit of Eq. (1a) that in the range 0 < V < V (U, S = 0), V enhances the BCDW25. The temperature at which χρ(2kF ) and χρ(4kF ) cross does not change significantly with larger g. Our results for g = 0.75 for V = 2.75 (not shown) are similar, except that the data are more noisy now (probably because all parameters in Fig. 5 are much too large for the larger g). We conclude therefore that in the intermediate V region, merely increasing g does not change the BCDW nature of the SP state. For g to have a qualitatively different effect, V should be much closer to V (U, S = 0) or perhaps even larger. 2. V > Vc(U, S = 0) In principle, calculations of low temperature instabili- ties here should be as straightforward as the weak inter- site Coulomb interaction V < 2t regime. The 4kF CO - AFM1 state ↑ 0 ↓ 0 would occur naturally for the case of weak e-p coupling. Obtaining the 4kF CO-SP state 1 = 0 = 1 · · · 0 · · · 1, with realistic V < 1 U is, however, difficult25. Previous work72, for example, finds this state for V > 1 U . Recent T-dependent mean-field calculations of Seo et al.39 also fail to find this state for nonzero V . There are two reasons for this. First, the spin exchange here involves charge-rich sites that are second neighbors, and are hence necessarily small. Energy gained upon alternation of this weak exchange interactions is even smaller, and thus the tendency to this particular form of the SP transition is weak to begin with. Second, this region of the parameter space involves either large U (for e.g., U = 10, for which Vc(U, S = 0) ≃ 2) or relatively large V (for e.g. Vc(U, S = 0) ≃ 3 for U = 8). In either case such strong Coulomb interactions make the applica- bility of mean-field theories questionable. Our QMC calculations do find the tendency to SP in- stability in this parameter region. In Fig. 6 we show QMC results for V > Vc(U = 8, S = 0). In contrast to Figs. 4 and 5, χρ(4kF ) now dominates over χρ(2kF ) at all T. The weaker peak at q = 2kF at low T is due to small differences in site charge populations between the charge-poor sites of the Wigner crystal that arises upon bond distortion25, and that adds a small period 4 compo- nent to the charge modulation. The bond susceptibility χB(q) has a strong peak at 2kF and a weaker peak at 4kF , exactly as expected for the 4kF CO-SP state. Previous work has shown that the difference in charge densities between the charge-rich and charge-poor sites in the 4kF CO-SP ground state is considerably larger than in the BCDW25. C. Spin excitations from the BCDW The above susceptibility calculations indicate that in the intermediate V region (Figs. 4 and 5) the BCDW ground state with · · · 1100· · · CO can evolve into the · · · 1010· · · CO as T increases. Our earlier work had shown that for weak V , the BCDW evolves into the 4kF BOW at high T18,25. Within the standard theory of the SP transition57,64,65, applicable to the 1 -filled band, the SP state evolves into the undistorted state for T > TSP . Thus the 4kF distortions, CO and BOW, take the role of the undistorted state in the 1 -filled band, and it appears paradoxical that the same ground state can evolve into two different high T states. We show here that this is inti- mately related to the nature of the spin excitations in the -filled BCDW. Spin excitations from the conventional SP state leave the site charge occupancies unchanged. We show below that not only do spin excitations from the 1 -filled BCDW ground state lead to changes in the site occupancies, two different kinds of site occupancies are possible in the localized defect states that characterize spin excited states here. We will refer to these as type I and type II defects, and depending on which kind of defect dominates at high T (which in turn depends on the relative magnitudes of V and the e-p couplings), the 4kF state is either the CO or the BOW. We will demonstrate the occurrence of type I and II defects in spin excitations numerically. Below we present a physical intuitive explanation of this highly unusual behavior, based on a configuration space picture. Very similar configuration space arguments can be found else- where in our discussion of charge excitations from the BCDW in the interacting 1 -filled band73. We begin our discussion with the standard 1 -filled band, for which the mechanism of the SP transition is well understood57,64,65. Fig. 7(a) shows in valence bond (VB) representation the generation of a spin triplet from the standard 1 -filled band. Since the two phases of bond alternation are isoenergetic, the two free spins can sepa- rate, and the true wavefunction is dominated by VB dia- grams as in Fig. 7(b), where the phase of the bond alter- nation in between the two unpaired spins (spin solitons) is opposite to that in the ground state. With increasing temperature and increasing number of spin excitations there occur many such regions with reversed bond al- ternations, and overlaps between regions with different phases of bond alternations leads ultimately to the uni- form state. The above picture needs to be modified for the dimer- ized dimer BCDW state in the 1 -filled band, in which the single site of the 1 -filled band system is replaced by a dimer unit with site populations 1 and 0 (or 0 and 1), and the stronger interdimer 1–1 bond (weaker interdimer 0· · · 0 bond) corresponds to the strong (weak) bond in the standard SP case. Fig. 7(c), showing triplet generation from the BCDW, is the 1 -filled analog of Fig. 7(a): a singlet bond between the dimer units has been broken to generate a localized triplet. The effective repulsion be- tween the free spins of Fig. 7(a) is due to the absence of binding between them, and the same is expected in Fig. 7(c). Because the site occupancies within the dimer units are nonuniform now, the repulsion between the spins is reduced from changes in intradimer as well as interdimer site occupancies: the site populations within the neighboring units can become uniform (0.5 each), or the site occupancies can revert to 1001 from 0110. There is no equivalent of this step in the standard SP case. The next steps in the separation of the resultant defects are identical to that in Fig. 7(b), and we have shown these possible final states in Fig. 7(d) and Fig. 7(e), for the two different intraunit spin defect populations. For V < Vc(U, S), defect units with site occupancies 0.5 occu- pancies (type I defects) are expected to dominate; while for V > Vc(U, S) site populations of 10 and 01 (type II defects) dominate. From the qualitative discussions it is difficult to predict whether the defects are free, in which case they are solitons, or if they are bound, in which case 0 1 0 1.5 .5 .5 .5 1 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 FIG. 7: (a) and (b) S = 1 VB diagrams in the Heisenberg SP chain. The SP bond order in between the S = 1 solitons in (b) is opposite to that elsewhere. (c) 1 -filled band equivalent of (a); the singlet bonds in the background BCDW are between units containing a pair of sites but a single electron (see text). (d) and (e) 1 -filled band equivalents of (b). Defect units with two different charge distributions are possible. The charge distribution will depend on V . two of them constitute a triplet. What is more relevant is that type I defects generate bond dimerization locally (recall that the 4kF BOW has uniform site charge densi- ties) while type II defects generate local site populations 1010, which will have the tendency to generate the 4kF The process by which the BCDW is reached from the 4kF BOW or the CO as T is decreased from T2kF is ex- actly opposite to the discussion of spin excitations from the BCDW in Fig. 7. Now · · · 1100· · · domain walls appear in the background bond-dimerized or charge- dimerized state. In the case of the · · · 1010· · · 4kF state, the driving force for the creation of such a domain wall is the energy gained upon singlet formation (for V < Vc(U,S)). Fig. 7(e) suggests the appearance of localized · · · 1010· · · regions in excitations from the · · · 1100· · · BCDW SP state for 2|t| ≤ V ≤ Vc(U, S = 0). We have verified this with exact spin-dependent calculations for N = 16 and 20 periodic rings with adiabatic e-p couplings, HASSH = t [1− α(ui − ui+1)](c i,σci+1,σ + h.c.) (ui − ui+1) 2 (4a) HAHol = g vini + v2i (4b) and Hee as in Eq. (1d). In the above ui is the displace- ment from equilibrium of a molecular unit and vi is a molecular mode. Note that due to the small sizes of the systems we are able to solve exactly, the e-p cou- plings in Eq. (4a) and Eq. (4b) cannot be directly com- pared to those in Eq. (1a). In the present case, we take the smallest e-p couplings necessary to generate the BCDW ground state self-consistently within the adia- batic Hamiltonian22,23,25. We then determine the bond distortions, site charge occupancies and spin-spin corre- 4 8 12 16 20 -0.04 -0.04 FIG. 8: (a) Self-consistent site charges (vertical bars) and spin-spin correlations (points, lines are to guide the eye) in the N=20 periodic ring with U = 8, V = 2.75, α2/KS = 1.5, g2/KH = 0.32 for Sz = 2. (b) Same parameters, except Sz = 3. Black and white circles at the bottoms of the panels denote charge-rich and charge-poor sites, respectively, while gray circles denote sites with population nearly exactly 0.5. The arrows indicate locations of the defect units with nonzero spin densities. lations self-consistently with the same parameters for ex- cited states with Sz > 0. In Fig. 8(a) we have plotted the deviations of the site charge populations from average density of 0.5 for the lowest Sz=2 state for N = 20, for U = 8, V = 2.75 and e-p couplings as given in the figure caption. Deviations of the charge occupancies from the perfect · · · 0110· · · sequence identify the defect centers, as seen from Fig. 7(c) - (e). In the following we refer to dimer units composed of sites i and j as [i,j]. Based on the charge occupancies in the Fig, we identify units [1,2], [7,8], [13,14] and [14,15] as the defect centers. Furthermore, based on site populations of nearly exactly 0.5, we identify defects on units [1,2] and [7,8] as type I; the populations on units [13,14] and [14,15] identify them type II. Type I defects appear to be free and soliton like, while type II defects appear to be bound into a triplet state, but both could be finite size effects. Fig. 7(d) and (e) suggest that the spin-spin z- component correlations, 〈Szi S j 〉, are large and positive only between pairs of sites which belong to the defect cen- ters, as all other spins are singlet-coupled. For our char- acterization of units [1,2], [7,8], [13,14] and [14,15] to be correct therefore, 〈Szi S j 〉 must be large and positive if sites i and j both belong to this set of sites, and small (close to zero) when either i or j does not belong to this set (as all sites that do not belong to the set are singlet-bonded). We have superimposed the calculated z-component spin-spin correlations between site 2 and all sites j = 1 – 20. The spin-spin correlations are in complete agreement with our characterization of units [1,2], [7,8], [13,14] and [14,15] as defect centers with free spins. The singlet 1–1 bonds between nearest neighbor charge-rich sites in Fig. 7(c) - (e) require that spin-spin couplings between such pairs of sites are large and neg- ative, while spin-spin correlations between one member of the pair any other site is small. We have verified such strong spin-singlet bonds between sites 4 and 5, 10 and 11, and 18 and 19, respectively (not shown). Thus mul- tiple considerations lead to the identification of the same sites as defect centers, and to their characterization as types I and II. We report similar calculations in Fig. 8(b) for Sz = 3. Based on charge occupancies, four out of the six defect units with unpaired spins in the Sz = 3 state are type II; these occupy units [3,4], [9,10], [13,14] and [19,20]. Type I defects occur on units [1,2] and [11,12]. As indicated in the figure, spin-spin correlations are again large between site 2 and all other sites that belong to this set, while they are again close to zero when the second site is not a defect site. As in the previous case, we have verified singlet spin couplings between nearest neighbor pairs of charge-rich sites. There occur then exact correspondences between charge densities and spin-spin correlations, exactly as for Sz = 2. The susceptibility calculations in Fig. 4 and Fig. 5 are consistent with the microscopic calculations of spin de- fects presented above. As 4kF defects are added to the 2kF background, the 2kF susceptibility peak is expected to broaden and shift towards higher q. This is exactly what is seen in the charge susceptibility, Fig. 4(a) and Fig. 5(a) as T is increased. A similar broadening and shift is seen in the bond order susceptibility as well. V. DISCUSSIONS AND CONCLUSIONS In summary, the SP state in the 1 -filled band CTS is unique for a wide range of realistic Coulomb interactions. Even when the 4kF state is the Wigner crystal, the SP state can be the · · · 1100· · · BOW. For U = 8, for exam- ple, the transition found here will occur for 2 < V < 3. This novel T-dependent transition from the Wigner crys- tal to the BCDW is a consequence of the spin-dependence of Vc. Only for V > 3 here can the SP phase be the tetramerized Wigner crystal 1 = 0 = 1 · · · 0 · · · 1 (note, however, that V ≤ 4 for U = 8). We have ignored the intrinsic dimerization along the 1D stacks in our reported results, but this increases Vc(U, S = 0) even further, and makes the Wigner crystal that much more unlikely24. Although even larger U (U = 10, for example) reduces Vc(U, S = 0), we believe that the Coulomb interactions in the (TMTTF)2X lie in the intermediate range. A Wigner crystal to BCDW transition would explain most of the experimental surprises discussed in Section II. The discovery of the charge redistribution upon en- tering the SP phase15,16 in X = AsF6 and PF6 is prob- ably the most dramatic illustration of this. Had the SP state maintained the same charge modulation pattern as the Wigner crystal that exist above T2kF , the dif- ference in charge densities between the charge-rich and the charge-poor sites would have changed very slightly25. The dominant effect of the SP transition leading to 1 = 0 = 1 · · · 0 · · · 1 is only on the charge-poor sites, which are now inequivalent (note that the charge-rich sites remain equivalent). The difference in charge den- sity of the charge-rich sites and the average of the charge density of the charge-poor sites thus remains the same25. In contrast, the difference in charge densities between the charge-rich and the charge-poor sites in the BCDW is considerably smaller than in the Wigner crystal25, and we believe that the experimentally observed smaller charge density difference in the SP phase simply reflects its BCDW character. The experiments of references 15,16 should not be taken in isolation: we ascribe the competition between the CO and the SP states in X = PF6 and AsF6, as reflected in the different pressure-dependences of these states8, to their having different site charge occupan- cies. The observation that the charge density difference in X = SbF6 decreases considerably upon entering the SP phase from the AFM1 phase9 can also be understood if the AFM1 and the SP phases are assigned to be Wigner crystal and the BCDW, respectively. The correlation be- tween larger TCO and smaller TSP 48 is expected. Larger TCO implies larger effective V , which would lower TSP . This is confirmed from comparing Figs. 4 and 5: the temperature at which χρ(2kF ) begins to dominate over χρ(4kF ) is considerably larger in Fig. 4 (smaller V ) than in Fig. 5 (larger V ). The isotope effect, strong enhance- ment of the TCO (from 69 K to 90 K) with deutera- tion of the methyl groups in X = PF6, and concomi- tant decrease in TSP 6,47 are explained along the same line. Deuteration decreases ωH in Eq. (1a), which has the same effect as increasing V . Thus from several dif- ferent considerations we come to the conclusion that the transition from the Wigner crystal to the BCDW that we have found here theoretically for intermediate V/|t| does actually occur in (TMTTF)2X that undergo SP transi- tion. This should not be surprising. Given that the 1:2 anionic CTS lie in the “weak” V/|t| regime, TMTTF with only slightly smaller |t| (but presumably very similar V , since intrastack intermolecular distances are comparable in the two families) lies in the “intermediate” as opposed to “strong” V/|t| regime. Within our theory, the two different antiferromagnetic regions that straddle the SP phase in Fig. 1, AFM1 and AFM2, have different charge occupancies. The Wigner crystal character of the AFM1 region is revealed from the similar behavior of TN and TCO in (TMTTF)2SbF6 un- der pressure9, indicating the absence of the competition of the type that exists between CO and SP, in agree- ment with our assignment. The occurrence of a Wigner crystal AFM1 instead of SP does not necessarily imply a larger V/|t| in the SbF6. A more likely reason is that the interaction with the counterions is strong here, and this interaction together with V pins the electrons on al- ternate sites. (TMTSF)2X, and possibly (TMTTF)2Br, belong to the AFM2 region. The observation that the CDW and the spin-density wave in the TMTSF have the same periodicities1 had led to the conclusion that the charge occupancy here is · · · 1100· · · 21. This conclusion remains unchanged. Finally, we comment that the ob- servation of Wigner crystal CO3 in (DI-DCNQI)2Ag is not against our theory, as the low T phase here is an- tiferromagnetic and not SP. We predict the SP system (DMe-DCNQI)2Ag to have the · · · 1100· · · charge order- In summary, it appears that the key concept of spin- dependent Vc within Eq. (1a) can resolve most of the mysteries associated with the temperature dependence of the broken symmetries in the 1 -filled band CTS. One interesting feature of our work involves demonstration of spin excitations from the BCDW state that necessar- ily lead to local changes in site charges. Even for weak Coulomb interactions, the 1 -filled band has the BCDW character18. 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0704.1657

Bubbling Surface Operators And S-Duality

arXiv:0704.1657v3 [hep-th] 28 May 2008 arXiv:0704.1657 Bubbling Surface Operators And S-Duality Jaume Gomis1 and Shunji Matsuura2 Perimeter Institute for Theoretical Physics Waterloo, Ontario N2L 2Y5, Canada1,2 Department of Physics University of Tokyo, 7-3-1 Hongo, Tokyo2 Abstract We construct smooth asymptotically AdS5×S5 solutions of Type IIB supergravity corresponding to all the half-BPS surface operators in N = 4 SYM. All the parameters labeling a half-BPS surface operator are identified in the corresponding bubbling geome- try. We use the supergravity description of surface operators to study the action of the SL(2, Z) duality group of N = 4 SYM on the parameters of the surface operator, and find that it coincides with the recent proposal by Gukov and Witten in the framework of the gauge theory approach to the geometrical Langlands with ramification. We also show that whenever a bubbling geometry becomes singular that the path integral description of the corresponding surface operator also becomes singular. 04/2007 [email protected] [email protected] http://arxiv.org/abs/0704.1657v3 1. Introduction and Summary Gauge invariant operators play a central role in the gauge theory holographically describing quantum gravity with AdS boundary conditions [1][2][3], as correlation functions of gauge invariant operators are the only observables in the boundary gauge theory. Finding the bulk description of all gauge invariant operators is necessary in order to be able to formulate an arbitrary bulk experiment in terms of gauge theory variables. In this paper we provide the bulk description of a novel class of half-BPS operators in N = 4 SYM which are supported on a surface Σ [4]. These nonlocal surface operators OΣ are defined by quantizing N = 4 SYM in the presence of a certain codimension two singularity for the classical fields of N = 4 SYM. The singularity characterizing such a surface operator OΣ depends on 4M real parameters, whereM is the number of U(1)’s left unbroken byOΣ. Surface operators are a higher dimensional generalization of Wilson and ’t Hooft operators, which are supported on curves and induce a codimension three singularity for the classical fields appearing in the Lagrangian. In this paper we extend the bulk description of all half-BPS Wilson loop operators found in [5] (see also1 [8][9][10][11][12]) to all half-BPS surface operators. We find the asymptotically AdS5×S5 solutions of Type IIB supergravity corresponding to all half-BPS surface operators OΣ in N = 4 U(N) SYM. The topology and geometry of the “bubbling” solution is completely determined in terms of some data, very much like in the case studied by Lin, Lunin and Maldacena (LLM) in the context of half-BPS local operators [13]2. In fact, we identify the system of equations determining the supergravity solution corresponding to the half-BPS surface operators inN = 4 SYM with that obtained by “analytic” continuation of the LLM equations [13][16]. The data determining the topology and geometry of a supergravity solution is charac- terized by the position of a collection of M point particles in a three dimensional space X , whereX is a submanifold of the ten dimensional geometry. Different particle configurations give rise to different asymptotically AdS5×S5 geometries. 1 The description of Wilson loops in the fundamental representation goes back to [6][7]. 2 The bubbling geometry description of half-BPS Wilson loops was found in [10][11] while that of half-BPS domain wall operators was found in [11][14]. For the bubbling Calabi-Yau geometries for Wilson loops in Chern-Simons, see [15]. Fig. 1: a) The metric and five-form flux is determined once the position of the particles in X – labeled by coordinates (~xl, yl) where y ≥ 0 – is given. The l-th particle is associated with a point Pl ∈ X. b) The configuration corresponding to the AdS5×S 5 vacuum. Even though the choice of a particle distribution in X completely determines the geometry and topology of the metric and the corresponding RR five-form field strength, further choices have to be made to fully characterize a solution of Type IIB supergravity on this geometry3. Given a configuration ofM particles inX , the corresponding ten dimensional geometry developsM non-trivial disks which end on the boundary4 of AdS5×S5 on a non-contractible S1. Since Type IIB supergravity has two two-form gauge fields, one from the NS-NS sector and one from the RR sector, a solution of the Type IIB supergravity equations of motion is fully determined only once the holonomy of the two-forms around the various disks is specified: l = 1, . . . ,M. (1.1) Therefore, an asymptotically AdS5×S5 solution depends on the position of theM particles in X – given by (~xl, yl) – and on the holonomies of the two-forms (1.1). A precise dictionary is given between all the 4M parameters that label a half-BPS surface operator OΣ and all the parameters describing the corresponding supergravity solution. We show that the supergravity solution describing a half-BPS surface operator is regular and that whenever the supergravity solution develops a singularity the N = 4 SYM path integral description of the corresponding surface operator also develops a singularity. We study the action of the SL(2, Z) symmetry of Type IIB string theory on the supergravity solutions representing the half-BPS surface operators in N = 4 SYM. By 3 This is on top of the obvious choice of dilaton and axion, which gets identified with the complexified coupling constant in N = 4 SYM. 4 The conformal boundary in this case is AdS3×S 1, where surface operators in N = 4 SYM can be studied by specifying non-trivial boundary conditions. using the proposed dictionary between the parameters of a supergravity solution and the parameters of the corresponding surface operator, we can show that the action of S-duality induced on the parameters of a surface operator coincides with the recent proposal by Gukov and Witten [4] in the framework of the gauge theory approach to the geometrical Langlands [17]5 with ramification. Whether surface operators can serve as novel order parameters in gauge theory remains an important open question. It is our hope that the viewpoint on these operators provided by the supergravity solutions in this paper may help shed light on this crucial question. The plan of the rest of the paper is as follows. In section 2 we study the gauge theory singularities corresponding to surface operators in N = 4 SYM, study the symmetries preserved by a half-BPS surface operator and review the proposal in [4] for the action of S-duality on the parameters that a half-BPS surface operator depends on. We also compute the scaling weight of these operators and show that it is invariant under Montonen-Olive duality. In section 3 we construct the solutions of Type IIB supergravity describing the half-BPS surface operators. We identify all the parameters that a surface operator depends on in the supergravity solution and show that the action of S-duality on surface operators proposed in [4] follows from the action of SL(2, Z) on the classical solutions of supergravity. The Appendices contain some details omitted in the main text. 2. Surface Operators in Gauge Theories A surface operator OΣ is labeled by a surface Σ in R1,3 and by a conjugacy class U of the gauge group G. The data that characterizes a surface operator OΣ, the surface Σ and the conjugacy class U , can be identified with that of an external string used to probe the theory. The surface Σ corresponds to the worldsheet of a string while the conjugacy class U is associated to the Aharonov-Bohm phase acquired by a charged particle encircling the string. The singularity6 in the gauge field produced by a surface operator is that of a non- abelian vortex. This singularity in the gauge field can be characterized by the phase 5 See e.g. [18] for a review of the geometric Langlands program. 6 Previous work involving codimension two singularities in gauge theory include [19][20][21]. acquired by a charged particle circumnavigating around the string. This gives rise to a group element7 U ⊂ U(N) U ≡ P exp i A ⊂ U(N), (2.1) which corresponds to the Aharonov-Bohm phase picked up by the wavefunction of the charged particle. Since gauge transformations act by conjugation U → gUg−1, a surface operator is labeled by a conjugacy class of the gauge group. By performing a gauge transformation, the matrix U can be diagonalized. If we demand that the gauge field configuration is scale invariant – so that OΣ has a well defined scaling weight – then the gauge field produced by a surface operator can then be written α1 ⊗ 1N1 0 . . . 0 0 α2 ⊗ 1N2 . . . 0 . . . 0 0 . . . αM ⊗ 1NM dθ, (2.2) where θ is the polar angle in the R2 ⊂ R1,3 plane normal to Σ and 1n is the n-dimensional unit matrix. We note that the matrix U takes values on the maximal torus TN = RN/ZN of the U(N) gauge group. Therefore the parameters αi take values on a circle of unit radius. The surface operator corresponding to (2.2) spontaneously breaks the U(N) gauge symmetry along Σ down to the so called Levi group L, where a group of Levi type is char- acterized by the subgroup of U(N) that commutes with (2.2). Therefore, L = l=1 U(Nl), where N = l=1Nl. Since the gauge group is broken down to the Levi group L = l=1 U(Nl) along Σ, there is a further choice [4] in the definition of OΣ consistent with the symmetries and equations of motion. This corresponds to turning on a two dimensional θ-angle for the unbroken U(1)’s along the string worldsheet Σ. The associated operator insertion into the N = 4 SYM path integral is given by: Tr Fl . (2.3) 7 We now focus on G = U(N) as it is the relevant gauge group for describing string theory with asymptotically AdS5×S 5 boundary conditions. The parameters ηi takes values in the maximal torus of the S-dual or Langlands dual gauge group LG [4]. Therefore, since LG = U(N) for G = U(N), we have that the matrix of θ-angles of a surface operator OΣ characterized by the Levi group L = l=1 U(Nl) is given by the L-invariant matrix: η1 ⊗ 1N1 0 . . . 0 0 η2 ⊗ 1N2 . . . 0 . . . 0 0 . . . ηM ⊗ 1NM . (2.4) The parameters ηi, being two dimensional θ-angles, also take values on a circle of unit radius. Therefore, a surface operator OΣ in pure gauge theory with Levi group L = l=1 U(Nl) is labeled by 2M L-invariant parameters (αl, ηl) up to the action of SM , which acts by permuting the different eigenvalues in (2.2) and (2.4). The operator is then defined by expanding the path integral with the insertion of the operator (2.3) around the singularity (2.2), and by integrating over connections that are smooth near Σ. In performing the path integral, we must divide [4] by the gauge transformations that take values in L = l=1 U(Nl) when restricted to Σ. This means that the operator becomes singular whenever the unbroken gauge symmetry near Σ gets enhanced, corresponding to when eigenvalues in (2.2) and (2.4) coincide. Surface Operators in N = 4 SYM In a gauge theory with extra classical fields like N = 4 SYM, the surface operator OΣ may produce a singularity for the extra fields near the location of the surface operator. The only requirement is that the singular field configuration solves the equations of motion of the theory away8 from the surface Σ. The global symmetries imposed on the operator OΣ determine which classical fields in the Lagrangian develop a singularity near Σ together with the type of singularity. A complementary viewpoint on surface operators is to add new degrees of freedom on the surface Σ. Such an approach to surface operators in N = 4 SYM has been considered 8 For pure gauge theory, the field configuration in (2.2) does satisfy the Yang-Mills equation of motion DmF mn = 0 away from Σ. Moreover, adding the two dimensional θ-angles (2.3) does not change the equations of motion. in [22][23] where the new degrees of freedom arise from localized open strings on a brane intersection. The basic effect of OΣ is to generate an Aharonov-Bohm phase corresponding to a group element U (2.1). If we let z be the complex coordinate in the R2 ⊂ R1,3 plane normal to Σ, the singularity in the gauge field configuration is then given by , (2.5) where AI are constant matrices. Scale invariance of the singularity – which we are going to impose – restricts AI = 0 for I ≥ 2. The operator OΣ can also excite a complex scalar field Φ of N = 4 SYM near Σ while preserving half of the Poincare supersymmetries of N = 4 SYM. Imposing that the singularity is scale invariant9 yields , (2.6) where Φ1 is a constant matrix. A surface operator OΣ is characterized by the choice of an unbroken gauge group L ⊂ G along Σ. Correspondingly, the singularity of all the fields excited by OΣ must be invariant under the unbroken gauge group L. For L = l=1 U(Nl) ⊂ U(N) the singularity in the gauge field is the non-abelian vortex configuration in (2.2) and the two dimensional θ-angles are given by (2.4). L-invariance together with scale invariance requires that Φ develops an L-invariant pole near Σ: β1 + iγ1 ⊗ 1N1 0 . . . 0 0 β2 + iγ2 ⊗ 1N2 . . . 0 . . . 0 0 . . . βM + iγM ⊗ 1NM . (2.7) Therefore, a half-BPS surface operator OΣ in N = 4 SYM with Levi group L = l=1 U(Nl) is labeled by 4M L-invariant parameters (αl, βl, γl, ηl) up to the action of SM , which permutes the different eigenvalues in (2.2)(2.4)(2.7). The operator is defined by the 9 If we relax the restriction of scale invariance, one can then get other supersymmetric singu- larities with higher order poles Φ = and A (2.5). The surface operators associated with these singularities may be relevant [4] for the gauge theory approach to the study of the geometric Langlands program with wild ramification. path integral of N = 4 SYM with the insertion of the operator (2.3) expanded around the L-invariant singularities (2.2)(2.7) and by integrating over smooth fields near Σ. As in the pure gauge theory case, we must mode out by gauge transformations that take values in L ⊂ U(N) when restricted to Σ. The surface operator OΣ becomes singular whenever the the parameters that label the surface operator (αl, βl, γl, ηl) for l = 1, . . . ,M are such that they are invariant under a larger symmetry than L, the group of gauge transformations we have to mode out when evaluating the path integral. S-duality of Surface Operators In N = 4 SYM the coupling constant combines with the four dimensional θ-angle into a complex parameter taking values in the upper half-plane: . (2.8) The group of duality symmetries ofN = 4 SYM is an infinite discrete subgroup of SL(2, R), which depends on the gauge group G. For N = 4 SYM with G = U(N) the relevant symmetry group is SL(2, Z): ∈ SL(2, Z). (2.9) Under S-duality τ → −1/τ and G gets mapped10 to the S-dual or Langlands dual gauge group LG. For G = U(N) the S-dual group is LG = U(N), and SL(2, Z) is a symmetry of the theory, which acts on the coupling of the theory by fractional linear transformations: τ → aτ + b cτ + d . (2.10) In [4], Gukov and Witten made a proposal of how S-duality acts on the parameters (αl, βl, γl, ηl) labeling a half-BPS surface operator. The proposed action is given by [4]: (βl, γl) → |cτ + d| (βl, γl) (αl, ηl) → (αl, ηl)M−1. (2.11) 10 For G not a simply-laced group, τ → −1/nτ , where n is the ratio of the length-squared of the long and short roots of G. With the aid of this proposal, it was shown in [4] that the gauge theory approach to the geometric Langlands program pioneered in [17] naturally extends to the geometric Langlands program with tame ramification. Symmetries of half-BPS Surface Operators in N = 4 SYM We now describe the unbroken symmetries of the half-BPS surface operators OΣ. These symmetries play an important role in determining the gravitational dual description of these operators, which we provide in the next section. In the absence of any insertions, N = 4 SYM is invariant under the PSU(2, 2|4) symmetry group. If we consider the surface Σ = R1,1 ⊂ R1,3, then Σ breaks the SO(2, 4) conformal group to a subgroup. A surface operator OΣ supported on this surface inserts into the gauge theory a static probe string. This surface is manifestly invariant under rotations and translations in Σ and scale transformations. It is also invariant under the action of inversion I : xµ → xµ/x2 and consequently11 invariant under special conformal transformations in Σ. Therefore, the symmetries left unbroken by Σ = R1,1 generate an SO(2, 2)× SO(2)23 subgroup of the SO(2, 4) conformal group, where SO(2)23 rotates the plane transverse to Σ in R1,3. In Euclidean signature, the surface Σ =S2 preserves an SO(1, 3) × SO(2)23 subgroup of the Euclidean conformal group. This surface can be obtained from the surface Σ = R2 ∈ R4 by the action of a broken special conformal generator and can also be used to construct a half-BPS surface operator OΣ in N = 4 Since the symmetry of a surface operator with Σ = R1,1 is SO(2, 2) × SO(2)23 one can study such an operator either by considering the gauge theory in R1,3 or in AdS3×S1, which can be obtained from R1,3 by a conformal transformation. Studying the gauge theory in AdS3×S1 has the advantage of making the symmetries of the surface operator manifest, as the conformal symmetries left unbroken by the surface act by isometries on AdS3×S1. Surface operators in R1,3 are described by a codimension two singularity while surface operators in AdS3×S1 are described by a boundary condition on the boundary of AdS3. A surface operator with Σ = R 1,1 corresponds to a boundary condition on AdS3 in Poincare coordinates while a surface operator on Σ =S2 corresponds to a boundary condition on global Euclidean AdS3. 11 We recall that a special conformal transformation Kµ is generated by IPµI, where Pµ is the translation generator and I is an inversion. The singularity in the classical fields produced by OΣ in (2.2)(2.7) is also invariant under SO(2, 2). The N = 4 scalar field Φ carries charge under an SO(2)R subgroup of the SO(6) R-symmetry and is therefore SO(4) invariant. The surface operator OΣ is therefore invariant under SO(2, 2)×SO(2)a×SO(4), where SO(2)a is generated by the anti-diagonal product12 of SO(2)23 × SO(2)R. N = 4 SYM has sixteen Poincare supersymmetries and sixteen conformal super- symmetries, generated by ten dimensional Majorana-Weyl spinors ǫ1 and ǫ2 of opposite chirality. As shown in the Appendix A, the surface operator OΣ for Σ = R1,1 preserves half of the Poincare and half of the conformal supesymmetries13 and is therefore half-BPS. With the aid of these symmetries we study in the next section the gravitational de- scription of half-BPS surface operators in N = 4 SYM. Scaling Weight of half-BPS Surface Operators in N = 4 SYM Conformal symmetry constraints the form of the OPE of the energy-energy tensor Tmn with the operators in the theory. For a surface operator OΣ supported on Σ = R1,1, SO(2, 2)× SO(2)23 invariance completely fixes the OPE of Tmn with OΣ: < Tµν(x)OΣ > < OΣ > < Tij(x)OΣ > < OΣ > [4ninj − 3δij ] ; < Tµi(x)OΣ >= 0. (2.12) Here xm = (xµ, xi), where xµ are coordinates along Σ and ni = xi/r, and r is the radial coordinate in the R2 transverse to R1,1. h is the scaling weight of OΣ, which generalizes [24] the notion of conformal dimension of local conformal fields to surface operators. In order to calculate the scaling dimension of a half-BPS surface operator OΣ in N = 4 SYM we evaluate the classical field configuration (2.2)(2.4)(2.7) characterizing a half-BPS surface operator on the classical energy-momentum tensor of N = 4 SYM: Tmn = Tr[DmφDnφ− gmn(Dφ) 2 − 1 (DmDn − gmnD2)φ2] Tr[−FmlFnl + gmnFlpFlp]. (2.13) 12 Since SO(2)a leaves Φ · z in (2.7) invariant. 13 For Σ =S2, the operator is also half-BPS, but it preserves a linear combination of Poincare and special conformal supersymmetries. A straightforward computation14 leads to: h = − 2 l + γ l ) = − i + γ i ). (2.14) The action of an SL(2, Z) transformation (2.10) on the coupling constant of N = 4 SYM implies that: Imτ → Imτ|cτ + d|2 . (2.15) Combining this with the action (2.11) of Montonen-Olive duality on the parameters of the surface operator, we find that the scaling weight (2.14) of a half-BPS surface operator OΣ is invariant under S-duality: h→ h. (2.16) In this respect half-BPS surface operators behave like the half-BPS local operators of N = 4 SYM, whose conformal dimension is invariant under SL(2, Z), and unlike the half-BPS Wilson-’t Hooft operators whose scaling weight is not S-duality invariant [24]. 3. Bubbling Surface Operators In this section we find the dual gravitational description of the half-BPS surface op- erators OΣ described in the previous section. The bulk description is given in terms of asymptotically AdS5×S5 and singularity free solutions of the Type IIB supergravity equa- tions of motion. The data from which the solution is uniquely determined encodes the corresponding data about the surface operator OΣ. The strategy to obtain these solutions is to make an ansatz for Type IIB supergravity which is invariant under all the symmetries preserved by the half-BPS surface operators OΣ. As discussed in the previous section, the bosonic symmetries preserved by a half- BPS surface operator OΣ are SO(2, 2) × SO(4) × SO(2)a. Therefore the most general ten dimensional metric invariant under these symmetries can be constructed by fibering AdS3×S3×S1 over a three manifold X , where the symmetries act by isometries on the fiber. The constraints imposed by unbroken supersymmetry on the ansatz are obtained by demanding that the ansatz for the supergravity background possesses a sixteen component 14 Contact terms depending on α, β, γ and proportional to the derivative of the two-dimensional δ-function appear when evaluating the on-shelll energy-momentum tensor. It would be interesting to understand the physical content of these contact terms. Killing spinor, which means that the background solves the Killing spinor equations of Type IIB supergravity. A solution of the Killing spinor equations and the Bianchi identity for the five-form field strength guarantee that the full set of equations of Type IIB supergravity are satisfied and that a half-BPS solution has been obtained. The problem of solving the Killing spinor equations of Type IIB supergravity with an SO(2, 2)× SO(4)× SO(2)a symmetry can be obtained by analytic continuation of the equations studied by LLM [13][16], which found the supergravity solutions describing the half-BPS local operators of N = 4 SYM, which have an SO(4)×SO(4)×R symmetry. The equations determining the metric and five-form flux can be read from [13][16], in which the analytic continuation that we need to construct the gravitational description of half-BPS surface operators OΣ was considered. The ten dimensional metric and five-form flux is completely determined in terms of data that needs to be specified on the three manifold X in the ten dimensional space. An asymptotically AdS5×S5 metric is uniquely determined in terms of a function z(x1, x2, y), where (x1, x2, y) ≡ (~x, y) are coordinates in X . The ten dimensional metric in the Einstein frame is given by15 ds2 = y 2z + 1 2z − 1ds 2z − 1 2z + 1 dΩ3 + 4z2 − 1 (dχ+ V )2 + 4z2 − 1 (dy2 + dxidxi), (3.1) where ds2X = dy 2+dxidxi with y ≥ 0 and V is a one-form in X satisfying dV = 1/y ∗X dz. AdS3 in Poincare coordinate corresponds to a surface operator on Σ = R 1,1 while AdS3 in global Euclidean coordinate corresponds to a surface operator on Σ =S2. The U(1)a symmetry acts by shifts on χ while SO(2, 2) and SO(4) act by isometries on the coordinates of AdS3 and S 3 respectively. A non-trivial solution to the equations of motion is obtained by specifying a config- uration of M point-like particles in X . The data from which the solution is determined is the “charge” Ql of the particles together with their positions (~xl, yl) in X (see Figure 1). Given a “charge” distribution, the function z(x1, x2, y) solves the following differential equation: ∂i∂iz(x1, x2, y) + y∂y ∂yz(x1, x2, y) Qlδ(y − yl)δ(2)(~x− ~xl). (3.2) 15 The “analytic” continuation from the bubbling geometries dual to the half-BPS local opera- tors is given by z → z, t → χ, y → −iy, ~x → i~x, dΩ3 → −ds [13][16]. Introducing a “charge” at the point (~xl, yl) in X has the effect of shrinking 16 the S1 with coordinate χ in (3.1) to zero size at that point. In order for this to occur in a smooth fashion the magnitude of the “charge” has to be fixed [13][16] so that Ql = 2πyl. Therefore, the independent data characterizing the metric and five-form of the solution is the position of the M “charges”, given by (~xl, yl). In summary, a smooth half-BPS SO(2, 2)× SO(4)× SO(2)a invariant asymptotically AdS5×S5 metric (3.1) solving the Type IIB supergravity equations of motion is found by solving (3.2) subject to the boundary condition z(x1, x2, 0) = 1/2 [13][16], so that the S in (3.1) shrinks in a smooth way at y = 0. The function z(x1, x2, y) is given by z(x1, x2, y) = zl(x1, x2, y), (3.3) where zl(x1, x2, y) = (~x− ~xl)2 + y2 + y2l ((~x− ~xl)2 + y2 + y2l )2 − 4y2l y2 , (3.4) and V can be computed from z(x1, x2, y) from dV = 1/y ∗X dz. Both the metric and five-form field strength are determined by an integer M and by (~xl, yl) for l = 1, . . . ,M . Topology of Bubbling Solutions And Two-form Holonomies The asymptotically AdS5×S5 solutions constructed from (3.3)(3.4) are topologically quite rich. In particular, a solution withM point “charges” hasM topologically non-trivial S5’s. We can associate to each point Pl ∈ X a corresponding five-sphere S5l . S5l can be constructed by fibering the S1×S3 in the geometry (3.1) over a straight line between the point (~xl, 0) and the point (~xl, yl) in X . The topology of this manifold is indeed an S as an S5 can be represented17 by an S1×S3 fibration over an interval where the S1 and S3 shrink to zero size at opposite ends of the interval, which is what happens in our geometry where the S3 shrinks at (~xl, 0) while the S 1 shrinks at the other endpoint (~xl, yl). 16 Near y = yl the form of the relevant part of the metric is that of the Taub-NUT space. Fixing the value of the “charge” at y = yl by imposing regularity of the metric coincides with the usual regularity constraint on the periodicity of the circle in Taub-NUT space. 17 This can be seen explicitly by writing dΩ5 = cos 2 θdΩ3 + dθ 2 + sin2 θdφ2. Fig. 2: A topologically non-trivial S5 can be constructed by fibering S1×S3 over an interval connecting the y = 0 plane and the location of the “charge” at the point Pl ∈ X with (~xl, yl) coordinates. Following [13][16] we can now integrate the five-form flux over the topologically non- trivial S5’s (see Appendix B): F5 = y l . (3.5) Since flux has to be quantized, the position in the y-axis of the l-th particle in X is also quantized y2l = 4πNll p Nl ∈ Z, (3.6) where lp is the ten dimensional Planck length. For an asymptotically AdS5×S5 geometry with radius of curvature R4 = 4πNl4p, which is dual to N = 4 U(N) SYM, we have that the total amount of five-form flux must be N : Nl. (3.7) The asymptotically AdS5×S5 solutions constructed from (3.3)(3.4) also contain non- trivial surfaces. In particular, a solution with M point “charges” has M non-trivial disks Dl. Just as in the case of the S 5’s, we can associate to each point Pl ∈ X a disk Dl. Inspection of the asymptotic form of the metric (3.1) given in (3.3)(3.4) reveals that the metric is conformal to AdS3×S1. This geometry on the boundary of AdS5×S5, which is where the dual N = 4 U(N) SYM lives, is the natural background geometry on which to study conformally invariant surface operators in N = 4 SYM. As explained in section 2, an SO(2, 2)× SO(2)23 invariant surface operator can be defined by specifying a codimension two singularity in R1,3 or by specifying appropriate boundary conditions for the classical fields in the gauge theory at the boundary of AdS3×S1. In the latter formulation, the worldsheet of the surface operator Σ is the boundary of AdS3. Therefore, in the boundary of AdS5×S5 we have a non-contractible S1. If we fiber the S1 parametrized by χ in (3.1) over a straight line connecting a point (~xl, yl) in X – where the S1 shrinks to zero size – to a point in X corresponding to the boundary of AdS5×S5 – given by ~x, y → ∞ – we obtain a surface Dl. This surface is topologically a disk18 and there are M of them for a “charge” distribution of M particles in X . Fig. 3: A disk D can be constructed by fibering S1 over an interval connecting the “charge” at the point Pl ∈ X with (~xl, yl) coordinates and the boundary of AdS5×S Due to the existence of the disks Di, the supergravity solution given by the metric and five-form flux alone is not unique. Type IIB supergravity has a two-form gauge field from the NS-NS sector and another one from the RR sector. In order to fully specify a solution of Type IIB supergravity in the bubbling geometry (3.1) we must complement the metric and the five-form with the integral of the two-forms around the disks19 αl = − l = 1, . . . ,M, (3.8) where we have used notation conducive to the later comparison with the parameters char- acterizing a half-BPS surface operator OΣ. Since both BNS and BR are invariant under large gauge transformations, the parameters (αl, ηl) take values on a circle of unit radius. Apart from the M disks Dl, the bubbling geometry constructed from (3.3)(3.4) also has topologically non-trivial S2’s. One can construct an S2 by fibering the S1 in (3.1) over 18 Such disks also appear in the study of the high temperature regime of N = 4 SYM, where the bulk geometry [25] is the AdS Schwarzschild black hole, which also has a non-contractible S1 in the boundary which is contractible in the full geometry. 19 The overall signs in the identification are fixed by demanding consistent action of S-duality of N = 4 SYM with that of Type IIB supergravity. a straight line connecting the points Pl and Pm in X . Since the S 1 shrinks to zero size in a smooth manner at the endpoints we obtain an S2. Therefore, to every pair of “charges” in X , characterized by different points Pl and Pm in X , we can construct a corresponding S2, which we label by S2l,m. The integral of BNS and BR over S l,m do not give rise to new parameters, since [S2l,m] = [Dl] − [Dm] in homology, and the periods can be determined from (3.8). Fig. 4: An S2 can be constructed by fibering S1 over an interval connecting the “charge” at the point Pl ∈ X with a different “charge” at point Pm ∈ X. Bubbling Geometries as Surface Operators As we discussed in section 2, a surface operator OΣ is characterized by an unbroken gauge group L ∈ U(N) along together with 4M L-invariant parameters (αl, βl, γl, ηl). On the other hand, the Type IIB supergravity solutions we have described depend on the positions (~xl, yl) of M “charged” particles in X and the two-form holonomies: . (3.9) We now establish an explicit dictionary between the parameters in gauge theory and the parameters in supergravity. For illustration purposes, it is convenient to start by considering the half-BPS surface operator OΣ with the largest Levi group L, which is L = U(N) for G = U(N). U(N) invariance requires that the singularity in the fields produced by OΣ take values in the center of U(N). Therefore, the gauge field and scalar field produced by OΣ is given by A = α01Ndθ (β0 + iγ0)1N , (3.10) where 1N is the identity matrix. We can also turn a two-dimensional θ-angle (2.3) for the overall U(1), so that: η = η01N . (3.11) We now identify this operator with the supergravity solution obtained by having a single point “charge” source in X (see Figure 1b). If we let the position of the “charge” be (~x0, y0) then z(x1, x2, y) = (~x− ~x0)2 + y2 + y20 ((~x− ~x0)2 + y2 + y20)2 − 4y20y2 VI = −ǫIJ (xJ − xJ0 )((~x− ~x0)2 + y2 − y20) 2(~x− ~x0)2 ((~x− ~x0)2 + y2 + y20)2 − 4y20y2 (3.12) where V = VIdx I . The metric (3.1) obtained using (3.12) is the metric of AdS5×S5. This can be seen by the following change of variables [16] x1 − x10 + i(x2 − x20) = rei(ψ+φ) r = y0 sinhu sin θ y = y0 coshu cos θ (ψ − φ), (3.13) which yields the AdS5×S5 metric with AdS5 foliated by AdS3×S1 slices: ds2 = y0 (cosh2 uds2AdS3 + du 2 + sinh2 udψ2) + (cos2 θdΩ3 + dθ 2 + sin2 θdφ2) . (3.14) We note that the U(1)a symmetry of the metric (3.1) – which acts by shifts on χ – identifies via (3.13) an SO(2)R subgroup of the the SO(6) symmetry of the S 5, acting by shifts on φ, with an SO(2)23 subgroup of the SO(2, 4) isometry group of AdS5, acting by opposite shifts on ψ. This is precisely the same combination of generators discussed in section 2 that is preserved by a half-BPS surface operator OΣ in N = 4 SYM. The radius of curvature of AdS5×S5 in (3.14) is given by R4 = y20 . Therefore using that R4 = 4πNl4p, where N is the rank of the N = 4 YM theory, we have that 4πl4p , (3.15) and the position of the “charge” in y gets identified with the rank of the unbroken gauge group and is therefore quantized. The residue of the pole in Φ (3.10) gets identified with the position of the “charge” in the ~x-plane. It follows from (3.1) that the coordinates ~x and y have dimensions of length2. Therefore, we identify the residue of the pole of Φ with the position of the “charge” in the ~x-plane in X via: (β0, γ0) = 2πl2s . (3.16) Unlike the position in y, the position in ~x is not quantized. The remaining parameters of the surface operator OΣ with U(N) Levi group – given by (α0, η0) – get identified with the holonomy of the two-forms of Type IIB supergravity over D α0 = − , (3.17) where D is the disk ending on the AdS5×S5 boundary on the S1. This identification properly accounts for the correct periodicity of these parameters, which take values on a circle of unit radius. The path integral which defines a half-BPS surface operator OΣ when L = U(N) is never singular as the gauge symmetry cannot be further enhanced by changing the parameters (α0, β0, γ0, η0) of the surface operator. Correspondingly, the dual supergravity solution with one “charge” also never acquires a singularity by changing the parameters of the solution. Let’s now consider the most general half-BPS surface operator OΣ. First we need to characterize the operator by its Levi group, which for a U(N) gauge group takes the l=1 U(Nl) with N = l=1Nl = N . The operator then depends on 4M L-invariant parameters (αl, βl, γl, ηl) for l = 1. . . . ,M up to the action of SM , which acts by permuting the parameters. The corresponding supergravity solution associated to such an operator is given by the metric (3.1). The number of unbroken gauge group factors – given by the integer M – corresponds to the number of point “charges” in (3.2). For M > 1, the metric that follows from (3.3)(3.4) is AdS5×S5 only asymptotically and not globally. The rank of the various gauge group factors in the Levi group l=1 U(Nl) – given by the integers Nl – correspond to the position of the “charges” along y ∈ X , given by the coordinates yl. The precise identification follows from (3.5)(3.6): 4πl4p l = 1, . . . ,M. (3.18) Nl also corresponds to the amount of five-form flux over S l , the S 5 associated with the l-th point charge: 4π4l4p F5 l = 1, . . . ,M. (3.19) This identification quantizes the y coordinate in X into lp size bits. Thus length is quan- tized as opposed to area, which is what happens for the geometry dual to the half-BPS local operators [13][16], where it can be interpreted as the quantization of phase space in the boundary gauge theory. A half-BPS surface operator OΣ develops a pole for the scalar field Φ (2.7). The pole is characterized by its residue, which is given by 2M real parameters (βl, γl). These parameters are identified with the position of the M “charges” in the ~x-plane in X via: (βl, γl) = 2πl2s . (3.20) All these parameters take values on the real line. The remaining parameters characterizing a half-BPS surface operator OΣ are the periodic variables (αl, ηl), which determine the holonomy produced by OΣ and the corre- sponding two-dimensional θ-angles. These parameters get identified with the holonomy of the two-forms of Type IIB supergravity over the M non-trivial disks Dl that the geometry generates in the presence of M “charges” in X : αl = − (3.21) The identification respects the periodicity of (αl, ηl), which in supergravity arises from the invariance of BNS and BR under large gauge transformations We have given a complete dictionary between all the parameters that a half-BPS surface operator in N = 4 SYM depends on and all the parameters in the corresponding bubbling geometry. We note that a surface operator OΣ depends on a set of parameters up to the action of the permutation group SM on the parameters, which is part of the U(N) gauge symmetry. The corresponding statement in supergravity is that the solution dual to a surface operator is invariant under the action of SM , which acts by permuting the “charges” in X . 20 Since the gauge invariant variables are e The supergravity solution is regular as long as the “charges” do not collide. A singu- larity arises whenever two point “charges” in X coincide (see Figure 1): (~xl, yl) → (~xm, ym) for l 6= m. (3.22) Whenever this occurs, there is a reduction in the number of independent disks since (see Figure 3): Dl → Dm for l 6= m, (3.23) and therefore for l 6= m. (3.24) In this limit of parameter space the non-trivial S2 connecting the points Pl and Pm in X shrinks to zero size as [S2l,m] = [Dl]− [Dm] → 0, and the geometry becomes singular. By using the dictionary developed in this paper, such a singular geometry corresponds to a limit when two of each of the set of parameters (αl, βl, γl, ηl) defining a half-BPS surface operator OΣ become equal: αl → αm, βl → βm, γl → γm, ηl → ηm for l 6= m. (3.25) In this limit the unbroken gauge group preserved by the surface operator OΣ is enhanced to L′ from the original Levi group l=1 U(Nl), where L ⊂ L′. As explained in section 2 the path integral from which OΣ is defined becomes singular. In summary, we have found the description of all half-BPS surface operators OΣ in N = 4 SYM in terms of solutions of Type IIB supergravity. The asymptotically AdS5×S5 solutions are regular and when they develop a singularity then the corresponding operator also becomes singular. S-Duality of Surface Operators from Type IIB Supergravity The group of dualities of N = 4 SYM acts non-trivially [4] on surface operators OΣ (see discussion in section 2). For G = U(N) the duality group is SL(2, Z) and its proposed action on the parameters on which OΣ depends on is [4]: (βl, γl) → |cτ + d| (βl, γl) (αl, ηl) → (αl, ηl)M−1, (3.26) where M is an SL(2, Z) matrix . (3.27) We now reproduce21 this transformation law by studying the action of the SL(2, Z) subgroup of the SL(2, R) classical symmetry of Type IIB supergravity, which is in fact the appropriate symmetry group of Type IIB string theory. For that we need to analyze the action of S-duality on our bubbling geometries. SL(2, Z) acts on the complex scalar τ = C0 + ie −φ of Type IIB supergravity in the familiar fashion τ → aτ + b cτ + d , (3.28) where as usual τ gets identified with the complexified coupling constant of N = 4 SYM (2.8). SL(2, Z) also rotates the two-form gauge fields22 of Type IIB supergravity , (3.29) while leaving the metric in the Einstein frame and the five-form flux invariant. Given that the metric in (3.1) is in the Einstein frame, SL(2, Z) acts trivially on the coordinates (~x, y). Nevertheless, since ls = g s lp with gs = e φ (3.30) the string scale transforms under SL(2, Z) as follows: l2s → |cτ + d| . (3.31) Therefore, under S-duality: 2πl2s → |cτ + d| ~xl 2πl2s . (3.32) Given our dictionary in (3.20), we find that the surface operator parameters (βl, γl) trans- form as in (3.26), agreeing with the proposal in [4]. 21 If we apply the same idea to the LLM geometries dual to half-BPS local operators in [13], we conclude that the half-BPS local operators are invariant under S-duality. 22 See e.g. [26][27]. The identification of the rest of the parameters is (3.21): αl = − (3.33) Using the action of SL(2, Z) on the two-forms (3.29) and the identification (3.33), it follows from a straightforward manipulation that the surface operator paramaters (αl, ηl) transform as in (3.26), agreeing with the proposal in [4]. Acknowledgements We would like to thank Xiao Liu for very useful discussions. Research at Perimeter In- stitute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. We also acknowledge further sup- port from an NSERC Discovery Grant. SM acknowledges support from JSPS Research Fellowships for Young Scientists. Appendix A. Supersymmetry of Surface Operator in N=4 SYM In this Appendix we study the Poincare and conformal supersymmetries preserved by a surface operator in N=4 SYM supported on R1,1. These symmetries are generated by ten dimensional Majorana-Weyl spinors ǫ1 and ǫ2 of opposite chirality. We determine the supersymmetries left unbroken by a surface operator by studying the supersymmetry variation of the gaugino in the presence of the surface operator singularity in (2.2)(2.7). The metric is given by: ds2 = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2 = −(dx0)2 + (dx1)2 + 2dzdz̄. (A.1) where z = 1√ (x2 + ix3) = |z|eiθ, while the singularity in the fields is Φω = Φω̄ = (Φ8 + iΦ9) = β + iγ√ A = αdθ, F = dA = 2παδD, (A.2) where [α, β] = [α, γ] = [β, γ] = 0 and δD = d(dθ) is a two form delta function. The relevant Γ-matrices are (Γ2 − iΓ3) Γz̄ = (Γ2 + iΓ3) {Γz,Γz̄} = 2. (A.3) A Poincare supersymmetry transformation is given by δλ = ( µν +∇µΦiΓµi + [Φi,Φj]Γ ij)ǫ1 (A.4) where µ runs from 0 to 3 and i runs from 4 to 9, while a superconformal supersymmetry transformation is given by δλ = [( µν +∇µΦiΓµi + [Φi,Φj]Γ ij)xσΓσ − 2ΦiΓi]ǫ2 (A.5) From (A.4), it follows that the unbroken Poincare supersymmetries are given by: Γω̄zǫ1 = 0 ⇔ Γ2389ǫ1 = −ǫ1. (A.6) The unbroken superconformal supersymmetries are given by: −β + iγ Γzω̄ − β − iγ 0 + x1Γ 1 + zΓz̄ + z̄Γz)− 2ΦωΓω − 2Φω̄Γω̄ ǫ2 = 0. (A.7) From the terms proportional to x0 and x1, we find that the unbroken superconformal supersymmetries are given by: Γzω̄ǫ2 = 0 ⇔ Γ2389ǫ2 = −ǫ2. (A.8) The rest of the conditions [Γz̄Γω̄Γz]ǫ2 = 0, (A.9) are automatically satisfied once (A.8) is imposed. We conclude that the singularity (2.2)(2.7) is half-BPS and that the preserved su- persymmetry is generated by ǫ1 and ǫ2 subject o the constraints Γ2389ǫ1 = −ǫ1 and Γ2389ǫ2 = −ǫ2. By acting with a broken special conformal transformation on Σ = R2 ⊂ R4 to get a surface operator supported on Σ =S2, one can show following [28] that such an operator also preserves half of the thirty-two supersymmetries, but are now generated by a linear combination of the Poincare and special conformal supersymmetries. Appendix B. Five form flux In this Appendix, we calculate (3.5) explicitly to evaluate the flux over a non-trivial The five-form flux is [13][16] {d[y2 2z + 1 2z − 1(dχ+ V )]− y 3 ∗3 d( z + 1 )} ∧ dV olAdS3 {d[y2 2z − 1 2z + 1 (dχ+ V )]− y3 ∗3 d( z − 1 )} ∧ dΩ3 (B.1) The five-cycle S5l in the bubbling geometry is spanned by coordinates Ω3, χ and y. 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0704.1658

Resolving Cosmic Gamma Ray Anomalies with Dark Matter Decaying Now

UCI-TR-2007-17 Resolving Cosmic Gamma Ray Anomalies with Dark Matter Decaying Now Jose A. R. Cembranos, Jonathan L. Feng, and Louis E. Strigari Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA Dark matter particles need not be completely stable, and in fact they may be decaying now. We consider this possibility in the frameworks of universal extra dimensions and supersymmetry with very late decays of WIMPs to Kaluza-Klein gravitons and gravitinos. The diffuse photon background is a sensitive probe, even for lifetimes far greater than the age of the Universe. Remarkably, both the energy spectrum and flux of the observed MeV γ-ray excess may be simultaneously explained by decaying dark matter with MeV mass splittings. Future observations of continuum and line photon fluxes will test this explanation and may provide novel constraints on cosmological parameters. PACS numbers: 95.35.+d, 11.10.Kk, 12.60.-i, 98.80.Cq The abundance of dark matter is now well known from observations of supernovae, galaxies and galactic clusters, and the cosmic mocriwave background (CMB) [1], but its identity remains elusive. Weakly-interacting massive particles (WIMPs) with weak-scale masses ∼ 0.1−1 TeV are attractive dark matter candidates. The number of WIMPs in the Universe is fixed at freeze-out when they decouple from the known particles about 1 ns after the Big Bang. Assuming they are absolutely stable, these WIMPs survive to the present day, and their number density is naturally in the right range to be dark matter. The standard signatures of WIMPs include, for example, elastic scattering off nucleons in underground laborato- ries, products fromWIMP annihilation in the galaxy, and missing energy signals at colliders [2]. The stability of WIMPs is, however, not required to preserve the key virtues of the WIMP scenario. In fact, in supersymmetry (SUSY) and other widely-studied sce- narios, it is just as natural for WIMPs to decay after freeze-out to other stable particles with similar masses, which automatically inherit the right relic density to be dark matter [3]. If the resulting dark matter interacts only gravitationally, the WIMP decay is very late, in some cases leading to interesting effects in structure for- mation [4] and other cosmological observables. Of course, the WIMP lifetime depends on ∆m, the mass splitting between the WIMP and its decay product. For high de- generacies, the WIMP lifetime may be of the order of or greater than the age of the Universe t0 ≃ 4.3 × 1017 s, leading to the tantalizing possibility that dark matter is decaying now. For very long WIMP lifetimes, the diffuse photon back- ground is a promising probe [3, 5]. Particularly interest- ing is the (extragalactic) cosmic gamma ray background (CGB) shown in Fig. 1. Although smooth, the CGB must be explained by multiple sources. For Eγ <∼ 1 MeV and Eγ >∼ 10 MeV, the CGB is reasonably well-modeled by thermal emission from obscured active galactic nu- clei (AGN) [9] and beamed AGN, or blazars [10], respec- tively. However, in the range 1 MeV <∼ Eγ <∼ 5 MeV, no astrophysical source can account for the observed CGB. Blazars are observed to have a spectral cut-off ∼ 10 MeV, FIG. 1: The CGB measured by HEAO-1 [6] (circles), COMP- TEL [7] (squares), and EGRET [8] (triangles), along with the known astrophysical sources: AGN (long-dash), SNIa (dot- ted), and blazars (short-dash, and dot-dashed extrapolation). and also only a few objects have been detected below this energy [11, 12]; a maximal upper limit [13] on the blazar contribution for Eγ <∼ 10 MeV is shown in Fig. 1. Dif- fuse γ-rays from Type Ia supernovae (SNIa) contribute below ∼ 5 MeV, but the most recent astronomical data show that they also cannot account for the entire spec- trum [14, 15]; previous calculations suggested that SNIa are the dominant source of γ-rays at MeV energies [16]. In this paper, we study the contribution to the CGB from dark matter decaying now. We consider simple models with extra dimensions or SUSY in which WIMP decays are highly suppressed by both the weakness of gravity and small mass splittings and are dependent on a single parameter, ∆m. We find that the CGB is an extremely sensitive probe, even for lifetimes τ ≫ t0. In- triguingly, we also find that both the energy spectrum and the flux of the gamma ray excess described above are naturally explained in these scenarios with ∆m ∼ MeV. As our primary example we consider minimal univer- sal extra dimensions (mUED) [17], one of the simplest imaginable models with extra dimensions. In mUED all http://arxiv.org/abs/0704.1658v3 particles propagate in one extra dimension compactified on a circle, and the theory is completely specified by mh, the Higgs boson mass, and R, the compactification ra- dius. (In detail, there is also a weak, logarithmic depen- dence on the cutoff scale Λ [18]. We present results for ΛR = 20.) Every particle has a Kaluza-Klein (KK) part- ner at every mass level ∼ m/R, m = 1, 2, . . ., and the lightest KK particle (LKP) is a dark matter candidate, with its stability guaranteed by a discrete parity. Astrophysical and particle physics constraints limit mUED parameters to regions of (R−1,mh) parameter space where the two lightest KK particles are the KK hypercharge gauge boson B1, and the KK graviton G1, with mass splitting ∆m <∼ O(GeV) [19]. This extreme degeneracy, along with the fact that KK gravitons inter- act only gravitationally, leads to long NLKP lifetimes b cos2 θW (∆m)3 4.7× 1022 s , (1) where MP ≃ 2.4× 1018 GeV is the reduced Planck scale, θW is the weak mixing angle, b = 10/3 for B 1 → G1γ, and b = 2 for G1 → B1γ [20]. Note that τ depends only on the single parameter ∆m. For 795 GeV <∼ R −1 <∼ 809 GeV and 180 GeV <∼ mh <∼ 215 GeV, the model is not only viable, but the B1 thermal relic abundance is consistent with that required for dark matter [21] and ∆m <∼ 30 MeV, leading to lifetimes τ(B 1 → G1γ) >∼ t0. We will also consider supersymmetric models, where small mass splittings are also possible, since the gravitino mass is a completely free parameter. If the two light- est supersymmetric particles are a Bino-like neutralino B̃ and the gravitino G̃, the heavier particle’s decay width is again given by Eq. (1), but with b = 2 for B̃ → G̃γ, and b = 1 for G̃ → B̃γ. As in mUED, τ depends only on ∆m, and ∆m <∼ 30 MeV yields lifetimes greater than t0. The present photon flux from two-body decays is δ (Eγ − aεγ) , (2) whereN(t) = N ine−t/τ andN in is the number of WIMPs at freeze-out, V0 is the present volume of the Universe, a is the cosmological scale factor with a(t0) ≡ 1, and εγ = ∆m is the energy of the produced photons. Pho- tons from two-body decays are observable in the diffuse photon background only if the decay takes place in the late Universe, when matter or vacuum energy dominates. In this case, Eq. (2) may be written as N in e−P (Eγ/εγ )/τ V0τEγH(Eγ/εγ) Θ(εγ − Eγ) , (3) where P (a) = t is the solution to (da/dt)/a = H(a) = ΩMa−3 +ΩDE a−3(1+w) with P (0) = 0, and ΩM and ΩDE are the matter and dark energy densities. If dark energy is a cosmological constant Λ with w = −1, P (a) ≡ ΩΛa3 + ΩM +ΩΛa3 . (4) The flux has a maximum at Eγ = εγ [ U(H20 τ 2ΩΛ)] where U(x) ≡ (x+ 1− 3x+ 1)/(x− 1). The energy spectrum is easy to understand for very long and very short decay times. For τ ≪ t0, H20 τ 2ΩDE ≪ 1, and the flux grows due to the deceler- ated expansion of the Universe as dΦ/dEγ ∝ E1/2 until it reaches its maximum at Emaxγ ≃ εγ(ΩMH20 τ2/4)1/3. Above this energy, the flux is suppressed exponentially by the decreasing number of decaying particles [3]. On the other hand, if τ ≫ t0, H20 τ2ΩDE ≫ 1, and the flux grows as dΦ/dEγ ∝ E1/2 only for photons that originated in the matter-dominated epoch. For decays in the vacuum-dominated Universe, the flux decreases asymptotically as dΦ/dEγ ∝ E(1+3w)/2 due to the accel- erated expansion. The flux reaches its maximal value at Emaxγ ≃ εγ [−ΩM/((1 + 3w)ΩDE)]−1/(3w) where photons were produced at matter-vacuum equality. Note that this value and the spectrum shape depend on the properties of the dark energy. Assuming ΩM = 0.25, ΩDE = 0.75, w = −1, and h = 0.7, and that these particles make up all of non-baryonic dark matter, so that = 1.0× 10−9 cm−3 ΩNBDM , (5) we find that the maximal flux is (Emaxγ ) = 1.33× 10 −3 cm−2 s−1 sr−1 MeV−1 1021 s ΩNBDM . (6) Fig. 2 shows example contributions to the CGB from decaying dark matter in mUED and SUSY. The mass splittings have been chosen to produce maximal fluxes at Eγ ∼ MeV. These frameworks are, however, highly constrained: once ∆m is chosen, τ and the flux are es- sentially fixed. It is thus remarkable that the predicted flux is in the observable, but not excluded, range and may explain the current excess above known sources. To explore this intriguing fact further, we relax model- dependent constraints and consider τ and ∆m to be in- dependent parameters in Fig. 3. The labeled curves give the points in (τ,∆m) parameter space where, for the WIMP masses indicated and assuming Eq. (5), the max- imal flux from decaying dark matter matches the flux of the observed photon background in the keV to 100 GeV range [6]. For a given WIMP mass, all points above the corresponding curve predict peak fluxes above the ob- served diffuse photon background and so are excluded. The shaded band in Fig. 3 is the region where the max- imal flux falls in the unaccounted for range of 1-5 MeV. FIG. 2: Data for the CGB in the range of the MeV excess, along with predicted contributions from extragalactic dark matter decay. The curves are for B1 → G1γ in mUED with lifetime τ = 103 t0 and mB1 = 800 GeV (solid) and B̃ → G̃γ in SUSY with lifetime τ = 5 × 103 t0 and mB̃ = 80 GeV (dashed). We have assumed ΩNBDM = 0.2 and smeared all spectra with energy resolution ∆E/E = 10%, characteristic of experiments such as COMPTEL. The dot-dashed curve is the upper limit to the blazar spectrum, as in Fig. (1). For τ >∼ t0, E γ ≃ 0.55∆m. However, for τ <∼ t0, E does not track ∆m, as the peak energy is significantly redshifted. For example, for a WIMP with mass 80 GeV, τ ∼ 1012 s and ∆m ∼ MeV, Emaxγ ∼ keV. The over- lap of this band with the labeled contours is where the observed excess may be explained through WIMP de- cays. We see that it requires 1020 s <∼ τ <∼ 10 22 s and 1 MeV <∼ ∆m <∼ 10 MeV. These two properties may be simultaneously realized by two-body gravitational de- cays: the diagonal line shows the relation between τ and ∆m given in Eq. (1) for B1 → G1γ, and we see that this line passes through the overlap region! Similar conclu- sions apply for all other decay models discussed above. These considerations of the diffuse photon background also have implications for the underlying models. For mUED, ∆m = 2.7− 3.2 MeV and τ = 4− 7× 1020 s can explain the MeV excess in the CGB. This preferred region is realized for the decay B1 → G1γ for R−1 ≈ 808 GeV. (See Fig. 4.) Lower R−1 predicts larger ∆m and shorter lifetimes and is excluded. The MeV excess may also be realized for G1 → B1γ for R−1 ≈ 810.5 GeV, though in this case the G1 must be produced non-thermally to have the required dark matter abundance [20, 22]. So far we have concentrated on the cosmic, or extra- galactic, photon flux, which is dependent only on cosmo- logical parameters. The Galactic photon flux depends on halo parameters and so is less robust, but it has the potential to be a striking signature, since these pho- tons are not redshifted and so will appear as lines with Eγ = ∆m. INTEGRAL has searched for photon lines within 13◦ from the Galactic center [23]. For lines with energyE ∼ MeV and width ∆E ∼ 10 keV, INTEGRAL’s FIG. 3: Model-independent analysis of decaying dark mat- ter in the (τ,∆m) plane. In the shaded region, the result- ing extragalactic photon flux peaks in the MeV excess range 1 MeV ≤ Emaxγ ≤ 5 MeV. On the contours labeled with WIMP masses, the maximal extragalactic flux matches the extragalactic flux observed by COMPTEL; points above these contours are excluded. The diagonal line is the predicted re- lation between τ and ∆m in mUED. On the dashed line, the predicted Galactic flux matches INTEGRAL’s sensitivity of 10−4 cm−2 s−1 for monoenergetic photons with Eγ ∼ MeV. energy resolution at these energies, INTEGRAL’s sensi- tivity is Φ ∼ 10−4 cm−2 s−1. The Galactic flux from decaying dark matter saturates this limit along the ver- tical line in Fig. 3, assuming mχ = 800 GeV. This flux is subject to halo uncertainties; we have assumed the halo density profiles of Ref. [24], which give a conservative up- per limit on the flux within the field of view. Remarkably, however, we see that the vertical line also passes through the overlap region discussed above. If the MeV CGB anomaly is explained by decaying dark matter, then, the Galactic flux is also observable, and future searches for photon lines will stringently test this scenario. In conclusion, well-motivated frameworks support the possibility that dark matter may be decaying now. We have shown that the diffuse photon spectrum is a sensi- tive probe of this possibility, even for lifetimes τ ≫ t0. This is the leading probe of these scenarios. Current bounds from the CMB [25] and reionization [26] do not exclude this scenario, but they may also provide comple- mentary probes in the future. We have also shown that dark matter with mass splittings ∆m ∼ MeV and life- times τ ∼ 103−104 Gyr can explain the current excess of observations above astrophysical sources at Eγ ∼ MeV. Such lifetimes are unusually long, but it is remarkable that these lifetimes and mass splittings are simultane- ously realized in simple models with extra dimensional or supersymmetric WIMPs decaying to KK gravitons and gravitinos. Future experiments, such as ACT [27], with large apertures and expected energy resolutions of ∆E/E = 1%, may exclude or confirm this explanation of Excluded Charged DM Excluded Overproduction R (GeV) 807 808 809 810 811 FIG. 4: Phase diagram of mUED. The top and bottom shaded regions are excluded for the reasons indicated [19]. In the yellow (light) shaded region, the B1 thermal relic density is in the 2σ preferred region for non-baryonic dark matter [21]. In the vertical band on the left (right) the decay B1 → G1γ (G1 → B1γ) can explain the MeV diffuse photon excess. the MeV excess through both continuum and line signals. 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0704.1659

Neutrino-cooled accretion and GRB variability

Astronomy & Astrophysics manuscript no. neutrino˙cooling January 2, 2019 (DOI: will be inserted by hand later) Neutrino-cooled accretion and GRB variability Dimitrios Giannios Max Planck Institute for Astrophysics, Box 1317, D-85741 Garching, Germany Received / Accepted Abstract. For accretion rates Ṁ ∼ 0.1 M⊙/s to a few solar mass black hole the inner part of the disk is expected to make a transition from advection dominance to neutrino cooling. This transition is characterized by sharp changes of the disk properties. I argue here that during this transition, a modest increase of the accretion rate leads to powerful enhancement of the Poynting luminosity of the GRB flow and decrease of its baryon loading. These changes of the characteristics of the GRB flow translate into changing gamma-ray spectra from the photosphere of the flow. The photospheric interpretation of the GRB emission explains the observed narrowing of GRB pulses with increasing photon energy and the luminosity-spectral peak relation within and among bursts. Key words. Gamma rays: bursts – Accretion, accretion disks 1. Introduction The commonly assumed model for the central engine of gamma-ray bursts (hereafter GRBs) consists of a compact ob- ject, most likely a black hole, surrounded by a massive ac- cretion disk. This configuration results naturally from the col- lapse of the core of a fast rotating, massive star (Woosley 1993; MacFadyen & Woosley 1999) or the coalescence of a neutron star-neutron star or a neutron star-black hole binary (for simu- lations see Ruffert et al. 1997). The accretion rates needed to power a GRB are in the range Ṁ ∼ 0.01 − 10 M⊙/s. Recently, much theoretical work has been done to understand the microphysics and the structure of the disk at this very high accretion-rate regime (e.g., Chen & Beloborodov 2007; hereafter CB07). These studies have shown that while for accretion rates Ṁ ≪ 0.1M⊙/s the disk is advec- tion dominated, when Ṁ ∼ 0.1M⊙/s it makes a sharp transition to efficient neutrino cooling. This transition results to a thiner, much denser and neutron rich disk. Here I show that, for reasonable scalings of the magnetic field strength with the properties of the inner disk, the advection dominance-neutrino cooling transition results in large changes in the Poynting flux output in the GRB flow. During this transi- tion, a moderate increase of the accretion rate is accompanied by large increase of the Poynting luminosity and decrease of the baryon loading of the GRB flow. This leads to powerful and “clean” ejections of material. The photospheric emission from these ejections explains the observed narrowing of GRB pulses with increasing photon energy (Fenimore et al. 1995) and the luminosity-spectral peak relation within and among bursts (Liang et al. 2004). Send offprint requests to: [email protected] 2. Disk transition to efficient neutrino cooling In accretion powered GRB models the outflow responsible for the GRB is launched in the polar region of the black-hole-disk system. This can be done by neutrino-antineutrino annihilation and/or MHD mechanisms of energy extraction. In either case, the power output in the outflow critically depends on the physi- cal properties of the inner part of the accretion disk. In this sec- tion, I focus on the disk properties around the transition from advection dominance to neutrino cooling. The implications of this transition on the energy output to the GRB flow are the topic of the next section. Recent studies have explored the structure of accretion disks that surround a black hole of a few solar masses for accre- tion rates Ṁ ∼ 0.01 − 10 M⊙/s. Most of these studies focus on 1-D “α”-disk models (where α relates the viscous stress to the pressure in the disk; Shakura & Sunyaev 1973) and put empha- sis on the treatment of the microphysics of the disks connected to the neutrino emission and opacity, nuclear composition and electron degeneracy (Di Matteo et al. 2002; Korhi & Mineshige 2002; Kohri et al. 2005; CB07; Kawanaka & Mineshige 2007; hereafter KM07) and on general relativistic effects on the hy- drodynamics (Popham et al. 1999; Pruet at al. 2003; CB07). These studies have shown that for Ṁ< 0.1 M⊙/s and viscos- ity parameter α ∼ 0.1 the disk is advection dominated since the large densities do not allow for photons to escape. The temper- ature at the inner parts of the disk is T > 1 MeV and the den- sity ρ ∼ 108 gr/cm3 which results in a disk filled with mildly degenerate pairs. In this regime of temperatures and densities the nucleons have dissociated and the disk consists of free pro- tons and neutrons of roughly equal number. The pressure in the disk is: P = Pγ,e± + Pb. The first term accounts for the pres- sure coming from radiation and pairs and the second for that http://arxiv.org/abs/0704.1659v1 2 Dimitrios Giannios: Neutrino-cooled accretion and GRB variability of the baryons. In the advection dominated regime the pressure is dominated by the “light particle” contribution (i.e. the first term in the last expression). For accretion rates Ṁ ∼ 0.1 M⊙/s, a rather sharp transition takes place in the inner parts of the disk. During this transition, the mean electron energy is high enough for electron capture by protons to be possible: e− + p → n + ν. As a result, the disk becomes neutron rich, enters a phase of efficient neutrino cooling and becomes thinner. The baryon density of the disk in- creases dramatically and the total pressure is dominated by the baryon pressure. After the transition is completed the neutron- to-proton ratio in the disk is ∼ 10. Hereafter, I refer to this transition as “neutronization” transition. The neutronization transition takes place at an approxi- mately constant disk temperature T ≈ several×1010 K and is completed for a moderate increase of the accretion rate by a factor of ≈ 2 − 3. During the transition the baryon density in- creases by ≈ 1.5 orders of magnitude and the disk pressure by a factor of several (see CB07; KM07). 2.1. Scalings of the disk properties with Ṁ Although the numbers quoted in the previous section hold quite generally, the range of accretion rates for which the neutroniza- tion transition takes place depends on the α viscosity parame- ter and on the spin of the black hole. For more quantitative statements to be made, I extract some physical quantities of the disk before and after transition from Figs. 13-15 of CB07 for disk viscosity α = 0.1 and spin parameter of the black hole a = 0.95. I focus at a fixed radius close to the inner edge of the disk (for convenience, I choose r = 6GM/c2). The quanti- ties before and after the transition are marked with the super- scripts “A” and “N” and stand for Advection dominance and Neutrino cooling respectively. At ṀA = 0.03 M⊙/s, the density of the disk is ρA ≃ 3 · 109gr/cm3 and has similar number of protons and neutrons, while at ṀN = 0.07 M⊙/s, the density is ρN ≃ 9 · 1010gr/cm3 and the neutron-to-proton ratio is ∼ 10. The temperature remains approximately constant for this range of accretion rates at T ≃ 5 · 1010 K. A factor of ṀN/ṀA ≃ 2.3 increase in the accretion rate in this specific example leads to the transition from advection dominance to neutrino cooling. Around the transition the (mildly degenerate) pairs con- tribute a factor of ∼ 2 more to the pressure w.r.t. radiation. The total pressure is: P = Pγ,e± + Pb ≈ arT 4 + ρkBT/mp, where ar and kB are the radiation and Boltzmann constants respectively (Beloborodov 2003; CB07). Using the last expression, the disk pressure before the transition is found: PA ≃ 6 · 1028 erg/cm3; dominated by the contribution of light particles as expected for an advection dominated disk. At the higher accretion rate ṀN, one finds for the pressure of the disk PN ≃ 4 · 1029 erg/cm3. Now the disk is baryon pressure supported. From the previous exercise one gets indicative scalings for the dependence of quantities in the disk as a function of Ṁ during the neutronization transition: ρ ∝ Ṁ4 and P ∝ Ṁ2.3. Doubling of the accretion rate during the transition leads to a factor of ∼ 16 and ∼ 5 increase of the density and pressure of the disk respectively. Similar estimates for the dependence of the disk density and pressure on the accretion rate can be done when the inner disk is in the advection dominance and neutrino cooling regime but fairly close to the transition. In these regimes, I estimate that ρ ∝ P ∝ Ṁ (see, for example Figs. 1-3 in KM07). Does this sharp change of the disk properties associated with the neutronization transition affect the rate of energy re- lease in the polar region of the disk where the GRB flow is expected to form? The answer depends on the mechanism re- sponsible for the energy release. 3. Changes in the GRB flow from the neutronization transition Gravitational energy released by the accretion of matter to the black hole can be tapped by neutrino-antineutrino annihilation or via MHD mechanisms and power the outflow responsible for the GRB. We consider both of these energy extraction mecha- nisms in turn. The neutrino luminosity of the disk just after the neutron- ization transition is of the order of Lν ∼ 10 52 erg/s and con- sists of neutrinos and antineutrinos of all flavors. The fraction of these neutrinos that annihilate and power the GRB flow de- pends on their spatial emission distribution which, in turn, de- pends critically on the disk microphysics. For Ṁ ∼ 0.1M⊙/s, this fraction is of the order of ∼ 10−3 (Liu et al. 2007), pow- ering an outflow of Lνν̄ ∼ 10 49 erg/s; most likely too weak to explain a cosmological GRB. The efficiency of the neutrino- antineutrino annihilation mechanism can be much higher for accretion rates Ṁ> 1M⊙/s (e.g., Liu et al. 2007; Birkl et al. 2007) which are not considered here. The second possibility is that energy is extracted by strong magnetic fields that thread the inner part of the disk (Blandford & Payne 1982) or the rotating black hole (Blandford & Znajek 1977) launching a Poynting-flux dominated flow. The Blandford-Znajek power output can be estimated to be (e.g. Popham et al. 1999) LBJ ≈ 10 50a2B215M 3 erg/s, (1) where B = 1015B15 Gauss and M = 3M3M⊙. taking into account that magnetic fields of similar strength are expected to thread the inner parts of the disk, the Poynting luminosity output from the disk is rather higher than LBJ because of the larger effective surface of the disk (Livio et al. 1999). In con- clusion, magnetic field strengths in the inner disk of the order of B ∼ 1015 erg/s are likely sufficient to power a GRB via MHD mechanisms of energy extraction. 3.1. Luminosity and baryon loading of the GRB flow as functions of Ṁ In this section, I estimate the Poynting luminosity of the GRB flow for different assumptions on the magnetic field-disk cou- pling. The mass flux in the GRB flow is harder to constrain since it depends on the disk structure and the magnetic field geometry on the disk’s surface. During the neutronization tran- sition, the disk becomes thinner and, hence, more bound grav- itationally. One can thus expect that a smaller fraction of Ṁ is Dimitrios Giannios: Neutrino-cooled accretion and GRB variability 3 injected in the outflow. Here, I make the, rather conservative, assumption that throughout the transition, the mass flux in the outflow is a fixed fraction of accretion rate Ṁ. How is the magnetic field strength related to the properties of the disk? The magneto-rotational instability (hereafter MRI; see Balbus & Hawley 1998 for a review) can amplify magnetic field with energy density up to a fraction ǫ of the pressure in the disk. This provides an estimate for the magnetic field: B2MRI = 8πǫP. This scaling leads to magnetic field strength of the order of∼ 1015 Gauss for the fiducial values of the pressure presented in the previous Sect. and for ǫ ≃ 0.2. The Poynting luminosity scales as Lp ∝ B MRI ∝ P ∝ Ṁ with the accretion rate during the neutronization transition (see previous Sect.). This leads to a rather large increase of the lu- minosity of the GRB flow by a factor of ∼ 7 for a moderate increase of the accretion rate by a factor of ≃ 2.3. Furthermore, if we assume that a fixed fraction of the accreting gas is chan- neled to the outflow, then the baryon loading of the Poynting- flux dominated flow scales as η ∝ Lp/Ṁ ∝ Ṁ 1.3. This means that during the transition the outflow becomes “cleaner” de- creasing its baryon loading by a factor of ∼ 3. The disk can support large-scale fields more powerful that those generated by MRI. These fields may have been advected with the matter during the core collapse of the star (or the bi- nary coalescence) or are captured by the disk in the form a mag- netic islands and brought in the inner parts of the disk (Spruit & Uzdensky 2005). These large scale fields can arguably provide much more promising conditions to launch a large scale jet. Stehle & Spruit (2001) have shown that a disk threaded by a large scale field becomes violently unstable once the radial tension force of the field contributes substantially against grav- ity. This instability is suppressed if the radial tension force is a faction δ ∼ a few % of the gravitational attraction. Large-scale magnetic fields with strength: B2LS = δ8πρcsvk ∝ (ρP) 1/2 can be supported over the duration of a GRB for δ ∼ a few %. In the last expression cs = P/ρ stands for the sound speed and vk is the Keplerian velocity at the inner boundary. The last estimate suggests that large scale field strong enough to power a GRB can be supported by the disk. The output Poynting luminosity scales, in this case , as Lp ∝ B (ρP)1/2. During the neutronization transition, the Poynting lu- minosity increases steeply as a function of the accretion rate: Lp ∝ (ρP) ∝ Ṁ3.2. This translates to a factor of ∼ 15 in- crease of the luminosity of the jet for a modest increase by ∼ 2.3 of the accretion rate. Assuming that the rate of ejection of material in the GRB flow is proportional to the mass accre- tion rate, the baryon loading of the flow is found to decrease by a factor of ∼ 6 during the transition (since η ∝ Lp/Ṁ ∝ Ṁ 2.2). Before and after the transition the disk is advection dom- inated and neutrino cooled respectively. When the disk is in either of these regimes the disk density and pressure scale roughly linearly with the accretion rate (at least for accretion rates fairly close to the neutronization transition; see previous Sect.), leading to Lp ∝ Ṁ and η ∼ constant. The Poynting lu- minosity and the baryon loading of the GRB flow around the neutronization transition are summarized by Fig. 1. Although the Poynting flux output depends on assumptions on the scaling of the magnetic field with the disk properties, 0.1 1 accretion rate (solar masses per second) Poynting luminosity L Baryon loading η I II III Fig. 1. Poynting luminosity and baryon loading (both in arbi- trary units) of the GRB flow around the neutronization transi- tion of the inner disk. In regions marked with I and III the inner disk is advection and neutrino cooling dominated respectively. In region II, the neutronization transition takes place. During the transition, the Poynting luminosity increases steeply with the accretion rate while the baryon loading of the flow is re- duced (i.e. η increases). the neutronization transition generally leads to steep increase of the Poynting luminosity as function of the accretion rate and to a “cleaner” (i.e. less baryon loaded) flow. Observational im- plications of the transition are discussed in the next section. 4. Connection to observations The mechanism I discuss here operates for accretion rates around the neutronization transition of the inner disk and pro- vides the means by which modest variations in the accretion rate give magnified variability in the Poynting flux output and baryon loading of the GRB flow. Since the transition takes place at Ṁ ∼ 0.1 M⊙/s which is close to the accretion rates ex- pected for the collapsar model (MacFadyen & Woosley 1999), it is particularly relevant for that model. To connect the flow variability to the observed properties of the prompt emission, one has to assume a model for the prompt emission. Here we discuss internal shock and photospheric models. Episodes of rapid increase of the luminosity of the flow can be viewed as the ejection of a distinct shells of material. These shells can collide with each other further out in the flow lead- ing to internal shocks that power the prompt GRB emission (Rees & Mészáros 1994). For the internal shocks to be efficient in dissipating energy, there must be a substantial variation of the baryon loading among shells. This may be achieved, in the context of the model presented here, if the accretion rate, at which the neutronization transition takes place, changes during the evolution of the burst. The accretion rate at the transition decreases, for example, with increasing spin of the black hole (CB07). Since the black hole is expected to be substantially span up because of accretion of matter during the evolution of the burst (e.g. MacFadyen & Woosley 1999), there is the possi- 4 Dimitrios Giannios: Neutrino-cooled accretion and GRB variability bility, though speculative at this level, that this leads to ejection of shells with varying baryon loading. 4.1. Photospheric emission Photospheric models for the origin of the prompt emission have been recently explored for both fireballs (Mészáros & Rees 2000; Ryde 2004; Rees & Mészáros 2005; Pe’er et al. 2006) and Poynting-flux dominated flows (Giannios 2006; Giannios & Spruit 2007; hereafter GS07). Here, I focus mainly to the photosphere of a Poynting-flux dominated flow since it is di- rectly applicable to this work. In the photospheric model, the observed variability of the prompt emission is direct manifestation of the central engine activity. Modulations of the luminosity and baryon loading of the GRB flow result in modulations of the location of the pho- tosphere of the flow and of the strength and the spectrum of the photospheric emission (Giannios 2006; GS07). In particular, in GS07 it is demonstrated that if the increase of the luminosity of the flow is accompanied by decrease of the baryon loading such that1 η ∝ L0.6, the photospheric model can explain the observed narrowing of the width of the GRB pulses with increasing pho- ton energy reported by Fenimore et al. (1995). The same η-L scaling also leads to the photospheric luminosity scaling with the peak of the ν · f (ν) spectrum as Lph ∝ E p during the burst evolution in agreement with observations (Liang et al. 2004). The simple model for the connection of the GRB flow to the properties of the central engine presented here predicts that L ∝ Ṁ2.3...3.2 and η ∝ Ṁ1.3...2.2 during the neutronization tran- sition. The range in the exponents comes from the different as- sumptions on the disk-magnetic field connection (see Sect. 3). This translates to η ∝ L0.6...0.7 which is very close that assumed by GS07 to explain the observed spectral and temporal proper- ties of the GRB light curves. Although the launched flow is Poynting-flux dominated, it is conceivable that it undergoes an initial phase of rapid mag- netic dissipation resulting to a fireball. The photospheric lu- minosity and the observed temperature of fireballs scale as Lph ∝ η 8/3L1/3, Tobs ∝ η 8/3L−5/12 respectively (Mészáros & Rees 2000). Using the scaling η ∝ L0.6...0.7 found in this work and identifying the peak of the photospheric component with the peak of the emission Ep one finds that Lph ∝ L 1.9...2.2 and Ep ∝ L 1.2...1.4. The last scalings suggest that the photospheric emission from a fireball can further enhance variations in the gamma-ray luminosity while Lph and Ep follow the Liang et al. relation. Still dissipative processes have to be considered in the fireball so that to explain the observed non-thermal spectra. 5. Conclusions In this work, a mechanism is proposed by which moderate changes of the accretion rate at around Ṁ ∼ 0.1 M⊙/s to a few solar mass black hole can give powerful energy release episodes to the GRB flow. This mechanism is directly applica- ble to the collapsar scenario for GRBs (Woosley; MacFadyen 1 In GS07, the parameterization of the baryon loading of the flow is done by the magnetization σ0 that is related to η through η = σ & Woosley 1999) and can explain how moderate changes in the accretion rate result in extremely variable GRB light curves. This mechanism operates when the inner part of the ac- cretion disk makes the transition from advection dominance to neutrino cooling. This, rather sharp, transition is accompanied by steep increase of the density and the pressure in the disk (CB07; KM07). This leads to substantial increase of the mag- netic field strength in the vicinity of the black hole and conse- quently boosts the Poynting luminosity of the GRB flow by a factor of ∼ 7 − 15. At the same time, assuming that the ejec- tion rate of material scales linearly with the accretion rate, the baryon loading of the flow decreases by a factor ∼ 3 − 6. This results in a luminosity-baryon loading anticorrelation. The changes of the characteristics of the GRB flow can be directly observed as modulations of the photospheric emission giving birth to pulses with spectral and temporal properties similar to the observed ones (GS07). The photospheric inter- pretation of the prompt emission is in agreement with the ob- served narrowing of the pulses with increasing photon energy (Fenimore et al. 1995) and the luminosity-peak energy corre- lation during the evolution of GRBs (Liang et al 2004). 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0704.1660

The VVDS type-1 AGN sample: The faint end of the luminosity function

Astronomy & Astrophysics manuscript no. ABongiorno c© ESO 2018 October 27, 2018 The VVDS type–1 AGN sample: The faint end of the luminosity function A. Bongiorno1, G. Zamorani2, I. Gavignaud3, B. Marano1, S. Paltani4,5, G. Mathez6, J.P. Picat6, M. Cirasuolo7, F. Lamareille2,6, D. Bottini8, B. Garilli8, V. Le Brun9, O. Le Fèvre9, D. Maccagni8, R. Scaramella10,11, M. Scodeggio8, L. Tresse9, G. Vettolani10, A. Zanichelli10, C. Adami9, S. Arnouts9, S. Bardelli2, M. Bolzonella2, A. Cappi2, S. Charlot12,13, P. Ciliegi2, T. Contini6, S. Foucaud14, P. Franzetti8, L. Guzzo15, O. Ilbert16, A. Iovino15, H.J. McCracken13,17, C. Marinoni18, A. Mazure9, B. Meneux8,15, R. Merighi2, R. Pellò6, A. Pollo9,19, L. Pozzetti2, M. Radovich20, E. Zucca2, E. Hatziminaoglou21 , M. Polletta22, M. Bondi10, J. Brinchmann23, O. Cucciati15,24, S. de la Torre9, L. Gregorini25, Y. Mellier13,17, P. Merluzzi20, S. Temporin15, D. Vergani8, and C.J. Walcher9 (Affiliations can be found after the references) Received; accepted Abstract In a previous paper (Gavignaud et al. 2006), we presented the type–1 Active Galactic Nuclei (AGN) sample obtained from the first epoch data of the VIMOS-VLT Deep Survey (VVDS). The sample consists of 130 faint, broad-line AGN with redshift up to z = 5 and 17.5 < IAB < 24.0, selected on the basis of their spectra. In this paper we present the measurement of the Optical Luminosity Function up to z = 3.6 derived from this sample, we compare our results with previous results from brighter samples both at low and at high redshift. Our data, more than one magnitude fainter than previous optical surveys, allow us to constrain the faint part of the luminosity function up to high redshift. By combining our faint VVDS sample with the large sample of bright AGN extracted from the SDSS DR3 (Richards et al., 2006b), we find that the model which better represents the combined luminosity functions, over a wide range of redshift and luminosity, is a luminosity dependent density evolution (LDDE) model, similar to those derived from the major X-surveys. Such a parameterization allows the redshift of the AGN space density peak to change as a function of luminosity and explains the excess of faint AGN that we find at 1.0 < z < 1.5. On the basis of this model we find, for the first time from the analysis of optically selected samples, that the peak of the AGN space density shifts significantly towards lower redshift going to lower luminosity objects. This result, already found in a number of X-ray selected samples of AGN, is consistent with a scenario of “AGN cosmic downsizing”, in which the density of more luminous AGN, possibly associated to more massive black holes, peaks earlier in the history of the Universe, than that of low luminosity ones. Key words. surveys-galaxies: high-redshift - AGN: luminosity function 1. Introduction Active Galactic Nuclei (AGN) are relatively rare objects that ex- hibit some of the most extreme physical conditions and activity known in the universe. A useful way to statistically describe the AGN activity along the cosmic time is through the study of their luminosity func- tion, whose shape, normalization and evolution can be used to derive constraints on models of cosmological evolution of black holes (BH). At z.2.5, the luminosity function of optically se- lected type–1 AGN has been well studied since many years (Boyle et al., 1988; Hewett et al., 1991; Pei, 1995; Boyle et al., 2000; Croom et al., 2004). It is usually described as a double power law, characterized by the evolutionary parameters L∗(z) and Φ∗(z), which allow to distinguish between simple evolution- ary models such as Pure Luminosity Evolution (PLE) and Pure Density Evolution (PDE). Although the PLE and PDE mod- els should be mainly considered as mathematical descriptions of the evolution of the luminosity function, two different phys- ical interpretations can be associated to them: either a small Send offprint requests to: Angela Bongiorno, e-mail: [email protected] fraction of bright galaxies harbor AGN, and the luminosities of these sources change systematically with time (‘luminosity evo- lution’), or all bright galaxies harbor AGN, but at any given time most of them are in ‘inactive’ states. In the latter case, the frac- tion of galaxies with AGN in an ‘active’ state changes with time (‘density evolution’). Up to now, the PLE model is the preferred description for the evolution of optically selected QSOs, at least at low redshift (z < 2). Works on high redshift type–1 AGN samples (Warren et al., 1994; Kennefick et al., 1995; Schmidt et al., 1995; Fan et al., 2001; Wolf et al., 2003; Hunt et al., 2004) have shown that the number density of QSOs declines rapidly from z ∼ 3 to z ∼ 5. Since the size of complete and well studied samples of QSOs at high redshift is still relatively small, the rate of this decline and the shape of the high redshift luminosity function is not yet as well constrained as at low redshift. For example, Fan et al. (2001), studying a sample of 39 luminous high redshift QSOs at 3.6 < z < 5.0, selected from the commissioning data of the Sloan Digital Sky Survey (SDSS), found that the slope of the bright end of the QSO luminosity function evolves with redshift, becoming flatter at high redshift, and that the QSO evolution from z = 2 to z = 5 cannot be described as a pure luminosity evolution. A similar result on the flattening at high redshift of the slope of http://arxiv.org/abs/0704.1660v1 2 Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function the luminosity function for luminous QSOs has been recently obtained by Richards et al. (2006b) from the analysis of a much larger sample of SDSS QSOs (but see Fontanot et al. (2007) for different conclusions drawn on the basis of combined analysis of GOODS and SDSS QSOs). At the same time, a growing number of observations at differ- ent redshifts, in radio, optical and soft and hard X-ray bands, are suggesting that also the faint end slope evolves, becoming flat- ter at high redshift (Page et al., 1997; Miyaji et al., 2000, 2001; La Franca et al., 2002; Cowie et al., 2003; Ueda et al., 2003; Fiore et al., 2003; Hunt et al., 2004; Cirasuolo et al., 2005; Hasinger et al., 2005). This evolution, now dubbed as “AGN cos- mic downsizing” is described either as a direct evolution in the faint end slope or as “luminosity dependent density evolution” (LDDE), and it has been the subject of many speculations since it implies that the space density of low luminosity AGNs peaks at lower redshift than that of bright ones. It has been observed that, in addition to the well known local scale relations between the black hole (BH) masses and the properties of their host galaxies (Kormendy & Richstone, 1995; Magorrian et al., 1998; Ferrarese & Merritt, 2000), also the galaxy spheroid population follows a similar pattern of “cos- mic downsizing” (Cimatti et al., 2006). Various models have been proposed to explain this common evolutionary trend in AGN and spheroid galaxies. The majority of them propose that the feedback from the black hole growth plays a key role in determining the BH-host galaxy relations (Silk & Rees, 1998; Di Matteo et al., 2005) and the co-evolution of black holes and their host galaxies. Indeed, AGN feedback can shut down the growth of the most massive systems steepening the bright end slope (Scannapieco & Oh, 2004), while the feedback-driven QSO decay determines the shape of the faint end of the QSO LF (Hopkins et al., 2006). This evolutionary trend has not been clearly seen yet with optically selected type–1 AGN samples. By combining results from low and high redshifts, it is clear from the studies of op- tically selected samples that the cosmic QSO evolution shows a strong increase of the activity from z ∼ 0 out to z ∼ 2, reaches a maximum around z ≃ 2 − 3 and then declines, but the shape of the turnover and the redshift evolution of the peak in activity as a function of luminosity is still unclear. Most of the optically selected type–1 AGN samples stud- ied so far are obtained through various color selections of candidates, followed by spectroscopic confirmation (e.g. 2dF, Croom et al. 2004 and SDSS, Richards et al. 2002), or grism and slitless spectroscopic surveys. These samples are expected to be highly complete, at least for luminous type–1 AGN, at either z ≤ 2.2 or z ≥ 3.6, where type–1 AGN show conspicuous colors in broad band color searches, but less complete in the redshift range 2.2 ≤ z ≤ 3.6 (Richards et al. 2002). An improvement in the multi-color selection in optical bands is through the simultaneous use of many broad and medium band filters as in the COMBO-17 survey (Wolf et al., 2003). This sur- vey is the only optical survey so far which, in addition to cov- ering a redshift range large enough to see the peak of AGN ac- tivity, is also deep enough to sample up to high redshift type–1 AGN with luminosity below the break in the luminosity func- tion. However, only photometric redshifts are available for this sample and, because of their selection criteria, it is incomplete for objects with a small ratio between the nuclear flux and the total host galaxy flux and for AGN with anomalous colors, such as, for example, the broad absorption line (BAL) QSOs , which have on average redder colors and account for ∼ 10 - 15 % of the overall AGN population (Hewett & Foltz, 2003). The VIMOS-VLT Deep Survey (Le Fèvre et al., 2005) is a spectroscopic survey in which the target selection is purely flux limited (in the I-band), with no additional selection criterion. This allows the selection of a spectroscopic type–1 AGN sample free of color and/or morphological biases in the redshift range z > 1. An obvious advantage of such a selection is the possi- bility to test the completeness of the most current surveys (see Gavignaud et al., 2006, Paper I), based on morphological and/or color pre-selection, and to study the evolution of type–1 AGN activity in a large redshift range. In this paper we use the type-1 AGN sample selected from the VVDS to derive the luminosity function in the redshift range 1 < z < 3.6. The VVDS type–1 AGN sample is more than one magnitude deeper than any previous optically selected sample and allow thus to explore the faint part of the luminosity func- tion. Moreover, by combining this LF with measurement of the LF in much larger, but very shallow, surveys, we find an analyt- ical form to dercribe, in a large luminosity range, the evolution of type-1 AGN in the redshift range 0< z <4. The paper is or- ganized as follows: in Section 2 and 3 we describe the sample and its color properties. In Section 4 we present the method used to derive the luminosity function, while in Section 5 we com- pare it with previous works both at low and high redshifts. The bolometric LF and the comparison with the results derived from samples selected in different bands (from X-ray to IR) is then presented in Section 6. The derived LF fitting models are pre- sented in Section 7 while the AGN activity as a function of red- shift is shown in Section 8. Finally in section 9 we summarize our results. Throughout this paper, unless stated otherwise, we assume a cosmology with Ωm = 0.3, ΩΛ = 0.7 and H0 = 70 km s−1 Mpc−1. 2. The sample Our AGN sample is extracted from the first epoch data of the VIMOS-VLT Deep Survey, performed in 2002 (Le Fèvre et al., 2005). The VVDS is a spectroscopic survey designed to measure about 150,000 redshifts of galaxies, in the redshift range 0 < z < 5, selected, nearly randomly, from an imaging survey (which consists of observations in U, B, V, R and I bands and, in a small area, also K-band) designed for this purpose. Full de- tails about VIMOS photometry can be found in Le Fèvre et al. (2004a), McCracken et al. (2003), Radovich et al. (2004) for the U-band and Iovino et al. (2005) for the K-band. In this work we will as well use the Galex UV-catalog (Arnouts et al., 2005; Schiminovich et al., 2005), the u∗,g′,r′,i′,z′ photometry obtained in the frame of the Canada-France-Hawaii Legacy Survey (CFHTLS)1, UKIDSS (Lawrence et al., 2006), and the Spitzer Wide-area InfraRed Extragalactic survey (SWIRE) (Lonsdale et al., 2003, 2004). The spectroscopic VVDS survey consists of a deep and a wide survey and it is based on a sim- ple selection function. The sample is selected only on the basis of the I band magnitude: 17.5 < IAB < 22.5 for the wide and 17.5 < IAB < 24.0 for the deep sample. For a detailed descrip- tion of the spectroscopic survey strategy and the first epoch data see Le Fèvre et al. (2005). Our sample consists of 130 AGN with 0 < z < 5, selected in 3 VVDS fields (0226-04, 1003+01 and 2217-00) and in the Chandra Deep Field South (CDFS, Le Fèvre et al., 2004b). All of them are selected as AGN only on the basis of their spectra, 1 www.cfht.hawaii.edu/Science/CFHLS Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function 3 Figure 1. Distribution of absolute magnitudes and redshifts of the total AGN sample. Open circles are the objects with am- biguous redshift, shown at all their possible z values. The dotted and dashed lines represent the magnitude limits of the samples: IAB < 22.5 for the wide sample and IAB < 24.0 for the deep sample. irrespective of their morphological or color properties. In partic- ular, we selected them on the basis of the presence of at least one broad emission line. We discovered 74 of them in the deep fields (62 in the 02h field and 12 in the CDFS) and 56 in the wide fields (18 in the 10h field and 38 in the 22h field). This represents an unprecedented complete sample of faint AGN, free of morpho- logical or color selection bias. The spectroscopic area covered by the First Epoch Data is 0.62 deg2 in the deep fields (02h field and CDFS) and 1.1 deg2 in the wide fields (10h and 22h fields). To each object we have assigned a value for the spectro- scopic redshift and a spectroscopic quality flag which quantifies our confidence level in that given redshift. As of today, we have 115 AGN with secure redshift, and 15 AGN with two or more possible values for the redshift. For these objects, we have two or more possible redshifts because only one broad emission line, with no other narrow lines and/or additional features, is detected in the spectral wavelength range adopted in the VVDS (5500 - 9500 Å) (see Figure 1 in Paper I). For all of them, however, a best solution is proposed. In the original VVDS AGN sample, the number of AGN with this redshift degeneracy was 42. To solve this problem, we have first looked for the objects already observed in other spectroscopic surveys in the same areas, solv- ing the redshift for 3 of them. For the remaining objetcs, we performed a spectroscopic follow-up with FORS1 on the VLT Unit Telescope 2 (UT2). With these additional observations we found a secure redshift for 24 of our AGN with ambiguous red- shift determination and, moreover, we found that our proposed best solution was the correct one in ∼ 80% of the cases. On the basis of this result, we decided to use, in the following analysis, our best estimate of the redshift for the small remaining fraction of AGN with ambiguous redshift determination (15 AGN). In Figure 1 we show the absolute B-magnitude and the redshift distributions of the sample. As shown in this Figure, our sample spans a large range of luminosities and consists of both Seyfert galaxies (MB >-23; ∼59%) and QSOs (MB <-23; ∼41%). A more detailed and exhaustive description of the prop- Figure 2. Composite spectra derived for our AGN with se- cure redshift in the 02h field, divided in a “bright” (19 objects at M1450 <-22.15, dotted curve) and a “faint” (31 objects at M1450 >-22.15, dashed curve) sample. We consider here only AGN with z > 1 (i.e. the AGN used in to compute the lumi- nosity function). The SDSS composite spectrum is shown with a solid line for comparison. erties of the AGN sample is given in Paper I (Gavignaud et al., 2006) and the complete list of BLAGN in our wide and deep samples is available as an electronic Table in Appendix of Gavignaud et al. (2006). 3. Colors of BLAGNs As already discussed in Paper I, the VVDS AGN sample shows, on average, redder colors than those expected by comparing them, for example, with the color track derived from the SDSS composite spectrum (Vanden Berk et al., 2001). In Paper I we proposed three possible explanations: (a) the contamination of the host galaxy is reddening the observed colors of faint AGN; (b) BLAGN are intrinsically redder when they are faint; (c) the reddest colors are due to dust extinction. On the basis of the sta- tistical properties of the sample, we concluded that hypothesis (a) was likely to be the more correct, as expected from the faint absolute magnitudes sampled by our survey, even if hypotheses (b) and (c) could not be ruled out. In Figure 2 we show the composite spectra derived from the sample of AGN with secure redshift in the 02h field, divided in a “bright” and a “faint” sample at the absolute magnitude M1450 = −22.15. We consider here only AGN with z > 1, which correspond to the AGN used in Section 4 to compute the lumi- nosity function. The choice of the reference wavelength for the absolute magnitude, λ = 1450 Å, is motivated by our photo- metric coverage. In fact, for most of the objects it is possible to interpolate M1450 directly from the observed magnitudes. In the same plot we show also the SDSS composite spectrum (solid curve) for comparison. Even if also the ”bright” VVDS compos- ite (dotted curve) is somewhat redder than the SDSS one, it is clear from this plot that the main differences occur for faintest objects (dashed curve). A similar result is shown for the same sample in the upper panel of Figure 3, where we plot the spectral index α as a func- tion of the AGN luminosity. The spectral index is derived here by fitting a simple power law f (ν) = ν−α to our photometric data points. This analysis has been performed only on the 02h deep 4 Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function Figure 3. Upper Panel: Distribution of the spectral index α as a function of M1450 for the same sample of AGN as in Figure 2. The spectral index is derived here by fitting a simple power law f (ν) = ν−α to our photometric data points. Asterisks are AGN morphologically classified as extended and the grey point is a BAL AGN. Bottom Panels: Distribution of the spectral in- dex α for the same sample of AGN. All the AGN in this sample are shown in the first of the three panels, while the AGN in the “bright” and “faint” sub–samples are shown in the second and third panel, respectively. The dotted curve in the second panel corresponds to the gaussian fit of the bright sub–sample and it is reported also in the third panel to highlight the differences in the α distributions of the two sub-samples. sample, since for the wide sample we do not have enough photo- metric coverage to reliably derive the spectral index. Most of the AGN with α > 1 are fainter than M1450 = −22.15, showing that, indeed, the faintest objects have on average redder colors than the brightest ones. The outlier (the brightest object with large α, i.e. very red colors, in the upper right corner of the plot) is a BAL The three bottom panels of Figure 3 show the histograms of the resulting power law slopes for the same AGN sample. The total sample is plotted in the first panel, while the bright and the faint sub-samples are plotted in the second and third panels, re- spectively. A Gaussian curve with < α >= 0.94 and dispersion σ = 0.38 is a good representation for the distribution of about 80% (40/50) of the objects in the first panel. In addition, there is a significant tail (∼ 20%) of redder AGN with slopes in the range from 1.8 up to ∼ 3.0. The average slope of the total sample (∼ 0.94) is redder than the fit to the SDSS composite (∼ 0.44). Moreover, the distribution of α is shifted toward much larger val- ues (redder continua) than the similar distribution in the SDSS sample (Richards et al., 2003). For example, only 6% of the ob- jects in the SDSS sample have α > 1.0, while this percentage is 57% in our sample. The differences with respect to the SDSS sample can be partly due to the differences in absolute magnitude of the two samples (Mi <-22.0 for the SDSS sample (Schneider et al., 2003) and MB <-20.0 for the VVDS sample). In fact, if we con- sider the VVDS “bright” sub-sample, the average spectral index < α > becomes ∼ 0.71, which is closer to the SDSS value (even if it is still somewhat redder), and only two objects (∼8% of the sample) show values not consistent with a gaussian distribution with σ ∼0.32. Moreover, only 30% of this sample have α > 1.0. Most of the bright SDSS AGNs with α > 1 are interpreted by Richards et al. (2003) to be dust-reddened, although a fraction of them is likely to be due to intrinsically red AGN (Hall et al., 2006). At fainter magnitude one would expect both a larger frac- tion of dust-reddened objects (in analogy with indications from the X-ray data (Brandt et al., 2000; Mushotzky et al., 2000) and a more significant contamination from the host galaxy. We have tested these possibilities by examining the global Spectral Energy Distribution (SED) of each object and fitting the observed fluxes fobs with a combination of AGN and galaxy emission, allowing also for the possibility of extinction of the AGN flux. Thanks to the multi-wavelength coverage in the deep field in which we have, in addition to VVDS bands, also data from GALEX, CFHTLS, UKIDSS and SWIRE, we can study the spectral energy distribution of the single objects. In particu- lar, we assume that: fobs = c1 fAGN · 10 −0.4·Aλ + c2 fGAL (1) and, using a library of galaxy and AGN templates, we find the best parameters c1, c2 and EB−V for each object. We used the AGN SED derived by Richards et al. (2006a) with an SMC- like dust-reddening law (Prevot et al., 1984) with the form Aλ/EB−V = 1.39λ µm , and a library of galaxies template by Bruzual & Charlot (2003). We found that for ∼37% of the objects, the observed flux is fitted by a typical AGN power law (pure AGN), while 44% of the sources require the presence of a contribution from the host galaxy to reproduce the observed flux. Only 4% of the ob- jects are fitted by pure AGN + dust, while the remaining 15% of objects require instead both contributions (host galaxy con- tamination and presence of dust). As expected, if we restrict the analysis to the bright sample, the percentage of pure AGN in- creases to 68%, with the rest of the objects requiring either some contribution from the host galaxy (∼21%) or the presence of dust oscuration (∼11%). In Figure 4 we show 4 examples of the resulting fits: (i) pure AGN; (ii) dust-extincted AGN; (iii) AGN contaminated by the host galaxy; (iv) dust-extincted AGN and contaminated by the host galaxy. The dotted line corresponds to the AGN template before applying the extinction law, while the solid blue line cor- responds to the same template, but extincted for the given EB−V ; the red line corresponds to the galaxy template and, finally, the black line is the resulting best fit to the SED. The host galaxy contaminations will be taken into account in the computation of the AGN absolute magnitude for the luminosity function. Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function 5 Figure 4. Four examples of different decompositions of the ob- served SEDs of our objects. Since for λ < 1216 Å, corresponding to the Lyα line, the observed flux is expected to decrease because of intervening absorption, all the photometric data at λ <1216 Å are not considered in the fitting. The only requested constraint is that they lie below the fit. The four fits shown in this Figure cor- respond, from top to bottom, to pure-AGN, dust-extincted AGN, AGN and host galaxy, dust-extincted AGN and host galaxy. The dotted line corresponds to the AGN template before applying the extinction law, while the solid blue line corresponds to the same template, but extincted for the given EB−V . The red line (third and fourth panel) corresponds to the galaxy template and, finally, the black line is the resulting best fit to the SED. Arrows correspond to 5σ upper limits in case of non detection in the IR. 4. Luminosity function 4.1. Definition of the redshift range For the study of the LF we decided to exclude AGN with z ≤ 1.0. This choice is due to the fact that for 0.5 ≤ z ≤ 1.0 the only visible broad line in the VVDS spectra is Hβ (see Figure 1 of Paper I). This means that all objects with narrow or almost narrow Hβ and broad Hα (type 1.8, 1.9 AGN; see Osterbrock 1981) would not be included in our sample, because we include in the AGN sample all the objects with at least one visible broad line. Since at low luminosities the number of intermediate type AGN is not negligible, this redshift bin is likely to be under- populated and the results would not be meaningful. At z < 0.5, in principle we have less problems, because also Hα is within the wavelength range of the VVDS spectra, but, since at this low redshift, our sampled volume is relatively small and QSOs rare, only 3 objects have secure redshifts in this red- shift bin in the current sample. For these reasons, our luminosity function has been computed only for z > 1.0 AGN. As already mentioned in Section 2, the small fraction of objects with an am- biguous redshift determination have been included in the compu- tation of the luminosity function assuming that our best estimate of their redshift is correct. The resulting sample used in the computation of the LF consists thus of 121 objects at 1< z <4. 4.2. Incompleteness function Our incompleteness function is made up of two terms linked, re- spectively, to the selection algorithm and to the spectral analysis: the Target Sampling Rate (TSR) and the Spectroscopic Success Rate (SSR) defined following Ilbert et al. (2005). The Target Sampling Rate, namely the ratio between the ob- served sources and the total number of objects in the photometric catalog, quantifies the incompleteness due to the adopted spec- troscopic selection criterion. The TSR is similar in the wide and deep sample and runs from 20% to 30%. The Spectroscopic Success Rate is the probability of a spec- troscopically targeted object to be securely identified. It is a com- plex function of the BLAGN redshift, apparent magnitude and intrinsic spectral energy distribution and it has been estimated by simulating 20 Vimos pointings, for a total of 2745 spectra. Full details on TSR and SSR can be found in Paper I (Gavignaud et al., 2006). We account for them by computing for each object the associated weights wtsr = 1/TS R and wssr = 1/S S R; the total weighted contribution of each object to the luminosity function is then the product of the derived weights (wtsr × wssr). 4.3. Estimate of the absolute magnitude We derived the absolute magnitude in the reference band from the apparent magnitude in the observed band as: M = mobs − 5log10(dl(z)) − 25 − k (2) where M is computed in the band in which we want to compute the luminosity function, mobs is the observed band from which we want to calculate it, dl(z) is the luminosity distance expressed in Mpc and k is the k-correction in the reference band. To make easier the comparison with previous results in the literature, we computed the luminosity function in the B-band. To minimize the uncertainties in the adopted k-correction, mobs for each object should be chosen in the observed band which is sampling the rest-wavelength closer to the band in which the luminosity function is computed. For our sample, which consists only of z > 1 objects, the best bands to use to compute the B-band absolute magnitudes should be respectively the I-, J- and K-bands going to higher redshift. Since however, the only observed band available for the entire sample (deep and wide), is the I-band, we decided to use it for all objects to com- pute the B-band magnitudes. This means that for z ∼ > 2, we introduce an uncertainty in the absolute magnitudes due to the k-correction. We computed the absolute magnitude considering the template derived from the SDSS sample (Vanden Berk et al., 2001). As discussed in Section 3, the VVDS AGN sample shows redder colors than those typical of normal, more luminous AGN and this can be due to the combination of the host galaxy contri- bution and the presence of dust. Since, in this redshift range, the fractional contribution from the host galaxies is expected to be more significant in the I-band than in bluer bands, the luminos- ity derived using the I-band observed magnitude could, in some cases, be somewhat overestimated due to the contribution of the host galaxy component. 6 Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function Figure 5. Real (full circles; AGN in the deep sample) and simu- lated (open triangles; AGN in the wide sample) B-band absolute magnitude differences as a function of MB(TOT) (upper panel) and redshift (bottom panel). MB(TOT) is the absolute magnitude computed considering the total observed flux, while MB(AGN) is the absolute magnitude computed after subtracting the host- galaxy contribution. We estimated the possible impact of this effect on our re- sults in the following way. From the results of the analysis of the SED of the single objects in the deep sample (see Section 3) we computed for each object the difference mI(TOT ) − mI(AGN) and, consequently, MB(TOT ) − MB(AGN). This could allow us to derive the LF using directly the derived MB(AGN), resolv- ing the possible bias introduced by the host galaxy contami- nation. These differences are shown as full circles in Figure 5 as a function of absolute magnitude (upper panel) and red- shift (lower panel). For most of the objects the resulting dif- ferences between the total and the AGN magnitudes are small (∆M≤0.2). However, for a not negligible fraction of the faintest objects (MB ≥-22.5, z ≤2.0) these differences can be signifi- cant (up to ∼1 mag). For the wide sample, for which the more restricted photometric coverage does not allow a detailed SED analysis and decomposition, we used simulated differences to derive the MB(AGN). These simulated differences have been de- rived through a Monte Carlo simulation on the basis of the bi- variate distribution ∆M(M,z) estimated from the objects in the deep sample. ∆M(M,z) takes into account the probability distri- bution of ∆M as a function of MB and z, between 0 and the solid line in Figure 5 derived as the envelope suggested by the black dots. The resulting simulated differences for the objects in the wide sample are shown as open triangles in the two panels of Figure 5. The AGN magnitudes and the limiting magnitudes of the samples have been corrected also for galactic extinction on the basis of the mean extinction values E(B−V) in each field derived from Schlegel et al. (1998). Only for the 22h field, where the ex- tinction is highly variable across the field, we used the extinction on the basis of the position of individual objects. The resulting corrections in the I-band magnitude are AI ≃ 0.027 in the 2h and 10h fields and AI = 0.0089 in the CDFS field, while the average value in the 22h field is AI = 0.065. These corrections have been applied also to the limiting magnitude of each field. 4.4. The 1/Vmax estimator We derived the binned representation of the luminosity function using the usual 1/Vmax estimator (Schmidt, 1968), which gives the space density contribution of individual objects. The lumi- nosity function, for each redshift bin (z − ∆z/2 ; z + ∆z/2), is then computed as: Φ(M) = M+∆M/2 M−∆M/2 wtsri w Vmax,i where Vmax,i is the comoving volume within which the i th ob- ject would still be included in the sample. wtsri and w i are re- spectively the inverse of the TSR and of the SSR, associated to the ith object. The statistical uncertainty on Φ(M) is given by Marshall et al. (1983): √M+∆M/2 M−∆M/2 (wtsri w V2max,i We combined our samples at different depths using the method proposed by Avni & Bahcall (1980). In this method it is assumed that each object, characterized by an observed red- shift zi and intrinsic luminosity Li, could have been found in any of the survey areas for which its observed magnitude is brighter than the corresponding flux limit. This means that, for our total sample, we consider an area of: Ωtot(m) = Ωdeep+Ωwide = 1.72 deg 2 for 17.5 < IAB < 22.5 Ωtot(m) = Ωdeep = 0.62 deg 2 for 22.5 < IAB < 24.0 The resulting luminosity functions in different redshift ranges are plotted in Figure 6 and 7, where all bins which contain at least one object are plotted. The LF values, together with their 1σ errors and the numbers of objects in each absolute magnitude bin are presented in Table 1. The values reported in Table 1 and plotted in Figures 6 and 7 are not corrected for the host galaxy contribution. We have in fact a posteriori verified that, even if the differences between the total absolute magnitudes and the mag- nitudes corrected for the host galaxy contribution (see Section 4.3) can be significant for a fraction of the faintest objects, the resulting luminosity functions computed by using these two sets of absolute magnitudes are not significantly different. For this reason and for a more direct comparison with previous works, the results on the luminosity function presented in the next sec- tion are those obtained using the total magnitudes. 5. Comparison with the results from other optical surveys We derived the luminosity function in the redshift range 1.0< z <3.6 and we compared it with the results from other surveys at both low and high redshift. 5.1. The low redshift luminosity function In Figure 6 we present our luminosity function up to z = 2.1. The Figure show our LF data points (full circles) derived in two redshift bins: 1.0 < z < 1.55 and 1.55 < z < 2.1 compared with the LF fits derived from the 2dF QSO sample by Croom et al. (2004) and by Boyle et al. (2000), with the COMBO-17 sample Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function 7 Figure 6. Our rest-frame B-band luminosity function, derived in the redshift bins 1.0 < z < 1.55 and 1.55 < z < 2.1, compared with the 2dFQRS (Croom et al., 2004; Boyle et al., 2000), COMBO-17 data (Wolf et al., 2003) and with the 2dF-SDSS (2SLAQ) data (Richards et al., 2005). The curves in the Figure show the PLE fit models derived by these authors. The thick parts of the curves correspond to the luminosity range covered by the data in each sample, while the thin parts are extrapolations based on the best fit parameters of the models. by Wolf et al. (2003), and with the 2dF-SDSS (2SLAQ) LF fit by Richards et al. (2005). In each panel the curves, computed for the average z of the redshift range, correspond to a double power law luminosity function in which the evolution with red- shift is characterized by a pure luminosity evolution modeled as M∗b(z) = M b(0)−2.5(k1z+ k2z 2). Moreover, the thick parts of the curves show the luminosity range covered by the data in each of the comparison samples, while the thin parts are extrapolation based on the the best fit parameters of the models. We start considering the comparison with the 2dF and the COMBO-17 LF fits. As shown in Figure 6, our bright LF data points connect rather smoothly to the faint part of the 2dF data. However, our sample is more than two magnitudes deeper than the 2dF sample. For this reason, a comparison at low luminosity is possible only with the extrapolations of the LF fit. At z > 1.55, while the Boyle’s model fits well our faint LF data points, the Croom’s extrapolation, being very flat, tends to underestimate our low luminosity data points. At z < 1.55 the comparison is worse: as in the higher redshift bin, the Boyle’s model fits our data better than the Croom’s one but, in this redshift bin, our data points show an excess at low luminosity also with respect to Boyle’s fit. This trend is similar to what shown also by the com- parison with the fit of the COMBO-17 data which, differently from the 2dF data, have a low luminosity limit closer to ours: at z > 1.55 the agreement is very good, but in the first redshift bin our data show again an excess at low luminosity. This excess is likely due to the fact that, because of its selection criteria, the COMBO-17 sample is expected to be significantly incomplete for objects in which the ratio between the nuclear flux and the total host galaxy flux is small. Finally, we compare our data with the 2SLAQ fits derived by Richards et al. (2005). The 2SLAQ data are derived from a sample of AGN selected from the SDSS, at 18.0 < g < 21.85 and z < 3, and observed with the 2-degree field instrument. Similarly to the 2dF sample, also for this sam- ple the LF is derived only for z < 2.1 and MB < −22.5. The plot- ted dot-dashed curve corresponds to a PLE model in which they fixed most of the parameters of the model at the values found by Croom et al. (2004), leaving to vary only the faint end slope and the normalization constant Φ∗. In this case, the agreement with our data points at z < 1.55 is very good also at low lu- minosity. The faint end slope found in this case is β = −1.45, which is similar to that found by Boyle et al. (2000) (β = −1.58) and significantly steeper than that found by Croom et al. (2004) (β = −1.09). At z > 1.55, the Richards et al. (2005) LF fit tends to overestimate our data points at the faint end of the LF, which suggest a flatter slope in this redshift bin. The first conclusion from this comparison is that, at low red- shift (i.e. z < 2.1), the data from our sample, which is ∼2 mag fainter than the previous spectroscopically confirmed samples, are not well fitted simultaneously in the two analyzed redshift bins by the PLE models derived from the previous samples. Qualitatively, the main reason for this appears to be the fact that our data suggest a change in the faint end slope of the LF, which appears to flatten with increasing redshift. This trend, already highlighted by previous X-ray surveys (La Franca et al., 2002; Ueda et al., 2003; Fiore et al., 2003) suggests that a simple PLE parameterization may not be a good representation of the evolu- tion of the AGN luminosity function over a wide range of red- shift and luminosity. Different model fits will be discussed in Section 7. 8 Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function Figure 7. Our luminosity function, at 1450 Å rest-frame, in the redshift range 2.1<z<3.6, compared with data from other high- z samples (Hunt et al. (2004) at z = 3; Combo-17 data from Wolf et al. (2003) at 2.4 < z < 3.6; data from Warren et al. (1994) at 2.2 < z < 3.5 and the SDSS data from Fan et al. (2001)). The SDSS data points at 3.6< z <3.9 have been evolved to z=3 using the luminosity evolution of Pei (1995) as in Hunt et al. (2004). The curves show some model fits in which the thick parts of the curves correspond to the luminosity range covered by the data samples, while the thin parts are model ex- trapolation. For this plot, anΩm = 1,ΩΛ = 0, h = 0.5 cosmology has been assumed for comparison with the previous works. 5.2. The high redshift luminosity function The comparison of our LF data points for 2.1< z <3.6 (full circles) with the results from other samples in similar redshift ranges is shown in Figure 7. In this Figure an Ωm = 1, ΩΛ = 0, h = 0.5 cosmology has been assumed for comparison with pre- vious works, and the absolute magnitude has been computed at 1450 Å. As before, the thick parts of the curves show the lu- minosity ranges covered by the various data samples, while the thin parts are model extrapolations. In terms of number of ob- jects, depth and covered area, the only sample comparable to ours is the COMBO-17 sample (Wolf et al., 2003), which, in this redshift range, consists of 60 AGN candidates over 0.78 square degree. At a similar depth, in terms of absolute mag- nitude, we show also the data from the sample of Hunt et al. (2004), which however consists of 11 AGN in the redshift range < z > ±σz =3.03±0.35 (Steidel et al., 2002). Given the small number of objects, the corresponding Hunt model fit was de- rived including also the Warren data points (Warren et al., 1994). Moreover, they assumed the Pei (1995) luminosity evolution model, adopting the same values for L∗ and Φ∗, leaving free to vary the two slopes, both at the faint and at the bright end of the LF. For comparison we show also the original Pei model fit derived from the empirical luminosity function estimated by Hartwick & Schade (1990) and Warren et al. (1994). In the same plot we show also the model fit derived from a sample of ∼100 z ∼ 3 (U-dropout) QSO candidates by Siana et al. (pri- vate comunication; see also Siana et al. 2006). This sample has been selected by using a simple optical/IR photometric selec- tion at 19< r′ <22 and the model fit has been derived by fix- ing the bright end slope at z=-2.85 as determined by SDSS data (Richards et al., 2006b). In general, the comparison of the VVDS data points with those from the other surveys shown in Figure 7 shows a satis- factory agreement in the region of overlapping magnitudes. The best model fit which reproduce our LF data points at z ∼ 3 is the Siana model with a faint end slope β = −1.45. It is interesting to note that, in the faint part of the LF, our data points appear to be higher with respect to the Hunt et al. (2004) fit and are instead closer to the extrapolation of the original Pei model fit. This dif- ference with the Hunt et al. (2004) fit is probably due to the fact that, having only 11 AGN in their faint sample, their best fit to the faint-end slope was poorly constrained. 6. The bolometric luminosity function The comparison between the AGN LFs derived from samples selected in different bands has been for a long time a critical point in the studies of the AGN luminosity function. Recently, Hopkins et al. (2007), combining a large number of LF measure- ments obtained in different redshift ranges, observed wavelength bands and luminosity intervals, derived the Bolometric QSO Luminosity Function in the redshift range z = 0 - 6. For each observational band, they derived appropriate bolometric correc- tions, taking into account the variation with luminosity of both the average absorption properties (e.g. the QSO column density NH from X-ray data) and the average global spectral energy dis- tributions. They show that, with these bolometric corrections, it is possible to find a good agreement between results from all different sets of data. We applied to our LF data points the bolometric corrections given by Eqs. (2) and (4) of Hopkins et al. (2007) for the B-band and we derived the bolometric LF shown as black dots in Figure 8. The solid line represents the bolometric LF best fit model de- rived by Hopkins et al. (2007) and the colored data points cor- respond to different samples: green points are from optical LFs, blue and red points are from soft-X and hard-X LFs, respec- tively, and finally the cyan points are from the mid-IR LFs. All these bolometric LFs data points have been derived following the same procedure described in Hopkins et al. (2007). Our data, which sample the faint part of the bolometric lu- minosity function better than all previous optically selected sam- ples, are in good agreement with all the other samples, selected in different bands. Only in the last redshift bin, our data are quite higher with respect to the samples selected in other wavelength bands. The agreement remains however good with the COMBO- 17 sample which is the only optically selected sample plotted here. This effect can be attributed to the fact that the conversions used to compute the Bolometric LF, being derived expecially for AGN at low redshifts, become less accurate at high redshift. Our data show moreover good agreement also with the model fit derived by Hopkins et al. (2007). By trying various an- alytic fits to the bolometric luminosity function Hopkins et al. (2007) concluded that neither pure luminosity nor pure density evolution represent well all the data. An improved fit can in- stead be obtained with a luminosity dependent density evolution model (LDDE) or, even better, with a PLE model in which both the bright- and the faint-end slopes evolve with redshift. Both Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function 9 Figure 8. Bolometric luminosity function derived in three redshift bins from our data (black dots), compared with Hopkins et al. (2007) best-fit model and the data-sets used in their work. The central redshift of each bin is indicated in each panel. Here, we adopted the same color-code as in Hopkins et al. (2007), but for more clarity we limited the number of samples presented in the Figure. Red symbols correspond to hard X-ray surveys (squares: Barger et al. 2005; circles: Ueda et al. 2003). Blue to soft X-ray surveys (squares: Silverman et al. 2005; cir- cles: Hasinger et al. 2005). Cyan to infra-red surveys (circles: Brown et al. 2006; squares: Matute et al. 2006). For the optical surveys we are showing here, with green circles, the data from the COMBO-17 survey (Wolf et al., 2003), which is comparable in depth to our sample. these models can reproduce the observed flattening with redshift of the faint end of the luminosity function. 7. Model fitting In this Section we discuss the results of a number of different fits to our data as a function of luminosity and redshift. For this pur- pose, we computed the luminosity function in 5 redshift bins at 1.0 < z < 4.0 where the VVDS AGN sample consists of 121 objects. Since, in this redshift range, our data cover only the faint part of the luminosity function, these fits have been per- formed by combining our data with the LF data points from the SDSS data release 3 (DR3) (Richards et al., 2006b) in the red- shift range 0 < z < 4. The advantage of using the SDSS sample, rather than, for example, the 2dF sample, is that the former sam- ple, because of the way it is selected, probes the luminosity func- tion to much higher redshifts. The SDSS sample contains more than 15,000 spectroscopically confirmed AGN selected from an effective area of 1622 sq.deg. Its limiting magnitude (i < 19.1 for z < 3.0 and i < 20.2 for z > 3.0) is much brighter than the VVDS and because of this it does not sample well the AGN in the faint part of the luminosity function. For this reason, Richards et al. (2006b) fitted the SDSS data using only a single power law, which is meant to describe the luminosity function above the break luminosity. Adding the VVDS data, which instead mainly sample the faint end of the luminosity function, and analyzing the two samples together, allows us to cover the entire luminosity range in the common redshift range (1.0 < z < 4.0), also extend- ing the analysis at z < 1.0 where only SDSS data are available. The goodness of fit between the computed LF data points and the various models is then determined by the χ2 test. For all the analyzed models we have parameterized the lu- minosity function as a double power law that, expressed in lumi- nosity, is given by: Φ(L, z) = (L/L∗)−α + (L/L∗)−β whereΦ∗L is the number of AGN per Mpc 3, L∗ is the characteris- tic luminosity around which the slope of the luminosity function is changing and α and β are the two power law indices. Equation 5 can be expressed in absolute magnitude 2 as: Φ(M, z) = 100.4(α+1)(M−M∗) + 100.4(β+1)(M−M∗) 7.1. The PLE and PDE models The first model that we tested is a Pure Luminosity Evolution (PLE) with the dependence of the characteristic luminosity de- scribed by a 2nd-order polynomial in redshift: M∗(z) = M∗(0) − 2.5(k1z + k2z 2). (7) Following the finding by Richards et al. (2006b) for the SDSS sample, we have allowed a change (flattening with redshift) of the bright end slope according to a linear evolution in redshift: α(z) = α(0) + A z. The resulting best fit parameters are listed in the first line of Table 2 and the resulting model fit is shown as a green short dashed line in Figure 9. The bright end slope α derived by our fit (αVVDS=-3.19 at z=2.45) is consistent with the one found by Richards et al. (2006b) (αSDSS = -3.1). This model, as shown in Figure 9, while reproduces well the bright part of the LF in the entire redshift range, does not fit the faint part of the LF at low redshift (1.0 < z < 1.5). This appears to be due to the fact that, given the overall best fit normalization, the derived faint end slope (β =-1.38) is too shallow to reproduce the VVDS data in this redshift range. Richards et al. (2005), working on a combined 2dF-SDSS (2SLAQ) sample of AGN up to z = 2.1. found that, fixing all of the parameters except β and the normalization, to those of Croom et al. (2004), the resulting faint end slope is β = −1.45 ± 0.03. This value would describe better our faint LF at low redshift. This trend suggests a kind of combined luminosity and density evolution not taken into account by the used model. 2 Φ∗M = Φ ∣ln10−0.4 3 in their parameterization A1=-0.4(α + 1) =0.84 10 Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function For this reason, we attempted to fit the data also including a term of density evolution in the form of: M(z) = Φ M(0) · 10 k1Dz+k2Dz In this model the evolution of the LF is described by both a term of luminosity evolution, which affects M∗, and a term of density evolution, which allows for a change in the global nor- malization Φ∗. The derived best fit parameters of this model are listed in the second line of Table 2 and the model fit is shown as a blue long dashed line in Figure 9. This model gives a better χ2 with respect to the previous model, describing the entire sam- ple better than a simple PLE (the reduced χ2 decreases from ∼ 1.9 to ∼ 1.35). However, it still does not satisfactorily reproduce the excess of faint objects in the redshift bin 1.0 < z < 1.5 and, moreover, it underestimates the faint end of the LF in the last redshift bin (3.0 < z < 4.0). 7.2. The LDDE model Recently, a growing number of observations at different red- shifts, in soft and hard X-ray bands, have found evidences of a flattening of the faint end slope of the LF towards high redshift. This trend has been described through a luminosity- dependent density evolution parameterization. Such a param- eterization allows the redshift of the AGN density peak to change as a function of luminosity. This could help in explain- ing the excess of faint AGN found in the VVDS sample at 1.0 < z < 1.5. Therefore, we considered a luminosity depen- dent density evolution model (LDDE), as computed in the major X-surveys (Miyaji et al. 2000; Ueda et al. 2003; Hasinger et al. 2005). In particular, following Hasinger et al. (2005), we as- sumed an LDDE evolution of the form: Φ(MB, z) = Φ(M, 0) ∗ ed(z,MB) (9) where: ed(z,MB) = (1 + z)p1 (z ≤ zc) ed(zc)[(1 + z)/(1 + zc)] p2 (z > zc) . (10) along with zc(MB) = zc,010 −0.4γ(MB−Mc) (MB ≥ Mc) zc,0 (MB < Mc) . (11) where zc corresponds to the redshift at which the evolution changes. Note that zc is not constant but it depends on the lu- minosity. This dependence allows different evolutions at differ- ent luminosities and can indeed reproduce the differential AGN evolution as a function of luminosity, thus modifying the shape of the luminosity function as a function of redshift. We also con- sidered two different assumptions for p1 and p2: (i) both param- eters constant and (ii) both linearly depending on luminosity as follows: p1(MB) = p1Mref − 0.4ǫ1 (MB − Mref) (12) p2(MB) = p2Mref − 0.4ǫ2 (MB − Mref) (13) The corresponding χ2 values for the two above cases are re- spectively χ2=64.6 and χ2=56.8. Given the relatively small im- provement of the fit, we considered the addition of the two fur- ther parameters (ǫ1 and ǫ2) unnecessary. The model with con- stant p1 and p2 values is shown with a solid black line in Figure Figure 10. Evolution of comoving AGN space density with red- shift, for different luminosity range: -22.0< MB <-20.0; -24.0< MB <-22.0; -26.0< MB <-24.0 and MB <-26.0. Dashed lines correspond to the redshift range in which the model has been extrapolated. 9 and the best fit parameters derived for this model are reported in the last line of Table 2. This model reproduces well the overall shape of the luminos- ity function over the entire redshift range, including the excess of faint AGN at 1.0 < z < 1.5. The χ2 value for the LDDE model is in fact the best among all the analyzed models. We found in fact a χ2 of 64.6 for 67 degree of freedom and, as the reduced χ2 is below 1, it is acceptable 4. The best fit value of the faint end slope, which in this model corresponds to the slope at z = 0, is β =-2.0. This value is consis- tent with that derived by Hao et al. (2005) studying the emission line luminosity function of a sample of Seyfert galaxies at very low redshift (0 < z < 0.15), extracted from the SDSS. They in fact derived a slope β ranging from -2.07 to -2.03, depending on the line (Hα, [O ii] or [O iii]) used to compute the nuclear lumi- nosity. Moreover, also the normalizations are in good agreement, confirming our model also in a redshift range where data are not available and indeed leading us to have a good confidence on the extrapolation of the derived model. 8. The AGN activity as a function of redshift By integrating the luminosity function corresponding to our best fit model (i.e the LDDE model; see Table 2), we derived the co- moving AGN space density as a function of redshift for different luminosity ranges (Figure 10). The existence of a peak at z∼ 2 in the space density of bright AGN is known since a long time, even if rarely it has been possi- ble to precisely locate the position of this maximum within a sin- gle optical survey. Figure 10 shows that for our best fit model the peak of the AGN space density shifts significantly towards lower 4 We note that the reduced χ2 of our best fit model, which in- cludes also VVDS data, is significantly better than that obtained by Richards et al. (2006b) in fitting only the SDSS DR3 data. Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function 11 Figure 9. Filled circles correspond to our rest-frame B-band luminosity function data points, derived in the redshift bins 1.0 < z < 1.5, 1.5 < z < 2.0, 2.0 < z < 2.5, 2.5 < z < 3.0 and 3.0 < z < 4.0. Open circles are the data points from the SDSS Data Release 3 (DR3) by Richards et al. (2006b). These data are shown also in two redshift bins below z = 1. The red dot-dashed line corresponds to the model fit derived by Richards et al. (2006b) only for the SDSS data. The other lines correspond to model fits derived considering the combination of the VVDS and SDSS samples for different evolutionary models, as listed in Table 2 and described in Section 7. redshift going to lower luminosity. The position of the maximum moves from z∼ 2.0 for MB <-26.0 to z∼ 0.65 for -22< MB <-20. A similar trend has recently been found by the analysis of several deep X-ray selected samples (Cowie et al., 2003; Hasinger et al., 2005; La Franca et al., 2005). To compare with X-ray results, by applying the same bolometric corrections used is Section 6, we derived the volume densities derived by our best fit LDDE model in the same luminosity ranges as La Franca et al. (2005). We found that the volume density peaks at z ≃ [0.35; 0.7; 1.1; 1.5] respectively for LogLX(2−10kev) = [42–43; 43–44; 44–44.5; 44.5–45]. In the same luminosity intervals, the values for the redshift of the peak obtained by La Franca et al. (2005) are z ≃ [0.5; 0.8; 1.1; 1.5], in good agree- ment with our result. This trend has been interpreted as evidence of AGN (i.e. black hole) “cosmic downsizing”, similar to what has recently been observed in the galaxy spheroid population (Cimatti et al., 2006). The downsizing (Cowie et al., 1996) is a term which is used to describe the phenomenon whereby lumi- 12 Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function 1.0 < z < 1.5 1.5 < z < 2.0 ∆M Nqso LogΦ(B) ∆LogΦ(B) ∆M Nqso LogΦ(B) ∆LogΦ(B) -19.46 -20.46 3 -5.31 +0.20 -0.38 -20.46 -21.46 11 -4.89 +0.12 -0.16 -20.28 -21.28 4 -5.29 +0.18 -0.30 -21.46 -22.46 17 -5.04 +0.09 -0.12 -21.28 -22.28 7 -5.18 +0.15 -0.22 -22.46 -23.46 9 -5.32 +0.13 -0.18 -22.28 -23.28 7 -5.54 +0.14 -0.20 -23.46 -24.46 3 -5.78 +0.20 -0.38 -23.28 -24.28 10 -5.34 +0.12 -0.17 -25.46 -26.46 1 -6.16 +0.52 -0.76 -24.28 -25.28 2 -5.94 +0.23 -0.53 2.0 < z < 2.5 2.5 < z < 3.0 ∆M Nqso LogΦ(B) ∆LogΦ(B) ∆M Nqso LogΦ(B) ∆LogΦ(B) -20.90 -21.90 1 -5.65 +0.52 -0.76 -21.90 -22.90 3 -5.48 +0.20 -0.38 -21.55 -22.55 3 -5.45 +0.20 -0.38 -22.90 -23.90 4 -5.76 +0.18 -0.30 -22.55 -23.55 4 -5.58 +0.19 -0.34 -23.90 -24.90 4 -5.81 +0.18 -0.30 -23.55 -24.55 3 -5.90 +0.20 -0.38 -24.90 -25.90 2 -5.97 +0.23 -0.53 -24.55 -25.55 2 -6.11 +0.23 -0.53 -25.90 -26.90 2 -6.03 +0.23 -0.55 -25.55 -26.55 1 -6.26 +0.52 -0.76 3.0 < z < 4.0 ∆M Nqso LogΦ(B) ∆LogΦ(B) -21.89 -22.89 4 -5.52 +0.19 -0.34 -22.89 -23.89 3 -5.86 +0.20 -0.40 -23.89 -24.89 7 -5.83 +0.14 -0.21 -24.89 -25.89 3 -6.12 +0.20 -0.38 Table 1. Binned luminosity function estimate for Ωm=0.3, ΩΛ=0.7 and H0=70 km · s−1 · Mpc−1. We list the values of Log Φ and the corresponding 1σ errors in five redshift ranges, as plotted with full circles in Figure 9 and in ∆MB=1.0 magnitude bins. We also list the number of AGN contributing to the luminosity function estimate in each bin Sample - Evolution Model α β M∗ k1L k2L A k1D k2D Φ ∗ χ2 ν VVDS+SDSS - PLE α var -3.83 -1.38 -22.51 1.23 -0.26 0.26 - - 9.78E-7 130.36 69 VVDS+SDSS - PLE+PDE -3.49 -1.40 -23.40 0.68 -0.073 - -0.97 -0.31 2.15E-7 91.4 68 Sample - Evolution Model α β M∗ p1 p2 γ zc,0 Mc Φ ∗ χ2 ν VVDS+SDSS - LDDE -3.29 -2.0 -24.38 6.54 -1.37 0.21 2.08 -27.36 2.79E-8 64.6 67 Table 2. Best fit models derived from the χ2 analysis of the combined sample VVDS+SDSS-DR3 in the redshift range 0.0 < z < 4.0 assuming a flat (Ωm + ΩΛ = 1) universe with Ωm = 0.3. nous activity (star formation and accretion onto black holes) ap- pears to be occurring predominantly in progressively lower mass objects (galaxies or BHs) as the redshift decreases. As such, it explains why the number of bright sources peaks at higher red- shift than the number of faint sources. As already said, this effect had not been seen so far in the analysis of optically selected samples. This can be due to the fact that most of the optical samples, because of their limiting magnitudes, do not reach luminosities where the difference in the location of the peak becomes evident. The COMBO-17 sample (Wolf et al., 2003), for example, even if it covers enough redshift range (1.2 < z < 4.8) to enclose the peak of the AGN activity, does not probe luminosities faint enough to find a significant indication for a difference between the space density peaks of AGN of different luminosities (see, for example, Figure 11 in Wolf et al. (2003), which is analogous to our Figure 10, but in which only AGN brighter than M ∼ -24 are shown). The VVDS sample, being about one magnitude fainter than the COMBO- 17 sample and not having any bias in finding faint AGN, allows us to detect for the first time in an optically selected sample the shift of the maximum space density towards lower redshift for low luminosity AGN. 9. Summary and conclusion In the present paper we have used the new sample of AGN, col- lected by the VVDS and presented in Gavignaud et al. (2006), to derive the optical luminosity function of faint type–1 AGN. The sample consists of 130 broad line AGN (BLAGN) se- lected on the basis of only their spectral features, with no mor- phological and/or color selection biases. The absence of these biases is particularly important for this sample because the typ- ical non-thermal AGN continuum can be significantly masked by the emission of the host galaxy at the low intrinsic luminos- ity of the VVDS AGN. This makes the optical selection of the faint AGN candidates very difficult using the standard color and morphological criteria. Only spectroscopic surveys without any pre-selection can therefore be considered complete in this lumi- nosity range. Because of the absence of morphological and color selec- tion, our sample shows redder colors than those expected, for example, on the basis of the color track derived from the SDSS composite spectrum and the difference is stronger for the intrin- sically faintest objects. Thanks to the extended multi-wavelength coverage in the deep VVDS fields in which we have, in addition Bongiorno, A. et al.: The VVDS type–1 AGN sample: The faint end of the luminosity function 13 to the optical VVDS bands, also photometric data from GALEX, CFHTLS, UKIDSS and SWIRE, we examined the spectral en- ergy distribution of each object and we fitted it with a combina- tion of AGN and galaxy emission, allowing also for the possi- bility of extinction of the AGN flux. We found that both effects (presence of dust and contamination from the host galaxy) are likely to be responsible for this reddening, even if it is not pos- sible to exclude that faint AGN are intrinsically redder than the brighter ones. We derived the luminosity function in the B-band for 1 < z < 3.6, using the usual 1/Vmax estimator (Schmidt, 1968), which gives the space density contributions of individual ob- jects. Moreover, using the prescriptions recently derived by Hopkins et al. (2007), we computed also the bolometric lumi- nosity function for our sample. This allows us to compare our results also with other samples selected from different bands. Our data, more than one magnitude fainter than previous op- tical surveys, allow us to constrain the faint part of the luminosity function up to high redshift. A comparison of our data with the 2dF sample at low redshift (1 < z < 2.1) shows that the VVDS data can not be well fitted with the PLE models derived by pre- vious samples. Qualitatively, our data suggest the presence of an excess of faint objects at low redshift (1.0 < z < 1.5) with respect to these models. Recently, a growing number of observations at different red- shifts, in soft and hard X-ray bands, have found in fact evi- dences of a similar trend and they have been reproduced with a luminosity-dependent density evolution parameterization. Such a parameterization allows the redshift of the AGN density peak to change as a function of luminosity and explains the excess of faint AGN that we found at 1.0 < z < 1.5. Indeed, by com- bining our faint VVDS sample with the large sample of bright AGN extracted from the SDSS DR3 (Richards et al., 2006b), we found that the evolutionary model which better represents the combined luminosity functions, over a wide range of red- shift and luminosity, is an LDDE model, similar to those derived from the major X-surveys. The derived faint end slope at z=0 is β = -2.0, consistent with the value derived by Hao et al. (2005) studying the emission line luminosity function of a sample of Seyfert galaxies at very low redshift. A feature intrinsic to these LDDE models is that the comov- ing AGN space density shows a shift of the peak with luminos- ity, in the sense that more luminous AGN peak earlier in the history of the Universe (i.e. at higher redshift), while the density of low luminosity ones reaches its maximum later (i.e. at lower redshift). In particular, in our best fit LDDE model the peak of the space density ranges from z ∼ 2 for MB < -26 to z∼ 0.65 for -22 < MB < -20. This effect had not been seen so far in the analysis of optically selected samples, probably because most of the optical samples do not sample in a complete way the faintest luminosities, where the difference in the location of the peak be- comes evident. Although the results here presented appear to be already ro- bust, the larger AGN sample we will have at the end of the still on-going VVDS survey (> 300 AGN), will allow a better sta- tistical analysis and a better estimate of the parameters of the evolutionary model. Acknowledgements. This research has been developed within the framework of the VVDS consortium. This work has been partially supported by the CNRS-INSU and its Programme National de Cosmologie (France), and by Italian Ministry (MIUR) grants COFIN2000 (MM02037133) and COFIN2003 (num.2003020150). Based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, program 070.A-9007(A), 272.A-5047, 076.A-0808, and on data obtained at the Canada-France-Hawaii Telescope, operated by the CNRS of France, CNRC in Canada, and the University of Hawaii. The VLT- VIMOS observations have been carried out on guaranteed time (GTO) allo- cated by the European Southern Observatory (ESO) to the VIRMOS consortium, under a contractual agreement between the Centre National de la Recherche Scientifique of France, heading a consortium of French and Italian institutes, and ESO, to design, manufacture and test the VIMOS instrument. 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S. 1994, ApJ, 421, 412 Wolf, C., Wisotzki, L., Borch, A., et al. 2003, A&A, 408, 499 1 Università di Bologna, Dipartimento di Astronomia - Via Ranzani 1, I-40127, Bologna, Italy 2 INAF-Osservatorio Astronomico di Bologna - Via Ranzani 1, I- 40127, Bologna, Italy 3 Astrophysical Institute Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany 4 Integral Science Data Centre, ch. d’Écogia 16, CH-1290 Versoix 5 Geneva Observatory, ch. des Maillettes 51, CH-1290 Sauverny, Switzerland 6 Laboratoire d’Astrophysique de Toulouse/Tabres (UMR5572), CNRS, Université Paul Sabatier - Toulouse III, Observatoire Midi- Pyriénées, 14 av. E. Belin, F-31400 Toulouse, France 7 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ 8 IASF-INAF - via Bassini 15, I-20133, Milano, Italy 9 Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS- Université de Provence, BP8, 13376 Marseille Cedex 12, France 10 IRA-INAF - Via Gobetti,101, I-40129, Bologna, Italy 11 INAF-Osservatorio Astronomico di Roma - Via di Frascati 33, I- 00040, Monte Porzio Catone, Italy 12 Max Planck Institut fur Astrophysik, 85741, Garching, Germany 13 Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd Arago, 75014 Paris, France 14 School of Physics & Astronomy, University of Nottingham, University Park, Nottingham, NG72RD, UK 15 INAF-Osservatorio Astronomico di Brera - Via Brera 28, Milan, Italy 16 Institute for Astronomy, 2680 Woodlawn Dr., University of Hawaii, Honolulu, Hawaii, 96822 17 Observatoire de Paris, LERMA, 61 Avenue de l’Observatoire, 75014 Paris, France 18 Centre de Physique Théorique, UMR 6207 CNRS-Université de Provence, F-13288 Marseille France 19 Astronomical Observatory of the Jagiellonian University, ul Orla 171, 30-244 Kraków, Poland 20 INAF-Osservatorio Astronomico di Capodimonte - Via Moiariello 16, I-80131, Napoli, Italy 21 Institute de Astrofisica de Canarias, C/ Via Lactea s/n, E-38200 La Laguna, Spain 22 Center for Astrophysics & Space Sciences, University of California, San Diego, La Jolla, CA 92093-0424, USA 23 Centro de Astrofsica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal 24 Universitá di Milano-Bicocca, Dipartimento di Fisica - Piazza delle Scienze 3, I-20126 Milano, Italy 25 Università di Bologna, Dipartimento di Fisica - Via Irnerio 46, I-40126, Bologna, Italy Introduction The sample Colors of BLAGNs Luminosity function Definition of the redshift range Incompleteness function Estimate of the absolute magnitude The 1/Vmax estimator Comparison with the results from other optical surveys The low redshift luminosity function The high redshift luminosity function The bolometric luminosity function Model fitting The PLE and PDE models The LDDE model The AGN activity as a function of redshift Summary and conclusion

0704.1661

Can a Black Hole Collapse to a Space-time Singularity?

Noname manuscript No. (will be inserted by the editor) Can a Black Hole Collapse to a Space-time Singularity? R. K. Thakur Received: date / Accepted: date Abstract A critique of the singularity theorems of Penrose, Hawking, and Geroch is given. It is pointed out that a gravitationally collapsing black hole acts as an ultrahigh energy particle accelerator that can accelerate particles to energies inconceivable in any terrestrial particle accelerator, and that when the energy E of the particles comprising matter in a black hole is ∼ 102GeV or more, or equivalently, the temperature T is ∼ 1015K or more, the entire matter in the black hole is converted into quark-gluon plasma permeated by leptons. As quarks and leptons are fermions, it is emphasized that the collapse of a black-hole to a space-time singularity is inhibited by Pauli’s exclusion principle. It is also suggested that ultimately a black hole may end up either as a stable quark star, or as a pulsating quark star which may be a source of gravitational radiation, or it may simply explode with a mini bang of a sort. Keywords black hole · gravitational collapse · space-time singularity · quark star 1 Introduction When all the thermonuclear sources of energy of a star are exhausted, the core of the star begins to contract gravitationally because, practically, there is no radiation pressure to arrest the contraction, the pressure of matter being inadequate for this purpose. If the mass of the core is less than the Chandrasekhar limit (∼ 1.44M⊙), the contraction stops when the density of matter in the core, ρ > 2 × 106 g cm−3; at this stage the pressure of the relativistically degenerate electron gas in the core is enough to withstand the force of gravitation. When this happens, the core becomes a stable white dwarf. However, when the mass of the core is greater than the Chandrasekhar limit, the pressure of the relativistically degenerate electron gas is no longer sufficient to arrest the gravitational contraction, the core continues to contract and becomes R. K. Thakur Retired Professor of Physics, School of Studies in Physics, Pt. Ravishankar Shukla University, Raipur, India Tel.: +91-771-2255168 E-mail: [email protected] http://arxiv.org/abs/0704.1661v2 denser and denser; and when the density reaches the value ρ ∼ 107 g cm−3, the process of neutronization sets in; electrons and protons in the core begin to combine into neutrons through the reaction p+ e− → n+ νe The electron neutrinos νe so produced escape from the core of the star. The gravi- tational contraction continues and eventually, when the density of the core reaches the value ρ ∼ 1014 g cm−3, the core consists almost entirely of neutrons. If the mass of the core is less than the Oppenheimer-Volkoff limit (∼ 3M⊙), then at this stage the contraction stops; the pressure of the degenerate neutron gas is enough to withstand the gravitational force. When this happens, the core becomes a stable neutron star. Of course, enough electrons and protons must remain in the neutron star so that Pauli’s exclusion principle prevents neutron beta decay n→ p+ e− + νe Where νe is the electron antineutrino (Weinberg 1972a). This requirement sets a lower limit ∼ 0.2M⊙ on the mass of a stable neutron star. If, however, after the end of the thermonuclear evolution, the mass of the core of a star is greater than the Chandrasekhar and Oppenheimer-Volkoff limit, the star may eject enough matter so that the mass of the core drops below the Chandrasekhar and Oppenheimer-Volkoff limit as a result of which it may settle as a stable white dwarf or a stable neutron star. If not, the core will gravitationally collapse and end up as a black hole. As is well known, the event horizon of a black hole of mass M is a spherical surface located at a distance r = rg = 2GM/c 2 from the centre, where G is Newton’s gravitational constant and c the speed of light in vacuum; rg is called gravitational radius or Schwarzschild radius. An external observer cannot observe anything that is happening inside the event horizon, nothing, not even light or any other electromagnetic signal can escape outside the event horizon from inside. However, anything that enters the event horizon from outside is swallowed by the black hole; it can never escape outside the event horizon again. Attempts have been made, using the general theory of relativity (GTR), to under- stand what happens inside a black hole. In so doing, various simplifying assumptions have been made. In the simplest treatment (Oppenheimer and Snyder 1939; Weinberg 1972b) a black hole is considered to be a ball of dust with negligible pressure, uniform density ρ = ρ(t), and at rest at t = 0. These assumptions lead to the unique solution of the Einstein field equations, and in the comoving co-ordinates the metric inside the black hole is given by 2 −R2(t) 1− k r2 in units in which speed of light in vacuum, c=1, and where k is a constant. The require- ment of energy conservation implies that ρ(t)R3(t) remains constant. On normalizing the radial co-ordinate r so that R(0) = 1 (2) one gets ρ(t) = ρ(0)R (t) (3) The fluid is assumed to be at rest at t = 0, so Ṙ(0) = 0 (4) Consequently, the field equations give ρ(0) (5) Finally, the solution of the field equations is given by the parametric equations of a cycloid : ψ + sin ψ (1 + cos ψ) (6) From equation (6) it is obvious that when ψ = π. i.e., when t = ts = 8πGρ(0) a space-time singularity occurs; the scale factor R(t) vanishes. In other words, a black hole of uniform density having the initial values ρ(0), and zero pressure collapses from rest to a point in 3 - subspace, i.e., to a 3 - subspace of infinite curvature and zero proper volume, in a finite time ts; the collapsed state being a state of infinite proper energy density. The same result is obtained in the Newtonian collapse of a ball of dust under the same set of assumptions (Narlikar 1978). Although the black hole collapses completely to a point at a finite co-ordinate time t = ts, any electromagnetic signal coming to an observer on the earth from the surface of the collapsing star before it crosses its event horizon will be delayed by its gravitational field, so an observer on the earth will not see the star suddenly vanish. Actually, the collapse to the Schwarzschild radius rg appears to an outside observer to take an infinite time, and the collapse to R = 0 is not at all observable from outside the event horizon. The internal dynamics of a non-idealized, real black hole is very complex. Even in the case of a spherically symmetric collapsing black hole with non-zero pressure the details of the interior dynamics are not well understood, though major advances in the understanding of the interior dynamics are now being made by means of nu- merical computations and analytic analyses. But in these computations and analyses no new features have emerged beyond those that occur in the simple uniform-density, free-fall collapse considered above (Misner,Thorne, and Wheeler 1973). However, us- ing topological methods, Penrose (1965,1969), Hawking (1996a, 1966b, 1967a, 1967b), Hawking and Penrose (1970), and Geroch (1966, 1967, 1968) have proved a number of singularity theorems purporting that if an object contracts to dimensions smaller than rg, and if other reasonable conditions - namely, validity of the GTR, positivity of energy, ubiquity of matter and causality - are satisfied, its collapse to a singularity is inevitable. 2 A critique of the singularity theorems As mentioned above, the singularity theorems are based, inter alia, on the assump- tion that the GTR is universally valid. But the question is : Has the validity of the GTR been established experimentally in the case of strong fields ? Actually, the GTR has been experimentally verified only in the limiting case of week fields, it has not been experimentally validated in the case of strong fields. Moreover, it has been demonstrated that when curvatures exceed the critical value Cg = 1/L where Lg = h̄ G/c3 = 1.6 × 10−33 cm corresponding to the critical density ρg = 5 × 1093 g cm−3, the GTR is no longer valid; quantum effects must enter the picture (Zeldovich and Novikov 1971). Therefore, it is clear that the GTR breaks down before a gravitationally collapsing object collapses to a singularity. Consequently, the conclusion based on the GTR that in comoving co-ordinates any gravitationally col- lapsing object in general, and a black hole in particular, collapses to a point in 3-space need not be held sacrosanct, as a matter of fact it may not be correct at all. Furthermore, while arriving at the singularity theorems attention has mostly been focused on the space-time geometry and geometrodynamics; matter has been tacitly treated as a classical entity. However, as will be shown later, this is not justified; quantum mechanical behavior of matter at high energies and high densities must be taken into account. Even if we regard matter as a classical entity of a sort, it can be easily seen that the collapse of a black hole to a space-time singularity is inhibited by Pauli’s exclusion principle. As mentioned earlier, a collapsing black hole consists, almost entirely, of neutrons apart from traces of protons and electrons; and neutrons as well as protons and electrons are fermions; they obey Pauli’s exclusion principle. If a black hole collapses to a point in 3-space, all the neutrons in the black hole would be squeezed into just two quantum states available at that point, one for spin up and the other for spin down neutron. This would violate Pauli’s exclusion principle, according to which not more than one fermion of a given species can occupy any quantum state. So would be the case with the protons and the electrons in the black hole. Consequently, a black hole cannot collapse to a space-time singularity in contravention to Pauli’s exclusion principle. Besides, another valid question is : What happens to a black hole after t > ts, i.e., after it has collapsed to a point in 3-space to a state of infinite proper energy density, if at all such a collapse occurs? Will it remain frozen forever at that point? If yes, then uncertainties in the position co-ordinates of each of the particles - namely, neutrons, protons, and electrons - comprising the black hole would be zero. Consequently, accord- ing to Heisenberg’s uncertainty principle, uncertainties in the momentum co-ordinates of each of the particles would be infinite. However, it is physically inconceivable how particles of infinite momentum and energy would remain frozen forever at a point. From this consideration also collapse of a black hole to a singularity appears to be quite unlikely. Earlier, it was suggested by the author that the very strong ’hard-core’ repulsive interaction between nucleons, which has a range lc ∼ 0.4× 10−13 cm, might set a limit on the gravitational collapse of a black hole and avert its collapse to a singularity (Thakur 1983). The existence of this hard-core interaction was pointed out by Jastro (1951) after the analysis of the data from high energy nucleon-nucleon scattering ex- periments. It has been shown that this very strong short range repulsive interaction arises due to the exchange of isoscalar vector mesons ω and φ between two nucleons ( Scotti and Wong 1965). Phenomenologically, that part of the nucleon-nucleon potential which corresponds to the repulsive hard core interaction may be taken as Vc(r) = ∞ for r < lc (8) where r is the distance between the two interacting nucleons. Taking this into account, the author concluded that no spherical object of mass M could collapse to a sphere of radius smaller than Rmin = 1.68 × 10−6M1/3 cm, or of the density greater than ρmax = 5.0× 1016 g cm−3. It was also pointed out that an object of mass smaller than Mc ∼ 1.21 × 1033 gm could not cross the event horizon and become a black hole; the only course left to an object of mass smaller than Mc was to reach equilibrium as either a white dwarf or a neutron star. However, one may not regard these conclusions as reliable because they are based on the hard core repulsive interaction (8) between nucleons which has been arrived at phenomenologically by high energy nuclear physi- cists while accounting for the high energy nucleon-nucleon scattering data; but it must be noted that, as mentioned above, the existence of the hard core interaction has been demonstrated theoretically also by Scotti and Wong in 1965. Moreover, it is interesting to note that the upper limitMc ∼ 1.21×1033 g = 0.69M⊙ on the masses of objects that cannot gravitationally collapse to form black holes is of the same order of magnitude as the Chandrasekhar and the Oppenheimer- Volkoff limits. Even if we disregard the role of the hard core, short range repulsive interaction in arresting the collapse of a black hole to a space-time singularity in comoving co- ordinates, it must be noted that unlike leptons which appear to be point-like particles - the experimental upper bound on their radii being 10−16 cm (Barber et al. 1979) -nucleons have finite dimensions. It has been experimentally demonstrated that the radius r0 of the proton is about 10 −13 cm(Hofstadter & McAllister 1955). Therefore, it is natural to assume that the radius r0 of the neutron is also about 10 −13 cm. This means the minimum volume vmin occupied by a neutron is 3. Ignoring the “mass defect” arising from the release of energy during the gravitational contraction (before crossing the event horizon), the number of neutrons N in a collapsing black hole of mass M is, obviously, Mmn where mn is the mass of the neutron. Assuming that neutrons are impregnable particles, the minimum volume that the black hole can occupy is Vmin = Nvmin = vmin , for neutrons cannot be more closely packed than this in a black hole. However, Vmin = where Rmin is the radius of the minimum volume to which the black hole can collapse. Consequently, Rmin = . On substituting 10−13 cm for r0 and 1.67× 10−24 g for mn one finds that Rmin = 8.40 × 10−6M1/3. This means a collapsing black hole cannot collapse to a density greater than ρmax = = Nmn 4/3πr3 = 3.99× 1014 g cm−3. The critical mass Mc of the object for which the gravitational radius Rg = Rmin is obtained from the equation This gives Mc = 1.35× 1034 g = 8.68M⊙ (10) Obviously, for M > Mc, Rg > Rmin, and for M < Mc, Rg < Rmin. Consequently, objects of mass M < Mc cannot cross the event horizon and become a black hole whereas those of mass M > Mc can. Objects of mass M < Mc will, depending on their mass, reach equilibrium as either white dwarfs or neutron stars. Of course, these conclusions are based on the assumption that neutrons are impregnable particles and have radius r0 = 10 −13cm each. Also implicit is the assumption that neutrons are fundamental particles; they are not composite particles made up of other smaller constituents. But this assumption is not correct; neutrons as well as protons and other hadrons are not fundamental particles; they are made up of smaller constituents called quarks as will be explained in section 4. In section 5 it will be shown how, at ultrahigh energy and ultrahigh density, the entire matter in a collapsing black hole is eventually converted into quark-gluon plasma permeated by leptons. 3 Gravitationally collapsing black hole as a particle accelerator We consider a gravitationally collapsing black hole. On neglecting mutual interactions the energy E of any one of the particles comprising the black hole is given by E2 = p2 + m2 > p2, in units in which the speed of light in vacuum c = 1, where p is the magnitude of the 3-momentum of the particle and m its rest mass. But p = h , where λ is the de Broglie wavelength of the particle and h Planck’s constant of action. Since all lengths in the collapsing black hole scale down in proportion to the scale factor R(t) in equation (1), it is obvious that λ ∝ R(t). Therefore it follows that p ∝ R−1(t), and hence p = aR−1(t), where a is the constant of proportionality. From this it follows that E > a/R. Consequently, E as well as p increases continually as R decreases. It is also obvious that E and p, the magnitude of the 3-momentum, → ∞ as R → 0. Thus, in effect, we have an ultra-high energy particle accelerator, so far inconceivable in any terrestrial laboratory, in the form of a collapsing black hole, which can, in the absence of any physical process inhibiting the collapse, accelerate particles to an arbitrarily high energy and momentum without any limit. What has been concluded above can also be demonstrated alternatively, without resorting to GTR, as follows. As an object collapses under its selfgravitation, the in- terparticle distance s between any pair of particles in the object decreases. Obviously, the de Broglie’s wavelength λ of any particle in the object is less than or equal to s, a simple consequence of Heisenberg’s uncertainty principle. Therefore, s ≥ h/p, where h is Planck’s constant and p the magnitude of 3-momentum of the particle. Consequently, p ≥ h/s and hence E ≥ h/s. Since during the collapse of the object s decreases, the energy E as well as the momentum p of each of the particles in the object increases. Moreover, from E ≥ h/s and p ≥ h/s it follows that E and p → ∞ as s → 0. Thus, any gravitationally collapsing object in general, and a black hole in particular, acts as an ultrahigh energy particle accelerator. It is also obvious that ρ, the density of matter in the black hole, increases as it collapses. In fact, ρ ∝ R−3, and hence ρ→ ∞ as R → 0. 4 Quarks: The building blocks of matter In order to understand eventually what happens to matter in a collapsing black hole one has to take into account the microscopic behavior of matter at high energies and high densities; one has to consider the role played by the electromagnetic, weak, and strong interactions - apart from the gravitational interaction - between the particles compris- ing the matter. For a brief account of this the reader is referred to Thakur(1995), for greater detail to Huang(1992), or at a more elementary level to Hughes(1991). As has been mentioned in Section 2, unlike leptons, hadrons are not point-like parti- cles, but are of finite size; they have structures which have been revealed in experiments that probe hadronic structures by means of electromagnetic and weak interactions. The discovery of a very large number of apparently elementary (fundamental) hadrons led to the search for a pattern amongst them with a view to understanding their nature. This resulted in attempts to group together hadrons having the same baryon number, spin, and parity but different strangeness S ( or equivalently hypercharge Y = B + S, where B is the baryon number) into I-spin (isospin) multiplets. In a plot of Y against I3 (z- component of isospin I), members of I-spin multiplets are represented by points. The existence of several such hadron (baryon and meson) multiplets is a manifestation of underlying internal symmetries. In 1961 Gell-Mann, and independently Neémann, pointed out that each of these multiplets can be looked upon as the realization of an irreducible representation of an internal symmetry group SU(3) ( Gell-Mann and Neémann 1964). This fact together with the fact that hadrons have finite size and inner structure led Gell-Mann, and independently Zweig, in 1964 to hypothesize that hadrons are not elementary particles, rather they are composed of more elementary constituents called quarks (q) by Gell- Mann (Zweig called them aces). Baryons are composed of three quarks (q q q) and antibaryons of three antiquarks (q q q) while mesons are composed of a quark and an antiquark each. In the beginning, to account for the multiplets of baryons and mesons, quarks of only three flavours, namely, u(up), d (down), and s(strange) were postulated, and they together formed the basic triplet  of the internal symmetry group SU(3). All these three quarks u, d, and s have spin 1/2 and baryon number 1/3. The u quark has charge 2/3 e whereas the d and s quarks have charge −1/3 e where e is the charge of the proton. The strangeness quantum number of the u and d quarks is zero whereas that of the s quark is -1. The antiquarks (u , d , s) have charges −2/3 e, 1/3 e, 1/3 e and strangeness quantum numbers 0, 0, 1 respectively. They all have spin 1/2 and baryon number -1/3. Both u and d quarks have the same mass, namely, one third that of the nucleon, i.e., ≃ 310MeV/c2 whereas the mass of the s quark is ≃ 500MeV/c2. The proton is composed of two up and one down quarks (p: uud) and the neutron of one up and two down quarks (n: udd). Motivated by certain theoretical considerations Glashow, Iliopoulos and Maiani (1970) proposed that, in addition to u, d, s quarks, there should be another quark flavour which they named charm (c). Gaillard and Lee (1974) estimated its mass to be ≃ 1.5GeV/c2. In 1974 two teams, one led by S.C.C. Ting at SLAC (Aubert et al. 1974) and another led by B. Richter at Brookhaven (Augustin et al. 1974) independently discovered the J/Ψ , a particle remarkable in that its mass (3.1GeV/c2) is more than three times that of the proton. Since then, four more particles of the same family, namely, ψ(3684), ψ(3950), ψ(4150), ψ(4400) have been found. It is now established that these particles are bound states of charmonium (cc), J/ψ being the ground state. On adopting non-relativistic independent quark model with a linear potential between c and c, and taking the mass of c to be approximately half the mass of J/ψ, i. e. , 1.5GeV/c2, one can account for the J/ψ family of particles. The c has spin 1/2, charge 2/3 e, baryon number 1/3, strangeness −1, and a new quantum number charm (c) equal to 1. The u, d, s quarks have c = 0. It may be pointed out here that charmed mesons and baryons, i. e. , the bound states like (cd), and (cdu) have also been found. Thus the existence of the c quark has been established experimentally beyond any shade of doubt. The discovery of the c quark stimulated the search for more new quarks. An ad- ditional motivation for such a search was provided by the fact that there are three generations of lepton weak doublets: , and where νe, νµ, and ντ are elec- tron (e), muon (µ), and tau lepton (τ ) neutrinos respectively. Hence, by analogy, one expects that there should be three generations of quark weak doublets also: . It may be mentioned here that weak interaction does not distinguish between the upper and the lower members of each of these doublets. In analogy with the isopin 1/2 of the strong doublet , the weak doublets are regarded as possessing weak isopin IW = 1/2, the third component (IW )3 of this weak isopin being + 1/2 for the upper components of these doublets and - 1/2 for the lower components. These statements ap- ply to the left-handed quarks and leptons, i. e. , those with negative helicity (i. e. , with the spin antiparallel to the momentum) only. The right-handed leptons and quarks, i. e. , those with positive helicity (i. e. , with the spin parallel to the momentum), are weak singlets having weak isopin zero. The discovery, at Fermi Laboratory, of a new family of vector mesons, the upsilon family, starting at a mass of 9.4GeV/c2 gave an evidence for a new quark flavour called bottom or beauty (b) (Herb et al. 1997; Innes et al. 1977). These vector mesons are in fact, bound states of bottomonium (bb). These states have since been studied in detail at the Cornell electron accelerator in an electron-positron storage ring of energy ideally matched to this mass range. Four such states with masses 9.46, 10.02, 10.35, and 10.58 GeV/c2 have been found, the state with mass 9.46GeV/c2 being the ground state (Andrews et al. 1980). This implies that the mass of the b quark is ≃ 4.73GeV/c2. The b quark has spin 1/2 and charge −1/3 e. Furthermore, the b flavoured mesons have been found with exactly the expected properties (Beherend et al. 1983). After the discovery of the b quark, the confidence in the existence of the sixth quark flavour called top or truth (t) increased and it became almost certain that, like leptons, the quarks also occur in three generations of weak isopin doublets, namely, . In view of this, intensive search was made for the t quark. But the discovery of the t quark eluded for eighteen years. However, eventually in 1995, two groups, the CDF (Collider Detector at Fermi lab) Collaboration (Abe et al. 1995) and the Dφ Collaboration (Abachi et al. 1995) succeeded in detecting toponium tt in very high energy pp collisions at Fermi Laboratory’s 1.8TeV Tevetron collider. The toponium tt is the bound state of t and t. The mass of t has been estimated to be 176.0±2.0GeV/c2 , and thus it is the most massive elementary particle known so far. The t quark has spin 1/2 and charge 2/3 e. Moreover, in order to account for the apparent breaking of the spin-statistics the- orem in certain members of the Jp = 3 decuplet (spin 3/2,parity even), e. g. , △++ (uuu), and Ω− (sss), Greenberg (1964) postulated that quark of each flavour comes in three colours, namely, red, green, and blue, and that real particles are always colour singlets. This implies that real particles must contain quarks of all the three colours or colour-anticolour combinations such that they are overall white or colourless. White or colourless means all the three primary colours are equally mixed or there should be a combination of a quark of a given colour and an antiquark of the corresponding anticolour. This means each baryon contains quarks of all the three colours(but not necessarily of the same flavour) whereas a meson contains a quark of a given colour and an antiquark having the corresponding anticolour so that each combination is overall white. Leptons have no colour. Of course, in this context the word ‘colour’ has noth- ing to do with the actual visual colour, it is just a quantum number specifying a new internal degree of freedom of a quark. The concept of colour plays a fundamental role in accounting for the interaction between quarks. The remarkable success of quantum electrodynamics (QED) in ex- plaining the interaction between electric charges to an extremely high degree of preci- sion motivated physicists to explore a similar theory for strong interaction. The result is quantum chromodynamics (QCD), a non-Abelian gauge theory (Yang-Mills theory), which closely parallels QED. Drawing analogy from electrodynamics, Nambu (1966) postulated that the three quark colours are the charges (the Yang-Mills charges) re- sponsible for the force between quarks just as electric charges are responsible for the electromagnetic force between charged particles. The analogue of the rule that like charges repel and unlike charges attract each other is the rule that like colours repel, and colour and anticolour attract each other. Apart from this, there is another rule in QCD which states that different colours attract if the quantum state is antisymmetric, and repel if it is symmetric under exchange of quarks. An important consequence of this is that if we take three possible pairs, red-green. green-blue, and blue-red, then a third quark is attracted only if its colour is different and if the quantum state of the resulting combination is antisymmetric under the exchange of a pair of quarks thus resulting in red-green-blue baryons. Another consequence of this rule is that a fourth quark is repelled by one quark of the same colour and attracted by two of different colours in a baryon but only in antisymmetric combinations. This introduces a factor of 1/2 in the attractive component and as such the overall force is zero, i.e., the fourth quark is neither attracted nor repelled by a combination of red-green-blue quarks. In spite of the fact that hadrons are overall colourless, they feel a residual strong force due to their coloured constituents. It was soon realized that if the three colours are to serve as the Yang-Mills charges, each quark flavour must transform as a triplet of SUc(3) that causes transitions between quarks of the same flavour but of different colours ( the SU(3) mentioned earlier causes transitions between quarks of different flavours and hence may more appropriately be denoted by SUf (3)). However, the SUc(3) Yang-Mills theory requires the introduction of eight new spin 1 gauge bosons called gluons. Moreover, it is reasonable to stipulate that the gluons couple to left-handed and right-handed quarks in the same manner since the strong interactions do not violate the law of conservation of parity. Just as the force between electric charges arise due to the exchange of a photon, a massless vector (spin 1) boson, the force between coloured quarks arises due to the exchange of a gluon. Gluons are also massless vector (spin 1) bosons. A quark may change its colour by emitting a gluon. For example, a red quark qR may change to a blue quark qB by emitting a gluon which may be thought to have taken away the red (R) colour from the quark and given it the blue (B) colour, or, equivalently, the gluon may be thought to have taken away the red (R) and the antiblue (B) colours from the quark. Consequently, the gluon GRB emitted in the process qR → qB may be regarded as the composite having the colour R B so that the emitted gluon GRB = qRqB . In general, when a quark qi of colour i changes to a quark qj of colour j by emitting a gluon Gij , then Gij is the composite state of qi and qj , i.e., Gij = qiqj . Since there are three colours and threeanticolours, there are 3×3 = 9 possible combinations (gluons)of the form Gij = qiqj . However, one of the nine combinations is a special combination corresponding to the white colour, namely, GW = qRqR = qGqG = qBqB . But there is no interaction between a coloured object and a white (colourless) object. Consequently, gluon GW may be thought not to exist. This leads to the conclusion that only 9−1 = 8 kinds of gluons exist. This is a heuristic explanation of the fact that SUc(3) Yang-Mills gauge theory requires the existence of eight gauge bosons, i.e., the gluons. Moreover, as the gluons themselves carry colour, gluons may also emit gluons. Another important consequence of gluons possessing colour is that several gluons may come together and form gluonium or glue balls. Glueballs have integral spin and no colour and as such they belong to the meson family. Though the actual existence of quarks has been indirectly confirmed by experiments that probe hardronic structure by means of electromagnetic and weak interactions, and by the production of various quarkonia (qq) in high energy collisions made possible by various particle accelerators, no free quark has been detected in experiments at these accelerators so far. This fact has been attributed to the infrared slavery of quarks, i.e., to the nature of the interaction between quarks responsible for their confinement inside hadrons. Perhaps enormous amount of energy , much more than what is available in the existing terrestrial accelerators, is required to liberate the quarks from confinement. This means the force of attraction between quarks increases with increase in their separation. This is reminiscent of the force between two bodies connected by an elastic string. On the contrary, the results of deep inelastic scattering experiments reveal an al- together different feature of the interaction between quarks. If one examines quarks at very short distances (< 10−13 cm ) by observing the scattering of a nonhadronic probe, e.g., an electron or a neutrino, one finds that quarks move almost freely inside baryons and mesons as though they are not bound at all. This phenomenon is called the asymp- totic freedom of quarks. In fact Gross and Wilczek (1973 a,b) and Politzer (1973) have shown that the running coupling constant of interaction between two quarks vanishes in the limit of infinite momentum (or equivalently in the limit of zero separation). 5 Eventually what happens to matter in a collapsing black hole? As mentioned in Section 3 the energy E of the particles comprising the matter in a collapsing black hole continually increases and so does the density ρ of the matter whereas the separation s between any pair of particles decreases. During the continual collapse of the black hole a stage will be reached when E and ρ will be so large and s so small that the quarks confined in the hadrons will be liberated from the infrared slavery and will enjoy asymptotic freedom, i.e., the quark deconfinement will occur. In fact, it has been shown that when the energy E of the particle ∼ 102 GeV (s ∼ 10−16 cm) corresponding to a temperature T ∼ 1015K all interactions are of the Yang-Mills type with SUc(3)×SUIW (2)×UYW (1) gauge symmetry, where c stands for colour, IW for weak isospin, and YW for weak hypercharge, and at this stage quark deconfinement occurs as a result of which matter now consists of its fundamental constituents : spin 1/2 leptons, namely, the electrons, the muons, the tau leptons, and their neutrinos, which interact only through the electroweak interaction(i.e., the unified electromagnetic and weak interactions); and the spin 1/2 quarks, u, d, s, c, b, t, which interact eletroweakly as well as through the colour force generated by gluons(Ramond, 1983). In other words, when E ≥ 102 GeV (s ≤ 10−16 cm) corresponding to T ≥ 1015K, the entire matter in the collapsing black hole will be in the form of qurak-gluon plasma permeated by leptons as suggested by the author earlier (Thakur 1993). Incidentally, it may be mentioned that efforts are being made to create quark-gluon plasma in terrestrial laboratories. A report released by CERN, the European Organi- zation for Nuclear Research, at Geneva, on February 10, 2000, said that by smashing together lead ions at CERN’s accelerator at temperatures 100,000 times as hot as the Sun’s centre, i.e., at T ∼ 1.5 × 1012K, and energy densities never before reached in laboratory experiments, a team of 350 scientists from institutes in 20 countries suc- ceeded in isolating tiny components called quarks from more complex particles such as protons and neutrons. “A series of experiments using CERN’s lead beam have pre- sented compelling evidence for the existence of a new state of matter 20 times denser than nuclear matter, in which quarks instead of being bound up into more complex particles such as protons and neutrons, are liberated to roam freely ” the report said. However, the evidence of the creation of quark gluon plasma at CERN is indirect, involving detection of particles produced when the quark-gluon plasma changes back to hadrons. The production of these particles can be explained alternatively without having to have quark-gluon plasma. Therefore, Ulrich Heinz at CERN is of the opinion that the evidence of the creation of quark-gluon plasma at CERN is not enough and conclusive. In view of this, CERN will start a new experiment, ALICE, soon (around 2007-2008) at its Large Hadron Collider (LHC) in order to definitively and conclusively creat QGP. In the meantime the focus of research on quark-gluon plasma has shifted to the Relativistic Heavy Ion Collider (RHIC), the worlds newest and largest particle accel- erator for nuclear research, at Brookhaven National Laboratory in Upton, New York. RHIC’s goal is to create and study quark-gluon plasma. RHIC’s aim is to create quark- gluon plasma by head-on collisions of two beams of gold ions at energies 10 times those of CERN’s programme, which ought to produce a quark-gluon plasma with higher temperature and longer lifetime thereby allowing much clearer and direct observation. RHIC’s quark-gluon plasma is expected to be well above the transition temperature for transition between the ordinary hadronic matter phase and the quark-gluon plasma phase. This will enable scientists to perform numerous advanced experiments in order to study the properties of the plasma. The programme at RHIC began in the summer of 2000 and after two years Thomas Kirk, Brookhaven’s Associate Laboratory Director for High Energy Nuclear Physics, remarked, “It is too early to say that we have dis- covered the quark-gulon plasma, but not too early to mark the tantalizing hints of its existence.” Other definitive evidence of quark-gluon plasma will come from experimen- tal comparisons of the behavior in hot, dense nuclear matter with that in cold nuclear matter. In order to accomplish this, the next round of experimental measurements at RHIC will involve collisions between heavy ions and light ions, namely, between gold nuclei and deuterons. Later, on June 18, 2003 a special scientific colloquium was held at Brcokhaven Natioal Laboratory (BNL) to discuss the latest findings at RHIC. At the colloquium, it was announced that in the detector system known as STAR ( Solenoidal Tracker AT RHIC ) head-on collision between two beams of gold nuclei of energies of 130 GeV per nuclei resulted in the phenomenon called “jet quenching“. STAR as well as three other experiments at RHIC viz., PHENIX, BRAHMS, and PHOBOS, detected suppression of “leading particles“, highly energetic individual particles that emerge from nuclear fireballs, in gold-gold collisions. Jet quenching and leading particle suppression are signs of QGP formation. The findings of the STAR experiment were presented at the BNL colloquium by Berkeley Laboratory’s NSD ( Nuclear Science Division ) physicist Peter Jacobs. 6 Collapse of a black hole to a space-time singularity is inhibited by Pauli’s exclusion principle As quarks and leptons in the quark-gluon plasma permeated by leptons into which the entire matter in a collapsing black hole is eventually converted are fermions, the collapse of a black hole to a space-time singularity in a finite time in a comoving co- ordinate system, as stipulated by the singularity theorems of Penrose, Hawking and Geroch, is inhibited by Pauli’s exclusion principle. For, if a black hole collapses to a point in 3-space, all the quarks of a given flavour and colour would be squeezed into just two quantum states available at that point, one for spin up and the other for spin down quark of that flavour and colour. This would violate Pauli’s exclusion principle according to which not more than one fermion of a given species can occupy any quantum state. So would be the case with quarks of each distinct combination of colour and flavour as well as with leptons of each species, namely, e, µ, τ, νe, νµ and ντ . Consequently, a black hole cannot collapse to a space-time singularity in contravention to Pauli’s exclusion principle. Then the question arises : If a black hole does not collapse to a space-time singularity, what is its ultimate fate? In section 7 three possibilities have been suggested. 7 Ultimately how does a black hole end up? The pressure P inside a black hole is given by P = Pr + Pij + P ij + P k (11) where Pr is the radiation pressure, Pij the pressure of the relativistically degenerate quarks of the ith flavour and jth colour, Pk the pressure of the relativistically degenerate leptons of the kth species, P ij the pressure of relativistically degenerate antiquarks of the ith flavour and jth colour, Pk that of the relativistically degenerate antileptons of the kth species. In equation (11) the summations over i and j extend over all the six flavours and the three colours of quarks, and that over k extend over all the six species of leptons. However, calculation of these pressures are prohibitively difficult for several reasons. For example, the standard methods of statistical mechanics for calculation of pressure and equation of state are applicable when the system is in thermodynamics equilibrium and when its volume is very large, so large that for practical purpose we may treat it as infinite. Obviously, in a gravitationally collapsing black hole, the photon, quark and lepton gases cannot be in thermodynamic equilibrium nor can their volume be treated as infinite. Moreover, at ultrahigh energies and densities, because of the SUIW (2) gauge symmetry, transitions between the upper and lower components of quark and lepton doublets occur very frequently. In addition to this, because of the SUf (3) and SUc(3) gauge symmetries transitions between quarks of different flavours and colours also occur. Furthermore, pair production and pair annihilation of quarks and leptons create additional complications. Apart from these, various other nuclear reactions may as well occur. Consequently, it is practically impossible to determine the number density and hence the contribution to the overall pressure P inside the black hole by any species of elementary particle in a collapsing black hole when E ≥ 102 Gev (s ≤ 10−16 cm), or equivalently, T ≥ 1015K. However, it may not be unreasonable to assume that, during the gravitational collapse, the pressure P inside a black hole increases monotonically with the increase in the density of matter ρ. Actually, it might be given by the polytrope, P = kρ (n+1) n , where K is a constant and n is polytropic index. Consequently, P → ∞ as ρ → ∞, i.e., P → ∞ as the scale factor R(t) → 0 (or equivalently s→ 0). In view of this, there are three possible ways in which a black hole may end up. 1. During the gravitational collapse of a black hole, at a certain stage, the pressure P may be enough to withstand the gravitational force and the object may become gravitationally stable. Since at this stage the object consists entirely of quark-gluon plasma permeated by leptons, it means it would end up as a stable quark star. Indeed, such a possibility seems to exist. Recently, two teams - one led by David Helfand of Columbia University, NewYork (Slane, Helfand, and Murray 2002) and another led by Jeremy Drake of Harvard-Smithsonian Centre for Astrophysics, Cambridge, Mass. USA (Drake et al. 2002) studied independently two objects, 3C58 in Cassiopeia, and RXJ1856.5-3754 in Corona Australis respectively by combining data from the NASA’s Chandra X-ray Observatory and the Hubble Space Telescope, that seemed, at first, to be neutron stars, but, on closer look, each of these objects showed evidence of being an even smaller and denser object, possibly a quark star. 2. Since the collapse of a black hole is inhibited by Pauli’s exclusion principle, it can collapse only upto a certain minimum radius, say, rmin. After this, because of the tremendous amount of kinetic energy, it would bounce back and expand, but only upto the event horizon, i.e., upto the gravitational (Schwarzschild ) radius rg since, according to the GTR, it cannot cross the event horizon. Thereafter it would collapse again upto the radius rmin and then bounce back upto the radius rg . This process of collapse upto the radius rmin and bounce upto the radius rg would occur repeatedly. In other words, the black hole would continually pulsate radially between the radii rmin and rg and thus become a pulsating quark star. However, this pulsation would cause periodic variations in the gravitational field outside the event horizon and thus produce gravitational waves which would propagate radially outwards in all directions from just outside the event horizon. In this way the pulsating quark star would act as a source of gravitational waves. The pulsation may take a very long time to damp out since the energy of the quark star (black hole) cannot escape outside the event horizon except via the gravitational radiation produced outside the event horizon. However, gluons in the quark-gluon plasma may also act as a damping agent. In the absence of damping, which is quite unlikely, the black hole would end up as a perpetually pulsating quark star. 3. The third possibility is that eventually a black hole may explode; amini bang of a sort may occur, and it may, after the explosion, expand beyond the event horizon though it has been emphasized by Zeldovich and Novikov (1971) that after a collapsing sphere’s radius decreases to r < rg in a finite proper time, its expansion into the external space from which the contraction originated is impossible, even if the passage of matter through infinite density is assumed. Notwithstanding Zeldovich and Novikov’s contention based on the very concept of event horizon, a gravitationally collapsing black hole may also explode by the very same mechanism by which the big bang occurred, if indeed it did occur. This can be seen as follows. At the present epoch the volume of the universe is ∼ 1.5 × 1085 cm3 and the density of the galactic material throughout the universe is ∼ 2×10−31 g cm−3 (Allen 1973). Hence, a conservative estimate of the mass of the universe is ∼ 1.5 × 1085 × 2× 10−31 g = 3× 1054 g. However, according to the big bang model, before the big bang, the entire matter in the universe was contained in an ylem which occupied very very small volume. The gravitational radius of the ylem of mass 3 × 1054g was 4.45× 1021 km (it must have been larger if the actual mass of the universe were taken into account which is greater than 3× 1054 g). Obviously, the radius of the ylem was many orders of magnitude smaller than its gravitational radius, and yet the ylem exploded with a big bang, and in due course of time crossed the event horizon and expanded beyond it upto the present Hubble distance c/H0 ∼ 1.5 × 1023 km where c is the speed of light in vacuum and H0 the Hubble constant at the present epoch. Consequently, if the ylem could explode in spite of Zeldovich and Novikov’s contention, a gravitationally collapsing black hole can also explode, and in due course of time expand beyond the event horizon. The origin of the big bang, i.e., the mechanism by which the ylem exploded, is not definitively known. However, the author has, earlier proposed a viable mechanism (Thakur 1992) based on supersymmetry/supergravity. But supersymmetry/supergravity have not yet been validated experimentally. 8 Conclusion From the foregoing three inferences may be drawn. One, eventually the entire matter in a collapsing black hole is converted into quark-gluon plasma permeated by leptons. Two, the collapse of a black hole to a space - time singularity is inhibited by Pauli’s exclusion principle. Three, ultimately a black hole may end up in one of the three possible ways suggested in section 7. Acknowledgements The author thanks Professor S. K. Pandey, Co-ordinator, IUCAA Ref- erence Centre, School of Studies in Physics, Pt. Ravishankar Shukla University, Raipur, for making available the facilities of the Centre. He also thanks Sudhanshu Barway, Mousumi Das for typing the manuscript. References 1. Abachi S., et al. , 1995, PRL, 74, 2632 2. Abe F., et al. , 1995, PRL, 74, 2626 3. Allen C. W., 1993, Astrophysical Quantities, The Athlone Press, University of London, 293 4. Andrew D., et al. , 1980, PRL, 44, 1108 5. Aubert J. J., et al. , 1974, PRL, 33, 1404 6. Augustin J. E., et al. , 1974, PRL, 33, 1406 7. Barber D.P.,et al. ,1979, PRL, 43, 1915 8. Beherend S., et al. , 1983, PRL, 50, 881 9. Drake J. et al. , 2002, ApJ, 572, 996 10. Gaillard M. K., Lee B. W., 1974, PRD, 10, 897 11. Gell-Mann M., Néeman Y., 1964, The Eightfold Way, W. A. Benjamin, NewYork 12. Geroch R. P., 1966, PRL, 17, 445 13. Geroch R. P., 1967, Singularities in Spacetime of General Relativity : Their Defination, Existence and Local Characterization, Ph.D. Thesis, Princeton University 14. Geroch, R. P., 1968, Ann. Phys., 48, 526 15. Galshow S. L., Iliopoulos J., Maiani L., 1970, PRD, 2,1285 16. Greenberg O. W., 1964, PRL, 13, 598 17. Gross D. J., Wilczek F., 1973a, PRL, 30, 1343 18. Gros, D. J., Wilczek F., 1973b, PRD, 8, 3633 19. Hawking S. W., 1966a, Proc. Roy. Soc., 294A, 511 20. Hawking S. W., 1966b, Proc. Roy. Soc., 295A, 490 21. Hawking S. W., 1967a, Proc. Roy. Soc., 300A, 187 22. Hawking S. W., 1967b, Proc. Roy. Soc., 308A, 433 23. Hawking S. W., Penrose R., 1970, Proc. Roy. Soc., 314A, 529 24. Herb S. W., et al. , 1977, PRL, 39, 252 25. Hofstadter R., McAllister R. W., PR, 98, 217 26. Huang K., 1982, Quarks, Leptons and Gauge Fields, World Scientific, Singapore 27. Hughes I. S., 1991, Elementry Particles, Cambridge Univ. Press, Cambridge 28. Innes W. R., et al. , 1977, PRL, 39, 1240 29. Jastrow R., 1951, PR, 81, 165 30. Misner C. W., Thorne K. S., Wheeler J. A., 1973, Gravtitation, Freemon, NewYork, 857 31. Nambu Y., 1966, in A. de Shalit (Ed.), Preludes in Theoretical Physics, North-Holland, Amsterdam 32. Narlikar J. V., 1978,Lectures on General Relativity and Cosmology, The MacMillan Com- pany of India Limited, Bombay, 152 33. Oppenheimer,J. R., Snyder H., 1939, PR 56, 455 34. Penrose R., 1965, PRL, 14, 57 35. Penrose R., 1969, Riv. Nuoro Cimento, 1, Numero Speciale, 252 36. Politzer H. D., 1973, PRL, 30, 1346 37. Ramond P., 1983, Ann. Rev. Nucl. Part. Sc., 33, 31 38. Scotti A., Wong D. W., 1965, PR, 138B, 145 39. Slane P. O., Helfand D. J., Murray, S. S., 2002, ApJL, 571, 45 40. Thakur R. K., 1983, Ap&SS, 91, 285 41. Thakur R. K., 1992, Ap&SS, 190, 281 42. Thakur R. K., 1993, Ap&SS, 199, 159 43. Thakur R. K., 1995, Space Science Reviews, 73, 273 44. Weinberg S., 1972a, Gravitation and Cosmology, John Wiley & Sons, New York, 318 45. Weinberg S., 1972b, Gravitation and Cosmology, John Wiley & Sons, New York. 342-349 46. Zeldovich Y. B., Novikov I. D., 1971, Relativistic Astrophysics, Vol. I, University of Chicogo Press, Chicago,144-148 47. Zweig G., 1964, Unpublished CERN Report Introduction A critique of the singularity theorems Gravitationally collapsing black hole as a particle accelerator Quarks: The building blocks of matter Eventually what happens to matter in a collapsing black hole? Collapse of a black hole to a space-time singularity is inhibited by Pauli's exclusion principle Ultimately how does a black hole end up? Conclusion

0704.1662

Right-Handed Quark Mixings in Minimal Left-Right Symmetric Model with General CP Violation

Right-Handed Quark Mixings in Minimal Left-Right Symmetric Model with General CP Violation Yue Zhang,1, 2 Haipeng An,2 Xiangdong Ji,2, 1 and R. N. Mohapatra2 1Center for High-Energy Physics and Institute of Theoretical Physics, Peking University, Beijing 100871, China 2Department of Physics, University of Maryland, College Park, Maryland 20742, USA (Dated: October 31, 2018) Abstract We present a systematic approach to solve analytically for the right-handed quark mixings in the minimal left-right symmetric model which generally has both explicit and spontaneous CP violations. The leading-order result has the same hierarchical structure as the left-handed CKM mixing, but with additional CP phases originating from a spontaneous CP-violating phase in the Higgs vev. We explore the phenomenology entailed by the new right-handed mixing matrix, particularly the bounds on the mass of WR and the CP phase of the Higgs vev. http://arxiv.org/abs/0704.1662v1 The physics beyond the standard model (SM) has been the central focus of high-energy phenomenology for more than three decades. Many proposals, including supersymmetry, technicolors, little Higgs, and extra dimensions, have been made and studied thoroughly in the literature; tests are soon to be made at the Large Hadron Collider (LHC). One of the earliest proposals, the left-right symmetric (LR) model [1], was motivated by the hypothesis that parity is a perfect symmetry at high-energy, and is broken spontaneously at low-energy due to an asymmetric vacuum. Asymptotic restoration of parity has a definite aesthetic appeal [2]. This model, based on the gauge group SU(2)L × SU(2)R × U(1)B−L, has a number of additional attractive features, including a natural explanation of the weak hyper- change in terms of baryon and lepton numbers [3], the existence of right-handed neutrinos, and the possibility of spontaneous CP (charge-conjugation-parity) violation (SCPV) [4]. The model can easily be constrained by low-energy physics and predict clear signatures at colliders [5]. It so far remains a decent possibility for new physics. The LR modes are best constrained at low-energies by flavor-changing mixings and de- cays, particularly the CP violating observables. In making theoretical predictions, the major uncertainty comes from the unknown right-handed quark mixing matrix, conceptually simi- lar to the left-handed quark Cabibbo-Kobayashi-Maskawa (CKM) mixing. The new mixing generally depends on 9 real parameters: 6 CP violation phases and 3 rotational angles. Over the years, two limiting cases of the model have usually been studied. The first case, “mani- fest left-right symmetry”, assumes that there is no SCPV, i.e., all Higgs vacuum expectation values (vev) are real. The quark mass matrices are then hermitian, and the left and right- handed quark mixings become identical, modulo the sign uncertainty from possible negative quark masses. The reality of the Higgs vev, however, does not survive radiative corrections which generate infinite renormalization. The second case, “pseudo-manifest left-right sym- metry”, assumes that the CP violation comes entirely from spontaneous symmetry breaking (SSB) and that all Yukawa couplings are real [6]. Here the quark mass matrices are complex but symmetric, the right-handed quark mixing is related to the complex conjugate of the CKM matrix multiplied by additional CP phases. There are few studies of the model with general CP violation in the literature [7], with the exception of an extensive numerical study in Ref. [8] where solutions were generated through a Monte Carlo method. In this paper, we report a systematic approach to solve analytically for the right-handed quark mixings in the minimal LR model with general CP violation. As is well-known, the model has a Higgs bi-doublet whose vev’s are complex, leading to both explicit and spontaneous CP violations. The approach is based on the fact that mt ≫ mb and hence the ratio of the two vev’s of the Higgs bi-doublet, ξ = κ′/κ, is small. In the leading-order in ξ, we find a linear matrix equation for the right-handed quark mixing which can readily be solved. We present an analytical solution of this equation valid to O(λ3), where λ = sin θC is the Cabibbo mixing parameter. The leading-order solution is very close to the left-handed CKM matrix, apart from additional phases that are fixed by ξ, the spontaneous CP phase α, and the quark masses. This explicit right-handed quark mixing allows definitive studies of the neutral meson mixing and CP-violating observables. We use the experimental data on kaon and B-meson mixings and neutron electrical dipole moment (EMD) to constrain the mass of WR and the SCPV phase α. The matter content of the LR model is the same as the standard model (SM), except for a right-handed neutrino for each family which, together with the right-handed charged lepton, forms a SU(2)R doublet. The Higgs sector contains a bi-doublet φ, which transforms like (2,2,0) of the gauge group, and the left and right triplets ∆L,R, which transform as (3, 1, 2) and (1, 3, 2), respectively. The gauge group is broken spontaneously into the SM group SU(2)L × U(1)Y at scale vR through the vev of ∆R. The breaking of the SM group is accomplished through vev’s of φ. The most general renormalizable Higgs potential can be found in Ref. [9]. Only one of the parameters, α2, which describes an interaction between the bi-doublet and triplet Higgs, is complex, and induces an explicit CP violation in the Higgs potential. It is known in the literature that when this parameter is real, SCPV does not occur if the SM group is to be recovered in the decoupling limit vR → ∞ [9]. Without SCPV, the Yukawa couplings in the quark sector are hermitian, and we have the manifest left-right symmetry limit. Here we are interested in the general case when α2 is complex. A complex α2 allows spontaneous CP violation as well, generating a finite phase α for the vevs of φ, 〈φ〉 = 0 κ′eiα . (1) In reference [10], a relation was derived between α and the phase δ2 of α2, α ∼ sin−1 2|α2| sin δ2 , (2) where α3 is another interaction parameter between the Higgs bi-doublet and triplets. The quark masses in the model are generated from the Yukawa coupling, LY = q̄(hφ+ h̃φ̃)q + h.c. . (3) Parity symmetry (φ → φ†, qL → qR) constrains h and h̃ be hermitian matrices. After SSB, the above lagrangian yields the following quark mass matrices, Mu = κh+ κ ′e−iαh̃ Md = κ ′eiαh+ κh̃ . (4) Because of the non-zero α, both Mu and Md are non-hermitian. And therefore, the right- handed quark mixing can in principle very different from that of the left-hand counter part. Since the top quark mass is much larger than that of down quark, one may assume, without loss of generality, κ′ ≪ κ, while at the same time h̃ is at most the same order as h. We parameterize κ′/κ = rmb/mt, where r is a parameter of order unity. As a consequence, Mu is nearly hermitian, and one may neglect the second term to leading order in ξ. One can account for it systematically in ξ expansion if the precision of a calculation demands. Now h can be diagonalized by a unitary matrix Uu, Mu = UuM̂uSU u = κh , (5) where M̂u is diag(mu, mc, mt), and S is a diagonal sign matrix, diag(su, sc, st), satisfying S2 = 1. Replacing the h-matrix in Md by the above expression, one finds eiαξM̂u + κUuh̃U uS = VLM̂dV R (6) where M̂d is diag(md, ms, mb), VL is the CKM matrix and VR is the right-handed mixing matrix that we are after. Two comments are in order. First, through redefinitions of quark fields, one can bring VL to the standard CKM form with four parameters (3 rotations and 1 CP violating phase) and the above equation remains the same. Second, all parameters in the unitary matrix VR are now physical, including 3 rotations and 6 CP-violating phases. To make further progress, one uses the hermiticity condition for Uuh̃U u, which yields the following equation, M̂dV̂ R − V̂RM̂d = 2iξ sinα V LM̂uSVL (7) where V̂R is the quotient between the left and right mixing VR = SVLV̂R. There are a total of 9 equations above, which are sufficient to solve 9 parameters in V̂R. It is interesting to note that if there is no SCPV, α = 0, the solution is simply VR = SVLS̃, where S̃ is another diagonal sign matrix, diag(sd, ss, sb), satisfying S̃ 2 = 1. We recover the manifest left-right symmetry case. The above linear equation can be solved using various methods. The simplest is to utilize the hierarchy between down-type-quark masses. Multiplying out the left-hand side and assuming V̂Rij and V̂ Rij are of the same order, which can be justified posteriori, the solution is (for ρ sinα ≤ 1) ImV̂R11 = −r sinα sc + st A2λ4((1− ρ)2 + η2) ImV̂R22 = −r sinα sc + st ImV̂R33 = −r sinα st (10) V̂R12 = 2ir sinα sc + st λ4A2(1− ρ+ iη) V̂R13 = −2ir sinαAλ 3(1− ρ+ iη)st (12) V̂R23 = 2ir sinαAλ 2st , (13) where ImV̂R11, ImV̂R22, ImV̂R33, V̂R12, V̂R23, V̂R13 are on the orders of λ, λ, 1, λ 2, λ2, and λ3, respectively. The above solution allows us to construct entirely the right-handed mixing to order O(λ3) VR = PUV PD , (14) where the factors PU =diag(su, sd exp(2iθ2), st exp(2iθ3)), PD =diag(sd exp(iθ1), ss exp(−iθ2), sb exp(−iθ3)), and 1− λ2/2 λ Aλ3(ρ− iη) −λ 1− λ2/2 Aλ2e−i2θ2 Aλ3(1− ρ− iη) −Aλ2e2iθ2 1  ; (15) with θi = s̃i sin −1 ImV̂Rii. The pseudo-manifest limit is recovered when η = 0. A few remarks about the above result are in order. First, the hierarchical structure of the mixing is similar to that of the CKM, namely 1-2 mixing is of order λ, 1-3 order λ3 and 2-3 order λ2. Second, every element has a significant CP phase. The elements involving the first two families have CP phases of order λ, and the phases involving the third family are of order 1. These phases are all related to the single SCPV phase α, and can produce rich phenomenology for K and B meson systems as well as the neutron EDM. Third, depending on signs of the quark masses, there are 25 = 32 discrete solutions. Finally, using the right- handed mixing at leading order in ξ, one can construct h̃ from Eq. (6) and solve Mu with a better approximation. The iteration yields a systematic expansion in ξ. In the remainder of this paper, we consider the kaon and B-meson mixing as well as the neutron EMD. We will first study the contribution to the KL − KS mass difference ∆MK and derive an improved bound on the mass of right-handed gauge boson WR, using the updated hadronic matrix elements and strange quark mass. Then we calculate the CP violation parameter ǫ in KL decay and the neutron EDM, deriving an independent bound on MWR. Finally, we consider the implications of the model in the B-meson system, deriving yet another bound on MWR. The leading non-SM contribution to the K0 −K0 mixing comes from the WL −WR box diagram and the tree-level flavor-changing, neutral-Higgs (FCNH) diagram[11, 12]. The latter contribution has the same sign as the former, and inversely proportional to the square of the FCNH boson masses. We assume large Higgs boson masses (> 20TeV) from a large α3 in the Higgs potential so that the contribution to the mixing is negligible. Henceforth we concentrate on the box diagram only. Because of the strong hierarchical structure in the left and right quark mixing, the WL− WR box contribution to the kaon mixing comes mostly from the intermediate charm quark, H12 = 4π sin2 θW 2ηλLRc λ c (16) × [4(1 + ln xc) + ln η] (d̄s)2 − (d̄γ5s) + h.c. where xc = m , η = M2WL/M , λRLc = V RcdVLcs, and λ c = V LcdVRcs. The above result is very similar to that from the manifest-symmetry limit because the phases in VRcd and VRcs are O(λ). Therefore, we expect a similar bound on MWR as derived in previous work [11]. However, the rapid progress in lattice QCD calculations warrants an update. When the QCD radiative corrections are taken into account explicitly, the above effective hamiltonian will be multiplied by an additional factor η4. [We neglect contributions of other operators with small coefficients.] In the leading-logarithmic approximation, η4 is about 1.4 when the the four-quark operators are defined at the scale of 2 GeV in MS scheme [13]. The hadronic matrix element of the above operator can be calculated in lattice QCD and expressed in terms of a factorized form 〈K0|d̄(1− γ5)sd̄(1 + γ5)s|K0〉 = 2MKf KB4(µ) ms(µ) +md(µ) . (17) Using the domain-wall fermion, one finds B4 = 0.81 at µ = 2 GeV in naive dimensional regularization (NDR) scheme [14]. In the same scheme and scale, the strange quark mass is ms = 98(6) MeV. Using the standard assumption that the new physics contribution shall be less than the experimental value, one finds MWR > 2.5 TeV , (18) which is now the bound in the model with the general CP violation. This bound is stronger than similar ones obtained before because of the new chiral-symmetric calculation of B4 and the updated value of the strange quark mass. The most interesting predictions of VR are for CP-violating observables. We first study the CP violating parameter ǫ in KL decay. When the SCPV phase α = 0, the WL−WR box diagram still makes a significant contribution to ǫ from the phase δCKM of the CKM matrix. The experimental data then requires WR be at least 20 TeV to suppress this contribution. When α 6= 0, it is possible to relax the constraint by cancelations. The most significant contribution due to α comes from the element VRcd which is naturally on the order of λ. In the presence of α, we have an approximate expression for ǫLR ǫLR = 0.77 1 TeV sssd Im g(MR, θ2, θ3)e −i(θ1+θ2) where the function g(MR, θ2, θ3) = −2.22+[0.076+(0.030+0.013i) cos 2(θ2−θ3)] ln 80 GeV The required value of r sinα for cancelation depends sensitively on the sign of quark masses. When ss = sd, there exist small r sinα solutions even when MWR is as low as 1 TeV, as shown by solid dots in Fig. 1. However, when ss = −sd, only large r sinα solutions are possible. -0.15 -0.1 -0.05 0 0.05 0.1 0.15 r sin Α FIG. 1: Constaints on the mass of WR and the spontaneous CP violating parameter r sinα from ǫ (red dots) and neutron EDM in two different limit: small r ∼ 0.1 (green dots) and large r ∼ 1 (blue trinagles). An intriguing feature appears if one considers the constraint from the neutron EMD as well. A calculation of EDM is generally complicated because of strongly interacting quarks inside the neutron. As an estimate, one can work in the quark models by first calculating the EDM of the constituent quarks. In our model, there is a dominant contribution from the WL − WR boson mixing [15]. Requiring the theoretical value be below the current experimental bound, the neutron EDM prefers small r sinα solutions as can be seen in Fig. 1. The combined constraints from ǫ and the neutron EMD (dn < 3×10 −26 e cm [16]) impose an independent bound on MWR MWR > (2− 6) TeV , (20) where the lowest bound is obtained for small r ∼ 0.05 and large CP phase α = π/2. Finally we consider the neutral B-meson mixing and CP-violating decays. In Bd − Bd and Bs − Bs mixing, due to the heavy b-quark mass, there is no chiral enhancement in the hadronic matrix elements of ∆B = 2 operators from WL −WR box diagram as in the kaon case. One generally expects the constraint from neutral B-meson mass difference to be weak. In fact, we find a lower bound on WR mass of 1-2 TeV from B-mixing. On the other hand, CP asymmetry in decay Bd → J/ψKS, SJ/ψKS = sin 2β in the standard model, receives a new contribution from both Bd −Bd and K0 −K0 mixing in the presence of WR [7] and is very sensitive to the relative sign sd and ss. By demanding the modified sin 2βeff within the experimental error bar, we find another independent bound on MWR , MWR > 2.4 TeV , (21) when sd = ss as required by the neutron EDM bound. To summarize, we have derived analytically the right-handed quark mixing in the minimal left-right symmetric model with general CP violation. Using this and the kaon and B-meson mixing and the neutron EMD bound, we derive new bounds on the mass of right-handed gauge boson, consistently above 2.5 TeV. To relax this constraint, one can consider models with different Higgs structure and/or supersymetrize the theory. A more detailed account of the present work, including direct CP observables, will be published elsewhere [17] This work was partially supported by the U. S. Department of Energy via grant DE- FG02-93ER-40762. Y. Z. acknowledges the hospitality and support from the TQHN group at University of Maryland and a partial support from NSFC grants 10421503 and 10625521. X. 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0704.1663

Dynamics of single polymers under extreme confinement

Dynamics of single polymers under extreme confinement Armin Rahmani1, Claudio Castelnovo1,2, Jeremy Schmit3 and Claudio Chamon1 1 Department of Physics, Boston University, Boston, MA 02215 USA, 2 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, UK, and 3 Department of Physics, Brandeis University, Waltham MA 02454 USA. (Dated: October 24, 2018) Abstract We study the dynamics of a single chain polymer confined to a two dimensional cell. We introduce a kinetically constrained lattice gas model that preserves the connectivity of the chain, and we use this kinetically constrained model to study the dynamics of the polymer at varying densities through Monte Carlo simulations. Even at densities close to the fully-packed configuration, we find that the monomers comprising the chain manage to diffuse around the box with a root mean square displacement of the order of the box dimensions over time scales for which the overall geometry of the polymer is, nevertheless, largely preserved. To capture this shape persistence, we define the local tangent field and study the two- time tangent-tangent correlation function, which exhibits a glass-like behavior. In both closed and open chains, we observe reptational motion and reshaping through local fingering events which entail global monomer displacement. http://arxiv.org/abs/0704.1663v2 I. INTRODUCTION In this paper we consider the situation of a single chain polymer confined within a space smaller than its radius of gyration. Such a situation is encountered within the nucleus of a cell where one or more chromosomes with a radius of gyration on the order of 10 µm are confined by the nuclear membrane to a space of order 1 µm. Even in the case of organisms where the genome is composed of many chromosomes, the situation is distinct from that of a polymer melt as each chromosome is effectively confined to a separate sub-volume of the nucleus [1]. Since many biological processes such as gene suppression and activation require a rearrangement of the DNA polymer, understanding the dynamics of confined polymers may yield insight to the dynamics of these cellular activities. Strongly confined polymers may also be encountered in the “lab on a chip” applications promised by microfluidic technology [2]. In these applications the reaction vessels are ∼ 10µm microdroplets. While the equilibrium properties of confined polymers may be understood based on scaling arguments [3], the dynamics of confined polymers are less well understood, and there has been growing interest in the problem. For instance, the transport of polymers in confined geometries has been studied in a variety of contexts, including translocation through pores [4, 5], diffusion through networks [6] and tubes [7], and the packing of DNA within viral capsids [8]. For these highly confined polymers the density profile strongly resembles that of a polymer melt. We might naively expect that since a given section of the polymer interacts primarily with segments that are greatly separated along the chain, each segment may be treated as a sub-chain embedded within a melt. However, this picture is troublesome for dynamical quantities as reptation theory says that the dynamics is governed by the time it takes for a given chain to vacate the tube defined by its immediate neighbors. With a system consisting of a single polymer, this would imply that the tube occupies the entire box. Therefore, we would be forced to conclude that the chain is completely immobile. We show here that reptation-like motion is, in fact, the dominant mode of deformation of confined polymers. In contrast to the situation in melts, however, reptational diffusion is not necessarily initiated by the chain ends, and therefore, cannot be always thought of as diffusion along a fixed tube. Polymers confined to thin films have been experimentally shown to have glassy characteris- tics [9, 10]. While this phenomenon has attracted considerable theoretical attention, it is not well understood [11, 12, 13]. It is also not known whether glassy behavior occurs in other confined geometries. Here we point out a connection between lattice polymer models and Kinetically Con- strained Models (KCM) with the chain connectivity as the analog of the kinetic constraint. Since many KCMs display glassy behavior at high density it is plausible that polymers do as well. In this paper we numerically explore the dynamics of confined polymers using a kinetically constrained lattice gas model. We find that monomer diffusion exhibits power law behavior up to densities very close to the close-packing limit. However, the overall shape of the chain, as quantified by a tangent-tangent correlation function, shows a broad plateau at high densities. This apparent paradox is due to the reptation-like nature of the chain movement. Because the monomer diffusion is primarily in the direction of the chain backbone, only relatively small rearrangements of the backbone are required for the monomers to move distances comparable to the system size. The outline of the paper is as follows. In Section II we define our model and employ Monte Carlo simulations to show that this model reproduces known results for the dynamic and static properties of unconfined polymers in two dimensions. In Section III we show that the individual monomers diffuse with a power law in time behavior up to the close-packing density. In Sec- tion IV we define the tangent-tangent correlation function and use it to show that the overall shape of the chain is essentially frozen within the time scale required by a monomer to diffuse across dis- tances much larger than the inter-particle seperation. In Section V we use a tangent-displacement correlation function to show that the discrepancy between the monomer diffusion and reshaping time scales is due to reptation-like diffusion of the polymer along the chain backbone. Finally, in Section VI we summarize our conclusions. II. A KINETICALLY CONSTRAINED LATTICE GAS MODEL Inspired by the bond fluctuation model [14] of polymer dynamics and kinetically constrained models (KCM) [15] such as the Kob-Andersen model [16, 17], we propose a KCM for the dy- namics of a self-avoiding polymer. The fact that the monomers constitute a polymer requires the connectivity to be preserved. Namely, connected (unconnected) monomers must remain connected (unconnected) during the polymer motion. We begin by introducing the model in two dimensions with monomers living on the sites of a square lattice for simplicity. Let us define the polymer connectivity in the following way. Consider a square box of linear size 2r whose center lies on a given monomer. Any other monomer that lies inside or on the boundary of this box is defined as a box-neighbor of the monomer at the center. Clearly, if monomer A is a box-neighbor of monomer B, then monomer B is also a box-neighbor of monomer A. We define two monomers as being connected by a bond if and only if they are box-neighbors. A monomer with no box neighbors is an isolated polymer of length one. A monomer with only one box-neighbor is the end-point of a polymer. A monomer with two box neighbors is a point in the middle of a polymer and a monomer with more than two box neighbors corresponds to a branching point along a polymer. Depending on the initial monomer positions, multiple open or closed chains can be modeled. Also by using a d-dimensional hyper-cube instead of a square, the model can be immediately generalized to higher dimensions. The dynamics is defined as follows. A monomer can hop to a nearest neighbor unoccupied site if it has exactly the same box-neighbors before and after the move, as in Fig. 1. If no monomer enters the box associated with the moving monomer and no monomer falls out of it during the move, the box will contain the exact same monomers before and after the move. In other words, all the 2(2r+1) sites (2(2r+1)(d−1) in d dimensions) that enter or exit the box as it is moved to the new position, must be unoccupied, as shown in Fig. 2. In our model as in many kinetically constrained lattice gas models [15], we take the energy to be independent of the configuration, resulting in constant hopping rates. We assume that the allowed moves take place at unit rate. So if n(x, y) is defined to assume the value 1 for occupied sites and 0 for empty sites, and n̄(x, y) ≡ 1− n(x, y), the rate of hopping to the right out of site (x, y) is given by w→(x, y) = n(x, y)n̄(x+ 1, y) n̄(x+ r + 1, y + j)n̄(x− r, y + j), (2.1) and by similar expressions for the other directions. The dynamics forbids monomers that are unconnected from getting too close to each other and therefore ensures self-avoidance. Notice that all the moves are reversible because, as seen in Fig. 2, any particle that was allowed to hop to a nearest neighbor empty site is allowed to hop back to its original position. We choose the smallest value of r for which the model behaves like a polymer while allowing for shorter simulation times. The r = 1 case is too restrictive to model different modes of motion. For example a polymer lying along a straight line is forbidden in the r = 1 model to undergo one-dimensional translation. We choose r = 2 as it is found to adequately describe the free polymer dynamics, as shown by our numerical simulations. Monte Carlo (MC) simulations are used to study the model. A particularly time-efficient al- gorithm is achieved by storing two representations of the system at each MC step. One consists of the position vector of all the monomers, and the other is the configuration matrix of the lattice, FIG. 1: An example of a configuration that satisfies the initial conditions discussed in the text to have a single closed polymer. This is illustrated explicitly for the particle in red: the box of size 2r (r = 2) is indicated by the dashed purple square, and the two other particles inside the box are colored in blue. The same condition holds for all the particles in the system. All the (nearest-neighbor) particle moves allowed by the kinetic constraint are shown for each particle with arrows along the corresponding lattice edge. One particle in this configuration is temporarily frozen (circled blue particle), and its move is subordinated to the move of one of the two particles in its box of size 2r. Notice that the initial sequence of particles, represented by the wiggly green line, is clearly preserved by the allowed moves. with unoccupied sites having value zero and occupied sites value one. This allows us to chose a monomer at random from the position vector (rather than a site at random from the whole lat- tice) and quickly determine if the monomer is allowed to move in a randomly chosen direction by checking the values of at most eleven elements (the nearest neighbor site plus the sites by which the old and new box differ) in the configuration matrix. Note that it is possible to define an al- ternative model by using a circle of radius r = 2 instead of a square box of side 2r = 4, which would require the same amount of computational effort because the same number of sites, namely FIG. 2: For a model with an r = 2 box, a monomer can hop in a given direction if the destination site, as well as the ten sites which the old and new box differ by, are empty. ten sites, would enter or leave the circle drawn around the moving monomer. The r = 2 model with a square box proved to give consistent results with the ones available in the literature. Namely, the time-averaged radius of gyration squared of the polymer computed for our model in an infinite box scales as R2 ∝ N1.451±0.084, which is consistent with Flory’s the- oretical result of R2 ∝ N 2 [18]. The dynamics of the polymer in unconfined environments is also compatible with the Rouse model to a good approximation. As shown in Fig. 3, the mean square displacement of the center of mass is diffusive with a diffusion constant that scales as N−0.96 com- pared to the N−1 theoretical value. Throughout the paper, time and length are measured in units of Monte-Carlo steps and lattice spacings respectively. Only four values of N = 128, 256, 512, 1024 were used for these consistency checks but the fits were very close to the theoretical predictions. The individual monomer mean square displacement is diffusive at very short times followed by an intermediate-time subdiffusive behavior and a cross-over to a final diffusive behavior at long times as each monomer begins to move with the center of mass. The subdiffusive MSD can be fit with an exponent of 0.5968±0.0008 over the two-decade interval of 101 < t < 103 which is consistent with the theoretical value of 3 and the bond fluctuation results [14]. Because of the cross-over to diffusive behavior at long times the curve fits well to a higher exponent of 0.6516 ± 0.0008 over the longer interval of 101 < t < 106. In the present paper we focus on open and closed polymers without any branching. The close- Individual Monomer Center of Mass ∝ t3/5 FIG. 3: Center of mass and individual monomer mean square displacement of a free polymer. The results are shown for N = 256. packing density of such polymers clearly has an upper limit. For a single open chain for ex- ample, each monomer except for the two end-points has exactly two box-neighbors. The maxi- mally packed configuration is achieved once the distance along the chain between the consecu- tive monomers alternates between one and two lattice spacings, and the distance between paral- lel segments of the folded polymer equals 3 lattice spacings, as depicted in Fig. 4. If we have N monomers on an L × L lattice with (L + 1)2 sites, the fully-packed configuration attains a thermodynamic-limit density ρ = N (L+1)2 in d dimensions). Note that except for fluctua- tions at the U-turns, the polymer is completely frozen at close-packing. III. MEAN SQUARE MONOMER DISPLACEMENT The question of whether or not placing a self-avoiding polymer in a highly confining envi- ronment can freeze its motion can be addressed by measuring the statistical average of the mean square displacement as a function of time c(t) = 〈 ~xi(t + tw)− ~xi(t) 〉, (3.1) FIG. 4: A fully packed configuration of the model described in the text. Notice that only the shaded monomers are allowed to move at any given time. where ~xi is the position of the i-th monomer and tw is the waiting time between the preparation of the sample and the measurement. Throughout the paper, we denote the ensemble average by 〈. . .〉. We prepare random samples with different densities by placing the polymer in a large box and gradually reducing the size of the box. This is achieved by forbidding the monomers to move to the edges of the box, which corresponds to an infinite repulsive potential at the boundary, and removing one vertical and one horizontal edge line once they have become completely empty. After each shrinking process, the system is confined to a smaller square box. Shrinking and measurements are done in series. Specifically, we start from a very low density of ρ = 0.00010 and after each shrinking step we let the system run for 1000 Monte-Carlo steps before trying to shrink further. For measurements involving a long waiting time (tw = 10 7 steps), i.e., for densities ρ ≃ 0.0010, 0.050, 0.10, 0.15 and ρ > 0.195, subsequent shrinking steps are preformed starting from the post-measurement configurations. With this method we are able to reach densities of approximately ρ = 0.21, compared to the limiting theoretical value of ρ = 2/9 ≃ 0.222. At high densities, as shown in Fig. 5, the overall geometry of the polymer resembles that of a compact polymer described by a Hamiltonian path, i.e., a path which visits all sites exactly once, exploring a lattice with lattice spacing three times larger than in the original one. At these high densities our model resembles a semi-flexible polymer because the chain is able to attain a higher packing density in the direction parallel to the backbone (average monomer spacing equal to 3 lattice spacings) than in the direction perpendicular to the backbone (average monomer spacing equal to 3 lattice spacings) (see Fig. 4). FIG. 5: Snapshots of a polymer with N = 1024 monomers. As the box shrinks, the polymer gets more and more confined. At densities close to full-packing, a geometry resembling that of a compact (Hamiltonian walk) polymer is formed. Measurements are done with two values of waiting time, tw = 10 7 and tw = 1.1× 10 8 Monte- Carlo steps over a period of t = 108 steps. (For the highest density we used t = 2 × 108 in- stead.) The mean square displacement shows time translation invariance up to the highest densities achieved. We study the behavior of c(t) for N = 128, 256, 512, 1024 at the densities listed above. The root mean square displacement c(t) is a measure of how much the monomers have moved. As shown in Fig. 6, c(t) increases with power law behavior and finally saturates with a limiting root mean square displacement of the order of the box size. For very large box sizes (lowest density) as well as very small box sizes (very high densities), the saturation plateau is not always reached within the measurement time. However, the maximum value of c(t) is still of the same order of the box dimensions. Although the dynamics slows down at high confinement, each monomer manages to move an average distance comparable to the box size over our measurement time t. Visually observing the polymer motion, however, clearly shows a more complex scenario where at high densities the overall geometry of the polymer is largely preserved (see Fig. 5). Indeed, we will see that the tangent-tangent correlation function, although time-translation invariant at low and intermediate densities, exhibits signs of aging at higher densities. In the following sections we discuss in detail the shape persistence as well as the mechanisms by which the polymer shape changes as the box size is reduced. ρ=0.209 ρ=0.198 ρ=0.152 ρ=0.102 ρ=0.051 ρ=0.001 FIG. 6: Mean square displacement for N = 256 and different box sizes (i.e., different densities) as a function of time. The ρ = 0.001 curve can be fit with a 0.63 exponent which is close to the intermediate- time regime of the Rouse model. The results for other values of N are qualitatively very similar. IV. TANGENT FIELD CORRELATION The large values reached by c(t) within our simulation times even for very high densities indicate that confinement does not freeze the motion of the monomers. The overall shape or ge- ometry of the polymer, however, exhibits global persistence at high densities as observed via direct visualization of the dynamics. In order to systematically study shape persistence and reshaping, we introduce the concept of a tangent field, a vector field defined on the entire lattice which captures the overall shape of the polymer. We define the instantaneous tangent field as ~sinst(~x, t) = ~xi+1(t)− ~xi−1(t) if ~x = ~xi(t) ~0 otherwise, (4.1) ~xi(t) being the position of monomer i at time t. The tangent field is defined in a symmetric way so that labeling the monomers in reverse order only changes the direction of the field. In an open chain, the definition needs to be modified at the end points. Note that the tangent field is indexed by a position in space and not by a monomer number; this allows us to compare the shapes at different times, independently of the monomer motion. Since local vibrations of the polymer do not change the overall geometry, we seek a quantity that is insensitive to these vibrations. Coarse-graining the field by time averaging over a carefully chosen interval removes the local vibrations and results in a smeared field ~s~x(t) which captures the overall geometry, as shown in Fig. 7. We have chosen the time interval to be 75 Monte Carlo steps (or 75×N single monomer attempts) which is sufficient to allow for several vibrations. Since the success rate of the Monte Carlo attempts at high density is found to be around 1/10, this interval corresponds to roughly 7 moves per monomer. In terms of FIG. 7: An example of the coarse-grained tangent field ~s~x(t) the coarse-grained tangent field defined above, we define the tangent-tangent correlation function cs(t, tw) = 〈 all ~x ~s~x(t + tw) · ~s~x(tw) 〉 (4.2) as a measure of the overlap of the tangent field at times t+ tw and tw. As shown in Fig. 8, cs(t, tw) decays as a power low in t for very low densities. As the density increases, a second time scale emerges and at the highest densities we clearly see an initial decay followed by a broad plateau and a secondary decay. (Notice the use of a logarithmic scale on both axes.) The time-averaging of the tangent field hides the fast mode responsible for the initial decay and causes the correlation to have a smaller initial value at lower densities. The correlation function (4.2) does not depend on the value of N at low densities, as seen in Fig. 9, while at higher densities we observe broader plateaux and longer decorrelation times as the number of monomers N is increased. ρ=0.209 ρ=0.198 ρ=0.152 ρ=0.102 ρ=0.051 ρ=0.001 FIG. 8: Tangent-tangent correlation function for 256 monomers and different box sizes. The ρ = 0.001 curve fits well to a power law of exponent 0.42. Note that the correlation functions are not normalized. To check for time-translation invariance, we ran the simulations two more times after increas- ing the waiting time tw by an order of magnitude each time. The mean square displacement exhibits time-translation invariance at all densities. For the tangent-tangent correlation function, N=128, ρ=0.205 N=512, ρ=0.205 N=256, ρ=0.209 N=1024, ρ=0.209 FIG. 9: The tangent-tangent correlation function for four different polymer sizes. Top: At low density ρ = 0.05, the curves are independent of N . Bottom: At the highest achieved density ρ ≃ 0.20 − 0.21, the width of the emergent plateau increases with N however, time-translation invariance is respected at low densities but violated at the highest densi- ties where a broad plateau emerges. It appears that at high densities the average distance between monomers slowly evolves with time, so that the initial value of the tangent-tangent correlation function cs(0, tw) depends on tw. If we normalize the correlation function using its value at the beginning of the measurement and plot cs(t, tw)/cs(0, tw) as a function of time, the violation of time-translation invariance suggests the existence of aging effects, a comprehensive study of which is beyond the scope of the present paper. For tw = 1.01× 10 9, the system has almost equilibrated N=1024, ρ=0.152, t N=1024, ρ=0.152, t =107+108 N=1024, ρ=0.192, t N=1024, ρ=0.192, t =107+108 N=1024, ρ=0.192, t =107+109 FIG. 10: The normalized tangent-tangent correlation function cs(t, tw)/cs(0, tw) vs. time for different waiting times shown in log-log plots. Top: Up to intermediate densities, before the appearance of a clear plateau, time translation invariance is not broken. Bottom: At higher densities, where a broad plateau has emerged, we observe that the second decay occurs at a longer time scale. With increasing the waiting times the correlation function approaches equilibrium but there is still a systematic shift between tw ≈ 10 9 and tw ≈ 10 8 curves indicating slower decay for the older system. but there is still a systematic shift toward longer times compared with the tw = 1.1 × 10 8 cor- relation function. Fig. 10 summarizes the above observations. Moreover, Fig. 9 suggests longer equilibration times for larger systems and in the thermodynamic limit (N → ∞) the aging effects are expected to survive for arbitrarily large tw. V. TANGENT-DISPLACEMENT CORRELATION As seen in the previous sections, confinement slows down the motion of individual monomers in a polymer chain, but it does not fundamentally change the characteristics of their mean square displacement. It does, however, have a profound effect on reshaping. Without any reshaping, the only possible motion can happen via reptation, i.e., when the monomers move back and forth along a fixed path. With the exception of the unlikely event of the two end-points finding each other, reptation without reshaping is not possible in open chains. Indeed, by studying closed loops in detail we do find that longitudinal diffusion is the main mechanism for motion at high densities. The existence of large root mean square displacements in strongly confined open chains and in the absence of major reshaping can be explained by noting that local reshaping events with only a minor contribution to the tangent-tangent decorrelation allow for global monomer motion through a reptation-like process. Fig. 11 shows one instance of such behavior in a particularly mobile realization. The mechanisms shown in Fig. 11, namely end-point initiated reptation and ”fingering” events, are observed in other realizations as well. A finger is formed when the chain makes a 180-degree bend resulting in two adjacent segments of the polymer running antiparallel to each other. A fingering event occurs when a finger retracts making room for the extension of another finger. Even in the case of closed loops where pure reptation is possible, reptation is usually accompanied by local fingering events as shown in Fig. 12. To explicitly quantify the contribution of reptation to highly confined motion, we define tangent-displacement and normal- displacement correlation functions as follows: ct(t) = 〈 ~s~xi(tw) |~s~xi(tw)| ~xi(tw + t)− ~xi(tw) 〉 (5.1) cn(t) = 〈 ~n~xi(tw) |~n~xi(tw)| ~xi(tw + t)− ~xi(tw) 〉, (5.2) where ~n is the normal field defined as ~n~x(t) = ẑ × ~s~x(t) and the polymer – and therefore ~s~x(t) – belongs to the xy plane. By comparison with Eq. (3.1), one can show that c(t) = ct(t) + cn(t). (5.3) Therefore, the correlation functions (5.2) and (5.1) are the contributions to the mean square dis- placement due to the transverse and longitudinal motion relative to the initial polymer orientation. At very high densities and up to time scales smaller than the beginning of the secondary decay FIG. 11: Snapshots of a particularly mobile realization of a 256-monomer chain at density ρ = 0.209. Fingering events are observed as well of end-point reptation accompanied by local rearrangements. An end-point initiated reptation is highlighted with a dashed ellipse at the moving end-point. The dashed rectangle shows the tip of a finger which participates in a fingering event during which the tagged monomer shown with a solid circle, for example, moves through reptation. in the tangent-tangent correlation function Eq. (4.2), ct is a good measure of the reptational con- tribution to the confined motion. This is due to the fact that over such time scales the polymer shape is largely preserved. Additionally, over the same time scales the mean square displacement FIG. 12: Snapshots of a realization with 256 monomers arranged in a single closed polymer chain, at density ρ = 0.209. Pure reptation and fingering events are visibly responsible for monomer motion. The dashed rectangle highlights the tip of a finger taking part in a fingering event. The dashed ellipse shows the reptational motion of a tagged monomer. is smaller than the average chain length between major bends, which as seen in Fig. 11 is a large fraction of the box size. If the polymer undergoes major reshaping or the monomers move through the bends of the folded polymer, reptation would no longer be equivalent to the motion along the original orientation. Let us define f as a measure of the anisotropy of the motion with respect to the longitudinal and transverse directions, f(t) = ct(t)− cn(t) . (5.4) We have ct(t)/c(t) = (1 + f(t))/2 and cn(t)/c(t) = (1 − f(t))/2, where f ranges from −1 to +1. A large value of f clearly indicates a motion primarily due to reptation. As shown in semi- logarithmic scale in Fig. 13, f reaches a maximum value of 0.8 at short time scales, demonstrating −0.15 −0.05 f(t)=0 N=128, ρ=0.205 N=256, ρ=0.209 N=512, ρ=0.205 N=1024, ρ=0.209 f(t)=0 FIG. 13: Top: At the lowest density studied ρ ≃ 0.001, the function f is independent of the number of monomers and approaches zero monotonically (i.e., the motion is isotropic at intermediate and large time scales). Bottom: At the highest densities achieved ρ ≃ 0.20 − 0.21, the function f reaches very large positive values before decreasing again towards zero. For short polymers f(t) goes down to negative values at large times. Here we ignore the causes of this observation because as explained in the text, f(t) is a good measure of the anisotropy of the motion only up to certain time scales. that reptation-like motion is the primary contributor to monomer displacement. We also observe that the maximum of f increases as the number of monomers N is increased, consistent with the fact that the width of the plateau becomes monotonically larger and larger with N . VI. CONCLUSIONS As the size of the confining box around a polymer is reduced, the monomer density makes it increasingly difficult for the polymer to move. However, the effect on the polymer movement is not isotropic. The transverse fluctuations are strongly suppressed due to the proximity of monomers that may be greatly separated along the chain backbone. This is in contrast to motion parallel to the chain backbone where, due to the connectivity constraint, the monomer density is very similar for the confined and unconfined chains. While longitudinal motion is sub-dominant in the free chain, it is the primary mode of monomer diffusion when the density becomes high enough to suppress the transverse fluctuations. The emergence of motion parallel to the chain backbone as the dominant mode of diffusion is similar to what occurs in a polymer solution when the density is increased to form a melt. However, the longitudinal diffusion observed here differs from the classic reptation picture in that the motion is not necessarily initiated at the chain ends but it can also be triggered by fingering events. The prevalence of fingering reptation over end-initiated reptation is due to three factors. First, the two-dimensional nature of our system imposes topological constraints that severely limit the mobility of the chain ends. Second, a single confined chain has only two end-points, while the number of fingers it can have grows with the system size. Third, the compact configuration due to the confinement forces the creation of more fingering structures relative to the extended polymer structures found in melts. The peculiarities of the dynamics of a single chain in extreme confinement (high density limit) leads to an interesting effect: monomers can diffuse through large distances comparable to the box size within time scales for which the overall shape of the polymer is, nevertheless, largely preserved. While monomer displacement exhibits a smooth power law behavior in time at all den- sities, the tangent-tangent correlation function develops a secondary decay at high densities. This decay takes place at longer time scales for older systems, suggestive of aging phenomena. We thus find glass-like behavior in the overall geometry concurrently with non-glassy monomer motion. Despite significant persistence of geometry, monomer displacement can reach large values relative to its saturation value over the same time scales because local rearrangements cause monomers to flow even in parts of the system where no reshaping is taking place. The two dimensional lattice model presented here is a largely simplified one. However, we believe that this model yields considerable insight into the generic properties of confined polymers. Namely, reptational or longitudinal motion is identified as the primary mechanism for motion at high densities and extreme confinement is found to primarily suppress changes in the overall geometry of the polymer rather than the monomer motion. Acknowledgments The simulations were carried out on Boston University supercomputer facilities (SCV). We thank B. Chakraborty, J. Kondev, D. Reichman and F. Ritort for useful discussions. This work is supported in part by the NSF Grant DMR-0403997 (AR, CC, CC, and JS) and by EPSRC Grant No. GR/R83712/01 (C. Castelnovo). Appendix: Closed chains Closed chains can be studied using the same correlation functions. The only subtlety with closed chains is the existence of a non-trivial background in finite systems. The background has to do with the topology of closed loops and must be subtracted from the tangent-tangent corre- lation function. Suppose that the monomers in the chain are initially indexed clockwise or anti- clockwise. The dynamics cannot change the chirality of the loop in two dimensions. Therefore, all the outer segments of the polymer running parallel to the walls of the box have correlated tangent fields. Using an ensemble with random chirality does not remove the problem because each real- ization, whether clockwise or anti-clockwise, would contribute a positive value to the correlation function. One can correct for this effect as follows. For an ensemble with a given chirality, let us call the equilibrium tangent-field background ~save(~x). We can then modify the tangent-tangent correlation function by subtracting this background field. cs,loop(t, tw) = 〈 all ~x ~s~x(t+ tw)− ~save(~x) ~s~x(tw)− ~save(~x) 〉. (6.1) The equilibrium background can be obtained at low densities via Monte Carlo simulations, using realizations with the same chirality and averaging over time and ensemble. This approach becomes less and less reliable as the density increases and glassy behavior arises, because each realization is essentially stuck in a small region of configuration space over the measurement time scales. Fig. 14 shows the background tangent field at an intermediate density ρ = 0.1. The modified tangent-tangent correlation function (6.1) and the mean square displacement were measured for a FIG. 14: The background tangent field for N = 256 monomers in a box of size L = 49. The relative scale between the field vectors reflects the actual values of the tangent-tangent field (an overall scale factor has been introduced to enhance visibility). Notice that the average field at the boundary does not vanish. closed loop of N = 256 monomers and no qualitative difference was observed in comparison to open chains. Also f(t) behaves similarly in the two cases, reaching high values at high densities for closed chains as well as open chains. As shown in Fig. 15, the mean square displacement for closed chains at high densities reaches its saturation value faster than for open chains, whereas at low and intermediate densities they are identical. [1] T. Cremer and C. Cremer, Nature Rev. Gen. 2, 292-301 (2001). [2] G.M. Whitesides, Nature 442, 368-373 (2006). [3] P.G. de Gennes, Scaling concepts in polymer physics, Cornell University Press (1979). [4] M. Muthukumar, Phys. Rev. Lett. 86, 3188 - 3191 (2001). [5] S.M. Bezrukov, I. Vodyanoy, R.A. Brutan, and J.J. Kasianowicz, Macromolecules 29, 8517 (1996). [6] D. Nykypanchuk, H.H. Strey, D.A. Hoagland, Science 297, 987 - 990 (2002). [7] J. Kalb, B. Chakraborty, cond-mat/0702152. http://arxiv.org/abs/cond-mat/0702152 N=256, ρ=0.197, OP N=256, ρ=0.197, CP N=256, ρ=0.152, OP N=256, ρ=0.152, CP FIG. 15: The mean square displacements of closed polymers (CP) and open polymers (OP) are identical at low and intermediate densities. At high densities closed polymers seem to reach the saturation value faster. [8] A. J. Spakowitz and Z.-G. Wang, Biophys. J., 88, 3912 (2005). [9] J.A. Forrest and K. Dalnoki-Veress, Adv. Colloid and Interface Sci. 94, 167 (2001). [10] J.L. Keddie, R.A.L. Jones, and R.A. Cory, Europhys. Lett. 27, 59 (1994). [11] P.G. de Gennes, Eur. Phys. J. E 2, 201 (2000). [12] K.L. Ngai, A.K. Rizos, and D.J. Plazek, J. Non-cryst. Sol. 235, 435 (1998). [13] Q. Jiang, H.X. Shi, and J.C. Li, Thin Solid Films 354, 283 (1999). [14] K. Kremer and I. Carmesin, Macromolecules 21, 2819 (1988). [15] F. Ritort and P. Sollich, Advances in Physics 52, 219 (2003). [16] W. Kob and H.C. Andersen, Phys. Rev. E 48, 4364 (1993). [17] C. Toninelli, G. Biroli and D. Fisher, Phys. Rev. Lett. 92, 185504 (2004). [18] P.J. Flory, J. Chem. Phys. 17, 303 (1949). [19] T. Odijk Macromolecules 16, 1340-1344 (1983) Introduction A kinetically constrained Lattice gas model Mean square monomer displacement Tangent field Correlation Tangent-Displacement correlation Conclusions Acknowledgments Appendix: Closed chains References

0704.1664

Diffuse Optical Light in Galaxy Clusters II: Correlations with Cluster Properties

Draft version November 9, 2018 Preprint typeset using LATEX style emulateapj v. 11/12/01 DIFFUSE OPTICAL LIGHT IN GALAXY CLUSTERS II: CORRELATIONS WITH CLUSTER PROPERTIES J.E. Krick 1,2 and R.A. Bernstein 1 [email protected], [email protected] Draft version November 9, 2018 ABSTRACT We have measured the flux, profile, color, and substructure in the diffuse intracluster light (ICL) in a sample of ten galaxy clusters with a range of mass, morphology, redshift, and density. Deep, wide- field observations for this project were made in two bands at the one meter Swope and 2.5 meter du Pont telescope at Las Campanas Observatory. Careful attention in reduction and analysis was paid to the illumination correction, background subtraction, point spread function determination, and galaxy subtraction. ICL flux is detected in both bands in all ten clusters ranging from 7.6× 1010 to 7.0× 1011 h−170 L�in r and 1.4× 10 10 to 1.2× 1011 h−170 L�in the B−band. These fluxes account for 6 to 22% of the total cluster light within one quarter of the virial radius in r and 4 to 21% in the B−band. Average ICL B− r colors range from 1.5 to 2.8 mags when k and evolution corrected to the present epoch. In several clusters we also detect ICL in group environments near the cluster center and up to 1h−170 Mpc distant from the cluster center. Our sample, having been selected from the Abell sample, is incomplete in that it does not include high redshift clusters with low density, low flux, or low mass, and it does not include low redshift clusters with high flux, mass, or density. This bias makes it difficult to interpret correlations between ICL flux and cluster properties. Despite this selection bias, we do find that the presence of a cD galaxy corresponds to both centrally concentrated galaxy profiles and centrally concentrated ICL profiles. This is consistent with ICL either forming from galaxy interactions at the center, or forming at earlier times in groups and later combining in the center. Subject headings: galaxies: clusters: individual (A4059, A3880, A2734, A2556, A4010, A3888, A3984, A0141, AC114, AC118) — galaxies: evolution — galaxies: interactions — galaxies: photometry — cosmology: observations 1. introduction A significant stellar component of galaxy clusters is found outside of the galaxies. The standard theory of cluster evolution is one of hierarchical collapse, as time proceeds, clusters grow in mass through the merging with other clusters and groups. These mergers as well as inter- actions within groups and within clusters strip stars out of their progenitor galaxies. The study of these intracluster stars can inform hierarchical formation models as well as tell us something about physical mechanisms involved in galaxy evolution within clusters. Paper I of this series (Krick et al. 2006) discusses the methods of ICL detection and measurement as well as the results garnered from one cluster in our sample. We re- fer the reader to that paper and the references therein for a summary of the history and current status of the field. This paper presents the remaining nine clusters in the sample and seeks to answer when and how intracluster stars are formed by studying the total flux, profile shape, color, and substructure in the ICL as a function of cluster mass, redshift, morphology, and density in the sample of 10 clusters. The advantage to having an entire sample of clusters is to be able to follow evolution in the ICL and use that as an indicator of cluster evolution. Strong evolution in the ICL fraction with mass of the cluster has been predicted in simulations by both Lin & Mohr (2004) and Murante et al. (2004). If ongoing stripping processes are dominant, ram pressure stripping (Abadi et al. 1999) or harassment (Moore et al. 1996), then high mass clusters should have a higher ICL fraction than low-mass clusters . If, however, most of the galaxy evolu- tion happens early on in cluster collapse by galaxy-galaxy merging, then the ICL should not correlate directly with current cluster mass. Because an increase in mass is tied to the age of the cluster under hierarchical formation, evolution has also been predicted in the ICL fraction as a function of red- shift (Willman et al. 2004; Rudick et al. 2006). Again, if ICL formation is an ongoing process then high redshift clusters will have a lower ICL fraction than low redshift clusters. Conversely, if ICL formation happened early on in cluster formation there will be no correlation of ICL with redshift. The stripping of stars (or even the gas to make stars) to create an intracluster stellar population requires an in- teraction between their original host galaxy and either an- other galaxy, the cluster potential, or possibly the hot gas in the cluster. Because all of these processes require an interaction, we expect cluster density to be a predictor of ICL fraction. Cluster density is linked to cluster morphol- ogy, which implies morphology should also be a predictor of ICL fraction. Specifically we measure morphology by the presence or absence of a cD galaxy. cD galaxies are the results of 2 - 5 times more mergers than the average cluster galaxy (Dubinski 1998). The added number of in- teractions that went into forming the cD galaxy will also mean an increased disruption rate in galaxies therefore morphological relaxed (dynamically old) clusters should 1 Astronomy Department, University of Michigan, Ann Arbor, MI 48109 2 Spitzer Science Center, Caltech, Pasadena, CA 91125 2 Krick, Bernstein have a higher ICL flux than dynamically young clusters. Observations of the color and fractional flux in the ICL over a sample of clusters with varying redshift and dynam- ical state will allow us to identify the timescales involved in ICL formation. If the ICL is the same color as the cluster galaxies, it is likely to be a remnant from ongoing interactions in the cluster. If the ICL is redder than the galaxies it is likely to have been stripped from galaxies at early times. Stripped stars will passively evolve toward red colors while the galaxies continue to form stars. If the ICL is bluer than the galaxies, then some recent star formation has made its way into the ICL, either from ellipticals with low metallicity or spirals with younger stellar populations, or from in situ formation. While multiple mechanisms are likely to play a role in the complicated process of formation and evolution of clus- ters, important constraints can come from ICL measure- ment in clusters with a wide range of properties. In addi- tion to directly constraining galaxy evolution mechanisms, the ICL flux and color is a testable prediction of cosmolog- ical models. As such it can indirectly be used to examine the accuracy of the physical inputs to these models. This paper is structured in the following manner. In §2 we discuss the characteristics of the entire sample. De- tails of the observations and reduction are presented in §3 and §4 including flat-fielding, sky background subtraction methods, object detection, and object removal and mask- ing. In §5 we lists the results for both cluster and ICL properties including a discussion of each individual clus- ter. Accuracy limits are discussed in §6. A discussion of the interesting correlations can be found in §7 followed by a summary of the conclusions in §8. Throughout this paper we use H0 = 70km/s/Mpc, ΩM = 0.3, ΩΛ = 0.7. 2. the sample The general properties of our sample of ten galaxy clus- ters have been outlined in paper I; for completeness we summarize them briefly here. Our choice of the 10 clusters both minimizes the observational hazards of the galactic and ecliptic plane, and maximizes the amount of informa- tion in the literature. All clusters were chosen to have published X–ray luminosities which guarantees the pres- ence of a cluster and provides an estimate of the clus- ter’s mass. The ten chosen clusters are representative of a wide range in cluster characteristics, namely red- shift (0.05 < z < 0.3), morphology (3 with no clear central dominant galaxy, and 7 with a central dominant galaxy as determined from this survey, §5.1.2, and not from Bautz Morgan morphological classifications), spatial projected density (richness class 0 - 3), and X–ray lumi- nosity (1.9 × 1044 ergs/s < Lx < 22 × 1044 ergs/s). We discuss results from the literature and this survey for each individual cluster in order of ascending redshift in §5. 3. observations The sample is divided into a “low” (0.05 < z < 0.1) and “high” (0.15 < z < 0.3) redshift range which we have observed with the 1 meter Swope and 2.5 meter du Pont telescope respectively. The du Pont observations were dis- cussed in detail in paper I. The Swope observations follow a similar observational strategy and data reduction pro- cess which we outline below. Observational parameters are listed in Table 2. We used the 2048 × 3150 “Site#5” CCD with a 3e−/count gain and 7e− readnoise on the Swope telescope. The pixel scale is 0.435′′/pixel (15µ pixels), so that the full field of view per exposure is 14.8′ × 22.8′. Data was taken in two filters, Gunn-r (λ0 = 6550 Å) and B (λ0 = 4300 Å). These filters were selected to provide some color con- straint on the stellar populations in the ICL by spanning the 4000Å break at the relevant redshifts, while avoid- ing flat-fielding difficulties at longer wavelengths and pro- hibitive sky brightness at shorter wavelengths. Observing runs occurred on October 20-26, 1998, September 2-11, 1999, and September 19-30, 2000. All observing runs took place within eight days of new moon. A majority of the data were taken under photometric con- ditions. Those images taken under non-photometric con- ditions were individually tied to the photometric data (see discussion in §4. Across all three runs, each cluster was observed for an average of 5 hours in each band. In addi- tion to the cluster frames, night sky flats were obtained in nearby, off-cluster, “blank” regions of the sky with total exposure times roughly equal to one third of the integra- tion times on cluster targets. Night sky flats were taken in all moon conditions. Typical B− and r−band sky levels during the run were 22.7 and 21.0 mag arcsec−2, respec- tively. Cluster images were dithered by one third of the field of view between exposures. The large overlap from the dithering pattern gives us ample area for linking back- ground values from the neighboring cluster images. Ob- serving the cluster in multiple positions on the chip reduces large-scale flat-fielding fluctuations upon combination. In- tegration times were typically 900 seconds in r and 1200 seconds in B. 4. reduction In order to create mosaiced images of the clusters with a uniform background level and accurate resolved–source fluxes, the images were bias and dark subtracted, flat– fielded, flux calibrated, background–subtracted, extinction corrected, and registered before combining. Methods for this are discussed in detail in paper I and summarized be- The bias level is roughly 270 counts which changed by approximately 8% throughout the night. This, along with the large-scale ramping effect in the first 500 columns of ev- ery row was removed in the standard manner using IRAF tasks. The mean dark level is 1.6 counts/900s, and there is some vertical structure in the dark which amounts to 1.4 counts/900s over the whole image. To remove this large-scale structure from the data images, a combined dark frame from the whole run was median smoothed over 9× 9 pixels (3.9′′), scaled by the exposure time, and sub- tracted from the program frames. Small scale variations were not present in the dark. Pixel–to–pixel sensitivity variations were corrected in all cluster and night-sky flat images using nightly, high S/N, median-combined dome flats with 70,000 – 90,000 total counts. After this step, a large-scale illumination pattern remains across the chip. This was removed using night-sky flats of “blank” regions of the sky, which, when combined using masking and re- ICL 3 jection, produced an image with no evident residual flux from sources but has the large scale illumination pattern intact. The illumination pattern was stable among images taken during the same moon phase. Program images were corrected only with night sky flats taken in conditions of similar moon. We find that the Site#3 CCD does have an approx- imately 7% non-linearity over the full range of counts, which we fit with a second order polynomial and corrected for in all the data. The same functional fit was found for both the 1998 and 1999 data, and also applied to the 2000 data. The uncertainty in the linearity correction is incor- porated in the total photometric uncertainty. Photometric calibration was performed in the usual manner using Landolt standards at a range of airmasses. Extinction was monitored on stars in repeat cluster images throughout the night. Photometric nights were analyzed together; solutions were found in each filter for an extinc- tion coefficient and common magnitude zero-point with a r− and B−band RMS of 0.04 & 0.03 magnitudes in Oc- tober 1998, 0.03 & 0.03 magnitudes in September 1999, and 0.05 & 0.05 magnitudes in September 2000. These uncertainties are a small contribution to our final error budget (§5.3). Those exposures taken in non-photometric conditions were individually tied to the photometric data using roughly 10 stars well distributed around each frame to find the effective extinction for that frame. Among those non-photometric images we find a standard devia- tion of 0.03 magnitudes within each frame. Two further problems with using non-photometric data for low surface brightness measurements are the scattering of light off of clouds causing a changing background illumination across the field and secondly the smoothing out of the PSF. We find no spatial gradient over the individual frame to the limit discussed in §5.3. The change in PSF is on small scales and will have no effect on the ICL measurement (see 4.2.1). Due to the temporal variations in the background, it is necessary to link the off-cluster backgrounds from adjacent frames to create one single background of zero counts for the entire cluster mosaic before averaging together frames. To determine the background on each individual frame we measure average counts in approximately twenty 20 × 20 pixel regions across the frame. Regions are chosen indi- vidually by hand to be a representative sample of all areas of the frame that are more distant than 0.8h−170 Mpc from the center of the cluster. This is well beyond the radius at which ICL components have been identified in other clusters (Krick et al. 2006; Feldmeier et al. 2002; Gonzalez et al. 2005; Zibetti et al. 2005). The average of these back- ground regions for each frame is subtracted from the data, bringing every frame to a zero background. The accuracy of the background subtraction is discussed in §5.3. The remaining flux in the cluster images after back- ground subtraction is corrected for atmospheric extinction by multiplying each individual image by 10τχ/2.5, where χ is the airmass and τ is the fitted extinction in magnitudes from the photometric solution. This multiplicative correc- tion is between 1.04 and 2.0 for an airmass range of 1.04 to 1.9. The IRAF tasks geomap and geotran were used to find and apply x and y shifts and rotations between all images of a single cluster. The geotran solution is accu- rate on average to 0.03 pixels (RMS). Details of the final combined image after pre-processing, background subtrac- tion, extinction correction, and registration are included in Table 2. 4.1. Object Detection Object detection follows the same methods as Paper I. We use SExtractor to both find all objects in the combined frames, and to determine their shape parameters. The de- tection threshold in the V , B, and r images was defined such that objects have a minimum of 6 contiguous pixels, each of which are greater than 1.5σ above the background sky level. We choose these parameters as a compromise be- tween detecting faint objects in high signal-to-noise regions and rejecting noise fluctuations in low signal-to-noise re- gions. This corresponds to minimum surface brightnesses which range from of 25.2 to 25.8 mag arcsec−2 in B, 25.9 to 26.9 mag arcsec−2 in V , and 24.7 to 26.4 mag arcsec−2 in r (see Table 2). This range in surface brightness is due to varying cumulative exposure time in the combined frames. Shape parameters are determined in SExtractor using only those pixels above the detection threshold. 4.2. Object Removal & Masking To measure the ICL we remove all detected objects from the frame by either subtraction of an analytical profile or masking. Details of this process are described below. 4.2.1. Stars Scattered light in the telescope and atmosphere pro- duce an extended point spread function (PSF) for all ob- jects. To correct for this effect, we determine the extended PSF using the profiles of a collection of stars from super- saturated 4th mag stars to unsaturated 14th magnitude stars. The radial profiles of these stars were fit together to form one PSF such that the extremely saturated star was used to create the profile at large radii and the unsat- urated stars were used for the inner portion of the profile. This allows us to create an accurate PSF to a radius of 7′, shown in Figure 1. The inner region of the PSF is well fit by a Moffat func- tion. The outer region is well fit by r−2.0 in the r−band and r−1.6 in the B−band. In the r−band there is a small additional halo of light at roughly 50 - 100′′(200-400pix) around stars imaged on the CCD. The newer, higher qual- ity, anti-reflection coated interference B−band filter does not show this halo, which implies that the halo is caused by reflections in the filter. To test the effect of clouds on the shape of the PSF we create a second deep PSF from stars in cluster fields taken under non-photometric conditions. There is a slight shift of flux in the inner 10 arcseconds of the PSF profile, which will have no impact on our ICL measurement. For each individual, non-saturated star, we subtract a scaled band–specific profile from the frame in addition to masking the inner 30′′ of the profile (the region which fol- lows a Moffat profile). For each individual saturated star, to be as cautious as possible with the PSF wings, we have subtracted a stellar profile given the USNO magnitude of that star, and produced a large mask to cover the inner re- gions and any bleeding. The mask size is chosen to be twice 4 Krick, Bernstein the radius at which the star goes below 30mag arcsec−2, and therefore goes well beyond the surface brightness limit at which we measure the ICL. We can afford to be liberal with our saturated star masking since most clusters have very few saturated stars which are not near the center of the cluster where we need the unmasked area to measure any possible ICL. In the specific case of A3880 there are two saturated stars (9th and 10th r−band magnitude) within two ar- cminutes of the core region of the cluster. If we used the same method of conservatively masking (twice the radius of the 30 mag arcsec−2 aperture), the entire central region of the image where we expect to find ICL would be lost. We therefore consider a less extreme method of removing the stellar profile by iteratively matching the saturated stars’ profiles with the known PSF shape. We measure the saturated star profiles on an image which has had ev- ery object except for those two saturated stars masked, as described in §4.2.2. We can then scale our measured PSF to the star’s profile, at radii where there is expected to be no contamination from the ICL, and the star’s flux is not saturated. Since the two stars are within an arcminute of each other, the scaled profiles of the stars are iteratively subtracted from the masked cluster image until the process converges on solutions for the scaling of each star. We still use a mask for the inner region (∼ 75′′) where saturation and seeing effect the profile shape. 4.2.2. Galaxies We want to remove all the flux in our images associated with galaxies. Although some galaxies might follow de- Vaucouleurs, Sersic, or exponential profiles, those galaxies which are near the centers of clusters can not be fit with these or other models either because of the overcrowding in the center or because their profiles really are different due to their location in a dense environment. A variety of strategies for modeling galaxies within the centers of clusters were explored in Paper 1 and were found to be in- adequate for these purposes. Since we can not fit and sub- tract the galaxies to remove their light, we instead mask all galaxies in our cluster images. By masking, we remove from our ICL measurements all pixels above a surface brightness limit which are centered on a galaxy as detected by SExtractor. For paper I, we chose to mask inside of 2 - 2.3 times the radius at which the galaxy light dropped below 26.4 mag arcsec−2 in r, akin to 2-2.3 times a Holmberg radius (Holmberg 1958). Holmberg radii are typically used to denote the outermost radii of the stellar populations in galaxies. Galaxy profiles will also have the characteristic underly- ing shape of the PSF, including the extended halo. How- ever for a 20th magnitude galaxy, the PSF is below 30 mag arcsec−2by a radius of 10′′. Each of the clusters has a different native surface bright- ness detection threshold based on the illumination correc- tion and background subtraction, and they are all at dif- ferent redshifts. However we want to mask galaxies at all redshifts to the same physical surface brightness to allow for a meaningful comparison between clusters at different redshifts. To do this we make a correction for (1+z)4 sur- face brightness dimming and a k correction for each cluster when calculating mask sizes. The masks sizes change by an average of 10% and at most 22% from what they would have been given the native detection threshold. Both the native and corrected surface brightness detection thresh- olds are listed in Table 2. To test the effect of mask size on the ICL profile and total flux, we also create masks which are 30% larger and 30% smaller in area than the calcu- lated mask size. The flux within the masked areas for these galaxies is on average 25% more than the flux iden- tified by SExtractor as the corrected isophotal magnitude for each object. 5. results Here we discuss our methods for measuring both cluster and ICL properties as well as a discussion of each individ- ual cluster in our sample. 5.1. Cluster Properties Cluster redshift, mass, and velocity dispersion are taken from the literature, where available, as listed in table 1. Additional properties that can be identified in our data, particularly those which may correlate with ICL proper- ties (cluster membership, flux, dynamical state, and global density), are discussed below and also summarized in Ta- ble 1. 5.1.1. Cluster Membership & Flux Cluster membership and galaxy flux are both deter- mined using a color magnitude diagram (CMD) of either B− r vs. r (clusters with z < 0.1) or V − r vs. r (clusters with z > 0.1). We create color magnitude diagrams for all clusters using corrected isophotal magnitudes as deter- mined by SExtractor. Membership is then assigned based on a galaxy’s position in the diagram. If a given galaxy is within 1σ of the red cluster sequence (RCS) determined with a biweight fit, then it is considered a member (fits are shown in Figure 2). All others are considered to be non- member foreground or background galaxies. This method selects the red elliptical galaxies as members. The ben- efits of this method are that membership can easily be calculated with 2 band photometry. The drawbacks are that it both does not include some of the bluer true mem- bers and does include some of the redder non-members. An alternative method of determining cluster flux with- out spectroscopy by integrating under a background sub- tracted luminosity function is discussed in detail in §5.3 of paper I. Due to the large uncertainties involved in both methods (∼ 30%), the choice of procedure will not greatly effect the conclusions. To determine the total flux in galaxies, we sum the flux of all member galaxies within the same cluster radius. The image size of our low-redshift clusters restricts that radius to one quarter of the virial radius of the cluster where virial radii are taken from the literature or calculated from X– ray temperatures as described in §A.1-A.10. From tests with those clusters where we do have some spectroscopic membership information from the literature (see §A.3 & §A.6), we expect the uncertainty in flux from using the CMD for membership to be ∼ 30%. Fits to the CMDs produce the mean color of the red ellipticals, the slope of the color versus magnitude relation (CMR) for each cluster, and the width of that distribution. ICL 5 Among our 10 clusters, the color of the red sequence is cor- related with redshift whereas the slopes of the relations are roughly the same across redshift, consistent with López- Cruz et al. (2004). The widths of the CMRs vary from 0.1 to 0.4 magnitudes. This is expected if these clusters are made up of multiple clumps of galaxies all at similar, but not exactly the same, redshifts. True background and foreground groups and clusters can also add to the width of the RCS. In order to compare fluxes from all clusters, we consider two correction factors. First, galaxies below the detection threshold will not be counted in the cluster flux as we have measured it, and will instead contribute to the ICL flux. Since each cluster has a different detection threshold based mainly on the quality of the illumination correction (see Table 2), we calculate individually for each cluster the flux contribution from galaxies below the detection threshold. Without luminosity functions for each cluster, we adopt the Goto et al. (2002) luminosity function based on 200 Sloan clusters (α′r = −0.85 ± 0.03). The flux from dwarf galaxies below the detection threshold ( M = −11 in r) is less than or equal to 0.1% of the flux from sources above the detection threshold (our assumed value of total flux). This is an extremely small contribution due to the faint end slope, and our deep, uniform images with detection thresholds in all cases more than 7 magnitudes dimmer than M∗. Our surface brightness detection thresholds are low enough that we don’t expect to miss galaxies of nor- mal surface brightness below our detection threshold at any redshift assuming that all galaxies at all redshifts have similar central surface brightnesses. Second, we apply k and evolutionary corrections to ac- count for the shifting of the bandpasses through which we are observing, and the evolution of the galaxy spectra due to the range in redshifts we observe. We use Poggianti (1997) for both of these corrections as calculated for sim- ple stellar population of elliptical galaxies in B, V , and 5.1.2. Dynamical Age Dynamical age is an important cluster characteristic for this work as dynamical age is tied to the number of past interactions among the galaxies. We discuss four methods for estimating cluster dynamical age based on optical and X–ray imaging. The first two methods are based on cluster morphology using Bautz Morgan type and an indication of the presence of a cD galaxy. We use morphology as a proxy for dynamical age since clusters with single large ellipti- cal galaxies at their centers (cD) have presumably been through more mergers and interactions than clusters that have multiple clumps of galaxies where none have settled to the center of the potential. Those clusters with more mergers are dynamically older, therefore clusters with cD galaxies should be dynamically older. Specifically Bautz Morgan type is a measure of cluster morphology defined such that type I clusters have cD galaxies, type III clusters do not have cD galaxies, and type II clusters may show cD-like galaxies which are not centrally located. Bautz Morgan type is not reliable as Abell did not have mem- bership information. To this we add our own binary in- dicator of cluster morphology; clusters which have single galaxy peaks in the centers of their ICL distributions (cD galaxies) versus clusters which have multiple galaxy peaks in the centers of their ICL distributions (no cD). We have more information about the dynamical age of the cluster beyond just the presence or absence of a cD galaxy, namely the difference in brightness of the bright- est cluster galaxy (BCG) relative to the next few bright- est galaxies in the cluster (the luminosity gap statistic Milosavljević et al. 2006), which is our third estimate of dynamical age. Clusters with one bright galaxy that is much brighter than any of the other cluster galaxies im- ply an old dynamic age because it takes time to form that bright galaxy through multiple mergers. Conversely, mul- tiple evenly bright galaxies imply a cluster that is dynam- ically young. For our sample we measure the magnitude differences between the first (M1) and second (M2) bright- est galaxies that are considered members based on color. We run the additional test of comparing M2-M1 with M3- M1, where consistency between these values insures a lack of foreground or background contamination. Values of M3- M1 range from 0.24 to 1.1 magnitudes and are listed in Table 1. This is the most reliable measure of dynamic age available to us in this dataset. In a sample of 12 galaxy groups from N-body hydrodynamical simulations, D’Onghia et al. (2005) find a clear, strong correlation be- tween the luminosity gap statistic and formation time of the group (spearman rank coefficient of 0.91) such that δmag increases by 0.69 ± 0.41(1σ) magnitudes for every one billion years of formation. We assume this simulation is also an accurate reflection of the evolution of clusters and therefore that M3-M1 is well correlated with forma- tion time and therefore dynamical age of the clusters. The fourth method for measuring dynamical state is based on the X–ray observations of the clusters. In a sim- ulation of 9 cluster mergers with mass ratios ranging from 1:1 to 10:1 with a range of orbital properties, Poole et al. (2006) show that clusters are virialized at or shortly af- ter they visually appear relaxed through the absence of structures (clumps, shocks, cavities) or centroid shifts (X– ray peak vs. center of the X–ray gas distribution). We then assume that spherically distributed hot gas as evi- denced by the X–ray morphologies of the clusters free from those structures and centroid shifts implies relaxed clusters which are therefore dynamically older clusters that have already been through significant mergers. With enough photons, X–ray spectroscopy can trace the metallicity of different populations to determine progenitor groups or clusters. X–ray observations are summarized in §A.1 - §A.10. 5.1.3. Global Density Current global cluster density is an important cluster characteristic for this work as density is correlated with the past interaction rate among galaxies. We would like a measure of the number of galaxies in each of the clusters within some well defined radius which encompasses the po- tentially dynamically active regions of the cluster. Abell chose to calculate global density as the number of galaxies with magnitudes between that of the third ranked member, M3, and M3+2 within 1.5 Mpc of the cluster, statistically correcting for foreground and background galaxy contami- nation with galaxy densities outside of 1.5Mpc (Abell et al. 1989). The cluster galaxy densities are then binned into 6 Krick, Bernstein richness classes with values of zero to three, where rich- ness three clusters are higher density than richness zero clusters. Cluster richnesses are listed in Table 1. In addition to richness class we use a measure of global density which has not been binned into coarse values and is not affected by sample completeness. To do this we count the number of member galaxies inside of 0.8 h−170 Mpc to the same absolute magnitude limit for all clusters. Mem- bership is assigned to those galaxies within 1σ of the color magnitude relation (CMR). The density may be affected by the width of the CMR if the CMR has been artificially widened due to foreground and background contamina- tion. We choose a magnitude limit of Mr = -18.5 which is deep enough to get many tens of galaxies at all clus- ters, but is shallow enough that our photometry is still complete. At the most distant clusters (z=0.31), an Mr = -18.5 galaxy is a 125σ detection. The numbers of galax- ies in each cluster that meet these criteria range from 62 - 288, and are in good agreement with the broader Abell richness determination. These density estimates are listed in Table 1. 5.2. ICL properties We detect an ICL component in all ten clusters of our sample. We describe below our methods for measuring the surface brightness profile, color, flux, and substructure in that component. 5.2.1. Surface brightness profile In eight out of 10 clusters the ICL component is central- ized enough to fit with a single set of elliptical isophotes. The exceptions are A0141 and AC118. We use the IRAF routine ellipse to fit isophotes to the diffuse light which gives us a surface brightness profile as a function of semi– major axis. The masked pixels are completely excluded from the fits. There are 3 free parameters in the isophote fitting: center, position angle (PA), and ellipticity. We fix the center and let the PA and ellipticity vary as a func- tion of radius. Average ICL ellipticities range from 0.3 to 0.7 and vary smoothly if at all within each cluster. The PA is notably coincident with that of the cD galaxy where present (discussed in §A.1 - A.10). We identify the surface brightness profile of the total cluster light (ie., including resolved galaxies) for compari- son with the ICL within the same radial extent. To do this, we make a new “cluster” image by masking non-member galaxies as determined from the color magnitude relation (§5.1.1). A surface brightness profile of the cluster light is then measured from this image using the same elliptical isophotes as were used in the ICL profile measurement. Figure 3 shows the surface brightness profiles of all eight clusters for which we can measure an ICL profile. Individ- ual ICL profiles in both r− and V− or B−bands are shown in Figures 4 - 13. Results based on all three versions of mask size (as discussed in §4.2.2) are shown via shading on those plots. Note that we are not able to directly measure the ICL at small radii (<∼ 70kpc) in any of the clusters because greater than 75% of those pixels are masked. The uncertainty in the ICL surface brightness is dominated by the accuracy with which the background level can be iden- tified, while the error on the mean within each elliptical isophote is negligible, as discussed in §5.3. Error bars in Figures 3 and 4 - 13 show the 1σ uncertainty based on the error budget for each cluster (see representative error budget in Table 3). The ICL surface brightness profiles have two interesting characteristics. First, in all cases they can be fit by both exponential and deVaucouleurs profiles. Both appear to perform equally well given the large error bars at low sur- face brightness. These profiles, in contrast to the galaxy profiles, are relatively smooth, only occasionally reflect- ing the clustering of galaxies. Second, the ICL is more concentrated than the galaxies, which is to say that the ICL falls off more rapidly with increased radius than the galaxy light. In all cases the ICL light is decreasing rapidly enough at large radii such that the additional flux beyond the radius at which we can reliable measure the surface brightness is at most 10% of the flux inside of that radius based on an extrapolation of the exponential fit. There are 2 clusters (A0141, Figure 11 & AC118, Fig- ure 13) for which there is no single centralized ICL profile. These clusters do not have a cD galaxy, and their giant el- lipticals are distant enough from each other that the ICL is not a continuous centralized structure. We therefore have no surface brightness profile for those clusters although we are still able to measure an ICL flux, as discussed below. We attempt to measure the profile of the cD galaxy where present in our sample. To do this we remove the mask of that galaxy and allow ellipse to fit isophotes all the way into the center. In 5 out of 7 clusters with a cD galaxy, the density of galaxies at the center is so great that just removing the mask for the cD galaxy is not enough to reveal the center of the cluster due to the other over- lapping galaxies. Only for A4059 and A2734 are we able to connect the ICL profile to the cD profile at small radii. These are shown in Figures 4 & 6. In both cases the entire profile of the cD plus ICL is well fit by a single DeVaucouleurs profile, although it can also be fit by 2 DeVaucouleurs profiles. The profiles can not be fit with single exponential functions. We do not see a break between the cD and ICL profiles as seen by Gonzalez et al. (2005). While those authors find that breaks in the extended BCG profile are common in their sample, ∼ 25% of the BCG’s in that sample did not show a clear pref- erence for a double deVaucouleurs model over the single deVaucouleurs model. In both clusters where we measure a cD profile, the color appears to start out with a blue color gradient, and then turn around and become increas- ingly redder at large radii as the ICL component becomes dominant (see Figures 4 & 6). 5.2.2. ICL Flux The total amount of light in the ICL and the ratio of ICL flux to total cluster flux can help constrain the impor- tance of galaxy disruption in the evolution of clusters. As some clusters have cD galaxies in the centers of their ICL distribution, we need a consistent, physically motivated method of measuring ICL flux in the centers of those clus- ters as compared to the clusters without a single central- ized galaxy. The key difference here is that in cD clusters the ICL stars will blend smoothly into the galaxy occupy- ing the center of the potential well, whereas with non-cD clusters the ICL stars in the center are unambiguous. Since our physical motivation is to understand galaxy interac- ICL 7 tions, we consider ICL to be all stars which were at some point stripped from their original host galaxies, regardless of where they are now. In the case of clusters with cD galaxies, although we cannot separate the ICL from the galaxy flux in the cen- ter of the cluster, we can measure the ICL profile outside of the cD galaxy. Gonzalez et al. (2005) have shown for a sample of 24 clusters that a BCG with ICL halo can be well fit with two deVaucouleurs profiles. The two profiles imply two populations of stars which follow different or- bits. We assume stars on the inner profile are cD galaxy stars and those stars on the outer profile are ICL stars. Gonzalez et al. (2005) find that the outer profile on aver- age accounts for 80% of the combined flux and becomes dominant at 40-100kpc from the center which is at sur- face brightness levels of 24 - 25 mag arcsec−2 in r. Since all of our profiles are well beyond this radius and well be- low this surface brightness level, we conclude that the ICL profile we identify is not contaminated by cD galaxy stars. Assuming that the stars on the outer profile have differ- ent orbits than the stars on the inner profile, we calculate ICL flux by summing all the light in the outer profile from a radius of zero to the radius at which the ICL becomes undetectable. Note that this method identifies ICL stars regardless of their current state as bound or unbound from the cD galaxy. We therefore calculate ICL flux by first finding the mean surface brightness in each elliptical annuli where all masked pixels are not included. This mean flux is then summed over all pixels within that annulus including the ones which were masked. This represents a difference from paper I where we performed an integration on the fit to the ICL profile; here we sum the profile values themselves. We are justified in using the area under the galaxy masks for the ICL sum since the galaxies only account for less than 3% of the volume of the cluster regardless of projected area. There are two non-cD clusters (A141 & A118) for which we could not recover a profile. We calculate ICL flux for those clusters by measuring a mean flux within three con- centric, manually–placed, elliptical annuli (again not uti- lizing masked pixels) in the mean, and then summing that flux over all pixels in those annuli. All ICL fluxes are subject to the same k and evolutionary corrections as dis- cussed in §5.1.1. 5.2.3. ICL Fraction In addition to fluxes, we present the ratio of ICL flux to total cluster flux, where total cluster flux includes ICL plus galaxy flux. Galaxy flux is taken from the CMDs out to 0.25rvirial, as discussed in §5.1.1. ICL fractions range from 6 to 22% in the r−band and 4 to 21% in the B−band where the smallest fraction comes from A2556 and the largest from A4059. All fluxes and fractions are listed in Table 1. As mentioned in §5.1.1, there is no perfect way of measuring cluster flux without a complete spectroscopic survey. Based on those clusters where we do have some spectroscopic information, we estimate the uncertainty in the cluster flux to be ∼ 30% . This includes both the ab- sence from the calculation of true member galaxies, and the false inclusion of non-member galaxies. All cluster fluxes as measured from the RCS do not in- clude blue member galaxies so those fluxes are potentially lower limits to the true cluster flux, implying that the ICL fractions are potentially biased high. This possible bias is made more complicated by the known fact that not all clusters have the same amount of blue member galaxies (Butcher & Oemler 1984). Less evolved clusters (at higher redshifts) will have higher fractions of blue galaxies than more evolved clusters (at lower redshifts). Therefore ICL fractions in the higher redshift clusters will be systemati- cally higher than in the lower redshift clusters since their fluxes will be systematically underestimated. We estimate the impact of this effect using blue fractions from Couch et al. (1998) who find maximal blue fractions of 60% of all cluster galaxies at z = 0.3 as compared to ∼ 20% at the present epoch. If none of those blue galaxies were included in our flux measurement for AC114 and AC118 (the two highest z clusters), this implies a drop in ICL fraction of ∼ 40% as compared to ∼ 10% at the lowest redshifts. This effect will strengthen the relations discussed below. Most simulations use a theoretically motivated defini- tion of ICL which determine its fractional flux within r200 or rvir. It is not straightforward to compare our data to those simulated values since our images do not extend to the virial radius nor do they extend to infinitely low surface brightness which keeps us from measuring both galaxy and ICL flux at those large radii. The change in fractional flux from 0.25rvir to rvir will be related to the relative slopes of the galaxies versus ICL. As the ICL is more centrally concentrated than the galaxies we expect the fractional flux to decrease from 0.25rvir to rvir since the galaxies will contribute an ever larger fraction to the total cluster flux at large radii. We estimate what the fraction at rvir would be for 2 clusters in our sample, A4059 and A3984 (steep profile and shallow profile respectively), by extrap- olating the exponential fits to both the ICL and galaxy profiles. Using the extrapolated flux values, the fractional flux decreases by 10% where ICL and galaxy profiles are steep and up to 90% where profiles are shallower. 5.2.4. Color For those clusters with an ICL surface brightness pro- file we measure a color profile as a function of radius by binning together three to four points from the surface brightness profile. All colors are k corrected and evolution corrected assuming a simple stellar population (Poggianti 1997). Color profiles range from flat to increasingly red or increasingly blue color gradients (see Figures 14). We fit simple linear functions to the color profiles with their cor- responding errors. To determine if the color gradients are statistically significant we look at the ±2σ values on the slope of the linear fit. If those values do not include zero slope, then we assume the color gradient is real. Color er- ror bars are quite large, so in most cases 2σ does include a flat profile. The significant color gradients (A4010, A3888, A3984) are discussed in §A.1 - A.10. For all clusters an average ICL color is used to compare with cluster properties. In the case where there is a color gradient, that average color is taken as an average of all points with error bars less than one magnitude. 5.2.5. ICL Substructure Using the technique of unsharp masking (subtracting a smoothed version of the image from itself) we scan each 8 Krick, Bernstein cluster for low surface brightness (LSB) tidal features as evidence of ongoing galaxy interactions and thus possible ongoing contribution to the ICL . All 10 clusters do have multiple LSB features which are likely from tidal interac- tions between galaxies, although some are possibly LSB galaxies seen edge on. For example we see multiple inter- acting galaxies and warped galaxies, as well as one shell galaxy. For further discussion see §6.5 of paper I. From the literature we know that the two highest redshift clusters in the sample (AC114 and AC118, z=0.31) have a higher fraction of interacting galaxies than other clusters (∼ 12% of galaxies, Couch et al. 1998). In two of our clusters, A3984 and A141, there appears to be plume-like structure in the diffuse ICL, which is to say that the ICL stretches from the BCG towards another set of galaxies. Of this sample, only A3888 has a large, hundred kpc scale, arc type feature, see Figure 9 and Table 2 of paper I. There are ∼ 4 examples of these large features in the literature (Gregg & West 1998; Calcáneo-Roldán et al. 2000; Feld- meier et al. 2004; Mihos et al. 2005). These structures are not expected to last longer than a few cluster cross- ing times, so we don’t expect that they must exist in our sample. Furthermore, it is possible that there is signifi- cant ICL substructure below our surface brightness limits (Rudick et al. 2006). 5.2.6. Groups In seven out of 10 clusters the diffuse ICL is determined by eye to be multi-peaked (A4059,A2734,A3888,A3984,A141,AC114,AC118). In some cases those excesses surround the clumps of galax- ies which appear to all be part of the same cluster, ie the clumps are within a few hundred kpc from the cen- ter but have obvious separations, and there is no central dominant galaxy (eg., A118). In other cases, the sec- ondary diffuse components are at least a Mpc from the cluster center (eg., A3888). In these cases, the secondary diffuse light component is likely associated with groups of galaxies which are falling in toward the center of the cluster, and may be at various different stages of merg- ing at the center. This is strong evidence for ICL cre- ation in group environments, which is consistent with re- cent measurements of a small amount of ICL in isolated galaxy groups (Castro-Rodŕıguez et al. 2003; Durrell et al. 2004; Rocha & de Oliveira 2005). This is also consistent with current simulations (Willman et al. 2004; Fujita 2004; Gnedin 2003b; Rudick et al. 2006; Sommer-Larsen 2006, and references therein). From the theory, we expect ICL formation to be linked with the number density of galax- ies. Since group environments can have high densities at their centers and have lower velocity dispersions, it is not surprising that groups have ICL flux associated with them. Sommer-Larsen (2006) find the intra-group light to have very similar properties to the ICL making up 12 − 45% of the group light, having roughly deVaucouleurs profiles, and in general varying in flux from group to group where groups with older dynamic ages (fossil groups D’Onghia et al. 2005) have a larger amount of ICL. Groups in indi- vidual clusters are discussed in §A.1 - A.10. 5.3. Accuracy Limits The accuracy of the ICL surface brightness is limited on small scales (< 10′′) by photon noise. On larger scales (> 10′′), structure in the background level (be it intrinsic or instrumental) will dominate the error budget. We de- termine the stability of the background level in each clus- ter image on large scales by first median smoothing the masked image by 20′′. We then measure the mean flux in thousands of random 1′′ regions more distant than 0.8 Mpc from the center of the cluster. The standard deviation of these regions represents the accuracy with which we can measure the background on 20′′ scales. We tested the accu- racy of this measure for even larger-scale uncertainties on two clusters (A3880 from the 40” data and A3888 from the 100” data). We find that the uncertainty remains roughly constant on scales equal to, or larger than, 20′′. These accuracies are listed for each cluster in Table 2. Regions from all around the frame are used to check that this es- timate of standard deviation is universal across the image and not affected by location in the frame. This empiri- cal measurement of the large-scale fluctuations across the image is dominated by the instrumental flat-fielding ac- curacy, but includes contributions from the bias and dark subtraction, physical variations in the sky level, and the statistical uncertainties mentioned above. We examine the effect of including data taken under non-photometric conditions on the large-scale background illumination. This noise is fully accounted for in the mea- surement described above. All B− and V− band data were taken on photometric nights. Five clusters include varying fractions of non-photometric r− band data; 47% of A3880, 12% of A3888, 15% of A3984, 48% of A141, and 14% of A114 are non-photometric. For A3880, the clus- ter with one of the largest fractions of non-photometric data, we compare the measured accuracy on the combined image which includes the non-photometric data with accu- racy measured from a combined image which includes only photometric frames. The resulting large-scale accuracy is 0.3 mag arcsec−2better on the frame which includes only photometric data. Although this does imply that the non- photometric frames are noisier, the added signal strength gained from having 4.5 more hours on source outweighs the extra noise. This empirical measurement of the large–scale back- ground fluctuations is likely to be a conservative estimate of the accuracy with which we can measure surface bright- ness on large scales because it is derived from the outer regions of the image where compared to the central re- gions on average a factor of ∼ 2 fewer individual exposures have been combined for the 100” data and a factor of 1.5 for the 40” (which has a larger field of view and requires less dithering). A larger number of dithered exposures at a range of airmass, lunar phase, photometric conditions, time of year, time of night, and distance to the moon has the effect of smoothing out large-scale fluctuations in the illumination pattern. We therefore expect greater accu- racy in the center of the image where the ICL is being measured. We include a list all sources of uncertainty for one cluster in our sample (A3888) in Table 3 (reproduced here from Paper I). In addition to the dominant uncertainty due to the large-scale fluctuations on the background as discussed above, we quantify the contributions from the photometry, masking, and the accuracy with which we can measure the mean in the individual elliptical isophotes. Errors for the ICL 9 other clusters are similarly dominated by background fluc- tuations, which are listed in Table 2. The errors on the total ICL fluxes in all bands range from 17% to 70% with an average of 39%. The exception is A2556 which reaches a flux error of 100% in the B−band due to its extremely faint profile (see §A.4). Assuming a 30% error in the galaxy flux (see §5.1.1), the errors on the ICL fraction are on average 48%. The errors plotted on the surface brightness profiles are the 1σ errors. 6. discussion We measure a diffuse intracluster component in all ten clusters in our sample. Clues to the physical mechanisms driving galaxy evolution come from comparing ICL prop- erties with cluster properties. We have searched for corre- lations between the entire set of properties. Pairs of prop- erties not explicitly discussed below showed no correla- tions. Limited by a small sample and non-parametric data, we use a Spearman rank test to determine the strength of any possible correlations where 1.0 or -1.0 indicate a definite correlation or anti–correlation respectively, and 0 indicates no correlation. Note that this test does not take into account the errors in the parameters, and instead only depends on their rank among the sample. Where a corre- lation is indicated we show the fit as well as ±2σ in both y-intercept and slope to graphically show the ranges of the fit, and give some estimate of the strength of the correla- tion. There are selection biases in our data between cluster parameters due to our use of an Abell selected sample. The Abell cluster sample is incomplete at high redshifts; it does not include low-mass, low-luminosity, low-density, high-redshift clusters because of the difficulty in obtaining the required sensitivity with increasing redshift. Although our 5 low-redshift clusters are not affected by this selection effect, and should be a random sampling, small numbers prevent those clusters from being fully representative of the entire range of cluster properties. Specifically we discuss the possibility that there is a real trend underlying the selection bias in the cases of lower luminosity (Figure 15) and lower density clusters (Figure 16) being preferentially found at lower redshift. Clusters in our sample with less total galaxy flux are preferentially found at low redshifts, however hierarchical formation pre- dicts the opposite trend; clusters should be gaining mass over time and hence light over time. Note that on size scales much larger than the virial radius mass does not change with time and therefore those systems can be con- sidered as closed boxes; but on the size scales of our data, a quarter of a virial radius, clusters are not closed boxes. We might expect a slight trend, as was found, such that lower density clusters are found at lower redshifts. As a cluster ages, it converts a larger number of galaxies into a smaller number of galaxies via merging and therefore has a lower density at lower redshifts despite being more massive than high redshift clusters. The infall of galaxies works against this trend. The sum total of merger and infall rates will control this evolution of density with red- shift. The observed density redshift relation for this sam- ple is strong; over the range z=0.3 - 0.05 (elapsed time of 3Gyr assuming standard ΛCDM) the projected number density of galaxies has to change by a factor of 5.5, imply- ing that every 5.5 galaxies in the cluster must have merged into 1 galaxy in the last 3 Gyr. This is well above a re- alistic merger rate for this timescale and this time period (Gnedin 2003a). Instead it is likely that we are seeing the result of a selection effect. An interesting correlation which may be indirectly due to the selection bias is that clusters with less total galaxy flux tend to have lower densities (Figure 17). While we ex- pect a smaller number of average galaxies to emit a smaller amount of total light, it is possible that the low density clusters are actually made up of a few very bright galaxies. So although the trend might be real, it is also likely that the redshift selection effect of both density and cluster flux is causing these two parameters to be correlated. A correlation which does not appear to be affected by sample selection is that lower density clusters in our sam- ple are weakly correlated with the presence of a cD galaxy, see Figure 18. A possible explanation for this is that as a cluster ages it will have made a cD galaxy out of many smaller galaxies, so the density will actually be lower for dynamically older clusters. Loh & Strauss (2006) find the same correlation by looking at a sample of environments around 2000 SDSS luminous red galaxies. In the remainder of this section we examine the inter- esting physics that can be gleaned from the combination of cluster properties and ICL properties given the above biases. The interpretation of ICL correlations with clus- ter properties is highly complicated due not only to small number statistics and the selection bias, but to the direc- tion of the selection bias. Biases in mass, density, and total galaxy flux with redshift will destructively combine to cancel the trends which we expect to find in the ICL (as described in the introduction). An added level of compli- cation is due to the fact that we expect the ICL flux to be evolving with time. We examine below each ICL property in turn, including how the selection bias will effect any conclusions drawn from the observed trends. 6.1. ICL flux We see a range in ICL flux likely caused by the differ- ing interaction rates and therefore differing production of tidal tails, streams, plumes, etc. in different clusters. Clus- ters include a large amount of tidal features at low surface brightness as evidenced by their discovery at low redshift where they are not as affected by surface brightness dim- ming (Mihos et al. 2005). It is therefore not surprising that we see a variation of flux levels in our own sample. ICL flux is apparently correlated with three cluster pa- rameters; M3-M1, density, and total galaxy flux (Figures 19, 17, & 20). There is no direct, significant correlation between ICL flux and redshift. As discussed above, the se- lection effects of density and mass with redshift will tend to cancel any expected trends in either density, mass, or redshift. We therefore are unable to draw conclusions from these correlations. Zibetti et al. (2005), who have a sam- ple of 680 SDSS clusters, are able to split their sample on both richness and magnitude of the BCG (as a proxy for mass). They find that both richer clusters and brighter BCG clusters have brighter ICL than poor or faint clus- ters. 6.1.1. ICL Flux vs. M3-M1 10 Krick, Bernstein Figure 19 shows the moderate correlation between ICL flux and M3-M1 such that clusters with cD galaxies have less ICL than clusters without cD galaxies (Spearman coef- ficient of -0.50). Although we choose M3-M1 to be cautious about interlopers, M2-M1 shows the same trend with a slightly more significant spearman coefficient of -0.61. Our simple binary indicator of the presence of a cD galaxy gives the same result. Clusters with cD galaxies (7) have an av- erage flux of 2.3±0.96×1011(1σ) whereas clusters without cD galaxies (3) have an average flux of 5.0±0.18×1011(1σ). Although density is correlated with M3-M1, and density is affected by incompleteness, this trend of ICL flux with M3-M1 is not necessarily caused by that selection effect. Furthermore, the correlation of M3-M1 with redshift is much weaker (if there at all) than trends of either density or cluster flux with redshift. If the observed relation is due to the selection effect then we are prevented from drawing conclusions from this relation. Otherwise, if this relation between ICL flux and the presence of a cD galaxy is not caused by a selection effect, then we conclude that the lower levels of measured ICL are a result of the ICL stars being indistinguishable form the cD galaxy and therefore the ICL is evolving in a similar way to a cD galaxy. By which physical mechanism can the ICL stars end up in the center of the cluster and therefore overlap with cD stars? cD galaxies indicate multiple major mergers of galaxies which have lost enough energy or angular mo- mentum to now reside in the center of the cluster potential well. ICL stars on their own will not be able to migrate to the center over any physically reasonable timescales unless they were stripped at the center, or are formed in groups and get pulled into the center along with their original groups(Merritt 1984). Assuming the ICL is observationally inseparable from the cD galaxy, we investigate how much ICL light the mea- sured relation implies is hidden amongst the stars of the cD galaxy. If 20% of the total cD + ICL light is added to the value of the ICL flux in the outer profile, then the observed trend of ICL flux with M3-M1 is weakened (Spearman co- efficient drops from 0.5 to 0.4). If 30% of the total cD + ICL light is hidden in the inner profile then the relation disappears (Spearman coefficient of 0.22). The measured relation between ICL r−band flux and dynamical age of the clusters may then imply that 25-40% of the ICL is coin- cident with the cD galaxy in dynamically relaxed clusters. 6.2. ICL fraction We focus now on the fraction of total cluster light which is in the diffuse ICL. If ICL and galaxy flux do scale to- gether (not just due to the selection effect), then the ICL fraction is the physically meaningful parameter in compar- ison to cluster properties. ICL fraction is apparently correlated with both mass and redshift (Figure 21 & 22) and not with density or total galaxy flux. The selection effect will again work against the predicted trend of ICL fraction to increase with in- creasing mass (Murante et al. 2004; Lin & Mohr 2004) and increasing density. Therefore the lack of trends of ICL fraction with mass and density could be attributable to the selection bias. 6.2.1. ICL fraction vs. Mass We find no trend in ICL fraction with mass. Our data for ICL fraction as a function of mass is inconsistent with the theoretical predictions of Murante et al. (2004), Mu- rante et al. (2007) (based on a cosmological hydrodynam- ical simulation including radiative cooling, star formation, and supernova feedback), and Lin & Mohr (2004)(based on a model of cluster mass and the luminosity of the BCG). However Murante et al. (2007) show a large scatter of ICL fractions within each mass bin. They also discuss the dependence of a simulations mass resolution on the ICL fraction. These theoretical predictions are over-plotted on Figure 21. Note that the simulations generally report the fractional light in the ICL out to much larger radii (rvirial or r200) than its surface brightness can be measured ob- servationally. To compare the theoretical predictions at rvirial to our measurement at 0.25rvirial, the predicted values should be raised by some significant amount which depends on the ICL and galaxy light profiles at large radii. This makes the predictions and the data even more incon- sistent than it first appears. As an example of the differ- ences, a cluster with the measured ICL fraction of A3888 would require a factor of greater than 100 lower mass than the literature values to fall along the predicted trend. Al- though these clusters are not dynamically relaxed, such large errors in mass are not expected. As an upper limit on the ICL flux, if we assumed the entire cD galaxy was made of intracluster stars, that flux plus the measured ICL flux would still not be enough to raise the ICL fractions to the levels predicted by these authors. There are no evident correlations between velocity dis- persion and ICL characteristics, although velocity disper- sion is a mass estimator. Large uncertainties are presum- ably responsible for the lack of correlation. 6.2.2. ICL fraction vs. Redshift Figure 22 is a plot of redshift versus ICL fraction for both the r− andB−or V−bands. We find a marginal anti– correlation between ICL fraction and redshift with a very shallow slope, if at all, in the direction that low redshift clusters have higher ICL fractions (Spearman rank coeffi- cient of -0.43). This relation is strengthened when assum- ing fractions of blue galaxies are higher in the higher red- shift clusters(spearman rank of -0.6) (see §5.2.3). A trend of ICL fraction with redshift tells us about the timescales of the mechanisms involved in stripping stars from galax- ies. This relation is possibly affected by the same redshift selection effects as discussed above. Over the redshift range of our clusters, 0.31 > z > 0.05, a chi–squared fit to our data gives a range of fractional flux of 11 to 14%. Willman et al. (2004) find the ICL fraction grows from 14 to 19%. over that same redshift range. Willman et al. (2004) measure the ICL fraction at r200 which means these values would need to be increased in order to directly compare with our values. While their normalization of the relation is not consistent with our data, the slopes are roughly consistent, with the caveat of the selection effect. The discrepancy is likely, at least in part, caused by different definitions of ICL. Simulations tag those particles which become unbound from galax- ies whereas in practice we do not have that information and instead use surface brightness cutoffs and ICL profile shapes. Rudick et al. (2006) do use a surface brightness ICL 11 cutoff in their simulations to tag ICL stars which is very similar to our measurement. They find on average from their 3 simulated clusters a change of ICL fraction of ap- proximately 2% over this redshift range. We are not able to observationally measure such a small change in fraction. Rudick et al. (2006) predict that in order to grow the ICL fraction by 10%, on average, we would need to track clus- ters as they evolve from a redshift of 2 to the present. However, both Willman et al. (2004) and Rudick et al. (2006) find that the ICL fraction makes small changes over short timescales (as major mergers or collisions occur). 6.3. ICL color The average color of the ICL, is roughly the same as the color of the red ellipticals in each of the clusters. In §8.1 of paper I we discuss the implications of this on ICL formation redshift and metallicity. Zibetti et al. (2005) have summed g−, r−, and i− band imaging of 680 clus- ters in a redshift range of 0.2 - 0.3. Similar to our results, they find that the summed ICL component has roughly the same g − r color at all radii as the summed cluster population including the galaxies. Since we have applied an evolutionary correction to the ICL colors, if there is only passive color evolution, the ICL will show no trend with redshift. Indeed we find no correlation between B−r color and the redshift of the cluster, as shown in Figure 23 (B − r = 2.3 ± 0.2(1σ)). ICL color may have the ability to broadly constrain the epoch at which these stars were stripped. In principle, as mentioned in the introduction, we could learn at which epoch the ICL had been stripped from the galaxies based on its color relative to the galaxies assuming passively evolving ICL and ongoing star forma- tion in galaxies. While this simple theory should be true, the color difference between passively evolving stars and low star forming galaxies may not be large enough to de- tect since clusters are not made up of galaxies which were all formed at a single epoch and we don’t know the star formation rates of galaxies once they enter a cluster. ICL color may have the ability to determine the types of galaxies from which the stars are being stripped. Un- fortunately the difference in color between stars stripped from ellipticals, and for example stars stripped from low surface brightness dwarfs is not large enough to confirm in our data given the large amount of scatter in the color of the ICL (see paper I for a more complete discussion). There is no correlation in our sample between the pres- ence or direction of ICL color gradients and any cluster properties. This is very curious since we see both blue- ward and red-ward color gradients. A larger sample with more accurate colors and without a selection bias might be able to determine the origin of the color gradients. 6.4. Profile Shape Figure 3 shows all eight surface brightness profiles for clusters that have central ICL components. To facilitate comparison, we have shifted all surface brightnesses to a redshift of zero, including a correction for surface bright- ness dimming, a k–correction, and an evolution correction. We see a range in ICL profile shape from cluster to cluster. This is consistent with the range of scale-lengths found in other surveys (Gonzalez et al. 2005, find a range of scale lengths from 18 - 480 kpc, fairly evenly distributed be- tween 30 and 250 kpc) . The profiles are equally well fit with the empirically motivated deVaucouleurs profiles and simple exponential profiles which are shown in the individual profile plots in Figures 4 - 13. The profiles can also be fit with a Hubble– Reynolds profile which is a good substitute for the more complicated surface brightness profile of an NFW density profile ( Lokas & Mamon 2001). An example of this profile shape is shown in Figure 3 with a 100 kpc scale length de- fined as the radius inside of which the profile contains 25% of the luminosity. This profile shape is what you would predict given a simple spherical collapse model. The phys- ically motivated Hubble–Reynolds profile gives acceptable fits to the ICL profiles with the exception of A4059, A2734, & A2556 which have steeper profiles. We explore causes of the differing profile shapes for these three clusters. A steeper profile is correlated with M3-M1, density, to- tal cluster flux, and redshift. These three clusters have an average M3-M1 value of 0.93 ± 0.27 as compared to the average of 0.49± 0.20 for the remaining 7 clusters. These three clusters are also three of the four lowest redshift clus- ters, have an average of 93 galaxies which is 45% smaller than the value for the remaining sample, and have an av- erage cluster flux of 12.3 × 1011L�which is 47% smaller than the value for the remaining sample. We have the same difficulties here in distinguishing be- tween the selection effects and the true physical correla- tions. The key difference is that the three clusters with the steepest profiles are the most relaxed clusters (which is not a redshift selection effect). We use “most relaxed” to de- scribe the three clusters with the most symmetric X–ray isophotes that have single, central, smooth ICL profiles. This is consistent with our finding that M3-M1 is a key in- dicator of ICL flux in §6.1.1 and that ICL can form either in groups at early times or at later times through galaxy in- teractions in the dense part of the cluster. If galaxy groups in which the ICL formed are able to get to the cluster cen- ter then their ICL will also be found in the cluster center, and can be hiding in the cD galaxy. If the galaxy groups in which the ICL formed have not coalesced in the center then the ICL will be less centrally distributed and there- fore have a shallower profile. This is consistent with the recent numerical work by Murante et al. (2007) who find that the majority of the ICL is formed by the merging processes which create the BCG’s in clusters. This pro- cess leads o the ICL having a steeper profile shape than the galaxies and having greater than half of the ICL be located inside of 250h−170 kpc, approaching radii where we do not measure the ICL due to the presence of the BCG. Their simulations also confirm that different clusters with different dynamical histories will have differing amounts and locations of ICL. 7. conclusion We have identified an intracluster light component in all 10 clusters which has fluxes ranging from 0.76 × 1011 to 7.0×1011 h−170 L�in r and 0.14×10 11 to 1.2×1011 h−170 L�in the B−band, ICL fractions of 6 to 22% of the total cluster light within one quarter of the virial radius in r and 4 to 21% in the B−band, and B−r colors ranging from 1.49 to 2.75 magnitudes. This work shows that there is detectable 12 Krick, Bernstein ICL in clusters and groups out to redshifts of at least 0.3, and in two bands including the shorter wavelength B− or V−band. The interpretation of our results is complicated by small number statistics, redshift selection effects of Abell clus- ters, and the fact that the ICL is evolving with time. Of the cluster properties (M3-M1, density, redshift, and clus- ter flux), only M3-M1 and redshift are not correlated. As a result of these selection effects ICL flux is apparently correlated with density and total galaxy flux but not with redshift or mass and ICL fraction is apparently correlated with redshift but not with M3-M1, density, total galaxy flux, or mass. However, we do draw conclusions from the ICL color, average values of the ICL fractions, the relation between ICL flux and M3-M1, and the ICL profile shape. We find a passively evolving ICL color which is similar to the color of the RCS at the redshift of each cluster. The relations between ICL fraction with redshift and ICL fraction with mass show the disagreement of our data with simulations since our fractional fluxes are lower than those predictions. These discrepancies do not seem to be caused by the details of our measurement. Furthermore we find evidence that clusters with sym- metric X–ray profiles and cD galaxies have both less ICL flux and significantly steeper profiles. The lower amount of flux can be explained if ICL stars have become in- distinguishable from cD stars. As the cluster formed a cD galaxy any groups which participated in the merging brought their ICL stars with them, as well as created more ICL through interactions. If a cD does not form, then the ICL already in groups or actively forming is also prevented from becoming very centralized as it has no way of loos- ing energy or angular momentum on its own. While the galaxies or groups are subject to tidal forces and dynam- ical friction, the ICL, once stripped, will not be able to loose energy and/or angular momentum to these forces, and instead will stay on the orbit on which it formed. Observed density may not be a good predictor of ICL properties since it does not directly indicate the density at the time in which the ICL was formed. We do indeed expect density at any one epoch to be linked to ICL pro- duction at that epoch through the interaction rates. The picture that is emerging from this work is that ICL is ubiquitous, not only in cD clusters, but in all clus- ters, and in group environments. The amount of light in the ICL is dependent upon cluster morphology. ICL forms from ongoing processes including galaxy–galaxy in- teractions and tidal interactions with the cluster potential (Moore et al. 1996; Gnedin 2003b) as well as in groups (Rudick et al. 2006). With time, as multiple interactions and dissipation of angular momentum and energy lead groups already containing ICL to the center of the cluster, the ICL moves with the galaxies to the center and be- comes indistinguishable from the cD’s stellar population. Any ICL forming from galaxy interactions stays on the orbit where it was formed. A large, complete sample of clusters, including a pro- portionate amount with high redshift and low density, will be able to break the degeneracies present in this work. Shifting to a lower redshift range will not be as benefi- cial because a shorter range than presented here will not be large enough to see the predicted evolution in the ICL fraction. In addition to large numbers of clusters it would be beneficial to go to extremely low surface brightness lev- els (<∼ 30 mag arcsec−2) to reduce significantly the error bars on the color measurement and thereby learn about the progenitor galaxies of the ICL and the timescales for strip- ping. It will not be easy to achieve these surface brightness limits for a large sample which includes high-redshift low- density clusters since those clusters will have very dim ICL due to both an expected lower amount as correlated with density, and due to surface brightness dimming. We acknowledge J. Dalcanton and V. Desai for observ- ing support and R. Dupke, E. De Filippis, and J. Kemp- ner for help with X–ray data. We thank the anonymous referee for useful suggestions on the manuscript. Par- tial support for J.E.K. was provided by the National Sci- ence Foundation (NSF) through UM’s NSF ADVANCE program. Partial support for R.A.B. was provided by a NASA Hubble Fellowship grant HF-01088.01-97A awarded by Space Telescope Science Institute, which is operated by the Association of Universities for Research in As- tronomy, Inc., for NASA under contract NAS 5-2655. This research has made use of data from the following sources: USNOFS Image and Catalogue Archive operated by the United States Naval Observatory, Flagstaff Station (http://www.nofs.navy.mil/data/fchpix/); NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Tech- nology, under contract with the National Aeronautics and Space Administration; the Two Micron All Sky Sur- vey, which is a joint project of the University of Mas- sachusetts and the Infrared Processing and Analysis Cen- ter/California Institute of Technology, funded by the Na- tional Aeronautics and Space Administration and the Na- tional Science Foundation; the SIMBAD database, oper- ated at CDS, Strasbourg, France; and the High Energy Astrophysics Science Archive Research Center Online Ser- vice, provided by the NASA/Goddard Space Flight Cen- REFERENCES Abadi, M. G., Moore, B., & Bower, R. G. 1999, MNRAS, 308, 947 Abell, G. O., Corwin, H. G., & Olowin, R. 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P., & Brinkmann, J. 2005, MNRAS, 358, 949 14 Krick, Bernstein ICL 15 16 Krick, Bernstein Table 3 Error Budget Source contribution to ICL uncertainty (%) 1σ uncertainty µ(0′′- 100′′) µ(100′′- 200′′) total ICL flux (V ) (r) (V ) (r) (V ) (r) (V ) (r) background levela 29.5 mag arcsec−2 28.8 mag arcsec−2 14 18 39 45 24 31 photometry 0.02 mag 0.03 mag 2 3 2 3 2 3 maskingb variation in mask area ±30 5 5 14 19 9 12 std. dev. in meanc 32.7 mag arcsec−2 32.7 mag arcsec−2 3 2 2 1 3 1 (total) 15 19 41 50 26 33 cluster fluxd 16% 16% · · · · · · · · · · · · · · · · · · Note. — a: Large scale fluctuations in background level are measured empirically and include instrumental calibration uncertainties as well as and true variations in background level (see §5.3). b: Object masks were scaled by ±30% in area to test the impact on ICL measurement (see §4.2.2). c: The statistical uncertainty in the mean surface brightness of the ICL in each isophote. d: Errors on the total cluster flux are based on errors in the fit to the luminosity function (see §5.1.1). Fig. 1.— The PSF of the 40-inch Swope telescope at Las Campanas Observatory. The y-axis shows surface brightness scaled to correspond to the total flux of a zero magnitude star. The profile within 5′′ was measured from unsaturated stars and can be affected by seeing. The outer profile was measured from two stars with super-saturated cores imaged in two different bands. The profile with the bump in it at 100′′ is the r−band profile, that without the bump is the B−band PSF. The bump in the profile at 100′′ is due to a reflection off the CCD which then bounces off of the filter, and back down onto the CCD. The outer surface brightness profile decreases as r−2 in the r−band and r−1.6 in the B, shown by the dashed lines. An r−3.9 profile is plotted to show the range in slopes. ICL 17 Fig. 2.— The color magnitude diagrams for all ten clusters in increasing redshift order from let to right, top to bottom; A4059, A3880, A2734, A2556, A4010, A3888, A3984, A014, AC114, AC118. All galaxies detected in our image are denoted with a gray star. Those galaxies which have membership information in the literature are over–plotted with open black triangles (members) or squares (non–members)(membership references are given in §A.1- A.10). Solid lines indicate a biweight fit to the red sequence with 1σ uncertainties. 18 Krick, Bernstein Fig. 3.— Surface brightness profiles for the eight clusters with a measurable profile. Profiles are listed on the plot in order of ascending redshift. To avoid crowding, error bars are only plotted on one of the profiles. Errors on the other profiles are similar at similar surface brightnesses. All surface brightnesses have been shifted to z = 0 using surface brightness dimming, k, and evolutionary corrections. The x-axis remains in arcseconds and not in Mpc since the y-axis is in reference to arcseconds. Physical scales are noted on the individual plots (4 - 13). In addition marks have been placed on each profile at the distances corresponding to 200kpc and 300kpc. Also included as the solid black line near the bottom of the plot is a Hubble Reynolds surface brightness profile as a proxy for an NFW density profile with a scale length of 100kpc. The ICL does not have a single uniform amount of flux or profile shape. Profile shape does correlate with dynamical age where those clusters with steeper profiles are dynamically more relaxed (see §6.4). ICL 19 Fig. 4.— A4059. The plots moving left to right and top to bottom are as follows. The first is our final combined r−band image zoomed in on the central cluster region. The second plot shows X–ray isophotes where available. Some clusters were observed during the ROSAT all sky survey, and so have X–ray luminosities, but have not had targeted observations to allow isophote fitting. Isophote levels are derived from quick-look images taken from HEASARC. X-ray luminosities of these clusters are listed in Table 1 of paper I and are discussed in the appendix. The third plot shows our background subtracted, fully masked r−band image of the central region of the cluster, smoothed to aid in visual identification of the surface brightness levels. Masks are shown in their intermediate levels which are listed in column 7 of Table 2. The six gray-scale levels show surface brightness levels of up to 28.5, 27.7,27.2,26.7 mag arcsec−2. The fourth plot shows the surface brightness profiles of the ICL (surrounded by shading;r−band on top, V− or B−band on the bottom) and cluster galaxies as a function of semi-major axis. The bottom axis is in arcseconds and the top axis corresponds to physical scale in Mpc. Error bars represent the 1σ background identification errors as discussed in §5.3. DeVaucouleurs fits to the entire cD plus ICL profile are over-plotted. 20 Krick, Bernstein Fig. 5.— A3880, same as Figure 4 ICL 21 Fig. 6.— A2734, same as Figure 4 22 Krick, Bernstein Fig. 7.— A2556, same as Figure 4 ICL 23 Fig. 8.— A4010, same as Figure 4 24 Krick, Bernstein Fig. 9.— A3888, same as Figure 4, except here we show the elliptical isophotes of the ICL over-plotted on the surface brightness image. ICL 25 Fig. 10.— A3984, same as Figure 4 26 Krick, Bernstein Fig. 11.— A141, same as Figure 4, except we are not able to measure a surface brightness profile or consequently a color profile. ICL 27 Fig. 12.— A114, same as Figure 4 28 Krick, Bernstein Fig. 13.— A118, same as Figure 4, except we are not able to measure a surface brightness profile or consequently a color profile. ICL 29 Fig. 14.— The color profile of the eight clusters where measurement was possible plotted as a function of semi-major axis in arcseconds on the bottom and Mpc on the top. The average color of the red cluster sequence is shown for comparison, as well as the best fit linear function to the data. 30 Krick, Bernstein Fig. 15.— Redshift versus total galaxy flux within one quarter of a virial radius. The Spearman rank coefficient is printed in the upper right corner. The best fit linear function as well as the lines representing ±2σ in both slope and y intercept are also plotted. The strong correlation between redshift and total galaxy flux shows the incompleteness of the Abell sample which does not include high-redshift, low-flux clusters ICL 31 Fig. 16.— Projected number of galaxies versus redshift. Galaxies brighter than Mr = −18.5 within 800 h−170 kpc are included in this count, which is used as a proxy for density. The Spearman rank coefficient is printed in the upper left corner. There is a strong correlation between density and redshift. The best fit linear function is included. While we do expect clusters to become less dense over time, this strong correlation is not expected. Instead this is due to an incompleteness at high redshift. See §6 for a discussion of the effects of this selection effect. 32 Krick, Bernstein Fig. 17.— Projected number of galaxies versus ICL luminosity. ICL luminosity shows 1σ error bars and has been K and evolution corrected. Galaxies brighter than Mr = −18.5 within 800 h−170 kpc are included in this count, which is used as a proxy for density. The Spearman rank coefficient is printed in the upper left corner. The best fit linear function as well as the lines representing ±2σ in both slope and y intercept are also plotted. There is a mild correlation between density and ICL luminosity such that higher density clusters have a larger amount of ICL flux. ICL 33 Fig. 18.— The difference in magnitude between the first and third ranked galaxy versus projected number of galaxies brighter than Mr = −18.5 within 800 h−170 kpc, which is used as a proxy for density. Clusters with cD galaxies will have larger M3-M1 values. This plot implies that over time galaxies merge in clusters to make a cD galaxy, and by the time the cD galaxy has formed, the global density is lower. As discussed in the §6, we assume this is not a selection bias. 34 Krick, Bernstein Fig. 19.— The difference in magnitude between the first and third ranked galaxy versus ICL luminosity. ICL luminosity shows 1σ error bars and has been K and evolution corrected. Clusters which have cD galaxies have larger M3 - M1 values and are dynamically older clusters. There is a mild correlation between dynamic age and ICL luminosity indicating that the ICL evolves at roughly the same rate as the cluster. ICL 35 Fig. 20.— The flux in galaxies versus the flux in ICL in units of solar luminosities. Errors on ICL luminosity are 1σ. Errors on galaxy luminosity are 30% as estimated in §5.1.1. Over-plotted is the best fit linear function as well as two lines which represent 2σ errors in both y-intercept and slope. The Spearman rank coefficient is printed in the upper right. Here galaxy luminosity is assumed to be a proxy for mass, so we find a significant correlation between mass and ICL flux such that more massive clusters have a larger amount of ICL flux. 36 Krick, Bernstein Fig. 21.— Cluster mass versus the ICL fraction measured at one quarter of the virial radius. Stars denote the r−band while squares show B− and diamonds show V−band. Errors on ICL fraction are 1σ as discussed in §5.3. Mass estimates and errors are taken from the literature as discussed in §A.1 - §A.10. The predictions of Lin & Mohr (2004) and Murante et al. (2004) at the virial radius are shown for comparison. These represent extrapolations beyond roughly 1×1015 M� in both cases (as marked by the crosses). The roughly constant ICL fraction with mass can be explained using hierarchical formation by the in-fall of groups with a similar ICL fraction as the main cluster, or by increased interaction rates with the infall of the groups, or both. ICL 37 Fig. 22.— Cluster redshift versus ICL fraction measured at one quarter of the virial radius. As in Figure 21, starred symbols denote the r−band, squares show B−band, and diamonds show V−band fractions. The prediction of Willman et al. (2004) for the ICL fraction as measured at r200 is shown for comparison. This prediction would increase if measured at smaller radii, such as was used in our measurement. There is mild evidence for a correlation between redshift and ICL fraction such that ICL fraction grows with decreasing redshift. This trend is consistent with ongoing ICL formation. 38 Krick, Bernstein Fig. 23.— Cluster redshift versus ICL color in B − r which has been k corrected and had simple passive evolution applied to it. If a color gradient is detected in a given cluster then the mean color plotted here is that measured near the center of the profile, weighted slightly toward the center. There is no trend in redshift with ICL color which leads to the conclusion that the ICL is simply passively reddening. ICL 39 APPENDIX the clusters In order of increasing redshift we discuss interesting characteristics of the clusters and their ICL components. Relevant papers are listed in Table 1. Relevant figures are 4 - 13. A4059 A4059 is a richness class 1, Bautz Morgan type I cluster at a redshift of 0.048. There is a clear cD galaxy which is however offset from the Abell center, likely due to the presence of at least two other bright elliptical galaxies. The cD galaxy is 0.91 ± .05 magnitudes brighter than the second ranked cluster galaxy. The cD galaxy is at the center of the Chandra and ASCA mass distributions. Those telescopes detect no hot gas around the other bright ellipticals. This cluster shows interesting features in it’s X–ray morphology. There appear to be large bubbles, or cavities in the hot gas, which is likely evidence of past radio galaxy interactions with the ICM (Choi et al. 2004). As additional evidence of past activity in this cluster, the cD galaxy contains a large dust lane (Choi et al. 2004). M500 (the mass within the radius where the mean mass density is equal to 500 times the critical density) is calculated by Reiprich & Böhringer (2002) for A4059 to be 2.82±0.370.34 ×1014h 70 M�. The color magnitude diagram shows a very tight red sequence. Membership information is taken from Collins et al. (1995), Colless et al. (2001), and Smith et al. (2004). Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 1.2± .35×1012L�in r and 4.2±1.3×1011L�in B inside of 0.65h−170 Mpc, which is one quarter of the virial radius of this cluster. In this particular cluster, since the Abell center is not at the true cluster center, and it is the nearest cluster in our sample, our image does not uniformly cover the entire one quarter of the virial radius. This estimate is therefore below the true flux in galaxies because we are missing area on the cluster. Figure 4 shows the relevant plots for this cluster. There is a strong ICL component ranging from 26 - 29 mag arcsec−2 in r centered on the cD galaxy. The total flux in the ICL is 3.4 ± 1.7 × 1011L�in r and 1.2 ± .24 × 1011L�in B, which makes for ICL fractions of 22± 12% in r and 21± 8% in B. The ICL has a flat color profile with B− r ' 1.7± .08, which is marginally bluer (0.2 magnitudes) than the RCS. One of the two other bright ellipticals at 0.7h−170 Mpc from the center has a diffuse component, the other bright elliptical is too close to a saturated star to detect a diffuse component. A3880 A3880 is a richness class 0, Bautz Morgan type II cluster at a redshift of 0.058. There is a clear cD galaxy in the center of this cluster, which is 0.52 ± .05 magnitudes brighter than the second ranked galaxy. This cluster is detected in the ROSAT All Sky Survey, however that survey is not deep enough to show us the shape of the mass distribution. Girardi et al. (1998b) find a mass for this cluster based on its velocity dispersion of 8.3+2.8−2.1 × 10 14h−170 M�. The color magnitude diagram shows a clear red sequence. There is possibly another red sequence at lower redshift adding to the width of the red sequence. Membership information is provided by Collins et al. (1995), Colless et al. (2001), and Smith et al. (2004). Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 8.6± 2.6× 1011L�in r and 3.8± 1.1× 1011L�in B inside of 0.62h−170 Mpc, which is one quarter of the virial radius of this cluster. Figure 5 shows the relevant plots for this cluster. Unfortunately this cluster has larger illumination problems than the other clusters which can be seen in the greyscale masked image. Nonetheless, there is clearly an r−band ICL component, although the B−band ICL is extremely faint. The total flux in the ICL is 1.4± 2.3× 1011L�in r and 4.4± 1.5× 1010L�in B, which makes for ICL fractions of 14±6% in r and 10±6% in B. The ICL has a flat color profile with B−r ' 2.4±1.1, which is 0.8 magnitudes redder than the RCS. A2734 A2734 is a richness class 1, Bautz Morgan type III cluster at a redshift of 0.062. The BCG by 0.51± .05 magnitudes is in the center of this cluster, however there are 2 other large elliptical galaxies 0.55h−170 Mpc and 0.85h 70 Mpc distant from the BCG. The X–ray gas does confirm the BCG as being at the center of the mass distribution. Those 2 other elliptical galaxies are not seen in the 44ks ASCA GIS observation of this cluster, however they are confirmed members based on spectroscopy (Collins et al. 1995; Colless et al. 2001; Smith et al. 2004). M500 is calculated by Reiprich & Böhringer (2002) for A2734 to be 2.49±0.890.63 ×1014h 70 M�. The color magnitude diagram shows a clear red sequence, which includes the 3 bright elliptical galaxies. 2df spectroscopy gives us roughly 80 galaxies in our field of view which we can use to estimate the effectiveness of the biweight fit to the RCS in finding true cluster members. Of those galaxies with confirmed membership, 94% are determined members with this method, however 86% of the confirmed non-members are also considered members. This is likely due to how galaxies were selected for spectroscopy in the 2df catalog. Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 1.2± .36× 1012L�in r and 3.4± 1.0× 1011L�in B inside of 0.60h−170 Mpc, which is one quarter of the virial radius of this cluster. Figure 6 shows the relevant plots for this cluster. There is a strong ICL component ranging from 26 - 29 mag arcsec−2 in r centered on the BCG. The total flux in the ICL is 2.8± .47× 1011L�in r and 7.0± 4.7× 1010L�in B, which makes for ICL fractions of 19± 6% in r and 17± 13% in B. The ICL has a flat to red-ward color profile with B − r ' 2.3± .03, 40 Krick, Bernstein which is marginally redder than the RCS (0.3 magnitudes). The cluster has a second diffuse light component around one of the giant elliptical galaxies, .55 h−170 Mpc from the center of the cD galaxy. The third bright elliptical has a saturated star just 40′′away, so we do not have a diffuse light map of that galaxy. A2556 A2556 is a richness class 1, Bautz Morgan type II-III cluster at a redshift of 0.087. Despite the Bautz Morgan classification, this cluster has a clear cD galaxy in the center of the X–ray distribution which is 0.93 ± .05 magnitudes brighter than any other galaxy in the cluster. The Chandra derived X–ray distribution is slightly elongated toward the NE where a second cluster, A2554, resides, 1.4h−170 Mpc from the center of A2556. The cD galaxy of A2554 is just on the edge of our images so we have no information about its low surface brightness component. A2556 and A2554 are a part of the Aquarius supercluster(Batuski et al. 1999), so they clearly reside in an overdense region of the universe. Given an X–ray luminosity from Ebeling et al. (1996) and a velocity dispersion from Reimers et al. (1996), we calculate the virial mass of A2556 to be 2.5± 1.1× 1015h−170 M�. The red sequence for this cluster is a bit wider than in other clusters. The one sigma width to a biweight fit is 0.38 magnitudes in B-r which is approximately 30% larger than in the rest of the low-z sample. This extra width is not caused by only a few galaxies, instead the entire red sequence appears to be inflated. This is probably caused by the nearby A2554 which is at z=0.11 (Struble & Rood 1999). This is close enough in redshift space that we cannot separate out the 2 red sequences in our CMD. We have roughly 30 redshifts for A2556 from Smith et al. (2004), Caretta et al. (2004), and Batuski et al. (1999) which are also unable to differentiate between the clusters. Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 1.3 ± .38 × 1012L�in r and 3.3 ± 1.0 × 1011L�in B inside of 0.65h−170 Mpc, which is one quarter of the virial radius of this cluster. Figure 7 shows the relevant plots for this cluster. There is an r−band ICL component ranging from 27 - 29 mag arcsec−2 in r centered on the cD galaxy. The B−band ICL is extremely faint, barely above or detection threshold. Although we were able to fit a profile to the B−band diffuse light, all points on the medium sized mask are below 29 mag arcsec−2. The total flux in the ICL is 7.6± 6.6× 1010L�in r and 1.4± 1.4× 1010L�in B, which makes for ICL fractions of 6± 5% in r and 4 ± 4% in B. Although Figure 7 shows a color profile, we do not assume anything about the profile shape due to the low SB level of the B−band. We take the B − r color from the innermost point to be 2.1 ± 0.4, which is fully consistent with the color of the RCS. A4010 A4010 is a richness class 1, Bautz Morgan type I-II cluster at a redshift of 0.096. This cluster has a cD galaxy in the center of the galaxy distribution, which is 0.7 ± .05 magnitudes brighter than the second ranked galaxy. There is only ROSAT All Sky Survey data for this cluster and no other sufficiently deep X–ray observations to show us the shape of the mass distribution. There are weak lensing maps which put the center of mass of the cluster at the same position as the cD galaxy, and elongated along the same position angle as the cD galaxy (Cypriano et al. 2004). Muriel et al. (2002) find a velocity dispersion of 743 ± 140 for this cluster which is 15% larger than found by Girardi et al. (1998b), where those authors find a virial mass of 3.8±1.61.2 ×1014h 70 M�. The color magnitude diagram for A4010 is typical among the sample with a clear red sequence. A few redshifts exist in the literature which help define the red sequence (Collins et al. 1995; Katgert et al. 1998). Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 1.2± .4× 1012L�in r and 3.5± 1.0× 1011L�in B inside of 0.75h−170 Mpc, which is one quarter of the virial radius of this cluster. Figure 8 shows the relevant plots for this cluster. There is an elongated ICL component ranging from 25.5 - 28 mag arcsec−2 in r centered on the cD galaxy. The total flux in the ICL is 3.2± 0.7× 1011L�in r and 7.7± 2.8× 1010L�in B, which makes for ICL fractions of 21 ± 8% in r and 18 ± 8% in B. The ICL has a significant red-ward trend in its color profile with an average color of B − r ' 2.1± 0.1, which is marginally redder (0.2 magnitudes) than the RCS. A3888 A3888 is discussed in great detail in paper I. In review, A3888 is a richness class 2, Bautz Morgan type I-II cluster at a redshift of 0.151. This cluster has no cD galaxy; instead the core is comprised of 3 distinct sub-clumps of multiple galaxies each. At least 2 galaxies in each of the subclumps are confirmed members based on velocities (Teague et al. 1990; Pimbblet et al. 2002). The brightest cluster galaxy is only 0.12± .04 magnitudes brighter than the second ranked galaxy. XMM contours show an elongated distribution centered roughly in the middle of the three clumps of galaxies. Reiprich & Böhringer (2002) estimate mass from the X–ray luminosity to be M200 = 25.5±10.57.4 × 1014h 70 M�, where r200 = 2.8h−170 Mpc. This is consistent with the mass estimate from the published velocity dispersion of 1102± 107 (Girardi & Mezzetti 2001). There is a clear red sequence of galaxies in the CMD of A3888. Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 3.0±0.9×1012L�in r and 7.2±2.2×1011L�in B inside of 0.92h−170 Mpc. We also determine galaxy flux using the Driver et al. (1998) luminosity distribution, which is based on the statistical background subtraction of non-cluster galaxies, to be 4.3 ± 0.7 × 1012L�in the r−band and 3.4 ± 0.6 × 1012L�in V . The difference in these two estimates is likely due to uncertainties in our membership identification (of order 30%) and difference in detection thresholds of the two surveys. ICL 41 Figure 9 shows the relevant plots for this cluster. There is a centralized ICL component ranging from 26 - 29 mag arcsec−2 in r despite the fact that there is no cD galaxy. The total flux in the ICL is 4.4 ± 2.1 × 1011L�in r and 8.6 ± 2.5 × 1010L�in B, which makes for ICL fractions of 13 ± 5% in r and 11 ± 3% in B. The ICL has a red color profile with an average color of V − r ' 0.5 ± 0.1, which is marginally redder (0.2 magnitudes) than the RCS. There is also a diffuse light component surrounding a group of galaxies that is 1.4 h−170 Mpc from the cluster center which totals 1.7± 0.5× 1010L�in V and 2.6± 1.2× 1010L�in r and has a color consistent with the main ICL component. A3984 A3984 is an interesting richness class 2, Bautz Morgan type II-III cluster at a redshift of 0.181. There appear to be 2 centers of the galaxy distribution. One around the BCG, and one around a semi-circle of ∼ 5 bright ellipticals which are 1h−170 Mpc north of the BCG. The BCG and at least one of the other bright ellipticals are at the same redshift (Collins et al. 1995). To determine if these 2 centers are part of the same redshift structure, we split the image in half perpendicular to the line bisecting the 2 regions, and plot the cumulative distributions of V − r galaxy colors. A KS test reveals that these 2 regions have an 89% probability of being drawn from the same distribution. Without X–ray observations we do not know where the mass in this cluster resides. There is a weak lensing map of just the northern region of the cluster which does show a centralized mass distribution, but does not include the southern clump (Cypriano et al. 2004). The BCG is 0.57± .04 magnitudes brighter than the second ranked galaxy. We use a velocity dispersion from the lensing measurement to determine a mass of 31± 10× 1014h−170 M�. There is a clear red sequence of galaxies in the CMD of A3984. Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 2.0± 0.6× 1012L�in r and 4.4± 1.3× 1011L�in B inside of 0.87h−170 Mpc, which is one quarter of the virial radius of this cluster. Figure 10 shows the relevant plots for this cluster. There are 2 clear groupings of diffuse light. We can only fit a profile to the ICL which is centered on the BCG. We stop fitting that profile before it extends into the other ICL group (∼ 600kpc) in an attempt to keep the fluxes separate. The total flux in the ICL is 2.2 ± 1.0 × 1011L�in r and 6.2 ± 2.1 × 1010L�in B, which makes for ICL fractions of 10± 6% in r and 12± 6% in B. The ICL becomes distinctly bluer with radius and is bluer at all radii than the RCS with an average color of V − r ' −0.2± 0.4 (0.5 magnitudes bluer than the RCS). A0141 A0141 is a richness class 3, Bautz Morgan type III cluster at a redshift of 0.23. True to its morphological type, this cluster has no cD galaxy, instead it has 4 bright elliptical galaxies, each at the center of a clump of galaxies, the brightest one of which is 0.42± .04 magnitudes brighter than the second brightest. The center of the cluster, as defined by ASCA observations and a weak lensing map (Dahle et al. 2002), is near the northernmost clumps of galaxies. The distribution is clearly elongated north-south, it is therefore possible that the other bright ellipticals are in-falling groups along a filament. M200 from the lensing map is 18.9±1.10.9 ×1014 h 70 M�. There is a clear red sequence of galaxies in the CMD of A0141. Using the CMD as an indication of membership, we estimate the flux in cluster galaxies to be 3.2± 1.0× 1012L�in r and 5.4± 1.6× 1011L�in B inside of 0.94 h−170 Mpc, which is one quarter of the virial radius of this cluster. Figure 11 shows the relevant plots for this cluster. There are 3 clear groupings of diffuse light which do not have a common center, although 1 of these ICL peaks does include 2 clumps of galaxies. We are unable to fit a single centralized profile to this ICL as the three clumps are too far separated. The total flux in the ICL as measured in manually placed elliptical annuli is 3.5± .9× 1011L�in r and 3.4± 1.1× 1010L�in B, which makes for ICL fractions of 10± 4% in r and 6 ± 3% in B. We estimate the color of the ICL to be V − r ' 1.0 ± 0.8, which is significantly redder (0.6 magnitudes) than the RCS. We have no color profile information. AC114 AC114 (AS1077) is a richness class 2, Bautz Morgan type II-III cluster at a redshift of 0.31. The brightest galaxy is only 0.28± .04 magnitudes brighter than the second ranked galaxy. The galaxy distribution is elongated southeast to northwest (Couch et al. 2001) as is the Chandra derived X–ray distribution. The X–ray gas shows a very irregular morphology, with a soft X–ray tail stretching toward a mass clump in the southeast which is also detected in a lensing map (De Filippis et al. 2004; Campusano et al. 2001). The X–ray gas is roughly centered on a bright elliptical galaxy, however the tail is an indication of a recent interaction. There is a clump of galaxies, 1.6h−170 Mpc northwest of the BCG, which looks like a group or cluster with its own cD-like galaxy which is not targeted in either the X–ray or lensing (strong) observations. Only one of these galaxies has redshifts in the literature, and it is a member of AC114. Without redshifts, we cannot know definitively if these galaxies are a part of the same structure, however their location along the probable filament might be evidence that they are part of the same velocity structure. As this cluster is not in dynamical equilibrium, mass estimates from the X–ray gas come from B-model fits to the surface brightness distribution. De Filippis et al. (2004) find a mass within 1h−170 Mpc of 4.5 ± 1.1 × 10 14h−170 M�. A composite strong and weak lensing analysis agree with the X–ray analysis within 500h−170 kpc, but they do not extend out to larger radii (Campusano et al. 2001). Within the virial radius, (Girardi & Mezzetti 2001) find a mass of 26.3+8.2−7.1 × 10 14h−170 M�. This cluster, in relation to lower-z clusters, is a prototypical example of the Butcher-Oemler effect. There is a higher fraction of blue, late-type galaxies at this redshift, than in our lower-z clusters, rising to 60% outside of the core region 42 Krick, Bernstein (Couch et al. 1998). This is not only evidenced in the morphologies, but in the CMD, which nicely shows these blue member galaxies. We adopt the Andreon et al. (2005) luminosity function for this cluster based on an extended likelihood distribution for background galaxies. Integrating the luminosity distribution from very dim dwarf galaxies (MR = −11.6) to infinity gives a total luminosity for AC114 of 1.5± 0.2× 1012L�in r and 1.9± 1.2× 1011L�in B inside of 0.9h−170 Mpc, which is one quarter of the virial radius of this cluster. For the purpose of comparison with other clusters, we adopt the cluster flux from the CMD, which gives 1.8 ± 0.5 × 1012L�in r and 2.3 ± 0.7 × 1011L�in B inside of one quarter of the virial radius of this cluster. The differences in these estimates are likely due to uncertainties in membership identification and differing detection thresholds of the two surveys. Figure 12 shows the relevant plots for this cluster. There is a centralized ICL component ranging from 27.5 - 29 mag arcsec−2 in r, in addition to a diffuse component around the group of galaxies to the northwest of the BCG. The total flux in the ICL is 2.2± 0.4× 1011L�in r and 3.8± 7.9× 1010L�in B, which includes the flux from the group as measured in elliptical annuli. The ICL fraction is 11±2% in r and 14±3% in B. The ICL has a flat color profile with V −r ' 0.1±0.1, which is marginally bluer (0.4 magnitudes) than the RCS. AC118 (A2744) AC118 (A2744) is a richness class 3, Bautz Morgan type III cluster at a redshift of 0.31. This cluster has 2 main clumps of galaxies separated by 1h−170 Mpc, with a third bright elliptical in a small group which is 1.2h 70 Mpc distant from the center of the other clumps. The BCG is 0.23 ± .04 magnitudes brighter than the second ranked galaxy. The Chandra X–ray data suggests that there are probably 3 clusters here, at least 2 of which are interacting. The gas distribution, along with abundance ratios, suggests that the third, smaller group might be the core of one of the interacting clusters which has moved beyond the scene of the interaction where the hot gas is detected. From velocity measurements Girardi & Mezzetti (2001) also find 2 populations of galaxies with distinctly different velocity dispersions. The presence of a large radio halo and radio relic are yet more evidence for dynamical activity in this cluster (Govoni et al. 2001). Mass estimates for this cluster range from ∼ 3×1013M� from X–ray data to ∼ 3×1015M� from the velocity dispersion data. This cluster clearly violates assumptions of sphericity and hydrostatic equilibrium, which is leading to the large variations. The two velocity dispersion peaks have a total mass of 38± 37× 1014 h−170 M�; we adopt this mass throughout the paper. AC118, at the same redshift as AC114, also shows a significant fraction of blue galaxies, which leads to a wider red cluster sequence(1σ = 0.3 magnitudes), than at lower redshifts. We adopt the Busarello et al. (2002) R and V−band luminosity distributions based on photometric redshifts and background counts from a nearby, large area survey. Integrating the luminosity distribution from very dim dwarf galaxies (MR = −11.6) to infinity gives a total luminosity for AC118 of 4.5 ± .2 × 1011L� in V and 4.2 ± .4 × 1012L� in the r−band inside of 0.25rvirial. For the purpose of comparison with other clusters, we adopt the cluster flux from the CMD, which gives 5.4 ± 1.6 × 1011L�in B and 4.4 ± 0.1 × 1012L�in r inside of 0.94h−170 Mpc, which is one quarter of the virial radius of this cluster. Figure 13 shows the relevant plots for this cluster. There are at least two, if not three groupings of diffuse light which do not have a common center. The possible third is mostly obscured behind the mask of a saturated star. We are unable to fit a centralized profile to this ICL. The total flux in the ICL as measured in manually placed elliptical annuli is 7.0± 1.0× 1011L�in r and 6.7± 1.7× 1010L�in B, which makes for ICL fractions of 14± 5% in r and 11± 5% in B. We estimate the color of the ICL to be V − r ' 1.0 ± 0.8, which is significantly redder (0.6 magnitudes) than the RCS. We have no color profile information. Introduction The Sample Observations Reduction Object Detection Object Removal & Masking Stars Galaxies Results Cluster Properties Cluster Membership & Flux Dynamical Age Global Density ICL properties Surface brightness profile ICL Flux ICL Fraction Color ICL Substructure Groups Accuracy Limits Discussion ICL flux ICL Flux vs. M3-M1 ICL fraction ICL fraction vs. Mass ICL fraction vs. Redshift ICL color Profile Shape Conclusion The Clusters A4059 A3880 A2734 A2556 A4010 A3888 A3984 A0141 AC114 AC118 (A2744)

0704.1665

Approach to Physical Reality: a note on Poincare Group and the philosophy of Nagarjuna

Approach to Physical Reality: a note on Poincaré Group and the philosophy of Nagarjuna. David Vernette, Punam Tandan, and Michele Caponigro We argue about a possible scenario of physical reality based on the parallelism between Poincaré group and the sunyata philosophy of Nagarjuna. The notion of ”relational” is the common denom- inator of two views. We have approached the relational concept in third-person perspective (ontic level). It is possible to deduce different physical consequence and interpretation through first-person perspective approach. This relational interpretation leave open the questions: i)we must abandon the idea for a physical system the possibility to extract the completeness information? ii)we must abandon the idea to infer a possible structure of physical reality? POINCARÉ GROUP There are two universal features of modern day physics regarding physical systems: all physical phenomena take place in 1)space-time and all phenomena are (in princi- ple) subject to 2)quantum mechanics. Are these aspects just two facets of the same underlying physical reality? The research is concentrate on this fundamental point. The notion of space-time is linked to the geometry, so an interesting question is what geometry is appropriate for quantum physics[3]. Can geometry give us any knowl- edge about the nature of physical space where the phys- ical laws take place? Can geometry give us the possible scenario of the physical reality? A fundamental aspect of a geometry is the group of transformations defined over it. Group theory is the necessary instruments for ex- pressing the laws of physics (the concept of symmetry is derived from group theory.)[4].Physics and the geometry in which it take place are not independent. We retain there is a close relationship between space-time struc- ture and physical theory. Space-time imposes univer- sally valid constraints on physical theories and the uni- versality of these laws starts to become less mysterious (i.e. various paradox). The invariance under the group of transformations is a fundamental criterion to classify mathematical structures. Poincaré introduced notion of invariance under continue transformations. The Poincaré Group is the group of translation, rotation, and boost operators in 4-dimensional space-time. Now, some nat- ural questions are: does space exist independently of phenomena? Itself has an intrinsic significance? A system defined in this space through physical law could exist by itself? We call ”absolute” reality the reality of a system that do not depend by its interaction with other system. The problem is that we have not a single system. In this brief note, we abandon the idea of absolute reality and we argue in favor of a relational reality, because re- lational reality is founded on the premise that an object is real only in relation to another object that it is inter- acting with. In the relational interpretation[2], the basic elements of objective reality are the measurement events themselves. This interpretation goes beyond the Copen- hagen interpretation by replacing the absolute reality with relational reality. In the relational interpreta- tion the wave function is merely a useful mathematical abstraction. Some authors proposes that the laws of na- ture are really the result of probabilities constrained by fundamental symmetries. Relational reality is asso- ciated with the fundamental concept of interac- tions. These later analysis of the ”relational” notion bring us to approach the same problem utilizing the sun- yata philosophy of Nagaujuna. CONCEPT OF REALITY IN THE PHILOSOPHY OF NAGARJUNA The Middle Way of Madhyamika refers to the teach- ings of Nagarjuna, very interesting are the implications between quantum physics and Madhyamika. The basic concept of reality in the philosophy of Nagarjuna is that the fundamental reality has no firm core but consists of systems of interacting objects. According to the middle way perspective, based on the notion of empti- ness, phenomena exist in a relative way, that is, they are empty of any kind of inherent and independent existence. Phenomena are regarded as dependent events existing relationally rather than permanent things, which have their own entity. Nagarjuna middle way perspective emerges as a relational approach, based on the insight of emptiness. Sunyata (emptiness) is the foundation of all things, and it is the basic principle of all phenomena. The emptiness implies the negation of unchanged, fixed sub- stance and thereby the possibility for relational existence and change. This suggests that both the ontological con- stitution of things and our epistemological schemes are just as relational as everything else. We are fundamen- tally relational internally and externally. In other words, Nagarjuna do not fix any ontological nature of the things: • 1)they do not arise. http://arxiv.org/abs/0704.1665v1 • 2)they do not exist. • 3)they are not to be found. • 4)they are not. • 5)and they are unreal In short, an invitation do not decide on either existence or non-existence(nondualism). According the theory of sunyata, phenomena exist in a relative state only, a kind of ’ontological relativity’. Phenomena are regarded as dependent(only in relation to something else) events rather than things which have their own inherent nature; thus the extreme of permanence is avoided. CONCLUSION We have seen the link between relational and in- teraction within a space-time governed by own geom- etry. Nagarjuna’s philosophy use the same basic con- cept of ”relational” in the interpretation of reality. We note that our parallelism between the scenario of physi- cal reality and the relational interpretation of the same reality is based on third-person perspective approach (i.e. the ontic level, relational view include the observer- device). Different considerations could be done thought first-person perspective approach, in this case we retain the impossibility to establish any parallelism. Finally, we note that probably the relational approach stimulate the interest to fundamental problems in physics like: the unification of laws and the discrete/continuum view[1]. —————— ⋄David Vernette, Punam Tandan, Michele Caponigro Quantum Philosophy Theories www.qpt.org.uk ⋄ [email protected] [1] Vernette-Caponigro: Continuum versus Discrete Physics physics/0701164 [2] Rovelli C. Relational quantum mechanics, Intl. J. theor. Phys. 35, 1637-1678 (1996) [3] note -1- It was suggested, for instance, that the universal symmetry group elements which act on all Hilbert spaces may be appropriate for constructing a physical geometry for quantum theory. [4] note -2- Some authors retain the symmetry is the ontic element, and the physical laws like the space-time are sec- ondary http://arxiv.org/abs/physics/0701164 Poincaré Group Concept of reality in the philosophy of Nagarjuna Conclusion References

0704.1666

Ultraviolet Observations of Supernovae

ULTRAVIOLET OBSERVATIONS OF SUPERNOVAE Nino Panagia STScI, Baltimore, MD, USA; [email protected] INAF - Observatory of Catania, Italy Supernova Ltd., Virgin Gorda, BVI Abstract. The motivations to make ultraviolet (UV) studies of supernovae (SNe) are reviewed and discussed in the light of the results obtained so far by means of IUE and HST observations. It appears that UV studies of SNe can, and do lead to fundamental results not only for our understanding of the SN phenomenon, such as the kinematics and the metallicity of the ejecta, but also for exciting new findings in Cosmology, such as the tantalizing evidence for "dark energy" that seems to pervade the Universe and to dominate its energetics. The need for additional and more detailed UV observations is also considered and discussed. Keywords: Supernovae: general, Ultraviolet: stars, Binaries: general, Cosmology: miscellaneous PACS: 97.60.Bw, 97.80.-d, 98.80.-k 1. INTRODUCTION Supernovae (SNe) are the explosive death of massive stars as well as moderate mass stars in binary systems. They enrich the interstellar medium of galaxies of most heavy elements (only C and N can efficiently be produced and ejected into the ISM by red giants winds and by planetary nebulae, as well as pre-SN massive star winds): nuclear detonation supernovae, i.e., Type Ia SNe (SNIa), provide mostly Fe and iron-peak elements, while core collapse supernovae, i.e., Type II (SNII) and Type Ib/c (SNIb/c), mostly O and alpha-elements (see below for type definitions). Therefore, they are the primary factors to determine the chemical evolution of the Universe. Moreover, SN ejecta carry approximately 1051 erg in the form of kinetic energy, which constitute a large injection of energy into the ISM of a galaxy (for a Milky Way class galaxy EMWkin ≃ 3×10 57 erg). This energy input is very important for the evolution of the entire galaxy, both dynamically and for star-formation through cloud compression/energetics. In addition SNe are bright events that can be detected and studied up to very large distances. Therefore: (1) SN observations can be used trace the evolution of the Uni- verse. (2) SNe can be used as measuring sticks to determine cosmologically interesting distances, either as "standard candles" (SNIa, which at maximum are about 10 billion times bright than the Sun, with a dispersion of the order of 10%) or employing a refined Baade-Wesselink method (SNII in which strong lines provide ideal conditions for the ap- plication of the method, with a distance accuracy of ±20%). (3) Their intense radiation can be used to study the ISM/IGM properties through measurements of the absorption lines. Since most of the strong absorption lines are found in the UV, this is best done observing SNII at early phases, when the UV continuum is still quite strong. Additional http://arxiv.org/abs/0704.1666v2 studies in the optical (mostly CaII and NaI lines) are possible using all bright SNe. How- ever, only combining optical and UV observations can one obtain the whole picture and, therefore, SNII are the preferred targets for these studies. (4) Finally, the strong light pulse provided by a SN explosion (the typical HPW of a light curve in the optical is about a month for SNIa and about two-three months for SNII; in the UV the light curve evolution is much faster) can used to probe the intervening ISM in a SN parent galaxy by observing the brightness, and the time evolution of associated light echoes. 2. ULTRAVIOLET OBSERVATIONS The launch of the International Ultraviolet Explorer (IUE) satellite in early 1978 marked the beginning of a new era for SN studies because of its capability of measuring the ul- traviolet emission from objects as faint as mB=15. Moreover, just around that time, other powerful astronomical instruments became available, such as the Einstein Observatory X-ray measurements, the VLA for radio observations, and a number of telescopes either dedicated to infrared observations (e.g. UKIRT and IRTF at Mauna Kea) or equipped with new and highly efficient IR instrumentation (e.g. AAT and ESO observatories). As a result, starting in the late 70’s a wealth of new information become available that, thanks to the coordinated effort of astronomers operating at widely different wavelengths, has provided us with fresh insights as for the properties and the nature of supernovae of all types. Eventually, the successful launch of the Hubble Space Telescope (HST) opened new possibilities for the study of supernovae, allowing us to study SNe with an accuracy unthinkable before and to reach the edge of the Universe. Even after 18 years of IUE observations and 16 more of HST observations, the number of SN events that have been monitored with UV spectroscopy is quite small and hardly include more than two objects per SN type and hardly with good quality spectra for more than three epochs each. As a consequence, we still know very little about the properties and the evolution of the ultraviolet emission of SNe. On the other hand, it is just the UV spectrum of a SN, especially at early epochs, that contains a wealth of valuable and crucial information that cannot be obtained with any other means. Therefore, we truly want to monitor many more SNe with much more frequent observations. We have learned at this Conference that SWIFT, in addition to doing UV photometry, is also able to obtain low resolution spectra of SNe at λ > 2000Å with a sensitivity comparable to that of IUE and that spectroscopic observations of SNe with SWIFT are in the planning (see, e.g., the contribution by F. Bufano). This is an exciting possibility that promises to provide very valuable results and to fill the gaps in our knowledge about UV properties of SNe. Here, I present a short summary of the UV observations of supernovae. A more detailed review on this subject can be found in Panagia (2003). 3. TYPE IA SUPERNOVAE Type Ia supernovae are characterized by a lack of hydrogen in their spectra at all epochs and by a number of typically broad, deep absorption bands, most notably the Si II 6150Å FIGURE 1. Ultraviolet spectra of ten Type Ia supernovae observed with IUE around maximum light. The dashed line is SN 1992A spectrum as measured with HST-FOS. (actually the blue-shifted absorption of the 6347-6371Å Si II doublet; see e.g. Filippenko 1997), which dominate their spectral distributions at early epochs. SNIa are found in all types of galaxies, from giant ellipticals to dwarf irregulars. However, the SNIa explosion rate, normalized relative to the galaxy H or K band luminosity and, therefore, relative to the galaxy mass, is much higher, up to a factor of 16 when comparing the extreme cases of irregulars and ellipticals (Della Valle & Livio 1994, Panagia 2000, Mannucci et al. 2005) in late type galaxies than in early type galaxies. This suggests that, contrary to common belief, a considerable fraction of SNIa belong to a relatively young (age much younger that 1∼Gyr), moderately massive (∼5M⊙< M(SNIa progenitor)< 8M⊙) stellar population (Mannucci, Della Valle & Panagia 2006), and that in present day ellipticals SNIa are mostly the result of capture of dwarf galaxies by massive ellipticals (Della Valle & Panagia, 2003, Della Valle et al. 2005) 3.1. Existing Samples of UV Spectra of SNIa Although 12 type Ia SNe were observed with IUE, only two events, namely SN1990N and SN1992A, had extensive time coverage, whereas all others were observed only around maximum light either because of their intrinsic UV faintness or because of satellite pointing constraints. Even so, one can reach important conclusions of general validity, which are confirmed by the detailed data obtained for a few SNIa. The UV spectra of type Ia SNe are found to decline rapidly with frequency, making it hard to detect any signal at short wavelengths. This aspect is illustrated in Fig. 1, which displays the UV long wavelength spectra of 10 type Ia SNe observed with IUE. It appears that the spectra do not have a smooth continuum but rather consist of a number of FIGURE 2. The spectral evolution of SN1992A [adapted from Kirshner et al. 1993]. "bands” that are observed with somewhat different strengths. The fact that the spectrum is so similar for most of the SNe supports the idea of an overall homogeneity in the properties of type Ia SNe. On the other hand, some clear deviations from “normal” can be recognized for some SNIa. In particular, one can notice that both SN1983G and SN1986G display excess flux around 2850 Å, and a deficient flux around 2950 Å. This suggests that the Mg II resonance line is much weaker, which may indicate a lower abundance of Mg in these fast-decline, under-luminous SNIa. On the other hand, SN1990N, SN1991T, and, possibly, SN1989M show excess flux around ∼2750 Å and ∼2950 Å and a clear deficit around ∼3100 Å, which may be ascribed to enhanced Mg II and Fe II features in these slow-decline, over-luminous SNIa. The best studied SNIa event so far is the "normal” type Ia supernova SN1992A in the S0 galaxy NGC1380 that was observed as a TOO by both IUE and HST (Kirshner et al. 1993). The HST-FOS spectra, from 5 to 45 days past maximum light, are the best UV spectra available for a SNIa (see Fig. 2) and reveal, with good signal to noise ratio, the spectral region blueward of ∼2650 Å. An LTE analysis of the SN1992A spectra shows that the features in the region shortward of ∼2650Å are P Cygni absorptions due to blends of iron peak element multiplets and the Mg II resonance multiplet. Newly synthesized Mg, S, and Si probably extend to velocities at least as high as ∼19,000 km s−1. Newly synthesized Ni and Co may dominate the iron peak elements out to ∼13,000 km s−1 in the ejecta of SN1992A. On the other hand, an analysis of the O I λ7773 line in SN1992A and other SNIa implies that the oxygen rich layer in typical SNIa extends over a velocity range of at least ∼11,000-19,000 km s−1, but none of the "canonical” models has an O-rich layer that completely covers this range. Even higher velocities were inferred by Jeffery et al. (1992) for the overluminous, slow-decline SNIa SN1990N and SN1991T through an LT analysis of their photospheric epoch optical and UV spectra. In particular, matter moving as fast as 40,000 and 20,000 km s−1 were found for SN1990N and SN1991T, respectively. It thus appears that type Ia supernovae are consistently weak UV emitters, and even at maximum light their UV spectra fall well below a blackbody extrapolation of their optical spectra. Broad features due to P Cygni absorption of Mg II and Fe II are present in all SNIa spectra, with remarkable constancy of properties for normal SNIa and systematic deviations for slow-decline, over-luminous SNIa (enhanced Mg II and Fe II absorptions) and fast-decline, under-luminous SNIa (weaker Mg II lines). 4. CORE COLLAPSE SUPERNOVAE: TYPES II AND IB/C Massive stars (M*>8M⊙) are believed to end their evolution collapsing over their inner Fe core and producing an explosion by a gigantic bounce that launches a shock wave that propagates through the star and eventually erupts through the progenitor photosphere, ejecting several solar masses of material at velocities of several thousand km s−1. The current view is that single stars (as well as stars in wide binary systems in which the companion does not affect the evolution of the primary star) explode as type II supernovae, while supernovae of types Ib and Ic originate from massive stars in interacting binary systems. Although the explosion mechanism is essentially the same in both types, the spectral characteristics and light curve evolution are markedly different among the different types. 4.1. Type Ib/c Supernovae Type Ib/c supernovae (SNIb/c) are similar to SNIa in not displaying any hydrogen lines in their spectra and are dominated by broad P Cygni-like metal absorptions, but they lack the characteristic SiII 6150Å trough of SNIa. The finer distinction into SNIb and SNIc was introduced by Wheeler and Harkness (1986) and is based on the strength of He I absorption lines, most importantly He I 5876Å, so that the spectra of SNIb display strong He I absorptions and those of SNIc do not. SNIb and SNIc are found only in late type galaxies, often (but not always) associated with spiral arms and/or H II regions. They are generally believed to be the result of the evolution of massive stars in close binary systems. Although the properties of some peculiarly red and under-luminous SNI (SN1962L and SN1964L) were already noticed by Bertola and collaborators in the mid-1960s (Bertola 1964, Bertola et al. 1965), the first widely recognized member and prototype of the SNIb class was SN1983N in NGC5236=M83. Because of its bright magnitude (B∼11.6 mag at maximum light), SN1983N is one of the best-studied SNe with IUE (see Panagia 1985). The UV spectrum of SN1983N closely resembles that of type Ia SNe at comparable epochs and, as such, only a minor fraction of the SN energy is radiated in the UV. In particular, only ∼13% of the total luminosity was emitted by SN1983N shortward of 3400Å at the time of the UV maxi- mum. Moreover, there is no indication of any stronger emission in the UV at very early epochs; this implies that the initial radius of the SN, i.e. the radius the stellar progenitor had when the shock front reached the photosphere, was probably < 1012cm, ruling out FIGURE 3. The spectrum of SN 1983N near maximum optical light, dereddened with E(B-V)=0.16. Both UV and optical spectra have been boxcar smoothed with a 100Å bandwith. The triangle is the IUE Fine Error Sensor (FES) photometric point, and the dots represent the J, H, and K data. The dash-dotted curve is a blackbody spectrum at T=8300K [adapted from Panagia 1985]. a RSG progenitor. From the bolometric light curve Panagia (1985) estimated that ∼0.15 M⊙ of 56Ni was synthesized in the explosion. The best observed SNIc is SN1994I that was discovered on 2 April 1994 in the grand design spiral galaxy M51 and was promptly observed both with IUE (as early as 3 April) and with HST- FOS (19 April). The UV spectra were remarkably similar to those obtained for SN1983N and, although they were taken only at two epochs well past maximum light (10 days and 35 days), they were of high quality. From synthetic spectra matching the observed spectra from 4 days before to 26 days after the time of maximum brightness, the inferred velocity at the photosphere decreased from 17,500 to 7,000 km s−1 (Millard et al. 1999). Simple estimates of the kinetic energy carried by the ejected mass gave values that were near the canonical supernova energy of 1051 erg. Such velocities and kinetic energies for SN1994I are "normal” for SNe and are much lower than those found for the peculiar type Ic SN1997ef and SN1998bw (see, e.g. Branch 2000) which appear to have been hyper-energetic. Thus, as type Ia, type Ib/c supernovae are weak UV emitters with their UV spectra much fainter than a blackbody extrapolation of both optical and NIR spectra, and their typical luminosity is about a factor of 4 lower than that of SNIa. The mass of 56Ni synthesized in a typical SNIb/c is, therefore, ∼0.15 M⊙. 4.2. Type II Supernovae Type II supernovae display prominent hydrogen lines in their spectra (Balmer series in the optical) and their spectral energy distributions are mostly a continuum with relatively few broad P Cygni-like lines superimposed, rather than being dominated by discrete features as is the case of all type I supernovae. SNII are believed to be the result of a core collapse of massive stars exploding at the end of their RSG phase. SN1987A was FIGURE 4. UV spectral evolution of SN1998S (SINS project, unpublished). Shown are spectra ob- tained near maximum light (March 16,1998), about two weeks past maximum (March 30, 1998), and about two months after maximum (May 13, 1998). both a confirmation and an exception to this model. It was clearly the product of the collapse of a massive star, but it exploded when it was a BSG, not an RSG. Since its properties are amply discussed in many detailed papers presented at this Conference, we do not include SN1987A in this summary of the UV properties of "normal" SNII. Among the other five SNII that were observed with IUE, only two, SN1979C and SN1980K, were bright enough to allow a detailed study of their properties in the UV (Panagia et al. 1980). They were both of the so-called "linear” type (SNIIL), which is characterized by an almost straight-line decay of the B and V-band light curves, rather than of the more common "plateau” type (SNIIP) which display a flattening in their light curves starting a few weeks after maximum light. The SNII studied best in the UV so far is possibly SN1998S in NGC3877, a type II with relatively narrow emission lines (SNIIn). SN1998S was discovered several days before maximum. Its first UV spectrum, obtained on 16 March 1998, near maximum light, was very blue and displayed lines with extended blue wings, which indicate expansion velocities up to 18,000 km s−1 (Panagia 2003). The UV spectral evolution of SN1998S (Fig. 5) showed the spectrum to gradually steepen in the UV, from near maximum light on 16 March 1998 to about two weeks past maximum on 30 March, and the blue absorptions to weaken or disappear completely. About two months after maximum (13 May 1998) the continuum was much weaker, although its UV slope had not changed appreciably, and it had developed broad emission lines, the most noticeable being the Mg II doublet at about 2800Å. This type of evolution is quite similar to that of SN1979C (Panagia 2003) and suggests that the two sub-types are related to each other, especially in their circumstellar interaction properties. A detailed analysis of early observations of SN1998S (Lentz et al. 2001) indicated that early spectra originated primarily in the circumstellar region itself, and later spectra are due primarily to the supernova ejecta. Intermediate spectra are affected by both regions. A mass-loss rate of order of ∼ 10−4[v/(100km s−1)] M⊙/yr was inferred from these calculations but with a fairly large uncertainty. Despite the fact that type II plateau (SNIIP) supernovae account for a large fraction of all SNII, so far SN1999em in NGC1637 is the only SNIIP that has been studied in some detail in the ultraviolet. Although caught at an early stage, SN1999em was already past maximum light (see, e.g. Hamuy et al. 2001). An early analysis of the optical and UV spectra (Baron et al. 2000) indicates that, spectroscopically, this supernova appears to be a normal type II. Also, the analysis suggests the presence of enhanced N as found in other SNII. Another sub-type of the SNII family is the so-called type IIb SNe, dubbed so because at early phases their spectra display strong Balmer lines, typical of type II SNe, but at more advanced phases the Balmer lines weaken significantly or disappear altogether (see, e.g. Filippenko et al. 1997) and their spectra become more similar to those of type Ib SNe. A prototypical member of this class is SN1993J that was discovered in early April 1993 in the nearby galaxy M81. An HST-FOS UV spectrum of SN1993J was obtained on 15 April 1993, about 18 days after explosion, and rather close to maximum light. The study of this spectrum (Jeffery et al. 1994) shows that the approximately 1650- 2900Å region is smoother than observed for SN1987A and SN1992A and lacks strong P Cygni lines absorptions caused by iron peak element lines. It is of interest to note that the UV spectrum of SN1993J is appreciably fainter than observed in most SNII, thus revealing its "hybrid” nature and some resemblance to a SNIb. Synthetic spectra calculated using a parameterized LT procedure and a simple model atmosphere do not fit the UV observations. Radio observations suggest that SN1993J is embedded in a thick circumstellar medium envelope (Van Dyk et al. 1994, Weiler et al. 2007). Interaction of supernova ejecta with circumstellar matter may be the origin of the smooth UV spectrum so that UV observations of supernovae could provide insight into the circumstellar environment of the supernova progenitors. Thus, despite their different characteristics in the detailed optical and UV spectra, all type II supernovae of the various sub-types appear to provide clear evidence for the presence of a dense CSM and, in many cases, enhanced nitrogen abundance. Their UV spectra at early phases are very blue, possibly with strong UV excess relative to a blackbody extrapolation of their optical spectra. 5. SUPERNOVAE AND COSMOLOGY SNIa have gained additional prominence because of their cosmological utility, in that one can use their observed light curve shape and color to standardize their luminosities. Thus, SNIa are virtually ideal standard candles (e.g. Macchetto and Panagia 1999) to measure distances of truly distant galaxies, currently up to redshift around 1 and, considerably more in the foreseeable future. In particular, Hubble Space Telescope observations of Cepheids in parent galaxies of SNIa (an international project lead by Allan Sandage) have produced very accurate determinations of their distances and the absolute magnitudes of normal SNIa at maximum light that, in turn, have lead to the most modern measure of the Hubble constant (i.e. the expansion rate of the local Universe), FIGURE 5. Top: Intermediate resolution (R=1500) spectra of three SNe near maximum light, SNIa 1992A (Kirshner et al. 1993), SNIIn/L 1998S (Lentz et al. 2001) and SNIIP 1999em (Baron et al. 2000) normalized in the V band. Bottom: Low-resolution rendition of the observed spectra convolved to a R=4 resolution showing the color differences in the UV. H0 = 62.3± 1.3(random)± 5.0(systematic) km s −1Mpc−1 (Sandage et al. 2006, and references therein). This value is lower than the determination obtained by the H0 key- project from a combination of various methods, (H0 = 72±8 km s −1Mpc−1; Freedman et al. 2001). The difference is well within the experimental uncertainties, and a weighted average of the two determinations would provide a compromise value of H0 = 65.2±4.3 km s−1Mpc−1. Observations of high redshift (i.e. z>0.1) SNIa have provided evidence for a recent (past several billion years) acceleration of the expansion of the Universe, pushed by some mysterious "dark energy". This is an exciting result that, if confirmed, may shake the foundations of physics. The results of two competing teams (Perlmutter et al. 1998, 1999, Riess et al. 1998, Knop et al. 2003, Tonry et al. 2003, Riess et al. 2004) appear to agree in indicating a non-empty inflationary Universe, which can be characterized by ΩM ≃ 0.3 and ΩΛ ≃ 0.7. Correspondingly, the age of the Universe can be bracketed within the interval 12.3-15.3 Gyrs to a 99.7% confidence level (Perlmutter et al. 1999). However, the uncertainties, especially the systematic ones, are still uncomfortably large and, therefore, the discovery and the accurate measurement of more high-z SNIa are absolutely needed. This is a challenging proposition, both for technical reasons, in that searching for SNe at high redshifts one has to make observations in the near IR (because of redshift) of increasingly faint objects (because of distance) and for more subtle scientific reasons, i.e. one has to verify that the discovered SNe are indeed SNIa and that these share the same properties as their local Universe relatives. One can discern Type I from Type II SNe on the basis of the overall properties of their UV spectral distributions (Panagia 2003), because Type II SNe are strong UV emitters, whereas all Type I SNe, irrespective of whether they are Ia or Ib/c, have spectra steeply declining at high frequencies (see Figure 5). This technique of recognizing SNIa from their steep UV spectral slope was devised by Panagia (2003), and has been successfully employed by Riess et al. (2004a,b) to select their best candidates for HST follow-up of high-z SNIa. However, we have to keep in mind that by using this technique one is barely separatiung the SNe with low UV emission (SNe Ia, Ib, Ic and, possibly, IIb) from the ones with high UV emission (most type II SNe). While it is a convenient approach to select interesting candidates, it cannot be a substitute for detailed spectroscopy, possibly at an R>100 resolution, to reliably characterize the SN type. REFERENCES . E. Baron et al. 2000, ApJ, 545, 444 . F. Bertola 1964, Ann.Ap, 27, 319 . F. Bertola, A. Mammano, M. Perinotto 1965, Asiago Contr., 174, 51 . D. Branch 2000, in "The Largest Explosions since the Big Bang: Supernovae and Gamma Ray Bursts", eds. M. Livio, N. Panagia, K. Sahu (Cambridge University Press, Cambridge) p. 96 . M. Della Valle , M. Livio, 1994, ApJ, 423, L31 . M. Della Valle , N. Panagia, 2003, ApJ, 587, L71 . M. Della Valle et al., 2005, ApJ, 629, 750 . A. Filippenko 1997, ARAAp, 35, 309 . W.L. Freedman et al. 2001, ApJ, 553, 47 . M. Hamuy et al. 2001, ApJ, 558, 615 . 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Soc., in press [astro-ph/0502247] . N. Panagia et al. 1980, MNRAS, 192, 861 . S. Perlmutter et al. 1998, Nature, 391, 51 . S. Perlmutter et al. 1999, ApJ, 517, 565 . A.G. Riess et al. 1998, AJ, 116, 1009 . A.G. Riess et al. 2004a, ApJ, 600, L163 . A.G. Riess et al. 2004b, ApJ, 607, 665 . A. Sandage, G.A. Tammann, A. Saha, B. Reindl, F.D. Macchetto, N. Panagia 2006, ApJ, 653, 843 . J.L. Tonry et al., 2003, ApJ, 594, 1 . S. Van Dyk et al. 1994, ApJ, 432, L115 . K.W. Weiler et al. 2007, ApJ, in press . J.C. Wheeler, R.P. Harkness 1986, in "Galaxy Distances and Deviations from Universal Expansion", eds. B.F. Madore, R.B. Tully (Reidel, Dordrecht) p. 45 http://arxiv.org/abs/astro-ph/0502247 Introduction Ultraviolet Observations Type Ia Supernovae Existing Samples of UV Spectra of SNIa Core Collapse Supernovae: Types II and Ib/c Type Ib/c Supernovae Type II Supernovae Supernovae and Cosmology

0704.1667

Stochastic fluctuations in metabolic pathways

Stochastic fluctuations in metabolic pathways Erel Levine and Terence Hwa Center for Theoretical Biological Physics and Department of Physics, University of California at San Diego, La Jolla, CA 92093-0374 Abstract. Fluctuations in the abundance of molecules in the living cell may affect its growth and well being. For regulatory molecules (e.g., signaling proteins or transcription factors), fluctuations in their expression can affect the levels of downstream targets in a network. Here, we develop an analytic framework to investigate the phenomenon of noise correlation in molecular networks. Specifically, we focus on the metabolic network, which is highly inter-linked, and noise properties may constrain its structure and function. Motivated by the analogy between the dynamics of a linear metabolic pathway and that of the exactly soluable linear queueing network or, alternatively, a mass transfer system, we derive a plethora of results concerning fluctuations in the abundance of intermediate metabolites in various common motifs of the metabolic network. For all but one case examined, we find the steady-state fluctuation in different nodes of the pathways to be effectively uncorrelated. Consequently, fluctuations in enzyme levels only affect local properties and do not propagate elsewhere into metabolic networks, and intermediate metabolites can be freely shared by different reactions. Our approach may be applicable to study metabolic networks with more complex topologies, or protein signaling networks which are governed by similar biochemical reactions. Possible implications for bioinformatic analysis of metabolimic data are discussed. http://arxiv.org/abs/0704.1667v1 Stochastic fluctuations in metabolic pathways 2 Due to the limited number of molecules for typical molecular species in microbial cells, random fluctuations in molecular networks are common place and may play important roles in vital cellular processes. For example, noise in sensory signals can result in pattern formation and collective dynamics [1], and noise in signaling pathways can lead to cell-to-cell variability [2]. Also, stochasticity in gene expression has implications on cellular regulation [3, 4] and may lead to phenotypic diversity [5, 6], while fluctuations in the levels of (toxic) metabolic intermediates may reduce metabolic efficiency [7] and impede cell growth. In the past several years, a great deal of experimental and theoretical efforts have focused on the stochastic expression of individual genes, at both the translational and transcriptional levels [8, 9, 10]. The effect of stochasticity on networks has been studied in the context of small, ultra-sensitivie genetic circuits, where noise at a circuit node (i.e., a gene) was shown to either attenuate or amplify output noise in the steady state [11, 12]. This phenomenon — termed ‘noise propagation’ — make the steady-state fluctuations at one node of a gene network dependent in a complex manner on fluctuations at other nodes, making it difficult for the cell to control the noisiness of individual genes of interest [13]. Several key questions which arise from these studies of genetic noise include (i) whether stochastic gene expression could further propagate into signaling and metabolic networks through fluctuations in the levels of key proteins controlling those circuits, and (ii) whether noise propagation occurs also in those circuits. Recently, a number of approximate analytical methods have been applied to analyze small genetic and signaling circuits; these include the independent noise approximation [14, 15, 16], the linear noise approximation [14, 17], and the self-consistent field approximation [18]. Due perhaps to the different approximation schemes used, conflicting conclusions have been obtained regarding the extent of noise propagation in various networks (see, e.g., [17].) Moreover, it is difficult to extend these studies to investigate the dependences of noise correlations on network properties, e.g., circuit topology, nature of feedback, catalytic properties of the nodes, and the parameter dependences (e.g., the phase diagram). It is of course also difficult to elucidate these dependences using numerical simulations alone, due to the very large degrees of freedoms involved for a network with even a modest number of nodes and links. In this study, we describe an analytic approach to characterize the probability distribution for all nodes of a class of molecular networks in the steady state. Specifically, we apply the method to analyze fluctuations and their correlations in metabolite concentrations for various core motifs of the metabolic network. The metabolic network consists of nodes which are the metabolites, linked to each other by enzymatic reactions that convert one metabolite to another. The predominant motif in the metabolic network is a linear array of nodes linked in a given direction (the directed pathway), which are connected to each other via converging pathways and diverging branch points [19]. The activities of the key enzymes are regulated allosterically by metabolites from other parts of the network, while the levels of many enzymes are controlled transcriptionally and are hence subject to deterministic as well as stochastic variations Stochastic fluctuations in metabolic pathways 3 in their expressions [20]. To understand the control of metabolic network, it is important to know how changes in one node of the network affect properties elsewhere. Applying our analysis to directed linear metabolic pathways, we predict that the distribution of molecule number of the metabolites at intermediate nodes to be statistically independent in the steady state, i.e., the noise does not propagate. Moreover, given the properties of the enzymes in the pathway and the input flux, we provide a recipe which specifies the exact metabolite distribution function at each node. We then show that the method can be extended to linear pathways with reversible links, with feedback control, to cyclic and certain converging pathways, and even to pathways in which flux conservation is violated (e.g., when metabolites leak out of the cell). We find that in these cases correlations between nodes are negligable or vanish completely, although nontrivial fluctuation and correlation do dominate for a special type of converging pathways. Our results suggest that for vast parts of the metabolic network, different pathways can be coupled to each other without generating complex correlations, so that properties of one node (e.g., enzyme level) can be changed over a broad range without affecting behaviors at other nodes. We expect that the realization of this remarkable property will shape our understanding of the operation of the metabolic network, its control, as well as its evolution. For example, our results suggest that correlations between steady-state fluctuations in different metabolites bare no information on the network structure. In contrast, temporal propagation of the response to an external perturbation should capture - at least locally - the morphology of the network. Thus, the topology of the metabolic network should be studied during transient periods of relaxation towards a steady-state, and not at steady-state. Our method is motivated by the analogy between the dynamics of biochemical reactions in metabolic pathways and that of the exactly solvable queueing systems [46] or, alternatively, as mass transfer systems [22, 47]. Our approach may be applicable also to analyzing fluctuations in signaling networks, due to the close analogy between the molecular processes underlying the metabolic and signaling networks. To make our approach accessible to a broad class of circuit modelers and bioengineers who may not be familiar with nonequilibrium statistical mechanics, we will present in the main text only the mathematical results supported by stochastic simulations, and defer derivations and illustrative calculations to the Supporting Materials. While our analysis is general, all examples are taken from amino-acid biosynthesis pathways in E. Coli [24]. 1. Individual Nodes 1.1. A molecular Michaelis-Menton model In order to set up the grounds for analyzing a reaction pathway and to introduce our notation, we start by analyzing fluctuations in a single metabolic reaction catalyzed by an enzyme. Recent advances in experimental techniques have made it possible to track the Stochastic fluctuations in metabolic pathways 4 enzymatic turnover of a substrate to product at the single-molecule level [26, 27], and to study instantaneous metabolite concentration in the living cell [28]. To describe this fluctuation mathematically, we model the cell as a reaction vessel of volume V , containing m substrate molecules (S) and NE enzymes (E). A single molecule of S can bind to a single enzyme E with rate k+ per volume, and form a complex, SE. This complex, in turn, can unbind (at rate k−) or convert S into a product form, P , at rate k2. This set of reactions is summarized by S + E k2→P + E . (1) Analyzing these reactions within a mass-action framework — keeping the substrate concentration fixed, and assuming fast equilibration between the substrate and the enzymes (k± ≫ k2) — leads to the Michaelis-Menten (MM) relation between the macroscopic flux c and the substrate concentration [S] = m/V : c = vmax[S]/([S] +KM) , (2) where KM = k−/k+ is the dissociation constant of the substrate and the enzyme, and vmax = k2[E] is the maximal flux, with [E] = NE/V being the total enzyme concentration. Our main interest is in noise properties, resulting from the discreteness of molecules. We therefore need to track individual turnover events. These are described by the turnover rate wm, defined as the inverse of the mean waiting time per volume between the (uncorrelated‡) synthesis of one product molecule to the next. Assuming again fast equilibration between the substrate and the enzymes, the probability of having NSE complexes given m substrate molecules and NE enzymes is simply given by the Boltzmann distribution, p(NSE|m,NE) = K−NSE Zm,NE m!NE ! NSE!(m−NSE)!(NE −NSE)! for NSE < NE and m. Here K −1 = V k+/k− is the Boltzmann factor associated with the formation of an SE complex, and the Zm,NE takes care of normalization (i.e., chosen such that p(NSE|m,NE) = 1.) Under this condition, the turnover rate NSE · p(NSE |m,NE) is given approximately by wm = vmax m+ (K +NE − 1) +O(K−3) , (4) with vmax = k2NE/V ; see Supp. Mat. We note that for a single enzyme (NE = 1), one has wm = vmaxm/(m+K), which was derived and verified experimentally [27, 29]. 1.2. Probability distribution of a single node In a metabolic pathway, the number of substrate molecules is not kept fixed; rather, these molecules are synthesized or imported from the environment, and at the same time ‡ We note in passing that some correlations do exist – but not dominate – in the presence of “dynamical disorder” [27], or if turnover is a multi-step process [29, 30]. Stochastic fluctuations in metabolic pathways 5 turned over into products. We consider the influx of substrate molecules to be a Poisson process with rate c. These molecules are turned into product molecules with rate wm given by Eq. (4). The number of substrate molecules is now fluctuating, and one can ask what is the probability π(m) of finding m substrate molecules at the steady-state. This probability can be found by solving the steady-state Master equation for this process (see Supp. Mat.), yielding π(m) = m+K + (NE − 1) (1− z)K+NEzm , (5) where z = c/vmax [31]. The form of this distribution is plotted in supporting figure 1 (solid black line). As expected, a steady state exists only when c ≤ vmax. Denoting the steady-state average by angular brackets, i.e., 〈xm〉 ≡ m xm π(m), the condition that the incoming flux equals the outgoing flux is written as c = 〈wm〉 = vmax s+ (K +NE) , (6) where s ≡ 〈m〉. Comparing this microscopically-derived flux-density relation with the MM relation (2) using the obvious correspondence [S] = s/V , we see that the two are equivalent with KM = (K +NE)/V . Note that this microscopically-derived form of MM constant is different by the amount [E] from the commonly used (but approximate form) KM = K/V , derived from mass-action. However, for typical metabolic reactions, KM ∼ 10 − 1000µM [24] while [E] is not more than 1000 molecules in a bacterium cell (∼ 1µM); so the numerical values of the two expressions may not be very different. We will characterize the variation of substrate concentration in the steady-state by the noise index η2s ≡ c · (K +NE) , (7) where σ2s is the variance of the distribution π(m). Since c ≤ vmax and increases with s towards 1 (see Eq. 6), ηs decreases with the average occupancy s as expected. It is bound from below by 1/ K +NE , which can easily be several percent. Generally, large noise is obtained when the reaction is catalyzed by a samll number of high-affinity enzymes (i.e., for low K and NE). 2. Linear pathways 2.1. Directed pathways We now turn to a directed metabolic pathway, where an incoming flux of substrate molecules is converted, through a series of enzymatic reactions, into a product flux [19]. Typically, such a pathway involves the order of 10 reactions, each takes as precursor the product of the preceding reaction, and frequently involves an additional side-reactant (such as a water molecule or ATP) that is abundant in the cell (and whose fluctuations can be neglected). As a concrete example, we show in figure 1(a) the tryptophan Stochastic fluctuations in metabolic pathways 6 Figure 1. Linear biosynthesis pathway. (a) Tryptophan biosynthesis pathway in E. Coli. (b) Model for a directed pathway. Dashed lines depict end-product inhibition. Abbreviations: CPAD5P, 1-O-Carboxyphenylamino 1-deoxyribulose-5-phosphate; NPRAN, N-5-phosphoribosyl-anthranilate; IGP, Indole glycerol phosphate; PPI, Pyrophosphate; PRPP, 5-Phosphoribosyl-1-pyrophosphate; T3P1, Glyceraldehyde 3- phosphate. biosynthesis pathway of E. Coli [24], where an incoming flux of chorismate is converted through 6 directed reactions into an outgoing flux of tryptophan, making use of several side-reactants. Our description of a linear pathway includes an incoming flux c of substrates of type S1 along with a set of reactions that convert substrate type Si to Si+1 by enzyme Ei (see figure 1(b)) with rate w mi = vimi/(mi +Ki − 1) according to Eq. (4). We denote the number of molecules of intermediate Si by mi, with m1 for the substrate and mL for the end-product. The superscript (i) indicates explicitly that the parameters vi = k E /V and Ki = (K (i) + N E ) describing the enzymatic reaction Si → Si+1 are expected to be different for different reactions. The steady-state of the pathway is fully described by the joint probability distribution π(m1, m2, . . . , mL) of having mi molecules of intermediate substrate type Si. Surprisingly, this steady-state distribution is given exactly by a product measure, π(m1, m2, . . . , mL) = πi(mi) , (8) where πi(m) is as given in Eq. (5) (with K +NE replaced by Ki and z by zi = c/vi), as we show in Supp. Mat. This result indicates that in the steady state, the number of molecules of one intermediate is statistically independent of the number of molecules of any other substrate§. The result has been derived previously in the context of queueing networks [46], and of mass-transport systems [47]. Either may serve as a useful analogy for a metabolic pathway. Since the different metabolites in a pathway are statistically decoupled in the steady state, the mean si = 〈mi〉 and the noise index η2si = c −1vi/Ki can be determined by Eq. (7) individually for each node of the pathway. It is an interesting consequence of the decoupling property of this model that both the mean concentration of each substrate and the fluctuations depend only on the properties of the enzyme immediately downstream. While the steady-state flux c is a constant throughout the pathway, the § We note, however, that short-time correlations between metabolites can still exist, and may be probed for example by measuring two-time cross-correlations; see discussion at the end of the text. Stochastic fluctuations in metabolic pathways 7 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Figure 2. Noise in metabolite molecular number (ηs = σs/s) for different pathways. Monte-Carlo simulations (bars) are compared with the analytic prediction (symbols) obtained by assuming decorrelation for different nodes of the pathways. The structure of each pathway is depicted under each panel. Parameter values were chosen randomly such that 103 < Ki < 10 4 and c < vi < 10c. SImilar decorrelation was obtained for 100 different random choices of parameters, and for 100 different sets with Ki 10-fold smaller (data not shown). The effect on the different metabolites of a change in the velocity of the first reaction, v1 = 1.1c (dark gray)→ 5c (light gray), is demonstrated. Similar results are obtained for changes in K1 (data not shown.) (a) Directed pathway. Here the decorrelation property is exact. (b) Directed pathway with two reversible reactions. For these reactions, v+ 3,4 = 8.4, 6.9c; v 3,4 = 1.6, 3.7, c;K 3,4 = 2500, 8000 and K− 3,4 = 7700, 3700. (c) Linear dilution of metabolites. Here β/c = 1/100. (d) End-product inhibition,where the influx rate is given by α = c0 [1 + (mL/KI)] KI = 1000. (e) Diverging pathways. Here metabolite 4 is being processed by two enzymes (with different affinities, KI = 810,KII = 370) into metabolites 5 and 7, resp. (f) Converging pathways. Here two independent 3-reaction pathways , with fluxes c and c′ = c/2, produce the same product, S4. parameters vi and Ki can be set separately for each reaction by the copy-number and kinetic properties of the enzymes (provided that c < vi). Hence, for example, in a case where a specific intermediate may be toxic, tuning the enzyme properties may serve to decrease fluctuations in its concentration, at the price of a larger mean. To illustrate the decorrelation between different metabolites, we examine the response of steady-state fluctuations to a 5-fold increase in the enzyme level [E1]. Typical time scale for changes in enzyme level much exceeds those of the enzymatic reactions. Hence, the enzyme level changes may be considered as quasi-steady state. In figure 2(a) we plot the noise indices of the different metabolites. While noise in the first node is significantly reduced upon a 5-fold increase in [E1], fluctuations at the other nodes are not affected at all. 2.2. Reversible reactions The simple form of the steady-state distribution (8) for the directed pathways may serve as a starting point to obtain additional results for metabolic networks with more elaborate features. We demonstrate such applications of the method by some examples below. In many pathways, some of the reactions are in fact reversible. Thus, a Stochastic fluctuations in metabolic pathways 8 metabolite Si may be converted to metabolite Si+1 with rate v maxmi/(mi + K i ) or to substrate Si−1 with rate v maxmi/(mi + K i ). One can show — in a way similar to Ref. [47] — that the decoupling property (5) holds exactly only if the ratio of the two rates is a constant independent of mi, i.e. when K i = K i . In this case the steady state probability is still given by (5), with the local currents obeying v+i zi − v−i+1zi+1 = c . (9) This is nothing but the simple fact that the overall flux is the difference between the local current in the direction of the pathway and that in the opposite direction. In general, of course, K+i 6= K−i . However, we expect the distribution to be given approximately by the product measure in the following situations: (a) K+i ≃ K−i ; (b) the two reactions are in the zeroth-order regime, s ≫ K±i ; (c) the two reactions are in the linear regime, s ≪ K±i . In the latter case Eq. (9) is replaced by zk+1 = c . Taken together, it is only for a narrow region (i.e., si ∼ Ki) where the product measure may not be applicable. This prediction is tested numerically, again by comparing two pathways (now containing reversible reactions) with 5-fold difference in the level of the first enzyme. From figure2(b), we see again that the difference in noise indices exist only in the first node, and the computed value of the noise index at each node is in excellent agreement with predictions based on the product measure (symbols). SImilar decorrelation was obtained for 100 different random choices of parameters, and for 100 different sets with Ki 10-fold smaller (data not shown). 2.3. Dilution of intermediates In the description so far, we have ignored possible catabolism of intermediates or dilution due to growth. This makes the flux a conserved quantity throughout the pathway, and is the basis of the flux-balance analysis [32]. One can generalize our framework for the case where flux is not conserved, by allowing particles to be degraded with rate um. Suppose, for example, that on top of the enzymatic reaction a substrate is subjected to an effective linear degradation, um = βm. This includes the effect of dilution due to growth, in which case β = ln(2)/(mean cell division time), and the effect of leakage out of the cell. As before, we first consider the dynamics at a single node, where the metabolite is randomly produced (or transported) at a rate c0. It is straightforward to generalize the Master equation for the microscopic process to include um, and solve it in the same way. With wm as before, the steady state distribution of the substrate pool size is then found to be π(m) = m+K − 1 (c0/β) (v/β +K)m , (10) where (a)m ≡ a(a+ 1) · · · (a+m− 1). This form of π(m) allows one to easily calculate moments of the molecule number from the partition function Z as in equilibrium statistical mechanics, e.g. s = 〈m〉 = c0dZ/dc0, and thence the outgoing flux, c = c0−βs. Using the fact that Z can be written explicitly in terms of hypergeometric functions, Stochastic fluctuations in metabolic pathways 9 we find that the noise index grows with β as η2s ≃ v/(Kc0) + β/c0. The distribution function is given in supporting figure 1 for several values of β. Generalizing the above to a directed pathway, we allow for β, as well as for vmax and K, to be i-dependent. The decoupling property (8) does not generally hold in the non-conserving case [33]. However, in this case the stationary distribution still seems to be well approximated by a product of the single-metabolite functions πi(m) of the form (10), with c0/β → ci−1/βi. This is supported again by the excellent agreement between noise indices obtained by numerical simulations and analytic calculations using the product measure Ansatz, for linear pathways with dilution of intermediates; see figure 2(c). In this case, change in the level of the first enzyme does ”propagate” to the downstream nodes. But this is not a “noise propagation” effect, as the mean fluxes 〈ci〉 at the different nodes are already affected. (To illustrate the effect of leakage, the simulation used parameters that corresponded to a huge leakage current which is 20% of the flux. This is substantially larger than typical leakage encountered, say due to growth-mediated dilution, and we do not expect propagation effects due to leakage to be significant in practice.) 3. Interacting pathways The metabolic network in a cell is composed of pathways of different topologies. While linear pathways are abundant, one can also find circular pathways (such as the TCA cycle), converging pathways and diverging ones. Many of these can be thought of as a composition of interacting linear pathways. Another layer of interaction is imposed on the system due to the allosteric regulation of enzyme activity by intermediate metabolites or end products. To what extent can our results for a linear pathway be applied to these more complex networks? Below we address this question for a few of the frequently encountered cases. To simplify the analysis, we will consider only directed pathways and suppress the dilution/leakage effect. 3.1. Cyclic pathways We first address the cyclic pathway, in which the metabolite SL is converted into S1 by the enzyme EL. Borrowing a celebrated result for queueing networks [34] and mass transfer models [35], we note that the decoupling property (8) described above for the linear directed pathway also holds exactly even for the cyclic pathways‖. This result is surprising mainly because the Poissonian nature of the “incoming” flux assumed in the analysis so far is lost, replaced in this case by a complex expression, e.g., w mL · πL(mL). In an isolated cycle the total concentration of the metabolites, stot – and not the ‖ In fact, the decoupling property holds for a general network of directed single-substrate reactions, even if the network contains cycles. Stochastic fluctuations in metabolic pathways 10 flux – is predetermined. In this case, the flux c is give by the solution to the equation stot = si(c) = vi − c . (11) Note that this equation can always be satisfied by some positive c that is smaller than all vi’s. In a cycle that is coupled to other branches of the network, flux may be governed by metabolites going into the cycle or taken from it. In this case, flux balance analysis will enable determination of the variables zi which specify the probability distribution 3.2. End-product inhibition Many biosynthesis pathways couple between supply and demand by a negative feedback [24, 19], where the end-product inhibits the first reaction in the pathway or the transport of its precursor; see, e.g., the dashed lines in figure 1. In this way, flux is reduced when the end-product builds up. In branched pathways this may be done by regulating an enzyme immediately downstream from the branch-point, directing some of the flux towards another pathway. To study the effect of end-product inhibition, we consider inhibition of the inflow into the pathway. Specifically, we model the probability at which substrate molecules arrive at the pathway by a stochastic process with exponentially-distributed waiting time, characterized by the rate α(mL) = c0 1 + (mL/KI) , where c0 is the maximal influx (determined by availability of the substrate either in the medium or in the cytoplasm), mL is the number of molecules of the end-product (SL), KI is the dissociation constant of the interaction between the first enzyme E0 and SL, and h is a Hill coefficient describing the cooperativity of interaction between E0 and SL. Because mL is a stochastic variable itself, the incoming flux is described by a nontrivial stochastic process which is manifestly non-Poissonian. The steady-state flux is now c = 〈α(mL)〉 = c0 · 1 + (mL/KI) . (12) This is an implicit equation for the flux c, which also appears in the right-hand side of the equation through the distribution π(m1, ..., mL). By drawing an analogy between feedback-regulated pathway and a cyclic pathway, we conjecture that metabolites in the former should be effectively uncorrelated. The quality of this approximation is expected to become better in cases where the ration between the influx rate α(mL) and the outflux rate wmL is typically mL idependent. Under this assumption, we approximate the distribution function by the product measure (8), with the form of the single node distributions given by (5). Note that the conserved flux then depends on the properties of the enzyme processing the last reaction, and in general should be influenced by the fluctuations in the controlling metabolite. In this sense, these fluctuations propagate throughout the pathway at the level of the mean Stochastic fluctuations in metabolic pathways 11 flux. This should be expected from any node characterized by a high control coefficient Using this approximate form, Eq. (12) can be solved self-consistently to yield c(c0), as is shown explicitly in Supp. Mat. for h = 1. The solution obtained is found to be in excellent agreement with numerical simulation (Supporting figure 2a). The quality of the product measure approximation is further scrutinized by comparing the noise index of each node upon increasing the enzyme level of the first node 5-fold. Figure 2(c) shows clearly that the effect of changing enzyme level does not propagate to other nodes. While being able to accurately predict the flux and mean metabolite level at each node, the predictions based on the product measure are found to be under-estimating the noise index by up to 10% (compare bars and symbols). We conclude that in this case correlations between metabolites do exist, but not dominate. Thus analytic expressions dervied from the decorrelation assumption can be useful even in this case (see supporting figure 2b). 3.3. Diverging pathways Many metabolites serve as substrates for several different pathways. In such cases, different enzymes can bind to the substrate, each catabolizes a first raction in a different pathway. Within our scheme, this can be modeled by allowing for a metabolite Si to be converted to metabolite SI1 with rate w = vImi/(mi +K I − 1) or to metabolite SII1 with rate wImi = v IImi/(mi +K II − 1). The paramters vI,II and KI,II characterize the two different enzymes. Similar to the case of reversible reactions, the steady-state distribution is given exactly by a product measure only if wImi/w is a constant, independent of mi (namely when KI = KII). Otherwise, we expect it to hold in a range of alternative scenarios, as described for reversible pathways. Considering a directed pathway with a single branch point, the distribution (5) describes exactly all nodes upstream of that point. At the branchpoint, one replaces wm by wm = w m + w m , to obtain the distribution function π(m) = (KI)m(K m!((KIvII +KIIvII)/(vI + vII))m . (13) From this distribution one can obtain the fluxes going down each one of the two branching pathway, cI,II = wI,IIm π(m). Both fluxes depend on the properties of both enzymes, as can be seen from (13), and thus at the branch-point the two pathways influence each other [36]. Moreover, fluctuations at the branch point to propagate into the branching pathways already at the level of the mean flux. This is consistent with the fact that the branch node is expected to be characterized by a high control coefficient While different metabolite upstream and including the branch point are uncorrelated, this is not exactly true for metabolites of the two branches. Nevertheless, since these pathways are still directed, we further conjecture that metabolites in the two Stochastic fluctuations in metabolic pathways 12 carbamoyl−phosphate L−ornithine citrulline L−arginino−succinate L−arginine L−threonine L−serine glycine (a) (b) Figure 3. Converging pathways. (a) Glycine is synthesized in two independent pathways. (b) Citrulline is synthesized from products of two pathways. Abbreviations: 2A3O, 2-Amino-3-oxobutanoate. branching pathways can still be described, independently, by the probability distribution (5), with c given by the flux in the relevant branch, as calculated from (13). Indeed, the numerical results of figure 2(e) strongly support this conjecture. We find that changing the noise properties of a metabolite in the upstream pathway do not propagte to those of the branching pathways. 3.4. Converging pathways – combined fluxes We next examine the case where two independent pathways result in synthesis of the same product, P . For example, the amino acid glycine is the product of two (very short) pathways, one using threonine and the other serine as precursors (figure 3(a)) [24]. With only directed reactions, the different metabolites in the combined pathway – namely, the two pathways producing P and a pathway catabolizing P – remain decoupled. The simplest way to see this is to note that the process describing the synthesis of P , being the sum of two Poisson processes, is still a Poisson process. The pathway which catabolizes P is therefore statistically identical to an isolated pathway, with an incoming flux that is the sum of the fluxes of the two upstream pathways. More generally, the Poissonian nature of this process allows for different pathways to dump or take from common metabolite pools, without generating complex correlations among them. 3.5. Converging pathways – reaction with two fluctuating substrates As mentioned above, some reactions in a biosynthesis pathway involve side-reactants, which are assumed to be abundant (and hence at a constant level). Let us now discuss briefly a case where this approach fails. Suppose that the two products of two linear pathways serve as precursors for one reaction. This, for example, is the case in the arginine biosynthesis pathway, where L-ornithine is combined with carbamoyl-phosphate by ornithine-carbamoyltransferase to create citrulline (figure 3(b)) [24]. Within a flux balance model, the net fluxes of both substrates must be equal to achieve steady state, in which case the macroscopic Michaelis-Menten flux takes the form c = vmax [S1][S2] (KM1 + [S1])(KM2 + [S2]) Stochastic fluctuations in metabolic pathways 13 0 2 4 6 8 10 Metabolite 1 Metabolite 2 Figure 4. Time course of a two-substrate enzymatic reaction, as obtained by a Gillespie simulation [44]. Here c = 3t−1, k+ = 5t −1 and k− = 2t −1 for both substrates, t being an arbitrary time unit. Here [S1,2] are the steady-state concentrations of the two substrates, and KM1,2 the corresponding MM-constants. However, flux balance provides only one constraint to a system with two degrees of freedom. In fact, this reaction exhibits no steady state. To see why, consider a typical time evolution of the two substrate pools (figure 4). Suppose that at a certain time one of the two substrates, say S1, is of high molecule-number compared with its equilibrium constant, m1 ≫ K1. In this case, the product synthesis rate is unaffected by the precise value of m1, and is given approximately by vmaxm2/(m2+K2). Thus, the number m2 of S2 molecules can be described by the single-substrate reaction analyzed above, while m1 performs a random walk (under the influence of a weak logarithmic potential), which is bound to return, after some time τ , to values comparable with K1. Then, after a short transient, one of the two substrates will become unlimiting again, and the system will be back in the scenario described above, perhaps with the two substrates changing roles (depending on the ratio between K1 and K2). Importantly, the probability for the time τ during which one of the substrates is at saturating concentration scales as τ−3/2 for large τ . During this time the substrate pool may increase to the order τ . The fact that τ has no finite mean implies that this reaction has no steady state. Since accumulation of any substrate is most likely toxic, the cell must provide some other mechanism to limit these fluctuations. This may be one interpretation for the fact that within the arginine biosynthesis pathway, L-ornithine is an enhancer of carbamoyl-phosphate synthesis (dashed line in figure 3(b)). In contrast, a steady-state always exists if the two metabolites experience linear degradation, as this process prevents indefinite accumulation. However, in general one expects enzymatic reactions to dominate over futile degradation. In this case, equal in-fluxes of the two substrates result in large fluctuations, similar to the ones described above [31]. Stochastic fluctuations in metabolic pathways 14 4. Discussion In this work we have characterized stochastic fluctuations of metabolites for dominant simple motifs of the metabolic network in the steady state. Motivated by the analogy between the directed biochemical pathway and the mass transfer model or, equivalently, as the queueing network, we show that the intermediate metabolites in a linear pawthway – the key motif of the biochemical netrowk – are statistically independent. We then extend this result to a wide range of pathway structures. Some of the results (e.g., the directed linear, diverging and cyclic pathways) have been proven previously in other contexts. In other cases (e.g., for reversible reaction, diverging pathway or with leakage/dilution), the product measure is not exact. Nevertheless, based on insights from the exactly solvable models, we conjecture that it still describes faithfuly the statistics of the pathway. Using the product measure as an Ansatz, we obtained quantitative predictions which turned out to be in excellent agreement with the numerics (figure 2). These results suggest that the product measure may be an effective starting point for quantitative, non-perturbative analysis of (the stochastic properties) of these circuit/networks. We hope this study will stimulate further analytical studies of the large variety of pathway topologies in metabolic networks, as well as in-depth mathematical analysis of the conjectured results. Moreover, it will be interesting to explore the applicability of the present approach to other cellular networks, in particular, stochasticity in protein signaling networks [2], whose basic mathematical structure is also a set of interlinked Michaelis-Menton reactions. Our main conclusion, that the steady-state fluctuations in each metabolite depends only on the properties of the reactions consuming that metabolite and not on fluctuations in other upstream metabolites, is qualitatively different from conclusions obtained for gene networks in recent studies, e.g., the “noise addition rule” [14, 15] derived from the Independent Noise Approximation, and its extension to cases where the singnals and the processing units interact [17]. The detailed analysis of [17], based on the Linear Noise Approximation found certain anti-correlation effects which reduced the extent of noise propagation from those expected by “noise addition” alone [14, 15]. While the specific biological systems studied in [17] were taken from protein signaling systems, rather than metabolic networks, a number of systems studied there are identical in mathematical structure to those considered in this work. It is reassuring to find that reduction of noise propagation becomes complete (i.e., no noise propagation) according to the analysis of [17], also, for Poissonian input noise where direct comparisons can be made to our work (ten Wolde, private communication). The cases in which residue noise propagation remained in [17], corresponded to certain “bursty” noises which is non- Poissonian. While bursty noise is not expected for metabolic and signaling reactions, it is nevertheless important to address the extent to which the main finding of this work is robust to the nature of stochasticity in the input and the individual reactions. The exact result on the cyclic pathways and the numerical result on the directed pathway with feedback inhibition suggest that our main conclusion on statistical independence Stochastic fluctuations in metabolic pathways 15 of the different nodes extends significantly beyond strict Poisson processes. Indeed, generalization that preserve this property include classes of transport rules and extended topologies [37, 38]. The absence of noise propagation for a large part of the metabolic network allows intermediate metabolites to be shared freely by multiple reactions in multiple pathways, without the need of installing elaborate control mechanisms. In these systems, dynamic fluctuations (e.g., stochasticity in enzyme expression which occurs at a much longer time scale) stay local to the node, and are shielded from triggering system-level failures (e.g., grid-locks). Conversely, this property allows convenient implementation of controls on specific node of pathways, e.g., to limit the pool of a specific toxic intermediate, without the concern of elevating fluctuations in other nodes. We expect this to make the evolution of metabolic network less constrained, so that the system can modify its local properties nearly freely in order to adapt to environmental or cellular changes. The optimized pathways can then be meshed smoothly into the overall metabolic network, except for junctions between pathways where complex fluctuations not constrained by flux conservation. In recent years, metabolomics, i.e., global metabolite profiling, has been suggested as a tool to decipher the structure of the metabolic network [39, 40]. Our results suggest that in many cases, steady-state fluctuations do not bare information about the pathway structure. Rather, correlations between metabolite fluctuations may be, for example, the result of fluctuation of a common enzyme or coenzyme, or reflect dynamical disorder [27]. Indeed, a bioinformatic study found no straightforward connection between observed correlation and the underlying reaction network [41]. Instead, the response to external perturbation [28, 39, 42] may be much more effective in shedding light on the underlying structure of the network, and may be used to study the morphing of the network under different conditions. It is important to note that all results described here are applicable only to systems in the steady state; transient responses such as the establishment of the steady state and the response to external perturbations will likely exhibit complex temporal as well as spatial correlations. Nevertheless, it is possible that some aspects of the response function may be attainable from the steady-state fluctuations through non- trivial fluctuation-dissipation relations as was shown for other related nonequilibrium systems [22, 43]. Acknowledgments We are grateful to Peter Lenz and Pieter Rein ten Wolde for discussions. This work was supported by NSF through the PFC-sponsored Center for Theoretical Biological Physics (Grants No. PHY-0216576 and PHY-0225630). TH acknowledges additional support by NSF Grant No. DMR-0211308. Stochastic fluctuations in metabolic pathways 16 Supporting Material Appendix A. Microscopic model Under the assumption of fast equlibration between the substrate and the enzyme, the probability of having NSE complexes given m substrate molecules and NE enzymes is given by equation (3) of the main text. To write the partition function explicitly, we define u(x) = U(x, 1 −m − NE;−K), where U denotes the Confluent Hypergeometric function [45]. One can then write the partition sum as Zm,NE = (−K)−NEu(−m). The turnover rate is then given by wm = [−mu(1 − m)]/[u(−m)], which can be approximated by Equation (4). Appendix B. Influx of metabolites Ametabolic reaction in vivo can be described as turnover of an incoming flux of substrate molecules, characterized by a Possion process with rate c, into an outgoing flux. To find the probability of having m substrate molecules we write down the Master equation, π(m) = [c(a− 1) + (â− 1)wm]π(m) = c[π(m−1)−π(m)]+[wm+1π(m+1)−wmπ(m)] , (B.1) where we took the opportunity to define the lowering and raising operators a and â, which – for any function h(n) – satisfy ah(n) = h(n−1), ah(0) = 0, and âh(n) = h(n+1). The first term in this equation is the influx, and the second is the biochemical reaction. The solution of this steady state equation is of the form π(m) ∼ cm/ k=1wk (up to a normalization constant), as can be verified by plugging it into the equation, π(m− 1) π(m+ 1) wm+1 − wm wm+1 − wm = 0.(B.2) Using the approximate form of wm, as given in (4), the probability π(m) takes the form, π(m) = m+K + (NE − 1) (1− z)K+NEzm , (B.3) as given in equation (5) of the main text. Appendix C. Directed linear pathway We now derive our key results, equation (8) (The result has been derived previously in the context of queueing networks [46], and of mass-transport systems [47]). To this end we write the Master equation for the joint probability function π ≡ π(m1, m2, · · · , mL), c(a1 − 1) + (âiai+1 − 1)w(i)mi + (âL − 1)w π , (C.1) which generalizes (B.1). As above, ai and âi are lowering and raising operators, acting on the number of Si molecules. The first term in this equation is the incoming flux c Stochastic fluctuations in metabolic pathways 17 of the substrate, and the last term is the flux of end product. Let us try to solve the steady-state equation by plugging a solution of the form π(m1, m2, · · · , mL) = gi(mi), yielding gi(m1 − 1) g1(m1) gi(mi + 1)gi+1(mi+1 − 1) gi(mi)gi+1(mi+1) −w(i)mi ]+[w gL(mL + 1) gL(mL) −w(L)mL ] = 0 .(C.2) Motivated by the solution to (B.1), we try to satisfy this equation by choosing gi(m) = c k . With this choice we have g(m + 1)/g(m) = c/wm+1 and g(m− 1)/g(m) = wm/c. It is now straightforward to verify that indeed w(i+1)mi+1 − w c− w(L)mL = 0 . (C.3) Finally, in our choice of gi(m) we replace w m by the MM- rate vimi/(mi+Ki), and find that in fact gi(m) = πi(m), namely π(m1, m2, . . . , mL) = πi(mi) , (C.4) as stated in (8). Appendix D. End-product inhibition Equation (13) of the main text is a self-consistent equation for the steady- state flux c through a pathway regulated via end-product inhibition. Using considerations analogous to what led to the exact result on the product measure distribution for the cyclic pathways, we conjecture that even for the present case of end-product inhibition, the distribution function can still be approximated by the product measure (C.4) with the form of the single node distributions given by (B.3). The flux c enters the calculation of the average on the right-hand side through the probability function π(m). Solving this equation for c yields the steady state current, and consequently determines the mean occupancy and standard deviation of all intermediates. To verify the validity of this conjecture, and to demonstrate its application, we consider the case h = 1. In this case one can carry the sum, and find 1 + (mL/KI) πL(mL) (D.1) = c0(1− z)KL2F1(KI , KL;KI + 1; z) with z = c/vL and 2F1 the hypergeometric function [45]. This equation was solved numerically, and plotted in supporting figure D2(a) for some values of KI and KL. Note that predictions based on the product measure (lines) are in excellent agreement with the results of numerical simulation (circles) for the different sets of parameters tried. Stochastic fluctuations in metabolic pathways 18 0 50 100 150 Figure D1. The steady-state distribution π(m) of a metabolite, that experiences enzymatic reaction (with rate wm = vm/(m +K − 1)) and linear degradation (with rate βm), as given by equation (10) of the main text. Here K = 100 and v = 2c0. Results obtained from equation (D.1) can be used, for example, to compare the flux that flows through the noisy pathway with the mean-field flux cMF, obtained when one ignores fluctuations in mL, i.e., cMF = 1 + (sL/KI)h . (D.2) The fractional difference δc = (c− cMF)/cMF is plotted in supporting figure D2(b). The results show that number fluctuations in the end-product always increase the flux in the pathway since δc > 0 always. Quantitatively, this increase can easily be several percent. For large c0, a simplifying expression can be derived by using an asymptotic expansion of the hypergeometric function [45]. For example, when KI < KL, (1− z)KL2F1(KI , KL;KI + 1; z) ∼ 1 +KL −KI , (D.3) which yields c− cMF . (D.4) Thus the effect of end-product fluctuations on the current is enhanced by stronger binding of the inhibitor (smaller KI), as one would expect. We note that obtaining these predictions from Monte-Carlo simulation is rather difficult, given the fact that one is interested here in sub-leading quantities. Stochastic fluctuations in metabolic pathways 19 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 =10, K =100, K =10, K =100, K Figure D2. Pathway with end-product inhibition. The influx rate is taken to be c0/(1+mL/KI), and thus the steady-state flux is given by equation (12) of the main text, with h = 1. (a) Assuming that different metabolites in the pathway remain decoupled even in the presence of feedback regulation, (12) can be approximated by (D.1). Numerical solutions of equation (D.1) (lines) are compared with Monte- Carlo simulations (symbols). Values of parameters are chosen randomly such that 100 < Ki < 1000 and c < vi < 10c. For the data presented here, vL = 2.4c. We find that (D.1) yields excellent prediction for the steady-state flux. (b) Neglecting fluctuations altogether yields a mean-field approximation for the flux, cMF, given in ( D.2). For the same data of (a), we plot the fractional difference δc = (c− cMF)/c. We find that steady-state flux is increased by fluctuations, and thus taking fluctuations into account (even in an approximate manner) better predicts the steady-state flux. 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0704.1668

