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0704.1943

The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes

The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes Pavle V. M. Blagojević∗ Mathematički Institut Knez Michailova 35/1 11001 Beograd, Serbia [email protected] Günter M. Ziegler∗∗ Inst. Mathematics, MA 6-2 TU Berlin D-10623 Berlin, Germany [email protected] December 9, 2010 Abstract We compute the complete Fadell–Husseini index of the dihedral group D8 = (Z2) 2 ⋊ Z2 acting on Sd × Sd for F2 and for Z coefficients, that is, the kernels of the maps in equivariant cohomology (pt,F2) −→ H (pt,Z) −→ H∗D8(S This establishes the complete cohomological lower bounds, with F2 and with Z coefficients, for the two-hyperplane case of Grünbaum’s 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably chosen hyperplanes in Rd? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of D8. Contents 1 Introduction 2 1.1 The hyperplane mass partition problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Statement of the main result (k = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Proof overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Evaluation of the index bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 F2-evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.2 Z-evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Configuration space/Test map scheme 8 2.1 Configuration space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Test map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The test space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ∗The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels. Also supported by the grant ON 174008 of the Serbian Ministry of Science and Environment. ∗∗The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels. http://arxiv.org/abs/0704.1943v4 [email protected] [email protected] 3 The Fadell–Husseini index theory 11 3.1 Equivariant cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 IndexG,R and Index G,R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 The restriction map and the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Basic calculations of the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4.1 The index of a product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4.2 The index of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 The cohomology of D8 and the restriction diagram 17 4.1 The poset of subgroups of D8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 The cohomology ring H∗(D8,F2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 The cohomology diagram of subgroups with coefficients in F2 . . . . . . . . . . . . . . . . 19 4.3.1 The Z2 × Z2-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.2 The D8-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 The cohomology ring H∗(D8,Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4.1 Evens’ view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4.2 The Bockstein spectral sequence view . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 The D8-diagram with coefficients in Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5.1 The Z2 × Z2-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5.2 The D8-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 IndexD8,F2S(R 4 ) 27 5.1 IndexD8,F2S(V−+ ⊕ V+−) = 〈w〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 IndexD8,F2S(V−−) = 〈y〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3 IndexD8,F2S(R 4 ) = 〈y jwj〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 IndexD8,ZS(R 4 ) 28 6.1 The case when j is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.2 The case when j is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7 IndexD8,F2S d×Sd 33 7.1 The d-th row as an H∗(D8,F2)-module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7.2 Indexd+2D8,F2S d × Sd = 〈πd+1, πd+2〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.3 IndexD8,F2S d × Sd = 〈πd+1, πd+2, w d+1〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.4 An alternative proof, sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8 IndexD8,ZS d×Sd 39 8.1 The d-th row as an H∗(D8,Z)-module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 8.1.1 The case when d is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 8.1.2 The case when d is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.2 Indexd+2D8,ZS d × Sd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 8.2.1 The case when d is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 8.2.2 The case when d is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1 Introduction 1.1 The hyperplane mass partition problem A mass distribution on Rd is a finite Borel measure µ(X) = fdµ determined by an integrable density function f : Rd → R. Every affine hyperplane H = {x ∈ Rd | 〈x, v〉 = α} in Rd determines two open halfspaces H− = {x ∈ Rd | 〈x, v〉 < α} and H+ = {x ∈ Rd | 〈x, v〉 > α}. An orthant of an arrangement of k hyperplanesH = {H1, H2, . . . , Hk} in R d is an intersection of halfspaces O = Hα11 ∩ · · · ∩ H k , for some αj ∈ Z2. Thus there are 2 k orthants determined by H and they are naturally indexed by elements of the group (Z2) An arrangement of hyperplanes H equiparts a collection of mass distributions M in Rd if for each orthant O and each measure µ ∈ M we have µ(O) = 1 µ(Rd). A triple of integers (d, j, k) is admissible if for every collectionM of j mass distributions in Rd there exists an arrangement of k hyperplanes H equiparting them. The general problem formulated by Grünbaum [16] in 1960 can be stated as follows. Problem 1.1. Determine the function ∆ : N2 → N given by ∆(j, k) = min{d | (d, j, k) is an admissible triple}. The case of one hyperplane, ∆(j, 1) = j, is the famous ham sandwich theorem, which is equivalent to the Borsuk–Ulam theorem. The equality ∆(2, 2) = 3, and consequently ∆(1, 3) = 3, was proven by Hadwiger [17]. Ramos [30] gave a general lower bound for the function ∆, ∆(j, k) ≥ 2 j. (1) Recently, Mani-Levitska, Vrećica and Živaljević [26] applied Fadell–Husseini index theory for an elemen- tary abelian subgroup (Z2) k of the Weyl group Wk = (Z2) k ⋊ Sk to obtain a new upper bound for the function ∆, ∆(2q + r, k) ≤ 2k+q−1 + r. (2) In the case of j = 2l+1 − 1 measures and k = 2 hyperplanes these bounds yield the equality ∆(j, 2) = ⌈ 3 1.2 Statement of the main result (k = 2) This paper addresses Problem 1.1 for k = 2 using two different but related Configuration Space/Test Map schemes (Section 2, Proposition 2.2). • The product scheme is the classical one, already considered in [32] and [26]. The problem is translated to the problem of the existence of a Wk-equivariant map, Yd,k := (R2k) where Wk = (Z2) k ⋊ Sk is the Weyl group. • The join scheme is a new one. It connects the problem with classical Borsuk–Ulam properties in the spirit of Marzantowicz [27]. It asks the question whether there exists a Wk-equivariant map Xd,k := Uk × (R2k) The Wk-representations R2k and Uk are introduced in Section 2.2. Obstruction theory methods cannot be applied to either scheme directly for k > 1, since the Wk-actions on the respective configuration spaces are not free (compare [26, Section 2.3.3], assumptions on the manifold Mn). Therefore we analyze the associated equivariant question for k = 2 via the Fadell–Husseini ideal index theory method. We show that the join scheme considered from the Fadell–Husseini point of view, with either F2 or Z coefficients, yields no obstruction to the existence of the equivariant map in question (Remarks 5.3 