A new search for planet transits in NGC 6791

arXiv:0704.1668v1 [astro-ph] 12 Apr 2007 Astronomy & Astrophysics manuscript no. n6791 c© ESO 2021 August 31, 2021 A new search for planet transits in NGC 6791. ⋆ M. Montalto1, G. Piotto1, S. Desidera2, F. De Marchi1, H. Bruntt3,4, P.B. Stetson5 A. Arellano Ferro6, Y. Momany1,2 R.G. Gratton2, E. Poretti7, A. Aparicio8, M. Barbieri2,9, R.U. Claudi2, F. Grundahl3, A. Rosenberg8. 1 Dipartimento di Astronomia, Università di Padova, Vicolo dell’Osservatorio 2, I-35122, Padova, Italy 2 INAF – Osservatorio Astronomico di Padova, Vicolo dell’ Osservatorio 5, I-35122, Padova, Italy 3 Department of Physics and Astronomy, University of Aarhus, Denmark 4 University of Sydney, School of Physics, 2006 NSW, Australia 5 Herzberg Institute of Astrophysics, Victoria, Canada 6 Instituto de Astronomı́a, Universidad Nacional Autónoma de México 7 INAF – Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy 8 Instituto de Astrofisica de Canarias, 38200 La Laguna, Tenerife, Canary Islands, Spain 9 Dipartimento di Fisica, Università di Padova, Italy ABSTRACT Context. Searching for planets in open clusters allows us to study the effects of dynamical environment on planet formation and evolution. Aims. Considering the strong dependence of planet frequency on stellar metallicity, we studied the metal rich old open cluster NGC 6791 and searched for close-in planets using the transit technique. Methods. A ten-night observational campaign was performed using the Canada-France-Hawaii Telescope (3.6m), the San Pedro Mártir tele- scope (2.1m), and the Loiano telescope (1.5m). To increase the transit detection probability we also made use of the Bruntt et al. (2003) eight-nights observational campaign. Adequate photometric precision for the detection of planetary transits was achieved. Results. Should the frequency and properties of close-in planets in NGC 6791 be similar to those orbiting field stars of similar metallicity, then detailed simulations foresee the presence of 2-3 transiting planets. Instead, we do not confirm the transit candidates proposed by Bruntt et al. (2003). The probability that the null detection is simply due to chance coincidence is estimated to be 3%-10%, depending on the metallicity assumed for the cluster. Conclusions. Possible explanations of the null-detection of transits include: (i) a lower frequency of close-in planets in star clusters; (ii) a smaller planetary radius for planets orbiting super metal rich stars; or (iii) limitations in the basic assumptions. More extensive photometry with 3–4m class telescopes is required to allow conclusive inferences about the frequency of planets in NGC 6791. Key words. open cluster: NGC 6791 – planetary systems – Techniques: photometric 1. Introduction During the last decade more than 200 extra-solar planets have been discovered. However, our knowledge of the formation and evolution of planetary systems remains largely incomplete. One crucial consideration is the role played by environment where planetary systems may form and evolve. More than 10% of the extra-solar planets so far discov- ered are orbiting stars that are members of multiple systems Send offprint requests to: M. Montalto, e-mail: [email protected] ⋆ Based on observation obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the Univesity of Hawaii and on observations obtained at San Pedro Mártir 2.1 m telescope (Mexico), and Loiano 1.5 m telescope (Italy). (Desidera & Barbieri 2007). Most of these are binaries with fairly large separations (a few hundred AU). However, in few cases, the binary separation reaches about 10 AU (Hatzes et al. 2003; Konacki 2005), indicating that planets can exist even in the presence of fairly strong dynamical interactions. Another very interesting dynamical environment is repre- sented by star clusters, where the presence of nearby stars or proto-stars may affect the processes of planet formation and evolution in several ways. Indeed, close stellar encoun- ters may disperse the proto-planetary disks during the fairly short (about 10 Myr, e.g., Armitage et al. 2003) epoch of giant planet formation or disrupt the planetary system after its forma- tion (Bonnell et al. 2001; Davies & Sigurdsson 2001; Woolfson 2004; Fregeau et al. 2006). Another possible disruptive effect is the strong UV flux from massive stars, which causes photo- evaporation of dust grains and thus prevents planet formation (Armitage 2000; Adams et al. 2004). These effects are expected http://arxiv.org/abs/0704.1668v1 2 M. Montalto et al.: A new search for planet transits in NGC 6791. to depend on star density, being much stronger for globular clusters (typical current stellar density ∼ 103 stars pc−3) than for the much sparser open clusters (≤ 102 stars pc−3). The recent discovery of a planet in the tight triple system HD188753 (Konacki 2005) adds further interest to the search for planets in star clusters. In fact, the small separation between the planet host HD188753A and the pair HD188753BC (about 6 AU at periastron) makes it very challenging to understand how the planet may have been formed (Hatzes & Wüchterl 2005). Portegies, Zwart & McMillan et al. (2005) propose that the planet formed in a wide triple within an open cluster and that dynamical evolution successively modified the configura- tion of the system. Without observational confirmation of the presence of planets in star clusters, such a scenario is purely speculative. On the observational side, the search for planets in star clusters is a quite challenging task. Only the closest open clus- ters are within reach of high-precision radial velocity surveys (the most successful planet search technique). However, the activity-induced radial velocity jitter limits significantly the detectability of planets in clusters as young as the Hyades (Paulson et al. 2004). Hyades red giants have a smaller activity level, and the first planet in an open cluster has been recently announced by Sato et al. (2007), around ǫ Tau. The search for photometric transits appears a more suitable technique: indeed it is possible to monitor simultaneously a large number of cluster stars. Moreover, the target stars may be much fainter. However, the transit technique is mostly sen- sitive to close-in planets (orbital periods ≤ 5 days). Space and ground-based wide-field facilities were also used to search for planets in the globular clusters 47 Tucanae and ω Centauri. These studies (Gilliland et al. 2000; Weldrake et al. 2005; Weldrake et al. 2006) reported not a single planet de- tection. This seemed to indicate that planetary systems are at least one order of magnitude less common in globular clusters than in Solar vicinity. The lack of planets in 47 Tuc and ω Cen may be due either to the low metallicity of the clusters (since planet frequency around solar type stars appears to be a rather strong function of the metallicity of the parent star: Fischer & Valenti 2005; Santos et al. 2004), or to environmental effects caused by the high stellar density (or both). One planet has been identified in the globular cluster M4 (Sigurdsson et al. 2003), but this is a rather peculiar case, as the planet is in a circumbinary orbit around a system including a pulsar and it may have formed in a different way from the planets orbiting solar type stars (Beer et al. 2004). Open clusters are not as dense as globular clusters. The dy- namical and photo-evaporation effects should therefore be less extreme than in globular clusters. Furthermore, their metallic- ity (typically solar) should, in principle, be accompanied by a higher planet frequency. In the past few years, some transit searches were specif- ically dedicated to open clusters: see e.g. von Braun et al. (2005), Bramich et al. (2005), Street et al. (2003), Burke et al. (2006), Aigrain et al. (2006) and references therein. However, in a typical open cluster of Solar metallicity with ∼ 1000 cluster members, less than one star is expected to show a planetary transit. This depends on the assumption that the planet frequency in open clusters is similar to that seen for nearby field stars 1. Considering the unavoidable transits de- tection loss due to the observing window and photometric er- rors, it turns out that the probability of success of such efforts is fairly low unless several clusters are monitored 2. On the other hand, the planet frequency might be higher for open clusters with super-solar metallicities. Indeed, for [Fe/H] between +0.2 and +0.4 the planet frequency around field stars is 2-6 times larger than at solar metallicity. However, only a few clusters have been reported to have metallicities above [Fe/H]= +0.2. The most famous is NGC 6791, a quite massive cluster that is at least 8 Gyr old (Stetson et al. 2003; King et al. 2005, and Carraro et al. 2006). As estimated by different au- thors, its metallicity is likely above [Fe/H]=+0.2 (Taylor 2001) and possibly as high as [Fe/H]=+0.4 (Peterson et al. 1998). The most recent high dispersion spectroscopy studies confirmed the very high metallicity of the cluster ([Fe/H]=+0.39, Carraro et al. 2006; [Fe/H]=+0.47, Gratton et al. 2006). Its old age im- plies the absence of significant photometric variability induced by stellar activity. Furthermore, NGC 6791 is a fairly rich clus- ter. All these facts make it an almost ideal target. NGC 6791 has been the target of two photometric cam- paigns aimed at detecting planets transits. Mochejska et al. (2002, 2005, hereafter M05) observed the cluster in the R band with the 1.2 m Fred Lawrence Whipple Telescope dur- ing 84 nights, over three observing seasons (2001-2003). They found no planet candidates, while the expected number of de- tections considering their photometric precision and observing window was ∼ 1.3. Bruntt et al. (2003, hereafter B03) observed the cluster for 8 nights using ALFOSC at NOT. They found 10 transit candidates, most of which (7) being likely due to instru- mental effects. Nearly continuous, multi-site monitoring lasting several days could strongly enhance the transit detectability. This idea inspired our campaigns for multi-site transit planet searches in the super metal rich open clusters NGC 6791 and NGC 6253. This paper presents the results of the observations of the cen- tral field of NGC 6791, observed at CFHT, San Pedro Mártir (SPM), and Loiano. We also made use of the B03 data-set [ob- tained at the Nordic Optical Telescope (NOT) in 2001] and re- duce it as done for our three data-sets. The analysis for the ex- ternal fields, containing mostly field stars, and of the NGC 6253 campaign, will be presented elsewhere. The outline of the paper is the following: Sect. 2 presents the instrumental setup and the observations. We then describe the reduction procedure in Sect. 3, and the resulting photomet- ric precision for the four different sites is in Sect. 4. The selec- tion of cluster members is discussed in Sect. 5. Then, in Sect. 6, we describe the adopted transit detection algorithm. In Sect. 7 we present the simulations performed to establish the transit detection efficiency (TDE) and the false alarm rate (FAR) of 1 0.75% of stars with planets with period less than 7 days (Butler et al. 2000), and a 10% geometric probability to observe a transit. 2 A planet candidate was recently reported by Mochejska et al. (2006) in NGC 2158, but the radius of the transiting object is larger than any planet known up to now (∼ 1.7 RJ). The companion is then most likely a very low mass star. M. Montalto et al.: A new search for planet transits in NGC 6791. 3 the algorithm for our data-sets. Sect. 8 illustrates the different approaches that we followed in the analysis of the data. Sect. 9 gives details about the transit candidates. In Sect. 10, we estimate the expected planet frequency around main sequence stars of the cluster, and the expected number of detectable transiting planets in our data-sets. In Sect. 11, we compare the results of the observations with those of the simulations, and discussed their significance. In Sect. 12 we discuss the different implications of our results, and, in Sect. 13, we make a comparison with other transit searches to- ward NGC 6791. In Sect. 14 we critically analyze all the ob- servations dedicated to the search for planets in NGC 6791 so far, and propose future observations of the cluster, and finally, in Sect. 15, we summarize our work. 2. Instrumental setup and observations The observations were acquired during a ten-consecutive-day observing campaign, from July 4 to July 13, 2002. Ideally, one should monitor the cluster nearly continuously. For this reason, we used three telescopes, one in Hawaii, one in Mexico, and the third one in Italy. In Table 1 we show a brief summary of our observations. In Hawaii, we used the CFHT with the CFH12K detector 3, a mosaic of 12 CCDs of 2048×4096 pixels, for a total field of view of 42×28 arcmin, and a pixel scale of 0.206 arcsec/pixel. We acquired 278 images of the cluster in the V filter. The see- ing conditions ranged between 0.′′6 to 1.′′9, with a median of 1.′′0. Exposure times were between 200 and 900 sec, with a median value of 600 sec. The observers were H. Bruntt and P.B. Stetson. In San Pedro Mártir, we used the 2.1m telescope, equipped with the Thomson 2k detector. However the data section of the CCD corresponded to a 1k × 1k pixel array. The pixel scale was 0.35 arcsec/pixel, and therefore the field of view (∼ 6 arcmin2) contained just the center of the cluster, and was smaller than the field covered with the other detectors. We made use of 189 images taken between July 6, 2002 and July 13, 2002. During the first two nights the images were acquired using the focal re- ducer, which increased crowding and reduced our photometric accuracy. All the images were taken in the V filter with ex- posure times of 480 − 1200 sec (median 660 sec), and seeing between 1.′′1 and 2.′′1 (median 1.′′4). Observations were taken by A. Arellano Ferro. In Italy, we used the Loiano 1.5m telescope4 equipped with BFOSC + the EEV 1300×1348B detector. The pixel scale was 0.52 arcsec/pixel, for a total field coverage of 11.5 arcmin2. We observed the target during four nights (2002 July 6−9). We ac- quired and reduced 63 images of the cluster in the V and Gunn i filters (61 in V , 2 in i). The seeing values were between 1.′′1 3 www.cfht.hawaii.edu/Instruments/Imaging/CFH12K/ 4 The observations were originally planned at the Asiago Observatory using the 1.82 m telescope + AFOSC. However, a ma- jor failure of instrument electronics made it impossible to perform these observations. We obtained four nights of observations at the Loiano Observatory, thanks to the courtesy of the scheduled observer M. Bellazzini and of the Director of Bologna Observatory F. Fusi Pecci. and 4.′′3 arcsec, with a median value of 1.′′4 arcsec. Exposure times ranged between 120 and 1500 sec (median 1080 sec). The observer was S. Desidera. We also make use of the images taken by B03 in 2001. We obtained these images from the Nordic Optical Telescope (NOT) archive. As explained in BO3, these data covered eight nights between July 9 and 17, 2001. The detector was ALFOSC5 a 2k×2k thinned Loral CCD with a pixel scale of 0.188 arcsec/pixel yielding a total field of view of 6.5 arcmin2. Images were taken in the I and V filters with a median seeing of 1 arcsec. We used only the images of the central part of the cluster (which were the majority) excluding those in the exter- nal regions. In total we reduced 227 images in the V filter and 389 in the I filter. It should be noted that ALFOSC and BFOSC are focal re- ducers, hence light concentration introduced a variable back- ground. These effects can be important when summed with flat fielding errors. In order to reduce these un-desired effects, the images were acquired while trying to maintain the stars in the same positions on the CCDs. Nevertheless, the precision of this pointing procedure was different for the four telescopes: the median values of the telescope shifts are of 0.′′5, 0.′′5, 3.′′4 and 2.′′1, respectively for Hawaii, NOT, SPM, and Loiano. This means that while for the CFHT and the NOT the median shift was at a sub-seeing level (half of the median seeing), for the other two telescopes it was respectively of the order of 2.4 and 1.5 times the median seeing. Hence, it is possible that flat- fielding errors and possible background variation have affected the NOT, SPM, and Loiano photometry, but the effects on the Hawaii photometry are expected to be smaller. Bad weather conditions and the limited time allocated at Loiano Observatory caused incomplete coverage of the sched- uled time interval. Moreover we did not use the images coming from the last night (eighth night) of observation in La Palma with the NOT because of bad weather conditions. Our observ- ing window, defined as the interval of time during which obser- vations were carried out, is shown in Tab. 2 for Hawaii, SPM and Loiano observations and in Tab. 3 for La Palma observa- tions. 3. The reduction process 3.1. The pre-reduction procedure For the San Pedro, Loiano and La Palma images, the pre- reduction was done in a standard way, using IRAF routines6. The images from the Hawaii came already reduced via the ELIXIR software7. 5 ALFOSC is owned by the Instituto de Astrofisica de Andalucia (IAA) and operated at the Nordic Optical Telescope under agreement between IAA and the NBIfAFG of the Astronomical Observatory of Copenhagen. 6 IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the 7 http://www.cfht.hawaii.edu/Instruments/Elixir 4 M. Montalto et al.: A new search for planet transits in NGC 6791. Table 1. Summary of the observations taken during 4-13 July, 2002 in Hawaii, San Pedro Màrtir, Loiano and from 9-17 July, 2001 in La Palma (by B03). Hawaii San Pedro Mártir Loiano NOT N. of Images 278 189 63 227(V), 389(I) Nights 8 8 4 8 Scale (arcsec/pix) 0.21 0.35 0.52 0.188 FOV (arcmin) 42 x 28 6 x 6 11.5 x 11.5 6.5 x 6.5 Table 2. The observing window relative to the July 2002 observations. The number of observing hours is given for each night. The last line shows the total number of observing hours for each site. Night Hawaii SPM Loiano 1st 3.58 − − 2nd 3.88 − − 3rd 2.68 6.31 3.58 4th 7.56 6.61 5.21 5th 5.23 6.58 6.08 6th 8.30 6.86 5.45 7th − 3.20 − 8th − 7.03 − 9th 8.34 7.27 − 10th 8.37 4.15 − Total 47.94 48.01 20.32 Table 3. The observing window relative to the July 2001 observations. Night La Palma 1st 5.45 2nd 7.06 3rd 8.04 4th 7.61 5th 7.57 6th 7.82 7th 7.72 8th 2.41 Total 53.68 3.2. Reduction strategies The data-sets described in Sec. 2 were reduced with three dif- ferent techniques: aperture photometry, PSF fitting photometry and image subtraction. An accurate description of these tech- niques is given in the next Sections. Our goal was to compare their performances to see if one of them performed better than the others. For what concerned aperture and PSF fitting pho- tometry we used the DAOPHOT (Stetson 1987) package. In particular the aperture photometry routine was slightly different from that one commonly used in DAOPHOT and was provided by P .B. Stetson. It performed the photometry after subtracting all the neighbors stars of each target star. Image subtraction was performed by means of the ISIS2.2 package (Alard & Lupton, 1998) except for what concerned the final photometry on the subtracted images which was performed with the DAOPHOT aperture routine for the reasons described in paragraph 3.4. 3.3. DAOPHOT/ALLFRAME reduction: aperture and PSF fitting photometry The DAOPHOT/ALLFRAME reduction package has been ex- tensively used by the astronomical community and is a very well tested reduction technique. The idea behind this stellar photometry package consists in modelling the PSF of each im- age following a semi-analytical approach, and in fitting the derived model to all the stars in the image by means of least square method. After some tests, we chose to calculate a vari- able PSF across the field (quadratic variability). We selected the first 200 brightest, unsaturated stars in each frame, and calcu- lated a first approximate PSF from them. We then rejected the stars to which DAOPHOT assigned anomalously high fitting χ values. After having cleaned the PSF star list, we re-calculated the PSF. This procedure was iterated three times in order to obtain the final PSF for each image. We then used DAOMATCH and DAOMASTER (Stetson 1992) in order to calculate the coordinate transformations among the frames, and with MONTAGE2 we created a refer- ence image by adding up the 50 best seeing images. We used this high S/N image to create a master list of stars, and applied ALLFRAME (Stetson 1994) to refine the estimated star posi- tions and magnitudes in all the frames. We applied a first selec- tion on the photometric quality of our stars by rejecting all stars with SHARP and CHI parameters (Stetson 1987) deviating by more than 1.5 times the RMS of the distribution of these pa- rameters from their mean values, both calculated in bins of 0.1 magnitudes. About 25% of the stars were eliminated by this se- lection. This was the PSF fitting photometry we used in further analysis. The aperture photometry with neighbor subtraction was obtained with a new version of the PHOT routine (devel- oped by P. B. Stetson). We used as centroids the same values used for ALLFRAME. The adopted apertures were equal to the FWHM of the specific image, and after some tests we set the annular region for the calculation of the sky level at a distance < r < 2.5 (both for the ALLFRAME and for the aperture photometry). Finally, we used again DAOMASTER for the cross- correlation of the final star lists, and to prepare the light curves. 3.4. Image Subtraction photometry In the last years the Image Subtraction technique has been largely used in photometric reductions. This method firstly im- plemented in the software ISIS did not assume any specific functional shape for the PSF of each image. Instead it mod- eled the kernel that convolved the PSF of the reference image M. Montalto et al.: A new search for planet transits in NGC 6791. 5 to match the PSF of a target image. The reference image is convolved by the computed kernel and then subtracted from the image. The photometry is then done on the resulting differ- ence image. Isolated stars were not required in order to model the kernel. This technique had rapidly gained an appreciable consideration across the astronomical community. Since its ad- vent, it appeared particularly well suited for the search for vari- able stars, and it has proved to be very effective in extremely crowded fields like in the case of globular clusters (e.g. Olech et al. 1999, Kaluzny et al. 2001, Clementini et al. 2004, Corwin et al. 2006). An extensive use of this approach has been applied also in long photometric surveys devoted to the search for ex- trasolar planet transits (eg. Mochejska et al., 2002, 2004). We used the standard reduction routines in the ISIS2.2 package. At first, the images were interpolated on the reference system of the best seeing image. Then we created a reference image from the 50 images with best seeing. We performed dif- ferent tests in order to set the best parameters for the subtrac- tion, and we checked the images to find which combination with lowest residuals. In the end, we decided to sub-divide the images in four sub-regions, and to apply a kernel and sky back- ground variable at the second order. Using ALLSTAR, we build up a master list of stars from the reference image. As in Bruntt et al. (2003), we were not able to obtain a reliable photometry using the standard pho- tometry routine of ISIS. We used the DAOPHOT aperture pho- tometry routine and slightly modified it in order to accept the subtracted images. Aperture, and sky annular region were set as for the aperture and PSF photometry. Then the magnitude of the stars in the subtracted images was obtained by means of the following formula: mi = mref − 2.5 log (Nref − Ni where mi was the magnitude of a generic star in a generic i subtracted image, mref was the magnitude of the correspondent star in the reference image, Nref were the counts obtained in the reference image and Ni is the i th subtracted image. 3.5. Zero point correction For what concerns psf fitting photometry and aperture photom- etry, we corrected the light curves taking into account the zero points magnitude (the mean difference in stellar magnitude) be- tween a generic target image, and the best seeing image. This was done by means of DAOMASTER and can be considered as a first, crude correction of the light curves. Image subtraction was able to handle these first order corrections automatically, and thus the resulting light curves were already free of large zero points offsets. Nevertheless, important residual correlations persisted in the light curves, and it was necessary to apply specific, and more refined post-reduction corrections, as explained in the next Section. 3.6. The post reduction procedure In general, and for long time series photometric studies, it has been commonly recognized that regardless of the adopted re- duction technique, important correlations between the derived magnitudes and various parameters like seeing, airmass, expo- sure time, etc. persist in the final photometric sequences. As put into evidence by Pont et al. (2006), the presence of correlated noise in real light curves, (red noise), can significantly reduce the photometric precision that can be obtained, and hence re- duce transit detectability. For example ground-based photomet- ric measurements are affected by color-dependent atmospheric extinction. This is a problem since, in general, photometric sur- veys employ only one filter and no explicit colour information is available. To take into account these effects, we used the method developed by Tamuz et al. (2005). The software was provided by the same authors, and an accurate description of it can be found in the referred paper. One of the critical points in this algorithm regarded the choice of systematic effects to be removed from the data. For each systematic effect, the algorithm performed an iteration af- ter which it removed the systematic passing to the next. To verify the efficiency of this procedure (and establish the number of systematic effects to be removed) we performed the following simulations. We started from a set of 4000 artifi- cial stars with constant magnitudes. These artificial light curves were created with a realistic modeling of the noise (which ac- counts for the intrinsic noise of the sources, of the sky back- ground, and of the detector electronics) but also for the pho- tometric reduction algorithm itself, as described in Sec. 7.1. Thus, they are fully representative of the systematics present in our data-set. At this point we also add 10 light curves in- cluding transits that were randomly distributed inside the ob- serving window. These spanned a range of (i) depths from a few milli-magnitudes to around 5 percents, (ii) durations and (iii) periods respectively of a few hours to days accordingly to our assumptionts on the distributions of these parameters for planetary transits, as accounted in Sec. 7.3. In a second step, we applied the Tamuz et al. (2005) algo- rithm to the entire light curves data-set. For a deeper under- standing of the total noise effects and the transit detection ef- ficiency, we progressively increased the number of systematic effects to be removed (eigenvectors in the analysis described by Tamuz 2005). Typically, we started with 5 eigenvectors and increased it to 30. Repeated experiments showed no significant RMS im- provement after the ten iterations. The final RMS was 15%- 20% lower than the original RMS . Thus, the number of eigen- vectors was set to 10. In no case the added transits were re- moved from the light curves and the transit depths remained unaltered. We conclude that this procedure, while reducing the RMS and providing a very effective correction for systematic effects, did not influence the uncorrelated magnitude variations associ- ated with transiting planets. 6 M. Montalto et al.: A new search for planet transits in NGC 6791. 4. Definition of the photometric precision To compare the performances of the different photometric al- gorithms we calculated the Root Mean Square, (RMS ), of the photometric measurements obtained for each star, which is de- fined as: RMS = Ii−<I> N − 1 Where Ii is the brightness measured in the generic i th image for a particular target star, < I > is the mean star brightness in the entire data-set, and N is the number of images in which the star has been measured. For constant stars, the relative variation of brightness is mainly due to the photometric measurement noise. Thus, the RMS (as defined above) is equal to the mean N/S of the source. In order to allow the detection of transiting jovian-planets eclipses, whose relative brightness variations are of the order of 1%, the RMS of the source must be lower than this level. 4.1. PSF photometry against aperture photometry The first comparison we performed was that between aper- ture photometry (having neighboring stars subtracted) and PSF fitting photometry. We started with aperture photometry. Figure 1, shows a comparison of the RMS dispersion of the light curves obtained with the new PHOT software with re- spect to the RMS of the PSF fitting photometry for the different sites. We also show the theoretical noise which was estimated considering the contribute to the noise coming from the pho- ton noise of the source and of the sky background as well as the noise coming from the instrumental noise (see Kjeldsen & Frandsen 1992, formula 31). In Fig. 1- 3, we also separated the contribution of the source’s Poisson noise from that of the sky (short dashed line) and of the detector noise (long dashed line). The total theoretical noise is represented as a solid line. It is clear that the data-sets from the different telescopes gave different results. In the case of the CFHT and SPM data-sets, aperture photometry does not reach the same level of preci- sion as PSF fitting photometry (for both bright and the faint sources). Moreover, it appears that the RMS of aperture pho- tometry reaches a constant value below V ∼ 18.5 for CFHT data and around V ∼ 17.5 for SPM data, while for PSF fitting photometry the RMS continues to decrease. For Loiano data, and with respect to PSF photometry, aperture photometry pro- vides a smaller RMS in the light curve, in particular for bright sources. The NOT observations on the other hand show that the two techniques are almost equivalent. Leaving aside these differences, it is clear that the CFHT provide the best photo- metric precision, and this is due to the larger telescope diam- eter, the smaller telescope pixel scale (0.206 arcsec/pixel, see Table 1), and the better detector performances at the CFHT. For this data-set, the photometric error remains smaller than 0.01 mag, from the turn-off of the cluster (∼ 17.5) to around mag- nitude V = 21, allowing the search for transiting planets over a magnitude range of about 3.5 magnitudes, (in fact, it is possible to go one magnitude deeper because of the expected increase in the transit depth towards the fainter cluster stars, see Sec. 5 for more details). Loiano photometry barely reaches 0.01 mag photometric precision even for the brightest stars, a photomet- ric quality too poor for the purposes of our investigation. The search for planetary transits is limited to almost 2 magnitudes below the turn-off for SPM data (in particular with the PSF fit- ting technique) and to 1.5 magnitude below the turn-off for the NOT data. In any case, the photometric precision for the SPM and NOT data-sets reaches the 0.004 mag level for the brightest stars, while, for the CFHT, it reaches the 0.002 mag level. It is clear that both the PSF fitting photometry and the aper- ture photometry tend to have larger errors with respect to the expected error level. This effect is much clearer for Loiano, SPM and NOT rather than for Hawaii photometry. As explained by Kjeldsen & Frandsen (1992), and more recently by Hartman et al. (2005), the PSF fitting approach in general results in poorer photometry for the brightest sources with respect to the aperture photometry. But, for our data-sets, this was true only for the case of Loiano photometry, as demonstrated above. The aperture photometry routine in DAOPHOT returns for each star, along with other information, the modal sky value asso- ciated with that star, and the rms dispersion (σsky) of the sky values inside the sky annular region. So, we chose to calcu- late the error associated with the random noise inside the star’s aperture with the formula: σAperture = σ2sky Area (3) where Area is the area (in pixels2) of the region inside which we measure the star’s brightness. This error automatically takes into account the sky Poissonian noise, instrumental effects like the Read Out Noise, (RON), or detector non-linearities, and possible contributions of neighbor stars. To calculate this error, we chose a representative mean-seeing image, and subdivide the magnitude range in the interval 17.5 < V < 22.0 into nine bins of 0.5 mag. Inside each of these bins, we took the minimum value of the stars’ sky variance as representative of the sky variance of the best stars in that magnitude bin. We over-plot this contribution in Fig. 4 which is relative to the San Pedro photometry. This error completely accounts for the ob- served photometric precision. So, the variance inside the star’s aperture is much larger than what is expected from simple pho- ton noise calculations. This can be the effect of neighbor stars or of instrumental problems. For CFHT photometry, as we have good seeing conditions and an optimal telescope scale, crowd- ing plays a less important role. Concerning the other sites, we noted that for Loiano the crowding is larger than for SPM and NOT, and this could explain the lower photometric precision of Loiano observations, along with the smaller telescope diameter. For NOT photometry, instead, the crowding should be larger than for San Pedro (since the scale of the telescope is larger) while the median seeing conditions are comparable for the two data-sets, as shown in Sect 2. Therefore this effect should be more evident for NOT rather than for SPM, but this is not the case, at least for what concerns aperture photometry. For PSF photometry, as it is seen in Fig. 2, the NOT photometric preci- sion appears more scattered than the SPM photometry. M. Montalto et al.: A new search for planet transits in NGC 6791. 7 Fig. 1. Comparison of the photometric precision for aperture photometry with neighbor subtraction (Ap) and for PSF fitting photometry (PSF) as a function of the apparent visual magnitude for CFHT, SPM, Loiano and NOT images. The short dashed line indicates the N/S considering only the star photon noise, the long dashed line is the N/S due to the sky photon noise and the detector noise. The continuous line is the total We are forced to conclude that poor flat fielding, optical distortions, non-linearity of the detectors and/or presence of saturated pixels in the brightest stars must have played a sig- nificant role in setting the aperture and PSF fitting photometric precisions. 4.2. PSF photometry against image subtraction photometry Applying the image subtraction technique we were able to improve the photometric precision with respect to that ob- tained by means of the aperture photometry and the PSF fitting techniques. This appears evident in Fig. 2, in which the im- age subtraction technique is compared to the PSF fitting tech- nique. Again, the best overall photometry was obtained for the CFHT , for the reasons explained in the previous subsection. For the image subtraction reduction, the photometric precision overcame the 0.001 mag level for the brightest stars in the CFHT data-set, and for the other sites it was around 0.002 mag (for the NOT ) or better (for S PM and Loiano). This clearly al- lowed the search for planets in all these different data-sets. In this case, it was possible to include also the Loiano observa- tions (up to 2 magnitudes below the turn-off), and, for the other sites, to extend by about 0.5-1 mag, the range of magnitudes over which the search for transits was possible, (see previous Section). The reason for which image subtraction gave better results could be that it is more suitable for crowded regions (as the center of the cluster), because it doesn’t need isolated stars in order to calculte the convolution kernel while the subtraction of stars by means of PSF fitting can give rise to higher residuals, 8 M. Montalto et al.: A new search for planet transits in NGC 6791. Fig. 2. Comparison of the photometric precision for image subtraction (ISIS) and for psf fitting photometry (PSF) as function of the apparent visual magnitude for CFHT, SPM, Loiano, and NOT images. because it’s much more difficult to obtain a reliable PSF from crowded stars. 4.3. The best photometric precision Given the results of the previous comparisons, we decided to adopt the photometric data set obtained with the image sub- traction technique. Figure 3, shows the photometric precision that we obtained for the four different sites. The photometric precision is very close to the theoretical noise for all the data- sets. The NOT data-set has a lower photometric precision with respect to SPM and even to Loiano, in particular for the bright- est stars. We observed that the mean S/N for the NOT images is lower than for the other sites because of the larger number of images taken (and consequently of their lower exposure times and S/N), see Tab. 1. 5. Selection of cluster members To detect planetary transits in NGC 6791 we selected the proba- ble main sequence cluster members as follows. Calibrated mag- nitudes and colors were obtained by cross-correlating our pho- tometry with the photometry by Stetson et al. (2003), based on the same NOT data-set used in the present paper. Then, as done in M05, we considered 24 bins of 0.2 magnitudes in the interval 17.3 < V < 22.1. For each bin, we calculated a robust mean of all (B-V) star colors, discarding outliers with colors differing by more than ∼ 0.06 mag from the mean. Our selected main sequence members are shown in Fig. 5. Overall, we selected 3311 main-sequence candidates in NGC 6791. These are the stars present in at least one of the four data-sets (see Sec. 8), and represent the candidates for our planetary transits search. Note that our selection criteria excludes stars in the bi- nary sequence of the cluster. These are blended objects, for which any transit signature should be diluted by the light of the unresolved companion(s) and then likely undetectable. M. Montalto et al.: A new search for planet transits in NGC 6791. 9 Fig. 3. The expected RMS noise for the observations taken at the different sites as a function of the visual apparent magnitude, is compared with the RMS of the observed light curves obtained with the image subtraction technique. Furthermore, a narrow selection range helps in reducing the field-star contamination. 6. Description of the transit detection technique 6.1. The box fitting technique To detect transits in our light curves we adopted the BLS algo- rithm by Kovács et al. (2002). This technique is based on the fitting of a box shaped transit model to the data. It assumes that the value of the magnitude outside the transit region is constant. It is applied to the phase folded light curve of each star span- ning a range of possible orbital periods for the transiting object, (see Table 4). Chi-squared minimization is used to obtain the best model solution. The quantity to be maximized in order to get the best solution is: )2[ 1 Nin Nout where mn = Mn − < M >. Mn is the n-th measurement of the stellar magnitude in the light curve, < M > is the mean mag- nitude of the star and thus mn is the n-th residual of the stellar magnitude. The sum at the numerator includes all photometric measurements that fall inside the transit region. Finally Nin and Nout are respectively the number of photometric measurements inside and outside the transit region. The algorithm, at first, folds the light curve assuming a par- ticular period. Then, it sub-divides the folded light curve in nb bins and starting from each one of these bins calculates the T index shown above spanning a range of transit lengths between qmi and qma fraction of the assumed period. Then, it provides the period, the depth of the brightness variation, δ, the transit length, and the initial and final bins in the folded light curve at which the maximum value of the index T occurs. We used a routine called ’eebls’ available on the web8. We applied also the so called directional correction (Tingley 2003a, 2003b) which 8 http://www.konkoly.hu/staff/kovacs/index.html 10 M. Montalto et al.: A new search for planet transits in NGC 6791. Fig. 4. RMS noise for the San Pedro Mártir observations with the aper- ture error (triangles) as estimated by Equation 3. Fig. 5. The NGC 6791 CMD highlighting the selection region of the main sequence stars (blue circles). consists in taking into account the sign of the numerator in the above formula in order to retain only the brightness variations which imply a positive increment in apparent magnitude. 6.2. Algorithm parameters The parameters to be set before running the BLS algorithm are the following: 1) nf, number of frequency points for which the spectrum is computed; 2) fmin, minimum frequency; 3) df, frequency step; 4) nb, number of bins in the folded time se- ries at any test frequency; 5) qmi, minimum fractional transit length to be tested; 6) qma, maximum fractional transit length to be tested; qmi and qma are given as the product of the tran- Table 4. Adopted parameters for the BLS algorithm: nf is the number of frequency steps adopted, fmin is the minimum frequency consid- ered, df is the increasing frequency step, nb is the number of bins in the folded time series at any test frequency, qmi and qma are the mini- mum and maximum fractional transit length to be tested, as explained in the text. nf fmin(days−1) df(days−1) nb qmi qma 3000 0.1 0.0005 1000 0.01 0.1 sit length to the test frequency. Table 4 displays our adopted parameters. 6.3. Algorithm transit detection criteria To characterize the statistical significance of a transit-like event detected by the BLS algorithm we followed the meth- ods by Kovács & Bakos (2005): deriving the Dip Significance Parameter (hereafter DSP) and the significance of the main pe- riod signal in the Out of Transit Variation (hereafter OOTV, given by the folded time series with the exclusion of the tran- sit). The Dip Significance Parameter is defined as DSP = δ(σ2/Ntr + A OOTV) 2 (5) where δ is the depth of the transit given by the BLS at the point at which the index T is maximum,σ is the standard deviation of the Ntr in-transit data points, AOOTV is the peak amplitude in the Fourier spectrum of the Out of Transit Variation. The threshold for the DSP set by Kovacs & Bakos (2005) is 6.0 and it was set on artificial constant light curves with gaussian noise. In real light curves the noise is not gaussian, as explained in Sec. 3.6, and, in general, the value of the DSP threshold should be set case by case. In Sec. 8, we presented the adopted thresholds, based on our simulations on artificial light curves, described in Sec. 7. The significance of the main periodic signal in the OOTV is defined as: SNROOTV = σ A (AOOTV− < A >) (6) where < A > and σA are the average and the standard deviation of the Fourier spectrum. This parameter accounts for the Out Of Transit Variation, and we impose it to be lower than 7.0, as in Kovacs & Bakos (2005). For our search we imposed a maximum transit duration of six hours; we also required that at least ten data points must be included in the transit region. 7. Simulations The Transit Detection Efficiency (TDE) of the adopted algo- rithm and its False Alarm Rate (FAR) were determined by means of detailed simulations. The TDE is a measure of the probability that an algorithm correctly identifies a transit in a light curve. The FAR is a measure of the probability that an M. Montalto et al.: A new search for planet transits in NGC 6791. 11 algorithm identifies a feature in a light curve that does not rep- resent a transit, but rather a spurious photometric effect. In the following discussion, we address the details of the simulations we performed, considering the case of the CFHT observations of NGC 6791. Because the CFHT data provided the best of our photometric sequences, the results on the algo- rithm performance is shown below, and should be considered as an upper limit for the other cases. 7.1. Simulations with constant light curves Artificial stars with constant magnitude were added to each im- age, according to an equally-spaced grid of 2*PSFRADIUS+1, (where the PSFRADIUS was the region over which the Point Spread Function of the stars was calculated, and was around 15 pixels for the CFHT images), as described in Piotto & Zoccali (1999). We took into account the photometric zero-point dif- ferences among the images, and the coordinate transformations from one image to another. 7722 stars were added on the CFHT images. In order to assure the homogeneity of these simula- tions, the artificial stars were added exactly in the same posi- tions, (relative to the real stars in the field), for the other sites. Because of the different field of views of the detectors, (see Tab. 1), the number of resulting added stars was 3660 for the NOT, 5544 for Loiano, and 3938 for SPM. The entire set of images was then reduced again with the procedure described in Sec.3. This way we got a set of constant light curves which is completely representative of many of the spurious artifacts that could have been introduced by the photometry procedure. This is certainly a more realistic test than simply considering Poisson noise on the light curves, as it is usually done. We then applied the algorithm, with the parameters described in Sec.6, to the constant light curves. The result is shown in Figure 6, where the DSP parameter is plotted against the mean magni- tude of the light curve. For the CFHT data, fixing the DSP threshold at 4.3 yielded a FAR of 0.1%. This was the FAR we adopted also when considering the other sites, which corre- sponded to different levels of the DSP parameter, as explained in Sec. 8. Repeating the whole procedure 4 times and slightly shift- ing the positions of the artificial stars, allowed us to better es- timate the FAR and its error, FAR=(0.10 ± 0.04)%. Therefore, running the transit search procedure on the 3311 selected main sequence stars, we expect (3.3 ± 1.3) false candidates. 7.2. Masking bad regions and temporal intervals We verified that, when stars were located near detector defects, like bad columns, saturate stars, etc., or, in correspondence of some instants of a particular night, (associated with sudden cli- matic variations, or telescope shifts), it was possible to have an over-production of spurious transit candidates. To avoid these effects, we chose to mask those regions of the detectors and the epochs which caused sudden changes in the photometric quality. This was done also for the simulations with the con- stant added stars, that were not inserted in detector defected regions, and in the excluded images that generating bad pho- Fig. 6. False Alarm Probability (FAR) in %, against the DSP parame- ter given by the algorithm. The points indicate the results of our sim- ulations on constant light curves, the solid line is our assumed best tometry. In particular, we observed these spurious effects for the NOT and SPM images. We further observed, when dis- cussing the candidates coming from the analysis of the whole data-set (as described in Sec.10.3) that the photometric varia- tions were concentrated on the first night of the NOT. This fact, which appeared from the simulations with the constant stars too, meant that this night was probably subject to bad weather conditions. had not we applied the Because we didn’t recog- nize it at the beginning, we retained that night, as long as those candidates, which were all recognized of spurious nature. Had not we applied any masking the number of false alarms would have almost quadruplicated. This fact probably can explain at least some of the candidates found by B03 (see Sec. 13) that were identified on the NOT observations. Even if some kind of masking procedure was applied by B03, many candidates ap- peared concentrated on the same dates, and were considered rather suspicious by the same authors. 7.3. Artificially added transits The transit detection efficiency (TDE) was determined by ana- lyzing light curves modified with the inclusion of random tran- sits. To properly measure the TDE and to estimate the number of transits we expect to detect it is mandatory to consider real- istic planetary transits. We proceeded as follows: 7.3.1. Stellar parameters The basic cluster parameters were determined by fitting the- oretical isochrones from Girardi et al. (2002) to the observed color-magnitude diagram (Stetson et al. 2003). Our best fit pa- rameters are (see Fig. 7): age = 10.0 Gyr, (m − M) = 13.30, E(B−V) = 0.12 for Z = 0.030 (corresponding to [Fe/H]= 12 M. Montalto et al.: A new search for planet transits in NGC 6791. Fig. 7. CMD diagram of NGC6791 with the best fit Z=0.030 isochrone (dashed line), and the best fit Z=0.046 isochrone (from Carraro et al. 2006, solid line). Photometry: Stetson et al. (2003) Fig. 8. Left: Mi/M⊙ vs visual apparent magnitude Right: Ri/R⊙ vs vi- sual apparent magnitude, from our best fit isochrone (dashed line) and from the Z=0.046 isochrone (solid line) applied to the stars of NGC 6791. +0.18), and age = 8.9 Gyr, (m−M) = 13.35 and E(B−V) = 0.09 for Z=0.046 (corresponding to [Fe/H]= +0.39). From the best-fit isochrones we then obtained the values of stellar mass and radius as a function of the visual magnitude (Fig. 8). 7.3.2. Planetary parameters The actual distribution of planetary radii has a very strong im- pact on the transit depth and therefore on the number of plan- etary transits we expect to be able to detect. The radius of the fourteen transiting planets discovered to date ranges from R=1.35 ± 0.07 RJ (HD209458b; Wittenmyrer et al. 2005) to R=0.725 ± 0.03 RJ (HD149026b; Sato et al. 2005), where J refers to the value for Jupiter. The observed distribution is likely biased towards larger radii. Gaudi (2005a) suggests for the close-in giant planets a mean radius Rp = 1.03 RJ. To eval- uate the efficiency of the algorithm we have considered three cases: Fig. 9. Continuous line: adopted distribution for planet peri- ods. Histogram: RV surveys data (from the Extrasolar Planets Encyclopaedia). – Rp = (0.7 ± 0.1) RJ – Rp = (1.0 ± 0.2) RJ – Rp = (1.4 ± 0.1) RJ assuming a Gaussian distribution for Rp. We fixed the planetary mass at Mp = 1 MJ , because the effect of planet mass on transit depth or duration is negligible. The period distribution was taken from the data for plan- ets discovered by radial velocity surveys, from the Extra-solar Planets Encyclopaedia9. We selected the planets discovered by radial velocity surveys with mass 0.3MJ ≤ Mpl sin i ≤ 10MJ (the upper limit was fixed to exclude brown dwarfs; the lower limit to ensure reasonable completeness of RV surveys and to exclude Hot Neptunes that might have radii much smaller than giant planets, Baraffe et al. 2005) and periods 1 ≤ P ≤ 9 days. We assumed that the period distribution of RV planets is un- biased in this period range. We then fitted the observed period distribution with a positive power law for the Very Hot Jupiters (VHJ, 1 ≤ P ≤ 3) and a negative power law for the Hot Jupiters (HJ, 3 < P ≤ 9, see Gaudi et al. 2005b for details) as shown in Fig. 9. 7.3.3. Limb-darkening To obtain realistic transit curves it is important to include the limb darkening effect. We adopted a non-linear law for the spe- cific intensity of a star: = 1 − ak (1 − µ k/2) (7) from Claret (2000). 9 http://exoplanet.eu/index.php M. Montalto et al.: A new search for planet transits in NGC 6791. 13 In this relation µ = cosγ is the cosine of the angle between the normal to the stellar surface and the line of sight of the observer, and ak are numerical coefficients that depend upon vturb (micro-turbulent velocity), [M/H], Te f f , and the spectral band. The coefficients are available from the ATLAS calcula- tions (available at CDS). We adopted the metallicity of the cluster for [M/H] and vturb=2 km s −1 for all the stars. For each star we adopted the appropriate V-band ak coefficients as a function of the values of log g and Te f f derived from the best fit isochrone. 7.3.4. Modified light curves In order to establish the TDE of the algorithm, we considered the whole sample of constant stars with 17.3 ≤ V ≤ 22.1, and each star was assigned a planet with mass, radius and period randomly selected from the distributions described above. The orbital semi-major axis a was derived from the 3rd Kepler’s law, assuming circular orbits. To each planet, we also assigned an orbit with a random inclination angle i, with 0 < cos i < 0.1, with a uniform dis- tribution in cos i. We infer that ∼ 85% of the planets result in potentially detectable transits. We also assigned a phase 2φ0 randomly chosen from 0 to 2π rad and a random direction of revolution s = ±1 (clockwise or counter-clockwise). Having fixed the planet’s parameters (P, i, φ0, Mp, Rp, a), the star’s parameters (M⋆, R⋆) and a constant light curve (ti , Vi) it is now possible to derive the position of the planet with respect to the star at every instant from the relation: φ = φ0 + where φ is the angle between the star-planet radius and the line of sight. The positions were calculated at all times ti corresponding to the Vi values of the light curve of the star. When the planet was transiting the star, the light curve was modified, calculating the brightness variation ∆V(ti) and adding this value to the Vi (see Fig. 10). 7.4. Calculating the TDE We then selected only the light curves for which there was at least a half transit inside an observing night and applied our transit detection algorithm. We considered not only central transits but also grazing ones. We considered the number of light curves that exceeded the thresholds, and also determined for how many of these the transit instants were correctly iden- tified on the unfolded light curves. We isolated three different outputs: 1. Missed candidates: the light curves for which the algorithm did not get the values of the parameters that exceeded the thresholds (DSP, OOTV, transit duration and number of in transit points, see Sec 6.3), or if it did, the epochs of the transits were not correctly recovered; 2. Partially recovered transit candidates: the parameters ex- ceeded the thresholds and at least one of the transits that fell in the observing window was correctly identified; Fig. 10. Top: constant light curve Bottom: the same light curve after inserting the simulated transit with limb-darkening (black points). The solid line shows the theoretical light curve of the transit. 3. Totally recovered transit candidates: the parameters ex- ceeded the thresholds and all the transits that were present were correctly recovered. The TDE was calculated as the sum of the totally and partially recovered transit candidates relative to the whole number of stars with transiting planets. We derive the TDE as a function of magnitude in Fig. 11. The TDE decreases with increasing magnitude because the lower photometric precision at fainter magnitudes is not fully compensated by the larger transit depth. The TDE depends strongly also on the assumptions concerning the planetary radii, and on the inclusion of the limb darkening effect. Fig. 11 is relative to a threshold equal to 4.3 for the DSP (cf. Fig. 6). The resulting TDE is about 11.5% around V = 18 and 1% around V = 21 for the case with R = (1.0 ± 0.2)RJ. Figure 12–14 show the histograms relative to the input tran- sit parameters and the recovered values of the BLS algorithm normalized to the whole number of transiting planets. For com- parison we also show in the upper left panel of each figure the recovered values of the BLS for the constant simulated light curves (normalized to the total number of constant light curves). We found that on average the BLS algorithm has un- derestimated the depth and duration of the transit by about 15- 20%. This is likely due to the deviation of the transit curves from the box shape assumed by the algorithm. For the periods, (Fig. 13), the recovered transit period distribution shown in the upper right panel of Fig. 13, had two clear peaks at 1.5 and 3 days, with the first one much more evident meaning that the algorithm tends to estimate half of the input transit period, as shown in the lower panels of the same Figure. The constant light curve period distribution of the upper left panel, instead, showed that the vast majority of constant stars were recovered with periods between 0.5 and 1 day, but residual peaks at 2.5 and 5 days were present. 14 M. Montalto et al.: A new search for planet transits in NGC 6791. Fig. 11. TDE as a function of the stellar magnitude for various as- sumptions on planetary radii distribution. From the top to the bottom: 1) Dashed line, R = (1.4 ± 0.1) RJ ; 2) Solid line, R = (1.0 ± 0.2) RJ ; 3) Dotted line, R = (0.7 ± 0.1) RJ . The adopted threshold for the DSP in this figure is 4.3. The normalization is respect to the whole number of transiting planets. Fig. 12. (Upper left) Distributions of transit depths measured by the BLS algorithm on the artificial constant-light-curves (lc); (upper right) transit depths measured on artificial light curves with transits added;(lower left) input transit depths used to generate artificial light curves with transits;(lower right) relative difference between the tran- sit depth recovered by BLS and its input value. Empty histograms refer to distributions relative to all light curves, filled ones to light curves with totally and partially recovered transits. Histograms are normal- ized to all light curves with transiting planets, or, for the upper left panel to all constant light curves. This Figure is relative to CFHT data, and the assumed planetary radii distribution is R = (1.0 ± 0.2). Fig. 13. The same as Fig.12 for the transit periods. Fig. 14. The same as Fig.12 for the transit durations. 8. Different approaches in the transit search The data we have acquired on NGC 6791 came from four dif- ferent sites and involved telescopes with different diameters and instrumentations. Moreover, the observing window of each site was clearly different with respect to the others as well as observing conditions like seeing, exposure times, etc. The first approach we tried consisted in putting together the observations coming from all the different telescopes. The most important complication we had to face regarded the dif- M. Montalto et al.: A new search for planet transits in NGC 6791. 15 Table 5. The different cases in which the data-sets analysis was split- ted into. The notation in the first column is explained in the text, the second column shows the number of stars in each case and the third column refers to the DSP values assumed, correspondent to a FAR= 0.1%. Case N.stars DSP threshold 11111 1093 7.5 10000 771 4.3 10100 870 5.5 11011 162 7.1 10001 112 7.1 11001 108 7.5 10111 99 7.2 10011 96 6.5 ferent field of views of the detectors. This had the consequence that some stars were measured only in a subset of the sites, and therefore these stars had in general different observing win- dows. Considering only the stars in common would reduce the number of candidates from 3311 to 1093 which means a re- duction of about 60% of the targets. We decided to distinguish eight different cases, which are shown in Tab. 5. In the first col- umn a simple binary notation identifies the different sites: each digit represents one site in the following order: CFHT, SPM, Loiano, NOT(V) and NOT(I). If the number correspondent to a generic site is 1, it indicates that the stars contained in that case have been observed, otherwise the value is set to 0. For exam- ple, the notation 11111 was used for the stars in common to all 4 sites. The notation 10000 indicates the number of stars which were present only on the CFHT field, and so on. Each one of these cases was treated as independent, and the resulting FAR and expected number of transiting planets were added together in order to obtain the final values. The second approach we followed was to consider only the CFHT data. As demonstrated in Section 3, overall we obtained the best photometric precision for this data-set. We considered the 3311 candidates which were recovered in the CFHT data- For the CFHT data-set, as shown in Tab. 5, the DSP value correspondent to a FAR= 0.1% is equal to 4.3, lower than the other cases reported in that Table. Thus, despite the reduced observing window of the CFHT data, it is possible to take ad- vantage of its increased photometric precision in the search for planets. In Section 9, and in Section 10, we presented the candidates and the different expected number of transiting planets for these two different approaches. 9. Presentation of the candidates Table 6 shows the characteristics of the candidates found by the algorithm, distinguishing those coming from the entire data-set analysis from those coming from the CFHT analysis. Fig. 15. Composite light curve of candidate 6598. In ordinate is re- ported the calibrated V magnitude and in abscissa the observing epoch, (in days), where 0 corresponds to JD = 52099. Filled circles indicate CFHT data, crosses SPM data, open triangles Loiano data, open circles NOT data in the V filter and open squares NOT data in the I filter. Light blue symbols highlight regions which were flagged by the BLS. 9.1. Candidates from the whole data-sets Applying the algorithm with the DSP thresholds shown in Tab. 5 on the real light curves we obtained four can- didates. Hereafter we adopt the S03 notation reported in Tab. 6. For what concerns candidates 6598, 4304, and 4699 (Fig. 15, 16, 17) we noted (see also Sec. 7.2) that the points contributing to the detected signal came from the first observ- ing night at the NOT, meaning that bad weather conditions deeply affected the photometry during that night. In particu- lar candidate 6598, was also found in the B03 transit search survey, (see Sec. 13), and flagged as a probable spurious can- didate. In none of the other observing nights we were able to confirm the photometric variations which are visible in the first night at the NOT. We concluded that these three candidates are of spurious nature. The fourth candidate corresponds to star 1239, that is lo- cated in the external regions of the cluster. For this reason we presented in Fig. 18 only the data coming from the CFHT. In this case, the data points appear irregularly scattered underly- ing a particular pattern of variability or simply a spurious pho- tometric effect. 9.2. Candidates from the CFHT data-set Considering only the data coming from the CFHT observing run we obtained three candidates. The star 1239 is in common with the list of candidates coming from the whole data-sets be- cause, as explained above, it is located in the external regions for which we had only the CFHT data. For candidate 4300, the algorithm identified two slight (∼ 0.004 mag) magnitude variations with duration of around one hour during the sixth and the tenth night, with a period of around 4.1 days. A jovian 16 M. Montalto et al.: A new search for planet transits in NGC 6791. Table 6. The candidates found in the two cases discussed in Sec. 8. The case of the whole data-sets put together is indicated with ALL (1st column), that one for the only CFHT data-set is indicated with CFHT (2nd column). A cross (x) indicates that the candidate was found in that case, a trait (-) that it is absent. In the 3rd column, the ID of the stars taken from S03 is shown. Follow the V calibrated magnitude, the (B − V) color, the right ascension, (α), and the declination, (δ), of the stars. ALL CFHT ID(S tetson) V (B − V) α(2000) δ(2000) x - 6598 18.176 0.921 19h 20m 48s.65 +37◦ 47 x - 4304 17.795 0.874 19h 20m 41s.39 +37◦ 43 x - 4699 17.955 0.846 19h 20m 42s.67 +37◦ 43 x x 1239 19.241 1.058 19h 20m 25s.42 +37◦ 47 - x 4300 18.665 0.697 19h 20m 41s.38 +37◦ 45 - x 7591 18.553 0.959 19h 20m 51s.51 +37◦ 48 Fig. 16. Composite light curve of candidate 4304. Fig. 17. Composite light curve of candidate 4699. planet around a main sequence star of magnitude V=18.665, (with R = 0.9 R⊙, see Fig. 8), should determine a transit with a maximum depth of around 1.2%, and maximum duration of 2.6 hours. Although compatible with a grazing transit, we ob- served that the two suspected eclipses are not identical, and, in Fig. 18. CFHT light curve for candidate 1239. Fig. 19. CFHT light curve for candidate 4300. Fig. 20. CFHT light curve for candidate 7591. any case, outside these regions, the photometry appears quite scattered. Star 7591, instead, does not show any significant fea- ture. From the analysis of these candidates we concluded that no transit features are detected for both the entire data-sets and the CFHT data. Moreover, we can say to have recovered the expected number of false alarm candidates which was (3.3 ± 1.3) as explained in Sec. 10.3. 10. Expected number of transiting planets 10.1. Expected frequency of close-in planets in NGC 6791 The frequency of short-period planets in NGC 6791 was es- timated considering the enhanced occurrence of giant planets M. Montalto et al.: A new search for planet transits in NGC 6791. 17 around metal rich stars and the fraction of hot Jupiters among known extrasolar planets. Fischer & Valenti (2005) derived the probability P of for- mation of giant planets with orbital period shorter than 4 yr and radial velocity semi-amplitude K > 30 ms−1 as a function of [Fe/H]: P = 0.03 · 10 2.0 [Fe/H] − 0.5 < [Fe/H] < 0.5 (8) The number of stars with a giant planet with P < 9 d was estimated considering the ratio between the number of the plan- ets with P < 9 days and the total number of planets from Table 3 of Fischer & Valenti 2005 (850 stars with uniform planet de- tectability). The result is 0.22+0.12 −0.09. Assuming for NGC 6791 [Fe/H]= +0.17 dex, a conserva- tive lower limit to the cluster metallicity, from Equation 8 we determined that the probability that a cluster star has a giant planet with P < 9 is 1.4%. Assuming [Fe/H]= +0.47, the metallicity resulting from the spectral analysis by Gratton et al. (2006), the probability rises to 5.7%. Our estimate assumes that the planet period and the metal- licity of the parent star are independent, as found by Fischer & Valenti (2005). If the hosts of hot Jupiters are even more metal rich than the hosts of planets with longer periods, as pro- posed by Society (2004), then the expected frequency of close- in planets at the metallicity of NGC 6791 should be slightly higher than our estimate. 10.2. Expected number of transiting planets In order to evaluate the expected number of transiting planets in our survey we followed this procedure: – From the constant stars of our simulations (see Sec. 7), tak- ing into account the luminosity function of main sequence stars of the cluster, we randomly selected a sample cor- responding to the probability that a star has a planet with P ≤ 9 d. – From the V magnitude of the star we calculated the mass and radius. – To each star in this sample we assigned a planet with mass, radius, period randomly chosen from the distributions de- scribed in Sec. 7.3.2, and cos i randomly chosen inside the range 0 - 1. The range spanned for the periods was 1 < P < 9 days, with a step size of 0.006 days. For plane- tary radii we considered the three distributions described in Sec. 7.3.2, sampled with a step size of 0.001 RJ, and incli- nations were varied of 0.005 degrees. – We selected only the stars with planets that can make tran- sits thanks to their inclination angle given by the relation: cos i ≤ Rpl + R⋆ – Finally, as described above, we assigned to each planet the initial phase φ0 and the revolution orbital direction s and modified the constant light curves inserting the transits. The initial phase was chosen randomly inside the range 0-360 degrees, with a step size of 0.3 degrees. – We applied the BLS algorithm to the modified light curves with the adopted thresholds. We performed 7000 different simulations and we calculated the mean values of these quantities: – The number of MS stars with a planet: Npl – The number of planets that make transits (thanks to their inclination angles): Ngeom – The number of planets that make one or more transits in the observing window: N+1 – The number of planets that make one single transit in the observing window: N1 – The number of transiting planets detected by the algo- rithm for the three different planetary radii distributions adopted, (as described in Sect. 7), R1 = (0.7 ± 0.1) RJ, R2 = (1.0 ± 0.2) RJ, and R 3 = (1.4 ± 0.1) RJ. 10.3. FAR and expected number of detectable transiting planets for the whole data-sets We followed the procedure reported in Sec. 7 to perform simu- lations with the artificial stars. It is important to note that artifi- cial stars were added exactly in the same positions in the fields of the different detectors. This is important because it assured the homogeneity of the artificial star tests. We decided to accept a FAR equal to 0.1%, which meant that we expected to obtain (3.3 ± 1.3) false alarms from the total number of 3311 clus- ter candidates. The DSP thresholds correspondent to this FAR value are different for each case, and is reported in Table 5. Table 7 displays the results for the simulations performed in order to obtain the expected numbers of detectable transit- ing candidates for three values of [Fe/H] (the values found by Carraro et al. 2006 and Gratton et al. 2006 and a conservative lower limit to the cluster metallicity). The columns listed as Ngeom, N1+ and N1 indicate respec- tively the number of planets which have a favorable geometric inclination for the transit, the number of expected planets that transit at least one time within the observing window and the number of expected planets that transit exactly one time in the observing window. The numbers of expected transiting planets in our observ- ing window detectable by the algorithm were calculated for the three different planetary radii distributions (see Sec. 7, and previous paragraph). On the basis of the current knowledge on giant planets the most likely case corresponds to R2 = (1.0 ± 0.2) RJ. Table 7 shows that, assuming the most likely planetary radii distribution R = (1.0 ± 0.2) RJ and the high metallicity result- ing from recent high dispersion studies (Carraro et al. 2006; Gratton et al. 2006), we expected to be able to detect 2 − 3 planets that exhibit at least one detectable transit in our observ- ing window. 10.4. FAR and expected number of detectable transiting planets for the CFHT data-set Table 8 shows the expected number of detectable planets in our observing window for the case of the CFHT data. A 18 M. Montalto et al.: A new search for planet transits in NGC 6791. Table 7. The Table shows the results of our simulations on the expected number of detectable transiting planets for the whole data-set (all the cases of Tab. 5) as explained in Sect. 10.3. Ngeom indicates planets with favorable inclination for transits, N1+, and N1, planets that transit respectively at least one time and only one time inside the observing window. R1, R2, R3, indicate the expected number of detectable transiting planets inside our observing window, for the three assumed planetary radii distributions, (see Sec. 7.3.2). [Fe/H] Ngeom N1+ N1 R 1 R2 R3 +0.17 5.39 3.08 1.68 0.0 ± 0.0 0.0 ± 0.0 1.8 ± 0.9 +0.39 15.13 8.32 4.60 0.1 ± 0.1 1.9 ± 0.8 3.6 ± 1.8 +0.47 21.92 11.95 6.62 0.2 ± 0.3 3.2 ± 1.9 5.4 ± 1.8 comparison with Table 7 revealed that, in general, except for the largest planetary radii distribution, R3, the number of ex- pected detections is not increasing considering all the sites to- gether instead of the CFH only. Moreover, for the cases of [Fe/H]= (+0.39,+0.47)dex, and the R = (0.7 ± 0.1)RJ radii distribution, we obtained significantly better results consider- ing only the CFHT data than putting together all the data-sets. We interpreted this result as the evidence that the transit signal is, in general, lower than the total scatter in the composite light curves and this didn’t allow the algorithm to take advantage of the increased observing window giving, for the cases of ma- jor interest, R = (1 ± 0.2)RJ and [Fe/H]= (+0.39,+0.47)dex, comparable results. 11. Significance of the results As explained in Sec. 9, on real data we obtained 4 candidates, considering the data coming from the entire data-sets, (all the cases of Tab. 5), and 3 candidates considering only the best photometry coming from the CFHT . None of these candidates shows clear transit features, and their number agrees with the expected number of false candidates coming from the simula- tions (3.3 ± 1.3) as explained in Sec. 7.1. Considering the case relative to the metallicity of Carraro et al. 2006 ([Fe/H]= +0.39) and the one relative to the metallic- ity of Gratton et al. 2006, ([Fe/H]= +0.47), and given the most probable planetary radii distribution with R = (1.0 ± 0.2)RJ, from Table 7 and Table 8 we expected between 2 and 3 planets with at least one detectable transit inside our observing win- Therefore, this study reveals a lack of transit detections. What is the probability that our survey resulted in no tran- siting planets just by chance? To answer this question we went back to the simulations described in Sect. 10.2 and calculated the ratio of the number of simulations for which we were not able to detect any planet relative to the total number of simula- tions performed. The resulting probabilities to obtain no tran- siting planets were respectively around 10% and 3% for the metallicities of Carraro et al. 2006 and Gratton et al. 2006 con- sidered above. 12. Implication of the results Beside the rather small, but not negligible probability of a chance result, (3-10%, see Sec. 11), different hypothesis can be invoked to explain the lack of observed transits. We have discussed them here. 12.1. Lower frequency of close-in planets in cluster environments The lack of observed transits might be due to a lower frequency of close-in planets in clusters compared to the field stars of sim- ilar metallicity. In general, two possible factors could prevent planet formation especially in clustered environments: – in the first million years of the cluster life, UV-flux can evaporate fragile embryonic dust disks from which planets are expected to form. Circumstellar disks associated with solar-type stars can be readily evaporated in sufficiently large clusters, whereas disks around smaller (M-type) stars can be evaporated in more common, smaller groups. In ad- dition, even though giant planets could still form in the disk region r = 5-15 AU, little disk mass (outside that region) would be available to drive planet migration.; – on the other hand, gravitational forces could strip nascent planets from their parent stars or, taking in mind that tran- sit planet searches are biased toward ’hot jupiter’ plan- ets, tidal effects could prevent the planetary migration pro- cesses which are essential for the formation of this kind of planets. These factors depend critically on the cluster size. Adams et al. (2006), show that for clusters with 100-1000 members modest effects are expected on forming planetary systems. The interaction rates are low, so that the typical solar system ex- periences a single encounter with closest approach distance of 1000 AU. The radiation exposure is also low, so that photo- evaporation of circumstellar disks is only important beyond 30 AU. For more massive clusters like NGC6791, these factors are expected to be increasingly important and could drastically affect planetary formation (Adams et al. 2004). 12.2. Smaller planetary radii for planets around very metal rich host stars Guillot et al. (2006) suggested that the masses of heavy ele- ments in planets was proportional to the metallicities of their parent star. This correlation remains to be confirmed, being still consistent with a no-correlation hypothesis at the 1/3 level in the least favorable case. A consequence of this would be a smaller radius for close-in planets orbiting super-metal rich stars. Since the transit depth scales with the square of the ra- dius, this would have important implications for ground-based transit detectability, (see Tables 8- 7). M. Montalto et al.: A new search for planet transits in NGC 6791. 19 Table 8. The same as 7, but for the case of the only CFHT data as explained in Sect. 10.2. [Fe/H] Ngeom N1+ N1 R 1 R2 R3 +0.17 5.39 2.49 1.98 0.2 ± 0.5 0.4 ± 0.7 0.6 ± 0.8 +0.39 15.13 7.01 5.39 1.6 ± 1.3 2.3 ± 1.6 2.6 ± 1.7 +0.47 21.92 10.12 7.94 2.5 ± 1.7 3.4 ± 2.0 4.0 ± 2.1 12.3. Limitations on the assumed hypothesis While we exploited the best available results to estimate the ex- pected number of transiting planets, it is possible that some of our assumptions are not completely realistic, or applicable to our sample. One possibility is that the planetary frequency no longer increases above a given metallicity. The small number of stars in the high metallicity range in the Fischer & Valenti sam- ple makes the estimate of the expected planetary frequency for the most metallic stars quite uncertain. Furthermore, the con- sistency of the metallicity scales of Fischer & Valenti (2005), Carraro et al. (2006) and Gratton et al. (2006) should be checked. Another possibility concerns systematic differences be- tween the stellar sample studied by Fischer & Valenti, and the present one. One relevant point is represented by binary sys- tems. The sample of Fischer & Valenti has some biases against binaries, in particular close binaries. As the frequency of plan- ets in close binaries appears to be lower than that of planets orbiting single stars and wide binaries (Bonavita & Desidera 2007, A&A, submitted), the frequency of planets in the Fischer & Valenti sample should be larger than that resulting in an un- biased sample. On the other hand, our selection of cluster stars excludes the stars in the binary sequence, partially compensat- ing this effect. Another possible effect is that of stellar mass. As shown in Fig. 8, the cluster’s stars searched for transits have mass be- tween 1.1 to 0.5 M⊙. On the other hand, the stars in the FV sample have masses between 1.6 to 0.8 M⊙. If the frequency of giant planets depends on stellar mass, the results by Fischer & Valenti (2005) might not be directly applicable to our sample. Furthermore, some non-member contamination is certainly present. As discussed in Section 5, the selection of cluster members was done photometrically around a fiducial main se- quence line. 12.4. Possibility of a null result being due to chance As shown in Sec. 11, the probability that our null result was simply due to chance was comprised between 3% and 10%, depending on the metallicity assumed for the cluster. This is a rather small, but not negligible probability, and other efforts must be undertaken to reach a firmer conclusion. 13. Comparison of the transit search surveys on NGC 6791 It is important to compare our results on the presence of planets with those of other photometric campaigns performed in past years. We consider in this comparison B03 and M05. 13.1. The Nordic Optical Telescope (NOT) transit search As already described in this paper, (e.g. see Sect. 2), in July 2001, at NOT, B03 undertook a transit search on NGC 6791 that lasted eight nights. Only seven of these nights were good enough to search for planetary transits. Their time coverage was thus comparable to the CFHT data presented here. The expected number of transits was obtained considering as can- didates all the stars with photometric precision lower than 2%, (they did not isolate cluster main sequence stars, as we did, but they then multiplied their resulting expected numbers for a factor equal to 85% in order to account for binarity), and as- suming that the probability that a local G or F-type field star harbors a close-in giant planet is around 0.7%. With these and other obvious assumptions B03 expected 0.8 transits from their survey. However, they made also the hypothesis that for metal- rich stars the fraction of stars harboring planets is ∼ 10 times greater than for general field stars, following Laughlin (2000). In this way, they would have expected to find “at least a few candidates with single transits”. In Section 3 we showed how the photometric precision for the NOT was in general of lower quality for the brightest stars with respect to that one of SPM and Loiano. This fact can be recognized also in Table 5 where the value of the threshold for the DSP was always bigger than 6.5 when the NOT observations were included. This demon- strates the higher noise level of this data-set. We did not per- form the accurate analysis of the expected number of transit- ing planets considering only the NOT data, but, on the basis of our assumptions, and on the photometric precision of the NOT data, the numbers showed in Table 8 for the CFHT should be considered as an upper limit for the expected transit from the NOT survey. B03 reported ten transit events, two of which, (identified in B03 as T6 and T10), showed double transit features, and the others were single transits. Except for candidate T2, which was recovered also in our analysis (see Sec. 9.1) our algorithm did not identify any other of the candidates reported by B03. B03 recognized that most of the candidates were likely spu- rious, while three cases, referred as T5, T7 and T8, were con- sidered the most promising ones. We noted that T8 lies off the cluster main sequence. Therefore, it can not be considered as a planet candidate for NGC 6791. Furthermore, from our CFHT images we noted that this candidate is likely a blended star. The other two candidates were on the main sequence of NGC 6791. Visual inspection of the light curves in Fig. 21 and Fig. 22 also show no sign of eclipse. Finally, candidate T9, (Fig. 23) lies off the cluster main sequence and it was recognized by B03 to be a long-period low-amplitude variable (V80). In our photometry, it shows clear signs of variability, and a ∼ 0.05 mag eclipse during the 20 M. Montalto et al.: A new search for planet transits in NGC 6791. Fig. 21. Composite light curve for candidate 3671 correspondent to T5 of BO3. Different symbols have the same meaning of Fig. 15. Fig. 22. Composite light curve for candidate 3723 correspondent to T7 of BO3. second night of the CFHT campaign at t = 361.8, and probably a partial eclipse at the end of the seventh night of the NOT data-set, at t = 6.4, ruling out the possibility of a planetary transit, because the magnitude depth of the eclipse is much larger than what is expected for a planetary transit. It is not surprising that almost all of the candidates reported by B03 were not confirmed in our work, even for the NOT pho- tometry itself. Even if the photometry reduction algorithm was the same, (image subtraction, see Sec. 3), all the other steps that followed, and the selection criteria of the candidates were in general different. This, in turn, reinforces the idea that they are due to spurious photometric effects. Fig. 23. Composite light curve for candidate 12390 correspondent to T9 of BO3. 13.2. The PISCES group extra-Solar planets search The PISCES group collected 84 nights of observations on NGC 6791, for a total of ∼ 300 hours of data collection from July 2001 to July 2003, at the 1.2m Fred Lawrence Whipple Observatory (M05). Starting from their 3178 cluster mem- bers (selected considering all the main sequence stars with RMS≤ 5%), assuming a distribution of planetary radii between 0.95 RJ and 1.5 RJ, and a planet frequency of 4.2%, M05 ex- pected to detect 1.34 transiting planets in the cluster. They didn’t identify any transiting candidate. Their planet frequency is within the range that we assumed (1.4%–5.7%). Our number of candidate main-sequence stars is slightly in excess relative to that of M05, even if their field of view is larger than our own (∼ 23 arcmin2 against ∼ 19 arcmin2 of S03 catalog), since we were able to reach ∼ 2 mag deeper with the same photomet- ric precision level. Their number of expected transiting plan- ets is of the same order of magnitude as our own because of their huge temporal coverage. In any case, looking at figure 7 of M05, one should recognize that their detection efficiency greatly favors planetary radii larger than 1 RJ. A more realistic planetary radius distribution, for example (1.0± 0.2) RJ, should significantly decrease their expectations, as recognized by the same authors. 14. Future investigations NGC 6791 has been recognized as one of the most promising targets for studying the planet formation mechanism in clus- tered environments, and for investigating the planet frequency as a function of the host star metallicity. Our estimate of the ex- pected number of transiting planets, (about 15–20 assuming the metallicity recently derived by means of high-dispersion spec- troscopy by Carraro et al. 2006, and Gratton et al. 2006, and the planet frequency derived by Fischer & Valenti 2005), confirms that this is the best open cluster for a planet search. M. Montalto et al.: A new search for planet transits in NGC 6791. 21 However, in spite of fairly ambitious observational efforts by different groups, no firm conclusions about the presence or lack of planets in the cluster can be reached. With the goal of understanding the implications of this re- sult and to try to optimize future observational efforts, we show, in Table 9, that the number of hours collected on this cluster with > 3 m telescopes is much lower than the time dedicated with 1 − 2 m class telescopes. Despite the fact that we were able to get adequate photometric precisions even with 1 − 2 m class telescopes, (see Sec. 3), in general smaller aperture tele- scopes are typically located on sites with poorer observing con- ditions, which limits the temporal sampling and their photom- etry is characterized by larger systematic effects. As a result, the number of cluster stars with adequate photometric precision for planet transit detections is quite limited. Our study suggests that more extensive photometry with wide field imagers at 3 to 4-m class telescopes (e.g. CFHT) is required to reach conclu- sive results on the frequency of planets in NGC 6791. We calculated that, extending the observing window to two transit campaigns of ten days each, providing that the same photometric precision we had at the CFHT could be reached, we could reduce the probability of null detection to 0.5%. 15. Conclusions The main purpose of this work was to investigate the problem of planet formation in stellar open clusters. We focused our at- tention on the very metal rich open cluster NGC 6791. The idea that inspired this work was that looking at more metal rich stars one should expect a higher frequency of planets, as it has been observed in the solar neighborhood (Santos et al. 2004, Fisher & Valenti, 2005). Clustered environments can be regarded as astrophysical laboratories in which to explore planetary fre- quency and formation processes starting from a well defined and homogeneous sample of stars with the advantage that clus- ter stars have common age, distance, and metallicity. As shown in Section 2, a huge observational effort has been dedicated to the study of our target cluster using four different ground based telescopes, (CFHT, SPM, Loiano, and NOT), and trying to take advantage from multi-site simultaneous observations. In Section 3, we showed how we were able to obtain adequate photometric precisions for the transit search for all the differ- ent data-sets (though in different magnitude intervals). From the detailed simulations described in Section 10, it was demon- strated that, with our best photometric sequence, and with the most realistic assumption that the planetary radii distribution is R = (1.0 ± 0.2)RJ, the expected number of detectable transiting planets with at least one transit inside our observing window was around 2, assuming as cluster metallicity [Fe/H]=+0.39, and around 3 for [Fe/H]= +0.47. Despite the number of ex- pected positive detections, no significant transiting planetary candidates were found in our investigation. There was a rather small, though not negligible probability that our null result can be simply due to chance, as explained in Sect. 11: we esti- mated that this probability is 10% for [Fe/H]= +0.39, and 3% for [Fe/H]= +0.47. Possible interpretations for the lack of ob- served transits (Sect. 12) are a lower frequency of close-in plan- ets around solar-type stars in cluster environments with respect to field stars, smaller planetary radii for planets around super metal rich stars, or some limitations in the assumptions adopted in our simulations. Future investigations with 3-4m class tele- scopes are required (Sect 14) to further constrain the planetary frequency in NGC 6791. Another twenty nights with this kind of instrumentation are necessary to reach a firm conclusion on this problem. The uniqueness of NGC 6791, which is the only galactic open cluster for which we expect more than 10 giant planets transiting main sequence stars if the planet frequency is the same as for field stars of similar metallicity, makes such an effort crucial for exploring the effects of cluster environment on planet formation. 22 M. Montalto et al.: A new search for planet transits in NGC 6791. Table 9. Number of nights and hours which have been devoted to the study of NGC 6791 as a function of the diameter of the telescope used for the survey. We adopted a mean of 5 hours of observations per night. Telescope Diameter(m) Nnights Hours Ref. FLWO 1.2 84 ∼300 M05 Loiano 1.5 4 20 This paper SPM 2.2 8 48 This paper NOT 2.54 7 35 B03 and this paper CFHT 3.6 8 48 This paper MMT 6.5 3 15 Hartmann et al. (2005) Acknowledgements. We warmly thank M. Bellazzini and F. Fusi Pecci for having made possible the run at Loiano Observatory. This work was partially funded by COFIN 2004 “From stars to plan- ets: accretion, disk evolution and planet formation” by Ministero Universitá e Ricerca Scientifica Italy. We thanks the referee, Dr. Mochejska, for useful comments and sug- gestions allowing the improvement of the paper. References Adams, F.C., Hollenbach, D., Laughlin, G. 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0704.1669

Possible polarisation and spin dependent aspects of quantum gravity

Possible polarisation and spin dependent aspects of quantum gravity D. V. Ahluwalia-Khalilova, N. G. Gresnigt, Alex B. Nielsen, D. Schritt, T. F. Watson Department of Physics and Astronomy, Rutherford Building University of Canterbury Private Bag 4800, Christchurch 8020, New Zealand E-mail: [email protected] We argue that quantum gravity theories that carry a Lie algebraic modification of the Poincaré and Heisenberg algebras inevitably provide inhomogeneities that may serve as seeds for cosmological structure formation. Furthermore, in this class of theories one must expect a strong polarisation and spin dependence of various quantum-gravity effects. I. Introduction— Quantum gravity proposals often come with a modification of the Heisenberg, and Poincaré, algebras. Confining ourselves to Lie algebraic modifications, we argue that the underlying physical space of all such theories must be inhomogeneous. In order to establish this result, we first review how, within a quantum framework, the homogeneity and continuity of physical space lead inevitably to the Heisenberg algebra. We then review general arguments that hint towards algebraic modifications encountered in quantum gravity proposals. Next, we argue that a natural extension of physical laws to the Planck scale can be obtained by a Lie algebraic modification of the Poincaré and Heisenberg algebras in such a way that the resulting algebra is immune to infinitesimal perturbations in its structure constants. With the context so chosen, we establish the main thesis: that quantum gravity theories of the aforementioned class inevitably provide inhomogeneities that may serve as seeds for structure formation; and that quantum gravity induced effects may carry a strong polarisation and spin dependence. The established results are not restricted to the chosen algebra but may easily be extended to all Lie algebraic modifications that alter the Heisenberg algebra.1 2. Homogeneity and continuity of physical space, and its imprint in the Heisenberg algebra— In order to understand the fundamental origin of primordial inhomogeneities we will first review the fundamental connection between the homogeneity and continuity of physical space and the Heisenberg algebra. It is in this spirit that we remind our reader of an argument that is presented, for example, by Isham in [1, Section 7.2.2]. There it is shown that, in the general quantum mechanical framework, and under the following two assumptions, — physical space is homogeneous, — any spatial distance r can be divided in to two equal parts, r = r/2 + r/2, it necessarily follows that the operator x associated with position measurements along the x-axis, and the generator of displacements dx along the x-direction, satisfy [x, dx] = i. If one now requires consistency with the elementary wave mechanics of Heisenberg, one must identify dx with px/h̄ (px is the operator associated with momentum mea- surements along the x-direction). This gives, [x, px] = ih̄. Without any additional assumptions, the argument easily generalises to yield the entire Heisenberg algebra [xj, pk] = ih̄δjk, [pj, pk] = 0, [xj, xk] = 0, where xj, j = 1, 2, 3, are the position operators associated with the three coordinate axes, where the observer is assumed to be located at the origin of the coordinate system. Thus it is evident that a quantum description of physical reality, with spatial homo- geneity and continuity, inevitably leads to the Heisenberg algebra. 3. On the need to go beyond the Heisenberg and Poincaré algebraic-based description of physical reality— From an algebraic point of view much of the success of modern physics can be traced back to the Poincaré and Heisenberg algebras. Had the latter algebra been discovered before the former, the conceptual formulation and evolution of theoretical physics would have been significantly different. For instance, it is a direct implication of Heisenberg’s fundamental commutator [xi, pj] = ih̄δij (with i, j = 1, 2, 3), that events should be characterised not only by their spatiotemporal location xµ, but also 1A slightly weaker argument can be constructed for non-Lie algebraic proposals when we confine ourselves to probed distances significantly larger than the length scale associated with the loss of spatial continuity. by the associated energy momentum pµ; and that should be done in a manner consistent with the fundamental measurement uncertainties inherent in the formalism. The reader may wish to come back to these remarks in the context of Eq. (16) where one shall find that in a specific sense the physical space that underlies the conformal algebra does indeed combine the notions of spacetime and energy momentum. Furthermore, as will be seen from Eq. (18) and the subsequent remarks, this interplay becomes increasingly important as we consider the early universe above ≈ 100 GeV. In the mentioned description the interplay of the general relativistic and quantum mechanical frameworks becomes inseparably bound. To see this, consider the well-known thought experiment to probe spacetime at spatial resolutions around the Planck length h̄G/c3. If one does that, one ends up creating a Planck mass mP h̄c/G black hole. This fleeting structure carries a temperature T ≈ 1030K and evaporates in a thermal explosion in ≈ 10−40 seconds. This, incidentally, is a long time – about ten thousand fold the Planck time τP h̄G/c5. The formation and evaporation of the black hole places a fundamental limit on the spatiotemporal resolution with which spacetime can be probed. The authors of [2, 3] have argued that once gravitational effects associated with the quantum measurement process are accounted for, the Heisenberg algebra, and in particular the commutator [xj, pk], must be modified. The role of gravity in the quantum measurement process was also emphasised by Penrose [4]. From the above discussion, we take it as suggestive that an operationally- defined view of physical space (or, its generalisation) shall inevitably ask for the length scale, `P to play an important role. In the context of the continuity of physical space we will take it as a working hypothesis that, just as a lack of commutativity of the x and px operators does not render the associated eigenvalues discrete, similarly the existence of a non-vanishing `P does not necessarily make the underlying space lose its continuum nature. This is a highly non- trivial issue requiring a detailed discussion from which we here refrain; yet, an element of justification shall become apparent below. From a dynamical point of view, as early as late 1800’s, the symmetries of Maxwell’s equations were already suggesting a merger of space and time into one physical entity, spacetime [5]. Algebraically, these symmetries are encoded in the Poincaré algebra. The emergent unification of space and time called for a new fundamental invariant, c, the speed of light (already contained in Maxwell’s equations). From an empirical point of view, the Michelson-Morley experiment established the constancy of the speed of light for all inertial observers, and thus re-confirmed, in the Einsteinian framework, the implications of the Poincaré spacetime symmetries. Concurrently, we note that while in classical statistical mechanics it is the volume that determines the number of accessible states and hence the entropy, the situation is dramat- ically different in a gravito-quantum mechanical setting. One example of this assertion may be found in the well-known Bekenstein-Hawking entropy result for a Schwarzschild black hole, SBH = (k/4)(A/` P ); where k is the Boltzmann constant, and A is the surface area of the sphere contained within the event horizon of the black hole. Thus quan- tum mechanical and gravitational realms conspire to suggest the holographic conjecture [6, 7, 8]. The underlying physics is perhaps two fold: (a) contributions from higher momenta in quantum fields to the number of accessible states is dramatically reduced because these are screened by the associated event horizons; and (b) the accessible states for a quantum system are severely influenced by the behaviour of the wave function at the boundary. From this discussion, we take it as suggestive that in quantum cosmology/gravity the new operationally-defined view of physical space shall inevitably ask for a cosmological length scale, `C. These observations prepare us to reach the next trail in our essay. In the immediate aftermath of cosmic creation with the big bang, the physical reality knew of no inertial frames of Einstein. This is due to the fact that massive particles had yet to appear on the scene. The spacetime symmetries at cosmic creation are encoded in the conformal algebra. So, whatever new operational view of spacetime emerges, it must somehow also incorporate a process by which one evolves from the “conformal phase” of the universe at cosmic creation to the present (see Fig. 1). Algebraically, we take it to suggest that there must be a mechanism that describes how the present day Poincaré-algebraic description relates to the conformal-algebraic description of the universe at its birth. We parenthetically note that in the conformal phase, where leptons and quarks were yet to acquire mass (through the Higg’s mechanism, or something of that nature), the operationally-accessible symmetries are not Poincaré but conformal. This is so because to define rest frames, so essential for operationally establishing the Poincaré algebra, one needs massive particles. In the transition when massive particles come to exist, the local algebraic symmetries of general relativity suffer an operational change. Consequently, for the cosmic epoch before ≈ 100 GeV general relativistic description of physical reality might require modification. 4. A new algebra for quantum gravity and the emergent inhomogeneity of physical space— Mathematically, a Lie algebra incorporating the three italicised items in Sec. 3 already exists. It was inspired by Faddeev’s mathematical analysis of the quantum and relativistic revolutions of the last century [9] and was followed up by Vilela Mendes in his 1994 paper [10]. The uniqueness of the said algebra was then explored through a Lie-algebraic investigation of its stability by Chryssomalakos and Okon, in 2004 [11]. Some of the physical implications were subsequently explored in Refs. [12, 13], and its Clifford-algebraic representation was provided by Gresnigt et al. [14]. Its importance was further noted in CERN Courier [15]. However, its candidacy for the algebra underlying quantum cosmology/gravity has been difficult to assert. This is essentially due to a perplexing observation made in Ref. [11] regarding the interpretation of the operators associated with the spacetime events. In this essay we overcome this interpretational hurdle and argue that it contains all the desired features for such an algebra. To this end we first write down what has come to be known as the Stabilised Poincaré- Heisenberg Algebra (SPHA) and then proceed with the interpretational issues. The SPHA contains the Lorentz sector (we follow the widespread physics convention which takes the Jµν as dimensionless and Pν as dimensionful) [Jµν ,Jρσ] = i (ηνρJµσ + ηµσJνρ − ηµρJνσ − ηνσJµρ) (1) This remains unchanged (as is strongly suggested by the analysis presented in [16]), as does the commutator [Jµν ,Pλ] = i (ηνλPµ − ηµλPν) (2) These are supplemented by the following modified sector [Jµν ,Xλ] = i (ηνλXµ − ηµλXν) (3) [Pµ,Pν ] = iqα1Jµν (4) [Xµ,Xν ] = iqα2Jµν (5) [Pµ,Xν ] = iqηµνI + iqα3 Jµν (6) [Pµ, I] = iα1Xµ − iα3Pµ (7) [Xµ, I] = iα3Xµ − iα2Pµ (8) [Jµν , I] = 0 (9) The metric ηµν is taken to have the signature (1,−1,−1,−1). The SPHA is stable, except for the instability surface defined by α23 = α1α2 (see Fig. 2). Away from the instability surface the SPHA is immune to infinitesimal perturbations in its structure constants. This distinguishes SPHA from many of the competing algebraic structures because a physical theory based on such an algebra is likely to be free from “fine tuning” problems. This is essentially self evident because if an algebraic structure does not carry this immunity, one can hardly expect the physical theory based upon such an algebra to enjoy the opposite. The SPHA involves three parameters α1, α2, α3. The c and h̄ arise in the process of the Lie algebraic stabilisation that takes us from the Galilean relativity to Einsteinian relativity, and from classical mechanics to quantum mechanics. Their specific values are fixed by experiment. Similarly, α1, α2, α3 owe their origin to a similar stabilisation of the combined Poincaré and Heisenberg algebra. Except for the fact that α1 must be a measure of the size of the observable universe (here assumed to be operationally determined from the Hubble parameter), the Lie algebraic procedure for obtaining SPHA does not determine α1, α2, α3. Dimensional and phenomenological considerations, along with the requirement that we obtain physically viable limits, suggest the following identifications:2 α1 := where `C is of the order of the Hubble radius, and therefore it depends on the cosmic epoch. The introductory remarks, and existing data suggest that [11] In the limit `P → 0, `C → ∞, β → 0, I → I, the identity operator, the SPHA splits into Heisenberg and Poincaré algebras. In that limit, the symbols Xµ → xµ,Pµ → pµ,Jµν → Jµν , and I → I. Thus xµ, pµ, Jµν , I acquire their traditional meaning, while 2In making the identifications it is understood that these may be true up to a multiplicative factor of the order of unity. Xµ,Pµ,Jµν , I are to be considered their generalisations. In particular xµ should then be interpreted as the generator of energy-momentum translation. The latter parallels the canonical interpretation of pµ as the generator of spacetime translation. This interpre- tation, we believe, removes the problematic interpretational aspects associated with Xµ in the analysis of Ref. [11]. The identification of q with h̄ is dictated by the demand that we recover the Heisen- berg algebra. It also suggests that at the present cosmic epoch α3 should not allow the second term in the right hand side of equation (6) to have a significant contribution. It will become apparent below that α3 is intricately connected to the conformal algebraic limit of SPHA. With these identifications, and with α3 renamed as the dimensionless parameter β, the SPHA takes the form [Jµν ,Jρσ] = i (ηνρJµσ + ηµσJνρ − ηµρJνσ − ηνσJµρ) (12) [Jµν ,Pλ] = i (ηνλPµ − ηµλPν) , [Jµν ,Xλ] = i (ηνλXµ − ηµλXν) (13) [Pµ,Pν ] = i h̄2/`2C Jµν , [Xµ,Xν ] = i`2PJµν , [Pµ,Xν ] = ih̄ηµνI + ih̄β Jµν (14) [Pµ, I] = i h̄/`2C Xµ − iβPµ, [Xµ, I] = iβXµ − i `2P/h̄ Pµ, [Jµν , I] = 0 (15) Since cosmic creation began with massless particles, it should be encouraging if in some limit SPHA reduced to the conformal algebra. This is indeed the case. It follows from a somewhat lengthy, though simple, exercise. Towards examining this question we introduce two new operators P̃µ = aPµ + bXµ, X̃µ = a′Xµ + b′Pµ (16) and find that if the introduced parameters a, b, a′, b′ satisfy the the following conditions , b = , a′ = `2C(1− β) with β2 restricted to the value 1 + (`2P/` C), then SPHA written in terms of P̃µ and X̃µ satisfies the conformal algebra [17, Sec. 4.1]. Using these results, we can re-express P̃µ and X̃µ in a fashion that supports the view taken in the opening paragraph of this section P̃µ = a (1− β)Xµ , X̃µ = a′ (1− β)Pµ β2 = 1 + . (19) Near the big bang, `C ≈ `P and thus β → ± 2 (see, Fig. 1). This results in a significant mixing of the Xµ and Pµ in the conformal algebraic description in terms of X̃µ and P̃µ. In contrast, hypothetically, had we been on the conformal surface at present then taking `C � `P makes β → ±1. Consequently, for β → +1, P̃µ becomes identical to Pµ up to a multiplicative scale factor a. Similarly, X̃µ becomes identical to Xµ up to a multiplicative scale factor a′. As is evident from Eq. (17), the multiplicative scale factors a and a′ are constrained by the relation aa′ = `2P/(` C(1− β)). We expect that similar modifications to spacetime symmetries would occur if we were to explore it at Planckian energies in the present epoch. For β → −1 ( `C � `P ), one again obtains significant mixing of the Xµ and Pµ. By containing `P and `C , the SPHA unifies the extreme microscopic with the extreme macroscopic, i.e., the cosmological. In the early universe it allows for the existence of conformal symmetry. The significant departure from the Heisenberg algebra at big bang, yields primordial inhomogeneities in the underlying physical space and the quantum fields that it supports. The latter is an unavoidable consequence of the discussion presented in Sec. 2.3 5. Polarisation and spin dependence of the cosmic inhomogeneities and other quantum gravity effects— A careful examination of SPHA presented in equations (12-15) reveals a strong Jµν dependence of the modifications to the Heisenberg algebra. Physically, this translates to the following representative implications — The induced primordial cosmic inhomogeneities are dependent on spin and polari- sation of the fields for which these are calculated. — The operationally-inferred commutativity/non-commutativity of the physical space depends on the spin and polarisation of the probing particle. — The just enumerated observation implies that a violation of equivalence principle is inherent in the SPHA based quantum gravity. 3Any one of the other suggestions in quantum gravity that modify the Heisenberg algebra (see, e.g., references [18]-[28]) carry similar implications for homogeneity and isotropy of the physical space. — Since Heisenberg algebra uniquely determines the nature of the wave particle du- ality [27, 28] (including the de Broglie result “λ = h/p”), it would undergo spin and polarisation dependent changes in quantum gravity based on SPHA. All these results carry over to any theory of quantum gravity that modifies the Heisenberg algebra with a Jµν dependence. 6. Conclusion— In this essay we have motivated a new candidate for the algebra which may underlie a physically viable and consistent theory of quantum cosmology/gravity. Besides yielding an algebraic unification of the extreme microscopic and cosmological scales, it generalises the notion of conformal symmetry. The modifications to the Heisen- berg algebra at the present cosmic epoch are negligibly small; but when `C and `P are of the same order (i.e, at, and near, the big bang), the induced inhomogeneities are intrinsic to the nature of physical space. These can then be amplified by the cosmic evolution and result in important back reaction effects [29, 30, 31, 32]. An important aspect of the SPHA-based quantum gravity is that it inevitably provides inhomogeneities that may serve as an important ingredient for structure formation [33]. Furthermore, in this class of theories one must expect a strong polarisation and spin dependence of various quantum-gravity effects. Acknowledgements— We wish to thank Daniel Grumiller and Peter West for their in- sightful questions and suggestions. References [1] C. J. Isham, Lectures on quantum theory: Mathematical and structural foundations (Imperial College Press, Singapore, 1995). [2] D. V. Ahluwalia, “Quantum measurements, gravitation, and locality,” Phys. Lett. B 339 (1994) 301 [arXiv:gr-qc/9308007]. [3] S. Doplicher, K. Fredenhagen and J. E. Roberts, “Space-time quantization induced by classical gravity,” Phys. Lett. B 331 (1994) 39. [4] R. Penrose, “On gravity’s role in quantum state reduction,” Gen. Rel. Grav. 28 (1996) 581. http://arxiv.org/abs/gr-qc/9308007 [5] H. R. Brown, Physical relativity: Space-time structure from a dynamical perspec- tive (Oxford University Press, Oxford, 2005). [6] G. ’t Hooft, “Dimensional reduction in quantum gravity,” arXiv:gr-qc/9310026. [7] L. Susskind, “The world as a hologram,” J. Math. Phys. 36 (1995) 6377 [arXiv:hep- th/9409089]. [8] S. de Haro, Quantum gravity and the holographic principle (Universal press - Science publishers, Veenendaal, 2001) [9] L. D. Faddeev, Mathematician’s view on the development of physics, Frontiers in Physics: High technology and mathematics ed H. A. Cerdeira and S. Lundqvist (Singapore: Word Scientific, 1989) pp. 238-46 [10] R. Vilela Mendes, “Deformations, stable theories and fundamental constants,” J. Phys. A 27 (1994) 8091. [11] C. Chryssomalakos and E. Okon, Int. J. Mod. Phys. D 13 (2004) 2003 [arXiv:hep- th/0410212]. [12] D. V. Ahluwalia-Khalilova, “A freely falling frame at the interface of gravitational and quantum realms,” Class. Quant. Grav. 22 (2005) 1433 [arXiv:hep-th/0503141]. [13] D. V. Ahluwalia-Khalilova, “Minimal spatio-temporal extent of events, neutrinos, and the cosmological constant problem,” Int. J. Mod. Phys. D 14 (2005) 2151 [arXiv:hep-th/0505124]. [14] N. G. Gresnigt, P. F. Renaud and P. H. Butler, “The stabilized Poincare-Heisenberg algebra: A Clifford algebra viewpoint,” arXiv:hep-th/0611034. [15] S. Reucroft and J. Swain, “Special relativity becomes more general”, CERN Courier, July/August 2005 p.9. [16] J. Collins, A. Perez, D. Sudarsky, L. Urrutia and H. Vucetich, “Lorentz invariance: An additional fine-tuning problem,” Phys. Rev. Lett. 93 (2004) 191301 [arXiv:gr- qc/0403053]. [17] P. Di Francesco, et al., Conformal field theory (Springer, New York, 1937). [18] L. J. Garay, “Quantum gravity and minimum length,” Int. J. Mod. Phys. 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Maggiore, “A Generalized uncertainty principle in quantum gravity,” Phys. Lett. B 304 (1993) 65 [arXiv:hep-th/9301067]. [24] N. Sasakura, “An uncertainty relation of space-time,” Prog. Theor. Phys. 102 (1999) 169 [arXiv:hep-th/9903146]. [25] F. Scardigli, “Generalized uncertainty principle in quantum gravity from micro- black hole gedanken experiment,” Phys. Lett. B 452 (1999) 39 [arXiv:hep- th/9904025]. [26] S. Capozziello, G. Lambiase and G. Scarpetta, “Generalized uncertainty principle from quantum geometry,” Int. J. Theor. Phys. 39 (2000) 15 [arXiv:gr-qc/9910017]. [27] A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52 (1995) 1108 [arXiv:hep- th/9412167]. [28] D. V. Ahluwalia, “ Wave particle duality at the Planck scale: Freezing of neutrino oscillations,” Phys. Lett. A 275 (2000) 31 [arXiv:gr-qc/0002005]. [29] T. Buchert, “A cosmic equation of state for the inhomogeneous universe: Can a global far-from-equilibrium state explain dark energy?,” Class. Quant. Grav. 22 (2005) L113 [arXiv:gr-qc/0507028]. [30] D. L. Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages,” arXiv:gr- qc/0702082. [31] S. Rasanen, “Accelerated expansion from structure formation,” JCAP 0611 (2006) 003 [arXiv:astro-ph/0607626]. http://arxiv.org/abs/gr-qc/9904026 http://arxiv.org/abs/gr-qc/9904026 http://arxiv.org/abs/gr-qc/9707042 http://arxiv.org/abs/gr-qc/9706007 http://arxiv.org/abs/hep-th/9301067 http://arxiv.org/abs/hep-th/9903146 http://arxiv.org/abs/hep-th/9904025 http://arxiv.org/abs/hep-th/9904025 http://arxiv.org/abs/gr-qc/9910017 http://arxiv.org/abs/hep-th/9412167 http://arxiv.org/abs/hep-th/9412167 http://arxiv.org/abs/gr-qc/0002005 http://arxiv.org/abs/gr-qc/0507028 http://arxiv.org/abs/gr-qc/0702082 http://arxiv.org/abs/gr-qc/0702082 http://arxiv.org/abs/astro-ph/0607626 [32] A. Ishibashi and R. M. Wald, “Can the acceleration of our universe be explained by the effects of inhomogeneities?,” Class. Quant. Grav. 23 (2006) 235 [arXiv:gr- qc/0509108]. [33] A. Perez, H. Sahlmann and D. Sudarsky, “On the quantum origin of the seeds of cosmic structure,” Class. Quant. Grav. 23 (2006) 2317 [arXiv:gr-qc/0508100]. http://arxiv.org/abs/gr-qc/0509108 http://arxiv.org/abs/gr-qc/0509108 http://arxiv.org/abs/gr-qc/0508100 Possible Scenario 1 Possible Scenario 2 Big bang Present FutureFuture Instability Conformal �!!!2 Figure 1: This figure is a cut, at `P = 1 (with h̄ set to unity), of Fig. 2 and it schemat- ically shows the cosmic evolution along two possible scenarios. For this purpose, only β ≥ 0 values have been taken. The β < 0 sector can easily be inferred from symmetry consideration. In one of the scenarios the conformal symmetry of the early universe is lost without crossing the instability surface, while in the other it crosses that surface. In the latter case the algebra changes [11] from so(2, 4) to so(1, 5). This crossover, we spec- ulate, may be related to the mass-generating process of spontaneous symmetry breaking (SSB) of the standard model of high energy physics. The big bang is here identified with `C ≈ `P . �����������{c2 Instability Cone Conformal Surface Poincaré-Heisenberg Algebra HunstableL Figure 2: The unmarked arrow is the `2P (= h̄α2) axis. The Poincaré-Heisenberg algebra corresponds to the origin of the parameters space, which coincides with the apex of the instability cone. In reference to Eq. (10), note that `2C = h̄/α1. Here, β is a dimensionless parameter that corresponds to a generalisation of the conformal algebra. The SPHA lives in the entire (`C , `P , β) space except for the surface of instability. The SPHA becomes conformal for all values of (`C , `P , β) that lie on the “conformal surface”.

0704.1670

On the support genus of a contact structure

ON THE SUPPORT GENUS OF A CONTACT STRUCTURE MEHMET FIRAT ARIKAN Abstract. The algorithm given by Akbulut and Ozbagci constructs an explicit open book decomposition on a contact three-manifold described by a contact surgery on a link in the three-sphere. In this article, we will improve this algorithm by using Giroux’s con- tact cell decomposition process. In particular, our algorithm gives a better upper bound for the recently defined “minimal supporting genus invariant” of contact structures. 1. Introduction Let (M, ξ) be a closed oriented contact 3-manifold, and let (Σ, h) be an open book (de- composition) of M which is compatible with the contact structure ξ (sometimes we also say that (Σ, h) supports ξ). Based on the correspondence theorem (see Theorem 2.3) between contact structures and their supporting open books, the topological invariant sg(ξ) was defined in [EO]. More precisely, we have sg(ξ) = min{ g(Σ) | (Σ, h) an open book decomposition supporting ξ} called supporting genus of ξ. There are some partial results for this invariant. For instance, we have: Theorem 1.1 ([Et1]). If (M, ξ) is overtwisted, then sg(ξ) = 0. Unlike the overtwisted case, there is not much known yet for sg(ξ) when ξ is tight. On the other hand, if we, furthermore, require that ξ is Stein fillable, then an algorithm to find an open book supporting ξ was given in [AO]. Although their construction is explicit, the pages of the resulting open books arise as Seifert surfaces of torus knots or links, and so this algorithm is far from even approximating the numbers sg(ξ). In [St], the same algorithm was generalized to the case where ξ need not to be Stein fillable (or even tight), but the pages are still of large genera. This article is organized as follows: After the preliminaries (Section 2), in Section 3 we will present an explicit construction of a supporting open book (with considerably less genus) for a given contact surgery diagram of any contact structure ξ. Of course, because of Theorem 1.1, our algorithm makes more sense for the tight structures than the overtwisted ones. Moreover, it depends on a choice of the contact surgery diagram describing ξ. Nevertheless, it gives better and more reasonable upper bound for sg(ξ) (when ξ is tight) as we will see from our examples in Section 4. Let L be any Legendrian link given in (R3, ξ0 = ker(α0 = dz + xdy)) ⊂ (S 3, ξst). L can be represented by a special diagram D called a square bridge diagram of L (see [Ly]). We will consider D as an abstract diagram such that (1) D consists of horizontal line segments h1, ..., hp, and vertical line segments v1, ..., vq for some integers p ≥ 2, q ≥ 2, The author was partially supported by NSF Grant DMS0244622. http://arxiv.org/abs/0704.1670v4 2 MEHMET FIRAT ARIKAN (2) there is no collinearity in {h1, . . . , hp}, and in {v1, . . . , vq}. (3) each hi (resp., each vj) intersects two vertical (resp., horizontal) line segments of D at its two endpoints (called corners of D), and (4) any interior intersection (called junction of D) is understood to be a virtual cross- ing of D where the horizontal line segment is passing over the vertical one. We depict Legendrian right trefoil and the corresponding D in Figure 1. Legendrian right trefoil p = q = 5 Figure 1. The square bridge diagram D for the Legendrian right trefoil Clearly, for any front projection of a Legendrian link, we can associate a square bridge diagram D. Using such a diagram D, the following two facts were first proved in [AO], and later made more explicit in [Pl]. Below versions are from the latter: Lemma 1.2. Given a Legendrian link L in (R3, ξ0), there exists a torus link Tp,q (with p and q as above) transverse to ξ0 such that its Seifert surface Fp,q contains L, dα0 is an area form on Fp,q, and L does not separate Fp,q. Proposition 1.3. Given L and Fp,q as above, there exist an open book decomposition of S3 with page Fp,q such that: (1) the induced contact structure ξ is isotopic to ξ0; (2) the link L is contained in one of the page Fp,q, and does not separate it; (3) L is Legendrian with respect to ξ; (4) there exist an isotopy which fixes L and takes ξ to ξ0, so the Legendrian type of the link is the same with respect to ξ and ξ0; (5) the framing of L given by the page Fp,q of the open book is the same as the contact framing. Being a Seifert surface of a torus link, Fp,q is of large genera. In Section 3, we will construct another open book OB supporting (S3, ξst) such that its page F arises as a subsurface of Fp,q (with considerably less genera), and given Legendrian link L sits on F as how it sits on the page Fp,q of the construction used in [AO] and [Pl]. The page F of the open book OB will arise as the ribbon of the 1-skeleton of an appropriate contact cell decomposition for (S3, ξst). As in [Pl], our construction will keep the given link L Legendrian with respect to the standard contact structure ξst. Our main theorem is: Theorem 1.4. Given L and Fp,q as above, there exists a contact cell decomposition ∆ of (S3, ξst) such that (1) L is contained in the Legendrian 1-skeleton G of ∆, ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 3 (2) The ribbon F of the 1-skeleton G is a subsurface of Fp,q (p and q as above), (3) The framing of L coming from F is equal to its contact framing tb(L), and (4) If p > 3 and q > 3, then the genus g(F ) of F is strictly less than the genus g(Fp,q) of Fp,q. As an immediate consequence (see Corollary 3.1), we get an explicit description of an open book supporting (S3, ξ) whose page F contains L with the correct framing. Therefore, if (M±, ξ±) is given by contact (±1)-surgery on L (such a surgery diagram exists for any closed contact 3-manifold by Theorem 2.1), we get an open book supporting ξ± with page F by Theorem 2.5. Hence, g(F ) improves the upper bound for sg(ξ) as g(F ) < g(Fp,q) (for p > 3, q > 3). It will be clear from our examples in Section 4 that this is indeed a good improvement. Acknowledgments. The author would like to thank Selman Akbulut, Selahi Durusoy, Cagri Karakurt, and Burak Ozbagci for their helpful conversations and comments on the draft of this paper. 2. Preliminaries 2.1. Contact structures and Open book decompositions. A 1-form α ∈ Ω1(M) on a 3-dimensional oriented manifold M is called a contact form if it satisfies α ∧ dα 6= 0. An oriented contact structure on M is then a hyperplane field ξ which can be globally written as the kernel of a contact 1-form α. We will always assume that ξ is a positive contact structure, that is, α ∧ dα > 0. Note that this is equivalent to asking that dα be positive definite on the plane field ξ, ie., dα|ξ > 0. Two contact structures ξ0, ξ1 on a 3-manifold are said to be isotopic if there exists a 1-parameter family ξt (0 ≤ t ≤ 1) of contact structures joining them. We say that two contact 3-manifolds (M1, ξ1) and (M2, ξ2) are contactomorphic if there exists a diffeomorphism f : M1 −→ M2 such that f∗(ξ1) = ξ2. Note that isotopic contact structures give contactomorphic contact manifolds by Gray’s Theorem. Any contact 3-manifold is locally contactomorphic to (R3, ξ0) where standard contact structure ξ0 on R 3 with coordinates (x, y, z) is given as the kernel of α0 = dz + xdy. The standard contact structure ξst on the 3-sphere S 3 = {(r1, r2, θ1, θ2) : r21 + r 2 = 1} ⊂ C 2 is given as the kernel of αst = r 1dθ1 + r 2dθ2. One basic fact is that (R3, ξ0) is contactomorphic to (S 3 \ {pt}, ξst). For more details on contact geometry, we refer the reader to [Ge], [Et3]. An open book decomposition of a closed 3-manifold M is a pair (L, f) where L is an oriented link in M , called the binding, and f : M \L → S1 is a fibration such that f−1(t) is the interior of a compact oriented surface Σt ⊂ M and ∂Σt = L for all t ∈ S 1. The surface Σ = Σt, for any t, is called the page of the open book. The monodromy of an open book (L, f) is given by the return map of a flow transverse to the pages (all diffeomorphic to Σ) and meridional near the binding, which is an element h ∈ Aut(Σ, ∂Σ), the group of (isotopy classes of) diffeomorphisms of Σ which restrict to the identity on ∂Σ . The group Aut(Σ, ∂Σ) is also said to be the mapping class group of Σ, and denoted by Γ(Σ). An open book can also be described as follows. First consider the mapping torus Σ(h) = [0, 1]× Σ/(1, x) ∼ (0, h(x)) where Σ is a compact oriented surface with n = |∂Σ| boundary components and h is an element of Aut(Σ, ∂Σ) as above. Since h is the identity map on ∂Σ, the boundary ∂Σ(h) of the mapping torus Σ(h) can be canonically identified with n copies of T 2 = S1 × S1, 4 MEHMET FIRAT ARIKAN where the first S1 factor is identified with [0, 1]/(0 ∼ 1) and the second one comes from a component of ∂Σ. Now we glue in n copies of D2 × S1 to cap off Σ(h) so that ∂D2 is identified with S1 = [0, 1]/(0 ∼ 1) and the S1 factor in D2 × S1 is identified with a boundary component of ∂Σ. Thus we get a closed 3-manifold M = M(Σ,h) := Σ(h) ∪n D 2 × S1 equipped with an open book decomposition (Σ, h) whose binding is the union of the core circles in the D2 × S1’s that we glue to Σ(h) to obtain M . To summarize, an element h ∈ Aut(Σ, ∂Σ) determines a 3-manifold M = M(Σ,h) together with an “abstract” open book decomposition (Σ, h) on it. For furher details on these subjects, see [Gd], and [Et2]. 2.2. Legendrian Knots and Contact Surgery. A Legendrian knot K in a contact 3-manifold (M, ξ) is a knot that is everywhere tangent to ξ. Any Legendrian knot comes with a canonical contact framing (or Thurston-Bennequin framing), which is defined by a vector field along K that is transverse to ξ. If K is null-homologous, then this framing can be given by an integer tb(K), called Thurston-Bennequin number. For any Legendrian knot K in (R3, ξ0), the number tb(K) can be computed as tb(K) = bb(K)−#left cusps of K where bb(K) is the blackboard framing of K. We call (M, ξ) (or just ξ) overtwisted if it contains an embedded disc D ≈ D2 ⊂ M with boundary ∂D ≈ S1 a Legendrian knot whose contact framing equals the framing it receives from the disc D. If no such disc exists, the contact structure ξ is called tight. For any p, q ∈ Z, a contact (r)-surgery (r = p/q) along a Legendrian knot K in a contact manifold (M, ξ) was first described in [DG1]. It is defined to be a special kind of a topological surgery, where surgery coefficient r ∈ Q∪∞ measured relative to the contact framing of K. For r 6= 0, a contact structure on the surgeried manifold (M − νK) ∪ (S1 ×D2), (νK denotes a tubular neighborhood of K) is defined by requiring this contact structure to coincide with ξ on Y − νK and its extension over S1 × D2 to be tight on (glued in) solid torus S1 ×D2. Such an extension uniquely exists (up to isotopy) for r = 1/k with k ∈ Z (see [Ho]). In particular, a contact (±1)-surgery along a Legendrian knot K on a contact manifold (M, ξ) determines a unique (up to contactomorphism) surgered contact manifold which will be denoted by (M, ξ)(K,±1). The most general result along these lines is: Theorem 2.1 ([DG1]). Every (closed, orientable) contact 3-manifold (M, ξ) can be ob- tained via contact (±1)-surgery on a Legendrian link in (S3, ξst). Any closed contact 3-manifold (M, ξ) can be described by a contact surgery diagram. Such a diagram consists of a front projection (onto the yz-plane) of a Legendrian link drawn in (R3, ξ0) ⊂ (S 3, ξst) with contact surgery coefficient on each link component. Theorem 2.1 implies that there is a contact surgery diagram for (M, ξ) such that the contact surgery coefficient of any Legendrian knot in the diagram is ±1. For more details see [Gm] and [OS]. ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 5 2.3. Compatibility and Stabilization. A contact structure ξ on a 3-manifold M is said to be supported by an open book (L, f) if ξ is isotopic to a contact structure given by a 1-form α such that (1) dα is a positive area form on each page Σ ≈ f−1(pt) of the open book and (2) α > 0 on L (Recall that L and the pages are oriented.) When this holds, we also say that the open book (L, f) is compatible with the contact structure ξ on M . Geometrically, compatibility means that ξ can be isotoped to be arbitrarily close (as oriented plane fields), on compact subsets of the pages, to the tangent planes to the pages of the open book in such a way that after some point in the isotopy the contact planes are transverse to L and transverse to the pages of the open book in a fixed neighborhood of L. Definition 2.2. A positive (resp., negative) stabilization S+K(Σ, h) (resp., S K(Σ, h)) of an abstract open book (Σ, h) is the open book (1) with page Σ′ = Σ ∪ 1-handle and (2) monodromy h′ = h ◦DK (resp., h ′ = h ◦D−1K ) where DK is a right-handed Dehn twist along a curve K in Σ′ that intersects the co-core of the 1-handle exactly once. Based on the result of Thurston and Winkelnkemper [TW], Giroux proved the following theorem which strengthened the link between open books and contact structures. Theorem 2.3 ([Gi]). Let M be a closed oriented 3-manifold. Then there is a one-to- one correspondence between oriented contact structures on M up to isotopy and open book decompositions of M up to positive stabilizations: Two contact structures supported by the same open book are isotopic, and two open books supporting the same contact structure have a common positive stabilization. For a given fixed open book (Σ, h) of a 3-manifold M , there exists a unique compatible contact structure up to isotopy on M = M(Σ,h) by Theorem 2.3. We will denote this contact structure by ξ(Σ,h). Therefore, an open book (Σ, h) determines a unique contact manifold (M(Σ,h), ξ(Σ,h)) up to contactomorphism. Taking a positive stabilization of an open book (Σ, h) is actually taking a special Murasugi sum of (Σ, h) with (H+, Dc) where H + is the positive Hopf band, and c is the core circle in H+. Taking a Murasugi sum of two open books corresponds to taking the connect sum of 3-manifolds associated to the open books. For the precise statements of these facts, and a proof of the following theorem, we refer the reader to [Gd], [Et2]. Theorem 2.4. (MS+ (Σ,h), ξS+ (Σ,h)) ∼= (M(Σ,h), ξ(Σ,h))#(S 3, ξst) ∼= (M(Σ,h), ξ(S,h)). 2.4. Monodromy and Surgery Diagrams. Given a contact surgery diagram for a closed contact 3-manifold (M, ξ), we want to construct an open book compatible with ξ. One implication of Theorem 2.1 is that one can obtain such a compatible open book by starting with a compatible open book of (S3, ξst), and then interpreting the effects of surgeries (yielding (M, ξ) ) in terms of open books. However, we first have to realize each surgery curve (in the given surgery diagram of (M, ξ) ) as a Legendrian curve sitting on a page of some open book supporting (S3, ξst). We refer the reader to Section 5 in [Et2] for a proof of the following theorem. 6 MEHMET FIRAT ARIKAN Theorem 2.5. Let (Σ, h) be an open book supporting the contact manifold (M, ξ). If K is a Legendrian knot on the page Σ of the open book, then (M, ξ)(K, ±1) = (M(Σ, h◦D∓ ), ξ(Σ, h◦D∓ 2.5. Contact Cell Decompositions and Convex Surfaces. The exploration of con- tact cell decompositions in the study of open books was originally initiated by Gabai [Ga], and then developed by Giroux [Gi]. We want to give several definitions and facts carefully. Let (M, ξ) be any contact 3-manifold, and K ⊂ M be a Legendrian knot. The twisting number tw(K,Fr) of K with respect to a given framing Fr is defined to be the number of counterclockwise 2π twists of ξ along K, relative to Fr. In particular, if K sits on a surface Σ ⊂ M , and FrΣ is the surface framing of K given by Σ, then we write tw(K,Σ) for tw(K,FrΣ). If K = ∂Σ, then we have tw(K,Σ) = tb(K) (by the definition of tb). Definition 2.6. A contact cell decomposition of a contact 3−manifold (M, ξ) is a finite CW-decomposition of M such that (1) the 1-skeleton is a Legendrian graph, (2) each 2-cell D satisfies tw(∂D,D) = −1, and (3) ξ is tight when restricted to each 3-cell. Definition 2.7. Given any Legendrian graph G in (M, ξ), the ribbon of G is a compact surface R = RG satisfying (1) R retracts onto G, (2) TpR = ξp for all p ∈ G, (3) TpR 6= ξp for all p ∈ R \G. For a proof of the following lemma we refer the reader to [Gd] and [Et2]. Lemma 2.8. Given a closed contact 3−manifold (M, ξ), the ribbon of the 1-skeleton of any contact cell decomposition is a page of an open book supporting ξ. The following lemma will be used in the next section. Lemma 2.9. Let ∆ be a contact cell decomposition of a closed contact 3-manifold (M, ξ) with the 1−skeleton G. Let U be a 3-cell in ∆. Consider two Legendrian arcs I ⊂ ∂U and J ⊂ U such that (1) I ⊂ G, (2) J ∩ ∂U = ∂J = ∂I, (3) C = I ∪∂ J is a Legendrian unknot with tb(C) = −1. Set G′ = G ∪ J . Then there exists another contact cell decomposition ∆′ of (M, ξ) such that G′ is the 1-skeleton of ∆′ Proof. The interior of the 3−cell U is contactomorphic to (R3, ξ0). Therefore, there exists an embedded disk D in U such that ∂D = C and int(D) ⊂ int(U) as depicted in Figure 2(a). We have tw(∂D,D) = −1 since tb(C) = −1. As we are working in (R3, ξ0), there exist two C∞-small perturbations of D fixing ∂D = C such that perturbed disks intersect each other only along their common boundary C. In other words, we can find two isotopies H1, H2 : [0, 1]×D −→ U such that for each i = 1, 2 we have (1) Hi(t, .) fixes ∂D = C pointwise for all t ∈ [0, 1], (2) Hi(0, D) = IdD where IdD is the identity map on D, ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 7 (3) Hi(1, D) = Di where each Di is an embedded disk in U with int(Di) ⊂ int(U), (4) D ∩D1 ∩D2 = C (see Figure 2(b)). int(U) Ext(U) int(U − U ′) (a) (b) Figure 2. Constructing a new contact cell decomposition Note that tw(∂Di, Di) = tw(C,Di) = −1 for i = 1, 2. This holds because each Di is a small perturbation of D, so the number of counterclockwise twists of ξ (along K) relative to FrDi is equal to the one relative to FrD. Next, we introduce G′ = G ∪ J as the 1-skeleton of the new contact cell decomposition ∆′. In M − int(U), we define the 2- and 3- skeletons of ∆′ to be those of ∆ . However, we change the cell structure of int(U) as follows: We add 2-cells D1, D2 to the 2-skeleton of ∆′ (note that they both satisfy the twisting condition in Definition 2.6). Consider the 2-sphere S = D1 ∪D2 where the union is taken along the common boundary C. Let U be the 3-ball with ∂U ′ = S. Note that ξ|U ′ is tight as U ′ ⊂ U and ξ|U is tight. We add U ′ and U −U ′ to the 3-skeleton of ∆′ (note that U −U ′ can be considered as a 3-cell because observe that int(U − U ′) is homeomorphic to the interior of a 3-ball as in Figure 2(b)). Hence, we established another contact cell decomposition of (M, ξ) whose 1-skeleton is G′ = G∪J . (Equivalently, by Theorem 2.4, we are taking the connect sum of (M, ξ) with (S3, ξst) along U ′.) � 3. The Algorithm 3.1. Proof of Theorem 1.4. Proof. By translating L in (R3, ξ0) if necessary (without changing its contact type), we can assume that the front projection of L onto the yz-plane lying in the second quadrant { (y, z) | y < 0, z > 0}. After an appropriate Legendrian isotopy, we can assume that L consists of the line segments contained in the lines ki = {x = 1, z = −y + ai}, i = 1, . . . , p, lj = {x = −1, z = y + bj}, j = 1, . . . , q for some a1 < a2 < · · · < ap, 0 < b1 < b2 < · · · < bq, and also the line segments (parallel to the x-axis) joining certain ki’s to certain lj ’s. In this representation, L seems to have 8 MEHMET FIRAT ARIKAN corners. However, any corner of L can be made smooth by a Legendrian isotopy changing only a very small neighborhood of that corner. Let π : R3 −→ R2 be the projection onto the yz-plane. Then we obtain the square bridge diagram D = π(L) of L such that D consists of the line segments hi ⊂ π(ki) = {x = 0, z = −y + ai}, i = 1, . . . , p, vj ⊂ π(lj) = {x = 0, z = y + bj}, j = 1, . . . , q. Notice that D bounds a polygonal region P in the second quadrant of the yz-plane, and divides it into finitely many polygonal subregions P1, . . . , Pm ( see Figure 3-(a) ). Throughout the proof, we will assume that the link L is not split (that is, the region P has only one connected component). Such a restriction on L will not affect the generality of our construction (see Remark 3.2). a1 a2 a3 a4 a5 P2 P3 h2 h3 Figure 3. The region P for right trefoil knot and its division into rectangles Now we decompose P into finite number of ordered rectangular subregions as follows: The collection {π(lj) | j = 1, . . . , q} cuts each Pk into finitely many rectangular regions R1k, . . . , R k . Consider the set P of all such rectangles in P . That is, we define = { Rlk | k = 1, . . . , m, l = 1, . . . , mk}. Clearly P decomposes P into rectangular regions ( see Figure 3-(b) ). The boundary of an arbitrary element Rlk in P consists of four edges: Two of them are the subsets of the lines π(lj(k,l)), π(lj(k,l)+1), and the other two are the subsets of the line segments hi1(k,l), hi2(k,l) where 1 ≤ i1(k, l) < i2(k, l) ≤ p and 1 ≤ j(k, l) < j(k, l) + 1 ≤ q (see Figure 4). Since the region P has one connected component, the following holds for the set P: ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 9 π(lj(k,l)) π(lj(k,l)+1) hi1(k,l) hi2(k,l) ai1(k,l) ai2(k,l) bj(k,l) bj(k,l)+1 Figure 4. Arbitrary element Rlk in P (⋆) Any element of P has at least one common vertex with some other element of P. By (⋆), we can rename the elements of P by putting some order on them so that any element of P has at least one vertex in common with the union of all rectangles coming before itself with respect to the chosen order. More precisely, we can write P = { Rk | k = 1, . . . , N} (N is the total number of rectangles in P) such that each Rk has at least one vertex in common with the union R1 ∪ · · · ∪ Rk−1. Equivalently, we can construct the polygonal region P by introducing the building rect- angles (Rk’s) one by one in the order given by the index set {1, 2, . . . , N}. In particular, this eliminates one of the indexes, i.e., we can use Rk’s instead of R k’s. In Figure 5, how we build P is depicted for the right trefoil knot (compare it with the previous picture given for P in Figure 3-(b)). π(k1) π(l5) π(l1) Figure 5. The region P for right trefoil knot 10 MEHMET FIRAT ARIKAN Using the representation P = R1 ∪ R2 ∪ · · · ∪ RN , we will construct the contact cell decomposition (CCD) ∆. Consider the following infinite strips which are parallel to the x-axis (they can be considered as the unions of “small” contact planes along ki’s and lj ’s): S+i = {1− ǫ ≤ x ≤ 1 + ǫ, z = y + ai}, i = 1, . . . , p, S−j = {−1− ǫ ≤ x ≤ −1 + ǫ, z = −y + bj}, j = 1, . . . , q. Note that π(S+i ) = π(ki) and π(S j ) = π(lj). Let Rk ⊂ P be given. Then we can write ∂Rk = C k ∪ C k ∪ C k ∪ C k where C k ⊂ π(ki1), C k ⊂ π(lj), C k ⊂ π(ki2), C k ⊂ π(lj+1) for some 1 ≤ i1 < i2 ≤ p and 1 ≤ j ≤ q. Lift C k , C k , C k , C k (along the x-axis) so that the resulting lifts (which will be denoted by the same letters) are disjoint Legendrian arcs contained in ki1 , lj, ki2, lj+1 and sitting on the corresponding strips S , S−j , S , S−j+1. For l = 1, 2, 3, 4, consider Legendrian linear arcs I lk (parallel to the x-axis) running between the endpoints of C lk’s as in Figure 6-(a)&(b). Along each I k the contact planes make a 90 left-twist. Let Blk be the narrow band obtained by following the contact planes along I Then define Fk to be the surface constructed by taking the union of the compact subsets of the above strips (containing corresponding C lk’s) with the bands B k’s (see Figure 6-(b)). C lk’s and I k’s together build a Legendrian unknot γk in (R 3, ξ0), i.e., we set γk = C k ∪ I k ∪ C k ∪ I k ∪ C k ∪ I k ∪ C k ∪ I Note that π(γk) = ∂Rk, γk sits on the surface Fk, and Fk deformation retracts onto γk. Indeed, by taking all strips and bands in the construction small enough, we may assume that contact planes are tangent to the surface Fk only along the core circle γk. Thus, Fk is the ribbon of γk. Observe that, topologically, Fk is a positive (left-handed) Hopf band. (a) (b) (c) 0−1 1 Dk ≈ fk(Rk) Figure 6. (a) The Legendrian unknot γk, (b) The ribbon Fk, (c) The disk Dk (shaded bands in (b) are the bands B Let fk : Rk −→ R 3 be a function modelled by (a, b) 7→ c = a2 − b2 (for an appropriate choice of coordinates). The image fk(Rk) is, topologically, a disk, and a compact subset of a saddle surface. Deform fk(Rk) to another “saddle” disk Dk such that ∂Dk = γk (see Figure 6-(c)). We observe here that tw(γk, Dk) = −1 because along γk, contact planes rotate 90◦ in the counter-clockwise direction exactly four times which makes one full left- twist (enough to count the twists of the ribbon Fk since Fk rotates with the contact planes along γk !). ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 11 We repeat the above process for each rectangle Rk in P and get the set D = { Dk | Dk ≈ fk(Rk), k = 1, . . . , N} consisting of the saddle disks. Note that by the construction of D, we have the property: (∗) If any two elements of D intersect each other, then they must intersect along a contractible subset (a contractible union of linear arcs) of their boundaries. For instance, if the corresponding two rectangles (for two intersecting disks in D) have only one common vertex, then those disks intersect each other along the (contractible) line segment parallel to the x-axis which is projected (by the map π) onto that vertex. For each k, let D′k be a disk constructed by perturbing Dk slightly by an isotopy fixing only the boundary of Dk. Therefore, we have (∗∗) ∂Dk = γk = ∂D k , int(Dk) ∩ int(D k) = ∅ , and tw(γk, D k) = −1 = tw(γk, Dk). In the following, we will define a sequence { ∆k | k = 1, . . . , N } of CCD’s for (S 3, ξst). ∆1k,∆ k, and ∆ k will denote the 1-skeleton, 2-skeleton, and 3-skeleton of ∆k, respectively. First, take ∆11 = γ1, and ∆ 1 = D1 ∪γ1 D 1. By (∗∗), ∆1 satisfies the conditions (1) and (2) of Definition 2.6. By the construction, any pair of disks Dk, D k (together) bounds a Darboux ball (tight 3-cell) Uk in the tight manifold (R 3, ξ0). Therefore, if we take ∆31 = U1 ∪∂ (S 3 − U1), we also achieve the condition (3) in Definition 2.6 ( the boundary union “ ∪∂” is taken along ∂U1 = S 2 = ∂(S3 − U1) ). Thus, ∆1 is a CCD for (S 3, ξst). Inductively, we define ∆k from ∆k−1 by setting ∆1k = ∆ k−1 ∪ γk = γ1 ∪ · · · ∪ γk−1 ∪ γk, ∆2k = ∆ k−1 ∪Dk ∪γk D k = D1 ∪γ1 D 1 ∪ · · · ∪Dk−1 ∪γk−1 D k−1 ∪Dk ∪γk D ∆3k = U1 ∪ · · · ∪ Uk−1 ∪ Uk ∪∂ (S 3 − U1 ∪ · · · ∪ Uk−1 ∪ Uk) Actually, at each step of the induction, we are applying Lemma 2.9 to ∆k−1 to get ∆k. We should make several remarks: First, by the construction of γk’s, the set (γ1 ∪ · · · ∪ γk−1) ∩ γk is a contractible union of finitely many arcs. Therefore, the union ∆1k−1 ∪ γk should be understood to be a set-theoretical union (not a topological gluing!) which means that we are attaching only the (connected) part (γk \ ∆ k−1) of γk to construct the new 1- skeleton ∆1k. In terms of the language of Lemma 2.9, we are setting I = ∆ k−1 \ γk and J = γk \∆ k−1. Secondly, we have to show that ∆ k = ∆ k−1 ∪Dk ∪γk D k can be realized as the 2-skeleton of a CCD: Inductively, we can achieve the twisting condition on 2-cells by using (∗∗). The fact that any two intersecting 2-cells in ∆2k intersect each other along some subset of the 1-skeleton ∆1k is guaranteed by the property (∗) if they have different index numbers, and guaranteed by (∗∗) if they are of the same index. Thirdly, we have to guarantee that 3-cells meet correctly: It is clear that U1, . . . , Uk meet with each other along subsets of the 1-skeleton ∆1k(⊂ ∆ k). Observe that ∂(U1 ∪ · · · ∪ Uk) = S 2 for any k = 1, . . . , N by (∗) and (∗∗). Therefore, we can always consider the complementary Darboux ball S3 − U1 ∪ · · · ∪ Uk−1 ∪ Uk, and glue it to U1 ∪ · · · ∪ Uk along their common boundary 2-sphere. Hence, we have seen that ∆k is a CCD for (S 3, ξst) with Legendrian 1-skeleton ∆1k = γ1 ∪ · · · ∪ γk. 12 MEHMET FIRAT ARIKAN To understand the ribbon, say Σk, of ∆ k, observe that when we glue the part γk \∆ k−1 of γk to ∆ k−1, actually we are attaching a 1-handle (whose core interval is (γk \∆ k−1)\Σk−1) to the old ribbon Σk−1 (indeed, this corresponds to a positive stabilization). We choose the 1-handle in such a way that it also rotates with the contact planes. This is equivalent to extending Σk−1 to a new surface by attaching the missing part (the part which retracts onto (γk \∆ k−1) \ Σk−1) of Fk given in Figure 6-(c). The new surface is the ribbon Σk of the new 1-skeleton ∆1k. By taking k = N , we get a CCD ∆N of (S 3, ξst). By the construction, γk’s are only piecewise smooth. We need a smooth embedding of L into the 1-skeleton ∆1N (the union of all γk’s). Away from some small neighborhood of the common corners of ∆ N and L (recall that L had corners before the Legendrian isotopies), L is smoothly embedded in ∆1N . Around any common corner, we slightly perturb ∆ N using the isotopy used for smoothing that corner of L. This guaranties the smooth Legendrian embedding of L into the Legendrian graph ∆1N = ∪ k=1γk. Similarly, any other corner in ∆ N (which is not in L) can be made smooth using an appropriate Legendrian isotopy. As L is contained in the 1-skeleton ∆1N , L sits (as a smooth Legendrian link) on the ribbon ΣN . Note that during the process we do not change the contact type of L, so the contact (Thurston-Bennequin) framing of L is still the same as what it was at the beginning. On the other hand, consider tubular neighborhood N(L) of L in ΣN . Being a subsurface of the ribbon ΣN , N(L) is the ribbon of L. By definition, the contact framing of any component of L is the one coming from the ribbon of that component. Therefore, the contact framing and the N(L)-framing of L are the same. Since N(L) ⊂ ΣN , the framing which L gets from the ribbon ΣN is the same as the contact framing of L. Finally, we observe that ΣN is a subsurface of the Seifert surface Fp,q of the torus link (or knot) Tp,q. To see this, note that P is contained in the rectangular region, say Pp,q, enclosed by the lines π(k1), π(kp), π(l1), π(lq). Divide Pp,q into the rectangular subregions using the lines π(ki), π(lj), i = 1, . . . , p, j = 1, . . . , q. Note that there are exactly pq rectangles in the division. If we repeat the above process using this division of Pp,q, we get another CCD for (S3, ξst) with the ribbon Fp,q. Clearly, Fp,q contains our ribbon ΣN as a subsurface (indeed, there are extra bands and parts of strips in Fp,q which are not in ΣN ). Thus, (1), (2) and (3) of the theorem are proved once we set ∆ = ∆N , (and so G = ∆ F = ΣN ). To prove (4), recall that we are assuming p > 3, q > 3. Then consider = total number of intersection points of all π(lj)’s with all hi’s. That is, we define κ = |{π(lj) | j = 1, . . . , q} ∩ {hi | i = 1, . . . , p} |. Notice that κ is the number of bands used in the construction of the ribbon F , and also that if D (so P ) is not a single rectangle (equivalently p > 2, q > 2), then κ < pq. Since there are p+ q disks in F , we compute the Euler characteristic and genus of F as χ(F ) = p+ q − κ = 2− 2g(F )− |∂F | =⇒ g(F ) = 2− p− q |∂F | Similarly, there are p + q disks and pq bands in Fp,q, so we get χ(Fp,q) = p+ q − pq = 2− 2g(Fp,q)− |∂Fp,q| =⇒ g(Fp,q) = 2− p− q |∂Fp,q| ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 13 Observe that |∂Fp,q| divides the greatest common divisor gcd(p, q) of p and q, so |∂Fp,q| ≤ gcd(p, q) ≤ p =⇒ g(Fp,q) ≥ 2− p− q Therefore, to conclude g(F ) < g(Fp,q), it suffices to show that pq−κ > p−|∂F |. To show the latter, we will show pq − κ− p ≥ 0 (this will be enough since |∂F | 6= 0). Observe that pq−κ is the number of bands (along x-axis) in Fp,q which we omit to get the ribbon F . Therefore, we need to see that at least p bands are omitted in the construction of F : The set of all bands (along x-axis) in Fp,q corresponds to the set {π(lj) | j = 1, . . . , q} ∩ {π(ki) | i = 1, . . . , p}. Notice that while constructing F we omit at least 2 bands corresponding to the intersec- tions of the lines π(k1), π(kp) with the family {π(lj) | j = 1, . . . , q} (in some cases, one of these bands might correspond to the intersection of the lines π(k2) or π(kp−1) with π(l1) or π(lq), but the following argument still works because in such a case we can omit at least 2 bands corresponding to two points on π(k2) or π(kp−1)). For the remaining p− 2 line segments h2, . . . , hp−1, there are two cases: Either each hi, for i = 2, . . . , p− 1 has at least one endpoint contained on a line other than π(l1) or π(lq), or there exists a unique hi, 1 < i < p, such that its endpoints are on π(l1) and π(lq) (such an hi must be unique since no two vj ’s are collinear !). If the first holds, then that endpoint corresponds to the intersection of hi with π(lj) for some j 6= 1, q. Then the band corresponding to either π(ki)∩π(lj−1) or π(ki)∩π(lj+1) is omitted in the construction of F (recall how we divide P into rectangular regions). If the second holds, then there is at least one line segment hi′ , which belongs to the same component of L containing hi, such that we omit at least 2 points on π(ki′) (this is true again since no two vj ’s are collinear). Hence, in any case, we omit at least p bands from Fp,q to get F . This completes the proof of Theorem 1.4. � Corollary 3.1. Given L and Fp,q as in Theorem 1.4, there exists an open book decompo- sition OB of (S3, ξst) such that (1) L lies (as a Legendrian link) on a page F of OB, (2) The page F is a subsurface of Fp,q (3) The page framing of L coming from F is equal to its contact framing tb(L), (4) If p > 3 and q > 3, then g(F ) is strictly less than g(Fp,q), (5) The monodromy h of OB is given by h = tγ1 ◦ · · · ◦ tγN where γk is the Legendrian unknot constructed in the proof of Theorem 1.4, and tγk denotes the positive (right- handed) Dehn twist along γk. Proof. The proofs of (1), (2), (3), and (4) immediately follow from Theorem 1.4 and Lemma 2.8. To prove (5), observe that by adding the missing part of each γk to the previous 1-skeleton, and by extending the previous ribbon by attaching the ribbon of the missing part of γk (which is topologically a 1-handle), we actually positively stabilize the old ribbon with the positive Hopf band (H+, tγk). Therefore, (5) follows. � With a little more care, sometimes we can decrease the number of 2-cells in the final 2-skeleton. Also the algorithm can be modified for split links: Remark 3.2. Under the notation used in the proof of Theorem 1.4, we have the following: (1) Suppose that the link L is split (so P has at least two connected components). Then we can modify the above algorithm so that Theorem 1.4 still holds. 14 MEHMET FIRAT ARIKAN (2) Let Tj denote the row (or set) of rectangles (or elements) in P (or in P) with bottom edges lying on the fixed line π(lj). Consider two consecutive rows Tj , Tj+1 lying between the lines π(lj), π(lj+1), and π(lj+2). Let R ∈ Tj and R ′ ⊂ Tj+1 be two rectangles in P with boundaries given as ∂R = C1 ∪ C2 ∪ C3 ∪ C4, ∂R ′ = C ′1 ∪ C 2 ∪ C 3 ∪ C Suppose that R and R′ have one common boundary component lying on π(lj+1), and two of the other components lie on the same lines π(ki1), π(ki2) as in Figure 7. Let γ, γ′ ⊂ ∆1N and D,D ′ ⊂ ∆N be the corresponding Legendrian unknots and 2-cells of the CCD ∆N coming from R,R ′. That is, ∂D = γ, ∂D′ = γ′, and π(D) = R, π(D′) = R′ Suppose also that L∩ γ ∩ γ′ = ∅. Then in the construction of ∆N , we can replace R,R′ ⊂ P with a single rectangle R′′ = R∪R′. Equivalently, we can take out γ∩γ′ from ∆1N , and replace D,D ′ by a single saddle disk D′′ with ∂D′′ = (γ∪γ′)\(γ∩γ′). π(lj) π(lj+1) π(ki1) π(ki2) ai1 ai2 R′′ C1 π(lj+2) Figure 7. Replacing R, R′ with their union R′′ Proof. To prove each statement, we need to show that CCD structure and all the conclu- sions in Theorem 1.4 are preserved after changing ∆N the way described in the statement. To prove (1), let P (1), . . . , P (m) be the separate components of P . After putting the corresponding separate components of L into appropriate positions (without changing their contact type) in (R3, ξ0), we may assume that the projection P = P (1) ∪ · · · ∪ P (m) of L onto the second quadrant of the yz-plane is given similar as the one which we illustrated in Figure 8. In such a projection, we require two important properties: (1) P (1), . . . , P (m) are located from left to right in the given order in the region bounded by the lines π(k1), π(l1), and π(lq). (2) Each of P (1), . . . , P (m) has at least one edge on the line π(l1). ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 15 π(lq) π(l1) π(k1) (m−1) Figure 8. Modifying the algorithm for the case when L is split If the components P (1) . . . P (m) remain separate, then our construction in Theorem 1.4 cannot work (the complement of the union of 3-cells corresponding to the rectangles in P would not be a Darboux ball; it would be a genus m handle body). So we have to make sure that any component P (l) is connected to the some other via some bridge consisting of rectangles. We choose only one rectangle for each bridge as follows: Let Al be the rectangle in T1 (the row between π(l1) and π(l2)) connecting P (l) to P (l+1) for l = 1, . . . , m − 1 (see Figure 8). Now, by adding 2- and 3-cells (corresponding to A1, . . . , Am−1), we can extend the CCD ∆N to get another CCD for (S 3, ξst). Therefore, we have modified our construction when L is split. To prove (2), if we replace D′′ in the way described above, then by the construction of ∆3N , we also replace two 3-cells with a single 3-cell whose boundary is the union of D and its isotopic copy. This alteration of ∆3N does not change the fact that the boundary of the union of all 3-cells coming from all pairs of saddle disks is still homeomorphic to a 2-sphere S2, Therefore, we can still complete this union to S3 by gluing a complementary Darboux ball. Thus, we still have a CCD. Note that γ ∩ γ′ is taken away from the 1- skeleton. However, since L∩ γ ∩ γ′ = ∅, the new 1-skeleton still contains L. Observe also that this process does not change the ribbon N(L) of L. Hence, the same conclusions in Theorem 1.4 are satisfied by the new CCD. � 16 MEHMET FIRAT ARIKAN 4. Examples Example I. As the first example, let us finish the one which we have already started in the previous section. Consider the Legendrian right trefoil knot L (Figure 1) and the corresponding region P given in Figure 5. Then we construct the 1-skeleton, the saddle disks, and the ribbon of the CCD ∆ as in Figure 9. All twists are left-handed Figure 9. (a) The page F for the right trefoil knot, (b) Construction of ∆ ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 17 In Figure 9-(a), we show how to construct the 1-skeleton G = ∆1 of ∆ starting from a single Legendrian arc (labelled by the number “ 0 ”). We add Legendrian arcs labelled by the pairs of numbers “1, 1”, . . . ,“8, 8” to the picture one by one (in this order). Each pair determines the endpoints of the corresponding arc. These arcs represent the cores of the 1-handles building the page F (the ribbon of G) of the corresponding open book OB. Note that by attaching each 1-handle, we (positively) stabilize the previous ribbon by the positive Hopf band (H+ , tγk) where γk is the boundary of the saddle disk Dk as before. Therefore, the monodromy h of OB supporting (S3, ξst) is given by h = tγ1 ◦ · · · ◦ tγ8 where tγk ∈ Aut(F, ∂F ) denotes the positive (right-handed) Dehn twist along γk. To compute the genus gF of F , observe that F is constructed by attaching eight 1-handles (bands) to a disk, and |∂F | = 3 where |∂F | is the number of boundary components of F . Therefore, χ(F ) = 1− 8 = 2− 2gF − |∂F | =⇒ gF = 3. Now suppose that (M±1 , ξ 1 ) is obtained by performing contact (±1)-surgery on L. Clearly, the trefoil knot L sits as a Legendrian curve on F by our construction, so by Theorem 2.5, we get the open book (F, h1) supporting ξ with monodromy h1 = tγ1 ◦ · · · ◦ tγ8 ◦ t L ∈ Aut(F, ∂F ). Hence, we get an upper bound for the support genus invariant of ξ1, namely, sg(ξ1) ≤ 3 = gF . We note that the upper bound, which we can get for this particular case, from [AO] and [St] is 6 where the page of the open book is the Seifert surface F5,5 of the (5, 5)-torus link (see Figure 10). z + y = 0x z − y = 0 All twists are left-handed Figure 10. Legendrian right trefoil knot sitting on F5,5 Example II. Consider the Legendrian figure-eight knot L, and its square bridge position given in Figure 11-(a) and (b). We get the corresponding region P in Figure 11-(c). Using Remark 3.2 we replace R5 and R8 with a single saddle disk. So this changes the set P. Reindexing the rectangles in P, we get the decomposition in Figure 12 which will be used to construct the CCD ∆. 18 MEHMET FIRAT ARIKAN a6a5a4a3a2a1 Figure 11. (a),(b) Legendrian figure-eight knot, (c) The region P π(l1) π(l6) π(k1) Figure 12. Modifying the region P In Figure 13-(a), similar to Example I, we construct the 1-skeleton G = ∆1 of ∆ again by attaching Legendrian arcs (labelled by the pairs of numbers “1, 1”, . . . , “10, 10”) to the initial arc (labelled by the number “0”) in the given order. Again each pair determines the endpoints of the corresponding arc, and the cores of the 1-handles building the page F (of the corresponding open book OB). Once again attaching each 1-handle is equivalent to (positively) stabilizing the previous ribbon by the positive Hopf band (H+ , tγk) for k = 1, . . . , 10. Therefore, the monodromy h of OB supporting (S3, ξst) is given by h = tγ1 ◦ · · · ◦ tγ10 To compute the genus gF of F , observe that F is constructed by attaching ten 1-handles (bands) to a disk, and |∂F | = 5. Therefore, χ(F ) = 1− 10 = 2− 2gF − |∂F | =⇒ gF = 3. ON THE SUPPORT GENUS OF A CONTACT STRUCTURE 19 D10(b) z + y = 0 z − y = 0 All twists are left-handed All twists are left-handed Figure 13. (a) The page F , (b) Construction of ∆, (c) The figure-eight knot on F6,6 20 MEHMET FIRAT ARIKAN Let (M±2 , ξ 2 ) be a contact manifold obtained by performing contact (±)-surgery on the figure-8 knot L. Since L sits as a Legendrian curve on F by our construction, Theorem 2.5 gives an open book (F, h2) supporting ξ2 with monodromy h2 = tγ1 ◦ · · · ◦ tγ10 ◦ t L ∈ Aut(F, ∂F ). Therefore, we get the upper bound sg(ξ2) ≤ 3 = gF . Once again we note that the smallest possible upper bound, which we can get for this particular case, using the method of [AO] and [St] is 10 where the page of the open book is the Seifert surface F6,6 of the (6, 6)-torus link (see Figure 13-(c)). References [AO] S. Akbulut, B. Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001), 319–334 (electronic). [DG1] F. Ding and H. Geiges, A Legendrian surgery presentation of contact 3-manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3, 583–598. [Et1] J. B. Etnyre, Planar open book decompositions and contact structures, IMRN 79 (2004), 4255– 4267. [Et2] J. B. Etnyre, Lectures on open book decompositions and contact structures, Floer homology, gauge theory, and low-dimensional topology, 103–141, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006. [Et3] J. Etnyre, Introductory Lectures on Contact Geometry, Topology and geometry of manifolds (Athens, GA, 2001), 81–107, Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Providence, RI, 2003. [EO] J. Etnyre, and B. Ozbagci, Invariants of Contact Structures from Open Books, arXiv:math.GT/0605441, preprint 2006. [Ga] D. Gabai, Detecting fibred links in S3, Comment. Math. Helv., 61(4):519-555, 1986. [Ge] H. Geiges, Contact geometry, Handbook of differential geometry. Vol. II, 315–382, Elsevier/North- Holland, Amsterdam, 2006. [Gd] N Goodman, Contact Structures and Open Books, PhD thesis, University of Texas at Austin (2003) [Gi] E. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, Pro- ceedings of the ICM, Beijing 2002, vol. 2, 405–414. [Gm] R. E. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998), 619–693. [GS] R. E. Gompf, A. I. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Math. 20, Amer. Math. Soc., Providence, RI, 1999. [Ho] K. Honda, On the classification of tight contact structures -I, Geom. Topol. 4 (2000), 309–368 (electronic). [LP] A. Loi, R. Piergallini, Compact Stein surfaces with boundary as branched covers of B4, Invent. Math. 143 (2001), 325–348. [Ly] H. Lyon, Torus knots in the complements of links and surfaces, Michigan Math. J. 27 (1980), 39-46. [OS] B. Ozbagci, A. I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, 13 (2004), Springer-Verlag, Berlin. [Pl] O. Plamenevskaya, Contact structures with distinct Heegaard Floer invariants, Math. Res. Lett., 11 (2004), 547-561. [St] A. I. Stipsicz, Surgery diagrams and open book decomposition of contact 3-manifolds, Acta Math. Hungar, 108 (1-2) (2005), 71-86. [TW] W. P. Thurston, H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. Department of Mathematics, MSU, East Lansing MI 48824, USA E-mail address : [email protected] http://arxiv.org/abs/math/0605441 1. Introduction 2. Preliminaries 2.1. Contact structures and Open book decompositions 2.2. Legendrian Knots and Contact Surgery 2.3. Compatibility and Stabilization 2.4. Monodromy and Surgery Diagrams 2.5. Contact Cell Decompositions and Convex Surfaces 3. The Algorithm 3.1. Proof of Theorem ?? 4. Examples References

0704.1671

Very Massive Stars in High-Redshift Galaxies

Mon. Not. R. Astron. Soc. 000, 1–11 (2006) Printed 11 November 2021 (MN LaTEX style file v2.2) Very Massive Stars in High-Redshift Galaxies Mark Dijkstra1⋆ and J. Stuart B. Wyithe1† 1School of Physics, University of Melbourne, Parkville, Victoria, 3010, Australia 11 November 2021 ABSTRACT A significant fraction of Lyα emitting galaxies (LAEs) at z > 5.7 have rest-frame equivalent widths (EW) greater than∼ 100Å. However only a small fraction of the Lyα flux produced by a galaxy is transmitted through the IGM, which implies intrinsic Lyα EWs that are in excess of the maximum allowed for a population-II stellar population having a Salpeter mass function. In this paper we study characteristics of the sources powering Lyα emission in high redshift galaxies. We propose a simple model for Lyα emitters in which galaxies undergo a burst of very massive star formation that results in a large intrinsic EW, followed by a phase of population-II star formation with a lower EW. We confront this model with a range of high redshift observations and find that the model is able to simultaneously describe the following eight properties of the high redshift galaxy population with plausible values for parameters like the efficiency and duration of star formation: i-iv) the UV and Lyα luminosity functions of LAEs at z=5.7 and 6.5, v-vi) the mean and variance of the EW distribution of Lyα selected galaxies at z=5.7, vii) the EW distribution of i-drop galaxies at z∼6, and viii) the observed correlation of stellar age with EW. Our modeling suggests that the observed anomalously large intrinsic equivalent widths require a burst of very massive star formation lasting no more than a few to ten percent of the galaxies star forming lifetime. This very massive star formation may indicate the presence of population- III star formation in a few per cent of i-drop galaxies, and in about half of the Lyα selected galaxies. Key words: cosmology–theory–galaxies–high redshift 1 INTRODUCTION Narrow band searches for redshifted Lyα lines have discovered a large number of Lyα emitting galaxies with redshifts between z = 4.5 and z = 7.0 (e.g. Hu & McMahon 1996; Hu et al. 2002; Malhotra & Rhoads 2002; Kodaira et al. 2003; Dawson et al. 2004; Hu et al. 2004; Stanway et al. 2004; Taniguchi et al. 2005; Westra et al. 2006; Kashikawa et al. 2006; Shimasaku et al. 2006; Iye et al. 2006; Stanway et al. 2007; Tapken et al. 2007). The Lyα line emitted by these galaxies is very prominent, often being the only observed feature. The prominence of the Lyα line is quantified by its equivalent width (EW), defined as the total flux of the Lyα line, FLyα divided by the flux density of the continuum at 1216 Å: EW≡ FLyα/f1216 . Throughout this paper we refer to the rest-frame EW of the Lyα line (which a factor of (1 + z) lower than the EW in the observers frame). Approximately 50% of Lyα emitters (hereafter LAEs) at z = 4.5 and z = 5.7 have lines with EW∼ 100 − 500 Å ⋆ E-mail:[email protected] † E-mail:[email protected] (Dawson et al. 2004; Hu et al. 2004; Shimasaku et al. 2006). For comparison, theoretical studies conclude that the maxi- mum EW which can be produced by a conventional popula- tion of stars is 200-300 Å. Moreover, this maximum EW can only be produced during the first few million years of a star- burst, while at later times the luminous phase of Lyα EW gradually fades (Charlot & Fall 1993; Malhotra & Rhoads 2002). Therefore, observed EWs lie near the upper envelope of values allowed by a normal stellar population. The quoted value for the upper envelope of EW∼ 200− 300 Å corresponds to the emitted Lyα flux. However not all Lyα photons are transmitted through the IGM, and we expect some attenuation. Within the framework of a Cold Dark Matter cosmology, gas surrounding galaxies is signifi- cantly overdense, and possesses an infall velocity relative to the mean IGM (Barkana 2004). As a net result, the IGM surrounding high redshift galaxies is significantly opaque to Lyα photons. Indeed it can be shown that for reasonable model assumptions, only ∼ 10− 30% of all Lyα photons are transmitted through the IGM (Dijkstra et al. 2007). As a result, the intrinsic Lyα EW emitted by high redshift LAEs is systematically larger than observed. Indeed, this observa- tion suggests that a significant fraction of LAEs at z > 4.5 c© 2006 RAS http://arxiv.org/abs/0704.1671v2 2 Mark Dijkstra & J. Stuart B. Wyithe have intrinsic EWs that are much larger than can possibly be produced by a conventional population of young stars. One possible origin for this large EW population is pro- vided by active galactic nuclei (AGN), which can have much larger EWs due to their harder spectra (e.g. Charlot & Fall 1993). However, large EW LAEs are not AGN for several reasons: (1) the Lyα lines are too narrow (Dawson et al. 2004) (2) these objects typically lack high–ionisation state UV emission lines, which are symptomatic of AGN activity (Dawson et al. 2004), and (3) deep X-Ray observations of 101 Lyα emitters by Wang et al. (2004, also see Malhotra et al. 2003, Lai et al, 2007) revealed no X-ray emission neither from any individual source, nor from their stacked X-Ray images. Several recent papers have investigated the stel- lar content of high-redshift LAEs by comparing stellar synthesis models with the observed broad band colors (Finkelstein et al. 2007). These comparisons are often aided by deep IRAC observations on Spitzer (Lai et al. 2007; Pirzkal et al. 2007). In this paper we take a different ap- proach. Instead of focusing on individual galaxies, our goal is to provide a simple model that describes the population of Lyα emitting galaxies as a whole. This population is de- scribed by the rest-frame ultraviolet (UV) and Lyα lumi- nosity functions (LFs) at z = 5.7 and z = 6.5 , and the Lyα EW distribution at z = 5.7 (Shimasaku et al. 2006; Kashikawa et al. 2006). The sample of high-redshift LAEs is becoming large enough that meaningful constraints can now be placed on simple models of galaxy formation. The outline of this paper is as follows: In § 2 -§ 5 we describe our models. In § 6 we discuss our results, and compare with results from stellar synthesis models, be- fore presenting our conclusions in § 7. The parameters for the background cosmology used throughout this paper are Ωm = 0.24, ΩΛ = 0.76, Ωb = 0.044, h = 0.73 and σ8 = 0.74 (Spergel et al. 2007). 2 THE MODEL Dijkstra et al. (2007b) found that the observed Lyα LFs at z = 5.7 and z = 6.5 are well described by a model in which the Lyα luminosity of a galaxy increases in proportion to the mass of its host dark matter Mtot. One can constrain quantities related to the star formation efficiency from such a model (also see Mao et al. 2007; Stark et al. 2007). However, it is also possible to obtain constraints from the rest-frame UV-LFs. In contrast to the Lyα LF, the UV- LF is not affected by attenuation by the IGM, which allows for more reliable constraints on quantities related to the star formation efficiency. In the first part of this paper (§ 3-§ 4) we present limited modeling to illustrate parameter depen- dences, using the UV-LF to constrain model parameters re- lated to star formation efficiency and lifetime. These model parameters may then be kept fixed, and the Lyα LFs and EW distributions used to constrain properties of high red- shift LAEs such as their intrinsic Lyα EW and the fraction of Lyα that is transmitted through the IGM. Later, in § 5, we present our most general model, and fit to both the UV and Lyα LFs, as well as the EW distribution, simultane- ously, treating all model parameters as free. 3 MODELING THE UV AND LYα LUMINOSITY FUNCTIONS. 3.1 Constraints from the UV-LF. We begin by presenting a simple model for the UV-LF (Wyithe & Loeb 2007; Stark et al. 2007). In Figure 1 we show the rest-frame UV-LFs of LAEs at z = 5.7 and z = 6.5 (Shimasaku et al. 2006; Kashikawa et al. 2006). We use the following simple prescription to relate the ultravi- olet flux density emitted by a galaxy, f1350, to the mass of its host dark matter Mtot. The total mass of baryons within a galaxy is (Ωb/Ωm)Mtot, of which a fraction f∗ is assumed to be converted into stars over a time scale of tsys = ǫDCthub. Here, ǫDC is the duty cycle and thub(z), the Hubble time at redshift z. This prescription yields a star formation rate of Ṁ∗ = f∗(Ωb/Ωm)M/tsys. The star forma- tion rate can then be converted into f1350 using the relation f1350 = 7 × 1027(Ṁ∗/[M⊙/yr]) erg s−1 Hz−1 (Kennicutt 1998). The precise relation is uncertain but differs by a fac- tor of less than 2 between a normal and a metal-free stel- lar population (see e.g. Loeb et al. 2005). Uncertainty in this conversion factor does not affect our main conclusions. The presence of dust would lower the ratio of f1350 and Ṁ∗, which could be compensated for by increasing f∗. However, Bouwens et al. (2006) found that dust in z = 6.0 Lyman Break Galaxies (LBGs) attenuates the UV-flux by an aver- age factor of only 1.4 (and dust obscuration may be even less important in LAEs, see § 6.3). Since this is within the uncertainty of the constraint we obtain on f∗, we ignore ex- tinction by dust. The number density of LAEs with UV-flux densities exceeding f1350 is then given by N(> f1350) = ǫDC , (1) where MUV is the mass that corresponds to the flux den- sity, f1350 (through the relations given above). The func- tion dn/dM is the Press-Schechter (1974) mass function (with the modification of Sheth et al. 2001), which gives the number density of halos of mass M (in units of comoving Mpc−3)1. The free parameters in our model are the duty cycle, ǫDC, of the galaxy, and the fraction of baryons that are converted into stars, f∗. We calculated the UV-LF for a grid of models in the (ǫDC, f∗)-plane, and generated like- lihoods L[P ] = exp[−0.5χ2], where χ2 = PNdata (modeli − datai) 2/σ2i , in which datai and σi are the i th UV-LF data point and its error, and modeli is the model evaluated at the ith luminosity bin. The sum is over Ndata = 8 data points. The inset in Figure 1 shows the resulting likelihood contours 1 Our model effectively states that the star formation rate in a galaxy increases linearly with halo mass. This is probably not cor- rect. To account for a different mass dependence we could write the star formation rate as Ṁ∗ ∝ M β , where β is left as a free pa- rameter. However, the range of observed luminosities span only 1 order of magnitude, and we will show that the choice β = 1 pro- vides a model that describes the observations well. Furthermore, the duty cycle ǫDC may be viewed as the fraction of dark mat- ter halos that are currently forming stars. The remaining fraction (1− ǫDC) of halos either have not formed stars yet, or are evolv- ing passively. In either case, the contribution of these halos to the UV-LF is set to be negligible. c© 2006 RAS, MNRAS 000, 1–11 Very Massive Stars in High-z Galaxies 3 Figure 1. Constraints on the star formation efficiency from the observed rest-frame UV-luminosity functions of LAEs at z = 5.7 (red squares) and z = 6.5 (blue circles) (Shimasaku et al. 2006; Kashikawa et al. 2006). In our best-fit model, a faction f∗ ∼ 0.06 of all baryons is converted into stars over a time-scale of ǫDCthub ∼ 0.03 Gyr (see text). The inset shows likelihood con- tours in the (ǫDC, f∗)-plane at 64%, 26% and 10% of the peak likelihood. Also shown on the upper horizontal axis is the mass corresponding to MAB,1350 in the best-fit model. in the (ǫDC, f∗)-plane at 64%, 26% and 10% of the peak like- lihood. The best fit model has (ǫDC, f∗) = (0.03, 0.06) and is plotted as the solid line. In the following sections we assume this combination of f∗ and ǫDC. 3.2 Constraints from the Lyα LF. We next model the Lyα LF, beginning with the best-fit model of the previous section. The number density of LAEs at redshift z with Lyα luminosities exceeding Tα × Lα is given by (Dijkstra et al. 2007b) N(> Tα × Lα, z) = ǫDC (z), (2) where the Lyα luminosity and host halo mass, Mα are re- lated by Tα × Lα = Lα Mα(M⊙) tsys(yr) Tα. (3) In this relation, Tα is the IGM transmission multiplied by the escape fraction of Lyα photons from the galaxy, and Lα = 2.0 × 1042 erg s−1/(M⊙ yr−1), is the Lyα luminos- ity emitted per unit of star formation rate (in M⊙ yr Throughout, Lα,42 denotes Lα in units of 1042 erg s−1/(M⊙ yr−1). We have taken Lα,42 = 2.0, which is appropriate for a metallicity of Z = 0.05Z⊙ and a Salpeter IMF (Dijkstra et al, 2007). Note that when comparing to observed lumi- nosities (Shimasaku et al. 2006; Kashikawa et al. 2006), we have replaced Lα with Tα×Lα. This is because the observed luminosities have been derived from the observed fluxes by assuming that all Lyα emerging from the galaxy was trans- mitted by the IGM, whereas there is substantial absorption (e.g. Dijkstra et al. 2007). The product Tα×Lα may be writ- ten as Tα×Lα = 4πd2L(z)Sα, where Sα is the total Lyα flux Figure 2. Joint constraints on the IGM transmission from the observed Lyα and rest-frame UV LFs of LAEs at z = 5.7 (Shimasaku et al. 2006) and z = 6.5 (Kashikawa et al. 2006). The red squares and blue circles represent the data at z = 5.7 and z = 6.5. Using the model that best describes the UV-LFs with (fstar, ǫDC) = (0.06, 0.03) we fit to the observed Lyα LF. The only free parameter of our model was the fraction of Lyα that was transmitted through the IGM at z = 5.7, Tα,57 (see text). Shown in the inset is the likelihood for Tα,57, normalised to a peak of unity. The figure shows that in order to simultaneously fit the Lyα and UV-LFs, only ∼ 30% of Lyα photons are trans- mitted through the IGM. The best fit models are overplotted as the solid lines. detected on earth and dL(z) is the luminosity distance to redshift z. The product Tα × Lα may therefore be viewed as an effective luminosity inferred at earth. Furthermore, the selection criteria used by Shimasaku et al. (2006) and Kashikawa et al. (2006) limits these surveys to be sensitive to LAEs with EW >∼10Å. In the reminder of this paper, the EW of model LAEs is always larger than this EWmin, and we need not worry about selection effects when comparing our model to the data. In this section, we set the transmission at z = 6.5 (de- noted by Tα,65) to be a factor of ∼ 1.2 lower2 than at z = 5.7 (denoted by Tα,57). This ratio is the median of the range found by Dijkstra et al. (2007). We then calculated the Lyα LF for a range of Tα,57, and generated likelihoods L[P ] = exp[−0.5χ2], where χ2 = Ndata (modeli − datai)2/σ2i , for each model. Here, datai and σi are the i th data point and its error, and modeli is the model evaluated at the i th luminos- ity bin. The sum is over Ndata = 6 points at each redshift. In Figure 2 we show the Lyα luminosity functions at z = 5.7 and z = 6.5. The red squares and blue circles represent data from Shimasaku et al. (2006, z = 5.7) and Kashikawa et al. (2006, z = 6.5), respectively. The likelihood for Tα,57 (normalised to a peak of unity) is shown in the inset. The best fit model is overplotted as the solid lines, for which the value of the transmission is Tα,57 = 0.30. The modeling presented in this and the 2 For our primary results in § 5 we allow this ratio to be a free parameter. The results presented in this section is not sensitive to the precise choice of the ratio of IGM transmission at z = 5.7 and z = 6.5. c© 2006 RAS, MNRAS 000, 1–11 4 Mark Dijkstra & J. Stuart B. Wyithe previous section therefore suggests that, in order to simul- taneously fit the Lyα and UV luminosity functions, only ∼ 30% of the Lyα can be transmitted through the IGM. This transmission is in good agreement with the results obtained by Dijkstra et al. (2007), who modeled the transmission di- rectly and found that for reasonable model parameters the transmission must lie in the range 0.1 <∼Tα <∼0.3. 3.3 The Predicted Equivalent Width While in agreement with the observed LFs, the model de- scribed in § 3.2 does not reproduce the very large ob- served equivalent widths. The Lyα luminosity can be rewrit- ten in terms of the star formation rate (Ṁ∗) and EW as Lα = 2.0 × 1042 erg s−1Ṁ∗(M⊙/yr)(EW/160 Å). Here we have used the relation Lα =EW×[ναf(να)/λα], where f(να) is the flux density in erg s−1 Hz−1 at να, and where we denoted the Lyα frequency and wavelength by να and λα respectively. Furthermore, we assumed the spectrum to be constant between 1216 Å and 1350 Å. For Lα,42 = 2.0, we find a best fit model that predicts LAEs to have an observed EW of Tα × 160 Å ∼ 50 Å. This value compares unfavor- ably with the observed sample that includes EWs exceed- ing ∼ 100 Å in ∼ 50% of cases for Lyα selected galaxies (Shimasaku et al. 2006). Thus although our simple model can successfully repro- duce the observed Lyα and UV luminosity functions, the model fails to reproduce the observed large EW LAEs. This discrepancy cannot be remedied by changing the intrinsic Lyα emissivity of a given galaxy, Lα: increasing Lα would be simply be compensated for by a lower Tα and vice versa. In the next section we discuss a simple modification of this model that aims to alleviate this discrepancy. 4 THE FLUCTUATING IGM MODEL The model described in § 3 assumed that the Lyα flux of galaxies was subject to uniform attenuation by the IGM. In this section we relax this assumption and investigate the predicted EWs in a more realistic IGM where transmission fluctuates between galaxies. We refer to this model as the ’fluctuating IGM’ model. In this model, a larger transmis- sion translates to a larger observed equivalent width. As a result, galaxies with large Tα are more easily detected, and the existence of these galaxies may therefore affect the ob- served EW-distribution for Lyα selected galaxies, even in cases where they comprise only a small fraction of the in- trinsic population. In this section we investigate whether this bias could explain the anomalously large observed EW. We assume a log-normal distribution for Tα,57, P (u)du = × exp “−(u− 〈u〉)2 du, (4) where 〈u〉 = log〈Tα,57〉 is the log (base 10) of the mean transmission and σu is the standard deviation in log-space. Throughout this section we drop the subscript ’57’. Eq (4) may be rewritten in the form f(> u) = u− 〈u〉√ , (5) Figure 3. Same as Figure 2. However, instead of assuming a single value of IGM transmission Tα,57, we assumed a log-normal distribution of IGM transmission with a mean of log Tα,57 and standard deviation of (in the log) σu (see text). This reflects the possibility that the IGM transmission fluctuates between galaxies. The inset shows likelihood contours for (log Tα,57, σu). Increasing σu flattens the luminosity function (and moves it upward), which is illustrated by the model LFs shown as solid lines, for which we used (Tα,57, σu) = (0.27, 0.2) (shown as the thick black dot in the inset). The best-fit model to the data has σu ∼ 0 (which corresponds to the model shown in Figure 2). which gives the fraction of LAEs with log Tα > u. The num- ber density of LAEs is then given by N(> Tα ×Lα) = ǫDC f(> u(Tα ×Lα,M)) (6) where u(Tα × Lα,M) = log “Tα × Lα LαṀ∗ . (7) Eq (6) differs from Eq (2) in two ways: (1) there is no lower integration limit, and (2) there is an additional term f(> u). These two differences reflect the facts that all masses contribute to the number density of LAEs brighter than Tα× Lα, and that lower mass systems require larger transmissions (Eq 7) which are less common (Eq 5). In the limit σu → 0, the function f(> u) ’jumps’ from 0 to 1 at Mmin (Eq 3), which corresponds to the original Eq (2). In this formalism we may also write the number density of LAEs with transmission in the range u±du/2 ≡ log Tα ± d log Tα , which is given by N(u)du = P (u)du Mmin(u) . (8) Here Mmin(u) is the minimum mass of galaxies that can be detected with a transmission in the range u± du/2 (for Mtot < Mmin the total flux falls below the detection thresh- old). The number density of LAEs with transmission in the range u ± du/2 may be used to find the number density of LAEs with equivalent widths in the range log EW± d log EW via the relation EW= 160Tα Å (for the choice Lα,42 = 2.0, see § 3.2). Eq (8) shows that the observed equivalent width distribution takes the shape of the original transmission dis- tribution, modulo a boost which increases towards larger As in § 3.2 we assume the best-fit model parameters c© 2006 RAS, MNRAS 000, 1–11 Very Massive Stars in High-z Galaxies 5 for ǫDC and f∗ derived from the UV-LFs determined in § 3.1. We calculate model Lyα LFs on a grid of models in the (σu, 〈Tα〉)-plane, and generate likelihoods following the procedure outlined in § 3.2. The results of this calculation are shown in Figure 3 where we plot the Lyα LFs together with likelihood contours in the (σu, 〈Tα〉)-plane (inset). The best-fit models favor no scatter in Tα (σu ∼ 0). The reason for this is that for any given model, a scatter in Tα serves to flatten the model LF. However, the observed Lyα LF is quite steep, and as a result the data prefers a model with no scatter. Furthermore, a scatter in Tα results in a model LF that lies above the original (constant transmission) LF at all Tα × Lα. This also explains the shape of the contours in the (σu, 〈Tα〉)-plane; increasing σu must be compensated for by lowering 〈Tα〉). To illustrate the impact of a fluctuating Tα, the model LFs shown in Figure 3 are not those of the best-fit model, but of a model with (〈Tα〉, σu) = (0.27, 0.2). Figure 4 shows the observed EW-distribution (solid line) at z = 5.7 associated with the Lyα LFs shown in Fig- ure 3. This model may be compared to the observed distri- bution (shown as the histogram, data from Shimasaku et al, 2006). The upper horizontal axis shows the transmission cor- responding to each equivalent width. The dotted line shows the distribution of transmission (given by Eq 4). Note that the range of transmissions shown in Figure 4 extends to Tα > 1, which of course is not physical. The model EW- distribution has a tail towards larger EWs. However, the model distribution peaks at EW∼ 50 Å. As was seen in the constant transmission model, this is clearly inconsistent with the observations, which favor values of EW∼ 100 Å. Shimasaku et al. (2006) also show a probability distribution of EW, which is probably closer to the actual distribution. Although this distribution does peak closer to EW∼ 50 Å, it is still significantly broader than that of the model (see § 6 for more arguments against the fluctuating IGM model). We point out that the model distribution shown in Figure 4 is independent of the assumed value of Lα,42. 5 LYα EMITTERS POWERED BY VERY MASSIVE STARS. In § 3 we demonstrated that a simple model where Lα and LUV were linearly related to halo mass can reproduce the UV and Lyα LFs, but not the observed EW distribution. In § 4 we showed that this situation is not remedied by a variable IGM transmission, and that favored models have a constant transmission. In this section we discuss an alternate model, which leads to consistency with both the observed Lyα LFs, UV-LFs, and the EW-distribution. In this model, galaxies are assumed to have a bright Lyα phase (hereafter the ’population III’-phase) which lasts a fraction fIII of the galaxies’ life-time. After this the galaxy’s Lyα luminosity drops to the ’normal’ value for population II star formation. This model may be viewed as an extension of the idea originally described by Malhotra & Rhoads (2002), that large EW LAEs are young galaxies in the early stages of their lives. In this picture, the sudden drop in Lyα luminos- ity could represent i) a sudden drop in the ionising lumi- nosity when the first O-stars died, or ii) an enhanced dust- opacity after enrichment by the first type-II supernovae. Al- ternatively, our parametrisation could represent a scenario Figure 4. Comparison of the observed equivalent width distribu- tion (EW, histogram), with the model prediction for a model in which we assumed a log-normal distribution of IGM transmission with (Tα,57, σu) = (0.27, 0.2) (see Fig 3). The EW is related to Tα via EW= 160Tα Å. The dotted line shows the fraction, f(> Tα) (shown on the right vertical axis), of galaxies with a transmission greater than Tα (Eq 5). Galaxies with large Tα are more easily detected, hence the large Tα (EW) end is boosted considerably, resulting in closer agreement (but not close enough) to the data. in which the population III phase ended after the first pop- ulation III stars enriched the surrounding interstellar gas from which subsequent generations of stars formed. Hence, we refer to this model as the ’population III’ model. We will show that to be consistent with the large values of the ob- served EW, a very massive population of stars is required during the early stages of star formation. To minimise the number of free parameters we modeled the time dependence of the Lyα EW as a step-function. The number density of LAEs is then given by N(> Tα × Lα, z) = fIII × ǫDC Mα,III (z) + (9) (1− fIII)× ǫDC Mα,II Here,Mα,II is the mass related to Tα×Lα through Tα×Lα = Lα×Tα×Ṁ∗, while Mα,III is the population III mass, which is calculated with Lα replaced by Lα = (EWIII/160 Å)×2× 1042 erg s−1. Whereas in previous sections we chose fiducial or best fit parameters for illustration, for the model described in this section we take the most general approach. We fit the model simultaneously to the UV-LF and Lyα LFs, as well as to the observed EW-distribution of Lyα selected galaxies. This model predicts two observed equivalent widths (Tα×EWIII and Tα×EWII) in various abundances. The associated mean and variance from the model are compared to the observed EW-distribution, which has a mean of 〈EW〉 = 120± 25 Å, and a standard deviation of σEW = 50± 10 Å. Our model has 6 free parameters (ǫDC, f∗, Tα,57, Tα,65, fIII,EWIII). We produce likelihoods for each parameter by marginalising over the others in this space. The lower set of panels in Figure 5 show likelihood contours for our model parameters at 64%, 26% and 10% of the peak likelihood. The best-fit models have EWIII ∼ 600 − 800 Å and fIII = 0.04 − 0.1 which c© 2006 RAS, MNRAS 000, 1–11 6 Mark Dijkstra & J. Stuart B. Wyithe Figure 5. Marginalised constraints on the 6 population-III model parameters, (ǫDC, f∗,Tα,57,Tα,65, fIII,EWIII), are shown in the lower three panels. In the best-fit population-III model, each galaxy goes through a luminous Lyα phase during which the equivalent width is EWIII = 600 − 800 Å for a fraction fIII = 0.04 − 0.1 of the galaxies life time. Although the bright phase only lasts a few to ten per cent of their life-time, galaxies in the bright phase are more easily detectable, and the number of galaxies detected in Lyα surveys in the bright phase is equal to that detected in the faint phase. This is also demonstrated by the thick solid line (with label ’1.0’) in the lower left panel, which shows the combination of fIII and EWIII that produces equal numbers of galaxies in the population III and II phase (the dashed lines are defined similarly, also see Fig 6). The best-fit intrinsic equivalent widths are in excess of the maximum allowed for a population-II stellar population having a Salpeter mass function. Therefore, this model requires a burst of very massive star formation lasting no more than a few to ten percent of the galaxies star forming lifetime, and may indicate the presence of population-III star formation in a large number of high-redshift LAEs. corresponds to a physical timescale for the population-III phase of fIII× ǫDC × thub ∼ 4−50 Myr (for 0.1 <∼ǫDC <∼0.5). The model Lyα luminosity functions at z=5.7 and z=6.5 described by (ǫDC, f∗, Tα,57, Tα,65, fIII,EWIII) = (0.2, 0.14, 0.22, 0.19, 0.08, 650 Å) are shown as solid lines and provide good fits to the data. The model produces two observed EWs, namely Tα×EWII = 35 Å and Tα× EWIII = 143 Å . It is worth emphasising that the emitted EW of the bright phase depends on the choice Lα,42 via EWIII = 650(Lα,42/2.0) Å. Hence, a lower/higher value of Lα,42 would decrease/increase the intrinsic brightness of the ’population III’ phase. Note that Lα,42 = 1.0 when LAEs formed out of gas of solar metallicity, which is unreasonable given the universe was only ∼ 1 Gyr old at z = 6. Furthermore, Lα,42 = 1 would have yielded a best-fit Tα,57 = 0.5, which is well outside the range calculated by Dijkstra et al. (2007). We there conclude that Lα,42 is in excess of unity. In performing fits we have fixed the value of EWII to correspond to a standard stellar population, and then ex- plored the possibility that there might be a second phase of SF producing a larger EW. Our modeling finds strong sta- tistical evidence for this early phase and rules out the null- hypothesis that properties can be described by population-II stars alone at high confidence (grey region in the lower left panel inset of Fig 5). Despite the fact that the best fit model has a bright phase which lasts only a few per cent of the total star formation lifetime, the two populations of LAEs are sim- ilarly abundant in model realisations of the observed sample in Lyα selected galaxies (see § 5.1 for a more detailed com- parison to the observed EW distributions). This is shown in the lower left panel in Figure 5 in which the solid line (with label ’1.0’) shows the combination of fIII and EWIII for which the observed number of galaxies in the popula- tion III phase (NIII) equals that in the population II phase (NII). The dashed lines show the cases NIII/NII = 0.3 and NIII/NII = 3.0. The duration of the bright Lyα phase meets theoretical expectations for a burst of star formation, while the large EW requires a very massive stellar population (e.g Schaerer 2003; Tumlinson et al. 2003). In summary, in order to reproduce both the UV and Lyα LFs, and the observed population of large EW galaxies, we require a burst of very massive star formation lasting <∼10 per cent of the galaxies lifetime. c© 2006 RAS, MNRAS 000, 1–11 Very Massive Stars in High-z Galaxies 7 5.1 EW Distribution of UV and Lyα Selected Galaxies Stanway et al. (2007) show that 11 out of 14 LAE candidates among i-drop galaxies in the Hubble Ultra Deep Field have EW< 100 Å. If galaxies are included for which only upper or lower limits on the EW are available, then this fraction be- comes 21 out of 26. Thus the distribution of EWs for i-drop selected galaxies differs strongly from the EW-distribution observed by Shimasaku et al. (2006). We next describe why this strong dependence of the observed Lyα EW distribu- tion on the precise galaxy selection criteria arises naturally in our population III model. We first assume that the Lyα selected and UV-selected galaxies were drawn from the same population (this assump- tion is discussed further in § 6.3). In our model a galaxy that is selected based on it’s rest-frame UV-continuum emission has a probability fIII of being observed in the Lyα bright phase, while the probability of finding a galaxies in the Lyα faint phase is 1 − fIII. In § 5 we found fIII ∼ 0.1, hence an i-drop galaxy is ∼ 10 times more likely to have a low than a high observed EW. If we denote the number of galaxies with EW> 100 Å by NIII, and the number of galaxies with EW< 100 Å by NII, then the model predicts NIII/NII =fIII/(1 − fIII) ∼ 0.1, while the observed fraction including the galaxies for which the EW is known as upper or lower limit is NIII/NII = 0.19± 0.05. Therefore the quali- tative difference in observed Lyα EW distribution among i- drop galaxies in the HUDF and among Lyα selected galaxies follows naturally from our two-phase star formation model. Note that our model predicts population III star formation to be observed in fIII/(1 − fIII) ∼ 10% of the z = 6.0 LBG population. The dependence of the observed EW distribution on the selection criteria used to construct the sample of galaxies is illustrated in Figure 6. To construct this figure, we have taken the best-fit population III model of § 5. For the pur- pose of presentation, we let the IGM fluctuate according to the prescription of § 4 with σu = 0.1, so that the model pre- dicts a finite range of EWs in each phase. The left and right panels show the predicted EW distribution for Lyα selected (left panel) and UV-selected (right panel) galaxies as the solid lines, respectively. For a UV-selected galaxy the prob- ability of being in the bright phase and having an observed EW in the range EWIII×(Tα±dTα/2) is fIIIP (Tα)dTα. Here P (Tα)dTα is the probability that the IGM transmission is in the range Tα±dTα/2, which is derived from Eq 4. The units on the vertical axis are arbitrary, and chosen to illustrate the different predicted and observed Lyα EW distribtions for the two samples at large EWs. The observed distribu- tions for Lyα and UV selected galaxies, shown as histograms, are taken from Shimasaku et al. (2006) and Stanway et al. (2007), respectively. Figure 6 clearly shows that both the predicted and observed Lyα selected samples contain signif- icantly more large EW LAEs than the UV-selected sample. Our model naturally explains the qualitative shape of these distributions and their differences. Before proceeding we mention a caveat to the distribu- tions shown in Figure 6. In our model all galaxies have an EW of Tα×EWII ∼ 25−35 Å during the population II phase, while in contrast Stanway et al. (2007) do not detect 10 out of 26 LBGs, which implies that ∼ 40% of LBGs have an Figure 6. Comparison of the predicted EW distribution for UV and Lyα selected galaxies. The best-fit population-III model (see § 5) was used. In order to get a finite range of observed EWs (in- stead of only two values at Tα×EWII and Tα×EWIII, we assumed the IGM transmission to fluctuate. The units on the vertical axis are arbitrary. The figure shows that in our population III model, the Lyα selected sample contains a larger relative fraction of large EW LAEs than the UV-selected (i-drop) sample, which is quali- tatively in good agreement with the observations (shown by the histograms). EW <∼6 Å. Thus there is a descrepancy between our model and the observations with respect to the value of EW in the population II phase. The resolution of this discrepancy lies in the fact that the very low EW emitters are drawn from the UV (i-drop) sample and not the Lyα selected sample our model was set up to describe. This issue is discussed in more detail in § 6.3. An EW distribution of dropout sources was also pre- sented by Dow-Hygelund et al. (2007). These authors per- formed an analysis similar to Stanway et al. (2007) and found 1 LAE with EW= 150 Å among 22 candidate z=6.0 LBGs. When interpreted in reference to our model, this translates to NIII/NII ∼ 5%, which is consistent with the model predictions. Therefore, when interpreted in light of a two-phase star formation history and differ- ent selection methods, the EW distribution observed by Dow-Hygelund et al. (2007) is consistent with that found by Shimasaku et al. (2006). If population III star formation does provide the ex- planation for the very large EW Lyα emitters, then we would expect the large EW emitters to become less common with time as the mean metallicity of the Universe increased. To test this idea, we can compare the EW-distribution at z = 5.7 with the results at lower redshift from Shapley et al. (2003) who found that <∼0.5% of z = 3 LBGs have Lyα EW> 150 Å , and that <∼2% of z = 3 LBGs to have Lyα EW> 100 Å. Dow-Hygelund et al. (2007) argue that the fraction of large EW Lyα lines at z = 6 is consistent with that observed at z = 3 (Shapley et al. 2003). However, if the EW distribution did not evolve with redshift, then the probability that a sample of 22 LBGs will contain at least 1 LAE with EW >∼150 Å is <∼10%. Thus the hypothesis that the observed EW distribution remains constant is ruled out at the ∼ 90% level. On the other hand, in a similar analysis Stanway et al. (2007) found 5 out of 26 LBGs to have an EW >∼100 Å. If the EW distribution did not evolve c© 2006 RAS, MNRAS 000, 1–11 8 Mark Dijkstra & J. Stuart B. Wyithe with redshift, then the probability of finding 5 EW >∼100 Å in this sample is only ∼ 10−4. Furthermore, Nagao et al. (2007) recently found at least 5 LAEs with EW> 100 Å at 6.0 <∼z <∼6.5, and conclude that 8% of i’-drop galaxies in the Subaru Deep Field have EW> 100 Å, which is signifi- cantly larger than the fraction of large EW LBGs at z = 3. Therefore, the observed EW distribution of LBGs at z = 6 is skewed more toward large EWs than at z = 3. The strength of this result is increased by the fact that the IGM is more opaque to Lyα photons at z = 6 than at z = 3. Thus we con- clude that the intrinsic EW distribution must have evolved with redshift. 6 DISCUSSION 6.1 Comparison with Population Synthesis Models Population synthesis models have suggested that the broad band colors of observed LAEs are best described with young stellar populations (Gawiser et al. 2006; Pirzkal et al. 2007; Finkelstein et al. 2007). Lai et al. (2007) found the stellar populations in three LAEs to be 5−100 Myr old, and possi- bly as old as 700 Myr (where the precise age upper limit de- pends on the assumed star formation history of the galaxies). However as was argued by Pirzkal et al. (2007), since these galaxies were selected based on their detection in IRAC, a selection bias towards older stellar populations may ex- ist (also see Lai et al. 2007). Furthermore, Finkelstein et al. (2007) found that LAEs with EW> 110 Å have ages <∼4 Myr, while LAEs with EW< 40 Å have ages between 20- 400 Myr. This latter result in particular agrees well with our population III model. On the other hand, in a fluctu- ating IGM model for example, the EW of LAEs should be uncorrelated with age. In models presented in this paper, on average f∗ ∼ 0.15 of all baryons are converted into stars within halos of mass Mtot ∼ 1010 − 1011M⊙, yielding stellar masses in the range M∗ = 10 8−109M⊙ (Dijkstra et al. 2007b; Stark et al. 2007). This compares unfavorably with the typical stellar masses found observationally in LAEs which can be as low as M∗ = 10 6 − 107M⊙ (Finkelstein et al. 2007; Pirzkal et al. 2007). However, the lowest stellar masses are found (natu- rally) for the younger galaxies. Indeed, the LAEs with the oldest stellar populations can have stellar masses as large as 1010M⊙. Thus, we do not find the derived stellar masses in LAEs to be at odds with the results of this paper. If significant very massive (or population III) star formation indeed occurred in high redshift LAEs, then one may ex- pect these stars to reveal themselves in unusual broad-band colors (e.g. Stanway et al. 2005). However, Tumlinson et al. (2003) have shown that the most distinctive feature in the spectrum of population III stars is the number of H and He ionising photons (also see Bromm et al. 2001). Since these are (mostly) absorbed in the IGM, the broad band spec- trum of population III stars is in practice difficult to dis- tinguish from a normal stellar population (Tumlinson et al. 2003), especially when nebular continuum emission is taken into account (Schaerer & Pelló 2005, see their Fig 1). Hence, population III stars would not necessarily be accompanied by unusually blue broad band colors. 6.2 Alternative Explanations for Large EW LAEs We have shown that a simple model in which high-redshift galaxies go through a population-III phase lasting <∼15 Myr can simultaneously explain the observed Lyα LFs at z = 5.7 and z = 6.5 (Kashikawa et al. 2006), and the ob- served EW-distribution of Lyα selected galaxies at z = 5.7 (Shimasaku et al. 2006). In addition, this model predicts the much lower EWs found in the population of UV selected galaxies (Stanway et al, 2007, see § 5.1). Moreover the con- straints on the population-III model parameters such as the duration and the equivalent width of the bright phase are physically plausible, and consistent with existing population synthesis work (see § 6.1). Are there other interpretations of the large observed EWs? One possibility was discussed in § 4, where we showed that the simple model in which the IGM transmission fluc- tuates between galaxies reproduces the LFs, but fails to simultaneously reproduce the Lyα LFs and the observed EW-distribution. In addition, this model fails to reproduce other observations. Dijkstra et al. (2007) calculated the im- pact of the high-redshift reionised IGM on Lyα emission lines and found the range of plausible transmissions to lie in the range 0.1 < Tα < 0.3. This work showed that it is possible to boost the transmission to (much) larger values but not without increasing the observed width of the Lyα line. Absorption in the IGM typically erases all flux blue- ward of the Lyα resonance, and when infall is accounted for, part of the Lyα redward of the Lyα resonance as well. This implies that Lyα lines that are affected by absorption in the IGM are systematically narrower than they would have been if no absorption in the IGM had taken place. It follows that in the fluctuating IGM model, Lyα EW should be strongly correlated with the observed Lyα line width (or FHWM). This correlation is not observed. In fact, observa- tions suggest that an anti-correlation exists between EW and FWHM (Shimasaku et al. 2006; Tapken et al. 2007). This anti-correlation provides strong evidence against the anomalously large EWs being produced by a fluctuating IGM transmission. A second possibility is the presence of galaxies with strong superwinds. The models of Dijkstra et al. (2007) did not study the impact of superwinds on the Lyα line pro- file. The presence of superwinds can cause the Lyα line to emerge with a systematic redshift relative to the Lyα reso- nance through back scattering of Lyα photons off the far side of the shell that surrounds the galaxy (Ahn et al. 2003; Ahn 2004; Hansen & Oh 2006; Verhamme et al. 2006). However, superwinds tend not only to redshift the Lyα line, they also make the Lyα line appear broader than when this scatter- ing does not occur. As in the case of the fluctuating IGM model, this results in a predicted correlation between EW and FWHM, which is not observed. Furthermore in wind- models, the overall redshift of the Lyα line, and thus Tα, increases with wind velocity, vw. This predicts that EW in- creases with wind velocity. However, observations of z = 3 LBGs by Shapley et al. (2003) show that EW correlates with v−1w (Ferrara & Ricotti 2006). We therefore conclude that the large EW in LAEs cannot be produced by superwind galaxies. A third possibility might be that within the Lyα emit- ting galaxy, cold, dusty clouds lie embedded in a hot inter- c© 2006 RAS, MNRAS 000, 1–11 Very Massive Stars in High-z Galaxies 9 Figure 7. Comparison of the best-fit population III model (shown in Figure 5) with the UV-LF constructed by Bouwens et al. (2006) using ∼ 300 LBGs in the Hubble Deep Fields. This good agreement is likely a coincidence since in our model all LBGs have an observed EW > 30 Å, while the obser- vations show that ∼ 40% of all LBGs are not detected in Lyα (EW < 6 Å). This overlap could possibly be (partly) due to a lower dust content of LAEs relative to their Lyα quiet counter- parts (see text). cloud medium of negligible Lyα opacity. Under such condi- tions, the continuum photons can suffer more attenuation than Lyα photons which bounce from cloud to cloud and mainly propagate through the hot, transparent inter-cloud medium (Neufeld 1991; Hansen & Oh 2006). This attenua- tion of continuum leads to a large EW. We point out that in this scenario, large EW LAEs are not intrinsically brighter in Lyα. At fixed Lyα flux, one is therefore equally likely to detect a low EW LAE. In other words, to produce the observed EW distribution one requires preferential destruc- tion of continuum flux by dust in ∼ 50% of the galaxies. Currently there is no evidence that this mechanism is at work even in one galaxy. Furthermore, the rest-frame UV colors of galaxies in the Hubble Ultra Deep Field imply that dust in high-redshift galaxies suppresses the continuum flux by only a factor of ∼ 1.4 (Bouwens et al. 2006). The maxi- mum boost of the EW in a multi-phase ISM is therefore 1.4, which is not nearly enough to produce intrinsic equivalent widths of EW∼ 600 − 800 Å. In summary, the only model able to simultaneously explain all observations calls for a short burst of very massive star formation. 6.3 Comparison with the LBG Population In § 5.1 we have shown that the observed EW distributions of Lyα selected and i-drop galaxies and their differences can be reproduced qualitatively with our population III model. However, in our model all high-redshift galaxies have an ob- served EW of at least Tα×EWII ∼ 30 Å, whereas many i-drop galaxies are not detected in Lyα. Kashikawa et al. (2006) show that the UV-LFs of LAEs at z = 6.5 and z = 5.7 overlap with that constructed by Bouwens et al. (2006) from a sample of ∼ 300 z = 6 LBGs discovered in the Hubble Deep Fields. Naively, this overlap implies that LBGs and LAEs are the same population and therefore that all LBGs should be detected by Lyα surveys. Since Lyα sur- veys only detect galaxies with EW >∼20 Å, this suggests all LBGs should have a Lyα EW >∼20 Å contrary to observa- tion. To illustrate this point further, we have taken the best- fit population III model shown in Figure 5 and compared the model predictions for the rest-frame UV-LF with that of Bouwens et al. (2006) in Figure 7. Clearly, our best-fit pop- ulation III model fits the data well. However, Stanway et al. (2007) found ∼ 40% of i-drop galaxies in the HUDF to have an observed EW <∼6 Å, and a similar result was presented by Dow-Hygelund et al. (2007). Two effects may help reconcile these two apparently conflicting sets of observations: (i) Dow-Hygelund et al. (2007) found Lyα emitting LBGs to be systematically smaller. That is, for a fixed angular size, the z850-band flux of LAEs is systematically higher with ∆z850 ∼ −1. If we assume that the angular scale of a galaxy is determined by the mass of its host halo, then this implies that for a fixed mass the z850-band flux of LAEs is systematically higher, and (ii) only a fraction of LBGS are LAEs. The drop-out technique used to select high redshift galaxies is known to introduce a bias against strong LAEs, as a strong Lyα line can affect the broad band colors of high-redshift galaxies. This may cause ∼ 10−46% of large EW LAEs to be missed using the i-drop technique (Dow-Hygelund et al. 2007). If only a fraction fα of all LBGs are detected in Lyα, then effect (i) would explain why the UV-LF of LAEs lies less than a factor of 1/fα below the observed UV-LF of the general population of LBGs. This is because a more abundant lower mass halo is required to produce the same UV-flux in LAEs, which would shift the LF upwards. In addition, effect (ii) may reduce this difference even further. It follows that these two effects combined may cause the LFs to overlap. Thus the overlap of the UV and Lyα selected UV- LFs appears to be a coincidence, and not evidence of their being the same population of galaxies. This implies that our model is valid for Lyα selected galaxies, but not the high- redshift population as a whole and explains the lack of very low EWs in UV selected samples discussed in § 5.1. The reason why LAEs may be brighter in the UV for a fixed halo mass is unclear. It is possibly related to dust content. Bouwens et al. (2006) found the average amount of UV exctinction to be 0.4 mag in the total sample of z = 6 LBGs. This value is close to the average excess z850- band flux detected from LAEs for a given angular scale (Dow-Hygelund et al. 2007). If LAEs contain less (or no) dust, then this would explain why they are brighter in the UV and thus why they appear more compact. The possi- bility that ’Lyα quiet’ LBG contain more dust than their Lyα emitting counterparts is not very surprising, as a low dust abundance has the potential to eliminate the Lyα line. Thus LAEs could be high redshift galaxies with a lower dust content. Shimasaku et al. (2006) and Ando et al. (2006) found that luminous LBGs, MUV <∼ − 21.0, typically do not con- tain large EW Lyα emission lines. This deficiency of large EW LAEs among UV-bright sources is not expected in our model, and may reflect that UV-bright sources are more massive, mature, galaxies that cannot go through a population-III phase anymore. It should be pointed out though that the absence of large EW LAEs among galax- c© 2006 RAS, MNRAS 000, 1–11 10 Mark Dijkstra & J. Stuart B. Wyithe ies with MUV <∼ − 21.0 in the survey of Shimasaku et al. (2006) is consistent with our model: The observed num- ber density of sources with MUV <∼ − 21.0 is ∼ 5 × 10 cMpc−3 (see Fig 1). In our best-fit pop-III model (shown in Fig 5), a fraction fIII ∼ 0.08 of these galaxies would be in the bright phase. This translates to a number density of large EW LAEs of ∼ 4× 10−6 cMpc−3. Given the survey volume of ∼ 2×105 cMpc3, the expected number of large EW LAEs with MUV <∼ − 21.0 is ∼ 0.8, and the absence of large EW LAE among UV bright sources is thus not surprising. 6.4 Clustering Properties of the LAEs In our model large EW LAEs are less massive by a factor of EWIII/EWII ∼ 4 at fixed Lyα luminosity. Since clus- tering of dark matter halos increases with mass, it follows that our model predicts large EW LAEs to be clustered less than their low EW counterparts (at a fixed Lyα luminos- ity). The clustering of LAEs is typically quantified by their angular correlation function (ACF), w(θ), which gives the excess (over random) probability of finding a pair of LAEs separated by an angle θ on the sky. The ACF depends on the square of the bias parameter (w(θ) ∝ b2(m)), which for galaxies in the population II phase is ∼ 1.24 − 1.4 times larger than for galaxies in the population III phase, for the mass range of interest. This implies that the clustering of low EW LAEs at fixed Lyα luminosity is enhanced by a factor of ∼ 1.5− 2.0. Existing determinations of the ACF of LAEs by Shimasaku et al. (2006) and Kashikawa et al. (2006) are still too uncertain to test this prediction. 7 CONCLUSIONS Observations of high redshift Lyα emitting galaxies (LAEs) have shown the typical equivalent width (EW) of the Lyα line to increase dramatically with redshift, with a signifi- cant fraction of the galaxies lying at z > 5.7 having an EW >∼100 Å. Recent calculations by Dijkstra et al. (2007) show that the IGM at z > 4.5 transmits only 10 − 30% of the Lyα photons emitted by galaxies. In this paper we have investigated the transmission using a model that re- produces the observed Lyα and UV LFs. This model re- sults in an empirically determined transmission of Tα ∼ 0.30(Lα,42/2.0)−1, where Lα,42 denotes the Lyα luminos- ity per unit star formation rate (in M⊙ yr −1) Lα in units of 1042 erg s−1 (§ 3). This value is in good agreement with earlier theoretical results. If only ∼ 30% of all Lyα that was emitted by high redshift galaxies reaches the observer, then this impies that the intrinsic EWs are systematically (much) larger than ob- served in many cases. To investigate the origin of these very high EWs, we have developed semi-analytic models for the Lyα and UV luminosity functions and the distribution of equivalent widths. In this model Lyα emitters undergo a burst of very massive star formation that results in a large intrinsic EW, followed by a phase of population-II star for- mation that produces a lower EW3. This model is referred 3 Technically, the model discussed in § 5 only specifies that galax- ies go through a ’population-III phase for a fraction fIII ∼ 0.1 of to as the ’population III model’ and is an extension of the idea originally described by Malhotra & Rhoads (2002), who proposed large EW Lyα emitters to be young galaxies. The population III model in which the Lyα equivalent width is EWIII ∼ 650(Lα,42/2.0) Å for <∼50 Myr, is able to simultaneously describe the following eight properties of the high redshift galaxy population: i-iv) the UV and Lyα luminosity functions of LAEs at z=5.7 and 6.5, v-vi) the mean and variance of the EW distribution of Lyα selected galaxies at z=5.7, vii) the EW distribution of UV- selected galaxies at z∼6 (§ 5), and viii) the observed correlation of stellar age and mass with EW (§ 6.1). Our modeling sug- gests that the anomalously large intrinsic equivalent widths observed in about half of the high redshift Lyα emitters require a burst of very massive star formation lasting no more than a few to ten percent of the galaxies star forming lifetime. This very massive star formation may indicate the presence of population-III star formation in a large number of high-redshift LAEs. The model parameters for the best-fit model are physically plausible where not previously known (e.g. those related to the efficiency and duration of star for- mation), and agree with estimates where those have been calculated directly (e.g the IGM transmission, EWIII, and fIII). In addition, we argued that the observed overlap of the UV-LFs of LAEs with that of z∼ 6 LBGs appears to be at odds the observed Lyα detection rate in high-redshift LBGs, suggesting that LAEs and LBGs are not the same popula- tion. A lower dust content of LAEs relative to their ‘Lyα quiet’ counterparts would partly remedy this discrepancy, and could also explain why LAEs appear to be typically more compact (§ 6.3). Semi-analytic modeling of the coupled reionisation and star formation histories of the universe suggests that popu- lation III star formation could still occur after the bulk of reionisation had been completed (Scannapieco et al. 2003; Schneider et al. 2006; Wyithe & Cen 2007). The observa- tion of anomalously large EWs in Lyα emitting galaxies at high redshift may therefore provide observational evidence for such a scenario. In the future, the He 1640 Å may be used as a complementary probe (e.g Tumlinson et al. 2001, 2003). The EW of this line is smaller by a factor of >∼20 for population III (Schaerer 2003). However, the He 1640Å will not be subject to a small transmission of ∼ 10− 30%, mak- ing it accessible to the next generation of space telescopes. On the other hand, it may also be possible to observe the He 1640 Å in a composite spectra of z = 6 LBGs. Indeed, the He 1640 Å line has already been observed in the com- posite spectrum of z= 3 LBGs (Shapley et al. 2003), which led Jimenez & Haiman (2006) to argue for population III star formation at redshifts as low as z = 3−4. If population III star formation was more widespread at higher redshifts, as predicted by our model, then the composite spectrum their lifetimes. Our model does not specify when this population- III phase occurs. Hypothetically, the population III phase could occur at an arbitrary moment in the galaxies’ lifetime when it is triggered by a merger of a regular star forming galaxy and a dark matter halo containing gas of primordial composition. Note however that such a model would probably have difficulties ex- plaining the apparent observed correlation between Lyα EW and the age of a stellar population (§ 6.1). c© 2006 RAS, MNRAS 000, 1–11 Very Massive Stars in High-z Galaxies 11 of LBGs at higher redshifts should exhibit an increasingly prominent He 1640 Å line. In particular, this line should be most prominent in the subset of LBGs that have large EW Lyα emission lines. Acknowledgments JSBW and MD thank the Aus- tralian Research Counsel for support. We thank Avi Loeb for useful discussions, and an anonymous referee for a helpful report that improved the content of this paper. REFERENCES Ahn, S.-H., Lee, H.-W., & Lee, H. 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EW Distribution of UV and Ly Selected Galaxies Discussion Comparison with Population Synthesis Models Alternative Explanations for Large EW LAEs Comparison with the LBG Population Clustering Properties of the LAEs Conclusions

0704.1672

Two center multipole expansion method: application to macromolecular systems

Two center multipole expansion method: application to macromolecular systems Ilia A. Solov’yov∗, Alexander V. Yakubovich*, Andrey V. Solov’yov*, and Walter Greiner Frankfurt Institute for Advanced Studies, Max von Laue Str. 1, 60438 Frankfurt am Main, Germany We propose a new theoretical method for the calculation of the interaction energy between macromolecular systems at large distances. The method provides a linear scaling of the computing time with the system size and is considered as an alternative to the well known fast multipole method. Its efficiency, accuracy and applicability to macromolecular systems is analyzed and discussed in detail. I. INTRODUCTION In recent years, there has been much progress in simulating the structure and dynamics of large molecules at the atomic level, which may include up to thousands and millions of atoms [1, 2, 3, 4]. For example, amorphous polymers may have segments each with 10000 atoms [4] which associate to form partially crystalline lamellae, random coil regions, and interfaces between these regions, each of which may contribute with special mechanical and chemical properties to the system. With increasing computer powers nowadays it became possible to study molecular systems of enormous sizes which were not imaginable just several years ago. For example in [1] a molecular dynamics simulations of the complete satellite tobacco mosaic virus was performed which includes up to 1 million of atoms. In that paper the stability of the whole virion and of the RNA core alone were demonstrated, and a pronounced instability was shown for the capsid without the RNA. The study of structure and dynamics of macromolecules often implies the calculation of the potential energy surface for the system. The potential energy surface of a macromolecule carries a lot of useful information about the system. For example from the potential energy landscape it is possible to estimate the characteristic times for the conformational changes [5, ∗ On leave from the A.F. Ioffe Institute, St. Petersburg, Russia. E-mail: [email protected] http://arxiv.org/abs/0704.1672v1 6, 7] and for fragmentation [8]. The potential energy surface of a macromolecular system can be used for studying the thermodynamical processes in the system such as phase transitions [9]. In proteins, the potential energy surface is related to one of the most intriguing problems of protein physics: protein folding [9, 10, 11, 12, 13]. The rate constants for complex biochemical reactions can also be established from the analysis of the potential energy surface [14, 15]. The calculation of the potential energy surface and molecular dynamics simulations often implies the evaluation of pairwise interactions. The direct method for evaluating these potentials is proportional to ∼ N2, where N is the number of particles in the system. This places a severe restraint on the treatable size of the system. During the last two decades many different methods have been suggested which provide a linear dependence of the computational cost with respect to N [16, 17, 18, 19, 20, 21]. The most widely used algorithm of this kind is the fast multipole method (FMM) [17, 18, 19, 20, 21, 22, 23, 24]. The critical size of the system at which this method becomes computationally faster than the exact method is accuracy dependent and is very sensitive to the slope in the N dependence of the computational cost. In refs. [18, 20, 25] critical sizes ranging from N ≈ 300 to N ≈ 30000 have been reported. Many discrepancies of the estimates in the critical size arise from differences in the effort of optimizing the algorithm and the underlying code. However, it is also important to optimize the methods themselves with respect to the required accuracy. The FMM is based on the systematic organization of multipole representations of a local charge distribution, so that each particle interacts with local expansions of the potential. Originally FMM was introduced in [21] by Greengard and Rokhlin. Later, Greengard’s method has been implemented in various forms. Schmidt and Lee [20] have produced a version based upon the spherical multipoles for both periodic and nonperiodic systems. Zhou and Johnson have implemented the FMM for use on parallel computers [26], while Board et al have reported both serial and parallel versions of the FMM [25]. Ding et al introduced a version of the FMM that relies upon Cartesian rather than spher- ical multipoles [18], which they applied to very large scale molecular dynamics calculations. Additionally they modified Greengard’s definition of the nearest neighbors to increase the proportion of interactions evaluated via local expansions. Shimada et al also developed a cartesian based FMM program [27], primarily to treat periodic systems described by molec- ular mechanics potentials. In both cases only low order multipoles were employed, since high accuracy was not sought. In the present paper we suggest a new method for calculating the interaction energy between macromolecules. Our method also provides a linear scaling of the computational costs with the size of the system and is based on the multipole expansion of the potential. However, the underlying ideas are quite different from the FMM. Assuming that atoms from different macromolecules interact via a pairwise Coulomb potential, we expand the potential around the centers of the molecules and build a two center multipole expansion using bipolar harmonics algebra. Finally, we obtain a general expression which can be used for calculating the energy and forces between the fragments. This approach is different from the one used in the FMM, where the so-called translational operators were used to expand the potential around a shifted center. Note that the final expression, which we suggest in our theory was not discussed before within the FMM. Similar expressions were discussed since the earlier 50’s (see e.g. [28, 29, 30, 31]). In these papers the two center multipole expansion was considered as a new form of Coulomb potential expansion, but the expansion was never applied to the study of macromolecular systems. We consider the interaction of macromolecules via Coulomb potential since this is the only long-range interaction in macromolecules, which is important for the description of the potential energy surface at large distances. Other interaction terms in macromolecular systems are of the short-range type and become important when macromolecules get close to each other [8]. At large distances these terms can be neglected. In the present paper we show that the method based on the two center multipole ex- pansion can be used for computing the interaction energy between complex macromolecular systems. In section II we present the formalism which lies behind the two center multipole expansion method. In subsection IIIA we analyze the behavior of the computation cost of this method and establish the critical sizes of the system, when the two center multipole expansion method demands less computer time than the exact energy calculation approach. In subsection IIIB we compare the results of our calculation with the results obtained within the framework of the FMM. In section IV we discuss the accuracy of the two center multipole expansion method. II. TWO CENTER MULTIPOLE EXPANSION METHOD In this section we present the formalism, which underlies the two center multipole expan- sion method, which will be further referred to as the TCM method. Let us consider two multi atomic systems, which we will denote as A and B. The pairwise Coulomb interaction energy of those systems can be written as follows: ∣R0 + r , (1) where NA and NB are the total number of atoms in systems A and B respectively, qi and qj are the charges of atoms i and j from the system A and B respectively, R0 is the vector interconnecting the center of system A with the center of system B, rAi and r j are the vectors describing the position of charges i and j with respect to the centers of the system A and B respectively. The centers of both systems can be any suitable points of each of the molecules. It is natural to define them as the centers of mass of the corresponding systems, but in some cases another choice might be more convenient (see for example [8], where we have applied the TCM method for studying fragmentation of alanine dipeptide). Expression (1) can be expanded into a series of spherical harmonics. The expansion depends on the vectors R0, r i and r j . In the present paper we consider the case when |R0| > ∣ (2) holds for all i and j. This particular case is important, because it describes well separated charge distributions, and can be used for modeling the interaction between complex objects at large distances. In this case the expansion of (1) reads as [32]: ∣R0 + r = qiqj 2L+ 1 ∣rBj − rAi RL+10 Y ∗LM YLM (ΘR0,ΦR0) (3) According to [32] the function j − rAi Y ∗LM can be expanded into series of bipolar harmonics: j − rAi Y ∗LM 4π(2L+ 1)! l1,l2=0 l1+l2=L (−1)l2 (2l1 + 1)!(2l2 + 1)! Yl1(ΘrA )⊗ Yl2(ΘrB where a bipolar harmonic is defined as follows: (Yl1(Θr1 ,Φr1) ⊗ Yl2(Θr2,Φr2))LM = m1,m2 CLMl1m1l2m2Yl1m1(Θr1 ,Φr1)Yl2m2(Θr2,Φr2). (5) Here CLMl1m1l2m2 are the Clebsch-Gordan coefficients, which can be transformed to the 3j- symbol notation as follows: CLMl1m1l2m2 = (−1) l1−l2+M 2L+ 1 l1 l2 L m1 m2 −M  (6) Using equations (4),(6) and (5) we can rewrite expansion (3) as follows: ∣R0 + r = qiqj l1,l2=0 l1+l2=L l1,l2 m1=−l1 m2=−l2 (−1)l1+M (4π)3(2L)! (2l1 + 1)!(2l2 + 1)! l1 l2 L m1 m2 −M RL+10 Yl1m1(ΘrAi ,Φr )Yl2m2(ΘrBj ,Φr )Y ∗LM (ΘR0,ΦR0) (7) The multipole moments of systems A and B are defined as follows: QAl1m1 = 2l1 + 1 Yl1m1(ΘrA ) (8) QBl2m2 = 2l2 + 1 Yl2m2(ΘrB Summing equation (7) over i and j, and accounting only for the first Lmax multipoles in both systems, we obtain: Umult = l1,l2=0 l1+l2=L l1,l2 m1=−l1 m2=−l2 (−1)l1+M RL+10 4π(2L)! (2l1)!(2l2)! l1 l2 L m1 m2 −M QAl1m1Q Y ∗LM (ΘR0 ,ΦR0) . (9) This expression describes the electrostatic energy of the system in terms of a two center multipole expansion. Note, that this expansion is only valid when the condition R0 > r holds for all i and j, otherwise more sophisticated expansions have to be considered, which is beyond the scope of the present paper. Summation in equation (9) is performed over l1, l2 ∈ [0...Lmax]; m1 ∈ [−l1...l1]; m2 ∈ [−l2...l2], and the condition M = m1 +m2 holds. Lmax is the principal multipole number, which determines the number of multipoles in the expansion. III. COMPUTATIONAL EFFICIENCY A. Comparison with direct Coulomb interaction method In this section we discuss the computational efficiency of the TCM method. For this purpose we have analyzed the time required for computing the Coulomb interaction energy between two systems of charges and the time required for the energy calculation within the framework of the TCM method for different system sizes, and for different values of the principal multipole number. For the study of the computational efficiency of the TCM method we have considered the interaction between two systems (we denote them as A and B) of randomly distributed charges, for which the condition eq. (2) holds. The charges in both systems were randomly distributed within the spheres of radiiRA = 1.0·N1/3A and RB = 1.0·N B respectively and the distance between the centers of mass of the two systems was chosen as R0 = 3/2(RA +RB). The computational time needed for the energy calculations is proportional to the number of operations required. Thus, the time needed for the Coulomb energy calculation (CE calculation) can be estimated as: τCoul = αCoulNANB ∼ N2 (10) where αCoul is a constant depending on the computer processor power and on the efficiency of the code, NA ∼ NB ∼ N . From equation (10) it follows that the computational cost of the CE calculation grows proportionally to the second power of the system size. For large systems the TCM method becomes more efficient because it provides a linear scaling with the system size. The time needed for the energy calculation reads as follows: τmult(Lmax) = βN + m1=−l1 m2=−l2 (NAτl1,m1 +NBτl2,m2) ≈ (11) ≈ αmultLmax(1 + Lmax)3N, where the first term, βN , corresponds to the computer time needed for allocating arrays in memory and tabulating the computationally expensive functions like cos(Φ) and exp(imΦ). τl,m is the time needed for evaluation of the spherical harmonic at given l and m, and αmult is a numerical coefficient, which depends on the processor power and on the efficiency of the code. In general it is different from αCoul. In Fig. 1 we present the dependencies of the computer time needed for the CE calculation (squares) and for the computation of energy within the TCM method for different values of the principal multipole number as a function of system size. This data was obtained on a 1.8 GHz 64-bit AMD Opteron-244 computer. From Fig. 1 it is clear that the time needed for the CE calculation has a prominent parabolic trend that is consistent with the analytical expression (10). The fitting expression which describes this dependance is given in the first row of Tab. I. At large N the N2 term becomes dominant and the other two terms can be neglected. Thus, αCoul ≈ 4.46 · 10−8 (sec). The fitting expressions which describe the time needed for the energy computations within the TCM method at different values of the principal multipole number are given in Tab. I, rows 2-10. These expressions were obtained by fitting the data shown in Fig. 1. Note the linear dependence on N . The numerical coefficient in all expressions correspond to the factor αmultLmax(1 + Lmax) 3 in equation (11). The fitting expressions in Tab. I were obtained by fitting of data obtained for systems with large number of particles (see Fig. 1). Therefore these expressions are applicable when N ≫ 1. From equations presented in Tab. I it is possible to determine the critical system sizes at which the TCM method becomes less computer time demanding then the CE calculation. FIG. 1: Time needed for energy calculation as a function of the system size. The critical system sizes calculated for different principal multipole numbers are shown in the third column of Tab. I. These sizes correspond to the intersection points of the parabola describing the time needed for the CE calculation with the straight lines describing the computational time needed for the TCM method. In Fig. 1 one can see six intersection points for Lmax = 2− 7. From equation (11), it follows that computation time of the energy within the framework of the TCM method grows as the power of 4 with increasing Lmax. To stress this fact, in Fig. 2 we present the dependencies of the computation time obtained within the TCM method at different system sizes as a function of principal multipole number. All curves shown in Fig. 2 can be perfectly fitted by the analytical expression (11). In the inset to Fig. 2, we plot the dependence of the fitting coefficient αmult as a function of the system size. From this plot it is seen that αmult varies only slightly for all system sizes considered, being equal to (1.982± 0.015) · 10−7 (sec). Thus, the expression for the time needed for the energy calculation within the framework TABLE I: Fitting expressions for the computational time needed for the CE calculation and for the energy computation within the TCM method at different values of the principal multipole number, Lmax (second column). System sizes, for which the Coulomb energy calculation becomes more computer time demanding at a given value of Lmax are shown in the third column. Lmax τ(N) (sec.) Nmax Coulomb 0.11736 − 0.0002N + 4.6768 · 10−8N2 - 2 −0.01986 + 3.0 · 10−5N 4223 3 −0.03159 + 5.0 · 10−5N 4662 4 −0.04714 + 1.0 · 10−4N 5809 5 −0.16054 + 2.1 · 10−4N 8026 6 −0.14710 + 3.7 · 10−4N 11704 7 −0.59675 + 7.4 · 10−4N 19308 8 −0.35383 + 10.9 · 10−4N 27212 9 −1.15856 + 1.9 · 10−3N 44286 10 −0.83688 + 2.71 · 10−3N 61892 of the TCM method reads as: τmult(Lmax) ≈ 1.98 · 10−7Lmax(1 + Lmax)3N. (12) Note, that αmult = 1.98 · 10−7 (sec) is larger than αCoul ≈ 4.46 · 10−8 (sec), since in one turn of the TCM method more algebraic operations have to be done, than in one turn of the CE calculation. From the analysis performed in this section it is clear that the TCM method can give a significant gain in the computation time. However, at larger principal multipole numbers (Lmax = 8, 9, 10) this method can compete with the CE calculation only at system sizes greater than 27000-61000 atoms. The accounting for higher multipoles is necessary if the distance between two interacting systems becomes comparable to the size of the systems. In the next section we discuss in detail the accuracy of the TCM method and identify situations in which higher multipoles should be accounted for. FIG. 2: Time needed for the calculation of energy of the systems of different sizes computed within the framework of the TCM method as a function of the principal multipole number Lmax. In the inset we plot αmult as a function of the system size. B. Comparison with the fast multipole method The fast multipole method (FMM) [21, 22, 23] is a well known method for calculating the electrostatic energy in a multiparticle system, which provides a linear scaling of the computing time with the system size. In order to stress the computational efficiency of the TCM method in this section we compare the time required for the energy calculation within the framework of the FMM and using the TCM method. To perform such a comparison we used an adaptive FMM library, which has been im- plemented for the Coulomb potential in three dimensions [24, 33]. We have generated two random charge distributions of different size and calculated the interaction energies between them as well as the required computation time using the FMM and the TCM methods. As in the previous section the charges in both systems were randomly distributed within the FIG. 3: Time needed for the calculation of the interaction energy between two systems as a function the total number of particles calculated within the framework of the TCM method (triangles) and within the framework of the FMM (squares). In the upper and lower insets we plot the relative error of the FMM and of the TCM methods as a function of the system size respectively. spheres of radii RA = 1.0 ·N1/3A and RB = 1.0 ·N B respectively and the distance between the center of mass of the two systems was chosen as R0 = 3/2(RA +RB). In Fig. 3 we present the comparison of the computer time needed for the FMM calculation (squares) and for the computation of energy within the TCMmethod (triangles) as a function of system size. These data were obtained on an Intel(R) Xeon(TM) CPU 2.40GHz computer. In the upper and lower insets of Fig. 3 we show the relative error of the FMM and of the TCM methods as a function of the system size respectively, which is defined as follows: ηmethod = |Ucoul − Umethod| |Ucoul| · 100%. (13) Here method indicates the FMM or the TCM methods. For comparing the efficiency of the two methods we have considered different charge distributions within the size range of 100 to 10000 particles. Each point in Fig. 3 corresponds to a particular charge distribution. For each system size ten different charge distributions were used. The time of the FMM calculation depends on the charge distribution, as is clearly seen in Fig. 3. Note that for a given system size the calculation time of the FMM can change by more than a factor of 5, depending on the charge distribution (see points for N = 10000 in Fig. 3). For all system sizes FMM requires some minimal computer time for calculating the energy of the system, which increases with the growth of system size (see Fig. 3). The comparison of the minimal FMM computation time with the computation time required for the TCM method shows that the TCM method appears to be significantly faster than the FMM. For N = 10000 FMM requires at least 2.15 seconds to compute the energy, while TCM method requires 0.53 seconds, being approximately 4 times faster. The results of the TCM method calculation shown in Fig. 3, were obtained for Lmax = 2. The analysis of relative errors presented in the inset to Fig. 3 shows that with this principal multipole number it is possible to calculate the energy between two systems with an error of less than 1 % for almost arbitrary charge distribution. Note that for the same charge distributions the error of the FMM is much more, being about 5 % in almost all of the considered systems. This allows us to conclude that the TCM method is more efficient and more accurate than the classical FMM. It is important to mention that in the traditional implementation, FMM calculates the total electrostatic energy of the system while TCM method was developed for studying interaction energy between system fragments. It is possible to modify the FMM to study only interaction energies between different parts of the system. However, the computation cost of the modified FMM is expected to be higher than of the TCMmethod. This happens because, within the framework of the modified FMM method, the field created by one fragment of the system should be expanded in the multipole series and the interactions of the resulting multipole moments with the charges from another fragment should be calculated. Thus the computation cost of this method will be proportional to NA ·NB, where NA and NB are the number of particles in two fragments, while the TCM method is proportional to NA +NB. The computation cost of the modified version of the FMM depends quadratically on the size of the system, because in this method the interacting fragments should be considered as two independent cells, while traditional FMM uses a hierarchical subdivision of the whole system into cells to achieve linear scaling. So far we have considered only the interaction between two multi particle systems in vacuo, and demonstrated the efficiency of the TCM method in this case, although the TCM method can also be applied to the larger number of interacting systems. The study of structure and dynamics of biomolecular systems consisting of several components (i.e an ensemble of proteins, DNA, macromolecules in solution) is a separate topic, which is beyond the scope of this paper and deserves a separate investigation. IV. ACCURACY OF THE TCM METHOD. POTENTIAL ENERGY SURFACE FOR PORCINE PANCREATIC TRYPSIN/SOYBEAN TRYPSIN INHIBITOR COMPLEX. We have calculated the interaction energy between two proteins within the framework of the TCM method and compared it with the exact Coulomb energy value. On the basis of this comparison we have concluded about the accuracy of the TCM method. In the present paper we have studied the interaction energy between the porcine pan- creatic trypsin and the soybean trypsin inhibitor proteins (Protein Data Bank (PDB) [36] entry 1AVW [37]). Trypsins are digestive enzymes produced in the pancreas in the form of inactive trypsinogens. They are then secreted into the small intestine, where they are activated by another enzyme into trypsins. The resulting trypsins themselves activate more trypsinogens (autocatalysis). Members of the trypsin family cleave proteins at the carboxyl side (or ”C-terminus”) of the amino acids lysine and arginine. Porcine pancreatic trypsin is a archetypal example. Its natural non-covalent inhibitor (porcine pancreatic trypsin inhibitor) inhibits the enzyme’s activity in the pancreas, protecting it from self-digestion. Trypsin is also inhibited non-covalently by the soybean trypsin inhibitor from the soya bean plant, although this inhibitor is unrelated to the porcine pancreatic trypsin inhibitor family of inhibitors. Although the biological function of the soybean trypsin inhibitor is mostly unknown it is assumed to help defend the plant from insect attack by interfering with the insect digestive system. The structure of both proteins is shown in Fig. 4. The coordinate frame used for our computations is marked in the figure. This coordinate frame is consistent with the standard coordinate frame used in the PDB. FIG. 4: Structure of the porcine pancreatic trypsin and soybean trypsin inhibitor with the coor- dinate frame used for the energy computation. Figure has been rendered with help of the VMD visualization package [38] We use this particular example as a model system in order to demonstrate the possible use of the TCM method. Therefore environmental effects are omitted and we consider only the protein-protein interaction in vacuo. The porcine pancreatic trypsin and the soybean trypsin inhibitor include 223 and 177 amino acids respectively. Both proteins include 5847 atoms. Thus for such system size the TCM method is faster than the CE calculation if Lmax ≤ 4 (see Tab. I). We have calculated the interaction energy between the porcine pancreatic trypsin and soybean trypsin inhibitor as a function of distance between the centers of masses of the fragments, ~R0, and the angle Θ, which is determined as the angle between the x-axis and the vector ~R0 (see Fig. 4). We have assumed that the porcine pancreatic trypsin is fixed in space at the center of the coordinate frame and have restricted ~R0 to the (xy)-plane. Of course, the two degrees of freedom considered are not sufficient for a complete description of the mutual interaction between the two systems. For this purpose at least six degrees of freedom are needed. However for our example of the energy calculation of the porcine pancreatic trypsin/soybean trypsin inhibitor complex within the framework of the TCM method the two coordinates ~R0 and Θ are sufficient. The interaction energy of the porcine pancreatic trypsin with the soybean trypsin in- hibitor as the function of R0 and Θ calculated within the framework of the TCM method is shown in Fig. 5. The Coulomb interaction energy between the two proteins is shown in the top-left plot. In [8] it has been shown that the interaction energy between two well sepa- rated biological fragments arises mainly due to the Coulomb forces. In the present paper we consider R0 ∈ [58, 100] Å and Θ ∈ [0, 360]◦, at which condition (2) holds and both proteins can be considered as separated. This means that the potential energy surface shown in the top-left plot of Fig. 5 describes the interaction energy between the porcine pancreatic trypsin and the soybean trypsin inhibitor on the level of accuracy of 90 % at least. The top-left plot of Fig. 5 shows that one can select several characteristic regions on the potential energy surface marked with numbers 1-4. The corresponding configurations (states) of the system are shown in Fig. 6. The potential energy surface is determined by the Coulomb interactions between atoms, thus at large distances it raises and asymptotically approaches to zero. State 1 has the maximum energy within the considered part of the potential energy surface because this state corresponds to the largest contact separation distance between porcine pancreatic trypsin and the soybean trypsin inhibitor being equal to 54.8 Å. At smaller distances the potential energy decreases due to the attractive forces acting between the two proteins. State 2 corresponds to the minimum on the potential energy sur- face. It arises because a positively charged polar arginine (R125) from the porcine pancreatic trypsin approaches the negatively charged site of the soybean trypsin inhibitor, which in- cludes negatively charged polar amino acids glutamic acid (E549) and aspartic acid (D551) (see state 2 in Fig. 6). The strong attraction between the amino acids leads to the formation of a potential well on the potential energy surface. This observation is essential for dynamics of the attachment process of two proteins, because it establishes the most probable angle at which the proteins stick in the (xy)-plane of the considered coordinate frame (Θ = 192◦). States 3 and 4 correspond to the saddle points on the potential energy surface and have energies higher than state 2. They are formed because at these configurations two positively FIG. 5: The interaction energies of the porcine pancreatic trypsin with the soybean trypsin inhibitor calculated as the function of R0 and Θ (see Fig. 4) within the framework of the TCM method at different values of the principal multipole number, Lmax. The principal multipole number is given above the corresponding image. The result of the CE calculation is shown in the top left plot. charged polar amino acids from the two proteins become closer in space providing a source of a local repulsive force. In state 3 these amino acids are lysines (K145 and K665) (see state 3 in Fig. 6), and in state 4 these are arginines (R62 and R563)(see state 4 in Fig. 6). FIG. 6: Relative orientations of the porcine pancreatic trypsin and the soybean trypsin inhibitor, corresponding to the selected points on the potential energy surface presented in Fig. 5. Below each image we give the corresponding values of R0 and Θ. Some important amino acids are marked according to their PDB id. Figure prepared with help of the VMD visualization package [38] In the top-right plot of Fig. 5 we present the potential energy surface obtained within the framework of the TCM method with Lmax = 2, i.e. with accounting for up to the quadrupole-quadrupole interaction term in the multipole expansion (9). From the figure it is seen that the TCM method describes correctly the major features of the potential energy landscape (i.e. the position of the minimum and maximum as well as their relative energies). However, the minor details of the landscape, such as the saddle points 3 and 4 (see top-left plot of Fig. 5) are missed. The relative error of the TCM method can be defined as follows: η(Lmax)(R0,Θ) = |Ucoul(R0,Θ)− ULmaxmult (R0,Θ)| |Ucoul(R0,Θ)| · 100%, (14) where Ucoul(R0,Θ) and U mult (R0,Θ) are the Coulomb energy and the energy calculated at given values of R0 and Θ within the TCM method respectively. In the top-left plot of Fig. 7 we present the relative error calculated according to (14) for Lmax = 2. From this plot it is clear that significant deviation from the exact result arise at Θ ∼ 50 − 60◦, 140 − 150◦, 245◦, 300− 310◦ and 350◦. The discrepancy at Θ ∼ 50− 60◦, Θ ∼ 300− 310◦ and Θ ∼ 350◦ arises because the saddle points 3 and 4, can not be described within the framework of TCM method with Lmax = 2. The discrepancy at Θ ∼ 140 − 150◦ and Θ ∼ 245◦ is due to the error in the calculation of the slopes of minimum 2 at R0 = 58 Å and Θ = 198 FIG. 7: Relative error of the interaction energies of the porcine pancreatic trypsin with the soybean trypsin inhibitor calculated as the function of R0 and Θ within the framework of the TCM method at different values of the principal multipole number, Lmax. The principal multipole number is given above the corresponding image. It is worth noting that the relative error of the TCM method with Lmax = 2 is less than 10 %. With increasing distance between the proteins, the relative error decreases, and becomes less than 5 % at R0 ≥ 72 Å and less than 3 % at R0 ≥ 86 Å. This means that already at Lmax = 2 the TCM method reproduces with a reasonable accuracy the essential features of the potential energy landscape. This observation is very important, because TCM method with Lmax = 2 requires less computer time then the CE calculation already at N = 4223 (see Tab. I). Thus, the TCM method can be used for the identification of major minima and maxima on the potential energy surface of macromolecules and modeling dynamics of complex molecular systems. Accounting for higher multipoles in the multipole expansion (9) leads to a more accurate calculation of the potential energy surface. In the second row of Fig. 5 we present the potential energy surfaces obtained within the framework of the TCM method with Lmax = 4 and 6. From these plots it is seen that all minor details of the Coulomb potential energy surface, such as the saddle points 3 and 4 are reproduced correctly. Figure 7 shows that the TCM method with Lmax = 4 gives the maximal relative error of about 5 % at R0 = 58 Å and Θ = 75◦, in the vicinity of the saddle point 3. The relative errors in the vicinity of the saddle point 4 and minimum 2 are equal to 4 % and 1 % respectively. The error becomes less then 1 % for all values of angle Θ at R0 ≥ 70 Å. For Lmax = 6, the largest relative error is equal to 1.5 % at R0 = 58 Å and Θ = 340 ◦ (saddle point 4), becoming less then 1 % at R0 ≥ 61 Å. By accounting for the higher multipoles in the multipole expansion (9) one can increase the accuracy of the method. Thus, with Lmax = 8 and 10 it is possible to calculate the potential energy surface with the error less then 1 % (see bottom row in Fig. 5 and Fig. 7). Although the time needed for computing the potential energy surfaces with Lmax = 8 and 10 is larger than the time needed for computing the Coulomb energy directly (see Tab. I), we present these surfaces in order to stress the convergence of the TCM method. V. CONCLUSION In the present paper we have proposed a new method for the calculation of the Coulomb interaction energy between pairs of macromolecular objects. The suggested method provides a linear scaling of the computational costs with the size of the system and is based on the two center multipole expansion of the potential. Analyzing the dependence of the required computer time on the system size, we have established the critical sizes at which our method becomes more efficient than the direct calculation of the Coulomb energy. The comparison of efficiency of the TCM method with the efficiency of FMM allows us to conclude that the TCM method has proved to be faster and more accurate than the classical The method based on the two center multipole expansion can be used for the efficient computation of the interaction energy between complex macromolecular systems. To de- termine that we have considered the interaction between two proteins: porcine pancreatic trypsin and the soybean trypsin inhibitor. The accuracy of the method has been discussed in detail. It has been shown that accounting of only four multipoles in both proteins gives an error in the interaction energy less than 5 %. The TCM method is especially useful for studying dynamics of rigid molecules, but it can also be adopted for studying dynamics of flexible molecules. In this work we have developed a method for the efficient calculation of the interaction energy between pairs of large multi particle systems, e.g. macromolecules, being in vacuo. The investigation of biomolecular systems consisting of several components (i.e complexes of proteins, DNA, macromolecules in solution) and the extension of the TCM method for these cases deserves a separate investigation. If a system of interest consists of several interacting molecules being placed in a solution, one can use the TCM method to describe the interaction between the molecules and then to take account of the solution as implicit solvent. This can be achieved using for example the formalism of the Poisson-Bolzmann [34, 35], similar to how it was implemented for the description of the antigen-antibody binding/unbinding process [14, 15]. The other possibility is to split the whole system into boxes and account for the solvent explicitly by calculating the interactions between the boxes and the molecules of interest. This can be achieved by using the TCM method or a combination of the FMM and the TCM methods. In this case the FMM can be used for the calculation of the resulting effective multipole moment of the solvent, while the TCM method is much better suitable for the description of the macromolecules energetics and dynamics. Note that all of the suggested methodologies provide linear scaling of the computation time on the system size. The results of this work can be utilized for the description of complex molecular systems such as viruses, DNA, protein complexes, etc and their dynamics. Many dynamical features and phenomena of these systems are caused by the electrostatic interaction between their various fragments and thus the use of the two center multipole expansion method should give a significant gain in their computation costs. VI. ACKNOWLEDGEMENTS This work is partially supported by the European Commission within the Network of Excellence project EXCELL and by INTAS under the grant 03-51-6170. We are grateful to Dr. Paul Gibbon for providing us with the FMM code. 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0704.1673

Holographic formula for Q-curvature

HOLOGRAPHIC FORMULA FOR Q-CURVATURE C. ROBIN GRAHAM AND ANDREAS JUHL Introduction In this paper we give a formula for Q-curvature in even-dimensional conformal geometry. The Q-curvature was introduced by Tom Branson in [B] and has been the subject of much research. There are now a number of characterizations of Q- curvature; see for example [GZ], [FG1], [GP], [FH]. However, it has remained an open problem to find an expression for Q-curvature which, for example, makes explicit the relation to the Pfaffian in the conformally flat case. Theorem 1. The Q-curvature of a metric g in even dimension n is given by (0.1) 2ncn/2Q = nv (n) + n/2−1∑ (n− 2k)p∗2kv (n−2k), where cn/2 = (−1) n/2 [2n(n/2)!(n/2− 1)!] Here the v(2j) are the coefficients appearing in the asymptotic expansion of the volume form of a Poincaré metric for g, the differential operators p2k are those which appear in the expansion of a harmonic function for a Poincaré metric, and p∗2k denotes the formal adjoint of p2k. These constructions are recalled in §1 below. We refer to the papers cited above and the references therein for background about Q-curvature. Each of the operators p∗2k for 1 ≤ k ≤ n/2− 1 can be factored as p 2k = δqk, where δ denotes the divergence operator with respect to g and qk is a natural operator from functions to 1-forms. So the second term on the right hand side is the divergence of a natural 1-form. In particular, integrating (0.1) over a compact manifold recovers the result of [GZ] that (0.2) 2cn/2 Qdvg = v(n)dvg. This quantity is a global conformal invariant; the right hand side occurs as the coeffi- cient of the log term in the renormalized volume expansion of a Poincaré metric (see [G]). The work of the first author was partially supported by NSF grant DMS-0505701. The work of the second author was supported by SFB 647 “Raum-Zeit-Materie” of DFG. http://arxiv.org/abs/0704.1673v1 2 C. ROBIN GRAHAM AND ANDREAS JUHL As we also discuss in §1, if g is conformally flat then v(n) = (−2)−n/2(n/2)!−1Pff , where Pff denotes the Pfaffian of g. So in the conformally flat case, Theorem 1 gives a decomposition of the Q-curvature as a multiple of the Pfaffian and the divergence of a natural 1-form. A general result in invariant theory ([BGP]) establishes the existence of such a decomposition, but does not produce a specific realization. We refer to (0.1) as a holographic formula because its ingredients come from the Poincaré metric, involving geometry in n + 1 dimensions. Our proof is via the char- acterization of Q-curvature presented in [FG1] in terms of Poincaré metrics; in some sense Theorem 1 is the result of making explicit the characterization in [FG1]. How- ever, passing from the construction in [FG1] to (0.1) involves a non-obvious appli- cation of Green’s identity. The transformation law of Q-curvature under conformal change, probably its most fundamental property, is not transparent from (0.1), but it is from the characterization in [FG1]. In §2, we derive another identity involving the p∗2kv (n−2k) which is used in §3 and we discuss relations to the paper [CQY]. In §3, we describe the relation between holographic formulae for Q-curvature and the theory of conformally covariant families of differential operators of [J], and in particular explain how this theory leads to the conjecture of a holographic formula for Q. We are grateful to the organizing committee of the 2007 Winter School ’Geome- try and Physics’ at Srńı, particularly to Vladimir Souc̆ek, for the invitation to this gathering, which made possible the interaction leading to this paper. We dedicate this paper to the memory of Tom Branson. His insights have led to beautiful new mathematics and have greatly influenced our own respective work. 1. Derivation Let g be a metric of signature (p, q) on a manifold M of even dimension n. In this paper, by a Poincaré metric for (M, g) we will mean a metric g+ on M × (0, a) for some a > 0 of the form (1.1) g+ = x −2(dx2 + gx), where gx is a smooth 1-parameter family of metrics on M satisfying g0 = g, such that g+ is asymptotically Einstein in the sense that Ric(g+) + ng+ = O(x n−2) and trg+(Ric(g+)+ng+) = O(x n+2). Such a Poincaré metric always exists and gx is unique up addition of a term of the form xnhx, where hx is a smooth 1-parameter family of symmetric 2-tensors on M satisfying trg(h0) = 0 on M . The Taylor expansion of gx is even through order n and the derivatives (∂x) 2kgx|x=0 for 1 ≤ k ≤ n/2− 1 and the trace trg((∂x) ngx|x=0) are determined inductively from the Einstein condition and are given by polynomial formulae in terms of g, its inverse, and its curvature tensor and covariant derivatives thereof. See [GH] for details. HOLOGRAPHIC FORMULA FOR Q-CURVATURE 3 The first ingredient in our formula for Q-curvature consists of the coefficients in the expansion of the volume form (1.2) dvg+ = x −n−1dvgxdx. Because the expansion of gx has only even terms through order n, it follows that dvgx = det gx det g = (1 + v(2)x2 + · · ·+ v(n)xn + · · · )dvg, (1.3) where each of the v(2k) for 1 ≤ k ≤ n/2 is a smooth function on M expressible in terms of the curvature tensor of g and its covariant derivatives. Set v(0) = 1. The second ingredient in our formula is the family of differential operators which appears in the expansion of a harmonic function for the metric g+. Given f ∈ C ∞(M), one can solve formally the equation ∆g+u = O(x n) for a smooth function u such that u|x=0 = f , and such a u is uniquely determined modulo O(x n). The Taylor expansion of u is even through order n − 2 and these Taylor coefficients are given by natural differential operators in the metric g applied to f which are obtained inductively by solving the equation ∆g+u = O(x n) order by order. See [GZ] for details. We write the expansion of u in the form (1.4) u = f + p2f x 2 + · · ·+ pn−2f x n−2 +O(xn); then p2k has order 2k and its principal part is (−1) Γ(n/2− k) 22k k! Γ(n/2) ∆k. (Our convention is ∆ = −∇i∇i.) Set p0f = f . We remark that the volume coefficients v(2k) and the differential operators p2k also arise in the context of an ambient metric associated to (M, [g]). If an ambient metric is written in normal form relative to g, then the same v(2k) are coefficients in the expansion of its volume form, and the same operators p2k appear in the expansion of a harmonic function homogeneous of degree 0 with respect to the ambient metric. Let g+ be a Poincaré metric for (M, g). In [FG1] it is shown that there is a unique solution U mod O(xn) to (1.5) ∆g+U = n +O(x n+1 log x) of the form (1.6) U = log x+ A+Bxn log x+O(xn) , A,B ∈ C∞(M × [0, a)) , A|x=0 = 0 . Also, A mod O(xn) is even in x and is formally determined by g, and (1.7) B|x=0 = −2cn/2Q. 4 C. ROBIN GRAHAM AND ANDREAS JUHL The proof of (1.7) presented in [FG1] used results from [GZ] about the scattering matrix, so is restricted to positive definite signature. However, a purely formal proof was also indicated in [FG1]. Thus (1.7) holds in general signature. Proof of Theorem 1. Let g+ be a Poincaré metric for g and let U be a solution of (1.5) as described above. Let f ∈ C∞(M) have compact support. Let u be a solution of ∆g+u = O(x n) with u|x=0 = f ; for definiteness we take u to be given by (1.4) with the O(xn) term set equal to 0. Let 0 < ǫ < x0 with ǫ, x0 small. Consider Green’s identity (1.8) ǫ<x<x0 (U∆g+u− u∆g+U) dvg+ = (U∂νu− u∂νU) dσ, where ν denotes the inward normal and dσ the induced volume element on the bound- ary, relative to g+. Both sides have asymptotic expansions as ǫ→ 0; we calculate the coefficient of log ǫ in these expansions. Using the form of the expansion of U and the fact that ∆g+u = O(x n), one sees that the expansion of U∆g+u dvg+ has no x −1 term, so ǫ<x<x0 U∆g+u dvg+ has no log ǫ term. Using (1.2), (1.3), (1.4), and (1.5), one finds that the log ǫ coefficient of ǫ<x<x0 u∆g+U dvg+ is (1.9) n n/2−1∑ v(n−2k)p2kf dvg. On the right hand side of (1.8), is independent of ǫ, and (U∂νu− u∂νU) dσ = ǫ (U∂xu− u∂xU) dvgǫ. A log ǫ term in the expansion of this quantity can arise only from the log x or xn log x terms in the expansion of U . Substituting the expansions, one finds without difficulty that the log ǫ coefficient is n/2−1∑ 2kv(n−2k)p2kf − nBf  dvg. Equating this to (1.9), using (1.7), and moving all derivatives off f gives the desired identity. � Since ∆g+1 = 0, it follows that p2k1 = 0 for 1 ≤ k ≤ n/2− 1. Thus these p2k have no constant term, so p∗2k = δqk for some natural operator qk from functions to 1-forms, where δ denotes the divergence with respect to the metric g. So in (0.1), the second term on the right hand side is the divergence of a natural 1-form. As mentioned in the introduction, integration gives (0.2). The proof of Theorem 1 presented above in the special case u = 1 is precisely the proof of (0.2) presented in [FG1]. HOLOGRAPHIC FORMULA FOR Q-CURVATURE 5 Theorem 1 provides an efficient way to calculate the Q curvature. Solving for the beginning coefficients in the expansion of the Poincaré metric and then expanding its volume form shows that the first few of the v(2k) are given by: v(2) = − v(4) = (J2 − |P |2) v(6) = P ijBij + 3J |P | 2 − J3 − 2P ijPi where Pij = Rij − 2(n− 1) 2(n− 1) = P ii Bij = Pij,k k − Pik,j k − P klWkijl and Wijkl denotes the Weyl tensor. Similarly, one finds that the operators p2 and p4 are given by: −2(n− 2)p2 = ∆ 8(n− 2)(n− 4)p4 = ∆ 2 + 2J∆+ 2(n− 2)P ij∇i∇j + (n− 2)J, (1.10) For n = 2, Theorem 1 states Q = −2v(2) = 1 R. For n = 4, substituting the above into Theorem 1 gives: Q = 2(J2 − |P |2) + ∆J, and for n = 6: Q = 8P ijBij + 16P kPkj − 24J |P | 2 + 8J3 +∆2J + 4∆(J2) + 8(P ijJ,i),j − 4∆(|P | In the formula for n = 6, the first line is (12c3) −16v(6) and the second line is (12c3) 4p∗2v (4) + 2p∗4v . Details of these calculations will appear in [J]. The expansion of the Poincaré metric g+ was identified explicitly in the case that g is conformally flat in [SS]. (Since we are only interested in local considerations, by conformally flat we mean locally conformally flat.) The two dimensional case is somewhat anomalous in this regard, but the identification of Q curvature is trivial when n = 2, so we assume n > 2 for this discussion. The conclusion of [SS] is that if g is conformally flat and n > 2 (even or odd), then the expansion of the Poincaré metric terminates at second order and (1.11) (gx)ij = gij − Pijx 6 C. ROBIN GRAHAM AND ANDREAS JUHL (The details of the computation are not given in [SS]. Details will appear in [FG2] and [J].) This easily yields Proposition 1. If g is conformally flat and n > 2, then v(2k) = (−2)−kσk(P ) 0 ≤ k ≤ n 0 n < k where σk(P ) denotes the k-th elementary symmetric function of the eigenvalues of the endomorphism Pi Proof. Write g−1P for Pi j. Then the σk(P ) are given by det(I + g−1P t) = σk(P )t Equation (1.11) can be rewritten as g−1gx = (I− g−1Px2)2. Taking the determinant and comparing with (1.3) gives the result. � We remark that for g conformally flat, gx given by (1.11) is uniquely determined to all orders by the requirement that g+ be hyperbolic. So in this case the v (2k) are invariantly determined and given by Proposition 1 for all k ≥ 0 in all dimensions n > 2. Returning to the even-dimensional case, we define the Pfaffian of the metric g by (1.12) 2n(n/2)! Pff = (−1)qµi1...inµj1...jnRi1i2j1j2 . . . Rin−1injn−1jn, where µi1...in = | det(g)| ǫi1...in is the volume form and ǫi1...in denotes the sign of the permutation. For a conformally flat metric, one has Rijkl = 2(Pi[kgl]j−Pj[kgl]i). Using this in (1.12) and simplifying gives Pff = (n/2)! σn/2(P ) (see Proposition 8 of [V] for details). Combining with Proposition 1, we obtain for conformally flat g: v(n) = (−2)−n/2(n/2)!−1 Pff . Hence in the conformally flat case, (0.1) specializes to 2Q = 2n/2(n/2− 1)! Pff +(ncn/2) n/2−1∑ (n− 2k)p∗2kv (n−2k), and again the second term on the right hand side is a formal divergence. HOLOGRAPHIC FORMULA FOR Q-CURVATURE 7 2. A Related Identity In this section we derive another identity involving the p∗2kv (n−2k). It is in gen- eral impossible to choose the O(xn) term in (1.4) to make ∆g+u = O(x n); in fact x−n∆g+u|x=0 is independent of the O(x n) term in (1.4) and is a conformally invariant operator of order n applied to f , namely a multiple of the critical GJMS operator Pn. Following [GZ], we consider the limiting behavior of the corresponding term in the expansion of an eigenfunction for ∆g+ as the eigenvalue tends to 0. Let g+ be a Poincaré metric as above. If 0 6= λ ∈ C is near 0, then for f ∈ C ∞(M), one can solve formally the equation (∆g+ − λ(n − λ))uλ = O(x n+λ+1) for uλ of the (2.1) uλ = x f + p2,λf x 2 + · · ·+ pn,λf x n +O(xn+1) where p2k,λ is a natural differential operator in the metric g of order 2k with principal part (−1)k Γ(n/2− k − λ) 22k k! Γ(n/2− λ) ∆k such that Γ(n/2− λ) Γ(n/2− k − λ) p2k,λ is polynomial in λ. Set p0,λf = f . The operators p2k,λ for k < n/2 extend analytically across λ = 0 and p2k,0 = p2k for such k, where p2k are the operators appearing in (1.4). But pn,λ has a simple pole at λ = 0 with residue a multiple of the critical GJMS operator Pn. Now Pn is self-adjoint, so it follows that pn,λ − p n,λ is regular at λ = 0. We denote its value at λ = 0 by pn − p n, a natural operator of order at most n − 2. Our identity below involves the constant term (pn−p n)1. Note that since Pn1 = 0, both pn,λ1 and p∗n,λ1 are regular at λ = 0. We denote their values at λ = 0 by pn1 and p n1; then (pn − p n)1 = pn1− p n1. Moreover, (4.7), (4.13), (4.14) of [GZ] show that (2.2) pn1 = −cn/2Q. It is evident that pn1 dvg = p∗n1 dvg. The next proposition expresses the differ- ence pn1− p n1 as a divergence. Proposition 2. (2.3) n (pn − p n) 1 = n/2−1∑ 2k p∗2kv (n−2k) Proof. Take f ∈ C∞(M) to have compact support, let 0 6= λ be near 0, and define uλ as in (2.1) with the O(x n+1) term taken to be 0. Define wλ by the corresponding expansion with f = 1: wλ = x 1 + p2,λ1 x 2 + · · ·+ pn,λ1 x As in the proof of Theorem 1, consider Green’s identity (2.4) ǫ<x<x0 (uλ∆g+wλ − wλ∆g+uλ)dvg+ = ǫ (uλ∂xwλ − wλ∂xuλ) dvgǫ + cx0 , 8 C. ROBIN GRAHAM AND ANDREAS JUHL where cx0 is the constant (in ǫ) arising from the boundary integral over x = x0. Consider the coefficient of ǫ2λ in the asymptotic expansion of both sides. The left hand side equals ǫ<x<x0 ∆g+ − λ(n− λ) wλ − wλ ∆g+ − λ(n− λ) dvg+. Now uλ ∆g+ − λ(n− λ) wλ dvg+ and wλ ∆g+ − λ(n− λ) uλ dvg+ are of the form x2λψ dxdvg where ψ is smooth up to x = 0. It follows that the asymptotic expansion of the left hand side of (2.4) has no ǫ2λ term. Consequently the coefficient of ǫn+2λ must vanish in the asymptotic expansion of (uλx∂xwλ − wλx∂xuλ) dvgǫ. This is the same as the coefficient of ǫn in the expansion of p2k,λf ǫ (2k + λ)p2k,λ1 ǫ p2k,λ1 ǫ (2k + λ)p2k,λf ǫ v(2k) ǫ2k  dvg. Evaluation of the ǫn coefficient gives 0≤k,l,m≤n/2 k+l+m=n/2 (2l − 2k)(p2k,λf)(p2l,λ1)v (2m) dvg = 0, and then moving the derivatives off f results in the pointwise identity (2.5) 0≤k,l,m≤n/2 k+l+m=n/2 (2l − 2k) p∗2k,λ (p2l,λ1)v The limit as λ → 0 exists of all p2l,λ1 with 0 ≤ l ≤ n/2 and all p 2k,λ with 0 ≤ k ≤ n/2− 1. Since k = n/2 forces l = m = 0, the operator p∗n,λ occurs only applied to 1. Thus we may let λ→ 0 in (2.5). Using p2l1 = 0 for 1 ≤ l ≤ n/2− 1 results in npn1− 0≤k,m≤n/2 k+m=n/2 2k p∗2kv (2m) = 0. Separating the k = n/2 term in the sum gives (2.3). � Proposition 2 may be combined with (0.1) and (2.2) to give other expressions for Q-curvature. However, (0.1) seems the preferred form, as the other expressions all involve some nontrivial linear combination of pn1 and p HOLOGRAPHIC FORMULA FOR Q-CURVATURE 9 We remark that the generalization of (2.5) obtained by replacing p2l,λ1 by p2l,λf remains true for arbitrary f ∈ C∞(M). This follows by the same argument, taking wλ to be given by the asymptotic expansion of the same form but with arbitrary leading coefficient. We conclude this section with some observations concerning relations to the paper [CQY]: (1) Recall that Theorem 1 was proven by consideration of the log ǫ term in (1.8), generalizing the proof of (0.2) in [FG1] where u = 1. In [CQY], it was shown that for a global conformally compact Einstein metric g+, consideration of the constant term in ∆g+U dvg+ = ∂νU dσ for U a global solution of ∆g+U = n gives a formula for the renormalized volume V (g+, g) of g+ relative to a metric g in the conformal infinity of g+. In our notation this formula reads (2.6) V (g+, g) = − (S(s)1) dvg + 2k ṗ∗2kv (n−2k) dvg, where ṗ2k = |λ=0p2k,λ (which exists for k = n/2 when applied to 1) and S(s) denotes the scattering operator relative to g. The operators ṗ2k arise in this context because the coefficient of x2k in the expansion of U is ṗ2k1 for 1 ≤ k ≤ n/2−1, and the coefficient of xn involves ṗn1. Likewise, consideration of the constant term in u∆g+U dvg+ = (u∂νU − U∂νu) dσ for harmonic u gives an analogous formula for the finite part of u dvg+ in terms of boundary data. (2) There is an analogue of Proposition 2 involving the ṗ∗2kv (n−2k). Differentiating (2.5) with respect to λ at λ = 0 and rearranging gives the identity ṗ∗2kv (n−2k) − (ṗ2k1)v (n−2k) (4l − 2k)p∗2k−2l (ṗ2l1)v (n−2k) which expresses the left hand side as a divergence. (3) In [CQY] it was also shown that under an infinitesimal conformal change, the scattering term S(g+, g) ≡ (S(s)1)dvg 10 C. ROBIN GRAHAM AND ANDREAS JUHL satisfies S(g+, e 2αΥg) = −2cn/2 ΥQdvg. Comparing with V (g+, e 2αΥg) = Υv(n) dvg (see [G]) and using (2.6) and Theorem 1, one deduces the curious conclusion that the infinitesimal conformal variation of 2k ṗ∗2kv (n−2k) dvg n/2−1∑ (n− 2k)p∗2kv (n−2k) dvg. This statement involves the conformal variation only of local expressions. For n = 2 this is the statement of conformal invariance of Rdvg, while for n = 4 it is the assertion that the infinitesimal conformal variation of J2 dvg Υ∆J dvg. 3. Q-curvature and families of conformally covariant differential operators In [J] one of the authors initiated a theory of one-parameter families of natural conformally covariant local operators (3.1) DN(X,M ; h;λ) : C ∞(X) → C∞(M), N ≥ 0 of orderN associated to a Riemannian manifold (X, h) and a hypersurface i :M → X , depending rationally on the parameter λ ∈ C. For such a family the conformal weights which describe the covariance of the family are coupled to the family parameter in the sense that (3.2) e−(λ−N)ωDN(X,M ; ĥ;λ)e λω = DN (X,M ; h;λ), ĥ = e for all ω ∈ C∞(X) (near M). Two families are defined in [J]: one via a residue construction which has its origin in an extension problem for automorphic functions of Kleinian groups through their limit set ([J2], chapter 8), and the other via a tractor construction. Whereas the tractor family depends on the choice of a metric h on X , the residue family depends on the choice of an asymptotically hyperbolic metric h+ and a defining function x, to which is associated the metric h = x2h+. HOLOGRAPHIC FORMULA FOR Q-CURVATURE 11 Fix an asymptotically hyperbolic metric h+ on one side X+ of X in M and choose a defining function x for M with x > 0 in X+. Set h = x 2h+. To an eigenfunction u on X+ satisfying ∆h+u = µ(n− µ)u, Reµ = n/2, µ 6= n/2 is associated the family 〈Tu(ζ, x), ϕ〉 ≡ xζ uϕ dvh, ϕ ∈ C c (X) of distributions on X . The integral converges for Re ζ > −n/2− 1 and the existence of a formal asymptotic expansion xµ+jaj(µ) + xn−µ+jbj(µ), x→ 0 with aj(µ), bj(µ) ∈ C ∞(M) implies the existence of a meromorphic continuation of Tu(ζ, x) to C with simple poles in the ladders −µ − 1− N0, −(n− µ)− 1− N0. For N ∈ N0, its residue at ζ = −µ− 1−N has the form a0δN(h;µ+N − n)(ϕ)dvi∗h, where δN (h;λ) : C ∞(X) → C∞(M) is a family of differential operators of order N depending rationally on λ ∈ C. If x̂ = eωx with ω ∈ C∞(X), then ĥ = e2ωh and it is easily checked that δN(h;λ) satisfies (3.2). (The family δN (h;λ) should more correctly be regarded as determined by x and h+, but we use this notation nonetheless.) If g is a metric on M , then we can take h+ = g+ to be a Poincaré metric for g on X+ =M×(0, a) and x to be the coordinate in the second factor, so that h = dx 2+gx. Then (assuming N ≤ n if n is even), the family δN(h;λ) depends only on the initial metric g. The residue can be evaluated explicitly and for even orders N = 2L one obtains (3.3) δ2L(h;µ+ 2L− n) = (2L− 2k)! p∗2l,µ ◦ v (2k−2l) ◦ i∗∂2L−2kx , where the p2l,µ are the operators appearing in (2.1) and the coefficients v (2j) are used as multiplication operators. The corresponding residue family is defined by (3.4) Dres2L (g;λ) = 2 Γ(−n/2 − λ+ 2L) Γ(−n/2 − λ+ L) δ2L(h;λ); 12 C. ROBIN GRAHAM AND ANDREAS JUHL the normalizing factor makes Dres2L (g;λ) polynomial in λ. We are interested in the critical case 2L = n for n even. Using Res0(pn,λ) = −cn/2Pn from [GZ], we see that (3.5) Dresn (g; 0) = (−1) n/2Pn(g)i Direct evaluation from (3.3), (3.4) gives Ḋresn (g; 0)1 = −(−1) n/2c−1 p∗n1 + n/2−1∑ p∗2kv (n−2k) where the dot refers to the derivative in λ. Suppose now that g is transformed conformally: ĝ = e2Υg with Υ ∈ C∞(M). By the construction of the normal form in §5 of [GL], the Poincaré metrics g+ and ĝ+ are related by Φ∗ĝ+ = g+ for a diffeomorphism Φ which restricts to the identity onM and for which the function Φ∗(x)/x restricts to eΥ. Using this the residue construction easily implies (3.6) e−(λ−n)ΥDresn (ĝ;λ) = D n (g;λ) (Φ ∗(x)/x) Applying (3.6) to the function 1, differentiating at λ = 0, and using (3.5) and Pn1 = 0 gives enΥḊresn (ĝ; 0)1 = Ḋ n (g; 0)1− (−1) n/2Pn/2Υ. This proves that the curvature quantity −(−1)n/2Ḋresn (g; 0)1 = c p∗n1 + n/2−1∑ p∗2kv (n−2k) satisfies the same transformation law as the Q-curvature. It is natural to conjecture that it equals the Q-curvature. Indeed, this follows from (0.1), (2.2), and (2.3): p∗n1 + n/2−1∑ p∗2kv (n−2k) = pn1 + p∗n1− pn1 + n/2−1∑ 2kp∗2kv (n−2k) n/2−1∑ p∗2kv (n−2k) − n/2−1∑ 2kp∗2kv (n−2k) The first term is −cn/2Q by (2.2), the second term is 0 by (2.3), and the last term is 2cn/2Q by (0.1). The relation Ḋresn (g; 0)1 = (−1) n/2+1Q and (3.5) show that both the critical GJMS operator Pn and the Q-curvature are contained in the one object D n (g;λ). In that HOLOGRAPHIC FORMULA FOR Q-CURVATURE 13 respect, Dresn (g;λ) resembles the scattering operator in [GZ]. However, the family Dresn (g;λ) is local and all operators in the family have order n. References [B] T. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. AMS 347 (1995), 3671-3742. [BGP] T. Branson, P. Gilkey, and J. Pohjanpelto, Invariants of locally conformally flat manifolds, Trans. AMS 347 (1995), 939–953. [CQY] S.-Y. A. Chang, J. Qing, and P. Yang, On the renormalized volumes for conformally com- pact Einstein manifolds, math.DG/0512376. [FG1] C. Fefferman and C.R. Graham, Q-curvature and Poincaré metrics, Math. Res. Lett. 9 (2002), 139–151. [FG2] C. Fefferman and C.R. Graham, The ambient metric, in preparation. [FH] C. Fefferman and K. Hirachi, Ambient metric construction of Q-curvature in conformal and CR geometries, Math. Res. Lett. 10 (2003), 819–832. [GP] A.R. Gover and L.J. Peterson, Conformally invariant powers of the Laplacian, Q-curvature and tractor calculus, Comm. Math. Phys. 235 (2003), 339–378. [G] C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo, Ser. II, Suppl. 63 (2000), 31–42. [GH] C.R. Graham and K. Hirachi, The ambient obstruction tensor and Q-curvature, in AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, IRMA Lec- tures in Mathematics and Theoretical Physics 8 (2005), 59–71, math.DG/0405068. [GJMS] C.R. Graham, R. Jenne, L.J. Mason, and G.A.J. Sparling Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), 557–565. [GL] C.R. Graham and J. M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186–225. [GZ] C.R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), 89–118. [J] A. Juhl, Families of conformally covariant differential operators, Q-curvature and holog- raphy, book in preparation. [J2] A. Juhl, Cohomological theory of dynamical zeta functions, Prog. Math. 194, Birkhäuser, 2001. [SS] K. Skenderis and S. Solodukhin, Quantum effective action from the AdS/CFT correspon- dence, Phys. Lett. B472 (2000), 316–322, hep-th/9910023. [V] J. Viaclovsky, Conformal Geometry, Contact Geometry and the Calculus of Variations, Duke Math. J. 101 (2000), 283–316. Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195 USA E-mail address : [email protected] Humboldt-Universität, Institut für Mathematik, Unter den Linden, 10099 Berlin E-mail address : [email protected] http://arxiv.org/abs/math/0512376 http://arxiv.org/abs/math/0405068 http://arxiv.org/abs/hep-th/9910023 Introduction 1. Derivation 2. A Related Identity 3. Q-curvature and families of conformally covariant differential operators References

0704.1674

The azimuth structure of nuclear collisions -- I

Version 1.6 The azimuth structure of nuclear collisions – I Thomas A. Trainor and David T. Kettler CENPA 354290, University of Washington, Seattle, WA 98195 (Dated: November 5, 2018) We describe azimuth structure commonly associated with elliptic and directed flow in the context of 2D angular autocorrelations for the purpose of precise separation of so-called nonflow (mainly minijets) from flow. We extend the Fourier-transform description of azimuth structure to include power spectra and autocorrelations related by the Wiener-Khintchine theorem. We analyze several examples of conventional flow analysis in that context and question the relevance of reaction plane estimation to flow analysis. We introduce the 2D angular autocorrelation with examples from data analysis and describe a simulation exercise which demonstrates precise separation of flow and nonflow using the 2D autocorrelation method. We show that an alternative correlation measure based on Pearson’s normalized covariance provides a more intuitive measure of azimuth structure. PACS numbers: 13.66.Bc, 13.87.-a, 13.87.Fh, 12.38.Qk, 25.40.Ep, 25.75.-q, 25.75.Gz I. INTRODUCTION A major goal of the RHIC is production of color- deconfined or QCD matter in heavy ion (HI) collisions, a bulk QCD medium extending over a nontrivial space- time volume which is in some sense thermalized and whose dynamics are dominated in some sense by quarks and gluons as the dominant degrees of freedom [1]. “Mat- ter” in this context means an aggregate of constituents in an equilibrium state, at least locally in space-time, such that thermodynamic state variables provide a nearly complete description of the system. Demonstration of thermalization is seen by many as a necessary part of the observation of QCD matter. A. Global variables One method proposed to demonstrate the existence of QCD matter is to measure trends of global event vari- ables, statistical measures formulated by analogy with macroscopic thermodynamic quantities and based on in- tegrals of particle yields over kinematically accessible mo- mentum space. E.g., temperature analogs include spec- trum inverse slope parameter T and ensemble-mean pt p̂t. Chemical analogs include particle yields and their ratios, such as the ensemble-mean K/π ratio [2]. Corre- sponding fluctuation measures have been formulated for the event-wise mean pt 〈pt〉 (“temperature” fluctuations) and K/π ratio (chemical or flavor fluctuations) [3, 4, 5]. Arguments by analogy are less appropriate when deal- ing with small systems (‘small’ in particle number, space and/or time) where large deviations from macroscopic thermodynamics may be encountered. B. Flow analysis One such global feature is the large-scale angular struc- ture of the event-wise particle distribution. The compo- nents of angular structure described by low-order spheri- cal or cylindrical harmonics are conventionally described as “flows.” The basic assumption is that such structure represents collective motion of a thermalized medium, and hydrodynamics is therefore an appropriate descrip- tion. Observation of larger flow amplitudes is therefore interpreted by many to provide direct evidence for event- wise thermalization in heavy ion collisions [6]. Given those assumptions each collision event is treated sepa- rately. Event-wise angular distributions are fitted with model functions associated with collective dynamics. The model parameters are interpreted physically in a thermo- dynamic (i.e., collective, thermalized) context. However, collective flow in many-body physics is a complex topic with longstanding open issues. There is conflict in the description of nuclear collisions between continuum hydrodynamics and discrete multiparticle sys- tems which echoes the state of physics prior to the study of Brownian motion by Einstein and Perrin [7, 8]. Be- yond the classical dichotomy between discrete and con- tinuous dynamics there is the still-uncertain contribu- tion of quantum mechanics to the early stages of nuclear collisions. Quantum transitions may play a major role in phenomena perceived to be “collective.” Premature imposition of hydrodynamic (hydro) models on collision data may hinder full understanding. C. Nonflow and multiplicity distortions A major concern for conventional flow analysis is the presence of “nonflow,” non-sinusoidal contributions to azimuth structure often comparable in amplitude to sinu- soid amplitudes (flows). Nonflow is treated as a system- atic error in flow analysis, reduced to varying degrees by analysis strategies. Another significant systematic issue is “multiplicity distortions” associated with small event multiplicities, also minimized to some degree by analysis strategies. Despite corrections nonflow and small multi- plicities remain a major limitation to conventional flow measurements. For those reasons flow measurements in peripheral heavy ion collisions are typically omitted. The http://arxiv.org/abs/0704.1674v1 opportunity is then lost to connect the pQCD physics of elementary collisions to nonperturbative, possibly collec- tive dynamics in heavy ion collisions. D. Minijets A series of recent experiments has demonstrated that the nonsinusoidal components of angular correlations at full RHIC energy are dominated by fragments from low- Q2 partons or minijets [9, 10, 11, 12]. Minijets in RHIC p-p and A-A collisions have been studied extensively via fluctuations [9, 10] and two-particle correlations [11, 12]. Minijets may dominate the production mechanism for the QCD medium [13], and may also provide the best probe of medium properties, including the extent of thermal- ization. Comparison of minijets in elementary and heavy ion collisions may relate medium properties and collec- tive motion to a theoretical QCD context. Ironically, demonstrating the existence and properties of collective flows and of jets (collective hadron motion from parton collisions and fragmentation) is formally equivalent. Identifying a partonic “reaction plane” and determining a nucleus-nucleus reaction plane require sim- ilar techniques. For example, sphericity has been used to obtain evidence for deformation of particle/pt/Et angu- lar distributions due to parton collisions [14] and collec- tive nucleon flow [15]. At RHIC we should ask whether final-state angular correlations depend on the geometry of parton collisions (minijets), N-N collisions or nucleus- nucleus collisions (flows), or all three. The analogy is important because to sustain a flow interpretation one has to prove that there is a difference: e.g., to what ex- tent do parton collisions contribute to flow correlations or mimic them? The phenomena coexist on a continuum. To resolve such ambiguities we require analysis meth- ods which treat flow and minijets on an equal footing and facilitate their comparison, methods which do not impose the hypothesis to be tested on the measurement scheme. To that end we should: 1) develop a consistent set of neutral symbols; 2) manipulate random variables with minimal approximations; 3) introduce proper statis- tical references so that nonstatistical correlations of any origin can be isolated unambiguously; 4) treat azimuth structure ab initio in a model-independent manner using standard mathematical methods (e.g., standard Fourier analysis); and 5) include what is known about minijets (“nonflow”) and “flow” in a more general analysis based on two-particle correlations. E. Structure of this paper An underlying theme of this paper is the formal re- lation between event-wise azimuth structure in nuclear collisions and Brownian motion, and how that relation can inform our study of heavy ion collisions. We begin with a review of Fourier transform theory and the rela- tion between power spectra and autocorrelations. That material forms a basis for analysis of sinusoidal compo- nents of angular correlations in nuclear collisions which is well-established in standard mathematics. We then review the conventional methods of flow anal- ysis from Bevalac to RHIC. Five papers are discussed in the context of Fourier transforms, power spectra and au- tocorrelations. To facilitate a more general description of angular asymmetries we set aside flow terminology (ex- cept as required to make connections with the existing lit- erature) and move to a model-independent description in terms of spherical and cylindrical multipole moments. We emphasize the relation of “flows” to multipole moments as model-independent correlation measures. Physical in- terpretation of multipole moments is an open question. We then consider whether event-wise estimation of the reaction plane is necessary for “flow” studies. The con- ventional method of flow analysis is based on such esti- mation, assuming that event-wise statistics are required to demonstrate collectivity, and hence thermalization, in heavy ion collisions. We define the 2D joint angular autocorrelation and de- scribe its properties. The autocorrelation is fundamental to time-series analysis, the Brownian motion problem and its generalizations and astrophysics, among many other fields. It is shown to be a powerful tool for separating “flow” from “nonflow.” The autocorrelation eliminates biases in conventional flow analysis stemming from finite multiplicities, and makes possible bias-free study of cen- trality variations in A-A collisions down to N-N collisions. “Nonflow” is dominated by minijets (minimum-bias parton fragments, mainly from low-Q2 partons) which can be regarded as Brownian probe particles for the QCD medium, offering the possibility to explore small-scale medium properties. Minijet systematics provide strong constraints on “nonflow” in the conventional flow con- text. Interaction of minijets with the medium, and par- ticularly its collective motion, is the subject of paper II of this series. Finally, we consider examples from RHIC data of au- tocorrelation structure. We show the relation between “flow” and minijets, how conventional flow analysis is biased by the presence of minijets, and how the autocor- relation method eliminates that bias and insures accurate separation of different collision dynamics. We include several appendices. In App. A we review Brownian motion and its formal connection to azimuth correlations in nuclear collisions. In App. B we review the algebra of random variables in relation to conven- tional flow analysis techniques. We make no approxima- tions in statistical analysis and invoke proper correlation references to obtain a minimally-biased, self-consistent analysis system in which flow and nonflow are precisely distinguished. In App. C we review the mathematics of spherical and cylindrical multipoles and sphericity. In App. D we review subevents, scalar products and event- plane resolution. In App. E we summarize some A-A centrality issues related to azimuth multipoles and mini- jets. II. FOURIER ANALYSIS The azimuth structure of nuclear collisions is part of a larger problem: angular correlations of number, pt and Et on angular subspace (η1, η2, φ1, φ2). There is a for- mal similarity between event-wise particle distributions on angle and the time series of displacements of a parti- cle in Brownian motion. In either case the distribution is discrete, combining a large random component with the possibility of a smaller deterministic component. The mathematical description of Brownian motion includes as a key element the autocorrelation density, related to the Fourier power spectrum through the Wiener-Khintchine theorem (cf. App. A). The Fourier series describes arbitrary distributions on bounded angular interval 2π or distributions periodic on an unbounded interval. The azimuth particle distribu- tion from a nuclear collision is drawn from (samples) a combination of sinusoids nearly invariant on rapidity near midrapidity, conventionally described as “flows,” and other azimuth structure localized on rapidity and conventionally described as “nonflow.” The two contri- butions are typically comparable in amplitude. We first consider the mathematics of the Fourier trans- form and power spectrum and their role in conventional flow analysis [16]. We assume for simplicity that the only angular structure in the data is represented by a few lowest-order Fourier terms. In conventional flow analysis the azimuth distribution is described solely by a Fourier series, and corrections are applied in an attempt to com- pensate for “nonflow” as a systematic error. We later re- turn to the more general angular correlation problem and consider non-sinusoidal (nonflow) structure described by non-Fourier model functions in the larger context of 2D (joint) angular autocorrelations. Precise description of the composite structure requires a hybrid mathematical model. Event-wise random variables are denoted by a tilde. Variables without tildes are ensemble averages, indicated in some cases explicitly by overlines. Event-wise averages are indicated by angle brackets. The algebra of random variables is discussed in App. B. Where possible we em- ploy notation consistent with conventional flow analysis. A. Azimuth densities The event-wise azimuth density (particle, pt or Et) is a set of n samples from a parent density integrated over some (pt, η) acceptance, a sum over Dirac delta functions (particle positions) ρ̃(φ) = ri δ(φ− φi) (1) The ri are weights (1, pt or Et) appropriate to a given physical context. We assume integration over one unit of pseudorapidity, so multiplicity n estimates dn/dη (simi- larly for pt and Et). The continuum parent density, not directly observable, is the object of analysis. Fixed parts of the parent density are estimated by a histogram aver- aged over an event ensemble. The discrete nature of the sample distribution and its statistical character present analysis challenges which are one theme of this paper. The correlation structure of the single-particle azimuth density is manifested in ensemble-averaged multiparticle (two-particle, etc.) densities. Accessing that structure by projection of multiparticle spaces to 2D or 1D with minimal distortion is the object of correlation analysis. In each event the two-particle density is the Cartesian product ρ̃(φ1, φ2) = ρ̃(φ1) ρ̃(φ2) ρ̃(φ1, φ2) = r2i δ(φ1 − φi)δ(φ2 − φi) (2) n,n−1 rirj δ(φ1 − φi)δ(φ2 − φj), where the first term represents self pairs. Ensemble- averaged two-particle distribution ρ(φ1, φ2) with correla- tions is not generally factorizable. By comparing the av- eraged two-particle distribution to a factorized (or mixed- pair) statistical reference two-particle correlations are re- vealed. In a later section we compare multipole moments from two-particle correlation analysis on azimuth to re- sults from conventional flow analysis methods. B. Fourier transforms on azimuth A Fourier series is an efficient representation if azimuth structure approximates a constant plus a few sinusoids whose wavelengths are integral fractions of 2π. A Fourier representation of a peaked distribution (e.g., jet cone) is not an efficient representation. We assume a simple combination of the lowest few Fourier terms. The Fourier forward transform (FT) is ρ̃(φ) = exp(imφ) (3) cos(m[φ−Ψm]), where boldface Q̃m is an event-wise complex amplitude, Q̃m is its magnitude and Ψm its phase angle. The second line arises because ρ̃(φ) is a real function, and the prac- tical upper limit on m is particle number n (wavelength ∼ mean interparticle spacing). Q̃m/2π = ρ̃m is the am- plitude of the density variation associated with the mth sinusoid. With ri → 1 Q̃m is the corresponding num- ber of particles in 2π if that density were uniform. Ψm is event-wise by definition and does not require a tilde. The reverse transform (RT) is Q̃m = dφ ρ̃(φ) exp(−imφ) (4) ri exp(−imφi) = Q̃m exp(imΨm). For the discrete transform φ ∈ [−π, π] is partitioned into M equal bins with bin contents r̃l and bin centers φl. We multiply Eq. (3) by bin width δφ = 2π/M , and the Fourier transform pair becomes r̃l = cos(m[φl −Ψm]) (5) Q̃m = l=−M/2 r̃l exp(−imφl), where the r̃l are also random variables. The upper limit M/2 in the first line is a manifestation of the Nyquist sampling theorem [17]. With ri → 1 r̃l → ñl and Qm/M is the maximum number of particles in a bin associated with the mth sinusoid. C. Autocorrelations and power spectra The azimuth autocorrelation density ρA(φ∆) is a pro- jection by averaging of the pair density on two-particle azimuth space (φ1, φ2) onto difference axis φ∆ = φ1−φ2. The autocorrelation concept is not restricted to peri- odic or bounded distributions or discrete Fourier trans- forms [16]. In what follows ρ̃A includes self pairs. The autocorrelation density is defined as [18] ρ̃A(φ∆) ≡ dφ ρ̃(φ) ρ̃(φ+ φ∆) (6) i,j=1 dφ δ(φ− φi) δ(φ− φj + φ∆) i,j=1 rirj δ(φi − φj + φ∆). Using Eq. (3) we obtain the FT as ρ̃A(φ∆) = dφ × (7) exp(imφ)× m′=−∞ Q̃∗m′ exp(−im′[φ+ φ∆]) [2π]2 exp(−i mφ∆) [2π]2 [2π]2 cos(mφ∆), and the RT as Q̃2m = n 2〈r cos(m[φ−Ψm])〉2 (8) dφ∆ ρ̃A(φ∆) cos(mφ∆) i,j=1 rirj cos(m[φi − φj ]) = n〈r2〉+ n(n− 1)〈r2 cos(mφ∆)〉. = n〈r2〉+ n(n− 1)〈r2 cos2(m[φ−Ψr])〉. Phase angle Ψm has been eliminated, and Ψr will be identified with the reaction plane angle. We can write the same relations for ensemble-averaged quantities because the terms are positive-definite ρA(φ∆) ≡ [2π]2 [2π]2 cos(mφ∆), (9) with RT Q2m = 2π dφ∆ ρA(φ∆) cos(mφ∆) (10) i,j=1 rirj cos(m[φi − φj ]) = n〈r2〉+ n(n− 1)〈r2 cos(mφ∆)〉. We have adopted the convention Q2m = Q̃ m to lighten the notation. Coefficients Q2m are power-spectrum ele- ments on wave-number indexm. That FT transform pair expresses the Wiener-Khintchine theorem which relates power-spectrum elements Q2m to autocorrelation density ρA(φ∆). The autocorrelation provides precise access to two-particle correlations given enough collision events, no matter how small the event-wise multiplicities. D. Autocorrelation structure The autocorrelation concept was developed in response to the Brownian motion problem and the Langevin equa- tion, a differential equation describing Brownian motion which contains a stochastic term. The concept is al- ready apparent in Einstein’s first paper on the subject (cf. App. A). The large-scale, possibly-deterministic mo- tion of the Brownian probe particle must be separated from its small-scale random motion due to thermal col- lisions with molecules. Similarly, we want to extract az- imuth correlation structure persisting in some sense over an event ensemble from event-wise random variations. The autocorrelation technique is designed for that pur- pose. The statistical reference for a power spectrum is the white-noise background representing an uncorrelated sys- tem. The reference is typically uniform up to large wave number or frequency (hence white noise), an in- evitable part of any power spectrum from a discrete pro- cess (point distribution). The “signal” is typically limited to a bounded region (signal bandwidth) of the spectrum at smaller frequencies or wave numbers. From Eq. (8) the power-spectrum elements are Q̃2m = r2i + n,n−1 rirj cos(m[φi − φj ]) (11) = n〈r2〉+ n(n− 1)〈r2 cos(mφ∆)〉 The first term in Eq. (11) is Q̃2ref , the white-noise back- ground component of the power spectrum common to all spectrum elements. The second term, which we de- note Ṽ 2m, represents true two-particle azimuth correla- tions. Note that Q̃20 = n 2〈r2〉, whereas Ṽ 20 = n(n−1)〈r2〉. In terms of complex (or vector) amplitudes we can write Q̃m = Q̃ref + Ṽm, (12) where Q̃ref represents a random walker. There is no cross term in Eq. (11) because Qref and Vm are uncor- related. Inserting the power-spectrum elements into Eq. (9) we obtain the ensemble-averaged autocorrelation density ρA(φ∆) = n〈r2〉 δ(φ∆) + n(n− 1)〈r2〉 [2π]2 [2π]2 cos(mφ∆). The first term is the self-pair or statistical noise term, which can be excluded from ρA by definition simply by excluding self pairs. The second term, with V 20 = n(n− 1)〈r2〉, is a uniform component, and the third term is the sinusoidal correlation structure. The self-pair term is referred to in conventional flow analysis as the “auto- correlation,” in the sense of a bias or systematic error, but that is a notional misuse of standard mathematical terminology. The true autocorrelation density is the en- tirety of Eq. (13), including (in this simplified case) the self-pair term, the uniform component and the sinusoidal two-particle correlations. In general, the single-particle ensemble-averaged dis- tribution ρ0 may be structured on (η, φ). We want to subtract the corresponding reference structure from the two-particle distribution to isolate the true correlations. In what follows we assume ri = 1 for simplicity, therefore describing number correlations. We subtract factorized reference autocorrelation ρA,ref (φ1, φ2) = ρ0(φ1) ρ0(φ2) representing a system with no correlations, with ρ0 = n̄/2π ≃ d2n/dηdφ in this simple example, to obtain the difference autocorrelation ∆ρA(φ∆) = ρA − ρA,ref (14) σ2n − n̄ [2π]2 [2π]2 cos(mφ∆). The first term measures excess (non-Poisson) multiplicity fluctuations in the full (pt, η, φ) acceptance. The second term is a sum over cylindrical multipoles. We now divide the autocorrelation difference by ρA,ref = ρ0 = n̄/2π to form the density ratio ρA,ref σ2n − n̄ 2π n̄ cos(mφ∆) (15) ≡ ∆ρA[0]√ ρA,ref ∆ρA[m]√ ρA,ref cos(mφ∆), The first term ∆ρA[0]/ ρA,ref is the density ratio av- eraged over acceptance (∆η,∆φ). Its integral at full acceptance is normalized variance difference ∆σ2 (σ2n− n̄)/n̄, which we divide by acceptance integral 1×2π to obtain the mean of the 2D autocorrelation density ∆ρA[0]/ ρA,ref = ∆σ (∆η,∆φ)/∆η∆φ. The sinusoid amplitudes are ∆ρA[m]/ ρA,ref = n(n− 1) ṽ2m/(2πn̄) ≡ m, defining unbiased vm. The event-wise ṽ 〈cos(mφ∆)〉 are related to conventional flow measures vm, but may be numerically quite different for small multiplicities due to bias in the latter. Different mean- value definitions result in different measured quantities (cf. App. B). Whereas the Q2m are power-spectrum ele- ments which include the white-noise reference, the V 2m (∝ squares of cylindrical multipole moments) represent the true azimuth correlation signal. This important result combines several measures of fluctuations and correla- tions within a comprehensive system. E. Azimuth vectors Power-spectrum elements Q2m are derived from com- plex Fourier amplitudes Qm. In an alternate representa- tion the complex Qm can be replaced by azimuth vectors ~Qm. The ~Qm are conventionally referred to as “flow vec- tors” [22], but they include a statistical reference as well as a “flow” sinusoid. The ~Vm defined in this paper are more properly termed “flow vectors,” to the extent that such terminology is appropriate. We refer to the ~Qm by the model-neutral term azimuth vector and define them by the following argument. The cosine of an angle difference—cos(m[φ1 − φ2]) = cos(mφ1) cos(mφ2) + sin(mφ1) sin(mφ2)—can be repre- sented in two ways, with complex unit vectors u(mφ) ≡ exp(imφ) or with real unit vectors ~u(mφ) ≡ cos(mφ)̂ı+ sin(mφ)̂ [complex plane (ℜz,ℑz) vs real plane (x, y)]. Thus, cos(m[φ1 − φ2]) = ℜ{u(mφ1)u∗(mφ2)} (16) = ~u(mφ1) · ~u(mφ2)). If an analysis is reducible to terms in cos(m[φ1 − φ2]) the same results are obtained with either representation. Thus, we can rewrite the first line of Eq. (3) as ρ̃(φ) = · ~u(mφ) (17) cos(m[φ−Ψm]), in which case ~̃Qm = dφ ρ̃(φ) ~u(mφ) = ri~u(mφi) (18) = n〈r[cos(mφ), sin(mφ)]〉 = Q̃m~u(mΨm), an event-wise random real vector. III. CONVENTIONAL METHODS We now use the formalism in Sec. II to review con- ventional flow analysis methods in a common framework. We consider five significant papers in chronological order. The measurement of angular correlations to detect col- lective dynamics (e.g., parton fragmentation and/or hy- drodynamic flow) proceeds from directivity (1983) at the Bevalac (1 - 2 GeV/u fixed target) to transverse spheric- ity predictions (1992) for the SPS/RHIC ( sNN = 17 - 200 GeV), then to Fourier analysis of azimuth distribu- tions (1994, 1998) and v2 centrality trends (1999). We pay special attention to the manipulation of random vari- ables (RVs). RVs do not follow the algebra of ordinary variables, and the differences are especially important for small sample numbers (multiplicities, cf. App. B). A. Directivity at the Bevalac An important goal of the Bevalac HI program was ob- servation of the collective response of projectile nucleons to compression during nucleus-nucleus collisions, called directed flow, which might indicate system memory of the initial impact parameter as opposed to isotropic ther- mal emission. Because of finite (small) multiplicities and large fluctuations relative to the measured quantity the geometry of a given collision may be poorly defined, but the final state may still contain nontrivial collective infor- mation. The analysis goal becomes separating possible collective signals from statistical noise. An initial search for collective effects was based on the 3D sphericity tensor S̃ = i ~pi~pi [14, 15] described in App. C 3. Alternatively, the directivity vector was defined in the transverse plane [19]. In the notation of Sec. II, including weights ri → wipti, directivity is ~̃Q1 ≡ wi~pti = wipti~u(φi) (19) = Q̃1~u(Ψ1), azimuth vector ~̃Qm with m = 1 (corresponding to di- rected flow). Event-plane (EP) angle Ψ1 estimates true reaction-plane (RP) angle Ψr. To maintain correspon- dence with SPS and RHIC analysis we simplify the de- scription in [19] to the case that n single nucleons are detected and no multi-nucleon clusters; thus a → 1 and A→ n. The terms in ~̃Q1 are weighted by wi = w(yi) = ±1 cor- responding to the forward or backward hemisphere rela- tive to the CM rapidity, with a region [−δ, δ] about mid- rapidity excluded from the sum (wi = 0). Q̃1 then ap- proximates quadrupole moment q̃21 derived from spher- ical harmonic ℜY 12 ∝ sin(2θ) cos(φ), as illustrated in Fig. 1 (left panel, dashed lines). In effect, a rotated quadrupole is modeled by two opposed dipoles, point- symmetric about the CM in the reaction plane. It is ini- tially assumed that EP angle Ψ1 of vector ~Q1 estimates RP angle Ψr, and magnitude Q1 measures directed flow. The first part of the analysis was based on subevents— nominally equivalent but independent parts of each event. The dot product ~Q1A · ~Q1B for subevents A and B of each event was used to establish the exis- tence of a significant flow phenomenon and the angu- lar resolution of the RP estimation via the distribu- tion on Ψ1A − Ψ1B. The EP resolution was defined by cos(Ψ1 − Ψr) = 2 cos(Ψ1A −Ψ1B). The magnitude of ~̃Q1 was then related to an estimate of the mean trans- verse momentum in the RP. Integrating over rapidity with weights wi we obtain event-wise quantities Q̃21 = w2i p n,n−1 wi wj ~pti · ~ptj (20) ≃ n〈p2t 〉+ n(n− 1)〈p2t cos(φ∆)〉 ≡ Q̃2ref + Ṽ 21 . The last line makes the correspondence with the notation of this paper. The initial analysis in [19] used Q̃21 − Q̃2ref = Ṽ 21 = n(n−1)〈p2x〉, assuming that x̂ is contained in the RP and there are no non-flow correlations. Note that n(n− 1) ≡ ∑n,n−1 i6=j |wiwj | contains weights wiwj implicitly. Since wi ∼ sin[2 θ(yi)] (cf. Fig. 1 – left panel), what is actu- ally calculated in [19] for the single-particle (no multi- nucleon clusters) case is the pt-weighted r.m.s. mean of the spherical harmonic ℜY 12 (θ, φ) ∝ quadrupole mo- ment q̃21, thereby connecting rank-1 tensor ~Q1 (with weights on rapidity) to rank-two sphericity tensor S (cf. App. C 3). The mean-square quantity calculated is p2x ≡ n(n− 1) ≃ sin 2(2θ)p2t cos 2(φ −Ψr) sin2(2θ) , (21) a minimally-biased statistical measure as discussed in App. B, from which we obtain px = p2x estimating the transverse momentum per particle in the reaction plane. The second part of the analysis sought to obtain px(y), the weighted-mean transverse momentum in the RP as a function of rapidity. It was decided to determine pxi = pti cos(φi−Ψr) for the ith particle relative to the RP, but with Ψr estimated by EP angle Ψ1. The initial attempt was based on xi ≡ wi~pti · ~u(Ψ1) = wi~pti · j wj ~ptj k wk ~ptk| . (22) Summing wi ~pti over all particles in a y bin gives 〈p′x〉 = ∑n,n−1 i6=j wiwj ~pti · ~ptj l |wl| | k wk ~ptk| n〈p2t 〉+ Ṽ 21 n Q̃1 = Q̃1/n = 〈p2t 〉+ 〈p2x〉, from which we obtain ensemble mean p′x = ˜〈p′x〉. That result can be compared directly with the linear speed in- ferred from a random walker trajectory, which is ‘infinite’ in the limit of zero time interval (cf. App. A). The first term in the radicand is said to produce “multiplicity dis- tortions” in conventional flow terminology. The second term contains the unbiased quantity. In contrast to the first part of the analysis the sec- ond method retains the statistical reference within Q̃1 as part of the result, so that Q̃1/n ∼ 〈p2t 〉/n for small multiplicities and/or flow magnitudes, a false signal com- parable in magnitude to the true flow signal. The unbi- ased directed flow px was said to be “distorted” by the presence of the statistical reference (called unwanted self- correlations or “autocorrelations”) to the strongly-biased value p′x dominated by the statistical reference. An attempt was made to remove statistical distortions arising from self pairs by redefining ~̃Q1 → ~̃Q1i, a vector complementary to each particle i with that particle omit- ted from the sum. The estimator of Ψr for particle i is then Ψ1i in ~̃Q1i ≡ j 6=i wj ~ptj = Q̃1i ~u(Ψ1i) and xi = wi~pti · ~u(Ψ1i) = wi~pti · j 6=i wj ~ptj k 6=i wk ~ptk| .(24) 0 1 2 vm/σ = √(2nvm 2 ) ~ √(2Vm 2 /n) n = 5 n = 10, ... , 50 √(n-1)Vm 0 1 2 3 FIG. 1: Left panel: Comparison of directed flow data from Fig. 3(a) of the event-plane analysis and the ℜY 12 ∝ sin(2θ[ylab]) spherical harmonic, with amplitude 95 MeV/c obtained from V 21 . Weights in the form w(ylab) are denoted by dashed lines. The correspondence with sin(2θ[ylab]) (solid curve) is apparent. Right panel: The EP resolution obtained from [22] (dashed curve) and from ratio (n− 1)V 2 /nQ′2 defined in this paper (solid and dotted curves for several n values). Summing over i within a rapidity bin one has 〈p′′x〉 = l |wl| j 6=i wiwj ~pti · ~ptj k 6=i wk ~ptk| 〈p2x〉 (n− 1)〈p2x〉 〈p2t 〉+ (n− 2)〈p2x〉 〈p2x〉 〈cos(Ψ′m −Ψr)〉 with Q̃′1 ≡ (n− 1)〈p2t 〉+ (n− 1)(n− 2)〈p2t cos(φ∆)〉. Since V 21 = n(n− 1)〈p2x〉 = n(n− 1)〈p2t cos(φ∆)〉 one sees that the division by Q̃′1 is incorrect, even though it seems to follow the chain of argument based on RP estimation and EP resolution with correction. The correct (minimally-biased) quantity is px ≡ p2x = /n(n− 1). The new EP definition removes the ref- erence term from the numerator, but Q̃′1 in the denom- inator retains the statistical reference in p′′x. There is the additional issue that x2 6= x̄. Two different mean values are represented by 〈px〉 and px = p2x. The dif- ference can be large for small event multiplicities. The remaining bias was attributed to the EP resolu- tion. The resolution correction factor derived from the initial subevent analysis (cf. App. D) was applied to 〈p′′x〉 to further reduce bias. In Fig. 1 (right panel) we compare the EP resolution correction from [22] (dashed curve) with factor (n− 1)Ṽ 2 /nQ̃′2 required to convert 〈p′′x〉 from Eq. (25) to px from Eq. (21). The agreement is very good. Eq. (21) is the least biased and most direct way to obtain px ≡ p2x, both globally over the detec- tor acceptance and locally in rapidity bins, without EP determination or resolution corrections. In Fig. 1 (left panel) we show the data (points) for px(ylab) from the EP-corrected analysis and the solid curve px sin[2θ(ylab)] ∝ q21Y21(θ(y), 0), where px = /n(n− 1) = 95 MeV/c. The agreement is good, and the similarity of sin[2θ(ylab)] (solid curve) to weights w(ylab) (dashed lines) noted above is apparent. Loca- tion of the sin[2θ(ylab)] extrema near the kinematic limits (vertical lines) is an accident of the collision energy and nucleon mass. These results are for Ecm = 1.32 GeV ∼√ 2. In the notation of this paper V 21 = 4.7 (GeV/c)2, /n(n− 1) = 0.095 GeV/c = wpx/a, and Qx ≡ n̄ /n(n− 1) = 2.17 GeV/c ≃ (not Q1). By direct and indirect means (directivity and RP esti- mation) quadrupole moment q21 ∝ V1 was measured. B. Transverse sphericity at higher energies The arguments and techniques in [20] suggest a smooth transition from relativistic Bevalac and AGS energies (collective nucleon and resonance flow) to intermediate SPS and ultra-relativistic RHIC energies (possible trans- verse flow, possibly QCD matter, collectivity manifested by correlations of produced hadrons, mainly pions). For all heavy ion collisions thermalization is a key issue. Clear evidence of thermalization is sought, and collective flow is expected to provide that evidence. Two limiting cases are presented in [20] for SPS flow measurements: 1) linear N-N superposition with no col- lective behavior (no flow); 2) thermal equilibrium – col- lective pressure in the reaction plane – fluid dynamics leading to “elliptic” flow. In a hydro scenario the initial space eccentricity transitions to momentum eccentricity through thermalization and early pressure. The paper considers flow measurement techniques appropriate for the SPS and RHIC, and in particular newly defines trans- verse sphericity St. According to [20] the 3D sphericity tensor introduced at lower energies [15] can be simplified in ultra-relativistic heavy ion collisions to a 2D transverse sphericity tensor. Sphericity is transformed to 2D by ~p → ~pt, omitting the momentum ẑ component near mid-rapidity. Transverse sphericity (in dyadic notation) is 2S̃t ≡ 2 ~pti~pti (26) p2ti {I + C(φi)} ≡ n〈p2t 〉 {I + α̃1 C(Ψ2)} defining α̃1 and Ψ2 in the tensor context, with C(φ) ≡ cos(2φ) sin(2φ) sin(2φ) − cos(2φ) . (27) This α̃ definition corresponds to Eq. (3.1) of [20]. We next form the contraction of S̃t with itself 2S̃t : S̃t = 2 (~pti · ~ptj)2 (28) p4ti + 2 p2tip tj cos 2(φi − φj) = 2n〈p4t 〉+ 2n(n− 1)〈p4t cos2(φ∆)〉 = n(n+ 1)〈p4t 〉+ n(n− 1)〈p4t cos(2φ∆)〉 using the dyadic contraction notation A : B ≡ AabBab, with the usual summation convention. That self- contraction of a rank-2 tensor can be compared to the more familiar self-contraction of a rank-1 tensor ~̃Q2 · ~̃Q2 = Q̃22 = n〈p2t 〉 + n(n − 1)〈p2t cos(2φ∆)〉. The quantity 2[S̃t : S̃t]ref = n(n + 1)〈p4t 〉 is the (uncorrelated) refer- ence for the rank-2 contraction, whereas Q̃2 2,ref = n〈p2t 〉 is the reference for the rank-1 contraction. Subtracting the rank-2 reference contraction gives S̃t : S̃t − [S̃t : S̃t]ref = n(n− 1)〈p4t cos(2φ∆)〉 ≃ 〈p2t 〉Ṽ 22 , (29) which relates transverse sphericity to two-particle corre- lations in the form Ṽ 22 = Q̃ 2 − Q̃22,ref , thus establishing the exact correspondence between S̃t and ~̃Q2. From the definition of α1 in Eq. 26 above and Eq. (3.1) of [20] we also have 2 S̃t : S̃t = n2〈p2t 〉2(1 + α̃21) (30) which implies n2〈p2t 〉2α̃21 = n2 σ̃2p2t + n〈p t 〉 (31) + n(n− 1)〈p4t cos(2φ∆)〉 ≃ n2 σ̃2p2t + 〈p t 〉Q̃22. (32) The first relation is exact, given the definition of α̃1, but produces a complex statistical object containing the event-wise variance of p2t in its numerator and random variable n2 in its denominator. For 1/n → 0 it is true that α̃1 → 〈cos(2[φ−Ψr])〉 (since Ψ2 → Ψr also), but α̃1 is a strongly biased statistic for finite n. The definition α̃2 = 〈p2x − p2y〉 〈p2x + p2y〉 from Eq. (2.5) of [20] seems to imply α̃2 = 〈p2t cos(2[φ− Ψr])〉/〈p2t 〉 → 〈cos(2[φ−Ψr])〉, assuming that x̂ lies in the RP. However, the latter relation fails for finite multiplic- ity [the effect of the statistical reference or self-pair term n in Eq. (31)] because each of event-wise 〈p2x〉 and 〈p2y〉 is a random variable, and their independent random varia- tions do not cancel in the numerator. The exact relation is the first line of n2〈p2t 〉2α̃22 = n2〈p2t cos(2[φ−Ψ2])〉2 (34) ≃ n2〈p2t 〉〈pt cos(2[φ−Ψ2])〉2 = 〈p2t 〉 Q̃22 α̃2 ≃ Q̃2/Q̃0 6= Ṽ2/Q̃0 The second line is an approximation which indicates that α̃2 is more directly related to Q̃2 than is α̃1. But Q 2 is a poor substitute for V 22 which represents true two-particle azimuth correlations in a minimally-biased way by incor- porating a proper statistical reference. The effect of the reference contribution is termed a ‘distortion’ in [20]. C. Fourier series I Application of Fourier series to azimuth particle dis- tributions was introduced in [21]. Fourier analysis is de- scribed as model independent, providing variables which are “easy to work with and have clear physical interpreta- tions.” Sinusoids or harmonics are associated with trans- verse collective flow, the model-dependent language fol- lowing [20] closely. To facilitate comparisons we convert notation in [21] to that used in this paper: r(φ) → ρ(φ), (xm, ym) → ~Qm, ψm → Ψm, vm → Qm and ṽm → Vm. According to the proposed method density ρ̃(φ) repre- sents, within some (pt, η) acceptance, an event-wise par- ticle distribution on azimuth φ including weights ri = 1, pti or Eti. The FT in terms of azimuth vectors is ρ̃(φ) = riδ(φ− φi) (35) · ~u(mφ) cos(m[φ−Ψm]), and the RT is ~̃Qm = ri~u(mφi) ≡ Q̃m~u(mΨm), (36) forming a conventional Fourier transform pair [cf. Eqs. (3) and (4)]. Scalar amplitude Q̃m = i ri~u(mφi) · ~u(mΨm) = n〈r cos(m[φ − Ψm])〉 is the proposed flow- analysis quantity. Q̃m is said to measure the flow mag- nitude, and Ψm estimates reaction-plane angle Ψr. It is proposed that Q̃m(η) evaluated within bins on η may characterize complex “event shapes” [densities on (η, φ)]. As with directivity and sphericity, multiplicity fluctu- ations are seen as a major obstacle to flow analysis with Fourier series. Finite multiplicity is described as a source of ‘bias’ which must be suppressed. It is stated that (Ṽm,Ψr) are the “parameter[s] relevant to the magnitude of flow,” whereas the observed (Q̃m,Ψm) are biased flow estimators. In the limit 1/n → 0 the two cases would be identical. A requirement is therefore placed on min- imum event-wise multiplicity in a “rapidity slice.” To solve the finite-number problem the paper proposes to use the event frequency distribution on Q̃2m from event- wise Fourier analysis to measure flow. If correlations are zero then Q̃2m → Q̃2ref = r2i = n〈r2〉 ≡ σ̃2, (37) with 〈r2〉 ≡ σ20 . By fitting the frequency distribution on Q̃2m with a model function it is proposed to obtain Ṽm as the unbiased flow estimator. The fitting procedure is said to require sufficiently large event multiplicities to obtain Ṽm unambiguously. The distribution on Q̃2m is derived as follows (cf. Fig. 2 – left panel). The magnitude of statistical ref- erence ~̃Qref (a random walker) has probability distri- bution ∝ exp(−Q̃2ref/2Q2ref), with Q2ref = n〈r2〉. But ~̃Qref = ~̃Qm − ~̃Vm, therefore (cf. Fig. 2 – left panel) Q̃2ref = Q̃ m + Ṽ m − 2Q̃mṼm cos(m[Ψm −Ψr]), (38) exp(−Q̃2ref/2Q2ref) → ρ(Q̃m,Ψm; Ṽm,Ψr). (39) When integrated over cos(m[Ψm −Ψr]) there results the required probability distribution on Q̃2m, with fit param- eter Vm. The distribution on Q̃ m is said to show a ‘non- statistical’ shape change from which Vm can be inferred by a model fit “free from uncertainties in event-wise de- termination of the reaction plane.” It is also proposed to use ρ(Q̃m,Ψm; Ṽm,Ψr) to determine the EP resolu- tion cos(Ψm − Ψr) by integrating over Q̃2m and using the resulting projection on cos(Ψm − Ψr) to determine the ensemble mean (Fig. 3 of [21]). While one could extract ensemble-average V 2m at some level of accuracy by fitting the frequency distribution on Q̃2m with a model function, we ask why go to that trou- ble when V 2m is easily obtained as a variance difference? Instead of Eq. (38) we simply write Q̃2m = Q̃ ref + Ṽ m, (40) where the cross term is zero on average and Q̃2ref = n〈r2〉 represents the power-spectrum white noise, which is the same for all m (i.e., ‘white’). If the vector mean values are zero that is a relation among variances. Ensemble mean V 2m = Q m −Q2ref is therefore simply determined. For the EP resolution we factor Ṽ 2m Ṽ 2m = n(n− 1)〈r2 cos2(m[φ−Ψr])〉 (41) = n(n− 1)〈r2 cos2(m[φ−Ψm])〉 cos2(m[Ψm −Ψr]). We use Q̃m = n〈r cos(m[φ − Ψm])〉 and the assumption that random variable Ψm − Ψr is uncorrelated with φ− Ψm to obtain cos2(m[Ψm −Ψr]) = n Ṽ 2m (n− 1) Q̃2m , (42) which defines the EP resolution of the full n-particle event in terms of power-spectrum elements (cf. App. D). The square root of that expression is plotted in Fig. 2 (right panel) as the solid curves for several values of n̄. The solid and dotted curves are nearly identical to those in Fig. 1. = - vm/σ = √(2nvm 2 ) ~ √(2Vm 2 /n) n = 5 n = 10, ... , 50 √(n-1) 0 1 2 3 FIG. 2: Left panel: Distribution of event-wise elements of ~̃Qm components determined by the gaussian-distributed random walker ~̃Qref and possible correlation component ~̃Vm. Right panel: Reaction-plane resolution estimator 〈cos(mδΨmr)〉, with δΨmr = Ψm−Ψr, determined from fits to a distribution on Q̃2m as in the left panel (dashed curve), and from Eq. 42 for several values of n (solid and dotted curves). As in other flow papers there is much emphasis on insuring adequate multiplicities to reduce bias to a man- agable level, because an easily-determined statistical ref- erence is not properly subtracted to reveal the contribu- tion from true two-particle correlations in isolation. For√ 2nvm > 1 in Fig. 2 (right panel) the EP is meaningful; bias of event-wise quantities relative to the EP is man- ageable. For 2nvm < 1/2 the EP is poorly defined, and ensemble-averaged two-particle correlations are the only reliable measure of azimuth correlations. In either case EP estimation is only justified when a non-flow phe- nomenon is to be studied relative to the reaction plane. D. Fourier series II A more elaborate review of flow analysis methods based on Fourier series is presented in [22]. The ap- proach is said to be general. The event plane is ob- tained for each event. The event-wise Fourier ampli- tude(s) q̃m ≡ Q̃m/Q̃0 relative to the EP are corrected for the EP resolution as obtained from subevents. We mod- ify the notation of the paper to vobsm → q̃m and wi → ri to maintain consistency within this paper. We distinguish between the unbiased ṽm ≡ Ṽm/Ṽ0 and the biased q̃m. According to [22], in the 1/n→ 0 limit (Q̃m → Vm, no tildes, no random variables) the dependence of the single- particle density on azimuth angle φ integrated over some (pt, y) acceptance can be expressed as a Fourier series of the form ρ(φ) = 1 + 2 vm cos [m(φ−Ψr)] , (43) with reaction-plane angle Ψr. As we have seen, the factor 2 comes from the symmetry on indexm for a real-number density, not an arbitrary choice as suggested in [22]. In this definition V0/2π has been factored from the Fourier series in [21]. The Fourier “coefficients” vm in this form (actually coefficient ratios) are not easily related to the power spectrum. In the 1/n → 0 limit the coefficients are vm = 〈cos(m[φ−Ψr])〉. In the analysis of finite-multiplicity events, reaction- plane angle Ψr defined by the collision (beam) axis and the collision impact parameter is estimated by event plane (EP) angle Ψm, with Ψm derived from event-wise azimuth vector ~̃Qm (conventional flow vector) ~̃Qm = ri~u(mφi) ≡ Q̃m~u(mΨm). (44) The finite-multiplicity event-wise FT is ρ̃(φ) = 1 + 2 q̃m cos [m(φ−Ψm)] . (45) with q̃m = 〈cos(m[φ−Ψm])〉; e.g., q̃2 ≃ α̃2 (cf. Eq. (34)). According to the conventional description the EP an- gle is biased by the presence of self pairs, unfortunately termed the “autocorrelation effect” or simply “autocor- relation” in conventional flow analysis [19, 22], whereas autocorrelations and cross-correlations are distributions on difference variables used for decades in statistical anal- ysis to measure correlation structure on time and space. As in [19], to eliminate “autocorrelations” EP angle Ψmi is estimated for each particle i from complementary flow vector ~Qmi = j 6=i rj~u(2φj) = Qmi~u(mΨmi), a form of subevent analysis with one particle vs n− 1 particles (cf. App. D). In the conventional description the event-plane res- olution results from fluctuations δΨr ≡ Ψm − Ψr of the event-plane angle Ψm (or Ψmi) relative to the true reaction-plane angle Ψr (e.g., due to finite particle num- ber). The EP resolution is said to reduce the observed q̃m relative to the true value ṽm: q̃m = ṽ2m (46) = 〈cos (m[Ψr −Ψm])〉 · ṽm. The EP resolution (first factor, second line) is obtained in a conventional flow analysis in two ways: the frequency distribution on Ψm − Ψr discussed in Sec. III C and the subevent method discussed in App. D. A parameteri- zation from the frequency-distribution method reported in [22] is plotted as the dashed curve in Fig. 2 (right panel). There is good agreement with the simple expres- n/(n− 1)Vm/Qm obtained in Eq. (42), which also follows from Eq. (46). Two methods are described for obtaining vm without an event-plane estimate, with the proviso that large event multiplicities in the acceptance are required. The first, in terms of the conventional flow vector, is expressed (with ri → 1) as (Eq. (26) of [22]) Q2m = n̄+ n 2 v̄2m. (47) That expression is constrasted with the exact event-wise treatment, where for each event we can write Q̃2m = n+ cos [m(φi − φj)] (48) = n+ n(n− 1)〈cos(mφ∆)〉 ≡ n+ n(n− 1)ṽ2m → Q2m = n̄+ n(n− 1)ṽ2m = n̄+ V 2m Note the similarity with fluctuations measured by num- ber variance σ2n = n̄+∆σ n, where the second term on the RHS is an integral over two-particle number correlations and the first is the uncorrelated Poisson reference (again, the self-pair term in the autocorrelation). There are substantial differences between the two Q2m formulations above, especially for smaller multiplicities. Eq. (48) is unbaised for all n and provides a simple way to obtain V 2m = n(n− 1)ṽ2m = Q2m− n̄. The conventional method uses a complex fit to the frequency distribution on Q̃2m to estimate Vm as in [21]. Why do that when such a simple alternative is available? E. v2 centrality dependence In [23] the expected trend of v2 with A-A centrality for different collision systems is discussed in a hydro context. It is stated that the v2 centrality trend should reveal the degree of equilibration in A-A collsions. The centrality dependence of v2/ǫ should be sensitive to the “physics of the collision”—the nature of the constituents (hadrons or partons) and their degree of thermalization or collec- tivity. “It is understood that such a state requires (at least local) thermalization of the system brought about by many rescatterings per particle during the system evolution...v2 is an indicator of the degree of equilibra- tion.” Thermalization is related to the number of rescatter- ings, which also strongly affects elliptic flow according to this hydro interpretation. In the full hydro limit, corre- sponding to full thermalization where the mean free path λ is much smaller than the flowing system, relation v2 ∝ ǫ is predicted, with ǫ the space eccentricity of the initial A-A overlap region [20]. Conversely, in the low-density limit (LDL), where λ is comparable to or larger than the system size, a different model predicts the relation v2 ∝ ǫA1/3/λ, where A1/3/λ estimates the mean num- ber of collisions per “particle” during system evolution to kinetic decoupling [24]. In the LDL case v2 ∝ ǫ 1S where S = πRxRy is the (weighted) cross-section area of the collision and ǫ = R2y−R R2y+R is the spatial eccentricity. Those trends are further discussed in App. E. According to the combined scenario, comparison of the centrality dependence of v2 at energies from AGS to RHIC may reveal a transition from hadronic to ther- malized partonic matter. The key expectation is that at some combination(s) of energy and centrality v2/ǫ tran- sitions from an LDL trend (monotonic increase) to hydro (saturation), indicating (partonic) equilibration. However, it is important to note two things: 1) That overall description is contingent on the strict hydro sce- nario. If the quadrupole component of azimuth correla- tions arises from some other mechanism then the descrip- tions in [20] and [24] are invalid, and v2 does not reveal the degree of thermalization. 2) v2 is a model-dependent and statistically-biased quantity motivated by the hydro scenario itself. The model-independent measure of az- imuth quadrupole structure is V 22 /n̄ ≡ n̄ v22 (defining an unbiased v2). It is important then to reconsider the az- imuth quadrupole centrality and energy trends revealed by that measure to determine whether a hydro interpre- tation is a) required by or even b) permitted by data. IV. IS THE EVENT PLANE NECESSARY? A key element of conventional flow analysis is estima- tion of the reaction plane and the resolution of the esti- mate. We stated above that determination of the event plane is irrelevant if averaged quantities are extracted from an ensemble of event-wise estimates. The reaction- plane angle is relevant only for study of nonflow (minijet) structure relative to the reaction plane on φΣ, the sum (pair mean azimuth) axis of (φ1, φ2). In contrast to con- cerns about low multiplicities in conventional flow analy- sis, proper autocorrelation techniques accurately reveal “flow” correlations (sinusoids) and any other azimuth correlations, even in p-p collisions and even within small kinematic bins. In this section we examine the necessity of the event plane in more detail. The reaction plane (RP), nominally defined by the beam axis and the impact parameter between centers of colliding nuclei, is determined statistically in each event by the distribution of participants. The RP is estimated by the event plane (EP), defined statistically in each event by the azimuth distribution of final-state particles in some acceptance. We now consider how to extract vm relative to a reaction plane estimated by an event plane in each event. Several different flow measures are implicitely defined in conventional flow analysis vm ≡ cos(m[φ−Ψr]) ideal case, 1/n→ 0 (49) ṽ2m ≡ 〈cos2(m[φ−Ψr])〉 unbiased estimate q̃m ≡ 〈cos(m[φ−Ψm])〉 self-pair bias ṽ′m ≡ 〈cos(m[φi −Ψmi])〉 reduced bias ṽ′m is the event-wise result of a conventional flow analysis. The ensemble average ṽ′m must be corrected for the “EP resolution” which we now determine. The basic event-wise quantities, starting with an inte- gral over two-particle azimuth space, are Ṽ 2m ≡ n,n−1 ~u(mφi) · ~u(mφj) (50) = n(n− 1)〈cos(mφ∆)〉 = n(n− 1)〈cos2(m[φ−Ψr])〉 ≡ n(n− 1)ṽ2m. For the limiting case of subevents A and B with A a single particle the subevent azimuth vector complementary to particle i is ~̃Qmi ≡ j 6=i ~u(mφj) = Q̃mi~u(mΨmi). (51) We make the following rearrangment Ṽ 2m = ~u(mφi) · j 6=i ~u(mφj) (52) Q̃mi cos(m[φi −Ψmi]) ≡ nQ̃′m〈cos(m[(φ −Ψ′m])〉 n(n− 1)ṽ2m = nQ̃′m ṽ′m ṽm = ṽ and, since Q̃′m ≃ n− 1 + (n− 1)(n− 2)〈cos(mφ∆)〉, we identify the EP resolution as 〈cos(m[Ψ′m −Ψr])〉 = where the primes refer to a subevent with n−1 particles. That expression, for full events with multiplicity n, is plotted in Fig. 2 (right panel) for several choices of n. An n-independent universal curve on V 2m/n̄ is multiplied by n-dependent factor n/(n− 1), where n is the number of samples in the event or subevent. The dashed curve is the parameterization from [22]. The right panel indicates that for large nṽ2m single- particle reaction-plane estimates can provide a “flow” measurement with manageable bias. For small nṽ2m the EP resolution averaged over many events is itself a “flow” measurement, even though the reaction plane is inacces- sible in any one event. Ṽ 2m is determined from the same underlying two-particle correlations by other means—the only difference is how pairs are grouped across events. In App. D the EP resolution is determined for the case of equal subevents A and B From this exercise we conclude that event-plane deter- mination is irrelevant for the measurement of cylindrical multipole moments (“flows”). Following a sequence of analysis steps in the conventional approach based on de- termination of and correction for the EP estimate, the event plane cancels out of the flow measurement. What results from the conventional method is approximations to signal components of power-spectrum elements which can be determined directly in the form V 2m/n̄ = n̄ v obtained with the autocorrelation method, which defines an unbiased version of vm. Event-plane determination can be useful for study of other event-wise phenomena in relation to azimuth multipoles. V. 2D (JOINT) AUTOCORRELATIONS We now return to the more general problem of angular correlations on (η1, η2, φ1, φ2). We consider the analysis of azimuth correlations based on autocorrelations, power spectra and cylindrical multipoles without respect to an event plane in the context of the general Fourier trans- form algebra presented in Sec. II. We seek a comprehen- sive method which treats η and φ equivalently. In conventional flow analysis there are two concerns beyond the measurement of flow in a fixed angular accep- tance: a) study flow phenomena in multiple narrow ra- pidity bins to characterize the overall “three-dimensional event shape,” analogous to the sphericity ellipsoid but admitting of more complex shapes over some rapidity in- terval, and b) remove nonflow contributions to flow mea- surements as a systematic error. Maintaining adequate bin multiplicities to avoid bias is strongly emphasized in connection with a). The contrast between such individ- ual Fourier decompositions on single-particle azimuth in single-particle rapidity bins and a comprehensive analy- sis in terms of two-particle joint angular autocorrelations is the subject of this section A. Stationarity condition In Fig. 3 we show two-particle pair-density ratios r̂ ≡ ρ/ρref on (η1, η2) (left panel) and (φ1, φ2) (right panel) for mid-central 130 GeV Au-Au collisions [12]. The hat on r̂ indicates that the number of mixed pairs in ρref has been normalized to the number of sibling pairs in ρ. In each case we observe approximate invariance along sum axes ηΣ = η1 + η2 and φΣ = φ1 + φ2. In time-series analysis the equivalent invariance of correlation structure on the mean time is referred to as stationarity, implying that averaging pair densities along the sum axes loses no information. The resulting averages are autocorrelation distributions on difference axes η∆ and φ∆. 0.9996 0.9998 1.0002 1.0004 1.0006 1.0008 1.001 0.999 0.9992 0.9995 0.9997 1.0002 1.0005 1.0007 1.001 1.0012 1.0015 FIG. 3: Normalized like-sign pair-number ratios r̂ = ρ/ρref from central Au-Au collisions at 130 GeV for (η1, η2) (left panel) and (φ1, φ2) (right panel) showing stationarity— approximate invariance along sum diagonal x1 + x2. In Fig. 3 (right panel) one can clearly see the cos(2φ∆) structure conventionally associated with elliptic flow (quadrupole component). However, there are other con- tributions to the angular correlations which should be distinguished from multipole components, accomplished accurately by combining the two angular correlations into one joint angular autocorrelation. B. Joint autocorrelation definition With the autocorrelation technique the dimensional- ity of pair density ρ(η1, η2, φ1, φ2) can be reduced from 4D to 2D without information loss provided the dis- tribution exhibits stationarity. Expressing pair density ρ(η1, η2, φ2, φ2) → ρ(ηΣ, η∆, φΣ, φ∆) in differential form d4n/dx4 we define the joint autocorrelation on (η∆, φ∆) ρA(η∆, φ∆) ≡ dη∆dφ∆ d2n(ηΣ, η∆, φΣ, φ∆) dηΣdφΣ ηΣ φΣ by averaging the 4D density over ηΣ and φΣ within a detector acceptance. The autocorrelation averaging on ηΣ is equivalent to 〈dn/dη〉η ≈ n(∆η)/∆η at η = 0. The magnitude is still the 4D density d4n/dη1dη2dφ1dφ2, but it varies only on the two difference axes (η∆, φ∆). In Fig. 4 we illustrate two averaging schemes [18]. In the left panel we show the averaging procedure applied to histograms on (x1, x2) as in Fig. 3. Index k denotes the position of an averaging diagonal on the difference axis. In the right panel we show a definition involving pair cuts applied on the difference axes. Pairs are his- togrammed directly onto (η∆, φ∆). Periodicity on φ im- plies that the averaging interval on φΣ/2 is 2π indepen- dent of φ∆. However, η is not periodic and the averaging interval on ηΣ/2 is ∆η − |η∆|, where ∆η is the single- particle η acceptance [18]. The autocorrelation value for average FIG. 4: Autocorrelation averaging schemes on xΣ = x1 + x2 for a prebinned single-particle space (left panel) and for pairs accumulated directly into bins on the two-particle difference axis (right panel). a given x∆ is the bin sum along that diagonal divided by the averaging interval. C. Autocorrelation examples In Fig. 5 we show pt joint angular autocorrelations for Hijing central Au-Au collisions at 200 GeV [25]. The left panel shows quench-on collisions. The right panel shows quench-off collisions. Aside from an amplitude change Hijing shows little change in correlation structure from N-N to central Au-Au collsions. Those results can be contrasted with examples from an analysis of real RHIC data [25] shown in Fig. 6. 0.002 0.004 0.006 0.008 0.012 0.002 0.004 0.006 0.008 0.012 FIG. 5: 2D pt angular autocorrelations from Hijing central Au-Au collisions at 200 GeV for quench on (left panel) and quench off (right panel) simulations. In N-N or p-p collisions there is no obvious quadrupole component. However, that possibility is not quanti- tatively excluded by the data and requires careful fit- ting techniques. For correlation structure which is not sinusoidal there is no point to invoking the Wiener- Khintchine theorem on φ∆ and transforming to a power spectrum. Instead, model functions specifically suited to such structure (e.g., 1D and 2D gaussians) are more ap- propriate. The power-spectrum model is appropriate for some parts of ρA, depending on the collision system. We consider hybrid decompositions in Sec. VII. D. Comparison with conventional methods We can compare the power of the autocorrelation tech- nique with the conventional approach based on EP es- timation in single-particle η bins. There are concerns in the single-particle approach about bias or distortion from low bin multiplicities. In contrast, the autocorrela- tion method is applicable to any collision system with any multiplicity, as long as some pairs appear in some events. The method is minimally biased, and the statistical error in ∆ρA/ ρref is determined only by the number of bins and the total number of events in an ensemble. There is interest within the conventional context in “correlating” flow results in different η bins. “A three- dimensional event shape can be obtained by correlat- ing and combining the Fourier coefficients in different longitudinal windows” [21]. That goal is automatically achieved with the joint autocorrelation technique but has not been implemented with the conventional method. The content of an autocorrelation bin at some η∆ rep- resents a covariance averaged over all bin pairs at ηa and ηb satisfying η∆ = ηa − ηb. If a flow component is iso- lated (e.g., V 22 /n̄), a given autocorrelation element rep- resents normalized covariance na nbṽ n̄an̄b, where ṽ2mab would be the event-wise result of a ‘subevent’ or ‘scalar-product’ analysis between bins a and b. Averaged over many events one obtains good resolution on rapid- ity and azimuth for arbitrary structures, without model dependence or bias. VI. FLOW AND MINIJETS (NONFLOW) The relation between azimuth multipoles and minijets is a critically important issue in heavy ion physics which deserves precise study. We should carefully compare multipole structures conventionally attributed to hydro- dynamic flows and parton fragmentation dominated by minijets in the same analysis context. The best arena for that comparison is the 2D (joint) angular autocorrelation and corresponding power-spectrum elements. Before pro- ceeding to autocorrelation structure we consider nonflow in the conventional flow context A. Nonflow and conventional flow analysis In conventional flow analysis azimuth correlation struc- ture is simply divided into ‘flow’ and ‘nonflow,’ where the latter is conceived of as non-sinusoidal structure of indeterminant origin. The premise is that all sinusoidal structure represents flows of hydrodynamic origin. It is speculated that nonflow is due to resonances, HBT and jets, including minijets. Various properties are assigned to nonflow which are said to distinguish it from flow [26]. Nonflow is by definition non-sinusoidal and is not corre- lated with the RP, thus it can appear perpendicular to the RP. The multiplicity dependence of nonflow is said to be quite different from flow, where “multiplicity depen- dence” can sometimes be read as centrality dependence. For instance, ~̃Qma · ~̃Qmb = Ṽma Ṽmb cos(Ψma−Ψmb) if A, B are disjoint, since there are no self pairs. The ensemble average then measures covariance V 2mab. For m = 2 it is claimed that the nonflow component of V 2 2ab is ∝ n̄ c [22]. Therefore, V 22 /n̄ ∝ c, a constant for nonflow—no cen- trality dependence. But V 2m/n̄ ∝ ∆ρA/ ρref which, for the minijets dominating nonflow, is very strongly depen- dent on centrality [9, 12]; the conventional assumption is incorrect. The above notation is inadequate because minijets (nonflow) should not be included in the Fourier power spectrum. They should be modeled by different functional forms which we consider in the next section. B. Cumulants Another strategy for isolating flow from nonflow is to use higher cumulants [27]. The basic assumption (a phys- ical correlation model) is that flow sinusoids are collec- tive phenomenon characteristic of almost all particles, whereas nonflow is a property only of pairs, termed “clus- ters.” That scenario is said to imply that vm should be the same no matter what the multiplicity, whereas non- flow should fall off as some inverse power of n. For instance, by subtracting v2[4] (four-particle cumu- lant) from v2[2] (two-particle cumulant) one should ob- tain “nonflow” as the difference (cf. Eq. (10) of [26]). In Fig. 31 of [26] we find a plot of g2 = Npart (v 2 [2]−v22 [4]) ∝ Npart/nch ×∆ρ/ ρref . Multiplying g2 by n part we obtain a measure of minijet correlations per participant pair. That ‘nonflow’ component increases rapidly with centrality (and therefore n), consistent with actual mea- surements of minijet centrality trends. The incorrect fac- tor in the definition of g2 removes a factor 2x increase from peripheral to central in the minijet centrality trend, thus suppressing the centrality dependence of ‘nonflow.’ C. Counterarguments The conventional flow analysis method requires a com- plex strategy to distinguish flow from nonflow in projec- tions onto 1D azimuth difference φ∆. An intricate and fragile system results, with multiple constraints and as- sumptions. The assumptions are not a priori justified, and must be tested. ‘Flow’ isolated with those assump- tions can and does contain substantial systematic errors. Claims about the multiplicity (centrality) dependence of ‘nonflow’ (independent of centrality or slowly varying) are unsupported speculations without basis in experiment. In fact, detailed measurements of minijet centrality de- pendence [9, 12] are quite inconsistent with typical as- sumptions about nonflow. Multiplicity (centrality) de- pendence of flow measurements is further compromised by biases resulting from improper statistical methods, especially true for small multiplicities or peripheral colli- sions. Such biases can masquerade as physical phenom- Finally, it is assumed that nonflow has no correlation with the RP, thus implying the ability of and need for the EP to distinguish flow from nonflow. But nonflow (minijets) should be strongly correlated with the EP (jet quenching), and such correlations should be measured. That is the main subject of paper II in this two-part se- ries. Precise decomposition of angular correlations into ‘flow’ sinusoids and minijet structure is realized with 2D joint angular autocorrelations combined with proper sta- tistical techniques. Non-flow is a suite of physical phenomena, each wor- thy of detailed study. In the conventional approach this physics is seen in limited ways by various projections and poorly-designed measures and described mainly by spec- ulation. With more powerful analysis methods it is pos- sibl to separate flow from the various sources of ‘nonflow’ reliably and identify those sources as interesting physical phenomena. VII. STRUCTURE OF THE JOINT ANGULAR AUTOCORRELATION IN A-A COLLISIONS We now return to the 2D angular autocorrelation. By separating its structure into a few well-defined compo- nents we obtain an accurate separation of multipoles, minijets and other phenomena. Minijets and “flows” can be compared quantitatively within the same analysis con- text. Each bin of an autocorrelation is a comparison of two “subevents.” The notional term “subevent” represents a partition element in conventional math terminology (e.g., topology, cf. Borel measure theory). An “event” is a dis- tribution in a bounded region of momentum space (de- tector acceptance), and a subevent is a partition element thereof. A distribution can be partitioned in many ways: by random selection, by binning the momentum space, by particle type, etc. A uniform partition is a binning, and the set of bin entries is a histogram. A bin in an angular autocorrelation represents an av- erage over all bin pairs in single-particle space separated by certain angular differences (η∆, φ∆). The bin contents represent normalized covariances averaged over all such pairs of bins. The notional “scalar product method,” re- lating two subevents in conventional flow analysis, is al- ready incorporated in conventional mathematical meth- ods developed over the past century as covariances in bins of an angular autocorrelation. Using 2D angular au- tocorrelations we easily and accurately separate nonflow from flow. “Nonflow” so isolated has revealed the physics of minijets—hadron fragments from the low-momentum partons which dominate RHIC collisions. A. Minijet angular correlations Minijet correlations are equal partners with multi- pole correlations on difference-variable space (η∆, φ∆). Minimum-bias jet angular correlations (dominated by minijets) have been studied extensively for p-p and Au- Au collisions at 130 and 200 GeV [9, 10, 11, 12]. Those structures dominate the “nonflow” of conventional flow analysis. In p-p collisions minijet structure—a same-side peak (jet cone) and away-side ridge uniform on η∆— are evident for hadron pairs down to 0.35 GeV/c for each hadron. Parton fragmentation down to such low hadron momenta is fully consistent with fragmentation studies over a broad range of parton energies (e.g., LEP, HERA) [28]. 80-90% 0.001 0.002 0.003 0.004 0.005 45-55% -0.005 0.005 0.015 80-90% 0.001 0.002 0.003 0.004 45-55% -0.002 0.002 0.004 0.006 0.008 0.012 FIG. 6: 2D pt angular autocorrelations from Au-Au collisions at 200 GeV for 80-90% central collisions (left panels) and 45- 55% central collisions (right panels). In the lower panels si- nusoids cos(φ∆) and cos(2φ∆) have been subtracted to reveal “nonflow” structure. In Fig. 6 we show autocorrelations obtained by inver- sion of pt fluctuation scale (bin-size) dependence [9]. The upper-left panel is 80-90% central and the upper-right panel is 45-55% central Au-Au collisions. Correlation structure is dominated by a same-side peak and mul- tipole structures (sinusoids). Subtracting the sinusoids reveals the minijet structure in the bottom panels and illustrates the precision with which flow and nonflow can be distinguished. The negative structure surrounding the same-side peak at lower right is an interesting and unan- ticipated new feature [9]. B. Decomposing 2D angular autocorrelations: a controlled comparison Based on extensive analysis [9, 10, 11, 25] we find three main contributions to angular correlations in RHIC nuclear collisions: 1) transverse fragmentation (mainly minijets), 2) longitudinal fragmentation (modeled as “string” fragmentation), 3) azimuth multipoles (flows). Longitudinal fragmentation plays a reduced role in heavy ion collisions. In this study we focus on the interplay be- tween 1) and 3), transverse parton fragmentation and azimuth multipoles, as the critical analysis issue for az- imuth correlations in A-A collisions. The 2D joint autocorrelation ρA(η∆, φ∆) is the basis for decomposition. The criteria for distinguishing az- imuth multipoles from minijet structure are η∆ depen- dence and sinusoidal φ∆ dependence. Structure with si- nusoidal φ∆ dependence and η∆ invariance is assigned to azimuth multipoles. Other structure, varying generally on (η∆, φ∆), is assigned in this exercise to minijets. We adopt the decomposition ρA(η∆, φ∆) = ρj(η∆, φ∆) + ρm(φ∆), (55) where j represents (mini)jets and m represents multi- poles. That decomposition is reasonable within a limited pseudorapidity acceptance, e.g., the STAR TPC accep- tance [29]. Over a larger acceptance other separation criteria must be added. To illustrate the separation process we construct artifi- cial autocorrelations combining flow sinusoids and mini- jet structure with centrality dependence taken from mea- surements. We add statistical noise appropriate to a typ- ical event ensemble of a few million Au-Au collisions. We then fit the autocorrelations with model functions and χ2 0.1994E-03/ 23 P1 0.1456E-01 P2 0.4811E-02 -0.02 -0.01 0.005 0.015 0.025 0 2 4 FIG. 7: Simulated three-component 2D angular autocorrela- tion for 80-90% central Au-Au collisions at 200 GeV (upper left), model of data distribution from fitting (upper-right), autocorrelation with eta-acceptance triangle imposed (lower left) and sinusoid fit to 1D projection of lower-left panel (lower-right). minimization. We compare the resulting fit parameters with the input parameters. We then project the 2D au- tocorrelations onto φ∆ and fit the results with a sinusoid. The result of the 1D fit represents the product of a con- ventional flow analysis if the EP resolution is perfectly corrected and there is no statistical bias in the method. We then compare the resulting sinusoid amplitudes. In Fig. 7 we show an analysis for 80-90% central Au-Au collisions, which are nearly N-N collisions. The upper- right panel is an accurate model of data from p-p col- lisions [11]. The upper-left panel is the simpler repre- sentation for this exercise with added statistical noise. The difference in constant offsets is not relevant to the exercise. At lower left is the distribution “seen” by a conventional flow analysis, which simply integrates over (projects) the η dependence. The projection includes a triangular acceptance factor on η∆ imposed on the joint autocorrelation, resulting in distortions that affect the si- nusoid fit. The lower-right panel is the projection onto φ∆ with χ 2 fit. The 1D fit gives ∆ρ[2]/ ρref = 0.0052, compared to 0.0013 from the 2D fit In Fig. 8 we show the same analysis applied to 5-10% Au-Au collisions. The model distribution, derived from observed data trends, is dominated by a dipole term ∝ cos(φ∆) and an elongated same-side jet peak, although the combination closely mimics a quadrupole ∝ cos(2φ∆) (elliptic flow). The lower-left panel shows the effect of the η-acceptance triangle on η∆ applied to the upper-right panel which is implicit in any projection onto φ∆ (obvious in this 2D plot). The resulting projection on φ∆ is shown 0.2331E-01/ 23 P1 0.8826E-01 P2 0.1026 0 2 4 FIG. 8: Same as the previous figure but for 5-10% central Au- Au collisions at 200 GeV. Note the pronounced effect of the η-acceptance triangle in the lower-left panel (relative to the upper-right panel) resulting from projection of space (η1, η2) onto its difference axis. at lower right. The 1D fit gives ∆ρ[2]/ ρref = 0.103, compared to 0.060 from the 2D fit. The differences be- tween 1D and 2D fits are much larger than the differences between input and fitted parameters in the previous ex- ercise. They reveal the limitations of conventional flow analysis. In Fig. 9 we give a summary of results for twelve cen- trality classes, nine 10% bins with mean values 95% · · · 15%, plus two 5% bins with mean values 7.5% and 2.5%. The twelfth class is b = 0 (0%), constructed by extrapo- lating the parameterizations from data. The solid curves and points represent the parameters inferred from 2D fits to the joint angular autocorrelations. The dashed curves represent the model parameters used to construct the simulated distributions. There is good agreement at the percent level. m = 1 projection onto φ∆ peak amplitude 2 4 6 2 4 6 FIG. 9: A parameter summary for the previous two figures. The solid curves are input model parameters for m = 1 and m = 2 sinusoids cos(mφ∆) and a same-side 2D gaussian with two widths and peak amplitude which approximate 200 GeV Au-Au collisions. The dashed curves are the results of fits to the model 2D autocorrelations exhibiting excellent accuracy. The dash-dot curve represents 1D fits to projections on φ∆ (previous lower-right panels) corresponding to conventional flow analysis. The fits to 1D projections on φ∆ (dash-dot curve) how- ever differ markedly from the 2D fit results and the in- put parameters. The differences are very similar to the changes of conventional flow measurements with different strategies to eliminate “nonflow.” This exercise demon- strates that with the 2D autocorrelation there is no guess- work. We can distinguish the multipole contributions from the minijet contributions. The 2D angular autocor- relation provides precise control of the separation. C. v2 in various contexts In Fig. 10 we contrast the results of 1D conven- tional flow analysis (dashed curves) and extraction of the quadrupole amplitude from the 2D angular autocorrela- tion (solid curves). We make the correspondence ∆ρA[2]√ 2π n̄ n̄v22 among variables, with n̄/2π modeling ρ0 = d2n/dηdφ. That expression defines a minimally-biased v2. We make the comparison in four plotting formats. The upper-right panel is most familiar from conventional flow analysis, where v2 is plotted vs participant nucleon number. The trends can be compared with Fig. 13 of [30] (open circles vs solid stars). Also included in that panel is the trend v2 ∼ 0.22 ǫ predicted by hydro for thermalized A-A col- lisions at 200 GeV (dotted curve, [31]). npart 0.22 ε projection onto φ∆ 2D fit 2 4 6 0 100 200 300 400 2 4 6 0 2 4 6 FIG. 10: The quadrupole component in various plotting con- texts derived from the previous model exercise. Conventional flow measure v2 is shown in the upper panels. ∆ρ[2]/ based on Pearson’s normalized covariance is shown in the lower panels. The upper-right panel shows v2 vs npart in the conventional plotting format. The lower-left panel repeats the left panel of the previous figure, with conventional 1D fit results shown as the dashed curve. Dashed and solid curves correspond in all panels. ν estimates the mean N-N encoun- ters per participant pair. The dotted curve at lower right is the error function used to generate the model quadrupole am- plitudes. It’s correspondent for v2 is at upper left. The dotted curve at upper right is eccentricity ǫ from a parameterization. The remaining panels are plotted on parameter ν = 2nbin/npart, the ratio of N-N binary encounters to par- ticipant pairs which estimates the mean participant path length in number of encountered nucleons, a geometri- cal measure. Comparing the upper panels we see that ν treats peripheral and central collisions equitably, whereas npart or ncharge compresses the important peripheral re- gion into a small interval. In the lower-left panel we plot per-particle density ra- tio ∆ρA[2]/ ρref vs ν. That quantity, when extracted from 2D fits, rises from near zero for peripheral collisions to a maximum for mid-central collisions, falling toward zero again for b = 0. In contrast to v2, which is the square root of a per-pair correlation measure, the per- particle density ratio reflects the trend for “flow” in the sense of a current density. “Flow” is small for periph- eral collisions and grows rapidly with increasing nucleon path length. The trend with centrality is intuitive. The values obtained from the 1D projection per conventional flow analysis (dashed curve) are consistently high, espe- cially for central collisions, exhibiting a strong systematic bias. The 1D fit procedure (identical to the “standard” and two-particle flow methods) confuses minijet structure with quadrupole structure. In the lower-right panel we show the density ratio di- vided by initial spatial eccentricity ǫ defined by a pa- rameterization derived from a Glauber simulation and plotted as the dotted curve in the upper-right panel [31]. The trend from 2D fits (solid curve) is closely approx- imated by a simple error function (dotted curve) with half-maximum point at the center of the ν range. In fact, the dotted curve is the basis for generating the input quadrupole amplitudes for our model, and the small devi- ation of the solid curve from the dotted curves in upper- left and lower-right panels reveals the systematic error or bias in the 2D fitting procedure (∼ 20% at ν ∼ 1, < 5% at ν = 5.8). The dashed curves from the conventional 1D fits show large relative deviations from the input trend, especially for peripheral collisions where the emergence of collectivity is of interest and for central collisions where the issue of thermalization is most important. Comparing the solid curve to existing v2 data in the format of the upper-right panel shown in [30] (Fig. 13) indicates that our simple formulation at lower right (dotted curve) is roughly consistent with analysis of flow data based on four-particle cumulants. The re- sult from data generated with the same model and ana- lyzed with the conventional flow analysis method (dashed curve in upper-right panel) also agrees with the conven- tional method applied to real RHIC data. Our model may therefore indicate an underlying simplicity to the quadrupole mechanism which is not hydrodynamic in ori- gin. The model centrality trend for v2 is certainly incon- sistent with the hydro expectation v2 ∝ ǫ [20, 23], as demonstrated in the upper-right panel (cf. App. E). VIII. DISCUSSION A. Conventional flow analysis The overarching premise of conventional flow analy- sis is that in the azimuth distribution of each collision event lies evidence of collective phenomena which must be discovered to establish event-wise thermalization. The 1/n → 0 hydro limit shapes the analysis strategy, and flow manifestations are the principal goal. Finite event multiplicities are seen as a major source of systematic error, as are correlation structures other than flow. Mul- tiple strategies are constructed to deal with non-flow and finite multiplicities. A stated advantage of the conven- tional method is that the Fourier coefficients can be cor- rected. The great disadvantage is theymust be corrected. From the perspective of two-particle correlation anal- ysis, especially in the context of autocorrelations and power spectra, the conventional program leaves much to be desired. The conventional analysis is essentially an attempt to measure two-particle correlations with single- particle methods combined with RP estimation, similar to the use of trigger particles in high-pt jet analysis. But, by analysis of the algebraic structure of conventional flow analysis we have demonstrated that RP estimation does not matter to the end result. Without a proper statistical reference conventional analysis results contain extrane- ous contributions from the statistical reference which are partially ‘corrected’ in a number of ways. The improper treatment of random variables incorporates sources of multiplicity-dependent bias in measurements, and the fi- nal results are questionable. Flow measure vm is nominally the square root of per- pair correlation measure V 2m/n(n− 1). vm centrality trends are thus nonintuitive and misleading (e.g., “el- liptic flow” decreases with increasing A-A centrality). The situation is similar to per-pair fluctuation measure Σpt , which provides a dramatically misleading picture of pt fluctuation dependence on collision energy [10]. In contrast, per-particle correlation measures provide intu- itively clear results and often make dynamical correla- tion mechanisms immediately obvious. In particular, the mechanisms behind “nonflow” in the form of minijets are clearly apparent when correlations are measured by per- particle normalized covariance density ∆ρ/ ρref in a 2D autocorrelation. B. Autocorrelations and nonflow In conventional flow analysis it is proposed to measure flow in narrow rapidity bins (“strips”) so as to develop a three-dimensional picture of event structure. However, there has been little implementation of that proposal. In contrast, the joint angular autocorrelation by construc- tion contains all possible covariances among pseudora- pidity bins within a detector acceptance. The ideal of full event characterization is thereby realized. The angular autocorrelation is the optimum solution to a geometry problem—how to reduce the six-dimensional two-particle momentum space to two-dimensional sub- spaces with minimum distortion or information loss. The autocorrelation is the unique solution to that problem, involving no model assumptions. The ensemble-averaged angular autocorrelation contains all the correlation infor- mation obtainable from a conventional flow analysis, but with negligible bias and no sensitivity to individual event multiplicities. Because it is a two-dimensional representation the angular autocorrelation is far superior for separating “flow” (multipoles) from “nonflow” (minijets), as we have demonstrated. A simple exercise demonstrates that sep- aration is complete at the percent level, whereas the con- ventional method admits crosstalk at the tens of percent level. Precise separation leads to new physics insights from the multipoles and minijets so revealed. C. Collision centrality dependence Collision centrality dependence is of critical impor- tance in the comparison of flow and minijets. Parton col- lisions and hydrodynamic response to early pressure have very different dependence on impact parameter and col- lision geometry, especially for peripheral collisions. Pe- ripheral A-A collisions should approach p-p (N-N) col- lisions, and correlation structure may change rapidly in mid-peripheral collisions as collective phenomena develop there. The possible onset of collective behavior in mid- peripheral collisions and reduction in more central col- lisions are of major importance for understanding the relation of minijets to flow. The conventional flow analy- sis method is severely limited for peripheral collisions. In contrast, correlation measure ∆ρ/ ρref , centrality measure ν and associated centrality techniques described in [32] are uniquely adapted to cover all centrality regions down to N-N with excellent accuracy. D. Physical interpretations Because similar flow measurement techniques have been applied at Bevalac and RHIC energies with sim- ilar motivations it is commonly assumed that azimuth multipoles have a common source over a broad collision energy range—hydrodynamic flows, collective response to early pressure. The hydro mechanism was proposed as the common element in [20] and persists as the lone interpretation of azimuth multipoles in HI collisions to date. At Bevalac and AGS energies it is indeed likely that azimuth multipoles result from ‘flow’ of initial-state nu- cleons in response to early pressure, with consequent final-state correlations of those nucleons—a true hydro phenomenon. However, at SPS and RHIC energies the source of azimuth multipoles inferred from final-state produced hadrons (mainly pions) may not be hydrody- namic, in contrast to arguments by analogy with lower energies. Other sources of multipole structure should be considered [34, 35]. Multipoles at higher energies could arise at the partonic or hadronic level, early or late in the collision, with collective motion or not, and if collective then implying thermalization or not. The chain of argument most often associated with elliptic flow asserts that observation of flow as a col- lective phenomenon demonstrates that a thermalized medium (QGP) has been formed which responds hy- drodynamically to early pressure and converts an ini- tial configuration-space eccentricity to a corresponding quadrupole moment in momentum space. However, nonflow in the form of minijets provides con- tradictory evidence. Minijet centrality trends indicate that thermalization is incomplete, and substantial mani- festations of initial-state parton scattering remain at ki- netic decoupling [9, 10, 12]. Precision studies of mini- jet centrality dependence (ν dependence) indicate that a large fraction of the minijet structure expected from lin- ear superposition of N-N collisions (no thermalization) persists in central Au-Au collisions. That contradic- tion requires more complete experimental characteriza- tion and careful theoretical study [36]. Arguments based on interpreting the quadrupole com- ponent as hydrodynamic flow exclude alternative phys- ical mechanisms. Aside from minijet systematics there are other hints that a different mechanism might be re- sponsible for azimuth multipoles. In Fig. 10 we showed that flow measurements based on four-particle cumulants (with bias sources and nonflow thereby reduced) are best described by a trend (solid curves) that is inconsistent with the hydro expectation v2 ∝ ǫ. The trend is in- stead simply described in terms of per-particle measure ρref and two shape parameters relative to ǫ. We question the theoretical assumption that ǫ should be simply related to v2 as opposed to some other mea- sure of the azimuth quadrupole component. We expect a priori and find experimentally that variance measures, integrals over two-particle momentum space, more typ- ically scale linearly with geometry parameters. Thus, ∆ρ[2]/ ρref ∝ n̄v22 may be more closely related to ǫ, and the relation may or may not be characteristic of a hydro scenario. IX. SUMMARY In conclusion, we have reviewed Fourier transform the- ory, especially the relation of autocorrelations to power spectra, essential for analysis of angular correlations in nuclear collisions. In that context we have reviewed five papers representative of conventional flow analysis and have related the methods and results to autocorre- lation structure and spherical and cylindrical multipole moments. We have examined the need for event-plane evaluation in correlationmeasurements and find that it is extraneous to measurement of azimuth multipole moments. The EP estimate drops out of the final ensemble average. We have introduced the definition of the 2D (joint) an- gular autocorrelation and considered the distinction be- tween flows (cylindrical multipoles) and nonflow (domi- nated by minijet structure) in conventional flow analysis and criticized the basic assumptions used to distinguish the two in that context. Based on measured minijet and flow centrality trends we have constructed a simulation exercise in which model autocorrelations of known composition are combined with statistical noise from a typical event ensemble and fit with a model function consisting of a few simple com- ponents, first as a 2D autocorrelation and second as a 1D projection on azimuth difference axis φ∆. We show that the 2D fit returns input parameters accurately at the percent level, whereas the 1D fit, representing conven- tional flow analysis, deviates systematically and strongly from the input. Comparisons with published flow data indicate that the observed bias in the simulation is ex- actly the difference attributed to “nonflow” in conven- tional measurements. By comparing our simple algebraic model of quadrupole centrality dependence to data we observe that the trend v2 ∝ ǫ is not met for any collision sys- tem, nor is there asymptotic approach to such a trend. That observation raises questions about the relevance of hydrodynamics to phenomena currently attributed to el- liptic flow at the SPS and RHIC. This work was supported in part by the Office of Sci- ence of the U.S. DoE under grant DE-FG03-97ER41020. APPENDIX A: BROWNIAN MOTION There is a close analogy between Brownian motion and the azimuth structure of nuclear collisions. The long his- tory of Brownian motion and its mathematical descrip- tion can thus provide critical guidance for the analysis of particle distributions. Brownian motion (more gen- erally, random motion of particles suspended in a fluid) was modeled by Einstein as a diffusion process (random walk) [7]. He sought to test the “kinetic-molecular” the- ory of thermodynamics and provide direct observation of molecules. Paul Langevin developed a differential equa- tion to describe such motion, which included a stochastic term representing random impulses delivered to the sus- pended particle by molecular collisions. Jean Perrin and collaborators performed extensive measurements which confirmed Einstein’s predictions and provided definitive evidence for the reality of molecules [8]. 1. The quasi-random walker We model a 2D quasi-random walker (including nonzero correlations) as follows. The walker position is recorded in equal time intervals δt. After n steps, with step-wise displacements r sampled randomly from a bounded distribution, the walker position relative to an arbitrary starting point is, in the notation of this paper, i ri~u(φi), where ri is the i th displacement. The squared total displacement is then R2 = n〈r2〉+ n(n− 1)〈r2 cos(φ∆)〉. (A1) The first term, linear in n (or t), was described by Ein- stein. The second term could represent “drift” of the walker due to deterministic response to an external in- fluence. The composite is then termed “Brownian motion with drift,” a popular model for stock markets and other quasi-random processes. Measuring multipole moments on azimuth in nuclear collisions is formally equivalent to measuring “drift” terms on time in the quasi-random walk of a charged particle suspended in a molecular fluid within a superposition of oscillating electric fields. There are many other applications for Eq. (A1). For a true random walk consisting of uncorrelated steps Einstein expressed 〈r2〉/δt ≡ d ·2D (random walk in d di- mensions) in terms of diffusion coefficient D. The second term 〈r2 cos(φ∆)〉 ≡ (δt)2v2x represents a possible deter- ministic component (correlations), with x̂ the direction of an applied “force.” In that case successive angles φi are correlated, and the result is a macroscopic nonstochastic drift of the walker trajectory. The fractal dimension of a random walk [first term in Eq. (A1)] is df = 2. The trajectory is therefore a “space-filling” curve in 2D configuration space. The appropriate measure of trajectory size is area, and the rate of size increase is the diffusion coefficient (rate of area increase). In contrast, the second term in Eq. (A1) represents a deterministic trajectory whose nominal di- mension is 1 (modulo the extent of curvature, which in- creases the dimension above 1). Therefore, the appropri- ate measure of trajectory size is length, and speed is the correct rate measure. For Brownian motion with drift the trajectory dimension is not well-defined, depending on the relative magnitudes of the drift and stochastic terms, and the concept of speed is therefore ambiguous. Attempts to measure the linear speed of Brownian mo- tion in the nineteenth century failed because of the frac- tal structure of random walks. From the structure of Eq. (A1) the average speed over interval ∆t = nδt is R2/(∆t)2 ∼ 〈r2/(δt)2〉/n, and the limiting case for ∆t = nδt → 0 is the so-called “infinite speed of diffu- sion.” That topological oddity is formally equivalent to the “multiplicity bias” of conventional flow analysis. 2. Brownian motion and nuclear collisions We now consider the close analogy between Einstein’s theory of Brownian motion and the measurement of p2x in a nuclear collision, using directivity as an example. Just as ~R is the vector total displacement of a quasi- randomwalker in 2D configuration space, ~Q1 is the vector total displacement of a quasi-random walker (event-wise particle ensemble) in 2D momentum space. After n steps the squared displacements are R2 = n2 δt2 v′2x = n δt 4D+ n(n− 1) δt2 v2x (A2) Q21 = n 2 p′2x = n 〈p2t 〉+ n(n− 1)p2x. 4Dδt is the increase in area per step of a random walker in 2D configuration space. 〈p2t 〉 is the increase in area per step (per particle) of a random walker in 2D momentum space, playing the same role as the diffusion coefficient. The RHS first term in the first line is the subject of Ein- stein’s 1905 Brownian motion paper. Its measurement by Perrin confirmed the reality of molecules and the validity of Boltzmann’s kinetic theory. As noted, attempts to measure mean speed v′x of a par- ticle in a fluid failed because speed is the wrong rate mea- sure for trajectory size increase. Speed measurements decreased with increasing sample number or observation time. It was not until Einstein’s formulation and later mathematical developments that the topology of the ran- dom walk and its consequences became apparent. Initial attempts at the Bevalac to measure px in the form p using directivity failed for the same reason. Corrections were developed to approximate the unbiased quantity px, and the failure was attributed to multiplicity bias or ‘au- tocorrelations.’ Ironically, the autocorrelation distribu- tion is the ideal method to access the unbiased quantity in either case. 3. Einstein and autocorrelations To provide a statistical description of Brownian motion Einstein introduced the autocorrelation concept with the following language [7]. Another important consideration can be re- lated to this method of development. We have assumed that the single particles are all referred to the same co-ordinate system. But this is unnecessary, since the movements of the single particles are mutually independent. We will now refer the motion of each parti- cle to a co-ordinate system whose origin co- incides at the [arbitrary] time t = 0 with the [arbitrary] position of the center of gravity of the particle in question; with this difference, that [probability distribution] f(x, t)dx now gives the number of the particles whose x co- ordinate has increased between the time t = 0 and the time t = t, by a quantity which lies between x and x+ dx. Einstein’s function f(ξ, τ) is a 2D autocorrelation which satisfies the diffusion equation. The solution is a gaussian on x relative to an arbitrary starting point (thus defining difference variables ξ = x − xstart and τ = t − tstart), with 1D variance σ2ξ = 2Dτ . The au- tocorrelation is sometimes called a two-point correlation function or two-point autocorrelation. The angular auto- correlation is a wide-spread and important analysis tool, e.g., in astrophysics, nuclear collisions and many other fields. 4. Wiener, Khintchine, Lévy and Kolmogorov The names Wiener, Lévy, Kolmogorov and Khintchine figure prominently in the copious mathematics derived from the Brownian motion problem. Norbert Wiener led efforts to provide a mathematical description of Brown- ian motion, abstracted to aWiener process, a special case of a Lévy process (generalization of a discrete random walk to a continuous random process) [37]. The Wiener- Khintchine theorem provides a power-spectrum represen- tation for stationary stochastic processes such as random walks, for which a Fourier transform does not exist. We have acknowledged the theorem with our Eq. (10). The analysis of azimuth structure in nuclear collisions in terms of angular autocorrelations is based on power- ful mathematics developed throughout the past century. Autocorrelations make it possible to study azimuth struc- ture for any event multiplicity down to p-p collisions with as little as two detected particles per event. The effects of “non-flow” can be eliminated from “flow” measurements (and vice versa) without model dependence or guesswork. The Brownian motion problem and Einstein’s fertile so- lution inform two central issues for studies of the correla- tion structure of nuclear collisions: analysis methodology and physics interpretation. APPENDIX B: RANDOM VARIABLES A random variable represents a set of samples from a parent distribution. The outcome of any one sample is unpredictable (i.e., random), but through statistical analysis an ensemble of samples can be used to infer prop- erties (statistics – results of algorithms applied to a set of samples) of the parent distribution. Sums over particles and particle pairs of kinematic quantities are the primary random variables in analysis of nuclear collision data. 1. The algebra of random variables Products and ratios of random variables behave non- intuitively because random variables don’t obey the alge- bra of ordinary variables. E.g., factorization of random variables results in the spawning of covariances. The approximation xy ≃ x̄ ȳ common in conventional flow analysis is a source of systematic error (bias) because xy = x̄ ȳ + xy − x̄ ȳ. The omitted term is a covariance. Such covariances play a role in statistics similar to QM commutators, with 1/n↔ ~. Conventional flow analysis assumes the 1/n → 0 limit for some random variables, and the results are undependable for small multiplicities. Similarly, improper treatment of ratios of random vari- ables results in infinite series of covariances. E.g., x/n = δx · δn x̄ n̄ x · (δn)2 x̄ n̄2 + · · · ), (B1) with (δn)2/n̄ ≡ σ2n/n̄ ∼ 1 − 2. Thus, the common ap- proximation x/n ≃ x̄/n̄ can result in significant n- and physics-dependent (x-n covariances) bias for small n. In this paper we distinguish between event-wise and ensemble-averaged quantities and do not employ en- semble averages of ratios of random variables. We in- clude event-wise factorizations and ratios only to sug- gest qualitative connections with conventional flow anal- ysis. E.g., we consider Ṽ 2m ≡ n(n − 1)〈cos(mφ∆)〉 with 〈cos(mφ∆)〉 = 〈cos2(m[φ − Ψr])〉 ≡ ṽ2m. But, ṽ2m 6= V 2m/n(n− 1) 6= v̄2m. vm as typically invoked in conven- tional flow analysis is not a well-defined statistic. 2. Statistical references The concept of a statistical reference is largely absent from conventional flow analysis. By ‘statistical reference’ we mean a quantity or distribution which represents an uncorrelated system, a system consistent with indepen- dent samples from a fixed parent distribution (central limit conditions [5]). Concerns about ‘bias’ from low mul- tiplicities [19, 21, 22] typically relate to the presence of an unsubtracted and unacknowledged statistical reference in the final result. Finite multiplicity fluctuations are then said to produce systematic errors, false azimuthal anisotropies, a problem masking true collective effects. In the limit 1/n → 0 the statistical reference may in- deed become negligible compared to the true correlation structure. However, its presence for nonzero 1/n is a po- tential source of systematic error which may block access to important small-multiplicity systems (peripheral col- lisions and/or small kinematic bins). In general, if the statistical reference is not correctly subtracted the result is increasingly biased with smaller multiplicities. Identi- fication and subtraction of the proper reference is one of the most important tasks in statistical analysis. Use of the term ‘statistical’ to mean ‘uncorrelated’ is misleading (e.g., ‘statistical’ vs ‘dynamical’). All ran- dom variables and their fluctuations about the mean are ‘statistical.’ Some random variables and their statistics are reference quantities, representing systems that are by construction uncorrelated (independent sampling from a fixed parent). We therefore label statistical reference quantities ‘ref,’ not ‘stat.’ 3. Random variables and Fourier analysis In the context of Fourier analysis the basic finite- number (Poisson) statistical reference is manifested as the delta-function component in the autocorrelation den- sity Eq. (13) and the white-noise constant term n〈r2〉 in the event-wise power spectrum. Other reference compo- nents may arise from two-particle correlations which are not of interest to the analysis (e.g., detector effects) and which may be revealed in mixed-pair distributions. A clear distinction should always be maintained between the reference and the sought-after correlation signal. Careful attention to random-variable algebra is es- pecially important in a Fourier analysis. The power- spectrum elements and autocorrelation density must sat- isfy the transform equations both for each event and after ensemble averaging. In conventional flow analysis that condition is often not satisfied. For instance, Ṽ 2m and V satisfy the FT transforms and Wiener-Khintchine the- orem before and after ensemble averaging respectively, whereas the vm do not. 4. Minimally-biased random variables It is frequently stated in the conventional flow liter- ature that flow analysis must insure sufficiently large multiplicities. The operating assumption in the design of conventional flow methods is the continuum limit 1/n → 0, with inevitable bias for smaller multiplicities. However, careful reference design and algebraic manipu- lation of random variables makes possible precise treat- ment of event-wise multiplicities down to n = 1. Some statistical measures perform consistently no matter what the sample number. The full multiplicity range is essen- tial to measure azimuth multipole evolution with central- ity down to N-N and p-p collisions, so that A-A “flow” phenomena may be connected to phenomena observed in elementary collisions and understood in a QCD context. Since multiplicity necessarily varies strongly with cen- trality, multiplicity-dependent bias in flow measurements is unacceptable, and every means should be used to in- sure minimally-biased statistics. To achieve that end analysis methods must carefully transition from safe event-wise factorizations (as featured in this paper) to ensemble averages minimally biased for all n. Linear combinations of powers of random variables, e.g., vari- ances and covariances, satisfy a linear algebra. Such in- tegrals of two-particle momentum space are nominally free of bias. APPENDIX C: MULTIPOLES AND SPHERICITY The 1D Fourier transform on azimuth is part of a larger representation of angular structure. The encompassing context is a 2D multipole decomposition on (θ, φ) repre- sented by the sphericity tensor, with the spherical har- monics Y m2 as elements. In limiting cases submatrices of the sphericity tensor reduce to “cylindrical harmon- ics” cos(mφ), part of the 1D Fourier representation on azimuth. The central premise of a multipole representation is that the final-state particle angular distribution on [θ(yz), φ] is efficiently represented by a few low-order spherical harmonics (SH) Y ml (θ, φ). At the Bevalac, sphericity tensor S containing spherical harmonics Y m2 as elements was introduced. Directivity ~Q1, simply related to Y 12 , was employed to represent a rotated quadrupole as a dipole pair antisymmetric about the collision mid- point. At lower energies (Bevalac, AGS) the quadrupole principal axis may be rotated to a large angle with re- spect to the collision axis and Y 12 dominates. At higher energies and near midrapidity (θ ∼ π/2) the dominant SH is Y 22 . 1. Spherical harmonics The spherical harmonics are defined as Ylm(Ω) = 2l + 1 · (l −m)! (l +m)! Pml (cos θ) e imφ, (C1) where Pml (θ) is an associated Legendre function [38]. An event-wise density on the unit sphere can be expanded ρ̃(Ω) = Q̃lm Ylm(Ω) (C2) Q̃lm = dΩY ∗lm(Ω)ρ̃(Ω) Y ∗lm(Ωi) = n〈Y ∗lm(Ω)〉, where Ω → (θ, φ) and dΩ ≡ d cos(θ)dφ. The FTs on φ form a special case of those relations when ρ̃(Ω) is peaked near θ ∼ π/2. The Ylm are orthonormal and complete: dΩYlm(θ, φ)Yl′m′(θ, φ) = δll′δmm′ (C3) Ylm(θ, φ)Y ′, φ′) = δ(Ω− Ω′). 2. Multipoles The spherical harmonics are model functions for single- particle densities on (θ, φ). The coefficients of the mul- tipole expansion of a distribution are complex spherical multipole moments describing 2l poles and defined as en- semble averages of the spherical harmonics over the unit sphere weighted by an angular density. The following relation is defined by analogy with the expansion of an electric potential in spherical harmonics, in this case on momentum space ~p rather than configu- ration space ~r [38] ρ̃(p′,Ω′) |~p− ~p ′| 2l+ 1 Ylm(Ω) . (C4) The coefficients are the event-wise spherical multipole moments q̃lm ≡ p2dp dΩ plρ̃(p,Ω)Y ∗lm(Ω) (C5) pli Y lm(Ωi) = n〈pl Y ∗lm(Ω)〉. Eq. (C2) is the special case for p restricted to unity (i.e., distribution on the unit sphere). In general, ℜY mm ∝ sinm(θ) cos(mφ), and the cos(mφ) are by analogy “cylindrical harmonics” [42]. The ensem- ble average of a cylindrical harmonic over the unit circle weighted by 1D density ρ(φ) results in complex cylindri- cal multipole moments Qm. The Fourier coefficients Qm obtained from analysis of SPS and RHIC data are there- fore cylindrical multipole moments describing 2m poles. E.g., m = 2 denotes a quadrupole moment and m = 4 denotes an octupole moment,. If nonflow contributions (i.e., structure rapidly vary- ing on η or y) are present, a multipole decomposition of ρ(θ, φ) is no longer efficient, and the inferred multipole moments are difficult to interpret physically (e.g., flow in- ferences per se are biased). In Sec. V we describe a more differential method for representing angular structure us- ing two-particle joint angular autocorrelations on differ- ence axes (η∆, φ∆). Given a decomposition of ρ(θ, φ) based on variations on η∆ we can distinguish cylindri- cal multipoles accurately from “nonflow” structure (cf. Sec. VII). 3. Sphericity The sphericity tensor has been employed in both jet physics and flow studies. A normalized 3D sphericity tensor was defined in [14] to search for initial evidence of jets in e+-e− collisions. A decade later sphericity was introduced to the search for collective nucleon flow in heavy ion collisions [15]. The close connection between flow and jets continues at RHIC, where we seek the rela- tion between minijets and “elliptic flow.” Event-wise sphericity S̃ is a measure of structure in single-particle density ρ(θ, φ) on the unit sphere. We use dyadic notation to reduce index complexity, analo- gous to vector notation ~̃Qm = i ri~u(mφi). S̃ (with ri → pi) describes a 3D quadrupole with arbitrary orien- tation. Given ~p = p [sin(θ) cos(φ), sin(θ) sin(φ), cos(θ)] ≡ p ~u(θ, φ) we have 2S̃ ≡ 2 ~pi~pi = 2 p2i ~u(θi, φi) ~u(θi, φi) (C6) p2i Ũ(θi, φi) = n〈p2 Ũ(θ, φ)〉, the last being an event-wise average, where U(θ, φ) = sin2(θ) I + Y(θ, φ) (C7) Y(θ, φ) = (C8) sin2(θ) cos(2φ) sin2(θ) sin(2φ) sin(2θ) cos(φ) sin2(θ) sin(2φ) − sin2(θ) cos(2φ) sin(2θ) sinφ sin(2θ) cos(φ) sin(2θ) sinφ 3 cos2(θ) − 1 In terms of event-wise quadrupole moments q̃2m derived from the Y2m 2S̃ = n p2 sin2(θ) I (C9) 2l+ 1 ℜq̃22 ℑq̃22 −ℜq̃21 ℑq̃22 −ℜq̃22 −ℑq̃21 −ℜq̃21 −ℑq̃21 an event-wise estimator of angular structure on the unit sphere, its reference defined by 2S̃ref = n〈p2 sin2(θ)〉 I, and p2 sin2(θ) = p2t . The sphericity tensor of [14] was normalized to Ŝ = S/n〈p2〉. Note that Q̃ = 3S̃ − n I (C10) is the traceless Cartesian quadrupole tensor appearing in the Taylor expansion of the (~p equivalent of the) electro- static potential [38]. We have defined instead Q̃′ = 3S̃ − I, (C11) an alternative quadrupole tensor wherein each element is a single spherical quadrupole moment. The difference lies in the diagonal elements: linear combinations of the ℜq̃2m in the diagonal elements of Q̃ are simplified to single moments in Q̃′. The ensemble mean of both ten- sors for an uncorrelated (spherically symmetric) system or system with event-wise quadrupole orientations ran- domly varying is the null tensor (all elements zero). APPENDIX D: SUBEVENTS The “subevent” is a notional re-invention of partition- ing/binning, the latter having a history of more than a century in mathematics. In conventional flow analysis subevents are groups of particles in an event segregated on the basis of random selection, charge, strangeness, PID or a kinematic variable such as pt, y or η. The scalar- product method [30] is based on a covariance between two single-particle bins (nominally equal halves of an event). The subevent method is thus a restricted reinvention of a common concept in multiparticle correlation analysis: determining covariances among all pairs of single-particle bins at some arbitrary binning scale – a two-particle cor- relation function. Diagonal averages of such distributions are the elements of autocorrelations. In the language of conventional flow analysis one way to eliminate statistical reference Q2ref from Q m is to par- tition events into a pair of disjoint (non-overlapping) subevents A, B [19]. In that case ~̃Qma· ~̃Qmb = ~̃Vma · ~̃Vmb = nanbṽ mab, a covariance. The partition may be asym- metric (unequal particle numbers) and may be as small as a pair of particles. In addition to eliminating the self-pair statistical reference such partitioning is said to reduce nonflow correlation sources, depending on their physical origins and the partition definition [30]. We as- sume for simplicity that there is no nonflow contribution. Subevent pairs can be used to determine the event-plane resolution for subevents A, B and full events. First, we consider the symmetric case, defining equiv- alent subevents A and B with multiplicities nA = nB = n/2 from an event with n particles. E.g., subevent A has azimuth vector ~QmA = i∈A ~u(mφi). The scalar product is a covariance ~̃Qma · ~̃Qmb ≡ Q̃a Q̃b〈cos(m[Ψa −Ψb])〉 (D1) na,nb i∈A,j∈B cos(m[φi − φj ]) ≡ na nb ṽ2mab = Ṽ 2mab, with e.g. Q̃2a = na + na(na − 1)ṽ2ma = na + Ṽ 2ma. Then cos(m[Ψma −Ψmb]) = Ṽ 2mab na + Ṽ 2ma nb + Ṽ .(D2) If subevents A and B are physically equivalent (e.g., a random partition of the total of n particles), then cos(m[Ψma −Ψmb]) = rab V 2ma n̄a + V 2ma = cos(m[Ψma −Ψr]) cos(m[Ψmb −Ψr]), where rab = V V 2ma V mb is Pearson’s normalized co- variance between subevents A and B for the mth power- spectrum elements. If A and B are perfectly correlated (rab = 1) then cos(m[Ψma −Ψmb]) = cos2(m[Ψma −Ψr]) (D4) In general, V 2m/n̄ = (1+ rab)V ma/n̄a, which provides the exact relation between the EP resolution for subevents and for composite events A + B. It is not generally cor- rect that cos(m[Ψm −Ψr]) = 2 · cos(m[Ψma −Ψr]). In this case cos2(m[Ψma −Ψr]) = n̄a − 1 n̄a + V 2ma and V 2ma = V m/4 for perfectly correlated subevents. Second, we consider the most asymmetric case A = one particle and B = n− 1 particles. 〈cos(m[φi −Ψr]) cos(m[Ψmi −Ψr])〉 = (D6) (n− 1)ṽ2mi n− 1 + (n− 1)(n− 2)ṽ2mi ṽm · Ṽ ′m where Q̃′2m = n− 1+ Ṽ ′2m describes a subevent with n− 1 particles. In general, the EP resolution for a full event of n particles is given by cos2(m[Ψm −Ψr]) ≃ nṼ 2m (n− 1)Q̃2m . (D7) Measurement of the EP resolution is simply a measure- ment of the corresponding power-spectrum element, since V 2m/n̄ ≃ cos2(m[Ψm −Ψr]) 1− cos2(m[Ψm −Ψr]) . (D8) In [22] the approximation 〈cos(m[Ψm −Ψr])〉2 ≈ V 2m/n̄ (D9) is given for V 2m/n̄≪ 1 or Q2m ∼ n̄. Equal subevents, as the largest possible event parti- tion, imply an expectation that only global (large-scale) variables are relevant to collision dynamics (e.g., to de- scribe thermalized events). The possibility of finer struc- ture in momentum space is overlooked, whereas autocor- relation studies with finer binnings and the covariances among those bins discover detailed event structure highly relevant to collision dynamics. APPENDIX E: CENTRALITY ISSUES Accurate A-A centrality determination and the cen- trality dependence of azimuth multipoles and related pa- rameters is critical to understanding heavy ion collisions. We must locate b = 0 accurately in terms of measured quantities to test theory expectations relative to hydro- dynamics and thermalization. And we must obtain accu- rate measurements for peripheral A-A collisions to pro- vide a solid connection to elementary collisions. 1. Centrality measures In [32] is described the power-law method of centrality determination. Because the minimum-bias distribution on participant-pair number npart/2 goes almost exactly as (npart/2) −3/4 the distribution on (npart/2) 1/4 is al- most exactly uniform, as is the experimental distribution ch , dominated by participant scaling. Those sim- ple forms can greatly improve the accuracy of central- ity determination, especially for peripheral and central collisions. The cited paper gives simple expressions for npart/2, nbin and ν relative to fraction of total cross sec- tion. In conventional centrality determination the minimum- bias distribution on nch is divided into several bins rep- resenting estimated fractions of the total cross section. The main source of systematic error is uncertainty in the fraction of total cross section which passes triggering and event reconstruction. The total efficiency is typi- cally 95%, the loss being mainly in the peripheral region, and the most peripheral 10 or 20% bins therefore have large systematic errors resulting in abandonment. Flow measurements with EP estimation are also excluded from peripheral collisions due to low event multiplicities. In contrast, with the power-law method running inte- grals of the Glauber parameters and nch can be brought into asymptotic coincidence for peripheral collisions re- gardless of the uncertainty in the total cross section. Pa- rameter ν measures the centrality and greatly reduces the cross-section error contribution. Centrality accuracy < 2% on ν is thereby achievable down to N-N collisions. That capability is essential to determine the correspon- dence of A-A quadrupole structure in elementary colli- sions, to test the LDL hypothesis for instance: is there “collective” behavior in N-N collisions? For central collisions the upper half-maximum point on the power-law minimum-bias distribution provides a precise determination of b = 0 on nch and therefore ν. The b = 0 point is critical for evaluation of correlation measures relative to Glauber parameter ǫ in the context of hydro expectations for v2/ǫ. 2. Geometry parameters and azimuth structure We consider the several A-A geometry parameters rel- evant to azimuth structure. In Fig. 11 (left panel) we plot npart/2 vs ν using the parameterization in [32]. The relation is very nonlinear. The dashed curve is npart/2 ≃ 2ν2.57. The most peripheral quarter of the centrality range is compressed into a small interval on npart/2. Mean path-length ν is the natural geometry measure for sensitive tests of departure from linear N- N superposition, whereas important minijet correlations (nonflow) ∝ ν are severely distorted on npart. 2 4 6 2 4 6 FIG. 11: Left panel: Participant pair number vs mean path- length ν for 200 GeV Au-Au collisions. Because of the nonlin- ear relation the peripheral third of collisions is compressed to a small interval on npart/2. Right panel: Impact parameter b vs ν. To good approximation the relation is linear over most of the centrality range. In Fig. 11 (right panel) we plot impact parameter b vs ν, again using the parameterization in [32] with fractional cross section σ/σ0 = (b/b0) 2. We note the interesting fact that over most of the centrality range b/b0 ≃ (R − ν)/(R − 1), with b0 ≡ 2R = 14.7 fm for Au-Au. Thus, any anticipated trends on b are also accessible on ν with minimal distortion. In Fig. 12 (left panel) we show the LDL parameter 1/S dnch/dη [24] vs ν for three collision energies. The energy dependence derives from the multiplicity factor, which we parameterize in terms of a two-component model [39]. Weighted cross-section area S(b/b0) (fm is an optical Glauber parameterization from [31]. Both ν and 1/S dnch/dη are pathlength measures. They can be compared with the inverse Knudson number K−1n intro- duced in [36] as a measure of collision number. The LDL measure is based on energy-dependent physical particle collisions, whereas ν is based on A-A geometry alone. The relation is monotonic and almost linear. Thus, struc- ture on one parameter should appear only slightly dis- torted on the other. 17 GeV 62 GeV 200 GeV 2 4 6 1/S dnch/dη hydro 0 10 20 FIG. 12: Left panel: Correspondence between LDL parame- ter 1/S dnch/dη and centrality measure ν for three energies. Right panel: Theory expectations for two limiting cases at 200 GeV. The solid curve is derived from the solid curve in Fig. 10 (upper-right panel) using the relation in the left panel. The hatched region is typically not measured in a con- ventional flow analysis, due to a combination of large sys- tematic uncertainty in the centrality determination and large biases in flow measurements due to small multi- plicities. However, peripheral collisions provide critical tests of flow models: e.g., how does collective behavior (if present) emerge with increasing centrality? In this paper we describe analysis methods which, when com- bined with the centrality methods of [32], make all A-A collisions accessible for accurate measurements down to In Fig. 12 (right panel) we show v2/ǫ vs 1/S dnch/dη for theory expectations (hatched bands) and the simula- tion in Sec. VII C. The latter is based on a simple error function on ν and is roughly consistent with four-particle cumulant results at 200 GeV [26]. We observe that the solid curve is not consistent with either the LDL trend for peripheral collisions (the LDL slope is arbitrary) or the hydro trend for central collisions. That provocative result suggests that accurate analysis of azimuth corre- lations over a broad range of energies and centralities with the methods introduced in this paper and [32] may produce interesting and unanticipated results. 3. Correlation measures If the centrality dependence of azimuth structure is to be accurately determined the correlation measure em- ployed must have little or no multiplicity bias, includ- ing statistical biases and irrelevant multiplicity factors which lead to incorrect physical inferences. The quantity ρref is the unique solution to a measurement prob- lem subject to multiple constraints. It is the only portable measure (density ratio) of two-particle correlations ap- plicable to collision systems with arbitrary multiplicity. ρref is invariant under linear superposition. If, ac- cording to that measure, central Au-Au is different from N-N the difference certainly indicates a unique physical aspect of Au-Au collisions relative to N-N, exactly what we require in a correlation measure. Conventional flow measures do not satisfy that basic requirement. Drawing a parallel with measures of 〈pt〉 fluctuations we compare v2 ↔ Σpt [40]. Both are square roots of per-pair correlation measures which tend to yield mis- leading systematic trends (on centrality and energy) [10]. In contrast V 2m ↔ ∆Σ2pt:n [5] (the total variance dif- ference for pt fluctuations) are integrals of two-particle correlations (azimuth number correlations vs pt correla- tions). The first is a measure of total azimuth corre- lations, the second a measure of total pt variance dif- ference [41], the integral of a two-particle distribution relative to its reference. In a minimally-biased con- text we then have vm = V 2m/n(n− 1), analogous to Σpt(CERES) = ∆Σ2pt:n/n(n− 1) p̂ t [10] (the similar- ity of notations is coincidental) as the square roots of per-pair measures. Both are physically misleading. [1] R. C. Hwa, Phys. Rev. D 32, 637 (1985); U. W. Heinz and P. F. Kolb, Nucl. Phys. A 702, 269 (2002). [2] F. Becattini, M. Gazdzicki and J. Sollfrank, Eur. Phys. J. C 5, 143 (1998); P. Braun-Munzinger, I. Heppe and J. Stachel, Phys. Lett. B 465, 15 (1999). [3] R. Stock, “Event by event analysis of ultrarelativistic nuclear collisions: A new method to search for criti- cal fluctuations,” Prepared for NATO Advanced Study Workshop on Hot Hadronic Matter: Theory and Exper- iment, Divonne-les-Bains, France, 27 Jun - 1 Jul 1994; R. Stock, Nucl. Phys. A661, 282c (1999); H. Heiselberg, Phys. Rep. 351, 161 (2001). [4] M. Gaździcki, St. Mrówczyński, Z. Phys. C 54, 127 (1992). [5] T. A. Trainor, hep-ph/0001148. [6] N. Borghini, P. M. Dinh and J. Y. 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0704.1675

Exploiting Social Annotation for Automatic Resource Discovery

Exploiting Social Annotation for Automatic Resource Discovery Anon Plangprasopchok and Kristina Lerman USC Information Sciences Institute 4676 Admiralty Way Marina del Rey, CA 90292, USA {plangpra,lerman}@isi.edu Abstract Information integration applications, such as mediators or mashups, that require access to information resources cur- rently rely on users manually discovering and integrating them in the application. Manual resource discovery is a slow process, requiring the user to sift through results obtained via keyword-based search. Although search methods have advanced to include evidence from document contents, its metadata and the contents and link structure of the referring pages, they still do not adequately cover information sources — often called “the hidden Web”— that dynamically gen- erate documents in response to a query. The recently popu- lar social bookmarking sites, which allow users to annotate and share metadata about various information sources, pro- vide rich evidence for resource discovery. In this paper, we describe a probabilistic model of the user annotation process in a social bookmarking system del.icio.us. We then use the model to automatically find resources relevant to a partic- ular information domain. Our experimental results on data obtained from del.icio.us show this approach as a promising method for helping automate the resource discovery task. Introduction As the Web matures, an increasing number of dynamic information sources and services come online. Unlike static Web pages, these resources generate their contents dynamically in response to a query. They can be HTML- based, searching the site via an HTML form, or be a Web service. Proliferation of such resources has led to a number of novel applications, including Web-based mashups, such as Google maps and Yahoo pipes, information integra- tion applications (Thakkar, Ambite, & Knoblock 2005) and intelligent office assistants (Lerman, Plangprasopchok, & Knoblock 2007) that com- pose information resources within the tasks they perform. In all these applications, however, the user must discover and model the relevant resources. Manual resource discovery is a very time consuming and laborious process. The user usually queries a Web search engine with appropriate keywords and additional parameters (e.g., asking for .kml or .wsdl files), and then must examine every resource returned by the search engine to evaluate whether it has the desired Copyright c© 2018, American Association for Artificial Intelli- gence (www.aaai.org). All rights reserved. functionality. Often, it is desirable to have not one but several resources with an equivalent functionality to ensure robustness of information integration applications in the face of resource failure. Identifying several equivalent resources makes manual resource discovery even more time consuming. The majority of the research in this area of in- formation integration has focused on automating modeling resources — i.e., understanding seman- tics of data they use (Heß & Kushmerick 2003; Lerman, Plangprasopchok, & Knoblock 2006) and the functionality they provide (Carman & Knoblock 2007). In comparison, the resource discovery problem has received much less attention. Note that traditional search engines, which index resources by their contents — the words or terms they contain — are not likely to be useful in this domain, since the contents is dynamically generated. At best, they rely on the metadata supplied by the resource authors or the anchor text in the pages that link to the resource. Woogle (Dong et al. 2004) is one of the few search engines to index Web services based on the syntactic metadata provided in the WSDL files. It allows a user to search for services with a similar functionality or that accept the same inputs as another services. Recently, a new generation of Web sites has rapidly gained popularity. Dubbed “social media,” these sites al- low users to share documents, including bookmarks, photos, or videos, and to tag the content with free-form keywords. While the initial purpose of tagging was to help users or- ganize and manage their own documents, it has since been proposed that collective tagging of common documents can be used to organize information via an informal classifica- tion system dubbed a “folksonomy” (Mathes 2004). Con- sider, for example, http://geocoder.us, a geocoding service that takes an input as address and returns its latitude and longitude. On the social bookmarking site del.icio.us1, this resource has been tagged by more than 1, 000 people. The most common tags associated by users with this resource are “map,” “geocoding,” “gps,” “address,” “latitude,” and “lon- gitude.” This example suggests that although there is gener- ally no controlled vocabulary in a social annotation system, tags can be used to categorize resources by their functional- 1http://del.icio.us http://arxiv.org/abs/0704.1675v1 We claim that social tagging can be used for information resource discovery. We explore three probabilistic gener- ative models that can be used to describe the tagging pro- cess on del.icio.us. The first model is the probabilistic Latent Semantic model (Hofmann 1999) which ignores in- dividual user by integrating bookmarking behaviors from all users. The second model, the three-way aspect model, was proposed (Wu, Zhang, & Yu 2006) to model del.icio.us users’ annotations. The model assumes that there exists a global conceptual space that generates the observed values for users, resources and tags independently. We propose an alternative third model, motivated by the Author-Topic model (Rosen-Zvi et al. 2004), which maintains that latent topics that are of interest to the author generate the words in the documents. Since a single resource on del.icio.us could be tagged differently by different users, we separate “top- ics”, as defined in Author-Topic model, into “(user) inter- ests” and “(resource) topics”. Together user interests and resource topics generate tags for one resource. In order to use the models for resource discovery, we describe each re- source by a topic distribution and then compare this distri- bution with that of all other resources in order to identify relevant resources. The paper is organized as follows. In the next section, we describe how tagging data is used in resource discovery. Subsequently we present the probabilistic model we have developed to aid in the resource discovery task. The section also describes two earlier related models. We then compares the performance of the three models on the datasets obtained from del.icio.us. We review prior work and finally present conclusions and future research directions. Problem Definition Suppose a user needs to find resources that provide some functionality: e.g., a service that returns current weather conditions, or latitude and longitude of a given address. In order to improve robustness and data coverage of an appli- cation, we often want more than one resource with the nec- essary functionality. In this paper, for simplicity, we assume that the user provides an example resource, that we call a seed, and wants to find more resources with the same func- tionality. By “same” we mean a resource that will accept the same input data types as the seed, and will return the same data types as the seed after applying the same operation to them. Note that we could have a more stringent requirement that the resource return the same data as the seed for the same input, but we don’t want to exclude resources that may have different coverage. We claim that users in a social bookmarking system such as del.icio.us annotate resources according to their function- ality or topic (category). Although del.icio.us and similar systems provide different means for users to annotate doc- ument, such as notes and tags, in this paper we focus on utilizing the tags only. Thus, the variables in our model are resources R, users U and tags T . A bookmark i of resource r by user u can be formalized as a tuple 〈r, u, {t1, t2, . . .}〉i, which can be further broken down into a co-occurrence of a triple of a resource, a user and a tag: 〈r, u, t〉. Figure 1: Graphical representations of the probabilistic La- tent Semantic Model (left) and Multi-way Aspect Model (right) R, U , T and Z denote variables “Resource”, “User”, “Tag” and “Topic” repectively. Nt represents a number of tag occurrences for a particular resource; D represents a number of resources. Meanwhile,Nb represents a number of all resource-user-tag co-occurrences in the social annotation system. Note that filled circles represent observed variables. We collect these triples by crawling del.icio.us. The sys- tem provides three types of pages: a tag page — listing all resources that are tagged with a particular keyword; a user page — listing all resources that have been bookmarked by a particular user; and a resource page — listing all the tags the users have associated with that resource. del.icio.us also provides a method for navigating back and forth between these pages, allowing us to crawl the site. Given the seed, we get what del.icio.us shows as the most popular tags as- signed by the users to it. Next we collect other resources annotated with these tags. For each of these we collect the resource-user-tag triples. We use these data to discover re- sources with the same functionality as the seed, as described below. Approach We use probabilistic models in order to find a compressed description of the collected resources in terms of topic de- scriptions. This description is a vector of probabilities of how a particular resource is likely to be described by dif- ferent topics. The topic distribution of the resource is sub- sequently used to compute similarity between resources us- ing Jensen-Shannon divergence (Lin 1991). For the rest of this section, we describe the probabilistic models. We first briefly describe two existing models: the probabilistic La- tent Semantic Analysis (pLSA) model and the Three-Way Aspect model (MWA). We then introduce a new model that explicitly takes into account users’ interests and resources’ topics. We compare performance of these models on the three del.icio.us datasets. Probabilistic Latent Semantic Model (pLSA) Hoffman (Hofmann 1999) proposed a probabilistic la- tent semantic model for associating word-document co- occurrences. The model hypothesized that a particular docu- ment is composed of a set of conceptual themes or topics Z . Words in a document were generated by these topics with some probability. We adapted the model to the context of social annotation by claiming that all users have common agreement on annotating a particular resource. All book- marks from all users associated with a given resource were aggregated into a single corpus. Figure 1 shows the graphi- cal representation of this model. Co-occurrences of a partic- ular resource-tag pair were computed by summing resource- user-tag triples 〈r, u, t〉 over all users. The joint distribution over resource and tag is p(r, t) = p(t|z)p(z|r)p(r) (1) In order to estimate parameters p(t|z), p(z|r), p(r) we define log likelihood L, which measures how the estimated parameters fit the observed data n(r, t)log(p(r, t)) (2) where n(r, t) is a number of resource-tag co-occurrences. The EM algorithm (Dempster, Laird, & Rubin 1977) was applied to estimate those parameters that maximize L. Three-way Aspect Model (MWA) The three-way aspect model (or multi-way aspect model, MWA) was originally applied to document recommenda- tion systems (Popescul et al. 2001), involving 3 entities: user, document and word. The model takes into account both user interest (pure collaborative filtering) and docu- ment content (content-based). Recently, the three-way as- pect model was applied on social annotation data in or- der to demonstrate emergent semantics in a social annota- tion system and to use these semantics for information re- trieval (Wu, Zhang, & Yu 2006). In this model, conceptual space was introduced as a latent variable, Z , which indepen- dently generated occurrences of resources, users and tags for a particular triple 〈r, u, t〉 (see Figure 1). The joint distribu- tion over resource, user, and tag was defined as follows p(r, u, t) = p(r|z)p(u|z)p(t|z)p(z) (3) Similar to pLSA, the parameters p(r|z), p(u|z), p(t|z) and p(z) were estimated by maximizing the log likelihood objective function, L = r,u,t n(r, u, t)log(p(r, u, t)). EM algorithm was again applied to estimate those parameters. Interest-Topic Model (ITM) The motivation to implement the model proposed in this pa- per comes from the observation that users in a social anno- tation system have very broad interests. A set of tags in a particular bookmark could reflect both users’ interests and resources’ topics. As in the three-way aspect model, using a single latent variable to represent both “interests” and “top- ics” may not be appropriate, as intermixing between these two may skew the final similarity scores computed from the topic distribution over resources. Figure 2: Graphical representation on the proposed model. R, U , T , I and Z denote variables “Resource”, “User”, “Tag”, “Interest” and “Topic” repectively. Nt represents a number of tag occurrences for a one bookmark (by a partic- ular user to a particular resource); D represents a number of all bookmarks in social annotation system. Instead, we propose to explicitly separate the latent vari- ables into two: one representing user interests, I; another representing resource topics, Z . According to the proposed model, the process of resource-user-tag co-occurrence could be described as a stochastic process: • User u finds a resource r interesting and she would like to bookmark it • User u has her own interest profile i; meanwhile the re- source has a set of topics z. • Tag t is then chosen based on users’s interest and re- source’s topic The process is depicted in a graphical form in Figure 2. From the process described above, the joint probability of resource, user and tag is written as P (r, u, t) = p(t|i, z)p(i|u)p(z|r)p(u)p(r) (4) Log likelihood L is used as the objective function to es- timate all parameters. Note that p(u) and p(r) could be obtained directly from observed data – the estimation thus involves three parameters p(t|i, z), p(i|u) and p(z|r). L is defined as r,u,t n(r, u, t)log(p(r, u, t)) (5) EM algorithm is applied to estimate these parameters. In the expectation step, the joint probability of hidden variables I and Z given all observations is computed as p(i, z|u, r, t) = p(t|i, z)p(i|u)p(z|r)∑ p(t|i, z)p(i|u)p(z|r) Subsequently, each parameter is re-estimated using p(i, z|u, r, t) we just computed from the E step p(t|i, z) = n(r, u, t)p(i, z|u, r, t) r,u,t n(r, u, t)p(i, z|u, r, t) p(i|u) = n(r, u, t) p(i, z|u, r, t) p(z|r) = n(r, u, t) p(i, z|u, r, t) The algorithm iterates between E and M step until the log likelihood or all parameter values converges. Once all the models are learned, we use the distribution of topics of a resource p(z|r) to compute similarity between resources and the seed using Jensen-Shannon divergence. Empirical Validation To evaluate our approach, we collected data for three seed resources: flytecomm2 geocoder3 and wunderground4. The first resource allows users to track flights given the airline and flight number or departure and arrival airports; the sec- ond resource returns coordinates of a given address; while, the third resource supplies weather information for a partic- ular location (given by zipcode, city and state, or airport). Our goal is to find other resources that provide flight track- ing, geocoding and weather information. Our approach is to crawl del.icio.us to gather resources possibly related to the seed; apply the probabilistic models to find the topic dis- tribution of the resources; then rank all gathered resources based on how similar their topic distribution is to the seed’s topic distribution. The crawling strategy is defined as fol- lows: for each seed • Retrieve the 20 most popular tags that users have applied to that resource • For each of the tags, retrieve other resources that have been annotated with that tag • For each resource, collect all bookmarks that have been created for it (i.e., resource-user-tag triples) We wrote special-purpose Web page scrapers to extract this information from del.icio.us. In principle, we could continue to expand the collection of resources by gathering tags and retrieving more resources that have been tagged with those tags, but in practice, even after the small traversal we do, we obtain more than 10 million triples for the wunderground seed. We obtained the datasets for the seeds flytecomm and geocoder in May 2006 and for the seed wunderground in January 2007. We reduced the dataset by omitting low (fewer than ten) and high (more than ten thousand) fre- quency tags and all the triples associated with those tags. After this reduction, we were left with (a) 2,284,308 triples with 3,562 unique resources; 14,297 unique tags; 34,594 unique users for the flytecomm seed; (b) 3,775,832 triples with 5,572 unique resources; 16,887 unique tags and 46,764 2http://www.flytecomm.com/cgi-bin/trackflight/ 3http://geocoder.us 4http://www.wunderground.com/ unique users for the geocoder seed; (c) 6,327,211 triples with 7,176 unique resources; 77,056 unique tags and 45,852 unique users for the wunderground seed. Next, we trained all three models on the data: pLSA, MWA and ITM. We then used the learned topic distributions to compute the similarity of the resources in each dataset to the seed, and ranked the resources by similarity. We evalu- ated the performance of each model by manually checking the top 100 resources produced by the model according to the criteria below: • same: the resource has the same functionality if it pro- vides an input form that takes the same type of data as the seed and returns the same type of output data: e.g., a flight tracker takes a flight number and returns flight status • link-to: the resource contains a link to a page with the same functionality as the seed (see criteria above). We can easily automate the step that check the links for the right functionality. Although evaluation is performed manually now, we plan to automate this process in the future by using the form’s metadata to predict semantic types of inputs (Heß & Kushmerick 2003), automatically query the source, extract data from it and classify it using the tools described in (Gazen & Minton 2005; Lerman, Plangprasopchok, & Knoblock 2006). We will then be able to validate that the resource has functionality similar to the seed by comparing its input and output data with that of the seed (Carman & Knoblock 2007). Note that since each step in the automatic query and data extraction process has some probability of failure, we will need to identify many more relevant resources than required in order to guarantee that we will be able to automatically verify some of them. Figure 3 shows the performance of different models trained with either 40 or 100 topics (and interests) on the three datasets. The figure shows the number of resources within the top 100 that had the same functionality as the seed or contained a link to a resource with the same func- tionality. The Interest-Topic model performed slightly bet- ter than pLSA, while both ITM and pLSA significantly out- performed the MWA model. Increasing the dimensionality of the latent variable Z from 40 to 100 generally improved the results, although sometimes only slightly. Google’s find “Similar pages” functionality returned 28, 29 and 15 re- sources respectively for the three seeds flytecomm, geocoder and wunderground, out of which 5, 6, and 13 had the same functionality as the seed and 3, 0, 0 had a link to a resource with the same functionality. The ITM model, in comparison, returned three to five times as many relevant results. Table 1 provides another view of performance of differ- ent resource discovery methods. It shows how many of the method’s predictions have to be examined before ten re- sources with correct functionality are identified. Since the ITM model ranks the relevant resources highest, fewer Web sites have to be examined and verified (either manually or automatically); thus, ITM is the most efficient model. One possible reason why ITM performs slightly better than pLSA might be because in the datasets we collected, flytecomm (100) (100) (100) link-to geocoder (100) (100) (100) wunderground (100) (100) (100) Figure 3: Performance of different models on the three datasets. Each model was trained with 40 or 100 topics. For ITM, we fix interest to 20 interests across all different datasets. The bars show the number of resources within the top 100 returned by each model that had the same functionality as the seed or contained a link to a resource with the same functionality as the seed. there is low variance of user interest. The resources were gathered starting from a seed and following related tag links; therefore, we did not obtain any resources that were anno- tated with different tags than the seed, even if they are tagged by the same user who bookmarks the seed. Hence user- resource co-occurrences are incomplete: they are limited by a certain tag set. pLSA and ITM would perform similarly if all users had the same interests. We believe that ITM would perform significantly better than pLSA when variation of user interest is high. We plan to gather more complete data to weigh ITM behavior in more detail. Although performances pLSA and ITM are only slightly different, pLSA is much better than ITM in terms of effi- ciency since the former ignores user information and thus reduces iterations required in its training process. However, for some applications, such as personalized resource discov- ery, it may be important to retain user information. For such applications the ITM model, which retains this information, may be preferred over pLSA. Previous Research Popular methods for finding documents relevant to a user query rely on analysis of word occurrences (including meta- data) in the document and across the document collection. Information sources that generate their contents dynamically in response to a query cannot be adequately indexed by con- ventional search engines. Since they have sparse metadata, PLSA MWA ITM GOOGLE* flytecomm 23 65 15 > 28 geocoder 14 44 16 > 29 wunderground 10 14 10 10 Table 1: The number of top predictions that have to be exam- ined before the system finds ten resources with the desired functionality (the same or link-to). Each model was trained with 100 topics. For ITM, we fixed the number of interests at 20. *Note that Google returns only 8 and 6 positive re- sources out of 28 and 29 retrieved resources for flytecomm and geocoder dataset respectively. the user has to find the correct search terms in order to get results. A recent research (Dong et al. 2004) proposed to utilize metadata in the Web services’ WSDL and UDDI files in or- der to find Web services offering similar operations in an unsupervised fashion. The work is established on a heuris- tic that similar operations tend to be described by similar terms in service description, operation name and input and output names. The method uses clustering techniques using cohesion and correlation scores (distances) computed from co-occurrence of observed terms to cluster Web service op- erations. In this approach, a given operation can only belong to a single cluster. Meanwhile, our approach is grounded on a probabilistic topic model, allowing a particular resource to be generated by several topics, which is more intuitive and robust. In addition, it yields a method to determine how the resource similar to others in certain aspects. Although our objective is similar, instead of words or metadata created by the authors of online resources, our ap- proach utilizes the much denser descriptive metadata gen- erated in a social bookmarking system by the readers or users of these resources. One issue to be considered is the metadata cannot be directly used for categorizing re- sources since they come from different user views, interests and writing styles. One needs algorithms to detect patterns in these data, find hidden topics which, when known, will help to correctly group similar resources together. We apply and extend the probabilistic topic model, in particular pLSA (Hofmann 1999) to address such issue. Our model is conceptually motivated by the Author-Topic model (Rosen-Zvi et al. 2004), where we can view a user who annotate a resource as an author who composes a docu- ment. The aim in that approach is to learn topic distribution for a particular author; while our goal is to learn the topic distribution for a certain resource. Gibbs sampling was used in parameter estimation for that model; meanwhile, we use the generic EM algorithm to estimate parameters, since it is analytically straightforward and ready to be implemented. The most relevant work, (Wu, Zhang, & Yu 2006), uti- lizes multi-way aspect model on social annotation data in del.icio.us. The model doesn’t explicitly separate user in- terests and resources topics as our model does. Moreover, the work focuses on emergence of semantic and personal- ized resource search, and is evaluated by demonstrating that it can alleviate a problem of tag sparseness and synonymy in a task of searching for resources by a tag. In our work, on the other hand, our model is applied to search for resources similar to a given resource. There is another line of researches on resource discov- ery that exploits social network information of the web graph. Google (Brin & Page 1998) uses visitation rate ob- tained from resources’ connectivity to measure their popu- larity. HITS (Kleinberg 1999) also use web graph to rate rel- evant resources by measuring their authority and hub values. Meanwhile, ARC (Chakrabarti et al. 1998) extends HITS by including content information of resource hyperlinks to improve system performance. Although the objective is somewhat similar, our work instead exploits resource meta- data generated by community to compute resources’ rele- vance score. Conclusion We have presented a probabilistic model that models social annotation process and described an approach to utilize the model in the resource discovery task. Although we can- not compare to performance to state-of-the-art search en- gine directly, the experimental results show the method to be promising. There remain many issues to pursue. First, we would like to study the output of the models, in particular, what the user interests tell us. We would also like to automate the source modeling process by identifying the resource’s HTML form and extracting its metadata. We will then use techniques de- scribed in (Heß & Kushmerick 2003) to predict the seman- tic types of the resource’s input parameters. This will enable us to automatically query the resource and classify the re- turned data using tools described in (Gazen & Minton 2005; Lerman, Plangprasopchok, & Knoblock 2006). We will then be able to validate that the resource has the same func- tionality as the seed by comparing its input and output data with that of the seed (Carman & Knoblock 2007). This will allow agents to fully exploit our system for integrating in- formation across different resources without human inter- vention. Our next goal is to generalize the resource discovery process so that instead of starting with a seed, a user can start with a query or some description of the information need. We will investigate different methods for translating the query into tags that can be used to harvest data from del.icio.us. In addition, there is other evidence potentially useful for resource categorization such as user comments, content and input fields in the resource. We plan to extend the present work to unify evidence both from annotation and resources’ content to improve the accuracy of resource dis- covery. Acknowledgements This research is based by work sup- ported in part by the NSF under Award No. CNS-0615412 and in part by DARPA under Contract No. NBCHD030010. References [Brin & Page 1998] Brin, S., and Page, L. 1998. The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems 30(1–7):107–117. [Carman & Knoblock 2007] Carman, M. J., and Knoblock, C. A. 2007. Learning semantic descriptions of web infor- mation sources. In Proc. of IJCAI. [Chakrabarti et al. 1998] Chakrabarti, S.; Dom, B.; Gibson, D.; Kleinberg, J.; Raghavan, P.; and Rajagopalan, S. 1998. Automatic resource list compilation by analyzing hyper- link structure and associated text. In Proceedings of the 7th International World Wide Web Conference. [Dempster, Laird, & Rubin 1977] Dempster, A. P.; Laird, N. M.; and Rubin, D. B. 1977. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 39(1):1–38. [Dong et al. 2004] Dong, X.; Halevy, A. Y.; Madhavan, J.; Nemes, E.; and Zhang, J. 2004. Simlarity search for web services. In Proc. of VLDB, 372–383. [Gazen & Minton 2005] Gazen, B. C., and Minton, S. N. 2005. Autofeed: an unsupervised learning system for gen- erating webfeeds. In Proc. of K-CAP 2005, 3–10. [Heß & Kushmerick 2003] Heß, A., and Kushmerick, N. 2003. Learning to attach semantic metadata to web ser- vices. In International Semantic Web Conference, 258– [Hofmann 1999] Hofmann, T. 1999. Probabilistic latent semantic analysis. In Proc. of UAI, 289–296. [Kleinberg 1999] Kleinberg, J. M. 1999. Authoritative sources in a hyperlinked environment. Journal of the ACM 46(5):604–632. [Lerman, Plangprasopchok, & Knoblock 2006] Lerman, K.; Plangprasopchok, A.; and Knoblock, C. A. 2006. Automatically labeling the inputs and outputs of web services. In Proc. of AAAI. [Lerman, Plangprasopchok, & Knoblock 2007] Lerman, K.; Plangprasopchok, A.; and Knoblock, C. A. 2007. Semantic labeling of online information sources. Interna- tional Journal on Semantic Web and Information Systems, Special Issue on Ontology Matching. [Lin 1991] Lin, J. 1991. Divergence measures based on the shannon entropy. IEEE Transactions on Information Theory 37(1):145–151. [Mathes 2004] Mathes, A. 2004. Folksonomies: coop- erative classification and communication through shared metadata. [Popescul et al. 2001] Popescul, A.; Ungar, L.; Pennock, D.; and Lawrence, S. 2001. Probabilistic models for uni- fied collaborative and content-based recommendation in sparse-data environments. In 17th Conference on Uncer- tainty in Artificial Intelligence, 437–444. [Rosen-Zvi et al. 2004] Rosen-Zvi, M.; Griffiths, T.; Steyvers, M.; and Smyth, P. 2004. The author-topic model for authors and documents. In AUAI ’04: Proceedings of the 20th conference on Uncertainty in artificial intelli- gence, 487–494. Arlington, Virginia, United States: AUAI Press. [Thakkar, Ambite, & Knoblock 2005] Thakkar, S.; Am- bite, J. L.; and Knoblock, C. A. 2005. Composing, op- timizing, and executing plans for bioinformatics web ser- vices. VLDB Journal 14(3):330–353. [Wu, Zhang, & Yu 2006] Wu, X.; Zhang, L.; and Yu, Y. 2006. Exploring social annotations for the semantic web. In WWW ’06: Proceedings of the 15th international confer- ence on World Wide Web, 417–426. New York, NY, USA: ACM Press. Introduction Problem Definition Approach Probabilistic Latent Semantic Model (pLSA) Three-way Aspect Model (MWA) Interest-Topic Model (ITM) Empirical Validation Previous Research Conclusion

0704.1676

Personalizing Image Search Results on Flickr

Personalizing Image Search Results on Flickr Kristina Lerman, Anon Plangprasopchok and Chio Wong University of Southern California Information Sciences Institute 4676 Admiralty Way Marina del Rey, California 90292 {lerman,plangpra,chiowong}@isi.edu Abstract The social media site Flickr allows users to upload their pho- tos, annotate them with tags, submit them to groups, and also to form social networks by adding other users as contacts. Flickr offers multiple ways of browsing or searching it. One option is tag search, which returns all images tagged with a specific keyword. If the keyword is ambiguous, e.g., “beetle” could mean an insect or a car, tag search results will include many images that are not relevant to the sense the user had in mind when executing the query. We claim that users express their photography interests through the metadata they add in the form of contacts and image annotations. We show how to exploit this metadata to personalize search results for the user, thereby improving search performance. First, we show that we can significantly improve search precision by filtering tag search results by user’s contacts or a larger social network that includes those contact’s contacts. Secondly, we describe a probabilistic model that takes advantage of tag information to discover latent topics contained in the search results. The users’ interests can similarly be described by the tags they used for annotating their images. The latent topics found by the model are then used to personalize search results by find- ing images on topics that are of interest to the user. Introduction The photosharing site Flickr is one of the earliest and more popular examples of the new generation of Web sites, la- beled social media, whose content is primarily user-driven. Other examples of social media include: blogs (personal online journals that allow users to share thoughts and re- ceive feedback on them), Wikipedia (a collectively writ- ten and edited online encyclopedia), and Del.icio.us and Digg (Web sites that allow users to share, discuss, and rank Web pages, and news stories respectively). The rise of so- cial media underscores a transformation of the Web as fun- damental as its birth. Rather than simply searching for, and passively consuming, information, users are collabora- tively creating, evaluating, and distributing information. In the near future, new information-processing applications en- abled by social media will include tools for personalized in- formation discovery, applications that exploit the “wisdom of crowds” (e.g., emergent semantics and collaborative in- Copyright c© 2018, American Association for Artificial Intelli- gence (www.aaai.org). All rights reserved. formation evaluation), deeper analysis of community struc- ture to identify trends and experts, and many others still dif- ficult to imagine. Social media sites share four characteristics: (1) Users create or contribute content in a variety of media types; (2) Users annotate content with tags; (3) Users evaluate con- tent, either actively by voting or passively by using content; and (4) Users create social networks by designating other users with similar interests as contacts or friends. In the pro- cess of using these sites, users are adding rich metadata in the form of social networks, annotations and ratings. Avail- ability of large quantities of this metadata will lead to the development of new algorithms to solve a variety of infor- mation processing problems, from new recommendation to improved information discovery algorithms. In this paper we show how user-added metadata on Flickr can be used to improve image search results. We claim that users express their photography interests on Flickr, among other ways, by adding photographers whose work they ad- mire to their social network and through the tags they use to annotate their own images. We show how to exploit this information to personalize search results to the individual user. The rest of the paper is organized as follows. First, we describe tagging and why it can be viewed as a useful ex- pression of user’s interests, as well as some of the challenges that arise when working with tags. In Section “Anatomy of Flickr” we describe Flickr and its functionality in greater de- tails, including its tag search capability. In Section “Data collections” we describe the data sets we have collected from Flickr, including image search results and user infor- mation. In Sections “Personalizing by contacts” and “Per- sonalizing by tags” we present the two approaches to per- sonalize search results for an individual user by filtering by contacts and filtering by tags respectively. We evaluate the performance of each method on our Flickr data sets. We conclude by discussing results and future work. Tagging for organizing images Tags are keyword-based metadata associated with some con- tent. Tagging was introduced as a means for users to orga- nize their own content in order to facilitate searching and browsing for relevant information. It was popularized by http://arxiv.org/abs/0704.1676v1 the social bookmarking site Delicious1, which allowed users to add descriptive tags to their favorite Web sites. In re- cent years, tagging has been adopted by many other so- cial media sites to enable users to tag blogs (Technorati), images (Flickr), music (Last.fm), scientific papers (CiteU- Like), videos (YouTube), etc. The distinguishing feature of tagging systems is that they use an uncontrolled vocabulary. This is in marked contrast to previous attempts to organize information via formal tax- onomies and classification systems. A formal classification system, e.g., Linnaean classification of living things, puts an object in a unique place within a hierarchy. Thus, a tiger (Panthera tigris) is a carnivorous mammal that belongs to the genus Panthera, which also includes large cats, such as lions and leopards. Tiger is also part of the felidae family, which includes small cats, such as the familiar house cat of the genus Felis. Tagging is a non-hierarchical and non-exclusive cat- egorization, meaning that a user can choose to high- light any one of the tagged object’s facets or proper- ties. Adapting the example from Golder and Huber- man (Golder and Huberman 2005), suppose a user takes an image of a Siberian tiger. Most likely, the user is not famil- iar with the formal name of the species (P. tigris altaica) and will tag it with the keyword “tiger.” Depending on his needs or mood, the user may even tag is with more general or spe- cific terms, such as “animal,” “mammal” or “Siberian.” The user may also note that the image was taken at the “zoo” and that he used his “telephoto” lens to get the shot. Rather than forcing the image into a hierarchy or multiple hierar- chies based on the equipment used to take the photo, the place where the image was taken, type of animal depicted, or even the animal’s provenance, tagging system allows the user to locate the image by any of its properties by filtering the entire image set on any of the tags. Thus, searching on the tag “tiger” will return all the images of tigers the user has taken, including Siberian and Bengal tigers, while searching on “Siberian” will return the images of Siberian animals, people or artifacts the user has photographed. Filtering on both “Siberian” and “tiger” tags will return the intersection of the images tagged with those keywords, in other words, the images of Siberian tigers. As Golder and Huberman point out, tagging systems are vulnerable to problems that arise when users try to attach semantics to objects through keywords. These problems are exacerbated in social media where users may use different tagging conventions, but still want to take advantage of the others’ tagging activities. The first problem is of homonymy, where the same tag may have different meanings. For exam- ple, the “tiger” tag could be applied to the mammal or to Apple computer’s operating system. Searching on the tag “tiger” will return many images unrelated the carnivorous mammals, requiring the user to sift through possibly a large amount of irrelevant content. Another problem related to homonymy is that of polysemy, which arises when a word has multiple related meanings, such as “apple” to mean the company or any of its products. Another problem is that 1http://del.icio.us of synonymy, or multiple words having the same or related meaning, for example, “baby” and “infant.” The problem here is that if the user wants all images of young children in their first year of life, searching on the tag “baby” may not return all relevant images, since other users may have tagged similar photographs with “infant.” Of course, plurals (“tigers” vs ”tiger”) and many other tagging idiosyncrasies (”myson” vs “son”) may also confound a tagging system. Golder and Huberman identify yet another problem that arises when using tags for categorization — that of the “ba- sic level.” A given item can be described by terms along a spectrum of specificity, ranging from specific to general. A Siberian tiger can be described as a “tiger,” but also as a “mammal” and “animal.” The basic level is the category people choose for an object when communicating to others about it. Thus, for most people, the basic level for canines is “dog,” not the more general “animal” or the more specific “beagle.” However, what constitutes the basic level varies between individuals, and to a large extent depends on the degree of expertise. To a dog expert, the basic level may be the more specific “beagle” or “poodle,” rather than “dog.” The basic level problem arises when different users choose to describe the item at different levels of specificity. For ex- ample, a dog expert tags an image of a beagle as “beagle,” whereas the average user may tag a similar image as “dog.” Unless the user is aware of the basic level variation and sup- plies more specific (and more general) keywords during tag search, he may miss a large number of relevant images. Despite these problems, tagging is a light weight, flexi- ble categorization system. The growing number of tagged images provides evidence that users are adopting tag- ging on Flickr (Marlow et al. 2006). There is specula- tion (Mika 2005) that collective tagging will lead to a com- mon informal classification system, dubbed a “folksonomy,” that will be used to organize all information from all users. Developing value-added systems on top of tags, e.g., which allow users to better browse or search for relevant items, will only accelerate wider acceptance of tagging. Anatomy of Flickr Flickr consists of a collection of interlinked user, photo, tag and group pages. A typical Flickr photo page is shown in Figure 1. It provides a variety of information about the im- age: who uploaded it and when, what groups it has been sub- mitted to, its tags, who commented on the image and when, how many times the image was viewed or bookmarked as a “favorite.” Clicking on a user’s name brings up that user’s photo stream, which shows the latest photos she has up- loaded, the images she marked as “favorite,” and her profile, which gives information about the user, including a list of her contacts and groups she belong to. Clicking on the tag shows user’s images that have been tagged with this key- word, or all public images that have been similarly tagged. Finally, the group link brings up the group’s page, which shows the photo group, group membership, popular tags, discussions and other information about the group. Figure 1: A typical photo page on Flickr Groups Flickr allows users to create special interest groups on any imaginable topic. There are groups for showcasing exceptional images, group for images of circles within a square, groups for closeups of flowers, for the color red (and every other color and shade), groups for rating sub- mitted images, or those used solely to generate comments. Some groups are even set up as games, such as The Infinite Flickr, where the rule is that a user post an image of her- self looking at the screen showing the last image (of a user looking at the screen showing next to last image, etc). There is redundancy and duplication in groups. For exam- ple, groups for child photography include Children’s Por- traits, Kidpix, Flickr’s Cutest Kids, Kids in Action, Tod- dlers, etc. A user chooses one, or usually several, groups to which to submit an image. We believe that group names can be viewed as a kind of publicly agreed upon tags. Contacts Flickr allows users to designate others as friends or contacts and makes it easy to track their activities. A single click on the “Contacts” hyperlink shows the user the latest images from his or her contacts. Tracking activities of friends is a common feature of many social media sites and is one of their major draws. Interestingness Flickr uses the “interestingness” criterion to evaluate the quality of the image. Although the algorithm that is used to compute this is kept secret to prevent gaming the system, certain metrics are taken into account: “where the clickthroughs are coming from; who comments on it and when; who marks it as a favorite; its tags and many more things which are constantly changing.”2 Browsing and searching Flickr offers the user a number of browsing and searching methods. One can browse by popular tags, through the groups directory, through the Explore page and the calen- dar interface, which provides access to the 500 most “inter- esting” images on any given day. A user can also browse geotagged images through the recently introduced map in- terface. Finally, Flickr allows for social browsing through the “Contacts” interface that shows in one place the recent images uploaded by the user’s designated contacts. Flickr allows searching for photos using full text or tag search. A user can restrict the search to all public photos, his or her own photos, photos she marked as her favorite, or photos from a specific contact. The advanced search inter- face currently allows further filtering by content type, date and camera. Search results are by default displayed in reverse chrono- logical order of being uploaded, with the most recent images on top. Another available option is to display images by their “interestingness” value, with the most “interesting” images on top. 2http://flickr.com/explore/interesting/ Personalizing search results Suppose a user is interested in wildlife photography and wants to see images of tigers on Flickr. The user can search for all public images tagged with the keyword “tiger.” As of March 2007, such a search returns over 55, 500 results. When images are arranged by their “interestingness,” the first page of results contains many images of tigers, but also of a tiger shark, cats, butterfly and a fish. Subsequent pages of search results show, in addition to tigers, children in striped suits, flowers (tiger lily), more cats, Mac OS X (tiger) screenshots, golfing pictures (Tiger Woods), etc. In other words, results include many false positives, images that are irrelevant to what the user had in mind when executing the search. We assume that when the search term is ambiguous, the sense that the user has in mind is related to her interests. For example, when a child photographer is searching for pictures of a “newborn,” she is most likely interested in photographs of human babies, not kittens, puppies, or ducklings. Simi- larly, a nature photographer specializing in macro photogra- phy is likely to be interested in insects when searching on the keyword “beetle,” not a Volkswagen car. Users express their photography preferences and interests in a number of ways on Flickr. They express them through their contacts (photographers they choose to watch), through the images they upload to Flickr, through the tags they add to these im- ages, through the groups they join, and through the images of other photographers they mark as their favorite. In this pa- per we show that we can personalize results of tag search by exploiting information about user’s preferences. In the sec- tions below, we describe two search personalization meth- ods: one that relies on user-created tags and one that exploits user’s contacts. We show that both methods improve search performance by reducing the number of false positives, or irrelevant results, returned to the user. Data collections To show how user-created metadata can be used to personal- ize results of tag search, we retrieved a variety of data from Flickr using their public API. Data sets We collected images by performing a single keyword tag search of all public images on Flickr. We specified that the returned images are ordered by their “interestingness” value, with most interesting images first. We retrieved the links to the top 4500 images for each of the following search terms: tiger possible senses include (a) big cat ( e.g., Asian tiger), (b) shark (Tiger shark), (c) flower (Tiger Lily), (d) golfing (Tiger Woods), etc. newborn possible senses include (a) a human baby, (b) kit- ten, (c) puppy, (d) duckling, (e) foal, etc. beetle possible senses include (a) a type of insect and (b) Volkswagen car model For each image in the set, we used Flickr’s API to retrieve the name of the user who posted the image (image owner), and all the image’s tags and groups. query relevant not relevant precision newborn 412 83 0.82 tiger 337 156 0.67 beetle 232 268 0.46 Table 1: Relevance results for the top 500 images retrieved by tag search Users Our objective is to personalize tag search results; therefore, to evaluate our approach, we need to have users to whose interests the search results are being tailored. We identi- fied four users who are interested in the first sense of each search term. For the newborn data set, those users were one of the authors of the paper and three other contacts within that user’s social network who were known to be interested in child photography. For the other datasets, the users were chosen from among the photographers whose images were returned by the tag search. We studied each user’s profile to confirm that the user was interested in that sense of the search term. We specifically looked at group membership and user’s tags. Thus, for the tiger data set, groups that pointed to the user’s interest in P. tigris were Big Cats, Zoo, The Wildlife Photography, etc. In addition to group mem- bership, tags that pointed to user’s interest in a topic, e.g., for the beetle data set, we assumed that users who used tags na- ture and macro were interested in insects rather than cars. Likewise, for the newborn data set, users who had uploaded images they tagged with baby and child were probably in- terested in human newborns. For each of the twelve users, we collected the names of their contacts, or Level 1 contacts. For each of these con- tacts, we also retrieved the list of their contacts. These are called Level 2 contacts. In addition to contacts, we also re- trieved the list of all the tags, and their frequencies, that the users had used to annotate their images. In addition to all tags, we also extracted a list of related tags for each user. These are the tags that appear together with the tag used as the search term in the user’s photos. In other words, suppose a user, who is a child photographer, had used tags such as “baby”, “child”, “newborn”, and “portrait” in her own im- ages. Tags related to newborn are all the tags that co-occur with the “newborn” tag in the user’s own images. This in- formation was also extracted via Flickr’s API. Search results We manually evaluated the top 500 images in each data set and marked each as relevant if it was related to the first sense of the search term listed above, not relevant or undecided, if the evaluator could not understand the image well enough to judge its relevance. In Table 1, we report the precision of the search within the 500 labeled images, as judged from the point of view of the searching users. Precision is defined as the ratio of relevant images within the result set over the 500 retrieved images. Precision of tag search on these sample queries is not very high due to the presence of false positives — images not rel- evant to the sense of the search term the user had in mind. In the sections below we show how to improve search per- formance by taking into consideration supplementary infor- mation about user’s interests provided by her contacts and tags. Personalizing by contacts Flickr encourages users to designate others as contacts by making is easy to view the latest images submitted by them through the “Contacts” interface. Users add contacts for a variety of reasons, including keeping in touch with friends and family, as well as to track photographers whose work is of interest to them. We claim that the latter reason is the most dominant of the reasons. Therefore, we view user’s contacts as an expression of the user’s interests. In this sec- tion we show that we can improve tag search results by filter- ing through the user’s contacts. To personalize search results for a particular user, we simply restrict the images returned by the tag search to those created by the user’s contacts. Table 2 shows how many of the 500 images in each data set came from a user’s contacts. The column labeled “# L1” gives the number of user’s Level 1 contacts. The follow- ing columns show how many of the images were marked as relevant or not relevant by the filtering method, as well as precision and recall relative to the 500 images in each data set. Recall measures the fraction of relevant retrieved im- ages relative to all relevant images within the data set. The last column “improv” shows percent improvement in preci- sion over the plain (unfiltered) tag search. As Table 2 shows, filtering by contacts improves the pre- cision of tag search for most users anywhere from 22% to over 100% when compared to plain search results in Ta- ble 1. The best performance is attained for users within the newborn set, with a large number of relevant images cor- rectly identified as being relevant, and no irrelevant images admitted into the result set. The tiger set shows an average precision gain of 42% over four users, while the beetle set shows an 85% gain. Increase in precision is achieved by reducing the number of false positives, or irrelevant images that are marked as rel- evant by the search method. Unfortunately, this gain comes at the expense of recall: many relevant images are missed by this filtering method. In order to increase recall, we en- large the contacts set by considering two levels of contacts: user’s contacts (Level 1) and her contacts’ contacts (Level 2). The motivation for this is that if the contact relation- ship expresses common interests among users, user’s inter- ests will also be similar to those of her contacts’ contacts. The second half of Table 2 shows the performance of filtering the search results by the combined set of user’s Level 1 and Level 2 contacts. This method identifies many more relevant images, although it also admits more irrele- vant images, thereby decreasing precision. This method still shows precision improvement over plain search, with pre- cision gain of 9%, 16% and 11% respectively for the three data sets. Personalizing by tags In addition to creating lists of contacts, users express their photography interests through the images they post on Flickr. We cannot yet automatically understand the content of images. Instead, we turn to the metadata added by the user to the image to provide a description of the image. The metadata comes in a variety of forms: image title, descrip- tion, comments left by other users, tags the image owner added to it, as well as the groups to which she submitted the image. As we described in the paper, tags are useful im- age descriptors, since they are used to categorize the image. Similarly, group names can be viewed as public tags that a community of users have agreed on. Submitting an image to a group is, therefore, equivalent to tagging it with a public In the section below we describe a probabilistic model that takes advantage of the images’ tag and group informa- tion to discover latent topics in each search set. The users’ interests can similarly be described by collections of tags they had used to annotate their own images. The latent top- ics found by the model can be used to personalize search results by finding images on topics that are of interest to a particular user. Model definition We need to consider four types of entities in the model: a set of users U = {u1, ..., un}, a set of images or photos I = {i1, ..., im}, a set of tags T = {t1, ..., to}, and a set of groups G = {g1, ..., gp}. A photo ix posted by owner ux is described by a set of tags {tx1, tx2, ...} and submitted to several groups {gx1, gx2, ...}. The post could be viewed as a tuple < ix, ux, {tx1, tx2, ...}, {gx1, gx2, ...} >. We as- sume that there are n users, m posted photos and p groups in Flickr. Meanwhile, the vocabulary size of tags is q. In order to filter images retrieved by Flickr in response to tag search and personalize them for a user u, we compute the conditional probability p(i|u), that describes the probability that the photo i is relevant to u based on her interests. Im- ages with high enough p(i|u) are then presented to the user as relevant images. As mentioned earlier, users choose tags from an uncon- trolled vocabulary according to their styles and interests. Images of the same subject could be tagged with different keywords although they have similar meaning. Meanwhile, the same keyword could be used to tag images of different subjects. In addition, a particular tag frequently used by one user may have a different meaning to another user. Proba- bilistic models offer a mechanism for addressing the issues of synonymy, polysemy and tag sparseness that arise in tag- ging systems. We use a probabilistic topic model (Rosen-Zvi et al. 2004) to model user’s image posting behavior. As in a typical probabilistic topic model, topics are hidden variables, representing knowledge cate- gories. In our case, topics are equivalent to image owner’s interests. The process of photo posting by a particular user could be described as a stochastic process: • User u decides to post a photo i. user # L1 rel. not rel. Pr Re improv # L2+L2 rel. not rel. Pr Re improv newborn user1 719 232 0 1.00 0.56 22% 49,539 349 62 0.85 0.85 4% user2 154 169 0 1.00 0.41 22% 10,970 317 37 0.9 0.77 10% user3 174 147 0 1.00 0.36 22% 13,153 327 39 0.89 0.79 9% user4 128 132 0 1.00 0.32 22% 8,439 310 29 0.91 0.75 11% tiger user5 63 11 1 0.92 0.03 37% 13,142 255 71 0.78 0.76 16% user6 103 78 3 0.96 0.23 44% 14,425 266 83 0.76 0.79 13% user7 62 65 1 0.98 0.19 47% 7,270 226 60 0.79 0.67 18% user8 56 30 0 0.97 0.09 44% 7,073 240 63 0.79 0.71 18% beetle user9 445 18 1 0.95 0.08 106% 53,480 215 221 0.49 0.93 7% user10 364 35 8 0.81 0.15 77% 41,568 208 217 0.49 0.90 7% user11 783 78 25 0.75 0.34 65% 62,610 218 227 0.49 0.94 7% user12 102 7 1 0.88 0.03 90% 14,324 163 152 0.52 0.70 13% Table 2: Results of filtering tag search by user’s contacts. “# L1” denotes the number of Level 1 contacts and “# L1+L2” shows the number of Level 1 and Level 2 contacts, with the succeeding columns displaying filtering results of that method: the number of images marked relevant or not relevant, as well as precision and recall of the filtering method relative to the top 500 images. The columns marked “improv” show improvement in precision over plain tag search results. Figure 2: Graphical representation for model-based infor- mation filtering. U , T , G and Z denote variables “User”, “Tag”, “Group”, and “Topic” respectively. Nt represents a number of tag occurrences for a one photo (by the photo owner); D represents a number of all photos on Flickr. Meanwhile, Ng denotes a number of groups for a particu- lar photo. • Based on user u’s interests and the subject of the photo, a set of topics z are chosen. • Tag t is then selected based on the set of topics chosen in the previous state. • In case that u decides to expose her photo to some groups, a group g is then selected according to the chosen topics. The process is depicted in a graphical form in Figure 2. We do not treat the image i as a variable in the model but view it as a co-occurrence of a user, a set of tags and a set of groups. From the process described above, we can represent the joint probability of user, tag and group for a particular photo as p(i) = p(ui, Ti, Gi) = p(ui) · p(zk|ui)p(ti|z) )ni(t) p(zk|ui)p(gi|z) )ni(g) Note that it is straightforward to exclude photo’s group information from the above equation simply by omitting the terms relevant to g. nt and ng is a number of all possible tags and groups respectively in the data set. Meanwhile, ni(t) and ni(g) act as indicator functions: ni(t) = 1 if an image i is tagged with tag t; otherwise, it is 0. Similarly, ni(g) = 1 if an image i is submitted to group g; otherwise, it is 0. k is the predefined number of topics. The joint probability of photos in the data set I is defined p(I) = p(im). In order to estimate parameters p(z|ui), p(ti|z), and p(gi|z), we define a log likelihood L, which measures how the esti- mated parameters fit the observed data. According to the EM algorithm (Dempster et al. 1977), L will be used as an objective function to estimate all parameters. L is defined as L(I) = log(p(I)). In the expectation step (E-step), the joint probability of the hidden variable Z given all observations is computed from the following equations: p(z|t, u) ∝ p(z|u) · p(t|z) (1) p(z|g, u) ∝ p(z|u) · p(g|z). (2) L cannot be maximized easily, since the summation over the hidden variable Z appears inside the logarithm. We in- stead maximize the expected complete data log-likelihood over the hidden variable, E[Lc], which is defined as E[Lc] = log(p(u)) ni(t) · p(z|u, t) (log(p(z|u)· (t|z)) ni(g) · p(z|u, g) (log(p(z|g)· (g|z)) Since the term log(p(ui)) is not related to parame- ters and can be computed directly from the observed data, we discard this term from the expected complete data log- likelihood. With normalization constraints on all parame- ters, Lagrange multipliers τ , ρ, ψ are added to the expected log likelihood, yielding the following equation H = E[Lc] + p(t|z) p(g|z) p(z|u) We maximize H with respect to p(t|zk), p(g|zk), and p(zk|u), and then eliminate the Lagrange multipliers to ob- tain the following equations for the maximization step: p(t|z) ∝ ni(t) · p(z|t, u) (3) p(g|z) ∝ ni(g) · p(z|g, u) (4) p(zk|um) ∝ nm(t) · p(zk|um, t) (5) nm(g) · p(zk|um, g) The algorithm iterates between E and M step until the log likelihood for all parameter values converge. Model-based personalization We can use the model developed in the previous section to find the images i most relevant to the interests of a partic- ular user u′. We do so by learning the parameters of the model from the data and using these parameters to compute the conditional probability p(i|u′). This probability can be factorized as follows: p(i|u′) = p(ui, Ti, Gi|z) · p(z|u ′) , (6) where ui is the owner of image i in the data set, and Ti and Gi are, respectively, the set of all the tags and groups for the image i. The former term in Equation 6 can be factorized further p(ui, Ti, Gi|z) ∝ p(Ti|z)· (Gi|z)· (z|ui) · p(ui) p(ti|z) p(gi|z) · p(z|ui) · p(ui) . We can use the learned parameters to compute this term di- rectly. We represent the interests of user u′ as an aggregate of the tags that u′ had used in the past for tagging her own images. This information is used to to approximate p(z|u′): p(z|u′) ∝ n(t′ = t) · p(z|t) where n(t′ = t) is a frequency (or weight) of tag t′ used by u′. Here we view n(t′ = t) is proportional to p(t′|u′). Note that we can use either all the tags u′ had applied to the images in her photostream, or a subset of these tags, e.g., only those that co-occur with some tag in user’s images. Evaluation We trained the model separately on each data set of 4500 images. We fixed the number of topics at ten. We then eval- uated our model-based personalization framework by using the learned parameters and the information about the in- terests of the selected users to compute p(i|u′) for the top 500 (manually labeled) images in the set. Information about user’s interests was captured either by (1) all tags (and their frequencies) that are used in all the images of the user’s pho- tostream or (2) related tags that occurred in images that were tagged with the search keyword (e.g., “newborn”) by the user. Computation of p(t|z) is central to the parameter estima- tion process, and it tells us something about how strongly a tag t contributes to a topic z. Table 3 shows the most prob- able 25 tags for each topic for the tiger data set trained on ten topics. Although the tag “tiger” dominates most topics, we can discern different themes from the other tags that ap- pear in each topic. Thus, topic z5 is obviously about domes- tic cats, while topic z8 is about Apple computer products. Meanwhile, topic z2 is about flowers and colors (“flower,” “lily,” “yellow,” “pink,” “red”); topic z6 is about about places (“losangeles,” “sandiego,” “lasvegas,” “stuttgard,”), presumably because these places have zoos. Topic z7 con- tains several variations of tiger’s scientific name, “panthera tigris.” This method appears to identify related words well. Topic z5, for example, gives synonyms “cat,” “kitty,” as well as the more general term “pet” and the more specific terms “kitten” and “tabby.” It even contains the Spanish version of the word: “gatto.” In future work we plan to explore using this method to categorize photos in a more abstract way. We also note that related terms can be used to increase search recall by providing additional keywords for queries. Table 4 presents results of model-based personalization for the case that uses information from all of user’s tags. The model was trained with ten topics. Results are pre- sented for different thresholds. The first two columns, for example, report precision and recall for a high threshold that z1 z2 z3 z4 z5 tiger tiger tiger tiger tiger zoo specanimal cat thailand cat animal animalkingdomelite kitty bengal animal nature abigfave cute animals animals animals flower kitten tigers zoo wild butterfly cats canon bigcat tijger macro orange d50 cats wildlife yellow eyes tigertemple tigre ilovenature swallowtail pet 20d animalplanet cub lily tabby white tigers siberiantiger green stripes nikon bigcats blijdorp canon whiskers kanchanaburi whitetiger london insect white detroit mammal australia nature art life wildlife portfolio pink feline michigan colorado white red fur detroitzoo stripes dierentuin flowers animal eos denver toronto orange gatto temple sumatrantiger stripes eastern pets park white amurtiger usa black asia feline nikonstunninggallery impressedbeauty paws ball mammals s5600 tag2 furry marineworld sumatran eyes specnature nose baseball exoticcats sydney black teeth detroittigers exoticcat cat streetart beautiful wild big z6 z7 z8 z9 z10 tiger nationalzoo tiger tiger tiger tigers tiger apple india lion dczoo sumatrantiger mac canon dog tigercub zoo osx wildlife shark california nikon macintosh impressedbeauty nyc lion washingtondc screenshot endangered cat cat smithsonian macosx safari man cc100 washington desktop wildanimals people florida animals imac wild arizona girl cat stevejobs tag1 rock wilhelma bigcat dashboard tag3 beach self tigris macbook park sand lasvegas panthera powerbook taggedout sleeping stuttgart bigcats os katze tree me d70s 104 nature forest baby pantheratigrissumatrae canon bravo puppy tattoo dc x nikon bird endangered sumatrae ipod asia portrait illustration animal computer canonrebelxt marwell ?? 2005 ibook bandhavgarh boy losangeles pantheratigris intel vienna fish portrait nikond70 keyboard schnbrunn panther sandiego d70 widget zebra teeth lazoo 2006 wallpaper pantheratigris brooklyn giraffe topv111 laptop d2x bahamas Table 3: Top tags ordered by p(t—z) for the ten topic model of the “tiger” data set. Pr Re Pr Re Pr Re Pr Re Pr Re newborn n=50 n=100 n=200 n=300 n=412* user1 1.00 0.12 1.00 0.24 1.00 0.49 0.94 0.68 0.89 0.89 user2 1.00 0.12 1.00 0.24 1.00 0.49 0.92 0.67 0.87 0.87 user3 1.00 0.12 0.88 0.21 0.84 0.41 0.85 0.62 0.89 0.89 user4 1.00 0.12 0.99 0.24 1.00 0.48 0.94 0.69 0.89 0.89 tiger n=50 n=100 n=200 n=300 n=337* user5 0.94 0.14 0.90 0.27 0.82 0.48 0.80 0.71 0.79 0.79 user6 0.76 0.11 0.80 0.24 0.79 0.47 0.77 0.69 0.77 0.77 user7 0.94 0.14 0.90 0.27 0.82 0.48 0.80 0.71 0.79 0.79 user8 0.90 0.13 0.88 0.26 0.82 0.49 0.79 0.71 0.79 0.79 beetle n=50 n=100 n=200 n=232* n=300 user9 1.00 0.22 0.99 0.43 0.77 0.66 0.70 0.70 0.66 0.85 user10 0.98 0.21 0.99 0.43 0.77 0.66 0.70 0.70 0.66 0.85 user11 0.98 0.21 0.93 0.40 0.50 0.43 0.51 0.51 0.50 0.65 user12 1.00 0.22 0.99 0.43 0.77 0.66 0.70 0.70 0.66 0.85 Table 4: Filtering results where a number of learned topics is 10, excluding group information, and user’s personal information obtained from all tags she used for her photos. Asterisk denotes R-precision of the method, or precision of the first n results, where n is the number of relevant results in the data set. marks only the 50 most probable images as relevant. The re- maining 450 images are marked as not relevant to the user. Recall is low, because many relevant images are excluded from the results for such a high threshold. As the thresh- old is decreased (n = 100, n = 200, . . .), recall relative to the 500 labeled images increases. Precision remains high in all cases, and higher than precision of the plain tag search reported in Table 1. In fact, most of the images in the top 100 results presented to the user are relevant to her query. The column marked with the asterisk gives the R-precision of the method, or precision of the first R results, where R is the number of relevant results. The average R-precision of this filtering method is 8%, 17% and 42% better than plain search precision on our three data sets. Performance results of the approach that uses related tags instead of all tags are given in Table 5. We explored this di- rection, because we believed it could help discriminate be- tween different topics that interest a user. Suppose, a child photographer is interested in nature photography as well as child portraiture. The subset of tags he used for tagging his “newborn” portraits will be different from the tags used for tagging nature images. These tags could be used to differen- tiate between newborn baby and newborn colt images. How- ever, on the set of users selected for our study, using related tags did not appear to improve results. This could be be- cause the tags a particular user used together with, for ex- ample, “beetle” do not overlap significantly with the rest of the data set. Including group information did not significantly improve results (not presented in this manuscript). In fact, group in- formation sometimes hurts the estimation rather than helps. We believe that this is because our data sets (sorted by Flickr according to image interestingness) are biased by the pres- ence of general topic groups (e.g., Search the Best, Spec- tacular Nature, Let’s Play Tag, etc.). We postulate that group information would help estimate p(i|z) in cases where the photo has few or no tags. Group information would help filling in the missing data by using group name as another tag. We also trained the model on the data with 15 topics, but found no significant difference in results. Previous research Recommendation or personalization systems can be cate- gorized into two main categories. One is collaborative fil- tering (Breese et al. 1998) which exploits item ratings from many users to recommend items to other like-minded users. The other is content-based recommendation, which relies on the contents of an item and user’s query, or other user information, for prediction (Mooney and Roy 2000). Our first approach, filtering by contacts, can be viewed as im- plicit collaborative filtering, where the user–contact rela- tionship is viewed as a preference indicator: it assumes that the user likes all photos produced by her contacts. In our previous work, we showed that users do indeed agree with the recommendations made by contacts (Lerman 2007; Lerman and Jones 2007). This is similar to the ideas imple- mented by MovieTrust (Golbeck 2006), but unlike that sys- tem, social media sites do not require users to rate their trust in the contact. Meanwhile, our second approach, filtering by tags (and groups), shares some characteristics with both methods. It is similar to collaborative filtering, since we use tags to rep- resent agreement between users. It is also similar to content- based recommendation, because we represent image content by the tags and group names that have been assigned to it by the user. Our model-based filtering system is technically similar to, but conceptually different from, probabilistic models pro- Pr Re Pr Re Pr Re Pr Re Pr Re newborn n=50 n=100 n=200 n=300 n=412* Pr Re Pr Re Pr Re Pr Re Pr Re user1 0.8 0.10 0.78 0.19 0.79 0.38 0.77 0.56 0.79 0.79 user2 0.8 0.10 0.82 0.20 0.80 0.39 0.77 0.56 0.83 0.83 user3 0.98 0.12 0.88 0.21 0.84 0.41 0.80 0.58 0.85 0.85 user4 0.98 0.12 0.88 0.21 0.84 0.41 0.85 0.62 0.88 0.88 tiger n=50 n=100 n=200 n=300 n=337* user5 0.84 0.12 0.86 0.26 0.78 0.46 0.78 0.69 0.77 0.77 user6 0.72 0.11 0.79 0.23 0.78 0.46 0.76 0.68 0.76 0.76 user7 0.72 0.11 0.78 0.23 0.78 0.46 0.76 0.68 0.76 0.76 user8 0.9 0.13 0.82 0.24 0.80 0.47 0.78 0.69 0.78 0.78 beetle n=50 n=100 n=200 n=232* n=300 user9 0.78 0.17 0.62 0.27 0.58 0.50 0.54 0.54 0.53 0.68 user10 0.98 0.21 0.88 0.38 0.77 0.66 0.72 0.72 0.65 0.84 user11 0.96 0.21 0.74 0.32 0.62 0.53 0.59 0.59 0.56 0.72 user12 0.98 0.21 0.99 0.43 0.77 0.66 0.70 0.70 0.66 0.85 Table 5: Filtering results where a number of learned topics is 10, excluding group information, and user’s personal information obtained from all tags she used for her photos, which are tagged by the search term posed by (Popescul et al. 2001). Both models are proba- bilistic generative models that describe co-occurrences of users and items of interest. In particular, the model assumes a user generates her topics of interest; then the topics gen- erate documents and words in those documents if the user prefers those documents. In our model, we metaphorically assume the photo owner generates her topics of interest. The topics, in turn, generate tags that the owner used to annotate her photo. However, unlike the previous work, we do not treat photos as variables, as they do for documents. This is because images are tagged only by their owners; meanwhile, in their model, all users who are interested in a document generate topics for that document. Our model-based approach is almost identical to the author-topic model(Rosen-Zvi et al. 2004). However, we extend their framework to address (1) how to exploit photo’s group information for personalized information filtering; (2) how to approximate user’s topics of interest from partially observed personal information (the tags the user used to de- scribe her own images). For simplicity, we use the classi- cal EM algorithm to train the model; meanwhile they use a stochastic approximation approach due to the difficulty in- volved in performing exact an inference for their generative model. Conclusions and future work We presented two methods for personalizing results of im- age search on Flickr. Both methods rely on the meta- data users create through their everyday activities on Flickr, namely user’s contacts and the tags they used for annotating their images. We claim that this information captures user’s tastes and preferences in photography and can be used to personalize search results to the individual user. We showed that both methods dramatically increase search precision. We believe that increasing precision is an important goal for personalization, because dealing with the information over- load is the main issue facing users, and we can help users by reducing the number of irrelevant results the user has to examine (false positives). Having said that, our tag-based approach can also be used to expand the search by suggest- ing relevant related keywords (e.g., “pantheratigris,” “big- cat” and ”cub” for the query tiger). In addition to tags and contacts, there exists other meta- data, favorites and comments, that can be used to aid infor- mation personalization and discovery. In our future work we plan to address the challenge of combing these heteroge- neous sources of evidence within a single approach. We will begin by combining contacts information with tags. The probabilistic model needs to be explored further. Right now, there is no principled way to pick the number of latent topics that are contained in a data set. We also plan to have a better mechanism for dealing with uninformative tags and groups. We would like to automatically identify general interest groups, such as the Let’s Play Tag group, that do not help to discriminate between topics. The approaches described here can be applied to other so- cial media sites, such as Del.icio.us. We imagine that in near future, all of Web will be rich with metadata, of the sort described here, that will be used to personalize information search and discovery to the individual user. Acknowledgements This research is based on work supported in part by the Na- tional Science Foundation under Award Nos. IIS-0535182 and in part by DARPA under Contract No. NBCHD030010. The U.S.Government is authorized to reproduce and dis- tribute reports for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclu- sions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of any of the above organizations or any person connected with them. References [Breese et al. 1998] John Breese, David Heckerman, and Carl Kadie. Empirical analysis of predictive algorithms for collaborative filtering. In Proceedings of the 14th An- nual Conference on Uncertainty in Artificial Intelligence (UAI-98), pages 43–52, San Francisco, CA, 1998. Morgan Kaufmann. [Dempster et al. 1977] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Soci- ety. Series B (Methodological), 39(1):1–38, 1977. [Golbeck 2006] J. Golbeck. Generating predictive movie recommendations from trust in social networks. In Pro- ceedings of the Fourth International Conference on Trust Management, Pisa, Italy, May 2006. [Golder and Huberman 2005] S. A. Golder and B. A. Huberman. The structure of collaborative tag- ging systems. Technical report, HP Labs, 2005. http://www.hpl.hp.com/research/idl/papers/tags/. [Lerman and Jones 2007] K. Lerman and Laurie Jones. So- cial browsing on flickr. In Proc. of International Confer- ence on Weblogs and Social Media (ICWSM-07), 2007. [Lerman 2007] K. Lerman. Social networks and social in- formation filtering on digg. In Proc. of International Con- ference on Weblogs and Social Media (ICWSM-07), 2007. [Marlow et al. 2006] C. Marlow, M. Naaman, d. boyd, and M. Davis. Ht06, tagging paper, taxonomy, flickr, academic article, toread. In Proceedings of Hypertext 2006, New York, 2006. ACM, New York: ACM Press. [Mika 2005] P. Mika. Ontologies are us: A unified model of social networks and semantics. In nternational Semantic Web Conference (ISWC-05), 2005. [Mooney and Roy 2000] Raymond J. Mooney and Loriene Roy. Content-based book recommending using learning for text categorization. In Proceedings of 5th ACM Con- ference on Digital Libraries, pages 195–204, San Antonio, US, 2000. ACM Press, New York, US. [Popescul et al. 2001] Alexandrin Popescul, Lyle Ungar, David Pennock, and Steve Lawrence. Probabilistic mod- els for unified collaborative and content-based recommen- dation in sparse-data environments. In 17th Conference on Uncertainty in Artificial Intelligence, pages 437–444, Seat- tle, Washington, August February–May 2001. [Rosen-Zvi et al. 2004] Michal Rosen-Zvi, Thomas Grif- fiths, Mark Steyvers, and Padhraic Smyth. The author- topic model for authors and documents. In AUAI ’04: Pro- ceedings of the 20th conference on Uncertainty in artificial intelligence, pages 487–494, Arlington, Virginia, United States, 2004. AUAI Press. Introduction Tagging for organizing images Anatomy of Flickr Browsing and searching Personalizing search results Data collections Data sets Users Search results Personalizing by contacts Personalizing by tags Model definition Model-based personalization Evaluation Previous research Conclusions and future work

0704.1677

Resummed Cross Section for Jet Production at Hadron Colliders

BNL-NT-07/17 hep-ph/yymmnnn Resummed Cross Section for Jet Production at Hadron Colliders Daniel de Florian Departamento de F́ısica, FCEYN, Universidad de Buenos Aires, (1428) Pabellón 1 Ciudad Universitaria, Capital Federal, Argentina Werner Vogelsang BNL Nuclear Theory, Brookhaven National Laboratory, Upton, NY 11973, USA Abstract We study the resummation of large logarithmic perturbative corrections to the single- inclusive jet cross section at hadron colliders. The corrections we address arise near the threshold for the partonic reaction, when the incoming partons have just enough energy to produce the high-transverse-momentum final state. The structure of the resulting logarithmic corrections is known to depend crucially on the treatment of the invariant mass of the produced jet at threshold. We allow the jet to have a non-vanishing mass at threshold, which most closely corresponds to the situation in experiment. Matching our results to available semi-analytical next-to-leading-order calculations, we derive resummed results valid to next-to-leading logarithmic accuracy. We present numerical results for the resummation effects at Tevatron and RHIC energies. hep-ph/yymmnnn November 2, 2018 http://arxiv.org/abs/0704.1677v1 1 Introduction High-transverse-momentum jet production in hadronic collisions, H1H2 → jetX , plays a fun- damental role in High-Energy Physics. It offers possibilities to explore QCD, for example the structure of the interacting hadrons or the emergence of hadronic final states, but is also inti- mately involved in many signals (and their backgrounds) for New Physics. At the heart of all these applications of jet production is our ability to perform reliable and precise perturbative cal- culations of the partonic short-distance interactions that generate the high-transverse-momentum final states. Up to corrections suppressed by inverse powers of the jet’s transverse momentum pT , the hadronic jet cross section factorizes into parton distribution functions that contain primar- ily long-distance information, and these short-distance cross sections. In the present paper, we address large logarithmic perturbative corrections to the latter. At partonic threshold, when the initial partons have just enough energy to produce the high- pT jet and an unobserved recoiling partonic final state, the phase space available for gluon bremsstrahlung vanishes, so that only soft and collinear emission is allowed, resulting in large logarithmic corrections to the partonic cross section. To be more specific, if we consider the cross section as a function of the jet transverse momentum, integrated over all jet rapidities, the par- tonic threshold is reached when s = 2pT , where s is the partonic center-of-mass (c.m.) energy. Defining x̂T ≡ 2pT/ s, the leading large contributions near threshold arise as αkS(pT ) ln 2m (1− x̂2T ) at the kth order in perturbation theory, where m ≤ k (the logarithms with m = k are leading) and αS is the strong coupling. Even if pT is large so that αS(pT ) is small, sufficiently close to threshold the logarithmic terms will spoil the perturbative expansion to any fixed order. Thresh- old resummation [1, 2, 3, 4, 5, 6, 7], however, allows to reinstate a useful perturbative series by systematically taking into account the terms αkS ln 2m (1− x̂2T ) to all orders in αS. This is achieved after taking a Mellin-transform of the hadronic cross section in xT = 2pT/ S, with S the hadronic c.m. energy. The threshold logarithms exponentiate in transform space. Regarding phenomenology, the larger xT , the more dominant will the threshold logarithms be, and hence the more important will threshold resummation effects be. In addition, because of the convoluted form of the partonic cross sections and the parton distribution functions (PDFs), the steep fall-off of the PDFs with momentum fraction x automatically enhances the contributions from the threshold regime to the cross section, because it makes it relatively unlikely that the initial partons have very high c.m. energy. This explains why partonic threshold effects often dominate the hadronic cross section even at not so high xT . Studies of cross sections for [8] pp → hX (with h a high-pT hadron) and [9, 10, 11, 12] pp → γX in the fixed-target regime, where typically 0.2 . xT . 0.7, indeed demonstrate that threshold-resummation effects dominate there and can be very large and important for phenomenology. They enhance the theoretical cross section with respect to fixed-order calculations. These observations suggest to study the resummation also for jet production at hadron col- liders, in particular when xT is rather large. An application of particular interest is the jet cross section at very high transverse momenta (pT ∼ several hundreds GeV) at the Tevatron [13, 14], for which initially an excess of the experimental data over next-to-leading order (NLO) theory was reported, which was later mostly attributed to an insufficient knowledge of the gluon distri- bution [15]. Similarly large values of xT are now probed in pp collisions at RHIC, where currently√ s = 200 GeV and jet cross section measurements by the STAR collaboration are already ex- tending to pT & 40 GeV [16]. In both these cases, one does expect threshold resummation effects to be smaller than in the case of related processes at similar xT in the fixed-target regime, just because (among other things) the strong coupling constant is smaller at these higher pT . On the other hand, as we shall see, the effects are still often non-negligible. Apart from addressing these interesting phenomenological applications, we believe we also improve in this paper the theoretical framework for threshold resummation in jet production. There has been earlier work in the literature on this topic [4, 17, 18]. In Ref. [4] the threshold resummation formalism for the closely related dijet production at large invariant mass of the jet pair was developed to next-to-leading logarithmic (NLL) order. In [17], these results were applied to the single-inclusive jet cross section at large transverse momentum, making use of earlier work [6] on the high-pT prompt-photon cross section, which is kinematically similar. As was emphasized in [4], there is an important subtlety for the resummed jet cross section related to the treatment of the invariant mass of the jet. The structure of the large logarithmic corrections that are addressed by resummation depends on whether or not the jet is assumed to be massless at partonic threshold, even at the leading-logarithmic (LL) level. This is perhaps surprising at first sight, because one might expect the jet mass to be generally inessential since it is typically much smaller than the jet’s transverse momentum pT and in fact vanishes for lowest-order partonic scattering. However, the situation can be qualitatively understood as follows [4]: let us assume that we are defining the jet cross section from the total four-momentum deposited in a cone of aperture R †. Considering for simplicity the next-to-leading order, we can have contributions by virtual 2 → 2 diagrams, or by 2 → 3 real-emission diagrams. For the former, a single particle produces the (massless) jet, in case of the latter, there are configurations where two particles in the final state jointly form the jet. Then, for a jet forced to be massless at partonic threshold, the contributions with two partons in the cone must either have one parton arbitrarily soft, or the two partons exactly collinear. The singularities associated with these configurations cancel against analogous ones in the virtual diagrams, but because the constraint on the real-emission diagrams is so restrictive, large double- and single-logarithmic contributions remain after the cancellation. This will happen regardless of the size R of the cone aperture, implying that the coefficients of the large logarithms will be independent of R. These final-state threshold logarithms arising from the observed jet suppress the cross section near threshold. Their structure is identical to that of the threshold logarithms generated by the recoiling “jet”, because the latter is not observed and is indeed massless at partonic threshold. The combined final-state logarithms then act against the threshold logarithms associated with initial-state radiation which are positive and enhance the cross section [1]. If, on the other hand, the jet invariant mass is not constrained to vanish near threshold, far more final states contribute– in fact, there will be an integration over the jet mass to an upper limit proportional to the aperture of the jet cone. As the 2 → 3 contributions are therefore much less restricted, the cancellations of infrared and collinear divergences between real and virtual diagrams leave behind only single logarithms [4], associated with soft, but not with collinear, emission. Compared to the previously discussed case, there is therefore no double-logarithmic suppression of the cross section by the observed jet, and one expects the calculated cross section to be larger. Also, the single-logarithmic terms will now depend on the jet cone size R. †Details of the jet definition do not matter for the present argument. The resummation for both these cases, with the jet massless or massive at threshold, has been worked out in [4]. The study [17] of the resummed single-inclusive high-pT jet cross section assumed massless jets at threshold. From a phenomenological point of view, however, we see no reason for demanding the jet to become massless at the partonic threshold. The experimental jet cross sections will, at any given pT , contain jet events with a large variety of jet invariant masses. NLO calculations of single-inclusive jet cross sections indeed reflect this: they have the property that jets produced at partonic threshold are integrated over a range of jet masses. This becomes evident in the available semi-analytical NLO calculations [19, 20, 21, 22]. For these, the jet cross section is obtained by assuming that the jet cone is relatively narrow, in which case it is possible to treat the jet definition analytically, so that collinear and infrared final-state divergences may be canceled by hand. This approximation is referred to as the “small-cone approximation (SCA)”. Section II.E in the recent calculation in [21] explicitly demonstrates for the SCA that the threshold double-logarithms associated with the observed final-state jet cancel, as described above. In light of this, we will study in this work the resummation in the more realistic case of jets that are massive at threshold. We will in fact make use of the NLO calculation in the SCA approximation of [21] to “match” our resummed cross sections to finite (next-to-leading) order. Knowledge of analytical NLO expressions allows one to extract certain hard-scattering coefficients that are finite at threshold and part of the full resummation formula. These coefficients will be presented and used in our paper for the first time. We emphasize that the use of the SCA in our work is not to be regarded as a limitation to the usefulness of our results. First, the SCA is known to be very accurate numerically even at relatively large jet cone sizes of R ∼ 0.7 [21, 20, 23]. In addition, one may use our results to obtain ratios of the resummed over the NLO cross sections. Such “K-factors” are then expected to be extremely good approximations for the effects of higher orders even when one goes away from the SCA and uses, for example, a full NLO Monte-Carlo integration code that allows to compute the jet cross section for larger cone aperture and for other jet definitions (see, for example, Ref. [24]). We will therefore in particular present K-factors for the resummed jet cross section in this paper. The paper is organized as follows: in Sec. 2 we provide the basic formulas for the single- inclusive-jet cross section at fixed order in perturbation theory, and discuss the SCA and the role of the threshold region. Section 3 presents details of the threshold resummation for the inclusive- jet cross section and describes the matching to the analytical expressions for the NLO cross section in the SCA. In Sec. 4 we give phenomenological results for the Tevatron and for RHIC. Finally, we summarize our results in Sec. 5. The Appendix collects the formulas for the hard-scattering coefficients in the threshold-resummed cross section mentioned above. 2 Next-to-leading order single-inclusive jet cross section Jets produced in high-energy hadronic scattering, H1(P1)H2(P2) → jet(PJ)X , are typically defined in terms of a deposition of transverse energy or four-momentum in a cone of aperture R in pseudo- rapidity and azimuthal-angle space, with detailed algorithms specifying the jet kinematic variables in terms of those of the observed hadron momenta [13, 25, 26, 27, 28, 29]. QCD factorization theorems allow to write the cross section for single-inclusive jet production in hadronic collisions in terms of convolutions of parton distribution functions with partonic hard-scattering functions [30]: dx1dx2 fa/H1 x1, µ fb/H2 x2, µ dσ̂ab(x1P1, x2P2, PJ , µF , µR) , (1) where the sum runs over all initial partons, quarks, anti-quarks, and gluons, and where µF and µR denote the factorization and renormalization scales, respectively. It is possible to use perturbation theory to describe the formation of a high-pT jet, as long as the definition of the jet is infrared- safe. The jet is then constructed from a subset of the final-state partons in the short-distance reaction ab → partons, and a “measurement function” in the dσ̂ab specifies the momentum PJ of the jet in terms of the momenta of the final-state partons, in accordance with the (experimental) jet definition. The computation of jet cross sections beyond the lowest order in perturbative QCD is rather complicated, due to the need for incorporating a jet definition and the ensuing complexity of the phase space, and due to the large number of infrared singularities of soft and collinear origin at intermediate stages of the calculation. Different methods have been introduced that allow the calculation to be performed largely numerically by Monte-Carlo “parton generators”, with only the divergent terms treated in part analytically (see, for example, Ref. [24]). A major simplification occurs if one assumes that the jet cone is rather narrow, a limit known as the “small-cone approximation (SCA)” [19, 20, 21, 22]. In this case, a semi-analytical computation of the NLO single-inclusive jet cross section can be performed, meaning that fully analytical expressions for the partonic hard-scattering functions dσ̂ab can be derived which only need to be integrated numerically against the parton distribution functions as shown in Eq. (1). The SCA may be viewed as an expansion of the partonic cross section for small δ ≡ R/ cosh η, where η is the jet’s pseudo-rapidity. Technically, the parameter δ is the half-aperture of a geometrical cone around the jet axis, when the four-momentum of the jet is defined as simply the sum of the four- momenta of all the partons inside the cone [19, 21]. At small δ, the behavior of the jet cross section is of the form A log(δ)+B+O(δ2), with both A and B known from Refs. [19, 21]. Jet codes based on the SCA have the virtue that they produce numerically stable results on much shorter time scales than Monte-Carlo codes. Moreover, as we shall see below, the relatively simple and explicit results for the NLO single-inclusive jet cross section obtained in the SCA are a great convenience for the implementation of threshold resummation, particularly for the matching needed to achieve full NLL accuracy. It turns out that the SCA is a very good approximation even for relatively large cone sizes of up toR ≃ 0.7 [21, 20, 23], the value used by both Tevatron collaborations. Figure 1 shows comparisons between the NLO cross sections for single-inclusive jet production obtained using a full Monte- Carlo code [24] and the SCA code of [21], for pp̄ collisions at c.m. energy S = 1800 GeV and very high pT . Throughout this paper we use the CTEQ6M [31] NLO parton distribution functions. We have chosen two different jet definitions in the Monte-Carlo calculation. One uses a conventional cone algorithm [25], the other the CDF jet definition [13]. One can see that the differences with respect to the SCA are of the order of only a few per cent. We note that similar comparisons in the RHIC kinematic regime have been shown in [21]. In their recent paper [16], the STAR collaboration used R = 0.4, for which the SCA is even more accurate. Encouraged by this good agreement, we will directly use the SCA analytical results when performing the threshold resummation. As stated in the Introduction, this is anyway not a Figure 1: Ratio between NLO jet cross sections at Tevatron at S = 1800 GeV, computed with a full Monte-Carlo code [24] and in the SCA. The solid line corresponds to the jet definition implemented by CDF [13] (with parameter Rsep = 1.3), and the dashed one to the standard cone definition [25]. In both cases the size of the jet cone is set to R = 0.7, and the CTEQ6M NLO [31] parton distributions, evaluated at the factorization scale µF = PT , were used. limitation, because we will also always provide the ratio of resummed over NLO cross sections (K-factors), which may then be used along with full NLO Monte-Carlo calculations to obtain resummed cross sections for any desired cone size or jet algorithm. A further simplification that we will make is to consider the cross section integrated over all pseudo-rapidities of the jet. As was discussed in [8], this considerably reduces the complexity of the resummed expressions. By simply rescaling the resummed prediction by an appropriate ratio of NLO cross sections one can nonetheless obtain a very good approximation also for the resummation effects on the non-integrated cross section, at central rapidities [11]. To perform the NLL threshold resummation for the full rapidity-dependence of the jet cross section remains an outstanding task for future work. From Eq. (1), we find for the single-inclusive jet cross section integrated over all jet pseudo- rapidity η, in the SCA: p3T dσ SCA(xT ) dx1 fa/H1 x1, µ dx2 fb/H2 x2, µ dx̂T δ x̂T − ∫ η̂+ x̂4T s dσ̂ab(x̂ T , η̂, R) dx̂2Tdη̂ , (2) where as before xT ≡ 2pT/ S is the customary scaling variable, and x̂T ≡ 2pT/ s with s = x1x2S is its partonic counterpart. η̂ is the partonic pseudo-rapidity, η̂ = η − 1 ln(x1/x2), which has the limits η̂+ = −η̂− = ln[(1 + 1− x̂2T )/x̂T ]. The dependence of the partonic cross sections on µF and µR has been suppressed for simplicity. The perturbative expansion of the dσ̂ab in the coupling constant αS(µR) reads dσ̂ab(x̂ T , η̂, R) =α S(µR) ab (x̂ T , η̂) + αS(µR) dσ̂ ab (x̂ T , η̂, R) +O(α2S) . (3) As indicated, the leading-order (LO) term dσ̂ab has no dependence on the cone size R, because for this term a single parton produces the jet. The analytical expressions for the NLO terms dσ̂ have been obtained in [19, 21]. It is customary to express them in terms of a different set of variables, v and w, that are related to x̂T and η̂ by x̂2T = 4vw(1− v) , e2η̂ = . (4) Schematically, the NLO corrections to the partonic cross section for each scattering channel then take the form s dσ̂ ab (w, v, R) dw dv = Aab(v, R) δ(1− w) +Bab(v, R) ln(1− w) +Cab(v, R) + Fab(w, v, R) , (5) where the “plus”-distributions are defined as usual by dwf(w)[g(w)]+ ≡ dw(f(w)− f(1))g(w) , (6) and where the Fab(w, v, δ) collect all terms without distributions in w. Partonic threshold corre- sponds to the limit w → 1. The “plus”-distribution terms in Eq. (5) generate the large logarithmic corrections that are addressed by threshold resummation. At order k of perturbation theory, the leading contributions are proportional to αkS ln(1−w) )2k−1 . Performing the integration of these terms over η̂, they turn into contributions ∝ αkS ln 2k (1− x̂2T ), as we anticipated in the Introduc- tion, and as we shall show below. Subleading terms are down by one or more powers of the logarithm. We will now turn to the NLL resummation of the threshold logarithms. 3 Resummed cross section The resummation of the soft-gluon contributions is carried out in Mellin-N moment space, where they exponentiate [1, 2, 3, 4, 5, 6, 7]. At the same time, in moment space the convolutions between the parton distributions and the partonic subprocess cross sections turn into ordinary products. For our present calculation, the appropriate Mellin moments are in the scaling variable xT σ(N) ≡ )(N−1) p T dσ(xT ) . (7) ‡We drop the superscript “SCA” from now on. In Mellin-N space the QCD factorization formula in Eq. (2) becomes σ(N) = (µ2F ) f (µ2F ) σ̂ab(N) , (8) where the fN are the moments of the parton distribution functions, fNa/H(µ F ) ≡ dxxN−1fa/H(x, µ F ) , (9) and where σ̂ab(N) ≡ dv [4v(1− v)w]N+1 s dσ̂ab(w, v) dw dv . (10) The threshold limit w → 1 corresponds to N → ∞, and the LL soft-gluon corrections contribute as αmS ln 2mN . The large-N behavior of the NLO partonic cross sections can be easily obtained by using [4v(1− v)]N+1 f(v) = [4v(1− v)]N+1 . (11) Up to corrections suppressed by 1/N it is therefore possible to perform the v-integration of the partonic cross sections by simply evaluating them at v = 1/2. According to Eq. (5), when v = 1/2 is combined with the threshold limit w = 1, one has x̂T = 1. It is worth mentioning that in the same limit one has η̂ = 0, and therefore the coefficients for the soft-gluon resummation for the rapidity-integrated cross section agree with those for the cross section at vanishing partonic rapidity. This explains why generally the resummation for the rapidity-integrated hadronic cross section yields a good approximation to the resummation of the cross section integrated over only a finite rapidity interval, as long as a region around η = 0 is contained in that interval [11]. The resummation of the large logarithms in the partonic cross sections is achieved by showing that they exponentiate in Sudakov form factors. The resummed cross section for each partonic subprocess is given by a formally rather simple expression in Mellin-N space: (res) ab (N) = Cab ∆ GIab→cd∆ (int)ab→cd (Born) ab→cd(N) , (12) where the first sum runs over all possible final state partons c and d, and the second over all possible color configurations I of the hard scattering. Except for the Born cross sections σ̂ (Born) ab→cd (which we have presented in earlier work [8]), each of the N -dependent factors in Eq. (12) is an exponential containing logarithms in N . The coefficients Cab collect all N -independent contributions, which partly arise from hard virtual corrections and can be extracted from comparison to the analytical expressions for the full NLO corrections in the SCA. Finally, the GIab→cd are color weights obeying ab→cd = 1. The color interferences expressed by the sum over I appear whenever the number of partons involved in the process at Born level is larger than three, as it is the case here §. Figure 2 gives a simple graphical mnemonic of the structure of the resummation formula, and of the origin of its various factors, whose expressions we know present. §In a more general case without rapidity integration, the terms GI ab→cd × σ̂ (Born) ab→cd (N) should be replaced by the color-correlated Born cross sections σ̂ (Born I) ab→cd (N) [4, 7, 17]. (int) Figure 2: Pictorial representation of the resummation formula in Eq. (12). Effects of soft-gluon radiation collinear to the initial-state partons are exponentiated in the functions ∆ N , which read: ln∆aN = zN−1 − 1 ∫ (1−z)2Q2 Aa(αS(q 2)) , (13) (and likewise for b) where Q2 = 2p2T . J N is the exponent associated with collinear, both soft and hard, radiation in the unobserved recoiling “jet”, ln JdN = zN−1 − 1 ∫ (1−z)Q2 (1−z)2Q2 Ad(αS(q 2)) + Bd(αS((1− z)Q2)) . (14) The function J ′cN describes radiation in the observed jet. As we reviewed in the Introduction, this function is sensitive to the assumption made about the jet’s invariant mass at threshold [4] even at LL level. Our choice is to allow the jet to be massive at partonic threshold, which is consistent with the experimental definitions of jet cross sections and with the available NLO calculations. In this case, J ′cN is given as ln J ′cN = zN−1 − 1 C ′c(αS((1− z)2Q2)) (jet massive at threshold) . (15) A similar exponent was derived for this case in [4]. The expression given in Eq.(15) agrees with the one of [4] to the required NLL accuracy. Notice that J ′cN contains only single logarithms, which arise from soft emission, whereas logarithms of collinear origin are absent. This is explicitly seen in the NLO calculations in the SCA [20, 21] in which there is an integration over the jet mass up to a maximum value of O(δ ∼ R), even when the threshold limit is strictly reached. Collinear contributions that would usually generate large logarithms are actually “regularized” by the cone size δ and give instead rise to log(δ) terms in the perturbative cross sections. If, however, the jet is forced to be massless at partonic threshold the jet-function is identical to the function for an “unobserved” jet given in Eq. (14) [4, 17]: ln J ′cN = lnJ N (jet massless at threshold) , (16) which produces a (negative) double-logarithm per emitted gluon, because it also receives collinear contributions due to the stronger restriction on the gluon phase space. There is then no dependence on log(δ) in this case. Finally, large-angle soft-gluon emission is accounted for by the factor ∆ (int)ab→cd I N , which reads (int)ab→cd I N = zN−1 − 1 DI ab→cd(αS((1− z)2Q2)) , (17) and depends on the color configuration I of the participating partons. The coefficients Aa, Ba, C a, DI ab→cd in Eqs. (13),(14),(15),(17) are free of large logarithmic contributions and are given as perturbative series in the coupling constant αS: F(αS) = F (1) + F (2) + . . . (18) for each of them. For the resummation to NLL accuracy we need the coefficients A a , A a , C a , and D I ab→cd. The last of these depends on the specifics of the partonic process under consideration; all the others are universal in the sense that they only distinguish whether the parton they are associated with is a quark or a gluon. The LL and NLL coefficients A a , A a and a are well known [32]: A(1) = Ca , A (2) = CaK , B a = γa (19) K = CA Nf , (20) where Cg = CA = Nc = 3, Cq = CF = (N c − 1)/2Nc = 4/3, γq = −3/2CF = −2, γg = −2π0¯, and Nf is the number of flavors. The coefficients C a needed in the case of jets that are massive at threshold, may be obtained by comparing the first-order expansion of the resummed formula to the analytic NLO results in the SCA. They are also universal and read C ′(1)a = −Ca log . (21) This coefficient contains the anticipated dependence on log(δ) that regularizes the final-state collinear configurations. As expected from a term of collinear origin, the exponent (15) hence provides one power of log(δ) for each perturbative order. The coefficients D I ab→cd governing the exponentiation of large-angle soft-gluon emission to NLL accuracy, and the corresponding “color weights” GI ab→cd, depend both on the partonic process and on its “color configuration”. They can be obtained from [4, 5, 17, 33, 34] where the soft anomalous dimension matrices were computed for all partonic processes. The results are given for the general case of arbitrary partonic rapidity η̂. As discussed above, the coefficients for the rapidity-integrated cross section may be obtained by setting η̂ = 0. We have presented the full set of the D I ab→cd and GI ab→cd in the Appendix of our previous paper [8]. Before we continue, we mention that one expects that a jet cross section defined by a cone algorithm will also have so-called “non-global” threshold logarithms [35, 36]. These logarithms arise when the observable is sensitive to radiation in only a limited part of phase space, as is the case in presence of a jet cone. For instance, a soft gluon radiated at an angle outside the jet cone may itself emit a secondary gluon at large angle that happens to fall inside the jet cone, thereby becoming part of the jet [35, 36]. Such configurations appear first at the next-to-next-to-leading order, but may produce threshold logarithms at the NLL level. One may therefore wonder if our NLL resummation formulas given above are complete. Fortunately, as an explicit example in [35] shows, it turns out that these effects are suppressed as R log(R) in the SCA. They may therefore be neglected at the level of approximation we are making here. Given their only mild suppression as R → 0, a study of non-global logarithms in hadronic jet cross sections is an interesting topic for future work. Returning to our resummed formulas, it is instructive to consider the structure of the leading logarithms. The LL expressions for the radiative factors are ∆aN = exp Ca ln JdN = exp Cd ln . (22) As discussed above, J ′c does not contribute at the double-logarithmic level. Therefore, for a given partonic channel, the leading logarithms are (res) ab→cd(N) ∝ exp Ca + Cb − ln2(N) . (23) The exponent is positive for each partonic channel, implying that the soft-gluon effects will increase the cross section. This enhancement arises from the initial-state radiation represented by the ∆a,b and is related to the fact that finite partonic cross sections are obtained after collinear (mass) factorization [1, 2]. In the MS scheme such an enhancing contribution is (for a given parton species) twice as large as the suppressing one associated with final-state radiation in Jd, for which no mass factorization is needed. For quark or anti-quark initiated processes, the color factor combination appearing in Eq. (23) ranges from 2CF − CF/2 = 2 for the qq → qq channel to 2CF −CA/2 = 7/6 for qq̄ → gg, while for those involving a quark-gluon initial state one has larger factors, CF + CA − CF/2 = 11/3 (for qg → qg) or CF + CA − CA/2 = 17/6 (for qg → gq). Yet larger factors are obtained for gluon-gluon scattering, with 2CA − CA/2 = 9/2 for gg → gg and 2CA−CF/2 = 16/3 for gg → qq̄. Initial states with more gluons therefore are expected to receive the larger resummation effects. We mention that if the observed jet is assumed strictly massless at threshold, an extra suppression term proportional to Jc arises (see Eq. (16)). It is customary to give the NLL expansions of the Sudakov exponents in the following way [2]: ln∆aN(αS(µ R), Q 2/µ2R;Q 2/µ2F ) = lnN h a (λ) + h a (λ,Q 2/µ2R;Q 2/µ2F ) +O αS(αS lnN) , (24) lnJaN (αS(µ R), Q 2/µ2R) = lnN f a (λ) + f a (λ,Q 2/µ2R) +O αS(αS lnN) , (25) ln J ′aN (αS(µ R)) = ln(1− 2λ) +O αS(αS lnN) , (26) (int)ab→cd I N (αS(µ R)) = I ab→cd ln(1− 2λ) +O αS(αS lnN) , (27) with λ = 0 R) lnN . The LL and NLL auxiliary functions h (1,2) a and f (1,2) a are h(1)a (λ) = + [2λ+ (1− 2λ) ln(1− 2λ)] , (28) h(2)a (λ,Q 2/µ2R;Q 2/µ2F ) =− [2λ+ ln(1− 2λ)]− ln(1− 2λ) 2λ+ ln(1− 2λ) + 1 ln2(1− 2λ) [2λ+ ln(1− 2λ)] ln Q , (29) f (1)a (λ) = − 2πb0λ (1− 2λ) ln(1− 2λ)− 2(1− λ) ln(1− λ) , (30) f (2)a (λ,Q 2/µ2R) = − 2πb30 ln(1− 2λ)− 2 ln(1− λ) + 1 ln2(1− 2λ)− ln2(1− λ) ln(1− λ)− ln(1− λ)− ln(1− 2λ) 2π2b20 2 ln(1− λ)− ln(1− 2λ) 2 ln(1− λ)− ln(1− 2λ) where 0 , b1 are the first two coefficients of the QCD β-function: (11CA − 2Nf ) , b1 = 17C2A − 5CANf − 3CFNf . (32) The N -independent coefficients Cab in Eq. (12), which include the hard virtual corrections, have the perturbative expansion Cab = 1 + ab +O(α S) . (33) The C ab we need to NLL are obtained by comparing the O(αS)-expansion (not counting the overall factor α2S of the Born cross sections) of the resummed expression with the fixed-order NLO result for the process, as given in [19, 21]. The full analytic expressions for the C ab are rather lengthy and will not be given here. For convenience, we present them in numerical form in the Appendix. We note that apart from being useful for extracting the coefficients C a and ab , the comparison of the O(αS)-expanded resummed result with the full NLO cross section also provides an excellent check of the resummation formula, since one can verify that all leading and next-to-leading logarithms are properly accounted for by Eq. (12). The improved resummed hadronic cross section is finally obtained by performing an inverse Mellin transformation, and by properly matching to the NLO cross section p3T dσ (NLO)(xT )/dpT as follows: p3T dσ (match)(xT ) a,b,c ∫ CMP+i∞ CMP−i∞ )−N+1 fNa/H1(µ F ) f (µ2F ) (res) ab→cd(N)− (res) ab→cd(N) (NLO) p3T dσ (NLO)(xT ) , (34) where σ̂ (res) ab→cd is given in Eq. (12) and (σ̂ (res) ab→cd)(NLO) represents its perturbative truncation at NLO. Thus, as a result of this matching procedure, in the final cross section in Eq. (34) the NLO cross section is exactly taken into account, and NLL soft-gluon effects are resummed beyond those already contained in the NLO cross section. The functions h (1,2) a (λ) and f (1,2) a (λ) in Eqs. (28)-(31) are singular at the points λ = 1/2 and/or λ = 1. These singularities are related to the divergent behavior of the perturbative running coupling αS near the Landau pole, and we deal with them by using the Minimal Prescription introduced in Ref. [2]. In the evaluation of the inverse Mellin transformation in Eq. (34), the constant CMP is chosen in such a way that all singularities in the integrand are to the left of the integration contour, except for the Landau singularities, that are taken to lie to its far right. We note that an alternative to such a definition one could choose to expand the resummed formula to a finite order, say, next-to-next-to-leading order (NNLO), and neglect all terms of yet higher order. This approach was adopted in Ref. [17]. We prefer to keep the full resummed formula in our phenomenological applications since, depending on kinematics, high orders in perturbation theory may still be very relevant [8]. It was actually shown in [2] that the results obtained within the Minimal Prescription converge asymptotically to the perturbative series. This completes the presentation of all ingredients to the NLL threshold resummation of the hadronic single-inclusive jet cross section. We will now turn to some phenomenological applica- tions. 4 Phenomenological Results We will study the effects of threshold resummation on the single-inclusive jet cross section in pp̄ collisions at S = 1.8 TeV and S = 630 GeV c.m. energies, and in pp collisions at 200 GeV. These choices are relevant for comparisons to Tevatron and RHIC data, respectively. Unless otherwise stated, we always set the factorization and renormalization scales to µF = µR = pT and use the NLO CTEQ6M [31] set of parton distributions, along with the two-loop expression for the strong coupling constant αS. We will first analyze the relevance of the different subprocesses contributing to single-jet pro- duction. The left part of Fig. 3 shows the relative contributions by “qq” (qq, qq′, qq̄ and qq̄′ combined), qg and gg initial states at Born level (dashed lines) and for the NLL resummed case (without matching, solid lines). Here we have chosen the case of pp̄ collisions at S = 1.8 TeV. As can be seen, the overall change in the curves when going from Born level to the resummed case is moderate. The main noticeable effect is an increase of the relative importance of processes with gluon initial states toward higher pT , compensated by a similar decrease in that of the qq channels. In the right part of Fig. 3 we show the enhancements from threshold resummation for each initial partonic state individually, and also for their sum. At the higher pT , where threshold resummation is expected to be best applicable, the enhancements are biggest for the gg channel, followed by the qg one. All patterns observed in Fig. 3 are straightforwardly understood from Eq. (23), which demonstrates that resummation yields bigger enhancements when the number of gluons in the initial state is larger. We note that results very similar to those shown in the figure hold also at√ S = 630 GeV, if the same value of xT = 2pT/ S is considered. This remains qualitatively true even when we go to pp collisions at S = 200 GeV, except for the larger enhancement in the Figure 3: Left: relative contributions of the various partonic initial states to the single-inclusive jet cross section in pp̄ collisions at S = 1.8 TeV, at Born level (dashed) and for the NLL resummed case (solid). We have chosen the jet cone size R = 0.7. Right: ratios between resummed and Born contributions for the various channels, and for the full jet cross section. Figure 4: Same as Fig. 3, but for pp collisions at S = 200 GeV and R = 0.4. quark contribution, due to the dominance of the qq channel (instead of the qq̄ as in pp̄ collisions) with a larger color factor combination in the Sudakov exponent (see the discussion after Eq. (22)). Figure 4 repeats the studies made for Fig. 3 for this case. As one can see, if the same xT as in Fig. 3 is considered, the qq scattering contributions are overall slightly less important. At the same time, resummation effects are overall somewhat larger because the pT values are now much smaller than in Fig. 3, so that the strong coupling constant that appears in the resummed exponents is larger. Figure 5: Ratio between the expansion to NLO of the (unmatched) resummed cross section and the full NLO one (in the SCA), for pp̄ collisions at s = 1.8 TeV (solid) and s = 630 GeV (dots), and for pp collisions at s = 200 GeV (dashed). Before presenting the results for the matched NLL resummed jet cross section and K-factors, we would like to identify the kinematical region where the logarithmic terms constitute the bulk of the perturbative corrections and subleading contributions are unimportant. Only in these is the resummation expected to provide an accurate picture of the higher-order terms. We can determine this region by comparing the resummed formula, expanded to NLO, to the full fixed- order (NLO) perturbative result, that is, by comparing the last two terms in Eq. (34). Figure 5 shows this comparison for both Tevatron energies and for the RHIC case, as function of the “scaling” variable xT . As can be observed, the expansion correctly reproduces the NLO result within at most a few per cent over a region corresponding to pT & 200 GeV for the higher Tevatron energy, and to pT & 30 GeV at RHIC. This demonstrates that, at this order, the perturbative corrections are strongly dominated by the terms of soft and/or collinear origin that are addressed by resummation. The accuracy of the expansion improves toward the larger values of the jet transverse momentum, were one approaches the threshold limit more closely. Having established the importance of the threshold corrections in a kinematic regime of interest for phenomenology, we show in Fig. 6 the impact of the resummation on the predicted single-jet cross section at S = 1.8 TeV. NLO and NLL resummed results are presented, computed at three different values of the factorization and renormalization scales, defined by µF = µR = ζpT (with ζ = 1, 2, 1/2). The most noticeable effect is a remarkable reduction in the scale dependence of the cross section. This observation was also made in the previous study [17]. If, as customary, one defines a theoretical scale “uncertainty” by ∆ ≡ (σ(ζ = 0.5) − σ(ζ = 2))/σ(ζ = 1), the improvement is considerable. While ∆ lies between 20 and 25% at NLO, it never exceeds 8% for the matched NLL result. The inset plot shows the NLL K-factor, defined as K(res) = dσ(match)/dpT dσ(NLO)/dpT , (35) at each of the scales. The corrections from resummation on top of NLO are typically very mod- erate, at the order of a few per cent, depending on the set of scales chosen. The higher-order corrections increase for larger values of the jet transverse momentum. These findings are again consistent with those of [17], even though more detailed comparisons reveal some quantitative differences that must be related to either the different choice of the resummed final-state jet func- tion in [17] (see discussion in Sec. 3), or to the fact that [17] uses only a NNLO expansion of the resummed cross section. The main features of our results remain unchanged when we go to the Tevatron-run II energy of S = 1.96 TeV, at which measured jet cross sections are now avail- able [37]. Quantitatively very similar results are also found for the lower Tevatron center-of-mass energy, as seen in Fig. 7. In the case of pp collisions at s = 200 GeV, presented in Fig. 8, a similar pattern emerges, even though the resummation effects tend to be overall somewhat more substantial here. Figure 6: NLO and NLL results for the single-inclusive jet cross section in pp̄ collisions at S = 1.8 TeV, for different values of the renormalization and factorization scales. We have chosen R = 0.7. The inset plot shows the corresponding K-factors as defined in Eq. (35). In Fig. 9 we analyze how the resummation effects build up order by order in perturbation theory. We expand the matched resummed formula beyond NLO and define the “partial” soft- gluon K-factors as dσ(match)/dpT dσ(NLO)/dpT , (36) which for n = 2, 3, . . . give the additional enhancement over full NLO due to the O(α2+nS ) terms in the resummed formula¶. Formally, K1 = 1 and K∞ = K(res) of Eq. (35). The results for K2,3,4,∞ are given in the figure, for the case of pp̄ collisions at S = 1.8 TeV. One can see that ¶We recall that the Born cross sections are of O(α2S), hence the additional power of two in this definition. Figure 7: Same as Fig. 6, but for S =630 GeV. contributions beyond N3LO (n = 3) are very small, and that the O(α6S) result can hardly be distinguished from the full NLL one. It is interesting to contrast the rather modest enhancement of the jet cross section by resum- mation to the dramatic resummation effects that we observed in [8] in the case of single-inclusive pion production, H1H2 → πX , in fixed-target scattering at typical c.m. energies of S ∼ 30 GeV. Even though in both cases the same partonic processes are involved at Born level, there are several important differences. First of all, the values of pT are much smaller in fixed-target scattering (even though roughly similar values of xT = 2pT/ S are probed), so that the strong coupling constant αS(pT ) is larger and resummation effects are bound to be more significant. Furthermore, for the process H1H2 → πX one needs to introduce fragmentation functions into the theoretical calculation that describe the formation of the observed hadron from a final-state parton. As the hadron takes only a certain fraction z & 0.5 of the parent parton’s momentum, the partonic hard- scattering necessarily has to be at the higher transverse momentum pT/z in order to produce a hadron with pT . Thus one is closer to partonic threshold than in case of a jet produced with transverse momentum pT which takes all of a final-state parton’s momentum. In addition, it turns out [8, 38] that due to the factorization of final-state collinear singularities associated with the fragmentation functions, the “jet” function J ′cN in the resummation formula Eq. (12) is to be replaced by a factor ∆cN , which has enhancing double logarithms. Finally, as one illustrative example, we compare our resummed jet cross section to data from CDF [13] at S = 1800 GeV. While so far we have always considered the cross section integrated over all jet rapidities, we here need to account for the fact that the data cover only a finite region in rapidity, 0.1 ≤ |η| ≤ 0.7. Also, we would like to properly match the jet algorithm chosen in experiment, rather than using the SCA. We have mentioned before that both these issues can be accurately addressed by “rescaling” the resummed cross section by an appropriate Figure 8: Same as Fig. 6, but for pp collisions at S =200 GeV and R = 0.4. ratio of NLO cross sections. We simply multiply our K-factors defined in Eq. (35) and shown in Fig. 6 by dσ(MC)(0.1 ≤ |η| ≤ 0.7)/dpT , the NLO cross section obtained with a full Monte- Carlo code [24], in the experimentally accessed rapidity regime. The comparison between the data and the NLO and NLL-resummed cross sections is shown in Fig. 10 in terms of the ratios “(data−theory)/theory”. As expected from Fig. 6, the impact of resummation is moderate and in fact smaller than the current uncertainties of the parton distributions [37]. Nonetheless, it does lead to a slight improvement of the comparison and, in particular, the plot again demonstrates the reduction of scale dependence by resummation. 5 Conclusions We have studied in this paper the resummation of large logarithmic threshold corrections to the partonic cross sections contributing to single-inclusive jet production at hadron colliders. Our study differs from previous work [17] mostly in that we allow the jet to have a finite invariant mass at partonic threshold, which is consistent with the experimental definitions of jet cross sections and with the available NLO calculations. Moreover, using semi-analytical expressions for the NLO partonic cross sections derived in the SCA [19, 21], we have extracted the N−independent coefficients that appear in the resummation formula, and properly matched our resummed cross section to the NLO one. We hope that with these improvements yet more realistic estimates of the higher-order corrections to jet cross sections in the threshold regime emerge. It is well known that the NLO description of jet production at hadron colliders is overall very successful, within the uncertainties of the theoretical framework and the experimental data. From that perspective, it is gratifying to see that the effects of NLL resummation are relatively moderate. Figure 9: Soft-gluon Kn factors as defined in Eq. (36), for pp̄ collisions at S = 1.8 TeV. On the other hand, resummation leads to a significant decrease of the scale dependence, and we expect that knowledge of the resummation effects should be useful in comparisons with future, yet more precise, data, and for extracting parton distribution functions. Given the general success of the NLO description, we have mostly focused on K-factors for the resummed cross section over the NLO one, and only given one example of a more detailed comparison with experimental data. We believe that these K-factors may be readily used in conjunction with other, more flexible NLO Monte-Carlo programs for jet production, to estimate threshold-resummation effects on cross sections for other jet algorithms and possibly for larger cone sizes. Acknowledgments We are grateful to Stefano Catani, Barbara Jäger, Nikolaos Kidonakis, Douglas Ross, George Sterman, and Marco Stratmann for helpful discussions. DdF is supported in part by UBACYT and CONICET. WV is supported by the U.S. Department of Energy under contract number DE-AC02-98CH10886. Appendix: First-order coefficients C ab in the SCA In this appendix we collect the process-dependent coefficients C ab for the various partonic channels in jet hadroproduction in the SCA. 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B 617, 253 (2001) [arXiv:hep-ph/0107138]. http://arxiv.org/abs/hep-ph/9706545 http://arxiv.org/abs/hep-ph/9808262 http://arxiv.org/abs/hep-ph/9208249 http://arxiv.org/abs/hep-ph/0409313 http://arxiv.org/abs/hep-ph/0201195 http://arxiv.org/abs/hep-ph/9902484 http://arxiv.org/abs/hep-ph/0606254 http://arxiv.org/abs/hep-ph/0607309 http://arxiv.org/abs/hep-ph/0104277 http://arxiv.org/abs/hep-ph/0203009 http://arxiv.org/abs/hep-ph/0110004 http://arxiv.org/abs/hep-ex/0701051 http://arxiv.org/abs/hep-ex/0609026 http://arxiv.org/abs/hep-ph/0107138 Introduction Next-to-leading order single-inclusive jet cross section Resummed cross section Phenomenological Results Conclusions

0704.1679

Equation of State in Relativistic Magnetohydrodynamics: variable versus constant adiabatic index

Mon. Not. R. Astron. Soc. 000, 1–14 (2007) Printed 11 February 2013 (MN LATEX style file v2.2) Equation of State in Relativistic Magnetohydrodynamics: variable versus constant adiabatic index A. Mignone1,2 ⋆ and Jonathan C. McKinney3⋆ 1INAF Osservatorio Astronomico di Torino, 10025 Pino Torinese, Italy 2Dipartimento di Fisica Generale dell’Università, Via Pietro Giuria 1, I-10125 Torino, Italy 3Institute for Theory and Computation, Center for Astrophysics, Harvard University, 60 Garden St., Cambridge, MA, 02138 Accepted 2007 April 12. Received 2007 April 12; in original form 2007 January 25 ABSTRACT The role of the equation of state for a perfectly conducting, relativistic magnetized fluid is the main subject of this work. The ideal constant Γ-law equation of state, commonly adopted in a wide range of astrophysical applications, is compared with a more realistic equation of state that better approximates the single-specie relativistic gas. The paper focus on three different topics. First, the influence of a more realis- tic equation of state on the propagation of fast magneto-sonic shocks is investigated. This calls into question the validity of the constant Γ-law equation of state in problems where the temperature of the gas substantially changes across hydromagnetic waves. Second, we present a new inversion scheme to recover primitive variables (such as rest-mass density and pressure) from conservative ones that allows for a general equa- tion of state and avoids catastrophic numerical cancellations in the non-relativistic and ultrarelativistic limits. Finally, selected numerical tests of astrophysical relevance (including magnetized accretion flows around Kerr black holes) are compared using different equations of state. Our main conclusion is that the choice of a realistic equa- tion of state can considerably bear upon the solution when transitions from cold to hot gas (or viceversa) are present. Under these circumstances, a polytropic equation of state can significantly endanger the solution. Key words: equation of state - relativity - hydrodynamics shock waves - methods: numerical - MHD 1 INTRODUCTION Recent developments in numerical hydrodynamics have made a breach in the understanding of astrophysical phe- nomena commonly associated with relativistic magnetized plasmas. Existence of such flows has nowadays been largely witnessed by observations indicating superluminal motion in radio loud active galactic nuclei and galactic binary systems, as well as highly energetic events occurring in proximity of X-ray binaries and super-massive black holes. Strong evi- dence suggests that the two scenarios may be closely related and that the production of relativistic collimated jets results from magneto-centrifugal mechanisms taking place in the in- ner regions of rapidly spinning accretion disks (Meier et al. 2001). Due to the high degree of nonlinearity present in the equations of relativistic magnetohydrodynamics (RMHD henceforth), analytical models are often of limited appli- cability, relying on simplified assumptions of time inde- ⋆ E-mail:[email protected](AM);[email protected](JCM) pendence and/or spatial symmetries. For this reason, they are frequently superseded by numerical models that appeal to a consolidated theory based on finite difference meth- ods and Godunov-type schemes. The propagation of rel- ativistic supersonic jets without magnetic field has been studied, for instance, in the pioneering work of van Putten (1993); Duncan & Hughes (1994) and, subsequently, by Mart́ı et al. (1997); Hardee et al. (1998); Aloy et al. (1999); Mizuta et al. (2004) and references therein. Similar inves- tigations in presence of poloidal and toroidal magnetic fields have been carried on by Nishikawa et al. (1997); Koide (1997); Komissarov (1999) and more recently by Leismann et al. (2005); Mignone et al. (2005). The majority of analytical and numerical models, in- cluding the aforementioned studies, makes extensive use of the polytropic equation of state (EoS henceforth), for which the specific heat ratio is constant and equal to 5/3 (for a cold gas) or to 4/3 (for a hot gas). However, the theory of relativistic perfect gases (Synge 1957) teaches that, in the limit of negligible free path, the ratio of specific heats can- not be held constant if consistency with the kinetic theory http://arxiv.org/abs/0704.1679v1 2 A. Mignone and J.C. McKinney is to be required. This was shown in an even earlier work by Taub (1948), where a fundamental inequality relating spe- cific enthalpy and temperature was proved to hold. Although these results have been known for many decades, only few investigators seem to have faced this im- portant aspect. Duncan et al. (1996) suggested, in the con- text of extragalactic jets, the importance of self-consistently computing a variable adiabatic index rather than using a constant one. This may be advisable, for example, when the dynamics is regulated by multiple interactions of shock waves, leading to the formation of shock-heated regions in an initially cold gas. Lately, Scheck et al. (2002) addressed similar issues by investigating the long term evolution of jets with an arbitrary mixture of electrons, protons and electron-positron pairs. Similarly, Meliani et al. (2004) con- sidered thermally accelerated outflows in proximity of com- pact objects by adopting a variable effective polytropic index to account for transitions from non-relativistic to relativis- tic temperatures. Similar considerations pertain to models of Gamma Ray Burst (GRB) engines including accretion discs, which have an EoS that must account for a combi- nation of protons, neutrons, electrons, positrons, and neu- trinos, etc. and must include the effects of electron degen- eracy, neutronization, photodisintegration, optical depth of neutrinos, etc. (Popham et al. 1999; Di Matteo et al. 2002; Kohri & Mineshige 2002; Kohri et al. 2005). However, for the disk that is mostly photodisintegrated and optically thin to neutrinos, a decent approximation of such EoS is a variable Γ-law with Γ = 5/3 when the temperature is be- low mec 2/kb and Γ = 4/3 when above mec 2/kb due to the production of positrons at high temperatures that gives a relativistic plasma (Broderick, McKinney, Kohri in prep.). Thus, the variable EoS considered here may be a reasonable approximation of GRB disks once photodisintegration has generated mostly free nuclei. The additional complexity introduced by more elabo- rate EoS comes at the price of extra computational cost since the EoS is frequently used in the process of obtaining numer- ical solutions, see for example, Falle & Komissarov (1996). Indeed, for the Synge gas, the correct EoS does not have a simple analytical expression and the thermodynamics of the fluid becomes entirely formulated in terms of the modified Bessel functions. Recently Mignone et al. (2005a, MPB henceforth) in- troduced, in the context of relativistic non-magnetized flows, an approximate EoS that differs only by a few percent from the theoretical one. The advantage of this approximate EoS, earlier adopted by Mathews (1971), is its simple analytical representation. A slightly better approximation, based on an analytical expression, was presented by Ryu et al. (2006). In the present work we wish to discuss the role of the EoS in RMHD, with a particular emphasis to the one pro- posed by MPB, properly generalized to the context of rel- ativistic magnetized flows. Of course, it is still a matter of debate the extent to which equilibrium thermodynamic prin- ciples can be correctly prescribed when significant deviations from the single-fluid ideal approximation may hold (e.g., non-thermal particle distributions, gas composition, cosmic ray acceleration and losses, anisotropy, and so forth). Nev- ertheless, as the next step in a logical course of action, we will restrict our attention to a single aspect - namely the use of a constant polytropic versus a variable one - and we will ignore the influence of such non-ideal effects (albeit poten- tially important) on the EoS. In §2, we present the relevant equations and discuss the properties of the new EoS versus the more restrictive con- stant Γ-law EoS. In §3, we consider the propagation of fast magneto-sonic shock waves and solve the jump conditions across the front using different EoS. As we shall see, this calls into question the validity of the constant Γ-law EoS in problems where the temperature of the gas substantially changes across hydromagnetic waves. In §4, we present nu- merical simulations of astrophysical relevance such as blast waves, axisymmetric jets, and magnetized accretion disks around Kerr black holes. A short survey of some existing models is conducted using different EoS’s in order to deter- mine if significant interesting deviations arise. These results should be treated as a guide to some possible avenues of research rather than as the definitive result on any individ- ual topic. Results are summarized in §5. In the Appendix, we present a description of the primitive variable inversion scheme. 2 RELATIVISTIC MHD EQUATIONS In this section we present the equations of motion for rel- ativistic MHD, discuss the validity of the ideal gas EoS as applied to a perfect gas, and review an alternative EoS that properly models perfect gases in both the hot (relativistic) and cold (non-relativistic) regimes. 2.1 Equations of Motion Our starting point are the relativistic MHD equations in conservative form: + ∇ · vv − bb + Ipt vB − Bv = 0 , (1) together with the divergence-free constraint ∇·B = 0, where v is the velocity, γ is the Lorentz factor, wt ≡ (ρh+p+b2) is the relativistic total (gas+magnetic) enthalpy, pt = p+ b is the total (gas+magnetic) fluid pressure, B is the lab-frame field, and the field in the fluid frame is given by = γ{v · B, B (v · B)}, (2) with an energy density of |b|2 = |B| + (v · B)2. (3) Units are chosen such that the speed of light is equal to one. Notice that the fluxes entering in the induction equation are the components of the electric field that, in the infinite conductivity approximation, become Ω = −v × B . (4) The non-magnetic case is recovered by letting B → 0 in the previous expressions. Equation of state in RMHD 3 Figure 1. Equivalent Γ (top left), specific enthalpy (top right), sound speed (bottom left) and specific internal energy (bottom right) as functions of temperature Θ = p/ρ. Different lines corre- spond to the various EoS mentioned the text: the ideal Γ = 5/3- law (dotted line), ideal Γ = 4/3-law (dashed line), TM EoS (solid line). For clarity the Synge-gas (dashed-dotted line) has been plotted only in the top left panel, where the “unphysical region” marks the area where Taub’s inequality is not fulfilled. The conservative variables are, respectively, the labora- tory density D, the three components of momentum mk and magnetic field Bk and the total energy density E: D = ργ , (5) mk = (Dhγ + |B|2)vk − (v · B)Bk , (6) E = Dhγ − p + |B| |v|2|B|2 − (v · B)2 , (7) The specific enthalpy h and internal energy ǫ of the gas are related by h = 1 + ǫ + , (8) and an additional equation of state relating two thermody- namical variables (e.g. ρ and ǫ) must be specified for proper closure. This is the subject of the next section. Equations (5)–(7) are routinely used in numerical codes to recover conservative variables from primitive ones (e.g., ρ, v, p and B). The inverse relations cannot be cast in closed form and require the solution of one or more non- linear equations. Noble et al. (2006) review several methods of inversion for the constant Γ-law, for which ρǫ = p/(Γ−1). We present, in Appendix A, the details of a new inversion procedure suitable for a more general EoS. 2.2 Equation of State Proper closure to the conservation law (1) is required in order to solve the equations. This is achieved by specifying an EoS relating thermodynamic quantities. The theory of relativistic perfect gases shows that the specific enthalpy is a function of the temperature Θ = p/ρ alone and it takes the form (Synge 1957) K3(1/Θ) K2(1/Θ) , (9) where K2 and K3 are, respectively, the order 2 and 3 modi- fied Bessel functions of the second kind. Equation (9) holds for a gas composed of material particles with the same mass and in the limit of small free path when compared to the sound wavelength. Direct use of Eq. (9) in numerical codes, however, results in time-consuming algorithms and alternative ap- proaches are usually sought. The most widely used and pop- ular one relies on the choice of the constant Γ-law EoS h = 1 + Γ − 1 Θ , (10) where Γ is the constant specific heat ratio. However, Taub (1948) showed that consistency with the relativistic kinetic theory requires the specific enthalpy h to satisfy (h − Θ) (h − 4Θ) > 1 , (11) known as Taub’s fundamental inequality. Clearly the con- stant Γ-law EoS does not fulfill (11) for an arbitrary choice of Γ, while (9) certainly does. This is better understood in terms of an equivalent Γeq, conveniently defined as Γeq = h − 1 h − 1 − Θ , (12) and plotted in the top left panel of Fig. 1 for different EoS. In the limit of low and high temperatures, the physically admissible region is delimited, respectively, by Γeq 6 5/3 (for Θ → 0) and Γeq 6 4/3 (for Θ → ∞). Indeed, Taub’s inequality is always fulfilled when Γ 6 4/3 while it cannot be satisfied for Γ > 5/3 for any positive value of the tem- perature. In a recent paper, Mignone et al. (2005a) showed that if the equal sign is taken in Eq. (11), an equation with the correct limiting values may be derived. The resulting EoS (TM henceforth), previously introduced by Mathews (1971), can be solved for the enthalpy, yielding Θ2 + 1 , (13) or, using ρh = ρ + ρǫ + p in (11) with the equal sign, ρǫ (ρǫ + 2ρ) 3 (ρǫ + ρ) ǫ + 2 ǫ + 1 . (14) Direct evaluation of Γeq using (13) shows that the TM EoS differs by less than 4% from the theoretical value given by the relativistic perfect gas EoS (9). The proposed EoS be- haves closely to the Γ = 4/3 law in the limit of high tem- peratures, whereas reduces to the Γ = 5/3 law in the cold gas limit. For intermediate temperatures, thermodynamical quantities (such as specific internal energy, enthalpy and sound speed) smoothly vary between the two limiting cases, as illustrated in Fig. 1. In this respect, Eq. (13) greatly im- proves over the constant Γ-law EoS and, at the same time, offers ease of implementation over Eq. (9). Since thermody- namics is frequently invoked during the numerical solution of (1), it is expected that direct implementation of Eq. (13) 4 A. Mignone and J.C. McKinney in numerical codes will result in faster and more efficient algorithms. Thermodynamical quantities such as sound speed and entropy are computed from the 2nd law of thermodynamics, − d log p , (15) where S is the entropy. From the definition of the sound speed, , (16) and using de = hdρ (at constant S), one finds the useful expression ḣ − 1 Γ-law EoS , 5h − 8Θ h − Θ TM EoS . where we set ḣ = dh/dΘ. In a similar way, direct integration of (15) yields S = k log σ with Γ-law EoS , (h − Θ) TM EoS . with h given by (13). 3 PROPAGATION OF FAST MAGNETO-SONIC SHOCKS Motivated by the previous results, we now investigate the role of the EoS on the propagation of magneto-sonic shock waves. To this end, we proceed by constructing a one- parameter family of shock waves with different velocities, traveling in the positive x direction. States ahead and be- hind the front are labeled with U 0 and U 1, respectively, and are related by the jump conditions vs [U ] = [F (U )] , (19) where vs is the shock speed and [q] = q1 − q0 is the jump across the wave for any quantity q. The set of jump condi- tions (19) may be reduced (Lichnerowicz 1976) to the fol- lowing five positive-definite scalar invariants [J ] = 0 , (20) [hη] = 0 , (21) [H] = = 0 , (22) p + b2/2 [h/ρ] = 0 , (23) + 2H [p] + 2 = 0 , (24) where J = ργγs(vs − vx) , (25) is the mass flux across the shock, and η = −J (v · B) + γs Bx . (26) Figure 2. Compression ratio (top panels), internal energy (mid- dle panels) and downstream Mach number (bottom panels) as functions of the shock four-velocity γsvs. The profiles give the so- lution to the shock equation for the non magnetic case. Plots on the left have zero tangential velocity ahead of the front, whereas plots on right are initialized with vy0 = 0.99. Axis spacing is logarithmic. Solid, dashed and dotted lines correspond to the so- lutions obtained with the TM EoS and the Γ = 4/3 and Γ = 5/3 laws, respectively. Here γs denotes the Lorentz factor of the shock. Fast or slow magneto-sonic shocks may be discriminated through the condition α0 > α1 > 0 (for the formers) or α1 < α0 < 0 (for the latters), where α = h/ρ −H. We consider a pre-shock state characterized by a cold (p0 = 10 −4) gas with density ρ = 1. Without loss of gen- erality, we choose a frame of reference where the pre-shock velocity normal to the front vanishes, i.e., vx0 = 0. Notice that, for a given shock speed, J2 can be computed from the pre-shock state and thus one has to solve only Eqns. (21)– (24). 3.1 Purely Hydrodynamical Shocks In the limit of vanishing magnetic field, only Eqns. (23) and (24) need to be solved. Since J2 is given, the problem sim- Equation of state in RMHD 5 plifies to the 2 × 2 nonlinear system of equations [h/ρ] = 0 , (27) = 0 . (28) We solve the previous equations starting from vs = 0.2, for which we were able to provide a sufficiently close guess to the downstream state. Once the p1 and ρ1 have been found, we repeat the process by slowly increasing the shock velocity vs and using the previously converged solution as the initial guess for the new value of vs. Fig. 2 shows the compression ratio, post-shock inter- nal energy ǫ1 and Mach number v1/cs1 as functions of the shock four velocity vsγs. For weakly relativistic shock speeds and vanishing tangential velocities (left panels), density and pressure jumps approach the classical (i.e. non relativistic) strong shock limit at γsvs ≈ 0.1, with the density ratio be- ing 4 or 7 depending on the value of Γ (5/3 or 4/3, respec- tively). The post-shock temperature keeps non-relativistic values (Θ ≪ 1) and the TM EoS behaves closely to the Γ = 5/3 case, as expected. With increasing shock velocity, the compression ratio does not saturate to a limiting value (as in the classical case) but keeps growing at approximately the same rate for the constant Γ-law EoS cases, and more rapidly for the TM EoS. This can be better understood by solving the jump conditions in a frame of reference moving with the shocked material and then transforming back to our origi- nal system. Since thermodynamics quantities are invariant one finds that, in the limit h1 ≫ h0 ≈ 1, the internal energy becomes ǫ1 = γ1 − 1 and the compression ratio takes the asymptotic value = γ1 + γ1 + 1 Γ − 1 , (29) when the ideal EoS is adopted. Since γ1 can take arbitrarily large values, the downstream density keeps growing indefi- nitely. At the same time, internal energy behind the shock rises faster than the rest-mass energy, eventually leading to a thermodynamically relativistic configuration. In absence of tangential velocities (left panels in Fig. 2), this transi- tion starts at moderately high shock velocities (γsvs & 1) and culminates when the shocked gas heats up to relativis- tic temperatures (Θ ∼ 1 ÷ 10) for γsvs & 10. In this regime the TM EoS departs from the Γ = 5/3 case and merges on the Γ = 4/3 curve. For very large shock speeds, the Mach number tends to the asymptotic value (Γ−1)−1/2, regardless of the frame of reference. Inclusion of tangential velocities (right panels in Fig. 2) leads to an increased mass flux (J2 ∝ γ20) and, consequently, to higher post-shock pressure and density values. Still, since pressure grows faster than density, temperature in the post- shock flow strains to relativistic values even for slower shock velocities and the TM EoS tends to the Γ = 4/3 case at even smaller shock velocities (γsvs & 2). Generally speaking, at a given shock velocity, density and pressure in the shocked gas attain higher values for lower Γeq. Downstream temperature, on the other hand, follows the opposite trend being higher as Γeq → 5/3 and lower when Γeq → 4/3. Figure 3. Compression ratio (top), downstream plasma β (mid- dle) and magnetic field strength (bottom) as function of the shock four-velocity γsvs with vanishing tangential component of the velocity. The magnetic field makes an angle π/6 (left) and π/2 (right) with the shock normal. The meaning of the different lines is the same as in Fig. 2. 3.2 Magnetized Shocks In presence of magnetic fields, we solve the 3 × 3 nonlin- ear system given by Eqns. (22), (23) and (24), and di- rectly replace η1 = η0h0/h1 with the aid of Eq. (21). The magnetic field introduces three additional parameters, namely, the thermal to magnetic pressure ratio (β ≡ 2p/b2) and the orientation of the magnetic field with respect to the shock front and to the tangential velocity. This is ex- pressed by the angles αx and αy such that Bx = |B| cos αx, By = |B| sin αx cos αy , Bz = |B| sin αx sin αy . We restrict our attention to the case of a strongly magnetized pre-shock flow with β0 ≡ 2p0/b20 = 10−2. Fig. 3 shows the density, plasma β and magnetic pres- sure ratios versus shock velocity for αx = π/6 (left panels) and αx = π/2 (perpendicular shock, right panels). Since there is no tangential velocity, the solution depends on one angle only (αx) and the choice of αy is irrelevant. For small shock velocities (γsvs . 0.4), the front is magnetically driven with density and pressure jumps attaining lower values than the non-magnetized counterpart. A similar behavior is found in classical MHD (Jeffrey & Taniuti 1964). Density and 6 A. Mignone and J.C. McKinney Figure 4. Density ratio (top), downstream plasma β (middle) and magnetic field strength (bottom) as function of γsvs when the tangential component of the upstream velocity is vt = 0.99. The magnetic field and the shock normal form an angle π/6. The tangential components of magnetic field and velocity are aligned (left) and orthogonal (right).Different lines have the same mean- ing as in Fig. 2. magnetic compression ratios across the shock reach the clas- sical values around γsvs ≈ 1 (rather than γsvs ≈ 0.1 as in the non-magnetic case) and increase afterwards. The mag- netic pressure ratio grows faster for the perpendicular shock, whereas internal energy and density show little dependence on the orientation angle αx. As expected, the TM EoS mim- ics the constant Γ = 5/3 case at small shock velocities. At γsvs . 0.46, the plasma β exceeds unity and the shock starts to be pressure-dominated. In other words, thermal pressure eventually overwhelms the Lorentz force and the shock be- comes pressure-driven for velocities of the order of vs ≈ 0.42. When γsvs & 1, the internal energy begins to become com- parable to the rest mass energy (c2) and the behavior of the TM EoS detaches from the Γ = 5/3 curve and slowly joins the Γ = 4/3 case. The full transition happens in the limit of strongly relativistic shock speeds, γsvs . 10. Inclusion of transverse velocities in the right state af- fects the solution in a way similar to the non-magnetic case. Relativistic effects play a role already at small velocities because of the increased inertia of the pre-shock state in- troduced by the upstream Lorentz factor. For αx = π/6 Figure 5. Density contrast (top), plasma β (middle) and mag- netic field strength (bottom) for vt = 0.99. The magnetic field is purely transverse and aligned with the tangential component of velocity on the left, while it is orthogonal on the right. Different lines have the same meaning as in Fig. 2. (Fig. 4), the compression ratio does not drop to small val- ues and keeps growing becoming even larger (. 400) than the previous case when vt = 0. The same behavior is re- flected on the growth of magnetic pressure that, in addi- tion, shows more dependence on the relative orientation of the velocity and magnetic field projections in the plane of the front. When αy = π/2, indeed, magnetic pressure at- tains very large values (b2/b20 . 10 4, bottom right panel in Fig. 4). Consequently, this is reflected in a decreased post- shock plasma β. For the TM EoS, the post-shock properties of the flow begin to resemble the Γ = 4/3 behavior at lower shock velocities than before, γsvs ≈ 2 ÷ 3. Similar consid- erations may be done for the case of a perpendicular shock (αx = π/2, see Fig. 5), although the plasma β saturates to larger values thus indicating larger post-shock pressures. Again, the maximum increase in magnetic pressure occurs when the velocity and magnetic field are perpendicular. 4 NUMERICAL SIMULATIONS With the exception of very simple flow configurations, the solution of the RMHD fluid equations must be carried out Equation of state in RMHD 7 Figure 6. Solution to the mildly relativistic blast wave (problem 1) at t = 0.4. From left to right, the different profiles give den- sity, thermal pressure, total pressure (top panels), the three com- ponents of velocity (middle panel) and magnetic fields (bottom panels). Computations with the TM EoS and constant Γ = 5/3 EoS are shown using solid and dotted lines, respectively. numerically. This allows an investigation of highly nonlinear regimes and complex interactions between multiple waves. We present some examples of astrophysical relevance, such as the propagation of one dimensional blast waves, the prop- agation of axisymmetric jets, and the evolution of magne- tized accretion disks around Kerr black holes. Our goal is to outline the qualitative effects of varying the EoS for some in- teresting astrophysical problems rather than giving detailed results on any individual topic. Direct numerical integration of Eq. (1) has been achieved using the PLUTO code (Mignone et al. 2007) in §4.1, §4.2 and HARM (Gammie et al. 2003) in §4.3. The new primitive variable inversion scheme presented in Appendix A has been implemented in both codes and the results pre- sented in §4.1 were used for code validation. The novel in- version scheme offers the advantage of being suitable for a more general EoS and avoiding catastrophic cancellation in the non-relativistic and ultrarelativistic limits. 4.1 Relativistic Blast Waves A shock tube consists of a sharp discontinuity separat- ing two constant states. In what follows we will be con- sidering the one dimensional interval [0, 1] with a discon- tinuity placed at x = 0.5. For the first test problem, states to the left and to the right of the discontinuity are given by (ρ, p,By , Bz)L = (1, 30, 6, 6) for the left state and (ρ, p, By , Bz)R = (1, 1, 0.7, 0.7) for the right state. This re- sults in a mildly relativistic configuration yielding a max- imum Lorentz factor of 1.3 6 γ 6 1.4. The second test Figure 7. Solution to the strong relativistic blast wave (problem 2) at t = 0.4. From left to right, the different profiles give den- sity, thermal pressure, total pressure (top panels), the three com- ponents of velocity (middle panel) and magnetic fields (bottom panels). Computations with the TM EoS and constant Γ = 5/3 EoS are shown using solid and dotted lines, respectively. consists of a left state given by (ρ, p,By , Bz)L = (1, 10 3, 7, 7) and a right state (ρ, p, By, Bz)R = (1, 0.1, 0.7, 0.7). This con- figuration involves the propagation of a stronger blast wave yielding a more relativistic configuration (3 6 γ 6 3.5). For both states, we use a base grid with 800 zones and 6 levels of refinement (equiv. resolution = 800 · 26) and evolve the solution up to t = 0.4. Computations carried with the ideal EoS with Γ = 5/3 and the TM EoS are shown in Fig. 6 and Fig. 7 for the first and second shock tube, respectively. From left to right, the wave pattern is comprised of a fast and slow rarefac- tions, a contact discontinuity and a slow and a fast shocks. No rotational discontinuity is observed. Compared to the Γ = 5/3 case, one can see that the results obtained with the TM EoS show considerable differences. Indeed, waves propagate at rather smaller velocities and this is evident at the head and the tail points of the left-going magneto-sonic rarefaction waves. From a simple analogy with the hydrody- namic counterpart, in fact, we know that these points prop- agate increasingly faster with higher sound speed. Since the sound speed ratio of the TM and Γ = 5/3 is always less than one (see, for instance, the bottom left panel in Fig. 1), one may reasonably predict slower propagation speed for the Riemann fans when the TM EoS is used. Furthermore, this is confirmed by computations carried with Γ = 4/3 that shows even slower velocities. Similar conclusions can be drawn for the shock velocities. The reason is that the opening of the Riemann fan of the TM equation state is smaller than the Γ = 5/3 case, because the latter always over-estimates the sound speed. The higher density peak behind the slow shock 8 A. Mignone and J.C. McKinney Figure 8. Jet velocity as a function of the Mach number for different values of the initial density contrast η. The beam Lorentz factor is the same for all plots, γb = 10. Solid, dashed and dotted lines correspond to the solutions obtained with the TM EoS and the Γ = 4/3 and Γ = 5/3 laws, respectively. follows from the previous considerations and the conserva- tion of mass across the front. 4.2 Propagation of Relativistic Jets Relativistic, pressure-matched jets are usually set up by injecting a supersonic cylindrical beam with radius rb into a uniform static ambient medium (see, for instance, Mart́ı et al. 1997). The dynamical and morphological prop- erties of the jet and its interaction with the surrounding are most commonly investigated by adopting a three parameter set: the beam Lorentz factor γb, Mach number Mb = vb/cs and the beam to ambient density ratio η = ρb/ρm. The presence of a constant poloidal magnetic field introduces a fourth parameter βb = 2pb/b 2, which specifies the thermal to magnetic pressure ratio. 4.2.1 One Dimensional Models The propagation of the jet itself takes place at the velocity Vj , defined as the speed of the working surface that sepa- rates shocked ambient fluid from the beam material. A one- dimensional estimate of Vj (for vanishing magnetic fields) can be derived from momentum flux balance in the frame of the working surface (Mart́ı et al. 1997). This yields ηhb/hm 1 + γb ηhb/hm , (30) where hb and hm are the specific enthalpies of the beam and the ambient medium, respectively. For given γb and den- sity contrast η, Eq. (30) may be regarded as a function of Figure 9. Computed results for the non magnetized jet at t = 90 for the ideal EoS (Γ = 5/3 and Γ = 4/3, top and middle panels) and the TM EoS (bottom panel), respectively. The lower and upper half of each panels shows the gray-scale map of density and internal energy in logarithmic scale. Figure 10. Position of the working surface as a function of time for Γ = 5/3 (circles), Γ = 4/3 (stars) and the TM EoS (dia- monds). Solid, dotted and dashed lines gives the one-dimensional expectation. Equation of state in RMHD 9 Figure 11. Density and magnetic field for the magnetized jet at t = 80 (first and second panels from top) and at t = 126 (third and fourth panels). Computations were carried with 40 zones per beam radius with the TM EoS. the Mach number alone that uniquely specifies the pres- sure pb through the definitions of the sound speed, Eq. (17). For the constant Γ-law EoS the inversion is straightforward, whereas for the TM EoS one finds, using the substitution Θ = 2/3 sinh x, pb = η 1 − t2m , (31) where tm satisfies the negative branch of the quadratic equa- 15 − 6M 24 − 10M + 9 = 0 , (32) with t = tanh x. In Fig. 8 we show the jet velocity for in- creasing Mach numbers (or equivalently, decreasing sound speeds) and different density ratios η = 10−5, 10−3, 10−1, 10. The Lorentz beam factor is γb = 10. Prominent discrepan- cies between the selected EoS arise at low Mach numbers, where the relative variations of the jet speed between the constant Γ and the TM EoS’s can be more than 50%. This regime corresponds to the case of a hot jet (Θ ≈ 10 in the η = 10−3 case) propagating into a cold (Θ ≈ 10−3) medium, for which neither the Γ = 4/3 nor the Γ = 5/3 approxima- tion can properly characterize both fluids. 4.2.2 Two Dimensional Models Of course, Eq. (30) is strictly valid for one-dimensional flows and the question remains as to whether similar conclusions can be drawn in more than one dimension. To this end we investigate, through numerical simulations, the propagation of relativistic jets in cylindrical axisymmetric coordinates (r, z). We consider two models corresponding to different sets of parameters and adopt the same computational do- main [0, 12] × [0, 50] (in units of jet radius) with the beam being injected at the inlet region (r 6 1, z = 0). Jets are in pressure equilibrium with the environment. In the first model, the density ratio, beam Lorentz fac- tor and Mach number are given, respectively, by η = 10−3, γb = 10 and Mb = 1.77. Magnetic fields are absent. Inte- gration are carried at the resolution of 20 zones per beam radius using the relativistic Godunov scheme described in MPB. Computed results showing density and internal en- ergy maps at t = 90 are given in Fig. 9 for Γ = 5/3, Γ = 4/3 and the TM EoS. The three different cases differ in several morphological aspects, the most prominent one being the position of the leading bow shock, z ≈ 18 when Γ = 5/3, z ≈ 48 for Γ = 4/3 and z ≈ 33 for the TM EoS. Smaller values of Γ lead to larger beam internal energies and there- fore to an increased momentum flux, in agreement with the one dimensional estimate (30). This favors higher propaga- tion velocities and it is better quantified in Fig. 10 where the position of the working surface is plotted as a function of time and compared with the one dimensional estimate. For the cold jet (Γ = 5/3), the Mach shock exhibits a larger cross section and is located farther behind the bow shock when compared to the other two models. As a result, the jet velocity further decreases promoting the formation of a thicker cocoon. On the contrary, the hot jet (Γ = 4/3) prop- agates at the highest velocity and the cocoon has a more elongated shape. The beam propagates almost undisturbed and cross-shocks are weak. Close to is termination point, the beam widens and the jet slows down with hot shocked gas being pushed into the surrounding cocoon at a higher rate. Integration with the TM EoS reveals morphological and dynamical properties more similar to the Γ = 4/3 case, although the jet is ≈ 40% slower. At t = 90 the beam does not seem to decelerate and its speed remains closer to the one-dimensional expectation. The cocoon develops a thin- ner structure with a more elongated conical shape and cross shocks form in the beam closer to the Mach disk. In the second case, we compare models C2-pol-1 and B1-pol-1 of Leismann et al. (2005) (corresponding to an ideal gas with Γ = 5/3 and Γ = 4/3, respectively) with the TM EoS adopting the same numerical scheme. For this model, η = 10−2, vb = 0.99, Mb = 6 and the ambient medium is threaded by a constant vertical magnetic field, 2pb. Fig. 11 shows the results at t = 80 and t = 126, corresponding to the final integration times shown in Leismann et al. (2005) for the selected values of Γ. For the sake of conciseness, integration pertaining to the TM EoS only are shown and the reader is reminded to the original work by Leismann et al. (2005) for a comprehensive descrip- tion. Compared to ideal EoS cases, the jet shown here pos- sesses morphological and dynamical properties intermediate between the hot (Γ = 4/3) and the cold (Γ = 5/3) cases. As expected, the jet propagates slower than in model B1-pol-1 (hot jet), but faster than the cold one (C2-pol-1). The head of the jet tends to form a hammer-like structure (although less prominent than the cold case) towards the end of the integration, i.e., for t & 100, but the cone remains more con- fined at previous times. Consistently with model C2-pol-1, the beam develops a series of weak cross shocks and outgo- ing waves triggered by the interaction of the flow with bent magnetic field lines. Although the magnetic field inhibits the formation of eddies, turbulent behavior is still observed in cocoon, where interior cavities with low magnetic fields are 10 A. Mignone and J.C. McKinney Figure 12. Magnetized accretion flow around a Kerr black hole for the ideal Γ-law EoS with Γ = 4/3. Shows the logarithm of the rest-mass density in colour from high (red) to low (blue) values. The magnetic field has been overlayed. This model demonstrates more vigorous turbulence and a thicker corona that leads to a more confined magnetized jet near the poles. Figure 13. As in figure 12 but for Γ = 5/3. Compared to the Γ = 4/3 model, there is less vigorous turbulence and the corona is more sharply defined. formed. In this respect, the jet seems to share more features with the cold case. Figure 14. As in figure 12 but for the TM EoS. This EoS leads to turbulence that is less vigorous than in the Γ = 4/3 model but more vigorous than in the Γ = 5/3 model. Qualitatively the TM EoS leads to an accretion disk that behaves somewhere between the behavior of the Γ = 4/3 and Γ = 5/3 models. 4.3 Magnetized Accretion near Kerr Black Holes In this section we study time-dependent GRMHD numerical models of black hole accretion in order to determine the ef- fect of the EoS on the behavior of the accretion disk, corona, and jet. We study three models similar to the models stud- ied by McKinney & Gammie (2004) for a Kerr black hole with a/M ≈ 0.94 and a disk with a scale height (H) to ra- dius (R) ratio of H/R ∼ 0.3. The constant Γ-law EoS with Γ = {4/3, 5/3} and the TM EoS are used. The initial torus solution is in hydrostatic equilibrium for the Γ-law EoS, but we use the Γ = 5/3 EoS as an initial condition for the TM EoS. Using the Γ = 4/3 EoS as an initial condition for the TM EoS did not affect the final quasi-stationary behavior of the flow. The simplest question to ask is which value of Γ will result in a solution most similar to the TM EoS model’s solution. More advanced questions involve how the structure of the accretion flow depends on the EoS. The previ- ous results of this paper indicate that the corona above the disk seen in the simulations (De Villiers et al. 2003; McKinney & Gammie 2004) will be most sensitive to the EoS since this region can involve both non-relativistic and relativistic temperatures. The corona is directly involved is the production of a turbulent, magnetized, thermal disk wind (McKinney & Narayan 2006a,b), so the disk wind is also expected to depend on the EoS. The disk inflow near the black hole has a magnetic pressure comparable to the gas pressure (McKinney & Gammie 2004), so the EoS may play a role here and affect the flux of mass, energy, and angular momentum into the black hole. The magnetized jet asso- ciated with the Blandford & Znajek solution seen in simu- lations (McKinney & Gammie 2004; McKinney 2006) is not expected to depend directly on the EoS, but may depend in- Equation of state in RMHD 11 directly through the confining action of the corona. Finally, the type of field geometries observed in simulations that thread the disk and corona (Hirose et al. 2004; McKinney 2005) might depend on the EoS through the effect of the stiffness (larger Γ leads to harder EoSs) of the EoS on the turbulent diffusion of magnetic fields. Figs. 12, 13 and 14 show a snapshot of the accretion disk, corona, and jet at t ∼ 1000GM/c3 . Overall the re- sults are quite comparable, as could be predicted since the Γ = {4/3, 5/3} models studied in McKinney & Gammie (2004) were quite similar. For all models, the field geometries allowed are similar to that found in McKinney (2005). The accretion rate of mass, specific energy, and specific angular momentum are similar for all models, so the EoS appears to have only a small effect on the flow through the disk near the black hole. The most pronounced effect is that the soft EoS (Γ = 4/3) model develops more vigorous turbulence due to the non-linear behavior of the magneto-rotational instability (MRI) than either the Γ = 5/3 or TM EoSs. This causes the coronae in the Γ = 4/3 model to be slightly thicker and to slightly more strongly confine the magnetized jet resulting in a slight decrease in the opening angle of the magnetized jet at large radii. Also, the Γ = 4/3 model develops a fast magnetized jet at slightly smaller radii than the other mod- els. An important consequence is that the jet opening angle at large radii might depend sensitively on the EoS of the ma- terial in the accretion disc corona. This should be studied in future work. 5 CONCLUSIONS The role of the EoS in relativistic magnetohydrodynamics has been investigated both analytically and numerically. The equation of state previously introduced by Mignone et al. (2005a) (for non magnetized flows) has been extended to the case where magnetic fields are present. The proposed equation of state closely approximates the single-specie per- fect relativistic gas, but it offers a much simpler analyti- cal representation. In the limit of very large or very small temperatures, for instance, the equivalent specific heat ratio reduces, respectively, to the 4/3 or 5/3 limits. The propagation of fast magneto-sonic shock waves has been investigated by comparing the constant Γ laws to the new equation of state. Although for small shock veloci- ties the shock dynamics is well described by the cold gas limit, dynamical and thermodynamical quantities (such as the compression ratio, internal energy, magnetization and so forth) substantially change across the wave front at moder- ately or highly relativistic speeds. Eventually, for increasing shock velocities, flow quantities in the downstream region smoothly vary from the cold (Γ = 5/3) to the hot (Γ = 4/3) regimes. We numerically studied the effect of the EoS on shocks, blast waves, the propagation of relativistic jets, and magne- tized accretion flows around Kerr black holes. Our results should serve as a useful guide for future more specific stud- ies of each topic. For these numerical studies, we formu- lated the inversion from conservative quantities to primitive quantities that allows a general EoS and avoids catastrophic numerical cancellation in the non-relativistic and ultrarela- tivistic limits. The analytical and numerical models confirm the general result that large temperature gradients cannot be properly described by a polytropic EoS with constant specific heat ratio. Indeed, when compared to a more re- alistic EoS, for which the polytropic index is a function of the temperature, considerable dynamical differences arises. This has been repeatedly shown in presence of strong dis- continuities, such shocks, across which the internal energy can change by several order of magnitude. We also showed that the turbulent behavior of magne- tized accretion flows around Kerr black holes depends on the EoS. The Γ = 4/3 EoS leads to more vigorous turbulence than the Γ = 5/3 or TM EoSs. This affects the thickness of the corona that confines the magnetized jet. Any study of turbulence within the accretion disk, the subsequent genera- tion of heat in the coronae, and the opening and acceleration of the jet (especially at large radii where the cumulative dif- ferences due to the EoS in the disc are largest) should use an accurate EoS. The effect of the EoS on the jet opening angle and Lorentz factor at large radii is a topic of future study. The proposed equation state holds in the limit where effects due to radiation pressure, electron degeneracies and neutrino physics can be neglected. It also omits potentially crucial physical aspects related to kinetic processes (such as suprathermal particle distributions, cosmic rays), plasma composition, turbulence effects at the sub-grid levels, etc. These are very likely to alter the equation of state by ef- fectively changing the adiabatic index computed on merely thermodynamic arguments. Future efforts should properly address additional physical issues and consider more gen- eral equations of state. ACKNOWLEDGMENTS We are grateful to our referee, P. Hughes, for his worthy considerations and comments that led to the final form of this paper. JCM was supported by a Harvard CfA Institute for Theory and Computation fellowship. AM would like to thank S. Massaglia and G. Bodo for useful discussions on the jet propagation and morphology. REFERENCES Aloy, M. A., Ibáñez, J. M. , Mart́ı, J. M. , Gómez, J.-L., Müller, E. 1999, ApJL, 523, L125 Aloy, M. A., Ibáñez, J. M., Mart́ı, J. M., Müller, E. 1999, ApJS, 122, 151 Anile, M., & Pennisi, S. 1987, Ann. Inst. Henri Poincaré, 46, 127 Anile, A. M. 1989, Relativistic Fluids and Magneto-fluids (Cambridge: Cambridge University Press), 55 Begelman, M. C., Blandford, R. D., & Rees, M. J. 1984, Reviews of Modern Physics, 56, 255 Bernstein, J.P., & Hughers, P.A. 2006, astro-ph/0606012 Blandford R. D., Znajek R. L., 1977, MNRAS, 179, 433 Einfeldt, B., Munz, C.D., Roe, P.L., and Sjögreen, B. 1991, J. Comput. 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L., Müller, E. 2002, MNRAS, 331, 615 Synge, J. L. 1957, The relativistic Gas, North-Holland Pub- lishing Company Taub, A. H. 1948, Physical Review, 74, 328 Tchekhovskoy, A, McKinney, J. C.,& Narayan, R. 2006, MNRAS, submitted Toro, E. F. 1997, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin van Putten, M. H. P. M. 1993, ApJL, 408, L21 APPENDIX A: PRIMITIVE VARIABLE INVERSION SCHEME We outline a new primitive variable inversion scheme that is used to convert the evolved conserved quantities into so- called primitive quantities that are necessary to obtain the fluxes used for the evolution. This scheme allows a gen- eral EoS by only requiring specification of thermodynamical quantities and it also avoids catastrophic cancellation in the non-relativistic and ultrarelativistic limits. Large Lorentz factors (up to 106) may not be uncommon in some astro- physical contexts (e.g. Gamma-Ray-Burst) and ordinary in- version methods can lead to severe numerical problems such as effectively dividing by zero and subtractive cancellation, see, for instance, Bernstein & Hughes (2006). First, we note that the general relativistic conserva- tive quantities can be written more like special relativis- tic quantities by choosing a special frame in which to mea- sure all quantities. A useful frame is the zero angular mo- mentum (ZAMO) observer in an axisymmetric space-time. See Noble et al. (2006) for details. From their expressions, it is useful to note that catastrophic cancellations for non- relativistic velocities can be avoided by replacing γ − 1 in any expression with (uαu α)/(γ + 1), where here uα is the relative 4-velocity in the ZAMO frame. From here on the expressions are in the ZAMO frame and appear similar to the same expressions in special relativity. A1 Inversion Procedure Numerical integration of the conservation law (1) proceeds by evolving the conservative state vector U = (D, m, B, E) in time. Computation of the fluxes, however, requires veloc- ity and pressure to be recovered from U by inverting Eqns. (5)–(7), a rather time consuming and challenging task. For the constant-Γ law, a recent work by Noble et al. (2006) examines several methods of inversion. In this section we discuss how to modify the equations of motion, interme- diate calculations, and the inversion from conservative to primitive quantities so that the RMHD method 1) permits a general EoS; and 2) avoids catastrophic cancellations in the non-relativistic and ultrarelativistic limits. Our starting relations are the total energy density (7), E = W − p + 1 + |v| |B|2 − S , (A1) and the square modulus of Eq. (6), http://arxiv.org/abs/astro-ph/0607575 http://arxiv.org/abs/astro-ph/0607576 Equation of state in RMHD 13 |m|2 = W + |B|2 )2 |v|2 − S 2W + |B|2 , (A2) where S ≡ m · B and W = Dhγ. Note that in order for this expression to be accurate in the non-relativistic limit, one should analytically cancel any appearance of E in this expression. Eq. (A2) can be inverted to express the square of the velocity in terms of the only unknown W : |v|2 = S 2(2W + |B|2) + |m|2W 2 (W + |B|2)2W 2 . (A3) After inserting (A3) into (A1) one has: E = W − p + |B| |B|2|m|2 − S2 2(|B|2 + W )2 . (A4) In order to avoid numerical errors in the non-relativistic limit one must modify the equations of motion and sev- eral intermediate calculations. One solves the conservation equations with the mass density subtracted from the en- ergy by defining a new conserved quantity (E′ = E − D) and similarly for the energy flux. In addition, operations based upon γ can lead to catastrophic cancellations since the residual γ − 1 is often requested and is dominant in the non-relativistic limit. A more natural quantity to consider is |v|2 or γ2|v|2. Also, in the ultrarelativistic limit calcula- tions based upon γ(|v|2) have catastrophic cancellation er- rors when |v| → 1. This can be avoided by 1) using instead |u|2 ≡ γ2|v|2 and 2) introducing the quantities E′ = E −D and W ′ = W − D, with W ′ properly rewritten as D|u|2 1 + γ to avoid machine accuracy problems in the nonrelativistic limit, where χ ≡ ρǫ+p. Thus our relevant equations become: ′ − p + |B| |B|2|m|2 − S2 2(|B|2 + W ′ + D)2 , (A6) |m|2 = (W + |B|2)2 |u| 1 + |u|2 − 2W + |B|2 , (A7) where W = W ′ + D. Equations (A6) and (A7) may be inverted to find W ′, p and |u|2. A one dimensional inversion scheme is derived by regarding Eq. (A6) as a single nonlinear equation in the only unknown W ′ and using Eq. (A7) to express |u|2 as a function of W ′. Using Newton’s iterative scheme as our root finder, one needs to compute the derivative = 1 − dp |B|2|m|2 − S2 (|B|2 + W ′ + D)3 . (A8) The explicit form of dp/dW ′ depends on the particular EoS being used. While prior methods in principle allow for a general EoS, one has to re-derive many quantities that in- volve kinematical expressions. This can be avoided by split- ting the kinematical and thermodynamical quantities. This also allows one to write the expressions so that there is no catastrophic cancellations in the non-relativistic or ultrarel- ativistic limits. Assuming that p = p(χ, ρ), we achieve this by applying the chain rule to the pressure derivative: . (A9) Partial derivatives involving purely thermodynamical quan- tities must now be supplied by the EoS routines. Derivatives with respect to W ′, on the other hand, involve purely kine- matical terms and do not depend on the choice of the EoS. Relevant expressions needed in our computations are given in the Appendix. Once W ′ has been determined to some accuracy, the inversion process is completed by computing the velocities from an inversion of equation (6) to obtain W + |B|2 , (A10) One then computes χ from an inversion of equation (A5) to obtain − D|u| (1 + γ)γ2 , (A11) from which p or ρǫ can be obtained for any given EoS. The rest mass density is obtained from , (A12) and the magnetic field is trivially inverted. In summary, we have formulated an inversion scheme that 1) allows a general EoS without re-deriving kinematical expressions; and 2) avoids catastrophic cancellation in the non-relativistic and ultrarelativistic limits. This inversion in- volves solving a single non-linear equation using, e.g., a one- dimensional Newton’s method. A similar two-dimensional method can be easily written with the same properties, and such a method may be more robust in some cases since the one-dimensional version described here involves more com- plicated non-linear expressions. One can show analytically that the inversion is accurate in the ultrarelativistic limit as long as γ.ǫ machine for γ and p/(ργ2)&ǫmachine for pressure, where ǫmachine ≈ 2.2 × 10−16 for double precision. The method used by Noble et al. (2006) requires γ.ǫ machine/10 due to the repeated use of the ex- pression γ = 1/ 1 − v2 in the inversion. Note that we use γ = 1 + |u|2 that has no catastrophic cancellation. The fundamental limit on accuracy is due to evolving en- ergy and momentum separately such that the expression E − |m| appears in the inversion. Only a method that evolves this quantity directly (e.g. for one-dimensional prob- lems one can evolve the energy with momentum subtracted) can reach higher Lorentz factors. An example test problem is the ultrarelativistic Noh test in Aloy et al. (1999) with p = 7.633 × 10−6, Γ = 4/3, 1 − v = 10−11 (i.e. γ = 223607) This test has p/(ργ2) ≈ 1.6 × 10−16, which is just below double precision and so the pressure is barely resolved in the pre-shock region. The post-shock region is insensitive to the pre-shock pressure and so is evolved accurately up to γ ≈ 6 × 107. These facts are have been also confirmed nu- merically using this inversion within HARM. Using the same error measures as in Aloy et al. (1999) we can evolve their test problem with an even higher Lorentz factor of γ = 107 and obtain similar errors of .0.1%. A2 Kinematical and Thermodynamical Expressions The kinematical terms required in equation (A9) may be easily found from the definition of W ′, ′ ≡ Dhγ − D = D(γ − 1) + χγ2 , (A13) 14 A. Mignone and J.C. McKinney by straightforward differentiation. This yields (D + 2γχ) d|v|2 , (A14) d(1/γ) = −Dγ d|v|2 , (A15) where d|v|2 = − 2 3W (W + |B|2) + |B|4 + |m|2W 3 (W + |B|2)3 , (A16) is computed by differentiating (A3) with respect to W (note that d/dW ′ ≡ d/dW ). Equation (A14) does not depend on the knowledge of the EoS. Thermodynamical quantities such as ∂p/∂χ, on the other hand, do require the explicit form of the EoS. For the ideal gas EoS one simply has p(χ, ρ) = Γ − 1 χ , (A17) where χ = ρǫ+ p. By taking the partial derivatives of (A17) with respect to χ (keeping ρ constant) and ρ (keeping χ constant) one has Γ − 1 = 0 . (A18) For the TM EoS, one can more conveniently rewrite (14) as 3p(ρ + χ − p) = (χ − p)(χ + 2ρ − p) , (A19) which, upon differentiation with respect to χ (keeping ρ con- stant) yields 2χ + 2ρ − 5p 5ρ + 5χ − 8p . (A20) Similarly, by taking the derivative with respect to ρ at con- stant χ gives 2χ − 5p 5ρ + 5χ − 8p . (A21) In order to use the above expressions and avoid catas- trophic cancellation in the non-relativistic limit, one must solve for the gas pressure as functions of only ρ and χ and then write the pressure that explicitly avoids catastrophic cancellation as {χ, p} → 0. One obtains: p(χ, ρ) = 2χ(χ + 2ρ) 5(χ + ρ) + 9χ2 + 18ρχ + 25ρ2 . (A22) Also, for setting the initial conditions it is useful to be able to convert from a given pressure to the internal energy by using ρǫ(ρ, p) = 9p2 + 4ρ2 , (A23) which also avoids catastrophic cancellation in the non- relativistic limit. A3 Newton-Raphson Scheme Equation (A6) may be solved using a Newton-Raphson it- erative scheme, where the (k + 1)-th approximation to the W ′ is computed as ′(k+1) ′(k) − f(W df(W ′)/dW ′ W ′=W ′(k) , (A24) where ) = W ′ − E′ − p + |B| |B|2|m|2 − S2 2(|B|2 + W ′ + D)2 , (A25) and df(W ′)/dW ′ ≡ dE′/dW ′ is given by Eq. (A8). The iteration process terminates when the residual ∣W ′(k+1)/W ′(k) − 1 ∣ falls below some specified tolerance. We remind the reader that, in order to start the iter- ation process given by (A24), a suitable initial guess must be provided. We address this problem by initializing, at the beginning of the cycle, W ′(0) = W̃+ − D, where W̃+ is the positive root of P(W, 1) = 0 , (A26) and P(W, |v|) is the quadratic function P(W, |v|) = |m|2−|v|2W 2+(2W+|B|2)(2W+|B|2−2E) .(A27) This choice guarantees positivity of pressure, as it can be proven using the relation P(W, |v|) 2(2W + |B|2) , (A28) which follows upon eliminating the (S/W )2 term in Eq. (A2) with the aid of Eq. (A1). Seeing that P(W, |v|) is a con- vex quadratic function, the condition p > 0 is equivalent to the requirement that the solution W must lie outside the interval [W−, W+], where P(W±, |v|) = 0. However, since P(W, |v|) > P(W, 1), it must follow that W̃+ > W+ and thus W̃+ lies outside the specified interval. We tacitly as- sume that the roots are always real, a condition that is al- ways met in practice. Introduction Relativistic MHD Equations Equations of Motion Equation of State Propagation of Fast Magneto-sonic Shocks Purely Hydrodynamical Shocks Magnetized Shocks Numerical Simulations Relativistic Blast Waves Propagation of Relativistic Jets Magnetized Accretion near Kerr Black Holes Conclusions Primitive Variable Inversion Scheme Inversion Procedure Kinematical and Thermodynamical Expressions Newton-Raphson Scheme

0704.1680

Carbon Nanotube Thin Film Field Emitting Diode: Understanding the System Response Based on Multiphysics Modeling

Carbon Nanotube Thin Film Field Emitting Diode: Understanding the System Response Based on Multiphysics Modeling N. Sinhaa, D. Roy Mahapatrab, J.T.W. Yeowa1, R.V.N. Melnikb and D.A. Jaffrayc aDepartment of Systems Design Engineering, University of Waterloo, ON, N2L3G1, Canada bMathematical Modeling and Computational Sciences, Wilfrid Laurier University, Waterloo, ON, N2L3C5, Canada cDepartment of Radiation Physics, Princess Margaret Hospital, Toronto, ON, M5G2M9, Canada Abstract In this paper, we model the evolution and self-assembly of randomly oriented carbon nanotubes (CNTs), grown on a metallic substrate in the form of a thin film for field emission under diode configuration. Despite high output, the current in such a thin film device often decays drastically. The present paper is focused on understanding this problem. A systematic, multiphysics based modelling approach is proposed. First, a nucleation coupled model for degradation of the CNT thin film is derived, where the CNTs are assumed to decay by fragmentation and formation of clusters. The random orientation of the CNTs and the electromechanical interaction are then modeled to explain the self-assembly. The degraded state of the CNTs and the electromechanical force are employed to update the orientation of the CNTs. Field emission current at the device scale is finally obtained by using the Fowler- Nordheim equation and integration over the computational cell surfaces on the anode side. The simulated results are in close agreement with the experimental results. Based on the developed model, numerical simulations aimed at understanding the 1Corresponding author: JTWY e-mail: [email protected]; Tel: 1 (519) 8884567 x 2152; Fax: 1 (519) 7464791 effects of various geometric parameters and their statistical features on the device current history are reported. Keywords: Field emission, carbon nanotube, degradation, electrodynamics, self- assembly. 1 Introduction The conventional mechanism used for electron emission is thermionic in nature where electrons are emitted from hot cathodes (usually heated filaments). The advantage of these hot cathodes is that they work even in environments that contain a large number of gaseous molecules. However, thermionic cathodes in general have slow response time and they consume high power. These cathodes have limited lifetime due to mechanical wear. In addition, the thermionic electrons have random spatial distribution. As a result, fine focusing of electron beam is very difficult. This adversely affects the performance of the devices such as X-ray tubes. An alternative mechanism to extract electrons is field emission, in which electrons near the Fermi level tunnel through the energy barrier and escape to the vacuum under the influence of a sufficiently high external electric field. The field emission cathodes have faster response time, consume less power and have longer life compared to thermionic cathodes. However, field emission cathodes require ultra-high vacuum as they are highly reactive to gaseous molecules during the field emission. The key to the high performance of a field emission device is the behavior of its cathode. In the past, the performance of cathode materials such as spindt-type emitters and nanostructured diamonds for field emission was studied by Spindt et al.1 , Gotoh et al.2 , and Zhu3 . However, the spindt type emitters suffer from high manufacturing cost and limited lifetime. Their failure is often caused by ion bombardment from the residual gas species that blunt the emitter cones2 . On the other hand, nanostructured diamonds are unstable at high current densities3 . Carbon nanotube (CNT), which is an allotrope of carbon, has potential to be used as cathode material in field emission devices. Since their discovery by Iijima in 19914 , extensive research on CNTs has been conducted. Field emission from CNTs was first reported in 1995 by Rinzler et al.5 , de Heer et al.6 , and Chernozatonskii et al.7 . Field emission from CNTs has been studied extensively since then. Currently, with significant improvement in processing technique, CNTs are among the best field emitters. Their applications in field emission devices, such as field emission displays, gas discharge tubes, nanolithography systems, electron microscopes, lamps, and X- ray tube sources have been successfully demonstrated8−9 . The need for highly controlled application of CNTs in X-ray devices is one of the main reasons for the present study. The remarkable field emission properties of CNTs are attributed to their geometry, high thermal conductivity, and chemical stability. Studies have reported that CNT sources have a high reduced brightness and their energy spread values are comparable to conventional field emitters and thermionic emitters10 . The physics of field emission from metallic surfaces is well understood. The current density (J) due to field emission from a metallic surface is usually obtained by using the Fowler-Nordheim (FN) equation11 CΦ3/2 , (1) where E is the electric field, Φ is the work function of the cathode material, and B and C are constants. The device under consideration in this paper is a X-ray source where a thin film of CNTs acts as the electron emitting surface (cathode). Under the influence of sufficiently high voltage at ultra high vacuum, the electrons are extracted from the CNTs and hit the heavy metal target (anode) to produce X-rays. However, in the case of a CNT thin film acting as cathode, the surface of the cathode is not smooth (like the metal emitters). In this case, the cathode consists of hollow tubes grown on a substrate. Also, some amount of carbon clusters may be present within the CNT-based film. An added complexity is that there is realignment of individual CNTs due to electrodynamic interaction between the neighbouring CNTs during field emission. At present, there is no adequate mathematical models to address these issues. Therefore, the development of an appropriate mathematical modeling approach is necessary to understand the behavior of CNT thin film field emitters. 1.1 Role of various physical processes in the degradation of CNT field emitter Several studies have reported experimental observations in favour of considerable degradation and failure of CNT cathodes. These studies can be divided into two categories: (i) studies related to degradation of single nanotube emitters12−17 and (ii) studies related to degradation of CNT thin films18−23 . Dean et al.13 found gradual decrease of field emission of single walled carbon nanotubes (SWNTs) due to “evaporation” when large field emitted current (300nA to 2µA) was extracted. It was observed by Lim et al.23 that CNTs are susceptible to damage by exposure to gases such as oxygen and nitrogen during field emission. Wei et al.14 observed that after field emission over 30 minutes at field emission current between 50 and 120nA, the length of CNTs reduced by 10%. Wang et al.15 observed two types of structural damage as the voltage was increased: a piece-by-piece and segment-by-segment splitting of the nanotubes, and a layer-by-layer stripping process. Occasional spikes in the current-voltage curves were observed by Chung et al.16 when the voltage was increased. Avouris et al.17 found that the CNTs break down when subjected to high bias over a long period of time. Usually, the breakdown process involves stepwise increases in the resistance. In the experiments performed by the present authors, peeling of the film from the substrate was observed at high bias. Some of the physics pertinent to these effects is known but the overall phenomenon governing such a complex system is difficult to explain and quantify and it requires further investigation. There are several causes of CNT failures: (i) In case of multi-walled carbon nanotubes (MWNTs), the CNTs undergo layer- by-layer stripping during field emission15 . The complete removal of the shells are most likely the reason for the variation in the current voltage curves16 ; (ii) At high emitted currents, CNTs are resistively heated. Thermal effect can sublime a CNT causing cathode-initiated vacuum breakdown24 . Also, in case of thin films grown using chemical vapor deposition (CVD), fewer catalytic metals such as nickel, cobalt, and iron are observed as impurities in CNT thin films. These metal particles melt and evaporate by high emission currents, and abruptly surge the emission current. This results in vacuum breakdown followed by the failure of the CNT film23 ; (iii) Gas exposure induces chemisorption and physisorption of gas molecules on the surface of CNTs. In the low-voltage regime, the gas adsorbates remain on the surface of the emitters. On the other hand, in the high-voltage regime, large emission currents resistively anneal the tips, and the strong electric field on the locally heated tips promotes the desorption of gas adsorbates from the tip sur- face. Adsorption of materials with high electronegativity hinders the electron emission by intensifying the local potential barriers. Surface morphology can be changed by an erosion of the cap of the CNT as the gases desorb reactively from the surface of the CNTs25 ; (iv) CVD-grown CNTs tend to show more defects in the wall as their radius in- creases. Possibly, there are rearrangements of atomic structures (for example, vacancy migration) resulting in the reduction of length of CNTs14 . In addition, the presence of defects may act as a centre for nucleation for voltage-induced oxidation, resulting in electrical breakdown16 ; (v) As the CNTs grow perpendicular to the substrate, the contact area of CNTs with the substrate is very small. This is a weak point in CNT films grown on planar substrates, and CNTs may fail due to tension under the applied fields20 . Small nanotube diameters and lengths are an advantage from the stability point of view. Although the degradation and failure of single nanotube emitters can be either abrupt or gradual, the degradation and failure of a thin film emitter with CNT clus- ter is mostly gradual. The gradual degradation occurs either during initial current- voltage measurement21 (at a fast time scale) or during measurements at constant applied voltage over a long period of time22 (at a slow time scale). Nevertheless, it can be concluded that the gradual degradation of thin films occurs due to the failure of individual emitters. Till date, several studies have reported experimental observations on CNT thin films26 . However, from mathematical, computational and design view points, the models and characterization methods are available only for vertically aligned CNTs grown on the patterned surface27−28 . In a CNT film, the array of CNTs may ideally be aligned vertically. However, in this case it is desired that the individual CNTs be evenly separated in such a way that their spacing is greater than their height to minimize the screening effect29 . If the screening effect is minimized following the above argument, then the emission properties as well as the lifetime of the cathodes are adversely affected due to the significant reduction in density of CNTs. For the cathodes with randomly oriented CNTs, the field emission current is produced by two types of sources: (i) small fraction of CNTs that point toward the anode and (ii) oriented and curved CNTs subjected to electromechanical forces causing reori- entation. As often inferred (see e.g., ref.29 ), the advantage of the cathodes with randomly oriented CNTs is that always a large number of CNTs take part in the field emission, which is unlikely in the case of cathodes with uniformly aligned CNTs. Such a thin film of randomly oriented CNTs will be considered in the present study. From the modeling point of view, its analysis becomes much more challenging. Al- though some preliminary works have been reported (see e.g., refs.30−31 ), neither a detailed model nor a subsequent characterization method are available that would allow to describe the array of CNTs that may undergo complex dynamics during the process of charge transport. In the detailed model, the effects of degradation and fragmentation of CNTs during field emission need to be considered. However, in the majority of analytical and design studies, the usual practice is to employ the classical Fowler-Nordheim equation11 to determine the field emission from the metallic surface, with correction factors to deal with the CNT tip geometry. Ideally, one has to tune such an empirical approach to specific materials and methods used (e.g. CNT geometry, method of preparation, CNT density, diode configuration, range of applied voltage, etc.). Also, in order to account for the oriented CNTs and interaction between themselves, it is necessary to consider the space charge and the electromechanical forces. By taking into account the evolution of the CNTs, a mod- eling approach is developed in this paper. In order to determine phenomenologically the concentration of carbon clusters due to degradation of CNTs, we introduce a homogeneous nucleation rate. This rate is coupled to a moment model for the evo- lution. The moment model is incorporated in a spatially discrete sense, that is by introducing volume elements or cells to physically represent the CNT thin film. Elec- tromechanical forces acting on the CNTs are estimated in time-incremental manner. The oriented state of CNTs are updated using a mechanics based model. Finally, the current density is calculated by using the details regarding the CNT orientation angle and the effective electric field in the Fowler-Nordheim equation. The remainder of this paper is organized as follows: in Sec. 2, a model is pro- posed, which combines the nucleation coupled model for CNT degradation with the electromechanical forcing model. Section 3 illustrates the computational scheme. Numerical simulations and the comparison of the simulated current-voltage charac- teristics with experimental results are presented in Sec. 4. 2 Model formulation The CNT thin film is idealized in our mathematical model by using the following simplifications. (i) CNTs are grown on a substrate to form a thin film. They are treated as aggregate while deriving the nucleation coupled model for degradation phe- nomenologically; (ii) The film is discretized into a number of representative volume element (cell), in which a number of CNTs can be in oriented forms along with an estimated amount of carbon clusters. This is schematically shown in Fig. 1. The car- bon clusters are assumed to be in the form of carbon chains and networks (monomers and polymers); (iii) Each of the CNTs with hexagonal arrangement of carbon atoms (shown in Fig. 2(a)) are treated as effectively one-dimensional (1D) elastic members and discretized by nodes and segments along its axis as shown in Fig. 2(b). Defor- mation of this 1D representation in the slow time scale defines the orientations of the segments within the cell. A deformation in the fast time scale (due to electron flow) defines the fluctuation of the sheet of carbon atoms in the CNTs and hence the resulting state of atomic arrangements. The latter aspect is ex- cluded from the present modeling and numerical simulations, however they will be discussed within a quantum-hydrodynamic framework in a forthcom- ing article. 2.1 Nucleation coupled model for degradation of CNTs Let NT be the total number of carbon atoms (in CNTs and in cluster form) in a cell (see Fig. 1). The volume of a cell is given by Vcell = ∆Ad, where ∆A is the cell surface interfacing the anode and d is distance between the inner surfaces of cathode substrate and the anode. Let N be the number of CNTs in the cell, and NCNT be the total number of carbon atoms present in the CNTs. We assume that during field emission some CNTs are decomposed and form clusters. Such degradation and fragmentation of CNTs can be treated as the reverse process of CVD or a similar growth process used for producing the CNTs on a substrate. Hence, NT = NNCNT +Ncluster , (2) where Ncluster is the total number of carbon atoms in the clusters in a cell at time t and is given by Ncluster = Vcell dn1(t) , (3) where n1 is the concentration of carbon cluster in the cell. By combining Eqs. (2) and (3), one has NT − Vcell dn1(t) . (4) The number of carbon atom in a CNT is proportional to its length. Let the length of a CNT be a function of time, denoted as L(t). Therefore, one can write NCNT = NringL(t) , (5) where Nring is the number of carbon atoms per unit length of a CNT and can be determined from the geometry of the hexagonal arrangement of carbon atoms in the CNT. By combining Eqs. (4) and (5), one can write NringL(t) NT − Vcell dn1(t) . (6) In order to determine n1(t) phenomenologically, we need to know the nature of evolution of the aggregate in the cell. From the physical point of view, one may expect the rate of formation of the carbon clusters from CNTs to be a function of thermodynamic quantities, such as temperature (T ), the relative distances (rij) between the carbon atoms in the CNTs, the relative distances between the clusters and a set of parameters (p∗) describing the critical cluster geometry. The relative distance rij between carbon atoms in CNTs is a function of the electromechanical forces. Modeling of this effect is discussed in Sec. 2.2. On the other hand, the relative distances between the clusters influence in homogenizing the thermodynamic energy, that is, the decreasing distances between the clusters (hence increasing densities of clusters) slow down the rate of degradation and fragmentation of CNTs and lead to a saturation in the concentration of clusters in a cell. Thus, one can write = f(T, rij, p ∗) . (7) To proceed further, we introduce a nucleation coupled model32−33 , which was origi- nally proposed to simulate aerosol formation. Here we modify this model according to the present problem which is opposite to the process of growth of CNTs from the gaseous phase. With this model the relative distance function is replaced by a collision frequency function (βij) describing the frequency of collision between the i-mers and j-mers, with βij = )1/6√6kT i1/3 + j1/3 , (8) and the set of parameters describing the critical cluster geometry by p∗ = {vj sj g∗ d∗p} , (9) where vj is the j-mer volume, sj is the surface area of j-mer, g ∗ is the normalized critical cluster size, d∗p is the critical cluster diameter, k is the Boltzmann constant, T is the temperature and ρp is the particle mass density. The detailed form of Eq. (7) is given by four nonlinear ordinary differential equations: dNkin = Jkin , (10) JkinSg − (S − 1) , (11) = Jkind p + (S − 1)B1Nkin , (12) JkinSg ∗2/3s1 2πB1S(S − 1)M1 , (13) where Nkin is the kinetic normalization constant, Jkin is the kinetic nucleation rate, S is the saturation ratio, An is the total surface area of the carbon cluster and M1 is the moment of cluster size distribution. The quantities involved are expressed as , M1 = ∫ dmaxp n(dp, t)dp d(dp) , (14) Nkin = exp(Θ) , Jkin = 27(lnS)2 , (15) , d∗p = kT lnS , B1 = 2nsv1 , (16) where ns is the equilibrium saturation concentration of carbon cluster, d p is the maximum diameter of the clusters, n(dp, t) is the cluster size distribution function, dp is the cluster diameter, mj is the mass of j-mer, Θ is the dimensionless surface tension given by , (17) σ is the surface tension. In this paper, we have considered i = 1 and j = 1 for numer- ical simulations, that is, only monomer type clusters are considered. In Eqs. (10)- (13), the variables are n1(t), S(t), M1(t) and An(t), and all other quantities are assumed constant over time. In the expression for moment M1(t) in Eq. (14), the cluster size distribution in the cell is assumed to be Gaussian, however, random distribution can be incorporated. We solve Eqs. (10)-(13) using a finite difference scheme as discussed in Sec. 3. Finally, the number of CNTs in the cell at a given time is obtained with the help of Eq. (6), where the reduced length L(t) is determined using geometric properties of the individual CNTs as formulated next. 2.2 Effect of CNT geometry and orientation It has been discussed in Sec. 1.1 that the geometry and orientation of the tip of the CNTs are important factors in the overall field emission performance of the film and must be considered in the model. As an initial condition, let L(0) = h at t = 0, and let h0 be the average height of the CNT region as shown in Fig. 1. This average height h0 is approximately equal to the height of the CNTs that are aligned vertically. If ∆h is the decrease in the length of a CNT (aligned vertically or oriented as a segment) over a time interval ∆t due to degradation and fragmentation, and if dt is the diameter of the CNT, then the surface area of the CNT decreased is πdt∆h. By using the geometry of the CNT, the decreased surface area can be expressed as πdt∆h = Vcelln1(t) s(s− a1)(s− a2)(s− a3) , (18) where Vcell is the volume of the cell as introduced in Sec. 2.1, a1, a2, a3 are the lattice constants, and s = 1 (a1 + a2 + a3) (see Fig. 2(a)). The chiral vector for the CNT is expressed as C h = n~a1 +m~a2 , (19) where n and m are integers (n ≥ |m| ≥ 0) and the pair (n,m) defines the chirality of the CNT. The following properties hold: ~a1.~a1 = a 1, ~a2.~a2 = a 2, and 2~a1.~a2 = a21 + a 2 − a23. With the help of these properties the circumference and the diameter of the CNT can be expressed as, respectively34 , |−→C h| = n2a21 +m 2a22 + nm(a 1 + a 2 − a23) , dt = |−→C h| , (20) Let us now introduce the rate of degradation of the CNT or simply the burning rate as vburn = lim ∆h/∆t. By dividing both side of Eq. (18) by ∆t and by applying limit, one has πdtvburn = Vcell dn1(t) s(s− a1)(s− a2)(s− a3) , (21) By combining Eqs. (20) and (21), the burning rate is finally obtained as vburn = Vcell dn1(t) [ s(s− a1)(s− a2)(s− a3) n2a21 +m 2a22 + nm(a 1 + a 2 − a23) . (22) In Fig. 3 we show a schematic drawing of the CNTs almost vertically aligned, that is along the direction of the electric field E(x, y). This electric field E(x, y) is assumed to be due to the applied bias voltage. However, there will be an additional but small amount of electric field due to several localized phenomena (e.g., electron flow in curved CNTs, field emission from the CNT tip etc.). Effectively, we assume that the distribution of the field parallel to z-axis is of periodic nature (as shown in Fig. 3) when the CNT tips are vertically oriented. Only a cross-sectional view in the xz plane is shown in Fig. 3 because only an array of CNTs across x-direction will be considered in the model for simplicity. Thus, in this paper, we shall restrict our attention to a two-dimensional problem, and out-of-plane motion of the CNTs will not be incorporated in the model. To determine the effective electric field at the tip of a CNT oriented at an angle θ as shown in Fig. 3, we need to know the tip coordinate with respect to the cell coordinate system. If it is assumed that a CNT tip was almost vertically aligned at t = 0 (as it is the desired configuration for the ideal field emission cathode), then its present height is L(t) = h0 − vburnt and the present distance between the tip and the anode is dg = d − L(t) = d − h0 + vburnt. We assume that the tip electric field has a z-dependence of the form E0L(t)/dg, where E0 = V/d and V is the applied bias voltage. Also, let (x, y) be the deflection of the tip with respect to its original location and the spacing between the two neighboring CNTs at the cathode substrate is 2R. Then the electric field at the deflected tip can be approximated as Ez′ = x2 + y2 (h0 − vburnt) (d− h0 + vburnt) E0 , θ(t) ≤ θc , (23) where θc is a critical angle to be set during numerical calculations along with the condition: Ez′ = 0 when θ(t) > θc. This is consistent with the fact that those CNTs which are low lying on the substrate do not contribute to the field emission. The electric field at the individual CNT tip derived here is defined in the local coordinate system (X ′, Z ′) as shown in Fig. 3. The components of the electric field in the cell coordinate system (X, Y, Z) is given by the following transformation:  = nz lz mz√ 1− n2z −lznz√ 1−n2z mznz√ 1−n2z 0 −lznz√ 1−n2z 1−n2z  , (24) where nz, lz, mz are the direction cosines. According to the cell coordinate system in Figs. 1 and 3, nz = cos θ(t), lz = sin θ(t), and mz = 0. Therefore, Eq. (24) can be rewritten as  = cos θ(t) sin θ(t) 0√ 1− cos2 θ(t) − cos θ(t) 0 0 − cos θ(t) −1  . (25) By simplifying Eq. (25), we get Ez = Ez′ cos θ(t) , Ex = Ez′ sin θ(t) . (26) Note that the identical steps of this transformation also apply to a generally oriented (θ 6= 0) segment of CNT as idealized in Fig. 2(b). The electric field components Ez and Ex are later used for calculation of the electromechanical force acting on the CNTs. Since in this study we aim at estimating the current density at the anode due to the field emission from the CNT tips, we also use Ez from Eq. (26) to compute the output current based on the Fowler-Nordheim equation (1). 2.3 Electromechanical forces For each CNT, the angle of orientation θ(t) is dependent on the electromechanical forces. Such dependence is geometrically nonlinear and it is not practical to solve the problem exactly, especially in the present situation where a large number of CNTs are to be dealt with. However, it is possible to solve the problem in time- dependent manner with an incremental update scheme. In this section we derive the components of the electromechanical forces acting on a generally oriented CNT segment. The numerical solution scheme based on an incremental update scheme will be discussed in Sec. 3. From the studies reported in published literature and based on the discussions made in Sec. 1.1, it is reasonable to expect that the major contribution is due to (i) the Lorentz force under electron gas flow in CNTs (a hydrodynamic formalism), (ii) the electrostatic force (background charge in the cell), (iii) the van der Waals force against bending and shearing of MWNT and (iv) the ponderomotive force acting on the CNTs. 2.3.1 Lorentz force It is known that the electrical conduction and related properties of CNTs depend on the mechanical deformation and the geometry of the CNT. In this paper we model the field emission behaviour of the CNT thin film by considering the time-dependent electromechanical effects, whereas the electronic properties and related effects are incorporated through the Fowler-Nordheim equation empirically. Electronic band- structure calculations are computationally prohibitive at this stage and at the same spatio-temporal scales considered for this study. However, a quantum-hydrodynamic formalism seems practical and such details will be dealt in a forthcoming article. Within the quantum-hydrodynamic formalism, one generally assumes the flow of electron gas along the cylindrical sheet of CNTs. The associated electron density distribution is related to the energy states along the length of the CNTs including the tip region. What is important for the present modeling is that the CNTs experience Lorentz force under the influence of the bias electric field as the electrons flow from the cathode substrate to the tip of a CNT. The Lorentz force is expressed as ~fl = e(n̂0 + n̂1) ~E ≈ en̂0 ~E , (27) where e is the electronic charge, n̂0 is the surface electron density corresponding to the Fermi level energy, n̂1 is the electron density due to the deformation in the slow time scale, and phonon and electromagnetic wave coupling at the fast time scale, and ~E is the electric field. The surface electron density corresponding to the Fermi level energy is expressed as35 n̂0 = , (28) where b is the interatomic distance and ∆ is the overlap integral (≈ 2eV for carbon). The quantity b can be related to the mechanical deformation of the 1D segments (See Fig. 2) and formulations reported by Xiao et al.36 can be employed. For simplicity, the electron density fluctuation n̂1 is neglected in this paper. Now, with the electric field components derived in Eq. 26, the components of the Lorentz force acting along z and x directions can now be written as, respectively, flz = πdten̂0Ez , flx = πdten̂0Ex ≈ 0 . (29) 2.3.2 Electrostatic force In order to calculate the electrostatic force, the interaction among two neighboring CNTs is considered. For such calculation, let us consider a segment ds1 on a CNT (denoted 1) and another segment ds2 on its neighboring CNT (denoted 2). These are parts of the representative 1D member idealized as shown in Fig. 2(b). The charges associated with these two segments can be expressed as q1 = en̂0πd t ds1 , q2 = en̂0πd t ds2 , (30) where d t and d t are diameters of two neighbouring CNTs (1) and (2). The electrostatic force on the segment ds1 by the segment ds2 is 4π��0 where � is the effective permittivity of the aggregate of CNTs and carbon clusters, �0 is the permittivity of free space, and r12 is the effective distance between the centroids of ds1 and ds2. The electrostatic force on the segment ds1 due to charge in the entire segment (s2) of the neighboring CNT (see Fig. 4) can be written as 4π��0 en̂0πd t ds1en̂0πd ds2 . The electrostatic force per unit length on s1 due to s2 is then 4π��0 (πen̂0) ds2 . (31) The differential of the force dfc acts along the line joining the centroids of the segments ds1 and ds2 as shown in Fig. 4. Therefore, the components of the total electrostatic force per unit length of CNT (1) in X and Z directions can be written as, respectively, fcx = dfc cosφ = 4π��0 (πen̂0) cosφ ds2 4π��0 h0/∆s2∑ (πen̂0) cosφ ∆s2 , (32) fcz = dfc sinφ = 4π��0 (πen̂0) sinφ ds2 4π��0 h0/∆s2∑ (πen̂0) sinφ ∆s2 , (33) where φ is the angle the force vector dfc makes with the X-axis. For numerical computation of the above integrals, we compute the angle φ = φ(sk1, s 2) and r12 = r12(s 2) at each of the centroids of the segments between the nodes k + 1 and k, where the length of the segments are assumed to be uniform and denoted as ∆s1 for CNT (1) and ∆s2 for CNT (2). As shown in Fig. 4, the distance r12 between the centroids of the segments ds1 and ds2 is obtained as r12 = (d1 − lx2 + lx1) 2 + (lz1 − lz2) , (34) where d1 is the spacing between the CNTs at the cathode substrate, lx1 and lx2 are the deflections along X-axis, and lz1 and lz2 are the deflections along Z-axis. The angle of projection φ is expressed as φ = tan−1 ( lz1 − lz2 d1 − lx2 + lx1 . (35) The deflections lx1 , lz1 , lx2 , and lz2 are defined as, respectively, lx1 = ds1 sin θ1 ≡ ∆s1 sin θ 1 (36) lz1 = ds1 cos θ1 ≡ ∆s1 cos θ 1 (37) lx2 = ds2 sin θ2 ≡ ∆s2 sin θ 2 (38) lz2 = ds2 cos θ2 ≡ ∆s2 cos θ 2 . (39) Note that the total electrostatic force on a particular CNT is to be obtained by summing up all the binary contributions within the cell, that is by summing up Eqs. (32) and (33) over the upper integer number of the quantity N − 1, where N is the number of CNTs in the cell as discussed in Sec. 2.1. 2.3.3 The van der Waals force Next, we consider the van der Waals effect. The van der Waals force plays important role not only in the interaction of the CNTs with the substrate, but also in the interaction between the walls of MWNTs and CNT bundles. Due to the overall effect of forces and flexibility of the CNTs (here assumed to be elastic 1D members), the cylindrical symmetry of CNTs is destroyed, leading to their axial and radial deformations. The change in cylindrical symmetry may significantly affect the the properties of CNTs37−38 . Here we estimate the van der Waals forces due to the interaction between two concentric walls of the MWCNTs. Let us assume that the lateral and the longitudinal displacements of a CNT be ux′ and uz′ , respectively. We use updated Lagrangian approach with local coordi- nate system for this description (similar to (X ′, Z ′) system shown in Fig. 3), where the longitudinal axis coincides with Z ′ and the lateral axis coincides with X ′. Such a description is consistent with the incremental procedure to update the CNT orien- tations in the cells as adopted in the computational scheme. Also, due to the large length-to-diameter ratio (L(t)/dt), let the kinematics of the CNTs, which are ideal- ized in this work as 1D elastic members, be governed by that of an Euler-Bernoulli beam. Therefore, the kinematics can be written as z′ = u z′0 − r (m)∂u , (40) where the superscript (m) indicates the mth wall of the MWNT with r(m) as its radius and uz′0 is the longitudinal displacement of the center of the cylindrical cross- section. Under tension, bending moment and lateral shear force, the elongation of one wall relative to its neighboring wall is z′ = u (m+1) z′ − u z′ = r (m+1)∂u (m+1) − r(m) ≈ (r(m+1) − r(m)) , (41) where we assume u x′ = u (m+1) x′ = ∆x′ as the lateral displacement as some function of tensile force or compression buckling or pressure in the thin film device. The lateral shear stress (τ vs ) due to the van der Waals effect can now be written as τ (m)vs = Cvs , (42) where Cvs is the van der Waals coefficient. Hence, the shear force per unit length can be obtained by integrating Eq. (42) over the individual wall circumferences and then by summing up for all the neighboring pair interactions, that is, fvs = reff dψ = (r(m+1) − r(m))∂∆x′ r(m+1) + r(m) ⇒ fvs = πCvs[(r (m+1))2 − (r(m))2] . (43) The components of van der Waals force in the cell coordinate system (X ′, Z ′) is then obtained as fvsz = fvs sin θ(t) , fvsx = fvs cos θ(t) . (44) 2.3.4 Ponderomotive force Ponderomotive force, which acts on free charges on the surface of CNTs, tends to straighten the bent CNTs under the influence of electric field in the Z-direction. Furthermore, the ponderomotive forces induced by the applied electric field stretch every CNT39 . We add this effect by assuming that the free charge at the tip region is subjected to Ponderomotive force, which is computed as40 fpz = 0∆A cos θ(t) , fpx = 0 , (45) where ∆A is the surface area of the cell on the anode side, fpz is the Z component of the Ponderomotive force and the X component fpx is assumed to be negligible. 2.4 Modelling the reorientation of CNTs The net force components acting on the CNTs along Z and X directions can be expressed as, respectively, (flz + fvzz) ds+ fcz + fpz , (46) (flx + fvsx) ds+ fcx + fpx . (47) For numerical computation, at each time step the force components obtained using Eqs. (46) and (47) are employed to update the curved shape S ′(x′ + ux′ , z ′ + uz′), where the displacements are approximated using simple beam mechanics solution: uz′ ≈ E ′A0 (f j+1z − f z )(z ′j+1 − z′j) , (48) ux′ ≈ 3E ′A2 (f j+1x − f x′j+1 − x′j , (49) where A0 is the effective cross-sectional area, A2 is the area moment, E ′ is the modulus of elasticity for the CNT under consideration. The angle of orientation, θ(t), of the corresponding segment of the CNT, that is between the node j + 1 and node j, is given by θ(t) = θ(t)j = tan−1 (xj+1 + uj+1x )− (xj + ujx) (zj+1 + u z )− (zj + ujz) , (50) Γ(θ(t−∆t)j) ]{ ujx′ , (51) where Γ is the usual coordinate transformation matrix which maps the displace- ments (ux′ , uz′) defined in the local (X ′, Z ′) coordinate system into the displace- ments (ux, uz) defined in the cell coordinate system (X,Z). For this transformation, we employ the angle θ(t−∆t) obtained in the previous time step and for each node j = 1, 2, . . .. 3 Computational scheme As already highlighted in the previous section, we model the CNTs as generally oriented 1D elastic members. These 1D members are represented by nodes and segments. With given initial distribution of the CNTs in the cell, we discretize the time into uniform steps ti+1 − ti = ∆t. The computational scheme involves three parts: (i) discretization of the nucleation coupled model for degradation of CNTs derived in Sec. 2.1, (ii) incremental update of the CNT geometry using the estimated electromechanical force and (iii) computation of the field emission current in the device. 3.1 Discretization of the nucleation coupled model for degra- dation With the help of Eqs. (14)-(16) and by eliminating the kinetic nucleation rate Nkin, we first rewrite the simplified form of Eqs. (10)-(13), which are given by, respectively, 27(lnS)2 , (52) 2β11nsΘS 2π(lnS)3 27(ln s)2 (S − 1)An , (53) 27(lnS)2 + 2n2sv1 exp(Θ) (S − 1) , (54) 2π(lnS)2 27(lnS)2 + 4πnsv1 M1(S − 1) . (55) By eliminating dS/dt from Eq. (52) with the help of Eq. (53) and by applying a finite difference formula in time, we get n1i − n1i−1 ti − ti−1 27(lnSi−1)2 Θ− 4Θ 27(lnSi−1)2 (lnSi−1)3 n21i(Si − 1)An(i) . (56) Similarly, Eqs. (53)-(55) are discretized as, respectively, Si − Si−1 ti − ti−1 Θ− 4Θ 27(lnSi−1)2 (lnSi−1)3 n1i(Si − 1)Ani , (57) M1i −M1i−1 ti − ti−1 27(lnSi−1)2 + 2v1 n21i(Si − 1) exp(Θ) , (58) Ani − Ani−1 ti − ti−1 β11s1Θ 5/2n1i Θ− 4Θ 27(lnSi−1)2 (lnSi−1)2 +4πv1 (Si−1)M1i . (59) By simplifying Eq. (56) with the help of Eqs. (57)-(59), we get a quadratic polyno- mial of the form (b1 − b2 − b3)n1i 2 − n1i + n1i−1 = 0 , (60) where b1 = ∆t 27(lnSi−1)2 , (61) b2 = ∆t Θ− 4Θ 27(lnSi−1)2 Si(lnSi−1)3 , (62) b3 = ∆t Si − 1 . (63) Solution of Eq. (60) yields two roots (denoted by superscripts (1, 2)): (1,2) 2(b1 − b2 − b3) 1− 4n1i−1(b1 − b2 − b3) 2(b1 − b2 − b3) . (64) For the first time step, the values of b1, b2 and b3 are obtained by applying the initial conditions: S(0) = S0, n10 = n0, and An0 = An0. Since the n1i must be real and finite, the following two conditions are imposed: 1−4n1i−1(b1− b2− b3) ≥ 0 and (b1 − b2 − b3) 6= 0. Also, it has been assumed that the degradation of CNTs is an irreversible process, that is, the reformation of CNTs from the carbon cluster does not take place. Therefore, an additional condition of positivity, that is, n1i > n1i−1 is introduced while performing the time stepping. Along with the above constraints, the n1 history in a cell is calculated as follows: • If n(1)1i > n1i−1 and n , then n1i = n • Else if n(2)1i > n1i−1 , then n1i = n • Otherwise the value of n1 remains the same as in the previous time step, that is, n1i = n1i−1 . Simplification of Eq. (57) results in the following equation: 2 + (c1 + c2 − Si−1)Si − c1 = 0 , (65) where c1 = ∆tn1iAni , (66) c2 = ∆t Θ− 4Θ 27(lnSi−1)2 (lnSi−1)3 . (67) Solution of Eq. (65) yields the following two roots: Si = − (c1 + c2 − Si−1)± c1 + c2 − S2i−1 + 4c1 . (68) For the first time step, c1 and c2 are calculated with the following conditions: n11 from the above calculation, S(0) = S0, and An0 = An0. Realistically, the saturation ratio S cannot be negative or equal to one. Therefore, Si > 0 yields c1 > 0. While solving for An, the Eq. (59) is solved with the values of n1 and S from the above calculations and the initial conditions An0 = An0, M10 = M0. The value of M10 was calculated by assuming n(dp, t) as a standard normal distribution function. 3.2 Incremental update of the CNT geometry At each time time step t = ti, once the n1i is solved, we are in a position to compute the net electromechanical force (see Sec. 2.3) as fi = fi(E0, n1i−1 , θ(ti−1)) . (69) Subsequently, the orientation angle for each segment of each CNT is then obtained as (see Sec. 2.4) θ(ti) j = θ(fi) j (70) and it is stored for future calculations. A critical angle, (θc), is generally employed with θc ≈ π/4 to π/2.5 for the present numerical simulations. For θ ≤ θc, the meaning of fz is the “longitudinal force” and the meaning of fx is the “lateral force” in the context of Eqs. (48) and (49). When θ > θc, the meanings of fz and fx are interchanged. 3.3 Computation of field emission current Once the updated tip angles and the electric field at the tip are obtained at a particular time step, we employ Eq. (1) to compute the current density contribution from each CNT tip, which can be rewritten as BE2zi CΦ3/2 , (71) with B = (1.4 × 10−6) × exp(9.8929 × Φ−1/2) and C = 6.5 × 107 taken from ref.41 . The device current (Ii) from each computational cell with surface area ∆A at the anode at the present time step ti is obtained by summing up the current density over the number of CNTs in the cell, that is, Ii = ∆A Ji . (72) Fig. 5 shows the flow chart of the computational scheme discussed above. At t = 0, in our model, the CNTs can be randomly oriented. This random distribution is parameterized in terms of the upper bound of the CNT tip deflection, which is given by ∆xmax = h/q, where h is the CNT length and q is a real number. In the numerical simulations which will be discussed next, the initial tip deflections can vary widely. The following values of the upper bound of the tip deflection have been considered: ∆xmax = h0/(5 + 10p), (p = 0, 1, 2, ..., 9). The tip deflection ∆x is randomized between zero and these upper bounds. Simulation for each initial input with a randomized distribution of tip deflections was run for a number of times and the maximum, minimum, and average values of the output current were obtained. In the first set, the simulations were run for a uniform height, radius and spacing of CNTs in the film. Subsequently, the height, the radius and the spacing were varied randomly within certain bounds, and their effects on the output current were analyzed. 4 Results and discussions The CNT film under study in this work consists of randomly oriented multi-walled nanotubes (MWNTs). The film samples were grown on a stainless steel substrate. The film has a surface area of 1cm2 and thickness of 10−14µm. The anode consists of a 1.59mm thick copper plate with an area of 49.93mm2. The current-voltage history is measured over a range of DC bias voltages for a controlled gap between the cathode and the anode. In the experimental set-up, the device is placed within a vacuum chamber of a multi-stage pump. The gap (d) between the cathode substrate and the anode is controlled from outside by a micrometer. 4.1 Degradation of the CNT thin films We assume that at t = 0, the film contains negligible amount of carbon cluster. To understand the phenomena of degradation and fragmentation of the CNTs, fol- lowing three sets of input are considered: n1(0) = 100, 150, 500. The other initial conditions are set as S(0) = 100, M1(0) = 2.12× 10−16, An(0) = 0, and T = 303K. Fig. 6 shows the three n1(t) histories over a small time duration (160s) for the three cases of n1(0), respectively. For n1(0) = 100 and 150, the time histories indicate that the rate of decay is very slow, which in turn implies longer lifetime of the device. For n1(0) = 500, the time history indicates that the CNTs decay comparatively faster, but still insignificant for the first 34s, and then the cluster concentration becomes constant. It can be concluded from the above three cases that the rate of decay of CNTs is generally slow under operating conditions, which implies stable performance and longer lifetime of the device if this aspect is considered alone. Next, the effect of variation in the initial saturation ratio S(0) on n1(t) history is studied. The value of n1(0) is set as 100, while other parameters are assumed to have identical value as considered previously. The following three initial conditions in S(0) are considered: S(0) = 50, 100, 150. Fig. 7 shows the n1(t) histories. It can be seen in this figure that for S(0) = 100 (moderate value), the carbon cluster concentration first increases and then tends to a steady state. This was also observed in Fig. (6). For higher values of S(0), n1 increases exponentially over time. For S(0) = 50, a smaller value, the decay is not observed at all. This implies that a small value of S(0) is favorable for longer lifetime of the cathode. However, a more detailed investigation on the physical mechanism of cluster formation and CNT fragmentation may be necessary, which is an open area of research. At t = 0, we assign random orientation angles (θ(0)j) to the CNT segments. For a cell containing 100 CNTs, Fig. 8 shows the terminal distribution of the CNT tip angles (at t = 160s corresponding to the n1(0) = 100 case discussed previously) compared to the initial distribution (at t = 0). The large fluctuations in the tip angles for many of the CNTs can be attributed to the significant electromechanical interactions. 4.2 Current-voltage characteristics In the present study, the quantum-mechanical treatment has not been explicitly carried out, and instead, the Fowler-Nordheim equation has been used to calculate the current density. In such a semi-empirical calculation, the work function Φ42 for the CNTs must be known accurately under a range of conditions for which the device-level simulations are being carried out. For CNTs, the field emission electrons originate from several excited energy states (non metallic electronic states)43−44 . Therefore, the the work function for CNTs is usually not well identified and is more complicated to compute than for metals. Several methodologies for calculating the work function for CNTs have been proposed in literature. On the experimental side, Ultraviolet Photoelectron Spectroscopy (UPS) was used by Suzuki et al.45 to calculate the work function for SWNTs. They reported a work function value of 4.8 eV for SWNTs. By using UPS, Ago et al.46 measured the work function for MWNTs as 4.3 eV. Fransen et al.47 used the field emission electronic energy distribution (FEED) to investigate the work function for an individual MWNT that was mounted on a tungsten tip. Form their experiments, the work function was found to be 7.3±0.5 eV. Photoelectron emission (PEE) was used by Shiraishi et al.48 to measure the work function for SWNTs and MWNTs. They measured the work function for SWNTs to be 5.05 eV and for MWNTs to be 4.95 eV. Experimental estimates of work function for CNTs were carried out also by Sinitsyn et al.49 . Two types were investigated by them: (i) 0.8-1.1 nm diameter SWNTs twisted into ropes of 10 nm diameter, and (ii) 10 nm diameter MWNTs twisted into 30-100 nm diameter ropes. The work functions for SWNTs and MWNTs were estimated to be 1.1 eV and 1.4 eV, respectively. Obraztsov et al.50 reported the work function for MWNTs grown by CVD to be in the range 0.2-1.0 eV. These work function values are much smaller than the work function values of metals (≈ 3.6 − 5.4eV ), silicon(≈ 3.30 − 4.30eV ), and graphite(≈ 4.6 − 5.4eV ). The calculated values of work function of CNTs by different techniques is summarized in Table 1. The wide range of work functions in different studies indicates that there are possibly other important effects (such as electromechanical interactions and strain) which also depend on the method of sample preparation and different experimental techniques used in those studies. In the present study, we have chosen Φ = 2.2eV . The simulated current-voltage (I-V) characteristics of a film sample for a gap d = 34.7µm is compared with the experimental measurement in Fig. 9. The average height, the average radius and the average spacing between neighboring CNTs in the film sample are taken as h0 = 12µm, r = 2.75nm, and d1 = 2µm. The simulated I-V curve in Fig. 9 corresponds to the average of the computed current for the ten runs. This is the first and preliminary simulation of its kind based on a multiphysics based modeling approach and the present model predicts the I-V characteristics which is in close agreement with the experimental measurement. However, the above comparison indicates that there are some deviations near the threshold voltage of ≈ 500 − 600V , which needs to be looked at by improving the model as well as experimental materials and method. 4.3 Field emission current history Next, we simulate the field emission current histories for the similar sample con- figuration as used previously, but for three different parametric variations: height, radius, and spacing. Current histories are shown for constant bias voltages of 440V , 550V and 660V . 4.3.1 Effects of uniform height, uniform radius and uniform spacing In this case, the values of height, radius, and the spacing between the neighboring CNTs are kept identical to the previous current-voltage calculation in Sec. 4.2. Fig. 10(a), (b) and (c) show the current histories for three different bias voltages of 440V , 550V and 660V . In the subfigures, we plot the minimum, the maximum and the average currents over time as post-processed from a number of runs with randomized input distributions. At a bias voltage of 440V , the average current decreases from 1.36× 10−8A to 1.25× 10−8A in steps. The maximum current varies between 1.86×10−8A to 1.68×10−8A, whereas the minimum current varies between 2.78 × 10−9A to 2.52 × 10−9A. Comparisons among the scales in the sub-figures indicate that there is an increase in the order of magnitude of current when the bias voltage is increased. The average current decreases from 1.25×10−5A to 1.06×10−5A in steps when the bias voltage is increased from 440V to 550V . At the bias voltage of 660V , the average value of the current decreases from 1.26×10−3A to 1.02×10−3A. The increase in the order of magnitude in the current at higher bias voltage is due to the fact that the electrons are extracted with a larger force. However, at a higher bias voltage, the current is found to decay faster (see Fig. 10(c)). 4.3.2 Effects of non-uniform radius In this case, the uniform height and the uniform spacing between the neighboring CNTs are taken as h0 = 12µm and d1 = 2µm, respectively. Random distribution of radius is given with bounds 1.5−4nm. The simulated results are shown in Fig. 11. At the bias voltage of 440V , the average current decreases from 1.37× 10−8A at t = 1s to 1.23× 10−8A at t = 138s in steps and then the current stabilizes. The maximum current varies between 1.87× 10−8A to 1.72× 10−8A, whereas the minimum current varies between 2.53 × 10−9A to 2.52 × 10−9A. The average current decreases from 1.26× 10−5A to 1.08× 10−5A in steps when the bias voltage is increased from 440V to 550V . At a bias voltage of 660V , the average current decreases from 1.26×10−3A to 1.02 × 10−3A. As expected, a more fluctuation between the maximum and the minimum current have been observed here when compared to the case of uniform radius. 4.3.3 Effects of non-uniform height In this case, the uniform radius and the uniform spacing between neighboring CNTs are taken as r = 2.75nm and d1 = 2µm, respectively. Random initial distribution of the height is given with bounds 10 − 14µm. The simulated results are shown in Fig. 12. At the bias voltage of 440V , the average current decreases from 1.79×10−6A to 1.53×10−6A. The maximum current varies between 6.33×10−6A to 5.89×10−6A, whereas the minimum current varies between 2.69× 10−10A to 4.18× 10−10A. The average current decreases from 0.495 × 10−3A to 0.415 × 10−3A in steps when the bias voltage is increased from 440V to 550V . At the bias voltage of 660V , the average current decreases from 0.0231A to 0.0178A. The device response is found to be highly sensitive to the height distribution. 4.3.4 Effects of non-uniform spacing between neighboring CNTs In this case, the uniform height and the uniform radius of the CNTs are taken as h0 = 12µm and r = 2.75nm, respectively. Random distribution of spacing d1 between the neighboring CNTs is given with bounds 1.5− 2.5µm. The simulated results are shown in Fig. 13. At the bias voltage of 440V , the average current decreases from 1.37× 10−8A to 1.26× 10−8A. The maximum current varies between 1.89× 10−8A to 1.76 × 10−8A, whereas the minimum current varies between 2.86 × 10−9A to 2.61× 10−9A. The average current decreases from 1.24× 10−5A to 1.08× 10−5A in steps when the bias voltage is increased from 440V to 550V . At the bias voltage of 660V , the average current decreases from 1.266 × 10−3A to 1.040 × 10−3A. There is a slight increase in the order of magnitude of current for non-uniform spacing. It can attributed to the reduction in screening effect at some emitting sites in the film where the spacing is large. 5 Conclusions In this paper, we have developed a multiphysics based modelling approach to analyze the evolution of the CNT thin film. The developed approach has been applied to the simulation of the current-voltage characteristics at the device scale. First, a phenomenological model of degradation and fragmentation of the CNTs has been derived. From this model we obtain degraded state of CNTs in the film. This information, along with electromechanical force, is then employed to update the initially prescribed distribution of CNT geometries in a time incremental manner. Finally, the device current is computed at each time step by using the semi-empirical Fowler-Nordheim equation and integration over the computational cell surfaces on the anode side. The model thus handles several important effects at the device scale, such as fragmentation of the CNTs, formation of the carbon clusters, and self- assembly of the system of CNTs during field emission. The consequence of these effects on the I-V characteristics is found to be important as clearly seen from the simulated results which are in close agreement with experiments. Parametric studies reported in the concluding part of this paper indicate that the effects of the height of the CNTs and the spacing between the CNTs on the current history is significant at the fast time scale. There are several other physical factors, such as the thermoelectric heating, interaction between the cathode substrate and the CNTs, time-dependent electronic properties of the CNTs and the clusters, ballistic transport etc., which may be important to consider while improving upon the model developed in the present paper. 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Sci. 111, 145 (1997). [50] A. N. Obraztsov, A. P. Volkov and I. Pavlovsky, Diam. Rel. Mater. 9, 1190 (2000). Table 1: Summary of work function values for CNTs. Type of CNT Φ (eV ) Method SWNT 4.8 Ultraviolet photoelectron spectroscopy45 MWNT 4.3 Ultraviolet photoelectron spectroscopy46 MWNT 7.3±0.5 Field emission electronic energy distribution47 SWNT 5.05 Photoelectron emission48 MWNT 4.95 Photoelectron emission48 SWNT 1.1 Experiments49 MWNT 1.4 Experiments49 MWNT 0.2-1.0 Numerical approximation50 Figure 1: Schematic drawing of the CNT thin film for model idealization. (a) (b) Figure 2: Schematic drawing showing (a) hexagonal arrangement of carbon atoms in CNT and (b) idealization of CNT as a one-dimensional elastic member. Figure 3: CNT array configuration. Figure 4: Schematic description of neighboring CNT pair interaction for calculation of electrostatic force. Figure 5: Computational flow chart for calculating the device current. Figure 6: Variation of carbon cluster concentration over time. Initial condition: S(0) = 100, T = 303K, M1(0) = 2.12× 10−16, An(0) = 0. Figure 7: Variation of carbon cluster concentration over time. Initial condition: n1(0) = 100m −3, T = 303K, M1(0) = 2.12× 10−16, An(0) = 0. Figure 8: Distribution of tip angles over the number of CNTs. Figure 9: Comparison of simulated current-voltage characteristics with experiments. (a) (b) (c) Figure 10: Simulated current histories for uniform radius, uniform height and uni- form spacing of CNTs at a bias voltage of (a) 440 V, (b) 550 V, and (c) 660 V. (a) (b) (c) Figure 11: Simulated current histories for non-uniform radius of CNTs at a bias voltage of (a) 440 V, (b) 550 V, and (c) 660 V. (a) (b) (c) Figure 12: Simulated current histories for non-uniform height of CNTs at a bias voltage of (a) 440 V, (b) 550 V, and (c) 660 V. (a) (b) (c) Figure 13: Simulated current histories for non-uniform spacing between neighboring CNTs at a bias voltage of (a) 440 V, (b) 550 V, and (c) 660 V. Introduction Role of various physical processes in the degradation of CNT field emitter Model formulation Nucleation coupled model for degradation of CNTs Effect of CNT geometry and orientation Electromechanical forces Lorentz force Electrostatic force The van der Waals force Ponderomotive force Modelling the reorientation of CNTs Computational scheme Discretization of the nucleation coupled model for degradation Incremental update of the CNT geometry Computation of field emission current Results and discussions Degradation of the CNT thin films Current-voltage characteristics Field emission current history Effects of uniform height, uniform radius and uniform spacing Effects of non-uniform radius Effects of non-uniform height Effects of non-uniform spacing between neighboring CNTs Conclusions

0704.1682

Modeling the Field Emission Current Fluctuation in Carbon Nanotube Thin Films

arXiv:0704.1682v1 [cond-mat.mtrl-sci] 13 Apr 2007 Modeling the Field Emission Current Fluctuation in Carbon Nanotube Thin Films N. Sinha*, D. Roy Mahapatra**, J.T.W. Yeow* and R. Melnik*** * Department of Systems Design Engineering, University of Waterloo,Waterloo, ON, Canada ** Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India *** Mathematical Modeling and Computational Sciences,Wilfrid Laurier University, Waterloo, ON, Canada [email protected] ABSTRACT Owing to their distinct properties, carbon nanotubes (CNTs) have emerged as promising candidate for field emission devices. It has been found experimentally that the results related to the field emission performance show variability. The design of an efficient field emit- ting device requires the analysis of the variabilities with a systematic and multiphysics based modeling approach. In this paper, we develop a model of randomly oriented CNTs in a thin film by coupling the field emission phe- nomena, the electron-phonon transport and the mechan- ics of single isolated CNT. A computational scheme is developed by which the states of CNTs are updated in time incremental manner. The device current is calcu- lated by using Fowler-Nordheim equation for field emis- sion to study the performance at the device scale. Keywords: carbon nanotube, field emission, electro- dynamics, current density. 1 INTRODUCTION Field emission from carbon nanotubes (CNTs) was first reported in 1995 [1],[2]. With advancement in syn- thesis techniques, application of CNTs in field emis- sion devices, such as field emission displays, gas dis- charge tubes and X-ray tube sources has been success- fully demonstrated [3], [4]. Field emission performance of a single isolated CNT is found to be remarkable due to its structural integrity, geometry, chemical stability and high thermal conductivity. One can use a single CNT to produce an electron beam in a single electron beam device. However, in many applications (such as X-ray imaging systems), a continuous or patterned film is re- quired to produce several independent electron beams. However, the situation in these cases becomes complex due to coupling among (i) the ballistic electron- phonon transport at moderate to high temperature range, (ii) field emission from each of the CNT tip and (iii) electro- dynamic forces causing mechanical srain and deforming CNTs (and thus changing the electron density and dy- namic conductivity). In such cases, the individual CNTs are not always inclined normal to the substrate surface (as shown in Fig. 1, where CNT tips are oriented in a random manner). This is the most common situation, which can evolve from an initially ordered state of uni- formly distributed and vertically oriented CNTs. Such evolution process must be analyzed accurately from the view point of long-term performance of the device. The interest of the authors’ towards such an analysis and de- sign studies stem from the problem of precision biomed- ical X-ray generation. In this paper, we focus on a diode configuration, where the cathode contains a CNT thin film grown on a metallic substrate and the anode is a copper plate acting as emission current collector. Here, the most important requirement is to have a stable field emission current without compromising the lifetime of the device. As the CNTs in the film deplete with time, which is due to burning and fragmentation that result in a decreas- ing number of emitting sites, one observes fluctuation in the output current. Small spikes in the current have also been observed experimentally [5], which can gener- ally be attributed to the change in the gap between the elongated CNT tip and the anode, and also possibly a dynamic contact of pulled up tip with the anode under high voltage. As evident from the reported studies [5], it is important to include various coupled phenomena in a numerical model, which can then be employed to un- derstand the effects of various material parameters and also geometric parameters (e.g. CNT geometry as well as thin film patterns) on the collective field emission performance of the thin film device. Another aspect of interest in this paper is the effect of the angle of orien- tation of the CNT tips on the collective performance. A physics based modeling approach has been developed here to analyze the device-level performance of a CNT thin film. 2 MODEL FORMULATION The physics of field emission from metallic surfaces is fairly well understood. The current density (J) due to field emission from a metallic surface is usually obtained by using the Fowler-Nordheim (FN) equation [6] CΦ3/2 , (1) where E is the electric field, Φ is the work function of the cathode material, and B and C are constants. How- http://arxiv.org/abs/0704.1682v1 Figure 1: SEM image showing randomly oriented tips of CNTs in a thin film. E(x,y) Cathode Anode Figure 2: CNT array configuration. ever, in the case of a CNT thin film acting as cathode, the surface of the cathode is not smooth (like the metal emitters) and consists of hollow tubes in curved shapes and with certain spacings. An added complexity is the realignment of individual CNTs due to electrodynamic interaction between the neighbouring CNTs during field emission. Analysis of these processes requires the de- termination of the current density by considering the individual geometry of the CNTs, their dynamic ori- entations and the variation in the electric field during electronic transport. Based on our previously developed model [7], which describes the degradation of CNTs and the CNT geom- etry and orientation, the rate of degradation of CNTs is defined as vburn = Vcell dn1(t) s(s− a1)(s− a2)(s− a3) n2a21 +m 2a22 + nm(a 1 + a 2 − a where Vcell is the representative volume element, n1 is the concentration of carbon atoms in the cluster form in the cell, a1, a2, a3 are lattice constants, s = (a1 + a2 + a3),n and m are integers (n ≥ |m| ≥ 0). The pair (n,m) defines the chirality of the CNT. Therefore, at a given time, the length of a CNT can be expressed as h(t) = h0− vburnt, where h0 is the initial average height of the CNTs and d is the distance between the cathode substrate and the anode (see Fig. 2). The effective electric field component for field emis- sion calculation in Eq. (1) is expressed as Ez = −e −1 dV(z) , (3) where e is the positive electronic charge and V is the electrostatic potential energy. The total electrostatic potential energy can be expressed as V(x, z) = −eVs−e(Vd−Vs) G(i, j)(n̂j−n) , (4) where Vs is the constant source potential (on the sub- strate side), Vd is the drain potential (on the anode side), G(i, j) is the Green’s function [8] with i being the ring position, n̂j denotes the electron density at node po- sition j on the ring, and (n,m) denotes the chirality parameter of the CNT. The field emission current (Icell) from the anode surface associated with the elemental volume Vcell of the film is obtained as Icell = Acell Jj , (5) where Acell is the anode surface area and N is the num- ber of CNTs in the volume element. The toal current is obtained by summing the cell-wise current (Icell). This formulation takes into account the effect of CNT tip ori- entations, and one can perform statistical analysis of the device current for randomly distributed and randomly oriented CNTs. However, due to the deformation of the CNTs due to electrodynamic forces, the evolution pro- cess requires a much more detailed treatment from the mechanics point of view. Based on the studies reported in published literature, it is reasonable to expect that a major contribution is by the Lorentz force due to the flow of electron gas along the CNT and the ponderomotive force due to electrons in the oscillatory electric field . The oscillatory electric field could be due to hopping of the electrons along the CNT surfaces and the changing relative distances be- tween two CNT surfaces. In addition, the electrostatic force and the van der Waals force are also important. The net force components acting on the CNTs parallel to the Z and the X directions are calculated as [9] (flz + fvsz )ds+ fcz + fpz , (6) (flx + fvsx)ds+ fcx + fpx . (7) where fl, fvs, fc and fp are Lorentz, van der Waals, coulomb and ponderomotive forces, respectively and ds is the length of a small segment of CNTs. Next, we em- ploy these force components in the expression of work done on the ensemble of CNTs and formulate an en- ergy conservation law. Due to their large aspect ratio, the CNTs have been idealized as one-dimensional elastic members (as in Euler-Bernoulli beam). By introducing the strain energy density, the kinetic energy density and the work density, and applying the Hamilton principle, we obtain the governing equations in (ux′ , uz′) for each CNT, which can be expressed as +ρA0ü x′ −ρA2 πCvs[(r (m+1))2 −(r(m))2] θ(z′) − flx′ − fcx′ = 0 , (8) −E′A0 E′A0α ∂∆T (z′) + ρA0ü z′0 − πCvs [(r(m+1))2−(r(m))2] θ(z′) −flz′−fcz′ = 0 , where ux′ and uz′ are lateral and longitudinal dispace- ments of the oriented CNTs, E′ is the effective modulus of elasticity of CNTs, A0 is the effective cross-sectional area, A2 is the second moment of cross-sectional area about Z-axis, ∆T (z′) = T (z′)− T0 is the difference be- tween the absolute temperature (T ) during field emis- sion and a reference temperature (T0), α is the effective coefficient of thermal expansion (longitudinal), Cvs is the van der Waals coefficient, superscript (m) indicates the mth wall of the MWNT with r(m) as its radius, ∆x′ is the lateral displacement due to pressure and ρ is the mass per unit length of CNT. We assume fixed bound- ary conditions (u = 0) at the substrate-CNT interface (z = 0) and forced boundary conditions at the CNT tip (z = h(t)). The governing equation in temperature is obtained by the thermodynamics of electron-phonon interaction. By considering the Fourier heat conduction and ther- mal radiation from the surface of CNT, the energy rate balance equation in T can be expressed as dqF − πdtσSB(T 4 − T 40 )dz ′ = 0 , (10) where dQ is the heat flux due to Joule heating over a segment of a CNT, qF is the Fourier heat conduction, dt is the diameter of the CNT and σSB is the Stefan- Boltzmann constant. Here, we assume the emissivity to be unity. At the substrate-CNT interface (z′ = 0), the boundary condition T = T0 is applied and at the tip we assign a reported estimate of the power dissipated by phonons exiting the CNT tip [10] to the conductive flux. We first compute the electric field at the nodes and then solve all the governing equations simultaneously at each time step and the curved shape s(x′ + ux′ , z ′ + uz′) of each of the CNTs is updated. The angle of orientation θ between the nodes j+1 and j at the two ends of segment ∆sj is expressed as θ(t) = tan−1 (xj+1 + uj+1x )− (x j + ujx) (zj+1 + u z )− (zj + u , (11) = [Γ(θ(t−∆t)j)] , (12) where Γ is the usual coordinate transformation matrix which maps the displacements (ux′ , uz′) defined in the local (X ′, Z ′) coordinate system into the displacements (ux, uz) defined in the cell coordinate system (X,Z). For this transformation, we employ the angle θ(t −∆t) obtained at the previous time step and for each node j = 1, 2, 3, . . .. 3 RESULTS AND DISCUSSIONS The CNT film considered in this study consists of randomly oriented multiwalled CNTs. The film was grown on a stainless steel substrate. The film surface area (projected on anode) is 49.93 mm2 and the aver- age thickness of the film (based on randomly distributed CNTs) is 10-14 µm. In the simulation and analysis, the constants B and C in Eq. (1) were taken as B = (1.4× 10−6)× exp((9.8929)×Φ−1/2) and C = 6.5× 107, respectively [11]. It has been reported in the literature (e.g., [11]) that the work function Φ for CNTs is smaller than the work functions for metal, silicon, and graphite. However, there are significant variations in the exper- imental values of Φ depending on the types of CNTs (i.e., SWNT/MWNT), geometric parameters. The type of substrate materials have also significant influence on the electronic band-edge potential. The results reported in this paper are based on computation with Φ = 2.2eV . Following sample configuration has been used in this study: average height of CNTs h0 = 12µm, uniform diameter dt = 3.92nm and uniform spacing between neighboring CNTs at the substrate contact region in the film d1 = 2µm. The initial height distribution h and the orientation angle θ are randomly distributed. The elec- trode gap (d) is maintained at 34.7µm. The orientation of CNTs is parametrized in terms of the upper bound of the CNT tip deflection (denoted by h0/m ′, m′ >> 1). Several computational runs are performed and the out- put data are averaged out at each sampling time step. For a constant bias voltage (650V in this case), as the initial state of deflection of the CNTs increases (from h0/50 to h0/25), the average current reduces until the initial state of deflection becomes large enough that the electrodynamic interaction among CNTs produces sud- den pull in the deflected tips towards the anode result- ing in current spikes (see Fig. 3). As mentioned earlier, 0 20 40 60 80 100 Time (s) Figure 3: Field emission current histories for various initial average tip deflections and under bias voltage of 650V. The current I is in Ampere unit. 0 20 40 60 80 100 CNT number t=100 Figure 4: Comparison of tip orientation angles at t=0 and t=100s. 0 20 40 60 80 100 CNT number Figure 5: Maximum temperature of CNT tips during 100s of field emission. spikes in the current have also been observed experimen- tally. Fig. 4 reveals that after experiencing the elec- trodynamic pull and Coulombic repulsion, some CNTs reorient themselves. In Fig. 5, maximum tip tempera- ture distribution over an array of 100 CNTs during field emission over 100 s duration is plotted. The maximum temperature rises up to approximately 350 K. 4 CONCLUSION In this paper, a model has been developed from the device design point of view, which sheds light on the coupling issues related to the mechanics, the thermo- dynamics, and the process of collective field emission from CNTs in a thin film, rather than a single isolated CNT. The proposed modeling approach handles several complexities at the device scale. While the previous works by the authors mainly dealt with decay, kinemat- ics and dynamics of CNTs during field emission, this work includes some more aspects that were assumed constant earlier. These include: (i) non-local nature of the electric field, (ii) non-linear relationship between the electronic transport and the electric field, and (iii) non- linear relationship between the electronic transport and the heat conduction. The trend in the simulated results matches qualitatively well with the results of published experimental studies. REFERENCES [1] A.G. Rinzler, J.H. Hafner, P. Nikolaev, L. Lou, S.G. Kim, D. Tomanek, D. Colbert and R.E. Smalley, Science 269, 1550, 1995. [2] W.A. de Heer, A. Chatelain, and D. Ugrate, Science 270, 1179, 1995. [3] J.M. Bonard, J.P. Salvetat, T. Stockli, L. Forro and A. Chatelain, Appl. Phys. A 69, 245, 1999. [4] Y. Saito and S. Uemura, Carbon 38, 169, 2000. [5] J.M. Bonard, J.P. Salvetat, T. Stockli, L. Forro and A. Chatelain, Phys. Rev. B 67, 115406, 2003. [6] R.H. Fowler and L. Nordheim, Proc. Royal Soc. London A 119, 173, 1928. [7] N. Sinha, D. Roy Mahapatra, J.T.W. Yeow, R. Mel- nik and D. A. Jaffray, Proc. IEEE Nano 2006. [8] A. Svizhenko, M.P. Anantram and T.R. Govindan, IEEE Trans. Nanotech. 4, 557, 2005. [9] N. Sinha, D. Roy Mahapatra, J.T.W. Yeow, R. Mel- nik and D. A. Jaffray, J. comp. Theor. Nanosci. (Accepted). [10] H.-Y. Chiu, V.V. Deshpande, H.W.Ch. Postma, C.N. Lau, C. Mikó, L. Forró and M. Bockrath, Phys. Rev. Lett. 95, 226101, 2005. [11] Z.P. Huang, Y. Tu, D.L. Carnahan and Z.F. Ren, “Field emission of carbon nanotubes,” Encyclope- dia of Nanoscience and Nanotechnology (Ed. H.S. Nalwa) 3, 401-416, 2004.

0704.1683

Spectral averaging for trace compatible operators

SPECTRAL AVERAGING FOR TRACE COMPATIBLE OPERATORS N.A.AZAMOV AND F.A. SUKOCHEV Abstract. In this note the notions of trace compatible operators and infin- itesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Krein’s formula is es- tablished. Some examples of trace compatible affine spaces of operators are given. 1. Introduction Let H0 be a self-adjoint operator, and let V be a trace class operator on a Hilbert space H. Then M.G.Krĕın’s famous result [13] says that there is a unique L1 -function ξH0+V,H0(λ), known as the Krein spectral shift function, such that for any C∞c (R) function f Tr(f(H0 + V )− f(H0)) = f ′(λ)ξH0+V,H0(λ) dλ.(1) The notion of the spectral shift function was discovered by the physicist I.M. Lifshits [15]. An excellent survey on the spectral shift function can be found in [6]. In 1975, Birman and Solomyak [7] proved the following remarkable formula for the spectral shift function ξ(λ) = Tr(V EHr (−∞,λ]) dr,(2) where Hr = H0 + rV, r ∈ R, and EHr(−∞,λ] is the spectral projection. Birman- Solomyak’s proof relies on double operator integrals. An elementary derivation of (2) was obtained in [10] (without using double operator integrals). Actually, this spectral averaging formula was discovered for the first time by Javrjan [12] in 1971, in case of a Sturm-Liouville operator on a half-line, perturba- tion being a perturbation of the boundary condition, so that in this case V was one-dimensional. An important contribution to spectral averaging was made by A.B.Alexandrov [1]. In 1998, B. Simon [18, Theorem 1] gave a simple short proof of the Birman-Solomyak formula. He also noticed, that this formula holds for the wide class of Schrödinger operators on Rn [18, Theorems 3,4]. The connection of 1991 Mathematics Subject Classification. Primary 47A11; Secondary 47A55 . Key words and phrases. Spectral shift function, spectral averaging, infinitesimal spectral flow, trace compatible operators, semifinite von Neumann algebra . http://arxiv.org/abs/0704.1683v2 2 N.A.AZAMOV AND F.A. SUKOCHEV this formula with the integral formula for spectral flow from non-commutative ge- ometry is outlined in [3]. An interesting approach to spectral averaging via Herglotz functions can be found in [11]. In this note we present an alternative viewpoint to the spectral shift function, and generalize the result of Simon so that it becomes applicable to a class of Dirac operators as well. The new point of view, which the Birman-Solomyak formula suggests, is that there is a more fundamental notion than that of the spectral shift function. We call this notion the speed of spectral flow or infinitesimal spectral flow of a self- adjoint operator H under perturbation by a bounded self-adjoint operator V. It was introduced in [3] in the case of operators with compact resolvent. It is defined by formula ΦH(V )(ϕ) = Tr(V ϕ(H)), ϕ ∈ C∞c (R),(3) whenever this definition makes sense. This naturally leads to the notion of trace compatibility of two operators. We say that operators H and H + V are trace compatible, if for all ϕ ∈ C∞c (R) the operator V ϕ(H) is trace class. The spectral shift function between two trace compatible operators is then considered as the integral of infinitesimal spectral flow. It turns out that the spectral shift function does not depend on the path connecting the initial and final operators, a fact which follows from the aforementioned result of B. Simon in the case of trace class perturbations. The results of this note are summarized in Theorem 2.9. This theorem extends formulae (1) and (2) to the setting of trace compatible pairs (H,H + V ) and also strengthens [18, Theorems 3,4] in the sense that it does not require H to be a positive operator and maximally weakens conditions on the path H+rV, r ∈ [0, 1]. Our results also hold for a more general setting, when H = H∗ is affiliated with a semifinite von Neumann algebra N and V = V ∗ ∈ N . Our investigation here also strengthens the link between the theory of the Krein spectral shift function and that of spectral flow firstly discovered in [2]. For exposi- tion of the latter theory we refer to [5] and a detailed discussion of the connection between the two theories in the situation where the resolvent of H is τ -compact (here, τ is an arbitrary faithful normal semifinite trace on N ) is contained in [3]. It should be pointed out here that the idea of viewing the spectral shift function as the integral of infinitesimal spectral flow is akin to I.M. Singer’s ICM-1974 pro- posal to define the η invariant (and hence spectral flow) as the integral of a one form. Very general formulae of that type have been produced in the framework of noncommutative geometry (see [5] and references therein). We believe that our present approach will have applications to noncommutative geometry, in particular, it may be useful in avoiding ”summability constraints” on H customarily used in that theory. In semifinite von Neumann algebras N Krein’s formula (1) was proved for the first time in [9] in case of a bounded self-adjoint operator H ∈ N and a trace class perturbation V = V ∗ ∈ L1(N , τ) and in [4] for self-adjoint operators H affiliated with N . SPECTRAL AVERAGING 3 An additional reason to call ΦH(V ) the speed of spectral flow is the follow- ing observation. Let H be the operator of multiplication by λ on L2(R, dρ(λ)) with some measure ρ and let the perturbation V be an integral operator with a sufficiently regular (for example C1 ) kernel k(λ′, λ). Then for any test function ϕ ∈ C∞c (R) ΦH(V )(ϕ) = Tr(V ϕ(H)) = k(λ, λ)ϕ(λ) dρ(λ). Hence, the infinitesimal spectral flow of H under perturbation by V is the measure on the spectrum of H with density k(λ, λ) dρ(λ). We note that this agrees with the classical formula [14, (38.6)] n = Vnn from formal perturbation theory. Here E n is the n -th eigenvalue of the unper- turbed operator H0, E n , j = 1, 2, . . . is the j -th correction term for the n -th eigenvalue En of the perturbed operator H = H0 + V in the formal perturbation series En = E n + E n + E n + . . . , and Vmn = m |V |ψ(0)n is the matrix element of the perturbation operator V with respect to the eigenfunctions ψ and ψ n of the unperturbed operator H0 [14]. Acknowledgement. We thank Alan Carey for useful comments and criticism. 2. Results Let N be a von Neumann algebra on a Hilbert space H with faithful normal semifinite trace τ. Let A = H0 + A0 be an affine space of self-adjoint operators affiliated with N , where H0 is a self-adjoint operator affiliated with N and A0 is a vector subspace of the real Banach space of all self-adjoint operators from N . We say that A is trace compatible, if for all ϕ ∈ C∞c (R), V ∈ A0 and H ∈ A V ϕ(H) ∈ L1(N , τ),(4) where L1(N , τ) is the ideal of trace class operators from N , and if A0 is endowed with a locally convex topology which coincides with or is stronger than the uniform topology, such that the map (V1, V2) ∈ A20 7→ V1ϕ(H0 + V2) is L1 continuous for all H0 ∈ A and ϕ ∈ C∞c (R). In particular, A is a locally convex affine space. The ideal property of L1(N , τ) and [16, Theorem VIII.20(a)] imply that, in the definition of trace compatibility, the condition ϕ ∈ C∞c (R) may be replaced by ϕ ∈ Cc(R). It follows from the definition of the topology on A0 that H ∈ A 7→ eitH is norm continuous. If A = H0+A0 is a trace compatible affine space then we define a (generalized) one-form (on A ) of infinitesimal spectral flow or speed of spectral flow by the formula ΦH(V ) = τ(V δ(H)), H ∈ A, V ∈ A0,(5) where δ is Dirac’s delta function. The last formula is to be understood in a generalized function sense, i.e. ΦH(V )(ϕ) = τ(V ϕ(H)), ϕ ∈ C∞c (R). Φ is a generalized function, since if ϕn → 0 in C∞c (R) such that supp(ϕn) ⊆ ∆ then |τ(V ϕn(H))| 6 ∥V EH∆ ‖ϕn(H)‖ → 0. Here ‖A‖1 = τ(|A|). 4 N.A.AZAMOV AND F.A. SUKOCHEV Since ϕ can be taken from Cc(R), for each V ∈ A0 the infinitesimal spectral flow ΦH(V ) is actually a measure on the spectrum of H. By a smooth path {Hr}r∈R in A, we mean a differentiable path, such that its derivative dHr ∈ A0 is continuous. Let Π = (s0, s1) ∈ R2 : s0s1 > 0, |s1| 6 |s0| , and let dνf (s0, s1) = sgn(s0) f̂(s0) ds0 ds1. If f ∈ C2c (R) then (Π, νf ) is a finite measure space [2]. For any H0, H1 ∈ A, any X ∈ A0 and any non-negative f ∈ C∞c (R) set by definition (6) T H1,H0 f [1] (X) = ei(s0−s1)H1 f(H1)Xe is1H0 + ei(s0−s1)H1X f(H0)e is1H0 dν√f (s0, s1), where the integral is taken in the so∗ -topology. For justification of this notation and details see [3]. Lemma 2.1. If {Hr} ⊂ A is a path, continuous (smooth) in the topology of A0, and if f ∈ C2c (R) then r 7→ f(Hr)− f(H0) takes values in L1(N , τ) and it is L1(N , τ) continuous (smooth). Proof. We can assume that f is non-negative and that f ∈ C2c (R). It is proved in [3] that f(Hr)− f(H0) = THr ,H0f [1] (Hr −H0).(7) Since ei(s0−s1)x f(x), eis1x f(x) ∈ C2c (R), trace compatibility implies that the integrand of the right hand side of (6) takes values in L1 and is L1 -continuous (smooth), so the dominated convergence theorem completes the proof. � If Γ = {Hr}r∈[0,1] is a smooth path in A, then we define the spectral shift function ξ along this path as the integral of infinitesimal spectral flow: ξ = ξ(ϕ) = ϕ(Hr) dr, ϕ ∈ C∞c (R).(8) Now we prove that the spectral shift function is well-defined in the sense that it does not depend on the path of integration. A one-form αH(V ) on an affine space A is called exact if there exists a zero- form θH on A such that dθ = α, i.e. αH(V ) = θH+rV We say that the generalized one-form Φ is exact if Φ(ϕ) is an exact form for any ϕ ∈ C∞c (R). SPECTRAL AVERAGING 5 The proof of the following proposition follows the lines of the proof of [3, Propo- sition 3.5]. Proposition 2.2. The infinitesimal spectral flow Φ is exact. Proof. Let V ∈ A0, Hr = H0 + rV, r ∈ [0, 1], and let f ∈ C∞c (R). By (7) (9) f(Hr)− f(H0) = THr ,H0f [1] (rV ) ei(s0−s1)Hr f(Hr)rV e is1H0 + ei(s0−s1)HrrV f(H0)e is1H0 dν√f (s0, s1), ei(s0−s1)H0 f(H0)rV e is1H0 + ei(s0−s1)H0rV f(H0)e is1H0 dν√f (s0, s1) (ei(s0−s1)Hr f(Hr)− ei(s0−s1)H0 f(H0))rV e is1H0 + (ei(s0−s1)Hr − ei(s0−s1)H0)rV f(H0)e is1H0 dν√f (s0, s1) H0,H0 f [1] (rV ) +R1 +R2. All three summands here are trace class by the trace compatibility assumption. So, for any S ∈ N τ(S(f(Hr)− f(H0))) = rτ H0,H0 f [1] + τ(SR1) + τ(SR2). Now, Duhamel’s formula and (7) show that τ(SR1) = o(r) and τ(SR2) = o(r). Hence, τ(S(f(Hr)− f(H0))) = τ Hr,Hr f [1] .(10) This implies that for any S ∈ N τ(S(f(H1)− f(H0))) = τ Hr ,Hr f [1] (V ) dr Now let H0 ∈ A be a fixed operator and for any f ∈ C∞c (R) let τ(V f(Hr)) dr, where Hr = H0 + rV, H = H1. We are going to show that dθ H(X) = ΦH(X)(f) for any X ∈ A0. Following the proof of [3, Proposition 3.5] we have (A) := dθ H(X) = lim Xf(Hr + srX) + lim f(Hr + srX)− f(Hr) By definition of A0 topology the integrand of the first summand is continuous with respect to r and s. So, the first summand is equal to Xf(Hr) 6 N.A.AZAMOV AND F.A. SUKOCHEV By [2, Theorem 5.3] the second summand is equal to Hr+srX,Hr f [1] (srX) dr = lim Hr+srX,Hr f [1] Hr,Hr f [1] Hr ,Hr f [1] r dr, where the second equality follows from the definition of A0 -topology and the last equality follows from [3, Lemma 3.2]. Now, using (10) and integrating by parts we (11) (A) = τ (Xf(H1)−Xf(H0)) + (τ(Xf(Hr))− τ(X [f(Hr)− f(H0)])) dr = τ (Xf(H1)) . The argument before [8, Proposition 1.5] now implies Corollary 2.3. The spectral shift function given by (8) is well-defined. Proposition 2.4. If r ∈ R 7→ Hr ∈ A is smooth then the equality df(Hr) ϕ(Hr) ′(Hr)ϕ(Hr) holds for any f ∈ C2c (R) and any bounded measurable function ϕ. Proof. Without loss of generality, we can assume that f > 0 and f ∈ C2c (R). We prove the above equality at r = 0. The formula (7) and the dominated convergence theorem imply that df(Hr) ϕ(H0) ϕ(H0) ei(s0−s1)Hr f(Hr) Hr −H0 eis1H0 + ei(s0−s1)Hr Hr −H0 f(H0)e is1H0 dν√f (s0, s1) where the limit is taken in L1(N , τ). By the A0 -smoothness of {Hr} , we have df(Hr) ϕ(H0) ϕ(H0) ei(s0−s1)H0 f(H0)Ḣr eis1H0 + ei(s0−s1)H0Ḣr f(H0)e is1H0 dν√f (s0, s1) so that by [2, Lemmas 3.7, 3.10] and letting A = Ḣr df(Hr) ϕ(H0) ϕ(H0)e is0H0 f(H0)A dν√f (s0, s1) Aϕ(H0) f(H0) eis0H0 f)(s0) ds0 Aϕ(H0)f ′(H0) SPECTRAL AVERAGING 7 Proposition 2.5. The spectral shift function given by (8) satisfies Krein’s formula, i.e. for any f ∈ C∞c , H0, H1 ∈ A τ (f(H1)− f(H0)) = ξ(f ′). Proof. Taking the integral of (12) with ϕ = 1 we have d(f(Hr)− f(H0)) ′(Hr) The right hand side is ξ(f ′) by definition. It follows from Lemma 2.1 that one can interchange the trace and the derivative in the left hand side. � Corollary 2.6. In case of trace class perturbations, the spectral shift function ξ defined by (8) coincides with classical definition, given by [4, Theorem 3.1]. Proof. This follows from Theorem 6.3 and Corollary 6.4 of [2]. � For trace class perturbations the absolute continuity of the spectral shift function is established in [13] (see also [11]). For the general semifinite case we refer to [4, 2]. Lemma 2.7. Let A be a trace compatible affine space and let f ∈ C∞c (R). Let H0, H1 ∈ A, let ξ and ξf be the spectral shift distributions of the pairs (H0, H1) and f(H0), f(H1) respectively. Then for any ϕ ∈ C∞c (R) ξf (ϕ) = ξ(ϕ ◦ f · f ′).(13) Proof. By Proposition 2.4 for any f, ϕ ∈ C∞c (R) df(Hr) ϕ(f(Hr)) ′(Hr)ϕ(f(Hr)) Ḣr(F ◦ f)′(Hr) where F ′ = ϕ. Hence, for any smooth path Γ = {Hr}r∈[0,1] ⊆ A df(Hr) ϕ(f(Hr)) Ḣr(F ◦ f)′(Hr) dr,(14) which is (13). � Proposition 2.8. Let A = H0 + A0 be a trace compatible affine space and let A0 be such that for any V ∈ A0 there exist positive V1, V2 ∈ A such that V = V1−V2. Then the spectral shift function ξ of any pair H,H+V ∈ A is absolutely continuous. Proof. Since the map (V1, V2) 7→ τ (V1ϕ(H + V2)) is L1(N , τ) -continuous (by definition), it follows that the infinitesimal spectral flow is a uniformly locally finite measure with respect to the path parameter. Hence, the spectral shift function is also a locally finite measure being the integral of locally finite measures, which are uniformly bounded on every segment. If, for H0, H1, H2 ∈ A, the spectral shift functions from H0 to H1 and from H1 to H2 are absolutely continuous, then evidently the spectral shift function from H0 to H2 is also absolutely continuous. Hence, if V = V1 − V2 with 0 6 V1, V2 ∈ A0, then representing the spectral shift function from H to H + V as the sum of 8 N.A.AZAMOV AND F.A. SUKOCHEV spectral shift function from H to H + V1 and from H + V1 to H + V, we see that we can assume that the perturbation V is positive. By Lemma 2.1 and [4, Theorem 3.1] the spectral shift function ξf of the pair (f(H), f(H + V )) is absolutely continuous. Let us suppose that the spectral shift function ξ of the pair (H,H + V ) has non-absolutely continuous part µ. Without loss of generality, we can assume that there exists a set of Lebesgue measure zero E ⊂ (ε, 1− ε) such that µ(E) > 0. For any a, b ∈ R with b− a > 2 let us consider a ”cap”-function fa,b, i.e. f is a smooth function which is zero on (−∞, a) and (b,∞), it is 1 on (a+1, b−1) and its derivatives on (a+ε, a+1−ε) and (b− 1 + ε, b− ε) is 1 and −1 respectively. Let U be an open set of Lebesgue measure < δ such that E ⊂ U and let ϕ be a smoothed indicator of U. Then (13), applied to functions ϕ and f0,b and to functions ϕ and fa,b (with big enough b ) implies that µ(E) = µ(a+ E), i.e. µ is translation invariant. Since it is also locally finite it is some multiple of Lebesgue measure. This yields a contradiction. � We summarize the results in the following theorem. Theorem 2.9. Let A be a trace compatible affine space of operators in a semifinite von Neumann algebra N with a normal semifinite faithful trace τ. Let H and H+ V be two operators from A. Let the spectral shift (generalized) function ξH,H+V be defined as the integral of infinitesimal spectral flow by the formula ξH,H+V (ϕ) = Φ(ϕ) = ΦHr (Ḣr)(ϕ) dr, ϕ ∈ C∞c , where Γ = {Hr}r∈[0,1] is any piecewise smooth path in A connecting H and H+V. Then the spectral shift function is well-defined in the sense that the integral does not depend on the choice of the piecewise smooth path Γ connecting H and H + V, and it satisfies Krein’s formula τ(f(H + V )− f(H)) = ξ(f ′), f ∈ C∞c . Moreover, if for any V ∈ A0 there exist V1, V2 ∈ A0 such that V = V1 −V2, then ξH,H+V is an absolutely continuous measure. Two extreme examples of trace compatible affine spaces are H0 + L1sa(N , τ), H0 = H 0ηN , with the topology induced by L1(N , τ) [2, 4], and D0 + Nsa, where (D0 − i)−1 is τ -compact, with the topology induced by operator norm [3]. In particular, the space −∆+C(M), where (M, g) is a compact Riemannian manifold, and ∆ is the Laplacian, is trace compatible. As an example of an intermediate trace compatible affine space one can consider Schrödinger operators −∆+ Cc(Rn) with the inductive topology of uniform con- vergence. It is proved in [19, Section B9] that for this example the condition (4) holds. It also follows from [19, Section B9] that ‖gf(H)‖1 6 C ‖g‖2 , where C depends only on f, on the support of g and on ‖V−‖∞ , where V− is the negative part of V, H = −∆+V. So, the condition on the topology of A0 is fulfilled by (7). SPECTRAL AVERAGING 9 Another example is given by Dirac operators of the form D +A0, where D = , α1, . . . , αn are m×m -matrices such that αjαk + αkαj = −2δjk, and a = a∗ ∈ Cc(Rn,Mm(R)) : ∃ϕ = ϕ∗ ∈ C1(Rn) iDϕ = a with the inductive topology of uniform convergence. A proof that the space D+A0 is trace compatible can be reduced to [17, Theorem 4.5] via the gauge transforma- tion ψ 7→ e−iϕ(x)ψ. We have (D + a)(e−iϕ(x)u) = −ie−iϕ(x) ∂ ϕ(x)αju+ e −iϕ(x)αj + ae−iϕ(x)u, where u is an m -column of C∞c -functions. So, if iDϕ = a then eiϕ(x)(D + a)(e−iϕ(x)u) = Du. Hence, (D+a)2 = e−iϕ(x)D2eiϕ(x). This shows that gf((D+a)2), g, a ∈ A0, f ∈ C∞c (R), is trace class iff gf(e −iϕ(x)D2eiϕ(x)) = e−iϕ(x)geiϕ(x)f(e−iϕ(x)D2eiϕ(x)) is trace class. But the last operator is unitarily equivalent to gf(D2), which is trace class by [17, Theorem 4.5]. So, if f > 0 then by the same theorem ‖gf(D + a)‖1 6 C ‖g‖∞ ‖f‖∞ ,(15) where C depends on supports of g and f. Hence, for g, g1, a, a1 ∈ A0, we have ‖gf(D + a)− g1f(D + a1)‖1 6 ‖(g − g1)f(D + a)‖1 + ‖g1(f(D + a)− f(D + a1)‖1 . So, the condition on the topology of A0 is fulfilled by (7) and (15). In case n = m = 1, we have D +A0 = 1i + Cc(R). References [1] A.B.Alexandrov, The multiplicity of the boundary values of inner functions, Sov. J. Comtemp. Math. Anal. 22 5 (1987), 74–87. [2] N.A.Azamov, A. L.Carey, P.G.Dodds, F.A. Sukochev, Operator integrals, spectral shift and spectral flow, to appear in Canad. J. Math, arXiv:math/0703442. [3] N.A.Azamov, A. L.Carey, F.A. Sukochev, The spectral shift function and spectral flow, to appear in Comm. Math. Phys., arXiv: 0705.1392. [4] N.A.Azamov, P.G.Dodds, F. A. Sukochev, The Krein spectral shift function in semifinite von Neumann algebras, Integral Equations Operator Theory 55 (2006), 347–362. [5] M-T.Benameur, A. L.Carey, J. Phillips, A.Rennie, F. A. Sukochev, K.P.Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, Analysis, geometry and topology of elliptic operators, 297–352, World Sci. Publ., Hackensack, NJ, 2006. [6] M. Sh. Birman, A.B. Pushnitski, Spectral shift function, amazing and multifaceted. Dedicated to the memory of Mark Grigorievich Krein (1907–1989), Integral Equations Operator Theory 30 (1998), 191–199. [7] M. Sh. Birman, M.Z. Solomyak, Remarks on the spectral shift function, J. Soviet math. 3 (1975), 408–419. [8] A. L. Carey, J. Phillips, Unbounded Fredholm modules and spectral flow, Canad. J. Math. 50 (1998), 673–718. [9] R.W.Carey, J. D.Pincus, Mosaics, principal functions, and mean motion in von Neumann algebras, Acta Math. 138 (1977), 153–218. [10] F.Gesztesy, K.A.Makarov, A.K.Motovilov, Monotonicity and concavity properties of the spectral shift function, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999), CMS Conf. Proc., 29, Amer. Math. Soc., Providence, RI, 2000, 207–222. http://arxiv.org/abs/math/0703442 10 N.A.AZAMOV AND F.A. SUKOCHEV [11] F.Gesztesy, K.A.Makarov, SL2(R), exponential Herglotz representations, and spectral aver- aging, Algebra i Analiz 15 (2003), 393–418. [12] V.A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), 246–251. [13] M.G.Krĕın, On the trace formula in perturbation theory, Mat. Sb., 33 75 (1953), 597–626. [14] L.D. Landau, E.M.Lifshitz, Quantum mechanics, 3rd edition, Pergamon Press. [15] I.M. Lifshits, On a problem in perturbation theory, Uspekhi Mat. Nauk 7 (1952), 171-180 (Russian). [16] M.Reed, B. Simon, Methods of modern mathematical physics: 1. Functional analysis, Aca- demic Press, New York, 1972. [17] B. Simon, Trace ideals and their applications, London Math. Society Lecture Note Series, 35, Cambridge University Press, Cambridge, London, 1979. [18] B. Simon, Spectral averaging and the Krein spectral shift, Proc. Amer. Math. Soc. 126 (1998), 1409–1413. [19] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526. School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042, SA Australia. E-mail address: [email protected], [email protected] 1. Introduction 2. Results References

0704.1684

The molecular environment of massive star forming cores associated with Class II methanol maser emission

Astrophysical masers and their environments Proceedings IAU Symposium No. 242, 2007 A.C. Editor, B.D. Editor & C.E. Editor, eds. c© 2007 International Astronomical Union DOI: 00.0000/X000000000000000X The molecular environment of massive star forming cores associated with Class II methanol maser emission S. N. Longmore1,2†, M. G. Burton1, P. J. Barnes3, T. Wong1,2,5, C. R. Purcell1,4, J. Ott2,6 1School of Physics, University of New South Wales, Kensington, NSW 2052, Sydney, Australia 2Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia 3School of Physics A28, University of Sydney, NSW 2006, Australia 4University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire SK11 9DL, UK 5Department of Astronomy, University of Illinois, Urbana IL 61801, USA 6National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA Abstract. Methanol maser emission has proven to be an excellent signpost of regions under- going massive star formation (MSF). To investigate their role as an evolutionary tracer, we have recently completed a large observing program with the ATCA to derive the dynamical and physical properties of molecular/ionised gas towards a sample of MSF regions traced by 6.7GHz methanol maser emission. We find that the molecular gas in many of these regions breaks up into multiple sub-clumps which we separate into groups based on their association with/without methanol maser and cm continuum emission. The temperature and dynamic state of the molecular gas is markedly different between the groups. Based on these differences, we attempt to assess the evolutionary state of the cores in the groups and thus investigate the role of class II methanol masers as a tracer of MSF. Keywords. stars: formation, ISM: evolution, ISM: molecules, line: profiles, masers, molecular data, stars: early-type, radio continuum: stars 1. Introduction In terms of luminosity, energetics and chemical enrichment, massive stars exert a dis- proportionate influence compared to their number on Galactic evolution. However, the collective effects of their rarity, short formation timescales and heavy obscuration due to dust, make it difficult to find large samples of massive young sources at well constrained evolutionary stages needed to develop an understanding of their formation mechanism. The 51→ 60 A + class II methanol (CH3OH) maser transition at 6.7GHz is one of the most readily observable signposts of MSF [Menten 1991]. The specific conditions required for the masers to exist makes them a powerful probe of the region’s evolutionary stage. While masers probe spatial scales much smaller than their natal cores, the numerous feedback processes from newly formed stars [e.g. turbu- lence injection from jets/outflows, ionisation (stars M>8M⊙) and heating from radiation etc.] must significantly alter the physical conditions of the surrounding region. We have completed an observing program with the Australia Telescope Compact Array (ATCA) to derive properties of molecular and ionised gas towards MSF regions traced by 6.7GHz methanol maser emission [Longmore et al. (2007)]. In this contribution, we use these re- † E-mail:[email protected] http://arxiv.org/abs/0704.1684v1 120 S. N. Longmore et al. sults to investigate the use of class II methanol masers as a diagnostic of the evolutionary stage of MSF. 2. Observations Observations of NH3(1,1), (2,2), (4,4) & (5,5) and 24GHz continuum were carried out using the ATCA towards 21 MSF regions traced by 6.7GHz methanol maser emission [selected from a similar sample to Hill et al. (this volume)]. The H168 [NH3(1,1)/(2,2)] and H214 [NH3(4,4)/(5,5)] antenna configurations with both East-West and North-South baselines, were used to allow for snapshot imaging. Primary and characteristic synthesised beam sizes were ∼2.′2 and ∼8 − 11′′ respectively. Each source was observed for 4×15 minute cuts in each transition separated over 8 hours to ensure the best possible sampling of the uv-plane. The data were reduced using the MIRIAD (see Sault et al. 1995) package. Characteristic spectra were extracted at every transition for each core at the position of the peak NH3(1,1) emission, baseline subtracted and fit using the gauss and nh3(1,1) methods in CLASS (see http://www.iram.fr/IRAMFR/GILDAS/). Continuum source fluxes and angular sizes were calculated in both the image domain and directly from the uv data. 3. Core separation NH3 detections within each region were separated into individual cores if offset by more than a synthesised beam width spatially, or more than the FWHM in velocity if at the same sky position. The same criteria were used to determine whether the NH3, continuum and methanol maser emission in each of the regions were associated. In all but three cases, these criteria were sufficient to both unambiguously separate cores and determine their association with continuum and maser emission. We find 41 NH3(1,1) cores (of which 3 are in absorption and 2 are separated in velocity) and 14 24GHz continuum cores. Observationally the cores fall in to 4 groups: NH3 only (Group 1); NH3 + methanol maser (Group 2); NH3 + methanol maser + 24GHz continuum (Group 3); NH3 + 24GHz continuum (Group 4). The cores were distributed with 16, 16, 6 and 2 cores in Groups 1 to 4, respectively. Based on this grouping, most of the NH3(1,1) cores are coincident with methanol maser emission (Groups 2 & 3), but there are a substantial fraction of NH3 cores with neither 24GHz continuum nor maser emission (Group 1). Having separated the cores into these groups, we then considered observational biases and selection effects which may affect their distribution. The biggest potential hindrance was the difference in linear resolution and sensitivity caused by the factor of ∼5 varia- tion in distance to the regions. Despite this, the NH3, continuum and methanol maser observations have the same sensitivity limit towards all the regions: therefore, the rela- tive flux densities of these tracers in a given region are directly comparable. In addition, no correlation was found between a region’s distance and the number of cores toward the region or their association with the different tracers. From this we conclude the dis- tance variation does not affect the distribution of cores into separate groups. However, it should be remembered that any conclusions drawn about the cores are limited by the observational parameters used to define the groups. 4. Deriving physical properties Properties of the molecular gas in each of the cores were derived from the NH3 obser- vations. The core size was calculated from the extent of the integrated NH3(1,1) emission http://www.iram.fr/IRAMFR/GILDAS/ The molecular environment associated with Class II methanol masers 121 after deconvolving the synthesised beam response. The dynamical state of the molecu- lar gas was derived from the line profiles of the high spectral resolution (0.197kms−1) NH3(1,1) observations after deconvolving the instrumental response. Preliminary gas kinetic temperatures were calculated by fitting the measured column densities of the multiple NH3 transitions to the LVG models described in Ott et al. (2005). Finally, properties of the ionised gas were derived from the 24GHz continuum emission follow- ing Mezger & Henderson (1967), assuming it was spherically symmetric and optically thin at an electron temperature of 104K. 5. Results In general the core properties are comparable to those derived from similar surveys towards young MSF regions. Below we outline differences, in particular between the cores in the different groups described in §3. 5.1. Molecular Gas The measured NH3(1,1) linewidth varies significantly between the groups, increasing from 1.43, 2.43, 3.00 kms−1 for Groups 1 to 3 respectively. This shows the NH3-only cores are more quiescent than those with methanol maser emission. The NH3(1,1) spectra of some cores deviate significantly from the predicted sym- metric inner and outer satellite brightness temperature ratios. These line profile asym- metries are often seen toward star forming cores and are understood to be caused by selective radiative trapping due to multiple NH3(1,1) sub-clumps within the beam [see Stuzki & Winnewisser (1985) and references therein]. The NH3-only cores (Group 1) have by far the strongest asymmetries. NH3(4,4) emission is detected toward the peak of 13 NH3(1,1) cores and 11 of these also have coincident NH3(5,5) emission. The higher spatial resolution of the NH3(4,4) and (5,5) images compared to the NH3(1,1) observations (8 ′′ vs 11′′) provides a stronger constraint to the criteria outlined in §3 as to whether this emission is associated with either methanol or continuum emission. In every case, the NH3(4,4) and (5,5) emission is unresolved, within a synthesised beam width of the methanol maser emission spatially and within the FWHM in velocity. This shows the methanol masers form at the warmest parts of the core. Significantly, no NH3(4,4) or (5,5) emission is detected toward NH3-only sources. As shown in Figure 1, cores with NH3 and methanol maser emission (Groups 2 and 3) are generally significantly warmer than those with only NH3 emission (Group 1). However, there are also a small number of cores with methanol maser emission that have very cool temperatures and quiescent gas, similar to the NH3-only cores. Modelling shows pumping of 6.7GHz methanol masers requires local temperatures sufficient to evaporate methanol from the dust grains (T&90K) and a highly luminous source of IR photons [Cragg et al. (2005)] i.e. an internal powering source. It is therefore plausible that the cold, quiescent sources with methanol maser emission are cores in which the feedback from the powering sources have not had time to significantly alter the larger scale properties of the gas in the cores. 5.2. Ionised Gas Of the 14 continuum cores detected at 24GHz, 10 are within two synthesised beams of the 6.7GHz methanol maser emission. This is contrary to the results of Walsh et al. (1998), who found the masers generally unassociated with 8GHz continuum emission. However, six of the 24GHz continuum sources found at the site of the methanol maser emission 122 S. N. Longmore et al. Figure 1. NH3(1,1) linewidth vs gas kinetic temperature. Cores with NH3 only (Group 1) are shown as crosses while those with NH3 and methanol maser emission (Groups 2 and 3) are shown as triangles. The dashed line shows the expected linewidth due to purely thermal motions. have no 8GHz counterparts. A possible explanation for this may be that the continuum emission is optically thick rather than optically thin between 8 and 24GHz and hence has a flux density proportional to ν2 rather than ν−0.1. The seemingly low emission mea- sures derived for the 24GHz continuum are unreliable due to the large beam size of the observations. Alternatively, the 24GHz continuum sources may have been too extended and resolved-out by the larger array configuration used at 8GHz by Walsh et al. (1998). Further high spatial resolution observations at ν > 24GHz are required to derive reliable emission measures and spectral indexes to unambiguously differentiate between the two explanations. 6. Towards an evolutionary sequence From the previous analysis, the core properties are seen to vary depending on their association with methanol maser and continuum emission. Making the reasonable as- sumption that cores heat up and becomes less quiescent with age, we now investigate what these physical conditions tell us about their evolutionary state. As the core sizes are similar, the measured linewidths can be reliably used to indi- cate how quiescent the gas is, without worrying about its dependence on the core size [Larson (1981)]. It then becomes obvious that the isolated NH3 cores (Group 1) con- tain the most quiescent gas. However, from the linewidths alone it is not clear if these cores will eventually form stars or if they are a transitory phenomenon. The fact that a large number of these Group 1 cores contain many dense sub-clumps (as evidenced by the NH3(1,1) asymmetries) suggests the former is likely for at least these cores. The linewidth of sources with methanol maser emission (Groups 2 and 3) are significantly larger, and hence contain less quiescent gas, than those of Group 1. The larger linewidths combined with generally higher temperatures, suggests cores in Groups 2 and 3 are more evolved than in Group 1. The detection of continuum emission suggests a massive star is already ionising the gas. With the current observations, the properties of the continuum sources are not well enough constrained to further separate their evolutionary stages. However, as all (with one possible exception) of the continuum sources only detected at 24GHz are associated with dense molecular gas and masers, this would suggest they are younger than those detected at both 8 + 24GHz, despite their seemingly small emission measures. In this The molecular environment associated with Class II methanol masers 123 scenario, the cores with only 8 + 24GHz continuum and no NH3 emission, may be sufficiently advanced for the UCHII region to have destroyed its natal molecular material. From this evidence, the cores in the different groups do appear to be at different evolutionary stages, going from most quiescent to most evolved according to the group number. 7. Conclusions What then, can we conclude about the role of methanol masers as a signpost of MSF? The observations show that 6.7GHz methanol masers: • are found at the warmest location within each core. • generally highlight significantly warmer regions with less quiescent gas (i.e. more evolved sources) than those with only NH3 emission. • may also highlight regions in which the internal pumping source is sufficiently young that it has not yet detectably altered the large scale core properties. Methanol masers therefore trace regions at stages shortly after a suitable powering source has formed right through to relatively evolved UCHII regions. While remaining a good general tracer of young MSF regions, the presence of a methanol maser does not single out any particular intermediate evolutionary stage. Finally, these data confirm and strengthen the results of Hill et al. (this volume), that the youngest MSF regions appear to be molecular cores with no detectable methanol maser emission. 8. Acknowledgements SNL is supported by a scholarship from the School of Physics at UNSW. We thank Andrew Walsh for comments on the manuscript. We also thank the Australian Research Council for funding support. The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. References Cragg D. M., Sobolev A. M., Godfrey P. D. 2005, MNRAS, 360, 533 Larson R. B., 1981, MNRAS, 194, 809 Longmore S. N., Burton M. G., Barnes P. J., Wong T., Purcell C. R., Ott J., 2007, MNRAS, in press. Menten K.M. 1991, ApJ, 380, 75 Mezger, P. G., Henderson, A. P. 1967, ApJ 147, 471 Ott J., Weiss A., Henkel C., Walter F., 2005, ApJ, 629, 767 Sault, R. J., Teuben, P. J., Wright, M. C. H. 1995, in: R. A. Shaw, H. E. Payne & J. J. E. Hayes (eds.), Astronomical Data Analysis Software and Systems IV, ASP Conf. Ser. 77, p. 433 Stutzki J., Winnewisser G. 1985, A&A, 144, 13 Walsh A. J., Burton M. G., Hyland A. R., Robinson G. 1998, MNRAS, 301, 640 Introduction Observations Core separation Deriving physical properties Results Molecular Gas Ionised Gas Towards an evolutionary sequence Conclusions Acknowledgements

0704.1685

Gravitating Global k-monopole

Gravitating Global k-monopole Xing-hua Jin, Xin-zhou Li†and Dao-jun Liu Shanghai United Center for Astrophysics(SUCA), Shanghai Normal University, Shanghai 200234, China School of Science, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China E-mail: †[email protected] Abstract. A gravitating global k-monopole produces a tiny gravitational field outside the core in addition to a solid angular deficit in the k-field theory. As a new feature, the gravitational field can be attractive or repulsive depending on the non-canonical kinetic term. PACS numbers: 11.27.+d, 11.10. Lm http://arxiv.org/abs/0704.1685v2 Gravitating Global k-monopole 2 1. Introduction The phase transition in the early universe could produce different kinds of topological defects which have some important implications in cosmology[1]. Domain walls are two- dimensional defects, and strings are one-dimensional defects. Point-like defects also arise in same theories which undergo the spontaneous symmetry breaking, and they appears as monopoles. The global monopole, which has divergent mass in flat spacetime, is one of the most interesting defects. The idea that monopoles ought to exist has been proved to be remarkably durable. Barriola and Vilenkin [2] firstly researched the characteristic of global monopole in curved spacetime, or equivalently, its gravitational effects. When one considers the gravity, the linearly divergent mass of global monopole has an effect analogous to that of a deficit solid angle plus that of a tiny mass at the origin. Harari and Loustò [3], and Shi and Li [4] have shown that this small gravitational potential is actually repulsive. Furthermore, Li et al [5, 6, 7] have proposed a new class of cold stars which are called D-stars (defect stars). One of the most important features of such stars, comparing to Q-stars, is that the theory has monopole solutions when the matter field is absent, which makes the D-stars behave very differently from the Q- stars. The topological defects are also investigated in the Friedmann-Robertson-Walker spacetime [8]. It is shown that the properties of global monopoles in asymptotically dS/AdS spacetime [9] and the Brans-Dicke theory [10] are very different from those of ordinary ones. The similar issue for the gravitational mass of composite monopole, i.e., global and local monopole has also been discussed [22]. The huge attractive force between global monopole M and antimonopole M̄ proposes that the monopole over-production problem does not exist, because the pair annihilation is very efficient. Barriola and Vilenkin have shown that the radiative lifetime of the pair is very short as they lose energy by Goldstone boson radiation [2]. No serious attempt has made to develop an analytical model of the cosmological evolution of a global monopole, so we are limited to the numerical simulations of evolution by Bennett and Rhie [11]. In the σ-model approximation, the average number of monopoles per horizon is NH ∼ 4. The gravitational field of global monopoles can lead to clustering in matter, and later evolve into galaxies and clusters. The scale-invariant spectrum of fluctuations has been given [11]. Furthermore, one can numerically obtain the microwave background anisotropy (δT/T )rms patterns [12]. Comparing theoretical value to the observed rms fluctuation, one can find the constraint of parameters in global monopole. On the other hand, non-canonical kinetic terms are rather ordinary for effective field theories. The k-field theory, in which the non-canonical kinetic terms are introduced in the Lagrangian, have been recently investigated to serve as the inflaton in the inflation scenario, which is so-called k-inflation [13], and to explain the current acceleration of the universe and the cosmic coincidence problem, k-essence [14]. Armendariz- Picon et al [15, 16] have discussed gravitationally bound static and spherically symmetric configurations of k-essence fields. Another interesting application of k- fields is topological defects, dubbed by k-defects [17]. Monopole [18] and vortex [19] Gravitating Global k-monopole 3 of tachyon field, which as an example of k-field comes from string/M theory, have also been investigated. The mass of global k-monopole diverges in flat spacetime, just as that of standard global monopole, therefore, it is of more physical significance to consider the gravitational effects of global k-monopole. In this paper, we study the gravitational field of global k-monopole and derive the solutions numerically and asymptotically. We find that the topological condition of vacuum manifold for the formation of a k-monopole is identical to that of an ordinary monopole, but their physical properties are disparate. Especially, we show that the mass of k-monopole can be positive in some form of the non-canonical kinetic terms. In other words, the gravitational field can be attractive or repulsive depending on the non-canonical kinetic term. 2. Equations of Motion We shall work within a particular model in units c = 1, where a global O(3) symmetry is broken down to U(1) in the k-field theory. Its action is given by M4K(X̃/M4)− λ2(φ̃aφ̃a − σ̃2 , (1) where κ = 8πG and λ is a dimensionless constant. In action (1), X̃ = 1 ∂̃µφ̃ a∂̃µφ̃a where φ̃a is the SO(3) triplet of goldstone field and σ̃0 is the symmetry breaking scale with a dimension of mass. After setting the dimensionless quantities: x = Mx̃, φa = φ̃a/M and σ0 = σ̃0/M , action (1) becomes −g [K(X)− V (φ)] , (2) where V (φ) = 1 λ2(φaφa − σ2 )2. The hedgedog configuration describing a global k- monopole is φa = σ0f(ρ) , (3) where xaxa = ρ2 and a = 1, 2, 3, so that we shall actually have a global k-monopole solution if f → 1 at spatial infinity and f → 0 near the origin. The static spherically symmetric metric can be written as ds2 = B(ρ)dt2 −A(ρ)dρ2 − ρ2(dθ2 + sin2 θdϕ2) (4) with the usual relation between the spherical coordinates ρ, θ, ϕ and the ”Cartesian” coordinate xa. Introducing a dimensionless parameter r = σ0ρ, from (2) and (3), we obtain the equations of motion for f as f ′′ + f ′ − 2 X ′f ′ − λ2f(f 2 − 1) = 0, (5) where the prime denotes the derivative with respect to r, the dot denotes the derivative with respect to X and X = −f2 − f ′2 . Since we only consider the static solution, positive X and negative X are irrelevant each other. In this paper, we will assume K(X) to be valid for negative X . Gravitating Global k-monopole 4 The Einstein equation for k-monopole is Gµν = κTµν (6) where Tµν is the energy-momentum tensor for the action (2). The tt and rr components of the Einstein equations now could be written as = ǫ2T 0 = ǫ2T 1 , (8) where = −K + λ (f 2 − 1)2 (9) = −K + λ (f 2 − 1)2 − K̇ f and ǫ2 = κσ2 = 8πGσ2 is a dimensionless parameter. 3. k-monopole Although the existence of global k-monopole, as well as the standard one, is guaranteed by the symmetry-breaking potential, there exist the non-canonical kinetic term in k- monopole which certainly leads to the appearance of a new scale in the action and the mass parameter in the potential term. However, the non-canonical kinetic term is non- trivial. At small gradients, it can be chosen to have a same asymptotical behavior with that of the standard one, so that it ensures the standard manner of a small perturbations. While at large gradient we choose it to have a different form from the standard one. In the small X case, we assume that the kinetic term has the asymptotically canonical behavior, which can avoid ”zero-kinetic problem”. If |X| ≪ 1, we have K(X) ∼ Xα, and α < 1 then there is a singularity at X = 0; and α > 1 then the system becomes non-dynamical at X = 0. For the monopole solution, it is easily found that K(X) ∼ X at r ≫ 1. On the other hand, we assume that the modificatory kinetic term K(X) ∼ Xα and α 6= 1 at |X| ≫ 1. One can easily obtain the equation of motion inside the core of a global monopole after assuming that |X| ≫ 1 in the core of the global monopole. The equations of motion are highly non-linear and cannot be solved analytically. Next, we investigate the asymptotic behaviors of global monopole with non-linear in X kinetic term. To be specific, we consider the following type of kinetic K(X) = X − βX2, (11) where β is a parameter of global k-monopole. It is easy to find that global k-monopole will reduce to be the standard one when β = 0. It is easy to check whether the kinetic term (11) satisfy the condition for the hyperbolicity [16, 20, 17] 2XK̈ + K̇ > 0, (12) Gravitating Global k-monopole 5 which leads to a positive definite speed of sound for the small perturbations of the field. The stability of solutions shows that for the case β < 0, the range 1 > X > 1 be excluded. However, this will not destroy the results which are carried out from the case β > 0. We here only consider the cases for β > 0. Using Eqs.(5)-(10), we get the asymptotic expression for A(r), B(r), and f(r) which is valid near r = 0, f(r) = f0r + 2 λ2 (−3 + ǫ2) + +42β ǫ2 f04 1 + 5β f0 ) r3 (13) (9 + 7β λ2) ǫ2 f0 2 + 36β2 ǫ2 f0 1 + 5β f0 ) r3 +O(r4) A(r) = 1 + λ2 + 6 f0 2 + 9β f0 r2 +O(r3) (14) B(r) = 1 + λ2 − 9β f04 r2 +O(r3), (15) where the undetermined coefficient f0 is characterized as the mass of k-monopole, which can be determined in the numerical calculation. In the region of r ≫ 1, similarly we can expand f(r), A(r) and B(r) as f(r) = 1− 1 − 3− 2 ǫ 2 + 4β λ2 +O(r−5) (16) A(r) = 1− ǫ2 (1− ǫ2)2 ǫ2(1− βλ2) (1− ǫ2)2 λ2 (1− ǫ2)3 + O(r−3) B(r) = (1− ǫ2) +M∞ 2 (1− βλ2) +O(r−3), (18) where the constant M∞ will be discussed in the following. Using shooting method for boundary value problems, we get the numerical results of the function f(r) which describes the configuration of global k-monopole. In Fig.1 we show the function f(r) for β = 0, β = 1, β = 5 and β = 10 respectively and for given values of λ and ǫ. Obviously, the configuration of field f is not impressible to the choice of the parameter β. From Eqs.(13) and (16), it is easy to construct global k-monopole which has the same asymptotic condition with standard global monopole, i.e., f will approach to zero when r ≪ 1 and approach to unity when r ≫ 1. Actually there is a general solution to Einstein equation with energy-momentum tensor Tµν which takes the form as (9) and (10) for spherically symmetric metric (4) A(r)−1 = 1− [−K + (f 2 − 1)2]r2dr (19) B(r) = A(r)−1 exp (K̇f ′2r)dr . (20) Gravitating Global k-monopole 6 0 5 10 15 20 Figure 1. The plot of f(r) as a function of r. Here we choose λ = 1, G = 1 and ǫ = 0.001. The four curved lines are plotted when β = 0, β = 1, β = 5 and β = 10 respectively. In terms of the dimensionless quantity ǫ, the metric coefficient A(r) and B(r) can be formally integrated and read as A(r)−1 = 1− ǫ2 − 2Gσ0MA(r) B(r) = 1− ǫ2 − 2Gσ0MB(r) . (22) The small dimensionless parameter ǫ arise naturally from Einstein equations and clearly ǫ2 describes a solid angular deficit of space-time. A global k-monopole solution f should approach unity when r ≫ 1. If this convergence is fast enough, then MA(r) and MB(r) will also rapidly converge to finite values. Therefore, from Eqs.(16)-(18) we have the asymptotic expansions: MA(r) = M∞ + 4πσ0 8πσ0 (−1 + ǫ2 + 2 β λ2) + O(r−5) MB(r) = M∞ + 4πσ0 4πσ0 (−1 + ǫ2 − 4 β λ2) + O(r−5), where M∞ ≡ limr→∞MA(r). One can easily find that the dependence on ǫ of the asymptotic expansion for f(r) is very weak, in other words, the asymptotic behavior is quite independent of the scale of symmetry breakdown σ0 up to value as large as the planck scale. On the contrary, MA(r) depends obviously on σ0. Gravitating Global k-monopole 7 0 5 10 15 20 Figure 2. The plot of MA(r)/σ0 as a function of r. Here we choose λ = 1, G = 1 and ǫ = 0.001. The four curved lines are plotted when β = 0, β = 1, β = 5 and β = 10 respectively. The numerical results of MA(r)/σ0 are shown in Fig.2 by shooting method for boundary value problems where we choose λ = 1, G = 1 and ǫ = 0.001. From the figure, we find that the mass of global k-monopole decrease to a negative asymptotic value when r approaches infinity in the case that β = 0 and β = 1. While the mass will be positive if β = 5 or β = 10. The asymptotic mass for the cases above are −19.15σ0, −13.83σ0, 3.62σ0 and 22.14σ0 respectively. It is clear that the presence of parameter β, which measures the degree of deviation of kinetic term from canonical one, affects the effective mass of the global k-monopole significantly. It is not difficult to understand this property. From Eqs.(11), (19) and (21), the mass function MA(r) can be expressed explicitly as MA(r) = −r + βX2 −X + (f 2 − 1)2 r2dr. (25) Obviously, the β-term in the integration has the positive contribution for the mass function. From Fig.1, f (then X) is not sensitive to the value of β, so the greater parameter β is chosen, the larger value MA(r) takes for a given r, and if β is greater than some value, MA(r) will be positive for large r as Fig.2 shows. However, inside the core, X varies slowly but f varies fast with respect to r, therefore, λ2-term in the integration will become dominant as r decreases. This leads to two characteristics of the mass curves which is also shown in Fig.2: (i) the mass curves with different β converge gradually in the region near r = 0; (ii) in the case that β is large enough, MA(r) has a minimum. Gravitating Global k-monopole 8 To show the effect of solid defect angle, we then investigate the motion test particles around a global k-monopole. It is a good approximation to take MA(r) as a constant in the region far away from the core of the global k-monopole, since the the mass MA(r) approaches very quickly to its asymptotic value. Therefore, we can consider the geodesic equation in the metric (4) with A(r)−1 = B(r) = 1− ǫ2 − where M = σ0M∞. Solving the geodesic equation and introducing a dimensionless quantaty u = GM/r, one will get the second order differentiating equation of u with respect to ϕ [9, 21] + (1− ǫ2)u = + 3u2, (27) where L is the angular momentum per unit of mass. When ϕ ≪ 1, one have the approximate solution of u 1− ǫ2 + e cos 1− 3√ (1− ǫ2)3 , (28) where e denotes the eccentricity. When a test particle rotates one loop around the global k-monopole, the precession of it will be ∆ϕ = 6π (1− ǫ2)3 ǫ2. (29) The last term in Eq.(29) is the modificaiton comparing this result with that for the precession around an ordinary star. 4. Conclusion In summary, k-monopole could arise during the phase transition in the early universe. We calculate the asymptotic solutions of global k-monopole in static spherically symmetric spacetime, and find that the behavior of a k-monopole is similar to that of a standard one. Although the choice of the parameter β, which measures the degree of deviation of kinetic term from canonical one, have little influence on the configuration of k-field φ, the effective mass of global k-monopole is affected significantly. The mass might be negative or positive when different parameters β are chosen. This shows that the gravitational field of the global k-monopole could be attractive or repulsive depending on the different non-canonical kinetic term. The configuration of a global k-monopole is more complicated than that of a standard one. As for its cosmological evolution, we should not attempt to get the analytical mode, instead we can only use numerical simulation. However, the energy dominance of global k-monopole is in the region outside the core. We can roughly estimate that global k-monopoles will result in the clustering in matter and evolve into galaxies and clusters in a way similar to that of standard monopoles. Gravitating Global k-monopole 9 Acknowledgement This work is supported in part by National Natural Science Foundation of China under Grant No. 10473007 and No. 10503002 and Shanghai Commission of Science and Technology under Grant No. 06QA14039. References [1] Vilenkin A and Shellard E P S, 1994 Cosmic Strings and Other Topological Defects (Cambridge Unversity Press, Cambridge, England) [2] Barriola M and Vilenkin A, 1989 Phys. Rev. Lett. 63, 341 [3] Harari D and Loustò C, 1990 Phys. Rev.D42, 2626 [4] Shi X and Li X Z, 1991 Class. Quantum Grav. 8, 761 [5] Li X Z and Zhai X H, 1995 Phys. Lett. B364, 212 [6] Li J M and Li X Z, 1998 Chin. Phys. Lett.15, 3; Li X Z, Liu D J and Hao J G, 2002 Science in China A45, 520 [7] Li X Z, Zhai X H and Chen G, 2000 Astropart. Phys. 13, 245 [8] Basu R, Guth A H and Vilenkin A, 1991 Phys. Rev. D44 340; Basu R and Vilenkin A, 1994 Phys. Rev. D50 7150; Chen C, Cheng H, Li X Z and Zhai X H, 1996 Class. Quantum Grav. 13, 701 [9] Li X Z and Hao J G, 2002 Phys. Rev. D66, 107701; Hao J G and Li X Z, 2003 Class. Quantum Grav. 20, 1703 [10] Li X Z and Lu J Z, 2000 Phys. Rev. D62, 107501 [11] Bennett D P and Rhie S H, 1990 Phys. Rev. Lett. 65, 1709 [12] Bennett D P and Rhie S H, 1993 Astrophys. J. 406, L7 [13] Armendariz-Picon C, Damour T, Mukhanov V, 1999 Phys.Lett. B458, 209 [14] Armendariz-Picon C, Mukhanov V, Steinhardt P J, 2000 Phys.Rev.Lett.85, 4438; Armendariz- Picon C, Mukhanov V, Steinhardt P J, 2001 Phys. Rev. D63: 103510 [15] Armendariz-Picon C and Lim E A, 2005 JCAP 0508, 007 [16] Bilic N, Tupper G B and Viollier R D, 2006 JCAP 0602 013; Diez-Tejedor A and Feinstein A, 2006 Phys. Rev. D74 023530; Nucamendi U, Salgado M and Sudarsky D, 2000 Phys. Rev. Lett. 84 3037; Nucamendi U, Salgado M and Sudarsky D, 2001 Phys. Rev. D63 125016. [17] Babichev E, 2006 Phys. Rev. D74 085004 [18] Li X Z and Liu D J, 2005 Int. J. Mod. Phys. A20, 5491 [19] Liu D J and Li X Z, 2003 Chin. Phys. Lett. 20, 1678 [20] Rendall A D, 2006 Class.Quant.Grav. 23, 1557. [21] Wald R M, 1984 General Ralitivity (The university of Chicago Press, Chicago) [22] Spinelly J, de Freitas U and Bezerra de Mello E R, 2002 Phys.Rev. D66, 024018. Introduction Equations of Motion k-monopole Conclusion

0704.1686

Effect of atomic beam alignment on photon correlation measurements in cavity QED

Effect of atomic beam alignment on photon correlation measurements in cavity QED L. Horvath and H. J. Carmichael Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand (Dated: November 3, 2018) Quantum trajectory simulations of a cavity QED system comprising an atomic beam traversing a standing-wave cavity are carried out. The delayed photon coincident rate for forwards scattering is computed and compared with the measurements of Rempe et al. [Phys. Rev. Lett. 67, 1727 (1991)] and Foster et al. [Phys. Rev. A 61, 053821 (2000)]. It is shown that a moderate atomic beam misalignment can account for the degradation of the predicted correlation. Fits to the experimental data are made in the weak-field limit with a single adjustable parameter—the atomic beam tilt from perpendicular to the cavity axis. Departures of the measurement conditions from the weak-field limit are discussed. PACS numbers: 42.50.Pq, 42.50.Lc, 02.70.Uu I. INTRODUCTION Cavity quantum electrodynamics [1, 2, 3, 4, 5, 6] has as its central objective the realization of strong dipole coupling between a discrete transition in matter (e.g., an atom or quantum dot) and a mode of an electromag- netic cavity. Most often strong coupling is demonstrated through the realization of vacuum Rabi splitting [7, 8]. First realized for Rydberg atoms in superconducting mi- crowave cavities [9, 10] and for transitions at optical wavelengths in high-finesse Fabry Perots [11, 12, 13, 14], vacuum Rabi splitting was recently observed in mono- lithic structures where the discrete transition is provided by a semiconductor quantum dot [15, 16, 17], and in a coupled system of qubit and resonant circuit engineered from superconducting electronics [18]. More generally, vacuum Rabi spectra can be observed for any pair of coupled harmonic oscillators [19] without the need for strong coupling of the one-atom kind. Prior to observations for single atoms and quantum dots, sim- ilar spectra were observed in many-atom [20, 21, 22] and -exciton [23, 24] systems where the radiative coupling is collectively enhanced. The definitive signature of single-atom strong coupling is the large effect a single photon in the cavity has on the reflection, side-scattering, or transmission of another photon. Strong coupling has a dramatic effect, for exam- ple, on the delayed photon coincidence rate in forwards scattering when a cavity QED system is coherently driven on axis [25, 26, 27, 28]. Photon antibunching is seen at a level proportional to the parameter 2C1 = 2g 2/γκ [27], where g is the atomic dipole coupling constant, γ is the atomic spontaneous emission rate, and 2κ is the pho- ton loss rate from the cavity; the collective parameter 2C = N2C1, with N the number of atoms, does not en- ter into the magnitude of the effect when N ≫ 1. In the one-atom case, and for 2κ ≫ γ, the size of the effect is raised to (2C1) 2 [25, 26] [see Eq. (30)]. The first demonstration of photon antibunching was made [29] for moderately strong coupling (2C1 ≈ 4.6) and N = 18, 45, and 110 (effective) atoms. The mea- surement has subsequently been repeated for somewhat higher values of 2C1 and slightly fewer atoms [30, 31], and a measurement for one trapped atom [32], in a slightly al- tered configuration, has demonstrated the so-called pho- ton blockade effect [33, 34, 35, 36, 37, 38]—i.e., the anti- bunching of forwards-scattered photons for coherent driv- ing of a vacuum-Rabi resonance, in which case a two-state approximation may be made [39], assuming the coupling is sufficiently strong. The early experiments of Rempe et al. [29] and those of Mielke et al. [30] and Foster et al. [31] employ systems designed around a Fabry-Perot cavity mode traversed by a thermal atomic beam. Their theoretical modeling therefore presents a significant challenge, since for the numbers of effective atoms used, the atomic beam car- ries hundreds of atoms—typically an order of magnitude larger than the effective number [40]—into the interac- tion volume. The Hilbert space required for exact calcu- lations is enormous (2100 ∼ 1030); it grows and shrinks with the number of atoms, which inevitably fluctuates over time; and the atoms move through a spatially vary- ing cavity mode, so their coupling strengths are chang- ing in time. Ideally, all of these features should be taken into account, although certain approximations might be made. For weak excitation, as in the experiments, the lowest permissible truncation of the Hilbert space—when cal- culating two-photon correlations—is at the two-quanta level. Within a two-quanta truncation, relatively simple formulas can be derived so long as the atomic motion is overlooked [27, 28]. It is even possible to account for the unequal coupling strengths of different atoms, and, through a Monte-Carlo average, fluctuations in their spa- tial distribution [29]. A significant discrepancy between theory and experiment nevertheless remains: Rempe et al. [29] describe how the amplitude of the Rabi oscillation (magnitude of the antibunching effect) was scaled down by a factor of 4 and a slight shift of the theoretical curve was made in order to bring their data into agreement with this model; the discrepancy persists in the experiments of Foster et al. [31], except that the required adjustment http://arxiv.org/abs/0704.1686v2 is by a scale factor closer to 2 than to 4. Attempts to account for these discrepancies have been made but are unconvincing. Martini and Schenzle [41] report good agreement with one of the data sets from Ref. [29]; they numerically solve a many-atom master equation, but under the unreasonable assumption of sta- tionary atoms and equal coupling strengths. The unlikely agreement results from using parameters that are very far from those of the experiment—most importantly, the dipole coupling constant is smaller by a factor of approx- imately 3. Foster et al. [31] report a rather good theoretical fit to one of their data sets. It is obtained by using the mentioned approximations and adding a detuning in the calculation to account for the Doppler broadening of a misaligned atomic beam. They state that “Imperfect alignment . . . can lead to a tilt from perpendicular of as much as 1◦”. They suggest that the mean Doppler shift is offset in the experiment by adjusting the driving laser frequency and account for the distribution about the mean in the model. There does appear to be a dif- ficulty with this procedure, however, since while such an offset should work for a ring cavity, it is unlikely to do so in the presence of the counter-propagating fields of a Fabry-Perot. Indeed, we are able to successfully simulate the procedure only for the ring-cavity case (Sec. IVC). The likely candidates to explain the disagreement be- tween theory and experiment have always been evident. For example, Rempe et al. [29] state: “Apparently the transient nature of the atomic mo- tion through the cavity mode (which is not included here or in Ref. [7]) has a profound effect in decorre- lating the otherwise coherent response of the sam- ple to the escape of a photon.” and also: “Empirically, we also know that |g(2)(0)− 1| is re- duced somewhat because the weak-field limit is not strictly satisfied in our measurements.” To these two observations we should add—picking up on the comment in [31]—that in a standing-wave cavity an atomic beam misalignment would make the decorrelation from atomic motion a great deal worse. Thus, the required improvements in the modeling are: (i) a serious accounting for atomic motion in a thermal atomic beam, allowing for up to a few hundred inter- acting atoms and a velocity component along the cavity axis, and (ii) extension of the Hilbert space to include 3, 4, etc. quanta of excitation, thus extending the model beyond the weak-field limit. The first requirement is entirely achievable in a quantum trajectory simulation [42, 43, 44, 45, 46], while the second, even with recent improvements in computing power, remains a formidable challenge. In this paper we offer an explanation of the discrepan- cies between theory and experiment in the measurements Parameter Set 1 Set 2 cavity halfwidth κ/2π 0.9MHz 7.9MHz dipole coupling constant gmax/κ 3.56 1.47 atomic linewidth γ/κ 5.56 0.77 mode waist w0 50µm 21.5µm wavelength 852nm (Cs) 780nm (Rb) effective atom number N̄eff 18 13 oven temperature 473K 430K mean speed in oven voven 274.5m/s 326.4m/s mean speed in beam vbeam 323.4m/s 384.5m/s TABLE I: Parameters used in the simulations. Set 1 is taken from Ref. [29] and Set 2 from Ref. [31]. of Refs. [29] and [31]. We perform ab initio quantum tra- jectory simulations in parallel with a Monte-Carlo sim- ulation of a tilted atomic beam. The parameters used are listed in Table I: Set 1 corresponds to the data dis- played in Fig. 4(a) of Ref. [29], and Set 2 to the data dis- played in Fig. 4 of Ref. [31]. All parameters are measured quantities— or are inferred from measured quantities— and the atomic beam tilt alone is varied to optimize the data fit. Excellent agreement is demonstrated for atomic beam misalignments of approximately 10mrad (a little over 1/2◦). These simulations are performed using a two- quanta truncation of the Hilbert space. Simulations based upon a three-quanta truncation are also carried out, which, although not adequate for the experimental conditions, can begin to address physics be- yond the weak-field limit. From these, an inconsistency with the intracavity photon number reported by Foster et al. [31] is found. Our model is described in Sec. II, where we formu- late the stochastic master equation used to describe the atomic beam, its quantum trajectory unraveling, and the two-quanta truncation of the Hilbert space. The previous modeling on the basis of a stationary-atom approxima- tion is reviewed in Sect. III and compared with the data of Rempe et al. [29] and Foster et al. [31]. The effects of atomic beam misalignment are discussed in Sec. IV; here the results of simulations with a two-quanta trunca- tion are presented. Results obtained with a three-quanta truncation are presented in Sec. V, where the issue of intracavity photon number is discussed. Our conclusions are stated in Sec. VI. II. CAVITY QED WITH ATOMIC BEAMS A. Stochastic Master Equation: Atomic Beam Simulation Thermal atomic beams have been used extensively for experiments in cavity QED [9, 10, 11, 12, 20, 21, 22, 29, 30, 31]. The experimental setups under consideration are described in detail in Refs. [47] and [48]. As typi- cally, the beam is formed from an atomic vapor created inside an oven, from which atoms escape through a colli- mated opening. We work from the standard theory of an effusive source from a thin-walled oriface [49], for which for an effective number N̄eff of intracavity atoms [11, 40] and cavity mode waist ω0 (N̄eff is the average number of atoms within a cylinder of radius w0/2), the average escape rate is R = 64N̄eff v̄beam/3π 2w0, (1) with mean speed in the beam v̄beam = 9πkBT/8M, (2) where kB is Boltzmann’s constant, T is the oven tem- perature, and M is the mass of an atom; the beam has atomic density ̺ = 4N̄eff/πw 0l, (3) where l is the beam width, and distribution of atomic speeds P (v)dv = 2u3(v)e−u 2(v)du(v), (4) u(v) ≡ 2v/ π v̄oven, where v̄oven = 8kBT/πM = (8/3π)v̄beam (5) is the mean speed of an atom inside the oven, as cal- culated from the Maxwell-Boltzmann distribution. Note that v̄beam is larger than v̄oven because those atoms that move faster inside the oven have a higher probability of escape. In an open-sided cavity, neither the interaction volume nor the number of interacting atoms is well-defined; the cavity mode function and atomic density are the well- defined quantities. Clearly, though, as the atomic dipole coupling strength decreases with the distance of the atom from the cavity axis, those atoms located far away from the axis may be neglected, introducing, in effect, a finite interaction volume. How far from the cavity axis, how- ever, is far enough? One possible criterion is to require that the interaction volume taken be large enough to give an accurate result for the collective coupling strength, or, considering its dependence on atomic locations (at fixed average density), the probability distribution over collec- tive coupling strengths. According to this criterion, the actual number of interacting atoms is typically an order of magnitude larger than N̄eff [40]. If, for example, one introduces a cut-off parameter F < 1, and defines the interaction volume by [40, 50, 51] VF ≡ {(x, y, z) : g(x, y, z) ≥ Fgmax}, (6) g(x, y, z) = gmax cos(kz) exp −(x2 + y2)/w20 the spatially varying coupling constant for a standing- wave TEM00 cavity mode [52]—wavelength λ = 2π/k— the computed collective coupling constant is [40] N̄eff gmax → N̄Feff gmax, N̄Feff = (2N̄eff/π) (1− 2F 2) cos−1 F + F 1− F 2 . (8) For the choice F = 0.1, one obtains N̄Feff = 0.98N̄eff, a reduction of the collective coupling strength by 1%, and the interaction volume—radius r ≈ 3(w0/2)—contains approximately 9N̄eff atoms on average. This is the choice made for the simulations with a three-quanta trunca- tion reported in Sec. V. When adopting a two-quanta truncation, with its smaller Hilbert space for a given number of atoms, we choose F = 0.01, which yields N̄Feff = 0.9998N̄eff and r ≈ 4.3(w0/2), and approximately 18N̄eff atoms in the interaction volume on average. In fact, the volume used in practice is a little larger than VF . In the course of a Monte-Carlo simulation of the atomic beam, atoms are created randomly at rate R on the plane x = −w0 | lnF |. At the time, tj0, of its creation, each atom is assigned a random position and velocity (j labels a particular atom), | lnF | , vj = vj cos θ sin θ , (9) where yj(t 0) and zj(t 0) are random variables, uniformly distributed on the intervals |yj(tj0)| ≤ w0 | lnF | and |zj(tj0)| ≤ λ/4, respectively, and vj is sampled from the distribution of atomic speeds [Eq. (4)]; θ is the tilt of the atomic beam away from perpendicular to the cavity axis. The atom moves freely across the cavity after its creation, passing out of the interaction volume on the plane x = w0 | lnF |. Thus the interaction volume has a square rather than circular cross section and measures | lnF |w0 on a side. It is larger than VF by approxi- mately 30%. Atoms are created in the ground state and returned to the ground state when they leave the interaction vol- ume. On leaving an atom is disentangled from the sys- tem by comparing its probability of excitation with a uniformly distributed random number r, 0 ≤ r ≤ 1, and deciding whether or not it will—anytime in the future— spontaneously emit; thus, the system state is projected onto the excited state of the leaving atom (the atom will emit) or its ground state (it will not emit) and propa- gated forwards in time. Note that the effects of light forces and radiative heat- ing are neglected. At the thermal velocities considered, typically the ratio of kinetic energy to recoil energy is of order 108, while the maximum light shift h̄gmax (assum- ing one photon in the cavity) is smaller than the kinetic energy by a factor of 107; even if the axial component of velocity only is considered, these ratios are as high as 104 and 103 with θ ∼ 10mrad, as in Figs. 10 and 11. In fact, the mean intracavity photon number is considerably less than one (Sec. V); thus, for example, the majority of atoms traverse the cavity without making a single spon- taneous emission. Under the atomic beam simulation, the atom number, N(t), and locations rj(t), j = 1, . . . , N(t), are chang- ing in time; therefore, the atomic state basis is dynamic, growing and shrinking with N(t). We assume all atoms couple resonantly to the cavity mode, which is coher- ently driven on resonance with driving field amplitude E . Then, including spontaneous emission and cavity loss, the system is described by the stochastic master equation in the interaction picture ρ̇ = E [↠− â, ρ] + g(rj(t))[â †σ̂j− − âσ̂j+, ρ] (2σ̂j−ρσ̂j+ − σ̂j+σ̂j−ρ− ρσ̂j+σ̂j−) 2âρ↠− â†âρ− ρâ†â , (10) with dipole coupling constants g(rj(t)) = gmax cos(kzj(t)) exp x2j (t) + y j (t) , (11) where ↠and â are creation and annihilation operators for the cavity mode, and σ̂j+ and σ̂j−, j = 1 . . .N(t), are raising and lowering operators for two-state atoms. B. Quantum Trajectory Unraveling In principle, the stochastic master equation might be simulated directly, but it is impossible to do so in prac- tice. Table I lists effective numbers of atoms N̄eff = 18 and N̄eff = 13. For cut-off parameter F = 0.01 and an interaction volume of approximately 1.3×VF [see the dis- cussion below Eq. (8)], an estimate of the number of in- teracting atoms gives N(t) ∼ 1.3×18N̄eff ≈ 420 and 300, respectively, which means that even in a two-quanta trun- cation the size of the atomic state basis (∼ 105 states) is far too large to work with density matrix elements. We therefore make a quantum trajectory unraveling of Eq. (10) [42, 43, 44, 45, 46], where, given our interest in delayed photon coincidence measurements, condition- ing of the evolution upon direct photoelectron counting records is appropriate: the (unnormalized) conditional state satisfies the nonunitary Schrödinger equation d|ψ̄REC〉 ĤB(t)|ψ̄REC〉, (12) with non-Hermitian Hamiltonian ĤB(t)/ih̄ = E(↠− â) + g(rj(t))(â †σ̂j− − âσ̂j+) − κâ†â− γ σ̂j+σ̂j−, (13) and this continuous evolution is interrupted by quantum jumps that account for photon scattering. There are N(t)+1 scattering channels and correspondinglyN(t)+1 possible jumps: |ψ̄REC〉 → â|ψ̄REC〉, (14a) for forwards scattering—i.e., the transmission of a photon by the cavity—and |ψ̄REC〉 → σ̂j−|ψ̄REC〉, j = 1, . . . , N(t), (14b) for scattering to the side (spontaneous emission). These jumps occur, in time step ∆t, with probabilities Pforwards = 2κ〈â†â〉REC∆t, (15a) side = γ〈σ̂j+σ̂j−〉REC∆t, j = 1, . . . , N(t); (15b) otherwise, with probability 1− Pforwards − side, the evolution under Eq. (12) continues. For simplicity, and without loss of generality, we as- sume a negligible loss rate at the cavity input mirror compared with that at the output mirror. Under this assumption, backwards scattering quantum jumps need not be considered. Note that non-Hermitian Hamiltonian (13) is explicitly time dependent and stochastic, due to the Monte-Carlo simulation of the atomic beam, and the normalized conditional state is |ψREC〉 = |ψ̄REC〉 〈ψ̄REC|ψ̄REC〉 . (16) C. Two-Quanta Truncation Even as a quantum trajectory simulation, a full im- plementation of our model faces difficulties. The Hilbert space is enormous if we are to consider a few hundred two-state atoms, and a smaller collective-state basis is inappropriate, due to spontaneous emission and the cou- pling of atoms to the cavity mode at unequal strengths. If, on the other hand, the coherent excitation is suffi- ciently weak, the Hilbert space may be truncated at the two-quanta level. The conditional state is expanded as |ψREC(t)〉 = |00〉+ α(t)|10〉+ βj(t)|0j〉+ η(t)|20〉+ ζj(t)|1j〉+ j>k=1 ϑjk(t)|0jk〉, (17) where the state |n0〉 has n = 0, 1, 2 photons inside the cavity and no atoms excited, |0j〉 has no photon inside the cavity and the j th atom excited, |1j〉 has one photon inside the cavity and the j th atom excited, and |0jk〉 is the two-quanta state with no photons inside the cavity and the j th and kth atoms excited. The truncation is carried out at the minimum level per- mitted in a treatment of two-photon correlations. Since each expansion coefficient need be calculated to domi- nant order in E/κ only, the non-Hermitian Hamiltonian (13) may be simplified as ĤB(t)/ih̄ = E ↠+ g(rj(t))(â †σ̂j− − âσ̂j+) − κâ†â− γ σ̂j+σ̂j−, (18) dropping the term −E â from the right-hand side. While this self-consistent approximation is helpful in the ana- lytical calculations reviewed in Sec. III, we do not bother with it in the numerical simulations. Truncation at the two-quanta level may be justified by expanding the density operator, along with the master equation, in powers of E/κ [25, 26, 53]. One finds that, to dominant order, the density operator factorizes as a pure state, thus motivating the simplification used in all previous treatments of photon correlations in many-atom cavity QED [27, 28]. The quantum trajectory formula- tion provides a clear statement of the physical conditions under which this approximation holds. Consider first that there is a fixed number of atoms N and their locations are also fixed. Under weak excitation, the jump probabilities (15a) and (15b) are very small, and quantum jumps are extremely rare. Then, in a time of order 2(κ+γ/2)−1, the continuous evolution (12) takes the conditional state to a stationary state, satisfying ĤB |ψss〉 = 0, (19) without being interrupted by quantum jumps. In view of the overall rarity of these jumps, to a good approximation the density operator is ρss = |ψss〉〈ψss|, (20) or, if we recognize now the role of the atomic beam, the continuous evolution reaches a quasi-stationary state, with density operator ρss = |ψqs(t)〉〈ψqs(t)|, (21) where |ψqs(t)〉 satisfies Eq. (12) (uninterrupted by quan- tum jumps) and the overbar indicates an average over the fluctuations of the atomic beam. This picture of a quasi-stationary pure-state evolution requires the time between quantum jumps to be much larger than 2(κ+ γ/2)−1, the time to recover the quasi- stationary state after a quantum jump has occurred. In terms of photon scattering rates, we require Rforwards +Rside ≪ 12 (κ+ γ/2), (22) where Rforwards = 2κ〈â†â〉REC, (23a) Rside = γ 〈σ̂j+σ̂j−〉REC. (23b) When considering delayed photon coincidences, after a first forwards-scattered photon is detected, let us say at time tk, the two-quanta truncation [Eq. (17)] is tem- porarily reduced by the associated quantum jump to a one-quanta truncation: |ψREC(tk)〉 → |ψREC(t+k )〉, where |ψREC(t+k )〉 = |00〉+ α(t k )|10〉+ N(tk) k )|0j〉, (24) α(t+k ) = 2η(tk) |α(tk)| , βj(t k ) = ζ(tk) |α(tk)| . (25) Then the probability for a subsequent photon detection at tk + τ is Pforwards = 2κ|α(tk + τ)|2∆t. (26) Clearly, if this probability is to be computed accurately (to dominant order) no more quantum jumps of any kind should occur before the full two-quanta truncation has been recovered in its quasi-stationary form; in the ex- periment a forwards-scattered “start” photon should be followed by a “stop” photon without any other scatter- ing events in between. We discuss how well this condi- tion is met by Rempe et al. [29] and Foster et al. [31] in Sec. V. Its presumed validity is the basis for com- paring their measurements with formulas derived for the weak-field limit. III. DELAYED PHOTON COINCIDENCES FOR STATIONARY ATOMS Before we move on to full quantum trajectory simula- tions, including the Monte-Carlo simulation of the atomic beam, we review previous calculations of the delayed photon coincidence rate for forwards scattering with the atomic motion neglected. Beginning with the original calculation of Carmichael et al. [27], which assumes a fixed number of atoms, denoted here by N̄eff , all cou- pled to the cavity mode at strength gmax, we then relax the requirement for equal coupling strengths [29]; finally a Monte-Carlo average over the spatial configuration of atoms, at fixed density ̺, is taken. The inadequacy of modeling at this level is shown by comparing the com- puted correlation functions with the reported data sets. A. Ideal Collective Coupling For an ensemble of N̄eff atoms located on the cavity axis and at antinodes of the standing wave, the non- Hermitian Hamiltonian (18) is taken over in the form ĤB/ih̄ = E ↠+ gmax(â†Ĵ− − âĴ+) − κâ†â− γ (Ĵz +Neff), (27) where Ĵ± ≡ σ̂j±, Ĵz ≡ σ̂jz (28) are collective atomic operators, and we have written 2σ̂j+σ̂j− = σ̂jz + 1. The conditional state in the two- quanta truncation is now written more simply as |ψREC(t)〉 = |00〉+ α(t)|10〉+ β(t)|01〉+ η(t)|20〉+ ζ(t)|11〉+ ϑ(t)|02〉, (29) where |nm〉 is the state with n photons in the cavity and m atoms excited, the m-atom state being a collective state. Note that, in principle, side-scattering denies the possibility of using a collective atomic state basis. While spontaneous emission from a particular atom results in the transition |n1〉 → σ̂j−|n1〉 → |n0〉, which remains within the collective atomic basis, the state σ̂j−|n2〉 lies outside it; thus, side-scattering works to degrade the atomic coherence induced by the interaction with the cav- ity mode. Nevertheless, its rate is assumed negligible in the weak-field limit [Eq. (22)], and therefore a calculation carried out entirely within the collective atomic basis is permitted. The delayed photon coincidence rate obtained from |ψREC(tk)〉 = |ψss〉 and Eqs. (24) and (26) yields the second-order correlation function [27, 28, 54] g(2)(τ) = 1− 2C1 1 + ξ 1 + 2C − 2C1ξ/(1 + ξ) (κ+γ/2)τ cos (Ωτ)+ (κ+ γ/2) sin (Ωτ) , (30) with vacuum Rabi frequency N̄effg2max − 14 (κ− γ/2)2, (31) where ξ ≡ 2κ/γ, (32) C ≡ N̄effC1, C1 ≡ g2max/κγ. (33) For N̄eff ≫ 1, as in Parameter Sets 1 and 2 (Table I), the deviation from second-order coherence—i.e., g(2)(τ) = 630-3-6 210-1-2 FIG. 1: Second-order correlation function for ideal coupling [Eq. (30)]: (a) Parameter Set 1, (b) Parameter Set 2. 1—is set by 2C1ξ/(1 + ξ) and provides a measure of the single-atom coupling strength. For small time delays the deviation is in the negative direction, signifying a photon antibunching effect. It should be emphasized that while second-order coherence serves as an unambiguous indica- tor of strong coupling in the single-atom sense, vacuum Rabi splitting—the frequency Ω—depends on the collec- tive coupling strength alone. Both experiments of interest are firmly within the strong coupling regime, with 2C1ξ/(1+ ξ) = 1.2 for that of Rempe et al. [29] (2C1 = 4.6), and 2C1ξ/(1+ ξ) = 4.0 for that of Foster et al. [31] (2C1 = 5.6). Figure 1 plots the correlation function obtained from Eq. (30) for Pa- rameter Sets 1 and 2. Note that since the expression is a perfect square, the apparent photon bunching of curve (b) is, in fact, an extrapolation of the antibunching ef- fect of curve (a); the continued nonclassicality of the correlation function is expressed through the first two side peaks, which, being taller than the central peak, are classically disallowed [26, 30]. A measurement of the in- tracavity electric field perturbation following a photon detection [the square root of Eq. (30)] presents a more unified picture of the development of the quantum fluctu- ations with increasing 2C1ξ/(1+ξ). Such a measurement may be accomplished through conditional homodyne de- tection [55, 56, 57]. In Fig. 1 the magnitude of the antibunching effect— the amplitude of the vacuum Rabi oscillation— is larger than observed in the experiments by approximately an order of magnitude (see Fig. 3). Significant improvement is obtained by taking into account the unequal coupling strengths of atoms randomly distributed throughout the cavity mode. B. Fixed Atomic Configuration Rempe et al. [29] extended the above treatment to the case of unequal coupling strengths, adopting the non- Hermitian Hamiltonian (18) while keeping the number of atoms and the atom locations fixed. For N atoms in a spatial configuration {rj}, the second-order correlation function takes the same form as in Eq. (30)—still a per- fect square—but with a modified amplitude of oscillation [29, 58]: (τ) = [1 + ξ(1 + C{rj})]S{rj} − 2C{rj} 1 + (1 + ξ/2)S{rj} (κ+γ/2)τ cos (Ωτ) + (κ+ γ/2) sin (Ωτ) , (34) C{rj} ≡ C1j , C1j ≡ g2(rj)/κγ, (35) S{rj} ≡ 1 + ξ(1 + C{rj})− 2ξC1j , (36) where the vacuum Rabi frequency is given by Eq. (31) with effective number of interacting atoms N̄eff → N{rj}eff ≡ g2(rj)/g max. (37) C. Monte-Carlo Average and Comparison with Experimental Results In reality the number of atoms and their configuration both fluctuate in time. These fluctuations are readily taken into account if the typical atomic motion is suf- ficiently slow; one takes a stationary-atom Monte-Carlo average over configurations, adopting a finite interaction volume VF and combining a Poisson average over the number of atoms N with an average over their uniformly distributed positions rj , j = 1, . . . , N . In particular, the effective number of interacting atoms becomes N̄eff = N eff , (38) where the overbar denotes the Monte-Carlo average. Although it is not justified by the velocities listed in Table I, a stationary-atom approximation was adopted when modeling the experimental results in Refs. [29] and [31]. The correlation function was computed as the Monte-Carlo average g(2)(τ) = g (τ), (39) with g (τ) given by Eq. (34). In fact, taking a Monte- Carlo average over normalized correlation functions in this way is not, strictly, correct. In practice, first the delayed photon coincidence rate is measured, as a sepa- rate average, then subsequently normalized by the aver- age photon counting rate. The more appropriate averag- ing procedure is therefore g(2)(τ) = 〈â†(0)â†(τ)â(τ)â(0)〉{rj} 〈â†â〉{rj} , (40) or, in a form revealing more directly the relationship to Eq. (34), the average is to be weighted by the square of the photon number: g(2)(τ) = 〈â†â〉{rj} 〈â†â〉{rj} , (41) where 〈â†â〉{rj} = 1 + 2C{rj} is the intracavity photon number expectation—in sta- tionary state |ψss〉 [Eq. (19)]—for the configuration of atoms {rj}. Note that the statistical independence of forwards- scattering events that are widely separated in time yields the limit (τ) → 1, (43) which clearly holds for the average (39) as well. Equa- tion (41), on the other hand, yields g(2)(τ) → 〈â†â〉{rj} 〈â†â〉{rj} ≥ 1. (44) A value greater than unity arises because while there are fluctuations in N and {rj}, their correlation time is in- finite under the stationary-atom approximation; the ex- pected decay of the correlation function to unity is there- fore not observed. The two averaging schemes are compared in the plots of Fig. 2, which suggest that atomic beam fluctuations should have at least a small effect in the experiments; although, just how important they turn out to be is not captured at all by the figure. The actual disagreement between the model and the data is displayed in Fig. 3. The measured photon antibunching effect is significantly 630-3-6 210-1-2 FIG. 2: Second-order correlation function with Monte-Carlo average over number of atoms N and configuration {rj}. The average is taken according to Eq. (39) (thin line) and Eq. (41) (thick line) for (a) Parameter Set 1, (b) Parameter Set 2. smaller than predicted in both experiments: smaller by a factor of 4 in Fig. 3(a), as the authors of Ref. [29] explicitly state, and by a factor of a little more than 2 in Fig. 3(b). The rest of the paper is devoted to a resolution of this disagreement. It certainly arises from a breakdown of the stationary-atom approximation as suggested by Rempe et al. [29]. Physics beyond the addition of a finite corre- lation time for fluctuations of N(t) and {rj(t)} is needed, however. We aim to show that the single most important factor is the alignment of the atomic beam. 52.50-2.5-5 0.90.50-0.5-0.9 ••••• ••••••••••• FIG. 3: Second-order correlation function with Monte-Carlo average, Eq. (41), over number of atoms N and configuration {rj} compared with the experimental data from (a) Fig. 4(a) of Ref. [29] (Parameter Set 1) and (b) Fig. 4 of Ref. [31] (Parameter Set 2). IV. DELAYED PHOTON COINCIDENCES FOR AN ATOMIC BEAM We return now to the full atomic beam simulation out- lined in Sec. II. With the beam perpendicular to the cavity axis, the rate of change of the dipole coupling con- stants might be characterized by the cavity-mode transit time, determined from the mean atomic speed and the cavity-mode waist. Taking the values of these quanti- ties from Table I, the experiment of Rempe et al. has w0/v̄source = 182nsec, which should be compared with a vacuum-Rabi-oscillation decay time 2(κ + γ/2)−1 = 94nsec, while Foster et al. have w0/v̄source = 66nsec and a decay time 2(κ+ γ/2)−1 = 29nsec. In both cases, the ratio between the transit time and decay time is ∼ 2; thus, we might expect the internal state dynamics to fol- low the atomic beam fluctuations adiabatically, to a good approximation at least, thus providing a justifying for the stationary-atom approximation. Figure 3 suggests that this is not so. Our first task, then, is to see how well in practice the adiabatic following assertion holds. A. Monte-Carlo Simulation of the Atomic Beam: Effect of Beam Misalignment Atomic beam fluctuations induce fluctuations of the intracavity photon number expectation, as illustrated by the examples in Figs. 4 and 5. Consider the two curves (a) in these figures first, where the atomic beam is aligned perpendicular to the cavity axis. The ring- ing at regular intervals along these curves is the tran- sient response to enforced cavity-mode quantum jumps— jumps enforced to sample the quantum fluctuations effi- ciently (see Sec. IVB). Ignoring these perturbations for the present, we see that with the atomic beam aligned perpendicular to the cavity axis the fluctuations evolve more slowly than the vacuum Rabi oscillation—at a simi- lar rate, in fact, to the vacuum Rabi oscillation decay. As anticipated, an approximate adiabatic following is plau- sible. Consider now the two curves (b); these introduce a 9.6mrad misalignment of the atomic beam, following up on the comment of Foster et al. [31] that misalignments as large as 1◦ (17.45mrad) might occur. The changes in the fluctuations are dramatic. First, their size increases, though by less on average than it might appear. The altered distributions of intracavity photon numbers are shown in Fig. 6. The means are not so greatly changed, but the variances (measured relative to the square of the mean) increase by a factor of 2.25 in Fig. 4 and 1.45 in Fig. 5. Notably, the distribution is asymmetric, so the most probable photon number lies below the mean. The asymmetry is accentuated by the tilt, especially for Parameter Set 1 [Fig. 6(a)]. More important than the change in amplitude of the fluctuations, though, is the increase in their frequency. Again, the most significant effect occurs for Parameter Set 1 (Fig. 4), where the frequency with a 9.6mrad tilt approaches that of the vacuum Rabi oscillation itself; clearly, there can be no adiabatic following under these conditions. Indeed, the net result of the changes from Fig. 4(a) to Fig. 4(b) is that the quantum fluctuations, initiated in the simulation by quantum jumps, are com- pletely lost in a background of classical noise generated by the atomic beam. It is clear that an atomic beam misalignment of sufficient size will drastically reduce the photon antibunching effect observed. For a more quantitative characterization of its effect, we carried out quantum trajectory simulations in a one- quantum truncation (without quantum jumps) and com- puted the semiclassical photon number correlation func- g(2)sc (τ) = 〈(â†â)(t)〉REC〈(â†â)(t+ τ)〉REC 〈(â†â)(t)〉REC , (45) where the overbar denotes a time average (in practice an average over an ensemble of sampling times tk). The photon number expectation was calculated in two ways: 740 760 780 FIG. 4: Typical trajectory of the intracavity photon number expectation for Parameter Set 1: (a) atomic beam aligned perpendicular to the cavity axis, (b) with a 9.6mrad tilt of the atomic beam. The driving field strength is E/κ = 2.5× 10−2. 560 590 620 FIG. 5: As in Fig. 4 but for Parameter Set 2. 〈a†a〉REC/〈a †a〉REC 31.50 FIG. 6: Distribution of intracavity photon number expecta- tion with the atom beam perpendicular to the cavity axis (thin line) and a 9.6mrad tilt of the atomic beam (thick line): (a) Parameter Set 1, (b) Parameter Set 2. first, by assuming that the conditional state adiabatically follows the fluctuations of the atomic beam, in which case, from Eq. (42), we may write 〈(â†â)(t)〉REC = 1 + 2C{rj(t)} , (46) and second, without the adiabatic assumption, in which case the photon number expectation was calculated from the state vector in the normal way. Correlation functions computed for different atomic beam tilts according to this scheme are plotted in Figs. 7 and 8. In each case the curves shown in the left column assume adiabatic following while those in the right col- umn do not. The upper-most curves [frames (a) and (e)] hold for a beam aligned perpendicular to the cavity axis and those below [frames (b)–(d) and (f)–(h)] show the effects of increasing misalignment of the atomic beam. A number of comments are in order. Consider first the aligned atomic beam. Correlation times read from the figures are in approximate agreement with the cavity- mode transit times computed above: the numbers are 191nsec and 167nsec from frames (a) and (e), respec- tively, of Fig. 7, compared with w0/v̄oven = 182nsec; and 68nsec and 53nsec from frames (a) and (e) of Fig. 8, re- spectively, compared with w0/v̄oven = 66nsec. The num- bers show a small decrease in the correlation time when the adiabatic following assumption is lifted (by 10-20%) but no dramatic change; and there is a corresponding small increase in the fluctuation amplitude. Consider now the effect of an atomic beam tilt. Here the changes are significant. They are most evident in frames (d) and (h) of each figure, but clear already in frames (c) and (g) of Fig. 7, and frames (b) and (f) of Fig. 8, where the tilts are close to the tilt used to generate Figs. 4(b) and 5(b) (also to those used for the data fits in Sec. IVB). There is first an increase in the magnitude of the fluctuations—the factors 2.25 and 1.45 noted above— but, more significant, a separation of the decay into two 40-4 40-4 FIG. 7: Semiclassical correlation function for Parameter Set 1, with adiabatic following of the photon number (left column) and without adiabatic following (right column); for atomic beam tilts of (a,e) 0mrad, (b,f) 4mrad, (c,g) 9mrad, (d,h) 13mrad. pieces: a central component, with short correlation time, and a much broader component with correlation time larger than w0/v̄oven. Thus, for a misaligned atomic beam, the dynamics become notably nonadiabatic. Our explanation of the nonadiabaticity begins with the observation that any tilt introduces a velocity compo- nent along the standing wave, with transit times through a quarter wavelength of λ/4v̄oven sin θ = 86nsec in the Rempe et al. [29] experiment and λ/4v̄oven sin θ = 60nsec in the Foster et al. [31] experiment. Compared with the transit time w0/v̄oven, these numbers have moved closer to the decay times of the vacuum Rabi oscillation— 94nsec and 29nsec, respectively. Note that the distances traveled through the standing wave during the cavity- mode transit, in time w0/v̄oven, are w0 sin θ = 0.53λ (Pa- rameter Set 1) and w0 sin θ = 0.28λ (Parameter Set 2). It is difficult to explain the detailed shape of the correla- tion function under these conditions. Speaking broadly, though, fast atoms produce the central component, the short correlation time associated with nonadiabatic dy- namics, while slow atoms produce the background com- ponent with its long correlation time, which follows from an adiabatic response. Increased tilt brings greater sep- aration between the responses to fast and slow atoms. Simple functional fits to the curves in frame (g) of Fig. 7 and frame (f) of Fig. 8 yield short correlation times 80-8 80-8 FIG. 8: As in Fig. 7 but for Parameter Set 2 and atomic beam tilts of (a,e) 0mrad, (b,f) 10mrad, (c,g) 17mrad, (d,h) 34mrad. of 40-50nsec and 20nsec, respectively. Consistent num- bers are recovered by adding the decay rate of the vac- uum Rabi oscillation to the inverse travel time through a quarter wavelength; thus, (1/94+1/86)−1nsec = 45nsec and (1/29+ 1/60)−1nsec = 20nsec, respectively, in good agreement with the correlation times deduced from the figures. The last and possibly most important thing to note is the oscillation in frames (g) and (h) of Fig. 7 and frame (h) of Fig. 8. Its frequency is the vacuum Rabi frequency, which shows unambiguously that the oscilla- tion is caused by a nonadiabatic response of the intra- cavity photon number to the fluctuations of the atomic beam. For the tilt used in frame (g) of Fig. 7, the tran- sit time through a quarter wavelength is approximately equal to the vacuum-Rabi-oscillation decay time, while it is twice that in frame (f) of Fig. 8. As the tilts used are close to those giving the best data fits in Sec. IVB, this would suggest that atomic beam misalignment places the experiment of Rempe et al. [29] further into the nonadi- abatic regime than that of Foster et al. [31], though the tilt is similar in the two cases. The observation is consis- tent with the greater contamination by classical noise in Fig. 4(b) than in Fig. 5(b) and with the larger departure of the Rempe et al. data from the stationary-atom model in Fig. 3. B. Simulation Results and Data Fits The correlation functions in the right-hand column of Figs. 7 and 8 account for atomic-beam-induced classi- cal fluctuations of the intracavity photon number. While some exhibit a vacuum Rabi oscillation, the signals are, of course, photon bunched; a correlation function like that of Fig. 7(g) provides evidence of collective strong cou- pling, but not of strong coupling of the one-atom kind, for which a photon antibunching effect is needed. We now carry out full quantum trajectory simulations in a two-quanta truncation to recover the photon antibunch- ing effect—i.e., we bring back the quantum jumps. In the weak-field limit the normalized photon corre- lation function is independent of the amplitude of the driving field E [Eqs. (30) and (34)]. The forwards photon scattering rate itself is proportional to (E/κ)2 [Eq. (42)], and must be set in the simulations to a value very much smaller than the inverse vacuum-Rabi-oscillation decay time [Eq. (22)]. Typical values of the intracavity pho- ton number were ∼ 10−7 − 10−6. It is impractical, un- der these conditions, to wait for the natural occurrence of forwards-scattering quantum jumps. Instead, cavity- mode quantum jumps are enforced at regular sample times tk [see Figs. 4(a) and 5(a)]. Denoting the record with enforced cavity-mode jumps by REC, the second- order correlation function is then computed as the ratio of ensemble averages g(2)(τ) = 〈(â†â)(tk)〉REC〈(â†â)(tk + τ)〉REC 〈(â†â)(tl)〉REC , (47) where the sample times in the denominator, tl, are cho- sen to avoid the intervals—of duration a few correlation times—immediately after the jump times tk; this ensures that both ensemble averages are taken in the steady state. With the cut-off parameter [Eq. (6)] set to F = 0.01, the number of atoms within the interaction volume typically fluctuates around N(t) ∼ 400-450 atoms for Parameter Set 1 and N(t) ∼ 280-320 atoms for Parameter Set 2; in a two-quanta truncation, the corresponding numbers of state amplitudes are ∼ 90, 000 (Parameter Set 1) and ∼ 45, 000 (Parameter Set 2). Figure 9 shows the computed correlation functions for various atomic beam tilts. We select from a series of such results the one that fits the measured correlation function most closely. Optimum tilts are found to be 9.7mrad for the Rempe et al. [29] experiment and 9.55mrad for the experiment of Foster et al. [31]. The best fits are displayed in Fig. 10. In the case of the Foster et al. data the fit is extremely good. The only obvious disagreement is that the fitted frequency of the vacuum Rabi oscillation is possibly a little low. This could be corrected by a small increase in atomic beam density—the parameter N̄eff— which is only known approximately from the experiment, in fact by fitting the formula (31) to the data. The fit to the data of Rempe et al. [29] is not quite so good, but still convincing with some qualifications. Note, in particular, that the tilt used for the fit might be judged a little too large, since the three central minima in Fig. 10(a) are almost flat, while the data suggest they should more closely follow the curve of a damped oscil- lation. As the thin line in the figure shows, increasing the tilt raises the central minimum relative to the two on the side; thus, although a better fit around κτ = 0 is obtained, the overall fit becomes worse. This trend results from the sharp maximum in the semiclassical cor- relation function of Fig. 7(g), which becomes more and more prominent as the atomic beam tilt is increased. The fit of Fig. 10(b) is extremely good, and, although it is not perfect, the thick line in Fig. 10(a), with a 9.7mrad tilt, agrees moderately well with the data once the un- certainty set by shot noise is included, i.e., adding error bars of a few percent (see Fig. 13). Thus, leaving aside possible adjustments due to omitted noise sources, such as spontaneous emission—to which we return in Sec. V— and atomic and cavity detunings, the results of this and the last section provide strong support for the proposal that the disagreement between theory and experiment presented in Fig. 3 arises from an atomic beam misalign- ment of approximately 0.5◦. One final observation should be made regarding the fit to the Rempe et al. [29] data. Figure 11 replots the comparison made in Fig. 10(a) for a larger range of time delays. Frame (a) plots the result of our sim- ulation for a perfectly aligned atomic beam, and frames (b) and (c) shows the results, plotted in Fig. 10(a), cor- responding to atomic beam tilts of θ = 9.7mrad and 10mrad, respectively. The latter two plots are overlayed by the experimental data. Aside from the reduced am- plitude of the vacuum Rabi oscillation, in the presence of the tilt the correlation function exhibits a broad back- ground arising from atomic beam fluctuations. Notably, the background is entirely absent when the atomic beam 52.50-2.5-5 210-1-2 FIG. 9: Second-order correlation function from full quantum trajectory simulations with a two-quanta truncation: (a) Pa- rameter Set 1 and θ = 0mrad (thick line), 7mrad (medium line), 12mrad (thin line); (b) Parameter Set 2 and θ = 0mrad (thick line), 10mrad (medium line), 17mrad (thin line). 52.50-2.5-5 0.90.50-0.5-0.9 FIG. 10: Best fits to experimental results: (a) data from Fig. 4(a) of Ref. [29] are fitted with Parameter Set 1 and θ = 9.7mrad (thick line) and 10mrad (thin line); (b) data from Fig. 4 of Ref. [31] are fitted with Parameter Set 2 and θ = 9.55mrad. Averages of (a) 200,000 and (b) 50,000 samples were taken with a cavity-mode cut-off F = 0.01. is aligned. The experimental data exhibit just such a background (Fig. 3(a) of Ref. [29]); moreover, an esti- mate, from Fig. 11, of the background correlation time yields approximately 400nsec, consistent with the exper- imental measurement. It is significant that this number is more than twice the transit time, w0/v̄oven = 182nsec, and therefore not explained by a perpendicular transit across the cavity mode. In fact the background mimics the feature noted for larger tilts in Figs. 7 and 8; as men- tioned there, it appears to find its origin in the separa- tion of an adiabatic (slowest atoms) from a nonadiabatic (fastest atoms) response to the density fluctuations of the atomic beam. Note, however, that a correlation time of 400nsec ap- pears to be consistent with a perpendicular transit across the cavity when the cavity-mode transit time is defined as 2w0/v̄oven = 364nsec, or, using the peak rather than average velocity, as 4w0/ πv̄oven = 411nsec; the latter definition was used to arrive at the 400nsec quoted in Ref. [29]. There is, of course, some ambiguity in how a transit time should be defined. We are assuming that the time to replace an ensemble of interacting atoms with a statistically independent one—which ultimately is what determines the correlation time—is closer to w0/v̄oven than 2w0/v̄oven. In support of the assumption we recall that the number obtained in this way agrees with the semiclassical correlation function for an aligned atomic beam [Figs. 7 and 8, frame (a)]. C. Mean-Doppler-Shift Compensation Foster et al. [31], in an attempt to account for the disagreement of their measurements and the stationary- atom model, extended the results of Sec. III B to include 52.50-2.5-5 FIG. 11: Second-order correlation function from full quantum trajectory simulations with a two-quanta basis for Parameter Set 1 and (a) θ = 0mrad, (b) θ = 9.7mrad, (c) θ = 10mrad. Averages of (a) 15,000, and (b) and (c) 200,000 samples were taken with a cavity-mode cut-off F = 0.01. an atomic detuning. They then fitted the data using the following procedure: (i) the component of atomic velocity along the cavity axis is viewed as a Doppler shift from the stationary-atom resonance, (ii) the mean shift is assumed to be offset by an adjustment of the driving field frequency (tuning to moving atoms) at the time the data are taken, and (iii) an average over residual detunings— deviations from the mean—is taken in the model, i.e., the detuning-dependent generalization of Eq. (34). The approach yields a reasonable fit to the data (Fig. 6 of Ref. [31]). The principal difficulty with this approach is that a standing-wave cavity presents an atom with two Doppler shifts, not one. It seems unlikely, then, that adjusting the driving field frequency to offset one shift and not the other could compensate for even the average effect of the atomic beam tilt. This difficulty is absent in a ring cavity, though, so we first assess the performance of the outlined prescription in the ring-cavity case. In a ring cavity, the spatial dependence of the coupling constant [Eq. (11)] is replaced by g(rj(t)) = gmax√ exp(ikzj(t)) exp x2j (t) + y j (t) where the factor 2 ensures that the collective coupling strength and vacuum Rabi frequency remain the same. Figure 12(a) shows the result of a numerical implemen- tation of the proposed mean-Doppler-shift compensation for an atomic beam tilt of 17.3mrad, as used in Fig. 6 of Ref. [31]. It works rather well. The compensated curve (thick line) almost recovers the full photon antibunching 630-3-6 FIG. 12: Doppler-shift compensation for a misaligned atomic beam in (a) ring and (b) standing-wave cavities (Parameter Set 2). The second-order correlation function is computed with the atomic beam perpendicular to the cavity axis (thin line), a 17.3mrad tilt of the atomic beam (medium line), and a 17.3mrad tilt plus compensating detuning of the cavity and stationary atom resonances ∆ω/κ = kv̄oven sin θ/κ = 0.916 (thick line). effect that would be seen with an aligned atomic beam (thin line). The degradation that remains is due to the uncompensated dispersion of velocities (Doppler shifts) in the atomic beam. For the case of a standing-wave cavity, on the other hand, the outcome is entirely different. This is shown by Fig. 12(b). There, offsetting one of the two Doppler shifts only makes the degradation of the photon antibunching effect worse. In fact, we find that any significant detuning of the driving field from the stationary atom resonance is highly detrimental to the photon antibunching effect and inconsistent with the Foster et al. data. V. INTRACAVITY PHOTON NUMBER The best fits displayed in Fig. 10 were obtained from simulations with a two-quanta truncation and premised upon the measurements being made in the weak-field limit. The strict requirement of the limit sets a severe constraint on the intracavity photon number. We con- sider now whether the requirement is met in the experi- ments. Working from Eqs. (23a) and (23b), and the solu- tion to Eq. (19), a fixed configuration {rj} of N atoms (Sec. III B) yields photon scattering rates [27, 28, 53] Rforwards = 2κ〈â†â〉REC = 2κ 1 + 2C{rj} , (49a) Rside = γ 〈σ̂k+σ̂k−〉 g(rk) 1 + 2C{rj} = 2C{rj}2κ〈â †â〉REC, (49b) with ratio Rside Rforwards = 2C{rj} = eff g ∼ 2N̄effg . (50) The weak-field limit [Eq. (22)] requires that the greater of the two rates be much smaller than 1 (κ + γ/2); it is not necessarily sufficient that the forwards scattering rate be low. The side scattering (spontaneous emission) rate is larger than the forwards scattering rate in both of the experiments being considered—larger by a large factor of 70–80. Thus, from Eqs. (49a) and (50), the constraint on intracavity photon number may be written as 〈â†â〉 ≪ 1 + γ/2κ 8N̄effg2max/κγ , (51) where, from Table I, the right-hand side evaluates as 1.2×10−2 for Parameter Set 1 and 4.7×10−3 for Param- eter Set 2, while the intracavity photon numbers inferred from the experimental count rates are 3.8×10−2 [29] and 7.6 × 10−3 [31]. It seems that neither experiment satis- fies condition (51). As an important final step we should therefore relax the weak-driving-field assumption (pho- ton number ∼ 10−7–10−6 in the simulations) and assess what effect this has on the data fits; can the simulations fit the inferred intracavity photon numbers as well? To address this question we extended our simulations to a three-quanta truncation of the Hilbert space with cavity-mode cut-off changed from F = 0.01 to F = 0.1. With the changed cut-off the typical number of atoms in the interaction volume is halved: N(t) ∼ 180–220 atoms for Parameter Set 1 and N(t) ∼ 150–170 atoms for Parameter Set 2, from which the numbers of state amplitudes (including three-quanta states) increase to 1, 300, 000 and 700, 000, respectively. The new cut-off introduces a small error in N̄eff , hence in the vacuum Rabi frequency, but the error is no larger than one or two percent. At this point an additional approximation must be made. At the excitation levels of the experiments, even a three-quanta truncation is not entirely adequate. Clumps of three or more side-scattering quantum jumps can oc- cur, and these are inaccurately described in a three- quanta basis. In an attempt to minimize the error, we ar- tificially restrict (through a veto) the number of quantum jumps permitted within some prescribed interval of time. The accepted number was set at two and the time inter- val to 1κ−1 for Parameter Set 1 and 3κ−1 for Parameter 52.50-2.5-5 0.90.50-0.5-0.9 FIG. 13: Second-order correlation function from full quan- tum trajectory simulations with a three-quanta truncation and atomic beam tilts as in Fig. 10: (a) Parameter Set 1, mean intracavity photon number 〈a†a〉 = 6.7×10−3; (b) Parameter Set 2, mean intracavity photon numbers 〈a†a〉 = 2.2 × 10−4, 5.7×10−4, 1.1×10−3, and 1.7×10−3 (thickest curve to thinest curve). Averages of 20,000 samples were taken with a cavity- mode cut-off F = 0.1. Shot noise error bars are added to the data taken from Ref. [29]. Set 2 (the correlation time measured in cavity lifetimes is longer for Parameter Set 2). With these settings approx- imately 10% of the side-scattering jumps were neglected at the highest excitation levels considered. The results of our three-quanta simulations appear in Fig. 13; they use the optimal atomic beam tilts of Fig. 10. Figure 13(a) compares the simulation with the data of Rempe et al. [29] at an intracavity photon number that is approximately six times smaller than what we estimate for the experiment (a more realistic simulation requires a higher level of truncation and is impossible for us to han- dle numerically). The overall fit in Fig. 13 is as good as that in Fig. 10, with a slight improvement in the relative depths of the three central minima. A small systematic disagreement does remain, however. We suspect that the atomic beam tilt used is actually a little large, while the contribution to the decoherence of the vacuum Rabi os- cillation from spontaneous emission should be somewhat more. We are satisfied, nevertheless, that the data of Rempe et al. [29] are adequately explained by our model. Results for the experiment of Foster et al [31] lead in a rather different direction. They are displayed in Fig. 13(b), where four different intracavity photon num- bers are considered. The lowest, 〈â†â〉 = 2.2 × 10−4, reproduces the weak-field result of Fig. 10(b). As the photon number is increased, the fit becomes progressively worse. Even at the very low value of 5.7× 10−4 intracav- ity photons, spontaneous emission raises the correlation function for zero delay by a noticeable amount. Then we obtain g(2)(0) > 1 at the largest photon number consid- ered. Somewhat surprisingly, even this photon number, 〈â†â〉 = 1.7×10−3, is smaller than that estimated for the experiment—smaller by a factor of five. Our simulations therefore disagree significantly with the measurements, despite the near perfect fit of Fig. 10(b). The simplest resolution would be for the estimated photon number to be too high. A reduction by more than an order of mag- nitude is needed, however, implying an unlikely error, considering the relatively straightforward method of in- ference from photon counting rates. This anomaly, for the present, remains unresolved. VI. CONCLUSIONS Spatial variation of the dipole coupling strength has for many years been a particular difficulty for cavity QED at optical frequencies. The small spatial scale set by the optical wavelength makes any approach to a resolution a formidable challenge. There has neverthe- less been progress made with cooled and trapped atoms [13, 14, 32, 59, 60, 61], and in semiconductor systems [15, 16, 17] where the participating ‘atoms’ are fixed. The earliest demonstrations of strong coupling at opti- cal frequencies employed standing-wave cavities and ther- mal atomic beams, where control over spatial degrees of freedom is limited to the alignment of the atomic beam. Of particular note are the measurements of photon anti- bunching in forwards scattering [29, 30, 31]. They pro- vide a definitive demonstration of strong coupling at the one-atom level; although many atoms might couple to the cavity mode at any time, a significant photon anti- bunching effect occurs only when individual atoms are strongly coupled. Spatial effects pose difficulties of a theoretical na- ture as well. Models that ignore them can point the direction for experiments, but fail, ultimately, to ac- count for experimental results. In this paper we have addressed a long-standing disagreement of this kind— disagreement between the theory of photon antibunch- ing in forwards scattering for stationary atoms in a cav- ity [25, 26, 27, 28, 29] and the aforementioned experi- ments [29, 30, 31]. Ab initio quantum trajectory sim- ulations of the experiments have been carried out, in- cluding a Monte-Carlo simulation of the atomic beam. Importantly, we allow for a misalignment of the atomic beam, since this was recognized as a critical issue in Ref. [31]. We conclude that atomic beam misalignment is, indeed, the most likely reason for the degradation of the measured photon antibunching effect from pre- dicted results. Working first with a two-quanta trunca- tion, suitable for the weak-field limit, data sets measured by Rempe et al. [29] and Foster et al. [31] were fitted best by atomic beam tilts from perpendicular to the cav- ity axis of 9.7mrad and 9.55mrad, respectively. Atomic motion is recognized as a source of decorrela- tion omitted from the model used to fit the measurements in Ref. [29]. We found that the mechanism is more com- plex than suggested there, however. An atomic beam tilt of sufficient size results in a nonadiabatic response of the intracavity photon number to the inevitable density fluctuations of the beam. Thus classical noise is writ- ten onto the forwards-scattered photon flux, obscuring the antibunched quantum fluctuations. The parameters of Ref. [29] are particularly unfortunate in this regard, since the nonadiabatic response excites a bunched vac- uum Rabi oscillation, which all but cancels out the anti- bunched oscillation one aims to measure. Although both of the experiments modeled operate at relatively low forwards scattering rates, neither is strictly in the weak-field limit. We have therefore extended our simulations—subject to some numerical constraints—to assess the effects of spontaneous emission. The fit to the Rempe et al. data [29] was slightly improved. We noted that the optimum fit might plausibly be obtained by adopting a marginally smaller atomic beam tilt and allowing for greater decorrelation from spontaneous emis- sion, though a more efficient numerical method would be required to verify this possibility. The fit to the Fos- ter et al. data [31] was highly sensitive to spontaneous emission. Even for an intracavity photon number five times smaller than the estimate for the experiment, a large disagreement with the measurement appeared. No explanation of the anomaly has been found. We have shown that cavity QED experiments can call for elaborate and numerically intensive modeling before a full understanding, at the quantitative level, is reached. Using quantum trajectory methods, we have significantly increased the scope for realistic modeling of cavity QED with atomic beams. While we have shown that atomic beam misalignment has significantly degraded the mea- surements in an important set of experiments in the field, this observation leads equally to a positive con- clusion: potentially, nonclassical photon correlations in cavity QED can be observed at a level at least ten times higher than so far achieved. Acknowledgements This work was supported by the NSF under Grant No. PHY-0099576 and by the Marsden Fund of the RSNZ. [1] P. R. Berman, ed., Cavity Quantum Electrodynamics, Advances in Atomic, Molecular, and Optical Physics, Supplement 2 (Academic Press, San Diego, 1994). [2] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (2001). [3] H. Mabuchi and A. Doherty, Nature 298, 1372 (2002). [4] K. J. Vahala, Nature 424, 839 (2003). [5] G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, Nature Physics 2, 81 (2006). [6] H. J. 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0704.1687

Studies on optimizing potential energy functions for maximal intrinsic hyperpolarizability

Studies on optimizing potential energy functions for maximal intrinsic hyperpolarizability Juefei Zhou, Urszula B. Szafruga, David S. Watkins* and Mark G. Kuzyk Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814; and *Department of Mathematics We use numerical optimization to study the properties of (1) the class of one-dimensional po- tential energy functions and (2) systems of point charges in two-dimensions that yield the largest hyperpolarizabilities, which we find to be within 30% of the fundamental limit. We investigate the character of the potential energy functions and resulting wavefunctions and find that a broad range of potentials yield the same intrinsic hyperpolarizability ceiling of 0.709. I. INTRODUCTION Materials with large nonlinear-optical suscep- tibilities are central for optical applications such as telecommunications,[1] three-dimensional nano- photolithography,[2, 3] and making new materials[4] for novel cancer therapies.[5] The fact that quantum calcu- lations show that there is a limit to the nonlinear-optical response[6, 7, 8, 9, 10, 11] is both interesting from the basic science perspective; and, provides a target for making optimized materials. In this work, we focus on the second-order susceptibility and the underlying molecular hyperpolarizability, which is the basis of electro-optic switches and frequency doublers. The fundamental limit of the off-resonance hyperpo- larizability is given by,[8] βMAX = , (1) where N is the number of electrons and E10 the energy difference between the first excited state and the ground state, E10 = E1 − E0. Using Equation 1, we can de- fine the off-resonant intrinsic hyperpolarizability, βint, as the ratio of the actual hyperpolarizability (measured or calculated), β, to the fundamental limit, βint = β/βMAX . (2) We note that since the dispersion of the fundamental limit of β is also known, [12] it is possible to calculate the intrinsic hyperpolarizability at any wavelength. In the present work, we treat only the zero-frequency limit. Until recently, the largest nonlinear susceptibilities of the best molecules fell short of the fundamental limit by a factor of 103/2, [10, 13, 14] so the very best molecules had a value of βint = 0.03. Since a Sum-Over-States (SOS) calculation of the hyperpolarizability[15] using the ana- lytical wavefunctions of the clipped harmonic oscillator yields a value βint = 0.57,[14] the factor-of-thirty gap is not of a fundamental nature. Indeed, recently, it was re- ported that a new molecule with asymmetric conjugation modulation has a measured value of βint = 0.048.[16] To investigate how one might make molecules with a larger intrinsic hyperpolarizability, Zhou and cowork- ers used a numerical optimization process where a trial potential energy function is entered as an input, and the code iteratively deforms the potential energy func- tion until the intrinsic hyperpolarizability, calculated from the resulting wavefunctions, converges to a local maximum.[17] In this work, a hyperbolic tangent func- tion was used as the starting potential due to the fact that it is both asymmetric yet relatively flat away from the ori- gin. This calculation was one-dimensional and included only one electron, so electron correlation effects were ignored. Furthermore, the intrinsic hyperpolarizability was calculated using the new dipole-free sum-over-states expression[18] and only 15 excited states were included. The resulting optimized potential energy function showed strong oscillations, which served to separate the spatial overlap between the energy eigenfunctions. This led Zhou and coworkers to propose that modulated conjugation in the bridge between donor and acceptor ends of such molecules may be a new paradigm for making molecules with higher intrinsic hyperpolarizability.[17] Based on this paradigm, Pérez Moreno reported mea- surements of a class of chromophores with varying de- gree of modulated conjugation.[16] The best measured intrinsic hyperpolarizability was βint = 0.048, about 50% larger than the best previously-reported. Given the mod- est degree of conjugation modulation for this molecule, this new paradigm shows promise for further improve- ments. In the present work, we extend Zhou’s calculations to a larger set of starting potentials. To circumvent trun- cation problems associated with sum-over-states calcula- tions, we instead determine the hyperpolarizability using a finite difference technique. The optimization procedure is then applied to this non-perturbative hyperpolarizabil- To study the effects of geometry on the hyperpo- larizability, Kuzyk and Watkins calculated the hyper- polarizability of various arrangements of point charges, representing nuclei, in two-dimensions using a two- dimensional Coulomb potential.[19] In the present con- tribution, we apply our numerical optimization technique to determine the arrangement and charges of the nuclei in a planar molecule that maximize the intrinsic hyper- polarizability. http://arxiv.org/abs/0704.1687v1 II. THEORY In our previous work, we used a finite-state SOS model of the hyperpolarizability that derives from perturbation theory (we used both the standard Orr andWard SOS ex- pression, βSOS ,[15] and the newer dipole free expression, βDF [18]). The use of a finite number of states in lieu of the full infinite sums can result in inaccuracies, so, in the present work, we use the non-perturbative approach, as follows. We begin by solving the 1-d Schrodinger Equa- tion on the interval a < x < b for the ground state wave- function ψ(x,E) of an electron in a potential well defined by V (x) and in the presence of a static electric field, E, that adds to the potential δV = −exE. From this, the off-resonant hyperpolarizability is calculated with numer- ical differentiation, i.e. using finite differences, yielding βNP = |ψ(x,E)|2 ex dx . (3) Equation 3 is evaluated using the standard second-order approximation to the second derivative: f ′′(z) ≈ f(z + h)− 2f(z) + f(z − h) with several h values h0, h0/5, h0/25, . . . . We then refine these values by Richardson extrapolation [20] and obtain our estimate from the two closest extrapolated values. Our computational mesh consists of 200 quadratic fi- nite elements with a total of 399 degrees of freedom. The potential energy function is a cubic spline with 40 degrees of freedom. Thus the numerical calculations in regions where the potential function is represented by 3 points in the spline are covered by 15 elements with a total of about 30 degrees of freedom. Calculating βint from Equations 3, 2 and 1 for a specific potential, we use an optimization algorithm that contin- uously varies the potential in a way that maximizes βint. We also compute the matrix[17, 26] τ (N)mp = δm,p − xmax10 xmax10 , (4) where xmax10 is the magnitude of the fundamental limit of the position matrix element x10 for a one electron system, and is given by, xmax10 = 2mE10 . (5) Each matrix element of τ (N), indexed by m and p, is a measure of how well the (m, p) sum rule is obeyed when truncated to N states. If the sum rules are ex- actly obeyed, τ mp = 0 for all m and p. We note that if the sum rules are truncated to an N-state model, the sum rules indexed by a large value of m or p (i.e. m, p ∼ N) are disobeyed even when the position matrix elements and energies are exact. We have found that the values mp are small for exact wavefunctions when m < N/2 and p < N/2. So, when evaluating the τ matrix to test our calculations, we consider only the components m≤N/2,p≤N/2 We observe that when using more than about 40 states in SOS calculations of the hyperpolarizability only a marginal increase of accuracy results when the poten- tial energy function is parameterized with 400 degrees of freedom. So, to ensure overkill, we use 80 states when calculating the τ matrix or the hyperpolarizability with an SOS expression so that truncation errors are kept to a minimum. Since the hyperpolarizability depends crit- ically on the transition dipole moment from the ground state to the excited states, we use the value of τ 00 as one important test of the accuracy of the calculated wave- functions. Additionally, we use the standard deviation of τ (N), ∆τ (N) = , (6) which quantifies, on average, how well the sum rules are obeyed in aggregate, making ∆τ (N) a broader test of the accuracy of a large set of wavefunctions. Our code is written in MATLAB. For each trial po- tential we use a quadratic finite element method [21] to approximate the Schrödinger eigenvalue problem and the implicitly restarted Arnoldi method [22] to compute the wave functions and energy levels. To optimize β we use the Nelder-Mead simplex algorithm [23]. As described in our previous work,[17] we perform op- timization, but this time using the exact intrinsic hyper- polarizability β = βNP /βMAX , where βMAX is the fun- damental limit of the hyperpolarizability, which is pro- portional to E 10 . To determine E10 ≡ E1 −E0, we also calculate the first excited state energy E1. III. RESULTS AND DISCUSSIONS Figure 1 shows an example of the optimized poten- tial energy function after 7,000 iterations when starting with the potential V (x) = 0 and optimizing the non- perturbative intrinsic hyperpolarizability βNP /βMAX as calculated with Equation 3. Also shown are the eigen- functions of the first 15 states computed from the opti- mized potential. First, we note that the potential energy function shows the same kinds of wiggles as in our original paper,[17] though not of sufficient amplitude to localize the wavefunctions. For the starting potentials we have investigated, our results fall into two broad classes. In the first, three common features are: (1) The best intrinsic hyperpolar- izabilities are near βint = 0.71; (2) the best potentials have a series of wiggles; and (3) the systems behave as a 0 5 10 15 20 FIG. 1: Optimized potential energy function and first 15 wavefunctions after 7,000 iterations. Starting potential is V (x) = 0, using the non-perturbative hyperpolarizability for optimization. limited-state model. In the second class of starting po- tentials, (2) the wiggles are much less pronounced and (3) more states contribute evenly. Figure 1 is an example of a Class II potential. However, in both classes, the max- imum calculated intrinsic hyperpolarizability appears to be bounded by βint = 0.71. Using the set of potentials from both classes that lead to optimized βNP /βMAX , we calculate the lowest 80 eigenfunctions and eigenvalues, from which we calculate βDF and βSOS . In most cases, we find that the three different formulas for β converge to the same value when only the first 20 excited states are used (using 80 states, the three are often the same to within at least 4 decimal places) and τ00 ≈ 10−4, showing that the ground state sum rules are well obeyed. Further- more, the rms deviation of the τ matrix when including 40 states leads to τ (80) < 0.001. Figure 2 shows an example of the optimized potential energy function when starting with the potential V (x) = tanhx and optimizing the exact (non-perturbative) in- trinsic hyperpolarizability. Also shown are the eigenfunc- tions of the first 15 states computed with the optimized potential. First, we note that the potential energy func- tion shows the same kinds of wiggles as in our original paper;[17] and only 2 excited state wavefunctions and the ground state are localized in the first deep well - placing this system in Class I. The observation that such potentials lead to hyper- polarizabilities that are near the fundamental limit mo- tivated Zhou and coworkers to suggest that molecules with modulated conjugation may have enhanced intrin- sic hyperpolarizabilities.[17] A molecule with a modu- lated conjugations bridge between the donor and accep- tor end was later shown to have record-high intrinsic hyperpolarizability.[16] As such, this result warrants a more careful analysis. It is worthwhile to compare our present results charac- 0 5 10 15 20 FIG. 2: Optimized potential energy function and first 15 wavefunctions after 8,000 iterations. Starting potential is V (x) = tanh(x), using the non-perturbative hyperpolarizabil- ity for optimization. terized by Figure 2 with our past work,[17] particularly for the purpose of examining the impact of the approx- imations used in the previous work.[17] Figure 3 shows the optimized potential and wavefunctions obtained by Zhou and coworkers using a 15-state model and opti- mizing the dipole-free intrinsic hyperpolarizability. Since only 15 states were used, the SOS expression for β did not fully converge; making the result inaccurate as suggested by the fact that βSOS and βDF did not agree. How- ever, since the code focused on optimizing the dipole-free form of β, and τ00 was small when βint was optimized, the dipole-free expression may have converged to a rea- sonably accurate value while the commonly-used SOS expression was inaccurate. Indeed, it was found that βDF ≈ 0.72 - in contrast to our more precise present calculations using the non-perturbative approach, which yields βNP < 0.71. So, the fact that our more precise calculations, which do not rely on a sum-over states ex- pression, agree so well with the 15-state model suggests that in both cases, the limit for a one-dimensional single electron molecule is just over β ≈ 0.7. This brute force calculation serves as a numerical illustration of the obser- vation that the limiting value of β is the same for an exact non-perturbation calculation and for a calculation that truncates the SOS expression, which presumedly should lead to large inaccuracies.[24, 25] At minimum, this result supports the existence of fundamental limits of nonlinear susceptibilities that are in line with past calculations. To state Zhou’s approach more precisely,[17] the cal- culations optimized the very special case of the intrin- sic hyperpolarizability for a 15 state model for a poten- tial energy function that is parameterized with 20 spline points. As such, the potential energy function can at most develop about 20 wiggles. As a consequence, there are enough degrees of freedom in the potential energy function to force the 15 states to be spatially well sepa- 0 5 10 15 20 V(x) FIG. 3: Optimized potential energy function using βDF and first 15 wavefunctions after 7,000 iterations. Starting po- tential is the tanh(x) potential. The final potential (shown above) we refer to as the Zhou potential. TABLE I: Evolution of Zhou’s Potential. βs is the hyperpo- larizability of the starting potential using 80 states while the other ones are after optimization of βNP . Number of βS βSOS βDF βNP τ 00 ∆τ Iterations (×10−5) (×10−4) 0 0.5612 0.5612 0.5607 0.5612 11.2 15 1000 0.5612 0.7087 0.6682 0.7083 1810 40 rated. Interestingly, after optimization, only two excited states overlap with ground state, allowing only these two states to have nonzero transition dipole moments with each other and the ground state – forcing the system into a three-level SOS model for βDF . This behavior is interesting in light of the three-level ansatz, which asserts that only three states determine the nonlinear response of a system when it is near the fundamental limits. It is interesting to compare the exact non-perturbation calculation, which does not depend on the excited state wavefunctions (Figure 2) and Zhou’s contrived system of 15 states (Figure 3). Both cases have wiggles and the wavefunctions appear to be mostly non-overlapping. So, for the first 15 states, the wavefunctions appear simi- larly localized. The situation becomes more interesting when 80 states are included in calculating the hyperpo- larizability for Zhou’s potential or when the exact non- perturbative approach is used. The first line in Table I summarizes the results with Zhou’s potential and 80 states. First, let’s focus on the sum-over-states results. Clearly, when 80 states are used in the calculation, it is impossible for the excited state wavefunctions to not overlap with each other, so the three-level approximation to β breaks down. According to the three-level ansatz, we would expect the hyperpolarizability to get smaller. Indeed, the additional excited states result in a smaller 0 5 10 15 20 FIG. 4: Optimized potential energy function and first 15 wavefunctions after 1,000 iterations. Starting potential is Zhou’s potential, using the non-perturbative hyperpolariz- ability for optimization. hyperpolarizability (≈ 0.56). Note that the exact and SOS expressions agree with each other and that τ and ∆τ (80) are small. Figure 4 shows the result after 1000 iterations, us- ing Zhou’s potential as the starting potential and us- ing the non-perturbative hyperpolarizability for opti- mization. First, the non-perturbative hyperpolarizabil- ity reaches just under 0.71, but, the SOS and dipole- free expressions do not agree with each other. Further- more, both convergence metrics (τ 00 and ∆τ (80)) are larger than before optimization. It would appear that for Zhou’s potential, even 80 states are not sufficient to characterize the nonlinear susceptibility when a sum- over-states expression is used (either dipole free or tradi- tional SOS expression - though the SOS expression agrees better with the non-perturbative approach). Interestingly, the optimized potential energy function still retains the wiggles and the wave functions are still well separated. This result is consistent with the sug- gestion of Zhou and coworkers that modulation of conju- gation may be a good design strategy for making large- hyperpolarizability molecules. We note that wiggles in the potential energy function are not required to get a large nonlinear-optical response; but, appears to be one way that Mother Nature optimizes the hyperpo- larizability. Since this idea has been used to identify molecules with experimentally measured record intrin- sic hyperpolarizability,[16] the concept of modulation of conjugation warrants further experimental studies. As a case in point that non-wiggly potentials can lead to a large nonlinear susceptibility is the clipped harmonic oscillator, which we calculated to have an intrinsic hyper- polarizability of about 0.57.[14] Figure 5 shows the opti- mized non-perturbative hyperpolarizability when using a clipped harmonic oscillator as the starting potential. The properties of all of the optimized potentials are summa- TABLE II: Summary of calculations with different starting potentials. βs is the hyperpolarizability of the starting po- tential while the other ones are after optimization. Function βS βSOS βDF βNP τ 00 ∆τ V (x) (×10−5) (×10−4) 0 0 0.7089 0.7089 0.7089 37.8 5.33 30 tanh(x) 0.67 0.7084 0.6918 0.7083 779 11.8 x 0.66 0.7088 0.7072 0.7088 78.7 8.79 x2 0.57 0.7089 0.7085 0.7088 18.6 703 x1/2 0.68 0.7087 0.7049 0.7087 190 9.76 x+ sin(x) 0.67 0.7088 0.7073 0.7088 75.0 8.46 x+ 10 sin(x) 0.04 0.7085 0.7085 0.7085 1.65 7.78 0 5 10 15 20 FIG. 5: Optimized potential energy function and first 15 wavefunctions after 8,000 iterations. Starting potential is V (x) = x2, using the non-perturbative hyperpolarizability for optimization. rized in Table II. The clipped square root function also has a large hyperpolarizability (0.69). The optimized po- tential is shown in Figure 6. In these cases, the amplitude of the wiggles are small and all the wavefunctions overlap. So, these fall into Class II. Note that the lack of wiggles shows that they are not an inevitable consequence of our numerical calculations. We may question whether small wiggles in the poten- tial energy function lead to large amplitude wiggles as an artifact of our numerical optimization technique. To test this hypothesis, we used the trial potential energy func- tion x + sin(x), where the wiggle amplitude is not large enough to cause the wavefunctions to localize at the min- ima. The optimized potential energy function retains an approximately linear from with only small fluctuation. In fact, the results are very similar to what we found for the linear starting potential and the wiggles do not affect the final result. The similarity between these cases can be seen in Table II. Next, we test a starting potential with large wiggles as shown in the upper portion of Figure 7. The lower 0 5 10 15 20 FIG. 6: Optimized potential energy function and first 15 wavefunctions after 8,000 iterations. Starting potential is V (x) = x, using the non-perturbative hyperpolarizability for optimization. energy eigenfunctions are found to be localized mostly in the first two wells. In fact, the lowest four energy eigen- functions are well approximated by harmonic oscillator wavefunctions, which are centrosymmetric. As a result, the first excited state holds most of the oscillator strength and the value of the intrinsic hyperpolarizability is only 0.04. After 3000 interactions, this Class I potential energy function has high amplitude wiggles at a wavelength that is significantly shorter than the wavelength of the ini- tial sine function (bottom portion of Figure 7). In com- mon with the optimized tanh(x) function, the wiggles are of large but almost chaotically varying amplitude. This leads to wavefunctions that are spatially separated. While the wavefunctions are not as well separated as we find for the tanh(x) starting potential, the optimized potential yields only two dominant transition from the ground state; so, this system is well approximated by a three-level model. As is apparent from Table II, the ground state sum rule (characterized by τ 00 ) is better obeyed in this optimized potential than in any others. So, the wavefunctions are accurate and all of the values of β have converged to the same value, suggesting that this calculation may be the most accurate of the set Our results bring up several interesting questions. First, all of our extensive numerical calculations, inde- pendent of the starting potential, yield an optimized in- trinsic hyperpolarizability with an upper bound of 0.71, which is about thirty percent lower than what the sum rules allow. Since numerical optimization can settle in to a local maximum, it is possible that all of the starting potentials are far from the global maximum of βint = 1. Indeed, since most potentials lead to systems that require more than three dominant states to express the hyper- polarizability, this may in itself be an indicator that we are not at the fundamental limit precisely because these 0 5 10 15 20 0 5 10 15 20 FIG. 7: Potential energy function and first 15 wavefunctions before (top) and after (bottom) 3,000 iterations. Starting potential is of the form V (x) = x+ 10 sin(x), using the non- perturbative hyperpolarizability for optimization. systems have more than three states. Indeed, the orig- inal results of Zhou and coworkers frames the problem in a way (i.e. a 15-level model in a potential limited to about 20 wiggles) that allows a solution to the optimiza- tion problem to lead to three dominant states. So, while it may be argued that this system is contrived and un- physical, we have found value in trying such toy models when testing various hypotheses. This toy model • leads to a three-level system as the three-level ansatz proposes • has the same qualitative properties as more precise methods • has given insights into making new molecules with record-breaking intrinsic hyperpolarizability Given the complexity of calculating nonlinear- susceptibilities, our semi-quantitative method may be a good way of generating new ideas. The three-level ansatz proposes that at the fundamen- tal limit, all transitions are negligible except between three dominant states. There appears to be no proof of the ansatz aside from the fact that it leads to an accurate prediction of the upper bound of nonlinear susceptibili- ties, both calculated and measured. To understand the motivation behind the ansatz, it is useful to understand how the two-level model optimizes the polarizability, α, without the need to rely on any assumptions. This is trivial to show by using the fact that the polarizability depends only on the positive-definite transition moments, 〈0|x |n〉 〈n|x |0〉, the same parameters that are found in the ground state sum rules.[26] For nonlinear susceptibilities, the situation is much more complicated because the SOS expression depends on quantities such as 〈0|x |n〉 〈n|x |m〉 〈m|x |0〉, where these terms can be both positive and negative. Fur- thermore, the sum rules that relate excited states mo- ments to each other allow for these moments to be much larger than transition moments to the ground state. So, it would seem plausible that one could design a system with many excited states in a way that all of the tran- sition moments between excited states would add con- structively to yield a larger hyperpolarizability than what we calculate with the three-level ansatz. None of our numerical calculations, independent of the potential en- ergy function, yield a value greater than 0.71. Since our potential energy functions are general 1-dimensional po- tentials (i.e. the potentials are not limited to Coulomb potentials, nor are the wavefunctions approximated as is common in standard quantum chemical computations), our calculations most likely span a broader range of pos- sible wavefunctions leading to a larger variety of states that contribute to the hyperpolarizability. However, there appear to be local maxima associated with systems that behave as a three-level system and oth- ers with many states, and, the maximum values both are 0.71. It is interesting that so may different sets of transi- tion moments and energies can yield the exact same local maximum. To gain a deeper appreciation of the under- lying physics, let’s consider the transition moments and energies in the sum-over-states expression for the hyper- polarizability as adjustable parameters. For a system with N states, there are N − 1 energy parameters of the form En − E0. The moment matrix xij has N2 com- ponents. If the matrix is real, there are (N2 − N)/2 unique off-diagonal terms and N diagonal dipole mo- ments. Since all dipole moments appear as differences of the form xnn − x00, there are only N − 1 dipole mo- ment parameters. Therefore, the dipole matrix is char- acterized by (N2 − N)/2 + N − 1 = (N + 2)(N − 1)/2 parameters. Combining the energy and dipole matrix pa- rameters, there are a total of (N + 2)(N − 1)/2 +N − 1 parameters. The N-state sum rules are of the form: (Em + Ep) 〈m|x |n〉 〈n|x |p〉 (7) δm,p, so the sum rules comprise a total of N2 equations (i.e. an equation for each (m, p)). If the sum rules are truncated to N states, the sum rule indexed by (m = N, p = N) is nonsensical because it contradicts the other sum rules. Furthermore, if the transition moments are real, then xmp = xpm, so only (N 2 − N)/2 of the equations are independent. As such, there are a total of (N2 −N)/2+ N − 1 = (N + 2)(N − 1)/2 independent equations. Since the SOS expression for the nonlinear- susceptibility has (N + 2)(N − 1)/2 +N − 1 parameters and the sum rules provide (N + 2)(N − 1)/2 equations, the SOS expression can be reduced to a form with N − 1 parameters. For example, the three-level model for the hyperpolarizability, which is expressed in terms of 7 parameters, can be reduced to two parameters using 5 sum rule equations. In practice, however, even fewer sum rule equations are usually available because some of them lead to physically unreasonable consequences. While the (N,N) sum rule is clearly unphysical due to truncation, sum rule equations that are near equa- tion (N,N) may also be unphysical. In the case of the three-level model, it is found that the equation (2, 1) allows for an infinite hyperpolarizability, so that equation is ignored on the grounds that it violates the principle of physical soundness.[12, 25, 26] This leads to a hyperpolarizability in terms of 3 variables, which are chosen to be E10, E = E10/E20, and X = x10/x The expression is then maximized with respect to the two parameters E and X , leaving the final result a function of E10. We conclude that the SOS expression for the hyperpo- larizability can be expressed in terms of at least N − 1 parameters; so, it would appear that as more levels are included in the SOS expression, there are more free pa- rameters that can be varied without violating the sum rules. As N → ∞, there are an infinite number of ad- justable parameters. So, it is indeed puzzling that the three-level ansatz yields a fundamental limit that is con- sistent with all of our calculations for a wide range of potentials, many of which have many excited states. It may be that we are only considering a small subset of potential energy functions; or, perhaps the expression for the hyperpolarizability depends on the parameters in such a way that large matrix elements contribute to the hyperpolarizability with alternating signs so that the big terms cancel. This is a puzzle that needs to be solved if we are to understand what makes β large. To investigate whether the limiting behavior is due to our use of 1-dimensional potentials, we have also optimized the intrinsic hyperpolarizability in two- dimensions. In this case, we focus on the largest ten- sor component, βxxx and describe the potential as a superposition of point charges. As described in the literature,[19] we solve the two-dimensional Schrödinger eigenvalue problem, ∇2Ψ+ VΨ = EΨ, (8) for the lowest ten to 25 energy eigenstates, depending on the degree of convergence of the resulting intrinsic hyper- polarizability. We use the two-dimensional logarithmic Coulomb potential, which for k nuclei with charges q1e, . . . , qke located at points s (1), . . . , s(k), is given by V (s) = qj log ‖s− s(j)‖, (9) where L is a characteristic length. With L = 2Å, the force due to a charge at distance 2Å is the same as it would be for a 3D Coulomb potential. We discretize the eigenvalue problem given by Equa- tion 8 using a quadratic finite element method [21, 27] and solve the resulting matrix eigenvalue problem for the ten to 25 smallest energy eigenvalues and corresponding eigenvectors by the implicitly-restarted Arnoldi method [22] as implemented in ARPACK [28]. Each eigenvector yields a wave function Ψn corresponding to energy level En. The moments xmn = s1Ψm(s1, s2)Ψn(s1, s2) ds1ds2 are computed, and these and the energy levels En are used to compute β Figure 8 shows the intrinsic hyperpolarizability of a two-nucleus molecule plotted as a function of the distance between the two nuclei and nuclear charge q1. The total nuclear charge is q1 + q2 = +e, and is expressed in units of the proton charge, e. Three extrema are observed. The positive peak parameters are βint = 0.649 for q1 = 0.58 and d = 4.36Å. The negative one yields βint = −0.649 for q1 = 0.42 and d = 4.36Å. The local negative peak that extends past the graph on the right reaches its maximum magnitude of βint = −0.405 at q1 = 2.959 and d = 2.0Å. Applying numerical optimization to the intrinsic hy- perpolarizability using the charges and separation be- tween the nuclei as parameters, we get βint = 0.654 at d = 4.539Å, and q1 = 0.430 when the starting param- eters are near the positive peak; and βint = −0.651, d = 4.443Å, and q1 = 0.572 when optimization gives the negative peak. The peak parameters are the same within roundoff errors when optimization or plotting is used, confirming that the optimization procedure yields the correct local extrema. Figure 9 shows the intrinsic hyperpolarizability of an octupolar-like molecule made of three evenly-spaced nu- clei on a circle plotted as a function of the circle’s diam- eter and charge fraction ǫ (q = ǫe). The charge on each of the other nuclei is e(1 − ǫ)/2. The positive peak at ǫ = 0.333 and diameter D = 6.9Å has a hyperpolariz- ability βint = 0.326, while βint = −0.605 for a charge fraction ǫ = 0.44 and a diameter D = 6.8Å. Charge (q Intrinsic Hyperpolarizability 0 0.5 1 1.5 2 FIG. 8: The intrinsic hyperpolarizability of two nuclei as a function of the distance between them and the charge of one nucleus, q1 where q1 + q2 = +e. Intrinsic Hyperpolarizability 0 0.2 0.4 0.6 0.8 1 FIG. 9: The intrinsic hyperpolarizability of three evenly- spaced nuclei on a circle as a function of the circle’s diameter and the charge, ǫ (in units of e), on one of the nuclei. The charge on each of the other nuclei is e(1− ǫ)/2. When the positions and magnitudes of the three charges are allowed to move freely, the best intrinsic hyperpolarizability obtained using numerical optimiza- tion is βint = 0.685 for charges located at ~r1 = (0, 0). ~r2 = (−4.87Å, 0.33Å), and ~r3 = (−9.57Å,−0.16Å); with charges q1 = 0.43e, q2 = 0.217e, and q3 = 0.351e. There are only small differences in the optimized values of βint depending on the starting positions and charges; and the best results are for a “molecule” that is nearly linear along the x-direction. This is not surprising given that the xxx-component of βint is the optimized quantity. The two-dimensional analysis illustrates that numer- ical optimization correctly identifies the local maxima (peaks and valleys) and that the magnitude of maximum intrinsic hyperpolarizability (0.65 vs 0.68) is close to the maximum we get for the one-dimensional optimization of the potential energy function (0.71). All computa- tions we have tried, including varying the potential en- ergy function in one dimension or moving around point charges in a plane all yield an intrinsic hyperpolarizabil- ity that is less than 0.71. An open question is the origin of the factor-of-thirty gap between the best molecules and the fundamental limit, which had remained firm for decades through the year 2006. Several of the common proposed explana- tions, such as vibronic dilution, have been eliminated.[14] Perhaps it is not possible to make large-enough varia- tions of the potential energy function without making the molecule unstable. Or, perhaps there are subtle is- sues with electron correlation, which prevents electrons from responding to light with their full potential. The fact that the idea of modulation of conjugation has lead to a 50% increase over the long-standing ceiling - reduc- ing the gap to a factor of twenty - makes it a promising approach for potential further improvements. Continued theoretical scrutiny, coupled with experiment, will be re- quired to confirm the validity of our approach. IV. CONCLUSIONS There appear to be many potential energy functions that lead to an intrinsic hyperpolarizability that is near the fundamental limit. These separate into two broad classes: one in which wiggles in the potential energy function forces the eigenfunctions to be spatially sepa- rated and a second class of monotonically varying wave- functions with small or no wiggles that allow for many strongly overlapping wavefunctions. Interestingly, all these one-dimensional “molecules” have the same max- imal intrinsic hyperpolarizability of 0.71. It is puzzling that the three-level ansatz correctly predicts the funda- mental limit even when the ansatz does not apply. A second open question pertains to the origin of the long- standing factor of 30 gap between the fundamental limit and the best molecules. The idea of conjugation modu- lation may be one promising approach for making wiggly potential energy profiles that lead to molecules that fall into the gap. Given that there are so many choices of potential energy functions that lead to maximal intrinsic hyperpolarizability, it may be possible to engineer many new classes of exotic molecules with record intrinsic hy- perpolarizability. Acknowledgements: MGK thanks the National Science Foundation (ECS-0354736) and Wright Paterson Air Force Base for generously supporting this work. [1] Q. Y. Chen, L. Kuang, Z. Y. Wang, and E. H. Sargent, Nano. Lett. 4, 1673 (2004). [2] B. H. Cumpston, S. P. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I.-Y. S. Lee, D. McCord-Maughon, et al., Nature 398, 51 (1999). [3] S. Kawata, H.-B. Sun, T. Tanaka, and K. Takada, Nature 412, 697 (2001). [4] A. Karotki, M. Drobizhev, Y. Dzenis, P. N. Taylor, H. L. Anderson, and A. Rebane, Phys. Chem. Chem. Phys. 6, 7 (2004). [5] I. Roy, O. T. Y., H. E. Pudavar, E. J. Bergey, A. R. Oseroff, J. Morgan, T. J. Dougherty, and P. N. Prasad, J. Am. Chem. Soc. 125, 7860 (2003). [6] M. G. Kuzyk, Opt. Lett. 25, 1183 (2000). [7] M. G. Kuzyk, IEEE Journal on Selected Topics in Quan- tum Electronics 7, 774 (2001). [8] M. G. Kuzyk, Phys. Rev. Lett. 85, 1218 (2000). [9] M. G. Kuzyk, Opt. Lett. 28, 135 (2003). [10] M. G. Kuzyk, Phys. Rev. Lett. 90, 039902 (2003). [11] M. G. Kuzyk, J. Nonl. Opt. Phys. & Mat. 13, 461 (2004). [12] M. G. Kuzyk, J. Chem Phys. 125, 154108 (2006). [13] M. G. Kuzyk, Optics & Photonics News 14, 26 (2003). [14] K. Tripathi, P. Moreno, M. G. Kuzyk, B. J. Coe, K. Clays, and A. M. Kelley, J. Chem. Phys. 121, 7932 (2004). [15] B. J. Orr and J. F. Ward, Molecular Physics 20, 513 (1971). [16] J. Pérez Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk, Opt. Lett. 32, 59 (2007). [17] J. Zhou, M. Kuzyk, and D. S. Watkins, Opt. Lett. 31, 2891 (2006). [18] M. G. Kuzyk, Phys. Rev. A 72, 053819 (2005). [19] M. G. Kuzyk and D. S. Watkins, J. Chem Phys. 124, 244104 (2006). [20] D. Kincaid and E. W. Cheney, Numerical Analy- sis: Mathematics of Scientific Computing (Brooks-Cole, 2002), 3rd ed. [21] O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals (Butterworth-Heinemanm, 2005), 6th ed. [22] D. C. Sorensen, SIAM J. Matrix Anal. Appl. 13, 357 (1992). [23] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. Wright, SIAM J. Optim. 9, 112 (1998). [24] B. Champagne and B. Kirtman, Phys. Rev. Lett. 95, 109401 (2005). [25] M. G. Kuzyk, Phys. Rev. Lett. 95, 109402 (2005). [26] M. G. Kuzyk, J. Nonl. Opt. Phys. & Mat. 15, 77 (2006). [27] K. Atkinson and W. Han, Theoretical Numerical Anal- ysis, a Functional Analysis Framework (Springer, New York, 2001). [28] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Prob- lems with Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia, 1998). 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

0704.1688

Critical test for Altshuler-Aronov theory: Evolution of the density of states singularity in double perovskite Sr$_2$FeMoO$_6$ with controlled disorder

Critical test for Altshuler-Aronov theory: Evolution of the density of states singularity in double perovskite Sr2FeMoO6 with controlled disorder M. Kobayashi,1 K. Tanaka,2 A. Fujimori,1 Sugata Ray,3, 4 and D. D. Sarma4, 5 Department of Physics and Department of Complexity Science and Engineering, University of Tokyo, Kashiwa, Chiba 277-8561, Japan Department of Applied Physics and Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, California 94305, USA Materials and Structures Laboratory, Tokyo Institute of Technology, Midori, Yokohama 226-8503, Japan Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India Centra for Advanced Materials, Indian Association for the Cultivation of Science, Kolkata 700032, India (Dated: October 31, 2018) With high-resolution photoemission spectroscopy measurements, the density of states (DOS) near the Fermi level (EF) of double perovskite Sr2FeMoO6 having different degrees of Fe/Mo antisite disorder has been investigated with varying temperature. The DOS near EF showed a systematic depletion with increasing degree of disorder, and recovered with increasing temperature. Altshuler- Aronov (AA) theory of disordered metals well explains the dependences of the experimental results. Scaling analysis of the spectra provides experimental indication for the functional form of the AA DOS singularity. PACS numbers: 71.20.-b, 71.23.-k, 71.27.+a, 79.60.-i Disordered electronic systems, which have random po- tentials deviating from an ideal crystal, have been inves- tigated from both fundamental and application points of view [1]. Ever since the finding of filling-control metal- insulator transitions (MIT) in transition-metal oxides known as strongly correlated system, disorder has at- tracted even more attention because not only electron- electron interaction but also disorder are supposed to play fundamentally important roles in the MIT. Altshuler and Aronov [2] studied the effect of electron-electron in- teraction in a disordered metallic medium, and predicted that the density of states (DOS) near the Fermi level (EF) shows a singularity of |E −EF|1/2 and the DOS at EF increases with increasing temperature in proportion T . The theory has been applied to the low tempera- ture conductivity of disordered metals such as disordered Au and Ag films [3], amorphous alloy Ge1−xAux [4], and transition metal chalcogenide Ni(S,Se)2 [5]. In a previous work, Sarma et al. [6] have reported photoemission (PES) measurements on B-site disordered perovskites LaNi1−xMxO3 (M=Mn and Fe), which show MIT as a function of x, and shown that the disor- der affects the DOS near EF in such a way that had been theoretically predicted by Altshuler and Aronov [2]. In a similar B-site substituted transition-metal ox- ide SrRu1−xTixO3, which demonstrates MIT at x ∼ 0.3 (SrRuO3 is metallic), the depletion of the DOS near EF has shown an unusual |E − EF|1.2 dependence in both metallic and insulating phases [7]. In addition, although it is believed that a disorder-induced insulator shows a soft Coulomb gap characterized by a (E − EF)2 de- pendence of the DOS near EF [8, 9], the unexpected |E − EF|3/2 dependence of the DOS near EF related to charge density wave has been observed in insulating BaIrO3 [10]. It is considered that fine structure in the DOS in the vicinity of EF is sensitive to both the degrees of disorder and electron correlation, and therefore experi- mental confirmation of a basic theory for disordered elec- tronic system such as the Altshuler-Aronov (AA) theory is necessary for understanding of the DOS singularity. While AA theory makes specific predictions about both E and T dependences, photoelectron spectroscopy has been used only to probe the E dependence with abso- lutely no reference to the T dependence. Therefore, de- tailed high-resolution temperature-dependent PES mea- surements are also highly desired to verify the AA theory. The present paper reports on high-resolution PES experiments on the B-site ordered double perovskite Sr2FeMoO6 (SFMO), where we have controlled the de- gree of Fe/Mo antisite disorder (AD) in the sample prepa- ration procedure. Through detailed analysis for the tem- perature and degree of disorder dependences of the PES spectra near EF, the results provide experimental confir- mation of the AA theory of disordered metals. SFMO has been investigated intensively due to the theoreti- cal prediction of half-metallic nature and the observa- tion of large magnetoresistance under low magnetic fields at room temperature [11]. In this system, there are characteristic defects known as Fe/Mo AD at the B- site, which remarkably affects the physical properties of SFMO [12, 13, 14, 15, 16]. By controlling the degree of AD, one can investigate disorder effects in the metal with- out changing the chemical composition and other condi- tions. Polycrystalline SFMO samples having different degrees of Fe/Mo AD prepared as follows; First, the samples with the highest degree of AD 45% were prepared. Then they were annealed at 1173 K, 1673 K, and 1523 K for a pe- riod of 5 hours under 2% H2/Ar to obtain the degrees of AD 40%, 25%, and 10%, respectively (SFMO having AD 50% is the same as ordinary perovskite SrFe0.5Mo0.5O3). Details of the sample preparation are given in Ref. [12] http://arxiv.org/abs/0704.1688v2 2.0 1.5 1.0 0.5 0 Binding Energy (eV) 0.4 0.2 0.0 1.5 1.0 0.5 0.0 Binding Energy (eV) 0.5 0.2 Fe eg↑ Fe t2g↓-Mo t2g↓ AD 40% T = 10 K fractured scraped AD 40% AD 10% background Fe t2g↓- Mo t2g↓ Fe eg↑ FIG. 1: Valence-band photoemission spectra of Sr2FeMoO6 taken at 10 K with He-I radiation. (a) Comparison between the spectra taken from the fractured and scraped surfaces. The spectra have been normalized to the area from 0 eV to 0.8 eV (Fe t2g↓-Mo t2g↓ states). (b) Comparison between different degrees of antisite disorder (AD) 10% and 40%. The spectra have been normalized to the area from 0.8 eV to 2 eV (Fe eg↑ states). Top: Background subtraction. The insets show enlarged plots near EF. and [13]. Using x-ray diffraction, the degree of disorder was quantified from the intensity of a supercell-reflection peak. Scanning electron microscopy in conjunction with energy dispersive x-ray analysis revealed no change in composition during the annealing. Transport measure- ments were performed on the AD 10% and 40% sam- ples by a standard four prove technique using a Physi- cal Property Measurement System (Quantum Design Co. Ltd). PES spectra were recorded using a spectrometer equipped with a monochromatized He resonance lamp (hν = 21.2 eV), where photoelectrons were collected with a Gammadata Scienta SES-100 hemispherical analyzer in the angle integrated mode. The total resolution of the spectrometer was ∼10 meV, and the base pressure was 1.0×10−8 Pa. Clean surfaces were obtained by repeated scraping in situ with a diamond file. The position of EF was determined by measuring PES spectra of evaporated gold which was electrically in contact with the sample. The line shapes of the PES spectra obtained were al- most the same as those in a previous report on single crystalline SFMO taken with synchrotron radiation [17]. In order to examine the influence of surface treatment, we measured valence-band spectra taken from fractured and scraped surfaces. In the binding-energy (EB) range from 2 eV to 10 eV, there were appreciable differences such as the sharpness of structures in the O 2p and the Fe t2g↑ bands (not shown), the background intensity, and to some extent the Fe eg↑ band. On the other hand, within ∼ 1 eV of EF, i.e., in the Fe t2g↓ +Mo t2g↓ conduc- tion bands, the line shapes of the fractured and scraped samples were similar to each other as shown in Fig. 1(a). 0.3 0.2 0.1 0 Binding Energy (eV) 0.01 0.00 -0.01 0.3 0.2 0.1 0.0 -0.1 Binding Energy (eV) 0.01 0.00 -0.01 AD 10% AD 25% AD 40% AD 45% T= 10 K T= 100 K T= 200 K T= 300 K (a) T=10 K (b) AD 10% Au 8 K 150 K 300 K FIG. 2: Photoemission spectra of Sr2FeMoO6 near the Fermi level. The spectra have been normalized to the area from EB = 0.3 eV to 0.6 eV. (a) Degree of Fe/Mo AD dependence at 10 K. (b) Temperature dependence of the AD 10% sample. As a reference, Au spectra are also shown. The insets show an enlarged plot in the vicinity of EF. Since in previous PES studies it has been reported that LDA+U calculation well explains the valence-band spec- tra taken from the fractured surface [17, 18], we consider that the different surface treatments have not affected the spectra near EF which reflect the bulk properties. In contrast, the spectra are intensively influenced by AD and temperature as we shall see below. Figure 1(b) shows valence-band spectra for different degrees of disorder, i.e., AD 40% and 10%. Although the peak due to the localized Fe eg↑ states was nearly identical between AD 10% and AD 40%, there was a clear difference in the Fe t2g↓-Mo t2g↓ conduction band between the two spectra. This result suggests that the disorder influences the DOS near EF. Figure 2 shows the temperature and degree of disorder dependences of the spectra near EF normalized to the area in the region from EB = 0.3 eV to 0.6 eV, in which the spectra were iden- tical and independent of the degree of AD as shown in Fig. 1(b). For a fixed temperature, the intensity of the spectra near EF was depleted with degree of disorder. This behavior is consistent with the previous report on a disordered metal system LaNi1−xMnxO3 [6]. Indeed, temperature dependent resistivity measurements on the AD 40% sample showed a minimum around 40 K, i.e., the resistivity increased with decreasing temperature below 40 K, while on the AD 10% one there was no minimum till the lowest measured temperature (20 K). The obser- vations are consistent with previous reports of transport measurements on SFMO having various degrees of dis- order [16, 19] and on SFMO with high Fe/Mo ordering [13, 20]. For a fixed degree of disorder, the intensity at EF increased with temperature as shown in Fig. 2(b). 3456789 23456789 √X (=√|²|/kBT ) AD 45% 10K, 100K, 200K, 300K AD 40% 10K, 100K, 200K, 300K AD 25% 10K, 100K, 200K, 300K AD 10% 10K, 100K, 200K, 300K j(X) = [X +(1.07) T=300 K 100 K FIG. 3: Scaling analysis for the spectral depletion. Scaled spectra as a function of X (X = |ǫ|/kBT ) are plotted on a bilogarithmic scale. An analytical form of ϕ = [X2 + (1.07)4]1/4 is also plotted. This behavior indicates that SFMO differs from normal metals such as Au in which PES spectra at various tem- peratures have temperature-independent intensity at EF and intersect at EF irrespective of temperature as shown in Fig 2(b), representing the simple Fermi-Dirac distri- bution function. Now, we analyze the temperature and degree of disor- der dependences of the spectra based on the theory for disordered metal suggested by Altshuler and Aronov [2]. The theory predicted that electron-electron interaction accompanied by impurity scattering leads to an anomaly in the DOS around EF and the resulting singular part of the DOS δD is given by δD(ǫ) (~D)3/2 ϕ (|ǫ|/kBT ) , ϕ (X) = X (X ≫ 1) 1.07 +O(X2) (X ≪ 1) where D0 is the original DOS at EF, ǫ is the energy mea- sured from EF, H is the inverse Debye radius, and D is the diffusion coefficient due to impurity scattering. In order to see whether experimental spectra satisfy Eq. (1), we have made the following scaling analysis. According to Eq. (1), the PES spectra I(ǫ, T ) near EF should be proportional to D′0+ δD(ǫ), where D′0 is a con- stant and D′0 < D0. Therefore, I(ǫ, T ) can be parame- terized as I(ǫ, T ) = A+G kBT ϕ , (2) 0.3 0.2 0.1 0 -0.1 Binding Energy (eV) 3002001000 Temperature (K) 403020100 Disorder (%) a + g [² +(1.07) Fitted AD 10% AD 25% AD 40% AD 45% AD 45% (b) (c) 300 K 200 K 100 K FIG. 4: Results of fitting of the photoemission spectra of Sr2FeMoO6 to the analytical curves of Altshuler-Aronov the- ory. (a) Fitted spectra for the AD 45% sample at various temperatures. (b) Degree of disorder dependence of the val- ues of γ/α. G/A is also plotted. The dotted line is a guide to the eye. (c) Temperature dependence of the DOS at EF. Solid curves demonstrate fitted curves using Eq. (4) with pa- rameters deduced from the spectral line-shape fitting of panel where A and G are only dependent on the degree of dis- order. (I − A)/G kBT plotted against X = |ǫ|/kBT should fall onto the same curve if Eq. (1) is valid, and can be used to evaluate the functional form of ϕ if A and G are chosen to satisfy the condition that ϕ(0) → 1.07 as ǫ/kBT → 0. In Fig. 3, the results are plotted on a log- arithmic scale, where the PES spectra have been divided by the Fermi-Dirac function convoluted with the experi- mental resolution estimated from the Au spectra. Notice that all the low energy part of the spectra fell onto the same curve, which we attribute to the scaling function ϕ(X). Deviation from the scaling function ϕ(X) occurs at high energies where the original DOS starts to deviate from the constant one. We find that ϕ(X) approaches√ X for X > 1, corresponding to AA theory. The results ensure the validness of analysis based on AA theory for the depletion of PES spectra near EF. In order to analyze the spectra using Eq. (1), we pro- pose an analytical form of ϕ(X) = [X2+(1.07)4]1/4 inter- polating both the limits of large and small X of Eq. (1). This form is shown to accurately reproduce the experi- mentally scaling function ϕ(X) deduced above as shown in Fig. 3. Therefore, we employ a model function {α+ γ [ǫ2 + (1.07)4(kBT )2]1/4} f(ǫ), (3) where f(ǫ) is the Fermi-Dirac function, α and γ are fit- ting parameters. γ/α depends only on degree of disorder and are independent of a way of intensity normalization. Figure 4(a) shows fitted results for the spectra of the AD 45% sample at various temperatures. The fitting function given by Eq. (3) well reproduced the spectra near EF, where the fitted ranges were chosen to the valid range of the scaling function ϕ(X) as shown in Fig. 3 [21]. Figure 4(b) shows the values of the coefficients which rep- resent the strength of the DOS singularity as a function of disorder. The γ/α value is independent of temperature and approximately linearly increases with degree of disor- der as shown in Fig. 4(b), indicating that the DOS singu- larity near EF is enhanced with increasing degree of dis- order as predicted by AA theory. The constant G will be relative to the value of γ. Actually, the G/A value shows the same dependence as γ/α [Fig. 4(b)]. Equation (3) as a function of X without f , i.e., α + γ[X2 + (1.07)4]1/4, well reproduced the line shape of the depletion as shown in Fig. 3, where the parameters were chosen α = 0 and γ = 1 to correspond with the analytical ϕ(X) [22]. The result demonstrates validness of our assumption for the functional form of ϕ given by Eq. (3). Equation (1) indicates that the singular contribution to the DOS δD/D0 at EF is proportional to T . In or- der to study the temperature dependence of the DOS at EF, comparison was made between the experimental and theoretical δD/D0 at EF. The PES intensity (or DOS) at EF is plotted as a function of temperature in Fig. 4(c). For a fixed temperature, the DOS at EF increased with decreasing degree of disorder. For a fixed degree of dis- order, the DOS at EF increased with increasing temper- ature. According to Eq. (1), temperature dependence of the DOS at EF is expressed as α+ 1.07 γ kBT , (4) corresponding Eq. (3) at ǫ = 0. Using the values of γ/α obtained by fitting to the PES spectra, Eq. (4) well re- produces the temperature-dependence of the DOS at EF as shown in Fig. 4(c). The result is consistent with theo- retically predicted temperature dependence of δD/D0 at EF. It follows from the arguments described above that AA theory applies to not only the disorder dependent depletion near EF but also the temperature dependent DOS at EF. As mentioned above, AA theory treats the scattering processes on general ground and does not depend on the functional form of the potential of the scattering center. This theory can be applied to the case that the mean free path of itinerant electrons is larger than or comparable with its wave length. In conclusion, we have performed high-resolution pho- toemission experiments on polycrystalline Sr2FeMoO6 samples having different degrees of Fe/Mo antisite dis- order. The photoemission spectra near the Fermi level depended on degree of the Fe/Mo antisite disorder as well as on temperature. The Altshuler-Aronov theory on disordered metal well explained both the dependences. Scaling analysis for the spectral depletion clarifies the functional form of the density of states singularity near the Fermi level. We believe that the findings will provide an indicator for degrees of disorder and electron-electron correlation, and promote spectral analysis for the index of the density of states depletion near the Fermi level. The present results point to a need for taking into ac- count both electron-electron interaction and disorder ef- fects for an understanding of the electronic structure of metallic correlated electron system. The authors thank H. Yagi and M. Hashimoto for help in experiments. This work was supported by a Grant- in-Aid for Scientific Research in Priority Area “Inven- tion of Anomalous Quantum Materials” (16076208) from MEXT, Japan. D.D.S. thanks DST and BRNS for fund- ing this research. S.R. thanks JSPS postdoctoral fellow- ship for foreign researchers. M.K. acknowledges support from the Japan Society for the Promotion of Science for Young Scientists. [1] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985). [2] B. L. Altshuler and A. G. Aronov, Solid State Commun. 30, 115 (1979). [3] S. Schmitz and S. Ewert, Solid State Commun. 74, 1067 (1990). [4] W. L. McMillan and J. Mochel, Phys. Rev. Lett. 46, 556 (1981). [5] A. Husmann, D. S. Jin, Y. V. Zastavker, T. F. Rosen- baum, X. Yao, and J. M. Honig, Science 274, 1874 (1996). [6] D. D. Sarma, A. Chainani, S. R. Krishnakumar, E. Vescovo, C. Carbone, W. Eberhardt, O. Rader, C. Jung, C. Hellwing, W. Gudat, H. Srikanth, and A. K. Ray- chaudhuri, Phys. Rev. Lett. 80, 4004 (1998). [7] J. Kim, J.-Y. Kim, B.-G. Park, and S.-J. Oh, Phys. Rev. B 73, 235109 (2006). [8] A. F. Efros and B. I. Shklovskii, J. Phys. C 8, L49 (1975). [9] J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266 (1995). [10] K. Maiti, R. S. Singh, V. R. R. Medicherla, S. Rayaprol, and E. V. Sampathkumaran, Phys. Rev. Lett. 95, 016404 (2005). [11] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature 395, 677 (1998). [12] D. D. Sarma, E. V. Sampathkumaran, S. Ray, R. Natara- jan, S. Majumdar, A. Kimar, G. Nalini, and T. N. G. Row, Solid State Commun. 114, 465 (2000). [13] D. D. Sarma, Sugata Ray, K. Tanaka, and A. Fujimori, PRL in press. [14] J. Navarro, J. Nogués, J. S. Muñoz, and J. Fontcuberta, Phys. Rev. B 67, 174416 (2003). [15] B. J. Park, H. Han, J. Kim, Y. J. Kim, C. S. Kim, and B. W. Lee, J. Magn. Magn. Mater. 272-276, 1851 (2004). [16] Y. H. Huang, M. Karppinen, H. Yamauchi, and J. B. Goodenough, Phys. Rev. B 73, 104408 (2006). [17] T. Saitoh, M. Nakatake, A. Kakizaki, H. Nakajima, O. Morimoto, S. Xu, Y. Moritomo, N. Hamada, and Y. Aiura, Phys. Rev. B 66, 035112 (2002). [18] J.-S. Kang, J. H. Kim, A. Sekiyama, S. Kasai, S. Suga, S. W. Han, K. H. Kim, T. Muro, Y. Saitoh, C. Hwang, C. G. Olson, B. J. Park, B. W. Lee, J. H. Shin, J. H. Park, and B. I. Min, Phys. Rev. B 66, 113105 (2002). [19] Y.-H. Huang, H. Yamauchi, and M. Karppinen, Phys. Rev. B 74, 174418 (2006). [20] Y. Tomioka, T. Okuda, Y. Okimoto, R. Kumai, K.-I. Kobayashi, and Y. Tokura, Phys. Rev. B 61, 422 (2000). [21] The starting points are about 76 meV, 109 meV, 129 meV, and 145 meV for 10 K, 100 K, 200 K, and 300 K, respectively. [22] The analytical form of ϕ(X) can be expressed as more general formula: ϕn(X) = [X 2n + (1.07)4n]1/4n, which also satisfies both the limits ofX. Using this function, the scaling function has been fitted well as n = 1.09 ± 0.14. This result emphasizes the validness of the analytical ϕ(X) because the exponent n is nearly 1.