and 6.3). In the case of the product scheme we give the ideal bounds obtained from the use of the full group of symmetries by proving the following theorem. Theorem 1.2. Let πd, d ≥ 0, be polynomials in F2[y, w] given by πd(y, w) = d− 1− i wiyd−2i and Πd, d ≥ 0, be polynomials in Z[Y,M,W ]/〈2Y, 2M, 4W ,M 2 −WY〉 given by Πd(Y,W) = d− 1− i W iYd−2i. (A) F2-bound: The triple (d, j, 2) ∈ N 3 is admissible if yjwj /∈ 〈πd+1, πd+2〉 ⊆ F2[y, w]. (B) Z-bound: The triple (d, j, 2) ∈ N3 is admissible if (j − 1)mod2 Y jmod2 Y 2 M, jmod 2 Y 〈 (d− 1)mod 2 Π d+2 , (d− 1)mod2 Π d+4 (d− 1)mod 2 MΠ d dmod 2 Π d+1 , dmod 2 Π d+3 in the ring Z[Y,M,W ]/〈2Y, 2M, 4W ,M2 −WY〉. Remark 1.3. Let Π̂d, d ≥ 0, be the sequence of polynomials in Z[Y,W ] defined by Π̂0 = 0, Π̂1 = Y and Π̂d+1 = Y Π̂d +W Π̂d−1 for d ≥ 2. Then the sequences of polynomials Πd and πd are reductions of the polynomials Π̂d. The polynomials Π̂d can be also described by the generating function (formal power series) Π̂d = 1− Y −W where Π̂d is homogeneous of degree 2d if we set deg(Y ) = 2 and deg (W ) = 4. Theorem 1.2 is a consequence of a topological result, the complete and explicit computation of the relevant Fadell–Husseini indexes of the D8-space S d × Sd and the D8-sphere S(R Theorem 1.4. (A) Index D8,F2 4 ) = IndexD8,F2S(R 4 ) = 〈y jwj〉. (B) Indexd+2D8,F2(S d × Sd) = 〈πd+1, πd+2〉. (C) Index 4 ) = 2 〉 , for j even, 2 M,Y 2 〉 , for j odd. (D) Indexd+2D8,ZS d × Sd = 〈Π d+2 ,Π d+4 ,MΠ d 〉 , for d even, 〈Π d+1 ,Π d+3 〉 , for d odd. The sequence of Fadell–Husseini indexes will be introduced in Section 3. The actions of the dihedral group D8 and the definition of the representation space R 4 are given in Section 2. Even though it does not seem to have any relevance to our study of Problem 1.1, the complete index IndexD8,F2(S d×Sd) will also be computed in the case of F2 coefficients, IndexD8,F2(S d × Sd) = 〈πd+1, πd+2, w d+1〉. (3) Final remark 1.5. The preprint versions of this paper, posted on the arXiv in April 2007 and July 2008, arXiv0704.1943v1–v2, have been referenced in diverse applications: see Gonzalez and Landweber [15], Adem and Reichstein [2], as well as [6]. 1.3 Proof overview The Problem 1.1 about mass partitions by hyperplanes can be connected with the problem of the existence of equivariant maps as discussed in Section 2, Proposition 2.2. The topological problems we face, about the existence of Wk = (Z2) k ⋊ Sk-equivariant maps, for the product / join schemes, Uk ×R have to be treated with care because the actions of the Weyl groups Wk are not free. Note that there is no naive Borsuk–Ulam theorem for fixed point free actions. Indeed, in the case k = 2 when W2 = D8 there exists a W2-equivariant map [5, Theorem 3.22, page 49] (V+− ⊕ V−+) (U2 ⊕ V−−) even though dim (V+− ⊕ V−+) > dim (U2 ⊕ V−−) . The W2 = D8-representations V+− ⊕ V−+, V−− and U2 are introduced in Section 2.2. In this paper we focus on the case of k = 2 hyperplanes. Theorem 1.2 gives the best possible answer to the question about the existence of W2 = D8-equivariant maps Sd × Sd −→ S(R from the point of view of Fadell–Husseini index theory (Section 3). We explicitly compute the relevant Fadell–Husseini indexes with F2 and Z coefficients (Theorem 1.4, Sections 5, 6, 7 and 8). Then Theorem 1.2 is a consequence of the basic index property, Proposition 3.2. The index of the sphere S(R 4 ), with F2 coefficients, is computed in Section 5 by • decomposition of the D8-representation R 4 into a sum of irreducible ones, and • computation of indexes of spheres of all irreducible D8-representations. The main technical tool is the restriction diagram derived in Section 4.3.2, which connects the indexes of the subgroups of D8. The index with Z coefficients is computed in Section 6 using • (for j even) the results for