0704.1689

Some Properties of and Open Problems on Hessian Nilpotent Polynomials

SOME PROPERTIES OF AND OPEN PROBLEMS ON HESSIAN NILPOTENT POLYNOMIALS WENHUA ZHAO Abstract. In the recent work [BE1], [M], [Z1] and [Z2], the well-known Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent) and their (deformed) inversion pairs. In this paper, we prove several results on HN polynomials, their (deformed) inversion pairs as well as on the associated symmetric polynomial or formal maps. We also propose some open problems for further study of these objects. 1. Introduction In the recent work [BE1], [M], [Z1] and [Z2], the well-known Jacobian conjecture (see [BCW] and [E]) has been reduced to a problem on HN (Hes- sian nilpotent) polynomials, i.e. the polynomials whose Hessian matrix are nilpotent, and their (deformed) inversion pairs. In this paper, we prove some properties of HN polynomials, the (deformed) inversion pairs of (HN) poly- nomial, the associated symmetric polynomial or formal maps, the graphs assigned to homogeneous harmonic polynomials, etc. Another purpose of this paper is to draw the reader’s attention to some open problems which we believe will be interesting and important for further study of these objects. In this section we first discuss some backgrounds and motivations in Sub- section 1.1 for the study of HN polynomials and their (deformed) inversion pairs. We also fix some terminology and notation in this subsection that will be used throughout this paper. Then in Subsection 1.2 we give an arrangement description of this paper. Date: November 17, 2021. 2000 Mathematics Subject Classification. 14R15, 32H02, 32A50. Key words and phrases. Hessian nilpotent polynomials, inversion pairs, harmonic polynomials, the Jacobian conjecture. The author has been partially supported by NSA Grant R1-07-0053. http://arxiv.org/abs/0704.1689v2 2 WENHUA ZHAO 1.1. Background and Motivation. Let z = (z1, z2, . . . , zn) be n free com- mutative variables. We denote by C[z] (resp.C[[z]]) the algebra of poly- nomials (resp. formal power series) of z over C. A polynomial or formal power series P (z) is said to be HN (Hessian nilpotent) if its Hessian matrix HesP := ( ∂ ∂zi∂zj ) are nilpotent. The study of HN polynomials is mainly mo- tivated by the recent progress achieved in [BE1], [M], [Z1] and [Z2] on the well-known JC (Jacobian conjecture), which we will briefly explain below. Recall that the JC first proposed by Keller [Ke] in 1939 claims: for any polynomial map F of Cn with the Jacobian j(F ) = 1, its formal inverse map G must also be a polynomial map. Despite intense study for more than half a century, the conjecture is still open even for the case n = 2. For more history and known results before 2000 on the Jacobian conjecture, see [BCW], [E] and references there. In 2003, M. de Bondt, A. van den Essen ([BE1]) and G. Meng ([M]) independently made the following breakthrough on the JC. Let Di := (1 ≤ i ≤ n) and D = (D1, . . . , Dn). For any P (z) ∈ C[[z]], denote by∇P (z) the gradient of P (z), i.e. ∇P (z) := (D1P (z), . . . , DnP (z)). We say a formal map F (z) = z − H(z) is symmetric if H(z) = ∇P (z) for some P (z) ∈ C[[z]]. Then, the symmetric reduction of the JC achieved in [BE1] and [M] is that, to prove or disprove the JC, it will be enough to consider only symmetric polynomial maps. Combining with the classical homogeneous reduction achieved in [BCW] and [Y], one may further assume that the symmetric polynomial maps have the form F (z) = z−∇P (z) with P (z) homogeneous (of degree 4). Note that, in this case the Jacobian condi- tion j(F ) = 1 is equivalent to the condition that P (z) is HN. For some other recent results on symmetric polynomial or formal maps, see [BE1]–[BE5], [EW], [M], [Wr1], [Wr2], [Z1], [Z2] and [EZ]. Based on the homogeneous reduction and the symmetric reduction of the JC discussed above, the author further showed in [Z2] that the JC is actually equivalent to the following so-called vanishing conjecture of HN polynomials. Conjecture 1.1. (Vanishing Conjecture) Let ∆ := i be the Laplace operator of C[z]. Then, for any HN polynomial P (z) (of homo- geneous of degree d = 4), ∆mPm+1(z) = 0 when m >> 0. Furthermore, the following criterion of Hessian nilpotency for formal power series was also proved in [Z2]. SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 3 Proposition 1.2. For any P (z) ∈ C[[z]] with o(P (z)) ≥ 2, the following statements are equivalent. (1) P (z) is HN. (2) ∆mPm = 0 for any m ≥ 1. (3) ∆mPm = 0 for any 1 ≤ m ≤ n. One crucial idea of the proofs in [Z2] for the results above is to study a special formal deformation of symmetric formal maps. More precisely, let t be a central formal parameter. For any P (z) ∈ C[[z]], we call F (z) = z−∇P (z) the associated symmetric maps of P (z). Let Ft(z) = z−t∇P (z). When the order o(P (z)) of P (z) with respect to z is greater than or equal to 2, Ft(z) is a formal map of C[[t]][[z]] with Ft=1(z) = F (z). Therefore, we may view Ft(z) as a formal deformation of the formal map F (z). In this case, one can also show (see [M] or Lemma 3.14 in [Z1]) that the formal inverse map Gt(z) := F t (z) of Ft(z) does exist and is also symmetric, i.e. there exists a unique Qt(z) ∈ C[[t]][[z]] with o(Qt(z)) ≥ 2 such that Gt(z) = z + t∇Qt(z). We call Qt(z) the deformed inversion pair of P (z). Note that, whenever Qt=1(z) makes sense, the formal inverse map G(z) of F (z) is given by G(z) = Gt=1(z) = z + ∇Qt=1(z), so in this case we call Q(z) := Qt=1(z) the inversion pair of P (z). Note that, under the condition o(P (z)) ≥ 2, the deformed inversion pair Qt(z) of P (z) might not be in C[t][[z]], so Qt=1(z) may not make sense. But, if we assume further that J(Ft)(0) = 1, or equivalently, (HesP )(0) is nilpotent, then Ft(z) is an automorphism of C[t][[z]], hence so is its inverse map Gt(z). Therefore, in this case Qt(z) lies in C[t][[z]] and Qt=1(z) makes sense. Throughout this paper, whenever the inversion pair Q(z) of a poly- nomial or formal power series P (z) ∈ C[[z]] (not necessarily HN) is under concern, our assumption on P (z) will always be o(P (z)) ≥ 2 and (HesP )(0) is nilpotent. Note that, for any HN P (z) ∈ C[[z]] with o(P (z)) ≥ 2, the condition that (HesP )(0) is nilpotent holds automatically. For later purpose, let us recall the following formula derived in [Z2] for the deformed inversion pairs of HN formal power series. Theorem 1.3. Suppose P (z) ∈ C[[z]] with o(P (z)) ≥ 2 is HN. Then, we Qt(z) = 2mm!(m+ 1)! ∆mPm+1(z),(1.1) 4 WENHUA ZHAO From the equivalence of the JC and the VC discussed above, we see that the study on the HN polynomials and their (deformed) inversion pairs becomes important and necessary, at least when the JC is concerned. Note that, due to the identity TrHesP = ∆P , HN polynomials are just a special family of harmonic polynomials which are among the most classical objects in mathematics. Even though harmonic polynomials had been very well studied since the late of the eighteenth century, it seems that not much has been known on HN polynomials. We believe that these mysterious (HN) polynomials deserve much more attentions from mathematicians. 1.2. Arrangement. Considering the length of this paper, we here give a more detailed arrangement description of the paper. In Section 2, we consider the following two questions. Let P, S, T ∈ C[[z]] with P = S + T and Q,U, V their inversion pairs, respectively. Q1: Under what conditions, P is HN iff both S and T are HN? Q2: Under what conditions, we have Q = U + V ? We give some sufficient conditions in Theorems 2.1 and 2.7 for the two questions above. In Section 3, we employ a recursion formula of inversion pairs derived in [Z1] and Eq. (1.1) above to derive some estimates for the radius of convergence of inversion pairs of homogeneous (HN) polynomials (see Propositions 3.1 and 3.3). For any P (z) ∈ C[[z]], we say it is self-inverting if its inversion pair Q(z) is P (z) itself. In Section 4, by using a general result on quasi-translations proved in [B], we derive some properties of HN self-inverting formal power series P (z). Another purpose of this section is to draw the reader’s attention to Open Problem 4.8 on classification of HN self-inverting polynomials or formal power series. In Section 5, we show in Proposition 5.1, when the base field has char- acteristic p > 0, the VC, unlike the JC, actually holds for any polynomials P (z) even without the HN condition on P (z). It also holds in this case for any HN formal power series. One interesting question (see Open Problem 5.2) is to see if the VC like the JC fails over C when P (z) is allowed to be any HN formal power series. In Section 6, we prove a criterion of Hessian nilpotency for homogeneous polynomials over C (see Theorem 6.1). Considering the criterion in Propo- sition 1.2, this criterion is somewhat surprising but its proof turns out to be very simple. SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 5 Section 7 is mainly motivated by the following question raised by M. Kumar ([K]) and D. Wright ([Wr3]). Namely, for a symmetric formal map F (z) = z −∇P (z), how to write f(z) := 1 σ2 − P (z) (where σ2 := i=1 z and P (z) itself as formal power series in F (z)? In this section, we derive some explicit formulas to answer the questions above and also for the same question for σ2 (see Proposition 7.2). From these formulas, we also show in Theorem 7.4 that, the VC holds for a HN polynomial P (z) iff one (hence, all) of σ2, P (z) and f(z) can be written as a polynomial in F , where F (z) = z −∇P (z) is the associated polynomial maps of P (z). Finally, in Section 8, we discuss a graph G(P ) assigned to each homo- geneous harmonic polynomials P (z). The graph G(P ) was first proposed by the author and later was further studied by Roel Willems in his master thesis [Wi] under direction of Professor Arno van den Essen. In Subsection 8.1 we give the definition of the graph G(P ) for any homogeneous harmonic polynomial P (z) and discuss the connectedness reduction (see Corollary 8.5) which says, to study the VC for homogeneous HN polynomials P (z), it will be enough to consider the case when the graph G(P ) is connected. In Sub- section 8.2 we consider a connection of G(P ) with the tree expansion formula derived in [M] and [Wr2] for the inversion pair Q(z) of P (z) (see also Propo- sition 8.9). As an application of the connection, we use it to give another proof for the connectedness reduction discussed in Corollary 8.5. One final remark on the paper is as follows. Even though we could have focused only on (HN) polynomials, at least when only the JC is concerned, we will formulate and prove our results in the more general setting of (HN) formal power series whenever it is possible. Acknowledgement: The author is very grateful to Professors Arno van den Essen, Mohan Kumar and David Wright for inspiring communications and constant encouragement. Section 7 was mainly motivated by some questions raised by Professors Mohan Kumar and David Wright. The author also would like to thank Roel Willems for sending the author his master thesis in which he has obtained some very interesting results on the graphs G(P ) of homogeneous harmonic polynomials. At last but not the least, the author thanks the referee and the editor for many valuable suggestions. 6 WENHUA ZHAO 2. Disjoint Formal Power Series and Their Deformed Inversion Pairs Let P, S, T ∈ C[[z]] with P = S + T , and Q, U and V their inversion pairs, respectively. In this section, we consider the following two questions: Q1: Under what conditions, P is HN if and only if both S and T are Q2: Under what conditions, we have Q = U + V ? We give some answers to the questions Q1 and Q2 in Theorems 2.1 and 2.7, respectively. The results proved here will also be needed in Section 8 when we consider a graph associated to homogeneous harmonic polynomials. To question Q1 above, we have the following result. Theorem 2.1. Let S, T ∈ C[[z]] such that 〈∇(DiS),∇(DjT )〉 = 0 for any 1 ≤ i, j ≤ n, where 〈·, ·〉 denotes the standard C-bilinear form of Cn. Let P = S + T . Then, we have (a) Hes (S)Hes (T ) = Hes (T )Hes (S) = 0. (b) P is HN iff both S and T are HN. Note that statement (b) in the theorem above was first proved by R. Willems ([Wi]) in a special setting as in Lemma 2.6 below for homogeneous harmonic polynomials. Proof: (a) For any 1 ≤ i, j ≤ n, consider the (i, j)th entry of the product Hes (S)Hes (T ): ∂zi∂zk ∂zk∂zj = 〈∇(DiS),∇(DjT )〉 = 0.(2.1) Hence Hes (S) Hes (T ) = 0. Similarly, we have Hes (T ) Hes (S) = 0. (b) follows directly from (a) and the lemma below. ✷ Lemma 2.2. Let A, B and C be n × n matrices with entries in any com- mutative ring. Suppose that A = B + C and BC = CB = 0. Then, A is nilpotent iff both B and C are nilpotent. Proof: The (⇐) part is trivial because B and C in particular commute with each other. To show (⇒), note that BC = CB = 0. So for any m ≥ 1, we have AmB = (B + C)mB = (Bm + Cm)B = Bm+1. SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 7 Similarly, we have Cm+1 = AmC. Therefore, if AN = 0 for some N ≥ 1, we have BN+1 = CN+1 = 0. ✷ Note that, for the (⇐) part of (b) in Theorem 2.1, we need only a weaker condition. Namely, for any 1 ≤ i, j ≤ n, 〈∇(DiS),∇(DjT )〉 = 〈∇(DjS),∇(DiT )〉, which will ensure that Hes (S) and Hes (T ) commute. To consider the second question Q2, let us first fix the following notation. For any P ∈ C[[z]], let A(P ) denote the subalgebra of C[[z]] generated by all partial derivatives of P (of any order). We also define a sequence {Q[m](z) |, m ≥ 1} by writing the deformed inversion pair Qt(z) of P (z) as Qt(z) = tm−1Q[m](z).(2.2) Lemma 2.3. For any P ∈ C[[z]], we have (a) A(P ) is closed under the action of any differential operator of C[z] with constant coefficients. (b) For any m ≥ 1, we have Q[m](z) ∈ A(P ). Proof: (a) Note that, by the definition of A(P ), a formal power series g(z) ∈ C[[z]] lies in A(P ) iff it can be written (not necessarily uniquely) as a polynomial in partial derivatives of P (z). Then, by the Leibniz Rule, it is easy to see that, for any g(z) ∈ A(P ), Dig(z) ∈ A(P ) (1 ≤ i ≤ n). Repeating this argument, we see that any partial derivative of g(z) is in A(P ). Hence (a) follows. (b) Recall that, by Proposition 3.7 in [Z1], we have the following recurrent formula for Q[m](z) (m ≥ 1) in general: Q[1](z) = P (z),(2.3) Q[m](z) = 2(m− 1) k,l≥1 k+l=m 〈∇Q[k](z),∇Q[l](z)〉.(2.4) for any m ≥ 2. By using (a), the recurrent formulas above and induction on m ≥ 1, it is easy to check that (b) holds too. ✷ Definition 2.4. For any S, T ∈ C[[z]], we say S and T are disjoint to each other if, for any g1 ∈ A(S) and g2 ∈ A(T ), we have 〈∇g1,∇g2〉 = 0. 8 WENHUA ZHAO This terminology will be justified in Section 8 when we consider a graph G(P ) associated to homogeneous harmonic polynomials P . Lemma 2.5. Let S, T ∈ C[[z]]. Then S and T are disjoint to each other iff, for any α, β ∈ Nn, we have 〈∇(DαS),∇(DβT )〉 = 0.(2.5) Proof: The (⇒) part of the lemma is trivial. Conversely, for any g1 ∈ A(S) and g2 ∈ A(T ) (i = 1, 2), we need show 〈∇g1,∇g2〉 = 0. But this can be easily checked by, first, reducing to the case that g1 and g2 are monomials of partial derivatives of S and T , respectively, and then applying the Leibniz rule and Eq. (2.5) above. ✷ A family of examples of disjoint polynomials or formal power series are given as in the following lemma, which will also be needed later in Section Lemma 2.6. Let I1 and I2 be two finite subsets of C n such that, for any αi ∈ Ii (i = 1, 2), we have 〈α1, α2〉 = 0. Denote by Ai (i = 1, 2) the completion of the subalgebra of C[[z]] generated by hα(z) := 〈α, z〉 (α ∈ Ii), i.e. Ai is the set of all formal power series in hα(z) (α ∈ Ii) over C. Then, for any Pi ∈ Ai (i = 1, 2), P1 and P2 are disjoint. Proof: First, by a similar argument as the proof for Lemma 2.3, (a), it is easy to check that Ai (i = 1, 2) are closed under action of any differen- tial operator with constant coefficients. Secondly, since Ai (i = 1, 2) are subalgebras of C[[z]], we have A(Pi) ⊂ Ai (i = 1, 2). Therefore, to show P1 and P2 are disjoint to each other, it will be enough to show that, for any gi ∈ Ai (i = 1, 2), we have 〈∇g1,∇g2〉 = 0. But this can be easily checked by first reducing to the case when gi (i = 1, 2) are monomials of hα(z) (α ∈ Ii), and then applying the Leibniz rule and the following identity: for any α, β ∈ Cn, 〈∇hα(z),∇hβ(z)〉 = 〈α, β〉. Now, for the second question Q2 on page 6, we have the following result. SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 9 Theorem 2.7. Let P, S, T ∈ C[[z]] with order greater than or equal to 2, and Qt, Ut, Vt their deformed inversion pairs, respectively. Assume that P = S + T and S, T are disjoint to each other. Then (a) Ut and Vt are also disjoint to each other, i.e. for any α, β ∈ N n, we ∇DαUt(z),∇D βVt(z) (b) We further have Qt = Ut + Vt.(2.6) Proof: (a) follows directly from Lemma 2.3, (b) and Lemma 2.5. (b) Let Q[m], U[m] and V[m] (m ≥ 1) be defined as in Eq. (2.2). Hence it will be enough to show Q[m] = U[m] + V[m](2.7) for any m ≥ 1. We use induction on m ≥ 1. When m = 1, Eq. (2.7) follows from the condition P = S + T and Eq. (2.3) . For any m ≥ 2, by Eq. (2.4) and the induction assumption, we have Q[m] = 2(m− 1) k,l≥1 k+l=m 〈∇Q[k],∇Q[l]〉 2(m− 1) k,l≥1 k+l=m 〈∇U[k] +∇V[k],∇U[l] +∇V[l]〉 Noting that, by Lemma 2.3, U[j] ∈ A(S) and V[j] ∈ A(T ) (1 ≤ j ≤ m): 2(m− 1) k,l≥1 k+l=m 〈∇U[k],∇U[l]〉+ 2(m− 1) k,l≥1 k+l=m 〈∇V[k],∇V[l]〉 Applying the recursion formula Eq. (2.4) to both U[m] and V[m]: = U[m] + V[m]. As later will be pointed out in Remark 8.11, one can also prove this theorem by using a tree expansion formula of inversion pairs, which was derived in [M] and [Wr2], in the setting as in Lemma 2.6. 10 WENHUA ZHAO From Theorems 2.1, 2.7 and Eqs. (1.1), (2.2), it is easy to see that we have the following corollary. Corollary 2.8. Let Pi ∈ C[[z]] (1 ≤ i ≤ k) which are disjoint to each other. Set P = i=1 Pi. Then, we have (a) P is HN iff each Pi is HN. (b) Suppose that P is HN. Then, for any m ≥ 0, we have ∆mPm+1 = ∆mPm+1i .(2.8) Consequently, if the VC holds for each Pi, then it also holds for P . 3. Local Convergence of Deformed Inversion Pairs of Homogeneous (HN) Polynomials Let P (z) be a formal power series which is convergent near 0 ∈ Cn. Then the associated symmetric map F (z) = z − ∇P is a well-defined analytic map from an open neighborhood of 0 ∈ Cn to Cn. If we further assume that JF (0) = In×n, the formal inverse G(z) = z +∇Q(z) of F (z) is also locally well-defined analytic map. So the inversion pair Q(z) of P (z) is also locally convergent near 0 ∈ Cn. In this section, we use the formulas Eqs. (2.4), (1.1) and the Cauchy estimates to derive some estimates for the radius of convergence of inversion pairs Q(z) of homogeneous (HN) polynomials P (z) (see Propositions 3.1 and 3.3). First let us fix the following notation. For any a ∈ Cn and r > 0, we denote by B(a, r) (resp.S(a, r)) the open ball (resp. the sphere) centered at a ∈ C with radius r > 0. The unit sphere S(0, 1) will also be denoted by S2n−1. Furthermore, we let Ω(a, r) be the polydisk centered at a ∈ Cn with radius r > 0, i.e. Ω(a, r) := {z ∈ n | |zi − ai| < r, 1 ≤ i ≤ n}. For any subset A ⊂ C n, we will use Ā to denote the closure of A in Cn. For any polynomial P (z) ∈ C[z] and a compact subset D ⊂ Cn, we set |P |D to be the maximum value of |P (z)| over D. In particular, when D is the unit sphere S2n−1, we also write |P | = |P |D, i.e. |P | := max{|P (z)| | z ∈ S2n−1}.(3.1) SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 11 Note that, for any r ≥ 0 and a ∈ B(0, r), we have Ω(a, r) ⊂ B(a, r) ⊂ B(0, 2r). Combining with the well-known Maximum Principle of holomor- phic functions, we get Ω(a,r) ≤ |P | B(a,r) ≤ |P | B(0,2r) = |P |S(0,2r).(3.2) For the inversion pairs Q of homogeneous polynomials P without HN condition, we have the following estimate for the radius of convergence at 0 ∈ Cn. Proposition 3.1. Let P (z) be a non-zero homogeneous polynomial (not necessarily HN) of degree d ≥ 3 and r0 = (n2 d−1|P |) 2−d . Then the inversion pair Q(z) converges over the open ball B(0, r0). To prove the proposition, we need the following lemma. Lemma 3.2. Let P (z) be any polynomial and r > 0. Then, for any a ∈ B(0, r) and m ≥ 1, we have ∣Q[m](a) nm−1|P |m S(0,2r) 2m−1r2m−2 .(3.3) Proof: We use induction on m ≥ 1. First, when m = 1, by Eq. (2.3) we have Q[1] = P . Then Eq. (3.3) follows from the fact B(a, r) ⊂ B(0, 2r) and the maximum principle of holomorphic functions. Assume Eq. (3.3) holds for any 1 ≤ k ≤ m − 1. Then, by the Cauchy estimates of holomorphic functions (e.g. see Theorem 1.6 in [R]), we have ∣(DiQ[k])(a) ∣Q[k] Ω(0,r) nk−1|P |k B(0,2r) 2k−1r2k−1 .(3.4) By Eqs. (2.4) and (3.4), we have |Q[m](a)| ≤ 2(m− 1) k,l≥1 k+l=m ∣〈∇Q[k],∇Q[l]〉 2(m− 1) k,l≥1 k+l=m nk−1|P |k S(0,2r) 2k−1r2k−1 nℓ−1|P |ℓ S(0,2r) 2ℓ−1r2ℓ−1 nm−1|P |m S(0,2r) 2m−1r2m−2 12 WENHUA ZHAO Proof of Proposition 3.1: By Eq. (2.2) , we know that, Q(z) = Q[m](z).(3.5) To show the proposition, it will be enough to show the infinite series above converges absolutely over B(0, r) for any r < r0. First, for any m ≥ 1, let Am be the RHS of the inequality Eq. (3.3). Note that, since P is homogeneous of degree d ≥ 3, we further have |P |mB(0,2r) = (2r)d|P |S2n−1 = (2r)dm|P |m.(3.6) Therefore, for any m ≥ 1, we have Am = 2 (d−1)m+1nm−1r(d−2)m+2|P |m,(3.7) and by Lemma 3.2, |Q[m](a)| ≤ Am(3.8) for any a ∈ B(0, r). Since 0 < r < r0 = (n2 d−1|P |)2−d, it is easy to see that = n2d−1rd−2|P | < 1. Therefore, by the comparison test, the infinite series in Eq. (3.5) converges absolutely and uniformly over the open ball B(0, r). ✷ Note that the estimate given in Proposition 3.1 depends on the number n of variables. Next we show that, with the HN condition on P , an estimate independent of n can be obtained as follows. Proposition 3.3. Let P (z) be a homogeneous HN polynomial of degree d ≥ 4 and set r0 := (2 d+1|P |) 2−d . Then, the inversion pair Q(z) of P (z) converges over the open ball B(0, r0). Note that, when d = 2 or 3, by Wang’s Theorem ([Wa]), the JC holds in general. Hence it also holds for the associated symmetric map F (z) = z −∇P when P (z) is HN. Therefore Q(z) in this case is also a polynomial of z and converges over the whole space Cn. To prove the proposition above, we first need the following two lemmas. SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 13 Lemma 3.4. Let P (z) be a homogeneous polynomial of degree d ≥ 1 and r > 0. For any a ∈ B(0, r), m ≥ 0 and α ∈ Nn, we have |(DαPm+1)(a)| ≤ (2r)d(m+1)|P |m+1.(3.9) Proof: First, by the Cauchy estimates and Eq. (3.2), we have |(DαPm+1)(a)| ≤ |Pm+1| Ω(a,r) |Pm+1| B(0,2r) .(3.10) On the other hand, by the maximum principle and the condition that P is homogeneous of degree d ≥ 3, we have |Pm+1| B(0,2r) = |P |m+1 B(0,2r) = |P |m+1 S(0,2r) = ((2r)d|P |)m+1(3.11) = (2r)d(m+1)|P |m+1. Then, combining Eqs. (3.10) and (3.11), we get Eq. (3.9). ✷ Lemma 3.5. For any m ≥ 1, we have |α|=m α! ≤ m! m+ n− 1 (m+ n− 1)! (n− 1)! .(3.12) Proof: First, for any α ∈ Nn with |α| = m, we have α! ≤ m! since the binomial is always a positive integer. Therefore, we have |α|=m α! ≤ m! |α|=m Secondly, note that |α|=m 1 is just the number of distinct α ∈ Nn with |α| = m, which is the same as the number of distinct monomials in n free commutative variables of degree m. Since the latter is well-known to be the binomial m+n−1 , we have |α|=m α! ≤ m! m+ n− 1 (m+ n− 1)! (n− 1)! Proof of Proposition 3.3: By Eq. (1.1) , we know that, Q(z) = ∆mPm+1 2mm!(m+ 1)! .(3.13) 14 WENHUA ZHAO To show the proposition, it will be enough to show the infinite series above converges absolutely over B(0, r) for any r < r0. We first give an upper bound for the general terms in the series Eq. (3.13) over B(0, r). Consider ∆mPm+1 = ( D2i ) mPm+1 = |α|=m D2αPm+1.(3.14) Therefore, we have |∆mPm+1(a)| ≤ |α|=m |D2αPm+1(a)| Applying Lemma 3.4 with α replaced by 2α: |α|=m (2α)! (2r)d(m+1)|P |m+1 Noting that (2α)! ≤ [(2α)!!]2 = 22m(α!)2: |α|=m 22m(α!)2 (2r)d(m+1)|P |m+1 = m!22m+d(m+1)rd(m+1)−2m|P |m+1 |α|=m Applying Lemma 3.5: m!(m+ n− 1)!22m+d(m+1)rd(m+1)−2m|P |m+1 (n− 1)! Therefore, for any m ≥ 1, we have ∆mPm+1 2mm!(m+ 1)! 2m+d(m+1)rd(m+1)−2m|P |m+1(m+ n− 1)! (m+ 1)!(n− 1)! .(3.15) For any m ≥ 1, let Am be the right hand side of Eq. (3.15) above. Then, by a straightforward calculation, we see that the ratio 2d+1rd−2|P |.(3.16) SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 15 Since r < r0 = (2 d+1|P |) 2−d , it is easy to see that = 2d+1rd−2|P | < 1. Therefore, by the comparison test, the infinite series in Eq. (3.13) con- verges absolutely and uniformly over the open ball B(0, r). ✷ 4. Self-Inverting Formal Power Series Note that, by the definition of inversion pairs (see page 3), Q ∈ C[[z]] is the inversion pair of P ∈ C[[z]] iff P is the inversion pair of Q. In other words, the relation that Q and P are inversion pair of each other in some sense is a duality relation. Naturally, one may ask, for which P (z), it is self-dual or self-inverting? In this section, we discuss this special family of polynomials or formal power series. Another purpose of this section is to draw the reader’s attention to the problem of classification of (HN) self-inverting polynomials (see Open Prob- lem 4.8). Even though the classification of HN polynomials seems to be out of reach at the current time, we believe that the classification of (HN) self- inverting polynomials is much more approachable. Definition 4.1. A formal power series P (z) ∈ C[[z]] with o(P (z)) ≥ 2 and (HesP )(0) nilpotent is said to be self-inverting if its inversion pair Q(z) = P (z). Following the terminology introduced in [B], we say a formal map F (z) = z − H(z) with H(z) ∈ C[[z]]×n and o(H(z)) ≥ 1 is a quasi-translation if j(F )(0) 6= 0 and its formal inverse map is given by G(z) = z +H(z). Therefore, for any P (z) ∈ C[[z]] with o(P (z)) ≥ 2 and (HesP )(0) nilpo- tent, it is self-inverting iff the associated symmetric formal map F (z) = z −∇P (z) is a quasi-translation. For quasi-translations, the following general result has been proved in Proposition 1.1 of [B] for polynomial quasi-translations. Proposition 4.2. A formal map F (z) = z − H(z) with o(H) ≥ 1 and JH(0) nilpotent is a quasi-translation if and only if JH ·H = 0. Even though the proposition above was proved in [B] only in the setting of polynomial maps, the proof given there works equally well for formal quasi-translations under the condition that JH(0) is nilpotent. Since it has 16 WENHUA ZHAO also been shown in Proposition 1.1 in [B] that, for any polynomial quasi- translations F (z) = z −H(z), JH(z) is always nilpotent, so the condition that JH(0) is nilpotent in the proposition above does not put any extra restriction for the case of polynomial quasi-translations. From Proposition 4.2 above, we immediately have the following criterion for self-inverting formal power series. Proposition 4.3. For any P (z) ∈ C[[z]] with o(P ) ≥ 2 and (HesP )(0) nilpotent, it is self-inverting if and only if 〈∇P,∇P 〉 = 0. Proof: Since o(P ) ≥ 2 and (HesP )(0) is nilpotent, by Proposition 4.2, we see that, P (z) ∈ C[[z]] is self-inverting iff J(∇P )·∇P = (HesP )·∇P = 0. But, on the other hand, it is easy to check that, for any P (z) ∈ C[[z]], we have the following identity: (HesP ) · ∇P = ∇〈∇P,∇P 〉. Therefore, (HesP ) · ∇P = 0 iff ∇〈∇P,∇P 〉 = 0, and iff 〈∇P,∇P 〉 = 0 because o(〈∇P,∇P 〉) ≥ 2. ✷ Corollary 4.4. For any P (z) ∈ C[[z]] with o(P ) ≥ 2 and (HesP )(0) nilpo- tent, if it is self-inverting, then so is Pm(z) for any m ≥ 1. Proof: Note that, for any m ≥ 2, we have o(Pm(z)) ≥ 2m > 2 and (HesP )(0) = 0. Then, the corollary follows immediately from Proposition 4.3 and the following general identity: 〈∇Pm,∇Pm〉 = m2P 2m−2〈∇P,∇P 〉.(4.1) Corollary 4.5. For any harmonic formal power series P (z) ∈ C[[z]] with o(P ) ≥ 2 and (HesP )(0) nilpotent, it is self-inverting iff ∆P 2 = 0. Proof: This follows immediately from Proposition 4.3 and the following general identity: ∆P 2 = 2(∆P )P + 2〈∇P,∇P 〉.(4.2) SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 17 Proposition 4.6. Let P (z) be a harmonic self-inverting formal power se- ries. Then, for any m ≥ 1, Pm is HN. Proof: First, we use the mathematical induction on m ≥ 1 to show that ∆Pm = 0 for any m ≥ 1. The case of m = 1 is given. For any m ≥ 2, consider ∆Pm = ∆(P · Pm−1) = (∆P )Pm−1 + P (∆Pm−1) + 2〈∇P,∇Pm−1〉 = (∆P )Pm−1 + P (∆Pm−1) + 2(m− 1)Pm−2〈∇P,∇P 〉. Then, by the mathematical induction assumption and Proposition 4.3, we get ∆Pm = 0. Secondly, for any fixed m ≥ 1 and d ≥ 1, we have ∆d[(Pm)d] = ∆d−1(∆P dm) = 0. Then, by the criterion in Proposition 1.2, Pm is HN. ✷ Example 4.7. Note that, in Section 5.2 of [Z2], a family of self-inverting HN formal power series has been constructed as follows. Let Ξ be any non-empty subset of Cn such that, for any α, β ∈ Ξ, 〈α, β〉 = 0. Let A be the completion of the subalgebra of C[[z]] generated by hα(z) := 〈α, z〉 (α ∈ Ξ), i.e. A is the set of all formal power series in hα(z) (α ∈ Ξ) over C. Then it is straightforward to check (or see Section 5.2 of [Z2] for details) that any element P (z) ∈ A is HN and self-inverting. It is unknown if all HN self-inverting polynomials or formal power series can be obtained by the construction above. More generally, we believe the following open problem is worth investigating. Open Problem 4.8. (a) Decide whether or not all self-inverting polyno- mials or formal power series are HN. (b) Classify all (HN) self-inverting polynomials and formal power series. Finally, let us point out that, for any self-inverting P (z) ∈ C[[z]], the deformed inversion pair Qt(z) (not just Q(z) = Qt=1(z)) is also same as P (z). Proposition 4.9. Let P (z) ∈ C[[z]] with o(P ) ≥ 2 and (HesP )(0) nilpo- tent. Then P (z) is self-inverting if and only if Qt(z) = P (z). 18 WENHUA ZHAO Proof: First, let us point out the following observations. Let t be a formal central parameter and Ft(z) = z − t∇P (z) as before. Since o(P ) ≥ 2 and (HesP )(0) is nilpotent, we have j(Ft)(0) = 1. Therefore, Ft(z) is an automorphism of the algebra C[t][[z]] of formal power series of z over C[t]. Since the inverse map of Ft(z) is given by Gt(z) = z + t∇Qt(z), we see that Qt(z) ∈ C[t][[z]]. Therefore, for any t0 ∈ C, Qt=t0(z) makes sense and lies in C[[z]]. Furthermore, by the uniqueness of inverse maps, it is easy to see that the inverse map of Ft0 = z− t0∇P of C[t][[z]] is given by Gt0(z) = z + t0∇Qt=t0 . Therefore the inversion pair of t0P (z) is given by t0Qt=t0(z). With the notation and observations above, by choosing t0 = 1, we have Qt=1(z) = Q(z) and the (⇐) part of the proposition follows immediately. Conversely, for any t0 ∈ C, we have 〈∇(t0P ),∇(t0P )〉 = t 0〈∇P,∇P 〉. Then, by Proposition 4.3, t0P (z) is self-inverting and its inversion pair t0Qt=t0(z) is same as t0P (z), i.e. t0Qt=t0(z) = t0P (z). Therefore, we have Qt=t0(z) = P (z) for any t0 ∈ C ×. But on the other hand, we have Qt(z) ∈ C[t][[z]] as pointed above, i.e. the coefficients of all monomials of z in Qt(z) are polynomials of t, hence we must have Qt(z) = P (z) which is the (⇒) part of the proposition. ✷ 5. The Vanishing Conjecture over Fields of Positive Characteristic It is well-known that the JC may fail when F (z) is not a polynomial map (e.g. F1(z1, z2) = e −z1; F2(z1, z2) = z2e z1). It also fails badly over fields of positive characteristic even in one variable case (e.g. F (x) = x − xp over a field of characteristic p > 0). However, the situation for the VC over fields of positive characteristic is dramatically different from the JC even through these two conjectures are equivalent to each other over fields of characteristic zero. Actually, as we will show in the proposition below, the VC over fields of positive characteristic holds for any polynomials (not even necessarily HN) and also for any HN formal power series. Proposition 5.1. Let k be a field of characteristic p > 0. Then (a) For any polynomial P (z) ∈ k[z] (not necessarily homogeneous nor HN) of degree d ≥ 1, ∆mPm+1 = 0 for any m ≥ d(p−1) SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 19 (b) For any HN formal power series P (z) ∈ k[[z]], i.e. ∆mPm = 0 for any m ≥ 1, we have, ∆mPm+1 = 0 for any m ≥ p− 1. In other words, over the fields of positive characteristic, the VC holds even for HN formal power series P (z) ∈ k[[z]]; while for polynomials, it holds even without the HN condition nor any other conditions. Proof: The main reason that the proposition above holds is because of the following simple fact due to the Leibniz rule and positiveness of the characteristics of the base field k, namely, for m ≥ 1, u(z), v(z) ∈ k[[z]] and any differential operator Λ of k[z], we have Λ(umpv) = umpΛv.(5.1) Now let P (z) be any polynomial or formal series as in the proposition. For any m ≥ 1, write m+1 = qmp+rm with qm, rm ∈ Z and 0 ≤ rm ≤ p−1. Then by Eq. (5.1) , we have ∆mPm+1 = ∆m(P qmpP rm) = P qmp∆mP rm.(5.2) If P (z) is a polynomial of degree d ≥ 1, we have ∆mP rm = 0 when d(p−1) , since in this case 2m > deg(P rm). If P (z) is a HN formal power series, we have ∆mP rm = 0 when m ≥ p− 1 ≥ rm. Therefore, (a) and (b) in the proposition follow from Eq. (5.2) and the observations above. ✷ One interesting question is whether or not the VC fails (as the JC does) for any HN formal power series P (z) ∈ C[[z]] but P (z) 6∈ C[z]? To our best knowledge, no such counterexample has been known yet. We here put it as an open problem. Open Problem 5.2. Find a HN formal power series P (z) ∈ C[[z]] but P (z) 6∈ C[z], if there are any, such that the VC fails for P (z). One final remark about Proposition 5.1 is as follows. Note that the crucial fact used in the proof is that any differential operator Λ of k[z] commutes with the multiplication operator by the pth power of any element of k[[z]]. Then, by a parallel argument as in the proof of Proposition 5.1, it is easy to see that the following more general result also holds. Proposition 5.3. Let k be a field of characteristics p > 0 and Λ a differ- ential operator of k[z]. Let f ∈ k[[z]]. Assume that, for any 1 ≤ m ≤ p− 1, there exists Nm > 0 such that Λ Nmfm = 0. Then, we have Λmfm+1 = 0 when m >> 0. 20 WENHUA ZHAO In particular, if Λ strictly decreases the degree of polynomials. Then, for any polynomial f ∈ k[z], we have Λmfm+1 = 0 when m >> 0. 6. A Criterion of Hessian Nilpotency for Homogeneous Polynomials Recall that 〈·, ·〉 denotes the standard C bilinear form of Cn. For any β ∈ Cn, we set hβ(z) := 〈β, z〉 and βD := 〈β,D〉. The main result of this section is the following criterion of Hessian nilpo- tency for homogeneous polynomials. Considering the criterion given in Proposition 1.2, it is somewhat surprising but the proof turns out to be very simple. Theorem 6.1. For any β ∈ Cn and homogeneous polynomial P (z) of degree d ≥ 2, set Pβ(z) := β D P (z). Then, we have HesPβ = (d− 2)! (HesP )(β).(6.1) In particular, P (z) is HN iff, for any β ∈ Cn, Pβ(z) is HN. To prove the theorem, we need first the following lemma. Lemma 6.2. Let β ∈ Cn and P (z) ∈ C[z] homogeneous of degree N ≥ 1. βNDP (z) = N !P (β).(6.2) Proof: Since both sides of Eq. (6.2) are linear on P (z), we may assume P (z) is a monomial, say P (z) = za for some a ∈ Nn with |a| = N . Consider βNDP (z) = ( βiDi) Nza = |k|=N βkDkza βaDaza = N !βa = N !P (β). Proof of Theorem 6.1: We consider HesPβ(z) = ∂2(βd−2D P ) ∂zi∂zj βd−2D ∂zi∂zj SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 21 Applying Lemma 6.2 to ∂ ∂zi∂zj = (d− 2)! ∂zi∂zj = (d− 2)! (HesP )(β). Let {ei | 1 ≤ i ≤ n} be the standard basis of C n. Applying the theorem above to β = ei (1 ≤ i ≤ n), we have the following corollary, which was first proved by M. Kumar [K]. Corollary 6.3. For any homogeneous HN polynomial P (z) ∈ C[z] of degree d ≥ 2, Dd−2i P (z) (1 ≤ i ≤ n) are also HN. The reason that we think the criteria given in Theorem 6.1 and Corollary 6.3 interesting is that, Pβ(z) = β D P (z) is homogeneous of degree 2, and it is much easier to decide whether a homogeneous polynomial of degree 2 is HN or not. More precisely, for any homogeneous polynomial U(z) of degree 2, there exists a unique symmetric n× n matrix A such that U(z) = zτAz. Then it is easy to check that HesU(z) = 2A. Therefore, U(z) is HN iff the symmetric matrix A is nilpotent. Finally we end this section with the following open question on the cri- terion given in Proposition 1.2. Recall that Proposition 1.2 was proved in [Z2]. We now sketch the argu- ment. For any m ≥ 1, we set um(P ) = TrHes m(P ),(6.3) vm(P ) = ∆ mPm.(6.4) For any k ≥ 1, we define Uk(P ) (resp.Vk(P )) to be the ideal in C[[z]] generated by {um(P )|1 ≤ m ≤ k} (resp. {vm(P )|1 ≤ m ≤ k}) and all their partial derivatives of any order. Then it has been shown (in a more general setting) in Section 4 in [Z2] that Uk(P ) = Vk(P ) for any k ≥ 1. It is well-known in linear algebra that, if um(P (z)) = 0 when m >> 0, then HesP is nilpotent and um(P ) = 0 for anym ≥ 1. One natural question is whether or not this is also the case for the sequence {vm(P ) |m ≥ 1}. More precisely, we believe the following conjecture which was proposed in [Z2] is worth investigating. 22 WENHUA ZHAO Conjecture 6.4. Let P (z) ∈ C[[z]] with o(P (z)) ≥ 2. If ∆mPm(z) = 0 for m >> 0, then P (z) is HN. 7. Some Results on Symmetric Polynomial Maps Let P (z) be any formal power series with o(P (z)) ≥ 2 and (HesP )(0) nilpotent, and F (z) and G(z) as before. Set σ2 : = z2i ,(7.1) f(z) : = σ2 − P (z).(7.2) Professors Mohan Kumar [K] and David Wright [Wr3] once asked how to write P (z) and f(z) in terms of F (z)? More precisely, find U(z), V (z) ∈ C[[z]] such that U(F (z)) = P (z),(7.3) V (F (z)) = f(z).(7.4) In this section, we first derive in Proposition 7.2 some explicit formulas for U(z) and V (z), and also for W (z) ∈ C[[z]] such that W (F (z)) = σ2(z).(7.5) We then show in Theorem 7.4 that, when P (z) is a HN polynomial, the VC holds for P or equivalently, the JC holds for the associated symmetric polynomial map F (z) = z −∇P , iff one of U , V and W is polynomial. Let t be a central parameter and Ft(z) = z− t∇P . Let Gt(z) = z+ t∇Qt be the formal inverse of Ft(z) as before. We set ft(z) : = σ2 − tP (z),(7.6) Ut(z) : = P (Gt(z)),(7.7) Vt(z) : = ft(Gt(z)),(7.8) Wt(z) : = σ2(Gt(z)).(7.9) Note first that, under the conditions that o(P (z)) ≥ 2 and (HesP )(0) is nilpotent, we have Gt(z) ∈ C[t][[z]] ×n as mentioned in the proof of Propo- sition 4.9. Therefore, we have Ut(z), Vt(z),Wt(z) ∈ C[t][[z]], and Ut=1(z), SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 23 Vt=1(z) and Wt=1(z) all make sense. Secondly, from the definitions above, we have Wt(z) = 2Vt(z) + 2tUt(z),(7.10) Ft(z) = ∇ft(z),(7.11) ft=1(z) = f(z).(7.12) Lemma 7.1. With the notations above, we have P (z) = Ut=1(F (z)),(7.13) f(z) = Vt=1(F (z)),(7.14) σ2(z) = Wt=1(F (z)).(7.15) In particular, f(z), P (z) and σ2(z) lie in C[F ] iff Ut=1(z), Vt=1(z) and Wt=1(z) lie in C[z]. In other words, by setting t = 1, Ut, Vt and Wt will give us U , V and W in Eqs. (7.3)–(7.5), respectively. Proof: From the definitions of Ut(z), Vt(z) and Wt(z) (see Eqs. (7.7)– (7.9), we have P (z) = Ut(Ft(z)), ft(z) = Vt(Ft(z)), σ2(z) = Wt(Ft(z)). By setting t = 1 in the equations above and noticing that Ft=1(z) = F (z), we get Eqs. (7.13)–(7.15). ✷ For Ut(z), Vt(z) and Wt(z), we have the following explicit formulas in terms of the deformed inversion pair Qt of P . Proposition 7.2. For any formal power series P (z) ∈ C[[z]] (not neces- sarily HN) with o(P (z)) ≥ 2 and (HesP )(0) nilpotent, we have Ut(z) = Qt + t ,(7.16) Vt(z) = σ2 + t(z −Qt),(7.17) Wt(z) = σ2 + 2tz + 2t2 .(7.18) Proof: Note first that, Eq. (7.18) follows directly from Eqs. (7.16), (7.17) and (7.10). 24 WENHUA ZHAO To show Eq. (7.16), by Eqs. (3.4) and (3.6) in [Z1], we have Ut(z) = P (Gt) = Qt + 〈∇Qt,∇Qt〉 = Qt + t .(7.19) To show Eq. (7.17), we consider Vt(z) = ft(Gt) 〈z + t∇Qt(z), z + t∇Qt(z)〉 − tP (Gt) σ2 + t〈z,∇Qt(z)〉 + 〈∇Qt,∇Qt〉 − tP (Gt) By Eq. (7.19), substituting Qt + 〈∇Qt,∇Qt〉 for P (Gt): σ2 + t〈z,∇Qt(z)〉 − tQt(z) σ2 + t(z −Qt). When P (z) is homogeneous and HN, we have the following more explicit formulas which in particular give solutions to the questions raised by Pro- fessors Mohan Kumar and David Wright. Corollary 7.3. For any homogeneous HN polynomial P (z) of degree d ≥ 2, we have Ut(z) = 2m(m!)2 ∆mPm+1(z)(7.20) Vt(z) = (dm − 1)t 2mm!(m+ 1)! ∆mPm+1(z) ,(7.21) Wt(z) = σ2 + (dm +m)t 2m−1m!(m+ 1)! ∆mPm+1(z) ,(7.22) where dm = deg (∆ mPm+1) = d(m+ 1)− 2m (m ≥ 0). Proof: We give a proof for Eq. (7.20). Eqs. (7.21) can be proved similarly. (7.22) follows directly from Eqs. Eq. (7.20), (7.21) and (7.10). By combining Eqs. (7.16) and (1.1), we have Ut(z) = tm∆mPm+1(z) 2mm!(m+ 1)! mtm∆mPm+1(z) 2mm!(m+ 1)! SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 25 = P (z) + 2m(m!)2 ∆mPm+1(z) 2m(m!)2 ∆mPm+1(z). Hence, we get Eq. (7.20). ✷ One consequence of the proposition above is the following result on sym- metric polynomials maps. Theorem 7.4. For any HN polynomial P (z) (not necessarily homogeneous) with o(P ) ≥ 2, the following statements are equivalent: (1) The VC holds for P (z). (2) P (z) ∈ C[F ]. (3) f(z) ∈ C[F ]. (4) σ2(z) ∈ C[F ]. Note that, the equivalence of the statements (1) and (3) was first proved by Mohan Kumar ([K]) by a different method. Proof: Note first that, by Lemma 7.1, it will be enough to show that, ∆mPm+1 = 0 when m >> 0 iff one of Ut(z), Vt(z) and Wt(z) is a polynomial in t with coefficients in C[z]. Secondly, when P (z) is homogeneous, the statement above follows directly from Eqs. (7.20)–(7.22). To show the general case, for any m ≥ 0 and Mt(z) ∈ C[t][[z]], we denote by [tm](Mt(z)) the coefficient of t m when we write Mt(z) as a formal power series of t with coefficients in C[[z]]. Then, from Eqs. (7.16)–(7.18) and Eq. (1.1), it is straightforward to check that the coefficients of tm (m ≥ 1) in Ut(z), Vt(z) and Wt(z) are given as follows. [tm](Ut(z)) = ∆mPm+1 2m(m!)2 ,(7.23) [tm](Vt(z)) = 2m−1(m− 1)!m! (∆m−1Pm)−∆m−1Pm ,(7.24) [tm](Wt(z)) = 2m−2(m− 1)!m! (∆m−1Pm) + (m− 1)∆m−1Pm (7.25) 26 WENHUA ZHAO From Eq. (7.23), we immediately have (1) ⇔ (2). To show the equiva- lences (1) ⇔ (3) and (1) ⇔ (4), note first that o(P ) ≥ 2, so o(∆m−1Pm) ≥ 2 for any m ≥ 1. While, on the other hand, for any polynomial h(z) ∈ C[z] with o(h(z)) ≥ 2, we have, h(z) = 0 iff (z ∂ − 1)h(z) = 0, and iff + (m − 1))h(z) = 0 for some m ≥ 1. This is simply because that, for any monomial zα (α ∈ Nn), we have (z ∂ − 1)zα = (|α| − 1)zα and + (m− 1))zα = (|α|+ (m− 1))zα. From this general fact, we see that (1) ⇔ (3) follows from Eq. (7.24) and (1) ⇔ (4) from Eq. (7.25). ✷ 8. A Graph Associated with Homogeneous HN Polynomials In this section, we would like to draw the reader’s attention to a graph G(P ) assigned to each homogeneous harmonic polynomials P (z). The graph G(P ) was first proposed by the author and later was further studied by R. Willems in his master thesis [Wi] under direction of Professor A. van den Essen. The introduction of the graph G(P ) is mainly motivated by a crite- rion of Hessian nilpotency given in [Z2] (see also Theorem 8.2 below), via which one hopes more necessary or sufficient conditions for a homogeneous harmonic polynomial P (z) to be HN can be obtained or described in terms of the graph structure of G(P ). We first give in Subsection 8.1 the definition of the graph G(P ) for any homogeneous harmonic polynomial P (z) and discuss the connectedness re- duction (see Corollary 8.5), i.e. a reduction of the VC to the homogeneous HN polynomials P such that G(P ) is connected. We then consider in Sub- section 8.2 a connection of G(P ) with the tree expansion formula derived in [M] and [Wr2] for the inversion pair Q(z) of P (z) (see Proposition 8.9). As an application of the connection, we give another proof for the connected- ness reduction given in Corollary 8.5. 8.1. Definition and the Connectedness Reduction. For any β ∈ Cn, set hβ(z) := 〈β, z〉 and βD := 〈β,D〉, where 〈·, ·〉 is the standard C-bilinear form of Cn. Let X(C) denote the set of all isotropic elements of Cn, i.e. the set of all elements α ∈ Cn such that 〈α, α〉 = 0. Recall that we have the following fundamental theorem on homogeneous harmonic polynomials. SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 27 Theorem 8.1. For any homogeneous harmonic polynomial P (z) of degree d ≥ 2, we have P (z) = (z)(8.1) for some ci ∈ C × and αi ∈ X(C n) (1 ≤ i ≤ k). Note that, replacing αi in Eq. (8.1) by c i αi, we may also write P (z) as P (z) = hdαi(z)(8.2) with αi ∈ X(C n) (1 ≤ i ≤ k). For the proof of Theorem 8.1, see, for example, [I] and [Wi]. We fix a homogeneous harmonic polynomial P (z) ∈ C[z] of degree d ≥ 2, and assume that P (z) is given by Eq. (8.2) for some αi ∈ X(C n) (1 ≤ i ≤ k). We may and will always assume {hdαi(z)|1 ≤ i ≤ k} are linearly independent in C[z]. Recall the following matrices had been introduced in [Z2]: AP = (〈αi, αj〉)k×k,(8.3) ΨP = (〈αi, αj〉h (z))k×k.(8.4) Then we have the following criterion of Hessian nilpotency for homoge- neous harmonic polynomials. For its proof, see Theorem 4.3 in [Z2]. Theorem 8.2. Let P (z) be as above. Then, for any m ≥ 1, we have TrHes m(P ) = (d(d− 1))mTrΨmP .(8.5) In particular, P (z) is HN if and only if the matrix ΨP is nilpotent. One simple remark on the criterion above is as follows. Let B be the k × k diagonal matrix with the ith (1 ≤ i ≤ k) diagonal entry being hαi(z). For any 1 ≤ j ≤ k, set ΨP ;j := B d−2−j = (hjαi〈αi, αj〉h d−2−j ).(8.6) Then, by repeatedly applying the fact that, for any two k× k matrices C and D, CD is nilpotent iff so is DC, it is easy to see that Theorem 8.2 can also be re-stated as follows. 28 WENHUA ZHAO Corollary 8.3. Let P (z) be given by Eq. (8.2) with d ≥ 2. Then, for any 1 ≤ j ≤ d− 2 and m ≥ 1, we have TrHes m(P ) = (d(d− 1))mTrΨmP ;j.(8.7) In particular, P (z) is HN if and only if the matrix ΨP ;j is nilpotent. Note that, when d is even, we may choose j = (d− 2)/2. So P is HN iff the symmetric matrix ΨP ;(d−2)/2(z) = ( h (d−2)/2 (z) 〈αi, αj〉 h (d−2)/2 (z) )(8.8) is nilpotent. Motivated by the criterion above, we assign a graph G(P ) to any homo- geneous harmonic polynomial P (z) as follows. We fix an expression as in Eq. (8.2) for P (z). The set of vertices of G(P ) will be the set of positive integers [k] := {1, 2, . . . , k}. The vertices i and j of G(P ) are connected by an edge iff 〈αi, αj〉 6= 0. In this case, we get a finite graph. Furthermore, we may also label edges of G(P ) by assigning 〈αi, αj〉 or (d−2)/2 αi 〈αi, αj〉h (d−2)/2 αi ), when d is even, for the edge connecting vertices i, j ∈ [k]. We then get a labeled graph whose adjacency matrix is exactly AP or ΨP,(d−2)/2 (depending on the labels we choose for the edges of G(P )). Naturally, one may also ask the following (open) questions. Open Problem 8.4. (a) Find some necessary or sufficient conditions on the (labeled) graph G(P ) such that the homogeneous harmonic polynomial P (z) is HN. (b) Find some necessary or sufficient conditions on the (labeled) graph G(P ) such that the VC holds for the homogeneous HN polynomial P (z). First, let us point out that, to approach the open problems above, it will be enough to focus on homogeneous harmonic polynomials P such that the graph G(P ) is connected. Suppose that the graph G(P ) is a disconnected graph with r ≥ 2 con- nected components. Let [k] = ⊔ri=1Ii be the corresponding partition of the set [k] of vertices of G(P ). For each 1 ≤ i ≤ r, we set Pi(z) := hdα(z). Note that, by Lemma 2.6, Pi (1 ≤ i ≤ r) are disjoint to each other, so Corollary 2.8 applies to the sum P = i=1 Pi. In particular, we have, (a) P is HN iff each Pi is HN. (b) if the VC holds for each Pi, then it also holds for P . SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 29 Therefore, we have the following connectedness reduction. Corollary 8.5. To study homogeneous HN polynomials P or the VC for homogeneous HN polynomials P , it will be enough to consider the case when G(P ) is connected. Note that, the property (a) above was first proved by R. Willems ([Wi]) by using the criterion in Theorem 8.2. (b) was first proved by the author by a different argument, and with the author’s permission, it had also been included in [Wi]. Finally, let us point out that R. Willems ([Wi]) has proved the following very interesting results on Open Problem 8.4. Theorem 8.6. ([Wi]) Let P be a homogeneous HN polynomial as in Eq.(8.2) with d ≥ 4. Let l(P ) be the dimension of the vector subspace of Cn spanned by {αi | 1 ≤ i ≤ k}. Then (1) If l(P ) = 1, 2, k−1 or k, the graph G(P ) is totally disconnected (i.e. G(P ) is the graph with no edges). (2) If l(P ) = k − 2 and G(P ) is connected, then G(P ) is the complete bi-graph K(4, k − 4). (3) In the case of (a) and (b) above, the VC holds. Furthermore, it has also been shown in [Wi] that, for any homogeneous HN polynomials P , the graph G(P ) can not be any path nor cycles of any positive length. For more details, see [Wi]. 8.2. Connection with the Tree Expansion Formula of Inversion Pairs. First let us recall the tree expansion formula derived in [M], [Wr2] for the inversion pair Q(z). Let T denote the set of all trees, i.e. the set of all connected and simply connected finite simple graphs. For each tree T ∈ T, denote by V (T ) and E(T ) the sets of all vertices and edges of T , respectively. Then we have the following tree expansion formula for inversion pairs. Theorem 8.7. ([M], [Wr2]) Let P ∈ C[[z]] with o(P ) ≥ 2 and Q its inver- sion pair. For any T ∈ T, set QT,P = ℓ:E(T )→[n] v∈V (T ) Dadj(v),ℓP,(8.9) where adj(v) is the set {e1, e2, . . . , es} of edges of T adjacent to v, and Dadj(v),ℓ = Dℓ(e1)Dℓ(e2) · · ·Dℓ(es). 30 WENHUA ZHAO Then the inversion pair Q of P is given by |Aut(T )| QT,P .(8.10) Now we assume P (z) is a homogeneous harmonic polynomial d ≥ 2 and has expression in Eq. (8.2). Under this assumption, it is easy to see that QT,P (T ∈ T) becomes QT,P = f :V (T )→[k] ℓ:E(T )→[n] v∈V (T ) Dadj(v),ℓh αf(v) (z).(8.11) The role played by the graph G(P ) of P is to restrict the maps f : V (T ) → V (G(P ))(= [k]) in Eq. (8.11) to a special family of maps. To be more precise, let Ω(T,G(P )) be the set of maps f : V (T ) → [k] such that, for any distinct adjoint vertices u, v ∈ V (T ), f(u) and f(v) are distinct and adjoint in G(P ). Then we have the following lemma. Lemma 8.8. For any f : V (T ) → [k] with f 6∈ Ω(T,G(P )), we have ℓ:E(T )→[n] v∈V (T ) Dadj(v),ℓh αf(v) (z) = 0.(8.12) Proof: Let f : V (T ) → [k] as in the lemma. Since f 6∈ Ω(T,G(P )), there exist distinct adjoint v1, v2 ∈ V (T ) such that, either f(v1) = f(v2) or f(v1) and f(v2) are not adjoint in the graph G(P ). In any case, we have 〈αf(v1), αf(v2)〉 = 0. Next we consider contributions to the RHS of Eq. (8.11) from the vertices v1 and v2. Denote by e the edge of T connecting v1 and v2, and {e1, . . . er} (resp. {ẽ1, . . . ẽs}) the set of edges connected with v1 (resp. v2) besides the edge e. Then, for any ℓ : E(T ) → [n], the factor in the RHS of Eq. (8.11) from the vertices v1 and v2 is the product Dℓ(e)Dℓ(e1) · · ·Dℓ(er)h αf(v1) Dℓ(e)Dℓ(ẽ1) · · ·Dℓ(ẽs)h αf(v2) .(8.13) Define an equivalent relation for maps ℓ : E(T ) → [n] by setting ℓ1 ∼ ℓ2 iff ℓ1, ℓ2 have same image at each edge of T except e. Then, by taking sum of the terms in Eq. (8.13) over each equivalent class, we get the factor ∇Dℓ(e1) · · ·Dℓ(er)h αf(v1) (z), ∇Dℓ(ẽ1) · · ·Dℓ(ẽs)h αf(v2) .(8.14) Note that Dℓ(e1) · · ·Dℓ(er)h αf(v1) (z) and Dℓ(ẽ1) · · ·Dℓ(ẽs)h αf(v2) (z) are con- stant multiples of some integral powers of hαf(v1)(z) and hαf(v2)(z), respec- tively. Therefore, 〈αf(v1), αf(v2)〉(= 0) appears as a multiplicative constant SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 31 factor in the term in Eq. (8.14), which makes the term zero. Hence the lemma follows. ✷ One immediate consequence of the lemma above is the following propo- sition. Proposition 8.9. With the setting and notation as above, we have QT,P = f∈Ω(T,G(P )) ℓ:E(T )→[n] v∈V (T ) Dadj(v),ℓh αf(v) (z).(8.15) Remark 8.10. (a) For any f ∈ Ω(T,G(P )), {f−1(j) | j ∈ Im(f)} gives a partition of V (T ) since no two distinct vertices in f−1(j) (j ∈ Im(f)) can be adjoint. In other words, f is nothing but a proper coloring for the tree T , which is also subject to certain more conditions from the graph structure of G(P ). It is interesting to see that the coloring problem of graphs also plays a role in the inversion problem of symmetric formal maps. (b) It will be interesting to see if more results can be derived from the graph G(P ) via the formulas in Eqs. (8.10) and (8.15). Remark 8.11. By similar arguments as those in proofs of Lemma 8.8, one may get another proof for Theorem 2.7 in the setting as in Lemma 2.6. Finally, as an application of Proposition 8.9 above, we give another proof for the connectedness reduction given in Corollary 8.5. Let P as given in Eq. (8.2) with the inversion pair Q. Suppose that there exists a partition [k] = I1 ⊔ I2 with Ii 6= ∅. Let Pi = hdα(z) (i = 1, 2) and Qi the inversion pair of Pi. Then we have P = P1 + P2 and G(P1)⊔G(P2) = G(P ). Therefore, to show the connectedness reduction discussed in the previous subsection, it will be enough to show Q = Q1+Q2. But this will follow immediately from Eqs. (8.10), (8.15) and the following lemma. Lemma 8.12. Let P , P1 and P2 as above, then, for any tree T ∈ T, we Ω(T,G(P )) = Ω(T,G(P1)) ⊔ Ω(T,G(P2)). Proof: For any f ∈ Ω(T,G(P )), f preserves the adjacency of vertices of G(P ). Since T as a graph is connected, Im(f) ⊂ V (G(P )) as a (full) subgraph of G(P ) must also be connected. Therefore, Im(f) ⊂ V (G(P1)) 32 WENHUA ZHAO or Im(f) ⊂ V (G(P2)). Hence Ω(T,G(P )) ⊂ Ω(T,G(P1)) ⊔ Ω(T,G(P2)). The other way of containess is obvious. ✷ References [BCW] H. Bass, E. Connell, D. Wright, The Jacobian Conjecture, Reduction of Degree and Formal Expansion of the Inverse. Bull. Amer. Math. Soc. 7, (1982), 287–330. [MR 83k:14028]. [Zbl.539.13012]. [B] M. de Bondt, Quasi-translations and Counterexamples to the Homogeneous De- pendence Problem. Proc. Amer. Math. 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Birkhuser Verlag, Basel, 2000. [MR1790619]. [EW] A. van den Essen and S. Washburn, The Jacobian Conjecture for Symmet- ric Jacobian Matrices, J. Pure Appl. Algebra, 189 (2004), no. 1-3, 123–133. [MR2038568] [EZ] A. van den Essen and W. Zhao, Two Results on Hessian Nilpotent Polynomials. To appear in J. Pure Appl. Algebra. See also arXiv:0704.1690v1 [math.AG]. [I] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathe- matics, 17. American Mathematical Society, Providence, RI, 1997. [MR1474964] [Ke] O. H. Keller, Ganze Gremona-Transformation, Monats. Math. Physik 47 (1939), no. 1, 299-306. [MR1550818]. [K] M. Kumar, Personal commucations. http://arxiv.org/abs/0704.1690 SOME PROPERTIES OF AND OPEN PROBLEMS ON HNPS 33 [M] G. Meng, Legendre Transform, Hessian Conjecture and Tree Formula, Appl. Math. Lett. 19 (2006), no. 6, 503–510. [MR2221506]. See also math-ph/0308035. [R] R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics, 108. Springer-Verlag New York Inc., 1986. [MR0847923]. [Wa] S. Wang, A Jacobian Criterion for Separability, J. Algebra 65 (1980), 453-494. [MR83e:14010]. [Wi] R. Willems, Graphs and the Jacobian Conjecture, Master Thesis, July 2005. Radboud University Nijmegen, The Netherlands. [Wr1] D. Wright, The Jacobian Conjecture: Ideal Membership Questions and Recent advances, Affine algebraic geometry, 261–276, Contemp. Math., 369, Amer. Math. Soc., Providence, RI, 2005. [MR2126667]. [Wr2] D. Wright, The Jacobian Conjecture as a Combinatorial Problem. Affine algebraic geometry, 483–503, Osaka Univ. Press, Osaka, 2007. See also math.CO/0511214. [MR2330486]. [Wr3] D. Wright, Personal communications. [Y] A. V. Jagžev, On a problem of O.-H. Keller. (Russian) Sibirsk. Mat. Zh. 21 (1980), no. 5, 141–150, 191. [MR0592226]. [Z1] W. Zhao, Inversion Problem, Legendre Transform and Inviscid Burgers’ Equa- tion, J. Pure Appl. Algebra, 199 (2005), no. 1-3, 299–317. [MR2134306]. See also math. CV/0403020. [Z2] W. Zhao, Hessian Nilpotent Polynomials and the Jacobian Conjecture, Trans. Amer. Math. Soc. 359 (2007), no. 1, 249–274 (electronic). [MR2247890]. See also math.CV/0409534. Department of Mathematics, Illinois State University, Normal, IL 61790-4520. E-mail: [email protected]. http://arxiv.org/abs/math-ph/0308035 http://arxiv.org/abs/math/0511214 http://arxiv.org/abs/math/0409534 1. Introduction 1.1. Background and Motivation 1.2. Arrangement 2. Disjoint Formal Power Series and Their Deformed Inversion Pairs 3. Local Convergence of Deformed Inversion Pairs of Homogeneous (HN) Polynomials 4. Self-Inverting Formal Power Series 5. The Vanishing Conjecture over Fields of Positive Characteristic 6. A Criterion of Hessian Nilpotency for Homogeneous Polynomials 7. Some Results on Symmetric Polynomial Maps 8. A Graph Associated with Homogeneous HN Polynomials 8.1. Definition and the Connectedness Reduction 8.2. Connection with the Tree Expansion Formula of Inversion Pairs References

0704.1690

Two Results on Homogeneous Hessian Nilpotent Polynomials

TWO RESULTS ON HOMOGENEOUS HESSIAN NILPOTENT POLYNOMIALS ARNO VAN DEN ESSEN∗ AND WENHUA ZHAO∗∗ Abstract. Let z = (z1, · · · , zn) and ∆ = the Laplace operator. A formal power series P (z) is said to be Hessian Nilpo- tent(HN) if its Hessian matrix HesP (z) = ( ∂ ∂zi∂zj ) is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjec- ture has been reduced to the following so-called vanishing conjec- ture(VC) of HN polynomials: for any homogeneous HN polynomial P (z) (of degree d = 4), we have ∆mPm+1(z) = 0 for any m >> 0. In this paper, we first show that, the VC holds for any homogeneous HN polynomial P (z) provided that the projective subvarieties ZP and Zσ2 of CP n−1 determined by the principal ideals generated by P (z) and σ2(z) := , respectively, intersect only at regular points of ZP . Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F = z−∇P with P (z) HN if F has no non-zero fixed point w ∈ Cn with = 0. Secondly, we show that the VC holds for a HN formal power series P (z) if and only if, for any polynomial f(z), ∆m(f(z)P (z)m) = 0 when m >> 0. 1. Introduction and Main Results Let z = (z1, z2, · · · , zn) be commutative free variables. Recall that the well-known Jacobian conjecture claims that: any polynomial map F (z) : Cn → Cn with the Jacobian j(F )(z) ≡ 1 is an autompophism of n and its inverse map must also be a polynomial map. Despite intense study from mathematicians in more than sixty years, the conjecture is still open even for the case n = 2. In 1998, S. Smale [S] included the Jacobian conjecture in his list of 18 important mathematical problems for 21st century. For more history and known results on the Jacobian conjecture, see [BCW], [E] and references there. Recently, M. de Bondt and the first author [BE1] and G. Meng [M] independently made the following remarkable breakthrough on the Ja- cobian conjecture. Namely, they reduced the Jacobian conjecture to 2000 Mathematics Subject Classification. 14R15, 31B05. Key words and phrases. Hessian nilpotent polynomials, the vanishing conjecture, symmetric polynomial maps, the Jacobian conjecture. http://arxiv.org/abs/0704.1690v1 2 ARNO VAN DEN ESSEN AND WENHUA ZHAO the so-called symmetric polynomial maps, i.e the polynomial maps of the form F = z − ∇P , where ∇P := ( ∂P , · · · , ∂P ), i.e. ∇P (z) is the gradient of P (z) ∈ C[z]. For more recent developments on the Jacobian conjecture for sym- metric polynomial maps, see [BE1]–[BE4]. Based on the symmetric reduction above and also the classical homo- geneous reduction in [BCW] and [Y], the second author in [Z] further reduced the Jacobian conjecture to the following so-called vanishing conjecture. Let ∆:= the Laplace operator and call a formal power series P (z) Hessian nilpotent(HN) if its Hessian matrix HesP (z) := ( ∂ ∂zi∂zj is nilpotent. It has been shown in [Z] that the Jacobian conjecture is equivalent to Conjecture 1.1. (Vanishing Conjecture of HN Polynomials) For any homogeneous HN polynomial P (z) (of degree d = 4), we have ∆mPm+1 = 0 when m >> 0. Note that, it has also been shown in [Z] that P (z) is HN if and only if ∆mPm = 0 for m ≥ 1. In this paper, we will prove the following two results on HN polyno- mials. Let P (z) be a homogeneous HN polynomial of degree d ≥ 3 and σ2(z):= i=1 z i . We denote by ZP and Zσ2 the projective subvarieties of CP n−1 determined by the principal ideals generated by P (z) and σ2(z), respectively. The first main result of this paper is the following theorem. Theorem 1.2. Let P (z) be a homogeneous HN polynomial of degree d ≥ 4. Assume that ZP intersects with Zσ2 only at regular points of ZP , then the vanishing conjecture holds for P (z). In particular, the vanishing conjecture holds if the projective variety ZP is regular. Remark 1.3. Note that, when degP (z) = d = 2 or 3, the Jacobian conjecture holds for the symmetric polynomial map F = z −∇P . This is because, when d = 2, F is a linear map with j(F ) ≡ 1. Hence F is an automorphism of Cn; while when d = 3, we have degF = 2. By Wang’s theorem [W], the Jacobian conjecture holds for F again. Then, by the equivalence of the vanishing conjecture for the homogeneous HN polynomial P (z) and the Jacobian conjecture for the symmetric map F = z −∇P established in [Z], we see that, when deg P (z) = d = 2 or 3, Theorem 1.2 actually also holds even without the condition on the projective variety ZP . HOMOGENEOUS HESSIAN NILPOTENT POLYNOMIALS 3 For any non-zero z ∈ Cn, denote by [z] its image in the projective space CP n−1. Set Z̃σ2 := {z ∈ C n | z 6= 0; [z] ∈ Zσ2}.(1.1) In other words, Z̃σ2 is the set of non-zero z ∈ C n such that i=1 z i = 0. Note that, for any homogeneous polynomial P (z) of degree d, it follows from the Euler’s formula dP = i=1 zi , that any non-zero w ∈ Cn, [w] ∈ CP n−1 is a singular point of ZP if and only if w is a fixed point of the symmetric map F = z−∇P . Furthermore, it is also well-known that, j(F ) ≡ 1 if and only if P (z) is HN. By the observations above and Theorem 1.2, it is easy to see that we have the following corollary on symmetric polynomial maps. Corollary 1.4. Let F = z − ∇P with P homogeneous and j(F ) ≡ 1 (or equivalently, P is HN). Assume that F does not fix any w ∈ Z̃σ2 . Then the Jacobian holds for F (z). In particular, if F has no non-zero fixed point, the Jacobian conjecture holds for F . Our second main result is following theorem which says that the vanishing conjecture is actually equivalent to a formally much stronger statement. Theorem 1.5. For any HN polynomial P (z), the vanishing conjec- ture holds for P (z) if and only if, for any polynomial f(z) ∈ C[z], ∆m(f(z)P (z)m) = 0 when m >> 0. 2. Proof of the Main Results Let us first fix the following notation. Let z = (z1, z2, · · · , zn) be free complex variables and C[z] (resp.C[[z]]) the algebra of polynomials (resp. formal power series) in z. For any d ≥ 0, we denote by Vd the vector space of homogeneous polynomials in z of degree d. For any 1 ≤ i ≤ n, we set Di = and D = (D1, D2, · · · , Dn). We define a C-bilinear map {·, ·} : C[z]× C[z] → C[z] by setting {f, g} := f(D)g(z) for any f(z), g(z) ∈ C[z]. Note that, for any m ≥ 0, the restriction of {·, ·} on Vm × Vm gives a C-bilinear form of the vector subspace Vm, which we will denote by Bm(·, ·). It is easy to check that, for any m ≥ 1, Bm(·, ·) is symmetric and non-singular. The following lemma will play a crucial role in our proof of the first main result. 4 ARNO VAN DEN ESSEN AND WENHUA ZHAO Lemma 2.1. For any homogeneous polynomials gi(z) (1 ≤ i ≤ k) of degree di ≥ 1, let S be the vector space of polynomial solutions of the following system of PDEs: g1(D) u(z) = 0, g2(D) u(z) = 0, ..... gk(D) u(z) = 0. (2.2) Then, dimS < +∞ if and only if gi(z) (1 ≤ i ≤ k) have no non-zero common zeroes. Proof: Let I the homogeneous ideal of C[z] generated by {gi(z)|1 ≤ i ≤ k}. Since all gi(z)’s are homogeneous, S is a homogeneous vector subspace S of C[z]. Write Sm,(2.3) Im.(2.4) where Im := I ∩ Vm and Sm := I ∩ Vm for any m ≥ 0. Claim: For any m ≥ 1 and u(z) ∈ Vm, u(z) ∈ Sm if and only if {u, Im} = 0, or in other words, Sm = I m with respect to the C-bilinear form Bm(·, ·) of Vm. Proof of the Claim: First, by the definitions of I and S, we have {Im, Sm} = 0 for any m ≥ 1, hence Sm ⊆ I m. Therefore, we need only show that, for any u(z) ∈ I⊥m ⊂ Vm, gi(D)u(z) = 0 for any 1 ≤ i ≤ n. We first fix any 1 ≤ i ≤ n. If m < di, there is nothing to prove. If m = di, then gi(z) ∈ Im, hence {gi, u} = gi(D)u = 0. Now suppose m > di. Note that, for any v(z) ∈ Vm−di , v(z)gi(z) ∈ Im. Hence we 0 = {v(z)gi(z), u(z)} = v(D)gi(D)u(z) = v(D) (gi(D)u) (z) = {v(z), (gi(D)u) (z)}. Therefore, we have Bm−di ((gi(D)u)(z), Vm−di) = 0. HOMOGENEOUS HESSIAN NILPOTENT POLYNOMIALS 5 Since Bm−di(·, ·) is a non-singular C-bilinear form of Vm−di , we have gi(D)u = 0. Hence, the Claim holds. ✷ By a well-known fact in Algebraic Geometry (see Exercise 2.2 in [H], for example), we know that the homogeneous polynomials gi(z) (1 ≤ i ≤ k) have no non-zero common zeroes if and only if Im = Vm when m >> 0. While, by the Claim above, we know that, Im = Vm when m >> 0 if and only if Sm = 0 when m >> 0, and if and only if the solution space S of the system (2.2) is finite dimensional. Hence, the lemma follows. ✷ Now we are ready to prove our first main result, Theorem 1.2. Proof of Theorem 1.2: Let P (z) be a homogeneous HN polynomial of degree d ≥ 4 and S the vector space of polynomial solutions of the following system of PDEs:   (D) u(z) = 0, (D) u(z) = 0, ..... (D) u(z) = 0, ∆ u(z) = 0. (2.5) First, note that the projective subvariety ZP intersects with Zσ2 only at regular points of ZP if and only if (z) (1 ≤ i ≤ n) and σ2 =∑n i=1 z i have no non-zero common zeros (agian use Euler’s formula). Then, by Lemma 2.1, we have dimS < +∞. On the other hand, by Theorem 6.3 in [Z], we know that ∆mPm+1 ∈ S for any m ≥ 0. Note that deg∆mPm+1 = (d−2)m+d for any m ≥ 0. So deg∆mPm+1 > deg∆kP k+1 for any m > k. Since dimS < +∞ (from above), we have ∆mPm+1 = 0 when m >> 0, i.e. the vanishing conjecture holds for P (z). ✷ Next, we give a proof for our second main result, Theorem 1.5. Proof of Theorem 1.5: The (⇐) part follows directly by choosing f(z) to be P (z) itself. To show (⇒) part, let d = deg f(z). If d = 0, f is a constant. Then, ∆m(f(z)P (z)m) = f(z)∆mPm = 0 for any m ≥ 1. So we assume d ≥ 1. By Theorem 6.2 in [Z], we know that, if the vanishing conjecture holds for P (z), then, for any fixed a ≥ 1, ∆mPm+a = 0 when m >> 0. Therefore there exists N > 0 such that, for any 0 ≤ b ≤ d and any m > N , we have ∆mPm+b = 0. 6 ARNO VAN DEN ESSEN AND WENHUA ZHAO By Lemma 6.5 in [Z], for any m ≥ 1, we have ∆m(f(z)P (z)m) =(2.6) k1+k2+k3=m k1,k2,k3≥0 k1, k2, k3 |s|=k2 ∂k2∆k1f(z) ∂k2∆k3Pm(z) where k1,k2,k3 denote the usual binomials. Note first that, the general term in the sum above is non-zero only if 2k1 + k2 ≤ d. But on the other hand, since 0 ≤ k1 + k2 ≤ 2k1 + k2 ≤ d,(2.7) by the choice of N ≥ 1, we have ∆k3Pm(z) = ∆k3P k3+(k1+k2)(z) is non-zero only if k3 ≤ N.(2.8) From the observations above and Eqs. (2.6), (2.7), (2.8) it is easy to see that, ∆m(f(z)P (z)m) 6= 0 only if m = k1 + k2 + k3 ≤ d + N . In other words, ∆m(f(z)P (z)m) = 0 for any m > d+N . Hence Theorem 1.5 holds. ✷ Note that all results used in the proof above for the (⇐) part of the theorem also hold for all HN formal power series. Therefore we have the following corollary. Corollary 2.2. Let P (z) be a HN formal power series such that the vanishing conjecture holds for P (z). Then, for any polynomial f(z), we have ∆m(f(z)P (z)m) = 0 when m >> 0. References [BCW] H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7, (1982), 287–330. [MR83k:14028], [Zbl.539.13012]. [BE1] M. de Bondt and A. van den Essen, A Reduction of the Jacobian Conjecture to the Symmetric Case, Proc. Amer. Math. Soc. 133 (2005), no. 8, 2201– 2205. [MR2138860]. [BE2] M. de Bondt and A. van den Essen, Nilpotent Symmetric Jacobian Matrices and the Jacobian Conjecture, J. Pure Appl. Algebra 193 (2004), no. 1-3, 61–70. [MR2076378]. [BE3] M. de Bondt and A. van den Essen, Nilpotent symmetric Jacobian matrices and the Jacobian conjecture II, J. Pure Appl. Algebra 196 (2005), no. 2-3, 135–148. [MR2110519]. [BE4] M. de Bondt and A. van den Essen, Singular Hessians, J. Algebra 282 (2004), no. 1, 195–204. [MR2095579]. HOMOGENEOUS HESSIAN NILPOTENT POLYNOMIALS 7 [E] A. van den Essen, Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, 190. Birkhuser Verlag, Basel, 2000. [MR1790619]. [H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York- Heidelberg-Berlin, 1977. [Ke] O. H. Keller, Ganze Gremona-Transformation, Monats. Math. Physik 47 (1939), 299-306. [M] G. Meng, Legendre Transform, Hessian Conjecture and Tree Formula, Appl. Math. Lett. 19 (2006), no. 6, 503–510. [MR2170971]. See also math-ph/0308035. [S] S. Smale, Mathematical Problems for the Next Century, Math. Intelligencer 20, No. 2, 7-15, 1998. [MR1631413 (99h:01033)]. [W] S. Wang, A Jacobian criterion for Separability, J. Algebra 65 (1980), 453- 494. [MR 83e:14010]. [Y] A. V. Jagžev, On a problem of O.-H. Keller. (Russian) Sibirsk. Mat. Zh. 21 (1980), no. 5, 141–150, 191. [MR0592226]. [Z] W. Zhao, Hessian Nilpotent Polynomials and the Jacobian Conjecture. Trans. Amer. Math. Soc. 359 (2007), 249-274. [MR2247890]. See also math.CV/0409534. ∗ Department of Mathematics, Radboud University Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands. E-mail: [email protected] ∗∗ Department of Mathematics, Illinois State University, Nor- mal, IL 61790-4520. E-mail: [email protected]. http://arxiv.org/abs/math-ph/0308035 http://arxiv.org/abs/math/0409534 1. Introduction and Main Results 2. Proof of the Main Results References

0704.1691

A Vanishing Conjecture on Differential Operators with Constant Coefficients

A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS WENHUA ZHAO Abstract. In the recent progress [BE1], [Me] and [Z2], the well- known JC (Jacobian conjecture) ([BCW], [E]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanish- ing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd order homogeneous differential operators Λ and Λ-nilpotent polynomials P (the polynomials P (z) satisfying ΛmPm = 0 for all m ≥ 1). We then transform some results in the literature on the JC, HN polynomials and the VC of the Laplace operators to certain results on Λ-nilpotent polynomials and the associated VC for 2nd order homogeneous differential operators Λ. This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of Λ-nilpotent polynomials in general with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homo- geneous Λ-nilpotent polynomials for 2nd order homogeneous full rank differential operators Λ with constant coefficients. 1. Introduction Let z = (z1, z2, . . . , zn, . . . ) be a sequence of free commutative vari- ables and D = (D1, D2, . . . , Dn, . . . ) with Di := (i ≥ 1). For any n ≥ 1, denote by An (resp. Ān) the algebra of polynomials (resp. formal power series) in zi (1 ≤ i ≤ n). Furthermore, we denote by D[An] or D[n] (resp.D[An] or D[n]) the algebra of differential operators of the polynomial algebra An (resp.with constant coefficients). Note that, for any k ≥ n, elements of D[n] are also differential operators of Ak and Date: November 7, 2018. 2000 Mathematics Subject Classification. 14R15, 33C45, 32W99. Key words and phrases. Differential operators with constant coefficients, Λ- nilpotent polynomials, Hessian nilpotent polynomials, classical orthogonal poly- nomials, the Jacobian conjecture. http://arxiv.org/abs/0704.1691v2 2 WENHUA ZHAO Āk. For any d ≥ 0, denote by Dd[n] the set of homogeneous differen- tial operators of order d with constants coefficients. We let A (resp. Ā) be the union of An (resp. Ān) (n ≥ 1), D (resp.D) the union of D[n] (resp.D[n]) (n ≥ 1), and, for any d ≥ 1, Dd the union of Dd[n] (n ≥ 1). Recall that JC (the Jacobian conjecture) which was first proposed by Keller [Ke] in 1939, claims that, for any polynomial map F of Cn with Jacobian j(F ) = 1, its formal inverse map G must also be a polynomial map. Despite intense study from mathematicians in more than sixty years, the conjecture is still open even for the case n = 2. For more history and known results before 2000 on JC, see [BCW], [E] and references there. Based on the remarkable symmetric reduction achieved in [BE1], [Me] and the classical celebrated homogeneous reduction [BCW] and [Y] on JC, the author in [Z2] reduced JC further to the following vanishing conjecture on the Laplace operators ∆n := i of the polynomial algebra An and HN (Hessian nilpotent) polynomials P (z) ∈ An, where we say a polynomial or formal power series P (z) ∈ Ān is HN if its Hessian matrix Hes (P ) := ( ∂ ∂zi∂zj )n×n is nilpotent. Conjecture 1.1. For any HN (homogeneous) polynomial P (z) ∈ An (of degree d = 4), we have ∆mn P m+1(z) = 0 when m >> 0. Note that, the following criteria of Hessian nilpotency were also proved in Theorem 4.3, [Z2]. Theorem 1.2. For any P (z) ∈ Ān with o(P (z)) ≥ 2, the following statements are equivalent. (1) P (z) is HN. (2) ∆mPm = 0 for any m ≥ 1. (3) ∆mPm = 0 for any 1 ≤ m ≤ n. Through the criteria in the proposition above, Conjecture 1.1 can be generalized to other differential operators as follows (see Conjecture 1.4 below). First let us fix the following notion that will be used throughout the paper. Definition 1.3. Let Λ ∈ D[An] and P (z) ∈ Ān. We say P (z) is Λ-nilpotent if ΛmPm = 0 for any m ≥ 1. Note that, when Λ is the Laplace operator ∆n, by Theorem 1.2, a polynomial or formal power series P (z) ∈ An is Λ-nilpotent iff it is HN. With the notion above, Conjecture 1.1 has the following natural generalization to differential operators with constant coefficients. A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 3 Conjecture 1.4. For any n ≥ 1 and Λ ∈ D[n], if P (z) ∈ An is Λ- nilpotent, then ΛmPm+1 = 0 when m >> 0. We call the conjecture above the vanishing conjecture for differential operators with constant coefficients and denote it by VC. The special case of VC with P (z) homogeneous is called the homogeneous vanish- ing conjecture and denoted by HVC. When the number n of variables is fixed, VC (resp.HVC) is called (resp. homogeneous) vanishing con- jecture in n variables and denoted by VC[n] (resp.HVC[n]). Two remarks on VC are as follows. First, due to a counter-example given by M. de Bondt (see example 2.4), VC does not hold in general for differential operators with non-constant coefficients. Secondly, one may also allow P (z) in VC to be any Λ-nilpotent formal power series. No counter-example to this more general VC is known yet. In this paper, we first apply certain linear automorphisms and Lef- schetz’s principle to show Conjecture 1.1, hence also JC, is equivalent to VC or HVC for all 2nd order homogeneous differential operators Λ ∈ D2 (see Theorem 2.9). We then in Section 3 transform some results on JC, HN polynomials and Conjecture 1.1 obtained in [Wa], [BE2], [BE3], [Z2], [Z3] and [EZ] to certain results on Λ-nilpotent (Λ ∈ D2) polynomials and VC for Λ. Another purpose of this section is to give a survey on recent study on Conjecture 1.1 and HN polynomials in the more general setting of Λ ∈ D2 and Λ-nilpotent polynomials. This is also why some results in the general setting, even though their proofs are straightforward, are also included here. Even though, due to M. de Bondt’s counter-example (see Example 2.4), VC does not hold for all differential operators with non-constant coefficients, it is still interesting to consider whether or not VC holds for higher order differential operators with constant coefficients; and if it also holds even for certain families of differential operators with non-constant coefficients. For example, when Λ = Da with a ∈ Nn and |a| ≥ 2, VC[n] for Λ is equivalent to a conjecture on Laurent polynomi- als (see Conjecture 3.21). This conjecture is very similar to a non-trivial theorem (see Theorem 3.20) on Laurent polynomials, which was first conjectured by O. Mathieu [Ma] and later proved by J. Duistermaat and W. van der Kallen [DK]. In general, to consider the questions above, one certainly needs to get better understandings on the Λ-nilpotency condition, i.e. ΛmPm = 0 for any m ≥ 1. One natural way to look at this condition is to consider the sequences of the form {ΛmPm |m ≥ 1} for general differential op- erators Λ and polynomials P (z) ∈ A. What special properties do these 4 WENHUA ZHAO sequences have so that VC wants them all vanish? Do they play any important roles in other areas of mathematics? The answer to the first question above is still not clear. The answer to the later seems to be ”No”. It seems that the sequences of the form {ΛmPm |m ≥ 1} do not appear very often in mathematics. But the answer turns out to be “Yes” if one considers the question in the setting of some localizationsB ofAn. Actually, as we will discuss in some detail in subsection 4.1, all classical orthogonal polynomials in one variable have the form {ΛmPm |m ≥ 1} except there one often chooses P (z) from some localizations B of An and Λ a differential operators of B. Some classical polynomials in several variables can also be obtained from sequences of the form {ΛmPm |m ≥ 1} by a slightly modified procedure. Note that, due to their applications in many different areas of math- ematics, especially in ODE, PDE, the eigenfunction problems and rep- resentation theory, orthogonal polynomials have been under intense study by mathematicians in the last two centuries. For example, in [SHW] published in 1940, about 2000 published articles mostly on one- variable orthogonal polynomials have been included. The classical ref- erence for one-variable orthogonal polynomials is [Sz] (see also [AS], [C], [Si]). For multi-variable orthogonal polynomials, see [DX], [Ko] and references there. It is hard to believe that the connection discussed above between Λ-nilpotent polynomials or formal power series and classical orthog- onal polynomials is just a coincidence. But a precise understanding of this connection still remains mysterious. What is clear is that, Λ- nilpotent polynomials or formal power series and the polynomials or formal power series P (z) ∈ Ān such that the sequence {ΛmPm |m ≥ 1} for some differential operator Λ provides a sequence of orthogonal poly- nomials lie in two opposite extreme sides, since, from the same sequence {ΛmPm |m ≥ 1}, the former provides nothing but zero; while the later provides an orthogonal basis for An. Therefore, one naturally expects that Λ-nilpotent polynomials P (z)∈ An should be isotropic with respect to a certain C-bilinear form of An. It turns out that, as we will show in Theorem 4.10 and Corollary 4.11, it is indeed the case when P (z) is homogeneous and Λ ∈ D2[n] is of full rank. Actually, in this case ΛmPm+1 (m ≥ 0) are all isotropic with respect to same properly defined C-bilinear form. Note that, Theorem 4.10 and Corollary 4.11 are just transformations of the isotropic prop- erties of HN nilpotent polynomials, which were first proved in [Z2]. But the proof in [Z2] is very technical and lacks any convincing inter- pretations. From the “formal” connection of Λ-nilpotent polynomials A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 5 and orthogonal polynomials discussed above, the isotropic properties of homogeneous Λ-nilpotent polynomials with Λ ∈ D2[n] of full rank become much more natural. The arrangement of the paper is as follows. In Section 2, we mainly show that Conjecture 1.1, hence also JC, is equivalent to VC or HVC for all Λ ∈ D2 (see Theorem 2.9). One consequence of this equivalence is that, to prove or disprove VC or JC, the Laplace operators are not the only choices, even though they are the best in many situations. Instead, one can choose any sequence Λnk ∈ D2 with strictly increasing ranks (see Proposition 2.10). For example, one can choose the “Laplace operators” with respect to the Minkowski metric or symplectic metric, or simply choose Λ to be the complex ∂̄-Laplace operator ∆∂̄,k (k ≥ 1) in Eq. (2.11). In Section 3, we transform some results on JC, HN polynomials and Conjecture 1.1 in the literature to certain results on Λ-nilpotent (Λ ∈ D2) polynomials P (z) and VC for Λ. In subsection 3.1, we discuss some results on the polynomial maps and PDEs associated with Λ-nilpotent polynomials for Λ ∈ D2[n] of full rank (see Theorems 3.1–3.3). The results in this subsection are transformations of those in [Z1] and [Z2] on HN polynomials and their associated symmetric polynomial maps. In subsection 3.2, we give four criteria of Λ-nilpotency (Λ ∈ D2) (see Propositions 3.4, 3.6, 3.7 and 3.10). The criteria in this subsection are transformations of the criteria of Hessian nilpotency derived in [Z2] and [Z3]. In subsection 3.3, we transform some results in [BCW], [Wa] and [Y] on JC; [BE2] and [BE3] on symmetric polynomial maps; [Z2], [Z3] and [EZ] on HN polynomials to certain results on VC for Λ ∈ D2. Finally, we recall a result in [Z3] which says, VC over fields k of characteristic p > 0, even under some conditions weaker than Λ- nilpotency, actually holds for any differential operators Λ of k[z] (see Proposition 3.22 and Corollary 3.23). In subsection 3.4, we consider VC for high order differential opera- tors with constant coefficients. In particular, we show in Proposition 3.18 VC holds for Λ = δk (k ≥ 1), where δ is a derivation of A. In particular, VC holds for any Λ ∈ D1 (see Corollary 3.19). We also show that, when Λ = Da with a ∈ Nn and |a| ≥ 2, VC is equivalent to a conjecture, Conjecture 3.21, on Laurent polynomials. This con- jecture is very similar to a non-trivial theorem (see Theorem 3.20) first conjectured by O. Mathieu [Ma] and later proved by J. Duistermaat and W. van der Kallen [DK]. 6 WENHUA ZHAO In subsection 4.1, by using Rodrigues’ formulas Eq. (4.1), we show that all classical orthogonal polynomials in one variable have the form {ΛmPm |m ≥ 1} for some P (z) in certain localizations B of An and Λ a differential operators of B. We also show that some classical polynomi- als in several variables can also be obtained from sequences of the form {ΛmPm |m ≥ 1} with a slight modification. Some of the most classical orthogonal polynomials in one or more variables are briefly discussed in Examples 4.2–4.5, 4.8 and 4.9. In subsection 4.2, we transform the isotropic properties of homogeneous HN homogeneous polynomials de- rived in [Z2] to homogeneous Λ-nilpotent polynomials for Λ ∈ D2[n] of full rank (see Theorem 4.10 and Corollary 4.11). Acknowledgment: The author is very grateful to Michiel de Bondt for sharing his counterexample (see Example 2.4) with the author, and to Arno van den Essen for inspiring personal communications. The author would also like to thank the referee very much for many valuable suggestions to improve the first version of the paper. 2. The Vanishing Conjecture for the 2nd Order Homogeneous Differential Operators with Constant Coefficients In this section, we apply certain linear automorphisms and Lef- schetz’s principle to show Conjecture 1.1, hence also JC, is equivalent to VC or HVC for all Λ ∈ D2 (see Theorem 2.9). In subsection 2.1, we fix some notation and recall some lemmas that will be needed through- out this paper. In subsection 2.2, we prove the main results of this section, Theorem 2.9 and Proposition 2.10. 2.1. Notation and Preliminaries. Throughout this paper, unless stated otherwise, we will keep using the notations and terminology in- troduced in the previous section and also the ones fixed as below. (1) For any P (z) ∈ An, we denote by ∇P the gradient of P (z), i.e. we set ∇P (z) := (D1P, D2P, . . . , DnP ).(2.1) (2) For any n ≥ 1, we let SM(n,C) (resp.SGL(n,C)) denote the symmetric complex n× n (resp. invertible) matrices. (3) For any A = (aij) ∈ SM(n,C), we set ∆A := i,j=1 aijDiDj ∈ D2[n].(2.2) A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 7 Note that, for any Λ ∈ D2[n], there exists a unique A ∈ SM(n,C) such that Λ = ∆A. We define the rank of Λ = ∆A simply to be the rank of the matrix A. (4) For any n ≥ 1, Λ ∈ D2[n] is said to be full rank if Λ has rank n. The set of full rank elements of D2[n] will be denoted by D 2[n]. (5) For any r ≥ 1, we set ∆r := D2i .(2.3) Note that ∆r is a full rank element in D2[r] but not in D2[n] for any n > r. For U ∈ GL(n,C), we define ΦU : Ān → Ān(2.4) P (z) → P (U−1z) ΨU : D[n] → D[n](2.5) Λ → ΦU ◦ Λ ◦ Φ−1U It is easy to see that, ΦU (resp.ΨU ) is an algebra automorphism of An (resp.D[n]). Moreover, the following standard facts are also easy to check directly. Lemma 2.1. (a) For any U = (uij) ∈ GL(n,C), P (z) ∈ Ān and Λ ∈ D[n], we have ΦU(ΛP ) = ΨU(Λ)ΦU(P ).(2.6) (b) For any 1 ≤ i ≤ n and f(z) ∈ An we have ΨU(Di) = ujiDj , ΨU(f(D)) = f(U In particular, for any A ∈ SM(n,C), we have ΨU(∆A) = ∆UAUτ .(2.7) The following lemma will play a crucial role in our later arguments. Actually the lemma can be stated in a stronger form (see [Hu], for example) which we do not need here. 8 WENHUA ZHAO Lemma 2.2. For any A ∈ SM(n,C) of rank r > 0, there exists U ∈ GL(n,C) such that A = U Ir×r 0 U τ(2.8) Combining Lemmas 2.1 and 2.2, it is easy to see we have the following corollary. Corollary 2.3. For any n ≥ 1 and Λ,Ξ ∈ D2[n] of same rank, there exists U ∈ GL(n,C) such that ΨU(Λ) = Ξ. 2.2. The Vanishing Conjecture for the 2nd Order Homoge- neous Differential Operators with Constant Coefficients. In this subsection, we show that Conjecture 1.1, hence also JC, is actu- ally equivalent to VC or HVC for all 2nd order homogeneous differ- ential operators Λ ∈ D2 (see Theorem 2.9). We also show that the Laplace operators are not the only choices in the study of VC or JC (see Proposition 2.10 and Example 2.11). First, let us point out that VC fails badly for differential opera- tors with non-constant coefficients. The following counter-example was given by M. de Bondt [B]. Example 2.4. Let x be a free variable and Λ = x d . Let P (x) = x. Then one can check inductively that P (x) is Λ-nilpotent, but ΛmPm+1 6= 0 for any m ≥ 1. Lemma 2.5. For any Λ ∈ D[n], U ∈ GL(n,C), A ∈ SM(n,C) and P (z) ∈ Ān, we have (a) P (z) is Λ-nilpotent iff ΦU (P ) is ΨU(Λ)-nilpotent. In particular, P (z) is ∆A-nilpotent iff ΦU (P ) = P (U −1z) is ∆UAUτ -nilpotent. (b) VC[n] (resp.HVC[n]) holds for Λ iff it holds for ΨU(Λ). In particular, VC[n] (resp.HVC[n]) holds for ∆A iff it holds for ∆UAUτ . Proof: Note first that, for any m, k ≥ 1, we have ΛmP k = (ΦUΛ mΦ−1U ) ΦUP = (ΦUΛΦ m(ΦUP ) = [ΨU(Λ)] m(ΦUP ) When Λ = ∆A, by Eq. (2.7), we further have = ΛmUAUτ (ΦUP ) A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 9 Since ΦU (resp.ΨU ) is an automorphism of Ān (resp.D[n]), it is easy to check directly that both (a) and (b) follow from the equations above. Combining the lemma above with Corollary 2.3, we immediately have the following corollary. Corollary 2.6. Suppose HVC[n] (resp.VC[n]) holds for a differential operator Λ ∈ D2[n] of rank r ≥ 1. Then HVC[n] (resp.VC[n]) holds for all differential operators Ξ ∈ D2[n] of rank r. Actually we can derive much more (as follows) from the conditions in the corollary above. Proposition 2.7. (a) Suppose HVC[n] holds for a full rank Λ ∈ D◦2[n]. Then, for any k ≤ n, HVC[k] holds for all full rank Ξ ∈ D◦2[k]. (b) Suppose VC[n] holds for a full rank Λ ∈ D◦2[n]. Then, for any m ≥ n, VC[m] holds for all Ξ ∈ D2[m] of rank n. Proof: Note first that, the cases k = n in (a) and m = n in (b) follow directly from Corollary 2.6. So we may assume k < n in (a) and m > n in (b). Secondly, by Corollary 2.6, it will be enough to show HVC[k] (k < n) holds for ∆k for (a) and VC[m] (m > n) holds for ∆n for (b). (a) Let P ∈ Ak a homogeneous ∆k-nilpotent polynomial. We view ∆k and P as elements of D2[n] and An, respectively. Since P does not depend on zi (k + 1 ≤ i ≤ n), for any m, ℓ ≥ 0, we have ∆mk P ℓ = ∆mn P Hence, P is also ∆n-nilpotent. Since HVC[n] holds for ∆n (as pointed out at the beginning of the proof), we have ∆mk P m+1 = ∆mn P m+1 = 0 when m >> 0. Therefore, HVC[k] holds for ∆k. (b) Let K be the rational function field C(zn+1, . . . , zm). We view Am as a subalgebra of the polynomial algebra K[z1, . . . , zn] in the standard way. Note that the differential operator ∆n = D2i of Am extends canonically to a differential operator of K[z1, . . . , zn] with constant co- efficients. Since VC[n] holds for ∆n over the complex field (as pointed out at the beginning of the proof), by Lefschetz’s principle, we know that VC[n] also holds for ∆n over the field K. Therefore, for any ∆n- nilpotent P (z) ∈ Am, by viewing ∆n as an element of D2(K[z1, . . . , zn]) and P (z) an element of K[z1, . . . , zn] (which is still ∆n-nilpotent in the new setting), we have ∆knP k+1 = 0 when k >> 0. Hence VC[m] holds for P (z) ∈ Am and ∆n ∈ D2[m]. ✷ 10 WENHUA ZHAO Proposition 2.8. Suppose HVC[n] holds for a differential operator Λ ∈ D2[n] with rank r < n. Then, for any k ≥ r, VC[k] holds for all Ξ ∈ D2[k] of rank r. Proof: First, by Corollary 2.6, we know HVC[n] holds for ∆r. To show Proposition 2.8, by Proposition 2.7, (b), it will be enough to show that VC[r] holds for ∆r. Let P ∈ Ar ⊂ An be a ∆r-nilpotent polynomial. If P is homoge- neous, there is nothing to prove since, as pointed out above, HVC[n] holds for ∆r. Otherwise, we homogenize P (z) to P̃ ∈ Ar+1 ⊆ An. Since ∆r is a homogeneous differential operator, it is easy to see that, for any m, k ≥ 1, ∆mr P k = 0 iff ∆mr P̃ k = 0. Therefore, P̃ ∈ An is also ∆r-nilpotent when we view ∆r as a differential operator of An. Since HVC[n] holds for ∆r, we have that ∆ m+1 = 0 when m >> 0. Then, by the observation above again, we also have ∆mr P m+1 = 0 when m >> 0. Therefore, VC[r] holds for ∆r. ✷ Now we are ready to prove our main result of this section. Theorem 2.9. The following statements are equivalent to each other. (1) JC holds. (2) HVC[n] (n ≥ 1) hold for the Laplace operator ∆n. (3) VC[n] (n ≥ 1) hold for the Laplace operator ∆n. (4) HVC[n] (n ≥ 1) hold for all Λ ∈ D2[n]. (5) VC[n] (n ≥ 1) hold for all Λ ∈ D2[n]. Proof: First, the equivalences of (1), (2) and (3) have been estab- lished in Theorem 7.2 in [Z2]. While (4) ⇒ (2), (5) ⇒ (3) and (5) ⇒ (4) are trivial. Therefore, it will be enough to show (3) ⇒ (5). To show (3) ⇒ (5), we fix any n ≥ 1. By Corollary 2.6, it will be enough to show VC[n] holds for ∆r (1 ≤ r ≤ n). But under the assumption of (3) (with n = r), we know that VC[r] holds for ∆r. Then, by Proposition 2.7, (b), we know VC[n] also holds for ∆r. ✷ Next, we show that, to study HVC, equivalently VC or JC, the Laplace operators are not the only choices, even though they are the best in many situations. Proposition 2.10. Let {nk | k ≥ 1} be a strictly increasing sequence of positive integers and {Λnk | k ≥ 1} a sequence of differential operators in D2 with rank (Λnk) = nk (k ≥ 1). Suppose that, for any k ≥ 1, HVC[Nk] holds for Λnk for some Nk ≥ nk. Then, the equivalent state- ments in Theorem 2.9 hold. A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 11 Proof: We show, under the assumption in the proposition, the state- ment (2) in Theorem 2.9 holds, i.e. for any n ≥ 1, HVC[n] (n ≥ 1) holds for the Laplace operator ∆n ∈ D2[n]. For any fixed n ≥ 1, let k ≥ 1 such that nk ≥ n. If Nk = nk, then, by Proposition 2.7, (a), we have HVC[n] (n ≥ 1) holds for the Laplace operator ∆n ∈ D2[n]. If Nk > nk, then, by Proposition 2.8, we know VC[nk] (hence also HVC[nk]) holds for ∆nk . Since nk ≥ n, by Proposition 2.7, (a) again, we know HVC[n] does hold for the Laplace operator ∆n. ✷ Example 2.11. Besides the Laplace operators, by Proposition 2.10, the following sequences of differential operators are also among the most natural choices. (1) Let nk = k (k ≥ 2) (or any other strictly increasing sequence of positive integers). Let Λnk be the “Laplace operator” with respect to the standard Minkowski metric of Rnk . Namely, choose Λk = D D2i .(2.9) (2) Choose nk = 2k (k ≥ 1) (or any other strictly increasing se- quence of positive even numbers). Let Λ2k be the “Laplace op- erator” with respect to the standard symplectic metric on R2k, i.e. choose Λ2k = DiDi+k.(2.10) (3) We may also choose the complex Laplace operators ∆∂̄ instead of the real Laplace operator ∆. More precisely, we choose nk = 2k for any k ≥ 1 and view the polynomial algebra of wi (1 ≤ i ≤ 2k) over C as the polynomial algebra C[zi, z̄i | 1 ≤ i ≤ k] by setting zi = wi + −1wi+k for any 1 ≤ i ≤ k. Then, for any k ≥ 1, we set Λk = ∆∂̄,k := ∂zi∂z̄i .(2.11) (4) More generally, we may also choose Λk = ∆Ank , where nk ∈ N and Ank ∈ SM(nk,C) (not necessarily invertible) (k ≥ 1) with strictly increasing ranks. 12 WENHUA ZHAO 3. Some Properties of ∆A-Nilpotent Polynomials As pointed earlier in Section 1 (see page 2), for the Laplace operators ∆n (n ≥ 1), the notion ∆n-nilpotency coincides with the notion of Hes- sian nilpotency. HN (Hessian nilpotent) polynomials or formal power series, their associated symmetric polynomial maps and Conjecture 1.1 have been studied in [BE2], [BE3], [Z1]–[Z3] and [EZ]. In this section, we apply Corollary 2.3, Lemma 2.5 and also Lefschetz’s principle to transform some results obtained in the references above to certain re- sults on Λ-nilpotent (Λ ∈ D2) polynomials or formal power series, VC for Λ and also associated polynomial maps. Another purpose of this section is to give a short survey on some results on HN polynomials and Conjecture 1.1 in the more general setting of Λ-nilpotent polynomials and VC for differential operators Λ ∈ D2. In subsection 3.1, we transform some results in [Z1] and [Z2] to the setting of Λ-nilpotent polynomials for Λ ∈ D2[n] of full rank (see Theo- rems 3.1–3.3). In subsection 3.2, we derive four criteria for Λ-nilpotency (Λ ∈ D2) (see Propositions 3.4, 3.6, 3.7 and 3.10). The criteria in this subsection are transformations of the criteria of Hessian nilpotency de- rived in [Z2] and [Z3]. In subsection 3.3, we transform some results in [BCW], [Wa] and [Y] on JC; [BE2] and [BE3] on symmetric polynomial maps; [Z2], [Z3] and [EZ] on HN polynomials to certain results on VC for Λ ∈ D2. In subsection 3.4, we consider VC for high order differential operators with constant coefficients. We mainly focus on the differential operators Λ = Da (a ∈ Nn). Surprisingly, VC for these operators is equivalent to a conjecture (see Conjecture 3.21) on Laurent polynomials, which is similar to a non-trivial theorem (see Theorem 3.20) first conjectured by O. Mathieu [Ma] and later proved by J. Duistermaat and W. van der Kallen [DK]. 3.1. Associated Polynomial Maps and PDEs. Once and for all in this section, we fix any n ≥ 1 and A ∈ SM(n,C) of rank 1 ≤ r ≤ n. We use z and D, unlike we did before, to denote the n-tuples (z1, z2, . . . , zn) and (D1, D2, . . . , Dn), respectively. We define a C-bilinear form 〈·, ·〉A by setting 〈u, v〉A := uτAv for any u, v ∈ Cn. Note that, when A = In×n, the bilinear form defined above is just the standard C-bilinear form of Cn, which we also denote by 〈·, ·〉. By Lemma 2.2, we may write A as in Eq. (2.8). For any P (z) ∈ Ān, we set P̃ (z) = Φ−1U P (z) = P (Uz).(3.1) A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 13 Note that, by Lemma 2.1, (b), we have Ψ−1U (∆A) = ∆r. By Lemma 2.5, (a), P (z) is ∆A-nilpotent iff P̃ (z) is ∆r-nilpotent. Theorem 3.1. Let t be a central parameter. For any P (z) ∈ An with o(P (z)) ≥ 2 and A ∈ SGL(n,C), set FA,t(z) := z − tA∇P (z). Then (a) there exists a unique QA,t(z) ∈ C[t][[z]] such that the formal inverse map GA,t(z) of FA,t(z) is given by GA,t(z) = z + tA∇QA,t(z).(3.2) (b) The QA,t(z) ∈ C[t][[z]] in (a) is the unique formal power series solution of the following Cauchy problem: ∂ QA,t (z) = 1 〈∇QA,t,∇QA,t〉A, QA,t=0(z) = P (z). (3.3) Proof: Let P̃ as given in Eq. (3.1) and set F̃A,t(z) = z − t∇P̃ (z).(3.4) By Theorem 3.6 in [Z1], we know the formal inverse map G̃A,t(z) of F̃A,t(z) is given by G̃A,t(z) = z + t∇Q̃A,t(z),(3.5) where Q̃A,t(z) ∈ C[t][[z]] is the unique formal power series solution of the following Cauchy problem: ∂ eQA,t (z) = 1 〈∇Q̃A,t,∇Q̃A,t〉, Q̃A,t=0(z) = P̃ (z). (3.6) From the fact that ∇P̃ (z) = (U τ∇P )(Uz), it is easy to check that (ΦU ◦ F̃A,t ◦ Φ−1U )(z) = z − tA∇P (z) = FA,t(z),(3.7) which is the formal inverse map of (ΦU ◦ G̃A,t ◦ Φ−1U )(z) = z + t(U∇Q̃A,t)(U −1z).(3.8) QA,t(z) := Q̃A,t(U −1z).(3.9) Then we have ∇QA,t(z) = (U τ )−1(∇Q̃A,t)(U−1z), U τ∇QA,t(z) = (∇Q̃A,t)(U−1z),(3.10) 14 WENHUA ZHAO Multiplying U to the both sides of the equation above and noticing that A = UU τ by Eq. (2.8) since A is of full rank, we get A∇QA,t(z) = (U∇Q̃A,t)(U−1z).(3.11) Then, combining Eq. (3.8) and the equation above, we see the formal inverse GA,t(z) of FA,t(z) is given by GA,t(z) = (ΦU ◦ G̃A,t ◦ Φ−1U )(z) = z + tA∇QA,t(z).(3.12) Applying ΦU to Eq. (3.6) and by Eqs. (3.9), (3.10), we see thatQA,t(z) is the unique formal power series solution of the Cauchy problem Eq. (3.3). By applying the linear automorphism ΦU of C[[z]] and employing a similar argument as in the proof of Theorem 3.1 above, we can gen- eralize Theorems 3.1 and 3.4 in [Z2] to the following theorem on ∆A- nilpotent (A ∈ SGL(n,C)) formal power series. Theorem 3.2. Let A, P (z) and QA,t(z) as in Theorem 3.1. We further assume P (z) is ∆A-nilpotent. Then, (a) QA,t(z) is the unique formal power series solution of the follow- ing Cauchy problem: ∂ QA,t (z) = 1 QA,t=0(z) = P (z). (3.13) (b) For any k ≥ 1, we have QkA,t(z) = 2mm!(m+ k)! m+1(z).(3.14) Applying the same strategy to Theorem 3.2 in [Z2], we get the fol- lowing theorem. Theorem 3.3. Let A, P (z) and QA,t(z) as in Theorem 3.2. For any non-zero s ∈ C, set Vt,s(z) := exp(sQt(z)) = skQkt (z) Then, Vt,s(z) is the unique formal power series solution of the following Cauchy problem of the heat-like equation: ∂Vt,s (z) = 1 ∆AVt,s(z), Ut=0,s(z) = exp(sP (z)). (3.15) A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 15 3.2. Some Criteria of ∆A-Nilpotency. In this subsection, with the notation and remarks fixed in the previous subsection in mind, we apply the linear automorphism ΦU to transform some criteria of Hessian nilpotency derived in [Z2] and [Z3] to criteria of ∆A-nilpotency (A ∈ SM(n,C)) (see Proposition 3.4, 3.6, 3.7 and 3.10 below). Proposition 3.4. Let A be given as in Eq. (2.8). Then, for any P (z) ∈ An, it is ∆A-nilpotent iff the submatrix of U τ (HesP )U consisting of the first r rows and r columns is nilpotent. In particular, when r = n, i.e. ∆A is full rank, any P (z) ∈ D2[n] is ∆A-nilpotent iff U τ (HesP )U is nilpotent. Proof: Let P̃ (z) be as in Eq. (3.1). Then, as pointed earlier, P (z) is ∆A-nilpotent iff P̃ (z) is ∆r-nilpotent. If r = n, then by Theorem 1.2 , P̃ (z) is ∆r-nilpotent iff Hes P̃ (z) is nilpotent. But note that in general we have Hes P̃ (z) = HesP (Uz) = U τ [(HesP )(Uz)]U.(3.16) Therefore, Hes P̃ (z) is nilpotent iff U τ [(HesP )(Uz)]U is nilpotent iff, with z replaced by U−1z, U τ [(HesP )(z)]U is nilpotent. Hence the proposition follows in this case. Assume r < n. We view Ar as a subalgebra of the polynomial algebra K[z1, . . . , zr], where K is the rational field C(zr+1, . . . , zn). By Theorem 1.2 and Lefschetz’s principle, we know that P̃ is ∆r-nilpotent iff the matrix ∂2 eP ∂zi∂zj 1≤i,j≤r is nilpotent. Note that the matrix ∂2 eP ∂zi∂zj 1≤i,j≤r is the submatrix of Hes P̃ (z) con- sisting of the first r rows and r columns. By Eq. (3.16), it is also the sub- matrix of U τ [HesP (Uz)]U consisting of the first r rows and r columns. Replacing z by U−1z in the submatrix above, we see ∂2 eP ∂zi∂zj 1≤i,j≤r nilpotent iff the submatrix of U τ [HesP (z)]U consisting of the first r rows and r columns is nilpotent. Hence the proposition follows. ✷ Note that, for any homogeneous quadratic polynomial P (z) = zτBz with B ∈ SM(n,C), we have HesP (z) = 2B. Then, by Proposition 3.4, we immediately have the following corollary. Corollary 3.5. For any homogeneous quadratic polynomial P (z) = zτBz with B ∈ SM(n,C), it is ∆A-nilpotent iff the submatrix of U τB U consisting of the first r rows and r columns is nilpotent. 16 WENHUA ZHAO Proposition 3.6. Let A be given as in Eq. (2.8). Then, for any P (z) ∈ Ān with o(P (z)) ≥ 2, P (z) is ∆A-nilpotent iff ∆mAPm = 0 for any 1 ≤ m ≤ r. Proof: Again, we let P̃ (z) be as in Eq. (3.1) and note that P (z) is ∆A-nilpotent iff P̃ (z) is ∆r-nilpotent. Since r ≤ n. We view Ar as a subalgebra of the polynomial algebra K[z1, . . . , zr], whereK is the rational field C(zr+1, . . . , zn). By Theorem 1.2 and Lefschetz’s principle (if r < n), we have P̃ (z) is ∆r-nilpotent iff ∆mr P̃ m = 0 for any 1 ≤ m ≤ r. On the other hand, by Eqs. (2.6) and (2.7), we have ΦU ∆mr P̃ = ∆mAP m for any m ≥ 1. Since ΦU is an automorphism of An, we have that, ∆ m = 0 for any 1 ≤ m ≤ r iff ∆mAP m = 0 for any 1 ≤ m ≤ r. Therefore, P̃ (z) is ∆A-nilpotent iff m = 0 for any 1 ≤ m ≤ r. Hence the proposition follows. ✷ Proposition 3.7. For any A ∈ SGL(n,C) and any homogeneous P (z) ∈ An of degree d ≥ 2, we have, P (z) is ∆A-nilpotent iff, for any β ∈ C, (βD)d−2P (z) is Λ-nilpotent, where βD := 〈β,D〉. Proof: Let A be given as in Eq. (2.8) and P̃ (z) as in Eq. (3.1). Note that, Ψ−1U (∆A) = ∆n (for ∆A is of full rank), and P (z) is ∆A-nilpotent iff P̃ (z) is ∆n-nilpotent. Since P̃ is also homogeneous of degree d ≥ 2, by Theorem 1.2 in [Z3], we know that, P̃ (z) is ∆n-nilpotent iff, for any β ∈ Cn, βd−2D P̃ is ∆n-nilpotent. Note that, from Lemma 2.1, (b), we have ΨU(βD) = 〈β, U τD〉 = 〈Uβ,D〉 = (Uβ)D, D P̃ ) = ΨU(βD) d−2ΦU(P̃ ) = (Uβ) D P.(3.17) Therefore, by Lemma 2.5, (a), βd−2D P̃ is ∆n-nilpotent iff (Uβ) D P is ∆A-nilpotent since ΨU(∆n) = ∆A. Combining all equivalences above, we have P (z) is ∆n-nilpotent iff, for any β ∈ Cn, (Uβ)d−2D P is ∆A- nilpotent. Since U is invertible, when β runs over Cn so does Uβ. Therefore the proposition follows. ✷ Let {ei | 1 ≤ i ≤ n} be the standard basis of Cn. Applying the propo- sition above to β = ei (1 ≤ i ≤ n), we have the following corollary. A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 17 Corollary 3.8. For any homogeneous ∆A-nilpotent polynomial P (z) ∈ An of degree d ≥ 2, Dd−2i P (z) (1 ≤ i ≤ n) are also ∆A-nilpotent. We think that Proposition 3.7 and Corollary 3.8 are interesting be- cause, due to Corollary 3.5, it is much easier to decide whether a qua- dratic form is ∆A-nilpotent or not. To state the next criterion, we need fix the following notation. For any A ∈ SGL(n,C), we letXA(Cn) be the set of isotropic vectors u ∈ Cn with respect to the C-bilinear form 〈·, ·〉A. When A = In×n, we also denote XA(C n) simply by of X(Cn). For any β ∈ Cn, we set hα(z) := 〈α, z〉. Then, by applying ΦU to a well-known theorem on classical harmonic polynomials, which is the following theorem for A = In×n (see, for example, [He] and [T]), we have the following result on homogeneous ∆A-nilpotent polynomials. Theorem 3.9. Let P be any homogeneous polynomial of degree d ≥ 2 such that ∆AP = 0. We have P (z) = hdαi(z)(3.18) for some k ≥ 1 and αi ∈ XA(Cn) (1 ≤ i ≤ k). Next, for any homogeneous polynomial P (z) of degree d ≥ 2, we introduce the following matrices: ΞP := (〈αi, αj〉A)k×k ,(3.19) ΩP := 〈αi, αj〉A hd−2αj (z) .(3.20) Then, by applying ΦU to Proposition 5.3 in [Z2] (the details will be omitted here), we have the following criterion of ∆A-nilpotency for homogeneous polynomials. Proposition 3.10. Let P (z) be as given in Eq. (3.18). Then P (z) is ∆A-nilpotent iff the matrix ΩP is nilpotent. One simple remark on the criterion above is as follows. Let B be the k × k diagonal matrix with hαi(z) (1 ≤ i ≤ k) as the ith diagonal entry. For any 1 ≤ j ≤ k, set ΩP ;j := B d−2−j = (hjαi〈αi, αj〉h d−2−j ).(3.21) Then, by repeatedly applying the fact that, for any C,D ∈ M(k,C), CD is nilpotent iff so is DC, it is easy to see that Proposition 3.10 can also be re-stated as follows. 18 WENHUA ZHAO Corollary 3.11. Let P (z) be given by Eq. (3.18) with d ≥ 2. Then, for any 1 ≤ j ≤ d − 2 and m ≥ 1, P (z) is ∆A-nilpotent iff the matrix ΩP ;j is nilpotent. Note that, when d is even, we may choose j = (d − 2)/2. So P is ∆A-nilpotent iff the symmetric matrix ΩP ;(d−2)/2 = (h (d−2)/2 〈αi, αj〉Ah(d−2)/2αj )(3.22) is nilpotent. 3.3. Some Results on the Vanishing Conjecture of the 2nd Order Homogeneous Differential Operators with Constants Coefficients. In this subsection, we transform some known results of VC for the Laplace operators ∆n (n ≥ 1) to certain results on VC for ∆A (A ∈ SGL(n,C)). First, by Wang’s theorem [Wa], we know that JC holds for any polynomial maps F (z) with degF ≤ 2. Hence, JC also holds for symmetric polynomials F (z) = z −∇P (z) with P (z) ∈ C[z] of degree d ≤ 3. By the equivalence of JC and VC for the Laplace operators established in [Z2], we know VC holds if Λ = ∆n and P (z) is a HN polynomials of degree d ≤ 3. Then, applying the linear automorphism ΦU , we have the following proposition. Theorem 3.12. For any A ∈ SGL(n,C) and ∆A-nilpotent P (z) ∈ An (not necessarily homogeneous) of degree d ≤ 3, we have ΛmPm+1 = 0 when m >> 0, i.e. VC[n] holds for Λ and P (z). Applying the classical homogeneous reduction on JC (see [BCW], [Y]) to associated symmetric maps, we know that, to show VC for ∆n (n ≥ 1), it will be enough to consider only homogeneous HN polyno- mials of degree 4. Therefore, by applying the linear automorphism ΦU of An, we have the same reduction for HVC too. Theorem 3.13. To study HVC in general, it will be enough to con- sider only homogeneous P (z) ∈ A of degree 4. In [BE2] and [BE3] it has been shown that JC holds for symmetric maps F (z) = z−∇P (z) (P (z) ∈ An) if the number of variables n is less or equal to 4, or n = 5 and P (z) is homogeneous. By the equivalence of JC for symmetric polynomial maps and VC for the Laplace operators established in [Z2], and Proposition 2.8 and Corollary 2.6, we have the following results on VC and HVC. Theorem 3.14. (a) For any n ≥ 1, VC[n] holds for any Λ ∈ D2 of rank 1 ≤ r ≤ 4. (b) HVC[5] holds for any Λ ∈ D2[5]. A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 19 Note that the following vanishing properties of HN formal power series have been proved in Theorem 6.2 in [Z2] for the Laplace operators ∆n (n ≥ 1). By applying the linear automorphism ΦU , one can show it also holds for any Λ-nilpotent (Λ ∈ D2) formal power series. Theorem 3.15. Let Λ ∈ D2[n] and P (z) ∈ Ān be Λ-nilpotent with o(P ) ≥ 2. The following statements are equivalent. (1) ΛmPm+1 = 0 when m >> 0. (2) There exists k0 ≥ 1 such that ΛmPm+k0 = 0 when m >> 0. (3) For any fixed k ≥ 1, ΛmPm+k = 0 when m >> 0. By applying the linear automorphism ΦU , one can transform Theo- rem 1.5 in [EZ] on VC of the Laplace operators to the following result on VC of Λ ∈ D2. Theorem 3.16. Let Λ ∈ D2[n] and P (z) ∈ Ān any Λ-nilpotent poly- nomial with o(P ) ≥ 2. Then VC holds for Λ and P (z) iff, for any g(z) ∈ An, we have Λm(g(z)Pm) = 0 when m >> 0. In [EZ], the following theorem has also been proved for Λ = ∆n. Next we show it is also true in general. Theorem 3.17. Let A ∈ SGL(n,C) and P (z) ∈ An a homogeneous ∆A-nilpotent polynomial with degP ≥ 2. Assume that σA−1(z) := zτA−1z and the partial derivatives ∂P (1 ≤ i ≤ n) have no non-zero common zeros. Then HVC[n] holds for ∆A and P (z). In particular, if the projective subvariety determined by the ideal 〈P (z)〉 of An is regular, HVC[n] holds for ∆A and P (z). Proof: Let P̃ as given in Eq. (3.1). By Theorem 1.2 in [EZ], we know that, when σ2(z) := i=1 z i and the partial derivatives (1 ≤ i ≤ n) have no non-zero common zeros, HVC[n] holds for ∆n and P̃ . Then, by Lemma 2.5, (b), HVC[n] also holds for ∆A and P . But, on the other hand, since U is invertible and, for any 1 ≤ i ≤ n, (Uz), σ2(z) and (1 ≤ i ≤ n) have no non-zero common zeros iff σ2(z) and (Uz) (1 ≤ i ≤ n) have no non-zero common zeros, and iff, with z replaced by U−1z, σ2(U −1z) = σA−1(z) and (z) (1 ≤ i ≤ n) have no non-zero common zeros. Therefore, the theorem holds. ✷ 20 WENHUA ZHAO 3.4. The Vanishing Conjecture for Higher Order Differential Operators with Constant Coefficients. Even though the most in- teresting case of VC is for Λ ∈ D2, at least when JC is concerned, the case of VC for higher order differential operators with constant coefficients is also interesting and non-trivial. In this subsection, we mainly discuss VC for the differential operators Da (a ∈ Nn). At the end of this subsection, we also recall a result proved in [Z3] which says that, when the base field has characteristic p > 0, VC, even under a weaker condition, actually holds for any differential operator Λ (not necessarily with constant coefficients). Let βj ∈ Cn (1 ≤ j ≤ ℓ) be linearly independent and set δj := 〈βj, D〉. Let Λ = j=1 δ j with aj ≥ 1 (1 ≤ j ≤ ℓ). When ℓ = 1, VC for Λ can be proved easily as follows. Proposition 3.18. Let δ ∈ D1[z] and Λ = δk for some k ≥ 1. Then (a) A polynomial P (z) is Λ-nilpotent if (and only if) ΛP = 0. (b) VC holds for Λ. Proof: Applying a change of variables, if necessary, we may assume δ = D1 and Λ = D Let P (z) ∈ C[z] such that ΛP (z) = Dk1P (z) = 0. Let d be the degree of P (z) in z1. From the equation above, we have k > d. Therefore, for any m ≥ 1, we have km > dm which implies ΛmP (z)m = Dkm1 Pm(z) = 0. Hence, we have (a). To show (b), let P (z) be a Λ-nilpotent polynomial. By the same notation and argument above, we have k > d. Choose a positive integer N > d . Then, for any m ≥ N , we have m > d , which is equivalent to (m + 1)d < km. Hence we have ΛmP (z)m+1 = Dkm1 P m+1(z) = 0. In particular, when k = 1 in the proposition above, we have the following corollary. Corollary 3.19. VC holds for any differential operator Λ ∈ D1. Next we consider the case ℓ ≥ 2. Note that, when ℓ = 2 and a1 = a2 = 1. Λ ∈ D2 and has rank 2. Then, by Theorem 3.14, we know VC holds for Λ. Besides the case above, VC for Λ = j=1 δ j with ℓ ≥ 2 seems to be non-trivial at all. Actually, we will show below, it is equivalent to a conjecture (see Conjecture 3.21) on Laurent polynomials. First, by applying a change of variables, if necessary, we may (and will) assume Λ = Da with a ∈ (N+)ℓ. Secondly, note that, for any A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 21 b ∈ Nn and h(z) ∈ C[z], Dbh(z) = 0 iff the holomorphic part of the Laurent polynomial z−bh(z) is zero. Now we fix a P (z) ∈ C[z] and set f(z) := z−aP (z). With the obser- vation above, it is easy to see that, P (z) is Da-nilpotent iff the holo- morphic parts of the Laurent polynomials fm(z) (m ≥ 1) are all zero; and VC holds for Λ and P (z) iff the holomorphic part of P (z)fm(z) is zero when m >> 0. Therefore, VC for Da can be restated as follows: Re-Stated VC for Λ = Da: Let P (z) ∈ An and f(z) as above. Suppose that, for any m ≥ 1, the holomorphic part of the Laurent poly- nomial fm(z) is zero, then the holomorphic part of P (z)fm(z) equals to zero when m >> 0. Note that the re-stated VC above is very similar to the following non-trivial theorem which was first conjectured by O. Mathieu [Ma] and later proved by J. Duistermaat and W. van der Kallen [DK]. Theorem 3.20. Let f and g be Laurent polynomials in z. Assume that, for any m ≥ 1, the constant term of fm is zero. Then the constant term gfm equals to zero when m >> 0. Note that, Mathieu’s conjecture [Ma] is a conjecture on all real com- pact Lie groups G, which is also mainly motivated by JC. The the- orem above is the special case of Mathieu’s conjecture when G the n-dimensional real torus. For other compact real Lie groups, Math- ieu’s conjecture seems to be still wide open. Motivated by Theorem 3.20, the above re-stated VC for Λ = Da and also the result on VC in Theorem 3.16, we would like to propose the following conjecture on Laurent polynomials. Conjecture 3.21. Let f and g be Laurent polynomials in z. Assume that, for any m ≥ 1, the holomorphic part of fm is zero. Then the holomorphic part gfm equals to zero when m >> 0. Note that, a positive answer to the conjecture above will imply VC for Λ = Da (a ∈ Nn) by simply choosing g(z) to be P (z). Finally let us to point out that, it is well-known that JC does not hold over fields of finite characteristic (see [BCW], for example), but, by Proposition 5.3 in [Z3], the situation for VC over fields of finite characteristic is dramatically different even though it is equivalent to JC over the complex field C. Proposition 3.22. Let k be a field of char. p > 0 and Λ any differential operator of k[z]. Let f ∈ k[[z]]. Assume that, for any 1 ≤ m ≤ p− 1, 22 WENHUA ZHAO there exists Nm > 0 such that Λ Nmfm = 0. Then, Λmfm+1 = 0 when m >> 0. From the proposition above, we immediately have the following corol- lary. Corollary 3.23. Let k be a field of char. p > 0. Then (a) VC holds for any differential operator Λ of k[z]. (b) If Λ strictly decreases the degree of polynomials. Then, for any polynomial f ∈ k[z] (not necessarily Λ-nilpotent), we have Λmfm+1 = 0 when m >> 0. 4. A Remark on Λ-Nilpotent Polynomials and Classical Orthogonal Polynomials In this section, we first in subsection 4.1 consider the “formal” con- nection between Λ-nilpotent polynomials or formal power series and classical orthogonal polynomials, which has been discussed in Section 1 (see page 4). We then in subsection 4.2 transform the isotropic prop- erties of homogeneous HN polynomials proved in [Z2] to isotropic prop- erties of homogeneous ∆A-nilpotent (A ∈ SGL(n,C)) polynomials (see Theorem 4.10 and Corollary 4.11). Note that, as pointed in Section 1, the isotropic results in subsection 4.2 can be understood as some natural consequences of the connection of Λ-nilpotent polynomials and classical orthogonal polynomials discussed in subsection 4.1. 4.1. Some Classical Orthogonal Polynomials. First, let us recall the definition of classical orthogonal polynomials. Note that, to be consistent with the tradition for orthogonal polynomials, we will in this subsection use x = (x1, x2, . . . , xn) instead of z = (z1, z2, . . . , zn) to denote free commutative variables. Definition 4.1. Let B be an open set of Rn and w(x) a real valued function defined over B such that w(x) ≥ 0 for any x ∈ B and 0 <∫ w(x)dx < ∞. A sequence of polynomials {fm(x) |m ∈ Nn} is said to be orthogonal over B if (1) deg fm = |m| for any m ∈ Nn. fm(x)fk(x)w(x) dx = 0 for any m 6= k ∈ Nn. The function w(x) is called the weight function. When the open set B ⊂ Rn and w(x) are clear in the context, we simply call the polynomials fm(x) (m ∈ Nn) in the definition above orthogonal poly- nomials. If the orthogonal polynomials fm(x) (m ∈ Nn) also satisfy∫ (x)w(x)dx = 1 for any m ∈ Nn, we call fm(x) (m ∈ Nn) or- thonormal polynomials. Note that, in the one dimensional case w(x) A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 23 determines orthogonal polynomials over B up to multiplicative con- stants, i.e. if fm(x) (m ≥ 0) are orthogonal polynomials as in Defi- nition 4.1, then, for any am ∈ R× (m ≥ 0), amfm (m ≥ 0) are also orthogonal over B with respect to the weight function w(x). The most natural way to construct orthogonal or orthonormal se- quences is: first to list all monomials in an order such that the degrees of monomials are non-decreasing; and then to apply Gram-Schmidt procedure to orthogonalize or orthonormalize the sequence of mono- mials. But, surprisingly, most of classical orthogonal polynomials can also be obtained by the so-called Rodrigues’ formulas. We first consider orthogonal polynomials in one variable. Rodrigues’ formula: Let fm(x) (m ≥ 0) be the orthogonal polyno- mials as in Definition 4.1. Then, there exist a function g(x) defined over B and non-zero constants cm ∈ R (m ≥ 0) such that fm(x) = cmw(x) (w(x)gm(x)).(4.1) Let P (x) := g(x) and Λ:= w(x)−1 w(x), where, throughout this paper any polynomial or function appearing in a (differential) operator always means the multiplication operator by the polynomial or function itself. Then, by Rodrigues’ formula above, we see that the orthogonal polynomials {fm(x) |m ≥ 0} have the form fm(x) = cmΛ mPm(x),(4.2) for any m ≥ 0. In other words, all orthogonal polynomials in one variable, up to multiplicative constants, has the form {ΛmPm |m ≥ 0} for a single differential operator Λ and a single function P (x). Next we consider some of the most well-known classical orthonor- mal polynomials in one variable. For more details on these orthogonal polynomials, see [Sz], [AS], [DX]. Example 4.2. (Hermite Polynomials) (a) B = R and the weight function w(x) = e−x (b) Rodrigues’ formula: Hm(x) = (−1)mex )me−x 24 WENHUA ZHAO (c) Differential operator Λ and polynomial P (x): Λ = ex − 2x, P (x) = 1, (d) Hermite polynomials in terms of Λ and P (x): Hm(x) = (−1)m ΛmPm(x). Example 4.3. (Laguerre Polynomials) (a) B = R+ and w(x) = xαe−x (α > −1). (b) Rodrigues’ formula: Lαm(x) = x−αex( )m(xm+αe−x). (c) Differential operator Λ and polynomial P (x): Λα = x −αex( d )(e−xxα) = d + (αx−1 − 1), P (x) = x, (d) Laguerre polynomials in terms of Λ and P (x): Lm(x) = ΛmPm(x). Example 4.4. (Jacobi Polynomials) (a) B = (−1, 1) and w(x) = (1− x)α(1 + x)β, where α, β > −1. (b) Rodrigues’ formula: P α,βm (x) = (−1)m (1− x)−α(1 + x)−β( d )m(1− x)α+m(1 + x)β+m. (c) Differential operator Λ and polynomial P (x): Λ = (1− x)−α(1 + x)−β( )(1− x)α(1 + x)β − α(1− x)−1 + β(1 + x)−1, P (x) = 1− x2. (d) Laguerre polynomials in terms of Λ and P (x): P α,βm (x) = (−1)m ΛmPm(x). A very important special family of Jacobi polynomials are the Gegen- bauer polynomials which are obtained by setting α = β = λ− 1/2 for some λ > −1/2. Gegenbauer polynomials are also called ultraspherical polynomials in the literature. A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 25 Example 4.5. (Gegenbauer Polynomials) (a) B = (−1, 1) and w(x) = (1− x2)λ−1/2, where λ > −1/2. (b) Rodrigues’ formula: P λm(x) = (−1)m 2m(λ+ 1/2)m (1− x2)1/2−λ( )m(1− x2)m+λ−1/2. where, for any c ∈ R and k ∈ N, (c)k = c(c+ 1) · · · (c+ k − 1). (c) Differential operator Λ and polynomial P (x): Λ = (1− x2)1/2−λ( )(1− x2)λ−1/2(4.3) (2λ− 1) x (1− x2) P (x) = 1− x2. (d) Laguerre polynomials in terms of Λ and P (x): P λm(x) = (−1)m 2m(λ+ 1/2)m ΛmPm(x). Note that, for the special cases with λ = 0, 1, 1/2, the Gegenbauer Polynomials P λm(x) are called the Chebyshev polynomial of the first kind, the second kind and Legendre polynomials, respectively. Hence all these classical orthogonal polynomials also have the form of ΛmPm (m ≥ 0) up to some scalar multiple constants cm with P (x) = 1 − x2 and the corresponding special forms of the differential operator Λ in Eq. (4.3). Remark 4.6. Actually, the Gegenbauer polynomials are more closely and directly related with VC in some different ways. See [Z4] for more discussions on connections of the Gegenbauer polynomials with VC. Next, we consider some classical orthogonal polynomials in several variables. We will see that they can also be obtained from certain sequences of the form {ΛmPm |m ≥ 0} in a slightly modified way. One remark is that, unlike the one-variable case, orthogonal polynomials in several variables up to multiplicative constants are not uniquely determined by weight functions. The first family of classical orthogonal polynomials in several vari- ables can be constructed by taking Cartesian products of orthogonal polynomials in one variable as follows. 26 WENHUA ZHAO Suppose {fm |m ≥ 0} is a sequence of orthogonal polynomials in one variable, say as given in Definition 4.1. We fix any n ≥ 2 and set W (x) := w(xi),(4.4) fm(x) := fmi(xi),(4.5) for any x ∈ B×n and m ∈ Nn. Then it is easy to see that the sequence {fm(x) |m ∈ Nn} are orthog- onal polynomials over B×n with respect to the weight function W (x) defined above. Note that, by applying the construction above to the classical one- variable orthogonal polynomials discussed in the previous examples, one gets the classical multiple Hermite Polynomials, multiple Laguerre polynomials, multiple Jacobi polynomials andmultiple Gegenbauer poly- nomials, respectively. To see that the multi-variable orthogonal polynomials constructed above can be obtained from a sequence of the form {ΛmPm(x) |m ≥ 0}, we suppose fm (m ≥ 0) have Rodrigues’ formula Eq. (4.1). Let s = (s1, . . . , sn) be n central formal parameters and set Λs :=W (x) W (x),(4.6) P (x) := g(xi).(4.7) Let Vm(x) (m ∈ Nn) be the coefficient of sm in Λ|m|s P |m|(x). Then, from Eqs. (4.1), (4.4)–(4.7), it is easy to check that, for any m ∈ Nn, we have fm(x) = cm Vm(x),(4.8) where cm = i=1 cmi . Therefore, we see that any multi-variable orthogonal polynomials constructed as above from Cartesian products of one-variable orthogo- nal polynomials can also be obtained from a single differential operator Λs and a single function P (x) via the sequence {Λms Pm |m ≥ 0}. Remark 4.7. Note that, one can also take Cartesian products of dif- ferent kinds of one-variable orthogonal polynomials to create more or- thogonal polynomials in several variables. By a similar argument as A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 27 above, we see that all these multi-variable orthogonal polynomials can also be obtained similarly from a single sequence {Λms Pm |m ≥ 0}. Next, we consider the following two examples of classical multi- variable orthogonal polynomials which are not Cartesian products of one-variable orthogonal polynomials. Example 4.8. (Classical Orthogonal Polynomials over Unit Balls) (a) Choose B to be the open unit ball Bn of Rn and the weight func- Wµ(x) = (1− ||x||2)µ−1/2, where ||x|| = i=1 x i and µ > 1/2. (b) Rodrigues’ formula: For any m ∈ Nn, set Um(x) := (−1)m(2µ)|m| 2|m|m!(µ+ 1/2)|m| ∂xm11 · · ·∂xmnn (1− ||x||2)|m|+µ−1/2. Then, by Proposition 2.2.5 in [DX], {Um(x) |m ∈ Nn} are orthonormal over Bn with respect to the weight function Wµ(x). (c) Differential operator Λs and polynomial P (x): Let s = (s1, . . . , sn) be n central formal parameters and set Λs :=Wµ(x) Wµ(x), P (x) :=1− ||x||2. Let Vm(x) (m ∈ Nn) be the coefficient of sm in Λ|m|s P |m|(x). Then from the Rodrigues type formula above, we have, for any m ∈ Nn, Um(x) = (−1)|m|(2µ)|m| 2|m||m|!(µ+ 1/2)|m| Vm(x). Therefore, the classical orthonormal polynomials {Um(x) |m ∈ Nn} over Bn can be obtained from a single differential operator Λs and P (x) via the sequence {Λms Pm |m ≥ 0}. Example 4.9. (Classical Orthogonal Polynomials over Sim- plices) (a) Choose B to be the simplex T n = {x ∈ Rn | xi < 1; x1, ..., xn > 0} 28 WENHUA ZHAO in Rn and the weight function Wκ(x) = x 1 · · ·xκnn (1− |x|1)κn+1−1/2,(4.9) where κi > −1/2 (1 ≤ i ≤ n+ 1) and |x|1 = i=1 xi. (b) Rodrigues’ formula: For any m ∈ Nn, set Um(x) := Wκ(x) ∂xm11 · · ·∂xmnn Wκ(x)(1 − |x|1)|m| Then, {Um(x) |m ∈ Nn} are orthonormal over T n with respect to the weight function Wκ(x). See Section 2.3.3 of [DX] for a proof of this claim. (c) Differential operator Λ and polynomial P (x): Let s = (s1, . . . , sn) be n central formal parameters and set Λs :=Wκ(x) Wκ(x), P (x) :=1− |x|1. Let Vm(x) (m ∈ Nn) be the coefficient of sm in Λ|m|s P |m|(x). Then from the Rodrigues type formula in (b), we have, for any m ∈ Nn, Um(x) = Vm(x). Therefore, the classical orthonormal polynomials {Um(x) |m ∈ Nn} over T n can be obtained from a single differential operator Λs and a function P (x) via the sequence {Λms Pm |m ≥ 0}. 4.2. The Isotropic Property of ∆A-Nilpotent Polynomials. As discussed in Section 1, the “formal” connection of Λ-nilpotent polyno- mials with classical orthogonal polynomials predicts that Λ-nilpotent polynomials should be isotropic with respect to a certain C-bilinear form of An. In this subsection, we show that, for differential operators Λ = ∆A (A ∈ SGL(n,C)), this is indeed the case for any homogeneous Λ-nilpotent polynomials (see Theorem 4.10 and Corollaries 4.11, 4.12). We fix any n ≥ 1 and let z and D denote the n-tuples (z1, . . . , zn) and (D1, D2, . . . , Dn), respectively. Let A ∈ SGL(n,C) and define the C-bilinear map {·, ·}A : An ×An → An(4.10) (f, g) → f(AD)g(z), Furthermore, we also define a C-bilinear form (·, ·)A : An ×An → C(4.11) A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 29 (f, g) → {f, g} |z=0, It is straightforward to check that the C-bilinear form defined above is symmetric and its restriction on the subspace of homogeneous poly- nomials of any fixed degree is non-singular. Note also that, for any homogeneous polynomials f, g ∈ An of the same degree, we have {f, g}A = (f, g)A. The main result of this subsection is the following theorem. Theorem 4.10. Let A ∈ SGL(n,C) and P (z) ∈ An a homogeneous ∆A-nilpotent polynomial of degree d ≥ 3. Let I(P ) be the ideal of An generated by σA−1(z) := z τA−1z and ∂P (1 ≤ i ≤ n). Then, for any f(z) ∈ I(P ) and m ≥ 0, we have {f,∆mAPm+1}A = f(AD)∆mAPm+1 = 0.(4.12) Note that, by Theorem 6.3 in [Z2], we know that the theorem does hold when A = In and ∆A = ∆n. Proof: Note first that, elements of An satisfying Eq. (4.12) do form an ideal. Therefore, it will be enough to show σA−1(z) and i ≤ n) satisfy Eq. (4.12). But Eq. (4.12) for σA−1(z) simply follows the facts that σA−1(Az) = z τAz and σA−1(AD) = ∆A. Secondly, by Lemma 2.2, we can write A = UU τ for some U = (uij) ∈ GL(n,C). Then, by Eq. (2.7), we have ΨU(∆n) = ∆A or Ψ U (∆A) = ∆n. Let P̃ (z) := Φ U (P ) = P (Uz). Then by Lemma 2.5, (a), P̃ is a homogeneous ∆n-nilpotent polynomial, and by Eq. (2.6), we also have Φ−1U (∆ m+1) = ∆mn P̃ m+1.(4.13) By Theorem 6.3 in [Z2], for any 1 ≤ i ≤ n and m ≥ 0, we have, ∆mn P̃ Since (z) = (Uz), we further have, ∆mn P̃ Since U is invertible, for any 1 ≤ i ≤ n, we have ∆mn P̃ = 0.(4.14) 30 WENHUA ZHAO Combining the equation above with Eq. (4.13), we get (UD)Φ−1U Φ−1U (ΦU (UD)Φ−1U ) (UD)Φ−1U ) = 0.(4.15) By Lemma 2.1, (b), Eq. (4.15) and the fact that A = UU τ , we get (UU τD) which is Eq. (4.12) for ∂P (1 ≤ i ≤ n). ✷ Corollary 4.11. Let A be as in Theorem 4.10 and P (z) be a homoge- neous ∆A-nilpotent polynomial of degree d ≥ 3. Then, for any m ≥ 1, m+1 is isotropic with respect to the C-bilinear form (·, ·)A, i.e. (∆mAP m+1,∆mAP m+1)A = 0.(4.16) In particular, we have (P, P )A = 0. Proof: By the definition Eq. (4.11) of the C-bilinear form (·, ·)A and Theorem 4.10, it will be enough to show that P and ∆mAP m+1 (m ≥ 1) belong to the ideal generated by the polynomials ∂P (1 ≤ i ≤ n) (here we do not need to consider the polynomial σA−1(z)). But this statement has been proved in the proof of Corollary 6.7 in [Z2]. So we refer the reader to [Z2] for a proof of the statement above. ✷ Theorem 4.10 and Corollary 4.11 do not hold for homogeneous HN polynomials P (z) of degree d = 2. But, by applying similar arguments as in the proof of Theorem 4.10 above to Proposition 6.8 in [Z2], one can show that the following proposition holds. Proposition 4.12. Let A be as in Theorem 4.10 and P (z) a homoge- neous ∆A-nilpotent polynomial of degree d = 2. Let J(P ) the ideal of C[z] generated by P (z) and σA−1(z). Then, for any f(z) ∈ J(P ) and m ≥ 0, we have {f,∆mAPm+1}A = f(AD)∆mAPm+1 = 0.(4.17) In particular, we still have (P, P )A = 0. A VANISHING CONJECTURE ON DIFFERENTIAL OPERATORS 31 References [AS] Handbook of mathematical functions with formulas, graphs, and mathemat- ical tables. Edited by Milton Abramowitz and Irene A. Stegun. Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. [MR0757537]. [BCW] H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7, (1982), 287–330. [MR 83k:14028]. Zbl.539.13012. [B] M. de Bondt, Personal Communications. 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[Z4] W. Zhao, A Conjecture on the Laplace-Beltrami Eigenfunctions. In prepa- ration. Department of Mathematics, Illinois State University, Nor- mal, IL 61790-4520. E-mail: [email protected]. http://arxiv.org/abs/math-ph/0308035 http://arxiv.org/abs/math/0409534 http://arxiv.org/abs/0704.1689 1. Introduction 2. The Vanishing Conjecture for the 2nd Order Homogeneous Differential Operators with Constant Coefficients 2.1. Notation and Preliminaries 2.2. The Vanishing Conjecture for the 2nd Order Homogeneous Differential Operators with Constant Coefficients 3. Some Properties of A-Nilpotent Polynomials 3.1. Associated Polynomial Maps and PDEs 3.2. Some Criteria of A-Nilpotency 3.3. Some Results on the Vanishing Conjecture of the 2nd Order Homogeneous Differential Operators with Constants Coefficients 3.4. The Vanishing Conjecture for Higher Order Differential Operators with Constant Coefficients 4. A Remark on -Nilpotent Polynomials and Classical Orthogonal Polynomials 4.1. Some Classical Orthogonal Polynomials 4.2. The Isotropic Property of A-Nilpotent Polynomials References

0704.1693

N_p N_n Scheme Based on New Empirical Formula for Excitation Energy

NpNn Scheme Based on New Empirical Formula for Excitation Energy Jin-Hee Yoon, Eunja Ha, and Dongwoo Cha∗ Department of Physics, Inha University, Incheon 402-751, Korea (Dated: July 25, 2007) Abstract We examine the NpNn scheme based on a recently proposed simple empirical formula which is highly valid for the excitation energy of the first excited natural parity even multipole states in even-even nuclei. We demonstrate explicitly that the NpNn scheme for the excitation energy emerges from the separate exponential dependence of the excitation energy on the valence nucleon numbers Np and Nn together with the fact that only a limited set of numbers is allowed for the Np and Nn of the existing nuclei. PACS numbers: 21.10.Re, 23.20.Lv ∗Electronic address: [email protected]; Fax: +82-32-866-2452 http://arxiv.org/abs/0704.1693v2 mailto:[email protected] The valence nucleon numbers Np and Nn have been frequently adopted in parameter- izing various nuclear properties phenomenologically over more than the past four decades. Hamamoto was the first to point out that the square roots of the ratios of the measured and the single particle B(E2) values were proportional to the product NpNn [1]. It was subsequently shown that a very simple pattern emerged whenever the nuclear data concern- ing the lowest collective states was plotted against NpNn [2]. This phenomenon has been called the NpNn scheme in the literature [3]. For example, when the measured excitation energies Ex(2 1 ) of the first excited 2 + states in even-even nuclei were plotted against the mass number A (A-plot), we got data points scattered irregularly over the Ex-A plane as seen in Fig. 1(a). However, we suddenly had a very neat rearrangement of the data points by just plotting them against the product NpNn (NpNn-plot) as shown in Fig. 1(b). A similar simplification was observed not only from Ex(2 1 ) but also from the ratio Ex(4 1 )/Ex(2 [5, 6, 7], the transition probability B(E2; 2+1 → 0 +) [8], and the quadrupole deformation parameter e2 [9]. The chief attraction of the NPNn scheme is twofold. One is the fact that the simplification in the graph occurs marvelously every time the NpNn plot is drawn. The other attraction 0 100 200 300 400 5000 100 200 4 (b) N -Plot (a) A-Plot Mass Number A FIG. 1: A typical example demonstrating the NpNn scheme. The excitation energies of the first 2 states in even-even nuclei are plotted (a) against the mass number A and (b) against the product NpNn. The dashed curve in part (a) represents the bottom contour line which is drawn by the first term αA−γ of Eq. (1). The excitation energies are quoted from Ref. 4. 0 100 200 0 100 200 Mass Number A (b) Empirical Formula(a) Data FIG. 2: Excitation energies of the first excited natural parity even multipole states. Part (a) shows the measured excitation energies while part (b) shows those calculated by the empirical formula given by Eq. (1). The measured excitation energies are quoted from the compilation in Raman et al. for 2+ states [4] and extracted from the Table of Isotopes, 8th-edition by Firestone et al. for other multipole states [20]. is the universality of the pattern, namely the exactly same sort of graphs appears even at different mass regions [2]. Since the performance of the NpNn scheme has been so impressive, many expected that the residual valence proton-neutron (p-n) interaction must have been the dominant controlling factor in the development of collectivity in nuclei and that the product NpNn may represent an empirical measure of the integrated valence p-n interaction strength [3]. Also, the importance of the p-n interaction in determining the structure of nuclei has long been pointed out by many authors [10, 11, 12, 13, 14, 15, 16]. In the meantime, we have recently proposed a simple empirical formula which describes the essential trends of the excitation energies Ex(2 1 ) in even-even nuclei throughout the periodic table [17]. This formula, which depends on the valence nucleon numbers, Np and Nn, and the mass number A, can be expressed as Ex = αA −γ + β [exp(−λNp) + exp(−λNn)] (1) where the parameters α, β, γ, and λ are fitted from the data. We have also shown that the source, which governs the 2+1 excitation energy dependence given by Eq. (1) on the valence nucleon numbers, is the effective particle number participating in the residual interaction from the Fermi level [18]. Furthermore, the same empirical formula can be applied quite successfully to the excitation energies of the lowest natural parity even multipole states such as 4+1 , 6 1 , 8 1 , and 10 1 [19]. It can be confirmed by Fig. 2 where the measured excitation energies in part (a) are compared with those in part (b) which are calculated by Eq. (1). The values of the parameters adopted for Fig. 2(b) are listed in Table I. 0 100 200 300 400 500 0 100 200 300 400 500 (I) Z=2~8 Data Empirical Formula (II) Z=10~20 (III) Z=22~28 (IV) Z=30~50 (V) Z=52~82 (VI) Z=84~126 FIG. 3: The NpNn-plot for the excitation energies of the first 2 + states using both the data (open triangles) and the empirical formula (solid circles). The plot is divided into six panels each of which contains plotted points that come from each one of the proton major shells. TABLE I: Values adopted for the four parameters in Eq. (1) for the excitation energies of the following multipole states: 2+ , and 10+ Multipole α(MeV) β(MeV) γ λ 34.9 1.00 1.19 0.36 94.9 1.49 1.15 0.30 441.4 1.51 1.31 0.25 1511.5 1.41 1.46 0.19 2489.0 1.50 1.49 0.17 In this study, we want to further elucidate about our examination of the NpNn scheme based on the empirical formula, Eq. (1), for Ex(2 1 ). Our goal is to clarify why Ex(2 complies with the NpNn scheme although the empirical formula, which reproduces the data quite well, does not depend explicitly on the product NpNn. First, we check how well the empirical formula does meet the requirements of the NpNn scheme. In Fig. 3, we display the NpNn-plot for the excitation energies of the first 2 states using both the data (empty triangles) and the empirical formula (solid circles). We show them with six panels. Each panel contains plotted points from nuclei which make up the following six different proton major shells: (I) 2 ≤ Z ≤ 8, (II) 10 ≤ Z ≤ 20, (III) TABLE II: The maximum value of NpNn and the minimum value of Ex for each major shell in Fig. 3 are indicated here. The numbers in the parenthesis represent Ex calculated by the empirical formula given by Eq. (1). Major Shell Z Max. NpNn Min. Ex (MeV) I 2 ∼ 8 8 1.59 (1.85) II 10 ∼ 20 36 0.67 (0.82) III 22 ∼ 28 16 0.75 (0.77) IV 30 ∼ 50 140 0.13 (0.18) V 52 ∼ 82 308 0.07 (0.08) VI 84 ∼ 126 540 0.04 (0.05) 0 50 100 150 200 250 0 50 100 150 200 250 Empirical Formula FIG. 4: Extract from Fig. 3 for some typical nuclei which belong to the rare earth elements. Different symbols are used to denote excitation energies of individual nuclei. 22 ≤ Z ≤ 28, (IV) 30 ≤ Z ≤ 50, (V) 52 ≤ Z ≤ 82, and (VI) 84 ≤ Z ≤ 126. From this figure, we can see an intrinsic feature of the NpNn-plot, namely, the plotted points have their own typical location in the Ex-NpNn plane according to which major shell they belong. For example, the plotted points of the first three major shells I, II, and III occupy the far left side part of the Ex-NpNn plane in Fig. 3 since their value of the product NpNn does not exceed several tens. On the contrary, the plotted points of the last major shell VI extend to the far right part of the Ex-NpNn plane along the lowest portion in Fig. 3. This is true since their value of the excitation energy Ex is very small and also their value of NpNn reaches more than five hundreds. We present specific information such as the maximum value ofNpNn and the minimum value of Ex in Table II for the plotted points which belong to each major shell in Fig. 3. There are two numbers for each major shell in the last column of Table II where one number is determined from the data and the other number in parenthesis is calculated by the empirical formula. We can find that those two numbers agree reasonably well. We also find in Fig. 3 that the results, calculated by the empirical formula (solid circles), meet the requirement of the NpNn scheme very well and agree with the data (empty triangles) satisfactorily for each and every panel. In order to make more detailed comparison between the measured and calculated excita- tion energies, we expand the largest two major shells V and VI of Fig. 3 and redraw them in Fig. 4 for some typical nuclei which belong to the rare earth elements. The upper part of Fig. 4 shows the data and the lower part of the same figure exhibits the corresponding cal- culated excitation energies. We can confirm that the agreement between them is reasonable even though the calculated excitation energies somewhat overestimate the data and also the empirical formula can not separate enough to distinguish the excitation energies of the two isotopes with the same value of the product NpNn for some nuclei. According to the empirical formula given by Eq. (1), the excitation energy Ex is deter- mined by two components: one is the first term αA−γ which depends only on the mass number A and the other is the second term β[exp(−λNp)+exp(−λNn)] which depends only on the valence nucleon numbers, Np and Nn. Let us first draw the NpNn-plot of Ex(2 1 ) by using only the first term αA−γ. The results are shown in Fig. 5(a) where we can find that the plotted points fill the lower left corner of the Ex-NpNn plane leaving almost no empty spots. These results simply reflect the fact that a large number of nuclei with different mass numbers, values of A, can have the same value of NpNn. Now we draw the same NpNn-plot by using both of the two terms in Eq. (1). We display the plot of the calculated excitation energies in Fig. 5(b) which is just the same sort of graph of the measured excitation energies shown in Fig. 1(b) except that the type of scale for Ex is changed from linear to log. By comparing Fig. 5 (a) and (b), we find that the second term of Eq. (1), which depends on the valence nucleon numbers, Np and Nn, pushes the plotted points up in the direction of higher excitation energies and arranges them to comply with the NpNn scheme. It is worthwhile to note the difference between the A-plot and the NpNn-plot. The graph drawn by using only the first term of Eq. (1) becomes a single curve in the A-plot as shown in Fig. 1(a) with the dashed curve. It becomes scattered plotted points in the NpNn-plot as can be seen from Fig. 5(a). Now, by adding the second term of Eq. (1) in the A-plot, the plotted points are dispersed as shown in the top graph of Fig. 2(b) which corresponds to the measured data points in Fig. 1(a); while by adding the same second term in NpNn-plot, we find a very neat rearrangement of the plotted points as shown in Fig. 5(b). Thus, the same second term plays the role of spreading plotted points in the A-plot while it plays the role of collecting them in the NpNn-plot. However, this mechanism of the second term alone is not sufficient to explain why the empirical formula given by Eq. (1) which obviously does not depend on NpNn at all, can 0 100 200 300 400 500 0 100 200 300 400 500 (a) Only the first term (b) Both terms FIG. 5: The NpNn-plot of the calculated first excitation energy Ex of 2 + states. The excitation energies Ex are calculated by (a) using only the first term and (b) using both terms of Eq. (1). show the characteristic feature of the NpNn scheme. In order to shed light on this question, we calculate the excitation energy Ex(2 1 ) by the following three different conditions on the exponents, Np and Nn, of the second term in Eq. (1). First, let Np and Nn have any even numbers as long as they satisfy Np +Nn ≤ A. The resulting excitation energy Ex is plotted against NpNn in Fig. 6(a). Next, let Np and Nn have any numbers that are allowed for the valence nucleon numbers. For example, suppose the three numbers of a plotted point are A = 90, Np = 40, and Nn = 50 in the previous case. For the fourth major shell IV in Table II, the valence proton number for the nucleus with the atomic number Z = 40 is 10 and the valence neutron number for the nucleus with the neutron number N = 50 is 0. Therefore, we assign Np = 10 and Nn = 0 instead of 40 and 50, respectively. The excitation energy Ex, calculated under such a condition, is plotted against NpNn in Fig. 6(b). Last, we take only those excitation energies which are actually measured among the excitation energies shown in Fig. 6(b). The results are shown in Fig. 6(c), which is, of course, exactly the same as shown in Fig. 5(b). From Fig. 6(d) where all the three previous plots (a), (b), and (c) are placed together, we can observe how the NpNn scheme emerges from the empirical formula given by Eq. (1) even though this equation does not depend on the product NpNn at all. On one hand, the two exponential terms which depend on Np and Nn separately push the excitation energy Ex upward as discussed with respect to Fig. 5. On the other hand, the restriction on the values of the valence nucleon numbers Np and Nn of the actually existing nuclei determines the upper bound of the excitation energy Ex as discussed regarding Fig. 6. Finally, we show the NpNn-plots of the first excitation energies for (a) 4 1 , (b) 6 1 , (c) 8+1 , and (d) 10 1 states in Fig. 7. The measured excitation energies are represented by the empty triangles and the calculated ones from the empirical formula, Eq. (1), are denoted by solid circles. These graphs are just the NpNn-plot versions of the A-plot shown in Fig. 2 with exactly the same set of plotted points. We can learn from Fig. 7 that the same kind of NpNn scheme observed in the excitation energy of 2 1 states is also functioning in the excitation energies of other natural parity even multipole states. We can also find from Fig. 7 that the calculated results, using the empirical formula, agree with the measured data quite well. Moreover, it is interesting to find from Fig. 7 that the width in the central part of the NpNn-plot is enlarged as the multipole of the state is increased. The origin of this enlargement in the empirical formula can be traced to the parameter α of the first term in Eq. (1). The value of α is monotonously increased from 34.9MeV for Ex(2 1 ) to 2489.0MeV for Ex(10 1 ) as can be seen in Table I. 0 100 200 300 400 5000 100 200 300 400 500 (a) All Possible Values (b) All Possible N 's and N (d) All Three Cases (c) Only Existing N 's and N FIG. 6: The NpNn-plot of the first excitation energy of the 2 + states calculated by the empirical formula given by Eq. (1) using the following three different conditions on the exponent Np and Nn: (a) Np and Nn can have any even numbers as long as they satisfy Np +Nn ≤ A. (b) Np and Nn can have any number that is allowed for the valence nucleon numbers. (c) Np and Nn can have numbers which are allowed for the actually existing nuclei. (d) All of the previous three cases are shown together. 0 100 200 300 400 500 0 100 200 300 400 500 (b) E (c) E (d) E ) (a) E Data Empirical Formula FIG. 7: The NpNn-plot for the first excitation energies of the natural parity even multipole states (a) 4+ , (b) 6+ , (c) 8+ , and (d) 10+ using both the measured data (open triangles) and the empirical formula (solid circles). These graphs are just the NpNn-plot versions of the A-plot shown in Fig. 2 with exactly the same set of data points. In summary, we have examined how the recently proposed empirical formula, Eq. (1), for the excitation energy Ex(2 1 ) of the first 2 1 state meets the requirement of the NpNn scheme even though it does not depend on the product NpNn at all. We have demonstrated explicitly that the structure of the empirical formula itself together with the restriction on the values of the valence nucleon numbers Np and Nn of the actually existing nuclei make the characteristic feature of the NpNn scheme appear. Furthermore, our result shows that the composition of the empirical formula, Eq. (1), is in fact ideal for revealing the NpNn scheme. Therefore it is better to regard the NpNn scheme as a strong signature suggesting that this empirical formula is indeed the right one. As a matter of fact, this study about the NpNn scheme has incidentally exposed the significance of the empirical formula given by Eq. (1) as a universal expression for the lowest collective excitation energy. A more detailed account of the empirical formula for the first excitation energy of the natural parity even multipole states in even-even nuclei will be published elsewhere [19]. However, it has been well established that the NpNn scheme holds not only for the lowest excitation energies 1 ) but also for the transition strength B(E2) [8]. Unfortunately, our empirical study intended to express only the excitation energies in terms of the valence nucleon numbers. The extension of our study to include the B(E2) values in our parametrization is in progress. Acknowledgments This work was supported by an Inha University research grant. [1] I. Hamamoto, Nucl. Phys. 73, 225 (1965). [2] R. F. Casten, Nucl. Phys. A443, 1 (1985). [3] For a review of the NpNn scheme, see R. F. Casten and N. V. Zamfir, J. Phys. G22, 1521 (1996). [4] S. Raman, C. W. Nestor, Jr., and P. Tikkanen, At. Data Nucl. Data Tables 78, 1 (2001). [5] R. F. Casten, Phys. Rev. Lett. 54, 1991 (1985). [6] R. F. Casten, D. S. Brenner, and P. E. Haustein, Phys. Rev. Lett. 58, 658 (1987). [7] R. B. Cakirli and R. F. Casten, Phys. Rev. Lett. 96, 132501 (2006). [8] R. F. Casten and N. V. Zamfir, Phys. Rev. Lett. 70, 402 (1993). [9] Y. M. Zhao, R. F. Casten, and A. Arima, Phys. Rev. Lett. 85, 720 (2000). [10] A. de-Shalit and M. Goldhaber, Phys. Rev. 92, 1211 (1953). [11] I. Talmi, Rev. Mod. Phys. 34, 704 (1962). [12] K. Heyde, P. Vanisacker, R. F. Casten, and J. L. Wood, Phys. Lett. 155B, 303 (1985). [13] R. F. Casten, K. Heyde, and A. Wolf, Phys. Lett. 208B, 33 (1988). [14] J.-Y. Zhang, R. F. Casten, and D. S. Brenner, Phys. Lett. 227B, 1 (1989). [15] P. Federmann and S. Pittel, Phys. Lett. 69B, 385 (1977). [16] J. Dobaczewski, W. Nazarewicz, J. Skalski, and T. Werner, Phys. Rev. Lett. 60, 2254 (1988). [17] E. Ha and D. Cha, J. Korean Phys. Soc. 50, 1172 (2007). [18] E. Ha and D. Cha, Phys. Rev. C 75, 057304 (2007). [19] D. Kim, E. Ha, and D. Cha, arXiv:0705.4620[nucl-th]. [20] R. B. Firestone, V. S. Shirley, C. M. Baglin, S. Y. Frank Chu, and J. Zipkin, Table of Isotopes (Wiley, New York, 1999). http://arxiv.org/abs/0705.4620 Acknowledgments References

0704.1694

Locally Decodable Codes From Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers

arXiv:0704.1694v1 [cs.CC] 13 Apr 2007 Locally Decodable Codes From Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers Kiran S. Kedlaya [email protected] Sergey Yekhanin [email protected] Abstract A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N -bit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently [34] introduced a novel technique for constructing locally decodable codes and vastly improved the upper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work of [34] and argue that further progress via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m = 2t − 1 that has a prime factor p > mγ yields a family of k(γ)-query locally decodable codes of length exp . Conversely, if for some fixed k and all ǫ > 0 one can use the technique of [34] to obtain a family of k-query LDCs of length exp (nǫ) ; then infinitely many Mersenne numbers have prime factors larger than known currently. 1 Introduction Classical error-correcting codes allow one to encode an n-bit string x into in N -bit codeword C(x), in such a way that x can still be recovered even if C(x) gets corrupted in a number of coordinates. It is well-known that codewords C(x) of length N = O(n) already suffice to correct errors in up to δN locations of C(x) for any constant δ < 1/4. The disadvantage of classical error-correction is that one needs to consider all or most of the (corrupted) codeword to recover anything about x. Now suppose that one is only interested in recovering one or a few bits of x. In such case more efficient schemes are possible. Such schemes are known as locally decodable codes (LDCs). Locally decodable codes allow reconstruction of an arbitrary bit xi, from looking only at k randomly chosen coordinates of C(x), where k can be as small as 2. Locally decodable codes have numerous applications in complexity theory [15, 29], cryptography [6, 11] and the theory of fault tolerant computation [24]. Below is a slightly informal definition of LDCs: A (k, δ, ǫ)-locally decodable code encodes n-bit strings to N -bit codewords C(x), such that for every i ∈ [n], the bit xi can be recovered with probability 1− ǫ, by a randomized decoding procedure that makes only k queries, even if the codeword C(x) is corrupted in up to δN locations. One should think of δ > 0 and ǫ < 1/2 as constants. The main parameters of interest in LDCs are the length N and the query complexity k. Ideally we would like to have both of them as small as possible. The concept of locally decodable codes was explicitly discussed in various papers in the early 1990s [2, 28, 21]. Katz and http://arxiv.org/abs/0704.1694v1 Trevisan [15] were the first to provide a formal definition of LDCs. Further work on locally decodable codes includes [3, 8, 20, 4, 16, 30, 34, 33, 14, 23]. Below is a brief summary of what was known regarding the length of LDCs prior to [34]. The length of optimal 2-query LDCs was settled by Kerenidis and de Wolf in [16] and is exp(n).1 The best upper bound for the length of 3-query LDCs was exp due to Beimel et al. [3], and the best lower bound is Ω̃(n2) [33]. For general (constant) k the best upper bound was exp nO(log log k/(k log k)) due to Beimel et al. [4] and the best lower bound is Ω̃ n1+1/(⌈k/2⌉−1) [33]. The recent work [34] improved the upper bounds to the extent that it changed the common perception of what may be achievable [12, 11]. [34] introduced a novel technique to construct codes from so-called nice subsets of finite fields and showed that every Mersenne prime p = 2t − 1 yields a family of 3-query LDCs of length . Based on the largest known Mersenne prime [9], this translates to a length of less than exp Combined with the recursive construction from [4], this result yields vast improvements for all values of k > 2. It has often been conjectured that the number of Mersenne primes is infinite. If indeed this conjecture holds, [34] gets three query locally decodable codes of length N = exp log log n for infinitely many n. Finally, assuming that the conjecture of Lenstra, Pomerance and Wagstaff [31, 22, 32] regarding the density of Mersenne primes holds, [34] gets three query locally decodable codes of length N = exp log1−ǫ log n for all n, for every ǫ > 1.1 Our results In this paper we address two natural questions left open by [34]: 1. Are Mersenne primes necessary for the constructions of [34]? 2. Has the technique of [34] been pushed to its limits, or one can construct better codes through a more clever choice of nice subsets of finite fields? We extend the work of [34] and answer both of the questions above. In what follows let P (m) denote the largest prime factor of m. We show that one does not necessarily need to use Mersenne primes. It suffices to have Mersenne numbers with polynomially large prime factors. Specifically, every Mersenne number m = 2t − 1 such that P (m) ≥ mγ yields a family of k(γ)-query locally decodable codes of length exp . A partial converse also holds. Namely, if for some fixed k ≥ 3 and all ǫ > 0 one can use the technique of [34] to (unconditionally) obtain a family of k-query LDCs of length exp (nǫ) ; then for infinitely many t we have P (2t − 1) ≥ (t/2)1+1/(k−2). (1) The bound (1) may seem quite weak in light of the widely accepted conjecture saying that the number of Mersenne primes is infinite. However (for any k ≥ 3) this bound is substantially stronger than what is currently known unconditionally. Lower bounds for P (2t − 1) have received a considerable amount of attention in the number theory literature [25, 26, 10, 27, 19, 18]. The strongest result to date is due to Stewart [27]. It says that for all integers t ignoring a set of asymptotic density zero, and for all functions ǫ(t) > 0 where ǫ(t) tends to zero monotonically and arbitrarily slowly: P (2t − 1) > ǫ(t)t (log t)2 / log log t. (2) 1Throughout the paper we use the standard notation exp(x) = eO(x). There are no better bounds known to hold for infinitely many values of t, unless one is willing to accept some number theoretic conjectures [19, 18]. We hope that our work will further stimulate the interest in proving lower bounds for P (2t − 1) in the number theory community. In summary, we show that one may be able to improve the unconditional bounds of [34] (say, by discovering a new Mersenne number with a very large prime factor) using the same technique. However any attempts to reach the exp (nǫ) length for some fixed query complexity and all ǫ > 0 require either progress on an old number theory problem or some radically new ideas. In this paper we deal only with binary codes for the sake of clarity of presentation. We remark however that our results as well as the results of [34] can be easily generalized to larger alphabets. Such generalization will be discussed in detail in [35]. 1.2 Outline In section 3 we introduce the key concepts of [34], namely that of combinatorial and algebraic niceness of subsets of finite fields. We also briefly review the construction of locally decodable codes from nice subsets. In section 4 we show how Mersenne numbers with large prime factors yield nice subsets of prime fields. In section 5 we prove a partial converse. Namely, we show that every finite field Fq containing a sufficiently nice subset, is an extension of a prime field Fp, where p is a large prime factor of a large Mersenne number. Our main results are summarized in sections 4.3 and 5.4. 2 Notation We use the following standard mathematical notation: • [s] = {1, . . . , s}; • Zn denotes integers modulo n; • Fq is a finite field of q elements; • dH(x, y) denotes the Hamming distance between binary vectors x and y; • (u, v) stands for the dot product of vectors u and v; • For a linear space L ⊆ Fm2 , L⊥ denotes the dual space. That is, L⊥ = {u ∈ Fm2 | ∀v ∈ L, (u, v) = 0}; • For an odd prime p, ord2(p) denotes the smallest integer t such that p | 2t − 1. 3 Nice subsets of finite fields and locally decodable codes In this section we introduce the key technical concepts of [34], namely that of combinatorial and algebraic niceness of subsets of finite fields. We briefly review the construction of locally decodable codes from nice subsets. Our review is concise although self-contained. We refer the reader interested in a more detailed and intuitive treatment of the construction to the original paper [34]. We start by formally defining locally decodable codes. Definition 1 A binary code C : {0, 1}n → {0, 1}N is said to be (k, δ, ǫ)-locally decodable if there exists a randomized decoding algorithm A such that 1. For all x ∈ {0, 1}n, i ∈ [n] and y ∈ {0, 1}N such that dH(C(x), y) ≤ δN : Pr[Ay(i) = xi] ≥ 1− ǫ, where the probability is taken over the random coin tosses of the algorithm A. 2. A makes at most k queries to y. We now introduce the concepts of combinatorial and algebraic niceness of subsets of finite fields. Our defini- tions are syntactically slightly different from the original definitions in [34]. We prefer these formulations since they are more appropriate for the purposes of the current paper. In what follows let F∗q denote the multiplicative group of Fq. Definition 2 A set S ⊆ F∗q is called t combinatorially nice if for some constant c > 0 and every positive integer m there exist two n = ⌊cmt⌋-sized collections of vectors {u1, . . . , un} and {v1, . . . , vn} in Fmq , such that • For all i ∈ [n], (ui, vi) = 0; • For all i, j ∈ [n] such that i 6= j, (uj , vi) ∈ S. Definition 3 A set S ⊆ F∗q is called k algebraically nice if k is odd and there exists an odd k′ ≤ k and two sets S0, S1 ⊆ Fq such that • S0 is not empty; • |S1| = k′; • For all α ∈ Fq and β ∈ S : |S0 ∩ (α+ βS1)| ≡ 0 mod (2). The following lemma shows that for an algebraically nice set S, the set S0 can always be chosen to be large. It is a straightforward generalization of [34, lemma 15]. Lemma 4 Let S ⊆ F∗q be a k algebraically nice set. Let S0, S1 ⊆ Fq be sets from the definition of algebraic niceness of S. One can always redefine the set S0 to satisfy |S0| ≥ ⌈q/2⌉. Proof: Let L be the linear subspace of Fq2 spanned by the incidence vectors of the sets α+ βS1, for α ∈ Fq and β ∈ S. Observe that L is invariant under the actions of a 1-transitive permutation group (permuting the coordinates in accordance with addition in Fq). This implies that the space L ⊥ is also invariant under the actions of the same group. Note that L⊥ has positive dimension since it contains the incidence vector of the set S0. The last two observations imply that L⊥ has full support, i.e., for every i ∈ [q] there exists a vector v ∈ L⊥ such that vi 6= 0. It is easy to verify that any linear subspace of Fq2 that has full support contains a vector of Hamming weight at least ⌈q/2⌉. Let v ∈ L⊥ be such a vector. Redefining the set S0 to be the set of nonzero coordinates of v we conclude the proof. We now proceed to the core proposition of [34] that shows how sets exhibiting both combinatorial and algebraic niceness yield locally decodable codes. Proposition 5 Suppose S ⊆ F∗q is t combinatorially nice and k algebraically nice; then for every positive integer n there exists a code of length exp(n1/t) that is (k, δ, 2kδ) locally decodable for all δ > 0. Proof: Our proof comes in three steps. We specify encoding and local decoding procedures for our codes and then argue the lower bound for the probability of correct decoding. We use the notation from definitions 2 and 3. Encoding: We assume that our message has length n = ⌊cmt⌋ for some value of m. (Otherwise we pad the message with zeros. It is easy to see that such padding does not not affect the asymptotic length of the code.) Our code will be linear. Therefore it suffices to specify the encoding of unit vectors e1, . . . , en, where ej has length n and a unique non-zero coordinate j. We define the encoding of ej to be a q m long vector, whose coordinates are labelled by elements of Fmq . For all w ∈ Fmq we set: Enc(ej)w = 1, if (uj , w) ∈ S0; 0, otherwise. It is straightforward to verify that we defined a code encoding n bits to exp(n1/t) bits. Local decoding: Given a (possibly corrupted) codeword y and an index i ∈ [n], the decoding algorithm A picks w ∈ Fmq , such that (ui, w) ∈ S0 uniformly at random, reads k′ ≤ k coordinates of y, and outputs the sum: yw+λvi . (4) Probability of correct decoding: First we argue that decoding is always correct if A picks w ∈ Fmq such that all bits of y in locations {w + λvi}λ∈S1 are not corrupted. We need to show that for all i ∈ [n], x ∈ {0, 1}n and w ∈ Fmq , such that (ui, w) ∈ S0: xj Enc(ej) w+λvi = xi. (5) Note that xj Enc(ej) w+λvi Enc(ej)w+λvi = I [(uj , w + λvi) ∈ S0] , (6) where I[γ ∈ S0] = 1 if γ ∈ S0 and zero otherwise. Now note that I [(uj , w + λvi) ∈ S0] = I [(uj , w) + λ(uj , vi) ∈ S0] = 1, if i = j, 0, otherwise. The last identity in (7) for i = j follows from: (ui, vi) = 0, (ui, w) ∈ S0 and k′ = |S1| is odd. The last identity for i 6= j follows from (uj , vi) ∈ S and the algebraic niceness of S. Combining identities (6) and (7) we get (5). Now assume that up to δ fraction of bits of y are corrupted. Let Ti denote the set of coordinates whose labels belong to w ∈ Fmq | (ui, w) ∈ S0 . Recall that by lemma 4, |Ti| ≥ qm/2. Thus at most 2δ fraction of coor- dinates in Ti contain corrupted bits. Let Qi = {w + λvi}λ∈S1 | w : (ui, w) ∈ S0 be the family of k′-tuples of coordinates that may be queried by A. (ui, vi) = 0 implies that elements of Qi uniformly cover the set Ti. Combining the last two observations we conclude that with probability at least 1 − 2kδ A picks an uncorrupted k′-tuple and outputs the correct value of xi. All locally decodable codes constructed in this paper are obtained by applying proposition 5 to certain nice sets. Thus all our codes have the same dependence of ǫ (the probability of the decoding error) on δ (the fraction of corrupted bits). In what follows we often ignore these parameters and consider only the length and query complexity of codes. 4 Mersenne numbers with large prime factors yield nice subsets of prime fields In what follows let 〈2〉 ⊆ F∗p denote the multiplicative subgroup of F∗p generated by 2. In [34] it is shown that for every Mersenne prime p = 2t − 1 the set 〈2〉 ⊆ F∗p is simultaneously 3 algebraically nice and ord2(p) combinatorially nice. In this section we prove the same conclusion for a substantially broader class of primes. Lemma 6 Suppose p is an odd prime; then 〈2〉 ⊆ F∗p is ord2(p) combinatorially nice. Proof: Let t = ord2(p). Clearly, t divides p− 1. We need to specify a constant c > 0 such that for every positive integer m there exist two n = ⌊cmt⌋-sized collections of m long vectors over Fp satisfying: • For all i ∈ [n], (ui, vi) = 0; • For all i, j ∈ [n] such that i 6= j, (uj , vi) ∈ 〈2〉. First assume that m has the shape m = m′−1+(p−1)/t (p−1)/t , for some integer m′ ≥ p − 1. In this case [34, lemma 13] gives us a collection of n = vectors with the right properties. Observe that n ≥ cmt for a constant c that depends only on p and t. Now assume m does not have the right shape, and let m1 be the largest integer smaller than m that does have it. In order to get vectors of length m we use vectors of length m1 coming from [34, lemma 13] padded with zeros. It is not hard to verify such a construction still gives us n ≥ cmt large families of vectors for a suitably chosen constant c. We use the standard notation F to denote the algebraic closure of the field F. Also let Cp ⊆ F 2 denote the multiplicative subgroup of p-th roots of unity in F2. The next lemma generalizes [34, lemma 14]. Lemma 7 Let p be a prime and k be odd. Suppose there exist ζ1, . . . , ζk ∈ Cp such that ζ1 + . . . + ζk = 0; (8) then 〈2〉 ⊆ F∗p is k algebraically nice. Proof: In what follows we define the set S1 ⊆ Fp and prove the existence of a set S0 such that that together S0 and S1 yield k algebraic niceness of 〈2〉. Identity 8 implies that there exists an odd integer k′ ≤ k and k′ distinct p-th roots of unity ζ ′1, . . . , ζ k ∈ Cp such that ζ ′1 + . . . + ζ k′ = 0. (9) Let t = ord2(p). Observe that Cp ⊆ F2t . Let g be a generator of Cp. Identity (9) yields gγ1 + . . .+ gγk′ = 0, for some distinct values of {γi}i∈[k′]. Set S1 = {γ1, . . . , γk′}. Consider a natural one to one correspondence between subsets S′ of Fp and polynomials φS′(x) in the ring F2[x]/(x p − 1) : φS′(x) = xs. It is easy to see that for all sets S′ ⊆ Fp and all α, β ∈ Fp, such that β 6= 0 : φα+βS′(x) = x αφS′(x Let α be a variable ranging over Fp and β be a variable ranging over 〈2〉. We are going to argue the existence of a set S0 that has even intersections with all sets of the form α+βS1, by showing that all polynomials φα+βS1 belong to a certain linear space L ∈ F2[x]/(xp − 1) of dimension less than p. In this case any nonempty set T ⊆ Fp such that φT ∈ L⊥ can be used as the set S0. Let τ(x) = gcd(xp−1, φS1(x)). Note that τ(x) 6= 1 since g is a common root of xp−1 and φS1(x). Let L be the space of polynomials in F2[x]/(xp−1) that are multiples of τ(x). Clearly, dimL = p− deg τ. Fix some α ∈ Fp and β ∈ 〈2〉. Let us prove that φα+βS1(x) is in L : φα+βS1(x) = x αφS1(x β) = xα(φS1(x)) The last identity above follows from the fact that for any f ∈ F2[x] and any integer i : f(x2 ) = (f(x))2 In what follows we present sufficient conditions for the existence of k-tuples of p-th roots of unity in F2 that sum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a more explicit conclusion. 4.1 A sufficient condition for the existence of three p-th roots of unity summing to zero Lemma 8 Let p be an odd prime. Suppose ord2(p) < (4/3) log2 p; then there exist three p-th roots of unity in F2 that sum to zero. Proof: We start with a brief review of some basic concepts of projective algebraic geometry. Let F be a field, and f ∈ F[x, y, z] be a homogeneous polynomial. A triple (x0, y0, z0) ∈ F3 is called a zero of f if f(x0, y0, z0) = 0. A zero is called nontrivial if it is different from the origin. An equation f = 0 defines a projective plane curve χf . Nontrivial zeros of f considered up to multiplication by a scalars are called F-rational points of χf . If F is a finite field it makes sense to talk about the number of F-rational points on a curve. Let t = ord2(p). Note that Cp ⊆ F2t . Consider a projective plane Fermat curve χ defined by t−1)/p + y(2 t−1)/p + z(2 t−1)/p = 0. (10) Let us call a point a on χ trivial if one of the coordinates of a is zero. Cyclicity of F∗ implies that χ contains exactly 3(2t − 1)/p trivial F2t-rational points. Note that every nontrivial point of χ yields a triple of elements of Cp that sum to zero. The classical Weil bound [17, p. 330] provides an estimate |Nq − (q + 1)| ≤ (d− 1)(d − 2) q (11) for the number Nq of Fq-rational points on an arbitrary smooth projective plane curve of degree d. (11) implies that in case 2t + 1 > 2t − 1 2t − 1 2t/2 + 3 2t − 1 there exists a nontrivial point on the curve (10). Note that (12) follows from 2t + 1 > 2t/2 − 23t/2+1 3 ∗ 2t , (13) and (13) follows from 2t > 22t+t/2/p2 and 2t/2+1 > 3. Now note that the first inequality above follows from t < (4/3) log2 p and the second follows from t > 1. Note that the constant 4/3 in lemma 8 cannot be improved to 2: there are no three elements of C13264529 that sum to zero, even though ord2(13264529) = 47 < 2 ∗ log2 13264529 ≈ 47.3. 4.2 A sufficient condition for the existence of k p-th roots of unity summing to zero Our argument in this section comes in three steps. First we briefly review the notion of (additive) Fourier coefficients of subsets of F2t . Next, we invoke a folklore argument to show that subsets of F2t with appropriately small nontrivial Fourier coefficients contain k-tuples of elements that sum to zero. Finally, we use a recent result of Bourgain and Chang [5] (generalizing the classical estimate for Gauss sums) to argue that (under certain constraints on p) all nontrivial Fourier coefficients of Cp are small. For x ∈ F2t let Tr(x) = x + x2 + . . . + x2 denote the trace of x. It is not hard to verify that for all x, Tr(x) ∈ F2. Characters of F2t are homomorphisms from the additive group of F2t into the multiplicative group {±1}. There exist 2t characters. We denote characters by χa, where a ranges in F2t , and set χa(x) = (−1)Tr(ax). Let C(x) denote the incidence function of a set C ⊆ F2t . For arbitrary a ∈ Ft2 the Fourier coefficient χa(C) is defined by χa(C) = χa(x)C(x), where the sum is over all x ∈ F2t . Fourier coefficient χ0(C) = |C| is called trivial, and other Fourier coefficients are called nontrivial. In what follows χ stands for summation over all 2 characters of F2t . We need the following two standard properties of characters and Fourier coefficients. χ(x) = 2t, if x = 0; 0, otherwise. χ2(C) = 2t|C|. (15) The following lemma is a folklore. Lemma 9 Let C ⊆ F2t and k ≥ 3 be a positive integer. Let F be the largest absolute value of a nontrivial Fourier coefficient of C. Suppose )1/(k−2) then there exist k elements of C that sum to zero. Proof: Let M(C) = # {ζ1, . . . , ζk ∈ C | ζ1 + . . .+ ζk = 0} . (14) yields M(C) = x1,...,xk∈F2t C(x1) . . . C(xk) χ(x1 + . . .+ xk). (17) Note that χ(x1 + . . .+ xk) = χ(x1) . . . χ(xk). Changing the order of summation in (17) we get M(C) = x1,...,xk∈F2t C(x1) . . . C(xk)χ(x1) . . . χ(xk) = χk(C). (18) Note that χk(C) = χ 6=χ0 χk(C) ≥ − F k−2 χ2(C) = − F k−2|C|, (19) where the last identity follows from (15). Combining (18) and (19) we conclude that (16) implies M(C) > 0. The following lemma is a special case of [5, theorem 1]. Lemma 10 Assume that n | 2t − 1 and satisfies the condition 2t − 1 ′ − 1 < 2t(1−ǫ)−t , for all 1 ≤ t′ < t, t′ | t, where ǫ > 0 is arbitrary and fixed. Then for all a ∈ F∗ x∈F2t (−1)Tr(axn) < c12 t(1−δ), (20) where δ = δ(ǫ) > 0 and c1 = c1(ǫ) are absolute constants. Below is the main result of this section. Recall that Cp denotes the set of p-th roots of unity in F2. Lemma 11 For every c > 0 there exists an odd integer k = k(c) such that the following implication holds. If p is an odd prime and ord2(p) < c log2 p then some k elements of Cp sum to zero. Proof: Note that if there exist k′ elements of a set C ⊆ F2 that sum to zero, where k′ is odd; then there exist k elements of C that sum to zero for every odd k ≥ k′. Also note that the sum of all p-th roots of unity is zero. Therefore given c it suffices to prove the existence of an odd k = k(c) that works for all sufficiently large p. Let t = ord2(p). Observe that p > 2 t/c. Assume p is sufficiently large so that t > 2c. Next we show that the precondition of lemma 10 holds for n = (2t − 1)/p and ǫ = 1/(2c). Let t′ | t and 1 ≤ t′ < t. Clearly gcd(2t ′ − 1, p) = 1. Therefore 2t − 1 2t − 1 ′ − 1 2t − 1 ′ − 1) 2t(1−1/c) ′ − 1 , (21) where the inequality follows from p > 2t/c. Clearly, t > 2c yields 2t/(2c)/2 > 1. Multiplying the right hand side of (21) by 2t/(2c)/2 and using 2(2t ′ − 1) > 2t′ we get 2t − 1 2t − 1 ′ − 1 < 2t(1−1/(2c))−t . (22) Combining (22) with lemma 10 we conclude that there exist δ > 0 and c1 such that for all a ∈ F∗2t x∈F2t (−1)Tr −1)/p < c12 t(1−δ). (23) Observe that x(2 t−1)/p takes every value in Cp exactly (2 t−1)/p times when x ranges over F∗ . Thus (23) implies (2t − 1)(F/p) < c12t(1−δ), (24) where F denotes that largest nontrivial Fourier coefficient of Cp. (24) yields F/p < (2c1)2 −δt. Pick k ≥ 3 to be the smallest odd integer such that (1 − 1/c)/(k − 2) < δ. We now have (1−1/c)t (k−2) (25) for all sufficiently large values of p. Combining p > 2t/c with (25) we get )1/(k−2) and the application of lemma 9 concludes the proof. 4.3 Summary In this section we summarize our positive results and show that one does not necessarily need to use Mersenne primes to construct locally decodable codes via the methods of [34]. It suffices to have Mersenne numbers with polynomially large prime factors. Recall that P (m) denotes the largest prime factor of an integer m. Our first theorem gets 3-query LDCs from Mersenne numbers m with prime factors larger than m3/4. Theorem 12 Suppose P (2t − 1) > 20.75t; then for every message length n there exists a three query locally decodable code of length exp(n1/t). Proof: Let P (2t − 1) = p. Observe that p | 2t − 1 and p > 20.75t yield ord2(p) < (4/3) log2 p. Combining lemmas 8,7 and 6 with proposition 5 we obtain the statement of the theorem. As an example application of theorem 12 one can observe that P (223−1) = 178481 > 2(3/4)∗23 ≈ 155872 yields a family of three query locally decodable codes of length exp(n1/23). Theorem 12 immediately yields: Theorem 13 Suppose for infinitely many t we have P (2t − 1) > 20.75t; then for every ǫ > 0 there exists a family of three query locally decodable codes of length exp(nǫ). The next theorem gets constant query LDCs from Mersenne numbers m with prime factors larger than mγ for every value of γ. Theorem 14 For every γ > 0 there exists an odd integer k = k(γ) such that the following implication holds. Suppose P (2t − 1) > 2γt; then for every message length n there exists a k query locally decodable code of length exp(n1/t). Proof: Let P (2t − 1) = p. Observe that p | 2t − 1 and p > 2γt yield ord2(p) < (1/γ) log2 p. Combining lemmas 22,7 and 6 with proposition 5 we obtain the statement of the theorem. As an immediate corollary we get: Theorem 15 Suppose for some γ > 0 and infinitely many t we have P (2t − 1) > 2γt; then there is a fixed k such that for every ǫ > 0 there exists a family of k query locally decodable codes of length exp(nǫ). 5 Nice subsets of finite fields yield Mersenne numbers with large prime factors Definition 16 We say that a sequence Si ⊆ F∗qi of subsets of finite fields is k-nice if every Si is k alge- braically nice and t(i) combinatorially nice, for some integer valued monotonically increasing function t. The core proposition 5 asserts that a subset S ⊆ F∗q that is k algebraically nice and t combinatorially nice yields a family of k-query locally decodable codes of length exp(n1/t). Clearly, to get k-query LDCs of length exp(nǫ) for some fixed k and every ǫ > 0 via this proposition, one needs to exhibit a k-nice sequence. In this section we show how the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime factors. Our argument proceeds in two steps. First we show that a k-nice sequence yields an infinite sequence of primes {pi}i≥1 , where every Cpi contains a k-tuple of elements summing to zero. Next we show that Cp contains a short additive dependence only if p is a large factor of a Mersenne number. 5.1 A nice sequence yields infinitely many primes p with short dependencies between p-th roots of unity We start with some notation. Consider a a finite field Fq = Fpl, where p is prime. Fix a basis e1, . . . , el of Fq over Fp. In what follows we often write (α1, . . . , αl) ∈ Flp to denote α = i=1 αiei ∈ Fq. Let R denote the ring F2[x1, . . . , xl]/(x 1 − 1, . . . , x l − 1). Consider a natural one to one correspondence between subsets S1 of Fq and polynomials φS1(x1, . . . , xl) ∈ R. φS1(x1, . . . , xl) = (α1,...,αl)∈S1 1 . . . x It is easy to see that for all sets S1 ⊆ Fq and all α, β ∈ Fq : φ(α1,...,αl)+βS1(x1, . . . , xl) = x 1 . . . x φβS1(x1, . . . , xl). (26) Let Γ be a family of subsets of Fq. It is straightforward to verify that a set S0 ⊆ Fq has even intersections with every element of Γ if and only if φS0 belongs to L ⊥, where L is the linear subspace of R spanned by {φS1}S1∈Γ . Combining the last observation with formula (26) we conclude that a set S ⊆ F∗q is k algebraically nice if and only if there exists a set S1 ⊆ Fq of odd size k′ ≤ k such that the ideal generated by polynomials {φβS1}{β∈S} is a proper ideal of R. Note that polynomials {f1, . . . , fh} ∈ R generate a proper ideal if an only if polynomials {f1, . . . , fh, xp1 − 1, . . . , x l − 1} generate a proper ideal in F2[x1, . . . , xl]. Also note that a family of polynomials generates a proper ideal in F2[x1, . . . , xl] if and only if it generates a proper ideal in F2[x1, . . . , xl]. Now an application of Hilbert’s Nullstellensatz [7, p. 168] implies that a set S ⊆ F∗q is k algebraically nice if and only if there is a set S1 ⊆ Fq of odd size k′ ≤ k such that the polynomials {φβS1}{β∈S} and {x i − 1}1≤i≤l have a common root in F2. Lemma 17 Let Fq = Fpl , where p is prime. Suppose Fq contains a nonempty k algebraically nice subset; then there exist ζ1, . . . , ζk ∈ Cp such that ζ1 + . . .+ ζk = 0. Proof: Assume S ⊆ F∗q is nonempty and k algebraically nice. The discussion above implies that there exists S1 ⊆ Fq of odd size k′ ≤ k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈ C p. Fix an arbitrary β0 ∈ S, and note that Cp is closed under multiplication. Thus, φβ0S1(ζ1, . . . , ζl) = 0 (27) yields k′ p-th roots of unity that add up to zero. It is readily seen that one can extend (27) (by adding an appropriate number of pairs of identical roots) to obtain k p-th roots of unity that add up to zero for any odd k ≥ k′. Note that lemma 17 does not suffice to prove that a k-nice sequence Si ⊆ F∗qi yields infinitely many primes p with short (nontrivial) additive dependencies in Cp. We need to argue that the set {charFqi}i≥1 can not be finite. To proceed, we need some more notation. Recall that q = pl and p is prime. For x ∈ Fq let Tr(x) = x+. . .+xp l−1 ∈ Fp denote the (absolute) trace of x. For γ ∈ Fq, c ∈ F∗p we call the set πγ,c = {x ∈ Fq | Tr(γx) = c} a proper affine hyperplane of Fq. Lemma 18 Let Fq = Fpl , where p is prime. Suppose S ⊆ F∗q is k algebraically nice; then there exist h ≤ pk proper affine hyperplanes {πγi,ci}1≤i≤h of Fq such that S ⊆ πγi,ci. Proof: Discussion preceding lemma 17 implies that there exists a set S1 = {σ1, . . . , σk′} ⊆ Fq of odd size k′ ≤ k such that all polynomials {φβS1}{β∈S} vanish at some (ζ1, . . . , ζl) ∈ C p. Let ζ be a generator of Cp. For every 1 ≤ i ≤ l pick ωi ∈ Zp such that ζi = ζωi . For every β ∈ S, φβS1(ζ1, . . . , ζl) = 0 yields µ=(µ1,...,µl)∈βS1 i=1 µiωi = 0. (28) Observe that for fixed values {ωi}1≤i≤l ∈ Zp the map D(µ) = i=1 µiωi is a linear map from Fq to Fp. It is not hard to prove that every such map can be expressed as D(µ) = Tr(δµ) for an appropriate choice of δ ∈ Fq. Therefore we can rewrite (28) as µ∈βS1 ζTr(δµ) = ζTr(δβσ) = 0. (29) Let W = (w1, . . . , wk′) ∈ Zk p | ζw1 + . . . + ζwk′ = 0 denote the set of exponents of k′-dependencies be- tween powers of ζ. Clearly, |W | ≤ pk. Identity (29) implies that every β ∈ S satisfies Tr((δσ1)β) = w1, Tr((δσk′)β) = wk′ ; for an appropriate choice of (w1, . . . , wk′) ∈ W. Note that the all-zeros vector does not lie in W since k′ is odd. Therefore at least one of the identities in (30) has a non-zero right-hand side, and defines a proper affine hyperplane of Fq. Collecting one such hyperplane for every element of W we get a family of |W | proper affine hyperplanes containing every element of S. Lemma 18 gives us some insight into the structure of algebraically nice subsets of Fq. Our next goal is to develop an insight into the structure of combinatorially nice subsets. We start by reviewing some relations between tensor and dot products of vectors. For vectors u ∈ Fmq and v ∈ Fnq let u⊗v ∈ Fmnq denote the tensor product of u and v. Coordinates of u⊗ v are labelled by all possible elements of [m]× [n] and (u⊗ v)i,j = uivj . Also, let u⊗l denote the l-the tensor power of u and u ◦ v denote the concatenation of u and v. The following identity is standard. For any u, x ∈ Fmq and v, y ∈ Fnq : (u⊗ v, x⊗ y) = i∈[m],j∈[n] uivjxiyj = i∈[m] j∈[n]  = (u, x)(v, y). (31) In what follows we need a generalization of identity (31). Let f(x1, . . . , xh) = i cix 1 . . . x h be a polynomial in Fq[x1, . . . , xh]. Given f we define f̄ ∈ Fq[x1, . . . , xh] by f̄ = 1 . . . x , i.e., we simply set all nonzero coefficients of f to 1. For vectors u1, . . . , uh in F q define f(u1, . . . , uh) = ◦i ciu 1 ⊗ . . .⊗ u h . (32) Note that to obtain f(u1, . . . , uh) we replaced products in f by tensor products and addition by concatenation. Clearly, f(u1, . . . , uh) is a vector whose length may be larger than m. Claim 19 For every f ∈ Fq[x1, . . . , xh] and u1, . . . , uh, v1, . . . , vh ∈ Fmq : f(u1, . . . , uh), f̄(v1, . . . , vh) = f((u1, v1), . . . , (uh, vh)). (33) Proof: Let u = (u1, . . . , uh) and v = (v1, . . . , vh). Observe that if (33) holds for polynomials f1 and f2 defined over disjoint sets of monomials then it also holds for f = f1 + f2 : f(u), f̄(v) (f1 + f2)(u), (f̄1 + f̄2)(v) f1(u) ◦ f2(u), f̄1(v) ◦ f̄2(v) f1 ((u1, v1), . . . , (uh, vh)) + f2 ((u1, v1), . . . , (uh, vh)) = f ((u1, v1), . . . , (uh, vh)) . Therefore it suffices to prove (33) for monomials f = cxα11 . . . x . It remains to notice identity (33) for monomi- als f = cxα11 . . . x h follows immediately from formula (31) using induction on i=1 αi. The next lemma bounds combinatorial niceness of certain subsets of F∗q. Lemma 20 Let Fq = Fpl, where p is prime. Let S ⊆ F∗q. Suppose there exist h proper affine hyperplanes {πγr ,cr}1≤r≤h of Fq such that S ⊆ πγr ,cr ; then S is at most h(p − 1) combinatorially nice. Proof: Assume S is t combinatorially nice. This implies that for some c > 0 and every m there exist two n = ⌊cmt⌋-sized collections of vectors {ui}i∈[n] and {vi}i∈[n] in Fmq , such that: • For all i ∈ [n], (ui, vi) = 0; • For all i, j ∈ [n] such that i 6= j, (uj , vi) ∈ S. For a vector u ∈ Fmq and integer e let ue denote a vector resulting from raising every coordinate of u to the power e. For every i ∈ [n] and r ∈ [h] define vectors u(r)i and v i in F i = (γrui) ◦ (γrui) p ◦ . . . ◦ (γrui)p and v i = vi ◦ v i ◦ . . . ◦ v i . (34) Note that for every r1, r2 ∈ [h], v i = v i . It is straightforward to verify that for every i, j ∈ [n] and r ∈ [h] : j , v = Tr(γr(uj , vi)). (35) Combining (35) with the fact that S is covered by proper affine hyperplanes πγi,ci we conclude that • For all i ∈ [n] and r ∈ [h], i , v • For all i, j ∈ [n] such that i 6= j, there exists r ∈ [h] such that j , v ∈ F∗p. Pick g(x1, . . . , xh) ∈ Fp[x1, . . . , xh] to be a homogeneous degree h polynomial such that for a = (a1, . . . , ah) ∈ p : g(a) = 0 if and only if a is the all-zeros vector. The existence of such a polynomial g follows from [17, Example 6.7]. Set f = gp−1. Note that for a ∈ Fhp : f(a) = 0 if a is the all-zeros vector, and f(a) = 1 otherwise. For all i ∈ [n] define u′i = f i , . . . , u ◦ (1) and v′i = f̄ i , . . . , v ◦ (−1). (36) Note that f and f̄ are homogeneous degree (p − 1)h polynomials in h variables. Therefore (32) implies that for all i vectors u′i and v i have length m ′ ≤ h(p−1)h(ml)(p−1)h. Combining identities (36) and (33) and using the properties of dot products between vectors discussed above we conclude that for every m there exist two n = ⌊cmt⌋-sized collections of vectors {u′i}i∈[n] and {v′i}i∈[n] in Fm q , such that: • For all i ∈ [n], (u′i, v′i) = −1; • For all i, j ∈ [n] such that i 6= j, (uj , vi) = 0. It remains to notice that a family of vectors with such properties exists only if n ≤ m′, i.e., ⌊cmt⌋ ≤ h(p−1)h(ml)(p−1)h. Given that we can pick m to be arbitrarily large, this implies that t ≤ (p− 1)h. The next lemma presents the main result of this section. Lemma 21 Let k be an odd integer. Suppose there exists a k-nice sequence; then for infinitely many primes p some k of elements of Cp add up to zero. Proof: Assume Si ⊆ F∗qi is k-nice. Let p be a fixed prime. Combining lemmas 18 and 20 we conclude that every k algebraically nice subset S ⊆ F∗ is at most (p − 1)pk combinatorially nice. Note that our bound on combinatorial niceness is independent of l. Therefore there are only finitely many extensions of the field Fp in the sequence {Fqi}i≥1 , and the set P = {charFqi}i≥1 is infinite. It remains to notice that according to lemma 17 for every p ∈ P there exist k elements of Cp that add up to zero. In what follows we present necessary conditions for the existence of k-tuples of p-th roots of unity in F2 that sum to zero. We treat the k = 3 case separately since in that case we can use a specialized argument to derive a slightly stronger conclusion. 5.2 A necessary condition for the existence of k p-th roots of unity summing to zero Lemma 22 Let k ≥ 3 be odd and p be a prime. Suppose there exist ζ1, . . . , ζk ∈ Cp such that i=1 ζi = 0; then ord2(p) ≤ 2p1−1/(k−1). (37) Proof: Let t = ord2(p). Note that Cp ⊆ F2t . Note also that all elements of Cp other than the multiplicative identity are proper elements of F2t. Therefore for every ζ ∈ Cp where ζ 6= 1 and every f(x) ∈ F2[x] such that deg f ≤ t− 1 we have: f(ζ) 6= 0. By multiplying i=1 ζi = 0 through by ζ , we may reduce to the case ζk = 1. Let ζ be the generator of Cp. For every i ∈ [k − 1] pick wi ∈ Zp such that ζi = ζwi . We now have i=1 ζ wi + 1 = 0. Set h = ⌊(t − 1)/2⌋. Consider the (k − 1)-tuples: (mw1 + i1, . . . ,mwk−1 + ik−1) ∈ Zk−1p , for m ∈ Zp and i1, . . . , ik−1 ∈ [0, h]. (38) Suppose two of these coincide, say (mw1 + i1, . . . ,mwk−1 + ik−1) = (m ′w1 + i 1, . . . ,m ′wk−1 + i k−1), with (m, i1, . . . , ik−1) 6= (m′, i′1, . . . , i′k−1). Set n = m−m′ and jl = i′l − il for l ∈ [k − 1]. We now have (nw1, . . . , nwk−1) = (j1, . . . , jl) with −h ≤ j1, . . . , jk−1 ≤ h. Observe that n 6= 0, and thus it has a multiplicative inverse g ∈ Zp. Consider a polynomial P (z) = zj1+h + . . .+ zjk−1+h + zh ∈ F2[z]. Note that degP ≤ 2h ≤ t − 1. Note also that P (1) = 1 and P (ζg) = 0. The latter identity contradicts the fact that ζg is a proper element of F2t . This contradiction implies that all (k−1)-tuples in (38) are distinct. This yields pk−1 ≥ p which is equivalent to (37). 5.3 A necessary condition for the existence of three p-th roots of unity summing to zero In this section we slightly strengthen lemma 22 in the special case when k = 3. Our argument is loosely inspired by the Agrawal-Kayal-Saxena deterministic primality test [1]. Lemma 23 Let p be a prime. Suppose there exist ζ1, ζ2, ζ3 ∈ Cp that sum up to zero; then ord2(p) ≤ ((4/3)p)1/2 . (39) Proof: Let t = ord2(p). Note that Cp ⊆ F2t . Note also that all elements of Cp other than the multiplicative identity are proper elements of F2t. Therefore for every ζ ∈ Cp where ζ 6= 1 and every f(x) ∈ F2[x] such that deg f ≤ t− 1 we have: f(ζ) 6= 0. Observe that ζ1 + ζ2 + ζ3 = 0 implies ζ1ζ 2 + 1 = ζ3ζ 2 . This yields 2 + 1 = 1. Put ζ = ζ1ζ Note that ζ 6= 1 and ζ, 1 + ζ ∈ Cp. Consider the products πi,j = ζ i(1 + ζ)j ∈ Cp for 0 ≤ i, j ≤ t− 1. Note that πi,j, πk,l cannot be the same if i ≥ k and l ≥ j, as then ζ i−k − (1 + ζ)l−j = 0, but the left side has degree less than t. In other words, if πi,j = πk,l and (i, j) 6= (k, l), then the pairs (i, j) and (k, l) are comparable under termwise comparison. In particular, either (k, l) = (i+a, j+b) or (i, j) = (k+a, l+b) for some pair (a, b) with πa,b = 1. We next check that there cannot be two distinct nonzero pairs (a, b), (a′, b′) with πa,b = πa′,b′ = 1. As above, these pairs must be comparable; we may assume without loss of generality that a ≤ a′, b ≤ b′. The equations πa,b = 1 and πa′−a,b′−b = 1 force a + b ≥ t and (a′ − a) + (b′ − b) ≥ t, so a′ + b′ ≥ 2t. But a′, b′ ≤ t − 1, contradiction. If there is no nonzero pair (a, b) with 0 ≤ a, b ≤ t − 1 and πa,b = 1, then all πi,j are distinct, so p ≥ t2. Otherwise, as above, the pair (a, b) is unique, and the pairs (i, j) with 0 ≤ i, j ≤ t − 1 and (i, j) 6≥ (a, b) are pairwise distinct. The number of pairs excluded by the condition (i, j) 6≥ (a, b) is (t− a)(t− b); since a+ b ≥ t, (t− a)(t− b) ≤ t2/4. Hence p ≥ t2 − t2/4 = 3t2/4 as desired. While the necessary condition given by lemma 23 is quite far away from the sufficient condition given by lemma 8, it nonetheless suffices for checking that for most primes p, there do not exist three p-th roots of unity summing to zero. For instance, among the 664578 odd primes p ≤ 108, all but 550 are ruled out by Lemma 23. (There is an easy argument that t must be odd if p > 3; this cuts the list down to 273 primes.) Each remaining p can be tested by computing gcd(xp + 1, (x + 1)p + 1); the only examples we found that did not satisfy the condition of lemma 8 were (p, t) = (73, 9), (262657, 27), (599479, 33), (121369, 39). 5.4 Summary In the beginning of this section 5 we argued that in order to use the method of [34], (i.e., proposition 5) to obtain k-query locally decodable codes of length exp(nǫ) for some fixed k and all ǫ > 0, one needs to exhibit a k-nice sequence of subsets of finite fields. In what follows we use technical results of the previous subsections to show that the existence of a k-nice sequence implies that infinitely many Mersenne numbers have large prime factors. Theorem 24 Let k be odd. Suppose there exists a k-nice sequence of subsets of finite fields; then for infinitely many values of t we have P (2t − 1) ≥ (t/2)1+1/(k−2). (40) Proof: Using lemmas 21 and 22 we conclude that a k-nice sequence yields infinitely many primes p such that ord2(p) ≤ 2p1−1/(k−1). Let p be such a prime and t = ord2(p). Then P (2t − 1) ≥ (t/2)1+1/(k−2). A combination of lemmas 21 and 23 yields a slightly stronger bound for the special case of 3-nice sequences. Theorem 25 Suppose there exists a 3-nice sequence of subsets; then for infinitely many values of t we have P (2t − 1) ≥ (3/4)t2. (41) We would like to remind the reader that although the lower bounds for P (2t − 1) given by (40) and (41) are extremely weak light of the widely accepted conjecture saying that the number of Mersenne primes is infinite, they are substantially stronger than what is currently known unconditionally (2). 6 Conclusion Recently [34] came up with a novel technique for constructing locally decodable codes and obtained vast im- provements upon the earlier work. The construction proceeds in two steps. First [34] shows that if there exist subsets of finite fields with certain ’nice’ properties then there exist good codes. Next [34] constructs nice subsets of prime fields Fp for Mersenne primes p. In this paper we have undertaken an in-depth study of nice subsets of general finite fields. We have shown that constructing nice subsets is closely related to proving lower bounds on the size of largest prime factors of Mersenne numbers. Specifically we extended the constructions of [34] to obtain nice subsets of prime fields Fp for primes p that are large factors of Mersenne numbers. This implies that strong lower bounds for size of the largest prime factors of Mersenne numbers yield better locally decodable codes. Conversely, we argued that if one can obtain codes of subexponential length and constant query complexity through nice subsets of finite fields then infinitely many Mersenne numbers have prime factors larger than known currently. Acknowledgements Kiran Kedlaya’s research is supported by NSF CAREER grant DMS-0545904 and by the Sloan Research Fel- lowship. Sergey Yekhanin would like to thank Swastik Kopparty for providing the reference [5] and outlining the proof of lemma 9. He would also like to thank Henryk Iwaniec, Carl Pomerance and Peter Sarnak for their feedback regarding the number theory problems discussed in this paper. References [1] M. Agrawal, N. Kayal, N. Saxena, “PRIMES is in P,” Annals of Mathematics, vol. 160, pp. 781-793, 2004. [2] L. Babai, L. Fortnow, L. Levin, and M. Szegedy, “Checking computations in polylogarithmic time,”. In Proc. of the 23th ACM Symposium on Theory of Computing (STOC), pp. 21-31, 1991. [3] A. Beimel, Y. Ishai and E. Kushilevitz,“General constructions for information-theoretic private information retrieval,” Journal of Computer and System Sciences, vol. 71, pp. 213-247, 2005. Preliminary versions in STOC 1999 and ICALP 2001. [4] A. Beimel, Y. Ishai, E. Kushilevitz, and J. F. Raymond. “Breaking the O n1/(2k−1) barrier for information- theoretic private information retrieval,” In Proc. of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 261-270, 2002. [5] J. Bourgain, M. Chang, “A Gauss sum estimate in arbitrary finite fields,” Comptes Rendus Mathematique, vol. 342, pp. 643-646, 2006. [6] B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan. “Private information retrieval,” In Proc. of the 36rd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 41-50, 1995. Also, in Journal of the ACM, vol. 45, 1998. [7] D. Cox, J. Little, D. O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer, 1996. [8] A. Deshpande, R. Jain, T. Kavitha, S. Lokam and J. Radhakrishnan, “Better lower bounds for locally decod- able codes,” In Proc. of the 20th IEEE Computational Complexity Conference (CCC), pp. 184-193, 2002. [9] Curtis Cooper, Steven Boone, http://www.mersenne.org/32582657.htm [10] P. Erdos and T. Shorey, “On the greatest prime factor of 2p − 1 for a prime p and other expressions,” Acta. Arith. vol. 30, pp. 257-265, 1976. [11] W. Gasarch, “A survey on private information retrieval,” The Bulletin of the EATCS, vol. 82, pp. 72-107, 2004. [12] O. Goldreich, “Short locally testable codes and proofs,” Technical Report TR05-014, Electronic Colloquim on Computational Complexity (ECCC), 2005. [13] O. Goldreich, H. Karloff, L. Schulman, L. Trevisan “Lower bounds for locally decodable codes and private information retrieval,” In Proc. of the 17th IEEE Computational Complexity Conference (CCC), pp. 175-183, 2002. [14] B. Hemenway and R. Ostrovsky, “Public key encryption which is simultaneously a locally-decodable error- correcting code,” In Cryptology ePrint Archive, Report 2007/083. [15] J. Katz and L. Trevisan, “On the efficiency of local decoding procedures for error-correcting codes,” In Proc. of the 32th ACM Symposium on Theory of Computing (STOC), pp. 80-86, 2000. [16] I. Kerenidis, R. de Wolf, “Exponential lower bound for 2-query locally decodable codes via a quantum argument,” Journal of Computer and System Sciences, 69(3), pp. 395-420. Earlier version in STOC’03. quant-ph/0208062. [17] R. Lidl and H. Niederreiter, Finite Fields. Cambridge: Cambridge University Press, 1983. [18] L. Murata, C. Pomerance, “On the largest prime factor of a Mersenne number,” Number theory, CRM Proc. Lecture Notes of American Mathematical Society vol. 36, pp. 209-218, 2004. [19] M. Murty and S. Wong, “The ABC conjecture and prime divisors of the Lucas and Lehmer sequences,” In Proc. of Milennial Conference on Number Theory III, (Urbana, IL, 2000) (A. K. Peters, Natick, MA, 2002) pp. 43-54. [20] K. Obata, “Optimal lower bounds for 2-query locally decodable linear codes,” In Proc. of the 6th RANDOM, vol. 2483 of Lecture Notes in Computer Science, pp. 39-50, 2002. [21] A. Polishchuk and D. Spielman, ”Nearly-linear size holographic proofs,” In Proc. of the 26th ACM Sympo- sium on Theory of Computing (STOC), pp. 194-203, 1994. [22] C. Pomerance, “Recent developments in primality testing,” Math. Intelligencer, 3:3, pp. 97-105, (1980/81). [23] P. Raghavendra, “A Note on Yekhanin’s locally decodable codes,” In Electronic Colloquium on Computa- tional Complexity Report TR07-016, 2007. [24] A. Romashchenko, “Reliable computations based on locally decodable codes,” In Proc. of the 23rd Inter- national Symposium on Theoretical Aspects of Computer Science (STACS), vol. 3884 of Lecture Notes in Computer Science, pp. 537-548, 2006. [25] A. Schinzel, “On primitive factors of an − bn,” In Proc. of Cambridge Philos. Soc. vol. 58, pp. 555-562, 1962. [26] C. Stewart, “The greatest prime factor of an − bn,” Acta Arith. vol. 26, pp. 427-433, 1974/75. [27] C. Stewart, “On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers,” In Proc. of London Math. Soc. vol. 35 (3), pp. 425-447, 1977. [28] M. Sudan, Efficient checking of polynomials and proofs and the hardness of approximation problems. PhD thesis, University of California at Berkeley, 1992. [29] L. Trevisan, “Some applications of coding theory in computational complexity,” Quaderni di Matematica, vol. 13, pp. 347-424, 2004. [30] S. Wehner and R. de Wolf, “Improved lower bounds for locally decodable codes and private information re- trieval,” In Proc. of 32nd International Colloquium on Automata, Languages and Programming (ICALP’05), LNCS 3580, pp. 1424-1436. [31] Lenstra-Pomerance-Wagstaff conjecture. (2006, May 22). In Wikipedia, The Free Encyclopedia. Re- trieved 00:18, October 3, 2006, from http://en.wikipedia.org/w/index.php?title=Lenstra-Pomerance- Wagstaff conjecture&oldid=54506577 [32] S. Wagstaff, “Divisors of Mersenne numbers,” Math. Comp., 40:161, pp. 385-397, 1983. [33] D. Woodruff, “New lower bounds for general locally decodable codes,” Electronic Colloquium on Computa- tional Complexity, TR07-006, 2007. [34] S. Yekhanin, “Towards 3-query locally decodable codes of subexponential length,” In Proc. of the 39th ACM Symposium on Theory of Computing (STOC), 2007. [35] S. Yekhanin, Locally decodable codes and private information retrieval schemes. PhD thesis, MIT, to appear.

0704.1695

Pair production with neutrinos in an intense background magnetic field

Pair production with neutrinos in an intense background magnetic field Duane A. Dicus,1 Wayne W. Repko,2 and Todd M. Tinsley3 Department of Physics, University of Texas at Austin, Austin, Texas 78712 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824 Physics and Astronomy Department, Rice University, Houston, Texas 77005 (Dated: July 7, 2021) We present a detailed calculation of the electron-positron production rate using neutrinos in an intense background magnetic field. The computation is done for the process ν → νeē (where ν can be νe, νµ, or ντ ) within the framework of the Standard Model. Results are given for various combinations of Landau-levels over a range of possible incoming neutrino energies and magnetic field strengths. PACS numbers: 13.15.+g, 12.15.-y, 13.40.Ks I. INTRODUCTION Neutrino interactions are of great importance in astrophysics because of their capacity to serve as mediators for the transport and loss of energy. Their low mass and weak couplings make neutrinos ideal candidates for this role. Therefore, the rates of neutrino interactions are integral in the evolution of all stars, particularly the collapse and subsequent explosion of supernovae, where the overwhelming majority of gravitational energy lost is radiated away in the form of neutrinos. Neutrinos have held a prominent place in models of stellar collapse ever since Gamow and Schoenberg suggested their role in 1941 [1]. While supernova models have progressed a great deal in the last 65 years, the precise mechanism for explosion is still uncertain. A common feature, however, among all models is the sensitivity to neutrino transport. Neutrino processes once thought to be negligible now become relevant, and this has inspired many authors to calculate rates for neutrino interactions beyond that of the fundamental “Urca” processes p e → n νe n → p e ν̄e . Recent examples include neutrino-electron scattering, neutrino-nucleus inelastic scattering, and electron-positron pair annihilation [2, 3]. Furthermore, the large magnetic field strengths associated with supernovae (1012–1017 G) are likely to cause significant changes in the behavior of neutrino transport. While the the electromagnetic field does not couple to the Standard Model neutrino, it does affect neutrino physics by altering the behavior of any charged particles, real or virtual, with which the neutrino may interact. A number of authors have considered such effects on Urca-type processes [4, 5, 6, 7] and on neutrino absorption by nucleons (and its reversed processes) [8, 9, 10]. Furthermore, Bhattacharya and Pal have prepared a very nice review of other processes involving neutrinos that are affected by the presence of a magnetic field [11]. The problem of interest in this work is the production of electron-positron pairs with neutrinos in an intense magnetic field ν → ν e ē . (1) Normally this process is kinematically forbidden, but the presence of the magnetic field changes the energy balance of the process, thereby permitting the interaction. Stimulation of this process with high-intensity laser fields has been shown to have an unacceptably low rate of production [12], but such an interaction could have important consequences in astrophysical phenomena where large magnetic field strengths exist. The process would most likely serve to transfer energy in core-collapse supernovae [13]. However, Gvozdev et al. have proposed that its role in magnetars could even help to explain observed gamma-ray ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] http://arxiv.org/abs/0704.1695v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] (a)Neutral current reaction (b)Charged current reaction FIG. 1: Possible diagrams considered for the process ν → νeē. Both diagrams contribute for electron-type neutrinos, but only the neutral current reaction (FIG. 1(a)) contributes for νµ and ντ . bursts [14]. The interest in this reaction has led to a previous treatment in the literature [15], but those authors present results for two special limiting cases: (1) when the generalized magnetic field strength eB is greater than the square of the initial neutrino energy E2, and (2) when the square of the initial neutrino energy E2 is much greater than the generalized magnetic field strength eB. In both cases the incoming neutrino energy E is much greater than the electron’s rest energy me. In this paper we present a more complete calculation of the production rate as mediated by the neutral and the charged-current processes (FIG. 1). We present the results of the calculation for varying Landau levels, neutrino energies, and magnetic field strengths. A comparison with the approximate method is also discussed. II. FIELD OPERATOR SOLUTIONS As we have pointed out in section I, the standard model neutrino can only be affected by the electromagnetic field through its interactions with charged particles. This means that for the process ν → νeē the Dirac field solution for the final state electron and positron must change relative to their free-field solutions. The magnetic field will also change the form of theW -boson’s field solution which can mediate the process when electron neutrinos are considered. However, in our analysis we take the limit that the momentum transfer for this reaction is much less than the mass of the W -boson (Q2 ≪ m2W ) and ignore any effects the magnetic field may have on this charged boson. Thus, in this section we review the results of our derivation of the Dirac field operator solutions for the electron and positron. We closely follow the conventions used by Bhattacharya and Pal and refer the reader to their work [9] for a detailed derivation. The reader who is familiar with these solutions may wish to begin with section III where we calculate the production rate. We choose our magnetic field to lie along the positive z-axis ~B = B0k̂ (2) which allows us some freedom in the choice of vector potential A(x). We make the choice Aµ(x) = (0,−yB, 0, 0) (3) both for its simplicity and its agreement with the choice found in reference [9]. This choice in vector potential leads us to assume that all of the y space-time coordinate dependence is within the spinors. The absence of any y dependence in, for instance, the phase leads us to define a notation such that y−µ = (t, x, 0, z) (4) ~Vy− = (Vx, 0, Vz) , (5) where ~V is any 3-vector. A. Electron field operator Solving the Dirac equation for our choice of vector potential (Eq. 3) results in the following electron field operator ψe(x) = d2~py− (2π)2 En +me us(~py−, n, y) e −ip·y− âse ~py−,n + v s(~py−, n, y) e +ip·y− b̂ e ~py−,n , (6) where the creation and annihilation operators obey the following anti-commutation relations âse ~py−,n, â e ~p′ b̂se ~py−,n, b̂ e ~p′ = (2π)2 δss δnn′ δ 2(~py− − ~py−′) . (7) In Eq. 6 we sum over all possible spins s and all Landau levels n where En is the energy of fermion occupying the nth Landau level p2z +m e + 2neB , n ≥ 0 . (8) The Dirac bi-spinors are u+(~py−, n, y) = In−1(ξ−) En+me In−1(ξ−) En+me In(ξ−) , u−(~py−, n, y) = In(ξ−) En+me In−1(ξ−) En+me In(ξ−) , (9a) v+(~py−, n, y) = En+me In−1(ξ+)√ En+me In(ξ+) In−1(ξ+) , v−(~py−, n, y) = En+me In−1(ξ+) En+me In(ξ+) In(ξ+) . (9b) The Im(ξ) are functions of the Hermite polynomials Im(ξ) = 2/2Hm(ξ) (10) where the dimensionless parameter ξ is defined by eB y ± px√ . (11) Recall that the Hermite polynomials Hm(ξ) are only defined for nonnegative values of m. Therefore, we must define I−1(ξ) = 0. This means that the electron in the lowest Landau energy level (n = 0) cannot exist in spin-up state and the positron in the lowest Landau energy level cannot exist in the spin-down state. The normalization in Eq. (10) has been chosen such that the functions Im(ξ) obey the following delta-function representation [16, p. 86] δ(y − y′) = δ(ξ − ξ |∂y/∂ξ| eB δ(ξ − ξ′) 2n n! 2/2Hn(ξ) e −ξ′2/2Hn(ξ δ(y − y′) = In(ξ)In(ξ ′) . (12) For convenience we choose to normalize our 1-particle states in a “box” with dimensions LxLyLz = V such that the states are defined as |e〉 = |~p1y−, n1, s1〉 = e ~p1y−,n1 |0〉 (13a) |ē〉 = |~p2y−, n2, s2〉 = e ~p2y−,n2 |0〉 , (13b) and the completeness relation for the states is d2~py− (2π)2 LxLz |~py−, n, s〉 〈~py−, n, s| . (14) B. Spin sums In order to evaluate the production rate for our process, we must derive the completeness relations for summations over the spin of the fermions. For a detailed calculation of the rules see reference [10]. The results of the calculation are as follows s=+,− us(~py−, n, y ′)ūs(~py−, n, y) = (2(En +me)) me(1− σ3)+ 6p‖+ 6q‖γ5 −)In(ξ−) me(1 + σ 3)+ 6p‖− 6q‖γ5 In−1(ξ −)In−1(ξ−) γ1 + iγ2 In−1(ξ −)In(ξ−) γ1 − iγ2 −)In−1(ξ−) (15a) vs(~py−, n, y)v̄ s(~py−, n, y ′) = (2(En +me)) −me(1− σ3)+ 6p‖+ 6q‖γ5 In(ξ+)In(ξ −me(1 + σ3)+ 6p‖− 6q‖γ5 In−1(ξ+)In−1(ξ γ1 + iγ2 In−1(ξ+)In(ξ γ1 − iγ2 In(ξ+)In−1(ξ , (15b) where ‖ = (E, 0, 0, pz) (16) ‖ = (pz, 0, 0, E) . (17) The above results have been derived using the standard “Bjorken and Drell” representation for the γ-matrices [17] , γi = −σi 0 . (18) C. Neutrino field operator Having no charge, the neutrino’s field operator solution ψν(x) is not modified due to the magnetic field. We present it here for easy reference ψν(x) = (2π)3 us(p) e−ip·x âsν + v s(p) e+ip·x b̂s †ν , (19) where the creation and annihilation operators obey the conventional anticommutation relations âsν , â b̂sν , b̂ = (2π)3 δss δ3(~p− ~p ′) . (20) The neutrino bi-spinors follow the standard spin sum rules us(p)ūs(p) = vs(p)v̄s(p) = 6p , (21) where we take the Standard Model neutrino mass to be zero. With “box” normalization the 1-particle states for the neutrino are |ν〉 = |~p, s〉 = 1√ ν ~p |0〉 , (22) satisfying the completeness relation (2π)3 V |~p, s〉 〈~p, s| . (23) III. THE PRODUCTION RATE The quantity of interest for the process ν → νeē in a background magnetic field is the rate at which the electron- positron pairs are produced Γ. The production rate is defined as the probability per unit time for creation of pairs Γ = lim . (24) where T is the timescale on which the process is normalized. We begin by finding the probability P of our reaction n1,n2=0 d3~p′ (2π)3 d2~p1y− (2π)2 d2~p2y− (2π)2 s,s′,s1,s2 ~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2 ~p, s . (25) In Eq. 25 quantities with the index 1 correspond to the electron, those with index 2 to the positron, the primed quantities to the final neutrino, and the unprimed quantities correspond to the initial neutrino. A. The scattering matrix The scattering matrix ~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2 ~p, s naturally depends on the flavor of the neutrino. While the process involving the electron neutrino can advance through either the charged (W ) or neutral (Z) current, the muon (or tau) neutrino can only proceed through the latter. For this reason we will break the scattering matrix into a neutral component ~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2 ~p, s (27a) and a charged component ~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2 ~p, s , (27b) where the scattering operators are defined by the Standard Model Lagrangian as ŜZ = 23 cos2 θW sin d4xψe(x)γ geV − geAγ5 ψe(x)Zµ(x) d4x′ ψνl(x 1− γ5 ψνl(x ′)Zσ(x ′) (28a) ŜW = 23 sin2 θW d4xψe(x)γ 1− γ5 ψνe(x)W µ (x) d4x′ ψνe(x 1− γ5 ′)W+σ (x ′) , (28b) and θW is the weak-mixing angle, νl indicates a neutrino of any flavor, νe refers to a electron neutrino, and the vector and axial vector couplings for the electron are geV = − + 2 sin2 θW (29a) geA = − . (29b) In our analysis we will be using incoming neutrino energies that are well below the rest energies of the Z and W bosons. Therefore, we can safely make the 4-fermion effective coupling approximation to the Z and W propagators 〈0 |T (Zµ(x)Zσ(x′))| 0〉 → δ4(x− x′) (30a) W−µ (x)W → δ4(x− x′) . (30b) After making this approximation our expressions for the scattering operators simplify to ŜZ = d4xψe(x)γ geV − geAγ5 ψe(x)ψνl(x)γµ 1− γ5 ψνl(x) (31a) ŜW = d4xψe(x)γ 1− γ5 ψνe(x)ψνe(x)γµ 1− γ5 ψe(x) , (31b) where GF / 2 = e2/(8 sin2 θW m W ), and we have made use of the fact that cos 2 θW = m After substituting of the scattering operators (Eqs. (31)) into the expressions for the components of the scattering matrix (Eqs. (27)), we can use our results from sections IIA and IIC to write the components in the form of SZ/W = i(2π)3 δ3 py− − p′y− − py−,1 − py−,2 Lx Lz V MZ/W , (32) where (En1 +me)(En2 +me) EE′En1En2 (p′)γµ 1− γ5 us(p) dy ei(py−p y)y ūs1 (~p1y−, n1, y) γ geV − geAγ5 vs2 (~p2y−, n2, y) (33a) (En1 +me)(En2 +me) EE′En1En2 (p′)γµ 1− γ5 vs2 (~p2y−, n2, y) dy ei(py−p y)y ūs1 (~p1y−, n1, y) γ 1− γ5 us(p) . (33b) The reversal of sign on Eq. (33b) relative to Eq. (33a) is from the anticommutation of the field operators. The scattering amplitude for the charged component MW can be transformed into the form of the neutral component MZ by making use of a Fierz rearrangement formula ū1γµ 1− γ5 u2 ū3γ 1− γ5 u4 = −ū1γµ 1− γ5 u4 ū3γ 1− γ5 u2 , (34) such that (En1 +me)(En2 +me) EE′En1En2 (p′)γµ 1− γ5 us(p) dy ei(py−p y)y ūs1 (~p1y−, n1, y) γ 1− γ5 vs2 (~p2y−, n2, y) . (35) With the rearrangement of MW in Eq. (35), we can now express the scattering amplitude in terms of the type of incoming neutrino. The muon neutrino can only proceed through exchange of a Z-boson, so its scattering amplitude is just that of MZ Mνµ = MZ Mνµ = (En1 +me)(En2 +me) EE′En1En2 (p′)γµ 1− γ5 us(p) dy ei(py−p y)y ūs1 (~p1y−, n1, y) γ G−V − γ vs2 (~p2y−, n2, y) . (36) The scattering matrix for a tau neutrino, and the subsequent decay rate, is exactly the same as the muon neutrino. We will keep the notation as νµ for simplicity. The electron neutrino has both a Z-boson exchange component and an W -boson exchange component. Therefore we must add the amplitudes to find its scattering amplitude Mνe = MZ +MW Mνe = (En1 +me)(En2 +me) EE′En1En2 (p′)γµ 1− γ5 us(p) dy ei(py−p )y ūs1 (~p1y−, n1, y)γ G+V − γ vs2 (~p2y−, n2, y) . (37) Note that the scattering amplitudes for electron (Eq. 37) and non-electron neutrinos (Eq. 36) depend on a generalized vector coupling GV defined by G±V = 1± 4 sin 2 θW . (38) We see that the scattering amplitudes for an incoming electron neutrino versus an incoming muon neutrino differ only in the value of the generalized vector coupling and an overall sign. And the overall sign will be rendered meaningless once the amplitude is squared. Therefore, we choose to make no distinction between the two processes, other than keeping the generalized vector coupling as G±V , until we discuss the results in section IV. B. The form of the production rate Having determined the scattering matrix S and scattering amplitude M in section IIIA, we can now make series of substitutions of those results to find the expression for the production rate Γ. We begin by substituting the form of the scattering matrix (Eq. (32)) into the expression for the production rate (Eq. (24) Γ = lim = lim T,V→∞ n1,n2=0 d3~p′ (2π)3 d2~p1y− (2π)2 d2~p2y− (2π)2 s,s′,s1,s2 ~p′, s′; ~p1y−, n1, s1; ~p2y−, n2, s2 ~p, s = lim T,V→∞ n1,n2=0 d3~p′ (2π)3 d2~p1y− (2π)2 d2~p2y− (2π)2 s,s′,s1,s2 i(2π)3 δ3 py− − p′y− − py−,1 − py−,2 Lx Lz V Γ = lim T,V→∞ (2πTV )−1 n1,n2=0 d3~p′ d2~p1y− d2~p2y− δ3(py− − p′y− − py−,1 − py−,2) )2 |M|2 , (39) where |M|2 is the square of the scattering amplitude after summing over spins |M|2 = s,s′,s1,s2 |M|2 . (40) We can simplify the square of the 3-dimensional delta function by expressing one of the 3-dimensional delta functions as a series of integrals over space-time coordinates δ3(py− − p′y− − py−,1 − py−,2) = δ3(py− − p′y− − py−,1 − py−,2) (2π)3 ei(p−p ′−p1−p2)· y− . (41) By using the remaining set of delta functions to reduce the exponential to unity, we can write the integrand in terms of the dimensions of our normalization “box” δ3(py− − p′y− − py−,1 − py−,2) = δ3(py− − p′y− − py−,1 − py−,2) (2π)3 δ3(py− − p′y− − py−,1 − py−,2) = δ3(py− − p′y− − py−,1 − py−,2) TLxLz (2π)3 . (42) With the above result for the square of the delta function, the production rate in Eq. (39) simplifies to Γ = lim n1,n2=0 d3~p′ d2~p1y− d2~p2y− δ 3(py− − p′y− − py−,1 − py−,2) (2π)4Ly . (43) The square of the scattering amplitude goes as the product of two traces |M|2 = s,s′,s1,s2 (En1 +me)(En2 +me) EE′En1En2 ūs(p)γσ 1− γ5 (p′)ūs (p′)γµ 1− γ5 us(p) dy ei(py−p dy′ e−i(py−p s1,s2 v̄s2 (~p2y−, n2, y ′) γσ G±V − γ us1 (~p1y−, n1, y ′) ūs1 (~p1y−, n1, y)γ G±V − γ vs2 (~p2y−, n2, y) |M|2 = G (EE′En1En2) dy ei(py−p dy′ e−i(py−p )y′Tr 1− γ5 6p′γµ 1− γ5 G±V − γ me(1 − σ3)+ 6p1‖+ 6q1‖γ5 In1 (ξ −,1)In1(ξ−,1) 2n1eB γ1 + iγ2 In1−1(ξ −,1)In1 (ξ−,1) 2n1eB γ1 − iγ2 In1(ξ −,1)In1−1(ξ−,1) me(1 + σ 3)+ 6p1‖− 6q1‖γ5 In1−1(ξ −,1)In1−1(ξ−,1) G±V − γ −me(1− σ3)+ 6p2‖+ 6q2‖γ5 In2(ξ +,2)In2 (ξ+,2) 2n2eB γ1 + iγ2 In2−1(ξ +,2)In2 (ξ+,2) 2n2eB γ1 − iγ2 In2(ξ +,2)In2−1(ξ+,2) −me(1 + σ3)+ 6p2‖− 6q2‖γ5 In2−1(ξ +,2)In2−1(ξ+,2) where we have used our result for the summations over spin from Eqs. (15) and (21). The space-time dependence of Eq. (44) can be factored into terms like In,m = dy ei(py−p y)y In(ξ−,1) Im(ξ+,2) (45) I∗n,m = dy′ e−i(py−p )y′ In(ξ −,1) Im(ξ +,2) , (46) where the In,m are functions of the momenta in the problem. We have included a detailed calculation for the general form of In,m in appendix A, but we only present the result In,m = 2/2 eiφ0 (ηx + iηy) Lm−nn , m ≥ n ≥ 0 2/2 eiφ0 (−ηx + iηy)n−m Ln−mm , n ≥ m ≥ 0 where p1x + p2x√ py − p′y√ (py − p′y)(p1 − p2) η2 = η2x + η y , (51) and Lm−nn (η 2) are the associated Laguerre polynomials. The full results of the traces and their subsequent contraction are nontrivial but have been included in appendix B. It is important to note, however, that the only dependence on the x-components of the electron and positron momentum is that which appears in Eq. (47) for In,m. Furthermore, we notice that all terms in the averaged square of the scattering amplitude have factors that go as a product of In,m and I n′,m′ . Therefore, the coefficient e i φ0 in Eq. (47) will vanish when this product is taken. The only remaining x-dependence of these two momenta appear as their sum in the parameter ηx = (p1x + p2x)/ 2eB. This helps to simplify the phase-space integral for our production rate (Eq. (43)) which is proportional to Γ ∝ lim dp2x . (52) If we make a change of variable from the x-component of the positron momentum p2x to the parameter ηx, the relationship in Eq. (52) is rewritten as Γ ∝ lim dp1 x dηx . (53) Because there is no longer any explicit dependence on the x-component of the electron’s momentum p1,x in the averaged square of our scattering amplitude, we can simply evaluate the integral dp1,x . To evaluate this integral we must determine its limits. As discussed previously, we have elected to use “box” normal- ization on our states. This means that our particle is confined to a large box with dimensions Lx, Ly, and Lz. The careful reader will note that we have already taken the limit that these dimensions go to infinity in some places, particularly in Eq. (A5), but it is imperative that we be cautious here, as we could naively evaluate the integral over p1,x to be infinite. Physically, the charged particles in our final state act as harmonic oscillators circling about the magnetic field lines. While they are free to slide about the lines along the z-axis, the particles are confined to circular orbits in the x and y-directions no larger than the dimensions of the box. For a charged particle undergoing circular motion in a constant magnetic field, the x-component of momentum is related to the y-position vector by px = −eQBy (54) where Q is the charge of the particle in units of the proton charge e = |e|. Therefore, the limits on p1,x are proportional to the limits on the size of our box in the y-direction. The integral over the electron’s momentum in the x-direction is ∫ eBLy/2 −eBLy/2 dp1,x = eBLy , (55) and the result helps to cancel the factor of Ly that already appears in the form of the production rate. We can now safely take the limit that our box has infinite size, and the production rate now has the form n1,n2=0 d3~p′ d~p1z d~p2z 2eB δ3(py− − p′y− − py−,1 − py−,2) eB |M|2 (2π)4 . (56) IV. RESULTS In our expression for the total production rate (Eq. (56)), one will notice is that there is a sum over all possible values of the Landau levels. As a consequence of energy conservation, upper limits do exist for the summation over the electron’s Landau level n1 E = E′ + En1 + En2 E ≥ En1 +me E −me ≥ m2e + 2n1eB E(E − 2me) , (57) and a similar one for the positron’s Landau level E = E′ + En1 + En2 m2e + 2n1eB + En2 m2e + 2n1eB ≥ m2e + 2n2eB m2e + 2n1eB . (58) These relationships help to constrain the extent of the summations. Physically, these constraints can be thought of as limits on the size of the electron’s (or positron’s) effective mass, where the electron (or positron) occupying the nth Landau level has an effective mass m2e + 2neB (59) and energy p2z +m∗ 2 . (60) For low incoming neutrino energies and large magnetic field strengths (eB > m2e), the constraints put very tight bounds on the limits of the summations. However, higher incoming energies and low magnetic field strengths impose limits that still require a great deal of computation time. For instance, at threshold (E = 2me) there can exist only one possible configuration of Landau levels (n1 = n2 = 0), while at an energy ten times that of threshold and a magnetic field equal to the critical field (B = Bc = m e/e = 4.414× 1013 G) there are nearly 7000 possible states. At the same magnetic field but an energy that is 100 times that of threshold, there are almost 70 million states. However, for incoming neutrino energies less than a certain value E < me + m2e + 2eB (61) only the lowest Landau level is occupied, n1 , n2 = 0. And even at energies above, yet near, this value we expect that production of electrons and positrons in the n1 , n2 = 0 level is still the dominant mode of production because it has more phase space available. Production rates at the 0, 0 Landau level are presented in FIG. 2 for both the electron and muon neutrinos. (All of the results for muon-type neutrinos are valid for tau-type neutrinos.) One interesting feature of these results is the flattening out of the rates at higher energies. The energy region at which this flattening begins increases with increasing magnetic field strength, and it appears to be in the neighborhood of energies just above the limit set in Eq. (61). At energies in this regime we expect that modes of production into other Landau levels are stimulated, which helps to explain why the behavior of the 0, 0 production rates change above this area. We should note that the results given in this work are all for an incoming neutrino traveling transversely to the magnetic field. The rates are maximized in this case as can be seen in the example found in FIG. 3 for an initial electron neutrino with energy Eνe = 20me in a magnetic field equal to the critical field B = Bc = m For comparison purposes, the production rates for other combinations of Landau levels have been calculated. These include the 1, 0 and 0, 1 cases (FIG. 4), the 20, 0 and 0, 20 cases (FIG. 5), and the 10, 10 case (FIG. 6). The first noteworthy feature of these results is that the production rates are decreasing at higher Landau levels. Because the energy required to create the pair goes as Epair = En1 + En2 p21 z + 2n1eB +m p22 z + 2n2eB +m Epair ≥ 2n1eB +m2e + 2n2eB +m2e , B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νe → νeeē Eνe (MeV) 1000100101 10−10 10−15 10−20 10−25 10−30 (a)Incoming electron neutrino B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νµ → νµeē Eνµ (MeV) 1000100101 10−10 10−15 10−20 10−25 10−30 (b)Incoming muon (tau) neutrino FIG. 2: Production rates for the n1 , n2 = 0 Landau levels where Γ is the rate of production, Eν is the energy of the incoming neutrino, and the magnetic field is measured relative to the critical field Bc = 4.414× 10 13 G. All plots are for a neutrino that is perpendicularly incident to the magnetic field. the available phase space for the process should decrease in the order 0, 0 → 0, 1 → 0, 20 → 10, 10. And as can be seen in FIGS. 2, 4, 5, and 6, the production rates fall off accordingly. Another interesting feature of these results is the apparent preference for the creation of electrons in the highest of the two Landau levels. That is, the rate of production is larger for the state n1 = i , n2 = 0 than for n1 = 0 , n2 = i (FIGS. 4 and 5). This behavior is especially significant over the range of incoming neutrino energies near its threshold value for creating pairs in the given states. Though the i, 0 production rate is larger and increases more quickly in this “near-threshold” range than its 0, i counterpart, both curves plateau at higher energies, and their difference approaches zero. This difference is presumably caused by the positron having to share the W ’s energy with the final electron-type neutrino. This also explains why such an effect is not seen for muon and tau-type neutrinos that only proceed through the neutral current reaction. It was mentioned in section I that previous authors have considered this process under two limiting cases [15]. One νµ → νµeē νe → νeeē Directional Dependence sin2 θ 10.90.80.70.60.50.40.30.20.10 10−15 10−16 10−17 10−18 FIG. 3: The production rate’s dependance on the direction of the incoming neutrino.. The production rate is for the 0, 0 Landau level with an electron of energy Eν = 20me traveling at an angle θ relative to a magnetic field of strength equal to the critical field B = Bc. Data is included for both an incoming electron-type neutrino (solid line) and a muon-type neutrino (dashed line). If we average over θ, then the average production rate is 1.38 × 10−16 cm−1 for electron-type neutrinos or 2.94 × 10−17 cm−1 for muon or tau-type. is when the square of the energy of the initial-state neutrino and the magnetic field strength satisfy the conditions Eν ≫ eB ≫ m2e. Under these conditions many possible Landau levels could be stimulated, offering a multitude of production modes. Therefore, it would be inappropriate to compare their expression to our results for a specific set of Landau levels. However, the second limiting case is for eB > E2ν ≫ m2e. This condition is slightly more restrictive than our condition for the energies below which only the lowest energy Landau levels are occupied (Eq. (61)). In this regime our results for the 0, 0 state are the total production rates, and we can compare our results to the expression derived by the previous authors [15] eB E3ν E2/eB , (62) where we have taken the direction of the incoming neutrino to be perpendicular to the magnetic field’s direction. Results of this comparison are shown in FIG. 7. The results in FIG. 7 demonstrate the drawbacks of using the approximation in Eq. (62). While the expression is very simple, it gives only reasonable agreement with the production rate at a magnetic field equal to 100 times that of the critical field (B = 100Bc). Here it overestimates, at the very least, by a factor of two, and the inclusion of higher order corrections makes no significant improvement. One reason for the disagreement at this field strength is that there is only a very small range of energies that satisfy the condition eB > E2ν ≫ m2e. Therefore at higher field strengths we should get better agreement, and we do. Closer inspection of FIG. 7 reveals that the differences are less than a factor of three for neutrino energies in the range 2 MeV < Eν < 20 MeV, and the expression successfully provides a good order of magnitude estimation. Though the estimate will improve at higher magnetic field strengths, it begins to loose relevance as there are only a handful of known objects (namely magnetars) that can conceivably possess fields as high as 1015 G. Even for these objects, fields stronger than 1015 G cause instability in the star and the field begins to diminish [13]. Probing the limiting case Eν ≫ eB is imperative because our present work has already demonstrated nontrivial deviation from approximate methods for realistic astrophysical magnetic field strengths and neutrino energies near and below the value eB. But, as was mentioned previously, the number of Landau level states which contribute to the total production rate grows very rapidly in this higher energy regime, and we need to sum over these states. Future work will attempt to do these sums by using an approximation routine that can interpolate between rates for known sets of Landau levels. This will provide a flexible way to balance accuracy with computation time while determining when the production rate deviates from its limiting behavior. The significance of these deviations will only be known when a more complete understanding of the role that neutrino processes play in events such as supernova core-collapse B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νe → νeeē Eνe (MeV) 1000100101 10−10 10−15 10−20 10−25 10−30 (a)Incoming electron neutrino B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νµ → νµeē Eνµ (MeV) 1000100101 10−10 10−15 10−20 10−25 10−30 (b)Incoming muon neutrino FIG. 4: Production rates for the n1 = 0 , n2 = 1 (solid) and n1 = 1 , n2 = 0 (dashed) Landau levels where Γ is the rate of production, Eν is the energy of the incoming neutrino, and the magnetic field is measured relative to the critical field Bc = 4.414 × 10 13 G. and in the formation of the resulting neutron star. This work aims to improve that understanding. Acknowledgments It is our pleasure to thank Craig Wheeler for several discussions about supernovae and Palash Pal for helping us to understand Ref. [11]. This work was supported in part by the U.S. Department of Energy under Grant No. DE- B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νe → νeeē Eνe (MeV) 104103102101100 10−10 10−12 10−14 10−16 10−18 10−20 10−22 10−24 10−26 (a)Incoming electron neutrino B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νµ → νµeē Eνµ (MeV) 104103102101100 10−10 10−12 10−14 10−16 10−18 10−20 10−22 10−24 10−26 10−28 (b)Incoming muon neutrino FIG. 5: Production rates for the n1 = 0 , n2 = 20 (solid) and n1 = 20 , n2 = 0 (dashed) Landau levels where Γ is the rate of production, Eν is the energy of the incoming neutrino, and the magnetic field is measured relative to the critical field Bc = 4.414 × 10 13 G. F603-93ER40757 and by the National Science Foundation under Grant PHY-0244789 and PHY-0555544. [1] G. Gamow and M. Schoenberg, Physical Review 59, 539547 (1941). [2] S. W. Bruenn and W. C. Haxton, Astrophysical Journal 376, 678 (1991). [3] A. Mezzacappa and S. W. Bruenn, Astrophysical Journal 410, 740 (1993). [4] O. F. Dorofeev, V. N. Rodionov, and I. M. Ternov, JETP Lett. 40, 917 (1984). [5] D. A. Baiko and D. G. Yakovlev, Astron. Astrophys. 342, 192 (1999), astro-ph/9812071. [6] A. A. Gvozdev and I. S. Ognev, JETP Lett. 69, 365 (1999), astro-ph/9909154. [7] P. Arras and D. Lai, Phys. Rev. D60, 043001 (1999), astro-ph/9811371. B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νe → νeeē Eνe (MeV) 104103102101100 10−10 10−12 10−14 10−16 10−18 10−20 10−22 10−24 10−26 (a)Incoming electron neutrino B/Bc = 1000 B/Bc = 100 B/Bc = 10 B/Bc = 1 B/Bc = 0.1 νµ → νµeē Eνµ (MeV) 104103102101100 10−10 10−12 10−14 10−16 10−18 10−20 10−22 10−24 10−26 (b)Incoming muon neutrino FIG. 6: Production rates for the n1 , n2 = 10 Landau levels where Γ is the rate of production, Eν is the energy of the incoming neutrino, and the magnetic field is measured relative to the critical field Bc = 4.414 × 10 13 G. [8] H. Duan and Y.-Z. Qian, Phys. Rev. D72, 023005 (2005), astro-ph/0506033. [9] K. Bhattacharya and P. B. Pal, Pramana 62, 1041 (2004), hep-ph/0209053. [10] K. Bhattacharya, Ph.D. thesis, Jadavpu University (2004), hep-ph/0407099. [11] K. Bhattacharya and P. B. Pal, Proc. Ind. Natl. Sci. Acad. 70, 145 (2004), hep-ph/0212118. [12] T. M. Tinsley, Phys. Rev. D71, 073010 (2005), hep-ph/0412014. [13] C. Thompson and R. C. Duncan, Astrophysical Journal 408, 194 (1993). [14] A. A. Gvozdev, A. V. Kuznetsov, N. V. Mikheev, and L. A. Vassilevskaya, Phys. Atom. Nucl. 61, 1031 (1998), hep- ph/9710219. [15] A. V. Kuznetsov and N. V. Mikheev, Phys. Lett. B394, 123 (1997), hep-ph/9612312. [16] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, San Diego, CA, 1995), 4th ed. [17] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, International Series in Pure and Applied Physics (McGraw- Hill, Inc., New York, 1964). B/Bc = 1000 B/Bc = 100 νe → νeeē 103102101100 10−10 10−12 10−14 10−16 (a)Incoming electron neutrino B/Bc = 1000 B/Bc = 100 νµ → νµeē 103102101100 10−10 10−11 10−12 10−13 10−14 10−15 10−16 10−17 10−18 (b)Incoming muon neutrino FIG. 7: The total production rate (solid lines) and its approximation (dashed lines)[15] for energies and magnetic field satisfying the condition E2ν < eB. Eν is the energy of the incoming neutrino, me is the mass of the electron, and the magnetic field is measured relative to the critical field Bc = 4.414 × 10 13 G. APPENDIX A: CALCULATION OF In,m In section III B we discuss the fact that the squared scattering amplitude has coefficients that are integrals over the space-time coordinate y In,m = dy ei(py−p )y In(ξ−,1) Im(ξ+,2) . (A1) In this appendix we will derive the result after integrating over y. By defining new parameters eB y (A2) ζi = pix/ eB (A3) ζ0 = (py − p′y)/ eB (A4) and using the definition of ξ (Eq. (11)) we can make a change of variable from y to ζ and rewrite In,m as In,m = eiζ0ζ In(ζ − ζ1) Im(ζ + ζ2) (A5) where the limits of integration are ±∞ because we have taken the limit of Ly as it approaches ∞. The In(ξ) in Eq. (10) depend on the Hermite polynomials Hn(ξ), which can be represented as a contour integral in the following way [16, Eq. (13.8)] Hn(ξ) = dt t−n−1 e−t 2+2tx . (A6) Substituting this definition of the Hermite polynomial into Eq. (10) allows us to write the In,m as In,m = eiζ0 ζ 2n n! e−(ζ−ζ1) 2/2Hn(ζ − ζ1) e−(ζ+ζ2) 2/2Hm(ζ + ζ2) 2n+m n!m!π )−1/2 dζ eiζ0 ζ e−(ζ−ζ1) 2/2Hn(ζ − ζ1)e−(ζ+ζ2) 2/2Hm(ζ + ζ2) 2n+m n!m!π )−1/2 dζ eiζ0 ζ e−(ζ−ζ1) 2/2 e−(ζ+ζ2) dt t−n−1 e−t 2+2t(ζ−ζ1) m! ds s−m−1 e−s 2+2s(ζ+ζ2) . (A7) Next, we isolate all of the ζ dependence, interchange the order of the integrals, and perform the integration over ζ Int1 = dζ exp −ζ2 + ζ(ζ1 − ζ2 + iζ0 + 2t+ 2s) π exp (ζ1 − ζ2 + iζ0 + 2t+ 2s)2 /4 . (A8) Substitution of this result back into Eq. (A7) gives In,m = 2n+m n!m! )−1/2 e−((ζ1+ζ2) 0)/4 eiζ0(ζ1−ζ2)/2 dt t−n−1 et(−ζ1−ζ2+iζ0) ds s−m−1 es(ζ1+ζ2+iζ0) e2st . If m ≥ n, then we can perform the integration over s first Int2 = ds s−m−1 es(ζ1+ζ2+iζ0) e2st . es(ζ1+ζ2+iζ0+2t) = (ζ1 + ζ2 + iζ0 + 2t) m , (A10) such that In,m = 2n+m n!m! )−1/2 (ζ1 + ζ2) 2 + ζ20 exp (iζ0(ζ1 − ζ2)/2) (ζ1 + ζ2 + iζ0 + 2t) exp (t(−ζ1 − ζ2 + iζ0)) . (A11) The integration over t is made easier by making the following changes of variable ζ1 + ζ2√ p1x + p2x√ (A12) py − p′y√ (A13) ζ0 (ζ1 − ζ2) (py − p′y)(p1 − p2) (A14) η± = ηx ± iηy (A15) η2 = η+ η− = η2x + η y (A16) u− η+ 2 . (A17) The integration over the variable t can now be written as Int3 = 2 u)m (u− η+) / e((u−η = 2(n+m)/2 eη (u− η+)n+1 = 2(n+m)/2 eη um e−uη = 2(n+m)/2 eη (η−)n−m d(η2) 2)m e−η = n! 2(n+m)/2 (η+)(m−n) Lm−nn (η where we have used the Rodrigues’ representation for Laguerre polynomials [16, Eq. (13.47)] Lkn(x) = ex x−k xn+k e−x , n, k ≥ 0 . (A18) With the result from Eq. (A18), we can now express the In,m as In,m = 2/2 eiφ0 (ηx + i ηy) Lm−nn (η 2) , m ≥ n ≥ 0 , (A19) For the case when n > m we first integrate over t in Eq. (A9) and follow a similar procedure to find In,m = 2/2 eiφ0 (−ηx + i ηy)n−m Ln−mm (η2) , n ≥ m ≥ 0 . (A20) APPENDIX B: RESULT OF TRACE We can express the trace result for the average of the squared scattering amplitude from from Eq. (44) as a sum of terms |M|2 = G 29EE′En1En2 Ai Ti , (B1) where the coefficients Ai depend on the products of In,m and I n′,m′ defined in Eq. (47) and presented in appendix A, and the Ti are the parts that depend on the contraction of the traces in Eq. (44). The results are as follows: A1 = In1,n2I n1,n2 T1 = Tr G±V − γ m(1− σ3)+ 6p1‖+ 6q1‖γ5 G±V − γ −m(1− σ3)+ 6p2‖+ 6q2‖γ5 T1 = −27 2 − 1 x + pyp G±V + 1 (E − pz)(E′ − p′z)(En1 + p1 z)(En2 + p2 z) G±V − 1 (E + pz)(E ′ + p′z)(En1 − p1 z)(En2 − p2 z) (B3) A2 = In1,n2−1I n1,n2 T2 = Tr G±V − γ m(1− σ3)+ 6p1‖+ 6q1‖γ5 G±V − γ 2n2eB) γ1 + iγ2 T2 = −26 G±V + 1 2n2eB (px + i py)(E ′ − p′z)(En1 + p1 z) G±V − 1 2n2eB (p x + i p y)(E + pz)(En1 − p1 z) (B5) A3 = In1,n2I n1,n2−1 = A 2 (B6) T3 = T 2 (B7) A4 = In1,n2−1I n1,n2−1 (B8) T4 = Tr G±V − γ m(1− σ3)+ 6p1‖+ 6q1‖γ5 G±V − γ −m(1 + σ3)+ 6p2‖− 6q2‖γ5 T4 = 2 2 − 1 m2e(E + pz)(E ′ − p′z) + 26 G±V + 1 (E + pz)(E ′ − p′z)(En1 + p1 z)(En2 − p2 z) G±V − 1 (E + pz)(E ′ − p′z)(En1 − p1 z)(En2 + p2 z) (B9) A5 = In1,n2I n1−1,n2 (B10) T5 = Tr G±V − γ 2n1eB) γ1 + iγ2 G±V − γ −m(1− σ3)+ 6p2‖+ 6q2‖γ5 T5 = 2 G±V + 1 2n1eB (p x + i p y)(E − pz)(En2 + p2 z) G±V − 1 2n1eB (px + i py)(E ′ + p′z)(En2 − p2 z) (B11) A6 = In1,n2−1I n1−1,n2 (B12) T6 = Tr G±V − γ 2n1eB) γ1 + iγ2 G±V − γ 2n2eB) γ1 + iγ2 T6 = −26 G±V + 1 2n1eB 2n2eB (px + i py)(p x + i p G±V − 1 2n1eB 2n2eB (px + i py)(p x + i p y) (B13) A7 = In1,n2I n1−1,n2−1 (B14) T7 = Tr G±V − γ 2n1eB) γ1 + iγ2 G±V − γ 2n2eB) γ1 − iγ2 T7 = −26 G±V + 1 2n1eB 2n2eB (px − i py)(p′x + i p′y) G±V − 1 2n1eB 2n2eB (px + i py)(p x − i p′y) (B15) A8 = In1,n2−1I n1−1,n2−1 (B16) T8 = Tr G±V − γ 2n1eB) γ1 + iγ2 G±V − γ −m(1 + σ3)+ 6p2‖− 6q2‖γ5 T8 = 2 G±V + 1 2n1eB (p x + i p y)(E + pz)(En2 − p2 z) G±V − 1 2n1eB (px + i py)(E ′ + p′z)(En2 + p2 z) (B17) A9 = In1−1,n2I n1,n2 = A∗5 (B18) T9 = T 5 (B19) A10 = In1−1,n2−1I n1,n2 = A7 (B20) T10 = T 7 (B21) A11 = In1,n2I n1,n2−1 = A 6 (B22) T11 = T 6 (B23) A12 = In1−1,n2−1I n1,n2−1 = A 8 (B24) T12 = T 8 (B25) A13 = In1−1,n2I n1−1,n2 (B26) T13 = Tr G±V − γ m(1 + σ3)+ 6p1‖− 6q1‖γ5 G±V − γ −m(1− σ3)+ 6p2‖+ 6q2‖γ5 T13 = 2 2 − 1 m2e(E − pz)(E′ + p′z) + 26 G±V + 1 (E − pz)(E′ + p′z)(En1 − p1 z)(En2 + p2 z) G±V − 1 (E − pz)(E′ + p′z)(En1 + p1 z)(En2 − p2 z) (B27) A14 = In1−1,n2−1I n1−1,n2 (B28) T14 = Tr G±V − γ m(1 + σ3)+ 6p1‖− 6q1‖γ5 G±V − γ 2n2eB) γ1 + iγ2 T14 = −26 G±V + 1 2n2eB)(px + ipy)(E ′ + p′z)(En1 − p1 z) G±V − 1 2n2eB)(p x + ip y)(E − pz)(En1 + p1 z) (B29) A15 = In1−1,n2I n1−1,n2−1 = A 14 (B30) T15 = T 14 (B31) A16 = In1−1,n2−1I n1−1,n2−1 (B32) T16 = Tr G±V − γ m(1 + σ3)+ 6p1‖− 6q1‖γ5 G±V − γ −m(1 + σ3)+ 6p2‖− 6q2‖γ5 T16 = −27 2 − 1 m2e(pxp x + pyp y) + 2 G±V + 1 (E + pz)(E ′ + p′z)(En1 − p1 z)(En2 − p2 z) G±V − 1 (E − pz)(E′ − p′z)(En1 + p1 z)(En2 + p2 z) . (B33) Introduction Field operator solutions Electron field operator Spin sums Neutrino field operator The production rate The scattering matrix The form of the production rate Results Acknowledgments References Appendices Calculation of In,m Result of trace

0704.1696

Theoretical Aspects of the SOM Algorithm

Theoretical Aspects of the SOM Algorithm M.Cottrell†, J.C.Fort‡, G.Pagès∗ † SAMOS/Université Paris 1 90, rue de Tolbiac, F-75634 Paris Cedex 13, France Tel/Fax : 33-1-40-77-19-22, E-mail: [email protected] ‡ Institut Elie Cartan/Université Nancy 1 et SAMOS F-54506 Vandœuvre-Lès-Nancy Cedex, France E-mail: [email protected] ∗ Université Paris 12 et Laboratoire de Probabilités /Paris 6 F-75252 Paris Cedex 05, France E-mail:[email protected] Abstract The SOM algorithm is very astonishing. On the one hand, it is very simple to write down and to simulate, its practical properties are clear and easy to observe. But, on the other hand, its theoretical properties still remain without proof in the general case, despite the great efforts of several authors. In this paper, we pass in review the last results and provide some conjectures for the future work. Keywords: Self-organization, Kohonen algorithm, Convergence of stochas- tic processes, Vectorial quantization. 1 Introduction The now very popular SOM algorithm was originally devised by Teuvo Kohonen in 1982 [35] and [36]. It was presented as a model of the self-organization of neu- ral connections. What immediatly raised the interest of the scientific community (neurophysiologists, computer scientists, mathematicians, physicists) was the abil- ity of such a simple algorithm to produce organization, starting from possibly total disorder. That is called the self-organization property. As a matter of fact, the algorithm can be considered as a generalization of the Competitive Learning, that is a Vectorial Quantization Algorithm [42], without any notion of neighborhood between the units. http://arxiv.org/abs/0704.1696v1 In the SOM algorithm, a neighborhood structure is defined for the units and is respected throughout the learning process, which imposes the conservation of the neighborhood relations. So the weights are progressively updated according to the presentation of the inputs, in such a way that neighboring inputs are little by little mapped onto the same unit or neighboring units. There are two phases. As well in the practical applications as in the theoretical studies, one can observe self-organization first (with large neighborhood and large adaptation parameter), and later on convergence of the weights in order to quantify the input space. In this second phase, the adaptation parameter is decreased to 0, and the neighborhood is small or indeed reduced to one unit (the organization is supposed not to be deleted by the process in this phase, that is really true for the 0-neighbor setting). Even if the properties of the SOM algorithm can be easily reproduced by simu- lations, and despite all the efforts, the Kohonen algorithm is surprisingly resistant to a complete mathematical study. As far as we know, the only case where a com- plete analysis has been achieved is the one dimensional case (the input space has dimension 1) for a linear network (the units are disposed along a one-dimensional array). A sketch of the proof was provided in the Kohonen’s original papers [35], [36] in 1982 and in his books [37], [40] in 1984 and 1995. The first complete proof of both self-organization and convergence properties was established (for uniform distribution of the inputs and a simple step-neighborhood function) by Cottrell and Fort in 1987, [9]. Then, these results were generalized to a wide class of input distributions by Bouton and Pagès in 1993 and 1994, [6], [7] and to a more general neighborhood by Erwin et al. (1992) who have sketched the extension of the proof of self-organization [21] and studied the role of the neighborhood function [20]. Recently, Sadeghi [59], [60] has studied the self-organization for a general type of stimuli distribution and neighborhood function. At last, Fort and Pagès in 1993, [26], 1995 [27], 1997 [3], [4] (with Benaim) have achieved the rigorous proof of the almost sure convergence towards a unique state, after self-organization, for a very general class of neighborhood functions. Before that, Ritter et al. in 1986 and 1988, [52], [53] have thrown some light on the stationary state in any dimension, but they study only the final phase after the self-organization, and do not prove the existence of this stationary state. In multidimensional settings, it is not possible to define what could be a well ordered configuration set that would be stable for the algorithm and that could be an absorbing class. For example, the grid configurations that Lo et al. proposed in 1991 or 1993, [45], [46] are not stable as proved in [10]. Fort and Pagès in 1996, [28] show that there is no organized absorbing set, at least when the stimuli space is continuous. On the other hand, Erwin et al. in 1992 [21] have proved that it is impossible to associate a global decreasing potential function to the algorithm, as long as the probability distribution of the inputs is continuous. Recently, Fort and Pagès in 1994, [26], in 1996 [27] and [28], Flanagan in 1994 and 1996 [22], [23] gave some results in higher dimension, but these remain incomplete. In this paper, we try to present the state of the art. As a continuation of previous paper [13], we gather the more recent results that have been published in different journals that can be not easily get-a-able for the neural community. We do not speak about the variants of the algorithm that have been defined and studied by many authors, in order to improve the performances or to facilitate the mathematical analysis, see for example [5], [47], [58], [61]. We do not either address the numerous applications of the SOM algorithm. See for example the Kohonen’s book [40] to have an idea of the profusion of these applications. We will only mention as a conclusion some original data analysis methods based on the SOM algorithm. The paper is organized as follows: in section 2, we define the notations. The section 3 is devoted to the one dimensional case. Section 4 deals with the multidi- mensional 0-neighbor case, that is the simple competitive learning and gives some light on the quantization performances. In section 5, some partial results about the multidimensional setting are provided. Section 6 treats the discrete finite case and we present some data analysis methods derived from the SOM algorithm. The conclusion gives some hints about future researches. 2 Notations and definitions The network includes n units located in an ordered lattice (generally in a one- or two-dimensional array). If I = {1, 2, . . . , n} is the set of the indices, the neighbor- hood structure is provided by a neighborhood function Λ defined on I × I. It is symmetrical, non increasing, and depends only on the distance between i and j in the set of units I, (e.g. | i − j | if I = {1, 2, . . . , n} is one-dimensional). Λ(i, j) decreases with increasing distance between i and j, and Λ(i, i) is usually equal to 1. The input space Ω is a bounded convex subset of Rd, endowed with the Eu- clidean distance. The inputs x(t), t ≥ 1 are Ω-valued, independent with common distribution µ. The network state at time t is given by m(t) = (m1(t), m2(t), . . . , mn(t)). where mi(t) is the d-dimensional weight vector of the unit i. For a given state m and input x, the winning unit ic(x,m) is the unit whose weight mic(x,m) is the closest to the input x. Thus the network defines a map Φm : x 7−→ ic(x,m), from Ω to I, and the goal of the learning algorithm is to converge to a network state such the Φm map will be “topology preserving”in some sense. For a given state m, let us denote Ci(m) the set of the inputs such that i is the winning unit, that is Ci(m) = Φ m (i). The set of the classes Ci(m) is the Euclidean Voronöı tessellation of the space Ω related to m. The SOM algorithm is recursively defined by : ic(x(t+ 1), m(t)) = argmin {‖x(t + 1)−mi(t)‖, i ∈ I} mi(t + 1) = mi(t)− εtΛ(i0, i)(mi(t)− x(t + 1)), ∀i ∈ I The essential parameters are • the dimension d of the input space • the topology of the network • the adaptation gain parameter εt, which is ]0, 1[-valued, constant or decreasing with time, • the neighborhood function Λ, which can be constant or time dependent, • the probability distribution µ. Mathematical available techniques As mentioned before, when dealing with the SOM algorithm, one has to separate two kinds of results: those related to self-organization, and those related to conver- gence after organization. In any case, all the results have been obtained for a fixed time-invariant neighborhood function. First, the network state at time t is a random Ωn-valued vector m(t) displaying m(t + 1) = m(t)− εt H(x(t+ 1), m(t)) (2) (where H is defined in an obvious way according to the updating equation) is a stochastic process. If εt and Λ are time-invariant, it is an homogeneousMarkov chain and can be studied with the usual tools if possible (and fruitful). For example, if the algorithm converges in distribution, this limit distribution has to be an invariant measure for the Markov chain. If the algorithm has some fixed point, this point has to be an absorbing state of the chain. If it is possible to prove some strong organization [28], it has to be associated to an absorbing class. Another way to investigate self-organization and convergence is to study the asso- ciated ODE (Ordinary Differential Equation) [41] that describes the mean behaviour of the algorithm : = − h(m) (3) where h(m) = E(H(x, m)) = H(x, m) dµ(x) (4) is the expectation of H(., m) with respect to the probability measure µ. Then it is clear that all the possible limit states m⋆ are solutions of the functional equation h(m) = 0 and any knowledge about the possible attracting equilibrium points of the ODE can give some light about the self-organizing property and the convergence. But actually the complete asymptotic study of the ODE in the multidimensional setting seems to be untractable. One has to verify some global assumptions on the function h (and on its gradient) and the explicit calculations are quite difficult, and perhaps impossible. In the convergence phase, the techniques depend on the kind of the desired con- vergence mode. For the almost sure convergence, the parameter εt needs to decrease to 0, and the form of equation (2) suggests to consider the SOM algorithm as a Robbins-Monro [57] algorithm. The usual hypothesis on the adaptation parameter to get almost sure results is then: εt = +∞ and ε2t < +∞. (5) The less restrictive conditions t εt = +∞ and εt ց 0 generally do not ensure the almost sure convergence, but some weaker convergence, for instance the convergence in probability. Let us first examine the results in dimension 1. 3 The dimension 1 3.1 The self-organization The input space is [0, 1], the dimension d is 1 and the units are arranged on a linear array. The neighborhood function Λ is supposed to be non increasing as a function of the distance between units, the classical step neighborhood function satisfies this condition. The input distribution µ is continuous on [0, 1]: this means that it does not weight any point. This is satisfied for example by any distribution having a density. Let us define F+n = {m ∈ R / 0 < m1 < m2 < . . . < mn < 1} F−n = {m ∈ R / 0 < mn < mn−1 < . . . < m1 < 1}. In [9], [6], the following results are proved using Markovian methods : Theorem 1 (i) The two sets F+n and F n are absorbing sets. (ii) If ε is constant, and if Λ is decreasing as a function of the distance (e.g. if there are only two neigbors) the entering time τ , that is the hitting time of F+n ∪ F n , is almost surely finite, and ∃λ > 0, s.t. supm∈[0,1]n Em(exp(λτ)) is finite, where Em denote the expectation given m(0) = m. The theorem 1 ensures that the algorithm will almost surely order the weights. These results can be found for the more particular case (µ uniform and two neigh- bors) in Cottrell and Fort [9], 1987, and the succesive generalisations in Erwin et al. [21], 1992, Bouton and Pagès [6], 1993, Fort and Pagès [27], 1995, Flanagan [23], 1996. The techniques are the Markov chain tools. Actually following [6], it is possible to prove that whenever ε ց 0 and εt = +∞, then ∀m ∈ [0, 1]n,Probam(τ < +∞) > 0, (that is the probability of self-organization is positive regardless the initial values, but not a priori equal to 1). In [60], Sadeghi uses a generalized definition of the winner unit and shows that the probability of self-organization is uniformly positive, without assuming a lower bound for εt. No result of almost sure reordering with a vanishing εt is known so far. In [10], Cot- trell and Fort propose a still not proved conjecture: it seems that the re-organization occurs when the parameter εt has a order. 3.2 The convergence for dimension 1 After having proved that the process enters an ordered state set (increasing or decreasing), with probability 1, it is possible to study the convergence of the process. So we assume that m(0) ∈ F+n . It would be the same if m(0) ∈ F 3.2.1 Decreasing adaptation parameter In [9] (for the uniform distribution), in [7], [27] and more recently in [3], [4], 1997, the almost sure convergence is proved in a very general setting. The results are gathered in the theorem below : Theorem 2 Assume that 1) (εt) ∈]0, 1[ satisfies the condition (5), 2) the neighborhood function satisfies the condition HΛ: there exists k0 < that Λ(k0 + 1) < Λ(k0), 3) the input distribution µ satisfy the condition Hµ: it has a density f such that f > 0 on ]0, 1[ and ln(f) is strictly concave (or only concave, with lim0+ f + lim1− f positive), (i) The mean function h has a unique zero m⋆ in F+n . (ii) The dynamical system dm = −h(m) is cooperative on F+n , i.e. the non diagonal elements of ∇h(m) are non positive. (iii) m⋆ is attracting. So if m(0) ∈ F+n , m(t) −→ m⋆ almost surely. In this part, the authors use the ODE method, a result by M.Hirsch on cooperative dynamical system [34], and the Kushner & Clark Theorem [41], [3]. A.Sadeghi put in light that the non-positivity of non-diagonal terms of ∇h is exactly the basic definition of a cooperative dynamical system and he obtained partial results in [59] and more general ones in [60]. We can see that the assumptions are very general. Most of the usual probability distributions (truncated on [0, 1]) have a density f such that ln(f) is strictly concave. On the other hand, the uniform distribution is not strictly ln-concave as well as the truncated exponential distribution, but both cumply the condition lim0+ f +lim1− f positive. Condition (5) is essential, because if εt ց 0 and t εt = +∞, there is only a priori convergence in probability. In fact, by studying the associated ODE, Flanagan [22] shows that before ordering, it can appear metastable equilibria. In the uniform case, it is possible to calculate the limit m⋆. Its coordinates are solutions of a (n × n)-linear system which can be found in [37] or [9]. An explicit expression, up to the solution of a 3 × 3 linear system is proposed in [6]. Some further investigations are made in [31]. 3.2.2 Constant adaptation parameter Another point of view is to study the convergence of m(t) when εt = ε is a constant. Some results are available when the neighborhood function corresponds to the two- neighbors setting. See [9], 1987, (for the uniform distribution) and [7], 1994, for the more general case. One part of the results also hold for a more general neighborhood function, see [3], [4]. Theorem 3 Assume that m(0) ∈ F+n , Part A: Assume that the hypotheses Hµ and HΛ hold as in Theorem 2, then For each ε ∈]0, 1[, there exists some invariant probability νε on F+n . Part B: Assume only that Λ(i, j) = 1 if and only if |i − j| = 0 or 1 (classical 2-neighbors setting), (i) If the input distribution µ has an absolutely continuous part (e.g. has a density), then for each ε ∈]0, 1[, there exists a unique probability distribution νε such that the distribution of mt weakly converges to νε when t −→ ∞. The rate of convergence is geometric. Actually the Markov chain is Doeblin recurrent. (ii) Furthermore, if µ has a positive density, ∀ε, νε is equivalent to the Lebesgue measure on F+n if and only if n is congruent with 0 or 1 modulo 3. If n is congruent with 2 modulo 3, the Lebesgue measure is absolutely continuous with respect to νε , but the inverse is not true, that is νε has a singular part. Part C: With the general hypotheses of Part A (which includes that of Part B), if m⋆ is the unique globally attractive equilibrium of the ODE (see Theorem 2), thus νε converges to the Dirac distribution on m⋆ when ε ց 0 . So when ε is very small, the values will remain very close to m⋆. Moreover, from this result we may conjecture that for a suitable choice of εt, certainly εt = , where A is a constant, both self-organization and convergence towards the unique m⋆ can be achieved. This could be proved by techniques very similar to the simulated annealing methods. 4 The 0 neighbor case in a multidimensional set- In this case, we take any dimension d, the input space is Ω ⊂ Rd and Λ(i, j) = 1 if i = j, and 0 elsewhere. There is no more topology on I, and reordering no makes sense. In this case the algorithm is essentially a stochastic version of the Linde, Gray and Buzo [44] algorithm (LBG). It belongs to the family of the vectorial quantization algorithms and is equivalent to the Competitive Learning. The mathematical results are more or less reachable. Even if this algorithm is deeply different from the usual Kohonen algorithm, it is however interesting to study it because it can be viewed as a limit situation when the neighborhood size decreases to 0. The first result (which is classical for Competitive learning), and can be found in [54], [50], [39] is: Theorem 4 (i) The 0-neighbor algorithm derives from the potential Vn(m) = 1≤i≤n ‖mi − x‖ 2dµ(x) (6) (ii) If the distribution probability µ is continuous (for example µ has a density f), Vn(m) = Ci(m) ‖mi − x‖ 2f(x)dx = 1≤i≤n ‖mi − x‖ 2f(x)dx (7) where Ci(m) is the Voronöı set related with the unit i for the current state m. The potential function Vn(m) is nothing else than the intra-classes variance used by the statisticians to characterize the quality of a clustering. In the vectorial quan- tization setting, Vn(m) is called distortion. It is a measure of the loss of information when replacing each input by the closest weight vector (or code vector). The po- tential Vn(m) has been extensively studied since 50 years, as it can be seen in the Special Issue of IEEE Transactions on Information Theory (1982), [42]. The expression (7) holds as soon as mi 6= mj for all i 6= j and as the borders of the Voronöı classes have probability 0, (µ(∪ni=1∂Ci(m)) = 0). This last condition is always verified when the distribution µ has a density f . With these two conditions, V (m) is differentiable at m and its gradient vector reads ∇Vn(m) = Ci(m) (mi − x)f(m)d(m) So it becomes clear ([50],[40]) that the Kohonen algorithm with 0 neighbor is the stochastic gradient descent relative to the function Vn(m) and can be written : m(t + 1) = m(t)− εt+11Ci(m(t))(x(t+ 1))(m(t)− x(t + 1)) where 1Ci(m(t))(x(t + 1)) is equal to 1 if x(t+ 1) ∈ Ci(m(t)), and 0 if not. The available results are more or less classical, and can be found in [44] and [8], for a general dimension d and a distribution µ satisfying the previous conditions. Concerning the convergence results, we have the following when the dimension d = 1, see Pagès ([50], [51]), the Special Issue in IEEE [42] and also [43] for (ii): The parameter ε(t) has to satisfy the conditions (5). Theorem 5 Quantization in dimension 1 (i) If ∇Vn has finitely many zeros in F n , m(t) converges almost surely to one of these local minima. (ii) If the hypothesis Hµ holds (see Theorem (2)), Vn has only one zero point in F+n , say m n. This point m n ∈ F n and is a minimum. Furthermore if m(0) ∈ F −→ m⋆n. (iii) If the stimuli are uniformly distributed on [0, 1], then m⋆n = ((2i− 1)/2n)1≤i≤n. The part (ii) shows that the global minimum de Vn(m) is reachable in the one- dimensional case and the part (iii) is a confirmation of the fact that the algorithm provides an optimal discretization of continous distributions. A weaker result holds in the d-dimensional case, because one has only the conver- gence to a local minimum of Vn(m). Theorem 6 Quantization in dimension d If ∇Vn has finitely many zeros in F n , and if these zeros have all their components pairwise distinct, m(t) converges almost surely to one of these local minima. In the d-dimensional case, we are not able to compute the limit, even in the uniform case. Following [48] and many experimental results, it seems that the minimum distortion could be reached for an hexagonal tesselation, as mentioned in [31] or [40]. In both cases, we can set the properties of the global minima of Vn(m), in the general d-dimensional setting. Let us note first that Vn(m) is invariant under any permutation of the integers 1, 2, . . . , n. So we can consider one of the global minima, the ordered one (for example the lexicographically ordered one). Theorem 7 Quantization property (i) The function Vn(m) is continuous on (R d)n and reaches its (global) minima inside Ωn. (ii) For a fixed n, a point m⋆n at which the function Vn is minimum has pairwise distinct components. (iii) Let n be a variable and m⋆n = (m n,1, m n,2, . . . , m n,n) the ordered minimum of Vn(m). The sequence min(Rd)n Vn(m) = Vn(m n) converges to 0 as n goes to +∞. More precisely, there exists a speed β = 2/d and a constante A(f) such that nβVn(m n) −→ A(f) when n goes to +∞. Following Zador [64], the constant A(f) can be computed, A(f) = ad ‖ f ‖ρ, where ad does not depend on f , ρ = d/(d+ 2) and ‖ f ‖ρ= [ f ρ(x)dx]1/ρ. (iv) Then, the weighted empirical discrete probability measure µ(Ci(m n))δm⋆n,i converges in distribution to the probability measure µ, when n → ∞. (v) If Fn (resp. F ) denotes the distribution function of µn (resp. µ), one has (Rd)n Vn(m) = min (Rd)n (Fn(x)− F (x)) so when n → ∞, Fn converges to F in quadratic norm. The convergence in (iv) properly defines the quantization property, and explains how to reconstruct the input distribution from the n code vectors after convergence. But in fact this convergence holds for any sequence y⋆n = y1,n, y2,n, . . . , yn,n, which “fills ” the space when n goes to +∞: for example it is sufficient that for any n, there exists an integer n′ > n such that in any interval yi,n, yi+1,n (in R d), there are some points of y⋆n′. But for any sequence of quantizers satisfying this condition, even if there is convergence in distribution, even if the speed of the convergence can be the same, the constant A(f) will differ since it will not realize the minimum of the distortion. For each integer n, the solution m⋆n which minimizes the quadratic distortion Vn(m) and the quadratic norm ‖ Fn − F ‖ 2 is said to be an optimal n-quantizer . It ensures also that the discrete distribution function associated to the minimum m⋆n suitably weighted by the probability of the Voronöı classes, converges to the initial distribution function F . So the 0-neighbor algorithm provides a skeleton of the input distribution and as the distortion tends to 0 as well as the quadratic norm distance of Fn and F , it provides an optimal quantizer. The weighting of the Dirac functions by the volume of the Voronöı classes implies that the distribution µn is usually quite different from the empirical one, in which each term would have the same weight 1/n. This result has been used by Pagès in [50] and [51] to numerically compute inte- grals. He shows that the speed of convergence of the approximate integrals is exactly d for smooth enough functions, which is faster than the Monte Carlo method while d ≤ 4. The difficulty remains that the optimal quantizer m⋆n is not easily reachable, since the stochastic process m(t) converges only to a local minimum of the distortion, when the dimension is greater than 1. Magnification factor There is some confusion [37], [52], between the asymptotic distribution of an optimal quantizer m⋆n when n −→ ∞ and that one of the best random quantizer, as defined by Zador [64] in 1982. The Zador’s result, extended to the multi-dimensional case, is as follows : Let f be the input density of the measure µ, and (Y1, Y2, . . . , Yn) a random quantizer, where the code vectors Yi are independent with common distribution of density g. Then, with some weak assumptions about f and g, the distortion tends to 0 when n −→ ∞, with speed β = 2/d, and it is possible to define the quantity A(f, g) = lim nβEg[ ‖Yi − x‖ 2f(x)dx] Then for any given input density f , the density g (assuming some weak condition) which minimises A(f, g) is g⋆ ∼ C f d/d+2. The inverse of the exponent d/(d + 2) is refered as Magnification Factor. Note that in any case, when the data dimension is large, this exponent is near 1 (it value is 1/3 when d = 1). Note also that this power has no effect when the density f is uniform. But in fact the optimal quantizer is another thing, with another definition. Namely the optimal quantizerm⋆n (formed with the code vectorsm 1,n, m 2,n, . . . , m n,n), minimizes the distortion Vn(m), and is got after convergence of the 0-neighbor al- gorithm (if we could ensure the convergence to a global minimum, that is true only in the one-dimensional case). So if we set An(f,m n) = n βVn(m n) = n ‖m⋆i,n − x‖ 2f(x)dx actually we have, A(f) = lim An(f,m n) < A(f, g and the limit of the discrete distribution of m⋆n is not equal to g ⋆. So there is no magnification factor, for the 0-neighbor algorithm as claimed in many papers. It can be an approximation, but no more. The problem comes from the confusion between two distinct notions: random quantizer and optimal quantizer. And in fact, the good property is the convergence of the weighted distribution function (7). As to the SOM algorithm in the one-dimensional case, with a neighborhood func- tion not reduced to the 0-neighbor case, one can find in [55] or [19] some result about a possible limit of the discrete distribution when the number of units goes to ∞. But actually, the authors use the Zador’s result which is not appropriate as we just see. 5 The multidimensional continuous setting In this section, we consider a general neighborhood function and the SOM algorithm is defined as in Section 2. 5.1 Self-organization When the dimension d is greater than 1, little is known on the classical Kohonen algorithm. The main reason seems to be the fact that it is difficult to define what can be an organized state and that no absorbing sets have been found. The configurations whose coordinates are monotoneous are not stable, contrary to the intuition. For each configuration set which have been claimed to be left stable by the Kohonen algorithm, it has been proved later that it was possible to go out with a positive probability. See for example [10]. Most people think that the Kohonen algorithm in dimension greater than 1 could correspond to an irreducible Markov chain, that is a chain for which there exists always a path with positive probability to go from anywhere to everywhere. That property imply that there is no absorbing set at all. Actually, as soon as d ≥ 2, for a constant parameter ε, the 0-neighbor algorithm is an Doeblin recurrent irreducible chain (see [7]), that cannot have any absorbing class. Recently, two apparently contradictory results were established, that can be col- lected together as follows. Theorem 8 (d = 2 and ε is a constant) Let us consider a n × n units square network and the set F++ of states whose both coordinates are separately increasing as function of their indices, i.e. F++ = ∀i1 ≤ n,m < m2i1,2 < . . . < m , ∀i2 ≤ n,m < m12,i2 < . . . < m (i) If µ has a density on Ω, and if the neighborhood function Λ is everywhere posi- tive and decreases with the distance, the hitting time of F++ is finite with positive probability (i.e. > 0, but possibly less than 1). See Flanagan ([22], [23]). (ii) In the 8-neighbor setting, the exit time from F++ is finite with positive proba- bility. See Fort and Pagès in ([28]). This means that (with a constant, even very small, parameter ε), the organi- zation is temporarily reached and that even if we guess that it is almost stable, dis-organization may occur with positive probability. More generally, the question is how to define an organized state. Many authors have proposed definitions and measures of the self-organization, [65], [18], [62], [32], [63], [33]. But none such “organized” sets have a chance to be absorbing. In [28], the authors propose to consider that a map is organized if and only if the Voronöı classes of the closest neighboring units are contacting. They also precisely define the nature of the organization (strong or weak). They propose the following definitions : Definition 1 Strong organization There is strong organization if there exists a set of organized states S such that (i) S is an absorbing class of the Markov chain m(t), (ii) The entering time in S is almost surely finite, starting from any random weight vectors (see [6]). Definition 2 Weak organization There is weak organization if there exists a set of organized states S such that all the possible attracting equilibrium points of the ODE defined in 3 belong to the set The authors prove that there is no strong organization at least in two seminal cases: the input space is [0, 1]2, the network is one-dimensional with two neighbors or two-dimensional with eight neighbors. The existence of weak organization should be investigated as well, but until now no exact result is available even if the simulations show a stable organized limit behavior of the SOM algorithm. 5.2 Convergence In [27], (see also [26]) the gradient of h is computed in the d-dimensional setting (when it exists). In [53], the convergence and the nature of the limit state is studied, assuming that the organization has occured, although there is no mathematical proof of the convergence. Another interesting result received a mathematical proof thanks to the computa- tion of the gradient of h: it is the dimension selection effect discovered by Ritter and Schulten (see [53]). The mathematical result is (see [27]: Theorem 9 Assume thatm⋆1 is a stable equilibrium point of a general d1-dimensional Kohonen algorithm, with n1 units, stimuli distribution µ1 and some neighborhood function Λ. Let µ2 be a d2-dimensional distribution with mean m 2 and covariance matrix Σ2. Consider the d1 + d2 Kohonen algorithm with the same units and the same neighborhood function. The stimuli distribution is now µ1 Then there exists some η > 0, such that if ‖Σ2‖ < η, the state m 1 in the subspace m2 = m 2 is still a stable equilibrium point for the d1 + d2 algorithm. It means that if the stimuli distribution is close to a d1-dimensional distribution in the d1 + d2 space, the algorithm can find a d1-space stable equilibrium point. That is the dimension selection effect. From the computation of the gradient ∇h, some partial results on the stability of grid equilibriums can also be proved: Let us consider I = I1×I2×. . .×Id a d-dimensional array, with Il = {1, 2, . . . , nl}, for 1 ≤ l ≤ d. Let us assume that the neighborhood function is a product function (for example 8 neighbors for d = 2) and that the input distributions in each coordi- nate are independent, that is µ = µ1 . . . µd. At last suppose that the support of each µl is [0,1]. Let us call grid states the states m⋆ = (m⋆ill, 1 ≤ il ≤ nl, 1 ≤ l ≤ d), such that for every 1 ≤ l ≤ d, (m⋆ill, 1 ≤ il ≤ nl) is an equilibrium for the one-dimensional algorithm. Then the following results hold [27] : Theorem 10 (i) The grid states are equilibrium points of the ODE (3) in the d- dimensional case. (ii) For d = 2, if µ1 and µ2 have strictly positive densities f1 and f2 on [0, 1], if the neighborhood functions are strictly decreasing, the grid equilibrium points are not stable as soon as n1 is large enough and the ratio is large (or small) enough (i.e. when n1 −→ +∞ and −→ +∞ or 0, see [27], Section 4.3). (iii) For d = 2, if µ1 and µ2 have strictly positive densities f1 and f2 on [0, 1], if the neighborhood functions are degenerated (0 neighbor case), m⋆ is stable if n1 and n2 are less or equal to 2, is not stable in any other case (may be excepted when n1 = n2 = 3). The (ii) gives a negative property for the non square grid which can be related with this one: the product of one-dimensional quantizers is not the correct vectorial quantization. But also notice that we have no result about the simplest case: the square grid equilibrium in the uniformly distributed case. Everybody can observe by simulation that this square grid is stable (and probably the unique stable “organized” state). Nevertheless, even if we can numerically verify that it is stable, using the gradient formula it is not mathematically proved even with two neighbors in each dimension! Moreover, if the distribution µ1 and µ2 are not uniform, generally the square grids are not stables, as it can be seen experimentally. 6 The discrete case In this case, there is a finite number N of inputs and Ω = {x1, x2, . . . , xN}. The input distribution is uniform on Ω that is µ(dx) = 1 l=1 δxl. It is the setting of many practical applications, like Classification or Data Analysis. 6.1 The results The main result ([39], [56]) is that for not time-dependent general neighborhood, the algorithm locally derives from the potential Vn(m) = xl∈Ci(m) Λ(i− j)‖mj − xl‖ Ci(m) Λ(i− j)‖mj − x‖ 2)µ(dx) i,j=1 Λ(i− j) Ci(m) ‖mj − x‖ 2µ(dx). When Λ(i, j) = 1 if i and j are neighbors, and if V(j) denotes the neighborhood of unit i in I, Vn(m) also reads Vn(m) = ∪i∈V(j)Ci(m) ‖mj − x‖ 2µ(dx). Vn(m) is an intra-class variance extended to the neighbor classes which is a gen- eralization of the distortion defined in Section 4 for the 0-neighbor setting. But this potential does have many singularities and its complete analysis is not achieved, even if the discrete algorithm can be viewed as a stochastic gradient descent proce- dure. In fact, there is a problem with the borders of the Voronöı classes. The set of all these borders along the process m(t) trajectories has measure 0, but it is difficult to assume that the given points xl never belong to this set. Actually the potential is the true measure of the self-organization. It measures both clustering quality and proximity between classes. Its study should provide some light on the Kohonen algorithm even in the continuous case. When the stimuli distribution is continuous, we know that the algorithm is not a gradient descent [21]. However the algorithm can be seen then as an approximation of the stochastic gradient algorithm derived from the function Vn(m). Namely, the gradient of Vn(m) has a non singular part which corresponds to the Kohonen algorithm and a singular one which prevents the algorithm to be a gradient descent. This remark is the base of many applications of the SOM algorithm as well in combinatorial optimization, data analysis, classification, analysis of the relations between qualitative classifying variables. 6.2 The applications For example, in [24], Fort uses the SOM algorithm with a close one-dimensional string, in a two dimensional space where are located M cities. He gets very quickly a very good sub-optimal solution. See also the paper [1]. The applications in data analysis and classification are more classical. The prin- ciple is very simple: after convergence, the SOM algorithm provides a two(or one)- dimensional organized classification which permit a low dimensional representation of the data. See in [40] an impressive list of examples. In [15] and [17], an application to forecasting is presented from a previous classi- fication by a SOM algorithm. 6.3 Analysis of qualitative variables Let us define here two original algorithms to analyse the relations between qualitative variables. The first one is defined only for two qualitative variables. It is called KORRESP and is analogous to the simple classical Correspondence Analysis. The second one is devoted to the analysis of any finite number of qualitative variables. It is called KACM and is similar to the Multiple Correspondence Analysis. See [11], [14], [16] for some applications. For both algorithms, we consider a sample of individuals and a number K of questions. Each question k, k = 1, 2, . . . , K has mk possible answers (or modalities). Each individual answers each question by choosing one and only one modality. If 1≤k≤mk is the total number of modalities, each individual is represented by a row M-vector with values in 0, 1. There is only one 1 between the 1st component and the m1-th one, only one 1 between the m1+1-th component and the m1+m2-th one and so on. In the general case whereM > 2, the data are summarized into a Burt Table which is a cross tabulation table. It is a M × M symmetric matrix and is composed of K×K blocks, such that the (k, l)-block Bkl (for k 6= l) is the (mk×ml) contingency table which crosses the question k and the question l. The block Bkk is a diagonal matrix, whose diagonal entries are the numbers of individuals who have respectively chosen the modalities 1, 2, . . . , mk for question k. In the following, the Burt Table is denoted by B. In the case M = 2, we only need the contingency table T which crosses the two variables. In that case, we set p (resp. q) for m1 (resp. m2). The KORRESP algorithm In the contingency table T , the first qualitative variable has p levels and corre- sponds with the rows. The second one has q levels and corresponds with the columns. The entry nij is the number of individuals categorized by the row i and the column j. From the contingency table, the matrix of relative frequencies (fij = nij/( ij nij)) is computed. Then the rows and the columns are normalized in order to have a sum equal to 1. The row profile r(i), 1 ≤ i ≤ p is the discrete probability distribution of the second variable given that the first variable has modality i and the column profile c(j), 1 ≤ j ≤ q is the discrete probability distribution of the first variable given that the second variable has modality j. The classical Correspondence Analysis is a simultaneous weighted Principal Component Analysis on the row profiles and on the column profiles. The distance is chosen to be the χ2 distance. In the simultaneous representation, related modalities are projected into neighboring points. To define the algorithm KORRESP, we build a new data matrix D : to each row profile r(i), we associate the column profile c(j(i)) which maximizes the probability of j given i, and conversely, we associate to each column profile c(j) the row profile r(i(j)) the most probable given j. The data matrix D is the ((p + q) × (q + p))- matrix whose first p rows are the vectors (r(i), c(j(i))) and last q rows are the vectors (r(i(j)), c(j)). The SOM algorithm is processed on the rows of this data matrix D. Note that we use the χ2 distance to look for the winning unit and that we alterna- tively pick at random the inputs among the p first rows and the q last ones. After convergence, each modality of both variables is classified into a Voronöı class. Re- lated modalities are classified into the same class or into neighboring classes. This method give a very quick, efficient way to analyse the relations between two quali- tative variables. See [11] and [12] for real-world applications. The KACM Algorithm When there are more than two qualitative variables, the above method does not work any more. In that case, the data matrix is just the Burt Table B. The rows are normalized, in order to have a sum equal to 1. At each step, we pick a normalized row at random according to the frequency of the corresponding modality. We define the winning unit according to the χ2 distance and update the weights vectors as usual. After convergence, we get an organized classification of all the modalities, where related modalities belong to the same class or to neighboring classes. In that case also, the KACM method provides a very interesting alternative to classical Multiple Correspondence Analysis. The main advantages of both KORRESP and KACM methods are their rapidity and their small computing time. While the classical methods have to use several representations with decreasing information in each, ours provide only one map, that is rough but unique and permit a rapid and complete interpretation. See [14] and [16] for the details and financial applications. 7 Conclusion So far, the theoretical study in the one-dimensional case is nearly complete. It remains to find the convenient decreasing rate to ensure the ordering. For the multidimensional setting, the problem is difficult. It seems that the Markov chain is irreducible and that further results could come from the careful study of the Ordinary Differential Equation (ODE) and from the powerful existing results about the cooperative dynamical systems. 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Introduction Notations and definitions The dimension 1 The self-organization The convergence for dimension 1 Decreasing adaptation parameter Constant adaptation parameter The 0 neighbor case in a multidimensional setting The multidimensional continuous setting Self-organization Convergence The discrete case The results The applications Analysis of qualitative variables Conclusion

0704.1697

Effect of the Spatial Dispersion on the Shape of a Light Pulse in a Quantum Well

Effect of the Spatial Dispersion on the Shape of a Light Pulse in a Quantum Well L. I. Korovin, I. G. Lang A. F. Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia S. T. Pavlov†‡ †Facultad de Fisica de la UAZ, Apartado Postal C-580, 98060 Zacatecas, Zac., Mexico; and ‡P.N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia; [email protected] Reflectance, transmittance and absorbance of a symmetric light pulse, the carrying frequency of which is close to the frequency of interband transitions in a quantum well, are calculated. Energy levels of the quantum well are assumed discrete, and two closely located excited levels are taken into account. A wide quantum well (the width of which is comparable to the length of the light wave, corresponding to the pulse carrying frequency) is considered, and the dependance of the interband matrix element of the momentum operator on the light wave vector is taken into account. Refractive indices of barriers and quantum well are assumed equal each other. The problem is solved for an arbitrary ratio of radiative and nonradiative lifetimes of electronic excitations. It is shown that the spatial dispersion essentially affects the shapes of reflected and transmitted pulses. The largest changes occur when the radiative broadening is close to the difference of frequencies of interband transitions taken into account. PACS numbers: 78.47. + p, 78.66.-w Irradiation of the low-dimensional semiconductor sys- tems by light pulses and analysis of reflected and trans- mitted pulses allow to obtain the information regarding the structure of energy levels as well as relaxation pro- cesses. The radiative mechanism of relaxation of excited en- ergy levels in quantum wells arises due to a violation of the translation symmetry perpendicular to the he quan- tum well plane1,2. At low temperatures, low impurity doping and perfect boundaries of quantum wells, the contributions of the radiative and nonradiative relax- ation can be comparable. In such situation, one can- not be limited by the linear approximation on the elec- tron - light interaction. All the orders of the interac- tion have to be taken into account3,4,5,6,7,8,9. Alterations of asymmetrical10,11,12,13 and symmetrical13,14,15 light pulses are valid for narrow quantum wells under condi- tions k d ≪ 1 (d is the quantum well width, k is the mag- nitude of the light wave vector corresponding to the car- rying frequency of the light pulse) and an independence of optical characteristics of a quantum well on d. How- ever, a situation is possible when the size quantization is preserved and for wide quantum wells if k d ≥ 1 (see corresponding estimates in16). In such a case, we have to take into account the spatial dispersion of a monochro- matic wave 9,19 and waves composing the light pulse16. Our investigation is devoted to the influence of the spa- tial dispersion on the optical characteristics (reflectance, transmittance and absorbance) of a quantum well irradi- ated by the symmetric light pulse. A system, consisting of a deep quantum well of type I, situated inside of the space interval 0 ≤ z ≤ d, and two semi-infinite barriers, is considered. A constant quantizing magnetic field is directed perpendicular to the quantum well plane what provides the discrete energy levels of the electron sys- tem. A stimulating light pulse propagates along the z axis from the side of negative values z. The barriers are transparent for the light pulse which is absorbed in the quantum well to initiate the direct interband transitions. The intrinsic semiconductor and zero temperature are assumed. The final results for two closely spaced energy levels of the electronic system in a quantum well are obtained. Effect of other levels on the optical characteristics may be neglected, if the carrying frequency ω ℓ of the light pulse is close to the frequencies ω 1 and ω 2 of the doublet levels, and other energy levels are fairly distant. It is assumed that the doublet is situated near the minimum of the conduction band, the energy levels may be considered in the effective mass approximation, and the barriers are infinitely high. In the case h̄ K⊥ = 0 (h̄ K⊥ is the vector of the quasi- momentum of electron-hole pair in the quantum well plane) in a quantum well, the discrete energy levels are the excitonic energy levels in a zero magnetic field or energy levels in a quantizing magnetic field directed per- pendicularly to the quantum well plane. As an example, the energy level of the electron-hole pair in a quantizing magnetic field directed along the z axis (without taking into account the Coulomb interaction between the elec- tron and hole which is a weak perturbation for the strong magnetic fields and not too wide quantum wells) is con- sidered. I. THE ELECTRIC FIELD Let us consider a situation when a symmetric excit- ing light pulse propagates through a single quantum well along the z axis from the side of negative values of z. Analogously to13,14,15, the electric field is chosen as E 0 (z, t) = e ℓE 0 e −iω ℓ p http://arxiv.org/abs/0704.1697v1 Θ(p)e−γ ℓ p/2 + [1−Θ(p)]eγ ℓ p/2 + c.c., (1) where E 0 is the real amplitude, p = t− ν z/ c, e ℓ = (ex ± i e y) are the unite vectors of the circular polarization, ex e y are the real unite vectors, Θ(p) is the Heaviside function, 1/γ ℓ determines the pulse width, c is the light velocity in vacuum, ν is the refraction index, which is assumed the same for the quantum well and barriers (the approxima- tion of a homogeneous media). The Fourier-transform of (1) is as follows E 0(z, ω) = e ikz [e ℓ E 0(ω) + e ℓ E 0(−ω)] , E 0(ω) = E 0 γ ℓ (ω − ω ℓ)2 + (γ ℓ/2)2 , k = . (2) The electric field in the region z ≤ 0 consists of the sum of the exciting and reflected pulses. The Fourier- transform may be written as ℓ (z, ω) = E 0 (z, ω) + ∆E ℓ (z, ω), where ∆E ℓ(z, ω) is the electric field of the reflected pulse ∆E ℓ(z, ω) = e ℓ ∆E ℓ(z, ω) + e ∗ℓ ∆E ℓ(z,−ω). (3) In the region z ≥ d, there is only the transmitted pulse, and its electric field is r(z, ω) = e ℓ E r(z, ω) + e ∗ℓ E r(z,−ω). (4) It is assumed below that the pulse, having absorbed in the quantum well, stimulates the interband transitions and, consequently, the appearance of a current. In barri- ers, the absorption is absent. Therefore, for the complex amplitudes ∆E ℓ(z, ω) and E r(z, ω) in barriers for z ≤ 0 and z ≥ d, we obtain the expression d z 2 + k 2 E = 0. (5) The expression for the electric field inside of the quantum well (0 ≤ z ≤ d) has a form d z 2 + k 2 E = −4πi ω J(z, ω), (6) where J(z, ω) is the Fourier-transform of the current den- sity, averaged on the ground state of the system. The current is induced by the monochromatic wave of the frequency ω. In the case of two excited energy levels, J(z, ω) is expressed as follows J(z, ω) = γ rj Φ j(z) dz′ Φ j(z ′)E(z ′, ω), ω̃ j = ω − ω j + iγ j/2. (8) where γ j is the nonradiative damping of the doublet, γ r j is the radiative damping of the levels of the doublet in the case of narrow quantum wells, when the spatial dispersion of electromagnetic waves may be neglected. In particular, the doublet system may be represented by a magnetopolaron state18. In such a case, γ r,j = γ r Q j , γ r = h̄ cν p 2c v h̄ ω g m 0 m 0 c where m 0 is the free electron mass, H is the magnetic field, e is the electron charge, p c v is the matrix element of the momentum, corresponding to the circular polar- ization, p2c v = |p c v x |2 + |p c v y |2. The factor Q j = ± h̄ (Ω c − ωLO) h̄2(Ω c − ωLO)2 + (∆E pol)2 determines the change of the radiative timelife at a deflec- tion of the magnetic field from the resonant value when the resonant condition Ω c = ωLO is carried out. ∆E pol is the polaron splitting, Ω c and ωLO are the cyclotron frequency and optical phonon frequency, respectively. In the resonance, Q j = 1/2 and γ r1 = γ r2. When calculating J(z, ω), it was assumed that the Lorentz force, determined by the external magnetic field, is large in comparison with the Coulomb and exchange forces in the electron-hole pair. In that case, the vari- ables z (along magnetic field) and r⊥ (in the quantum well plane) in the wave function of the electron-hole pair may be separated. This condition is carried out for the quantum well on basis of GaAs for the magnetic field, corresponding to the magnetopolaron formation9. Be- sides, if the energy of the size quantization exceeds the Coulomb and exchange energies, the electron-hole pair may be considered as a free particle. Then, in the ap- proximation of the effective mass and infinitely high bar- riers, the wave function, describing the dependence on z , accepts a simple form Φ j(z) = πm cz πm vz , 0 ≤ z ≤ d, (9) and Φ j(z) = 0 in barriers, where m c (m v) are the quantum numbers of the size-quantization of an electron (hole). In the real systems, the approximation (9) is not al- ways carried out. However, the taking into account the Coulomb and exchange interactions will result only into some changes of the function Φ j(z), what does not change qualitatively the optical characteristics, as it was shown for the monochromatic irradiation20. Indices j = 1 and j = 2 in Φ j(z) correspond to the pairs of quantum numbers of the size-quantization in a direct interband transition. m c ( v) corresponds to the index j = 1, and m c ( v) corresponds to the index j = 2. In interband transitions, the Landau quantum numbers are conserved. The total electric field E is included into the RHS of (7), what is connected with the refuse from the perturbation theory on the coupling constant e2/h̄c. In further calculations, an equality of quantum num- bers m c v = m c v is assumed. Then, Φ 1(z) = Φ 2(z) = Φ(z), and the current density in the RHS of (7) takes the form J(z, ω) = i ν c ω − ω 1 + iγ 1 ω − ω 2 + iγ 2 ×Φ(z) dz′Φ(z′)E(z′). (10) With the help of the indicated simplifications, as it was shown in9,18,20, the field amplitudes in the Fourier- representation ∆E ℓ(z, ω) and E r(z, ω) result in ∆E ℓ(z, ω) = −iE0(ω)(−1)m c+m ve−ik(z−d)N , E r(z, ω) = E0(ω)e ikz(1− iN ), (11) where E0(ω) is given in (3). Here, the frequency depen- dence is determined by the function N = ε (γ r1 ω̃2 + γ r2 ω̃1)/2 ω̃1 ω̃2 + iε(γ r1 ω̃2 + γ r2 ω̃1)/2 . (12) The function N includes the value ε = ε′ + iε′′, (13) which determines influence of the spatial dispersion on the radiative broadening (ε′ γ r) and shift (ε ′′ γ r) of the doublet levels. ε′ and ε′′ are equal9,20: ε′ = Re ε = 2B2 1− (−1)m c+m v cos kd , (14) ε′′ = Imε = 2B (1 + δmc,mv )(mc +mv) 2 + (mc −mv)2 8mcmv −(−1)mc+mvB sinkd− (2 + δmc,mv)(kd) 8π2mcmv , (15) B = 4π 2 m c m v kd [π 2 (m c +m v)2 − (kd)2 ] [(kd)2 − π 2 (m c −m v)2 ] (if kd → 0, ε → 1 (m c = m v, an allowed transition ) and ε → 0 (m c 6= m v, a forbidden transition)). II. THE TIME-DEPENDENCE OF THE ELECTRIC FIELD OF REFLECTED AND TRANSMITTED LIGHT PULSES With the help of the standard formulas, let us go to the time-representation ∆E ℓ(z, t) ≡ ∆E ℓ(s) = dω e−iω s ∆E ℓ(z, ω), s = t+ νz/c, (16) E r(z, t) ≡ E r(p) = 1 dω e−iω p E r(z, ω), p = t− νz/c. (17) The vectors ∆E ℓ(s) and E r(p) have the form ∆E ℓ(s) = e ℓ ∆E ℓ(s) + c.c., r(p) = e ℓ E r(p) + c.c.. (18) It is seen from expressions (11) and (12) that the denom- inator in integrands of (16) and (17) is the same. It may be transformed conveniently to the form ω̃1 ω̃2 + i(ε/2) (γ r1 ω̃2 + γ r2 ω̃1) = (ω − Ω 1) (ω − Ω 2), (19) where Ω 1 and Ω 2 determine the poles of the integrand in the complex plane ω. They are equal Ω1,2 = ω1 + ω2 − (γ1 + γ2)− (γr1 + γr2) [ω1 − ω2 − (γ1 − γ2)− (γr1 − γr2)]2 −ε2γr1γr2 . (20) Thus, in the integrands of 16) and (17), there are 4 poles: ω = ω ℓ± iγ ℓ/2 and ω = Ω 1,2. The pole ω = ω ℓ+ iγ ℓ/2 is situated in the upper half plane, others are situated in the lower half plane. Integrating in the complex plane ω, we obtain that the function ∆E ℓ(z, t), determining, according to (17), the electric field vector of the reflected pulse ∆E ℓ(z, t), has the form ∆E ℓ(z, t) = −iE 0(−1)m c+m veikd ×{R1[1−Θ(s)] + (R2 +R3 +R4)Θ(s)}, (21) where R1 = exp(−iω ℓs+ γ ℓs/2) γ̄ r1/2 ω ℓ − Ω1 + iγ ℓ/2 γ̄ r2/2 ω ℓ − Ω2 + iγ ℓ/2 R2 = exp(−iω ℓ − γ ℓs/2) γ̄ r1/2 ω ℓ − Ω1 − iγ ℓ/2 γ̄ r2/2 ω ℓ − Ω2 − iγ ℓ/2 R3 = − exp(−iΩ1s)(γ̄ r1/2) ω ℓ − Ω1 − iγ ℓ/2 ω ℓ − Ω1 + iγ ℓ/2 R4 = − exp(−iΩ2s)(γ̄ r2/2) ω ℓ − Ω2 − iγ ℓ/2 ω ℓ − Ω2 + iγ ℓ/2 , (22) where γ̄ r1 = ε ′ γ r1 +∆γ, γ̄ r2 = ε ′ γ r2 −∆γ, ε′ γ r1(Ω2 − ω2 + iγ2/2) Ω1 − Ω2 ε′ γ r2(Ω1 − ω1 + iγ1/2) Ω1 − Ω2 . (23) The function E r(z, t), corresponding to a transmitted light pulse, is represented in the form E r(z, t) = E 0 T1[1−Θ(p)] + (T2 + T3 + T4)Θ(p) Ω1 − Ω2 where T1 = exp(−iω ℓp+ γ ℓp/2)M(ω ℓ + iγ ℓ/2) ω ℓ − Ω1 + iγ ℓ/2 ω ℓ − Ω2 + iγ ℓ/2 T2 = exp(−iω ℓp− γ ℓp/2)M(ω ℓ − iγ ℓ/2) ω ℓ − Ω1 − iγ ℓ/2 ω ℓ − Ω2 − iγ ℓ/2 T3 = − exp(−iΩ1 p)M(Ω1) ω ℓ − Ω1 − iγ ℓ/2 ω ℓ − Ω1 + iγ ℓ/2 T4 = exp(−iΩ2 p)M(Ω2) ω ℓ − Ω2 − iγ ℓ/2 ω ℓ − Ω1 + γ ℓ/2 . (25) The function M has the structure M(ω) = (ω − ω1 + iγ1/2)(ω − ω2 + iγ2/2)− (ε′′/2)[γr1(ω − ω2 + iγ2/2) + γr2(ω − ω1 + iγ1/2)]. When the electric field of stimulating light pulse E 0(z, t) (determined in (2)) is extracted from E r(z, t) , i.e., it is assumed r(z, t) = E 0(z, t) + ∆E r(z, t), (26) then, ∆E r(z, t) will differ from ∆E ℓ(z, t) only by substi- tution of the variable s = t + νz/c by p = t − νz/c and by absence of the factor (−1)m c+m v exp(ikd). Thus, being taken into account, the spatial dispersion provides a renormalization of radiative damping γ r i. In nominators of formulas (21), the renormalization leads to multiplication of γ r i on the real factor ε ′, i.e., decreases the value γ r i (diagrams of functions ε ′ and ε′′ are rep- resented in9). In denominators, γ r i is multiplied on the complex function ε, that means the appearance, together with the change of the radiative broadening, of a shift of resonant frequencies. In the limit kd = 0, expressions (21) - (23) coincide with obtained in (14). =1 =0.0005 eVeV =0.00005 eV =0.0005 eV t, s, p {ps) 6420-2 FIG. 1: The reflectance R, transmittance T , absorbance A, and stimulating pulse P as time dependent functions for three magnitudes of the parameter kd in the case of a long stimulat- ing pulse (γ ℓ ≪ ∆ω) γ r ≪ γ, γ ℓ .∆ω = 6.65.10 eV, ω ℓ = ReΩ1 = Ω res. t (s, p) ps =0.005 eV =0.00005 eV =0.0005 eV 2.10-4 1.10-4 -1.10-4 -2.10-4 -1 0 21 FIG. 2: Same as in Fig.1 for an exciting pulse of a middle duration (γ ℓ ≃ ∆ω) γ r ≪ γ ≪ γ ℓ . III. THE REFLECTANCE, TRANSMITTANCE AND ABSORBANCE OF STIMULATING LIGHT PULSE The energy flux S(p), corresponding to the electric field of stimulating light pulse, is equal S(p) = (E 0(z, t))2 = ezS0P(p), (27) where S0 = cE 0/2πν, ez is the unite vector along the light pulse. The dimensionless function P(p) = (E 0(z, t))2 = Θ(p)e−γℓp + [1−Θ(p)]eγℓp (28) determines the spatial and time dependence of the energy flux of stimulating pulse. The flux, transmitted through the quantum well, has a form (E r(z, t))2 = ezS0T (p), (29) the reflected energy flux has a form ℓ = − ezc (E ℓ(z, t))2 = −ezS0R(s). (30) The dimensionless functions T (p) and R(s) correspond to parts of transmitted and reflected energy fluxes of the stimulating pulse. The dimensionless absorbance is de- fined as A(p) = P(p)−R(p) − T (p) (31) (since for reflection z ≤ 0, the variable in R is s = t − |z|/c). The dependencies of the reflectance R, transmittance T , absorbance A, and stimulated momentum P on the variable p (or s for R) for the case m c = m v = 1 are represented in figures. It was assumed also that γ r 1 = γ r 2 = γ r, γ 1 = γ 2 = γ. (32) It follows from (21) and (24) that the resonant frequen- cies are ω ℓ = ReΩ1 and ω ℓ = ReΩ2. The calculations were performed for ω ℓ = ReΩ1 = Ωres. (33) Let us go from the frequency ω ℓ to Ω = ω ℓ − ω1, (34) then the resonant frequency is Ωres = −∆ω + ε′γ r +Re (∆ω)2 − ε2γ2r . (35) It depends on three parameters: ∆ω = ω1 − ω2, γ r and kd, since the complex function ε depends on kd (see (15)). Functions R, T ,A and P are homogeneous functions of the inverse lifetimes and frequencies ω1, ω2, ω ℓ. There- fore, a choice of the measurement units is arbitrary. For the sake of certainty, all these values are expressed in eV . The time dependence of the optical characteristics of a quantum well is represented in figures for the different magnitudes of kd. The curves, corresponding to kd = 0, were obtained in14. It was assumed in calculations that ∆ω = 0.065eV , what corresponds to the magnetopolaron state in a quantum well on basis of GaAs and to the width d = 300Å of the quantum well18,19,21. t (p, s) ps = 0.0005 eV = 0.00005 eV = 0 151050 FIG. 3: Same as in Fig.1 for an exciting pulse of a middle duration (γ ℓ ≃ ∆ω) γ r ≪ γ ≪ γ ℓ . t (s, p) ps = 0.005 eV = 0.00005 eV = 0 4.10-4 -6.10-4 -4.10-4 2.10-4 -2.10-4 FIG. 4: Same as in Fig.1 for an exciting pulse of a middle duration (γ ℓ ≃ ∆ω) γ = 0. IV. THE DISCUSSION OF RESULTS Fig.1 corresponds to a long (wide in comparison to ∆ω) stimulating pulse and a small radiative broadening (γ r ≪ γ, γ ℓ). In this case, the transmittance T domi- nates. The shape of the curve weakly differs from P and weakly depends on kd. The dependence on the spatial dispersion is seen at the curves R and A. For example, the reflectance R at kd = 3 is two times less than at kd = 0. However, the magnitude R is shares of percent. Fig.2 corresponds to a stimulating light pulse of a mid- dle length, when γ ℓ ≃ ∆ω and γ r ≪ γ ≪ γ ℓ. There appear peculiarities: a light generation (negative absorp- = 0.00666 t (p, s) ps 21-1 0 FIG. 5: Same as in Fig.1 for four magnitudes of the parameter kd in the case, when ∆ω is close to γ r, γ ℓ ≫ γ r ≫ γ. tion) after the light pulse transmission and oscillations of R,A and T . The generation is a consequence of the fact that the electronic system has no time to irradiate the energy during the propagation of such pulse. Oscil- lations is a consequence of beatings with the frequency (under condition ω ℓ = ReΩ1) Re (Ω1 − Ω2) = Re (ω1 − ω2)2 − (ε′ + iε′′)2γ2r. (36) A noticeable effect of the spatial dispersion takes place in the reflectance R during transmission of the pulse, as well as after its transmission. The spatial dispersion af- fects the transmittance T and absorbanceA after passing through a quantum well, when these values are small. In Fig. 3 and 4, the optical characteristics are repre- sented at γ = 0 and a long stimulating pulse (γ ℓ ≪ ∆ω, Fig.3) and a pulse of a middle duration, when γ ℓ ≃ ∆ω (Fig.4). Since, in that case, the real absorption is ab- sent, one have to accept the function A, defined in (31), as an energy part, stored up by a quantum well for the time being due to the interband transitions (if A > 0), or an energy part, which is generated by the quantum well during and after propagation of the pulse (A < 0). The same concerns to Fig.2, however, the part of the stored energy there, which disappears if γ → 0, corresponds to the real absorption. The oscillation period in Fig.2 and 4 does not depend on the parameter kd, since, at chosen magnitudes of the parameters ∆ω and γ r the beating frequency (36) is almost equal to ω1 − ω2, and compara- tively small changes of the functions ε′ and ε′′ does not affect practically on the beating frequency. In Fig.5, where ∆ω is near γ r (6.65.10 −3eV and 6.66.10−3eV , respectively), the stimulating light pulse is 5 times shorter, than in Fig.3 and 4, and γ ℓ ≫ γ r ≫ γ. In that case, the spatial dispersion affects strongly on the optical characteristics. In the interval 0 ≤ kd ≤ 3, the reflectance increases 8 times approximately, and the transmittance decreases 6 times. Such a sharp change is due to the dependence of γ̄ r1 and γ̄ r2 on kd. For ex- ample, at kd = 0 γ̄ r1 = −17303.9, γ̄ r2 = 193066, 6, and at kd = 3 Re γ̄ r1 = 1960.21, Re γ̄ r2 = 442, 718. And at the same time, R and T ≤ 1, since they are the result of substraction of large magnitudes and therefore these differences are sensitive to changes of kd. In14, it was shown that, at kd = 0, there are the sin- gular points on the time axis, where T = A = 0 and R = P , or R = A = 0 and T = P (total reflection or total transmission). It is seen from the figures that the singular points are preserved and in the case kd 6= 0, there is only a small shift of them. In Fig.5 the point of the total transmission appears at kd = 0. At kd = 0.5, this point disappears, and at kd = 1.5 and kd = 3.0 the point of the total reflection appears. If kd = 1.5, then R = P , A+ T = 0 (A < 0). If kd = 3.0, then as before R = P , but A = T = 0. Thus, growing of the parameter kd changes the type of a singular point. Thus, the spatial dispersion of the electromagnetic waves, forming the light pulse, noticeably affect the op- tical characteristics of a quantum well. This influence is especially strong, when γ r ≃ ∆ω. Let us note in conclusion that the results obtained above are valid at equal refraction indices of barriers and quantum well. Otherwise, one is to take into account re- flection of boundaries of a quantum well. However, this problem is outside the scopes of present article. 1 L. C. Andreani, F. Tassone, F. Bassani. Sol. State Com- mun. 77, 9, 641 (1991). 2 L. C. Andreani. In: Confined electrons and phonons. Eds E. Burstein, C. Weisbuch, Plenum Press, N. Y. (1995), p. 3 E. L. Ivchenko. Fiz. Tverd. Tela, 1991, 33, N 8, 2388 (Physics of the Solid State (St. Petersburg), 33, 2182 (1991)). 4 F. Tassone, F. Bassani, L. C. Andreani. Phys. Rev. B 45, 11, 6023 (1992). 5 T.Stroucken, A. Knorr, C. Anthony, P. Thomas, S. W. Koch, M. Koch, S. T. Gundiff, J. Feldman, E. O. Göbel. Phys. Rev. Lett. 74, 9, 2391 (1996). 6 T.Stroucken, A. Knorr, P. Thomas, S. W. Koch. Phys. Rev. B 53, 4, 2026 (1996). 7 L. C. Andreani, G. Panzarini, A. V. Kavokin, M. R. Vladimirova. Phys. Rev. B 57, 8, 4670 (1998). 8 M. Hübner, T. Kuhl, S. Haas, T.Stroucken, S. W. Koch, R. Hey, K. Ploog. Sol. State Commun., 105, 2, 105 (1998). 9 L. I. Korovin, I. G. Lang, D. A. Contreras-Solorio, S. T. Pavlov. Fiz. Tverd. Tela, 43, 2091 (2001) (Physics of the Solid State (St. Petersburg), 43, 2182 (2001)); cond-mat/0104262. 10 I. G Lang, V. I. Belitsky, M. Cardona. Phys. Stat. Sol. (a) 164, 1, 307 (1997). 11 I. G Lang, V. I. Belitsky. Solid. State Commun. 107, 10, 577 (1998). 12 I. G Lang, V. I. Belitsky. Phys. Lett. A 245, 329 (1998). 13 I. G. Lang, L. I. Korovin, A. Contreras-Solorio, S. T. Pavlov. Fiz. Tverd. Tela, 43, 1117 (2001) (Physics of the Solid State (St. Petersburg), 43, 1159 (2001)); cond-mat/ 0004178. 14 D. A. Contreras-Solorio, S. T. Pavlov, L. I. Korovin, I. G. Lang. Phys. Rev 62, 24, 16815 (2000); cond-mat/0002229. 15 I. G. Lang, L. I. Korovin, A. Contreras-Solorio, S. T. Pavlov. Fiz. Tverd. Tela, 42, 2230 (2000) ( Physics of the Solid State (St. Petersburg) , 42, N 12, 2300 (2000)); cond-mat/0006364. 16 L. I. Korovin, I. G. Lang, D. A. Contreras-Solorio, S. T. Pavlov. Fiz. Tverd. Tela, 44, 1681 (2002) (Physics of the Solid State (St. Petersburg), 44, 1759 (2002)); cond-mat/0203390. 17 I.V.Lerner, J.E.Lozovik. Zh. Eksp. Teor. Fiz., 78, N 3, 1167 (1980) (JETP, 51,588 (1980)). 18 I. G. Lang, L. I. Korovin, A. Contreras-Solorio, S. T. Pavlov, Fiz. Tverd.Tela, 44, 2084 (2002) (Physics of the Solid State (St. Petersburg), 44, 2181 (2002)); cond-mat/ 0001248. 19 I. G. Lang, L. I. Korovin, A. Contreras-Solorio, S. T. Pavlov. Fiz. Tverd.Tela, 48, 1795 (2006) (Physics of the Solid State (St. Petersburg), 48, 1693 (2006)); cond-mat/ 0403302. 20 L. I. Korovin, I. G. Lang, S. T. Pavlov. Fiz. Tverd.Tela, 48, 2208 (2006) (Physics of the Solid State (St. Petersburg), 48, 2337 (2006)); cond-mat/ 0001248. 21 L. I. Korovin, I. G. Lang, S. T. Pavlov. Zh. Eksp. Teor. Fiz., 115, 187 (1999) (JETP, 88, 105 (1999)). http://arxiv.org/abs/cond-mat/0104262 http://arxiv.org/abs/cond-mat/0004178 http://arxiv.org/abs/cond-mat/0002229 http://arxiv.org/abs/cond-mat/0006364 http://arxiv.org/abs/cond-mat/0203390 http://arxiv.org/abs/cond-mat/0001248 http://arxiv.org/abs/cond-mat/0403302 http://arxiv.org/abs/cond-mat/0001248

0704.1698

The origin of the anomalously strong influence of out-of-plane disorder on high-Tc superconductivity

The origin of the anomalously strong influence of out-of-plane disorder on high-T superconductivity Y. Okada,1 T. Takeuchi,2 T. Baba,3 S. Shin,3 and H. Ikuta1 1Department of Crystalline Materials Science, Nagoya University, Nagoya 464-8603, Japan 2EcoTopia Science Institute, Nagoya University, Nagoya 464-8603, Japan 3Institute for Solid State Physics (ISSP), University of Tokyo, Kashiwa 277-8581, Japan (Dated: ) Abstract The electronic structure of Bi2Sr2−xRxCuOy (R=La, Eu) near the (π,0) point of the first Bril- louin zone was studied by means of angle-resolved photoemission spectroscopy (ARPES). The temperature T ∗ above which the pseudogap structure in the ARPES spectrum disappears was found to have an R dependence that is opposite to that of the superconducting transition tempera- ture Tc. This indicates that the pseudogap state is competing with high-Tc superconductivity, and the large Tc suppression caused by out-of-plane disorder is due to the stabilization of the pseudogap state. http://arxiv.org/abs/0704.1698v2 High temperature superconductivity occurs with doping carriers to a Mott insulator. Carriers are usually doped either by varying the oxygen content or by an element substitu- tion. Unavoidably, these procedures introduce disorder that influences the superconducting transition temperature Tc even though only sites outside the CuO2 plane are chemically modified. For instance, Tc of the La2CuO4 family depends on the size of the cation that substitutes for La,1 and Tc of Bi2Sr1.6R0.4CuOy depends on the R element. 2,3 Recently, some of the present authors have studied extensively the Bi2Sr2−xRxCuOy system using single crystals and varied the R content x over a wide range for R=La, Sm, and Eu.4 The results clearly show that Tc at the optimal doping T c depends strongly on the R element and decreases with the decrease in the ionic radius of R, in other words, with increasing disorder. By plotting Tc as a function of the thermopower at 290 K S(290), it was found that the range of S(290) values for samples with a non-zero Tc becomes narrower with increasing disorder (see Fig. 1). Because S(290) correlates well with hole doping in many high-Tc cuprates, this suggests that the doping range where superconductivity occurs decreases with increas- ing out-of-plane disorder, in contrast to the naive expectation that the plot of Tc/T c vs. doping would merge into a universal curve for all high-Tc cuprates. Despite the strong influence on Tc and on the doping range where superconductivity can be observed, out-of-plane disorder affects only weakly the conduction along the CuO2 plane. According to Fujita et al.,6 out-of-plane disorder suppresses Tc more than Zn when samples with a similar residual resistivity are compared. This means that out-of-plane disorder influences Tc without being a strong scatterer, and that this type of disorder has an unexplained effect on Tc. To elucidate the reason of this puzzling behavior and why the carrier range of high-Tc superconductivity is affected by out-of-plane disorder, we studied the electronic structure of R=La and Eu crystals by means of angle-resolved photoemission spectroscopy (ARPES) measurements. We particularly focused on the so-called antinodal position, the point where the Fermi surface crosses the (π,0)-(π,π) zone boundary (M̄-Y cut), due to the following reasons. It is generally accepted that in-plane resistivity is sensitive to the electronic structure near the nodal point of the Fermi surface.7,8,9 The small influence of out-of-plane disorder on residual resistivity hence suggests that the electronic structure of this region is not much affected, as Fujita et al. mentioned.6 Therefore, if out-of-plane disorder causes any influence on the electronic structure, it would be more likely to occur at the antinodal point of the Fermi surface. The single crystals used in this study were grown by the floating zone method as re- ported previously.4 As mentioned in that work and commonly observed for Bi-based high-Tc cuprates, the composition of the grown crystal is not the same as the starting one and depends on the position within the boule. Accordingly, the hole doping level can not be determined from the starting composition of the crystal. On the other hand, it has been shown for many cuprates that S(290) correlates well with hole doping. Although S(290) is not directly related to the amount of carriers and should depend on the detail of the electronic structure, this empirical connection provides a reasonable indicator for the hole doping level. We note that we have confirmed in a separate experiment that the Fermi surface of a R=La and a R=Eu crystal with similar S(290) values coincided quite well,10 implying that their hole doping was similar. Therefore, we use S(290) as a measure of doping in the following.11 All crystals were annealed at 750◦C for 72 hours in air. The ARPES spectra were accu- mulated using a Scienta SES2002 hemispherical analyzer with the Gammadata VUV5010 photon source (He Iα) at the Institute of Solid State Physics (ISSP), the University of Tokyo, and at beam-line BL5U of UVSOR at the Institute for Molecular Science, Okazaki with an incident photon energy of 18.3 eV. The energy resolution was 10-20 meV for all measure- ments, which was determined by the intensity reduction from 90% to 10% at the Fermi edge of a reference gold spectrum. Thermopower was measured by a four-point method using a home-built equipment. S(290) was determined using crystals that were cleaved from those used for ARPES measurements except the R=La sample that had the largest doping in Fig. 4(a). For that particular sample, the S(290) value was estimated from the c-axis length deduced from x-ray diffraction based on the data shown in the inset to Fig. 1. Figures 2(a) and (c) show the ARPES intensity plots along the (π,0)-(π,π) direction at 100 K for R=La and Eu crystals that have a similar hole concentration. The samples were cleaved in situ at 250 K in a vacuum of better than 5×10−11 Torr. The S(290) values were 4.7 µV/K and 4.8 µV/K for the R=La and Eu samples, respectively, indicating that they are slightly underdoped (see Fig. 1). Figures 2(b) and (d) show momentum distribution curves (MDCs) of the R=La and Eu samples, respectively. We fitted the MDC curves to a Lorentz function to determine the peak position. The thus extracted dispersion is superimposed by white small circles on Figs. 2(a) and (c). The momentum where the dispersion curve crosses the Fermi energy EF corresponds to the Fermi wave vector kF on the (π,0)-(π,π) cut, and -2002040 S(290) (µV/K) -2002040 S(290) (µV/K) FIG. 1: (color online) The critical temperature Tc as a function of S(290), the thermopower at 290 K. Tc was determined from the temperature dependence of resistivity, which was measured simultaneously with thermopower. Data are based on our previous work,4 and some new data points are included. Inset: Lattice constant c plotted as a function of S(290). Fig. 2(e) shows the energy distribution curves (EDCs) of the two samples at kF . Obviously, the R=La sample has a larger spectral weight at EF , although the doping level of the two samples is very similar. Figure 3 shows the EDCs of the two samples of Fig. 2 at various temperatures. To remove the effects of the Fermi function on the spectra, we applied the symmetrization method Isym(ω) = I(ω) + I(−ω), where ω denotes the energy relative to EF . 12 As shown in Figs. 3(a) and (b), the symmetrized spectra of both samples show clearly a gap structure at the lowest measured temperature, 100 K. Because we are probing the antinodal direction at a temperature that is higher than Tc, we attribute this gap structure to the pseudogap. With increasing the temperature, the gap structure fills up without an obvious change in the gap size. At 250 K, only a small suppression of the spectral weight was observed for the R=La -0.08 -0.06 -0.04 -0.02 Momentum T=100KR=Eu -0.08 -0.06 -0.04 -0.02 Momentum R=La T=100K (0,0) (π,π) Momentum −40meV E-EF=−80meV Momentum −40meV E-EF=−80meV -0.4 -0.3 -0.2 -0.1 0.0 0.1 E-EF (eV) (a) (c) (e) (f) (b) (d) FIG. 2: (color online) Intensity plots in the energy-momentum plane of the ARPES spectra at 100 K of slightly underdoped Bi2Sr2−xRxCuOy samples that have a similar doping level with (a) R=La and (c) R=Eu along the momentum line indicated by the arrow in (f). (b), (d) Momentum distribution curves (MDCs) of the two samples. (e) The energy distribution curves (EDCs) of the two samples at kF . (f) Schematic drawing of the underlying Fermi surface. sample. On the other hand, a clear pseudogap structure can be observed for the R=Eu sample even at 250 K. This means that the temperature T ∗ up to which the pseudogap structure can be observed is certainly different despite the closeness of the doping level. The thin solid lines Ifit(ω) of Figs. 3(a) and (b) are the results of fitting a Lorentz function to the symmetrized spectrum in the energy range of EF ± 150 meV. The dashed lines are, on the other hand, the background spectra Ibkg(ω), which were determined by making a similar fit in the energy range of 150 meV≤ |ω| ≤400 meV. It can be seen that the difference between the two fitted curves at EF increases with decreasing the temperature, reflecting the growth of the pseudogap. To quantify how much the spectral weight at EF is depressed, we define IPG as the difference between unity and the spectral weight of the fitted spectrum at EF divided by that of the background curve (1 − Ifit(0)/Ibkg(0)). Figure 3(c) shows the temperature dependence of IPG of the two samples. Obviously, the depression of the spectral weight is larger for the R=Eu sample at all measured temperatures, and T ∗ is higher. IPG is roughly linear temperature dependent for both samples with a very similar slope. Therefore, we extrapolated the data with the same slope as shown by the dashed lines, and estimated T ∗ to be 282 K and 341 K for the R=La and Eu samples, respectively. We also measured ARPES spectra of four other samples at 150 K. Assuming that the temperature dependence of IPG is the same as that of Fig. 3(c), we can estimate T ∗ from the IPG value at 150 K. The thus estimated T ∗ values are included in Fig. 4(a), which shows T ∗ of all samples studied in this work as a function of S(290). It can be seen that T ∗ is higher for R=Eu than R=La when compared at the same S(290) value. Because Tc at the same hole doping decreases with changing the R element to one with a smaller ionic radius (Fig. 1), it is clear that Tc and T ∗ have an opposite R dependence. This important finding is summarized on the schematic phase diagram shown in Fig. 4(b). As shown, both Tmaxc and the carrier range where superconductivity takes place on the phase diagram decrease with decreasing the ionic radius of R, while T ∗ at the same hole concentration increases.11 One of the most important issues of pseudogap has been whether such state is a compet- itive one or a precursor state of high-Tc superconductivity. 14 Figure 4(b) clearly shows that whatever the microscopic origin of the pseudogap is, it is competing with high-Tc supercon- ductivity. In contrast, some other studies have concluded that pseudogap is closely related to the superconducting state because the momentum dependence of the gap is the same above and below Tc and the evolution of the gap structure through Tc is smooth. 12,15,16 We point out, however, that several recent experiments revealed the existing of two energy gaps at a temperature well below Tc for underdoped cuprate superconductors 17,18 as well as for optimally doped and overdoped (Bi,Pb)2(Sr,La)2CuO6+δ. 19,20 The energy gap observed in the antinodal region was attributed to the pseudogap while the one near the nodal direction 0 100 200 300 T (K) -0.2 0 0.2 E-EF (eV) -0.2 0 0.2 E-EF (eV) R=La(a) FIG. 3: (color online) Temperature dependence of the symmetrized ARPES spectrum at the antin- odal point for the (a) R=La and (b) R=Eu crystals of Fig. 2. (c) Temperature dependence of the amount of spectral weight suppression IPG. to the superconducting gap. We think that the conflict encountered in the pseudogap issue arose because distinguishing these two gaps would be not easy when their magnitudes are similar. We next discuss why Tc decreases when T ∗ increases. We think that the ungapped Carrier Concentration (a) (b) S(290) (µV/K) carrier doping FIG. 4: (color online) (a) Pseudogap temperature T ∗ plotted as a function of S(290). (b) A schematic phase diagram of Bi2Sr2−xRxCuOy based on the results of Figs. 1 and 4(a). portion of the Fermi surface above Tc is smaller for R=Eu when compared at the same doping and at the same temperature because pseudogap opens at a temperature that is higher than R=La. Indeed, the results of our ARPES experiment on optimally doped Bi2Sr2−xRxCuOy confirmed this assumption by demonstrating that the momentum region where a quasiparticle or a coherence peak was observed was narrower for the R=Eu sample than the R=La sample.10 In other words, the Fermi arc shrinks with changing the R element from La to Eu, which mimics the behavior observed when doping is decreased.8,13 Because the superfluid density ns decreases with underdoping, 21 it is reasonable to assume that only the states on the Fermi arc can participate to superconductivity. If we can think in analogy to the carrier underdoping case therefore, ns would be smaller forR=Eu than R=La, and the decrease of Tc is quite naturally explained from the Uemura relation. 21 The faster disappearance of superconductivity with carrier underdoping for R=Eu (see Fig. 1) is also a straightforward consequence of this model. Furthermore, the opening of the pseudogap at the antinodal direction would not increase much the residual in-plane resistivity because the in-plane conduction is mainly governed by the nodal carriers.7,8,9 Hence, the observation that out-of-plane disorder largely suppresses Tc with only a slight increase in residual resistivity can also be immediately understood. Finally, we discuss our results in conjunction with the reported data of scanning tunneling microscopy/spectroscopy (STM/STS) experiments, which unveiled a strong inhomogeneity in the local electronic structure for Bi2Sr2CaCu2Oy. 22,23,24 It was demonstrated that the volume fraction of the pseudogapped region increases with underdoping. The ARPES ex- periments, on the other hand, show that the spectral weight at the chemical potential of the antinodal region decreases with carrier underdoping.14 Hence, the antinodal spectral weight at EF is likely to correlate with the volume fraction of the pseudogap region. In the present work, we increased the degree of out-of-plane disorder while the hole doping was unaltered, and observed that the spectral weight at the chemical potential is lower for the R=Eu sample when compared at the same temperature, as shown in Fig. 3(c). We thus expect that the fraction of superconducting region in real space is smaller for R=Eu. Indeed, quite recent STM/STS experiments on optimally doped Bi2Sr2−xRxCuOy report that the averaged gap size is larger when the ionic radius of R is smaller,25 which is attributable to an increase of the pseudogapped region. Moreover, our results complement the STM/STS data and indi- cate that not only the area where a pseudogap is observed at low temperature but also T ∗ increases with disorder which means that the pseudogap state is stabilized and persists up to higher temperatures. Further, while the STM/STS studies on Bi2Sr2−xRxCuOy investigated only optimally doped samples,25,26 we have varied hole doping, which further corroborates the conclusion that the pseudogap state competes with high-Tc superconductivity. In summary, we have studied the mechanism why Tmaxc and the carrier range where high-Tc superconductivity occurs strongly depend on the R element in the Bi2Sr2−xRxCuOy system by investigating the electronic structure at the antinodal direction of the Fermi sur- face of R=La and Eu samples. We observed a pseudogap structure in the ARPES spectrum up to a higher temperature for R=Eu samples when samples with a similar hole doping are compared, which clearly indicates that the pseudogap state is competing with high-Tc su- perconductivity. This result suggests that out-of-plane disorder increases the pseudogapped region and reduces the superconducting fluid density, which explains its strong influence on high-Tc superconductivity. 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0704.1699

Relativistic Hydrodynamics at RHIC and LHC

Relativistic Hydrodynamics at RHIC and LHC Tetsufumi Hirano1,∗) 1 Department of Physics, The University of Tokyo, Tokyo 113–0033, Japan Recent development of a hydrodynamic model is discussed by putting an emphasis on realistic treatment of the early and late stages in relativistic heavy ion collisions. The model, which incorporates a hydrodynamic description of the quark-gluon plasma with a kinetic approach of hadron cascades, is applied to analysis of elliptic flow data at the Relativistic Heavy Ion Collider energy. It is predicted that the elliptic flow parameter based on the hybrid model increases with the collision energy up to the Large Hadron Collider energy. §1. Introduction One of the important discoveries made at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven National Laboratory is that the elliptic flow parameter,1) namely, the second Fourier coefficient v2 = 〈cos(2φ)〉 of the azimuthal momentum distribution dN/dφ,2) is quite large in non-central Au+Au collisions.3) Over the past years, many studies have been devoted to understanding the elliptic flow from dynamical models: (1) The observed v2 values near midrapidity at low transverse momentum (pT ) in central and semi-central collisions are consistent with predictions from ideal hydrodynamics.4) (2) The v2 data cannot be interpreted by hadronic cas- cade models.5), 6) (3) A partonic cascade model7) can reproduce these data only with significantly larger cross sections than the ones obtained from the perturbative cal- culation of quantum chromodynamics. The produced dynamical system is beyond the description of naive kinetic theories. Thus, a paradigm of the strongly cou- pled/interacting/correlated matter is being established in the physics of relativistic heavy ion collisions.8) The agreement between hydrodynamic predictions and the data suggests that the heavy ion collision experiment indeed provides excellent op- portunities for studying matter in local equilibrium at high temperature and for drawing information of the bulk and transport properties of the quark-gluon plasma (QGP). These kinds of phenomenological studies closely connected with experimen- tal results, so to say, the “observational QGP physics”, will be one of the main trends in modern nuclear physics in the eras of the RHIC and the upcoming Large Hadron Collider (LHC). Then it is indispensable to sophisticate hydrodynamic mod- eling of heavy ion collisions for making quantitative statements on properties of the produced matter with estimation of uncertainties. In fact, the ideal fluid dynamical description gradually breaks down as one studies peripheral collisions4) or moves away from midrapidity.9), 10) This requires a more realistic treatment of the early and late stages11)–14) in dynamical modeling of relativistic heavy ion collisions. In this paper, recent studies of the state-of-the-art hydrodynamic simulations are highlighted with emphases on the importance of the final decoupling stage (Sec. 3) ∗) e-mail address: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1699v1 2 T. Hirano and of much better understanding of initial conditions (Sec. 4). A prediction of the v2 parameter at the LHC energy will be made in Sec. 5. See also other reviews 15) to complement other topics of hydrodynamics in heavy ion collisions at RHIC. §2. A QGP fluid + hadronic cascade model We have formulated a dynamical and unified model,13) based on fully three- dimensional (3D) ideal hydrodynamics,9), 10) toward understanding the bulk and transport properties of the QGP. During the fluid dynamical evolution one assumes local thermal equilibrium. However, this assumption can be expected to hold only during the intermediate stage of the collision. In order to extract properties of the QGP from experimental data one must therefore supplement the hydrodynamic de- scription by appropriate models for the beginning and end of the collision. In Sec. 3, we employ the Glauber model for initial conditions in hydrodynamic simulations. Initial entropy density is parametrized as a superposition of terms scaling with the densities of participant nucleons and binary collisions, suitably generalized to ac- count for the longitudinal structure of the initial fireball.13) Instead, in Secs. 4 and 5, we employ the Color Glass Condensate (CGC) picture17) for colliding nuclei and calculate the produced gluon distributions18) as input for the initial conditions in the hydrodynamical calculation.19) During the late stage, local thermal equilibrium is no longer maintained due to expansion and dilution of the matter. We treat this gradual transition from a locally thermalized system to free-streaming hadrons via a dilute interacting hadronic gas by employing a hadronic cascade model20) below a switching temperature of T sw=169 MeV. A massless ideal parton gas equation of state (EOS) is employed in the QGP phase (T >Tc = 170 MeV) while a hadronic resonance gas model is used at T <Tc. When we use the hydrodynamic code all the way to final decoupling, we take into account10) chemical freezeout of the hadron abundances at T ch = 170 MeV, separated from thermal freezeout of the momen- tum spectra at a lower decoupling temperature T th, as required to reproduce the experimentally measured yields.21) §3. Success of a hybrid approach Initial conditions in 3D hydrodynamic simulations are put so that centrality and pseudorapidity dependences of charged particle yields are reproduced. A linear combination of terms scaling with the number of participants and that of binary collisions enables us to describe centrality dependence of particle yields at midra- pidity. This agreement with the data still holds when the ideal fluid description is replaced by a more realistic hadronic cascade below T sw. See also Fig. 3. When ideal hydrodynamics is utilized all the way to kinetic freezeout, T th = 100 MeV is needed to generate enough radial flow for reproduction of proton pT spectrum at midra- pidity. One major advantage of the hybrid model over the ideal hydrodynamics is that the hybrid model automatically describes freezeout processes without any free parameters. The hybrid model works remarkably well in reproduction of pT spectra for identified hadrons below pT ∼ 1.5 GeV/c. Relativistic Hydrodynamics at RHIC and LHC 3 partN 0 50 100 150 200 250 300 350 400 QGP + hadron fluids QGP fluid and hadron gas Hadron gas PHOBOS(hit) PHOBOS(track) -6 -4 -2 0 2 4 6 hydro+cascade =100MeV, hydrothT =169MeV, hydrothT PHOBOS 25-50% b=8.5fm Fig. 1. (Left) Centrality dependence of v2. The solid (dashed) line results from a full ideal fluid dynamic approach (a hybrid model). For reference, a result from a hadronic cascade model6) is also shown (dash-dotted line). (Right) Pseudorapidity dependence of v2. The solid (dashed) line is the result from a full ideal hydrodynamic approach with T th = 100 MeV (T th = 169 MeV). Filled circles are the result from the hybrid model. All data are from the PHOBOS Collaboration.22) The centrality dependences of v2 at midrapidity (| η |< 1) from (1) a hadronic cascade model6) (dash-dotted), (2) a QGP fluid with hadronic rescatterings taken through a hadronic cascade model (dashed), and (3) a QGP+hadron fluid with T th = 100 MeV (solid) are compared with the PHOBOS data.22) A hadronic cascade model cannot generate enough elliptic flow to reproduce the data. This is observed also in other hadronic cascade calculations.5) Thus it is almost impossible to interpret the v2 data from a hadronic picture only. Models based on a QGP fluid generate large elliptic flow and gives v2 values which are comparable with the data. When a hadronic matter is also treated by ideal hydrodynamics, v2 is overpredicted in peripheral collisions. This is improved by dissipative effects in the hadronic matter. Note that there could exist effects of eccentricity fluctuation,23) which is not taken into account in the current approach. Deviation between the data and the QGP- fluid-based results above Npart ∼ 200 could be attributed to these effects. From the integrated elliptic flow data at midrapidity, initial push from QGP pressure turns out to be important at midrapidity. In Fig. 1 (right), the pseu- dorapidity dependence of v2 data in 25-50% centrality observed by PHOBOS 22) are compared with QGP fluid models. Ideal hydrodynamics with T th = 169 MeV, which is just below the transition temperature Tc = 170 MeV, underpredicts the data in the whole pseudorapidity region. Hadronic rescatterings after QGP fluid evolution generate the right amount of elliptic flow and, consequently, the triangle pattern of the data is reproduced well. If the hadronic matter is also assumed to be described by ideal hydrodynamics until T th = 100 MeV, v2 overshoots in forward/backward rapidity regions (| η |∼ 4). This is simply due to the fact that, in ideal hydrody- namics, v2 is approximately proportional to the initial eccentricity which is almost independent of space-time rapidity. So the hadronic dissipation is quite important in forward/backward rapidity regions as well as at midrapidity in peripheral collisions (Npart < 100). From these studies, the perfect fluidity of the QGP is needed to 4 T. Hirano obtain enough amount of the integrated v2, while the dissipation (or finite values of the mean free path among hadrons) in the hadronic matter is also important to obtain less elliptic flow coefficients when the multiplicity is small at midrapidity in peripheral collisions and/or in forward/backward rapidity regions. This is exactly the novel picture of dynamics in relativistic heavy ion collisions, namely, the nearly perfect fluid QGP core and the highly dissipative hadronic corona, addressed in Ref. 16) (GeV/c)Tp 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.05 πSTAR, STAR, K STAR, p Fig. 2. Transverse momentum dependence of v2 for pions, kaons and protons. Filled plots are the results from the hybrid model. The impact parameter in the model simulation is 7.2 fm which corresponds to 20-30% centrality. Data (open plots) for pions, kaons and protons are obtained by the STAR Collaboration.24) As a cross-check on the picture, we also study pT dependence of v2 for identi- fied hadrons in semi-central collisions to see whether the hybrid model works. We correctly reproduce mass ordering behavior of differential elliptic flow below pT ∼ 1 GeV/c as shown in Fig. 2. Here experimental data are from STAR.24) Although we also reproduce the data in 10-20% and 30-40% well (not shown), it is hard to reproduce data in very central collisions (0-5%) due to a lack of initial eccentricity fluctuation in this model. It is worth mentioning that, recently, the hybrid model succeeds to describes differential elliptic flow data for identified hadrons at forward rapidity observed by BRAHMS.25) §4. Challenge for a hydrodynamic approach So far, an ideal hydrodynamic description of the QGP fluid with the Glauber type initial conditions followed by an kinetic description of the hadron gas describes the space-time evolution of bulk matter remarkably well. The CGC,17) whose cases are growing both in deep inelastic scatterings and in d+Au collisions recently, is one of the relevant pictures to describe initial colliding nuclei in high energy collisions. In this section, novel hydrodynamic initial conditions19) based on the CGC are employed for an analysis of elliptic flow. We first calculate the centrality dependence of the multiplicity to see that the CGC indeed correctly describes the initial entropy production and gives proper initial conditions for the fluid dynamical calculations. Both CGC and Glauber model initial Relativistic Hydrodynamics at RHIC and LHC 5 partN 0 50 100 150 200 250 300 350 400 =100MeVthGlauber+hydro, T =100MeVthCGC+hydro, T CGC+hydro+cascade PHOBOS Fig. 3. Centrality dependence of charged particle multiplicity per number of participant nucleons. The solid (dashed) line results from Glauber-type (CGC) initial conditions. The dash-dotted line results from our hybrid model. Experimental data are from PHOBOS.27) conditions, propagated with ideal fluid dynamics, reproduce the observed centrality dependence of the multiplicity,27) see Fig. 3. In the hydrodynamic simulations, the numbers of stable hadrons below T ch are designed to be fixed by introducing chemical potential for each hadron.10) On the other hand, the number of charged hadrons is approximately conserved during hadronic cascades. So the centrality dependence of charged particle yields is also reasonably reproduced by the hybrid approach. In the left panel of Fig. 4 we show the impact parameter dependence of the eccentricity of the initial energy density distributions at τ0 = 0.6 fm/c. We neglect event-by-event eccentricity fluctuations although these might be important for very central and peripheral events.23) Even though both models correctly describe the centrality dependence of the multiplicity as shown in Fig. 3, they exhibit a signifi- cant difference: The eccentricity from the CGC is 20-30% larger than that from the Glauber model.13), 28) The situation does not change even when we employ the “uni- versal” saturation scale29) in calculation of gluon production. The initial eccentricity is thus quite sensitive to model assumptions about the initial energy deposition which can be discriminated by the observation of elliptic flow. The centrality dependence of v2 from the CGC initial conditions followed by the QGP fluid plus the hadron gas is shown in Fig. 4 (right). With Glauber model initial conditions,4) the predicted v2 from ideal fluid dynamics overshoots the peripheral collision data. 22) Hadronic dissipative effects within hadron cascade model reduce v2 and, in the Glauber model case, are seen to be sufficient to explain the data (Fig. 1 (left)).13) Initial conditions based on the CGC model, however, lead to larger elliptic flows which overshoot the data even after hadronic dissipation is accounted for,13) unless one additionally as- sumes significant shear viscosity also during the early QGP stage. Therefore precise understanding of the bulk and transport properties of QGP from the elliptic flow data requires a better understanding of the initial stages in heavy ion collisions. 6 T. Hirano b (fm) 0 2 4 6 8 10 12 14 no diffuseness CGC (IC-e) =0.85)αGlauber ( partN 0 50 100 150 200 250 300 350 400 hydro+cascade, CGC hydro+cascade, Glauber PHOBOS(hit) PHOBOS(track) Fig. 4. (Left) Impact parameter dependence of the eccentricity of the initial energy density dis- tributions. The solid (dashed) line results from Glauber-type (CGC) initial conditions. The dotted line assumes a box profile for the initial energy density distribution. (Right) Centrality dependence of v2. The solid (dashed) line results from CGC (Glauber model) initial conditions followed by ideal fluid QGP dynamics and a dissipative hadronic cascade. The data are from PHOBOS.22) §5. Elliptic flow at LHC The elliptic flow parameter plays a very important role in understanding global aspects of dynamics in heavy ion collisions at RHIC. It must be also important to measure elliptic flow parameter at the LHC energy toward comprehensive under- standing of the degree and mechanism of thermalization and the bulk and transport properties of the QGP. Figure 5 shows the excitation function of the charged particle elliptic flow v2, scaled by the initial eccentricity ε, for Au+Au collisions at b = 6.3 fm impact pa- rameter, using three different models: (i) a pure 3D ideal fluid approach with a typical kinetic freezeout temperature T th = 100 MeV where both QGP and hadron gas are treated as ideal fluids (dash-dotted line); (ii) 3D ideal fluid evolution for the QGP, with kinetic freezeout at T th = 169 MeV and no hadronic rescattering (dashed line); and (iii) 3D ideal fluid QGP evolution followed by hadronic rescattering below T sw = 169MeV (solid line). Although applicability of the CGC model for SPS ener- gies might be questioned, we use it here as a systematic tool for obtaining the energy dependence of the hydrodynamic initial conditions. By dividing out the initial eccen- tricity ε, we obtain an excitation function for the scaled elliptic flow v2/ε whose shape should be insensitive to the facts that CGC initial conditions produce larger eccen- tricities and the resulting integrated v2 overshoots the data at RHIC and also that experiments with different collision system (Pb+Pb) will be performed at the LHC. Figure 5 shows the well-known bump in v2/ε at SPS energies ( sNN ∼ 10GeV) predicted by the purely hydrodynamic approach, as a consequence of the softening of the equation of state (EOS) near the quark-hadron phase transition region,30) and that this structure is completely washed out by hadronic dissipation,12) consistent with the experimental data.31), 32) Even at RHIC energies, hadronic dissipation still reduces v2 by ∼ 20%. The hybrid model predicts a monotonically increasing excita- Relativistic Hydrodynamics at RHIC and LHC 7 tion function for v2/ε which keeps growing from RHIC to LHC energies, 12) contrary to the ideal fluid approach whose excitation function almost saturates above RHIC energies.30) (GeV)NNs 10 210 310 410 =100MeV CGC+hydro, T =169MeV CGC+hydro, T CGC+hydro+cascade Au+Au Charged, b=6.3fm LHCRHIC Fig. 5. Excitation function of v2/ε in Au+Au collisions at b = 6.3 fm. The solid line results from CGC initial conditions followed an ideal QGP fluid and a dissipative hadronic cascade. The dashed (dash-dotted) line results from purely ideal fluid dynamics with thermal freezeout at T th = 169MeV (100MeV). §6. Conclusions We have studied the recent elliptic flow data at RHIC by using a hybrid model in which an ideal hydrodynamic treatment of the QGP is combined with a hadronic cascade model. With the Glauber-type initial conditions, the space-time evolution of the bulk matter created at RHIC is well described by the hybrid model. The agree- ment between the model results and the data includes v2(Npart), v2(η), pT spectra for identified hadron below pT ∼ 1.5 GeV/c and v2(pT ) for identified hadrons at midrapidity and in the forward rapidity region. If the Glauber type initial condi- tions are realized, we can establish a picture of the nearly perfect fluid of the QGP core and the highly dissipative hadronic corona. However, in the case of the CGC initial conditions, the energy density profile in the transverse plane is more “eccen- tric” than that from the conventional Glauber model. This in turn generates large elliptic flow, which is not consistent with the experimental data. Without viscous effects even in the QGP phase, we cannot interpret the integrated elliptic flow at RHIC. 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0704.1700

Retract rationality and Noether's problem

arXiv:0704.1700v1 [math.AC] 13 Apr 2007 RETRACT RATIONALITY AND NOETHER’S PROBLEM Ming-chang Kang Department of Mathematics National Taiwan University Taipei, Taiwan, Rep. of China E-mail: [email protected] Abstract. Let K be any field and G be a finite group. Let G act on the rational function fields K(xg : g ∈ G) by K-automorphisms defined by g · xh = xgh for any g, h ∈ G. Denote by K(G) the fixed field K(xg : g ∈ G) G. Noether’s problem asks whether K(G) is rational (=purely transcendental) over K. We will prove that, if K is any field, p an odd prime number, and G is a non-abelian group of exponent p with |G| = p3 or p4 satisfying [K(ζp) : K] ≤ 2, then K(G) is rational over K. A notion of retract rationality is introduced by Saltman in case K(G) is not rational. We will also show that K(G) is retract rational if G belongs to a much larger class of p-groups. In particular, generic G-polynomials of G-Galois extensions exist for these groups. Mathematics Subject Classification 2000: Primary 12F12, 13A50, 11R32, 14E08. Keywords and Phrases: Noether’s problem, rationality problem, retract rational, generic polynomial, p-groups of exponent p, flabby class maps, projective modules. http://arxiv.org/abs/0704.1700v1 §1. Introduction Let K be any field and G be a finite group. Let G act on the rational function field K(xg : g ∈ G) by K-automorphisms defined by g · xh = xgh for any g, h ∈ G. Denote K(G) := K(xg : g ∈ G) G the fixed subfield under the action of G. Noether’s problem asks whether K(G) is rational (= purely transcendental) over K. Noether’s problem for abelian groups was studied by Fischer, Furtwängler, K. Ma- suda, Swan, Voskresenskii, S. Endo and T. Miyata, Lenstra, etc. It was known that K(Zp) is rational if Zp is a cyclic group of order p with p = 3, 5, 7 or 11. The first counter-example was found by Swan: Q(Z47) is not rational over Q [Sw1]. However Saltman showed that Q(Zp) is retract rational over K for any prime number p, which is enough to ensure the existence of a generic Galois G-extension and will fulfill the original purpose of Emmy Noether [Sa1]. For the convenience of the reader, we recall the definition of retract rationality. Definition 1.1. ([Sa3]) Let K ⊂ L be a field extension. We say that L is re- tract rational over K, if there is a K-algebra R contained in L such that (i) L is the quotient field of R, and (ii) the identity map 1R : R → R factors through a local- ized polynomial K-algebra, i.e. there is an element f ∈ K[x1, . . . , xn] the polynomial ring over K and there are K-algebra homomorphisms ϕ : R → K[x1, . . . , xn][1/f ] and ψ : K[x1, . . . , xn][1/f ] → R satisfying ψ ◦ ϕ = 1R. It is not difficult to see that “rational” ⇒ “stably rational” ⇒ “retract rational”. One of the motivation to study Noether’s problem arises from the inverse Galois problem. If K is an infinite field, it is known that K(G) is retract rational over K if and only if there exists a generic Galois G-extension over K [Sa1, Theorem 5.3; Sa3, Theorem 3.12], which guarantees the existence of a Galois G-extension of K, provided that K is a Hilbertian field. On the other hand, the existence of a generic Galois G-extension over K is equivalent to the existence of a generic polynomial for G. For the relationship among these notions, see [DM]. For a survey of Noether’s problem the reader is referred to articles of Swan and Kersten [Sw2; Ke]. Although Noether’s problem for abelian groups was investigated extensively, our knowledge for the non-abelian Noether’s problem was amazingly scarce (see, for exam- ple, [Ka2]). We will list below some previous results of non-abelian Noether’s problem, which are relevant to the theme of this article. Theorem 1.2. (Saltman [Sa2]) For any prime number p and for any field K with char K 6= p (in particular, K may be an algebraically closed field), there is a meta-abelian p-group G of order p9 such that K(G) is not retract rational over K. In particular, it is not rational. Theorem 1.3. (Hajja [Ha]) Let G be a finite group containing an abelian normal subgroup N such that G/N is a cyclic group of order < 23. Then C(G) is rational over Theorem 1.4. (Chu and Kang [CK]) Let p be a prime number, G be a p-group of order ≤ p4 with exponent pe. Let K be any field such that either char K = p or char K 6= p and K contains a primitive pe-th root of unity. Then K(G) is rational over K. Theorem 1.5. (Kang [Ka4]) Let G be a metacyclic p-group with exponent pe, and let K be any field such that (i) char K = p, or (ii) char K 6= p and K contains a primitive pe-th root of unity. Then K(G) is rational over K. Note that, in Theorems 1.3—1.5, it is assumed that the ground field contains enough roots of unity. We may wonder whether Q(G) is rational if G is a non-abelian p-group of small order. The answer is rather optimistic when G is a group of order 8 or 16. Theorem 1.6. ([CHK; Ka3]) Let K be any field and G be any non-abelian group of order 8 or 16 other than the generalized quaternion group of order 16. Then K(G) is always rational over K. However Serre was able to show that Q(G) is not rational when G is the generalized quaternion group [Se, p.441–442; GMS, Theorem 33.26 and Example 33.27, p.89–90]. On the other hand, if p is an odd prime number, Saltman proves the following theorem. Theorem 1.7. (Saltman [Sa4]) Let p be an odd prime number and G be a non- abelian group of order p3. If K is a field containing a primitive p-th root of unity, then K(G) is stably rational. The above theorem may be generalized to the case of p-groups containing a maximal cyclic subgroup, namely, Theorem 1.8. (Hu and Kang [HuK]) Let p be a prime number and G be a non- abelian p-group of order pn containing a cyclic subgroup of index p. If K is any field containing a primitive pn−2-th root of unity, then K(G) is rational over K. In this article we will prove the following theorem. Theorem 1.9. Let p be an odd prime number, G be the non-abelian group of ex- ponent p and of order p3 or p4. If K is a field with [K(ζp) : K] ≤ 2, then K(G) is rational over K. The rationality problem of K(G) seems rather intricate if the ground field K has no enough root of unity. We don’t know the answer to the rationality of K(G) when the assumption that [K(ζp) : K] ≤ 2 is waived in the above theorem. On the other hand, as to the retract rationality of K(G), a lot of information may be obtained. Before stating our results, we recall a theorem of Saltman first. Theorem 1.10. (Saltman [Sa1, Theorem 3.5]) Let K be a field, G = A ⋊ G0 be a semi-direct product group where A is an abelian normal subgroup of G. Assume that gcd{|A|, |G0|} = 1 and both K(A) and K(G0) are retract rational over K. Then K(G) is retract rational over K. Thus the main problem is to investigate the retract rationality for p-groups. We will prove K(G) is retract rational for many p-groups G of exponent p. Theorem 1.11. Let p be a prime number, K be any field, and G = A ⋊ G0 be a semi-direct product group where A is a normal elementary p-group of G and G0 is a cyclic group of order pm. If p = 2 and charK 6= 2, assume furthermore that K(ζ2m) is a cyclic extension of K. Then K(G) is retract rational over K. If p is an odd prime number, a p-group of exponent p containing an abelian normal subgroup of index p certainly satisfies the assumption in Theorem 1.11. In particular, a p-group of exponent p and of order p3 or p4 belongs to this class of p-groups (see [CK]). There are six p-groups of exponent p and of order p5; only four of them contain abelian normal subgroups of index p. Previously the retract rationality of K(G) for non-abelian p-groups, i.e. the existence of generic polynomials for such groups G, is known only when G is of order p3 and of exponent p. Similarly if G = A⋊G0 is a semi-direct product of p-groups such that A is a normal subgroup of order p, and G0 is a direct product of an elementary p-group with a cyclic group of order pm, then G also satisfies the assumption in Theorem 1.11 provided that the assumption that K(ζ2m) is a cyclic extension of K remains in force. The above Theorem 1.11 is deduced from the following theorem. Theorem 1.12. Let K be any field, and G = A⋊G0 be a semi-direct product group where A is a normal abelian subgroup of exponent e and G0 is a cyclic group of order m. Assume that (i) either charK = 0 or charK > 0 with charK ∤ em, and (ii) both K(ζe) and K(ζm) are cyclic extensions of K such that gcd{m, [K(ζe) : K]} = 1. Then K(G) is retract rational over K. The idea of the proof of Theorem 1.12 is to add a primitive e-th root of unity to the ground field and the question is reduced to a question of multiplicative group actions. It is Voskresenskii who realizes that the multiplicative group action is related to the birational classification of algebraic tori [Vo]. However, the multiplicative group action arising in the present situation is not the function field of an algebraic torus; it is a new type of multiplicative group actions. Thus we need a new criterion for retract rationality. It is the following theorem. Theorem 1.13. Let π1 and π2 be finite abelian groups, π = π1 × π2, and L be a Galois extension of the field K with π1 = Gal(L/K). Regard L as a π-field through the projection π → π1. Assume that (i) gcd{|π1|, |π2|} = 1, (ii) char K = 0 or charK > 0 with charK ∤ |π2|, and (iii) K(ζm) is a cyclic extension of K where m is the exponent of π2. If M is π-lattice such that ρπ(M) is an invertible π-lattice, than L(M) π is retract rational over K. The reader will find that the above theorem is an adaptation of Saltman’s criterion for retract rational algebraic tori [Sa3, Theorem 3.14] (see Theorem 2.5). We also formulate another criterion for retract rationality of L(M)π when π is a semi-direct product group (see Theorem 4.3). An amusing consequence of this criterion (when compared with Theorem 1.3) is that, if G = A ⋊ H is a semi-direct product of an abelian normal subgroup A and a cyclic subgroup H , then C(G) is always retract rational (see Proposition 5.2). We will organize this paper as follows. We recall some basic facts of multiplicative group actions in Section 2. In particular, the flabby class map which was mentioned in Theorem 1.13 will be defined. We will give additional tools for proving Theorem 1.9 and Theorems 1.11–1.13 in Section 3. In Section 4 Theorem 1.13 and its variants will be proved. The proof of Theorem 1.11 and Theorem 1.12 will be given in Section 5. Section 6 contains the proof of Theorem 1.9. Acknowledgements. I am indebted to Prof. R. G. Swan for providing a simplified proof in Step 7 of Case 1 of Theorem 1.9 (see Section 6). The proof in a previous version of this paper was lengthy and complicated. I thank Swan’s generosity for allowing me to include his proof in this article. Notations and terminology. A field extension L over K is rational if L is purely transcendental over K; L is stably rational over K if there exist y1, . . . , yN such that y1, . . . , yN are algebraically independent over L and L(y1, . . . , yN) is rational over K. More generally, two fields L1 and L2 are called stably isomorphic if L1(x1, . . . , xm) is isomorphic to L2(y1, . . . , yn) where x1, . . . , xm and y1, . . . , yn are algebraically indepen- dent over L1 and L2 respectively. Recall the definition of K(G) at the beginning of this section: K(G) = K(xg : g ∈ G)G. If L is a field with a finite group G acting on it, we will call it a G-field. Two G-fields L1 and L2 are G-isomorphic if there is an isomorphism ϕ : L1 → L2 satisfying ϕ(σ · u) = σ · ϕ(u) for any σ ∈ G, any u ∈ L1. We will denote by ζn a primitive n-th root of unity in some extension field of K when char K = 0 or char K = p > 0 with p ∤ n. All the groups in this article are finite groups. Zn will be the cyclic group of order n or the ring of integers modulo n depending on the situation from the context. Z[π] is the group ring of a finite group π over Z. Z(G) is the center of the group G. The exponent of a group G is the least common multiple of the orders of elements in G. The representation space of the regular representation of G over K is denoted by W = g∈GK · x(g) where G acts on W by g · x(h) = x(gh) for any g, h ∈ G. §2. Multiplicative group actions Let π be a finite group. A π-latticeM is a finitely generated Z[π]-module such that M is a free abelian group when it is regarded as an abelian group. For any field K and a π-lattice M , K[M ] will denote the Laurent polynomial ring and K(M) is the quotient field of K[M ]. Explicitly, if M = 1≤i≤m Z · xi as a free abelian group, then K[M ] = K[x±11 , . . . , x m ] and K(M) = K(x1, . . . , xm). Since π acts on M , it will act on K[M ] and K(M) by K-automorphisms, i.e. if σ ∈ π and σ · xj = 1≤i≤m aijxi ∈ M , then we define the action of σ in K[M ] and K(M) by σ · xj = 1≤i≤m x The multiplicative action of π onK(M) is called a purely monomial action in [HK1]. If π is a group acting on the rational function field K(x1, . . . , xm) by K-automorphism such that σ · xj = cj(σ) · 1≤i≤m x i where σ ∈ π, aij ∈ Z and cj(σ) ∈ K\{0}, such a multiplicative group action is called a monomial action. Monomial actions arise when studying Noether’s problem for non-split extension groups [Ha; Sa5]. We will introduce another kind of multiplicative actions. Let K ⊂ L be fields and π be a finite group. Suppose that π acts on L by K-automorphisms (but it is not assumed that π acts faithfully on L). Given a π-lattice M , the action of π on L can be extended to an action of π on L(M) (= L(x1, . . . , xm) if M = 1≤i≤m Z · xi) by K-automorphisms defined as follows: If σ ∈ π and σ · xj = 1≤i≤m aijxi ∈ M , then the multiplication action in L(M) is defined by σ · xj = 1≤i≤m x i for 1 ≤ j ≤ m. When L is a Galois extension of K and π = Gal(L/K) (and therefore π acts faithfully on L), the fixed subfield L(M)π is the function field of the algebraic torus defined over K, split by L and with character group M (see [Vo]). We recall some basic facts of the theory of flabby (flasque) π-lattices developed by Endo and Miyata, Voskresenskii, Colliot-Thélène and Sansuc, etc. [Vo; CTS]. We refer the reader to [Sw2; Sw3; Lo] for a quick review of the theory. In the sequel, π denotes a finite group unless otherwise specified. Definition 2.1. A π-lattice M is called a permutation lattice if M has a Z-basis permuted by π. M is called an invertible (or permutation projective) lattice, if it is a direct summand of some permutation lattice. A π-lattice M is called a flabby (or flasque) lattice if H−1(π′,M) = 0 for any subgroup π′ of π. (Note that H−1(π′,M) denotes the Tate cohomology group.) Similarly, M is called coflabby if H1(π′,M) = 0 for any subgroup π′ of π. It is known that an invertible π-lattice is necessarily a flabby lattice [Sw2, Lemma 8.4; Lo, Lemma 2.5.1]. Theorem 2.2. (Endo and Miyata [Sw3, Theorem 3.4; Lo, 2.10.1]) Let π be a finite group. Then any flabby π-lattice is invertible if and only if all Sylow subgroups of π are cyclic. Denote by Lπ the class of all π-lattices, and by Fπ the class of all flabby π-lattices. Definition 2.3. We define an equivalence relation ∼ on Fπ: Two π-lattices E1 and E2 are similar, denoted by E1 ∼ E2, if E1 ⊕ P is isomorphic to E2 ⊕ Q for some permutation lattices P and Q. The similarity class containing E will be denoted by [E]. Define Fπ = Fπ/ ∼, the set of all similarity classes of flabby π-lattices. Fπ becomes a commutative monoid if we define [E1]+[E2] = [E1⊕E2]. The monoid Fπ is called the flabby class monoid of π. Definition 2.4. We define a map ρ : Lπ → Fπ as follows. For any π-lattice M , there exists a flabby resolution, i.e. a short exact sequence of π-lattices 0 → M → P → E → 0 where P is a permutation lattice and E is a flabby lattice [Sw2, Lemma 8.5]. We define ρπ(M) = [E] ∈ Fπ. The map ρπ : Lπ → Fπ is well-defined [Sw2, Lemma 8.7]; it is called the flabby class map. We will simply write ρ instead of ρπ, if the group π is obvious from the context. Theorem 2.5. (Saltman [Sa3, Theorem 3.14]) Let L be a Galois extension of K with π = Gal(L/K) and M be a π-lattice. Then ρπ(M) is invertible if and only if L(M)π is retract rational over K. §3. Generalities We recall several results which will be used later. Theorem 3.1. ([HK2, Theorem 1]) Let L be a field and G be a finite group acting on L(x1, . . . , xm), the rational function field of m variables over L. Suppose that (i) for any σ ∈ G, σ(L) ⊂ L; (ii) The restriction of the action of G to L is faithful; (iii) for any σ ∈ G,  σ(x1) σ(xm)  = A(σ)   +B(σ) where A(σ) ∈ GLm(L) and B(σ) is an m× 1 matrix over L. Then L(x1, . . . , xm) = L(z1, . . . , zm) where σ(zi) = zi for all σ ∈ G, and for any 1 ≤ i ≤ m. In fact, z1, . . . , zm can be defined by   = A ·   for some A ∈ GLm(L) and for some B which is an m× 1 matrix over L. Moreover, if B(σ) = 0 for all σ ∈ G, we may choose B = 0 in defining z1, . . . , zm. Theorem 3.2. (Kuniyoshi [CHK, Theorem 2.5]) LetK be a field with charK = p > 0 and G be a p-group. Then K(G) is always rational over K. Proposition 3.3. Let π be a finite group and L be a π-field. Suppose that 0 → M1 → M2 → N → 0 is a short exact sequence of π-lattices satisfying (i) π acts faithfully on L(M1), and (ii) N is an invertible π-lattice. Then the π-fields L(M2) and L(M1 ⊕N) are π-isomorphic. Proof. We follow the proof of [Le, Proposition 1.5]. Denote L(M1) × = L(M1)\{0}. Consider the exact sequence of π-modules: 0 → L(M1) × → L(M1) × ·M2 → N → 0. By Hilbert Theorem 90, we find that H1(π′, L(M1) ×) = 0 for any subgroup π′ ⊂ π. Applying [Le, Proposition 1.2] we find that the above exact sequence splits. The resulting π-morphism N → L(M1) × · M2 provides the required π-isomorphism form L(M2) to L(M1 ⊕N). � Lemma 3.4. Let the assumptions be the same as in Proposition 3.3. Assume fur- thermore that N is a permutation π-lattice. Then L(M2) π is rational over L(M1) Proof. By Proposition 3.3, L(M2) = L(M1)(N). Since π acts faithfully on L(M1), we may apply Theorem 3.1 and find u1, . . . , un ∈ L(M2) such that L(M2) = L(M1)(u1, . . . , un) with σ(ui) = ui for any σ ∈ π, any 1 ≤ i ≤ n where n = rank(N). Hence L(M2) π = L(M1) π(u1, . . . , un). � Lemma 3.5. Let π be a finite abelian group of exponent e and K be a field such that char K = 0 or charK > 0 with charK ∤ e. If P is a permutation π-lattice, then K(P )π = K(ζe)(M) π0 where π0 = Gal(K(ζe)/K) and M is some π0-lattice. Proof. We follow the standard approach to solving Noether’s problem for abelian groups [Sw1; Sw2; Le]. Note that K(P )π = {K(ζe)(P ) 〈π0〉}〈π〉 = K(ζe)(P ) 〈π,π0〉 where the action of π is extended to K(ζe)(P ) by defining g(ζe) = ζe for any g ∈ π, and the action of π0 is extended to K(ζe)(P ) by requiring that π0 acts trivially on P . Since π is abelian of exponent e, we may diagonalize its action on P , i.e. we may find x1, . . . , xn ∈ K(ζe)(P ) such that n = rank(P ), g(xi)/xi ∈ 〈ζe〉 for any g ∈ π, and K(ζe)(P ) = K(ζe)(x1, . . . , xn). Thus K(ζe)(P ) 〈π〉 = K(ζe)(y1, . . . , yn) where y1, . . . , yn are monomials in x1, . . . , xn. LetM be the multiplicative subgroup generated by y1, . . . , yn in K(ζe)(y1, . . . , yn)\{0}. Then M is a π0-lattice and K(ζe)(y1, . . . , yn) π0 = K(ζe)(M) π0. � Proposition 3.6. Let π and K be the same as in Lemma 3.5. If K(ζe) is a cyclic extension of K, then K(π) is retract rational over K. Proof. We may regard the regular representation of π is given by a permutation π-lattice. Thus K(π) = K(ζe)(M) π0 where π0 = Gal(K(ζe)/K). Since π0 is assumed cyclic, thus we may apply Theorem 2.2 and Theorem 2.5. � §4. Proof of Theorem 1.13 Lemma 4.1. Let π be a finite group, M be a π-lattice. Suppose that π0 is a normal subgroup of π and π0 acts trivially on M . Thus we may regard M as a lattice over π/π0. (1) M is a permutation π-lattice ⇔ So is it as a π/π0-lattice. (2) M is an invertible π-lattice ⇔ So is it as a π/π0-lattice. (3) M is a flabby π-lattice ⇔ So is it as a π/π0-lattice. (4) If 0 → M → P → E → 0 is a flabby resolution of M as a π/π0-lattice, this short exact sequence is also a flabby resolution of M as a π-lattice. (5) ρπ(M) is an invertible π-lattice ⇔ ρπ/π0(M) is an invertible π/π0-lattice. Proof. The properties (1)–(4) can be found in [CTS, Lemma 2, p.179–180]. As to (5), the direction “⇐” is obvious by applying (4). For the other direction, assume ρπ(M) is an invertible π-lattice. Let 0 → M → P → E → 0 be a flabby resolution of M as a π-lattice. Then 0 → Mπ0 → P π0 → Eπ0 → 0 is a flabby resolution of M =Mπ0 in the category of π/π0-lattices by [CTS, Lemma (xi), p.180]. It remains to show that Eπ0 is invertible. Since [E] = ρπ(M) is invertible, we can find a π-lattice N such that E ⊕N = N ′ is a permutation π-lattice. Note that N ′ π0 is a permutation π/π0-lattice by [CTS, Lemma 2(i), p.180]. We find that E π0 is invertible because Eπ0 ⊕Nπ0 = (E ⊕N)π0 = N ′ π0 . � Lemma 4.2. Let the assumptions be the same as in Theorem 1.13. If P is a permutation π-lattice, then L(P )π is retract rational over K. Proof. Since π1 and π2 are abelian groups with gcd{|π1|, |π2|} = 1, every subgroup π′ of π can be written as π′ = ρ × λ where ρ is a subgroup of π1 and λ is a subgroup of π2. As a permutation π-lattice, we may write P = Z[π/π(i)] where π(i) is a subgroup of π. Write π(i) = ρi × λi where ρi ⊂ π1, λi ⊂ π2. Hence Z[π/π (i)] = Z[π/(ρi × λi)] = Z[(π1/ρi)× (π2/λi)]. It is not difficult to see that Z[π/π(i)] = Z · u kl where 1 ≤ k ≤ t = |π1/ρi|, 1 ≤ l ≤ r = |π2/λi|. Moreover, if g ∈ π1 and g ′ ∈ π2, then g · u kl = u g(k),l , g′ · kl = u k,g′(l) and the homomorphisms π1 → St and π2 → Sr are induced from the permutation representations associated to π/π(i) = (π1/ρi)× (π2/λi) where St and Sr are the symmetric groups of degree t and r respectively. Since π1 is faithful on L, we may apply Theorem 3.1. Explicitly, for any 1 ≤ l ≤ r, we may find A(i) ∈ GLt(L) and define v kl by   = A(i)   such that g · v k,l = v k,l for any g ∈ π1, and L(Z[π/π (i)]) = L(u kl : 1 ≤ k ≤ t, 1 ≤ l ≤ r. If g′ ∈ π2, from the relation g ′ · u kl = u k,g′(l) and Formula (1), we find that g′ · v k,g′(l) Since L(P )π1 = L(u π1 = L(v π1 = K(v kl ) where i, k, l runs over index sets which are understood, it follows that L(P )π = K(v π2. Note that π2 acts on {v by permutations. By Lemma 3.5 K(v π2 = K(ζm)(M) π0 where m is the exponent of π2, π0 = Gal(K(ζm)/K) and M is some π0-lattice. By our assumption, π0 is a cyclic group. Hence ρπ0(M) is invertible by Theorem 2.2. Apply Theorem 2.5. We find that K(ζm)(M) π0 is retract rational over K. � Proof of Theorem 1.13 ———————————– Step 1. Suppose that M is a π-lattice such that ρπ(M) is invertible. Define π0 = {g ∈ π2 : g acts trivially on L(M)}. Then π/π0 acts faithfully on L(M). Moreover, ρπ/π0(M) is invertible by Lemma 4.1. In other words, without loss of generality we may assume that π is faithfully on L(M) . Thus we will keep in force this assumption in the sequel. Step 2. Since ρπ(M) is invertible, by [Sa3, Theorem 2.3, p.176], we may find π- lattice M ′, P , Q such that P and Q are permutation lattices, 0 →M →M ′ → Q→ 0 is exact, and the inclusion map M →M ′ factors through P , i.e. the following diagram commutes 0 ✲ M ✲ M ′ ✲ Q ✲ 0. The remaining proof proceeds quite similar to that of [Sa3, Theorem 3.14, p.189]. Step 3. We get a commutative diagram of K-algebra morphisms from the diagram in (2), i.e. L[M ]π L[P ]π L[M ]π ✲ L[M ′]π ✲ L[Q]π Step 4. The quotient field of L[P ]π is L(P )π, which is retract rational over K by Lemma 4.2. Thus the identity map 1 : L[P ]π → L[P ]π factors rationally by [Sa3, Lemma 3.5], i.e. there is a localized polynomial ring K[x1, . . . , xn][1/f ] and K-algebra maps ϕ : L[P ]π → K[x1, . . . , xn][1/f ], ψ : K[x1, . . . , xn][1/f ] → L[P ] π such that ψ ◦ ϕ = 1. It follows that the composite map g : L[M ]π → L[P ]π → L[M ′]π also factors ratio- nally, i.e. there areK-algebra ϕ′ : L[M ]π → K[x1, . . . , xn][1/f ], ψ ′ : K[x1, . . . , xn][1/f ] → L[M ′]π such that g = ψ′ ◦ ϕ′. Step 5. By Lemma 3.4 L(M ′)π is rational over L(M)π. (This is the only one step we use the assumption that π is faithful on L(M).) Now we may apply [Sa3, Proposition 3.6(b), p.183] where, in the notation of [Sa3], we take S = T = L[M ]π, ϕ is the identity map on L(M)π. We conclude that 1 : L[M ]π → L[M ]π factors rationally, i.e. L(M)π is retract rational over K. � Here is a variant of Theorem 1.13. Theorem 4.3. Let π be a finite group, 0 → π1 → π → π2 → 1 is a group extension, and L be a Galois extension of the field K with π2 = Gal(L/K). Let π act on L through the projection π → π2. Assume that (i) π1 is an abelian group of exponent e with ζe ∈ L, (ii) the extension 0 → π1 → π → π2 → 1 splits, and (iii) every Sylow subgroup of π2 is cyclic. If M is a π-lattice such that ρπ(M) is an invertible lattice, then L(M) π is retract rational over K. Proof. The proof is very similar to the proof of Theorem 1.13. We claim that L(P )π is retract rational for any permutation π-lattice P . For the proof, we will use [Sa5, Theorem 2.1, p.546]. We will show that (c) of [Sa5, Theorem 2.1] is valid, which will guarantee that L(P )π is retract rational. By assumption (iii), part (d) of [Sa5, Theorem 2.1] is valid by Theorem 2.2. It remains to check that the embedding problem of L/K and the extension 0 → π1 → π → π2 → 1 is solvable. But this is the well-known split embedding problem [ILF, Theorem 1.9, p.12]. Now define π0 = {g ∈ π : g acts trivially on L(M)}. Note that π0 ⊂ π1. The remaining proof is the same as in the proof of Theorem 1.13 and is omitted. � Corollary 4.4. Let π be an abelian group of exponent e, K be a field with ζe ∈ K. Suppose that M is a π-lattice and π acts on K(M) by K-automorphisms. If ρπ(M) is an invertible module, than K(M)π is retract rational over K. §5. Proof of Theorems 1.11 and 1.12 Proof of Theorem 1.12 ———————————- Step 1. Write G = A ⋊ G0 where G0 =< σ > is a cyclic group of order m, and A = A1 ×A2 × · · · ×Ar with each Ai =< ρi > being a cyclic group of order ei, e = e1 and er | er−1, · · · , e2 | e1. Define Bi =< ρi, · · · , ρ̂i, · · · , ρr > to be the subgroup of A by deleting the i-th direct factor. Let Gal(K(ζe)/K) =< τ > be a cyclic group of order n. Write ζ = ζe and τ ·ζ = ζ for some integer a. For 1 ≤ i ≤ r, choose ζi ∈< ζ > such that ζi is a primitive ei-th root of unity. Note that τ · ζi = ζ Let ni = [K(ζi) : K] for 1 ≤ i ≤ r. Note that n = n1. Step 2. Let W = g∈GK · x(g) be the representation space of the regular repre- sentation of G. Then K(G) = K(W )G = {K(ζ)(W )<τ>}G = K(ζ)(W )<G,τ> where the action of G and τ are extended to K(ζ)(W ) by requiring that G and τ act trivially on K(ζ) and W respectively. Step 3. For 1 ≤ i ≤ r, define l∈Zei , g∈Bi ζ li · x(ρ i · g). Note that ρi · ui = ζ i ui and, if j 6= i, then ρi · ui = ui. For 1 ≤ i ≤ r, write χi to be the character χi : A −→ K(ζ) × such that g · ui = χi(g) · ui. Since χ1, · · · , χr are distinct characters on A, it follows that u1, · · · , ur are linearly independent vectors in K(ζ)⊗K W . Moreover, the subgroup A acts faithfully on 1≤i≤rK(ζ)ui. Step 4. For 1 ≤ i ≤ r, j ∈ Zni , define vij = l∈Zei , g∈Bi i · x(ρ i · g). Thus we have elements v1,1, · · · , v1,n1 , v2,1, · · · , v2,n2, · · · , vr,nr ; in total we have n1 + n2 + · · ·+ nr such elements. As in Step 3, these elements vij are linearly independent. Note that τ · vij = vi,j+1. Step 5. For 1 ≤ i ≤ r, j ∈ Zni , k ∈ Zm. define xi,j,k = l∈Zei , g∈Bi i · x(σ k · ρli · g). We have in total m(n1 + n2 + · · ·+ nr) such elements xi,j,k. Note that xi,j,0 = vij , and the vector xi,j,k is just a “translation” of the vector xi,j,0 in the space g∈GK(ζ)x(g) (with basis x(g), g ∈ G). Thus these vectors xi,j,k are linearly independent. Note that σ · vi,j,k = vi,j,k+1. Moreover, the group < G, τ > acts faithfully on K(ξ) · xi,j,k. Apply Theorem 3.1. We find K(G) = K(ζ)(xijk : 1 ≤ i ≤ r, j ∈ Zni, k ∈ Zm)(w1, · · · , ws) where s = |G| −m(n1 + n2 + · · ·+ nr), and λ(wi) = wi for any i, any λ ∈< G, τ >. Step 6. We will consider the fixed fieldK(ζ)(xijk : 1 ≤ i ≤ r, j ∈ Zni , k ∈ Zm) <G,τ>. Let π =< σ, τ >, Λ = Z[π]. Let N =< xijk : 1 ≤ i ≤ r, j ∈ Zni , k ∈ Zm > be the multiplicative subgroup generated by these xijk in K(ζ)(xijk : 1 ≤ i ≤ r, j ∈ Zni , k ∈ Zm) \ {0}. Note that N is a Λ-module. In fact, N is a permutation π-lattice. Define Φ : N −→ Zn1 × Zn2 × · · · × Znr by Φ(x) = (l̄1, l̄2, · · · , l̄r) ∈ Zn1 × Zn2 × · · · × Znr , if x = ijk (with bijk ∈ Z) and ρ1(x) = ζ 1 x, ρ2(x) = ζ 2 x, · · ·, ρr(x) = ζ Define M = Ker(Φ). We find that K(ζ)(xijk : 1 ≤ i ≤ r, j ∈ Zni , k ∈ Zm) K(ζ)(M). Thus it remains to find K(ζ)(M)π. Note that M is a π-lattice. Step 7. Since gcd{m, n} = 1, it follows that π is a cyclic group. Hence ρπ(M) is invertible by Theorem 2.2. Apply Theorem 1.13. We find that K(ζ)(M)π is retract rational over K. Since K(G) is rational over K(ζ)(M)π, it follows that K(G) is also retract rational over K by [Sa3, Proposition 3.6(a), p.183]. � Proof of Theorem 1.11 ——————————- If charK = p, apply Theorem 3.2. Thus K(G) is rational. In particular it is retract rational. ¿From now on we will assume that charK 6= p. It is not difficult to verify that all the assumptions of Theorem 1.12 are valid in the present situation. Hence the result. Corollary 5.1. Let K be a field, p be any prime number, G = A⋊ G0 be a semi- direct product of p-groups where A is a normal abelian subgroup of exponent pe and G0 is a cyclic group of order pm. If charK 6= p, assume that (i) both K(ζpe) and K(ζpm) are cyclic extensions of K and (ii) p ∤ [K(ζpe) : K]. Then K(G) is retract rational over Proposition 5.2. Let K be a field and G = A ⋊ G0 be a semi-direct product. Assume that (i) A is an abelian normal subgroup of exponent e in G and G0 is a cyclic group of order m. (ii) either charK = 0 or charK > 0 with charK ∤ em, and (iii) ζe ∈ K and K(ζm) is a cyclic extension of K. If G −→ GL(V ) is a finite-dimensional linear representation of G over K, then K(V )G is retract rational over K. Proof. Decompose V into V = Vχ where χ : A −→ K \ {0} runs over linear characters of A and Vχ = {v ∈ V : g · v = χ(g) · v for all g ∈ A} It is easy to see that σ(Vχ) = Vχ′ for some χ ′. Suppose Vχ1, Vχ2, · · · , Vχs is an orbit of σ, i.e. σ(Vχj ) = Vχj+1 and σ(Vχs) = Vχ1. Choose a basis v1, · · · , vt of Vχ1. It follows that {σj(vi) : 1 ≤ i ≤ t, 0 ≤ j ≤ s− 1} is a basis of ⊕1≤j≤sVχj . In this way, we may find a basis w1, · · · , wn in V such that (i) for any g ∈ A, any 1 ≤ i ≤ r, g · wi = αwi for some α ∈ K; and (ii) σ · wi = wj for some j. It follows that K(V )A = K(w1, · · · , wn) A = K(u1, · · · , un) where u1, · · · , un are monomials in w1, · · · , wn. Thus K(u1, · · · , un) = K(M) for some G0-lattice M . It follows that K(V )G = {K(V )A}G0 = K(M)<σ>. By Theorem 2.2, ρG0(M) is invertible. Apply Theorem 1.12. � Remark. In the above theorem it is essential that G is a is a split extension group. For non-split extension groups, monomial actions (instead of merely purely monomial actions) may arise; see the proof of Theorem 1.3 [Ha] and also [Sa5]. §6. Proof of Theorem 1.9 We will prove Theorem 1.9 in this section. If charK = p, apply Theorem 3.2. We find that K(G) is rational. In particular, it is retract rational. Thus we will assume that charK 6= p from now on till the end of this section. If ζp ∈ K, we apply Theorem 1.4 to find that K(G) is rational. Hence we will consider only the situation [K(ζp) : K] = 2 in the sequel. Since p be an odd prime number, there is only one non-abelian p-group of order p3 with exponent p, and there are precisely two non-isomorphic non-abelian p-groups of order p4 with exponent p (see [CK, Section 2 and Section 3]). We will solve the rationality problem of these three groups separately. Let ζ = ζp, Gal(K(ζ)/K) = 〈τ〉 with τ · ζ = ζ Case 1. G = 〈σ1, σ2, σ3 : σ i = 1, σ1 ∈ Z(G), σ2σ3 = σ3σ1σ2〉. Step 1. Let W = g∈GK · x(g) be the representation space of the regular repre- sentation of G. Note that K(G) = K(x(g) : g ∈ G)G = {K(ζ)(x(g) : g ∈ G)〈τ〉}G = K(ζ)(x(g) : g ∈ G)〈G,τ〉. Step 2. For i ∈ Zp, define x0,i, x1,i ∈ g∈GK(ζ) · x(g) by x0,i = j,k∈Zp ζ−j−kx(σi3σ x1,i = j,k∈Zp ζj+k · x(σi3σ We find that σ1 : x0,i 7→ ζx0,i, x1,i 7→ ζ −1x1,i, σ2 : x0,i 7→ ζ i+1x0,i, x1,i 7→ ζ −i−1x1,i, σ3 : x0,i 7→ x0,i+1, x1,i 7→ x1,i+1, τ : x0,i ↔ x1,i. The restriction of the action of 〈σ1, σ2〉 toK(ζ)·x0,i andK(ζ)·x1,i is given by distinct characters of 〈σ1, σ2〉 to K(ζ) \ {0}. Hence x0,0, x0,1, x0,2, . . . , x0,p−1, x1,0, . . . , x1,p−1 are linearly independent vectors. Moreover, the action of 〈G, τ〉 on K(ζ)(x0,i, x1,i : i ∈ Zp) is faithful. By Theorem 3.1 K(ζ)(x(g) : g ∈ G) = K(ζ)(x0,i, x1,i : i ∈ Zp)(X1, . . . , Xl) where l = p3 − 2p and ρ(Xj) = Xj for any 1 ≤ j ≤ l, any ρ ∈ 〈G, τ〉. Step 3. Define x0 = x 0,0, y0 = x0,0x1,0 and xi = x0,i · x 0,i−1, yi = y0,i · y 0,i−1 for 1 ≤ i ≤ p − 1. It follows that K(ζ)(x0,i, x1,i : i ∈ Zp) 〈σ1〉 = K(ζ)(xi, yi : i ∈ Zp). Moreover, the actions of σ2, σ3, τ are given as σ2 : x0 7→ x0, y0 7→ y0, xi 7→ ζxi, yi 7→ ζ −1yi for 1 ≤ i ≤ p− 1, σ3 : x0 7→ x0x 1, x1 7→ x2 7→ · · · 7→ xp−1 7→ (x1x2 · · ·xp−1) y0 7→ y0y1x1, y1 7→ y2 7→ · · · 7→ yp−1 7→ (y1y2 · · · yp−1) τ : x0 7→ y 0 , y0 7→ y0, xi ↔ yi for 1 ≤ i ≤ p− 1. Step 4. Define X = x0y −(p−1)/2 0 , Y = x (p+1)/2 Then K(ζ)(xi, yi : i ∈ Zp) = K(ζ)(X, Y, xi, yi : 1 ≤ i ≤ p − 1) and σ2(X) = X , σ2(Y ) = Y , σ3(X) = αX , σ3(Y ) = βY , τ : X ↔ Y where α, β ∈ K(ζ)(xi, yi : 1 ≤ i ≤ p− 1)\{0}. Apply Theorem 3.1. We may find X̃ , Ỹ so that K(ζ)(xi, yi : i ∈ Zp) = K(ζ)(xi, yi : 1 ≤ i ≤ p − 1)(X̃, Ỹ ) with ρ(X̃) = X̃ , ρ(Ỹ ) = Ỹ for any ρ ∈ 〈σ2, σ3, τ〉. Thus it remains to consider K(ζ)(xi, yi : 1 ≤ i ≤ p− 1) 〈σ2,σ3,τ〉. Step 5. Define u0 = x 1, v0 = x1y1, ui = xi+1x i , vi = yi+1y i for 1 ≤ i ≤ p− 2. It follows that K(ζ)(xi, yi : 1 ≤ i ≤ p − 1) 〈σ2〉 = K(ζ)(ui, vi : 0 ≤ i ≤ p − 2). The actions of σ3 and τ are given by σ3 : u0 7→ u0u 1, v0 7→ v0v1u1, u1 7→ u2 7→ · · · 7→ up−2 7→ (u0u 2 · · ·u v1 7→ v2 7→ · · · 7→ vp−2 7→ u0(v 1 · · · v τ : u0 7→ u 0, v0 7→ v0, ui ↔ vi for 1 ≤ i ≤ p− 2. Step 6. Define up−1 = (u0u 2 · · ·u −1, wi = v0v1 · · · viu1u2 · · ·ui for 1 ≤ i ≤ p − 2, and wp−1 = (v 1 · · · vp−2u 2 · · ·up−2) −1. We find that K(ui, vi : 0 ≤ i ≤ p− 2) = K(ui, wi : 1 ≤ i ≤ p− 1) and σ3 : u1 7→ u2 7→ · · · 7→ up−1 7→ (u1u2 · · ·up−1) w1 7→ w2 7→ · · · 7→ wp−1 7→ (w1w2 · · ·wp−1) τ : ui 7→ wi(uiwi−1) −1, wi 7→ wi for 1 ≤ i ≤ p− 1. where we write w0 = v0 for convenience. (Granting that w0 is defined as above, we have a relation w0w1 · · ·wp−1 = 1. But we don’t have the relation u0u1 · · ·up−1 = 1 because we define u0, u1, . . . , up−2 first and up−1 is defined by another way.) We will study whether K(ζ)(ui, wi : 1 ≤ i ≤ p− 1) 〈σ3,τ〉 is rational or not. Step 7. The multiplicative action in Step 6 can be formulated in terms of π-lattices where π = 〈τ, σ3〉 as follows. LetM = ( 1≤i≤p−1 Z·ui)⊕( 1≤i≤p−1Z·wi) and define w0 = −w1−w2−· · ·−wp−1. Define a Z[π]-module structure on M by σ3 : u1 7→ u2 7→ · · · 7→ up−1 7→ −u1 − u2 − · · · − up−1, w1 7→ w2 7→ · · · 7→ wp−1 7→ −w1 − w2 − · · · − wp−1, τ : ui 7→ −ui + wi − wi−1, wi 7→ wi for 1 ≤ i ≤ p− 1. We claim that M ≃ Z[π1] Z[π2]/Φp(σ3) with π1 = 〈τ〉, π2 = 〈σ3〉. Throughout this step, we will write σ = σ3 and π =< σ, τ >. Define ρ = στ . Then ρ is a generator of π with order 2p where p is an odd prime number. Let Φp(T ) ∈ Z[T ] be the p-th cyclotomic polynomial. Note that Φp(σ)(ui) = Φp(σ)(wi) = 0. It follows that Φp(σ) ·M = 0. Since Φp(σ 0≤i≤p−1 σ i = Φp(σ), we know that Φp(σ 2) ·M = 0. From ρ2 = σ2, we find that Φp(ρ 2) ·M = 0. It is well-known that Φp(T 2) = Φp(T )Φ2p(T ) (see [Ka1, Theorem 1.1] for example). We conclude that Φp(ρ)Φ2p(ρ) ·M = 0. In other words, we may regard M as a module over Λ = Z[π]/ < Φp(ρ)Φ2p(ρ) >. Clearly Λ is isomorphic to Z[π1] Z[π2]/Φp(σ) where π1 = 〈τ〉, π2 = 〈σ〉. It remains to show that M is isomorphic to Λ as a Λ-module. It is not difficult to verify that Φ2p(ρ)(u1) = f(ρ)(w1) where f(T ) = Φ2p(T )−T Define v = u1 − w1. We find that Φ2p(ρ)(v) = −w0. Since M is a Λ-module generated by u1 and w0, it follows that, as a Λ-module, M =< u1, w0 >=< u1 − σw0, w0 > =< u1 − w1, w0 >=< v,w0 > =< v,w0 + Φ2p(ρ)(v) >=< v,w0 − w0 >=< v >, i.e. M is a cyclic Λ-module generated by v. Thus we get an epimorphism Λ → M , which is an isomorphism by counting the Z-ranks of both sides. Hence the result. Step 8. By Step 7, M ≃ Z[π1] Z[π2]/Φp(σ3) where π1 = 〈τ〉, π2 = 〈σ3〉. Thus we may choose a Z-basis Y1, Y2, . . . , Yp−1, Z1, . . . , Zp−1 for M such that σ3 : Y1 7→ Y2 7→ · · · 7→ Yp−1 7→ −Y1 − · · · − Yp−1, Z1 7→ Z2 7→ · · · 7→ Zp−1 7→ −Z1 − · · · − Zp−1, τ : Yi ↔ Zi for 1 ≤ i ≤ p− 1. Hence K(ζ)(M) = K(ζ)(Yi, Zi : 1 ≤ i ≤ p− 1). We emphasize that σ3 acts on Yi, Zi by multiplicative actions on the field K(ζ)(M) and σ3 · ζ = ζ , τ · ζ = ζ Step 9. In the field K(ζ)(M), define s0 = 1 + Y1 + Y1Y2 + Y1Y2Y3 + · · ·+ Y1Y2 · · ·Yp−1, t0 = 1 + Z1 + Z1Z2 + · · ·+ Z1Z2 · · ·Zp−1, s1 = (1/s0)− (1/p), s2 = (Y1/s0)− (1/p), . . . , sp−1 = (Y1Y2 · · ·Yp−2/s0)− (1/p), t1 = (1/t0)− (1/p), t1 = (Z1/t0)− (1/p), . . . , tp−1 = (Z1Z2 · · ·Zp−2/t0)− (1/p). It is easy to verify that K(ζ)(M) = K(ζ)(si, ti : 1 ≤ i ≤ p− 1) and σ3 : s1 7→ s2 7→ · · · 7→ sp−1 7→ −s1 − s2 − · · · − sp−1, t1 7→ t2 7→ · · · 7→ tp−1 7→ −t1 − t2 − · · · − tp−1, τ : si ↔ ti. Step 10. Define ri = si + ti for 1 ≤ i ≤ p − 1. Then K(ζ)(si, ti : 1 ≤ i ≤ p− 1) = K(ζ)(si, ri : 1 ≤ i ≤ p− 1). Note that σ3 : r1 7→ r2 7→ · · · 7→ rp−1 7→ −r1 − r2 − · · · − rp−1, τ : ri 7→ ri, si 7→ −si + ri. Apply Theorem 3.1. We find that K(ζ)(si, ri : 1 ≤ i ≤ p− 1) = K(ζ)(ri : 1 ≤ i ≤ p−1)(A1, . . . , Ap−1) for some A1, . . . , Ap−1 with σ3(Ai) = τ(Ai) = Ai for 1 ≤ i ≤ p−1. Thus K(ζ)(ri, si : 1 ≤ i ≤ p − 1) 〈τ〉 = K(ζ)(ri : 1 ≤ i ≤ p − 1) 〈τ〉(A1, . . . , Ap−1) = K(r1, . . . , rp−1, A1, . . . , Ap−1). It remains to find K(r1, . . . , rp−1) 〈σ3〉. Step 11. Write π2 = 〈σ3〉. The π2-fieldsK(r1, . . . , rp−1, A1) andK(B0, B1, . . . , Bp−1) are π2-isomorphic where σ3 : B0 7→ B1 7→ · · · 7→ Bp−1 7→ B0. For, we may define B = B0 +B1 + · · ·+Bp−1 and Ci = Bi − (B/p) for 0 ≤ i ≤ p− 1. In other words K(r1, . . . , rp−1, A1, . . . , Ap−1) 〈σ3〉 = K(B0, B1, . . . , Bp−1) 〈σ3〉(A2, . . ., Ap−1) = K(Zp)(A2, . . . , Ap−1). By Lemma 3.5, K(Zp) = K(ζ)(N) π1 where ζ = ζp, π1 = Gal(K(ζ)/K) = 〈τ〉 with τ · ζ = ζ−1, and N is some π1-lattice. By Reiner’s Theorem [Re], the π1-lattice N is a direct sum of lattices of three types: (1) τ : z 7→ −z, (2) τ : z 7→ z, (3) τ : z1 ↔ z2. Thus we find K(ζ)(N) 〈τ〉 = K(ζ)(z1, . . . , za, z 2, . . . , z 1 , w 1 , . . . , z c , w c ) where τ : z1 7→ 1/z1, . . ., za 7→ 1/za, z 1 7→ z . . ., z′b 7→ z 1 ↔ w 1 , . . ., z c ↔ z By Theorem 3.1 we may “neglect” the roles of z′1, . . . , z 1 , w 1 , . . . , z c , w c . We may linearize the action on z1, . . . , za by defining wi = 1/(1 + zi) when 1 ≤ i ≤ a. Then τ : wi 7→ −wi + 1. Thus we may “neglect” the roles of wi by Theorem 3.1 again. In conclusion, K(ζ)(z1, . . . , za, z 1, . . . , z 1 , w 1 , . . . , z c , w 〈τ〉 is rational over K. � Case 2. G = 〈σ1, σ2, σ3, σ4 : σ i = 1, σ1, σ2 ∈ Z(G), σ 4 σ3σ4 = σ1σ3〉. The proof is very similar to Case 1. Step 1. For i ∈ Zp, define x0,i, x1,i, y0,i, y1,i ∈ g∈GK(ζ) · x(g) by x0,i = j,k,l∈Zp ζ−j−k−lx(σi4σ x1,i = j,k,l∈Zp ζj+k+lx(σi4σ y0,i = j,k,l∈Zp ζ−j−k+lx(σi4σ y1,i = j,k,l∈Zp ζj+k−lx(σi4σ The action of 〈G, τ〉 is given by σ1 : x0,i 7→ ζx0,i, x1,i 7→ ζ −1x1,i, y0,i 7→ ζy0,i, y1,i 7→ ζ −1y1,i, σ2 : x0,i 7→ ζx0,i, x1,i 7→ ζ −1x1,i, y0,i 7→ ζ −1y0,i, y1,i 7→ ζy1,i, σ3 : x0,i 7→ ζ i+1x0,i, x1,i 7→ ζ −i−1x1,i, y0,i 7→ ζ i+1y0,i, y1,i 7→ ζ −i−1y1,i, σ4 : x0,i 7→ x0,i+1, x1,i 7→ x1,i+1, y0,i 7→ y0,i+1, y1,i 7→ y1,i+1, τ : x0,i ↔ x1,i, y0,i ↔ y1,i. Note that x0,i, x1,i, y0,i, y1,i where i ∈ Zp are linearly independent vectors in g∈GK(ζ) · x(g). Apply Theorem 3.1. It suffices to consider the rationality prob- lem of K(ζ)(x0,i, x1,i, y0,i, y1,i : i ∈ Zp) 〈G,τ〉. Step 2. Define x0 = x 0,0, y0 = x0,0x1,0, X0 = y0,0(x0,0) −1, Y0 = y1,0(x1,0) −1; for 1 ≤ i ≤ p − 1, define xi = x0,i(x0,i−1) −1, yi = x1,i(x1,i−1) −1, Xi = y0,i(y0,i−1) −1, Yi = y1,i(y1,i−1) −1. Then K(ζ)(x0,i, x1,i, y0,i, y1,i : i ∈ Zp) 〈σ1〉 = K(ζ)(xi, yi, Xi, Yi : i ∈ Zp) and the actions are given by σ2 : x0 7→ x0, y0 7→ y0, X0 7→ ζ −2X0, Y0 7→ ζ All the other generators are fixed by σ2; σ3 : x0 7→ x0, y0 7→ y0, X0 7→ X0, Y0 7→ Y0, xi 7→ ζxi, yi 7→ ζ −1yi, Xi 7→ ζXi, Yi 7→ ζ −1Yi, σ4 : x0 7→ x0x 1, y0 7→ y0y1x1, X0 7→ X0X1x 1 , Y0 7→ Y0Y1y x1 7→ x2 7→ x3 7→ · · · 7→ xp−1 7→ (x1x2 · · ·xp−1) Similarly for y1, y2, . . . , yp−1, X1, . . . , Xp−1, Y1, . . . , Yp−1; τ : x0 7→ y 0 , y0 7→ y0, X0 7→ Y0 7→ X0, xi ↔ yi, Xi ↔ Yi for 1 ≤ i ≤ p− 1. Step 3. Define x̃ = x0y −(p−1)/2 0 , ỹ = x (p+1)/2 Since x̃, ỹ are fixed by both σ2 and σ3, while σ4 : x̃ 7→ αx̃, ỹ 7→ βỹ, τ : x̃ ↔ ỹ where α, β ∈ K(x1, y1, X1, Y1)\{0}, we may apply Theorem 3.1. Thus the roles of x̃, ỹ may be “neglected”. It suffices to consider whether K(ζ)(X0, Y0, xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1)〈σ2,σ3,σ4,τ〉 is rational over K. Step 4. Define X̃ = X 0 , Ỹ = X0Y0. Then K(ζ)(X0, Y0, xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1)〈σ2〉 = K(ζ)(X̃, Ỹ , xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1). Moreover, the actions on X̃ and Ỹ are given by σ3 : X̃ 7→ X̃, Ỹ 7→ Ỹ , σ4 : X̃ 7→ X̃X 1 , Ỹ 7→ Ỹ X1Y1x τ : X̃ 7→ X̃−1Ỹ p, Ỹ 7→ Ỹ . Define X ′ = X̃Ỹ −(p−1)/2, Y ′ = X̃−1Ỹ (p+1)/2. As in Step 3, we may apply The- orem 3.1 and “neglect” the roles of X ′ and Y ′. It remains to make sure whether K(ζ)(xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1) 〈σ3,σ4,τ〉 is rational. Define u0 = x 1, v0 = x1y1; for 1 ≤ i ≤ p−2, define ui = xi+1(x i ), vi = yi+1 · (y and for 1 ≤ i ≤ p−1, define Ui = Xix i , Vi = Yiy i . It follows that K(ζ)(xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1)〈σ3〉 = K(ζ)(u0, v0, Up−1, Vp−1, ui, vi, Ui, Vi : 1 ≤ i ≤ p− 2). Note that the actions of σ4 and τ are given by σ4 : u0 7→ u0u 1, v0 7→ v0v1u1, u1 7→ u2 7→ · · · 7→ up−2 7→ (u0u 2 · · ·u v1 7→ v2 7→ · · · 7→ vp−2 7→ u0(v 1 · · · v U1 7→ U2 7→ · · · 7→ Up−1 7→ (U1U2 · · ·Up−1) V1 7→ V2 7→ · · · 7→ Vp−1 7→ (V1V2 · · ·Vp−1) τ : u0 7→ u 0, v0 7→ v0, ui ↔ vi for 1 ≤ i ≤ p− 2, Ui ↔ Vi for 1 ≤ i ≤ p− 1. Note that the actions of σ4 and τ on U1, U2, . . . , Up−1, V1, . . . , Vp−1 may be linearized by the same method as in Step 9 of Case 1. Hence we may apply Theorem 3.1 and “neglect” the roles of Ui, Vi for 1 ≤ i ≤ p− 1. In conclusion, all we need is to prove that K(ζ)(ui, vi : 0 ≤ i ≤ p−2) 〈σ4,τ〉 is rational over K. Compare the present situation with that of Step 5 of Case 1. We have the same generators and the same actions (and even the same notation). Thus we finish the proof. � Case 3. G = 〈σ1, σ2, σ3, σ4 : σ i = 1, σ1 ∈ Z(G), σ2σ3 = σ3σ2, σ 4 σ2σ4 = σ1σ2, σ 4 σ3σ4 = σ2σ3〉. Step 1. For i ∈ Zp, define x0,i, x1,i, y0,i, y1,i ∈ g∈GK(ζ) · x(g) by x0,i = j,k,l∈Zp ζ−j−k−lx(σi4σ x1,i = j,k,l∈Zp ζj+k+lx(σi4σ y0,i = j,k,l∈Zp ζ−j−k+lx(σi4σ y1,i = j,k,l∈Zp ζj+k−lx(σi4σ The action of 〈G, τ〉 is given by σ1 : x0,i 7→ ζx0,i, x1,i 7→ ζ −1x1,i, y0,i 7→ ζ −1y0,i, y1,i 7→ ζy1,i, σ2 : x0,i 7→ ζ i+1x0,i, x1,i 7→ ζ −i−1x1,i, y0,i 7→ ζ −i+1y0,i, y1,i 7→ ζ i−1y1,i, σ3 : x0,i 7→ ζ i+1x0,i, x1,i 7→ ζ −i−1x1,i, y0,i 7→ ζ i+1y0,i, y1,i 7→ ζ −i−1y1,i, σ4 : x0,i 7→ x0,i+1, x1,i 7→ x1,i+1, y0,i 7→ y0,i+1, y1,i 7→ y1,i+1, τ : x0,i ↔ x1,i, y0,i ↔ y1,i. Checking the restriction of 〈σ1, σ2, σ3〉 as in Step 1 of Case 2, we find that these vectors x0,i, x1,i, y0,i, y1,i are linearly independent except possibly for the case x0,p−2 and y1,0, and the case x1,p−2 and y0,0. But it is easy to see that these vectors are linearly independent, because their ”supports” are disjoint. Apply Theorem 3.1. It suffices to consider K(ζ)(x0,i, x1,i, y0,i, y1,i : i ∈ Zp) 〈G,τ〉. Step 2. Define x0 = x 0,0, y0 = x0,0x1,0, X0 = x0,0 · y0,0, Y0 = x1,0y1,0; and for 1 ≤ i ≤ p − 1, define xi = x0,i(x0,i−1) −1, yi = x1,i(x1,i−1) −1, Xi = y0,i(y0,i−1) −1, Yi = y1,i(y1,i−1) −1. Then K(ζ)(x0,i, x1,i, y0,i, y1,i : i ∈ Zp) 〈σ1〉 = K(ζ)(xi, yi, Xi, Yi : i ∈ Zp) and the actions are given by σ2 : x0 7→ x0, y0 7→ y0, X0 7→ ζ 2X0, Y0 7→ ζ −2Y0, xi 7→ ζxi, yi 7→ ζ −1yi, Xi 7→ ζ −1Xi, Yi 7→ ζYi, σ3 : x0 7→ x0, y0 7→ y0, X0 7→ ζ 2X0, Y0 7→ ζ −2Y0, xi 7→ ζxi, yi 7→ ζ −1yi, Xi 7→ ζXi, Yi 7→ ζ −1Yi, σ4 : x0 7→ x0x 1, y0 7→ y0y1X1, X0 7→ X0X1x1, Y0 7→ Y0Y1y1, x1 7→ x2 7→ x3 7→ · · · 7→ xp−1 7→ (x1x2 · · ·xp−1) Similarly for y1, y2, . . . , yp−1, X1, . . . , Xp−1, Y1, . . . , Yp−1. τ : x0 7→ y 0 , y0 7→ y0, X0 7→ Y0 7→ X0, xi ↔ yi, Xi ↔ Yi for 1 ≤ i ≤ p− 1. Step 3. Define x̃ = x0y −(p−1)/2 0 , ỹ = x (p+1)/2 0 . Apply Theorem 3.1 again. It suffices to consider K(ζ)(X0, Y0, xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1) 〈σ2,σ3,σ4,τ〉. We may apply Theorem 3.1 again to “neglect” X0 and Y0. Thus it suffices to consider K(ζ)(xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1) 〈σ2,σ3,σ4,τ〉. Step 4. Define u0 = x 1, v0 = x1y1, U0 = x1X1, V0 = y1Y1; and for 1 ≤ i ≤ p−2, de- fine ui = x i xi+1, vi = y i yi+1, Ui = X i Xi+1, Vi = Y i Yi+1. ThenK(ζ)(xi, yi, Xi, Yi : 1 ≤ i ≤ p− 1)〈σ2〉 = K(ζ)(ui, vi, Ui, Vi : 0 ≤ i ≤ p− 2). Note that the actions of σ3, σ4, τ are given by σ3 : u0 7→ u0, v0 7→ v0, U0 7→ ζ 2U0, V0 7→ ζ −2U0, All the other generators are fixed by σ3. σ4 : u0 7→ u0u 1, v0 7→ v0v1u1, U0 7→ U0U1u1, V0 7→ V0V1v1, u1 7→ u2 7→ · · · 7→ up−2 7→ (u0u 2 · · ·u v1 7→ v2 7→ · · · 7→ vp−2 7→ u0(v 1 · · · v U1 7→ U2 7→ · · · 7→ Up−2 7→ u0(U 1 · · ·U V1 7→ V2 7→ · · · 7→ Vp−2 7→ u 1 · · ·V τ : u0 7→ v 0 , v0 7→ v0, U0 7→ V0 7→ U0, ui ↔ vi, Ui ↔ Vi for 1 ≤ i ≤ p− 2. Step 5. Define R0 = U 0 , S0 = U0V0; and for 1 ≤ i ≤ p− 2, define Ri = Ui, Si = Vi. Then K(ζ)(ui, vi, Ui, Vi : 0 ≤ i ≤ p − 2) 〈σ3〉 = K(ζ)(ui, vi, Ri, Si : 0 ≤ i ≤ p − 2). We will write the actions of σ4 and τ on Ri, Si as follows. σ4 : R0 7→ R0R 1, S0 7→ S0S1R1u1v1, R1 7→ R2 7→ · · · 7→ Rp−2 7→ u0(R0R 2 · · ·R S1 7→ S2 7→ · · · 7→ Sp−2 7→ u 0R0(S 1 · · ·S τ : R0 7→ S 0 , S0 7→ S0, Ri ↔ Si for 1 ≤ i ≤ p− 2. Step 6. Imitate the change of variables in Step 6 of Case 1. We define up−1 = 2 · · ·u −1, Rp−1 = u0(R0R 2 · · ·R −1; and for 1 ≤ i ≤ p − 2, define wi = v0v1 · · · viu1u2 · · ·ui, Ti = S0S1 · · ·SiR1R2 · · ·Ri; define wp−1 = (v 1 · · · vp−2u 2 · · ·up−2) −1, Tp−1 = u1v1v 1 · · ·Sp−2R 2 · · ·Rp−2) We find that K(ui, vi, Ri, Si : 0 ≤ i ≤ p− 2) = K(ui, wi, Ri, Ti : 1 ≤ i ≤ p− 1) and σ4 : u1 7→ u2 7→ · · · 7→ up−1 7→ (u1u2 · · ·up−1) w1 7→ w2 7→ · · · 7→ wp−1 7→ (w1w2 · · ·wp−1) R1 7→ R2 7→ · · · 7→ Rp−1 7→ (R1R2 · · ·Rp−1) T1 7→ T2 7→ · · · 7→ Tp−1 7→ (T1T2 · · ·Tp−1) τ : wi 7→ wi, Ti 7→ Ti, ui 7→ wi(uiwi−1) −1, Ri 7→ Ti(RiTi−1) where we write w0 = v0, T0 = S0 for convenience. Step 7. The multiplicative action in Step 6 can be formulated as follows. Let π = 〈τ, σ4〉 and define a π-lattice as in Step 7 of Case 1 (but σ3 should be replaced by σ4 in the present situation). Then K(ζ)(ui, wi, Ri, Ti : 1 ≤ i ≤ p − 1) K(ζ)(M ⊕M)π where M is the same lattice in Step 7 of Case 1. The structure ofM has been determined in Step 7 of Case 1. Thus M⊕M ≃ Λ⊕Λ where Λ ≃ Z[π1] ⊗Z Z[π2]/Φp(σ4) where π1 = 〈τ〉 and π2 = 〈σ4〉. It follows that we can find elements Y1, . . . , Yp−1, Z1, . . . , Zp−1, W1, . . . ,Wp−1, Q1, . . . , Qp−1 in the field K(ζ)(ui, wi, Ri, Ti : 1 ≤ i ≤ p − 1) so that K(ζ)(ui, wi, Ri, Ti : 1 ≤ i ≤ p − 1) = K(ζ)(Y1, . . . , Yp−1, Z1, . . . , Zp−1,W1, . . . ,Wp−1, Q1, . . . , Qp−1) and the actions of σ4 and τ are given by σ4 : Y1 7→ · · · 7→ Yp−1 7→ (Y1Y2 · · ·Yp−1) Similarly for Z1, . . . , Zp−1, W1, . . . ,Wp−1, Q1, . . . , Qp−1; τ : Yi ↔ Zi, Wi ↔ Qi. Step 8. We can linearize the actions of σ4 and τ by the same method in Step 9 of Case 1. Apply Theorem 3.1. We may “neglect” the roles of Wi, Qi. 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0704.1801

A calculation of the shear viscosity in SU(3) gluodynamics

MIT-CTP 3830 A calculation of the shear viscosity in SU(3) gluodynamics Harvey B. Meyer∗ Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A. (Dated: October 22, 2018) We perform a lattice Monte-Carlo calculation of the two-point functions of the energy-momentum tensor at finite temperature in the SU(3) gauge theory. Unprecedented precision is obtained thanks to a multi-level algorithm. The lattice operators are renormalized non-perturbatively and the clas- sical discretization errors affecting the correlators are corrected for. A robust upper bound for the shear viscosity to entropy density ratio is derived, η/s < 1.0, and our best estimate is η/s = 0.134(33) at T = 1.65Tc under the assumption of smoothness of the spectral function in the low-frequency region. PACS numbers: 12.38.Gc, 12.38.Mh, 25.75.-q Introduction.— Models treating the system produced in heavy ion collisions at RHIC as an ideal fluid have had significant success in describing the observed flow phe- nomena [1, 2]. Subsequently the leading corrections due to a finite shear viscosity were computed [3], in parti- cular the flattening of the elliptic flow coefficient v2(pT) above 1GeV. It is therefore important to compute the QCD shear and bulk viscosities from first principles to establish this description more firmly. Small transport coefficients are a signature of strong interactions, which lead to efficient transmission of momentum in the system. Strong interactions in turn require non-perturbative com- putational techniques. Several attempts have been made to compute these observables on the lattice in the SU(3) gauge theory [4, 5]. The underlying basis of these calcu- lations are the Kubo formulas, which relate each trans- port coefficient to a spectral function ρ(ω) at vanishing frequency. Even on current computers, these calcula- tions are highly non-trivial, due to the fall-off of the re- levant correlators in Euclidean time (as x−50 at short dis- tances), implying a poor signal-to-noise ratio in a stan- dard Monte-Carlo calculation. The second difficulty is to solve the ill-posed inverse problem for ρ(ω) given the Euclidean correlator at a finite set of points. Mathemati- cally speaking, the uncertainty on a transport coefficient χ is infinite for any finite statistical accuracy, because adding ǫωδ(ω) to ρ(ω) merely corresponds to adding a constant to the Euclidean correlator of order ǫ, while ren- dering χ infinite. Therefore smoothness assumptions on ρ(ω) have to be made, which are reasonable far from the one-particle energy eigenstates, and can be proved in the hard-thermal-loop framework [6]. In this Letter we present a new calculation which dra- matically improves on the statistical accuracy of the Eu- clidean correlator relevant to the shear viscosity through the use of a two-level algorithm [16]. This allows us to derive a robust upper bound on the viscosity and a use- ∗Electronic address: [email protected] ful estimate of the ratio η/s, which has acquired a special significance since its value 1/4π in a class of strongly cou- pled supersymmetric gauge theories [7] was conjectured to be an absolute lower bound for all substances [8]. Methodology.— In the continuum, the energy- momentum tensor Tµν(x) = F να − 14δµνF ρσ , be- ing a set of Noether currents associated with translations in space and time, does not renormalize. With L0 = 1/T the inverse temperature, we consider the Euclidean two- point function (0 < x0 < L0) C(x0) = L d3x 〈T 12(0)T 12(x0,x)〉. (1) The tree-level expression is Ct.l.(x0) = f(τ)− π with τ = 1 − 2x0 , dA = 8 the number of gluons and f(z) = ds s4 cosh2(zs)/ sinh2 s. The correlator C(x0) is thus dimensionless and, in a conformal field theory, would be a function of Tx0 only. The spectral function is defined by C(x0) = L coshω(1 L0 − x0) sinh ωL0 dω. (2) The shear viscosity is given by [4, 13] η(T ) = π . (3) Important properties of ρ are its positivity, ρ(ω)/ω ≥ 0 and parity, ρ(−ω) = −ρ(ω). The spectral function that reproduces Ct.l.(x0) is ρt.l.(ω) = At.l. ω tanh 1 +BL−40 ωδ(ω), (4) At.l. = (4π)2 , B = dA. (5) While the ω4 term is expected to survive in the inter- acting theory with only logarithmic corrections, the δ- function at the origin corresponds to the fact that gluons http://arxiv.org/abs/0704.1801v1 mailto:[email protected] are asymptotic states in the free theory and implies an infinite viscosity. On the lattice, translations only form a discrete group, so that a finite renormalization is necessary, Tµν(g0) = Z(g0)T (bare) µν . We employ the Wilson action [9], Sg = x,µ6=ν Tr {1 − Pµν(x)}, on an L0 · L3 hypertoroidal lattice, and the following discretized expression of the Euclidean energy: (bare) 00 (x) ≡ a4g20 ReTrPkl(x)− ReTrP0k(x) One of the lattice sum rules [10] can be interpreted as a non-perturbative renormalization condition for this par- ticular discretization, from which we read off Z(g0) = g20(cσ − cτ ). The definition of the anisotropy coeffi- cients cσ,τ can be found in [12], where they are computed non-perturbatively. With a precision of about 1%, a Padé fit constrained by the one-loop result [11] yields Z(g0) = 1− 1.0225g20 + 0.1305g40 1− 0.8557g20 , (6/g20 ≥ 5.7). (6) Numerical results.— We report results obtained on a β ≡ 6/g20 = 6.2, 8 · 203 lattice and on a β = 6.408, 8 · 283 lattice. The first is thus at a temperature of 1.24Tc, the second at T = 1.65Tc. We use the results for aTc obtained in [14] and the non-perturbative lattice β-function of [15] to determine this. We employ the two-level algorithm described in [16]. The computing time invested into the 1.65Tc simulation is about 860 PC days. Following [4], we discretize 1 〈(T 11 −T 22)(T 11 −T 22)〉 instead of 〈T 12T 12〉 (the two are equal in the continuum) to write C(x0) = 〈Oη(0)Oη(x0)〉+O(a2), where Oη(x0) ≡ 12a {T 11 − T 22}(g0, x) 2Z(g0) a g20 ReTr {P10 + P13 − P20 − P23}(x). The three electric-electric, magnetic-magnetic and electric-magnetic contributions to C(x0) are computed separately and shown on Fig. 1. We apply the follow- ing technique to remove the tree-level discretization er- rors [17] separately to CBB, CEE and CEB . Firstly, x̄0 is defined such that Ct.l.cont(x̄0) = C lat (x0). The improved correlator is defined at a discrete set of points through C(x̄0) = C(x0), and then augmented to a continuous function via C(x̄ 0 ) = α + βC cont(x̄ 0 ), i = 1, 2, where 0 and x̄ 0 correspond to two adjacent measurements. The resulting improved correlator, normalized by the continuum tree-level result, is shown on Fig. 2. One ob- serves that the deviations from the tree-level result are surprisingly small, while deviations from conformality are visible. The latter is not unexpected at these tempera- tures, where p/T 4 is still strongly rising [18]. Finite- volume effects on the T = 1.65Tc lattice are smaller than 1000 0 0.1 0.2 0.3 0.4 0.5 C**(x0) FIG. 1: The correlators that contribute to C(x0) = (CBB + CEE + 2CEB). Filled symbols correspond to T = 1.65Tc, open symbols to 1.24Tc. Error bars are smaller than the data symbols. 0.2 0.3 0.4 0.5 C(x0) / C t.l.(x0) 1.65Tc 1.24Tc FIG. 2: The tree-level improved correlator C(x0) normalized to the tree-level continuum infinite-volume prediction. The four points in each sequence are strongly correlated, but their covariance matrix is non-singular. one part in 103 at tree-level. Non-perturbatively, at the same temperature with resolution L0/a = 6, increasing L/a from 20 to 30 reduces C(L0/2) by a factor 0.922(73). While not statistically compelling as it stands, the effect deserves further investigation. The entropy density is obtained from the relation s = (ǫ + p)/T and the standard method to compute ǫ + p ([12], Eq.1.14). We find s/T 3 = 4.72(3)(5) and 5.70(2)(6) respectively at T/Tc = 1.24 and 1.65 (the first error is statistical and the second is the uncertainty on Z(g0)). The Stefan-Boltzmann value is 32π 2/45 in the continuum and 1.0867 times that value [12] at L0/a = 8. Unsatisfactory attempts to extract the viscosity.— In order to compare with previous studies [4, 5], we fit C(x0) with a Breit-Wigner ansatz ρ(ω)/ω = 1 + b2(ω − ω0)2 1 + b2(ω + ω0)2 , (7) although it clearly ignores asymptotic freedom, which im- plies that ρ(ω) ∼ ω4 at ω ≫ T [6]. The result of a cor- related fit at T = 1.65Tc using the points at Tx0 = 0.5, 0.35 and 0.275 is a3F = 0.78(4), (b/a)2 = 240(30) and aω0 = 2.36(4), and hence η/s|T=1.65Tc = 0.33(3). A comparison of this to the results of Ref. [5] illustrates the progress made in statistical accuracy. An ansatz motivated by the hard-thermal-loop frame- work is [6] ρ(ω)/ω = 1 + b2ω2 + θ(ω − ω1) tanhω/4T . (8) It is capable of reproducing the tree-level prediction, Eq. 4, and it allows for a thermal broadening of the delta function at the origin. Fitting the T = 1.65Tc points shown on Fig. 2, the χ2 is minimized for b = 0 (effec- tively eliminating a free parameter), A/At.l. = 0.996(8), ω1/T = 7.5(2) and η/s = 0.25(3), with χ min = 4.0. Thus while the ansatz is hardly compatible with the data, it shows that the data tightly constrains the coefficient A to assume its tree-level value. A bound on the viscosity.— The positivity property of ρ(ω) allows us to derive an upper bound on the viscosity, based on the following assumptions: 1. the contribution to the correlator from ω > Λ is correctly predicted by the tree-level formula 2. the width of any potential peak in the region ω < T is no less than O(T ). The standard QCD sum rule practice is to use perturba- tion theory from the energy lying midway between the lightest state and the first excitation. With this in mind we choose Λ = max(1 [M2 +M2∗ ] ≈ 2.6GeV, 5T ), where M2(∗) are the masses of the two lightest tensor glueballs. Perturbation theory predicts a Breit-Wigner centered at the origin of width Γ = 2γ [6], where γ ≈ αsNT is the gluon damping rate. To derive the upper bound we con- servatively assume that for ω < 2T , ρ(ω)/ω is a Breit- Wigner of width Γ = T centered at the origin. From L0) ≥ L50 ρBW (ω) + ρt.l.(ω) sinhωL0/2 obtain (with 90% statistical confidence level) η/s < 0.96 (T = 1.65Tc) 1.08 (T = 1.24Tc). The spectral function.— As illustrated above, it is rather difficult to find a functional form for ρ(ω) that is both physically motivated and fits the data. In a more model- independent approach, ρ(ω) is expanded in an orthogo- nal set of functions, which grows as the lattice resolution on the correlator increases, and becomes complete in the 0 5 10 15 20 25 ρ(ω) K(x0=1/2T,ω)/T T=1.24Tc T=1.65Tc FIG. 3: The result for ρ(ω). The meaning of the error bands and the curves is described in the text. The area under them equals C(L0/2) = 8.05(31) and 9.35(42) for 1.24Tc and 1.65Tc respectively. limit of L0/a → ∞. We proceed to determine the func- tion ρ̄(ω) ≡ ρ(ω)/ tanh(1 ωL0) by making the ansatz ρ̄(ω) = m(ω) [1 + a(ω)], (10) where m(ω) > 0 has the high-frequency behavior of Eq. 4, and correspondingly define K̄(x0, ω) = coshω(x0− L0)/ cosh ωL0. Suppose that m(ω) already is a smooth approximate solution to ρ̄(ω); inserting (10) into (Eq. 2), one requires that a(ω) = ℓ cℓaℓ(ω), with {aℓ} a basis of functions which is as sensitive as pos- sible to the discrepancy between the lattice correla- tor and the correlator generated by m(ω). These are the eigenfunctions of largest eigenvalue of the symmet- ric kernel G(ω, ω′) ≡ M(x0, ω)M(x0, ω ′), where M(x0, ω) ≡ K̄(x0, ω)m(ω). These functions satisfy dωuℓ(ω)uℓ′(ω) = δℓℓ′ and have an increasing number of nodes as their eigenvalue decreases. Thus the more data points available, the larger the basis and the finer details of the spectral function one is able to determine. To determine the spectral function from N points of the correlator, we proceed by first discretizing the ω vari- able into an Nω-vector. The final spectral function is given by the last member ρ(N) of a sequence whose first member is ρ(0) = m and whose general member ρ(n) re- produces n points (or linear combinations) of the lattice correlator. For n ≥ 1, ρ(n) = ρ(n−1)[1 + ℓ=1 c and the functions a ℓ (ω) are found by the SVD decom- position [19] of the Nω × n matrix M (n)t, where M (n)ij ≡ 0 , ωj)ρ̄ (n−1)(ωj). The ‘model’ m(ω) is thus up- dated and agrees with ρ(ω) at the end of the procedure. We first performed this procedure on coarser lattices with L0/a = 6 at the same temperatures, starting from m(ω) = At.l.ω 4/(tanh(1 ωL0) tanh( ωL0) tanh 2(cωL0)) with 1 ≤ c ≤ 1 , and then recycled the output as seed for the L0/a = 8 lattices. On the latter we used the N = 4 points shown on Fig. 2. The next question to address is the uncertainty on ρ(ω). It is important to realize that even in the absence of statistical errors, a systematic uncertainty subsists due to the finite number of basis functions we can afford to describe ρ(ω) with. A reasonable measure of this uncer- tainty is by how much ρ(ω) varies if one doubles the re- solution on C(x0). This can be estimated by ‘generating’ new points by using the computed ρ(N)(ω). On the other hand we perform a two-point interpolation in x0-space (we chose the form (α + β(x0 − 12L0) 2)/ sin5(πx0/L0)), and take the difference between these and the generated ones as their systematic uncertainty. In practice this dif- ference is added in quadrature with the statistical uncer- tainty. Next we repeat the procedure to find ρ described above with N → 2N : if we use as seed ρ(N), then by construction it is left invariant by the iterative proce- dure, but the derivatives of ρ(2N) with respect to the 2N points of the correlator can be evaluated. The er- ror on ρ(ω) is then obtained from a formula of the type (δρ)2 = )2(δCi) 2 which however keeps track of correlations in x0 and Monte-Carlo time. This is the er- ror band shown on Fig. 3 and the corresponding shear viscosity values are η/s = 0.134(33) (T = 1.65Tc) 0.102(56) (T = 1.24Tc). It is also interesting to check for the stability of the so- lution under the use of a larger basis of functions. If in- stead of starting from ρ(N)(ω) we restart from ρ(0) (the output of the L0/a = 6 lattice) and fit the 2N (depen- dent) points using 2N basis functions {aℓ}, we obtain the curves drawn on Fig. 3. As one would hope, the oscilla- tions of ρ(2N)(ω) are covered by the error band. Conclusion.— Using state-of-the-art lattice techniques, we have computed the correlation functions of the energy- momentum tensor to high accuracy in the SU(3) pure gauge theory. We have calculated the leading high- temperature cutoff effects and removed them from the correlator relevant to the shear viscosity, and we nor- malized it non-perturbatively, exploiting existing results. We obtained the entropy density with an accuracy of 1%. The most robust result obtained on the shear viscosity is the upper bound Eq. (9), which comes from lumping the area under the curve on Fig. 3 in the interval [0, 6T ] into a peak of width Γ = T centered at the origin. Sec- ondly, our best estimate of the shear viscosity is given by Eq. (11), using a new method of extraction of the spectral function. The errors contain an estimate of the systema- tic uncertainty associated with the limited resolution in Euclidean time. We are extending the calculation to finer lattice spacings and larger volumes to further consolidate our findings. The values (11) are intriguingly close to saturating the KSS bound [8] η/s ≥ 1/4π. We note that in perturba- tion theory the ratio η/s does not depend strongly on the number of quark flavors [20]. Our results thus corrobo- rate the picture of a near-perfect fluid that has emerged from the RHIC experiments, with the magnitude of the anisotropic flow incompatible with η/s & 0.2 [3]. Acknowledgments.— I thank Krishna Rajagopal and Philippe de Forcrand for their encouragement and many useful discussions. This work was supported in part by funds provided by the U.S. Department of Energy under cooperative research agreement DE-FC02-94ER40818. [1] P. F. Kolb, P. Huovinen, U. W. Heinz and H. Heiselberg, Phys. Lett. B 500, 232 (2001); P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuskanen and S. A. Voloshin, Phys. Lett. B 503, 58 (2001). [2] D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev. Lett. 86, 4783 (2001). [3] D. Teaney, Phys. Rev. C 68, 034913 (2003). [4] F. Karsch and H.W.Wyld, Phys. Rev. D 35, 2518 (1987). [5] A. Nakamura and S. Sakai, Phys. Rev. Lett. 94, 072305 (2005). [6] G. Aarts and J.M. Martinez Resco, JHEP 0204, 053 (2002). [7] G. Policastro, D.T. Son and A.O. Starinets, Phys. Rev. Lett. 87, 081601 (2001). [8] P. Kovtun, D.T. Son and A.O. Starinets, Phys. Rev. Lett. 94, 111601 (2005). [9] K.G. Wilson, Phys. Rev. D 10, 2445 (1974). [10] C. Michael, Phys. Rev. D 53 (1996) 4102. [11] F. Karsch, Nucl. Phys. B 205, 285 (1982). [12] J. Engels, F. Karsch and T. Scheideler, Nucl. Phys. B 564 (2000) 303 [arXiv:hep-lat/9905002]. [13] A. Hosoya, M.A. Sakagami and M. Takao, Annals Phys. 154, 229 (1984). [14] B. Lucini, M. Teper and U. Wenger, JHEP 0401, 061 (2004). [15] S. Necco and R. Sommer, Nucl. Phys. B 622 (2002) 328. [16] H.B. Meyer, JHEP 0401, 030 (2004). [17] R. Sommer, Nucl. Phys. B 411 (1994) 839. [18] G. Boyd, J. Engels, F. Karsch, E. 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0704.1803

Regularity properties in the classification program for separable amenable C*-algebras