F2 coefficients and comparison of Serre spectral sequences, and • (for j odd) the Bockstein spectral sequence combined with known results for F2 coefficients and comparison of Serre spectral sequences. The index of the product Sd × Sd is computed in Sections 7 and 8 by an explicit study of the Serre spectral sequence associated with the fibration Sd × Sd → ED8 ×D8 (S d × Sd)→ BD8. The major difficulty comes from non-triviality of the local coefficients in the Serre spectral sequence. The computation of the spectral sequence with non-trivial local coefficients is done by an independent study of H∗(D8,F2)-module and H ∗(D8,Z)-module structures of relevant rows in the Serre spectral sequence (Sections 7.1 and 8.1). 1.4 Evaluation of the index bounds 1.4.1 F2-evaluation It was pointed out to us by Sinǐsa Vrećica that, with F2-coefficients, the D8 index bound gives the same bounds as the H1 = (Z2) index bound. This observation follows from the implication ajbj(a+ b)j ∈ 〈ad+1, (a+ b)d+1〉 ⇒ ajbj(a+ b)j ∈ 〈ad+1 + (a+ b)d+1, ad+2 + (a+ b)d+2〉. By introducing a new variable c := a+ b, it is enough to prove the implication ajcj(a+ c)j ∈ 〈ad+1, cd+1〉 ⇒ ajcj(a+ c)j ∈ 〈ad+1 + cd+1, ad+2 + cd+2〉. (4) Let us assume that ajcj(a + c)j ∈ 〈ad+1, cd+1〉. The monomials in the expansion of ajcj(a + c)j always come in pairs ad+kc3j−d−k + cd+ka3j−d−k. This is also true when j is even since =mod2 0 implies there are no middle terms. The sequence of equations ad+1c3j−d−1 + cd+1a3j−d−1 = (ad+1 + cd+1)(c3j−d−1 + a3j−d−1) + a3j + c3j ad+2c3j−d−2 + cd+2a3j−d−2 = (ad+1 + cd+1)(ac3j−d−2 + a3j−d−2c) + a3j−1c+ ac3j−1 . . . . a3j + c3j = (ad+2 + cd+2)(a3j−d−2 + c3j−d−2) + ad+2c3j−d−2 + cd+2a3j−d−2 shows that all the binomials ad+1c3j−d−1 + cd+1a3j−d−1, ad+2c3j−d−2 + cd+2a3j−d−2, . . . , a3j + c3j belong to the ideal 〈ad+1 + cd+1, ad+2 + cd+2〉 or none of them do. Since for 3j − d− 1 even 3j−d−1 3j−d−1 2 + c 3j−d−1 3j−d−1 2 = (ad+1 + cd+1)a 3j−d−1 3j−d−1 ∈ 〈ad+1 + cd+1, ad+2 + cd+2〉 and for 3j − d− 1 odd ad+2+ 3j−d−2 3j−d−2 2 + cd+2+ 3j−d−2 3j−d−2 2 = (ad+2 + cd+2)a 3j−d−2 3j−d−2 ∈ 〈ad+1 + cd+1, ad+2 + cd+2〉 the implication (4) is proved. 1.4.2 Z-evaluation More is true, even the complete D8 index bound, now with Z-coefficients, implies the same bounds as does the subgroup H1 = (Z2) for the k = 2 hyperplanes mass partition problem. Lemma 1.6. Let a = i and b = i be the dyadic expansions. Then mod 2 This classical fact [25] about binomial coefficients mod 2 yields the following property for the sequence of polynomials Πd, d ≥ 0. Lemma 1.7. Let q > 0 and i be integers. Then 2q−1−i 0, i 6= 0 1, i = 0 (B) Π2q = Y Proof. The statement (B) is a direct consequence of the fact (A) and the definition of polynomials Πd. For i /∈ {1, . . . , 2q−1} the statement (A) is true from boundary conditions on binomial coefficients. Let i ∈ {1, . . . , 2q−1} and i = k∈I⊆{0,...,q−1} 2k. Then 2q − 1− i = 20 + 21 + 22 + · · ·+ 2q−1 − k∈I⊆{0,...,q−1} k∈Ic⊆{0,...,q−1} where Ic is the complementary index set in {0, . . . , q−1}. The statement (A) follows from Lemma 1.6 Let j be an integer such that j = 2q + r where 0 ≤ r < 2q and d = 2q+1 + r − 1. Let us introduce the following ideals 2 〉, for j even, 2 M,Y 2 〉, for j odd, and Bd = 〈Π d+2 ,Π d+4 ,MΠ d 〉, for d even, 〈Π d+1 ,Π d+3 〉, for d odd. The fact that theD8 index bound with Z-coefficients does not improve the mass partition bounds obtained by using the subgroup H1 = (Z2) is a consequence of the following facts: • r = 0 ⇒ Aj ⊆ Bd, • (r 6= 2q − 1 and Aj ⊆ Bd) =⇒ Aj+1 ⊆ Bd+1, that are proved in Lemma 1.8 and 1.9, respectively. Lemma 1.8. 