REGULARITY PROPERTIES IN THE CLASSIFICATION PROGRAM FOR SEPARABLE AMENABLE C∗-ALGEBRAS GEORGE A. ELLIOTT AND ANDREW S. TOMS Abstract. We report on recent progress in the program to classify separable amenable C∗-algebras. Our emphasis is on the newly apparent role of regular- ity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup. 1. Introduction Rings of bounded operators on Hilbert space were first studied by Murray and von Neumann in the 1930s. These rings, later called von Neumann algebras, came to be viewed as a subcategory of a more general category, namely, C∗-algebras. (The C∗-algebra of compact operators appeared for perhaps the first time when von Neumann proved the uniqueness of the canonical commutation relations.) A C∗-algebra is a Banach algebra A with involution x 7→ x∗ satisfying the C∗-algebra identity: ||xx∗|| = ||x||2, ∀x ∈ A. Every C∗-algebra is isometrically ∗-isomorphic to a norm-closed sub-∗-algebra of the ∗-algebra of bounded linear operators on some Hilbert space, and so may still be viewed as a ring of operators on a Hilbert space. In 1990, the first named author initiated a program to classify amenable norm- separable C∗-algebras via K-theoretic invariants. The graded and (pre-)ordered group K0 ⊕ K1 was suggested as a first approximation to the correct invariant, as it had already proved to be complete for both approximately finite-dimensional (AF) algebras and approximately circle (AT) algebras of real rank zero ([15], [17]). It was quickly realised, however, that more sensitive invariants would be required if the algebras considered were not sufficiently rich in projections. The program was refined, and became concentrated on proving that Banach algebra K-theory and positive traces formed a complete invariant for simple separable amenable C∗- algebras. Formulated as such, it enjoyed tremendous success throughout the 1990s and early 2000s. Date: November 1, 2021. 2000 Mathematics Subject Classification. Primary 46L35, Secondary 46L80. Key words and phrases. C∗-algebras, classification. This work was partially supported by NSERC. http://arxiv.org/abs/0704.1803v3 2 GEORGE A. ELLIOTT AND ANDREW S. TOMS Recent examples based on the pioneering work of Villadsen have shown that the classification program must be further revised. Two things are now appar- ent: the presence of a dichotomy among separable amenable C∗-algebras dividing those algebras which are classifiable via K-theory and traces from those which will require finer invariants; and the possibility—the reality, in some cases—that this di- chotomy is characterised by one of three potentially equivalent regularity properties for amenable C∗-algebras. (Happily, the vast majority of our stock-in-trade simple separable amenable C∗-algebras have one or more of these properties, including, for instance, those arising from directed graphs or minimal C∗-dynamical systems.) Our plan in this article is to give a brief account of the activity in the classifica- tion program over the past decade, with particular emphasis on the now apparent role of regularity properties. After reviewing the successes of the program so far, we will cover the work of Villadsen on rapid dimension growth AH algebras, the examples of Rørdam and the second named author which have necessitated the present re-evaluation of the classification program, and some recent and sweeping classification results of Winter obtained in the presence of the aforementioned regu- larity properties. We will also discuss the possible consequences to the classification program of including the Cuntz semigroup as part of the invariant (as a refinement of the K0 and tracial invariants). 2. Preliminaries Throughout the sequel K will denote the C∗-algebra of compact operators on a separable infinite-dimensional Hilbert space H. For a C∗-algebra A, we let Mn(A) denote the algebra of n × n matrices with entries from A. The cone of positive elements of A will be denoted by A+. 2.1. The Elliott invariant and the original conjecture. The Elliott invariant of a C∗-algebra A is the 4-tuple (1) Ell(A) := (K0A,K0A +,ΣA),K1A,T +A, ρA where the K-groups are the topological ones, K0A + is the image of the Murray- von Neumann semigroup V(A) under the Grothendieck map, ΣA is the subset of K0A corresponding to projections in A, T +A is the space of positive tracial linear functionals on A, and ρA is the natural pairing of T +A and K0A given by evaluating a trace at a K0-class. The reader is referred to Rørdam’s monograph [45] for a detailed treatment of this invariant. In the case of a unital C∗-algebra the invariant becomes (K0A,K0A +, [1A]),K1A,TA, ρA where [1A] is the K0-class of the unit, and TA is the (compact convex) space of tracial states. We will concentrate on unital C∗-algebras in the sequel in order to limit technicalities. The original statement of the classification conjecture for simple unital separable amenable C∗-algebras read as follows: 2.1. Let A and B be simple unital separable amenable C∗-algebras, and suppose that there exists an isomorphism φ : Ell(A) → Ell(B). It follows that there is a ∗-isomorphsim Φ : A→ B which induces φ. REGULARITY PROPERTIES FOR AMENABLE C -ALGEBRAS 3 It will be convenient to have an abbreviation for the statement above. Let us call it (EC). 2.2. Amenability. We will take the following deep theorem, which combines re- sults of Choi and Effros ([7]), Connes ([9]), Haagerup ([27]), and Kirchberg ([32]), to be our definition of amenability. Theorem 2.2. A C∗-algebra A is amenable if and only if it has the following property: for each finite subset G of A and ǫ > 0 there are a finite-dimensional C∗-algebra F and completely positive contractions φ, ψ such that the diagram idA // commutes up to ǫ on G. The property characterising amenability in Theorem 2.2 is known as the completely positive approximation property. Why do we only consider separable and amenable C∗-algebras in the classification program? It stands to reason that if one has no good classification of the weak closures of the GNS representations for a class of C∗-algebras, then one can hardly expect to classify the C∗-algebras themselves. These weak closures have separable predual if the C∗-algebra is separable. Connes and Haagerup gave a classification of injective von Neumann algebras with separable predual (see [10] and [28]), while Choi and Effros established that a C∗-algebra is amenable if and only if the weak closure in each GNS representation is injective ([8]). Separability and amenability are thus natural conditions which guarantee the existence of a good classification theory for the weak closures of all GNS representations of a given C∗-algebra. The assumption of amenability has been shown to be necessary by Dădărlat ([14]). The reader new to the classification program who desires a fuller introduction is referred to Rørdam’s excellent monograph [45]. 2.3. The Cuntz semigroup. One of the three regularity properties alluded to in the introduction is defined in terms of the Cuntz semigroup, an analogue for positive elements of the Murray-von Neumann semigroup V(A). It is known that this semigroup will be a vital part of any complete invariant for separable amenable C∗-algebras ([51]). Given its importance, we present both its original definition, and a modern version which makes the connection with classical K-theory more transparent. Definition 2.3 (Cuntz-Rørdam—see [12] and [49]). Let M∞(A) denote the alge- braic limit of the direct system (Mn(A), φn), where φn : Mn(A) → Mn+1(A) is given by Let M∞(A)+ (resp. Mn(A)+) denote the positive elements in M∞(A) (resp. Mn(A)). Given a, b ∈ M∞(A)+, we say that a is Cuntz subequivalent to b (written a - b) if there is a sequence (vn) n=1 of elements in some Mk(A) such that ||vnbv n − a|| −→ 0. 4 GEORGE A. ELLIOTT AND ANDREW S. TOMS We say that a and b are Cuntz equivalent (written a ∼ b) if a - b and b - a. This relation is an equivalence relation, and we write 〈a〉 for the equivalence class of a. The set W(A) := M∞(A)+/ ∼ becomes a positively ordered Abelian semigroup when equipped with the operation 〈a〉+ 〈b〉 = 〈a⊕ b〉 and the partial order 〈a〉 ≤ 〈b〉 ⇔ a - b. Definition 2.3 is slightly unnatural, as it fails to consider positive elements in A ⊗ K. This defect is the result of mimicking the construction of the Murray- von Neumann semigroup in letter rather than in spirit. Each projection in A ⊗ K is equivalent to a projection in some Mn(A), whence M∞(A) is large enough to encompass all possible equivalence classes of projections. The same is not true, however, of positive elements and Cuntz equivalence. The definition below amounts essentially to replacing M∞(A) with A⊗K in the definition above (this is a theorem), and also gives a new and very useful characterisation of Cuntz subequivalence. We refer the reader to [35] and [39] for background material on Hilbert C∗-modules. Consider A as a (right) Hilbert C∗-module over itself, and let HA denote the countably infinite direct sum of copies of this module. There is a one-to-one corre- spondence between closed countably generated submodules of HA and hereditary subalgebras of A ⊗ K: the hereditary subalgebra B corresponds to the closure of the span of BHA. Since A is separable, B is singly hereditarily generated, and it is fairly routine to prove that any two generators are Cuntz equivalent in the sense of Definition 2.3. Thus, passing from positive elements to Cuntz equivalence classes factors through the passage from positive elements to the hereditary subalgebras they generate. Let X and Y be closed countably generated submodules of HA. Recall that the compact operators on HA form a C ∗-algebra isomorphic to A⊗ K. Let us say that X is compactly contained in Y if X is contained in Y and there is compact self-adjoint endomorphism of Y which fixes X pointwise. Such an endomorphism extends naturally to a compact self-adjoint endomorphism of HA, and so may be viewed as a self-adjoint element of A⊗K. We write X - Y if each closed countably generated compactly contained submodule of X is isomorphic to such a submodule of Y . Theorem 2.4 (Coward-Elliott-Ivanescu, [11]). The relation - on Hilbert C∗ mod- ules defined above, when viewed as a relation on positive elements in M∞(A), is precisely the relation - of Definition 2.3. Let [X ] denote the Cuntz equivalence class of the module X . One may con- struct a positive partially ordered Abelian semigroup Cu(A) by endowing the set of countably generated Hilbert C∗-modules over A with the operation [X ] + [Y ] := [X ⊕ Y ] and the partial order [X ] ≤ [Y ] ⇔ X - Y. The semigroup Cu(A) coincides with W(A) whenever A is stable, i.e., A ⊗ K ∼= A, and has some advantages over W(A) in general. First, suprema of increasing REGULARITY PROPERTIES FOR AMENABLE C -ALGEBRAS 5 sequences always exist in Cu(A). This leads to the definition of a category including this structure in which Cu(A) sits as an object, and as a functor into which it is continuous with respect to inductive limits. (Definition 2.3 casts W(A) as a functor into just the category of partially ordered Abelian semigroups with zero. This functor fails to be continuous with respect to inductive limits.) Second, it allows one to prove that if A has stable rank one, then Cuntz equivalence of positive elements simply amounts to isomorphism of the corresponding Hilbert C∗-modules. This has led, via recent work of Brown, Perera, and the second named author, to the complete classification of all countably generated Hilbert C∗-modules over A via K0 and traces, and to the classification of unitary orbits of positive operators in A⊗K through recent work of Ciuperca and the first named author ([4], [5], [6]). Essentially, W(A) and Cu(A) contain the same information, but we have chosen to maintain separate notation both to avoid confusion and because many results in the literature are stated only for W(A). Cuntz equivalence is often described roughly as the Murray-von Neumann equiv- alence of the support projections of positive elements. This heuristic is, modulo accounting for projections, precise in C∗-algebras for which the Elliott invariant is known to be complete ([42]). In the stably finite case, one recovers both K0, the tracial simplex, and the pairing ρ (see (1)) from the Cuntz semigroup, whence the invariant (Cu(A),K1A) is finer than Ell(A) in general. Remarkably, these two invariants determine each other in a natural way for the largest class of simple separable amenable C∗-algebras in which (EC) can be expected to hold ([4], [5]). 3. Three regularity properties Let us now describe three agreeable properties which a C*-algebra may enjoy. We will see later how virtually all classification theorems for separable amenable C∗-algebras via the Elliott invariant assume, either explicitly or implicitly, one of these properties. 3.1. Strict comparison. Our first regularity property—strict comparison—is one that guarantees, in simple C∗-algebras, that the heuristic view of Cuntz equivalence described at the end of Section 2 is in fact accurate for positive elements which are not Cuntz equivalent to projections (see [42]). The property is K-theoretic in character. Let A be a unital C∗-algebra, and denote by QT(A) the space of normalised 2- quasitraces on A (v.[2, Definition II.1.1]). Let S(W(A)) denote the set of additive and order preserving maps d from W(A) to R+ having the property that d(〈1A〉) = 1. Such maps are called states. Given τ ∈ QT(A), one may define a map dτ : M∞(A)+ → R (2) dτ (a) = lim τ(a1/n). This map is lower semicontinuous, and depends only on the Cuntz equivalence class of a. It moreover has the following properties: (i) if a - b, then dτ (a) ≤ dτ (b); (ii) if a and b are orthogonal, then dτ (a+ b) = dτ (a) + dτ (b). 6 GEORGE A. ELLIOTT AND ANDREW S. TOMS Thus, dτ defines a state on W(A). Such states are called lower semicontinuous dimension functions, and the set of them is denoted by LDF(A). If A has the property that a - b whenever d(a) < d(b) for every d ∈ LDF(A), then let us say that A has strict comparison of positive elements or simply strict comparison. A theorem of Haagerup asserts that every element of QT(A) is in fact a trace if A is exact ([29]). All amenable C∗-algebras are exact, so we dispense with the consideration of quasi-traces in the sequel. 3.2. Finite decomposition rank. Our second regularity property, introduced by Kirchberg and Winter, is topological in flavour. It is based on a noncommutative version of covering dimension called decomposition rank. Definition 3.1 ([34], Definitions 2.2 and 3.1). Let A be a separable C∗-algebra. (i) A completely positive map ϕ : i=1Mri → A is n-decomposable, if there is a decomposition {1, . . . , s} = j=0 Ij such that the restriction of ϕ to preserves orthogonality for each j ∈ {0, . . . , n}. (ii) A has decomposition rank n, drA = n, if n is the least integer such that the following holds: Given {b1, . . . , bm} ⊂ A and ǫ > 0, there is a completely positive approximation (F, ψ, ϕ) for b1, . . . , bm within ǫ (i.e., ψ : A → F and ϕ : F → A are completely positive contractions and ‖ϕψ(bi) − bi‖ < ǫ) such that ϕ is n-decomposable. If no such n exists, we write drA = ∞. Decomposition rank has good permanence properties. It behaves well with re- spect to quotients, inductive limits, hereditary subalgebras, unitization and sta- bilization. Its topological flavour comes from the fact that it generalises covering dimension in the commutative case: if X is a locally compact second countable space, then drC0(X) = dimX . We refer the reader to [34] for details. The regularity property that we are interested in is finite decomposition rank, expressed by the inequality dr < ∞. This can only occur in a stably finite C∗- algebra. 3.3. Z-stability. The Jiang-Su algebra Z is a simple separable amenable and infinite-dimensional C∗-algebra with the same Elliott invariant as C ([30]). We say that a second algebra A is Z-stable if A ⊗ Z ∼= A. Z-stability is our third regularity property. It is very robust with respect to common constructions (see [56]). The next theorem shows Z-stability to be highly relevant to the classification program. Recall that a pre-ordered Abelian group (G,G+) is said to be weakly unperforated if nx ∈ G+\{0} implies x ∈ G+ for any x ∈ G and n ∈ N. Theorem 3.2 (Gong-Jiang-Su, [26]). Let A be a simple unital C∗-algebra with weakly unperforated K0-group. Then, Ell(A) ∼= Ell(A⊗Z). Thus, modulo a mild restriction on K0, the completeness of Ell(•) in the simple unital case of the classification program would imply Z-stability. Remarkably, there exist algebras satisfying the hypotheses of the above theorem which are not Z-stable ([46], [51], [52]). REGULARITY PROPERTIES FOR AMENABLE C -ALGEBRAS 7 3.4. Relationships. In general, no two of the regularity properties above are equivalent. The most important general result connecting them is the following theorem of M. Rørdam ([47]): Theorem 3.3. Let A be a simple, unital, exact, finite, and Z-stable C∗-algebra. Then, A has strict comparison of positive elements. We shall see in the sequel that for a substantial class of simple, separable, amenable, and stably finite C∗-algebras, all three of our regularity properties are equivalent. Moreover, the algebras in this class which do satisfy these three properties also satisfy (EC). There is good reason to believe that the equivalence of these three properties will hold in much greater generality, at least in the stably finite case; in the general case, strict comparison and Z-stability may well prove to be equiva- lent characterisations of those simple, unital, separable, and amenable C∗-algebras which satisfy (EC). 4. A brief history We will now take a short tour of the classification program’s biggest successes, and also the fascinating algebras of Villadsen. We have two goals in mind: to edify the reader unfamiliar with the classification program, and to demonstrate that the regularity properties of Section 3 pervade the known confirmations of (EC). This is a new point of view, for when these results were originally proved, there was no reason to think that anything more than simplicity, separability, and amenability would be required to complete the classification program. We have divided our review of known classification results into three broad cat- egories according to the types of algebras covered: purely infinite algebras, and two formally different types of stably finite algebras. It is beyond the scope of this article to provide and exhaustive list of of known classification results, much less demonstrate their connections to our regularity properties. We will thus choose, from each of the three categories above, the classification theorem with the broadest scope, and indicate how the algebras it covers satisfy at least one of our regularity properties. 4.1. Purely infinite simple algebras. We first consider a case where the theory is summarised with one beautiful result. Recall that a simple separable amenable C∗-algebra is purely infinite if every hereditary subalgebra contains an infinite pro- jection (a projection is infinite if it is equivalent, in the sense of Murray and von Neumann, to a proper subprojection of itself—otherwise the projection is finite). Theorem 4.1 (Kirchberg-Phillips, 1995, [31] and [43]). Let A and B be separable amenable purely infinite simple C∗-algebras which satisfy the Universal Coefficient Theorem. If there is an isomorphism φ : Ell(A) → Ell(B), then there is a ∗-isomorphism Φ : A→ B. In the theorem above, the Elliott invariant is somewhat simplified. The hypothe- ses on A and B guarantee that they are traceless, and that the order structure on K0 is irrelevant. Thus, the invariant is simply the graded group K0⊕K1, along with the K0-class of the unit if it exists. The assumption of the Universal Coefficient Theorem (UCT) is required in order to deduce the theorem from a result which is 8 GEORGE A. ELLIOTT AND ANDREW S. TOMS formally more general: A and B as in the theorem are ∗-isomorphic if and only if they are KK-equivalent. The question of whether every amenable C∗-algebra satisfies the UCT is open. Which of our three regularity properties are present here? As noted earlier, finite decomposition rank is out of the question. The algebras we are considering are traceless, and so the definition of strict comparison reduces to the following statement: for any two non-zero positive elements a, b ∈ A, we have a - b. This, in turn, is often taken as the very definition of pure infiniteness, and can be shown to be equivalent to the definition preceding Theorem 4.1 without much difficulty. Strict comparison is thus satisfied in a slightly vacuous way. As it turns out, A and B are also Z-stable, although this is less obvious. One first proves that A and B are approximately divisible (again, this does not require Theorem 4.1), and then uses the fact, due to Winter and the second named author, that any separable and approximately divisible C∗-algebra is Z-stable ([57]). 4.2. The stably finite case, I: inductive limits. We now move on to the case of stably finite C∗-algebras, i.e., those algebras A such that that every projection in the (unitization of) each matrix algebra Mn(A) is finite. (The question of whether a simple amenable C∗-algebra must always be purely infinite or stably finite was recently settled negatively by Rørdam. We will address his example again later.) Many of the classification results in this setting apply to classes of C∗-algebras which can be realised as inductive limits of certain building block algebras. The original classification result for stably finite algebras is due to Glimm. Recall that a C∗-algebra A is uniformly hyperfinite (UHF) if it is the limit of an inductive sequence −→ Mn2 −→ Mn3 −→ · · · , where each φi is a unital ∗-homomorphism. We will state his result here as a con- firmation of the Elliott conjecture, but note that it predates both the classification program and the realisation that K-theory is the essential invariant. Theorem 4.2 (Glimm, 1960, [24]). Let A and B be UHF algebras, and suppose that there is an isomorphsim φ : Ell(A) → Ell(B). It follows that there is a ∗-isomorphism Φ : A→ B which induces φ. Again, the invariant is dramatically simplified here. Only the ordered K0-group is non-trivial. The strategy of Glimm’s proof (which did not use K-theory explicitly) was to “intertwine” two inductive sequences (Mni , φi) and (Mmi , ψi), i.e., to find sequences of ∗-homomorphisms ηi and γi making the diagram φ1 // φ2 // φ3 // · · · ψ1 // ;;xxxxxxxx ψ2 // ;;xxxxxxxx ψ3 // ==zzzzzzzzz · · · commute. One then gets an isomorphism between the limit algebras by extending the obvious morphism between the inductive sequences by continuity. The intertwining argument above can be pushed surprisingly far. One replaces the inductive sequences above with more general inductive sequences (Ai, φi) and REGULARITY PROPERTIES FOR AMENABLE C -ALGEBRAS 9 (Bi, ψi), where the Ai and Bi are drawn from a specified class (matrix algebras over circles, for instance), and seeks maps ηi and γi as before. Usually, it is not possible to find ηi and γi making the diagram commute, but approximate commutativity on ever larger finite sets can be arranged for, and this suffices for the existence of an isomorphism between the limit algebras. This generalised intertwining is known as the Elliott Intertwining Argument. The most important classification theorem for inductive limits covers the so- called approximately homogeneous (AH) algebras. An AH algebra A is the limit of an inductive sequence (Ai, φi), where each Ai is semi-homogeneous: pi,j(C(Xi,j)⊗K)pi,j for some natural number ni, compact metric spaces Xi,j , and projections pi,j ∈ C(Xi,j) ⊗ K. We refer to the sequence (Ai, φi) as a decomposition for A; such decompositions are not unique. All AH algebras are separable and amenable. Let A be a simple unital AH algebra. Let us say that A has slow dimension growth if it has a decomposition (Ai, φi) satisfying lim sup dim(Xi,1) rank(pi,1) , . . . , dim(Xi,ni) rank(pi,ni) Let us say that A has very slow dimension growth if it has a decomposition satisfying the (formally) stronger condition that lim sup dim(Xi,1) rank(pi,1) , . . . , dim(Xi,ni) rank(pi,ni) Finally, let us say that A has bounded dimension if there is a constant M > 0 and a decomposition of A satisfying {dim(Xi,l)} ≤M. Theorem 4.3 (Elliott-Gong and Dădărlat, [20] and [13]). (EC) holds among simple unital AH algebras with slow dimension growth and real rank zero. Theorem 4.4 (Elliott-Gong-Li and Gong, [22] and [25]). (EC) holds among simple unital AH algebras with very slow dimension growth. All three of our regularity properties hold for the algebras of Theorems 4.3 and 4.4, but some are easier to establish than others. Let us first point out that an algebra from either class has stable rank one and weakly unperforated K0-group (cf.[1]), and that these facts predate Theorems 4.3 and 4.4. A simple unital C∗- algebra of real rank zero and stable rank one has strict comparison if and only if its K0-group is weakly unperforated (cf.[41]), whence strict comparison holds for the algebras covered by Theorem 4.3. A recent result of the second named author shows that strict comparison holds for any simple unital AH algebra with slow dimension growth ([55]), and this result is independent of the classification theorems above. Thus, strict comparison holds for the algebras of Theorems 4.3 and 4.4, and the proof of this fact, while not easy, is at least much less complicated than the proofs of the classification theorems themselves. Establishing finite decomposition rank requires the full force of the classification theorems: a consequence of both theorems is that the algebras they cover are all in fact simple unital AH algebras of bounded 10 GEORGE A. ELLIOTT AND ANDREW S. TOMS dimension, and such algebras have finite decomposition rank by [34, Corollary 3.12 and 3.3 (ii)]. Proving Z-stability is also an application of Theorems 4.3 and 4.4: one may use the said theorems to prove that the algebras in question are approximately divisible ([21]), and this entails Z-stability for separable C∗-algebras ([57]). Why all the interest in inductive limits? Initially at least, it was surprising to find that any classification of C∗-algebras by K-theory was possible, and the earliest theorems to this effect covered inductive limits (see, for instance, the first named author’s classification of AF algebras and AT-algebras of real rank zero []). But it was the realisation by Evans and the first named author that a very natural class of C∗-algebras arising from dynamical systems—the irrational rotation algebras— were in fact inductive limits of elementary building blocks that began the drive to classify inductive limits of all stripes ([19]). This theorem of Elliott and Evans has recently been generalised in sweeping fashion by Lin and Phillips, who prove that virtually every C∗-dynamical system giving rise to a simple algebra is an inductive limit of fairly tractable building blocks. This result continues to provide strong motivation for the study of inductive limit algebras. 4.3. The stably finite case, II: tracial approximation. Natural examples of separable amenable C∗-algebras are rarely equipped with obvious and useful induc- tive limit decompositions. Even the aforementioned theorem of Lin and Phillips, which gives an inductive limit decomposition for each minimal C∗-dynamical sys- tem, does not produce inductive sequences covered by existing classification theo- rems. It is thus desirable to have theorems confirming the Elliott conjecture under hypotheses that are (reasonably) straightforward to verify for algebras not given as inductive limits. Lin in [36] introduced the concept of tracial topological rank for C∗-algebras. His definition, in spirit if not in letter, is this: a unital simple tracial C∗-algebra A has tracial topological rank at most n ∈ N if for any finite set F ⊆ A, tolerance ǫ > 0, and positive element a ∈ A there exist unital subalgebras B and C of A such (i) 1A = 1B ⊕ 1C , (ii) F is almost (to within ǫ) contained in B ⊕ C, (iii) C is isomorphic to F⊗C(X), where dim(X) ≤ n and F is finite-dimensional, (iv) 1B is dominated, in the sense of Cuntz subequivalence, by a. One denotes by TR(A) the least integer n for which A satisfies the definition above; this is the tracial topological rank, or simply the tracial rank, of A. The most important value of the tracial rank is zero. Lin proved that simple unital separable amenable C∗-algebras of tracial rank zero satisfy the Elliott con- jecture, modulo the ever present UCT assumption ([37]). The great advantage of this result is that its hypotheses can be verified for a wide variety of C∗-dynamical systems and all simple non-commutative tori, without ever having to prove that the latter have tractable inductive limit decompositions (see [44], for instance). Indeed, the existence of such decompositions is a consequence of Lin’s theorem! One can also verify the hypotheses of Lin’s classification theorem for many real rank zero C∗-algebras with unique trace ([3]), always with the assumption, indirectly, of strict comparison. REGULARITY PROPERTIES FOR AMENABLE C -ALGEBRAS 11 Simple unital C∗-algebras of tracial rank zero can be shown to have stable rank one and weakly unperforated K0-groups, whence they have strict comparison of positive elements by a theorem of Perera ([41]). (There is a classification theorem for algebras of tracial rank one ([38]), but this has been somewhat less useful—it is difficult to verify tracial rank one in natural examples. Also, Niu has recently proved a classification theorem for some C∗-algebras which are approximated in trace by certain subalgebras of Mn ⊗ C[0, 1] ([40]).) And what of our regularity properties? Lin proved in [36] that every unital simple C∗-algebra of tracial rank zero has stable rank one and weakly unperforated K0- group. These facts, by the results reviewed at the end of the preceding subsection, entail strict comparison, and are not nearly so difficult to prove as the tracial rank zero classification theorem. In a further analogy with the case of AH algebras, finite decomposition rank and Z-stability can only be verified by applying Lin’s classification theorem—a consequence of this theorem is that the algebras it covers are in fact AH algebras of bounded dimension! 4.4. Villadsen’s algebras. Until the mid 1990s we had no examples of simple separable amenable C∗-algebras where one of our regularity properties failed. To be fair, two of our regularity properties had not yet even been defined, and strict comparison was seen as a technical version of the more attractive Second Fun- damental Comparability Question for projections (this last condition, abbreviated FCQ2, asks for strict comparison for projections only). This all changed when Vil- ladsen produced a simple separable amenable and stably finite C∗-algebra which did not have FCQ2, answering a long-standing question of Blackadar ([59]). The techniques introduced by Villadsen were subsequently used by him and others to answer many open questions in the theory of nuclear C∗-algebras including the following: (i) Does there exist a simple separable amenable C∗-algebra containing a finite and an infinite projection? (Solved affirmatively by Rørdam in [46].) (ii) Does there exist a simple and stably finite C∗-algebra with non-minimal stable rank? (Solved affirmatively by Villadsen in [60].) (iii) Is stability a stable property for simple C∗-algebras? (Solved negatively by Rørdam in [48].) (iv) Does a simple and stably finite C∗-algebra with cancellation of projections necessarily have stable rank one? (Solved negatively by the second named author in [54].) Of the results above, (i) was (and is) the most significant. In addition to showing that simple separable amenable C∗-algebras do not have a factor-like type classifi- cation, Rørdam’s example demonstrated that the Elliott invariant as it stood could not be complete in the simple case. This and other examples due to the second named author have necessitated a revision of the classification program. It is to the nature of this revision that we now turn. 5. The way(s) forward 5.1. New assumptions. (EC) does not hold in general, and this justifies new as- sumptions in efforts to confirm it. In particular, one may assume any combination of our three regularity properties. We will comment on the aptness of these new assumptions in the next subsection. For now we observe that, from a certain point 12 GEORGE A. ELLIOTT AND ANDREW S. TOMS of view, we have been making these assumptions all along. Existing classification theorems for C∗-algebras of real rank zero are accompanied by the crucial assump- tions of stable rank one and weakly unperforated K-theory; as has already been pointed out, unperforated K-theory can be replaced with strict comparison in this setting. How much further can one get by assuming the (formally) stronger condition of Z-stability? What role does finite decomposition rank play? As it turns out, these two properties both alone and together produce interesting results. Let RR0 denote the class of simple unital separable amenable C∗-algebras of real rank zero. The following subclasses of RR0 satisfy (EC): (i) algebras which satisfy the UCT, have finite decomposition rank, and have tracial simplex with compact and zero-dimensional extreme boundary; (ii) Z-stable algebras which satisfy the UCT and are approximated locally by subalgebras of finite decomposition rank. These results, due to Winter ([61], [62]), showcase the power of our regularity properties: included in the algebras covered by (ii) are all simple separable unital Z-stable ASH (approximately subhomogeneous) algebras of real rank zero. Another advantage to the assumptions of Z-stability and strict comparison is that they allow one to recover extremely fine isomorphism invariants for C∗-algebras from the Elliott invariant alone. (This recovery is not possible in general.) We will be able to give precise meaning to this comment below, but first require a further dicussion of Cuntz semigroup. 5.2. New invariants. A natural reaction to an incomplete invariant is to enlarge it: include whatever information was used to prove incompleteness. This is not always a good idea. It is possible that one’s distinguishing information is ad hoc, and unlikely to yield a complete invariant. Worse, one may throw so much new information into the invariant that the impact of its potential completeness is se- verely diminished. The revision of an invariant is a delicate business. In this light, not all counterexamples are equal. Rørdam’s finite-and-infinite-projection example is distinguished from a simple and purely infinite algebra with the same K-theory by the obvious fact that the latter contains no finite projections. The natural invariant which captures this dif- ference is the semigroup of Murray-von Neumann equivalence classes of projections in matrices over an algebra A, denoted by V(A). After the appearance of Rørdam’s example, the second named author produced a pair of simple, separable, amenable, and stably finite C∗-algebras which agreed on the Elliott invariant, but were not isomorphic. In this case the distinguishing invariant was Rieffel’s stable rank. It was later discovered that these algebras could not be distinguished by their Murray- von Neumann semigroups, but it was not yet clear which data were missing from the Elliott invariant. More dramatic examples were needed, ones which agreed on most candidates for enlarging the invariant, and pointed the way to the “missing information”. In [52], the second named author constructed a pair of simple unital AH algebras which, while non-isomorphic, agreed on a wide swath of invariants including the Elliott invariant, all continuous (with respect to inductive sequences) and homotopy invariant functors from the category of C∗-algebras (a class which includes the Murray-von Neumann semigroup), the real and stable ranks, and, as was shown REGULARITY PROPERTIES FOR AMENABLE C -ALGEBRAS 13 later in [], stable isomorphism invariants (those invariants which are insensitive to tensoring with a matrix algebra or passing to a hereditary subalgebra). It was thus reasonable to expect that the distinguishing invariant in this example—the Cuntz semigroup—might be a good candidate for enlarging the invariant. At least, it was an object which after years of being used sparingly as a means to other ends, merited study for its own sake. Let us collect some evidence supporting the addition of the Cuntz semigroup to the usual Elliott invarariant. First, in the biggest class of algebras where (EC) can be expected to hold—Z-stable algebras, as shown by Theorem 3.2—it is not an addition at all! Recent work of Brown, Perera, and the second named author shows that for a simple unital separable amenable C∗-algebra which absorbs Z tensorially, there is a functor which recovers the Cuntz semigroup from the Elliott invariant ([4], [42]). This functorial recovery also holds for simple unital AH algebras of slow dimension growth, a class for which Z-stability is not known and yet confirmation of (EC) is expected. (It should be noted that the computation of the Cuntz semigroup for a simple approximately interval (AI) algebra was essentially carried out by Ivanescu and the first named author in [23], although one does require [11, Corollary 4] to see that the computation is complete.) Second, the Cuntz semigroup unifies the counterexamples of Rørdam and the second named author. One can show that the examples of [45], [51], and [52] all consist of pairs of algebras with different Cuntz semigroups; there are no counterexamples to the conjecture that simple separable amenable C∗-algebras will be classified up to ∗-isomorphism by the Elliott invariant and the Cuntz semigroup. Third, the Cuntz semigroup provides a bridge to the classification of non-simple algebras. Ciuperca and the first named author have recently proved that AI algebras—limits of inductive sequences of algebras of the Mmi(C[0, 1]) — are classified up to isomorphism by their Cuntz semigroups. This is accomplished by proving that the approximate unitary equivalence classes of positive operators in the unitization of a stable C∗-algebra of stable rank one are determined by the Cuntz semigroup of the algebra, and then appealing to a theorem of Thomsen ([50]). (These approximate unitary equivalence classes of positive operators can be endowed with the structure of a topological partially ordered semigroup with functional calculus. This invariant, known as Thomsen’s semigroup, is recovered functorially from the Cuntz semigroup for separable algebras of stable rank one, and so from the Elliott invariant in algebras which are moreover simple, unital, exact, finite, and Z-stable by the results of [4]. This new semigroup is the fine invariant alluded to at the end of subsection 5.1.) There is one last reason to suspect a deep connection between the classification program and the Cuntz semigroup. Let us first recall a theorem of Kirchberg, which is germane to the classification of purely infinite C∗-algebras (cf. Theorem 4.1). Theorem 5.1 (Kirchberg, c. 1994; see [33]). Let A be a separable amenable C∗- algebra. The following two properties are equivalent: (i) A is purely infinite; (ii) A⊗O∞ ∼= A. 14 GEORGE A. ELLIOTT AND ANDREW S. TOMS A consequence of Kirchberg’s theorem is that among simple separable amenable C∗-algebras which merely contain an infinite projection, there is a two-fold char- acterisation of the (proper) subclass which satisfies the original form of the Elliott conjecture (modulo UCT). If one assumes a priori that A is simple and unital with no tracial state, then a theorem of Rørdam (see [47]) shows that property (ii) above — known as O∞-stability—is equivalent to Z-stability. Under these same hypotheses, property (i) is equivalent to the statement that A has strict compar- ison. Kirchberg’s theorem can thus be rephrased as follows in the simple unital case: Theorem 5.2. Let A be a simple separable unital amenable C∗-algebra without a tracial state. The following two properties are equivalent: (i) A has strict comparison; (ii) A⊗Z ∼= A. The properties (i) and (ii) in the theorem above make perfect sense in the presence of a trace. We moreover have that (ii) implies (i) even in the presence of traces (this is due to Rørdam—see [47]). It therefore makes sense to ask whether the theorem might be true without the tracelessness hypothesis. Remarkably, this appears to be the case. Winter and the second named author have proved that for a substantial class of stably finite C∗-algebras, strict comparison and Z-stability are equivalent, and that these properties moreover characterise the (proper) subclass which satisfies (EC) ([58]). In other words, Kirchberg’s theorem is quite possibly a special case of a more general result, one which will give a unified two-fold characterisation of those simple separable amenable C∗-algebras which satisfy the original form of the Elliott conjecture. It is too soon to know whether the Cuntz semigroup together with Elliott in- variant will suffice for the classification of simple separable amenable C∗-algebras, or indeed, whether such a broad classification can be hoped for at all. But there is already cause for optimism. Zhuang Niu has recently obtained some results on lifting maps at the level of the Cuntz semigroup to ∗-homomorphisms. This type of lifting result is a key ingredient in proving classification theorems of all stripes. His results suggest the algebras of [52] as the appropriate starting point for any effort to establish the Cuntz semigroup as a complete isomorphism invariant, at least in the absence of K1. We close our survey with a few questions for the future, both near and far. (i) When do natural examples of simple separable amenable C∗-algebras satisfy one or more of the regularity properties of Section 3? In particular, do simple unital inductive limits of recursive subhomogeneous algebras have strict comparison whenever they have strict slow dimension growth? (ii) Can the classification of positive operators up to approximate unitary equiv- alence via the Cuntz semigroup in algebras of stable rank one be extended to normal elements, provided that one accounts for K1? (iii) Let A be a simple, unital, separable, and amenable C∗-algebra with strict comparison of positive elements. Is A Z-stable? Less ambitiously, does A have stable rank one whenever it is stably finite? (iv) Can one use Thomsen’s semigroup to prove new classification theorems? (The attraction here is that Thomsen’s semigroup is already implicit in the Elliott invariant for many classes of C∗-algebras.) 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Anal. 154 (1998), 110-116 [60] Villadsen, J.: On the stable rank of simple C∗-algebras, J. Amer. Math. Soc. 12 (1999), 1091-1102 [61] Winter, W.: On topologically finite-dimensional simple C∗-algebras, Math. Ann. 332 (2005), 843-878 [62] Winter, W.: Simple C∗-algebras with locally finite decomposition rank, arXiv preprint math.OA/0602617 (2006) http://arxiv.org/abs/math/0601478 http://arxiv.org/abs/math/0609783 http://arxiv.org/abs/math/0509103 http://arxiv.org/abs/math/0609214 http://arxiv.org/abs/math/0607099 http://arxiv.org/abs/math/0502211 http://arxiv.org/abs/math/0508218 http://arxiv.org/abs/math/0611059 http://arxiv.org/abs/math/0602617 REGULARITY PROPERTIES FOR AMENABLE C -ALGEBRAS 17 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 2E4 E-mail address: [email protected] Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario, Canada, M3J 1P3 E-mail address: [email protected] 1. Introduction 2. Preliminaries 2.1. The Elliott invariant and the original conjecture 2.2. Amenability 2.3. The Cuntz semigroup 3. Three regularity properties 3.1. Strict comparison 3.2. Finite decomposition rank 3.3. Z-stability 3.4. Relationships 4. A brief history 4.1. Purely infinite simple algebras 4.2. The stably finite case, I: inductive limits 4.3. The stably finite case, II: tracial approximation 4.4. Villadsen's algebras 5. The way(s) forward 5.1. New assumptions 5.2. New invariants References

0704.1804

Equation of state of atomic systems beyond s-wave determined by the lowest order constrained variational method: Large scattering length limit

Equation of state of atomic systems beyond s-wave determined by the lowest order constrained variational method: Large scattering length limit Ryan M. Kalas(1) and D. Blume(1,2) (1)Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814 (2)INFM-BEC, Dipartimento di Fisica, Università di Trento, I-38050 Povo, Italy Dilute Fermi systems with large s-wave scattering length as exhibit universal properties if the interparticle spacing ro greatly exceeds the range of the underlying two-body interaction potential. In this regime, ro is the only relevant length scale and observables such as the energy per particle depend only on ro (or, equivalently, the energy EFG of the free Fermi gas). This paper investigates Bose and Fermi systems with non-vanishing angular momentum l using the lowest order constrained variational method. We focus on the regime where the generalized scattering length becomes large and determine the relevant length scales. For Bose gases with large generalized scattering lengths, we obtain simple expressions for the energy per particle in terms of a l-dependent length scale ξl, which depends on the range of the underlying two-body potential and the average interparticle spacing. We discuss possible implications for dilute two-component Fermi systems with finite l. Furthermore, we determine the equation of state of liquid and gaseous bosonic helium. PACS numbers: I. INTRODUCTION The experimental realization of dilute degenerate Bose and Fermi gases has led to an explosion of activities in the field of cold atom gases. A particularly intriguing feature of atomic Bose and Fermi gases is that their interaction strengths can be tuned experimentally through the ap- plication of an external magnetic field in the vicinity of a Feshbach resonance [1, 2]. This external knob allows dilute systems with essentially any interaction strength, including infinitely strongly attractive and repulsive in- teractions, to be realized. Feshbach resonances have been experimentally observed for s-, p- and d-wave interacting gases [3, 4, 5, 6, 7] and have been predicted to exist also for higher partial waves. A Feshbach resonance arises due to the coupling of two Born-Oppenheimer potential curves coupled through a hyperfine Hamiltonian, and requires, in general, a multi- channel description. For s-wave interacting systems, Fes- hbach resonances can be classified as broad or narrow [8]. Whether a resonance is broad or narrow depends on whether the energy width of the resonance is large or small compared to the characteristic energy scale, such as the Fermi energy or the harmonic oscillator energy, of the system. In contrast to s-wave resonances, higher partial wave resonances are necessarily narrow due to the presence of the angular momentum barrier [9]. This pa- per uses an effective single channel description to investi- gate the behaviors of strongly-interacting Bose and Fermi systems with different orbital angular momenta. In dilute homogeneous Bose and Fermi gases with large s-wave scattering length as, a regime has been identi- fied in which the energy per particle takes on a universal value which is set by a single length scale, the average interparticle spacing ro [10, 11, 12]. In this so-called uni- tary regime, the length scales of the s-wave interacting system separate according to |as| ≫ ro ≫ R, where R denotes the range of the two-body potential. The en- ergy per particle EB,0/N (the subscripts “B” and “0” stand respectively for “boson” and “s-wave interacting”) for a homogeneous one-component gas of bosons with mass m in the unitary regime has been calculated to be EB,0/N ≈ 13.3 ~ B /m using the lowest order con- strained variational (LOCV) method [12]. The energy EB,0/N at unitarity is thus independent of as and R, and depends on the single length scale ro through the boson number density nB, ro = (4πnB/3) −1/3. How- ever, Bose gases in the large scattering length limit are expected to be unstable due to three-body recombina- tion [13, 14, 15, 16]. On the other hand, the Fermi pressure prevents the collapse of two-component Fermi gases with equal masses and equal number of “spin-up” and “spin-down” fermions with large interspecies s-wave scattering length [10, 11, 17, 18]. At unitarity, the energy per particle is given by EF,0/N ≈ 0.42EFG, where EFG = (3/10)(~ 2k2F /m) de- notes the energy per particle of the non-interacting Fermi gas [19, 20, 21, 22]. The Fermi wave vector kF is related to the number density of the Fermi gas by nF = k F /3π which implies that EF,0/N depends on ro but is indepen- dent of as and R. We note that the inequality |as| ≫ ro is equivalent to 1/(kF |as|) ≪ 1. This paper investigates Bose and Fermi systems with large generalized scattering lengths using the LOCV method. For p- and d-wave interacting Bose systems, we define the unitary regime [23] through the inequali- ties |al(Erel)| ≫ ξl ≫ R, where ξl denotes a l-dependent length scale given by the geometric combination of ro and R, i.e., ξl = r (1−l/4) l/4, and Erel the relative scatter- ing energy. The generalized energy-dependent scattering length al(Erel) [24, 25, 26] characterizes the scattering strength (see below). We find that the energy of p-wave interacting two-component Bose gases and d-wave inter- acting one- and two-component Bose gases at unitary is determined by the combined length ξl. While Bose gases with higher angular momentum in the unitary regime are http://arxiv.org/abs/0704.1804v1 of theoretical interest, they are, like their s-wave cousin, expected to be unstable. We comment that the energet- ics of two-component Fermi gases with large generalized scattering length may depend on the same length scales. Furthermore, we consider s-wave interacting Bose sys- tems over a wide range of densities. Motivated by two re- cent studies by Gao [27, 28], we determine the energy per particle EB,0/N of the Bose system characterized by two atomic physics parameters, the s-wave scattering lengh as and the van der Waals coefficient C6. Our results lead to a phase diagram of liquid helium in the low-density regime that differs from that proposed in Ref. [28]. Section II describes the systems under study and in- troduces the LOCV method. Section III describes our results for dilute s-wave interacting Bose and Fermi sys- tems and for liquid helium. Section IV considers Bose and Fermi systems interacting through l-wave (l > 0) scattering. Finally, Section V concludes. II. LOCV METHOD FOR BOSONS AND FERMIONS This section introduces the three-dimensional Bose and Fermi systems under study and reviews the LOCV method [29, 30, 31, 32]. The idea of the LOCV method is to explicitly treat two-body correlations, but to ne- glect three- and higher-body correlations. This allows the many-body problem to be reduced to solving an effective two-body equation with properly chosen con- straints. Imposing these constraints makes the method non-variational, i.e., the resulting energy does not place an upper bound on the many-body energy. The LOCV method is expected to capture some of the key physics of dilute Bose and Fermi systems. The Hamiltonian HB for a homogeneous system con- sisting of identical mass m bosons is given by HB = − ∇2i + v(rij), (1) where the spherically symmetric interaction potential v depends on the relative distance rij , rij = |ri−rj|. Here, ri denotes the position vector of the ith boson. The Hamiltonian HF for a two-component Fermi system with equal masses and identical spin population is given by HF = − ∇2i − ∇2i′ + v(rii′ ), (2) where the unprimed subscripts label spin-up and the primed subscripts spin-down fermions. Throughout, we take like fermions to be non-interacting. Our primary interest in this paper is in the description of systems for which many-body observables are insensitive to the short-range behavior of the atom-atom potential v(r). This motivates us to consider two simple model poten- tials: an attractive square well potential vsw with depth Vo (Vo ≥ 0), vsw(r) = −Vo for r < R 0 for r > R ; and an attractive van der Waals potential vvdw with hard- core rc, vvdw(r) = ∞ for r < rc −C6/r 6 for r > rc . In all applications, we choose the hardcore rc so that the inequality rc ≪ β6, where β6 = (mC6/~ 2)1/4, is satisfied. The natural length scale of the square well potential is given by the range R and that of the van der Waals po- tential by the van der Waals length β6. The solutions to the two-body Schrödinger equation for vsw are given in terms of spherical Bessel and Neumann functions (impos- ing the proper continuity conditions of the wave function and its derivations at those r values where the poten- tial exhibits a discontinuity), and those for vvdw in terms of convergent infinite series of spherical Bessel and Neu- mann functions [33]. The interaction strength of the short-range square well potential can be characterized by the generalized energy- dependent scattering lengths al(k), al(k) = sgn[− tan δl(k)] tan δl(k) k2l+1 1/(2l+1) , (5) where δl(k) denotes the phase shift of the lth partial wave calculated at the relative scattering energy Erel, mErel/~2. This definition ensures that al(k) ap- proaches a constant as k → 0 [34, 35]. For the van der Waals potential vvdw, the threshold behavior changes for higher partial waves and the definition of al(k) has to be modified accordingly [34, 35]. In general, for a poten- tial that falls off as −r−n at large interparticle distances, al(k) is defined by Eq. (5) if 2l < n− 3 and by al(k) = sgn[− tan δl(k)] tan δl(k) 1/(n−2) if 2l > n− 3. For our van der Waals potential, n is equal to 6 and al(k) is given by Eq. (5) for l ≤ 1 and by Eq. (6) for l ≥ 2. The zero-energy generalized scattering lengths al can now be defined readily through al = lim al(k). (7) We note that a new two-body l-wave bound state ap- pears at threshold when |al| → ∞. The unitary regime for higher partial waves discussed in Sec. IV is thus, as in the s-wave case, closely related to the physics of ex- tremely weakly-bound atom-pairs. To uncover the key behaviors at unitarity, we assume in the following that the many-body system under study is interacting through a single partial wave l. While this may not be exactly realized in an experiment, this situation may be approx- imated by utilizing Feshbach resonances. We now outline how the energy per particle EB,l/N of a one-component Bose system with l-wave interac- tions [36] is calculated by the LOCV method [29, 30, 31, 32]. The boson wave function ΨB is taken to be a product of pair functions fl, ΨB(r1, . . . , rN ) = fl(rij), (8) and the energy expectation value ofHB, Eq. (1), is calcu- lated using ΨB. If terms depending on the coordinates of three or more different particles are neglected, the result- ing energy is given by the two-body term in the cluster expansion, fl(r) ∇2 + v(r) fl(r) d r. (9) The idea of the LOCV method is now to introduce a heal- ing distance d beyond which the pair correlation function fl is constant, fl(r > d) = 1. (10) To ensure that the derivative of fl is continuous at r = d, an additional constraint is introduced, f ′l (r = d) = 0. (11) Introducing a constant average field λl and varying with respect to fl while using that fl is constant for r > d, gives the Schrödinger-like two-body equation for r < d, ∇2 + v(r) (rfl(r)) = λlrfl(r). (12) Finally, the condition f2l (r) d r = 1 (13) enforces that the average number of particles within d equals 1. Using Eqs. (9), (10) and (12), the energy per particle becomes, v(r)d3r. (14) The second term on the right hand side of Eq. (14) is identically zero for the square well potential vsw but con- tributes a so-called tail or mean-field energy for the van der Waals potential vvdw [27, 28]. We determine the three unknown nB, λl and d by simultaneously solving Eqs. (12) and (13) subject to the boundary condition given by Eq. (11). Note that nB and d depend, just as fl and λl, on the angular momentum; the subscript has been dropped, however, for notational convenience. In addition to one-component Bose systems, Sec. IV considers two-component Bose systems, characterized by l-wave interspecies and vanishing intraspecies interac- tions. The Hamiltonian for the two-component Bose sys- tem is given by Eq. (2), with the sum of the two-body interactions restricted to unlike bosons. Correspondingly, the product wave function is written as a product of pair functions, including only correlations between unlike bosons. The LOCV equations are then given by Eqs. (10) through (13) with nB in Eq. (13) replaced by nB/2. Next, we discuss how to determine the energy EF,l/N per particle for a two-component Fermi system within the LOCV method [29, 30, 31, 32]. The wavefunction is taken to be ΨF (r1, . . . , r1′ , . . . ) = ΦFG fl(rij′ ), (15) where ΦFG denotes the ground state wavefunction of the non-interacting Fermi gas. The product of pair functions fl accounts for the correlations between unlike fermions. In accord with our assumption that like fermions are non- interacting, Eq. (15) treats like fermion pairs as uncor- related. Neglecting exchange effects, the derivation of the LOCV equations parallels that outlined above for the bosons. The boundary conditions, given by Eqs. (10) and (11), and the Schrödinger-like differential equation for λl, Eq. (12), are unchanged. The “normalization condition,” however, becomes f2l (r)d r = 1, (16) where the left-hand side is the number of fermion pairs within d. The fermion energy per particle is then the sum of the one-particle contribution from the non-interacting Fermi gas and the pair correlation energy λl [20], EF,l/N = EFG + . (17) This equation excludes the contribution from the tail of the potential, i.e., the term analogous to the second term on the right hand side of Eq. (14), since this term is neg- ligible for the fermion densities considered in this paper. The LOCV solutions for fl, λl and d for the ho- mogeneous one-component Bose system and the two- component Fermi system are formally identical if the bo- son density is chosen to equal half the fermion density, i.e., if nB = nF /2. This relation can be understood by realizing that any given fermion (e.g., a spin-up particle) interacts with only half of the total number of fermions (e.g., all the spin-down fermions). Consequently, the two- component Fermi system appears twice as dense as the one-component Bose system. The fact that the LOCV solutions for bosons can be converted to LOCV solutions for fermions suggests that some physics of the bosonic system can be understood in terms of the fermionic sys- tem and vice versa. In fact, it has been shown previ- ously [20] that the LOCV energy for the first excited gas-like state of s-wave interacting fermions at unitarity can be derived from the LOCV energy of the energet- ically lowest-lying gas-like branch of s-wave interacting bosons [12]. Here, we extend this analysis and show that the ground state energy of the Fermi gas at unitarity can 0 1 2 3 4 FIG. 1: Energy per particle EB,0/N as a function of the density nB , both plotted as dimensionless quantities, for a one-component Bose system interacting through the van der Waals potential with s-wave scattering length as = 16.9β6. The dotted line shows the gas branch and the dashed line the liquid branch. The minimum of the liquid branch is discussed in reference to liquid 4He in the text. be derived from the energetically highest-lying liquid-like branch of the Bose system. Furthermore, we extend this analysis to higher angular momentum scattering. III. s-WAVE INTERACTING BOSE AND FERMI SYSTEMS Figure 1 shows the energy per particle EB,0/N , Eq. (14), obtained by solving the LOCV equations for a one-component Bose system interacting through the van der Waals potential with s-wave scattering length as = 16.9β6. The dotted line in Fig. 1 has positive energy and increases with increasing density; it describes the energetically lowest-lying “gas branch” for the Bose sys- tem with as = 16.9β6 and corresponds to the metastable gaseous condensate studied experimentally. The dashed line in Fig. 1 has negative energy at small densities, de- creases with increasing density, and then exhibits a mini- mum; this dashed line describes the energetically highest- lying “liquid branch” for a Bose system with as = 16.9β6. Within the LOCV framework, these two branches arise because the Schrödinger-like equation, Eq. (12), permits for a given interaction potential solutions f0 with differ- ing number of nodes, which in turn give rise to a host of liquid and gas branches [27, 28]. Throughout this work we only consider the energetically highest-lying liquid branch with n nodes and the energetically lowest-lying gas branch with n + 1 nodes. To obtain Fig. 1, we con- sider a class of two-body potentials with fixed as/β6, and decrease the value of the ratio rc/β6 till EB,0/N , Eq. (14), no longer changes over the density range of interest, i.e., the number of nodes n of the energetically highest-lying liquid branch is increased till convergence is reached. 0.0 4.0×10 8.0×10 1.2×10 -0.003 0.003 0.006 FIG. 2: Energy per particle EB,0/N as a function of the den- sity nB for a one-component Bose system interacting through the van der Waals potential with s-wave scattering lengths as = 16.9β6 (open circles) and as = 169β6 (filled circles). To guide the eye, dashed and dotted lines connect the data points of the liquid and gas branches, respectively. The liq- uid branches go to Edimer/2 as the density goes to zero. The solid lines show EB,0/N at unitarity; see text for discussion. Compared to Fig. 1, the energy and density scales are greatly enlarged. In Fig. 1, the two-body van der Waals potential is cho- sen so that the scattering length of as = 16.9β6 coin- cides with that of the 4He pair potential [37]. The liquid branch in Fig. 1 can hence be applied to liquid 4He, and has previously been considered in Refs. [27, 28]. The min- imum of the liquid branch at a density of nB = 2.83β or 1.82 × 1022cm−3, agrees quite well with the experi- mental value of 2.18× 1022cm−3 [38]. The corresponding energy per particle of −6.56 K deviates by 8.5 % from the experimental value of −7.17 K [38]. This shows that the LOCV framework provides a fair description of the strongly interacting liquid 4He system, which is charac- terized by interparticle spacings comparable to the range of the potential. This is somewhat remarkable consid- ering that the LOCV method includes only pair corre- lations and that the van der Waals potential used here contains only two parameters. Open circles connected by a dashed line in Fig. 2 show the liquid branch for as = 16.9β6 in the small density region. As the density goes to zero, the energy per par- ticle EB,0/N does not terminate at zero but, instead, goes to Edimer/2, where Edimer denotes the energy of the most weakly-bound s-wave molecule of vvdw. In this small density limit, the liquid branch describes a gas of weakly-bound molecules, in which the interparticle spac- ing between the molecules greatly exceeds the size of the molecules, and Edimer is to a very good approximation given by −~2/(ma2s). As seen in Fig. 2, we find solu- tions in the whole density range considered. In contrast to our findings, Ref. [28] reports that the LOCV solu- tions of the liquid branch disappear at densities smaller than a scattering length dependent critical density, i.e., FIG. 3: Scaled interparticle spacing ro/β6 as a function of the scaled density nBβ 6 for the gas branch of a one-component Bose system interacting through the van der Waals poten- tial with as = 169β6. The horizontal lines show the scaled s-wave scattering length as = 169β6 and the range of the van der Waals potential, which is one in scaled units (almost in- distinguishable from the x-axis). This graph shows that the unitary inequalities as ≫ ro ≫ β6 hold for nB larger than about 10−5β−36 . at a critical density of 8.68 × 10−7β−36 for as = 16.9β6. Thus we are not able to reproduce the liquid-gas phase diagram proposed in Fig. 2 of Ref. [28], which depends on this termination of the liquid branch. We note that the liquid branch is, as indicated by its imaginary speed of sound, dynamically unstable at sufficiently small den- sities. The liquid of weakly-bound bosonic molecules discussed here can, as we show below, be related to weakly-bound molecules on the BEC side of the BEC- BCS crossover curve for two-component Fermi gases. We now discuss the gas branch in more detail. Open and filled circles connected by dotted lines in Fig. 2 show the energy per particle for as = 16.9β6 and 169β6, respec- tively. These curves can be applied, e.g., to 85Rb, whose scattering length can be tuned by means of a Feshbach resonance and which has a β6 value of 164abohr, where abohr denotes the Bohr radius. For this system, a scat- tering length of as = 16.9β6 corresponds to 2770abohr, a comparatively large value that can be realized experi- mentally in 85Rb gases. As a point of reference, a density of 10−5β−36 corresponds to a density of 1.53× 10 13cm−3 for 85Rb. The solid curve with positive energy in Fig. 2 shows the energy per particle EB,0/N at unitarity, EB,0/N ≈ 13.3~2n B /m [12]. As seen in Fig. 2, this unitary limit is approached by the energy per particle for the Bose gas with as = 169β6 (filled circles connected by a dotted line). To illustrate this point, Fig. 3 shows the scaled average interparticle spacing ro/β6 as a function of the scaled density nBβ 6 for as = 169β6. This plot indicates that the unitary requirement, as ≫ ro ≫ R, is met for values of nBβ 6 larger than about 10 −5. Similarly, we find -1 -0.5 0 0.5 1 1/(kFas) FIG. 4: Scaled energy per particle (EF,0/N)/EFG as a func- tion of 1/(kF as) for a two-component s-wave Fermi gas inter- acting through the square well potential for nF = 10 −6R−3. The combined dashed and dash-dotted curve corresponds to the BEC-BCS crossover curve and the dotted curve corre- sponds to the first excited state of the Fermi gas. The dashed and dotted linestyles are chosen to emphasize the connection to the gas and liquid branches of the Bose system in Figs. 2 and 3 (see text for more details). that the family of liquid curves converges to EB,0/N ≈ −2.46~2n B /m (see Sec. IV for details), plotted as a solid line in Fig. 2, when the inequalities as ≫ ro ≫ β6 are fullfilled. We note that the unitarity curve with negative energy is also approached, from above, for systems with large negative scattering lengths (not shown in Fig. 2). Aside from the proportionality constant, the power law relation for the liquid and gas branches at unitarity is the same. In addition to a Bose system interacting through the van der Waals potential, we consider a Bose system in- teracting through the square well potential with range R. For a given scattering length as and density nB, the energy per particle EB,0/N for these two two-body po- tentials is essentially identical for the densities shown in Fig. 2. This agreement emphasizes that the details of the two-body potential become negligible at low density, and in particular, that the behavior of the Bose gas in the unitary limit is governed by a single length scale, the average interparticle spacing ro. As discussed in Sec. II, the formal parallels between the LOCV method applied to bosons and fermions allows the energy per particle EF,0/N for a two-component Fermi gas, Eq. (17), to be obtained straightforwardly from the energy per particle EB,0/N of the Bose system. Fig- ure 4 shows the dimensionless energy (EF,0/N)/EFG as a function of the dimensionless quantity 1/(kFas) for the square well potential for nF = 10 −6R−3. We find essen- tially identical results for the van der Waals potential. The crossover curve shown in Fig. 4 describes any dilute Fermi gas for which the range R of the two-body poten- tial is very small compared to the average interparticle spacing ro. In converting the energies for the Bose sys- tem to those for the Fermi system, the gas branches of the Bose system (dotted lines in Figs. 2 and 3) “turn into” the excited state of the Fermi gas (dotted line in Fig. 4); the liquid branches of the Bose system with positive as (dashed lines in Figs. 2 and 3) “turn into” the part of the BEC-BCS crossover curve with positive as (dashed line in Fig. 4); and the liquid branches of the Bose system with negative as (not shown in Figs. 2 and 3) “turn into” the part of the BEC-BCS crossover curve with negative as (dash-dotted line in Fig. 4). To emphasize the connection between the Bose and Fermi systems further, let us consider the BEC side of the crossover curve. If 1/(kFas) & 1, the fermion energy per particle EF,0/N is approximately given by Edimer/2, which indicates that the Fermi gas forms a molecular Bose gas. Similarly, the liquid branch of the Bose sys- tem with positive scattering length is made up of bosonic molecules as the density goes to zero. The formal analogy between the Bose and Fermi LOCV solutions also allows the energy per particle EF,0/N at unitarity, i.e., in the 1/(kF |as|) → 0 limit, to be calculated from the energies for large as of the gas and liquid branches of the Bose system (solid lines in Fig. 2). For the excited state of the Fermi gas we find EF,0/N ≈ 3.92EFG, and for the low- est gas state we find EF,0/N ≈ 0.46EFG. These results agree with the LOCV calculations of Ref. [20], which use an attractive cosh-potential and a δ-function potential. The value of 0.46EFG is in good agreement with the en- ergy of 0.42EFG obtained by fixed-node diffusion Monte Carlo calculations [21, 22]. IV. BOSE AND FERMI SYSTEMS BEYOND s-WAVE AT UNITARITY This section investigates the unitary regime of Bose and Fermi systems interacting through higher angular momentum resonances. These higher angular momen- tum resonances are necessarily narrow [9], and we hence expect the energy-dependence of the generalized scatter- ing length al(k) to be particularly important in under- standing the many-body physics of dilute atomic systems beyond s-wave. In the following we focus on the strongly- interacting limit. Figure 5 shows 1/al(k) as a function of the relative scattering energy Erel for the square-well po- tential with infinite zero-energy scattering length al for three different angular momenta, l = 0 (solid line), l = 1 (dashed line), and l = 2 (dotted line). Figure 5 shows that the energy-dependence of al(k) increases with in- creasing l. Our goal is to determine the energy per particle EB,l/N for Bose systems with finite angular momentum l in the strongly-interacting regime. For s-wave interac- tions, the only relevant length scale at unitarity is the av- erage interparticle spacing ro (see Sec. III). In this case, the energy per particle at unitarity can be estimated an- alytically by evaluating the LOCV equations subject to -0.01 -0.005 0 0.005 0.01 rel /(h- FIG. 5: R/al(Erel) as a function of the scaled relative scatter- ing energy Erel/(~ 2/mR2) for the square well potential vsw with infinite zero-energy scattering length al, i.e., 1/al = 0, for three different partial waves [l = 0 (solid line), l = 1 (dashed line), and l = 2 (dotted line)]. the boundary condition implied by the zero-range s-wave pseudo-potential [12]. Unfortunatey, a similarly simple analysis that uses the boundary condition implied by the two-body zero-range pseudo-potential for higher partial waves fails. This combined with the following arguments suggests that EB,l/N depends additionally on the range of the underlying two-body potential for finite l: i) The probability distribution of the two-body l-wave bound state, l > 0, remains finite as al approaches infinity and depends on the interaction potential [39, 40]. ii) A de- scription of l-wave resonances (l > 0) that uses a coupled channel square well model depends on the range of the square well potential [41]. iii) The calculation of struc- tural expectation values of two-body systems with finite l within a zero-range pseudo-potential treatment requires a new length scale to be introduced [26]. Motivated by these two-body arguments (see also Refs. [42, 43, 44] for a treatment of p-wave interacting Fermi gases) we propose the following functional form for the energy per particle EB,l/N of a l-wave Bose system at unitarity interacting through the square-well potential vsw with range R, mRxl/2 2/3−xl/6 B . (18) Here, Cl denotes a dimensionless l-dependent proportion- ality constant. The dimensionless parameter xl deter- mines the powers of the range R and the density nB, and ensures the correct units of the right hand side of Eq. (18). To test the validity of Eq. (18), we solve the LOCV equations, Eqs. (11) through (14), for l = 0 to 2 for the one-component Bose system. Note that the one- component p-wave system is unphysical since it does not obey Bose symmetry; we nevertheless consider it here since its LOCV energy determines the energy of two- component p-wave Bose and Fermi systems (see below). Figure 6 shows the energy per particle EB,l/N for 0 5×10 FIG. 6: Scaled energy per particle (EB,l/N)/(~ 2/mR2) for a one-component Bose system for the energetically lowest-lying gas branch as a function of the scaled density nBR 3 obtained by solving the LOCV equations [Eqs. (11) through (14)] for vsw for three different angular momenta l [l = 0 (crosses), l = 1 (asterisks) and l = 2 (pluses)]. The depth V0 of vsw is adjusted so that 1/al(k) = 0. Solid, dotted and dashed lines show fits of the LOCV energies at low densities to Eq. (18) for l = 0, l = 1 and l = 2 (see text for details). Note that the system with l = 1 is of theoretical interest but does not describe a physical system. l xl C 0 0.00 −2.46 13.3 1 1.00 −3.24 9.22 2 2.00 −3.30 6.98 TABLE I: Dimensionless parameters xl, C l and C l for l = 0 to 2 for a one-component Bose system obtained by fitting the LOCV energies EB,l/N for small densities to the functional form given in Eq. (18) (see text for details). a one-component Bose system, obtained by solving the LOCV equations for the energetically lowest-lying gas branch, as a function of the density nB for l = 0 (crosses), l = 1 (asterisks), and l = 2 (pluses) for the square well potential, whose depth V0 is adjusted for each l so that the energy-dependent generalized scattering length al(k) diverges, i.e., 1/al(k) = 0. Setting al(k) to infinity en- sures that the l-wave interacting Bose system is infinitely strongly interacting over the entire density regime shown in Fig. 6. Had we instead set the zero-energy scattering length al to infinity, the system would, due to the strong energy-dependence of al(k) [see Fig. 5], “effectively” in- teract through a finite scattering length. Table I summarizes the values for xl and C l , which we obtain by performing a fit of the LOCV energies EB,l/N for the one-component Bose system for small densities to the functional form given in Eq. (18). In particular, we find xl = l, which implies that EB,l/N varies as n B and n B for l = 0, 1 and 2, respectively. Table I uses the superscript “G” to indicate that the proportion- ality constant is obtained for the energetically lowest- lying gas branch. The density ranges used in the fit are chosen so that Eq. (18) describes the low-density or uni- versal regime accurately. Solid, dotted and dashed lines in Fig. 6 show the results of these fits for l = 0, 1 and 2, respectively; in the low density regime, the lines agree well with the symbols thereby validating the functional form proposed in Eq. (18). We repeat the LOCV calculations for the energeti- cally highest-lying liquid branch of the one-component Bose system. By fitting the LOCV energies of the liquid branches for small densities to Eq. (18), we determine xl and Cl [45]. We find the same xl as for the gas branch but different Cl than for the gas branch (the proportionality constants obtained for the liquid branch are denoted by CLl ; see Table I). Our values for x0, C 0 and C 0 agree with those reported in the literature [12, 20, 46]. Equation (18) can be rewritten in terms of the com- bined length ξl, )(l/6−2/3) ~ , (19) where ξl = r (1−l/4) l/4. (20) For the s-wave case, ξl reduces to ro and the conver- gence to the unitary regime can be seen by plotting (EB,0/N)/(~ 2/mr2o) as a function of as/ro [12]. To investigate the convergence to the unitary regime for higher partial waves, Fig. 7 shows the energy per parti- cle EB,l/N for the energetically lowest-lying gas branch as a function of al(Erel)/ξl for fixed energy-dependent scattering lengths al(k), i.e., for as(k) = 10 10R, ap(k) = 1010R, and ad(k) = 10 6R [the different values of al(k) are chosen for numerical reasons]. Figure 7 shows that the inequality |al(Erel)| ≫ ξl ≫ R (21) is fulfilled when (EB,l/N)/(~ 2/mξ2l ) is constant. Note that this inequality is written in terms of the energy- dependent scattering length (see above). We find sim- ilar results for the liquid branches for l = 0 to 2. For higher partial waves, we hence use the inequality given by Eq. (21) to define the unitary regime. In the uni- tary regime, the energy per particle EB,l/N of the Bose system depends only on the combined length scale ξl. For s-wave interacting systems, we have ξs = ro and as(k) ≈ as, and Eq. (21) reduces to the well known s- wave unitary condition, i.e., to |as| ≫ ro ≫ R. We now discuss those regions of Fig. 7, where the energy per particle (EB,l/N)/(~ 2/mξ2l ) for the one- component Bose system deviates from a constant. For sufficiently large densities, the characteristic length ξl be- comes of the order of the range R of the square well potential. In this “high density” regime the system exhibits non-universal behaviors. In Fig. 7, e.g., the energy-dependent scattering length ad(k) equals 10 ) /ξl FIG. 7: Scaled energy per particle (EB,l/N)/(~ 2/mξ2l ) for the energetically lowest-lying gas branch of l-wave interacting one-component Bose systems obtained by solving the LOCV equations [Eqs. (11) through (14)] for vsw as a function of al(Erel)/ξl. The depth V0 of vsw is adjusted so that al(Erel) = 1010R for l = 0 and l = 1, and al(Erel) = 10 6R for l = 2. Note that the system with l = 1 is of theoretical interest but does not describe a physical system. In the regime where the inequality R ≪ ξl ≪ al(Erel) is fulfilled, the scaled energy per particle is constant; this defines the unitary regime. correspondingly, ξd equals R when ad(k)/ξd = 10 6. As ad(k)/ξd approaches 10 6 from below, the system becomes non-universal, as indicated in Fig. 7 by the non-constant dependence of the scaled energy per particle on ad(k)/ξd. On the left side of Fig. 7, where al(Erel)/ξl becomes of order 1, the “low density” end of the unitary regime is reached. When ap(Erel)/ξp equals 10, e.g., the inter- particle spacing ro equals 10 2ap(Erel), i.e., the system exhibits universal behavior even when the interparticle spacing is 100 times larger than the scattering length ap(Erel). This is in contrast to the s-wave case, where the universal regime requires |as| ≫ ro. The different behavior of the higher partial wave systems compared to the s-wave system can be understood by realizing that ξl is a combined length, which contains both the range R of the two-body potential and the average interparticle spacing ro. For a given al(Erel)/R, the first inequality in Eq. (21) is thus satisfied for larger average interparti- cle spacings ro/R, or smaller scaled densities nBR 3, as l increases from 0 to 2. In addition to investigating l-wave Bose gases interact- ing through the square well potential vsw , we consider the van der Waals potential vvdw. For the energetically lowest-lying gas branch of the one-component “p-wave Bose” system we find the same results as for the square well potential if we replace R in Eqs. (18) to (20) by β6. We believe that the same replacement needs to be done for the liquid branch with l = 1 and for the liquid and gas branches of d-wave interacting bosons, and that the scaling at unitarity derived above for the square well potential holds for a wide class of two-body potentials. Within the LOCV framework, the results obtained for l xl C 0 0.00 −1.55 8.40 1 1.00 −2.29 6.52 2 2.00 −2.62 5.54 TABLE II: Dimensionless parameters xl, C l and C l for l = 0 to 2 for a two-component Bose system (see text for details). xl, C l and C l for the one-component Bose systems can be applied readily to the corresponding two-component system by scaling the Bose density appropriately (see Sec. II). The resulting parameters xl, C l and C l for the two-component Bose systems are summarized in Ta- ble II. The energy per particle EF,l/N for l-wave interact- ing two-component Fermi systems can be obtained from Eq. (17) using the LOCV solutions for the liquid and gas branches discussed above for l-wave interacting one- component Bose systems. In the unitary limit, we find F +Bl 2/3−l/6 , (22) where A = (3/10)(3π2)2/3 ≈ 2.87 and Bl = 2/3−l/6 (the C l are given in Table I). The first term on the right hand side of Eq. (22) equals EFG, and the second term, which is obtained from the LOCV solu- tions, equals λl/2. The energy per particleEF,l/N at uni- tarity is positive for all densities for Bl = C 2/3−l/6. For Bl = C 2/3−l/6, however, the energy per particle EF,l/N at unitarity is negative for l > 0 for small den- sities, and goes through a minimum for larger densities. This implies that this branch is always mechanically un- stable in the dilute limit for l > 0. The LOCV treatment for fermions relies heavily on the product representation of the many-body wave function, Eq. (15), which in turn gives rise to the two terms on the right hand side of Eq. (22). It is the competition of these two energy terms that leads to the energy minimum discussed in the previous paragraph. Future work needs to investigate whether the dependence of EF,l/N on two length scales as implied by Eq. (22) is correct. In contrast to the LOCV method, mean-field treatments predict that the energy at unitarity is proportional to EFG|kF re,l|, where re,l denotes a range parameter that characterizes the underlying two-body potential [42, 43, 44]. V. CONCLUSION This paper investigates Bose and Fermi systems us- ing the LOCV method, which assumes that three- and higher-order correlations can be neglected and that the behaviors of the many-body system are governed by two- body correlations. This assumption allows the many- body problem to be reduced to an effective two-body problem. Besides the reduced numerical effort, this for- malism allows certain aspects of the many-body physics to be interpreted from a two-body point of view. Further- more, it allows parallels between Bose and Fermi systems to be drawn. In agreement with previous studies, we find that the energy per particle “corrected” by the dimer binding en- ergy, i.e., EF,0/N − Edimer/2, of dilute two-component s-wave Fermi gases in the whole crossover regime de- pends only on the s-wave scattering length and not on the details of the underlying two-body potential. Fur- thermore, at unitarity the energy per particle is given by EF,0/N = 0.46EFG. This LOCV result is in good agree- ment with the energy per particle obtained from fixed- node diffusion Monte Carlo calculations, which predict EF,0/N = 0.42EFG [19, 20, 21]. This agreement may be partially due to the cancellation of higher-order cor- relations, and thus somewhat fortuitous. In contrast to Ref. [28], we find that the liquid branch of bosonic he- lium does not terminate at low densities but exists down to zero density. For higher angular momentum interactions, we deter- mine the energy per particle of one- and two-component Bose systems with infinitely large scattering lengths. For these systems, we expect the LOCV formalism to pre- dict the dimensionless exponent xl, which determines the functional dependenc of EB,l/N on the range R of the two-body potential and on the average interparticle spacing ro, correctly. The values of the proportionality constants CGl and C l , in contrast, may be less accurate. We use the LOCV energies to generalize the known uni- tary condition for s-wave interacting systems to systems with finite angular momentum. Since higher angular mo- mentum resonances are necessarily narrow, leading to a strong energy-dependence of the scattering strength, we define the universal regime using the energy-dependent scattering length al(k). In the unitary regime, the en- ergy per particle can be written in terms of the length ξl, which is given by a geometric combination of ro and R. 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0704.1805

On Charge Conservation and The Equivalence Principle in the Noncommutative Spacetime

On Charge Conservation and The Equivalence Principle in the Noncommutative Spacetime Youngone Lee Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea. Abstract We investigate one of the consequences of the twisted Poincaré symmetry. We derive the charge conservation law and show that the equivalence principle is satisfied in the canonical non- commutative spacetime. We applied the twisted Poincaré symmetry to the Weinberg’s analysis [11]. To this end, we generalize our earlier construction of the twisted S matrix [10], which apply the noncommutativity to the fourier modes, to the massless fields of integer spins. The transforma- tion formula for the twisted S matrix for the massless fields of integer spin has been obtained. For massless fields of spin 1, we obtain the conservation of charge, and the universality of coupling constant for massless fields of spin 2, which can be interpreted as the equality of gravitational mass and inertial mass, i.e., the equivalence principle. PACS numbers: 11.10.Nx, 11.30.-j, 11.30.Cp, 11.55.-m [email protected] http://arxiv.org/abs/0704.1805v3 1 Introduction In effort to construct an effective theory of quantum gravity at the Planck scale, noncommutativity of the spacetime has been considered. The canonical noncommutative spacetime has the commutation relations between the coordinates [1], [xµ, xν ] = iθµν , (1) where θµν(µ, ν = 0 to 3) is a constant antisymmetric matrix. Field theories in the canonical noncommutative spacetime can be replaced by field theories in the commutative spacetime with the Moyal product (The Weyl-Moyal correspondence [2]). One of the significant problems of those theories is that they violate the Lorentz symmetry. One finds that the symmetry group is SO(1, 1)×SO(2) instead of the Lorentz group, SO(1, 3). Since there is no spinor or vector representations in that symmetry group, most of the earlier studies performed by using the spinor, vector representations of the Lorentz group can not be justfied. Moreover, the factors 1(−1) are multiplied for a boson(fermion)loop without knowing the spin-statistics relation. To get around this, Chaichian et.al.∗ have deformed the Poincaré symmetry as well as its module space to which the symmetry acts [3]. The twisted symmetry group has the same representations as the original Poincaré group and at the same time they successfully retain the physical information of the canonical noncommutativity. The main idea was that one can change a classical symmetry group to a quantum group, ISOθ(1, 3) in this case, and twist-deform the module algebra consistently to reproduce the noncommutativity. In their approach, the noncommutative parameter θµν transform as an invariant tensor. This reminds us the situation that Einstein had to change the symmetry group of the spacetime and its module space(to the Minkowski spacetime) when the speed of light is required to be constant for any observer in an inertial frame. Similarly, Chaichian et.al. have required the change of the Hopf algebra with its module algebra so that any observer in an inertial frame feel the noncommutativity in the same way. For the κ-deformed noncommutativity, Majid and Ruegg found the κ-deformed spacetime [6] as a module space of the κ-deformed symmetry after Lukierski et.al. discovered the symmetry [7]. The real benefit of the twist is in the use of the same irreducible representations of original theories unlike general deformed theories, as in the case of the κ-deformed theory. Recently, groups of physicists have constructed the quantum field theory in the noncommutative spacetime by twisting the quantum space as a module space [8],[9],[10]. Espcially, Bu et.al. have proposed a twisted S-matrix as well as a twisted Fock space for consistency [10]. There we have obtained the twisted algebra of the creation and annihilation operators and the spin-statistics relation by applying the twisted Poincaré symmetry on the quantum space consistently. The analysis of this paper is mainly based on this work. These works justify the use of the irreducible representations of the Poincaré group and the sign factors being used in the earlier studies. Are these works merely a change of viewpoint? Mathemat- ically they look equivalent and seem to have equal amounts of information. But when the physics ∗ Oeckl [4], Wess [5] have proposed the same deformed Poincaré algebra. is concerned the action of symmetry becomes more subtle than it seems because it confines possible configurations of physical systems. In this article, we present an example showing the role of the twisted symmetry for solving physics problems, especially in the canonical noncommutative space- time. As an example, we derive the conservation law of charge and show the fact that the equivalence principle is satisfied even in the noncommutative spacetime. In this derivation we consider spin 1 and 2 massless fields for the photon and the graviton, respectively. For this purpose, we extend our ear- lier study on the scalar field theory to more general field theories and investigate a noncommutative version of Weinberg’s analysis [11, 12, 13]. Actually, there are many studies for the relation between noncommutativity and gauge theory [14], [15], [16], [17] or between the noncommutativity and the gravity [18],[19],[20] and between them [21]. And there were many argument whether the equivalence principle is satisfied at the quantum level. Some people argued that the equivalence principle is violated in quantum regime [22],[23], while there are studies which show non-violation of the equivalence principle [24]. Whether the principle of equivalence is violated or not is an important issue for quantum gravity because the principle is the core of the general relativity. The paper is organized as follows. We extend our previous construction of the S⋆ matrix to the massless fields of integer spin after giving a brief review on the construction and the properties of the S⋆ matrix. We give an exact transformation formula for the S⋆ matrix elements in section 2. We give the consequence of requiring the twist invariance to the S⋆ matrix elements for the scattering process. These results lead to the charge conservation law for the spin 1 field theory and the universality of the coupling constant for spin 2 field in noncommutative spacetime in section 3. Finally, we discuss the implication of the twisted symmetry and its applicability to other issues in section 4. We give some related calculations of the polarization vector, the noncommutative definition of the invariant M function, and a twisted transformation formula for the S⋆ matrix in the Appendices. 2 Properties of the general S⋆ matrix 2.1 A short introduction of useful properties of the twist-deformation An algebra with a product · and a coalgebra with a coproduct ∆ constitute a Hopf algebra if it has an invertible element S called antipode and with some compatibility relations. For a Lie algebra g, there is a unique universal enveloping algebra U(g) which preserves the Lie algebra properties in terms of unital associative algebra. The Hopf algebra of a Lie algebra g is denoted as H ≡ {U(g), ·,∆, ǫ, S}, where U(g) is an universal enveloping algebra of the corresponding algebra g and we denotes the counit as ǫ. The Sweedler notation is being widely used for a shorthand notation of the coproduct, Y(1) ⊗ Y(2) [25]. The action of a Hopf algebra H to a module algebra A is defined as Y ⊲ (a · b) = (Y(1) ⊲ a) · (Y(2) ⊲ b), (2) where a, b ∈ A, the symbol · is a multiplication in the module algebra A, and the symbol ⊲ denotes an action of the Lie generators Y ∈ U(g) on the module algebra A. The product · in H and the multiplication · in A should be distinguished. If there is an invertible ’twist element’, F = F(1) ⊗ F(2) ∈ H ⊗H, which satisfies (F ⊗ 1) · (∆⊗ id)F = (1⊗F) · (id ⊗∆)F , (3) (ǫ⊗ id)F = 1 = (id ⊗ ǫ)F , (4) one can obtain a new Hopf algebra HF ≡ {UF (g), ·,∆F , ǫF , SF} from the original one. The relations between them are ∆FY = F ·∆Y · F−1 , ǫF (Y ) = ǫ(Y ), SF (Y ) = u · S(Y ) · u−1 , u = F(1) · S(F(2)), (5) with the same product in the algebra sector. The ’covariant’ multiplication of the module algebra AF for the twisted Hopf algebra HF which maintain the form of Eq.(2) is given as (a ⋆ b) = ·[F−1 (a⊗ b)]. (6) From the above relations, one can derive an important property of the twist such that it does not change the representations of the algebra: DF(Y )(a ⋆ b) = ⋆ [∆FY (a⊗ b)] = · [F−1 · F∆0YF−1(a⊗ b)] = · [∆0Y F−1(a⊗ b)] = D0(Y )(a ⋆ b), (7) where representations of the coproduct and the twist element is implied, i.e., D[∆Y ] = D(Y(1))⊗D(Y(2)), D[F ] = D(F(1))⊗D(F(2)). (8) The above considerations lead us to the golden rule: The irreducible representations are not changed by a twist and one can regard the covariant action of a twisted Hopf algebra on a twisted module algebra as the action of the original algebra on the twisted module algebra. 2.2 The S⋆ matrix and its twist invariance Recently, a quantum field theory has been constructed in such a way to preserve the twisted Poincaré symmetry [10]. There we confined the construction for the space-space noncommutativity. It is hard to know whether one can construct a consistent twist Poincaré invariant field theory satisfying the causality in the case of space-time noncommutativity. They have tried to apply the twisted symmetry to quantum spaces consistently, especially to the algebra of the creation and annihilation operators (a†p and ap). As a result, they obtained the twisted algebra of quantum operators. If we use a shorthand notation, p ∧ q = pµθµνqν , the twisted algebra of a†p and ap can be denoted as: cp ⋆ cq = e ep∧eq cp · cq, (9) where cp can be ap or a p, p̃ ≡ −p(p) for cp = ap(a†p) and · denotes the ordinary multiplication of operators in the commutative theories. This twisted algebra naturally leads to the twisted form of Fock space, S-matrix and quantities related to the creation and annihilation operators. Thus, we obtain the twisted basis of Fock space and S⋆-matrix: |q1, · · · , qn〉 → |q1, · · · , qn〉⋆ = E(q1, · · · , qn)|q1, · · · , qn〉, (10) where E(q1, · · · , qn) = exp i<j qi ∧ qj is a phase factor which has the interesting properties [10], and S → S⋆ = (−i)k d4x1 · · · d4xk T {H⋆I(x1) ⋆ · · · ⋆H⋆I(xk)} , (11) where T denotes the time ordering and H⋆I(x) is an interaction Hamiltonian density in the Dyson formalism. The explicit form of the S⋆ matrix elements for the scalar φ n theoryin the momentum space is: ⋆〈β|S⋆|α〉⋆ = E(−β, α) (−ig)k · · · ···cQk E(Q̃1) · · · E(Q̃k)〈β|Sk(Q̃1 · · · , Q̃k)|α〉, (12) where Q̃ is the shorthand notation for (q̃1, . . . , q̃n) [10]. In the above, 〈β|Sk(Q̃1 · · · , Q̃k)|α〉 is a gk order term of the S-matrix element of the commutative theory where g is the coupling constant of the theory. From the momentum conservation, i.e., delta functions in the 〈β|Sk(Q̃1 · · · , Q̃k)|α〉, one can show that the S⋆ matrix element, ⋆〈β|S⋆|α〉⋆, can be represented by Feynman diagrams with extra phase factors E(Q̃) for each vertex. The phase factors drastically change the predictions of the theory. This result agrees with Filk’s result[26], but we have overall factors E(−β, α) corresponding to external lines in the Feynman diagram which originated from the twisted Fock space. From the above considerations, the new modified Feynman diagrams can be obtained from the untwisted ones by changing the phase factors from 1 to E(Q̃i) at each vertex. The twist invariance of this prescription of the S⋆ matrix is not manifest because non-locality of the interactions may violate the twist invariance of the S⋆ matrix, in general, i.e., [H⋆I(x),H⋆I(y)]⋆ 6= 0 for spacelike (x− y). (13) However, we see from the form of S⋆ matrix in Eq.(12), that the proposed S⋆ matrix is clearly twist invariant since it is constructed from phase factors which are twist invariant, and the Feynman propa- gators. Twisted product of fields operators satisfy 〈0|ψ(x) ⋆ ψ(y)|0〉 = 〈0|ψ(y) ⋆ ψ(x)|0〉, for spacelike (x− y). (14) Hence we see that the Feynman propagator, same as the twist Feynman propagator 〈0|T [ψ(x) ⋆ ψ(y)]|0〉, is twist invariant. From this, the invariance of the S⋆ matrix elements follows immediately. 2.3 Generalization to arbitrary fields We need to get the S⋆ matrix for massless field theories of integer spin for the analysis of this paper. In the previous work [10], we have constructed the S⋆ matrix for scalar field theory and we have expected that the same formulation could be possible for general field theories. In this section, we generalize our argument used in that paper to obtain the form of the S⋆ matrix elements for massless field theories with spin 1 and 2. In the analysis of this paper, we use the (s/2, s/2) representation for massless integer spin fields. The reason we use it and the considerations for the other representations are given in section 4. We have used the perturbation theory in our formulation of the S⋆ matrix in Eq.(11). Another assumption was the particle interpretation. That is, the field operators are represented as linear com- binations of the creation and annihilation operators and the fields of spin s transform(twist) as† : [Dsθ(Λ −1)]AB Ψ̂ θ (Λx+ a) = Uθ(Λ, a) Ψ̂ θ (x) U θ (Λ, a), (15) where Ds denotes the irreducible representation for spin s. Since translations act homogenously on the fields, twisted tensor fields can be written as Ψ̂ Aθ (x) = [aσ(p) ǫ θ (p, σ) e ip·x + b†σ(p) χ θ (p, σ) e −ip·x], (16) where A denotes the tensor index, A ≡ (a1 · · · as) for massless spin s fields and σ for the helicity indices. Thus, the transformation relation of the fields, Eq.(15), can be reduced to [Dsθ(Λ −1)]AB Ψ̂ θ (Λx) = Uθ(Λ) Ψ̂ θ (x) U θ (Λ). (17) In order that the field operators transform correctly, the ’polarization’ tensor ǫ Aθ (p, σ) are to be trans- formed as ǫ Aθ (p, σ) = (2π)3 [Dsθ(L(p))]AB ǫ Bθ (k, σ), χ Aθ (p, σ) = (2π)3 [Dsθ(L(p))]AB χ Bθ (k, σ′) C−1σ′σ, (18) where L(p) is the Lorentz transformation‡ which reflects the little group and C is a matrix with the properties [12], C∗ C = (−1)2s, C† C = 1. (19) † Recently, Joung and Mourad [27] have argued that a covariant field linear in creation and annihilation operators does not exist. But the definition of creation and annihilation operators in that paper is different from ours. We try to follow the view that we use the same fundamental quantities as of those in the untwisted theories changing the algebras only. The field operators in linear combinations of the creation and annihilation operators can be justified in this basis, because the free fields are the same as the commutative case. ‡ L(p) is a Lorentz transformation that takes the standard momentum kµ to pµ ≡ (|p|,p) for massless fields. For the massless case L(p) satisfies L(p) = R(p̂)B(|p|) where R(p̂) is a rotation which takes the direction of the standard momentum k to the direction of p and B(|p|) is the boost along the p direction [11]. Since the most important properties of the twist is that the representations are the same as in the original group, the Poincaré group in this case, the above Dsθ(Λ −1) and Dsθ(L) can be replaced by Ds(Λ−1) and Ds(L), respectively. Thus the θ-dependance remains only in the polarization tensors. Furthermore, one can see that there’s no θ-dependance in ǫ Aθ (σ) and χ ′). We can show this by considering the case of ǫ Aθ (σ). From the group property of the twisted Lorentz transformations, the related transformation to the polarization tensor ǫ Aθ (σ) can be written as: [Dsθ(Λ −1)]AB [D θ(L)]CD ǫ Dθ (σ) = [Dsθ(Λ−1 · L)]AB ǫ Bθ (σ). (20) In order for this transformation to be a twist transformation, it is to be satisfied [Dsθ(Λ −1 · L)]AB ǫ Bθ (σ) = [Ds(Λ−1 · L)]AB ǫ Bθ (σ), (21) i.e., twisted transformation has the same representation as the untwisted one. From the primary relation between the twist and the module algebra, one can see that the form of ǫ Aθ (σ), twisted version of the commutative polarization tensor, should be ǫ a1···anθ (σ) = · [F n ⊲ ǫ a1(σ)⊗ · · · ⊗ ǫan(σ)], (22) where F−1n can be obtain from§ the Fθ, and Fθ is the very twist element corresponding to the canonical noncommutativity: Fθ = exp θαβPα ⊗ Pβ . (23) Since Pα ⊲ ǫ a(σ) = 0, we obtain ǫ a1a2···asθ (σ) ≡ ǫ a1a2···as(σ) = ǫ(a1σ ǫ σ · · · ǫas)σ , (24) where ǫ(a1a2···as)(σ) is the polarization tensor in the corresponding commutative field theory. This relation leads to the relation Ψ̂ Aθ (x) ≡ Ψ̂A(x). (25) This relation is expected because the twist does not change the representations and we write the field operators by using an irreducible representations of the symmetry group. Explicit calculations for massless (s/2, s/2) tensor representation which shows the non-dependance on the θ is given in the Appendix A. Therefore, one can safely use the same representations for the field operators in the twisted theory as those of the untwisted ones. What really be twisted are the multiplications of the creation and an- nihilation operators only. Since the multiplication of the creation and annihilation operators between different species of particles act like composite mappings on the Hilbert space, one can construct a § The associativity of the twisted product ∗ guaranty that it can be obtained by successive applications of Eq.(6). module algebra from them. That is, when cp’s and dp’s are the creation and annihilation operators corresponding to different species, their twisted multiplications are cp ⋆ dq = e ep∧eq cp · dq, (26) where cp(dp) can be ap(bp) or a p), and p̃ and q̃ are defined as in section 2.2. Consequently, the scheme for twisting S-matrix used in [10] applies for general field theories. Twist invariance of the S⋆ matrix for general field theories follows immediately. As in the scalar field theory, the amplitudes can be obtained by multiplying the phase factor E(q1, · · · , qn) for each vertex in the Feynman diagram of the untwisted theories. 2.4 Exact transformation formula for the S⋆ matrix element for massless fields Transformation formula for the S⋆ matrix element corresponding to the process in which a massless particle is emitted with momentum q and helicity s can be inferred as (Appendix C) S±s⋆ (q, p) = exp [±isΘ(q,Λ)]S±s⋆ (Λq,Λp). (27) The S⋆ matrix can be written as the scalar product of a polarization tensor and theM⋆ function(Appendix S±s⋆ (q, p) = ± (q̂) · · · ǫ ± (q̂)(M ⋆ )µ1···µs(q, p), (28) where the M⋆ function twist-transform covariantly as M±µ1···µs⋆ (q, p) = Λ · · ·Λ µjνs M ±ν1···νs ⋆ (Λq,Λp). (29) The form of the S⋆ matrix element in Eq.(28) appears to break the twisted Poincaré symmetry because the polarization vectors do not satisfy the Lorentz covariance, rather they satisfy (Λ µν − qµΛ 0ν /|q|)ǫ ν± (Λq) = exp {±iΘ(q,Λ)} ǫ ± (q̂). (30) Hence requiring the twist invariance of the S⋆ matrix would lead to a constraint relation between the momentum, the polarization vectors and the M⋆ function. From Eq.(27) and Eq.(30), the S⋆ matrix element in Eq.(28) can be written as S±s⋆ (q, p) = exp [±isΘ(q,Λ)] [ǫ µ1± (Λq)− (Λq)µ1Λ 0ν ǫν±(Λq)/|q|]∗ · · · [ǫ µs± (Λq)− (Λq)µsΛ 0ν ǫν±(Λq)/|q|]∗(M⋆)±µ1···µs(Λq,Λp). (31) Requiring the twist invariance of the S⋆ matrix element results in: ± (q̂) · · · ǫ ± (q̂)M ⋆ µ1···µs = 0. (32) This leads us to the desired identities: ±ρµ2···µs ⋆ (q, p) = 0. (33) 3 Charge conservation and the equivalence principle Since the analysis of the conservation law is just a noncommutative generalization of Weinberg’s work [11], the derivation of this section will be fairly straightforward. As we saw in section 2, the differences between the noncommutative field theory and the commutative one are the phase factors at each vertex of the Feynman diagrams. 3.1 Dynamical definition of charge and gravitational mass We define the charge and the gravitational mass dynamically as the coupling of the vertex amplitudes for the soft(very low energy) photon and graviton, respectively. Consider the vertex amplitude for the process that a soft¶ massless particle of momentum q and spin s is emitted by a particle of momentum p and spin J . Since the only tensor which can be used to form the invariantM function is known to be pµ1 · · ·pµs , the noncommutative M function, M⋆ function, is given by the commutative M function with phase factor E(q̃, p̃, p̃′) multiplied at each vertex. That is, the vertex amplitude can be written as E(q̃, p̃, p̃′) 2E(p) pµ1 · · · pµs ǫ ± (q̂) · · · ǫ ± (q̂). (34) When the emitting particle has spin J we have to multiply to the vertex amplitude δσσ′ [11]. Thus, the explicit form of the vertex amplitudes can be written as: 2i(2π)4 · es · δσσ′ [pµ ǫ µ∗± (q̂)]s (2π)9/2[2E(p)] · E(q̃, p̃, p̃′), (35) where es is a coupling constant for emitting a soft massless particle of spin s (e.g., photon and gravi- ton). These coupling constants for emitting a soft particle can be interpreted as: e1 ≡ e as the elec- tric charge, and e2 ≡ 8πGg, with g the ratio of the gravitational mass and the inertial mass. Let us consider a near forward scattering of the two particles A and B with the coupling esA and B , respectively. From the properties of the phase factors [10], the phase factor for this scattering, E(q̃, p̃A, p̃A′) · E(−q̃, p̃B, p̃B′), goes to: E(q̃, p̃A, p̃A′) · E(−q̃, p̃B, p̃B′) = E(p̃A, p̃A′, q̃) · E(−q̃, p̃B, p̃B′) = E(p̃A, p̃A′) · E(p̃B, p̃B′). (36) However, in the forward scattering limit, the direction of the particles does not change, i.e. pA ‖ p′A. For space-space noncommutativity pA ∧ p′A goes to zero, i.e., the phase factor goes to 1. Thus, when the invariant momentum transfer t = −(p′A − pA)2 goes to zero, using the properties of the polarization vectors in the Appendix B, the S⋆ matrix element can be shown to approach the same form as in the corresponding commutative quantity which is easily calculated in a well chosen‖ ¶ This is to define the charge and gravitational mass as a monopole, not as a multipole moments. ‖ The coordinate system in which q · pA = q · pB = 0 [11]. coordinate system, δσAσB′δσBσB′ 4π2EAEBt eAeB(pA · pB) + 8πG gAgB (pA · pB)2 − m 2Am . (37) This coincidence is quite special in the sense that the S⋆ matrix elements are quite different from the S matrix elements in the commutative theory when there is a momentum transfer. If particle B is at rest, this gives δσAσA′δσBσB′ −eAeB +G · gA 2EA − · gBmB . (38) Hence, one can interpret the coupling constant eA as the usual charge of the particle A. Moreover one can identify the effective gravitational mass of A as (mg)A = gA 2EA − . (39) For nonrelativistic limit the gA can be interpreted as a ratio of the gravitational mass and the inertial mass, i.e., (mg)A = gA ·mA, EA ≃ mA. (40) Consequently, if gA does not depend on the species of A, it suggests that the equivalence principle holds. 3.2 Conservation law Consider a S matrix element S(α→ β) for some reaction α→ β, where the states α and β consist of various species of particles. The same reaction with emitting a soft massless particle of spin s (photon or graviton for s = 1 or 2), momentum q and helicity ±s can occur. We denote the corresponding S matrix element as S±s(q, α → β). Each amplitude of this process breaks the Lorentz symmetry because a massless fields of (s/2, s/2) representation break the symmetry [28]. However, a real physical reaction, to which a S-matrix element (the sums of each amplitude) correspond, should be Lorentz invariant. By requiring this condition, Weinberg could obtain the conservation relations. In this section, we investigate the similar relations by requiring the twist invariance of the S⋆ matrix elements. Suppose that a soft particle is emitted by ith particle (i = 1, · · · , n). Then by the polology of the conventional field theory, the S matrix elements will have poles at |q| = 0 when an extra soft massless particle is emitted by one of the external lines: (pi + ηi · q)2 +m 2i 2pi · q , (41) where ηi = 1(−1) for the emission of a soft particle from an out(in) particle, respectively. By utilizing the above relation, one can obtain the S matrix elements for the soft massless particles in the |q| → 0 limit [11]: S±s(q, α→ β) = (2π)3/2 ηiesi [pi · ǫ ∗±(q̂)]s (pi · q) S(α→ β). (42) In the noncommutative case, from the result of the section (2) and by using the above relation, one can deduce the S⋆ matrix elements for the same process in the noncommutative spacetime: S ±s⋆ (q, α→ β) = Ei(q̃, α → β) · S±i (q, α → β) E(q̃, p̃i, p̃i + q̃) · EI(p̃1, · · · , p̃i + ηiq̃, · · · , p̃n) · S±i (q, α→ β) E(q, p̃i) (2π)3/2 ηiesi [pi · ǫ ∗±(q̂)]s (pi · q) × EI(p̃1, · · · , p̃i + ηiq̃, · · · , p̃n) · S±i (α→ β), (43) where E iI = EI(p̃1, · · · , p̃i + q̃, · · · , p̃n) are the phase factors in the internal process∗∗. Since all the E iI’s are the same when q → 0 (E iI = EI), one obtains in that limit, S ±s⋆ (q, α→ β) = (2π)3/2 ηiesi [pi · ǫ ∗±(q̂)]s (pi · q) S⋆(α → β), (44) where we used the relation S⋆(α → β) = EI · S(α → β). From the transformation properties of the S⋆ matrix elements, Eq.(28), we obtain, S ±s⋆ (q, α→ β) → ± (q̂) · · · ǫ ± (q̂)(M ⋆ )µ1···µs(q, α→ β). (45) Identifying the S⋆ matrix elements in Eq.(44) and Eq.(45) gives the invariant M⋆ functions for spin 1 and 2: Mµ⋆ (q, α→ β) = (2π)3/2 ei · ηip µi (pi · q) S⋆(α→ β), Mµν⋆ (q, α→ β) = (2π)3/2 gi · ηip µi p νi (pi · q) S⋆(α → β). (46) Requiring the twist invariance to the S⋆ matrix elements, Eq.(33), gives: 0 = qµM ⋆ (q, α→ β) → ηiei = 0, 0 = qµM ⋆ (q, α→ β) → gi · ηipνi = 0, (47) ∗∗ The integrals over the internal loop momenta have been suppressed because they do not affect the final result since the change in the phase factor occurs only in the external lines. in general. Hence, one obtains the charge conservation law for the spin 1 fields. For s = 2, in order to satisfy the two relations, ηipi = 0 (4-momentum conservation) and i gi · ηipνi = 0, gi should be constants (i.e. independence of the particle species). The universality of this coupling constant as a ratio of gravitational mass and inertial mass shows that the equivalence principle is satisfied even in the noncommutative spacetime. 4 Discussion We have found that the conservation of charge and the equivalence principle are satisfied even in the canonical noncommutative spacetime. The derivation was fairly straightforward, once we have con- structed the S⋆ matrix for general field theories. The assumption was that the quanta of the gravitation is massless spin 2, massless spin 1 for photon. We extended the construction of the S⋆ matrix to general fields, especially for the massless fields of integer spin. The twisted Feynman diagrams can be constructed by the same irreducible represen- tations as those in the untwisted theories with the same rule except for the different phase factors at each vertex. Hence, we can say that the same reasoning apply to the massive fields. We use the (s/2, s/2) representation for the field operators mainly because one can obtain the condition, Eq.(33), requiring the twist Lorentz invariance. In this representation, since the polarization vectors are not Lorentz four vectors, each amplitude of emitting a real soft massless particle violate the symmetry. We have used this property to show the conservation law in this paper. Another representation for the fields, the (s, 0)⊕ (0, s) representation, which has the parity symmetry, can be made Lorentz covariant. One realizes that due to these properties one cannot derive the conservation law by the method used in this paper. Charge conservation in the noncommutative spacetime is expected from the gauge symmetry and the noncommutative Noether theorem. However, it is not quite certain whether the equivalence princi- ple is satisfied in the noncommutative spacetime. Since the noncommutative spacetime is not locally Minkowski nor the local symmetry group is SO(1, 3), one can not guarantee if the principle is satis- fied. But in our twisted symmetry context, one can expect that the equivalence principle is satisfied because the algebra structure is the same as the conventional theory though the coalgebra structure is different. The applicability of the S matrix theoretic proof of the equivalence principle given in this paper is restrictive because the analysis is perturbative in nature. We expect that the conclusion of this paper is to be one of the stepping stone towards the further understanding of the nature of the principle in quantum gravity. This paper shows an example for the usefulness of the twisted symmetry to derive the physically important relations in the noncommutative spacetime. We think that the twist analysis is adequate for the schematic approaches to the noncommutative physics, while the other approaches are suitable for the explicit calculations though they are equivalent. The future applications of the twisted symmetry to the other issues in this direction of approach are expected. Acknowledgement I express my deep gratitude to Prof. J. H. Yee for his support. I would like to thank Dr. Jake Lee for helpful discussions and Dr. A. Tureanu for his comments on the draft. This work was supported in part by Korea Science and Engineering Foundation Grant No. R01-2004-000-10526-0, and by the Science Research Center Program of the Korea Science and Engineering Foundation through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number R11 - 2005 - 021. A Exact calculation of the polarization tensor for massless fields The creation and annihilation operators of the massless field transform under the Lorentz transforma- tion as [28] a†(Λp, σ) = e−iσΘ[W (Λ,p)] U(Λ) a†(p, σ) U−1(Λ), a(Λp, σ) = e+iσΘ[W (Λ,p)] U(Λ) a(p, σ) U−1(Λ), (48) up to phase factors, where W (Λ, p) = L−1(p)Λ−1L(Λp) denotes the Wigner rotation defined as in [28] (Appendix C), and the Θ[W (Λ, p)] is the related angle to which the little group corresponds. Hereafter, we abbreviate Θ[W (Λ, p)] as Θ(Λ, p). It is to be noted that we use U(Λ) instead of Uθ(Λ) in here. This follows immediately from the properties of the twist in section 2.1. The field Ψ̂ Aθ transform as [Ds(Λ−1)]AB Ψ̂ θ (Λx) = U(Λ) Ψ̂ θ (x) U −1(Λ). (49) In order to satisfy the two relations (48), (49), the polarization tensor should transform as ǫ Aθ (p, σ) = e −iσΘ(Λ,p) [Ds(Λ−1)]AB ǫ θ (Λp, σ). (50) When Λ = W and p = k, the above relation goes to [Ds(W )]AB ǫ θ (k, σ) = e −iσΘ(W,p)ǫ Aθ (k, σ), (51) where k denotes the standard momentum. The Wigner rotation can be written as W (φ, α, β) = R(φ)T (α, β) because the little group is isomorphic to ISO(2) for massless fields [28]. Suppose that the field transforms as (m,n) representation of spin s (m + n = s). When the φ, α and β are infinitesimal, Ds(W ) can be written as Ds(W ) ≃ 1− iφ(M3 +N3) + (α + iβ)(M1 − iM2) + (α− iβ)(N1 + iN2). (52) For M− ≡M1 − iM2, N+ ≡M1 + iM2, and Θ → φ we have: (M3 +N3) ǫθ(k, σ) = σ ǫθ(k, σ), M− ǫθ(k, σ) = 0, N+ ǫθ(k, σ) = 0, (53) which gives M3 ǫθ(k, σ) = −m ǫθ(k, σ), N3 ǫθ(k, σ) = +n ǫθ(k, σ). (54) The ǫθ(k, σ) satisfys the same equation as ǫ(k, σ). Since the highest or lowest weight corresponds to a unique state, one obtains the same polarization tensor as a solution, ǫθ(k, σ) = ǫ(k, σ). B Properties of the polarization vectors Here, we summarize the properties of the polarization vector. The properties of the polarization tensors of other rank, Eq.(24), follows from it. Solving the (53) for σ = ±1 gives the explicit form of the polarization vector for the standard momentum as ± (k) ≡ (1,±i, 0, 0), (55) where we made the conventional choice of the phase. The polarization vector of momentum p is defined as ± (p) = [L(p)]µν ǫ ν± (k), (56) where L(p) is the Lorentz transformation which takes k to p, i.e., pµ = [L(p)]µν kν . Then the well known properties of the polarization vectors can be deduced [11]: ± (p̂) = 0, (57) ǫ ∗±µ(p̂)ǫ ± (p̂) = 1, ǫ±µ(p̂)ǫ ± (p̂) = 0, (58) ± (p̂) = ǫ ∓ (p̂), ǫ ± (p̂) = 0, (59)∑ ± (p̂)ǫ ± (p̂) = η µν + (p̃µpν + p̃νpµ)/2|p|2 ≡ Πµν , p̃ ≡ (|p|,−p), (60) ± (p̂)ǫ ± (p̂)ǫ ± (p̂)ǫ ± (p̂) = {Πµ1ν1(p̂)Πµ2ν2(p̂) + Πµ1ν2(p̂)Πµ2ν1(p̂) −Πµ1µ2(p̂)Πν1ν2(p̂)} . (61) The polarization ’vectors’ are not the Lorentz four vectors††, rather they transform as ±(p) = e ∓iΘ(Λ,p)[Dsǫ (Λ, p)] ±(Λp), [Dsǫ (Λ, p)] ν = (Λ −1)µν − (Λ−1)0ν . (62) For general spin s, the polarization tensor can be written as ǫAσ (p) = e ∓iΘ(Λ,p)[Dsǫ (Λ, p)] σ (Λp), (63) up to a phase factor. ††It comes from the fact that the little group is not semisimple for massless fields. Translation in the little group isomorphic to ISO(2) generates the gradient term in the transformations. C Invariant M⋆ function for massless field Let P denotes a shorthand notation for the external lines, P ≡ (p1, . . . , pn), and K denotes a standard momenta‡‡ K ≡ (k1, . . . , kn). There exists a unique Lorentz transformation satisfying P ≡ LPK. Then the relation between the Lorentz group and the little group can be described symbolically as: W (Λ,P ) // ΛP , where W (Λ, P ) is the Wigner transformation to which a Lorentz transformation Λ and the momenta P correspond. Since the twist do not change the group properties, above relation also holds for the twisted symmetry group. The S⋆ matrix elements transform as S⋆[P ] = D θ[W (Λ, P )] S⋆[ΛP ] = Dsθ[L P · Λ −1 · LΛP ] S⋆[ΛP ] (64) where the indices for the external lines are suppressed. From the golden rule and (48), the explicit transformation formular for S⋆[P ] can be obtained: S⋆[P ] = N(ΛP ) N(P ) e±isΘ(Λ,P ) S⋆[ΛP ], (65) where N denotes the corresponding normalization factor. If one defines M⋆[P ] as S⋆[P ] = D P ]M⋆[P ], one can show that M⋆[P ] transform as M⋆[P ] = D −1]M⋆[ΛP ], (66) i.e., it is twist invariant. We will call it as an invariant M⋆ function as in [29, 30]. Let us find out the invariant M⋆ function for massless fields. First, we define the quanties for the standard momenta K. The choice for M⋆[K] is MA⋆ [K] = N(K) ǫ A[K] S⋆[K]. (67) If we define M⋆[P ] as MA⋆ [P ] = [D θ(LP )] ⋆ [K], (68) then one can easily show that MA⋆ [P ] is twist invariant. By using the explicit form of Dsǫ in (62) and kbjM b1···bs ⋆ [K] = 0, one obtains the relation ǫC [P ]∗ (M⋆)C [P ] = e±isΘ(L ,P )[Dǫθ(L A[K]∗ [Dθ(LP )] C (M⋆)B[K] = e±isΘ(L ,P ) ǫA[K]∗ [Dθ(L P ) ·D A (M⋆)B[K] = e±isΘ(L ,P ) ǫA[P ]∗ (M⋆)A[P ] = e±isΘ(L ,P )N(K) S⋆[K], (69) ‡‡ see [11],[29]. where we used the relation [Dθ(L P ) ·D A (M⋆)B[K] ≡ [D(L−1P ) ·Dǫ(L A (M⋆)B[K] δb1a1 − (LP ) · · · δbsas − (LP ) × (M⋆)b1···bs [K] = (M⋆)a1···as [K]. (70) By setting Λ = L−1P in (65), one can see that (69) is S⋆[P ]. Thus, the desired form of S⋆[P ] is S⋆[P ] = N(P ) ǫ A[P ]M ⋆ [P ]. (71) References [1] S. Doplicher, K. Fredenhagen, and J. E. Roberts, Comm. Math. Phys. 172, 187 (1995). [2] H. J. Groenewold, Physica 12, 405 (1946); J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949); H. Weyl, Quantum mechanics and group theory, Z. Phys. 46, 1 (1927). [3] M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu, Phys. Lett. B 604, 98(2004). [4] R. Oeckl, Nucl. Phys. B 581, 559 (2000). [5] J. 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B 558, 191 (2003). [19] M. Banados, O. Chandia, N. Grandi, F. A. Schaposnik, G. A. Silva, Phys. Rev. D64, 084012 (2001). [20] P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, J. Wess, Class. Quant. Grav. 22, 3511 (2005). [21] R. Banerjee, H. S. Yang, Nucl. Phys. B708, 434 (2005). [22] J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, A. S. Sakharov, Int. J. Mod. Phys. A19, 4413 (2004). [23] G. Z. Adunas, E. Rodriguez-Milla, D. V. Ahluwalia, Gen. Rel. Grav. 33, 183 (2001). [24] Y. N. Obukhov, Phys. Rev. Lett. 86, 192 (2001). [25] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, (1995). [26] T. Filk, Phys. Lett. B 376, 53 (1996). [27] E. Joung, J. Mourad, hep-th/0703245. [28] S. Weinberg, The Quantum Theory of Fields, Vol.I, Cambridge University Press, (1996). [29] H. Stapp, Phys. Rev. 125, 2139 (1962). [30] A. O. Barut, I. Muzinich, D. N. Williams, Phys. Rev. 130, 442 (1963). http://arxiv.org/abs/hep-th/0703245 Introduction Properties of the general S matrix A short introduction of useful properties of the twist-deformation The S matrix and its twist invariance Generalization to arbitrary fields Exact transformation formula for the S matrix element for massless fields Charge conservation and the equivalence principle Dynamical definition of charge and gravitational mass Conservation law Discussion Exact calculation of the polarization tensor for massless fields Properties of the polarization vectors Invariant M function for massless field

0704.1806

Critical Scaling of Shear Viscosity at the Jamming Transition

Critical Scaling of Shear Viscosity at the Jamming Transition Peter Olsson1 and S. Teitel2 Department of Physics, Ume̊a University, 901 87 Ume̊a, Sweden Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 (Dated: July 30, 2021) We carry out numerical simulations to study transport behavior about the jamming transition of a model granular material in two dimensions at zero temperature. Shear viscosity η is computed as a function of particle volume density ρ and applied shear stress σ, for diffusively moving particles with a soft core interaction. We find an excellent scaling collapse of our data as a function of the scaling variable σ/|ρc − ρ| ∆, where ρc is the critical density at σ = 0 (“point J”), and ∆ is the crossover scaling critical exponent. We define a correlation length ξ from velocity correlations in the driven steady state, and show that it diverges at point J. Our results support the assertion that jamming is a true second order critical phenomenon. PACS numbers: 45.70.-n, 64.60.-i, 83.80.Fg Keywords: In granular materials, or other spatially disordered systems such as colloidal glasses, gels, and foams, in which thermal fluctuations are believed to be negligible, a jamming transition has been proposed: upon increas- ing the volume density (or “packing fraction”) of parti- cles ρ above a critical ρc, the sudden appearance of a finite shear stiffness signals a transition between flowing liquid and rigid (but disordered) solid states [1]. It has further been proposed by Liu and Nagel and co-workers [2, 3] that this jamming transition is a special second or- der critical point (“point J”) in a wider phase diagram whose axes are volume density ρ, temperature T , and ap- plied shear stress σ (the latter parameter taking one out of equilibrium to non-equilibrium driven steady states). A surface in this three dimensional parameter space then separates jammed from flowing states, and the intersec- tion of this surface with the equilibrium ρ − T plane at σ = 0 is related to the structural glass transition. Several numerical [3, 4, 5, 6, 7, 8, 9, 10], theoretical [11, 12, 13, 14] and experimental [5, 15, 16, 17, 18] works have investigated the jamming transition, mostly by con- sidering behavior as the transition is approached from the jammed side. In this work we consider the flowing state, computing the shear viscosity η under applied uniform shear stress. Previous works have simulated the flow- ing response to applied shear in glassy systems at finite temperature [19, 20, 21], and in foams [4] and granular systems [10] at T = 0, ρ > ρc. Here we consider the ρ − σ plane at T = 0, showing for the first time that, near point J, η−1(ρ, σ) collapses to a universal scaling function of the variable σ/|ρc − ρ| ∆ for both ρ < ρc and ρ > ρc. We further define a correlation length ξ from steady state velocity correlations, and show that it di- verges at point J. Our results support that jamming is a true second order critical phenomenon. Following O’Hern et al. [3], we simulate frictionless soft disks in two dimensions (2D) using a bidisperse mixture with equal numbers of disks of two different radii. The radii ratio is 1.4 and the interaction between the particles V (rij) = ǫ(1− rij/dij) 2/2 for rij < dij 0 for rij ≥ dij where rij is the distance between the centers of two par- ticles i and j, and dij is the sum of their radii. Particles are non-interacting when they do not touch, and inter- act with a harmonic repulsion when they overlap. We measure length in units such that the smaller diameter is unity, and energy in units such that ǫ = 1. A system of N disks in an area Lx ×Ly thus has a volume density ρ = Nπ(0.52 + 0.72)/(2LxLy) . (2) To model an applied uniform shear stress, σ, we first use Lees-Edwards boundary conditions [22] to introduce a uniform shear strain, γ. Defining particle i’s position as ri = (xi+γyi, yi), we apply periodic boundary conditions on the coordinates xi and yi in an Lx × Ly system. In this way, each particle upon mapping back to itself under the periodic boundary condition in the ŷ direction, has displaced a distance ∆x = γLy in the x̂ direction, re- sulting in a shear strain ∆x/Ly = γ. When particles do not touch, and hence all mutual forces vanish, xi and yi are constant and a time dependent strain γ(t) produces a uniform shear flow, dri/dt = yi(dγ/dt)x̂. When particles touch, we assume a diffusive response to the inter-particle forces, as would be appropriate if the particles were im- mersed in a highly viscous liquid or resting upon a rough surface with high friction. This results in the following equation of motion, which was first proposed as a model for sheared foams [4], dV (rij) x̂ . (3) The strain γ is then treated as a dynamical variable, http://arxiv.org/abs/0704.1806v2 obeying the equation of motion, LxLyσ − dV (rij)  , (4) where the applied stress σ acts like an external force on γ and the interaction terms V (rij) depend on γ via the particle separations, rij = ([xi−xj]Lx +γ[yi−yj]Ly , [yi− yj ]Ly), where by [. . .]Lµ we mean that the difference is to be taken, invoking periodic boundary conditions, so that the result lies in the interval (−Lµ/2, Lµ/2]. The constants D and Dγ are set by the dissipation of the medium in which the particles are embedded; we take units of time such that D = Dγ ≡ 1. In a flowing state at finite σ > 0, the sum of the inter- action terms is of order O(N) so that the right hand side of Eq. (4) is O(1). The strain γ(t) increases linearly in time on average, leading to a sheared flow of the particles with average velocity gradient dvx/dy = 〈dγ/dt〉, where vx(y) is the average velocity in the x̂ direction of the par- ticles at height y. We then measure the shear viscosity, defined by, dvx/dy 〈dγ/dt〉 . (5) We expect η−1 to vanish in a jammed state. We integrate the equations of motion, Eqs. (3)-(4), starting from an initial random configuration, using the Heuns method. The time step ∆t is varied according to system size to ensure our results are independent of ∆t. We consider a fixed number of particles N , in a square system L ≡ Lx = Ly, and vary the volume density ρ by adjusting the length L according to Eq. (2). We simulate for times ttot such that the total relative displacement per unit length transverse to the direction of motion is typ- ically γ(ttot) ∼ 10, with γ(ttot) ranging between 1 and 200 depending on the particular system parameters. In Fig. 1 we show our results for η−1 using a fixed small shear stress, σ = 10−5, representative of the σ → 0 limit. Our raw results are shown in Fig. 1a for several different numbers of particles N from 64 to 1024. Com- paring the curves for different N as ρ increases, we see that they overlap for some range of ρ, before each drops discontinuously into a jammed state. As N increases, the onset value of ρ for jamming increases to a limiting value ρc ≃ 0.84 (consistent with the value for random close packing [3]) and η−1 vanishes continuously. For finite N , systems jam below ρc because there is always a fi- nite probability to find a configuration with a force chain spanning the width of the system, thus causing it to jam; and at T = 0, once a system jams, it remains jammed for all further time. As the system evolves dynamically with increasing simulation time, it explores an increasing region of configuration space, and ultimately finds a con- figuration that causes it to jam. The statistical weight 10-3 10-2 10-1 eta^-1 eta^-1 eta^-1 eta^-1 eta^-1 beta=1.6 volume density #c " # (b)$ = 10 #c = 0.8415 % = 1.65 !"1~ |# " #c| 0.78 0.79 0.80 0.81 0.82 0.83 0.84 eta^-1 eta^-1 eta^-1 eta^-1 eta^-1 volume density # (a)$ = 10"5 FIG. 1: (color online) a) Plot of inverse shear viscosity η−1 vs volume density ρ for several different numbers of particles N , at constant small applied shear stress σ = 10−5. As N increases, one see jamming at a limiting value of the density ρc ∼ 0.84. b) Log-log replot of the data of (a) as η −1 vs ρc − ρ, with ρc = 0.8415. The dashed line has slope β = 1.65 indicating the continuous algebraic vanishing of η−1 at ρc with a critical exponent β. of such jamming configurations decreases, and hence the average time required to jam increases, as one either de- creases ρ, or increases N [3]. In the limit N → ∞, we expect jamming will occur in finite time only for ρ ≥ ρc. In Fig. 1b we show a log-log plot of η−1 vs ρc − ρ, us- ing a value ρc = 0.8415. We see that the data in the unjammed state is well approximated by a straight line of slope β = 1.65, giving η−1 ∼ |ρ − ρc| β in agreement with the expectation that point J is a second order phase transition. If point J is indeed a true critical point, one expects that its influence will be felt also at finite values of the stress σ, with η−1 obeying a typical scaling law, η−1(ρ, σ) = |ρ− ρc| |ρ− ρc|∆ . (6) Here z ≡ σ/|ρ−ρc| ∆ is the crossover scaling variable, ∆ is the crossover scaling critical exponent, and f−(z), f+(z) are the two branches of the crossover scaling function for ρ < ρc and ρ > ρc respectively. In Fig. 2 we show a log-log plot of inverse shear vis- cosity η−1 vs applied shear stress σ, for several different values of volume density ρ. Our results are for systems large enough that we believe finite size effects are negli- gible. We use N = 1024 for ρ < 0.844 and N = 2048 for ρ ≥ 0.844. Again we see that ρc ≃ 0.8415 separates two limits of behavior. For ρ < ρc, log η −1 is convex in log σ, decreasing to a finite value as σ → 0. For ρ > ρc, log η is concave in log σ, decreasing towards zero as σ → 0. The dashed straight line, separating the two regions of behavior, indicates the power law dependence that is ex- pected exactly at ρ = ρc (see below). Similar power law behavior at ρc was recently found in simulations of a three dimensional granular material [23]. 10-5 10-4 10-3 10-2 shear stress # $=0.830 $=0.834 $=0.836 $=0.838 $=0.840 $=0.841 $=0.842 $=0.844 $=0.848 $=0.852 $=0.856 $=0.860 $=0.864 $=0.868 #=0.0012 FIG. 2: (color online) Plot of inverse shear viscosity η−1 vs applied shear stress σ for several different values of the vol- ume density ρ. The dashed line represents the power law dependence expected exactly at ρ = ρc and has a slope β/∆ = 1.375. Solid lines are guides to the eye. Points la- beled σ = 0.0012 correspond to densities ρ = 0.870, 0.872, 0.874, 0.876, and 0.878. In Fig. 3 we replot the data of Fig. 2 in the scaled variables η−1/|ρ−ρc| β vs σ/|ρ−ρc| ∆. Using ρc = 0.8415, β = 1.65 (the same values used in Fig. 1b) and ∆ = 1.2, we find an excellent scaling collapse in agreement with the prediction of Eq. (6). As the scaling variable z → 0, f−(z) → constant; this gives the vanishing of η −1 ∼ |ρ− β at σ = 0. As z → ∞, both branches of the scaling function approach a common curve, f±(z) ∼ z β/∆, so that precisely at ρ = ρc, η −1 ∼ σβ/∆ as σ → 0 [24]. This is shown as the dashed line in both Figs. 3 and 2. A similar scaling collapse of η has been found in simulations [20] of a sheared Lennard-Jones glass, as a function of temperature and applied shear strain rate γ̇, but only above the glass transition, T > Tc. By comparing the goodness of the scaling collapse as parameters are varied, we estimate the accuracy of the critical exponents to be roughly, β = 1.7± 0.2 and ∆ = 1.2± 0.2. That the crossover scaling exponent ∆ > 0, implies that σ is a relevant variable in the renormalization group sense, and that critical behavior at finite σ should be in a different universality class from the jamming tran- sition at point J (i.e. σ = 0). The nature of jamming at finite σ > 0 will be determined by the behavior of the branch of the crossover scaling function f+(z), that describes behavior for ρ > ρc. From Fig. 3 we see that f+(z) is a decreasing function of z. If f+(z) vanishes only when z → 0, then Eq. (6) implies that η−1 vanishes for ρ > ρc only when σ = 0, and so there will be no jamming at finite σ > 0. If, however, f+(z) vanishes at some fi- nite z0, then η −1 will vanish whenever σ/(ρ− ρc) ∆ = z0; there will then be a line of jamming transitions emanat- ing from point J in the ρ − σ plane given by the curve ρ∗(σ) = ρc + (σ/z0) 1/∆. If f+(z) vanishes continuously 10-2 10-1 100 z = %/|# " #c| #c = 0.8415 $ = 1.65 & = 1.2 f"(z) # < #c f+(z) # > #c~ z$/& FIG. 3: (color online) Plot of scaled inverse viscosity η−1/|ρ− β vs scaled shear stress z ≡ σ/|ρ − ρc| ∆ for the data of Fig. 2. We find an excellent collapse to the scaling form of Eq. (6) using values ρc = 0.8415, β = 1.65 and ∆ = 1.2. The dashed line represents the large z asymptotic dependence, ∼ zβ/∆. Data point symbols correspond to those used in Fig. 2. at z0, jamming at finite σ will be like a second order transition; if f+(z) jumps discontinuously to zero at z0, it will be like a first order transition. Such a first order like transition has been reported in simulations [20, 21] of sheared glasses at finite temperature below the glass transition, T < Tc. However, recent simulations [10] of a granular system at T = 0, ρ > ρc, showed that a sim- ilar first order like behavior was a finite size effect that vanished in the thermodynamic limit. With these obser- vations, we leave the question of criticality at finite σ to future work The critical scaling found in Fig. 3 strongly suggests that point J is indeed a true second order phase transi- tion, and thus implies that there ought to be a diverg- ing correlation length ξ at this point. Measurements of dynamic (time dependent) susceptibilities have been used to argue for a divergent length scale in both the thermally driven glass transition [25], and the density driven jamming transition [17]. Here we consider the equal time transverse velocity correlation function in the shear driven steady state, g(x) = 〈vy(xi, yi)vy(xi + x, yi)〉 , (7) where vy(xi, yi) is the instantaneous velocity in the ŷ direction, transverse to the direction of the average shear flow, for a particle at position (xi, yi). The average is over particle positions and time. In the inset to Fig. 4 we plot g(x)/g(0) vs x for three different values of ρ at fixed σ = 10−4 and number of particlesN = 1024. We see that g(x) decreases to negative values at a well defined minimum, before decaying to zero as x increases. We define ξ to be the position of this minimum. That g(ξ) < 0, indicates 10-2 10-1 100 101 z = %/|# " #c| #c = 0.8415 $ = 0.6 & = 1.2h"(z) # < #c h+(z) # > #c ~ z$/& 0 5 10 15 20 rho=0.83 rho=0.834 rho=0.838 # = 0.830 # = 0.834 # = 0.838 N = 1024 % = 10-4 FIG. 4: (color online) Inset: Normalized transverse velocity correlation function g(x)/g(0) vs longitudinal position x for N = 1024 particles, applied shear stress σ = 10−4, and vol- ume densities ρ = 0.830, 0.834 and 0.838. The position of the minimum determines the correlation length ξ. Main fig- ure: Plot of scaled inverse correlation length ξ−1/|ρ− ρc| scaled shear stress z ≡ σ/|ρ− ρc| ∆ for the data of Fig. 2. We find a good scaling collapse using values ρc = 0.8415, ∆ = 1.2 (the same as in Fig. 3) and ν = 0.6. Data point symbols correspond to those used in Fig. 2. that regions separated by a distance ξ are anti-correlated. We can thus interpret the sheared flow in the unjammed state as due to the rotation of correlated regions of length ξ. Similar behavior, leading to a similar definition of ξ, has previously been found [26] in correlations of the nonaffine displacements of particles in a Lennard-Jones glass, in response to small elastic distortions. As with viscosity, we expect the correlation length ξ(ρ, σ) to obey a scaling equation similar to Eq. (6). We consider here the inverse correlation length ξ−1, which like η−1 should vanish at the jamming transition, obey- ing the scaling equation, ξ−1(ρ, σ) = |ρ− ρc| |ρ− ρc|∆ . (8) The correlation length critical exponent is ν, but the crossover exponent ∆ remains the same as for the vis- cosity. In Fig. 4 we plot the scaled inverse correlation length, ξ−1/|ρ−ρc| ν vs the scaled stress, σ/|ρ−ρc| ∆. Using ρc = 0.8415 and ∆ = 1.2, as was found for the scaling of η−1, we now find a good scaling collapse for ξ−1 by taking the value ν = 0.6. By comparing the goodness of the collapse as ν is varied, we estimate ν = 0.6±0.1. From the scaling equation Eq. (8) we expect both branches of the scaling function to approach the power law h±(z) ∼ z ν/∆ as z → ∞, so that ξ−1 ∼ σν/∆ as σ → 0 at ρ = ρc [24]. This is shown as the dashed line in Fig. 4. Our result is consistent with the conclusion “ν is between 0.6 and 0.7” of Drocco et al. [7] for the flowing phase, ρ < ρc. It also agrees with ν = 0.71 ± 0.08 found by O’Hern et al. [3] from a finite size scaling argument. Wyart et al. [14] have proposed a diverging length scale with exponent ν = 0.5 by considering the vibrational spectrum of soft modes approaching point J from the jammed side, ρ > ρc. While our results cannot rule out ν = 0.5, our scaling collapse in Fig. 4 does seem somewhat better when using the larger value 0.6. This work was supported by Department of Energy grant DE-FG02-06ER46298 and by the resources of the Swedish High Performance Computing Center North (HPC2N). We thank J. P. Sethna, L. Berthier, M. Wyart, J. M. Schwarz, N. Xu, D. J. Durian, A. J. Liu and S. R. Nagel for helpful discussion. [1] Jamming and Rheology, edited by A. J. Liu and S. R. Nagel (Taylor & Francis, New York, 2001). [2] A. J. Liu and S. R. Nagel, Nature 396, 21 (1998). [3] C. S. O’Hern et al. Phys. Rev. E 68, 011306 (2003). [4] D. J. Durian, Phys. Rev. Lett. 75, 4780 (1995) and Phys. Rev. E 55, 1739 (1997). [5] H. A. Makse, D. L. Johnson and L. M. Schwartz, Phys. Rev. Lett. 84, 4160 (2000). [6] C. S. O’Hern et al., Phys. Rev. Lett. 86, 000111 (2001) and 88, 075507 (2002). [7] J. A. Drocco et al., Phys. Rev. Lett. 95, 088001 (2005). [8] L. E. Silbert, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 95, 098301 (2005) and Phys. Rev. E 73, 041304 (2006). [9] W. G. Ellenbroek et al., Phys. Rev. Lett. 97, 258001 (2006). [10] N. Xu and C. S. O’Hern, Phys. Rev. E 73, 061303 (2006). [11] J. M. Schwarz, A. J. Liu and L. Q. Chayes, Europhys. Lett. 73, 560 (2006). [12] C. Toninelli, G. Biroli and D. S. Fisher, Phys. Rev. Lett. 96, 035702 (2006). [13] S. Henkes and B. Chakraborty, Phys. Rev. Lett. 95, 198002 (2005). [14] M. Wyart, S. R. Nagel, T. A. Witten, Europhys. Lett. 72, 486-492 (2005); M. Wyart et al., Phys. Rev. E 72, 051306 (2005); C. Brito and M. Wyart, Europhys. Lett. 76, 149 (2006). [15] V. Trappe et al., Nature (London) 411, 772 (2001). [16] T. S. Majmudar et al., Phys. Rev. Lett. 98, 058001 (2007). [17] A. .S. Keys et al., Nature physics 3, 260 (2007). [18] M. Schröter et al., Europhys Lett. 78, 44004 (2007). [19] R. Yamamoto and A. Onuki, Phys. Rev. E 58, 3515 (1998). [20] L. Berthier and J.-L. Barat, J. Chem. Phys. 116, 6228 (2002). [21] F. Varnik, L. Bocquet and J.-L.Barrat, J. Chem. Phys. 120, 2788, (2004). [22] D. J. Evans and G. P. Morriss, Statistical Mechanics of Non-equilibrium Liquids (Academic, London, 1990). [23] T. Hatano, M. Otsuki and S. Sasa, condmat/0607511. [24] In general, one should consider nonlinear scaling vari- ables. In our case, the most important correction would be to replace ρ− ρc in Eq. (6) by gρ(ρ, σ) ≡ ρ− ρc + cσ this could lead to noticeable corrections to our scaling equation near ρ = ρc. However, since we find ∆ > 0.5, our conclusion that η−1 ∼ σβ/∆ at ρ = ρc remains valid. See, A. Aharony and M. E. Fisher, Phys. Rev. B 27, 4394 (1983). [25] L. Berthier et al., Science 310, 1797 (2005). [26] A. Tanguy et al., Phys. Rev. B 66, 174205 (2002).

0704.1807

Polar actions on compact Euclidean hypersurfaces

Polar actions on compact Euclidean hypersurfaces. Ion Moutinho & Ruy Tojeiro Abstract: Given an isometric immersion f : Mn → Rn+1 of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n+ 1, we prove that the identity com- ponent Iso0(Mn) of the isometry group Iso(Mn) of Mn admits an orthogonal representation Φ: Iso0(Mn) → SO(n + 1) such that f ◦ g = Φ(g) ◦ f for every g ∈ Iso0(Mn). If G is a closed connected subgroup of Iso(Mn) acting locally polarly on Mn, we prove that Φ(G) acts polarly on Rn+1, and we obtain that f(Mn) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several suf- ficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure. MSC 2000: 53 A07, 53 C40, 53 C42. Key words: polar actions, rotation hypersurfaces, isoparametric submanifolds, rigidity of hypersurfaces, warped products. 1 Introduction Let G be a connected subgroup of the isometry group Iso(Mn) of a compact Riemannian manifold Mn of dimension n ≥ 3, which we always assume to be connected. Given an isometric immersion f : Mn → RN into Euclidean space of dimension N , in general one can not expect G to be realizable as a group of rigid motions of RN that leave f(Mn) invariant. Nevertheless, a fundamental fact for us is that in codimension N −n = 1 this is indeed the case. Theorem 1 Let f : Mn → Rn+1, n ≥ 3, be a compact hypersurface. Then the identity component Iso0(Mn) of the isometry group of Mn admits an orthogonal representation Φ: Iso0(Mn) → SO(n+ 1) such that f ◦ g = Φ(g) ◦ f for all g ∈ Iso0(Mn). Theorem 1 may be regarded as a generalization of a classical result of Kobayashi [Ko], who proved that a compact homogeneous hypersurface of Euclidean space must http://arxiv.org/abs/0704.1807v1 be a round sphere. In fact, the crucial step in the proof of Kobayashi’s theorem is to show that the isometry group of the hypersurface can be realized as a closed subgroup of O(n+ 1). The idea of the proof of Theorem 1 actually appears already in [MPST], where Euclidean G-hypersurfaces of cohomogeneity one, i.e., with principal orbits of codimension one, are considered. We apply Theorem 1 to study compact Euclidean hypersurfaces f : Mn → Rn+1, n ≥ 3, on which a connected closed subgroup G of Iso(Mn) acts locally polarly. Recall that an isometric action of a compact Lie group G on a Riemannian manifoldMn is said to be locally polar if the distribution of normal spaces to principal orbits on the regular part of Mn is integrable. If Mn is complete, this implies the existence of a connected complete immersed submanifold Σ of Mn that intersects orthogonally all G-orbits (cf. [HLO]). Such a submanifold is called a section, and it is always a totally geodesic submanifold of Mn. In particular, any isometric action of cohomogeneity one is locally polar. The action is said to be polar if there exists a closed and embedded section. Clearly, for orthogonal representations there is no distinction between polar and locally polar actions, for in this case sections are just affine subspaces. It was shown in [BCO], Proposition 3.2.9 that if a closed subgroup of SO(N) acts polarly on RN and leaves invariant a submanifold f : Mn → RN , then its restricted action on Mn is locally polar. Our next result states that any locally polar isometric action of a compact connected Lie group on a compact Euclidean hypersurface of dimension n ≥ 3 arises in this way. Theorem 2 Let f : Mn → Rn+1, n ≥ 3, be a compact hypersurface and let G be a closed connected subgroup of Iso(Mn) acting locally polarly onMn with cohomogeneity k. Then there exists an orthogonal representation Ψ: G→ SO(n+1) such that Ψ(G) acts polarly on Rn+1 with cohomogeneity k + 1 and f ◦ g = Ψ(g) ◦ f for every g ∈ G. A natural problem that emerges is how to explicitly construct all compact hy- persurfaces f : Mn → Rn+1 that are invariant under a polar action of a closed sub- group G ⊂ SO(n + 1). This is accomplished by the following result. Recall that the Weyl group of the G-action is defined as W = N(Σ)/Z(Σ), where Σ is a section, N(Σ) = {g ∈ G | gΣ = Σ} and Z(Σ) = {g ∈ G | gp = p, ∀ p ∈ Σ} is the intersection of the isotropy subgroups Gp, p ∈ Σ. Theorem 3 Let G ⊂ SO(n + 1) be a closed subgroup that acts polarly on Rn+1, let Σ be a section and let L be a compact immersed hypersurface of Σ which is invariant under the Weyl group of the G-action. Then G(L) is a compact G-invariant immersed hypersurface of Rn+1. Conversely, any compact hypersurface f : Mn → Rn+1 that is invariant under a polar action of a closed subgroup of SO(n+ 1) can be constructed in this way. The simplest examples of hypersurfaces f : Mn → Rn+1 that are invariant under a polar action on Rn+1 of a closed subgroup of SO(n+ 1) are the rotation hypersurfaces. These are invariant under a polar representation which is the sum of a trivial represen- tation on a subspace Rk ⊂ Rn+1 and one which acts transitively on the unit sphere of its orthogonal complement Rn−k+1. In this case any subspace Rk+1 containing Rk is a section, and the Weyl group of the action is just the group Z2 generated by the reflection with respect to the “axis” Rk. Thus, the condition that a hypersurface Lk ⊂ Rk+1 be invariant under the Weyl group reduces in this case to Lk being symmetric with respect to Rk. We now give several sufficient conditions for a compact Euclidean hypersurface as in Theorem 2 to be a rotation hypersurface. Corollary 4 Under the assumptions of Theorem 2, any of the following additional con- ditions implies that f is a rotation hypersurface: (i) there exists a totally geodesic (in Mn) G-principal orbit; (ii) k = n− 1; (iii) the G-principal orbits are umbilical in Mn; (iv) there exists a G-principal orbit with nonzero constant sectional curvatures; (v) there exists a G-principal orbit with positive sectional curvatures. Moreover, G is isomorphic to one of the closed subgroups of SO(n − k + 1) that act transitively on Sn−k. For a list of all closed subgroups of SO(n) that act transitively on the sphere, see, e.g., [EH], p. 392. Corollary 4 generalizes similar results in [PS], [AMN] and [MPST] for compact Euclidean G-hypersurfaces of cohomogeneity one. In part (v), weakening the assumption to non-negativity of the sectional curvatures of some principal G-orbit implies f to be a multi-rotational hypersurface in the sense of [DN]: Corollary 5 Under the assumptions of Theorem 2, suppose further that there exists a G-principal orbit Gp with nonnegative sectional curvatures. Then there exist an or- thogonal decomposition Rn+1 = ni into G̃-invariant subspaces, where G̃ = Ψ(G), and connected Lie subgroups G1, . . . , Gk of G̃ such that Gi acts on R ni, the action being transitive on Sni−1 ⊂ Rni, and the action of Ḡ = G1 × . . .×Gk on R n+1 given by (g1 . . . gk)(v0, v1, . . . , vk) = (v0, g1v1, . . . , gkvk) is orbit equivalent to the action of G̃. In particular, if Gp is flat then ni = 2 and Gi is isomorphic to SO(2) for i = 1, . . . , k. Finally, we apply some of the previous results to a problem that at first sight has no relation to isometric actions whatsoever. Let f : Mn → Rn+1 be a rotation hypersurface as described in the paragraph following Theorem 3. Then the open and dense subset of Mn that is mapped by f onto the complement of the axis Rk is isometric to the warped product Lk ×ρN n−k, where Nn−k is the orbit of some fixed point f(p) ∈ Lk under the action of G̃, and the warping function ρ : Lk → R+ is a constant multiple of the distance to R k. Recall that a warped product N1 ×ρ N2 of Riemannian manifolds (N1, 〈 , 〉N1) and (N2, 〈 , 〉N2) with warping function ρ: N1 → R+ is the product manifold N1 ×N2 endowed with the metric 〈 , 〉 = π∗1〈 , 〉N1 + (ρ ◦ π1) 2π∗2〈 , 〉N2, where πi: N1 ×N2 → Ni, 1 ≤ i ≤ 2, denote the canonical projections. We prove that, conversely, compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their warped product structure. Theorem 6 Let f : Mn → Rn+1, n ≥ 3, be a compact hypersurface. If there exists an isometry onto an open and dense subset U ⊂ Mn of a warped product Lk ×ρ N with Nn−k connected and complete (in particular if Mn is isometric to a warped product Lk ×ρ N n−k) then f is a rotation hypersurface. Theorem 6 can be seen as a global version in the hypersurface case of the local classification in [DT] of isometric immersions in codimension ℓ ≤ 2 of warped products Lk ×ρ N n−k, n− k ≥ 2, into Euclidean space. Acknowledgment. We are grateful to C. Gorodski for helpful discussions. 2 Proof of Theorem 1 The proof of Theorem 1 relies on a result of Sacksteder [Sa] (see also the proof in [Da], Theorem 6.14, which is based on an unpublished manuscript by D. Ferus) according to which a compact hypersurface f : Mn → Rn+1, n ≥ 3, is rigid whenever the subset of totally geodesic points of f does not disconnect Mn. Recall that f is rigid if any other isometric immersion f̃ : Mn → Rn+1 differs from it by a rigid motion of Rn+1. The proof of Sacksteder’s theorem actually shows more than the preceding statement. Namely, let f, f̃ : Mn → Rn+1 be isometric immersions into Rn+1 of a compact Riemannian manifold Mn, n ≥ 3, and let φ: T⊥Mnf → T be the vector bundle isometry between the normal bundles of f and f̃ defined as follows. Given a unit vector ξx ∈ T x ∈M , let φ(ξx) be the unique unit vector in T such that {f∗X1, . . . , f∗Xn, ξx} and {f̃∗X1, . . . , f̃∗Xn, φ(ξx)} determine the same orientation in R n+1, where {X1, . . . , Xn} is any ordered basis of TxM n. Then it is shown that (x) = ±φ(x) ◦ αf(x) (1) at each point x ∈Mn, where αf and αf̃ denote the second fundamental forms of f and f̃ , respectively, with values in the normal bundle. The proof is based on a careful analysis of the distributions on Mn determined by the relative nullity subspaces ∆(x) = ker αf (x) and ∆̃(x) = ker α (x) of f and f̃ , respectively. Rigidity of f under the assumption that the subset of totally geodesic points of f does not disconnect Mn then follows immediately from the Fundamental Theorem of Hypersurfaces (cf. [Da], Theorem 1.1). Proof of Theorem 1. Given g ∈ Iso0(Mn), let αf◦g denote the second fundamental form of f ◦ g. We claim that αf◦g(x) = φg(x) ◦ αf(x) (2) for every g ∈ Iso0(Mn) and x ∈ Mn, where φg denotes the vector bundle isometry between T⊥Mnf and T ⊥Mnf◦g defined as in the preceding paragraph. On one hand, αf◦g(x)(X, Y ) = αf(gx)(g∗X, g∗Y ) (3) for every g ∈ Iso0(Mn), x ∈ Mn and X, Y ∈ TxM n. In particular, this implies that for any fixed x ∈ Mn the map Θx: Iso 0(Mn) → Sym(TxM n × TxM n → T⊥x M f ) into the vector space of symmetric bilinear maps of TxM n × TxM n into T⊥x M f , given by Θx(g)(X, Y ) = φg(x) −1(αf◦g(x)(X, Y )) = φg(x) −1(αf (gx)(g∗X, g∗Y )) for any X, Y ∈ TxM n, is continuous. On the other hand, by the preceding remarks on Sacksteder’s theorem, either αf◦g(x) = φg(x) ◦αf (x) or αf◦g(x) = −φg(x) ◦αf (x). Thus Θx is a continuous map taking values in {αf(x),−αf (x)}, hence it must be constant because Iso0(Mn) is connected. Since Θx(id) = αf(x), our claim follows. We conclude that for each g ∈ Iso0(Mn) there exists a rigid motion g̃ ∈ Iso(Rn+1) such that f ◦g = g̃◦f . It now follows from standard arguments that g 7→ g̃ defines a Lie- group homomorphism Φ: Iso0(Mn) → Iso(Rn+1), whose image must lie in SO(n + 1) because it is compact and connected. Remarks 7 (i) Theorem 1 is also true for compact hypersurfaces of dimension n ≥ 3 of hyperbolic space, as well as for complete hypersurfaces of dimension n ≥ 4 of the sphere. It also holds for complete hypersurfaces of dimension n ≥ 3 of both Euclidean and hyperbolic spaces, under the additional assumption that they do not carry a com- plete leaf of dimension (n− 1) or (n− 2) of their relative nullity distributions. In fact, the proof of Theorem 1 carries over in exactly the same way for these cases, because so does equation (1) (cf. [Da], p. 96-100). (ii) Clearly, Theorem 1 does not hold for isometric immersions f : Mn → Rn+ℓ of codi- mension ℓ ≥ 2. Namely, counterexamples can be easily constructed, for instance, by means of compositions f = h ◦ g of isometric immersions g: Mn → Rn+1 and h: V → Rn+ℓ, with V ⊂ Rn+1 an open subset containing g(Mn). The next result gives a sufficient condition for rigidity of a compact hypersurface f : Mn → Rn+1 as in Theorem 1 in terms of G̃ = Φ(Iso0(Mn)). Proposition 8 Under the assumptions of Theorem 1, suppose that G̃ = Φ(Iso0(Mn)) does not have a fixed vector. Then f is free of totally geodesic points. In particular, f is rigid. Proof: Suppose that the subset B of totally geodesic points of f is nonempty. Since αf◦g(x) = φg(x) ◦ αf(x) for every g ∈ G = Iso 0(Mn) and x ∈ Mn by (2), B coincides with the set of totally geodesic points of f ◦ g for every g ∈ G. In view of (3), this amounts to saying that B is G-invariant. Thus, if Gp is the orbit of a point p ∈ B then Gp ⊂ B. Since Gp is connected, it follows from [DG], Lemma 3.14 that f(Gp) is contained in a hyperplane H that is tangent to f along Gp. Therefore, a unit vector v orthogonal to H spans T⊥gpM f for every gp ∈ Gp. Since g̃T f = T f for every g̃ = Φ(g) ∈ G̃, because f is equivariant with respect to Φ, the connectedness of G̃ implies that it must fix v. Proposition 8 implies, for instance, that if a closed connected subgroup of Iso(Mn) acts onMn with cohomogeneity one then either f is a rotation hypersurface over a plane curve or it is free of totally geodesic points, and in particular it is rigid (see [MPST], Theorem 1). As another consequence we have: Corollary 9 Let f : M3 → R4 be a compact hypersurface. If f has a totally geodesic point (in particular, if it is not rigid) then either M3 has finite isometry group or f is a rotation hypersurface. Proof: Let Φ: Iso0(M3) → SO(4) be the orthogonal representation given by Theorem 1. By Proposition 8, if f has a totally geodesic point then G̃ = Φ(Iso0(M3)) has a fixed vector v, hence it can be regarded as a subgroup of SO(3). Therefore, either the re- stricted action of G̃ on {v}⊥ has also a fixed vector or it is transitive on the sphere. In the first case, either Iso0(M3) is trivial, that is, Iso(M3) is finite, or G̃ fixes a two dimensional subspace R2 of R4, in which case f is a rotation hypersurface over a surface in a half-space R3+ with R 2 as boundary. In the latter case, f is a rotation hypersurface over a plane curve in a half-space R2+ having span{v} as boundary. 3 Proof of Theorem 2 For the proof of Theorem 2, we recall from Theorem 1 that there exists an orthogonal representation Φ: Iso0(Mn) → SO(n + 1) such that f ◦ g = Φ(g) ◦ f for every g ∈ Iso0(Mn). Since G is connected we have G ⊂ Iso0(Mn), hence it suffices to prove that G̃ = Φ(G) acts polarly on Rn+1 with cohomogeneity k + 1 and then set Ψ = Φ|G. We claim that there exists a principal orbit Gp such that the position vector f is nowhere tangent to f(Mn) along Gp, that is, f(g(p)) 6∈ f∗(g(p))Tg(p)M n for any g ∈ G. In order to prove our claim we need the following observation. Lemma 10 Let f : Mn → Rn+1 be a hypersurface. Assume that the position vector is tangent to f(Mn) on an open subset U ⊂ Mn. Then the index of relative nullity νf(x) = dim ∆f (x) of f is positive at any point x ∈ U . Proof: Let Z be a vector field on U such that f∗(p)Z(p) = f(p) for any p ∈ U and let η be a local unit normal vector field to f . Differentiating 〈η, f〉 = 0 yields 〈AX,Z〉 = 0 for any tangent vector field X on U , where A denotes the shape operator of f with respect to η. Thus AZ = 0. Going back to the proof of the claim, since Mn is a compact Riemannian manifold isometrically immersed in Euclidean space as a hypersurface, there exists an open subset V ⊂ Mn where the sectional curvatures of Mn are strictly positive. In particular, the index of relative nullity of f vanishes at every x ∈ V . If the position vector were tangent to f(Mn) at every regular point of V , it would be tangent to f(Mn) everywhere on V , because the set of regular points is dense onMn. This is in contradiction with Lemma 10 and shows that there must exist a regular point p ∈ V such that f(p) 6∈ f∗(p)TpM Since f is equivariant, we must have in fact that f(g(p)) 6∈ f∗(g(p))Tg(p)M n for any g ∈ G and the claim is proved. Now let Gp be a principal orbit such that the position vector f is nowhere tangent to f(Mn) along Gp. Then the normal bundle of f |Gp splits (possibly non-orthogonally) as span{f}⊕ f∗T ⊥Gp. Let ξ be an equivariant normal vector field to Gp in Mn and let η = f∗ξ. Then, denoting Φ(g) by g̃ and identifying g̃ with its derivative at any point, because it is linear, we have g̃η(p) = (f ◦ g)∗(p)ξ(p) = f∗(gp)g∗(p)ξ(p) = f∗(gp)ξ(gp) = η(gp) (4) for any g ∈ G. In particular, 〈η(gp), f(gp)〉 = 〈g̃η(p), g̃f(p)〉 = 〈η(p), f(p)〉 for every g ∈ G, that is, 〈η, f〉 is constant on Gp. It follows that X〈η, f〉 = 0, for any X ∈ TGp, (5) and hence 〈∇̃Xη, f〉 = X〈η, f〉 − 〈ξ,X〉 = 0, (6) where ∇̃ denotes the derivative in Rn+1. On the other hand, since G acts locally polarly on Mn, we have that ξ is parallel in the normal connection of Gp in M . Therefore (∇̃Xη)f∗T⊥Gp = f∗(∇Xξ)T⊥Gp = 0, (7) where ∇ is the Levi-Civita connection of Mn; here, writing a vector subbundle as a subscript of a vector field indicates taking its orthogonal projection onto that subbundle. It follows from (6) and (7) that η is parallel in the normal connection of f |Gp. On the other hand, the position vector f is clearly also an equivariant normal vector field of f |Gp which is parallel in the normal connection. Thus, we have shown that there exist equivariant normal vector fields to G̃(f(p)) = f(Gp) that form a basis of the normal spaces at each point and which are parallel in the normal connection. The statement now follows from the next known result, a proof of which is included for completeness. Lemma 11 Let G̃ ⊂ SO(n+ 1) have an orbit G̃(q) along which there exist equivariant normal vector fields that form a basis of the normal spaces at each point and which are parallel in the normal connection. Then G̃ acts polarly. Proof: Since there exist equivariant normal vector fields to G̃(q) that form a basis of the normal spaces at each point, the isotropy group acts trivially on each normal space, hence G̃(q) is a principal orbit. We now show that the normal space T⊥q G̃(q) is a section, for which it suffices to show that any Killing vector field X induced by G̃ is everywhere orthogonal to T⊥q G̃(q). Given ξq ∈ T q G̃(q), let ξ be an equivariant normal vector field to G̃(q) extending ξq, which is also parallel in the normal connection. Then, denoting by φt the flow of X and setting c(t) = φt(q) we have |t=0φt(ξq) T⊥q G̃(q) |t=0ξ(c(t)) T⊥q G̃(q) = ∇⊥X(q)ξ = 0, where denotes the covariant derivative in Euclidean space along c(t). Remark 12 A closed subgroup G ⊂ SO(n + ℓ), ℓ ≥ 2, that acts non-polarly on Rn+ℓ may leave invariant a compact submanifold f : Mn → Rn+ℓ and induce a locally polar action on Mn. For instance, consider a compact submanifold f : Mn → Rn+2 that is invariant by the action of a closed subgroup G ⊂ SO(n+2) that acts non-polarly on Rn+2 with cohomogeneity three. Then the induced action of G onMn has cohomogeneity one, whence is locally polar. Moreover, taking f as a compact hypersurface of Sn+1 shows also that Theorem 2 is no longer true if Rn+1 is replaced by Sn+1. Theorem 2 yields the following obstruction for the existence of an isometric immer- sion in codimension one into Euclidean space of a compact Riemannian manifold acted on locally polarly by a closed connected Lie subgroup of its isometry group. Corollary 13 Let Mn be a compact Riemannian manifold of dimension n ≥ 3 acted on locally polarly by a closed connected subgroup G of its isometry group. If G has an exceptional orbit then Mn can not be isometrically immersed in Euclidean space as a hypersurface. Proof: Let f : Mn → Rn+1 be an isometric immersion of a compact Riemannian manifold acted on locally polarly by a closed connected subgroup G of its isometry group. We will prove that G can not have any exceptional orbit. By Theorem 2 there exists an orthogonal representation Ψ: G → SO(n + 1) such that G̃ = Ψ(G) acts polarly on n+1 with cohomogeneity k + 1 and f ◦ g = Ψ(g) ◦ f for every g ∈ G. Let Gp be a nonsingular orbit. Then Gp has maximal dimension among all G-orbits, and hence G̃f(p) = f(Gp) has maximal dimension among all G̃-orbits. Since polar representations are known to admit no exceptional orbits (cf. [BCO], Corollary 5.4.3), it follows that G̃f(p) is a principal orbit. Then, for any g in the isotropy subgroup Gp we have that g̃ = Ψ(g) ∈ G̃f(p), thus for any ξp ∈ T p Gp we obtain f∗(p)ξp = g̃∗f∗(p)ξp = (g̃ ◦ f)∗(p)ξp = (f ◦ g)∗(p)ξp = f∗(gp)g∗ξp = f∗(p)g∗ξp. Since f∗(p) is injective, then g∗ξp = ξp. This shows that the slice representation, that is, the action of the isotropy group Gp on the normal space T p G(p) to the orbit G(p) at p, is trivial. Thus Gp is a principal orbit. 4 Isoparametric submanifolds We now recall some results on isoparametric submanifolds and derive a few additional facts on them that will be needed for the proofs of Theorem 3 and Corollaries 4 and 5. Given an isometric immersion f : Mn → RN with flat normal bundle, it is well- known (cf. [St]) that for each point x ∈Mn there exist an integer s = s(x) ∈ {1, . . . , n} and a uniquely determined subset Hx = {η1, . . . , ηs} of T f such that TxM n is the orthogonal sum of the nontrivial subspaces Eηi(x) = {X ∈ TxM n : α(X, Y ) = 〈X, Y 〉 ηi, for all Y ∈ TxM n}, 1 ≤ i ≤ s. Therefore, the second fundamental form of f has the simple representation α(X, Y ) = 〈Xi, Yi〉 ηi, (8) or equivalently, AξX = 〈ξ, ηi〉Xi, (9) where X 7→ Xi denotes orthogonal projection onto Eηi . Each ηi ∈ Hx is called a principal normal of f at x. The Gauss equation takes the form R(X, Y ) = i,j=1 〈ηi, ηj〉Xi ∧ Yj, (10) where (Xi ∧ Yj)Z = 〈Z, Yj〉Xi − 〈Z,Xi〉Yj. Lemma 14 Let f : Mn → RN be an isometric immersion with flat normal bundle of a Riemannian manifold with constant sectional curvature c. Let η1, . . . , ηs be the distinct principal normals of f at x ∈ Mn. Then (i) There exists at most one ηi such that |ηi| 2 = c. (ii) For all i, j, k ∈ {1, . . . , s} with i 6= j 6= k 6= i the vectors ηi − ηk and ηj − ηk are linearly independent. Proof: It follows from (10) that 〈ηi, ηj〉 = c for all i, j ∈ {1, . . . , s} with i 6= j. If 2 = |ηj| 2 = c, this gives |ηi − ηj | 2 = 0. (iii) Assume that there exist λ 6= 0 and i, j, k ∈ {1, . . . , s} with i 6= j 6= k 6= i such that ηi − ηk = λ(ηj − ηk). Then 2 − c = 〈ηi − ηk, ηi〉 = λ〈ηj − ηk, ηi〉 = 0, and similarly |ηj| 2 = c, in contradiction with (i). Let f : Mn → RN be an isometric immersion with flat normal bundle and let Hx = {η1, . . . , ηs} be the set of principal normals of f at x ∈M n. If the mapMn → {1, . . . , n} given by x 7→ #Hx has a constant value s on an open subset U ⊂M n, there exist smooth normal vector fields η1, . . . , ηs on U such that Hx = {η1(x), . . . , ηs(x)} for any x ∈ U . Furthermore, each Eηi = (Eηi(x))x∈U is a C ∞- subbundle of TU for 1 ≤ i ≤ s. The following result is contained in [DN], Lemma 2.3. Lemma 15. Let f : Mn → Rn+p be an isometric immersion with flat normal bundle and a constant number s of principal normals η1, . . . , ηs everywhere. Assume that for a fixed i ∈ {1, . . . , s} all principal normals ηj, j 6= i, are parallel in the normal connection along Eηi and that the vectors ηi− ηj and ηi− ηℓ are everywhere pairwise linearly independent for any pair of indices 1 ≤ j 6= ℓ ≤ s with j, ℓ 6= i. Then E⊥ηi is totally geodesic. Proof: The Codazzi equation yields 〈∇XℓXj , Xi〉(ηi − ηj) = 〈∇XℓXj , Xi〉(ηi − ηℓ), i 6= j 6= k 6= i, (11) ∇⊥Xiηj = 〈∇XjXj , Xi〉(ηj − ηi), i 6= j, (12) for all unit vectors Xi ∈ Eηi , Xj ∈ Eηj and Xℓ ∈ Eηℓ . An isometric immersion f : Mn → RN is called isoparametric if it has flat normal bundle and the principal curvatures of f with respect to every parallel normal vector field along any curve in Mn are constant (with constant multiplicities). The following facts on isoparametric submanifolds are due to Strübing [St]. Theorem 16 Let f : Mn → RN be an isoparametric isometric immersion. Then (i) The number of principal normals is constant on Mn. (ii) The first normal spaces, i.e, the subspaces of the normal spaces spanned by the im- age of the second fundamental form, determine a parallel subbundle of the normal bundle. (iii) The subbundles Eηi, 1 ≤ i ≤ s, are totally geodesic and the principal normals η1, . . . , ηs are parallel in the normal connection. (iv) The subbundles Eηi, 1 ≤ i ≤ s, are parallel if and only if f has parallel second fundamental form. (v) If the principal normals η1, . . . , ηs satisfy 〈ηi, ηj〉 ≥ 0 everywhere then f has parallel second fundamental form and 〈ηi, ηj〉 = 0 everywhere. The next result will be used in the proofs of Corollaries 4 and 5. Proposition 17 Let f : Mn → RN , n ≥ 2, be a compact isoparametric submanifold. (i) If Mn has constant sectional curvature c, then either c > 0 and f(Mn) is a round sphere or c = 0 and f(Mn) is an extrinsic product of circles. (ii) If Mn has nonnegative sectional curvatures, then f(Mn) is an extrinsic product of round spheres or circles. In particular, if Mn has positive sectional curvatures then f(Mn) is a round sphere. Proof: By Theorem 16, the number of distinct principal normals η1, . . . , ηs of f is con- stant on Mn and all of them are parallel in the normal connection. Moreover, the subbundles Eηi , 1 ≤ i ≤ s, are totally geodesic. If M n has constant sectional cur- vature c, then it follows from Lemmas 14 and 15 that also E⊥ηi is totally geodesic for 1 ≤ i ≤ s, and hence the sectional curvatures along planes spanned by vectors in dif- ferent subbundles vanish. Therefore c = 0, unless s = 1 and f is umbilic, in which case c > 0 and f(Mn) is a round sphere. Furthermore, if c = 0 and Eηi has rank at least 2 for some 1 ≤ i ≤ s then the sectional curvature along a plane tangent to Eηi is |ηi| 2 = 0, in contradiction with the compactness of Mn. Hence Eηi has rank 1 for 1 ≤ i ≤ s. We conclude that the universal covering of Mn is isometric to Rn, and that f ◦ π splits as a product of circles by Moore’s Lemma [Mo], where π: Rn → Mn is the covering map. Assume now that Mn has nonnegative sectional curvatures. It follows from (10) that 〈ηi, ηj〉 ≥ 0 for 1 ≤ i 6= j ≤ s, whence f has parallel second fundamental form and all subbundles Eηi , 1 ≤ i ≤ s, are parallel by parts (iv) and (v) of Theorem 16. We obtain from the de Rham decomposition theorem that the universal covering of Mn splits isometrically as Mn11 × · · ·×M s , where each factor M i is either R if ni = 1 or a sphere Snii of curvature |ηi| 2 if ni ≥ 2. Moreover, if π: M 1 × · · · ×M s → M n denotes the covering map, then Moore’s Lemma implies that f ◦π splits as f ◦π = f1×· · ·× fs, where fi(M i ) is a round sphere or circle for 1 ≤ i ≤ s. To every compact isoparametric submanifold f : Mn → RN one can associate a finite group, its Weyl group, as follows. Let η1, . . . , ηg denote the principal normal vector fields of f . For p ∈ Mn, let Hj(p), 1 ≤ j ≤ g, be the focal hyperplane of T n given by the equation 〈ηj(p), 〉 = 1. Then one can show that the reflection on the affine normal space p+T⊥p M n with respect to each affine focal hyperplane p+Hi(p) leaves j=1(p+Hj(p)) invariant, and thus the set of all such reflections generate a finite group, the Weyl group of f at p. Moreover, the Weyl groups of f at different points are conjugate by the parallel transport with respect to the normal connection, hence a well-defined Weyl group W can be associated to f . We refer to [PT2] for details. In the proof of Theorem 3 we will need the following property of the Weyl group of an isoparametric submanifold. Proposition 18 Let f : Mn → RN be a compact isoparametric submanifold and let W (p) be its Weyl group at p ∈ Mn. Assume that W (p) leaves invariant an affine hyperplane H orthogonal to ξ ∈ T⊥p M . Then f(M n) is contained in the affine hyperplane of RN through p orthogonal to ξ. Proof: It follows from the assumption that H is orthogonal to every focal hyperplane p + Hj(p), 1 ≤ j ≤ g, of f at p. For q ∈ H, let Q = g∈W (p) gq ∈ H. Then Q is a fixed point of W (p), hence it lies in the intersection (p + Hj(p)) of all affine focal hyperplanes of f at p. We obtain that the line through Q orthogonal to H lies in j=1(p +Hj(p)). Therefore 〈ηj, Q+ λξ〉 = 1 for every λ ∈ R, 1 ≤ j ≤ g, which implies that 〈ηj , ξ〉 = 0, for every 1 ≤ j ≤ g. Now extend ξ to a parallel vector field along M with respect to the normal connection. Since the principal normal vector fields η1, . . . , ηg of f are parallel with respect to the normal connection by Theorem 16-(iii), it follows that 〈ηj, ξ〉 = 0 everywhere, and hence the shape operator Aξ of f with respect to ξ is identically zero by (9). Then ξ is constant in RN and the conclusion follows. A rich source of isoparametric submanifolds is provided by the following result of Palais and Terng (see [PT1], Theorem 6.5). Proposition 19 If a closed subgroup G ⊂ SO(N) acts polarly on RN then any of its principal orbits is an isoparametric submanifold of RN . To conclude this section, we point out that if G ⊂ SO(N) acts polarly on RN and Σ is a section, then the the Weyl group W = N(Σ)/Z(Σ) of the G-action coincides with the Weyl group W (p) just defined of any principal orbit Gp, p ∈ Σ, as an isoparametric submanifold of RN (cf. [PT2]). 5 Proof of Theorem 3 We first prove the converse. Let G ⊂ SO(n+1) act polarly on Rn+1, let Σ be a section of the G-action and let Mn ⊂ Rn+1 be a G-invariant immersed hypersurface. It suffices to prove that Σ is transversal to Mn, for then L = Σ ∩Mn is a compact hypersurface of Σ that is invariant under the Weyl group W of the action and Mn = G(L). Assume, on the contrary, that transversality does not hold. Then there exists p ∈ Σ ∩Mn with TpΣ ⊂ TpM n. Fix v ∈ TpΣ in a principal orbit of the slice representation at p and let γ: (−ǫ, ǫ) → Mn be a smooth curve with γ(0) = p and γ′(0) = v. Since Mn is G-invariant, it contains {gγ(t) | g ∈ Gp, t ∈ (−ǫ, ǫ)}. Therefore, TpM n contains Gpv, and hence Rv⊕TvGpv. Recall that TpΣ is a section for the slice representation at p (see [PT1], Theorem 4.6). Therefore TvGpv is a subspace of T p G(p) orthogonal to TpΣ with dimTvGpv = dim T p G(p)− dimΣ. (13) Moreover, again by the G-invariance of Mn, we have that G(p) ⊂ Mn, and hence TpG(p) ⊂ TpM n. Using (13), we conclude that dimTpM n ≥ dimG(p) + dimΣ + dimTvGpv = dimG(p) + dimT⊥p G(p) = n + 1, a contradiction. In order to prove the direct statement, it suffices to show that at each point p ∈ L which is a singular point of the G-action the subset H = {γ′(0) | γ: (−ǫ, ǫ) → Rn+1 is a curve in Rn+1 with γ(−ǫ, ǫ) ⊂Mn and γ(0) = p} is an n-dimensional subspace of Rn+1, the tangent space of Mn at p. Clearly, we have H = TpG(p) gTpL. We use again that TpΣ is a section of the slice representation at p and that, in addition, the Weyl group for the slice representation is W (Σ)p = W ∩Gp ([PT1], Theorem 4.6). Since L is invariant under W by assumption, it follows that TpL is invariant under W (Σ)p. Let ξ ∈ Σ be a unit vector normal to L at p. Then, for any principal vector v ∈ TpL ⊂ TpΣ of the slice representation at p it follows from Proposition 18 that Gpv lies in the affine hyperplane π of RN through p orthogonal to ξ. Therefore gTpL ⊂ π for every g ∈ Gp. Since TpG(p) is orthogonal to Σ, we conclude that H ⊂ π, and hence H = π by dimension reasons. Remark 20 Theorems 1 and 3 yield as a special case the main theorem in [MPST]: any compact hypersurface of Rn+1 with cohomogeneity one under the action of a closed connected subgroup of its isometry group is given as G(γ), where G ⊂ SO(n + 1) acts on Rn+1 with cohomogeneity two (hence polarly) and γ is a smooth curve in a (two- dimensional) section Σ which is invariant under the Weyl group W of the G-action. We take the opportunity to point out that starting with a smooth curve β in a Weyl chamber σ of Σ (which is identified with the orbit space of the G-action) which is orthogonal to the boundary ∂σ of σ is not enough to ensure smoothness of γ = W (β), or equivalently, of G(β), as claimed in [MPST]. One should require, in addition, that after expressing β locally as a graph z = f(x) over its tangent line at a point of intersection with ∂σ, the function f be even, that is, all of its derivatives of odd order vanish and not only the first one. 6 Proofs of Corollaries 4 and 5. For the proof of Corollary 4 we need another known property of polar representations, a simple proof of which is included for the sake of completeness Lemma 21 Let G̃ ⊂ SO(n+ 1) act polarly and have a principal orbit G̃(q) that is not full, i.e., the linear span V of G̃(q) is a proper subspace of Rn+1. Then G̃ acts trivially on V ⊥. Proof: Let v ∈ V ⊥. Then v belongs to the normal spaces of G̃(q) at any point, hence admits a unique extension to an equivariant normal vector field ξ along G̃(q). Moreover, the shape operator Aξ of G̃(q) with respect to ξ is identically zero, for Av is clearly zero and ξ is equivariant. Now, since G̃ acts polarly, the vector field ξ is parallel in the normal connection. It follows that ξ is a constant vector in Rn+1, which means that G̃ fixes v. Proof of Corollary 4. We know from Theorem 2 that there exists an orthogonal represen- tation Ψ: G→ SO(n+1) such that G̃ = Ψ(G) acts polarly on Rn+1 and f ◦g = Ψ(g)◦f for every g ∈ G. We claim that any of the conditions in the statement implies that f immerses some G-principal orbit Gp in Rn+1 as a round sphere (circle, if k = n − 1). Assuming the claim, it follows from Lemma 21 that G̃ fixes the orthogonal complement V ⊥ of the linear span V of G̃p, hence f is a rotation hypersurface with V ⊥ as axis. Moreover, if k 6= n − 1 then f |Gp must be an embedding by a standard covering map argument. From this and f ◦ g = Ψ(g) ◦ f for every g ∈ G it follows that any g in the kernel of Ψ must fix any point of Gp. Since Gp is a principal orbit, this easily implies that g = id. Therefore Ψ is an isomorphism of G onto G̃. We now prove the claim. By Proposition 19 we have that f immerses Gp as an isoparametric submanifold. If condition (i) holds then the first normal spaces of f |Gp in Rn+1 are one-dimensional. By Theorem 16-(ii) and a well-known result on reduction of codimension of isometric immersions (cf. [Da], Proposition 4.1), we obtain that f(Gp) = G̃(f(p)) is contained as a hypersurface in some affine hyperplane H ⊂ Rn+1, and hence f(Gp) must be a round hypersphere of H (a circle if k = n − 1). Moreover, condition (i) is automatic if k = n − 1, for in this case the principal orbit of maximal length must be a geodesic. Now, if (iii) holds, let Gp be a principal orbit such that the position vector of f is nowhere tangent to f(Mn) along Gp. Then any normal vector ξ̃ to f |Gp at gp ∈ Gp can be written as ξ̃ = af(gp) + f∗(gp)ξgp, with ξgp normal to Gp in Mn. Therefore the shape operator A f |Gp is a multiple of the identity tensor, hence f |Gp is umbilical. Since, by (ii), the dimension of Gp can be assumed to be at least two, the claim is proved also in this case. As for conditions (iv) and (v), the claim follows from Proposition 17-(i) and the last assertion in Proposition 17-(ii), respectively. Proof of Corollary 5. By Proposition 19 and part (ii) of Proposition 17 we have that the orbit G̃(f(p)) = f(Gp) of G̃ is an extrinsic product Sn1−11 ×. . .×S of round spheres or circles. In particular, this implies that the orthogonal decomposition Rn+1 = where Rni is the linear span of S i for 1 ≤ i ≤ k, is G̃-stable, that is, R ni is G̃-invariant for 0 ≤ i ≤ k (cf. [GOT], Lemma 6.2). By [D], Theorem 4 there exist connected Lie subgroups G1, . . . , Gk of G̃ such that Gi acts on R ni and the action of Ḡ = G1× . . .×Gk on Rn+1 given by (g1 . . . gk)(v0, v1, . . . , vk) = (v0, g1v1, . . . , gkvk) is orbit equivalent to the action of G̃. Moreover, writing q = f(p) = (q0, . . . , qk) then G̃(q) = {q0} × G1(q1) × . . . × Gk(qk), and hence Gi(qi) is a hypersphere of R ni for 1 ≤ i ≤ k. The last assertion is clear. 7 Proof of Theorem 6. Let Lk×ρN n−k be a warped product with Nn−k connected and complete and let ψ: Lk×ρ Nn−k → U be an isometry onto an open dense subset U ⊂Mn. Since Mn is a compact Riemannian manifold isometrically immersed in Euclidean space as a hypersurface, there exists an open subset W ⊂Mn with strictly positive sectional curvatures. The subset U being open and dense inMn, W ∩U is a nonempty open set. Let L1×N1 be a connected open subset of Lk×Nn−k that is mapped intoW∩U by ψ. Then the sectional curvatures of Lk ×ρ N n−k are strictly positive on L1 ×N1. For a fixed x ∈ L1, choose a unit vector Xx ∈ TxL1. For each y ∈ N n−k, let X(x,y) be the unique unit horizontal vector in T(x,y)(L1 × N n−k) that projects onto Xx by (π1)∗(x, y). Then the sectional curvature of L k ×ρ N n−k along a plane σ spanned by X(x,y) and any unit vertical vector Z(x,y) ∈ T(x,y)(L1 ×N n−k) is given by K(σ) = −Hess ρ(x)(Xx, Xx)/ρ(x). Observe that K(σ) depends neither on y nor on the vector Z(x,y). Since K(σ) > 0 if y ∈ N1, the same holds for any y ∈ N n−k. In particular, L1×ρN n−k is free of flat points. If n − k ≥ 2, it follows from [DT], Theorem 16 that f ◦ ψ immerses L1 ×ρ N either as a rotation hypersurface or as the extrinsic product of an Euclidean factor Rk−1 with a cone over a hypersurface of Sn−k+1. The latter possibility is ruled out by the fact that the sectional curvatures of Lk ×ρ N n−k are strictly positive on L1 ×N1. Thus the first possibility holds, and in particular f ◦ ψ immerses each leaf {x} × Nn−k, x ∈ L1, isometrically onto a round (n−k)-dimensional sphere. It follows that Nn−k is isometric to a round sphere. In any case, Iso0(Nn−k) acts transitively on Nn−k and each g ∈ Iso0(Nn−k) induces an isometry ḡ of Lk ×ρ N n−k by defining ḡ(x, y) = (x, g(y)), for all (x, y) ∈ Lk ×Nn−k. The map g ∈ Iso0(Nn−k) 7→ ḡ ∈ Iso(Lk ×ρ N n−k) being clearly continuous, its image Ḡ is a closed connected subgroup of Iso(Lk ×ρ N n−k). For each ḡ ∈ Ḡ, the induced isometry ψ ◦ ḡ ◦ ψ−1 on U extends uniquely to an isometry of Mn. The orbits of the induced action of Ḡ on U are the images by ψ of the leaves {x} ×Nn−k, x ∈ Lk, hence are umbilical in Mn. Moreover, the normal spaces to the (principal) orbits of Ḡ on U are the images by ψ∗ of the horizontal subspaces of L k ×ρ N n−k. Therefore, they define an integrable distribution on U , whence on the whole regular part of Mn. Thus, the action of Ḡ on Mn is locally polar with umbilical principal orbits. We conclude from Corollary 4-(iii) that f is a rotation hypersurface. References [AMN] ASPERTI, A.C., MERCURI, F., NORONHA, M.H.: Cohomogeneity one man- ifolds and hypersurfaces of revolution. Bolletino U.M.I. 11-B (1997), 199-215. [BCO] BERNDT, J., CONSOLE, S., OLMOS, C.: Submanifolds and Holonomy, CRC/Chapman and Hall Research Notes Series in Mathematics 434 (2003), Boca Ratton. [D] DADOK, J.: Polar coordinates induced by actions of compact Lie groups. Trans. Amer. Math. Soc. 288 (1985), 125-137. [Da] DAJCZER, M. et al.: Submanifolds and isometric immersions. Matematics Lec- ture Series 13, Publish or Perish Inc., Houston-Texas, 1990. [DG] DAJCZER, M., GROMOLL, D.: Rigidity of complete Euclidean hypersurfaces. J.Diff. Geom. 31 (1990), 401-416. [DT] DAJCZER, M., TOJEIRO, R.: Isometric immersions in codimension two of warped products into space forms. Illinois J. Math. 48 (3) (2004), 711-746. [DN] DILLEN, F., NÖLKER, S.: Semi-parallel submanifolds, multi-rotation surfaces and the helix-property. J. reine angew. Math. 435 (1993), 33-63. [EH] ESCHENBURG, J.-H., HEINTZE, E.: On the classification of polar representa- tions. Math. Z. 232 (1999), 391-398. [HLO] HEINTZE, E., LIU, X., OLMOS, C.: Isoparametric submanifolds and a Chevalley-type restriction theorem. Integrable systems, geometry and topology, 151-190, AMS/IP Stud. Adv. Math. 36, Amer. Math. Soc., Providence, RI, 2006. [GOT] GORODSKI, C., OLMOS, C., TOJEIRO, R.: Copolarity of isometric actions. Trans. Amer. Math. Soc. 356 (2004), 1585-1608. [Ko] KOBAYASHI, S.: Compact homogeneous hypersurfaces. Trans. Amer. Math. Soc. 88 (1958), 137-143. [MPST] MERCURI, F., PODESTÀ, F., SEIXAS, J. A., TOJEIRO, R.: Cohomogene- ity one hypersurfaces of Euclidean spaces. Comment. Math. Helv. 81 (2) (2006), 471-479. [Mo] MOORE, J. D.: Isometric immersions of Riemannian products, J. Diff. Geom. 5 (1971), 159–168. [PT1] PALAIS, R., TERNG, C.-L.: A general theory of canonical forms. Trans. Amer. Math. Soc. 300 (1987), 771-789. [PT2] PALAIS, R., TERNG, C.-L.: Critical Point Theory and Submanifold Geometry, Lecture Notes in Mathematics 1353 (1988), Springer-Verlag. [PS] PODESTÀ, F., SPIRO, A.: Cohomogeneity one manifolds and hypersurfaces of Euclidean space. Ann. Global Anal. Geom. 13 (1995), 169-184. [Sa] SACKSTEDER, R.: The rigidity of hypersurfaces, J. Math. Mech. 11 (1962), 929-939. [St] STRÜBING, W.: Isoparametric submanifolds. Geom. Dedicata 20 (1986), 367- Universidade Federal Fluminense Universidade Federal de São Carlos 24020-140 – Niteroi – Brazil 13565-905 – São Carlos – Brazil E-mail: [email protected] E-mail: [email protected] Introduction Proof of Theorem ?? Proof of Theorem ?? Isoparametric submanifolds Proof of Theorem ?? Proofs of Corollaries ?? and ??. Proof of Theorem ??.

0704.1808

Tests of Bayesian Model Selection Techniques for Gravitational Wave Astronomy

Tests of Bayesian Model Selection Techniques for Gravitational Wave Astronomy Neil J. Cornish and Tyson B. Littenberg Department of Physics, Montana State University, Bozeman, MT 59717 The analysis of gravitational wave data involves many model selection problems. The most important example is the detection problem of selecting between the data being consistent with instrument noise alone, or instrument noise and a gravitational wave signal. The analysis of data from ground based gravitational wave detectors is mostly conducted using classical statistics, and methods such as the Neyman-Pearson criteria are used for model selection. Future space based detectors, such as the Laser Interferometer Space Antenna (LISA), are expected to produced rich data streams containing the signals from many millions of sources. Determining the number of sources that are resolvable, and the most appropriate description of each source poses a challenging model selection problem that may best be addressed in a Bayesian framework. An important class of LISA sources are the millions of low-mass binary systems within our own galaxy, tens of thousands of which will be detectable. Not only are the number of sources unknown, but so are the number of parameters required to model the waveforms. For example, a significant subset of the resolvable galactic binaries will exhibit orbital frequency evolution, while a smaller number will have measurable eccentricity. In the Bayesian approach to model selection one needs to compute the Bayes factor between competing models. Here we explore various methods for computing Bayes factors in the context of determining which galactic binaries have measurable frequency evolution. The methods explored include a Reverse Jump Markov Chain Monte Carlo (RJMCMC) algorithm, Savage-Dickie density ratios, the Schwarz-Bayes Information Criterion (BIC), and the Laplace approximation to the model evidence. We find good agreement between all of the approaches. I. BACKGROUND Bayesian statistical techniques are becoming increas- ingly popular in gravitational wave data analysis, and have shown great promise in tackling the various difficul- ties of gravitational wave (GW) source extraction from modeled data for the Laser Interferometer Space Antenna (LISA). A powerful tool in the suite of Bayesian methods is that of quantitative model selection [1, 2]. To under- stand why this is a valuable feature consider a scenario where one is attempting to fit data with two competing models of differing dimension. In general, a higher di- mensional model will produce a better fit to a given set of data. This can be taken to the limit where there are as many model parameters as there are data points allowing one to perfectly match the data. The problem then is to decide how many parameters are physically meaningful and to select the model containing only those parameters. In the context of GW detection these extra parameters could be additional physical parameters used to model the source or additional sources in the data. If a model is over-parameterized it will over-fit the data and produce spurious results. Many of the model selection problems associated with LISA astronomy involve nested models, where the sim- pler model forms a subset of the more complicated model. The problem of determining the number of resolvable galactic binaries, and the problem of determining the number of measurable source parameters, are both ex- amples of nested model selection. One could argue that the later is better described as “approximation selection” since we are selecting between different parameterizations of the full 17 dimensional physical model that describes the signals from binary systems of point masses in general relativity. However, many similar modeling problems in astrophysics and cosmology [2], as well as in other fields such as geophysics [3], are considered to be examples of model selection, and we will adopt that viewpoint here. The LISA observatory [4] is designed to explore the low frequency portion of the gravitational wave spectrum between ∼ 0.1 → 100 mHz. This frequency region will be heavily populated by signals from galactic binary systems composed of stellar mass compact objects (e.g. white dwarfs and neutron stars), of which millions are theorized to exist. Tens of thousands of these GW sources will be resolvable by LISA and the remaining sources will contribute to a confusion-limited background [5]. This is expected to be the dominant source of low frequency noise for LISA. Detection and subsequent regression of the galactic foreground is of vital importance in order to then pur- sue dimmer sources that would otherwise be buried by the foreground. Because of the great number of galac- tic sources, and the ensuing overlap between individ- ual sources, a one-by-one detection/regression is inaccu- rate [6]. Therefore a global fit to all of the galactic sources is required. Because of the uncertainty in the number of resolvable sources one can not fix the model dimension a priori which presents a crucial model selection problem. Over-fitting the data will result in an inaccurate regres- sion which would then remove power from other sources in the data-stream, negatively impacting their detection and characterization. The Reverse Jump Markov Chain Monte Carlo approach to Bayesian model selection has been used to determine the number of resolvable sources in the context of a toy problem [7, 8] which shares some of the features of the LISA foreground removal prob- lem. Meanwhile the Laplace approximation to Bayesian http://arxiv.org/abs/0704.1808v3 model selection has been employed to estimate the num- ber of resolvable sources as part of a MCMC based al- gorithm to extracting signals from simulated LISA data streams [6, 9]. Another important problem for GW astronomy is the determination of which parameters need to be included in the description of the waveforms. For example, the GW signal from a binary inspiral, as detected by LISA, may involve as many as 17 parameters. However, for massive black hole binaries of comparable mass we expect the ec- centricity to be negligible, reducing the model dimension to 15, while for extreme mass ratio systems we expect the spin of the smaller body to have little impact on the waveforms, reducing the model dimension to 14. In many cases the inspiral signals may be described by even fewer parameters. For low mass galactic binaries spin effects will be negligible (removing six parameters), and various astrophysical processes will have circularized the orbits of the majority of systems (removing two param- eters). Of the remaining nine parameters, two describe the frequency evolution - e.g. the first and second time derivatives of the GW frequency, which may or may not be detectable [27]. Here we investigate the application of Bayesian model selection to LISA data analysis in the context of deter- mining the conditions under which the first time deriva- tive of the GW frequency, ḟ , can be inferred from the data. We parameterize the signals using the eight pa- rameters ~λ→ (A, f, θ, φ, ψ, ι, ϕ0, ḟ) (1) where A is the amplitude, f is the initial gravitational wave frequency, θ and φ are the ecliptic co-latitude and longitude, ψ is the polarization angle, ι is the orbital inclination of the binary and ϕ0 is the GW phase. The parameters f , ḟ and ϕ0 are evaluated at some fiducial time (e.g. at the time the first data is taken). For our analysis only a single source is injected into the simulated data streams. In the frequency domain the output s(f) in channel α can be written as s̃α(f) = h̃α(~λ) + ñα(f) (2) where h̃α(~λ) is the response of the detector to the incident GW and ñα(f) is the instrument noise. For our work we will assume stationary Gaussian instrument noise with no contribution from a confusion background. In our anal- ysis we use the noise orthogonal A,E, T data streams, which can be constructed from linear combinations of the Michelson type X,Y, Z signals: (2X − Y − Z) , (Z − Y ) , (X + Y + Z) . (3) This set of A,E, T variables differ slightly from those constructed from the Sagnac signals [10]. We do not use the T channel in our analysis as it is insensitive to GWs at the frequencies we are considering. The noise spectral density in the A,E channels has the form Sn(f) = (1− cos(2f/f∗)) (2 + cos(f/f∗))Ss +2(3 + 2 cos(f/f∗) + cos(2f/f∗))Sa Hz−1 (4) where f∗ = 1/2πL, and the acceleration noise Sa and shot noise Ss are simulated at the levels 10−22 9× 10−30 (2πf)4L2 Hz−1 . Here L is the LISA arm length (≈ 5× 109 m). Of central importance to Bayesian analysis is the pos- terior distribution function (PDF) of the model parame- ters. The PDF p(~λ|s) describes the probability that the source is described by parameters ~λ given the signal s. According to Bayes’ Theorem, p(~λ|s) = p( ~λ)p(s|~λ) d~λ p(~λ)p(s|~λ) where p(~λ) is the a priori, or prior, distribution of the parameters ~λ and p(s|~λ) is the likelihood that we measure the signal s if the source is described by the parameters ~λ. We define the likelihood using the noise weighted inner product (A|B) ≡ 2 Ã∗α(f)B̃α(f) + Ãα(f)B̃ Sαn (f) p(s|~λ) = C exp s− h(~λ) s− h(~λ) where the normalization constant C depends on the noise, but not the GW signal. One goal of the data anal- ysis method is to find the parameters ~λ which maximizes the posterior. Markov Chain Monte Carlo (MCMC) methods are ideal for this type of application [11]. The MCMC algorithm will simultaneously find the param- eters which maximize the posterior and accurately map out the PDF of the parameters. This is achieved through the use of a Metropolis-Hastings [12, 13] exploration of the parameter space. A brief description of this process is as follows: The chain begins at some random position ~x in the parameter space and subsequent steps are made by randomly proposing a new position in the parame- ter space ~y. This new position is determined by drawing from some proposal distribution q(~x|~y). The choice of whether or not adopt the new position ~y is made by cal- culating the Hastings ratio (transition probability) α = min p(~y)p(s|~y)q(~y|~x) p(~x)p(s|~x)q(~x|~y) and comparing α to a random number β taken from a uniform draw in the interval [0:1]. If α exceeds β then the chain adopts ~y as the new position. This process is repeated until some convergence criterion is satisfied. The MCMC differs from a Metropolis extremization by forbidding proposal distributions that depend on the past history of the chain. This ensures that the progress of the chain is Markovian and therefore statistically unbiased. Once the chain has stabilized in the neighborhood of the best fit parameters all previous steps of the chain are excluded from the statistical analysis (these early steps are referred to as the “burn in” phase of the chain) and henceforth the number of iterations the chain spends at different parameter values can be used to infer the PDF. The power of the MCMC is two-fold: Because the al- gorithm has a finite probability of moving away from a favorable location in the parameter space it can avoid getting trapped by local features of the likelihood sur- face. Meanwhile, the absence of any “memory” within the chain of past parameter values allows the algorithm to blindly, statistically, explore the region in the neigh- borhood of the global maximum. It is then rigorously proven that an MCMC will (eventually) perfectly map out the PDF, completely removing the need for user in- put to determine parameter uncertainties or thresholds. The parameter vector that maximizes the posterior dis- tribution is stored as the maximum a posteriori (MAP) value and is considered to be the best estimate of the source parameters. Note that because of the prior weight- ing in the definition of the PDF this is not necessarily the ~λ that yields the greatest likelihood. Upon obtaining the MAP value for a particular model X the PDF, now written as p( ~λ,X |s), can be employed to solve the model selection problem. II. BAYES FACTOR ESTIMATES The Bayes Factor BXY [14] is a comparison of the ev- idence for two competing models, X and Y , where pX(s) = d~λ p(~λ,X |s) (10) is the marginal likelihood, or evidence, for model X. The Bayes Factor can then be calculated by BXY (s) = pX(s) pY (s) . (11) The Bayes Factor has been described as the Holy Grail of model selection: It is a powerful entity that is very difficult to find. The quantity BXY can be thought of as the odds ratio for a preference of model X over model Y . Apart from carefully concocted toy problems, direct cal- culation of the evidence, and therefore BXY , is imprac- tical. To determine BXY the integral required to com- pute pX can not generally be solved analytically and for high dimension models Monte-Carlo integration proves BXY 2 logBXY Evidence for model X < 1 < 0 Negative (supports model Y ) 1 to 3 0 to 2 Not worth more than a bare mention 3 to 12 2 to 5 Positive 12 to 150 5 to 10 Strong > 150 > 10 Very Strong TABLE I: BXY ‘confidence’ levels taken from [1] to be inefficient. To employ this powerful statistical tool various estimates for the Bayes Factor have been devel- oped that have different levels of accuracy and computa- tional cost [1, 2]. We have chosen to focus on four such methods: Reverse Jump Markov Chain Monte Carlo and Savage-Dickie density ratios, which directly estimate the Bayes factor, and the Schwarz-Bayes Information Cri- terion (BIC) and Laplace approximations of the model evidence. A. RJMCMC Reverse JumpMarkov Chain Monte Carlo (RJMCMC) algorithms are a class of MCMC algorithms which admit “trans-dimensional” moves between models of different dimension [3, 15, 16]. For the trans-dimensional imple- mentation applicable to the LISA data analysis problem the choice of model parameters becomes one of the search parameters. The algorithm proposes parameter ‘birth’ or ‘death’ moves (proposing to include or discard the ‘extra’ parameter(s)) while holding all other parameters fixed. The priors in the RJMCMC Hastings ratio α = min p(~λY )p(s|~λY )g(~uY ) p(~λX)p(s|~λX)g(~uX) automatically penalizes the posterior density of the higher dimensional model, which compensate for its generically higher likelihood, serving as a built in ‘Occam Factor.’ The g(~u) which appears in (12) is the distribu- tion from which the random numbers ~u are drawn and |J| is the Jacobian |J| ≡ ∂(~λY , ~uY ) ∂(~λX , ~uX) The chain will tend to spend more iterations using the model most appropriately describing the data, making the decision of which model to favor a trivial one. To quantitatively determine the Bayes Factor one simply computes the ratio of the iterations spent within each model. BXY ≃ # of iterations in model X # of iterations in model Y Like the simpler MCMC methods, the RJMCMC is guar- anteed to converge on the correct value of BXY making 0 10000 20000 30000 40000 50000 iterations FIG. 1: 50 000 iteration segment of an RJMCMC chain mov- ing between models with and without frequency evolution. This particular run was for a source with q = 1 and SNR=10 and yielded BXY ∼ 1. it the ‘gold standard’ of Bayes Factor estimation. And, like regular MCMCs, the convergence can be very slow, so that in practice the Bayes Factor estimates can be inaccu- rate. This is especially true when the trans-dimensional moves involve many parameters, such as the 7 or 8 di- mensional jumps that are required to transition between models with differing numbers of galactic binaries. Figure 1 shows the output of a RJMCMC search of a simulated LISA data stream containing the signal from a galactic binary with q = ḟT 2obs = 1 and and observation time of Tobs = 2 years. The chain moved freely between the 7-dimensional model with no frequency evolution and the 8-dimensional model which included the frequency evolution. B. Laplace Approximation A common approach to model selection is to approx- imate the model evidence directly. Working under the assumption that the PDF is Gaussian, the integral in equation (10) can be estimated by use of the Laplace approximation. This is accomplished by comparing the volume of the models parameter space V to that of the parameter uncertainty ellipsoid ∆V pX(s) ≃ p(~λMAP, X |s) . (15) The uncertainty ellipsoid can be determined by calculat- ing the determinant of the Hessian H of partial deriva- tives of the posterior with respect to the model parameter evaluated at the MAP value for the parameters. pX(s) ≃ p(~λMAP, X |s) (2π)D/2√ The Fisher Information Matrix (FIM) Γ with compo- nents Γij ≡ (h,i |h,j ) where h,i≡ can be used as a quadratic approximation to H yielding pX(s) ≃ p(~λMAP, X |s) (2π)D/2√ We will refer to this estimate of the evidence as the Laplace-Fisher (LF) approximation. The LF approxima- tion breaks down if the priors have large gradients in the vicinity of the MAP parameter estimates. The FIM esti- mates can also be poor if some of the source parameters are highly correlated, or if the quadratic approximation fails. In addition, the FIM approximation gets progres- sively worse as the SNR of the source decreases. A more accurate (though more costly) method for esti- mating the evidence is the Laplace-Metropolis (LM) ap- proximation which employs the PDF as mapped out by the MCMC exploration of the likelihood surface to es- timate H [16]. This can be accomplished by fitting a minimum volume ellipsoid (MVE) to the D-dimensional posterior distribution function. The principle axes of the MVE lie in eigen-directions of the distribution which gen- erally do not lie along the parameter directions. Here we employ the MVE.jar package which utilizes a genetic al- gorithm to determine the MVE of the distribution and returns the covariance matrix of the PDF [17]. The de- terminant of the covariance matrix can then be used to determine the evidence via pX(s) ≃ p(~λMAP, X |s)(2π)D/2 detC. (19) In the MCMC literature the LM approximation is gen- erally considered to be second only to the RJMCMC method for estimating Bayes Factors. C. Savage Dickie Density Ratio Both RJMCMC and LM require exploration of the pos- terior for each model under consideration. The Savage- Dickie (SD) approximation estimates the Bayes Factor directly while only requiring exploration of the highest di- mensional space [2, 18]. This approximation requires that two conditions are met: Model X must be nested within Model Y (adding and subtracting parameters clearly sat- isfies this condition) and the priors for any given model must be separable (i.e. p(~λ) = p(λ1)×p(λ2)×. . .×p(λD)) which is, to a good approximation, satisfied in our exam- ple. The Bayes Factor BXY is then calculated by com- paring the weight of the marginalized posterior to the weight of the prior distribution for the ‘extra’ parameter at the default, lower-dimensional, value for the parameter in question. BXY (s) ≃ p(λ0|s) p(λ0) It is interesting to note that if the above conditions are precisely satisfied it can then be shown that this is an exact calculation of BXY (assuming sufficient sampling of the PDF), as opposed to an approximation. D. Schwarz-Bayes Information Criterion All of the approximations discussed so far depend on the supplied priors p(~λ). The Schwarz-Bayes Informa- tion Criterion (BIC) method is an approximation to the model evidence which makes its own assumptions about the priors - namely that they take the form of a multi- variate Gaussian with covariance matrix derived from the Hessian H [16, 19]. The BIC estimate for the evidence is ln pX(s) ≃ ln p(~λMAP, X |s)− lnNeff (21) where DX is the dimension of model X and Neff is the effective number of samples in the data. For our tests we defined Neff to be the number of data points required to return a (power) signal-to-noise ratio of SNR2−D, where SNR is the signal-to-noise one gets by summing over the entire LISA band. This choice was motivated by the fact that the variance in SNR2 is equal to D2, so Neff accounts for the data points that carry significant weight in the model fit. The BIC estimate has the advantage of being very easy to calculate, but is generally considered less reliable than the other techniques we are using. III. CASE STUDY To compare the various approximations to the Bayes Factor we simulated a ‘typical’ galactic binary. The in- jected parameters for our test source can be found in ta- ble II. Since ḟ ∼ f11/3, higher frequency sources are more likely to have a measurable ḟ . On the other hand, the number of binaries per frequency bin falls of as ∼ f−11/3, so high frequency systems are fairly rare. As a compro- mise, we selected a system with a GW frequency of 5 mHz. To describe the frequency evolution we introduced the dimensionless parameter q ≡ ḟT 2obs, (22) which measures the change in the Barycenter GW fre- quency in units of 1/Tobs frequency bins. For q ≫ 1 it is reasonable to believe that a search algorithm will have no difficulty detecting the frequency shift. Likewise, for q ≪ 1 it is unlikely that the frequency evolution can be detected (at least for sources with modest SNR). There- fore we have selected q ∼ 1 to test the model selection techniques. Achieving q = 1 for typical galactic binaries at 5 mHz requires observation times of approximately two years. A range of SNRs were explored by varying the distance to the source. TABLE II: Source parameters f (mHz) cos θ φ (deg) ψ (deg) cos ι ϕ0 (deg) q Tobs (yr) 5.0 1.0 266.0 51.25 0.17 204.94 1 2 We can rapidly calculate accurate waveform templates using the fast-slow decomposition described in the Ap- pendix. Our waveform algorithm has been used in the second round of Mock LISA Data Challenges [20] to sim- ulate the response to a galaxy containing some 26 million sources. The simulation takes just a few hours to run on a single 2 GHz processor. We simulated a 1024 frequency bin snippet of LISA data around 5 mHz that included the injected signal and stationary Gaussian instrument noise. The Markov chains were initialized at the injected source parameters as the focus of this study is the statistical character of the detection, and not the initial detection (a highly ef- ficient solution to the detection problem is described in Ref. [9]). We used uniform priors for all of the parame- ters, with the angular parameters taking their standard ranges. We took the frequency to lie somewhere within the frequency snippet, and lnA to be uniform across the range 1 ln(Sn/(2T )) and ln(1000Sn/(2T ), which roughly corresponds to signal SNRs in the range 1 to 1000. We took the frequency evolution parameter q to be uniformly distributed between -3 and 3 and adopted q = 0 as the default value when operating under the 7-dimensional model. In reality, astrophysical considera- tions yield a very strong prior for q (see Section V) that will significantly impact model selection. We decided to use a simple uniform prior to compare the various ap- proximations to the Bayes Factor, before moving on to consider the effects of the astrophysical prior in Section The choice of proposal distribution q(~x|~y) from which to draw new parameter values has no effect on the asymp- totic form of the recovered PDFs, but the choice is cru- cially important in determining the rate of convergence to the stationary distribution. We took q(~x|~y) to be a mul- tivariate Gaussian with covariance matrix given by the inverse of the FIM. In addition to the source parameters we included two additional parameters, kA and kE , that describe the noise levels in the A and E data channels: SAn (f) = kASn(f) SEn (f) = kESn(f). (23) In a given realization of the instrument noise kA and kE will differ from unity by some random amount δ. The quantity δ will have a Gaussian distribution with variance σ2 = 1/N , whereN is the number of frequency bins being analyzed. The likelihood p(s|~λ) can then be written as p(s|~λ) = C′ exp −N ln (kAkE) where the constant C′ is independent of the signal param- eters ~λ and the noise parameters kA and kE . We explored the noise level estimation capabilities of our MCMC al- gorithm by starting kA and kE far from unity. As can be seen in Figure 2 the chain quickly identified the correct noise level. 0 500 1000 1500 2000 2500 iterations FIG. 2: Demonstration of the MCMC algorithm’s rapid de- termination of the injected noise level. The parameters kA and kE were initialized at 10 and 0.1 respectively. IV. COMPARISON OF TECHNIQUES We compared the Bayes Factor estimates obtained us- ing the various methods in two ways. First, we fixed the frequency derivative of the source at q = 1 and varied the SNR between 5 and 20 in unit increments. Second, we fixed the signal to noise ratio at SNR = 12 and varied the frequency derivative of the source. Favor 7D model Not worth more than a bare mention Positive Strong Very StrongV RJMCMC 0.001 0.01 1000 10000 6 8 10 12 14 16 18 20 FIG. 3: Plot of the Bayes Factor estimates as a function of SNR for each of the approximation schemes described in the text. The results of the first test are shown in Figure 3 We see that all five methods agree very well for SNR > 7. As expected, the Laplace-Metropolis and Savage-Dicke methods provide the best approximation to the “Gold Standard” RJMCMC estimate, showing good agreement all the way down to SNR = 5. Most importantly, all five methods agree on when the 8-dimensional source model is favored over the 7-dimensional model, placing the tran- sition point at SNR ≃ 12.2. To get a rough estimate for the numerical error in the various Bayes Factor es- timates we repeated the SNR= 15 case 10 times using different random number seeds. We found that the nu- merical error was enough to account for any quantitative differences between the estimates returned by the various approaches. It is interesting to compare the Bayesian model se- lection results to the frequentist “3-σ” rule for positive detection: |q̄| > 3σq, (25) where q̄ is the MAP estimate for the frequency change and σq is the standard deviation in q as determined by the FIM. For the source under consideration we found the “3-σ” rule to require SNR ≃ 13 for a detection, in good agreement with the Bayesian analysis. This lends support to Seto’s [21] earlier FIM based study of the de- tectability of the frequency evolution of galactic bina- ries, but we should caution that the literature is replete with examples where the “3-σ” rule yields results in dis- agreement with Bayesian model selection and common sense [22]. Favor 7D model Not worth more than a bare mention Positive Strong 0.7 0.8 0.9 1 1.1 1.2 1.3 RJMCMC FIG. 4: Plot of the Bayes Factor estimates as a function of q for each of the approximation schemes described in the text. The signal to noise ratio was held fixed at SNR = 12. The results of the second test are displayed in Figure 4 In this case all five methods produced results that agree to within numerical error. While the results shown here are for a particular choice of source parameters, we found similar results for other sets of source parameters. In general all five methods for estimating the Bayes Factor gave consistent results for signals with SNR > 7. One exception to this general trend were sources with inclinations close to zero, as then the PDFs tend to be highly non-gaussian. The Laplace- Metropolis and Laplace-Fisher approximations suffered the most in those cases. V. ASTROPHYSICAL PRIORS Astrophysical considerations lead to very strong pri- ors for the frequency evolution of galactic binaries. The detached systems, which are expected to account for the majority of LISA sources, will evolve under gravita- tional radiation reaction in accord with the leading order quadrapole formula: 3(8π)8/3 f11/3M5/3 , (26) where M is the chirp mass. Contact binaries undergoing stable mass transfer from the lighter to the heavier com- ponent are driven to longer orbital periods by angular momentum conservation. The competition between the effects of mass transfer and gravitational wave emission lead to a formula for ḟ with the same frequency and mass scaling as (26), but with the opposite sign and a slightly lower magnitude [23]. Population synthesis models, calibrated against obser- vational data, yield predictions for the distribution of chirp masses M as a function of orbital frequency. These distributions can be converted into priors on q. In con- structing such priors one should also fold in observational selection effects, which will favor systems with larger chirp mass (the GW amplitude scales as M5/6). To get some sense of how such priors will affect the model selec- tion we took the chirp mass distribution for detached sys- tems at f ∼ 5 mHz from the population synthesis model described in Ref. [24], (kindly provided to us by Gijs Nele- mans), and used (26) to construct the prior on q shown in Figures 5 and 6 (observation selection effects were ig- nored). The prior has been modified slightly to give a small but no-vanishing weight to sources with q = 0. The astrophysically motivated prior has a very sharp peak at q = 0.64, and we use this value when fixing the frequency derivative for the 7-dimensional model. To explore the impact on model selection when such a strong prior has been adopted we simulated a source with q = 1 and varied the SNR. The RJMCMC algorithm was applied using chains of length 107 in conjunction with a fixed 8-dimensional MCMC (also allowed to run for 107 iterations) in order to compare the RJMCMC results with the Savage-Dickie density ratio. The results of this first exploration are shown in Fig- ure 5. We found that for SNR < 15 the marginalized PDF very closely resembled the prior distribution. This demonstrates that the information content of the data is insufficient to change our prior belief about the value of the frequency derivative. As the SNR increased, how- ever, the PDF began to move away from the prior until we reached SNR=30 when the astrophysical prior had a 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 SNR=5 SNR=10 SNR=15 SNR=20 SNR=25 SNR=30 q q q FIG. 5: Comparison between astrophysically motivated prior distribution of q for f = 5 mHz and Tobs = 2 years (dashed, blue) to marginalized PDF (solid, red) for sources injected with q = 1 and SNRs varying from 5 to 30. SNR BXY (SD) BXY (RJMCMC) 5 0.926 1.015 10 0.977 0.996 15 0.749 0.742 20 0.427 0.427 25 0.176 0.177 30 0.060 0.056 TABLE III: Savage-Dickie density ratio estimates of BXY for sources with q = 1 and SNRs varying from 5 to 30. Compar- isons with RJMCMC explorations of the same data set show excellent agreement between the two methods. negligible effect on the shape of the posterior, signaling confidence in the quoted measurement of q. This qualita- tive assessment of model preference is strongly supported by the Bayes factor estimation made by the RJMCMC algorithm as can be seen in Table III. It should also be noted that the excellent agreement between the RJM- CMC and S-D estimates for Bayes factor BXY . Both methods indicate that for the chosen value of q = 1, the signal-to-noise needs to exceed SNR ∼ 25 for the 8-dimensional model to be favored. This is in contrast to the case discussed earlier where a uniform prior was adopted for the frequency derivative, and the model se- lection methods began showing a preference for the 8- dimensional model around SNR=12. Figure 6 shows the impact of the astrophysically moti- vated prior when the SNR was held at 15 and four differ- ent injected values for q were adopted, corresponding to the full width at half maximum (FWHM) and full width at quarter maximum (FWQM) of the prior distribution. The Bayes factors listed in Table IV indicate that for modestly loud sources with SNR=15 the model selection techniques do not favor updating our estimate of the fre- 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 1 .5 2 0.005 0.01 0.015 0.02 0.025 0.03 .5 2 q=0.35 q=0.83 q=0.48 q=1.15 FIG. 6: Marginalized PDF (solid, red) for fixed SNR=15 in- jected sources with q corresponding to FWHM and FWQM of the astrophysical prior (dashed, blue) q BXY (SD) BXY (RJMCMC) 0.35 1.412 1.414 0.48 1.381 1.388 0.83 1.059 1.052 1.15 0.432 0.428 TABLE IV: Savage-Dickie and RJMCMC density ratio esti- mates of BXY for sources with SNR=15 and q at FWHM and FWQM of astrophysical prior quency derivative until the frequency derivative exceeds q = 1.2. VI. DISCUSSION We have found that the several common methods for estimating Bayes Factors give good agreement when ap- plied to the the model selection problem of deciding when the data from the LISA observatory can be used to detect the orbital evolution of a galactic binary. The methods studied require varying degrees of effort to implement and calculate, and although found to be accurate in this test case, it is clear that some of these methods would be in- appropriate approximations for more physically relevant examples. If a RJMCMC algorithm is used as the sole model selection technique, the resistance of the algorithm to change dimension, especially when making multi- dimensional jumps, can result in invalid model selection unless the chains are run for a very large numbers of steps. In the examples we studied the transdimensional jumps only had to span one dimension, and our basic RJMCMC algorithm performed well. However, a more sophisticated implementation, using e.g. rejection sam- pling or coupled chains, will be required to select the number of sources, as this requires jumps that span seven or more dimensions. The Laplace-Metropolis method for approximating the model evidence is more robust than the commonly used Fisher Information Matrix approximation of the Hessian of the PDF. Implementing an LM evidence estimation is a somewhat costly because of the need to fit the posterior to a minimum volume ellipsoid. The Savage-Dickie approximation is more economical than the RJMCMC or LM methods, but is limited by the requirement that the competing models must be nested. The Bayes Information Criterion approximation to the evidence is by far the cheapest to implement, and is able to produce reliable results when the SNR is high. It has therefore shown the most promise as an ‘on the fly’ model determination scheme. More thorough (and therefore more costly) methods such as RJMCMC and LM could then be used to refine the conclusions initially made by the BIC. Our investigation using a strong astrophysical prior indicated that the gravitational wave signals will need to have high signal-to-noise (SNR > 25), or moderate signal-to-noise (SNR > 15) and frequency derivatives far from the peak of the astrophysical distribution, in order to update our prior belief in the value of the frequency derivative. In other words, the frequency derivative will only been needed as a search parameter for a small num- ber of bright high frequency sources. Acknowledgments This work was supported by NASA Grant NNG05GI69G. We are most grateful to Gijs Nele- mans for providing us with data from his population synthesis studies. Appendix A To leading order in the eccentricity, e, the Cartesian coordinates of the ith LISA spacecraft are given by [25] xi(t) = R cos(α) + cos(2α− βi)− 3 cos(β) yi(t) = R sin(α) + sin(2α− βi)− 3 sin(β) zi(t) = − 3eR cos(α− βi) . (27) In the above R = 1 AU, is the radial distance of the guiding center, α = 2πfmt+ κ is the orbital phase of the guiding center, and βi = 2π(i − 1)/3 + λ (i = 1, 2, 3) is the relative phase of the spacecraft within the constel- lation. The parameters κ and λ give the initial ecliptic longitude and orientation of the constellation. The dis- tance between the spacecraft is L = 2 3eR. Setting e = 0.00985 yields L = 5× 109 m. An arbitrary gravitational wave traveling in the k̂ di- rection can be written as the linear sum of two indepen- dent polarization states h(ξ) = h+(ξ)ε + + h×(ξ)ε × (28) where the wave variable ξ = t − k̂ · x gives the surfaces of constant phase. The polarization tensors can be ex- panded in terms of the basis tensors e+ and e× as ε+ = cos(2ψ)e+ − sin(2ψ)e× ε× = sin(2ψ)e+ + cos(2ψ)e× , (29) where ψ is the polarization angle and + = û⊗ û− v̂ ⊗ v̂ × = û⊗ v̂ + v̂ ⊗ û . (30) The vectors (û, v̂, k̂) form an orthonormal triad which may be expressed as a function of the source location on the celestial sphere û = cos θ cosφ x̂+ cos θ sinφ ŷ − sin θ ẑ v̂ = sinφ x̂− cosφ ŷ k̂ = − sin θ cosφ x̂− sin θ sinφ ŷ − cos θ ẑ . (31) For mildly chirping binary sources we have h(ξ) = ℜ + + ei3π/2A×ε eiΨ(ξ) where 2M(πf)2/3 1 + cos2 ι A× = − 4M(πf)2/3 cos ι . (33) Here M is the chirp mass, DL is the luminosity dis- tance and ι is the inclination of the binary to the line of sight. Higher post-Newtonian corrections, eccentric- ity of the orbit, and spin effects will introduce additional harmonics. For chirping sources the adiabatic approxi- mation requires that the frequency evolution ḟ occurs on a timescale long compared to the light travel time in the interferometer: f/ḟ ≪ L. The gravitational wave phase can be approximated as Ψ(ξ) = 2πf0ξ + πḟ0ξ 2 + ϕ0 , (34) where ϕ0 is the initial phase. The instantaneous fre- quency is given by 2πf = ∂tΨ: f = (f0 + ḟ0ξ)(1 − k̂ · v) . (35) The general expression for the path length variation caused by a gravitational wave involves an integral in ξ from ξi to ξf . Writing ξ = ξi + δξ we have Ψ(ξ) ≃ 2π(f0 + ḟ0ξi)δξ + const . (36) Thus, we can treat the wave as having fixed frequency f0 + ḟ0ξi for the purposes of the integration, and then increment the frequency forward in time in the final ex- pression [26]. The path length variation is then given by [25, 26] δℓij(ξ) = Lℜ d(f, t, k̂) : h(ξ) , (37) where a : b = aijbij . The one-arm detector tensor is given by d(f, t, k̂) = r̂ij(t)⊗ r̂ij(t) T (f, t, k̂) , (38) and the transfer function is T (f, t, k̂) = sinc 1− k̂ · r̂ij(t) × exp 1− k̂ · r̂ij(t) , (39) where f∗ = 1/(2πL) is the transfer frequency and f = f0 + ḟ0ξ. The expression can be attacked in pieces. It is useful to define the quantities d+ij(t) ≡ (r̂ij(t)⊗ r̂ij(t)) : e + (40) d×ij(t) ≡ (r̂ij(t)⊗ r̂ij(t)) : e × . (41) and yij(t) = δℓij(t)/(2L). Then yij(t) = ℜ yslowij (t)e 2πif0t , (42) where yslowij (t) = T (f, t, k̂) d+ij(t)(A+(t) cos(2ψ) +e3πi/2A×(t) sin(2ψ)) +d×ij(t)(e 3πi/2A×(t) cos(2ψ) −A+(t) sin(2ψ))) e(πiḟ0ξ 2+iϕ0−2πif0k̂·x) It is a simple exercise to derive explicit expressions for the antenna functions and the transfer function appearing in yslowij (t) using (27) and (31). In the Fourier domain the response can be written as yij(t) = ℜ 2πint/Tobs e2πif0t , (44) where the coefficients an can be found by a numerical FFT of the slow terms yslowij (t). Note that the sum over n should extend over both negative and positive values. The number of time samples needed in the FFT will de- pend on f0 and ḟ0 and Tobs, but is less than 2 9 = 512 for any galactic sources we are likely to encounter when Tobs ≤ 2yr. The bandwidth of a source can be estimated B = 2 (4 + 2πf0R sin(θ)) fm + ḟ0Tobs . (45) The number of samples should exceed 2BTobs. The Fourier transform of the fast term can be done analyti- cally: e2πif0t = 2πimt/Tobs (46) where bm = Tobs sinc(xm)e ixm (47) xm = f0Tobs −m. (48) The cardinal sine function in (46) ensures that the Fourier components bm away from resonance, xm ≈ 0, are quite small. It is only necessary to keep ∼ 100 → 1000 terms either side of p = [f0Tobs], depending on how bright the source is, and how far f0Tobs is from an integer. We now have yij(t) = ℜ 2πijt/Tobs , (49) where anbj−n . (50) The final step is to ensure that our Fourier transform yields a real yij(t). This is done by setting the final an- swer for the Fourier coefficients equal to dj = (cj+c −j)/2. But since xm never hits resonance for positive j (we are not interested in the negative frequency components j < 0), we can neglect the second term and simply write dj = cj/2. Basically what we are doing is hetrodyning the signal to the base frequency f0, then Fourier transforming the slowly evolving hetrodyned signal numerically. We then convolve these Fourier coefficients with the analytically derived Fourier coefficients of the carrier wave. The Michelson type TDI variables are given by X(t) = y12(t− 3L)− y13(t− 3L) + y21(t− 2L) −y31(t− 2L) + y13(t− L)− y12(t− L) +y31(t)− y21(t), (51) Y (t) = y23(t− 3L)− y21(t− 3L) + y32(t− 2L) −y12(t− 2L) + y21(t− L)− y23(t− L) +y12(t)− y32(t), (52) Z(t) = y31(t− 3L)− y32(t− 3L) + y13(t− 2L) −y23(t− 2L) + y32(t− L)− y31(t− L) +y23(t)− y13(t). (53) Note that in the Fourier domain X(f) = ỹ12(f)e −3if/f∗ − ỹ13(f)e−3if/f∗ + ỹ21(f)e−2if/f∗ −ỹ31(f)e−2if/f∗ + ỹ13(f)e−if/f∗ − ỹ12(f)e−if/f∗ +ỹ31(f)− ỹ21(f) . (54) This saves us from having to interpolate in the time do- main. We just combine phase shifted versions of our orig- inal Fourier transforms. [1] Raftery, A.E., Practical Markov Chain Monte Carlo, (Chapman and Hall, London, 1996). [2] Trotta, R., astro-ph/0504022 (2005). [3] Sambridge, M., Gallagher, K., Jackson, A. & Rickwood, P., Geophys. J. Int. 167 528-542 (2006). [4] Bender, P. et al., LISA Pre-Phase A Report, (1998). [5] S. Timpano, L. J. Rubbo & N. J. Cornish, Phys. Rev. D73 122001 (2006). [6] Cornish, N.J. & Crowder, J., Phys. Rev. D 72, 043005 (2005). [7] C. Andrieu & A. Doucet, IEEE Trans. Signal Process. 47 2667 (1999). [8] R. Umstatter, N. Christensen, M. Hendry, R. Meyer, V. Simha, J. Veitch, S. Viegland & G. Woan, gr-qc/0503121 (2005). [9] J. Crowder & N. J. Cornish, Phys. Rev. D75 043008 (2007). [10] T. A. Prince, M. Tinto, S. L. Larson & J. W. Armstrong, Phys. Rev. D66, 122002 (2002). [11] Gamerman, D., Markov Chain Monte Carlo: Stochas- tic Simulation of Bayesian Inference, (Chapman & Hall, London, 1997). [12] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H.,& Teller E., J. Chem. Phys. 21, 1087 (1953). [13] Hastings, W.K., Biometrika 57, 97 (1970). [14] Jeffreys, H. Theory of Probability, Third Edition. (Oxford University Press 1961). [15] Green, P.J., Biometrika 82 711-32 (1995). [16] Lopes, H.F., & West M., Statistica Sinica 14, 41-67 (2004). [17] Kim van der Linde (2004) MVE: Minimum Vol- ume Ellipsoid estimation for robust outlier detec- tion in multivariate space, Java version. Website: http://www.kimvdlinde.com/professional/mve.html. [18] Dickey, J.M., Ann. Math. Stat., 42, 204 (1971). [19] Schwarz, G., Ann. Stats. 5, 461 (1978). [20] K. Arnaud et al., preprint gr-qc/0701170 (2007). [21] Seto, N., Mon. Not. Roy. Astron. Soc. 333, 469-474 (2002). [22] D. Lindley, Biometrica 44 187 (1957). [23] G. Nelemans, L. R. Yungelson & S. F. Portegies Zwart, Mon. Not. Roy. Astron. Soc. 349 181, (2004). [24] G. Nelemans, L. R. Yungelson & S. F. Portegies Zwart, A&A 375, 890 (2001). [25] Cornish, N.J. & Rubbo, L.J., Phys. Rev. D 67, 022001 (2003). [26] Cornish, N.J., Rubbo, L.J. & Poujade, O., Phys. Rev. D 69, 082003 (2004). http://arxiv.org/abs/astro-ph/0504022 http://arxiv.org/abs/gr-qc/0503121 http://www.kimvdlinde.com/professional/mve.html http://arxiv.org/abs/gr-qc/0701170 [27] While this count is only strictly correct for point-like masses, frequency evolution due to tides and mass trans- fer can also be described by the same two parameters for the majority of sources in the LISA band.

0704.1810

A Cascade Model for Particle Concentration and Enstrophy in Fully Developed Turbulence with Mass Loading Feedback

A Cascade Model for Particle Concentration and Enstrophy in Fully Developed Turbulence with Mass Loading Feedback R. C. Hogan Bay Area Environmental Research Institute; MS 245-3 Moffett Field, CA 94035-1000 J. N. Cuzzi NASA Ames Research Center; MS 245-3 Moffett Field, CA 94035-1000 (Dated: November 4, 2018) A cascade model is described based on multiplier distributions determined from 3D direct numer- ical simulations (DNS) of turbulent particle laden flows, which include two-way coupling between the phases at global mass loadings equal to unity. The governing Eulerian equations are solved using psuedo-spectral methods on up to 5123 computional grid points. DNS results for particle concentration and enstrophy at Taylor microscale Reynolds numbers in the range 34 - 170 were used to directly determine multiplier distributions on spatial scales 3 times the Kolmogorov length scale. The multiplier probability distribution functions (PDFs) are well characterized by the β dis- tribution function. The width of the PDFs, which is a measure of intermittency, decreases with increasing mass loading within the local region where the multipliers are measured. The functional form of this dependence is not sensitive to Reynolds numbers in the range considered. A partition correlation probability is included in the cascade model to account for the observed spatial anti- correlation between particle concentration and enstrophy. Joint probability distribution functions of concentration and enstrophy generated using the cascade model are shown to be in excellent agreement with those derived directly from our 3D simulations. Probabilities predicted by the cas- cade model are presented at Reynolds numbers well beyond what is achievable by direct simulation. These results clearly indicate that particle mass loading significantly reduces the probabilities of high particle concentration and enstrophy relative to those resulting from unloaded runs. Particle mass density appears to reach a limit at around 100 times the gas density. This approach has promise for significant computational savings in certain applications. PACS numbers: 47.61.Jd, 47.27.E-, 47.27.eb Keywords: Turbulence, Multiphase Flows, Statistical Distributions I. INTRODUCTION The study of turbulent flows incorporating heavy par- ticles in suspension (particles with finite stopping times) is an important endeavor that has both fundamental and practical relevance to many scientific and engineering problems. Such flows have been investigated mainly in numerical simulations where detailed statistical analysis of the flow fields is possible [1, 2, 3, 4] These simulations, limited to relatively low Taylor microscale Reynolds num- bers Reλ (∼ 40), demonstrated that particles whose fluid response times are comparable to the lifetime of the smallest turbulent eddies produce a highly nonuniform field with intense regions of concentration. Preliminary indications were that the feedback from such concentra- tions of particles could locally damp turbulence - how- ever, the role of this “mass loading” effect in determining the statistical distributions of particle density and vari- ous fluid scalars has not been thoroughly studied. Ex- perimental investigations of turbulence modification by particles have demonstrated that the degree of turbu- lence damping increases with particle mass loading and ∗Electronic address: [email protected] †Electronic address: [email protected] concentration [4]. The phenomenon known as intermittency can be de- scribed as intense fluctuations, on small spatial and tem- poral scales in the turbulent field, that contribute to the exponential tails of probability distribution functions (PDFs) of scalars such as velocity increments and gradi- ents [5, 6, 7], dissipation [8], pressure [9, 10], enstrophy [11, 12] and velocity circulation [13]. Intermittency in the density field of preferentially concentrated particles has also been observed and studied [14, 15]. Although intermittency in turbulence still lacks a com- plete theoretical understanding, progress has been made with phenomenological models that capture intermit- tency in a cascade process. Richardson [16] and later Kol- mogorov [17] suggested that such models might be used to explain the process of eddy fragmentation initiated by unstable large scale structures in a turbulent fluid. Intermittency in the context of fragmentation though a cascading process has been studied for large-scale gravi- tating masses [18] and velocity increments in turbulence [19]. Simple cascade models were explored by Meneveau and Sreenivasan [20] and were reviewed by Sreenivasan and Stolovitzky [21] The scale similarity of random fields was explored by Novikov [22, 23], with a focus on the energy dissipation cascade. In Novikov’s work, the ratio of dissipation averaged over two spheres, one embedded within the other, served as a measure of enstrophy par- http://arxiv.org/abs/0704.1810v2 mailto:[email protected] mailto:[email protected] titioning between larger and smaller scales. The prob- ability distribution of these ratios, known as multipliers or breakdown coefficients, was shown to relate to multi- fractal and statistical measures (moments) of the velocity and dissipation fields. A recent review of intermittency in multiplicative cascades stresses that this theory is a kinematic description and its connection with the real dynamics remains unclear [24]. Our previous numerical study of particle concentration in turbulent flows showed that the particle density field is a multifractal on scales comparable to the Kolmogorov length scale [14]. This result suggests that a deeper de- scription of the statistical properties of the particle con- centration field, based on multiplier PDFs, may also be possible. Analytical efforts have suggested that dissipa- tion and vorticity in the fluid phase should be locally linked with particle concentration [25]. Numerical work in this regard has demonstrated that preferential concen- tration is statistically anticorrelated with low vorticity: particles tend to concentrate in regions where enstrophy is relatively weak [26, 27]. In this paper we present a cascade model in the spirit of Novikov [22, 23] that follows the partitioning of pos- itive definite scalars associated with both the fluid and the particles. Multipliers controlling the partitioning of enstrophy and particle density at each step in the cascade are drawn from probability distribution functions (PDFs) which are determined empirically from direct numerical simulations (DNS). Moreover, the multiplier PDFs are dependent on, or conditioned by, the particle mass den- sity or mass loading. The cascade model then generates joint PDFs for particle concentration and enstrophy at arbitrary cascade levels. A partitioning correlation prob- ability is also applied at each cascade level to account for the observed spatial anticorrelation between enstrophy and particle concentration [26, 28]. In Section II we describe the cascade model and its parameters, which are empirically determined from DNS calculations. Details of the DNS equations, and our nu- merical methods, are discussed in the Appendix. Results are shown in section III, including comparisons of joint PDFs of enstrophy and particle concentration as pre- dicted by the cascade model with those obtained directly from the DNS results. Cascade model PDF predictions at Reynolds numbers well beyond the DNS values are also presented. In section IV, we summarize our results and discuss their implications. II. CASCADE MODEL A turbulent cascade can be envisioned as an hierar- chical breakdown of larger eddies into smaller ones that halts when the fluid viscosity alone can dissipate eddy ki- netic energy. Eddies or similar turbulent structures such as vortex tubes are bundles of energy containing vorticity and dissipation. These structures start with a size com- parable to the integral scale Λ of the flow, and break down in steps to a size comparable to the Kolmogorov scale η before being dissipated away by viscosity. The fluid vorticity and dissipation exhibit spatial fluctuations that increase in intensity as the spatial scale decreases. This phenonemon is known as intermittency and has been ob- served in a variety of processes with strong nonlinear in- teractions. In previous numerical and experimental studies, locally averaged intermittent dissipation fields with scale at or near η were used to quantify the statistical properties of multiplier distributions [21]. Multipliers are random variables that govern the partitioning of a positive defi- nite scalar as turbulent structures break down along the cascade. In these studies the statistical distribution of multipliers (their PDF) were shown to be invariant over spatial scales that fall within the turbulent inertial range. Multifractal properties of the cascading field are deriv- able from such multiplier distributions [23], and cascade models based on the iterative application of multipliers to a cascading variable have been shown to mimic inter- mittency. While invariant with level in the inertial range of a cascade, multiplier PDFs might depend on local proper- ties of the environment. For instance, Sreenivasan and Stolovitzky [21] showed that the degree of intermittency in dissipation increases with the degree of local strain rate, and constructed multiplier distributions for local energy dissipation conditioned on the local strain rate. The physical mechanism behind this effect is believed to be related to vortex stretching dynamics creating intense bursts of dissipation. All the multiplier PDFs measured by Sreenivasan and Stolovitzky [21], whether conditioned or unconditioned by local properties, are well characterized by the β dis- tribution function, p(m) = Γ(2β) Γ(β)2 mβ−1(1−m)β−1 (1) where m is the multiplier variable and β is a shape con- trolling parameter. A large β produces a narrow, delta- function-like curve centered at m = 0.5, whereas β = 1 produces a flat distribution between m = 0 and 1. These limits for β correspond to uniform and highly intermit- tent processes respectively. In conditioned multipliers, the value of β varies with some local property of the fluid. Concentration of particles in turbulence is a result of the active dynamics of eddies on all scales. The process depends on the scale of the eddies and the corresponding particle response to those eddies. Intense particle den- sity fluctuations, akin to intermittency, were observed in a previous numerical study where it was also shown that nonuniform particle concentrations have multifrac- tal scaling properties [14]. These results strongly suggest that a phenomenological cascade model based on multi- pliers may adequately describe the particle density field. Simulations that have included particle feedback on the fluid through the mass loading effect show that damp- ing of local turbulence occurs [2, 29]. The latter have shown that vorticity dynamics is affected locally by par- ticle feedback. This interplay between the phases could attenuate vortex stretching and, thereby, diminish local turbulent intermittency. Multiplier distributions condi- tioned on local mass loading should therefore be an inte- gral part of a realistic fluid-particle cascade model. A. Two-Phase Cascade model Below we describe a two-phase cascade model that in- corporates simultaneous multiplier processes for parti- cle concentration C and fluid enstrophy S, in addition to a process that models their spatial anticorrelation. The multiplier distributions are conditioned by the local particle concentration, as determined empirically from DNS fields equilibrated to Reλ = 34, 60, 107, and 170. The spatial anticorrelation was also quantified from these fields. Local measures of particle concentration (C) and enstrophy (S) used are defined in the Appendix. A schematic illustration of our two-phase partition- ing process is shown in FIG. 1. The cascading vector (S,C) has components representing enstrophy and parti- cle concentration. Initially the components are assigned the value unity and are associated with a common cell having a volume of unity. Each component is partitioned into two parts; (mSS, (1−mS)S) and (mCC, (1−mC)C), respectively, where mS ,mC are multipliers for S and C whose values are between zero and one inclusive and are random members of the corresponding multiplier distri- butions. The parts are associated with two daughter cells each containing half the volume of the starting cell. In the example shown in FIG. 1, mS and mC are assumed to be greater than 0.5. The largest parts of S and C are placed in the same daughter cell with probability Γ (and in different cells with probability 1− Γ). This partition- ing process is repeated for each daughter cell down the cascade until the ratio of the daughter cell size to the initial cell size equals a specified cutoff. When this cutoff is set to the ratio of the turbulent lengthscales Λ and η, the cascade corresponds to turbulence characterized by Reλ ∼ (Λ/η)2/3 [30]. B. Conditioned Multipliers The parameters of the cascade model are empirically derived from the particle density and enstrophy fields C and S as calculated by DNS (see Appendix). The simu- lation parameters for four DNS runs representing Reλ = 36, 60, 104, and 170 are shown in Table I. The turbu- lence kinetic energy q, the volume averaged dissipation ǫ, and Λ are calculated from the 3-D turbulent energy spectrum E(k) and kinematic viscosity ν, E(k)dk (2) mC mC *C 1−Γ = .7 Γ = .3 mS *S mC *C (1− )*CmC (1− )*S (1− )*SmS *S FIG. 1: Figure depicting the breakdown of a parcel of en- strophy (S) and particle concentration (C) into two parcels each with half the volume of the parent. The corresponding multipliers mS and mC are assumed to be greater than 0.5 in this figure. These measures are broken down and distributed between the two parcels in one of two ways - the larger por- tions are partitioned together with probability Γ= 0.3 (upper figure), or in opposite directions with probability 1− Γ= 0.7 (lower figure). ǫ = 2ν E(k)k2dk (3) dk (4) where k is wavenumber. kmax = times the number of computational nodes per side is the maximum effective wavenumber. Thus kmaxη > 1 indicates an adequate resolution of the Kolmogorov scale. Parameter Case I Case II Case III Case IV Nodes/side 64 128 256 512 ν .01 .003 .0007 .0002 Reλ 34 60. 104 170 q 1.5 .65 .28 .14 23. 22.8 22.4 23 kmaxη 1.4 1.5 1.45 1.56 14.1 23.3 45.8 86.2 Γ .31 .29 .27 .32 D .0001 .00003 .000007 .000002 νp .001 .0003. .00007 .00002 TABLE I: Case Parameters for DNS runs. The quantities D and νp are defined in the Appendix. Other quantities above are defined in Section II. The 3-D DNS computational box is uniformly subdi- vided into spatial cells 3η on a side, and the average value of C and S is determined for each cell ( see Appendix ). The cells are divided into groups associated with disjoint ranges of C. Each cell is then divided into two parts of equal volume and averages for C and S are determined for each part. The C and S multipliers for each cell are evaluated as the ratio of these averages to the averages in the parent cell. A conditional multiplier distribution p(m) is then determined for each binned value of C from the corresponding set of cell multipliers. Plots of p(m) for three values of C are shown in FIG. 2. The points repre- sent distributions derived from all DNS runs and the solid lines are least squares fits to the β distribution function (Eq. 1). For the lower values of C, Reλ-independence is apparent; only the Reλ = 170 case provided data for the largest C range. The plots clearly indicate that the intermittency in C is reduced (multiplier PDFs narrow) as C is increased. Derived values of βC(C) and βS(C) are shown as a function of C in FIG. 3. Least squares fits to the functional form p1 exp(p2C p3) are drawn as solid lines and the best fit parameter values for this func- tion are tabulated in Table II. Bounding curves (dashed lines) are defined by setting p2 and p3 to their 2σ lim- its, to establish a plausible range of uncertainty in the predictions. Scalar p1 p2 p3 C 2.7 .045 1.02 S 9. .03 1.06 TABLE II: β model parameters It is certainly of interest that such large solid/gas mass loadings as C = 100 appear in the DNS runs at all, given published reports that particle mass loading significantly dampens turbulent intensity even for mass loadings on the order of unity [1, 4]. These diverse results might be reconciled since the particles we study herein are all far smaller than the Kolmogorov scale and also have only a very small lag velocity relative to the gas. Recall that we force the turbulence, as might be the case if it were FIG. 2: Empirically determined conditional multiplier distri- butions p(m|C) for particle concentration at three different mass loading values, C = 1, 20 and 50. The distributions are obtained from bifurcations of cells with a spatial scale equal to 3η. Results at Reλ = 34 ( square ), 60 (triangle), 107 (cir- cle) and 170 ( cross ) are overlain. Only the simulation with Reλ = 170 provided results for C = 50. At each mass loading the p(m) at all Reynolds numbers are very well approximated with the β distribution function ( solid line ). The distribu- tion widths narrow as the mass loading increases, indicating a decrease in the intermittency. being constantly forced by energetic sources operating on larger scales than our computational volume. However, FIG. 3 strongly suggests an upper limit for C ( ∼ 100 ) for both βS and βC . The cascade anticorrelation parameter Γ was deter- mined by counting the number of parent cells within which the larger partitions of C and S were found to share the same daughter cell. This number divided by the total number of parent cells defines Γ. The derived Γ value is approximately constant across the DNS cases, as indicated in Table I. Operationally, the Γ used in the cascade model was determined by taking a simple average of the Γ values in Table I. Overall, the invariance of Γ and the βC(C) and βS(C) functions across our range of Reλ justifies their treat- ment as level independent parameters in the two-phase cascade model. One caveat remains, which would be of interest to address in future work. While it has been shown that multiplier distributions leading to βC and βS are level-invariant over a range of scales within an inertial range [21], our simulations were numerically restricted to values of Re in which the inertial range has not yet become fully developed. Our reliance on the smallest available scales of 3η to 1.5η (those providing the largest available intermittency) might lead to some concern that FIG. 3: The β parameters as functions of local mass loading C for enstrophy and particle concentration at 3η. Results for all DNS cases are indicated as described in FIG. 2. A least squares fit of an exponential function to the points over the entire mass loading range is shown ( solid line ). Dashed lines correspond to the upper and lower limits of the function, and are derived using the 2σ errors of p2 and p3. they were already sampling the dissipation range of our calculations, and thus may not be appropriate for a cas- cade code. We tested this possibility by calculating mul- tipliers for the next largest level bifurcation (6η to 3η) for the Reλ = 170 case. The β values for those multi- plier distributions are slightly larger in value, but consis- tent with the C-dependence shown in FIG. 2 (6η scales don’t provide good distribution functions beyond C ∼ 15). Thus we believe that for the purpose of demonstrat- ing this technique, and for the purpose of estimating the occurrence statistics of C under particle mass loading, our results are satisfactory. For applications requiring quantitatively detailed and/or more accurate P (S,C), it would certainly be of interest to extend the DNS calcu- lations to larger Re, at which a true inertial range might be found. III. MODEL RESULTS The 2D joint probability distribution function or PDF of concentration and enstrophy, a fractional volume mea- sure, was generated from the cascade model and com- pared with results derived directly from numerical DNS simulations. The basic probability density P (S,C) gives the fractional volume occupied by cells having enstrophy S and concentration C, per unit S and C; thus the frac- tional volume having C and S in some range ∆S,∆C is P (S,C)∆S∆C. For quantities varying over orders of magnitude, it is convenient to adopt ∆S = S and ∆C = C, and we will present the results in the form P (S,C)SC. We started by binning results at spatial scale 3η, ob- tained from the semi-final level of a cascade model run, into a uniform logarithmic grid of S,C bins each having width ∆(logS) = ∆(logC) = δ, with corresponding val- ues of ∆S and ∆C. The number of 3η cells accumulated in each bin was normalized by the total number of such cells in the sample to convert it to a fractional volume ∆V (S,C) = P (S,C)∆S∆C. Then ∆V (S,C) P (S,C)∆S∆C ∆(logS)∆(logC) → P (S,C)SC as δ → 0. In practice of course, the binning ranges δ are not van- ishingly small. The plots in FIGs. 4 5 and 7 then, show the PDF as the volume fraction P (S,C)SC. Cascade levels 9, 12, 15, and 18 correspond approximately to the Reλ of the four simulation cases shown in Table I. These levels were determined from the ratio of Λ and η for each case: level = 3log (Λ/η). The factor 3 accounts for cascade bifurcations of 3D cells, because it takes three partition- ings, along three orthogonal planes, to generate eight subvolumes of linear dimension one-half that of the par- ent volume. That is, 2level is equal to the number of η cells within a 3D volume having linear dimension Λ and (2level/3)2/3 is the corresponding Reλ. The number of cascade realizations is, in turn, equal to the product of the number of Λ-size volumes in the computational box and the number of simulation snapshots processed. In general it is difficult to generate DNS results with a ratio of Λ and η that is an exact power of two. In or- der to correctly compare DNS simulations with the cas- cade model it was necessary to interpolate between two cascade generated P (S,C)SC computed at scale ratios (levels) that bracketed the ratios that were actually sim- ulated. In FIG. 4 we compare iso-probability contours of P (S,C)SC predicted by cascade models representing the four DNS cases with the same contours derived directly from the simulated S and C fields. The agreement is very good. A. Predictions at higher Reynolds number The cascade model was used to generate PDFs at deeper levels in order to assess the effect of mass loading on the probabilities of high C and S. We generated 256 realizations of a level 24 cascade, 20 realizations of a level 30 cascade, and one realization of a level 36 cascade. FIG. 5(a) shows the average of 256 realizations of a 24 level cascade, taken to lower probability values. The pro- nounced crowding of the contours at the top of the figure indicates the effect of particle mass loading on reducing the intermittency of C at high values of C. For compar- FIG. 4: Comparisons of cascade model predictions of P (S,C)SC with DNS results at Reλ = 34 (a), 60 (b), 107 (c) , and 170 (d). Contours indicate probabilities .001, .01, .1 and .3. Dashed contours are cascade model predictions and solid ones are DNS results. ison, FIG. 5(b) shows a control run of a 24 level cascade with all conditioning turned off. In this control case, the exponential tails characterizing intermittent fluctuations are seen at both low and high C. In order to evaluate the effect of the uncertainties in the extrapolations of the β curves for C and S on the PDF, two cascade runs to level 24 were generated using the pa- rameters for the upper and lower dotted curves in FIG. 3. In FIG. 6 we show cross-sections of the PDFs produced by these runs along the C axis through the distribution modes to compare with the same cross-section for a run using the nominal parameters in Table II. Both models diverge from the mean model beyond C > 40, with the upper (lower) curve corresponding to the outside (inside) βC(C) and βS(C) bounds in FIG. 3. Figure 6 indicates that the sensitivity of the PDF to the β model parame- ters at the 2σ level is only apparent at large C, and all models show a sharp dropoff in the probability for C > A crowding effect similar to the one seen in FIG. 5(a) is shown in FIG. 7 for iso-probability contours equal to 5× 10−4, for cascade levels 6, 12, 18, 24, 30 and 36. Figures 8(a) and 8(b) compare 1D cuts through the modes of the PDFs for cascades of 18 - 36 levels, indi- cating that going to deeper levels (higher Reλ) results in larger intermittency at the low-C end (as expected), re- taining the exponential tail characteristic of intermittent processes, but the highest particle concentration end of the distribution is extended more slowly. Certainly at the order of magnitude level, a particle mass loading ra- tio of 100 times the gas density appears to be as high as preferential concentration can produce. This result could be inferred directly from inspection of the conditioned β distributions of FIG. 3. FIG. 5: (a) Cascade model predictions for a 24 level case, taken to lower probability levels, using 256 realizations of the cascade. Contours are labeled by log(P (S,C)SC). Note the crowding of contours at high C values, indicating the high-C limit of the process under conditions of mass loading.(b) A control cascade to level 24, as in FIG. 5(a), with conditioning turned off. The difference between (a) and (b) clearly shows the “choking” effects of particle mass loading on intermittency in C. IV. SUMMARY A two-phase cascade model for enstrophy and parti- cle concentration in 3-D, isotropic, fully developed tur- bulence with particle loading feedback has been devel- oped and tested. Multiplier distributions for enstrophy and particle concentration were empirically determined from direct numerical simulation fields at Taylor scale Reynolds numbers between 34 and 170. These simula- tions included ‘two-way’ coupling between the phases at global particle/gas mass loadings equal to unity. The shape of all multiplier distributions is well characterized by the β distribution function, with a value of β that depends systematically on the local degree of mass load- ing. The values of β increase monotonically with mass loading and begin to rapidly increase at mass loadings FIG. 6: 1D cuts through the mode of the PDF of FIG. 5(a) parallel to the C axis, showing the effects of uncertainty in the conditioning curve βC(C). The solid curve is the nominal model and the dashed curves are obtained by allowing the parameters p2 and p3 to take their 2σ extreme values. FIG. 7: Cascade model predictions for P (S,C)SC = 5×10−4 for levels 6, 12, 18, 24, 30, and 36. Contour labels indicate the cascade levels. greater than 100. The C-dependent multiplier distributions were used as input to a cascade model that simulates the breakdown, or cascade, of enstrophy S and particle concentration C from large to small spatial scales. The spatial anticorre- lation between enstrophy and particle concentration was empirically determined from 3D DNS models and shown to be constant with Reλ. This constant was used as a correlation probability governing the relative spatial distribution of S and C at each bifurcation step in the cascade model. The cascade model we have developed clearly repro- duces the statistical distributions and spatial correlations observed in our DNS calculations. The cascade parame- ter values we have derived appear to be universal within FIG. 8: (a) 1D global cuts through the cascade model PDFs P (S,C)SC for runs with 18, 24, 30, and 36 levels. (b) closeup of 1-D cuts through high-C regime. the range of Reλ of our simulations. We thus specu- late that they can be used to predict approximate joint probabilities of enstrophy and particle concentration at higher Reynolds numbers, at great savings in computer time. For example, a typical DNS run to Reλ = 170 takes about 170 cpu hours on an Origins 3000 machine, while a cascade model to an equivalent level takes 0.1 cpu hours. We have presented joint probabilites of S and C de- rived from cascade runs up to level 36. The contours shown in FIG. 5(a) and FIG. 6 clearly show the effects of particle mass loading on the probability distribution functions of C in the regimes where C is large. It appears that particle mass loadings greater than 100 are rare in turbulent flows. The properties of the cascade rest on the physics of our DNS simulations, and we speculate that two separate ef- fects are involved. First, particle mass loading dampens fluid motions of all types, decreasing vorticity stretching and all other forms of ongoing eddy bifurcation which are needed to produce intermittency. Second, as a byproduct of this, particle mass loading may alter the Kolmogorov timescale locally and shift the most effectively concen- trated particle Stokes number St to a larger value than that characterizing particles already lying in the local volume, reducing the probability of preferentially con- centrating the local particles any further. Caveats and Future Work: As described in section II, our multiplier distributions were taken from the most numerous cells, with the largest intermittency, which are at the smallest scales possible (furthest from the forcing scale). At Reynolds numbers accessible to DNS, a true inertial range is only beginning to appear, and while, sampling at the smallest spatial scales possible, we are as closely approaching the asymp- totic values within the true inertial range as possible, where level-independence has been demonstrated in the past [21], it is possible that our values are subject to inaccuracy by virtue of being sampled too close to the dissipation scale. Any such inaccuracy will affect our cascade results quantitatively but not qualitatively. As computer power increases, it would be a sensible thing to continue experiments like these at higher Reλ. A more general model that treats enstrophy and strain as independent cascading scalarsmight allow for a higher- fidelity particle concentration cascade, since C is known to be linked to the difference between these two scalars [25] (the so-called second invariant tensor II). However, such an effort would introduce further complexity of its own, as II is no longer positive definite. We consider the development of such a model a suitable task for future work. APPENDIX We used an Eulerian scheme developed by Dr. Alan Wray to solve the coupled set of fluid/particle equations used in this study. This was done to maximize the computational efficiency of the calculations and, more importantly, to accurately evaluate multipliers over the wide range of particle concentrations and enstrophies ex- pected. In this study the effects of particle collisions and external forces on the particles (e.g., gravity) are not con- sidered. The turbulence is spectrally forced at k = such that moments of the Fourier coefficients of the force satisfy isotropy up to the fourth order. The instanta- neous Navier-Stokes equations describing the conserva- tion of mass and momentum for an incompressible fluid ∇ ·U = 0 (A.1) +(U · ∇)U = − +ν∇2U−α (U −V) (A.2) where U is fluid velocity, V is particle velocity, ρf and ρp are the fluid and particle mass densities, ν is fluid viscosity, P is pressure, and α is the inverse of the particle gas drag stopping time τp. The compressible equations for the particles are +∇(ρpV) = D∇2ρp (A.3) ∂(ρpV) +∇(ρpVV) = νp∇2(ρpV)+αρp(U−V) (A.4) where νp is a “particle viscosity”, and D is a “particle diffusivity”. The particle diffusivity and viscosity terms numerically smooth out particle mass and momentum, alleviating the formation of steep gradients of ρp that can lead to numerical instabilities eg. [31]. The right hand sides of Eqs. A.2 and A.4 contain phase coupling terms which are linearly dependent on (U−V). The linear form of the coupling follows from the assump- tions that the particle size is much less than η, and that the material density of the particles is much greater than ρf [2]. Additional contributions to the particle-gas cou- plings involving pressure, viscous and Basset forces [29] have not been added since they are expected to be weak in our size regime of interest. The particle field is intro- duced with a constant mass density and an initial veloc- ity given by the local gas velocity in a field of statisti- cally stationary turbulence. All runs are continued until the particle statistics (RMS of conentration distribution) have equilibrated. The particle Stokes number St is defined relative to the Kolmogorov time scale τη as St = τp/τη, and Φ = Mp/Mf is the global mass loading, whereMp and Mf are the total mass of particles and fluid respectively. In this study ρf , St, and Φ are set to unity, D/ν = 0.01, and νp/ν = 0.1. Explicitly setting St = 1 guarantees that the particles are preferentially concentrated. When Φ is unity, ρp is a surrogate for the local mass loading or local concentration factor C. The values of νp and D minimize the diluting effects of numerical particle diffusion while preventing numerical blowups; their values were deter- mined from a set of DNS runs in which their values were decreased systematically until numerical instabilities set Eqs. A.1 - A.4 are solved using psuedo-spectral meth- ods commonly used to solve Naviers-Stokes equations for a turbulent fluid. The Fast Fourier Transform (FFT) al- gorithm is used to efficiently evaluate the dynamical vari- ables U, V and ρp on a 3D uniform grid of computional nodes with periodic boundary conditions. The computa- tional algorithm is parallelized using MPI and is written in Fortran 90. All runs for this study were executed on SGI Origins supercomputers with up to 1024 processors. Enstrophy is defined as (∂iUj − ∂jUi)2 (A.5) where i, j are summed over the three coordinate dimen- sions of U. The local spatial average of a scalar over a sample vol- ume is estimated as, Fidv (A.6) where Fi is the scalar’s value on computational node i centered within a cube of volume dv and the sum is over all n nodes covering the sample volume. We normalized this average by the global average value to get a quan- tity that measures the scalar’s local value relative to its mean. In this paper C and S will denote normalized spa- tial averages of particle concentration and enstrophy over cubes 3η on a side. ACKNOWLEDGMENTS We are very grateful to Dr. Alan Wray for providing the 3-D code and for useful comments on its use. We thank Robert Last for parallelizing the cascade code on the SGI Origins 3000. We also would like to thank the consultants and support staff at the NAS facility for pro- viding invaluable assistance, and the Science Mission Di- rectorate of NASA for generous grants of computer time. We thank Prof. K. Sreenivasan for several helpful con- versations in the preliminary stages of this project and the internal reviewers Drs. Alan Wray and Denis Richard for their suggestions for improving the manuscript. This research has been made possible by a grant from NASA’s Planetary Geology and Geophysics program. [1] K. D. Squires and J. K. Eaton, Phys. Fluids A 2, 1191 (1990). [2] K. D. Squires and J. K. Eaton, Tech. Rep. MD-55, Stan- ford University (1990). [3] K. D. Squires and J. K. Eaton, Phys. Fluids. A 3, 1159 (1990). [4] J. D. Kulick, J. R. Fessler, and J. K. Eaton, J. Fluid Mech. 227, 109 (1994). [5] B. Castaing, Y. Gagne, and E. J. Hopfinger, Physica D 46, 177 (1990). [6] S. P. G. Dinavahi, K. S. Breuer, and L. Sirovich, Phys. Fluids 7, 1122 (1995). [7] P. Kailasnath, K. R. Sreenivasan, and G. Stolovitzky, Phys. Rev. Lett. 68, 2766 (1992). [8] A. Vincent and M. Meneguzzi, J. Fluid Mech. 225, 1 (1991). [9] A. Pumir, Phys. Fluids 6, 2071 (1994). [10] E. Lamballais, M. Lesieur, and O. Métais, Phys. Rev. E 56, 6761 (1997). [11] J. Jiménez, A. A. Wray, P. G. Saffman, and R. S. Rogallo, J. Fluid Mech. 255, 65 (1993). [12] G. He, S. Chen, R. H. Kraichnan, R. Zhang, and Y. Zhou, Phys. Rev. Lett. 81, 4636 (1998). [13] N. Cao, S. Chen, and K. R. Sreenivasan, Phys. Rev. Lett. 76, 616 (1996). [14] R. C. Hogan, J. N. Cuzzi, and A. R. Dobrovolskis, Phys. Rev. E 60, 1674 (1999). [15] E. Balkovsky, G. Falkovich, and A. Fouxon, Phys. Rev. Lett. 86, 2790 (2001). [16] L. F. Richardson, Weather Prediction by Numerical Pro- cess. (Cambridge University Press, Cambridge U.K., 1922). [17] A. N. Komolgorov, J. Fluid Mech. 13, 82 (1962). [18] T. Chiueh, Chin. J. Phys. 32, 319 (1994). [19] M. Gorokhovski, Tech. Rep., Center for Turbulence Re- search, Annual Research Briefs (2003). [20] C. Meneveau and K. R. Sreenivasan, Phys. Rev. Lett. 59, 1424 (1987). [21] K. R. Sreenivasan and G. Stolovitzky, J. Fluid Mech. 379, 105 (1995). [22] E. A. Novikov, Phys. Fluids A 2, 814 (1990). [23] E. A. Novikov, Phys. Rev. E 50, R3303 (1994). [24] J. Jiménez, J. Fluid Mech. 409, 99 (2000). [25] M. R. Maxey, Phys. Fluids 30, 1915 (1987). [26] K. D. Squires and J. K. Eaton, J. Fluid Mech. 226, 1 (1991). [27] A. M. Ahmed and S. Elghobashi, Phys. Fluids 13, 3346 (2001). [28] J. K. Eaton and J. R. Fessler, Int. J. Multiphase Flow 20, Suppl., 169 (1994). [29] S. Elghobashi and G. C. Truesdell, Phys. Fluids A 5, 1790 (1993). [30] U. Frisch, Turbulence (Cambridge University Press, Cambridge, U.K., 1995), chap. 8. [31] A. Johansen, A. C. Anderson, and A. Brandenburg, As- tron. Astrophys. (2004).

0704.1811

Unifying Evolutionary and Network Dynamics

APS/123-QED Unifying Evolutionary and Network Dynamics Samarth Swarup∗ Department of Computer Science, University of Illinois at Urbana-Champaign Les Gasser Graduate School of Library and Information Science, and Department of Computer Science, University of Illinois at Urbana-Champaign. (Dated: November 17, 2018) Many important real-world networks manifest “small-world” properties such as scale-free degree distributions, small diameters, and clustering. The most common model of growth for these networks is “preferential attachment”, where nodes acquire new links with probability proportional to the number of links they already have. We show that preferential attachment is a special case of the process of molecular evolution. We present a new single-parameter model of network growth that unifies varieties of preferential attachment with the quasispecies equation (which models molecular evolution), and also with the Erdõs-Rényi random graph model. We suggest some properties of evolutionary models that might be applied to the study of networks. We also derive the form of the degree distribution resulting from our algorithm, and we show through simulations that the process also models aspects of network growth. The unification allows mathematical machinery developed for evolutionary dynamics to be applied in the study of network dynamics, and vice versa. PACS numbers: 89.75.Hc, 89.75.Da, 87.23.Kg Keywords: Evolutionary dynamics, Small-world networks, Scale-free networks, Preferential attachment, Quasi-species, Urn models. I. INTRODUCTION The study of networks has become a very active area of research since the discovery of “small-world” networks [1, 2]. Small-world networks are characterized by scale- free degree distributions, small diameters, and high clus- tering coefficients. Many real networks, such as neuronal networks [2], power grids [3], the world wide web [4] and human language [5], have been shown to be small-world. Small-worldness has important consequences. For exam- ple, such networks are found to be resistant to random attacks, but susceptible to targeted attacks, because of the power-law nature of the degree distribution. The process most commonly invoked for the genera- tion of such networks is called “preferential attachment” [6, 7]. Briefly, new links attach preferentially to nodes with more existing links. Simon analyzed this stochas- tic process, and derived the resulting distribution [8]. This simple process has been shown to generate networks with many of the characteristics of small-world networks, and has largely replaced the Erdõs-Rényi random graph model [9] in modeling and simulation work. Another major area of research in recent years has been the consolidation of evolutionary dynamics [10], and its application to alternate areas of research, such as lan- guage [11]. This work rests on the foundation of quasi- species theory [12, 13], which forms the basis of much subsequent mathematical modeling in theoretical biology. ∗Electronic address: [email protected] In this paper we bring together network generation models and evolutionary dynamics models (and partic- ularly quasi-species theory) by showing that they have a common underlying probabilistic model. This unified model relates both processes through a single parameter, called a transfer matrix. The unification allows mathe- matical machinery developed for evolutionary dynamics to be applied in the study of network dynamics, and vice versa. The rest of this paper is organized as follows: first we describe the preferential attachment algorithm and the quasispecies model of evolutionary dynamics. Then we show that we can describe both of these with a single probabilistic model. This is followed by a brief analy- sis, and some simulations, which show that power-law degree distributions can be generated by the model, and that the process can also be used to model some aspects of network growth, such as densification power laws and shrinking diameters. II. PREFERENTIAL ATTACHMENT The Preferential Attachment algorithm specifies a pro- cess of network growth in which the addition of new (in- )links to nodes is random, but biased according to the number of (in-)links the node already has. We identify each node by a unique type i, and let xi indicate the proportion of the total number of links in the graph that is already assigned to node i. Then equation 1 gives the probablity P (i) of adding a new link to node i [6]. P (i) = αx i . (1) http://arxiv.org/abs/0704.1811v1 mailto:[email protected] where α is a normalizing term, and γ is a constant. As γ approaches 0 the preference bias disappears; γ > 1 causes exponentially greater bias from the existing in-degree of the node. III. EVOLUTIONARY DYNAMICS AND QUASISPECIES Evolutionary dynamics describes a population of types (species, for example) undergoing change through repli- cation, mutation, and selection[28]. Suppose there are N possible types, and let si,t denote the number of individ- uals of type i in the population at time t. Each type has a fitness, fi which determines its probability of repro- duction. At each time step, we select, with probability proportional to fitness, one individual for reproduction. Reproduction is noisy, however, and there is a probability qij that an individual of type j will generate an individ- ual of type i. The expected value of the change in the number of individuals of type i at time t is given by, ∆si,t = j fjsjqij j fjsj This is known as the quasispecies equation [13]. The fit- ness, fi, is a constant for each i. Fitness can also be frequency-dependent, i.e. it can depend on which other types are present in the population. In this case the above equation is known as the replicator-mutator equa- tion (RME) [10],[14]. IV. A GENERALIZED POLYA’S URN MODEL THAT DESCRIBES BOTH PROCESSES Urn models have been used to describe both prefer- ential attachment [15], and evolutionary processes [16]. Here we describe an urn process derived from the quasis- pecies equation that also gives a model of network gener- ation. In addition, this model of network generation will be seen to unify the Erdõs-Rényi random graph model [9] with the preferential attachment model. Our urn process is as follows: • We have a set of n urns, which are all initially empty except for one, which has one ball in it. • We add balls one by one, and a ball goes into urn i with probability proportional to fimi, where fi is the “fitness” of urn i, and mi is the number of balls already in urn i. • If the ball is put into urn j, then a ball is taken out of urn j, and moved to urn k with probability qkj . The matrix Q = [qij ], which we call the transfer matrix, is the same as the mutation matrix in the quasispecies equation. This process describes the preferential attachment model if we set the fitness, fi, to be proportional tom where γ is a constant (as in equation 1). Now we get a network generation algorithm in much the same way as Chung et al. did [15], where each ball corresponds to a half-edge, and each urn corresponds to a node. Placing a ball in an urn corresponds to linking to a node, and moving a ball from one urn to another corresponds to rewiring. We call this algorithm Noisy Preferential At- tachment (NPA). If the transfer matrix is set to be the identity matrix, Noisy Preferential Attachment reduces to pure preferential attachment. In the NPA algorithm, just like in the preferential at- tachment algorithm, the probability of linking to a node depends only on the number of in-links to that node. The “from” node for a new edge is chosen uniformly randomly. In keeping with standard practice, the graphs in the next section show only the in-degree distribution. However, since the “from” nodes are chosen uniformly randomly, the total degree distribution has the same form. Consider the case where the transfer matrix is almost diagonal, i.e. qii is close to 1, and the same ∀i, and all the qij are small and equal, ∀i 6= j. Let qii = p and qij = = q, ∀i 6= j. (3) Then, the probability of the new ball being placed in bin P (i) = αm i p+ (1− αm i )q, (4) where α is a normalizing constant. That is, the ball could be placed in bin i with probability αm i and then replaced in bin i with probability p, or it could be placed in some other bin with probability (1 − αm i ), and then trans- ferred to bin i with probability q. Rearranging, we get, P (i) = αm i (p− q) + q. (5) In this case, NPA reduces to preferential attachment with initial attractiveness [17], where the initial attractiveness (q, here) is the same for each node. We can get differ- ent values of initial attractiveness by setting the transfer matrix to be non-uniform. We can get the Erdõs-Rényi model by setting the transfer matrix to be entirely uni- form, i.e. qij = 1/n, ∀i, j. Thus the Erdõs-Rényi model and the preferential attachment model are seen as two ex- tremes of the same process, which differ with the transfer matrix, Q. This process also obviously describes the evolutionary process when γ = 1. In this case, we can assume that at each step we first select a ball from among all the balls in all the urns with probability proportional to the fitness of the ball (assuming that the fitness of a ball is the same as the fitness of the urn in which it is). The probability that we will choose a ball from urn i is proportional to fimi. We then replace this ball and add another ball to the same urn. This is the replication step. This is followed by a mutation step as before, where we choose a ball from the urn and either replace it in the urn with with probability p or move it to any one of the remaining urns. If we assume that all urns (i.e. all types or species) have the same intrinsic fitness, then this process reduces to the preferential attachment process. Having developed the unified NPA model, we can now point towards several concepts in quasi-species theory that are missing from the study of networks, that NPA makes it possible to investigate: • Quasi-species theory assumes a genome, a bit string for example. This allows the use of a distance mea- sure on the space of types. • Mutations are often assumed to be point mutations, i.e. they can flip one bit. This means that a mu- tation cannot result in just any type being intro- duced into the population, only a neighbor of the type that gets mutated. • This leads to the notion of a quasi-species, which is a cloud of mutants that are close to the most-fit type in genome space. • Quasi-species theory also assumes a fitness land- scape. This may in fact be flat, leading to neutral evolution [18]. Another (toy) fitness landscape is the Sharply Peaked Landscape (SPL), which has only one peak and therefore does not suffer from problems of local optima. In general, though, fit- ness landscapes have many peaks, and the rugged- ness of the landscape (and how to evaluate it) is an important concept in evolutionary theory. The notion of (node) fitness is largely missing from net- work theory (with a couple of exceptions: [19], [20]), though the study of networks might benefit greatly from it. • The event of a new type entering the population and “taking over” is known as fixation. This means that the entire population eventually consists of this new type. Typically we speak of gene fixa- tion, i.e. the probability that a single new gene gets incorporated into all genomes present in the population. Fixation can occur due to drift (neu- tral evolution) as well as due to selection. V. ANALYSIS AND SIMULATIONS We next derive the degree distribution of the network. Since there is no “link death” in the NPA algorithm and the number of nodes is finite, the limiting behavior in our model is not the same as that of the preferential attach- ment model (which allows introduction of new nodes). This means that we cannot re-use Simon’s result [8] di- rectly to derive the degree distribution of the network that results from NPA. A. Derivation of the degree distribution Suppose there are N urns and n balls at time t. Let xi,t denote the fraction of urns with i balls at time t. We choose a ball uniformly at random and “replicate” it, i.e. we add a new ball (and replace the chosen ball) into the same urn. Uniformly random choice corresponds to a model where all the urns have equal intrinsic fitness. We follow this up by drawing another ball from this urn and moving it to a uniformly randomly chosen urn (from the N − 1 other urns) with probability q = (1− p)/(N − 1), where p is the probability of putting it back in the same urn. Let P1(i) be the probability that the ball to be replicated is chosen from an urn with i balls. Let P2(i) be the probability that the new ball is placed in an urn with i balls. The net probability that the new ball ends up in an urn with i balls, P (i) = P1(i) and P2(i) or P̄1(i) and P2(i). (6) The probability of selecting a ball from an urn with i balls, P1(i) = Nxi,ti n0 + t where n0 is the number of balls in the urns initially. P2(i) depends on the outcome of the first step. P2(i) = p+ (Nxi,t − 1)q when step 1 is “successful”, Nxi,tq when step 1 is a “failure”. Putting these together, we get, P (i) = Nxi,ti n0 + t (p+ (Nxi,t − 1)q) + Nxi,ti n0 + t Nxi,tq Nxi,ti n0 + t (p− q) +Nxi,tq. Now we calculate the expected value of xi,t+1. xi,t will increase if the ball goes into an urn with i−1 balls. Sim- ilarly it will decrease if the ball ends up in an urn with i balls. Otherwise it will remain unchanged. Remember- ing that xi,t is the fraction of urns with i balls at time t, we write, Nxi,t+1 = Nxi,t + 1 w. p. Nxi−1,t(i−1) (p− q) +Nxi−1,tq, Nxi,t − 1 w. p. Nxi,ti (p− q) +Nxi,tq, Nxi,t otherwise. From this, the expected value of xi,t+1 works out to be, xi,t+1 = i(p− q) n0 + t xi,t+ [(i− 1)(p− q) n0 + t xi−1,t. We can show the approximate solution for xi,t to be, xi,t = ri−1Γ(i) k=1(kr + 1) (t+ 1)(1− q)t−1, (8) 0 200 400 600 800 1000 0.5t(0.99)t t(0.99)t FIG. 1: Example xi,t curves. 1 10 100 r=0.33 FIG. 2: The form of the degree distribution. where r = (p − q)/(1 − q). This approximation is valid while t << N . See Appendix A for details. For any particular i, the shape of this curve is given by t(1− q)t. An example curve is shown in fig 1. This matches our intuition. Initially, xi,t = 0 for i > 1. As t increases, xi,t increases through mutations. However, since N is finite and we keep adding balls, eventually the number of bins with i balls must go to zero for any particular i. Thus xi,t must eventually start decreasing, which is what we see in figure 1. The middle term can be simplified further k=1(kr + 1) ∏i+1/r k=1+1/r ∏i+1/r k=1+1/r Γ(1/r) r2Γ(i + 1+ 1/r) Therefore, in terms of i, equation 8 can be written as (for fixed t), xi = C Γ(i+ 1 + 1 , (9) where C is a constant. This is the form of the degree distribution. This is a power law, because as i → ∞, equation 9 tends to i−(1+1/r) (see discussion of eq. 1.4 in [8, pg 426]). This is also demonstrated in the sample plots in figure 2. These results are confirmed through simulation. We did an experiment where the number of possible nodes was set to 100000, and 10000 links were added. The experiment was repeated for values of p ranging from 0.01 to 0.99, in steps of 0.01. Figure 3 shows a plot of 0 10 20 30 40 50 60 70 80 90 100 p (= 1 - mutation probability) FIG. 3: N = 100000, number of edges = 10000. 1000 10000 1 10 100 1000 10000 Indegree FIG. 4: p = 0.8, N = 100000, number of edges = 10000. coherence, φ, which is defined as, x2i . (10) Coherence is a measure of the non-uniformity of the de- gree distribution. It is 1 when a single node has all the links. When all nodes have one link each, coherence has its lowest value, 1/N . We see that as p increases (i.e. mutation rate decreases), coherence also increases. This is borne out by the degree distribution plots (figures 4 through 6). The degree distribution is steeper for lower values of p. B. Stability We can rewrite equation 2 as ∆si = j fjsj (fisiqii + j 6=i fjsjqij) (11) The first term in the parentheses represents the change in si due to selection. Some of the copies of type i are lost due to mutation. The fraction that are retained are given 1000 10000 1 10 100 1000 10000 Indegree FIG. 5: p = 0.6, N = 100000, number of edges = 10000. 1000 10000 1 10 100 1000 10000 Indegree FIG. 6: p = 0.4, N = 100000, number of edges = 10000. by the product fiqii. If this product is greater than 1, the proportion of type i will increase due to selection, oth- erwise it will decrease. The second term represents the contribution to type i due to mutation from all the other types in the population. Thus, if si decreases towards zero due to a selective disadvantage, it will be maintained in the population at “noise” level due to mutations. This leads to the notion of an error threshold. Sup- pose that the fitness landscape has only one peak. This is known as the Sharply Peaked Landscape, or SPL. Sup- pose further that mutations only alter one position on the genome at a time. Then it can be shown that if the mu- tation rate is small enough the population will be closely clustered about the fittest type. The fittest type keeps getting regenerated due to selection, and mutations gen- erate a cloud of individuals with genomes very close to the genome of the fittest type. This cloud is known as a quasi-species [21]. If, on the other hand, the mutation rate is above a certain threshold (essentially 1/fi, where i is the fittest type) then all types will persist in the population in equal proportions. This threshold is known as the error thresh- VI. FITNESS LANDSCAPES AND NEUTRAL EVOLUTION We have seen above that noisy preferential attachment is equivalent to molecular evolution where all intrinsic fitnesses are equal. If node fitnesses are allowed to be different, we get standard quasi-species behavior. If the mutation rate is low enough, the fittest node dominates the network and acquires nearly all the links. If the mu- tation rate is high enough to be over the error threshold, no single node dominates. Figures 7 and 8 show simulations where nodes are as- signed intrinsic fitness values uniformly randomly in the range (0, 1), for different values of p. We see that when p is high (0.9), i.e. mutation rate is low, the degree distri- bution stretches out along the bottom, and one or a few nodes acquire nearly all the links. When p = 0.4, though, we don’t get this behavior, because the mutation rate is over the error threshold. Since we generally don’t see a single node dominating in real-world networks, we are led to one of two conclu- sions: either mutation rates in real-world networks are 1000 10000 1 10 100 1000 10000 Indegree FIG. 7: p = 0.4, N = 100000, number of edges = 10000, node fitnesses are uniformly randomly distributed between 0 and 1000 1 10 100 1000 10000 Indegree dominant node FIG. 8: p = 0.9, N = 100000, number of edges = 10000, node fitnesses are uniformly randomly distributed between 0 and rather high, or the intrinsic fitnesses of the nodes are all equal. The former seems somewhat untenable. The latter suggests that most networks undergo neutral evo- lution [18]. Fitness landscapes can also be dynamic. Golder and Huberman give examples of short term dynamics in col- laborative tagging systems (in particular Del.icio.us) [22]. Figures 9 and 10, which are taken from their paper, show two instances of the rate at which two different web sites acquired bookmarks. The first one shows a peak right after it appears, before the rate of bookmarking drops to a baseline level. The second instance shows a web site existing for a while before it suddenly shows a peak in the rate of bookmarking. Both are examples of dynamic, i.e. changing, fitness. Wilke et al. have shown that in the case of molecular evolution a rapidly changing fit- FIG. 9: This is figure 6a from [22]. It shows number of book- marks received against time (day number). This particular site acquires a lot of bookmarks almost immediately after it appears, but thereafter receives few bookmarks. FIG. 10: This is figure 6b from [22]. It shows number of bookmarks received against time (day number). This par- ticular site suddenly acquires a lot of bookmarks in a short period of time, though it has existed for a long time. ness landscape is equivalent to the time-averaged fitness landscape [23]. Thus while short term dynamics show peaks in link (or bookmark) acquisition, the long-term dynamics could still be neutral or nearly neutral. VII. DYNAMICAL PROPERTIES OF REAL-WORLD NETWORKS Leskovec et al. point out that though models like pref- erential attachment are good at generating networks that match static “snapshots” of real-world networks, they do not appropriately model how real-world networks change over time [24]. They point out two main properties which are observed for several real-world networks over time: densification power laws, and shrinking diameters. The term densification power law refers to the fact that the number of edges grows super-linearly with respect to the number of nodes in the network. In particular, it grows as a power law. This means that these networks are getting more densely connected over time. The second surprising property of the dynamics of growing real-world networks is that the diameter (or 90th percentile distance, which is called the effective diameter) decreases over time. In most existing models of scale-free network generation, it has been shown that the diameter increases very slowly over time [25]. Leskovec et al. stress the importance of modeling these dynamical aspects of network growth, and they present an alternate algorithm that displays both the above properties. Noisy preferential attachment can also show these properties if we slowly decrease the mutation rate over time. Figures 11 and 12 show the effective diameter of the network and the rate of change of the number of nodes with respect to the number of edges for a simulation in which the mutation rate was changed from 0.3 to 0.01 over the course of the simulation run. 0 20 40 60 80 100 120 140 160 180 200 Number of edges (x100) FIG. 11: The effective diameter of the network when the mu- tation rate decreases over time from 0.3 to 0.01. It increases quickly at first and then decreases slowly over time. 1000 10000 100 1000 10000 100000 Number of edges FIG. 12: The number of nodes grows as a power law with re- spect to the number of edges (or time, since one edge is added at each time step). The slope of the line is approximately 0.86. VIII. CONCLUSIONS We have shown that, when modeled appropriately, the preferential attachment model of network generation can be seen as a special case of the process of molecular evo- lution because they share a common underlying proba- bilistic model. We have presented a new, more general, model of network generation, based on this underlying probabilistic model. Further, this new model of network generation, which we call Noisy Preferential Attachment, unifies the Erdõs-Rényi random graph model with the preferential attachment model. The preferential attachment algorithm assumes that the fitness of a node depends only on the number of links it has. This is not true of most real networks. On the world wide web, for instance, the likelihood of linking to an existing webpage depends also on the content of that webpage. Some websites also experience sudden spurts of popularity, after which they may cease to acquire new links. Thus the probability of acquiring new links de- pends on more than the existing degree. This kind of behavior can be modeled by the Noisy Preferential At- tachment algorithm by including intrinsic fitness values for nodes. The Noisy Preferential Attachment algorithm can also be used to model some dynamical aspects of network growth such as densification power laws and shrinking di- ameters by gradually decreasing mutation rate over time. If true, this brings up the intriguing question of why mu- tation rate would decrease over time in real-world net- works. On the world wide web, for example, this may have to do with better quality information being avail- able through the emergence of improved search engines etc. However, the fact that many different kinds of net- works exhibit densification and shrinking diameters sug- gests that there may be some deeper explanation to be found. From a design point of view, intentional modulation of the mutation rate can provide a useful means of trad- ing off between exploration and exploitation of network structure. We have been exploring this in the context of convergence in a population of artificial language learners [26]. The larger contribution of this work, however, is to bring together the fields of study of networks and evo- lutionary dynamics, and we believe that many further connections can be made. IX. ACKNOWLEDGEMENTS We appreciate the helpful comments of Roberto Al- dunate and Jun Wang. Work supported under NSF Grant IIS-0340996. APPENDIX A Here we solve the difference equation, xi,t+1 = i(p− q) n0 + t xi,t+ [(i− 1)(p− q) n0 + t xi−1,t. x0,t is a special case. Nx0,t+1 = Nx0,t − 1 w. p. Nx0,tq, Nx0,t otherwise. Expanding and simplifying as above, we get, x0,t+1 = (1− q)x0,t. The solution to this difference equation is simply, x0,t = (1− q) tx0,0, (A2) where x0,0 = (N−1)/N is the initial value of the number of empty urns. Note that here, and henceforth, we are assuming that initially all the urns are empty except for one, which has one ball in it. Therefore x1,0 = 1, and xi,0 = 0 ∀i > 1. This also means that n0 = 1. These conditions together specify the entire initial state of the system. Equation A1 is difficult to solve directly, so we shall take the approach of finding the solution to x1,t and x2,t and then simply guessing the solution to xi,t. Substituting i = 1 in equation 7 gives us, x1,t+1 = (p− q) n0 + t x1,t + qx0,t. Substituting the solution for x0,t from equation A2 gives x1,t+1 = (p− q) n0 + t x1,t + q(1 − q) tx0,0. (A3) The complete solution for x1,t is (see Appendix B), x1,t = (1 − q) A(t+ 1) + , (A4) where A = qx0,0 1+p−2q and B = 2(p−q) (1+p−2q)NΓ(1−r) are con- stants. Let us now use this result to derive the solution for x2,t. Substituting i = 2 in equation A1, we get, x2,t+1 = 2(p− q) n0 + t x2,t + [ p− q n0 + t x1,t. Substituting the solution for x1,t from equation A4 and replacing n0 by 1 for convenience gives us, x2,t+1 = 2(p− q) 1 + t x2,t+ (1− q)t A(t+ 1) + ][p− q 1 + t . (A5) The solution to this (after some work) turns out to be (see Appendix B), x2,t = (1− q) A(t+ 1) 2r + 1 q(1− q)t 1 + p− 2q A(t+ 1) 2rt+ t+ 2r 2(2r + 1) (t+ 2) In the above expression, compared to the first term, the remaining terms are negligible. To see this, consider that B/tr can be at most B (as r → 0), and at least B/t (as r → 1). B itself is less than 1/N . Therefore the con- tribution of the second term is upper-bounded by 1/N . A similar observation will hold for D/t2r. This is far less than the contribution due to the first term, since A (which is also close to 1/N) is multiplied by (t+1). The remaining terms are approximately of the form t2/N2 (and higher i will contain higher powers). We can ignore these as long as t << N . Thus, we can write the solution for x2,t approximately as, x2,t = 2r + 1 (t+ 1)(1− q)t 1 + p− 2q N − 1 (t+ 1)(1− q)t (r + 1)(2r + 1) (t+ 1)(1− q)t−1. We can continue on with x3,t: x3,t+1 = 3(p− q) 1 + t x3,t + [2(p− q) 1 + t x2,t. If we follow through with this as for x2,t, we will see the 2 from the constant in the second term ( ) appear as a factor in the first term of the solution for x3,t. In the general expression for the solution, this appears as Γ(i). Therefore, we can guess the approximate expression for xi,t to be, xi,t = ri−1Γ(i) k=1(kr + 1) (t+ 1)(1− q)t−1, (A7) which is the same as equation 8 APPENDIX B Equation A3 is, x1,t+1 = (p− q) n0 + t x1,t + q(1− q) tx0,0. This equation is of the form y(t + 1) = p(t)y(t) + r(t). The general form of the solution is, y(t) = u(t) ∑ r(t) Eu(t) , (B1) where u(t) is the solution of the homogeneous part of the above equation, i.e. u(t + 1) = p(t)u(t), and E is the time-shift operator, i.e. Eu(t) = u(t+ 1). Now, the homogeneous part of equation A3 is, u(t+ 1) = 1− q − n0 + t ( (1− q)t+ (1 − q)n0 − (p− q) n0 + t = (1− q) t+ n0 − t+ n0 u(t). The solution to this difference equation is, u(t) = C(1− q)t Γ(t+ n0 − r) Γ(t+ n0) , (B2) where r = (p − q)/(1 − q), C is a constant, and Γ(·) is the gamma-function, which is a “generalization” of the factorial to the complex plane. It is defined recursively as Γ(n + 1) = nΓ(n). The derivation of equation B2 is given in Appendix C. From equations A3, B1, and B2, we get, x1,t = C(1− q)t Γ(t+ n0 − r) Γ(t+ n0) ∑ qx0,0(1− q) tΓ(t+ 1 + n0) C(1 − q)t+1Γ(t+ 1 + n0 − r) C(1 − q)t (t+ n0 − 1)r [ qx0,0 C(1 − q) (t+ n0) r +D1 (tr is read as “t to the r falling”) q(1− q)t−1x0,0 (t+ n0 − 1)r (t+ n0) r + 1 D(1− q)t (t+ n0 − 1)r (where D = CD1 is another constant) q(1− q)tx0,0 1 + p− 2q Γ(t+ n0 − r) Γ(t+ n0) Γ(t+ n0 + 1) Γ(t+ n0 − r) D(1− q)t (t+ n0 − 1)r q(1− q)tx0,0(t+ n0) 1 + p− 2q D(1 − q)t (t+ n0 − 1)r Let us evaluate the constant by applying the initial con- ditions t = 0, x0,0 = (N−1)/N , x1,0 = 1/N , and n0 = 1. We get, 1 + p− 2q +DΓ(1− r) q(N − 1) 1 + p− 2q +NDΓ(1− r). Therefore, D = 2(p− q) (1 + p− 2q)NΓ(1− r) . (B3) This gives us the complete solution for x1,t as, x1,t = (1 − q) A(t+ 1) + where A = qx0,0 1+p−2q and B = D = 2(p−q) (1+p−2q)NΓ(1−r) constants. This is the same as equation A4. 1. Solution to equation A5 Equation A5 is, x2,t+1 = 2(p− q) 1 + t +(1− q)t A(t+ 1) + ][p− q 1 + t Again, this equation is of the form of equation B1. The solution to the homogeneous part in this case is, u(t) = C(1− q)t Γ(t+ 1− 2(p−q) Γ(t+ 1) . (B4) This is found in exactly the same way as equation B2 (see Appendix B). Now, from equations B1, A5, and B4, we get, x2,t = C(1 − q)t ∑ (1− q)t(A(t+ 1) + B )(p−q C(1− q)t+1 1 (t+1)2r C(1 − q)t C(1− q) A(p− q) (t+ 1)2r (t+ 1)(t+ 1)2r +B(p− q) ∑ (t+ 1)2r tr(t+ 1) ∑ (t+ 1)2r Solving the summations (see Appendix C), we get, x2,t = C(1 − q)t C(1− q) [A(p− q)(t+ 1)2r+1 2r + 1 ( t(t+ 1)2r+1 2r + 1 (t+ 1)2r+2 (2r + 1)(2r + 2) +B(p− q) (t+ 2)t2r (1 + r)tr Simplifying, x2,t = (1− q) [Ar(t + 1) 2r + 1 Aq(t+ 1)(2rt+ t+ 2r) (1 − q)(2r + 1)(2r + 2) Bq(t+ 2) (1− q)(1 + r)tr D(1 − q)t = (1− q)t A(t+ 1) 2r + 1 q(1− q)t 1 + p− 2q A(t+ 1) 2rt+ t+ 2r 2(2r + 1) (t+ 2) This is the same as equation B5. APPENDIX C 1. Derivation of equation B2 Equation B2 is the solution to the following difference equation: u(t+ 1) = (1 − q) t+ n0 − t+ n0 u(t). Note that all the factors in this equation are positive. Taking log, we get, log u(t+ 1) = log (1− q) ( t+ n0 − r t+ n0 + log u(t), ∆log u(t) = log (1− q) ( t+ n0 − r t+ n0 log u(t) = log(1− q) + log(t+ n0 − r) −log(t+ n0) Remembering that a = ta, and log(t+a) = logΓ(t+ a), we get, log u(t) = tlog(1− q) + logΓ(t+ n0 − r) −logΓ(t+ n0) +D, Therefore, u(t) = C(1− q)t Γ(t+ n0 − r) Γ(t+ n0) This is the same as equation B2. 2. Derivation of equation B5 Equation B5 is the solution to the following difference equation: x2,t = C(1 − q)t C(1− q) A(p− q) (t+ 1)2r (t+ 1)(t+ 1)2r +B(p− q) ∑ (t+ 1)2r tr(t+ 1) ∑ (t+ 1)2r We shall solve each of the summations individually. At several points, we will use the summation by parts for- mula, Ey(t)∆z(t) = y(t)z(t)− z(t)∆y(t) . (C1) The first summation term can be obtained directly: (t+ 1)2r = (t+ 1)2r+1 2r + 1 + C1. (C2) The second summation term can be obtained using the summation by parts formula. Let Ey(t) = t + 1. Then y(t) = t, and ∆y(t) = 1. Let ∆z(t) = (t + 1)2r. Then z(t) = (t+1)2r+1 . We get, (t+1)(t+1)2r = (t+ 1)(t+ 1)2r+1 2r + 1 ∑ (t+ 1)2r+1 2r + 1 (t+1)(t+1)2r = (t+ 1)(t+ 1)2r+1 2r + 1 (t+ 1)2r+2 (2r + 1)(2r + 2) Before proceeding, we pause to calculate (1/tr). Note that, (t+ 1)r t+ 1− r (t+ 1)tr (t+ 1)tr Taking summation, we get, Using the summation by parts formula, we get, (1− r)tr We now proceed to the third summation term in the dif- ference equation for x2,t. ∑ (t+ 1)2r tr(t+ 1) ∑ t2r−1 We shall again use the summation by parts formula. Let Ey(t) = t2r−1. Therefore y(t) = (t−1)2r−1, and ∆y(t) = (2r − 1)(t − 1)2r−2. Let ∆z(t) = 1/tr. Therefore z(t) = t/(1− r)tr (from equation C4). We get, ∑ t2r−1 t(t− 1)2r−1 (1− r)tr ∑ 2r − 1 t(t− 1)2r−2 t(t− 1)2r−1 (1− r)tr 2r − 1 ∑ t2r−1 2r − 1 ∑ t2r−1 (t− 1)2r−1 ∑ t2r−1 Therefore, ∑ (t+ 1)2r tr(t+ 1) The fourth summation term in the difference equation for x2,t is similar to the third one. ∑ (t+ 1)2r ∑ (t+ 1)2r tr(t+ 1) (t+ 1) Let Ey(t) = (t + 1). Then y(t) = t, and ∆y(t) = 1. Let ∆z(t) = ∑ (t+1)2r tr(t+1) . 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0704.1812

The LuckyCam Survey for Very Low Mass Binaries II: 13 new M4.5-M6.0 Binaries

Mon. Not. R. Astron. Soc. 000, 1–11 () Printed 2 November 2021 (MN LATEX style file v2.2) The LuckyCam Survey for Very Low Mass Binaries II: 13 new M4.5-M6.0 Binaries⋆ N.M. Law1,2†, S.T. Hodgkin2 and C.D. Mackay2 1Department of Astronomy, Mail Code 105-24, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA 2Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK ABSTRACT We present results from a high-angular-resolution survey of 78 very low mass (VLM) binary systems with 6.0 6 V-K colour 6 7.5 and proper motion > 0.15 arcsec/yr. Twenty-one VLM binaries were detected, 13 of them new discoveries. The new binary systems range in separa- tion between 0.18 arcsec and 1.3 arcsec. The distance-corrected binary fraction is 13.5+6.5 −4 %, in agreement with previous results. Nine of the new binary systems have orbital radii > 10 AU, including a new wide VLM binary with 27 AU projected orbital separation. One of the new systems forms two components of a 2300 AU separation triple system. We find that the orbital radius distribution of the binaries with V-K < 6.5 in this survey appears to be different from that of redder (lower-mass) objects, suggesting a possible rapid change in the orbital radius distribution at around the M5 spectral type. The target sample was also selected to investigate X-ray activity among VLM binaries. There is no detectable correlation between excess X-Ray emission and the frequency and binary properties of the VLM systems. Key words: Binaries: close - Stars: low-mass, brown dwarfs - Instrumentation: high angular resolution - Methods: observational - Techniques: high angular resolution 1 INTRODUCTION Multiple star systems offer a powerful way to constrain the pro- cesses of star formation. The distributions of companion masses, orbital radii and thus binding energies provide important clues to the systems’ formation processes. In addition, binaries provide us with a method of directly determining the masses of the stars in the systems. This is fundamental to the calibration of the mass- luminosity relation (Henry & McCarthy 1993; Henry et al. 1999; Ségransan et al. 2000). A number of recent studies have tested the stellar multiplic- ity fraction of low-mass and very-low-mass (VLM) stars. The fraction of known directly-imaged companions to very-low-mass stars is much lower than that of early M-dwarfs and solar type stars. Around 57% of solar-type stars (F7–G9) have known stellar companions (Abt & Levy 1976; Duquennoy & Mayor 1991), while imaging and radial velocity surveys of early M dwarfs suggest that between 25% & 42% have companions (Henry & McCarthy 1990; Fischer & Marcy 1992; Leinert et al. 1997; Reid & Gizis 1997). For M6–L1 primary spectral types direct imaging studies find bi- nary fractions of only 10–20% (Close et al. 2003; Siegler et al. ⋆ Based on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. † E-mail: [email protected] 2005; Law et al. 2006; Montagnier et al. 2006), and similar binary fractions have been found for still later spectral types (Bouy et al. 2003; Gizis et al. 2003; Burgasser et al. 2003). Recent radial- velocity work has, however, suggested that a large fraction of ap- parently single VLM stars are actually very close doubles, and the VLM multiplicity fraction may thus be comparable to higher mass stars (Jeffries & Maxted 2005; Basri & Reiners 2006). Very low mass M, L and T systems appear to have a tighter and closer distribution of orbital separations, peaking at around 4 AU compared to 30 AU for G dwarfs (Close et al. 2003). How- ever, the relatively few known field VLM binaries limit the sta- tistical analysis of the distribution, in particular for studying the frequency of the rare large-orbital-radii systems which offer strong constraints on some formation theories (eg. Bate & Bonnell 2005; Phan-Bao et al. 2005; Law et al. 2006; Close et al. 2006; Caballero 2007; Artigau et al. 2007). We have been engaged in a programme to image a large and carefully selected sample of VLM stars, targeting separations greater than 1 AU (Law et al. 2005, 2006). The programme has yielded a total of 18 new VLM binary systems, where VLM is de- fined as a primary mass <0.11 M⊙. This paper presents the second of the surveys, targeting field stars in the range M4.5–M6.0. The spectral type range of this survey is designed to probe the transition between the properties of the 30 AU median-radius binaries of the early M-dwarfs and the 4 AU median-radius late M-dwarf binaries. We observed 78 field M-dwarf targets with estimated spec- tral types between M4.5 and M6.0, searching for companions with c© RAS http://arxiv.org/abs/0704.1812v1 2 N.M. Law et al. separations between 0.1 and 2.0 arcsec. The surveyed primary stel- lar masses range from 0.089 M⊙ to 0.11 M⊙ using the models in Baraffe et al. (1998). It has been suggested in Makarov (2002) that F & G field stars detected in the ROSAT Bright Source Catalogue are 2.4 times more likely to be members of wide (> 0.3 arcsec) multiple sys- tems than those not detected in X-Rays. There is also a well-known correlation between activity and stellar rotation rates (eg. Simon 1990; Soderblom et al. 1993; Terndrup et al. 2002). A correlation between binarity and rotation rate would thus be detectable as a correlation between activity and binarity. To test these ideas, we di- vided our targets into two approximately equal numbered samples on the basis of X-ray activity. All observations used LuckyCam, the Cambridge Lucky Imaging system. The system has been demonstrated to reliably achieve diffraction-limited images in I-band on 2.5m telescopes (Law 2007; Law et al. 2006; Mackay et al. 2004; Tubbs et al. 2002; Baldwin et al. 2001). A Lucky Imaging system takes many rapid short-exposure images, typically at 20-30 frames per second. The turbulence statistics are such that a high-quality, near-diffraction- limited frame is recorded a few percent of the time; in Lucky Imag- ing only those frames are aligned and co-added to produce a final high-resolution image. Lucky Imaging is an entirely passive pro- cess, and thus introduces no extra time overheads beyond those re- quired for standard CCD camera observations. The system is thus very well suited to rapid high-angular-resolution surveys of large numbers of targets. In section 2 we describe the survey sample and the X-Ray activity selection. Section 3 describes the observations and their sensitivity. Section 4 describes the properties of the 13 new VLM binaries, and section 5 discusses the results. 2 THE SAMPLE We selected a magnitude and colour limited sample of nearby late M-dwarfs from the LSPM-North High Proper motion catalogue (Lépine & Shara 2005). The LSPM-North catalogue is a survey of the Northern sky for stars with annual proper motions greater than 0.15”/year. Most stars in the catalogue are listed with both 2MASS IR photometry and V-band magnitudes estimated from the photo- graphic BJ and RF bands. The LSPM-North high proper motion cut ensures that all stars are relatively nearby, and thus removes contaminating distant giant stars from the sample. We cut the LSPM catalogue to include only stars with V-K colour >6 and 67.5, and K-magnitude brighter than 10. The colour cut selects approximately M4.5 to M6.0 stars; its effectiveness is confirmed in Law et al. (2006). 2.1 X-ray selection After the colour and magnitude cuts the sample contained 231 late M-dwarfs. We then divide the stars into two target lists on the basis of X-ray activity. We mark a star as X-ray active if the target star has a ROSAT All-Sky Survey detection from the Faint Source Cat- alogue (Voges 2000) or the Bright Source catalogue (Voges 1999) within 1.5× the 1σ uncertainty in the X-ray position. Known or high-probability non-stellar X-Ray associations noted in the QORG catalogue of radio/X-ray sources (Flesch & Hardcastle 2004) are removed. Finally, we manually checked the Digitized Sky Survey (DSS) field around each star to remove those stars which did not show an unambiguous association with the position of the X-ray 4 5 6 7 8 9 10 11 12 2MASS K magnitude Non-activeN X-ray-active 5.5 6 6.5 7 7.5 8 LSPM V-K colour Non-active 16 X-ray-active 0 5 10 15 20 25 30 Photometric Estimated Distance / pc Non-activeN X-ray-active Figure 1. The 2MASS K-magnitude, V-K colour and distance distributions of the X-ray-active and non-X-ray-active samples. Distances are estimated from the LSPM V-K colours of the samples and the V-K photometric ab- solute magnitude relations detailed in Leggett (1992). The distances shown in this figure have a precision of approximately 30%, and assume that all targets are single stars. detection. The completeness and biases of the X-Ray selection are discussed in section 5.2. It should be noted that the fraction of stars which show mag- netic activity (as measured in Hα) reaches nearly 100% at a spectral type of M7, and so the X-ray selection here picks only especially active stars (Gizis et al. 2000; Schmitt & Liefke 2004). However, for convenience, we here denote the stars without ROSAT evidence for X-Ray activity as “non-X-ray active”. One star in the remaining sample, LSPM J0336+3118, is listed as a T-Tauri in the SIMBAD database, and was therefore re- moved from the sample. We note that in the case of the newly de- tected binary LSPM J0610+2234, which is ∼0.7σ away from the ROSAT X-Ray source we associate with it, there is another bright star at 1.5σ distance which may be the source of the X-Ray emis- sion. GJ 376B is known to be a common-proper-motion companion to the G star GJ 376, located at a distance of 134 arcsec (Gizis et al. 2000). Since the separation is very much greater than can detected in the LuckyCam survey, we treat it as a single star in the following analysis. 2.2 Target distributions These cuts left 51 X-ray active stars and 179 stars without evidence for X-Ray activity. We drew roughly equal numbers of stars at ran- dom from these both these lists to form the final observing target set c© RAS, MNRAS 000, 1–11 13 New VLM Binaries 3 LSPM ID Other Name K V-K Est. SpT PM/”/yr LSPM ID Other Name K V-K Est. SpT PM/”/yr LSPM J0023+7711 LHS 1066 9.11 6.06 M4.5 0.839 LSPM J0722+7305 9.44 6.20 M4.5 0.178 LSPM J0035+0233 9.54 6.82 M5.0 0.299 LSPM J0736+0704 G 89-32 7.28 6.01 M4.5 0.383 LSPM J0259+3855 G 134-63 9.52 6.21 M4.5 0.252 LSPM J0738+4925 LHS 5126 9.70 6.34 M4.5 0.497 LSPM J0330+5413 9.28 6.92 M5.0 0.151 LSPM J0738+1829 9.81 6.58 M5.0 0.186 LSPM J0406+7916 G 248-12 9.19 6.43 M4.5 0.485 LSPM J0810+0109 9.74 6.10 M4.5 0.194 LSPM J0408+6910 G 247-12 9.40 6.08 M4.5 0.290 LSPM J0824+2555 9.70 6.10 M4.5 0.233 LSPM J0409+0546 9.74 6.34 M4.5 0.255 LSPM J0825+6902 LHS 246 9.16 6.47 M4.5 1.425 LSPM J0412+3529 9.79 6.25 M4.5 0.184 LSPM J0829+2646 V* DX Cnc 7.26 7.48 M5.5 1.272 LSPM J0414+8215 G 222-2 9.36 6.13 M4.5 0.633 LSPM J0841+5929 LHS 252 8.67 6.51 M5.0 1.311 LSPM J0417+0849 8.18 6.36 M4.5 0.405 LSPM J0849+3936 9.64 6.25 M4.5 0.513 LSPM J0420+8454 9.46 6.10 M4.5 0.279 LSPM J0858+1945 V* EI Cnc 6.89 7.04 M5.5 0.864 LSPM J0422+3900 9.67 6.10 M4.5 0.840 LSPM J0859+2918 LP 312-51 9.84 6.26 M4.5 0.434 LSPM J0439+1615 9.19 7.05 M5.5 0.797 LSPM J0900+2150 8.44 7.76 M6.5 0.782 LSPM J0501+2237 9.23 6.21 M4.5 0.248 LSPM J0929+2558 LHS 269 9.96 6.67 M5.0 1.084 LSPM J0503+2122 NLTT 14406 8.89 6.28 M4.5 0.177 LSPM J0932+2659 GJ 354.1 B 9.47 6.33 M4.5 0.277 LSPM J0546+0025 EM* RJHA 15 9.63 6.50 M4.5 0.309 LSPM J0956+2239 8.72 6.06 M4.5 0.533 LSPM J0602+4951 LHS 1809 8.44 6.20 M4.5 0.863 LSPM J1848+0741 7.91 6.72 M5.0 0.447 LSPM J0604+0741 9.78 6.15 M4.5 0.211 LSPM J2215+6613 7.89 6.02 M4.5 0.208 LSPM J0657+6219 GJ 3417 7.69 6.05 M4.5 0.611 LSPM J2227+5741 NSV 14168 4.78 6.62 M5.0 0.899 LSPM J0706+2624 9.95 6.26 M4.5 0.161 LSPM J2308+0335 9.86 6.18 M4.5 0.281 LSPM J0711+4329 LHS 1901 9.13 6.74 M5.0 0.676 Table 1. The observed non-X-ray-emitting sample. The quoted V & K magnitudes are taken from the LSPM catalogue. K magnitudes are based on 2MASS photometry; the LSPM-North V-band photometry is estimated from photographic BJ and RF magnitudes and is thus approximate only, but is sufficient for spectral type estimation – see section 4.2. Spectral types and distances are estimated from the V & K photometry (compared to SIMBAD spectral types) and the young-disk photometric parallax relations described in Leggett (1992). Spectral types have a precision of approximately 0.5 spectral classes and distances have a precision of ∼30%. LSPM ID Other Name K V-K ST PM/as/yr ROSAT BSC/FSC ID ROSAT CPS LSPM J0045+3347 9.31 6.50 M4.5 0.263 1RXS J004556.3+334718 2.522E-02 LSPM J0115+4702S 9.31 6.04 M4.5 0.186 1RXS J011549.5+470159 4.323E-02 LSPM J0200+1303 6.65 6.06 M4.5 2.088 1RXS J020012.5+130317 1.674E-01 LSPM J0207+6417 8.99 6.25 M4.5 0.283 1RXS J020711.8+641711 8.783E-02 LSPM J0227+5432 9.33 6.05 M4.5 0.167 1RXS J022716.4+543258 2.059E-02 LSPM J0432+0006 9.43 6.37 M4.5 0.183 1RXS J043256.1+000650 1.557E-02 LSPM J0433+2044 8.96 6.47 M4.5 0.589 1RXS J043334.8+204437 9.016E-02 LSPM J0610+2234 9.75 6.68 M5.0 0.166 1RXS J061022.8+223403 8.490E-02 LSPM J0631+4129 8.81 6.34 M4.5 0.212 1RXS J063150.6+412948 4.275E-02 LSPM J0813+7918 LHS 1993 9.13 6.07 M4.5 0.539 1RXS J081346.5+791822 1.404E-02 LSPM J0921+4330 GJ 3554 8.49 6.21 M4.5 0.319 1RXS J092149.3+433019 3.240E-02 LSPM J0953+2056 GJ 3571 8.33 6.15 M4.5 0.535 1RXS J095354.6+205636 2.356E-02 LSPM J0958+0558 9.04 6.17 M4.5 0.197 1RXS J095856.7+055802 2.484E-02 LSPM J1000+3155 GJ 376B 9.27 6.86 M5.0 0.523 1RXS J100050.9+315555 2.383E-01 LSPM J1001+8109 9.41 6.20 M4.5 0.363 1RXS J100121.0+810931 3.321E-02 LSPM J1002+4827 9.01 6.57 M5.0 0.426 1RXS J100249.7+482739 6.655E-02 LSPM J1125+4319 9.47 6.16 M4.5 0.579 1RXS J112502.7+431941 5.058E-02 LSPM J1214+0037 7.54 6.33 M4.5 0.994 1RXS J121417.5+003730 9.834E-02 LSPM J1240+1955 9.69 6.08 M4.5 0.307 1RXS J124041.4+195509 2.895E-02 LSPM J1300+0541 7.66 6.02 M4.5 0.959 1RXS J130034.2+054111 1.400E-01 LSPM J1417+3142 LP 325-15 7.61 6.19 M4.5 0.606 1RXS J141703.1+314249 1.145E-01 LSPM J1419+0254 9.07 6.29 M4.5 0.233 1RXS J141930.4+025430 2.689E-02 LSPM J1422+2352 LP 381-49 9.65 6.38 M4.5 0.248 1RXS J142220.3+235241 2.999E-02 LSPM J1549+7939 G 256-25 8.86 6.11 M4.5 0.251 1RXS J154954.7+793949 2.033E-02 LSPM J1555+3512 8.04 6.02 M4.5 0.277 1RXS J155532.2+351207 1.555E-01 LSPM J1640+6736 GJ 3971 8.95 6.91 M5.0 0.446 1RXS J164020.0+673612 7.059E-02 LSPM J1650+2227 8.31 6.38 M4.5 0.396 1RXS J165057.5+222653 6.277E-02 LSPM J1832+2030 9.76 6.28 M4.5 0.212 1RXS J183203.0+203050 1.634E-01 LSPM J1842+1354 7.55 6.28 M4.5 0.347 1RXS J184244.9+135407 1.315E-01 LSPM J1926+2426 8.73 6.37 M4.5 0.197 1RXS J192601.4+242618 1.938E-02 LSPM J1953+4424 6.85 6.63 M5.0 0.624 1RXS J195354.7+442454 1.982E-01 LSPM J2023+6710 9.17 6.60 M5.0 0.296 1RXS J202318.5+671012 2.561E-02 LSPM J2059+5303 GSC 03952-01062 9.12 6.34 M4.5 0.170 1RXS J205921.6+530330 4.892E-02 LSPM J2117+6402 9.18 6.62 M5.0 0.348 1RXS J211721.8+640241 3.628E-02 LSPM J2322+7847 9.52 6.97 M5.0 0.227 1RXS J232250.1+784749 2.631E-02 LSPM J2327+2710 9.42 6.07 M4.5 0.149 1RXS J232702.1+271039 4.356E-02 LSPM J2341+4410 5.93 6.48 M4.5 1.588 1RXS J234155.0+441047 1.772E-01 Table 2. The observed X-ray emitting sample. The star properties are estimated as described in the caption to table 1. ST is the estimated spectral type; the ROSAT flux is given in units of counts per second. c© RAS, MNRAS 000, 1–11 4 N.M. Law et al. 0 2 4 6 8 10 X-ray active Non-X-ray active Figure 2. The observed samples, plotted in a V/V-K colour-magnitude dia- gram. The background distribution shows all stars in the LSPM-North cat- alogue. Name Ref. GJ 3417 Henry et al. (1999) G 89-32B Henry et al. (1997) V* EI Cnc Gliese & Jahreiß (1991) LP 595-21 Luyten (1997) GJ 1245 McCarthy et al. (1988) GJ 3928 McCarthy et al. (2001) GJ 3839 Delfosse et al. (1999) LHS 1901 Montagnier et al. (2006) Table 3. The previously known binaries which were re-detected by Lucky- Cam in this survey. of 37 X-Ray active stars and 41 non-X-ray active stars (described in tables 1 and 2). Four of the X-Ray active stars and 4 of the non- X-ray stars were previously known to be binary systems (detailed in table 3), but were reimaged with LuckyCam to ensure a uniform survey sensitivity in both angular resolution and detectable com- panion contrast ratio. Figure 1 shows the survey targets’ distributions in K magni- tude, V-K colour and photometrically estimated distance. Figure 2 compares the targets to the rest of the stars in the LSPM catalogue. The X-ray and non-X-ray samples are very similar, although the non-X-ray sample has a slightly higher median distance, at 15.4pc rather than 12.2pc (the errors on the distance determination are about 30%). 3 OBSERVATIONS We imaged all 78 targets in a total of 11 hours of on-sky time in June and November 2005, using LuckyCam on the 2.56m Nordic Optical Telescope. Each target was observed for 100 seconds in both i’ and the z’ filters. Most of the observations were performed through varying cloud cover with a median extinction on the order of three magnitudes. This did not significantly affect the imaging performance, as all these stars are 3-4 magnitudes brighter than the LuckyCam guide star requirements, but the sensitivity to faint objects was reduced and no calibrated photometry was attempted. 3.1 Binary detection and photometry Companions were detected according to the criteria described in detail in Law et al. (2006). We required 10σ detections above both photon and speckle noise; the detections must appear in both i’ and z’ images. Detection is confirmed by comparison with point spread function (PSF) reference stars imaged before and after each target. In this case, because the observed binary fraction is only ∼30%, other survey sample stars serve as PSF references. We measured resolved photometry of each binary system by the fitting and sub- traction of two identical PSFs to each image, modelled as Moffat functions with an additional diffraction-limited core. 3.2 Sensitivity The sensitivity of the survey was limited by the cloud cover. Be- cause of the difficulty of flux calibration under very variable extinc- tion conditions we do not give an overall survey sensitivity. How- ever, a minimum sensitivity can be estimated. LuckyCam requires an i’=+15.5m guide star to provide good correction; all stars in this survey must appear to be at least that bright during the obser- vations1. The sensitivity of the survey around a i=+15.5m star is calculated in Law et al. (2006) and the sensitivity as a function of companion separation is discussed in section 5.4. The survey is also sensitive to white dwarf companions around all stars in the sample. However, until calibrated resolved photom- etry or spectroscopy is obtained for the systems it is not possi- ble to distinguish between M-dwarf and white-dwarf companions. Since a large sample of very close M-dwarf companions to white dwarf primaries have been found spectroscopically (for example, Delfosse et al. 1999; Raymond et al. 2003), but very few have been resolved, it is unlikely that the companions are white dwarfs. It will, however, be of interest to further constrain the frequency of white-dwarf M-dwarf systems. 4 RESULTS & ANALYSIS We found 13 new very low mass binaries. The binaries are shown in figure 3 and the observed properties of the systems are detailed in table 4. In addition to the new discoveries, we also confirmed eight previously known binaries, detailed in tables 3 and 4. 4.1 Confirmation of physical association Seven of the newly discovered binaries have moved more than one DSS PSF-radius between the acquisition of DSS images and these observations (table 5). With a limiting magnitude of iN ∼ +20.3m (Gal et al. 2004), the DSS images are deep enough for clear de- tection of all the companions found here, should they actually be stationary background objects. None of the DSS images show an object at the present position of the detected proposed companion, confirming the common proper motions of these companions with their primaries. The other binaries require a probabilistic assessment. In the entire LuckyCam VLM binary survey, covering a total area of (22′′ × 14.4′′) × 122 fields, there are 10 objects which would have 1 LSPM J2023+6710 was observed though ∼5 magnitudes of cloud, much more than any other target in the survey, and was too therefore faint for good performance Lucky Imaging. However, its bright companion is at 0.9 arcsec separation and so was easily detected. c© RAS, MNRAS 000, 1–11 13 New VLM Binaries 5 LSPM J0035+0233 0.25" LSPM J0409+0546 NLTT 14406 0.26" LSPM J0610+2234 0.18" LHS 1901 0.26" LHS 5126 LP 312-51 0.26" LSPM J0045+3347 0.27" LSPM J0115+4702 0.68" LSPM J0227+5432 0.90" G 134-63 GJ 3554 0.90" LSPM J2023+6710 1.30" LSPM J1832+2030 Figure 3. The newly discovered binaries. All images are orientated with North up and East to the left. The images are the results of a Lucky Imaging selection of the best 10% of the frames taken in i’, with the following exceptions: LSPM J0409+0546, LSPM J0610+2234 and LP 312-51 are presented in the z’ band, as the cloud extinction was very large during their i’ observations. The image of LSPM LHS 5126 uses the best 2% of the frames taken and LSPM J0115+4702S uses the best 1%, to improve the light concentration of the secondary. LSPM J2023+6710 was observed through more than 5 magnitudes of cloud extinction, and was thus too faint for Lucky Imaging; a summed image with each frame centroid-centred is presented here, showing clear binarity. LHS 1901 was independently found by Montagnier et al. (2006) during a similar M-dwarf survey. We present our image here to confirm its binarity. been detected as companions if they had happened to be close to the target star. One of the detected objects is a known wide common proper motion companion; others are due to random alignments. For the purposes of this calculation we assume that all detected widely separated objects are not physically associated with the tar- get stars. Limiting the detection radius to 2 arcsec around the target star (we confirm wider binaries by testing for common proper motion against DSS images) 0.026 random alignments would be expected in our dataset. This corresponds to a probability of only 2.5 per cent that one or more of the apparent binaries detected here is a chance alignment of the stars. We thus conclude that all the detected binaries are physically associated systems. 4.2 Constraints on the nature of the target stars Clouds unfortunately prevented calibrated resolved photometry for the VLM systems. However, unresolved V & K-band photometry listed in the LSPM survey gives useful constraints on the spectral types of the targets. About one third of the sample has a listed spec- tral type in the SIMBAD database (from Jaschek 1978). To obtain estimated spectral types for the VLM binary systems, we fit the LSPM V-K colours to those spectral types. The fit has a 1σ un- certainty of ∼0.5 spectral types. The colour-magnitude relations in Leggett (1992) show the unresolved system colour is dominated by the primary for all M2–M9 combinations of primary and sec- ondary. We then estimate the secondaries’ spectral types by: 1/ as- suming the estimated primary spectral type to be the true value and 2/ using the spectral type vs. i’ and z’ absolute magnitude relations in Hawley et al. (2002) to estimate the difference in spectral types between the primary and secondary. This procedure gives useful constraints on the nature of the systems, although resolved spec- troscopy is required for definitive determinations. c© RAS, MNRAS 000, 1–11 6 N.M. Law et al. Name ∆i′ ∆z′ Sep. (arcsec) P.A. (deg) Epoch X-ray emitter? LSPM J0035+0233 1.30 ± 0.30 · · · 0.446 ± 0.01 14.3 ± 1.4 2005.9 LSPM J0409+0546 < 1.5 < 1.5 0.247 ± 0.01 40.0 ± 3.2 2005.9 NLTT 14406 1.30 ± 0.30 0.77 ± 0.30 0.310 ± 0.01 351.6 ± 1.1 2005.9 LSPM J0610+2234 < 1.0 < 1.0 0.255 ± 0.01 268.4 ± 2.7 2005.9 * LHS 5126 0.50 ± 0.20 0.50 ± 0.30 0.256 ± 0.02 235.1 ± 3.4 2005.9 LP 312-51 0.74 ± 0.10 0.51 ± 0.10 0.716 ± 0.01 120.5 ± 1.1 2005.9 LSPM J0045+3347 0.80 ± 0.35 0.77 ± 0.35 0.262 ± 0.01 37.6 ± 1.9 2005.9 * LSPM J0115+4702S 0.55 ± 0.25 0.73 ± 0.25 0.272 ± 0.01 249.8 ± 1.3 2005.9 * LSPM J0227+5432 0.60 ± 0.10 0.59 ± 0.10 0.677 ± 0.01 275.8 ± 1.1 2005.9 * G 134-63 1.55 ± 0.10 1.35 ± 0.10 0.897 ± 0.01 13.6 ± 1.1 2005.9 GJ 3554 0.51 ± 0.20 0.57 ± 0.20 0.579 ± 0.01 44.0 ± 1.1 2005.9 * LSPM J2023+6710 0.55 ± 0.20 · · · 0.900 ± 0.15 232.5 ± 3.2 2005.9 * LSPM J1832+2030 0.48 ± 0.10 0.45 ± 0.10 1.303 ± 0.01 20.6 ± 1.1 2005.4 * GJ 3417 1.66 ± 0.10 1.42 ± 0.10 1.526 ± 0.01 −39.8 ± 1.0 2005.9 LHS 1901 1.30 ± 0.70 1.30 ± 0.70 0.177 ± 0.01 51.4 ± 1.6 2005.9 G 89-32 0.43 ± 0.10 0.38 ± 0.10 0.898 ± 0.01 61.3 ± 1.0 2005.9 V* EI Cnc 0.62 ± 0.10 0.49 ± 0.10 1.391 ± 0.01 76.6 ± 1.0 2005.9 LP 595-21 0.74 ± 0.10 0.60 ± 0.10 4.664 ± 0.01 80.9 ± 1.0 2005.9 * GJ 1245AC 2.95 ± 0.20 2.16 ± 0.20 1.010 ± 0.01 −11.3 ± 1.0 2005.4 * GJ 3928 2.32 ± 0.20 2.21 ± 0.20 1.556 ± 0.01 −10.7 ± 1.0 2005.4 * LP 325-15 0.36 ± 0.10 0.33 ± 0.10 0.694 ± 0.01 −21.5 ± 1.0 2005.4 * Table 4. The observed properties of the detected binaries. The top group are stars with newly detected companions; the bottom group are the previously known systems. LSPM J0409+0546 and LSPM J0610+2234 were observed though thick cloud and in poor seeing, and so only upper limits on the contrast ratio are given. LSPM J2023+6710 was not observed in z’, and cloud prevented useful z’ observations of LSPM J0035+0233. LSPM ID Years since DSS obs. Dist. moved 1RXS J004556.3+334718 16.2 4.3” G 134-63 16.2 4.1” NLTT 14406 19.1 3.4” LHS 5126 6.8 3.4” LP 312-51 7.6 3.3” GJ 3554 15.8 5.0” LSPM J2023+6710 14.2 4.2” Table 5. The newly discovered binaries which have moved far enough since DSS observations to allow confirmation of the common proper motion of their companions. 4.3 Distances The measurement of the distances to the detected binaries is a vital step in the determination of the orbital radii of the systems. None of the newly discovered binaries presented here has a measured par- allax (although four2 of the previously known systems do) and cal- 2 G 132-25 (NLTT 2511) is listed in Reid & Cruz (2002) and the SIM- BAD database as having a trigonometric parallax of 14.7 ± 4.0 mas, based on the Yale General Catalogue of Trigonometric Stellar Parallaxes (van Altena et al. 2001). However, this appears to be a misidentification, as the star is not listed in the Yale catalogue. The closest star listed, which does have the parallax stated for G 132-25 in Reid & Cruz (2002), is LP 294-2 (NLTT 2532). This star has a very different proper motion speed and direction to G 132-25 (0.886 arcsec/yr vs. 0.258 arcsec/yr in the LSPM cat- alogue & SIMBAD). In addition, the G 132-25 LSPM V and K photometry is inconsistent with that of an M-dwarf at a distance of 68pc. We thus do not use the stated parallax for G 132-25. ibrated resolved photometry is not available for almost all the sys- tems. We therefore calculate distances to the newly discovered sys- tems using the V-K colour-absolute magnitude relations described in Leggett (1992). Calculation of the distances in this manner re- quires care, as the V and K-band photometry is unresolved, and so two luminous bodies contribute to the observed colours and mag- nitudes. The estimated distances to the systems, and the resulting or- bital separations, are given in table 6. The stated 1σ distance ranges include the following contributions: • A 0.6 magnitude Gaussian-distributed uncertainty in the V-K colour of the system (a combination of the colour uncertainty noted in the LSPM catalogue and the maximum change in the V-K colour of the primary induced by a close companion). • A 0.3 magnitude Gaussian-distributed uncertainty in the abso- lute K-band magnitude of the system, from the uncertainty in the colour-absolute magnitude relations from Leggett 1992. • A 0.75 magnitude flat-distributed uncertainty in the absolute K-band magnitude of the system, to account for the unknown K- band contrast ratio of the binary system. The resulting distances have 1σ errors of approximately 35%, with a tail towards larger distances due to the K-band contrast ratio uncertainties. 4.4 NLTT 14406 – A Newly Discovered Triple System We found NLTT 14406 to have a 0.31 arcsec separation companion. NLTT 14406 is identified with LP 359-186 in the NLTT catalogue (Luyten 1995), although it is not listed in the revised NLTT cat- alogue (Salim & Gould 2003). LP 359-186 is a component of the common-proper-motion (CPM) binary LDS 6160 (Luyten 1997), with the primary being LP 359-216 (NLTT 14412), 167 arcsec dis- tant and listed in the SIMBAD database as a M2.5 dwarf. c© RAS, MNRAS 000, 1–11 13 New VLM Binaries 7 Name Parallax / mas Distance / pc Orbital rad. / AU Prim. ST (est.) Sec. ST (est.) LSPM J0035+0233 · · · 14.5+6.3 −2.4 6.8 −1.0 M5.0 M6.0 LSPM J0409+0546 · · · 19.9+9.1 −3.8 4.9 −0.7 M4.5 6M6.0 NLTT 14406 · · · 13.7+6.5 4.4+2.3 −0.7 M4.5 M5.5 LSPM J0610+2234 · · · 17.0+7.5 −2.9 4.6 −0.8 M5.0 6M6.0 LHS 5126 · · · 19.5+8.9 −3.7 4.9 −0.6 M4.5 M5.0 LP 312-51 · · · 21.5+10.1 −4.0 16.1 −2.7 M4.5 M5.0 LSPM J0045+3347 · · · 14.9+7.0 −2.6 4.0 −0.6 M4.5 M5.5 LSPM J0115+4702S · · · 18.7+9.3 −3.6 5.2 −0.9 M4.5 M5.0 LSPM J0227+5432 · · · 18.6+9.5 −3.4 13.2 −2.2 M4.5 M5.0 G 134-63 · · · 18.8+9.3 −3.4 17.6 −2.8 M4.5 M5.5 GJ 3554 · · · 11.8+5.6 −2.2 7.1 −1.2 M4.5 M4.5 LSPM J2023+6710 · · · 13.6+5.9 −2.5 12.8 −2.6 M5.0 M5.0 LSPM J1832+2030 · · · 20.6+9.6 −3.9 27.0 +14.6 −4.0 M4.5 M5.0 GJ 3417 87.4 ± 2.3 11.4+0.3 −0.3 17.5 −0.5 M4.5 M6.5 G 89-32 · · · 7.3+3.9 −1.3 6.5 −1.1 M4.5 M5.0 LHS 1901 · · · 12.3+5.6 −2.0 2.3 −0.4 M4.5 M6.0 V* EI Cnc 191.2 ± 2.5 5.23+0.07 −0.07 7.27 +0.11 −0.11 M5.5 M6.0 LP 595-21 · · · 16.5+8.2 −2.7 76.2 +38.7 −11.8 M4.5 M5.5 GJ 1245AC 220.2 ± 1.0 4.54+0.02 −0.02 4.6 +0.05 −0.05 M5.0 M8.5 GJ 3928 · · · 10.2+5.6 −1.7 15.7 −2.5 M4.5 M6.5 LP 325-15 62.2 ± 13.0 16.1+3.4 −3.4 11.2 −2.4 M4.5 M4.5 Table 6. The derived properties of the binary systems. The top group are stars with newly detected companions; the bottom group are the previously known binaries. All parallaxes are from the Yale General Catalogue of Trigonometric Stellar Parallaxes (van Altena et al. 2001). Distances and orbital radii are estimated as noted in the text; the stated errors are 1σ. The primaries’ spectral types have a 1σ uncertainty of ∼0.5 subtypes (section 4.2); the difference in spectral types is accurate to ∼0.5 spectral subtypes. The identification of these high proper motion stars can be occasionally problematic when working over long time baselines. As a confirmatory check, the LSPM-listed proper motion speeds and directions of these candidate CPM stars agree to within 1σ (using the stated LSPM proper motion errors). In the LSPM catalogue, the two stars are separated by 166.3 arcsec at the J2000.0 epoch. We thus identify our newly discovered 4.4 AU separation companion to NLTT 14406 as a member of a triple system with an M2.5 primary located at 22801080 −420 AU separa- tion. 5 DISCUSSION 5.1 The binary frequency of stars in this survey We detected 13 new binaries in a sample of 78 VLM stars, as well as a further 8 previously known binaries. This corresponds to a bi- nary fraction of 26.9+5.5 −4.4%, assuming Poisson errors. However, the binaries in our sample are brighter on average than single stars of the same colour and so were selected from a larger volume. Cor- recting for this, assuming a range of contrast ratio distributions be- tween all binaries being equal magnitude and all constrast ratios being equally likely (Burgasser et al. 2003), we find a distance- corrected binary fraction of 13.5+6.5 −4 %. However, because the binaries are more distant on average than the single stars in this survey, they also have a lower aver- age proper motion. The LSPM proper motion cut will thus pref- erentially remove binaries from a sample which is purely selected on magnitude and colour. The above correction factor for the in- creased magnitude of binary systems does not include this effect, and so will underestimate the true binary fraction of the survey. 5.2 Biases in the X-ray sample Before testing for correlations between X-ray emission and binary parameters, it is important to assess the biases introduced in the selection of the X-ray sample. The X-ray flux assignment criteria described in section 2.1 are conservative. To reduce false associa- tions, the X-ray source must appear within 1.5σ of the candidate star, which implies that ∼13% of true associations are rejected. The requirement for an unambiguous association will also reject some fraction of actual X-ray emitters (10% of the candidate emitting systems were rejected on this basis). The non-X-ray emitting sam- ple will thus contain some systems that do actually meet the X-ray flux-emitting limit. The X-ray source detection itself, which cuts only on the de- tection limit in the ROSAT Faint Source catalogue, is biased both towards some sky regions (the ROSAT All-Sky Survey does not have uniform exposure time (Voges 1999)) and towards closer stars. However, these biases have only a small effect: all but three of the target stars fall within the relatively constant-exposure area of the ROSAT survey, where the brightness-cutoff is constant to within c© RAS, MNRAS 000, 1–11 8 N.M. Law et al. 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 Figure 4. The fraction of stellar luminosity which appears as X-Ray emis- sion. Empty circles denote single stars; filled circles denote the binaries detected in this survey. No binarity correction is made to either the X-Ray flux or K-magnitude. The two high points are likely to be due to flaring. about 50%. The samples also do not show a large bias in distance – the X-ray stars’ median distance is only about 10% smaller than that of the non-X-ray sample (figure 1). Finally, the X-Ray active stars also represent a different stel- lar population from the non-active sample. In particular, the X-ray active stars are more likely to be young (eg. Jeffries (1999) and ref- erences therein). It may thus be difficult to disentangle the biases introduced by selecting young stars from those intrinsic to the pop- ulation of X-ray emitting older stars. As the results below show, there are no detectable correlations between binarity and X-ray emission. If correlations are detected in larger samples, constraints on the ages of the targets would have to be found to investigate the causes of the correlations. 5.3 Is X-ray activity an indicator of binarity? 11 of the 21 detected binaries are X-ray active. The non-distance- corrected binary fraction of X-Ray active targets in our survey is thus 30+8 −6%, and that of non-X-ray-active targets is 24 −5%. X-Ray activity therefore does not appear to be an indicator of binarity. The fraction of the X-ray target’s bolometric luminosity which is in the X-Ray emission (Lx/Lbol) is shown in figure 4, and again no correlation with binarity is found. The two targets with very large Lx/Lbol (GJ 376B and LSPM J1832+2030) are listed as flar- ing sources in Fuhrmeister & Schmitt (2003) and thus were prob- ably observed during flare events (although Gizis et al. (2000) ar- gues that GJ 376B is simply very active). This contrasts with the 2.4 times higher binarity among the similarly-selected sample of F & G type X-ray active stars in Makarov (2002). However, the binary fractions themselves are very similar, with a 31% binary fraction among X-ray active F & G stars, compared with 13% for X-ray mute F & G stars. Since the fraction of stars showing X-Ray activity increases towards later types, it is possible that the Makarov sample preferentially selects systems containing an X-ray emitting late M-dwarf. However, most of the stellar components detected in Makarov (2002) are F & G types. The much longer spin-down timescales of late M-dwarfs, in combination with the rotation-activity paradigm, may ex- plain the lack of activity-binarity correlation in late M-dwarfs. Delfosse et al. (1998) show that young disk M dwarfs with spectral types later than around M3 are still relatively rapidly rotating (with vsini’s up to 40 km/s and even 60 km/s in one case), while earlier spectral types do not have detectable rotation periods to the limit of their sensitivity (around 2 km/s). Indeed solar type stars spin down 0 10 20 30 40 50 Separation / AU Figure 5. The i-band contrast ratios of the detected binaries, plotted as a function of binary separation in AU. For reasons of clarity, the 76AU binary and the contrast ratio errorbars (table 4) have been omitted. Filled circles are X-ray emitters. 0 10 20 30 40 50 60 70 80 90 100 Separation / AU Figure 6. The histogram distribution of the orbital radii of the detected binaries in the sample. on relatively short timescales, for example in the 200 Myr old open cluster M34 Irwin et al. (2006) find that the majority of solar type stars have spun down to periods of around 7 days (Vrot ∼ 7 km/sec). The M-dwarfs thus have a high probability of fast rotation and thus activity even when single, which could wash-out any obvious bina- rity correlation with X-ray activity. 5.4 Contrast ratios In common with previous surveys, the new systems have low con- trast ratios. All but two of the detected systems have contrast ratios <1.7 mags. This is well above the survey sensitivity limits, as illus- trated by the two binaries detected at much larger contrast ratios. Although those two systems are at larger radii, they would have been detected around most targets in the survey at as close as 0.2- 0.3 arcsec. It is difficult to obtain good constraints on the mass contrast ratio for these systems because of the lack of calibrated photome- try, and so we leave the determination of the individual component masses for future work. However, we note that an interesting fea- ture of the sample is that no binaries with contrast ratios consistent with equal mass stars are detected. There is no obvious correlation between the orbital radius and the i-band contrast ratios, nor between X-ray emission and the con- trast ratios (figure 5). 5.5 The distribution of orbital radii Early M-dwarfs and G-dwarfs binaries have a broad or- bital radius peak of around 30 AU (Fischer & Marcy 1992; Duquennoy & Mayor 1991), while late M-dwarfs have a peak at around 4 AU (eg. Close et al. 2005). Our survey covers a narrow c© RAS, MNRAS 000, 1–11 13 New VLM Binaries 9 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 V-K colour Figure 7. Orbital radius in the detected binaries as a function of colour. V- K=6 corresponds approximately to M4.5, and V-K=7 to M5.5. Filled circles are X-ray emitters. For clarity, the ∼0.3 mags horizontal error bars have been omitted. There is no obvious correlation between X-ray emission and orbital radius. (0.02M⊙) mass range in the region between the two populations and so allows us to test the rapidity of the transition in binary prop- erties. The orbital radius distribution derived in this survey (figure 6) replicates the previously known VLM-star 4 AU orbital radius peak. However, 9 of the 21 systems are at a projected separation of more than 10 AU. These wide VLM binaries are known to be rare – for example, in the 36 M6-M7.5 M-dwarf sample of Siegler et al. (2005) 5 binaries are detected but none are found to be wider than 10 AU. To test for a rapid transition between the low-mass and very- low-mass binary properties in the mass range covered by our sur- vey, we supplemented the V-K > 6.5 systems from the LuckyCam sample with the known VLM binaries from the Very Low Mass Bi- naries archive3 (which, due to a different mass cut, all have a lower system mass than the LuckyCam sample). To reduce selection ef- fects from the instrumental resolution cut-offs we only considered VLM binaries with orbital radius > 3.0 AU. The resulting cumulative probability distributions are shown in figure 8. There is a deficit in wider systems in the redder sample compared to the more massive, bluer systems. A K-S test between the two orbital radius distributions gives an 8% probability that they are derived from the same underlying distribution. This suggests a possibly rapid change in the incidence of systems with orbital radii > 10AU, at around the M5-M5.5 spectral type. However, confirma- tion of the rapid change will require a larger number of binaries and a more precise mass determination for each system. 5.6 The LuckyCam surveys in the context of formation mechanisms VLM star formation is currently usually modelled as fragmentation of the initial molecular cloud core followed by ejection of the low mass stellar embryos before mass accretion has completed – the ejection hypothesis (Reipurth & Clarke 2001). Multiple systems formed by fragmentation are limited to be no smaller than 10AU by the opacity limit (eg. Boss 1988), although closer binaries can 3 collated by Nick Siegler; VLM there is defined at the slightly lower cutoff of total system mass of < 0.2M⊙ 0 10 20 30 40 50 Orbital radius / AU V-K < 6.5 V-K > 6.5 Figure 8. The cumulative distribution of orbital radii of the detected bina- ries in the sample with V-K < 6.5 (dashed line). The solid line shows those with V-K > 6.5, with the addition of the full sample of known VLM binaries with total system masses < 0.2M⊙, collated by Siegler. Neither distribution reaches a fraction of 1.0 because of a small number of binaries wider than 50 AU. be formed by dynamical interactions and orbital decay (Bate et al. 2002). The ejection hypothesis predicted binary frequency is about 8% (Bate & Bonnell 2005); few very close (< 3AU) binaries are expected (Umbreit et al. 2005) without appealing to orbital decay. Few wide binaries with low binding energies are expected to sur- vive the ejection, although recent models produce some systems wider than 20AU when two stars are ejected simultaneously in the same direction (Bate & Bonnell 2005). The standard ejecton hy- pothesis orbital radius distribution is thus rather tight and centered at about 3-5 AU, although its width can be enlarged by appealing to the above additional effects. The LuckyCam VLM binary surveys (this work and Law et al. 2006) found several wide binary systems, with 11 of the 24 de- tected systems at more than 10 AU orbital radius and 3 at more than 20 AU. With the latest models, the ejection hypothesis can- not be ruled out by these observations, and indeed (as suggested in Bate & Bonnell 2005) the frequency of wider systems will be very useful for constraining more sophisticated models capable of predicting the frequency in detail. The observed distance-bias- corrected binary frequency in the LuckyCam survey is consistent with the ejection hypothesis models, but may be inconsistent when the number of very close binaries undetected in the surveys is taken into account (Maxted & Jeffries 2005; Jeffries & Maxted 2005). For fragmentation to reproduce the observed orbital radius dis- tribution, including the likely number of very close systems, dy- namical interactions and orbital decay must be very important pro- cesses. However, SPH models also predict very low numbers of close binaries. An alternate mechanism for the production of the closest binaries is thus required (Jeffries & Maxted 2005), as well as modelling constraints to test against the observed numbers of wider binaries. Radial velocity observations of the LuckyCam sam- ples to test for much closer systems would offer a very useful in- sight into the full orbital radius distribution that must be reproduced by the models. 6 CONCLUSIONS We found 21 very low mass binary systems in a 78 star sample, including one close binary in a 2300 AU wide triple system and c© RAS, MNRAS 000, 1–11 10 N.M. Law et al. one VLM system with a 27 AU orbital radius. 13 of the binary sys- tems are new discoveries. All of the new systems are significantly fainter than the previously known close systems in the sample. The distance-corrected binary fraction is 13.5+6.5 −4 %, in agreement with previous results. There is no detectable correlation between X-Ray emission and binarity. The orbital radius distribution of the binaries appears to show characteristics of both the late and early M-dwarf distributions, with 9 systems having an orbital radius of more than 10 AU. We find that the orbital radius distribution of the binaries with V-K < 6.5 in this survey appears to be different from that of lower-mass objects, suggesting a possible sharp cutoff in the num- ber of binaries wider than 10 AU at about the M5 spectral type. ACKNOWLEDGEMENTS The authors would like to particularly thank the staff members at the Nordic Optical Telescope. We would also like to thank John Baldwin and Peter Warner for many helpful discussions. NML ac- knowledges support from the UK Particle Physics and Astronomy Research Council (PPARC). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. We also made use of NASA’s Astrophysics Data System Bibliographic Ser- vices. 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0704.1813

The Discovery of a Companion to the Lowest Mass White Dwarf

The Discovery of a Companion to the Lowest Mass White Dwarf1 Mukremin Kilic2, Warren R. Brown3, Carlos Allende Prieto4, M. H. Pinsonneault2, and S. J. Kenyon3 ABSTRACT We report the detection of a radial velocity companion to SDSS J091709.55+463821.8, the lowest mass white dwarf currently known with M ∼ 0.17M⊙. The radial velocity of the white dwarf shows variations with a semi- amplitude of 148.8 ± 6.9 km s−1 and a period of 7.5936 ± 0.0024 hours, which implies a companion mass of M ≥ 0.28M⊙. The lack of evidence of a companion in the optical photometry forces any main-sequence companion to be smaller than 0.1M⊙, hence a low mass main sequence star companion is ruled out for this sys- tem. The companion is most likely another white dwarf, and we present tentative evidence for an evolutionary scenario which could have produced it. However, a neutron star companion cannot be ruled out and follow-up radio observations are required to search for a pulsar companion. Subject headings: stars: individual (SDSS J091709.55+463821.8) – low-mass – white dwarfs 1. Introduction Recent discoveries of several extremely low mass white dwarfs (WDs) in the field (Kilic et al. 2007; Eisenstein et al. 2006; Kawka et al. 2006) and around pulsars (Bassa et al. 2006; van Kerkwijk et al. 1996) show that WDs with mass as low as 0.17M⊙ are formed in the Galaxy. No galaxy is old enough to produce such extremely low mass WDs through single star evolution. The oldest globular clusters in our Galaxy are currently producing 2Ohio State University, Department of Astronomy, 140 West 18th Avenue, Columbus, OH 43210 3Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138 4McDonald Observatory and Department of Astronomy, University of Texas, Austin, TX 78712 1Observations reported here were obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. http://arxiv.org/abs/0704.1813v1 – 2 – ∼ 0.5M⊙ WDs (Hansen et al. 2007), therefore lower mass WDs must experience significant mass loss. The most likely explanation is a close binary companion. If a WD forms in a close binary, it can lose its outer envelope without reaching the asymptotic giant branch and without ever igniting helium, ending up as a low mass, helium core WD. Confirmation of the binary nature of several low mass WDs by Marsh et al. (1995) supports this binary formation scenario. White dwarf binaries provide an important tool for testing binary evolution, specifically the efficiency of the mass loss process and the common envelope phase. Since WDs can be created only at the cores of giant stars, their properties can be used to reconstruct the properties of the progenitor binary systems. Using a simple core mass - radius relation for giants and the known orbital period, the initial orbital parameters of the binary system can be determined. For a review of binary evolution involving WDs, see e.g. Sarna et al. (1996), Iben et al. (1997), Sandquist et al. (2000), Yungelson et al. (2000), Nelemans & Tout (2005), and Benvenuto & De Vito (2005). Known companions to low mass WDs include late type main sequence stars (Farihi et al. 2005; Maxted et al. 2007), helium or carbon/oxygen core WDs (Marsh et al. 1995; Marsh 2000; Napiwotzki et al. 2001), and in some cases neutron stars (Nice et al. 2005). Late type stellar companions to low mass WDs have a distribution of masses with median 0.27M⊙ (Nelemans & Tout 2005). This median companion mass is nearly identical to the peak companion mass of 0.3M⊙(spectral type M3.5) observed in the field population of late type main sequence stars within 20 pc of the Earth (Farihi et al. 2005). Low mass WD - WD binaries, on the other hand, tend to have equal mass WDs. The median mass for both the brighter and dimmer components of the known low mass WD binary systems is 0.44M⊙ (Nelemans & Tout 2005). The discovery of extremely low mass WDs around pulsars suggests that neutron star companions may be responsible for creating white dwarfs with masses of about 0.2M⊙. PSR J0437-4715, J0751+1807, J1012+5307, J1713+0747, B1855+09, and J1909-3744 are pulsars in pulsar – He-WD binary systems with circular orbits and orbital periods of ∼0.2-100 days (Nice et al. 2005). So far, only two of these companions are spectroscopically confirmed to be helium-core WDs. Van Leeuwen et al. (2007) searched for radio pulsars around 8 low mass WDs, but did not find any companions. They concluded that the fraction of low mass helium-core WDs with neutron star companions is less than 18% ± 5%. Kilic et al. (2007) reported the discovery of the lowest mass WD currently known: SDSS J091709.55+463821.8 (hereafter J0917+46). With an estimated mass of 0.17M⊙, J0917+46 provides a unique opportunity to search for a binary companion and to test our understanding of the formation scenarios for extremely low mass WDs. Do extremely low mass WDs form in – 3 – binaries with neutron stars, WDs, or late type stars? If the companion is a neutron star, the mass of the neutron star can be used to constrain the neutron star equation of state. In case of a WD or a late type star companion, the orbital parameters can be used to constrain the common-envelope phase of binary evolution in these systems. In this Letter, we present new optical spectroscopy and radial velocity measurements for J0917+46. Our observations are discussed in §2, while an analysis of the spectroscopic data and the discovery of a companion are discussed in §3. The nature of the companion is discussed in §4. 2. Observations We used the 6.5m MMT telescope equipped with the Blue Channel Spectrograph to obtain moderate resolution spectroscopy of SDSS J0917+46 nine times over the course of five nights between UT 2006 December 22-27 and five times on UT 2007 March 19. The spectrograph was operated with the 832 line mm−1 grating in second order, providing a wavelength coverage of 3650 - 4500 Å. Most spectra were obtained with a 1.0′′ slit yielding a resolving power of R = 4300, however a 1.25′′ slit was used on 2006 December 24, which resulted in a resolving power of 3500. Exposure times ranged from 15 to 22 minutes and yielded signal-to-noise ratio S/N > 20 in the continuum at 4000 Å. All spectra were obtained at the parallactic angle, and comparison lamp exposures were obtained after every exposure. The spectra were flux-calibrated using blue spectrophotometric standards (Massey et al. 1988). Heliocentric radial velocities were measured using the cross-correlation package RVSAO (Kurtz & Mink 1998). We obtained preliminary velocities by cross-correlating the obser- vations with bright WD templates of known velocity. However, greater velocity precision comes from cross-correlating J0917+46 with itself. Thus we shifted the individual spectra to rest-frame and summed them together into a high S/N template spectrum. Our final veloc- ities come from cross-correlating the individual observations with the J0917+46 template, and are presented in Table 1. 3. The Discovery of a Companion The radial velocity of the lowest mass WD varies by as much as 263 km s−1 between different observations, revealing the presence of a companion object. We solve for the best- fit orbit using the code of Kenyon & Garcia (1986). We find that the heliocentric radial velocities of the WD are best fitted with a circular orbit and a radial velocity amplitude K = – 4 – 148.8 ± 6.9 km s−1. We used the method of Lucy & Sweeney (1971) to show that there is no evidence for an eccentric orbit from our data and that the 1σ upper limit to the eccentricity is e = 0.06. We have assumed that the orbit is circular. The orbital period is 7.5936 ± 0.0024 hours with spectroscopic conjunction at HJD 2454091.73 ± 0.002. Figure 1 shows the observed radial velocities and our best fit model for SDSS J0917+46. Measurement of the orbital period P and the semi-amplitude of the radial velocity variations K allows us to calculate the mass function M3 sin3i (MWD +M)2 = 0.108± 0.018M⊙, (1) where i is the orbital inclination angle, MWD is the WD mass (0.17M⊙ for SDSS J0917+46), and M is the mass of the companion object. We can put a lower limit on the mass of the companion by assuming an edge-on orbit (sin i = 1), for which the companion would be an M = 0.28M⊙ object at an orbital separation of 1.5R⊙. Therefore, the companion mass is M ≥ 0.28M⊙. 4. The Nature of the Companion To understand the nature of the companion, first we need to understand the properties of the WD. Kilic et al.’s (2007) analysis of J0917+46 was based on a single 15 minute exposure with S/N = 20 at 4000 Å. Here we use 14 different spectra of J0917+46 (4.3 hours total exposure time) and repeat our spectroscopic analysis. We also combine all of these spectra into a composite (weighted average) spectrum with R = 3500 that results in S/N = 80 at 4000 Å. Figure 2 shows the composite spectrum and our fits using the entire spectrum (excluding the Ca K line) and also using only the Balmer lines (see Kilic et al. 2007 for the details of the spectroscopic analysis). We find a best-fit solution of Teff = 11855 K and log g = 5.55 if we use the observed composite spectrum. If we normalize (continuum-correct) the composite spectrum and fit just the Balmer lines, then we obtain Teff = 11817 K and log g = 5.51. Since we have 14 different spectra, we also fit each spectrum individually to obtain a robust estimate of the errors in our analysis. For the first case where we fit the observed spectra, we obtain a best fit solution of Teff = 11984 ± 168 K and log g = 5.57 ±0.05. For the second case where we fit only the Balmer lines, we obtain Teff = 11811± 67 K and log g = 5.51 ±0.02. Our results are consistent with each other and also with Kilic et al.’s analysis. We confirm that our temperature and gravity estimates are robust; SDSS J0917+46 is still the lowest gravity/mass WD currently known. – 5 – We adopt our best fit solution of Teff = 11855 ± 168 K and log g = 5.55 ±0.05 for the remainder of the paper. Using our new temperature and gravity measurements and Althaus et al. (2001) models, we estimate the absolute magnitude of the WD to be MV ∼ 7.0, corresponding to a distance modulus of 11.8 (at 2.3 kpc) and a cooling age of about 500 Myr. At a Galactic latitude of +44◦, J0917+46 is located at 1.6 kpc above the plane. The radial velocity of the binary system is ∼29 km s−1. J0917+46 displays a proper motion of µRAcosδ = −2 ± 3.4 mas yr −1 and µDEC = 2 ± 3.4 mas yr −1 measured from its SDSS and USNO-B positions (kindly provided by J. Munn). These measurements correspond to a tangential velocity of 31 ± 52 km s−1. J0917+46 has disk kinematics, and its location above the Galactic plane is consistent with a thick disk origin. Therefore the main sequence age of the progenitor star of the lowest mass WD needs to be on the order of 10 Gyr; the progenitor was a ∼1M⊙ main sequence star. The broad-band spectral energy distribution of J0917+46 is shown in panel c in Figure 2. The de-reddened SDSS photometry and the fluxes predicted for the parameters derived from our spectroscopic analysis are shown as error bars and circles, respectively. The SDSS photometry is consistent with our spectroscopic solution within the errors. The g-band photometry is slightly brighter than expected, however it is within 1.7σ of our spectroscopic solution and therefore the excess is not significant. We also note that many low mass WD candidates analyzed by Kilic et al. (2007) had discrepant g-band photometry. A similar problem may cause the observed slight g-band excess. The mass function for our target plus the MMT spectroscopy and the SDSS photometry can be used to constrain the nature of the companion star. Since the companion mass has to be ≥ 0.28M⊙, it can be a low mass star, another WD, or a neutron star. 4.1. A Low Mass Star 4.1.1. Constraints from the SDSS Photometry If the orbital inclination angle of the binary system is between 47◦ and 90◦, the compan- ion mass would be 0.28-0.50M⊙, consistent with being an M dwarf. However, an M dwarf companion would show up as an excess in the SDSS photometry. For example, an M6 dwarf would cause a 10% excess in the z-band photometry of J0917+46 at its distance, and hence any star with M ≥ 0.1M⊙ would be visible in the SDSS photometry. Since the mass function derived from the observed orbital parameters of the binary limits the companion mass to ≥ 0.28M⊙ (earlier than M3.5V spectral type), and any such companion would be visible in the SDSS photometry (see panel c in Figure 2), a main sequence star companion is ruled – 6 – 4.1.2. Constraints from the MMT Spectroscopy We search for spectral features from a companion by subtracting all 14 individual spectra from our best-fit WD model after shifting the individual spectra to the rest-frame. The only significant feature that we detect is a Ca K line. This same calcium line was also present in the discovery spectrum of J0917+46 (Kilic et al. 2007). It had the same radial velocity as the Balmer lines and hence, it was predicted to be photospheric. The Ca K line is visible in all of our new spectra of this object, and its radial velocity changes with the radial velocity of the Balmer lines; it is confirmed to be photospheric. The Ca H line overlaps with the saturated Hǫ line and it is not detected in our spectrum. If J0917+46 had a low mass star companion, it could contribute an additional calcium line in the spectrum. Marsh et al. (1995) were able to find the companion to the low mass white dwarf WD1713+332 by searching for asymmetries in the Hα line profile. We perform a similar analysis for J0917+46 using the calcium line. We combine the spectra near maximum radial velocity (V ≥ 125 km s−1) and near minimum radial velocity (V ≤ −80 km s−1) into two composite spectra. If there is a contribution from a companion object, it should be visible as an asymmetry in the line profile. This asymmetry should be on the blue side of the red-shifted composite spectrum, and on the red side of the blue-shifted composite spectrum. Figure 3 shows the red-shifted (solid line) and blue-shifted (dotted line) composite spectra of J0917+46. The rest-frame wavelegth of the Ca K line is shown as a dashed line. This figure demonstrates that there is an asymmetry in the Ca K line profile of the red-shifted spectrum, however the additional Ca K feature is close to the rest-frame velocity, and it is not detected in the blue-shifted spectrum. The equivalent width of the main Ca K feature in the red-shifted spectrum is 0.42Å, and the additional feature has an equivalent width of ∼0.16 Å. The blue shifted spectrum has a stronger Ca K feature with an equivalent width of 0.57 Å, and it is consistent with a blend of the two Ca K features seen in the red-shifted spectrum. The observed additional calcium feature seems to be stationary and it seems to originate from interstellar absorption. Therefore, we conclude that our optical spectroscopy does not reveal any spectral features from a companion object. – 7 – 4.2. Another White Dwarf Using the most likely inclination angle for a random stellar sample, i = 60◦, we estimate that the companion mass is most likely to be 0.36M⊙, another low mass WD. The orbital separation of the system would be about 1.6R⊙. If the inclination angle is between 47 ◦ and 27◦ (21% likelihood), the companion mass would be 0.5-1.4M⊙, consistent with a normal carbon/oxygen or a massive WD. Liebert et al. (2004) argue that it is possible to create a 0.16-0.19M⊙ WD from a 1M⊙ progenitor star, if the binary separation is appropriate and the common-envelope phase is sufficiently unstable so that the envelope can be lost quickly from the system. We re-visit their claim to see if J0917+46 can be a low mass WD formed from such a progenitor system. Close WD pairs can be created by two consecutive common envelope phases or an Algol-like stable mass transfer phase followed by a common envelope phase (Iben et al. 1997). In the first scenario, due to orbital shrinkage the recently formed WD is expected to be less massive than its companion by a factor of ≤0.55. On the other hand, the expected mass ratio for the second scenario involving a stable mass transfer and a common-envelope phase is around 1.1 (Nelemans et al. 2000). The mass ratio of the J0917+46 binary is Mbright/Mdim ≤ 0.61, therefore the progenitor binary star system has probably gone through two common envelope phases. Nelemans & Tout (2005) found that the common-envelope evolution of close WD bina- ries can be reconstructed with an algorithm (γ-algorithm) equating the angular momentum balance. The final orbital separation of a binary system that went through a common enve- lope phase is afinal ainitial Mgiant Mcore Mcore +Mcompanion Mgiant +Mcompanion Menvelope Mgiant +Mcompanion , (2) where a is the orbital separation, M are the masses of the companion, core, giant, and the envelope, and γ = 1.5 (Nelemans & Tout 2005). We assume that the mass of the WD is the same as the mass of the core of the giant at the onset of the mass transfer. Assuming a giant mass of 0.8-1.3M⊙, a core mass of 0.17M⊙, and possible WD companion masses of 0.28-1.39M⊙, we estimate the initial orbital separation. Using the core-mass - radius relation for giants, R = 103.5M4core (Iben & Tutukov 1985), we find that the radius of the giant star that created J0917+46 was R = 2.6R⊙. For M = 0.28-1.39M⊙ companions, we use the size of the Roche lobe RL as given by Eggleton (1983) to determine the separation at the onset of mass transfer assuming Rgiant = RL. The – 8 – size of the Roche lobe depends on the mass ratio and the orbital separation of the system. The initial separation and the size of the Roche lobe gives us a unique solution for the binary mass ratio. Table 2 presents the companion mass, initial orbital separation and the orbital period for 0.8-1.3M⊙ giants that could create the lowest mass WD with the observed orbital parameters. The mass function for J0917+46 favors a low mass WD companion, therefore the first four scenarios in Table 2 seem to be more likely. For example, a 0.8M⊙ giant and a 0.33M⊙ WD companion with an initial orbital separation of 5.7R⊙ and an orbital period of 36 hr would create a 0.17M⊙ WD with the observed orbital parameters. The same procedures can be used to re-create the first common-envelope phase of the binary evolution. The progenitor of the unseen companion to J0917+46 must be in the range 0.8-2.3M⊙. The lower mass limit is set by the fact that the unseen companion has to be more massive than the lowest mass WD. The upper limit is set by the fact that more massive stars do not form degenerate helium cores (Nelemans et al. 2000) and that a common envelope phase with a more massive giant would end up in a merger and not in a binary system. Assuming 0.8-2.3M⊙ giants and 0.8-1.3M⊙ main sequence companions, we calculate possible evolutionary scenarios to create the orbital parameters of the binary system before the last common envelope phase. We find that a 2.2M⊙ star and a 0.8M⊙ companion (which is the progenitor of the lowest mass WD) with an orbital period of 51 days and orbital separation of 0.4 AU would create a 0.33M⊙ WD. In addition, we find that the progenitor of the lowest mass WD has to be less massive than 0.9M⊙ since a M ≥ 0.9M⊙ progenitor would require a companion more massive than 2.3M⊙ to match the binary properties of the system before the last common envelope phase. Therefore, the most likely evolutionary scenario for a WD + WD binary involving J0917+46 would be: 2.2M⊙ giant + 0.8M⊙ star −→ 0.33M⊙ WD + 0.8M⊙ star −→ 0.33M⊙ WD + 0.8M⊙ giant −→ 0.33M⊙ WD + 0.17M⊙ WD. The main sequence lifetime of a 2.2M⊙ thick disk ([Fe/H] ≃ −0.7) star is about 650 Myr (Bertelli et al. 1994), and a 0.33M⊙ white dwarf created from such a system would be a ∼10 Gyr old WD. According to Althaus et al. (2001) models, a 0.33M⊙ He-core WD would cool down to 3700 K in 10 Gyr and it would have MV ∼ 15.8. This companion would be several orders of magnitude fainter than the 0.17M⊙ WD observed today, and therefore the lack of evidence of a companion in the SDSS photometry and our optical spectroscopy is consistent with this formation scenario. – 9 – 4.3. A Neutron Star The formation scenarios for low mass helium WDs in close binary systems also include neutron star companions. There are already several low mass WDs known to have neutron star companions (van Kerkwijk et al. 1996; Nice et al. 2005; Bassa et al. 2006). According to the theoretical calculations by Benvenuto & De Vito (2005), a 1M⊙ star with a neutron star companion in an initial orbital period of 0.5-0.75 days would end up as a 0.05-0.21M⊙ WD with a 1.8-1.9M⊙ neutron star in an orbital period of 0.03-3.5 days. They expect a 0.17M⊙ WD + neutron star binary to have an orbital period of 9.6 hours (see their Figure 16). The orbital period of J0917+46 is consistent with their analysis. If the orbital inclination angle of the J0917+46 binary system is less than 27◦, the companion mass would be ≥1.4M⊙, consistent with being a neutron star. The probability of observing a binary system at an angle less than 27◦ is only 11%. This is unlikely, but cannot be definitely ruled out. 5. Discussion Our radial velocity measurements of J0917+46 shows that it is in a binary system with an orbital period of 7.59 hours. Short period binaries may merge within a Hubble time by losing angular momentum through gravitational radiation. The merger time for such binaries is given by (M1 +M2) P 8/3 × 107yr (3) where the masses are in solar units and the period is in hours (Landau & Lifshitz 1958; Marsh et al. 1995). For the J0917+46 binary, if the companion is a low mass WD with M ≤ 0.5M⊙, the merger time is longer than 23 Gyr. If the companion is a 1.4M⊙ neutron star, then the merger time would be 10.8 Gyr. J0917+46 binary system will not merge within the next 10 Gyr. J0917+46 has nearly solar calcium abundance, and it has more calcium than many of the metal-rich WDs with circumstellar debris disks. The extremely short timescales for gravitational settling of Ca implies that the WD requires an external source for the observed metals (Koester & Wilken 2006; Kilic & Redfield 2007). The star is located far above the Galactic plane where accretion from the ISM is unlikely. A possible scenario for explaining the photospheric calcium is accretion from a circumbinary disk created during the mass loss phase of the giant. Since J0917+46 went through a common envelope phase with a companion, a left-over circumbinary disk is possible. An accretion disk is observed around – 10 – the companion to the giant Mira A (Ireland et al. 2007). In addition, a fallback disk is observed around a young neutron star (Wang et al. 2006). A similar mechanism could create a circumbinary disk around J0917+46. Several calcium-rich WDs are known to host disks (Kilic et al. 2006; von Hippel et al. 2007). These WDs are 0.2 - 1.0 Gyr old. The presumed origin of these disks is tidal disruption of asteroids or comets, however a leftover disk from the giant phase of the WD is not completely ruled out (von Hippel et al. 2007). Su et al. (2007) detected a disk around the central star of the Helix planetary nebula. This disk could form from the left-over material from the giant phase that is ejected with less than the escape speed. The majority of the metal-rich white dwarfs with disks show 30-100% excess in the K-band (Kilic & Redfield 2007). J0917+46 is expected to be fainter than 19th magnitude in the near-infrared, and it is not detected in the Point Source Catalog (PSC) part of the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006). The PSC 99% completeness limits are 15.8, 15.1 and 14.3 in J,H and Ks filters, respectively. These limits do not provide any additional constraints on the existence of a disk or a companion object. Follow-up near-infrared observations with an 8m class telescope are required to search for the signature of a debris disk which could explain the observed calcium abundance in this WD. 6. Conclusions SDSS J0917+46, the lowest gravity/mass WD currently known, has a radial velocity companion. The lack of excess in the SDSS photometry and the orbital parameters of the system rule out a low mass star companion. We find that the companion is likely to be another WD, and most likely to be a low mass WD. We show that if the binary separation is appropriate and the common-envelope phase is efficient, it is possible to create a 0.17M⊙ WD in a 7.59 hr orbit around another WD. A neutron star companion is also possible if the inclination angle is smaller than 27◦; the likelihood of this is 11%. If the companion is a neutron star, it would be a milli-second pulsar. Radio observations of J0917+46 are needed to search for such a companion. M. Kilic thanks A. Gould for helpful discussions. Facilities: MMT (Blue Channel Spectrograph) – 11 – REFERENCES Althaus, L. G., Serenelli, A. M., & Benvenuto, O. G. 2001, MNRAS, 323, 471 Bassa, C. G., van Kerkwijk, M. H., Koester, D., & Verbunt, F. 2006, A&A, 456, 295 Benvenuto, O. G., & De Vito, M. A. 2005, MNRAS, 362, 891 Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., & Nasi, E. 1994, A&AS, 106, 275 Brown, W. R., Geller, M. J., Kenyon, S. J., & Kurtz, M. 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R. 1996, ApJ, 467, L89 van Leeuwen, J., Ferdman, R. D., Meyer, S., & Stairs, I. 2007, MNRAS, 374, 1437 von Hippel, T., Kuchner, M. J., Kilic, M., Mullally, F., & Reach, W. T. 2007, ApJ, in press (astro-ph/0703473) Wang, Z., Chakrabarty, D., & Kaplan, D. L. 2006, Nature, 440, 772 Yungelson, L. R., Nelemans, G., Zwart, S. F. P., & Verbunt, F. 2000, “The influence of binaries on stellar population studies”, Brussels, (Kluwer, D. Vanbeveren ed) This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0702005 http://arxiv.org/abs/astro-ph/0703473 – 13 – Table 1. Radial Velocity Measurements for SDSS J0917+46 Julian Date Heliocentric Radial Velocity (km s−1) 2454091.77060 134.29 ± 5.45 2454091.85741 124.83 ± 8.56 2454093.82302 −82.91 ± 4.22 2454093.94566 41.98 ± 4.65 2454094.04293 152.42 ± 3.79 2454095.83998 39.88 ± 3.45 2454095.89902 172.60 ± 3.57 2454096.04373 −79.87 ± 3.76 2454096.95394 20.45 ± 7.49 2454178.62439 −86.78 ± 6.08 2454178.69885 −90.08 ± 3.90 2454178.72963 −11.16 ± 6.32 2454178.77125 106.28 ± 6.53 2454178.83047 160.16 ± 5.48 Table 2. The Last and the First Common-Envelope Phases CE Phase Mgiant Mcompanion ainitial afinal Pinitial Pfinal (M⊙) (M⊙) (R⊙) (R⊙) (hr) (hr) 2 0.8 0.33 5.70 1.55 35.6 7.6 2 0.9 0.39 5.73 1.61 33.6 7.6 2 1.0 0.45 5.78 1.66 32.1 7.6 2 1.1 0.50 5.78 1.71 30.5 7.6 2 1.2 0.56 5.85 1.75 29.7 7.6 2 1.3 0.61 5.84 1.80 28.4 7.6 1 2.2 0.80 83.52 5.70 1222.9 35.6 – 14 – Fig. 1.— The radial velocities of the white dwarf SDSS J0917+46 (black dots) observed in 2006 December (top panel) and 2007 March (bottom left panel). The bottom right panel shows all of these data points phased with the best-fit period. The solid line represents the best-fit model for a circular orbit with a radial velocity amplitude of 148.8 km s−1 and a period of 7.5936 hours. – 15 – Fig. 2.— Spectral fits (solid lines) to the observed composite spectrum of SDSS J0917+46 (jagged lines, panel a) and to the flux-normalized line profiles (panel b). The Ca K line region (3925 - 3940 Å) is not included in our fits. The SDSS photometry (error bars) and the predicted fluxes from our best fit solution to the spectra (circles) are shown in panel c. The dashed line shows the effect of adding an M3.5V (0.3M⊙) companion to our best-fit white dwarf model. – 16 – Fig. 3.— The spectra averaged around the maximum and minimum radial velocity for J0917+46. The red-shifted spectrum (solid line) is a combination of five spectra with V = 125, 134, 152, 160, and 173 km s−1 shifted to an average velocity of 149 km s−1. The blue- shifted spectrum (dotted line) is a combination of four spectra with V = −80,−83,−87, and −90 km s−1 shifted to an average velocity of −85 km s−1. The dashed line marks the rest wavelength of the Ca K line. Introduction Observations The Discovery of a Companion The Nature of the Companion A Low Mass Star Constraints from the SDSS Photometry Constraints from the MMT Spectroscopy Another White Dwarf A Neutron Star Discussion Conclusions

0704.1814

A Measure of de Sitter Entropy and Eternal Inflation

arXiv:0704.1814v1 [hep-th] 13 Apr 2007 A Measure of de Sitter Entropy and Eternal Inflation Nima Arkani-Hameda, Sergei Dubovskya,b, Alberto Nicolisa, Enrico Trincherinia, and Giovanni Villadoroa a Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA b Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, 117312 Moscow, Russia Abstract We show that in any model of non-eternal inflation satisfying the null energy condition, the area of the de Sitter horizon increases by at least one Planck unit in each inflationary e-folding. This observation gives an operational meaning to the finiteness of the entropy SdS of an inflationary de Sitter space eventually exiting into an asymptotically flat region: the asymptotic observer is never able to measure more than eSdS independent inflationary modes. This suggests a limitation on the amount of de Sitter space outside the horizon that can be consistently described at the semiclassical level, fitting well with other examples of the breakdown of locality in quantum gravity, such as in black hole evaporation. The bound does not hold in models of inflation that violate the null energy condition, such as ghost inflation. This strengthens the case for the thermodynamical interpretation of the bound as conventional black hole thermodynamics also fails in these models, strongly suggesting that these theories are incompatible with basic gravitational principles. http://arxiv.org/abs/0704.1814v1 1 Introduction String theory appears to have a landscape of vacua [1, 2], and eternal inflation [3, 4] is a plausible mechanism for populating them. In this picture there is an infinite volume of spacetime undergoing eternal inflation, nucleating bubbles of other vacua that either themselves eternally inflate, or end in asymptotically flat or AdS crunch space-times. These different regions are all space-like separated from each other and are therefore naively completely independent. The infinite volumes and infinite numbers of bubbles vex simple attempts to define a “measure” on the space of vacua, since these involve ratios of infinite quantities. This picture relies on an application of low-energy effective field theory to inflation and bubble nucleation. On the face of it this is totally justified, since everywhere curvatures are low compared to the Planck or string scales. However, we have long known that effective field theory can break down dramatically even in regions of low curvature, indeed it is precisely the application of effective field theory within its putative domain of validity that leads to the black hole information paradox. Complementarity [5, 6] suggests that regions of low-curvature spacetime that are space- like separated may nonetheless not be independent. How can we transfer these relatively well- established lessons to de Sitter space and eternal inflation [7]? In this note we begin with a brief discussion of why locality is necessarily an approximate concept in quantum gravity, and why the failure of locality can sometimes manifest itself macro- scopically as in the information paradox (see, e.g., [8, 9, 10] for related discussions with somewhat different accents). Much of this material is review, though some of the emphasis is novel. The conclusion is simple: effective field theory breaks down when it relies on the presence of eS states behind a horizon of entropy S. Note that if the spacetime geometry is kept fixed as gravity is decoupled G→ 0, the entropy goes to infinity and effective field theory is a perfectly valid descrip- tion. In attempting to extend these ideas to de Sitter space, there is a basic confusion. It is very natural to assign a finite number of states to a black hole, since it occupies a finite region of space [11]. De Sitter space also has a finite entropy [12], but its spatially flat space-like surfaces have infinite volume, and it is not completely clear what this finite entropy means operationally, though clearly it must be associated with the fact that any given observer only sees a finite volume of de Sitter space. We regulate this question by considering approximate de Sitter spaces which are non-eternal inflation models, exiting into asymptotically flat space-times. We show that for a very broad class of inflationary models, as long the null-energy condition is satisfied, the area of the de Sitter horizon grows by at least one Planck unit during each e-folding, so that dSdS/dNe ≫ 1, and so the number of e-foldings of inflation down to a given value of inflationary Hubble is bounded as Ne ≪ SdS (limits on the effective theory of inflation have also been considered in e.g. [13, 14]). This provides an operational meaning to the finiteness of the de Sitter entropy: the asymptotic observer detects a spectrum of scale-invariant perturbations that she associates with the early de Sitter epoch; however, she never measures more than eSdS of these modes. The bound is vio- lated when the conditions for eternal inflation are met; indeed, dSdS/dNe . 1 thereby provides a completely macroscopic characterization of eternal inflation. This bound suggests that no more than eSdS spacetime Hubble volumes can be consistently described within an effective field theory. Our bound does not hold in models of inflation that violate the null-energy condition. Of course most theories that violate this energy condition are obviously pathological, with instabili- ties present even at the long distances. However in the last number of years, a class of theories have been studied [15, 16, 17], loosely describing various “Higgs” phases of gravity, which appear to be consistent as long-distance effective theories, and which (essentially as part of their raison d’etre) violate the null energy condition. Our result suggests that these theories violate the thermody- namic interpretation of de Sitter entropy—an asymptotic observer exiting into flat space from ghost inflation [18] could, for instance, measure parametrically more than eSdS inflationary modes. This fits nicely with other recent investigations [19, 20] that show that the second law of black hole thermodynamics also fails for these models. Taken together these results strongly suggest that, while these theories may be consistent as effective theories, they are in the “swampland” of effective theories that are incompatible with basic gravitational principles [21, 22]. 2 Locality, gravity, and black holes 2.1 Locality in gravity Since the very early days of quantum gravity it has been appreciated that the notion of local off-shell observables is not sharply well-defined (see e.g. [23]). It is important to realize that it is dynamical gravity that is crucial for this conclusion, and not just the reparameterization invariance of the theory. The existence of local operators clashes with causality in a theory with a dynamical metric. Indeed, causality tells that the commutator of local operators taken at space-like separated points should be zero, [O(x),O(y)] = 0 if (x− y)2 > 0 However, whether two points are space-like separated or not is determined by the metric, and is not well defined if the metric itself fluctuates. Clearly, this argument crucially relies on the ability of the metric to fluctuate, i.e. on the non-trivial dynamics of gravity. Another argument is that Green’s functions of local field operators, such as 〈φ(x)φ(y) · · · 〉 are not invariant (as opposed to covariant) under coordinate changes. Consequently, they cannot represent physical quantities in a theory of gravity, where coordinate changes are gauge transformations. Related to this, there is no standard notion of time evolution in gravity. Indeed, as a consequence of time reparameterization invariance, the canonical quantization of general relativity leads to the Wheeler-de Witt equation [24], which is analogous to the Schroedinger equation in ordinary quantum mechanics, but does not involve time, HΨ = 0 (1) These somewhat formal arguments seem to rely only on the reparametrization invariance of the theory, but of course this is incorrect—it is the dynamical gravity that is the culprit. To see this, let us take the decoupling limit MPl → ∞, so that gravity becomes non-dynamical. If we are in flat space, in this limit the metric gαβ must be diffeomorphic to ηαβ : gαβ = ηµν (2) where ηµν is the Minkowski metric and Y µ’s are to be thought of as the component functions of the space-time diffeomorphism (diff), xµ → Y µ(x). The resulting theory is still reparameterization invariant, with matter fields transforming in the usual way under the space-time diffs x → ξ(x), and the transformation rule of the Y µ fields is Y µ → ξ−1 ◦ Y where ◦ is the natural multiplication of two diffeomorphisms. Nevertheless, there are local diff- invariant observables now, such as 〈φ(Y (x))φ(Y (y)) · · · 〉. Of course this theory is just equivalent to the conventional flat space field theory, which is recovered in the “unitary” gauge Y µ = xµ. Conversely, any field theory can be made diff invariant by introducing the “Stueckelberg” fields Y µ according to (2). Diff invariance by itself, like any gauge symmetry, is just a redundancy of the description and cannot imply any physical consequences. Conventional time evolution is also recovered in the decoupling limit; the Hamiltonian constraint (1) still holds as a consequence of time reparameterization invariance, and in the decoupling limit the Hamiltonian H is pµ +HM Ψ[Y µ, matter] = 0 (3) where HM is the matter Hamiltonian. Noting that the canonical conjugate momenta act as pµ = i we find that the Hamiltonian constraint reduces to the conventional time-dependent Schroedinger equation with Ψ depending on time through Y µ. In a sense, the gauge degrees of freedom Y µ’s play the role of clocks and rods in the decoupling limit. This is to be contrasted with what happens for finite MPl. In this case it is not possible to explicitly disentangle the gauge degrees of freedom from the metric. As a result to recover the conventional time evolution from the Wheeler-de Witt equation one has to specify some physical clock field (for instance, it can be the scale factor of the Universe, or some rolling scalar field), and use this field similarly to how we used Y µ’s in (3) to recover the time-dependent Schroedinger equation [25, 26, 27]. This strongly suggests that with dynamical gravity one is forced to consider whether there exist physical clocks that can resolve a given physical process. In particular, this means that in a region of size L it does not make sense to discuss time evolution with resolution better than δt ∼ (LM2Pl)−1, as any physical clocks aiming to measure time with that precision by the uncertainty principle would collapse the whole region into a black hole. What does the formal absence of local observables in gravity mean operationally? There must be an intrinsic obstacle to measuring local observables with arbitrary precision; what is this intrinsic uncertainty? Imagine we want to determine the value of the 2-point function 〈φ(x)φ(y)〉 of a scalar field φ(x) between two space-like separated points x and y. We have to set up an apparatus that measures φ(x) and φ(y), repeat the experiment N times and collect the outcomes φi(x), φi(y) for i = 1, · · · , N . We can then plot the values for the product φi(x)φi(y), which will be peaked around some value. The width of the distribution will represent the uncertainty due to quantum fluctuations. Without gravity there is no limit to the precision we can reach, just by increasing N the width of the distribution decreases as 1/ N . The presence of gravity, however, sets an intrinsic systematic uncertainty in the measurement. The Bekenstein bound [11], indeed, limits the number of states in a localized region of space-time. This is due to the fact that, in a theory with gravity, the object with the largest density of states is a black hole, whose size RS grows with its entropy (SBH = R S /4G), or equivalently, with the number of states it can contain (∼ eSBH). This means that an apparatus of finite size has a finite number of degrees of freedom (d.o.f.), thus can reach only a finite precision, limited by the number of states. For an apparatus with size smaller than r = |x − y|, the number of d.o.f. is bounded by S = rD−2/G. Without gravity there is no limit to the number of d.o.f. a compact apparatus can have so that the indetermination in the two-point function is only limited by the statistical error, which can be reduced indefinitely by increasing the number of measurements N . With gravity instead this is no longer true; an intrinsic systematic error (which must be a decreasing function of S) is always present to fuzz the notion of locality. The only two ways to eliminate such indetermination are: a) by switching off gravity (G → 0); b) by giving up with local observables and considering only S-matrix elements (for asymptotically Minkowski spaces) where r → ∞: in this sense there are no local (off-shell) observables in gravity. Let us now try to quantify the amount of indetermination due to quantum gravity. The parameter controlling the uncertainty 1/S = G/rD−2 is always tiny for distances larger than the Planck length, which signals the fact that quantum gravity becomes important at this scale. We do not expect the low-energy effective theory to break down at any order in perturbation theory, i.e. at any order in 1/S. This is what perturbative string theory suggests by providing, in principle, a well defined higher-derivative low-energy expansion at all order in G. Also, in our 2-point function example, the natural limit on the resolution should be set by the number of states of the apparatus (eS) instead of its number of d.o.f. (S). We thus expect the irreducible error due to quantum gravity to be non-perturbative in the coupling G, δ〈φ(x)φ(y)〉 ∼ e−S ∼ e− (x−y) The smallness and the non-perturbative nature of this effect suggest that it becomes important only at very short distances, with the low-energy field theory remaining a very good approximation at long distances. This is true except in special situations where the effective theory breaks down when it is not naively expected to. However, before discussing this point further, let us examine the issue of locality from another angle by looking at what it means in S-matrix language. As is well-known, the S-matrix associated with a local theory enjoys analyticity properties. For instance, for the 2 → 2 scattering, the amplitude must be an analytic function of the Maldelstam’s variables s and t away from the real axis. It must also be exponentially bounded in energy—at fixed angles, the amplitude can not fall faster than e− s log s [28]. In local QFT, both of these requirements follow directly from the sharp vanishing of field commutators outside the light-cone in position space. A trivial example illustrates the point: consider a function f(x) that vanishes sharply outside the interval [x1, x2]. What does this imply for the Fourier transform f̃(p)? Since the integral for f̃(p) is over a finite range [x1, x2] and e ipx is analytic in p, f̃(p) must be both analytic and exponentially bounded in the complex p plane. Now amplitudes in UV complete local quantum field theories certainly satisfy these requirements—they are analytic and fall off as powers of energy. More significantly, amplitudes in perturbative string theory also satisfy these bounds. That they are analytic is no surprise, since after all the Veneziano amplitude arose in the context of the analytic S-matrix program. More non-trivially they are also exponentially bounded—high energy amplitudes for E ≫ Ms are dominated by genus g = E/Ms and fall off precisely as e−E logE, saturating the locality bound [29] (see also [30] and referenced therein for discussion of high-energy scattering in string theory). Thus despite naive appearances, the finite extent of the string does not in itself give rise to any violations of locality. Indeed, we now know of non-gravitational string theories—little string theories in six dimensions. These theories have a definition in terms of four-dimensional gauge theories via deconstruction and are manifestly local in this sense [31]. It is possible that violations of locality do show up in the S-matrix when black hole production becomes important. At high enough energies relative to the Planck scale, the two-particle scatter- ing is dominated by black hole production, when the energy becomes larger than MP l divided by some power of gs so the would-be BH becomes larger than the string scale. The 2 → 2 scattering amplitude therefore cannot be smaller than e−S(E), and it is natural to conjecture that this lower bound is met: A2→2(E ≫MPl) ∼ e−S(E) ∼ e−ER(E) (5) where R(E) ∼ (GE)1/(D−3) is the radius of the black hole formed with center of mass energy E and S(E) is the associated entropy. Note that since R(E) grows as a power of energy, saturating this lower bound leads to an amplitude falling faster than exponentially at high energies, so that the only sharp mirror of locality in the scattering amplitude is lost. A heuristic measure of the size of these non-local effects in position space can be obtained by Fourier transforming the analytically continued A2→2 back to position space; a saddle point approximation using the black-hole dominated amplitude gives a Fourier transform of order e−r D−2/G ∼ e−S, in accordance with our general expectations. Of course this asymptotic form of the scattering amplitude is a guess; it is hard to imagine that the amplitude is smaller than this but one might imagine that it can be larger (we thank J. Maldacena for pointing this out to us). The point is that there is no reason to expect perturbative string effects to violate notions of locality—they certainly do not in the S-matrix—while gravitational effects can plausibly do it. Naively one would expect that the breakdown of locality only shows up when scales of order of the Planck length or shorter are probed, while for IR physics the corrections are ridiculously tiny (e−S) with no observable effects. This is however not true. There are several important cases where the loss of locality by quantum gravity give O(1) effects. This happens when in processes with O(eS) states, the tiny O(e−S) corrections sum to give O(1) effects. This is similar to renormalon contributions in QCD. Independently of the value of αs, or equivalently of the energy considered, every QCD amplitude is indeed affected by non-perturbative power corrections Λ2QCD β0 αs(Q2) (6) which limit the power of the “asymptotic” perturbative expansion. Because in the N -loop order contributions, and equivalently in the N -point functions, combinatorics produce enhancing N ! factors, they start receiving O(1) corrections when N ∼ 1/αs. Analogously in gravity, we must expect O(1) corrections from “non-perturbative” quantum gravity in processes with N -point func- tions with N ≃ S. These contributions are not captured by the perturbative expansion, they show the very nature of quantum gravity and its non-locality, which is usually thought to be confined at the Planck scale. Indeed in eq. (5) it is the presence of eS states (the inclusive amplitude is an S-point function) that suppresses exponentially the 2 → 2 amplitude, thus violating the locality bound. An example where this effect becomes macroscopic is well-known as the black hole information paradox [32], and will be reviewed more extensively below in section 2.2. Notice however that only for specific questions e−S effects become relevant, in all other cases, where less than O(S) quanta are involved, the low energy effective theory of gravity (or perturbative string theory) remains an excellent tool for describing gravity at large distances. 2.2 The black hole information paradox Since an effective field theory analysis of black hole information and evaporation leads to dramat- ically incorrect conclusions, it is worth reviewing this well-worn territory in some detail, in order to draw a lesson that can then be applied to cosmology. Schwarzschild black hole solutions of mass M and radius RS (with R S ∼ GM) exist for any D > 3 spacetime dimensions. Black holes lose mass via Hawking radiation [33] with a rate dM/dt ∼ −R−2, so that the evaporation time is ∼MRS ∼ SBH (7) where SBH ∼ RD−2S is the black hole entropy, the large dimensionless parameter in the problem. Note that there is a natural limit where the geometry (RS) is kept fixed, M → ∞ but G → 0 so that SBH → ∞. In this limit there is still a black hole together with its horizon and singularity, and it emits Hawking radiation with temperature TH ∼ R−1S , but tev → ∞ so the black hole never evaporates. Hawking radiation can certainly be computed using effective field theory, after all the horizon of a macroscopic black hole is a region of spacetime with very small curvature and as a consequence there should be a description of the evaporation where only low-energy degrees of freedom are excited. In order to derive Hawking radiation, one has to be able to describe the evolution of r = 0 t = 0 infalling matter Figure 1: Nice slices in Kruskal coordinates (left) and in the Penrose diagram (right). The singularity is at T 2 −X2 = 1. an initial state on the black hole semiclassical background to some final state that has Hawking quanta. Following the laws of quantum mechanics, all that is needed is a set of spatial slices and the corresponding—in general time dependent—Hamiltonian. However, because the aim is to compute the final state within a long distance effective field theory, the curvature of the sliced region of spacetime must be low everywhere (the slices can also cross the horizon if they stay away from the singularity) and the extrinsic curvature of the slices themselves has to be small as well. Spatial surfaces with these properties are called “nice slices” [34, 35]. One can easily arrange for this slicing to cover also most of the collapsing matter that forms the black hole. To be specific we can take the first (t = 0) slice to be T = c0 for X > 0 and the hyperbola T 2 − X2 = c20 for X < 0, where X and T are Kruskal coordinates; this slice has small extrinsic curvature by construction. Then we take a second slice with c1 > c0 and we boost it in such a way that the asymptotic Schwarzschild time on this slice is larger than the asymptotic time on the previous one. We can build in this way a whole set of slices c0 < . . . < cn, all with small extrinsic curvature; if the region they cover inside the horizon is still far away from the singularity, while outside they can be boosted arbitrarily far in the future (Fig. 1) so that they can intercept most of the outgoing Hawking quanta. When the black hole evaporates the background geometry changes and the slices can be smoothly adjusted with the change in the geometry until very late in the evaporation process, when the curvature becomes Planckian and the black hole has lost most of its mass. Starting with a pure state |ψi〉 at t = 0, one can now evolve it using the Hamiltonian HNS defined on this set of slices, never entering the regime of high curvature. We can now imagine dividing the slices in a portion that is outside the horizon and one inside it; even if the state on the entire slice is pure, we can consider the effective density operator outside the black hole defined as ρout(t) = Trin |ψ(t)〉〈ψ(t)|. In principle we can measure ρout. As usual in quantum mechanics, this is done by repeating exactly the same experiments an infinite number of times, and measuring all the mutually commuting observables that are possible. We should certainly expect that at early times ρout is a mixed state, representing the entanglement between infalling matter and Hawking radiation along the early nice-slices. This can be quantified by looking at the entanglement entropy associated with ρout: Sent = −Tr ρout log ρout (9) Clearly at early times Sent is non-vanishing. What happens at late times? Should we expect the final state of the evolution to be |ψf〉 = |ψout〉 ⊗ |ψin〉, with no entanglement between inside and outside and Sent = 0? The answer is negative because of the quantum Xerox principle [36]. If this decomposition were correct, two different states |A〉 and |B〉 should evolve into |A〉 → |Aout〉 ⊗ |Ain〉, |B〉 → |Bout〉 ⊗ |Bin〉 (10) but a linear superposition of them |A〉+ |B〉 |Aout〉 ⊗ |Ain〉+ |Bout〉 ⊗ |Bin〉 cannot be of the form (|A〉+ |B〉)out⊗ (|A〉+ |B〉)in unless the states behind the horizon are equal |Ain〉 = |Bin〉 for every A and B, and this is clearly impossible. No mystery then that the outgoing Hawking radiation ρout looks thermal, being correlated with states behind the horizon. Using nice slices one can compute in the low energy theory the entanglement entropy associated with the density matrix ρout: while the horizon area shrinks, the number of emitted quanta increases, the entanglement entropy of these thermal states grows monotonically as a function of time until the black hole becomes Planckian, the effective field theory is no longer valid and we don’t know what happens next without a UV completion (Fig. 2). This seems a generic prediction of low energy EFT. It implies a peculiar fate for black hole evaporation: either the evolution of a pure state ends in a mixed state, violating unitarity, or the black hole doesn’t evaporate completely, a Planckian remnant is left and the information remains stored in the correlations between Hawking radiation and the remnant. What cannot be is that the purity of the final state is recovered in the last moments of black hole evaporation, because the number of remaining quanta is not large enough to carry all the information. This is the black hole information paradox. It suggests that in order to preserve unitarity, effective field theory should break down earlier than expected. If we believe in the holographic principle, the total dimension of the Hilbert space of the region inside the black hole has to be bounded by the exponential of the horizon area in Planckian units. Since the entropy of any density operator is always smaller than the logarithm of the dimension of the Hilbert space, and since the entanglement entropy for a pure state divided into two subsystems is the same for each of them, the correct value of the entanglement entropy that is measured from ρout should start decreasing at a time of order tev, finally becoming zero when the black hole evaporates and a pure state is recovered. tt ~ t Planckianev EFTarea horizon correct Figure 2: The entanglement entropy for an evaporating black hole as a function of time. After a time of order of the evaporation time the EFT prediction (blue line) starts violating the holographic bound (dashed line). The correct behavior (red line) must reduce to the former at early times and approach the latter at late times. At the final stages, t ≃ tPlanckian, curvatures are large and EFT breaks down. According to this picture, the difference between the prediction of EFT and the right answer is of O(1) in a regime where curvature is low and there is no reason why effective field theory should be breaking down. However, the way this O(1) difference manifests itself is rather subtle. To understand this point let us first consider N spins σi = ±12 and take the following state: |ψ〉 = |σ1 . . . σN 〉eiθ(σ1,...,σN ) (12) where θ(σ1, ..., σN) are random phases. If only k of the N spins are measured, the density matrix ρk can be computed taking the trace over the remaining N − k spins |σ1 . . . σk〉〈σ1 . . . σk|+O(2− 2 )off-diagonal (13) the off-diagonal exponential suppression comes from averaging 2N−k random phases. When k ≪ N this density matrix looks diagonal and maximally mixed. Let us now study the entanglement entropy: for small k we can expand Sent = −Tr ρk log ρk = k log 2 +O(2−N+2k) (14) and conclude that the effect of correlations becomes important only when k ≃ N/2 spins are measured; finally when k ∼ N the entanglement entropy goes to zero as expected for a pure state. A state that looks thermal instead of maximally mixed is |ψ〉 = 2 |En〉eiθ(En) with random phases θ(En). This is of course why common pure states in nature, like the proverbial “lump of coal” entangled with the photons it has emitted, look thermal when only a subset of the states is observed. This is a simple illustration of a general result due to Page [37], showing how the difference between a pure and a mixed state is exponentially small until a number of states of order of the dimensionality of the Hilbert space is measured. Suppose we have to verify if the black hole density operator has an entanglement entropy of order S, then we need to measure an eS × eS matrix—the entropy of any N×N matrix is bounded by logN—with entries of order e−S; in order to see O(1) deviations from thermality in the spectrum, a huge number of Hawking states must be measured with incredibly fine accuracy. Because it takes a time scale of order of the evaporation time tev = RSSBH to emit order SBH quanta, before that time effective field theory predictions are correct up to tiny e−S effects (Fig. 2). In particular this means that when looking at the N -point functions of the theory, the exact value is the one obtained using EFT plus corrections that are exponentially small until N ≃ S: 〈φ1 . . . φN〉correct = 〈φ1 . . . φN〉EFT +O(e−(S−N)) (15) This can be explicitly seen with large black holes in AdS, as discussed by Maldacena [38] and Hawking [41]. The semiclassical boundary two-point function for a massless scalar field falls off as e−t/R. Its vanishing as t → ∞ is the information paradox in this context, while the CFT ensures that this two-point function never drops below e−S; but the discrepancy of the semiclassical approximation relative to the exact unitary CFT result for the two-point function is of order e−S. There is another heuristic observation that supports the idea that the whole process of black hole evaporation cannot be described within a single effective field theory. There is actually a limitation in the slicing procedure that we described at the beginning of this section. In order for the slices to extend arbitrarily in the future outside the black hole, they have to be closer and closer inside the horizon. However, quantum mechanics plus gravitation put a strong constraint: to measure shorter time intervals heavier clocks are needed. Of course they must be larger than their own Schwarzschild radius but a clock has also to be smaller than the black hole itself. This gives a bound on the shortest interval of time δt (the difference ck − ck−1 between two subsequent slices in (Fig. 1)) that makes sense to talk about inside the horizon Mclock RD−3S In this equation we have temporarily restored ~ to highlight the fact that whenever ~ or G goes to zero the bound becomes trivial. On the other hand, the proper time inside the black hole is finite τin . RS. These two conditions imply a striking bound: the maximum number of slices inside the black hole is also finite, Nmax ≃ τin/δt ≃ RD−2S /G ≃ SBH. How large is then the time interval that we can cover outside? With a spacing between the slices of the order of the Planck length (ℓPl) the total time interval is τout ≃ NmaxℓPl ≃ RD−2S G(3−D)/(D−2). Note, however, that if we are only interested in the Hawking quanta we may allow for a much less dense slicing: the spacing outside can be of order of the typical wavelength of the radiation δtout ∼ 1/TBH ∼ RS. In this way we can cover at most τout . NmaxRS ≃ SBHRS (17) which is precisely the evaporation time tev. Summarizing, the system of slices we need to define the Hamiltonian evolution cannot cover enough space-time to describe the process of black hole evaporation for time intervals parametrically larger than tev. With this argument we find that effective field theory should break down exactly when it starts giving the wrong prediction for the entanglement entropy (Fig. 2). Most previous estimates instead accounted for a much shorter regime of validity, up to time-scales of order RS logRS [39, 40]. This would imply that the EFT breakdown originates at some finite order in perturbation theory while in our case it comes from non-perturbative O(e−S) effects. Because the EFT description of states is incorrect for late times t ≫ tev, also the association of commuting operators to the 2-point function is wrong. It can well be that two observables evaluated on the same nice slice, one inside the horizon and the other outside, will no longer commute, even if they are at large spatial separation. This is consistent with the principle of black hole complementarity [5, 6]. Notice however that this breaking down is not merely a kinematical effect due to the presence of the horizon, after all EFT is perfectly good for computing Hawking radiation. In fact in the limit described at the beginning of this section, when we keep the geometry fixed and we decouple dynamical gravity, there still is an horizon but effective field theory now gives the right answer for arbitrarily long time scale: the black hole doesn’t evaporate and information is entangled with states behind the horizon. The limitation on the validity of EFT comes from dynamical gravity. There is nothing wrong in talking about both the inside and the outside of the horizon for time intervals parametrically smaller than tev and even if one goes past that point, O(S) quanta have to be measured to see a deviation of O(1). 3 Limits on de Sitter space We now consider de Sitter space. According to the covariant entropy bound, de Sitter space should have a finite maximum entropy given in 4D by the horizon area in 4G units, SdS = πH −2/G. For a black hole in asymptotically flat space it makes sense that the number of internal quantum states should be finite. After all for an external observer a black hole is a localized object, occupying a limited region of space. But for de Sitter space it is less clear how to think about the finiteness of the number of quantum states: de Sitter has infinitely large spatial sections, at least in flat FRW slicing, and continuous non-compact isometries—features that seem to clash with the idea of a finite-dimensional Hilbert space. In particular the de Sitter symmetry group SO(n, 1) has no finite-dimensional representations, so it cannot be realized in the de Sitter Hilbert space (see however Ref. [42] for a discussion on this point). However the fact that no single observer can ever experience what is beyond his or her causal horizon makes it tempting to postulate some sort of ‘complementarity’ between the outside and the inside of the horizon, in the same spirit as the black hole complementarity. From this point of view the global picture of de Sitter space would not make much sense at the quantum level. It is plausible that the global picture of de Sitter space is only a semiclassical approximation, which becomes strictly valid only in the limit where gravity is decoupled while the geometry is kept fixed. In the same limit the entropy SdS diverges, and one recovers the infinite-dimensional Hilbert space of a local QFT in a fixed de Sitter geometry. With dynamical gravity we expect tiny non- perturbative effects of order e−SdS to put fundamental limitations on how sharply one can define local observables, in the spirit of sect. 2.1. These tiny effects can have dramatic consequences in situations where they are enhanced by huge ∼ e+SdS multiplicative factors. For instance it is widely believed that on a timescale of order H−1eSdS—the Poincaré recurrence time—de Sitter space necessarily suffers from instabilities and no consistent theory of pure de Sitter space is possible; although this view has been seriously challenged by Banks [43, 44, 45]. Notice however that the near-horizon geometry of de Sitter space is identical to that of a black hole—they are both equivalent to Rindler space. As we discussed in sect. 2.2, in the black hole case the local EFT description must break down at a time tev ∼ RS · SBH after the formation of the black hole itself. It is natural to conjecture that a similar breakdown of EFT occurs in de Sitter space after a time of order H−1 · SdS. This is an extremely shorter timescale than the Poincaré recurrence time, which is instead exponential in the de Sitter entropy. 3.1 Slow-roll inflation Is there a way to be more concrete? In pure de Sitter any observer has access only to a small portion of the full spacetime, and it is not even clear what the observables are [46]. But we can make better sense of de Sitter space if we regulate it by making it a part of inflation. If inflation eventually ends in a flat FRW cosmology with zero cosmological constant, then asymptotically in the future every observer will have access to the whole of spacetime. In particular an asymptotic observer can detect—in the form of density perturbations—modes that exited the cosmological horizon during the near-de Sitter inflationary epoch. Notice that from this point of view it looks perfectly sensible to talk about what is outside the early de Sitter horizon—we even have experimental evidence that computing density perturbations by following quantum fluctuations outside the horizon is reliable—and a strict complementarity between the inside and the outside of the de Sitter horizon seems too restrictive. Now, the interesting point is that the fact that an asymptotic observer can detect modes coming from the early inflationary phase gives an operational meaning to the de Sitter degrees of freedom, and to their number. Every detectable mode corresponds to a state in the de Sitter Hilbert space. Let’s consider for instance an early phase of ordinary slow-roll inflation. Classically the inflaton φ rolls down its potential V (φ) with a small velocity φ̇cl ∼ V ′/H . On top of this classical motion there are small quantum fluctuations. Modes get continuously stretched out of the de Sitter horizon, and quantum fluctuations get frozen at their typical amplitude at horizon-crossing, δφq ∼ H (18) For a future observer, who makes observations in an epoch when the inflaton is no longer an important degree of freedom, these fluctuations are just small fluctuations of the space-like hyper- surface that determines the end of inflation. That is, since with good approximation inflation ends at some fixed value of φ, small fluctuations in φ curve this hypersurface by perturbing the local scale factor a, ∼ Hδt ∼ H δφq Such a perturbation is locally unobservable as long as its wavelength is larger than the cosmological horizon. But eventually every mode re-enters the horizon, and when this happens a perturbation in the local a translates into a perturbation in the local energy density ρ, where we made use of eq. (18). By observing density perturbations in the sky an asymptotic observer is able to assign states to the approximately de Sitter early phase. If we believe the finiteness of de Sitter entropy, the maximum number of independent modes from inflation an observer can ever detect should be bounded by the dimensionality of the de Sitter Hilbert space, dim(H) = eS. Of course slow-roll inflation has a finite duration, thus only a finite number of modes can exit the horizon during inflation and re-enter in the asymptotic future. Roughly speaking, if inflation lasts for a total of Ntot e-foldings, the number of independent modes coming from inflation is of order e 3Ntot—it is the number of different Hubble volumes that get populated starting from a single inflationary Hubble patch. If the number of e-foldings during inflation gets larger than the de Sitter entropy, Ntot & S, this operational definition of de Sitter degrees of freedom starts violating the entropy bound. In slow-roll inflation the Hubble rate slowly changes with time, Ḣ = −(4πG) φ̇2 (21) and so does the associated de Sitter entropy S = πH−2/G. In particular, the rate of entropy change per e-folding is 8π2φ̇2 where we made use of eq. (20). By integrating this equation we get a bound on the total number of e-foldings, Ntot . · Send (23) where Send is the de Sitter entropy at the end of inflation. We thus see that since δρ/ρ is smaller than one, the total number of e-foldings is bounded by the de Sitter entropy. As a consequence a future observer will never be able to associate more than eS states to the near-de Sitter early phase! By adjusting the model parameters one can make the inflationary potential flatter and flatter, thus enhancing the amplitude of density perturbations δρ/ρ. In this way, according to eq. (23) for a fixed de Sitter entropy the allowed number of e-foldings can be made larger and larger. When δρ/ρ becomes of order one we start saturating the de Sitter entropy bound, Ntot ∼ S. However exactly when δρ/ρ is of order one we enter the regime of eternal inflation. Indeed quantum fluctuations in the inflaton field, δφq ∼ H , are so large that they are of the same order as the classical advancement of the inflaton itself in one Hubble time, ∆φcl ∼ φ̇cl ·H−1, ∼ 1 (24) Now in principle there is no limit to the total number of e-foldings one can have in an inflationary patch—the field can fluctuate up the potential as easily as it is classically rolling down. Still when a future observer starts detecting modes coming from an eternal-inflation phase, precisely because they correspond to density perturbations of order unity the Hubble volume surrounding the observer will soon get collapsed into a black hole [47, 48]. Therefore a future observer will not be able to assign more than eS states to the inflationary phase. Notice that when dealing with eternal inflation we are pushing the semiclassical analysis beyond its regime of validity, by applying it to a regime of large quantum fluctuations. This is to be contrasted with standard (i.e., non-eternal) slow-roll inflation, where the semiclassical computation is under control and quantitatively reliable. This matches nicely with what we postulated above by analogy with the black hole system—that in de Sitter space the local EFT description should break down after a time of order H−1 ·S. Indeed in standard slow-roll inflation the near-de Sitter phase cannot be kept for longer than Ntot ∼ S e-foldings. Normally whether inflation is eternal or not is controlled by the microscopic parameters of the inflaton potential. For slow-roll inflation we have just given instead a macroscopic characterization of eternal inflation, involving geometric quantities only: an observer living in an inflationary Universe can in principle measure the local H and Ḣ with good accuracy, and determine the rate of entropy change per e-folding. If such a quantity is of order one, the observer lives in an eternally inflating Universe. Indeed we will see that this macroscopic characterization of eternal inflation is far more general than the simple single-field slow-roll inflationary model we are discussing here. By now we know several alternative mechanisms for driving inflation, well known examples being for instance DBI inflation [49], locked inflation [50], k-inflation [51]. These models can be thought of as different regularizations of de Sitter space—different ways of sustaining an approximately de Sitter early phase for a finite period of time before matching onto an ordinary flat FRW cosmology, thus allowing an asymptotic observer to gather information about de Sitter space. We will show in a model-independent fashion that the absence of eternal inflation requires that the Hubble rate decrease faster than a critical speed, |Ḣ| ≫ GH4 (25) This is a necessary condition for the classical motion not to be overwhelmed by quantum fluctua- tions, so that the semiclassical analysis is trustworthy. In terms of the de Sitter entropy the above inequality reads ≫ 1 (26) which once integrated limits the total number of e-folds an inflationary model can achieve without entering an eternal-inflation regime, Ntot ≪ Send (27) As pointed out by Bousso, Freivogel and Yang, the bound (26) is necessarily violated [48] in slow-roll eternal inflation, thereby avoiding conflict with the second law of thermodynamics. Indeed, during eternal inflation the evolution of the horizon area is dominated by quantum jumps of the inflaton field and can go either way during each e-folding. From ∣ < 1 one infers that the entropy changes by less than one unit during each e-folding and, consequently, its decrease is unobservable. One notable exception is ghost inflation [18]. There φ̇ and Ḣ are not tightly bound to each other like in eq. (21). Indeed there exists an exactly de Sitter solution with vanishing Ḣ but constant, non-vanishing φ̇. This is because the stress-energy tensor of the ghost condensate vacuum is that of a cosmological constant, even though the vacuum itself breaks Lorentz invariance through a non-zero order parameter 〈φ̇〉 [15]. Therefore, the requirement of not being eternally inflating still gives a lower bound on φ̇ but now this does not translate into a lower bound on |Ḣ|. Ḣ can be strictly zero, still inflation is guaranteed to end by the incessant progression of the scalar, which will eventually trigger a sudden drop in the cosmological constant [18]. Thus in ghost inflation there is no analogue of the local bounds (25) and (26), nor there is any upper bound on the total number of e-foldings. Notice however that the ghost condensate is on the verge of violating the null energy condition, having ρ + p = 0. Indeed small perturbations about the condensate do violate it. In the next subsection we will prove that our bounds are guaranteed to hold for all inflationary systems that do not admit violations of the null energy condition. This matches with the general discussion of sect. 4: the NEC is known to play an important role in the holographic bound and in general in limiting the accuracy with which one can define local observables in gravity. The fact that all reliable NEC-respecting semiclassical models of inflation obey our bounds, suggests that the latter really limit the portion of de Sitter space one can consistently talk about within local EFT. 3.2 General case Let us consider a generic inflationary cosmology driven by a collection of matter fields ψm. We want to see under what conditions the time-evolution of the system is mainly classical, with quantum fluctuations giving only negligible corrections. We could work with a completely generic matter Lagrangian, function of the matter fields and their first derivatives, and possibly including higher-derivative terms, which in specific models like ghost inflation can play a significant role. We should then: take the proper derivatives with respect to the metric to find the stress-energy tensor; plug it into the Friedmann equations and solve them; expand the action at quadratic order in the fluctuations around the classical solution; compute the size of typical quantum fluctuations; impose PSfrag repla ements end of in ation Figure 3: Love in an inflationary Universe. that they do not overcome the classical evolution. This procedure would be quite cumbersome, at the very least. Fortunately we can answer our question in general, with no reference to the actual system that is driving inflation. To this purpose it is particularly convenient to work with the effective theory for adiabatic scalar perturbations of a generic FRW Universe. This framework has been developed in Ref. [52], to which we refer for details. The idea is to focus on a scalar excitation that is present in virtually all expanding Universes: the Goldstone boson of broken time-translations. That is, given the background solution for the matter fields ψm(t), we consider the matter fluctuation δψm(x) ≡ ψm t+ π(x) − ψm(t) (28) parameterized by π(x), and the corresponding scalar perturbation of the metric as enforced by Einstein equations (after fixing, e.g., Newtonian gauge). This fluctuation corresponds to a com- mon, local shift in time for all matter fields and is what in the long wavelength limit is called an ‘adiabatic’ perturbation. As for all Goldstone bosons, its Lagrangian is largely dictated by sym- metry considerations. This is clearly the relevant degree of freedom one has to consider to decide whether eternal inflation is taking place or not. Minimally, a sufficient condition for having eternal inflation is to have large quantum fluctuations back and forth along the classical trajectory. In the presence of several matter fields other fluctuation modes will be present. For the moment we concentrate on the Goldstone alone. As we will see at the end of this section, our conclusions are unaltered by the presence of large mixings between π and extra degrees of freedom. The situation is schematically depicted in Fig. 4. Of course with dynamical gravity time-translations are gauged and formally there is no Gold- stone boson at all—it is “eaten” by the gravitational degrees of freedom and one can always fix the gauge π(x) = 0 (‘unitary gauge’). Still it remains a convenient parametrization of a particular scalar fluctuation at short distances, shorter than the Hubble scale, which plays the role of the field space classical solution Figure 4: A given cosmological history is a classical trajectory in field space (red line), parameterized by time. The Goldstone field π describes small local fluctuations along the classical solution. In general other light oscillation modes, transverse to the trajectory will also be present, and π can be mixed with them. In the picture ϕ1 and ϕ2 are the modes that locally diagonalize the quadratic Lagrangian of perturbations. The blue ellipsoid gives the typical size of quantum fluctuations. graviton Compton wavelength. This is completely analogous to the case of massive gauge theories, where the dynamics of longitudinal gauge bosons is well described by the “eaten” Goldstones at energies higher than the mass. This approach allows us to analyze essentially any model of inflation. The reason is that, no matter what the underlying model is, it produces some a(t), and in unitary gauge the effective Lagrangian breaks time diffs but as we will see is still quite constrained by preserving spatial diffs, so a completely general model can be characterized in a systematic derivative expansion with only a few parameters. The inside-horizon dynamics of the “clock” field can be simply obtained from the unitary gauge Lagrangian by re-introducing the time diff Goldstone à la Stückelberg. The construction of the Lagrangian for π is greatly simplified in ‘unitary’ gauge, π = 0. That is, by its very definition eq. (28), π(x) can always be gauged away from the matter sector through a time redefinition, t → t − π(x). Then the scalar fluctuation appears only in the metric, thus its Lagrangian only involves the metric variables. We can reintroduce π at any stage of the computation simply by performing the opposite time diffeomorphism t → t + π(x). Notice that by construction π has dimension of length. All Lagrangian terms must be invariant under the symmetries left unbroken by the background solution and by the unitary gauge choice. These are time- and space-dependent spatial diffeomorphisms, xi → xi + ξi(t, ~x). At the lowest derivative level the only such invariant is g00. Notice that, given the residual symmetries, the Lagrangian terms will have explicitly time-dependent coefficients. From the top-down viewpoint this time- dependence arises because we are expanding around the time-dependent background matter fields ψm(t) and metric a(t). Because of this, we expect the typical time-variation rate to be of order H , so that at frequencies larger than H it can be safely ignored. The matter Lagrangian in unitary gauge takes the form [52] Smatter = Ḣ g00 − 1 (3H2 + Ḣ) + F g00 + 1 where the first two terms are fixed by imposing that the background a(t) solves Friedmann equa- tions, since they contribute to ‘tadpole’ terms. F instead can be a generic function that starts quadratic in its argument δg00 ≡ g00+1, so that it doesn’t contribute to the background equations of motion, with time-dependent coefficients, F (δg00) =M4(t) (δg00)2 + M̃4(t) (δg00)3 + . . . (30) To match this description with a familiar situation, consider for instance the case of an ordinary scalar φ with a potential V driving the expansion of the Universe. If we perturb the scalar and the metric around the background solution φ0(t), a(t) and choose unitary gauge, φ(x) = φ0(t), the Lagrangian is gµν ∂µφ∂νφ− V (φ) φ̇20 g 00 − V φ0(t) which, upon using the background Friedmann equations, reproduces exactly the first two terms in eq. (29). Therefore an ordinary scalar corresponds to the case F (δg00) = 0. We can now reintroduce the Goldstone π. This amounts to performing in eq. (29) the time diffeomorphism t→ t+ π g00 → −1− 2π̇ + (∂π)2 (32) Notice that we should really evaluate all explicit functions of time like H , etc., at t+π rather than at t. However, after expanding in π, this would give rise only to non-derivative terms suppressed by H , Ḣ, etc., that can be safely neglected as long as we consider frequencies faster than H . Of course in the end we are interested in the physics at freeze-out, i.e. exactly at frequencies of order H . A correct analysis should then include these non-derivative terms for π, as well as the effect of mixing with gravity—the Goldstone is a convenient parameterization only at high frequencies. However, being only interested in orders of magnitude we can use the high-frequency Lagrangian for π and simply extrapolate our estimates down to frequencies of order H . From eq. (29) we get Lπ = M2PlḢ (∂π)2 + F − 2π̇ + (∂π)2 = (4M4 −M2PlḢ) π̇2 +M2PlḢ (~∇π)2 + higher orders (34) where we neglected a total derivative term and we expanded F as in eq. (30). At the lowest derivative level, the quadratic Lagrangian for π only has one free parameter, M4. The only constraint onM4 is that it must be positive for the propagation speed of π fluctuations (the ‘speed of sound’, from now on) c2 ≡ M2Pl|Ḣ|/(4M4 +M2Pl|Ḣ|) to be smaller than one. For instance, a relativistic scalar with c2 = 1 corresponds to M4 = 0; a perfect fluid with constant equation of state 0 < w < 1 corresponds to M4 =M2Pl|Ḣ| (1− w)/w. IfM4 .M2Pl|Ḣ| the speed of sound is of order one and we can repeat exactly the same analysis as in the case of slow roll inflation, modulo straightforward changes in the notation. Therefore, let us concentrate on the case c2 ≪ 1, M4 ≫ M2Pl|Ḣ|; the Lagrangian further simplifies to Lπ = 4M4 π̇2 +M2PlḢ (~∇π)2 + higher orders (35) We now want to use the Lagrangian (35) to estimate the size of quantum fluctuations, and to impose that they don’t overcome the classical evolution of the system. For the latter requirement the π language is particularly convenient: π is the perturbation of the classical ‘clock’ t, directly in time units, so we just have to impose π̇ ≪ 1 at freeze-out, that is at frequencies of order H . Alternatively, in unitary gauge we can look at the dimensionless perturbation in the metric, = H π (36) so that imposing ζ ≪ 1 at freeze-out we get the same condition for π as above. The typical size of the vacuum quantum fluctuations for a non-relativistic, canonically nor- malized field φ with a generic speed of sound c ∼ ω/k at frequencies of order ω is 〈φ2〉ω ∼ where the ω in the denominator comes from the canonical wave-function normalization, and the k3 in the numerator from the measure in Fourier space. Taking into account the non-canonical normalization of π, at frequencies of order H we have 〈π2〉H ∼ M4 c3 The size of quantum fluctuation is enhanced for smaller sound speeds c. And since c2 is pro- portional to |Ḣ|, clearly there will be a lower bound on |Ḣ| below which the system is eternally inflating. Indeed imposing 〈π̇2〉H ≪ 1 and using c2 =M2Pl|Ḣ|/M4 we directly get |Ḣ| ≫ 1 GH4 (39) which in the limit c ≪ 1 is even stronger than eq. (25). From this the constraint dS ≫ 1 immediately follows. This proves our bounds for all models in which the physics of fluctuations is correctly described by the Goldstone two-derivative Lagrangian, eq. (35). This class includes for instance all single- field inflationary models where the Lagrangian is a generic function of the field and its first derivatives, L = P (∂φ)2, φ , from slow-roll inflation to k-inflation models [51]. It is however useful to consider an even stronger bound that comes from taking into account non-linear interactions of π. This bound will be easily generalizable to theories with sizable higher-derivative corrections to the quadratic π Lagrangian, like the ghost condensate. This is where the null energy condition comes in. The null energy condition requires that the stress-energy tensor contracted with any null vector nµ be non-negative, Tµν n µnν ≥ 0. We can read off the stress energy tensor from the matter action in unitary gauge eq. (29) by performing the appropriate derivatives with respect to the metric. Given a generic null vector nµ = (n0, ~n) the relevant contraction is Tµν n µnν = −2 (n0)2 M2PlḢ + F ′(δg00) where δg00 = g00+1 is the fluctuation in g00 around the background. In a more familiar notation, for a scalar field with a generic Lagrangian L = P (X, φ), X ≡ (∂φ)2, the above contraction is just Tµν n µnν = 2 (nµ ∂µφ) 2 ∂XP , so the NEC is equivalent to ∂XP ≥ 0. On the background solution δg00 vanishes and since F ′(0) vanishes by construction, the NEC is satisfied—of course as long as Ḣ is negative, as we are assuming. However F ′′(0) = M4 is positive, making F ′ positive for positive δg00. As a consequence the r.h.s. of eq. (40) is pushed towards negative values for positive δg00. So the NEC tends to be violated in the vicinity of the background solution unless higher order terms in the expansion of F , eq. (30), save the day, see Fig. 5. But this can only happen if their coefficient is large enough. For instance in order for the n-th order term to keep eq. (40) positive definite its coefficient must be at least as large as M4 (M4/M2PlḢ) n−2. The smaller |Ḣ|, the closer is the background solution to violate the NEC, and so the larger is the ‘correction’ needed not to violate it. But then if higher derivatives of F on the background solution are large, self-interactions of π are strong. Minimally, we don’t want π fluctuations to be strongly coupled at frequencies of order H . If this happened the semiclassical approximation would break down, and the classical background solution could not be trusted at all—quantum effects would be as important as the classical dynamics in determining the evolution of the system, much like in the usual heuristic picture of eternal inflation. Recall that the argument of F expressed in terms of the Goldstone is δg00 = −2π̇ − π̇2 + (~∇π)2. Given an interaction term (δg00)n, it is easy to check that for fixed n the most relevant π interactions come from taking only the linear π̇ term in δg00, i.e. (δg00)n → π̇n. Therefore, if eq. (40) is kept positive definite thanks to the n-th order term in the Taylor expansion of F , the ratio of the π self-interaction induced by this term and the free kinetic energy of π is M4 (M4/M2Pl|Ḣ| )n−2 π̇n M4 π̇2 M4 π̇ M2Pl|Ḣ| M2Pl|Ḣ| · c3/2 M2Pl|Ḣ| · c5 where we plugged in the size of typical quantum fluctuations at frequencies of orderH , eq. (38), and we used the fact thatM2Pl|Ḣ| = c2M4. From eq. (41) it is evident that if we require that quantum fluctuations be weakly coupled at frequencies of order H we automatically get the constraint |Ḣ| ≫ 1 GH4 , dS ≫ 1 dN (42) on the background classical solution. strong coupling − M 2Pl Figure 5: The null energy condition is violated whenever F ′(δg00) enters the shaded region, F ′+M2P lḢ > 0. Since F ′ starts with a strictly positive slope at the origin, to avoid this one needs that higher derivatives of F bend F ′ away from the NEC-violating region. The smaller |Ḣ|, the stronger the needed ‘bending’. This can make π fluctuations strongly coupled at H. The above proof holds in all cases where the Goldstone two-derivative Lagrangian, eq. (35) is a good description of the physics of fluctuations. However, when |Ḣ| is very small the (~∇π)2 term appears in the Lagrangian with a very small coefficient, and one can worry that higher derivative corrections to the π quadratic Lagrangian start dominating the gradient energy. This is exactly what happens in ghost inflation, where the (~∇π)2 term is absent—in agreement with the vanishing of Ḣ—and the spatial-gradient part of the quadratic Lagrangian is dominated by the (∇2π)2 term, which enters the Lagrangian with an arbitrary coefficient [15, 18]. In such cases, at all scales where the gradient energy is dominated by higher derivative terms one has M2Pl|Ḣ| < c2M4, where c is the propagation speed, simply because the (∇π)2 term of eq. (35) is not the dominant source of gradient energy, thus the sound speed is dominated by other sources. So the last equality in eq. (41) becomes a ‘>’ sign, and our bound gets even stronger. Therefore our results equally apply to theories where higher derivative corrections can play a significant role, like the ghost condensate. In summary: imposing that the NEC is not violated in the vicinity of the background solution implies sizable non-linearities in the system. For smaller |Ḣ| the system is closer to violating the NEC—Ḣ = 0 saturates the NEC. So the smaller |Ḣ|, the larger the non-linearities needed to make the system healthy. Requiring that fluctuations not be strongly coupled at the scale H—a necessary condition for the applicability of the semiclassical description—sets a lower bound on |Ḣ|, eq. (42). So far we neglected possible mixings of π with other light fluctuation modes. However our conclusion are unaltered by the presence of such mixings. At any give moment of time t the quadratic Lagrangian for fluctuations can be diagonalized, L = 1 ϕ̇2i − c2i (~∇ϕi)2 (43) Typical quantum fluctuations now define an ellipsoid in the ϕi’s space, whose semi-axes depend on the individual speeds ci (see Fig. 4). The Goldstone π corresponds to some specific direction in field space, and in any direction quantum fluctuations are bounded from below by the shortest semi-axis. By requiring that the system does not enter eternal inflation it is straightforward to show that our bound (39) generalizes to |Ḣ| ≫ 1 GH4 , dS ≫ 1 dN (44) where cmax ≤ 1 is the maximum of the ci’s. The generalization to theories in which higher spatial derivative terms are important proceeds along the same lines as in the case of the π alone, by imposing that the NEC is not violated along π and that π fluctuations are not strongly coupled at H . 4 Null energy condition and thermodynamics of horizons The proof of our central result (26) and the related interpretation of what finite de Sitter entropy means crucially relies on the null energy condition, µnν ≥ 0 (45) where nµ is null. The history of general relativity knows many examples when the assumed “energy conditions”—assumptions about the properties of physically allowed energy-momentum tensors— turned out to be wrong. In the end, the NEC is also known to be violated both by quantum effects (Casimir energy, Hawking evaporation) and by non-perturbative objects (orientifold planes in string theory). So it is important to clarify to what extent the violation of the NEC needed to get around the bound (26) is qualitatively different from these examples, and why the relevance of the NEC in our proof is more than just a technicality. Note first that all qualitative arguments of section 2.1, indicating that sharply defined local observables are absent in quantum gravity, implicitly rely on the notion of positive gravitational energy. Indeed, schematically these arguments reduce to saying that, by the uncertainty principle, preparing arbitrarily precise clocks and rods requires concentrating indefinitely large energy in a small volume. Then the self-gravity of clocks and rods themselves causes the volume to collapse into a black hole and screws up the result of the measurement. Clearly this problem would not be there if there were some negative gravitational energy available around. Using this energy one would be able to screen the self-gravity of clocks and rods and to perform an arbitrarily precise local measurement. NEC is a natural candidate to define what the positivity of energy means; at the end it is the only energy condition in gravity that cannot be violated by just changing the vacuum part of the energy-momentum, Tµν → Tµν+Λgµν . Indeed the NEC is a crucial assumption in proving the positivity of the ADM mass in asymptotically flat spaces [53, 54]. Generically, classical field theoretic systems violating NEC suffer from either ghost or rapid gradient instabilities. In a very broad class of systems, including conventional relativistic fluids, these instability can be proven [55] to originate from the“clock and rod” sector of the system—one of the Goldstones of the spontaneously broken space-time translations is either a ghost or has an imaginary propagation speed. For instance, if space translations are not spontaneously broken and only the Goldstone of time translations (the “clock” field π of section 3.2) is present, then the instability is due to the wrong-sign gradient energy in the Goldstone Lagrangian (35) in the NEC violating case Ḣ > 0. The examples of stable NEC violations we mentioned above avoid this problem by either being quantum and non-local effects (Casimir energy and Hawking process) or by projecting out the corresponding Goldstone mode (orientifold planes). This allows to avoid the instability, but simultaneously makes these systems incapable of providing the non-gravitating clocks and rods. Nevertheless, stable effective field theories describing non-gravitating systems of clocks and rods can be constructed. This is the ghost condensate model [15] where space diffs are unbroken, and so only the clock field appears, as well as more general models describing gravity in the Higgs phase where Goldstones of the space diffs are present as well [16, 17]. All these setups provide constructions of de Sitter space with intrinsic clock variable and thus allow to get around our bound (26). Related to that, all these theories describe systems on the verge of violating NEC, and small perturbations around their vacuum violate it. Nevertheless these effective theories avoid rapid instabilities as a combined result of taking into account the higher derivative operators in the Goldstone sector and of imposing special symmetries. Does the existence of these counterexamples cause problems in relating the bound (26) to the fundamental properties of de Sitter space in quantum gravity? We believe that the answer is no, and that actually the opposite is true—this failure of the bound (26) provides a quite non-trivial support to the idea that the bound is deeply related to de Sitter thermodynamics. The reason is that the conventional black hole thermodynamics also fails in these models [19]. To see how this can be possible, note that, more or less by construction, all these models spontaneously break Lorentz invariance. For instance, in the ghost condensate Minkowski or de Sitter vacuum a non-vanishing time-like vector—the gradient of the ghost condensate field ∂µφ—is present. As usual in Lorentz violating theories the maximum propagation velocities need not be universal for different fields, now as a consequence of the direct interactions with the ghost condensate. Being a consistent non-linear effective theory, ghost condensate allows to study the consequences of the velocity differences in a black hole background. The result is very simple— the effective metric describing propagation of a field with v 6= 1 in a Schwarzschild background has the Schwarzschild form with a different value of the mass. As one could have expected, the black hole horizon appears larger for subluminal particles and smaller for superluminal ones. As a consequence, the temperature of the Hawking radiation is not universal any longer; “slow” fields are radiated with lower temperatures than “fast” fields. Figure 6: In the presence of the ghost condensate black holes can have different temperatures for different fields. This allows to perform thermodynamic transformations whose net effect is the transfer of heat Q2 from a cold reservoir at temperature T2 to a hotter one at temperature T1 (left). Then one can close a cycle by feeding heat Q1 at the higher temperature T1 into a machine that produces work W and as a byproduct releases heat Q2 at the lower temperature T2 (right). The net effect of the cycle is the conversion of heat into mechanical work. Also the horizon area does not have a universal meaning any longer, making it impossible to define the black hole entropy just as a function of mass, angular momentum and gauge charges. To make the conflict with thermodynamics explicit, let us consider a black hole radiating two different non-interacting species with different Hawking temperatures TH1 > TH2. Let us bring the black hole in thermal contact with two thermal reservoirs containing species 1 and 2 and having temperatures T1 and T2 respectively. By tuning these temperatures one can arrange that they satisfy TH1 > T1 > T2 > TH2 and the thermal flux from the black hole to the first reservoir is exactly equal in magnitude to the flux from the second reservoir to the black hole. As a result the mass of the black hole remains unchanged and the heat is transferred from the cold to the hot body in contradiction with the second law of thermodynamics, see Fig. 6. The case for violation of the second law of black hole thermodynamics in models with sponta- neous Lorentz violation is even strengthened by the observation [20] that the same conclusion can be achieved purely at the classical level and without neglecting the interaction between the two species. This classical process is analogous to the Penrose process. Namely, in a region between the two horizons the energy of the “slow” field can be negative similarly to what happens in the ergosphere of a Kerr black hole. The fast field can escape from this region making it possible to arrange an analogue of the Penrose process. In the case at hand, this process just extracts energy from the black hole by decreasing its mass. The mass decrease can be compensated by throwing in more entropic stuff, which again results in an entropy decrease outside with the black hole parameters remaining unchanged (this does not happen in the conventional Penrose process because the angular momentum of the black hole changes). Actually, it is not surprising at all that a violation of the NEC implies the breakdown of black hole thermodynamics, as the NEC is needed in the proof [56] of the covariant entropy bound [57], which is one of the basic ingredients of black hole thermodynamics and holography. Also note that the above conflict with thermodynamics is just a consequence of spontaneous breaking of Lorentz invariance (existence of non-gravitating clocks); in particular, it is there even if one assumes that all fields propagate subluminally. The second law of thermodynamics is a consequence of a few very basic properties, such as unitarity, so it is expected to hold in any sensible quantum theory. Hence, the only chance for Lorentz violating models to be embedded in a consistent microscopic theory is if black holes are not actually black in these theories, so that the observer can measure both the inside and the outside entropy and there is no need for a purely outside counting as provided by the Bekenstein formula (this is indeed what happens if space diffs are broken as well, due to the existence of instantaneous interactions). In any case, this definitely puts the ghost condensate with other Lorentz violating models in a completely different ballpark from GR as far as the physics of horizons goes. That is why we find it encouraging for a thermodynamical interpretation of the bound (26) that is also violated by the ghost condensate. 5 Open questions We have seen that all NEC-obeying models of inflation that do not eternally inflate increase the de Sitter at a minimal rate, dS/dN ≫ 1, and therefore cannot sustain an approximate de Sitter phase for longer that N ∼ S e-foldings. This gives an observational way of determining whether or not inflation is eternal. For instance, if our current accelerating epoch lasts for longer than ∼ 10130 years, or if (1+w) is smaller than 10−120, our current inflationary epoch is eternal. While these are somewhat challenging measurements, they can at least be done at timescales shorter than the recurrence time! This bound implies that an observer exiting into flat space in the asymptotic future cannot detect more than eS independent modes coming from inflation, which matches nicely with the idea of de Sitter space having a finite-dimensional Hilbert space of dimension ∼ eS. Although we are not able to provide a microscopic counting of de Sitter entropy, we can at least give an operational meaning to the number of de Sitter degrees of freedom. The NEC is very important in proving our bound; indeed the NEC is crucial in existing derivations of various holographic bounds, and indeed consistent EFTs that violate the NEC like the ghost condensate are also known to violate the thermodynamics of black hole horizons. This suggests that our bound is related to holography. We can view different inflationary models as possible regularizations of pure de Sitter space in which a semiclassical analysis in terms of a local EFT is reliable. Then our universal bound suggests that any semiclassical, local description of de Sitter space cannot be trusted past ∼ S Hubble times and further than ∼ eS Hubble radii in space—perhaps a more covariant statement Figure 7: A possible covariant generalization of our bound. Given an observer’s worldline and a“start” and an “end” times (red dots), one identifies the portion of de Sitter spacetime that is detectable by the observer in this time interval (shaded regions). Then EFT properly describes such a region only for spacetime volumes smaller than ∼ eSH−4. If applied to eternal de Sitter (left) this gives the Poincaré recurrence time eSH−1 times the causal patch volume H−3. If applied to an FRW observer after inflation (right) it gives S e-foldings times eS Hubble volumes. is that the largest four-volume one can consistently describe in terms of a local EFT is of order eSH−4, see Fig. 7. Notice that this is analogous to what happens for a black hole: in order not to violate unitarity the EFT description must break down after a time of order S Schwarzschild times, when more than eS modes must be invoked behind the horizon to accommodate the entanglement entropy. Ultimately we are interested in eternal inflation, in particular in its effectiveness in populating the string landscape. In this case the relevant mechanism is false vacuum eternal inflation, in which there is no classically rolling scalar to begin with, and the evolution of the Universe is governed by quantum tunneling. Our analysis does not directly apply here—there is no classical non-eternal version of this kind of inflation. In particular, in the slow roll eternal inflation case an asymptotic future observer only has access to the late phase of inflation, when the Universe is not eternally inflating. The eternal inflation part corresponds to density perturbations of order unity, thus making the Hubble volume surrounding the observer collapse when they become observable. As a consequence the number of possible independent measurements such an observer can make is always bounded by eS. In the false vacuum eternal inflation case instead there can be asymptotic observers who live in a zero cosmological constant bubble. This is the case if the theory does not have negative energy vacua, or if the zero energy ones are supersymmetric, and therefore perfectly stable. Such zero- energy bubbles are occasionally hit from outside by small bubbles that form in their vicinity, but these collisions are not very energetic and do not perturb significantly the bubble evolution—the observer Figure 8: (Left) In false vacuum eternal inflation there seems to be no limit to the spacetime volume of the outside de Sitter space an asymptotic flat-space observer can detect. The spacetime volumes diverges in the shaded corners. Bubble collisions don’t alter this conclusion; the pattern of collisions is simply depicted on a Poincaré disk representation of the hyperbolic FRW spatial slices (Right). Maloney, Shenker and Susskind argue that observers in the bubbles can make an infinite number of observations and arrive at sharply defined observables. total probability of being hit and eaten by a large bubble is small, of order ΓH−4 ≪ 1, where Γ is the typical transition rate per unit volume. By measuring the remnants of such collisions the observer inside the bubble can gather information about the outside de Sitter space and the landscape of vacua [58]. Then, in this case these measurements play the same role in giving an operational definition of de Sitter degrees of freedom as density perturbations did in slow- roll inflation. But now there seems to be no limit to how many independent measurements an asymptotic observer can make. The expected total number of bubble collisions experienced by a zero-energy bubble is infinite, and with very good probability none of these collisions destroys the bubble. It is true that as time goes on for such an observer it becomes more and more difficult to perform these measurements—collisions get rarer and rarer, and their observational consequences get more and more redshifted. Still we have not been able to find a physical reason why these observations cannot be done, at least in principle. The asymptotic observer in the bubble can in principle perform infinitely many independent measurements, and Maloney, Shenker and Susskind argue that these might give sharply defined observables [58]. The case of collisions with negative vacuum energy supersymmetric bubbles is particularly interesting; in this case, as the boundary of the zero energy bubble is covered by an infinite fractal of domain-wall horizons [59], the pattern of bubble collisions with other supersymmetric vacua as seen on the hyperbolic spatial slices of the bubble FRW Universe is shown in Fig. (8) where the hyperbolic space is represented as a Poincaré disk; at early times the walls are at the boundary while at infinite time they asymptote to fixed Poincaré co-ordinates as shown. The pattern of collisions is scale-invariant, reflecting the origin of the bubbles in the underlying de Sitter space. Still, it appears that an observer away from these walls can make an infinite number of observations. This apparently violates the expectation that one should not be able to assign more than eS independent states to de Sitter space. Perhaps false vacuum eternal inflation is a qualitatively different regularization of de Sitter space than offered by the class of inflationary models we studied for our bound. There may be some more subtle effect that prevents the bubble observer from making observations with better than e−S accuracy of the ambient de Sitter space. Or perhaps the limitation is correct, and it is the effective field theory description that is breaking down when more than eS observations are allowed, much as in black hole evaporation. We believe these issues deserve further investigation. Acknowledgments We thank Tom Banks, Raphael Bousso, Ben Freivogel, Steve Giddings, David Gross, Don Marolf, Joe Polchinski, Leonardo Senatore, and Andy Strominger for stimulating discussions. We es- pecially thank Juan Maldacena for clarifying many aspects of the information paradox and de Sitter entropy, and also Alex Maloney and Steve Shenker for extensive discussions of their work in progress with Susskind. 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0704.1815

Kondo-lattice screening in a d-wave superconductor

Kondo-lattice screening in a d-wave superconductor Daniel E. Sheehy1,2 and Jörg Schmalian2 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803 Department of Physics and Astronomy, Iowa State University and Ames Laboratory, Ames IA 50011 (Dated: November 28, 2018) We show that local moment screening in a Kondo lattice with d-wave superconducting conduction electrons is qualitatively different from the corresponding single Kondo impurity case. Despite the conduction-electron pseudogap, Kondo-lattice screening is stable if the gap amplitude obeys ∆ <√ TKD, in contrast to the single impurity condition ∆ < TK (where TK is the Kondo temperature for ∆ = 0 and D is the bandwidth). Our theory explains the heavy electron behavior in the d-wave superconductor Nd2−xCexCuO4. I. INTRODUCTION The physical properties of heavy-fermion metals are commonly attributed to the Kondo effect, which causes the hybridization of local 4-f and 5-f electrons with itin- erant conduction electrons. The Kondo effect for a single magnetic ion in a metallic host is well understood1. In contrast, the physics of the Kondo lattice, with one mag- netic ion per crystallographic unit cell, is among the most challenging problems in correlated electron systems. At the heart of this problem is the need for a deeper un- derstanding of the stability of collective Kondo screen- ing. Examples are the stability with respect to com- peting ordered states (relevant in the context of quan- tum criticality2) or low conduction electron concentra- tion (as discussed in the so-called exhaustion problem3). In these cases, Kondo screening of the lattice is believed to be more fragile in comparison to the single-impurity case. In this paper, we analyze the Kondo lattice in a host with a d-wave conduction electron pseudogap4. We demonstrate that Kondo lattice screening is then sig- nificantly more robust than single impurity screening. The unexpected stabilization of the state with screened moments is a consequence of the coherency of the hy- bridized heavy Fermi liquid, i.e. it is a unique lattice ef- fect. We believe that our results are of relevance for the observed large low temperature heat capacity and sus- ceptibility of Nd2−xCexCuO4, an electron-doped cuprate superconductor5. The stability of single-impurity Kondo screening has been investigated by modifying the properties of the con- duction electrons. Most notably, beginning with the work of Withoff and Fradkin (WF)6, the suppression of the single-impurity Kondo effect by the presence of d-wave superconducting order has been studied. A variety of an- alytic and numeric tools have been used to investigate the single impurity Kondo screening in a system with conduc- tion electron density of states (DOS) ρ (ω) ∝ |ω|r, with variable exponent r (see Refs. 6,7,8,9,10,11,12). Here, r = 1 corresponds to the case of a d-wave superconduc- tor, i.e. is the impurity version of the problem discussed in this paper. For r ≪ 1 the perturbative renormaliza- tion group of the ordinary13 Kondo problem (r = 0), can be generalized6. While the Kondo coupling J is marginal, 0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2 local moment screened Kondo lattice screened impurity FIG. 1: The solid line is the critical pairing strength ∆c for T → 0 [Eq. (33)] separating the Kondo screened (shaded) and local moment regimes in the Kondo-lattice model Eq. (4). Following well-known results6,7 (see also Ap- pendix A), the single-impurity Kondo effect is only stable for ∆ . D exp(−2D/J) ∼ TK (dashed). a fixed point value J∗ = r/ρ0 emerges for finite but small r. Here, ρ0 is the DOS for ω = D with bandwidth D. Kondo screening only occurs for J∗ and the transition from the unscreened doublet state to a screened singlet ground state is characterized by critical fluctuations in time. Numerical renormalization group (NRG) calculations demonstrated the existence of a such an impurity quan- tum critical point even if r is not small but also revealed that the perturbative renormalization group breaks down, failing to correctly describe this critical point9. For r = 1, Vojta and Fritz demonstrated that the universal properties of the critical point can be un- derstood using an infinite-U Anderson model where the level crossing of the doublet and singlet ground states is modified by a marginally irrelevant hybridization be- tween those states10,11. NRG calculations further demon- strate that the non-universal value for the Kondo cou- pling at the critical point is still given by J∗ ≃ r/ρ0, even if r is not small8. This result applies to the case of broken particle-hole symmetry, relevant for our compari- son with the Kondo lattice. In the case of perfect particle http://arxiv.org/abs/0704.1815v3 hole symmetry it holds that8 J∗ → ∞ for r ≥ 1/2. The result J∗ ≃ r/ρ0 may also be obtained from a large N mean field theory6, which otherwise fails to properly describe the critical behavior of the transition, in partic- ular if r is not small. The result for J∗ as the transition between the screened and unscreened states relies on the assumption that the DOS behaves as ρ (ω) ∝ |ω|r all the way to the bandwidth. However, in a superconductor with nodes we expect that ρ (ω) ≃ ρ0 is essentially con- stant for |ω| > ∆, with gap amplitude ∆, altering the predicted location of the transition between the screened and unscreened states. To see this, we note that, for en- ergies above ∆, the approximately constant DOS implies the RG flow will be governed by the standard metallic Kondo result1,13 with r = 0, renormalizing the Kondo coupling to J̃ = J/ (1− Jρ0 lnD/∆) with the effective bandwidth ∆ (see Ref. 9). Then, we can use the above result in the renormalized system, obtaining that Kondo screening occurs for J̃ρ0 & r which is easily shown to be equivalent to the condition ∆ . ∆∗ with ∆∗ = e 1/rTK, (1) where TK = D exp , (2) is the Kondo temperature of the system in the absence of pseudogap (which we are using here to clarify the typ- ical energy scale for ∆∗). Setting r = 1 to establish the implication of Eq. (1) for a d-wave superconductor, we see that, due to the d-wave pseudogap in the density of states, the conduction electrons can only screen the im- purity moment if their gap amplitude is smaller than a critical value of order the corresponding Kondo temper- ature TK for constant density of states. In particular, for ∆ large compared to the (often rather small) energy scale TK, the local moment is unscreened, demonstrating the sensitivity of the single impurity Kondo effect with respect to the low energy behavior of the host. Given the complexity of the behavior for a single im- purity in a conduction electron host with pseudogap, it seems hopeless to study the Kondo lattice. We will show below that this must not be the case and that, moreover, Kondo screening is stable far beyond the single-impurity result Eq. (1), as illustrated in Fig. 1 (the dashed line in this plot is Eq. (1) with ρ0 = 1/2D). To do this, we utilize a the large-N mean field theory of the Kondo lattice to demonstrate that the transition between the screened and unscreened case is discontinuous. Thus, at least within this approach, no critical fluctuations oc- cur (in contrast to the single-impurity case discussed above). More importantly, our large-N analysis also finds that the stability regime of the Kondo screened lattice is much larger than that of the single impurity. Thus, the screened heavy-electron state is more robust and the local-moment phase only emerges if the conduction elec- tron d-wave gap amplitude obeys ∆ > ∆c ≃ TKD ≫ TK, (3) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 local moment screened Kondo lattice FIG. 2: (Color online) The solid line is a plot of the Kondo temperature TK(∆), above which V = 0 (and Kondo screen- ing is destroyed), normalized to its value at ∆ = 0 [Eq. (14)], as a function of the d-wave pairing amplitude ∆, for the case of J = 0.3D and µ = −0.1D. With these parameters, TK(0) = 0.0014D, and ∆c, the point where TK(∆) reaches zero, is 0.14D [given by Eq. (33)] The dashed line indicates a spinodal, along which the term proportional to V 2 in the free energy vanishes. At very small ∆ < 2.7 × 10−4D, where the transition is continuous, the dashed line coincides with the solid line. with D the conduction electron bandwidth. Below, we shall derive a more detailed expression for ∆c; in Eq. (3) we are simply emphasizing that ∆c is large compared to TK [and, hence, Eq. (1)]. In addition, we find that for ∆ < ∆c, the renormalized mass only weakly depends on ∆, except for the region close to ∆c. We give a detailed explanation for this en- hanced stability of Kondo lattice screening, demonstrat- ing that it is a direct result of the opening of a hybridiza- tion gap in the heavy Fermi liquid state. Since the re- sult was obtained using a large-N mean field theory we stress that such an approach is not expected to properly describe the detailed nature close to the transition. It should, however, give a correct order of magnitude result for the location of the transition. To understand the resilience of Kondo-lattice screen- ing, recall that, in the absence of d-wave pairing, it is well known that the lattice Kondo effect (and concomi- tant heavy-fermion behavior) is due a hybridization of the conduction band with an f -fermion band that rep- resents excitations of the lattice of spins. A hybridized Fermi liquid emerges from this interaction. We shall see that, due to the coherency of the Fermi liquid state, the resulting hybridized heavy fermions are only marginally affected by the onset of conduction-electron pairing. This weak proximity effect, with a small d-wave gap ampli- tude ∆f ≃ ∆TK/D for the heavy fermions, allows the Kondo effect in a lattice system to proceed via f -electron- dominated heavy-fermion states that screen the local mo- ments, with such screening persisting up to much larger values of the d-wave pairing amplitude than implied by the single impurity result6,7, as depicted in Fig. 1 (which applies at low T ). A typical finite-T phase diagram is shown in Fig. 2. Our theory directly applies to the electron-doped cuprate Nd2−xCexCuO4, possessing both d-wave superconductivity14,15 with Tc ≃ 20K and heavy fermion behavior below5 TK ∼ 2 − 3K. The latter is exhibited in a large linear heat capacity coefficient γ ≃ 4J/(mol×K2) together with a large low-frequency susceptibility χ with Wilson ratio R ≃ 1.6. The lowest crystal field state of Nd3+ is a Kramers doublet, well separated from higher crystal field levels16, supporting Kondo lattice behavior of the Nd-spins. The supercon- ducting Cu-O-states play the role of the conduction electrons. Previous theoretical work on Nd2−xCexCuO4 discussed the role of conduction electron correlations17. Careful investigations show that the single ion Kondo temperature slightly increases in systems with elec- tronic correlations18,19, an effect essentially caused by the increase in the electronic density of states of the conduction electrons. However, the fact that these con- duction electrons are gapped has not been considered, even though the Kondo temperature is significantly smaller than the d-wave gap amplitude ∆ ≃ 3.7meV (See Ref. 20). We argue that Kondo screening in Nd2−xCexCuO4 with TK ≪ ∆ can only be understood in terms of the mechanism discussed here. We add for completeness that an alternative sce- nario for the large low temperature heat capacity of Nd2−xCexCuO4 is based on very low lying spin wave excitations21. While such a scenario cannot account for a finite value of C (T ) /T as T → 0, it is consistent with the shift in the overall position of the Nd-crystal field states upon doping. However, an analysis of the spin wave contribution of the Nd-spins shows that for realistic parameters C (T ) /T vanishes rapidly below the Schottky anomaly22, in contrast to experiments. Thus we believe that the large heat capacity and susceptibility of Nd2−xCexCuO4 at low temperatures originates from Kondo screening of the Nd-spins. Despite its relevance for the d-wave superconductor Nd2−xCexCuO4, we stress that our theory does not ap- ply to heavy electron d-wave superconductors, such as CeCoIn5 (see Ref. 23), in which the d-wave gap is not a property of the conduction electron host, but a more subtle aspect of the heavy electron state itself. The latter gives rise to a heat capacity jump at the superconducing transition ∆C (Tc) that is comparable to γTc, while in our theory ∆C (Tc) ≪ γTc holds. II. MODEL The principal aim of this paper is to study the screen- ing of local moments in a d-wave superconductor. Thus, we consider the Kondo lattice Hamiltonian, possessing lo- cal spins (Si) coupled to conduction electrons (ckα) that are subject to a pairing interaction: kαckα + i,α,β Si · c†iασαβciβ + Upair. (4) Here, J is the exchange interaction between conduction electrons and local spins and ξk = ǫk − µ with ǫk the conduction-electron energy and µ the chemical potential. The pairing term Upair = − Ukk′c c−k′↓ck′↑, (5) is characterized by the attractive interaction between conduction electrons Ukk′ . We shall assume the latter stabilizes d-wave pairing with a gap ∆k = ∆cos 2θ with θ the angle around the conduction-electron Fermi surface. We are particularly interested in the low-temperature strong-coupling phase of this model, which can be studied by extending the conduction-electron and local-moment spin symmetry to SU(N) and focusing on the large-N limit24. In case of the single Kondo impurity, the large- N approach is not able to reproduce the critical behavior at the transition from a screened to an unscreeened state. However, it does correctly determine the location of the transition, i.e. the non-universal value for the strength of the Kondo coupling where the transition from screened to unscreened impurity takes place8. Since the location of the transition and not the detailed nature of the tran- sition is the primary focus of this paper, a mean field theory is still useful. Although the physical case corresponds to N = 2, the large-N limit yields a valid description of the heavy Fermi liquid Kondo-screened phase25. We thus write the spins in terms of auxiliary f fermions as Si · σαβ → f †iαfiβ − δαβ/2, subject to the constraint iαfiα = N/2. (6) To implement the large-N limit, we rescale the ex- change coupling via J/2 → J/N and the conduction- electron interaction as Uk,k′ → s−1Uk,k′ [where N ≡ (2s + 1)]. The utility of the large-N limit is that the (mean-field) stationary-phase approximation to H is be- lieved to be exact at large N . Performing this mean field decoupling of H yields k,m=−s kmckm + V kmckm + h.c. kmfkm k,m=1/2 c−k−mckm + h.c. + E0, (7) with E0 a constant in the energy that is defined below. The pairing gap, ∆k, and the hybridization between con- duction and f -electrons, V , result from the mean field de- coupling of the pairing and Kondo interactions, respec- tively. The hybridization V (that we took to be real) measures the degree of Kondo screening (and can be di- rectly measured experimentally26) and λ is the Lagrange multiplier that implements the above constraint, playing the role of the f -electron level. The free energy F of this single-particle problem can now be calculated, and has the form: F (V, λ,∆k) = ∆k∆k′U k,α=± (ξk + λ)− Ekα − T ln 1 + e−βEkα where T = β−1 is the temperature. The first three terms are the explicit expressions for E0 in Eq. (7), and Ek± is Ek± = + λ2 + 2V 2 + ξ2 Sk, (9) Sk = (∆ − λ2)2 + 4V 2 (ξk + λ) 2 +∆2 describing the bands of our d-wave paired heavy-fermion system. The phase behavior of this Kondo lattice system for given values of T , J and µ is determined by finding points at which F is stationary with respect to the variational parameters V , λ, and ∆k. For simplicity, henceforth we take ∆k as given (and having d-wave symmetry as noted above) with the goal of studying the effect of nonzero pairing on the formation of the heavy-fermion metal char- acterized by V and λ that satisfy the stationarity condi- tions = 0, (10a) = 0, (10b) with the second equation enforcing the constraint, Eq. (6). We shall furthermore restrict attention to µ < 0 (i.e., a less than half-filled conduction band). Before we proceed we point out that the magnitude of the pairing gap near the unpaired heavy-fermion Fermi surface (located at ξ = V 2/λ) is remarkably small. Tay- lor expanding Ek− near this point, we find Ek− ≃ ξ − V 2/λ− λ∆2 , (11) giving a heavy-fermion gap ∆fk = (λ/V ) ∆k [with am- plitude ∆f = ∆(λ/V ) ]. We show below that (λ/V ) ≪ 1 such that ∆fk ≪ ∆k. In Fig. 3, we plot the lower heavy-fermion band for the unpaired case ∆k = 0 (dashed line) along with ±Ek− for the case of finite ∆k (solid lines) in the vicinity of the unpaired heavy-fermion Fermi surface, showing the small heavy-fermion gap ∆fk. Thus, we find a weak proximity effect in which the heavy- fermion quasiparticles, which are predominantly of f - character, are only weakly affected by the presence of d-wave pairing in the conduction electron band. 0.6 0.8 1.0 1.2 1.4 FIG. 3: The dashed line is the lower heavy-fermion band (crossing zero at the heavy-fermion Fermi surface) for the unpaired (∆ = 0) case and the solid lines are ±Ek− for ∆k = 0.1D, showing a small f-electron gap ∆fk ≃ .014D. -1 -0.5 0.5 1 FIG. 4: Plot of the energy bands E+(ξ) (top curve) and (ξ) (bottom curve), defined in Eq. (13), in the heavy Fermi liquid state (for ∆ = 0), for the case V = 0.2D and λ = 0.04D, that has a heavy-fermion Fermi surface near ξ = D and an experimentally-measurable hybridization gap26 (the minimum value of E+ − E−, i.e., the direct gap) equal to 2V ∼ TKD. Note, however, the indirect gap is λ ∼ TK. III. KONDO LATTICE SCREENING A. Normal conduction electrons A useful starting point for our analysis is to recall the well-known27 unpaired (∆ = 0) limit of our model. By minimizing the correpsonding free energy [simply the ∆ = 0 limit of Eq. (8)], one obtains, at low temperatures, that the Kondo screening of the local moments is repre- sented by the nontrivial stationary point of F at V = V0 and λ = λ0 = V 0 /D, with D + µ , (12) Here we have taken the conduction electron density of states to be a constant, ρ0 = (2D) −1, with 2D the bandwidth. The resulting phase is a metal accommo- dating both the conduction and f -electrons with a large density of states ∝ λ0−1 near the Fermi surface at ǫk ≃ µ + V 20 /λ0, revealing its heavy-fermion character. In Fig. 4, we plot the energy bands E± (ξk) = ξk + λ± (ξk − λ)2 + 4V 2 , (13) of this heavy Fermi liquid in the low-T limit. With increasing T , the stationary V and λ decrease monotonically, vanishing at the Kondo temperature D2 − µ2 exp , (14) D − µ D + µ λ0. (15) Here, the second line is meant to emphasize that TK is of the same order as the T = 0 value of the f -fermion chemical potential λ0, and therefore TK ≪ V0, i.e., TK is small compared to the zero-temperature hybridization energy V0. It is well established that the phase transition-like be- havior of V at TK is in fact a crossover onceN is finite 1,24. Nevertheless, the large-N approach yields the correct or- der of magnitude estimate for TK and provides a very use- ful description of the strong coupling heavy-Fermi liquid regime, including the emergence of a hybridization gap in the energy spectrum. B. d-wave paired conduction electrons Next, we analyze the theory in the presence of d-wave pairing with gap amplitude ∆. Thus, we imagine contin- uously turning on the d-wave pairing amplitude ∆, and study the stability of the Kondo-screened heavy-Fermi liquid state characterized by the low-T hybridization V0, Eq. (12). As we discussed in Sec. I, in the case of a single Kondo impurity, it is well known that Kondo screening is qualitatively different in the case of d-wave pairing, and the single impurity is only screened by the conduction electrons if the Kondo coupling exceeds a critical value 1 + lnD/∆ . (16) For J < J∗, the impurity is unscreened. This result for J∗ can equivalently be expressed in terms of a critical pairing strength ∆∗, beyond which Kondo screening is destroyed for a given J : ∆∗ = D exp , (17) [equivalent to Eq. (1) for r = 1], which is proportional to the Kondo temperature TK. This result, implying 0.02 0.04 0.06 0.08 FIG. 5: (Color online) Main: Mean-field Kondo parameter V as a function of the d-wave pairing amplitude ∆, for exchange coupling J = 0.30D and chemical potential µ = −0.1D, ac- cording to the approximate formula Eq. (31) (solid line) and via a direct minimization of Eq. (8) at T = 10−4D (points), the latter exhibiting a first-order transition near ∆ = 0.086D. that a d-wave superconductor can only screen a local spin if the pairing strength is much smaller than TK, can also be derived within the mean-field approach to the Kondo problem, as shown in Appendix A (see also Ref. 7). Within this approach, a continuous transition to the unscreened phase (where V 2 → 0 continuously) takes place at ∆ ≃ ∆∗. Thus, calculations for the single impurity case indi- cate that Kondo screening is rather sensitive to a d-wave pairing gap. The question we wish to address is, how does d-wave pairing affect Kondo screening in the lattice case? In fact, we will see that the results are quite differ- ent in the Kondo lattice case, such that Kondo screening persists beyond the point ∆∗. To show this, we have nu- merically studied the ∆-dependence of the saddle point of the free energy Eq. (8), showing that, at low temper- atures, V only vanishes, in a discontinuous manner, at much larger values of ∆, as shown in Fig. 5 (solid dots) for the case of J = 0.30D, µ = −0.1D and T = 10−4D (i.e., T/TK ≃ .069). In Fig. 2, we plot the phase diagram as a function of T and ∆, for the same values of J and µ, with the solid line denoting the line of discontinuous transitions. The dashed line in Fig. 2 denotes the spinodal Ts of the free energy F at which the quadratic coefficient of Eq. (8) crosses zero. The significance of Ts is that, if the Kondo- to-local moment transition were continuous (as it is for ∆ = 0), this would denote phase boundary; the T → 0 limit of this quantity coincides with the single-impurity critical pairing Eq. (17). An explicit formula for Ts can be easily obtained by finding the quadratic coefficient of Eq. (8): tanhEk/2Ts(∆) , (18) with Ek ≡ , and where we set λ = 0 [which must occur at a continuous transition where V → 0, as can be seen by analyzing Eq. (10b)]. As seen in Fig. 2, the spinodal temperature is generally much smaller than the true transition temperature; however, for very small ∆ → 0, Ts(∆) coincides with the actual transition (which becomes continuous), as noted in the figure caption. Our next task is to understand these results within an approximate analytic analysis of Eq. (8); before do- ing so, we stress again that the discontinuous transition from a screened to an unscreened state as function of T becomes a rapid crossover for finite N . The large N the- ory is, however, expected to correctly determine where this crossover takes place. 1. Low-T limit According to the numerical data (points) plotted in Fig. 5, the hybridization V is smoothly suppressed with increasing pairing strength ∆ before undergoing a discon- tinuous jump to V = 0. To understand, analytically, the ∆-dependence of V at low-T , we shall analyze the T = 0 limit of F , i.e., the ground-state energy E. The essen- tial question concerns the stability of the Kondo-screened state with respect to a d-wave pairing gap, characterized by the following ∆-dependent hybridization V (∆) = V0 ∆2typ , (19) with ∆typ an energy scale, to be derived, that gives the typical value of ∆ for which the heavy-fermion state is affected by d−wave pairing. To show that Eq. (19) correctly describes the smooth suppression of the hybrization with increasing ∆, and to obtain the scale ∆typ, we now consider the dimensionless quantity χ∆ ≡ − , (20) that characterizes the change of the ground state en- ergy with respect to the pairing gap. Separating the amplitude of the gap from its momentum dependence, i.e. writing ∆k = ∆φk, we obtain from the Hellmann- Feynman theorem that: χ∆ = − . (21) For ∆ → 0 this yields Gcc (k,iω)Gcc (−k,−iω) . (22) Here, Gcc (k,iω) is the conduction electron propagator. As expected, χ∆ is the particle-particle correlator of the conduction electrons. Thus, for T = 0 the particle- particle response will be singular. This is the well known Cooper instability. For V = 0 we obtain for example χ∆ (V = 0) = D2 − µ2 , (23) where we used ∆ as a lower cut off to control the Cooper logarithm. Below we will see that, except for extremely small values of ∆, the corresponding Cooper logarithm is overshadowed by another logarithmic term that does not have its origin in states close to the Fermi surface, but rather results from states with typical energy V ≃√ In order to evaluate χ∆ in the heavy Fermi liquid state, we start from: Gcc (k,ω) = ω − E+ (ξk) ω − E− (ξk) , (24) where E± is given in Eq. (13) and the coherence factors of the hybridized Fermi liquid are: 1− ξk − λ√ (ξk − λ)2 + 4V 2 ξk − λ√ (ξk − λ)2 + 4V 2  . (25) Inserting Gcc (k,ω) into the above expression for χ∆ yields ∫ D−µ 4v2u2θ (E−) E+ + E− We used that E+ > 0 is always fulfilled, as we consider a less than half filled conduction band. Considering first the limit λ = 0, it holds E− (ξ) < 0 and the last term in the above integral disappears. The remaining terms simplify to χ∆ (λ = 0) = ∫ D−µ ξ2 + 4V 2 D2 − µ2 . (27) Even for λ nonzero, this is the dominant contribution to χ∆ in the relevant limit λ ≪ V ≪ D. To demonstrate this we analyze Eq. (26) for nonzero λ, but assuming λ ≪ V as is indeed the case for small ∆. The calculation is lengthy but straightforward. It follows: D2 − µ2 D |µ| . (28) The last term is the Cooper logarithm, but now in the heavy fermion state. The prefactor λ/D ≃ TK/D is a result of the small weight of the conduction electrons on the Fermi surface (i.e. where ξ ≃ V 2/λ) as well as the reduced velocity close to the heavy electron Fermi sur- face. Specifically it holds u2 ξ ≃ V 2/λ ≃ λ2/V 2 as well as E− ξ ≃ V 2/λ ξ − V Thus, except for extremely small gap values where ∆2 < D2 )−D/TK , χ∆ is dominated by the λ = 0 result, Eq. (27), and the Cooper logarithm plays no role in our analysis. The logarithm in Eq. (27) is not origi- nating from the heavy electron Fermi surface (i.e. it is not from ξ ≃ r ). Instead, it has its origin in the inte- gration over states where E− < 0. The important term in Eq. (26) is peaked for ξ ≃ 0 i.e. where E± (ξ ≃ 0) = ±V and is large as long as |ξ| . V . For ξ ≃ 0 holds v ≃ − u . This peak at ξ ≃ 0 has its origin in the competition between two effects. Usu- ally, u or v are large when E± ≃ ξ. The only regime where u or v are still sizable while E± remain small is close to the bare conduction electron Fermi surface at |ξ| ≃ V (the position of the level repulsion between the two hybridizing bands). Thus, the logarithm is caused by states that are close to the bare conduction electron Fermi surface. Although these states have the strongest response to a pairing gap, they don’t have much to do with the heavy fermion character of the system. It is in- teresting that this heavy fermion pairing response is the same even in case of a Kondo insulator where λ = 0 and the Fermi level is in the middle of the hybridization gap. The purpose of the preceding analysis was to derive an accurate expression for the ground-state energy E at small ∆. Using Eq. (20) gives: E = E(∆ = 0)− χ∆ρ0∆2, (29) which, using Eq. (27) and considering the leading order in λ ≪ V and ∆ ≪ V , safely neglecting the last term of Eq. (28) according to the argument of the previous paragraph, and dropping overall constants, yields +V 2ρ0 ln D + µ − ρ0∆ D2 − µ2 . (30) Using Eq. (10), the stationary value of the hybridization (to leading order in ∆2) is then obtained via minimization with respect to V and λ. This yields V (∆) ≃ V0 − , (31) with the stationary value of λ = 2ρ0V 2, which estab- lishes Eq. (19). A smooth suppression of the Kondo hybridization from the ∆ = 0 value V0 [Eq. (12)] oc- curs with increasing d-wave pairing amplitude ∆ at low T . This result thus implies that the conduction electron gap only causes a significant reduction of V and λ for ∆ ≃ ∆typ ∝ In Fig. 5 we compare V (∆) of Eq. (31) (solid line) with the numerical result (solid dots). As long as V 0.0001 0.0002 0.0003 0.0004 0.0005 C/γ0T FIG. 6: Plot of the low-temperature specific heat coefficient = − ∂ , for the case of λ = 10−2D, V = 10−1D, and µ = −0.1D, for the metallic case (∆ = 0, dashed line) and the case of nonzero d-wave pairing (∆ = 0.1D, solid line). This shows that, even with nonzero ∆, the specific heat coef- ficient will appear to saturate at a large value at low T (thus exhibiting signatures of a heavy fermion metal), before van- ishing at asymptotically low T ≪ ∆f (= ∆(λ/V )2 = 10−4D) Each curve is normalized to the T = 0 value for the metallic case, γ0 ≃ 23π 2/λ2. stays finite, the simple relation Eq. (31) gives an ex- cellent description of the heavy electron state. Above the small f -electron gap ∆f , these values of V and λ yield a large heat capacity coefficient (taking N = 2) γ ≃ 2 π2ρ0V 2/λ2 and susceptibility χ ≃ 2ρ0V 2/λ2, re- flecting the heavy-fermion character of this Kondo-lattice system even in the presence of a d-wave pairing gap. Ac- cording to our theory, this standard heavy-fermion be- havior (as observed experimentally5 in Nd2−xCexCuO4) will be observed for temperatures that are large com- pared to the f -electron gap ∆f . However, for very small T ≪ ∆f , the temperature dependence of the heat capac- ity changes (due to the d-wave character of the f -fermion gap), behaving as C = AT 2/∆ with a large prefactor A ≃ (D/TK)2. This leads to a sudden drop in the heat capacity coefficient at low T , as depicted in Fig. 6. The surprising robustness of the Kondo screening with respect to d-wave pairing is rooted in the weak proximity effect of the f -levels and the coherency as caused by the formation of the hybridization gap. Generally, a pairing gap affects states with energy ∆k from the Fermi en- ergy. However, low energy states that are within TK of the Fermi energy are predominantly of f -electron charac- ter (a fact that follows from our large-N theory but also from the much more general Fermi liquid description of the Kondo lattice28) and are protected by the weak prox- imity. These states only sense a gap ∆fk ≪ ∆k and can readily participate in local-moment screening. Furthermore, the opening of the hybridization gap co- herently pushes conduction electrons to energies ≃ V from the Fermi energy. Only for ∆ ≃ V ≃ TKD will the conduction electrons ability to screen the local mo- ments be affected by d-wave pairing. This situation is very different from the single impurity Kondo problem where conduction electron states come arbitrarily close to the Fermi energy. 2. First-order transition The result Eq. (31) of the preceding subsection strictly applies for ∆ → 0, although as seen in Fig. 5, in practice it agrees quite well with the numerical minimization of the free energy until the first-order transition. To under- stand the way in which V is destroyed with increasing ∆, we must consider the V → 0 limit of the free energy. We start with the ground-state energy. Expanding E [the T → 0 limit of Eq. (8)] to leading order in V and zeroth order in λ (valid for V → 0), we find (dropping overall constants) ≃ −4ρ0V 2 ln V 3, (32) where we defined the quantity ∆c ∆c = 4 D2 − µ2 exp , (33) at which the minimum value of V in Eq. (32) vanishes continuously, with the formula for V (∆) given by V (∆) ≃ 1 , (34) near the transition. According to Eq. (33), the equilib- rium hybridization V vanishes (along with the destruc- tion of Kondo screening) for pairing amplitude ∆c ∼√ TKD, of the same order of magnitude as the T = 0 hybridization V0, as expected [and advertised above in Eq. (3)]. Equation (33) strictly applies only at T = 0, appar- ently yielding a continuous transition at which V → 0 for ∆ → ∆c. What about T 6= 0? We find that, for small but nonzero T , Eq. (33) approximately yields the correct location of the transition, but that the nature of the transition changes from continuous to first-order. Thus, for ∆ near ∆c, there is a discontinuous jump to the local-moment phase that is best obtained numeri- cally, as shown above in Figs. 5 and 2. However, we can get an approximate analytic understanding of this first- order transition by examining the low-T limit. Since ex- citations are gapped, at low T the free energy FK of the Kondo-screened (V 6= 0) phase is well-approximated by inserting the stationary solution Eq. (34) into Eq. (32): 2 ln3 , (35) for FK at ∆ → ∆c. The discontinuous Kondo-to-local moment transition occurs when the Kondo free energy Eq. (35) is equal to the local-moment free energy. For the latter we set V = λ = 0 in Eq. (8), obtaining (recall ρ0(D + µ) 2 − 1 D2 − µ2 −T ln 2− T 1 + e−βEk , (36) where we dropped an overall constant depending on the conduction-band interaction. The term proportional to T in Eq. (36) comes from the fact that Ek− = 0 for V = λ = 0, and corresponds to the entropy of the local moments. At low T , the gapped nature of the d-wave quasiparticles implies the last term in Eq. (36) can be neglected (although the nodal quasi- particles give a subdominant power-law contribution). In deriving the Kondo free energy FK, Eq. (35), we dropped overall constant terms; re-establishing these to allow a comparison to FLM , and setting FLM = FK, we find 2 ln3 = T ln 2, (37) that can be solved for temperature to find the transition temperature TK for the first-order Kondo screened-to- local moment phase transition: TK(∆) = 6 ln 2 , (38) that is valid for ∆ → ∆c, providing an accurate ap- proximation to the numerically-determined TK curve in Fig. 2 (solid line) in the low temperature regime (i.e., near ∆c = 0.14D in Fig. 2). Equation (38) yields the temperature at which, within mean-field theory, the screened Kondo lattice is destroyed by the presence of nonzero d-wave pairing; thus, as long as ∆ < TK(∆), heavy-fermion behavior is compatible with d-wave pairing in our model. The essential feature of this result is that TK(∆) is only marginally reduced from the ∆ = 0 Kondo temperature Eq. (2), establishing the stability of this state. In comparison, according to ex- pectations based on a single-impurity analysis, one would expect the Kondo temperature to follow the dashed line in Fig. 2. Away from this approximate result valid at large N , the RKKY interaction between moments is expected to lower the local-moment free energy, altering the predicted location of the phase boundary. Then, even for T = 0, a level crossing between the screened and unscreened ground states occurs for a finite V . Still, as long as the ∆ = 0 heavy fermion state is robust, it will remain stable at low T for ∆ small compared to ∆c, as summarized in Figs. 1 and 2. IV. CONCLUSIONS We have shown that a lattice of Kondo spins coupled to an itinerant conduction band experiences robust Kondo screening even in the presence of d-wave pairing among the conduction electrons. The heavy electron state is pro- tected by the large hybridization energy V ≫ TK. The d-wave gap in the conduction band induces a relatively weak gap at the heavy-fermion Fermi surface, allowing Kondo screening and heavy-fermion behavior to persist. Our results demonstrate the importance of Kondo-lattice coherency, manifested by the hybridization gap, which is absent in case of dilute Kondo impurities. As pointed out in detail, the origin for the unexpected robustness of the screened heavy electron state is the coherency of the Fermi liquid state. With the opening of a hybridization gap, conduction electron states are pushed to energies of order TKD away from the Fermi energy. Whether or not these conduction electrons open up a d-wave gap is therefore of minor importance for the stability of the heavy electron state. Our conclusions are based on a large-N mean field the- ory. In case of a single impurity, numerical renormaliza- tion group calculations demonstrated that such a mean field approach fails to reproduce the correct critical be- havior where the transition between screened and un- screened impurity takes place. However the mean field theory yields the correct value for the strength of the Kondo coupling at the transition. In our paper we are not concerned with the detailed nature in the near vicin- ity of the transition. Our focus is solely the location of the boundary between the heavy Fermi liquid and un- screened local moment phase, and we do expect that a mean field theory gives the correct result. One possibility to test the results of this paper is a combination of dy- namical mean field theory and numerical renormalization group for the pseudogap Kondo lattice problem. In case where Kondo screening is inefficient and ∆ >√ TKD, i.e., the “local moment” phase of Figs. 1 and 2, the ground state of the moments will likely be magneti- cally ordered. This can have interesting implications for the superconducting state. Examples are reentrance into a normal phase (similar to ErRh4B4, see Ref. 29) or a modified vortex lattice in the low temperature magnetic phase. In our theory we ignored these effects. This is no problem as long as the superconducting gap amplitude ∆ is small compared to TKD and the Kondo lattice is well screened. Thus, the region of stability of the Kondo screened state will not be significantly affected by includ- ing the magnetic coupling between the f -electrons. Only the nature of the transition and, of course, the physics of the unscreened state will depend on it. Finally, our theory offers an explanation for the heavy fermion state in Nd2−xCexCuO4, where ∆ ≫ TK. Acknowledgments — We are grateful for useful discus- sions with A. Rosch and M. Vojta. This research was supported by the Ames Laboratory, operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. DES was also sup- ported at the KITP under NSF grant PHY05-51164. APPENDIX A: SINGLE IMPURITY CASE For a single Kondo impurity a critical value J∗ for the coupling between conduction electron and impurity spin emerges, separating Kondo-screened from local mo- ment behavior for a single spin impurity in a d-wave su- perconductor, see Eq. (16). As discussed in the main text, this is equivalent to a critical pairing Eq. (17) above which Kondo screening does not occur. The re- sult was obtained in careful numerical renormalization group calculations8,9. In the present section, we demon- strate that the same result also follows from a simple large-N mean field approach. It is important to stress that this approach fails to describe the detailed critical behavior. However, here we are only concerned with the approximate value of the non-universal quantity J∗. In- deed, mean field theory is expected to give a reasonable value for the location of the transition. Our starting point is the model Hamiltonian kmckm ++ m,m′,k,k′ f †mfmc kmck′m′ Ukk′c . (A1) with the corresponding mean-field action S = Sf + Sb + Sint with (introducing the Lagrange multiplier λ and hy- bridization V as usual, and making the BCS mean-field approximation for the conduction fermions): km(∂τ + ǫk)ckm + f m(∂τ + λ)fm V †V − λNq0 , (A2) Sint = f †mckmV + V †ckmfm m=1/2 −k−mckm + c −k−m∆k , (A3) where the λ integral implements the constraint Nq0 =∑ mfm, with q0 = 1/2. Here, we have taken the large N limit, with N = 2J + 1. The mean-field approximation having been made, it is now straightforward to trace over the fermionic degrees of freedom to yield N |V |2 − λNq0 − (iω − λ− Γ1(iω))(iω + λ+ Γ1(−iω))− Γ2(iω)Γ̄2(iω) , (A4) for the free energy contribution due to a single impurity in a d-wave superconductor. Here, we dropped an overall constant due to the conduction fermions only, as well as the quadratic term in ∆k (which of course determines the equilibrium value of ∆k; here, as in the main text, we’re interested in the impact of a given ∆k on the degree of Kondo screening), and defined the functions Γ1(iω) = |V |2 iω + ǫk (iω)2 − E2 , (A5) Γ2(iω) = V (iω)2 − E2 , (A6) Γ̄2(iω) = (V (iω)2 − E2 . (A7) At this point we note that, for a d-wave superconduc- tor, Γ2 = Γ̄2 = 0 due to the sign change of the d-wave order parameter. The self-energy Γ1(iω) is nonzero and essentially measures the density of states (DOS) ρd(ω) of the d-wave superconductor. In fact, one can show that the corresponding retarded function Γ1R(ω) satisfies Γ1R(ω) = |V |2 ρd(z) ω + iδ − z , (A8) with δ = 0+, so that the imaginary part Γ′′1R(ω) = −π|V |2ρd(ω) measures the DOS. Writing Γ1R(ω) ≡ |V |2G(ω), we have for the free energy N |V |2 − λNq0 (A9) nF(z) tan ( −|V |2G′′(z) z − λ− |V |2G′(z) and for the stationarity conditions, Eq. (10), nF(z)G ′′(z)(z − λ) (z − λ− |V |2G′(z))2 + |V |4(G′′(z))2 ,(A10) q0 = − nF(z)|V |2G′′(z) (z − λ− |V |2G′(z))2 + |V |4(G′′(z))2 ,(A11) which can be evaluated numerically to determine V and λ as a function of T and ∆. The Kondo temperature TK is defined by the temper- ature at which V 2 → 0 continuously; at such a point, the constraint Eq. (A11) requires λ → 0. Here, we are in- terested in finding the pairing ∆ at which TK → 0; thus, this is obtained by setting T = V = λ = 0 in Eq. (A10): −πρd(z) , (A12) = −ρ0 log D + µ + ρ0, (A13) where, for simplicity, in the final line we approximated ρd(z) to be given by ρd(ω) ≃ ρ0|ω|/∆, for |ω| < ∆, ρ0, for |ω| > ∆, (A14) that captures the essential features (except for the nar- row peak near ω = ∆) of the true DOS of a d-wave superconductor, with ρ0 the (assumed constant) DOS of the underlying conduction band. The solution to Eq. (A13) is: ∆∗ = (D + µ) exp , (A15) showing a destruction of the Kondo effect for ∆ → ∆∗, as V → 0 continuously, thus separating the Kondo-screened (for ∆ < ∆∗) from the local moment (for ∆ > ∆∗) phases. 1 See, e.g., A.C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, Eng- land, 1993). 2 P. Coleman, C. Pepin, Q. Si and R. Ramazashvili, Journ. of Phys: Cond. Mat. 13, R723 (2001). 3 P. Nozières, Eur. Phys. J. B 6, 447 (1998). 4 By “pseudogap”, we are of course referring to the nodal structure of d-wave pairing, not the pseudogap regime of the high-Tc cuprates. 5 T. Brugger, T. Schreiner, G. Roth, P. Adelmann, and G. Czjzek, Phys. Rev. Lett. 71, 2481 (1993). 6 D. Withoff and E. Fradkin, Phys. Rev. Lett. 64, 1835 (1990). 7 L.S. Borkowski and P.J. Hirschfeld, Phys. Rev. B 46, 9274 (1992). 8 K. Ingersent, Phys. Rev. B 54, 11936 (1996). 9 C. Gonzalez-Buxton and K. Ingersent, Phys. Rev. B 57, 14254 (1998). 10 M. Vojta and L. Fritz, Phys. Rev. B 70, 094502 (2004). 11 L. Fritz and M. Vojta, Phys. Rev. B 70, 214427 (2004). 12 M. Vojta, Philos. Mag. 86, 1807 (2006). 13 P.W. Anderson, J. Phys. C 3, 2439 (1970). 14 C.C. Tsuei and J.R. Kirtley, Phys. Rev. Lett. 85, 182 (2000). 15 R. Prozorov, R.W. Giannetta, P. Fournier, R.L. Greene, Phys. Rev. Lett. 85, 3700 (2000). 16 N.T. Hien, V.H.M. Duijn, J.H.P. Colpa, J.J.M. Franse, and A.A. Menovsky, Phys. Rev. B 57, 5906 (1998). 17 P. Fulde, V. Zevin, and G. Zwicknagl, Z. Phys. B 92, 133 (1993). 18 G. Khaliullin and P. Fulde, Phys. Rev. B 52, 9514 (1995). 19 W. Hofstetter, R. Bulla, and D. Vollhardt, Phys. Rev. Rev. Lett. 84, 4417 (2000). 20 Q. Huang, J.F. Zasadzinsky, N. Tralshawala, K.E. Gray, D.G. Hinks, J.L. Peng and R.L. Greene, Nature 347, 369 (1990). 21 W. Henggeler and A.Furrer, Journ. of Phys. Cond. Mat. 10, 2579 (1998). 22 J. Ba la, Phys. Rev. B 57, 14235 (1998). 23 C. Petrovic, P.G. Pagliuso, M.F. Hundley, R. Movshovich, J.L. Sarrao, J.D. Thompson, Z. Fisk and P. Monthoux, J. Phys. Condens. Matter 13, L337 (2001). 24 See, e.g., D.M. Newns and N. Read, Adv. Phys. 36, 799 (1987); P. Coleman, Phys. Rev. B 29, 3035 (1984); A. Auerbach and K. Levin, Phys. Rev. Lett. 57, 877 (1986); A.J. Millis and P.A. Lee, Phys. Rev. B 35, 3394 (1987). 25 H. Shiba and P. Fazekas, Prog. Theor. Phys. Suppl. 101, 403 (1990). 26 S. V. Dordevic, D. N. Basov, N. R. Dilley, E. D. Bauer, and M. B. Maple, Phys. Rev. Lett. 86, 684 (2001); L. Degiorgi, F.B.B. Anders, and G. Grüner, Eur. Phys. J. B 19, 167 (2001); J.N. Hancock. T. McKnew, Z. Schlesinger, J.L. Sarrao, and Z. Fisk, Phys. Rev. Lett. 92, 186405 (2004). 27 P. Coleman, in Lectures on the Physics of Highly Correlated Electron Systems VI , F. Mancini, Ed., American Institute of Physics, New York (2002), p. 79 - 160. 28 K. Yamada and K. Yosida, Prog. Theor. Phys. 76, 621 (1986). 29 W. A. Fertig, D. C. Johnston, L. E. DeLong, R. W. McCal- lum, M. B. Maple, and B. T. Matthias, Phys. Rev. Lett. 38, 987 (1977).

0704.1817

Redefining the Missing Satellites Problem

Draft version May 29, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 REDEFINING THE MISSING SATELLITES PROBLEM Louis E. Strigari , James S. Bullock , Manoj Kaplinghat , Juerg Diemand , Michael Kuhlen , Piero Madau Draft version May 29, 2018 ABSTRACT Numerical simulations of Milky-Way size Cold Dark Matter (CDM) halos predict a steeply rising mass function of small dark matter subhalos and a substructure count that greatly outnumbers the observed satellites of the Milky Way. Several proposed explanations exist, but detailed comparison between theory and observation in terms of the maximum circular velocity (Vmax) of the subhalos is hampered by the fact that Vmax for satellite halos is poorly constrained. We present comprehensive mass models for the well-known Milky Way dwarf satellites, and derive likelihood functions to show that their masses within 0.6 kpc (M0.6) are strongly constrained by the present data. We show that the M0.6 mass function of luminous satellite halos is flat between ∼ 107 and 108M⊙. We use the “Via Lactea” N-body simulation to show that the M0.6 mass function of CDM subhalos is steeply rising over this range. We rule out the hypothesis that the 11 well-known satellites of the Milky Way are hosted by the 11 most massive subhalos. We show that models where the brightest satellites correspond to the earliest forming subhalos or the most massive accreted objects both reproduce the observed mass function. A similar analysis with the newly-discovered dwarf satellites will further test these scenarios and provide powerful constraints on the CDM small-scale power spectrum and warm dark matter models. Subject headings: Cosmology: dark matter, theory–galaxies 1. INTRODUCTION It is now well-established that numerical simulations of Cold Dark Matter (CDM) halos predict many orders of magnitude more dark matter subhalos around Milky Way-sized galaxies than observed dwarf satellite galax- ies (Klypin et al. 1999; Moore et al. 1999; Diemand et al. 2007). Within the context of the CDM paradigm, there are well-motivated astrophysical solutions to this ‘Miss- ing Satellites Problem’ (MSP) that rely on the idea that galaxy formation is inefficient in the smallest dark matter halos (Bullock et al. 2000; Benson et al. 2002; Somerville 2002; Stoehr et al. 2002; Kravtsov et al. 2004; Moore et al. 2006; Gnedin & Kravtsov 2006). However, from an observational perspective, it has not been possi- ble to distinguish between these solutions. A detailed understanding of the MSP is limited by our lack of a precise ability to characterize the dark matter halos of satellite galaxies. From an observational per- spective, the primary constraints come from the veloc- ity dispersion of ∼ 200 stars in the population of dark matter dominated dwarf spheroidal galaxies (dSphs) (Wilkinson et al. 2004; Lokas et al. 2005; Munoz et al. 2005, 2006; Westfall et al. 2006; Walker et al. 2005, 2006; Sohn et al. 2006; Gilmore et al. 2007). However, the ob- served extent of the stellar populations in dSphs are ∼ kpc, so these velocity dispersion measurements are only able to probe properties of the halos in this limited radial 1 Center for Cosmology, Department of Physics & Astronomy, University of California, Irvine, CA 92697 2 Department of Astronomy & Astrophysics, University of Cali- fornia, Santa Cruz, CA 95064 3 School of Natural Sciences, Institute for Advanced Study, Ein- stein Drive, Princeton, NJ 08540 4 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85740 Garching, Germany. 5 Hubble Fellow regime. From the theoretical perspective, dissipationless nu- merical simulations typically characterize subhalo counts as a function of the total bound mass or maximum cir- cular velocity, Vmax. While robustly determined in sim- ulations, global quantities like Vmax are difficult to con- strain observationally because dark halos can extend well beyond the stellar radius of a satellite. Indeed stellar kinematics alone provide only a lower limit on the halo Vmax value (see below). This is a fundamental limita- tion of stellar kinematics that cannot be remedied by increasing the number of stars used in the velocity dis- persion analysis (Strigari et al. 2007). Thus determining Vmax values for satellite halos requires a theoretical ex- trapolation. Any extrapolation of this kind is sensitive to the predicted density structure of subhalos, which de- pends on cosmology, power-spectrum normalization, and the nature of dark matter (Zentner & Bullock 2003). Our inability to determinate Vmax is the primary lim- itation to test solutions to the MSP. One particular so- lution states that the masses of the dSphs have been systematically underestimated, so that the ∼ 10 bright- est satellites reside systematically in the ∼ 10 most mas- sive subhalos (Stoehr et al. 2002; Hayashi et al. 2003). A byproduct of this solution is that there must be a sharp mass cutoff at some current subhalo mass, below which galaxy formation is suppressed. Other solutions, based on reionization suppression, or a characteristic halo mass scale prior to subhalo accretion predict that the sup- pression comes in gradually with current subhalo mass (Bullock et al. 2000; Kravtsov et al. 2004; Moore et al. 2006). In this paper, we provide a systematic investigation of the masses of the Milky Way satellites. We highlight that in all cases the total halo masses and maximum http://arxiv.org/abs/0704.1817v2 2 Strigari et al. circular velocities are not well-determined by the data. We instead use the fact that the total cumulative mass within a characteristic radius ∼ 0.6 kpc is much bet- ter determined by the present data (Strigari et al. 2007). We propose using this mass, which we define as M0.6, as the favored means to compare to the dark halo pop- ulation in numerical simulations. Unlike Vmax, M0.6 is measured directly and requires no cosmology-dependent or dark-matter-model-dependent theoretical prior. In the following sections, we determine the M0.6 mass function for the Milky Way satellites, and compare it to the corresponding mass function measured directly in the high-resolution “Via Lactea” substructure simulation of Diemand et al. (2007). We rule out the possibility that there is a one-to- one correspondence between the 11 most luminous satellites and the most massive subhalos in Via Lactea. We find that MSP solutions based on reionization and/or characteristic halo mass scales prior to accretion are still viable. 2. MILKY WAY SATELLITES Approximately 20 satellite galaxies can be classified as residing in MW subhalos. Of these, ∼ 9 were discovered within the last two years and have very low luminosities and surface brightnesses (Willman et al. 2005; Willman et al. 2005; Belokurov et al. 2006, 2007; Zucker et al. 2006a,b). The lack of precision in these numbers reflects the ambiguity in the classification of the newly-discovered objects. The nine ‘older’ galaxies clas- sified as dSphs are supported by their velocity dispersion, and exhibit no rotational motion (Mateo 1998). Two satellite galaxies, the Small Megallanic Cloud (SMC) and Large Megallanic Cloud (LMC), are most likely sup- ported by some combination of rotation and dispersion. Stellar kinematics suggest that the LMC and SMC are likely the most massive satellite systems of the Milky We focus on determining the masses of the nine most well-studied dSphs. The dark matter masses of the dSphs are determined from the line-of-sight velocities of the stars, which trace the total gravitational potential. We assume a negligible contribution of the stars to the grav- itational potential, which we find to be an excellent ap- proximation. The dSph with the smallest mass-to-light ratio is Fornax, though even for this case we find that the stars generally do not affect the dynamics of the system (see Lokas 2002; Wu 2007, and below). Under the assumptions of equilibrium and spherical symmetry, the radial component of the stellar velocity dispersion, σr, is linked to the total gravitational poten- tial of the system via the Jeans equation, d(ν⋆σ = −ν⋆V 2c − 2βν⋆σ r . (1) Here, ν⋆ is the stellar density profile, V c = GM(r)/r includes the total gravitating mass of the system, and the parameter β(r) = 1 − σ2θ/σ r characterizes the dif- ference between the radial (σr) and tangential (σθ) ve- locity dispersions. The observable velocity dispersion is constructed by integrating the three- dimensional stellar radial velocity dispersion profile along the line-of-sight, σ2los(R) = I⋆(R) rrdr√ r2 −R2 , (2) where R is the projected radius on the sky. The surface density of stars in all dSphs are reasonably well-fit by a spherically-symmetric King profile (King 1962), I⋆(R) ∝ r2king )−1/2 r2king )−1/2 where rt and rking are fitting parameters denoted as the tidal and core radii. 6 The spherically symmetric stellar density can be obtained with an integral transformation of the surface density. From the form of Eq. (2), the normalization in Eq. (3) is irrelevant. Some dSphs show evidence for multiple, dynamically distinct stellar populations, with each population de- scribed by its own surface density and velocity dispersion (Harbeck et al. 2001; Tolstoy et al. 2004; Westfall et al. 2006; McConnachie et al. 2006; Ibata et al. 2006). In a dSph with i = 1...Np populations of stars, standard ob- servational methods will sample a density-weighted av- erage of the populations: νi (4) σ2r = r,ı , (5) where νi and σi are the density profile and radial stellar velocity dispersion of stellar component i. In principle, each component has its own stellar velocity anisotropy profile, βi(r). In this case, Equation (1) is valid for ν⋆ and σr as long as an effective velocity anisotropy is defined β(r)= ν⋆σ2r βiνiσ i . (6) From these definitions, we also have ı I⋆,ıσ los,ı = The conclusion we draw from this argument is that the presence of multiple populations will not affect the inferred mass structure of the system, provided that the velocity anisotropy is modeled as a free function of ra- dius. Since the functional form of the stellar velocity anisotropy is unknown, we allow for a general, three pa- rameter model of the velocity anisotropy profile, β(r) = β∞ r2β + r + β0. (7) A profile of this form allows for the possibility for β(r) to change from radial to tangential orbits within the halo, and a constant velocity anisotropy is recovered in the limit β∞ → 0, and β0 → const. In Equations (1) and (2), the radial stellar velocity dis- persion, σr , depends on the total mass distribution, and thus the parameters describing the dark matter density profile. Dissipation-less N-body simulations show that the density profiles of CDM halos can be characterized ρ(r̃) = r̃γ(1 + r̃)δ ; r̃ = r/rs, (8) 6 Our results are insensitive to this particular parameterization of the light profile. Redefining the Missing Satellites Problem 3 where rs and ρs set a radial scale and density normaliza- tion and γ and δ parameterize the inner and outer slopes of the distribution. For dark matter halos unaffected by tidal interactions, the most recent high-resolution simulations find δ + γ ≈ 3 works well for the outer slope, while 0.7 . γ . 1.2 works well down to ∼ 0.1% of halo virial radii (Navarro et al. 2004; Diemand et al. 2005). This interior slope is not altered as a subhalo loses mass from tidal interactions with the MW potential (Kazantzidis et al. 2004). The outer slope, δ, depends more sensitively on the tidal interactions in the halo. The majority of the stripped material will be from the outer parts of halos, and thus δ of subhalo density profiles will become steeper than those of field halos. Subhalos are characterized by outer slopes in the range 2 . δ . 3. Given the uncertainty in the β(r) and ρ(r) profiles, we are left with nine shape parameters that must be constrained via line-of-sight velocity dispersion measure- ments: ρs, rs, β0, β∞, rβ , γ, δ, rking , and rt. While the problem as posed may seem impossible, there are a num- ber of physical parameters, which are degenerate between different profile shapes, that are well constrained. Specif- ically, the stellar populations constrain Vc(r) within a radius comparable to the stellar radius rt ∼ kpc. As a result, quantities like the local density and integrated mass within the stellar radius are determined with high precision (Strigari et al. 2006), while quantities that de- pend on the mass distribution at larger radii are virtually unconstrained by the data. It is useful to determine the value of the radius where the constraints are maximized. The location of this char- acteristic radius is determined by the form of the integral in Eq. (2). We can gain some insight using the exam- ple of a power-law stellar distribution ν⋆(r), power-law dark matter density profile ρ ∝ r−γ⋆ , and constant ve- locity anisotropy. The line-of-sight velocity dispersion depends on the three-dimensional stellar velocity disper- sion, which can be written as σ2r (r) = ν Gν⋆(r)M(r)r 2β−2dr ∝ r2−γ⋆ (9) From the shape of the King profile, the majority of stars reside at projected radii rking . R . rt, where the stellar distribution is falling rapidly ν⋆ ∼ r−3.5. In this case, for β = 0, the LOS component scales as σ2los(R) ∝ r−0.5−γ⋆(r2 −R2)−1/2dr and is dominated by the mass and density profile at the smallest relevant radii, r ∼ rking. For R . rking, ν⋆ ∝ r−1 and σ2los is simi- larly dominated by r ∼ rking contributions. We note that although the scaling arguments above hold for constant velocity anisotropies, they can be extended by consider- ing anisotropies that vary significantly in radius. They are also independent of the precise form of I⋆, provided there is a scale similar to rking . In Strigari et al. (2007), it was shown that typical ve- locity dispersion profiles best constrain the mass (and density) within a characteristic radius ≃ 2 rking . For example, the total mass within 2rking is constrained to within ∼ 20% for dSphs with ∼ 200 line-of-sight veloc- ities. Note that when deriving constraints using only the innermost stellar velocity dispersion and fixing the anisotropy to β = 0, the characteristic radius for best constraints decreases to∼ 0.5rking (e.g. Penarrubia et al. 2007). As listed in Table 1, the Milky Way dSphs are ob- served to have variety of rking values, but rking ∼ 0.3 kpc is typical. The values of rking and rt are taken from Mateo (1998). In order to facilitate comparison with simulated subhalos, we chose a single characteristic ra- dius of 0.6 kpc for all the dwarfs, and we represent the mass within this radius as M0.6 = M(< 0.6 kpc). The relative errors on the derived masses are unaffected for small variations in the characteristic radius in the range ∼ 1.5− 2.5 rking. Deviations from a true King profile at large radius (near rt) do not affect these arguments, as long as there is a characteristic scale similar to rking in the surface density. The only dSph significantly affected by the choice of 0.6 kpc as the characteristic radius is Leo II, which has rt = 0.5 kpc. Since the characteristic radius is greater than twice rking , the constraints on its mass will be weakest of all galaxies (with the exception of Sagittarius, as discussed below). 3. DARK MATTER HALO MASSES AT THE CHARACTERISTIC RADIUS We use the following data sets: Wilkinson et al. (2004); Munoz et al. (2005, 2006); Westfall et al. (2006); Walker et al. (2005, 2006); Sohn et al. (2006); Siegal et al. in preparation. These velocity dispersions are deter- mined from the line-of-sight velocities of ∼ 200 stars in each galaxy, although observations in the coming years will likely increase this number by a factor ∼ 5 − 10. From the data, we calculate the χ2, defined in our case (σobs,ı − σth,ı)2 . (10) Here σ2obs is the observed velocity dispersion in each bin, σ2th is the theoretical value, obtained from Eq. (2), and ǫ2ı are errors as determined from the observations. It is easy to see that, when fitting to a single data set of ∼ 200 stars, parameter degeneracies will be signifi- cant. However, from the discussion in section 2, M0.6 is well-determined by the LOS data. To determine how well M0.6 is constrained, we construct likelihood func- tions for each galaxy. When thought of as a function of the theoretical parameters, the likelihood function, L, is defined as the probability that a data set is acquired given a set of theoretical parameters. In our case L is a function of the parameters γ, δ, rs, ρs, and β0, β∞, rβ , and is defined as L = e−χ 2/2. In writing this likelihood function, we assume that the errors on the measured ve- locity dispersions are Gaussian, which we find to be an excellent approximation to the errors for a dSph system (Strigari et al. 2007). We marginalize over the parame- ters γ, δ, rs, ρs, and β0, β∞, rβ at fixed M0.6, and the optimal values for M0.6 are determined by the maximum of L . We determine L for all nine dSphs with velocity dis- persion measurements. For all galaxies we use the full published velocity dispersion profiles. The only galaxy that does not have a published velocity dispersion profile is Sagittarius, and for this galaxy we use the central ve- locity dispersion from Mateo (1998). The mass modeling of Sagittarius is further complicated by the fact that it is experiencing tidal interactions with the MW (Ibata et al. 4 Strigari et al. Fig. 1.— The likelihood functions for the mass within 0.6 kpc for the nine dSphs, normalized to unity at the peak. 1997; Majewski et al. 2003), so a mass estimate from the Jeans equation is not necessarily reliable. We caution that in this case the mass we determine is likely only an approximation to the total mass of the system. We determine the likelihoods by marginalizing over the following ranges of the velocity anisotropy, inner and outer slopes: −10 < β0 < 1, −10 < β∞ < 1, 0.1 < rβ < 10 kpc, 0.7 < γ < 1.2, and 2 < δ < 3. As discussed above, these ranges for the asymptotic in- ner and outer slopes are appropriate because we are con- sidering CDM halos. It is important to emphasize that these ranges are theoretically motivated and that obser- vations alone do not demand such restrictive choices. It is possible to fit all of the dSphs at present with a constant density cores with scale-lengths ∼ 100 pc (Strigari et al. 2006; Gilmore et al. 2007), although the data by no means demand such a situation. Though we consider inner and outer slopes in the ranges quoted above, our results are not strongly affected if we widen these intervals. For example, we find that if we allow the inner slope to vary down to γ = 0, the widths of the likelihoods are only changed by ∼ 10%. This reflects the fact that there is a negligible degeneracy between M0.6 and the inner and outer slopes. We are left to determine the regions of ρs and rs pa- rameter space to marginalize over. In all dSphs, there is a degeneracy in this parameter space, telling us that it is not possible to derive an upper limit on this pair of parameters from the data alone (Strigari et al. 2006). While this degeneracy is not important when determin- ing constraints on M0.6, it is the primary obstacle in determining Vmax. From the fits we present below, we find that the lowest rs value that provides an acceptable fit is ∼ 0.1 kpc, and we use this as the lower limit in all cases. In our fiducial mass models, we conservatively re- strict the maximum value of rs using the known distance to each dSph. In this case, we use 0.1 kpc < rs < D/2, where D is the distance to the dSph. In Figure 1 we show the M0.6 likelihood functions for all of the dSphs. As is shown, we obtain strong con- Fig. 2.— The velocity dispersion for Ursa Minor as a function of radial distance, along with the model that maximizes the likelihood function. straints on M0.6 in all cases except Sagittarius, for which we use only a central velocity dispersion. Table 1 sum- marizes the best fitting M0.6 values for each dwarf. The quoted errors correspond to the points where the likeli- hood falls to 10% of its peak value. The upper panel of Figure 3 shows M0.6 values for each dwarf as a function of luminosity. In Figure 2 we show an example of the velocity dispersion data as a function of radial distance for Ursa Minor, along with the model that maximizes the likelihood function. For all galaxies, we find χ2 per degree of freedom values . 1. The maximum likelihood method also allows us to con- strain the mass at other radii spanned by the stellar dis- tribution. The sixth column of Table 1 provides the inte- grated mass within each dwarf’s King tidal radius. This radius roughly corresponds to the largest radius where a reasonable mass constraint is possible. As expected, the mass within rt is not as well determined as the mass within 2 rking. From these masses we are able to deter- mine the mass-to-light ratios within rt, which we present in the seventh column of Table 1. In the bottom panel of Figure 3, we show mass-to-light ratios within rt as a function of dwarf luminosity. We see the standard result that the observable mass-to-light ratio increases with de- creasing luminosity (Mateo 1998). Note, however, that our results are inconsistent with the idea that all of the dwarfs have the same integrated mass within their stellar extent. We note that for Sagittarius, we can only obtain a lower limit on the total mass-to-light ratio. The last two columns in Table 1 list constraints on Vmax for the dSphs. Column 8 shows results for an anal- ysis with limits on rs as described above. In this case, the integrated mass within the stellar radius is constrained by the velocity dispersion data, but the halo rotation ve- locity curve, Vc(r), can continue to rise as r increases beyond the stellar radius in an unconstrained manner. The result is that the velocity dispersion data alone pro- vide only a lower limit on Vmax. Stronger constraints on Vmax can be obtained if we limit the range of rs by imposing a cosmology-dependent prior on the dark matter mass profile. CDM simula- Redefining the Missing Satellites Problem 5 TABLE 1 Parameters Describing Milky Way Satellites. Galaxy rking rt LV Mass < 0.6 kpc Mass < rt M(< rt)/L Vmax [km s −1] Vmax [km s [kpc] [kpc] [106 L⊙] [10 7 M⊙] [10 7 M⊙] [M⊙/L⊙] (w/o prior) (with theory prior) Draco 0.18 0.93 0.26 4.9 530 > 22 28 Ursa Minor 0.30 1.50 0.29 5.3 790 > 21 26 Leo I 0.20 0.80 4.79 4.3 106 > 14 19 Fornax 0.39 2.70 15.5 4.3 28 > 20 25 Leo II 0.19 0.52 0.58 2.1 128 > 17 9 Carina 0.26 0.85 0.43 3.4 82 > 13 15 Sculptor 0.28 1.63 2.15 2.7 68 > 20 14 Sextans 0.40 4.01 0.50 0.9 260 > 8 9 Sagittarius 0.3 4.0 18.1 20 > 20 > 11 > 19 — Note. — Determination of the mass within 0.6 kpc and the maximum circular velocity for the dark matter halos of the dSphs. The errors are determined as the location where the likelihood function falls off by 90% from its peak value. For Sagittarius, no reliable estimate of Vmax with the CDM prior could be determined. The CDM prior is determined using the concordance cosmology with σ8 = 0.74, n = 0.95 (see text for details). Fig. 3.— The mass within 0.6 kpc (upper) and the mass-to-light ratios within the King tidal radius (lower) for the Milky Way dSphs as a function of dwarf luminosity. The error-bars here are defined as the locations where the likelihoods fall to 40% of the peak values (corresponding to ∼ 1σ errors). The lines denote, from top to bottom, constant values of mass of 107, 108, 109 M⊙. tions have shown that there is a correlation between Vmax and rmax for halos, where rmax is the radius where the circular velocity peaks. Because subhalo densities will depend on the collapse time and orbital evolution of each system, the precise Vmax-rmax relation is sen- sitive to cosmology (e.g. σ8) and the formation his- tory of the host halo itself (e.g. Zentner & Bullock 2003; Power 2003; Kazantzidis et al. 2004; Bullock & Johnston 2005; Bullock & Johnston 2006). When converted to the relevant halo parameters, the imposed Vmax-rmax rela- tion can be seen as a theoretical prior on CDM ha- los, restricting the parameter space we need to inte- grate over. In order to illustrate the technique, we adopt log10(rmax) = 1.35(log10(Vmax/kms −1) − 1)− 0.196 kpc with a scatter of 0.2 in log10, as measured from simulated subhalos within the Via Lactea host halo (Diemand et al. 2007). This simulation is for a LCDM cosmology with σ8 = 0.74 and n = 0.95. The scatter in the subhalo mass function increases at the very high mass end, which re- flects the fact that these most massive subhalos are those that are accreted most recently (Zentner & Bullock 2003; van den Bosch et al. 2005). However, as we show below our results are not strongly dependent on the large scat- ter at the high mass end. Column 9 in Table 1 shows the allowed subhalo Vmax values for the assumed prior. Note that in most cases, this prior degrades the quality of the fit, and the likeli- hood functions peak at a lower overall value. The mag- nitude of this effect is not large except for the cases of Leo II and Sagittarius. For Leo II, the peak likelihood with the prior occurs at a value that is below the 10% likelihood for the case without a prior on rs (i.e. the data seem to prefer a puffier subhalo than would be ex- pected in CDM). For Sagittarius, we are unable to obtain a reasonable fit within a subhalo that is typical of those expected. This is not too surprising. Sagittarius is be- ing tidally disrupted and its dark matter halo is likely atypical. We emphasize that the Vmax determinations listed in Column 9 are driven by theoretical assumptions, and can only be fairly compared to predictions for this specific cosmology (LCDM, σ8 = 0.74). The M0.6 values in Col- umn 5 are applicable for any theoretical model, including non-CDM models, or CDM models with any normaliza- tion or power spectrum shape. 4. COMPARISON TO NUMERICAL SIMULATIONS The recently-completed Via Lactea run is the highest- resolution simulation of galactic substructure to date, containing an order of magnitude more particles than its predecessors (Diemand et al. 2007). As mentioned above, Via Lactea assumes a LCDM cosmology with σ8 = 0.74 and n = 0.95. For a detailed description of the simulation, see Diemand et al. (2007). For our pur- poses, the most important aspect of Via Lactea is its ability to resolve the mass of subhalos on length scales of the characteristic radius 0.6 kpc. In Via Lactea, the force resolution is 90 pc and the smallest well-resolved length scale is 300 pc, so that the mass within 0.6 kpc is well-resolved in nearly all subhalos. Due to the choice of time steps we expect the simulation to underestimate local densities in the densest regions (by about 10% at 6 Strigari et al. Fig. 4.— The mass within 0.6 kpc versus the maximum circular velocity for the mass ranges of Via Lactea subhalos corresponding to the population of satellites we study. densities of 9 × 107M⊙/kpc3). There is only one sub- halo with a higher local density than this at 0.6 kpc. For this subhalo, ρ(r = 0.6 kpc) = 1.4 × 108M⊙/kpc3, so its local density might be underestimated by up to 10%, and the errors in the enclosed mass might be ∼ 20% (Diemand et al. 2005). For all other subhalos the densi- ties at 0.6 kpc are well below the affected densities, and the enclosed mass should not be affected by more than 10% by the finite numerical resolution. We define subhalos in Via Lactea to be the self-bound halos that lie within the radius R200 = 389 kpc, where R200 is defined to enclose an average density 200 times the mean matter density. We note that in comparing to the observed MW dwarf population, we could have conservatively chosen subhalos that are restricted to lie within the same radius as the most distant MW dSph (250 kpc). We find that this choice has a negligible effect on our conclusions – it reduces the count of small halos by ∼ 10%. In Figure 4, we show how M0.6 relates to the more familiar quantity Vmax in Via Lactea subhalos. We note that the relationship between subhaloM0.6 and Vmax will be sensitive to the power spectrum shape and normaliza- tion, as well as the nature of dark matter (Bullock et al. 2001; Zentner & Bullock 2003). The relationship shown is only valid for the Via Lactea cosmology, but serves as a useful reference for this comparison. Given likelihood functions for the dSph M0.6 values, we are now in position to determine the M0.6 mass func- tion for Milky Way (MW) satellites and compare this to the corresponding mass function in Via Lactea. For both the observations and the simulation, we count the num- ber of systems in four mass bins from 4 × 106 < M0.6 < 4× 108M⊙. This mass range is chosen to span the M0.6 values allowed by the likelihood functions for the MW satellites. We assume that the two non-dSph satellites, the LMC and SMC, belong in the highest mass bin, cor- responding to M0.6 > 10 8 M⊙ (Harris & Zaritsky 2006; van der Marel et al. 2002). In Figure 5 we show resulting mass functions for MW satellites (solid) and for Via Lactea subhalos (dashed, Fig. 5.— The M0.6 mass function of Milky Way satellites and dark subhalos in the Via Lactea simulation. The red (short-dashed) curve is the total subhalo mass function from the simulation. The black (solid) curve is the median of the observed satellite mass function. The error-bars on the observed mass function represent the upper and lower limits on the number of configurations that occur with a probability of > 10−3. with Poisson error-bars). For the MW satellites, the cen- tral values correspond to the median number of galaxies per bin, which are obtained from the maximum values of the respective likelihood functions. The error-bars on the satellite points are set by the upper and lower configurations that occur with a probability of > 10−3 after drawing 1000 realizations from the respective like- lihood functions. As seen in Figure 5, the predicted dark subhalo mass function rises as ∼ M−20.6 while the visi- ble MW satellite mass function is relatively flat. The lowest mass bin (M0.6 ∼ 9 × 106M⊙) always contains 1 visible galaxy (Sextans). The second-to-lowest mass bin (M0.6 ∼ 2.5×107M⊙) contains between 2 and 4 satellites (Carina, Sculptor, and Leo II). The fact that these two lowest bins are not consistent with zero galaxies has im- portant implications for the Stoehr et al. (2002) solution to the MSP: specifically, it implies that the 11 well-known MW satellites do not reside in subhalos that resemble the 11 most massive subhalos in Via Lactea. To further emphasize this point, we see from Figure 5 that the mass of the 11th most massive subhalo in Via Lactea is 4 × 107M⊙. From the likelihood functions in Figure 1, Sextans, Carina, Leo II, and Sculptor must have values of M0.6 less than 4 × 107M⊙ at 99% c.l., implying a negligible probability that all of these dSphs reside in halos with M0.6 > 4× 107M⊙. Using the M0.6 mass function of MW satellites, we can test other CDM-based solutions to the MSP. Two models of interest are based on reionization suppres- sion (Bullock et al. 2000; Moore et al. 2006) and on there being a characteristic halo mass scale prior to subhalo accretion (Diemand et al. 2007). To roughly represent these models, we focus on two subsamples of Via Lactea subhalos: the earliest forming (EF) halos, and the largest mass halos before they were accreted (LBA) into the host halo. As described in Diemand et al. (2007), the LBA sample is defined to be the 10 subhalos that had Redefining the Missing Satellites Problem 7 Fig. 6.— The solid and dashed curves show the MW satellites and dark subhalos in Via Lactea, respectively. These lines are reproduced from Figure 5, with error-bars suppressed for clarity. The blue (dotted) curve represents the ten earliest forming halos in Via Lactea, and the green (long-dashed) curve represents the 10 most massive halos before accretion into the Milky Way halo. the highest Vmax value throughout their entire history. These systems all had Vmax > 37.3 kms −1 at some point in their history. The EF sample consists of the 10 sub- halos with Vmax > 16.2 kms −1 (the limit of atomic cool- ing) at z = 9.6. The Kravtsov et al. (2004) model would correspond to a selection intermediate between EF and LBA. In Figure 6 we show the observed mass function of MW satellites (solid, squares) along with the EF (dotted, triangles) and LBA (long-dashed, circles) samples. We conclude that both of these models are in agreement with the MW satellite mass function. Future observations and quantification of the masses of the newly-discovered MW satellites will enable us to do precision tests of the viable MSP solutions. Additionally, once the capability to do numerical simulations of substructure in warm dark mat- ter models becomes a reality, the M0.6 mass function will provide an invaluable tool to place constraints on WDM models. 5. SUMMARY AND DISCUSSION We have provided comprehensive dark matter mass constraints for the 9 well-studied dSph satellite galax- ies of the Milky Way and investigated CDM-based solu- tions for the missing satellite problem in light of these constraints. While subhalo Vmax values are the tradi- tional means by which theoretical predictions quantify substructure counts, this is not the most direct way to confront the observational constraints. Specifically, Vmax is poorly constrained by stellar velocity dispersion measurements, and can only be estimated by adopting cosmology-dependent, theoretical extrapolations. We ar- gue the comparison between theory and observation is best made using the integrated mass within a fixed phys- ical radius comparable to the stellar extent of the known satellites, ∼ 0.6 kpc. This approach is motivated by Strigari et al. (2007) who showed that the mass within two stellar King radii is best constrained by typical ve- locity dispersion data. Using M0.6 to represent the dark matter mass within a radius of 0.6 kpc, we computed M0.6 likelihood functions for the MW dSphs based on published velocity dispersion data. Our models allow for a wide range of underlying dark matter halo profile shapes and stellar velocity dis- persion profiles. With this broad allowance, we showed that the M0.6 for most dwarf satellites is constrained to within ∼ 30%. We derived the M0.6 mass function of MW satellites (with error bars) and compared it to the same mass func- tion computed directly from the Via Lactea substruc- ture simulation. While the observed M0.6 mass function of luminous satellites is relatively flat, the comparable CDM subhalo mass function rises as ∼ M−20.6 . We rule out the hypothesis that all of the well-known Milky Way satellites strictly inhabit the most massive CDM subha- los. If luminosity does track current subhalo mass, this would only be possible if the subhalo population of the Milky Way were drastically different than that predicted in CDM. However, we show that other plausible CDM so- lutions are consistent with the observed mass function. Specifically, the earliest forming subhalos have a flatM0.6 mass function that is consistent with the satellite subhalo mass function. This would be expected if the population of bright dwarf spheroidals corresponds to the residual halo population that accreted a significant mount of gas before the epoch of reionization (Bullock et al. 2000). We also tested the hypothesis that the present dwarf spheroidal population corresponds to the subhalos that were the most massive before they fell into the MW halo (Kravtsov et al. 2004). This hypothesis is also consistent with the current data. In deriving the M0.6 mass function for this paper we have set aside the issue of the most- recently discovered population of MW dwarfs. We aim to return to this issue in later work, but it is worth speculating on the expected impact that these systems would have on our conclusions. If we had included the new systems, mak- ing ∼ 20 satellites in all, would it be possible to place these systems in the ∼ 20 most massive subhalos in Via Lactea? Given the probable mass ranges for the new dwarfs, we find that this is unlikely. We can get a rough estimate of their masses from their observed luminosi- ties. We start by considering the mass-to-light ratios of the known dSph population from figure 3 and from Mateo (1998). If we assume that the other dwarfs have similar M/L range, we can assign a mass range for each of them. In all cases, the new MW dwarfs are approxi- mately 1 to 2 orders of magnitude smaller in luminosity than the well-known dSph population. Using the central points for the known dSphs, we obtain M0.6/L spanning the range from 3 − 230. Considering the width of the likelihoods, we can allow a slightly larger range, 2− 350. If we place the new dwarfs in this latter range, the uncer- tainty in their masses is (2 − 350)LM⊙/L⊙. Even with this generous range we expect most of the new dwarfs have M0.6 . 10 The discovery of more members of the MW group, 7 These estimates are in rough agreement with recent de- terminations from stellar velocity dispersion measurements in the new dwarfs, as presented by N. Martin and J. Simon at the 3rd Irvine Cosmology Workshop, March 22-24, 2007, http://www.physics.uci.edu/Astrophysical-Probes/ http://www.physics.uci.edu/Astrophysical-Probes/ 8 Strigari et al. and the precise determination of the M0.6 mass func- tion, could bring the status of the remaining viable MSP solutions into sharper focus. These measurements would also provide important constraints on warm dark matter models or on the small scale power spectrum in CDM. 6. ACKNOWLEDGMENTS We thank Jason Harris, Tobias Kaufmann, Savvas Koushiappas, Andrey Kravtsov, Steve Majewski, Nicolas Martin, Josh Simon, and Andrew Zentner for discussions on this topic. We thank Mike Siegal for sharing his Leo II data. LES is supported in part by a Gary McCue post- doctoral fellowship through the Center for Cosmology at the University of California, Irvine. L.E.S., J.S.B., and M.K. are supported in part by NSF grant AST-0607746. M.K. acknowledges support from PHY-0555689. J. D. acknowledges support from NASA through Hubble Fel- lowship grant HST-HF-01194.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. P.M. ac- knowledges support from NASA grants NAG5-11513 and NNG04GK85G, and from the Alexander von Humboldt Foundation. The Via Lactea simulation was performed on NASA’s Project Columbia supercomputer system. REFERENCES Belokurov, V. et al. 2006, Astrophys. J., 647, L111 —. 2007, Astrophys. J., 654, 897 Benson, A. J., Lacey, C. G., Baugh, C. M., Cole, S., & Frenk, C. S. 2002, Mon. Not. Roy. Astron. Soc., 333, 156 Bullock, J. S. & Johnston, K. 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0704.1818

Low-density graph codes that are optimal for source/channel coding and binning

Low-density graph codes that are optimal for source/channel coding and binning Martin J. Wainwright Emin Martinian Dept. of Statistics, and Tilda Consulting, Inc. Dept. of Electrical Engineering and Computer Sciences Arlington, MA University of California, Berkeley [email protected] wainwrig@{eecs,stat}.berkeley.edu Technical Report 730, Department of Statistics, UC Berkeley, April 2007 Abstract We describe and analyze the joint source/channel coding properties of a class of sparse graphical codes based on compounding a low-density generator matrix (LDGM) code with a low-density parity check (LDPC) code. Our first pair of theorems establish that there exist codes from this ensemble, with all degrees remaining bounded independently of block length, that are simultaneously optimal as both source and channel codes when encoding and decoding are performed optimally. More precisely, in the context of lossy compression, we prove that finite degree constructions can achieve any pair (R,D) on the rate-distortion curve of the binary symmetric source. In the context of channel coding, we prove that finite degree codes can achieve any pair (C, p) on the capacity-noise curve of the binary symmetric channel. Next, we show that our compound construction has a nested structure that can be exploited to achieve the Wyner- Ziv bound for source coding with side information (SCSI), as well as the Gelfand-Pinsker bound for channel coding with side information (CCSI). Although the current results are based on optimal encoding and decoding, the proposed graphical codes have sparse structure and high girth that renders them well-suited to message-passing and other efficient decoding procedures. Keywords: Graphical codes; low-density parity check code (LDPC); low-density generator matrix code (LDGM); weight enumerator; source coding; channel coding; Wyner-Ziv problem; Gelfand- Pinsker problem; coding with side information; information embedding; distributed source coding. 1 Introduction Over the past decade, codes based on graphical constructions, including turbo codes [3] and low- density parity check (LDPC) codes [17], have proven extremely successful for channel coding prob- lems. The sparse graphical nature of these codes makes them very well-suited to decoding using efficient message-passing algorithms, such as the sum-product and max-product algorithms. The asymptotic behavior of iterative decoding on graphs with high girth is well-characterized by the density evolution method [25, 39], which yields a useful design principle for choosing degree dis- tributions. Overall, suitably designed LDPC codes yield excellent practical performance under iterative message-passing, frequently very close to Shannon limits [7]. http://arxiv.org/abs/0704.1818v1 However, many other communication problems involve aspects of lossy source coding, either alone or in conjunction with channel coding, the latter case corresponding to joint source-channel coding problems. Well-known examples include lossy source coding with side information (one variant corresponding to the Wyner-Ziv problem [45]), and channel coding with side information (one variant being the Gelfand-Pinsker problem [19]). The information-theoretic schemes achieving the optimal rates for coding with side information involve delicate combinations of source and channel coding. For problems of this nature—in contrast to the case of pure channel coding—the use of sparse graphical codes and message-passing algorithm is not nearly as well understood. With this perspective in mind, the focus of this paper is the design and analysis sparse graphical codes for lossy source coding, as well as joint source/channel coding problems. Our main contribution is to exhibit classes of graphical codes, with all degrees remaining bounded independently of the blocklength, that simultaneously achieve the information-theoretic bounds for both source and channel coding under optimal encoding and decoding. 1.1 Previous and ongoing work A variety of code architectures have been suggested for lossy compression and related problems in source/channel coding. One standard approach to lossy compression is via trellis-code quantization (TCQ) [26]. The advantage of trellis constructions is that exact encoding and decoding can be performed using the max-product or Viterbi algorithm [24], with complexity that grows linearly in the trellis length but exponentially in the constraint length. Various researchers have exploited trellis-based codes both for single-source and distributed compression [6, 23, 37, 46] as well as information embedding problems [5, 15, 42]. One limitation of trellis-based approaches is the fact that saturating rate-distortion bounds requires increasing the trellis constraint length [43], which incurs exponential complexity (even for the max-product or sum-product message-passing algorithms). Other researchers have proposed and studied the use of low-density parity check (LDPC) codes and turbo codes, which have proven extremely successful for channel coding, in application to various types of compression problems. These techniques have proven particularly successful for lossless distributed compression, often known as the Slepian-Wolf problem [18, 40]. An attractive feature is that the source encoding step can be transformed to an equivalent noisy channel de- coding problem, so that known constructions and iterative algorithms can be leveraged. For lossy compression, other work [31] shows that it is possible to approach the binary rate-distortion bound using LDPC-like codes, albeit with degrees that grow logarithmically with the blocklength. A parallel line of work has studied the use of low-density generator matrix (LDGM) codes, which correspond to the duals of LDPC codes, for lossy compression problems [30, 44, 9, 35, 34]. Focusing on binary erasure quantization (a special compression problem dual to binary erasure channel coding), Martinian and Yedidia [30] proved that LDGM codes combined with modified message- passing can saturate the associated rate-distortion bound. Various researchers have used techniques from statistical physics, including the cavity method and replica methods, to provide non-rigorous analyses of LDGM performance for lossy compression of binary sources [8, 9, 35, 34]. In the limit of zero-distortion, this analysis has been made rigorous in a sequence of papers [12, 32, 10, 14]. Moreover, our own recent work [28, 27] provides rigorous upper bounds on the effective rate- distortion function of various classes of LDGM codes. In terms of practical algorithms for lossy binary compression, researchers have explored variants of the sum-product algorithm [34] or survey propagation algorithms [8, 44] for quantizing binary sources. 1.2 Our contributions Classical random coding arguments [11] show that random binary linear codes will achieve both channel capacity and rate-distortion bounds. The challenge addressed in this paper is the design and analysis of codes with bounded graphical complexity, meaning that all degrees in a factor graph representation of the code remain bounded independently of blocklength. Such sparsity is critical if there is any hope to leverage efficient message-passing algorithms for encoding and decoding. With this context, the primary contribution of this paper is the analysis of sparse graphical code ensembles in which a low-density generator matrix (LDGM) code is compounded with a low-density parity check (LDPC) code (see Fig. 2 for an illustration). Related compound constructions have been considered in previous work, but focusing exclusively on channel coding [16, 36, 41]. In contrast, this paper focuses on communication problems in which source coding plays an essential role, including lossy compression itself as well as joint source/channel coding problems. Indeed, the source coding analysis of the compound construction requires techniques fundamentally different from those used in channel coding analysis. We also note that the compound code illustrated in Fig. 2 can be applied to more general memoryless channels and sources; however, so as to bring the primary contribution into sharp focus, this paper focuses exclusively on binary sources and/or binary symmetric channels. More specifically, our first pair of theorems establish that for any rate R ∈ (0, 1), there exist codes from compound LDGM/LDPC ensembles with all degrees remaining bounded independently of the blocklength that achieve both the channel capacity and the rate-distortion bound. To the best of our knowledge, this is the first demonstration of code families with bounded graphical complexity that are simultaneously optimal for both source and channel coding. Building on these results, we demonstrate that codes from our ensemble have a naturally “nested” structure, in which good channel codes can be partitioned into a collection of good source codes, and vice versa. By exploiting this nested structure, we prove that codes from our ensembles can achieve the information-theoretic limits for the binary versions of both the problem of lossy source coding with side information (SCSI, known as the Wyner-Ziv problem [45]), and channel coding with side information (CCSI, known as the Gelfand-Pinsker [19] problem). Although these results are based on optimal encoding and decoding, a code drawn randomly from our ensembles will, with high probability, have high girth and good expansion, and hence be well-suited to message-passing and other efficient decoding procedures. The remainder of this paper is organized as follows. Section 2 contains basic background material and definitions for source and channel coding, and factor graph representations of binary linear codes. In Section 3, we define the ensembles of compound codes that are the primary focus of this paper, and state (without proof) our main results on their source and channel coding optimality. In Section 4, we leverage these results to show that our compound codes can achieve the information-theoretic limits for lossy source coding with side information (SCSI), and channel coding with side information (CCSI). Sections 5 and 6 are devoted to proofs that codes from the compound ensemble are optimal for lossy source coding (Section 5) and channel coding (Section 6) respectively. We conclude the paper with a discussion in Section 7. Portions of this work have previously appeared as conference papers [28, 29, 27]. 2 Background In this section, we provide relevant background material on source and channel coding, binary linear codes, as well as factor graph representations of such codes. 2.1 Source and channel coding A binary linear code C of block length n consists of all binary strings x ∈ {0, 1}n satisfying a set of m < n equations in modulo two arithmetic. More precisely, given a parity check matrix H ∈ {0, 1}m×n, the code is given by the null space C : = {x ∈ {0, 1}n | Hx = 0} . (1) Assuming the parity check matrix H is full rank, the code C consists of 2n−m = 2nR codewords, where R = 1− m is the code rate. Channel coding: In the channel coding problem, the transmitter chooses some codeword x ∈ C and transmits it over a noisy channel, so that the receiver observes a noise-corrupted version Y . The channel behavior is modeled by a conditional distribution P(y | x) that specifies, for each transmitted sequence Y , a probability distribution over possible received sequences {Y = y}. In many cases, the channel is memoryless, meaning that it acts on each bit of C in an independent manner, so that the channel model decomposes as P(y | x) = i=1 fi(xi; yi) Here each function fi(xi; yi) = P(yi | xi) is simply the conditional probability of observing bit yi given that xi was transmitted. As a simple example, in the binary symmetric channel (BSC), the channel flips each transmitted bit xi with probability p, so that P(yi | xi) = (1 − p) I[xi = yi] + p (1− I[xi 6= yi]), where I(A) represents an indicator function of the event A. With this set-up, the goal of the re- ceiver is to solve the channel decoding problem: estimate the most likely transmitted codeword, given by x̂ : = argmax P(y | x). The Shannon capacity [11] of a channel specifies an upper bound on the rate R of any code for which transmission can be asymptotically error-free. Continuing with our example of the BSC with flip probability p, the capacity is given by C = 1− h(p), where h(p) := −p log2 p− (1− p) log2(1− p) is the binary entropy function. Lossy source coding: In a lossy source coding problem, the encoder observes some source se- quence S ∈ S, corresponding to a realization of some random vector with i.i.d. elements Si ∼ PS . The idea is to compress the source by representing each source sequence S by some codeword x ∈ C. As a particular example, one might be interested in compressing a symmetric Bernoulli source, consisting of binary strings S ∈ {0, 1}n, with each element Si drawn in an independent and identically distributed (i.i.d.) manner from a Bernoulli distribution with parameter p = 1 One could achieve a given compression rate R = m by mapping each source sequence to some codeword x ∈ C from a code containing 2m = 2nR elements, say indexed by the binary sequences z ∈ {0, 1}m. In order to assess the quality of the compression, we define a source decoding map x 7→ Ŝ(x), which associates a source reconstruction Ŝ(x) with each codeword x ∈ C. Given some distortion metric d : S × S → R+, the source encoding problem is to find the codeword with min- imal distortion—namely, the optimal encoding x̂ : = argmin d(Ŝ(x), S). Classical rate-distortion theory [11] specifies the optimal trade-offs between the compression rate R and the best achievable average distortion D = E[d(Ŝ, S)], where the expectation is taken over the random source sequences S. For instance, to follow up on the Bernoulli compression example, if we use the Hamming metric d(Ŝ, S) = 1 i=1 |Ŝi − Si| as the distortion measure, then the rate-distortion function takes the form R(D) = 1− h(D), where h is the previously defined binary entropy function. We now provide definitions of “good” source and channel codes that are useful for future reference. Definition 1. (a) A code family is a good D-distortion binary symmetric source code if for any ǫ > 0, there exists a code with rate R < 1− h (D) + ǫ that achieves Hamming distortion less than or equal to D. (b) A code family is a good BSC(p)-noise channel code if for any ǫ > 0 there exists a code with rate R > 1− h (p)− ǫ with error probability less than ǫ. 2.2 Factor graphs and graphical codes Both the channel decoding and source encoding problems, if viewed naively, require searching over an exponentially large codebook (since |C| = 2nR for a code of rate R). Therefore, any practically useful code must have special structure that facilitates decoding and encoding operations. The success of a large subclass of modern codes in use today, especially low-density parity check (LDPC) codes [17, 38], is based on the sparsity of their associated factor graphs. PSfrag replacements y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 c1 c2 c3 c4 c5 c6 PSfrag replacements x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 z1 z2 z3 z4 z5 z6 z7 z8 z9 (a) (b) Figure 1. (a) Factor graph representation of a rate R = 0.5 low-density parity check (LDPC) code with bit degree dv = 3 and check degree d = 6. (b) Factor graph representation of a rate R = 0.75 low-density generator matrix (LDGM) code with check degree dc = 3 and bit degree dv = 4. Given a binary linear code C, specified by parity check matrix H, the code structure can be captured by a bipartite graph, in which circular nodes (◦) represent the binary values xi (or columns of H), and square nodes (�) represent the parity checks (or rows of H). For instance, Fig. 1(a) shows the factor graph for a rate R = 1 code in parity check form, with m = 6 checks acting on n = 12 bits. The edges in this graph correspond to 1’s in the parity check matrix, and reveal the subset of bits on which each parity check acts. The parity check code in Fig. 1(a) is a regular code with bit degree 3 and check degree 6. Such low-density constructions, meaning that both the bit degrees and check degrees remain bounded independently of the block length n, are of most practical use, since they can be efficiently represented and stored, and yield excellent performance under message-passing decoding. In the context of a channel coding problem, the shaded circular nodes at the top of the low-density parity check (LDPC) code in panel (a) represent the observed variables yi received from the noisy channel. Figure 1(b) shows a binary linear code represented in factor graph form by its generator matrix G. In this dual representation, each codeword x ∈ {0, 1}n is generated by taking the matrix-vector product of the form Gz, where z ∈ {0, 1}m is a sequence of information bits, and G ∈ {0, 1}n×m is the generator matrix. For the code shown in panel (b), the blocklength is n = 12, and information sequences are of length m = 9, for an overall rate of R = m/n = 0.75 in this case. The degrees of the check and variable nodes in the factor graph are dc = 3 and dv = 4 respectively, so that the associated generator matrix G has dc = 3 ones in each row, and dv = 4 ones in each column. When the generator matrix is sparse in this setting, then the resulting code is known as a low-density generator matrix (LDGM) code. 2.3 Weight enumerating functions For future reference, it is useful to define the weight enumerating function of a code. Given a binary linear code of blocklength m, its codewords x have renormalized Hamming weights w : = range in the interval [0, 1]. Accordingly, it is convenient to define a function Wm : [0, 1] → R+ that, for each w ∈ [0, 1], counts the number of codewords of weight w: Wm(w) := x ∈ C | w = ⌉ }∣∣∣∣ , (2) where ⌈·⌉ denotes the ceiling function. Although it is typically difficult to compute the weight enu- merator itself, it is frequently possible to compute (or bound) the average weight enumerator, where the expectation is taken over some random ensemble of codes. In particular, our analysis in the sequel makes use of the average weight enumerator of a (dv, d c)-regular LDPC code (see Fig. 1(a)), defined as Am(w; dv , d c) := logE [Wm(w)] , (3) where the expectation is taken over the ensemble of all regular (dv , d c)-LDPC codes. For such regular LDPC codes, this average weight enumerator has been extensively studied in previous work [17, 22]. 3 Optimality of bounded degree compound constructions In this section, we describe the compound LDGM/LDPC construction that is the focus of this paper, and describe our main results on their source and channel coding optimality. 3.1 Compound construction Our main focus is the construction illustrated in Fig. 2, obtained by compounding an LDGM code (top two layers) with an LDPC code (bottom two layers). The code is defined by a factor graph with three layers: at the top, a vector x ∈ {0, 1}n of codeword bits is connected to a set of n parity checks, which are in turn connected by a sparse generator matrix G to a vector y ∈ {0, 1}m of information bits in the middle layer. The information bits y are also codewords in an LDPC code, defined by the parity check matrix H connecting the middle and bottom layers. In more detail, considering first the LDGM component of the compound code, each codeword x ∈ {0, 1}n in the top layer is connected via the generator matrix G ∈ {0, 1}n×m to an information sequence y ∈ {0, 1}m in the middle layer; more specifically, we have the algebraic relation x = Gy. Note that this LDGM code has rate RG ≤ . Second, turning to the LDPC component of the compound construction, its codewords correspond to a subset of information sequences y ∈ {0, 1}m in the middle layer. In particular, any valid codeword y satisfies the parity check relation Hy = 0, where H ∈ {0, 1}m×k joins the middle and bottom layers of the construction. Overall, this defines an LDPC code with rate RH = 1− , assuming that H has full row rank. The overall code C obtained by concatenating the LDGM and LDPC codes has blocklength n, and rate R upper bounded by RGRH . In algebraic terms, the code C is defined as C : = {x ∈ {0, 1}n | x = Gy for some y ∈ {0, 1}m such that Hy = 0} , (4) where all operations are in modulo two arithmetic. PSfrag replacements k1 k2 H1 H2 Figure 2. The compound LDGM/LDPC construction analyzed in this paper, consisting of a (n,m) LDGM code over the middle and top layers, compounded with a (m, k) LDPC code over the middle and bottom layers. Codewords x ∈ {0, 1}n are placed on the top row of the construction, and are associated with information bit sequences z ∈ {0, 1}m in the middle layer. The LDGM code over the top and middle layers is defined by a sparse generator matrix G ∈ {0, 1}n×m with at most dc ones per row. The bottom LDPC over the middle and bottom layers is represented by a sparse parity check matrix H ∈ {0, 1}k×m with dv ones per column, and d ones per row. Our analysis in this paper will be performed over random ensembles of compound LDGM/LDPC ensembles. In particular, for each degree triplet (dc, dv, d c), we focus on the following random ensemble: (a) For each fixed integer dc ≥ 4, the random generator matrix G ∈ {0, 1} n×m is specified as follows: for each of the n rows, we choose dc positions with replacement, and put a 1 in each of these positions. This procedure yields a random matrix with at most dc ones per row, since it is possible (although of asymptotically negligible probability for any fixed dc) that the same position is chosen more than once. (b) For each fixed degree pair (dv, d c), the random LDPC matrix H ∈ {0, 1} k×m is chosen uni- formly at random from the space of all matrices with exactly dv ones per column, and exactly d′c ones per row. This ensemble is a standard (dv , d c)-regular LDPC ensemble. We note that our reason for choosing the check-regular LDGM ensemble specified in step (a) is not that it need define a particularly good code, but rather that it is convenient for theoretical purposes. Interestingly, our analysis shows that the bounded degree dc check-regular LDGM ensemble, even though it is sub-optimal for both source and channel coding in isolation [28, 29], defines optimal source and channel codes when combined with a bottom LDPC code. 3.2 Main results Our first main result is on the achievability of the Shannon rate-distortion bound using codes from LDGM/LDPC compound construction with finite degrees (dc, dv , d c). In particular, we make the following claim: Theorem 1. Given any pair (R,D) satisfying the Shannon bound, there is a set of finite degrees (dc, dv , d c) and a code from the associated LDGM/LDPC ensemble with rate R that is a D-good source code (see Definition 1). In other work [28, 27], we showed that standard LDGM codes from the check-regular ensemble cannot achieve the rate-distortion bound with finite degrees. As will be highlighted by the proof of Theorem 1 in Section 5, the inclusion of the LDPC lower code in the compound construction plays a vital role in the achievability of the Shannon rate-distortion curve. Our second main result of this result is complementary in nature to Theorem 1, regarding the achievability of the Shannon channel capacity using codes from LDGM/LDPC compound construc- tion with finite degrees (dc, dv , d c). In particular, we have: Theorem 2. For all rate-noise pairs (R, p) satisfying the Shannon channel coding bound R < 1 − h (p), there is a set of finite degrees (dc, dv , d c) and a code from the associated LDGM/LDPC ensemble with rate R that is a p-good channel code (see Definition 1). To put this result into perspective, recall that the overall rate of this compound construction is given by R = RGRH . Note that an LDGM code on its own (i.e., without the lower LDPC code) is a special case of this construction with RH = 1. However, a standard LDGM of this variety is not a good channel code, due to the large number of low-weight codewords. Essentially, the proof of Theorem 2 (see Section 6) shows that using a non-trivial LDPC lower code (with RH < 1) can eliminate these troublesome low-weight codewords. 4 Consequences for coding with side information We now turn to consideration of the consequences of our two main results for problems of coding with side information. It is well-known from previous work [47] that achieving the information- theoretic limits for these problems requires nested constructions, in which a collection of good source codes are nested inside a good channel code (or vice versa). Accordingly, we begin in Section 4.1 by describing how our compound construction naturally generates such nested ensembles. In Sec- tions 4.2 and 4.3 respectively, we discuss how the compound construction can be used to achieve the information-theoretic optimum for binary source coding with side information (a version of the Wyner-Ziv problem [45]), and binary information embedding (a version of “dirty paper coding”, or the Gelfand-Pinsker problem [19]). 4.1 Nested code structure The structure of the compound LDGM/LDPC construction lends itself naturally to nested code constructions. In particular, we first partition the set of k lower parity checks into two disjoint subsets K1 and K2, of sizes k1 and k2 respectively, as illustrated in Fig. 2. Let H1 and H2 denote the corresponding partitions of the full parity check matrix H ∈ {0, 1}k×m. Now let us set all parity bits in the subset K2 equal to zero, and consider the LDGM/LDPC code C(G,H1) defined by the generator matrix G and the parity check (sub)matrix H1, as follows C(G,H1) := {x ∈ {0, 1} n | x = Gy for some y ∈ {0, 1}m such that H1 y = 0} . (5) Note that the rate of C(G,H1) is given by R ′ = RGRH1 , which can be suitably adjusted by modifying the LDGM and LDPC rates respectively. Moreover, by applying Theorems 1 and 2, there exist finite choices of degree such that C(G,H1) will be optimal for both source and channel coding. Considering now the remaining k2 parity bits in the subset K2, suppose that we set them equal to a fixed binary sequence m ∈ {0, 1}k2 . Now consider the code C(m) := x ∈ {0, 1}n | x = Gy for some y ∈ {0, 1}m such that . (6) Note that for each binary sequencem ∈ {0, 1}k2 , the code C(m) is a subcode of C(G,H1); moreover, the collection of these subcodes forms a disjoint partition as follows C(G,H1) = m∈{0,1}k2 C(m). (7) Again, Theorems 1 and 2 guarantee that (with suitable degree choices), each of the subcodes C(m) is optimal for both source and channel coding. Thus, the LDGM/LDPC construction has a natural nested property, in which a good source/channel code—namely C(G,H1)—is partitioned into a disjoint collection {C(m), m ∈ {0, 1}k1} of good source/channel codes. We now illustrate how this nested structure can be exploited for coding with side information. 4.2 Source coding with side information We begin by showing that the compound construction can be used to perform source coding with side information (SCSI). 4.2.1 Problem formulation Suppose that we wish to compress a symmetric Bernoulli source S ∼ Ber(1 ) so as to be able to reconstruct it with Hamming distortion D. As discussed earlier in Section 2, the minimum achievable rate is given by R(D) = 1− h (D). In the Wyner-Ziv extension of standard lossy com- pression [45], there is an additional source of side information about S—say in the form Z = S ⊕W where W ∼ Ber(δ) is observation noise—that is available only at the decoder. See Fig. 3 for a block diagram representation of this problem. Decoder PSfrag replacements Figure 3. Block diagram representation of source coding with side information (SCSI). A source S is compressed to rate R. The decoder is given the compressed version, and side information Z = S⊕W , and wishes to use (Ŝ, Z) to reconstruct the source S up to distortion D. For this binary version of source coding with side information (SCSI), it is known [2] that the minimum achievable rate takes the form RWZ(D, p) = l. c. e. h (D ∗ p)− h (D) , (p, 0) , (8) where l. c. e. denotes the lower convex envelope. Note that in the special case p = 1 , the side information is useless, so that the Wyner-Ziv rate reduces to classical rate-distortion. In the discussion to follow, we focus only on achieving rates of the form h (D ∗ p)−h (D), as any remaining rates on the Wyner-Ziv curve (8) can be achieved by time-sharing with the point (p, 0). 4.2.2 Coding procedure for SCSI In order to achieve rates of the form R = h (D ∗ p)− h (D), we use the compound LDGM/LDPC construction, as illustrated in Fig. 2, according to the following procedure. Step #1, Source coding: The first step is a source coding operation, in which we transform the source sequence S to a quantized representation S. In order to do so, we use the code C(G,H1), as defined in equation (5) and illustrated in Fig. 4(a), composed of the generator matrix G and the parity check matrix H1. Note that C(G,H1), when viewed as a code with blocklength n, has rate R1 : = 1− k1 = m−k1 . Suppose that we choose1 the middle and lower layer sizes m and k1 respectively such that m− k1 = 1− h (D) + ǫ/2, (9) where ǫ > 0 is arbitrary. For any such choice, Theorem 1 guarantees the existence of finite degrees (dc, dv , d c) such that that C(G,H1) is a good D-distortion source code. Consequently, for the speci- fied rate R1, we can use C(G,H1) in order to transform the source to some quantized representation Ŝ such that the error Ŝ ⊕ S has average Hamming weighted bounded by D. Moreover, since Ŝ is a codeword of C(G,H1), there is some sequence of information bits Ŷ ∈ {0, 1} m such that Ŝ = GŶ and H1Ŷ = 0. PSfrag replacements 0 0 0 PSfrag replacements 0 0 0 1 0 k1 k2 H1 H2 (a) (b) Figure 4. (a) Source coding stage for Wyner-Ziv procedure: the C(G,H1), specified by the generator matrix G ∈ {0, 1}n×m and parity check matrix H1 ∈ {0, 1} k1×m, is used to quantize the source vector S ∈ {0, 1}n, thereby obtaining a quantized version Ŝ ∈ {0, 1}n and associated vector of information bits Ŷ ∈ {0, 1}m, such that Ŝ = G Ŷ and H1 Ŷ = 0. Step #2. Channel coding: Given the output (Ŷ , Ŝ) of the source coding step, consider the sequence H2Ŷ ∈ {0, 1} k2 of parity bits associated with the parity check matrix H2. Transmitting this string of parity bits requires rate Rtrans = . Overall, the decoder receives both these k2 parity bits, as well as the side information sequence Z = S ⊕W . Using these two pieces of information, the goal of the decoder is to recover the quantized sequence Ŝ. Viewing this problem as one of channel coding, the effective rate of this channel code is m−k1−k2 . Note that the side information can be written in the form Z = S ⊕W = Ŝ ⊕ E ⊕W, 1Note that the choices of m and k1 need not be unique. where E : = S⊕Ŝ is the quantization noise, andW ∼ Ber(p) is the channel noise. If the quantization noise E were i.i.d. Ber(D), then the overall effective noise E ⊕W would be i.i.d. Ber(D ∗ p). (In reality, the quantization noise is not exactly i.i.d. Ber(D), but it can be shown [47] that it can be treated as such for theoretical purposes.) Consequently, if we choose k2 such that m− k1 − k2 = 1− h (D ∗ p)− ǫ/2, (10) for an arbitrary ǫ > 0, then Theorem 2 guarantees that the decoder will (w.h.p.) be able to recover a codeword corrupted by (D ∗ p)-Bernoulli noise. Summarizing our findings, we state the following: Corollary 1. There exist finite choices of degrees (dc, dv, d c) such that the compound LDGM/LDPC construction achieves the Wyner-Ziv bound. Proof. With the source coding rate R1 chosen according to equation (9), the encoder will return a quantization Ŝ with average Hamming distance to the source S of at most D. With the channel coding rate R2 chosen according to equation (10), the decoder can with high probability recover the quantization Ŝ. The overall transmission rate of the scheme is Rtrans = m− k1 m− k1 − k2 = R1 − R2 = (1− h (D) + ǫ/2)− (1− h (D ∗ p)− ǫ/2) = h (D ∗ p)− h (D) + ǫ. Since ǫ > 0 was arbitrary, we have established that the scheme can achieve rates arbitrarily close to the Wyner-Ziv bound. 4.3 Channel coding with side information We now show how the compound construction can be used to perform channel coding with side information (CCSI). 4.3.1 Problem formulation In the binary information embedding problem, given a specified input vector V ∈ {0, 1}n, the channel output Z ∈ {0, 1}n is assumed to take the form Z = V ⊕ S ⊕W, (11) where S is a host signal (not under control of the user), and W ∼ Ber(p) corresponds to channel noise. The encoder is free to choose the input vector V ∈ {0, 1}n, subject to an average channel constraint E [‖V ‖1] ≤ w, (12) for some parameter w ∈ (0, 1 ]. The goal is to use a channel coding scheme that satisfies this constraint (12) so as to maximize the number of possible messages m that can be reliably commu- nicated. Moreover, We write V ≡ V to indicate that each channel input is implicitly identified with some underlying message m. Given the channel output Z = V ⊕ S ⊕W , the goal of the de- coder is to recover the embedded message m. The capacity for this binary information embedding problem [2] is given by RIE(w, p) = u. c. e. h (w)− h (p) , (0, 0) , (13) where u. c. e. denotes the upper convex envelope. As before, we focus on achieving rates of the form h (w) − h (p), since any remaining points on the curve (13) can be achieved via time-sharing with the (0, 0) point. Encoder Decoder PSfrag replacements Figure 5. Block diagram representation of channel coding with side information (CCSI). The encoder embeds a message m into the channel input V , which is required to satisfy the average channel constraint 1 ‖1] ≤ w. The channel produces the output Z = Vm⊕S⊕W , where S is a host signal known only to the encoder, and W ∼ Ber(p) is channel noise. Given the channel output Y , the decoder outputs an estimate m̂ of the embedded message. 4.3.2 Coding procedure for CCSI In order to achieve rates of the form R = h (w)− h (p), we again use the compound LDGM/LDPC construction in Fig. 2, now according to the following two step procedure. Step #1: Source coding: The goal of the first stage is to embed the message into the transmitted signal V via a quantization process. In order to do so, we use the code illustrated in Fig. 6(a), specified by the generator matrix G and parity check matrices H1 and H2. The set K1 of k1 parity bits associated with the check matrix H1 remain fixed to zero throughout the scheme. On the other hand, we use the remaining k2 lower parity bits associated with H2 to specify a particular message m ∈ {0, 1}k2 that the decoder would like to recover. In algebraic terms, the resulting code C(m) has the form C(m) := x ∈ {0, 1}n | x = Gy for some y ∈ {0, 1}m such that .(14) Since the encoder has access to host signal S, it may use this code C(m) in order to quantize the host signal. After doing so, the encoder has a quantized signal Ŝ ∈ {0, 1}n and an associated sequence Ŷ ∈ {0, 1}m of information bits such that Ŝ = GŶ . Note that the quantized signal ) specifies the message m in an implicit manner, since m = H2 Ŷm by construction of the code C(m). Now suppose that we choose n,m and k such that m− k1 − k2 = 1− h (w) + ǫ/2 (15) for some ǫ > 0, then Theorem 1 guarantees that there exist finite degrees (dc, dv , d c) such that the resulting code is a good w-distortion source code. Otherwise stated, we are guaranteed that w.h.p, the quantization error E : = S ⊕ Ŝ has average Hamming weight upper bounded by wn. Consequently, we may set the channel input V equal to the quantization noise (V = E), thereby ensuring that the average channel constraint (12) is satisfied. Step #2, Channel coding: In the second phase, the decoder is given a noisy channel observation of the form Z = E ⊕ S ⊕W = Ŝ ⊕W, (16) and its task is to recover Ŝ. In terms of the code architecture, the k1 lower parity bits remain set to zero; the remaining k2 parity bits, which represent the message m, are unknown to the coder. The resulting code, as illustrated illustrated in Fig. 6(b), can be viewed as channel code with effective PSfrag replacements 0 0 0 1 0 H1 H2 PSfrag replacements 0 0 0 (a) (b) Figure 6. (a) Source coding step for binary information embedding. The message m ∈ {0, 1}k2 specifies a particular coset; using this particular source code, the host signal S is compressed to Ŝ, and the quantization error E = S ⊕ Ŝ is transmitted over the constrained channel. (b) Channel coding step for binary information embedding. The decoder receives Z = Ŝ ⊕W where W ∼ Ber(p) is channel noise, and seeks to recover Ŝ, and hence the embedded message m specifying the coset. rate m−k1 . Now suppose that we choose k1 such that the effective code used by the decoder has m− k1 = 1− h (p)− ǫ/2, (17) for some ǫ > 0. Since the channel noise W is Ber(p) and the rate R2 chosen according to (17), Theorem 2 guarantees that the decoder will w.h.p. be able to recover the pair Ŝ and Ŷ . Moreover, by design of the quantization procedure, we have the equivalence m = H2 Ŷ so that a simple syndrome-forming procedure allows the decoder to recover the hidden message. Summarizing our findings, we state the following: Corollary 2. There exist finite choices of degrees (dc, dv, d c) such that the compound LDGM/LDPC construction achieves the binary information embedding (Gelfand-Pinsker) bound. Proof. With the source coding rate R1 chosen according to equation (15), the encoder will return a quantization Ŝ of the host signal S with average Hamming distortion upper bounded by w. Consequently, transmitting the quantization error E = S ⊕ Ŝ will satisfy the average channel constraint (12). With the channel coding rate R2 chosen according to equation (17), the decoder can with high probability recover the quantized signal Ŝ, and hence the message m. Overall, the scheme allows a total of 2k2 distinct messages to be embedded, so that the effective information embedding rate is Rtrans = m− k1 m− k1 − k2 = R2 − R1 = (1− h (p)− ǫ/2) − (1− h (w) + ǫ/2) = h (w)− h (p) + ǫ, for some ǫ > 0. Thus, we have shown that the proposed scheme achieves the binary information embedding bound (13). 5 Proof of source coding optimality This section is devoted to the proof of the previously stated Theorem 1 on the source coding optimality of the compound construction. 5.1 Set-up In establishing a rate-distortion result such as Theorem 1, perhaps the most natural focus is the random variable dn(S,C) := ‖x− S‖1, (18) corresponding to the (renormalized) minimum Hamming distance from a random source sequence S ∈ {0, 1}n to the nearest codeword in the code C. Rather than analyzing this random variable directly, our proof of Theorem 1 proceeds indirectly, by studying an alternative random variable. Given a binary linear code with N codewords, let i = 0, 1, 2, . . . , N − 1 be indices for the different codewords. We say that a codeword Xi is distortion D-good for a source sequence S if the Hamming distance ‖Xi⊕S‖1 is at most Dn. We then set the indicator random variable Z i(D) = 1 when codeword Xi is distortion D-good. With these definitions, our proof is based on the following random variable: Tn(S,C;D) := Zi(D). (19) Note that Tn(S,C;D) simply counts the number of codewords that are distortion D-good for a source sequence S. Moreover, for all distortions D, the random variable Tn(S,C;D) is linked to dn(S,C) via the equivalence P[Tn(S,C;D) > 0] = P[dn(S,C) ≤ D]. (20) Throughout our analysis of P[Tn(S,C;D) > 0], we carefully track only its exponential behavior. More precisely, the analysis to follow will establish an inverse polynomial lower bound of the form P[Tn(S,C;D) > 0] ≥ 1/f(n) where f(·) collects various polynomial factors. The following concentration result establishes that the polynomial factors in these bounds can be ignored: Lemma 1 (Sharp concentration). Suppose that for some target distortion D, we have P[Tn(S,C;D) > 0] ≥ 1/f(n), (21) where f(·) is a polynomial function satisfying log f(n) = o(n). Then for all ǫ > 0, there exists a fixed code C̄ of sufficiently large blocklength n such that E[dn(S; C̄)] ≤ D + ǫ. Proof. Let us denote the random code C as (C1,C2), where C1 denotes the random LDGM top code, and C2 denotes the random LDPC bottom code. Throughout the analysis, we condition on some fixed LDPC bottom code, say C2 = C̄2. We begin by showing that the random variable (dn(S,C) | C̄2) is sharply concentrated. In order to do so, we construct a vertex-exposure martin- gale [33] of the following form. Consider a fixed sequential labelling {1, . . . , n} of the top LDGM checks, with check i associated with source bit Si. We reveal the check and associated source bit in a sequential manner for each i = 1, . . . , n, and so define a sequence of random variables {U0, U1, . . . , Un} via U0 : = E[dn(S,C) | C̄2], and Ui : = E dn(S,C) | S1, . . . , Si, C̄2 , i = 1, . . . , n. (22) By construction, we have Un = (dn(S,C) | C̄2). Moreover, this sequence satisfies the following bounded difference property: adding any source bit Si and the associated check in moving from Ui−1 to Ui can lead to a (renormalized) change in the minimum distortion of at most ci = 1/n. Consequently, by applying Azuma’s inequality [1], we have, for any ǫ > 0, [∣∣(dn(S,C) | C̄2)− E[dn(S,C) | C̄2] ∣∣ ≥ ǫ ≤ exp . (23) Next we observe that our assumption (21) of inverse polynomial decay implies that, for at least one bottom code C̄2, P[dn(S,C) ≤ D | C̄2] = P[Tn(S,C;D) > 0 | C̄2] ≥ 1/g(n), (24) for some subexponential function g. Otherwise, there would exist some α > 0 such that P[Tn(S,C;D) > 0 | C̄2] ≤ exp(−nα) for all choices of bottom code C̄2, and taking averages would violate our assumption (21). Finally, we claim that the concentration result (23) and inverse polynomial bound (24) yield the result. Indeed, if for some ǫ > 0, we had D < E[dn(S,C) | C̄2] − ǫ, then the concentration bound (23) would imply that the probability P[dn(S,C) ≤ D | C̄2] ≤ P[dn(S,C) ≤ E[dn(S,C) | C̄2]− ǫ | C̄2] [∣∣(dn(S,C) | C̄2)− E[dn(S,C) | C̄2] ∣∣ ≥ ǫ decays exponentially, which would contradict the inverse polynomial bound (24) for sufficiently large n. Thus, we have shown that assumption (21) implies that for all ǫ > 0, there exists a sufficiently large n and fixed bottom code C̄2 such that E[dn(S,C) | C̄2] ≤ D + ǫ. If the average over LDGM codes C1 satisfies this bound, then at least one choice of LDGM top code must also satisfy it, whence we have established that there exists a fixed code C̄ such that E[dn(S; C̄)] ≤ D+ǫ, as claimed. 5.2 Moment analysis In order to analyze the probability P[Tn(S,C;D) > 0], we make use of the moment bounds given in the following elementary lemma: Lemma 2 (Moment methods). Given any random variable N taking non-negative integer values, there holds (E[N ]) E[N2] ≤ P[N > 0] ≤ E[N ]. (25) Proof. The upper bound (b) is an immediate consequence of Markov’s inequality, whereas the lower bound (a) follows by applying the Cauchy-Schwarz inequality [20] as follows (E[N ]) N I[N > 0] ≤ E[N2] E 2[N > 0] = E[N2] P[N > 0]. The remainder of the proof consists in applying these moment bounds to the random variable Tn(S,C;D), in order to bound the probability P[Tn(S,C;D) > 0]. We begin by computing the first moment: Lemma 3 (First moment). For any code with rate R, the expected number of D-good codewords scales exponentially as logE[Tn] = [R− (1− h (D))] ± o(1). (26) Proof. First, by linearity of expectation E[Tn] = ∑2nR−1 i=0 P[Z i(D) = 1] = 2nRP[Z0(D) = 1], where we have used symmetry of the code construction to assert that P[Zi(D) = 1] = P[Z0(D) = 1] for all indices i. Now the event {Z0(D) = 1} is equivalent to an i.i.d Bernoulli(1 ) sequence of length n having Hamming weight less than or equal to Dn. By standard large deviations theory (either Sanov’s theorem [11], or direct asymptotics of binomial coefficients), we have logP[Z0(D) = 1] = 1− h (D) ± o(1), which establishes the claim. Unfortunately, however, the first moment E[Tn] need not be representative of typical behavior of the random variable Tn, and hence overall distortion performance of the code. As a simple illustration, consider an imaginary code consisting of 2nR copies of the all-zeroes codeword. Even for this “code”, as long as R > 1− h (D), the expected number of distortion-D optimal codewords grows exponentially. Indeed, although Tn = 0 for almost all source sequences, for a small subset of source sequences (of probability mass ≈ 2−n [1−h(D)]), the random variable Tn takes on the enormous value 2nR, so that the first moment grows exponentially. However, the average distortion incurred by using this code will be ≈ 0.5 for any rate, so that the first moment is entirely misleading. In order to assess the representativeness of the first moment, one needs to ensure that it is of essentially the same order as the variance, hence the comparison involved in the second moment bound (25)(a). 5.3 Second moment analysis Our analysis of the second moment begins with the following alternative representation: Lemma 4. E[T 2n(D)] = E[Tn(D)] j 6=0 P[Zj(D) = 1 | Z0(D) = 1] . (27) Based on this lemma, proved in Appendix C, we see that the key quantity to control is the condi- tional probability P[Zj(D) = 1 | Z0(D) = 1]. It is this overlap probability that differentiates the low-density codes of interest here from the unstructured codebooks used in classical random coding arguments.2 For a low-density graphical code, the dependence between the events {Zj(D) = 1} and {Z0(D) = 1} requires some analysis. Before proceeding with this analysis, we require some definitions. Recall our earlier definition (3) of the average weight enumerator associated with an (dv , d c) LDPC code, denoted by Am(w). Moreover, let us define for each w ∈ [0, 1] the probability Q(w;D) := P [‖X(w) ⊕ S‖1 ≤ Dn | ‖S‖1 ≤ Dn] , (28) where the quantity X(w) ∈ {0, 1}n denotes a randomly chosen codeword, conditioned on its under- lying length-m information sequence having Hamming weight ⌈wm⌉. As shown in Lemma 9 (see Appendix A), the random codeword X(w) has i.i.d. Bernoulli elements with parameter δ∗(w; dc) = 1− (1− 2w)dc . (29) With these definitions, we now break the sum on the RHS of equation (27) intom terms, indexed by t = 1, 2, . . . ,m, where term t represents the contribution of a given non-zero information sequence y ∈ {0, 1}m with (Hamming) weight t. Doing so yields j 6=0 P[Zj(D) = 1 | Z0(D) = 1] = Am(t/m)Q(t/m;D) ≤ m max 1≤t≤m {Am(t/m) Q(t/m;D)} ≤ m max w∈[0,1] {Am(w) Q(w;D)} . Consequently, we need to control both the LDPC weight enumerator Am(w) and the probability Q(w;D) over the range of possible fractional weights w ∈ [0, 1]. 5.4 Bounding the overlap probability The following lemma, proved in Appendix D, provides a large deviations bound on the probability Q(w;D). Lemma 5. For each w ∈ [0, 1], we have logQ(w;D) ≤ F (δ∗(w; dc);D) + o(1), (30) 2In the latter case, codewords are chosen independently from some ensemble, so that the overlap probability is simply equal to P[Zj(D) = 1]. Thus, for the simple case of unstructured random coding, the second moment bound actually provides the converse to Shannon’s rate-distortion theorem for the symmetric Bernoulli source. where for each t ∈ (0, 1 ] and D ∈ (0, 1 ], the error exponent is given by F (t;D) := D log (1− t)eλ + (1−D) log (1− t) + teλ − λ∗D. (31) Here λ∗ : = log b2−4ac , where a : = t (1− t) (1−D), b : = (1− 2D)t2, and c : = −t (1− t)D. In general, for any D ∈ (0, 1 ], the function F ( · ;D) has the following properties. At t = 0, it achieves its maximum F (0 ;D) = 0, and then is strictly decreasing on the interval (0, 1 ], ap- proaching its minimum value − [1− h (D)] as t → 1 . Figure 7 illustrates the form of the function F (δ∗(ω; dc);D) for two different values of distortion D, and for degrees dc ∈ {3, 4, 5}. Note that 0 0.1 0.2 0.3 0.4 0.5 Weight Decay of overlap probability: D = 0.1101 0 0.1 0.2 0.3 0.4 0.5 Weight Decay of overlap probability: D = 0.3160 (a) (b) Figure 7. Plot of the upper bound (30) on the overlap probability 1 logQ(w;D) for different choices of the degree dc, and distortion probabilities. (a) Distortion D = 0.1100. (b) Distortion D = 0.3160. increasing dc causes F (δ ∗(ω; dc);D) to approach its minimum −[1− h (D)] more rapidly. We are now equipped to establish the form of the effective rate-distortion function for any compound LDGM/LDPC ensemble. Substituting the alternative form of E[T 2n ] from equation (27) into the second moment lower bound (25) yields logP[Tn(D) > 0] ≥ logE[Tn(D)]− log j 6=0 P[Zj(D) = 1 | Z0(D) = 1] ≥ R− (1− h (D))− max w∈[0,1] logAm(w) + logQ(w;D) − o(1) ≥ R− (1− h (D))− max w∈[0,1] logAm(w) + F ( δ∗(w; dc),D) − o(1), (32) 0 0.1 0.2 0.3 0.4 0.5 Weight Minimum achievable rates: (R,D) = (0.50, 0.1100) Compound Naive LDGM 0 0.1 0.2 0.3 0.4 0.5 0.025 0.075 0.125 Weight Minimum achievable rates: (R,D) = (0.10, 0.3160) Compound Naive LDGM (a) (b) Figure 8. Plot of the function defining the lower bound (33) on the minimum achievable rate for a specified distortion. Shown are curves with LDGM top degree dc = 4, comparing the uncoded case (no bottom code, dotted curve) to a bottom (4, 6) LDPC code (solid line). (a) Distortion D = 0.1100. (b) Distortion D = 0.3160. where the last step follows by applying the upper bound on Q from Lemma 5, and the relation m = RGn = n. Now letting B(w; dv , d c) be any upper bound on the log of average weight enumerator logAm(w) , we can then conclude that 1 log P[Tn(D) > 0] is asymptotically non-negative for all rate-distortion pairs (R,D) satisfying R ≥ max w∈[0,1] 1− h (D) + F (δ∗(w; dc),D) B(w;dv,d′c) . (33) Figure 8 illustrates the behavior of the RHS of equation (33), whose maximum defines the effective rate-distortion function, for the case of LDGM top degree dc = 4. Panels (a) and (b) show the cases of distortion D = 0.1100 and D = 0.3160 respectively, for which the respective Shannon rates are R = 0.50 and R = 0.10. Each panel shows two plots, one corresponding the case of uncoded information bits (a naive LDGM code), and the other to using a rate RH = 2/3 LDPC code with degrees (dv , dc) = (4, 6). In all cases, the minimum achievable rate for the given distortion is obtained by taking the maximum for w ∈ [0, 0.5] of the plotted function. For any choices of D, the plotted curve is equal to the Shannon bound RSha = 1 − h (D) at w = 0, and decreases to 0 for w = 1 Note the dramatic difference between the uncoded and compound constructions (LDPC-coded). In particular, for both settings of the distortion (D = 0.1100 and D = 0.3160), the uncoded curves rise from their initial values to maxima above the Shannon limit (dotted horizontal line). Con- sequently, the minimum required rate using these constructions lies strictly above the Shannon optimum. The compound construction curves, in contrast, decrease monotonically from their max- imum value, achieved at w = 0 and corresponding to the Shannon optimum. In the following section, we provide an analytical proof of the fact that for any distortion D ∈ [0, 1 ), it is al- ways possible to choose finite degrees such that the compound construction achieves the Shannon optimum. 5.5 Finite degrees are sufficient In order to complete the proof of Theorem 1, we need to show that for all rate-distortion pairs (R,D) satisfying the Shannon bound, there exist LDPC codes with finite degrees (dv , d c) and a suitably large but finite top degree dc such that the compound LDGM/LDPC construction achieves the specified (R,D). Our proof proceeds as follows. Recall that in moving from equation (32) to equation (33), we assumed a bound on the average weight enumerator Am of the form logAm(w) ≤ B(w; dv , d c) + o(1). (34) For compactness in notation, we frequently write B(w), where the dependence on the degree pair (dv , d c) is understood implicitly. In the following paragraph, we specify a set of conditions on this bounding function B, and we then show that under these conditions, there exists a finite degree dc such that the compound construction achieves specified rate-distortion point. In Appendix F, we then prove that the weight enumerator of standard regular LDPC codes satisfies the assumptions required by our analysis. Assumptions on weight enumerator bound We require that our bound B on the weight enumerator satisfy the following conditions: A1: the function B is symmetric around 1 , meaning that B(w) = B(1− w) for all w ∈ [0, 1]. A2: the function B is twice differentiable on (0, 1) with B′(1 ) = 0 and B′′(1 ) < 0. A3: the function B achieves its unique optimum at w = 1 , where B(1 ) = RH . A4: there exists some ǫ1 > 0 such that B(w) < 0 for all w ∈ (0, ǫ1), meaning that the ensemble has linear minimum distance. In order to establish our claim, it suffices to show that for all (R,D) such that R > 1− h (D), there exists a finite choice of dc such that w∈[0,1] + F (δ∗(w; dc),D) ︸ ︷︷ ︸  ≤ R− [1− h (D)] : = ∆ (35) K(w; dc) Restricting to even dc ensures that the function F is symmetric about w = ; combined with as- sumption A2, this ensures that K is symmetric around 1 , so that we may restrict the maximization to [0, 1 ] without loss of generality. Our proof consists of the following steps: (a) We first prove that there exists an ǫ1 > 0, independent of the choice of dc, such that K(w; dc) ≤ ∆ for all w ∈ [0, ǫ1]. (b) We then prove that there exists ǫ2 > 0, again independent of the choice of dc, such that K(w; dc) ≤ ∆ for all w ∈ [ − ǫ2, (c) Finally, we specify a sufficiently large but finite degree d∗c that ensures the conditionK(w; d c) ≤ ∆ for all w ∈ [ǫ1, ǫ2]. 5.5.1 Step A By assumption A4 (linear minimum distance), there exists some ǫ1 > 0 such that B(w) ≤ 0 for all w ∈ [0, ǫ1]. Since F (δ ∗(w; dc);D) ≤ 0 for all w, we have K(w; dc) ≤ 0 < ∆ in this region. Note that ǫ1 is independent of dc, since it specified entirely by the properties of the bottom code. 5.5.2 Step B For this step of the proof, we require the following lemma on the properties of the function F : Lemma 6. For all choices of even degrees dc ≥ 4, the function G(w; dc) = F (δ ∗(w; dc),D) is differentiable in a neighborhood of w = 1 , with ; dc) = − [1− h (D)] , G ; dc) = 0, and G ; dc) = 0. (36) See Appendix E for a proof of this claim. Next observe that we have the uniform bound G(w; dc) ≤ G(w; 4) for all dc ≥ 4 and w ∈ [0, ]. This follows from the fact that F (u;D) is decreasing in u, and that δ∗(w; 4) ≤ δ∗(w; dc) for all dc ≥ 4 and w ∈ [0, ]. Since B is independent of dc, this implies that K(w; dc) ≤ K(w; 4) for all w ∈ [0, ]. Hence it suffices to set dc = 4, and show that K(w; 4) ≤ ∆ for all w ∈ [1 − ǫ2, ]. Using Lemma 6, Assumption A2 concerning the derivatives of B, and Assumption A4 (that B(1 ) = RH), we have ; 4) = R− [1− h (D)] = ∆, ; 4) = R B′(1 ; 4) = 0, and K ′′( ; 4) = R B′′(1 +G′′( ; 4) = R B′′(1 By the continuity of K ′′, the second derivative remains negative in a region around 1 , say for all w ∈ [1 − ǫ2, ] for some ǫ2 > 0. Then, for all w ∈ [ − ǫ2, ], we have for some w̃ ∈ [w, 1 ] the second order expansion K(w; 4) = K( ; 4) +K ′( ; 4)(w − K(w̃; 4) K(w̃; 4) Thus, we have established that there exists an ǫ2 > 0, independent of the choice of dc, such that for all even dc ≥ 4, we have K(w; dc) ≤ K(w, 4) ≤ ∆ for all w ∈ [ − ǫ2, ]. (37) 5.5.3 Step C Finally, we need to show that K(w; dc) ≤ ∆ for all w ∈ [ǫ1, ǫ2]. From assumption A3 and the continuity of B, there exists some ρ(ǫ2) > 0 such that B(w) ≤ RH [1− ρ(ǫ2)] for all w ≤ − ǫ2. (38) From Lemma 6, limu→ 1 F (u;D) = F (1 ;D) = − [1− h (D)]. Moreover, as dc → +∞, we have δ∗(ǫ1; dc) → . Therefore, for any ǫ3 > 0, there exists a finite degree d c such that F (δ∗(ǫ1; d c);D) ≤ − [1− h (D)] + ǫ3. Since F is non-increasing in w, we have F (δ∗(w; d∗c);D) ≤ − [1− h (D)] + ǫ3 for all w ∈ [ǫ1, ǫ2]. Putting together this bound with the earlier bound (38) yields that for all w ∈ [ǫ1, ǫ2]: K(w; dc) = R + F (δ∗(w; d∗c),D) ≤ R [1− ρ(ǫ2)]− [1− h (D)] + ǫ3 = {R− [1− h (D)]}+ (ǫ3 − Rρ(ǫ2)) = ∆ + (ǫ3 − Rρ(ǫ2)) Since we are free to choose ǫ3 > 0, we may set ǫ3 = Rρ(ǫ2) to yield the claim. 6 Proof of channel coding optimality In this section, we turn to the proof of the previously stated Theorem 2, concerning the channel coding optimality of the compound construction. If the codeword x ∈ {0, 1}n is transmitted, then the receiver observes V = x ⊕ W , where W is a Ber(p) random vector. Our goal is to bound the probability that maximum likelihood (ML) decoding fails where the probability is taken over the randomness in both the channel noise and the code construction. To simplify the analysis, we focus on the following sub-optimal (non-ML) decoding procedure. Let ǫn be any non-negative sequence such that ǫn/n → 0 but ǫ n/n → +∞—say for instance, ǫn = n Definition 2 (Decoding Rule:). With the threshold d(n) := pn+ ǫn, decode to codeword xi ⇐⇒ ‖xi ⊕ V ‖1 ≤ d(n), and no other codeword is within d(n) of V . The extra term ǫn in the threshold d(n) is chosen for theoretical convenience. Using the following two lemmas, we establish that this procedure has arbitrarily small probability of error, whence ML decoding (which is at least as good) also has arbitrarily small error probability. Lemma 7. Using the suboptimal procedure specified in the definition (2), the probability of decoding error vanishes asymptotically provided that RG B(w)−D (p||δ ∗(w; dc) ∗ p) < 0 for all w ∈ (0, ], (39) where B is any function bounding the average weight enumerator as in equation (34). Proof. Let N = 2nR = 2mRH denote the total number of codewords in the joint LDGM/LDPC code. Due to the linearity of the code construction and symmetry of the decoding procedure, we may assume without loss of generality that the all zeros codeword 0n was transmitted (i.e., x = 0n). In this case, the channel output is simply V = W and so our decoding procedure will fail if and only if one the following two conditions holds: (i) either ‖W‖1 > d(n), or (ii) there exists a sequence of information bits y ∈ {0, 1}m satisfying the parity check equation Hy = 0 such that the codeword Gy satisfies ‖Gy ⊕W‖1 ≤ d(n). Consequently, using the union bound, we can upper bound the probability of error as follows: perr ≤ P[‖W‖1 > d(n)] + ‖Gyi ⊕W‖1 ≤ d(n) . (40) Since E[‖W‖1] = pn, we may apply Hoeffdings’s inequality [13] to conclude that P[‖W‖1 > d(n)] ≤ 2 exp → 0 (41) by our choice of ǫn. Now focusing on the second term, let us rewrite it as a sum over the possible Hamming weights ℓ = 1, 2, . . . ,m of information sequences (i.e., ‖y‖1 = ℓ) as follows: ‖Gyi ⊕W‖1 ≤ d(n) ‖Gy ⊕W‖1 ≥ d(n) ∣∣ ‖y‖1 = ℓ where we have used the fact that the (average) number of information sequences with fractional weight ℓ/m is given by the LDPC weight enumerator Am( ). Analyzing the probability terms in this sum, we note Lemma 9 (see Appendix A) guarantees that Gy has i.i.d. Ber(δ∗( ℓ ; dc)) elements, where δ∗( · ; dc) was defined in equation (29). Consequently, the vector Gy⊕W has i.i.d. Ber(δ( ℓ ) ∗ p) elements. Applying Sanov’s theorem [11] for the special case of binomial variables yields that for any information bit sequence y with ℓ ones, we have ‖Gy ⊕W‖1 ≥ d(n) ∣∣ ‖y‖1 = ℓ ≤ f(n)2−nD(p||δ( )∗p), (42) for some polynomial term f(n). We can then upper bound the second term in the error bound (40) ‖Gyi ⊕W‖1 ≤ d(n) ≤ f(m) exp 1≤ℓ≤m ) + o(m)− nD p||δ( ) ∗ p where we have used equation (42), as well as the assumed upper bound (34) on Am in terms of B. Simplifying further, we take logarithms and rescale by m to assess the exponential rate of decay, thereby obtaining ‖Gyi ⊕W‖1 ≤ d(n) ≤ max 1≤ℓ≤m p||δ( ) ∗ p + o(1) ≤ max w∈[0,1] B(w)− D (p||δ(w) ∗ p) + o(1), and establishing the claim. Lemma 8. For any p ∈ (0, 1) and total rate R : = RGRH < 1 − h (p), there exist finite choices of the degree triplet (dc, dv , d c) such that (39) is satisfied. Proof. For notational convenience, we define L(w) := RGB(w)−D (p||δ ∗(w; dc) ∗ p) . (43) First of all, it is known [17] that a regular LDPC code with rate RH = < 1 and dv ≥ 3 has linear minimum distance. More specifically, there exists a threshold ν∗ = ν∗(dv, dc) such that B(w) ≤ 0 for all w ∈ [0, ν∗]. Hence, since B(w)−D (p||δ∗(w; dc) ∗ p) ≥ 0 for all w ∈ (0, 1), for w ∈ (0, ν ∗], we have L(w) < 0. Turning now to the interval [ν∗, 1 ], consider the function L̃(w) := Rh (w)−D (p||δ∗(w; dc) ∗ p) . (44) Since B(w) ≤ RHh (w), we have L(w) ≤ L̃(w), so that it suffices to upper bound L̃. Observe that ) = R − (1 − h (p)) < 0 by assumption. Therefore, it suffices to show that, by appropriate choice of dc, we can ensure that L̃(w) ≤ L̃( ). Noting that L̃ is infinitely differentiable, calculating derivatives yields L̃′(1 ) = 0 and L̃′′(1 ) < 0. (See Appendix G for details of these derivative calculations.) Hence, by second order Taylor series expansion around w = 1 , we obtain L̃(w) = L̃( L̃′′(w̄)(w − where w̄ ∈ [w, 1 ]. By continuity of L̃′′, we have L̃′′(w) < 0 for all w in some neighborhood of 1 so that the Taylor series expansion implies that L̃(w) ≤ L̃(1 ) for all w in some neighborhood, say (µ, 1 It remains to bound L̃ on the interval [ν∗, µ]. On this interval, we have L̃(w) ≤ Rh (µ) − D (p||δ∗(ν∗; dc) ∗ p). By examining equation (29) from Lemma 9, we see that by choosing dc sufficiently large, we can make δ∗(ν∗; dc) arbitrarily close to , and hence D (p||δ∗(ν∗; dc) ∗ p) arbitrarily close to 1 − h (p). More precisely, let us choose dc large enough to guarantee that D (p||δ∗(ν∗; dc) ∗ p) < (1 − ǫ) (1 − h (p)), where ǫ = R (1−h(µ)) 1−h(p) . With this choice, we have, for all w ∈ [ν∗, µ], the sequence of inequalities L̃(w) ≤ Rh (µ)−D (p||δ∗(ν∗; dc) ∗ p) < Rh (µ)− (1− h (p))−R(1− h (µ)) = R− (1− h (p)) < 0, which completes the proof. 7 Discussion In this paper, we established that it is possible to achieve both the rate-distortion bound for symmetric Bernoulli sources and the channel capacity for the binary symmetric channel using codes with bounded graphical complexity. More specifically, we have established that there exist low-density generator matrix (LDGM) codes and low-density parity check (LDPC) codes with finite degrees that, when suitably compounded to form a new code, are optimal for both source and channel coding. To the best of our knowledge, this is the first demonstration of classes of codes with bounded graphical complexity that are optimal as source and channel codes simultaneously. We also demonstrated that this compound construction has a naturally nested structure that can be exploited to achieve the Wyner-Ziv bound [45] for lossy compression of binary data with side information, as well as the Gelfand-Pinsker bound [19] for channel coding with side information. Since the analysis of this paper assumed optimal decoding and encoding, the natural next step is the development and analysis of computationally efficient algorithms for encoding and decoding. Encouragingly, the bounded graphical complexity of our proposed codes ensures that they will, with high probability, have high girth and good expansion, thus rendering them well-suited to message- passing and other efficient decoding procedures. For pure channel coding, previous work [16, 36, 41] has analyzed the performance of belief propagation when applied to various types of compound codes, similar to those analyzed in this paper. On the other hand, for pure lossy source coding, our own past work [44] provides empirical demonstration of the feasibility of modified message-passing schemes for decoding of standard LDGM codes. It remains to extend both these techniques and their analysis to more general joint source/channel coding problems, and the compound constructions analyzed in this paper. Acknowledgements The work of MJW was supported by National Science Foundation grant CAREER-CCF-0545862, a grant from Microsoft Corporation, and an Alfred P. Sloan Foundation Fellowship. A Basic property of LDGM codes For a given weight w ∈ (0, 1), suppose that we enforce that the information sequence y ∈ {0, 1}m has exactly ⌈wm⌉ ones. Conditioned on this event, we can then consider the set of all codewords X(w) ∈ {0, 1}n, where we randomize over low-density generator matrices G chosen as in step (a) above. Note for any fixed code, X(w) is simply some codeword, but becomes a random variable when we imagine choosing the generator matrix G randomly. The following lemma characterizes this distribution as a function of the weight w and the LDGM top degree dc: Lemma 9. Given a binary vector y ∈ {0, 1}m with a fraction w of ones, the distribution of the random LDGM codeword X(w) induced by y is i.i.d. Bernoulli with parameter δ∗(w; dc) = 1− (1− 2w)dc Proof. Given a fixed sequence y ∈ {0, 1}m with a fraction w ones, the random codeword bit Xi(w) at bit i is formed by connecting dc edges to the set of information bits. 3 Each edge acts as an i.i.d. Bernoulli variable with parameter w, so that we can write Xi(w) = V1 ⊕ V2 ⊕ . . .⊕ Vdc , (45) where each Vk ∼ Ber(w) is independent and identically distributed. A straightforward calculation using z-transforms (see [17]) or Fourier transforms over GF (2) yields that Xi(w) is Bernoulli with parameter δ∗(w; dc) as defined. B Bounds on binomial coefficients The following bounds on binomial coefficients are standard (see Chap. 12, [11]): log(n+ 1) . (46) Here, for α ∈ (0, 1), the quantity h(α) := −α logα − (1 − α) log(1 − α) is the binomial entropy function. 3In principle, our procedure allows two different edges to choose the same information bit, but the probability of such double-edges is asymptotically negligible. C Proof of Lemma 4 First, by the definition of Tn(D), we have E[T 2n(D)] = E Zi(D)Zj(D) = E[Tn] + j 6=i P[Zi(D) = 1, Zi(D) = 1]. To simplify the second term on the RHS, we first note that for any i.i.d Bernoulli(1 ) sequence S ∈ {0, 1}n and any codeword Xj , the binary sequence S′ : = S ⊕ Xj is also i.i.d. Bernoulli(1 Consequently, for each pair i 6= j, we have Zi(D) = 1, Zj(D) = 1 ‖Xi ⊕ S‖1 ≤ Dn, ‖X j ⊕ S‖1 ≤ Dn ‖Xi ⊕ S′‖1 ≤ Dn, ‖X j ⊕ S′‖1 ≤ Dn ‖Xi ⊕Xj ⊕ S‖1 ≤ Dn, ‖S‖1 ≤ Dn Note that for each j 6= i, the vector Xi ⊕Xj is a non-zero codeword. For each fixed i, summing over j 6= i can be recast as summing over all non-zero codewords, so that i 6=j Zi(D) = 1, Zj(D) = 1 j 6=i ‖Xi ⊕Xj ⊕ S‖1 ≤ Dn, ‖S‖1 ≤ Dn k 6=0 ‖Xk ⊕ S‖1 ≤ Dn, ‖S‖1 ≤ Dn = 2nR k 6=0 ‖Xk ⊕ S‖1 ≤ Dn, ‖S‖1 ≤ Dn = 2nRP Z0(D) = 1 k 6=0 Zk(D) = 1 | Z0(D) = 1 = E[Tn] k 6=0 Zk(D) = 1 | Z0(D) thus establishing the claim. D Proof of Lemma 5 We reformulate the probability Q(w,D) as follows. Recall that Q involves conditioning the source sequence S on the event ‖S‖1 ≤ Dn. Accordingly, we define a discrete variable T with distribution P(T = t) = ) for t = 0, 1, . . . ,Dn, representing the (random) number of 1s in the source sequence S. Let Ui and Vj denote Bernoulli random variables with parameters 1 − δ∗(w; dc) and δ ∗(w; dc) respectively. With this set-up, con- ditioned on codeword j having a fraction wn ones, the quantity Q(w,D) is equivalent to the probability that the random variable W : = i=1 Uj + j=1 Vj if T ≥ 1 j=1 Vj if T = 0 is less than Dn. To bound this probability, we use a Chernoff bound in the form logP[W ≤ Dn] ≤ inf logMW (λ)− λD . (48) We begin by computing the moment generating function MW . Taking conditional expectations and using independence, we have MW (λ) = P[T = t] [MU (λ)] [MV (λ)] Here the cumulant generating functions have the form logMU(λ) = log (1− δ)eλ + δ , and (49a) logMV (λ) = log (1− δ) + δeλ , (49b) where we have used (and will continue to use) δ as a shorthand for δ∗(w; dc). Of interest to us is the exponential behavior of this expression in n. Using the standard entropy approximations to the binomial coefficient (see Appendix B), we can bound MW (λ) as − h (D) + logMU (λ) + logMV (λ) ︸ ︷︷ ︸ , (50) where f(n) denotes a generic polynomial factor. Further analyzing this sum, we have g(t) ≤ 0≤t≤Dn log g(t) + log f(n) log(nD) = max 0≤t≤Dn − h (D) + logMU (λ) + logMV (λ) + o(1) ≤ max u∈[0,D] {h (u)− h (D) + u logMU (λ) + (1− u) logMV (λ)} + o(1). Combining this upper bound on 1 logMW (λ) with the Chernoff bound (48) yields that logP[W ≤ Dn] ≤ inf u∈[0,D] G(u, λ; δ) + o(1) (51) where the function G takes the form G(u, λ; δ) := h (u)− h (D) + u logMU (λ) + (1− u) logMV (λ)− λD. (52) Finally, we establish that the solution (u∗, λ∗) to the min-max saddle point problem (51) is unique, and specified by u∗ = D and λ∗ as in Lemma 5. First of all, observe that for any δ ∈ (0, 1), the function G is continuous, strictly concave in u and strictly convex in λ. (The strict concavity follows since h (u) is strictly concave with the remaining terms linear; the strict convexity follows since cumulant generating functions are strictly convex.) Therefore, for any fixed λ < 0, the maximum over u ∈ [0,D] is always achieved. On the other hand, for any D > 0, u ∈ [0,D] and δ ∈ (0, 1), we have G(u;λ; t) → +∞ as λ → −∞, so that the infimum is either achieved at some λ∗ < 0, or at λ∗ = 0. We show below that it is always achieved at an interior point λ∗ < 0. Thus far, using standard saddle point theory [21], we have established the existence and uniqueness of the saddle point solution (u∗, λ∗). To verify the fixed point conditions, we compute partial derivatives in order to find the optimum. First, considering u, we compute (u, λ; δ) = log + logMU (λ)− logMV (λ) = log + log (1− δ)eλ + δ − log (1− δ) + δeλ Solving the equation ∂G (u, λ; δ) = 0 yields exp(λ) 1 + exp(λ) 1 + exp(λ) (1−D) ≥ 0. (53) Since D ≤ 1 , a bit of algebra shows that u′ ≥ D for all choices of λ. Since the maximization is constrained to [0,D], the optimum is always attained at u∗ = D. Turning now to the minimization over λ, we compute the partial derivative to find (u, λ; δ) = u (1− δ) exp(λ) (1− δ) exp(λ) + δ + (1− u) δ exp(λ) (1− δ) + δ exp(λ) Setting this partial derivative to zero yields a quadratic equation in exp(λ) with coefficients a = δ (1− δ) (1 −D) (54a) b = u(1− δ)2 + (1− u)δ2 −D δ2 + (1− δ)2 . (54b) c = −Dδ(1− δ). (54c) The unique positive root ρ∗ of this quadratic equation is given by ρ∗(δ,D, u) := b2 − 4ac . (55) It remains to show that ρ∗ ≤ 1, so that λ∗ : = log ρ∗ < 0. A bit of algebra (using the fact a ≥ 0) shows that ρ∗ < 1 if and only if a+ b+ c > 0. We then note that at the optimal u∗ = D, we have b = (1− 2D)δ2, whence a+ b+ c = δ (1− δ) (1 −D) + (1− 2D)δ2 −Dδ(1 − δ) = (1− 2D) δ > 0 since D < 1 and δ > 0. Hence, the optimal solution is λ∗ : = log ρ∗ < 0, as specified in the lemma statement. E Proof of Lemma 6 A straightforward calculation yields that ) = F (δ∗( ; dc);D) = F ( ;D) = − (1− h (D)) as claimed. Turning next to the derivatives, we note that by inspection, the solution λ∗(t) defined in Lemma 5 is twice continuously differentiable as a function of t. Consequently, the function F (t,D) is twice continuously differentiable in t. Moreover, the function δ∗(w; dc) is twice continuously differentiable in w. Overall, we conclude that G(w) = F (δ∗(w; dc);D) is twice continuously differ- entiable in w, and that we can obtain derivatives via chain rule. Computing the first derivative, we have ) = δ′( )F ′(δ∗( ; dc);D) = 0 since δ′(w) = −dc (1− 2w) dc−1, which reduces to zero at w = 1 . Turning to the second derivative, we have ) = δ′′( )F ′(δ∗( ; dc);D) + F ′′(δ∗( ; dc);D) = δ )F ′(δ∗( ; dc);D). We again compute δ′′(w) = 2dc (dc − 1)(1 − 2w) dc−2, which again reduces to zero at w = 1 since dc ≥ 4 by assumption. F Regular LDPC codes are sufficient Consider a regular (dv, d c) code from the standard Gallager LDPC ensemble. In order to complete the proof of Theorem 1, we need to show for suitable choices of degree (dv , d c), the average weight enumerator of these codes can be suitably bounded, as in equation (34), by a function B that satisfies the conditions specified in Section 5.5. It can be shown [17, 22] that for even degrees d′c, the average weight enumerator of the regular Gallager ensemble, for any block length m, satisfies the bound logAm(w) = B(w; dv , d c) + o(1). The function B in this relation is defined for w ∈ [0, 1 B(w; dv , d c) := (1− dv)h (w)− (1−RH) + dv inf (1 + eλ)d c + (1− eλ)d , (56) and by B(w) = B(w − 1 ) for w ∈ [1 , 1]. Given that the minimization problem (56) is strictly convex, a straightforward calculation of the derivative shows the optimum is achieved at λ∗, where λ∗ ≤ 0 is the unique solution of the equation (1 + eλ)d c−1 − (1− eλ)d (1 + eλ)d c + (1− eλ)d = w. (57) Some numerical computation for RH = 0.5 and different choices (dv, d c) yields the curves shown in Fig. 9. We now show that for suitable choices of degree (dv , d c), the function B defined in equation (56) satisfies the four assumptions specified in Section 5.5. First, for even degrees d′c, the function B 0 0.1 0.2 0.3 0.4 0.5 Weight LDPC weight enumerators = 10 Figure 9. Plots of LDPC weight enumerators for codes of rate RH = 0.5, and check degrees ∈ {6, 8, 10}. is symmetric about w = 1 , so that assumption (A1) holds. Secondly, we have B(w) ≤ RH , and moreover, for w = 1 , the optimal λ∗(1 ) = 0, so that B(1 ) = RH , and assumption (A3) is satisfied. Next, it is known from the work of Gallager [17], and moreover is clear from the plots in Fig. 9, that LDPC codes with dv > 2 have linear minimum distance, so that assumption (A4) holds. The final condition to verify is assumption (A2), concerning the differentiability of B. We summarize this claim in the following: Lemma 10. The function B is twice continuously differentiable on (0, 1), and in particular we ) = 0, and B′′( ) < 0. (58) Proof. Note that for each fixed w ∈ (0, 1), the function f(λ) = (1 + eλ)d c + (1− eλ)d (e−λ + 1)d c + (e−λ − 1)d is strictly convex and twice continuously differentiable as a function of λ. Moreover, the function f∗(w) := infλ≤0 {f(λ)− λw} corresponds to the conjugate dual [21] of f(λ) + I≤0(λ). Since the optimum is uniquely attained for each w ∈ (0, 1), an application of Danskin’s theorem [4] yields that f∗ is differentiable with d f∗(w) = −λ∗(w), where λ∗ is defined by equation (57). Putting together the pieces, we have B′(w) = (1 − dv)h ′(w) − dvλ ∗(w). Evaluating at w = 1 yields ) = 0− dvλ ∗(0) = 0 as claimed. We now claim that λ∗(w) is differentiable. Indeed, let us write the defining relation (57) for λ∗(w) as F (λ,w) = 0 where F (λ,w) := f ′(λ)−w. Note that F is twice continuously differentiable in both λ and w; moreover, ∂F exists for all λ ≤ 0 and w, and satisfies ∂F (λ,w) = f ′′(λ) > 0 by the strict convexity of f . Hence, applying the implicit function theorem [4] yields that λ∗(w) is differentiable, and moreover that dλ (w) = 1/f ′′(λ∗(w)). Hence, combined with our earlier calculation of B′, we conclude that B′′(w) = (1 − dv)h ′′(w) − dv f ′′(λ(w)) . Our final step is to compute the second derivative f ′′. In order to do so, it is convenient to define g = log f ′, and exploit the relation g′f ′ = f ′′. By definition, we have g(λ) = λ+ log (1 + eλ)d c−1 − (1− eλ)d − log (1 + eλ)d c + (1− eλ)d whence g′(λ) = 1 + eλ(d′c − 1) (1 + eλ)d c−2 + (1− eλ)d (1 + eλ)d c−1 − (1− eλ)d − eλd′c (1 + eλ)d c−1 − (1− eλ)d (1 + eλ)d c + (1− eλ)d Evaluating at w = 1 corresponds to λ(0) = 0, so that f ′′(λ( )) = f ′(0) g′(0) = 1 + (d′c − 1) − d′c Consequently, combining all of the pieces, we have B′′(w) = (1− dv)h )− dv f ′′(λ(1 dv − 1 − 4dv < 0 as claimed. G Derivatives of L̃ Here we calculate the first and second derivatives of the function L̃ defined in equation (44). The first derivative takes the form L̃′(v) = R log δ′(v; dc) δ(v; dc) − (1− p) δ′(v; dc) 1− δ(v; dc) where δ′(v; dc) = dc(1 − 2v) dc−1. Since δ′(1 ; dc) = 0, we have L̃ ) = 0 as claimed. Second, using chain rule, we calculate L̃′′(v) = −R δ′′(v; dc)δ(v; dc)− [δ ′(v; dc)] [δ(v; dc)] − (1− p) δ′′(v; dc) 1− δ(v; dc) + [δ′(v; dc)] [1− δ(v; dc)]2 and δ′′(v; dc) = −dc (dc − 1) (1 − 2v) dc−2. Now for dc > 2, we have δ ) = 0, so that L̃′′(1 −4R < 0 as claimed. References [1] N. Alon and J. Spencer. The Probabilistic Method. Wiley Interscience, New York, 2000. [2] R. J. Barron, B. Chen, and G. W. Wornell. The duality between information embedding and source coding with side information and some applications. IEEE Trans. Info. Theory, 49(5):1159–1180, 2003. [3] C. Berroux and A. Glavieux. Near optimum error correcting coding and decoding: Turbo codes. IEEE Trans. Commun., 44:1261–1271, October 1996. [4] D. Bertsekas. Nonlinear programming. Athena Scientific, Belmont, MA, 1995. [5] J. Chou, S. 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Theory, 6(48):1250–1276, 2002. http://arxiv.org/abs/cs/0508068 Introduction Previous and ongoing work Our contributions Background Source and channel coding Factor graphs and graphical codes Weight enumerating functions Optimality of bounded degree compound constructions Compound construction Main results Consequences for coding with side information Nested code structure Source coding with side information Problem formulation Coding procedure for SCSI Channel coding with side information Problem formulation Coding procedure for CCSI Proof of source coding optimality Set-up Moment analysis Second moment analysis Bounding the overlap probability Finite degrees are sufficient Step A Step B Step C Proof of channel coding optimality Discussion Basic property of LDGM codes Bounds on binomial coefficients Proof of Lemma ?? Proof of Lemma ?? Proof of Lemma ?? Regular LDPC codes are sufficient Derivatives of L"0365L

0704.1819

Comments on Charges and Near-Horizon Data of Black Rings

arXiv:0704.1819v3 [hep-th] 17 Dec 2007 Preprint typeset in JHEP style - HYPER VERSION TIT/HEP-570 arXiv:0704.1819 Comments on Charges and Near-Horizon Data of Black Rings Kentaro Hanaki1, Keisuke Ohashi2 and Yuji Tachikawa3 1 Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA E-mail: [email protected] 2 DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3OWA, UK E-mail: [email protected] 3 School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA E-mail: [email protected] Abstract: We study how the charges of the black rings measured at the asymptotic infinity are encoded in the near-horizon metric and gauge potentials, independent of the detailed structure of the connecting region. Our analysis clarifies how different sets of four-dimensional charges can be assigned to a single five-dimensional object under the Kaluza-Klein reduction. Possible choices are related by the Witten effect on dyons and by a large gauge transformation in four and five dimensions, respectively. Keywords: Black Rings, Page Charges. http://arxiv.org/abs/0704.1819v3 mailto:[email protected] mailto:[email protected] mailto:[email protected] http://jhep.sissa.it/stdsearch Contents 1. Introduction 1 2. Near-Horizon Data and Conserved Charges 3 2.1 Electric charges 4 2.2 Angular momenta 6 2.3 Example 1: the black ring 9 2.3.1 Geometry 9 2.3.2 Electric charge 11 2.3.3 Angular momenta 11 2.4 Example 2: concentric black rings 12 2.5 Generalization 14 3. Relation to Four-Dimensional Charges 16 3.1 Mapping of the fields 17 3.2 Mapping of the charges 18 3.3 Reduction and the attractor 19 3.4 Gauge dependence and monodromy 20 3.5 Monodromy and Taub-NUT 21 4. Summary 24 A. Geometry of Concentric Black Rings 24 1. Introduction One of the achievements of string/M theory is the microscopic explanation for the Bekenstein-Hawking entropy for a class of four-dimensional supersymmetric black holes [1, 2]. The microscopic counting predicts subleading corrections to the entropy, which can also be calculated from the macroscopic point of view, i.e. from stringy modi- fications to the Einstein-Hilbert Lagrangian [3]. Comparison of the two approaches has proven to be very fruitful, e.g. it has led to the relation to the partition function – 1 – of topological strings [4]. Beginning in Ref. [5], it has been also generalized to non- supersymmetric extremal black holes using the fact that the near-horizon geometry has enhanced symmetry. The analysis has also been extended to rotating black holes [6]. There is a richer set of supersymmetric black objects in five dimensions, including black rings [7], on which we focus. The entropy is still given by the area law macro- scopically to leading order, and it can be understood microscopically using a D-brane construction [8, 9]. The understanding of higher-derivative corrections remains more elusive [10, 11, 12]. One reason for this is that the supersymmetric higher-derivative terms were not known until quite recently [13]. Even with this supersymmetric higher- derivative action, it has been quite difficult to construct the black ring solution embed- ded in the asymptotically flat spacetime, and it is preferable if we can only study the near horizon geometry. Then the problem is to find the charges carried by the black ring from its data at the near-horizon region. The usual approach taken in the literature so far is to consider the dimensional reduction along a circle down to four dimensions, and to study the charges there [12, 14, 15, 16]. Then, the attractor mechanism fixes the scalar vacuum expectation values (vevs) and the metric at the horizon by the electric and magnetic charges [17, 18]. Conversely, the magnetic charge can be measured by the flux, and the electric charge can be found by taking the variation of the Lagrangian by the gauge potential. In this way, the entropy as a function of charges can be obtained from the analysis of the near-horizon region alone [5, 6]. Nevertheless, it has not been clarified how to reconcile the competing proposals [8, 9, 19, 20, 21] of the mapping between the four- and five-dimensional charges of the black rings embedded in the asymptotically flat spacetime. Thus we believe it worthwhile to revisit the identification of the charges directly in five dimensions, with local five-dimensional Lorentz symmetry intact. It poses two related problems because of the presence of the Chern-Simons interaction in the La- grangian. One is that, in the presence of the Chern-Simons interaction, the equation of motion of the gauge field is given by d ⋆ F = F ∧ F, (1.1) which means that the topological density of the gauge field itself becomes the source of electric charge. To put it differently, the attractor mechanism for the black rings [22] determines the scalar vevs at the near-horizon region via the magnetic dipole charges only, and the information about the electric charges seems to be lost. Then the electric charge of a black ring seems to be diffusely distributed throughout the spacetime. Eq.(1.1) can be rewritten in the form d(⋆F −A ∧ F ) = 0, (1.2) – 2 – (∗F − A ∧ F ) is independent of Σ. This integral is called the Page charge. Similar analysis can be done for angular momenta, and Suryanarayana and Wapler [23] obtained a nice formula for them using the Noether charge of Wald. There is a second problem remaining for black rings, which stems from the fact that A is not a well-defined one-form there because of the presence of the magnetic dipole. It makes (⋆F − A ∧ F ) ill-defined, because in the integral all the forms are to be well-defined. The same can be said for the angular momenta. The aim of this paper is then to show how this second problem can be overcome, and to see how the near-horizon region of a black ring encodes its charges measured at the asymptotic infinity. In Section 2, we use elementary methods to convert the integral at the asymptotic infinity to the one at the horizon. We apply our formalism to the supersymmetric black ring and check that it correctly reproduces known values for the conserved charges. We will show how the gauge non-invariance of A∧F can be solved by using two coordinate patches and a compensating term along the boundary of the patches. Then in Section 3 we will see that our viewpoint helps in identifying the relation of the charges under the Kaluza-Klein reduction along S1. We will see that the change in the charges under a large gauge transformation in five dimensions maps to the Witten effect on dyons [24] in four dimensions. Proposals in the literature [8, 9, 19, 20, 21] will be found equivalent under the transformation. We conclude with a summary in Section 4. In Appendix A the geometry of the concentric rings is briefly reviewed. 2. Near-Horizon Data and Conserved Charges To emphasize essential physical ideas, we discuss the problem first for the minimal supergravity in five dimensions. Later in this section we will apply the technique to the case with vector multiplets. The bosonic part of the Lagrangian of the minimal supergravity theory is ⋆ R − F ∧ ⋆F − 4 A ∧ F ∧ F . (2.1) Our metric is mostly plus, and Rµν is defined to be positive for spheres. We define the Hodge star operator for an n-form as ⋆ (dxµ0 ∧ · · · ∧ dxµn−1) = (5− n)!ǫ µ0···µn−1 µn···µ4dx µn ∧ · · · ∧ dxµ4 . (2.2) – 3 – with the Levi-Civita symbol ǫ01234 = +1 and ǫ 01234 = −1 defined in local Lorentz coordinates. The equations of motion are Rµν = − gµνFρσF ρσ + 2FµρFν ρ, (2.3) d ⋆ F = − 2√ F ∧ F. (2.4) 2.1 Electric charges From the equation of motion of the gauge field (2.4), we see that F ∧ F is the electric current for the charge ⋆F . Thus, the charge is distributed diffusely in the spacetime as was emphasized e.g. in [25]. However, the equation (2.4) can also be cast in the form A ∧ F = 0. (2.5) At the asymptotic infinity, A ∧ F decays sufficiently rapidly, so that we have A ∧ F A ∧ F , (2.6) where the subscript ∞ indicates that the integral is taken at S3 at the asymptotic infinity, and Σ is an arbitrary three-cycle surrounding the black object. Thus we can think of the electric charge as the integral of the quantity inside the bracket, which is called the Page charge. One problem about the Page charge is that, even in the case where A is a glob- ally defined one-form, it changes its value under a large gauge transformation. It is completely analogous to the fact that A for an uncontractible circle C is only de- fined up to an integral multiple of 2π under a large gauge transformation. Indeed, let us parametrize C by 0 ≤ θ ≤ 2π and we perform a gauge transformation by g(θ) ∈ U(1), i.e. we change A to A + i−1g−1dg. Such a continuous g(θ) can be writ- ten as g(θ) = exp(iφ(θ)). Then A changes by dφ(θ) = φ(2π) − φ(0), which can jump by a multiple of 2π. Thus, A is invariant under a small gauge transformation φ(0) = φ(2π) but is not under a large gauge transformation φ(0) 6= φ(2π). Exactly the same analysis can be done for A ∧ F , and it changes under a large gauge transfor- mation along C if Σ contains intersecting one-cycle C and two-cycle S and F 6= 0. However, this non-invariance under a large gauge transformation poses no problem if Σ is at the asymptotic infinity of the flat space, because we usually demand that A should decay sufficiently rapidly there, which removes the freedom to do a large gauge transformation. – 4 – These facts are well-known, and have been utilized previously e.g. in [14]. It is the manifestation of the fact that there are several notions of electric charges in the presence of Chern-Simons interactions, as clearly discussed by Marolf in Ref. [26]. One is the Maxwell charge which is gauge-invariant but not conserved, and another is the charge which is conserved but not gauge-invariant. In our case ⋆F is the Maxwell charge and (⋆F + (2/ 3)A∧F ) is the Page charge. Yet another notion of the charge is the quantity which generates the symmetry in the Hamiltonian framework, which can be constructed using Noether’s theorem and its generalization by the work of Wald and collaborators [27, 28, 29, 30]. The charge thus constructed is called the Noether charge, and in our case it agrees with the Page charge. Unfortunately, the manipulation above cannot be directly applied to the black rings with dipole charges. It is because A is not a globally well-defined one-form, and the integrals are not even well-defined. The way out is to generalize the definition of A ∧ F to the case A is a U(1) gauge field defined using two coordinate patches, so F ∧ F = “ A ∧ F ” (2.7) holds. Then the manipulation (2.6) makes sense. The essential idea is to introduce a term localized in the boundary of the patches which compensates the gauge variation. Copsey and Horowitz [31] used similar subtlety associated to the gauge transformation between patches to study how the magnetic dipole enters in the first law of the black rings. Figure 1: Coordinate patches used to define A ∧ F consistently without ambiguity. Let us assume the whole spacetime is covered by two coordinate patches, S and T , see Figure 1. We denote the boundary of two regions by D = ∂S = −∂T . The gauge field A is represented as well-defined one-forms AS and AT on the patches S and T , respectively. These two are related by a gauge transformation, AS = AT + β with dβ = 0 on the boundary D. Suppose the region B has the boundary C = ∂B. Then – 5 – we have F ∧ F = F ∧ F + F ∧ F (2.8) C∩S+D∩B AS ∧ F + C∩T−D∩B AT ∧ F (2.9) AS ∧ F + AT ∧ F ) + (AS ∧ F − AT ∧ F ) (2.10) AS ∧ F + AT ∧ F ) + AS ∧ β. (2.11) Now we define the symbol A ∧ F for a three-cycle M to mean A ∧ F ” ≡ AS ∧ F + AT ∧ F + AS ∧ β, (2.12) then the relation (2.7) holds as is. The important point here is that we need a term D∩M AS ∧ β which compensates the gauge variation localized at the boundary of the coordinate patches. One immediate concern might be the gauge invariance of the definition (2.12), but it is guaranteed for C = ∂B from the very fact the relation (2.7) holds. It is because its left hand side is obviously gauge invariant. For illustration, consider the case ∂B = C1 − C2. The Page charges measured at C1, C2 themselves are affected by a large gauge transformation, but their difference is not. When one takes C1 as the asymptotic infinity, it is conventional to set the gauge potential to be zero there, thus fixing the gauge freedom. Then the Page charge at the cycle C2 is defined without ambiguity. In the following, we drop the quotation marks around the generalized integral A ∧ F ”. We believe it does not cause any confusion. 2.2 Angular momenta The technique similar to the one we used for electric charges can be applied to the angular momenta, and we can obtain a formula which expresses them by the inte- gral at the horizon. There is a general formalism, developed by Lee and Wald [27], which constructs the appropriate integrand from a given arbitrary generally-covariant Lagrangian, and the expression for the angular momenta was obtained in [23, 32]. In- stead, here we will construct a suitable quantity in a more down-to-earth and direct method. We will see that the integrand contains the gauge field A without the exterior derivative, and that it is ill-defined in the presence of magnetic dipole. We will use the technique developed in the last section to make it well-defined. – 6 – Firstly, the angular momentum corresponding to the axial Killing vector ξ can be measured at the asymptotic infinity by Komar’s formula Jξ = − ⋆∇ξ, (2.13) where ∇ξ is an abbreviation for the two-form ∇µξνdxµ ∧ dxν = dξ. Using the Killing identity, the divergence of the integrand is given by d ⋆∇ξ = 2 ⋆ Rµνξµdxν , (2.14) which vanishes in the pure gravity. Thus, the angular momentum of a black object of the pure gravity theory can be measured by ⋆∇ξ for any surface S which surrounds the object. Let us analyze our case, where the equations of motion are given by (2.3) and (2.4). We need to introduce some notations: £ξ denotes the Lie derivative along the vector field ξ, ιξω denotes the interior product of a vector ξ to a differential form ω, i.e. the contraction of the index of ξ to the first index of ω. Then £ξ = dιξ + ιξd when it acts on the forms. For a vector ξ and a one-form A, we abbreviate ιξA as (ξ · A). We will take the gauge where gauge potentials are invariant under the axial isometry £ξA = 0. It can be achieved by averaging over the orbit of the isometry ξ. We furthermore assume that every chain or cycle we use is invariant under the isometry ξ, then any term of the form ιξ(· · · ) vanishes upon integration on such a chain or cycle. Under these assumptions, the difference of the integral of ⋆∇ξ at the asymptotic infinity and at C is evaluated with the help of the Einstein equation (2.3) to be ⋆∇ξ − ⋆∇ξ = 2 ⋆Rµνξ µdxν = 4 (ιξF ) ∧ ⋆F (2.15) where B is a hypersurface connecting the asymptotic infinity and C. We dropped the ιξ(⋆F 2) because it vanishes upon integration. The right hand side can be partially-integrated using the following relations: one d [⋆(ξ · A)F ] = −(ιξF ) ∧ ⋆F − (ξ · A) F ∧ F (2.16) and another is d [(ξ · A)A ∧ F ] = (ξ · A)F ∧ F − (ιξF ) ∧ A ∧ F (2.17) (ξ · A)F ∧ F − 1 ιξ(A ∧ F ∧ F ) (2.18) – 7 – of which the last term vanishes upon integration. Thus we have dXξ[A] = −(ιξF ) ∧ ⋆F (2.19) modulo the term of the form ιξ(· · · ), where Xξ[A] ≡ ⋆(ξ · A)F + (ξ ·A)A ∧ F. (2.20) Xξ[A] is not a globally well-defined form. Thus, to perform the partial integration of the right hand side of (2.19), compensating terms along the boundary of the coordinate patches need to be introduced, just as we did in the previous section in the analysis of the Page charge. Let S and T be two coordinate patches, D = ∂S = −∂T be their common bound- ary, and AS = AT + β as before. Let us call the correction term Yξ[β,AS] and we define Xξ[A] ≡ Xξ[AS] + Xξ[AT ] + Yξ[β,AT ]. (2.21) We demand that it satisfies Xξ[A] = (ιξF ) ∧ ⋆F. (2.22) Then Y [β,A] should solve dYξ[β,AT ] = Xξ[AS]−Xξ[AT ]. (2.23) The right hand side is automatically closed since dXξ[A] is gauge invariant. Thus the equation above should have a solution if there is no cohomological obstruction. Indeed, substituting (2.20) in the above equation, we get Yξ[β,AT ] = (ξ · β)Z − 2(ξ · β)β ∧AT + (ξ · AT )β ∧ AT (2.24) modulo ιξ(· · · ), where dZ should satisfy dZ = ⋆F + AT ∧ F, (2.25) the right hand side of which is closed using the equation of motion (2.4). Unfortunately there seems to be no general way to write Z as a functional of A and β. We need to choose Z by hand for each on-shell configuration. With these preparation, we can finally integrate the right hand side of (2.15) partially and conclude that (⋆∇ξ + 4Xξ[A]) . (2.26) – 8 – is independent under continuous deformation of C. Taking C to be the 3-sphere at the asymptotic infinity, the terms X [A] vanish too fast to contribute to the integral. Then, the integral above is proportional to the Komar integral at the asymptotic infinity. Thus we arrive at the formula Jξ = − ⋆∇ξ + 4 ⋆ (ξ ·A)F + 16 (ξ · A)A ∧ F , (2.27) where Σ is any surface enclosing the black object. The right hand side is precisely the Noether charge of Wald as constructed in [23, 32]. The contribution ⋆∇ξ to the angular momentum is gauge invariant but is not conserved. It is expected, since the matter energy-momentum tensor carries the angular momentum. The rest of the terms in (2.27) was obtained by the partial integral of the contribution from the matter energy-momentum tensor, and can also be obtained by constructing the Noether charge. The price we paid is that it is now not invariant under a gauge transformation. 2.3 Example 1: the black ring Let us check our formulae against known examples. First we consider the celebrated supersymmetric black ring in five dimensions [7]. 2.3.1 Geometry It has been known [33] that any supersymmetric solution of the minimal supergravity in the asymptotically flat R1,4 can be written in the form ds2 = −f 2(dt+ ω)2 + f−1ds2(R4) (2.28) where f and ω is a function and a one-form on R4, respectively. For the supersymmetric black ring [7], we use a coordinate system adopted for a ring of radius R in the R4 given ds2(R4) = (x− y)2 1− x2 + (1− x 2)dφ21 + y2 − 1 + (y 2 − 1)dφ22 (2.29) with the ranges −1 ≤ x ≤ 1, −∞ < y ≤ −1 and 0 ≤ φ1,2 < 2π.1 φ1, φ2 were denoted by φ, ψ in Ref. [7]. 1We fix the orientations so that dx∧ dφ1 ∧ dφ2 > 0 and dx∧ dφ1 < 0 for S2 surrounding the ring. – 9 – The solution for the single black ring is parametrized by the radius R in the base 4 above, and two extra parameter q and Q. More details can be found in Appendix A. q controls the magnetic dipole through S2 surrounding the ring, q. (2.30) Conserved charges measured at the asymptotic infinity are as follows: Q, (2.31) J1 = − ⋆∇ξ1 = q(3Q− q2), (2.32) J2 = − ⋆∇ξ2 = q(6R2 + 3Q− q2) (2.33) where ξ1, ξ2 are the vector fields ∂φ1 , ∂φ2 respectively. There is a magnetic flux through S2 surrounding the ring, so we need to introduce two patches S, T . We choose S to cover the region x < 1−ǫ and T to cover 1−ǫ < x < 1, with infinitesimal ǫ. The boundary D is at x = ǫ and parametrized by 0 ≤ φ1, φ2 < 2π. We choose the gauge transformation between the two patches to be AT = AS + qdφ1 (2.34) which is chosen to make AT smooth at the origin of R The horizon is located at y → −∞ and has the topology S1 × S2. The gauge potential near the horizon is AS = − q(x+ 1)dχ, (2.35) while the geometry near the horizon is given as ds2 = 2dvdr + rdvdψ + ℓ2dψ2 + (dθ2 + sin2 θdχ2) (2.36) where r = r(y) is chosen so that r → 0 corresponds to y → −∞, v is a combination of t and y, x = cos θ, ψ = φ2 + C1/r + C0 for suitably chosen C0,1, χ = φ1 − φ2, and ℓ2 = 3 (Q− q2)2 . (2.37) It is a direct product of an extremal Bañados-Teitelboim-Zanelli (BTZ) black hole with horizon length 2πℓ and curvature radius q and of a round two-sphere with radius q/2. – 10 – ℓ is a more physical quantity characterizing the ring than R is, so it is preferable to express J2, (2.33), using ℓ in the form −2ℓ2 + 3Q . (2.38) Our objective is to reproduce the conserved charges, (2.31), (2.32) and (2.38), purely from the near-horizon data, (2.35) and (2.36). 2.3.2 Electric charge We use the formula (2.6) to get the electric charge. Using the form of the gauge field near the horizon (2.34) and (2.35), we obtain A ∧ F AS ∧ F + AS ∧ β Q+ q2 Q− q2 Q, (2.39) which correctly reproduces the charge measured at the asymptotic infinity. Vanishing ⋆F at the horizon means that all the Maxwell charge of the system is carried outside of the horizon in the form of F ∧F , while all of the Page charge is still inside the horizon. One important fact behind the gauge invariance of the calculation above is that the integral AS along the ψ ′ direction is not just defined mod integer, but is well-defined as a real number. It is because the circle along ψ, which is not contractible in the near-horizon region, becomes contractible in the full geometry. 2.3.3 Angular momenta The integral of the right hand side of (2.25) can be made arbitrarily small by choosing very small ǫ, so that we can forget the complication coming from the choice of Z. Then for ξ1 = ∂φ1 = ∂χ, we have −1<x<1−ǫ (ξ · AS)AS ∧ F + x=1−ǫ (ξ · β)β ∧ AT (2π)2 (q3 + qQ) + (−q3 + qQ) q(3Q− q2), (2.40) – 11 – reproducing (2.32). For ξψ = ∂ψ = ∂φ1 + ∂φ2 , we have a contribution from ⋆∇ξψ = 4π2qℓ2. Adding contribution from X [A], we obtain −2qℓ2 − q + 3qQ+ (2.41) which matches with J1 + J2, see (2.32) and (2.38). The second and the third terms in the expression above are obtained by the partial integration of the contribution from the angular part of the energy-momentum tensor of the gauge field. In this sense, a part of the angular momentum is carried outside of the horizon and the part proportional to ℓ2 is carried inside the horizon. However, the Noether charge of the black ring resides purely inside of the horizon. 2.4 Example 2: concentric black rings The concentric black-ring solution constructed in Ref. [34] is a superposition of the single black ring we discussed in the last subsection. We focus on the case where all the rings lie on a plane in the base R4. For the superposition of N rings, the full geometry is parametrized by 3N parameters qi, Qi and Ri, (i = 1, . . . , N). qi is the dipole charge and Ri is the radius in the base R 4 of the i-th ring. For more details, see Appendix A. We order the rings so that R1 < R2 < · · · < RN . The conserved charges measured at infinity are known to be Qi − q2i , (2.42) 2s3 + 3s (Qj − q2j ) , (2.43) 2s3 + 3s (Qj − q2j ) + 6 (2.44) where s is an abbreviation for the sum of the magnetic charges, i.e. s = i=1 qi. Our aim is to reproduce these results from the near-horizon data. The near-horizon metric of i-th ring has the form (2.36) with q, Q, R replaced with qi, Qi and Ri, respectively. The horizon radius ℓi is given by ℓ2i = 3 (Qi − q2i )2 − R2i . (2.45) Since each ring has a magnetic dipole charge, we introduce coordinate patches S and Ti so that the gauge field is non-singular in each patch. Let Ti be the patch covering – 12 – the region between (i− 1)-th and i-th ring and S be a patch covering the outer region. More precisely, we introduce the ring coordinate (2.29) for each of the ring, and choose S to cover −1 + ǫ < xi < 1 − ǫ for each ring while Ti to cover 1 − ǫ < xi < 1 for the i-th ring and −1 < xi−1 < −1 + ǫ for the (i − 1)-th ring. Then, near the i-th horizon the gauge field on S is given by AS = − − qi + 2s qi(1 + x) + 2 j=i+1 . (2.46) Its ψ component is determined in Appendix A, while the coefficient for dχ is determined so that the field strength is reproduced, the gauge field is non-singular except for x = ±1 for the 1st to (N − 1)-th rings and non-singular except for x = −1 for the N -th ring. The gauge field on Ti is given by ATi = AS + qjdφ1. (2.47) The electric charge is given by using (2.6) and βi = AS −ATi = − j=i qjdφ1 as AS ∧ F + Σi∩∂S AS ∧ βi + Σi−1∩∂S AS ∧ βi Qi − q2i + 2sqi Qi − q2i Qi − q2i . (2.48) This correctly reproduces the known result (2.42). Let us move onto the evaluation of the angular momenta. Note that for certain configurations of charges, the concentric black rings develop singularities on the rotation axes. While the condition for the absence of singularities has not been known fully, it was pointed out in Ref. [34] that there is no singularity on the rotation axes if all Qi − q2i (2.49) are equal. We will show that we can obtain the correct angular momenta if this condi- tion is satisfied. – 13 – The angular momentum associated with ξ1 = ∂φ1 = ∂χ is given by J1 = − (ξ1 · AS)AS ∧ F Σi∩∂Ti Σi−1∩∂Ti (ξ1 · βi)βi ∧ATi . (2.50) After summing up terms, we have 2s3 + 6 (Qi − q2i ) j=i+1 qj + 3 (qi(Qi − q2i )) . (2.51) If the condition (2.49) is satisfied, J1 computed above matches (2.43) and we have 2s3 + 3Λis . (2.52) Finally, let us consider the angular momentum associated with ξψ = ∂ψ = ∂φ1+∂φ2 . In addition to (2.50) with ξ1 being replaced by ξψ, here we have to consider two more contributions. Namely, ⋆∇ξψ − Σi∩∂Ti Σi−1∩∂Ti (ξψ ·ATi)βi ∧ATi . (2.53) It is easy to check that the sum of each term is given by i + 4s 3 + 6s (Qi − q2i ) . (2.54) When evaluated under the condition (2.49), this gives i + 4s 3 + 6Λis (2.55) and agrees with Jψ given as the sum of (2.43) and (2.44). 2.5 Generalization It is straightforward to generalize the techniques we developed so far to the supergravity theory with n of U(1) vector fields AI , (I = 1, . . . , n). There are (n−1) vector multiplets because the gravity multiplet also contains the graviphoton field which is a vector field. – 14 – The scalars in the vector multiplet are denoted by M I , which are constrained by the condition N ≡ cIJKM IMJMK = 1. (2.56) cIJK is a set of constants. The action for the boson fields is given by ⋆R− aIJdM I ∧ ⋆dMJ − aIJF I ∧ ⋆F J − cIJKAI ∧ F J ∧ FK (2.57) where R is the Ricci scalar, and aIJ = − (NIJ −NINJ) . (2.58) In the last expression, NI = ∂N /∂M I and NIJ = ∂2N /∂M I∂MJ . This is the low- energy action of M-theory compactified on a Calabi-Yau manifoldM with n = h1,1(M), 6cIJK = ωI ∧ ωJ ∧ ωK (2.59) is the triple intersection of integrally-quantized two-forms ωI on M . The action for the minimal supergavity (2.1) is obtained by setting n = 1, c111 = (2/ 3)3, and a11 = 2. As for the calculation of the electric charges, one only needs to put the indices I, J,K to the vector fields and the result is ⋆aIJF cIJKA J ∧ FK . (2.60) As for the angular momenta, there is extra terms coming from the energy-momentum tensor of the scalar fields in the right hand side of (2.15). Its contribution to the angular momenta vanishes upon integration, so that the result is Jξ = − ⋆∇ξ + 2 ⋆ aIJ(ξ · AI)F J + 2cIJK(ξ ·AI)AJ ∧ FK . (2.61) For a more complicated Lagrangian, e.g. with charged hypermultiplets and/or with higher-derivative corrections, it is easier to utilize the general framework set up by Wald, than to find the partial integral in (2.15) by inspection. The charge constructed by this technique has an important property [27] that it acts as the Hamiltonian for the corresponding local symmetry in the Hamiltonian formulation of the theory, and it reproduces the Page charge and the angular momenta (2.61). Consequently, the charge as the generator of the symmetry is not the gauge-invariant Maxwell charge, but the Page charge which depends on a large gauge transformation. The integrands in the expressions above are not well-defined as differential forms when there are magnetic fluxes, thus it needs to be defined appropriately as we did – 15 – in the previous sections. Generically, we would like to rewrite the integral of a gauge invariant form ω on a region B to the integral of ω(1) satisfying dω(1) = ω (2.62) on its boundary ∂B. The problem is that ω(1) may depend on the gauge. On two patches S and T , it is represented by differential forms ωS(1) and ω (1) respectively. Since ω is gauge-invariant, we have dωS = dωT . Thus, if we take a sufficiently small coordinate patch, we can choose ω (S,T ) such that (S,T ) = ωS(1) − ωT(1). (2.63) Then one defines the integral of ω(1) on C = ∂B via ω(1) ≡ ωS(1) + ωT(1) + , (2.64) where D = ∂S = −∂T . The equations (2.62), (2.63) are the so-called descent relation which is important in the understanding of the anomaly. It will be interesting to generalize our analysis to the case where there are more than two patches and multiple overlaps among them. Presumably we need to include higher descendants ω (S1,...,Sn) the correction term at the boundary of n patches S1, . . . , Sn in the definition of the integral (2.64). 3. Relation to Four-Dimensional Charges We have seen how the near-horizon data of the black rings encode the charges measured at the asymptotic infinity. We can also consider rings in the Taub-NUT space [19, 20, 21] instead in the five-dimensional Minkowski space. Then the theory can also be thought of as a theory in four dimensions, via the Kaluza-Klein reduction along S1 of the Taub-NUT space. It has been established [35] that supersymmetric solutions for five dimensional supergravity nicely reduces to supersymmetric solutions for the corresponding four dimensional theory. In four dimensions, there are no problems in defining the charges, because the equations of motion and Bianchi identities yield the relations dF I = 0, dGI = d(⋆(g IJ )F J + θIJF J) = 0 (3.1) where (g−2)IJ are the inverse coupling constants and θIJ are the theta angles. The electric and magnetic charges can be readily obtained by integrating GI and F I over – 16 – the horizon. Then it is natural to expect that our formulae for the charges will yield the four-dimensional ones after the Kaluza-Klein reduction. One apparent problem is that the Page charges changes under a large gauge transformation, whereas the four- dimensional charges are seemingly well-defined as is. We will see that a large gauge transformation corresponds to the Witten effect on dyons in four-dimensions. 3.1 Mapping of the fields First let us recall the well-known mapping of the fields in four and five dimensions. The details can be found e.g. in [11, 12, 15, 16]. When we reduce a five-dimensional N = 2 supergravity with n vector fields along S1, it results in a four-dimensional N = 2 supergravity with (n+1) vector fields. The metrics in respective dimensions are related ds25d = e 2ρ(dψ − A0)2 + e−ρds24d, (3.2) where we take the periodicity of ψ to be 2π so that eρ is the five-dimensional radius of the Kaluza-Klein circle. The factor in front of the four-dimensional metric is so chosen that the four-dimensional Einstein-Hilbert term is canonical. The gauge fields in four and five dimensions are related by AI5d = a I(dψ − A0) + AI4d (3.3) where I = 1, . . . , n. It is chosen so that a gauge transformation of A0 do not affect AI4d. We need to introduce coordinate patches when there is a flux for AI5d. We demand that gauge transformations used between patches should not depend on ψ so that aI are globally well-defined scalar fields. Then, by the reduction of the five-dimensional action (2.57), the action of four- dimensional gauge fields is determined to be 2 L = − e3ρ + eρaIJa F 0 ∧ ⋆F 0 − cIJKaIaJaKF 0 ∧ F 0 + 2eρaIJa IF 0 ∧ ⋆F J + 3cIJKaIaJF 0 ∧ FK − eρaIJF I ∧ ⋆F J − 3cIJKaIF J ∧ FK . (3.4) Partial integrations are necessary to bring the naive Kaluza-Klein reduction to the form above. The resulting Lagrangian above follows from the prepotential F (X) = cIJKX IXJXK , (3.5) 2We take the following conventions in four dimensions: The orientations in four and five dimensions are related such that dx0 ∧ dx1 ∧ dx2 ∧ dx3 ∧ dψ = 2π dx0 ∧ dx1 ∧ dx2 ∧ dx3. The Levi-Civita symbol in four dimensions is defined by ǫ0123 = +1 and ǫ 0123 = −1 in local Lorentz coordinates. – 17 – if one defines special coordinates zI = XI/X0 by zI = aI + ieρM I . (3.6) This relation can be checked without the detailed Kaluza-Klein reduction. Indeed, the ratio of aI and M I in (3.6) can be fixed by inspecting the mass squared of a hypermultiplet, and the fact aI should enter in zI linearly with unit coefficient is fixed by the monodromy. 3.2 Mapping of the charges In many references including Ref. [12, 16, 23], the charge of the black object in five di- mensions is defined to be the charges in four dimensions after the dimensional reduction determined from the Lagrangian (3.4). It was motivated partly because the analysis of the charge in five dimensions was subtle due to the presence of the Chern-Simons interaction, whereas we studied how we can obtain the formula for the charges which has five-dimensional general covariance in Section 2. Now let us compare the charges thus defined in four- and five- dimensions. Firstly, the magnetic charge F 0 (3.7) in four dimensions counts the number of the Kaluza-Klein monopole inside C. It is also called the nut charge. The other magnetic charges in four dimensions F I (3.8) come directly from the dipole charges in five dimensions, as long as the surface C does not enclose the nut. When C does contain a nut, the Kaluza-Klein circle is non-trivially fibered over C. Thus, the surface C cannot be lifted to five dimensions. We will come back to this problem in Section 3.5. The formulae for the electric charges follow from the Lagrangian : ⋆2eρaIJ(F J − aJF 0) + 6cIJKaJFK − 3cIJKaJaKF 0 , (3.9) ⋆e3ρF 0 − ⋆2eρaIJaI(F J − aJF 0) + 2cIJKaIaJaKF 0 − 3cIJKaIaJFK (3.10) It is easy to verify that the five-dimensional Page charges (2.60) and the Noether charge Jψ (2.61) for the isometry ∂ψ along the Kaluza-Klein circle are related to the – 18 – four-dimensional electric charges via QI = − QI , Q0 = − Jψ. (3.11) An important point in the calculation is that the compensating term on the boundary of the coordinate patches vanishes, since aI and F J4d are globally well-defined. Thus we see that the four-dimensional charges are not the reduction of the gauge- invariant Maxwell charges ⋆F or that of the gauge-invariant “Maxwell-like” part of the angular momentum, ⋆∇ξ. They are rather the reduction of the Page or the Noether charges, which change under a large gauge transformation. 3.3 Reduction and the attractor In the literature, the attractor equation is often analyzed after the reduction to four dimensions [12, 15, 16], while the five-dimensional attractor mechanism for the black rings in [22] only determines the scalar vacuum expectation values via the magnetic dipoles. As we saw in the previous sections, the electric charges at the asymptotic infinity are encoded by the Wilson lines along the horizon. We show that how these five-dimensional consideration reproduces the known attractor solution [36, 37] in four- dimensions. The five-dimensional metric is characterized by the magnetic charges qI through the horizon, and the physical radius of the horizon ℓ = eρ there. From the attractor mechanism for the black rings [22], the near-horizon geometry is of the form AdS3×S2, and the curvature radii are q and q/2 in each factor, where q3 = cIJKq IqJqK . The scalar vevs are fixed to be proportional to the magnetic dipoles, i.e. M I = qI/q. For the calculation of electric charges the Wilson lines aI along the horizon are also important. Then we can evaluate the Page charges and angular momenta on the horizon to obtain QI = 6cIJKa JqK , Q0 = qℓ 2 − 3cIJKaIaJqK . (3.12) We can solve the equations above for ℓ and aI so that we have the formula for the four-dimensional special coordinates zI in terms of the charges. The result is zI = aI + ieρM I = DIJQI + i qI (3.13) where DIJ = cIJKq K , DIJDJK = δ K (3.14) – 19 – D = q3 = cIJKq IqJqK , Q̂0 = qℓ 2 = Q0 + DIJQIQJ . (3.15) It is the well-known solution of the attractor equation in four-dimensions with q0 = 0 [36, 37]. Thus, the combination of the attractor mechanism in five dimensions and the tech- nique of Page charges yield the attractor mechanism in four dimensions. The point is that the Wilson lines aI along the horizon of the black string carry the information of its electric charges. Conversely, the Wilson line at the horizon is determined by the electric charge. The horizon length is also determined by the angular momentum. In this sense, the attractor mechanism for the black rings also fixes all the relevant near-horizon data by means of the charges, angular momenta and dipoles. 3.4 Gauge dependence and monodromy Let us now come back to the question of the variation of the Page charges under large gauge transformations. The problem is that the integral A∧F depends on the shift A→ A+β for dβ = 0 if C has a non-contractible loop ℓ and β 6= 0. In the spacetime which asymptotes to R4,1, the large gauge transformation can be fixed by demanding that the gauge potential vanishes at the asymptotic infinity. In the present case of reduction to four dimensions, however, the gauge potential along the Kaluza-Klein circle is one of the moduli and is not a thing to be fixed. More precisely, if the ψ direction is non-contractible, a large gauge transformation associated to the Kaluza-Klein circle corresponds to a shift aI → aI + tI where tI are integers. In four-dimensional language it is the shift zI → zI + tI , (3.16) and the gauge variation of the Page charge translates to the variation of the electric charge under the transformation (3.16). It is precisely the Witten effect on dyons [24] if one recalls the fact that the dynamical theta angles of the theory depends on zI . In the terminology of N = 2 supergravity and special geometry, it is called the monodromy transformation associated to the shift (3.16), which acts symplectically on the charges (qI , QI) and on the projective special coordinates (X I , FI) For the M-theory compactification on the product of S1 and a Calabi-Yau, electric charges QI and q I correspond to the number of M2-branes and M5-branes wrapping two-cycles ΠI and four-cycles ΣI , respectively. The relation (2.59) translates to 6cIJK = #(ΣI ∩ ΣJ ∩ ΣK) in this language. The gauge fields AI arise from the Kaluza-Klein reduction of the M-theory three-form C on ΠI . Thus, the results above imply that – 20 – the M2-brane charges transform non-trivially in the presence of M5-branes under large gauge transformations of the C-field. It might sound novel, but it can be clearly seen from the point of view of Type IIA string theory on the Calabi-Yau. Consider a soliton without D6-brane charge. There, the D2-brane charge QI of the soliton is induced by the world-volume gauge field F on the D4 brane wrapped on a four-cycle Σ = qIΣI through the Chern-Simons coupling (F +B) ∧ C (3.17) where B is the NSNS two-form and C is the RR three-form. In this description, aI is given by B. The induced brane charge in the presence of the non-zero B-field is an intricate problem in itself, but the end result is that the large gauge transformation B → B + ω with ω = tI changes the D2-brane charge of the system by 6cIJKq ItJ . It will be interesting to derive the same effect from the worldvolume Lagrangian [38] of the M5 brane, which is subtle because the worldvolume tensor field is self-dual. The change in the M2-brane charge induce a change in the Kaluza-Klein momentum carried by the zero-mode on the black strings wrapped on S1, so that Q0 also changes [2]. The point is that the momentum carried by non-zero modes, Q̂0 defined in (3.15), is a monodromy-invariant quantity. Before leaving this section, it is worth noticing that if an M2-brane has the world- volume V , it enters in the equation of motion for G = dC in the following way: d ⋆ G+G ∧G = δV (3.18) where δV is the delta function supported on V . Thus, the quantized M2-brane charge is not the source of the Maxwell charge. It is rather the source of the Page charge. Essentially the same argument in five dimensions, using the specific decomposition (2.28), was made in Ref. [39]. 3.5 Monodromy and Taub-NUT If we use the Taub-NUT space in the dimensional reduction, in other words if there is a Kaluza-Klein monopole in the system, the Kaluza-Klein circle shrinks at the nut of the monopole. As the circle is now contractible, one might think that one can no longer do a large gauge transformation and that it is natural to choose aI = 0 at the nut. Nevertheless, from a four-dimensional standpoint the monodromy transformation should be always possible. How can these two points of view be reconciled? Firstly, the fact that the five-dimensional spacetime is smooth at the nut only requires that the gauge field strength is zero there and that the integral of the gauge – 21 – potential is an integer. There should be a patch around the nut in the five-dimensional spacetime in which AI should be smooth, but it is not the patch connected to the asymptotic region of the Taub-NUT space where aI is defined. A similar problem was studied in Ref. [40]. There, it was shown how the winding number can still be conserved in the background with the nut, where the circle on which strings are wound degenerates. A crucial role is played by the normalizable self- dual two-form ω localized at the nut, which gives the worldvolume gauge field A of the D6-brane realized as the M-theory Kaluza-Klein monopole via C = A ∧ ω. It should enter in the worldvolume Lagrangian in the combination dA+B, and the large gauge transformation affects the contribution from B. Indeed, the Kaluza-Klein ansatz of the gauge fields (3.3), one can make the com- bined shift aI → aI + tI , AI4d → AI4d + tIA0 (3.19) without changing the five-dimensional gauge field strengths. Therefore, the magnetic charge also transforms as qI → qI + tIq0. (3.20) The action of the transformation (3.16) on the electric charges then becomes QI → QI + 6cIJKtJqK + 3cIJKtJ tKq0, (3.21) Q0 → Q0 −QItI − 3cIJKtItJqK − cIJKtItJtKq0, (3.22) which is exactly how the projective coordinates X0, XI , FI = 3cIJKX JXK/X0, F0 = −cIJKXIXJXK/(X0)2. (3.23) get transformed by the monodromy aI → aI + tI . It was already noted in Ref. [21] that the same symmetry acts on the functions which characterize the supersymmetric solution on the Taub-NUT, (V,KI , LI ,M) in their notation. The point is that it modifies the five-dimensional Page charges, and hence the four-dimensional charges. If we neglect quantum corrections coming from instantons wrapping the Kaluza- Klein circle, it is allowed to do the monodromy transformation zI → zI + tI even with continuous parameters tI . It maps a solution of the equations of motion to another, and the electric charges in four-dimensions depends continuously on the vevs for the moduli aI at the asymptotic infinity. The issue concerning the stability of the solitons can be safely ignored. In the analyses in Refs. [19, 20, 21], their proposals for the identification of four-dimensional electric charges QI and of five-dimensional ones QI were different from one another. The source of the discrepancy in the identification is now clear after our long discussion. It can be readily checked that the differing – 22 – proposals for the identification can be connected by the monodromy transformation with tI = 1 qI . Namely, the charges in the five-dimensional language are transformed QI − 3cIJKqJqK , Jψ → Jψ − Jφ (3.24) for Q0 ≫ q3 limit.3 Thus they are equivalent under a large gauge transformation. The analysis above also answers the question raised in Section 3.2 how the dipole charges in five dimensions are related in the magnetic charges in four dimensions in the presence of the nut. It is instructive to consider the case of a black ring in the Taub- NUT space. From a five-dimensional viewpoint, the dipole charge is not a conserved quantity measurable at the asymptotic infinity. Correspondingly, the surface of the Dirac string necessary to define the gauge potential can be chosen to fill the disc inside the black ring only, and not to extend to the asymptotic infinity. It was what we did in Section 2.3.1 in defining the coordinate patches. However, the gauge transformation required to achieve it necessarily depends on the ψ coordinate, which is the direction along the Kaluza-Klein circle. Hence it is not allowed if one carries out the reduction to four dimensions. In this case, the Dirac string emanating from the black ring necessarily extends all the way to the spatial infinity, thus making the magnetic charge measurable at the asymptotic infinity. A related point is that dipole charges enter in the first law of black objects because of the existence of two patches [31]4. It is easier to understand it after the reduction because now it is a conserved quantity measurable at the asymptotic infinity. As a final example to illustrate the subtlety in the identification of the four- and five-dimensional charges, let us consider a two-centered Taub-NUT space with centers x1 and x2. There is an S 2 between two centers, and one can introduce a self-dual mag- netic fluxes qI through it. Although the Chern-Simons interactions put some constraint on the allowed qI , there is a supersymmetric solution of this form [44]. In this configu- ration, the Wilson lines aI at x1 and x2 necessarily differ by the amount proportional to the flux, and one cannot simultaneously make them zero. An important consequence is that the magnetic charges F I4d of the nuts at x2 and x2 necessarily differ, in spite of the fact that the geometry and the gauge fields in five dimensions are completely symmetric under the exchange of x1 and x2. 3We noticed that a small discrepancy proportional to cIJKq IqJqK remains, which is related to the zero-point energy of the conformal field theory of the black string. Its effect on the entropy is subleading in the large Q0 limit. 4The authors of [31] used the approach to the first law developed in [41]. There is another un- derstanding of appearance of the dipole charges in the first law [42] if one follows the approach in [43]. – 23 – 4. Summary In this paper, we have first clarified how the near-horizon data of black objects encode the conserved charges measured at asymptotic infinity. Namely, the existence of the Chern-Simons coupling means that F ∧ F is a source of electric charges, thus it was necessary to perform the partial integration once to rewrite the asymptotic electric charge by the integral of A ∧ F over the horizon. Since F has magnetic flux through the horizon, A∧F cannot be naively defined, and we showed how to treat it consistently. Likewise, we obtained the formula for the angular momenta using the near-horizon data. Then, we saw how our formula for the charges in five dimensions is related to the four-dimensional formula under Kaluza-Klein reduction. We studied how the ambiguity coming from large gauge transformations in five dimensions corresponds to the Witten effect and the associated monodromy transformation in four dimensions. It is now straightforward to obtain the correction to the entropy of the black rings, since we now have the supersymmetric higher-derivative action [13], the near-horizon geometry [45, 46, 47], and also the formulation developed in this paper to obtain con- served charges from the near-horizon data alone. It would be interesting to see if it matches with the microscopic calculation. Acknowledgments YT would like to thank Juan Maldacena, Masaki Shigemori and Johannes Walcher for discussions. KH is supported by the Center-of-Excellence (COE) Program “Nanometer- Scale Quantum Physics” conducted by Graduate Course of Solid State Physics and Graduate Course of Fundamental Physics at Tokyo Institute of Technology. The work of KO is supported by Japan Society for the Promotion of Science (JSPS) under the Post-doctoral Research Program. YT is supported by the United States DOE Grant DE-FG02-90ER40542. A. Geometry of Concentric Black Rings Any supersymmetric solution in the asymptotically flat R1,4 is known to be of the form ds2 = −f 2(dt+ ω)2 + f−1ds2(R4) (A.1) where f and ω is a function and a one-form on R4, respectively. We parametrize the base R4 in the Gibbons-Hawking coordinate system ds2(R4) = H [dr2 + r2(dθ2 + sin2 θdχ2)] +H−1(2dψ + cos θdχ)2 (A.2) – 24 – where (r, θ, φ) parametrize a flat R3, the periodicity of ψ is 2π and H = 1/r. Our notation mostly follows the one in Ref. [34], with the change ψthere = 2ψhere. The quantities f , ω and the gauge field F = dA are determined by three functions K, L and M on the flat R3. The relations we need are f−1 = H−1K2 + L, ι∂ψω = 2H −2K3 + 3H−1KL+ 2M, (A.3) d[f(dt+ ω)]− 1√ G+, ι∂ψG + = −3d(H−1K) (A.4) where G+ = f(dω + ⋆dω)/2 is a self-dual two-form on R4. To construct the concentric black ring solutions, we take N points xi, (i = 1, . . . , N) at r = R2i /4, θ = π on R 3. The orbit of xi along the coordinate ψ is a ring of radius Ri embedded in R4. We choose functions K, L and M by K = −1 qihi, L = 1 + (Qi − q2i )hi, M = qi(1− |xi|hi) (A.5) where hi(x) = 1/|x−xi| are harmonic functions on R3. For the case with a single ring, conversion to the ring coordinate used in (2.29) can be achieved via φ1 = ψ + χ/2, φ2 = ψ − χ/2 (A.6) y2 − 1 x− y = 2 r sin 1− x2 x− y = 2 r cos . (A.7) The behavior of ω and F at the asymptotic infinity, and the near-horizon metric (2.36) are well-known and are not repeated here. The reader is referred to the orig- inal article Ref. [34]. The gauge potential near the horizon can be obtained by the combination of (A.3) and (A.4). First we have ι∂ψF = (−dι∂ψ)[f(dt+ ω)] + 3d(KH−1). (A.8) which can be integrated by inspection. Hence the ψ component of the gauge field is given by ι∂ψA = H−1KL/2 +M H−1K2 + L (A.9) for some constant c. By demanding ιψA→ 0 as r → ∞, we obtain c = −1 qi. (A.10) – 25 – Thus, we have ι∂ψA = − Qi − q2i . 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0704.1820

Odd Triplet Pairing in clean Superconductor/Ferromagnet heterostructures

Odd Triplet Pairing in clean Superconductor/Ferromagnet heterostructures Klaus Halterman,1, ∗ Paul H. Barsic,2, † and Oriol T. Valls2, ‡ Physics and Computational Sciences, Research and Engineering Sciences Department, Naval Air Warfare Center, China Lake, California 93555 School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 (Dated: October 27, 2018) We study triplet pairing correlations in clean Ferromagnet (F)/Superconductor (S) nanojunctions, via fully self consistent solution of the Bogoliubov-de Gennes equations. We consider FSF trilayers, with S being an s-wave superconductor, and an arbitrary angle α between the magnetizations of the two F layers. We find that contrary to some previous expectations, triplet correlations, odd in time, are induced in both the S and F layers in the clean limit. We investigate their behavior as a function of time, position, and α. The triplet amplitudes are largest at times on the order of the inverse “Debye” frequency, and at that time scale they are long ranged in both S and F. The zero temperature condensation energy is found to be lowest when the magnetizations are antiparallel. PACS numbers: 74.45.+c, 74.25.Bt, 74.78.Fk The proximity effects in superconductor/ferromagnet (SF) heterostructures lead to the coexistence of ferromag- netic and superconducting ordering and to novel trans- port phenomena[1, 2]. Interesting effects that arise from the interplay between these orderings have potential tech- nological applications in fields such as spintronics[3]. For example, the relative orientation of the magnetizations in the F layers in FSF trilayers can have a strong influence on the conductivity[4, 5, 6, 7, 8], making them good spin valve candidates. Such trilayers were first proposed[9] for insulating F layers and later for metallic[10, 11] ones. This interplay also results in fundamental new physics. An outstanding example is the existence of “odd” triplet superconductivity. This is an s-wave pairing triplet state that is even in momentum, and therefore not destroyed by nonmagnetic impurities, but with the triplet corre- lations being odd in frequency, so that the equal time triplet amplitudes vanish as required by the Pauli prin- ciple. This exotic pairing state with total spin one was proposed long ago [12] as a possible state in superfluid 3He. Although this type of pairing does not occur there, it is possible in certain FSF systems[1, 2, 13, 14] with or- dinary singlet pairing in S. This arrangement can induce, via proximity effects, triplet correlations with m = 0 and m = ±1 projections of the total spin. If the magnetiza- tion orientations in both F layers are unidirectional and along the quantization axis, symmetry arguments show that only the m = 0 projection along that axis can exist. Odd triplet pairing in F/S structures has been studied in the dirty limit through linearized Usadel-type quasi- classical equations [2, 13, 14, 15]. In this case, it was found that m = 0 triplet pairs always exist. They are suppressed in F over short length scales, just as the sin- glet pairs. The m = ±1 components, for which the ex- change field is not pair-breaking, can be long ranged, and were found to exist for nonhomogeneous magnetization. For FSF trilayers[2, 16, 17], the quasiclassical methods predict that the structure contains a superposition of all FIG. 1: Schematic of FSF junction. The left ferromagnetic layer F1 has a magnetization oriented at an angle −α/2 in the x− z plane, while the other ferromagnet, F2, has a mag- netization orientation at an angle α/2 in the x− z plane. three spin triplet projections except when the magneti- zations of the F layers are collinear, in which case the m = ±1 components along the magnetization axis van- ish. It is noted in Ref. [1] that the existence of such effects in the clean limit has not been established and may be doubted. This we remedy in the present work, where we establish that, contrary to the doubts voiced there, induced, long-ranged, odd triplet pairing does occur in clean FSF structures. Experimental results that may argue for the existence of long range triplet pairing of superconductors through a ferromagnet have been obtained in superlattices[18] with ferromagnetic spacers, and in two superconduc- tors coupling through a single ferromagnet[19, 20]. Measurements[19] on a SQUID, in which a phase change of π in the order parameter is found after inversion, in- dicate an odd-parity state. Very recently, a Josephson current through a strong ferromagnet was observed, in- dicating the existence of a spin triplet state[20] induced http://arxiv.org/abs/0704.1820v1 by NbTiN, an s-wave superconductor. In this paper, we study the induced odd triplet superconductivity in FSF trilayers in the clean limit through a fully self-consistent solution of the microscopic Bogoliubov-de Gennes (BdG) equations. We consider ar- bitrary relative orientation of the magnetic moments in the two F layers. We find that there are indeed induced odd triplet correlations which can include both m = 0 and m = ±1 projections. We directly study their time dependence and we find that they are largest for times of order of the inverse cutoff “Debye” frequency. The corre- lations are, at these time scales, long ranged in both the S and F regions. We also find that the condensation en- ergy depends on the relative orientation of the F layers, being a minimum when they are antiparallel. To find the triplet correlations arising from the non- trivial spin structure in our FSF system, we use the BdG equations with the BCS Hamiltonian, Heff : Heff = ψδ(r) + (iσy)δβ∆(r)ψ (r) + h.c.]− (r)(h · σ)δβ ψβ(r) where ∆(r) is the pair potential, to be determined self- consistently, ψ , ψδ are the creation and annihilation op- erators with spin δ, EF is the Fermi energy, and σ are the Pauli matrices. We describe the magnetism of the F layers by an effective exchange field h(r) that vanishes in the S layer. We will consider the geometry depicted in Fig. 1, with the y axis normal to the layers and h(r) in the x − z plane (which is infinite in extent) forming an angle ±α/2 with the z axis in each F layer. Next, we expand the field operators in terms of a Bo- goliubov transformation which we write as: ψδ(r) = unδ(r)γn + ηδvnδ(r)γ , (1) where ηδ ≡ 1(−1) for spin down (up), unδ and vnδ are the quasiparticle and quasihole amplitudes. This transforma- tion diagonalizes Heff : [Heff , γn] = −ǫnγn, [Heff , γ n. By taking the commutator [ψδ(r),Heff ], and with h(r) in the x − z plane as explained above, we have the following: [ψ↑(r),Heff ] = (He − hz)ψ↑(r)− hxψ↓(r) + ∆(r)ψ [ψ↓(r),Heff ] = (He + hz)ψ↓(r)− hxψ↑(r)−∆(r)ψ Inserting (1) into (2) and introducing a set ρ of Pauli-like matrices in particle-hole space, yields the spin-dependent BdG equations: H01̂− hzσz ∆(y)ρx − hx1̂ Φn = ǫnΦn, where Φn ≡ (un↑(y), un↓(y), vn↑(y), vn↓(y)) T and H0 ≡ −∂2y/(2m) + ε⊥ − EF . Here ε⊥ is the transverse kinetic energy and a factor of eik⊥·r has been suppressed. In deriving Eq. (3) care has been taken to consistently use the phase conventions in Eq. (1). To find the quasiparti- cle amplitudes along a different quantization axis in the x−z plane, one performs a spin rotation: Φn → Û(α ′)Φn, where Û(α′) = cos(α′/2)1̂⊗ 1̂− i sin(α′/2)ρz ⊗ σy. When the magnetizations of the F layers are collinear, one can take hx = 0. For the general case shown in Fig. 1 one has in the F1 layer, hx = h0 sin(−α/2) and hz = h0 cos(−α/2), where h0 is the magnitude of h, while in F2, hx = h0 sin(α/2), and hz = h0 cos(α/2). With an appropriate choice of basis, Eqs. (3) are cast into a matrix eigenvalue system that is solved itera- tively with the self consistency condition, ∆(y) = g(y)f3 (f3 = [〈ψ↑(r)ψ↓(r)〉 − 〈ψ↓(r)ψ↑(r)〉]). In the F layers we have g(y) = 0, while in S, g(y) = g, g being the usual BCS singlet coupling constant there. Through Eqs. (1), the self-consistency condition becomes a sum over states restricted by the factor g to within ωD from the Fermi surface. Iteration is performed until self-consistency is reached. The numerical process is the same that was used in previous work[24, 25], with now the hx term requiring larger four-component matrices to be diagonalized. We now define the following time dependent triplet am- plitude functions in terms of the field operators, f̃0(r, t) = [〈ψ↑(r, t)ψ↓(r, 0)〉+ 〈ψ↓(r, t)ψ↑(r, 0)〉] , (4a) f̃1(r, t) = [〈ψ↑(r, t)ψ↑(r, 0)〉 − 〈ψ↓(r, t)ψ↓(r, 0)〉] , (4b) which, as required by the Pauli principle for these s-wave amplitudes, vanish at t = 0, as we shall verify. Making use of Eq. (1) and the commutators, one can derive and formally integrate the Heisenberg equation of the motion for the operators and obtain: f̃0(y, t) = [un↑(y)vn↓(y)− un↓(y)vn↑(y)]ζn(t), (5a) f̃1(y, t) =− [un↑(y)vn↑(y) + un↓(y)vn↓(y)]ζn(t), FIG. 2: (Color online) The real part, f0, of the triplet ampli- tude f̃0, for a FSF trilayer at 7 different times. We normalize f0 by the singlet bulk pair amplitude, ∆0/g. The coordinate y is scaled by the Fermi wavevector, Y ≡ kF y, and time by the Debye frequency, τ ≡ ωDt. At τ = 0, f0 ≡ 0 as required by the Pauli principle. The interface is marked by the verti- cal dashed line, with an F region to the left and the S to the right. Half of the S region and part of the left F layer are shown. The inset shows the maximum value of f0 versus τ . where ζn(t) ≡ cos(ǫnt)− i sin(ǫnt) tanh(ǫn/2T ). The amplitudes in Eqs. (5) contain all information on the space and time dependence of induced triplet correla- tions throughout the FSF structure. The summations in Eqs. (5) are over the entire self-consistent spectrum, en- suring that f0 and f1 vanish identically at t = 0 and thus obey the exclusion principle. Using a non self consistent ∆(y) leads to violations of this condition, particularly near the interface where proximity effects are most pro- nounced. Geometrically, the indirect coupling between magnets is stronger with fairly thin S layers and rela- tively thick F layers. We thus have chosen dS = (3/2)ξ0 and dF1 = dF2 = ξ0, with the BCS correlation length ξ0 = 100k . We consider the low T limit and take ωD = 0.04EF . The magnetic exchange is parametrized via I ≡ h0/EF . Results shown are for I = 0.5 (unless otherwise noted) and the magnetization orientation an- gle, α, is swept over the range 0 ≤ α ≤ π. No triplet amplitudes arise in the absence of magnetism (I = 0). For the time scales considered here, the imaginary parts of f̃0(y, t) and f̃1(y, t) at t 6= 0 are considerably smaller than their real parts, and thus we focus on the latter, which we denote by f0(y, t) and f1(y, t). In Fig. 2, the spatial dependence of f0 is shown for parallel mag- netization directions (α = 0) at several times τ ≡ ωDt. The spatial range shown includes part of the F1 layer (to the left of the dashed line) and half of the S layer (to the right). At finite τ , the maximum occurs in the ferromagnet close to the interface, after which f0 under- goes damped oscillations with the usual spatial length FIG. 3: (Color online) Spatial and angular dependence of f1, at τ = 4 ≈ τc and several α. Normalizations and ranges are as in Fig. 2. Inset: maxima of f0 and f1 in F1 versus α. scale ξf ≈ (kF↑ − kF↓) −1 ≈ k−1F /I. The height of the main peak first increases with time, but drops off after a characteristic time, τc ≈ 4, as seen in the inset, which depicts the maximum value of f0 as a function of τ . As τ increases beyond τc, the modulating f0 in F develops more complicated atomic scale interference patterns and becomes considerably longer ranged. In S, we see imme- diately that f0 is also larger near the interface. Since the triplet amplitudes vanish at τ = 0, short time scales exhibit correspondingly short triplet penetration. The figure shows, however, that the value of f0 in S is sub- stantial for τ & τc, extending over length scales on the order of ξ0 without appreciable decay. In contrast, the usual singlet correlations were found to monotonically drop off from their τ = 0 value over τ scales of order unity. In the main plot of Fig. 3 we examine the spatial de- pendence of the real part of the m = ±1 triplet ampli- tude, f1. Normalizations and spatial ranges are as in Fig. 2 but now the time is fixed at τ = 4 ≈ τc, and five equally spaced magnetization orientations are con- sidered. At α = 0, f1 vanishes identically at all τ , as expected. For nonzero α, correlations in all triplet chan- nels are present. As was found for f0, the plot clearly shows that f1 is largest near the interface, in the F re- gion. Our geometry and conventions imply (see Fig. 1) that the magnetization has opposite x-components in the F1 and F2 regions. The f1 triplet pair amplitude profile is thus antisymmetric about the origin, in contrast to the symmetric f0, implying the existence of one node in the superconductor. Nevertheless, the penetration of the f1 correlations in S can be long ranged. We find that f1 and f0 oscillate in phase and with the same wavelength, re- gardless of α. The inset illustrates the maximum attained values of f0 and f1 in F1 as α varies. It shows that for FIG. 4: (Color online) The T = 0 condensation energy, ∆E0, normalized by N(0)∆20 (N(0) is the usual density of states), vs. the angle α for two values of I . When the two magne- tizations are antiparallel (α = π) ∆E0 is lowest. The inset shows the ordinary (singlet) pair potential averaged over the S region, normalized to the bulk ∆0. a broad range of α, α . 3π/4, the maximum of f0 varies relatively little, after which it drops off rapidly to zero at α = π. This is to be expected as the anti-parallel orienta- tion corresponds to the case in which the magnetization is in the x direction, which is perpendicular to the axis of quantization (see Fig. 1). The rise in the maximum of f1 is monotonic, cresting at α = π, consistent with the main plot. At this angle the triplet correlations ex- tend considerably into the superconductor. At α = π/2 the maxima coincide since the two triplet components are then identical throughout the whole space because the magnetization vectors have equal projections on the x and z axes. At α = π both magnetizations are normal to the axis of quantization z (see Fig. 1). By making use of the rotation matrix Û (see below Eq. 3) one can verify that the m = ±1 components with respect to the axis x along the magnetizations are zero. We next consider the condensation energy, ∆E0, cal- culated by subtracting the zero temperature supercon- ducting and normal state free energies. The calculation uses the self consistent spectra and ∆(y), and methods explained elsewhere [25, 26]. In the main plot of Fig. 4, we show ∆E0 (normalized at twice its bulk S value) at two different values of I. The condensation energy results clearly demonstrate that the antiparallel state (α = π) is in general the lowest energy ground state. These results are consistent with previous studies[8] of FSF structures with parallel and antiparallel magnetizations. The inset contains the magnitude of the spatially averaged pair po- tential, normalized by ∆0, at the same values of I. The inset correlates with the main plot, as it shows that the singlet superconducting correlations in S increase with α and are larger at I = 1 than at I = 0.5. The half- metallic case of I = 1 illustrates that by having a single spin band populated at the Fermi surface, Andreev reflec- tion is suppressed, in effect keeping the superconductivity more contained within S. Thus, we have shown that in clean FSF trilayers in- duced odd triplet correlations, with m = 0 and m = ±1 projections of the total spin, exist. We have used a mi- croscopic self-consistent method to study the time and angular dependence of these triplet correlations. The correlations in all 3 triplet channels were found, at times τ ≡ ωDt & τc, where τc ≈ 4, to be long ranged in both the F and S regions. Finally, study of the condensation energy revealed that the ground state energy is always lowest for antiparallel magnetizations. This project was supported in part by a grant of HPC resources from the ARSC at the University of Alaska Fairbanks (part of the DoD HPCM program) and by the University of Minnesota Graduate School. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]; Also at Minnesota Supercomputer Institute, University of Minnesota, Min- neapolis, Minnesota 55455 [1] A.I. Buzdin, Rev. Mod. 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0704.1821

The S-parameter in Holographic Technicolor Models

The S-parameter in Holographic Technicolor Models Kaustubh Agashea, Csaba Csákib, Christophe Grojeanc,d, and Matthew Reeceb a Department of Physics, Syracuse University, Syracuse, NY 13244, USA b Institute for High Energy Phenomenology Newman Laboratory of Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA d CERN, Theory Division, CH 1211, Geneva 23, Switzerland c Service de Physique Théorique, CEA Saclay, F91191, Gif-sur-Yvette, France [email protected], [email protected], [email protected], [email protected] Abstract We study the S parameter, considering especially its sign, in models of electroweak symmetry breaking (EWSB) in extra dimensions, with fermions localized near the UV brane. Such models are conjectured to be dual to 4D strong dynamics triggering EWSB. The motivation for such a study is that a negative value of S can significantly ameliorate the constraints from electroweak precision data on these models, allowing lower mass scales (TeV or below) for the new particles and leading to easier discovery at the LHC. We first extend an earlier proof of S > 0 for EWSB by boundary conditions in arbitrary metric to the case of general kinetic functions for the gauge fields or arbitrary kinetic mixing. We then consider EWSB in the bulk by a Higgs VEV showing that S is positive for arbitrary metric and Higgs profile, assuming that the effects from higher-dimensional operators in the 5D theory are sub-leading and can therefore be neglected. For the specific case of AdS5 with a power law Higgs profile, we also show that S ∼ +O(1), including effects of possible kinetic mixing from higher-dimensional operator (of NDA size) in the 5D theory. Therefore, our work strongly suggests that S is positive in calculable models in extra dimensions. http://arxiv.org/abs/0704.1821v2 1 Introduction One of the outstanding problems in particle physics is to understand the mechanism of electroweak symmetry breaking. Broadly speaking, models of natural electroweak symmetry breaking rely either on supersymmetry or on new strong dynamics at some scale near the electroweak scale. However, it has long been appreciated that if the new strong dynamics is QCD-like, it is in conflict with precision tests of electroweak observables [1]. Of particular concern is the S parameter. It does not violate custodial symmetry; rather, it is directly sensitive to the breaking of SU(2). As such, it is difficult to construct models that have S consistent with data, without fine-tuning. The search for a technicolor model consistent with data, then, must turn to non-QCD- like dynamics. An example is “walking” [2], that is, approximately conformal dynamics, which can arise in theories with extra flavors. It has been argued that such nearly-conformal dynamics can give rise to a suppressed or even negative contribution to the S parameter [3]. However, lacking nonperturbative calculational tools, it is difficult to estimate S in a given technicolor theory. In recent years, a different avenue of studying dynamical EWSB models has opened up via the realization that extra dimensional models [4] may provide a weakly coupled dual description to technicolor type theories [5]. The most studied of these higgsless models [6] is based on an AdS5 background in which the Higgs is localized on the TeV brane and has a very large VEV, effectively decoupling from the physics. Unitarization is accomplished by gauge KK modes, but this leads to a tension: these KK modes cannot be too heavy or perturbative unitarity is lost, but if they are too light then there are difficulties with electroweak precision: in particular, S is large and positive [7]. In this argument the fermions are assumed to be elementary in the 4D picture (dual to them being localized on the Planck brane). A possible way out is to assume that the direct contribution of the EWSB dynamics to the S-parameter are compensated by contributions to the fermion-gauge boson vertices [8, 9]. In particular, there exists a scenario where the fermions are partially composite in which S ≈ 0 [10], corresponding to almost flat wave functions for the fermions along the extra dimension. The price of this cancellation is a percent level tuning in the Lagrangian parameter determining the shape of the fermion wave functions. Aside from the tuning itself, this is also undesirable because it gives the model-builder very little freedom in addressing flavor problems: the fermion profiles are almost completely fixed by consistency with electroweak precision. While Higgsless models are the closest extra-dimensional models to traditional technicolor models, models with a light Higgs in the spectrum do not require light gauge KK modes for unitarization and can be thought of as composite Higgs models. Particularly appealing are those where the Higgs is a pseudo-Nambu-Goldstone boson [11, 12]. In these models, the electroweak constraints are less strong, simply because most of the new particles are heavy. They still have a positive S, but it can be small enough to be consistent with data. Unlike the Higgsless models where one is forced to delocalize the fermions, in these models with a higher scale the fermions can be peaked near the UV brane so that flavor issues can be addressed. Recently, an interesting alternative direction to eliminating the S-parameter constraint has been proposed in [13]. There it was argued, that by considering holographic models of EWSB in more general backgrounds with non-trivial profiles of a bulk Higgs field one could achieve S < 0. The aim of this paper is to investigate the feasibility of this proposal. We will focus on the direct contribution of the strong dynamics to S. In particular, we imagine that the SM fermions can be almost completely elementary in the 4D dual picture, corresponding to them being localized near the UV brane. In this case, a negative S would offer appealing new prospects for model-building since such values of S are less constrained by data than a positive value [14]. Unfortunately we find that the S > 0 quite generally, and that backgrounds giving negative S appear to be pathological. The outline of the paper is as follows. We first present a general plausibility argument based purely on 4D considerations that one is unlikely to find models where S < 0. This argument is independent from the rest of the paper, and the readers interested in the holo- graphic considerations may skip directly to section 3. Here we first review the formalism to calculate the S parameter in quite general models of EWSB using an extra dimension. We also extend the proof of S > 0 for BC breaking [7] in arbitrary metric to the case of arbitrary kinetic functions or localized kinetic mixing terms. These proofs quite clearly show that no form of boundary condition breaking will result in S < 0. However, one may hope that (as argued in [13]) one can significantly modify this result by using a bulk Higgs with a profile peaked towards the IR brane to break the electroweak symmetry. Thus, in the crucial section 4, we show that S > 0 for models with bulk breaking from a scalar VEV as well. Since the gauge boson mass is the lowest dimensional operator sensitive to EWSB one would expect that this is already sufficient to cover all interesting possibilities. However, since the Higgs VEV can be very strongly peaked, one may wonder if other (higher dimensional) operators could become important as well. In particular, the kinetic mixing operator of L,R after Higgs VEV insertion would be a direct contribution to S. To study the effect of this operator in section 5, it is shown that the bulk mass term for axial field can be converted to kinetic functions as well, making a unified treatment of the effects of bulk mass terms and the effects of the kinetic mixing from the higher-dimensional operator possible. Although we do not have a general proof that S > 0 including the effects of the bulk kinetic mixing for a general metric and Higgs profile, in section 5.2 we present a detailed scan for AdS metric and for power-law Higgs vev profile using the technique of the previous section for arbitrary kinetic mixings. We find S > 0 once we require that the higher-dimensional operator is of NDA size, and that the theory is ghost-free. We summarize and conclude in section 6. 2 A plausibility argument for S > 0 In this section we define S and sketch a brief argument for its positivity in a general techni- color model. The reader mainly interested in the extra-dimensional constructions can skip this section since it is independent from the rest of the paper. However, we think it is worth- while to try to understand why one might expect S > 0 on simple physical grounds. The only assumptions we will make are that we have some strongly coupled theory that sponta- neously breaks SU(2)L×SU(2)R down to SU(2)V , and that at high energies the symmetry is restored. With these assumptions, S > 0 is plausible. S < 0 would require more complicated dynamics, and might well be impossible, though we cannot prove it.1 Consider a strongly-interacting theory with SU(2) vector current V aµ and SU(2) axial vector current Aaµ. We define (where J represents V or A): d4x e−iq·x Jaµ(x)J = δab qµqν − gµνq2 2). (2.1) We further define the left-right correlator, denoted simply Π(q2), as ΠV (q 2)−ΠA(q2). In the usual way, ΠV and ΠA are related to positive spectral functions ρV (s) and ρA(s). Namely, the Π functions are analytic functions of q2 everywhere in the complex plane except for Minkowskian momenta, where poles and branch points can appear corresponding to physical particles and multi-particle thresholds. The discontinuity across the singularities on the q2 > 0 axis is given by a spectral function. In particular, there is a dispersion relation ΠV (q ρV (s) s− q2 + iǫ , (2.2) with ρV (s) > 0, and similarly for ΠA. Chiral symmetry breaking establishes that ρA(s) contains a term πf πδ(s). This is the massless particle pole corresponding to the Goldstone of the spontaneously broken SU(2) axial flavor symmetry. (The corresponding pions, of course, are eaten once we couple the theory to the Standard Model, becoming the longitudinal components of the W± and Z bosons. However, for now we consider the technicolor sector decoupled from the Standard Model.) We define a subtracted correlator by Π̄(q2) = Π(q2) + and a subtracted spectral function by ρ̄A(s) = ρA(s)− πf 2πδ(s). Now, the S parameter is given by S = 4πΠ̄(0) = 4 (ρV (s)− ρ̄A(s)) . (2.3) Interestingly, there are multiple well-established nonperturbative facts about ΠV − ΠA, but none are sufficient to prove that S > 0. There are the famous Weinberg sum rules [17] ds (ρV (s)− ρ̄A(s)) = f 2π , (2.4) ds s (ρV (s)− ρ̄A(s)) = 0. (2.5) Further, Witten proved that Σ(Q2) = −Q2(ΠV (Q2)−ΠA(Q2)) > 0 for all Euclidean momenta Q2 = −q2 > 0 [18]. However, the positivity of S seems to be more difficult to prove. Our plausibility argument is based on the function Σ(Q2). In terms of this function, S = −4πΣ′(0). (Note that in Σ(Q2) the 1/Q2 pole from ΠA is multiplied by Q2, yielding a constant that does not contribute when we take the derivative. Thus when considering 1For a related discussion of the calculation of S in strongly coupled theories, see [15]. Σ we do not need to subtract the pion pole as we did in Π̄.) We also know that Σ(0) = f 2π > 0. On the other hand, we know something else that is very general about theories that spontaneously break chiral symmetry: at very large Euclidean Q2, we should see symmetry restoration. More specifically, we expect behavior like Σ(Q2) → O , (2.6) where k is associated with the dimension of some operator that serves as an order parameter for the symmetry breaking. (In some 5D models the decrease of ΠA − ΠV will actually be faster, e.g. in Higgsless models one has exponential decrease.) While we are most familiar with this from the OPE of QCD, it should be very general. If a theory did not have this property and ΠV and ΠA differed significantly in the UV, we would not view it as a sponta- neously broken symmetry, but as an explicitly broken one. Now, in this context, positivity of S is just the statement that, because Σ(Q2) begins at a positive value and eventually becomes very small, the smoothest behavior one can imagine is that it simply decreases monotonically, and in particular, that Σ′(0) < 0 so that S > 0.2 The alternative would be that the chiral symmetry breaking effects push Σ(Q2) in different directions over different ranges of Q2. We have not proved that this is impossible in arbitrary theories, but it seems plausible that the simpler case is true, namely that chiral symmetry restoration always acts to decrease Σ(Q2) as we move to larger Q2. Indeed, we will show below that in a wide variety of perturbative holographic theories S is positive. 3 Boundary-effective-action approach to oblique cor- rections. Simple cases with boundary breaking In this section we review the existing results and calculational methods for the electroweak precision observables (and in particular the S-parameter) in holographic models of elec- troweak symmetry breaking. There are two equivalent formalisms for calculating these parameters. One is using the on-shell wave function of the W/Z bosons [19], and the electroweak observables are calculated from integrals over the extra dimension involving these wave functions. The advantage of this method is that since it uses the physical wave functions it is easier to find connections to the Z and the KK mass scales. The alternative formalism proposed by Barbieri, Pomarol and Rattazzi [7] (and later extended in [20] to in- clude observables off the Z-pole) uses the method of the boundary effective action [21], and involves off-shell wave functions of the boundary fields extended into the bulk. This latter method leads more directly to a general expression of the electroweak parameters, so we will be applying this method throughout this paper. Below we will review the basic expressions from [7]. A theory of electroweak symmetry breaking with custodial symmetry has an SU(2)L× SU(2)R global symmetry, of which the SU(2)L×U(1)Y subgroup is gauged (since the S- parameter is unaffected by the extra B − L factor we will ignore it in our discussion). At 2For a related discussion of the behaviour of Σ in the case of large-Nc QCD, see [16]. low energies, the global symmetry is broken to SU(2)D. In the holographic picture of [7] the elementary SU(2)×U(1) gauge fields are extended into the bulk of the extra dimension. The bulk wave functions are determined by solving the bulk EOM’s as a function of the boundary fields, and the effective action is just the bulk action in terms of the boundary fields. In order to first keep the discussion as general as possible, we use an arbitrary background metric over an extra dimension parametrized by 0 < y < 1, where y = 0 corresponds to the UV boundary, and y = 1 to the IR boundary. In order to simplify the bulk equations of motion it is preferential to use the coordinates in which the metric takes the form 1 [7] ds2 = e2σdx2 + e4σdy2 . (3.1) The bulk action for the gauge fields is given by S = − 1 (FLMN) 2 + (FRMN) . (3.2) The bulk equations of motion are given by µ − p2e2σAL,Rµ = 0, (3.3) or equivalently the same equations for the combinations Vµ, Aµ = (AµL ± AµR)/ We assume that the (light) SM fermions are effectively localized on the Planck brane and that they carry their usual quantum numbers under SU(2)L × U(1)Y that remains unbroken on the UV brane. The values of these fields on the UV brane have therefore a standard couplings to fermion and they are the 4D interpolating fields we want to compute an effective action for. This dictates the boundary conditions we want to impose on the UV brane ALaµ (p 2, 0) = ĀLaµ (p 2), AR 3µ (p 2, 0) = ĀR 3µ (p 2), AR 1,2µ (p 2, 0) = 0. (3.4) R are vanishing because they correspond to ungauged symmetry generators. The solutions of the bulk equations of motion satisfying these UV BC’s take the form 2, y) = v(y, p2)V̄µ(p 2), Aµ(p 2, y) = a(y, p2)Āµ(p 2). (3.5) where the interpolating functions v and a satisfy the bulk equations ∂2yf(y, p 2)− p2e2σf(y, p2) = 0 (3.6) and the UV BC’s v(0, p2) = 1, a(0, p2) = 1. (3.7) The effective action for the boundary fields reduces to a pure boundary term since by integrating by parts the bulk action vanishes by the EOM’s: Seff = d4x(Vµ∂yV µ + Aµ∂yA µ)|y=0 = d4p(V̄ 2µ ∂yv + Ā µ∂ya)|y=0 (3.8) 1In this paper, we use a (−+ . . .+) signature. 5D bulk indices are denoted by capital Latin indices while we use Greek letters for 4D spacetime indices. 5D indices will be raised and lowered using the 5D metric while the 4D Minkowski metric is used for 4D indices. And we obtain the non-trivial vacuum polarizations for the boundary vector fields ΣV (p 2) = − ∂yv(0, p 2), ΣA(p 2) = − ∂ya(0, p 2). (3.9) The various oblique electroweak parameters are then obtained from the momentum ex- pansion of the vacuum polarizations in the effective action, Σ(p2) = Σ(0) + p2Σ′(0) + Σ′′(0) + . . . (3.10) For example the S-parameter is given by S = 16πΣ′3B(0) = 8π(Σ V (0)− Σ′A(0)). (3.11) A similar momentum expansion can be performed on the interpolating functions v and a: v(y, p2) = v(0)(y) + p2v(1)(y) + . . ., and similarly for a. The S-parameter is then simply expressed as S = −8π (1) − ∂ya(1))|y=0. (3.12) The first general theorem was proved in [7]: for the case of boundary condition breaking in a general metric, S ≥ 0. The proof uses the explicit calculation of the functions v(n), a(n), n = 0, 1. First, the bulk equations (3.3) write (0) = ∂2ya (0) = 0, ∂2yv (1) = e2σv(0), ∂2ya (1) = e2σa(0). (3.13) And the p2-expanded UV BC’s are v(0) = a(0) = 1, v(1) = a(1) = 0 at y = 0 (3.14) Finally, we need to specify the BC’s on the IR brane that correspond to the breaking SU(2)L×SU(2)R → SU(2)D ∂yVµ = 0, Aµ = 0, (3.15) which translates into simple BC’s for the interpolating functions (n) = a(n) = 0, n = 0, 1. (3.16) The solution of these equations are v(0) = 1, a(0) = 1 − y, v(1) = dy′′e2σ(y ′′) − dy′e2σ(y ′), a(1) = dy′′e2σ(y ′′)(1−y′′)−y dy′′e2σ(y ′′)(1−y′′). Consequently dye2σ(y)dy − dy′(1− y′)e2σ(y′) (3.17) which is manifestly positive. 3.1 S > 0 for BC breaking with boundary kinetic mixing The first simple generalization of the BC breaking model is to consider the same model but with an additional localized kinetic mixing operator added on the TeV brane (the effect of this operator has been studied in flat space in [7] and in AdS space in [19]). The localized Lagrangian is −gV 2µν . (3.18) This contains only the kinetic term for the vector field since the axial gauge field is set to zero by the BC breaking. In this case the BC at y = 1 for the vector field is modified to ∂yVµ + τp 2Vµ = 0. In terms of the wave functions expanded in small momenta we get (1)+τv(0) = 0. The only change in the solutions will be that now v(1) = −τ− e2σ(y ′)dy′, resulting in e2σ(y)dy − (1− y′)e2σ(y′)dy′ + τ (3.19) Thus as long as the localized kinetic term has the proper sign, the shift in the S-parameter will be positive. If the sign is negative, there will be an instability in the theory since fields localized very close to the TeV brane will feel a wrong sign kinetic term. Thus we conclude that for the physically relevant case S remains positive. 3.2 S > 0 for BC breaking with arbitrary kinetic functions The next simple extension of the BPR result is to consider the case when there is an arbitrary y-dependent function in front of the bulk gauge kinetic terms. These could be interpreted as effects of gluon condensates modifying the kinetic terms in the IR. In this case the action is S = − φ2L(y)(F 2 + φ2R(y)(F . (3.20) φL,R(y) are arbitrary profiles for the gauge kinetic terms, which are assumed to be the consequence of some bulk scalar field coupling to the gauge fields. Note that this case also covers the situation when the gauge couplings are constant but g5L 6= g5R. The only assumption we are making is that the gauge kinetic functions for L,R are strictly positive. Otherwise one could create a wave packet localized around the region where the kinetic term is negative which would have ghost-like behavior. Due to the y-dependent kinetic terms it is not very useful to go into the V,A basis. Instead we will directly solve the bulk equations in the original basis. The bulk equations of motion for L,R are given by L,R∂yA µ )− p2e2σφ2L,RAL,Rµ = 0 (3.21) To find the boundary effective action needed to evaluate the S-parameter we perform the following decomposition: ALµ(p 2, y) = L̄µ(p 2)LL(y, p 2) + R̄µ(p 2)LR(y, p ARµ (p 2, y) = L̄µ(p 2)RL(y, p 2) + R̄µ(p 2)RR(y, p 2). (3.22) Here L̄, R̄ are the boundary fields, and the fact that we have four wave functions expresses the fact that these fields will be mixing due to the BC’s on the IR brane. The UV BC’s (3.4) and the IR BC’s (3.15) can be written in terms of the interpolating functions as (UV) LL(0, p 2) = 1, LR(0, p 2) = 0, RL(0, p 2) = 0, RR(0, p 2) = 1. (3.23) LL(1, p 2) = RL(1, p 2), LR(1, p 2) = RR(1, p ∂y(LL(1, p 2) +RL(1, p 2)) = 0, ∂y(LR(1, p 2) +RR(1, p 2)) = 0. (3.24) The solution of these equations with the proper boundary conditions and for small values of p2 is rather lengthy, so we have placed the details in Appendix A. The end result is that S = −8π φ2L∂yL R + φ |y=0 = − (aLR + aRL), (3.25) where the constants aRL are negative as their explicit expressions shows it. Therefore S is positive. 4 S > 0 in models with bulk Higgs Having shown than S > 0 for arbitrary metric and EWSB through BC’s, in this section, we switch to considering breaking of electroweak symmetry by a bulk scalar (Higgs) vev. We begin by neglecting the effects of kinetic mixing between SU(2)L and SU(2)R fields coming from higher-dimensional operator in the 5D theory, expecting that their effect, being suppressed by the 5D cut-off, is sub-leading. We will return to a consideration of such kinetic mixing effects in the following sections. We will again use the metric (3.1) and the bulk action (3.2). Instead of BC breaking we assume that EWSB is caused by a bulk Higgs which results in a y-dependent profile for the axial mass term A2M . (4.1) Here M2 is a positive function of y corresponding to the background Higgs VEV. The bulk equations of motion are: (∂2y − p2e2σ)Vµ = 0, (∂2y − p2e2σ −M2e4σ)Aµ = 0. (4.2) On the IR brane, we want to impose regular Neumann BC’s that preserve the full SU(2)L× SU(2)R gauge symmetry (IR) ∂yVµ = 0, ∂yAµ = 0. (4.3) As in the previous section, the BC’s on the UV brane just define the 4D interpolating fields (UV ) Vµ(p 2, 0) = V̄µ(p 2), Aµ(p 2, 0) = Āµ(p 2). (4.4) The solutions of the bulk equations of motion satisfying these BC’s take the form 2, y) = v(y, p2)V̄µ(p 2), Aµ(p 2, y) = a(y, p2)Āµ(p 2), (4.5) where the interpolating functions v and a satisfy the bulk equations ∂2yv − p2e2σv = 0, ∂2ya− p2e2σa−M2e4σa = 0. (4.6) As before, these interpolating functions are expanded in powers of the momentum: v(y, p2) = v(0)(y) + p2v(1)(y) + . . ., and similarly for a. The S-parameter is again given by the same expression S = −8π (1) − ∂ya(1))|y=0. (4.7) We will not be able to find general solutions for a(1) and v(1) but we are going to prove that (1) > ∂yv (1) on the UV brane, which is exactly what is needed to conclude that S > 0. First at the zeroth order in p2, the solution for v(0) is simply constant, v(0) = 1, as before. And a(0) is the solution of (0) = M2e4σa(0), a(0)|y=0 = 1, ∂ya(0)|y=1 = 0. (4.8) In particular, since a(0) is positive at y = 0, this implies that a(0) remains positive: if a(0) crosses through zero it must be decreasing, but then this equation shows that the derivative will continue to decrease and can not become zero to satisfy the other boundary condition. Now, since a(0) is positive, the equation of motion shows that it is always concave up, and then the condition that its derivative is zero at y = 1 shows that it is a decreasing function of y. In particular, we have for all y a(0)(y) ≤ v(0)(y), (4.9) with equality only at y = 0. Next consider the order p2 terms. What we wish to show is that ∂ya (1) > ∂yv (1) at the UV brane. First, let’s examine the behavior of v(1): the boundary conditions are v(1)|y=0 = 0 and ∂yv = 0. The equation of motion is: (1) = e2σv(0) = e2σ > 0, (4.10) so the derivative of v(1) must increase to reach zero at y = 1. Thus it is negative everywhere except y = 1, and v(1) is a monotonically decreasing function of y. Since v(1)|y=0 = 0, v(1) is strictly negative on (0, 1]. For the moment suppose that a(1) is also strictly negative; we will provide an argument for this shortly. The equation of motion for a(1) is: (1) = e2σa(0) +M2e4σa(1). (4.11) Now, we know that a(0) < v(0), so under our assumption that a(1) < 0, this means that (1) ≤ ∂2yv(1), (4.12) with equality only at y = 0. But we also know that ∂yv (1)∂ya (1) at y = 1, since they both satisfy Neumann boundary conditions there. Since the derivative of ∂ya (1) is strictly smaller over (0, 1], it must start out at a higher value in order to reach the same boundary condition. Thus we have that > ∂yv . (4.13) The assumption that we made is that a(1) is strictly negative over the interval (0, 1]. The reason is the following: suppose that a(1) becomes positive at some value of y. Then as it passes through zero it is increasing. But then we also have that ∂2ya (1) = e2σa(0)+M2e4σa(1), and we have argued above that a(0) > 0. Thus if a(1) is positive, ∂ya (1) remains positive, because ∂2ya (1) cannot become negative. In particular, it becomes impossible to reach the boundary condition ∂ya (1) = 0 at y = 1. This fills the missing step in our argument and shows that the S parameter must be positive. In the rest of this section we show that the above proof for the positivity of S remains essentially unchanged in the case when the bulk gauge couplings for the SU(2)L and SU(2)R gauge groups are not equal. In this case (in order to get diagonal bulk equations of motion) one needs to also introduce the canonically normalized gauge fields. We start with the generic action (metric factors are understood when contracting indices) 4g25L (FLMN) 4g25R (FRMN) h2(z) (LM −RM )2 (4.14) To get to a canonically normalized diagonal basis we redefine the fields as g25L + g (L−R) , Ṽ = g25L + g . (4.15) To get the boundary effective action, we write the fields Ṽ , à as Ã(p2, z) = g25L + g L̄(p2)− R̄(p2) ã(p2, z) , (4.16) Ṽ (p2, z) = g25L + g L̄(p2) + R̄(p2) ṽ(p2, z) . (4.17) Here L̄, R̄ are the boundary effective fields (with non-canonical normalization exactly as in [7]), while the profiles ã, ṽ satisfy the same bulk equations and boundary conditions as a, v in (4.2)–(4.4) with an appropriate replacement for M2 = (g25L + g 2. In terms of the canonically normalized fields, the boundary effective action takes its usual form Seff = Ṽ ∂yṼ + Ã∂yà . (4.18) And we deduce the vacuum polarization ΣL3B(p 2) = − g25L + g (∂y ṽ(0, p 2)− ∂yã(0, p2)) (4.19) And finally the S-parameter is equal to S = − 16π g25L + g (∂y ṽ (1) − ∂yã(1)) (4.20) Since ã(n), ṽ(n), n = 0, 1 satisfy the same equations (4.2)–(4.4) as before, the proof goes through unchanged and we conclude that S > 0. 5 Bulk Higgs and bulk kinetic mixing Next, we wish to consider the effects of kinetic mixing from higher-dimensional operator in the bulk involving the Higgs VEV – as mentioned earlier, this kinetic mixing is suppressed by the 5D cut-off and hence expected to be a sub-leading effect. The reader might wonder why we neglected it before, but consider it now? The point is that, although the leading effect on S parameter is positive as shown above, it can be accidentally suppressed so that the formally sub-leading effects from the bulk kinetic mixing can be important, in particular, such effects could change the sign of S. Also, the Higgs VEV can be large, especially when the Higgs profile is “narrow” such that it approximates BC breaking, and thus the large VEV can (at least partially) compensate the suppression from the 5D cut-off. Of course, in this limit of BC breaking (δ-function VEV), we know that kinetic mixing gives S < 0 only if tachyons are present in the spectrum, but we would like to cover the cases intermediate between BC breaking limit and a broad Higgs profile as well. In this section, we develop a formalism, valid for arbitrary metric and Higgs profile, to treat the bulk mass term and kinetic mixing on the same footing and then we apply this technique to models in AdS space and with power-law profiles for Higgs VEV in the next section. We first present a discussion of how a profile for the y-dependent kinetic term is equivalent to a bulk mass term. This is equivalent to the result [13] that a bulk mass term can be equivalent to an effective metric. However, we find the particular formulation that we present here to be more useful when we deal with the case of a kinetic mixing. Assume we have a Lagrangian for a gauge field that has a kinetic term S = − 1 −gφ2(y)F 2MN (5.1) We work in the axial gauge A5 = 0 and again the metric takes the form (3.1). We redefine the field to absorb the function φ: Ã(y) = φ(y)A(y). The action in terms of the new field is then written as S = − 1 e2σF̃ 2µν + 2(∂yõ) 2 + 2 Ã2µ − 4(∂yõ)õ (5.2) To see that the kinetic profile φ is equivalent to a mass term, we integrate by parts in the second term S = − 1 F̃ 2MN + 2e (5.3) Thus we find that a bulk kinetic profile is equivalent to a bulk mass plus a boundary mass. The bulk equations of motion for the new variables will then be ∂2yõ − e2σp2õ − õ = 0, (5.4) and the boundary conditions become ∂yõ = õ. (5.5) Note, that despite the bulk mass term, there is still a massless mode whose wavefunction is simply φ(z). Now we can reverse the argument and say that a bulk mass must be equivalent to a profile for the bulk kinetic term plus a boundary mass term. 5.1 The general case We have seen above how to go between a bulk mass terms and a kinetic function. We will now use this method to discuss the general case, when there is electroweak symmetry breaking due to a bulk higgs with a sharply peaked profile toward the IR brane, and the same Higgs introduces kinetic mixing between L and R fields corresponding to a higher dimensional operator from the bulk. For now we assume that the Higgs fields that breaks the electroweak symmetry is in a (2,2) of SU(2)L×SU(2)R, with a VEV 〈H〉 = diag(h(z), h(z))/ 2.1 This Higgs profile h has dimension 3/2. The 5D action is given by (FLMN) 2 + (FRMN) − (DMH)†(DMH) + Tr(FLMNH †HFMN R) (5.6) Here α is a coefficient of O(1) and Λ is the 5D cutoff scale, given approximately by Λ ∼ 24π3/g25. The kinetic mixing term just generates a shift in the kinetic terms of the vector and axial vector field, and we will write the bulk mass term also as a shift in the kinetic term for the axial vector field. The exact form of the translation between the two forms is given by answering the question of how to redefine the field with an action (note that m2 has a mass dimension 3) wF 2MN +m 22g25AµA (5.7) to a theory with only a modified kinetic term. The appropriate field redefinition A = ρà will be canceling the mass term if ρ satisfies ∂y(w∂yρ) = m 2g25e 4σρ, (5.8) 1An alternative possibility would be to consider a Higgs in the (3,3) representation of SU(2)L×SU(2)R. together with the boundary conditions ρ′|y=1 = 0, ρ|y=0 = 1. The relation between the new and the old expression for w will be w̃ = ρ2w. The action in this case is given by −gw̃F̃ 2MN + w̃(0) (∂yρ)à 2|y=0 (5.9) This last boundary term is actually irrelevant for the S-parameter: since it does not contain a derivative on the field it can not get an explicit p-dependence so it will not contribute to S, so for practical purposes this boundary term can be neglected. With this expression we now can calculate S. For this we need the modified version of the formula from [13], where the breaking is not by boundary conditions but by a bulk Higgs. The expression is dye2σ(wV − w̃A). (5.10) In our case wV = 1− αh 2(y)2g2 while w̃A = wAρ 2 = (1 + αh2(y)2g2 This formula also gives another way to see that S > 0 in the absence of kinetic mixing, without analyzing the functions v(1) and a(1) from Section 4 in detail. Without kinetic mixing, wV = 1 and w̃A = ρ 2, and the equation of motion for ρ is simply ∂2yρ = m 2g25e In that case ρ is just the function we called a(0) in Section 4. Since we showed there that a(0) ≤ 1, we see that our expression 5.10 gives an alternative argument that S > 0 without kinetic mixing, because it is simply an integral of e2σ(1− ρ2) ≥ 0. 5.2 Scan of the parameter space for AdS backgrounds Having developed the formalism for a unified treatment of bulk mass terms and bulk kinetic mixing, we then apply it to the AdS case with a power-law profile for the Higgs vev. Requiring (i) calculability of the 5D theory, i.e., NDA size of the higher-dimensional operator, (ii) that excited W/Z’s are heavier than a few 100 GeV, and (iii) a ghost-free theory, i.e., positive kinetic terms for both V and A fields, we find that S is always positive in our scan for this model. We do not have a general proof that S > 0 for an arbitrary background with arbitrary Higgs profiles, if we include the effects of the bulk kinetic mixing, but we feel that such a possibility is quite unlikely based on our exhaustive scan. For this scan we will take the parametrization of the Higgs profile from [22]. Here the metric is taken as AdS space ds2 = ηµνdx µdxν − dz2 , (5.11) where as usual R < z < R′. The bulk Higgs VEV is assumed to be a pure monomial in z (rather than a combination of an increasing and a decreasing function). The reason for this is that we are only interested in the effect of the strong dynamics on the electroweak precision parameters. A term in the Higgs VEV growing toward the UV brane would mean that the value of bulk Higgs field evaluated on the UV brane gets a VEV, implying that there is EWSB also by a elementary Higgs (in addition to the strong dynamics) in the 4D dual. We do not want to consider such a case. The form of the Higgs VEV is then assumed to be v(z) = 2(1 + β) logR′/R (1− (R/R′)2+2β) , (5.12) where the parameter β characterizes how peaked the Higgs profile is toward the TeV brane (β → −1 corresponds to a flat profile, β → ∞ to an infinitely peaked one). The other parameter V corresponds to an “effective Higgs VEV”, and is normalized such that for V → 246 GeV we recover the SM and the KK modes decouple (R′ → ∞ irrespective of β). For more details about the definitions of these parameters see [22].2 We first numerically fix the R′ parameter for every given V, β and kinetic mixing pa- rameter α by requiring that the W -mass is reproduced. We do this approximately, since we assume the simple matching relation 1/g2 = R log(R′/R)/g25 to numerically fix the value of g5, which is only true to leading order, but due to wave function distortions and the extra kinetic term will get corrected. Then, ρ can be numerically calculated by solving (5.8), and from this S can be obtained via (5.10). We see that S decreases as we increase α. On the the hand, the kinetic function for vector field (wV ) also decreases in this limit. So, in order to find the minimal value of S consistent with the absence of ghosts in the theory, we find numerically the maximal value of α for every value of V, β for which the kinetic function of the vectorlike gauge field is still strictly positive. We then show contour plots for the minimal value of S taking this optimal value of α as a function of V, β in Fig. 5.2. In the first figure we fix R′ = 10−8 GeV−1, which is the usual choice for higgsless models with light KK W’ and Z’ states capable of rendering the model perturbative. In the second plot we choose the more conventional value R = 10−18 GeV−1. We can see the S is positive in both cases over all of the physical parameter space. We can estimate the corrections to the above matching relation from the wavefunction distortion and kinetic mixing as follows. The effect from wavefunction distortion is expected to be ∼ g2S/(16π) which is <∼ 10% if we restrict to regions of parameter space with S <∼ 10. Similarly, we estimate the effect due to kinetic mixing by simply integrating the operator over the extra dimension to find a deviation ∼ g6(V R′)2 log2 (R′/R) / (24π3)2. So, if restrict <∼ 1 TeV and 1/R′ >∼ 100 GeV, then this deviation is also small enough. We see that both effects are small due to the deviation being non-zero only near IR brane – even though it is O(1) in that region, whereas the zero-mode profile used in the matching relation is spread throughout the extra dimension. In order to be able to make a more general statement (and to check that the neglected additional contributions to the gauge coupling matching from the wave function distortions and the kinetic mixing indeed do not significantly our results) we have performed an addi- tional scan over AdS space models where we do not require the correct physical value of MW to be reproduced. In this scan we then treat R′ as an independent free parameter. In this case the correct matching between g and g5 is no longer important for the sign of S, since at 2Refs. [23] also considered similar models. 400 500 600 700 800 900 1000 400 500 600 700 800 900 1000 Figure 1: The contours of models with fixed values of the S-parameter due to the electroweak breaking sector. In the left panel we fix 1/R = 108 GeV, while in the right 1/R = 1018 GeV. The gauge kinetic mixing parameter α is fixed to be the maximal value corresponding to the given V, β (and R′ chosen such that the W mass is approximately reproduced). In the left panel the contours are S = 1, 2, 3, 4, 5, 6, while in the right S = 1, 1.5, 2. every place where g5 appears it is multiplied by a parameter we are scanning over anyway (V or α). We performed the scan again for two values of the AdS curvature, 1/R = 108 and 1018 GeV. For the first case we find that if we restrict α < 10, 1/R′ < 1 TeV there is no case with S < 0. However, there are some cases with S < 0 for α > 10, although in these cases the theory is likely not predictive. For 1/R = 1018 GeV we find that S < 0 only for V ∼ 250 GeV and β ∼ 0, 1/R′ ∼ 1 TeV. In this case α is of order one (for example α ∼ 5). This case corresponds to the composite Higgs model of [11] and it is quite plausible that at tree-level S < 0 if a large kinetic mixing is added in the bulk. However in this case EWSB is mostly due to a Higgs, albeit a composite particle of the strong dynamics, rather than directly by the strong dynamics, so it does not contradict the expectation that when EWSB is triggered directly via strong dynamics, then S is always large and positive. However, it shows that any general proof for S > 0 purely based on analyzing the properties of Eqs. (5.8)-(5.10) is doomed to failure, since these equations contain physical situations where EWSB is not due to the strong dynamics but due to a light Higgs in the spectrum. Thus any general proof likely needs to include more physical requirements on the decoupling of the physical Higgs. 6 Conclusions In this paper, we have studied the S parameter in holographic technicolor models, focusing especially on its sign. The motivation for our study was as follows. An alternative (to SUSY) solution to the Planck-weak hierarchy involves a strongly interacting 4D sector spontaneously breaking the EW symmetry. One possibility for such a strong sector is a scaled-up version of QCD as in the traditional technicolor models. In such models, we can use the QCD data to “calculate” S finding S ∼ +O(1) which is ruled out by the electroweak precision data. Faced by this constraint, the idea of a “walking” dynamics was proposed and it can be then argued that S < 0 is possible which is much less constrained by the data, but the S parameter cannot be calculated in such models. In short, there is a dearth of calculable models of (non-supersymmetric) strong dynamics in 4D. Based on the AdS/CFT duality, the conjecture is that certain kinds of theories of strong dynamics in 4D are dual to physics of extra dimensions. The idea then is to construct models of EWSB in an extra dimension. Such constructions allow more possibilities for model-building, at the same time maintaining calculability if the 5D strong coupling scale is larger than the compactification scale, corresponding to large number of technicolors in the 4D dual. It was already shown that S > 0 for boundary condition breaking for arbitrary metric (a proof for S > 0 for the case of breaking by a localized Higgs vev was recently studied in reference [24]). In this paper, we have extended the proof for boundary condition breaking to the case of arbitrary bulk kinetic functions for gauge fields or gauge kinetic mixing. Throughout this paper, we have assumed that the (light) SM fermions are effectively localized near the UV brane so that flavor violation due to higher-dimensional operators in the 5D theory can be suppressed, at the same time allowing for a solution to the flavor hierarchy. Such a localization of the light SM fermions in the extra dimension is dual to SM fermions being “elementary”, i.e., not mixing with composites from the 4D strong sector. It is known that the S parameter can be suppressed (or even switch sign) for a flat profile for SM fermions (or near the TeV brane) – corresponding to mixing of elementary fermions with composites in the 4D dual, but in such a scenario flavor issues could be a problem. We also considered the case of bulk breaking of the EW symmetry motivated by recent arguments that S < 0 is possible with different effective metrics for vector and axial fields. For arbitrary metric and Higgs profile, we showed that S > 0 at leading order, i.e., neglect- ing effects from all higher-dimensional operators in the 5D theory (especially bulk kinetic mixing), which are expected to be sub-leading effects being suppressed by the cut-off of the 5D theory. We also note that boundary mass terms can generally be mimicked to arbitrary precision by localized contributions to the bulk scalar profile, so we do not expect a more general analysis of boundary plus bulk breaking to find new features. Obtaining S < 0 must then require either an unphysical Higgs profile or higher-dimensional operators to contribute effects larger than NDA size, in which case we lose calculability of the 5D theory. To make our case for S > 0 stronger, we then explored effects of the bulk kinetic mixing between SU(2)L,R gauge fields due to Higgs vev coming from a higher-dimensional operator in the 5D theory. Even though, as mentioned above, this effect is expected to be sub-leading, it can nevertheless be important (especially for the sign of S) if the leading contribution to S is accidentally suppressed. Also, the large Higgs VEV, allowed for narrow profiles in the extra dimension (approaching the BC breaking limit), can compensate the suppression due to the cut-off in this operator. For this analysis, we found it convenient to convert bulk (mass)2 for gauge fields also to kinetic functions. Although a general proof for S > 0 is lacking in such a scenario, using the above method of treating the bulk mass for axial fields, we found that S ∼ +O(1) for AdS5 model with power-law Higgs profile in the viable (ghost-free) and calculable regions of the parameter space. In summary, our results combined with the previous literature strongly suggests that S is positive for calculable models of technicolor in 4D and 5D. We also presented a plausibility argument for S > 0 which is valid in general, i.e., even for non-calculable models. 7 Acknowledgments We thank Giacomo Cacciapaglia, Cédric Delaunay, Antonio Delgado, Guido Marandella, Riccardo Rattazzi, Matthew Schwartz and Raman Sundrum for discussions. We also thank Johannes Hirn and Veronica Sanz for comments on the manuscript. As we were finishing the paper, we learned that Raman Sundrum and Tom Kramer have also obtained results similar to ours [25]. C.C. thanks the theory group members at CERN for their hospitality during his visit. K.A. is supported in part by the U. S. DOE under Contract no. DE-FG-02-85ER 40231. The research of C.C. is supported in part by the DOE OJI grant DE-FG02-01ER41206 and in part by the NSF grant PHY-0355005. The research of M.R. is supported in part by an NSF graduate student fellowship and in part by the NSF grant PHY-0355005. C.G. is supported in part by the RTN European Program MRTN-CT-2004-503369 and by the CNRS/USA exchange grant 3503. Note added After submitting our paper to the arXiv, we learned of [26] which gives a proof for S > 0 for an arbitrary bulk Higgs profile in AdS background that is similar to our proof in Section 4. However, our proof of S > 0 is valid for a general metric and we have also included the effect of kinetic mixing between SU(2)L and SU(2)R fields via higher-dimensional operator (with Higgs vev) for the calculation of S in AdS background. We thank Deog-Ki Hong and Ho-Ung Yee for pointing out their paper to us. A Details of BC breaking with arbitrary kinetic func- tions Here we present the detailed calculation of S in the case with boundary breaking and arbi- trary kinetic functions described in Section 3.2. Recall that we had the following decompo- sition: ALµ(p 2, y) = L̄µ(p 2)LL(y, p 2) + R̄µ(p 2)LR(y, p ARµ (p 2, y) = L̄µ(p 2)RL(y, p 2) + R̄µ(p 2)RR(y, p 2), (A.1) with boundary conditions (UV) LL(0, p 2) = 1, LR(0, p 2) = 0, RL(0, p 2) = 0, RR(0, p 2) = 1. (A.2) LL(1, p 2) = RL(1, p 2), LR(1, p 2) = RR(1, p ∂y(LL(1, p 2) +RL(1, p 2)) = 0, ∂y(LR(1, p 2) +RR(1, p 2)) = 0. (A.3) The action again reduces to a boundary term Seff = φ2L(0)Lµ∂L µ + φ2R(0)Rµ∂R , (A.4) so we find that S = −8π φ2L∂yL R + φ |y=0 (A.5) where we have done an expansion in terms of the momentum for all the wave functions as usual as LL(y, p 2) = L L (y) + p L (y) + . . .. The lowest order wave functions satisfy the following bulk equations: J ) = 0, (A.6) where I and J can refer to L or R. Imposing the BC’s these equations can be simply solved in terms of the integrals fL(y) = , fR(y) = (A.7) L = 1− fL(y), L R = fL(y), R L = fR(y), R R = 1− fR(y). (A.8) In order to actually find S we need to go one step further, that is calculate the next order terms in the wave functions I J . These will satisfy the equations J ) = e 2σφ2II J , (A.9) where for the I J we use the solutions in (A.8). The form of the solutions will be given by J (y) = φ2I(y due2σφ2I(u)I J (u) + aIJ , (A.10) where aIJ are constants. In terms of these quantities the S-parameter is just given by S = −8π (aLR + aRL) (A.11) One can again solve the boundary conditions to find the constants aRL , aLR . These turn out to be aRL = − φ2L(y) dy′e2σ(y ′)φ2L(y ′)(1− fL(y′)) φ2L(1) φ2R(1) dye2σ(y)φ2R(y)fR(y) φ2L(y) φ2R(y) dy′e2σ(y ′)φ2R(y ′)fR(y (A.12) where φ2R(y) φ2L(1) φ2R(1) φ2L(y) . (A.13) A similar expressions applies for aLR with L ↔ R everywhere. Since 0 < fL,R < 1, we can see that every term in the expression is manifestly positive, so S is definitely positive. References [1] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990); B. Holdom and J. 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D 74, 015011 (2006) [arXiv:hep-ph/0602177]. http://arxiv.org/abs/hep-ph/0409126 http://arxiv.org/abs/hep-ph/0409266 http://arxiv.org/abs/hep-ph/0502162 http://arxiv.org/abs/hep-ph/0306259 http://arxiv.org/abs/hep-ph/0412089 http://arxiv.org/abs/hep-ph/0510164 http://arxiv.org/abs/hep-ph/0703164 http://arxiv.org/abs/hep-ph/0606086 http://arxiv.org/abs/hep-ph/0612239 http://arxiv.org/abs/hep-ph/0607231 http://arxiv.org/abs/hep-ph/0408281 http://arxiv.org/abs/hep-ph/0203034 http://arxiv.org/abs/hep-ph/0401160 http://arxiv.org/abs/hep-ph/0405040 http://arxiv.org/abs/hep-th/9802109 http://arxiv.org/abs/hep-th/9802150 http://arxiv.org/abs/hep-ph/0611358 http://arxiv.org/abs/hep-ph/0508279 http://arxiv.org/abs/hep-ph/0612242 http://arxiv.org/abs/hep-ph/0609104 http://arxiv.org/abs/hep-ph/0608241 http://arxiv.org/abs/hep-ph/0702234 http://arxiv.org/abs/hep-ph/0602177 Introduction A plausibility argument for S > 0 Boundary-effective-action approach to oblique corrections. Simple cases with boundary breaking S>0 for BC breaking with boundary kinetic mixing S>0 for BC breaking with arbitrary kinetic functions S>0 in models with bulk Higgs Bulk Higgs and bulk kinetic mixing The general case Scan of the parameter space for AdS backgrounds Conclusions Acknowledgments Details of BC breaking with arbitrary kinetic functions

0704.1822

Energy Functionals for the Parabolic Monge-Ampere Equation

ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION ZUOLIANG HOU AND QI LI 1. Introduction Because of its close connection with the Kähler-Ricci flow, the parabolic complex Monge-Ampère equation on complex manifolds has been studied by many authors. See, for instance, [Cao85, CT02, PS06]. On the other hand, theories for complex Monge-Ampère equation on both bounded domains and complex manifolds were developed in [BT76, Yau78, CKNS85, Ko l98]. In this paper, we are going to study the parabolic complex Monge-Ampère equation over a bounded domain. Let Ω ⊂ Cn be a bounded domain with smooth boundary ∂Ω. Denote QT = Ω × (0, T ) with T > 0, B = Ω × {0}, Γ = ∂Ω × {0} and ΣT = ∂Ω × (0, T ). Let ∂pQT be the parabolic boundary of QT , i.e. ∂pQT = B∪Γ∪ΣT . Consider the following boundary value problem: − log det = f(t, z, u) in QT , u = ϕ on ∂pQT . where f ∈ C∞(R× Ω̄×R) and ϕ ∈ C∞(∂pQT ). We will always assume that Then we will prove that Theorem 1. Suppose there exists a spatial plurisubharmonic (psh) function u ∈ C2(Q̄T ) such that u t − log det u αβ̄ ≤ f(t, z, u) in QT , u ≤ ϕ on B and u = ϕ on ΣT ∩ Γ. Then there exists a spatial psh solution u ∈ C∞(Q̄T ) of (1) with u ≥ u if following compatibility condition is satisfied: ∀ z ∈ ∂Ω, ϕt − log det = f(0, z, ϕ(z)), ϕtt − log det(ϕαβ̄) = ft(0, z, ϕ(z)) + fu(0, z, ϕ(z))ϕt . Date: October 29, 2018. http://arxiv.org/abs/0704.1822v1 2 ZUOLIANG HOU AND QI LI Motivated by the energy functionals in the study of the Kähler-Ricci flow, we introduce certain energy functionals to the complex Monge-Ampère equation over a bounded domain. Given ϕ ∈ C∞(∂Ω), denote (5) P(Ω, ϕ) = u ∈ C2(Ω̄) | u is psh, and u = ϕ on ∂Ω then define the F 0 functional by following variation formula: (6) δF 0(u) = δudet We shall show that the F 0 functional is well-defined. Using this F 0 func- tional and following the ideas of [PS06], we prove that Theorem 2. Assume that both ϕ and f are independent of t, and (7) fu ≤ 0 and fuu ≤ 0. Then the solution u of (1) exists for T = +∞, and as t approaches +∞, u(·, t) approaches the unique solution of the Dirichlet problem = e−f(z,v) in QT , v = ϕ on ∂pQT , in C1,α(Ω̄) for any 0 < α < 1. Remark : Similar energy functionals have been studied in [Bak83, Tso90, Wan94, TW97, TW98] for the real Monge-Ampère equation and the real Hessian equation with homogeneous boundary condition ϕ = 0, and the convergence for the solution of the real Hessian equation was also proved in [TW98]. Our construction of the energy functionals and the proof of the con- vergence also work for these cases, and thus we also obtain an independent proof of these results. Li [Li04] and Blocki [B lo05] studied the Dirichlet problems for the complex k-Hessian equations over bounded complex do- mains. Similar energy functional can also be constructed for the parabolic complex k-Hessian equations and be used for the proof of the convergence. 2. A priori C2 estimate By the work of Krylov [Kry83], Evans [Eva82], Caffarelli etc. [CKNS85] and Guan [Gua98], it is well known that in order to prove the existence and smoothness of (1), we only need to establish the a priori C2,1(Q̄T )1 estimate, i.e. for solution u ∈ C4,1(Q̄T ) of (1) with (9) u = u on ΣT ∪ Γ and u ≥ u in QT , (10) ‖u‖C2,1(QT ) ≤ M2, where M2 only depends on QT , u, f and ‖u(·, 0)‖C2(Ω̄). m,n(QT ) means m times and n times differentiable in space direction and time di- rection respectively, same for Cm,n-norm. ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 3 Proof of (10). Since u is spatial psh and u ≥ u, so u ≤ u ≤ sup (11) ‖u‖C0(QT ) ≤ M0. Step 1. |ut| ≤ C1 in Q̄T . Let G = ut(2M0−u)−1. If G attains its minimum on Q̄T at the parabolic boundary, then ut ≥ −C1 where C1 depends on M0 and u t on Σ. Otherwise, at the point where G attains the minimum, Gt ≤ 0 i.e. utt + (2M0 − u)−1u2t ≤ 0, Gα = 0 i.e. utα + (2M0 − u)−1utuα = 0, Gβ̄ = 0 i.e. utβ̄ + (2M0 − u) −1utuβ̄ = 0, and the matrix Gαβ̄ is non-negative, i.e. (13) utαβ̄ + (2M0 − u) −1utuαβ̄ ≥ 0. Hence (14) 0 ≤ uαβ̄ utαβ̄ + (2M0 − u) −1utuαβ̄ = uαβ̄utαβ̄ + n(2M0 − u) −1ut, where (uαβ̄) is the inverse matrix for (uαβ̄), i.e. uαβ̄uγβ̄ = δ Differentiating (1) in t, we get (15) utt − uαβ̄utαβ̄ = ft + fu ut, (2M0 − u)−1u2t ≤ −utt = −uαβ̄utαβ̄ − ft − fu ut ≤ n(2M0 − u)−1ut − fu ut − ft, hence u2t − (n− (2M0 − u)fu)ut + ft(2M0 − u) ≤ 0. Therefore at point p, we get (16) ut ≥ −C1 where C1 depends on M0 and f . Similarly, by considering the function ut(2M0 + u) −1 we can show that (17) ut ≤ C1. Step 2. |∇u| ≤ M1 4 ZUOLIANG HOU AND QI LI Extend u|Σ to a spatial harmonic function h, then (18) u ≤ u ≤ h in QT and u = u = h on ΣT . (19) |∇u|ΣT ≤ M1. Let L be the linear differential operator defined by (20) Lv = − uαβ̄vαβ̄ − fuv. L(∇u + eλ|z|2) = L(∇u) + Leλ|z|2 ≤ ∇f − eλ|z|2 uαᾱ − fu). Noticed that and both u and u̇ are bounded and = eu̇−f , (22) 0 < c0 ≤ det ≤ c1, where c0 and c1 depends on M0 and f . Therefore uαᾱ ≥ nc−1/n1 . Hence after taking λ large enough, we can get L(∇u + eλ|z|2) ≤ 0, (24) |∇u| ≤ sup |∇u| + C2 ≤ M1. Step 3. |∇2u| ≤ M2 on Σ. At point (p, t) ∈ Σ, we choose coordinates z1, · · · , zn for Ω, such that at z1 = · · · = zn = 0 at p and the positive xn axis is the interior normal direction of ∂Ω at p. We set s1 = y1, s2 = x1, · · · , s2n−1 = yn, s2n = xn and s′ = (s1, · · · , s2n−1). We also assume that near p, ∂Ω is represented as a graph (25) xn = ρ(s j,k<2n Bjksjsk + O(|s′|3). Since (u− u)(s′, ρ(s′), t) = 0, we have for j, k < 2n, (26) (u− u)sjsk(p, t) = −(u− u)xn(p, t)Bjk, hence (27) |usjsk(p, t)| ≤ C3, where C3 depends on ∂Ω, u and M1. ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 5 We will follow the construction of barrier function by Guan [Gua98] to estimate |uxnsj |. For δ > 0, denote Qδ(p, t) = Ω ∩Bδ(p) × (0, t). Lemma 3. Define the function (28) d(z) = dist(z, ∂Ω) (29) v = (u− u) + a(h− u) −Nd2. Then for N sufficiently large and a, δ sufficiently small, Lv ≥ ǫ(1 + uαᾱ) in Qδ(p, t) v ≥ 0 on ∂(Bδ(p) ∩ Ω) × (0, t) v(z, 0) ≥ c3|z| for z ∈ Bδ(p) ∩ Ω where ǫ depends on the uniform lower bound of he eigenvalues of {u αβ̄}. Proof. See the proof of Lemma 2.1 in [Gua98]. � For j < 2n, consider the operator + ρsj Tj(u− u) = 0 on ∂Ω ∩Bδ(p) × (0, t) |Tj(u− u)| ≤ M1 on Ω ∩ ∂Bδ(p) × (0, t) |Tj(u− u)(z, 0)| ≤ C4|z| for z ∈ Bδ(p) So by Lemma 3 we may choose C5 independent of u, and A >> B >> 1 so Av + B|z|2 − C5(uyn − u yn) 2 ± Tj(u− u) ≥ 0 in Qδ(p, t), Av + B|z|2 − C5(uyn − u yn) 2 ± Tj(u− u) ≥ 0 on ∂pQδ(p, t). Hence by the comparison principle, Av + B|z|2 − C5(uyn − u yn) 2 ± Tj(u− u) ≥ 0 in Qδ(p, t), and at (p, t) (33) |uxnyj | ≤ M2. To estimate |uxnxn |, we will follow the simplification in [Tru95]. For (p, t) ∈ Σ, define λ(p, t) = min{uξξ̄ | complex vector ξ ∈ Tp∂Ω, and |ξ| = 1} Claim λ(p, t) ≥ c4 > 0 where c4 is independent of u. Let us assume that λ(p, t) attains the minimum at (z0, t0) with ξ ∈ Tzo∂Ω. We may assume that λ(z0, t0) < u ξξ̄(z0, t0). 6 ZUOLIANG HOU AND QI LI Take a unitary frame e1, · · · , en around z0, such that e1(z0) = ξ, and Re en = γ is the interior normal of ∂Ω along ∂Ω. Let r be the function which defines Ω, then (u− u )11̄(z, t) = −r11̄(z)(u − u )γ(z, t) z ∈ ∂Ω Since u11̄(z0, t0) < u 11̄(z0, t0)/2, so −r11̄(z0)(u− u )γ(z0, t0) ≤ − u 11̄(z0, t0). Hence r11̄(z0)(u− u )γ(z0, t) ≥ u 11̄(z0, t) ≥ c5 > 0. Since both ∇u and ∇u are bounded, we get r11̄(z0) ≥ c6 > 0, and for δ sufficiently small ( depends on r11̄ ) and z ∈ Bδ(z0) ∩ Ω, r11̄(z) ≥ So by u11̄(z, t) ≥ u11̄(z0, t0), we get u 11̄(z, t) − r11̄(z)(u− u )γ(z, t) ≥ u 11̄(z0, t0) − r11̄(z0)(u− u )γ(z0, t0). Hence if we let Ψ(z, t) = r11̄(z) r11̄(z0)(u− u )γ(z0, t0) + u 11̄(z, t) − u 11̄(z0, t0) (u− u )γ(z, t) ≤ Ψ(z, t) on ∂Ω ∩Bδ(z0) × (0, T ) (u− u )γ(z0, t0) = Ψ(z0, t0). Now take the coordinate system z1, · · · , zn as before. Then (u− u )xn(z, t) ≤ γn(z) Ψ(z, t) on ∂Ω ∩Bδ(z0) × (0, T ) (u− u )xn(z0, t0) = γn(z0) Ψ(z0, t0). where γn depends on ∂Ω. After taking C6 independent of u and A >> B >> 1, we get Av + B|z|2 − C6(uyn − u yn) Ψ(z, t) γn(z) − Tj(u− u) ≥ 0 in Qδ(p, t), Av + B|z|2 − C6(uyn − u yn) Ψ(z, t) γn(z) − Tj(u− u) ≥ 0 on ∂pQδ(p, t). Av + B|z|2 − C6(uyn − u yn) Ψ(z, t) γn(z) − Tj(u− u) ≥ 0 in Qδ(p, t), |uxnxn(z0, t0)| ≤ C7. ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 7 Therefore at (z0, t0), uαβ̄ is uniformly bounded, hence u11̄(z0, t0) ≥ c4 with c4 independent of u. Finally, from the equation detuαβ̄ = e we get |uxnxn | ≤ M2. Step 4. |∇2u| ≤ M2 in Q. By the concavity of log det, we have L(∇2u + eλ|z|2) ≤ O(1) − eλ|z|2 uαᾱ − fu So for λ large enough, L(∇2u + eλ|z|2) ≤ 0, (35) sup |∇2u| ≤ sup |∇2u| + C8 with C8 depends on M0, Ω and f . 3. The Functionals I, J and F 0 Let us recall the definition of P(Ω, ϕ) in (5), P(Ω, ϕ) = u ∈ C2(Ω̄ | u is psh, and u = ϕ on ∂Ω Fixing v ∈ P, for u ∈ P, define (36) Iv(u) = − (u− v)( −1∂∂̄u)n. Proposition 4. There is a unique and well defined functional Jv on P(Ω, ϕ), such that (37) δJv(u) = − −1∂∂̄u)n − ( −1∂∂̄v)n and Jv(v) = 0. Proof. Notice that P is connected, so we can connect v to u ∈ P by a path ut, 0 ≤ t ≤ 1 such that u0 = v and u1 = u. Define (38) Jv(u) = − −1∂∂̄ut)n − ( −1∂∂̄v)n We need to show that the integral in (38) is independent of the choice of path ut. Let δut = wt be a variation of the path. Then w1 = w0 = 0 and wt = 0 on ∂Ω, 8 ZUOLIANG HOU AND QI LI −1∂∂̄u)n − ( −1∂∂̄v)n −1∂∂̄u)n − ( −1∂∂̄v)n + u̇ n −1∂∂̄w( −1∂∂̄u)n−1 Since w0 = w1 = 0, an integration by part with respect to t gives −1∂∂̄u)n − ( −1∂∂̄v)n −1∂∂̄u)n dt = − −1nw∂∂̄u̇( −1∂∂̄u)n−1 dt. Notice that both w and u̇ vanish on ∂Ω, so an integration by part with respect to z gives −1nw∂∂̄u̇( −1∂∂̄u)n−1 = − −1n∂w ∧ ∂̄u̇( −1∂∂̄u)n−1 −1nu̇∂∂̄w( −1∂∂̄u)n−1. (39) δ −1∂∂̄u)n − ( −1∂∂̄v)n dt = 0, and the functional J is well defined. � Using the J functional, we can define the F 0 functional as (40) F 0v (u) = Jv(u) − −1∂∂̄v)n. Then by Proposition 4, we have (41) δF 0v (u) = − −1∂∂̄u)n. Proposition 5. The basic properties of I, J and F 0 are following: (1) For any u ∈ P(Ω, ϕ), Iv(u) ≥ Jv(u) ≥ 0. (2) F 0 is convex on P(Ω, ϕ), i.e. ∀u0, u1 ∈ P, (42) F 0 (u0 + u1 F 0(u0) + F 0(u1) (3) F 0 satisfies the cocycle condition, i.e. ∀u1, u2, u3 ∈ P(Ω, ϕ), (43) F 0u1(u2) + F (u3) = F (u3). ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 9 Proof. Let w = (u− v) and ut = v + tw = (1 − t)v + tu, then Iv(u) = − −1∂∂̄u)n − ( −1∂∂̄v)n −1∂∂̄ut)n dt −1nw∂∂̄w( −1∂∂̄ut)n−1 −1n∂w ∧ ∂̄w ∧ ( −1∂∂̄ut)n−1 ≥ 0, Jv(u) = − −1∂∂̄ut)n − ( −1∂∂̄v)n −1∂∂̄us)n ds −1nw∂∂̄w( −1∂∂̄us)n−1 ds dt (1 − s) −1n∂w ∧ ∂̄w ∧ ( −1∂∂̄us)n−1 ds ≥ 0. Compare (44) and (45), it is easy to see that Iv(u) ≥ Jv(u) ≥ 0. To prove (42), let ut = (1 − t)u0 + tu1, then F 0(u1/2) − F 0(u0) = − (u1 − u0) ( −1∂∂̄ut)n dt, F 0(u1) − F 0(u1/2) = − (u1 − u0) ( −1∂∂̄ut)n dt. Since (u1 − u0) ( −1∂∂̄ut)n dt− (u1 − u0) ( −1∂∂̄ut)n dt. (u1 − u0) −1∂∂̄ut)n − ( −1∂∂̄ut+1/2)n (ut+1/2 − ut) −1∂∂̄ut)n − ( −1∂∂̄ut+1/2)n dt ≥ 0. F 0(u1) − F 0(u1/2) ≥ F 0(u1/2) − F 0(u0). The cocycle condition is a simple consequence of the variation formula 41. 10 ZUOLIANG HOU AND QI LI 4. The Convergence In this section, let us assume that both f and ϕ are independent of t. For u ∈ P(Ω, ϕ), define (46) F (u) = F 0(u) + G(z, u)dV, where dV is the volume element in Cn, and G(z, s) is the function given by G(z, s) = e−f(z,t) dt. Then the variation of F is (47) δF (u) = − det(uαβ̄) − e−f(z,u) Proof of Theorem 2. We will follow Phong and Sturm’s proof of the conver- gence of the Kähler-Ricci flow in [PS06]. For any t > 0, the function u(·, t) is in P(Ω, ϕ). So by (47) F (u) = − det(uαβ̄) − e −f(z,u) log det(uαβ̄) − (−f(z, u)) det(uαβ̄) − e −f(z,u) Thus F (u(·, t)) is monotonic decreasing as t approaches +∞. On the other hand, u(·, t) is uniformly bounded in C2(Ω) by (10), so both F 0(u(·, t)) and f(z, u(·, t)) are uniformly bounded, hence F (u) is bounded. Therefore log det(uαβ̄) + f(z, u) det(uαβ̄) − e −f(z,u) dt < ∞. Observed that by the Mean Value Theorem, for x, y ∈ R, (x + y)(ex − e−y) = (x + y)2eη ≥ emin(x,−y)(x− y)2, where η is between x and −y. Thus log det(uαβ̄) + f det(uαβ̄) − e log det(uαβ̄) + f = C9|u̇|2 where C9 is independent of t. Hence ‖u̇‖2L2(Ω) dt ≤ ∞ (50) Y (t) = |u̇(·, t)|2 det(uαβ̄) dV, 2üu̇ + u̇2uαβ̄ u̇αβ̄ det(uαβ̄) dV. Differentiate (1) in t, (51) ü− uαβ̄ u̇αβ̄ = fuu̇, ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 11 2u̇u̇αβ̄u αβ̄ + u̇2 2fu + ü− fuu̇ det(uαβ̄) dV 2fu + ü− fuu̇ − 2u̇αu̇β̄u det(uαβ̄) dV From (51), we get u − uαβ̄ üαβ̄ − fuü ≤ fuuu̇ Since fu ≤ 0 and fuu ≤ 0, so ü is bounded from above by the maximum principle. Therefore Ẏ ≤ C10 u̇2 det(uαβ̄) dV = C10Y, (52) Y (t) ≤ Y (s)eC10(t−s) for t > s, where C10 is independent of t. By (49), (52) and the uniform boundedness of det(uαβ̄), we get ‖u(·, t)‖L2(Ω) = 0. Since Ω is bounded, the L2 norm controls the L1 norm, hence ‖u(·, t)‖L1(Ω) = 0. Notice that by the Mean Value Theorem, |ex − 1| < e|x||x| |eu̇ − 1| dV ≤ esup |u̇| |u̇| dV Hence eu̇ converges to 1 in L1(Ω) as t approaches +∞. Now u(·, t) is bounded in C2(Ω), so u(·, t) converges to a unique function ũ, at least sequentially in C1(Ω), hence f(z, u) → f(z, ũ) and det(ũαβ̄) = lim det(u(·, t)αβ̄) = lim eu̇−f(z,u) = e−f(z,ũ), i.e. ũ solves (8). References [Bak83] Ilya J. Bakelman. Variational problems and elliptic Monge-Ampère equations. J. Differential Geom., 18(4):669–699 (1984), 1983. [B lo05] Zbigniew B locki. Weak solutions to the complex Hessian equation. Ann. Inst. Fourier (Grenoble), 55(5):1735–1756, 2005. [BT76] Eric Bedford and B. A. Taylor. The Dirichlet problem for a complex Monge- Ampère equation. Invent. Math., 37(1):1–44, 1976. [Cao85] Huai Dong Cao. Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math., 81(2):359–372, 1985. 12 ZUOLIANG HOU AND QI LI [CKNS85] L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck. The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math., 38(2):209–252, 1985. [CT02] X. X. Chen and G. Tian. Ricci flow on Kähler-Einstein surfaces. Invent. Math., 147(3):487–544, 2002. [Eva82] Lawrence C. Evans. Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math., 35(3):333–363, 1982. [Gua98] Bo Guan. The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function. Comm. Anal. Geom., 6(4):687– 703, 1998. [Ko l98] S lawomir Ko lodziej. The complex Monge-Ampère equation. Acta Math., 180(1):69–117, 1998. [Kry83] N. V. Krylov. Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat., 47(1):75–108, 1983. [Li04] Song-Ying Li. On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian. Asian J. Math., 8(1):87–106, 2004. [PS06] Duong H. Phong and Jacob Sturm. On stability and the convergence of the Kähler-Ricci flow. J. Differential Geom., 72(1):149–168, 2006. [Tru95] Neil S. Trudinger. On the Dirichlet problem for Hessian equations. Acta Math., 175(2):151–164, 1995. [Tso90] Kaising Tso. On a real Monge-Ampère functional. Invent. Math., 101(2):425– 448, 1990. [TW97] Neil S. Trudinger and Xu-Jia Wang. Hessian measures. I. Topol. Methods Non- linear Anal., 10(2):225–239, 1997. Dedicated to Olga Ladyzhenskaya. [TW98] Neil S. Trudinger and Xu-Jia Wang. A Poincaré type inequality for Hessian integrals. Calc. Var. Partial Differential Equations, 6(4):315–328, 1998. [Wan94] Xu Jia Wang. A class of fully nonlinear elliptic equations and related function- als. Indiana Univ. Math. J., 43(1):25–54, 1994. [Yau78] Shing Tung Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978. Mathematics Department, Columbia University, New York, NY 10027 E-mail address: [email protected] Mathematics Department, Columbia University, New York, NY 10027 E-mail address: [email protected] 1. Introduction 2. A priori C2 estimate 3. The Functionals I, J and F0 4. The Convergence References

0704.1823

Compatible Actions and Cohomology of Crystallographic Groups

COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS ALEJANDRO ADEM∗, JIANQUAN GE, JIANZHONG PAN, AND NANSEN PETROSYAN Abstract. We compute the cohomology of crystallographic groups Γ = Zn ⋊Z/p with holonomy of prime order by establishing the collapse at E2 of the spectral sequence associated to their defining extension. As an application we compute the group of gerbes associated to many six–dimensional toroidal orbifolds arising in string theory. 1. Introduction Given a finite group G and an integral representation L for G (i.e. a homomorphism G → GLn(Z), where L is the underlying ZG–module), we can define the semi–direct product Γ = L⋊G. Calculating the cohomology of these groups is a problem of instrinsic algebraic interest; indeed if the representation is faithful then these groups can be thought of as crystallographic groups (see [7], page 74). From the geometric point of view, the action on L gives rise to a G–action on the n–torus X = Tn; this approach can be used to derive important examples of orbifolds, known as toroidal orbifolds (see [2]). In the case when n = 6 these are of particular interest in string theory (see [4], [11]). Given the split group extension 0 → L → Γ → G → 1, the basic problem which we address is that of providing conditions which imply the collapse (without extension problems) of the associated Lyndon–Hochshild–Serre spectral sequence. The conditions which we establish are representation–theoretic, namely depending solely on the structure of the integral representation L. This can be a difficult problem (see [15] for further background) and there are well–known examples where the spectral sequence does not collapse. Our approach is to systematically apply the methods used in [2]. The key idea is to construct a free resolution F for the semidirect product L ⋊ G such that the Lyndon– Hochschild–Serre spectral sequence of the group extension collapses at E2. This requires Date: October 30, 2018. Key words and phrases. spectral sequence, group cohomology. ∗The first author was partially supported by the NSF and by NSERC. http://arxiv.org/abs/0704.1823v1 COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 2 a chain–level argument, more specifically the construction of a compatible G–action on a certain free resolution for the torsion–free abelian group L (see §2 for details). We concentrate on the case of G = Z/p, a cyclic group of prime order p, as the representation theory is well–understood. Our main algebraic result is the following Theorem 1.1. Let G = Z/p, where p is any prime. If L is any finitely generated ZG– lattice1, and Γ = L⋊G is the associated semi–direct product group, then for each k ≥ 0 Hk(Γ,Z) ∼= i+j=k H i(G,∧j(L∗)) where ∧j(L∗) denotes the j-th exterior power of the dual module L∗ = Hom(L,Z). Expressed differently: these results imply a complete calculation for the integral coho- mology of crystallographic groups Zn ⋊ Z/p where p is prime. These calculations can be made explicit. The theorem has an interesting geometric application: Theorem 1.2. Let G = Z/p, where p is any prime. Suppose that G acts on a space X homotopy equivalent to (S1)n with XG 6= ∅, then for each k ≥ 0 Hk(EG×G X,Z) ∼= i+j=k H i(G,Hj(X,Z)) ∼= Hk(Γ,Z). where Γ = π1(X)⋊G. On the other hand, the explicit computation for torsion–free crystallographic groups with holonomy of prime order was carried out long ago by Charlap and Vásquez (see [8], page 556). Combining the two results we obtain a complete calculation: Theorem 1.3. Let Γ denote a crystallographic group with holonomy of prime order p, expressed as an extension 1 → L → Γ → Z/p → 1 where L is a free abelian group of finite rank. (1) If Γ is torsion–free, then L ∼= N ⊕ Z (it splits off a trivial direct summand) and Hk(Γ,Z) ∼= H0(Z/p,∧k(L∗))⊕H1(Z/p,∧k−1(N∗)) for 0 ≤ k ≤ rk(L); Hk(Γ,Z) = 0 for k > rk(L). (2) If Γ is not torsion–free, then H∗(Γ,Z) can be computed using Theorem 1.1. 1A ZG–lattice is a ZG–module which happens to be a free abelian group. COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 3 In this paper we also consider the situation for the cyclic group of order four; some partial results are obtained but a general collapse has not been established. However, based on these and other computations we conjecture that for G any cyclic group, the spectral sequence associated to a semi–direct product of the form Zn ⋊ G must collapse at E2. In the last section we give an application of our methods to calculations for six– dimensional toroidal orbifolds, showing that among the 18 inequivalent N = 1 supersym- metric string theories on symmetric orbifolds of (2, 2)–type without discrete background, only two of them cannot be analyzed using our methods i.e. we cannot show the existence of compatible actions for the associated modules. If X = [T6/G] is an orbifold arising this way, then our results provide a complete calculation for its associated group of gerbes Gb(X ) ∼= H3(EG×G T 6,Z) (see [2] for more details). 2. Preliminary Results The notion of a compatible action was first introduced in [9]. If such an action exists it allows one to construct practical projective resolutions and from these to compute the cohomology of the group. We will give the basic definition and the main theorem that follows. More details can be found in [2]. Let Γ = L ×ρ G = L ⋊ G be the semidirect product of a finite group G and a finite dimensional Z-lattice L via a representation ρ : G → GL(L). G acts on the group L by the homomorphism ρ, and this extends linearly to an action on the group algebra R[L], where R denotes a commutative ring with unit. We write lg for ρ(g)l where l ∈ R[L], g ∈ G. In the rest of this paper R will represent Z (the integers) or Z(p) (the ring of integers localized at a fixed prime p). Definition 2.1. Given a free resolution ǫ : F → R of R over R[L], we say that it admits an action of G compatible with ρ if for all g ∈ G there is an augmentation-preserving chain map τ(g) : F → F such that (1) τ(g)[l · f ] = lg · [τ(g)f ] for all l ∈ R[L] and f ∈ F , (2) τ(g)τ(g′) = τ(gg′) for all g, g′ ∈ G, (3) τ(1) = 1F . The following two lemmas (see [2]) reduce the construction of compatible actions to the case of faithful indecomposable representations. Lemma 2.2. If ǫi : Fi → R is a projective R[Li]-resolution of R for i = 1, 2, then ǫ1 ⊗ ǫ2 : F1 ⊗ F2 → R is a projective R[L1 × L2]-resolution of R. Furthermore, if G acts COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 4 compatibly on Fi by τi for i = 1, 2, then a compatible action of G on ǫ1⊗ ǫ2 : F1⊗F2 → R is given by τ(g)(f1 ⊗ f2) = τ1(g)(f1)⊗ τ2(g)(f2). Lemma 2.3. If L is a R[G1]-module, π : G2 → G1 a group homomorphism, and ǫ : F → R is a R[L]-resolution of R such that G1 acts compatibly on it by τ ′, then G2 also acts compatibly on it by τ(g)f = τ ′(π(g))f for any g ∈ G. If a compatible action exists, we can give F a Γ-module structure as follows. An element γ ∈ Γ can be expressed uniquely as γ = lg, with l ∈ L and g ∈ G. We set γ · f = (lg) · f = l · τ(g)f . Note that given any G–module M , this inflates to a Γ–action on M via the projection Γ → G. We can always construct a special free resolution F of R over L, characterized by the property that the cochain complex HomL(F,R) for computing the cohomology H ∗(L,R) has all coboundary maps zero (more details will be provided in the next section). Using this fact, the following was proved in [2]: Theorem 2.4. (Adem-Pan) Let ǫ : F → R be a special free resolution of R over L and suppose that there is a compatible action of G on F . Then for all integers k ≥ 0, we have Hk(L⋊G,R) = i+j=k H i(G,Hj(L,R)). This result can be interpreted as saying that the Lyndon-Hochschild-Serre spectral sequence 2 = H p(G,Hq(L,R)) ⇒ Hp+q(L⋊G,R) collapses at E2 without extension problems. Note that this is not always the case; in fact there are examples of semi–direct products of the form Zn ⋊ (Z/p)2 where the associated spectral sequence has non–trivial differentials (see [15]). This will be discussed in §5. 3. Construction of Compatible Actions Let R[L] denote the group ring of L, a free abelian group with basis {x1, . . . , xn}. Then the elements x1 − 1, . . . , xn − 1 form a regular sequence in R[L], hence the Koszul complex K∗ = K(x1−1, . . . , xn−1) is a free resolution of the trivial module R. It has the additional property of being a differential graded algebra (or DGA). We briefly recall how it looks. There are generators a1, . . . , an in degree one, and the graded basis for K∗ can be identified with the usual basis for the exterior algebra they generate. The differential is given by the following formula: if ai1...ip = ai1 . . . aip is a basis element in Kp, then COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 5 d(ai1...ip) = (−1)j−1(xij − 1)ai1...îj ...ip. Now the cohomology of the free abelian group L is precisely an exterior algebra on n one- dimensional generators, which in fact can be identified with the dual elements a∗1, . . . , a In particular we see that the cochain complex HomR[L](K∗, R) has zero differentials, and hence K∗ is a special free resolution of R over R[L] (this resolution also appears in [7] pp. 96–97). We now consider how to construct a compatible G–action on K∗, given a G–module structure on L. Theorem 3.1. If G acts on the lattice L, let K∗ = K(x1 − 1, . . . xn − 1) denote the special free resolution of R over R[L] defined using the Koszul complex associated to the elements x1 − 1, . . . , xn − 1, where {x1, . . . , xn} form a basis for L. Suppose that there is a homomorphism τ : G → Aut(K1) such that for every g ∈ G and a ∈ K1 it satisfies dτ(g)(a) = d(a)g where d : K1 → K0 is the usual Koszul differential, and d(a) g ∈ K0 = R[L]. Then τ extends to K∗ using its DGA structure and so defines a compatible G–action on K∗. Proof. First we observe that τ(g) acts on K0 = R[L] via the original G–action, i.e. τ(g)(x) = xg for any x ∈ K0. Next we define the action on the basis of K∗ as a graded R[L]–module, namely: τ(g)(ai1 . . . aip) = τ(g)(ai1) . . . τ(g)(aip). If α ∈ R[L] and u ∈ K∗, we define τ(g)(αu) = α gτ(g)(x). By linearity and the DGA structure of K∗, this will define τ : G → Aut(K∗), with the desired properties. � Generally speaking it can be quite difficult to construct a compatible action; however there is an important special case where it is quite straightforward. Theorem 3.2. Let φ : G → Σn denote a group homomorphism, where Σn denotes the symmetric group on n elements. Let G act on Zn via2 this homomorphism. Then the associated Koszul complex K∗ admits a compatible G–action. 2Such a module is called a permutation module. COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 6 Proof. By Lemma 2.3 we can assume that G is a subgroup of Σn, hence it will suffice to prove this for Σn itself. If we take generators a1, . . . , an for the Koszul complex corre- sponding to the elements x1, . . . , xn in the underlying module L, then we can define τ as follows: τ(σ)(ai) = aσ(i). This obviously defines a permutation representation onK1, and compatibility follows from the fact that for all ai, 1 ≤ i ≤ n and σ ∈ Σ we have dτ(σ)(ai) = d(aσ(i)) = xσ(i) − 1 = (xi − 1) This completes the proof. � Aside from permutation representations, it is difficult to construct general examples of compatible actions. However if G is a cyclic group then we can handle an important additional type of module. Proposition 3.3. Let the cyclic group G = 〈t|tn = 1〉 act on Zn−1 by: ξ1 : t 7→ 0 1 0 . . . 0 0 0 0 1 . . . 0 0 . . . 0 0 0 . . . 1 0 −1 −1 −1 . . . −1 −1 ∈ GLn−1(Z). If x1, . . . , xn is the canonical basis under which the action is represented by the matrix above, then the free resolution K∗ = K∗(x1−1, . . . , xn−1) admits an action of G compatible with ξ1, which can be defined by: τ(t)(a1) = −x n−1an−1 τ(t)(ak) = −x n−1(an−1 − ak−1), 1 < k ≤ n− 1. Proof. The proof is a straightforward calculation verifying that τ defines a compatible action. First we verify that τn = 1. For this we observe that if A is the matrix in GLn−1(Z) representing the generator t, then expressed in terms of the basis {a1, . . . , an} we have that τ(t) = x−1n−1A. If we iterate this action and use the fact that τ(g)(αu) = α gτ(g)(u) then we obtain τ(t)n = (x−1n−1) tn−1(x−1n−1) tn−2 . . . (x−1n−1) t(x−1n−1)A n = 1. COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 7 This follows from the fact that the characteristic polynomial of A is the cyclotomic poly- nomial p(z) = 1+ z + · · ·+ zn−1, hence we have that p(A) = 0 on the underlying module L and so in multiplicative notation we have that ut · · · · ·ut ·u = 1 for any u ∈ L. Next we verify compatibility: τ(t)d(a1) = τ(t)(x1 − 1) = x n−1 − 1 dτ(t)(a1) = d(−x n−1an−1) = −x n−1(xn−1 − 1) = x n−1 − 1. Similarly for all 1 < k ≤ n− 1 we have that: τ(t)d(ak) = τ(t)(xk − 1) = xk−1x n−1 − 1 = −x−1n−1(xn−1 − 1− xk−1 − 1) = d(−x n−1(an−1 − ak−1)) = dτ(t)(ak). For G = Z/n, the module which gives rise to the matrix in 3.3 is the augmentation ideal IG, which has rank equal to n − 1. The following proposition is an application of the results in this section. Proposition 3.4. Let G = Z/n, and assume that L is a ZG–lattice such that L ∼= M ⊕ IGt where M is a permutation module. Then, for any coefficient ring R, the special free resolution K∗ over R[L] admits a compatible G–action. Proof. This follows from applying Lemma 2.2 to Theorem 3.2 and Proposition 3.3. � For our cohomology calculations it will be practical to use the coefficient ring R = Z(p), where p is a prime. In this situation, for G = Z/p (see [10]) there are only three distinct isomorphism classes of indecomposable RG–lattices, namely R (the trivial module), IG (the augmentation ideal) and RG, the group ring. Moreover, if L is any finitely generated ZG–lattice, we can construct a ZG-homomorphism f : L′ → L such that • L′ ∼= Zr ⊕ ZGs ⊕ IGt • f is an isomorphism after tensoring with R. We shall call L′ a representation of type (r, s, t). COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 8 4. Applications to Cohomology We are now ready to prove our main result. Theorem 4.1. Let G = Z/p, where p is any prime. If L is any finitely generated ZG– lattice, and Γ = L⋊G is the associated semi–direct product group, then for each k ≥ 0 Hk(Γ,Z) ∼= i+j=k H i(G,∧j(L∗)) where ∧j(L∗) denotes the j-th exterior power of the dual module L∗ = Hom(L,Z). Proof. First, let us prove the analogous result for the cohomology with coefficients in R = Z(p). We make the assumption that L is a module of type (r, s, t). We need to verify: Hk(Γ, R) ∼= i+j=k H i(G,∧j(L∗R)) where L∗R = L ∗ ⊗ R. In fact, we see that in the associated Lyndon-Hochschild-Serre spectral sequence for the extension 0 → L → Γ → G → 1 with 2 (R) = H i(G,Hj(L,R)) ⇒ Hk(Γ, R) there are no differentials and no extension problems. This follows from applying Theorem 2.4 and the fact that the module L gives rise to a special resolution with a compatible action by Proposition 3.4. Now let us consider the case when L is not of type (r, s, t). As observed previously, we can construct a ZG–lattice L′ and a map f : L′ → L such that L′ is of type (r, s, t) and f is an isomorphism after tensoring with R. Under these conditions f will induce a map between the spectral sequences with R–coefficients for the extensions corresponding to L and L′. However by our hypotheses, ∧k(L′∗R) and ∧ k(L∗R) are isomorphic as RG–modules for all k ≥ 0, with the isomorphism induced by f . Hence the corresponding E2–terms are isomorphic, and so the spectral sequences both collapse and the result follows. It now remains to prove the result with coefficients in the integers Z. Note that by the universal coefficient theorem, we have H∗(Γ,Z(p)) ∼= H ∗(Γ,Z) ⊗ Z(p) hence the only relevant discrepancy between H∗(Γ,Z(p)) and H ∗(Γ,Z) might arise from the presence of torsion prime to p in the integral cohomology of Γ. However, a quick inspection of the spectral sequence of the extension 0 → L → Γ → G → 1 with Z(q) coefficients shows that there is no torsion prime to p in the cohomology, as L is free abelian and G is a p–group. This completes our proof. � COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 9 In [2], Corollary 3.3, it was observed that the spectral sequence for the extension L⋊G satisfies a collapse at E2 wthout extension problems if the same is true for all the restricted extensions L ⋊ Gp, where the Gp ⊂ G are the p–Sylow subgroups of G. We obtain the following Corollary 4.2. Let G denote a finite group of square–free order, and L any finitely generated ZG–lattice. Then for all k ≥ 0 we have Hk(L⋊G,Z) ∼= i+j=k H i(G,∧j(L∗)). We now consider a more geometric situation. Suppose that the group G = Z/p acts on a space X which has the homotopy type of a product of circles. Theorem 4.3. Let G = Z/p, where p is any prime. Suppose G acts on a space X homotopy equivalent to (S1)n with XG 6= ∅, then for each k ≥ 0 Hk(EG×G X,Z) ∼= i+j=k H i(G,Hj(X,Z)) ∼= Hk(Γ,Z) where Γ = π1(X)⋊G. Proof. The space EG×GX fits into a fibration X →֒ EG×GX → BG which has a section due to the fact that XG 6= ∅. Let Γ denote the fundamental group of EG×GX . The long exact sequence for the homotopy groups of the fibration gives rise to a split extension 1 → π1(X) → Γ → G → 1. Since π1(X) ∼= L, a ZG–lattice, this shows that Γ ∼= L ⋊G, where the G action is induced on L via the action on the fiber. Note that EG ×G X is an Eilenberg-MacLane space of type K(Γ, 1). Hence, H∗(EG ×G X,Z) ∼= H ∗(Γ,Z) and the result follows from Theorem 4.1. Note that a special case of this result was proved in [1], namely for actions where π1(X)⊗ Z(p) is isomorphic to a direct sum of indecomposables of rank p− 1. The terms H i(Z/p,∧j(L∗)) can be computed if L is known up to isomorphism. In fact, all we need is to know L up to Z/p cohomology, as this will determine its indecomposable factors (at least up to Z(p)–equivalence). As we mentioned in the introduction, our results complete the calculation for the co- homology of crystallographic groups with prime order holonomy when combined with previous work on the torsion–free case (Bieberbach groups). The terms appearing in the formulas in Theorem 1.3 can be explicitly computed, as was observed in [8]. COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 10 5. Extensions to Other Groups In this section we explore to what degree our results can be extended to other groups. In the case of the cyclic group of order four, the indecomposable integral representations are easy to describe, so it is a useful test case. Let G = Z/4, from [3] we can give a complete list of all (nine) indecomposable pair- wise nonequivalent integral representations by the following adopted table, where a is a generator for Z/4: ρ1 : a → 1; ρ2 : a → −1; ρ3 : a → ; ρ4 : a → ρ5 : a → 0 0 −1 1 0 −1 0 1 −1 ; ρ6 : a → 0 1 0 −1 0 1 0 0 1 ρ7 : a → 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 ; ρ8 : a → 0 0 −1 1 1 0 −1 1 0 1 −1 0 0 0 0 1 ρ9 : a → 0 1 0 0 −1 0 0 1 0 0 −1 1 0 0 0 1 Theorem 5.1. Let G = Z/4 and L a finitely generated ZG–lattice. If L is a direct sum of indecomposables of type ρi for i ≤ 7, and i 6= 6, then there is a compatible action and Hk(L⋊G,Z) = i+j=k H i(G,Hj(L,Z)). Proof. For the indecomposables ρ1, ρ2, ρ3, ρ4, compatible actions are known to exist on the associated resolutions by the results in [2]. The same is true3 for ρ5 and ρ7 by 3.3 and 3.2. Hence if L is any integral representation expressed as a direct sum of ρi, i ≤ 7 and i 6= 6, the result follows from 2.2 and 2.4. � 3In fact ρ5 corresponds to the dual module IG ∗, but for cyclic groups there is an isomorphism IG ∼= COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 11 In the case of ρ6, ρ8 and ρ9, a compatible action is not known to exist. However, via an explicit computation done in [14], we can establish the collapse of the spectral sequence for the extension Γ6 associated to ρ6, yielding H i(Γ6,Z) = Z if i = 0, 1 Z/4⊕ Z if i = 2 Z/2⊕ Z if i = 3 Z/4⊕ Z/2 if i ≥ 4 which verifies the statement analogous to 5.1 for the cohomology of Γ6. Indeed, for all the examples of semidirect products we have considered so far, there is a collapse at E2 in the Lyndon-Hochschild-Serre spectral sequence of the group extension 0 → L → G⋊ L → G → 1 and therefore we can make the following: Conjecture 5.2. Suppose that G is a finite cyclic group and L a finitely generated ZG– lattice; then for any k ≥ 0 we have Hk(G⋊ L,Z) = i+j=k H i(G,Hj(L,Z)). In [15] examples were given of semi-direct products of the form L ⋊ (Z/p)2 where the associated mod p Lyndon-Hochschild-Serre spectral sequence has non–zero differentials. This relies on the fact that for G = Z/p × Z/p, there exist ZG–modules M which are not realizable as the cohomology of a G–space. These are the counterexamples to the Steenrod Problem given by G.Carlsson (see [5]), where M can be identified with L∗. No such counterexamples exist for finite cyclic groups Z/N , which means that disproving our conjecture will require a different approach. We should also mention that by using results due to Nakaoka (see [12], pages 19 and 50) we know that the spectral sequence for a wreath product Zn ⋊ G, where G ⊂ Σn (the symmetric group) acting on Zn via permutations will always collapse at E2, without extension problems. This can be interpreted as the fact that a strong collapse theorem holds for all permutation modules and all finite groups G. A simple proof of this result can be obtained by applying Theorem 2.4 to Proposition 3.2. As suggested by [15], the results here can be considered part of a very general problem, which is both interesting and quite challenging: Problem: Given a finite group G, find suitable conditions on a ZG–lattice L so that the spectral sequence for L⋊G collapses at E2. COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 12 6. Application to Computations for Toroidal Orbifolds Interesting examples arise from calculations for six–dimensional orbifolds, where the usual spectral sequence techniques become rather complicated. Here our methods provide an important new ingredient that allows us to compute rigorously beyond the known range. An important class of examples in physics arises from actions of a cyclic group G = Z/N on T6. In our scheme, these come from six-dimensional integral representations of Z/N . However, the constraints from physics impose certain restrictions on them (see [11], [16]). If θ ∈ GL6(Z) is an element of order N , then it can be diagonalized over the complex numbers. The associated eigenvalues, denoted α1, α2, α3, should satisfy α1α2α3 = 1, and in addition all of the αi 6= 1. The first condition implies that the orbifold T 6 → T6/G is a Calabi–Yau orbifold, and so admits a crepant resolution. These more restricted representations have been classified4 in [11], where it is shown that there are precisely 18 inequivalent lattices of this type. It turns out that calculations are focused on computing the equivariant cohomology H∗(EG ×G T 6,Z) (see [2] and [4] for more details). As was observed in [2], we can compute the group of gerbes associated to the orbifold X = [T6/G] via the isomorphism Gb(X ) ∼= H3(EG×G T 6,Z) ∼= H3(Z6 ⋊G,Z), whence our methods can be used to obtain some fairly complete results in this setting. Before proceeding we recall that as in Corollary 4.2 the collapse of the spectral sequence for an extension L ⋊ G will follow from the existence of compatible Sylp(G) actions on the Koszul complex K∗ for every prime p dividing |G|. If these exist we shall say that K∗ admits a local compatible action. Theorem 6.1. Among the 18 inequivalent integral representations associated to the six– dimensional orbifolds T6/Z/N described above, only two of them are not known to admit (local) compatible actions. Hence for those 16 examples there is an isomorphism Hk(EZ/N ×Z/N T 6,Z) ∼= i+j=k H i(Z/N,Hj(T6,Z)) Proof. Consider the defining matrix of an indecomposable action of Z/N on Zn with determinant one, expressed in canonical form as 4In the language of physics, they show that there exist 18 inequivalent N = 1 supersymmetric string theories on symmetric orbifolds of (2, 2)–type without discrete background. COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 13 0 . . . 0 v1 1 0 . . . 0 v2 0 1 0 . . . 0 v3 . . . 0 . . . 1 vn where v1 = ±1. In [11] it was determined that the matrices that specify the indecomposable modules appearing as summands for the N = 1 supersymmetric Z/N -orbifolds can be given as follows, where the vectors represent the values (v1, v2, . . . , vn): Indecomposable matrices relevant for N = 1 supersymmetry n = 1 n = 2 n = 3 n = 4 Z/2(1) : (−1) Z/3(2) : (−1,−1) Z/4(3) : (−1,−1,−1) Z/6(4) : (−1, 0,−1, 0) Z/4(2) : (−1, 0) Z/6(3) : (−1, 0, 0) Z/8(4) : (−1, 0, 0, 0) Z/6(2) : (−1, 1) Z/12(4) : (−1, 0, 1, 0) n = 5 n = 6 Z/6(5) : (−1,−1,−1,−1,−1) Z/7(6) : (−1,−1,−1,−1,−1,−1) Z/8(5) : (−1,−1, 0, 0,−1) Z/8(6) : (−1, 0,−1, 0,−1, 0) Z/12(6) : (−1,−1, 0, 1, 0,−1) We will show that all of these, except possibly Z/8(5) and Z/12(6), admit local compat- ible actions. The examples of rank two or less were dealt with in [2]; for N = 2, 3, 6, 7 the result follows directly from 4.1 and 4.2. The case Z/4(3) was covered in 5.1. We will deal explicitly with the cases Z/8(4), Z/12(4) and Z/8(6). (1) The group Z/8 acts on Z4 with generator represented by the matrix: 0 0 0 −1 1 0 0 0 0 1 0 0 0 0 1 0 COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 14 We define a compatible action by the following formulas: τ(t)(a1) = −x 4 a4, τ(t)(a2) = a1, τ(t)(a3) = a2, τ(t)(a4) = a3. (2) The group Z/12 acts on Z4 with generator represented by the matrix: 0 0 0 −1 1 0 0 0 0 1 0 1 0 0 1 0 For this example it suffices to construct a compatible action for Z/4 (with generator rep- resented by the matrix T 3) as we already know that a compatible action exists restricted to Z/3. Now we have that Z/4 acts on Z4 with T 3 = 0 −1 0 −1 0 0 −1 0 0 1 0 0 1 0 1 0 which is a matrix whose square is −I. This implies that the module is a sum of two copies of the faithful rank two indecomposable (see §5), for which a compatible action is known to exist (as explained in Theorem 5.1), and so this case is taken care of. (3) The group Z/8 acts on Z6 with generator represented by the matrix: 0 0 0 0 0 −1 1 0 0 0 0 0 0 1 0 0 0 −1 0 0 1 0 0 0 0 0 0 1 0 −1 0 0 0 0 1 0 The formulas for a compatible action are given by τ(t)(a1) = −x 6 a6, τ(t)(a2) = a1, τ(t)(a3) = x 6 (a2 − a6) τ(t)(a4) = a3, τ(t)(a5) = x 6 (a4 − a6), τ(t)(a6) = a5. We have shown that (local) compatible actions exist for all representations constructed using indecomposables other than Z/8(5) and Z/12(6). However, these indecomposables can only appear once in the list due to dimensional constraints, namely in the form Z/8(5) ⊕ Z/2(1) and Z/12(6) itself. Thus our proof is complete. � COMPATIBLE ACTIONS AND COHOMOLOGY OF CRYSTALLOGRAPHIC GROUPS 15 References [1] Adem, A., Z/pZ actions on (Sn)k, Trans. Amer. Math. Soc. 300 (1987), no. 2, 791–809. [2] Adem, A. and Pan, J., Toroidal Orbifolds, Gerbes and Group Cohomology, Trans. Amer. Math. Soc. 358 (2006), 3969-3983. [3] Berman, S. and Gudikov, P. Indecomposable representations of finite groups over the ring of p-adic integers, Izv. Akad. Nauk. SSSR 28 (1964), 875–910. [4] de Boer, J., Dijkgraaf, R., Hori, K., Keurentjes, A., Morgan, J., Morrison, D. and Sethi, S., Triples, Fluxes, and Strings, Adv. Theor. Math. Phys. 4 (2000), no. 5, 995–1186. [5] Carlsson, G., A counterexample to a conjecture of Steenrod, Inv. Math. 64 (1981), no. 1, 171–174. [6] Cartan, H. and Eilenberg, S., Homological Algebra, Oxford University Press, Oxford, 1956. [7] Charlap, L., Bieberbach Groups and Flat Manifolds, Universitext, Springer–Verlag, Berlin, 1986. [8] Charlap, L. and Vasquez, A., Compact Flat Riemannian Manifolds II: the Cohomology of Zp– manifolds, Amer. J. Math. 87 (1965), 551–563. Trans, Amer. Math. Soc. [9] Brady, T., Free resolutions for semi-direct products, Tohoku Math. J. (2) 45 (1993), no. 4, 535– [10] Curtis, C.W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras Wiley-Interscience (1987). [11] Erler, J. and Klemm, A., Comment on the generation number in orbifold compactifications, Comm. Math. Phys. 153 (1993), 579–604. [12] Evens, L., Cohomology of groups, Oxford Mathematical Monographs, Oxford University Press (1991). [13] Joyce, D., Deforming Calabi-Yau orbifolds, Asian J. Math. 3 (1999), no. 4, 853–867. [14] Petrosyan, N. Jumps in cohomology of groups, periodicity, and semi–direct products, Ph.D. Dis- sertation, University of Wisconsin–Madison (2006). [15] Totaro, B., Cohomology of Semidirect Product Groups, J. of Algebra 182 (1996), 469–475. [16] Vafa, C. and Witten, E., On orbifolds with discrete torsion, J. Geom. Phys. 15 (1995), no. 3, 189–214. Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada E-mail address : [email protected] Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China E-mail address : [email protected] Institute of Mathematics, Academia Sinica, Beijing 100080, China E-mail address : [email protected] Department of Mathematics, Indiana University, Bloomington IN 47405, USA E-mail address : [email protected] 1. Introduction 2. Preliminary Results 3. Construction of Compatible Actions 4. Applications to Cohomology 5. Extensions to Other Groups 6. Application to Computations for Toroidal Orbifolds References

0704.1824

Stochastic Heat Equation Driven by Fractional Noise and Local Time

Stochastic Heat Equation Driven by Fractional Noise and Local Time Yaozhong Hu∗ and Nualart† Department of Mathematics , University of Kansas 405 Snow Hall , Lawrence, Kansas 66045-2142 [email protected] and [email protected] Abstract The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0, 1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos development for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion. 1 Introduction This paper deals with the d-dimensional stochastic heat equation ∆u+ u ⋄ ∂ (1.1) driven by a Gaussian noise WH which is a white noise in the spatial variable and a fractional Brownian motion with Hurst parameter H ∈ (0, 1) in the time variable (see (2.1) in the next section for a precise definition of this noise). The initial condition u0 is a bounded continuous function on R d, and the solution will be a random field {ut,x, t ≥ 0, x ∈ Rd}. The symbol ⋄ in Equation (1.1) denotes the Wick product. For H = 1 is a space-time white noise, and in this case, Equation (1.1) coincides with the stochastic heat equation considered by Walsh (see [17]). We know that in this case the solution exists only in dimension one (d = 1). There has been some recent interest in studying stochastic partial differential equations driven by a fractional noise. Linear stochastic evolution equations in ∗Y. Hu is supported by the National Science Foundation under DMS0504783 †D. Nualart is supported by the National Science Foundation under DMS0604207 http://arxiv.org/abs/0704.1824v1 a Hilbert space driven by an additive cylindrical fBm with Hurst parameter H were studied by Duncan et al. in [3] in the case H ∈ (1 , 1) and by Tindel et al. in [15] in the general case, where they provide necessary and sufficient conditions for the existence and uniqueness of an evolution solution. In particular, the heat equation on Rd has a unique solution if and only if H > d . The same result holds when one adds to the above equation a nonlinearity of the form b(t, x, u), where b satisfies the usual linear growth and Lipschitz conditions in the variable u, uni- formly with respect to (t, x) (see Maslowski and Nualart in [9]). The stochastic heat equation on [0,∞) × Rd with a multiplicative fractional white noise of Hurst parameter H = (H0, H1, . . . , Hd) has been studied by Hu in [6] under the conditions 1 < Hi < 1 for i = 0, . . . , d and i=0Hi < d− 2H0−1 The main purpose of this paper is to find conditions on H and d for the solution to Equation (1.1) to exist as a real-valued stochastic process, and to relate the moments of the solution to the exponential moments of weighted in- tersection local times. This relation is based on Feynman-Kac’s formula applied to a regularization of Equation (1.1). In order to illustrate this fact, consider the particular case d = 1 and H = 1 . It is known that there is no Feynman- Kac’s formula for the solution of the one-dimensional stochastic heat equation driven by a space-time white noise. Nevertheless, using an approximation of the solution by regularizing the noise we can establish the following formula for the moments: ukt,x u0(x+B t ) exp i,j=1,i<j s −Bjs)ds  , (1.2) for all k ≥ 2, where Bt is a k-dimensional Brownian motion independent of the spaced-time white noise W 2 . In the case H > 1 and d ≥ 1, a similar formula holds but s − Bjs)ds has to be replaced by the weighted intersection local time Lt = H(2H − 1) |s− r|2H−2 δ0(Bis −Bjr)dsdt, (1.3) where Bj , j ≥ 1 are independent d-dimensional Brownian motions (see The- orem 5.3). The solution of Equation (1.1) has a formal Wiener chaos expansion ut,x =∑∞ n=0 In(fn(·, t, x)). Then, for the existence of a real-valued square integrable solution we need n! ‖fn(·, t, x)‖2H⊗n <∞, (1.4) whereHd is the Hilbert space associated with the covariance of the noiseWH . It turns out that, if H > 1 , the asymptotic behavior of the norms ‖fn(·, t, x)‖H⊗n is similar to the behavior of the nth moment of the random variable Lt defined in (1.3). More precisely, if u0 is a constant K, for all n ≥ 1 we have 2 ‖fn(·, t, x)‖2H⊗n = K2E(Lnt ). These facts leads to the following results: i) If d = 1 and H > 1 , the series (1.4) converges, and there exists a solution to Equation (1.1) which has moments of all orders that can be expressed in terms of the exponential moments of the weighted intersection local times Lt. In the case H = we just need the local time of a one-dimensional standard Brownian motion (see (1.2)). ii) If H > 1 and d < 4H , the norms ‖fn(·, t, x)‖H⊗n are finite and E(Lnt ) < ∞ for all n. In the particular case d = 2, the series (1.4) converges if t is small enough, and the solution exists in a small time interval. Similarly, if d = 2 the random variable Lt satisfies E(expλLt) < ∞ if λ and t are small enough. iii) If d = 1 and 3 < H < 1 , the norms ‖fn(·, t, x)‖H⊗n are finite and E(Lnt ) < ∞ for all n. A natural problem is to investigate what happens if we replace the Wick product by the ordinary product in Equation (1.1), that is, we consider the equation ∆u + u . (1.5) In terms of the mild formulation, the Wick product leads to the use of Itô- Skorohod stochastic integrals, whereas the ordinary product requires the use of Stratonovich integrals. For this reason, if we use the ordinary product we must assume d = 1 and H > 1 . In this case we show that the solution exists and its moments can be computed in terms of exponential moments of weighted intersection local times and weighted self-intersection local times in the case H > 3 The paper is organized as follows. Section 2 contains some preliminaries on the fractional noiseWH and the Skorohod integral with respect to it. In Section 3 we present the results on the moments of the weighted intersection local times assuming H ≥ 1 . Section 4 is devoted to study the Wiener chaos expansion of the solution to Equation (1.1). The case H < 1 is more involved because it requires the use of fractional derivatives. We show here that if 3 < H < 1 the norms ‖fn(·, t, x)‖H⊗n are finite and they are related to the moments of a fractional derivative of the intersection local time. We derive the formulas for the moments of the solution in the case H ≥ 1 in Section 5. Finally, Section 6 deals with equations defined using ordinary product and Stratonovich integrals. 2 Preliminaries Suppose that WH = {WH(t, A), t ≥ 0, A ∈ B(Rd), |A| < ∞}, where B(Rd) is the Borel σ-algebra of Rd, is a zero mean Gaussian family of random variables with the covariance function E(WH(t, A)WH(s,B)) = (t2H + s2H − |t− s|2H)|A ∩B|, (2.1) defined in a complete probability space (Ω,F , P ), where H ∈ (0, 1), and |A| denotes the Lebesgue measure of A. Thus, for each Borel set A with finite Lebesgue measure, {WH(t, A), t ≥ 0} is a fractional Brownian motion (fBm) with Hurst parameter H and variance t2H |A|, and the fractional Brownian motions corresponding to disjoint sets are independent. Then, the multiplicative noise ∂ appearing in Equation (1.1) is the for- mal derivative of the random measure WH(t, A): WH(t, A) = dsdx. We know that there is an integral representation of the form WH(t, A) = KH(t, s)W (ds, dx), where W is a space-time white noise, and the square integrable kernel KH is given by KH(t, s) = cHs (u− s)H− 2 du, for some constant cH . We will set KH(t, s) = 0 if s > t. Denote by E the space of step functions on R+. Let H be the closure of E with respect to the inner product induced by 1[0,t],1[0,s] = KH(t, s). The operator K∗H : E → L2(R+) defined by K∗H(1[0,t])(s) = KH(t, s) provides a linear isometry between H and L2(R+). The mapping 1[0,t]×A →WH(t, A) extends to a linear isometry between the tensor product H ⊗ L2(Rd), denoted by Hd, and the Gaussian space spanned by WH . We will denote this isometry by WH . Then, for each ϕ ∈ Hd we have WH(ϕ) = (K∗H ⊗ I)ϕ(t, x)W (dt, dx). We will make use of the notation WH(ϕ) = ϕdWH . If H = 1 , then H = L2(R+), and the operator K∗H is the identity. In this case, we have Hd = L2(R+ × Rd). Suppose now that H > 1 . The operator K∗H can be expressed as a fractional integral operator composed with power functions (see [11]). More precisely, for any function ϕ ∈ E with support included in the time interval [0, T ] we have (K∗Hϕ) (t) = c ϕ(s)sH− where I T− is the right-sided fractional integral operator defined by T− f(t) = Γ(H− 1 (s− t)H− 2 f(s)ds. In this case the space H is not a space of functions (see [14]) because it contains distributions. Denote by |H| the space of measurable functions on [0, T ] such that ∫ ∞ |r − u|2H−2|ϕr||ϕu|drdu <∞. Then, |H| ⊂ H and the inner product in the space H can be expressed in the following form for ϕ, ψ ∈ |H| 〈ϕ, ψ〉H = φ(r, u)ϕrϕudrdu, (2.2) where φ(s, t) = H(2H − 1)|t− s|2H−2. Using Hölder and Hardy-Littlewood inequalities, one can show (see [10]) ‖ϕ‖Hd ≤ βH ‖ϕ‖L 1H (R+;L2(Rd)) , (2.3) and this easily implies that ‖ϕ‖H⊗n ≤ βnH ‖ϕ‖L 1H (Rn ;L2(Rnd)) . (2.4) If H < 1 , the operator K∗H can be expressed as a fractional derivative operator composed with power functions (see [11]). More precisely, for any function ϕ ∈ E with support included in the time interval [0, T ] we have (K∗Hϕ) (t) = c ϕ(s)sH− where D T− is the right-sided fractional derivative operator defined by T− f(t) = Γ(H+ 1 (T − t) 12−H f(s)− f(t) (s− t)H− 32 Moreover, for any γ > 1 − H and any T > 0 we have Cγ([0, T ]) ⊂ H = T− (L 2([0, T ]). If ϕ is a function with support on [0, T ], we can express the operator K∗H in the following form K∗Hϕ(t) = KH(T, t)ϕ(t) + [ϕ(s) − ϕ(t)]∂KH (s, t)ds. (2.5) We are going to use the following notation for the operator K∗H : K∗Hϕ = [0,T ] ϕ(t)K∗H(dt, r). (2.6) Notice that if H > 1 , the kernel KH vanishes at the diagonal and we have K∗H(dt, r) = (t, r)1[r,T ](t)dt. Let us now present some preliminaries on the Skorohod integral and the Wick product. The nth Wiener chaos, denoted by Hn, is defined as the closed linear span of the random variables of the form Hn(W H(ϕ)), where ϕ is an element of Hd with norm one and Hn is the nth Hermite polynomial. We denote by In the linear isometry between H⊗nd (equipped with the modified norm n! ‖·‖H⊗n and the nth Wiener chaos Hn, given by In(ϕ ⊗n) = n!Hn(W H(ϕ)), for any ϕ ∈ Hd with ‖ϕ‖Hd = 1. Any square integrable random variable, which is measurable with respect to the σ-field generated by WH , has an orthogonal Wiener chaos expansion of the form F = E(F ) + In(fn), where fn are symmetric elements of H⊗nd , uniquely determined by F . Consider a random field u = {ut,x, t ≥ 0, x ∈ Rd} such that E u2t,x for all t, x. Then, u has a Wiener chaos expansion of the form ut,x = E(ut,x) + In(fn(·, t, x)), (2.7) where the series converges in L2(Ω). Definition 2.1 We say the random field u satisfying (2.7) is Skorohod inte- grable if E(u) ∈ Hd, for all n ≥ 1, fn ∈ H⊗(n+1)d , and the series WH(E(u)) + In+1(f̃n) converges in L2(Ω), where f̃n denotes the symmetrization of fn.We will denote the sum of this series by δ(u) = uδWH . The Skorohod integral coincides with the adjoint of the derivative operator. That is, if we define the space D1,2 as the closure of the set of smooth and cylindrical random variables of the form F = f(WH(h1), . . . ,W H(hn)), hi ∈ Hd, f ∈ C∞p (Rn) (f and all its partial derivatives have polynomial growth) under the norm ‖DF‖1,2 = E(F 2) + E(‖DF‖2Hd), where (WH(h1), . . . ,W H(hn))hj , then, the following duality formula holds E(δ(u)F ) = E 〈DF, u〉Hd , (2.8) for any F ∈ D1,2 and any Skorohod integrable process u. If F ∈ D1,2 and h is a function which belongs to Hd, then Fh is Skorohod integrable and, by definition, the Wick product equals to the Skorohod integral of Fh: δ(Fh) = F ⋄WH(h). (2.9) This formula justifies the use of the Wick product in the formulation of Equation (1.1). Finally, let us remark that in the case H = 1 , if ut,x is an adapted stochastic process such that E u2t,xdxdt <∞, then u is Skorohod integrable and δ(u) coincides with the Itô stochastic integral: δ(u) = ut,xW (dt, dx). 3 Weighted intersection local times for standard Brownian motions In this section we will introduce different kinds of weighted intersection local times which are relevant in computing the moments of the solutions of stochastic heat equations with multiplicative fractional noise. Suppose first that B1 and B2 are independent d-dimensional standard Brow- nian motions. Consider a nonnegative measurable function η(s, t) on R2+. We are interested in the weighted intersection local time formally defined by η(s, t)δ0(B s −B2t )dsdt. (3.1) We will make use of the following conditions on the weight η: C1) For all T > 0 ‖η‖1,T := max 0≤t≤T η(s, t)ds, sup 0≤s≤T η(s, t)dt C2) For all T > 0 there exist constants γT > 0 and H ∈ (0, 1) such that η(s, t) ≤ γT |s− t|2H−2 , for all s, t ≤ T . Clearly, C2) is stronger than C1). We will denote by pt(x) the d-dimensional heat kernel pt(x) = (2πt) 2t . Consider the approximation of the inter- section local time (3.1) defined by η(s, t)pε(B s −B2t )dsdt. (3.2) Let us compute the kth moment of Iε, where k ≥ 1 is an integer. We can write [0,T ]2k η(si, ti)ψε(s, t)dsdt, (3.3) where s = (s1, . . . , sk), t = (t1, . . . , tk) and ψε(s, t) = E −B2t1) · · · pε(B −B2tk) . (3.4) Using the Fourier transform of the heat kernel we can write ψε(s, t) = ξj , b − b2tj |ξj |2 j,l=1 ξjCov , (3.5) where ξ = (ξ1, . . . , ξk) and b t, i = 1, 2, are independent one-dimensional Brow- nian motions. Then ψε(s, t) ≤ ψ(s, t), where ψ(s, t) = (2π)− 2 [det (sj ∧ sl + tj ∧ tl)]− 2 . (3.6) [0,T ]2k η(si, ti)ψ(s, t)dsdt. (3.7) Then, if αk <∞ for all k ≥ 1, the family Iε converges in Lp, for all p ≥ 2, to a limit I and E(Ik) = αk. In fact, ε,δ↓0 E(IεIδ) = α2, so Iε converges in L 2, and the convergence in Lp follows from the boundedness in Lq for q > p. Then the following result holds. Proposition 3.1 Suppose that C1) holds and d = 1. Then, for all λ > 0 the random variable defined in (3.2) satisfies E (exp (λIε)) ≤ 1 + Φ ‖η‖1,T λ , (3.8) where Φ(x) = Γ(k+1 . Also, Iε converges in L p for all p ≥ 2, and the limit, denoted by I, satisfies the estimate (3.8). Proof The term ψ(s, t) defined in (3.6) can be estimated using Cauchy- Schwarz inequality: ψ(s, t) ≤ (2π)−k [det (sj ∧ sl)] 4 [det (tj ∧ tl)] = 2−kπ− 2 [β(s)β(t)] 4 , (3.9) where for any element (s1, . . . , sk) ∈ (0,∞)k with si 6= sj if i 6= j, we denote by σ the permutation of its coordinates such that sσ(1) < · · · < sσ(n) and β(s) = sσ(1)(sσ(2) − sσ(1)) · · · (sσ(k) − sσ(k−1)). Therefore, from (3.9) and (3.7) we obtain αk ≤ 2−kπ− [0,T ]2k η(si, ti) [β(s)β(t)] 4 dsdt. (3.10) Applying again Cauchy-Schwarz inequality yields αk ≤ 2−kπ− [0,T ]2k η(si, ti) [β(s)] 2 dsdt [0,T ]2k η(si, ti) [β(t)] 2 dsdt 2−1π− 2 ‖η‖1,T [β(s)] k!2−kT 2 ‖η‖k1,T Γ(k+1 , (3.11) where Tk = {s = (s1, . . . , sk) : 0 < s1 < · · · < sk < T }, which implies the estimate (3.8). This result can be extended to the case of a d-dimensional Brownian motion under the stronger condition C2): Proposition 3.2 Suppose that C2) holds and d < 4H. Then, limε↓0 Iε = I, exists in Lp, for all p ≥ 2. Moreover, if d = 2 and λ < λ0(T ), where λ0(T ) = H(2H − 1)4π , (3.12) and βH is the constant appearing in the inequality (2.3), then E (exp (λIε)) <∞, (3.13) and I satisfies E (exp (λI)) <∞. Proof As in the proof of Proposition 3.1, using condition C2) and inequality (2.4) we obtain the estimates αk ≤ γkT 2−dkπ− [0,T ]2k |ti − si|2H−2 [β(s)β(t)]− 4 dsdt ≤ γkT 2−dkπ− 2 αkH [0,T ]k [β(s)] 4H ds γTαH2 −2π−1 2H Γ(1− d4H ) k2HT k(1− Γ(k(1− d ) + 1)2H γTαH2 −2π−1Γ(1− d )2HT 2H− where αH = H(2H−1) . This allows us to conclude the proof. If d = 2 and η(s, t) = 1 it is known that the intersection local time B2t )dsdt exists and it has finite exponential moments up to a critical exponent λ0 (see Le Gall [7] and Bass and Chen [1]). Consider now a one-dimensional standard Brownian motion B, and the weighted self-intersection local time η(s, t)δ0(Bs −Bt)dsdt. As before, set η(s, t)pε(Bs −Bt)dsdt. Proposition 3.3 Suppose that C2) holds. If H > 1 , then we have E (exp (λ [Iε − E (Iε)])) <∞, (3.14) for all λ > 0. Moreover, the normalized local time I −E (I) exists as a limit in Lp of Iε − E (Iε), for all p ≥ 2, and it has exponential moments of all orders. If H > 3 , then we have for all λ > 0 E (exp (λIε)) <∞, (3.15) for all λ > 0, and the local time I exists as a limit in Lp of Iε, for all p ≥ 2, and it is exponentially integrable. Proof We will follow the ideas of Le Gall in [7]. Suppose first that H > 1 let us show (3.14). To simplify the proof we assume T = 1. It suffices to show these results for Jε := η(s, t)pε(Bs −Bt)dsdt. Denote, for n ≥ 1, and 1 ≤ k ≤ 2n−1 An,k = 2k − 2 2k − 1 2k − 1 αεn,k = η(s, t)pε(Bs −Bt)dsdt ᾱεn,k = α n,k − E αεn,k Notice that the random variables αεn,k, 1 ≤ k ≤ 2n−1, are independent. We 2n−1∑ αεn,k, Jε − E (Jε) = 2n−1∑ ᾱεn,k. We can write αεn,k = 2 2k − 1 2k − 1 ×pε(B 2k−1 −B 2k−1 )dsdt ≤ γ12−2n−(2H−2)n |t+ s|2H−2pε(B 2k−1 −B 2k−1 )dsdt, which has the same distribution as βεn,k = γ12 n−(2H−2)n |t+ s|2H−2pε2n(B1s −B2t )dsdt, where B1 and B2 are independent one-dimensional Brownian motions. Hence, using the estimate (3.11), we obtain ᾱεn,k = 1 + ᾱεn,k ≤ 1 + βεn,k ≤ 1 + n−(2H−2)nλ Γ( j+2 for some constant CT . Hence, ᾱεn,k ≤ 1 + cλ2−3n−2(2H−2)nλ2, (3.16) for some function cλ. Fix a > 0 such that a < 2(2H − 1)̇. For any N ≥ 2 define (1− 2−a(j−1)), and notice that limN→∞ bN = b∞ > 0. Then, by Hölder’s inequality, for all N ≥ 2 we have 2n−1∑ ᾱεn,k  λbN 1− 2−a(N−1) 2n−1∑ ᾱεn,k 1−2−a(N−1) λbN2a(N−1) 2N−1∑ ᾱεN,k 2−a(N−1) λbN−1 2n−1∑ ᾱεn,k a(N−1)ᾱεN,k )]}2(1−a)(N−1) Using (3.16), the second factor in the above expression can be dominated by a(N−1)ᾱN,k )]}2(1−a)(N−1) 1 + cλλ 2b222 2a(N−1)2−3N−2(2H−2)N )2(1−a)(N−1) ≤ exp 22(a−2−2(2H−2))N where κ = b222 −a−1. Thus by induction we have 2n−1∑ ᾱn,k  ≤ exp 22(a−2−2(2H−2))n ×E (exp ᾱ1,1) ≤ exp(κcλλ2(1 − 2a+2−4H)−1) ×E (exp(ᾱ1,1)) <∞, because a < 2(2H − 1). By Fatou lemma we see that E (exp (λb∞ (Jε − E (Jε)))) <∞, and (3.14) follows. On the other hand, one can easily show that ε,δ↓0 E((Jε − E (Jε)) (Jδ − E (Jδ))) = s<t<1,s′<t′<1 η(s, t)η(s′, t′) t− s |[s, t] ∩ [s′, t′]| |[s, t] ∩ [s′, t′]| t′ − s′ ])− 1 − ((t− s)(t′ − s′))− dsdtds′dt′ <∞, which implies the convergence of Iε in L 2. The convergence in Lp for p ≥ 2 and the estimate (3.14) follow immediately. The proof of the inequality (3.15) is similar. The estimate (3.16) is replaced αεn,k ≤ 1 + dλ2−3n−2(2H−2)nλ, (3.17) for a suitable function dλ, and we obtain 2n−1∑ ≤ exp κdλλ2 (− 52−2H)n E (exp (α1,1)) ≤ exp( 2(1− 2(− −2H)n)−1)E (exp (α1,1)) <∞, because H > 3 . By Fatou lemma we see that E (exp (λb∞ (Jε − E (Jε)))) <∞, which implies (3.15). The convergence in Lp of Iε is proved as usual. Notice that condition H > 3 cannot be improved because |t− s|− 2 δ0(Bs −Bt)dsdt |t− s|−1dsdt = ∞. 4 Stochastic heat equation in the Itô-Skorohod sense In this section we study the stochastic partial differential equation (1.1) on Rd, where WH is a zero mean Gaussian family of random variables with the covariance function (2.1), defined on a complete probability space (Ω,F , P ), and the initial condition u0 belongs to Cb(R d). First we give the definition of a solution using the Skorhohod integral, which corresponds formally to the Wick product appearing in Equation (1.1). For any t ≥ 0, we denote by Ft the σ-field generated by the random variables {W (s, A), 0 ≤ s ≤ t, A ∈ B(Rd), |A| < ∞} and the P -null sets. A random field u = {ut,x, t ≥ 0, x ∈ R} is adapted if for any (t, x), ut,x is Ft-measurable. For any bounded Borel function ϕ on R we write ptϕ(x) = pt(x − y)ϕ(y)dy. Definition 4.1 An adapted random field u = {ut,x, t ≥ 0, x ∈ Rd} such that E(u2t,x) < ∞ for all (t, x) is a solution to Equation (1.1) if for any (t, x) ∈ [0,∞) × Rd, the process {pt−s(x − y)us,y1[0,t](s), s ≥ 0, y ∈ Rd} is Skorohod integrable, and the following equation holds ut,x = ptu0(x) + pt−s(x − y)us,yδWHs,y . (4.1) The fact that Equation (1.1) contains a multiplicative Gaussian noise allows us to find recursively an explicit expression for the Wiener chaos expansion of the solution. This approach has extensively used in the literature. For instance, we refer to the papers by Hu [6], Buckdahn and Nualart [2], Nualart and Zakai [13], Nualart and Rozovskii [12], and Tudor [16], among others. Suppose that u = {ut,x, t ≥ 0, x ∈ Rd} is a solution to Equation (1.1). Then, for any fixed (t, x), the random variable ut,x admits the following Wiener chaos expansion ut,x = In(fn(·, t, x)), (4.2) where for each (t, x), fn(·, t, x) is a symmetric element in H⊗nd . To find the explicit form of fn we substitute (4.2) in the Skorohod integral appearing in (4.1) we obtain pt−s(x− y)us,yδWHs,y = In(pt−s(x− y)fn(·, s, y)) δWHs,y In+1( ˜pt−s(x− y)fn(·, s, y)) . Here, ( ˜pt−s(x− y)fn(·, s, y) denotes the symmetrization of the function pt−s(x− y)fn(s1, x1; . . . ; sn, xn; s, y) in the variables (s1, x1), . . . , (sn, xn), (s, y), that is, ˜pt−s(x− y)fn(·, s, y) = [pt−s(x− y)fn(s1, x1, . . . , sn, xn, s, y) pt−sj (x − yj) ×fn(s1, x1, . . . , sj−1, xj−1, s, y, sj+1, xj+1, . . . , sn, yn, sj , yj)]. Thus, Equation (4.1) is equivalent to say that f0(t, x) = ptu0(x), and fn+1(·, t, x) = ˜pt−s(x− y)fn(·, s, y) (4.3) for all n ≥ 0. Notice that, the adaptability property of the random field u implies that fn(s1, x1, . . . , sn, xn, t, x) = 0 if sj > t for some j. This leads to the following formula for the kernels fn, for n ≥ 1 fn(s1, x1, . . . , sn, xn, t, x) = ×pt−sσ(n)(x − xσ(n)) · · · psσ(2)−sσ(1)(xσ(2) − xσ(1))psσ(1)u0(xσ(1)), (4.4) where σ denotes the permutation of {1, 2, . . . , n} such that 0 < sσ(1) < · · · < sσ(n) < t. This implies that there is a unique solution to Equation (4.1), and the kernels of its chaos expansion are given by (4.4). In order to show the existence of a solution, it suffices to check that the kernels defined in (4.4) determine an adapted random field satisfying the conditions of Definition 4.1. This is equivalent to show that for all (t, x) we have n! ‖fn(·, t, x)‖2H⊗n <∞. (4.5) It is easy to show that (4.5) holds ifH = 1 and d = 1. In fact, we have, assuming |u0| ≤ K, and with the notation x = (x1, . . . , xn), and s = (s1, . . . , sn): ‖fn(·, t, x)‖2H⊗n1 [0,t]n pt−sσ(n)(x− xσ(n)) 2 · · · psσ(2)−sσ(1)(xσ(2) − xσ(1)) ×psσ(1)u0(xσ(1)) 2 dxds ≤ K2 (4π) [0,t]n (sσ(j+1) − sσ(j))− K2 (4π) (sj+1 − sj)− 2 ds, where Tn = {(s1, . . . , sn) ∈ [0, t]n : 0 < s1 < · · · < sn < t} and by convention sn+1 = t. Hence, ‖fn(·, t, x)‖2H⊗n1 ≤ K22−nt n!Γ(n+1 which implies (4.5). On the other hand, if H = 1 and d ≥ 2, these norms are infinite. Notice that if u0 = 1, then (n!) 2 ‖fn(·, t, x)‖2H⊗n1 coincides with the moment of order n of the local time at zero of the one-dimensional Brownian motion with variance 2t, that is, (n!)2 ‖fn(·, t, x)‖2H⊗n1 = E [(∫ t δ0(B2s)ds To handle the case H > 1 , we need the following technical lemma. Lemma 4.2 Set gs(x1, . . . , xn) = pt−sσ(n)(x− xσ(n)) · · · psσ(2)−sσ(1)(xσ(2) − xσ(1))). (4.6) Then, 〈gs, gt〉L2(Rnd) = ψ(s, t), where ψ(s, t) is defined in (3.4). Proof By Plancherel’s identity 〈gs, gt〉L2(Rnd) = (2π) −dn 〈Fgs,Fgt〉L2(Rnd) , where F denotes the Fourier transform, given by Fgs(ξ1, . . . , ξn) = (2π)− (sσ(j+1) − sσ(j))− i 〈ξj , xj〉 − ∣∣xσ(j+1) − xσ(j) sσ(j+1) − sσ(j) with the convention xn+1 = x and sn+1 = t. Making the change of variables uj = xσ(j+1) − xσ(j) if 1 ≤ j ≤ n− 1, and un = x− xσ(n), we obtain Fgs(ξ1, . . . , ξn) = (2π)− (sσ(j+1) − sσ(j))− ξσ(j), x− un − · · · − uj − |uj | sσ(j+1) − sσ(j) ξσ(j), x−Bt −Bsσ(j) ξj , x−Bt −Bsj As a consequence, 〈gs, gt〉L2(Rnd) = (2π) ξj , B −B2tj  dξ, which implies the desired result. In the case H > 1 , and assuming that u0 = 1, the next proposition shows that the norm (n!)2 ‖fn(·, t, x)‖2H⊗n coincides with the nth moment of the in- tersection local time of two independent d-dimensional Brownian motions with weight φ(t, s). Proposition 4.3 Suppose that H > 1 and d < 4H. Then, for all n ≥ 1 (n!)2 ‖fn(·, t, x)‖2H⊗n ≤ ‖u0‖2∞E [(∫ t φ(s, r)δ0(B s −B2r )dsdr (4.7) with equality if u0 is constant. Moreover, we have: 1. If d = 1, there exists a unique solution to Equation (4.1). 2. If d = 2 , then there exists a unique solution in an interval [0, T ] provided T < T0, where βHΓ(1− )−1/(2H−1) . (4.8) Proof We have (n!)2 ‖fn(·, t, x)‖2H⊗n ≤ ‖u0‖2∞ [0,t]n φ(sj , tj) 〈gs, gt〉L2(Rnd) dsdt, (4.9) where gs is defined in (4.6). Then the results follow easily from from Lemma 4.2 and Proposition 3.2. In the two-dimensional case and assuming H > 1 , the solution would exists in any interval [0, T ] as a distribution in the Watanabe space Dα,2 for any α > 0 (see [18]). 4.1 Case H < 1 and d = 1 We know that in this case, the norm in the space H is defined in terms of fractional derivatives. The aim of this section is to show that ‖fn(·, t, x)‖2H⊗n1 is related to the nth moment of a fractional derivative of the self-intersection local time of two independent one-dimensional Brownian motions, and these moments are finite for all n ≥ 1, provided 3 < H < 1 Consider the operator (K∗H) on functions of two variables defined as the action of the operator K∗H on each coordinate. That is, using the notation (2.5) we have (K∗H) f(r1, r2) = KH(T, r1)KH(T, r2)f(r1, r2) +KH(T, r1) (s, r2) (f(r1, s)− f(r1, r2)) ds +KH(T, r2) (v, r1) (f(v, r2)− f(r1, r2)) dv (s, r2) (v, r1) [f(v, s)− f(r1, s)− f(v, r2)− f(r1, r2)] dsdv. Suppose that f(s, t) is a continuous function on [0, T ]2. Define the Hölder norms ‖f‖1,γ = sup |f(s1, t)− f(s2, t)| |s1 − s2|γ , s1, s2, t ∈ T, s1 6= s2 ‖f‖2,γ = sup |f(s, t1)− f(s, t2)| |t1 − t2|γ , t1, t2, s ∈ T, t1 6= t2 ‖f‖1,2,γ = sup |f(s1, t1)− f(s1, t2)− f(s2, t1) + f(s2, t2)| |s1 − s2|γ |t1 − t2|γ where the supremum is taken in the set {t1, t2, s2, s2 ∈ T, s1 6= s2, t1 6= t2}. Set ‖f‖0,γ = ‖f‖1,γ + ‖f‖2,γ + ‖f‖1,2,γ Then, (K∗H) f is well defined if ‖f‖0,γ < ∞ for some γ > − H . As a consequence, if B1 and B2 are two independent one-dimensional Brownian motions, the following random variable is well defined for all ε > 0 (K∗H) · −B2· )(r, r)dr. (4.10) The next theorem asserts that Jε converges in L p for all p ≥ 2 to a fractional derivative of the intersection local time of B1 and B2. Proposition 4.4 Suppose that 3 < H < 1 .Then, for any integer k ≥ 1 and, T > 0 we have E ≥ 0 and Moreover, for all p ≥ 2, Jε converges in Lp as ε tends to zero to a random variable denoted by (K∗H) · −B2· )(r, r)dr. Proof Fix k ≥ 1. Let us compute the moment of order k of Jε. We can write [0,T ]k (K∗H) p2ε(B 1 −B2)(ri, ri) dr. (4.11) Using the expression (2.6) for the operator K∗H , and the notation (3.4) yields [0,T ]3k ψε(s, t) K∗H(dsi, ri)K H(dti, ri)dr. (4.12) As a consequence, using (3.5) we obtain = (2π)−k [0,T ]k [0,T ]2k j,l=1 ξjξlCov −B2tj ,B1sl −B2tl K∗H(dsi, ri)K H(dti, ri)e j=1 ξ j dξdr ≤ (2π)−k [0,T ]k [0,T ]k K∗H(dti, ri) dξdr. Then, it suffices to show that for each k the following quantity is finite [sj − sj−1 + tj − tj−1]− K∗H(dsi, ri) K∗H(dti, ri)dr, (4.13) where Tk = {0 < t1 < · · · < tk < T }. Fix a constant a > 0. We are going to compute [tj − tj−1 + a]−2 K∗H(dti, ri). To do this we need some notation. Let ∆j and Ij be the operators defined on a function f(t1, . . . , tk) by ∆jf = f − f |tj=rj , Ijf = f |tj=rj . The operatorK∗H(dti, ri) is the sum of two components (see (2.5)), and it suffices to consider only the second one because the first one is easy to control. In this way we need to estimate the following term [0,T ]k ∆1 · · ·∆k j [tj − tj−1 + a] 2 1{tj−1<tj} (tj − rj)H− j 1{rj<tj} Because t j ≤ 1, we can disregard the factors r j and t j . Using the rule ∆j(FG) = F (tj)G(tj)− F (rj)G(rj) = [F (tj)− F (rj)]G(tj) + F (rj) [G(tj)−G(rj)] = ∆jFG+ IjF∆jG, we obtain ∆1 · · ·∆k [tj − tj−1 + a]− 2 1{tj−1<tj} [tj − tj−1 + a]− 2 1{tj−1<tj} where Sj is an operator of the form: IIj , I∆j ,∆j−1Ij ,∆j−1∆j , and for each j, ∆j must appear only once in the product j=1 Sj . Let us estimate each one of the possible four terms. Fix ε > 0 such that H − 3 > 2ε. 1. Term IIj : [tj − tj−1 + a]− 2 1{tj−1<tj} = [rj − tj−1 + a]− 2 1{tj−1<rj}, 2. Term I∆j : ∣∣∣I∆j [tj − tj−1 + a]− 2 1{tj−1<tj} ∣∣∣[tj − tj−1 + a]− 2 1{tj−1<tj} − [rj − tj−1 + a] 2 1{tj−1<rj} ≤ C [tj − rj ] [rj − tj−1 + a]H−1−ε 1{tj−1<rj} +C [tj − tj−1 + a]− 2 1{rj<tj−1}. 3. Term ∆j−1I: ∣∣∣∆j−1I [tj − tj−1 + a]− 2 1{tj−1<tj} ∣∣∣[tj − tj−1 + a]− 2 1{tj−1<tj} − [tj − rj−1 + a] 2 1{rj−1<tj} ≤ C [tj−1 − rj−1] [tj − tj−1 + a]H−1−ε 1{rj−1<tj−1<tj}. 4. Term ∆j−1∆j : ∣∣∣∆j−1∆j [tj − tj−1 + a]− 2 1{tj−1<tj} ∣∣∣[tj − tj−1 + a]− 2 1{tj−1<tj} − [rj − tj−1 + a] 2 1{tj−1<rj} − [tj − rj−1 + a]− 2 1{rj−1<tj} + [rj − rj−1 + a] 2 1{rj−1<rj} ≤ C [tj − rj ] [tj−1 − rj−1] [rj − tj−1 + a]2H− 1{tj−1<rj<tj} +C [tj−1 − rj−1] [tj − tj−1 + a]H−1−ε 1{rj<tj−1<tj} +C [rj − rj−1 + a]− 2 1{rj−1<rj<tj−1<tj}. If we replace the constant a by sj−sj−1 and we treat the the term sj−sj−1 in the same way, using the inequality (a+ b)−α ≤ a− we obtain the same estimates as if we had started with [tj − tj−1]− K∗H(dtj , rj) instead of (4.13). As a consequence, it suffices to control the following integral j (t, r)dt dr, (4.14) where a, b ∈ {0, 1}, and Aj has one of the following forms j = [rj − tj−1] 4 1{tj−1<rj}, j,1 = [tj − rj ] [rj − tj−1]H− 1{tj−1<rj} j,2 = [tj − tj−1] 4 [tj − rj ]H− 2 1{rj<tj−1}, j = [tj−1 − rj−1] [tj − tj−1]H− 1{rj−1<tj−1<tj}, j,1 = [tj − rj ] [tj−1 − rj−1]−1+ε [rj − tj−1]2H− 1{tj−1<rj<tj}, j,2 = [tj−1 − rj−1] [tj − tj−1]H− [tj − rj ]H− 2 1{rj<tj−1<tj}, j,3 = [rj − rj−1] 4 [tj − rj ]H− 2 [tj−1 − rj−1]H− 2 1{rj−1<rj<tj−1<tj}, and with the convention that any term of the form A j or A j must be followed j or A j and any term of the form A j or A j must be followed by j or A j . It is not difficult to check that the integral (4.14) is finite. For instance, for a product of the form A j,1 we get {rj−1<tj−1<rj<tj} [rj−1 − tj−2]− 4 [tj−1 − rj−1]−1+ε [rj − tj−1]2H− × [tj − rj ]−1+ε dtj−1 = [rj−1 − tj−2]− 4 [rj − rj−1]2H− −ε [tj − rj ]−1+ε , and the integral in the variable rj of the square of this expression will be finite because 4H − 5 − 2ε > −1. So, we have proved that supε E(J ε ) < ∞ for all k. Notice that all these moments are positive. It holds that limε,δ↓0 E(JεJδ) exists, and this implies the convergence in L2, and also in Lp, for all p ≥ 2. On the other hand, if the initial condition of Equation (1.1) is a constant K, then for all n ≥ 1 we have (n!)2 ‖fn(·, t, x)‖2H⊗n1 = K [(∫ T (K∗H) · −B2· )(r, r)dr provided H ∈ . In fact, by Lemma 4.2 we have (n!)2 ‖fn(·, t, x)‖2H⊗n1 = K [0,t]2n 〈gs, gt〉L2(Rn) K∗H(dti, ri) K∗H(dsi, ri)dsdt [0,t]2n ψ(s, t) K∗H(dti, ri) K∗H(dsi, ri)dsdt, and it suffices to apply the above proposition. However, we do not know the rate of convergence of the sequence ‖fn(·, t, x)‖2H⊗n1 as n tends to infinity, and for this reason we are not able to show the existence of a solution to Equation (1.1) in this case. 5 Moments of the solution In this section we introduce an approximation of the Gaussian noise WH by means of an approximation of the identity. In the space variable we choose the heat kernel to define this approximation and in the time variable we choose a rectangular kernel. In this way, for any ε > 0 and δ > 0 we set t,x = ϕδ(t− s)pε(x− y)dWHs,y, (5.1) where ϕδ(t) = 1[0,δ](t). Now we consider the approximation of Equation (1.1) defined by t,x + u t,x ⋄ Ẇ t,x . (5.2) We recall that the Wick product u t,x⋄Ẇ t,x is well defined as a square integrable random variable provided the random variable u t,x belongs to the space D (see (2.9)), and in this case we have uε,δs,y ⋄ Ẇ ε,δs,y = ϕδ(s− r)pε(y − z)uε,δs,yδWHr,z . (5.3) The mild or evolution version of Equation (5.2) will be t,x = ptu0(y) + pt−s(x− y)uε,δs,y ⋄ Ẇ ε,δs,y dsdy. (5.4) Substituting (5.3) into (5.4), and formally applying Fubini’s theorem yields t,x = ptu0(y)+ pt−s(x− y)ϕδ(s− r)pε(y − z)uε,δs,ydsdy δWHr,z . (5.5) This leads to the following definition. Definition 5.1 An adapted random field uε,δ = {uε,δt,x, t ≥ 0, x ∈ Rd} is a mild solution to Equation (5.2) if for each (r, z) ∈ R+ × Rd the integral Y t,xr,z = pt−s(x − y)ϕδ(s− r)pε(y − z)uε,δs,ydsdy exists and Y t,x is a Skorohod integrable process such that (5.5) holds for each (t, x). The above definition is equivalent to saying that u t,x ∈ L2(Ω), and for any random variable F ∈ D1,2 , we have t,x) = E(F )ptu0(y) 〈(∫ t pt−s(x− y)ϕδ(s− ·)pε(y − ·)uε,δs,ydsdy Our aim is to construct a solution of Equation (5.2) using a suitable version of Feynman-Kac’s formula. Suppose that B = {Bt, t ≥ 0} is a d-dimensional Brownian motion starting at 0, independent of W . Set t−s,x+Bs ϕδ(t− s− r)pε(Bs + x− y)dWHr,yds Aε,δr,ydW r,y , where Aε,δr,y = ϕδ(t− s− r)pε(Bs + x− y)ds. (5.6) Define t,x = E u0(x+Bt) exp Aε,δr,ydW r,y − , (5.7) where αε,δ = ∥∥Aε,δ Proposition 5.2 The random field u t,x given by (5.7) is a solution to Equation (5.2). Proof The proof is based on the notion of S transform from white noise analysis (see [5]). For any element ϕ ∈ H1 we define St,x(ϕ) = E t,xFϕ where Fϕ = exp WH(ϕ)− 1 ‖ϕ‖2Hd From (5.7) we have St,x(ϕ) = E u0(x+Bt) exp WH(Aε,δ + ϕ)− αε,δ − ‖ϕ‖2Hd u0(x+Bt) exp Aε,δ, ϕ u0(x+Bt) exp 〈ϕδ(t− s− ·)pε(Bs + x− ·), ϕ〉Hd ds By the classical Feynman-Kac’s formula, St,x(ϕ) satisfies the heat equation with potential V (t, x) = 〈ϕδ(t− ·)pε(x− ·), ϕ〉Hd , that is, ∂St,x(ϕ) ∆St,x(ϕ) + St,x(ϕ) 〈ϕδ(t− ·)pε(x − ·), ϕ〉Hd . As a consequence, St,x(ϕ) = ptu0(x) + pt−s(x− y)Ss,y(ϕ) 〈ϕδ(s− ·)pε(y − ·), ϕ〉Hd dsdy. Notice that DFϕ = ϕFϕ. Hence, for any exponential random variable of this form we have t,xFϕ) = ptu0(x) + pt−s(x− y)E t,x 〈ϕδ(s− ·)pε(y − ·), DFϕ〉Hd and we conclude by the duality relationship between the Skorohod integral and the derivative operator. The next theorem says that the random variables u t,x have moments of all orders, uniformly bounded in ε and δ, and converge to the solution to Equation (1.1) as δ and ǫ tend to zero. Moreover, it provides an expression for the moments of the solution to Equation (1.1). Theorem 5.3 Suppose that H ≥ 1 and d = 1. Then, for any integer k ≥ 1 we [∣∣∣uε,δt,x <∞, (5.8) and the limit limε↓0 limδ↓0 u t,x exists in L p, for all p ≥ 1, and it coincides with the solution ut,x of Equation (1.1). Furthermore, if U 0 (t, x) = j=1 u0(x+B where Bj are independent d-dimensional Brownian motions, we have for any k ≥ 2 ukt,x UB0 (t, x) exp s −Bjs)ds  . (5.9) if H = 1 , and ukt,x UB0 (t, x) exp φ(s, r)δ0(B s −Bjr)dsdr  . (5.10) if H > 1 In the case d = 2, for any integer k ≥ 2 there exists t0(k) > 0 such that for all t < t0(k) (5.8) holds. If t < t0(M) for some M ≥ 3 then the limit limε↓0 limδ↓0 u t,x exists in L p for all 2 ≤ p < M , and it coincides with the solution ut,x of Equation (1.1). Moreover, (5.10) holds for all 1 ≤ k ≤M − 1. Proof Fix an integer k ≥ 2. Suppose that Bi = Bit , t ≥ 0 , i = 1, . . . , k are independent d-dimensional standard Brownian motions starting at 0, indepen- dent of WH . Then, using (5.7) we have u0(x +B t ) exp Aε,δ,B r,y dW r,y − αε,δ,B where Aε,δ,B r,y and α ε,δ,Bj are computed using the Brownian motion Bj . There- fore, = E B  exp ∥∥∥∥∥∥ Aε,δ,B ∥∥∥∥∥∥ αε,δ,B u0(x+B  exp Aε,δ,B , Aε,δ,B u0(x+B That is, the correction term 1 αε,δ in (5.7) due to the Wick product produces a cancellation of the diagonal elements in the square norm of j=1 A ε,δ,Bj . The next step is to compute the scalar product Aε,δ,B , Aε,δ,B for i 6= j. We consider two cases. Case 1. Suppose first that H = 1 and d = 1. In this case we have Aε,δ,B , Aε,δ,B ϕδ(t− s1 − r)pε(Bis1 + x− y) ×ϕδ(t− s2 − r)pε(Bjs2 + x− y)ds1ds2drdy ϕδ(t− s1 − r)ϕδ(t− s2 − r) ×p2ε(Bis1 −B )ds1ds2dr. We have ϕδ(t− s1 − r)ϕδ(t− s2 − r)dr = δ−2 (t− s1) ∧ (t− s2)− (t− s1 − δ)+ ∨ (t− s2 − δ)+ = ηδ(s1, s2). It it easy to check that ηδ is a a symmetric function on [0, t] 2 such that for any continuous function g on [0, t]2, ηδ(s1, s2)g(s1, s2)ds1ds2 = g(s, s)ds. As a consequence the following limit holds almost surely Aε,δ,B , Aε,δ,B p2ε(B s −Bjs)ds, and by the properties of the local time of the one-dimensional Brownian motion we obtain that, almost surely. Aε,δ,B , Aε,δ,B s −Bjs)ds. The function ηδ satisfies 0≤r≤t ηδ(s, r)ds ≤ 1, and, as a consequence, the estimate (3.8) implies that for all λ > 0 λ exp Aε,δ,B , Aε,δ,B Hence (5.8) holds and limε↓0 limδ↓0 u t,x := vt,x exists in L p, for all p ≥ 1. Moreover, E(vkt,x) equals to the right-hand side of Equation (5.9). Finally, Equation (5.5) and the duality relationship (2.8) imply that for any random variable F ∈ D1,2 with zero mean we have pt−s(x− y)ϕδ(s− ·)pε(y − ·)uε,δs,ydsdy and letting δ and ε tend to zero we get Fvt,x pt−s(x − y)ϕδ(s− ·)pε(y − ·)vs,ydsdy which implies that the process v is the solution of Equation (1.1), and by the uniqueness vt,x = ut,x. Case 2. Consider now the case H > 1 and d = 2. We have Aε,δ,B , Aε,δ,B ϕδ(t− s1 − r1)pε(Bis1 + x− y) ×ϕδ(t− s2 − r2)pε(Bjs2 + x− y)ds1ds2 φ(r1, r2)dr1dr2dy ϕδ(t− s1 − r1)ϕδ(t− s2 − r2) ×p2ε(Bis1 −B )ds1ds2φ(r1, r2)dr1dr2. This scalar product can be written in the following form Aε,δ,B , Aε,δ,B ηδ(s1 − s2)p2ε(Bis1 −B )ds1ds2, where ηδ(s1, s2) = ϕδ(t− s1 − r1)ϕδ(t− s2 − r2) φ(r1, r2)dr1dr2. (5.11) We claim that there exists a constant γ such that ηδ(s1, s2) ≤ γ|s1 − s2|2H−2. (5.12) In fact, if |s2 − s1| = s we have ηδ(s1, s2) ≤ H(2H − 1)δ−2 ∫ s+δ |u− v|2H−2dudv (s+ δ)2H − (s− δ)2H − 2s2H ∫ s+δ y2H−1 − (y − δ)2H−1 dy ≤ Hδ2H−2 ≤ H22−2Hs2H−2, if s ≤ 2δ. On the other hand, if s ≥ 2δ, we have (s+ δ)2H − (s− δ)2H − 2s2H s2H−1 − (s− δ)2H−1 ≤ H(2H − 1)(s− δ)2H−2 ≤ H(2H − 1)22−2Hs2H−2. It it easy to check that for any continuous function g on [0, t]2, ηδ(s1, s2)g(s1, s2)ds1ds2 = φ(s1, s2)g(s1, s2)ds1ds2. As a consequence the following limit holds almost surely Aε,δ,B , Aε,δ,B φ(s1, s2)δ0(B −Bjs2)ds1ds2. From (5.12) and the estimate (3.13) we get Aε,δ,B , Aε,δ,B <∞, (5.13) if λ < λ0(t), where λ0(t) is defined in (3.12) with gammaT replaced by γ. Hence, for any integer k ≥ 2, if t < t0(k), where k(k−1)2 = λ0(t0(k)), then (5.8) holds because ≤ ‖u0‖k k(k − 1) Aε,δ,B , Aε,δ,B )]) 2 k(k−1) Finally, if t < t0(M) and M ≥ 3, the limit limε↓0 limδ↓0 uε,δt,x := vt,x exists in Lp, for all 2 ≤ p < M and it is equal to the right-hand side of Equation (5.10). As in the case H = 1 we show that vt,x = ut,x. 6 Pathwise heat equation In this section we consider the one-dimensional stochastic partial differential equation ∆u+ uẆHt,x, (6.1) where the product between the solution u and the noise ẆHt,x is now an ordinary product. We first introduce a notion of solution using the Stratonovich integral and a weak formulation of the mild solution. Given a random field v = {vt,x, t ≥ 0, x ∈ R} such that |vt,x| dxdt < ∞ a.s. for all T > 0, the Stratonovich integral ∫ T vt,xdW is defined as the following limit in probability if it exists vt,xẆ t,x dxdt, where W t,x is the approximation of the noise W H introduced in (5.1). Definition 6.1 A random field u = {ut,x, t ≥ 0, x ∈ R} is a weak solution to Equation (6.1) if for any C∞ function ϕ with compact support on R, we have ut,xϕ(x)dx = u0(x)ϕ(x)dx+ us,xϕ ′′(x)dxds+ us,xϕ(x)dW Consider the approximating stochastic heat equation ∂uε,δ ∆uε,δ + uε,δẆ t,x . (6.2) Theorem 6.2 Suppose that H > 3 . For any p ≥ 2, the limit t,x = ut,x exists in Lp, and defines a weak solution to Equation (6.2) in the sense of Definition 6.1. Furthermore, for any positive integer k ukt,x UB0 (t, x) exp i,j=1 φ(s1, s2)δ(B −Bjs2)ds1ds2 where UB0 (t, x) has been defined in Theorem (5.3). Proof By Feynman-Kac’s formula we can write t,x = E u0(x+Bt) exp Aε,δr,ydW , (6.3) where Aε,δr,y has been defined in (5.6). We will first show that for all k ≥ 1 [∣∣∣uε,δt,x <∞. (6.4) Suppose that Bi = Bit , t ≥ 0 , i = 1, . . . , k are independent standard Brownian motions starting at 0, independent of WH .Then, we have, as in the proof of Theorem 5.3 = E B  exp i,j=1 Aε,δ,B , Aε,δ,B UB0 (t, x) Notice that Aε,δB , Aε,δB ηδ(s1, s2)p2ε(B −Bjs2)ds1ds2, where ηδ(s1, s2) satisfies (5.12). As a consequence, the inequalities (3.8) and (3.15) imply that for all λ > 0, and all i,j we have Aε,δB , Aε,δB Thus, (6.4) holds, and = EB exp UB0 (t, x) exp i,j=1 φ(s1, s2)δ0(B −Bjs2)ds1ds2 In a similar way we can show that the limit limε,ε′↓0 limδ,δ′↓0E ε′,δ′ ists. Therefore, the iterated limit limε↓0 limδ↓0E exists in L2. Finally we need to show that us,xϕ(x)dW s,x − uε,δs,xϕ(x)Ẇ s,xdsdx in probability. We know that uε,δs,xϕ(x)Ẇ s,xdsdx converges in L 2 to some random variable G. Hence, if Bε,δ = uε,δs,x − us,x ϕ(x)ẆHs,xdsdx (6.5) converges in L2 to zero, us,xϕ(x) will be Stratonovich integrable and us,xϕ(x)Ẇ s,xdsdx = G. The convergence to zero of (6.5) is done as follows. First we remark that Bε,δ = δ(φε,δ), where φε,δr,z = uε,δs,x − us,x ϕ(x)ϕδ(s− r)pε(x− z)dsdx. Then, from the properties of the divergence operator, it suffices to show that (∥∥Dφε,δ H1⊗H1 = 0. (6.6) It is clear that limε↓0 limδ↓0E (∥∥φε,δ = 0. On the other hand, φε,δr,z uε,δs,x −D (us,x) ϕ(x)ϕδ(s− r)pε(x − z)dsdx, uε,δs,x = E B u0(x+Bt) exp Aε,δs,ydW Then, as before we can show that ε,ε′↓0 δ.δ′↓0 uε,δs,x = E B u0(x+B1t )u0(x+B t ) exp i,j=1 φ(s1, s2)δ0(B −Bjs2)ds1ds2 φ(s1, s2)δ0(B −B2s2)ds1ds2 This implies that uε,δs,x converges in the space D 1,2 to us,x as δ ↓ 0 and ε ↓ 0. Actually, the limit is in the norm of the space D1,2(H1). Then, (6.6) follows easily. Since the solution is square integrable it admits a Wiener-Itô chaos expan- sion. The explicit form of the Wiener chaos coefficients are given below. Theorem 6.3 The solution to (6.1) is given by ut,x = In(fn(·, t, x)) (6.7) where fn(t1, x1, . . . , tn, xn, t, x) u0(x+Bt) exp φ(s1, s2)δ0(Bs1 −Bs2)ds1ds2 ×δ0(Bt1 + x− x1) · · · δ0(Btn + x− xn)] . (6.8) Proof From the Feynman-Kac formula it follows that t,x = E u0(x +Bt) exp Aε,δr,ydW u0(x +Bt) exp ‖Aε,‖2H1 Aε,δr,ydW r,y − ‖Aε,δ‖2H1 n (t, x)), where f ε,δn (t1, x1, . . . , tn, xn, t, x) = E u0(x+Bt) exp ‖Aε,δ‖2H1 t1,x1 · · ·Aε,δtn,xn Letting δ and ε go to 0, we obtain the chaos expansion of ut,x. Consider the stochastic partial differential equation (6.1) and its approxima- tion (6.2). The initial condition is u0(x). We shall study the strict positivity of the solution. In particular we shall show that E [|ut(x)|−p] <∞. Theorem 6.4 Let H > 3/4. If E (|u0(Bt)|) > 0, then for any 0 < p < ∞, we have that |ut,x|−p <∞ (6.9) and moreover, |ut(x)|−p ≤ (E|u0(x +Bt)|)−p−1E B |u0(x+Bt)| × exp δ(Bs1 −Bs2)φ(s1, s2)ds1ds2 . (6.10) Proof Denote κp = E B (|u0(x+Bt)|) )−p−1 . Then, Jensen’s inequality ap- plied to the equality u t,x = E u0(x +Bt) exp Aε,δr,ydW implies |uε,δt,x|−p ≤ κpE B |u0(x +Bt)| exp Aε,δr,ydW Therefore |uε,δt,x|−p ≤ κpE B |u0(x+Bt)|E Aε,δr,ydW = κpE |u0(x+Bt)|E ∥∥Aε,δ and we can conclude as in the proof of Theorem 6.2. Using the theory of rough path analysis (see [8]) and p-variation estimates, Gubinelli, Lejay and Tindel [4] have proved that for H > 3 , the equation ∆u+ σ(u)ẆHt,x had a unique mild solution up to a random explosion time T > 0, provided σ ∈ C2b (R). In this sense, the restriction H > 34 , that we found in the case σ(x) = x is natural, and in this particular case, using chaos expansion and Feynman-Kac’s formula we have been able to show the existence of a solution for all times. Acknowledgment. We thank Carl Mueller for discussions. References [1] R. F. Bass and Xia Chen: Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm. Ann. Probab. 32 (2004) 3221-3247. [2] R. Buckdahn and D. Nualart: Linear stochastic differential equations and Wick products. Probab. Theory Related Fields 99 (1994) 501–526. [3] T. E. Duncan, B.Maslowski and B. Pasik-Duncan: Fractional Brownian Motion and Stochastic Equations in Hilbert Spaces. Stochastics and Dy- namics 2 (2002) 225-250. [4] M. Gubinelli, A. Lejay and S. Tindel: Young integrals and SPDE. To appear in Potential Analysis, 2006. [5] T. Hida, H. H. Kuo, J. Potthoff, and L. Streit: White noise. An infinite- dimensional calculus. Mathematics and its Applications, 253. Kluwer Aca- demic Publishers Group, Dordrecht, 1993. [6] Y. Hu: Heat equation with fractional white noise potentials. Appl. Math. Optim. 43 (2001) 221-243. [7] Le Gall, J.-F. Exponential moments for the renormalized self-intersection local time of planar Brownian motion. Séminaire de Probabilités, XXVIII, 172–180, Lecture Notes in Math., 1583, Springer, Berlin, 1994. [8] T. Lyons and Z. Qian: System control and rough paths. Oxford Mathemat- ical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2002. [9] B. Maslowski and D. Nualart: Evolution equations driven by a fractional Brownian motion. Journal of Functional Analysis. 202 (2003) 277-305. [10] J. Memin, Y. Mishura and E. Valkeila: Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51 (2001) 197–206. [11] D. Nualart: The Malliavin Calculus and related topics. 2nd edition. Springer-Verlag 2006. [12] D. Nualart and B. Rozovskii: Weighted stochastic Sobolev spaces and bi- linear SPDEs driven by space-time white noise. J. Funct. Anal. 149 (1997) 200–225. [13] D. Nualart and M. Zakai: Generalized Brownian functionals and the solu- tion to a stochastic partial differential equation. J. Funct. Anal. 84 (1989) 279–296 [14] V. Pipiras and M. Taqqu: Integration questions related to fractional Brow- nian motion. Probab. Theory Related Fields 118 (2000) 251–291. [15] S. Tindel, C. Tudor and F. Viens: Stochastic evolution equations with fractional Brownian motion. Probab. Theory Related Fields 127 (2003) 186– [16] C. Tudor: Fractional bilinear stochastic equations with the drift in the first fractional chaos. Stochastic Anal. Appl. 22 (2004) 1209–1233. [17] J. B. Walsh: An introduction to stochastic partial differential equations. In: Ecole d’Ete de Probabilites de Saint Flour XIV, Lecture Notes in Mathematics 1180 (1986) 265-438. [18] S. Watanabe: Lectures on stochastic differential equations and Malliavin calculus. Published for the Tata Institute of Fundamental Research, Bom- bay; by Springer-Verlag, Berlin, 1984. Introduction Preliminaries Weighted intersection local times for standard Brownian motions Stochastic heat equation in the Itô-Skorohod sense Case H<12 and d=1 Moments of the solution Pathwise heat equation

0704.1825

Extended envelopes around Galactic Cepheids III. Y Oph and alpha Per from near-infrared interferometry with CHARA/FLUOR

Extended envelopes around Galactic Cepheids III. Y Oph and α Per from near-infrared interferometry with CHARA/FLUOR Antoine Mérand Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3965, Atlanta, Georgia 30302-3965, USA [email protected] Jason P. Aufdenberg Embry-Riddle Aeronautical University, Physical Sciences Department, 600 S. Clyde Morris Blvd, Daytona Beach, FL 32114, USA Pierre Kervella and Vincent Coudé du Foresto LESIA, UMR 8109, Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon, France Theo A. ten Brummelaar, Harold A. McAlister, Laszlo Sturmann, Judit Sturmann and Nils H. Turner Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3965, Atlanta, Georgia 30302-3965, USA ABSTRACT Unbiased angular diameter measurements are required for accurate distances to Cepheids using the interferometric Baade Wesselink method (IBWM). The precision of this technique is currently limited by interferometric measurements at the 1.5% level. At this level, the center-to-limb darkening (CLD) and the presence of circumstellar envelopes (CSE) seem to be the two main sources of bias. The observations we performed aim at improving our knowledge of the interferometric visibility profile of Cepheids. In particular, we assess the systematic presence of CSE around Cepheids in order determine accurate distances with the IBWM free from CSE biased angular diameters. We observed a Cepheid (Y Oph) for which the pulsation is well resolved and a non-pulsating yellow supergiant (α Per) using long-baseline near-infrared interferometry. We interpreted these data using a simple CSE model we previously developed. We found that our observations of α Per do not provide evidence for a CSE. The measured CLD is explained by an hydrostatic photospheric model. Our observations of Y Oph, when compared http://arxiv.org/abs/0704.1825v1 – 2 – to smaller baseline measurements, suggest that it is surrounded by a CSE with similar characteristics to CSE found previously around other Cepheids. We have determined the distance to Y Oph to be d = 491 ± 18 pc. Additional evidence points toward the conclusion that most Cepheids are surrounded by faint CSE, detected by near infrared interferometry: after observing four Cepheids, all show evidence for a CSE. Our CSE non-detection around a non-pulsating supergiant in the instability strip, α Per, provides confidence in the detection technique and suggests a pulsation driven mass-loss mechanism for the Cepheids. Subject headings: stars: variables: Cepheid - stars: circumstellar matter - stars: individual (Y Oph) - stars: individual (α Per) - techniques: interferometric 1. Introduction In our two previous papers, (Kervella et al. 2006; Mérand et al. 2006b), hereafter Paper I and Paper II, we reported the discovery of faint circumstellar envelops (CSE) around Galactic classical Cepheids. Interestingly, all the Cepheids we observed (ℓ Car in Paper I, α UMi, and δ Cep in Paper II) were found to harbor CSE with similar characteristics: a CSE 3 to 4 times larger than the star which accounts for a few percent of the total flux in the infrared K band. The presence of CSE was discovered in our attempt to improve our knowledge of Cepheids in the context of distance determination via the interferometric Baade-Wesselink method (IBWM). Part of the method requires the measurement of the angular diameter variation of the star during its pulsation. The determination of the angular diameters from sparse interferometric measurements is not straightforward because optical interferometers gather high angular resolution data only at a few baselines at a time, thus good phase and angular resolution coverage cannot be achieved in a short time. For Cepheids, the main uncertainty in the IBWM was thought to be the center-to-limb darkening (CLD), which biases the interferometric angular diameter measurements (Marengo et al. 2004). The direct measurement of CLD is possible using an optical interferometer, given suffi- cient angular resolution and precision. Among current optical interferometers, CHARA/FLUOR (ten Brummelaar et al. 2005; Mérand et al. 2006a) is one of the few capable of such a mea- surement for Cepheids. The only Cepheid accessible to CHARA/FLUOR, i.e. large enough in angular diameter, for such a measurement is Polaris (α UMi), which we observed and found to have a CLD compatible with hydrostatic photospheric models, though surrounded by a CSE (Paper II). Polaris, however, is a very low amplitude pulsation Cepheid: 0.4% in diameter, compared to 15 to 20% for type I Cepheids (Moskalik & Gorynya 2005), thus the agreement is not necessarily expected for large amplitude Cepheids, whose photospheres – 3 – are more out of equilibrium. The direct measurement of CLD of a high amplitude Cepheid during its pulsation phase remains to be performed. Hydrodynamic simulations (Marengo et al. 2003) suggest that the CLD variations dur- ing the pulsation do not account for more than a 0.2% bias in distance determination in the near infrared using the IBWM, where most of the IBWM observing work has been done in recent years: the best formal distance determination to date using the IBWM is of the order of 1.5% (Mérand et al. 2005b). Whereas the near infrared IBWM seems to be relatively immune to bias from CLD, the recent discovery of CSEs raises the issue of possible bias in angular diameter measurements, hence bias in distance estimations at the 10% level (Paper II). It is therefore important to continue the study of CSE around Cepheids. We present here interferometric observa- tions of the non-pulsating supergiant α Per and the low amplitude Cepheid Y Oph. We obtained these results in the near infrared K-band, using the Fiber Linked Unit for Optical Recombination — FLUOR — (Mérand et al. 2006a), installed at Georgia State University’s Center for High Angular Resolution Astronomy (CHARA) Array located on Mount Wilson, California (ten Brummelaar et al. 2005). 2. The low amplitude Cepheid Y Oph In the General Catalog of Variable Stars (Kholopov et al. 1998), Y Oph is classified in the DCEPS category, i.e. low amplitude Cepheids with almost symmetrical light curves and with periods less than 7 days. The GCVS definition adds that DCEPS are first overtone and/or crossing the instability strip for the first time. A decrease in photometric variation amplitude over time has been measured, as well as a period change (Fernie et al. 1995b). Using this period change rate, 7.2± 1.5 syr−1 and the period of 17.1207 days, the star can be identified as crossing the instability strip for the third time, according to models (Turner et al. 2006). The fact that Y Oph belongs to the DCEPS category is questionable: its period is longer than 7 days, by almost three times, though its light curve is quasi-symmetric and with a low amplitude compared to other type I Cepheids of similar periods (Vinko et al. 1998). Indeed, Y Oph is almost equally referred to in publications as being a fundamental-mode Cepheid or a first overtone. In this context, a direct determination of the linear diameter can settle whether Y Oph belongs to the fundamental mode group or not. This is of prime importance: because of its brightness and the large amount of observational data available, Y Oph is often used to cali- brate the Period-Luminosity (PL) or the Period-Radius (PR) relations. The interferometric – 4 – Baade-Wesselink method offers this opportunity to geometrically measure the average linear radius of pulsating stars: if Y Oph is not a fundamental pulsator, its average linear diameter should depart from the classical PR relation. 2.1. Interferometric observations The direct detection of angular diameter variations of a pulsating star has been achieved for many stars now using optical interferometers (Lane et al. 2000; Kervella et al. 2004a; Mérand et al. 2005b). We showed (Mérand et al. 2006b) that for a given average diameter, one should use a baseline that maximizes the amplification factor between the variation in angular diameter and observed squared visibility. This baseline provides an angular resolu- tion of the order of Bθ/λ ≈ 1, in other words in the first lobe, just before the first minimum (Bθ/λ ≈ 1.22 for a uniform disk model), where B is the baseline (in meters), θ the angular diameter (in radians) and λ the wavelength of observation (in meters). According to pre- vious interferometric measurements (Kervella et al. 2004a), the average angular diameter of Y Oph is of the order of 1.45 mas (milli arcsecond). Ideally, that would mean using a baseline of the order of 300 m, which is available at the CHARA Array. Because of a trade we made with other observing programs, we used only a 250 m baseline provided by telescopes S1 and E2. The fringes squared visibility is estimated using the integration of the fringes power spectrum. A full description of the algorithm can be found in Coude Du Foresto et al. (1997) and Mérand et al. (2006a). The raw squared visibilities have been calibrated using resolved calibrator stars, chosen from a specific catalog (Mérand et al. 2005a) using criteria defined to minimize the calibra- tion bias and maximize signal to noise. The error introduced by the uncertainty on each calibrator’s estimated angular diameter has been properly propagated. Among the three main calibrators (Tab. 1), one, HR 6639, turned out to be inconsistent with the others. The raw visibilility of this star was found to vary too much to be consistent with the expected statistical dispersion. The quantity to calibrate, the interferometric efficiency (also called instrument visibility), is very stable for an instrument using single mode fibers, such as FLUOR. If this quantity is assumed to be constant over a long period of time, and if obser- vations of a given simple star are performed several times during this period, one can check whether or not the variation of the raw visibilities with respect to the projected baseline is consistent with a uniform disk model. Doing so, HR 6639 was found inconsistent with the over stars observed during the same night (Fig 1). The unconsistency may be explained – 5 – by the presence of a faint companion with a magnitude difference of 3 or 4 with respect to the primary. Two over calibrators, from another program, were also used as check stars: HR 7809 and ρ Aql (Tab. 1). This latter calibrator is not part of the catalog by Mérand et al. (2005a). Its angular diameter has been determined using the Kervella et al. (2004b) surface brightness calibration applied to published photometric data in the visible and near infrared. For each night we observed Y Oph, we determined a uniform disk diameter (Tab. 3) based on several squared visibility measurements (Tab. 2). Each night was assigned a unique pulsation phase established using the average date of observation and the Fernie et al. (1995b) ephemeris, including the measured period change: D = JD − 2440007.720 (1) E = 0.05839D − 3.865× 10−10D2 (2) P = 17.12507 + 3.88× 10−6E (3) where E is the epoch is the epoch of maximum light (the fractional part is the pulsation phase) and P the period at this epoch. 2.2. Pulsation 2.2.1. Radial Velocity integration In order to measure the distance to a pulsating star, the IBWM makes use of radial velocities and angular diameters. The latter is the integral other time of the former. The radial velocities, which have been acquired at irregular intervals during the pulsation phase, must be numerically integrated. This process is highly sensitive to noisy data and the best way to achieve a robust integration is to interpolate the data before integration. For this purpose, we use a periodic cubic spline function, defined by floating nodes (Fig. 2). The coordinates of these nodes are adjusted such that the cubic spline function going through these nodes provides the best fit to the data points. The phase positions φi of these nodes are forced to be between 0 and 1, but they are replicated every φi+n, where n is an integer, in order to obtain a periodic function of period 1. Among published Y Oph radial velocities data, we chose Gorynya et al. (1998) because of the uniform phase coverage and the algorithm used to extract radial velocities: the cross- correlation method. As shown by Nardetto et al. (2004), the method used can influence the distance determination via the choice of the so-called projection factor, which we shall introduce in the following section. The pulsation phases have been also determined using Eq. 2. – 6 – The data presented by Gorynya et al. (1998) were acquired between June 1996 and August 1997. As we already mentioned, Y Oph is known for its changing period and photo- metric amplitude. Based on Fernie et al. (1995b), the decrease in amplitude observed for the photometric B and V bands does not have a measurable counterpart in radial velocity. This is why we did not apply any correction in amplitude to the radial velocity data in order to take into account the ten years between the spectroscopic and interferometric measurements. 2.2.2. Distance determination method Once radial velocities vrad are interpolated (Fig. 2) and integrated, the distance d is determined by fitting the radial displacement to the measured angular diameters (Fig. 3): θUD(T )− θUD(0) = −2 vrad(t)dt (4) where θUD is the interferometric uniform disk diameter, and k is defined as the ratio between θUD and the true stellar angular diameter. The projection factor, p, is the ratio between the pulsation velocity and the spectroscopically measured radial velocity. The actual parameters of the fit are the average angular diameter θUD(0) and the biased distance This formalism assumes that both k and p do not vary during the pulsation. There is evidence that this might be true for k, based on hydrodynamic simulation (Marengo et al. 2003), at the 0.2% level. Observational evidence exists as well: when we measured the p- factor of δ Cep (Mérand et al. 2005b) we did not find any difference between the shapes of the left and right parts of Eq. 4, therefore kp is probably constant other a pulsation period, at least at the level of precision we have available. For this work, we will adopt the value for p we determined observationally for near infrared interferometry/ cross correlation radial velocity: p = 1.27. This result has been established for δ Cep (Mérand et al. 2005b). This is also the latest value computed from hydrodynamical photospheric models (Nardetto et al. 2004). The IBWM fit yields a biased distance d/k = 480± 18 pc and an average angular uniform disk diameter θUD(0) = 1.314± 0.005 mas. Note that we had to allow a phase shift between interferometric and radial velocity observations: −0.074 ± 0.005 (Fig. 3). The final reduced χ2 is of the order of 3, mostly due to one data point (φ = 0.887). – 7 – 2.2.3. Choice of k Usually, the choice of k is made assuming the star is a limb-darkened disk. The strength of the CLD is computed using photospheric models, then a value of k is computed. This approach is sometimes confusing because, even for a simple limb darkened disk, there is no unique value of k, in the sense that this value varies with respect to angular resolution. The uniform disk angular size depends upon which portion of the visibility curve is measured. However, it is mostly unambiguous in the first lobe of visibility, i.e. at moderate angular resolution: Bθ/λ ≤ 1. However, as shown in Paper II, the presence of a faint CSE around Cepheids biases k up to 10%, particularly when the angular resolution is moderate and the star is not well resolved (V 2 ∼ 0.5). Under these conditions, the CSE is largely resolved, leading to a strong bias if the CSE is omitted. On the other hand, at greater angular resolution (Bθ/λ ∼ 1), the star is fully resolved (V 2 approaches it first null) and the bias from the CSE is minimized. In any cases, it is critical to determine whether or not Y Oph is surrounded by a CSE if an accurate distance is to be derived. 2.3. Interferometric evidence of a CSE around Y Oph We propose here to compare the uniform disk diameters obtained by VLTI/VINCI (Kervella et al. 2004a) and CHARA/FLUOR (this work). This makes sense because these two instruments are very similar. Both observe in the near infrared K-band. Moreover, both instruments observed Y Oph in the first lobe of the visibility profile, though at different baselines. If Y Oph is truly a uniform disk, or even a limb-darkened disk, the two instruments should give similar results. That is because the star’s first lobe of squared visibility is insen- sitive to the CLD and only dependent the on the size. Conversely, if Y Oph is surrounded by a CSE, we expect a visibility deficit at smaller baseline (VLTI/VINCI), hence a larger apparent uniform disk diameter (see Fig. 5). This is because the CSE is fully resolved at baselines which barely resolve the star. This is indeed the case, as seen on Fig. 4: VLTI/VINCI UD diameters are larger than CHARA/FLUOR’s. Even if the angular resolution of VLTI/VINCI is smaller than for CHARA/FLUOR, leading to less precise angular diameter estimations, the disagreement is still statistically significant, of the order of 3 sigmas. Using the CSE model we can predict the correct differential UD size between the two instruments, consistant with the presence of a CSE around Y Oph. The amount of discrepancy can be used to estimate the flux ra- – 8 – tio between the CSE and the star. In the case of Y Oph, we find that the CSE amounts for 5 ± 2% of the stellar flux. Note that for this comparison, we recomputed the phase of VLTI/VINCI data using Fernie et al. (1995b) ephemeris presented in Eq. 2. In Fig. 5, we plot k as a function of the observed squared visibility for different models: hydrostatic CLD, 2% CSE and 5% CSE (K-Band flux ratio). For the hydrostatic model we have k = 0.983. For the 5% CSE models, for CHARA/FLUOR (0.20 < V 2 < 0.35), θUD/θ⋆ = 1.023. This is the value we shall adopt. If we ignore the presence of the CSE, the bias introduced is 1.023/0.983 ≈ 1.04, or 4%. 2.4. Unbiased distance and linear radius This presence of a 5% CSE leads to an unbiased distance of d = 491 ± 18 pc, which corresponds to a 3.5% uncertainty on the distance. This is to be compared with the bias we corrected for if one omits the CSE, of the order of 4%. Ignoring the CSE leads to a distance of d = 472± 18 pc We note that k biases only the distance, so one can form the following quantity: [θUD(0)]× [d/k], which is the product of the two adjusted parameters in the fit, both biased. This quantity is by definition the linear diameter of the star, and does not depend on the factor k, even if it is still biased by the choice of p. If θ is in mas and d in parsecs, then the average linear radius in solar radii is: R = 0.1075θd. In the case of Y Oph, this leads to R = 67.8± 2.5 R⊙. 2.5. Conclusion YOph appears larger (over 2 sigma) in the infrared K-band at a 140 m baseline comapred to a 230 m baseline. Using a model of a star surrounded by a CSE we developed based on observations of other Cepheids, this disagreement is explained both qualitatively and quantitatively by a CSE accounting of 5% of the stellar flux in the near infrared K-Band. This model allows us to unbias the distance estimate: d = 491 ± 18 pc. The linear radius estimate is not biased by the presence of CSE and we found R = 67.8± 2.5 R⊙. Our distance is not consistent with the estimation using the Barnes-Evans method: Barnes et al. (2005) found d = 590 ± 42 pc (Bayesian) and d = 573 ± 8 pc (least squares). For this work, they used the same set of radial velocities we used. Our estimate is even more inconsistent with the other available interferometric estimate, by Kervella et al. (2004a): d = 690±50 pc. This later result has been established using an hybrid version of the BWmethod: – 9 – a value of the linear radius is estimated using the Period-Radius relation calibrated using classical Cepheids, not measured from the data. This assumption is questionable, as we noted before, since Y Oph is a low amplitude Cepheid. Kervella et al. (2004a) deduced R = 100± 8 R⊙ from PR relation, whereas we measured R = 67.8±2.5 R⊙. Because Y Oph’s measured linear radius is not consistent with the PR relation for classical, fundamental mode Cepheids, it is probably safe to exclude it from further calibrations. Interestingly, Barnes et al. (2005) observationally determined Y Oph’s linear radius to be also slightly larger (2.5 sigma) than what we find: R = 92.1±6.6 R⊙ (Bayesian) and R = 89.5±1.2 R⊙ (least squares). They use a surface brightness technique using a visible-near infrared color, such as V-K. This method is biased if the reddening is not well known. If the star is reddened, V magnitudes are increased more than K-band magnitudes. This leads to an underestimated surface brightness, because the star appears redder, thus cooler than it is. The total brightness (estimated from V) is also underestimated. These two underestimates have opposing effects on the estimated angular diameter: an underestimated surface brigthness leads to an overestimated angular diameter, whereas an underestimated luminosity leads to an underestimated angular diameter. In the case of a reddening law, the two effects combine and give rise to a larger angular diameter: the surface brightness effect wins over the total luminosity. Based on their angular diameter, θ ≈ 1.45 mas, it appears that Barnes et al. (2005) overestimated Y Oph’s angular size. Among Cepheids brighter than mV = 6.0, Y Oph has the strongest B-V color excess, E(B − V ) = 0.645 (Fernie et al. 1995a) and one of the highest fraction of polarized light, p = 1.34±0.12 % (Heiles 2000). Indeed, Y Oph is within the galactic plane: this means that it has a probably a large extinction due to the interstellar medium. 3. The non-pulsating yellow super giant α Per α Per (HR 1017, HD 20902, F5Ib) is among the most luminous non-pulsating stars inside the Cepheids’ instability strip. The Doppler velocity variability has been found to be very weak, of the order of 100 m/s in amplitude (Butler 1998). This amplitude is ten times less than what is observed for the very low amplitude Cepheid Polaris (Hatzes & Cochran 2000). α Per’s apparent angular size, approximately 3 mas (Nordgren et al. 2001), makes it a perfect candidate for direct center-to-limb darkening detection with CHARA/FLUOR. Following the approach we used for Polaris (Paper II), we observed α Per using three different baselines, including one sampling the second lobe, in order to measure its CLD strength, but also in order to be able to assess the possible presence of a CSE around this star. – 10 – 3.1. Interferometric observations If the star is purely a CLD disk, then only two baselines are required to measure the angular diameter and the CLD strength. Observations must take place in the first lobe of the squared visibility profile, in order to set the star’s angular diameter θ. The first lobe is defined by BθUD/λ < 1.22. Additional observations should taken in the second lobe, in particular near the local maximum (Bθ/λ ∼ 3/2), because the strength of the CLD is directly linked to the height of the second lobe. To address the presence of a CSE, observations should be made at a small baseline. Because the CSEs that were found around Cepheids are roughly 3 times larger that the star itself (Paper I and II), we chose a small baseline where Bθ/λ ∼ 1/3. As demonstrated by our Polaris measurements, the presence of CSE is expected to weaken the second lobe of visibility curve, mimicking stronger CLD. FLUOR operates in the near infrared K-band, with a mean wavelength of λ0 ≈ 2.13 µm. This sets the small, first-lobe and second-lobe baselines at approximatively 50, 150 and 220 meters, which are well-matched to CHARA baselines W1-W2 (∼ 100 m), E2-W2 (∼ 150 m) and E2-W1 (∼ 250 m). See Fig. 6 for a graphical representation of the baseline coverage. The data reduction was performed using the same pipeline as for Y Oph. Squared visibilities were calibrated using a similar strategy we adopted for Y Oph. We used two sets of calibrators: one for the shorter baselines, W1-W2 and E2-W2, and one for the longest baseline, E2-W1 (Tab. 4). 3.2. Simple Model 3.2.1. Limb darkened disk To probe the shape of the measured visibility profile, we first used an analytical model which includes the stellar diameter and a CLD law. Because of its versatility, we adopt here a power law (Hestroffer 1997): I(µ)/I(0) = µα, with µ = cos(θ), where θ is the angle between the stellar surface and the line of sight. The uniform disk diameter case corresponds to α = 0.0, whereas an increasing value of α corresponds to a stronger CLD. All visibility models for this work have been computed taking into account instrumental bandwidth smearing (Aufdenberg et al. 2006). From this two-parameter fit, we deduce a stellar diameter of θα = 3.130 ± 0.007 mas and a power law coefficient α = 0.119 ± 0.016. The fit yields a reduced χ2 = 1.0. There is a strong correlation between the diameter and CLD coefficent: 0.9. This is a well known effect, that a change in CLD induces a change in apparent diameter. – 11 – This fit is entirely satisfactory, as the reduced χ2 is of the order unity and the residuals of the fit do not show any trend (Fig. 7). We note that this is not the case for Polaris (Paper II). Polaris, with very similar u-v coverage to α Per, does not follow a simple LD disk model because it is surrounded by a faint CSE. 3.2.2. Hydrostatic models We computed a small grid of models which consists of six one-dimensional, spherical, hydrostatic models using the PHOENIX general-purpose stellar and planetary atmosphere code version 13.11; for a description see Hauschildt et al. (1999) and Hauschildt & Baron (1999). The range of effective temperatures and surface gravities is based on a summary of α Per’s parameters by Evans et al. (1996): • effective temperatures, Teff = 6150 K, 6270 K, 6390 K; • log(g) = 1.4, 1.7; • radius, R = 3.92× 1012 cm; • depth-independent micro-turbulence, ξ = 4.0 km s−1; • mixing-length to pressure scale ratio, 1.5; • solar elemental abundance LTE atomic and molecular line blanketing, typically 106 atomic lines and 3× 105 molecular lines dynamically selected at run time; • non-LTE line blanketing of H I (30 levels, 435 bound-bound transitions), He I (19 levels, 171 bound-bound transitions), and He II (10 levels, 45 bound-bound transitions); • boundary conditions: outer pressure, 10−4 dynes cm−2, extinction optical depth at 1.2 µm: outer 10−10, inner 102. For this grid of models the atmospheric structure is computed at 50 radial shells (depths) and the radiative transfer is computed along 50 rays tangent to these shells and 14 additional rays which impact the innermost shell, the so-called core-intersecting rays. The intersection of the rays with the outermost radial shell describes a center-to-limb intensity profile with 64 angular points. Power law fits to the hydrostatic model visibilities range from α = 0.132 to α = 0.137 (Tab. 6), which correspond to 0.8 to 1.1 sigma above the value we measured. Our measured – 12 – CLD is below that predicted by the the models, or in other words the predicted darkening is slightly overestimated. 3.3. Possible presence of CSE Firstly, we employ the CSE model used for Cepheids (Paper I, Paper II). This model is purely geometrical and it consists of a limb darkened disk surrounded by a faint CSE whose two-dimensional geometric projection (on the sky) is modeled by a ring. The ring parameters consist of the angular diameter, width and flux ratio with respect to the star (Fs/F⋆). We adopt the same geometry found for Cepheids: a ring with a diameter equal to 2.6 times the stellar diameter (Paper II) with no dependency with respect to the width (fixed to a small fraction of the CSE angular diameter, say 1/5). This model restriction is justified because testing the agreement between a generic CSE model and interferometric data requires a complete angular resolution coverage, from very short to very large baselines. Though our α Per data set is diverse regarding the baseline coverage, we lack a very short baseline (B ∼ 50m) which was not available at that time at the CHARA Array. The simplest fit consists in adjusting the geometrical scale: the angular size ratio be- tween the CSE and the star is fixed. This yields a reduced χ2 of 3, compared to 1.1 for a hydrostatic LD and 1.0 for an adjusted CLD law (Tab. 7). We can force the data to fit a CSE model by relaxing the CLD of the star or the CSE flux ratio. A fit of the size of the star and the brightness of the CSE leads to a reduced χ2 of 1.4 and results in a very small flux ratio between the CSE and the star of 0.0006± 0.0026, which provides an upper limit for the CSE flux of 0.26% (compared to 1.5% for Polaris and δ Cep for example, and 5% for Y Oph). On the other hand, forcing the flux ratio to match that for Cepheids with CSEs and leaving the CLD free leads to a reduced χ2 of 1.4 but also to a highly unrealistic α = 0.066± 0.004 (Tab. 7 and Fig. 7). This leads to the conclusion that the presence of a CSE around α Per similar to the measured Cepheid CSE is highly improbable. As shown above, the measured CLD is slightly weaker (1 sigma) than one predicted by model atmospheres. A compatible CSE model exists if the CLD is actually even weaker but in unrealistic proportions. – 13 – 3.4. Conclusion The observed α Per visibilities are not compatible with the presence of a CSE similar to that detected around Cepheids. The data are well explained by an adjusted center-to-limb darkening. The strength of this CLD is compatible with the hydrostatic model within one sigma. 4. Discussion Using the interferometric Baade-Wesselink method, we determined the distance to the low amplitude Cepheid Y Oph to be d = 491± 18 pc. This distance has been unbiased from the presence of the circumstellar envelope. This bias was found to be of the order of 4% for our particular angular resolution and the amount of CSE we measured. Y Oph’s average linear diameter has also been estimated to be R = 67.8 ± 2.5 R⊙. This latter quantity in intrinsically unbiased by the center-to-limb darkening or the presence of a circumstellar envelope. This value is found to be substantially lower, almost 4 sigmas, than the Period- Radius relation prediction: R = 100 ± 8 R⊙. Among other peculiarities, we found a very large phase shift between radial velocity measurements and interferometric measurements: ∆φ = −0.074 ± 0.005, which corresponds to more that one day. For these reasons, we think Y Oph should be excluded from future calibrations of popular relations for classical Cepheids, such as Period-Radius, Period-Luminosity relations. So far, the four Cepheids we looked at have a CSE: ℓ Car (Paper I), Polaris, δ Cep (Paper II) and Y Oph (this work). On the other hand, we have presented here similar observations of α Per, a non-pulsating supergiant inside the instability strip, which provides no evidence for a circumstellar envelope. This non detection reinforces the confidence we have in our CSE detection technique and draws more attention to a pulsation driven mass loss mechanism. Interestingly, there seems to be a correlation between the period and the CSE flux ratio in theK-Band: short-period Cepheids seem to have a fainter CSE compared to longer-period Cepheids (Tab. 8). Cepheids with long periods have higher masses and larger radii, thus, if we assume that the CSE K-Band brightness is an indicator of the mass loss rate, this would mean that heavier stars experience higher mass loss rates. This is of prime importance in the context of calibrating relations between Cepheids’ fundamental parameters with respect to their pulsation periods. If CSEs have some effects on the observational estimation of these fundamental parameters (luminosity, mass, radius, etc.), a correlation between the period and the relative flux of the CSE can lead to a biased calibration. – 14 – Research conducted at the CHARA Array is supported at Georgia State University by the offices of the Dean of the College of Arts and Sciences and the Vice President for Research. Additional support for this work has been provided by the National Science Foundation through grants AST 03-07562 and 06-06958. We also wish to acknowledge the support received from the W.M. Keck Foundation. This research has made use of SIMBAD database and the VizieR catalogue access tool, operated at CDS, Strasbourg, France. – 15 – REFERENCES Aufdenberg, J. P., Mérand, A., Foresto, V. C. d., Absil, O., Di Folco, E., Kervella, P., Ridgway, S. T., Berger, D. H., Brummelaar, T. A. t., McAlister, H. A., Sturmann, J., Sturmann, L., & Turner, N. H. 2006, ApJ, 645, 664 Barnes, III, T. G., Storm, J., Jefferys, W. H., Gieren, W. P., & Fouqué, P. 2005, ApJ, 631, Butler, R. P. 1998, ApJ, 494, 342 Coude Du Foresto, V., Ridgway, S., & Mariotti, J.-M. 1997, A&AS, 121, 379 Evans, N. R., Teays, T. J., Taylor, L. L., Lester, J. B., & Hindsley, R. B. 1996, AJ, 111, Fernie, J. D., Evans, N. R., Beattie, B., & Seager, S. 1995a, Informational Bulletin on Variable Stars, 4148, 1 Fernie, J. 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Calibrated squared visibility as a function of baseline for UT 2006/07/10. Left is Y Oph and right is HR 6639. The interferometric efficiency was supposed constant during the night and established using HR 7809 and ρ Aql with a reduced χ2 of 1.3. A UD fit, shown as a continuous line, works for Y Oph (χ2r = 0.6) whereas it fails for HR 6639 (χ r = 7.2). This could the sign that HR 6639 has a faint companion: the visibility variation has a 3-5% amplitude, which corresponds to a magnitude difference of 3 to 4 between the main star and its companion. – 19 – 0.0 0.5 1.0 0.0 0.5 1.0 Fig. 2.— Y Oph: Radial velocities from Gorynya et al. (1998). The continuous line is the periodic spline function defined by 3 adjustable floating nodes (large filled circles). The systematic velocity, Vγ = −7.9 ± 0.1 kms −1, has been evaluated using the interpolation function and removed. The lower panel displays the residuals of the fit. – 20 – 0.0 0.5 1.0 0.0 0.5 1.0 −0.02 Fig. 3.— Y Oph: angular diameter variations. Upper panel: CHARA/FLUOR uniform disk angular diameters as a function of phase. Each data point corresponds to a given night, which contains several individual squared visibility measurements (Tab. 3). The solid line is the integration of the radial velocity (Fig. 2) with distance, average angular diameter and phase shift adjusted. The dashed line has a phase shift set to zero. The lower panel shows the residuals to the continuous line. – 21 – CSE 2% CSE 5% 0.0 0.5 1.0 Fig. 4.— Y Oph: Comparison between CHARA/FLUOR and VLTI/VINCI observations. Uniform disk angular diameter as a function of phase. Small data points (with small error bars) and lower continuous line: CHARA/FLUOR observations and distance fit. Large open squares: VLTI/VINCI observations. The distance fit and it uncertainty are reprensenteb by the shaded band. Dashed and dotted lines: VLTI/VINCI expected biased observations based on CHARA/FLUOR and our CSE model with a flux ratio of 2% (dashed) and 5% (dotted). – 22 – CSE 5% CSE 2% 0.0 0.2 0.4 0.6 0.8 squared visibility Fig. 5.— Y Oph: angular diameter correction factor. We plot here θUD/θ⋆ for three different Cepheid models as a function of observed squared visibility (first lobe): the dashed line is a simple limb darkened disk with the appropriate CLD strength; the continuous line is a similar LD disk surrounded by a CSE with a 2% K-band flux (short period: Polaris and δ Cep, see Paper II) and 5% (long period: ℓ Car, see paper I). Note that, in the presence of CSE, the bias is stronger at large visibilities (hence smaller angular resolution). The shaded regions represents near infrared Y Oph observations: from CHARA (this work) and the VLTI (Kervella et al. 2004a). – 23 – −200 0 200 u (m) Fig. 6.— α Per. Projected baselines, in meters. North is up East is right. The shaded area corresponds to the squared visibility’s second lobe. The baselines are W1-W2, E2-W2 and E2-W1, from the shortest to the longest. – 24 – adjusted CLD CSE "Polaris" CSE (1) 50 100 150 200 250 B (m) Fig. 7.— α Per squared visibility models: UD, adjusted CLD, PHOENIX and 2 different CSE models are plotted here as the residuals to the PHOENIX CLD with respect to baseline. The common point at B ≈ 175 m is the first minimum of the visibility function. The top of the second lobe is reached for B ≈ 230 m. See Tab. 7 for parameters and reduced χ2 of each model. – 25 – α UMi δ Cep Y Oph l Car α Per 0 10 20 30 40 Period (days) Fig. 8.— measured relative K-band CSE fluxes (in percent) around Cepheids, as a function of the pulsation period (in days). The non-pulsating yellow supergiant α Per is plotted with – 26 – Table 1. Y Oph calibrators. Star S.T. UD Diam. Notes (mas) HD 153033 K5III 1.100± 0.015 HD 175583 K2III 1.021± 0.014 HR 6639 K0III 0.904± 0.012 rejected HR 7809 K1III 1.055± 0.015 ρ Aql A2V 0.370± 0.005 not in M05 Note. — “S.T.” stands for spectral type. Uniform Disk diameters, given in mas, are only intended for computing the expected squared visibility in the K-band. All stars but ρ Aql are from M05 catalog (Mérand et al. 2005a). Refer to text for an explanation why HR 6639 has been rejected. – 27 – Table 2. Journal of observations: Y Oph. MJD-53900 B P.A. V 2 (m) (deg) 18.260 229.808 63.540 0.2348± 0.0061 18.296 215.049 71.242 0.2823± 0.0065 18.321 207.247 77.913 0.3030± 0.0075 21.245 232.792 62.353 0.2511± 0.0065 21.283 216.576 70.243 0.2954± 0.0085 21.309 208.313 76.765 0.3194± 0.0085 22.212 245.894 58.023 0.2092± 0.0071 22.234 236.338 61.050 0.2352± 0.0078 22.251 228.941 63.902 0.2615± 0.0085 24.188 253.988 55.914 0.2108± 0.0062 24.212 243.641 58.680 0.2427± 0.0060 24.231 235.388 61.389 0.2634± 0.0065 26.168 259.377 54.714 0.2185± 0.0041 26.194 249.210 57.113 0.2526± 0.0039 26.224 235.772 61.251 0.2983± 0.0034 26.242 227.859 64.367 0.3171± 0.0035 27.194 247.788 57.495 0.2650± 0.0040 27.211 240.340 59.704 0.2854± 0.0038 27.228 232.806 62.348 0.3177± 0.0045 27.250 223.415 66.429 0.3385± 0.0045 28.241 225.978 65.207 0.3332± 0.0045 28.257 219.413 68.547 0.3629± 0.0061 29.219 234.329 61.775 0.2857± 0.0049 29.237 226.405 65.012 0.3148± 0.0053 29.253 219.709 68.380 0.3320± 0.0054 30.192 245.267 58.203 0.2510± 0.0035 30.216 234.738 61.624 0.2893± 0.0041 30.235 226.418 65.007 0.3112± 0.0044 30.244 222.544 66.866 0.3226± 0.0046 30.261 215.937 70.653 0.3462± 0.0048 – 28 – Table 2—Continued MJD-53900 B P.A. V 2 (m) (deg) Note. — Date of observation (modified julian day), telescope projected separation (m), baseline projection angle (degrees) and squared visibility – 29 – Table 3. Y Oph angular diameters. MJD-53900 φ Nobs. B θUD χ (m) (mas) 18.292 0.248 3 207-230 1.3912± 0.0067 0.55 21.279 0.423 3 208-233 1.3516± 0.0074 1.06 22.232 0.478 3 229-246 1.3387± 0.0077 0.05 24.210 0.594 3 235-254 1.2929± 0.0059 0.19 26.207 0.710 4 228-259 1.2488± 0.0029 0.61 27.221 0.770 4 223-248 1.2374± 0.0032 0.76 28.249 0.830 2 219-226 1.2394± 0.0055 0.30 29.237 0.887 3 220-234 1.2728± 0.0047 0.26 30.229 0.945 5 216-245 1.2715± 0.0030 0.56 Note. — Data points have been reduced to one uniform disk diameter per night. Avger- age date of observation (modified julian day), pulsation phase, number of calibrated V 2, projected baseline range, uniform disk angular diameter and reduced χ2. – 30 – Table 4. α Per calibrators. Name S.T. UD Diam. Baselines (mas) HD 18970 G9.5III 1.551± 0.021 W1-W2, E2-W2 HD 20762 K0II-III 0.881± 0.012 E2-W1 HD 22427 K2III-IV 0.913± 0.013 E2-W1 HD 31579 K3III 1.517± 0.021 W1-W2, E2-W2 HD 214995 K0III 0.947± 0.013 W1-W2 Note. — “S.T.” stands for spectral type. Uniform Disk diameters, given in mas, are only intended for computing the expected squared visibility in the K-band. These stars come from our catalog of interferometric calibrators (Mérand et al. 2005a) – 31 – Table 5. Journal of observations: α Per. MJD-54000 B P.A. V 2 (m) (deg) 46.281 147.82 79.37 0.02350± 0.00075 46.321 153.87 69.04 0.01368± 0.00067 46.347 155.79 62.17 0.01093± 0.00060 46.372 156.27 55.30 0.01134± 0.00039 46.398 155.65 48.10 0.01197± 0.00051 46.421 154.41 41.32 0.01367± 0.00052 46.442 152.94 34.84 0.01581± 0.00045 46.466 151.13 27.03 0.01824± 0.00064 46.488 149.58 19.40 0.02167± 0.00071 46.510 148.49 12.06 0.02250± 0.00058 46.539 147.77 1.53 0.02370± 0.00070 47.225 96.66 -44.51 0.27974± 0.00282 47.233 97.75 -47.34 0.27321± 0.00274 47.252 100.37 -53.97 0.24858± 0.00258 47.274 103.03 -60.91 0.23529± 0.00242 47.327 249.17 81.65 0.01363± 0.00032 47.352 251.22 74.84 0.01340± 0.00032 47.374 251.04 68.91 0.01330± 0.00035 47.380 250.67 67.10 0.01316± 0.00035 Note. — Date of observation (modified julian day), telescope projected separation, base- line projection angle and squared visibility – 32 – Table 6. α Per PHOENIX models. Teff log g = 1.4 log g = 1.7 6150 0.137 0.136 6270 0.135 0.134 6390 0.133 0.132 Note. — Models tabulated for different effective temperatures and surface gravities. The K-band CLD is condensed into a power law coefficient α: I(µ) ∝ µα. – 33 – Table 7. Models for α Per. Model θ⋆ α Fs/F⋆ χ (mas) (%) UD 3.080±0.004 0.000 - 5.9 PHOENIX CLD 3.137±0.004 0.135 - 1.1 adjusted CLD 3.130±0.007 0.119±0.016 - 1.0 CSE “Polaris” 3.086±0.007 0.135 1.5 3.0 CSE (1) 3.048±0.007 0.066±0.004 1.5 1.4 CSE (2) 3.095±0.010 0.135 0.06±0.26 1.4 Note. — The parameters are: θ⋆ the stellar angular diameter, the CLD power law coef- ficient α and, if relevant, the brightness ratio between the CSE and the star Fs/F⋆. The first line is the uniform disk diameter, the second one expected CLD from the PHOENIX model, the third one is the adjusted CLD. The fourth line is a scaled Polaris CSE model (Paper II). The last two lines are attempts to force a CSE model to the data. Lower script are uncertainties, the absence of lower script means that the parameter is fixed – 34 – Table 8. Relative flux in K-Band for the CSE discovered around Cepheids and the non pulsating α Per. Star Period CSE flux (d) (%) α UMi 3.97 1.5± 0.4 δ Cep 5.37 1.5± 0.4 Y Oph 17.13 5.0± 2.0 ℓ Car 35.55 4.2± 0.2 α Per - < 0.26 Introduction The low amplitude Cepheid Y Oph Interferometric observations Pulsation Radial Velocity integration Distance determination method Choice of k Interferometric evidence of a CSE around Y Oph Unbiased distance and linear radius Conclusion The non-pulsating yellow super giant Per Interferometric observations Simple Model Limb darkened disk Hydrostatic models Possible presence of CSE Conclusion Discussion