〈Y2 〉 = A2q ⊆ B2q+1−1 = 〈Π2q ,Π2q+1〉. Proof. Since Y2 = Π2q−1 by Lemma 1.7, W = Π2q−1W = Π2q−1+2 + YΠ2q−1+1 ∈ 〈Π2q−1+1,Π2q−1+2〉. By induction on the power i of W in Y2 W i ∈ 〈Π2q−1+i,Π2q−1+i+1〉, and consequently ∈ 〈Π2q ,Π2q+1〉. Lemma 1.9. If r 6= 2q − 1 and Aj ⊆ Bd then Aj+1 ⊆ Bd+1. Proof. We distinguish two cases depending on the parity of j. (A) Let j be even and Y 2 ∈ 〈Π d+1 ,Π d+3 〉. There are polynomials α and β such that 2 = αΠ d+1 + βΠ d+3 (j+1)+1 (j+1)−1 2 M = Y 2M = YM αΠ d+1 + βΠ d+3 ∈ 〈Π (d+1)+2 ,MΠ d+1 〉 ⊆ 〈Π d+3 ,Π d+5 ,MΠ d+1 〉 = Bd+1, (j+1)+1 (j+1)+1 2 = YW αΠ d+1 + βΠ d+3 = αM2Π d+1 + βYWΠ d+3 ∈ 〈MΠ d+1 ,Π d+3 〉 ⊆ 〈Π d+3 ,Π d+5 ,MΠ d+1 〉 = Bd+1. Thus Aj+1 ⊆ Bd+1. (B) Let j be odd and 2 M,Y 2 〉 = Aj ⊆ Bd = 〈Π d+2 ,Π d+4 ,MΠ d There are polynomials α, β and γ such that 2 = αΠ d+2 + βΠ d+4 + γMΠ d and no occurrence of the defining relation Π d+4 = YΠ d+2 +WΠ d , Remark 1.3, can be subtracted from the presentation. Then γMΠ d ∈ 〈Π d+2 ,Π d+4 〉, and sinceM is of odd degree γ =Mγ′. In the first case the inclusion Aj+1 ⊆ Bd+1 follows directly. Consider γ =Mγ ′. Since (Y+X )WΠi = YWΠi for every i > 0, we have that 2 = αΠ d+2 + βΠ d+4 + γ′M2Π d = αΠ d+2 + βΠ d+4 + γ′YWΠ d = αΠ d+2 + βΠ d+4 + γ′Y(YΠ d +1 +Π d +2) ∈ 〈Π d+2 ,Π d+4 〉 = Bd+1. Thus Aj+1 ⊆ Bd+1. Acknowledgements We are grateful to Jon Carlson and to Carsten Schultz for useful comments and insightful observations. The referee provided many useful suggestions and comments that are incorporated in the latest version of the manuscript. Some of this work was done in the framework of the MSRI program “Computational Applications of Algebraic Topology” in the fall semester 2006. 2 Configuration space/Test map scheme The Configuration Space/Test Map (CS/TM) paradigm (formalized by Živaljević in [31], and also beau- tifully exposited by Matoušek in [28]) has been very powerful in the systematic derivation of topological lower bounds for problems of Combinatorics and of Discrete Geometry. In many instances, the problem suggests natural configuration spacesX , Y , a finite symmetry groupG, and a test set Y0 ⊂ Y , where one would try to show that every G-equivariant map f : X → Y must hit Y0. The canonical tool is then Dold’s theorem, which says that if the group actions are free, then the map f must hit the test set Y0 ⊂ Y if the connectivity of X is higher than the dimension of Y \ Y0. For the success of this “canonical approach” one crucially needs that a result such as Dold’s theorem is applicable. Thus the group action must be free, so one often reduces the group action to a prime order cyclic subgroup of the full symmetry group, and results may follow only in “the prime case” , or with more effort and deeper tools in the prime power case. The main example for this is the Topological Tverberg Problem, which is still not resolved for (d, q) if d > 1 and q is not a prime power [28, Section 6.4, page 165]. So in general one has to work much harder when the “canonical” approach fails. In the following, we present configuration spaces and test maps for the mass partition problem. 2.1 Configuration space The space of all oriented affine hyperplanes in Rd can be naturally identified with the subspace of the sphere Sd obtained by removing two points, namely the “oriented hyperplanes at infinity”. Indeed, let Rd be embedded in Rd+1 by (x1, . . . , xd) 7−→ (x1, . . . , xd, 1). Then every oriented affine hyperplane H in Rd determines a unique oriented hyperplane H̃ through the origin in Rd+1 such that H̃ ∩ Rd = H , and conversely if the hyperplane at infinity is included. The oriented hyperplane uniquely determined by the unit vector v ∈ Sd is denoted by Hv and the assumed orientation is determined by the half-space H Then H−−v = H v . The obvious and classically used candidate for the configuration space associated with the problem of testing admissibility of (d, j, k) is Yd,k = The relevant group acting on this space is the Weyl group Wk = (Z2) k ⋊ Sk. Each