0704.1826

Squark and Gaugino Hadroproduction and Decays in Non-Minimal Flavour Violating Supersymmetry

Squark and Gaugino Hadroproduction and Decays in Non-Minimal Flavour Violating Supersymmetry Giuseppe Bozzi Institut für Theoretische Physik, Universität Karlsruhe, Postfach 6980, D-76128 Karlsruhe, Germany Benjamin Fuks, Björn Herrmann, and Michael Klasen∗ Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier/CNRS-IN2P3, 53 Avenue des Martyrs, F-38026 Grenoble, France (Dated: October 29, 2018) We present an extensive analysis of squark and gaugino hadroproduction and decays in non- minimal flavour violating supersymmetry. We employ the so-called super-CKM basis to define the possible misalignment of quark and squark rotations, and we use generalized (possibly complex) charges to define the mutual couplings of (s)quarks and gauge bosons/gauginos. The cross sections for all squark-(anti-)squark/gaugino pair and squark-gaugino associated production processes as well as their decay widths are then given in compact analytic form. For four different constrained supersymmetry breaking models with non-minimal flavour violation in the second/third generation squark sector only, we establish the parameter space regions allowed/favoured by low-energy, elec- troweak precision, and cosmological constraints and display the chirality and flavour decomposition of all up- and down-type squark mass eigenstates. Finally, we compute numerically the dependence of a representative sample of production cross sections at the LHC on the off-diagonal mass matrix elements in the experimentally allowed/favoured ranges. I. INTRODUCTION KA-TP-07-2007 LPSC 07-023 SFB/CPP-07-09 Weak scale supersymmetry (SUSY) remains a both theoretically and phenomenologically attractive extension of the Standard Model (SM) of particle physics [1, 2]. Apart from linking bosons with fermions and unifying internal and external (space-time) symmetries, SUSY allows for a stabilization of the gap between the Planck and the electroweak scale and for gauge coupling unification at high energies. It appears naturally in string theories, includes gravity, and contains a stable lightest SUSY particle (LSP) as a dark matter candidate. Spin partners of the SM particles have not yet been observed, and in order to remain a viable solution to the hierarchy problem, SUSY must be broken at low energy via soft mass terms in the Lagrangian. As a consequence, the SUSY particles must be massive in comparison to their SM counterparts, and the Tevatron and the LHC will perform a conclusive search covering a wide range of masses up to the TeV scale. If SUSY particles exist, they should also appear in virtual particle loops and affect low-energy and electroweak precision observables. In particular, flavour-changing neutral currents (FCNC), which appear only at the one-loop level even in the SM, put severe constraints on new physics contributions appearing at the same perturbative order. Extended technicolour and many composite models have thus been ruled out, while the Minimal Supersymmetric Standard Model (MSSM) has passed these crucial tests. This is largely due to the assumption of constrained Minimal Flavour Violation (cMFV) [3, 4] or Minimal Flavour Violation (MFV) [5, 6, 7], where heavy SUSY particles may appear in the loops, but flavour changes are either neglected or completely dictated by the structure of the Yukawa couplings and thus the CKM-matrix [8, 9]. The squark mass matrices M2 , and M2 are usually expressed in the super-CKM flavour basis [10]. In MFV SUSY scenarios, their flavour violating non-diagonal entries ∆ij , where i, j = L,R refer to the helicity of the (SM partner of the) squark, stem from the trilinear Yukawa couplings of the fermion and Higgs supermultiplets and the resulting different renormalizations of the quark and squark mass matrices, which induce additional flavour violation at the weak scale through renormalization group running [11, 12, 13, 14], while in cMFV scenarios, these flavour violating off-diagonal entries are simply neglected at both the SUSY-breaking and the weak scale. When SUSY is embedded in larger structures such as grand unified theories (GUTs), new sources of flavour violation can appear [15]. For example, local gauge symmetry allows for R-parity violating terms in the SUSY Lagrangian, but these terms are today severely constrained by proton decay and collider searches. In non-minimal flavour violating (NMFV) SUSY, additional sources of flavour violation are included in the mass matrices at the weak scale, and ∗[email protected] http://arxiv.org/abs/0704.1826v2 mailto:[email protected] their flavour violating off-diagonal terms cannot be simply deduced from the CKM matrix alone. NMFV is then conveniently parameterized in the super-CKM basis by considering them as free parameters. The scaling of these entries with the SUSY-breaking scale MSUSY implies a hierarchy ∆LL ≫ ∆LR,RL ≫ ∆RR [15]. Squark mixing is expected to be largest for the second and third generations due to the large Yukawa couplings involved [16]. In addition, stringent experimental constraints for the first generation are imposed by precise mea- surements of K0 − K̄0 mixing and first evidence of D0 − D̄0 mixing [17, 18, 19]. Furthermore, direct searches of flavour violation depend on the possibility of flavour tagging, established experimentally only for heavy flavours. We therefore consider here only flavour mixings of second- and third-generation squarks. The direct search for SUSY particles constitutes a major physics goal of present (Tevatron) and future (LHC) hadron colliders. SUSY particle hadroproduction and decay has therefore been studied in detail theoretically. Next- to-leading order (NLO) SUSY-QCD calculations exist for the production of squarks and gluinos [20], sleptons [21], and gauginos [22] as well as for their associated production [23]. The production of top [24] and bottom [25] squarks with large helicity mixing has received particular attention. Recently, both QCD one-loop and electroweak tree-level contributions have been calculated for non-diagonal, diagonal and mixed top and bottom squark pair production [26]. However, flavour violation has never been considered in the context of collider searches apart from the CKM-matrix appearing in the electroweak stop-sbottom production channel [26]. It is the aim of this paper to investigate for the first time the possible effects of non-minimal flavour violation at hadron colliders. To this end, we re-calculate all squark and gaugino production and decay helicity amplitudes, keeping at the same time the CKM-matrix and the quark masses to account for non-diagonal charged-current gaugino and Higgsino Yukawa interactions, and generalizing the two-dimensional helicity mixing matrices, often assumed to be real, to generally complex six-dimensional helicity and generational mixing matrices. We keep the notation compact by presenting all analytical expressions in terms of generalized couplings. In order to obtain numerical predictions for hadron colliders, we have implemented all our results in a flexible computer program. In our phenomenological analysis of NMFV squark and gaugino production, we concentrate on the LHC due to its larger centre-of-mass energy and luminosity. We pay particular attention to the interesting interplay of parton density functions (PDFs), which are dominated by light quarks, strong gluino contributions, which are generally larger than electroweak contributions and need not be flavour-diagonal, and the appearance of third-generation squarks in the final state, which are easily identified experimentally and generally lighter than first- and second-generation squarks. After reviewing the MSSM with NMFV and setting up our notation in Sec. II, we define in Sec. III generalized couplings of quarks, squarks, gauge bosons, and gauginos. We then use these couplings to present our analytical calculations in concise form. In particular, we have computed partonic cross sections for NMFV squark-antisquark and squark-squark pair production, squark and gaugino associated and gaugino pair production as well as NMFV two-body decay widths of all squarks and gauginos. Section IV is devoted to a precise numerical analysis of the experimentally allowed NMFV SUSY parameter space, an investigation of the corresponding helicity and flavour decomposition of the up- and down-type squarks, and the definition of four collider-friendly benchmark points. These points are then investigated in detail in Sec. V so as to determine the possible sensitivity of the LHC experiments on the allowed NMFV parameter regions in the above-mentioned production channels. Our conclusions are presented in Sec. VI. II. NON-MINIMAL FLAVOUR VIOLATION IN THE MSSM Within the SM, the only source of flavour violation arises through the rotation of the up-type (down-type) quark interaction eigenstates u′L,R (d L,R) to the basis of physical mass eigenstates uL,R (dL,R), such that dL,R = V L,R d L,R and uL,R = V L,R u L,R. (1) The four bi-unitary matrices V L,R diagonalize the quark Yukawa matrices and render the charged-current interactions proportional to the unitary CKM-matrix [8, 9] V = V uL V Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb  . (2) In the super-CKM basis [10], the squark interaction eigenstates undergo the same rotations at high energy scale as their quark counterparts, so that their charged-current interactions are also proportional to the SM CKM-matrix. However, different renormalizations of quarks and squarks introduce a mismatch of quark and squark field rotations at low energies, so that the squark mass matrices effectively become non-diagonal [11, 12, 13, 14]. NMFV is then conveniently parameterized by non-diagonal entries ∆ ij with i, j = L,R in the squared squark mass matrices M2ũ =  ∆ucLL ∆ LL muXu ∆ ∆cuLL M ∆ctLL ∆ RL mcXc ∆ ∆tuLL ∆ ∆tuRL ∆ RL mtXt muXu ∆ ∆ucRR ∆ ∆cuLR mcXc ∆ ∆ctRR ∆tuLR ∆ LR mtXt ∆   ∆dsLL ∆ LL mdXd ∆ ∆sdLL M ∆sbLL ∆ RL msXs ∆ ∆bdLL ∆ ∆bdRL ∆ RL mbXb mdXd ∆ ∆dsRR ∆ ∆sdLR msXs ∆ ∆sbRR ∆bdLR ∆ LR mbXb ∆  , (4) where the diagonal elements are given by +m2q + cos 2βM q − eqs2W ), (5) +m2q + cos 2βM W for up− type squarks, (6) +m2q + cos 2βM W for down− type squarks, (7) while the well-known squark helicity mixing is generated by the elements Xq = Aq − µ cotβ for up− type squarks, tanβ for down− type squarks. (8) Here, mq, T q , and eq denote the mass, weak isospin quantum number, and electric charge of the quark q. mZ is the Z-boson mass, and sW (cW ) is the sine (cosine) of the electroweak mixing angle θW . The soft SUSY-breaking mass terms are MQ̃q and MŨq,D̃q for the left- and right-handed squarks. Aq and µ are the trilinear coupling and off-diagonal Higgs mass parameter, respectively, and tanβ = vu/vd is the ratio of vacuum expectation values of the two Higgs doublets. The scaling of the flavour violating entries ∆ ij with the SUSY-breaking scale MSUSY implies a hierarchy ∆ LL ≫ ∆ LR,RL ≫ ∆ RR among them [15]. They are usually normalized to the diagonal entries [18], so that ij = λ ij MĩqMj̃q′ . (9) Note also that SU(2) gauge invariance relates the (numerically largest) ∆ LL of up- and down-type quarks through the CKM-matrix, implying that a large difference between them is not allowed. The diagonalization of the mass matrices M2ũ and M requires the introduction of two additional 6 × 6 matrices Ru and Rd with diag (m2ũ1 , . . . ,m ) = RuM2ũ R u† and diag (m2 , . . . ,m2 ) = RdM2 Rd†. (10) By convention, the masses are ordered according to mq̃1 < . . . < mq̃6 . The physical mass eigenstates are given by         . (11) In the limit of vanishing off-diagonal parameters ∆ ij , the matrices R q become flavour-diagonal, leaving only the well-known helicity mixing already present in cMFV. III. ANALYTICAL RESULTS FOR PRODUCTION CROSS SECTIONS AND DECAY WIDTHS In this section, we introduce concise definitions of generalized strong and electroweak couplings in NMFV SUSY and compute analytically the corresponding partonic cross sections for squark and gaugino production as well as their decay widths. The cross sections of the production processes aha(pa) bhb(pb) → i (p1) q̃ j (p2), χ̃±j (p1) q̃ i (p2), i (p1) χ̃ j (p2) are presented for definite helicities ha,b of the initial partons a, b = q, q̄, g and expressed in terms of the squark, chargino, neutralino, and gluino masses mq̃k , mχ̃± , mχ̃0 , and mg̃, the conventional Mandelstam variables, s = (pa + pb) 2, t = (pa − p1)2, and u = (pa − p2)2, (13) and the masses of the neutral and charged electroweak gauge bosons mZ and mW . Propagators appear as mass- subtracted Mandelstam variables, sw = s−m2W , sz = s−m2Z , = t−m2 , uχ̃0 = u−m2 tχ̃j = t−m2χ̃± , uχ̃j = u−m2χ̃± tg̃ = t−m2g̃ , ug̃ = u−m2g̃ , tq̃i = t−m2q̃i , uq̃i = u−m Unpolarized cross sections, averaged over initial spins, can easily be derived from the expression dσ̂ = dσ̂1,1 + dσ̂1,−1 + dσ̂−1,1 + dσ̂−1,−1 , (15) while single- and double-polarized cross sections, including the same average factor for initial spins, are given by d∆σ̂L = dσ̂1,1 + dσ̂1,−1 − dσ̂−1,1 − dσ̂−1,−1 and d∆σ̂LL = dσ̂1,1 − dσ̂1,−1 − dσ̂−1,1 + dσ̂−1,−1 , (16) so that the single- and double-spin asymmetries become d∆σ̂L and ALL = d∆σ̂LL . (17) A. Generalized Strong and Electroweak Couplings in NMFV SUSY Considering the strong interaction first, it is well known that the interaction of quarks, squarks, and gluinos, whose coupling is normally just given by gs = 4παs, can in general lead to flavour violation in the left- and right-handed sectors through non-diagonal entries in the matrices Rq, Lq̃jqkg̃, Rq̃jqk g̃ jk,−R j(k+3) . (18) Of course, the involved quark and squark both have to be up- or down-type, since the gluino is electrically neutral. For the electroweak interaction, we define the square of the weak coupling constant g2W = e 2/ sin2 θW in terms of the electromagnetic fine structure constant α = e2/(4π) and the squared sine of the electroweak mixing angle xW = sin 2 θW = s W = 1− cos2 θW = 1− c2W . Following the standard notation, the W± − χ̃0i − χ̃ j , Z − χ̃ i − χ̃ j , and Z − χ̃0i − χ̃0j interaction vertices are proportional to [2] OLij = − j2 +Ni2V j1 and O N∗i3Uj2 +N i2Uj1, O′Lij = −Vi1V ∗j1 − j2 + δijxW and O ij = −U∗i1Uj1 − U∗i2Uj2 + δijxW , O′′Lij = − j4 and O N∗i3Nj3 − N∗i4Nj4. (19) In NMFV, the coupling strengths of left- and right-handed (s)quarks to the electroweak gauge bosons are given by {Lqq′Z , Rqq′Z} = (2T 3q − 2 eq xW )× δqq′ , {Lq̃iq̃jZ , Rq̃iq̃jZ} = (2T 3q̃ − 2 eq̃ xW )× {Ruik Ru∗jk , Rui(3+k) Ru∗j(3+k)}, {Lqq′W , Rqq′W } = { 2 cW Vqq′ , 0}, {Lũid̃jW , Rũid̃jW } = k,l=1 2 cW Vukdl R jl , 0}, (20) where the weak isospin quantum numbers are T 3 {q,q̃} = ±1/2 for left-handed and T 3 {q,q̃} = 0 for right-handed up- and down-type (s)quarks, their fractional electromagnetic charges are denoted by e{q,q̃}, and Vkl are the elements of the CKM-matrix defined in Eq. (2). To simplify the notation, we have introduced flavour indices in the latter, d1 = d, d2 = s, d3 = b, u1 = u, u2 = c, and u3 = t. The SUSY counterparts of these vertices correspond to the quark-squark-gaugino couplings, Ld̃jdkχ̃0i (eq − T 3q ) sW Ni1 + T 3q cW Ni2 Rd∗jk + mdk cW Ni3R j(k+3) 2mW cosβ d̃jdkχ̃ = eq sW Ni1 R j(k+3) − mdk cW Ni3R 2mW cosβ Lũjukχ̃0i = (eq − T 3q ) sW Ni1 + T 3q cW Ni2 Ru∗jk + muk cW Ni4 R j(k+3) 2mW sinβ −R∗ũjukχ̃0i = eq sW Ni1 R j(k+3) − muk cW Ni4 R 2mW sinβ Ld̃julχ̃±i Ui1 R mdk Ui2R j(k+3)√ 2mW cosβ Vuldk , d̃julχ̃ mul Vi2 V Rdjk√ 2mW sinβ Lũjdlχ̃±i V ∗i1 R muk V j(k+3)√ 2mW sinβ Vukdl , ũjdlχ̃ mdl U Ru∗jk√ 2mW cosβ , (21) where the matrices N , U and V relate to the gaugino/Higgsino mixing (see App. A). All other couplings vanish due to (electromagnetic) charge conservation (e.g. Lũjulχ̃±i ). These general expressions can be simplified by neglecting the Yukawa couplings except for the one of the top quark, whose mass is not small compared to mW . For the sake of simplicity, we use the generic notation C1abc, C2abc = {Labc, Rabc} (22) in the following. B. Squark-Antisquark Pair Production The production of charged squark-antisquark pairs q(ha, pa) q̄ ′(hb, pb) → ũi(p1) d̃∗j (p2), (23) where i, j = 1, ..., 6 label up- and down-type squark mass eigenstates, ha,b helicities, and pa,b,1,2 four-momenta, proceeds from an equally charged quark-antiquark initial state through the tree-level Feynman diagrams shown in Fig. 1. The corresponding cross section can be written in a compact way as FIG. 1: Tree-level Feynman diagrams for the production of charged squark-antisquark pairs in quark-antiquark collisions. ha,hb = (1− ha)(1 + hb) k,l=1,...,4 N kl11 k=1,...,4 [NW ]k [GW ] tg̃ sw + (1− ha)(1 − hb) k,l=1,...,4 N kl12 + (1 + ha)(1 + hb) k,l=1,...,4 N kl21 + (1 + ha)(1 − hb) k,l=1,...,4 N kl22 thanks to the form factors W = π α 16 x2W (1− xW )2 s2 ∣∣∣L∗qq′W Lũid̃jW u t−m2ũi m N klmn = x2W (1− xW )2 s2 d̃jq′χ̃ Cm∗ũiqχ̃0k C d̃jq′χ̃ Cmũiqχ̃0l u t−m2ũi m δmn + (1− δmn) Gmn = 2 π α2s ∣∣∣Cn∗ d̃jq′g̃ Cmũiqg̃ u t−m2ũi m δmn + m2g̃ s (1− δmn) [NW ]k = π α 6 x2W (1− xW )2 s2 L∗qq′W Lũid̃jW Lũiqχ̃0k L∗q̃jq′χ̃0k u t−m2ũi m [GW ] = 4 π αs α 18 xW (1− xW ) s2 L∗ũiqg̃ Ld̃jq′g̃ L qq′W Lũid̃jW u t−m2ũi m , (25) which combine coupling constants and Dirac traces of the squared and interference diagrams. In cMFV, superpartners of heavy flavours can only be produced through the purely left-handed s-channel W -exchange, since the t-channel diagrams are suppressed by the small bottom and negligible top quark densities in the proton, and one recovers the result in Ref. [26]. In NMFV, t-channel exchanges can, however, contribute to heavy-flavour final state production from light-flavour initial states and even become dominant, due to the strong gluino coupling. Neutral squark-antisquark pair production proceeds either from equally neutral quark-antiquark initial states q(ha, pa) q̄ ′(hb, pb) → q̃i(p1) q̃∗j (p2), (26) through the five different gauge-boson/gaugino exchanges shown in Fig. 2 (top) or from gluon-gluon initial states g(ha, pa) g(hb, pb) → q̃i(p1) q̃∗i (p2) (27) through the purely strong couplings shown in Fig. 2 (bottom). The differential cross section for quark-antiquark FIG. 2: Tree-level Feynman diagrams for the production of neutral squark-antisquark pairs in quark-antiquark (top) and gluon-gluon collisions (bottom). scattering ha,hb = (1− ha)(1 + hb) [YZ]1 [G̃Y]1 tg̃ s [G̃Z]1 tg̃ sz [G̃G]1 tg̃ s k,l=1,...,4 N kl11 k=1,...,4 [NY]k1 [NZ]k1 [NG]k1 + (1 + ha)(1 − hb) [YZ]2 [G̃Y]2 tg̃ s [G̃Z]2 tg̃ sz [G̃G]2 tg̃ s k,l=1,...,4 N kl22 k=1,...,4 [NY]k2 [NZ]k2 [NG]k2 + (1− ha)(1 − hb) k,l=1,...,4 N kl12 + (1 + ha)(1 + hb) k,l=1,...,4 N kl21 involves many different form factors, π α2 e2q e q̃ δij δqq′ u t−m2q̃i m 16 s2 x2W (1 − xW )2 ∣∣Lq̃iq̃jZ +Rq̃i q̃jZ ∣∣2 (Cmqq′Z u t−m2q̃i m G = 2 π α s δij δqq′ u t−m2q̃i m N klmn = x2W (1 − xW )2 s2 Cm∗q̃iqχ̃0k C q̃iqχ̃ Cnq̃jq′χ̃0kC q̃jq′χ̃ u t−m2q̃i m δmn + (1− δmn) G̃mn = 2 π α2s ∣∣∣Cmq̃iqg̃ C q̃jq′g̃ u t−m2q̃i m δmn + m2g̃ s (1− δmn) [YZ]m = π α2 eq eq̃ δij δqq′ 2 s2 xW (1− xW ) Lq̃iq̃jZ +Rq̃i q̃jZ Cmqq′Z u t−m2q̃i m [NY]km = 2 π α2 eq eq̃ δij δqq′ 3 xW (1 − xW ) s2 Cmq̃iqχ̃0k C q̃jq′χ̃ u t−m2q̃i m FIG. 3: Tree-level Feynman diagrams for the production of one down-type squark (q̃i) and one up-type squark (q̃ j) in the collision of an up-type quark (q) and a down-type quark (q′). [NZ]km = 6 x2W (1− xW )2 s2 Cmq̃iqχ̃0k C q̃jq′χ̃ Lq̃iq̃jZ +Rq̃iq̃jZ Cmqq′Z u t−m2q̃i m [NG]km = 8 π ααs δij δqq′ 9 xW (1− xW ) s2 Cmq̃iqχ̃0k C q̃jq′χ̃ u t−m2q̃i m = −4 π α s δij δqq′ 27 s2 Cm∗q̃iqg̃ C q̃jq′g̃ u t−m2q̃i m [G̃Y]m = 8 π ααs eq eq̃ δij δqq′ Cm∗q̃iqg̃ C q̃jq′g̃ u t−m2q̃i m [G̃Z]m = 2 π ααs 9 xW (1− xW ) s2 Cm∗q̃iqg̃ C q̃jq′g̃ Lq̃iq̃jZ +Rq̃i q̃jZ Cmqq′Z u t−m2q̃i m , (29) since only very few interferences (those between strong and electroweak channels of the same propagator type) are eliminated due to colour conservation. On the other hand, the gluon-initiated cross section ha,hb 128s2 1− 2 tq̃iuq̃i (1− hahb)− 2 sm2q̃i tq̃iuq̃i (1− hahb)− sm2q̃i tq̃iuq̃i involves only the strong coupling constant and is thus quite compact. In the case of cMFV, but diagonal or non- diagonal squark helicity, our results agree with those in Ref. [26]. Diagonal production of identical squark-antisquark mass eigenstates is, of course, dominated by the strong quark-antiquark and gluon-gluon channels. Their relative im- portance depends on the partonic luminosity and thus on the type and energy of the hadron collider under considera- tion. Non-diagonal production of squarks of different helicity or flavour involves only electroweak and gluino-mediated quark-antiquark scattering, and the relative importance of these processes depends largely on the gluino mass. C. Squark Pair Production While squark-antisquark pairs are readily produced in pp̄ collisions, e.g. at the Tevatron, from valence quarks and antiquarks, pp colliders have a larger quark-quark luminosity and will thus more easily lead to squark pair production. The production of one down-type and one up-type squark q(ha, pa) q ′(hb, pb) → d̃i(p1) ũj(p2), (31) in the collision of an up-type quark q and a down-type quark q′ proceeds through the t-channel chargino or u-channel neutralino and gluino exchanges shown in Fig. 3. The corresponding cross section FIG. 4: Tree-level Feynman diagrams for the production of two up-type or down-type squarks. ha,hb = (1−ha)(1−hb) k=1,2 l=1,2 Ckl11 tχ̃k tχ̃l k=1,...,4 l=1,...,4 N kl11 k=1,2 l=1,...,4 [CN ]kl11 tχ̃k uχ̃0 k=1,2 [CG]k11 tχ̃k ug̃ + (1+ha)(1+hb) k=1,2 l=1,2 Ckl22 tχ̃k tχ̃l k=1,...,4 l=1,...,4 N kl22 k=1,2 l=1,...,4 [CN ]kl22 tχ̃k uχ̃0l k=1,2 [CG]k22 tχ̃k ug̃ + (1−ha)(1+hb) k=1,2 l=1,2 Ckl12 tχ̃k tχ̃l k=1,...,4 l=1,...,4 N kl12 k=1,2 l=1,...,4 [CN ]kl12 tχ̃k uχ̃0 k=1,2 [CG]k12 tχ̃k ug̃ + (1+ha)(1−hb) k=1,2 l=1,2 Ckl21 tχ̃k tχ̃l k=1,...,4 l=1,...,4 N kl21 k=1,2 l=1,...,4 [CN ]kl21 tχ̃k uχ̃0 k=1,2 [CG]k21 tχ̃k ug̃ involves the form factors Cklmn = 4 x2W s ũjq′χ̃ d̃iqχ̃ ũjq′χ̃ d̃iqχ̃ u t−m2 m2ũj (1− δmn) +mχ̃± s δmn N klmn = x2W (1− xW )2 s2 Cm∗ũjqχ̃0k C d̃iq′χ̃ Cmũjqχ̃0l C d̃iq′χ̃ u t−m2 m2ũj (1− δmn) +mχ̃0 s δmn Gmn = 2 π α2s ∣∣∣Cmũjqg̃ C d̃iq′g̃ u t−m2 m2ũj (1− δmn) +m2g̃ s δmn [CN ]klmn = 3 x2W (1− xW ) s2 ũjq′χ̃ d̃iqχ̃ Cmũjqχ̃0l C d̃iq′χ̃ u t−m2 m2ũj (δmn − 1) +mχ̃± s δmn [CG]kmn = 4 π ααs 9 s2 xW ũjq′χ̃ d̃iqχ̃ Cm∗ũjqg̃ C d̃iq′g̃ u t−m2 m2ũj (δmn − 1) +mχ̃± mg̃ s δmn , (33) where the neutralino-gluino interference term is absent due to colour conservation. The cross section for the charge- conjugate production of antisquarks from antiquarks can be obtained from the equations above by replacing ha,b → −ha,b. Heavy-flavour final states are completely absent in cMFV due to the negligible top quark and small bottom quark densities in the proton and can thus only be obtained in NMFV. The Feynman diagrams for pair production of two up- or down-type squarks q(ha, pa) q ′(hb, pb) → q̃i(p1) q̃j(p2) (34) are shown in Fig. 4. In NMFV, neutralino and gluino exchanges can lead to identical squark flavours for different quark initial states, so that both t- and u-channels contribute and may interfere. The cross section ha,hb = (1 − ha)(1− hb) k=1,...,4 l=1,...,4 [NT ]kl11 [NU ]kl11 [NT U ]kl11 [GT ]11 [GU ]11 [GT U ]11 ug̃tg̃ k=1,...,4 [NGA]k11 [NGB]k11 1 + δij + (1 + ha)(1 + hb) k=1,...,4 l=1,...,4 [NT ]kl22 [NU ]kl22 [NT U ]kl22 [GT ]22 [GU ]22 [GT U ]22 ug̃tg̃ k=1,...,4 [NGA]k22 [NGB]k22 1 + δij + (1 − ha)(1 + hb) k=1,...,4 l=1,...,4 [NT ]kl12 [NU ]kl12 [NT U ]kl12 [GT ]12 [GU ]12 [GT U ]12 ug̃tg̃ k=1,...,4 [NGA]k12 [NGB]k12 1 + δij + (1 + ha)(1− hb) k=1,...,4 l=1,...,4 [NT ]kl21 [NU ]kl21 [NT U ]kl21 [GT ]21 [GU ]21 [GT U ]21 ug̃tg̃ k=1,...,4 [NGA]k21 [NGB]k21 1 + δij depends therefore on the form factors [NT ]klmn = x2W (1− xW )2 s2 Cn∗q̃jq′χ̃0k C q̃iqχ̃ Cnq̃jq′χ̃0l C q̃iqχ̃ u t−m2q̃i m (1− δmn) +mχ̃0 s δmn [NU ]klmn = x2W (1− xW )2 s2 Cn∗q̃iq′χ̃0k C q̃jqχ̃ Cnq̃iq′χ̃0l C q̃jqχ̃ u t−m2q̃i m (1− δmn) +mχ̃0 s δmn [NT U ]klmn = 2 π α2 3 x2W (1− xW )2 s2 Cm∗q̃iqχ̃0k C q̃jq′χ̃ Cnq̃iq′χ̃0l C q̃jqχ̃ u t−m2q̃i m (1− δmn) +mχ̃0 s δmn [GT ]mn = 2 π α2s ∣∣∣Cnq̃jq′g̃ C q̃iqg̃ u t−m2q̃i m (1− δmn) +m2g̃ s δmn [GU ]mn = 2 π α2s ∣∣∣Cmq̃iq′g̃ C q̃jqg̃ u t−m2q̃i m (1− δmn) +m2g̃ s δmn [GT U ]mn = −4 π α2s 27 s2 Cmq̃iqg̃ C q̃jq′g̃ Cm∗q̃iq′g̃ C q̃jqg̃ u t−m2q̃i m (1− δmn) +m2g̃ s δmn [NGA]kmn = 8 π ααs 9 s2 xW (1− xW ) Cn∗q̃jq′χ̃0k C q̃iqχ̃ Cm∗q̃iq′g̃ C q̃jqg̃ ] [ ( u t−m2q̃i m (1− δmn) +mχ̃0 mg̃ s δmn [NGB]kmn = 8 π ααs 9 s2 xW (1− xW ) Cn∗q̃iq′χ̃0k C q̃jqχ̃ Cn∗q̃jq′g̃ C q̃iqg̃ ] [ ( u t−m2q̃i m (1− δmn) +mχ̃0 mg̃ s δmn . (36) Gluinos will dominate over neutralino exchanges due to their strong coupling, and the two will only interfere in the mixed t- and u-channels due to colour conservation. At the LHC, up-type squark pair production should dominate over mixed up-/down-type squark production and down-type squark pair production, since the proton contains two valence up-quarks and only one valence down-quark. As before, the charge-conjugate production of antisquark pairs FIG. 5: Tree-level Feynman diagrams for the associated production of squarks and gauginos. is obtained by making the replacement ha,b → −ha,b. If we neglect electroweak contributions as well as squark flavour and helicity mixing and sum over left- and right-handed squark states, our results agree with those of Ref. [27]. D. Associated Production of Squarks and Gauginos The associated production of squarks and neutralinos or charginos q(ha, pa) g(hb, pb) → χ̃j(p1) q̃i(p2) (37) is a semi-weak process that originates from quark-gluon initial states and has both an s-channel quark and a t-channel squark contribution. They involve both a quark-squark-gaugino vertex that can in general be flavour violating. The corresponding Feynman diagrams can be seen in Fig. 5. The squark-gaugino cross section ha,hb π ααs nχ̃ s2 −uχ̃j (1− ha)(1 − hb) ∣∣Lq̃iqχ̃j ∣∣2 + (1 + ha)(1 + hb) ∣∣Rq̃iqχ̃j t+m2q̃i t2q̃i (1 − ha) ∣∣Lq̃iqχ̃j ∣∣2 + (1 + ha) ∣∣Rq̃iqχ̃j 2 (u t−m2q̃i m s tq̃i (1 − ha)(1− hb) ∣∣Lq̃iqχ̃j ∣∣2 + (1 + ha)(1 + hb) ∣∣Rq̃iqχ̃j tχ̃j (tχ̃j − uq̃i) s tq̃i (1 − ha) ∣∣Lq̃iqχ̃j ∣∣2 + (1 + ha) ∣∣Rq̃iqχ̃j , (38) where nχ̃ = 6xW (1− xW ) for neutralinos and nχ̃ = 12xW for charginos, is sufficiently compact to be written without the definition of form factors. Note that the t-channel diagram involves the coupling of the gluon to scalars and does thus not depend on its helicity hb. The cross section of the charge-conjugate process can be obtained by taking ha → −ha. Third-generation squarks can only be produced in NMFV, preferably through a light (valence) quark in the s-channel. For non-mixing squarks and gauginos, we agree again with the results of Ref. [27]. E. Gaugino Pair Production Finally, we consider the purely electroweak production of gaugino pairs q(ha, pa) q̄ ′(hb, pb) → χ̃i(p1) χ̃j(p2) (39) from quark-antiquark initial states, where flavour violation can occur via the quark-squark-gaugino vertices in the t- and u-channels (see Fig. 6). However, if it were not for different parton density weights, summation over complete squark multiplet exchanges would make these channels insensitive to the exchanged squark flavour. Furthermore there are no final state squarks that could be experimentally tagged. The cross section can be expressed generically as FIG. 6: Tree-level Feynman diagrams for the production of gaugino pairs. ha,hb (1− ha)(1 + hb) |QuLL| uχ̃iuχ̃j + ∣∣QtLL ∣∣2 tχ̃itχ̃j + 2Re[Q LL]mχ̃imχ̃js (1 + ha)(1− hb) |QuRR| uχ̃iuχ̃j + ∣∣QtRR ∣∣2 tχ̃itχ̃j + 2Re[Q RR]mχ̃imχ̃js (1 + ha)(1 + hb) |QuRL| uχ̃iuχ̃j + ∣∣QtRL ∣∣2 tχ̃itχ̃j +Re[Q RL](ut−m2χ̃im (1− ha)(1− hb) |QuLR| uχ̃iuχ̃j + ∣∣QtLR ∣∣2 tχ̃itχ̃j +Re[Q LR](ut−m2χ̃im , (40) i.e. in terms of generalized charges. For χ̃−i χ̃ j -production, these charges are given by Qu−+LL = eqδijδqq′ Lqq′ZO 2 xW (1 − xW ) sz Ld̃kq′χ̃±i d̃kqχ̃ 2 xW ud̃k Qt−+LL = eqδijδqq′ Lqq′ZO 2 xW (1 − xW ) sz ũkq′χ̃ Lũkqχ̃±i 2 xW tũk Qu−+RR = eqδijδqq′ Rqq′ZO 2 xW (1 − xW ) sz Rd̃kq′χ̃±i d̃kqχ̃ 2 xW ud̃k Qt−+RR = eqδijδqq′ Rqq′ZO 2 xW (1 − xW ) sz ũkq′χ̃ Rũkqχ̃±i 2 xW tũk Qu−+LR = Rd̃kq′χ̃±i d̃kqχ̃ 2 xW ud̃k Qt−+LR = ũkq′χ̃ Lũkqχ̃±i 2 xW tũk Qu−+RL = Ld̃kq′χ̃±i d̃kqχ̃ 2 xW ud̃k Qt−+RL = ũkq′χ̃ Rũkqχ̃±i 2 xW tũk . (41) Note that there is no interference between t- and u-channel diagrams due to (electromagnetic) charge conservation. The cross section for chargino-pair production in e+e−-collisions can be deduced by setting eq → el = −1, Lqq′Z → LeeZ = (2T l − 2 el xW ) and Rqq′Z → ReeZ = −2 el xW . Neglecting all Yukawa couplings, we can then reproduce the calculations of Ref. [28]. The charges of the chargino-neutralino associated production are given by Qu+0LL = 2 (1− xW )xW OL∗ji L qq′W√ ũkq′χ̃ ũkqχ̃ Qt+0LL = 2 (1− xW )xW OR∗ji L qq′W√ d̃kqχ̃ Ld̃kq′χ̃0j Qu+0RR = 2 (1− xW )xW ũkq′χ̃ ũkqχ̃ Qt+0RR = 2 (1− xW )xW d̃kqχ̃ Rd̃kq′χ̃0j Qu+0LR = 2 (1− xW )xW ũkq′χ̃ ũkqχ̃ Qt+0LR = 2 (1− xW )xW d̃kqχ̃ Rd̃kq′χ̃0j Qu+0RL = 2 (1− xW )xW ũkq′χ̃ ũkqχ̃ Qt+0RL = 2 (1− xW )xW d̃kqχ̃ Ld̃kq′χ̃0j . (42) The charge-conjugate process is again obtained by making the replacement ha,b → −ha,b in Eq. (40). In the case of non-mixing squarks with neglected Yukawa couplings, we agree with the results of Ref. [22], provided we correct a sign in their Eq. (2) as described in Ref. [29]. Finally, the charges for the neutralino pair production are given by Qu00LL = xW (1− xW ) 1 + δij Lqq′ZO LQ̃kq′χ̃0i Q̃kqχ̃ Qt00LL = xW (1− xW ) 1 + δij Lqq′ZO Q̃kqχ̃ LQ̃kq′χ̃0j Qu00RR = xW (1− xW ) 1 + δij Rqq′ZO RQ̃kq′χ̃0i Q̃kqχ̃ Qt00RR = xW (1− xW ) 1 + δij Rqq′ZO Q̃kqχ̃ RQ̃kq′χ̃0j Qu00LR = xW (1− xW ) 1 + δij RQ̃kq′χ̃0i Q̃kqχ̃ Qt00LR = xW (1− xW ) 1 + δij Q̃kqχ̃ RQ̃kq′χ̃0j Qu00RL = xW (1− xW ) 1 + δij LQ̃kq′χ̃0i Q̃kqχ̃ Qt00RL = xW (1− xW ) 1 + δij Q̃kqχ̃ LQ̃kq′χ̃0j , (43) which agrees with the results of Ref. [30] in the case of non-mixing squarks. FIG. 7: Tree-level Feynman diagrams for squark decays into gauginos and quarks (top) and into electroweak gauge bosons and lighter squarks (bottom). F. Squark Decays We turn now from SUSY particle production to decay processes and show in Fig. 7 the possible decays of squarks into gauginos and quarks (top) as well as into electroweak gauge bosons and lighter squarks (bottom). Both processes can in general induce flavour violation. The decay widths of the former are given by Γq̃i→χ̃0jqk = 2m3q̃i xW (1− xW ) m2q̃i −m −m2qk )( ∣∣∣Lq̃iqkχ̃0j ∣∣∣Rq̃iqkχ̃0j − 4mχ̃0 mqk Re Lq̃iqkχ̃0jR q̃iqkχ̃ λ1/2(m2q̃i ,m ,m2qk), (44) Γq̃i→χ̃±j q 4m3q̃i xW m2q̃i −m −m2q′ )( ∣∣∣Lq̃iq′kχ̃±j ∣∣∣Rq̃iq′kχ̃±j − 4mχ̃± Lq̃iq′kχ̃ λ1/2(m2q̃i ,m ,m2q′ ), (45) Γq̃i→g̃qk = 3m3q̃i xW m2q̃i −m g̃ −m2qk |Lq̃iqkg̃| + |Rq̃iqk g̃| − 4mg̃mqk Re Lq̃iqkg̃R q̃iqkg̃ × λ1/2(m2q̃i ,m ), (46) while those of the latter are given by Γq̃i→Zq̃k = 16m3q̃i m Z xW (1− xW ) |Lq̃i q̃kZ +Rq̃i q̃kZ | λ3/2(m2q̃i ,m ), (47) Γq̃i→W±q̃′k = 16m3q̃i m W xW (1− xW ) ∣∣∣Lq̃iq̃′kW λ3/2(m2q̃i ,m ). (48) The usual Källen function is λ(x, y, z) = x2 + y2 + z2 − 2(x y + y z + z x). (49) In cMFV, our results agree with those of Ref. [31]. G. Gluino Decays Heavy gluinos can decay strongly into squarks and quarks as shown in Fig. 8. The corresponding decay width FIG. 8: Tree-level Feynman diagram for gluino decays into squarks and quarks. FIG. 9: Tree-level Feynman diagrams for gaugino decays into squarks and quarks (left) and into lighter gauginos and electroweak gauge bosons (centre and right). Γg̃→q̃∗ 8m3g̃ m2g̃ −m2q̃j +m )( ∣∣Lq̃jqkg̃ ∣∣2 + ∣∣Rq̃jqk g̃ + 4mg̃mqk Re Lq̃jqk g̃R q̃jqk g̃ × λ1/2(m2g̃,m2q̃j ,m ) (50) can in general also induce flavour violation. In cMFV, our result agrees again with the one of Ref. [31]. H. Gaugino Decays Heavier gauginos can decay into squarks and quarks as shown in Fig. 9 (left) or into lighter gauginos and electroweak gauge bosons (Fig. 9 centre and right). The analytical decay widths are →q̃j q̄ −m2q̃j +m )( ∣∣∣Lq̃jq′kχ̃±i ∣∣∣Rq̃jq′kχ̃±i + 4mχ̃± Lq̃jq′kχ̃ λ1/2(m2 ,m2q̃j ,m ) (51) m2W xW +m4χ̃0 − 2m4W +m2χ̃± m2W +m m2W − 2m2χ̃± m2χ̃0 ( ∣∣OLij ∣∣2 + ∣∣ORij − 12mχ̃± m2W mχ̃0 OLijO λ1/2(m2 ,m2χ̃0 ,m2W ), (52) m2Z xW (1− xW ) − 2m4Z +m2χ̃± m2Z +m m2Z − 2m2χ̃± ( ∣∣O′Lij ∣∣2 + ∣∣O′Rij − 12mχ̃± m2Z mχ̃± O′Lij O λ1/2(m2 ,m2Z) (53) for charginos and →q̃j q̄k xW (1− xW ) m2χ̃0 −m2q̃j +m )( ∣∣∣Lq̃jqkχ̃0i ∣∣∣Rq̃jqkχ̃0i + 4mχ̃0 mqk Re Lq̃jqkχ̃0iR q̃jqkχ̃ λ1/2(m2χ̃0 ,m2q̃j ,m ) (54) m2W xW m4χ̃0 − 2m4W +m2χ̃0 m2W +m m2W − 2m2χ̃0 ( ∣∣OLij ∣∣2 + ∣∣ORij − 12mχ̃0 m2W mχ̃± OLijO λ1/2(m2χ̃0 ,m2W ), (55) m2Z xW (1− xW ) m4χ̃0 +m4χ̃0 − 2m4Z +m2χ̃0 m2Z +m m2Z − 2m2χ̃0 m2χ̃0 ( ∣∣O′′Lij ∣∣2 + ∣∣O′′Rij − 12mχ̃0 m2Z mχ̃0j Re O′′Lij O λ1/2(m2χ̃0 ,m2χ̃0 ,m2Z) (56) for neutralinos, respectively. Chargino decays into a slepton and a neutrino (lepton and sneutrino) can be deduced from the previous equations by taking the proper limits, i.e. by removing colour factors and up-type masses in the coupling definitions. Our results agree then with those of Ref. [32] in the limit of non-mixing sneutrinos. Note that the same simplifications also permit a verification of our results for squark decays into a gaugino and a quark in Eqs. (44) and (45) when compared to their leptonic counterparts in Ref. [32]. IV. EXPERIMENTAL CONSTRAINTS, SCANS AND BENCHMARKS IN NMFV SUSY In the absence of experimental evidence for supersymmetry, a large variety of data can be used to constrain the MSSM parameter space. For example, sparticle mass limits can be obtained from searches of charginos (mχ̃± GeV for heavier sneutrinos at LEP2), neutralinos (mχ̃0 ≥ 59 GeV in minimal supergravity (mSUGRA) from the combination of LEP2 results), gluinos (mg̃ ≥ 195 GeV from CDF), stops (mt̃1 ≥ 95 . . . 96 GeV for neutral- or charged-current decays from the combination of LEP2 results), and other squarks (mq̃ ≥ 300 GeV for gluinos of equal mass from CDF) at colliders [33]. Note that all of these limits have been obtained assuming minimal flavour violation. For non-minimal flavour violation, rather strong constraints can be obtained from low-energy, electroweak precision, and cosmological observables. These are discussed in the next subsection, followed by several scans for experimentally allowed/favoured regions of the constrained MSSM parameter space and the definition of four NMFV benchmark points/slopes. Finally, we exhibit the corresponding chirality and flavour decomposition of the various squark mass eigenstates. A. Low-Energy, Electroweak Precision, and Cosmological Constraints In a rather complete analysis of FCNC constraints more than ten years ago [18], upper limits from the neutral kaon sector (on ∆mK , ε, ε ′/ε), on B- (∆mB) and D-meson oscillations (∆mD), various rare decays (BR(b → sγ), BR(µ → eγ), BR(τ → eγ), and BR(τ → µγ)), and electric dipole moments (dn and de) were used to impose constraints on non-minimal flavour mixing in the squark and slepton sectors. The limit obtained for the absolute value in the left-handed, down-type squark sector was rather weak (|λsbLL| < 4.4 . . .26 for varying gluino-to-squark mass ratio), while the limits for the mixed/right-handed, imaginary or sleptonic parts were already several orders of magnitude smaller. In the meantime, many of the experimental bounds have been improved or absolute values for the observables have been determined, so that an updated analysis could be performed [34]. The results for the down-type squark sector are cited in Tab. I. As can be seen and as has already been hinted at in the introduction, only mixing between second- and third-generation squarks can be substantial, and this only in the left-left or right-right chiral sectors, the latter being disfavoured by its scaling with the soft SUSY-breaking mass. Independent analyses focusing TABLE I: The 95% probability bounds on |λ ij | obtained in Ref. [34]. ij LL LR RL RR 12 1.4×10−2 9.0×10−5 9.0×10−5 9.0×10−3 13 9.0×10−2 1.7×10−2 1.7×10−2 7.0×10−2 23 1.6×10−1 4.5×10−3 6.0×10−3 2.2×10−1 on this particular sector, i.e. on BR(b → sγ), BR(b → sµµ), and ∆mBs , have been performed recently by two other groups [35, 36] with very similar results. In our own analysis, we take implicitly into account all of the previously mentioned constraints by restricting ourselves to the case of only one real NMFV parameter, λ ≡ λsbLL = λctLL. Allowed regions for this parameter are then obtained by imposing explicitly a number of low-energy, electroweak precision, and cosmological constraints. We start by imposing the theoretically robust inclusive branching ratio BR(b→ sγ) = (3.55± 0.26)× 10−4, (57) obtained from the combined measurements of BaBar, Belle, and CLEO [37], at the 2σ-level on the two-loop QCD/one- loop SUSY calculation [36, 38], which affects directly the allowed squark mixing between the second and third generation. A second important consequence of NMFV in the MSSM is the generation of large splittings between squark-mass eigenvalues. The splitting within isospin doublets influences the Z- and W -boson self-energies at zero-momentum ΣZ,W (0) in the electroweak ρ-parameter ∆ρ = ΣZ(0)/M Z − ΣW (0)/M2W (58) and consequently theW -boson massMW and the squared sine of the weak mixing angle sin 2 θW . The latest combined fits of the Z-boson mass, width, pole asymmetry, W -boson and top-quark mass constrain new physics contributions to T = −0.13± 0.11 [33] or ∆ρ = −αT = 0.00102± 0.00086, (59) where we have used α(MZ) = 1/127.918. This value is then imposed at the 2σ-level on the one-loop NMFV and two-loop cMFV SUSY calculation [39]. A third observable sensitive to SUSY loop-contributions is the anomalous magnetic moment aµ = (gµ − 2)/2 of the muon, for which recent BNL data and the SM prediction disagree by [33] ∆aµ = (22± 10)× 10−10. (60) In our calculation, we take into account the SM and MSSM contributions up to two loops [40, 41] and require them to agree with the region above within two standard deviations. For cosmological reasons, i.e. in order to have a suitable candidate for non-baryonic cold dark matter [42], we require the lightest SUSY particle (LSP) to be stable, electrically neutral, and a colour singlet. The dark matter relic density is then calculated using a modified version of DarkSUSY 4.1 [43], that takes into account the six-dimensional squark helicity and flavour mixing, and constrained to the region 0.094 < ΩCDMh 2 < 0.136 (61) at 95% (2σ) confidence level. This limit has recently been obtained from the three-year data of the WMAP satellite, combined with the SDSS and SNLS survey and Baryon Acoustic Oscillation data and interpreted within an eleven- parameter inflationary model [44], which is more general than the usual six-parameter “vanilla” concordance model of cosmology. Note that this range is well compatible with the older, independently obtained range of 0.094 < ΩCDMh 2 < 0.129 [45]. B. Scans of the Constrained NMFV MSSM Parameter Space The above experimental limits are now imposed on the constrained MSSM (cMSSM), or minimal supergravity (mSUGRA), model with five free parameters m0, m1/2, tanβ, A0, and sgn(µ) at the grand unification scale. Since [GeV]1/2m 500 1000 1500 =0λ<0, µ=10, β=0, tan [GeV]1/2m 500 1000 1500 =0.03λ<0, µ=10, β=0, tan [GeV]1/2m 500 1000 1500 =0.05λ<0, µ=10, β=0, tan [GeV]1/2m 500 1000 1500 =0.1λ<0, µ=10, β=0, tan FIG. 10: The (m0,m1/2)-planes for tan β = 10, A0 = 0 GeV, µ < 0, and λ = 0, 0.03, 0.05 and 0.1. We show WMAP (black) favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in minimal (λ = 0) and non-minimal (λ > 0) flavour violation. our scans of the cMSSM parameter space in the common scalar mass m0 and the common fermion mass m1/2 depend very little on the trilinear coupling A0, we set it to zero in the following. Furthermore, we fix a small (10), intermediate (30), and large (50) value for the ratio of the Higgs vacuum expectation values tanβ. The impact of the sign of the off-diagonal Higgs mass parameter µ is investigated for tanβ = 10 only, before we set it to µ > 0 for tanβ = 30 and 50 (see below). With these boundary conditions at the grand unification scale, we solve the renormalization group equations numer- ically to two-loop order using the computer program SPheno 2.2.3 [46] and compute the soft SUSY-breaking masses at the electroweak scale with the complete one-loop formulas, supplemented by two-loop contributions in the case of the neutral Higgs bosons and the µ-parameter. At this point we generalize the squark mass matrices as described in Sec. II in order to account for flavour mixing in the left-chiral sector of the second- and third-generation squarks, diagonalize these mass matrices, and compute the low-energy, electroweak precision, and cosmological observables with the computer programs FeynHiggs 2.5.1 [47] and DarkSUSY 4.1 [43]. For the masses and widths of the electroweak gauge bosons and the mass of the top quark, we use the current values of mZ = 91.1876 GeV, mW = 80.403 GeV, mt = 174.2 GeV, ΓZ = 2.4952 GeV, and ΓW = 2.141 GeV. The CKM-matrix elements are computed using the parameterization c12c13 s12c13 s13e −s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13 s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13  , (62) where sij = sin θij and cij = cos θij relate to the mixing of two specific generations i and j and δ is the SM CP -violating complex phase. The numerical values are given by s12 = 0.2243, s23 = 0.0413, s13 = 0.0037, and δ = 1.05. (63) The squared sine of the electroweak mixing angle sin2 θW = 1 − m2W /m2Z and the electromagnetic fine structure constant α = W sin 2 θW /π are calculated in the improved Born approximation using the world average value of GF = 1.16637 · 10−5 GeV−2 for Fermi’s coupling constant [33]. Typical scans of the cMSSM parameter space in m0 and m1/2 with a relatively small value of tanβ = 10 and A0 = 0 are shown in Figs. 10 and 11 for µ < 0 and µ > 0, respectively. All experimental limits described in Sec. IVA are imposed at the 2σ-level. The b→ sγ excluded region depends strongly on flavour mixing, while the regions favoured by gµ − 2 and the dark matter relic density are quite insensitive to variations of the λ-parameter. ∆ρ constrains the parameter space only for heavy scalar masses m0 > 2000 GeV and heavy gaugino masses m1/2 > 1500 GeV, so that the corresponding excluded regions are not shown here. The dominant SUSY effects in the calculation of the anomalous magnetic moment of the muon come from induced quantum loops of a gaugino and a slepton. Squarks contribute only at the two-loop level. This reduces the dependence on flavour violation in the squark sector considerably. Furthermore, the region µ < 0 is disfavoured in all SUSY models, since the one-loop SUSY contributions are approximatively given by [48] aSUSY, 1−loopµ ≃ 13× 10−10 100 GeV MSUSY tanβ sgn(µ), (64) if all SUSY particles (the relevant ones are the smuon, sneutralino, chargino, and neutralino) have a common mass MSUSY. Negative values of µ would then increase, not decrease, the disagreement between the experimental measure- ments and the theoretical SM value of aµ. Furthermore, the measured b → sγ branching ratio excludes virtually all [GeV]1/2m 500 1000 1500 =0λ>0, µ=10, β=0, tan [GeV]1/2m 500 1000 1500 =0.03λ>0, µ=10, β=0, tan [GeV]1/2m 500 1000 1500 =0.05λ>0, µ=10, β=0, tan [GeV]1/2m 500 1000 1500 =0.1λ>0, µ=10, β=0, tan FIG. 11: The (m0,m1/2)-planes for tanβ = 10, A0 = 0 GeV, µ > 0, and λ = 0, 0.03, 0.05 and 0.1. We show aµ (grey) and WMAP (black) favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in minimal (λ = 0) and non-minimal (λ > 0) flavour violation. [GeV]1/2m 500 1000 1500 =0λ>0, µ=30, β=0, tan [GeV]1/2m 500 1000 1500 =0.03λ>0, µ=30, β=0, tan [GeV]1/2m 500 1000 1500 =0.05λ>0, µ=30, β=0, tan [GeV]1/2m 500 1000 1500 =0.1λ>0, µ=30, β=0, tan FIG. 12: The (m0,m1/2) planes for tan β = 30, A0 = 0 GeV, µ > 0, and λ = 0, 0.03, 0.05 and 0.1. We show aµ (grey) and WMAP (black) favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in minimal (λ = 0) and non-minimal (λ > 0) flavour violation. of the region favoured by the dark matter relic density, except for very high scalar SUSY masses. We therefore do not consider negative values of µ in the rest of this work. As stated above, we have also checked that the shape of the different regions depends extremely weakly on the trilinear coupling A0. In Figs. 12 and 13, we show the (m0,m1/2)-planes for larger tanβ, namely tanβ = 30 and tanβ = 50, and for µ > 0. The regions which are favoured both by the anomalous magnetic moment of the muon and by the cold dark matter relic density, and which are not excluded by the b→ sγ measurements, are stringently constrained and do not allow for large flavour violation. [GeV]1/2m 500 1000 1500 =0λ>0, µ=50, β=0, tan [GeV]1/2m 500 1000 1500 =0.03λ>0, µ=50, β=0, tan [GeV]1/2m 500 1000 1500 =0.05λ>0, µ=50, β=0, tan [GeV]1/2m 500 1000 1500 =0.1λ>0, µ=50, β=0, tan FIG. 13: The (m0,m1/2) planes for tan β = 50, A0 = 0 GeV, µ > 0, and λ = 0, 0.03, 0.05 and 0.1. We show aµ (grey) and WMAP (black) favoured as well as b → sγ (blue) and charged LSP (beige) excluded regions of mSUGRA parameter space in minimal (λ = 0) and non-minimal (λ > 0) flavour violation. TABLE II: Benchmark points allowing for flavour violation among the second and third generations for A0 = 0, µ > 0, and three different values of tanβ. For comparison we also show the nearest pre-WMAP SPS [49, 50] and post-WMAP BDEGOP [51] benchmark points and indicate the relevant cosmological regions. m0 [GeV] m1/2 [GeV] A0 [GeV] tanβ sgn(µ) SPS BDEGOP Cosmol. Region A 700 200 0 10 1 2 E’ Focus Point B 100 400 0 10 1 3 C’ Co-Annihilation C 230 590 0 30 1 1b I’ Co-Annihilation D 600 700 0 50 1 4 L’ Bulk/Higgs-funnel C. (c)MFV and NMFV Benchmark Points and Slopes Restricting ourselves to non-negative values of µ, we now inspect the (m0,m1/2)-planes in Figs. 11-13 for cMSSM scenarios that • are allowed/favoured by low-energy, electroweak precision, and cosmological constraints, • permit non-minimal flavour violation among left-chiral squarks of the second and third generation up to λ ≤ 0.1, • and are at the same time collider-friendly, i.e. have relatively low values of m0 and m1/2. Our choices are presented in Tab. II, together with the nearest pre-WMAP Snowmass Points (and Slopes, SPS) [49, 50] and the nearest post-WMAP scenarios proposed in Ref. [51]. We also indicate the relevant cosmological region for each point and attach a model line (slope) to it, given by A : 180 GeV ≤ m1/2 ≤ 250 GeV , m0 = − 1936 GeV + 12.9m1/2, B : 400 GeV ≤ m1/2 ≤ 900 GeV , m0 = 4.93 GeV + 0.229m1/2, C : 500 GeV ≤ m1/2 ≤ 700 GeV , m0 = 54 GeV + 0.297m1/2, D : 575 GeV ≤ m1/2 ≤ 725 GeV , m0 = 600 GeV. (65) These slopes trace the allowed/favoured regions from lower to higher masses and can, of course, also be used in cMFV scenarios with λ = 0. We have verified that in the case of MFV [7] the hierarchy ∆ LL ≫ ∆ LR,RL ≫ ∆ RR and the equality of λsbLL = λ LL are still reasonably well fulfilled numerically with the values of λ LL ≈ λctLL ranging from zero to 5× 10−3 . . . 1× 10−2 for our four typical benchmark points. Starting with Fig. 11 and tanβ = 10, the bulk region of equally low scalar and fermion masses is all but excluded by the b → sγ branching ratio. This leaves as a favoured region first the so-called focus point region of low fermion massesm1/2, where the lightest neutralinos are relatively heavy, have a significant Higgsino component, and annihilate dominantly into pairs of electroweak gauge bosons. Our benchmark point A lies in this region, albeit at smaller masses than SPS 2 (m0 = 1450 GeV, m1/2 = 300 GeV) and BDEGOP E’ (m0 = 1530 GeV, m1/2 = 300 GeV), which lie outside the region favoured by aµ (grey-shaded) and lead to collider-unfriendly squark and gaugino masses. The second favoured region for small tanβ is the co-annihilation branch of low scalar masses m0, where the lighter tau-slepton mass eigenstate is not much heavier than the lightest neutralino and the two have a considerable co- annihilation cross section. This is where we have chosen our benchmark point B, which differs from the points SPS 3 (m0 = 90 GeV, m1/2 = 400 GeV) and BDEGOP C’ (m0 = 85 GeV, m1/2 = 400 GeV) only very little in the scalar mass. This minor difference may be traced to the fact that we use DarkSUSY 4.1 [43] instead of the private dark matter program SSARD of Ref. [51]. At the larger value of tanβ = 30 in Fig. 12, only the co-annihilation region survives the constraints coming from b → sγ decays. Here we choose our point C, which has slightly higher masses than both SPS 1b (m0 = 200 GeV, m1/2 = 400 GeV) and BDEGOP I’ (m0 = 175 GeV, m1/2 = 350 GeV), due to the ever more stringent constraints from the above-mentioned rare B-decay. For the very large value of tanβ = 50 in Fig. 13, the bulk region reappears at relatively heavy scalar and fermion masses. Here, the couplings of the heavier scalar and pseudo-scalar Higgses H0 and A0 to bottom quarks and tau- leptons and the charged-Higgs coupling to top-bottom pairs are significantly enhanced, resulting e.g. in increased dark matter annihilation cross sections through s-channel Higgs-exchange into bottom-quark final states. So as tanβ increases further, the so-called Higgs-funnel region eventually makes its appearance on the diagonal of large scalar and fermion masses. We choose our point D in the concentrated (bulky) region favoured by cosmology and aµ at masses, that are slightly higher than those of SPS 4 (m0 = 400 GeV, m1/2 = 300 GeV) and BDEGOP L’ (m0 = 300 GeV, m1/2 = 450 GeV). We do so in order to escape again from the constraints of the b → sγ decay, which are stronger today than they were a few years ago. In this scenario, squarks and gluinos are very heavy with masses above 1 TeV. 0 0.2 0.4 0.6 0.8 1 0.0001 0.001 0 0.2 0.4 0.6 0.8 1 γ s →b 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Up-type squark masses ~ ≈ 4u ~ ≈ 3u 0 0.2 0.4 0.6 0.8 1 Down-type squark masses ≈ 4d ≈ 2d FIG. 14: Dependence of the precision variables BR(b → sγ), ∆ρ, and the cold dark matter relic density ΩCDMh 2 (top) as well as of the lightest SUSY particle, up- and down-type squark masses (bottom) on the NMFV parameter λ in our benchmark scenario A. The experimentally allowed ranges (within 2σ) are indicated by horizontal dashed lines. D. Dependence of Precision Observables and Squark-Mass Eigenvalues on Flavour Violation Let us now turn to the dependence of the precision variables discussed in Sec. IVA on the flavour violating parameter λ in the four benchmark scenarios defined in Sec. IVC. As already mentioned, we expect the leptonic observable aµ to depend weakly (at two loops only) on the squark sector, and this is confirmed by our numerical analysis. We find constant values of 6, 14, 16, and 13×10−10 for the benchmarks A, B, C, and D, all of which lie well within 2σ (the latter three even within 1σ) of the experimentally favoured range (22± 10)× 10−10. The electroweak precision observable ∆ρ is shown first in Figs. 14-17 for the four benchmark scenarios A, B, C, and D. On our logarithmic scale, only the experimental upper bound of the 2σ-range is visible as a dashed line. While the self-energy diagrams of the electroweak gauge bosons depend obviously strongly on the helicities, flavours, and mass eigenvalues of the squarks in the loop, the SUSY masses in our scenarios are sufficiently small and the experimental error is still sufficiently large to allow for relatively large values of λ ≤ 0.57, 0.52, 0.38, and 0.32 for the benchmark points A, B, C, and D, respectively. As mentioned above, ∆ρ conversely constrains SUSY models in cMFV only for masses above 2000 GeV for m0 and 1500 GeV for m1/2. The next diagram in Figs. 14-17 shows the dependence of the most stringent low-energy constraint, coming from the good agreement between the measured b → sγ branching ratio and the two-loop SM prediction, on the NMFV parameter λ. The dashed lines of the 2σ-bands exhibit two allowed regions, one close to λ = 0 (vertical green line) and a second one around λ ≃ 0.57, 0.75, 0.62, and 0.57, respectively. As is well-known, the latter are, however, disfavoured by b→ sµ+µ− data constraining the sign of the b→ sγ amplitude to be the same as in the SM [52]. We will therefore limit ourselves later to the regions λ ≤ 0.05 (points A, C, and D) and λ ≤ 0.1 (point B) in the vicinity of (c)MFV (see also Tab. I). The 95% confidence-level (or 2σ) region for the cold dark matter density was given in absolute values in Ref. [44] and is shown as a dashed band in the upper right part of Figs. 14-17. However, only the lower bound (0.094) is of relevance, as the relic density falls with increasing λ. This is not so pronounced in our model B as in our model A, where squark masses are light and the lightest neutralino has a sizable Higgsino-component, so that squark exchanges contribute significantly to the annihilation cross sections. For models C and D there is little sensitivity of ΩCDMh (except at very large λ ≤ 1), as the squark masses are generally larger. 0 0.2 0.4 0.6 0.8 1 0.0001 0.001 0 0.2 0.4 0.6 0.8 1 γ s →b 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Up-type squark masses ~, 4u ~, 3u ~, 2u 0 0.2 0.4 0.6 0.8 1 Down-type squark masses ≈ 4d ≈ 2d FIG. 15: Same as Fig. 14 for our benchmark scenario B. 0 0.2 0.4 0.6 0.8 1 0.00001 0.0001 0.001 0 0.2 0.4 0.6 0.8 1 γ s →b 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Up-type squark masses ~, 4u ~, 3u 0 0.2 0.4 0.6 0.8 1 Down-type squark masses FIG. 16: Same as Fig. 14 for our benchmark scenario C. 0 0.2 0.4 0.6 0.8 1 0.00001 0.0001 0.001 0 0.2 0.4 0.6 0.8 1 γ s →b 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Up-type squark masses ~ ,4u ~, 3u 0 0.2 0.4 0.6 0.8 1 Down-type squark masses ≈ 4d FIG. 17: Same as Fig. 14 for our benchmark scenario D. The rapid fall-off of the relic density for very large λ ≤ 1 can be understood by looking at the resulting lightest up- and down-type squark mass eigenvalues in the lower left part of Figs. 14-17. For maximal flavour violation, the off- diagonal squark mass matrix elements are of similar size as the diagonal ones, leading to one squark mass eigenvalue that approaches and finally falls below the lightest neutralino (dark matter) mass. Light squark propagators and co-annihilation processes thus lead to a rapidly falling dark matter relic density and finally to cosmologically excluded NMFV SUSY models, since the LSP must be electrically neutral and a colour singlet. An interesting phenomenon of level reordering between neighbouring states can be observed in the lower central diagrams of Figs. 14-17 for the two lowest mass eigenvalues of up-type squarks. The squark mass eigenstates are, by definition, labeled in ascending order with the mass eigenvalues, so that ũ1 represents the lightest, ũ2 the second- lightest, and ũ6 the heaviest up-type squark. As λ and the off-diagonal entries in the mass matrix increase, the splitting between the lightest and highest mass eigenvalues naturally increases, whereas the intermediate squark masses (of ũ3,4,5) are practically degenerate and insensitive to λ. These remarks also hold for the down-type squark masses shown in the lower right diagrams of Figs. 14-17. However, for up-type squarks it is first the second-lowest mass that decreases up to intermediate values of λ = 0.2...0.5, whereas the lowest mass is constant, and only at this point the second-lowest mass becomes constant and takes approximately the value of the until here lowest squark mass, whereas the lowest squark mass starts to decrease further with λ. These “avoided crossings” are a common phenomenon for Hermitian matrices and reminiscent of meta-stable systems in quantum mechanics. At the point where the two levels should cross, the corresponding squark eigenstates mix and change character, as will be explained in the next subsection. For scenario C (Fig. 16), the phenomenon occurs even a second time with an additional avoided crossing between the states ũ2 and ũ3 at λ ≃ 0.05. For scenario B (Fig. 15), this takes place at λ ≃ 0.1, and there is even another crossing at λ ≃ 0.02. For down-type squarks, the level-reordering phenomenon is not so pronounced. E. Chirality and Flavour Decomposition of Squark Mass Eigenstates In NMFV, squarks will not only exhibit the traditional mixing of left- and right-handed helicities of third-generation flavour eigenstates, but will in addition exhibit generational mixing. As discussed before, we restrict ourselves here to the simultaneous mixing of left-handed second- and third-generation up- and down-type squarks. For our benchmark scenario A, the helicity and flavour decomposition of the six up-type (left) and down-type (right) squark mass eigen- 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 FIG. 18: Dependence of the chirality (L, R) and flavour (u, c, t; d, s, and b) content of up- (ũi) and down-type (d̃i) squark mass eigenstates on the NMFV parameter λ ∈ [0; 1] for benchmark point A. 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 FIG. 19: Same as Fig. 18 for λ ∈ [0; 0.1]. 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 FIG. 20: Same as Fig. 18 for benchmark point B. 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 FIG. 21: Same as Fig. 20 for λ ∈ [0; 0.1]. 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 FIG. 22: Same as Fig. 18 for benchmark point C. 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 FIG. 23: Same as Fig. 22 for λ ∈ [0; 0.1]. 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 FIG. 24: Same as Fig. 18 for benchmark point D. 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 FIG. 25: Same as Fig. 24 for λ ∈ [0; 0.1]. states is shown in Fig. 18 for the full range of the parameter λ ∈ [0; 1] and in Fig. 19 for the experimentally favoured range in the vicinity of (c)MFV, λ ∈ [0; 0.1]. First-generation and right-handed second-generation squarks remain, of course, helicity- and flavour-diagonal, ũ3 = c̃R , d̃2 = s̃R, ũ4 = ũR , d̃3 = d̃R, ũ5 = ũL , d̃5 = d̃L, (66) with the left-handed and first-generation squarks being slightly heavier due their weak isospin coupling (see Eqs. (5)- (7)) and different renormalization-group running effects. Production of these states will benefit from t- and u-channel contributions of first- and second-generation quarks with enhanced parton densities in the external hadrons, but they will not be identified easily with heavy-flavour tagging and are of little interest for our study of flavour violation. The lightest up-type squark ũ1 remains the traditional mixture of left- and right-handed stops over a large region of λ ≤ 0.4, but it shows at this point an interesting flavour transition, which is in fact expected from the level reordering phenomenon discussed in the lower central plot of Fig. 14. The transition happens, however, above the experimental limit of λ ≤ 0.1. Below this limit, it is the states ũ2, ũ6, d̃1, and in particular d̃4 and d̃6 that show, in addition to helicity mixing, the most interesting and smooth variation of second- and third-generation flavour content (see Fig. 19). Note that at very low λ ≃ 0.002 the states d̃L and s̃L rapidly switch levels. This numerically small change was not visible on the linear scale in Fig. 14. For the benchmark point B, whose helicity and flavour decomposition is shown in Fig. 20, level reordering occurs at λ ≃ 0.4 for the intermediate-mass up-type squarks, ũ3,4 = c̃R , d̃2 = s̃R, ũ4,3 = ũR , d̃3 = d̃R, ũ5 = ũL , d̃5 = d̃L (67) whereas the ordering of down-type squarks is very similar to scenario A. Close inspection of Fig. 21 shows, however, that also d̃R and s̃R switch levels at low values of λ ≃ 0.02. At λ ≃ 0.01, in addition s̃R and b̃L switch levels, and at λ ≃ 0.002 it is the states ũL and c̃L. The lightest up-type squark is again nothing but a mix of left- and right-handed stops up to λ ≤ 0.4. Phenomenologically smooth transitions below λ ≤ 0.1 involving taggable third-generation squarks are observed for ũ4, ũ6, d̃1, and d̃6. The helicity and flavour decomposition for our scenario D, shown in Fig. 24, is rather similar to the one in scenario A, i.e. ũ3 = c̃R , d̃3 = s̃R, ũ4 = ũR , d̃4 = d̃R, ũ5 = ũL , d̃5 = d̃L (68) are exactly the same in the up-squark sector, and only the mixed down-type state d̃4 is now lighter and becomes d̃2. The lightest up-type squark, ũ1, is again mostly a mix of left- and right-handed top squarks up to λ ≃ 0.4, where the level reordering and generation mixing occurs (see lower central part of Fig. 17). At the experimentally favoured lower values of λ ≤ 0.1, the states ũ2, ũ6, d̃1, d̃2, and d̃6 exhibit some smooth variations, shown in detail in Fig. 25, albeit to a lesser extent than in scenario A. At very low λ ≃ 0.004, it is now the up-type squarks ũL and c̃L that rapidly switch levels. This numerically small change was again not visible on a linear scale (see Fig. 17). For our scenario C, shown in Fig. 22, the assignment of the intermediate states ũ3 = c̃R , d̃3 = s̃R, ũ4 = ũR , d̃4 = d̃R, ũ5 = ũL , d̃5 = d̃L (69) is the same as for scenario D above λ ≥ 0.1. Just below, ũR and c̃R as well as d̃R and s̃R rapidly switch levels, whereas ũL and c̃L switch levels at very low λ ≃ 0.002. These changes were already visible upon close inspection of the lower central and right plots in Fig. 16. On the other hand, the lightest squarks ũ1 and d̃1 only acquire significant flavour admixtures at relatively large λ ≃ 0.2...0.4, whereas they are mostly superpositions of left- and right-handed stops and sbottoms in the experimentally favourable range of λ ≤ 0.1 shown in Fig. 23. Here, the heaviest states ũ6 and d̃6 show already smooth admixtures of third-generation squarks as it was the case for the scenarios A and D discussed above. The most interesting states are, however, ũ2, ũ4, d̃2, and d̃4, respectively, since they represent combinations of up to four different helicity and flavour states and have a significant, taggable third-generation flavour content. V. NUMERICAL PREDICTIONS FOR NMFV SUSY PARTICLE PRODUCTION AT THE LHC In this section, we present numerical predictions for the production cross sections of squark-antisquark pairs, squark pairs, the associated production of squarks and gauginos, and gaugino pairs in NMFV SUSY at the CERN LHC, i.e. for pp-collisions at S = 14 TeV centre-of-mass energy. Thanks to the QCD factorization theorem, total unpolarized hadronic cross sections 4m2/S ∫ 1/2 ln τ −1/2 ln τ ∫ tmax dt fa/A(xa,M a ) fb/B(xb,M can be calculated by convolving the relevant partonic cross sections dσ̂/dt, computed in Sec. III, with universal parton densities fa/A and fb/B of partons a, b in the hadrons A,B, which depend on the longitudinal momentum fractions of the two partons xa,b = τe±y and on the unphysical factorization scales Ma,b. For consistency with our leading order (LO) QCD calculation in the collinear approximation, where all squared quark masses (except for the top-quark mass)m2q ≪ s, we employ the LO set of the latest CTEQ6 global parton density fit [53], which includes nf = 5 “light” (including the bottom) quark flavours and the gluon, but no top-quark density. Whenever it occurs, i.e. for gluon initial states and gluon or gluino exchanges, the strong coupling constant αs(µR) is calculated with the corresponding LO value of Λ LO = 165 MeV. We identify the renormalization scale µR with the factorization scales Ma =Mb and set the scales to the average mass of the final state SUSY particles i and j, m = (mi +mj)/2. The numerical cross sections for charged squark-antisquark and squark-squark production, neutral up- and down- type squark-antisquark and squark-squark pair production, associated production of squarks with charginos and neutralinos, and gaugino pair production are shown in Fig. 26 for our benchmark scenario A, in Fig. 27 for scenario B, in Fig. 28 for scenario C, and in Fig. 29 for scenario D. The magnitudes of the cross sections vary from the barely visible level of 10−2 fb for weak production of heavy final states over the semi-strong production of average squarks and gauginos and quark-gluon initial states to large cross sections of 102 to 103 fb for the strong production of diagonal squark-(anti)squark pairs or weak production of very light gaugino pairs. Unfortunately, these processes, whose cross sections are largest (top right, center left, and lower right parts of Figs. 26-29), are practically insensitive to the flavour violation parameter λ, as the strong gauge interaction is insensitive to quark flavours and gaugino pair production cross sections are summed over exchanged squark flavours. Some of the subleading, non-diagonal cross sections show, however, sharp transitions, in particular down-type squark-antisquark production at the benchmark point B (centre-left part of Fig. 27), but also the other squark- antisquark and squark-squark production processes. At λ = 0.02, the cross sections for d̃1d̃ 6 and d̃3d̃ 6 switch places. Since the concerned sstrange and sbottom mass differences are rather small, this is mainly due to the different strange and bottom quark densities in the proton. The cross section is mainly due to the exchange of strongly coupled gluinos despite their larger mass. At λ = 0.035 the cross sections for d̃3d̃ 6 and d̃1d̃ 3 increase sharply, since d̃3 = d̃R can then be produced from down-type valence quarks. The cross section of the latter process increases with the strange squark content of d̃1. At the benchmark point C (Fig. 28), sharp transitions occur between the ũ2/ũ4 and d̃2/d̃4 states, which are pure charm/strange squarks below/above λ = 0.035, for all types of charged and neutral squark-antisquark and squark- squark production and also squark-gaugino associated production. As a side-remark we note that an interesting perspective might be the exploitation of these t-channel contributions to second- and third-generation squark pro- duction for the determination of heavy-quark densities in the proton. This requires, of course, efficient experimental techniques for heavy-flavour tagging. Smooth transitions and semi-strong cross sections of about 1 fb are observed for the associated production of third- generation squarks with charginos (lower left diagrams) and neutralinos (lower centre diagrams) and in particular for the scenarios A and B. For benchmark point A (Fig. 26), the cross section for d̃4 production decreases with its strange squark content, while the bottom squark content increases at the same time. For benchmark point B (Fig. 27), the same (opposite) happens for d̃6 (d̃1), while the cross sections for ũ6 increase/decrease with its charm/top squark content. Even in minimal flavour violation, the associated production of stops and charginos is a particularly useful channel for SUSY particle spectroscopy, as can be seen from the fact that cross sections vary over several orders of magnitude among our four benchmark points (see also Ref. [54]). An illustrative summary of flavour violating hadroproduction cross section contributions for third-generation squarks and/or gauginos is presented in Tab. III, together with the competing flavour-diagonal contributions. λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 * + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 * + c.c. →p p ≈* 4d λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ →p p ∼ ≈ 1u λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 10000 + c.c. χ∼ →p p FIG. 26: Cross sections for charged squark-antisquark (top left) and squark-squark (top centre) production, neutral up-type (top right) and down-type (centre left) squark-antisquark and squark-squark pair (centre and centre right) production, associated production of squarks with charginos (bottom left) and neutralinos (bottom centre), and gaugino pair production (bottom right) at the LHC in our benchmark scenario A. VI. CONCLUSIONS In conclusion, we have performed an extensive analysis of squark and gaugino hadroproduction and decays in non- minimal flavour violating supersymmetry. Within the super-CKM basis, we have taken into account the possible misalignment of quark and squark rotations and computed all squared helicity amplitudes for the production and the decay widths of squarks and gauginos in compact analytic form, verifying that our results agree with the literature in the case of non-mixing squarks whenever possible. Flavour violating effects have also been included in our analysis of dark matter (co-)annihilation processes. We have then analyzed the NMFV SUSY parameter space for regions allowed by low-energy, electroweak precision, and cosmological data and defined four new post-WMAP benchmark points and slopes equally valid in minimal and non-minimal flavour violating SUSY. We found that left-chiral mixing of second- and third-generation squarks is slightly stronger constrained than previously believed, mostly due to smaller experimental errors on the b → sγ branching ratio and the cold dark matter relic density. For our four benchmark points, we have presented the dependence of squark mass eigenvalues and the flavour and helicity decomposition of λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 + c.c. u~ →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 + c.c. u~ →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 * + c.c. u~ →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 * + c.c. →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 + c.c. u~ →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 + c.c. →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 + c.c. χ∼ →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 + c.c. χ∼ →p p λ0 0.02 0.04 0.06 0.08 0.1λ0 0.02 0.04 0.06 0.08 0.1 + c.c. χ∼ →p p FIG. 27: Same as Fig. 26 for our benchmark scenario B. TABLE III: Dominant s-, t-, and u-channel contributions to the flavour violating hadroproduction of third-generation squarks and/or gauginos and the competing dominant flavour-diagonal contributions. Exchange s t u Final State t̃b̃∗ W NMFV-g̃ - b̃s̃∗ NMFV-Z NMFV-g̃ - t̃c̃∗ NMFV-Z NMFV-g̃ - t̃b̃ - - NMFV-g̃ b̃b̃ - g̃ g̃ t̃t̃ - NMFV-g̃ NMFV-g̃ χ̃0b̃ b b̃ - χ̃±b̃ NMFV-c NMFV-b̃ - χ̃0 t̃ NMFV-c NMFV-t̃ - t̃ b t̃ - χ̃χ̃ γ, Z,W q̃ q̃ λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 * + c.c. u~ iu ~ →p p u~ ≈* λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 * + c.c. →p p ≈* 2d λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ iχ ∼ →p p FIG. 28: Same as Fig. 26 for our benchmark scenario C. the squark mass eigenstates on the flavour violating parameter λ. We have computed numerically all production cross sections for the LHC and discussed in detail their dependence on flavour violation. A full experimental study including heavy-flavour tagging efficiencies, detector resolutions, and background processes would, of course, be very interesting in order to establish the experimental significance of NMFV. While the implementation of our analytical results in a general-purpose Monte Carlo generator should now be straight-forward, such a detailed experimental study represents a research project of its own [55] and is beyond the scope of the work presented here. Acknowledgments A large part of this work has been performed in the context of the CERN 2006/2007 workshop on “Flavour in the Era of the LHC”. The authors also acknowledge interesting discussions with J. Debove, A. Djouadi, W. Porod, J.M. Richard, and P. Skands. This work was supported by two Ph.D. fellowships of the French ministry for education and research. λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 * + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 * + c.c. →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. u~ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ →p p λ0 0.01 0.02 0.03 0.04 0.05λ0 0.01 0.02 0.03 0.04 0.05 + c.c. χ∼ →p p FIG. 29: Same as Fig. 26 for our benchmark scenario D. APPENDIX A: GAUGINO AND HIGGSINO MIXING The soft SUSY-breaking terms in the minimally supersymmetric Lagrangian include a term L ⊃ −1 (ψ0)T Y ψ0 + h.c., (A1) which is bilinear in the (2-component) fermionic partners ψ0j = (−iB̃,−iW̃ 3, H̃01 , H̃02 )T (A2) of the neutral electroweak gauge and Higgs bosons and proportional to the, generally complex and symmetric, neu- tralino mass matrix M1 0 −mZ sW cβ mZ sW sβ 0 M2 mZ cW cβ −mZ cW sβ −mZ sW cβ mZ cW cβ 0 −µ mZ sW sβ −mZ cW sβ −µ 0  . (A3) Here, M1, M2, and µ are the SUSY-breaking bino, wino, and off-diagonal higgsino mass parameters with tanβ = sβ/cβ = vu/vd being the ratio of the vacuum expectation values vu,d of the two Higgs doublets, while mZ is the SM Z- boson mass and sW (cW ) is the sine (co-sine) of the electroweak mixing angle θW . After electroweak gauge-symmetry breaking and diagonalization of the mass matrix Y , one obtains the neutralino mass eigenstates χ0i = Nij ψ j , i, j = 1, . . . , 4, (A4) where N is a unitary matrix satisfying the relation N∗ Y N−1 = diag (mχ̃0 ,mχ̃0 ,mχ̃0 ,mχ̃0 ). (A5) In 4-component notation, the Majorana-fermionic neutralino mass eigenstates can be written as χ̃0i = , i = 1, . . . , 4. (A6) Their mass eigenvalues mχ̃0 can, e.g., be found in analytic form in [56] and can be chosen to be real and non-negative. The chargino mass term in the SUSY Lagrangian L ⊃ −1 (ψ+ψ−) + h.c. (A7) is bilinear in the (2-component) fermionic partners ψ±j = (−iW̃ ±, H̃±2,1) T (A8) of the charged electroweak gauge and Higgs bosons and proportional to the, generally complex, chargino mass matrix M2 mW 2 cβ µ , (A9) where mW is the mass of the SM W -boson. 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D 45 (1992) 4345. http://arxiv.org/abs/hep-ph/0703200 http://arxiv.org/abs/hep-ph/0703204 http://arxiv.org/abs/hep-ph/0702144 http://arxiv.org/abs/hep-ph/0512315 http://arxiv.org/abs/hep-ex/0603003 Introduction Non-Minimal Flavour Violation in the MSSM Analytical Results for Production Cross Sections and Decay Widths Generalized Strong and Electroweak Couplings in NMFV SUSY Squark-Antisquark Pair Production Squark Pair Production Associated Production of Squarks and Gauginos Gaugino Pair Production Squark Decays Gluino Decays Gaugino Decays Experimental Constraints, Scans and Benchmarks in NMFV SUSY Low-Energy, Electroweak Precision, and Cosmological Constraints Scans of the Constrained NMFV MSSM Parameter Space (c)MFV and NMFV Benchmark Points and Slopes Dependence of Precision Observables and Squark-Mass Eigenvalues on Flavour Violation Chirality and Flavour Decomposition of Squark Mass Eigenstates Numerical Predictions for NMFV SUSY Particle Production at the LHC Conclusions Acknowledgments Gaugino and Higgsino Mixing References

0704.1827

Transaction-Oriented Simulation In Ad Hoc Grids

Microsoft Word - MSc thesis - G Krafft - 2007.doc TRANSACTION-ORIENTED SIMULATION IN AD HOC GRIDS Gerald Krafft This report is submitted in partial fulfilment of the requirements of the M.Sc. degree in Advanced Computer Science at the Westminster University. Supervisor: Vladimir Getov Submitted on: 24th January 2007 Abstract Abstract Computer Simulation is an important area of Computer Science that is used in many other research areas like for instance engineering, military, biology and climate research. But the growing demand for more and more complex simulations can lead to long runtimes even on modern computer systems. Performing complex Computer Simulations in parallel, distributed across several processors or computing nodes within a network has proven to reduce the runtime of such complex simulations. Large-scale parallel computer systems are usually very expensive. Grid Computing is a cost-effective way to perform resource intensive computing tasks because it allows several organisations to share their computing resources. Besides more traditional Computing Grids the concept of Ad Hoc Grids has emerged that offers a dynamic and transient resource-sharing infrastructure, suitable for short-term collaborations and with a very small administrative overhead to allow even small organisations or individual users to form Computing Grids. A Grid framework that fulfils the requirements of Ad Hoc Grids is ProActive. This paper analyses the possibilities of performing parallel transaction-oriented simulations with a special focus on the space-parallel approach and discrete event simulation synchronisation algorithms that are suitable for transaction-oriented simulation and the target environment of Ad Hoc Grids. To demonstrate the findings a Java-based parallel transaction-oriented simulator is implemented on the basis of the promising Shock Resistant Time Warp synchronisation algorithm and using the Grid framework ProActive. The validation of this parallel simulator shows that the Shock Resistant Time Warp algorithm can successfully reduce the number of rolled back Transaction moves but it also reveals circumstances in which the Shock Resistant Time Warp algorithm can be outperformed by the normal Time Warp algorithm. The conclusion of this paper suggests possible improvements to the Shock Resistant Time Warp algorithm to avoid such problems. Table of Content Table of Content 1 INTRODUCTION ...................................................................................................1 2 FUNDAMENTAL CONCEPTS .............................................................................4 2.1 GRID COMPUTING...............................................................................................4 2.1.1 Ad Hoc Grids .............................................................................................5 2.2 GRANULARITY AND HARDWARE ARCHITECTURE................................................6 2.3 SIMULATION TYPES.............................................................................................7 2.3.1 Transaction-Oriented Simulation and GPSS.............................................8 2.4 PARALLELISATION OF DISCRETE EVENT SIMULATION .........................................9 2.5 SYNCHRONISATION ALGORITHMS .....................................................................10 2.5.1 Conservative Algorithms .........................................................................11 2.5.2 Optimistic Algorithms..............................................................................11 3 AD HOC GRID ASPECTS ...................................................................................16 3.1 CONSIDERATIONS..............................................................................................16 3.1.1 Service Deployment .................................................................................16 3.1.2 Service Migration ....................................................................................17 3.1.3 Fault Tolerance........................................................................................17 3.1.4 Resource Discovery .................................................................................17 3.2 PROACTIVE ......................................................................................................18 3.2.1 Descriptor-Based Deployment.................................................................19 3.2.2 Peer-to-Peer Infrastructure.....................................................................19 3.2.3 Active Object Migration ..........................................................................20 3.2.4 Transparent Fault Tolerance ...................................................................20 4 PARALLEL TRANSACTION-ORIENTED SIMULATION............................21 4.1 PAST RESEARCH WORK......................................................................................21 4.1.1 Transactions as events.............................................................................22 4.1.2 Accessing objects in other LPs ................................................................22 4.1.3 Analysis of GPSS language .....................................................................24 4.2 SYNCHRONISATION ALGORITHM .......................................................................26 4.2.1 Requirements ...........................................................................................26 Table of Content 4.2.2 Algorithm selection..................................................................................27 4.2.3 Shock resistant Time Warp Algorithm......................................................29 4.3 GVT CALCULATION .........................................................................................33 4.4 END OF SIMULATION.........................................................................................35 4.5 CANCELLATION TECHNIQUES ...........................................................................36 4.6 LOAD BALANCING ............................................................................................37 4.7 MODEL PARTITIONING ......................................................................................38 5 IMPLEMENTATION............................................................................................40 5.1 IMPLEMENTATION CONSIDERATIONS.................................................................40 5.1.1 Overall Architecture ................................................................................40 5.1.2 Transaction Chain and Scheduling..........................................................42 5.1.3 Generation and Termination of Transactions..........................................43 5.1.4 Supported GPSS Syntax...........................................................................45 5.1.5 Simulation Termination at Specific Simulation Time...............................46 5.2 IMPLEMENTATION PHASES ................................................................................47 5.2.1 Model Parsing .........................................................................................47 5.2.2 Basic GPSS Simulation Engine ...............................................................50 5.2.3 Time Warp Parallel Simulation Engine ...................................................51 5.2.4 Shock Resistant Time Warp......................................................................53 5.3 SPECIFIC IMPLEMENTATION DETAILS ................................................................56 5.3.1 Scheduling ...............................................................................................56 5.3.2 GVT Calculation and End of Simulation.................................................64 5.3.3 State Saving and Rollbacks......................................................................66 5.3.4 Memory Management ..............................................................................68 5.3.5 Logging....................................................................................................69 5.4 RUNNING THE PARALLEL SIMULATOR...............................................................70 5.4.1 Prerequisites ............................................................................................70 5.4.2 Files .........................................................................................................70 5.4.3 Configuration...........................................................................................72 5.4.4 Starting a Simulation...............................................................................73 5.4.5 Increasing Memory Provided by JVM.....................................................73 Table of Content 6 VALIDATION OF THE PARALLEL SIMULATOR.........................................75 6.1 VALIDATION 1 ...................................................................................................76 6.2 VALIDATION 2 ...................................................................................................78 6.3 VALIDATION 3 ...................................................................................................80 6.4 VALIDATION 4 ...................................................................................................84 6.5 VALIDATION 5 ...................................................................................................86 6.6 VALIDATION 6 ...................................................................................................91 6.7 VALIDATION ANALYSIS .....................................................................................93 7 CONCLUSIONS....................................................................................................96 REFERENCES ..............................................................................................................98 APPENDIX A: DETAILED GPSS SYNTAX............................................................102 APPENDIX B: SIMULATOR CONFIGURATION SETTINGS............................107 APPENDIX C: SIMULATOR LOG4J LOGGERS .................................................110 APPENDIX D: STRUCTURE OF THE ATTACHED CD ......................................114 APPENDIX E: DOCUMENTATION OF SELECTED CLASSES.........................115 APPENDIX F: VALIDATION OUTPUT LOGS......................................................147 Abbreviations Abbreviations GFT Global Furthest Time GPSS General Purpose Simulation System GPW Global Progress Window GVT Global Virtual Time J2SE Java 2 Platform, Standard Edition JRE Java Runtime Environment JVM Java Virtual Machine LP Logical Process LPCC Logical Process Control Component LVT Local Virtual Time NAT Network Address Translation Figures Figures Figure 1: Ad Hoc Grid architecture overview [27]............................................................6 Figure 2: Comparison of fine grained and coarse grained granularity ..............................7 Figure 3: Classification of simulation types ......................................................................7 Figure 4: Parallelisation of independent simulation runs ..................................................9 Figure 5: Event execution with rollback [21]..................................................................12 Figure 6: Comparison of aggressive and lazy cancellation .............................................13 Figure 7: Accessing objects in other LPs.........................................................................23 Figure 8: Global Progress Window with its zones [33]...................................................28 Figure 9: Overview of LP and LPCC in Shock Resistant Time Warp [8] .......................30 Figure 10: Cancellation in transaction-oriented simulation ............................................37 Figure 11: Architecture overview....................................................................................41 Figure 12: Single synchronised Termination Counter .....................................................45 Figure 13: Simulation model class hierarchies for parsing and simulation runtime .......49 Figure 14: Parallel simulator main class overview (excluding LPCC) ...........................51 Figure 15: Main communication sequence diagram........................................................53 Figure 16: Parallel simulator main class overview..........................................................54 Figure 17: Scheduling flowchart - part 1.........................................................................57 Figure 18: Scheduling flowchart - part 2.........................................................................58 Figure 19: Scheduling flowchart - part 3.........................................................................59 Figure 20: Extended parallel simulation scheduling flowchart .......................................62 Figure 21: GVT calculation sequence diagram ...............................................................64 Figures Figure 22: State saving example......................................................................................67 Figure 23: Validation 5.1 Actuator value graph...............................................................89 Tables Tables Table 1: Change of Transaction execution path...............................................................25 Table 2: Access to objects................................................................................................25 Table 3: Shock Resistant Time Warp sensors ..................................................................31 Table 4: Shock Resistant Time Warp indicators ..............................................................31 Table 5: Transaction-oriented sensor names....................................................................33 Table 6: Transaction-oriented indicator names................................................................33 Table 7: Overview of supported GPSS block types.........................................................46 Table 8: Methods implementing basic GPSS scheduling functionality...........................60 Table 9: Methods implementing extended parallel simulation scheduling .....................63 Table 10: Validation 5.1 LPCC Actuator values..............................................................89 Table 11: LP2 processing statistics of validation 5..........................................................90 Introduction 1 Introduction Computer Simulation is one of the oldest areas in Computer Science. It provides answers about the behaviour of real or imaginary systems that otherwise could only be gained under great expenditure of time, with high costs or that could not be gained at all. Computer Simulation uses simulation models that are usually simpler than the systems they represent but that are expected to behave as analogue as possible or as required. The growing demand of complex Computer Simulations for instance in engineering, military, biology and climate research has also lead to a growing demand in computing power. One possibility to reduce the runtime of large, complex Computer Simulations is to perform such simulations distributed on several CPUs or computing nodes. This has induced the availability of high-performance parallel computer systems. Even so the performance of such systems has constantly increased, the ever-growing demand to simulate more and more complex systems means that suitable high- performance systems are still very expensive. Grid computing promises to provide large-scale computing resources at lower costs by allowing several organisations to share their resources. But traditional Computing Grids are relatively static environments that require a dedicated administrative authority and are therefore less well suited for transient short-term collaborations and small organisations with fewer resources. Ad Hoc Grids provide such a dynamic and transient resource-sharing infrastructure that allows even small organisations or individual users to form Computing Grids. They will make Grid computing and Grid resources widely available to small organisations and mainstream users allowing them to perform resource intensive computing tasks like Computer Simulations. There are several approaches to performing Computer Simulations distributed across a parallel computer system. The space-parallel approach [12] is one of these approaches that is robust, applicable to many different simulation types and that can be used to speed up single simulation runs. It requires the simulation model to be partitioned into relatively independent sub-systems that are then performed in parallel on several nodes. Synchronisation between these nodes is still required because the model sub-systems are not usually fully independent. A lot of past research has concentrated on different Introduction synchronisation algorithms for parallel simulation. Some of these are only suitable for certain types of parallel systems, like for instance shared memory systems. This work investigates the possibility of performing parallel transaction-oriented simulation in an Ad Hoc Grid environment with the main focus on the aspects of parallel simulation. Potential synchronisation algorithms and other simulation aspects are analysed in respect of their suitability for transaction-oriented simulation and Ad Hoc Grids as the target environment and the chosen solutions are described and reasons for their choice given. A past attempt to investigate the parallelisation of transaction- oriented simulation was presented in [19] with the result that the synchronisation algorithm employed was not well suited for transaction-oriented simulation. Lessons from this past attempt have been learned and included in the considerations of this work. Furthermore this work outlines certain requirements that a Grid environment needs to fulfil in order to be appropriate for Ad Hoc Grids. The proposed solutions are demonstrated by implementing a Java-based parallel transaction-oriented simulator using the Grid middleware ProActive [15], which fulfils the requirements described before. The specific simulation type transaction-oriented simulation was chosen because it is still taught at many universities and is therefore well known. It uses a relatively simple language for the modelling that does not require extensive programming skills and it is a special type of discrete event simulation so that most findings can also be applied to this wider simulation classification. The remainder of this report is organised as follows. Section 2 introduces the fundamental concepts and terminology essential for the understanding of this work. In section 3 the specific requirements of Ad Hoc Grids are outlined and the Grid middleware ProActive is briefly described as an environment that fulfils these requirements. Section 4 focuses on the aspects of parallel simulation and their application to transaction-oriented simulation. Past research results are discussed, requirements for a suitable synchronisation algorithm outlined and the most promising algorithm selected. This section also addresses other points related to parallel transaction-oriented simulation like GVT calculation, handling of the simulation end, suitable cancellation techniques and the influence of the model partitioning. Section 5, which is the largest section of this report, describes the implementation of the parallel Introduction transaction-oriented simulator, starting from the initial implementation considerations and the implementation phases to specific details of the implementation and how the simulator is used. The functionality of the implemented parallel simulator is then validated in section 6 and the final conclusions presented in section 7. Fundamental Concepts 2 Fundamental Concepts This main section introduces the fundamental concepts and terminology essential for the understanding of this work. It covers areas like Grid Computing and the relation between the granularity of parallel algorithms and their expected target hardware architecture. It also describes the classification of simulation models as well as different approaches to parallel discrete event simulation and the main groups of synchronisation algorithms. 2.1 Grid Computing The term “the Grid” first appeared in the mid-1990s in connection with a proposed distributed computing infrastructure for advanced science and engineering [9]. Today Grid computing is commonly used for a „distributed computing infrastructure that supports the creation and operation of virtual organizations by providing mechanisms for controlled, cross-organizational resource sharing“ [9]. Similar to electric power grids Grid computing provides computational resources to clients using a network of multi organisational resource providers establishing a collaboration. In the context of Grid computing resource sharing means the “access to computers, software, data, and other resources” [9]. Control is needed for the sharing of resources that describes who is providing and who is consuming resources, what is shared and what are the conditions for the resource sharing to occur. These sharing rules and the group of organisations or individuals that are defined by it form a so-called Virtual Organisation (VO). Grid computing technology has evolved and gone through several phases since it’s beginning [9]. The first phase was characteristic for custom solutions to Grid computing problems. These were usually built directly on top of Internet protocols with limited functionality for security, scalability and robustness and interoperability was not considered to be important. From 1997 the emerging open source Globus Toolkit version 2 (GT2) became the de facto standard for Grid computing. It provided usability and interoperability via a set of protocols, APIs and services and was used in many Grid deployments worldwide. With the Open Grid Service Architecture (OGSA), which is a true community standard, came the shift of Grid computing towards a service-oriented architecture. In addition to a set of standard interfaces and services OGSA provides the framework in which a wide range of interoperable and portable services can be defined. Fundamental Concepts 2.1.1 Ad Hoc Grids Traditional computing Grids share certain characteristics [1]. They usually use a dedicated administrative authority, which often consists of a group of trained professionals to regulate and control membership and sharing rules of the Virtual Organisations. This includes administration of policy enforcement, monitoring and maintenance of the Grid resources. Well-defined policies are used for access privileges and the deployment of Grid applications and services. It can be seen that these common characteristics are not ideal for a transient short-term collaboration with a dynamically changing structure because the administrative overhead for establishing and maintaining such a Virtual Organisation could outweigh its benefits [2]. Ad Hoc Grids provide this kind of dynamic and transient resource sharing infrastructure. According to [27] “An Ad Hoc Grid is a spontaneous formation of cooperating heterogeneous computing nodes into a logical community without a preconfigured fixed infrastructure and with only minimal administrative requirements”. The transient dynamic structure of an Ad Hoc Grid means that new nodes can join or leave the collaboration at almost any time but Ad Hoc Grids can also contain permanent nodes. Figure 1 [27] at the next page shows two example Ad Hoc Grids structures. Ad hoc Grid A is a collaboration of nodes from two organisations. It contains permanent nodes in form of dedicated high-performance computers but also transient nodes in form of non- dedicated workstations. Compared to this Ad hoc Grids B is an example for a more personal Grid system. It consists entirely of transient individual nodes. A practical example for the application of an Ad Hoc Grid is a group of scientists that for a specific scientific experiment want to collaborate and share computing resources. Using Ad Hoc Grid technology they can establish a short-term collaboration lasting only for the time of the experiment. These scientists might be part of research organisations but as the example of Ad hoc Grid B from Figure 1 shows Ad Hoc Grids allow even individuals to form Grid collaborations without the resources of large organisations. This way Ad Hoc Grids offer a way to more mainstream and personal Grid computing. Fundamental Concepts Figure 1: Ad Hoc Grid architecture overview [27] 2.2 Granularity and Hardware Architecture When evaluating the suitability of different parallel algorithms for a specific parallel hardware architecture it is important to consider the granularity of the parallel algorithms and to compare the granularity to the processing and communication performance provided by the hardware architecture. Definition: granularity The granularity of a parallel algorithm can be defined as the ratio of the amount of computation to the amount of communication performed [18]. According to this definition parallel algorithms with a fine grained granularity perform a large amount of communication compared to the actual computation as apposed to parallel algorithms with a coarse grained granularity which only perform a small Fundamental Concepts amount of communication compared to the computation. The following diagram illustrates the difference between fine grained and coarse grained granularity. Granularity fine grained coarse grained Chronological sequence of the algorithms computation communication Figure 2: Comparison of fine grained and coarse grained granularity Independent of the exact performance figures of a parallel hardware architecture it can be seen that for a hardware architecture with a high communication performance a fine grained parallel algorithm is well suited and that a hardware architecture with a low communication performance will require a coarse grained parallel algorithms [23]. 2.3 Simulation Types Simulation models are classified into continuous and discrete simulation according to when state transitions can occur [14]. Figure 3 below illustrates the classification of simulation types. Simulation DiscreteContinuous Discrete-eventTime-controlled Event-oriented Activity-oriented Process-oriented Transaction-oriented Figure 3: Classification of simulation types Fundamental Concepts In continuous simulation the state can change continuously with the time. This type of simulation uses differential equations, which are solved numerically to calculate the state. Continuous models are for instance used to simulate the streaming of liquids or Discrete simulation models allow the changing of the state only at discrete time intervals. They can be divided further according to whether the discrete time intervals are of fixed or variable length. In a time-controlled simulation model the time advances in fixed steps changing the state after each step as required. But for many systems the state only changes in variable intervals, which are determined during the simulation. For these systems the discrete event simulation model is used. In discrete event simulation the state of the entities in the system is changed by events. Each event is linked to a specific simulated time and the simulation system keeps a list of events sorted by their time [35]. It then selects the next event according to its time stamp and executes it resulting in a change of the system state. The simulated time then jumps to the time of the next event that will be executed. The execution of an event can create new events with a time greater than the current simulated time that will be sorted into the event list according to their time stamp. Discrete event simulation is very flexible and can be applied to many groups of systems, which is why many general- purpose simulation systems use the discrete event model and a lot of research has gone into this model. 2.3.1 Transaction-Oriented Simulation and GPSS A special case of the discrete event simulation is the transaction-oriented simulation. Transaction-oriented simulation uses two types of objects. There are stationary objects that make up the model of the system and then there are mobile objects called Transactions that move through the system and that can change the state of the stationary objects. The movement of a Transaction happens at a certain time (i.e. the time does not progress while the Transaction is moved), which is equivalent to an event in the discrete event model. But stationary objects can delay the movement of a Transaction by a random or fixed time. They can also spawn one Transaction into several or assemble several sub-Transactions back to one. The fact that transaction- oriented simulation systems are usually a bit simpler than full discrete event simulation Fundamental Concepts systems makes them very useful for teaching purpose and academic use, especially because most discrete event simulation aspects can be applied to transaction-oriented simulation and vice versa. The best-known transaction-oriented simulation language is GPSS, which stand for General Purpose Simulation System. GPSS was developed by Geoffrey Gordon at IBM around 1960 and has contributed important concepts to discrete event simulation. Later improved versions of the GPSS language were implemented in many systems, two of which are GPSS/H [36] and GPSS/PC. A detailed description of transaction-oriented simulation and the improved GPSS/H language can be found in [26]. 2.4 Parallelisation of Discrete Event Simulation Parallelisation of computer simulation is important because the growing performance of modern computer systems leads to a demand for the simulation of more and more complex systems that still result in excessive simulation time. Parallelisation reduces the time required for simulating such complex systems by performing different parts of the simulation in parallel on multiple CPUs or multiple computers within a network. There are different approaches for the parallelisation of discrete event simulation that also cover different levels of parallelisation. One approach is to perform independent simulation runs in parallel [21]. There is only little communication needed for this approach, as it is limited to sending the model and a set of parameters to each node and collecting the simulation results after the simulation runs have finished. But this approach is relatively trivial and does not reduce the simulation time of a single simulation run. It can be used for simulations that consist of many shorter simulation runs. But these simulation runs have to be independent from each other (i.e. parameters for the simulation runs do not depend on results from each other). Simulation runs Simulation runs Simulation runs Parameter generation Distributing parameters Collecting results Analysation and visualisation Figure 4: Parallelisation of independent simulation runs Fundamental Concepts Two other approaches for the parallelisation of discrete event simulation are the time- parallel approach and the space-parallel approach [12]. Both can be used to reduce the simulation time of single simulation runs. The time-parallel approach partitions the simulated time into intervals [T1, T2], [T2, T3], …, [Ti, Ti+1]. Each of these time intervals is then run on separate processors or nodes. This approach relies on being able to determine the starting state of each time interval before the simulation of the earlier time interval has been completed, e.g. it has to be possible to determine the state T2 before the simulation of the time interval [T1, T2] has been completed which is only possible for certain systems to be simulated, e.g. systems with state recurrences. For the space-parallel approach the system model is partitioned into relatively independent sub-systems. Each of these sub-systems is then assigned and performed by a logical process (LP) with different LPs running on separate processors or nodes. In most cases these sub-systems will not be completely independent from each other, which is why the LPs will have to communicate with each other in order to exchange events. The space-parallel approach offers greater robustness and is applicable to most discrete event systems but the resulting speedup will depend on how the system is partitioned and how relative independent the resulting sub-systems are. A high dependency between the sub-systems will result in an increased synchronisation and communication overhead between the LPs. It will further depend on the synchronisation algorithm used. 2.5 Synchronisation Algorithms The central problem for the space-parallel simulation approach is the synchronisation of the event execution. This synchronisation is also called time management. In discrete event simulation each event has a time stamp, which is the simulated time at which the event occurs. If two events are causal dependent on each other then they have to be performed in the correct order. Because causal dependent events could originate in different LPs synchronisation between the LPs becomes very important. There are two main classes of algorithms for the event synchronisation between LPs, which are the classes of conservative and optimistic algorithms. Fundamental Concepts 2.5.1 Conservative Algorithms Conservative algorithms prevent that causal dependent events are executed out of order by executing only “safe” events [12]. An LP will consider an event to be “safe” if it is guaranteed that the LP cannot later receive an event with an earlier time stamp. The main task of conservative algorithms is to provide such guarantees so that LPs can determine which of the events are guaranteed and can be executed. Definition: guaranteed event An event e with the timestamp t which is to be executed in LPi is called guaranteed event if LPi knows all events with a timestamp t’ < t that it will need to execute during the whole simulation. One drawback of conservative algorithms is that LPs will have to wait or block if they don’t have any “safe” events. This can even lead to deadlocks where all LPs are waiting for guarantees so that they can execute their events. Many of the conservative algorithms also require additional information about the simulation model like the communication topology1 or lookahead2 information. Further details about conservative algorithms can be found in [12], [34]. 2.5.2 Optimistic Algorithms Optimistic algorithms allow causal dependent events to be executed out of order first but they provide mechanisms to later recover from possible violations of the causal order. The best-known and most analysed optimistic algorithm is Time Warp [16] on which many other optimistic algorithms are based. In Time Warp an LP will first execute all events in its local event list but if it receives an event from another LP with a time stamp smaller than the ones already executed then it will rollback all its events that should have been executed after the event just received. State checkpointing3 (also 1 describes which LP can send events to which other LP 2 is the models ability to predict the future course of events [5] 3 the state of the simulation is saved into a state list together with the current simulation time after the execution of each event or in other defined intervals Fundamental Concepts known as state saving) is used in order to be able to rollback the state of the LP if required. e1 5 e2 10 rollback e3 8 e2 10 simulated Figure 5: Event execution with rollback [21] Figure 5 shows an example LP that performs the local events e1 and e2 with the time stamps of 5 and 10 but then receives another event e3 with a time stamp of 8 from a different LP. At this point LP1 will rollback the execution of event e2 then execute the newly received event e3 and afterwards execute the event e2 again in order to retain the causal order of the events. The rollback of already executed events can result in having to rollback events that have already been sent to other LPs. To archive this anti-events are sent to the same LPs like the original events, which will result in the original event being deleted if it has not been executed yet or in a rollback of the received event and all later events. These rollbacks and anti-events can lead to large cascaded rollbacks and many events having to be executed again. It is also possible that after the rollback the same events that have been rolled back are executed again sending out the same events to other LPs for which anti-events were sent during the rollback. In order to avoid this a different mechanism for cancelling events exists which is called lazy cancellation [13]. Compared to the original cancellation mechanism that is also called aggressive cancellation and was suggested by Jefferson [16], the lazy cancellation mechanism does not send out anti-events immediately during the rollback but instead keeps a history of the events sent that have been rolled back and only sends out anti-events when the event that was sent and rolled back is not re-executed. If for instance the LP is rolled back from the simulation time t’ to the new simulation time t’’ ≤ t’ then the lazy cancellation mechanism will re-execute the events in the interval [t’’,t’] and will only sent anti- events for events that had been sent during the first execution of that time interval but that were not generated during the re-execution. The difference between aggressive Fundamental Concepts cancellation and lazy cancellation can be seen in the following diagram. In this diagram the event index is describing the scheduled time of the event. e3 e3e2 e4 e4 e4 e4- e4 e3 e3e2 Lazy cancellationAggressive cancellation timetime ex ex- Event for other LP Rollback Anti-event for other LP Figure 6: Comparison of aggressive and lazy cancellation As shown in Figure 6 lazy cancellation can reduce cascaded rollbacks but it can also allow false events to propagate further and therefore lead to longer cascaded rollbacks when such false events are cancelled. The concept of Global Virtual Time (GVT) is used to regain memory and to control the overall global progress of the simulation. The GVT is defined as the minimum of the local simulation time, also called Local Virtual Time (LVT), of all LPs and of the time stamps of all events that have been send but not yet processed by the receiving LP [16]. The GVT describes the minimum simulation time any unexecuted event can have at a particular point in real time. It therefore acts as a guarantee for all executed events with a time stamp smaller than the GVT, which can now be deleted. Further memory is freed by removing all state checkpoints with a virtual time less than the GVT except the one closest to the GVT. Both conservative and optimistic algorithms have their advantages and disadvantages. The speedup of conservative algorithms can be limited because only guarantied events are executed. Compared to conservative algorithm optimistic algorithms can offer grater exploitation of parallelism [5] and they are less reliant on application specific information [11] or information about the communication topology. But optimistic algorithms have the overhead of maintaining rollback information and over-optimistic Fundamental Concepts event execution in some LPs can lead to frequent and cascaded rollbacks and result in a degradation of the effective processing rate of events. Therefore research has focused on combining the advantages of conservative and optimistic algorithms creating so-called hybrid algorithms and on controlling the optimism in Time Warp. Such attempts to limit the optimism in Time Warp can be grouped into non-adaptive algorithms, adaptive algorithms with local state and adaptive algorithms with global state. Carl Tropper [34] and Samir R. Das [5] both give a good overview on algorithms in these categories. The group of non-adaptive algorithms for instance contains algorithms that use time windows in order to limit how far ahead of the current GVT single LPs can process their events, which limits the frequency and length of rollbacks. Other algorithms in this group add conservative ideas to Time Warp. One example for this is the Breathing Time Buckets algorithm (also known as SPEEDES algorithm) [30]. Like Time Warp this algorithm executes all local events immediately and performs local rollbacks if required but it only sends events to other LPs that have been guaranteed by the current GVT and by doing so avoids cascaded rollbacks. The problem of all these algorithms is that either the effectiveness depends on finding the optimum value for static parameters like the window size or conservative aspects of the algorithm limit its effectiveness for models with certain characteristics. Finding the optimum value for such control parameters can be difficult for simulation modellers and many simulation models show a very dynamic behaviour of their characteristics, which would require different parameters at different times of the simulation. Adaptive algorithms solve this problem by dynamically adapting the control parameters of the synchronisation algorithm according to “selected aspects of the state of the simulation” [24]. Some of these algorithms use mainly global state information like the Adaptive Memory Management algorithm [6], which uses the total amount of memory used by all LPs or the Near Perfect State Information algorithms [28] that are based on the availability of a reduced information set that almost perfectly describes the current global state of the simulation. Adaptive algorithms based on local state use only local information available to each LP in order to change the control parameters. They collect historic local state information and from these try to predict future local states and the required control parameter. Some examples for adaptive algorithms using local state Fundamental Concepts information are Adaptive Time Warp [4], Probabilistic Direct Optimism Control [7] and the Shock Resistant Time Warp algorithm [8]. Ad Hoc Grid Aspects 3 Ad Hoc Grid Aspects There are certain requirements that a Grid environment needs to fulfil in order to be suitable for Ad Hoc Grids. These are outlined in this main section of the report and the Grid middleware ProActive [15] is chosen for the planned implementation of a Grid- based parallel simulator because it fulfils the requirements mentioned. 3.1 Considerations In section 2.1.1 Ad Hoc Grids where described as dynamic, spontaneous and transient resource sharing infrastructures. The dynamic and transient structure of Ad Hoc Grids and the fact that Ad Hoc Grids should only have a minimal administrative overhead compared to traditional Grids creates special requirements that a Grid environment needs to fulfil in order to be suitable for Ad Hoc Grids. These requirements include automatic service deployment, service migration, fault tolerance and the discovery of resources. 3.1.1 Service Deployment In a traditional Grid environment the deployment of Grid services is performed by an administrative authority that is also responsible for the usage policy and the monitoring of the Grid services. Grid services are usually deployed by installing a service factory onto the nodes. A service is then instantiated by calling the service factory for that service which will return a handle to the newly created service instance. In traditional Grid environments the deployment of service factories requires special access permissions and is performed by administrators. Because of their dynamically changing structure Ad Hoc Grids need different ways of deploying Grid services that impose less administrative overhead. Automatic or hot service deployment has been suggested as a possible solution [10]. A Grid environment suitable for Ad Hoc Grids will have to provide means of installing services onto nodes either automatically or with very little administrative overhead. Ad Hoc Grid Aspects 3.1.2 Service Migration Because Ad Hoc Grids allow a transient collaboration of nodes and the fact that nodes can join or leave the collaboration at different times a Grid application cannot rely on the discovered resources to be available for the whole runtime of the application. One solution to reach some degree of certainty about the availability of resources within an Ad Hoc Grid is the introduction of a scheme where individual nodes of the Grid guarantee the availability of the resources provided by them for a certain time as suggested in [1]. But such guaranties might not be possible for all transient nodes, especially for personal individual nodes as shown in the example Ad hoc Grid B in section 2.1.1. Whether or not guarantees are used for the availability of resources an application running within an Ad Hoc Grid will have to migrate services or other resources from a node that wishes to leave the collaboration to another node that is available. The migration of services or processes within distributed systems is a known problem and a detailed description can be found in [32]. A Grid environment for Ad Hoc Grids will have to support service migration in order to adapt to the dynamically changing structure of the Grid. 3.1.3 Fault Tolerance Ad Hoc Grids can contain transient nodes like personal computer and there might be no guarantee for how long such nodes are available to the Grid application. In addition Ad Hoc Grids might be based on off-the-shelf computing and networking hardware that is more susceptible to hardware faults than special purpose build hardware. A Grid environment suitable for Ad Hoc Grids will therefore have to provide mechanisms that offer fault tolerance and that can handle the loss of the connection to a node or the unexpected disappearing of a node in a manner that is transparent to Grid applications using the Ad Hoc Grid. 3.1.4 Resource Discovery Resource discovery is one of the main tasks of Grid environments. It is often implemented by a special resource discovery service that keeps a directory of available Ad Hoc Grid Aspects resources and their specifications. But in an Ad Hoc Grid this becomes more of a challenge because of its dynamically changing structure. The task of the resource discovery can be divided further into the sub tasks of node discovery and node property assessment [27]. The node discovery task deals with the detection of new nodes that are joining and existing nodes that are leaving the collaboration. In an Ad Hoc Grid this detection has to be optimised towards the detection of frequent changes in the Grid structure. When a new node has joined the collaboration then its properties and shared resources will have to be discovered which is described by the node property assessment task. In addition to this high-level resource information some Grid environments also provide low-level resource information about the nodes. Such low- level resource information can include properties like the operating system type and available hardware resources. But depending on the abstraction level implemented by the Grid environment such low-level resource information might not be needed nor be accessible for Grid applications. The minimum resource discovery functionality that an Ad Hoc Grid environment has to provide is the node discovery and more specifically the detection of new nodes joining the Grid structure and existing nodes that are leaving the structure. 3.2 ProActive ProActive is a Grid middleware implemented in Java that supports parallel, distributed, and concurrent computing including mobility and security features within a uniform framework [15]. It is developed and maintained as an open-source project at INRIA4 and uses the Active Object pattern to provide remotely accessible objects that can act as Grid services or mobile agents. Calls to such active objects can be performed asynchronous using a future-based synchronisation scheme known as wait-by-necessity for return values. A detailed documentation including programming tutorials as well as the full source code can be found at the ProActive Web site [15]. 4 Institut national de recherche en informatique et en automatique (National Institute for Research in Computer Science and Control) Ad Hoc Grid Aspects ProActive was chosen as the Grid environment for the implementation of this project because it fulfils the specific requirements of Ad Hoc Grids as outlined in 3.1. As such it is very well suited for the dynamic and transient structure of Ad Hoc Grids and allows the setup of Grid structures with very little administrative overhead. The next few sections will briefly describe the features of ProActive that make it especially suited for Ad Hoc Grids. 3.2.1 Descriptor-Based Deployment ProActive uses a deployment descriptor XML file to separate Grid applications and their source code from deployment related information. The source code of such Grid applications will only refer to virtual nodes. The actual mapping from a virtual node to real ProActive nodes is defined by the deployment descriptor file. When a Grid application is started ProActive will read the deployment descriptor file and will provide access to the actual nodes within the Grid application. The deployment descriptor file includes information about how the nodes are acquired or created. ProActive supports the creation of its nodes on physical nodes via several protocols, these include for instance ssh, rsh, rlogin as well as other Grid environments like Globus Toolkit or glite. Alternatively ProActive nodes can be started manually using the startNode.sh script provided. For the actual communication between Grid nodes, ProActive can use a variety of communication protocols like for instance rmi, http or soap. Even file transfer is supported as part of the deployment process. Further details about the deployment functionality provided by ProActive can be found in its documentation at the ProActive Web site [15]. 3.2.2 Peer-to-Peer Infrastructure ProActive provides a self-organising Peer-to-Peer functionality that can be used to discover new nodes, which are not defined within the deployment descriptor file of a Grid application. The only thing required is an entry point into an existing ProActive- based Peer-to-Peer network, for instance through a known node that is already part of that network. Further nodes from the Peer-to-Peer network can then be discovered and used by the Grid application. The Peer-to-Peer functionality of ProActive is not limited to sub-networks, it can communicate through firewalls and NAT routers and is therefore suitable for Internet-based Peer-to-Peer infrastructures. It is also self-organising which Ad Hoc Grid Aspects means that an existing Peer-to-Peer network tries to keep itself alive as long as there are nodes belonging to it. 3.2.3 Active Object Migration In ProActive Active Objects can easily be migrated between different nodes. This can either be triggered by the Active Object itself or by an external tool. The migration functionality is based on standard Java serialisation, which is why Active Objects that need to be migrated and their private passive objects have to be serialisable. A detailed description of the migration functionality including examples can be found in the ProActive documentation. 3.2.4 Transparent Fault Tolerance ProActive can provide fault tolerance to Grid applications that is fully transparent. Fault tolerance can be enabled for Grid applications just by configuring it within the deployment descriptor configuration. The only requirement is that Active Objects for which fault tolerance is to be enabled need to be serialisable. There are currently two fault tolerance protocols provided by ProActive. Both protocols use checkpointing and are based on the standard Java serialisation functionality. Further details about how the fault tolerance works and how it is configured can be found in the ProActive documentation. Parallel Transaction-oriented Simulation 4 Parallel Transaction-oriented Simulation 4.1 Past research work Past research performed by the author looked at the parallelisation of transaction- oriented simulation using an existing Matlab-based5 GPSS simulator and Message- Passing for the communication [19]. It was shown that the Breathing Time Buckets algorithm, which is also known as SPEEDES algorithm (a description can be found in section 2.5.2), can be applied to transaction-oriented simulation. This algorithm uses a relatively simple communication scheme without anti-events and cancellations. But further evaluation has revealed that the Breathing Time Buckets algorithm is not well suited for transaction-oriented simulation. The reason for this is that the Breathing Time Buckets algorithm makes use of what is know as the event horizon [29]. This event horizon is the time stamp of the earliest new event generated by the execution of the current events. Using this event horizon the Breathing Time Buckets algorithm can execute local events until it reaches the time of a new event that needs to be sent to another LP. At this point a GVT calculation is required because only events guaranteed by the GVT can be sent. The Breathing Time Buckets algorithm works well for discrete event models that have a large event horizon, i.e. where current events create new events that are relatively far in the future so that many local events can be executed before a GVT calculation is required. This is where the Breathing Time Buckets algorithm fails when it is applied to transaction-oriented simulation. In transaction- oriented simulation the simulation time does not change while a Transaction is moved. Whenever a Transaction moves from one LP to another this results in an event horizon of zero because the time stamp of the Transaction in the new LP will be the same like the time stamp it had in the LP from which it was sent. The validation of the parallel transaction-oriented simulator based on Breathing Time Buckets (alias SPEEDES) showed that a GVT calculation was required each time a Transaction needed to be sent to another LP. 5 MATLAB is a numerical computing environment and programming language created by The MathWorks Inc. Parallel Transaction-oriented Simulation The described past research comes to the conclusion that the Breathing Time Buckets algorithm does not perform well for transaction-oriented simulation but the research still provides some useful findings about the application of discrete event simulation algorithms to transaction-oriented simulation. Some of these findings that also apply to this work are outlined in the following sections. 4.1.1 Transactions as events An event can be described as a change of the state at a specified time. From the simulation perspective this change of state is always caused by an action (e.g. the execution of an event procedure). Therefore an event can also be seen as an action that is performed at a specific point in time. In transaction-oriented simulation the state of the simulation system is changed by the execution of blocks through Transactions. Transactions are moved from block to block at a specific point in time as long as they are movable, i.e. not advanced and not terminated. Therefore this movement of a Transaction for a specific point in time and as long as the Transaction is movable describes an action, which is equivalent to the event describing an action in the discrete event model. Considering this equivalence it is generally possible to apply synchronisation algorithms and other techniques for discrete event simulation also to transaction- oriented simulation. But because transaction-oriented simulation has specific properties certain algorithms are more and other less well suited for transaction-oriented simulation. 4.1.2 Accessing objects in other LPs In a simulation that performs partitions of the simulation model on different LPs it is possible that the simulation of the model partition within one LP needs to access an object in another LP. For instance this could be a TEST block within one LP trying to access the properties of a STORAGE entity within another LP. The main problem for accessing objects like this in other LPs is that at a certain point of real time each LP can have a different simulation time. Figure 7 shows an example for this problem. In this example the LP1 that contains object o1 has already reached the simulation time 12 and the LP2, which is trying to access the object o1 has reached the simulation time 5. It can be seen that event e4 from LP2 would potentially read the wrong value for object o1 Parallel Transaction-oriented Simulation because this read access should happen at the simulation time 5 which is before the event e2 at LP1 overwrote the value of o1. Instead event e4 reads the value of o1 as it appears at the simulation time 12. e1 3, read o1 e2 8, write o1 e3 12, ... Past events Future events e4 5, read o1 Current event position Event list Event list Object Current simulation time = 12 Current event = e3 Current simulation time = 5 Current event = e4 Figure 7: Accessing objects in other LPs Because accessing an object within another LP is an action that is linked to a specific point of simulation time it can also be viewed as an event according to the description of events in section 4.1.1. Like other events they have to be executed in the correct causal order. This means that event e4 that is reading the value of object o1 has to happen at the simulation time 5. Sending this event to LP1 would cause LP1 to roll back to the simulation time 5 so that e4 would read the correct value. Treating the access to objects as a special kind of event solves the problem mentioned above. Such a solution can also be applied to transaction-oriented simulation by implementing a simulation scheduler that besides Transactions can also handle these kinds of object access events. Alternatively the object access could be implemented as a pseudo Transaction that does not point to its next block but instead to an access method that when executed performs the object access and for a read access returns the value. Such a pseudo Transaction would send the value back to the originating LP and then be deleted. Depending on the synchronisation algorithm it can also be useful treat read and write access differently. If for instance an optimistic synchronisation algorithm is used that saves system states for possible required rollbacks then rollbacks as a result of Parallel Transaction-oriented Simulation object read access events can be avoided if the LP that contains the object has passed the time of the read access. In this case the object value for the required point in simulation time could be read from the saved system state instead of rolling back the whole LP. Another even simpler solution to the problem of accessing objects in other LPs is to prevent the access of objects in other LPs all together, i.e. to allow only access to objects within the same LP. This sounds like a contradiction but by preventing one LP from accessing local objects in another LP the event that wants to access a particular object needs to be moved to the LP that holds that object. For the example from Figure 7 this means that instead of synchronizing the object access from event e4 on LP2 to object o1 held by LP1 the event e4 is moved to LP1 that holds object o1 so that accessing the object can be performed as a local action. This solution reduces the problem to the general problem of moving events and synchronisation between LPs as solved by discrete event synchronisation algorithms (see section 2.5). 4.1.3 Analysis of GPSS language A synchronisation strategy is a requirement for parallel discrete event simulation because LPs cannot predict the correct causal order of the events they will execute as they can receive further events from other LPs at any time. When applying discrete event synchronisation algorithms to transaction-oriented simulation based on GPSS/H it is first of interest to analyse which of the GPSS/H blocks6 can actually cause the transfer of Transactions to another LP or which of them require access to objects that might be located at a different LP. Because Transactions usually move from one block to the next a transfer to a different LP can only be the result of a block that causes the execution of a Transaction to jump to a different block than the next following including blocks that can cause a conditional branching of the execution path. The following two tables list GPSS/H blocks that can change the execution path of a Transaction or that access other objects within the model. 6 A detailed description of the GPSS/H language and its block types can be found in [26]. Parallel Transaction-oriented Simulation Blocks that can change the execution path Block Change of execution path TRANSFER Jump to specified block SPLIT Jump of Transaction copy to specified block GATE Jump to specified block depending on Logic Switch TEST Jump to specified block depending on condition LINK Jump to specified block depending on condition UNLINK Jump of the unlinked Transactions to specified block and possible jump of Transaction causing the unlink operation Table 1: Change of Transaction execution path Blocks that can access objects Block Access to object SEIZE RELEASE Access to Facility object ENTER LEAVE Access to Storage object QUEUE DEPART Access to Queue object LOGIC Access to Logic Switch UNLINK Access to User Chain TERMINATE Access to Termination Counter Table 2: Access to objects Parallel Transaction-oriented Simulation 4.2 Synchronisation algorithm An important conclusion from section 4.1 is that the choice of synchronisation algorithm has a large influence on how much of the parallelism that exists in a simulation model can be utilised by the parallel simulation system. A basic overview of the classification of synchronisation algorithms for discrete event simulation was given in section 2.5. Conservative algorithms utilise the parallelism less well than optimistic algorithms because they require guarantees, which are often derived from additional knowledge about the behaviour of the simulation model, like for instance the communication topology or lookahead attributes of the model. For this reason conservative algorithms are often used to simulate very specific systems where such knowledge is given or can easily be derived from the model. For general simulation systems optimistic algorithms are better suited as they can utilise the parallelism within a model to a higher degree without requiring any guarantees or additional knowledge. Another important aspect for choosing the right synchronisation algorithm is the relation between the performance properties of the expected parallel hardware architecture and the granularity of the parallel algorithm as outlined in section 2.2. In order for the parallel algorithm to perform well in general on the target hardware environment the granularity of the algorithm, i.e. the ratio between computation and communication has to fit the ratio of the computation performance and communication performance of the parallel hardware. The goal of this work is to provide a basic parallel transaction-oriented simulation system for Ad Hoc Grid environments. Ad Hoc Grids can make use of special high performance hardware but more likely will be based on standard hardware machines using Intranet or Internet as the communication channel. It can therefore be expected that Ad Hoc Grids will mostly be targeted at parallel systems with reasonable computation performance but relatively poor communication performance. 4.2.1 Requirements Considering the target environment of Ad Hoc Grids and the goal of designing and implementing a general parallel simulation system based on the transaction-oriented Parallel Transaction-oriented Simulation simulation language GPSS it can be concluded that the best suitable synchronisation algorithm is an optimistic or hybrid algorithm that has a coarse grained granularity. The algorithm should require only little communication compared to the amount of computation it performs. At the same time the algorithm should be flexible enough to adapt to a changing environment, as this is the case in Ad Hoc Grids. A further requirement is that the algorithm can be adapted to and is suitable for transaction- oriented simulation. Finding such an algorithm is a condition for achieving the outlined goals. 4.2.2 Algorithm selection Most optimistic algorithms are based on the Time Warp algorithm but attempt to limit the optimism. As described in section 2.5.2 these algorithms can be grouped into non- adaptive algorithms, adaptive algorithms with local state and adaptive algorithms with global state. Non-adaptive algorithms usually rely on external parameters (e.g. the window size for window based algorithms) to specify how strongly the optimism is limited. Such algorithms are not ideal for a general simulation system as it can be difficult for a simulation modeller to find the optimum parameters for each simulation model. It is also common that simulation models change their behaviour during the runtime of the simulation. As a result later research has focused more on the adaptive algorithms, which qualify for a general simulation system. They are also better suited for dynamically changing environments like Ad Hoc Grids. Two interesting adaptive algorithms are the Elastic Time algorithm [28] and the Adaptive Memory Management algorithm [6]. The Elastic Time algorithm is based on Near Perfect State Information (NPSI). It requires a feedback system that constantly receives input state vectors from all LPs, processes these using several functions and then returns output vectors to all LPs that describe how the optimism of each LP needs to be controlled. As described in [28] for a shared memory system such a near-perfect state information feedback system can be implemented using a dedicated set of processes and processors but for a distributed memory system a high speed asynchronous reduction network would be needed. This shows that the Elastic Time algorithm is not suited for a parallel simulation system based on Grid environments Parallel Transaction-oriented Simulation where communication links between nodes might use the Internet and nodes might not be physically close to each other. Similar to the Elastic Time algorithm the Adaptive Memory Management algorithm is also best suited for shared memory systems. The Adaptive Memory Management algorithm is based on the link between optimism and memory usage in optimistic algorithms. The more over-optimistic an LP is the more memory does it use to store the executed events and state information which cannot be committed and fossil collected as they are far ahead of the GVT. It is shown that by limiting the overall memory available to the optimistic simulation artificially, the optimism can also be controlled. For this the Adaptive Memory Management algorithm uses a shared memory pool providing the memory used by all LPs. The algorithm then dynamically changes the size of the memory pool and therefore the total amount of memory available to the simulation based on several parameters like frequency of rollbacks, fossil collections and cancel backs in order to find the optimum amount of memory for the best performance. The required shared memory pool can easily be provided in a shared memory system but in a distributed memory system implementing it would require extensive synchronisation and communication between the nodes which makes this algorithm unsuitable for this work. An algorithm that is more applicable to Grid environments as it does not need a shared memory or a high speed reduction network is the algorithm suggested in [33]. This algorithm uses a Global Progress Window (GPW) described by the GVT and the Global Furthest Time (GFT). Because the GVT is equivalent to the LVT of the slowest LP and the GFT is the LVT of the LP furthest ahead in simulation time the GPW represents the window in simulation time in which all LPs are located. This time window is then divided further into the slow zone, the fast zone and the hysteresis zone as shown in Figure 8. GVT GFT hysteresis hl hu Slow Zone Fast Zone Figure 8: Global Progress Window with its zones [33] Parallel Transaction-oriented Simulation The algorithm will slow down LPs in the fast zone and try to accelerate LPs in the slow zone with the hysteresis zone acting as a buffer between the other two zones. This algorithm could be implemented without any additional communication overhead because the GFT can be determined and passed back to the LPs by the same process that performs the GVT calculation. It is therefore well suited for very loosely coupled systems based on relatively slow communication channels. The only small disadvantage is that similar to many other algorithms the fast LPs will always be penalized even if they don’t actually contribute to the majority of the cascaded rollback. In [33] the authors also explore how feasible LP migration and load balancing is for reducing the runtime of a parallel simulation. The most promising algorithm regards the requirements outlined in 4.2.1 is the Shock Resistant Time Warp algorithm [8]. This algorithm follows similar ideas like the Elastic Time algorithm and the Adaptive Memory Management algorithm mentioned above but at the same time is very different. Similar to the Elastic Time algorithm state vectors are used to describe the current states of all LPs plus a set of functions to determine the output vector but the Shock Resistant Time Warp algorithm does not require a global state. Instead each LP tries to optimise its parameters towards the best performance. And similar to the Adaptive Memory Management algorithm the optimism is controlled indirectly be setting artificial memory limits but each LP will artificially limit its own memory instead of using an overall memory limit for the whole simulation. The Shock Resistant Time Warp algorithm was chosen for the implementation of the parallel transaction-oriented simulator because it promises to be very adaptable and at the same time is very flexible regards changes in the environment and it does not create any additional communication overhead compared to Time Warp. The following section will describe this algorithm in more detail. 4.2.3 Shock resistant Time Warp Algorithm The Shock Resistant Time Warp algorithm [8] is a fully distributed approach to controlling the optimism in Time Warp LPs that requires no additional communication between the LPs. It is based on the Time Warp algorithm but extends each LP with a control component called LPCC that constantly collects information about the current Parallel Transaction-oriented Simulation state of the LP using a set of sensors. These sets of sensor values are then translated into sets of indicator values representing state vectors for the LP. The LPCC will keep a history of such state vectors so that it can search for past state vectors that are similar to the current state vector but provide a better performance indicator. An actuator value will be derived from the most similar of such state vectors that is subsequently used to control the optimism of the LP. Figure 9 gives an overview of the interaction between LPCC and LP. Figure 9: Overview of LP and LPCC in Shock Resistant Time Warp [8] The specific sensors used by the LPCC are described in Table 3 but other or additional sensors could be used if appropriate. There are two types of sensors. The point sample sensors describe a momentary value of a performance or state metric, which can fluctuate significantly whereas the cumulative sensors characterise metrics that contain a sum value produced over the runtime of the simulation. The indicator for each sensor is calculated depending on which type of sensor it is. For cumulative sensors the rate of increase over a specified time period is used as the indicator values and for point sample Parallel Transaction-oriented Simulation sensors the arithmetic mean value over the same time period. Table 4 shows the corresponding indicators. Sensor Type Description CommittedEvents cumulative total number of events committed SimulatedEvents cumulative total number of events simulated MsgsSent cumulative total number of (positive) messages sent AntiMsgsSent cumulative total number of anti-messages sent MsgsRcvd cumulative total number of (positive) messages received AntiMsgsRcvd cumulative total number of anti-messages received EventsRollback cumulative total number of events rolled back EventsUsed point sample momentary number of events in use Table 3: Shock Resistant Time Warp sensors Indicator Description EventRate number of events committed per second SimulationRate number of events simulated per second MsgsSentRate number of (positive) messages sent per second AntiMsgsSentRate number of anti-messages sent per second MsgsRcvdRate number of (positive) messages received per second AntiMsgsRcvd number of anti-messages received per second EventsRollbackRate number of events rolled back per second MemoryConsumption average number of events in use Table 4: Shock Resistant Time Warp indicators Two of these indicators are slightly special. The EventRate indicator, which describes the number of events committed per second during a time period, is the performance indicator used to identify how much useful work has been performed. And the actuator value MemoryLimit is derived from the MemoryConsumption indicator. For a state vector with n different indicator values the LPCC will use an n-dimensional state vector Parallel Transaction-oriented Simulation space to store and compare the state vectors. The similarity of two state vectors within this state vector space is characterised by the Euclidean distance between the vectors. When searching for the most similar historic state vector that has a higher performance indicator then the Euclidean distance is calculated by ignoring the indicators EventRate and MemoryConsumption because the EventRate is the indicator that the LPCC is trying to optimise and the MemoryConsumption is directly linked to the MemoryLimit actuator controlled by the LPCC. Keeping a full history of the past state vectors would require a large amount of memory and would create an exponentially increasing performance overhead. For these reasons the Shock Resistant Time Warp algorithm uses a clustering mechanism to cluster similar state vectors. The algorithm will keep a defined number of clusters. At first each new state is stores as a new cluster but when the cluster limit is reached then new states are added to existing clusters if the distance between the state and the cluster is smaller than any of the inter-cluster distances and otherwise the two closest clusters are merged into one and the second cluster is replaced with the new state vector. The clustering mechanism limits the total number of clusters stored and at the same time clusters will move their location within the state space to reflect the mean position of the state vectors they represent. The Shock Resistant Time Warp algorithm as described in [8] is specific to discrete event simulation but it can also be applied to transaction-oriented simulation because of the equivalence between events in discrete event simulation and the movement of Transactions in transaction-oriented simulation as outlined in 4.1.1. Because the transaction-oriented simulation does not know events as such the names of the sensors and indicators described above need to be changed to avoid confusion when applying the Shock Resistant Time Warp algorithm to transaction-oriented simulation. The two tables below show the sensor and indicator names that will be used for this work. Parallel Transaction-oriented Simulation Discrete event sensor Transaction-oriented sensor CommittedEvents CommittedMoves EventsUsed UncommittedMoves SimulatedEvents SimulatedMoves MsgsSent XactsSent AntiMsgsSent AntiXactsSent MsgsRcvd XactsReceived AntiMsgsRcvd AntiXactsReceived EventsRollback MovesRolledback Table 5: Transaction-oriented sensor names Discrete event indicator Transaction-oriented indicator EventRate CommittedMoveRate MemoryConsumption AvgUncommittedMoves SimulationRate SimulatenRate MsgsSentRate XactSentRate AntiMsgsSentRate AntiXactSentRate MsgsRcvdRate XactReceivedRate AntiMsgsRcvd AntiXactReceivedRate EventsRollbackRate MovesRolledbackRate Table 6: Transaction-oriented indicator names 4.3 GVT Calculation The concept of Global Virtual Time (GVT) was mentioned and briefly explained in 2.5.2. GVT is a fundamental concept of optimistic synchronisation algorithms and describes a lower bound on the simulation times of all LPs. Its main purpose is to guarantee past simulation states as being correct so that the memory for these saved Parallel Transaction-oriented Simulation states can be reclaimed through fossil collection. Another important purpose is to determine the overall progress of the simulation, which includes the detection of the simulation end. Besides these reasons optimistic parallel simulations can often run without any additional GVT calculations for long time periods or even until they reach the simulation end if enough memory for the required state saving is available. In environments with a relatively low communication performance like Computing Grids it is desirable to minimise the need for GVT calculations because the GVT calculation process is based on the exchange of messages and adds a communication overhead. The best-known GVT calculation algorithm was suggested by Jefferson [16]. It defines the GVT as the minimum of all local simulation times and the time stamps of all events sent but not yet acknowledged as being handled by the receiving LP. The planned parallel simulator will use this algorithm for the GVT calculation because it is relatively easy to implement and well studied. Future work could also look at alternative GVT algorithms that might be suitable for Grid environments, like the one suggested in [20]. The movement of a Transaction in transaction-oriented simulation can be seen as equivalent to an event being executed in discrete event simulation as concluded in 4.1.1. But in transaction-oriented simulation the causal order is not only determined by the movement time of a Transaction but also by its priority because if several Transactions exist that have the same move time then they are moved through the system in order of their priority, i.e. Transactions with higher priority first. As a result the priority had to be included in the GVT calculation in [19] because the Breathing Time Buckets algorithm (SPEEDES algorithm) used there needs the GVT to guarantee outgoing Transactions. For a parallel transaction-oriented simulator based on the Time Warp algorithm or the Shock Resistant Time Warp algorithm it is not necessary to include the Transaction priority in the GVT calculation because the GVT is only used to determine the progress of the overall simulation and to regain memory through fossil collection. For the Shock Resistant Time Warp algorithm one additional use of the GVT is to determine realistic values for the CommittedEvents sensor. Events are committed when receiving a GVT that is greater than the event’s time. As a result the number of committed events during a certain period of time is only known if GVT calculations have been performed. The suggested parallel simulator based on the Shock Resistant Time Warp algorithm will therefore synchronise the processing of its LPCC with GVT calculations. Parallel Transaction-oriented Simulation 4.4 End of Simulation In transaction-oriented simulation a simulation is complete when the defined end state is reached, i.e. the termination counter reaches a value less or equal to zero. When using an optimistic synchronisation algorithm for the parallelisation of transaction-oriented simulation it is crucial to consider that optimistic algorithms will first execute all local events without guarantee that the causal order is correct. They will recover from wrong states by performing a rollback if it later turns out that the causal order was violated. Therefore any local state reached by an optimistic LP has to be considered provisional until a GVT has been received that guarantees the state. In addition it needs to be considered that at any point in real time it is most likely that each of the LPs has reached a different local simulation time so that after an end state has been reached by one of the LPs that is guaranteed by a GVT it is important to synchronise the states of all LPs so that the combined end state from all model partitions is equivalent to the model end state that would have been reached in a sequential simulator. To summarise, a parallel transaction-oriented simulation based on an optimistic algorithm is only complete when the defined end state has been reached in one of the LPs and when this state has been confirmed by a GVT. Furthermore if the confirmed end of the simulation has been reached by one of the LPs then the states of all the other LPs need to be synchronised so that they all reflect the state that would exist within the model when the Transaction causing the simulation end executed its TERMINATE block. These significant aspects regarding the simulation end of a parallel transaction- oriented simulation that had not been considered in [19]. A mechanism is suggested for this work that leads to a consistent and correct global end state of the simulation considering the problems mentioned above. For this mechanism the LP reaching a provisional end state is switched into the provisional end mode. In this mode the LP will stop to process any further Transactions leaving the local model partition in the same state but it will still respond to and process control messages like GVT parameter requests and it will receive Transactions from other LPs that might cause a rollback. The LP will stay in this provisional end mode until the end of the simulation is confirmed by a GVT or a received Transaction causes a rollback with a potential re-execution that is not resulting in the same end state. While the LP is in the provisional end mode additional GVT parameters are passed on for every GVT Parallel Transaction-oriented Simulation calculation denoting the fact that a provisional end state has been reached and the simulation time and priority of the Transaction that caused the provisional end. The GVT calculation process can then assess whether the earliest current provisional end state is guaranteed by the GVT. If this is the case then all other LPs are forced to synchronise to the correct end state by rolling back using the simulation time and priority of the Transaction that caused the provisional end and the simulation is stopped. 4.5 Cancellation Techniques Transaction-oriented simulation has some specific properties compared to discrete event simulation. One of these properties is that Transactions do not consume simulation time while they are moving from block to block. This has an influence on which of the synchronisation algorithms are suitable for transaction-oriented simulation as described in 4.1 but also on the cancellation techniques used. If a Transaction moves from LP1 to LP2 then it will arrive at LP2 with the same simulation time that it had at LP1. A Transaction moving from one LP to another is therefore equivalent to an event in discrete event simulation that when executed creates another event for the other LP with exactly the same time stamp. Because simulation models can contain loops as it is common for the models of quality control systems where an item failing the quality control needs to loop back through the production process (see [26] for an example) this specific behaviour of transaction-oriented simulation can lead to endless rollback loops if aggressive cancellation is used (cancellation techniques were briefly described in 2.5.2). The example in Figure 10 demonstrates this effect. It shows the movement of a Transaction x1 from LP1 to LP2 but without a delay in simulation time the Transaction is transferred back to LP1. As a result LP1 will be rolled back to the simulation time just before x1 was moved. At this point two copies of Transaction x1 will exist in LP1. The first one is x1 itself which needs to be moved again and the second is x1’ which is the copy that was send back from LP2. This is the point from where the execution differs between lazy cancellation and aggressive cancellation. In lazy cancellation x1 would be moved again resulting in the same transfer to LP2. But because x1 was sent to LP1 already it will not be transferred again and no anti-transaction will be sent. From here Parallel Transaction-oriented Simulation LP1 just proceeds moving the Transactions in its Transaction chain according to their simulation time (Transaction priorities are ignored for this example). Apposed to that the rollback in aggressive cancellation would result in an anti-Transaction being sent out for x1 immediately which would cause a second rollback in LP2 and another anti- Transaction for x1’ being sent back to LP1. At the end both LPs will end up in the same state in which they were before x1 was moved by LP1. The same cycle of events would start again without any actual simulation progress. Lazy cancellation Aggressive cancellation Transaction transferred to other LP Rollback Anti-Transaction for other LP x1 x1' x1- x1'- cycle 1 repeat of cycle 1 x2 x2 Figure 10: Cancellation in transaction-oriented simulation It can therefore be concluded that lazy cancellation needs to be used for a parallel transaction-oriented simulation based on an optimistic algorithm in order to avoid such endless loops. 4.6 Load Balancing Load balancing and the automatic migration of slow LPs to nodes or processors that have a lighter work load has been suggested in order to reduce the runtime of parallel simulations. This has also been explored by the authors of [33]. They concluded that the Parallel Transaction-oriented Simulation migration of LPs involves a “substantial amount of overheads in saving the process context, flushing the communication channels to prevent loss of messages”. And especially on loosely coupled systems with relatively slow communication channels sending the full process context of the LP from one node to another can add a significant performance penalty to the overall simulation. This penalty would depend on the size of the process context as well as the communication performance between the nodes involved in the migration. The gained performance on the other hand depends on the difference in processing performance and other workload on these nodes. To determine reliably when such an automatic migration is beneficial within a loosely coupled, dynamically changing Ad Hoc Grid environment would be difficult and it is likely that the performance penalty outweighs the gains. This work will therefore not investigate the load balancing and automatic LP migration for performance reasons but only support automatic LP migration as part of the fault tolerance functionality provided by ProActive and described in 3.1.3. Manual LP migration will be supported by the parallel simulator using ProActive tools. 4.7 Model Partitioning Besides the chosen synchronisation algorithm the partitioning of the simulation model also has a large influence on the performance of the parallel simulation because the communication required between the Logical Processes depends to a large degree on how independent the partitions of a simulation model are. Looking at the requirements of a general-purpose transaction-oriented simulation system for Ad Hoc Grid environments in 4.2 the conclusion was drawn that the required communication needs to be kept to a minimum in order to reach acceptable performance results through parallelisation in such environments. The communication required for the exchange of Transactions between the Logical Processes is part of this overall communication. A simulation model that is supposed to be run in a Grid based parallel simulation system therefore needs to be partitioned in such a way that the expected amount of Transactions moved within the partitions is significantly larger than the amount of Transactions that need to be transferred between these partitions. This means that Grid based parallel Parallel Transaction-oriented Simulation simulation systems are best suited for the simulation of systems that contain relatively independent sub-systems. In practice the ratio of computation performance to communication performance provided by the underlying hardware architecture of the Grid environment will have to match the ratio of computation performance to communication performance required by the parallel simulation as reasoned in 2.2. Whether a partitioned simulation model will perform well will therefore also depend on the underlying hardware architecture. Implementation 5 Implementation The GPSS based parallel transaction-oriented simulator will be implemented using the JavaTM 2 Platform Standard Edition 5.0, also known as J2SE5.0 [31] and ProActive version 3.1 [15] as the Grid environment. An object-oriented design will be applied for the implementation of the simulator and resulting classes will be grouped into a hierarchy of packages according to the functional parts of the parallel simulator and the implementation phases. The parallel simulator will use the logging library log4j [3] for all its output, which will provide very flexible means to enable or disable specific parts of the output as required. The log4j library is the same logging library that is used by ProActive so that only one configuration file will be needed to configure the logging of ProActive and the parallel simulator. 5.1 Implementation Considerations 5.1.1 Overall Architecture Figure 11 shows the suggested architecture of the parallel simulator including its main components. The main parts of the parallel simulator will be the Simulation Controller and the Logical Processes. The Simulation Controller controls the overall simulation. It is created when the user starts the simulation and will use the Model Parser component to read the simulation model file and parse it into an in memory object structure representation of the model. After the model is parsed the Simulation Controller will create Logical Process instances, one for each model partition contained within the simulation model. The Simulation Controller and the Logical Processes will be implemented as ProActive Active Objects so that they can communicate with each other via method calls. Communication will take place between the Simulation Controller and the Logical Processes but also between the Logical Processes for instance in order to exchange Transactions. Note that the communication between the Logical Processes is not illustrated in Figure 11. After the Logical Process instances have been created, they will be initialised, they will receive the model partitions from the Simulation Controller that they are going to simulate and the simulation is started. Implementation Simulation Controller Model Parser GVT Calculation Reporting Logical Process State List State Cluster Space Simulation Engine Model Partition Transaction Chain Figure 11: Architecture overview Each Logical Process implements an LP according to the Shock Resistant Time Warp algorithm. The main component of the Logical Process is the Simulation Engine, which contains the Transaction chain and the model partition that is simulated. The Simulation Engine is the part that is performing the actual simulation. It is moving the Transactions from block to block by executing the block functionality using the Transactions. Another important part of the Logical Process is the State List. It contains historic simulation states in order to allow rollbacks as required by optimistic synchronisation algorithms. Note that there will be other lists like for instance the list of Transactions received and the list of Transactions sent to other Logical Processes, which are not shown in Figure 11. Furthermore the Logical Process will contain the Logical Process Control Component (LPCC) according to the Shock Resistant Time Warp algorithm described in 4.2.3. Using specific sensors within the Logical Process the LPCC will limit the optimism by the means of an artificial memory limit if this promises a better simulation performance at the current circumstances. The Simulation Controller will perform GVT calculations in order to establish the overall progress of the simulation and if requested by one of the Logical Processes that needs to reclaim memory using fossil collection. GVT calculation will also be used to Implementation confirm a provisional simulation end state that might be reached by one of the Logical Processes. When the end of the simulation is reached then the Simulation Controller will ensure that the partial models in all Logical Processes are set to the correct and consistent end state and it will collect information from all Logical Processes in order to assemble and output the post simulation report. 5.1.2 Transaction Chain and Scheduling Thomas J. Schriber gives an overview in section 4. and 7. of [26] on how the Transaction chains and the scheduling work in the original GPSS/H implementation. The scheduling and the Transaction chains for the parallel transaction-oriented simulator of this work will be based on his description but with some significant differences. Only one Transaction chain is used containing Transactions for current and future simulation time in a sorted order. The Transactions within the chain are first sorted ascending by their next moving time and Transactions for the same time are also sorted descending by their priority. Transactions will be taken out of the chain before they are moved and put back into the chain after they have been moved (unless the Transaction has been terminated). The functionality to put a Transaction back into the chain will do so at the right position ensuring the correct sort order of the Transactions within the chain. The scheduling will be slightly simpler than described in [26] because no “Model’s Status changed flag” will be needed. Another difference is that in order to keep the time management as simple as possible the proposed parallel simulator will restrict the simulation time and time related parameters to integer values instead of floating point values. At first this might seem like a major restriction but it is not because decimal places can be represented using scaling. If for instance a simulation time with three decimal places is needed for a simulation then a scale of 1000:1 can be used, which means 1000 time units of the parallel simulator represent one second of simulation time or a single time unit represents 1ms. The Java integer type long will be used for time values providing a large value range that allows flexible scaling for different required precisions. The actual scheduling will be implemented using a SimulationEngine class. This class will for instance contain the functionality for moving Transactions, updating the Implementation simulation time and also the Transaction chain itself. The SimulationEngine class will be implemented as part of the Basic GPSS Simulation Engine implementation phase detailed in section 5.2.2 and a description of how the scheduling was implemented can be found in 5.3.1. 5.1.3 Generation and Termination of Transactions Performing transaction-oriented simulation in a parallel simulator also has an influence on how Transactions are generated and how they are terminated. Generating Transactions Transactions are generated in the GENERATE blocks of the simulation. During the generation each Transaction receives a unique numerical ID that identifies the Transaction during the rest of the simulation. In a parallel transaction-oriented simulator GENERATE blocks can exist in any of the model partitions and therefore in any of the LPs. This requires a scheme, which ensures that the Transaction IDs generated in each LP are unique across the overall parallel simulation. Ideally such a scheme requires as little communication between the LPs as possible. The scheme used for this parallel simulator will generate unique Transaction IDs without any additional communication overhead. This is achieved by partitioning the value range of the numeric IDs according to the number of partitions in the simulation model. The only requirement of this scheme is that all LPs are aware of the total number of partitions and LPs within the simulation. This information will be passed to them during the initialisation. The used scheme is based on an offset that depends on the total number of LPs. Each LP has its own counter that is used to generate unique Transaction IDs. These counters are initialised with different starting values and the same offset is used for incrementing the counters when one of their values has been used for a Transaction ID. A simulation with n LPs (i.e. n partitions) will use the offset n to increment the local Transaction ID counters and each LP will initialise its counter with its own number in the list of LPs. In a simulation with 3 LPs, LP1 would initialise its counter to the value 1, LP2 to 2 and LP3 to 3 and the increment offset used would be the total number of LPs which is 3. The sequence of IDs generated by these LPs would be LP1: 1, 4, 7, … and by LP2: 2, 5, 8, … and by LP3: 3, 6, 9, … and so forth. Further advantages of this scheme are that it partitions the possible ID value range into equally Implementation large numerical partitions independent of the total size of the value range and it also makes it possible to determine in which partition a Transaction was generated using their ID. Terminating Transactions The termination of Transactions raises similar problems like their generation. According to GPSS Transactions are terminated in TERMINATE blocks. Each time a Transaction is terminated a global Termination Counter is decremented by the decrement parameter of the TERMINATE block involved. A GPSS simulation initialises the Termination Counter with a positive value at the start of the simulation and the counter is then used to detect the end of the simulation, which is reached as soon as the Termination Counter has a value of zero or less. The required global Termination Counter could be located in one of the LPs but accessing it form other LPs would require additional synchronisation. The problem of accessing such a Termination Counter is the same like accessing other objects from different LPs as outlined in 4.1.2. In order to avoid the additional complexity and communication overhead of implementing a global Termination Counter the parallel simulator will use a separate local Termination Counter in each LP. This solution will perform simulations that don’t require a global Termination Counter without additional communication overhead. For simulations that do require a global Termination Counter the problem can be reduced to the synchronisation and the movement of Transactions between LPs as solved by the synchronisation algorithm. In this case all TERMINATE blocks of such a simulation need to be located within the same partition. This will result in additional communication and synchronisation when Transactions are moved from other LPs to the one containing the TERMINATE blocks. Figure 12 below shows how a simulation model with two partitions that needs a single Termination Counter can be converted into equivalent simulation models so that the simulation will effectively use a single Termination Counter. Implementation TERMINATE TERMINATE Original partitioning Partition 1 Partition 2 Partitioning using a single synchronised Termination Counter Option 1 Option 2 TRANSFER TRANSFER Partition 1 Partition 2 TERMINATE TERMINATE Partition 3 TRANSFER TERMINATE Partition 1 Partition 2 TERMINATE ...... Figure 12: Single synchronised Termination Counter 5.1.4 Supported GPSS Syntax In order to ease the migration of existing GPSS models to this new parallel GPSS simulator the GPSS syntax supported will be kept as close as possible to the original GPSS/H language described in [26]. At the same time only a sub-set of the full GPSS/H language will be implemented but this sub-set will include all the main GPSS functionality including functionality needed to demonstrate the parallel simulation of partitioned models on more than one LP. The simulator will not support Transaction cloning, Logic Switches, User Chains and user defined properties for Transactions but such functionality can easily be added in future if required. Implementation A detailed description of the GPSS syntax expected by the parallel GPSS simulator can be found in Appendix A. In particular the simulator will support the generating, delay and termination of Transactions as well as their transfer to other partitions of the model. It will also support Facilities, Queues, Storages and labels. The following table gives an overview of the supported GPSS block types. Block type Short description GENERATE Generate Transactions TERMINATE Terminate Transactions ADVANCE Delay the movement of Transactions SEIZE Capture Facility RELEASE Release Facility ENTER Capture Storage units LEAVE Release Storage units QUEUE Enter Queue DEPART Leave Queue TRANSFER Jump to a different block than the next following Table 7: Overview of supported GPSS block types In addition to the GPSS functionality described above the new reserved word PARTITION is introduced. This reserved word marks the start of a new partition within the model. If the model does not start with such a partition definition then a default partition is created for the following blocks. All block definitions following a partition definition are automatically assigned to that partition. During the simulation each partition will be performed on a separate LP. 5.1.5 Simulation Termination at Specific Simulation Time The parallel simulator will not provide any syntax or configuration options for terminating a simulation when a specific simulation time is reached but such behaviour Implementation can easily be modelled in any simulation model using an additional GENERATE and TERMINATE block. For instance if a simulation is supposed to be terminated when it reaches a simulation time of 10,000 then an additional GENERATE block is added that generates a single Transaction for the simulation time 10,000 immediately followed by a TERMINATE block that stops the simulation when that Transaction is terminated. This additional set of GENERATE and TERMINATE block can either be added to the end of an existing partition or as an additional partition. All other TERMINATE blocks in such a simulation will need to have a decrement parameter of 0. The following GPSS code shows an example model that will terminate at the simulation time 10,000. PARTITION Partition1,1 sets Termination Counter to 1 … original model partition GENERATE 1,0,10000 generates a Transaction for time 10000 TERMINATE 1 end of simulation after 1 Transaction 5.2 Implementation Phases The following sections will describe the four main development phases of the parallel simulator. 5.2.1 Model Parsing The classes for parsing and validating the GPSS model read from the model file can be found in the package parallelJavaGpssSimulator.gpss.parser. A GPSS model file is parsed by calling the method ModelFileParser.parseFile(). This method returns an instance of the class Model from the package parallelJavaGpssSimulator.gpss that contains the whole GPSS model as an object structure. The Model instance contains a list of model partitions represented by instances of the class Partition and each Partition instance contains a list of GPSS blocks and lists of other entities like labels, queues, facilities and storages that make up the model partition. Global GPSS block references GPSS simulators require a way of referencing GPSS blocks. A TRANSFER block for instance needs to reference the block it should transfer Transactions to. Sequential simulators often just use the block index within the model to refer to a specific block. Implementation But in a parallel GPSS simulator each Logical Process only knows its own model partition. Still a block reference needs to uniquely identify a GPSS block within the whole simulation model and ideally it should also be possible to determine the target partition from a block reference. The GlobalBlockReference class in the package parallelJavaGpssSimulator.gpss implements a block reference that fulfils these criteria and it is used to represent global block references and also block labels within the runtime model object structure and other parts of the simulator. Parser class hierarchy A parallel class hierarchy and the Builder design pattern [22] is used in order to separate the code for parsing and validating the GPSS model from the code that represents the model at the runtime of the simulation. This second class hierarchy is found in the parallelJavaGpssSimulator.gpss.parser package and contains a builder class for all element types that can make up the model structure at runtime. Figure 13 shows the UML diagram of the two class hierarchies including some of the relevant methods. When loading and parsing a GPSS model file the instance of the ModelFileParser class internally creates an instance of the ModelBuilder class, which for each partition found in the model file holds an instance of a PartitionBuilder class and the PartitionBuilder class holds builder classes for all blocks and other entities that are found in the partition. The parsing and validation of the different model elements is delegated to these builder classes. In addition the Factory design pattern [22] is used by the BlockBuilderFactory class that creates the correct builder class for a GPSS block depending on the block type keyword found in the model file. All builder classes have a build() method that returns an instance of the corresponding simulation runtime class for that element. These build() methods are called recursively so that the ModelBuilder.build() method calls the build() method of the PartitionBuilder instances it contains and each PartitionBuilder instance calls the build() method of all builder classes it contains. This delegation of responsibility within the class hierarchy makes it possible to return an instance of the Model class representing the whole GPSS model just by calling the ModelBuilder.build() method. As mention above the package parallelJavaGpssSimulator.gpss.parser is only used to load, parse and verify a GPSS model from file into the object structure used at simulation runtime. For this reason the only class from this package with public Implementation visibility is ModelFileParser. All other classes of this package are only visible within the package itself. Model Partition Block AdvanceBlock TransferBlock FacilityEntity StorageEntity #build() : Model ModelBuilder #build() : Partition PartitionBuilder #build() : Block BlockBuilder #build() : AdvanceBlock AdvanceBlockBuilder #build() : TransferBlock TransferBlockBuilder #build() : FacilityEntity FacilityEntityBuilder #build() : StorageEntity StorageEntityBuilder +parseFile(in fileName : String) : Model ModelFileParser Runtime model object structure classes in parallelJavaGpssSimulator.gpss Model parsing object structure classes in parallelJavaGpssSimulator.gpss.parser Figure 13: Simulation model class hierarchies for parsing and simulation runtime Test and Debugging of the Model Parsing In order to test and debug the parsing of the GPSS model file and the correct creation of the object structure representing the GPSS model at simulation run time, the toString() methods of all classes from the runtime class hierarchy were implemented to output their properties in textual form and to recursively call the toString() methods of any sub- element contained. A test application class with a main() method was implemented to load a GPSS model from a file using the ModelFileParser.parseFile() method and to output the whole structure of the model in textual form. Using this application different Implementation GPSS test models were parsed that contained all of the supported GPSS entities and the textual output of the resulting object structures was checked. These tests included checking the default values of the different GPSS block types and parsing errors for invalid model files. 5.2.2 Basic GPSS Simulation Engine The second implementation phase focused on the development of the basic GPSS simulation functionality. A GPSS simulation engine was implemented that can perform the sequential simulation of one model partition. This sequential simulation engine will be the basis of the parallel simulation engine implemented in the third phase. The classes for the basic GPSS simulation functionality can be found in the package parallelJavaGpssSimulator.gpss. The main class in this package is the SimulationEngine class that encapsulates the GPSS simulation engine functionality. It uses the runtime model object structure class hierarchy mentioned in 5.2.1 to represent the model partition and the model state in memory. The runtime model object structure class hierarchy contains classes for the GPSS block types plus some additional classes to represent other GPSS entities like Facilities, Queues and Storages. Each of these classes implements the functionality that will be performed when a block of that type is executed or the GPSS entity is used by a Transaction. Two further classes in this package are the Transaction class representing a single Transaction and the GlobalBlockReference class introduced in 5.2.1. The basic GPSS simulation scheduling is also implemented using the SimulationEngine class (a detailed description of the scheduling can be found in 5.3.1). It will generate Transactions and move them through the simulation model partition using the runtime model object structure. All block instances within this runtime model object structure inherit from the abstract Block class and therefore have to implement the execute() method. When a Transaction is moved through the model it will call this execute() method for each block it enters. Test and Debugging of Basic GPSS Simulation functionality The test application class TestSimulationApp was used to test and debug the basic GPSS simulation functionality implemented during this phase. This class contains a main() Implementation method and can therefore be run as an application. It allows the simulation of a single model partition using the SimulationEngine class. The exact simulation processing can be followed using the log4j logger parallelJavaGpssSimulator.gpss (details of the logging can be found in 5.3.5). In debug mode this logger outputs detailed steps of the simulation. Several test models where used to test the correct implementation of the basic scheduling and the GPSS blocks and other entities. They will not be described in more detail here because the same functionality will be tested again in the final version of the simulator (see validation phase in section 6). 5.2.3 Time Warp Parallel Simulation Engine During the third development phase the parallel simulator was implemented based on the Time Warp synchronisation algorithm. The functionality of the parallel simulator is split into the Simulation Controller side and the Logical Process side, each found in a different package. Figure 14 shows the general architecture and the main classes involved at each side during this development phase. Classes marked with (AO) are instantiated as ProActive Active Objects. Instances of the LogicalProcess class also communicate with each other, for instance in order to exchange Transactions, which is not displayed in Figure 14. LogicalProcess (AO) ParallelSimulationEngine Partition Simulate SimulationController (AO) Configuration ModelFileParser LogicalProcess (AO) ParallelSimulationEngine Partition Figure 14: Parallel simulator main class overview (excluding LPCC) Implementation Simulation Controller side At the Simulation Controller side is the root package parallelJavaGpssSimulator. It contains the class Simulate, which is the application class that is used to start the parallel simulator. When run the Simulate class will load the configuration from command line arguments or the configuration file, it will also load and parse the simulation model file and then create an Active Object instance of the SimulationController class (the JavaDoc documentation for this class can be found in Appendix E). The SimulationController instance receives the configuration settings and the simulation model when its simulate() method is called. As a result of this call the SimulationController class will read the deployment descriptor file and create the required number of LogicalProcess instances at the specified nodes. Logical Process side The functionality of the Logical Processes is found in the package parallelJavaGpssSimulator.lp. This package contains the LogicalProcess class, the ParallelSimulationEngine class (the JavaDoc documentation for both can be found in Appendix E) and a few helper classes. The LogicalProcess instances are created as Active Objects by the Simulation Controller. After their creation the LogicalProcess instances receive the simulation model partitions and the configuration when their initialize() method is called. When all LogicalProcess instances are initialised then the Simulation Controller calls their startSimulation() method to start the simulation. Figure 15 illustrates the communication flow between the Simulation Controller and the Logical Processes before and at the end of the simulation. The method calls just described can be found at the start of this communication flow. When the Simulation Controller detects that a confirmed simulation end has been reached then all Logical Processes are requested to end the simulation with a consistent state matching that confirmed simulation end using the endOfSimulationByTransaction() method. The Logical Processes will confirm when they reached the consistent simulation state after which the Simulation Controller will request the post simulation report details from each Logical Process. Further specific details about the implementation of the parallel simulator can be found in section 5.3. Implementation Simulation Controller LP 1 LP n initialize() startSimulation() endOfSimulationByTransaction() getSimulationReport() Initialization Simulation End of simulation Post simulation reporting ... ... Communication phases Figure 15: Main communication sequence diagram Test and Debugging of the Time Warp parallel simulator The parallel simulator resulting from this development phase was tested and debugged with extensive logging enabled and using different models. The functionality was tested again in the final version of the parallel simulator as part of the validation phase, of which details can be found in section 6. 5.2.4 Shock Resistant Time Warp This development phase extended the Time Warp based parallel simulator from the former development phase to support the Shock Resistant Time Warp algorithm by adding the LPCC and the required sensor value functionality to the LogicalProcess class. The functionality for the Shock Resistant Time Warp algorithm is found in the package parallelJavaGpssSimulator.lp.lpcc. The main class in this package is the class LPControlComponent that implements the LPCC (see Appendix E for the JavaDoc documentation of this class). The package also contains two classes that represent the sets of sensor and indicator values and the class StateClusterSpace that encapsulates the functionality to store and retrieve past indicator state information using the cluster Implementation technique described in [8]. The Shock Resistant Time Warp algorithm of the parallel simulator is implemented so that it can be enabled and disabled by a configuration setting of the parallel simulator as required. If the LPCC and therefore the Shock Resistant Time Warp algorithm is disabled then the parallel simulator will simulate according to the normal Time Warp algorithm, if it is enabled then the Shock Resistant Time Warp algorithm will be used. The option to enable/disable the LPCC makes it possible to compare the performance of both algorithms for specific simulation models and hardware setup using the same parallel simulator. The Logical Process Control Component (LPCC) implemented by the LPControlComponent class is used by the LogicalProcess instances during a simulation according to the Shock Resistant Time Warp algorithm. It is the main component of this algorithm that attempts to steer the parameters of the LPs towards values of past states that promise better performance using an actuator that limits the number of uncommitted Transaction moves allowed. Figure 16 shows the architecture and main classes used by the final version of the parallel simulator including the LPControlComponent class representing the LPCC. LogicalProcess (AO) ParallelSimulationEngine Partition LPControlComponent StateClusterSpace LogicalProcess (AO) ParallelSimulationEngine Partition LPControlComponent StateClusterSpace Simulate SimulationController (AO) Configuration ModelFileParser Figure 16: Parallel simulator main class overview Implementation The LPCC receives the current sensor values with each simulation time update cycle (details of the scheduling can be found in 5.3.1) but the main processing of the LPCC is only called during specified time intervals as set in the configuration file of the simulator. When the main processing of the LPCC is called using its processSensorValues() method then the LPCC will create a set of indicator values for the sensor values cumulated. Using the State Cluster Space it will search for a similar indicator set that promises better performance and it will set the actuator according to the indicator set found. Finally the current indicator set will be added to the State Cluster Space. The LPCC is also used to check whether the current number of uncommitted Transaction moves exceeds the current actuator limit. Within the scheduling cycle the LP will call the isUncommittedMovesValueWithinActuatorRange() method of its LPControlComponent instance to perform this check. As a result the number of uncommitted Transaction moves passed in is compared to the maximum actuator limit determined by the mean actuator value and the standard deviation with a confidence level of 95% as described in [8]. The method will return false if the number of uncommitted Transaction moves exceeds the maximum actuator limit forcing the LP into cancelback mode (see 5.3.4). State Cluster Space The StateClusterSpace class encapsulates the functionality to store sets of indicator values and to return a similar indicator set for a given one. Each stored indicator set is treated as a vector in an n-dimensional vector space with n being the number of indicators per set. The similarity between two indicator sets is determined by their Euclidean vector distance. A clustering technique is used that groups similar indicator sets into clusters to limit the amount of memory required when large numbers of indicator sets are stored. The two main public methods provided by the StateClusterSpace class are addIndicatorSet() and getClosestIndicatorSetForHigherCommittedMoveRate(). The first method adds a new indicator set to the State Cluster Space and the second returns the indicator set most similar to the one passed in that has a higher CommittedMoveRate indicator value. Note that the two indicators AvgUncommittedMoves and CommittedMoveRate are ignored when determining the similarity by calculating the Implementation Euclidean distance because AvgUncommittedMoves is directly linked to the actuator and CommittedMoveRate is the performance indicator that is hoped to be maximized. Test and Debugging of the Shock Resistant Time Warp and the State Cluster Space The State Cluster Space was tested and debugged using the test application class TestStateClusterSpaceApp, which allows for the StateClusterSpace class to be tested outside the parallel simulator. Using this class the detailed functionality of the State Cluster Space was tested using specific scenarios that would have been difficult to create within the parallel simulator. The test application class is left in the project so that possible future changes or enhancements to the StateClusterSpace class can also be tested outside the parallel simulator. The implementation of the Shock Resistant Time Warp algorithm was tested and debugged in the final version of the parallel simulator using a selection of different models of which a significant one was chosen for validation 5 in section 6.5. 5.3 Specific Implementation Details The following sections describe some specific implementation details of the parallel simulator. 5.3.1 Scheduling The scheduling of the parallel simulator was implemented in two phases. The first part is the basic scheduling of the GPSS simulation that was implemented using the SimulationEngine class as described in section 5.2.2. This scheduling algorithm was later extended for the parallel simulation by the LogicalProcess class and the ParallelSimulationEngine class, which inherits from the SimulationEngine class as part of the Time Warp parallel simulator implementation phase described in 5.2.3. Basic GPSS Scheduling The basic GPSS scheduling is implemented using the functionality provided by the SimulationEngine class. A flowchart diagram of the scheduling algorithm is shown in Figure 17. As seen from this diagram the scheduling algorithm will first initialise the GENERATE blocks in order to create the first Transactions. Subsequent Transactions Implementation are created whenever a Transaction leaves a GENERATE block. The algorithm then updates the simulation time to the move time of the earliest movable Transaction. After the simulation time has been updated all movable Transactions with a move time of the current simulation time are moved through the model as far as possible. Unless this results in the simulation being completed the algorithm will repeat the cycle of updating the simulation time and moving the Transactions. Start Initialise GENERATE blocks Update simulation time Move all Transactions for current simulation time Is simulation finished? Figure 17: Scheduling flowchart - part 1 Figure 18 shows the flowchart of the Move all Transactions for current simulation time processing block from Figure 17. The algorithm for this block will retrieve the first movable Transaction for the current simulation time and take this Transaction out of the Transaction chain. If no such Transaction is found then the processing block is left. Otherwise the Transaction is moved through the model as far as possible. If the Transaction is not terminated as a result then it is chained back into the Transaction Implementation chain at the correct position according to its move time and priority (note that the move time and priority could have changed while the Transaction was moved). Start Chain out next movable Transaction for current time Transaction found? Move Transaction as far as possible Transaction terminated? Chain in Transaction Move all Transactions for current simulation time Figure 18: Scheduling flowchart - part 2 Implementation The Move Transaction as far as possible processing block is split down further and its algorithm illustrated in the flowchart shown in Figure 19. Start Is current block GENERATE? Execute GENERATE block Execute next block Has Transaction time changed? Is Transaction terminated? Is next block in same partition? Move Transaction as far as possible Figure 19: Scheduling flowchart - part 3 Implementation The algorithm will first check whether the Transaction is currently within a GENERATE block and if so the GENERATE block is execute. Then the Transaction is moved into the next following block by executing it. Unless the move time of the Transaction changed, the Transaction got terminated or the next block of the Transaction lays within a different partition the algorithm will repeatedly execute the next block for the Transaction in a loop and therefore move the Transaction from block to block. From this flowchart it can be seen that the execution of GENERATE blocks is treated different to the execution of other blocks. The reason is that GENERATE blocks are the only blocks that are executed when a Transaction leaves the block where as all other blocks are executed when the Transaction enters them. This allows a GENERATE block to create the next Transaction when the last one created leaves it. The table below mentions the different methods that implement the flowchart processing blocks described. Flowchart processing block Method Initialise GENERATE blocks SimulationEngine.initializeGenerateBlocks() Update simulation time SimulationEngine.updateClock() Move all Transactions for current simulation time SimulationEngine. moveAllTransactionsAtCurrentTime() Chain out next movable Transaction for current time SimulationEngine. chainOutNextMovableTransactionForCurrentTime() Move Transaction as far as possible SimulationEngine.moveTransaction() Chain in Transaction SimulationEngine.chainIn() Execute GENERATE block GenerateBlock.execute() Execute next block Calls the execute() method of the next block instance for the Transaction Table 8: Methods implementing basic GPSS scheduling functionality Implementation Extended parallel simulation scheduling For the parallel simulator the simulation scheduling is implemented in the Logical Processes. It integrates the Active Object request processing of the LogicalProcess class and the synchronisation algorithm of the parallel simulation. This results in a scheduling algorithm that looks quite different to the one for the basic GPSS simulation. A slightly simplified flowchart of this algorithm can be found in Figure 20 (note that the darker flowchart processing blocks are blocks that already existed in the basic GPSS scheduling algorithm). Because the LogicalProcess class is used as an Active Object its scheduling algorithm is implemented in the runActivity() method inherited from the org.objectweb.proactive. RunActive interface that is part of the ProActive library. The algorithm first checks whether the body of the Active Object is still active and then processes any Active Object method requests received. If the Logical Process is not in the mode SIMULATING then the algorithm will return and loop through checking the body and processing Active Object requests. If the mode is changed to SIMULATING then it will proceed to update the simulation time. This step existed already in the basic GPSS scheduling algorithm. Note that the functionality to initialize the GENERATE blocks is not part of the actual scheduling algorithm any more as it is performed when the LogicalProcess class is initialized using the initialize() method. After the simulation time has been updated the start state for the new simulation time will be saved. The state saving and rollback process is described in detail in section 5.3.3. The next step is to handle received Transactions, which includes anti-Transactions and cancelbacks. They are received via ProActive remote method calls and stored in an input list during the Process Active Object requests step. Normal received Transactions are handled by chaining them into the Transaction chain. This might require a rollback if the local simulation time has already passed the move time of the new Transaction. In order to handle a received anti-Transaction the matching normal Transaction has to be found and deleted. If the normal Transaction has been moved through the model already then a rollback is required as well. Cancelback requests are also handled by performing a rollback (see section 5.3.4 for details of the memory management and cancelback). Implementation Start Process Active Object requests Update simulation time Move all Transactions for current simulation time Save current state Handle received Transactions Send lazy-cancellation anti-Transactions Do movable Transactions exist? Is in Cancel Back mode? Send outgoing Transactions Update LPCC Is Active Object body active? Mode = SIMULATING? Figure 20: Extended parallel simulation scheduling flowchart Implementation Following the handling of received Transactions and anti-Transactions the scheduling algorithm will send out any anti-Transactions required by the lazy-cancellation mechanism. It will identify all Transactions that have been sent out for an earlier simulation time and which have been rolled back and subsequently not sent again. Such Transactions need to be cancelled by sending out anti-Transactions. If following the lazy-cancellation handling the Simulation Engine has movable Transactions and is not in cancelback mode then all movable Transactions for the current simulation time are moved through the simulation model. Any outgoing Transactions are sent to their destination Logical Process and the LPCC sensors are updated. The whole scheduling algorithm will be repeated until the LogicalProcess instance is terminated and its Active Object body becomes inactive. The methods implementing the flowchart processing blocks described are shown below. Flowchart processing block Method Process Active Object requests LogicalProcess.processActiveObjectRequests() Update simulation time SimulationEngine.updateClock() Save current state LogicalProcess.saveCurrentState() Handle received Transactions LogicalProcess.handleReceivedTransactions() Send lazy-cancellation anti- Transactions LogicalProcess. sendLazyCancellationAntiTransactions() Move all Transactions for current simulation time ParallelSimulationEngine. moveAllTransactionsAtCurrentTime() Send outgoing Transactions LogicalProcess. sendTransactionsFromSimulationEngine() Update LPCC LogicalProcess.updateLPControlComponent() Table 9: Methods implementing extended parallel simulation scheduling Implementation 5.3.2 GVT Calculation and End of Simulation Details of why and how the GVT is calculated during the simulation have already been described in 4.3 but here the focus lies on the actual implementation. The GVT calculation is performed by the SimulationController class within the private method performGvtCalculation(). During the GVT calculation the Simulation Controller will request the required parameters from each LP, determine the GVT and pass the GVT back to the LPs so that these can perform the fossil collection. Figure 21 shows the sequence diagram of the GVT calculation process. Simulation Controller LP 1 LP n requestGvtParameter() receiveGvt() performGvtCalculation() Figure 21: GVT calculation sequence diagram There are different circumstances that can cause a GVT calculation within the parallel simulator. First LPs can request a GVT calculation from the Simulation Controller by calling its requestGvtCalculation() method. This happens when an LP reached certain defined memory limits (as described in 5.3.4) or when a provisional simulation end is reached by one of the LPs, which is described in more detail further below. Another reason for a GVT calculation is that the LPCC processing is required because the defined processing time interval has passed. For the Shock Resistant Time Warp algorithm the LPCC processing is linked to a GVT calculation so that the sensor and indicator for the number of committed Transaction moves have realistic values that reflect the simulation progress made during the time interval. For this reason the LPCC processing times are controlled by the Simulation Controller and linked to GVT Implementation calculations that are triggered when the next LPCC processing is needed. An additional parameter for the method receiveGvt() of the LogicalProcess class indicates to the LP that an LPCC processing is needed after the new GVT has been received. Finally the user can also trigger a GVT calculation, which is useful for simulations in normal Time Warp mode that might not require any GVT calculation for large parts of the simulation. Forcing a GVT calculation allows the user to check what progress the simulation has made so far as the GVT is an approximation for the confirmed simulation times that has been reached by all LPs. End of simulation The detection of the simulation end is closely linked to the GVT calculation because a provisional simulation end reached by one of the LPs can only be confirmed by a GVT. The background of detecting the simulation end has already been discussed in 4.4 but the actual implementation will be explained here. When an LP reaches a provisional simulation end then the parallel simulator will attempt to confirm this simulation end as soon as possible if at all possible. First the LP reaching the provisional simulation end will request a GVT calculation from the Simulation Controller. But the resulting GVT might not confirm the provisional simulation end if the LP is ahead of other LPs in respect of the simulation time. For this case a scheme is introduced in which the LP reaching the provisional simulation end tells all other LPs to request a GVT calculation themselves if they pass the simulation time of that provisional simulation end. The method forceGvtAt() of the LogicalProcess class is used to tell other LPs about the provisional simulation end time. Because it is possible for more than one LP to reach a provisional simulation end before any of them is confirmed this method will keep a list of the times at which the LPs need to request GVT calculations. Whether or not a provisional simulation end reached by one of the LPs is confirmed, is detected by the private method performGvtCalculation() of the SimulationController class that also performs the calculation of the GVT. In order to make this possible the method requestGvtCalculation() of the LogicProcess class returns additional information about a possible simulation end reached by that LP. This way the GVT calculation process described above is also used to confirm a provisional simulation end. Such a simulation end is confirmed during the GVT calculation when it is found that all other LPs have reached a later simulation time than the one that reported the Implementation provisional simulation end. In this case no future rollback could occur that can undo the provisional simulation end, which is therefore guaranteed. If the GVT calculation confirms a simulation end then no GVT is send back to the LPs but instead the Simulation Controller calls the method endOfSimulationByTransaction() of all LPs as shown in Figure 15 of section 5.2.3. 5.3.3 State Saving and Rollbacks Optimistic synchronisation algorithms execute all local events without guarantee that additional events received later will not violate the causal order of events already execute. In order to correct such causal violations they have to provide means to restore a past state before the causal violation occurred so that the new event can be inserted into the event chain and the events be executed again in the correct order. A common technique to allow the restoration of past states is called State Saving or State Checkpointing where an LP saves the state of the simulation into a list of simulation states each time the state changes or in defined intervals. The parallel simulator implemented employs a relatively simple state saving scheme. Each time the simulation time is incremented the LP serialises the state of the Simulation Engine and saves it together with the corresponding simulation time into a state list. Each state record therefore describes the simulation state at the beginning of that time, i.e. before any Transactions were moved. To keep the complexity of this solution low the standard object serialisation functionality provided by Java is used to serialise and deserialise the state to and from a Stream object that is then stored in the state list. The state list keeps all states sorted by their time. The saving of the state is implemented in the method saveCurrentState() of the LogicalProcess class. The purpose of saving the simulation state is to allow LPs to rollback to past simulation states if required. The functionality to rollback to a past simulation state is implemented in the method rollbackState(). Using the example shown in Figure 22 the principle of rolling back to a past simulation state is briefly explained. Implementation t = 0 t = 3 t = 8 t = 12 S(0) S(3) S(8) S(12) Reale time Simulation time Saved states S(0) S(12) List of saved states Current simulation time: 12 Figure 22: State saving example Figure 22 shows the state information of an LP that has gone through the simulation times 0, 3, 8 and 12 and that has saved the simulation state at the beginning of each of these times. There are two possible options for a rollback depending on whether a state for the simulation time that needs to be restored exists in the state list or not. If for instance the LP receives a Transaction for the time 3 then the LP will just restore the state of the time 3, chain in the new Transaction and proceed moving the Transactions in their required order. But if a Transaction for the simulation time 5 is received, which implies that a rollback to the simulation time 5 is needed then the state of the time 8 is restored because this is the same state that would have existed at the simulation time 5. Recapitulating it can be said that if no saved state exists for the simulation time to that a rollback is needed then the rollback functionality will restore the state with the next higher simulation time. In addition to the basic task of restoring the correct simulation state the rollbackState() method also performs a few related tasks like chaining in any Transactions that were received after that restored state was saved or marking any Transactions sent out after the rollback simulation time for the lazy-cancellation mechanism. A further task related to the state management is performing the fossil collection which is implemented by the commitState() method of the LogicalProcess class. This method is called when the LP receives a new GVT. It will remove any past simulation states and other information, for instance about Transactions received or sent, that are not needed any more. Because of the time scale of this project and in order to keep the complexity of the implementation low, the state saving scheme used by the parallel simulator is a Implementation relatively basic periodic state saving scheme. Future work on the simulator could look at enhancing the state saving using an adaptive periodic checkpointing scheme with variable state saving intervals as suggested in [25]. Alternatively an incremental state saving scheme could be used but this would drastically increase the complexity of the state saving because the standard Java serialisation functionality could not be used or would need to be extended. An incremental state saving scheme would also add an additional overhead for restoring a specific state so that an adaptive periodic checkpointing scheme appears to be the best option for future enhancements. 5.3.4 Memory Management Optimistic synchronisation algorithms make extensive use of the available memory in order to save state information that allow the restoration and the rollback to a past simulation state required if an LP receives an event or Transaction that would violate the causal order of events or Transactions already executed. At the same time a parallel simulator has to avoid running out of available memory completely as this would mean the abortion of the simulation. The parallel simulator implemented here will therefore use a relatively simple mechanism to avoid reaching the given memory limits. It will monitor the memory available to the LP within the JVM during the simulation and perform defined actions if the available memory drops below certain limits. The first limit is defined at 5MB. If the amount of available memory goes below this limit then the LP will request a GVT calculation from the Simulation Controller in the expectation that a new GVT will confirm some of the uncommitted Transaction moves and saved simulation states so that fossil collection can free up some of the memory currently used. In some circumstances GVT calculations will not free up any memory used by the LP or not enough. This is for instance the case when the LP is far ahead in simulation time compared to the other LPs. If none if its uncommitted Transaction moves or saved simulation states are confirmed by the new GVT then no memory will be freed by fossil collection. Otherwise it is also possible that only very few uncommitted Transaction moves and saved simulation states are confirmed by the new GVT resulting in very little memory being free. The parallel simulator defines a second memory limit of 1MB for the case that GVT calculations did not help in freeing memory. When the memory available to the LP drops below this second limit then the LP switches into cancelback Implementation mode. A cancelback strategy was already mentioned by David Jefferson [17] but the cancelback strategy used here will differ slightly from the one suggested by him. When the LP operates in cancelback mode then it will still respond to control messages and will still receive Transactions from other LPs but it will stop moving or processing any local Transaction so that no simulation progress is made by the LP and no simulation state information are saved as a result. Further the LP will attempt to cancel back Transactions that it received from other LPs in order to free memory or at least stop memory usage growing further. To cancelback a Transaction means that all local traces that a Transaction was received are removed and the Transaction is sent back to its original sender that will rollback to the move time of that Transaction. The main methods involved with the cancelback mechanism are the method LogicalProcess.needToCancelBackTransactions() which is called by an LP that is in cancelback mode and the method LogicalProcess.cancelBackTransaction() which is used to send a cancelled Transaction back to the sender LP. This cancelback mechanism of the parallel simulator is not only used for the general memory management but also when the Actuator value of the LPCC has been exceeded. 5.3.5 Logging The parallel simulator uses the Java logging library log4j [3] for its logging and standard user output. It is the same logging library that is used by ProActive. The log4j library makes it possible to enable or disable parts or all of the logging or to change the detail of logging by means of a configuration file without any changes to the Java code. To utilise the same logging library for ProActive and the parallel simulator means that only a single configuration file can be used to configure the logging output for both. A hierarchical structure of loggers combined with inheritance between loggers makes it very easy and fast to configure the logging of the simulator. A detailed description of the log4j library and its configuration can be found at [3]. The specific loggers used by the parallel simulator are described in Appendix C. As mentioned above the parallel simulator will use the same log4j configuration file like ProActive. By default this is the file proactive-log4j but a different file can be specified as described in the ProActive documentation [15]. The log4j root logger for all output from the parallel simulator is parallelJavaGpssSimulator (in the log4j configuration file all loggers have to be prefixed with “log4j.logger.” so that this logger would appear as Implementation log4j.logger.parallelJavaGpssSimulator). A hierarchy of lower level loggers allow the configuration of which information will be output or logged by the parallel simulator. The log4j logging library supports the inheritance of logger properties, which means that a lower level logger that is not specifically configured will inherit the configuration from a logger at a higher level within the same name space. For example if only the logger parallelJavaGpssSimulator is configured then all other loggers of the parallel simulator would inherit the same configuration settings from it. 5.4 Running the Parallel Simulator 5.4.1 Prerequisites The parallel GPSS simulator was implemented using the JavaTM 2 Platform Standard Edition 5.0, also known as J2SE5.0 or Java 1.5 [31] and ProActive version 3.1 [15] as the Grid environment. J2SE5.0 or the JRE of the same version plus ProActive 3.1 need to be installed on all nodes that are supposed to be used by the parallel GPSS simulator. The parallel simulator might also work with later versions of the Java Runtime Environment and ProActive as long as these are backwards compatible but the author of this work can give no guarantees in this respect. Because the parallel simulator and the libraries it uses are written in Java it can be run on many different platforms. But the main target platforms of this work are Unix based systems because the scripts that are part of the parallel simulator are only provided as Unix shell scripts. These relatively basic scripts will need to be rewritten before the parallel simulator can be used on Windows or other non-Unix based platforms. 5.4.2 Files The following files are required or are optionally needed in order to run the parallel simulator. They can be found in the folder /ParallelJavaGpssSimulator/ on the attached CD and will briefly be described here. deploymentDescriptor.xml This is the ProActive deployment descriptor file mentioned in 3.2.1. It is read by ProActive to determine which nodes the parallel simulator should use and how these Implementation need to be accessed. A detailed description of this file and the deployment configuration of ProActive can be found at the ProActive project Web site [15]. ProActive uses the concept of virtual nodes for its deployment. For the parallel simulator the ProActive deployment descriptor file needs to contain the virtual node ParallelJavaGpssSimulator. If this virtual node is not found then the parallel simulator will abort with an error message. In addition the deployment descriptor file needs to define enough JVM nodes linked to this virtual node for the number of partitions within the simulation model to be simulated. DescriptorSchema.xsd This is the XML schema file that describes the structure of the deployment descriptor XML file. It is used by ProActive to verify that the XML structure of the file deploymentDescriptor.xml mentioned above is correct. env.sh This Unix shell script is part of ProActive and is only included because it is needed by the file startNode.sh described further down. It can also be found in the ProActive installation. Together with the file startNode.sh it is used to start ProActive nodes directly from this folder. But first the environment variable PROACTIVE defined in the beginning of this file might have to be changed to point to the installation location of the ProActive library. ParallelJavaGpssSimulator.jar This is the JAR file (Java archive) that contains the Java class files, which make up the parallel simulator. It is required by the script simulate.sh described further down in order to start and run the parallel simulator. proactive.java.policy This is a copy of the default security policy file provided by ProActive. It can also be found in the ProActive installation and is provided here so that the parallel simulator can be run straight from this folder. This security policy file basically disables any access restrictions by granting all permissions. It should only be used when no security and access restrictions are needed. Please refer to the ProActive documentation [15] regards defining a proper security policy for a ProActive Grid application like the parallel simulator. Implementation proactive-log4j This is the log4j logging configuration file used by the parallel simulator and ProActive. A description of this file and how logging is configured for the parallel simulator can be found in section 5.3.5. simulate.config This is the default configuration file for the parallel simulator. The configuration of the parallel simulator is explained in detail in 5.4.3. simulate.sh This Unix shell script is used to start the parallel simulator. It defines the two environment variables PROACTIVE and SIMULATOR. Both might need to be changed before the parallel simulator can be run so that PROACTIVE points to the ProActive installation directory and SIMULATOR points to the directory containing the parallel simulator JAR file ParallelJavaGpssSimulator.jar. Further details about how to run the parallel simulator can be found in 5.4.4. startNode.sh This Unix shell script is part of ProActive and is used to start a ProActive node. It is a copy of the of the same file found in the ProActive installation and is only provided here so that ProActive nodes for the LPs of the parallel simulator can be started straight from the same directory. The file env.sh is called be this script to setup all environment variables needed by ProActive. 5.4.3 Configuration The parallel simulator can be configured using command line arguments or by a configuration file. The reading of the configuration settings from the command line arguments or from the configuration file is handled by the Configuration class in the root package. If the parallel simulator is started with no further command line arguments after the simulation model file name then the default configuration file simulate.config is used for the configuration. If the next command line argument after the simulation model file name has the format ConfigFile=… then the specified configuration file is used. Otherwise the configuration is read from the existing Implementation command line arguments and default values are used for any configuration settings not specified. Configuration settings have the format <parameter name>=<value> and Boolean configuration settings can be specified without value and equal sign in which case they are set to true. This is useful when specifying configuration settings as command line arguments. For instance to get the parallel simulator to output the parsed simulation model it is enough to add the command line argument ParseModelOnly instead of ParseModelOnly=true. A detailed description of the configuration settings can be found in Appendix B. 5.4.4 Starting a Simulation Before a simulation model can be simulated using the parallel simulator the deploymentDescriptor.xml needs to contain enough node definitions linked to the virtual node ParallelJavaGpssSimulator for the number of partitions within the simulation model. If the deployment descriptor file does not define how ProActive can automatically start the required nodes then the ProActive nodes have to be created manually on the relevant machines using the startNode.sh script before the parallel simulator can be started. The parallel simulator is started using the shell script simulate.sh. The exact syntax is: simulate.sh <simulation model file> [<command line argument>] […] The configuration of the parallel simulator and possible command line arguments are described in 5.4.3 and the files required to run the parallel simulator and their meaning are explained in 5.4.2. 5.4.5 Increasing Memory Provided by JVM By default the JVM of J2SE5.0 provides only a maximum of 64MB of memory to the Java applications that run inside it (Maximum Memory Allocation Pool Size). Considering that at the time of this paper standard PCs already come with a physical memory of around 1GB and dedicated server machines even more, the Maximum Memory Allocation Pool Size of the JVM does not seem appropriate. Therefore in order to make the best possible use of the memory provided by the Grid nodes the Maximum Implementation Memory Allocation Pool Size of the JVM needs to be increased to the amount of memory available. This is especially important for long running simulations and complex simulation models. The Maximum Memory Allocation Pool Size of the JVM can be set using the command line argument –Xmxn of the java command (see Java documentation for more details [31]). If the ProActive nodes running the LPs of the parallel simulator are started using the startNode.sh script then this command line argument with the appropriate memory size can be added to this script, otherwise if the nodes are started via the deployment descriptor file then the command line argument has to be added there. The following example shows how the startNode.sh script needs to be changed in order to increase the Maximum Memory Allocation Pool Size from its default value to 512MB. $JAVACMD org.objectweb.proactive.core.node.StartNode $1 $2 $3 $4 $5 $6 $7 Extract of the startNode.sh script with default memory pool size $JAVACMD –Xmx512m org.objectweb.proactive.core.node.StartNode $1 $2 $3 $4 $5 $6 $7 Extract of the startNode.sh script with memory pool size of 512MB Validation of the Parallel Simulator 6 Validation of the Parallel Simulator The functionality of the parallel simulator was validated using a set of example simulation models. These simulation models were deliberately kept very simple in order to evaluate specific aspects of the parallel simulator as complex models would possibly hide some of the findings and would make the analysis of the results more difficult. Each of the validations evaluates a particular part of the overall functionality and the example simulation models were specifically chosen for that evaluation. They therefore don’t represent any real live systems. Of course it cannot be expected that this validation using example simulation models will prove the absolute correctness of the implemented functionality. But instead the different validation runs performed provide a sufficient level of confidence that the functionality of the parallel simulator is correct. All files required to perform these validations including the specific configuration files and the resulting validation output log files can be found in specific sub folders of the attached CD. For further details about the CD see Appendix D. The relevant output log files of the validation runs performed are also included in Appendix F. Line numbers in brackets were added to all lines of the output log files in order to make it possible to refer to a particular line. The log4j logging system [3] was specifically configured for each validation run to include certain details or exclude details that were not relevant to that particular validation. The Termination Counters for the validation runs were chosen so that the simulation runs were long enough to evaluate the specific aspects but also kept as short as possible in order to avoid unnecessary long output log files. Nevertheless some of the validations still resulted in long output log files. In these cases some of the lines that were not relevant to the validation have been removed from the output logs listed in Appendix F. The complete output log files can still be found on the attached CD. The validation runs were performed on a standard PC with a single CPU (Intel Pentium 4 with 3.2GHz, 1GB RAM) running SuSE Linux 10.0. As the validation was performed only on a single CPU it should be noted that it does not represent a detailed investigation into the performance of the parallel simulator. Such an investigation would exceed the expected time scale of this project because the performance of a parallel simulation depends on a lot of different factors besides the simulation system (e.g. Validation of the Parallel Simulator simulation model, computation and communication performance of the hardware used) and would need to be analysed using a variety of simulation models and on different systems in order to draw any reliable conclusions. Nevertheless some basic performance conclusions where made as part of Validation 5 and 6. 6.1 Validation 1 The first validation checks the correct parsing of the supported GPSS syntax elements. Two models are used to evaluate the parser component of the parallel simulator. Both include examples of all GPSS block types and other GPSS entities but in the first model all possible parameters of the blocks and entities are used whereas in the second model all optional parameters are left out in order to test the correct defaulting by the parser. For both models the simulator was started using the ParseModelOnly command line argument option. When this option is specified then the simulator will not actually perform the simulation but instead parse the specified simulation model and either output the parsed in memory object structure representation of the simulation model or parsing errors if found. Validation 1.1 The first simulation model used is shown below: PARTITION Partition1,5 STORAGE Storage1,2 GENERATE 1,0,100,50,5 ENTER Storage1,1 ADVANCE 5,3 LEAVE Storage1,1 TRANSFER 0.5,Label1 TERMINATE 1 PARTITION Partition2,10 Label1 QUEUE Queue1 DEPART Queue1 SEIZE Facility1 RELEASE Facility1 TERMINATE 1 Simulation model file model_validation1.1.gps Validation of the Parallel Simulator The output log for this simulation model can be found in Appendix F. A comparison of the original simulation model file and the in memory object structure representation that was output by the simulator shows that they are equivalent and that the parser correctly parsed all lines of the simulation model. Validation 1.2 The simulation model for this validation is based on the earlier simulation model but all optional elements of the model were removed. STORAGE Storage1 GENERATE ENTER Storage1 ADVANCE LEAVE Storage1 TRANSFER Label1 TERMINATE PARTITION Partition2 Label1 QUEUE Queue1 DEPART Queue1 SEIZE Facility1 RELEASE Facility1 TERMINATE Simulation model file model_validation1.2.gps As described this simulation model tests the parser regards setting default values for optional elements and parameters. Comparing the simulation model file to corresponding output log in Appendix F it can be found that the parser automatically created a new partition before parsing the first line of the model so that the in memory representation of the model contains two partitions (see line 2 of output log). The default name given to this partition by the parser is ‘Partition 1’ (see line 6 of output log). Line 9 shows that the Storage size was set to its maximum value of 2147483647. The GENERATE block at line 10 was parsed with all its parameters set to its default values as described in Appendix A. This also applies to the ADVANCED block at line 12 and the TERMINATE blocks at line 15 and 27 of the output log. The usage count of the ENTER and LEAVE block at the lines 11 and 13 were set to the expected default value of 1 and the TRANSFER block at line 14 of the output log also has the default transfer probability of 1 so that all Transactions would be transferred to the specified Validation of the Parallel Simulator label. It can be seen that all the missing parameters were set to their expected default values. 6.2 Validation 2 This validation evaluates the basic GPSS functionality of the parallel simulator. This includes the basic scheduling and the movement of Transactions as well as the correct processing of the GPSS blocks. The simulation model used for this contains only a single partition but otherwise all possible GPSS block types and entities. There is even a TRANSFER block that transfers Transactions with a probability of 0.5. The model is shown below: PARTITION Partition1,4 STORAGE Storage1,2 GENERATE 3,2 QUEUE Queue1 ENTER Storage1,1 ADVANCE 5,3 LEAVE Storage1,1 DEPART Queue1 TRANSFER 0.5,Label1 SEIZE Facility1 RELEASE Facility1 Label1 TERMINATE 1 Simulation model file model_validation2.gps The model is simulated with the log4j loggers parallelJavaGpssSimulator.simulation and parallelJavaGpssSimulator.gpss set to DEBUG (see configuration file proactive- log4j at the corresponding sub folder on the attached CD). The last of these two loggers will result in a very detailed logging of the GPSS processing and Transaction movement. For this reason the Termination Counter is kept very small, i.e. set to 4 so that the simulation is stopped after 4 Transactions have been terminated. Otherwise the output log would be too long to be useful. The deployment descriptor XML file is set to a single ProActive node as the model contains exactly one partition and the simulation will require only one LP. Validation of the Parallel Simulator The interesting output for this validation is the output log of the LP. Following this output log the simulation starts with initialising the GENERATE block (line 4 to 6). This results in a new Transaction with the ID 1 being chained in for the move time 4. The model above shows the GENERATE block with an average interarrival time of 3 and a half range of 2. This means that the interarrival times of the generated Transactions will lie in the open interval (1,5) with possible values of 2, 3 or 4. The current block of the new Transaction is (1,1) which is the GENERATE block itself as this Transaction has not been moved yet (in the logging of the parallel simulator a block reference is shown as a comma separated set of the partition number and the block number within that partition). The next step of the simulator found in the log is the updating of the simulation time to the value 4 at line 7 because the first movable Transaction (the one just generated) has a move time of 4. The lines 8 to 16 show how this Transaction is moved through the model until it reaches the ADVANCED block where it is delayed. The first block to be executed by the Transaction is the GENERATE block which results in a second Transaction being created when the first one is leaving this block as shown in line 10 and 11. The lines 12 and 13 show the first Transaction executing the QUEUE and ENTER block until it reaches the ADVANCE block at line 14. The ADVANE block changes the move time of the Transaction from 4 to 9 (delay by a value of 5), which means that, this Transaction is no longer movable at the simulation time of 4. At line 16 the Transaction is therefore chain back into the Transaction chain and because there is no other movable Transaction for the time of 4 the current simulation time is updated to the move time of the next movable Transaction, which is the one with an ID of 2 and a move time of 7. In the lines 18 to 26 the second Transaction is going through the same move process like the first Transaction before and when it is leaving the GENERATE block this results in a third Transaction with a move time of 10 being created and chained in. When the ADVANCE block changes the move time of the second Transaction from 7 to 13 as shown in line 24 the current simulation time is updated to the value of 9 and the first Transaction starts moving again (see line 27 to 35). It will execute the LEAVE and DEPART block before reaching the TRANSFER block at line 34. Here it is transferred directly to the TERMINATE block which can be seen from the next block property of the Transaction jumping from the block (1,7) to block (1,10). After executing the TERMINATE block the Transaction stops moving but is not chained back into the Transaction chain as it has been Validation of the Parallel Simulator terminated (see line 35 and 36). The simulator proceeds with updating the simulation time and moving the next Transaction. The rest of the output log can be followed analogue to before. The output log of the Simulate process is also found in Appendix F. This log contains the post simulation report and shows the interesting fact that all of the 4 Transactions that executed the TRANSFER block were transferred directly to the TERMINATE block. There should have been a ratio of 50% of the Transactions transferred but because the number of Transactions is very low this results in a large statistical error. Nevertheless Validation 3 will show that for a large number of Transactions the statistical behaviour of the TRANSFER block is correct. The validation has shown that the Transaction scheduling and movement as well as the processing of the blocks is performed by the simulator as expected. 6.3 Validation 3 The third validation focuses on the exchange of Transaction between LPs. It evaluates that the sending of Transactions from one LP to another works correctly including the correct functioning of the TRANSFER block. In addition it shows that an LP can correctly handle the situation where it has no movable Transactions left. In a sequential simulator this would lead to an error and the abortion of the simulation but in a parallel simulator this is a valid state as long as at least one of the LPs has movable Transactions. Further this validation shows the correct processing of the simulation end by the Simulation Controller and the LPs. Validation 3.1 This validation run uses a very simple model with two partitions. The first partition contains a GENERATE block and a TRANSFER block and the second partition the TERMINATE block. When run, the model will generate Transactions in the first partition and then transfer them to the second partition where they are terminated. Detailed GPSS logging is used again in order to follow the Transaction processing. The loggers enabled for debug output are shown below. Validation of the Parallel Simulator log4j.logger.parallelJavaGpssSimulator.gpss=DEBUG log4j.logger.parallelJavaGpssSimulator.lp=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.rollback=DEBUG log4j.logger.parallelJavaGpssSimulator.simulation=DEBUG Extract of the used log4j configuration file proactive-log4j To avoid unnecessary long output log files the Termination Counter for the partitions is set to 4 again so that the simulation will be finished after 4 Transactions have been terminated. In addition the GENERATE block has a limit count parameter of 10 so that it will only create a maximum of 10 Transactions. This limit is used because otherwise LP1 simulating the first partition would create more Transactions before the Simulation Controller has established the confirmed end of the simulation resulting in a longer output log with details that are not relevant to the simulation. The whole simulation model is shown below. PARTITION Partition1,4 GENERATE 3,2,,10 TRANSFER Label1 PARTITION Partition2,4 Label1 TERMINATE 1 Simulation model file model_validation3.1.gps Looking at the output log of LP1 the start of the Transaction processing is similar to the one of Validation 2. The initialisation of the GENERATE block creates the first Transaction (see line 5 to 7) and when the Transaction leaves the GENERATE block it creates the next Transaction and so forth. After the first Transaction with the ID 1 executed the TRANSFER block at line 15 it stops moving but is not chained back into the Transaction chain because it has been transferred to LP2 that simulates partition 2. The next block property of the Transaction shown in that line now points to the first block of partition two, i.e. has the value (2,1). Swapping to the output log of LP2 it can be seen at line 10 that LP2 just received the Transaction with the ID 1 and that this Transaction is chained in. Because of the communication delay and LP1 being ahead in simulation time LP2 already receives the next few Transactions as well as shown in line Validation of the Parallel Simulator 12 and 13. In the lines 14 to 16 the first Transaction received is now chained out and moved into the TERMINATE block where it is terminated. The processing of any subsequent Transactions within LP1 and LP2 follows the same pattern. The correct handling of the situation when an LP has no movable Transaction can be seen at line 9 of the output log of LP2. The LP will just stay in a waiting position not moving any Transaction until it either receives a Transaction from another LP or until the Simulation Controller establishes that the end of the simulation has been reached. Using all three output log files from LP1, LP2 and the simulate process the correct processing of the simulation end can be followed. The first step of the simulation end is the forth Transaction being terminated in LP2 (see line 32 of the output log of LP2). The LP detects that a provisional simulation end has been reached (line 33) and requests a GVT calculation from the Simulation Controller (line 34). Subsequently it is still receiving Transactions from LP1 (e.g. line 35, 38 and 41) but no Transactions are moved because the LP is in the provisional simulation end mode. The output log of the simulate process shows at the lines 14 to 16 that the Simulation Controller performs a GVT calculation and receives the information that LP1 has reached the simulation time 26 and LP2 has reached a provisional simulation end at the time 11. Because all other LPs except LP2 have passed the provisional simulation end time the Simulation Controller concludes that the simulation end is confirmed. It now informs the LPs of the simulation end which can be seen in line 55 of the output log of LP2 and line 83 of the output log of LP1. Because LP1 is ahead of the simulation end time this information causes it to rollback to the state at the start of simulation time 11 (line 84). The rollback leads to the Transaction with ID 7 being moved to the TRANSFER block again to reach the exact state needed for the simulation end. It can be seen from the output log of LP1 that the lines 32 to 37 are identical to the lines 85 to 90 which is a result of the rollback and re-execution in order to reach a state that is consistent with the simulation end in LP2. Both LPs confirm to the Simulation Controller that they reached the consistent simulation end state, which then outputs the combined post simulation report showing the correct counts as seen in line 23 to 30 of the simulate process output log. This post simulation report confirms that four Transactions were moved through all blocks and a 5th is already waiting in the GENERATE. The GENERATE block has not yet been Validation of the Parallel Simulator executed by the 5th Transaction because GENERATE blocks are executed when a Transaction leaves them. The output logs confirm that the transfer of Transactions between LPs and the handling of the simulation end reached by one of the LPs works correctly as expected. Validation 3.2 The second validation of this validation group looks at the correct statistical behaviour of the TRANSFER block when it is used with a transfer probability parameter. The model used for the validation run is similar to the model used by validation 3.1 but differs in the fact that this time the partition 1 has its own TERMINATE block and that the TRANSFER block only transfers 25% of the Transactions to partition 2. Below is the complete simulation model used for this run. PARTITION Partition1,750 GENERATE 3,2 TRANSFER 0.25,Label1 TERMINATE 1 PARTITION Partition2,750 Label1 TERMINATE 1 Simulation model file model_validation3.2.gps Another difference is that larger Termination Counter values are used in order to get reliable values for the statistical behaviour. With such large Termination Counter values the output logs of the LPs would be very long and not of much use, which is why, they are not included in Appendix F. The interesting output log for this validation run is the one of the simulate process and specifically the post simulation report. The expected simulation behaviour from the model shown above would be that LP1 reaches the simulation end after 750 Transactions have been terminated in it’s TERMINATE block. Because the TRANSFER block transfers 25% of the Transactions to LP2 this means that about 1000 Transactions would need to be generated in LP1 of which around 250 should end up in LP2. The output log of the simulate process confirms in the lines 27 to 31 that this is the case. In fact the number of Transactions that reached LP2 and the overall count is only short of two Transactions. This proves that the statistical behaviour of the TRANSFER block is correct. Validation of the Parallel Simulator 6.4 Validation 4 Evaluating the memory management of the parallel simulator is the subject of this validation. It will show that the simulator performs the correct actions when the two different defined memory limits are reached. The simulation model used for this validation is shown below. PARTITION Partition1,2000 GENERATE 1,0 Label1 QUEUE Queue1 DEPART Queue1 TERMINATE 1 PARTITION Partition2 GENERATE 1,0,2000 TRANSFER Label1 Simulation model file model_validation4.gps The simulation model contains two partitions. The first partition has a GENERATE block that will create a Transaction for each time unit. All Transactions will be terminated in the TERMINATE block of the first partition. The only purpose of the QUEUE and DEPART block in between is to slightly slow down the processing of the Transactions by the LP. The second partition also generates Transactions for each time unit but with an offset of 2000 so that its Transactions will start from the simulation time 2000. The second partition will therefore always be ahead in simulation time compared to the first partition and all its Transactions are transferred to the QUEUE block within the first partition. From the design of this model it can be seen that the first partition will become a bottleneck because on top of its own Transactions it will also receive Transactions from the second partition. The number of Transactions in its Transaction chain, i.e. Transactions that still need to be moved will constantly grow. In order to reach the memory limits of the parallel simulator more quickly the script startnode.sh, which is used to start the ProActive nodes for the LPs, is changed so that the command line argument –Xmx12m is passed to the Java Virtual Machine. This instructs the JVM to make only 12MB of memory available to its Java programs. The LPs for this validation are therefore run with a memory limit of 12MB. To avoid any memory management side effects introduced by the LPCC, the LPCC is switched off in Validation of the Parallel Simulator the config file simulate.config. The Termination Counter for the simulation model was chosen as small as possible but just about large enough for the simulation to reach the desired effects on the hardware used. The logging configuration was changed to include the current time and the debug output was enabled only for the loggers shown below. log4j.logger.parallelJavaGpssSimulator.lp=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.commit=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.rollback=DEBUG log4j.logger.parallelJavaGpssSimulator.simulation=DEBUG Extract of the used log4j configuration file proactive-log4j The output log of LP1 shows at line 7 that the memory limit 1 of 5MB available memory left is already reached after around 2 minutes of simulating. As expected the LP requests a GVT calculation so that some of its past states can be confirmed and fossil collected in order to free up memory. The GVT is received from the Simulation Controller and possible states are committed and fossil collection at line 9 and 10. Because Java uses garbage collection the memory freed by the LP does not become available immediately. As a result the LP is still operating pass the memory limit 1 and requests a few further GVT calculations until between line 65 and 66 there is more than one minute of simulation without GVT calculation because the garbage collector has freed enough memory for the LP to be out of memory limit 1. This pattern repeats itself several time as the memory used by the LP keeps growing until at line 376 of the output log LP1 reaches the memory limit 2 of 1MB of available memory left. At this point LP1 turns into cancelback mode and cancels back 25 of the Transactions it received from LP2 in order to free up memory. This can be seen at line 377. After the cancelback of these Transactions and another GVT calculation the memory available to LP1 raises above the memory limit 2 and the LP changes back from cancelback mode into normal simulation mode as shown in line 381. The effects of the Transactions cancelled back on LP2 can be seen in line 294, 295 and the following lines of the output log of LP2. Because Transactions are cancelled back one by one as they might have been received from different LPs they do not all arrive at LP2 at once. The log shows that the 25 Transactions are cancelled back by LP2 in groups of 9, 9, 5 and 2 Transactions. It also Validation of the Parallel Simulator shows that LP2 has reached the memory limit 2 even earlier than LP1. This fact can be explained by looking at the output log of the simulate process. The GVT calculation shown at the lines 311 to 313 indicates that LP2 has a lot more saved simulation states then LP1. LP1 hast started simulating at the time 1, has created one Transaction for every time unit and has reached a simulation time of 1520. That makes it 1520 saved simulation states of which some will have been fossil collected already. LP2 has started simulating from the time 2000 and has reached a simulation time of 4481 which means 2481 saves simulation states of which non will have been fossil collected as the GVT has not yet reached 2000. Saved simulation states require more memory than outstanding Transactions. This explains why LP2 had reached the memory limit 2 and the cancelback mode before LP1. But because LP2 does not receive Transactions from other LPs it has no Transactions that it can cancelback. The validation showed that the memory management of the parallel simulator works as expected. The LPs perform the required actions when they reach the defined memory limits and avoided Out Of Memory Exceptions by not running completely out of memory. 6.5 Validation 5 The fifth validation evaluates the correct functioning of the Shock Resistant Time Warp algorithm and its main component, the LPCC. It will show that the LPCC within the LPs is able to steer the simulation towards an actuator value that results in less rolled back Transaction moves compared to normal Time Warp. The simulation model used for this validation contains two partitions. Both partitions have a GENERATE block and a TERMINATE block but in addition partition 1 also contains a TRANSFER block that with a very small probability of 0.001 sends some of its Transactions to partition 2. The whole model is constructed so that partition 2 is usually ahead in simulation time compared to partition 1, achieved through the different configuration of the GENERATE blocks, and that occasionally partition 2 receives a Transaction from the first partition. Because partition 2 is usually ahead in simulation time this will lead to rollbacks in this partition. The simulation stops when 20000 Transactions have been terminated in partition 2. This model attempts to emulate the Validation of the Parallel Simulator common scenario where a distributed simulation uses nodes with different performance parameters or partitions that create different loads so that during the simulation the LPs drift apart and some of them are further ahead in simulation time than others leading to rollbacks and re-execution. The details of the model used can be seen below. PARTITION Partition1,20000 GENERATE 1,0 TRANSFER 0.001,Label1 TERMINATE 0 PARTITION Partition2,20000 GENERATE 4,0,5000 Label1 TERMINATE 1 Simulation model file model_validation5.gps In order to reduce the influence of the general memory management on this validation the amount of memory available to the LPs was increase from the default value of 64MB to 128MB by adding the JVM command line argument -Xmx128m in the startNode.sh script used to start the ProActive nodes for the LPs. The logging configuration was extended to get additional debug output relevant to the processing of the LPCC and some additional LP statistics at the end of the simulation. The loggers for which debug logging was enabled are shown below. log4j.logger.parallelJavaGpssSimulator.lp=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.commit=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.rollback=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.stats=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.lpcc=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.lpcc.statespace=DEBUG log4j.logger.parallelJavaGpssSimulator.simulation=DEBUG Extract of the used log4j configuration file proactive-log4j Validation 5.1 The first validation run of this model was performed with the LPCC enabled and the LPCC processing time interval set to 5 seconds. The extract of the simulation configuration file below shows all the configuration settings relevant to the LPCC. Validation of the Parallel Simulator LpccEnabled=true LpccClusterNumber=500 LpccUpdateInterval=5 Extract of the configuration file simulate.config for validation 5.1 From the output log of LP2 it can be seen that LP2 constantly has to rollback to an earlier simulation time because of Transactions it receives from LP1. For instance in line 6 of this output log LP2 has to roll back from simulation time 13332 to the time 1133 and in line 7 from time 6288 to 1439. The LPCC is processing the indicator set around every 5 seconds. Such a processing step is shown for instance in the lines 18 to 25 and lines 37 to 44. During these first two LPCC processing steps no actuator is being set (9223372036854775807 is the Java value of Long.MAX_VALUE meaning no actuator is set) because no past indicator set that promises a better performance indicator could be found. But at the third LPCC indicator processing a better indicator set is found as shown from line 55 to 62 and the actuator was set to 4967 as a result. At this point the number of uncommitted Transaction moves does not reach this limit but slightly later during the simulation, when the actuator limit is 4388, the LP reaches a number of uncommitted Transaction moves that exceeds the current actuator limit forcing the LP into cancelback mode. This can be found in the output log from line 109. While in cancelback mode LP2 is not cancelling back any Transactions received from LP1 as these are earlier than any Transaction generated within LP2 and therefore have been executed and terminated already but being in cancelback mode also means that no local Transactions are processed reducing the lead in simulation time of LP2 compared to LP1. LP2 stays in cancelback mode until Transaction moves are committed during the next GVT calculation reducing the number of uncommitted Transaction moves below the actuator limit. The following table shows the Actuator values set by the LPCC during the simulation and whether the Actuator limit was exceeded resulting in the cancelback mode. Validation of the Parallel Simulator LPCC processing step Time Actuator limit Limit exceeded 1 19:37:10 no limit No 2 19:37:15 no limit No 3 19:37:20 4967 No 4 19:37:25 4267 No 5 19:37:30 4388 Yes 6 19:37:35 4396 No 7 19:37:40 3135 Yes 8 19:37:45 3146 Yes 9 19:37:50 3762 No 10 19:37:55 2817 No Table 10: Validation 5.1 LPCC Actuator values From this table it is possible to see that the LPCC is limiting the number of uncommitted Transactions and therefore the progress of LP2 in order to reduce the number of rolled back Transaction moves and increase the number of committed Transaction moves per second, which is the performance indicator. The graph below shows the same Actuator values in graphical form. Actuator limit 1 2 3 4 5 6 7 8 9 10 LPCC processing steps Actuator limit Figure 23: Validation 5.1 Actuator value graph Validation of the Parallel Simulator Validation 5.2 Exactly the same simulation model and logging was used for the second simulation run but this time the LPCC was switched off so that the normal Time Warp algorithm was used to simulate the model. The only configuration setting changed for this simulation is the following: LpccEnabled=false Extract of the configuration file simulate.config for validation 5.2 Regards rollbacks the output log for LP2 looks similar compared to Validation 5.1 but does not contain any logging from the LPCC as this was switched off. Therefore the LP does not reach any Actuator limit and does not switch into cancelback mode. Comparison of validation 5.1 and 5.2 Comparing the output logs of validation 5.1 and 5.2 it becomes visible that performing the given simulation model using the Shock Resistant Time Warp algorithm reduces the number of Transaction moves rolled back. This can be seen especially when comparing the statistic information output by LP2 at the end of both simulation runs as found in the output log files for LP2 or in the table below. LP statistic item Validation 5.1 Validation 5.2 Total committed Transaction moves 19639 19953 Total Transaction moves rolled back 70331 77726 Total simulated Transaction moves 90330 97725 Table 11: LP2 processing statistics of validation 5 Table 11 shows that the simulation run of validation 5.1 using the Shock Resistant Time Warp algorithm required around 7400 less rolled back Transaction moves, which is about 10% less compared to the simulation run of validation 5.2 using the Time Warp algorithm. As a result the total number of Transaction moves performed by the simulation was reduced as well. The simulation run using the Shock Resistant Time Warp algorithm also performed slightly better than the simulation run using the Time Warp algorithm. This can be seen from the simulation performance information output Validation of the Parallel Simulator as part of the post simulation report found in the output logs of the simulate process for both simulation runs. For validation 5.1 the simulation performance was 1640 time units per second real time and for validation 5.2 1607 time units per second real time. The performance difference is quite small which suggests that for the example model used the processing saved on rolled back Transaction moves just about out weights the extra processing required for the LPCC, additional GVT calculations and the extra logging for the LPCC (the LP2 output log size of validation 5.2 is only around 3% of the one of validation 5.1). But for more complex simulation models where rollbacks in one LP lead to cascaded rollbacks in other LPs a much larger saving on the number of rolled back Transaction moves can be expected. It also needs to be considered that the hardware setup used (i.e. all nodes being run on a single CPU machine) is not ideal for a performance evaluation as the main purpose of this validation is to evaluate the functionality of the parallel simulator. 6.6 Validation 6 During the testing of the parallel simulator it became apparent that in same cases the normal Time Warp algorithm can outperform the Shock Resistant Time Warp algorithm. This last validation is showing this in an example. The simulation model used is very similar to the one used for validation 5. It contains two partitions with the first partition transferring some of its Transactions to the second partition but this time the GENERATE blocks are configured so that the first partition is ahead in simulation time compared to the second. The simulation is finished when 3000 Transactions have been terminated in one of the partitions. The complete simulation model can be seen here: PARTITION Partition1,3000 GENERATE 1,0,2000 TRANSFER 0.3,Label1 TERMINATE 1 PARTITION Partition2,3000 GENERATE 1,0 Label1 TERMINATE 1 Simulation model file model_validation6.gps Validation of the Parallel Simulator As a result of the changed GENERATE block configuration and the first partition being ahead of the second partition in simulation time, all Transactions received by partition 2 from partition 1 are in the future for partition 2 and no rollbacks will be caused. But it will lead to an increase of the number of outstanding Transactions within partition 2 pushing up the number of uncommitted Transaction moves during the simulation. The logging configuration for this validation is also similar to the one used for validation 5 except that the LP statistic is not needed this time and is therefore switched off. The extract below shows the loggers for which debug output was enabled. log4j.logger.parallelJavaGpssSimulator.lp=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.commit=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.rollback=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.lpcc=DEBUG log4j.logger.parallelJavaGpssSimulator.lp.lpcc.statespace=DEBUG log4j.logger.parallelJavaGpssSimulator.simulation=DEBUG Extract of the used log4j configuration file proactive-log4j Like for validation 5 the script startNode.sh used to run the LP nodes is changed to extend the memory limit of the JVM to 128MB. Validation 6.1 For the first validation run the LPCC was enabled using the same configuration like for validation 5.1 resulting in the model being simulated using the Shock Resistant Time Warp algorithm. The significant effect of the simulation run is that the LPCC in LP2 starts setting actuator values in order to steer the local simulation processing towards a state that promises better performance but because the number of uncommitted Transaction moves within the second partition increases as a result of the Transactions received from partition 1 the actuator limits set by the LPCC are reached and the LP is switched into cancelback mode leading to its simulation progress being slowed down. In addition LP1 is also slowed down by the Transactions cancelled back from LP 2 as indicated by the output log of LP1. The actuator being set for LP2 can for instance be seen in line 25 to 28 of the output log of LP2. Subsequently the actuator limit is reached Validation of the Parallel Simulator as shown in line 32, 36 and 42 of the output log and the LP turned into cancelback mode in the lines below. LP2 keeps switching into cancel back mode and keeps cancelling back Transactions to LP1 for large parts of the simulation resulting in a significant slowdown of the overall simulation progress. Validation 6.2 The output logs for validation 6.2 are very short compared to the former simulation run because the simulation is performed using the normal Time Warp algorithm with the LPCC being switched off. Therefore no actuator values are set and none of the LPs is switching into cancelback mode. There are also no rollbacks so that the simulation progresses with the optimum performance for the model and setup used. Comparison of validation 6.1 and 6.2 The simulation model for this validation is processed with the optimum simulation performance by the normal Time Warp algorithm. As a result no performance increase can be expected from the Shock Resistant Time Warp algorithm. But the Shock Resistant Time Warp Algorithm performs significantly worse than the normal Time Warp algorithm. This is caused by the Shock Resistant Time Warp algorithm slowing down the LP that is already behind in simulation time, i.e. the slowest LP. The validation shows that the approach of the Shock Resistant Time Warp algorithm to optimise the parameters of each LP by only considering local status information within these LPs does not always work. 6.7 Validation Analysis The first few validations evaluate basic functional aspects of the parallel simulator. For instance validation 1 focuses on the GPSS parser component of the simulator and validation 2 on the basic GPSS simulation engine functionality. The transfer of Transactions between LPs is the main subject matter of validation 3 and the memory management is evaluated by validation 4. Validation 5 examines the correct functioning of the LPCC as the main component of the Shock Resistant Time Warp algorithm. Using specific simulation models and configurations these validations demonstrate with Validation of the Parallel Simulator a certain degree of confidence that the parallel simulator is functionally correct and working as expected. In addition validation 5 and 6 give some basic idea about the performance of the parallel transaction-oriented simulation based on the Shock Resistant Time Warp algorithm and about the performance of the parallel simulator even so the validations performed here cannot be seen as proper performance validations. Validation 5 shows that the Shock Resistant Time Warp algorithm can successfully reduce the number of rolled back Transaction moves resulting in more useful processing during the simulation and possibly better performance. But validation 6 revealed that the Shock Resistant Time Warp algorithm can also perform significantly worse than normal Time Warp. This is usually the case when the LPCC of the LP that is already furthest behind in simulation time decides to set an actuator value that limits the simulation progress resulting in the LP and the overall simulation progress being slowed down further. The problem of the Shock Resistant Time Warp algorithm found is a direct result of the fact that if implemented as described in [8] the control decisions of each LPCC are only based on local sensor values within each LP and not on an overall picture of the simulation progress. Such problems could possibly be avoided by combining the Shock Resistant Time Warp algorithm with ideas from the adaptive throttle scheme suggested in [33] which is also briefly described in section 4.2.2. The GFT needed by this algorithm in order to describe the Global Progress Window could easily be determined and passed back to the LPs together with the GVT after the GVT calculation without much additional processing being required or communication overhead being created. Using the information of such a global progress window one option to improve the Shock Resistant Time Warp algorithm would be to add another sensor to the LPCC that describes the position of the current LP within the Global Progress Window. But another option that promises greater influence of the global progress information on the local LPCC control decisions would be to change the function that determines the new actuator value in a way that makes direct use of the global progress information. Such a function could for instance ignore the actuator value resulting from the best past state found if the position of the LP within the Global Progress Window is very close towards the GVT. It could also increase or decrease the actuator value returned by the original function depending on whether the LP is located within the slow or the fast zone of the Validation of the Parallel Simulator Global Progress Window (see Figure 8 in 4.2.2). And a finer influence of the position within the Global Progress Window could be reached by dividing this window into more than 2 zones, each resulting in a slightly different effect on the actuator value and the future progress of the LP. Future work on this parallel simulator could investigate and compare these options with a prospect of creating a synchronisation algorithm that combines the advantages of both these algorithms. Conclusions 7 Conclusions Even so the performance of modern computer systems has steadily increased during that last decades the ever growing demand for the simulation of more and more complex systems can still lead to long runtimes. The runtime of a simulation can often be reduced by performing its parts distributed across several processors or nodes of a parallel computer system. Purpose-build parallel computer systems are usually very expensive. This is where Computing Grids provide a cost-saving alternative by allowing several organisations to share their computing resources. A certain type of Computing Grids called Ad Hoc Grid offers a dynamic and transient resource-sharing infrastructure, suitable for short-term collaborations and with a very small administrative overhead that makes it even for small organisations or individual users possible to form Computing Grids. In the first part of this paper the requirements of Ad Hoc Grids are outlined and the Grid framework ProActive [15] is identified as a Grid environment that fulfils these requirements. The second part analyses the possibilities of performing parallel transaction-oriented simulation with a special focus on the space-parallel approach and synchronisation algorithms for discrete event simulation. From the algorithms considered the Shock Resistant Time Warp algorithm [8] was chosen as the most suitable for transaction-oriented simulation as well as the target environment of Ad Hoc Grids. This algorithm was subsequently applied to transaction-oriented simulation, considering further aspects and properties of this simulation type. These additional considerations included the GVT calculation, detection of the simulation end, cancellation techniques suitable for transaction-oriented simulation and the influence of the model partitioning. Following the theoretical decisions a Grid-based parallel transaction-oriented simulator was then implemented in order to demonstrate the decisions made. Finally the functionality of the simulator was evaluated using different simulation models in several validation runs in order to show the correctness of the implementation. The main contribution of this work is to provide a Grid-based parallel transaction- oriented simulator that can be used for further research, for educational purpose or even for real live simulations. The chosen Grid framework ProActive ensures its suitability Conclusions for Ad Hoc Grids. The parallel simulator can operate according to the normal Time Warp or the Shock Resistant Time Warp algorithm allowing large-scale performance comparisons of these two synchronisation algorithms using different simulation models and on different hardware environments. It was shown that the Shock Resistant Time Warp algorithm can successfully reduce the number of rolled back Transaction moves, which for simulations with many or long cascaded rollbacks will lead to a better simulation performance. But this work also revealed a problem of the Shock Resistant Time Warp algorithm, implemented as described in [8]. Because according to this algorithm all LPs try to optimise their properties based only on local information it is possible for the Shock Resistant Time Warp algorithm to perform significantly worse than the normal Time Warp algorithm. Future work on this simulator could improve the Shock Resistant Time Warp algorithm by making the LPs aware of their position within the GPW as suggested in [33]. Combining these two synchronisation algorithms would create an algorithm that has the advantages of both without any major additional communication and processing overhead. Future work on improving this parallel transaction-oriented simulator could also look at employing different GVT algorithms and state saving schemes. Possible options were suggested in 4.3 and 5.3.3. This work also discussed the aspect of accessing objects in a different LP including a single shared Termination Counter. As mentioned in the report these options were not implemented in the parallel simulator and could be considered for future enhancements. Finally the simulator does not support the full GPSS/H language but only a large sub-set of the most important entities, which leaves further room for improvements. References References [1] Amin K. An Integrated Architecture for Ad Hoc Grids [online]. 2004 [cited 8 Jan 2007]. Available from: URL: http://students.csci.unt.edu/~amin/publications/ phd-thesis-proposal/proposal.pdf [2] Amin K, von Laszewski G and Mikler A. Toward an Architecture for Ad Hoc Grids. In: 12th International Conference on Advanced Computing and Communications (ADCOM 2004); 15-18 Dec 2004; Ahmedabad Gujarat, India [online]. 2004 [cited 8 Jan 2007]. 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Washington, DC: IEEE Computer Society; 1997. [34] Tropper C. Parallel discrete-event simulation applications. Journal of Parallel and Distributed Computing, Mar 2002; 62(3):327-335. [35] Wikipedia. Computer simulation [online]. [cited 3 Dec 2005]. Available from: URL: http://en.wikipedia.org/wiki/Computer_simulation [36] Wolverine Software. GPSS/H - Serving the simulation community since 1977 [online]. [cited 3 Dec 2005]. Available from: URL: http://www.wolverinesoftware.com/GPSSHOverview.htm Appendix A: Detailed GPSS Syntax Appendix A: Detailed GPSS Syntax This is a detailed description of the GPSS syntax supported by the parallel GPSS simulator. The correct syntax of the simulation models loaded into the simulator is validated by the GPSS parser within the simulator (see section 5.2.1). GPSS simulation model files are line-based text files. Each line contains a block definition, a Storage definition or a partition definition. Comment lines starting with the sign * or empty lines are ignored by the parser. A block definition line starts with an optional label followed by the reserved word of a block type and an optional comma separated list of parameters. The label is used to reference the block definition for jumps or a branched execution path from other blocks. The label, the block type and the parameter list need to be separated by at least one space. Note that a comma separated parameter list cannot contain any spaces. Any other characters following the comma separated parameter list are considered to be comments and are ignored. Labels as well as other entity names (i.e. for Facilities, Queues and Storages) can contain any alphanumerical characters but no spaces and they must be different to the defined reserved words. Labels for two different block definitions within the same model cannot be same. But labels are case sensitive so that two labels that only differ in the case of their characters are considered different. Apposed to that reserved words are not case sensitive but is an accepted convention in GPSS to use upper case letters for GPSS reserved words. Storage definitions and partition definitions cannot start with a label. They therefore start with the reserved word STORAGE or PARTITION. Below the syntax of the different GPSS definitions is explained in more detail. For a detailed description of the actual GPSS functionality see [26]. Partition definition: Reserved word: PARTITION Syntax: PARTITION [<partition name>[,<termination counter>]] Description: The partition definition declares the beginning of a new partition. If the optional partition name parameter is not specified then it will default to ‘Partition x’ with x being the number of the partition within the model. The partition name cannot contain any Appendix A: Detailed GPSS Syntax spaces. If the optional termination counter parameter is not specified then the default termination counter value from the simulator configuration will be used. If specified then the termination counter parameter has to be a positive integer value. Storage definition: Reserved word: STORAGE Syntax: STORAGE <Storage name>[,<Storage capacity>] Description: The Storage definition declares a Storage entity. The Storage name parameter is required. If the optional Storage capacity parameter is not specified then it will default to the maximum value of the Java int type. If specified then the Storage capacity parameter needs to be a positive integer value. The Storage definition has to appear in the simulation model before any block that uses the specific Storage. Block definitions: Reserved word: ADVANCE Syntax: [<label>] ADVANCE [<average holding time>[,<half range>]] Description: The ADVANCE block delays Transactions by a fixed or random amount of time. The optional average holding time parameter describes the average time by which the Transaction is delayed defaulting to zero. Together with the average holding time the second parameter which is the half range parameter describe the uniformly distributed random value range from which the actually delay time is drawn. If the half range parameter is not specified then the delay time will always have the deterministic value of the average holding time parameter. Both parameters can either be a positive integer value or zero. In addition the half range parameter cannot be greater than the average holding time parameter. Appendix A: Detailed GPSS Syntax Reserved word: DEPART Syntax: [<label>] DEPART <Queue name> Description: The DEPART block causes the Transaction to leave the specified Queue entity. The required Queue name parameter describes the Queue entity that the Transaction will leave. Reserved word: ENTER Syntax: [<label>] ENTER <Storage name>[,<usage count>] Description: Through the ENTER block a Transaction will capture a certain number of units from the specified Storage entity. The Storage name parameter defines the Storage entity used and the second optional usage count parameter specifies how many Storage units will be captured by the Transaction. If not specified this parameter will default to 1. Otherwise this parameter has to have a positive integer value that is less or equal to the size of the Storage. If the specified number of units are not available for that Storage then the Transaction will be blocked until they become available. Reserved word: GENERATE Syntax: [<label>] GENERATE [<average interarrival time>[,<half range> [,<time offset>[,<limit count>[,<priority>]]]]] Description: The GENERATE block generates new Transactions that enter the simulation. The first two parameter average interarrival time and half range describe the uniformly distributed random value range from which the interarrival time is drawn. The interarrival time is the time between the last Transaction entering the simulation through this block and the next one. Both these parameters default to zero if not specified. The next parameter is the time offset parameter that describes the arrival time of the first Transaction. If it is not specified then the arrival time of the first Transaction will be determined via a uniformly distributed random sample using the first two parameters. The limit count parameter specifies the total count of Transactions that will enter the simulation through this GENERATE block. If the count is reached then no further Transactions are generated. If this parameter is not specified then no limit applies. The Appendix A: Detailed GPSS Syntax priority parameter specifies the priority value assigned to the generated Transactions and will default to 0. If specified all these parameters are required to have a positive integer value or zero. In addition the half range parameter cannot be greater than the average interarrival time parameter. Reserved word: LEAVE Syntax: [<label>] LEAVE <Storage name>[,<usage count>] Description: The LEAVE block will release the specified number of Storage units held by the Transaction. The Storage name parameter defines the Storage entity from which units will be released and the second optional usage count parameter specifies how many Storage units will be released by the Transaction. If not specified this parameter will default to 1. Otherwise this parameter has to have a positive integer value that is less or equal to the size of the Storage. If the specified number of units is greater than the number of units currently held by the Transaction then a runtime error will occur. Reserved word: QUEUE Syntax: [<label>] QUEUE <Queue name> Description: The QUEUE block causes the Transaction to enter the specified Queue entity. The required Queue name parameter describes the Queue entity that the Transaction will enter. Reserved word: RELEASE Syntax: [<label>] RELEASE <Facility name> Description: The RELEASE block will release the specified Facility entity held by the Transaction. The Facility name parameter defines the Facility entity that will be released. If the Facility entity is not currently held by the Transaction then a runtime error will occur. Appendix A: Detailed GPSS Syntax Reserved word: SEIZE Syntax: [<label>] SEIZE <Facility name> Description: Through the SEIZE block a Transaction will capture the specified Facility entity. The Facility name parameter defines the Facility entity that will be captured. If the Facility entity is already held by another Transaction then the Transaction will be blocked until it becomes available. Reserved word: TERMINATE Syntax: [<label>] TERMINATE [<Termination Counter decrement>] Description: TERMINATE blocks are used to destroy Transactions. When a Transaction enters a terminate block then the Transaction is removed from the model and not chained back into the Transaction chain. Each time a Transaction is destroyed by a TERMINATE block the local Termination counter is decremented by the decrement specified for that TERMINATE block. The Termination Counter decrement parameter is optional and will default to zero if it is not specified. TERMINATE blocks with a zero decrement parameter will not change the Termination Counter when they destroy a Transaction. Reserved word: TRANSFER Syntax: [<label>] TRANSFER [<transfer probability>,]<destination label> Description: A TRANSFER block changes the execution path of a Transaction based on the specified probability. Normally a Transaction is moved from one block to the next but when it executes a TRANSFER block the Transaction can be transferred to a different block than the next following. This destination can even be located in a different partition of the model. For the decision of whether a Transaction is transferred or not a random value is drawn and compared to the specified probability. If the random value is less than or equal to the probability then the Transaction is transferred. The transfer probability parameter needs to be a floating point number between 0 and 1 (inclusive). It is optional and will default to 1, which will transfer all Transactions, if not specified. The destination label parameter has to be a valid block label within the model. Appendix B: Simulator configuration settings Appendix B: Simulator configuration settings This appendix describes the configuration settings that can be used for the parallel simulator. Most of these settings can be applied as command line arguments or as settings within the simulate.config file. A general description of the simulator configuration can be found in 5.4.3. Setting: ConfigFile Default value: simulate.config Description: This configuration setting can only be used as a command line argument and it has to follow straight after the simulation model file name. It specifies the name of the configuration file used by the parallel simulator. Setting: DefaultTC Default value: none Description: This is a configuration setting that defines the default Termination Counter used for partitions that do not have a Termination Counter defined in the simulation model file. When specified then it needs to have a non-negative numeric value. Setting: DeploymentDescriptor Default value: ./deploymentDescriptor.xml Description: The ProActive deployment descriptor file used by the parallel simulator is specified using this configuration setting. Setting: LogConfigDetails Default value: false Description: If this Boolean configuration setting is switched on then the parallel simulator will always output the current configuration setting used including default ones at the start of a simulation. This can be useful for debugging purposes. Appendix B: Simulator configuration settings Setting: LpccClusterNumber Default value: 1000 Description: This numeric configuration setting sets the maximum number of clusters stored in the State Cluster Space used by the LPCC. If a new indicator set is added to the State Cluster Space and the maximum number of clusters has been reached already then two clusters or a cluster and the new indicator set are merged. The larger the value of this setting the more distinct state indicator sets can be stored and used by the Shock Resistant Time Warp algorithm but the more memory is also need to store such information. Setting: LpccEnabled Default value: true Description: If this Boolean configuration setting is set to true which is also the default of this setting then the LPCC is enabled and the simulation is performed according to the Shock Resistant Time Warp algorithm. Otherwise the LPCC is switched off and the normal Time Warp algorithm is used for the parallel simulation. Setting: LpccUpdateInterval Default value: 10 Description: This is a configuration setting that defines the LPCC processing time interval. Its value has to be a positive number greater than zero and describes the number of seconds between LPCC processing steps. It also specifies how often the LPCC tries to find and set a new actuator value. For long simulation runs on systems with large memory pools larger LPCC processing intervals can be beneficial because less GVT calculation are needed. On the other hand if the systems used have frequently changing additional loads and the simulation model is known to have a frequently changing behaviour pattern then small values might give better results. Appendix B: Simulator configuration settings Setting: ParseModelOnly Default value: false Description: If this configuration setting is enabled then the parallel simulator will parse the simulation model file and output the in memory representation of the model but no simulation will be performed. This setting can therefore be used to evaluate whether the simulation model was parsed correctly and to check which defaults have been set for optional GPSS parameter. Appendix C: Simulator log4j loggers Appendix C: Simulator log4j loggers The following loggers of the parallel simulator can be configured in its log4j configuration file. In this section each logger is briefly described and its supported log levels are mentioned. Logger: parallelJavaGpssSimulator.gpss Log levels used: debug Description: This logger is used to output debug information of the GPSS block and Transaction processing during the simulation. It creates a detailed log of when a Transaction is moved, which blocks it executes and when it is chained in or out. It is also the root logger for any logging related to the GPSS simulation processing. Logger: parallelJavaGpssSimulator.gpss.facility Log levels used: debug Description: Whenever a Transaction releases a Facility entity this logger outputs detailed information about the Transaction and when it captured and released the Facility. Logger: parallelJavaGpssSimulator.gpss.queue Log levels used: debug Description: Whenever a Transaction leaves a Queue entity this logger outputs detailed information about the Transaction and when it entered and left the Queue. Logger: parallelJavaGpssSimulator.gpss.storage Log levels used: debug Description: Whenever a Transaction releases a Storage entity this logger outputs detailed information about the Transaction and when it captured and released the Storage. Appendix C: Simulator log4j loggers Logger: parallelJavaGpssSimulator.lp Log levels used: debug, info, error, fatal Description: This logger is the root logger for any output of the LPs. The default log level is info which outputs some basic information about what partition was assigned to the LP and when the simulation is completed. Errors within the LP are also output using this logger and if the debug log level is enabled then it outputs detailed information about the communication and the processing of the LP related to the synchronisation algorithm. Logger: parallelJavaGpssSimulator.lp.commit Log levels used: debug Description: This logger outputs information about when simulation states are committed and for which simulation time. Logger: parallelJavaGpssSimulator.lp.rollback Log levels used: debug Description: A logger that outputs information about the rollbacks performed. Logger: parallelJavaGpssSimulator.lp.memory Log levels used: debug Description: This logger outputs detailed information about the current memory usage of the LP and the amount of available memory within the JVM. It is called with each scheduling cycle and can therefore create very large logs if enabled. Logger: parallelJavaGpssSimulator.lp.stats Log levels used: debug Description: This logger outputs the values of the sensor counters that are also user by the LPCC as a statistic of the overall LP processing at the end of the simulation. Appendix C: Simulator log4j loggers Logger: parallelJavaGpssSimulator.lp.lpcc Log levels used: debug Description: A logger that outputs detailed information about the processing of the LPCC, including for instance any actuator values set or when an actuator limit has been exceeded. Logger: parallelJavaGpssSimulator.lp.lpcc.statespace Log levels used: debug Description: This logger outputs information about the processing of the State Cluster Space. This includes details of new indicator sets added or possible past indicator sets found that promises better performance. Logger: parallelJavaGpssSimulator.simulation Log levels used: debug, info, error, fatal Description: This is the root logger for all general output about the simulation, the Simulation Controller and the simulate process. The default log level is info, which outputs the standard information about the simulation. The logger also outputs errors thrown during the simulation and in debug mode gives detailed information about the processing of the Simulation Controller. Logger: parallelJavaGpssSimulator.simulation.gvt Log levels used: debug, info Description: A logger that outputs detailed information about GVT calculations if in debug level. If the logger is set to the info log level then only basic information about the GVT reached is logged. Logger: parallelJavaGpssSimulator.simulation.report Log levels used: info Description: This logger is the root logger for the post simulation report. It can be used to switch off the output of the post simulation report by setting the log level to off. Appendix C: Simulator log4j loggers Logger: parallelJavaGpssSimulator.simulation.report.block Log levels used: info Description: This is the logger that is used for the block section of the post simulation report. It allows this section to be switched off if required. Logger: parallelJavaGpssSimulator.simulation.report.summary Log levels used: info Description: This is the logger that is used for the summary section of the post simulation report. It allows this section to be switched off if required. Logger: parallelJavaGpssSimulator.simulation.report.chain Log levels used: info Description: This is the logger that is used for the Transaction chain section of the post simulation report. It allows this section to be switched off if required. Appendix D: Structure of the attached CD Appendix D: Structure of the attached CD The folder structure of the attached CD is briefly explained in this section. The root folder of the CD also contains this report as a Microsoft Word and PDF document. /ParallelJavaGpssSimulator This is the main folder of the parallel simulator. It contains all the files required to run the simulator as described in 5.4.2. It also contains some of the folders mentioned below. /ParallelJavaGpssSimulator/bin The folder structure within this folder contains all the binary Java class files of the parallel simulator. The same Java class files are also included in the main JAR file of the simulator. /ParallelJavaGpssSimulator/doc This folder contains the full JavaDoc documentation of the parallel simulator. The JavaDoc documentation can be viewed by opening the index.html file within this folder in a Web browser. It describes the source code of the parallel simulator and is generated from comments within the source code using the JavaDoc tool. /ParallelJavaGpssSimulator/src The src folder contains the actual source code of the parallel simulator, i.e. all the java files. /ParallelJavaGpssSimulator/validation This folder contains a sub-folder for each validation. All files required to repeat the validation runs can be found in these sub-folders, including simulation models, configuration and all the output log files of the validation runs described in section 6. The validation runs can be performed directly from these folders. /ProActive This additional folder contains the compressed archive of the ProActive version used. Appendix E: Documentation of selected classes Appendix E: Documentation of selected classes This section contains the JavaDoc documentation of the following selected classes. • parallelJavaGpssSimulator.SimulationController • parallelJavaGpssSimulator.lp.LogicalProcess • parallelJavaGpssSimulator.lp.ParallelSimulationEngine • parallelJavaGpssSimulator.lp.lpcc.LPControlComponent The full JavaDoc documentation of all classes can be found on the attached CD, see Appendix D for further details. Appendix E: Documentation of selected classes – SimulationController �������������� �� ����� ����� ��� �� ������ ���� ����������� � �������������� �� ������ �� ���������������� All Implemented Interfaces: java.io.Serializable, org.objectweb.proactive.Active, org.objectweb.proactive.RunActive �� ��� ����� �� ���������������� ��� ����������� � ��� ����� � ��������� ��������� � ��������� ������� � ��� ���� ��� �� ��� �� � �� ��� ���� � ��� ���� � �� ��� � ��� ����� ��������� � ��� ���� ���� ��� � �� �� ��� ��� �� ��� � � ���� �� �� ��� � ��������� ������ �� ��� � ����� �� ��� !��� ��� ������������ � �� ��� �� ��������� ������ ������ � Author: Gerald Krafft See Also: Serialized Form Field Summary ������ ���������������� VIRTUAL_NODE_NAME Name of the virtual node that needs to be defined in the deployment descriptor file. Appendix E: Documentation of selected classes – SimulationController Constructor Summary SimulationController�� Main constructor Method Summary ������ SimulationController createActiveInstance������ � ��������� Static method that creates an Active Object SimulationController instance on the specified node SimulationState getSimulationState�� Returns the state of the simulation ���� reportException����������� �� ����� � ��� �������!��� ��"�� Called by logical process instances to report exceptions thrown by the simulation. ���� requestGvtCalculation�� Called by LPs to request a GVT calculation by the SimulationController. ���� runActivity������ � ��������� �#��$ ��$� Implements the main activity loop of the Active Object ���� simulate�Model ��� �� Configuration ���%���������� Starts parallel simulation of the specified model and using the specified configuration ���� terminateLPs�� This method terminates all LPs. Appendix E: Documentation of selected classes – SimulationController Methods inherited from class java.lang.Object &����� %������ �'����� (��('�� � ����%$� ����%$���� ��������� ����� ����� ���� Field Detail ����������������� �� ��� ������ %���� ���������������� ����������������� Name of the virtual node that needs to be defined in the deployment descriptor file. Its value is "ParallelJavaGpssSimulator". See Also: Constant Field Values Constructor Detail �� ���������������� �� ��� �� ������������������ Main constructor Method Detail � �� �����! �� ��� ���� � �� �����!������ � ��������� �#��$ ��$� Implements the main activity loop of the Active Object Specified by: ����������$ in interface ����� � ��������� ��������� Appendix E: Documentation of selected classes – SimulationController Parameters: ��$ - body of the Active Object See Also: �������� �����������$������ � ��������� �#��$� ������ ������ ��� � �� ��� ������ ����������'������� ������ ������ ��� ������� � ��������� �(���� ����� � ��������� ������ ����� �� ������ ����� � ��������� ����� Static method that creates an Active Object SimulationController instance on the specified node Parameters: - node at which the instance will be created or within current JVM if null Returns: active instance of SimulationController Throws: ����� � ��������� ������ ����� �� ����� ����� � ��������� ����� �� ���� �� ��� ���� �� �����)�� � ��� �� '��%��������� ���%���������� �(���� ����� � ��������� �!������� ������ ����� � ��������� ������ '���������������� �� ������ ����� � ��������� �����'�������� �%�� � ������ Appendix E: Documentation of selected classes – SimulationController ����������'�������*���� �� ����� Starts parallel simulation of the specified model and using the specified configuration Parameters: � - GPSS model that will be simulated ���%��������� - configuration settings Throws: ����� � ��������� �!������� ����� - can be thrown by ProActive ����� � ��������� ����� - can be thrown by ProActive CriticalSimulatorException - Critical error that makes simulation impossible ����� � ��������� �����'�������� �%�� � ����� - can be thrown by ProActive ����������'�������*���� �� ����� - can be thrown by ProActive ����������" �� ��� ���� ����������" �� �(���� ��������"� �� ����� This method terminates all LPs. It is called by the main application Simulate class. It returns an exception if terminating the LPs fails which automatically forces calls to this method to be synchronous. Throws: ��������"� �� ����� Appendix E: Documentation of selected classes – SimulationController �������# ������ �� ��� ���� �������# ����������������� �� ����� � ��� �������!��� ��"�� Called by logical process instances to report exceptions thrown by the simulation. This method is used for exceptions that occur within runActivity() of these instances and not within remote method calls. Exceptions thrown within remote method calls are automatically passed back by ProActive. Parameters: - Exception that was thrown in LP �������!��� ��"�� � - index of the LP that reports the exception �� ������ �� ��� �������������� �� ������ ������ Returns the state of the simulation Returns: state of the simulation ��% � ������� ������ �� ��� ���� ��% � ������� �������� Called by LPs to request a GVT calculation by the SimulationController. Appendix E: Documentation of selected classes – LogicalProcess �������������� �� �����&�� ����� �������� ����� ����������� � �������������� �� �����������$� ��"�� � All Implemented Interfaces: java.io.Serializable, org.objectweb.proactive.Active, org.objectweb.proactive.RunActive �� ��� ����� ��$� ��"�� � ��� ����������� � ��� ����� � ��������� ��������� � ��������� ������� � ���� � �� ��� ��� �� ����� ������� "��# ������ � �� ��� � ���� ��� �� ���� �� ��� �������� $ �� ����� ������� ���� �����%�� ������ �% ��� ���� ��� ���� � ����� ��������� �� � �� ��������� ������ �� ����� ���� ����� �� �� !��� � �� ����� � ����� �� �� �� ���� � � !��� ��� ���� ��� �� ��� �� � �� �� %�� ��� �� ��� �% ��� ���� ��� � Author: Gerald Krafft See Also: Serialized Form Constructor Summary LogicalProcess�� Main constructor (also used for serialization purpose) Appendix E: Documentation of selected classes – LogicalProcess Method Summary ���� cancelBackTransaction�Transaction ����� Called by other Logical Processes to force a cancel back of the specified Transaction sent by this LP. � ���� commitState����� ���� Performs fossil collection for changes earlier than the ������ LogicalProcess createActiveInstance������ � ��������� This static method is called by the Simulation Controller in order to create a ProActive Active Object instance of the LogicalProcess class. ����� � ��������� ���������� ��#��� ��+���� endOfSimulationByTransaction�Transaction ����� Requests the Logical Process to end the simulation at the specified Transaction. ���� forceGvtAt����� ��� Calling this method will force the Logical Process to request a GVT calculation as soon as it passes the specified simulation time. SimulationReportSet getSimulationReport� ��� �� ������ '(���� ����� Returns the simulation report. � ���� handleReceivedTransactions�� Goes through the list of received Transactions and anti- Transactions and either chains the new Transaction in or undoes the original Transaction for received anti-Transaction. Appendix E: Documentation of selected classes – LogicalProcess ���� initialize�Partition ���������� ����� � ��������� �������,���� �������!��� ��,����� SimulationController ����������'������� Configuration ���%���������� Initializes the Logical Process. � ���� needToCancelBackTransactions����� ������ Cancel back a certain number of Transactions. ���� receiveGvt����� ���� ��� �� ����!��� ������ Called by SimulationController to send the calculated GVT (global virtual time). ����� � ��������� ������ ����� ��#��� ��+���� receiveTransaction�Transaction ����� ��� �� ����� Public method that is used by other Logical Processes to send a Transaction or anti-Transaction to this Logical Process. LocalGvtParameter requestGvtParameter�� Returns the parameters of this Logical Process required for the GVT calculation. � ���� rollbackState����� ��� Rolls the state of the simulation engine back to the state for the given time or the next later state. ���� runActivity������ � ��������� �#��$ ��$� Implements the main activity loop of the Active Object � ���� saveCurrentState�� Saves the current state of the simulation engine into the local state list (unless an unconfirmed end of simulation has Appendix E: Documentation of selected classes – LogicalProcess been reached by this LP or the LP is in Cancelback mode, in both cases the local simulation time would have the value of Long.MAX_VALUE). � ���� sendLazyCancellationAntiTransactions�� This method performs the main lazy-cancellation for Transactions that have been sent and subsequently rolled back. ���� startSimulation�� Tells the LP to start simulating the local partition of the simulation model. Methods inherited from class java.lang.Object &����� %������ �'����� (��('�� � ����%$� ����%$���� ��������� ����� ����� ���� Constructor Detail ��$� ��"�� � �� ��� ��$� ��"�� � �� Main constructor (also used for serialization purpose) Method Detail ������ ������ ��� � �� ��� ������ -������!��� ������ ������ ��� ������� � ��������� �(���� ����� � ��������� ������ ����� �� ������ ����� � ��������� ����� Appendix E: Documentation of selected classes – LogicalProcess This static method is called by the Simulation Controller in order to create a ProActive Active Object instance of the LogicalProcess class. The Active Object instance is created at the specified node. Parameters: - node at which the Active Object LogicalProcess instance will be created Returns: the Active Object LogicalProcess instance (i.e. a stub of the LogicalProcess instance) Throws: ����� � ��������� ������ ����� �� ����� ����� � ��������� ����� ��������'� �� ��� ���� ��������&��!�������� ���������� ����� � ��������� �������,���� �������!��� ��,����� ����������'������� � ����������'������� '��%��������� ���%���������� Initializes the Logical Process. This method is called by the Simulation Controller. The initialization is done outside the constructor because it requires the group of all Logical Process active objects to be passed in. The LogicalProcess instance cannot be used before it is initialized. Parameters: ��������� - the simulation model partition that this LP will process �������!��� ��,���� - a group containing all LPs (i.e. stubs to all LPs) Appendix E: Documentation of selected classes – LogicalProcess � �� �����! �� ��� ���� � �� �����!������ � ��������� �#��$ ��$� Implements the main activity loop of the Active Object Specified by: ����������$ in interface ����� � ��������� ��������� Parameters: ��$ - body of the Active Object See Also: �������� �����������$������ � ��������� �#��$� ���� �� ������ �� ��� ���� ���� �� �������� Tells the LP to start simulating the local partition of the simulation model. The LP needs to be initialized by calling initialize() before the simulation can be started. This method is called by the Simulation Controller after it created and initialized all LogicalProcess instances. �� �������� � ���� �� ��� ����� � ��������� ����������� ��#��� ��+���� �� �������� � �����.���������� ����� ��� �� ����� Public method that is used by other Logical Processes to send a Transaction or anti-Transaction to this Logical Process. Parameters: ���� - Transaction received from other LP ���� - true if an anti-Transaction has been received Appendix E: Documentation of selected classes – LogicalProcess Returns: Returns a Future object which allows the send to verify that is has been received �� ��(� )���� � ���� �� ��� ���� �� ��'� (���� � �����.���������� ����� Called by other Logical Processes to force a cancel back of the specified Transaction sent by this LP. Parameters: ���� - Transaction that needs to be cancelled back *��+���� ����+���� � ���� � ���� )��*���� ����*���� � ���� �� Goes through the list of received Transactions and anti-Transactions and either chains the new Transaction in or undoes the original Transaction for received anti-Transaction. This method also handles received cancelbacks. ���+����� ��(� )���� � ���� � ���� ���*����� ��'� (���� � ���� ����� ������ Cancel back a certain number of Transactions. This method is called by the Logical Process if it is in CancelBack mode and it will attempt to cancel back the specified number of received Transactions from the end of the Transaction chain, i.e. the Transactions that are furthest ahead in simulation time and that where received from other LPs. Parameters: ����� - number of Transactions to cancel back Appendix E: Documentation of selected classes – LogicalProcess ��+��'!��� ���������������� � ���� � ���� ��*��&!��� ���������������� � ���� �� This method performs the main lazy-cancellation for Transactions that have been sent and subsequently rolled back. The method is called after the simulation time has been updated (increased). It looks for any past sent and rolled back Transactions that still exist in rolledBackSentHistoryList (i.e. that had not been re-sent in identical form after the rollback) and sends out anti-Transactions for these. ����� � ���� ����� ��������� ���� Performs fossil collection for changes earlier than the GVT. This will remove any saved state information and any records in the sent and received history lists that are not needed any more. Parameters: ������� - time until which all Transaction movements are guarantied, this means there cannot be any rollback to a time before this time ����,� ) � ���� ����+� ( ��������� ��� Rolls the state of the simulation engine back to the state for the given time or the next later state. This also changes some of the information within the Logical Process back to what it was at the time to which the simulation engine is rolled back. Parameters: - time to which the simulation state will be rolled back Appendix E: Documentation of selected classes – LogicalProcess ���� ����� � ���� ���� ����� ������ Saves the current state of the simulation engine into the local state list (unless an unconfirmed end of simulation has been reached by this LP or the LP is in Cancelback mode, in both cases the local simulation time would have the value of Long.MAX_VALUE). ��% � ����"�������� �� ��� -����,��!���� � ��% � ����"���������� Returns the parameters of this Logical Process required for the GVT calculation. This method is called by the Simulation Controller when it performs a GVT calculation. The parameters include the minimum time of all received and not executed Transactions (i.e. either in receivedList or in the simulation engine queue) and the minimum time of any Transaction in transit (i.e. sent but not yet received). Returns: GVT parameter object �� ������� �� ��� ���� �� ������������ ���� ��� �� ����!��� ������ Called by SimulationController to send the calculated GVT (global virtual time). This time guarantees all executed Transactions and state changes with a time smaller than the GVT and as a result the Logical Process can perform fossil collection by committing any changes that happened before the GVT. Parameters: ��� - GVT (global virtual time) Appendix E: Documentation of selected classes – LogicalProcess -�� ������ �� ��� ���� ,�� ����������� ��� Calling this method will force the Logical Process to request a GVT calculation as soon as it passes the specified simulation time. If the specified time has been passed already then a GVT calculation is requested at the next simulation scheduling cycle. This method is called by a Logical Process that reached an unconfirmed End of Simulation in order to force other LPs to request a GVT calculation when they pass the provisional End of Simulation time. Parameters: - simulation time after which a GVT calculation should be requested ��+�- �� ������(!���� � ���� �� ��� ����� � ��������� ����������� ��#��� ��+���� ��*�, �� ������'!���� � �����.���������� ����� Requests the Logical Process to end the simulation at the specified Transaction. This method is called by the Simulation Controller when a GVT calculation confirms a provisional End of Simulation reached by one of the LPs. If this is the LP that reported the unconfirmed End of Simulation by this Transaction then it will have stopped simulating already. All other LPs will be rolled back to the time of this Transaction and then they will simulate any Transactions for the same time that in a sequential simulator would have been executed before the specified Transaction. Afterwards the simulation is stopped and completion is reported back to the SimulationController. Parameters: ���� - Transaction that finished the simulation Returns: BooleanWrapper to indicate to the SimulationController that the LP completed the simulation at the specified end Appendix E: Documentation of selected classes – LogicalProcess �� ������������ �� ��� ����������� ����� � $�� �� ������������� ��� �� ������ '(���� ����� Returns the simulation report. This method is called by the Simulation Controller after the simulation has finished in order to output the combined simulation report from all LPs. The simulation report can optionally contain the Transaction chain report section. This additional section is optional because it can be very large. It is therefore only returned if needed, i.e. requested by the user. This method will be called by the Simulation Controller after the simulation was completed in order to output the combined reports from all LPs. Parameters: ������ '(���� ���� - include Transaction chain report section Returns: populated instance of SimulationReportSet Appendix E: Documentation of selected classes – ParallelSimulationEngine �������������� �� �����&�� ����� �� �������� �� ��������� ����������� � ������ �/���,���������������������������� ���� �������������� �� ���������"������� �� ��������$��� All Implemented Interfaces: java.io.Serializable �� ��� ����� "������� �� ��������$��� ��� ���������� ���� ��� ��������� ������� � ��� � � ���� ��� � �� � ���� �� ��� ���� ���� ���� � �� �&�� �� ��� ��� '��� ���� ��� � �� � � ����� �� ������� ������ %� ���� ��(����� �� ��� � � ���� Author: Gerald Krafft See Also: ���������� ���� , Serialized Form Constructor Summary ParallelSimulationEngine�� Constructor for serialization purpose ParallelSimulationEngine�Partition ���������� Main constructor Appendix E: Documentation of selected classes – ParallelSimulationEngine Method Summary � ���� chainIn�Transaction ��0���� Overrides chainIn() from class parallelJavaGpssSimulator.gpss.SimulationEngine. � ���� deleteLaterTransactions�Transaction ����� Removes all Transactions from the local chain that would be executed/moved after the specified Transaction, i.e. all Transactions that have a move time later than the specified Transaction or with the same move time but a lower priority. � ��� �� deleteTransaction�Transaction ����� Removes the specified Transaction from the Transaction chain. � ���� getMinChainTime�� Returns the minimum time of all movable Transactions in the Transaction chain. � ���� getNoOfTransactionsInChain�� Returns the number of Transactions currently in the chain ���� getTotalTransactionMoves�� Returns the total number of Transaction moves performed since the start of the simulation. ��������������$-���1Transaction2 getTransactionChain�� Gives access to the Transaction chain for classes that inherit from SimulationEngine and makes this visible Appendix E: Documentation of selected classes – ParallelSimulationEngine within the current package. ��������������$-���1Transaction2 getTransactionToSendList�� Returns the out list of Transactions that need to be sent to other LPs. � Transaction getUnconfirmedEndOfSimulationXact�� Returns the Transaction that caused an unconfirmed end of simulation within this simulation engine. ���� moveAllTransactionsAtCurrentTime�� Moves all Transactions that are movable at the current simulation time. � ���� moveTransaction�Transaction ����� Overrides the inherited method in order to add some sensor information used by the LP Control Component. � ���� setCurrentSimulationTime����� ���� ������������.�� Sets the simulation time to the specified value �� unconfirmedEndOfSimulationReached�� Returns whether an unconfirmed end of simulation has been reached by this engine �� updateClock�� Overrides the inherited method. Methods inherited from class parallelJavaGpssSimulator.gpss.SimulationEngine chainOutNextMovableTransactionForCurrentTime� getBlockForBlockReference� getBlockReferenceForLocalBlock� getBlockReport� getChainReport� getCurrentSimulationTime� Appendix E: Documentation of selected classes – ParallelSimulationEngine getFacilitySummaryReport� getNextTransactionId� getNoOfTransactionsAtBlock� getPartition� getQueueSummaryReport� getStorageSummaryReport� initializeGenerateBlocks� isTransactionBlocked� setBlockReferenceToLocalBlock Methods inherited from class java.lang.Object &����� %������ �'����� (��('�� � ����%$� ����%$���� ��������� ����� ����� ���� Constructor Detail "������� �� ��������$��� �� ��� "������� �� ��������$����� Constructor for serialization purpose "������� �� ��������$��� �� ��� "������� �� ��������$����!�������� ���������� Main constructor Parameters: ��������� - model partition that will be simulated by this simulation engine Method Detail *����� � ���� )������.���������� ��0���� Overrides chainIn() from class parallelJavaGpssSimulator.gpss.SimulationEngine. If the next block of the Transaction to be chained in lies in a different partition then the Transaction is Appendix E: Documentation of selected classes – ParallelSimulationEngine stored in the out list so that it can later be sent to the LP of that partition, otherwise the inherited chainIn() method is called. Overrides: chainIn in class SimulationEngine Parameters: ��0��� - Transaction to be added to the chain See Also: ���������� ���� ��(���"�������� �/���,������������������.����������� ����������� � ���� ��� ��������� �� ��� ���� ����������� � ���� ��� ����������� �(���� "������#���3� ����� Moves all Transactions that are movable at the current simulation time. This method overrides the same method from class parallelJavaGpssSimulator.gpss.SimulationEngine in order to implement end of simulation detection for parallel simulation. Overrides: moveAllTransactionsAtCurrentTime in class SimulationEngine Throws: InvalidBlockReferenceException See Also: ���������� ���� ���.�������������'��� ��.�� �������� � ���� � ������������ � �����.���������� ����� Appendix E: Documentation of selected classes – ParallelSimulationEngine Overrides the inherited method in order to add some sensor information used by the LP Control Component. In addition it calls the inherited method to perform the actual movement of the Transaction. Overrides: moveTransaction in class SimulationEngine Parameters: ���� - Transaction to move See Also: ���������� ���� .�����������.���������� ����� �+������ ) �� ��� ��� �� �*������ (�� Overrides the inherited method. The inherited method is only called and the simulation time updated if no provisional End of Simulation has been reached. Overrides: updateClock in class SimulationEngine Returns: true if a movable Transaction was found, otherwise false See Also: ���������� ���� ������ '���3�� +��������� � ���� � ��� �� *��������� � �����.���������� ����� Removes the specified Transaction from the Transaction chain. Appendix E: Documentation of selected classes – ParallelSimulationEngine Parameters: ����"� - Id of the Transaction Returns: true if the Transaction was found and removed, otherwise false +�������������� � ���� � ���� *�������������� � ���� �.���������� ����� Removes all Transactions from the local chain that would be executed/moved after the specified Transaction, i.e. all Transactions that have a move time later than the specified Transaction or with the same move time but a lower priority. Parameters: ���� - Transaction for which any later Transactions will be removed $������ � �����*��� � ��������������$-���1.����������2 $������ � �����)����� Gives access to the Transaction chain for classes that inherit from SimulationEngine and makes this visible within the current package. Overrides: getTransactionChain in class SimulationEngine Returns: Returns the Transaction chain. $������*������� � ���� $������)��������� Appendix E: Documentation of selected classes – ParallelSimulationEngine Returns the minimum time of all movable Transactions in the Transaction chain. This is the current simulation time unless there are no Transactions in the chain or an unconfirmed end of simulation has been reached in which case Long.MAX_VALUE is returned. This method is used by the LP to determine the local time that will be sent to the GVT calculation. Returns: minimum local chain time $�����-���� � ���� ���*��� � ���� $�����,���� � ���� ���)����� Returns the number of Transactions currently in the chain Returns: number of Transactions in the chain ��� ����� �� ���������� � ���� ��� ����� �� ��������������� ���� ������������.�� Description copied from class: SimulationEngine Sets the simulation time to the specified value Overrides: setCurrentSimulationTime in class SimulationEngine Parameters: ������������.�� - new current simulation time $���� ��-����+��+�- �� ������.� � � .���������� $���� ��,����*��*�, �� ������-� ��� Appendix E: Documentation of selected classes – ParallelSimulationEngine Returns the Transaction that caused an unconfirmed end of simulation within this simulation engine. Returns: Returns Transaction that caused an unconfirmed end of simulation � ��-����+��+�- �� ��������� *�+ �� ��� ��� �� � ��,����*��*�, �� ��������� )�*�� Returns whether an unconfirmed end of simulation has been reached by this engine Returns: true if unconfirmed end of simulation has been reached $����������� � �������� �� ��� ���� $����������� � �������� �� Returns the total number of Transaction moves performed since the start of the simulation. This information is required by the LPCC as a sensor value. Returns: total number of Transaction moves performed $������ � ������ ��+�� � �� ��� ��������������$-���1.����������2 $������ � ������ ��*�� ��� Returns the out list of Transactions that need to be sent to other LPs. Returns: outgoing list of Transactions Appendix E: Documentation of selected classes – LPControlComponent �������������� �� �����&��&�� ����� ����� ���������� ����������� � �������������� �� ����������� ��"���������������� All Implemented Interfaces: java.io.Serializable �� ��� ����� �"���������������� ��� ����������� � ��� ��������� ������� � ���� � �� ��� ��� �� ��� �� �� ��� ����� � �� !���� �� �� � �� �% ��� ����) *����� � ���� + �������� �� �� ���� �� �� ��� ��� �������� �% ���� �� �� ���� �� � ��� ����� � ��� �% �� ��� � ��� ��� � ��� �� � �� �� ������ � ��� ���� ��� ��! �� �� � �������� ������ ���%��� Author: Gerald Krafft See Also: Serialized Form Constructor Summary LPControlComponent�� Constructor for serialization LPControlComponent���� ����� �'����� Main constructor, initializes the cluster space with the maximum number of clusters to be held. Appendix E: Documentation of selected classes – LPControlComponent Method Summary ���� getCurrentUncommittedMovesMeanLimit�� Returns the mean limit for uncommitted Transaction moves (AvgUncommittedMoves) that is based on the indicators passed in the last time processSensorValues() was called. ���� getCurrentUncommittedMovesUpperLimit�� Returns the current upper limit for uncommitted Transaction moves as determined by the LPCC. ���� getLastSensorProcessingTime�� Returns the last time the sensor values were processed in milliseconds. SensorSet getSensorSet�� Returns the sensor set with the current sensor values. �� isUncommittedMovesValueWithinActuatorRange����� ��������� Returns true if the value is within the actuator limit using the UncommittedMoves standard deviation and a confidence level of 95%. ���� processSensorValues�� This method performs the main processing of the sensor values which will result in a new actuator value. Methods inherited from class java.lang.Object &����� %������ �'����� (��('�� � ����%$� ����%$���� ��������� ����� ����� ���� Appendix E: Documentation of selected classes – LPControlComponent Constructor Detail �"���������������� �� ��� �"������������������ Constructor for serialization �"���������������� �� ��� �"�������������������� ����� �'����� Main constructor, initializes the cluster space with the maximum number of clusters to be held. Parameters: ����� �'���� - maximum number of clusters to be held Method Detail �� �� �� ��� � ����� � $�� �� �� Returns the sensor set with the current sensor values. Returns: sensor set $���� � �� ��"�� � ��$���� �� ��� ���� $���� � �� ��"�� � ��$������ Returns the last time the sensor values were processed in milliseconds. Returns: the last time the sensor values were processed. Appendix E: Documentation of selected classes – LPControlComponent ��� � �� ����� � �� ��� ���� ��� � �� ����� � �� This method performs the main processing of the sensor values which will result in a new actuator value. It generates an indicator set for the sensor values, determines the closest (most similar) past indicator set with a higher performance indicator (CommittedMoveRate) using a state cluster space and then adds the current indicator set to the state cluster space. $��� ������� �������+���� ��������� �� ��� ���� $��� ������� �������*���� ����������� Returns the mean limit for uncommitted Transaction moves (AvgUncommittedMoves) that is based on the indicators passed in the last time processSensorValues() was called. Returns: mean actuator limit $��� ������� �������+���� ���������� �� ��� ���� $��� ������� �������*���� ������������ Returns the current upper limit for uncommitted Transaction moves as determined by the LPCC. This is the upper limit based on the average uncommitted moves limit, the standard deviation and a confidence level of 95%. Returns: upper actuator value Appendix E: Documentation of selected classes – LPControlComponent � �� �������+���� ��� �/��*��� � �������$� �� ��� ��� �� � �� �������*���� ��� �.��)��� � �������$������ ��������� Returns true if the value is within the actuator limit using the UncommittedMoves standard deviation and a confidence level of 95%. Parameters: ��������� � - current UncommittedMoves sample value Returns: true if UncommittedMoves sample value is within actuator limits, otherwise false Appendix F: Validation output logs Appendix F: Validation output logs This appendix contains the relevant output log files resulting from the validation runs performed as part of the validation in section 6. Line numbers in brackets were added to all lines of the output log files in order to make it possible to refer to a specific line. For very long output log files non-relevant lines where removed and replaced with “...”. But the complete output log files can still be found on the attached CD. Appendix F: Validation output logs – validation 1 Appendix F: Validation output logs – validation 1 Appendix F: Validation output logs – validation 2 Appendix F: Validation output logs – validation 2 Appendix F: Validation output logs – validation 2 Appendix F: Validation output logs – validation 2 Appendix F: Validation output logs – validation 2 Appendix F: Validation output logs – validation 2 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 3 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 4 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 5 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6 Appendix F: Validation output logs – validation 6

0704.1828

Gauge invariance in gravity-like descriptions of massive gauge field theories

Gauge invariance in gravity-like descriptions of massive gauge field theories Dennis D. Dietrich Institut für Theoretische Physik, Universität Heidelberg, Heidelberg, Germany (Dated: January 13, 2019) We discuss gravity-like formulations of massive Abelian and non-Abelian gauge field theories in four space-time dimensions with particular emphasis on the issue of gauge invariance. Alternative descriptions in terms of antisymmetric tensor fields and geometric variables, respectively, are anal- ysed. In both approaches Stückelberg degrees of freedom factor out. We also demonstrate, in the Abelian case, that the massless limit for the gauge propagator, which does not exist in the vector potential formulation, is well-defined for the antisymmetric tensor fields. PACS numbers: 11.15.-q, 11.15.Ex, 11.30.Qc, 12.15.-y I. INTRODUCTION Massive gauge bosons belong to the fundamental con- cepts we use for picturing nature. Prominent exam- ples are found in the physics of electroweak interac- tions, superconductivity, and confinement. Even more than in the massless case, gauge invariance is a severe constraint for the construction of massive gauge field theories. Usually additional fields beyond the original gauge field have to be included in order to obtain gauge invariant expressions.[26] Technically, this is linked to the fact that the aforementioned gauge field—the Yang– Mills connection—changes inhomogeneously under gauge transformations and encodes also spurious degrees of freedom arising from the construction principle of gauge invariance. This complicates the extraction of physical quantities. A variety of approaches has been developed in order to deal with this situation. Wilson loops [1] represent gauge invariant but non-local variables.[27] Al- ternatively, there exist decomposition techniques like the one due to Cho, Faddeev, and Niemi [2]. Here we first pursue a reformulation of massive Yang–Mills theories in terms of antisymmetric gauge algebra valued tensor fields Baµν (Sect. II) and subsequently continue with a represen- tation in terms of geometric variables (Sect. III). In Sect. II A we review the massless case. It is re- lated to gravity [3] formulated as BF gravity [4] and thus linked to quantum gravity. The antisymmetric ten- sor field can be seen as dual field strength and transforms homogeneously under gauge transformations. This fact already makes it simpler to keep track of gauge invari- ance. In Sect. II B the generalisation to the massive case is presented. In the Baµν field representation the (non- Abelian) Stückelberg fields, which are commonly present in massive gauge field theories and needed there in or- der to keep track of gauge invariance, factor out com- pletely. In other words, no scalar fields are needed for a gauge invariant formulation of massive gauge field theo- ries in terms of antisymmetric tensor fields. The case of a constant mass is linked to sigma models (gauged and ungauged) in different respects. Sect. II B 1 contains the generalisation to a position dependent mass, which cor- responds to introducing the Higgs degree of freedom. In Sect. II B 2 non-diagonal mass terms are admitted. This is necessary to accommodate the Weinberg–Salammodel, which is studied as particular case. Sect. III presents a description of the massive case, with constant and varying mass, in terms of geometric variables. In this step the remaining gauge degrees of freedom are eliminated. The emergent description is in terms of local colour singlet variables. Finally, Sect. III A is concerned with the geometric representation of the Weinberg–Salam model. The Appendix treats the Abelian case. It allows to better interpret and understand several of the findings in the non-Abelian settings. Of course, in the Abelian case already the Bµν field is gauge invariant. Among other things, we demonstrate that them→ 0 limit of the gauge propagator for the Bµν fields is well-defined as opposed to the ill-defined limit for the Aµ field propagator. Sect. IV summarises the paper. II. ANTISYMMETRIC TENSOR FIELDS A. Massless Before we investigate massive gauge field theories let us recall some details about the massless case. The parti- tion function of a massless non-Abelian gauge field theory without fermions is given by [dA] exp{i xL}, (1) with the Lagrangian density L = L0 := − 14g2F aµν (2) and the field tensor µν := ∂µA ν − ∂νAaµ + fabcAbµAcν . (3) Aaµ stands for the gauge field, f abc for the antisymmetric structure constant, and g for the coupling constant. [28] Variation of the classical action with respect to the gauge field gives the classical Yang–Mills equations Dabµ (A)F bµν = 0, (4) http://arxiv.org/abs/0704.1828v1 where the covariant derivative is defined as Dabµ (A) := δab∂µ + f acbAcµ. The partition function in the first-order formalism can be obtained after multiplying Eq. (1) with a prefactor in form of a Gaussian integral over an anti- symmetric tensor field Baµν , [dA][dB] exp{i d4x[L0 − g BaµνB aµν ]}. (5) (”∼=” indicates that in the last step the normalisation of the partition function has been changed.) Subsequently, the field Baµν is shifted by F̃ aµν , where the dual field tensor is defined as F̃ aµν := ǫµνκλF [dA][dB] × × exp{i x[− 1 aµν − g aµν ]}. (6) In this form the partition function is formulated in terms of the Yang–Mills connection Aaµ and the antisymmetric tensor field Baµν as independent variables. Variation of the classical action with respect to these variables leads to the classical equations of motion µν = −F̃ aµν and Dabµ (A)B̃bµν = 0, (7) where B̃aµν := ǫκλµνB aκλ. By eliminating Baµν the origi- nal Yang–Mills equation (4) is reproduced. Every term in the classical action in the partition function (6) contains at most one derivative as opposed to two in Eq. (1). This explains the name ”first-order” formalism. The classical action in Eq. (6) is invariant under simultaneous gauge transformations of the independent variables according AaµT a =: Aµ → AµU := U [A µ − iU †(∂µU)]U † (8) BaµνT a =: Bµν → Bµν := UBµνU †, (9) or infinitesimally, δAaµ = ∂µθ a + fabcAbµθ δBaµν = f abcBbµνθ c. (10) The T a stand for the generators of the gauge group. From the Bianchi identity Dabµ (A)F̃ bµν = 0 follows a second symmetry of the BF term alone: Infinitesimally, for un- changed Aaµ, µν = ∂µϑ ν − ∂νϑaµ + fabc(Abµϑcν −Abνϑcµ). (11) A particular combination of the transformations (10) and (11), θa = nµAaµ and ϑ ν = n µBaµν , corresponds to the transformation of a tensor under an infinitesimal local coordinate transformation xµ → xµ − nµ(x), δBµν = Bλν∂µn λ +Bµλ∂νn λ + nλ∂λBµν , (12) that is a diffeomorphism. Hence, the BF term is dif- feomorphism invariant, which explains why this theory is also known as BF gravity. The BB term is not dif- feomorphism invariant and, hence, imposes a constraint. The combination of the two terms amounts to an action of Plebanski type which are studied in the context of quantum gravity [3, 4]. We now would like to eliminate the Yang–Mills connec- tion by integrating it out. For fixed Baµν the integrand of the path integral is not gauge invariant with respect to gauge transformations of the gauge field Aaµ alone; the field tensor F aµν transforms homogeneously and the corre- sponding gauge transformations are not absorbed if Baµν is held fixed. Therefore, the integral over the gauge group is in general not cyclic which otherwise would render the path integral ill-defined. The term in the exponent linear in the gauge field Aaµ, A ν∂µB̃ aµν , is obtained by carry- ing out a partial integration in which surface terms are ignored. Afterwards it is absorbed by shifting Aaµ by (B−1)abµν(∂λB̃ bλν), where Babµν := f abcB̃aµν . In general its inverse (B−1)abµν , defined by (B −1)abµνB νκ = δacgµλ exists in three or more space-time dimensions [5]. We are left with a Gaussian integral in Aaµ giving the inverse square-root of the determinant of Babµν , [da] exp{− i bν}. (13) In the last expression Babµν appears in the place of an in- verse gluon propagator, that is sandwiched between two gauge fields. This analogy carries even further: Interpret- ing ∂µB̃ aµν as a current, (B−1)abµν(∂λB̃ bλν), the current together with the ”propagator” (B−1)abµν , is exactly the abovementioned term to be absorbed in the gauge field Aaµ. Finally, we obtain, [dB]Det− B exp{i d4x[− g BaµνB aµν − (∂κB̃ aκµ)(B−1)abµν(∂λB̃ bλν)]}. (14) This result is known from [5, 6, 7]. The exponent in the previous expression corresponds to the value of the [dA] integral at the saddle-point value Ăaµ of the gauge field. It obeys the classical field equation (7). Using Ăaµ(B) = (B −1)abµν(∂λB̃ bλν) the second term in the above exponent can be rewritten as − i d4xB̃aµνF aµν [Ă(B)], which involves an integration by parts and makes its gauge invariance manifest. The fluctuations aaµ around the saddle point Ăaµ, contributing to the partition func- tion (6), are Gaussian because the action in the first-order formalism is only of second order in the gauge field Aaµ. They give rise to the determinant (13). What happens if a zero of the determinant is encountered can be un- derstood by looking at the Abelian case discussed in Ap- pendix A. There the BF term does not fix a gauge for the integration over the gauge field Aµ because the Abelian field tensor Fµν is gauge invariant. If it is performed nevertheless one encounters a functional δ distribution which enforces the vanishing of the current ∂µB̃ µν . In this sense the zeros of the determinant in the non-Abelian case arise if B̃aµν is such that the BF term does not totally fix a gauge for the [dA] integration, but leaves behind a residual gauge invariance. It in turn corresponds to van- ishing components of the current ∂µB̃ aµν . (Technically, there then is at least one flat direction in the otherwise Gaussian integrand. The flat directions are along those eigenvectors of B possessing zero eigenvalues.) When incorporated with the exponent, which requires a regularisation [8], the determinant contributes a term proportional to 1 ln detB to the action. This term to- gether with the BB term constitutes the effective poten- tial, which is obtained from the exponent in the partition function after dropping all terms containing derivatives of fields. The effective potential becomes singular for field configurations for which detB = 0. It is gauge invari- ant because all contributing addends are gauge invariant separately. The classical equations of motion obtained by varying the action in Eq. (14) with respect to the dual antisym- metric tensor field B̃aµν are given by g2B̃aµν = (g µ − gρµgσν )∂ρ(B−1)abσκ(∂λB̃bλκ)− −(∂ρB̃dρκ)(B−1)dbκµfabc(B−1)ceνλ(∂σB̃eσλ), which coincides with the first of Eqs. (7) with the field tensor evaluated at the saddle point of the action, F aµν [Ă(B)]. Taking into account additionally the effect due to fluctuations of Aaµ contributes a term proportional to δDetB δB̃aµν det−1B to the previous equation. B. Massive In the massive case the prototypical Lagrangian is of the form L = L0 +Lm, where Lm := m aµ. (Due to our conventions the physical mass is given by mphys := mg.) This contribution to the Lagrangian is of course not gauge invariant. Putting it, regardlessly, into the partition function, gives [dA][dU ] exp{i x[L0 + m aµ]}, (16) which can be interpreted as the unitary gauge represen- tation of an extended theory. In order to see this let us split the functional integral over Aaµ into an integral over the gauge group [dU ] and gauge inequivalent field config- urations [dA]′. Usually this separation is carried out by fixing a gauge according to [dA]′ := [dA]δ[fa(A) − Ca]∆f (A). (17) fa(A) = Ca is the gauge condition and ∆f (A) stands for the Faddeev–Popov determinant defined through [dA]δ[fa(A) − Ca]∆f (A) [29]. Introducing this reparametrisation into the partition function (16) yields, [dA]′[dU ] exp(i d4x{− 1 F aµνF aµν + [Aµ − iU †(∂µU)]a[Aµ − iU †(∂µU)]a}). L0 is gauge invariant in any case and remains thus un- affected. In the mass term the gauge transformations appear explicitly [9]. We now replace all of these gauge transformations with an auxiliary (gauge group valued) scalar field Φ, U † → Φ, obeying the constraint !≡ 1. (19) The field Φ can be expressed as Φ =: e−iθ, where θ =: θaT a is the gauge algebra valued non-Abelian gen- eralisation of the Stückelberg field [10]. For a massive gauge theory they are a manifestation of the longitudi- nal degrees of freedom of the gauge bosons. In the con- text of symmetry breaking they arise as Goldstone modes (”pions”). In the context of the Thirring model these ob- servations have been made in [11]. There it was noted as well that the θ is also the field used in the canonical Hamiltonian Batalin–Fradkin–Vilkovisky formalism [12]. We can extract the manifestly gauge invariant classical Lagrangian Lcl := − 14g2F aµν +m2tr[(DµΦ) †(DµΦ)], (20) where the scalars have been rearranged making use of the product rule of differentiation and the cyclic property of the trace and whereDµΦ := ∂µΦ−iAµΦ. Eq. (20) resem- bles the Lagrangian density of a non-linear gauged sigma model. In the Abelian case the fields θ decouple from the dynamics. For non-Abelian gauge groups they do not and one would have to deal with the non-polynomial coupling to them. In the following we show that these spurious degrees of freedom can be absorbed when making the transition to a formulation based on the antisymmetric tensor field Baµν . Introducing the antisymmetric tensor field into the corre- sponding partition function, like in the previous section, results in, [dA][dΦ][dB] exp(i ×{− g aµν − 1 aµν + [Aµ − iΦ(∂µΦ†)]a[Aµ − iΦ(∂µΦ†)]a}). Removing the gauge scalars Φ from the mass term by a gauge transformation of the gauge field Aaµ makes them explicit in the BF term, [dA][dΦ][dB] exp{i d4x[− g BaµνB aµν − −tr(ΦF̃µνΦ†Bµν) + m aµ]}. (22) In the next step we would like to integrate over the Yang– Mills connection Aaµ. Already in the previous expression, however, we can perceive that the final result will only depend on the combination of fields Φ†BµνΦ. [The Φ field can also be made explicit in the BB term in form of the constraint (19).] Therefore, the functional inte- gral over Φ only covers multiple times the range which is already covered by the [dB] integration. Hence the degrees of freedom of the field Φ have become obsolete in this formulation and the [dΦ] integral can be factored out. Thus, we could have performed the unitary gauge calculation right from the start. In either case, the final result reads, [dB]Det− M exp{i x[− g aµν − (∂κB̃ aκµ)(M−1)abµν(∂λB̃ bλν)]}, (23) where Mabµν := B µν − m2δabgµν , which coincides with [13]. Mabµν and hence (M −1)abµν transform homogeneously under the adjoint representation. In Eq. (14) the cen- tral matrix (B−1)abµν in the analogous term transformed in exactly the same way. There this behaviour ensured the gauge invariance of this term’s contribution to the classical action. Consequently, the classical action in the massive case has the same invariance properties. In par- ticular, the aforementioned gauge invariant classical ac- tion describes a massive gauge theory without having to resort to additional scalar fields. For detB 6= 0, the limit m → 0 is smooth. For detB = 0 the conserved current components alluded to above would have to be separated appropriately in order to recover the corresponding δ dis- tributions present in these situations in the massless case. Again the effective action is dominated by the term proportional to 1 detM. The contribution from the mass to M shifts the eigenvalues from the values obtained for B. Hence the singular contributions are typically ob- tained for eigenvalues of B of the order of m2. The ef- fective potential is again gauge invariant, for the same reason as in the massless case. The classical equations of motion obtained by variation of the action in Eq. (21) are given by, µν = −F̃ aµν , Dabµ (A)B̃ bµν = −m2[Aν − iΦ(∂νΦ†)]a, 0 = δ x{[Aaµ − iΦ(∂µΦ†)]a}2. (24) In these equations a unique solution can be chosen, that is a gauge be fixed, by selecting the scalar field Φ. Φ ≡ 1 gives the unitary gauge, in which the last of the above equations drops out. The general non-Abelian case is difficult to handle already on the classical level, which is one of the main motivations to look for an alternative formulation. In the non-Abelian case, the equation of motion obtained from Eq. (23) resembles strongly the massless case, g2B̃aµν = (g µ − gρµgσν )∂ρ(M−1)abσκ(∂λB̃bλκ)− −(∂ρB̃dρκ)(M−1)dbκµfabc(M−1)ceνλ(∂σB̃eσλ), insofar as all occurrences of (B−1)abµν have been replaced by (M−1)abµν . Incorporation of the effect of the Gaussian fluctuations of the gauge field Aaµ would give rise to a con- tribution proportional to δB δB̃aµν det−1M in the previous equation. Before we go over to more general cases of massive non-Abelian gauge field theories, let us have a look at the weak coupling limit: There the BB term in Eq. (21) is neglected. Subsequently, integrating out the Baµν field enforces F aµν ≡ 0. [This condition also arises from the classical equation of motion (24) for g=0.] Hence, for vanishing coupling exclusively pure gauge configurations of the gauge field Aaµ contribute. They can be combined with the Φ fields and one is left with a non-linear reali- sation of a partition function, g=0∼= [dΦ] exp{im2 d4x tr[(∂µΦ †)(∂µΦ)]}, (26) of a free massless scalar [13]. Setting g = 0 interchanges with integrating out the Baµν field from the partition func- tion (21). Thus, the partition function (23) with g = 0 is equivalent to (26). That a scalar degree of freedom can be described by means of an antisymmetric tensor field has been noticed in [14]. 1. Position-dependent mass and the Higgs One possible generalisation of the above set-up is ob- tained by softening the constraint (19). This can be seen as allowing for a position dependent mass. The new degree of freedom ultimately corresponds to the Higgs. When introducing the mass m as new degree of freedom (as ”mass scalar”) we can restrict its variation by in- troducing a potential term V (m2), which remains to be specified, and a kinetic term K(m), which we choose in its canonical form K(m) = 1 (∂µm)(∂ µm). It gives a penalty for fast variations of m between neighbouring space-time points. The fixed mass model is obtained in the limit of an infinitely sharp potential with its mini- mum located at a non-zero value for the mass. Putting together the partition function in unitary gauge leads to, [dA][dm] exp{i d4x[− 1 F aµνF aµν + aµ +K(m) + V (m2)]}, (27) where we have introduced the normalisation constant N := dim R, with R standing for the representation of the scalars. This factor allows us to keep the canonical normalisation of the mass scalar m. We can now repeat the same steps as in the previous section in order to iden- tify the classical Lagrangian, Lcl := − 14g2F aµν +N−1tr[(Dµφ)†(Dµφ)] + V (|φ|2), where now φ := mΦ. In order to reformulate the parti- tion function in terms of the antisymmetric tensor field we can once more repeat the steps in the previous sec- tion. Again the spurious degrees of freedom represented by the field Φ can be factored out. Finally, this gives [15], [dB][dm]Det− M exp{i d4x[− g BaµνB aµν − (∂κB̃ aκµ)(M−1)abµν(∂λB̃ bλν) + +K(m) + V (m2)]}, (28) whereMabµν = B µν −m2N−1δbgµν depends on the space- time dependent mass m. The determinant can as usual be included with the exponent in form of a term pro- portional to 1 detM, the pole of which will dominate the effective potential. As just mentioned, however,M is also a function of m. Hence, in order to find the minimum, the effective potential must also be varied with respect to the mass m. Carrying the representation in terms of antisymmetric tensor fields another step further, the partition function containing the kinetic term K(m) of the mass scalar can be expressed as Abelian version of Eq. (26), [db][da] exp{i d4x[− 1 b̃µνf µν + 1 µ]} = [dm] exp{i d4x[ 1 (∂µm)(∂ µm)]}, (29) where here the mass scalar m is identified with the Abelian gauge parameter. Combining the last equation with the partition function (28) all occurrences of the mass scalar m can be replaced by the phase integral dxµaµ. The bf term enforces the curvature f to vanish which constrains aµ to pure gauges ∂µm and the aforementioned integral becomes path-independent. 2. Non-diagonal mass term and the Weinberg–Salam model The mass terms investigated so far had in common that all the bosonic degrees of freedom they described pos- sessed the same mass. A more general mass term would be given by Lm := m ab. Another similar ap- proach is based on the Lagrangian Lm := m tr{AµAµΨ} where Ψ is group valued and constant. We shall begin our discussion with this second variant and limit ourselves to a Ψ with real entries and trΨ = 1, which, in fact, does not impose additional constraints. Using this expression in the partition function (27) and making explicit the gauge scalars yields, [dA][dm] exp{i d4x[− 1 F aµνF aµν + tr{(Dµφ)†(Dµφ)Ψ} + V (m2)]}. (30) Expressed in terms of the antisymmetric tensor field Baµν , the corresponding partition function coincides with Eq. (28) but with Mabµν replaced by M µν := B m2tr{T aT bΨ}gµν. Let us now consider directly the SU(2)× U(1) Weinberg–Salam model. Its partition function can be expressed as, [dA][dψ] exp{i x[− 1 aµν + ∂ µ + iAµ)( ∂ µ − iAµ)ψ + V (|ψ|2)]}, where ψ is a complex scalar doublet, Aµ := A a, with a ∈ {0; . . . ; 3}, T a here stands for the generators of SU(2) in fundamental representation, and, accordingly, T 0 for g0 times the 2 × 2 unit matrix, with the U(1) coupling constant g0. The partition function can be reparametrised with ψ = mΦψ̂, where m = |ψ|2, Φ is a group valued scalar field as above, and ψ̂ is a con- stant doublet with |ψ̂|2 = 1. The partition function then becomes, [dA][dΦ][dm] exp(i x{− 1 aµν + tr[Φ†( ∂ µ + iAµ)( µ − iAµ)ΦΨ] + (∂µm)(∂ µm) + V (m2)}), (32) where Ψ = ψ̂ ⊗ ψ̂†. (33) Making the transition to the first order formalism leads [dA][dB][dΦ][dm] exp(i d4x{− g BaµνB aµν − F aµνB̃ aµν +K(m) + V (m2) + tr[Φ†( ∂ µ + iAµ)( ∂ µ − iAµ)ΦΨ]}). (34) As in the previous case, a gauge transformation of the gauge field Aaµ can remove the gauge scalar Φ from the mass term (despite the matrix Ψ). Thereafter Φ only appears in the combination Φ†BµνΦ and the integral [dΦ] merely leads to repetitions of the [dB] integral. [The U(1) part drops out completely right away.] Therefore the [dΦ] integration can be factored out, [dA][dB][dm] exp{i d4x[− g BaµνB aµν − F aµνB̃ aµν + m tr(AµA µΨ) + +K(m) + V (m2)]}. (35) The subsequent integration over the gauge fields A µ leads [dB][dm]Det− M exp{i d4x[− g BaµνB aµν − (∂κB̃ aκµ)(M−1)abµν(∂λB̃ bλν) + +K(m) + V (m2)]}, (36) where M µν := B µν −m2tr(T aT bΨ)gµν . From hereon we continue our discussion based on the mass matrix ab := 1 tr({T a, T b}Ψ), (37) which had already been mentioned at the beginning of Sect. II B 2. mab is real and has been chosen to be sym- metric. (Antisymmetric parts are projected out by the contraction with the symmetric A bµ.) Thus it pos- sesses a complete orthonormal set of eigenvectors µ j with the associated real eigenvalues mj , m = 6Σjmjµ With the help of these normalised eigenvectors one can construct projectors π j := 6Σjµ j and decompose the mass matrix, mab = mjπ j . The projectors are com- plete, 1ab = Σjπ j , idempotent 6Σjπ j = π j , and satisfy π j 6=k = 0. The matrix B µν , the antisym- metric tensor field Baµν , and the gauge field A µ can also be decomposed with the help of the eigenvectors: µν = µ , where bjkµν := µ µν = b where bjµν := B j ; and A µ = a j , where a µ := A Using this decomposition in the partition function (36) leads to, [db][dm]Det− m exp{i d4x[− g bjµνb jµν − (∂κb̃ jκµ)(m−1)jkµν(∂λb̃ kλν) + +K(m) + V (m2)]}, (38) where mjkµν := b µν −m2 jlδklgµν . Making use of the concrete form of mab given in Eq. (37), inserting Ψ from Eq. (33), and subsequent diag- onalisation leads to the eigenvalues 0, 1 and 1 These correspond to the photon, the two W bosons and the heavier Z boson, respectively. The thus obtained tree- level Z to W mass ratio squared consistently reproduces the cosine of the Weinberg angle in terms of the coupling constants, cos2 ϑw = g2+g2 . Due to the masslessness of the photon one addend in the sum over l in the expression µν above does not contribute. Still, the totalm µν does not vanish like in the case of a single massless Abelian gauge boson (see Appendix A). Physically this corre- sponds to the coupling of the photon to the W and Z bosons. III. GEOMETRIC REPRESENTATION The fact that the antisymmetric tensor field Baµν trans- forms homogeneously represents already an advantage over the formulation in terms of the inhomogeneously transforming gauge fields Aaµ. Still, B µν contains de- grees of freedom linked to the gauge transformations (9). These can be eliminated by making the transition to a formulation in terms of geometric variables. In this section we provide a classically equivalent description of the massive gauge field theories in terms of geometric variables in Euclidean space for two colours by adapt- ing Ref. [16] to include mass. The first-order action is quadratic in the gauge-field Aaµ.[30] Thus the evaluation of the classical action at the saddle point yields the ex- pression equivalent to the different exponents obtained after integrating out the gauge field Aaµ in the various partition functions in the previous section. In Euclidean space the classical massive Yang–Mills action in the first order formalism reads d4x(LBB + LBF + LAA), (39) where LBB = − g BaµνB µν , (40) LBF = + i4ǫ µνκλBaµνF κλ, (41) LAA = −m µ. (42) At first we will investigate the situation for the unitary gauge mass term LAA and study the role played by the scalars Φ afterwards. As starting point it is important to note that a metric can be constructed that makes the tensor Baµν self-dual [7]. In order to exploit this fact, it is convenient to define the antisymmetric tensor (j ∈ {1; 2; 3}) µν := η ν , (43) with the self-dual ’t Hooft symbol η [17] [31] and the tetrad eAµ . From there we construct a metric gµν in terms of the tensor T jµν gµν ≡ eAµ eAν = 16ǫ jklT jµκT kκλT lλν , (44) where jµν := 1 g)3 := 1 (ǫjklT T kµ2ν2T ×(ǫj′k′l′T j ×ǫµ1ν1κ1λ1ǫµ2ν2κ2λ2ǫµ3ν3κ3λ3 (46) Subsequently, we introduce a triad daj such that Baµν =: d µν . (47) This permits us to reexpress the BB term of the classical Lagrangian, LBB = − g µνhjkT µν , (48) where hjk := d k. Putting Eqs. (47) and (45) into the saddle point condition ǫκλµνDabµ (Ă)B κλ = +im 2Ăaν (49) gives µ (Ă)( gdbjT jµν) = +im2Ăaν . (50) In the following we define the connection coefficients γµ|kj as expansion parameters of the covariant derivative of the triads at the saddle point in terms of the triads, Dabµ (Ă)d j =: γµ|kj dak. (51) This would not be directly possible for more than two colours, as then the set of triads is not complete. The connection coefficients allow us to define covariant deriva- tives according to ∇µ|kj := ∂µδkj + γµ|kj . (52) These, in turn, permit us to rewrite the saddle point condition (49) as dak∇µ|kj ( gT jµν) = im2Ăaν , (53) and the mass term in the classical Lagrangian becomes LAA = 12m2 [∇µ| gT iµν)]hkl[∇κ|lj( gT jκν)]. (54) In the limit m→ 0 this term enforces the covariant con- servation condition ∇µ|ki ( gT iµν) ≡ 0, known for the massless case. It results also directly from the saddle point condition (53). Here dak∇µ|ki ( gT iµν) are the di- rect analogues of the Abelian currents ǫµνκλ∂µBκλ, which are conserved in the massless case [see Eq. (A6)] and dis- tributed following a Gaussian distribution in the massive case [see Eq. (A10)]. The commutator of the above covariant derivatives yields a Riemann-like tensor Rkjµν Rkjµν := [∇µ,∇ν ]kj . (55) By evaluating, in adjoint representation (marked by )̊, the following difference of double commutators [D̊µ(Ă), [D̊ν(Ă), d̊j ]]− (µ↔ ν) in two different ways, one can show that i[d̊j , F̊µν(Ă)] = d̊kR jµν , (56) or in components, F aµν(Ă) = ǫabcdbjdckR jµν , (57) where dajdak := δ defines the inverse triad, daj = hjkdak. Hence, we are now in the position to rewrite the remain- ing BF term of the Lagrangian density. Introducing Eqs. (47) and (57) into Eq. (41) results in LBF = i4 gT jµνRklµνǫjmkh lm. (58) Let us now repeat the previous steps with a mass term in which the gauge scalars Φ are explicit, LΦAA := − [Aµ − iΦ(∂µΦ†)]a[Aµ − iΦ(∂µΦ†)]a. (59) In that case the saddle point condition (49) is given by, µ (Ă)B̃ κλ = im 2[Ăν − iΦ(∂νΦ†)]a, (60) or in the form of Eq. (53), that is with the left-hand side replaced, dak∇µ|kj ( gT jµν) = im2[Ăν − iΦ(∂νΦ†)]a. (61) Reexpressing LΦAA with the help of the previous equation reproduces exactly the unitary gauge result (54) for the mass term. Finally, the tensor B appearing in the determinant (13), which accounts for the Gaussian fluctuations of the gauge field Aaµ, formulated in the new variables reads gfabcdai T iµν . Now all ingredients are known which are needed to express the equivalent of the parti- tion function (16) in terms of the new variables. For a position-dependent mass the discussion does not change materially. The potential and kinematic term for the mass scalar m have to be added to the action. Contrary to the massless case the Aaµ dependent part of the Euclidean action is genuinely complex. Without mass only the T-odd and hence purely imaginary BF term was Aaµ dependent. With mass there contributes the additional T-even and thus real mass term. There- fore the saddle point value Ăaµ for the gauge field becomes complex. This is a known phenomenon and in this con- text it is essential to deform the integration contour of the path integral in the partition function to run through the saddle point [18]. For the Gaussian integrals which are under consideration here, in doing so, we do not re- ceive additional contributions. The imaginary part IĂaµ of the saddle point value of the gauge field transforms homogeneously under gauge transformations. The com- plex valued saddle point of the gauge field which is in- tegrated out does not affect the real-valuedness of the remaining fields, here Baµν . In this sense the field B represents a parameter for the integration over Aaµ. The tensor T jµν is real-valued by definition and therefore the same holds also for the triad daj [see Eq. (47)]. hkl is composed of the triads and, consequently, real-valued as well. The imaginary part of the saddle point value of the gauge field, IĂaµ, enters the connection coefficients (51). Through them it affects the covariant derivative (52) and the Riemann-like tensor (55). More concretely the con- nection coefficients γµ|kj can be decomposed according to Dabµ (RĂ)dbj = (Rγµ|kj )dak, (62) abc(IĂcµ)dbj = (Iγµ|kj )dak, (63) with the obvious consequences for the covariant deriva- tive, ∇µ|kj = R∇µ|kj + iI∇µ|kj , (64) R∇µ|kj = ∂µδkj +Rγµ|kj , (65) I∇µ|kj = Iγµ|kj . (66) This composition reflects in the mass term, RLAA = 12m2 {[R∇µ| gT iµν)]hkl[R∇κ|lj( gT jκν)]− −[Iγµ|kj ( gT iµν)]hkl[Iγκ|lj( gT jκν)]} ILAA = 22m2 [R∇µ| gT iµν)]hkl[I∇κ|lj( jκν)] on one hand, and in the Riemann-like tensor, RRkjµν = [R∇µ,R∇ν ]kj − [I∇µ, I∇ν ]kj (67) IRkjµν = [R∇µ, I∇ν ]kj + [I∇µ,R∇ν ]kj . (68) on the other. The connection to the imaginary part of Ăaµ is more direct in Eq. (57) which yields, RF aµν (Ă) = 12ǫ abcdbjdckRRkjµν , (69) IF aµν (Ă) = 12ǫ kIRkjµν , (70) Finally, the BF term becomes, RLBF = − 14 gT jµνǫjmkh lmIRklµν , (71) ILBF = + 14 gT jµνǫjmkh lmRRklµν . (72) Summing up, at the complex saddle point of the [dA] in- tegration the emerging Euclidean LAA and LBF are both complex, whereas before they were real and purely imag- inary, respectively. Both terms together determine the saddle point value Ăaµ. Therefore, they become coupled and cannot be considered separately anymore. This was already to be expected from the analysis in Minkowski space in Sect. II, where the matrixMabµν combines T-odd and T-even contributions, which originate from LAA and LBF , respectively. There the different contributions be- come entangled when the inverse (M−1)abµν is calculated. A. Weinberg–Salam model Finally, let us reformulate the Weinberg–Salam model in geometric variables. We omit here the kinematic term K(m) and the potential term V (m2) for the sake of brevity because they do not interfere with the calcula- tions and can be reinstated at every time. The remaining terms of the classical action are x(LAbelBB + LAbelBF + LBB + LBF + LAA), LAA := −m µ, (73) LAbelBB := − g µν , (74) LAbelBF := + i4ǫ µνκλB0µνF κλ, (75) and LBB as well as LBF have been defined in Eqs. (40) and (41), respectively. The saddle point conditions for the [dA] integration with this action are given by ǫκλµνDabµ (Ă)B κλ = +im abAbν , (76) κλ = +im ν . (77) For the following it is convenient to use linear combina- tions of these equations, which are obtained by contrac- tion with the eigenvectors µ of the matrix mab—defined between Eqs. (37) and (38)—, ǫκλµν [µalD µ (Ă)B κλ + µ l ∂µB κλ] = im abAbν .(78) The non-Abelian term on the left-hand side can be rewritten using the results from the first part of Sect. III. The right-hand side may be expressed in terms of eigen- values of the matrix mab. We find (no summation over Xaν = im2mla ν , (79) where aν := daj∇µ| gT kµν) + 1 l ∂µB κλ. (80) The mass term can be decomposed in the eigenbasis of ab as well and, subsequently, be formulated in terms of the geometric variables, LAA = −m (m̄−1)abXaνXbν , (81) where (m̄−1)ab := ∑∀ml 6=0 . (82) With the help of these relations and the results from the beginning of Sect. III we are now in the posi- tion to express the classical action in geometric vari- ables: The mass term is given in the previous expres- sion. It describes a Gaussian distribution of a com- posite current. The components of the current are su- perpositions of Abelian and non-Abelian contributions. This mixture is caused by the symmetry breaking pat- tern SU(2)L × U(1)Y → U(1)em which leaves unbroken U(1)em and not the U(1)Y which is a symmetry in the unbroken phase. The Abelian antisymmetric fields B0µν in LAbelBB are gauge invariant and we leave LAbelBB as de- fined in Eq. (74). In geometric variables LBB is given by Eq. (48) and LBF by Eq. (58). At the end the ki- netic term K(m) and the potential term V (m2) should be reinstated. Additional contributions from fluctuations give rise to an addend (on the level of the Lagrangian) proportional ln detm, wherem can be expressed in the new vari- ables, mjkµν = f abcdal µ gT lµν −m2 ljδklgµν . Repeating the entire calculation not in unitary gauge, but with explicit gauge scalars Φ, yields exactly the same result because the mass term and the saddle point con- dition change in unison, such that Eq. (79) is obtained again. This has already been demonstrated explicitly for a massive Yang–Mills theory just before Sect. III A. IV. SUMMARY We have discussed the formulation of massive gauge field theories in terms of antisymmetric tensor fields (Sect. II) and of geometric variables (Sect. III). The description in terms of an antisymmetric tensor field Baµν has the advantage that it transforms homogeneously under gauge transformations, whereas the usual gauge field Aaµ transforms inhomogeneously, which complicates a gauge-independent treatment of massive gauge field theories. In fact, the (Stückelberg-like) degrees of free- dom needed for a gauge-invariant formulation in terms of a Yang–Mills connections are directly absorbed in the antisymmetric tensor fields. No scalar field is re- quired in order to construct a gauge invariant massive theory in terms of the new variables. After recapitu- lating the massless case in Sect. IIA, we have treated the massive setting in Sect. IIB. After the fixed mass case, at the beginning of Sect. IIB, this section encom- passes also a position dependent mass (Sect. IIB1), that is the Higgs degree of freedom, and a non-diagonal mass term (Sect. IIB2). This is required for describing the Weinberg–Salam model. In this context, we have identi- fied the degrees of freedom which represent the different electroweak gauge bosons in the Baµν representation by a gauge-invariant eigenvector decomposition. The Abelian section (App. A) serves as basis for an easier understanding of some issues arising in the non- Abelian case, like for example vanishing conserved cur- rents. In that section we also address the massless limits of propagators in the Aµ and Bµν representations, respec- tively. We notice that while the limit is ill-defined for the Aµ fields it is well-defined for the Bµν fields. That is due to the consistent treatment of gauge degrees of freedom in the latter case. In Sect. III we continue with a description of massive gauge field theories in terms of geometric variables in four space-time dimensions and for two colours. Thereby we can eliminate the remaining degrees of freedom which are still encoded in the Baµν fields. After deriving the expressions for a fixed mass and in the presence of the Higgs degree of freedom, respectively, we also investigate the Weinberg–Salam model (Sect. III A). Acknowledgments DDD would like to thank Gerald Dunne and Stefan Hofmann for helpful, informative and inspiring discus- sions. Thanks are again due to Stefan Hofmann for read- ing the manuscript. APPENDIX A: ABELIAN 1. Massless The partition function of an Abelian gauge field theory without fermions is given by [dA] exp{i xL} (A1) with the Lagrangian density L = L0 := − 14g2FµνF µν (A2) and the field tensor Fµν := ∂µAν − ∂νAµ. (A3) g stands for the coupling constant. The transition to the first-order formalism can be performed just like in the non-Abelian case, which is treated in the main body of the paper. We find the partition function, [dA][dB] × × exp{i d4x[− 1 F̃µνB µν − g µν ]}. (A4) Here the antisymmetric tensor field Bµν , like the field tensor Fµν , is gauge invariant. The classical equations of motion are given by µν = 0 and g2Bµν = −F̃µν , (A5) which after elimination of Bµν reproduce the Maxwell equations one would obtain from Eq. (A2). Now we can formally integrate out the gauge field Aµ. As no gauge is fixed by the BF term because the Abelian field ten- sor Fµν is gauge invariant this gives rise to a functional δ distribution. This constrains the allowed field configu- rations to those for which the conserved current ∂µB̃ vanishes, [dB]δ(∂µB̃ µν) exp{i d4x[− g µν ]}. 2. Massive In the massive case the Lagrangian density becomes L = L0 + Lm, where Lm := m µ. First, we here repeat some steps carried out above in the non-Abelian case: We can directly write down the partition function in unitary gauge. Regauging like in Eq. (18) leads to [dA]′[dU ] exp(i d4x{− 1 [Aµ − iU †(∂µU)][Aµ − iU †(∂µU)]}).(A7) The corresponding gauge-invariant Lagrangian then reads, Lcl := − 14g2FµνF µν + m (DµΦ) †(DµΦ), (A8) with the constraint Φ†Φ = 1. Constructing a partition function in the first-order formalism from the previous Lagrangian yields, [dA][dΦ][dB] × × exp(i d4x{− 1 Bµν F̃ µν − g [Aµ − iΦ(∂µΦ†)][Aµ − iΦ(∂µΦ†)]}). (A9) The Φ fields can be absorbed entirely in a gauge- transformation of the gauge field Aµ. The integration over Φ decouples. This can also be seen by putting the parametrisation Φ = e−iθ into the previous equation and carrying out the [dA] integration, [dB][dθ] exp{i d4x[− g (∂κB̃ κµ)gµν(∂λB̃ λν)− (∂µθ)(∂κBκµ)]}. (A10) The only θ dependent term in the exponent is a total derivative and drops out, leading to a factorisation of the θ integral. A third way which yields the same final result, starts by integrating out the θ field first. This gives a transverse mass term∼ Aµ(gµν− ∂µ∂ν� )A ν . Integration overAµ then leads to the same result as before. Instead of a vanishing current ∂µB̃ µν like in the mass- less case, in the massive case the current has a Gaussian distribution. The distribution’s width is proportional to the mass of the gauge boson. m → 0 limit In the gauge-field representation the massless limit for the classical actions discussed above are smooth. In terms of the Bµν field the mass m ends up in the denom- inator of the corresponding term in the action. Together with the m dependent normalisation factors arising form the integrations over the gauge-field in the course of the derivation of the Bµν representation, however, the limit m → 0 still yields the m = 0 result for the partition function (A6). Still, it is known that the perturbative propagator for a massive photon is ill-defined if the mass goes to zero: Variation of the exponent of the Abelian massive parti- tion function in unitary gauge with respect to Aκ and Aλ gives the inverse propagator for the gauge fields, (G−1)κλ = [(p2 −m2phys)gκλ − pκpλ], (A11) which here is already transformed to momentum space. The corresponding equation of motion, (G−1)κλGλµ = gκµ, (A12) is solved by Gλµ = p2 −m2phys m2phys p2 −m2phys , (A13) with boundary conditions (an ǫ prescription) to be spec- ified and mphys := mg. This propagator diverges in the limit m→ 0. In the representation based on the antisymmetric ten- sor fields, variation of the exponent of the partition func- tion (A10) with respect to the fields B̃µν and B̃κλ yields the inverse propagator (G−1)µν|κλ = gµκgνλ − gνκgµλ + +m−2phys(∂ µ∂κgνλ − ∂ν∂κgµλ − − ∂µ∂λgνκ + ∂ν∂λgµκ), (A14) already expressed in momentum space. Variation with respect to B̃µν instead of Bµν corresponds only to a reshuffling of the Lorentz indices and gives an equiva- lent description. The antisymmetric structure of the in- verse propagator is due to the antisymmetry of B̃µν . The equation of motion is then given by (G−1)µν|κλGκλ|ρσ = gµρ g σ − gµσgνρ (A15) and solved by 2Gκλ|ρσ = (gκρgλσ − gκσgλρ)− p2 −m2phys ×(pκpρgλσ − pκpσgλρ − pλpρgκσ + pλpσgκρ). (A16) Here we observe that the limit m→ 0 is well-defined, 2Gκλ|ρσ m→0−−−→ gκρgλσ − gκσgλρ − (pκpρgλσ − pκpσgλρ − − pλpρgκσ + pλpσgκρ). (A17) This is due to the consistent treatment of the gauge de- grees of freedom in the second approach. [1] A. M. Polyakov, Nucl. Phys. B 164 (1980) 171; Yu. M. Makeenko and A. A. Migdal, Phys. Lett. B 88 (1979) 135 [Erratum-ibid. B 89 (1980) 437]; Nucl. Phys. B 188 (1981) 269 [Sov. J. Nucl. Phys. 32 (1980) 431; Yad. Fiz. 32 (1980) 838]. [2] Y. M. Cho, Phys. Rev. D 21 (1980) 1080; L. D. Faddeev and A. J. Niemi, Phys. Rev. Lett. 82 (1999) 1624 [arXiv:hep-th/9807069]; K.-I. Kondo, Phys. Rev. D 74 (2006) 125003 [arXiv:hep-th/0609166]. [3] S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38 (1977) 739 [Erratum-ibid. 38 (1977) 1376]. [4] J. F. Plebanski, J. Math. Phys. 12 (1977) 2511; J. C. Baez, Lect. Notes Phys. 543 (2000) 25 [arXiv:gr-qc/9905087]; T. Thiemann, Lect. Notes Phys. 631 (2003) 41 [arXiv:gr-qc/0210094]. [5] S. Deser and C. Teitelboim, Phys. 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0704.1830

An Electromagnetic Calorimeter for the JLab Real Compton Scattering Experiment

An Electromagnetic Calorimeter for the JLab Real Compton Scattering Experiment D. J. Hamiltona,∗, A. Shahinyanb, B. Wojtsekhowskic, J. R. M. Annanda, T.-H. Changd, E. Chudakovc, A. Danagouliand, P. Degtyarenkoc, K. Egiyanb,1, R. Gilmane, V. Gorbenkof, J. Hinesg,2, E. Hovhannisyanb,1, C. E. Hyde-Wrighth, C.W. de Jagerc, A. Ketikyanb, V. H. Mamyanb,c,3, R. Michaelsc, A. M. Nathand, V. Nelyubini, I. Rachekj, M. Roedelbromd, A. Petrosyanb,1, R. Pomatsalyukf, V. Popovc, J. Segalc, Y. Shestakovj, J. Templong,4, H. Voskanyanb aUniversity of Glasgow, Glasgow G12 8QQ, Scotland, UK bYerevan Physics Institute, Yerevan 375036, Armenia cThomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA dUniversity of Illinois, Urbana-Champaign, IL 61801, USA eRutgers University, Piscataway, NJ 08855, USA fKharkov Institute of Physics and Technology, Kharkov 61108, Ukraine gThe University of Georgia, Athens, GA 30602, USA hOld Dominion University, Norfolk, VA 23529, USA i St. Petersburg Nuclear Physics Institute, Gatchina, 188350, Russia jBudker Institute for Nuclear Physics, Novosibirsk 630090, Russia Abstract A lead-glass hodoscope calorimeter that was constructed for use in the Jefferson Lab Real Compton Scattering experiment is described. The detector provides a measurement of the coordinates and the energy of scattered photons in the GeV energy range with resolutions of 5 mm and 6%/ Eγ [GeV]. Features of both the detector design and its performance in the high luminosity environment during the experiment are presented. Keywords: Calorimeters, Čerenkov detectors PACS: 29.40Vj, 29.40.Ka ‘ 1. Introduction A calorimeter was constructed as part of the instrumentation of the Jefferson Lab (JLab) Hall A experiment E99-114, “Ex- clusive Compton Scattering on the Proton” [1], the schematic layout for which is shown in Fig. 1. The study of elastic pho- ton scattering provides important information about nucleon structure, which is complementary to that obtained from elastic electron scattering [2]. Experimental data on the Real Comp- ton Scattering (RCS) process at large photon energies and large scattering angles are rather scarce, due mainly to the absence of high luminosity facilities with suitable high-resolution photon detectors. Such data are however crucial, as the basic mecha- nism of the RCS reaction is the subject of active debate [3, 4, 5]. The only data available before the JLab E99-114 experiment were obtained at Cornell about 30 years ago [6]. The construction of the CEBAF (Continuous Electron Beam Accelerator Facility) accelerator has led to an extension of many experiments with electron and photon beams in the GeV energy range and much improved precision. This is the result of a number of fundamental improvements to the electron beam, ∗Tel.: +44-141-330-5898; Fax: +44-141-330-5889 Email address: [email protected] (D. J. Hamilton) 1deceased 2present address: Applied Biosystems/MDS, USA 3present address: University of Virginia, Charlottesville, VA 22901, USA 4present address: NIKHEF, 1009 DB Amsterdam, The Netherlands ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� ��������� Beam dump Cable lines Calorimeter Front−end electronics Beam line Magnet Radiator and target Figure 1: Layout of the RCS experiment in Hall A. An electron beam incident on a radiator produces an intense flux of high energy photons. including a 100% duty cycle, low emittance and high polariza- tion, in addition to new dedicated target and detector systems. The CEBAF duty factor provides an improvement of a factor of 15 compared to the best duty factor of a beam extracted from a synchrotron, at a similar instantaneous rate in the detectors. In 1994 work began on the development of a technique for an RCS experiment at JLab, leading in 1997 to the instigation of a large-scale prototyping effort. The results of the subsequent test runs in 1998 and 1999 [7] provided sufficient information Preprint submitted to NIM A August 16, 2018 http://arxiv.org/abs/0704.1830v3 Figure 2: A photograph of the experimental set-up for E99-114, showing the calorimeter (center) and part of the proton spectrometer (rear). for the final design of the apparatus presented in the present article. The fully realized physics experiment took place in 2002 (see Fig. 1) at a photon-nucleon luminosity which was a factor of 1300 higher than in the previous Cornell experi- ment. The experimental technique involves utilizing a mixed electron-photon beam which is incident on a liquid hydrogen target and passes to a beam dump. The scattered photons are detected in the calorimeter, while the recoiling protons are de- tected in a high resolution magnetic spectrometer (HRS-L). A magnet between the hydrogen target and the calorimeter de- flects the scattered electrons, which then allows for clean sep- aration between Compton scattering and elastic e-p scattering events. The Data Acquisition Electronics (DAQ) is shielded by a 4 inch thick concrete wall from the beam dump and the target. Figure 2 shows a photograph of the experimental set-up with the calorimeter in the center. The experiment relied on a proton-photon time coincidence and an accurate measurement of the proton-photon kinematic correlation for event selection. The improvement in the event rate over the previous measurement was achieved through the use of a mixed electron-photon beam, which in turn required a veto detector in front of the calorimeter or the magnetic de- flection of the scattered electron [1]. In order to ensure redun- dancy and cross-checking, both a veto and deflection magnet were designed and built. The fact that a clean photon beam was not required meant that the photon radiator could be situated very close to the hydrogen target, leading to a much reduced background near the beam line and a dramatic reduction of the photon beam size. This small beam size in combination with the large dispersion in the HRS-L proton detector system [8] resulted in very good momentum and angle resolution for the recoiling proton without the need for a tracking detector near the target, where the background rate is high. Good energy and coordinate resolutions were key features of the photon detector design goals, both of which were sig- nificantly improved in the JLab experiment as compared to the Cornell one. An energy resolution of at least 10% is required to separate cleanly RCS events from electron bremsstrahlung and neutral pion events. In order to separate further the background from neural pion photo-production, which is the dominant com- ponent of the high-energy background in this measurement, a high angular resolution between proton and photon detectors is crucial. This was achieved on the photon side by constructing a highly segmented calorimeter of 704 channels. The RCS exper- iment was the first instance of a calorimeter being operated at an effective electron-nucleon luminosity of 1039 cm2/s [9, 10] (a 40 µA electron beam on a 6% Cu radiator upstream of a 15 cm long liquid hydrogen target). It was observed in the test runs that the counting rate in the calorimeter fell rapidly as the threshold level was increased, which presented an opportunity to maintain a relatively low trigger rate even at high luminosity. However, on-line use of the calorimeter signal required a set of summing electronics and careful equalizing and monitoring of the individual channel outputs during the experiment. As the RCS experiment represented the first use of such a calorimeter at very high luminosity, a detailed study of the calorimeter performance throughout the course of the experi- ment has been conducted. This includes a study of the rela- tionship between luminosity, trigger rate, energy resolution and ADC pedestal widths. An observed fall-off in energy resolu- tion as the experiment progressed allowed for characterization of radiation damage sustained by the lead-glass blocks. It was possible to mitigate this radiation damage after the experiment by annealing, with both UV curing and heating proving effec- tive. We begin by discussing the various components which make up the calorimeter and the methods used in their construction. This is followed by a description of veto hodoscopes which were used for particle identification purposes. An overview of the high-voltage and data acquisition systems is then pre- sented, followed, finally, by a discussion on the performance of the calorimeter in the unique high-luminosity environment during the RCS experiment. 2. Calorimeter The concepts and technology associated with a fine- granularity lead-glass Čerenkov electromagnetic calorimeter (GAMS) were developed by Yu. Prokoshkin and collaborators at the Institute of High Energy Physics (IHEP) in Serpukhov, Russia [11]. The GAMS type concept has since been employed for detection of high-energy electrons and photons in several experiments at JLab, IHEP, CERN, FNAL and DESY (see for example [12]). Many of the design features of the calorimeter presented in this article are similar to those of Serpukhov. A schematic showing the overall design of the RCS calorimeter can be seen in Fig. 3. The main components are: • the lead-glass blocks; • a light-tight box containing the PhotoMultiplier Tubes (PMTs); • a gain-monitoring system; • a doubly-segmented veto hodoscopes; • the front-end electronics; • an elevated platform; • a lifting frame. The calorimeter frame hosts a matrix of 22×32 lead-glass blocks together with their associated PMTs and High Voltage (HV) dividers. Immediately in front of the lead-glass blocks is a sheet of UltraViolet-Transmitting (UVT) Lucite, which is used to distribute calibration light pulses for gain-monitoring purposes uniformly among all 704 blocks. The light-tight box provides protection of the PMTs from ambient light and con- tains an air-cooling system as well as the HV and signal cable systems. Two veto hodoscopes, operating as Čerenkov coun- ters with UVT Lucite as a radiator, are located in front of the calorimeter. The front-end electronics located a few feet be- hind the detector were assembled in three relay racks. They are comprised of 38 analog summers, trigger logic and patch pan- els. The elevated platform was needed to bring the calorimeter to the level of the beam line, while the lifting frame was used to re-position the calorimeter in the experimental hall by means of an overhead crane. This procedure, which took on average around two hours, was performed more than 25 times during the course of the experiment. platform �������� Calorimeter Elevation route Cable boxframe Lifting Light−tight Front−end electronics system monitoring Beam line Veto counters ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ Figure 3: Schematic side view of the RCS calorimeter detector system. 2.1. Calorimeter Design The main frame of the calorimeter is made of 10 inch wide steel C-channels. A thick flat aluminum plate was bolted to the bottom of the frame, with a second plate installed vertically and aligned to 90◦ with respect to the first one by means of align- ment screws (see Fig. 4). Another set of screws, mounted inside and at the top of the main frame on the opposite side of the ver- tical alignment plate, was used to compress all gaps between the lead-glass modules and to fix their positions. The load was ap- plied to the lead-glass blocks through 1 inch× 1 inch × 0.5 inch plastic plates and a 0.125 inch rubber pad. In order to further as- sist block alignment, 1 inch wide stainless steel strips of 0.004 inch thickness running from top to bottom of the frame were inserted between every two columns of the lead-glass modules. 2.1.1. Air Cooling All PMTs and HV dividers are located inside a light-tight box, as shown in Fig. 5. As the current on each HV divider is 1 mA, simultaneous operation of all PMTs would, without cooling, lead to a temperature rise inside the box of around 50-70◦C. An air-cooling system was developed to prevent the PMTs from overheating, and to aid the stable operation of the calorimeter. The air supply was provided by two parallel oil- less regenerative blowers of R4110-2 type5, which are capable 5Manufactured by S&F Supplies, Brooklyn, NY 11205, USA. Plastic plate ������ �������������� ������ ������ ������ ��������������� ����� ����� ����� �������������� ������ ������ ������ ��������������� ����� ����� ����� �������������������������������� ������������������������ �������������������� ������������������ �������������������������������������������������������� Plastic Alignment screw alignment Vertical plate Lead−glass force Compacting Horizontal alignment plate Main frame SS strip Rubber ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������ ������������ ������������ ������������ ������������ �������������������������������������������������������������������� �������� �������� ������������������������������������������ ������������������������������������������ ������������������������������������������ ������������������������������������������ Figure 4: Front cross-section of the calorimeter, showing the mechanical com- ponents. of supplying air at a maximum pressure of 52 inches water and a maximum flow of 92 CFM. The air is directed toward the HV divider via vertical collector tubes and numerous outlets. When the value on any one of the temperature sensors installed in sev- eral positions inside the box exceeds a preset limit, the HV on the PMTs is turned off by an interlock system. The air line is equipped with a flow switch of type FST-321-SPDT which was included in the interlock system. The average temperature in- side the box during the entire experimental run did not exceed the preset limit of 55◦C. 2.1.2. Cabling System A simple and reliable cabling system is one of the key fea- tures of multichannel detectors, with easy access to the PMTs and HV dividers for installation and repair being one of the key features. The cabling system includes: • 1 foot long HV and signal pig-tails soldered to the HV divider; • patch panels for Lemo and HV connectors; • 10 feet long cables from those patch panels to the front- end electronics and the HV distribution boxes; • the HV distribution boxes themselves; • BNC-BNC patch panels for the outputs of the front-end modules; �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� �������������������� ������������������ ������������������ ������������������ ������������������ Patch panel Air outlet Pressure Gauge Flow switch Blower R4110−2 Cable bundle Lead Glass Air supplyPMT tube HV, Signal cables Figure 5: A schematic showing the calorimeter air cooling and cabling systems. • BNC-BNC patch panels on the DAQ side for the analog signals; • BNC-Lemo patch panels on the DAQ side for the veto- counter lines. Figure 6 shows the cabling arrangement inside the PMT box. The patch panels, which are custom-built and mounted on the air supply tubes, have the ability to swing to the side in order to allow access to the PMTs and the HV dividers. The box has two moving doors, the opening of which leads to activation of an in- terlock system connected to the HV supply. In order to reduce the diameter of the cable bundle from the PMT box, RG-174 cable (diameter 0.1 inch) was used for the PMT signals, and a twisted pair for the HV connection (two individually insulated inner 26 AWG conductors with an overall flame-retardant PVC jacket, part number 001-21803 from the General Wire Product company). The box patch panels used for the HV lines each convert 24 of the above twisted pairs (single HV line) to the multi-wire HV cables (the part 001-21798 made by General Wire Product), which run to the HV power supply units located in the shielded area near DAQ racks. 2.2. Lead-Glass Counter The basic components of the segmented calorimeter are the TF-1 lead-glass blocks and the FEU 84-3 PMTs. In the 1980s the Yerevan Physics Institute (YerPhI) purchased a consignment of TF-1 lead-glass blocks of 4 cm × 4 cm ×40 cm and FEU 84-3 PMTs of 34 mm diameter (with an active photo-cathode diam- eter of 25 mm) for the construction of a calorimeter to be used Lemo connector from HV divider HV connector Patch panel Al T−channel HV cables Signal cables HV&Sig. cables Figure 6: A photograph of the cabling inside the PMT box. in several experiments at the YerPhI synchrotron. In January of 1998 the RCS experiment at JLab was approved and soon after these calorimeter components were shipped from Yerevan to JLab. This represented the YerPhi contribution to the exper- iment, as the properties of the TF-1 lead-glass met the require- ments of the experiment in terms of photon/electron detection with reasonable energy and position resolution and radiation hardness. The properties of TF-1 lead-glass [12, 13] are given in Table 1. Table 1: Important properties of TF-1 lead-glass. Density 3.86 gcm−3 Refractive Index 1.65 Radiation Length 2.5 cm Moliére Radius 3.50 cm Critical Energy 15 MeV All PMTs had to pass a performance test with the follow- ing selection criteria: a dark current less than 30 nA, a gain of 106 with stable operation over the course of the experiment (2 months), a linear dependence of the PMT response (within 2 %) on an incident optical pulse of 300 to 30000 photons. 704 PMTs out of the 900 available were selected as a result of these per- formance tests. Furthermore, the dimensional tolerances were checked for all lead-glass blocks, with strict requirements de- manded on the length (400±2 mm) and transverse dimensions (40±0.2 mm). 2.2.1. Design of the Counter In designing the individual counters for the RCS calorime- ter, much attention was paid to reliability, simplicity and the possibility to quickly replace a PMT and/or HV divider. The individual counter design is shown in Fig. 7. A titanium flange is glued to one end of the lead-glass block by means of EPOXY- 190. Titanium was selected because its thermal expansion coef- ficient is very close to that of the lead glass. The PMT housing, which is bolted to the Ti flange, is made of an anodized Al flange and an Al tube. The housing contains the PMT and a µ- metal shield, the HV divider, a spring, a smaller Al tube which transfers a force from the spring to the PMT, and a ring-shaped spring holder. The optical contact between the PMT and the lead-glass block is achieved by use of optical grease, type BC- 630 (Bicron), which was found to increase the amount of light detected by the PMT by 30-40% compared to the case without grease. The PMT is pressed to the lead-glass block by means of a spring, which pushes the HV base with a force of 0.5-1 lbs. Such a large force is essential for the stability of the optical contact over time at the elevated temperature of the PMTs. The glue-joint between the lead glass and the Ti flange, which holds that force, failed after several months in a significant fraction (up to 5%) of the counters. An alternative scheme of force compensation was realized in which the force was applied to the PMT housing from the external bars placed horizontally be- tween the PMT housing and the patch-panel assembly. Each individual lead-glass block was wrapped in aluminized Mylar film and black Tedlar (a polyvinyl fluoride film from DuPont) for optimal light collection and inter-block isolation. Single- side aluminized Mylar film was used with the Al layer on the opposite side of the glass. Such an orientation of the film limits the diffusion of Al atoms into the glass and the non-oxidized surface of aluminum, which is protected by Mylar, provides a better reflectivity. The wrapping covers the side surface of the lead-glass block, leaving the front face open for the gain mon- itoring. The signal and the HV cables are each one foot long. They are soldered to the HV divider on one end and terminated with Lemo c©00 and circular plastic connectors (cable mount re- ceptacle from Hypertronics) on the other end. The cables leave the PMT housing through the open center of the spring holder. 2.2.2. HV Divider At the full luminosity of the RCS experiment (0.5 × 1039 cm2/s) and at a distance of 6 m from the target the back- ground energy load per lead-glass block reaches a level of 108 MeVee (electron equivalent) per second, which was found from the average value of anode current in the PMTs and the shift of the ADC pedestals for a 150 ns gate width. At least 30% of this energy flux is due to high energy particles which define the counting rate. The average energy of the signals for that component, according to the observed rate distribution, is in the range of 100-300 MeVee, depending on the beam en- ergy and the detector angle. The corresponding charge in the PMT pulse is around 5-15 pC collected in 10-20 ns. The elec- tronic scheme and the selected scale of 1 MeV per ADC chan- nel (50 fC) resulted in an average anode current of 5 µA due to background load. A high-current HV base (1 mA) was there- fore chosen to reduce the effect of the beam intensity variation on the PMT amplitude and the corresponding energy resolu- tion to the level of 1%. The scheme of the HV base is shown in Fig. 8. According to the specification data for the FEU 84- 3 PMTs the maximum operation voltage is 1900 V. Therefore a nominal voltage value of 1800 V and a current value in the voltage divider of 1 mA were chosen. ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������� ������������������������������������������������������������������ grease metal Al tube Al tube housing Ti flange Lead−glass Al flange Al mylarTedlar FEU 84−3 Optical Spring Signal Glue joint Spring holder �������������������������������������������������������������� ������������������������������� ������������������������������� ������������������������������� ������������������������������� Figure 7: Schematic of the lead-glass module structure. 7.5ns VME Scalers Fastbus TDC Fastbus ADC Fastbus ADC Fastbus ADC DAQ Platform100m cable lines Calorimeter Platform LG PMT signals 600 central channels LG PMT signals 104 rim channels SUM−32 analog signals 56 channels SUM−32 logic signals 56 channels LRS428F PS706D LRS3412 PHOTON TRIGGER To Trigger Superviser2ns6ns Sum−8 PS755 PS757 Figure 9: A block diagram of the calorimeter electronics. R1 R2 R9 R10 D9 D10 R11 R12 Figure 8: Schematic of the high-voltage divider for the FEU 84-3 PMT. The values of the resistors are R(1 − 10) = 100 kΩ, R11 = 130 kΩ,R12 = 150 kΩ, R13 = 200 kΩ, R14 = 150 kΩ, R15 = 10 kΩ,R16 = 10 MΩ, R17 = 4 kΩ. The capacitance C is 10 nF. 2.3. Electronics The calorimeter electronics were distributed over two loca- tions; see the block diagram in Fig. 9. The first group of modules (front-end) is located in three racks mounted on the calorimeter platform in close vicinity to the lead-glass blocks. These are the trigger electronics modules which included a mix of custom-built and commercially available NIM units: • 38 custom-built analog summing modules used for level- one signal summing 6; • 14 linear fan-in/fan-out modules (LeCroy model 428F) for a second-level signal summation; • 4 discriminator units (Phillips Scientific model 706); • a master OR circuit, realized with Phillips Scientific logic units (four model 755 and one model 757 modules); • several additional NIM modules used to provide auxiliary trigger signals for the calorimeter calibration with cosmics and for the PMT gain-monitoring system. The second group of electronic modules, which include charge and time digitizers as well as equipment for the Data Acquisition, High Voltage supply and slow-control systems, is 6This module was designed by S. Sherman, Rutgers University. placed behind a radiation-protecting concrete wall. All 704 lead-glass PMT signals and 56 SUM-32 signals are digitized by LeCroy 1881M FastBus ADC modules. In addition, 56 SUM- 32 discriminator pulses are directed to scalers and to LeCroy 1877 FastBus TDCs. Further detailed information about the electronics is presented in Section 5. The signals between these locations are transmitted via patch-panels and coaxial cables, consisting of a total number of 1040 signal and 920 HV lines. The length of the signal ca- bles is about 100 m, which serve as delay lines allowing the timing of the signals at the ADC inputs to be properly set with respect to the ADC gate, formed by the experiment trigger. The width of the ADC gate (150 ns) was made much wider than the duration of PMT pulse in order to accommodate the wider pulses caused by propagation in the 500 ns delay RG-58 signal cables. The cables are placed on a chain of bogies, which per- mits the calorimeter platform to be moved in the experimental hall without disconnecting the cables. This helped allow for a quick change of kinematics. 2.3.1. Trigger Scheme The fast on-line photon trigger is based on PMT signals from the calorimeter counters. The principle of its operation is a sim- ple constant-threshold method, in which a logic pulse is pro- duced if the energy deposition in the calorimeter is above a given magnitude. Since the Molière radius of the calorimeter material is RM ≈ 3.5 cm, the transverse size of the electro- magnetic shower in the calorimeter exceeds the size of a single lead-glass block. This enables a good position sensitivity of the device, while at the same time making it mandatory for the trigger scheme to sum up signals from several adjacent coun- ters to get a signal proportional to the energy deposited in the calorimeter. From an electronics point of view, the simplest realization of such a trigger would be a summation of all blocks followed by a single discriminator. However, such a design is inappropri- ate for a high-luminosity experiment due to the very high back- ground level. The opposing extreme approach would be to form a summing signal for a small group including a single counter hit and its 8 adjacent counters, thus forming a 3 × 3 block struc- ture. This would have to be done for every lead-glass block, ex- cept for those at the calorimeter’s edges, leading to an optimal signal-to-background ratio, but an impractical 600 channels of analog splitter→analog summer→discriminator circuitry fol- lowed by a 600-input fan-in module. The trigger scheme that was adopted and is shown in Fig. 10 is a trade-off between the above extreme cases. This scheme contains two levels of ana- log summation followed by appropriate discriminators and an OR-circuit. It involved the following functions: • the signals from each PMT in the 75 2×4 sub-arrays of ad- jacent lead-glass blocks, excluding the outer-most blocks, are summed in a custom-made analog summing module to give a SUM-8 signal (this module duplicates the signals from the PMTs with less then 1% integral nonlinearity); • these signals, in turn, are further summed in overlapping groups of four in LeCroy LRS428F NIM modules to pro- duce 56 SUM-32 signals. Thus, each SUM-32 signal is proportional to the energy deposition in a subsection of the calorimeter of 4 blocks high and 8 blocks wide, i.e. 16×32 cm2. Although this amounts to only 5% of the calorime- ter acceptance, for any photon hit (except for those at the edges) there will be at least one segment which contains the whole electromagnetic shower. • the SUM-32 signals are sent to constant-threshold discrim- inators, from which the logical pulses are OR-ed to form the photon singles trigger T1 (see Section 5). The discrim- inator threshold is remotely adjustable, and was typically set to around half of the RCS photon energy for a given kinematic setting. ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������ ������ ������ ������ 141312 56 57 58 59 60 73 75747271 66 67 07 08 09 01 0403 S06 S07 S08 S50 S51 S11S09 S10 S03S01 Figure 10: The principle of two-level summation of signals for the hardware trigger: 75 eight-block sub-arrays and 56 overlapping groups of four sub-arrays forming SUM-32 signals labeled as S01-S56. In the highlighted example the sums 02,03,07, and 08 form a S02 signal. 2.4. Gain Monitoring System The detector is equipped with a system that distributes light pulses to each calorimeter module. The main purpose of this system is to provide a quick way to check the detector opera- tion and to calibrate the dependence of the signal amplitudes on the applied HV. The detector response to photons of a given en- ergy may drift with time, due to drifts in the PMT gains and to changes in the glass transparency caused by radiation damage. For this reason, the gain monitoring system also allowed mea- surements of the relative gains of all detector channels during the experiment. In designing the gain-monitoring system ideas developed for a large lead-glass calorimeter at BNL[14] were used. The system includes two components: a stable light source and a system to distribute the light to all calorimeter modules. The light source consists of an LN300 nitrogen laser7, which provides 5 ns long, 300 µJ ultraviolet light pulses of 337 nm 7 Manufactured by Laser Photonics, Inc, FL 32826, USA. wavelength. The light pulse coming out of the laser is atten- uated, typically by two orders of magnitude, and monitored using a silicon photo-diode S1226-18BQ8 mounted at 150◦ to the laser beam. The light passes through an optical filter, sev- eral of which of varying densities are mounted on a remotely controlled wheel with lenses, before arriving at a wavelength shifter. The wavelength shifter used is a 1 inch diameter semi- spherical piece of plastic scintillator, in which the ultraviolet light is fully absorbed and converted to a blue (∼ 425 nm) light pulse, radiated isotropically. Surrounding the scintillator about 40 plastic fibers (2 mm thick and 4 m long) are arranged, in or- der to transport the light to the sides of a Lucite plate. This plate is mounted adjacent to the front face of the lead-glass calorime- ter and covers its full aperture (see Fig.11). The light passes through the length of the plate, causing it to glow due to light scattering in the Lucite. Finally, in order to eliminate the cross- talk between adjacent counters a mask is inserted between the Lucite plate and the detector face. This mask, which reduces the cross-talk by at least a factor of 100, is built of 12.7 mm thick black plastic and contains a 2 cm × 2 cm hole in front of each module. Light distributor Optic fibers Lead−glass modules Light source Figure 11: Schematic of the Gain-monitoring system. Such a system was found to provide a rather uniform light collection for all modules, and proved useful for detector test- ing and tuning, as well as for troubleshooting during the exper- iment. However, it was found that monitoring over extended periods of time proved to be less informative than first thought. The reason for this is due to the fact that the main radiation dam- age to the lead-glass blocks occurred at a depth of about 2-4 cm 8Manufactured by Hamamatsu Photonics, Hamamatsu, Japan. from the front face. The monitoring light passes through the damaged area, while an electromagnetic shower has its maxi- mum at a depth of about 10 cm. Therefore, as a result of this radiation damage the magnitude of the monitoring signals drops relatively quicker than the real signals. Consequently, the re- sulting change in light-output during the experiment was char- acterized primarily through online analysis of dedicated elastic e-p scattering runs. This data was then used for periodic re- calibration of the individual calorimeter gains. 3. Veto Hodoscopes In order to ensure clean identification of the scattered pho- tons through rejection of high-energy electrons in the compli- cated environment created by the mixed electron-photon beam, a veto detector which utilizes UVT Lucite as a Čerenkov radi- ator was developed. This veto detector proved particularly use- ful for low luminosity runs, where its use made it possible to take data without relying on the deflection magnet (see Fig. 1). The veto detector consists of two separate hodoscopes located in front of the calorimeter’s gain monitoring system. The first hodoscope has 80 counters oriented vertically, while the second has 110 counters oriented horizontally as shown in Fig. 12. The segmentation scheme for the veto detector was chosen so that it was consistent with the position resolution of the lead-glass calorimeter. An effective dead time of an individual counter is about 100 ns due to combined double-pulse resolution of the PMT, the front-end electronics, the TDC, and the ADC gate- width. ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� ������������������������������������� PMT XP2971 Honeycomb plate Lucite radiator Figure 12: Cut-off view of the “horizontal” veto hodoscope. Each counter is made of a UVT Lucite bar with a PMT glued directly to one of its end, which can be seen in Fig. 13. The Lu- cite bar of 2×2 cm2 cross section was glued to a XP2971 PMT and wrapped in aluminized Mylar and black Tedlar. Counters are mounted on a light honeycomb plate via an alignment groove and fixed by tape. The counters are staggered in such a way so as to allow for the PMTs and the counters to overlap. The average PMT pulse generated by a high-energy electron corresponds to 20 photo-electrons. An amplifier, powered by the HV line current, was added to the standard HV divider, in order that the PMT gain could be reduced by a factor of 10 XP2971 �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� Lucite radiator Figure 13: Schematic of the veto counter. [15, 16]. After gain-matching by using cosmic ray data a good rate uniformity was achieved, as can be seen in the experimental rate distribution of the counters shown in Fig. 14. The regular variation in this distribution reflects the shielding effect result- ing from the staggered arrangement of the counters. A signif- 0 25 50 75 100 125 0 20 40 60 80 Horizontal Veto Detector Vertical Veto Detector Counter number Figure 14: The counting rate in the veto counters observed at luminosity of 1.5 · 1038 cm−2/s. icant reduction of the rate (by a factor of 5) was achieved by adding a 2 inch polyethylene plate in front of the hodoscopes. Such a reduction as a result of this additional shielding is con- sistent with the observed variation of the rate (see Fig. 14) and indicates that the typical energy of the dominant background is around a few MeV. The veto plane efficiency measured for different beam intensities is shown in Table 2. It drops signif- icantly at high rate due to electronic dead-time, which limited the beam intensity to 3-5 µA in data-taking runs with the veto. An analysis of the experimental data with and without veto de- tectors showed that the deflection of the electrons by the magnet Table 2: The efficiency of the veto hodoscopes and the rate of a single counter at different beam currents. The detector was installed at 30◦ with respect to the beam at a distance 13 m from the target. The radiator had been removed from the beam path, the deflection magnet was off and the 2 inch thick polyethylene protection plate was installed. Run Beam Rate of the Efficiency Efficiency current counter V12 horizontal vertical [µA] [MHz] hodoscope hodoscope 1811 2.5 0.5 96.5% 96.8% 1813 5.0 1.0 95.9% 95.0% 1814 7.5 1.5 95.0% 94.0% 1815 10. 1.9 94.4% 93.0% 1816 14. 2.5 93.4% 91.0% 1817 19 3.2 92.2% 89.3% provided a sufficiently clean photon event sample. As a result the veto hodoscopes were switched off during most high lu- minosity data-taking runs, although they proved important in analysis of low luminosity runs and in understanding various aspects of the experiment. 4. High Voltage System Each PMT high-voltage supply was individually monitored and controlled by the High Voltage System (HVS). The HVS consists of six power supply crates of LeCroy type 1458 with high-voltage modules of type 1461N, a cable system, and a set of software programs. The latter allows to control, monitor, download and save the high-voltage settings and is described below in more detail. Automatic HV monitoring provides an alarm feature with a verbal announcement and a flashing signal on the terminal. The controls are implemented over an Ethernet network using TCP/IP protocol. A Graphical User Interface (GUI) running on a Linux PC provides access to all features of the LeCroy system, loading the settings and saving them in a file. A sample distribution of the HV settings is shown in Fig. 15. The connections between the outputs of the high-voltage modules and the PMT dividers were arranged using 100 m long multi-wire cables. The transition from the individual HV sup- ply outputs to a multi-wire cable and back to the individual PMT was arranged via high-voltage distribution boxes that are located inside the DAQ area and front-end patch panels outside the PMT box. These boxes have input connectors for individual channels on one side and two high-voltage multi-pin connec- tors (27 pins from FISCHER part number D107 A051-27) on the other. High-voltage distribution boxes were mounted on the side of the calorimeter stand and on the electronics rack. 5. Data Acquisition System Since the calorimeter was intended to be used in Hall A at JLab together with the standard Hall A detector devices, the Data Acquisition System of the calorimeter is part of the standard Hall A DAQ system. The latter uses CODA (CE- BAF On-line Data Acquisition system) [17] developed by the -1800 -1700 -1600 -1500 -1400 -1300 -1200 -1100 -1000 At the end of the experiment High Voltage [V] -1800 -1700 -1600 -1500 -1400 -1300 -1200 -1100 -1000 At the start of the experiment Figure 15: The HV settings for the calorimeter PMTs. JLab data-acquisition group. The calorimeter DAQ includes one Fastbus crate with a single-board VME computer installed using a VME-Fastbus interface and a trigger supervisor mod- ule [18], which synchronizes the read-out of all the information in a given event. The most important software components are a Read-Out Controller (ROC), which runs on the VME com- puter under the VxWorks OS, and an Event Builder and Event Recorder which both run on a Linux workstation. For a detailed description of the design and operation of the Hall A DAQ sys- tem see [8] and references therein. All 704 PMT signals and 56 SUM-32 signals are digitized by LeCroy 1881M FastBus ADC modules. The 56 SUM-32 dis- criminator pulses are also read-out by scalers and LeCroy 1877 FastBus TDCs. During the RCS experiment the calorimeter was operating in conjunction with one of the High Resolution Spectrometers (HRS), which belong to the standard Hall A de- tector equipment [8]. The Hall A Data Acquisition System is able to accumulate data involving several event types simulta- neously. In the RCS experiment there were 8 types of trigger signals and corresponding event types. Trigger signals from the HRS are generated by three scintillator planes: S0, S1 and S2 (see Fig. 8 in [8]). In the standard configuration the main sin- gle arm trigger in the spectrometer is formed by a coincidence of signals from S1 and S2. An alternative trigger, logically de- scribed by (S0 AND S1) OR (S0 AND S2), is used to measure the trigger efficiency. In the RCS experiment one more proton arm trigger was used, defined as being a single hit in the S0 plane. As this is the fastest signal produced in the proton arm, it was better suited to form a fast coincidence trigger with the photon calorimeter. The logic of the Photon Arm singles trigger was described in detail in Section 2.3. Besides this singles trigger there are two auxiliary triggers that serve to monitor the calorimeter blocks and electronics. The first is a photon arm cosmics trigger, which was defined by a coincidence between signals from two plas- tic scintillator paddles, placed on top and under the bottom of photo tube Laser calibration trigger Cosmics trigger top paddle bottom paddle 1024 Hz Pulser Retiming Retiming ADC gates TDC Common Stop Proton Arm ADC gates TDC Common Stop Photon Arm Proton Arm Main photon trigger Photon Arm Figure 16: Schematic diagram of the DAQ trigger logic. the calorimeter. The other trigger is the light-calibration (laser) trigger which was used for gain monitoring purposes. The two-arm coincidence trigger is formed by a time overlap of the main calorimeter trigger and the signal from the S0 scin- tillator plane in the HRS. The width of the proton trigger pulse is set to 100 ns, while the photon trigger pulse, which is delayed in a programmable delay line, is set to 10 ns. As a result, the co- incidence events are synchronized with the photon trigger, and a correct timing relation between trigger signals from two arms is maintained for all 25 kinematic configurations of the RCS experiment. Finally, a 1024 Hz puls generator signal forms a pulser trigger, which was used to measure the dead time of the electronics. All 8 trigger signals are sent to the Trigger Supervisor mod- ule which starts the DAQ readout. Most inputs of the Trigger Supervisor can be individually pre-scaled. Triggers which are accepted by the DAQ are then re-timed with the scintillators of a corresponding arm to make gates for ADCs and TDCs. This re-timing removes trigger time jitter and ensures the timing is independent of the trigger type. Table 3 includes information on the trigger and event types used in the RCS experiment and shows typical pre-scale factors used during the data-taking. A schematic diagram of the overall RCS experiment DAQ trigger logic is shown in Fig.16. 6. Calorimeter Performance The calorimeter used in the RCS experiment had three related purposes. The first purpose is to provide a coincidence trigger Table 3: A list of triggers used in the RCS experiment. Typical pre-scale factors which were set during a data-taking run (run #1819) are shown. Trigger Trigger Description pre-scale ID factor T1 Photon arm singles trigger 100,000 T2 Photon arm cosmics trigger 100,000 T3 Main Proton arm trigger: (S1 AND S2) T4 Additional Proton arm trigger: (S0 AND S1) OR (S0 AND S2) T5 Coincidence trigger 1 T6 Calorimeter light-calibration trigger T7 Signal from the HRS S0 scintil- lator plane 65,000 T8 1024 Hz pulser trigger 1,024 Coincidence Time (ns) 10 20 30 40 Figure 17: The time of the calorimeter trigger relative to the recoil proton trig- ger for a production run in kinematic 3E at maximum luminosity (detected Eγ = 1.31 GeV). The solid curve shows all events, while the dashed curve shows events with a cut on energy in the most energetic cluster > 1.0 GeV. signal for operation of the DAQ. Fig. 17 shows the coincidence time distribution, where one can see a clear relation between en- ergy threshold and time resolution. The observed resolution of around 8 ns (FWHM) was sufficient to identify cleanly coinci- dence events over the background, which meant that no off-line corrections were applied for variation of the average time of in- dividual S UM − 32 summing modules. The second purpose is determination of the energy of the scattered photon/electron to within an accuracy of a few percent, while the third is rea- sonably accurate reconstruction of the photon/electron hit co- ordinates in order that kinematic correlation cuts between the scattered photon/electron and the recoil proton can be made. The off-line analysis procedure and the observed position and energy resolutions are presented and discussed in the following two sections. 6.1. Shower Reconstruction Analysis and Position Resolution The off-line shower reconstruction involves a search for clus- ters and can be characterized by the following definitions: 1. a cluster is a group of adjacent blocks; 2. a cluster occupies 9 (3 × 3) blocks of the calorimeter; 3. the distribution of the shower energy deposition over the cluster blocks (the so-called shower profile) satisfies the following conditions: (a) the maximum energy deposition is in the central block; (b) the energy deposition in the corner blocks is less than that in each of two neighboring blocks; (c) around 50% of the total shower energy must be de- posited in the central row (and column) of the cluster. For an example in which the shower center is in the middle of the central block, around 84% of the total shower energy is in the central block, about 14% is in the four neighboring blocks, and the remaining 2% is in the corner blocks. Even at the largest luminosity used in the RCS experiment the proba- bility of observing two clusters with energies above 50% of the elastic value was less than 10%, so for the 704 block hodoscope a two-cluster overlap was very unlikely. The shower energy reconstruction requires both hardware and software calibration of the calorimeter channels. On the hardware side, equalization of the counter gains was initially done with cosmic muons, which produce 20 MeV energy equiv- alent light output per 4 cm path (muon trajectories perpendic- ular to the long axis of the lead-glass blocks). The calibration was done by selecting cosmic events for which the signals in both counters above and below a given counter were large. The final adjustment of each counter’s gain was done by using cali- bration with elastic e-p events. This calibration provided PMT gain values which were on average different from the initial cos- mic set by 20% The purpose of the software calibration is to define the co- efficients for transformation of the ADC amplitudes to energy deposition for each calorimeter module. These calibration co- efficients are obtained from elastic e-p data by minimizing the function: Ci · (A i − Pi) − E where: n = 1 ÷ N — number of the selected calibration event; i — number of the block, included in the cluster; Mn — set of the blocks’ numbers, in the cluster; Ani — amplitude into the i-th block; Pi — pedestal of the i-th block; Ene — known energy of electron; Ci — calibration coefficients, which need to be fitted. The scattered electron energy Ene is calculated by using the en- ergy of the primary electron beam and the scattered electron angle. A cut on the proton momentum-angle correlation is used to select clean elastic events. Following calculation of the calibration coefficients, the total energy deposition E, as well as the X and Y coordinates of the shower center of gravity are calculated by the formulae: Ei , X = Ei · Xi/E , Y = Ei · Yi/E where M is the set of blocks numbers which make up the clus- ter, Ei is the energy deposition in the i-th block, and Xi and Yi are the coordinates of the i-th block center. The coordinates cal- culated by this simple center of gravity method are then used for a more accurate determination of the incident hit position. This second iteration was developed during the second test run [7], in which a two-layer MWPC was constructed and positioned directly in front of the calorimeter. This chamber had 128 sen- sitive wires in both X and Y directions, with a wire spacing of 2 mm and a position resolution of 1 mm. In this more refined procedure, the coordinate xo of the shower center of gravity in- side the cell (relative to the cell’s low boundary) is used. An estimate of the coordinate xe can be determined from a polyno- mial in this coordinate (P(xo)): xe = P(xo) = a1 · xo + a3 · x o + a5 · x o + a7 · x o + a9 · x For symmetry reasons, only odd degrees of the polynomial are used. The coefficients an are calculated by minimizing the func- tional: P(an, x o) − x where: i = 1 ÷ N — number of event; xio — coordinate of the shower center of gravity inside the cell; xit — coordinate of the track (MWPC) on the calorimeter plane; an — coordinate transformation coefficients to be fitted. The resulting resolution obtained from such a fitting proce- dure was found to be around 5.5 mm for a scattered electron energy of 2.3 GeV. For the case of production data, where the MWPC was not used, Fig. 18 shows a scatter plot of events on the front face of the calorimeter. The parameter plotted is the differences between the observed hit coordinates in the calorimeter and the coordinates calculated from the proton pa- rameters and an assumed two-body kinematic correlation. The dominant contribution to the widths of the RCS and e-p peaks that can be seen in this figure is from the angular resolution of the detected proton, which is itself dominated by multiple scat- tering. As the calorimeter distance varied during the experiment between 5.5 m and 20 m, the contribution to the combined an- gular resolution from the calorimeter position resolution of a few millimeters was minimal. 6.2. Trigger Rate and Energy Resolution At high luminosity, when a reduction of the accidental co- incidences in the raw trigger rate is very important, the trigger threshold should be set as close to the signal amplitude for elas- tic RCS photons as practical. However, the actual value of the threshold for an individual event has a significant uncertainty due to pile-up of the low-amplitude signals, fluctuations of the signal shape (mainly due to summing of the signals from the PMTs with different HV and transit time), and inequality of the x (cm)δ y (cm) ’p)γ,γp(p(e,e’p) p)0π,γp( Figure 18: The scatter plot of p − γ(e) events in the plane of the calorimeter front face. gain in the individual counters. Too high a threshold, therefore, can lead to a loss in detection efficiency. The counting rate of the calorimeter trigger, f , which defines a practical level of operational luminosity has an exponential dependence on the threshold, as can be seen in Fig. 19. It can be described by a function of Ethr: f = A × exp(−B × Ethr/Emax), where Emax is the maximum energy of an elastically scat- tered photon/electron for a given scattering angle, A an angle- dependent constant, and B a universal constant ≈ 9±1. The an- 0.3 0.6 0.9 1.2 1.5 1.8 2.1 40200 60 80 100 120 140 elastic Threshold [GeV] Threshold [mV] Beam energy: Beam current: Target: Calorimeter angle: Calorimeter to target: Solid angle: 10 µ A 3.3 GeV Figure 19: Calorimeter trigger rate vs threshold level. gular variation of the constant A, after normalization to a fixed luminosity and the calorimeter solid angle, is less than a factor of 2 for the RCS kinematics. The threshold for all kinemat- ics was chosen to be around half of the elastic energy, thereby balancing the need for a low trigger rate without affecting the detection efficiency. In order to ensure proper operation and to monitor the perfor- mance of each counter the widths of the ADC pedestals were used (see Fig. 20). One can see that these widths vary slightly with block number, which reflects the position of the block in the calorimeter and its angle with respect to the beam direction. This pedestal width also allows for an estimate of the contri- bution of the background induced base-line fluctuations to the overall energy resolution. For the example shown in Fig. 20 the width of 6 MeV per block leads to energy spectrum noise of about 20 MeV because a 9-block cluster is used in the off-line analysis. 0 100 200 300 400 500 600 700 LG block number Figure 20: The width of the ADC pedestals for the calorimeter in a typical run. The observed reduction of the width vs the block number reflects the lower background at larger detector angle with respect to the beam direction. The energy resolution of the calorimeter was measured by using elastic e-p scattering. Such data were collected many times during the experiment for kinematic checks and calorime- ter gain calibration. Table 4 presents the observed resolution and the corresponding ADC pedestal widths over the course of the experiment. For completeness, the pedestal widths for cos- mic and production data are also included. At high luminosity the energy resolution degrades due to fluctuations of the base line (pedestal width) and the inclusion of more accidental hits during the ADC gate period. However, for the 9-block cluster size used in the data analysis the contribution of the base line fluctuations to the energy resolution is just 1-2%. The measured widths of ADC pedestals confirmed the results of Monte Carlo simulations and test runs that the radiation background is three times higher with the 6% Cu radiator upstream of the target than without it. The resolution obtained from e-p calibration runs was cor- rected for the drift of the gains so it could be attributed directly to the effect of lead glass radiation damage. It degraded over the course of the experiment from 5.5% (for a 1 GeV photon energy) at the start to larger than 10% by the end. It was es- timated that this corresponds to a final accumulated radiation dose of about 3-10 kRad, which is in agreement with the known level of radiation hardness of the TF-1 lead glass [19]. This observed radiation dose corresponds to a 500 hour experiment with a 15 cm LH2 target and 50 µA beam. 6.3. Annealing of the radiation damage The front face of the calorimeter during the experiment was protected by plastic material with an effective thickness of 10 g/cm2. For the majority of the time the calorimeter was lo- cated at a distance of 5-8 m and an angle of 40-50◦ with respect to the electron beam direction. The transparency of 20 lead- glass blocks was measured after the experiment, the results of which are shown in Fig. 21. This plot shows the relative trans- mission through 4 cm of glass in the direction transverse to the block length at different locations. The values were nor- malized to the transmission through similar lead-glass blocks which were not used in the experiment. The transmission mea- surement was done with a blue LED (λmax of 430 nm) and a Hamamatsu photo-diode (1226-44). 0 10 20 30 40 Distance from the calorimeter face [cm] Figure 21: The blue light attenuation in 4 cm of lead-glass vs distance from the front face of calorimeter measured before (solid) and after (dashed) UV irradiation. A UV technique was developed and used in order to cure ra- diation damage. The UV light was produced by a 10 kW total power 55-inch long lamp9, which was installed vertically at a distance of 45 inches from the calorimeter face and a quartz plate (C55QUARTZ) was used as an infrared filter. The inten- sity of the UV light at the face of the lead-glass blocks was found to be 75 mW/cm2 by using a UVX digital radiometer10. In situ UV irradiation without disassembly of the lead-glass stack was performed over an 18 hour period. All PMTs were removed before irradiation to ensure the safety of the photo- cathode. The resultant improvement in transparency can be seen in Fig. 21. An alternative but equally effective method to restore the lead-glass transparency, which involved heating of the lead-glass blocks to 250◦C for several hours, was also 9Type A94551FCB manufactured by American Ultraviolet, Lebanon, IN 46052, USA 10 Manufactured by UVP, Inc., Upland, CA 91786, USA Table 4: Pedestal widths and calorimeter energy resolution at different stages of the RCS experiment for cosmic (c), electron (e) and production (γ) runs in order of increasing effective luminosity. Runs L e f f Beam Current Accumulated Detected Ee/γ σE /E σE /E at Eγ=1 GeV Θcal σped (1038 cm−2/s) (µA) Beam Charge (C) (GeV) (%) (%) (degrees) (MeV) 1517 (c) - - - - - - - 1.5 1811 (e) 0.1 2.5 2.4 2.78 4.2 7.0 30 1.7 1488 (e) 0.2 5 0.5 1.32 4.9 5.5 46 1.75 2125 (e) 1.0 25 6.6 2.83 4.9 8.2 34 2.6 2593 (e) 1.5 38 14.9 1.32 9.9 11.3 57 2.0 1930 (e) 1.6 40 4.4 3.39 4.2 7.7 22 3.7 1938 (γ) 1.8 15 4.5 3.23 - - 22 4.1 2170 (γ) 2.4 20 6.8 2.72 - - 34 4.0 1852 (γ) 4.2 35 3.0 1.63 - - 50 5.0 tested. The net effect of heating on the transparency of the lead- glass was similar to the UV curing results. In summary, operation of the calorimeter at high luminos- ity, particularly when the electron beam was incident on the bremsstrahlung radiator, led to a degradation in energy resolu- tion due to fluctuations in the base-line and a higher accidental rate within the ADC gate period. For typical clusters this effect was found to be around a percent or two. By far the largest con- tributor to the observed degradation in resolution was radiation damage sustained by the lead-glass blocks, which led to the res- olution being a factor of two larger at the end of the experiment. The resulting estimates of the total accumulated dose were con- sistent with expectations for this type of lead-glass. Finally, it was found that both UV curing and heating of the lead-glass were successful in annealing this damage. 7. Summary The design of a segmented electromagnetic calorimeter which was used in the JLab RCS experiment has been de- scribed. The performance of the calorimeter in an unprece- dented high luminosity, high background environment has been discussed. Good energy and position resolution enabled a suc- cessful measurement of the RCS process over a wide range of kinematics. 8. Acknowledgments We acknowledge the RCS collaborators who helped to oper- ate the detector and the JLab technical staff for providing out- standing support, and specially D. Hayes, T. Hartlove, T. Hun- yady, and S. Mayilyan for help in the construction of the lead- glass modules. We appreciate S. Corneliussen’s careful reading of the manuscript and his valuable suggestions. This work was supported in part by the National Science Foundation in grants for the University of Illinois University and by DOE contract DE-AC05-84ER40150 under which the Southeastern Universi- ties Research Association (SURA) operates the Thomas Jeffer- son National Accelerator Facility for the United States Depart- ment of Energy. References [1] C. Hyde-Wright, A. Nathan, and B. Wojtsekhowski, spokespersons, JLab experiment E99-114. [2] Charles Hyde-Wright and Kees de Jager Ann.Rev.Nucl.Part.Sci. 54, 217 (2004). [3] A.V. Radyushkin, Phys. Rev. D 58, 114008 (1998). [4] H.W. Huang, P. Kroll, T. Morii, Eur. Phys. J. C 23, 301 (2002); erratum ibid., C 31, 279 (2003). [5] R. Thompson, A. Pang, Ch.-R. Ji, Phys. Rev. D 73, 054023 (2006). [6] M.A. Shupe et al., Phys. Rev. D 19, 1929 (1979). [7] E. Chudakov et al., Study of Hall A Photon Spectrometer. Hall A internal report, 1998. [8] J. Alcorn et al., Nucl. Instr. Meth. A 522, (2004) 294. [9] D.J. Hamilton et al., Phys. Rev. Lett. 94, 242001 (2005). [10] A. Danagoulian et al., Phys. Rev. Lett. 98, 152001 (2007). [11] Yu.D. Prokoshkin et al., Nucl. Instr. Meth. A 248, 86102 (1986). [12] M.Y. Balatz et al., Nucl. Instr. Meth. A 545, 114 (2005). [13] R.G. Astvatsaturov et al., Nucl. Instr. Meth. 107, 105 (1973). [14] R.R. Crittenden et al., Nucl. Instr. Meth. A 387, 377 (1997). [15] V. Popov et al., Proceedings of IEEE 2001 Nuclear Science Symposium (NSS) And Medical Imaging Conference (MIC). Ed. J.D. Valentine IEEE (2001) p. 634-637. [16] V. Popov, Nucl. Instr. Meth. A 505, 316 (2003). [17] W.A. Watson et al., CODA: a scalable, distributed data acquisition sys- tem, in: Proceedings of the Real Time 1993 Conference, p. 296; [18] E. Jastrzembski et al., The Jefferson Lab trigger supervisor system, 11th IEEE NPSS Real Time 1999 Conference, JLab-TN-99-13, 1999. [19] A.V. Inyakin et al., Nucl. Instr. Meth. 215, 103 (1983). 1 Introduction 2 Calorimeter 2.1 Calorimeter Design 2.1.1 Air Cooling 2.1.2 Cabling System 2.2 Lead-Glass Counter 2.2.1 Design of the Counter 2.2.2 HV Divider 2.3 Electronics 2.3.1 Trigger Scheme 2.4 Gain Monitoring System 3 Veto Hodoscopes 4 High Voltage System 5 Data Acquisition System 6 Calorimeter Performance 6.1 Shower Reconstruction Analysis and Position Resolution 6.2 Trigger Rate and Energy Resolution 6.3 Annealing of the radiation damage 7 Summary 8 Acknowledgments

0704.1831

Multiplicity Fluctuations in Au+Au Collisions at RHIC

January 4, 2019 Multiplicity Fluctuations in Au+Au Collisions at RHIC V.P. Konchakovski,1, 2 M.I. Gorenstein,1, 3 and E.L. Bratkovskaya3 1Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine 2Helmholtz Research School, University of Frankfurt, Frankfurt, Germany 3Frankfurt Institute for Advanced Studies, Frankfurt, Germany Abstract The preliminary data of the PHENIX collaboration for the scaled variances of charged hadron multiplicity fluctuations in Au+Au at s = 200 GeV are analyzed within the model of independent sources. We use the HSD transport model to calculate the participant number fluctuations and the number of charged hadrons per nucleon participant in different centrality bins. This combined picture leads to a good agreement with the PHENIX data and suggests that the measured multi- plicity fluctuations result dominantly from participant number fluctuations. The role of centrality selection and acceptance is discussed separately. PACS numbers: 24.10.Lx, 24.60.Ky, 25.75.-q Keywords: nucleus-nucleus collisions, fluctuations, transport models http://arxiv.org/abs/0704.1831v1 The event-by-event fluctuations in high energy nucleus-nucleus (A+A) collisions (see e.g., the reviews [1, 2]) are expected to provide signals of the transition between different phases (see e.g., Refs. [3, 4]) and the QCD critical point [5]. In the present letter we study the charged multiplicity fluctuations in Au+Au collisions at RHIC energies. The preliminary data of the PHENIX collaboration [6] at s = 200 GeV are analyzed within the model of independent sources while employing the microscopic Hadron-String-Dynamics (HSD) transport model [7, 8] to define the centrality selection and to calculate the properties of hadron production sources. The centrality selection is an important aspect of fluctuation studies in A+A collisions. At the SPS fixed target experiments the samples of collisions with a fixed number of projectile participants N P can be selected to minimize the participant number fluctuations in the sample of collision events. This selection is possible due to a measurement of the number of nucleon spectators from the projectile, N S , in each individual collision by a calorimeter which covers the projectile fragmentation domain. However, even in the sample with N const the number of target participants fluctuates considerably. In the following the variance, V ar(n) ≡ 〈n2〉 − 〈n〉2, and scaled variance, ω ≡ V ar(n)/〈n〉, where n stands for a given random variable and 〈· · · 〉 for event-by-event averaging, will be used to quantify fluctuations. In each sample with N P = const the number of target participants fluctuates around its mean value, 〈N targP 〉 = N P , with the scaled variance ω P . Within the HSD and UrQMD transport models it was found in Ref. [9] that the fluctuations of N P strongly influence the charged hadron fluctuations. The constant values of N P and fluctuations of N P lead also to an asymmetry between the fluctuations in the projectile and target hemispheres. The consequences of this asymmetry depend on the A+A dynamics as discussed in Ref. [10]. In Au+Au collisions at RHIC a different centrality selection is used. There are two kinds of detectors which define the centrality of Au+Au collision, Beam-Beam Counters (BBC) and Zero Degree Calorimeters (ZDC). At the c.m. pair energy s = 200 GeV, the BBC measure the charged particle multiplicity in the pseudorapidity range 3.0 < |η| < 3.9, and the ZDC – the number of neutrons with |η| > 6.0 [6]. These neutrons are part of the nucleon spectators. Due to technical reasons the neutron spectators can be only detected by the ZDC (not protons and nuclear fragments), but in both hemispheres. The BBC distribution will be used in the HSD calculations to divide Au+Au collision events into 5% centrality samples. HSD does not specify different spectator groups – neutrons, protons, and nuclear fragments such that we can not use the ZDC information. In Fig. 1 (left) the HSD results for the BBC distribution and centrality classes in Au+Au collisions at s =200 GeV are shown. We find a good agreement between the HSD shape of the BBC distribution and the PHENIX data [6]. The experimental estimates of 〈NP 〉 for different centrality classes are based on the Glauber model. These estimates vary by less than 0.5% depending on the shape of the cut in the ZDC/BBC space or whether the BBC alone is used as a centrality measure [6]. Note, however, that the HSD 〈NP 〉 numbers are not exactly equal to the PHENIX values. It is also not obvious that different definitions for the 5% centrality classes give the same values of the scaled variance ωP for the participant number fluctuations. 0 200 400 600 800 Charge in BBC 0 100 200 300 400 FIG. 1: HSD model results for Au+Au collisions at s = 200 GeV. Left: Centrality classes defined via the BBC distribution. Right: The average number of participants, 〈NP 〉, and the scaled variance of the participant number fluctuations, ωP , calculated for the 5% BBC centrality classes. Defining the centrality selection via the HSD transport model (which is similar to the BBC in the PHENIX experiment) we calculate the mean number of nucleon participants, 〈NP 〉, and the scaled variance of its fluctuations, ωP , in each 5% centrality sample. The results are shown in Fig. 1, right. The Fig. 2 (left) shows the HSD results for the mean number of charged hadrons per nucleon participant, ni = 〈Ni〉/〈NP 〉, where the index i stands for “−”, “+”, and “ch”, i.e negatively, positively, and all charged final hadrons. Note that the centrality dependence of ni is opposite to that of ωP : ni increases with 〈NP 〉, whereas ωP decreases. The PHENIX detector accepts charged particles in a small region of the phase space with pseudorapidity |η| < 0.26 and azimuthal angle φ < 245o and the pT range from 0.2 to 2.0 GeV/c [6]. The fraction of the accepted particles qi = 〈Nacci 〉/〈Ni〉 calculated within the HSD model is shown in the r.h.s. of Fig. 2. According to the HSD results only 3÷ 3.5% of charged particles are accepted by the mid-rapidity PHENIX detector. 0 100 200 300 400 0 100 200 300 400 0.030 0.035 FIG. 2: HSD results for different BBC centrality classes in Au+Au collisions at s = 200 GeV. Left: The mean number of charged hadrons per participant, ni = 〈Ni〉/〈NP 〉. Right: The fraction of accepted particles, qi = 〈Nacci 〉/〈Ni〉. To estimate the role of the participant number event-by-event fluctuations we use the model of independent sources (see e.g., Refs [1, 9, 10]), ωi = ω i + ni ωP . (1) The first term in the r.h.s. of Eq. (1) corresponds to the fluctuations of the hadron mul- tiplicity from one source, and the second term, ni ωP , gives additional fluctuations due to the fluctuations of the number of sources. As usually, we have assumed that the number of sources is proportional to the number of nucleon participants. The value of ni in Eq. (1) is then the average number of i’th particles per participant, ni = 〈Ni〉/〈NP 〉. We also assume that nucleon-nucleon collisions define the fluctuations ω∗i from a single source. To calculate the fluctuations ωacci in the PHENIX acceptance we use the acceptance scaling formula (see e.g., Refs. [1, 9, 10]): ωacci = 1 − qi + qi ωi , (2) where qi is the fraction of the accepted i’th hadrons by the PHENIX detector. Using Eq. (1) for ωi one finds, ωacci = 1 − qi + qi ω∗i + qi ni ωP . (3) The HSD results for ωP (Fig. 1, right), ni (Fig. 2, left), qi (Fig. 2, right), together with the HSD nucleon-nucleon values, ω∗ = 3.0, ω∗+ = 2.7, and ω ch = 5.7 at s = 200 GeV, define completely the results for ωacci according to Eq. (3). We find a surprisingly good agreement of the results given by Eq. (3) with the PHENIX data shown in Fig. 3. Note that the centrality dependence of ωacci stems from the product, ni · ωP , in the last term of the r.h.s. of Eq. (3). 100 200 300 400 PHENIX Eq. (3) FIG. 3: The scaled variance of charged particle fluctuations in Au+Au collisions at s = 200 GeV with the PHENIX acceptance. The circles are the PHENIX data [6] while the open points (con- nected by the solid line) correspond to Eq. (3) with the HSD results for ωP , ni, and qi. In summary, the preliminary PHENIX data [6] for the scaled variances of charged hadron multiplicity fluctuations in Au+Au collisions at s = 200 GeV have been analyzed within the model of independent sources. Assuming that the number of hadron sources are propor- tional to the number of nucleon participants, the HSD transport model was used to calculate the scaled variance of participant number fluctuations, ωP , and the number of i’th hadrons per nucleon accepted by the mid-rapidity PHENIX detector, qini, in different Beam-Beam Counter centrality classes. The HSD model for nucleon-nucleon collisions was also used to estimate the fluctuations from a single source, ω∗i . We find that this model description is in a good agreement with the PHENIX data [6]. In different (5%) centrality classes ωP goes down and qini goes up with increasing 〈NP 〉. This results in non-monotonic dependence of ωacci on 〈NP 〉 as seen in the PHENIX data. We conclude that both qualitative and quantitative features of the centrality dependence of the fluctuations seen in the present PHENIX data are the consequences of participant number fluctuations. To avoid a dominance of the participant number fluctuations one needs to analyze most central collisions with a much more rigid (≤ 1%) centrality selection. The statistical model then predicts ω± < 1 [11], whereas the HSD transport model predicts the values of ω± much larger than unity at s = 200 GeV [12]. To allow for a clear distinction between these predictions it is mandatory to enlarge the acceptance of the mid-rapidity detector up to about 10% (see the discussion in Ref. [12]). Acknowledgments: We like to thank V.V. Begun, W. Cassing, M. Gaździcki, W. Greiner, M. Hauer, B. Lungwitz, I.N. Mishustin and J.T. Mitchell for useful discus- sions. One of the author (M.I.G.) is thankful to the Humboldt Foundation for financial support. [1] H. Heiselberg, Phys. Rep. 351, 161 (2001). [2] S. Jeon and V. Koch, Review for Quark-Gluon Plasma 3, eds. R.C. Hwa and X.-N. Wang, World Scientific, Singapore, 430-490 (2004) [arXiv:hep-ph/0304012]. [3] M. Gaździcki, M. I. Gorenstein, and S. Mrowczynski, Phys. Lett. B 585, 115 (2004); M. I. Gorenstein, M. Gaździcki, and O. S. Zozulya, Phys. Lett. B 585, 237 (2004). [4] I.N. Mishustin, Phys. Rev. Lett. 82, 4779 (1999); Nucl. Phys. A 681, 56 (2001); H. Heiselberg and A.D. Jackson, Phys. Rev. C 63, 064904 (2001). [5] M.A. Stephanov, K. Rajagopal, and E.V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998); Phys. Rev. D 60, 114028 (1999); M.A. Stephanov, Acta Phys. Polon. B 35, 2939 (2004); [6] S.S. Adler et al., [PHENIX Collaboration], arXiv: nucl-ex/0409015; J.T. Mitchell [PHENIX http://arxiv.org/abs/hep-ph/0304012 http://arxiv.org/abs/nucl-ex/0409015 Collaboration], J. Phys. Conf. Ser. 27, 88 (2005); J.T. Mitchell, private communications. [7] W. Ehehalt and W. Cassing, Nucl. Phys. A 602, 449 (1996); E.L. Bratkovskaya and W. Cassing, Nucl. Phys. A 619 413 (1997); W. Cassing and E.L. Bratkovskaya, Phys. Rep. 308, 65 (1999); W. Cassing, E.L. Bratkovskaya, and A. Sibirtsev, Nucl. Phys. A 691, 753 (2001); W. Cassing, E. L. Bratkovskaya, and S. Juchem, Nucl. Phys. A 674, 249 (2000). [8] H. Weber, et al., Phys. Rev. C67, 014904 (2003); E.L. Bratkovskaya, et al., Phys. Rev. C 67, 054905 (2003); ibid, 69, 054907 (2004); Prog. Part. Nucl. Phys. 53, 225 (2004). [9] V.P. Konchakovski, et al., Phys. Rev. C 73, 034902 (2006); ibid. C 74, 064911 (2006). [10] M. Gaździcki and M.I. Gorenstein, Phys. Lett. B 640, 155 (2006). [11] V.V. Begun, et al., Phys. Rev. C 74, 044903 (2006); nucl-th/0611075. [12] V.P. Konchakovski, E.L. Bratkovskaya, and M.I. Gorenstein, nucl-th/0703052. http://arxiv.org/abs/nucl-th/0611075 http://arxiv.org/abs/nucl-th/0703052 References

0704.1832

Near and Mid-IR Photometry of the Pleiades, and a New List of Substellar Candidate Members

ApJS, in press; version with embedded figures can be obtained at http://spider.ipac.caltech.edu/staff/stauffer/ Near and Mid-IR Photometry of the Pleiades, and a New List of Substellar Candidate Members1,2 John R. Stauffer Spitzer Science Center, Caltech 314-6, Pasadena, CA 91125 [email protected] Lee W. Hartmann Astronomy Department, University of Michigan Giovanni G. Fazio, Lori E. Allen, Brian M. Patten Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138 Patrick J. Lowrance, Robert L. Hurt, Luisa M. Rebull Spitzer Science Center, Caltech , Pasadena, CA 91125 Roc M. Cutri, Solange V. Ramirez Infrared Processing and Analysis Center, Caltech 220-6, Pasadena, CA 91125 Erick T. Young, George H. Rieke, Nadya I. Gorlova3, James C. Muzerolle Steward Observatory, University of Arizona, Tucson, AZ 85726 Cathy L. Slesnick Astronomy Department, Caltech, Pasadena, CA 91125 Michael F. Skrutskie Astronomy Department, University of Virginia, Charlottesville, VA 22903 1This work is based (in part) on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under NASA contract 1407. 2This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. 3Current address: University of Florida, 211 Bryant Space Center, Gainesville, FL 32611 http://arxiv.org/abs/0704.1832v1 http://spider.ipac.caltech.edu/staff/stauffer/ – 2 – ABSTRACT We make use of new near and mid-IR photometry of the Pleiades cluster in order to help identify proposed cluster members. We also use the new photometry with previously published photometry to define the single-star main sequence locus at the age of the Pleiades in a variety of color-magnitude planes. The new near and mid-IR photometry extend effectively two magnitudes deeper than the 2MASS All-Sky Point Source catalog, and hence allow us to select a new set of candidate very low mass and sub-stellar mass members of the Pleiades in the central square degree of the cluster. We identify 42 new candi- date members fainter than Ks =14 (corresponding to 0.1 Mo). These candidate members should eventually allow a better estimate of the cluster mass function to be made down to of order 0.04 solar masses. We also use new IRAC data, in particular the images obtained at 8 um, in order to comment briefly on interstellar dust in and near the Pleiades. We confirm, as expected, that – with one exception – a sample of low mass stars recently identified as having 24 um excesses due to debris disks do not have significant excesses at IRAC wavelengths. However, evidence is also presented that several of the Pleiades high mass stars are found to be impacting with local condensations of the molecular cloud that is passing through the Pleiades at the current epoch. Subject headings: stars: low mass — young; open clusters — associations: individual (Pleiades) 1. Introduction Because of its proximity, youth, richness, and location in the northern hemisphere, the Pleiades has long been a favorite target of observers. The Pleiades was one of the first open clusters to have members identified via their common proper motion (Trumpler 1921), and the cluster has since then been the subject of more than a dozen proper motion studies. Some of the earliest photoelectric photometry was for members of the Pleiades (Cummings 1921), and the cluster has been the subject of dozens of papers providing additional optical photometry of its members. The youth and nearness of the Pleiades make it a particularly attractive target for identifying its substellar population, and it was the first open cluster studied for those purposes (Jameson & Skillen 1989; Stauffer et al. 1989). More than 20 pa- pers have been subsequently published, identifying additional substellar candidate members of the Pleiades or studying their properties. – 3 – We have three primary goals for this paper. First, while extensive optical photometry for Pleiades members is available in the literature, photometry in the near and mid-IR is relatively spotty. We will remedy this situation by using new 2MASS JHKs and Spitzer IRAC photometry for a large number of Pleiades members. We will use these data to help identify cluster non-members and to define the single-star locus in color-magnitude diagrams for stars of 100 Myr age. Second, we will use our new IR imaging photometry of the center of the Pleiades to identify a new set of candidate substellar members of the cluster, extending down to stars expected to have masses of order 0.04 M⊙. Third, we will use the IRAC data to briefly comment on the presence of circumstellar debris disks in the Pleiades and the interaction of the Pleiades stars with the molecular cloud that is currently passing through the cluster. In order to make best use of the IR imaging data, we will begin with a necessary digression. As noted above, more than a dozen proper motion surveys of the Pleiades have been made in order to identify cluster members. However, no single catalog of the cluster has been published which attempts to collect all of those candidate members in a single table and cross-identify those stars. Another problem is that while there have been many papers devoted to providing optical photometry of cluster members, that photometry has been bewilderingly inhomogeneous in terms of the number of photometric systems used. In Sec. 3 and in the Appendix, we describe our efforts to create a reasonably complete catalog of candidate Pleiades members and to provide optical photometry transformed to the best of our ability onto a single system. 2. New Observational Data 2.1. 2MASS “6x” Imaging of the Pleiades During the final months of Two Micron All Sky Survey (2MASS; Skrutskie et al. (2006)) operations, a series of special observations were carried out that employed exposures six times longer than used for the the primary survey. These so-called “6x” observations targeted 30 regions of scientific interest including a 3 deg x 2 deg area centered on the Pleiades cluster. The 2MASS 6x data were reduced using an automated processing pipeline similar to that used for the main survey data, and a calibrated 6x Image Atlas and extracted 6x Point and Extended Source Catalogs (6x-PSC and 6x-XSC) analogous to the 2MASS All-Sky Atlas, PSC and XSC have been released as part of the 2MASS Extended Mission. A description of the content and formats of the 6x image and catalog products, and details about the 6x – 4 – observations and data reduction are given by Cutri et al. (2006; section A3). 1 The 2MASS 6x Atlas and Catalogs may be accessed via the on-line services of the NASA/IPAC Infrared Science Archive (http://irsa.ipac.caltech.edu). Figure 1 shows the area on the sky imaged by the 2MASS 6x observations in the Pleiades field. The region was covered by two rows of scans, each scan being one degree long (in declination) and 8.5’ wide in right ascension. Within each row, the scans overlap by approx- imately one arcminute in right ascension. There are small gaps in coverage in the declination boundary between the rows, and one complete scan in the southern row is missing because the data in that scan did not meet the minimum required photometric quality. The total area covered by the 6x Pleiades observations is approximately 5.3 sq. degrees. There are approximately 43,000 sources extracted from the 6x Pleiades observations in the 2MASS 6x-PSC, and nearly 1,500 in the 6x-XSC. Because there are at most about 1000 Pleiades members expected in this region, only ∼2% of the 6x-PSC sources are cluster members, and the rest are field stars and background galaxies. The 6x-XSC objects are virtually all resolved background galaxies. Near infrared color-magnitude and color-color diagrams of the unresolved sources from the 2MASS 6x-PSC and all sources in the 6x-XSC sources from the Pleiades region are shown in Figures 2 and 3, respectively. The extragalactic sources tend to be redder than most stars, and the galaxies become relatively more numerous towards fainter magnitudes. Unresolved galaxies dominate the point sources that are fainter than Ks > 15.5 and redder than J −Ks > 1.2 mag. The 2MASS 6x observations were conducted using the same freeze-frame scanning tech- nique used for the primary survey (Skrutskie et al. 2006). The longer exposure times were achieved by increasing the “READ2-READ1” integration to 7.8 sec from the 1.3 sec used for primary survey. However, the 51 ms “READ1” exposure time was not changed for the 6x observations. As a result, there is an effective “sensitivity gap” in the 8-11 mag region where objects may be saturated in the 7.8 sec READ2-READ1 6x exposures, but too faint to be detected in the 51 ms READ1 exposures. Because the sensitivity gap can result in incom- pleteness and/or flux bias in the photometric overlap regime, the near infrared photometry for sources brighter than J=11 mag in the 6x-PSC was taken from the 2MASS All-Sky PSC during compilation of the catalog of Pleiades candidate members presented in Table 2 (c.f. Section 3). 1http://www.ipac.caltech.edu/2mass/releases/allsky/doc/explsup.html http://irsa.ipac.caltech.edu – 5 – 2.2. Shallow IRAC Imaging Imaging of the Pleiades with Spitzer was obtained in April 2004 as part of a joint GTO program conducted by the IRAC instrument team and the MIPS instrument team. Initial results of the MIPS survey of the Pleiades have already been reported in Gorlova et al. (2006). The IRAC observations were obtained as two astronomical observing requests (AORs). One of them was centered near the cluster center, at RA=03h47m00.0s and Dec=24d07m (2000), and consisted of a 12 row by 12 column map, with “frametimes” of 0.6 and 12.0 seconds and two dithers at each map position. The map steps were 290′′ in both the column and row direction. The resultant map covers a region of approximately one square degree, and a total integration time per position of 24 sec over most of the map. The second AOR used the same basic mapping parameters, except it was smaller (9 rows by 9 columns) and was instead centered northwest from the cluster center at RA=03h44m36.0s and Dec=25d24m. A two-band color image of the AOR covering the center of the Pleiades is shown in Figure 4. A pictorial guide to the IRAC image providing Greek names for a few of the brightest stars, and Hertzsprung (1947) numbers for several stars mentioned in Section 6 is provided in Figure 5. We began our analysis with the basic calibrated data (BCDs) from the Spitzer pipeline, using the S13 version of the Spitzer Science Center pipeline software. Artifact mitigation and masking was done using the IDL tools provided on the Spitzer contributed software website. For each AOR, the artifact-corrected BCDs were combined into single mosaics for each channel using the post-BCD “MOPEX” package (Makovoz & Marleau 2005). The mosaic images were constructed with 1.22×1.22 arcsecond pixels (i.e., approximately the same pixel size as the native IRAC arrays). We derived aperture photometry for stars present in these IRAC mosaics using both APEX (a component of the MOPEX package) and the “phot” routine in DAOPHOT. In both cases, we used a 3 pixel radius aperture and a sky annulus from 3 to 7 pixels (except that for Channel 4, for the phot package we used a 2 pixel radius aperture and a 2 to 6 pixel annulus because that provided more reliable fluxes at low flux levels). We used the flux for zero magnitude calibrations provided in the IRAC data handbook (280.9, 179.7, 115.0 and 64.1 Jy for Ch 1 through Ch 4, respectively), and the aperture corrections provided in the same handbook (multiplicative flux correction factors of 1.124, 1.127, 1.143 and 1.584 for Ch 1-4, inclusive. The Ch4 correction factor is much bigger because it is for an aperture radius of 2 rather than 3 pixels.). Figure 6 and Figure 7 provide two means to assess the accuracy of the IRAC photometry. The first figure compares the aperture photometry from APEX to that from phot, and shows that the two packages yield very similar results when used in the same way. For this reason, – 6 – we have simply averaged the fluxes from the two packages to obtain our final reported value. The second figure shows the difference between the derived 3.6 and 4.5 µm magnitudes for Pleiades members. Based on previous studies (e.g. Allen et al. (2004)), we expected this difference to be essentially zero for most stars, and the Pleiades data corroborate that expectation. For [3.6]<10.5, the RMS dispersion of the magnitude difference between the two channels is 0.024 mag. Assuming that each channel has similar uncertainties, this indicates an internal 1-σ accuracy of order 0.017 mag. The absolute calibration uncertainty for the IRAC fluxes is currently estimated at of order 0.02 mag. Figure 7 also shows that fainter than [3.6]=10.5 (spectral type later than about M0), the [3.6]−[4.5] color for M dwarfs departs slightly from zero, becoming increasingly redder to the limit of the data (about M6). 3. A Catalog of Pleiades Candidate Members If one limits oneself to only stars visible with the naked eye, it is easy to identify which stars are members of the Pleiades – all of the stars within a degree of the cluster center that have V < 6 are indeed members. However, if one were to try to identify the M dwarf stellar members of the cluster (roughly 14 < V < 23), only of order 1% of the stars towards the cluster center are likely to be members, and it is much harder to construct an uncontaminated catalog. The problem is exacerbated by the fact that the Pleiades is old enough that mass segregation through dynamical processes has occurred, and therefore one has to survey a much larger region of the sky in order to include all of the M dwarf members. The other primary difficulty in constructing a comprehensive member catalog for the Pleiades is that the pedigree of the candidates varies greatly. For the best studied stars, astrometric positions can be measured over temporal baselines ranging up to a century or more, and the separation of cluster members from field stars in a vector point diagram (VPD) can be extremely good. In addition, accurate radial velocities and other spectral indicators are available for essentially all of the bright cluster members, and these further allow membership assessment to be essentially definitive. Conversely, at the faint end (for stars near the hydrogen burning mass limit in the Pleiades), members are near the detection limit of the existing wide-field photographic plates, and the errors on the proper motions become correspondingly large, causing the separation of cluster members from field stars in the VPD to become poor. These stars are also sufficiently faint that spectra capable of discriminating members from field dwarfs can only be obtained with 8m class telescopes, and only a very small fraction of the faint candidates have had such spectra obtained. Therefore, any comprehensive catalog created for the Pleiades will necessarily have stars ranging from certain members to candidates for which very little is known, and where the fraction of – 7 – spurious candidate members increases to lower masses. In order to address the membership uncertainties and biases, we have chosen a sliding scale for inclusion in our catalog. For all stars, we require that the available photometry yields location in color-color and color-magnitude diagrams consistent with cluster membership. For the stars with well-calibrated photoelectric photometry, this means the star should not fall below the Pleiades single-star locus by more than about 0.2 mag or above that locus by more than about 1.0 mag (the expected displacement for a hierarchical triple with three nearly equal mass components). For stars with only photographic optical photometry, where the 1-σ uncertainties are of order 0.1 to 0.2 mag, we still require the star’s photometry to be consistent with membership, but the allowed displacements from the single star locus are considerably larger. Where accurate radial velocities are known, we require that the star be considered a radial velocity member based on the paper where the radial velocities were presented. Where stars have been previously identified as non-members based on photometric or spectroscopic indices, we adopt those conclusions. Two other relevant pieces of information are sometimes available. In some cases, in- dividual proper motion membership probabilities are provided by the various membership surveys. If no other information is available, and if the membership probability for a given candidate is less than 0.1, we exclude that star from our final catalog. However, often a star appears in several catalogs; if it appears in two or more proper motion membership lists we include it in the final catalog even if P < 0.1 in one of those catalogs. Second, an entirely different means to identify candidate Pleiades members is via flare star surveys towards the cluster (Haro et al. 1982; Jones 1981). A star with a formally low membership probability in one catalog but whose photometry is consistent with membership and that was identified as a flare star is retained in our catalog. Further details of the catalog construction are provided in the appendix, as are details of the means by which the B, V , and I photometry have been homogenized. A full discussion and listing of all of the papers from which we have extracted astrometric and photometric information is also provided in the appendix. Here we simply provide a very brief description of the inputs to the catalog. We include candidate cluster members from the following proper motion surveys: Trumpler (1921), Hertzsprung (1947), Jones (1981), Pels and Lub – as reported in van Leeuwen, Alphenaar, & Brand (1986), Stauffer et al. (1991), Artyukhina (1969), Hambly et al. (1993), Pinfield et al. (2000), Adams et al. (2001) and Deacon & Hambly (2004). Another important compilation which provides the initial identification of a significant number of low mass cluster members is the flare star catalog of Haro et al. (1982). Table 1 provides a brief synopsis of the characteristics of the candidate member catalogs from these papers. The Trumpler paper is listed twice – 8 – in Table 1 because there are two membership surveys included in that paper, with differing spatial coverages and different limiting magnitudes. In our final catalog, we have attempted to follow the standard naming convention whereby the primary name is derived from the paper where it was first identified as a cluster member. An exception to this arises for stars with both Trumpler (1921) and Hertzsprung (1947) names, where we use the Hertzsprung numbers as the standard name because that is the most commonly used designation for these stars in the literature. The failure for the Trumpler numbers to be given precedence in the literature perhaps stems from the fact that the Trumpler catalog was published in the Lick Observatory Bulletins as opposed to a refereed journal. In addition to providing a primary name for each star, we provide cross- identifications to some of the other catalogs, particularly where there is existing photometry or spectroscopy of that star using the alternate names. For the brightest cluster members, we provide additional cross-references (e.g., Greek names, Flamsteed numbers, HD numbers). For each star, we attempt to include an estimate for Johnson B and V , and for Cousins I (IC). Only a very small fraction of the cluster members have photoelectric photometry in these systems, unfortunately. Photometry for many of the stars has often been obtained in other systems, including Walraven, Geneva, Kron, and Johnson. We have used previously published transformations from the appropriate indices in those systems to Johnson BV or Cousins I. In other cases, photometry is available in a natural I band system, primarily for some of the relatively faint cluster members. We have attempted to transform those I band data to IC by deriving our own conversion using stars for which we already have a IC estimate as well as the natural I measurement. Details of these issues are provided in the Appendix. Finally, we have cross-correlated the cluster candidates catalog with the 2MASS All-Sky PSC and also with the 6x-PSC for the Pleiades. For every star in the catalog, we obtain JHKs photometry and 2MASS positions. Where we have both main survey 2MASS data and data from the 6x catalog, we adopt the 6x data for stars with J >11, and data from the standard 2MASS catalog otherwise. We verified that the two catalogs do not have any obvious photometric or astrometric offsets relative to each other. The coordinates we list in our catalog are entirely from these 2MASS sources, and hence they inherit the very good and homogeneous 2MASS positional accuracies of order 0.1 arcseconds RMS. We have then plotted the candidate Pleiades members in a variety of color-magnitude diagrams and color-color diagrams, and required that a star must have photometry that is consistent with cluster membership. Figure 8 illustrates this process, and indicates why (for example) we have excluded HII 1695 from our final catalog. – 9 – Table 2 provides the collected data for the 1417 stars we have retained as candidate Pleiades members. The first two columns are the J2000 RA and Dec from 2MASS; the next are the 2MASS JHKs photometry and their uncertainties, and the 2MASS photometric quality flag (“ph-qual”). If the number following the 2MASS quality flag is a 1, the 2MASS data come from the 2MASS All-Sky PSC; if it is a 2, the data come from the 6x-PSC. The next three columns provide the B, V and IC photometry, followed by a flag which indicates the provenance of that photometry. The last column provides the most commonly used names for these stars. The hydrogen burning mass limit for the Pleiades occurs at about V=22, I=18, Ks=14.4. Fifty-three of the candidate members in the catalog are fainter than this limit, and hence should be sub-stellar if they are indeed Pleiades members. Table 3 provides the IRAC [3.6], [4.5], [5.8] and [8.0] photometry we have derived for Pleiades candidate members included within the region covered by the IRAC shallow survey of the Pleiades (see section 2). The brightest stars are saturated even in our short integration frame data, particularly for the more sensitive 3.6 and 4.5 µm channels. At the faint end, we provide photometry only for 3.6 and 4.5 µm because the objects are undetected in the two longer wavelength channels. At the “top” and “bottom” of the survey region, we have incomplete wavelength coverage for a band of width about 5′, and for stars in those areas we report only photometry in either the 3.6 and 5.8 bands or in 4.5 and 8.0 bands. Because Table 2 is an amalgam of many previous catalogs, each of which have different spatial coverage, magnitude limits and other idiosyncrasies, it is necessarily incomplete and inhomogeneous. It also certainly includes some non-members. For V < 12, we expect very few non-members because of the extensive spectroscopic data available for those stars; the fraction of non-members will likely increase to fainter magnitudes, particularly for stars located far from the cluster center. The catalog is simply an attempt to collect all of the available data, identify some of the non-members and eliminate duplications. We hope that it will also serve as a starting point for future efforts to produce a “cleaner” catalog. Figure 9 shows the distribution on the sky of the stars in Table 2. The complete spatial distribution of all members of the Pleiades may differ slightly from what is shown due to the inhomogeneous properties of the proper motion surveys. However, we believe that those effects are relatively small and the distribution shown is mostly representative of the parent population. One thing that is evident in Figure 9 is mass segregation – the highest mass cluster members are much more centrally located than the lowest mass cluster members. This fact is reinforced by calculating the cumulative number of stars as a function of distance from the cluster center for different absolute magnitude bins. Figure 10 illustrates this fact. Another property of the Pleiades illustrated by Figure 9 is that the cluster appears to be elongated parallel to the galactic plane, as expected from n-body simulations of galactic – 10 – clusters (Terlevich 1987). Similar plots showing the flattening of the cluster and evidence for mass segregation for the V < 12 cluster members were provided by (Raboud & Mermilliod 1998). 4. Empirical Pleiades Isochrones and Comparison to Model Isochrones Young, nearby, rich open clusters like the Pleiades can and should be used to provide template data which can help interpret observations of more distant clusters or to test theoretical models. The identification of candidate members of distant open clusters is often based on plots of stars in a color-magnitude diagram, overlaid upon which is a line meant to define the single-star locus at the distance of the cluster. The stars lying near or slightly above the locus are chosen as possible or probable cluster members. The data we have collected for the Pleiades provide a means to define the single-star locus for 100 Myr, solar metallicity stars in a variety of widely used color systems down to and slightly below the hydrogen burning mass limit. Figure 11 and Figure 12 illustrate the appearance of the Pleiades stars in two of these diagrams, and the single-star locus we have defined. The curve defining the single-star locus was drawn entirely “by eye.” It is displaced slightly above the lower envelope to the locus of stars to account for photometric uncertainties (which increase to fainter magnitudes). We attempted to use all of the information available to us, however. That is, there should also be an upper envelope to the Pleiades locus in these diagrams, since equal mass binaries should be displaced above the single star sequence by 0.7 magnitudes (and one expects very few systems of higher multiplicity). Therefore, the single star locus was defined with that upper envelope in mind. Table 4 provides the single-star loci for the Pleiades for BV IcJKs plus the four IRAC channels. We have dereddened the empirical loci by the canonical mean extinction to the Pleiades of AV = 0.12 (and, correspondingly, AB = 0.16, AI = 0.07, AJ = 0.03, AK = 0.01, as per the reddening law of Rieke & Lebofsky (1985)). The other benefit to constructing the new catalog is that it can provide an improved comparison dataset to test theoretical isochrones. The new catalog provides homogeneous photometry in many photometric bands for stars ranging from several solar masses down to below 0.1 M⊙. We take the distance to the Pleiades as 133 pc, and refer the reader to Soderblom et al. (2005) for a discussion and a listing of the most recent determinations. The age of the Pleiades is not as well-defined, but is probably somewhere between 100 and 125 Myr (Meynet, Mermilliod, & Maeder 1993; Stauffer et al. 1998). We adopt 100 Myr for the purposes of this discussion; our conclusions relative to the theoretical isochrones would not be affected significantly if we instead chose 125 Myr. As noted above, we adopt AV=0.12 as the – 11 – mean Pleiades extinction, and apply that value to the theoretical isochrones. A small number of Pleiades members have significantly larger extinctions (Breger 1986; Stauffer & Hartmann 1987), and we have dereddened those stars individually to the mean cluster reddening. Figures 13 and 14 compare theoretical 100 Myr isochrones from Siess et al. (2000) and Baraffe et al. (1998) to the Pleiades member photometry from Table 2 for stars for which we have photoelectric photometry. Neither set of isochrones are a good fit to the V − I based color-magnitude diagram. For Baraffe et al. (1998) this is not a surprise because they illustrated that their isochrones are too blue in V−I for cool stars in their paper, and ascribed the problem as likely the result of an incomplete line list, resulting in too little absorption in the V band. For Siess et al. (2000), the poor fit in the V − I CMD is somewhat unexpected in that they transform from the theoretical to the observational plane using empirical color- temperature relations. In any event, it is clear that neither model isochrones match the shape of the Pleiades locus in the V vs. V − I plane, and therefore use of these V − I based isochrones for younger clusters is not likely to yield accurate results (unless the color-Teff relation is recalibrated, as described for example in Jeffries & Oliveira (2005)). On the other hand, the Baraffe et al. (1998) model provides a quite good fit to the Pleiades single star locus for an age of 100 Myr in the K vs. I − K plane.2. This perhaps lends support to the hypothesis that the misfit in the V vs. V − I plane is due to missing opacity in their V band atmospheres for low mass stars (see also Chabrier et al. (2000) for further evidence in support of this idea). The Siess et al. (2000) isochrones do not fit the Pleiades locus in the K vs. I − K plane particularly well, being too faint near I − K=2 and too bright for I −K > 2.5. 5. Identification of New Very Low Mass Candidate Members The highest spatial density for Pleiades members of any mass should be at the cluster center. However, searches for substellar members of the Pleiades have generally avoided the cluster center because of the deleterious effects of scattered light from the high mass cluster members and because of the variable background from the Pleiades reflection nebulae. The deep 2MASS and IRAC 3.6 and 4.5 µm imaging provide accurate photometry to well below the hydrogen burning mass limit, and are less affected by the nebular emission than shorter wavelength images. We therefore expect that it should be possible to identify a new 2These isochrones are calculated for the standard K filter, rather than Ks. However, the difference in location of the isochrones in these plots because of this should be very slight, and we do not believe our conclusions are significantly affected. – 12 – set of candidate Pleiades substellar members by combining our new near and mid-infrared photometry. The substellar mass limit in the Pleiades occurs at about Ks=14.4, near the limit of the 2MASS All-Sky PSC. As illustrated in Figure 2, the deep 2MASS survey of the Pleiades should easily detect objects at least two magnitudes fainter than the substellar limit. The key to actually identifying those objects and separating them from the background sources is to find color-magnitude or color-color diagrams which separate the Pleiades members from the other objects. As shown in Figure 15, late-type Pleiades members separate fairly well from most field stars towards the Pleiades in a Ks vs. Ks − [3.6] color-magnitude diagram. However, as illustrated in Figure 2, in the Ks magnitude range of interest there is also a large population of red galaxies, and they are in fact the primary contaminants to identi- fying Pleiades substellar objects in the Ks vs. Ks − [3.6] plane. Fortunately, most of the contaminant galaxies are slightly resolved in the 2MASS and IRAC imaging, and we have found that we can eliminate most of the red galaxies by their non-stellar image shape. Figure 15 shows the first step in our process of identifying new very low mass members of the Pleiades. The red plus symbols are the known Pleiades members from Table 2. The red open circles are candidate Pleiades substellar members from deep imaging surveys published in the literature, mostly of parts of the cluster exterior to the central square degree, where the IRAC photometry is from Lowrance et al. (2007). The blue, filled circles are field M and L dwarfs, placed at the distance of the Pleiades, using photometry from Patten et al. (2006). Because the Pleiades is ∼100 Myr, its very low mass stellar and substellar objects will be displaced about 0.7 mag above the locus of the field M and L dwarfs according to the Baraffe et al. (1998) and Chabrier et al. (2000) models, in accord with the location in the diagram of the previously identified, candidate VLM and substellar objects. The trapezoidal shaped region outlined with a dashed line is the region in the diagram which we define as containing candidate new VLM and substellar members of the Pleiades. We place the faint limit of this region at Ks=16.2 in order to avoid the large apparent increase in faint, red objects for Ks> 16.2, caused largely by increasing errors in the Ks photometry. Also, the 2MASS extended object flags cease to be useful fainter than about Ks= 16. We took the following steps to identify a set of candidate substellar members of the Pleiades: • keep only objects which fall in the trapezoidal region in Figure 15. • remove objects flagged as non-stellar by the 2MASS pipeline software; • remove objects which appear non-stellar to the eye in the IRAC images; – 13 – • remove objects which do not fall in or near the locus of field M and L dwarfs in a J−H vs. H −Ks diagram; • remove objects which have 3.6 and 4.5 µm magnitudes that differ by more than 0.2 • remove objects which fall below the ZAMS in a J vs. J −Ks diagram. As shown in Figure 15, all stars earlier than about mid-M have Ks − [3.6] colors bluer than 0.4. This ensures that for most of the area of the trapezoidal region, the primary contaminants are distant galaxies. Fortunately, the 2MASS catalog provides two types of flags for identifying extended objects. For each filter, a chi-square flag measures the match between the objects shape and the instrumental PSF, with values greater than 2.0 generally indicative of a non-stellar object. In order not to be misguided by an image artifact in one filter, we throw out the most discrepant of the three flags and average the other two. We discard objects with mean χ2 greater than 1.9. The other indicator is the 2MASS extended object flag, which is the synthesis of several independent tests of the objects shape, surface brightness and color (see Jarrett, T. et al (2000) for a description of this process). If one simply excludes the objects classified as extended in the 2MASS 6x image by either of these techniques, the number of candidate VLM and substellar objects lying inside the trapezoidal region decreases by nearly a half. We have one additional means to demonstrate that many of the identified objects are probably Pleiades members, and that is via proper motions. The mean Pleiades proper motion is ∆RA = 20 mas yr−1 and ∆Dec = −45 mas yr−1 (Jones 1973). With an epoch difference of only 3.5 years between the deep 2MASS and IRAC imaging, the expected motion for a Pleiades member is only 0.07 arcseconds in RA and −0.16 arcseconds in Dec. Given the relatively large pixel size for the two cameras, and the undersampled nature of the IRAC 3.6 and 4.5 µm images, it is not a priori obvious that one would expect to reliably detect the Pleiades motion. However, both the 2MASS and IRAC astrometric solutions have been very accurately calibrated. Also, for the present purpose, we only ask whether the data support a conclusion that most of the identified substellar candidates are true Pleiades members (i.e., as an ensemble), rather than that each star is well enough separated in a VPD to derive a high membership probability. Figure 16 provides a set of plots that we believe support the conclusion that the majority of the surviving VLM and substellar candidates are Pleiades members. The first plot shows the measured motions between the epoch of the 2MASS and IRAC observations for all known Pleiades members from Table 2 that lie in the central square degree region and have 11 < Ks < 14 (i.e., just brighter than the substellar candidates). The mean offset of the Pleiades – 14 – stellar members from the background population is well-defined and is quantitatively of the expected magnitude and sign (+0.07 arcsec in RA and −0.16 arcsec in Dec). The RMS dispersion of the coordinate difference for the field population in RA and Dec is 0.076 and 0.062 arcseconds, supportive of our claim that the relative astrometry for the two cameras is quite good. Because we expect that the background population should have essentially no mean proper motion, the non-zero mean “motion” of the field population of about < ∆RA>=0.3 arcseconds is presumably not real. Instead, the offset is probably due to the uncertainty in transferring the Spitzer coordinate zero-point between the warm star-tracker and the cryogenic focal plane. Because it is simply a zero-point offset applicable to all the objects in the IRAC catalog, it has no effect on the ability to separate Pleiades members from the field star population. The second panel in Figure 16 shows the proper motion of the candidate Pleiades VLM and substellar objects. While these objects do not show as clean a distribution as the known members, their mean motion is clearly in the same direction. After removing 2-σ deviants, the median offsets for the substellar candidates are 0.04 and −0.11 arcseconds in RA and Dec, respectively. The objects whose motions differ significantly from the Pleiades mean may be non-members or they may be members with poorly determined motions (since a few of the high probability members in the first panel also show discrepant motions). The other two panels in Figure 16 show the proper motions of two possible control samples. The first control sample was defined as the set of stars that fall up to 0.3 magnitudes below the lower sloping boundary of the trapezoid in Figure 15. These objects should be late type dwarfs that are either older or more distant than the Pleiades or red galaxies. We used the 2MASS data to remove extended or blended objects from the sample in the same way as for the Pleiades candidates. If the objects are nearby field stars, we expect to see large proper motions; if galaxies, the real proper motions would be small – but relatively large apparent proper motions due to poor centroiding or different centroids at different effective wavelengths could be present. The second control set was defined to have −0.1 < K − [3.6] < 0.1 and 14.0 < K < 14.5, and to be stellar based on the 2MASS flags. This control sample should therefore be relatively distant G and K dwarfs primarily. Both control samples have proper motion distributions that differ greatly from the Pleiades samples and that make sense for, respectively, a nearby and a distant field star sample. Figure 17 shows the Pleiades members from Table 2 and the 55 candidate VLM and substellar members that survived all of our culling steps. We cross-correlated this list with the stars from Table 2 and with a list of the previously identified candidate substellar members of the cluster from other deep imaging surveys. Fourteen of the surviving objects correspond to previously identified Pleiades VLM and substellar candidates. We provide the new list – 15 – of candidate members in Table 5. The columns marked as µ(RA) and µ(DEC) are the measured motions, in arcsec over the 3.5 year epoch difference between the 2MASS-6x and IRAC observations. Forty-two of these objects have Ks> 14.0, and hence inferred masses less than about 0.1 M⊙; thirty-one of them have Ks> 14.4, and hence have inferred masses below the hydrogen burning mass limit. Our candidate list could be contaminated by foreground late type dwarfs that happen to lie in the line of sight to the Pleiades. How many such objects should we expect? In order to pass our culling steps, such stars would have to be mid to late M dwarfs, or early to mid L dwarfs. We use the known M dwarfs within 8 pc to estimate how many field M dwarfs should lie in a one square degree region and at distance between 70 and 100 parsecs (so they would be coincident in a CMD with the 100 Myr Pleiades members). The result is ∼3 such field M dwarf contaminants. Cruz et al. (2006) estimate that the volume density of L dwarfs is comparable to that for late-M dwarfs, and therefore a very conservative estimate is that there might also be 3 field L dwarfs contaminating our sample. We regard this (6 contaminating field dwarfs) as an upper limit because our various selection criteria would exclude early M dwarfs and late L dwarfs. Bihain et al. (2006) made an estimate of the number of contaminating field dwarfs in their Pleiades survey of 1.8 square degrees; for the spectral type range of our objects, their algorithm would have predicted just one or two contaminating field dwarfs for our survey. How many substellar Pleiades members should there be in the region we have surveyed? That is, of course, part of the question we are trying to answer. However, previous studies have estimated that the Pleiades stellar mass function for M < 0.5 M⊙ can be approximated as a power-law with an exponent of -1 (dN/dM ∝ M−1). Using the known Pleiades members from Table 2 that lie within the region of the IRAC survey and that have masses of 0.2 < M/M⊙< 0.5 (as estimated from the Baraffe et al. (1998) 100 Myr isochrone) to normalize the relation, the M−1 mass function predicts about 48 members in our search region and with 14 < K < 16.2 (corresponding to 0.1 < M/M⊙< 0.035). Other studies have suggested that the mass function in the Pleiades becomes shallower below 0.1 M⊙, dN/dM ∝ M −0.6. Using the same normalization as above, this functional form for the Pleiades mass function for M < 0.1 M⊙ yields a prediction of 20 VLM and substellar members in our survey. The number of candidates we have found falls between these two estimates. Better proper motions and low-resolution spectroscopy will almost certaintly eliminate some of these candidates as non-members. – 16 – 6. Mid-IR Observations of Dust and PAHS in the Pleiades Since the earliest days of astrophotography, it has been clear that the Pleiades stars are in relatively close proximity to interstellar matter whose optical manifestation is the spider-web like network of filaments seen particularly strongly towards several of the B stars in the cluster. High resolution spectra of the brightest Pleiades stars as well as CO maps towards the cluster show that there is gas as well as dust present, and that the (primary) interstellar cloud has a significant radial velocity offset relative to the Pleiades (White 2003; Federman & Willson 1984). The gas and dust, therefore, are not a remnant from the forma- tion of the cluster but are simply evidence of a a transitory event as this small cloud passes by the cluster in our line of sight (see also Breger (1986)). There are at least two claimed mor- phological signatures of a direct interaction of the Pleiades with the cloud. White & Bally (1993) provided evidence that the IRAS 60 and 100 µm image of the vicinity of the Pleiades showed a dark channel immediately to the east of the Pleiades, which they interpreted as the “wake” of the Pleiades as it plowed through the cloud from the east. Herbig & Simon (2001) provided a detailed analysis of the optically brightest nebular feature in the Pleiades – IC 349 (Barnard’s Merope nebula) – and concluded that the shape and structure of that nebula could best be understood if the cloud was running into the Pleiades from the south- east. Herbig & Simon (2001) concluded that the IC 349 cloudlet, and by extension the rest of the gas and dust enveloping the Pleiades, are relatively distant outliers of the Taurus molecular clouds (see also Eggen (1950) for a much earlier discussion ascribing the Merope nebulae as outliers of the Taurus clouds). White (2003) has more recently proposed a hybrid model, where there are two separate interstellar cloud complexes with very different space motions, both of which are colliding simultaneously with the Pleiades and with each other. Breger (1986) provided polarization measurements for a sample of member and back- ground stars towards the Pleiades, and argued that the variation in polarization signatures across the face of the cluster was evidence that some of the gas and dust was within the clus- ter. In particular, Figure 6 of that paper showed a fairly distinct interface region, with little residual polarization to the NE portion of the cluster and an L-shaped boundary running EW along the southern edge of the cluster and then north-south along the western edge of the cluster. Stars to the south and west of that boundary show relatively large polarizations and consistent angles (see also our Figure 5 where we provide a few polarization vectors from Breger (1986) to illustrate the location of the interface region and the fact that the position angle of the polarization correlates well with the location in the interface). There is a general correspondence between the polarization map and what is seen with IRAC, in the sense that the B stars in the NE portion of the cluster (Atlas and Alcyone) have little nebular emission in their vicinity, whereas those in the western part of the cluster – 17 – (Maia, Electra and Asterope) have prominent, filamentary dust emission in their vicinity. The L-shaped boundary is in fact visible in Figure 4 as enhanced nebular emission running between and below a line roughly joining Merope and Electra, and then making a right angle and running roughly parallel to a line running from Electra to Maia to HII1234 (see Figure 5). 6.1. Pleiades Dust-Star Encounters Imaged with IRAC The Pleiades dust filaments are most strongly evident in IRAC’s 8 µm channel, as evidenced by the distinct red color of the nebular features in Figure 4. The dominance at 8 µm is an expected feature of reflection nebulae, as exemplified by NGC 7023 (Werner et al. 2004), where most of the mid-infrared emission arises from polycyclic aromatic hydrocarbons (PAHs) whose strongest bands in the 3 to 10 µm region fall at 7.7 and 8.6 µm. One might expect that if portions of the passing cloud were particularly near to one of the Pleiades members, it might be possible to identify such interactions by searching for stars with 8.0 µm excesses or for stars with extended emission at 8 µm. Figure 18 provides two such plots. Four stars stand out as having significant extended 8 µm emission, with two of those stars also having an 8 µm excess based on their [3.6]−[8.0] color. All of these stars, plus IC 349, are located approximately along the interface region identified by Breger (1986). We have subtracted a PSF from the 8 µm images for the stars with extended emission, and those PSF-subtracted images are provided in Figure 19. The image for HII 1234 has the appearance of a bow-shock. The shape is reminiscent of predictions for what one should expect from a collision between a large cloud or a sheet of gas and an A star as described in Artymowicz & Clampin (1997). The Artymowicz & Clampin (1997) model posits that A stars encountering a cloud will carve a paraboloidal shaped cavity in the cloud via radi- ation pressure. The exact size and shape of the cavity depend on the relative velocity of the encounter, the star’s mass and luminosity and properties of the ISM grains. For typical parameters, the predicted characteristic size of the cavity is of order 1000 AU, quite compa- rable to the size of the structures around HII 652 and HII 1234. The observed appearance of the cavity depends on the view angle to the observer. However, in any case, the direction from which the gas is moving relative to the star can be inferred from the location of the star relative to the curved rim of the cavity; the “wind” originates approximately from the direction connecting the star and the apex of the rim. For HII 1234, this indicates the cloud which it is encountering has a motion relative to HII 1234 from the SSE, in accord with a Taurus origin and not in accord for where a cloud is impacting the Pleiades from the west as posited in White (2003). The nebular emission for HII 652 is less strongly bow-shaped, – 18 – but the peak of the excess emission is displaced roughly southward from the star, consistent with the Taurus model and inconsistent with gas flowing from the west. Despite being the brightest part of the Pleiades nebulae in the optical, IC 349 appears to be undetected in the 8 µm image. This is not because the 8 µm image is insensitive to the nebular emission - there is generally good agreement between the structures seen in the optical and at 8 µm, and most of the filaments present in optical images of the Pleiades are also visible on the 8 µm image (see Figures 4 and 19) and even the psf-subtracted image of Merope shows well-defined nebular filaments. The lack of enhanced 8 µm emission from the region of IC 349 is probably because all of the small particles have been scoured away from this cloudlet, consistent with Herbig’s model to explain the HST surface photometry and colors. There is no PAH emission from IC 349 because there are none of the small molecules that are the postulated source of the PAH emission. IC349 is very bright in the optical, and undetected to a good sensitivity limit at 8 µm; it must be detectable via imaging at some wavelength between 5000 Å and 8 µm. We checked our 3.6 µm data for this purpose. In the standard BCD mosaic image, we were unable to discern an excess at the location of IC349 either simply by displaying the image with various stretches or by doing cuts through the image. We performed a PSF subtraction of Merope from the image in order to attempt to improve our ability to detect faint, extended emission 30” from Merope - unfortunately, bright stars have ghost images in IRAC Ch. 1, and in this case the ghost image falls almost exactly at the location of IC349. IC349 is also not detected in visual inspection of our 2MASS 6x images. 6.2. Circumstellar Disks and IRAC As part of the Spitzer FEPS (Formation and Evolution of Planetary Systems) Legacy program, using pointed MIPS photometry, Stauffer et al. (2005) identified three G dwarfs in the Pleiades as having 24 µm excesses probably indicative of circumstellar dust disks. Gorlova et al. (2006) reported results of a MIPS GTO survey of the Pleiades, and identified nine cluster members that appear to have 24 µm excesses due to circumstellar disks. However, it is possible that in a few cases these apparent excesses could be due instead to a knot of the passing interstellar dust impacting the cluster member, or that the 24 µm excess could be flux from a background galaxy projected onto the line of sight to the Pleiades member. Careful analysis of the IRAC images of these cluster members may help confirm that the MIPS excesses are evidence for debris disks rather than the other possible explanations. Six of the Pleiades members with probable 24 µm excesses are included in the region – 19 – mapped with IRAC. However, only four of them have data at 8 µm – the other two fall near the edge of the mapped region and only have data at 3.6 and 5.8 µm. None of the six stars appear to have significant local nebular dust from visual inspection of the IRAC mosaic images. Also, none of them appear problematic in Figure 18. For a slightly more quantitative analysis of possible nebular contamination, we also constructed aperture growth curves for the six stars, and compared them to other Pleiades members. All but one of the six show aperture growth curves that are normal and consistent with the expected IRAC PSF. The one exception is HII 489, which has a slight excess at large aperture sizes as is illustrated in Figure 20. Because HII 489 only has a small 24 µm excess, it is possible that the 24 µm excess is due to a local knot of the interstellar cloud material and is not due to a debris disk. For the other five 24 µm excess stars we find no such problem, and we conclude that their 24 µm excesses are indeed best explained as due to debris disks. 7. Summary and Conclusions We have collated the primary membership catalogs for the Pleiades to produce the first catalog of the cluster extending from its highest mass members to the substellar limit. At the bright end, we expect this catalog to be essentially complete and with few or no non-member contaminants. At the faint end, the data establishing membership are much sparser, and we expect a significant number of objects will be non-members. We hope that the creation of this catalog will spur efforts to obtain accurate radial velocities and proper motions for the faint candidate members in order to eventually provide a well-vetted membership catalog for the stellar members of the Pleiades. Towards that end, it would be useful to update the current catalog with other data – such as radial velocities, lithium equivalent widths, x-ray fluxes, Hα equivalent widths, etc. – which could be used to help accurately establish membership for the low mass cluster candidates. It is also possible to make more use of “negative information” present in the proper motion catalogs. That is, if a member from one catalog is not included in another study but does fall within its areal and luminosity coverage, that suggests that it likely failed the membership criteria of the second study. For a few individual stars, we have done this type of comparison, but a systematic analysis of the proper motion catalogs should be conducted. We intend to undertake these tasks, and plan to establish a website where these data would be hosted. We have used the new Pleiades member catalog to define the single-star locus at 100 Myr for BV IcKs and the four IRAC bands. These curves can be used as empirical calibration curves when attempting to identify members of less well-studied, more distant clusters of similar age. We compared the Pleiades photometry to theoretical isochrones from Siess et al. – 20 – (2000) and Baraffe et al. (1998). The Siess et al. (2000) isochrones are not, in detail, a good fit to the Pleiades photometry, particularly for low mass stars. The Baraffe et al. (1998) 100 Myr isochrone does fit the Pleiades photometry very well in the I vs. I −K plane. We have identified 31 new substellar candidate members of the Pleiades using our com- bined seven-band infrared photometry, and have shown that the majority of these objects appear to share the Pleiades proper motion. We believe that most of the objects that may be contaminating our list of candidate brown dwarfs are likely to be unresolved galaxies, and therefore low resolution spectroscopy should be able to provide a good criterion for culling our list of non-members. The IRAC images, particularly the 8 µm mosaic, provide vivid evidence of the strong in- teraction of the Pleiades stars and the interstellar cloud that is passing through the Pleiades. Our data are supportive of the model proposed by Herbig & Simon (2001) whereby the pass- ing cloud is part of the Taurus cloud complex and hence is encountering the Pleiades from the SSE direction. White & Bally (1993) had proposed a model whereby the cloud was encountering the Pleiades from the west and used this to explain features in the IRAS 60 and 100 µm images of the region as the wake of the Pleiades moving through the cloud. Our data appear to not be supportive of that hypothesis, and therefore leaves the apparent structure in the IRAS maps as unexplained. Most of the support for this work was provided by the Jet Propulsion Laboratory, Cal- ifornia Institute of Technology, under NASA contract 1407. This research has made use of NASA’s Astrophysics Data System (ADS) Abstract Service, and of the SIMBAD database, operated at CDS, Strasbourg, France. This research has made use of data products from the Two Micron All-Sky Survey (2MASS), which is a joint project of the University of Mas- sachusetts and the Infrared Processing and Analysis Center, funded by the National Aero- nautics and Space Administration and the National Science Foundation. These data were served by the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The research described in this paper was partially carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. – 21 – A. APPENDIX A.1. Membership Catalogs Membership lists of the Pleiades date back to antiquity if one includes historical and literary references to the Seven Sisters (Alcyone, Maia, Merope, Electra, Taygeta, Asterope and Celeno) and their parents (Atlas and Pleione). The first paper discussing relative proper motions of a large sample of stars in the Pleiades (based on visual observations) was published by Pritchard (1884). The best of the early proper motion surveys of the Pleiades derived from photographic plate astrometry was that by Trumpler (1921), based on plates obtained at Yerkes and Lick observatories. The candidate members from that survey were presented in two tables, with the first being devoted to candidate members within about one degree from the cluster center (operationally, within one degree from Alcyone) and the second table being devoted to candidates further than one degree from the cluster center. Most of the latter stars were denoted by Trumpler by an S or R, followed by an identification number. We use Tr to designate the Trumpler stars (hence Trnnn for a star from the 1st table and the small number of stars in the second table without an “S” or an “R”, and TrSnnn or TrRnnn for the other stars). For the central region, Trumpler’s catalog extends to V ∼ 13, while the outer region catalog includes stars only to about V ∼ 9. The most heavily referenced proper motion catalog of the Pleiades is that provided by Hertzsprung (1947). That paper makes reference to two separate catalogs: a photometric catalog of the Pleiades published by Hertzsprung in 1923 (Hertzsprung 1923), whose members are commonly referred to by HI numbers, and the new proper motion catalog from the 1947 paper, commonly referenced as the HII catalog. While both HI and HII numbers have been used in subsequent observational papers, it is the HII identification numbers that predominate. That catalog – derived from Carte du Ciel blue-sensitive plates from 14 observatories – includes stars in the central 2×2 degree region of the cluster, and has a faint limit of about V = 15.5. Johnson system BV I photometry is provided for most of the proposed Hertzsprung members in Johnson & Mitchell (1958) and Iriarte (1967). Additional Johnson B and V photometry plus Kron I photometry for a fairly large number of the Hertzsprung members can be found in Stauffer (1980), Stauffer (1982), and Stauffer (1984). Other Johnson BV photometry for a scattering of stars can be found in Jones (1973), Robinson & Kraft (1974), Messina (2001). Spectroscopic confirmation, primarily via radial velocities, that these are indeed Pleiades members has been provided in Soderblom et al. (1993); Queloz et al. (1998) and Mermilliod et al. (1997). Two other proper motion surveys provide relatively bright candidate members relatively far from the cluster center: Artyukhina & Kalinina (1970) and van Leeuwen, Alphenaar, & Brand – 22 – (1986). Stars from the Artyukhina catalog are designated as AK followed by the region from which the star was identified followed by an identification number. The new members pro- vided in the van Leeuwen paper were taken from an otherwise unpublished proper motion study by Pels, where the first 118 stars were considered probable members and the remaining 75 stars were considered possible members. Van Leeuwen categorized a number of the Pels stars as non-members based on the Walraven photometry they obtained, and we adopt those findings. Radial velocities for stars in these two catalogs have been obtained by Rosvick et al (1992), Mermilliod et al. (1997), and Queloz et al. (1998), and those authors identified a list of the candidate members that they considered confirmed by the high resolution spectroscopy. For these outlying candidate members, to be included in Table 2 we require that the star be a radial velocity member from one of the above three surveys, or be indicated as having “no dip” in the Coravel cross-correlation (indicating rapid rotation, which at least for the later type stars is suggestive of membership). Geneva photometry of the Artyukhina stars considered as likely members was provided by Mermilliod et al. (1997). The magnitude limit of these surveys was not well-defined, but most of the Artyukhina and Pels stars are brighter than V=13. Jones (1973) provided proper motion membership probabilities for a large sample of proposed Pleiades members, and for a set of faint, red stars towards the Pleiades. A few star identification names from the sources considered by Jones appear in Table 2, including MT (McCarthy & Treanor 1964), VM (van Maanen 1946), ALR (Ahmed et al. 1965), and J (Jones 1973). The chronologically next significant source of new Pleiades candidate members was the flare star survey of the Pleiades conducted at several observatories in the 1960s, and summarized in Haro et al. (1982), hereafter HCG. The logic behind these surveys was that even at 100 Myr, late type dwarfs have relatively frequent and relatively high luminosity flares (as demonstrated by Johnson & Mitchell (1958) having detected two flares during their photometric observations of the Pleiades), and therefore wide area, rapid cadence imaging of the Pleiades at blue wavelengths should be capable of identifying low mass cluster members. However, such surveys also will detect relatively young field dwarfs, and therefore it is best to combine the flare star surveys with proper motions. Dedicated proper motion surveys of the HCG flare stars were conducted by Jones (1981) and Stauffer et al. (1991), with the latter also providing photographic V I photometry (Kron system). Photoelectric photometry for some of the HCG stars have been reported in Stauffer (1982), Stauffer (1984), Stauffer & Hartmann (1987), and Prosser et al. (1991). High resolution spectroscopy of many of the HCG stars is reported in Stauffer (1984), Stauffer & Hartmann (1987) and Terndrup et al. (2000). Because a number of the papers providing additional observational data for the flare stars were obtained prior to 1982, we also include in Table 2 the original – 23 – flare star names which were derived from the observatory where the initial flare was detected. Those names are of the form an initial letter indicating the observatory – A (Asiago), B (Byurakan), K (Konkoly), T (Tonantzintla) – followed by an identification number. Stauffer et al. (1991) conducted two proper motion surveys of the Pleiades over an approximately 4×4 degree region of the cluster based on plates obtained with the Lick 20′′ astrographic telescope. The first survey was essentially unbiased, except for the requirement that the stars fall approximately in the region of the V vs. V − I color-magnitude diagram where Pleiades members should lie. Candidate members from this survey are designated by SK numbers. The second survey was a proper motion survey of the HCG stars. Photo- graphic V I photometry of all the stars was provided as well as proper motion membership probabilities. Photoelectric photometry for some of the candidate members was obtained as detailed above in the section on the HCG catalog stars. The faint limit of these surveys is about V=18. Hambly et al. (1991) provided a significantly deeper, somewhat wider area proper mo- tion survey, with the faintest members having V ≃ 20 and the total area covered being of order 25 square degrees. The survey utilized red sensitive plates from the Palomar and UK Schmidt telescopes. Due to incomplete coverage at one epoch, there is a vertical swath slightly east of the cluster center where no membership information is available. Stars from this survey are designated by their HHJ numbers . Hambly et al. (1993) provide RI photo- graphic photometry on a natural system for all of their candidate members, plus photoelectric Cousins RI photometry for a small number of stars and JHK photometry for a larger sam- ple. Some spectroscopy to confirm membership has been reported in Stauffer et al. (1994), Stauffer et al. (1995), Oppenheimer et al. (1997), Stauffer et al. (1998), and Steele et al. (1995), though for most of the HHJ stars there is no spectroscopic membership confirma- tion. Pinfield et al. (2000) provide the deepest wide-field proper motion survey of the Pleiades. That survey combines CCD imaging of six square degrees of the Pleiades obtained with the Burrell Schmidt telescope (as five separate, non-overlapping fields near but outside the cluster center) with deep photographic plates which provide the 1st epoch positions. Candidate members are designated by BPL numbers (for Burrell Pleiades), with the faintest stars having I ≃ 19.5, corresponding to V > 23. Only the stars brighter than about I= 17 have sufficiently accurate proper motions to use to identify Pleiades members. Fainter than I= 17, the primary selection criteria are that the star fall in an appropriate place in both an I vs. I − Z and an I vs. I −K CMD. Adams et al. (2001) combined the 2MASS and digitized POSS databases to produce a very wide area proper motion survey of the Pleiades. By design, that survey was very inclu- – 24 – sive - covering the entire physical area of the cluster and extending to the hydrogen burning mass limit. However, it was also very “contaminated”, with many suspected non-members. The catalog of possible members was not published. We have therefore not included stars from this study in Table 2; we have used the proper motion data from Adams et al. (2001) to help decide cases where a given star has ambiguous membership data from the other surveys. Deacon & Hambly (2004) provided another deep and very wide area proper motion survey of the Pleiades. The survey covers a circular area of approximately five degrees radius to R ∼ 20, or V ∼ 22. Candidate members are designated by DH. Deacon & Hambly (2004) also provide membership probabilities based on proper motions for many candidate cluster members from previous surveys. For stars where Deacon & Hambly (2004) derive P < 0.1 and where we have no other proper motion information or where another proper motion survey also finds low membership probability, we exclude the star from our catalog. For cases where two of our proper motion catalogs differ significantly in their membership assessment, with one survey indicating the star is a probable member, we retain the star in the catalog as the conservative choice. Examples of the latter where Deacon & Hambly (2004) derive P < 0.1 include HII 1553, HII 2147, HII 2278 and HII 2665 – all of which we retain in our catalog because other surveys indicate these are high probability Pleiades members. A.2. Photometry Photometry for stars in open cluster catalogs can be used to help confirm cluster mem- bership and to help constrain physical properties of those stars or of the cluster. For a variety of reasons, photometry of stars in the Pleiades has been obtained in a panoply of different photometric systems. For our own goals, which are to use the photometry to help verify membership and to define the Pleiades single-star locus in color magnitude diagrams, we have attempted to convert photometry in several of these systems to a common sys- tem (Johnson BV and Cousins I). We detail below the sources of the photometry and the conversions we have employed. Photoelectric photometry of Pleiades members dates back to at least 1921 (Cummings 1921). However, as far as we are aware the first “modern” photoelectric photometry for the Pleiades, using a potassium hydride photoelectric cell, is that of Calder & Shapley (1937). Eggen (1950) provided photoelectric photometry using a 1P21 phototube (but calibrated to a no-longer-used photographic system) for most of the known Pleiades members within one degree of the cluster center and with magnitudes < 11. The first phototube photom- – 25 – etry of Pleiades stars calibrated more-or-less to the modern UBV system was provided by Johnson & Morgan (1951). An update of that paper, and the oldest photometry included here was reported in Johnson & Mitchell (1958), which provided UBV Johnson system pho- tometry for a large sample of HII and Trumpler candidate Pleiades members. Iriarte (1967) later reported Johnson system V − I colors for most of these stars. We have converted Iriarte’s V − I photometry to estimated Cousins V − I colors using a formula from Bessell (1979): V − I(Cousins) = 0.778× V − I(Johnson). (A1) BV RI photometry for most of the Hertzsprung members fainter than V= 10 has been published by Stauffer (1980), Stauffer (1982), Stauffer (1984), and Stauffer & Hartmann (1987). The BV photometry is Johnson system, whereas the RI photometry is on the Kron system. The Kron V − I colors were converted to Cousins V − I using a transformation provided by Bessell & Weis (1987): V − I(Cousins) = 0.227 + 0.9567(V − I)k + 0.0128(V − I) k − 0.0053(V − I) k (A2) Other Kron system V−I colors have been published for Pleiades candidates in Stauffer et al. (1991) (photographic photometry) and in Prosser et al. (1991). These Kron-system colors have also been converted to Cousins V − I using the above formula. Johnson/Cousins UBV R photometry for a set of low mass Pleiades members was pro- vided by Landolt (1979). We only use the BV magnitudes from that study. Additional John- son system UBV photometry for small numbers of stars is provided in Robinson & Kraft (1974), Messina (2001) and Jones (1973). van Leeuwen, Alphenaar, & Meys (1987) provided Walraven V BLUW photometry for nearly all of the Hertzsprung members brighter than V ∼ 13.5 and for the Pels candidate members. Van Leeuwen provided an estimated Johnson V derived from the Walraven V in his tables. We have transformed the Walraven V − B color into an estimate of Johnson B − V using a formula from Rosvick et al (1992): B − V (Johnson) = 2.571(V − B)− 1.02(V −B)2 + 0.5(V − B)3 − 0.01 (A3) Hambly et al. (1993) provided photographic V RI photometry for all of the HHJ candidate members, and V RI Cousins photoelectric photometry for a small fraction of those stars. We took all of the HHJ stars with photographic photometry for which we also have photoelectric V I photometry on the Cousins system, and plotted V (Cousins) vs. V (HHJ) and I(Cousins) vs. I(HHJ). While there is some evidence for slight systematic departures of the HHJ photo- graphic photometry from the Cousins system, those departures are relatively small and we have chosen simply to retain the HHJ values and treat them as Cousins system. – 26 – Pinfield et al. (2000) reported their I magnitudes in an instrumental system which they designated as Ikp. We identified all BPL candidate members for which we had photoelectric Cousins I estimates, and plotted Ikp vs. IC. Figure 21 shows this correlation, and the piecewise linear fit we have made to convert from Ikp to IC. Our catalog lists these converted IC measures for the BPL stars for which we have no other photoelectric I estimates. Deacon & Hambly (2004) derived RI photometry from the scans of their plates, and calibrated that photometry by reference to published photometry from the literature. When we plotted their the difference between their I band photometry and literature values (where available), we discovered a significant dependence on right ascension. Unfortunately, because the DH survey extended over larger spatial scales than the calibrating photometry, we could not derive a correction which we could apply to all the DH stars. We therefore developed the following indirect scheme. We used the stars for which we have estimated IC magnitudes (from photoelectric photometry) to define the relation between J and (IC−J) for Pleiades members. For each DH star, we combined that relation and the 2MASS J magnitude to yield a predicted IC. Figure 22 shows a plot of the difference of this predicted IC and I(DH) with right ascension. The solid line shows the relation we adopt. Figure 23 shows the relation between the corrected I(DH) values and Table 2 IC measures from photoelectric sources. There is still a significant amount of scatter but the corrected I(DH) photometry appears to be accurately calibrated to the Cousins system. In a very few cases (specifically, just five stars), we provide an estimate of Ic based on data from a wide-area CCD survey of Taurus obtained with the Quest-2 camera on the Palomar 48 inch Samuel Oschin telescope (Slesnick et al. 2006). That survey calibrated their photometry to the Sloan i system, and we have converted the Sloan i magnitudes to Ic. We intend to make more complete use of the Quest-2 data in a subsequent paper. When we have multiple sources of photometry for a given star, we consider how to com- bine them. In most cases, if we have photoelectric data, that is given preference. However, if we have photographic V and I, and only a photoelectric measurement for I, we do not replace the photographic I with the photoelectric value because these stars are variable and the photographic measurements are at least in some cases from nearly simultaneous expo- sures. Where we have multiple sources for photoelectric photometry, and no strong reason to favor one measurement or set of measurements over another, we have averaged the pho- tometry for a given star. In most cases, where we have multiple photometry the individual measurements agree reasonably well but with the caveat that the Pleiades low mass stars are in many cases heavily spotted and “active” chromospherically, and hence are photometrically variable. In a few cases, even given the expectation that spots and other phenomena may affect the photometry, there seems to be more discrepancy between reported V magnitudes – 27 – than we expect. We note two such cases here. We suspect these results indicate that at least some of the Pleiades low mass stars have long-term photometric variability larger than their short period (rotational) modulation. HII 882 has at least four presumably accurate V magnitude measurements reported in the literature. Those measures are: V=12.66 Johnson & Mitchell (1958); V=12.95 Stauffer (1982); V=12.898 van Leeuwen, Alphenaar, & Brand (1986); and V=12.62 Messina (2001). HII 345 has at least three presumably accurate V magnitude measurements. Those measurements are: V=11.65 Landolt (1979); V=11.73 van Leeuwen, Alphenaar, & Brand (1986); V=11.43 Messina (2001). At the bottom of Table 2, we provide a key to the source(s) of the optical photometry provided in the table. This research made use of the SIMBAD database operated at CDS, Strasbourg, France, and also of the NED and NStED databases operated at IPAC, Pasadena, USA. 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(1991) 16 14< V <18 225 SK Hambly et al. (1993) 23 10< I <17.5 440 HHJ Pinfield et al. (2000) 6 13.5< I <19.5 339 BPL Adams et al. (2001) 300 8< Ks <14.5 1200 ... Deacon & Hambly (2004) 75 10< R <19 916 DH aThe Trumpler paper is listed twice because there are two membership surveys included in that paper, with differing spatial coverages and different limiting magnitudes. Fig. 1.— Spatial coverage of the six times deeper “2MASS 6x” observations of the Pleiades. The 2MASS survey region is approximately centered on Alcyone, the most massive member of the Pleiades. The trapezoidal box roughly indicates the region covered with the shallow IRAC survey of the cluster core. The star symbols correspond to the brightest B star members of the cluster. The red points are the location of objects in the 2MASS 6x Point Source Catalog. Fig. 2.— Color-magnitude diagram for the Pleiades derived from the 2MASS 6x obser- vations. The red dots correspond to objects identified as unresolved, whereas the green dots correspond to extended sources (primarily background galaxies). The lack of green dots fainter than K = 16 is indicative that there is too few photons to identify sources as extended - the extragalactic population presumably increases to fainter magnitudes. Fig. 3.— As for Figure 2, except in this case the axes are J − H and H − Ks. The extragalactic objects are very red in both colors. Fig. 4.— FIGURE REMOVED TO FIT WITHIN ASTRO-PH FILESIZE GUIDELINES. See http://spider.ipac.caltech.edu/staff/stauffer/pleiades07/ for full-res version. Two-color (4.5 and 8.0 micron) mosaic of the central square degree of the Pleiades from the IRAC survey. North is approximately vertical, and East is approximately to the left. The bright star nearest the center is Alcyone; the bright star at the left of the mosaic is Atlas; and the bright star at the right of the mosaic is Electra. http://spider.ipac.caltech.edu/staff/stauffer/pleiades07/ Fig. 5.— Finding chart corresponding approximately to the region imaged with IRAC. The large, five-pointed stars are all of the Pleiades members brighter than V= 5.5. The small open circles correspond to other cluster members. Several stars with 8 µm excesses are labelled by their HII numbers, and are discussed further in Section 6. The short lines through several of the stars indicate the size and position angle of the residual optical polarization (after subtraction of a constant foreground component), as provided in Figure 6 of Breger (1986). Fig. 6.— Comparison of aperture photometry for Pleiades members derived from the IRAC 3.6 µm mosaic using the Spitzer APEX package and the IRAF implementation of DAOPHOT. Fig. 7.— Difference between aperture photometry for Pleiades members for IRAC channels 1 and 2. The [3.6]−[4.5] color begins to depart from essentially zero at magnitudes ∼10.5, corresponding approximately to spectral type M0 in the Pleiades. Fig. 8.— Ks vs. Ks −[4.5] CMD for Pleiades candidate members, illustrating why we have excluded HII 1695 from the final catalog of cluster members. The “X” symbol marks the location of HII 1695 in this diagram. Fig. 9.— Spatial plot of the candidate Pleiades members from Table 2. The large star symbols are members brighter than Ks= 6; the open circles are stars with 6 < Ks < 9; and the dots are candidate members fainter than Ks= 9. The solid line is parallel to the galactic plane. Fig. 10.— The cumulative radial density profiles for Pleiades members in several magnitude ranges: heavy, long dash – Ks < 6; dots – 6 < Ks < 9; short dash – 9 < Ks < 12; light, long dash – Ks > 12. Fig. 11.— V vs. (V − I)c CMD for Pleiades members with photoelectric photometry. The solid curve is the “by eye” fit to the single-star locus for Pleiades members. Fig. 12.— Ks vs. Ks − [3.6] CMD for Pleiades candidate members from Table 2 (dots) and from deeper imaging of a set of Pleiades VLM and brown dwarf candidate members from Lowrance et al. (2007) (squares). The solid curve is the single-star locus from Table 3. Fig. 13.— V vs. (V − I)c CMD for Pleiades candidate members from Table 2 for which we have photoelectric photometry, compared to theoretical isochrones from Siess et al. (2000) (left) and from Baraffe et al. (1998) (right). For the left panel, the curves correspond to 10, 50, 100 Myr and a ZAMS; the right panel includes curves for 50, 100 Myr and a ZAMS. Fig. 14.— K vs. (I −K) CMD for Pleiades candidate members from Table 2, compared to theoretical isochrones from Siess et al. (2000) (left) and from Baraffe et al. (1998) (right). The curves correspond to 50 Myr, 100 Myr and a ZAMS. Fig. 15.— Ks vs. Ks−[3.6] CMD for the objects in the central one square degree of the Pleiades, combining data from the IRAC shallow survey and 2MASS. The symbols are defined within the figure (and see text for details). The dashed-line box indicates the region within which we have searched for new candidate Pleiades VLM and substellar members. The solid curve is a DUSTY 100 Myr isochrone from Chabrier et al. (2000), for masses from 0.1 to 0.03 M⊙. Fig. 16.— Proper motion vector point diagrams (VPDs) for various stellar samples in the central one degree field, derived from combining the IRAC and 2MASS 6x observations. Top left: VPD comparing all objects in the field (small black dots) to Pleiades members with 11 < Ks < 14 (large blue dots). Top right: same, except the blue dots are the new candidate VLM and substellar Pleiades members. Bottom left: same, except the blue dots are a nearby, low mass field star sample from a box just blueward of the trapezoidal region in 15. Bottom right: VPD just showing a second, distant field star sample as described in the text. Fig. 17.— Same as Fig. 15, except that the new candidate VLM and substellar objects from Table 4 are now indicated as small, red squares. Fig. 18.— Two plots intended to isolate Pleiades members with excess and/or extended 8 µm emission. The plot with [3.6]−[8.0] micron colors shows data from Table 3 (and hence is for aperture sizes of 3 pixel and 2 pixel radius, respectively). The increased vertical spread in the plots at faint magnitudes is simply due to decreasing signal to noise at 8 µm. The numbers labelling stars with excesses are the HII identification numbers for those stars. Fig. 19.— Postage stamp images extracted from individual, 8 µm BCDs for the stars with extended 8 µm emission, from which we have subtracted an empirical PSF. Clockwise from the upper left, the stars shown are HII1234, HII859, Merope and HII652. The five-pointed star indicates the astrometric position of the star (often superposed on a few black pixels where the 8 µm image was saturated. The circle in the Merope image is centered on the location of IC349 and has diameter about 25” (the size of IC349 in the optical is of order 10” x 10”). Fig. 20.— Aperture growth curves from the 8 µm mosaic for stars with 24 µm excesses from Gorlova et al. (2006) and for a set of control objects (dashed curves). All of the objects have been scaled to common zero-point magnitudes for 9 pixel apertures, with the 24 µm excess stars offset from the control objects by 0.1 mag. The three Gorlova et al. (2006) stars with no excess at 8 µm are HII 996, HII 1284 and HII 2195. The Gorlova et al. (2006) star with a slight excess at 8 µm is HII 489. Fig. 21.— Calibration derived relating Ikp from Pinfield et al. (2000) and IC. The dots are stars for which we have both Ikp and IC measurements (small dots: photographic IC; large dots: photoelectric IC), and the solid line indicates the piecewise linear fit we use to convert the Ikp values to IC for stars for which we only have Ikp. Fig. 22.— Difference between the predicted IC and Deacon & Hambly (2004) I magnitude as a function of right ascension for the DH stars. No obvious dependence is present versus declination. Fig. 23.— Comparison of the recalibrated DH I photometry with estimates of IC for stars in Table 2 with photoelectric data. This figure "f1.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f2.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f3.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f4.jpg" is available in "jpg" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f5.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f6.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f7.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f8.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f9.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f10.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f11.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f12.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f13.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f14.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f15.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f16.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f17.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f18.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f19.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f20.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f21.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f22.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 This figure "f23.gif" is available in "gif" format from: http://arxiv.org/ps/0704.1832v1 http://arxiv.org/ps/0704.1832v1 Introduction New Observational Data 2MASS ``6x" Imaging of the Pleiades Shallow IRAC Imaging A Catalog of Pleiades Candidate Members Empirical Pleiades Isochrones and Comparison to Model Isochrones Identification of New Very Low Mass Candidate Members Mid-IR Observations of Dust and PAHS in the Pleiades Pleiades Dust-Star Encounters Imaged with IRAC Circumstellar Disks and IRAC Summary and Conclusions APPENDIX Membership Catalogs Photometry

0704.1833

Analysis of the 802.11e Enhanced Distributed Channel Access Function

Analysis of the 802.11e Enhanced Distributed Channel Access Function † Inanc Inan, Feyza Keceli, and Ender Ayanoglu Center for Pervasive Communications and Computing Department of Electrical Engineering and Computer Science The Henry Samueli School of Engineering University of California, Irvine, 92697-2625 Email: {iinan, fkeceli, ayanoglu}@uci.edu Abstract The IEEE 802.11e standard revises the Medium Access Control (MAC) layer of the former IEEE 802.11 standard for Quality-of-Service (QoS) provision in the Wireless Local Area Networks (WLANs). The Enhanced Distributed Channel Access (EDCA) function of 802.11e defines multiple Access Categories (AC) with AC-specific Contention Window (CW) sizes, Arbitration Interframe Space (AIFS) values, and Transmit Opportunity (TXOP) limits to support MAC-level QoS and prioritization. We propose an analytical model for the EDCA function which incorporates an accurate CW, AIFS, and TXOP differentiation at any traffic load. The proposed model is also shown to capture the effect of MAC layer buffer size on the performance. Analytical and simulation results are compared to demonstrate the accuracy of the proposed approach for varying traffic loads, EDCA parameters, and MAC layer buffer space. I. INTRODUCTION The IEEE 802.11 standard [1] defines the Distributed Coordination Function (DCF) which provides best-effort service at the Medium Access Control (MAC) layer of the Wireless Local Area Networks (WLANs). The recently ratified IEEE 802.11e standard [2] specifies the Hybrid Coordination Function (HCF) which enables prioritized and parameterized Quality-of-Service (QoS) services at the MAC layer, on top of DCF. The HCF combines a distributed contention-based channel access mechanism, referred to as Enhanced Distributed Channel Access (EDCA), and a centralized polling-based channel access mechanism, referred to as HCF Controlled Channel Access (HCCA). † This work is supported by the Center for Pervasive Communications and Computing, and by National Science Foundation under Grant No. 0434928. Any opinions, findings, and conclusions or recommendations expressed in this material are those of authors and do not necessarily reflect the view of the National Science Foundation. http://arXiv.org/abs/0704.1833v3 We confine our analysis to the EDCA scheme, which uses Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) and slotted Binary Exponential Backoff (BEB) mechanism as the basic access method. The EDCA defines multiple Access Categories (AC) with AC-specific Contention Window (CW) sizes, Arbitration Interframe Space (AIFS) values, and Transmit Opportunity (TXOP) limits to support MAC-level QoS and prioritization [2]. In order to assess the performance of these functions, simulations or mathematical analysis can be used. Although simulation models may capture system dynamics very closely, they lack explicit mathematical relations between the network parameters and performance measures. A number of networking functions would benefit from the insights provided by such mathematical relations. For example, analytical modeling is a more convenient way to assist embedded QoS-aware MAC scheduling and Call Admission Control (CAC) algorithms. Theoretical analysis can provide invaluable insights for QoS provisioning in the WLAN. On the other hand, analytical modeling can potentially be complex, where the effect of multiple layer network parameters makes the task of deriving a simple and accurate analytical model highly difficult. However, a set of appropriate assumptions may lead to simple yet accurate analytical models. The majority of analytical work on the performance of 802.11e EDCA (and of 802.11 DCF) assumes that every station has always backlogged data ready to transmit in its buffer anytime (in saturation) as will be discussed in Section III. Analysis of the system in this state (saturation analysis) provides accurate and practical asymptotic figures. However, the saturation assumption is unlikely to be valid in practice given the fact that the demanded bandwidth for most of the Internet traffic is variable with significant idle periods. Our main contribution is an accurate EDCA analytical model which releases the saturation assumption. The model is shown to predict EDCA performance accurately for the whole traffic load range from a lightly loaded non-saturated channel to a heavily congested saturated medium for a range of traffic models. Similarly, the majority of analytical work on the performance of 802.11e EDCA (and of 802.11 DCF) assumes constant collision probability for any transmitted packet at an arbitrary backoff slot independent of the number of retransmissions it has experienced. A complementary assumption is the constant transmission probability for any AC at an arbitrary backoff slot independent of the number of retransmissions it has experienced. As will be discussed in Section III, these approximations lead to accurate analysis in saturation. Our analysis shows that the slot homogeneity assumption leads to accurate performance prediction even when the saturation assumption is released. Furthermore, the majority of analytical work on the performance of 802.11e EDCA (and of 802.11 DCF) in non-saturated conditions assumes either a very small or an infinitely large MAC layer buffer space. Our analysis removes such assumptions by incorporating the finite size MAC layer queue (interface queue between Link Layer (LL) and MAC layer) into the model. The finite size queue analysis shows the effect of MAC layer buffer space on EDCA performance which we will show to be significant. A key contribution of this work is that the proposed analytical model incorporates all EDCA QoS parameters, CW, AIFS, and TXOP. The model also considers varying collision probabilities at different AIFS slots which is a direct result of varying number of contending stations. Comparing with simulations, we show that our model can provide accurate results for an arbitrary selection of AC-specific EDCA parameters at any load. We present a Markov model the states of which represent the state of the backoff process and MAC buffer occupancy. To enable analysis in the Markov framework, we assume constant probability of packet arrival per state (for the sake of simplicity, Poisson arrivals). On the other hand, we have also shown that the results also hold for a range of traffic types. II. EDCA OVERVIEW The IEEE 802.11e EDCA is a QoS extension of IEEE 802.11 DCF. The major enhancement to support QoS is that EDCA differentiates packets using different priorities and maps them to specific ACs that are buffered in separate queues at a station. Each ACi within a station (0 ≤ i ≤ imax, imax = 3 in [2]) having its own EDCA parameters contends for the channel independently of the others. Following the convention of [2], the larger the index i is, the higher the priority of the AC is. Levels of services are provided through different assignments of the AC specific EDCA parameters; AIFS, CW, and TXOP limits. If there is a packet ready for transmission in the MAC queue of an AC, the EDCA function must sense the channel to be idle for a complete AIFS before it can start the transmission. The AIFS of an AC is determined by using the MAC Information Base (MIB) parameters as AIFS = SIFS + AIFSN × Tslot, (1) where AIFSN is the AC-specific AIFS number, SIFS is the length of the Short Interframe Space and Tslot is the duration of a time slot. If the channel is idle when the first packet arrives at the AC queue, the packet can be directly transmitted as soon as the channel is sensed to be idle for AIFS. Otherwise, a backoff procedure is completed following the completion of AIFS before the transmission of this packet. A uniformly distributed random integer, namely a backoff value, is selected from the range [0, W ]. Should the channel be sensed busy at any time slot during AIFS or backoff, the backoff procedure is suspended at the current backoff value. The backoff resumes as soon as the channel is sensed to be idle for AIFS again. When the backoff counter reaches zero, the packet is transmitted in the following slot. The value of W depends on the number of retransmissions the current packet experienced. The initial value of W is set to the AC-specific CWmin. If the transmitter cannot receive an Acknowledgment (ACK) packet from the receiver in a timeout interval, the transmission is labeled as unsuccessful and the packet is scheduled for retransmission. At each unsuccessful transmission, the value of W is doubled until the maximum AC-specific CWmax limit is reached. The value of W is reset to the AC-specific CWmin if the transmission is successful, or the retry limit is reached thus the packet is dropped. The higher priority ACs are assigned smaller AIFSN. Therefore, the higher priority ACs can either transmit or decrement their backoff counters while lower priority ACs are still waiting in AIFS. This results in higher priority ACs enjoying a lower average probability of collision and relatively faster progress through backoff slots. Moreover, in EDCA, the ACs with higher priority may select backoff values from a comparably smaller CW range. This approach prioritizes the access since a smaller CW value means a smaller backoff delay before the transmission. Upon gaining the access to the medium, each AC may carry out multiple frame exchange sequences as long as the total access duration does not go over a TXOP limit. Within a TXOP, the transmissions are separated by SIFS. Multiple frame transmissions in a TXOP can reduce the overhead due to contention. A TXOP limit of zero corresponds to only one frame exchange per access. An internal (virtual) collision within a station is handled by granting the access to the AC with the highest priority. The ACs with lower priority that suffer from a virtual collision run the collision procedure as if an outside collision has occured [2]. III. RELATED WORK In this section, we provide a brief summary of the theoretical DCF and EDCA function performance analysis in the literature. The majority of previous work carries out performance analysis for asymptotical conditions assuming each station is in saturation. Three major saturation performance models have been proposed for DCF; i) assuming constant collision probability for each station, Bianchi [3] developed a simple Discrete-Time Markov Chain (DTMC) and the saturation throughput is obtained by applying regenerative analysis to a generic slot time, ii) Cali et al. [4],[5] employed renewal theory to analyze a p-persistent variant of DCF with persistence factor p derived from the CW, and iii) Tay et al. [6] instead used an average value mathematical method to model DCF backoff procedure and to calculate the average number of interruptions that the backoff timer experiences. Having the common assumption of slot homogeneity (for an arbitrary station, constant collision or transmission probability at an arbitrary slot), these models define all different renewal cycles all of which lead to accurate saturation performance analysis. Similarly, Medepalli et al. [7] provided explicit expressions for average DCF cycle time and system throughput. Pointing out another direction for future performance studies, Hui et al. [8] recently proposed the application of metamodeling techniques in order to find approximate closed-form mathematical models. These major methods are modified by several researchers to include the extra features of the EDCA function in the saturation analysis. Xiao [9],[10] extended [3] to analyze only the CW differentiation. Kong et al. [11] took AIFS differentiation into account via a 3-dimensional DTMC. On the other hand, these EDCA extensions miss the treatment of varying collision probabilities at different AIFS slots due to varying number of contending stations. Robinson et al. [12],[13] proposed an average analysis on the collision probability for different contention zones during AIFS and employed calculated average collision probability on a 2-dimensional DTMC. Hui et al. [14],[15] unified several major approaches into one approximate average model taking into account varying collision probability in different backoff subperiods (corresponds to contention zones in [12]). Zhu et al. [16] proposed another analytical EDCA Markov model averaging the transition probabilities based on the number and the parameters of high priority flows. Inan et al. [17] proposed a simple DTMC which provides accurate treatment of AIFS and CW differentiation between the ACs for the constant transmission probability assumption. Another 3-dimensional DTMC is proposed by Tao et al. [18],[19] in which the third dimension models the state of backoff slots between successive transmission periods. In [18],[19], the fact that the number of idle slots between successive transmissions can be at most the minimum of AC-specific CWmax values is considered. Independent from [18],[19], Zhao et al. [20] had previously proposed a similar model for the heterogeneous case where each station has traffic of only one AC. Banchs et al. [21],[22] proposed another model which considers varying collision probability among different AIFS slots due to a variable number of stations. Chen et al. [23], Kuo et al. [24], and Lin et al. [25] extended [6] in order to include mean value analysis for AIFS and CW differentiation. Although it has not yet received much attraction, the research that releases the saturation assumption basically follows two major methods; i) modeling the non-saturated behavior of DCF or EDCA function via Markov analysis, ii) employing queueing theory [26] and calculating certain quantities through average or Markov analysis. Our approach in this work falls into the first category. Markov analysis for the non-saturated case still assumes slot homogeneity and extends [3] with necessary extra Markov states and transitions. Duffy et al. [27] and Alizadeh-Shabdiz et al. [28],[29] proposed similar extensions of [3] for non-saturated analysis of 802.11 DCF. Due to specific structure of the proposed DTMCs, these extensions assume a MAC layer buffer size of one packet. We show that this assumption may lead to significant performance prediction errors for EDCA in the case of larger buffers. Cantieni et al. [30] extended the model of [28] assuming infinitely large station buffers and the MAC queue being empty with constant probability regardless of the backoff stage the previous transmission took place. Li et al. [31] proposed an approximate model for non-saturation where only CW differentiation is considered. Engelstad et al. [32] used a DTMC model to perform delay analysis for both DCF and EDCA considering queue utilization probability as in [30]. Zaki et al. [33] proposed yet another Markov model with states that are of fixed real-time duration which cannot capture the pre-saturation DCF throughput peak. A number of models employing queueing theory have also been developed for 802.11(e) performance analysis in non-saturated conditions. These models are assisted by independent analysis for the calculation of some quantities such as collision and transmission probabilities. Tickoo et al. [34],[35] modeled each 802.11 node as a discrete time G/G/1 queue to derive the service time distribution, but the models are based on an assumption that the saturated setting provides good approximation for certain quantities in non-saturated conditions. Chen et al. [36] employed both G/M/1 and G/G/1 queue models on top of [10] which only considers CW differentiation. Lee et al. [37] analyzed the use of M/G/1 queueing model while employing a simple non-saturated Markov model to calculate necessary quantities. Medepalli et al. [38] built upon the average cycle time derivation [7] to obtain individual queue delays using both M/G/1 and G/G/1 queueing models. Foh et al. [39] proposed a Markov framework to analyze the performance of DCF under statistical traffic. This framework models the number of contending nodes as an M/Ej/1/k queue. Tantra et al. [40] extended [39] to include service differentiation in EDCA. However, such analysis is only valid for a restricted scenario where all nodes have a MAC queue size of one packet. There are also a few studies that investigated the effect of EDCA TXOPs on 802.11e performance for a saturated scenario. Mangold et al. [41] and Suzuki et al. [42] carried out the performance analysis through simulation. The efficiency of burst transmissions with block acknowledgements is studied in [43]. Tinnirello et al. [44] also proposed different TXOP managing policies for temporal fairness provisioning. Peng et al. [45] proposed an analytical model to study the effect of burst transmissions and showed that improved service differentiation can be achieved using a novel scheme based on TXOP thresholds. A thorough and careful literature survey shows that an EDCA analytical model which incorporates all EDCA QoS parameters, CW, AIFS, and TXOP, for any traffic load has not been designed yet. IV. EDCA DISCRETE-TIME MARKOV CHAIN MODEL Assuming slot homogeneity, we propose a novel DTMC to model the behavior of the EDCA function of any AC at any load. The main contribution of this work is that the proposed model considers the effect of all EDCA QoS parameters (CW, AIFS, and TXOP) on the performance for the whole traffic load range from a lightly-loaded non-saturated channel to a heavily congested saturated medium. Although we assume constant probability of packet arrival per state (for the sake of simplicity, Poisson arrivals), we show that the model provides accurate performance analysis for a range of traffic types. The state of the EDCA function of any AC at an arbitrary time t depends on several MAC layer events that may have occured before t. We model the MAC layer state of an ACi, 0 ≤ i ≤ 3, with a 3-dimensional Markov process, (si(t), bi(t), qi(t)). The stochastic process si(t) represents the value of the backoff stage at time t, i.e., the number of retransmissions that the packet to be transmitted currently has experienced until time t. The stochastic process bi(t) represents the state of the backoff counter at time t. Up to this point, the definition of the first two dimensions follows [3] which is introduced for DCF. In order to enable the accurate non-saturated analysis considering EDCA TXOPs, we introduce another dimension which models the stochastic process qi(t) denoting the number of packets buffered for transmission at the MAC layer. Moreover, as the details will be described in the sequel, in our model, bi(t) does not only represent the value of the backoff counter, but also the number of transmissions carried out during the current EDCA TXOP (when the value of backoff counter is actually zero). Using the assumption of independent and constant collision probability at an arbitrary backoff slot, the 3-dimensional process (si(t), bi(t), qi(t)) is represented as a Discrete-Time Markov Chain (DTMC) with states (j, k, l) and index i. We define the limits on state variables as 0 ≤ j ≤ ri − 1, −Ni ≤ k ≤ Wi,j and 0 ≤ l ≤ QSi. In these inequalities, we let ri be the retransmission limit of a packet of ACi; Ni be the maximum number of successful packet exchange sequences of ACi that can fit into one TXOPi; Wi,j = 2 min(j,mi)(CWi,min + 1) − 1 be the CW size of ACi at the backoff stage j where CWi,max = 2mi(CWi,min +1)−1, 0 ≤ mi < ri; and QSi be the maximum number of packets that can buffered at the MAC layer, i.e., MAC queue size. Moreover, it is important to note that a couple of restrictions apply to the state indices. • When there are not any buffered packets at the AC queue, the EDCA function of the corresponding AC cannot be in a retransmitting state. Therefore, if l = 0, then j = 0 should hold. Such backoff states represent the postbackoff process [1],[2], therefore called as postbackoff slots in the sequel. The postbackoff procedure ensures that the transmitting station waits at least another backoff between successive TXOPs. Note that, when l > 0 and k ≥ 0, these states are named backoff slots. • The states with indices −Ni ≤ k ≤ −1 represent the negation of the number of packets that are successfully transmitted at the current TXOP rather than the value of the backoff counter (which is zero during a TXOP). For simplicity, in the design of the Markov chain, we introduced such states in the second dimension. Therefore, if −Ni ≤ k ≤ −1, we set j = 0. As it will be clear in the sequel, the addition of these states enables EDCA TXOP analysis. Let pci denote the average conditional probability that a packet from ACi experiences either an external or an internal collision after the EDCA function decides on the transmission. Let pnt(l ′, T |l) be the probability that there are l′ packets in the MAC buffer at time t + T given that there were l packets at t and no transmissions have been made during interval T . Similarly, let pst(l ′, T |l) be the probability that there are l′ packets in the MAC buffer at time t + T given that there were l packets at time t and a transmission has been made during interval T . Note that since we assume Poisson arrivals, the exponential interarrival distributions are independent, and pnt and pst only depend on the interval length T and are independent of time t. Then, the nonzero state transmission probabilities of the proposed Markov model for ACi, denoted as Pi(j ′, k′, l′|j, k, l) adopting the same notation in [3], are calculated as follows. 1) The backoff counter is decremented by one at the slot boundary. Note that we define the postbackoff or the backoff slot as Bianchi defines the slot time [3]. Then, for 0 ≤ j ≤ ri − 1, 1 ≤ k ≤ Wi,j, and 0 ≤ l ≤ l′ ≤ QSi, Pi(j, k − 1, l ′|j, k, l) = pnt(l ′, Ti,bs|l). (2) It is important to note that the proposed DTMC’s evolution is not real-time and the state duration varies depending on the state. The average duration of a backoff slot Ti,bs is calculated by (29) which will be derived. Note that, in (2), we consider the probability of packet arrivals during Ti,bs (buffer size l′ after the state transition depends on this probability). 2) We assume the transmitted packet experiences a collision with constant probability pci (slot homo- geneity). In the following, note that the cases when the retry limit is reached and when the MAC buffer is full are treated separately, since the transition probabilities should follow different rules. Let Ti,s and Ti,c be the time spent in a successful transmission and a collision by ACi respectively which will be derived. Then, for 0 ≤ j ≤ ri − 1, 0 ≤ l ≤ QSi − 1, and max(0, l − 1) ≤ l ′ ≤ QSi, Pi(0,−1, l ′|j, 0, l) = (1 − pci) · pst(l ′, Ti,s|l) (3) Pi(0,−1, QSi − 1|j, 0, QSi) = 1 − pci. (4) For 0 ≤ j ≤ ri − 2, 0 ≤ k ≤ Wi,j+1, and 0 ≤ l ≤ l ′ ≤ QSi, Pi(j + 1, k, l ′|j, 0, l) = pci · pnt(l ′, Ti,c|l) Wi,j+1 + 1 . (5) For 0 ≤ k ≤ Wi,0, 0 ≤ l ≤ QSi − 1, and max(0, l − 1) ≤ l ′ ≤ QSi, Pi(0, k, l ′|ri − 1, 0, l) = Wi,0 + 1 · pst(l ′, Ti,s|l) (6) Pi(0, k, QSi − 1|ri − 1, 0, QSi) = Wi,0 + 1 Note that we use pnt in (5) although a transmission has been made. On the other hand, the packet has collided and is still at the MAC queue for retransmission as if no transmission has occured. This is not the case in (3) and (6), since in these transitions a successful transmission or a drop occurs. When the MAC buffer is full, any arriving packet is discarded as (4) and (7) imply. 3) Once the TXOP is started, the EDCA function may continue with as many packet SIFS-separated exchange sequences as it can fit into the TXOP duration. Let Ti,exc be the average duration of a successful packet exchange sequence for ACi which will be derived in (24). Then, for −Ni + 1 ≤ k ≤ −1, 1 ≤ l ≤ QSi, and max(0, l − 1) ≤ l ′ ≤ QSi, Pi(0, k − 1, l ′|0, k, l) = pst(l ′, Ti,exc|l). (8) When the next transmission cannot fit into the remaining TXOP, the current TXOP is immediately concluded and the unused portion of the TXOP is returned. By design, our model includes maximum number of packets that can fit into one TXOP. Then, for 0 ≤ k ≤ Wi,0 and 1 ≤ l ≤ QSi, Pi(0, k, l|0,−Ni, l) = Wi,0 + 1 . (9) The TXOP ends when the MAC queue is empty. Then, for 0 ≤ k′ ≤ Wi,0 and −Ni ≤ k ≤ −1, Pi(0, k ′, 0|0, k, 0) = Wi,0 + 1 . (10) Note that no time passes in (9) and (10), so the definition of these states and transitions is actually not necessary for accuracy. On the other hand, they simplify the DTMC structure and symmetry. 4) If the queue is still empty when the postbackoff counter reaches zero, the EDCA function enters the idle state until another packet arrival. Note (0,0,0) also represents the idle state. We make two assumptions; i) At most one packet may arrive during Tslot with constant probability ρi (considering the fact that Tslot is in the order of microseconds, the probability that multiple packets can arrive in this interval is very small), ii) if the channel is idle at the slot the packet arrives at an empty queue, the transmission will be successful at AIFS completion without any backoff. The latter assumption is due to the following reason. While the probability of the channel becoming busy during AIFS or a collision occuring for the transmission at AIFS is very small at a lightly loaded scenario, the probability of a packet arrival to an empty queue is very small at a highly loaded scenario. As observed via simulations, these assumptions do not lead to any noticeable changes in the results while simplifying the Markov chain structure and symmetry. Then, for 0 ≤ k ≤ Wi,0 and 1 ≤ l ≤ QSi, Pi(0, 0, 0|0, 0, 0) = (1 − pci) · (1 − ρi) + pci · pnt(0, Ti,b|0), (11) Pi(0, k, l|0, 0, 0) = Wi,0 + 1 · pnt(l, Ti,b|0), (12) Pi(0,−1, l|0, 0, 0) = (1 − pci) · ρi · pnt(l, Ti,s|0). (13) Let Ti,b in (11) and (12) be the length of a backoff slot given it is not idle. Note that actually a successful transmission occurs in the state transition in (13). On the other hand, the transmitted packet is not reflected in the initial queue size state which is 0. Therefore, pnt is used instead of pst. Parts of the proposed DTMC model are illustrated in Fig. 1 for an arbitrary ACi with Ni = 2. Fig. 1(a) shows the state transitions for l = 0. Note that in Fig. 1(a) the states with −Ni ≤ k ≤ −2 can only be reached from the states with l = 1. Fig. 1(b) presents the state transitions for 0 < l < QSi and 0 ≤ j < ri. Note that only the transition probabilities and the states marked with rectangles differ when j = ri − 1 (as in (6)). Therefore, we do not include an extra figure for this case. Fig. 1(c) shows the state transitions when l = QSi. Note also that the states marked with rectangles differ when j = ri − 1 (as in (7)). The combination of these small chains for all j, k, l constitutes our DTMC model. A. Steady-State Solution Let bi,j,k,l be the steady-state probability of the state (j, k, l) of the proposed DTMC with index i which can be solved using (2)-(13) subject to l bi,j,k,l = 1 (the proposed DTMC is ergodic and irreducible). Let τi be the probability that an ACi transmits at an arbitrary backoff or postbackoff slot ∑ri−1 l=1 bi,j,0,l + bi,0,0,0 · ρi · (1 − pci) ∑ri−1 ∑Wi,j l=0 bi,j,k,l . (14) Note that −Ni ≤ k ≤ −1 is not included in the normalization in (14), since these states represent a continuation in the EDCA TXOP rather than a contention for the access. The value of τi depends on the values of the average conditional collision probability pci , the various state durations Ti,bs, Ti,b, Ti,s and Ti,c, and the conditional queue state transition probabilities pnt and pst. 1) Average conditional collision probability pci: The difference in AIFS of each AC in EDCA creates the so-called contention zones as shown in Fig. 2 [12]. In each contention zone, the number of contending stations may vary. The collision probability cannot simply be assumed to be constant among all ACs. We can define pci,x as the conditional probability that ACi experiences either an external or an internal collision given that it has observed the medium idle for AIFSx and transmits in the current slot (note AIFSx ≥ AIFSi should hold). For the following, in order to be consistent with the notation of [2], we assume AIFS0 ≥ AIFS1 ≥ AIFS2 ≥ AIFS3. Let di = AIFSi − AIFS3. Also, let the total number ACi flows be fi. Then, for the heterogeneous scenario in which each station has only one AC pci,x = 1 − i′:di′≤dx (1 − τi′) (1 − τi) . (15) When each station has multiple ACs that are active, internal collisions may occur. Then, for the scenario in which each station has all 4 ACs active pci,x = 1 − i′:di′≤dx (1 − τi′) fi′−1 i′′>i (1 − τi′′). (16) Similar extensions when the number of active ACs are 2 or 3 are straightforward. We use the Markov chain shown in Fig. 3 to find the long term occupancy of contention zones. Each state represents the nth backoff slot after completion of the AIFS3 idle interval following a transmission period. The Markov chain model uses the fact that a backoff slot is reached if and only if no transmission occurs in the previous slot. Moreover, the number of states is limited by the maximum idle time between two successive transmissions which is Wmin = min(CWi,max) for a saturated scenario. Although this is not the case for a non-saturated scenario, we do not change this limit. As the comparison with simulation results show, this approximation does not result in significant prediction errors. The probability that at least one transmission occurs in a backoff slot in contention zone x is ptrx = 1 − i′:di′≤dx (1 − τi′) fi′ . (17) Note that the contention zones are labeled with x regarding the indices of d. In the case of equal AIFS values, the contention zone is labeled with the index of the AC with higher priority. Given the state transition probabilities as in Fig. 3, the long term occupancy of the backoff slots b′n can be obtained from the steady-state solution of the Markov chain. Then, the AC-specific average collision probability pci is found by weighing zone specific collision probabilities pci,x according to the long term occupancy of contention zones (thus backoff slots) pci = ∑Wmin n=di+1 pci,x · b ∑Wmin n=di+1 where x = max y | dy = max (dz | dz ≤ n) which shows x is assigned the highest index value within a set of ACs that have AIFS smaller than or equal to n+AIFS3. This ensures that at backoff slot n, ACi has sensed the medium idle for AIFSx. Therefore, the calculation in (18) fits into the definition of pci,x . Note that the average collision probability calculation in [12, Section IV-D] is a special case of our calculation for two ACs. 2) The state duration Ti,s and Ti,c: Let Ti,p be the average payload transmission time for ACi (Ti,p includes the transmission time of MAC and PHY headers), δ be the propagation delay, Tack be the time required for acknowledgment packet (ACK) transmission. Then, for the basic access scheme, we define the time spent in a successful transmission Ti,s and a collision Ti,c for any ACi as Ti,s =Ti,p + δ + SIFS + Tack + δ + AIFSi (19) Ti,c =Ti,p∗ + ACK Timeout + AIFSi (20) where Ti,p∗ is the average transmission time of the longest packet payload involved in a collision [3]. For simplicity, we assume the packet size to be equal for any AC, then Ti,p∗ = Ti,p. Being not explicitly specified in the standards, we set ACK Timeout, using Extended Inter Frame Space (EIFS) as EIFSi− AIFSi. The extensions of (19) and (20) for the Request-to-Send/Clear-to-Send (RTS/CTS) scheme are Ti,s =Trts + δ + SIFS + Tcts + δ + SIFS + Ti,p + δ + SIFS + Tack + δ + AIFSi (21) Ti,c =Trts + CTS Timeout + AIFSi (22) where Trts and Tcts are the time required for RTS and CTS packet transmissions respectively. Being not explicitly specified in the standards, we set CTS Timeout as we set ACK Timeout. 3) The state duration Ti,bs and Ti,b: The average time between successive backoff counter decrements is denoted by Ti,bs. The backoff counter decrement may be at the slot boundary of an idle backoff slot or the last slot of AIFS following an EDCA TXOP or a collision period. We start with calculating the average duration of an EDCA TXOP for ACi Ti,txop as Ti,txop = l=0 bi,0,−Ni,l · ((Ni − 1) · Ti,exc + Ti,s) + k=−Ni+1 bi,0,k,0 · ((−k − 1) · Ti,exc + Ti,s) k=−Ni+1 bi,0,k,0 + l=0 bi,0,−Ni,l where Ti,exc is defined as the duration of a successful packet exchange sequence within a TXOP. Since the packet exchanges within a TXOP are separated by SIFS rather than AIFS, Ti,exc = Ti,s − AIFSi + SIFS, (24) Ni = max(1, ⌊(TXOPi + SIFS)/Ti,exc⌋). (25) Given τi and fi, simple probability theory can be used to calculate the conditional probability of no transmission (pidlex,i ), only one transmission from ACi′ (p suci′ x,i ), or at least two transmissions (p x,i) at the contention zone x given one ACi is in backoff. pidlex,i = i′:di′≤dx (1 − τi′) fi′ , if di > dx i′:di′≤dx (1 − τi′) 1 − τi , if di ≤ dx. suci′ x,i = 0, if dx < di′ fi′τi′(1 − τi′) fi′−1 i′′:di′′≤dx (1 − τi′′) fi′′ , if di > dx and di′ ≤ dx fi′τi′(1 − τi′) fi′−1 1 − τi i′′:di′′≤dx (1 − τi′′) fi′′ , if di ≤ dx and di′ ≤ dx. pcolx,i =1 − p x,i − suci′ x,i (28) Let xi be the first contention zone in which ACi can transmit. Then, Ti,bs = xi<x′≤3 (pidlex′,i · Tslot + p x′,i · Ti,c + suci′ x′,i · Ti′,txop) · pzx′ (29) where pzx denotes the stationary distribution for a random backoff slot being in zone x. Note that, in (29), the fractional term before summation accounts for the busy periods experienced before AIFSi is completed. Therefore, if we let d−1 = Wmin, pzx = min(dx′ |dx′>dx) n=dx+1 b′n. (30) The expected duration of a backoff slot given it is busy and one ACi is in idle state is calculated as Ti,b = pcolx′,i 1 − pidlex′,i · Ti,c + suci′ 1 − pidlex′,i · Ti′,txop · pzx′ . (31) 4) The conditional queue state transition probabilities pnt and pst: We assume the packets arrive at the AC queue with size QSi according to a Poisson process with rate λi packets per second. Using the probability distribution function of the Poisson process, the probability of k arrivals occuring in time interval t can be calculated as Pr(Nt,i = k) = exp−λit(λit) . (32) Then, pnt(l ′, T |l) and pst(l ′, T |l) can be calculated as follows. Note that the finite buffer space is considered throughout calculations since the number of packets that may arrive during T can be more than the available queue space. pnt(l ′, T |l) = Pr(NT,i = l ′ − l), if l′ < QSi QSi−1 Pr(NT,i = l ′ − l), if l′ = QSi. pst(l ′, T |l) = Pr(NT,i = l ′ − l + 1), if l′ < QSi QSi−1 l′=l−1 Pr(NT,i = l ′ − l + 1), if l′ = QSi. Note that in (11)-(14), ρi = 1−Pr(NTslot,i = 0). Together with the steady-state transition probabilities, (14)-(34) represent a nonlinear system which can be solved using numerical methods. B. Normalized Throughput Analysis The normalized throughput of a given ACi, Si, is defined as the fraction of the time occupied by the successfully transmitted information. Then, psiNi,txopTi,p pITslot + i′ psi′Ti′,txop + (1 − pI − i′ psi′ )Tc pI is the probability of the channel being idle at a backoff slot, psi is the conditional successful transmission probability of ACi at a backoff slot, and Ni,txop = (Ti,txop−AIFSi+SIFS)/Ti,exc. Note that, we consider Ni,txop and Ti,txop in (35) to define the generic slot time and the time occupied by the successfully transmitted information in the case of EDCA TXOPs. The probability of a slot being idle, pI , depends on the state of previous slots. For example, conditioned on the previous slot to be busy (pB = 1−pI ), pI only depends on the transmission probability of the ACs with the smallest AIFS, since others have to wait extra AIFS slots. Generalizing this to all AIFS slots, pI can be calculated as γnpB(pI) γnpBp I + γd0p I (36) where γn denotes the probability of no transmission occuring at the (n + 1) th AIFS slot after AIFS3. Substituting γn = γd0 for n ≥ d0, and releasing the condition on the upper limit of summation, Wmin, to ∞, pI can be approximated as in (36). According to the simulation results, this approximation works well. Note that γn = 1 − p x where x = max y | dy = max (dz | dz ≤ n) The probability of successful transmission psi is conditioned on the states of the previous slots as well. This is again because the number of stations that can contend at an arbitrary backoff slot differs depending on the number of previous consecutive idle backoff slots. Therefore, for the heterogeneous case, in which each station only has one AC, psi can be calculated as psi = (1 − τi) n=di+1 (n−1) i′:0≤di′≤(n−1) (1 − τi′) + (pI) (1 − τi′)  . (37) Similarly, for the scenario, in which each station has four active ACs, psi = (1 − τi) n=di+1 (n−1) i′:0≤di′≤(n−1) (1 − τi′) fi′−1 i′′>i (1 − τi′′) + (pI) (1 − τi′) fi′−1 i′′>i (1 − τi′′) . (38) C. Average Delay Analysis Our goal is to find total average delay E[Di] which is defined as the average time from when a packet enters the MAC layer queue of ACi until it is successfully transmitted. Di has two components; i) queueing time Qi and ii) access time Ai. Qi is the period that a packet waits in the queue for other packets in front to be transmitted. Ai is the period a packet waits at the head of the queue until it is transmitted successfully (backoff and transmission period). We carry out a recursive calculation as in [11] to find E[Ai] for ACi. Then, using E[Ai] and bi,j,k,l, we calculate E[Di]=E[Qi]+E[Ai]. Note that, E[Ai] differs depending on whether the EDCA function is idle or not when the packet arrives. We will treat these cases separately. In the sequel, Ai,idle denotes the access delay when the EDCA function is idle at the time a packet arrives. The recursive calculation is carried out in a bottom-to-top and left-to-right manner on the AC-specific DTMC. For the analysis, let Ai(j, k) denote the time delay from the current state (j, k, l) until the packet at the head of the ACi queue is transmitted successfully (l ≥ 1). The initial condition on the recursive calculation is Ai(ri − 1, 0) = Ti,s. (39) Recursive delay calculations for 0 ≤ j ≤ ri − 1 are Ai(j, k) = Ai(j, k − 1) + Ti,bs, if 1 ≤ k ≤ Wi,j (1 − pci)Ti,s + pci PWi,j+1 Ai(j+1,k Wi,j+1+1 + Ti,c , if k = 0 and j 6= ri − 1. Then, E[Ai] = ∑Wi,0 k=0 Ai(0, k) Wi,0 + 1 Following the assumptions made in (11)-(13) and considering the packet loss probability due to the retry limit as pl,r = (pci) ri (note that the delay a dropped packet experiences cannot be considered in a total delay calculation), E[Ai,idle] can be calculated as E[Ai,idle] = Ti,s · (1 − pci) + (E[Ai] + Ti,b) · pci · (1 − pl,r). (42) In this case, the average access delay is equal to the total average delay, i.e., Di(0, 0, 0) = E[Ai,idle]. We perform another recursive calculation to calculate the total delay a packet experiences Di(j, k, l) (given that the packet arrives while the EDCA function is at state (j, k, l)). In the calculations, we account for the remaining access delay for the packet at the head of the MAC queue and the probability that this packet may be dropped due to the retry limit. Let Ai,d(j, k) be the access delay conditioned that the packet drops. Ai,d(j, k) can easily be calculated by modifying the recursive method of calculating Ai(j, k). The initial condition on this recursive calculation Ai,d(ri − 1, 0) = Ti,c. (43) Recursive delay calculations for 0 ≤ j ≤ ri − 1 are Ai,d(j, k) = Ai(j, k − 1) + Ti,bs, if 1 ≤ k ≤ Wi,j Wi,j+1 Ai,d(j + 1, k) + Ti,c, if k = 0 and j 6= ri − 1. Then, E[Ai,d] = ∑Wi,0 k=0 Ai,d(0, k) Wi,0 + 1 If a packet arrives during the backoff of another packet, it is delayed at least for the remaining access time. Depending on the queue size, it may be transmitted at the current TXOP, or may be delayed till further accesses are gained. Then, for 0 ≤ j ≤ ri − 1, 0 ≤ k ≤ Wi,j, and 1 ≤ l ≤ QSi, Di(j, k, l) =(1 − pl,r) · (Ai(j, k) + min(Ni − 1, l − 1) · Ti,exc + Di(−1,−1, l − Ni)) + pl,r · (Ai,d(j, k) + Di(−1,−1, l − 1)) . (46) When the packet arrives during postbackoff, the total delay is equal to the access delay. Then, for 0 ≤ k ≤ Wi,j and l = 0, Di(j, k, l) = Ai(j, k). (47) When the packet arrives during a TXOP, it may be transmitted at the current TXOP, or it may wait for further accesses. Then, for −Ni + 1 ≤ k ≤ −1 and 1 ≤ l ≤ QSi, Di(j, k, l) = min(k − 1, l) · Ti,exc + Di(−1,−1, l − k + 1). (48) Di(−1,−1, l) is calculated recursively according to the value of l Di(−1,−1, l) = 0, if l ≤ 0 E[Ai] · (1 − pl,r), if l = 1 χ, if l > 1 where χ =(1 − pl,r) · (E[Ai] + min(Ni − 1, l − 1) · Ti,exc +Di(−1,−1, l − Ni)) + pl,r · (E[Ai,d] + Di(−1,−1, l − 1)) . (50) Let the probability of any arriving packet seeing the EDCA function at state (j, k, l) be b̄i,j,k,l. Since we assume independent and exponentially distributed packet interarrivals, b̄i,j,k,l can simply be calculated by normalizing bi,j,k,l excluding the states in which no time passes, i.e., ∀(j, k, l) such that (0,−Ni, 1 ≤ l ≤ QSi) or (0,−Ni ≤ k ≤ −1, 0). Note that b̄i,j,k,l is zero for these states b̄i,j,k,l = bi,j,k,l l=1 bi,0,−Ni,l − k=−Ni bi,0,k,0 . (51) Then, the total average delay a successful packet experiences E[Di] can be calculated averaging Di(j, k, l) over all possible states E[Di] = E[Ai,idle] · b̄i,0,0,0 + ∀(j,k,l)/(0,0,0) Di(j, k, l) · b̄i,j,k,l. (52) D. Average Packet Loss Ratio We consider two types of packet losses; i) the packet is dropped when the MAC layer retry limit is reached, ii) the packet is dropped if the MAC queue is full at the time of packet arrival. Let plri denote the average packet loss ratio for ACi. We use the steady-state probability bi,j,k,l to find the probability whether the MAC queue is full or not at the time of packet arrival. If the queue is full, the arriving packet is dropped (second term in (53)). Otherwise, the packet is dropped with probability prici , i.e. only if the retry limit is reached (first term in (53)). Note that we consider packet retransmissions only due to packet collisions. Then, plri = QSi−1 bi,j,k,l · p bi,j,k,QSi. (53) E. Queue Size Distribution Due to the specific structure of the proposed model, it is straightforward to calculate the MAC queue size distribution for ACi. Note that we use queue size distribution in the calculation of average packet loss ratio. Pr(l = l′) = bi,j,k,l′. (54) V. NUMERICAL AND SIMULATION RESULTS We validate the accuracy of the numerical results calculated via the proposed EDCA model by comparing them with the simulations results obtained from ns-2 [46]. For the simulations, we employ the IEEE 802.11e HCF MAC simulation model for ns-2.28 that we developed [47]. This module implements all the EDCA and HCCA functionalities stated in [2]. As in all work on the subject in the literature, we consider ACs that transmit fixed-size User Datagram Protocol (UDP) packets. In simulations, we consider two ACs, one high priority and one low priority. Each station runs only one traffic class. Unless otherwise stated, the packets are generated according to a Poisson process with equal rate for both ACs. We set AIFSN1 = 3, AIFSN3 = 2, CW1,min = 15, CW3,min = 7, m1 = m3 = 3, r1 = r3 = 7. For both ACs, the payload size is 1034 bytes. Again, as in most of the work on the subject, the simulation results are reported for the wireless channel which is assumed to be not prone to any errors during transmission. The errored channel case is left for future study. All the stations have 802.11g Physical Layer (PHY) using 54 Mbps and 6 Mbps as the data and basic rate respectively (Tslot = 9 µs, SIFS = 10 µs) [48]. The simulation runtime is 100 seconds. Fig. 4 shows the differentiation of throughput for two ACs when EDCA TXOP limits of both are set to 0 (1 packet exchange per EDCA TXOP). In this scenario, there are 5 stations for both ACs and they are transmitting to an AP. The normalized throughput per AC as well as the total system throughput is plotted for increasing offered load per AC. We have carried out the analysis for maximum MAC buffer sizes of 2 packets and 10 packets. The comparison between analytical and simulation results shows that our model can accurately capture the linear relationship between throughput and offered load under low loads, the complex transition in throughput between under-loaded and saturation regimes, and the saturation throughput. Although we do not present here, considerable inaccuracy is observed if the postbackoff procedure, varying collision probability among different AIFS zones, and varying service time among different backoff stages are not modeled correctly as proposed. The results also present that the slot homogeneity assumption works accurately in a non-saturated model for throughput estimation. The proposed model can also capture the throughput variation with respect to the size of the MAC buffer. The results reveal how significantly the size of the MAC buffer affects the throughput in the transition period from underloaded to highly loaded channel. This also shows small interface buffer assumptions of previous models [27],[28],[29],[40] can lead to considerable analytical inaccuracies. Although the total throughput for the small buffer size case has higher throughput in the transition region for the specific example, this cannot be generalized. The reason for this is that AC1 suffers from low throughput for QS1 = 10 due to the selection of EDCA parameters, which affects the total throughput. It is also important to note that the throughput performance does not differ significantly (around %1- %2) for buffer sizes larger than 10 packets for the given scenarios. Therefore, we do not include such cases in order not to complicate the figures. Since the complexity of the mathematical solution increases with the increasing size of the third dimension of DTMC, it may be preferable to implement the model for smaller queue sizes when the throughput performance is not expected to be affected by the selection. Fig. 5 depicts the differentiation of throughput for two ACs when EDCA TXOP limits are set to 1.504 ms and 3.008 ms for high and low priority ACs respectively. For TXOP limits, we use the suggested values for voice and video ACs in [2]. It is important to note that the model works for an arbitrary selection of the TXOP limit. According to the selected TXOP limits, N1 = 5 and N2 = 11. The normalized throughput per AC as well as the total system throughput is plotted while increasing offered load per AC. We have done the analysis for maximum MAC buffer sizes of 2 packets and 10 packets. The model accurately captures the throughput for any traffic load. As expected, increasing maximum buffer size to 10 packets increases the throughput both in the transition and the saturation region. Note that when more than a packet fits into EDCA TXOPs, this decreases contention overhead which in turn increases channel utilization and throughput (comparison of Fig. 5 with Fig. 4). Although corresponding results are not presented here, the model works accurately for higher queue sizes in the case of EDCA TXOPs as well. Fig. 6 displays the differentiation of throughput for two ACs when packet arrival rate is fixed to 2 Mbps and the station number per AC is increased. We have done the analysis for the MAC buffer size of 10 packets with EDCA TXOPs enabled. The analytical and simulation results are well in accordance. As the traffic load increases, the differentiation in throughput between the ACs is observed. Fig. 7 shows the normalized throughput for two ACs when offered load per AC is not equal. In this scenario, we set the packet arrival rate per AC1 to 2 Mbps and the packet arrival rate per AC3 to 0.5 Mbps. The analytical and simulation results are well in accordance. As the traffic load increases, AC3 maintains linear increase with respect to offered load, while AC1 experiences decrease in throughput due to larger settings of AIFS and CW if the total number of stations exceeds 22. In the design of the model, we assume constant packet arrival probability per state. The Poisson arrival process fits this definition because of the independent exponentially distributed interarrival times. We have also compared the throughput estimates obtained from the analytical model with the simulation results obtained using an On/Off traffic model in Fig. 8. A similar study has first been made for DCF in [27]. We modeled the high priority with On/Off traffic model with exponentially distributed idle and active intervals of mean length 1.5 s. In the active interval, packets are generated with Constant Bit Rate (CBR). The low priority traffic uses Poisson distributed arrivals. Note that we leave the packet size unchanged, but normalize the packet arrival rate according to the on/off pattern so that total offered load remains constant to have a fair comparison. The analytical predictions closely follow the simulation results for the given scenario. We have observed that the predictions are more sensitive if the transition region is entered with a few number of stations (5 stations per AC). Our model also provides a very good match in terms of the throughput for CBR traffic. In Fig. 9, we compare the throughput prediction of the proposed model with simulations using CBR traffic. The packet arrival rate is fixed to 2 Mbps for both ACs and the station number per AC is increased. MAC buffer size is 10 packets and EDCA TXOPs are enabled. Fig. 10 depicts the total average packet delay with respect to increasing traffic load per AC. We present the results for two different scenarios. In the first scenario, TXOP limits are set to 0 ms for both ACs. In the second scenario, TXOP limits are set to 1.504 ms and 3.008 ms for high and low priority ACs respectively. The analysis is carried out for a buffer size of 10 packets. As the results imply, the analytical results closely follow the simulation results for both scenarios. In the lightly loaded region, the delays are considerably small. The increase in the transition region is steeper when TXOP limits are 0. In the specific example, enabling TXOPs decreases the total delay where the decrease is more considerable for the low priority AC (due to selection of parameters). Since the buffer size is limited, the total average delay converges to a specific value as the load increases. Still this limit is not of interest, since the packet loss rate at this region is unpractically large. Note that this limit will be higher for larger buffers. The region of interest is the start of the transition region (between 2 Mbps and 3 Mbps for the example in Fig. 10). On the other hand, we also display other data points to show the performance of the model for the whole load span. Fig. 11 depicts the average packet loss ratio with respect to increasing traffic load per AC. We present the results for two different scenarios. In the first scenario, TXOP limits are set to 0 ms for both ACs. In the second scenario, TXOP limits are set to 1.504 ms and 3.008 ms for high and low priority ACs respectively. The analysis is carried out for a buffer size of 10 packets. As the results imply, the analytical results closely follow the simulation results for both scenarios. Although it is not presented in Fig. 11, the packet loss ratio drops exponentially to 0 when the offered load per AC is lower than 2.5 Mbps. The results presented in this paper fixes the AIFS and CW parameters for each AC. The results are compared for different TXOP values at varying traffic load. Therefore, the presented results can mainly indicate the effects of TXOP on the maximum throughput. The model can also be used in order to investigate the effects of AIFS and CW on the maximum throughput. As the comparison of Fig. 4 and Fig. 5 reveals, the total throughput can be maximized with the introduction of EDCA TXOPs which enable multiple frame transmissions in one channel access (note that MAC buffer sizes for each AC should be equal to or larger than the number of packets that can fit to the AC-specific TXOP in order to efficiently utilize each TXOP gained). EDCA TXOPs decrease the channel contention overhead and the ACs can efficiently utilize the resources. Note also that the effects of EDCA TXOPs in the lightly loaded region is marginal compared to highly loaded region. This is expected since the MAC queues do not build up in the lightly loaded scenario where stations usually have just one packet to send at their access to the channel. As Fig. 4 shows the saturation throughput is usually less than the maximum throughput that can be obtained. This is also observed for DCF in [3]. Similarly, in Fig. 6-Fig. 9, the total throughput slightly decreases as the total load increases. As the load in the system increases the collision overhead becomes significant which decreases the total channel utilization. On the other hand, as also discussed in [3], the point where the maximum throughput is observed is unstable in a random access system. Therefore, a good admission control algorithm should be defined to operate the system at the point right before the lightly loaded to highly loaded transition region starts. VI. CONCLUSION We have presented an accurate Markov model for analytically calculating the EDCA throughput and delay for the whole traffic load range from a lightly loaded non-saturated channel to a heavily congested saturated medium. The presented model shows the accuracy of the homogeneous slot assumption (constant collision and transmission probability at an arbitrary backoff slot) that is extensively studied in saturation scenarios for the whole traffic range. The presented model accurately captures the linear relationship between throughput and offered load under low loads and the limiting behavior of throughput at saturation. The key contribution of this paper is that the model accounts for all of the differentiation mechanisms EDCA proposes. The analytical model can incorporate any selection of AC-specific AIFS, CW, and TXOP values for any number of ACs. The model also considers varying collision probabilities at different con- tention zones which provides accurate AIFS differentiation analysis. Although not presented explicitly in this paper, it is straightforward to extend the presented model for scenarios where the stations run multiple ACs (virtual collisions may take place) or RTS/CTS protection mechanism is used. The approximations made for the sake of DTMC simplicity and symmetry may also be removed easily for increased accuracy, although they are shown to be highly accurate. We also show that the MAC buffer size affects the EDCA performance significantly between underloaded and saturation regimes (including saturation) especially when EDCA TXOPs are enabled. The presented model captures this complex transition accurately. This analysis also points out the fact that including an accurate queue treatment is vital. Incorporating MAC queue states also enables EDCA TXOP analysis so that the EDCA TXOP continuation process is modeled in considerable detail. To the authors’ knowledge this is the first demonstration of an analytic model including EDCA TXOP procedure for finite load. It is also worth noting that our model can easily be simplified to model DCF behavior. Moreover, after modifying our model accordingly, the throughput analysis for the infrastructure WLAN where there are transmissions both in the uplink and downlink can be performed (note that in a WLAN downlink traffic load may significantly differ from uplink traffic load). Although the Markov analysis assumes the packets are generated according to Poisson process, the comparison with simulation results shows that the throughput analysis is valid for a range of traffic types such as CBR and On/Off traffic (On/Off traffic model is a widely used model for voice and telnet traffic). The non-existence of a closed-form solution for the Markov model limits its practical use. 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Sohraby, “On Use of Traditional M/G/1 Model for IEEE 802.11 DCF in Unsaturated Traffic Conditions,” in Proc. IEEE WCNC ’06, May 2006. [38] K. Medepalli and F. A. Tobagi, “System Centric and User Centric Queueing Models for IEEE 802.11 based Wireless LANs,” in Proc. IEEE Broadnets ’05, October 2005. [39] C. H. Foh and M. Zukerman, “A New Technique for Performance Evaluation of Random Access Protocols,” in Proc. European Wireless ’02, February 2002. [40] J. W. Tantra, C. H. Foh, I. Tinnirello, and G. Bianchi, “Analysis of the IEEE 802.11e EDCA Under Statistical Traffic,” in Proc. IEEE ICC ’06, June 2006. [41] S. Mangold, S. Choi, P. May, and G. Hiertz, “IEEE 802.11e - Fair Resource Sharing Between Overlapping Basic Service Sets,” in Proc. IEEE PIMRC ’02, September 2002. [42] T. Suzuki, A. Noguchi, and S. Tasaka, “Effect of TXOP-Bursting and Transmission Error on Application-Level and User-Level QoS in Audio-Video Transmission with 802.11e EDCA,” in Proc. IEEE PIMRC ’06, September 2006. [43] I. Tinnirello and S. Choi, “Efficiency Analysis of Burst Transmissions with Block ACK in Contention-Based 802.11e WLANs,” in Proc. IEEE ICC ’05, May 2005. [44] ——, “Temporal Fairness Provisioning in Multi-Rate Contention-Based 802.11e WLANs,” in Proc. IEEE WoWMoM ’05, June 2005. [45] F. Peng, H. M. Alnuweiri, and V. C. M. Leung, “Analysis of Burst Transmission in IEEE 802.11e Wireless LANs,” in Proc. IEEE ICC ’06, June 2006. [46] (2006) The Network Simulator, ns-2. [Online]. Available: http://www.isi.edu/nsnam/ns [47] IEEE 802.11e HCF MAC model for ns-2.28. [Online]. Available: http://newport.eecs.uci.edu/$\sim$fkeceli/ns.htm [48] IEEE Standard 802.11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications: Further Higher Data Rate Extension in the 2.4 GHz Band, IEEE 802.11g Std., 2003. http://www.isi.edu/nsnam/ns http://newport.eecs.uci.edu/$\sim $fkeceli/ns.htm 0,-1,0 0,0,0 0,1,0 0,2,0 0,Wi,0-1,0 0,Wi,0,0 0,1,l’ 0,Wi,0-1,l’0,0,l’ 0,Wi,0-2,l’ pnt(0,Ti,bs|0)pnt(0,Ti,bs|0)pnt(0,Ti,bs|0) 0,-1,l’ .pnt(0,Ti,s|0) .pnt(l’,Ti,s|0) pnt(l’,Ti,bs|0)pnt(l’,Ti,bs|0)pnt(l’,Ti,bs|0) )(1-ρ 0,0BO0,l’ pnt(l’,Ti,b|0) 0,-2,0 0,-1,l j,0,l j,1,l j,2,l j,Wi,j-1,l j,Wi,j,l j,1,l’ j,Wi,j-1,l’j,0,l’ j,Wi,j-2,l’ pnt(l,Ti,bs|l)pnt(l,Ti,bs|l) 0,-1,l’ pnt(l’,Ti,bs|l)pnt(l’,Ti,bs|l)pnt(l’,Ti,bs|l) pnt(l,Ti,bs|l) 0,-2,l 0,-2,l’ pst(l’,Ti,exc|l) pst(l,Ti,exc|l) j+1,l’ (1-pc )pst(l,Ti,s|l) (1-pc )pst(l’,Ti,s|l) pnt(l’,Ti,c|l) 0,-1,l-1 j,0,l j,1,l j,2,l j,Wi,j-1,l j,Wi,j,l j,lBOj+1,l 1111-pc Fig. 1. Parts of the proposed DTMC model for Ni=2. The combination of these small chains for all j, k, l constitutes the proposed DTMC model. (a) l = 0. (b) 0 < l < QSi. (c) l = QSi. Remarks: i) the transition probabilities and the states marked with rectangles differ when j = ri − 1 (as in (6) and (7)), ii) the limits for l ′ follow the rules in (2)-(13). Transmission/ Collision period AIFSN AIFSN AIFSN AIFSN No Tx Zone 3 Zone 2 Zone 1 AC3 in Backoff AC2 in Backoff AC1 in Backoff AC0 in Backoff Fig. 2. EDCA backoff after busy medium. 1 d2 d1 Wmin tr 1-p tr 1-p Fig. 3. Transition through backoff slots in different contention zones for the example given in Fig.2. 1 2 3 4 5 6 7 8 9 10 Offered traffic rate per AC (Mbps) QS=2, AC − analysis QS=2, AC − analysis QS=2, total − analysis QS=2, AC − sim QS=2, AC − sim QS=2, total −sim QS=10, AC − analysis QS=10, AC − analysis QS=10, total − analysis QS=10, AC − sim QS=10, AC − sim QS=10, total − sim Fig. 4. Normalized throughput prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing load per AC at each station and varying MAC buffer size in basic access mode (TXOP3 = 0, TXOP1 = 0). Simulation results are also added for comparison. 1 2 3 4 5 6 7 8 9 10 Offered traffic rate per AC (Mbps) QS=2, AC − analysis QS=2, AC − analysis QS=2, total − analysis QS=2, AC − sim QS=2, AC − sim QS=2, total − sim QS=10, AC − analysis QS=10, AC − analysis QS=10, total − analysis QS=10, AC − sim QS=10, AC − sim QS=10, total − sim Fig. 5. Normalized throughput prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing load per AC at each station and varying MAC buffer size in basic access mode (TXOP3 = 1504ms, TXOP1 = 3008ms). Simulation results are also added for comparison. 2 3 4 5 6 7 8 9 10 11 12 Number of stations per AC − analysis − analysis total − analysis − sim − sim total −sim Fig. 6. Normalized throughput prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing number of stations when MAC buffer size is 10 packets and total offered load per AC is 2 Mbps (TXOP3 = 1504ms, TXOP1 = 3008ms). Simulation results are also added for comparison. 8 9 10 11 12 13 14 15 16 Number of stations per AC − analysis − analysis total − analysis − sim − sim total − sim Fig. 7. Normalized throughput prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing number of stations when MAC buffer size is 10 packets (TXOP3 = 1504ms, TXOP1 = 3008ms). Total offered load per AC3 is 0.5 Mbps while total offered load per AC3 is 2 Mbps. Simulation results are also added for comparison. 18 19 20 21 22 23 24 25 26 27 Number of stations per AC t AC1 − analysis − analysis total − analysis , Poisson − sim , On/Off − sim total − sim Fig. 8. Normalized throughput prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing number of stations when total offered load per AC is 0.5 Mbps (TXOP3 = 1504ms, TXOP1 = 3008ms). Simulation results are also added for the scenario when AC3 uses On/Off traffic with exponentially distributed idle and active times both with mean 1.5s. AC1 uses Poisson distribution for packet arrivals. 2 3 4 5 6 7 8 9 10 11 12 Number of stations per AC − analysis − analysis total − analysis , CBR − sim , CBR − sim total, CBR − sim Fig. 9. Normalized throughput prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing number of stations when MAC buffer size is 10 packets and total offered load per AC is 2 Mbps (TXOP3 = 1504ms, TXOP1 = 3008ms). Simulation results are also added for the scenario when both AC1 and AC3 uses CBR traffic. 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Offered traffic rate per AC (Mbps) − analysis − analysis − sim − sim + TXOP − analysis + TXOP − analysis + TXOP − sim + TXOP − sim Fig. 10. Total average delay prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing load per AC at each station. In the first scenario, TXOP limits are set to 0 ms for both ACs. In the second scenario, TXOP limits are set to 1.504 ms and 3.008 ms for high and low priority ACs respectively. Simulation results are also added for comparison. 2.5 3 3.5 4 4.5 5 Offered traffic rate per AC (Mbps) − analysis − analysis − sim − sim + TXOP − analysis + TXOP − analysis + TXOP − sim + TXOP − sim Fig. 11. Average packet loss ratio prediction of the proposed model for 2 AC heterogeneous scenario with respect to increasing load per AC at each station. In the first scenario, TXOP limits are set to 0 ms for both ACs. In the second scenario, TXOP limits are set to 1.504 ms and 3.008 ms for high and low priority ACs respectively. Simulation results are also added for comparison. Introduction EDCA Overview Related Work EDCA Discrete-Time Markov Chain Model Steady-State Solution Average conditional collision probability pci The state duration Ti,s and Ti,c The state duration Ti,bs and Ti,b The conditional queue state transition probabilities pnt and pst Normalized Throughput Analysis Average Delay Analysis Average Packet Loss Ratio Queue Size Distribution Numerical and Simulation Results Conclusion References

0704.1834

Charge Ordering in Half-Doped Manganites: Weak Charge Disproportion and Leading Mechanisms

Charge Ordering in Half-Doped Manganites: Weak Charge Dis- proportion and Leading Mechanisms Dmitri Volja1,2, Wei-Guo Yin1 (a) and Wei Ku1,2 (b) 1 Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA 2 Physics Department, State University of New York, Stony Brook, New York 11790, USA PACS 75.47.Lx – Manganites (magnetotransport materials) PACS 71.45.Lr – Charge-density waves - collective excitations PACS 71.10.Fd – Lattice fermion models (Hubbard model, etc.) PACS 71.30.+h – Metal-insulator transitions and other electronic transitions Abstract. - The apparent contradiction between the recently observed weak charge disproportion and the traditional Mn3+/Mn4+ picture of the charge-orbital orders in half-doped manganites is resolved by a novel Wannier states analysis of the LDA+U electronic structure. Strong electron itinerancy in this charge-transfer system significantly delocalizes the occupied low-energy “Mn3+” Wannier states such that charge leaks into the “Mn4+”-sites. Furthermore, the leading mechanisms of the charge order are quantified via our first-principles derivation of the low-energy effective Hamiltonian. The electron-electron interaction is found to play a role as important as the electron- lattice interaction. Introduction. – The exploration of interplay among distinct orders lies in the heart of condensed matter physics and materials science, as this interplay often give rises to tunable properties of practical applications, such as exotic states and colossal responses to external stimuli. Manganese oxides such as La1−xCaxMnO3, which host rich charge, orbital, spin, and lattice degrees of freedom, have thus attracted great attention [1]. In particular, the vastly interesting colossal magnetoresistance (CMR) ef- fect for x ∼ 0.2 − 0.4 exemplifies rich physics originating from proximity of competing orders. In a slightly more doped system (x = 0.5), all these orders coexist in an in- sulating state [2], providing a unique opportunity for a clean investigation of the strength and origin of each or- der. Therefore, the study of half-doped manganites is key to a realistic understanding of the physics of manganites in general and the CMR effect in particular [3, 4]. The peculiar multiple orders in half-doped manganites have long been understood in the Goodenough model [2] of a Mn 3+/4+ checkerboard charge order (CO) with the occupied Mn3+ eg orbitals zigzag ordered in the CE-type antiferromagnetic background [2]. Pertaining to CMR, it is broadly believed that a key is the emerging of nanoscale E-mail: [email protected] E-mail: [email protected] charge-ordered insulating regions of Goodenough type at intermediate temperatures, which could melt rapidly in the magnetic field [4, 5]. Nonetheless, the simple yet profound Goodenough pic- ture has been vigorously challenged thanks to recent ex- perimental observations of nearly indiscernible charge dis- proportion (CD) in a number of half-doped manganites [6–13]. Such weak CD has also been observed in first- principles computations [14, 15] and charge transfer be- tween Mn and O sites was reported as well [15]. In essence, these findings have revived a broader discussion on the substantial mismatch of valence and charge in most charge-transfer insulators. Indeed, extensive experimen- tal and theoretical effort has been made in light of the novel Zener-polaron model [9,33] in which all the Mn sites become equivalent with valence being +3.5. Amazingly, most of these investigations concluded in favor of two distinct Mn sites as predicted in the Goodenough model [11–13,16, 17], calling for understanding the emergence of weak CD within the 3+/4+ valence picture. Another closely related crucial issue is the roles of dif- ferent microscopic interactions in the observed charge- orbital orders, in particular the relevance of electron- electron (e-e) interactions in comparison with the well- accepted electron-lattice (e-l) interactions. For example, http://arxiv.org/abs/0704.1834v2 D. Volja et al. ∆n, the difference in the electron occupation number be- tween Mn3+ and Mn4+ states, was shown to be small in an e-e only picture [18]; this is however insufficient to explain the observed weak CD, since the established e-l interac- tions will cause a large ∆n [19]. Moreover, despite the common belief that e-l interactions dominate the general physics of eg electrons in the manganites [2, 20], a recent theoretical study [21] showed that e-e interaction plays an essential and leading role in ordering the eg orbitals in the parent compound. It is thus important to quantify the leading mechanisms in the doped case and uncover the ef- fects of the additional charge degree of freedom in general. In this Letter, we present a general, simple, yet quanti- tative picture of doped holes in strongly correlated charge- transfer systems, and apply it to resolve the above con- temporary fundamental issues concerning the charge order in half-doped manganites. Based on recently developed first-principles Wannier states (WSs) analysis [21,22,34] of the LDA+U electronic structure in prototypical Ca-doped manganites, the doped holes are found to reside primarily in the oxygen atoms. They are entirely coherent in short range [23], forming a Wannier orbital of Mn eg symme- try at low-energy scale. This hybrid orbital, together with the unoccupied Mn 3d orbital, forms the effective “Mn eg” basis in the low-energy theory with conventional 3+/4+ valence picture, but simultaneously results in a weak CD owing to the similar degree of mixing with the intrinsic Mn orbitals, thus reconciling the current conceptual con- tradictions. Moreover, our first-principles derivation of the low-energy interacting Hamiltonian reveals a surprisingly essential role of e-e interactions in the observed charge or- der, contrary to the current lore. Our theoretical method and the resulting simple picture provide a general frame- work to utilize the powerful valence picture even with weak CD, and can be directly applied to a wide range of doped charge-transfer insulators. Small CD vs. 3+/4+ valence Picture. – To pro- ceed with our WSs analysis, the first-principles electronic structure needs to reproduce all the relevant experimental observations, including a band gap of ∼ 1.3 eV, CE-type magnetic and orbital orders, and weak CD, as well as two distinct Mn sites. We find that the criteria are met by the LDA+U (8 eV) [14,24] band structure of the prototypical half-hoped manganite La1/2Ca1/2MnO3 based on the re- alistic crystal structure [25] supplemented with assumed alternating La/Ca order. Hence, a proper analysis of this LDA+U electronic structure is expected to illustrate the unified picture of weak CD with the Mn3+/Mn4+ assign- ment, which can be easily extended to other cases. In practice, we shall focus on the most relevant low-energy (near the Fermi level EF) bands—they are 16 “eg” spin- majority bands (corresponding to 8 “spin-up” Mn atoms in our unit cell) spanning an energy window of 3.2 eV, as clearly shown in Fig. 1(a). For short notation, the Mn bridge- and corner-sites in the zigzag ferromagnetic chain are abbreviated to B- and C-sites, respectively. Mn spin minority Mn spin majority 2 2x y−2 23z r− Oxygen Fig. 1: (Color online) (a) LDA+U Band structures (dots). The (red) lines result from the Wannier states analysis of the four occupied spin-majority Mn 3d-derived bands. (b) An occupied B-site (“Mn3+”) Wannier orbital in a spin-up (up arrow) zig- zag chain, showing remarkable delocalization to the neighbor- ing Mn C-sites. (c) Low-energy Mn atomic-like Wannier states containing in their tails the integrated out O 2p orbitals. The simplest yet realistic picture of the CO can be ob- tained by constructing occupation-resolvedWSs (ORWSs) from the four fully occupied bands, each centered at one B-site as illustrated in Fig. 1(a). This occupied B-site eg Wannier orbital of 3x2 − r2 or 3y2 − r2 symmetry (so for- mal valency is 3+) contains in its tail the integrated out O 2p orbitals with considerable weight, indicative of the Charge Ordering in Half-Doped Manganites charge-transfer nature [15]. Moreover, this “molecular or- bital in the crystal” extends significantly to neighboring C-sites on the same zig-zag chain. Therefore, although by construction the C-site eg ORWSs (not shown) are com- pletely unoccupied (so formal valency is 4+), appreciable charge is still accumulated within the C-site Mn atomic sphere owing to the large tails of the two occupied OR- WSs centered at the two neighboring B-sites. Integrating the charges within the atomic spheres around the B- and C-site Mn atoms leads to a CD of mere 0.14 e, in agree- ment with experimental 0.1− 0.2 e [7–12]. In this simple picture, one finds a large difference in the occupation num- bers of the ORWSs at the B- and C- sites (∆n = 1), but a small difference in real charge. That is, the convenient 3+/4+ picture is perfectly applicable and it allows weak CD, as long as the itinerant nature of manganites is incor- porated via low-energy WSs rather than standard “atomic states.” In comparison, to make connection with the conven- tional atomic picture and to formulate the spontaneous symmetry breaking with a symmetric starting point (c.f. the next section), we construct from the 16 low-energy bands “atomic-like” WSs (AWSs) of Mn d3z2−r2 and dx2−y2 symmetry, as shown in Fig. 1(b). In this pic- ture, both B- and C-site AWSs are partially occupied with ∆n = 0.6. Now weak CD results from large hybridization with O 2p orbitals, which significantly decreases the charge within the Mn atomic sphere. Obviously, this picture is less convenient for an intuitive and quantitative under- standing of weak CD compatible with the 3+/4+ picture than the previous one, as the latter builds the information of the Hamiltonian and the resulting reduced density ma- trix into the basis. On the other hand, it indeed implies that strong charge-transfer in the system renders it highly inappropriate to associate CD with the difference in the occupation numbers of atomic-like states. Furthermore, we find that the above conclusions are generic in manganites by also looking into the two end lim- its of La1−xCaxMnO3 (x = 0, 1). As shown in Fig. 2, Mn d charge (within the Mn atomic sphere) is found to change only insignificantly upon doping, in agreement with ex- periments [7, 26]. This indicates that doped holes reside primarily in the oxygen atoms, but are entirely coherent in short-range and form additional effective “Mn eg” or- bitals in order to gain the most kinetic energy from the d-p hybridization, as shown in Fig. 1(a)-(b), in spirit sim- ilar to hole-doped cuprates [23]. This justifies the present simplest description of the charge-orbital orders with only the above Mn-centering eg WSs [27]. Leading Mechanisms. – To identify the leading mechanisms of the charge-orbital orders in a rigorous for- malism, we proceed to derive a realistic effective low- energy Hamiltonian, Heff , following our recently devel- oped first-principles WS approach [21, 34]. As clearly shown in Fig. 1, the low-energy physics concerning charge and orbital orders is mainly the physics of one zig-zag FM 0.0 0.5 1.0 spin minority spin majority total C-site C-site C-site B-site B-site B-site Level of Ca doping Fig. 2: The calculated d charge within the Mn atomic sphere for La1−xCaxMnO3 (x = 0, 1/2, and 1). The results are shown in terms of (i) B-site (filled symbols), C-site (empty symbols), and site-average (half-filled symbols); (ii) spin-majority (up tri- angles), spin-minority (down triangles), and total charge (dia- monds). The lines are guides to eye. chain, since electron hopping between the antiferromag- netically arranged chains is strongly suppressed by the double-exchange effect [18, 28, 29]. Our unbiased first- principles analysis of the 16-band one-particle LDA+U Hamiltonian in the above AWS representation reveals [18, 28, 29, 36] Heff = − 〈ij〉γγ′ jγ′diγ + h.c.)− Ez + Ueff ni↑ni↓ + V niQ1i + T i Q2i + T i Q3i) (1) in addition to the elastic energy K({Qi}). Here d†iγ and diγ are electron creation and annihilation operators at site i with “pseudo-spin” γ defined as | ↑〉 = |3z2 − r2〉 and | ↓〉 = |y2−x2〉 AWSs, corresponding to pseudo-spin oper- ator T x i↑di↓+d i↓di↑)/2 and T i↑di↑−d i↓di↓)/2. ni = d i↑di↑ + d i↓di↓ is the electron occupation number. The in-plane hoppings are basically symmetry related: = t/4, t = 3t/4, t 3t/4 where the signs depend on hopping along the x or y direction. Ez stands for the oxygen octahedral-tilting induced crystal field. Ueff and V are effective on-site and nearest-neighbor e-e in- teractions, respectively. g is the e-l coupling constant. Qi = (Q1i, Q2i, Q3i) is the standard octahedral-distortion vector, where Q1i is the breathing mode (BM), and Q2i and Q3i are the Jahn-Teller (JT) modes [18,19,28,29,36]. In Eq. (1) the electron-lattice couplings have been con- strained to be invariant under the transformation of the cubic group [36]. The effective Hamiltonian are determined by match- ing its self-consistent Hartree-Fock (HF) expression with D. Volja et al. Table 1: Contributions of different terms to the energy gain (in units of meV per Mn) due to the CO formation in self- consistent mean-field theory. BM (JT) denotes the contribu- tion from electronic coupling to the breathing (Jahn-Teller) mode. K denotes that from the elastic energy. Qi Total Ueff V t BM JT K 0 -13 -11 -15 12 0 0 0 realistic -127 -22 -42 71 -42 -113 24 HLDA+U [21, 34] owing to the analytical structure of the LDA+U approximation [30]. An excellent mapping re- sults from t = 0.6 eV, Ez = −0.08 eV, Ueff = 1.65 eV, V = 0.44 eV, and g = 2.35 eV/Å. These numbers are close to those obtained for undoped LaMnO3 [21] (exclud- ing V , which is inert in the undoped case), and indicates that the spin-majority eg electrons in the manganites are still in the intermediate e-e interaction regime with compa- rable e-l interaction. Note that Ueff should be understood as an effective repulsion of corresponding Mn-centered WS playing the role of “d” states, rather than the “bare” d-d interaction U = 8 eV [21]. Furthermore, the rigidity of the low-energy parameters upon significant (x = 0.5) doping verifies the validity of using a single set of parameters for a wide range of doping levels, a common practice that is not a priori justified for low-energy effective Hamiltoni- ans. Clearly, the observed optical gap energy scale of ∼ 2 eV originates mainly from Ueff instead of the JT splitting widely assumed in existing theories [5, 20]. Now based on the AWSs, CO is measured by ∆n = 〈ni∈B〉 − 〈nj∈C〉 = 0.6. It deviates from unity because the kinetic energy ensures that the ground state is a hybrid of both B- and C-site AWSs, like in the usual tight-binding modeling based on atomic orbitals. However, since the AWSs considerably extend to neighboring oxygen atoms, the actual CD is much smaller than ∆n. With the successful derivation of Heff , the microscopic mechanisms of the charge-orbital orders emerge. First of all, note that the kinetic term alone is able to produce the orbital ordered insulating phase [18,29]: Since the intersite interorbital hoppings of the Mn eg electrons along the x and y directions have opposite signs, the occupied bonding state is gapped from the unoccupied nonbonding and an- tibonding states (by t and 2t, respectively) in the enlarged unit cell. As for orbital ordering, the Mn dy2−z2 (dx2−z2) orbital on any B-site bridging two C-sites along the x (y) direction is irrelevant, as the hopping integrals involving it is vanishing. That is, only the d3x2−r2 (d3y2−r2) orbital on that B-site is active and the B-sites on the zigzag FM chains have to form an “ordered” pattern of alternating (3x2 − r2)/(3y2 − r2) orbitals. However, the kinetic term alone give only ∆n = 0. Clearly, CO is induced by the interactions, Ueff , V , or g. To quantify their relative importance for CO, we calcu- late their individual contributions to the total energy gain with respect to the aforementioned ∆n = 0 but orbital- ordered insulating state in the self-consistent mean-field theory. The results are listed in Table 1. The first row obtained without lattice distortions provides a measure of the purely electronic mechanisms for CO. Interestingly, al- though the tendency is weak (−13 meV), e-e interactions all together are sufficient to induce a CO, consistent with the results of our first-principles calculations. The second row is obtained for the realistic lattice distortions, which shows a dramatic enhancement of CO by the JT distor- tions (−113 meV), given that only half of the Mn atoms are JT active. Together with a −42 meV gain from the BM distortion, the e-l interactions overwhelm the cost of the kinetic (71 meV) and elastic (24 meV) energy by −63 meV, further stabilizing the observed CO. Nevertheless, the −64 meV energy gain from the overall e-e interac- tions accounts for half of the total energy gain, illustrating clearly their importance to the realization of the resulting ∆n = 0.6. Indeed, a further analysis reveals that e-l cou- plings alone (Ueff = V = 0) produce ∆n ∼ 0.3, only half of the realistic ∆n, manifesting the necessity of including e-e interactions. Further insights can be obtained by considering the in- dividual microscopic roles of these interactions acting to the kinetic-only starting point. First, infinitesimal Ueff or g can induce CO, as a result of exploiting the fact that the B- (C-) site has one (two) active eg orbital: (i) Ueff has no effect on the B-sites; therefore, Ueff pushes the eg electrons to the B-sites, in order to lower the Coulomb energy on the C-sites [18]. This is opposite to its normal behavior of favoring charge homogeneity in systems such as straight FM chains (realized in the C-type antiferro- magnet). (ii) It is favorable to cooperatively induce the (3x2 − r2)/(3y2 − r2)-type JT distortions on the B-sites and the BM distortions on the C-sites in order to mini- mize elastic energy [2]. These lattice distortions lower the relative potential energy of the active orbitals in the B- sites, also driving the eg electrons there. Hence, Ueff and e-l interactions work cooperatively in the CO formation. Unexpectedly, we find that V alone must be larger than Vc ≃ 0.8 eV to induce CO. This is surprising in compar- ison to the well-known Vc = 0 for straight FM chains. The existence of Vc is in fact a general phenomenon in a “pre-gapped” system. Generally speaking, in a sys- tem with a charge gap, ∆0, before CO takes place (e.g. the zigzag chain discussed here), forming CO always costs non-negligible kinetic energy due to the mixing of states across the gap. As a consequence, unlike the first order energy gain from Ueff and g, the second order energy gain from V is insufficient to overcome this cost until V is large enough (of order ∆0). In this specific case, V = 0.44 eV is insufficient to induce CO by itself, but it does contribute significantly to the total energy gain once CO is triggered by Ueff or g, as discussed above. It is worth mentioning that the contribution from the EzTz term is neglected from Table 1 because of the small coefficient of Ez ≃ 8 meV, consistent with the previous Charge Ordering in Half-Doped Manganites study for undoped La3MnO3 [21]. In perovskites, the tilt- ing of the oxygen octahedra could yield the Jahn-Teller- like distortion of GdFeO3 type, which can be mathemat- ically described by the EzTz or ExTx term. However, in the perovskite manganites these terms are shown here and in Ref. [21] to be negligibly small. In addition, in half- doped manganites, the pattern of the orbital order is pre- dominantly pined by the zig-zag pathway of the itinerant electrons and thus the effect of tilting is less relevant. On the other hand, the EzTz is very effective to explain the zig-zag orbital ordering of (x2 − z2)/(y2 − z2) pattern in half-doped layered manganites, such as La0.5Sr1.5MnO4 Ref. [35], where pseudo-spin-up (the 3z2 − r2 orbital) is favored by a much stronger Ez due to the elongation of the oxygen octahedral along the c-axis. Described via the pseudo-spin angel, θ = arctan(Tx/Tz) [21], Ez signif- icantly reduces θ from ±120◦ [i.e., (3x2 − r2)/(3y2 − r2)] to ±30◦ [i.e., (x2 − z2)/(y2 − z2). The present results would impose stringent constraints on the general understanding of the manganites. For ex- ample, Zener polarons were shown to coexist with the CE phase within a purely electronic modeling of near half- doped manganites [31]. However, to predict the realistic phase diagram of the manganites, one must take into ac- count the lattice degree of freedom. In fact, when propos- ing the CE phase, Goodenough considered its advantage of minimizing the strain energy cost. Note that the pre- vious HF theory indicated a decrease of the total energy by 0.5 eV per unit cell with Zener-polaron-like displace- ment [15]. The present LDA+U calculations reveal an increase of the total energy by 1.07 eV per unit cell with the same Zener-polaron-like displacement. This discrep- ancy is quite understandable from the characteristics of the LDA functional, which favors covalent bond, while the HF approximation tends to over localize the orbital and disfavor bonding. To resolve the competition between the CE phase and the Zener polaron phase, real structural optimization is necessary and will be presented elsewhere. As another example, in the absence of e-l interactions the holes were predicted to localize in the B-site-type region (i.e., the straight segment portion of the zigzag FM chain) in the CxE1−x phase of doped E-type manganites [32]. However, since the B-sites are susceptible to the JT dis- tortion, they are more likely to favor electron localization instead; future experimental verification is desirable. Summary. – A general first-principles Wannier func- tion based method and the resulting valence picture of doped holes in strongly correlated charge-transfer systems are presented. Application to the charge order in half- doped manganites reconciles the current fundamental con- tradictions between the traditional 3+/4+ valence picture and the recently observed small charge disproportion. In essence, while the doped holes primarily resides in the oxy- gen atoms, the local orbital are entirely coherent following the symmetry of Mn eg orbital, giving rise to an effective valence picture with weak CD. Furthermore, our first- principles derivation of realistic low-energy Hamiltonian reveals a surprisingly important role of electron-electron interactions in ordering charges, contrary to current lore. Our theoretical method and the resulting flexible valence picture can be applied to a wide range of doped charge- transfer insulators for realistic investigations and interpre- tations of the rich properties of the doped holes. ∗ ∗ ∗ We thank E. Dagotto for stimulating discussions and V. Ferrari and P. B. Littlewood for clarifying their Hartree- Fock results [15]. The work was supported by U.S. Depart- ment of Energy under Contract No. DE-AC02-98CH10886 and DOE-CMSN. 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[29] Solovyev I. V. and Terakura K., Phys. Rev. Lett., 83 (1999) 2825. D. Volja et al. [30] Unlike the local interactions, the intersite interactions in LDA+U is treated via LDA functional, which has dominant Hartree contribution that we used for this mapping. [31] Efremov D. V. et al., Nature Materials, 3 (2004) 853. [32] Hotta T. et al., Phys. Rev. Lett., 90 (2003) 247203. [33] Ch. Jooss et al., PNAS, 104 (2007) 13597. [34] Yin W.-G. and Ku W., Phys. Rev. B, 79 (2009) 214512. [35] D. J. Huang et al., Phys. Rev. Lett., 92 (2004) 087202. [36] P. B. Allen and V. Perebeinos, Phys. Rev. B, 60 (1999) 10747. Introduction. – Small CD vs. 3+/4+ valence Picture. – Leading Mechanisms. – Summary. –

0704.1835

Direct extraction of one-loop integral coefficients

arXiv:0704.1835v1 [hep-ph] 15 Apr 2007 SLAC–PUB–12455 UCLA/07/TEP/12 Direct extraction of one-loop integral coefficients ∗ Darren Forde Stanford Linear Accelerator Center Stanford University Stanford, CA 94309, USA, Department of Physics and Astronomy, UCLA Los Angeles, CA 90095–1547, USA. (Dated: 12th April 2007) Abstract We present a general procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted by considering two-particle and triple unitarity cuts of the corresponding bubble and triangle integral functions. After choosing a specific parameterisation of the cut loop momentum we can uniquely identify the coefficients of the desired integral functions simply by examining the behaviour of the cut integrand as the unconstrained parameters of the cut loop momentum approach infinity. In this way we can produce compact forms for scalar integral coefficients. Applications of this method are presented for both QCD and electroweak processes, including an alternative form for the recently computed three- mass triangle coefficient in the six-photon amplitude A6(1 −, 2+, 3−, 4+, 5−, 6+). The direct nature of this extraction procedure allows for a very straightforward automation of the procedure. PACS numbers: 11.15.Bt, 11.25.Db, 12.15.Lk, 12.38.Bx ∗ Research supported in part by the US Department of Energy under contracts DE–FG03–91ER40662 and DE–AC02–76SF00515. http://arXiv.org/abs/0704.1835v1 I. INTRODUCTION Maximising the discovery potential of future colliders such as CERN’s Large Hadron Collider (LHC) will rely upon a detailed understanding of Standard Model processes. Dis- tinguishing signals of new physics from background processes requires precise theoretical calculations. These background processes need to be known to at least a next-to-leading order (NLO) level. This in turn entails the need for computation of one-loop amplitudes. Whilst much progress has been made in calculating such processes, the feasibility of produc- ing these needed higher multiplicity amplitudes, such as one-loop processes with one or more vector bosons (W’s, Z’s and photons) along with multiple jets, strains standard Feynman diagram techniques. Direct calculations using Feynman diagrams are generally inefficient; the large number of terms and diagrams involved has by necessity demanded (semi)numerical approaches be taken when dealing with higher multiplicity amplitudes. Much progress has been made in this way, numerical evaluations of processes with up to six partons have been performed [1, 2, 3, 4, 5]. On assembling complete amplitudes from Feynman diagrams it is commonly found that large cancellations take place between the various terms. The remaining result is then far more compact than would naively be expected from the complexity of the original Feynman diagrams. The greater simplicity of these final forms has spurred the development of alternative more direct and efficient techniques for calculating these processes. The elegant and efficient approach of recursion relations has long been a staple part of the tree level calculational approach [6, 7]. Recent progress, inspired by developments in twistor string theory [8, 9], builds upon the idea of recursion relations, but centred around the use of gauge-independent or on-shell intermediate quantities and hence negating a potential source of large cancellations between terms. Britto, Cachazo and Feng [10] initially wrote down a set of tree level recursion relations utilising on-shell amplitudes with complex values of external momenta. Then, along with Witten [11], they proved these on-shell recursion relations using just a knowledge of the factorisation properties of the amplitudes and Cauchy’s theorem. The generality of the proof has led to their application in many diverse areas beyond that of massless gluons and fermions in gauge theory [10, 13]. There have been extensions to theories with massive scalars and fermions [14, 15, 16] as well as amplitudes in gravity [12]. Similarly “on-shell” approaches can also be constructed at loop level. The unitarity of the perturbative S-matrix can be used to produce compact analytical results by “glu- ing” together on-shell tree amplitudes to form the desired loop amplitude. This unitarity approach has been developed into a practical technique for the construction of loop ampli- tudes [17, 18, 19], initially, for computational reasons, for the construction of amplitudes where the loop momentum was kept in D = 4 dimensions. This limited its applicability to computations of the “cut-constructible” parts of an amplitude only, i.e. (poly)logarithmic containing terms and any associated π2 constants. Amplitudes consisting of only such terms, such as supersymmetric amplitudes, can therefore be completely constructed in this way. QCD amplitudes contain in addition rational pieces which cannot be derived using such cuts. The “missing” rational parts are constructible directly from the unitarity approach only by taking the cut loop momentum to be in D = 4 − 2ǫ dimensions [20]. The greater difficulty of such calculations has, with only a few exceptions [21, 22], restricted the application of this approach, although recent developments [23, 24, 25] have provided new promise for this direction. The generality of the foundation of on-shell recursion relation techniques does not limit their applicability to tree level processes only. The “missing” rational pieces at one-loop, in QCD and other similar theories, can be constructed in an analogous way to (rational) tree level amplitudes [26, 27]. The “unitarity on-shell bootstrap” technique combines unitarity with on-shell recursion, and provides, in an efficient manner, the complete one-loop ampli- tude. This approach has been used to produce various new analytic results for amplitudes containing both fixed numbers as well as arbitrary numbers of external legs [28, 29, 30]. Other newly developed alternative methods have also proved fruitful for calculating rational terms [31, 32, 33, 34]. In combination with the required cut-containing terms [35, 36, 37] these new results for the rational loop contributions combine to give the complete analytic form for the one-loop QCD six-gluon amplitude. The development of efficient techniques for calculating, what were previously difficult to derive rational terms, has emphasised the need to optimise the derivation of the cut- constructible pieces of the amplitude. One-loop amplitudes can be decomposed entirely in terms of a basis of scalar bubble, scalar triangle and scalar box integral functions. Deriving cut-constructible terms therefore reduces to the problem of finding the coefficients of these basis integrals. For the coefficients of scalar box integrals it was shown in [38] that a combination of generalised unitarity [19, 39, 40, 41], quadruple cuts in this case, along with the use of complex momenta could be used, within a purely algebraic approach, to extract the desired coefficient from the cut integrand of the associated box topology. Extracting triangle and bubble coefficients presents more of a problem. Unlike for the case of box coefficients, cutting all the propagators associated with the desired integral topology does not uniquely isolate a single integral coefficient. Inside a particular two-particle or triple cut lie multiple scalar integral coefficients corresponding to integrals with topologies sharing not only the same cuts but also additional propagators. These coefficients must therefore be disentangled in some way. There are multiple directions within the literature which have been taken to effect this separation. The pioneering work by Bern, Dixon, Dunbar and Kosower related unitarity cuts to Feynman diagrams and thence to the scalar integral basis, this then allowed for the derivation of many important results [17, 18, 19]. More recently the technique of Britto et. al. [23, 24, 25, 35, 36] has for two-particle cuts and the its extension to triple cuts by Mastrolia [42], highlighted the benefits of working in a spinor formalism, where the cut integrals can be integrated directly. Important results obtained in this way include the most difficult of the cut-constructable pieces for the one-loop amplitude for six gluons with the helicity configurations A6(+−+−+−) and A6(−+−−++). The cut-constructible parts of Maximum-Helicity-Violating (MHV) one-loop amplitudes were found by joining MHV amplitudes together in a similar manner to at tree level [43]. This method has been applied by Bedford, Brandhuber, Spence and Travaglini to produce new QCD results [37]. In the approach of Ossola, Papadopoulos and Pittau [44, 45] it is possible to avoid the need to perform any integration or use any integral reduction techniques. Coefficients are instead extracted by solving sets of equations. The solutions of these equations include the desired coefficients, along with additional “spurious” terms corresponding to coefficients of terms which vanish after integrating over the loop momenta. The many-fold different processes and their differing parton contents that will be needed at current and future collider experiments suggests that some form of automation, even of the more efficient “on-shell” techniques, will be required. From an efficiency standpoint, therefore, we would ideally wish to minimise the degree of calculation required for each step of any such process. Here we propose a new method for the extraction of scalar integral coefficients which aims to meet this goal. The technique follows in the spirit of the simplicity of the derivation of scalar box coefficients given in ref. [38]. Desired coefficients can be constructed directly using two-particle or triple cuts. The complete one-loop amplitude can then be obtained by summing over all such cuts and adding any box terms and rational pieces. Alternatively our technique can be used to extract the bubble and triangle coefficients from a one-loop amplitude, generated for example from a Feynman diagram. Hence the technique is acting as an efficient way to perform the integration. We use unitarity cuts to freeze some of the degrees of freedom of the integral loop mo- mentum, whilst leaving others unconstrained. This then isolates a specific single bubble or triangle integral topology and hence its coefficient. Within each cut there remain ad- ditional coefficients. In the triangle case those of scalar box integrals. In the bubble case both scalar box and scalar triangle integrals contribute. Disentangling our desired coefficient from these extra contributions is a straightforward two step procedure. First one rewrites the loop momentum inside the cut integrand in terms of its unconstrained parameters. In the triangle case there is a single parameter, and in the bubble case there are a pair of parameters. Examining the behaviour of the integrand as these unconstrained parameters approach infinity then allows for a straightforward separation of the desired coefficient from any extra contributions. The coefficient of each basis integral function can therefore be extracted individually in an efficient manner with no further computation. This paper is organised as follows. In section II we outline the notation used throughout this paper. In section III we proceed to present the basic structure of a one-loop amplitude in terms of a basis of scalar integral functions. We describe in section IV our procedure for extracting the coefficients of scalar triangle coefficients through the use of a particular loop-momentum parameterisation for the triple cuts along with the properties of the cut as the single free integral parameter tends to infinity. Section V extends this formalism to include the extraction of scalar bubble coefficients. The two-particle cut used in this case contains an additional free parameter and requires an additional step in our procedure. Finally in section VI we conclude by providing some applications which act as checks of our method. Initially we examine the extraction of various basis integral coefficients from some common one-loop integral functions. We then turn our attention to the construction of the coefficients of some more phenomenologically interesting processes. These include the three-mass triangle coefficient for the six photon amplitude A6(− + − + −+), as well as a representative three-mass triangle coefficient of the process e+e− → q+q−g−g+. Finally we construct the complete cut-containing part of the amplitude A 1−loop −, 2−, 3+, 4+, 5+) and discuss further comparisons against coefficients of more complicated gluon amplitudes contained in the literature. II. NOTATION In this section we summarise the notation used in the remainder of the paper. We will use the spinor helicity formalism [47, 48], in which the amplitudes are expressed in terms of spinor inner-products, 〈j l〉 = 〈j−|l+〉 = ū−(kj)u+(kl) , [j l] = 〈j+|l−〉 = ū+(kj)u−(kl) , (2.1) where u±(k) is a massless Weyl spinor with momentum k and positive or negative chirality. The notation used here follows the QCD literature, with [i j] = sign(k0i k j )〈j i〉∗ for real momenta so that, 〈i j〉[j i] = 2ki · kj = sij . (2.2) Our convention is that all legs are outgoing. We also define, λi ≡ u+(ki), λ̃i ≡ u−(ki) . (2.3) We denote the sums of cyclicly-consecutive external momenta by i...j ≡ k i + k i+1 + · · ·+ k j−1 + k j , (2.4) where all indices are mod n for an n-gluon amplitude. The invariant mass of this vector is si...j ≡ K2i...j . (2.5) Special cases include the two- and three-particle invariant masses, which are denoted by sij ≡ K2ij ≡ (ki + kj)2 = 2ki · kj, sijk ≡ (ki + kj + kk)2 . (2.6) We also define spinor strings, ∣ (/a ± /b) = 〈i a〉[a j] ± 〈i b〉[b j] , ∣ (/a + /b)(/c + /d) = [i a] ∣ (/c + /d) + [i b] ∣ (/c + /d) . (2.7) III. UNITARITY CUTTING TECHNIQUES AND THE ONE-LOOP INTEGRAL BASIS Our starting point will be the general dimensionally-regularised decomposition of a one- loop amplitude into a basis of scalar integral functions [18, 53] A1−loopn =Rn+rΓ (µ2)ǫ (4π)2−ǫ biB0(K cijC0(K i , K dijkD0(K i , K j , K .(3.1) The scalar bubble, triangle and box integral functions are denoted by B0, C0 and D0 respec- tively, and along with rΓ their explicit forms can be found in Appendix C. The bi, cij and dijk are their corresponding rational coefficients. Any ǫ dependence within these coefficients has been removed and placed into the rational, Rn, term. The problem of deriving the one-loop amplitude is therefore reduced to that of finding the coefficients of these scalar integral functions and any rational terms when working in D = 4 dimensions. We are going to consider obtaining these coefficients via the application of various cuts within the framework of generalised unitarity [19, 39, 40, 41]. In general our cut momenta will be complex, so for our purposes we define a “cut” as the replacement (l + Ki)2 → (2π)δ((l + Ki)2). (3.2) By systematically constructing all possible unitarity cuts we can reproduce every integral coefficient of a particular amplitude. Alternatively, application of the same procedure of “cutting” legs can be used to extract from a one-loop integral the corresponding coefficients of the standard basis integrals making up that particular integral, in a sense acting as a form of specialised integral reduction. This approach follows in a similar vein to that adopted by Ossola, Papadopoulos and Pittau [44]. The most straightforward implementation of the technique we present here is when the cut loop momentum is massless and kept in D = 4 dimensions. Eq. 3.1 therefore contains, within the term Rn, any rational terms missed by performing cuts in only D = 4. Approaches for deriving such terms independently of unitarity cuts exist and so we do not concern ourselves with these here [23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 44, 45]. As was demonstrated in [38], the application of a quadruple cut, as shown in figure 1, to A1−loopn uniquely identifies a particular box integral topology D0(K i , K j , K k) and hence its Kk−Ki − Kj − Kk FIG. 1: A generic quadruple cut used to isolate the scalar box integral D0(K coefficient. This coefficient is then given by dijk = A1(lijk;a)A2(lijk;a)A3(lijk;a)A4(lijk;a), (3.3) where lijk;a is the a th solution of the cut loop momentum l that isolates the scalar box function D0(K i , K j , K k), there are 2 such solutions. Eq. 3.3 applies as well to the cases when one or more of the four legs of the box is massless. This is a result of the existence, for complex momenta, of a well-defined three-point tree amplitude corresponding to any corner of a box containing a massless leg. Applying a triple cut to the amplitude A1−loopn does not isolate a single basis integral. Instead we have a triangle integral plus a sum of box integrals obtained by “opening” a fourth propagator. This can be represented schematically via (µ2)ǫ (4π)2−ǫ cijC0(K i , K j ) + dijkD0(K i , K j , K k) + . . . , (3.4) where the additional terms correspond to “opening” the Ki leg or the Kj leg instead of the −(Ki + Kj) leg. Similarly in the case of a two-particle cut we again cannot isolate a single basis integral by itself. Instead we get additional triangle and box integrals corresponding to “opening” third and forth propagators. Schematically this is given by (µ2)ǫ (4π)2−ǫ biB0(K i ) + cijC0(K i , K j ) + dijkD0(K i , K j , K k) + . . . , (3.5) where again the additional terms are boxes with the Ki leg or the Kj legs “opened”. Whilst not isolating a single integral each of the above cuts does single out either one scalar triangle, in the triple cut case, or one scalar bubble, in the two-particle cut case. Disentangling these single bubble or triangle integral functions from the contributions of the remaining basis integrals will allow us to directly read off the corresponding coefficient. Applying all possible two-particle, triple and quadruple cuts then enables us to derive the coefficients of every basis integral function. IV. TRIPLE CUTS AND SCALAR TRIANGLE COEFFICIENTS A triple cut contains not only contributions for the corresponding scalar triangle integral, but also contributions from scalar box integrals which share the same three cuts as the triangle. Of the four propagators of a scalar box integral, three will be given by the three cut legs of the triple cut loop integral. The forth propagator will be contained inside the cut integrand in a denominator factor of the form (l − P )2, which corresponds to a propagator pole. Ideally we want to separate terms containing such poles from the remainder of the cut integrand. The remaining term will be the scalar triangle integral multiplied by its coefficient for that particular cut. The three delta functions of a triple cut constrain the cut loop momentum such that only a single free parameter of the integral remains, which we label t. We can express the loop momentum in terms of this parameter using the orthogonal null four-vectors, a i , with i = 1, 2, 3, specific forms for these basis vectors are presented in section IVA. The loop momentum is then given by lµ = a 0 t + 1 + a 2 . (4.1) Denominator factors of the cut integrand depending upon the cut loop momentum, can be written as propagators of the general form, (l − P )2. When these propagators go on-shell they will correspond to poles in t. These poles will be solutions of the following equation (l − P )2 = 0 ⇒ 2(a0 · P )t + 2(a1 · P ) + 2(a2 · P ) − P 2 = 0. (4.2) If we consider t to be a complex parameter then we can use a partial fraction decomposi- tion in terms of t to rewrite an arbitrary triple-cut integral. For the extraction of integral coefficients we need only work with integrals in D = 4 dimensions. We also drop an overall denominator factor of 1/(2π)4 which multiplies all integrals. The partial fraction decompo- sition is therefore given, in the case when we have applied a triple cut on the legs l2, (l−K1)2 and (l − K2)2, by (2π)3 δ(l2i )A1A2A3 =(2π)3 δ(l2i ) [InftA1A2A3] (t) + poles {j} Rest=tj A1A2A3 t − tj  , (4.3) where li = l −Ki and l0 = l. This is a sum of all possible poles of t, labelled here as the set {j}, contained in the cut integrand denoted by A1A2A3. Pieces of the integrand without a pole are contained in the Inf term, originally given in [30], and defined such that ([InftA1A2A3] (t) − A1(t)A2(t)A3(t)) = 0. (4.4) In general [InfzA1A2A3](t) will be some polynomial in t, [Inf tA1A2A3] (t) = i, (4.5) where m is the leading degree of large t behaviour and depends upon the specific integrand in question. After applying the three delta functions constraints we see that taking the residue of A1A2A3 at a particular pole, t = t0, removes any remaining dependence upon the loop momentum. Hence we can write δ(l2i ) Rest=t0 A1A2A3 t − t0 ∼ lim [(t − t0)A1A2A3] δ(l2i ) t − t0 . (4.6) Where on the right hand side of this we understand the integral, d4l, as over the parame- terised form of l in terms of t and the three other degrees of freedom. In the cut integrand the only source of poles in t is from propagator terms of the type 1/(l − P )2. Generally each such propagator, when on-shell, contains two poles due to the quadratic nature, in t, of eq. (4.2). If we label these solutions t± then we can write a triple-cut scalar box in terms of these poles as δ(l2i ) (l − P )2 ∼ t+ − t− δ(l2i ) t − t+ δ(l2i ) t − t− . (4.7) From comparing this to eq. (4.6) we see that all residue terms of eq. (4.3) simply correspond to pieces of triple-cut scalar box functions multiplied by various coefficients. Therefore we can associate all residue terms with scalar boxes, meaning that our triple cut amplitude can be written simply as (2π)3 δ(l2i )A1A2A3 = (2π) boxes {l} 0 . (4.8) This is a sum over the set {l} of possible cut scalar boxes, Dcut0 , and their associated coefficients, dl, along with a power series in positive powers of t. In eq. (4.8) we have integrated over the three delta functions after performing the integral transformation from lµ to t, the Jacobian of which, and any additional factors picked up from the integration is contained in the factor Jt. The limit m of the summation is the maximum power of t appearing in the integrand, which in turn is the maximum power of l appearing in the numerator of the integrand. In general for renormalisable theories, such as QCD amplitudes, m ≤ 3. We must now turn our attention to answering the question of what do the remaining terms correspond to? To do this we need to understand the behaviour of the integrals over positive powers of t. There is a freedom in our choice of the parameterisation of the cut-loop momentum. This freedom extends, as we will prove in section IVA, to choosing a parametrisation where the integrals over all positive powers of t vanish. Doing this then reduces the cut integrand to (2π)3 δ(l2i )A1A2A3 = (2π) dtJt + boxes {l} 0 . (4.9) The remaining integral is now simply that of a triple-cut scalar triangle, multiplied by the co- efficient f0. For the triple-cut scalar triangle integral, C i , K j ), given by −(2π)3 dtJt, the triple cut form of eq. (C4), we find that its corresponding coefficient is given simply by cij = − [InftA1A2A3] (t) , (4.10) which is just the first term in the series expansion in t of the cut-integrand at infinity. The simplicity of this result relies crucially upon two facts. The first is that on the triple cut the integral is sufficiently simple that it can be decomposed into either a triangle contribution or a box contribution. This is important as it allows us to easily distinguish between the two types of term. As an example consider a linear box which contains a numerator factor constructed such that it vanishes at the pole contained in the denominator, but without being proportional to the denominator itself. To which basis integral does this term contribute to? In the simplest case such a term would look like δ(l2i ) 〈lW 〉 〈lP 〉 = δ(l2i ) 〈aW 〉(t − t0) 〈aP 〉(t− t0) 〈aW 〉 〈aP 〉 δ(l2i ), and hence must contribute entirely to the triangle integral, it contains no box terms. Here we have chosen a simplified loop momentum parameterisation in terms of two basis spinors |a+〉 and |a+〉 such that 〈lP 〉 = t〈aP 〉+ 〈aP 〉. This then contains a pole in t at t0 = −〈aP 〉/〈aP 〉 and we have chosen the spinor |W+〉 such that 〈aW 〉 = −t0〈aW 〉. The second crucial fact is the vanishing of the other integrals over t so that the complete scalar triangle integral is given by only the remaining integral over t0. Hence the coefficient is given by a single term. Furthermore, the use of a complex loop momentum also means that we can apply this formalism to the extraction of scalar coefficients corresponding to one- and two-mass triangles as well as three-mass triangles. As discussed above for the case of box coefficients, this is a result of the possibility of a well-defined three-point vertex when using complex momentum, enabling in these cases the construction of non-vanishing cut integrands. A. The momentum parameterisation We wish to compute the coefficient of the scalar triangle singled out by the triple cut given in figure 2. The cut integral when written in terms of tree amplitudes is l1 = l − K1 l2 = l − K2 c3 − 1 c2 − 1 c1 − 1 FIG. 2: The triple cut used to compute the scalar triangle coefficient of C0(K (2π)3 δ(l2i )A c3−c1+2 (−l, c1, . . . , (c3 − 1), l1)Atreec2−c3+2(−l1, c3, . . . , (c2 − 1), l2) × Atreen−c2+c1+2(−l2, c2, . . . , (c1 − 1), l), (4.11) with l1 = l − K1 = l − Kc1...c3−1 and l2 = l − K2 = l + Kc2...c1−1, so that K1 = Kc1...c3−1 and K2 = −Kc2...c1−1. Our first step will be to find a parameterisation of l in terms of the single free integral parameter remaining after satisfying all three of the cut delta functions constraints, l2 = 0, l21 = (l − K1)2 = 0, and l22 = (l − K2)2 = 0. (4.12) Each of the three legs can be massive or massless. We will deal with the general case of three massive legs explicitly here. The cases with massless legs are then easily found by setting the relevant mass in the parameterisation to zero. We will find it very convenient to express lµ in terms of a basis of momentum identical to the momenta l1 and l2 used by Ossola, Papadopoulos and Pittau [44]. We will write these momenta in the suggestive notation K♭1 and K♭2 and define them via 1 = K 2 = K 1 , (4.13) with γ = 〈K♭,−1 | /K 1 〉 ≡ 〈K 2 | /K 2 〉 and Si = K2i . Each momentum K♭1, K♭2 is the massless projection of one of the massive legs in the direction of the other masslessly projected leg. A more practical definition of K♭1 and K 2, in terms of the external momenta alone, can be found by solving the above equations for K♭1 and K 2, so that in terms of S1, S2, K 1 and K 2 we have 1 − (S1/γ)K 1 − (S1S2/γ2) 2 − (S2/γ)K 1 − (S1S2/γ2) . (4.14) In addition γ can be expressed in terms of the external momenta, γ± = (K1 · K2) ± ∆, ∆ = (K1 · K2)2 − K21K22 . (4.15) When using eq. (4.10) we must average over the number of solutions of γ. In the three-mass case there are a pair of solutions. For the one- and two-mass cases, when either K21 = 0 or K22 = 0, then there is only a single solution. After satisfying the three constraints given by eq. (4.12) we write the spinor components of lµ in terms of our basis K♭1 and K 〈l−| = t〈K♭,−1 | + α01〈K 〈l+| = α02 〈K♭,+1 | + 〈K 2 |, (4.16) where α01 = S1 (γ − S2) (γ2 − S1S2) , α02 = S2 (γ − S1) (γ2 − S1S2) . (4.17) Written as a four-vector, lµ is given by lµ = α02K 1 + α01K 〈K♭,−1 |γµ|K 2 〉 + α01α02 〈K♭,−2 |γµ|K 1 〉. (4.18) We can also use momentum conservation to write component forms for the other two cut momenta li with i = 1, 2, 〈l−i | = t〈K 1 | + αi1〈K 〈l+i | = 〈K♭,+1 | + 〈K 2 |, (4.19) where the αij are given in Appendix A. A final point is that after having integrated over the three delta function constraints and performed the change of variables to the momentum parameterisation of eq. (4.16) we have the factor Jt = 1/(tγ) contained in eq. (4.8). We always associate this factor with the scalar triangle integral and so its explicit form does not play a role in our formalism. B. Vanishing integrals As we have remarked previously, the simplicity of the method outlined here rests crucially upon the properties of the momentum parameterisation we have used. The key feature is the vanishing of the integrals over t. It can easily be shown that within our chosen momentum parameterisation, of section IVA, any integral of a positive or negative power of t vanishes. Following an argument very similar to that used by Ossola, Papadopoulos and Pittau [44] we use 〈K♭,±1 | /K1|K 2 〉 = 0, 〈K 1 | /K2|K 2 〉 = 0 and 〈K 1 |γµ|K 2 〉〈K 1 |γµ|K 2 〉 = 0, to show that 〈K♭,−1 |/l|K l2l21l = 0 ⇒ = 0 for n ≥ 1, 〈K♭,−2 |/l|K l2l21l = 0 ⇒ dtJtt n = 0 for n ≥ 1. (4.20) The vanishing of these terms then leads directly to our general procedure, encapsulated in eq. (4.10), which is to simply express the triple cut of the desired scalar triangle in the momentum parameterisation given by eq. (4.16) and then take the t0 component of a series expansion in t around infinity. V. TWO-PARTICLE CUTS AND SCALAR BUBBLE COEFFICIENTS In the same spirit as the triangle case we now wish to extract the coefficients of scalar bubble terms using, in this case, a two-particle cut. Now a two-particle cut will contain in addition to our desired scalar bubble both scalar boxes and triangles, all of which need to be disentangled. What we will find, though, is that naively applying the technique as given for the scalar triangle coefficients will not give us the complete scalar bubble contribution. The reason for this is straightforward to see. A two-particle cut places only two constraints on the loop momentum and so we can parameterise it in terms of two free variables, which we will label t and y. Consider rewriting the cut integrand in a partial fraction decomposition in terms of y. Schematically, therefore, the two-particle cut of the legs l2 and (l −K1)2 can be written as (2π)2 δ(l2i )A1A2 =(2π) dtdyJt,y [InfyA1A2] (y) + poles {j} Resy=yj A1A2 y − yj  ,(5.1) where again {j} is the sum over all possible poles, this time in y, and Jt,y contains any terms from the change into the parameterisation of y and t as well as any pieces picked up by integrating over the two delta functions. So far this seems to be similar to the triangle case, but with the residue terms now corresponding to triangles as well as boxes. As we have two parameters though we can consider a further partial fraction decomposition, this time with t, giving (2π)2 δ(l2i )A1A2 = (2π)2 dtdyJt,y [Inft [InfyA1A2] (y)] (t) + Inft poles {j} Resy=yj A1A2 y − yj  (t) poles {l} Rest=tl [InfyA1A2] (y) t − tl poles {j},{l} Rest=tl Resy=yj A1A2 t − tl  , (5.2) where here {l} is the sum over all possible poles in t. The general dependence of the cut integral momentum, lµ, on the free integral parameters t and y can be written in terms of null four-vectors a i with i = 0, 1, 2, 3, 4 such that l 2 = 0. An explicit form for these will be presented in section VA. We then define lµ by 1 + ya 2 + ta 3 + a 4 . (5.3) Again residues of pole terms will correspond to the solutions of (l − P )2 = 0 and hence it is straightforward to see that the final term of eq. (5.2), containing the sum of residues in both y and t, has both of these free parameters fixed. Any such terms must contain at least one propagator pole. Also the numerator will be independent of any integration variables, as both y and t are fixed. Thus all such terms will correspond to purely scalar triangle and scalar box terms. Looking at the second and third terms of eq. (5.2) we might also, at least initially, want to associate these terms with contributions to scalar triangle terms only and hence naively conclude that only the first term of eq. (5.2) contributes to the scalar bubble coefficient. This assumption though would be wrong. The crucial difference between the single residue terms of eq. (5.2) and those of eq. (4.3) is the parameterisation of the loop momentum which is being used. Taking the residue of a pole term at a particular point y freezes y such that we force a particular momentum parameterisation upon these triple-cut terms. Importantly, in general this particular forced momentum parameterisation is such that the integrals over t in the second and third terms of eq. (5.2) now no longer vanish. If only scalar triangle contributions came from the integrals over t then this would not be an issue; we could just discard these terms as not relevant for the extraction of our bubble coefficient. What we find though, through a simple application of Passarino-Veltman reduction techniques, is that these integrals contain scalar bubble contributions, B0, with coefficients b, dtJ ′tt n = bB0 + c C0, (5.4) where J ′t is the relevant Jacobian for this parameterisation of the loop momentum and c is the coefficient corresponding to the scalar triangle contribution, C0. We cannot therefore simply discard the residue pieces of eq. (5.2), as we could in the triangle case, if we want to derive the full scalar bubble coefficient. Furthermore, there is an additional complication. We will see that the integrals over powers of y contained in the first term of eq. (5.2) also do not vanish in general and hence must also be taken into account. There is a limit to the maximum positive powers of y and t that appear in the rewritten partial-fractioned decomposition of the integral. For renormalisable theories, such as QCD, up to three powers of t appear for triangle coefficients and up to four powers of y for bubble coefficients. Therefore the power series in y and t of the Inf operators will always terminate at these fixed points. It is then straightforward, as we will discuss in section VD and section VB, to derive the general form for all possible non-vanishing contributing integrals, over powers of y and t, in terms of their scalar bubble contributions. Calculation of the scalar bubble coefficient therefore requires a two stage process. First take the Infy and Inft pieces of the cut integrand and replace any integrals over y with their known general forms, as we shall see integrals proportional to t will vanish. Secondly compute all possible triple cuts that could be generated by applying a third cut to the two- particle cut we are considering. To these terms then apply, not the parameterisation we used in section IV, but the parameterisation forced upon us by taking the residues of the poles in y, which we will derive in section VC. This is equivalent to calculating all the contributions from the residues of the partial fraction decomposed cut integrand of eq. (5.2). Within these terms we then replace any integrals of powers of t with their known general forms. Finally we sum all the contributing pieces together to get the full scalar bubble contribution and hence its coefficient. Our final result for assembling the bubble coefficient is then given by eq. (5.28). A. The momentum parameterisation for the two-particle cut We want to extract the scalar bubble coefficient obtainable from the application of the two-particle cut given in figure 3. This two-particle cut can be expressed in terms of tree amplitudes as (2π)2 δ(l2i )A c2−c1+2 (−l, (c1 + 1), . . . , c2, l1)Atreen−c2+c1+2(−l1, (c2 + 1), . . . , c1, l),(5.5) with l1 = l − Kc1+1...c2 = l − K1. A bubble can be classified entirely in terms of the momentum of one of its two legs, which we label K1, and so we will find it useful to express the cut loop momentum l in terms of c2 + 1 c1 + 1 l1 = l − K1 FIG. 3: The two-particle cut for computing the scalar bubble coefficient of B0(K the pair of massless momenta K♭1 and χ defined via 1 = K χµ, (5.6) here γ = 〈χ±| /K1|χ±〉 ≡ 〈χ±| /K♭1|χ±〉. The arbitrary vector χ can be chosen independently for each bubble coefficient as a result of the independence of the choice of basis representation for the cut momentum. In the two-particle cut case we have only two momentum constraints l2 = 0, and l21 = (l − K1)2 = 0, (5.7) and so we have two free parameters which we will label y and t. The loop momentum can then be expressed in terms of spinor components as 〈l−| = t〈K♭,−1 | + (1 − y) 〈χ−|, 〈l+| = y 〈K♭,+1 | + 〈χ+|. (5.8) Written as a four-vector lµ is lµ = yK (1 − y)χµ + t 〈K♭,−1 |γµ|χ−〉 + (1 − y)〈χ−|γµ|K♭,−1 〉. (5.9) We can also use momentum conservation to write a component form for the other cut momentum. We have 〈l−1 | = 〈K 1 | − 〈χ−|, 〈l+1 | = (y − 1) 〈K 1 | + t〈χ+|. (5.10) Furthermore after rewriting the integral in this cut-momentum parameterisation and integrating over the two delta function constraints we find the following simple result for the constant Jt,y contained in eq. (5.1), namely Jt,y = 1. B. Non-vanishing integrals In the case of the scalar triangles of section IVB crucial simplifications occurred as a result of our chosen cut momentum parameterisation. Any integral over a power of t vanished, leaving only a single contribution corresponding to the desired coefficient. For the scalar bubble coefficient things are not quite as simple. We can use 〈K♭,±1 | /K1|χ±〉 = 0 as well as 〈K 1 |γµ|χ±〉〈K 1 |γµ|χ±〉 = 0 to show that 〈χ−|/l|K♭,−1 〉n l2l21 = 0 ⇒ dtdy tn = 0, 〈K♭,−1 |/l|χ−〉n l2l21 = 0 ⇒ (1 − y)n = 0. (5.11) Hence the integrals over all positive and negative powers of t vanish, dtdy tn = 0 for n 6= 0. (5.12) Integrals over positive powers of y, contained within the double Inf piece of the first term of eq. (5.2), will not vanish. These integrals are straightforwardly derivable with the aid of identities involving the four vector nµ = K 1 − (S1/γ)χµ which satisfies the constraints (K1 · n) = 0 and n2 = −S1. It is then possible to show the following relations in D = 4 dimensions, and remembering that Jt,y = 1, (l · n)2m−1 l2l21 = 0 ⇒ )2m−1 (l · n)2m l2l21 = S2m1 B = S2m1 B̃ (l · K1)2m l2l21 = (2m + 1)S2m1 B dtdy = (2m + 1)S2m1 B̃ PV , (5.13) where BmPV and B̃ PV are Passarino-Veltman reduction coefficients, the explicit forms of which are not needed. Solving these equations for the integral of ym leads to the result dtdy ym = m + 1 dtdy for m ≥ 0. (5.14) Contributions to our desired scalar bubble coefficient from the double Inf piece of eq. (5.2) therefore come not only from the single constant t0y0 term but also from terms proportional to integrals of t0ym. This is not the end of the story. As described above, there can be further contributions from the second and third residue terms generated in the decomposition of eq. (5.2). We could proceed from the cut integrand to explicitly calculate these residue terms. However as we will shall see, a more straightforward approach is to derive these terms by relating them to triple cuts. C. The momentum parameterisation for triple cut contributions We wish to relate the contributions to the bubble coefficient of the residue pieces, sep- arated in the decomposition of eq. (5.2), to triple cuts in a specific basis of the cut-loop momentum. To find this basis we will apply the additional constraint (l + K2) 2 = 0, (5.15) to the two-particle cut momentum of section VA. Note that here we label the “K2” leg as K2 in contrast to (−K2) as we did in the triangle coefficient case of section IVA. This constraint corresponds to the application of an additional cut which would appear as δ((l+K2) 2) inside the integral. This additional constraint, applied to the starting point of the two-particle cut loop momentum, forces us to use K♭1 and χ as the momentum basis vectors of l. Importantly, this differs from the basis choice for the triple cut momenta developed in section IVA, which leads to the differing behaviour of these triple-cut contributions. The presence of y in both 〈l−| and 〈l+| directs us for reasons of efficiency to choose to use eq. (5.15) to first constrain y, leaving t free. Looking at eq. (5.9) we see that as lµ is quadratic in y then there are two solutions to this constraint, y±, which are given by 2S1〈χ−| /K2|K♭,−1 〉 γ〈K♭,−1 | /K2|K 1 〉 − S1〈χ−| /K2|χ−〉 t + S1〈χ−| /K2|K♭,−1 〉 S1〈χ−| /K2|K♭,−1 〉 + 2tγ (K1 · K2) − 4S1S2γt tγ − 〈χ−| /K2|K♭,−1 〉 . (5.16) On substituting these two solutions into the two-particle cut momentum of eq. (5.8) we obtain our desired triple-cut momentum parameterisation. Our final step is then to relate the triple-cut integrals defined in this basis to the residue terms of eq. (5.2). Rewriting the triple cut integral after the change of momentum parame- terisation and integrating over all but the third delta function gives the general form (2π)3 dtdy J ′t (δ(y − y+) + δ(y − y−))M(y, t), (5.17) where M(y, t) is a general cut integrand and J ′t = S1〈χ−| /K2|K♭,−1 〉 + 2tγ (K1 · K2) − 4S1S2γt tγ − 〈χ−| /K2|K♭,−1 〉 . (5.18) Upon examination of a general residue term we find that it corresponds to an integral of the (2π)2i dtdyJt,y Res M(y, t) (l + K2)2 ≡ −(2π) dtdy J ′t (δ(y − y+) + δ(y − y−))M(y, t),(5.19) and hence that residue contributions are given, up to a factor of (−1/2), by the triple cut. This result applies equally when S2 = K 2 = 0, corresponding to a one or two-mass triangle, when the appropriate scale is set to zero in eq. (5.16) and eq. (5.18). The momentum parameterisation in this simplified case is contained in Appendix B. D. More non-vanishing integrals and bubble coefficients There is a direct correspondence between a triple cut contribution and a residue contri- bution. The sum of all possible triple cuts, which contain the original two-particle cut, will therefore correspond to the sum of all residue terms. We must now examine how such terms contribute to the bubble coefficient itself. Unlike for the case of triple cut integrands as parameterised in section IVA we will find that there are contributions, specifically in this case bubble coefficient contributions, coming from the integrals over t. To see this let us investigate the integrals over t in more detail. As an example consider extracting the scalar bubble term coming from a two-mass linear triangle (with the massless leg K2 so that S2 = 0). We would start from a two-particle cut which, after decomposing as eq. (5.1), would give (2π)2 δ(l2i ) 〈K−2 |/l|a−〉 (l + K2)2 (5.20) = (2π)2 δ(l2i ) [lK2] = (2π)2 dtdyJ ′t [K♭1a] [K♭1K2] [χK2] y[K♭1a]+t[χa] [χK2] [K♭1K2] [K♭1K2] + [χK2] The first term of this is clearly not the complete coefficient, and so we need to obtain the bubble contribution contained within the second term. Consider reconstructing this term using a triple cut with the cut loop momentum parameterised in a form given by setting y equal to its value at the residue of the pole of this second term. This triple cut term is given −(2π) δ(l2i )〈K−2 |/l|a−〉 = −(2π) dtJ ′t [χK♭1][K2a] [K♭1K2] 〈K−2 | /K1|K−2 〉t+ 〈χ−| /K2|K♭,−1 〉 , (5.21) where we have used the parameterisation of lµ given by eq. (B9) and added an extra overall factor of i which would come from the additional tree amplitude in a triple cut. Of this triple cut integrand only the first, t dependent, term can give anything other than a scalar triangle contribution. To derive the result of this integral over t we will, as we have done previously, use our parameterisation of the cut momentum, eq. (5.9), to pick out the integral as follows 〈χ−|/l|K♭,−1 〉 l2l21(l + K2) ≡ (2π)3γ dtJ ′tt. (5.22) Using Passarino-Veltman reduction on the single tensor integral on the left hand side of this as well as dropping anything but the contributing bubble integrals of our particular cut leaves us with the result dtJ ′tt = (2π)3 S1〈χK2〉[K2K♭1] γ〈K−2 | /K1|K−2 〉2 Bcut0 (K 1 ), (5.23) where Bcut0 (K 1 ) is the cut form of the scalar bubble integral of eq. (C1). This non-vanishing result for the integral over t, in contrast to that of section IVA, is a direct consequence of the cut momentum parameterisation forced upon us when taking the residues contained in the two-particle cut integrand with which we started. On substituting the result of eq. (5.23) into eq. (5.21) we find that we can write eq. (5.20), using the bubble integral given in eq. (C1), as (2π)2 δ(l2i ) 〈K−2 |/l|a−〉 (l + K2)2 =−i [χK 1][K2a] [K♭1K2] 〈χ−| /K2|K♭,−1 〉 〈K−2 | /K1|K−2 〉 Bcut0 (K 1 )+i [K♭1a] [K♭1K2] Bcut0 (K 〈K−2 | /K1|a−〉 〈K−2 | /K1|K−2 〉 Bcut0 (K 1), (5.24) which is the known coefficient of the scalar bubble contained inside the linear triangle. Of course, if we had chosen χ = K2 from the beginning, then the first term on the left hand side of eq. (5.20) would have been the complete bubble coefficient. In general, if we are able to rewrite a two-particle cut integrand such that each term contains only a single propagator then we can always choose a different χ = K♭2, defined via 2 = K 〈K♭,−1 | /K2|K 1 , (5.25) for each term individually such that there are no contributions from the residue terms. Whether this is both feasible and a more computationally effective approach than calculating the residue contributions through the use of triple cuts would depend upon the cut integrand in question. In general we will be considering processes which contain terms with powers of up to t3, so we will need to know these integrals. Again these can be found using a straightforward application of tensor reduction techniques. When all three legs in the cut are massive these integrals over t are given, after dropping an overall factor of 1/(2π)3 witch always cancels out of the final coefficient, by T (j) = dtJ ′tt )j〈χ−| /K2|K♭,−1 〉j(K1 · K2)j−1 Sl−12 (K1 · K2)l−1 Bcut0 (K 1). (5.26) Simply taking the relevant mass to zero gives the forms in the one and two mass cases. ∆ was previously defined in eq. (4.15) and we have C11 = C21 = − , C22 = − C31 = − (K1 · K2)2 , C32 = , C33 = . (5.27) Also for later use we define T (0) = 0. E. The bubble coefficient We have now assembled all the pieces necessary to compute our desired scalar bubble coefficient, bj , corresponding to the cut scalar bubble integral B j ). It is given in general not as the coefficient of a single term but by summing together the t0ym terms from both the double Inf in y followed by t as well as residue contributions which we derive by considering all possible triple cuts contained in the two-particle cut. The coefficient is given by bj = −i [Inft [InfyA1A2] (y)] (t) t→0, ym→ 1 {Ctri} [InftA1A2A3] (t) tj→T (j) , (5.28) where T (j) is defined in eq. (5.26) and the sum over the set {Ctri} is a sum over all triple cuts obtainable by cutting one more leg of the two-particle cut integrand A1A2. When computing with eq. (5.28) there is a freedom in the choice of χ. A suitable choice of which can simplify the degree of computation involved in extracting a particular coefficient. Particular choices of χ can eliminate the need to calculate the second term of eq. (5.28) completely, as discussed in section VD. We also note that there are choices of χ which eliminate the need to evaluate the first term of eq. (5.28), so that the coefficient comes entirely from the second term of eq. (5.28) instead. VI. APPLICATIONS To demonstrate our method we now present the recalculation of some representative triangle and bubble integral coefficients. We also discuss checks we have made against other various state-of-the-art cut-constructable coefficients contained in the literature. A. Extracting coefficients To highlight the application of our procedure to the extraction of basis integral coefficients we consider deriving the coefficients of some simple integral functions which commonly appear, for example, in one-loop Feynman diagrams. 1. The triangle coefficient of a linear two-mass triangle First we consider deriving the scalar triangle coefficient of a linear two-mass triangle with massive leg K1, massless leg K2, and a and b arbitrary massless four-vectors not equal to K2. This is given by the integral 〈a−|/l|b−〉 l2(l − K1)2(l + K2)2 . (6.1) Extracting the triangle coefficient requires cutting all three propagators of the integrand. We do this here by simply removing the “cut” propagator as we are interested only in the integrand. This leaves only 〈a−|/l|b−〉. (6.2) Rewriting this integrand in terms of the parameterisation of eq. (4.16) gives α01〈a−| /K2|b−〉 + t〈aK♭1〉[χb] . (6.3) As S2 = 0 we see that α01 = S1/γ and that γ = 2(K1 · K2). Then taking the t0 component of the [Inft] of this in accordance with eq. (4.10) leaves us with our desired coefficient 〈K−2 | /K1|K−2 〉 〈a−| /K2|b−〉, (6.4) which matches the expected result. 2. The bubble contributions of a three-mass linear triangle Consider a linear triangle with in this case three massive legs, so now K2 is massive but again a and b are arbitrary massless four-vectors, 〈a−|/l|b−〉 l2(l − K1)2(l + K2)2 . (6.5) Extracting the bubble coefficient of the integral B0(K 1) is done by cutting the two propa- gators l2 and (l−K1)2. Again cutting the legs is done by removing the relevant propagators from the integrand so that it is given by 〈a−|/l|b−〉 (l + K2)2 . (6.6) As this contains a single propagator, and therefore a single pole, we could choose to set χ = K♭2 (as defined in eq. (5.25)), before performing the series expansions in y and t. For this choice of χ the bubble coefficient comes entirely from the two-particle cut. Using the first term of eq. (5.28) gives directly γ〈a−| /K♭1|b−〉 γ2 − S1S2 − S1〈a −| /K♭2|b−〉 γ2 − S1S2 , (6.7) where γ = 〈K♭,−2 | /K♭1|K 2 〉, a result which is equivalent to the expected answer. In order to demonstrate the procedure of using triple cut contributions in extracting a bubble coefficient we will now reproduce this by assuming χ 6= K♭2. For this case the first term of eq. (5.28) then gives − i 〈aχ〉[K 〈χ−| /K2|K♭,−1 〉 , (6.8) which upon choosing χ = a vanishes and so the complete contribution will come from the triple cut pieces of eq. (6.5). Cutting the remaining propagator in eq. (6.6) gives us the single triple cut term which will contribute. The integrand of this is given, after multiplying by an additional factor of i which would come from the third tree amplitude if this was a triple cut, by 〈al〉[lb] + 〈al〉[lb] (y+ + y−)〈aK♭1〉[K♭1b] + 2t〈aK♭1〉[ab] , (6.9) where we have set χ = a. From eq. (5.16) we have y+ + y− = 〈K♭,−1 | /K2|K 1 〉 − S1〈a−| /K2|a−〉 t + S1〈a−| /K2|K♭,−1 〉 S1〈a−| /K2|K♭,−1 〉 , (6.10) Hence taking the [Inft] of the cut integrand, eq. (6.9), and dropping any terms not propor- tional to t leaves it〈a−| /K1|b−〉 γ〈K♭,−1 | /K2|K 1 〉 − S1〈a−| /K2|a−〉 S1〈a−| /K2|K♭,−1 〉 [K♭1b] , (6.11) which after inserting the result for the t integral given by eq. (5.26) and substituting this into the second term of eq. (5.28) gives for our desired coefficient −| /K1|b−〉 〈K♭,−1 | /K2|K 〈a−| /K2|a−〉 + 〈a−| /K2|K♭,−1 〉 [K♭1b] = −i 1 (K1 · K2)〈a−| /K1|b−〉 − S1〈a−| /K2|b−〉 , (6.12) where ∆ was given in eq. (4.15). This matches both the expected result and eq. (6.7). B. Constructing the one-loop six-photon amplitude A6(1 −, 2+, 3−, 4+, 5−, 6+) Recently an analytic form for the last unknown six-photon one-loop amplitude was ob- tained by Binoth, Heinrich, Gehrmann and Mastrolia in ref. [46]. This result was used to confirm a previous numerical result [50]. More recently still further corroboration has been provided by [45]. Here we reproduce, as an example, the calculation of the three-mass triangle and bubble coefficients, again confirming part of these results. Firstly it is a very simple exercise to demonstrate by explicit computation that all bubble coefficients vanish. If we were to use the basis of finite box integrals, as defined in [35], then there is only a single unique three-mass triangle coefficient, a complete explicit derivation of which we now present. Starting from the cut in the 12 : 34 : 56 channel shown in figure 4 we can write the cut integrand as FIG. 4: Triple cut six-photon amplitude in the 12 : 34 : 56 channel. 16A4(−l−hq , 1−, 2+, lh22,q)A4(−l−h22,q , 3−, 4+, lh11,q)A4(−l−h11,q , 5−, 6+, lhq ), (6.13) with all unlabelled legs photons and l1 = l − K56 and l2 = l + K12. The overall factor of 16 comes from the differing normalisation conventions between QCD colour-ordered ampli- tudes and QED photon amplitudes. Both helicity choices h = h1 = h2 = ± give identical contributions. Written explicitly, eq. (6.13) is 〈l1〉2〈l23〉2〈l15〉2 〈l2〉〈l22〉〈l14〉〈l24〉〈l6〉〈l16〉 . (6.14) After inserting the momentum parameterisation of eq. (4.16) this becomes t〈K♭12〉 + α01〈K♭22〉 t〈K♭12〉 + α21〈K♭22〉 t〈K♭14〉 + α11〈K♭24〉 t〈K♭11〉 + α01〈K♭21〉 t〈K♭13〉 + α21〈K♭23〉 t〈K♭15〉 + α11〈K♭25〉 t〈K♭14〉 + α21〈K♭24〉 t〈K♭16〉 + α01〈K♭26〉 t〈K♭16〉 + α11〈K♭26〉 ) . (6.15) Applying eq. (4.10) implies taking only the t0 piece of the [Inft] of this expression. Averaging over both solutions leaves us with our form for the three mass triangle coefficient − 16i 〈K♭11〉2〈K♭13〉2〈K♭15〉2 〈K♭12〉2〈K♭14〉2〈K♭16〉2 , (6.16) where K♭1 depends upon the form of γ± as given in eq. (4.14). Numerical comparison with the analytic result of [46] shows complete agreement. C. Contributions to the one-loop A6(1 q , 2 q , 3 −, 4+; 5−e , 6 e ) amplitude This particular amplitude was originally obtained by Bern, Dixon and Kosower in [19]. Making up this amplitude are many box, triangle and bubble integrals along with rational terms. Here we will recompute one particular representative three-mass triangle coefficient in order to highlight the application of our technique to a phenomenologically interesting process. Following the notation of [19], we wish to calculate the three-mass triangle coefficient of I3m3 (s14, s23, s56) ≡ C0(s14, s56) of the F cc term. The only contributing cut is shown in figure 5. We begin by writing down the triple cut integrand for this case FIG. 5: Triple cut in the 14 : 23 : 56 channel. A4(−l−h11,q̄ , 5−e , 6+e , lh22,q)A4(−l−h22,q , 4+, 1+q , lhg )A4(−l−hg , 2−q , 3−, lh11,q), (6.17) where l1 = l − K23 and l2 = l + K14. Only when h = −, h1 = + and h2 = + do we get a contribution. It can be written explicitly as 〈l25〉2〈ll2〉2〈23〉2 〈14〉〈56〉〈4l2〉〈2l〉〈ll1〉〈l1l2〉 . (6.18) Rewriting this in terms of the loop momentum parametrisation of eq. (4.16) gives t〈K♭15〉 + α21〈K♭25〉 )2 〈23〉2 1 − s23 〈14〉〈56〉 t〈4K♭1〉 + α21〈4K♭2〉 t〈2K♭1〉 + α01〈2K♭2〉 . (6.19) The two solutions of γ are given by γ± = −(K23 · K14) ± (K23 · K14)2 − s23s14, the αij ’s are given in Appendix A. The application of eq. (4.10) involves taking [Inft] of eq. (6.19), dropping all but the t component of the result and then averaging over both solutions of γ giving the coefficient γ〈K♭15〉2〈23〉2 1 − S1 〈14〉〈56〉〈4K♭1〉〈2K♭1〉 , (6.20) where again K♭1 depends upon γ±. Numerical comparison against the solution for this coefficient presented in [19], 〈2−| /K14 /K23|5+〉2 − 〈25〉2s14s23 〈14〉[23]〈56〉〈2−| /K14|3−〉〈2−| /K34|1−〉 + flip, (6.21) shows complete agreement, where the operation flip is defined as the exchanges 1 ↔ 2, 3 ↔ 4, 5 ↔ 6, 〈ab〉 ↔ [ab]. The remaining triangle and bubble coefficients can be derived in an analogous way. We have computed a selection of these coefficients for A6(1 q , 2 q , 3 −, 4+; 5−e , 6 e ), along with coefficients of other amplitudes given in [19], and find complete agreement. D. Bubble coefficients of the one-loop 5-gluon QCD amplitude A5(1 −, 2−, 3+, 4+, 5+) This result for the 1-loop 5 gluon QCD amplitude A5(1 −, 2−, 3+, 4+, 5+) was originally calculated by Bern, Dixon, Dunbar and Kosower in [18]. It contains neither box nor triangle integrals, only bubbles. We need therefore only compute bubble coefficients. There are only a pair of such coefficients, with masses s23 and s234 = s51. For the first cut in the channel K1 = K23 we have, for the sum of the two possible helicity configurations, the two-particle cut integrand 〈23〉〈45〉〈51〉 〈1l1〉2〈1l〉〈2l〉〈2l1〉2 〈4l1〉〈3l1〉〈ll1〉2 , (6.22) and for the second, in the channel K1 = K234, 〈23〉〈34〉〈51〉 〈1l1〉2〈1l〉〈2l〉〈2l1〉2 〈4l1〉〈5l1〉〈ll1〉2 . (6.23) Focus upon the K1 = K23 cut initially. There are two pole-containing terms in the denominator of this cut. We could choose to partial fraction these terms and then pick χ = K2 in each case to extract the coefficient. Instead though we will derive the coefficient using triple cut contributions. Choosing χ = k1 so that after inserting the cut loop momentum parameterisation of eq. (5.8) the cut integrand becomes 2γ2〈1K♭1〉 S21〈23〉〈45〉〈51〉 〈2K♭1〉 − S1γ t〈2K♭1〉 + S1γ (1 − y) 〈21〉 〈3K♭1〉 − S1γ 〈4K♭1〉 − S1γ ) , (6.24) and hence produces no [Infy[Inft]] term. Consequentially the two-particle cut contribution to the bubble coefficient vanishes. The same choice of χ similarly removes all two-particle cut contributions in the channel K1 = K234 from the corresponding scalar bubble coefficient. Examining the triple cuts of the bubble in the K23 channel shows only two possible contributions, again after summing over both contributing helicities, given by 〈45〉〈51〉 [3l][3l2]〈1l1〉〈1l〉2〈2l1〉〈2l〉 〈ll1〉〈l1l2〉[ll2]〈l4〉 , (6.25) when K2 = k3 and − 2i〈23〉〈51〉 [4l][4l2]〈1l1〉〈1l2〉2〈2l1〉〈2l〉2 〈ll1〉〈l1l2〉[ll2]〈5l2〉〈3l〉 , (6.26) when K2 = k4. In both cases K2 is massless and is of positive helicity so we use the parameterisation of the triple cut momenta for y+ given in eq. (B2). Then along with setting χ = k1 gives for the first triple cut integrand 2i〈1K♭1〉2〈23〉 〈13〉〈34〉〈45〉〈51〉 〈1−|/2|3−〉 〈1K♭1〉 〈3−| /K23|3−〉 〈1K♭1〉 〈13〉 〈23〉+ , (6.27) and for the second − 2i〈1K 1〉2〈24〉2 〈23〉〈34〉〈45〉〈51〉〈14〉 〈4−| /K23|4−〉 〈14〉 − 〈1−| /K23|4−〉 〈1K♭1〉 〈1K♭1〉 〈14〉 〈24〉+ S1〈21〉 .(6.28) Applying these integrands to the second term of eq. (5.28) by taking [Inft], dropping any terms not proportional to t and then performing the substitution ti → T (i) gives for the coefficient of the first triple cut simply 1 Atree5 , and for the second triple cut 〈1+|/2/4 /K23|1+〉2 〈4−| /K23|4−〉2 s12 − 〈1+|/2/4 /K23|1+〉 〈4−| /K23|4−〉 . (6.29) After following the same series of steps as above for the second bubble coefficient with K1 = K234 we find only a single triple cut contributing term corresponding to K2 = k4. This is related to the second triple cut coefficient derived above via the replacement K23 → K234 and swapping the overall sign. After combining the three triple cut pieces above we arrive at the following form for the cut constructable pieces of this amplitude (4π)2−ǫ Atree5 B0(s23) 〈1+|/2/4 /K23|1+〉2 〈4−| /K23|4−〉2 〈1+|/2/4 /K23|1+〉 〈4−| /K23|4−〉 (B0(s23)−B0(s234)) , (6.30) which can easily be shown to match the result given in [18]. While this example is particularly simple we have also performed additional compar- isons against other results in the literature. Such tests include the cut constructible pieces of all two-minus gluon amplitudes with up to seven external legs, originally obtained in [18, 37]. Additionally we find agreement for the case when, with six gluon legs, three are of negative helicity and adjacent to each other and the remainder are positive helic- ity, which was originally obtained in [49]. We have also successfully reproduced the known three mass triangle coefficients in N = 1 supersymmetry for A6(1−, 2+, 3−, 4+, 5−, 6+) and −, 2−, 3+, 4−, 5+, 6+), originally obtained in [35]. VII. CONCLUSIONS The calculation of Standard Model background processes at the LHC requires efficient techniques for the production of amplitudes. The large numbers of processes involved along with their differing partonic makeups suggests that as much automation as possible is de- sired. In this paper we have presented a new formalism which directs us towards this goal. Coefficients of the basis scalar integrals making up a one-loop amplitude are constructed in a straightforward manner involving only a simple change of variables and a series expansion, thus avoiding the need to perform any integration or calculate any extraneous intermediate quantities. The main results of this paper can be encapsulated simply by eq. (4.10) and eq. (5.28) along with the cut loop momentum given by eq. (4.16), eq. (5.8) and eq. (5.16). Although this technique has been presented mainly in the context of using generalised unitarity [19, 39, 40, 41] to construct coefficients, and hence the cut-constructible part of the amplitude, it can also be used as an efficient method of performing one-loop integration. Using the idea of “cutting” two, three or four of the propagators inside an integral, we isolate and then extract scalar basis coefficients. This procedure then allows us to rewrite the integral in terms of the scalar one-loop basis integrals, hence giving us a result for the integral. Different unitarity cuts isolate particular basis integrals. For the extraction of triangle integral coefficients this means triple cuts and for bubble coefficients we use a combination of two-particle and triple cuts. Extracting the desired coefficients from these cut integrands is then a two step process. The first step is to rewrite the cut loop momentum in terms of a parameterisation which depends upon the remaining free parameters of the integral after all the cut delta functions have been applied. Triangle coefficients are then found by taking the terms independent of the sole free integral parameter as this parameter is taken to infinity. Bubble coefficients are calculated in a similar if slightly more complicated way. The pres- ence of a second free parameter in the bubble case means that we must take into account, not only the constant term in the expansion of the cut integrand as the free integral parameters are taken to infinity, but also powers of one of these parameters. The limit on the maximum power of lµ appearing in the cut integral restricts the appearance of such terms and hence we need consider only finite numbers of powers of these free parameters. Additionally it can also be necessary to take into account contributions from terms generated by applying an additional cut to the bubble integral. The flexibility in our choice of the cut-loop momentum parameterisation allows us to directly control whether we need compute any of these triple cut terms. Furthermore we can control which of these triple cut terms appears, in cases when their computation is necessary. As we consider the application of this procedure to more diverse processes than those detailed here, we should also investigate the “complexity” of the generated coefficients. In the applications we have presented we can see that we produce “compact” forms with minimal amounts of simplification required. This is important if we are to consider further automation. The straightforward nature of this technique combined with the minimal need for simplification means that efficient computer implementations can easily be produced. As a test of this assertion we have implemented the formalism within a Mathematica program which has been used to perform checks against state-of-the-art results contained in the literature. Such checks have included various helicity configurations of up to seven external gluons as well as the bubble and three-mass triangle coefficients of the six photon A6(−+−+ −+) amplitude. In addition representative coefficients of processes of the type e+e− → qqgg have been successfully obtained. Our procedure as presented has mainly been in the context of massless theories. Funda- mentally there is no restriction to the application of this to theories also involving massive fields circulating in the loop. Extensions to include masses should require only a suitable momentum parameterisation for the cut loop momentum; the procedure is then expected to apply as before. In conclusion therefore we believe that the technique presented here shows great potential for easing the calculation of needed one-loop integrals for current and future colliders. Acknowledgements I would like to thank David Kosower for collaboration in the early stages of this work and also Zvi Bern and Lance Dixon for many interesting and productive discussions as well as for useful comments on this manuscript. I would also like to thank the hospitality of Saclay where early portions of this work were carried out. The figures were generated using Jaxodraw [51], based on Axodraw [52]. APPENDIX A: THE TRIPLE CUT PARAMETERISATION In this appendix we give the complete detail of the triple cut parameterisation along with some other useful results. The three cut momenta are given by 〈l−i | = t〈K 1 | + αi1〈K 2 |, 〈l+i | = 〈K♭,+1 | + 〈K 2 |, (A1) α01 = S1 (γ − S2) (γ2 − S1S2) , α02 = S2 (γ − S1) (γ2 − S1S2) α11 = α01 − = −S1S2 (1 − (S1/γ)) γ2 − S1S2 , α12 = α02 − 1 = γ(S2 − γ) γ2 − S1S2 α21 = α01 − 1 = γ(S1 − γ) γ2 − S1S2 , α22 = α02 − = −S1S2 (1 − (S2/γ)) γ2 − S1S2 ,(A2) along with the identities α01α02 = α11α12 and α01α02 = α21α22. When written as four-vectors the cut momentum are given by i = αi2K 1 + αi1K 〈K♭,−1 |γµ|K 2 〉 + αi1αi2 〈K♭,−2 |γµ|K 1 〉. (A3) From these parameterised forms we have the following spinor product identities [ll1] = α12 − α02 [K♭1K 2] = − [K♭2K 〈ll1〉 = t(α11 − α01)〈K♭1K♭2〉 = − 〈K♭1K♭2〉, [ll2] = α22 − α02 [K♭1K 2] = − [K♭2K 〈ll2〉 = t(α21 − α01)〈K♭1K♭2〉 = −t〈K♭1K♭2〉, [l1l2] = α22 − α12 [K♭1K 1 − S2 [K♭2K 〈l1l2〉 = t(α11 − α21)〈K♭1K♭2〉 = −t 1 − S1 〈K♭1K♭2〉. (A4) and we note that 1 − S2 1 − S1 γ = −γ − S1S2 + S1 + S2 = (K1 − K2)2 = S3, (A5) and so with l ≡ l0 we have 〈lilj〉[ljli] = Si+j, as expected. APPENDIX B: THE TRIPLE CUT BUBBLE CONTRIBUTION MOMENTUM PARAMETERISATION WHEN K22 = 0 In this appendix we give the forms for the triple cut momentum of section VC in the case when S2 = 0, i.e. we have a one or two mass triangle. Firstly in these cases the K2 leg is attached to a three-point vertex and so the amplitude for this will contain either [K2l] or 〈K2l〉 depending upon the helicity of K2. This means that in the positive helicity case only the delta function solution δ(y − y+) survives and for a negative helicity K2 the δ(y − y−) survives. We have for both solutions J ′t = S1〈χ−| /K2|K♭,−1 〉 + tγ〈K−2 | /K1|K−2 〉 ) . (B1) The momentum parameterisation for the y+ solution is given in spinor components by 〈l−| = 〈χK 〈χK2〉 〈K−2 |, 〈l+| = 〈K 1 | − 〈χK2〉 〈K−2 | /K1, (B2) and as a 4-vector by 〈χK♭1〉 2〈χK2〉 S1〈χK2〉 〈K−2 |γµ /K1|K+2 〉 + 〈K−2 |γµ|K . (B3) The other momenta are given by 〈l−1 | = t 〈χK♭1〉 〈χK2〉 〈K−2 | − 〈χ−|, 〈l+1 | = − S1〈χK2〉 〈K−2 | /K1, 〈l−2 | = 〈χK♭1〉 〈χK2〉 〈K−2 |, 〈l+2 | = − 〈χ−| /K3 〈χK♭1〉 〈χK2〉 〈K−2 | /K1. (B4) where we have moved the overall factor of t from 〈l−1 | to 〈l+1 | to avoid the presence of a 1/t term for aesthetical reasons. The spinor products formed from these are given by 〈ll1〉 = 〈χK♭1〉, [ll1] = [K♭1χ], 〈ll2〉 = 0, [ll2] = − 〈χK2〉 〈χK♭1〉 [lK2], 〈l1l2〉 = − 〈χK♭1〉, [l1l2] = − [K♭1χ]. (B5) and we see that again, as expected, with l = l0, we have 〈lilj〉[ljli] = Si+j . As we have massless legs some spinor products will consequentially vanish. In the two-mass case these 〈ll2〉 = 0, 〈lK2〉 = 0, [l2K2] = 0, (B6) and for the one-mass case [l1l2] = 0, 〈ll2〉 = 0, 〈lK2〉 = 0, 〈l2K2〉 = 0, [l1K3] = 0, [l2K3] = 0, (B7) where K3 is the momentum of the third leg. The momentum parameterisation for the y− solution is given in spinor components by 〈l−| = t 〈K+2 | /K1 + 〈χ−|, 〈l+| = [χK 〈K+2 |, (B8) and as a 4-vector by [χK♭1] 2[K2K 〈K+2 | /K1γµ|K−2 〉 + 〈χ−|γµ|K−2 〉 . (B9) The other momenta are given by 〈l−1 | = 〈K+2 | /K1, 〈l+1 | = −t [K♭1χ] 〈K+2 | − 〈K 〈l−2 | = [χK♭1] 〈K♭,+1 | /K3 + 〈K+2 | /K1, 〈l+2 | = [χK♭1] 〈K+2 |. (B10) The spinor products formed from these are given by 〈ll1〉 = 〈χK♭1〉, [ll1] = [K♭1χ], 〈ll2〉 = [χK♭1] 〈K2l〉, [ll2] = 0, 〈l1l2〉 = 〈K♭1χ〉, [l1l2] = [χK♭1]. (B11) and again 〈lilj〉[ljli] = Si+j as expected. The vanishing spinor products in the two mass case [ll2] = 0, [lK2] = 0, [l2K2] = 0, (B12) and in the one mass case 〈l1l2〉 = 0, [ll2] = 0, [lK2] = 0, [l2K2] = 0, 〈l1K3〉 = 0, 〈l2K3〉 = 0. (B13) APPENDIX C: THE SCALAR INTEGRAL FUNCTIONS The scalar bubble integral with massive leg K1 given in figure 6 is defined as FIG. 6: The scalar bubble integral with a leg of mass K21 . 1 ) = (−i)(4π)2−ǫ d4−2ǫl (2π)4−2ǫ l2(l − K1)2 , (C1) and is given by ǫ(1 − 2ǫ)(−K −ǫ = rΓ − ln(−K21 ) + 2 + O(ǫ), (C2) Γ(1 + ǫ)Γ2(1 − ǫ) Γ(1 − 2ǫ) . (C3) The general form of the scalar triangle integral with the masses of its legs labelled K21 , K22 and K 3 given in figure 7 is defined as FIG. 7: The scalar triangle with its three legs of mass K21 , K 2 and K 1 , K 2) = i(4π) d4−2ǫl (2π)4−2ǫ l2(1 − K1)2(l − K2)2 , (C4) and separates into three cases depending upon the masses of these external legs. In the one mass case we have K22 = 0 and K 3 = 0 and the corresponding integral is given by 1 , K 2 ) = (−K21 )−1−ǫ = (−K21 ) − ln(−K ln2(−K21 ) + O(ǫ), (C5) If two legs are massive the integral, assuming K23 = 0, is given by 1 , K (−K21 )−ǫ − (−K22 )−ǫ (−K21 ) − (−K22 ) (−K21 ) − (−K22 ) − ln (−K 1 ) − ln (−K22 ) ln2 (−K21 ) − ln2 (−K22 ) .(C6) Finally if all three legs are massive then the integral is as given in [53, 54] 1 , K 1 + iδj 1 − iδj − Li2 1 − iδj 1 + iδj + O(ǫ), (C7) where K21 − K22 − (K1 + K2)2√ −K21 + K22 − (K1 + K2)2√ −K21 − K22 + (K1 + K2)2√ , (C8) ∆3 = −(K22 )2 − (K22)2 − (K23)2 + 2(K21K22 + K23K21 + K22K23) = −4∆, (C9) with ∆ given by eq. (4.15). The general form for a scalar box function is given by 1 , K 2 , K 3) = (−i)(4π)2−ǫ d4−2ǫl (2π)4−2ǫ l2(l − K1)2(l − K2)2(l − K3)2 . (C10) The solution of this integral is split up into classes depending upon the masses of the external legs. 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0704.1836

Comment on Electroweak Higgs as a Pseudo-Goldstone Boson of Broken Scale Invariance

arXiv:0704.1836v1 [hep-ph] 13 Apr 2007 CSULB–PA–07–4 Comment on Electroweak Higgs as a Pseudo-Goldstone Boson of Broken Scale Invariance Hitoshi NISHINO1) and Subhash RAJPOOT2) Department of Physics & Astronomy California State University 1250 Bellflower Boulevard Long Beach, CA 90840 Abstract The first model of Foot, Kobakhidze and Volkas described in their work in arXiv:0704.1165 [hep-ph] is a tailored version of our model on broken scale invari- ance in the standard model presented in hep-th/0403039. 1) E-Mail: [email protected] 2) E-Mail: [email protected] http://arxiv.org/abs/0704.1836v1 The merits of implementing scale invariance in the standard model of particle interactions were enunciated by us in [1]. An extended version of this work that also addresses the important issue of unification of elementary particle interactions will appear in [2]. The salient features of our model were recently recapitulated in our comment [3]. Here we point out that in the work of Foot, Kobakhidze and Volkas [4] on broken scale invariance in the standard model, their first model corresponds to a tailored version of our model. R. Foot et. al. [4] are fully aware of the fact that scale invariance symmetry can be realised as a local symmetry,3) in which case breaking it results in an additional neutral gauge boson. In our work [1] we dubbed this gauge boson the Weylon, named after Herman Weyl [5]. References [1] H. Nishino and S. Rajpoot, ‘Broken Scale Invariance in the Standard Model’, CSULB- PA-04-2, hep-th/0403039. [2] H. Nishino and S. Rajpoot, ‘Standard Model and SU(5) GUT with Local Scale Invariance and the Weylon’, CSULB-PA-06-4, to appear in CICHEP-II Conference Proceedings, 2006, published by AIP. [3] H. Nishino and S. Rajpoot, ‘Comment on Shadow and Non-Shadow Extensions of the Standard Model’, hep-th/0702080. [4] R. Foot, A. Kobakhidze and R.R. Volkas, ‘Electroweak Higgs as a Pseudo-Goldstone Boson of Broken Scale Invariance’, arXiv:0704.1165 [hep-ph]. [5] H. Weyl, S.-B. Preuss. Akad. Wiss. 465 (1918); Math. Z. 2 (1918) 384; Ann. Phys. 59 (1919) 101; Raum, Zeit, Materie, vierte erweiterte Auflage: Julius Springer (1921). 3) Cf. Footnote #2 in [4].

0704.1837

Hard x-ray photoemission study of LaAlO3/LaVO3 multilayers

Hard x-ray photoemission study of LaAlO3/LaVO3 multilayers H. Wadati,1, ∗ Y. Hotta,2 A. Fujimori,1 T. Susaki,2 H. Y. Hwang,2, 3 Y. Takata,4 K. Horiba,4 M. Matsunami,4 S. Shin,4, 5 M. Yabashi,6, 7 K. Tamasaku,6 Y. Nishino,6 and T. Ishikawa6, 7 1Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 2Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan 3Japan Science and Technology Agency, Kawaguchi 332-0012, Japan 4Soft X-ray Spectroscopy Laboratory, RIKEN/SPring-8, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan 5Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 6Coherent X-ray Optics Laboratory, RIKEN/SPring-8, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan 7JASRI/SPring-8, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5198, Japan (Dated: November 28, 2018) We have studied the electronic structure of multilayers composed of a band insulator LaAlO3 (LAO) and a Mott insulator LaVO3 (LVO) by means of hard x-ray photoemission spectroscopy, which has a probing depth as large as ∼ 60 Å. The Mott-Hubbard gap of LVO remained open at the interface, indicating that the interface is insulating unlike the LaTiO3/SrTiO3 multilayers. We found that the valence of V in LVO were partially converted from V3+ to V4+ only at the interface on the top side of the LVO layer and that the amount of V4+ increased with LVO layer thickness. We suggest that the electronic reconstruction to eliminate the polarity catastrophe inherent in the polar heterostructure is the origin of the highly asymmetric valence change at the LVO/LAO interfaces. PACS numbers: 71.28.+d, 73.20.-r, 79.60.Dp, 71.30.+h I. INTRODUCTION The interfaces of hetero-junctions composed of transition-metal oxides have recently attracted great in- terest. For example, it has been suggested that the inter- face between a band insulator SrTiO3 (STO) and a Mott insulator LaTiO3 (LTO) shows metallic conductivity. 1,2,3 Recently, Takizawa et al.4 measured photoemission spec- tra of this interface and observed a clear Fermi cut-off, indicating that an electronic reconstruction indeed oc- curs at this interface. In the case of STO/LTO, elec- trons penetrate from the layers of the Mott insulator to the layers of the band insulator, resulting in the inter- mediate band filling and hence the metallic conductivity of the interfaces. It is therefore interesting to investi- gate how electrons behave if we confine electrons in the layers of the Mott insulator. In this paper, we inves- tigate the electronic structure of multilayers consisting of a band insulator LaAlO3 (LAO) and a Mott insula- tor LaVO3 (LVO). LAO is a band insulator with a large band gap of about 5.6 eV. LVO is a Mott-Hubbard in- sulator with a band gap of about 1.0 eV.5 This material shows G-type orbital ordering and C-type spin ordering below the transition temperature TOO = TSO = 143 K. From the previous results of photoemission and inverse photoemission spectroscopy, it was revealed that in the valence band there are O 2p bands at 4 − 8 eV and V 3d bands (lower Hubbard bands; LHB) at 0 − 3 eV and that above EF there are upper Hubbard bands (UHB) of V 3d origin separated by a band gap of about 1 eV from the LHB.7 Since the bottom of the conduction band of LAO has predominantly La 5d character and its energy position is well above that of the LHB of LVO,8 the V 3d electrons are expected to be confined within the LVO layers as a “quantum well” and not to penetrate into the LAO layers, making this interface insulating unlike the LTO/STO case.1,2,3,4 Recently, Hotta et al.9 investi- gated the electronic structure of 1-5 unit cell thick layers of LVO embedded in LAO by means of soft x-ray (SX) photoemission spectroscopy. They found that the V 2p core-level spectra had both V3+ and V4+ components and that the V4+ was localized in the topmost layer. However, due to the surface sensitivity of SX photoemis- sion, the information about deeply buried interfaces in the multilayers is still lacking. Also, they used an un- monochromatized x-ray source, whose energy resolution was not sufficient for detailed studies of the valence band. In the present work, we have investigated the electronic structure of the LAO/LVO interfaces by means of hard x-ray (HX) photoemission spectroscopy (hν = 7937 eV) at SPring-8 BL29XU. HX photoemission spectroscopy is a bulk-sensitive experimental technique compared with ultraviolet and SX photoemission spectroscopy, and is very powerful for investigating buried interfaces in mul- tilayers. From the valence-band spectra, we found that a Mott-Hubbard gap of LVO remained open at the in- terface, indicating the insulating nature of this interface. From the V 1s and 2p core-level spectra, the valence of V in LVO was found to be partially converted from V3+ to V4+ at the interface, confirming the previous study.9 Fur- thermore, the amount of V3+ was found to increase with LVO layer thickness. We attribute this valence change to the electronic reconstruction due to polarity of the layers. http://arxiv.org/abs/0704.1837v1 SampleB: LAO (3ML)/LVO (50ML) LaAlO3 (3ML) LaVO3 (50ML) SrTiO3 (5ML) SrTiO3 (100) Sample A: LAO (3ML)/LVO (3ML)/LAO (30ML) LaAlO3 (3ML) LaVO3 (3ML) SrTiO3 (5ML) SrTiO3 (100) LaAlO3 (30ML) Sample C: LVO (50ML) LaVO3 (50ML) SrTiO3 (100) FIG. 1: Schematic view of the LaAlO3/LaVO3 mul- tilayer samples. Sample A: LaAlO3 (3ML)/LaVO3 (50ML)/SrTiO3. Sample B: LaAlO3 (3ML)/LaVO3 (50ML)/LaAlO3 (30ML)/SrTiO3. Sample C: LaVO3 (50ML)/SrTiO3. II. EXPERIMENT The LAO/LVOmultilayer thin films were fabricated on TiO2-terminated STO(001) substrates 10 using the pulsed laser deposition (PLD) technique. An infrared heat- ing system was used for heating the substrates. The films were grown on the substrates at an oxygen pres- sure of 10−6 Torr using a KrF excimer laser (λ = 248 nm) operating at 4 Hz. The laser fluency to ablate LaVO4 polycrystalline and LAO single crystal targets was ∼ 2.5 J/cm2. The film growth was monitored us- ing real-time reflection high-energy electron diffraction (RHEED). Schematic views of the fabricated thin films are shown in Fig. 1. Sample A consisted of 3ML LVO capped with 3ML LAO. Below the 3ML LVO, 30ML LAO was grown, making LVO sandwiched by LAO. Sam- ple B consisted of 50ML LVO capped with 3ML LAO. Sample C was 50ML LVO without LAO capping lay- ers. Details of the fabrication and characterization of the films were described elsewhere.11 The characterization of the electronic structure of uncapped LaVO3 thin films by x-ray photoemission spectroscopy will be described elsewhere.12 HX photoemission experiments were per- formed at an undulator beamline, BL29XU, of SPring-8. The experimental details are described in Refs. 13,14,15. The total energy resolution was set to about 180 meV. All the spectra were measured at room temperature. The Fermi level (EF ) position was determined by measuring gold spectra. III. RESULTS AND DISCUSSION Figure 2 shows the valence-band photoemission spectra of the LAO/LVO multilayer samples. Figure 2 (a) shows the entire valence-band region. Compared with the pre- vious photoemission results,7 structures from 9 to 3 eV are assigned to the O 2p dominant bands, and emission from 3 eV to EF to the V 3d bands. The energy posi- tions of the O 2p bands were almost the same in these three samples, indicating that the band bending effect at the interface of LAO and LVO was negligible. Fig- 3.0 2.0 1.0 0 Binding Energy (eV) hn = 7937 eV A: LAO (3ML)/LVO (3ML) /LAO (30ML) B: LAO (3ML)/LVO (50ML) C: LVO (50ML) 10 8 6 4 2 0 -2 -4 Binding Energy (eV) hn = 7937 eV O 2p bands V 3d bands A: LAO (3ML)/LVO (3ML) /LAO (30ML) C: LVO (50ML) B: LAO (3ML)/LVO (50ML) FIG. 2: Valence-band photoemission spectra of the LaAlO3/LaVO3 multilayer samples. (a) Valence-band spec- tra over a wide energy range. (b) V 3d band region. ure 2 (b) shows an enlarged plot of the spectra in the V 3d-band region. A Mott-Hubbard gap of LVO remained open at the interface between LAO and LVO, indicat- ing that this interface is insulating unlike the STO/LTO interfaces.1,2,3,4 The line shapes of the V 3d bands were almost the same in these three samples, except for the energy shift in sample A. We estimated the value of the band gap from the linear extrapolation of the rising part of the peak as shown in Fig. 2 (b). The gap size of sample B was almost the same (∼ 100 meV) as that of sample C, while that of sample A was much larger (∼ 400 meV) due to the energy shift of the V 3d bands. The origin of the enhanced energy gap is unclear at present, but an increase of the on-site Coulomb repulsion U in the thin LVO layers compared to the thick LVO layers or bulk LVO due to a decrease of dielectric screening may explain the experimental observation. Figure 3 shows the V 1s core-level photoemission spec- tra of the LAO/LVO multilayer samples. The V 1s spec- tra had a main peak at 5467 eV and a satellite structure at 5478 eV. The main peaks were not simple symmet- ric peaks but exhibited complex line shapes. We there- fore consider that the main peaks consisted of V3+ and V4+ components. In sample C, there is a considerable amount of V4+ probably due to the oxidation of the sur- face of the uncapped LVO. A satellite structure has also been observed in the V 1s spectrum of V2O3 16 and inter- preted as a charge transfer (CT) satellites arising from the 1s13d3L final state, where L denotes a hole in the O 2p band. Screening-derived peaks at the lower-binding- energy side of V 1s, which have been observed in the metallic phase of V2−xCrxO3, 16,17 were not observed in the present samples, again indicating the insulating na- ture of these interfaces. Figure 4 shows the O 1s and V 2p core-level photoemis- sion spectra of the LAO/LVO multilayer samples. The O 1s spectra consisted of single peaks without surface contamination signal on the higher-binding-energy side, indicating the bulk sensitivity of HX photoemission spec- troscopy. The energy position of the O 1s peak of sample 5490 5480 5470 5460 5450 Binding Energy (eV) La 2p3/2 CT satellite hn = 7937 eV B: LAO (3ML)/LVO (50ML) A: LAO (3ML)/LVO (3ML) /LAO (30ML) C: LVO (50ML) FIG. 3: V 1s core-level photoemission spectra of the LaAlO3/LaVO3 multilayer samples. 520 518 516 514 512 510 508 Binding Energy (eV) hn = 7937 eV V 2p3/2 B: LAO (3ML)/LVO (50ML) A: LAO (3ML)/LVO (3ML) /LAO (30ML) C: LVO (50ML) 535 530 525 520 515 510 Binding Energy (eV) V 2p1/2 V 2p3/2 hn = 7937 eV(a) B: LAO (3ML)/LVO (50ML) A: LAO (3ML)/LVO (3ML) /LAO (30ML) C: LVO (50ML) FIG. 4: O 1s and V 2p core-level photoemission spectra of the LaAlO3/LaVO3 multilayer samples. (a) shows wide energy region and (b) is an enlarged plot of the V 2p3/2 spectra. A, whose LVO layer thickness was only 3 ML, was differ- ent from those of the rest because LAO and LVO have different energy positions of the O 1s core levels. Fig- ure 4 (b) shows an enlarged plot of the V 2p3/2 spectra. Here again, the V 2p3/2 photoemission spectra showed complex line shapes consisting of V3+ and V4+ com- ponents, and no screening-derived peaks on the lower- binding-energy side of V 2p3/2 were observed. The line shapes of the V 2p3/2 spectra were very similar for sam- ples A and B. The amount of V4+ was larger in sample C, consistent with the case of V 1s and again shows the effect of the oxidation of the uncapped LVO. We have fitted the core-level spectra of samples A and B to a Gaussian convoluted with a Lorentzian to esti- mate the amount of V3+, V4+ and V5+ at the interface following the procedure of Ref. 9. Figure 5 shows the fitting results of the V 1s and V 2p3/2 core-level spec- tra. Here, the spectra were decomposed into the V3+ and V4+ components, and the V5+ component was not necessary. The full width at half maximum (FWHM) of the Lorentzian has been fixed to 1.01 eV for V 1s and to 0.24 eV for V 2p3/2 according to Ref. 18. The FWHM of the Gaussian has been chosen 0.90 eV for V 1s and 1.87 eV for V 2p3/2, reflecting the larger multiplet splitting for V 2p than for V 1s. In Fig. 6, we summarize the ratio of the V3+ component thus estimated, together with the results of the emission angle (θe) dependence of the V experiment A: LAO (3ML) /LVO (3ML) /LAO (30ML) hn = 7937 eV 5472 5470 5468 5466 5464 Binding Energy (eV) experiment hn = 7937 eV B: LAO (3ML) /LVO (50ML) experiment A: LAO (3ML) /LVO (3ML) /LAO (30ML) hn = 7937 eV V 2p3/2 520 518 516 514 512 Binding Energy (eV) experiment hn = 7937 eV B: LAO (3ML) /LVO (50ML) V 2p3/2 FIG. 5: Fitting results for the V 1s and 2p3/2 core-level spec- tra. (a) V 1s core level of sample A (LaVO3 3ML), (b) V 2p3/2 core level of sample A (LaVO3 3ML), (c) V 1s core level of sample B (LaVO3 50ML), (d) V 2p3/2 core level of sample B (LaVO3 50ML). 3 4 5 6 7 8 9 2 3 4 5 6 7 Mean free path (Å) (V 2p) (V 1s)(30 SX SX SX surface bulk B: LAO (3ML)/LVO (50ML) Experiment Model (B) 3 4 5 6 7 8 9 2 3 4 5 6 7 Mean free path (Å) (V 2p) (V 1s) SX SX SX surface bulk A: LAO (3ML)/LVO (3ML) /LAO (30ML) Experiment Asymetric model (A-1) Symetric model (A-2) FIG. 6: Ratio of V4+ or V4++ V5+ determined under various experimental conditions using hard x-rays and soft x-rays.9 (a) Sample A (3ML LaVO3), (b) Sample B (50ML LaVO3). Here, SX is a result of soft x-ray photoemission, and HX is of hard x-ray photoemission. In the case of SX, the values in the parenthesis denote the values of θe. 2p core-level SX photoemission spectra measured using a laboratory SX source.9 In order to interpret those results qualitatively, first we have to know the probing depth of photoemission spec- troscopy under various measurement conditions. From the kinetic energies of photoelectrons, the mean free paths of the respective measurements are obtained as de- scribed in Ref. 19.20 When we measure V 2p3/2 spectra with the Mg Kα line (hν = 1253.6 eV), the kinetic en- ergy of photoelectrons is about 700 eV, and the mean free path is estimated to be about 10 Å. Likewise, we also estimate the mean free path in the HX case. The values are summarized in Table I. In the SX case, these values are 10 cos θe Å. One can obtain the most surface- sensitive spectra under the condition of SX with θe = 70 [denoted by SX(70o)] and the most bulk-sensitive spectra for HX measurements of the V 2p3/2 core level [denoted TABLE I: Mean free path of photoelectrons (in units of Å) SX SX SX HX HX (70◦) (55◦) (30◦) (V 1s) (V 2p) 3.4 5.7 8.7 30 60 A-2: Symmetric model :50%V B: LAO (3ML)/LVO (50ML) :85%V A: LAO (3ML)/LVO (3ML)/LAO (30ML) :70%V A-1: Asymmetric model FIG. 7: Models for the V valence distributions in the LaAlO3/LaVO3 multilayer samples. A: LaVO3 3ML. A-1 is an asymmetric model, whereas A-2 is a symmetric model. B: LaVO3 50ML. by HX(V 2p)]. From Fig. 6 and Table I, one observes a larger amount of V4+ components under more surface- sensitive conditions. These results demonstrate that the valence of V in LVO is partially converted from V3+ to V4+ at the interface. In order to reproduce the present experimental result and the result reported in Ref. 9 (shown in Fig. 6), we propose a model of the V valence distribution at the in- terface as shown in Fig. 7. For sample A, we consider two models, that is, an asymmetric model and a symmetric model. In the asymmetric model (A-1), no symmetry is assumed between the first and the third layers. As shown in Fig. 6, the best fit result was obtained for the valence distribution that 70 % of the first layer is V4+ and there are no V4+ in the second and third layers, assuming the above-mentioned mean free paths in Table I and expo- nential decrease of the number of photoelectrons. In the symmetric model (A-2), it is assumed that the electronic structures are symmetric between the first and the third layers. The best fit was obtained when 50 % of the first and third layers are V4+. In sample B, a model (B) where 85 % of the first layer and 50 % of the second layer are V4+ best reproduced the experimental result. As shown in Fig. 6, for the 3ML case, the model (A-2) did not reproduce the experimental results well compared to (A- 1), which demonstrates that the valence distribution of V was highly asymmetric at these interfaces. The origin of this highly asymmetric valence change from V3+ to V4+ at the interfaces can be interpreted in two ways. One possible scenario is a simple chemical effect during the fabrication process of the PLD tech- nique. The topmost LVO layer spends a longer time be- fore the next deposition of LAO than the rest LVO lay- ers, and therefore, oxidation process may easily proceed at the topmost layer. In this scenario, if we make sam- ples under different experimental conditions, especially under different oxygen pressures, the amount of V4+ at the interface may change greatly. In the other scenario, we consider that the polarity of the LAO/LVO multilay- ers plays an essential role. In the present samples, both the LAO and LVO layers are polar, and do not consist of charge neutral layers, that is, they consist of alternating stack of LaO+ and AlO−2 or VO 2 layers. As recently discussed by Nakagawa et al.,23 electronic reconstruction occurs during the fabrication of the polar layers in or- der to prevent the divergence of Madelung potential, i.e., so-called polar catastrophe.24 We consider that the elec- tronic reconstruction occurs in the present samples, and that the valence change of V at the interface is a result of this reconstruction. This effect explains 0.5 ML of V4+, but we cannot explain the total amount of V4+ exceed- ing 0.5 ML, and we must also consider some chemical effects that V atoms are relatively easily oxidized at the topmost layer. Similar studies on samples with different termination layers will be necessary to test this scenario. Recently, Huijben et al.25 studied STO/LAO multilayers and found a critical thickness of LAO and STO, below which a decrease of the interface conductivity and car- rier density occurs. Therefore, changing the numbers of LAO capping layers may also change the valence of V at the interface. Further systematic studies, including other systems like LTO/STO1,2,3,4 and LAO/STO23,25,26, will reveal the origin of the valence change at the interface. IV. CONCLUSION We have investigated the electronic structure of the multilayers composed of a band insulator LaAlO3 and a Mott insulator LaVO3 (LVO) by means of HX photoe- mission spectroscopy. The Mott-Hubbard gap of LVO remained open at the interface, indicating that the inter- face is insulating and the delocalization of 3d electrons does not occur unlike the LaTiO3/SrTiO3 multilayers. From the V 1s and 2p core-level photoemission intensi- ties, we found that the valence of V in LVO was partially converted from V3+ to V4+ at the interface only on the top side of the LVO layer and that the amount of V4+ increased with LVO layer thickness. We constructed a model for the V valence redistribution in order to ex- plain the experimental result and found that the V4+ is preferentially distributed on the top of the LVO layers. We suggest that the electronic reconstruction to elimi- nate polar catastrophe may be the origin of the highly asymmetric valence change at the interfaces. V. ACKNOWLEDGMENTS The HX photoemission experiments reported here have benefited tremendously from the efforts of Dr. D. Miwa of the coherent x-ray optics laboratory RIKEN/SPring- 8, Japan and we dedicate this work to him. This work was supported by a Grant-in-Aid for Scientific Research (A16204024) from the Japan Society for the Promotion of Science (JSPS) and a Grant-in-Aid for Scientific Re- search in Priority Areas “Invention of Anomalous Quan- tum Materials” from the Ministry of Education, Culture, Sports, Science and Technology. H. W. acknowledges fi- nancial support from JSPS. Y. H. acknowledges support from QPEC, Graduate School of Engineering, University of Tokyo. ∗ Electronic address: [email protected]; URL: http://www.geocities.jp/qxbqd097/index2.htm; Present address: Department of Physics and Astron- omy, University of British Columbia, Vancouver, British Columbia V6T-1Z1, Canada 1 A. Ohtomo, D. A. Muller, J. L. Grazul, and H. Y. Hwang, Nature 419, 378 (2002). 2 K. Shibuya, T. Ohnishi, M. Kawasaki, H. Koinuma, and M. Lippmaa, Jpn. J. Appl. Phys. 43, L1178 (2004). 3 S. Okamoto and A. J. Millis, Nature 428, 630 (2004). 4 M. Takizawa, H. Wadati, K. Tanaka, M. Hashimoto, T. Yoshida, A. Fujimori, A. Chikamatsu, H. Kumigashira, M. Oshima, K. Shibuya, T. Mihara, T. Ohnishi, M. Lippmaa, M. Kawasaki, H. Koinuma, S. Okamoto, and A. J. Millis, Phys. Rev. Lett. 97, 057601 (2006). 5 T. Arima, Y. Tokura, and J. B. Torrance, Phys. Rev. B 48, 17006 (1993). 6 S. Miyasaka, Y. Okimoto, M. Iwama, and Y. Tokura, Phys. Rev. B 68, 100406(R) (2003). 7 K. Maiti and D. D. Sarma, Phys. Rev. B 61, 2525 (2000). 8 S.-G. Lim, S. Kriventsov, T. N. Jackson, J. H. Haeni, D. G. Schlom, A. M. Balbashov, R. Uecker, P. Reiche, J. L. Freeouf, and G. Lucovsky, J. Appl. Phys. 91, 4500 (2002). 9 Y. Hotta, H. Wadati, A. Fujimori, T. Susaki, and H. Y. Hwang, Appl. Phys. Lett. 89, 251916 (2006). 10 M. Kawasaki, K. Takahashi, T. Maeda, R. Tsuchiya, M. Shinohara, O. Ishihara, T. Yonezawa, M. Yoshimoto, and H. Koinuma, Science 266, 1540 (1994). 11 Y. Hotta, Y. Mukunoki, T. Susaki, H. Y. Hwang, L. Fit- ting, and D. A. Muller, Appl. Phys. Lett. 89, 031918 (2006). 12 H. Wadati, Y. Hotta, M. Takizawa, A. Fujimori, T. Susaki, and H. Y. Hwang (unpublished). 13 K. Tamasaku, Y. Tanaka, M. Yabashi, H. Yamazaki, N. Kawamura, M. Suzuki, and T. Ishikawa, Nucl. Instrum. Methods A 467/468, 686 (2001). 14 T. Ishikawa, K. Tamasaku, and M. Yabashi, Nucl. Instrum. Methods A 547, 42 (2005). 15 Y. Takata, M. Yabashi, K. Tamasaku, Y. Nishino, D. Miwa, T. Ishikawa, E. Ikenaga, K. Horiba, S. Shin, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, H. No- hira, T. Hattori, S. Sodergren, B. Wannberg, and K. Kobayashi, Nucl. Instrum. Methods A 547, 50 (2005). 16 N. Kamakura, M. Taguchi, A. Chainani, Y. Takata, K. Horiba, K. Yamamoto, K. Tamasaku, Y. Nishino, D. Miwa, E. Ikenaga, M. Awaji, A. Takeuchi, H. Ohashi, Y. Senba, H. Namatame, M. Taniguchi, T. Ishikawa, K. Kobayashi, and S. Shin, Europhys. Lett. 68, 557 (2004). 17 M. Taguchi, A. Chainani, N. Kamakura, K. Horiba, Y. Takata, M. Yabashi, K. Tamasaku, Y. Nishino, D. Miwa, T. Ishikawa, S. Shin, E. Ikenaga, T. Yokoya, K. Kobayashi, T. Mochiku, K. 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0704.1838

Performance Analysis of the IEEE 802.11e Enhanced Distributed Coordination Function using Cycle Time Approach

Performance Analysis of the IEEE 802.11e Enhanced Distributed Coordination Function using Cycle Time Approach † Inanc Inan, Feyza Keceli, and Ender Ayanoglu Center for Pervasive Communications and Computing Department of Electrical Engineering and Computer Science The Henry Samueli School of Engineering University of California, Irvine, 92697-2625 Email: {iinan, fkeceli, ayanoglu}@uci.edu Abstract The recently ratified IEEE 802.11e standard defines the Enhanced Distributed Channel Access (EDCA) function for Quality-of-Service (QoS) provisioning in the Wireless Local Area Networks (WLANs). The EDCA uses Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) and slotted Binary Exponential Backoff (BEB) mechanism. We present a simple mathematical analysis framework for the EDCA function. Our analysis considers the fact that the distributed random access systems exhibit cyclic behavior where each station successfully transmits a packet in a cycle. Our analysis shows that an AC-specific cycle time exists for the EDCA function. Validating the theoretical results via simulations, we show that the proposed analysis accurately captures EDCA saturation performance in terms of average throughput, medium access delay, and packet loss ratio. The cycle time analysis is a simple and insightful substitute for previously proposed more complex EDCA models. I. INTRODUCTION The IEEE 802.11e standard [1] specifies the Hybrid Coordination Function (HCF) which enables prioritized and parameterized Quality-of-Service (QoS) services at the MAC layer. The HCF combines a distributed contention-based channel access mechanism, referred to as Enhanced Distributed Channel Access (EDCA), and a centralized polling-based channel access mechanism, referred to as HCF Con- trolled Channel Access (HCCA). We confine our analysis to the EDCA scheme, which uses Carrier † This work is supported by the Center for Pervasive Communications and Computing, and by Natural Science Foundation under Grant No. 0434928. Any opinions, findings, and conclusions or recommendations expressed in this material are those of authors and do not necessarily reflect the view of the Natural Science Foundation. http://arxiv.org/abs/0704.1838v1 Sense Multiple Access with Collision Avoidance (CSMA/CA) and slotted Binary Exponential Backoff (BEB) mechanism as the basic access method. The EDCA defines multiple Access Categories (AC) with AC-specific Contention Window (CW) sizes, Arbitration Interframe Space (AIFS) values, and Transmit Opportunity (TXOP) limits to support MAC-level QoS and prioritization. We evaluate the EDCA performance for the saturation (asymptotic) case. The saturation analysis provides the limits reached by the system throughput and protocol service time in stable conditions when every station has always backlogged data ready to transmit in its buffer. The analysis of the saturation provides in-depth understanding and insights into the random access schemes and the effects of different contention parameters on the performance. The results of such analysis can be employed in access parameter adaptation or in a call admission control algorithm. Our analysis is based on the fact that a random access system exhibits cyclic behavior. A cycle time is defined as the duration in which an arbitrary tagged user successfully transmits one packet on average [2]. We will derive the explicit mathematical expression of the AC-specific EDCA cycle time. The derivation considers the AIFS and CW differentiation by employing a simple average collision probability analysis. We will use the EDCA cycle time to predict the first moments of the saturation throughput, the service time, and the packet loss probability. We will show that the results obtained using the cycle time model closely follow the accurate predictions of the previously proposed more complex analytical models and simulation results. Our cycle time analysis can serve as a simple and practical alternative model for EDCA saturation throughput analysis. II. EDCA OVERVIEW The IEEE 802.11e EDCA is a QoS extension of IEEE 802.11 Distributed Coordination Function (DCF). The major enhancement to support QoS is that EDCA differentiates packets using different priorities and maps them to specific ACs that are buffered in separate queues at a station. Each ACi within a station (0 ≤ i ≤ imax, imax = 3 in [1]) having its own EDCA parameters contends for the channel independently of the others. Following the convention of [1], the larger the index i is, the higher the priority of the AC is. Levels of services are provided through different assignments of the AC-specific EDCA parameters; AIFS, CW, and TXOP limits. If there is a packet ready for transmission in the MAC queue of an AC, the EDCA function must sense the channel to be idle for a complete AIFS before it can start the transmission. The AIFS of an AC is determined by using the MAC Information Base (MIB) parameters as AIFS = SIFS+AIFSN ×Tslot, where AIFSN is the AC-specific AIFS number, SIFS is the length of the Short Interframe Space, and Tslot is the duration of a time slot. If the channel is idle when the first packet arrives at the AC queue, the packet can be directly transmitted as soon as the channel is sensed to be idle for AIFS. Otherwise, a backoff procedure is completed following the completion of AIFS before the transmission of this packet. A uniformly distributed random integer, namely a backoff value, is selected from the range [0,W ]. The backoff counter is decremented at the slot boundary if the previous time slot is idle. Should the channel be sensed busy at any time slot during AIFS or backoff, the backoff procedure is suspended at the current backoff value. The backoff resumes as soon as the channel is sensed to be idle for AIFS again. When the backoff counter reaches zero, the packet is transmitted in the following slot. The value of W depends on the number of retransmissions the current packet experienced. The initial value of W is set to the AC-specific CWmin. If the transmitter cannot receive an Acknowledgment (ACK) packet from the receiver in a timeout interval, the transmission is labeled as unsuccessful and the packet is scheduled for retransmission. At each unsuccessful transmission, the value of W is doubled until the maximum AC-specific CWmax limit is reached. The value of W is reset to the AC-specific CWmin if the transmission is successful, or the retry limit is reached thus the packet is dropped. The higher priority ACs are assigned smaller AIFSN. Therefore, the higher priority ACs can either transmit or decrement their backoff counters while lower priority ACs are still waiting in AIFS. This results in higher priority ACs facing a lower average probability of collision and relatively faster progress through backoff slots. Moreover, in EDCA, the ACs with higher priority may select backoff values from a comparably smaller CW range. This approach prioritizes the access since a smaller CW value means a smaller backoff delay before the transmission. Upon gaining the access to the medium, each AC may carry out multiple frame exchange sequences as long as the total access duration does not go over a TXOP limit. Within a TXOP, the transmissions are separated by SIFS. Multiple frame transmissions in a TXOP can reduce the overhead due to contention. A TXOP limit of zero corresponds to only one frame exchange per access. An internal (virtual) collision within a station is handled by granting the access to the AC with the highest priority. The ACs with lower priority that suffer from a virtual collision run the collision procedure as if an outside collision has occured. III. RELATED WORK In this section, we provide a brief summary of the studies in the literature on the theoretical DCF and EDCA function saturation performance analysis. Three major saturation performance models have been proposed for DCF; i) assuming constant collision probability for each station, Bianchi [3] developed a simple Discrete-Time Markov Chain (DTMC) and the saturation throughput is obtained by applying regenerative analysis to a generic slot time, ii) Cali et al. [4] employed renewal theory to analyze a p-persistent variant of DCF with persistence factor p derived from the CW, and iii) Tay et al. [5] instead used an average value mathematical method to model DCF backoff procedure and to calculate the average number of interruptions that the backoff timer experiences. Having the common assumption of slot homogeneity (for an arbitrary station, constant collision or transmission probability at an arbitrary slot), these models define all different renewal cycles all of which lead to accurate saturation performance analysis. These major methods (especially [3]) are modified by several researchers to include the extra features of the EDCA function in the saturation analysis. Xiao [6] extended [3] to analyze only the CW differentiation. Kong et al. [7] took AIFS differentiation into account. On the other hand, these EDCA extensions miss the treatment of varying collision probabilities at different AIFS slots due to varying number of contending stations. Robinson et al. [8] proposed an average analysis on the collision probability for different contention zones during AIFS. Hui et al. [9] unified several major approaches into one approximate average model taking into account varying collision probability in different backoff subperiods (corresponds to contention zones in [8]). Zhu et al. [10] proposed another analytical EDCA Markov model averaging the transition probabilities based on the number and the parameters of high priority flows. Inan et al. [11] proposed a 3-dimensional DTMC which provides accurate treatment of AIFS and CW differentiation. Another 3-dimensional DTMC is proposed by Tao et al. [12] in which the third dimension models the state of backoff slots between successive transmission periods. The fact that the number of idle slots between successive transmissions can be at most the minimum of AC-specific CWmax values is considered. Independently, Zhao et al. [13] had previously proposed a similar model for the heterogeneous case where each station has traffic of only one AC. Banchs et al. [14] proposed another model which considers varying collision probability among different AIFS slots due to a variable number of stations. Lin et al. [15] extended [5] in order to carry out mean value analysis for approximating AIFS and CW differentiation. Our approach is based on the observation that the transmission behavior in the 802.11 WLAN follows a pattern of periodic cycles. Previously, Medepalli et al. [2] provided explicit expressions for average DCF cycle time and system throughput. Similarly, Kuo et al. [16] calculated the EDCA transmission cycle assuming constant collision probability for any traffic class. On the other hand, such an assumption leads to analytical inaccuracies [7]-[15]. The main contribution is that we incorporate accurate AIFS and CW differentiation calculation in the EDCA cycle time analysis. We show that the cyclic behavior is observed on a per AC basis in the EDCA. To maintain the simplicity of the cycle time analysis, we employ averaging on the AC-specific collision probability. The comparison with more complex and detailed theoretical and simulation models reveals that the analytical accuracy is preserved. IV. EDCA CYCLE TIME ANALYSIS In this section, we will first derive the AC-specific average collision probability. Next, we will calculate the AC-specific average cycle time. Finally, we will relate the average cycle time and the average collision probability to the average normalized throughput, EDCA service time, and packet loss probability. A. AC-specific Average Collision Probability The difference in AIFS of each AC in EDCA creates the so-called contention zones or periods as shown in Fig. 1 [8],[9]. In each contention zone, the number of contending stations may vary. We employ an average analysis on the AC-specific collision probability rather than calculating it separately for different AIFS and backoff slots as in [11]-[14]. We calculate the AC-specific collision probability according to the long term occupancy of AIFS and backoff slots. We define pci,x as the conditional probability that ACi experiences either an external or an internal collision given that it has observed the medium idle for AIFSx and transmits in the current slot (note AIFSx ≥ AIFSi should hold). For the following, in order to be consistent with the notation of [1], we assume AIFS0 ≥ AIFS1 ≥ AIFS2 ≥ AIFS3. Let di = AIFSNi − AIFSN3. Following the slot homogeneity assumption of [3], assume that each ACi transmits with constant probability, τi. Also, let the total number ACi flows be Ni. Then, for the heterogeneous scenario in which each station has only one AC pci,x = 1− i′≤dx (1− τi′) (1− τi) . (1) We only formulate the situation when there is only one AC per station, therefore no internal collisions can occur. Note that this simplification does not cause any loss of generality, because the proposed model can be extended for the case of higher number of ACs per station as in [7],[11]. We use the Markov chain shown in Fig. 2 to find the long term occupancy of the contention zones. Each state represents the nth backoff slot after the completion of the AIFS3 idle interval following a transmission period. The Markov chain model uses the fact that a backoff slot is reached if and only if no transmission occurs in the previous slot. Moreover, the number of states is limited by the maximum idle time between two successive transmissions which is Wmin = min(CWi,max) for a saturated scenario. The probability that at least one transmission occurs in a backoff slot in contention zone x is ptrx = 1− i′:di′≤dx (1− τi′) Ni′ . (2) Note that the contention zones are labeled with x regarding the indices of d. In the case of an equality in AIFS values of different ACs, the contention zone is labeled with the index of AC with higher priority. Given the state transition probabilities as in Fig. 2, the long term occupancy of the backoff slots b′n can be obtained from the steady-state solution of the Markov chain. Then, the AC-specific average collision probability pci is found by weighing zone specific collision probabilities pci,x according to the long term occupancy of contention zones (thus backoff slots) pci = ∑Wmin n=di+1 pci,xb ∑Wmin n=di+1 where x = max y | dy = max (dz | dz ≤ n) which shows x is assigned the highest index value within a set of ACs that have AIFSN smaller than or equal to n+AIFSN3. This ensures that at backoff slot n, ACi has observed the medium idle for AIFSx. Therefore, the calculation in (3) fits into the definition of pci,x . B. AC-Specific Average Cycle Time Intuitively, it can be seen that each user transmitting at the same AC has equal cycle time, while the cycle time may differ among ACs. Our analysis will also mathematically show this is the case. Let Ei[tcyc] be average cycle time for a tagged ACi user. Ei[tcyc] can be calculated as the sum of average duration for i) the successful transmissions, Ei[tsuc], ii) the collisions, Ei[tcol], and iii) the idle slots, Ei[tidle] in one cycle. In order to calculate the average time spent on successful transmissions during an ACi cycle time, we should find the expected number of total successful transmissions between two successful transmissions of ACi. Let Qi represent this random variable. Also, let γi be the probability that the transmitted packet belongs to an arbitrary user from ACi given that the transmission is successful. Then, n=di+1 psi,n/Ni psj,n where psi,n = (1− τi) i′:di′≤n−1 (1− τi′)Ni′ , if n ≥ di + 1 0, if n < di + 1. Then, the Probability Mass Function (PMF) of Qi is Pr(Qi = k) = γi(1− γi) k, k ≥ 0. (6) We can calculate expected number of successful transmissions of any ACj during the cycle time of ACi, STj,i, as STj,i = NjE[Qi] 1− γi . (7) Inserting E[Qi] = (1 − γi)/γi in (7), our intuition that each user from ACi can transmit successfully once on average during the cycle time of another ACi user, i.e., STi,i = Ni, is confirmed. Therefore, the average cycle time of any user belonging to the same AC is equal in a heterogeneous scenario where each station runs only one AC. Including the own successful packet transmission time of tagged ACi user in Ei[tsuc], we find Ei[tsuc] = STj,iTsj (8) where Tsj is defined as the time required for a successful packet exchange sequence. Tsj will be derived in (16). To obtain Ei[tcol], we need to calculate average number of users that involve in a collision, Ncn , at the nth slot after last busy time for given Ni and τi, ∀i. Let the total number of users transmitting at the n th slot after last busy time be denoted as Yn. We see that Yn is the sum of random variables, Binomial(Ni, τi), ∀i : di ≤ n− 1. Employing simple probability theory, we can calculate Ncn = E[Yn|Yn ≥ 2]. After some simplification, Ncn = i:di≤n−1 (Niτi − psi,n) i:di≤n−1 (1− τi)Ni − i:di≤n−1 psi,n If we let the average number of users involved in a collision at an arbitrary backoff slot be Nc, then b′nNcn . (10) We can also calculate the expected number of collisions that an ACj user experiences during the cycle time of an ACi, CTj,i, as CTj,i = 1− pcj STj,i. (11) Then, defining Tcj as the time wasted in a collision period (will be derived in (17), Ei[tcol] = CTj,iTcj . (12) Given pci , we can calculate the expected number of backoff slots Ei[tbo] that ACi waits before attempting a transmission. Let Wi,k be the CW size of ACi at backoff stage k [11]. Note that, when the retry limit ri is reached, any packet is discarded. Therefore, another Ei[tbo] passes between two transmissions with probability prici Ei[tbo] = pk−1ci (1− pci) . (13) Noticing that between two successful transmissions, ACi also experiences CTi,i collisions, Ei[tidle] = Ei[tbo](CTi,i/Ni + 1)tslot. (14) As shown in [9], the transmission probability of a user using ACi, Ei[tbo] + 1 . (15) Note that, in [9], it is proven that the mean value analysis for the average transmission probability as in (15) matches the Markov analysis of [3]. The fixed-point equations (1)-(15) can numerically be solved for τi and pci , ∀i. Then, each component of the average cycle time for ACi, ∀i, can be calculated using (4)-(14). C. Performance Analysis Let Tpi be the average payload transmission time for ACi (Tpi includes the transmission time of MAC and PHY headers), δ be the propagation delay, Tack be the time required for acknowledgment packet (ACK) transmission. Then, for the basic access scheme, we define the time spent in a successful transmission Tsi and a collision Tci for any ACi as Tsi =Tpi + δ + SIFS + Tack + δ + AIFSi (16) Tci =Tp∗i + ACK Timeout+ AIFSi (17) where Tp∗ is the average transmission time of the longest packet payload involved in a collision [3]. For simplicity, we assume the packet size to be equal for any AC, then Tp∗ = Tpi . Being not explicitly specified in the standards, we set ACK Timeout, using Extended Inter Frame Space (EIFS) as EIFSi −AIFSi. Note that the extensions of (16) and (17) for the RTS/CTS scheme are straightforward [3]. The average cycle time of an AC represents the renewal cycle for each AC. Then, the normalized throughput of ACi is defined as the successfully transmitted information per renewal cycle NiTpi Ei[tsuc] + Ei[tcol] + Ei[tidle] . (18) The AC-specific cycle time is directly related but not equal to the mean protocol service time. By definition, the cycle time is the duration between successful transmissions. We define the average protocol service time such that it also considers the service time of packets which are dropped due to retry limit. On the average, 1/pi,drop service intervals correspond to 1/pi,drop − 1 cycles. Therefore, the mean service time µi can be calculated as µi = (1− pi,drop)Ei[tcyc]. (19) Simply, the average packet drop probability due to MAC layer collisions is pi,drop = p . (20) V. NUMERICAL AND SIMULATION RESULTS We validate the accuracy of the numerical results by comparing them to the simulation results obtained from ns-2 [17]. For the simulations, we employ the IEEE 802.11e HCF MAC simulation model for ns-2.28 [18]. This module implements all the EDCA and HCCA functionalities stated in [1]. In simulations, we consider two ACs, one high priority (AC3) and one low priority (AC1). Each station runs only one AC. Each AC has always buffered packets that are ready for transmission. For both ACs, the payload size is 1000 bytes. RTS/CTS handshake is turned on. The simulation results are reported for the wireless channel which is assumed to be not prone to any errors during transmission. The errored channel case is left for future study. All the stations have 802.11g Physical Layer (PHY) using 54 Mbps and 6 Mbps as the data and basic rate respectively (Tslot = 9 µs, SIFS = 10 µs) [19]. The simulation runtime is 100 seconds. In the first set of experiments, we set AIFSN1 = 3, AIFSN3 = 2, CW1,min = 31, CW3,min = 15, m1 = m3 = 3, r1 = r3 = 7. Fig. 3 shows the normalized throughput of each AC when both N1 and N3 are varied from 5 to 30 and equal to each other. As the comparison with a more detailed analytical model [11] and the simulation results reveal, the cycle time analysis can predict saturation throughput accurately. Fig. 4 and Fig. 5 display the mean protocol service time and packet drop probability respectively for the same scenario of Fig. 3. As comparison with [11] and the simulation results show, both performance measures can accurately be predicted by the proposed cycle time model. Although not included in the figures, a similar discussion holds for the comparison with other detailed and/or complex models of [12]-[14]. In the second set of experiments, we fix the EDCA parameters of one AC and vary the parameters of the other AC in order to show the proposed cycle time model accurately captures the normalized throughput for different sets of EDCA parameters. In the simulations, both N1 and N3 are set to 10. Fig. 6 shows the normalized throughput of each AC when we set AIFSN3 = 2, CW3,min = 15, and vary AIFSN1 and CW1,min. Fig. 7 shows the normalized throughput of each AC when we set AIFSN1 = 4, CW1,min = 127, and vary AIFSN3 and CW3,min. As the comparison with simulation results show, the predictions of the proposed cycle time model are accurate. We do not include the results for packet drop probability and service time for this experiment. No discernable trends toward error are observed. VI. CONCLUSION We have presented an accurate cycle time model for predicting the EDCA saturation performance analytically. The model accounts for AIFS and CW differentiation mechanisms of EDCA. We employ a simple average collision probability calculation regarding AIFS and CW differentiation mechanisms of EDCA. Instead of generic slot time analysis of [3], we use the AC-specific cycle time as the renewal cycle. We show that the proposed simple cycle time model performs as accurate as more detailed and complex models previously proposed in the literature. The mean saturation throughput, protocol service time and packet drop probability are calculated using the model. This analysis also highlights some commonalities between approaches in EDCA saturation performance analysis. The simple cycle time analysis can provide invaluable insights for QoS provisioning in the WLAN. REFERENCES [1] IEEE Standard 802.11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications: Medium access control (MAC) Quality of Service (QoS) Enhancements, IEEE 802.11e Std., 2005. [2] K. Medepalli and F. A. Tobagi, “Throughput Analysis of IEEE 802.11 Wireless LANs using an Average Cycle Time Approach,” in Proc. IEEE Globecom ’05, November 2005. [3] G. Bianchi, “Performance Analysis of the IEEE 802.11 Distributed Coordination Function,” IEEE Trans. Commun., pp. 535–547, March 2000. [4] F. Cali, M. Conti, and E. Gregori, “Dynamic Tuning of the IEEE 802.11 Protocol to Achieve a Theoretical Throughput Limit,” IEEE/ACM Trans. Netw., pp. 785–799, December 2000. [5] J. C. Tay and K. C. Chua, “A Capacity Analysis for the IEEE 802.11 MAC Protocol,” Wireless Netw., pp. 159–171, July 2001. [6] Y. Xiao, “Performance Analysis of Priority Schemes for IEEE 802.11 and IEEE 802.11e Wireless LANs,” IEEE Trans. Wireless Commun., pp. 1506–1515, July 2005. [7] Z. Kong, D. H. K. Tsang, B. Bensaou, and D. Gao, “Performance Analysis of the IEEE 802.11e Contention-Based Channel Access,” IEEE J. Select. Areas Commun., pp. 2095–2106, December 2004. [8] J. W. Robinson and T. S. Randhawa, “Saturation Throughput Analysis of IEEE 802.11e Enhanced Distributed Coordination Function,” IEEE J. Select. Areas Commun., pp. 917–928, June 2004. [9] J. Hui and M. Devetsikiotis, “A Unified Model for the Performance Analysis of IEEE 802.11e EDCA,” IEEE Trans. Commun., pp. 1498–1510, September 2005. [10] H. Zhu and I. Chlamtac, “Performance Analysis for IEEE 802.11e EDCF Service Differentiation,” IEEE Trans. Wireless Commun., pp. 1779–1788, July 2005. [11] I. Inan, F. Keceli, and E. Ayanoglu, “Saturation Throughput Analysis of the 802.11e Enhanced Distributed Channel Access Function,” to appear in Proc. IEEE ICC ’07. [12] Z. Tao and S. Panwar, “Throughput and Delay Analysis for the IEEE 802.11e Enhanced Distributed Channel Access,” IEEE Trans. Commun., pp. 596–602, April 2006. [13] J. Zhao, Z. Guo, Q. Zhang, and W. Zhu, “Performance Study of MAC for Service Differentiation in IEEE 802.11,” in Proc. IEEE Globecom ’02, November 2002. [14] A. Banchs and L. Vollero, “Throughput Analysis and Optimal Configuration of IEEE 802.11e EDCA,” Comp. Netw., pp. 1749–1768, August 2006. [15] Y. Lin and V. W. Wong, “Saturation Throughput of IEEE 802.11e EDCA Based on Mean Value Analysis,” in Proc. IEEE WCNC ’06, April 2006. [16] Y.-L. Kuo, C.-H. Lu, E. H.-K. Wu, G.-H. Chen, and Y.-H. Tseng, “Performance Analysis of the Enhanced Distributed Coordination Function in the IEEE 802.11e,” in Proc. IEEE VTC ’03 - Fall, October 2003. [17] (2006) The Network Simulator, ns-2. [Online]. Available: http://www.isi.edu/nsnam/ns [18] IEEE 802.11e HCF MAC model for ns-2.28. [Online]. Available: http://newport.eecs.uci.edu/$\sim$fkeceli/ns.htm [19] IEEE Standard 802.11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications: Further Higher Data Rate Extension in the 2.4 GHz Band, IEEE 802.11g Std., 2003. http://www.isi.edu/nsnam/ns http://newport.eecs.uci.edu/$\sim $fkeceli/ns.htm Transmission/ Collision period AIFSN AIFSN AIFSN AIFSN No Tx Zone 3 Zone 2 Zone 1 AC3 in Backoff AC2 in Backoff AC1 in Backoff AC0 in Backoff Fig. 1. EDCA backoff after busy medium. 1 2 d2 d2+1 d1 d1+1 tr 1-p tr 1-p Fig. 2. Transition through backoff slots in different contention zones for the example given in Fig.1. 5 10 15 20 25 30 Number of AC and AC Cycle−AC Cycle−AC Cycle−total [11]−AC [11]−AC [11]−total Sim−AC Sim−AC Sim−total Fig. 3. Analyzed and simulated normalized throughput of each AC when both N1 and N3 are varied from 5 to 30 and equal to each other for the cycle time analysis. Analytical results of the model proposed in [11] are also added for comparison. 5 10 15 20 25 30 Number of AC and AC Cycle−AC Cycle−AC [11]−AC [11]−AC Sim−AC Sim−AC Fig. 4. Analyzed and simulated mean protocol service time of each AC when both N1 and N3 are varied from 5 to 30 and equal to each other for the proposed cycle time analysis and the model in [11]. 5 10 15 20 25 30 Number of AC and AC Cycle−AC Cycle−AC [11]−AC [11]−AC Sim−AC Sim−AC Fig. 5. Analyzed and simulated mean packet drop probability of each AC when both N1 and N3 are varied from 5 to 30 and equal to each other for the proposed cycle time analysis and the model in [11]. 0 50 100 150 200 250 1,min A=0−AC A=0−AC A=1−AC A=1−AC A=2−AC A=2−AC sim−AC sim−AC Fig. 6. Analytically calculated and simulated performance of each AC when AIFSN3 = 2, CW3,min = 15, N1 = N3 = 10, AIFSN1 varies from 2 to 4, and CW1,min takes values from the set {15, 31, 63, 127, 255}. Note that AIFSN1 − AIFSN3 is denoted by A. 0 20 40 60 80 100 120 3,min A=0−AC A=0−AC A=1−AC A=1−AC A=2−AC A=2−AC sim−AC sim−AC Fig. 7. Analytically calculated and simulated performance of each AC when AIFSN1 = 4, CW1,min = 127, N1 = N3 = 10, AIFSN3 varies from 2 to 4, and CW3,min takes values from the set {15, 31, 63, 127}. Note that AIFSN1 − AIFSN3 is denoted by A. Introduction EDCA Overview Related Work EDCA Cycle Time Analysis AC-specific Average Collision Probability AC-Specific Average Cycle Time Performance Analysis Numerical and Simulation Results Conclusion References

0704.1839

ALHEP symbolic algebra program for high-energy physics

ALHEP symbolic algebra program for high-energy physics V. Makarenko 1 NC PHEP BSU, 153 Bogdanovicha str., 220040 Minsk, Belarus Abstract ALHEP is the symbolic algebra program for high-energy physics. It deals with amplitudes calculation, matrix element squaring, Wick theorem, dimensional regu- larization, tensor reduction of loop integrals and simplification of final expressions. The program output includes: Fortran code for differential cross section, Mathe- matica files to view results and intermediate steps and TeX source for Feynman diagrams. The PYTHIA interface is available. The project website http://www.hep.by/alhep contains up-to-date executa- bles, manual and script examples. 1 Introduction The analytical calculations in high-energy physics are mostly impossible with- out a powerful computing tool. The big variety of packages is commonly used [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Some are general-purpose symbolic algebra programs with specific HEP-related plug-ins (REDUCE [15], Mathematica [16]), some are designed especially for particle physics (CompHEP [1], SANC [2], GRACE [3] etc.) and some are created for specific interaction class or spe- cific task. Many of them uses external symbolic algebra core (Form [17], Math- Link [16]). They can deal with matrix elements squaring (FeynCalc [7]) or cal- culate helicity amplitudes directly (MadGraph [4], CompHEP [1], O’Mega [5]). Some packages provide numerical calculations, some require external Monte- Carlo generator to be linked. Some programs contain also one-loop calculation routines (FormCalc [6], GRACE [3], SANC [2]). Nevertheless, there is no uni- form program that meets all the user requirements. Email address: [email protected] (V. Makarenko ). 1 Supported by INTAS YS Grant no. 05-112-5429 Preprint submitted to Elsevier 17 June 2021 http://arxiv.org/abs/0704.1839v1 Every calculation requires the program-independent check. The optimal tac- tics is the simultaneous usage of two (or more) different symbolic algebra packages. ALHEP is a symbolic algebra program for performing the way from Standard Model Lagrangian to amplitude or squared matrix element for the specified scattering process. It can also be useful for loop diagrams analysis. The basic features are: • Diagrams generation using Wick theorem and SM Lagrangian. • Amplitude calculation or matrix element squaring. • Bondarev functions method for traces calculation. • Tensor reduction of loop integrals. • Dimensional regularization scheme. • Generation of Fortran procedures for numerical analysis (PYTHIA [18] and LoopTools [19] interfaces are implemented). The current ALHEP version have several implementation restrictions, that will be lifted in future. The following features are in progress of implementation: • Bremsstrahlung part of radiative correction (the integration over real photon phase space). • Complete one-loop renormalization scheme including renormalization con- stants derivation. • Arbitrary Lagrangian assignment. After these methods implementation the complete one-loop analysis will be available. Please refer to project website for program updates. ALHEP website http://www.hep.by/alhep contains the up-to-date executa- bles (for both Linux & Win32 platforms), manual and script examples. The mirror at http://cern.ch/~makarenko/alhep is also updated. 2 ALHEP Review. Program Structure The ALHEP program internal structure can be outlined as follows: • The native symbolic algebra core. • Common algebra libraries: Dirac matrices, tensor and spinor algebra, field operators & particle wave functions zoo. • Specific HEP functions and libraries. It include Feynman diagrams generation, trace calculations, helicity am- plitudes method, HEP-specific simplification procedures, tensor integrals reduction and others. • Interfaces to Mathematica, Fortran, TeX and internal I/O format. Fortran code is used for further numerical analysis. Mathematica code can be used for viewing any symbolic expression in program. But no backward interface from Mathematica is currently implemented. TeX output can be generated for Feynman diagrams view. Internal I/O format is implemented for the most of symbolic expressions, allowing save and restore calculation at intermediate steps. • Command script processor. User interface is implemented in terms of command scripts. The ALHEP script language have C-like syntax, variables, arithmetic operations, func- tion calls. All HEP-related tasks are implemented as build-in functions. 2.1 Getting Started To use ALHEP one should download the pre-compiled executable for appro- priate platform (Linux and Win32 are available) and write a control script to describe your task. ALHEP program should be launched with the single argument: script file name to be invoked The following steps are required to create a workspace: • Download ALHEP executables from project website: alhep.gz (for Linux) or alhep.zip (Win32). Unpack executable, e.g. gzip -d alhep.gz • Download up-to date command list: ALHEPCommands.txt. The set of com- mands (or options) may be changed in future versions and this manual may be somewhat obsolete. Please refer to ALHEPCommands.txt file that always corresponds to the latest ALHEP version. The last changes are outlined in RecentChanges.txt file at website. • Create some working directory and compose command script file therein. For example consider the uū → W+W−γ process with {−+}-helicities of initial quarks. To calculate amplitude we create the following script file (call it "test.al"): SetKinematics(2, 3 // 2->3 process ,QUARK_U,"p\_1","e\_1" // u ,-QUARK_U,"p\_2","e\_2" // u-bar ,WBOZON,"f\_1","g\_1" // W{+} ,PHOTON,"f\_0","g\_0" // photon ,-WBOZON, "f\_2", "g\_2" ); // W{-} SetDiagramPhysics(PHYS_SM_Q1GEN);// SM with 2 quarks only SetMassCalcOrder(QUARK_U, 0); // consider massless SetMassCalcOrder(QUARK_D, 0); // consider massless diags = ComposeDiagrams(3); // create diagrams, e^3 DrawDiagrams(diags, "res.tex", DD_SMALL|DD_SWAP_TALES, FILE_START); SetFermionHelicity(1, -1); SetFermionHelicity(2, 1); SetParameter(PAR_TRACES_BONDEREV, 1); ampl = CalcAmplitude(RetrieveME(diags)); ampl = KinArrange(ampl); // arrange result ampl = Minimize(ampl); // minimize result SaveNB("res.nb",ampl,"",FILE_START|FILE_CLOSE); f = NewFortranFile("res.F", CODE_F77); CreateFortranProc(f, "UUWWA", ampl, CODE_IS_AMPLITUDE|CODE_COMPLEX16 |CODE_CHECK_DENOMINATORS|CODE_PYTHIA_VECTORS); The SetKinematics() function declares particles, momenta and polariza- tion symbols. The physics is declared with PHYS_SM_Q1GEN option to restrict diagrams number. The amplitude for all (d, c, b) internal quarks can be ob- tained from generated here by quark mixing matrix replacement: U2ud → U2ud+U ub (chiral limit). Diagrams are created by ComposeDiagrams() call. CalcAmplitude() function creates the symbolic value for process am- plitude. For detailed discussion of this example see sec. 15.1. • Create simple batch file "run.me" like: ~/alhep_bin_path/alhep test.al The ALHEP program creates some console output (current commands, scroll bars and some debugging data). If it is not allowed one should redirect console output to file here. The batch execution with test.al command file takes about 1 minute at 1.8GHz P4 processor. The following files are created: res.nb: Mathematica file containing symbolic expression of amplitude. Cre- ated by SaveNB() function. res.F: F77 code for numerical analysis created by NewFortranFile() and CreateFortranProc() calls. The library file alhep_lib.F is required for code compilation and should be downloaded from project website. See sec. 11.1 for Fortran generation and compilation review. res.tex: TeX source for 19 Feynman graphs generated. The AxoDraw [23] Latex package is used. The res.tex file should be included into your La- TeX document using \input res.tex command. The template document to include your diagrams can be found at program website. See fig. 1 in sec. 15.1 for diagrams generated. debug.nb: Mathematica file with debugging information and some interme- diate steps. The amount of debugging information is declared in debug.ini file in working directory. See sec. 13 for further details. See sec. 15 and project website for another examples. It is convenient to use some example script as template and modify it for your purposes. 2.2 Calculation scheme The usual ALHEP script contains several steps: (1) Initialization section Declaration of process kinematics: initial and final-state particles, titles for particle momenta & polarization vectors (SetKinematics()). Physics model definition. The SM physics or part of SM Hamiltonian should be specified (SetDiagramPhysics()). The shorter Hamiltonian is selected, the faster is Wick theorem invocation. Polarization declaration. Every particle is considered as polarized with abstract polarization vector by default. The specific helicity value can be set manually or particle can be marked as unpolarized. The several ways of polarization involving are inplemented, see sec. 4.4 (SetPolarized(), SetFermionHelicity(), SetPhotonHelicity(), ...). Setting mass-order rules for specific particles (SetMassCalcOrder()). One can demand the massless calculation for light particle, that greatly saves evaluation time. One can also demand keeping particle mass with specific order Mn and drop out the higher-order expressions like Mn+1. It allows to consider the leading mass contribution without calculating precisely. (2) Diagrams generation Feynman diagrams are generated for specific en order using the Wick theorem algorithm (ComposeDiagrams()) User may draw diagrams here (DrawDiagrams()), halt the program (Halt()) and check out if diagrams are generated correctly. After the diagram set is generated one may cut-off not interesting di- agrams to work with the shorter set or select the single diagram to work with (SelectDiagrams()). The loop corrections are calculating faster when processed by single diagrams. Then matrix element is retrieved from diagrams set (RetrieveME()). Before any operation with loop matrix element one should declare the N−dimensional space (SetNDimensionSpace()). Some procedures involve N−dimensional mode automatically for loop objects, but most functions (arranging, simplification etc.) don’t know the nature of expres- sion they work with. Therefore the dimensional mode should be forced. The diagrams set and any symbolic expression may be saved and re- stored at next session to refrain from job repetition (Save(), Load()). (3) Amplitude calculation The amplitude evaluation (CalcAmplitude()) is the faster way for multi-diagram process analysis. All the particles are considered polarized. The spinor objects are pro- jected to abstract basis spinor. The basis vectors are generated in nu- merical code to meet the non-zero denominators condition. See sec. 7 for method details. (4) Matrix element squaring (coupling to other) The squaring procedure (SquareME()) is controlled by plenty of op- tions, intended mostly for the performance tuning and debugging. It basically includes reduction of gamma-matrix sequences coupled to kinematically dependent vectors. It reduces the number of matrices in every product to minimum. The item-by-item squaring is followed. For loop × born∗ couplings the virtual integrals are involved. See sec. 8 for details. Amplitude calculation is fast, but its result may be more complicated than squaring expressions. Amplitude depends on particle momenta, po- larization vectors and additional basis vectors. For unpolarized process the averaging cycle is generated in Fortran code, and complex-numbers calculation should be performed. The squaring result for unpolarized pro- cess is the polynomial of momenta couplings only. Hence there is no unique answer what result is simpler for 10-15 diagrams reaction. One should definitely use amplitude method for more than 10-15 diagrams squaring. (5) Loop diagrams analysis The tensor virtual integrals are reduced to scalar ones (Evaluate(), sec. 9). The scalar coefficients of tensor integral decompositions are also reduced to scalar integrals in the most cases (for 1− 4 point integrals). For scalar loop integrals the tabulated values are used. There is no reason to tabulate integrals with complicated structure. Hence scalar in- tegrals table contain the A0 and B0 integrals with different mass configu- ration. It also contains a useful D0 chiral decomposition. Other integrals should to be resolved using LoopTools-like [19,20] numerical programs. The renormalization procedure is under construction now. The counter- terms (CT) part of Lagrangian leads to CT diagrams generating. In the nearest future the abstract renormalization constants (δm, δf etc.) will be involved and tabulated for the minimal on-shell scheme. The automatic derivation of constants is supposed to be implemented further. Please refer ALHEP website for implementation progress. (6) Simplification The kinematic simplification procedure is available (KinSimplify(), sec. 12). It reduces expression using all the possible kinematic relations between momenta and invariants. The minimization of +/* operations in huge expressions can also be performed (Minimize()). (7) Fortran procedure creation for numerical analysis F90 or F77-syntaxes for generated procedures are used. Generated code can be linked to PYTHIA, LoopTools and any Monte-Carlo generator for numerical analysis. 3 ALHEP script language The script syntax is similar to C/Java languages. Command line breaks at ”;” symbol only. Comments are marked as // or /*...*/. Operands may be variables or function calls. If no function with some title is defined, the operand is considered as variable. The notation is case-sensitive. The script language have no user-defined functions, classes or loop operators. It seems to be useless in current version. All ALHEP features are implemented as build-in functions. The execution starts from the first line and finishes at the end of file (or at Halt() command). Variable types are casted and checked automatically (in run-time mode). No manual type specifying or casting are available. List of ALHEP script internal types: • Abstract Symbolic Expression (expr). Result of function operations. Can be stored to file(Save()) and loaded back(Load()). Basic operations: +,-,*. The division is implemented using Frac(a,b) function. Are not supposed to be inputed manually, although a few commands for manual input exist. • Integer Value (int). Any number parameter will be casted to integer value. Basic operations: +,-,*,|. The division is performed using Frac(a,b) function. No fractional values are currently available. • String (str). String parameters are started and closed by double quotes (”string vari- able”). Used to specify symbol notation (e.g. momenta titles), file names, etc. Basic operations: +(concatenate strings). • Set of Feynman Diagrams (diagrams). Result of diagrams composing function. Basic operations: Save(), Load() and SelectDiagrams(). Diagrams can also be converted into TeX graphics. • Matrix Element (me). Contains symbolic expression for matrix element and information on it’s use: list of virtual momenta (integration list) etc. The total up-to-date functions list can be found in ALHEPCommands.txt file at project website. 4 Initialization section 4.1 Particles Particles are determined by integer number, called particle kind (PK). The following integer constants are defined: ELECTRON, MUON, TAULEPTON, PHOTON, ZBOZON, WBOZON, QUARK_D, QUARK_U, QUARK_S, QUARK_C, QUARK_B, QUARK_T, NEUTRINO_ELECTRON, NEUTRINO_MUON, NEUTRINO_TAU. The ghost and scalar particles are not supposed to be external and their codes are unavailable. Antiparticles have negative PK that is obtained by ”-PK” operation. If kinematics is declared the particles can be secelted using the particle ID (PID) number. Initial particles have negeative ID (-1, -2) and final are pointed by positive numbers (1, 2, 3...). 4.2 Kinematic selection Before any computations may be performed one needs to declare the kinematic conditions. They are: number of initial and final state particles, PK codes, symbols for momentum and polarization vector for every particle. SetKinematics((int)N Initials, (int)N Finals, [(int)PK I, (str)momentum I, (str)polarization I, ...]); Here N_Initials and N_Finals – numbers of initial and final particles in kine- matics. The next parameters are particle kind, momentum and polarization symbols, repeated for every particle. For example, the e−(k1, e1)e +(k2, e2) → µ−(p1, e3)µ +(p2, e4) process should be declared as follows: SetKinematics(2, 2, ELECTRON,"k\_1","e\_1", -ELECTRON,"k\_2","e\_2", MUON, "p\_1", "e\_3", -MUON, "p\_2", "e\_4"); 4.3 Particle masses All the particles are considered massive by default. Particle can be declared massless using the SetMassCalcOrder function. SetMassCalcOrder((int)PK, (int)order); PK: particle kind, order: maximum order of particle mass to be kept in calculations. Zero value means massless calculations for specified particle. Negative value declares the mass-exact operations. The mass symbols are generated automatically and look like me, mW etc. Hence, all the electrons in process will have the same mass symbol. To declare unique mass symbols for specific particles the SetMassSym() function is used. The different masses for unique particles are often required to involve the Breit-Wigner distribution for particles masses. SetMassSym((int)PID, (str or expr)mass); PID: particle ID in kinematics (< 0 for initial and > 0 for final particles), mass: new mass symbol, like "m\_X" for mX . 4.4 Polarization data All the particles are considered as polarized initially. The default polarization vector symbols are set together with kinematic data (in SetKinematics()). To declare particle unpolarized the SetPolarized() function is used: SetPolarized((int)PID, (int)polarized); PID: particle ID in kinematics, polarized: 1 (polarized) or 0 (unpolarized). One can declare the specific polarization state (helicity value) for particles. The photon (k,e) helicities are involved in terms of two outer physical mo- menta: e±µ = (p+ · k)p−µ − (p− · k)p+µ ± iǫµαβνpα+p /Nk,p±. (1) SetPhotonHelicity((int)PID, (int)h, (str)”pP”, (str)”pM”]); PID: photon particle ID in kinematics, h: helicity value: ±1 or 0. Zero value clears the helicity information pP, pM: p± base vectors in (1) formula. The γ-coupled forms of (1) (precise or chiral [26]) are used automatically if available. One may select the transverse unpolarized photon density matrix (instead of usual −(1/2)gµν): ν → −gµν + (P · k) (P · k) kµkνk (P · k)2 . (2) SetPhotonDMBase((int)PID, (str)”P”); PID: photon particle ID in kinematics, "P": Basis vector P in (2). The fermion (k,e) helicity is declared using: SetFermionHelicity((int)PID, (int)h); SetFermionHelicity((int)PID, (srt)”h”); PID: fermion particle ID in kinematics, h: helicity value: ±1 or 0. Zero value clears the helicity information. Density matrix is usual: uū → p̂γ± (massless fermion). "h": symbol for scalar parameter in the following density matrix (massless fermion): uū → (1/2)p̂(1 + hγ5). Notes: • SetPolarized(PID, 0) call will also clear helicity data. • SetXXXHelicity(PID, 1, ...) also sets particle as polarized (previous SetPolarized(PID,0) call is canceled). • SetXXXHelicity(PID, 0, ...) clears helicity data but does not set parti- cle unpolarized. 4.5 Physics selection One can either declare the full Standard Model (in unitary or Feynman gauge) or select the parts of SM Lagrangian to be used. The QED physics is used by default. The Feynman rules corresponds to [27] paper. SetDiagramPhysics( (int)physID ); physID: physics descriptor, to be constructed from the following flags: PHYS_QED: Pure QED interactions, PHYS_Z and PHYS_W: Z- and W-boson vertices, PHYS_MU, PHYS_TAU: Muons and tau leptons (and corresponding neutrinos), PHYS_SCALARS: Scalar particles including Higgs bosons, PHYS_GHOSTS: Faddeev-Popov ghosts, PHYS_GAUGE_UNITARY: Use unitary gauge, PHYS_QUARKS, PHYS_QUARKS_2GEN and PHYS_QUARKS_3GEN: {d, u}, {d, u, s, c} and {d, u, s, c, b, t} sets of quarks, PHYS_RARE_VERICES: vertices with 3 or more SCALAR/GHOST tales, PHYS_CT: Renormalization counter-terms (implementation in progress), PHYS_SM: full SM physics in unitary gauge (all the flags above except for PHYS_CT), PHYS_SM_Q1GEN: the SM physics in unitary gauge with only first generation of quarks (d, u), PHYS_ELW: SM in Feynman gauge with no rare 3-scalar vertice, PHYS_EONLY: no muons, tau-leptons and adjacent neutrinos, PHYS_NOQUARKS: no quarks, PHYS_4BOSONS_ANOMALOUS: anomalous quartic gauge boson interactions (see [28]), affects on AAWW and AZWW vertices. The less items are selected in Lagrangian, the faster diagram generation is performed. 5 Bondarev functions The Bondarev method of trace calculation is implemented according to [25] paper. The trace of γµ-matrices product becomes much shorter in terms of F -functions. The number of items for Tr[(1 − γ5)â1â2 · · · â2n] occurs 2n. For example, the 12 matrices trace contains 10395 items usually while the new method leads to 64-items sum. The 8 complex functions are introduced: F1(a, b) = 2[(aq−)(bq+)− (ae+)(be−)], F2(a, b) = 2[(ae+)(bq−)− (aq−)(be+)], F3(a, b) = 2[(aq+)(bq−)− (ae−)(be+)], F4(a, b) = 2[(ae−)(bq+)− (aq+)(be−)], (3) F5(a, b) = 2[(aq−)(bq+)− (ae−)(be+)], F6(a, b) = 2[(ae−)(bq−)− (aq−)(be−)], F7(a, b) = 2[(aq+)(bq−)− (ae+)(be−)], F8(a, b) = 2[(ae+)(bq+)− (aq+)(be+)]. The basis vectors q± and e± are selected as follows: (1, ±1, 0, 0), eµ± = (0, 0, 1,±i). (4) The results for traces evaluation looks as follows: Tr[(1− γ5)â1â2] = F1(a1, a2) + F3(a1, a2), T r[(1− γ5)â1â2â3â4] = F1(a1, a2)F1(a3, a4) + F2(a1, a2)F4(a3, a4) + +F3(a1, a2)F3(a3, a4) + F4(a1, a2)F2(a3, a4). Please refer to [25] paper for method details. SetParameter(PAR TRACES BONDEREV, (int)par); par: 1 or 0 – allow or forbid Bondarev functions usage. The numerical code for F -functions (3) is contained in alhep_lib.F library file. The code is available for scalar couplings only: Fµνp µqν . If vector Bondarev functions remain in result, the Fortran-generation procedure fails. One should repeat the whole calculation without Bondarev functions in that case. 6 Diagrams generation The diagrams are generated after the kinematics and physics model are de- clared. The Wick-theorem-based method is implemented. The only distinct from the usual Feynman diagrams is the following: the vertex rules have additional 1/2 factors for every identical lines pair. It makes two effects: - The crossing diagrams are usually involved if they have different topology from original. I.e. if two external photon lines starts from single vertex, they are not crossed. Nevertheless all the similar external lines have crossings in ALHEP. - Some ALHEP diagrams have 2n factors due to identical internal lines. For example, the W+W− → {γγWWvertex} → γγ → {γγWWvertex} → W+W− diagram will have additional factor 4. The result remains correct due to 1/2 factors at every γγW+W− vertex. The diagrams are generated without any crossings. The crossed graphs are added automatically during the squaring or amplitude calculation procedures. diagrams ComposeDiagrams((int)n); n: order of diagrams to be created, MX→Y ∼ en. Uses current physics and kinematics information. Returns generated diagrams One may select specified diagrams into another diagrams set: diagrams SelectDiagrams( (diagrams)d, (int)i0 [, i1, i2...]); d: initial diagrams set. Remains unaffected during the procedure. i0..iN: numbers of diagrams to be selected. First diagram is ”0”. The new diagrams set is returned. To retrieve matrix element from the diagram set the RetrieveME() is used. me RetrieveME( (diagrams)d ); d: diagrams list. 7 Helicity amplitudes The amplitudes are calculated according to [24] paper. Every spinor in matrix element is projected to common abstract spinor: ūp = ūQupūp ūQup = eiC ūQPp (Tr[PpPQ])1/2 , up = upūpuQ ūpuQ = eiC (Tr[PQPp])1/2 The eiC factor is equal for all diagrams and may be neglected. The projection operator PQ = upūp is choosen as follows: • PQ = Q̂(1 + γ5)/2 – for massive external fermions, • PQ = Q̂(1 + ÊQ)/2 – if one of fermions is massless. The value of vector Q is selected arbitrary in Fortran numerical procedure. The additional basis vector EQ (if exists) is selected meet the polarization requirements ((EQ.Q) = 0, (EQ.EQ) = −1). The fractions like 1/Tr[PQPp] may turn 1/0 at some Q and EQ values. The denominators check procedures are generated in Fortran code (the CODE_CHECK_DENOMINATORS key should be used in CreateFortranProc() call). If the | Tr[PQPp] |> δ check is failed, the another Q and EQ values are generated. expr CalcAmplitude( (me)ME ); ME: matrix element retrieved from the whole diagrams set. The result expression is a function of all the particle helicity vectors. The averaging over polarization vectors is performed numerically. The numerical averaging procedure is automatically generated in Fortran output if unpolar- izaed particles are declared. The CODE_IS_AMPLITUDE key in CreateFortranProc() procedure declares expression as amplitude and leads to proper numerical code (Ampl×Amlp∗). 8 Matrix Element squaring The squaring procedure have the following steps: • matrix elements simplification to minimize the γ-matrices number, • denominators caching to make procedure faster, • item-by-item squaring (coupling to other conjugated), • saving memory mechanism to avoid huge sums arranging (SQR_SAVE_MEMORY option), • virtual integrals reconstruction. expr SquareME((me)ME1, [(me)ME2,] [(int)flags ]); ME1: matrix element #1, ME2: matrix element #2 (should be omitted for squaring), flags: method options (defauls is ”0”): SQR_CMS: c.m.s. consideration (initial momenta are collinear). The additional pseudo-covariant relations (p1.ε2 → 0, p2.ε1 → 0) appear that simplify work with abstract polarization vectors. SQR_NO_CROSSING_1: do not involve crossings for ME #1, SQR_NO_CROSSING_2: do not involve crossings for ME #2, SQR_MANDELSTAMS: allow Mandelstam variable usage, SQR_PH_GAMMA_CHIRAL: tries to involve photons helicities in short chiral ê± form (see sec. 4.4, [26]). SQR_PH_GAMMA_PRECISE: involve precise ê± form for photon helicities. If the polarization vectors are not coupled to γµ, the vector form e µ is used (see sec. 4.4). SQR_SAVE_MEMORY: save memory and processor time for huge matrix ele- ment squaring. Minimizes sub-results by every 1000 items and skips final arranging of the whole sum. No huge sums occurs in calculation in this mode, but the result is also not minimal. If result expression contains a sum of 105 − 106 items (when expanded), the arranging time is significant and SQR_SAVE_MEMORY flag should be involved. If no results are calculated in reasonable time the CalcAmplitude() (see sec. 7) procedure should be used. If two matrix elements are given, the first will be conjugated, i.e. the result is ME1∗ ×ME2. 9 Virtual integrals operations The tensor virtual integrals are reduced to scalar ones using two methods. If tensor integral is coupled to external momentum Iµp µ and p-vector can be decomposed by integral vector parameters, the fast reduction is involved. The Dx integrals for 2 → 2 process contain the whole basis of 4-dimension space and Dx-couplings to any external momentum can be decomposed. It works well if all the polarization vectors are constructed in terms of external momenta. The common tensor reduction scheme is involved elsewhere. Tensor integrals are decomposed by the vector basis like Iµν (p, q) = I pµpν , qνqµ, p(µqν), gµν The linear system for scalar coefficients is composed and solved. Implemented forBj and Ci,ij integrals only. The other scalar coefficients should be calculated numerically [19]. expr Evaluate( (expr)src ); Evaluate: Reduction of tensor virtual integrals to scalar ones. The scalar coefficients in tensor VI decomposition are also evaluated (not for all integrals). expr ConvertInvariantVI( (expr)src ); ConvertInvariantVI: Vector parameters are substituted by scalars. I.e.: C0(k1, k2, m1, m2, m3) → C0(k21, (k1 − k2)2, k22, m1, m2, m3). Note: A- and B- integrals are converted automatically during arrangement. expr CalcScalarVI( (expr)src ); CalcScalarVI: Substitutes known scalar integrals with its values. The most of UV-divergent integrals (A0,B0) are substituted. The chiral decomposition for D0-integral is also applied. The complicated integrals should be calculated numerically [19]. The source expressions are unaffected in all the functions above. 10 Regularization The dimensional regularization scheme is implemented. One may change the space-time dimension before every operation: SetNDimensionSpace( (int)val ); val: 1 (n-dimensions) or 0 (4-dimensions space). expr SingularArrange( (expr)src ); SingularArrange: turns expression to 4-dimensional form. Calculates (n−4)i factor in every item and drops out all the neglecting contributions. SetDRMassPK( (int)PK ) SetDRMassPK: set the particle to be used as DR mass regulator. PK: particle kind (see sec. 4.1). ”0” value declares the default ”µ” DR mass. 11 ALHEP interfaces 11.1 Fortran numerical code The numerical analysis in particle physics is commonly performed using the Fortran programming language. Hence, we should provide the Fortran code to meet the variety of existing Monte-Carlo generators. To start a new Fortran file the NewFortranFile() function is used: int NewFortranFile( (str)fn [, (int)type]); fn: output file name type: Fortran compiler conventions: CODE_F77 or CODE_F90. The FORTRAN 77 conventions are presumed by default. Function returns the ID of created file. Then we can add a function to FORTAN file: CreateFortranProc( (int)fID, (str)name, (expr)src [, (int)keys]); fID: file ID returned by NewFortranFile() call. name: Fortran function name. src: symbolic expression to be calculated. The source may be | M |2, dσ/dΓ (use CODE_IS_DIFF_CS flag) or amplitude (CODE_IS_AMPLITUDE flag). The result is always dσ/dΓ calculation procedure. keys: option flags for code generation (default is CODE_REAL8): CODE_REAL8: mean all the symbols in expression as REAL*8 values. CODE_COMPLEX16: declare variables type as COMPLEX*16. CODE_IS_DIFF_CS: the source expression is differential cross section. CODE_IS_AMPLITUDE the source expression is amplitude and a squaring code should be generated: AMPL*DCONJG(AMPL). CODE_CHECK_DENOMINATORS check denominators for zero. Used to re-generate free basis vectors in amplitude code (see sec. 7). CODE_LOOPTOOLS: Create LoopTools [19] calls for virtual integrals. CODE_SEPARATE_VI create unique title for every virtual integral function (ac- cording to parameter values). Do not use with CODE_LOOPTOOLS. CODE_PYTHIA_VECTORS retrieve vector values from PYTHIA [18] user process PUP(I,J) array. CODE_POWER_PARAMS factorize and pre-calculate powers if possible. CODE_NO_4VEC_ENTRY do not create a 4-vector entry for function. CODE_NO_CONSTANTS do not use predefined physics constants. All variables becomes external parameters. CODE_NO_SPLIT do not split functions by 100 lines. CODE_NO_COMMON: don’t use CONNON-block to keep internal variables. The scalar vector couplings, Bondarev functions (see sec. 5) and εabcdp objects are replaced with scalar parameters to be calculated once. These func- tions are calculated using alhep_lib.F library procedures. Some compilers works extremely slow with long procedures. Therefore Fortran functions are automatically splitted after every 100 lines for faster compilation. The internal functions functions have TEMPXXX() notation. To avoid problems with several ALHEP-genarated files linking we should rename the TEMP-prefix to make internal functions unique. SetParameterS(PAR STR FORTRAN TEMP, (str)prefix); prefix: "TMP1" or another unique prefix for temporary functions name. To obtain the better performance of numerical calculations ALHEP provides the mechanism for minimization of +/× operations in the expression. We recommend to invoke the Minimize() function before Fortran generation (see sec. 12). 11.1.1 PYTHIA interface The PYTHIA [18] interface is implemented in terms of UPINIT/UPEVNT proce- dures and 2 → N phase-space generator. The momenta of external particles are retrieved from PYTHIA user process event common block (PUP(I,J)). The order of particles in kinematics should meet the order of generated particles, and the CODE_PYTHIA_VECTORS option should be used. The template UPINIT/UPEVNT procedures for ALHEP → PYTHIA junction are found at ALHEP website. However one should modify them by adjusting the generated particles sequence, including user cutting rules, using symmetries for calculation similar processes in single function etc. The plane 2 → N phase-space generator in alhep_lib.F library file is written by V. Mossolov. It is desirable to replace the plain phase-space generator by the adaptive one for multiparticle production process. The more automation will be implemented in future. Please refer to ALHEP website for details. 11.2 Mathematica The output interface to Mathematica Notebook [16] file is basically used to view the expressions in the convenient form. Implemented for all the symbolic objects in ALHEP. SaveNB( (str)fn, (expr or me)val [, (str)comm ][, (int)flags ]); MarkNB( (str)fn [, (str)comm ][, (int)flags ]); fn: Mathematica output file name. val: expression to be stored. comm: comment text to appear in output file ("" if no comments are required). flags: file open flags (default is ”0”): 0: append to existing file and do not close it afterward, FILE_START: delete previous, start new file and add Mathematica header, FILE_CLOSE: add closing Mathematica block. The MarkNB function is used to add comments only. The Mathematica program can open valid files only. The valid x.nb file should be started (FILE_START) and closed once (FILE_CLOSE). It is convenient to in- sert MarkNB("x.nb","",FILE_START) and MarkNB("x.nb","",FILE_CLOSE) calls to start and end of your script file. No backward interface (Mathematica → ALHEP) is currently available. 11.3 LaTeX The LaTeX interface in ALHEP is only implemented for Feynman diagrams drawing. The diagrams are illustrated in terms of AxoDraw [23] package. DrawDiagrams( (diagrams)d, (str)fn [, (int)flags][, (int)draw]); MarkTeX( (str)fn, [, (str)comm][, (int)flags]); d: diagrams set. fn: output LaTeX file name. flags: file open flags (0(default): append, FILE_START: truncate old): comm: comment text to appear in output file ("" if no comments are required). draw: draw options (default is DD_SMALL): DD_SMALL: diagrams with small font captions. DD_LARGE: diagrams with large font captions. DD_MOMENTA: print particles momenta. DD_SWAP_TALES: allow arbitrary order for final-state lines (option is included automatically for 2 → N kinematics). DD_DONT_SWAP_TALES: deny the DD_SWAP_TALES option. The order of final lines is same to kinematics declaration. 11.4 ALHEP native save/load operations The native ALHEP serialization format is XML-structured. It may be edited outside the ALHEP for some debug purposes. Save( (str)fn, (diagrams or expr)val); object Load( (str)fn ); fn: output XML file name. val: diagrams set or symbolic expression to be stored. 12 Common algebra utilities The following set of common symbolic operations is available: • Expand(): expands all the brackets and arranges result. Works slowly with huge expressions. • Arrange(): arranges expression (makes alphabetic order in commutative se- quences). The most of ALHEP functions performs arranging automatically and there is no need to call Arrange() directly. • Minimize(): reduces the number of ”sum-multiply” operations in expres- sion. Should be used for simplification of big sums before numerical calcu- lations. • Factor(): factorize expression. • KinArrange(): arranges expression using kinematic relations, • KinSimplify(): simplify expression using the kinematic relations. Works very slowly with large expressions. Mostly useful for 2 → 2 process. expr Expand( (expr)src ); expr Arrange( (expr)src ); expr Minimize( (expr)src [, (int)flags]); expr Factor( (expr)src [, (int)flags]); expr KinArrange( (expr)src [, (int)flags]); expr KinSimplify( (expr) src [, (int)flags]); src: source expression (remains unaffected). Minimize() function flags (default is MIN_DEFAULT): MIN_DEFAULT = MIN_FUNCTIONS|MIN_DENS|MIN_NUMERATORS, MIN_FUNCTIONS: factorize functions, MIN_DENS: factorize denominators, MIN_NUMERATORS: factorize numerators, MIN_NUMBERS: factorize numbers, MIN_ALL_DENOMINATORS: factorize all denominators 1/(a + b) + 1/x → (x+ a+ b)/(x ∗ (a+ b))), MIN_ALL_SINGLE_DENS: factorize single denominators (but not sums, prod- ucts etc.), MIN_VERIFY: verify result (self-check: expand result back and compare to source). Factor() function flags (default is 0): FACT_NO_NUMBERS: do not factorize numbers, FACT_NO_DENS: do not factorize fraction denominators, FACT_ALL_DENS: factorize all denominators, FACT_ALL_SINGLE_DENS: factorize all single denominators (but not sums, products etc.), FACT_VERIFY: verify result (self-check: expand result back and compare to source). KinArrange() function flags (default is 0): KA_MASS_EXACT: do not truncate masses (neglecting SetMassCalcOrder() settings), KA_MANDELSTAMS: involve Mandelstam variables (for 2 → 2 kinematics), KA_NO_EXPAND: do not expand source. KinSimplify() function flags (default is KS_FACTORIZE_DEFAULT): KS_FACTORIZE_DEFAULT: factorize functions and denominators (the first two flags below), KS_FACTORIZE_FUNCTIONS: factorize functions before simplification, KS_FACTORIZE_DENS: factorize denominators (including partial factoriza- tion) before simplification, KS_FACTORIZE_ALL_DENS: factorize all denominators, KS_MASS_EXACT: do not truncate masses (neglecting SetMassCalcOrder() settings), KS_MANDELSTAMS: involve Mandelstam variables (for 2 → 2 kinematics), KS_NO_EXPAND: do not expand source (if no simplification are found). 13 Debugging tools ALHEP allows user to control the most of internal calculation flow. The debug info is stored to debug.nb file (critical messages will also appear in console). The debug.ini file contains numerical criteria for messages to be logged, the debug levels for different internal classes. Warning: raising the debug.ini val- ues leads to sufficient performance drop and enormous debug.nb file growth. If one feels some problems with ALHEP usage, please contact author for assis- tance (attaching your script file). Do not waste the time for manual debugging using debug.ini. There are some specific commands to view internal data. For example, the whole list of tensor virtual integrals reduction results is kept in internal data storage and can be dumped to Mathematica file for viewing: • str ViewParticleData((int)PID) returns the brief information on parti- cle settings. Puts the information string to console and returns it also as a string variable. PID: particle ID in kinematics. • ViewFeynmanRules((str)nb_file, (int)flags) stores Feynman rules of current physics to Mathematica file. • ViewTensorVITable((str)nb_file, (int)flags)) stores tensor integrals reduction table to Mathematica file. The VI reduction table is filled during the Evaluate() operation. • ViewScalarVICache((str)nb_file, (int)flags)) stores scalar loop in- tegrals values cache. The scalar VI cache is filled during the CalcScalarVI() invocation. nb file: output Mathematica file name, flags: access flags to Mathematica file: 0,FILE_START and/or FILE_CLOSE. Call without parameters turns output to debug.nb file. 14 System commands The two system commands are useful: • Halt(): stop further script processing. May be used to test the first part of script and save (Save()) internal result. It first part finished successfully, it may be commented (\*...*\) and followed by loading procedure (Load()). Then script execution is restarted. • Timer(): view time elapsed since the last Timer() call (from program start for first call). 15 Examples 15.1 Amplitude for qq̄ → W+W−γ Let’s consider the example from sec. 2.1 in details. We also extend it for anoma- lous quartic gauge boson interactions [28]. And we don’t use the Bondarev method for traces calculation this time. Please refer to sec. 2.1 for ALHEP installation notes. We start test.al script from output files creation: nbfile = "uuWWA_MPXXX.nb"; // Mathenatica file name MarkNB(nbfile, "", FILE_START); // Create file texfile = "res.tex"; // LaTeX file name MarkTeX(texfile, "", FILE_START); // Create file Then 2 → 3 process kinematics and physics are declared: u(p1, e1) ū(p2, e2) → γ(f0, g0) W+(f1, g1) W−(f2, g2). (5) SetKinematics(2, 3 // 2->3 process ,QUARK_U,"p\_1","e\_1" // u ,-QUARK_U,"p\_2","e\_2" // u-bar ,WBOZON,"f\_1","g\_1" // W{+} ,PHOTON,"f\_0","g\_0" // photon ,-WBOZON, "f\_2", "g\_2" ); // W{-} SetDiagramPhysics(PHYS_SM_Q1GEN|PHYS_4BOSONS_ANOMALOUS); We declare physics with u- and d-quarks only. The amplitude will be sum- marized for all the possible internal quarks numerically. It requires the simple replacing of quark mixing matrix in resulting Fortran code: U2ud → U2ud+U2us+ U2ub). Next we declare the u- and d-quarks massless: SetMassCalcOrder(QUARK_U, 0); // consider massless SetMassCalcOrder(QUARK_D, 0); // consider massless Set polarizations to ”-+UUU”: SetFermionHelicity(1, -1); // u SetFermionHelicity(2, 1); // u-bar Create diagrams set and store it to LaTeX file: diags = ComposeDiagrams(3); //e^3 order DrawDiagrams(diags, texfile); Next we include the following lines: Save("diags.xml",diags); //save to XML file //Halt(); //stop execution //diags = Load("diags.xml"); //load from XML file We can save diagrams, stop the program now and view diagrams generated. To stop ALHEP session the Halt() line should be uncommented. Then we modify our script as follows: Fig. 1. The diagrams generated for uū → W−W+γ process (see res.tex file). The anomalous quartic gauge boson interaction affects the first two diagrams. /* diags = ComposeDiagrams(3); // commented ... // commented Halt(); */ // commented diags = Load("diags.xml"); // uncommented If we run the script again, it will skip the diagrams generation step and load diagrams from XML file. Matrix element retrieval: me = RetrieveME(diags); //get matrix element SaveNB(nbfile, me, "Matrix element"); //view Calculate helicity amplitude, arrange result and minimize the +/× operations number: ampl = CalcAmplitude(me); SaveNB(nbfile, ampl, "Amplitude after CalcAmplitude()"); ampl = KinArrange(ampl); SaveNB(nbfile, ampl, "Amplitude after KinArrange()"); ampl = Minimize(ampl); SaveNB(nbfile, ampl, "Amplitude after Minimize()"); The another breakpoint can be inserted here. The result for amplitude is saved, the Halt and Load commands are commented for further use: Save("ampl.xml", ampl); // save amplitude //MarkNB(nbfile, FILE_CLOSE); Halt(); // close NB and exit //ampl = Load("ampl.xml"); // load amplitude This breakpoint allows to repeat the next Fortran creation step without re- calculating of matrix element. Let’s average over final state polarizations in further numerical procedure. Set final particles unpolarized: SetPolarized(-1, 0); // set unpolarized SetPolarized(-2, 0); // set unpolarized SetPolarized(-3, 0); // set unpolarized The Fortran output for differential cross section: SetParameterS(PAR_STR_FORTRAN_TEMP, "TMP1"); f = NewFortranFile("uuWWA.F", CODE_F77); //f77 file CreateFortranProc(f, "uuWWA", ampl, CODE_IS_AMPLITUDE| //square amplitude CODE_CHECK_DENOMINATORS| //check 1/0 limits CODE_COMPLEX16| //complex values CODE_POWER_PARAMS| //F(M^2) instead of F(M) CODE_PYTHIA_VECTORS); //use PYTHIA PUP(I,J) vectors The SetParameterS call sets the unique notation for internal variables and functions. Please do not make it too long. The complex-type code is required for proper amplitude calculation. Close Mathematica output file at the end of script: MarkNB(nbfile, FILE_CLOSE); The execution of this script takes less than 2 minutes at 1.8GHz P4 processor. We will not discuss the structure of generated uuWWA.F file in details. But some remarks should be done: Line 5: The main function call. The following parameters are declared (order is changed here): All the parameters (except the kQOrig) are of COMPLEX*16 type. Ones the CODE_COMPLEX16 option is set, all the real objects are treated as complex. kQOrig (INTEGER): The ID of u(first)-quark in PYTHIA PUP(I,J) array. Possible values: 1 or 2. PAR a 0, PAR a c, PAR a n, PAR ah c, PAR ahat n: Anomalous quartic gauge boson interaction constants a0, ac, an, âc, ân [28]. PAR CapitalLambda: Scale factor Λ for anomalous interaction [28]. PAR VudP2: Quark mixing matrix element squared |Uud|2. The U2ud +U2us + U2ub value may be passed to summarize the whole diagrams (neglecting quarks masses). The numbers for mixing matrix elements may be obtained using QMIX_VAL(ID1,ID2), QMIX_SQR_SUM(ID) and QMIX_PROD_SUM func- tions of alhep_lib.F library. Line 29: Internal COMMON-block with PAR(XX) array. All the scalar cou- plings and other compound objects are precalculated and stored in PAR(XX). Lines 49-55: External momenta initialization from PYTHIA PUP(I,J) ar- ray. The order of external vectors is expected as follows: PUP(I,1),PUP(I,2): initial particles. If kQOrig=2 the order is backward: PUP(I,2), PUP(I,1). PUP(I,3..N): final particles in the same order as in SetKinematics() call. One should modify this section (or SetKinematics() parameters) to make the proper particles order. DO 10 I=1,4 p_1(I) = DCMPLX(PUP(I,kQ1Orig),0D0) // kQ1Orig = kQOrig f_0(I) = DCMPLX(PUP(I,4),0D0) p_2(I) = DCMPLX(PUP(I,kQ2Orig),0D0) // kQ2Orig = 3-kQOrig f_1(I) = DCMPLX(PUP(I,3),0D0) f_2(I) = DCMPLX(PUP(I,5),0D0) 10 CONTINUE Lines 69-231: Polarization averaging and basis vector generation cycle. For any momenta set the PAR(XXX) array is filled. Then denominator checks and amplitude averaging are performed. Lines 244-252, 438-444, ... Interaction constants and particle masses defi- nitions in sub-procedures. The constants can be declared as main function parameters using CODE_NO_CONSTANTS option in CreateFortranProc func- tion. For the complete pp̄ → W+W−γ analysis the following steps are required: • The PYTHIA client program should be written. The template files are avail- able at ALHEP website. • The another helicity configuration +-UUU should be calculated separately. • The another channels qiq̄j → W+W−γ (i 6= j) should be calculated and included into generator. 15.2 Z-boxes for e−e+ → µ−µ+ Let’s calculate some box diagrams now. Consider the following process: e−(p1, e1) e +(p2, e2) → µ−(f1, g1) µ+(f2, g2). (6) As in previous example, we start command script from files initialization: nbfile = "Zbox.nb"; // Mathenatica file name MarkNB(nbfile, "", FILE_START); // Create file texfile = "res.tex"; // LaTeX file name MarkTeX(texfile, "", FILE_START); // Create file The 2 → 2 kinematics declaration: SetKinematics(2, 2, // 2->2 process ELECTRON, "p\_1", "e\_1" , // e^{-} -ELECTRON, "p\_2", "e\_2", // e^{+} MUON, "f\_1", "g\_1", // mu^{-} -MUON, "f\_2", "g\_2"); // mu^{+} The Standard model physics in Feynman gauge (we omit quarks for faster diagrams generation): SetDiagramPhysics(PHYS_ELW|PHYS_NOQUARKS); Set leptons massless and declare the N -dimensional space: SetMassCalcOrder(ELECTRON, 0); // massless electrons SetMassCalcOrder(MUON, 0); // massless muons SetNDimensionSpace(1); Use Mandelstam variables throughout the calculation: SetParameter(PAR_MANDELSTAMS,1); Consider unpolarized particles: SetPolarized(1, 0); // unpolarized e^{-} SetPolarized(2, 0); // unpolarized e^{+} SetPolarized(-1, 0); // unpolarized mu^{-} SetPolarized(-2, 0); // unpolarized mu^{+} Compose born and one-loop diagrams: diags_born = ComposeDiagrams(2); //e^2 order DrawDiagrams(diags_born, texfile); diags_loop = ComposeDiagrams(4); //e^4 order Save("diags_loop.xml",diags_loop); //diags_loop=Load("diags_loop.xml"); DrawDiagrams(diags_loop, texfile); The 220 loop diagrams are created, saved to internal format and TeX file. The ComposeDiagrams(4) procedure takes 5-10 minutes here. It is convenient to comment ComposeDiagrams(4)-Save() lines at second run and Load loop diagrams from disk. Fig. 2. Born level diagrams for e−e+ → µ−µ+. Fig. 3. Part of 220 loop diagrams stored to res.tex file Let’s select the double Z-exchange box graphs from the whole set (194 and 195 diagrams at fig. 3): diag_box = SelectDiagrams(diags_loop,194,195); Next we couple the loop and born matrix elements: me_born = RetrieveME(diags_born); me_box = RetrieveME(diag_box); me_sqr = SquareME(me_box, me_born); The simplification procedures are not included into SquareME implementation. It may take much time to arrange items in huge expression. Therefore all the simplification procedures are optional and should be called manually: me_sqr = KinArrange(me_sqr); me_sqr = KinSimplify(me_sqr); SaveNB(nbfile, me_sqr, "squared & simplified"); The reduction of tensor virtual integrals follows: me_sqr = Evaluate(me_sqr); me_sqr = KinArrange(me_sqr); me_sqr = KinSimplify(me_sqr); SaveNB(nbfile, me_sqr, "VI evaluated"); Next we convert scalar integrals to invariant-dependent form and replace with tabulated values: me_sqr = ConvertInvariantVI(me_sqr); me_sqr = CalcScalarVI(me_sqr); // use pre-calculated values me_sqr = KinArrange(me_sqr); SaveNB(nbfile, me_sqr, "VI scalars "); Turn to 4-dimensional space, drop out (n− 4)i items and final simplification: me_sqr = SingularArrange(me_sqr); SetNDimensionSpace(0); me_sqr = KinArrange(me_sqr); me_sqr = KinSimplify(me_sqr); Save result and create Fortran code with LoopTools [19] interface: Save("ZBox.xml",me_sqr); // save result //me_sqr = Load("ZBox.xml"); // reload result SaveNB(nbfile, me_sqr, "Z boxes result"); // view result f = NewFortranFile("ZBOX.F", CODE_F77); CreateFortranProc(f, "ZBOX", me_sqr, CODE_POWER_PARAMS|CODE_LOOPTOOLS); View tensor integrals reduction table and close Mathematica output file: ViewTensorVITable(nbfile); MarkNB(nbfile, FILE_CLOSE); The script runs about 15 minutes on 1.8GHz P4 processor. The half of this time takes the ComposeDiagrams(4) procedure. ������� jjjjjjjj jjjjjjjj 2,t,me 2,s,me 2,mZ,m�,me,mZ< jjjjjjjj jjjjjjjj jjjjjjjj jjjjjjj ����������������������HmZ2 − sL JmZ 3N s − JmZ2 t2 + mZ4 t + 3N e2 zzzzzzz gm ����������������������HmZ2 − sL s t zzzzzzzz zzzzzzzz jjjjjjjj ����������������������HmZ2 − sL JmZ 3N s − JmZ2 t2 + mZ4 t + 1���� 3N e2 zzzzzzzz zzzzzzzz 2,s,m ,mZ,mZ< jjjjjjjj jjjjjjjj J 1���� ����������������������HmZ2 − sL s − J uN e2 zzzzzzzz jjjjjjj jjjjjjj jjjjjjjJ ����������������������HmZ2 − sL s − J uN e2 zzzzzzz gm HmZ2 + 1����2 s + uL ������������������������������������HmZ2 − sL s zzzzzzz gm 2 JmZ2 + 1���� s + uN e2 zzzzzzz gm zzzzzzzz 2,s,me 2,me,mZ,mZ< jjjjjjjj jjjjjjjj J 1���� ����������������������HmZ2 − sL s − J uN e2 zzzzzzzz jjjjjjj jjjjjjj jjjjjjjJ ����������������������HmZ2 − sL s − J uN e2 zzzzzzz gm HmZ2 + 1����2 s + uL ������������������������������������HmZ2 − sL s zzzzzzz gm 2 JmZ2 + 1���� s + uN e2 zzzzzzz gm zzzzzzzz s + D0 2,u,me 2,s,me 2,mZ,m�,me,mZ< jjjjjjjj jjjjjjj jjjjjjJ2 mZ t + 2 mZ ����������������������HmZ2 − sL s − J2 mZ t + 2 mZ 3N e2 zzzzzz jjjjjjj ����������������������HmZ2 − sL s − e zzzzzzz gm zzzzzzz gm jjjjjjjj ����������������������HmZ2 − sL s − e zzzzzzzz zzzzzzzz 2,t,me 2,mZ,m HmZ2 + 1����2 t − 1����2 uL �������������������������������������������HmZ2 − sL s − JmZ uN e2y zzz g HmZ2 + 1����2 t − 1����2 uL �������������������������������������������HmZ2 − sL s − JmZ uN e2y zzz g zzz t − 8s,mZ,mZ< jjjjjjjj jjjjjjjj jjjjjjjj jjjjjjj ����������������������HmZ2 − sL s − e zzzzzzz gm ����������������������HmZ2 − sL s t zzzzzzzz zzzzzzzz jjjjjjjj ����������������������HmZ2 − sL s − e zzzzzzzz zzzzzzzz 2,u,me 2,mZ,m�,me< i jjjjjjH2 mZ + s + 2 uL g ����������������������HmZ2 − sL s − H2 mZ + s + 2 uL e2 zzzzzz g LogA 1������� jjjjjjjj jjjjjjj ����������������������HmZ2 − sL s − e zzzzzzz gm jjjjjjjj ����������������������HmZ2 − sL s − e zzzzzzzz zzzzzzzz LogA 1������� jjjjjjg ����������������������HmZ2 − sL s − e zzzzzz g jjjjjjjj H2 + χL g jjjjjjjj H2 + χL ����������������������HmZ2 − sL s t − jjjjjjjJ1 + ����������������������HmZ2 − sL s − J1 + χN e2 zzzzzzz gm zzzzzzzz zzzzzzzz jjjjjjjj J1 + 1���� ����������������������HmZ2 − sL s + e 2 J−J1 + 1���� zzzzzzzz zzzzzzzz zzzzzzzz Fig. 4. The result expression in Zbox.nb file. This result contains UV-regulator term χ, that should cancel one in B0(s,MZ ,MZ) integral. It can be checked using GetUVTerm() function. Code remarks: Line 5: The main function call. The 3 parameters are usual Mandelstam variables (s, t, u, type is complex). The current ALHEP version does not care about interdependent parameters in Fortran output. And all the three Mandelstam variables may occur in parameters list. The future versions will be saved from this trouble. Line 14,50: Include LoopTools header file ("looptools.h"). See LoopTools manual [19] for details. Line 20,21: Retrieve LoopTools values for UV-regulator getdelta() and DR-mass squared getmudim(). The complete code of examples including scripts, batches and output files are available at ALHEP website. 16 Conclusions The new program for symbolic computations in high-energy physics is pre- sented. In spite of several restrictions remained in current version, it can be useful for computation of observables in particle collision experiments. It con- cerns both multiparticle production amplitudes and loop diagrams analysis. The nearest projects are: • Bondarev functions method improvement, • Complete renormalization scheme for SM, • Complete covariant analysis of the one-loop radiative corrections including the hard bremsstrahlung scattering contribution. • Arbitrary Lagrangian assignment. Refer ALHEP project websites for program updates: http://www.hep.by/alhep , http://cern.ch/~makarenko/alhep . References [1] CompHEP, http://theory.sinp.msu.ru/comphep, E. Boos et al., Nucl.Instrum.Meth. A534 (2004) 250, hep-ph/0403113. [2] SANC, http://sanc.jinr.ru/ (or http://pcphsanc.cern.ch/), A. Andonov et al., hep-ph/0411186, to appear in Comp.Phys.Comm.; D. Bardin, P. Christova, L. Kalinovskaya, Nucl.Phys.Proc.Suppl. B116 (2003) 48. [3] GRACE, http://minami-home.kek.jp/, G. 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[28] A.Denner, S.Dittmaier, M.Roth,D.Wackeroth, Eur.Phys.J.C 20 (2001) 201. http://www.feynarts.de/looptools/ http://arxiv.org/abs/hep-ph/9807565 http://arxiv.org/abs/hep-ph/0005029 http://www.feynarts.de/ http://arxiv.org/abs/hep-ph/0012260 http://arxiv.org/abs/hep-ph/0105349 http://www-ttp.physik.uni-karlsruhe.de/whizard/ http://arxiv.org/abs/hep-ph/0511324 http://arxiv.org/abs/hep-ph/0504223 Introduction ALHEP Review. Program Structure Getting Started Calculation scheme ALHEP script language Initialization section Particles Kinematic selection Particle masses Polarization data Physics selection Bondarev functions Diagrams generation Helicity amplitudes Matrix Element squaring Virtual integrals operations Regularization ALHEP interfaces Fortran numerical code Mathematica LaTeX ALHEP native save/load operations Common algebra utilities Debugging tools System commands Examples Amplitude for q "7016q W+ W- Z-boxes for e- e+ - + Conclusions References

0704.1840

Simulations of aging and plastic deformation in polymer glasses

APS/123-QED Simulations of Aging and Plastic Deformation in Polymer Glasses Mya Warren∗ and Jörg Rottler Department of Physics and Astronomy, The University of British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1, Canada (Dated: August 14, 2018) We study the effect of physical aging on the mechanical properties of a model polymer glass using molecular dynamics simulations. The creep compliance is determined simultaneously with the structural relaxation under a constant uniaxial load below yield at constant temperature. The model successfully captures universal features found experimentally in polymer glasses, including signatures of mechanical rejuvenation. We analyze microscopic relaxation timescales and show that they exhibit the same aging characteristics as the macroscopic creep compliance. In addition, our model indicates that the entire distribution of relaxation times scales identically with age. Despite large changes in mobility, we observe comparatively little structural change except for a weak logarithmic increase in the degree of short-range order that may be correlated to an observed decrease in aging with increasing load. PACS numbers: 81.40.Lm, 81.40.Lg, 83.10.Rs I. INTRODUCTION Glassy materials are unable to reach equilibrium over typical experimental timescales [1, 2, 3]. Instead, the presence of disorder at temperatures below the glass tran- sition permits only a slow exploration of the configura- tional degrees of freedom. The resulting structural re- laxation, also known as physical aging [4], is one of the hallmarks of glassy dynamics and leads to material prop- erties that depend on the wait time tw since the glass was formed. While thermodynamic variables such as energy and density typically evolve only logarithmically, the re- laxation times grow much more rapidly with wait time [3, 4, 5]. Aging is a process observed in many different glassy systems, including colloidal glasses [6], microgel pastes [7], and spin glasses [8], but is most frequently studied in polymers due to their good glass-forming ability and ubiquitous use in structural applications. Of particular interest is therefore to understand the effect of aging on their mechanical response during plastic deformation [5]. In a classic series of experiments, Struik [4] studied many different polymer glasses and determined that their stiff- ness universally increases with wait time. However, it has also been found that large mechanical stimuli can alter the intrinsic aging dynamics of a glass. Cases of both decreased aging (rejuvenation) [4] and increased aging (overaging) [9, 10] have been observed, but the interpre- tation of these findings in terms of the structural evolu- tion remains controversial [11, 12]. The formulation of a comprehensive molecular model of the non-equilibrium dynamics of glasses has been im- peded by the fact that minimal structural change oc- curs during aging. Traditional interpretations of aging presume that structural relaxation is accompanied by a ∗Electronic address: [email protected] decrease in free volume available to molecules and an as- sociated reduction in molecular mobility [4]. While this idea is intuitive, it suffers from several limitations. First, the free volume has been notoriously difficult to define ex- perimentally. Also, this model does not seem compatible with the observed aging in glassy solids under constant volume conditions [13], and cannot predict the aging be- havior under complex thermo-mechanical histories. Mod- ern energy landscape theories describe the aging process as a series of hops between progressively deeper traps in configuration space [14, 15]. These models have had some success in capturing experimental trends, but have yet to directly establish a connection between macroscopic ma- terial response and the underlying molecular level pro- cesses. Recent efforts to formulate a molecular theory of aging are promising but require knowledge of how local density fluctuations control the relaxation times in the glass [16]. Molecular simulations using relatively simple models of glass forming solids, such as the binary Lennard-Jones glass [17] or the bead spring model [18] for polymers, have shown rich aging phenomenology. For instance, calculations of particle correlation functions have shown explicitly that the characteristic time scale for particle relaxations increases with wait time [19]. Recent work [13, 20] has focused on the effect of aging on the mechan- ical properties; results showed that the shear yield stress (defined as the overshoot or maximum of the stress-strain curve) in deformation at constant strain rate generally in- creases logarithmically with tw. Based on a large number of simulations at different strain rates and temperatures, a phenomenological rate-state model was developed that describes the combined effect of rate and age on the shear yield stress for many temperatures below the glass tran- sition [21]. In contrast to the strain-controlled studies described above, experiments on aging typically impose a small, constant stress and measure the resulting creep as a func- tion of time and tw [4]. In this study, we perform molecu- http://arxiv.org/abs/0704.1840v1 mailto:[email protected] lar dynamics simulations on a coarse grained, glass form- ing polymer model in order to investigate the relation- ship between macroscopic creep response and microscopic structure and dynamics. In Section IIIA, we determine creep compliance curves for different temperatures and applied loads (in the sub-yield regime) and find that, as in experiments, curves for different ages can be super- imposed by rescaling time. The associated shift factors exhibit a power-law dependence on the wait time, and the effect of aging can be captured by an effective time as originally envisioned by Struik [4]. In Section III B, we compute microscopic mobilities and the full spectrum of relaxation times and show their relationship to the creep response. Additionally, we study several parameters that are sensitive to the degree of short-range order in Section III C. We detect very little evolution toward increased local order in our polymer model, indicating that short range order is not a sensitive measure of the mechanical relaxation times responsible for the creep compliance of glassy polymers. II. SIMULATIONS We perform molecular dynamics (MD) simulations with a well-known model polymer glass on the bead- spring level. The beads interact via a non-specific van der Waals interaction given by a 6-12 Lennard-Jones po- tential, and the covalent bonds are modeled with a stiff spring that prevents chain crossing [22]. This level of modeling does not include chemical specificity, but al- lows us to study longer aging times than a fully atomistic model and seems appropriate to examine a universal phe- nomenon found in all glassy polymers. All results will be given in units of the diameter a of the bead, the mass m, and the Lennard-Jones energy scale, u0. The characteris- tic timescale is therefore τLJ = ma2/u0, and the pres- sure and stress are in units of u0/a 3. The Lennard-Jones interaction is truncated at 1.5a and adjusted vertically for continuity. All polymers have a length of 100 beads, and unless otherwise noted, we analyze 870 polymers in a periodic simulation box. Results are obtained either with one large simulation containing the full number of poly- mers, or with several smaller simulations, each starting from a unique configuration, whose results are averaged. The large simulations and the averaged small simulations provide identical results. The small simulations are used to estimate uncertainties caused by the finite size of the simulation volume. To create the glass, we begin with a random dis- tribution of chains and relax in an ensemble at con- stant volume and at a melt temperature of 1.2u0/kB. Once the system is fully equilibrated, it is cooled over 750τLJ to a temperature below the glass transition at Tg ≈ 0.35u0/kB [18]. The density of the melt is cho- sen such that after cooling the pressure is zero. We then switch to an NPT ensemble - the pressure and temperature are controlled via a Nosé-Hoover thermo- 750 2250 7500 22500 75000 500 1500 5000 15000 50000 FIG. 1: Simulated creep compliance J(t, tw) at a glassy tem- perature of T = 0.2u0/kB for various wait times tw (indi- cated in the legend in units of τLJ ). A uniaxial load of (a) σ = 0.4u0/a 3 and (b) σ = 0.5u0/a 3 is applied to the aged glasses. The strain during creep is monitored over time to give the creep compliance. stat/barostat - with zero pressure and age for various wait times (tw) between 500 to 75,000τLJ. The aged samples undergo a computer creep experiment where a uniaxial tensile stress (in the z-direction) is ramped up quickly over 75τLJ , and then held constant at a value of σ, while the strain ǫ = ∆Lz/Lz is monitored. After an initial elastic deformation, the glass slowly elongates in the direction of applied stress due to structural relax- ations. In the two directions perpendicular to the applied stress, the pressure is maintained at zero. III. RESULTS A. Macroscopic Mechanical Deformation Historically, measurements of the creep compliance have been instrumental in probing the relaxation dynam- ics of glasses, and continue to be the preferred tool in investigating the aging of glassy polymers [15, 23, 24]. In his seminal work on aging in polymer glasses, Struik [4] performed an exhaustive set of creep experiments on dif- ferent materials, varying the temperature and the applied load. In this section, we perform a similar set of exper- iments with our model polymer glass. The macroscopic creep compliance is defined as J(t, tw) = ǫ(t, tw) . (1) short time long time FIG. 2: The same data as Fig. 1 is shown with the curves shifted by aJ (tw) to form a master curve. The dashed lines are fits to the master curves using the effective time formulation, and the dotted line is a short-time fit for comparison (see text). Compliance curves J(t, tw) for several temperatures and stresses were obtained as a function of wait time since the quench; representative data is shown in Figure 1. The curves for different wait times appear similar and agree qualitatively with experiment. An initially rapid rise in compliance crosses over into a slower, logarith- mic increase at long times. The crossover between the two regimes increases with increasing wait time. Struik showed that experimental creep compliance curves for different ages can be superimposed by rescaling the time variable by a shift factor, aJ , J(t, tw) = J(taJ , t w). (2) This result is called time-aging time superposition [4, 5]. Simulated creep compliance curves from Fig. 1 can simi- larly be superimposed, and the resulting master curve is shown in Fig. 2. Shift factors required for this data collapse are plot- ted versus the wait time in Fig. 3. All data fall along a straight line in the double-logarithmic plot, clearly indi- cating power law behavior: w . (3) This power law in the shift factor is characteristic of ag- ing. µ is called the aging exponent, and has been found experimentally to be close to unity for a wide variety of glasses in a temperature range near Tg [4]. Figure 4 shows the effect of stress and temperature on the aging exponent, as determined from linear fits to the data in Fig. 3. At T = 0.2u0/kB, µ is close to one for small stresses, but decreases strongly with stress. This apparent erasure of aging by large mechanical deforma- tions has been called “mechanical rejuvenation” [25]. Ex- periments have frequently found a stress dependence of 3 4 5 3 4 5 σ = 0.2 σ = 0.4 σ = 0.5 σ = 0.1 σ = 0.2 σ = 0.4 T = 0.2 T = 0.3 FIG. 3: Plot of the shift factors found by superimposing the creep compliance curves, aJ (circles), the mean-squared dis- placement curves, aMSD (triangles), and the incoherent scat- tering function curves, aC (×) at different wait times (see text). The solid lines are linear fits to the data. the aging exponent [4], although it is not always the case that the aging process slows down with applied stress; stress has been known to increase the rate of aging in some circumstances as well [9, 10]. The structural ori- gins of this effect are not well understood [11, 12]. At T = 0.3u0/kB, we find that the aging exponent is somewhat smaller than at T = 0.2u0/kB and varies much less with applied stress. This behavior is most likely due to the fact that the temperature is approaching Tg. Indeed, experiments show that µ rapidly drops to zero above Tg. compliance is an order of magnitude larger at T = 0.3u0/kB than at T = 0.2u0/kB and the data does not fully superimpose in a master curve for long times where J > 0.2u0/a 3. Shift factors were obtained from the small creep portion of the curves. The relatively simple relationship between shift factors and wait time permits construction of an expression that describes the entire master curve in Fig. 2. For creep times that are short compared to the wait time - such that minimal physical aging occurs over the timescale of the experiment - experimental creep compliance curves can be fit to a stretched exponential (typical of processes with a spectrum of relaxation times), J(t) = J0 exp[(t/t0) m] (4) where t0 is the retardation factor, and the exponent, m, has been found to be close to 1/3 for most glasses [4]. A fit of this expression to our simulated creep compli- ance curves is shown in Fig. 2 (dotted line). This ex- 0 0.1 0.2 0.3 0.4 0.5 0.6 T = 0.2 T = 0.3 FIG. 4: The aging exponent, µ, determined from the slopes of log(aJ ) versus log(tw) (from Fig. 3) plotted versus stress (open symbols). The solid symbols at zero stress refer to shift factors determined from aMSD (eq. 7) and aC (eq. 6) data only. The dashed lines are guides to the eye. pression is clearly only consistent with the data at times t < tw. At times much longer than the wait time, the creep compliance varies more slowly due to the stiffen- ing caused by aging during the course of the experiment. Struik suggested that eq. (4) could be extended to the long-time creep regime, where the experimental timescale may be longer than the wait time, by introducing an ef- fective time to account for the slowdown in the relaxation timescales: teff = tw + t′ dt′ (5) Upon replacing t with teff , eq. (4) may be used to de- scribe the entire creep curve. Creep compliance curves from Fig. 2 can indeed be fit to this form (dashed lines) for a known wait time, tw, and aging exponent, µ, as obtained from the master curve. We find m ≈ 0.5 ± 0.1 for all stresses at T = 0.2u0/kB, and a relatively broad range of values for J0 and t0 are consistent with the data. For the simple thermo-mechanical history prescribed by the creep experiment, Struik’s effective time formulation appears to work quite well. The present results parallel those of a recent simula- tion study of the shear yield stress in glassy solids [21]. In this work, the glassy solid was deformed at constant strain rate, and two different regimes of strong and weak rate dependence emerged depending on the time to reach the yield point relative to the wait time. In order to ra- tionalize these results, a rate-state model was developed that accounted for the internal evolution of the material with age through a single state variable Θ(t). This formu- lation successfully collapses yield stress data for different ages and strain rates in a universal curve by adapting the evolution of the state variable during the strain interval. We note here that this state variable is closely related to Struik’s effective time, as it tries to subsume the modified aging dynamics during deformation in a single variable and in particular easily includes the effects of overaging or rejuvenation. B. Microscopic Dynamics The aging behavior of the simulated mechanical re- sponse functions agrees remarkably well with experiment. Additional microscopic information from simulations al- lows us to obtain more directly the relevant timescales of the system, and the relevant microscopic processes re- sponsible for aging. One parameter which has been use- ful in studying glassy dynamics is the “self” part of the incoherent scattering factor [19], Cq(t, tw) = exp(i~q · [~rj(tw + t)− ~rj(tw)]) (6) where ~rj is the position of the j th atom, and ~q is the wave-vector. Cq curves as a function of age are shown in Fig. 5 and exhibit three distinct regions. Initially, Cq decreases as particles make very small unconstrained excursions about their positions. There follows a long plateau, where the correlation function does not change considerably. In this regime, atoms are not free to dif- fuse, but are trapped in local cages formed by their near- est neighbours. For this reason, the time spent in the plateau regime is often associated with a “cage time”. The plateau region ends when particles finally escape from local cages (α-relaxation), and larger atomic rear- rangements begin to take place. The cage time corre- sponds closely to the transition from short-time to long- time regime observed in the creep compliance. Structural rearrangements taking place in the α-relaxation regime are clearly associated with the continued aging observed in the creep compliance, as well as plastic deformations occurring in that region. The correlation functions for different ages are similar in form, but the time spent in the plateau region in- creases with age. Just as creep compliance curves can be shifted in time to form a master curve, we may overlap the long-time, cage-escape regions of Cq by rescaling the time variable of the correlation data at different ages (see inset of Fig. 5). The corresponding shift factors aC(tw) are also shown in Fig. 3, where we see that the increase in cage time with age follows the same power law as the shift factors of the creep compliance. These results are qualitatively similar to the scaling of the relaxation times with age found in [19] with no load. The real space quantity corresponding to Cq is the mean squared displacement, 〈~r(t, tw) ∆~rj(t, tw) 2 (7) FIG. 5: Incoherent scattering factor (eq. 6) for different wait times measured under the same loading conditions as in Fig. 1(a) for q = (0, 0, 2π). The inset shows the master curve created by rescaling the time variable by aC . Symbols as in Fig. 1(b). FIG. 6: Mean-squared displacement (eq. 7) for different wait times measured under the same loading conditions as in Fig. 1(a). The inset shows the master curve created by rescal- ing the time variable by aMSD. Symbols as in Fig. 1(b). where ∆~rj(t, tw) = ~rj(tw + t) − ~rj(tw). This function is shown in Fig. 6. Again we see three characteristic re- gions of unconstrained (ballistic), caged, and cage-escape behavior. The departure from the cage plateau likewise increases with age, and a master curve can be constructed by shifting the curves with a factor aMSD (see inset of Fig. 6). Shift factors aMSD are plotted in Fig. 3, along with shifts for creep compliance and incoherent scatter- ing function. As anticipated, the shifts versus wait time fully agree with those obtained from Cq and J . This clearly demonstrates that for our model, the cage escape time is indeed the controlling factor in the aging dynamics of the mechanical response functions. 0 0.5 1 1.5 2 2.5 3 3.5 FIG. 7: The displacement probability distribution versus time measured under the same loading conditions as in Fig. 1(a), with a wait time of 500τLJ . The solid lines from left to right are obtained at times t of 75, 750, 7500, and 75000τLJ . The dashed lines show fits to the double Gaussian distribution (see text, eq. 8). Additional information about microscopic processes can be obtained by studying not only the mean of the dis- placements, but also the full spectrum of relaxation dy- namics as a function of time and wait time. To this end, we measure the probability distribution P (∆r(t, tw) 2) of atomic displacements during time intervals, t, for glasses at various ages, tw. This quantity is complementary to the measurements of dynamical heterogeneities detailed in [26], where the spatial variations of the vibrational am- plitudes were measured at a snapshot in time to show the correlations of mobile particles in space. In our study, we omit the spatial information, but retain all of the dynam- ical information. Representative distribution functions are shown in Fig. 7 for a constant wait time of tw = 500τLJ and var- ious time intervals t. The distributions were obtained from a smaller system of only 271 polymer chains due to memory constraints. The data does not reflect a simple Gaussian distribution, but clearly exhibits the presence of two distinct scales: there is a narrow distribution of caged particles and a wider distribution of particles that have escaped from their cages. This behavior can be de- scribed by the sum of two Gaussian peaks, P (∆r2) = N1 exp +N2 exp . (8) Deviations from purely Gaussian behavior are common in glassy systems and are a signature of dynamical hetero- geneities [26, 27]. Experiments on colloidal glasses [28] show a similar separation of displacement distributions into fast and slow particles. A fit of the normalized distributions to eq. (8) (dashed lines in Fig. 7) requires adjustment of three parameters: the variance of caged and mobile particle distributions, 1 2 3 4 5 FIG. 8: The Gaussian fit parameters for the distribution of displacements (see text, eq. 8), (a) N1/N , (b) σ 1 , and (c) σ measured under the same loading conditions as in Fig. 1(a). The curves are for wait times increasing from left to right from 500τLJ to 15000τLJ . and σ2 , as well as their relative contributions N1/N , where N = N1 +N2. These parameters are sufficient to describe the full evolution of the displacement distribu- tion during aging. In Fig. 8, we show the fit parameters as a function of time and wait time. Again two distinct time scales are evident. At short times, most of the parti- cles are caged (N1/N ≈ 1), and the variance of the cage peak is also changing very little. There are few rear- rangements in this regime, however Fig. 8(c) shows that a small fraction of particles are mobile at even the short- est times. At a time corresponding to the onset of cage escape, the number of particles in the cage peak begins to rapidly decay, and the variance of the cage peak in- creases. This indicates that the cage has become more malleable - small, persistent rearrangements occur lead- ing to eventual cage escape. In this regime, the variance of the mobile peak increases very little. Note that the typical length scale of rearrangements is less than a par- ticle diameter even in the cage escape regime, but the number of particles undergoing rearrangements changes by more than 50%. Similar to the compliance and mean-squared displace- ment curves, the data in Fig. 8(a) and (b) can also be 0 0.5 1 1.5 2 500 1500 5000 15000 FIG. 9: The displacement probability distribution measured under the same loading conditions and wait times as in Fig. 1(a) plotted at times corresponding to < r2(t, tw) >= 0.7, shown in the inset as a dashed line. The legend indicates the wait time. superimposed by shifting time. Shift factors for N1/N and σ2 coincide exactly with shifts for the mean; how- ever, data for σ2 (Fig. 8(c)) seems to be much less af- fected by the wait time. The aging dynamics appears to be entirely determined by the cage dynamics, and not by larger rearrangements within the glass. Since the fit parameters exhibit the same scaling with wait time as the mean, one might expect that the en- tire distribution of displacements under finite load scales with the evolution of the mean. In Fig. 9, we plot dis- placement distributions for several wait times at time in- tervals chosen such that the mean squared displacements are identical (see inset). Indeed, we find that all curves overlap, indicating that the entire relaxation spectrum ages in the same way. A similar observation was recently made in simulations of a model for a metallic glass aging at zero stress [29], although in this study the tails of the distribution were better described by stretched exponen- tials. In order to study the effect of load on the relaxation dynamics, we compare in Figure 10 the fit parameters for a sample undergoing creep (replotted from Fig. 8) and a reference sample without load. It is clear that the dy- namics are strongly affected by the applied stress only in the region characterized by α-relaxations. For the stress applied here, the onset of cage-escape does not appear to be greatly modified by the stress, however the decay in N1/N and the widening of the cage peak are accelerated. The stress does not modify the variance of the mobile peak, confirming again the importance of local rearrange- ments as compared to large-scale motion in the dynamics of the system. The accelerated structural rearrangements caused by the stress result in creep on the macroscopic scale, but may also be responsible for the modification of the aging dynamics with stress as observed in Fig. 4. 1 2 3 4 5 σ = 0 σ = 0.4 (a) FIG. 10: The Gaussian fit parameters to the displacement distributions (see text, eq. 8) (a) N1/N , (b) σ 1 , and (c) σ a sample aged at T = 0.2u0/kB for tw = 500τLJ , and then either undergoing a creep experiment at σ = 0.4u0/a 3 (black), or simply aging further at zero stress (red). C. Structural evolution The connection between the dynamics and the struc- ture of a glass during aging remains uncertain, mostly because no structural parameter has been found that strongly depends on wait time. Recent simulation stud- ies of metallic glasses have shown the existence of sev- eral short range order parameters that can distinguish between glassy states created through different quench- ing protocols [30, 31, 32]. A strong correlation has been found between “ordered” regions of the glass and strain localization. Many metallic glasses are known to form quasi-crystalline structures that optimize local packing. It remains to be seen whether the short-range order evolves in the context of aging and in other glass for- mers such as polymers and colloids. A recent experimen- tal study of aging in colloidal glasses found no change in the distribution function of a tetrahedral order parame- ter [33]. Below, we investigate several measures of local order in our model as they evolve with age and under load. Since Lennard-Jones liquids are known to condense into a crystal with fcc symmetry at low temperatures, it is reasonable to look for the degree of local fcc order in our polymer model. The level of fcc order can be quan- tified via the bond orientational parameter [34], . (9) This parameter has been successfully used to character- ize the degree of order in systems of hard sphere glasses. Q6 is determined for each atom by projecting the bond angles of the nearest neighbours onto the spherical har- monics, Y6m(θ, φ). The overbar denotes an average over all bonds. Nearest neighbours are defined as all atoms within a cutoff radius, rc, of the central atom. For all of the order parameters discussed here, the cutoff radius is defined by the first minimum in the pair correlation func- tion, in this case 1.45a. Q6 is approximately 0.575 for a perfect fcc crystal; for jammed structures, it can exhibit a large range of values less than about 0.37 [34]. The full distribution of Q6 for our model glass is shown for several ages as well as an initial melt state in Fig. 11(a). We see that there is very little difference even between melt and glassy states in our model, and no discernible difference at all with increasing age. Locally, close-packing is achieved by tetrahedral order- ing and not fcc ordering, however, tetrahedral orderings cannot span the system. The glass formation process has been described in terms of frustration between optimal local and global close-packing structures. To investigate the type of local ordering in this model, we investigate a 3-body angular correlation function, P (θ). θ is de- fined as the internal angle created by a central atom and individual pairs of nearest-neighbours, and P (θ) is the probability of occurrence of θ. Results for this corre- lation are shown in Fig. 11(b). Two peaks at approxi- mately 60◦ and 110◦ indicate tetrahedral ordering. The peaks sharpen under quenching from the melt, but the distribution does not evolve significantly during aging. In contrast, simulations of metallic glass formers showed a stronger sensitivity of this parameter to the quench protocol [31], but most of those changes may be due to rearrangements in the supercooled liquid state and not in the glassy state. Another parameter that has been successful in classi- fying glasses is the triangulated surface order parameter [32], (6 − q)νq (10) which measures the degree of quasi-crystalline order. The surface coordination number, q, is defined for each atom of the coordination shell as the number of neighbouring atoms also residing in the coordination shell; νq is the number of atoms in the coordination shell with surface coordination q. Ordered systems have been identified with S equal to 12 (icosahedron), 14, 15 and 16. Figure 11(c) shows the probability distribution for P (S) for the melt and for glassy states with short and long wait times. 0.1 0.2 0.3 0.4 0.5 0 50 100 150 θ (degrees) 5 10 15 20 25 30 35 FIG. 11: Short-range order parameters: (a) the bond- orientational parameter, (b) the three-body angular corre- lations, (c) the surface triangulated order (see text for dis- cussion). x’s show the melt state, circles show the sample aged for tw = 500τLJ , and triangles show the sample aged for 500, 000τLJ . The peak of the distribution moves toward lower S (more ordered) upon cooling, and continues to evolve slowly in the glass. The mean of S relative to the as-quenched state, 〈S〉, is shown in Fig. 12 as a function of wait time at two temperatures. We see that 〈S〉 is a logarithmi- cally decreasing function of wait time. Even though this is not a strong dependence, this order parameter is sig- nificantly more sensitive to age than the others that have been investigated. Figure 12 also shows the order parameter 〈S〉 after the ramped-up stress has been applied to the aged samples. We can see that at T = 0.2u0/kB, some of the order that developed during age is erased, while no appreciable change occurs at the higher temperature T = 0.3u0/kB. These results correlate well with the behavior of the ag- ing exponent found in Fig. 4, where mechanical rejuve- nation was found at lower temperatures but was much less pronounced at higher T . The load is applied very quickly, and most of the deformation in this regime is affine, however, the strain during this time was similar for both temperatures, therefore the effect is not simply due to a change in density. More work is needed to clarify the nature of the structural changes during rejuvenation. T = 0.2 T = 0.3 FIG. 12: Precent change in the triangulated surface order parameter with wait time as compared to the just-quenched sample. Circles are for samples aged at zero pressure for the time tw. Triangles are for the same samples immediately after ramping up to the creep stress. For T = 0.2u0/kB this stress is 0.4u0/a 3, and for T = 0.3u0/kB the stress is 0.1u0/a IV. CONCLUSIONS We investigate the effects of aging on both macroscopic creep response and underlying microscopic structure and dynamics through simulations on a simple model polymer glass. The model qualitatively reproduces key experi- mental trends in the mechanical behavior of glasses under sustained stress. We observe a power-law dependence of the relaxation time on the wait time with an aging expo- nent of approximately unity, and a decrease in the aging exponent with increasing load that indicates the presence of mechanical rejuvenation. The model creep compliance curves can be fit in the short and long-time regimes using Struik’s effective time formulation. Additionally, inves- tigation of the microscopic dynamics through two-time correlation functions has shown that, for our model glass, the aging dynamics of the creep compliance exactly corre- sponds to the increase in the cage escape or α-relaxation time. A detailed study of the entire distribution of parti- cle displacements yields an interesting picture of the mi- croscopic dynamics during aging. The distribution can be described by the sum of two Gaussians, reflecting the presence of caged and mobile particles. The frac- tion of mobile particles and the amplitude of rearrange- ments in the cage strongly increase at the cage escape time. However, in analogy with results in colloidal glasses [35], structural rearrangements occur even for times well within the “caged” regime, and fairly independent of wait time and stress. For our model glass, we find that the entire distribution of displacements scales with age in the same way as the mean. At times when the long- time portion of the mean squared displacement overlaps, the distribution of displacements at different wait times completely superimpose, confirming that all of the me- chanical relaxation times scale in the same way with age. To characterize the evolution of the structure during aging, we investigate several measures of short-range or- der in our model glass. We find that the short-range order does not evolve strongly during aging. The triangulated surface order [32], however, shows a weak logarithmic de- pendence on age. Results also show a change in structure when a load is rapidly applied, and this seems to be cor- related with an observed decrease in the aging exponent under stress. This study has characterized the dynamics of a model glass prepared by a rapid quench below Tg, followed by aging at constant T and subsequent application of a con- stant load. For such simple thermo-mechanical histo- ries, existing phenomenological models work well, how- ever, the dynamics of glasses are in general much more complex. For instance, large stresses in the non-linear regime modify the aging dynamics and cause nontrivial effects such as mechanical rejuvenation and over-aging [10, 11]. Also, experiments have shown that the time-age time superposition no longer holds when polymer glasses undergo more complex thermal histories such as a tem- perature jump [23]. The success of our study in analyzing simple aging situations indicates that the present simula- tion methodology will be able to shed more light on these topics in the near future. Acknowledgments We thank the Natural Sciences and Engineering Coun- cil of Canada (NSERC) for financial support. 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[36] http://lammps.sandia.gov.

0704.1841

Gaseous Inner Disks

Gaseous Inner Disks Joan R. Najita National Optical Astronomy Observatory John S. Carr Naval Research Laboratory Alfred E. Glassgold University of California, Berkeley Jeff A. Valenti Space Telescope Science Institute As the likely birthplaces of planets and an essential conduit for the buildup of stellar masses, inner disks are of fundamental interest in star and planet formation. Studies of the gaseous component of inner disks are of interest because of their ability to probe the dynamics, physical and chemical structure, and gas content of this region. We review the observational and theoretical developments in this field, highlighting the potential of such studies to, e.g., measure inner disk truncation radii, probe the nature of the disk accretion process, and chart the evolution in the gas content of disks. Measurements of this kind have the potential to provide unique insights on the physical processes governing star and planet formation. 1. INTRODUCTION Circumstellar disks play a fundamental role in the for- mation of stars and planets. A significant fraction of the mass of a star is thought to be built up by accretion through the disk. The gas and dust in the inner disk (r <10 AU) also constitute the likely material from which planets form. As a result, observations of the gaseous component of in- ner disks have the potential to provide critical clues to the physical processes governing star and planet formation. From the planet formation perspective, probing the structure, gas content, and dynamics of inner disks is of interest, since they all play important roles in establish- ing the architectures of planetary systems (i.e., planetary masses, orbital radii, and eccentricities). For example, the lifetime of gas in the inner disk (limited by accretion onto the star, photoevaporation, and other processes) places an upper limit on the timescale for giant planet formation (e.g., Zuckerman et al., 1995). The evolution of gaseous inner disks may also bear on the efficiency of orbital migration and the eccentricity evo- lution of giant and terrestrial planets. Significant inward orbital migration, induced by the interaction of planets with a gaseous disk, is implied by the small orbital radii of extra- solar giant planets compared to their likely formation dis- tances (e.g., Ida and Lin, 2004). The spread in the orbital radii of the planets (0.05–5 AU) has been further taken to in- dicate that the timing of the dissipation of the inner disk sets the final orbital radius of the planet (Trilling et al., 2002). Thus, understanding how inner disks dissipate may impact our understanding of the origin of planetary orbital radii. Similarly, residual gas in the terrestrial planet region may play a role in defining the final masses and eccentricities of terrestrial planets. Such issues have a strong connection to the question of the likelihood of solar systems like our own. An important issue from the perspective of both star and planet formation is the nature of the physical mechanism that is responsible for disk accretion. Among the proposed mechanisms, perhaps the foremost is the magnetorotational instability (Balbus and Hawley, 1991) although other pos- sibilities exist. Despite the significant theoretical progress that has been made in identifying plausible accretion mech- anisms (e.g., Stone et al., 2000), there is little observational evidence that any of these processes are active in disks. Studies of the gas in inner disks offer opportunities to probe the nature of the accretion process. For these reasons, it is of interest to probe the dynami- cal state, physical and chemical structure, and the evolution of the gas content of inner disks. We begin this Chapter with a brief review of the development of this field and an overview of how high resolution spectroscopy can be used to study the properties of inner disks (Section 1). Previ- ous reviews provide additional background on these top- ics (e.g., Najita et al., 2000). In Sections 2 and 3, we re- view recent observational and theoretical developments in this field, first describing observational work to date on the gas in inner disks, and then describing theoretical models for the surface and interior regions of disks. In Section 4, we look to the future, highlighting several topics that can be explored using the tools discussed in Sections 2 and 3. http://arxiv.org/abs/0704.1841v1 1.1 Historical Perspective One of the earliest studies of gaseous inner disks was the work by Kenyon and Hartmann on FU Orionis objects. They showed that many of the peculiarities of these sys- tems could be explained in terms of an accretion outburst in a disk surrounding a low-mass young stellar object (YSO; cf. Hartmann and Kenyon, 1996). In particular, the varying spectral type of FU Ori objects in optical to near-infrared spectra, evidence for double-peaked absorption line pro- files, and the decreasing widths of absorption lines from the optical to the near-infrared argued for an origin in an optically thick gaseous atmosphere in the inner region of a rotating disk. Around the same time, observations of CO vibrational overtone emission, first in the BN object (Scov- ille et al., 1983) and later in other high and low mass objects (Thompson, 1985; Geballe and Persson, 1987; Carr, 1989), revealed the existence of hot, dense molecular gas plausibly located in a disk. One of the first models for the CO over- tone emission (Carr, 1989) placed the emitting gas in an optically-thin inner region of an accretion disk. However, only the observations of the BN object had sufficient spec- tral resolution to constrain the kinematics of the emitting The circumstances under which a disk would produce emission or absorption lines of this kind were explored in early models of the atmospheres of gaseous accretion disks under the influence of external irradiation (e.g., Calvet et al., 1991). The models interpreted the FU Ori absorption features as a consequence of midplane accretion rates high enough to overwhelm external irradiation in establishing a temperature profile that decreases with disk height. At lower accretion rates, the external irradiation of the disk was expected to induce a temperature inversion in the disk atmo- sphere, producing emission rather than absorption features from the disk atmosphere. Thus the models potentially pro- vided an explanation for the FU Ori absorption features and CO emission lines that had been detected. By PPIV (Najita et al., 2000), high-resolution spec- troscopy had demonstrated that CO overtone emission shows the dynamical signature of a rotating disk (Carr et al., 1993; Chandler et al., 1993), thus confirming theoreti- cal expectations and opening the door to the detailed study of gaseous inner disks in a larger number of YSOs. The detection of CO fundamental emission (Section 2.3) and emission lines of hot H2O (Section 2.2) had also added new probes of the inner disk gas. Seven years later, at PPV, we find both a growing number of diagnostics available to probe gaseous inner disks as well as increasingly detailed information that can be gleaned from these diagnostics. Disk diagnostics familiar from PPIV have been used to infer the intrinsic line broaden- ing of disk gas, possibly indicating evidence for turbulence in disks (Section 2.1). They also demonstrate the differen- tial rotation of disks, provide evidence for non-equilibrium molecular abundances (Section 2.2), probe the inner radii of gaseous disks (Section 2.3), and are being used to probe the gas dissipation timescale in the terrestrial planet region (Section 4.1). Along with these developments, new spectral line diagnostics have been used as probes of the gas in inner disks. These include transitions of molecular hydrogen at UV, near-infrared, and mid-infrared wavelengths (Sections 2.4, 2.5) and the fundamental ro-vibrational transitions of the OH molecule (Section 2.2). Additional potential diag- nostics are discussed in Section 2.6. 1.2 High Resolution Spectroscopy of Inner Disks The growing suite of diagnostics can be used to probe in- ner disks using standard high resolution spectroscopic tech- niques. Although inner disks are typically too small to resolve spatially at the distance of the nearest star form- ing regions, we can utilize the likely differential rotation of the disk along with high spectral resolution to separate disk radii in velocity. At the warm temperatures (∼100 K– 5000 K) and high densities of inner disks, molecules are ex- pected to be abundant in the gas phase and sufficiently ex- cited to produce rovibrational features in the infrared. Com- plementary atomic transitions are likely to be good probes of the hot inner disk and the photodissociated surface lay- ers at larger radii. By measuring multiple transitions of different species, we should therefore be able to probe the temperatures, column densities, and abundances of gaseous disks as a function of radius. With high spectral resolution we can resolve individual lines, which facilitates the detection of weak spectral fea- tures. We can also work around telluric absorption fea- tures, using the radial velocity of the source to shift its spec- tral features out of telluric absorption cores. This approach makes it possible to study a variety of atomic and molecular species, including those present in the Earth’s atmosphere. Gaseous spectral features are expected in a variety of situations. As already mentioned, significant vertical vari- ation in the temperature of the disk atmosphere will pro- duce emission (absorption) features if the temperature in- creases (decreases) with height (Calvet et al., 1991; Mal- bet and Bertout, 1991). In the general case, when the disk is optically thick, observed spectral features measure only the atmosphere of the disk and are unable to probe directly the entire disk column density, a situation familiar from the study of stellar atmospheres. Gaseous emission features are also expected from re- gions of the disk that are optically thin in the continuum. Such regions might arise as a result of dust sublimation (e.g., Carr, 1989) or as a consequence of grain growth and planetesimal formation. In these scenarios, the disk would have a low continuum opacity despite a potentially large gas column density. Optically thin regions can also be produced by a significant reduction in the total column density of the disk. This situation might occur as a consequence of giant planet formation, in which the orbiting giant planet carves out a “gap” in the disk. Low column densities would also be characteristic of a dissipating disk. Thus, we should be able to use gaseous emission lines to probe the properties of inner disks in a variety of interesting evolutionary phases. 2. OBSERVATIONS OF GASEOUS INNER DISKS 2.1 CO Overtone Emission The CO molecule is expected to be abundant in the gas phase over a wide range of temperatures, from the tem- perature at which it condenses on grains (∼20 K) up to its thermal dissociation temperature (∼4000 K at the densities of inner disks). As a result, CO transitions are expected to probe disks from their cool outer reaches (>100 AU) to their innermost radii. Among these, the overtone transitions of CO (∆v=2, λ=2.3µm) were the emission line diagnostics first recognized to probe the gaseous inner disk. CO overtone emission is detected in both low and high mass young stellar objects, but only in a small fraction of the objects observed. It appears more commonly among higher luminosity objects. Among the lower luminosity stars, it is detected from embedded protostars or sources with energetic outflows (Geballe and Persson, 1987; Carr, 1989; Greene and Lada, 1996; Hanson et al., 1997; Luh- man et al., 1998; Ishii et al., 2001; Figueredo et al., 2002; Doppmann et al., 2005). The conditions required to excite the overtone emission, warm temperatures (& 2000 K) and high densities (>1010 cm−3), may be met in disks (Scoville et al., 1983; Carr, 1989; Calvet et al., 1991), inner winds (Carr, 1989), or funnel flows (Martin, 1997). High resolution spectroscopy can be used to distinguish among these possibilities. The observations typically find strong evidence for the disk interpretation. The emission line profiles of the v=2–0 bandhead in most cases show the characteristic signature of bandhead emission from sym- metric, double-peaked line profiles originating in a rotating disk (e.g., Carr et al., 1993; Chandler et al., 1993; Na- jita et al., 1996; Blum et al., 2004). The symmetry of the observed line profiles argues against the likelihood that the emission arises in a wind or funnel flow, since inflowing or outflowing gas is expected to produce line profiles with red- or blue-shifted absorption components (alternatively line asymmetries) of the kind that are seen in the hydrogen Balmer lines of T Tauri stars (TTS). Thus high resolution spectra provide strong evidence for rotating inner disks. The velocity profiles of the CO overtone emission are normally very broad (>100km s−1). In lower mass stars (∼1M⊙), the emission profiles show that the emission ex- tends from very close to the star, ∼0.05 AU, out to ∼0.3 AU (e.g., Chandler et al., 1993; Najita et al., 2000). The small radii are consistent with the high excitation temperatures measured for the emission (∼1500–4000K). Velocity re- solved spectra have also been modeled in a number of high mass stars (Blum et al., 2004; Bik and Thi, 2004), where the CO emission is found to arise at radii ∼ 3AU. The large near-infrared excesses of the sources in which CO overtone emission is detected imply that the warm emit- ting gas is located in a vertical temperature inversion re- gion in the disk atmosphere. Possible heating sources for the temperature inversion include: external irradiation by the star at optical through UV wavelengths (e.g., Calvet et al., 1991; D’Alessio et al., 1998) or by stellar X-rays (Glassgold et al., 2004; henceforth GNI04); turbulent heat- ing in the disk atmosphere generated by a stellar wind flow- ing over the disk surface (Carr et al., 1993); or the dissi- pation of turbulence generated by disk accretion (GNI04). Detailed predictions of how these mechanisms heat the gaseous atmosphere are needed in order to use the observed bandhead emission strengths and profiles to investigate the origin of the temperature inversion. The overtone emission provides an additional clue that suggests a role for turbulent dissipation in heating disk at- mospheres. Since the CO overtone bandhead is made up of closely spaced lines with varying inter-line spacing and optical depth, the emission is sensitive to the intrinsic line broadening of the emitting gas (as long as the gas is not op- tically thin). It is therefore possible to distinguish intrinsic line broadening from macroscopic motions such as rotation. In this way, one can deduce from spectral synthesis model- ing that the lines are suprathermally broadened, with line widths approximately Mach 2 (Carr et al., 2004; Najita et al., 1996). Hartmann et al. (2004) find further evidence for turbulent motions in disks based on high resolution spec- troscopy of CO overtone absorption in FU Ori objects. Thus disk atmospheres appear to be turbulent. The tur- bulence may arise as a consequence of turbulent angular momentum transport in disks, as in the magnetorotational instability (MRI; Balbus and Hawley, 1991) or the global baroclinic instability (Klahr and Bodenheimer, 2003). Tur- bulence in the upper disk atmosphere may also be generated by a wind blowing over the disk surface. 2.2 Hot Water and OH Fundamental Emission Water molecules are also expected to be abundant in disks over a range of disk radii, from the temperature at which water condenses on grains (∼150 K) up to its thermal dissociation temperature (∼2500 K). Like the CO overtone transitions, the rovibrational transitions of water are also ex- pected to probe the high density conditions in disks. While the strong telluric absorption produced by water vapor in the Earth’s atmosphere will restrict the study of cool wa- ter to space or airborne platforms, it is possible to observe from the ground water that is much hotter than the Earth’s atmosphere. Very strong emission from hot water can be detected in the near-infrared even at low spectral resolution (e.g., SVS-13; Carr et al., 2004). More typically, high reso- lution spectroscopy of individual lines is required to detect much weaker emission lines. For example, emission from individual lines of water in the K- and L-bands have been detected in a few stars (both low and high mass) that also show CO overtone emission (Carr et al., 2004; Najita et al., 2000; Thi and Bik, 2005). Velocity resolved spectra show that the widths of the water lines are consistently narrower than those of the CO emis- sion lines. Spectral synthesis modeling further shows that the excitation temperature of the water emission (typically ∼1500 K), is less than that of the CO emission. These re- sults are consistent with both the water and CO originat- ing in a differentially rotating disk with an outwardly de- creasing temperature profile. That is, given the lower dis- sociation temperature of water (∼2500 K) compared to CO (∼4000 K), CO is expected to extend inward to smaller radii than water, i.e., to higher velocities and temperatures. The ∆v=1 OH fundamental transitions at 3.6µm have also been detected in the spectra of two actively accreting sources, SVS-13 and V1331 Cyg, that also show CO over- tone and hot water emission (Carr et al., in preparation). As shown in Fig. 1, these features arise in a region that is crowded with spectral lines of water and perhaps other species. Determining the strengths of the OH lines will, therefore, require making corrections for spectral features that overlap closely in wavelength. Spectral synthesis modeling of the detected CO, H2O and OH features reveals relative abundances that depart sig- nificantly from chemical equilibrium (cf. Prinn, 1993), with the relative abundances of H2O and OH a factor of 2–10 below that of CO in the region of the disk probed by both diagnostics (Carr et al., 2004; Carr et al., in preparation; see also Thi and Bik, 2005). These abundance ratios may arise from strong vertical abundance gradients produced by the external irradiation of the disk (see Section 3.4). 2.3 CO Fundamental Emission The fundamental (∆v=1) transitions of CO at 4.6µm are an important probe of inner disk gas in part because of their broader applicability compared, e.g., to the CO overtone lines. As a result of their comparatively small A-values, the CO overtone transitions require large column densities of warm gas (typically in a disk temperature inversion region) in order to produce detectable emission. Such large column densities of warm gas may be rare except in sources with the largest accretion rates, i.e., those best able to tap a large Fig. 1.— OH fundamental ro-vibrational emission from SVS-13 on a relative flux scale. Fig. 2.— Gaseous inner disk radii for TTS from CO fundamental emission (filled squares) compared with corotation radii for the same sources. Also shown are dust inner radii from near-infrared interferometry (filled circles; Akeson et al., 2005a,b) or spectral energy distributions (open circles; Muzerolle et al., 2003). The solid and dashed lines indicate an inner radius equal to, twice, and 1/2 the corotation radius. The points for the three stars with measured inner radii for both the gas and dust are connected by dotted lines. Gas is observed to extend inward of the dust inner radius and typically inward of the corotation radius. accretion energy budget and heat a large column density of the disk atmosphere. In contrast, the CO fundamental transitions, with their much larger A-values, should be de- tectable in systems with more modest column densities of warm gas, i.e., in a broader range of sources. This is borne out in high resolution spectroscopic surveys for CO funda- mental emission from TTS (Najita et al., 2003) and Herbig AeBe stars (Blake and Boogert, 2004) which detect emis- sion from essentially all sources with accretion rates typical of these classes of objects. In addition, the lower temperatures required to excite the CO v=1–0 transitions make these transitions sensitive to cooler gas at larger disk radii, beyond the region probed by the CO overtone lines. Indeed, the measured line profiles for the CO fundamental emission are broad (typically 50– 100km s−1 FWHM) and centrally peaked, in contrast to the CO overtone lines which are typically double-peaked. These velocity profiles suggest that the CO fundamental emission arises from a wide range of radii, from .0.1 AU out to 1–2 AU in disks around low mass stars, i.e., the ter- restrial planet region of the disk (Najita et al., 2003). CO fundamental emission spectra typically show sym- metric emission lines from multiple vibrational states (e.g., v=1–0, 2–1, 3–2); lines of 13CO can also be detected when the emission is strong and optically thick. The ability to study multiple vibrational states as well as isotopic species within a limited spectral range makes the CO fundamental lines an appealing choice to probe gas in the inner disk over a range of temperatures and column densities. The relative strengths of the lines also provide insight into the excitation mechanism for the emission. In one source, the Herbig AeBe star HD141569, the excitation temperature of the rotational levels (∼200 K) is much lower than the excitation temperature of the vibra- tional levels (v=6 is populated), which is suggestive of UV pumping of cold gas (Brittain et al., 2003). The emis- sion lines from the source are narrow, indicating an origin at &17 AU. The lack of fluorescent emission from smaller radii strongly suggests that the region within 17 AU is de- pleted of gaseous CO. Thus detailed models of the fluores- cence process can be used to constrain the gas content in the inner disk region (S. Brittain, personal communication). Thus far HD141569 appears to be an unusual case. For the majority of sources from which CO fundamental is de- tected, the relative line strengths are consistent with emis- sion from thermally excited gas. They indicate typical ex- citation temperatures of 1000–1500K and CO column den- sities of ∼1018 cm−2 for low mass stars. These temper- atures are much warmer than the dust temperatures at the same radii implied by spectral energy distributions (SEDs) and the expectations of some disk atmosphere models (e.g., D’Alessio et al., 1998). The temperature difference can be accounted for by disk atmosphere models that allow for the thermal decoupling of the gas and dust (Section 3.2). For CTTS systems in which the inclination is known, we can convert a measured HWZI velocity for the emission to an inner radius. The CO inner radii, thus derived, are typ- ically ∼0.04 AU for TTS (Najita et al., 2003; Carr et al., in preparation), smaller than the inner radii that are mea- sured for the dust component either through interferometry (e.g., Eisner et al., 2005; Akeson et al., 2005a; Colavita et al., 2003; see chapter by Millan-Gabet et al.) or through the interpretation of SEDs (e.g., Muzerolle et al., 2003). This shows that gaseous disks extend inward to smaller radii than dust disks, a result that is not surprising given the relatively low sublimation temperature of dust grains (∼1500–2000 K) compared to the CO dissociation temperature (∼4000 K). These results are consistent with the suggestion that the inner radius of the dust disk is defined by dust sublimation rather than by physical truncation (Muzerolle et al., 2003; Eisner et al., 2005). Perhaps more interestingly, the inner radius of the CO emission appears to extend up to and usually within the corotation radius (i.e., the radius at which the disk rotates at the same angular velocity as the star; Fig. 2). In the cur- rent paradigm for TTS, a strong stellar magnetic field trun- cates the disk near the corotation radius. The coupling be- tween the stellar magnetic field and the gaseous inner disk regulates the rotation of the star, bringing the star into coro- tation with the disk at the coupling radius. From this re- gion emerge both energetic (X-)winds and magnetospheric accretion flows (funnel flows; Shu et al., 1994). The ve- locity extent of the CO fundamental emission shows that gaseous circumstellar disks indeed extend inward beyond Fig. 3.— The distribution of gaseous inner radii, measured with the CO fundamental transitions, compared to the distribution of orbital radii of short-period extrasolar planets. A minimum plane- tary orbital radius of ∼0.04 AU is similar to the minimum gaseous inner radius inferred from the CO emission line profiles. the dust destruction radius to the corotation radius (and be- yond), providing the material that feeds both X-winds and funnel flows. Such small coupling radii are consistent with the rotational rates of young stars. It is also interesting to compare the distribution of inner radii for the CO emission with the orbital radii of the “close- in” extrasolar giant planets (Fig. 3). Extra-solar planets discovered by radial velocity surveys are known to pile-up near a minimum radius of 0.04 AU. The similarity between these distributions is roughly consistent with the idea that the truncation of the inner disk can halt the inward orbital migration of a giant planet (Lin et al., 1996). In detail, how- ever, the planet is expected to migrate slightly inward of the truncation radius, to the 2:1 resonance, an effect that is not seen in the present data. A possible caveat is that the wings of the CO lines may not trace Keplerian motion or that the innermost gas is not dynamically significant. It would be interesting to explore this issue further since the results impact our understanding of planet formation and the origin of planetary architectures. In particular, the ex- istence of a stopping mechanism implies a lower efficiency for giant planet formation, e.g., compared to a scenario in which multiple generations of planets form and only the last generation survives (e.g., Trilling et al., 2002). 2.4 UV Transitions of Molecular Hydrogen Among the diagnostics of inner disk gas developed since PPIV, perhaps the most interesting are those of H2. H2 is presumably the dominant gaseous species in disks, due to high elemental abundance, low depletion onto grains, and robustness against dissociation. Despite its expected ubiq- uity, H2 is difficult to detect because permitted electronic transitions are in the far ultraviolet (FUV) and accessible only from space. Optical and rovibrational IR transitions have radiative rates that are 14 orders of magnitude smaller. 1208 1212 1216 1220 Vacuum Wavelength (Å) TW Hya v"=0 v"=1 v"=2 v"=3 E/k = 20,000 K Fig. 4.— Lyα emission from TW Hya, an accreting T Tauri star, and a reconstruction of the Lyα profile seen by the circumstel- lar H2. Each observed H2 progression (with a common excited state) yields a single point in the reconstructed Lyα profile. The wavelength of each point in the reconstructed Lyα profile corre- sponds to the wavelength of the upward transition that pumps the progression. The required excitation energies for the H2 before the pumping is indicated in the inset energy level diagram. There are no low excitation states of H2 with strong transitions that overlap Lyα. Thus, the H2 must be very warm to be preconditioned for pumping and subsequent fluorescence. Considering only radiative transitions with spontaneous rates above 107 s−1, H2 has about 9000 possible Lyman- band (B-X) transitions from 850-1650 Å and about 5000 possible Werner-band (C-X) transitions from 850-1300 Å (Abgrall et al., 1993a,b). However, only about 200 FUV transitions have actually been detected in spectra of accret- ing TTS. Detected H2 emission lines in the FUV all origi- nate from about two dozen radiatively pumped states, each more than 11 eV above ground. These pumped states of H2 are the only ones connected to the ground electronic configuration by strong radiative transitions that overlap the broad Lyα emission that is characteristic of accreting TTS (see Fig. 4). Evidently, absorption of broad Lyα emission pumps the H2 fluorescence. The two dozen strong H2 tran- sitions that happen to overlap the broad Lyα emission are all pumped out of high v and/or high J states at least 1 eV above ground (see inset in Fig. 4). This means some mech- anism must excite H2 in the ground electronic configura- tion, before Lyα pumping can be effective. If the excitation mechanism is thermal, then the gas must be roughly 103 K to obtain a significant H2 population in excited states. H2 emission is a ubiquitous feature of accreting TTS. Fluoresced H2 is detected in the spectra of 22 out of 24 ac- creting TTS observed in the FUV by HST/STIS (Herczeg et al., 2002; Walter et al., 2003; Calvet et al., 2004; Bergin et al., 2004; Gizis et al., 2005; Herczeg et al., 2005; unpub- lished archival data). Similarly, H2 is detected in all 8 ac- creting TTS observed by HST/GHRS (Ardila et al., 2002) and all 4 published FUSE spectra (Wilkinson et al., 2002; Herczeg et al., 2002; 2004; 2005). Fluoresced H2 was even detected in 13 out of 39 accreting TTS observed by IUE, despite poor sensitivity (Brown et al., 1981; Valenti et al., 2000). Fluoresced H2 has not been detected in FUV spec- tra of non-accreting TTS, despite observations of 14 stars with STIS (Calvet et al., 2004; unpublished archival data), 1 star with GHRS (Ardila et al., 2002), and 19 stars with IUE (Valenti et al., 2000). However, the existing observa- tions are not sensitive enough to prove that the circumstellar H2 column decreases contemporaneously with the dust con- tinuum of the inner disk. When accretion onto the stellar surface stops, fluorescent pumping becomes less efficient because the strength and breadth of Lyα decreases signifi- cantly and the H2 excitation needed to prime the pumping mechanism may become less efficient. COS, if installed on HST, will have the sensitivity to set interesting limits on H2 around non-accreting TTS in the TW Hya association. The intrinsic Lyα profile of a TTS is not observable at Earth, except possibly in the far wings, due to absorption by neutral hydrogen along the line of sight. However, observa- tions of H2 line fluxes constrain the Lyα profile seen by the fluoresced H2. The rate at which a particular H2 upward transition absorbs Lyα photons is equal to the total rate of observed downward transitions out of the pumped state, corrected for missing lines, dissociation losses, and propa- gation losses. If the total number of excited H2 molecules before pumping is known (e.g., by assuming a temperature), then the inferred pumping rate yields a Lyα flux point at the wavelength of each pumping transition (Fig. 4). Herczeg et al. (2004) applied this type of analysis to TW Hya, treating the circumstellar H2 as an isothermal, self-absorbing slab. Fig. 4 shows reconstructed Lyα flux points for the upward pumping transitions, assuming the fluoresced H2 is at 2500 K. The smoothness of the recon- structed Lyα flux points implies that the H2 level popula- tions are consistent with thermal excitation. Assuming an H2 temperature warmer or cooler by a few hundred degrees leads to unrealistic discontinuities in the reconstructed Lyα flux points. The reconstructed Lyα profile has a narrow absorption component that is blueshifted by −90 kms−1, presumably due to an intervening flow. The spatial morphology of fluoresced H2 around TTS is diverse. Herczeg et al. (2002) used STIS to observe TW Hya with 50 mas angular resolution, corresponding to a spa- tial resolution of 2.8 AU at a distance of 56 pc, finding no evidence that the fluoresced H2 is extended. At the other extreme, Walter et al. (2003) detected fluoresced H2 up to 9 arcsec from T Tau N, but only in progressions pumped by H2 transitions near the core of Lyα. Fluoresced H2 lines have a main velocity component at or near the stellar radial velocity and perhaps a weaker component that is blueshifted by tens of km s−1 (Herczeg et al., 2006). These two compo- nents are attributed to the disk and the outflow, respectively. TW Hya has H2 lines with no net velocity shift, consistent with formation in the face-on disk (Herczeg et al., 2002). On the other hand, RU Lup has H2 lines that are blueshifted by 12 kms−1, suggesting formation in an outflow. In both of these stars, absorption in the blue wing of the C II 1335 Å wind feature strongly attenuates H2 lines that happen to overlap in wavelength, so in either case H2 forms inside the warm atomic wind (Herczeg et al., 2002; 2005). The velocity widths of fluoresced H2 lines (after re- moving instrumental broadening) range from 18 km s−1 to 28 km s−1 for the 7 accreting TTS observed at high spec- tral resolution with STIS (Herczeg et al., 2006). Line width does not correlate well with inclination. For example, TW Hya (nearly face-on disk) and DF Tau (nearly edge-on disk) both have line widths of 18 km s−1. Thermal broadening is negligible, even at 2000 K. Keplerian motion, enforced corotation, and outflow may all contribute to H2 line width in different systems. More data are needed to understand how velocity widths (and shifts) depend on disk inclination, accretion rate, and other factors. 2.5 Infrared Transitions of Molecular Hydrogen Transitions of molecular hydrogen have also been stud- ied at longer wavelengths, in the near- and mid-infrared. The v=1–0 S(1) transition of H2 (at 2µm) has been de- tected in emission in a small sample of classical T Tauri stars (CTTS) and one weak T Tauri star (WTTS; Bary et al., 2003 and references therein). The narrow emission lines (.10kms−1), if arising in a disk, indicate an origin at large radii, probably beyond 10 AU. The high temperatures required to excite these transitions thermally (1000s K), in contrast to the low temperatures expected for the outer disk, suggest that the emission is non-thermally excited, possibly by X-rays (Bary et al., 2003). The measurement of other rovibrational transitions of H2 is needed to confirm this. The gas mass detectable by this technique depends on the depth to which the exciting radiation can penetrate the disk. Thus, the emission strength may be limited either by the strength of the radiation field, if the gas column density is high, or by the mass of gas present, if the gas column density is low. While it is therefore difficult to measure to- tal gas masses with this approach, clearly non-thermal pro- cesses can light up cold gas, making it easier to detect. Emission from a WTTS is surprising since WTTS are thought to be largely devoid of circumstellar dust and gas, given the lack of infrared excesses and the low accre- tion rates for these systems. The Bary et al. results call this assumption into question and suggest that longer lived gaseous reservoirs may be present in systems with low ac- cretion rates. We return to this issue in Section 4.1. At longer wavelengths, the pure rotational transitions of H2 are of considerable interest because molecular hydrogen carries most of the mass of the disk, and these mid-infrared transitions are capable of probing the ∼100 K temperatures that are expected for the giant planet region of the disk. These transitions present both advantages and challenges as probes of gaseous disks. On the one hand, their small A- values make them sensitive, in principle, to very large gas masses (i.e., the transitions do not become optically thick until large gas column densities NH=10 − 1024 cm−2 are reached). On the other hand, the small A-values also im- ply small critical densities, which allows the possibility of contaminating emission from gas at lower densities not as- sociated with the disk, including shocks in outflows and UV excitation of ambient gas. In considering the detectability of H2 emission from gaseous disks mixed with dust, one issue is that the dust continuum can become optically thick over column densi- ties NH ≪ 10 − 1024 cm−2. Therefore, in a disk that is optically thick in the continuum (i.e., in CTTS), H2 emis- sion may probe smaller column densities. In this case, the line-to-continuum contrast may be low unless there is a strong temperature inversion in the disk atmosphere, and high signal-to-noise observations may be required to detect the emission. In comparison, in disk systems that are op- tically thin in the continuum (e.g., WTTS), H2 could be a powerful probe as long as there are sufficient heating mech- anisms (e.g., beyond gas-grain coupling) to heat the H2. A thought-provoking result from ISO was the report of approximately Jupiter-masses of warm gas residing in ∼20 Myr old debris disk systems (Thi et al., 2001) based on the detection of the 28 µm and 17 µm lines of H2. This result was surprising because of the advanced age of the sources in which the emission was detected; gaseous reser- voirs are expected to dissipate on much shorter timescales (Section 4.1). This intriguing result is, thus far, uncon- firmed by either ground-based studies (Richter et al., 2002; Sheret et al., 2003; Sako et al., 2005) or studies with Spitzer (e.g., Chen et al., 2004). Nevertheless, ground-based studies have detected pure rotational H2 emission from some sources. Detections to date include AB Aur (Richter et al., 2002). The narrow width of the emission in AB Aur (∼10 kms−1 FWHM), if arising in a disk, locates the emission beyond the giant planet region. Thus, an important future direction for these studies is to search for H2 emission in a larger number of sources and at higher velocities, in the giant planet region of the disk. High resolution mid-IR spectrographs on >3- m telescopes will provide the greater sensitivity needed for such studies. 2.6 Potential Disk Diagnostics In a recent development, Acke et al. (2005) have reported high resolution spectroscopy of the [OI] 6300 Å line in Her- big AeBe stars. The majority of the sources show a narrow (<50 km s−1 FWHM), fairly symmetric emission compo- nent centered at the radial velocity of the star. In some cases, double-peaked lines are detected. These features are interpreted as arising in a surface layer of the disk that is irradiated by the star. UV photons incident on the disk sur- face are thought to photodissociate OH and H2O, produc- ing a non-thermal population of excited neutral oxygen that decays radiatively, producing the observed emission lines. Fractional OH abundances of ∼10−7 − 10−6 are needed to account for the observed line luminosities. Another recent development is the report of strong ab- sorption in the rovibrational bands of C2H2, HCN, and CO2 in the 13–15 µm spectrum of a low-mass class I source in Ophiuchus, IRS 46 (Lahuis et al., 2006). The high excita- tion temperature of the absorbing gas (400-900 K) suggests an origin close to the star, an interpretation that is consis- tent with millimeter observations of HCN which indicate a source size ≪100 AU. Surprisingly, high dispersion obser- vations of rovibrational CO (4.7 µm) and HCN (3.0 µm) show that the molecular absorption is blueshifted relative to the molecular cloud. If IRS 46 is similarly blueshifted relative to the cloud, the absorption may arise in the at- mosphere of a nearly edge-on disk. A disk origin for the absorption is consistent with the observed relative abun- dances of C2H2, HCN, and CO2 (10 −6–10−5), which are close to those predicted by Markwick et al. (2002) for the inner region of gaseous disks (.2 AU; see Section 3). Alter- natively, if IRS 46 has approximately the same velocity as the cloud, then the absorbing gas is blueshifted with respect to the star and the absorption may arise in an outflowing wind. Winds launched from the disk, at AU distances, may have molecular abundances similar to those observed if the chemical properties of the wind change slowly as the wind is launched. Detailed calculations of the chemistry of disk winds are needed to explore this possibility. The molecular abundances in the inner disk midplane (Section 3.3) provide the initial conditions for such studies. 3. THERMAL-CHEMICAL MODELING 3.1 General Discussion The results discussed in the previous section illustrate the growing potential for observations to probe gaseous in- ner disks. While, as already indicated, some conclusions can be drawn directly from the data coupled with simple spectral synthesis modeling, harnessing the full diagnos- tic potential of the observations will likely rely on detailed models of the thermal-chemical structure (and dynamics) of disks. Fortunately, the development of such models has been an active area of recent research. Although much of the effort has been devoted to understanding the outer re- gions of disks (∼100 AU; e.g., Langer et al., 2000; chap- ters by Bergin et al. and Dullemond et al.), recent work has begun to focus on the region within 10 AU. Because disks are intrinsically complex structures, the models include a wide array of processes. These encom- pass heating sources such as stellar irradiation (including far UV and X-rays) and viscous accretion; chemical pro- cesses such as photochemistry and grain surface reactions; and mass transport via magnetocentrifugal winds, surface evaporative flows, turbulent mixing, and accretion onto the star. The basic goal of the models is to calculate the den- sity, temperature, and chemical abundance structures that result from these processes. Ideally, the calculation would be fully self-consistent, although approximations are made to simplify the problem. A common simplification is to adopt a specified density distribution and then solve the rate equations that define the chemical model. This is justifiable where the thermal and chemical timescales are short compared to the dynamical timescale. A popular choice is the α-disk model (Shakura and Sunyaev, 1973; Lynden-Bell and Pringle, 1974) in which a phenomenological parameter α characterizes the efficiency of angular momentum transport; its vertically av- eraged value is estimated to be ∼10−2 for T Tauri disks on the basis of measured accretion rates (Hartmann et al., 1998). Both vertically isothermal α-disk models and the Hayashi minimum mass solar nebula (e.g., Aikawa et al., 1999) were adopted in early studies. An improved method removes the assumption of vertical isothermality and calculates the vertical thermal structure of the disk including viscous accretion heating at the midplane (specified by α) and stellar radiative heating under the as- sumption that the gas and dust temperatures are the same (Calvet et al., 1991; D’Alessio et al., 1999). Several chemi- cal models have been built using the D’Alessio density dis- tribution (e.g., Aikawa and Herbst, 1999; GNI04; Jonkheid et al., 2004). Starting about 2001, theoretical models showed that the gas temperature can become much larger than the dust tem- perature in the atmospheres of outer (Kamp and van Zadel- hoff, 2001) and inner (Glassgold and Najita, 2001) disks. This suggested the need to treat the gas and dust as two in- dependent but coupled thermodynamic systems. As an ex- ample of this approach, Gorti and Hollenbach (2004) have iteratively solved a system of chemical rate equations along with the equations of hydrostatic equilibrium and thermal balance for both the gas and the dust. The chemical models developed so far are character- ized by diversity as well as uncertainty. There is diver- sity in the adopted density distribution and external radi- ation field (UV, X-rays, and cosmic rays; the relative im- portance of these depends on the evolutionary stage) and in the thermal and chemical processes considered. The rel- evant heating processes are less well understood than line cooling. One issue is how UV, X-ray, and cosmic rays heat the gas. Another is the role of mechanical heating associated with various flows in the disk, especially accre- tion (GNI04). The chemical processes are also less cer- tain. Our understanding of astrochemistry is based mainly on the interstellar medium, where densities and tempera- tures are low compared to those of inner disks, except per- haps in shocks and photon-dominated regions. New reac- tion pathways or processes may be important at the higher densities (> 107 cm−3) and higher radiation fields of in- ner disks. A basic challenge is to understand the thermal- chemical role of dust grains and PAHs. Indeed, perhaps the most significant difference between models is the treatment of grain chemistry. The more sophisticated models include adsorption of gas onto grains in cold regions and desorption Fig. 5.— Temperature profiles from GNI04 for a protoplanetary disk atmosphere. The lower solid line shows the dust temperature of D’Alessio et al. (1999) at a radius of 1 AU and a mass accretion rate of 10−8M⊙ yr −1. The upper curves show the corresponding gas temperature as a function of the phenomenological mechani- cal heating parameter defined by Equation 1, αh = 1 (solid line), 0.1 (dotted line), and 0.01 (dashed line). The αh = 0.01 curve closely follows the limiting case of pure X-ray heating. The lower vertical lines indicate the major chemical transitions, specifically CO forming at ∼ 1021cm−2, H2 forming at ∼ 6 × 10 and water forming at higher column densities. in warm regions. Yet another level of complexity is intro- duced by transport processes which can affect the chemistry through vertical or radial mixing. An important practical issue in thermal-chemical mod- eling is that self-consistent calculations become increas- ingly difficult as the density, temperature, and the number of species increase. Almost all models employ truncated chemistries with with somewhere from 25 to 215 species, compared with 396 in the UMIST data base (Le Teuff et al., 2000). The truncation process is arbitrary, determined largely by the goals of the calculations. Wiebe et al., (2003) have an objective method for selecting the most important reactions from large data bases. Additional insights into disk chemistry are offered in the chapter by Bergin et al. 3.2 The Disk Atmosphere As noted above, Kamp and van Zadelhoff (2001) con- cluded in their model of debris disks that the gas and dust temperature can differ, as did Glassgold and Najita (2001) for T Tauri disks. The former authors developed a compre- hensive thermal-chemical model where the heating is pri- marily from the dissipation of the drift velocity of the dust through the gas. For T Tauri disks, stellar X-rays, known to be a universal property of low-mass YSOs, heat the gas to temperatures thousands of degrees hotter than the dust temperature. Fig. 5 shows the vertical temperature profile obtained by Glassgold et al. (2004) with a thermal-chemical model based on the dust model of D’Alessio et al. (1999) for a generic T Tauri disk. Near the midplane, the densities are high enough to strongly couple the dust and gas. At higher altitudes, the disk becomes optically thin to the stellar opti- cal and infrared radiation, and the temperature of the (small) grains rises, as does the still closely-coupled gas tempera- ture. However, at still higher altitudes, the gas responds strongly to the less attenuated X-ray flux, and its tempera- ture rises much above the dust temperature. The presence of a hot X-ray heated layer above a cold midplane layer was obtained independently by Alexander et al. (2004). GNI04 also considered the possibility that the surface layers of protoplanetary disks are heated by the dissipation of mechanical energy. This might arise through the interac- tion of a wind with the upper layers of the disk or through disk angular momentum transport. Since the theoretical un- derstanding of such processes is incomplete, a phenomeno- logical treatment is required. In the case of angular momen- tum transport, the most widely accepted mechanism is the MRI (Balbus and Hawley, 1991; Stone et al., 2000), which leads to the local heating formula, Γacc = 2Ω, (1) where ρ is the mass density, c is the isothermal sound speed, Ω is the angular rotation speed, and αh is a phenomeno- logical parameter that depends on how the turbulence dis- sipates. One can argue, on the basis of simulations by Miller and Stone (2000), that midplane turbulence gener- ates Alfvén waves which, on reaching the diffuse surface regions, produce shocks and heating. Wind-disk heating can be represented by a similar expression on the basis of dimensional arguments. Equation 1 is essentially an adap- tation of the expression for volumetric heating in an α-disk model, where α can in general depend on position. GNI04 used the notation αh to distinguish its value in the disk at- mosphere from the usual midplane value. In the top layers fully exposed to X-rays, the gas tem- perature at 1 AU is ∼5000 K. Further down, there is a warm transition region (500–2000 K) composed mainly of atomic hydrogen but with carbon fully associated into CO. The conversion from atomic H to H2 is reached at a column density of ∼6× 1021 cm−2, with more complex molecules such as water forming deeper in the disk. The location and thickness of the warm molecular region depends on the strength of the surface heating. The curves in Fig. 5 illus- trate this dependence for a T Tauri disk at r = 1AU. With αh = 0.01, X-ray heating dominates this region, whereas with αh > 0.1, mechanical heating dominates. Gas temperature inversions can also be produced by UV radiation operating on small dust grains and PAHs, as demonstrated by the thermal-chemical models of Jonkheid et al. (2004) and Kamp and Dullemond (2004). Jonkheid et al. use the D’Alessio et al. (1999) model and focus on the disk beyond 50 AU. At this radius, the gas temperature can rise to 800 K or 200 K, depending on whether small grains are well mixed or settled. For a thin disk and a high stel- lar UV flux, Kamp and Dullemond obtain temperatures that asymptote to several 1000 K inside 50 AU. Of course these results are subject to various assumptions that have been made about the stellar UV, the abundance of PAHs, and the growth and settling of dust grains. Many of the earlier chemical models, oriented towards outer disks (e.g., Willacy and Langer, 2000; Aikawa and Herbst, 1999; 2001; Markwick et al., 2002), adopt a value for the stellar UV radiation field that is 104 times larger than Galactic at a distance of 100 AU. This choice can be traced back to early IUE measurements of the stellar UV beyond 1400 Å for several TTS (Herbig and Goodrich, 1986). Although the UV flux from TTS covers a range of values and is undoubtedly time-variable, detailed stud- ies with IUE (e.g., Valenti et al., 2000; Johns-Krull et al., 2000) and FUSE (e.g., Wilkinson et al., 2002; Bergin et al., 2003) indicate that it decreases into the FUV domain with a typical value ∼10−15erg cm−2s−1 Å−1, much smaller than earlier estimates. A flux of ∼10−15erg cm−2s−1 Å−1 at Earth translates into a value at 100 AU of ∼100 times the traditional Habing value for the interstellar medium. The data in the FUV range are sparse, unfortunately, as a func- tion of age or the evolutionary state of the system. More measurements of this kind are needed since it is obviously important to use realistic fluxes in the crucial FUV band be- tween 912 and 1100 Å where atomic C can be photoionized and H2 and CO photodissociated (Bergin et al., 2003 and the chapter by Bergin et al.). Whether stellar FUV or X-ray radiation dominates the ionization, chemistry, and heating of protoplanetary disks is important because of the vast difference in photon energy. The most direct physical consequence is that FUV photons cannot ionize H, and thus the abundance of carbon provides an upper limit to the ionization level produced by the pho- toionization of heavy atoms, xe ∼10 −4–10−3. Next, FUV photons are absorbed much more readily than X-rays, al- though this depends on the size and spatial distribution of the dust grains, i.e, on grain growth and sedimentation. Us- ing realistic numbers for the FUV and X-ray luminosities of TTS, we estimate that LFUV ∼ LX. The rates used in many early chemical models correspond to LX ≪ LFUV. This suggests that future chemical modeling of protoplan- etary disks should consider both X-rays and FUV in their treatment of ionization, heating, and chemistry. 3.3 The Midplane Region Unlike the warm upper atmosphere of the disk, which is accessible to observation, the optically thick midplane is much more difficult to study. Nonetheless, it is extremely important for understanding the dynamics of the basic flows in star formation such as accretion and outflow. The impor- tant role of the ionization level for disk accretion via the MRI was pointed out by Gammie (1996). The physical rea- son is that collisional coupling between electrons and neu- trals is required to transfer the turbulence in the magnetic field to the neutral material of the disk. Gammie found that Galactic cosmic rays cannot penetrate beyond a surface layer of the disk. He suggested that accretion only occurs in the surface of the inner disk (the “active region”) and not in the much thicker midplane region (the “dead zone”) where the ionization level is too small to mediate the MRI. Glassgold et al. (1997) argued that the Galactic cosmic rays never reach the inner disk because they are blown away by the stellar wind, much as the solar wind excludes Galac- tic cosmic rays. They showed that YSO X-rays do almost as good a job as cosmic rays in ionizing surface regions, thus preserving the layered accretion model of the MRI for YSOs. Igea and Glassgold (1999) supported this con- clusion with a Monte Carlo calculation of X-ray transport through disks, demonstrating that scattering plays an impor- tant role in the MRI by extending the active surface layer to column densities greater than 1025 cm−2, approaching the Galactic cosmic ray range used by Gammie (1996). This early work showed that the theory of disk ionization and chemistry is crucial for understanding the role of the MRI for YSO disk accretion and possibly for planet formation. Indeed, Glassgold, Najita, and Igea suggested that Gam- mie’s dead zone might provide a good environment for the formation of planets. These challenges have been taken up by several groups (e.g., Sano et al., 2000; Fromang et al., 2002; Semenov et al., 2004; Kunz and Balbus, 2004; Desch, 2004; Matsumura and Pudritz, 2003, 2005; and Ilgner and Nelson, 2006a,b). Fromang et al. discussed many of the issues that affect the size of the dead zone: differences in the disk model, such as a Hayashi disk or a standard α-disk; temporal evolution of the disk; the role of a small abundance of heavy atoms that recombine mainly radiatively; and the value of the mag- netic Reynolds number. Sano et al. (2000) explored the role played by small dust grains in reducing the electron fraction when it becomes as small as the abundance of dust grains. They showed that the dead zone decreases and even- tually vanishes as the grain size increases or as sedimenta- tion towards the midplane proceeds. More recently, Inut- suka and Sano (2005) have suggested that a small fraction of the energy dissipated by the MRI leads to the produc- tion of fast electrons with energies sufficient to ionize H2. When coupled with vertical mixing of highly ionized sur- face regions, Inutsuka and Sano argue that the MRI can self generate the ionization it needs to be operative throughout the entire disk. Recent chemical modeling (Semenov et al., 2004; Ilgner and Nelson, 2006a,b) confirms that the level of ionization in the midplane is affected by many microphysical processes. These include the abundances of radiatively-recombining atomic ions, molecular ions, small grains, and PAHs. The proper treatment of the ions represents a great challenge for disk chemistry, one made particularly difficult by the lack of observations of the dense gas at the midplane of the in- ner disk. Thus the uncertainties in inner disk chemistry pre- clude definitive quantitative conclusions about the midplane ionization of protoplanetary disks. Perhaps the biggest wild card is the issue of grain growth, emphasized anew by Se- menov et al., (2004). If the disk grain size distribution were close to interstellar, then the small grains would be effective in reducing the electron fraction and producing dead zones. But significant grain growth is expected and observed in the disks of YSOs, limiting the extent of dead zones (e.g., Sano et al., 2002). The broader chemical properties of the inner midplane region are also of great interest since most of the gas in the disk is within one or two scale heights. The chemical com- position of the inner midplane gas is important because it provides the initial conditions for outflows and for the for- mation of planets and other small bodies; it also determines whether the MRI operates. Relatively little work has been done on the midplane chemistry of the inner disk. For ex- ample, GNI04 excluded N and S species and restricted the carbon chemistry to species closely related to CO. However, Willacy et al. (1998), Markwick et al. (2002), and Ilgner et al. (2004) have carried out interesting calculations that shed light on a possible rich organic chemistry in the inner disk. Using essentially the same chemical model, these au- thors follow mass elements in time as they travel in a steady accretion flow towards the star. At large distances, the gas is subject to adsorption, and at small distances to thermal desorption. In between it reacts on the surface of the dust grains; on being liberated from the dust, it is processed by gas phase chemical reactions. The gas and dust are assumed to have the same temperature, and all effects of stellar radi- ation are ignored. The ionizing sources are cosmic rays and 26Al. Since the collisional ionization of low ionization po- tential atoms is ignored, a very low ionization level results. Markwick et al. improve on Willacy et al. by calculating the vertical variation of the temperature, and Ilgner et al. con- sider the effects of mixing. Near 1 AU, H2O and CO are very abundant, as predicted by simpler models, but Mark- wick et al. find that CH4 and CO have roughly equal abun- dances. Nitrogen-bearing molecules, such as NH3, HCN, and HNC are also predicted to be abundant, as are a vari- ety of hydrocarbons such as CH4, C2H2, C2H3, C2H4, etc. Markwick et al. also simulate the presence of penetrating X- rays and find increased column densities of CN and HCO+. Despite many uncertainties, these predictions are of interest for our future understanding of the midplane region. Infrared spectroscopic searches for hydrocarbons in disks may be able to test these predictions. For example, Gibb et al. (2004) searched for CH4 in absorption toward HL Tau. The upper limit on the abundance of CH4 relative to CO (<1%) in the absorbing gas may contradict the pre- dictions of Markwick et al. (2002) if the absorption arises in the disk atmosphere. However, some support for the Mark- wick et al. (2002) model comes from a recent report by Lahuis et al. (2006) of a rare detection by Spitzer of C2H2 and HCN in absorption towards a YSO, with ratios close to those predicted for the inner disk (Section 2.6). 3.4 Modeling Implications An interesting implication of the irradiated disk atmo- sphere models discussed above is that the region of the at- mosphere over which the gas and dust temperatures differ includes the region that is accessible to observational study. Indeed, the models have interesting implications for some of the observations presented in Section 2. They can ac- count roughly for the unexpectedly warm gas temperatures that have been found for the inner disk region based on the CO fundamental (Section 2.3) and UV fluorescent H2 tran- sitions (Section 2.4). In essence, the warm gas temperatures arise from the direct heating of the gaseous component and the poor thermal coupling between the gas and dust com- ponents at the low densities characteristic of upper disk at- mospheres. The role of X-rays in heating disk atmospheres has some support from the results of Bergin et al. (2004); they suggested that some of the UV H2 emission from TTS arises from excitation by fast electrons produced by X-rays. In the models, CO is found to form at a column den- sity NH≃10 21 cm−2 and temperature ∼1000 K in the ra- dial range 0.5–2 AU (GNI04; Fig. 5), conditions similar to those deduced for the emitting gas from the CO fundamen- tal lines (Najita et al., 2003). Moreover, CO is abundant in a region of the disk that is predominantly atomic hydrogen, a situation that is favorable for exciting the rovibrational transitions because of the large collisional excitation cross section for H + CO inelastic scattering. Interestingly, X- ray irradiation alone is probably insufficient to explain the strength of the CO emission observed in actively-accreting TTS. This suggests that other processes may be important in heating disk atmospheres. GNI04 have explored the role of mechanical heating. Other possible heating processes are FUV irradiation of grains and or PAHs. Molecular hydrogen column densities comparable to the UV fluorescent column of ∼5 × 1018cm−2 observed from TW Hya are reached at 1 AU at a total vertical hydrogen column density of ∼5 × 1021cm−2, where the fractional abundance of H2 is ∼10 −3 (GNI04; Fig. 5). Since Lyα photons must traverse the entire ∼5×1021cm−2 in order to excite the emission, the line-of-sight dust opacity through this column must be relatively low. Observations of this kind, when combined with atmosphere models, may be able to constrain the gas-to-dust ratio in disk atmospheres, with consequent implications for grain growth and settling. Work in this direction has been carried out by Nomura and Millar (2005). They have made a detailed thermal model of a disk that includes the formation of H2 on grains, destruction via FUV lines, and excitation by Lyα photons. The gas at the surface is heated primarily by the photoelec- tric effect on dust grains and PAHs, with a dust model ap- propriate for interstellar clouds, i.e., one that reflects little grain growth. Both interstellar and stellar UV radiation are included, the latter based on observations of TW Hya. The gas temperature at the surface of their flaring disk model reaches 1500 K at 1 AU. They are partially successful in ac- counting for the measurements of Herczeg et al. (2002), but their model fluxes fall short by a factor of five or so. A likely defect in their model is that the calculated tempera- ture of the disk surface is too low, a problem that might be remedied by reducing the UV attenuation by dust and by including X-ray or other surface heating processes. The relative molecular abundances that are predicted by these non-turbulent, layered model atmospheres are also of interest. At a distance of 1 AU, the calculations of GNI04 indicate that the relative abundance of H2O to CO is ∼10 in the disk atmosphere for column densities <1022 cm−2; only at column densities >1023 cm−2 are H2O and CO comparably abundant. The abundance ratio in the atmo- sphere is significantly lower than the few relative abun- dances measurements to date (0.1–0.5) at <0.3 AU (Carr et al., 2004; Section 2.2). Perhaps layered model atmo- spheres, when extended to these small radii, will be able to account for the abundant water that is detected. If not, the large water abundance may be evidence of strong vertical (turbulent) mixing that carries abundant water from deeper in the disk up to the surface. Thus, it would be of great in- terest to develop the modeling for the sources and regions where water is observed in the context of both layered mod- els and those with vertical mixing. Work in this direction has the potential to place unique constraints on the dynam- ical state of the disk. 4. CURRENT AND FUTURE DIRECTIONS As described in the previous sections, significant progress has been made in developing both observational probes of gaseous inner disks as well as the theoretical models that are needed to interpret the observations. In this section, we describe some areas of current interest as well as future directions for studies of gaseous inner disks. 4.1 Gas Dissipation Timescale The lifetime of gas in the inner disk is of interest in the context of both giant and terrestrial planet formation. Since significant gas must be present in the disk in order for a gas giant to form, the gas dissipation timescale in the gi- ant planet region of the disk can help to identify dominant pathways for the formation of giant planets. A short dissi- pation time scale favors processes such as gravitational in- stabilities which can form giant planets on short time scales (< 1000 yr; Boss, 1997; Mayer et al., 2002). A longer dissi- pation time scale accommodates the more leisurely forma- tion of planets in the core accretion scenario (few–10 Myr; Bodenheimer and Lin, 2002). Similarly, the outcome of terrestrial planet formation (the masses and eccentricities of the planets and their con- sequent habitability) may depend sensitively on the resid- ual gas in the terrestrial planet region of the disk at ages of a few Myr. For example, in the picture of terrestrial planet formation described by Kominami and Ida (2002), if the gas column density in this region is ≫1 g cm−2 at the epoch when protoplanets assemble to form terrestrial plan- ets, gravitational gas drag is strong enough to circularize the orbits of the protoplanets, making it difficult for them to collide and build Earth-mass planets. In contrast, if the gas column density is ≪1 g cm−2, Earth-mass planets can be produced, but gravitational gas drag is too weak to re- circularize their orbits. As a result, only a narrow range of gas column densities around ∼1 g cm−2 is expected to lead to planets with the Earth-like masses and low eccentricities that we associate with habitability on Earth. From an observational perspective, relatively little is known about the evolution of the gaseous component. Disk lifetmes are typically inferred from infrared excesses that probe the dust component of the disk, although processes such as grain growth, planetesimal formation, and rapid grain inspiraling produced by gas drag (Takeuchi and Lin, 2005) can compromise dust as a tracer of the gas. Our un- derstanding of disk lifetimes can be improved by directly probing the gas content of disks and using indirect probes of disk gas content such as stellar accretion rates (see Na- jita, 2006 for a review of this topic). Several of the diagnostics decribed in Section 2 may be suitable as direct probes of disk gas content. For example, transitions of H2 and other molecules and atoms at mid- through far-infrared wavelengths are thought to be promis- ing probes of the giant planet region of the disk (Gorti and Hollenbach, 2004). This is a important area of investigation currently for the Spitzer Space Telescope and, in the future, for Herschel and 8- to 30-m ground-based telescopes. Studies of the lifetime of gas in the terrestrial planet region are also in progress. The CO transitions are well suited for this purpose because the transitions of CO and its isotopes probe gas column densities in the range of inter- est (10−4 − 1 g cm−2). A current study by Najita, Carr, and Mathieu, which explores the residual gas content of optically thin disks (Najita, 2004), illustrates some of the challenges in probing the residual gas content of disks. Firstly, given the well-known correlation between IR ex- cess and accretion rate in young stars (e.g., Kenyon and Hartmann, 1995), CO emission from sources with optically thin inner disks may be intrinsically weak if accretion con- tributes significantly to heating disk atmospheres. Thus, high signal-to-noise spectra may be needed to detect this emission. Secondly, since the line emission may be intrin- sically weak, structure in the stellar photosphere may com- plicate the identification of emission features. Fig. 6 shows an example in which CO absorption in the stellar photo- sphere of TW Hya likely veils weak emission from the disk. Correcting for the stellar photosphere would not only am- plify the strong v=1–0 emission that is clearly present (cf. Rettig et al., 2004), it would also uncover weak emission in the higher vibrational lines, confirming the presence of the warmer gas probed by the UV fluorescent lines of H2 (Herczeg et al., 2002). Stellar accretion rates provide a complementary probe of the gas content of inner disks. In a steady accretion disk, the column density Σ is related to the disk accretion rate Ṁ by a relation of the form Σ ∝ Ṁ/αT , where T is the disk temperature. A relation of this form allows us to infer Σ from Ṁ given a value for the viscosity parameter α. Alter- natively, the relation could be calibrated empirically using measured disk column densities. Fig. 6.— (Top) Spectrum of the transitional disk system TW Hya at 4.6 µm (histogram). The strong emission in the v=1–0 CO fundamental lines extend above the plotted region. Although the model stellar photospheric spectrum (light solid line) fits the weaker features in the TW Hya spectrum, it predicts stronger ab- sorption in the low vibrational CO transitions (indicated by the lower vertical lines) than is observed. This suggests that the stellar photosphere is veiled by CO emission from warm disk gas. (Bot- tom) CO fundamental emission from the transitional disk system V836 Tau. Vertical lines mark the approximate line centers at the velocity of the star. The velocity widths of the lines locate the emission within a few AU of the star, and the relative strengths of the lines suggest optically thick emission. Thus, a large reservoir of gas may be present in the inner disk despite the weak infrared excess from this portion of the disk. Accretion rates are available for many sources in the age range 0.5–10 Myr (e.g., Gullbring et al., 1998; Hart- mann et al., 1998; Muzerolle et al., 1998, 2000). A typical value of 10−8M⊙ yr −1 for TTS corresponds to a(n active) disk column density of ∼100 g cm−2 at 1 AU for α=0.01 (D’Alessio et al., 1998). The accretion rates show an overall decline with time with a large dispersion at any given age. The existence of 10 Myr old sources with accretion rates as large as 10−8M⊙ yr −1 (Sicilia-Aguilar et al., 2005) sug- gests that gaseous disks may be long lived in some systems. Even the lowest measured accretion rates may be dy- namically significant. For a system like V836 Tau (Fig. 6), a ∼3 Myr old (Siess et al., 1999) system with an optically thin inner disk, the stellar accretion rate of 4 × 10−10M⊙ yr (Hartigan et al., 1995; Gullbring et al., 1998) would corre- spond to ∼4 g cm−2 at 1 AU. Although the accretion rate is irrelevant for the buildup of the stellar mass, it corresponds to a column density that would favorably impact terrestrial planet formation. More interesting perhaps is St34, a TTS with a Li depletion age of 25 Myr; its stellar accretion rate of 2× 10−10M⊙ yr −1 (White and Hillenbrand, 2005) sug- gests a dynamically significant reservoir of gas in the inner disk region. These examples suggest that dynamically sig- nificant reservoirs of gas may persist even after inner disks become optically thin and over the timescales needed to in- fluence the outcome of terrestrial planet formation. The possibility of long lived gaseous reservoirs can be confirmed by using the diagnostics described in Section 2 to measure total disk column densities. Equally important, a measured the disk column density, combined with the stel- lar accretion rate, would allow us to infer a value for viscos- ity parameter α for the system. This would be another way of constraining the disk accretion mechanism. 4.2 Nature of Transitional Disk Systems Measurements of the gas content and distribution in in- ner disks can help us to identify systems in various states of planet formation. Among the most interesting objects to study in this context are the transitional disk systems, which possess optically thin inner and optically thick outer disks. Examples of this class of objects include TW Hya, GM Aur, DM Tau, and CoKu Tau/4 (Calvet et al., 2002; Rice et al., 2003; Bergin et al., 2004; D’Alessio et al., 2005; Calvet et al., 2005). It was suggested early on that optically thin in- ner disks might be produced by the dynamical sculpting of the disk by orbiting giant planets (Skrutskie et al., 1990; see also Marsh and Mahoney, 1992). Indeed, optically thin disks may arise in multiple phases of disk evolution. For example, as a first step in planet for- mation (via core accretion), grains are expected to grow into planetesimals and eventually rocky planetary cores, produc- ing a region of the disk that has reduced continuum opac- ity but is gas-rich. These regions of the disk may there- fore show strong line emission. Determining the fraction of sources in this phase of evolution may help to establish the relative time scales for planetary core formation and the accretion of gaseous envelope. If a planetary core accretes enough gas to produce a low mass giant planet (∼1MJ), it is expected to carve out a gap in its vicinity (e.g., Takeuchi et al., 1996). Gap crossing streams can replenish an inner disk and allow further accre- tion onto both the star and planet (Lubow et al., 1999). The small solid angle subtended by the accretion streams would produce a deficit in the emission from both gas and dust in the vicinity of the planet’s orbit. We would also expect to detect the presence of an inner disk. Possible examples of systems in this phase of evolution include GM Aur and TW Hya in which hot gas is detected close to the star as is accre- tion onto the star (Bergin et al., 2004; Herczeg et al., 2002; Muzerolle et al., 2000). The absence of gas in the vicinity of the planet’s orbit would help to confirm this interpretation. Once the planet accretes enough mass via the accretion streams to reach a mass ∼5–10MJ , it is expected to cut off further accretion (e.g., Lubow et al., 1999). The inner disk will accrete onto the star, leaving a large inner hole and no trace of stellar accretion. CoKu Tau/4 is a possible example of a system in this phase of evolution (cf. Quillen et al., 2004) since it appears to have a large inner hole and a low to negligible accretion rate (<few×10−10M⊙ yr −1). This interpretation predicts little gas anywhere within the orbit of the planet. At late times, when the disk column density around 10 AU has decreased sufficiently that the outer disk is being photoevaporated away faster than it can resupply material to the inner disk via accretion, the outer disk will decouple from the inner disk, which will accrete onto the star, leav- ing an inner hole that is devoid of gas and dust (the “UV Switch” model; Clarke et al., 2001). Measurements of the disk gas column density and the stellar accretion rate can be used to test this possibility. As an example, TW Hya is in the age range (∼10 Myr) where photoevaporation is likely to be significant. However, the accretion rate onto star, gas content of the inner disk (Sections 2 and 4), as well as the column density inferred for the outer disk (32 g cm−2 at 20 AU based on the dust SED; Calvet et al., 2002) are all much larger than is expected in the UV switch model. Al- though this mechanism is, therefore, unlikely to explain the SED for TW Hya, it may explain the presence of inner holes in less massive disk systems of comparable age. 4.3 Turbulence in Disks Future studies of gaseous inner disks may also help to clarify the nature of the disk accretion process. As indicated in Section 2.1, evidence for suprathermal line broadening in disks supports the idea of a turbulent accretion process. A turbulent inner disk may have important consequences for the survival of terrestrial planets and the cores of giant plan- ets. An intriguing puzzle is how these objects avoid Type-I migration, which is expected to cause the object to lose an- gular momentum and spiral into the star on short timescales (e.g., Ward, 1997). A recent suggestion is that if disk accre- tion is turbulent, terrestral planets will scatter off turbulent fluctuations, executing a “random walk” which greatly in- creases the migration time as well as the chances of survival (Nelson et al., 2000; see chapter by Nelson et al.). It would be interesting to explore this possible connec- tion further by extending the approach used for the CO over- tone lines to a wider range of diagnostics to probe the intrin- sic line width as a function of radius and disk height. By comparing the results to the detailed predictions of theoret- ical models, it may be possible to distinguish between the turbulent signature, produced e.g., by the MRI instability, from the turbulence that might be produced by, e.g., a wind blowing over the disk. A complementary probe of turbulence may come from exploring the relative molecular abundances in disks. As noted in Section 3.4, if relative abundances cannot be ex- plained by model predictions for non-turbulent, layered ac- cretion flows, a significant role for strong vertical mixing produced by turbulence may be implied. Although model- dependent, this approach toward diagnosing turbulent ac- cretion appears to be less sensitive to confusion from wind- induced turbulence, especially if one can identify diagnos- tics that require vertical mixing from deep down in the disk. Another complementary approach toward probing the ac- cretion process, discussed in Section 4.1, is to measure to- tal gas column densities in low column density, dissipating disks in order to infer values for the viscosity parameter α. 5. SUMMARY AND CONCLUSIONS Recent work has lent new insights on the structure, dy- namics, and gas content of inner disks surrounding young stars. Gaseous atmospheres appear to be hotter than the dust in inner disks. This is a consequence of irradiative (and pos- sibly mechanical) heating of the gas as well as the poor ther- mal coupling between the gas and dust at the low densities of disk atmospheres. In accreting systems, the gaseous disk appears to be turbulent and extends inward beyond the dust sublimation radius to the vicinity of the corotation radius. There is also evidence that dynamically significant reser- voirs of gas can persist even after the inner disk becomes optically thin in the continuum. These results bear on im- portant star and planet formation issues such as the origin of winds, funnel flows, and the rotation rates of young stars; the mechanism(s) responsible for disk accretion; and the role of gas in the determining the architectures of terres- trial and giant planets. Although significant future work is needed to reach any conclusions on these issues, the fu- ture for such studies is bright. Increasingly detailed studies of the inner disk region should be possible with the advent of powerful spectrographs and interferometers (infrared and submillimeter) as well as sophisticated models that describe the coupled thermal, chemical, and dynamical state of the disk. Acknowledgments. We thank Stephen Strom who con- tributed significantly to the discussion on the nature of tran- sitional disk systems. We also thank Fred Lahuis and Matt Richter for sharing manuscripts of their work in advance of publication. 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