Z2 = ({+1,−1}, ·) acts antipodally on the appropriate copy of Sd (changing the orientation of hyperplanes), while Sk acts by permuting copies. The second configuration space that we can use is Xd,k = S d ∗ · · · ∗ Sd ︸ ︷︷ ︸ k copies ∼= Sdk+k−1. The elements of Xd,k are denoted by t1v1 + · · ·+ tkvk, with ti ≥ 0, t1 = 1, vi ∈ S d. The Weyl group Wk acts on Xd,k by εi · (t1v1 + · · ·+ tivi + · · ·+ tkvk) = t1v1 + · · ·+ ti(−vi) + · · ·+ tkvk, π · (t1v1 + · · ·+ tivi + · · ·+ tkvk) = tπ−1(1)vπ−1(1) + · · ·+ tπ−1(i)vπ−1(i) + · · ·+ tπ−1(k)vπ−1(k), where εi is the generator of the i-th copy of Z2 and π ∈ Sk is an arbitrary permutation. 2.2 Test map LetM = {µ1, . . . , µj} be a collection of mass distributions in R d. Let the coordinates of R2 be indexed by the elements of the group (Z2) k. The Weyl group Wk acts on R 2k by acting on its coordinate index set (Z2) k in the following way: ((β1, . . . , βk)⋊ π) · (α1, . . . , αk) = β1απ−1(1), . . . , βkαπ−1(k) The test map φ : Yd,k → (R 2k)j used with the configuration space Yd,k is a Wk-equivariant map given by φ (v1, . . . , vk) = ∩ · · · ∩Hαkvk )− (α1,...,αk)∈(Z2)k i∈{1,...,j} Denote the i-th component of φ by φi, i = 1, . . . , j. To define a test map associated with the configuration space Xd,k, we discuss the (Z2) k- and Wk- module structures on R2 All irreducible representations of the group (Z2) k are 1-dimensional. They are in bijection with the homomorphisms (characters) χ : (Z2) k → Z2. These homomorphisms are completely determined by the values on generators ε1,. . . ,εk of (Z2) k, i.e. by the vector (χ(ε1), . . . , χ(εk)). For (α1, . . . , αk) ∈ (Z2) k let Vα1...αk = span{vα1...αk} ⊂ R 2k denote the 1-dimensional representation given by εi · vα1...αk = αi vα1...αk The vector vα1...αk ∈ {+1,−1} 2k is uniquely determined up to a scalar multiplication by −1. Note that 〈vα1...αk , vβ1...βk〉 = 0 for α1 . . . αk 6= β1 . . . βk. For k = 2, with the abbreviation + for +1, − for −1, the coordinate index set for R4 is {++,+−,−+,−−}. Then v++ = (1, 1, 1, 1) , v+− = (1,−1, 1,−1), v−+ = (1, 1,−1,−1) , v−− = (1,−1,−1, 1). The following decomposition of (Z2) k-modules holds, with the index identification (Z2) k = {+,−}k, k ∼= V+···+ ⊕ α1...αk∈(Z2)k\{+···+} Vα1...αk where V+···+ is the trivial (Z2) k-representation. Let R2k denote the orthogonal complement of V+···+ and π : R2 → R2k the associated (equivariant) projection. Explicitly R2k = {(x1, . . . , x2k) ∈ R xi = 0} = α1...αk∈(Z2)k\{+···+)} Vα1...αk , (5) x = (x1, . . . , x2k) 7−→ 1 〈x, vα1...αk)〉 α1...αk∈(Z2)k\{+···+} where 〈·, ·〉 denotes the standard inner product of R2 . Observe that im φ = φ(Yd,k) ⊆ (R2k) Let α1 . . . αk ∈ (Z2) k and let η(α1 . . . αk) = αi). The following decomposition of Wk-modules holds k ∼= V+···+ ⊕ n=η(α1,...,αk) Vα1...αk ∼= V+···+ ⊕R2k . (6) The test map τ : Xd,k → Uk × (R2k) is defined by τ (t1v1 + · · ·+ tkvk) = , . . . , tk − 1 · · · t k 〈φi (v1, . . . , vk) , vα1...αk〉 α1...αk∈(Z2)k\{+···+} Here Uk = {(ξ1, . . . , ξk) ∈ R ξi = 0} is a Wk-module with an action given by ((β1, . . . , βk)⋊ π) · (ξ1, . . . , ξk) := ξπ−1(1), . . . , ξπ−1(k) The subgroup (Z2) k acts trivially on Uk. The action on Uk× (R2k) is assumed to be the diagonal action. The test map τ is well defined, continuous and Wk-equivariant. Example 2.1. The test map τ : Xd,k → Uk × (R2k) is in the case of k = 2 hyperplanes and j = 1 measure given by τ : Xd,2 → U2 ×R4 = U2 × ((V+− ⊕ V−+)⊕ V−−) and τ (t1v1 + t2v2) = , t2 − t1〈φ (v1, v2) , v−+〉, t2〈φ (v1, v2) , v+−〉, t1t2〈φ (v1, v2) , v−−〉) where φ (v1, v2) = ∩Hα2v2 )− µ(Rd) α1α2∈(Z2)2 ∈ R4. 2.3 The test space The test spaces for the maps φ and τ are the origins of (R2k) and Uk × (R2k) , respectively. The constructions that we perform in this section satisfy the usual hypotheses for the CS/TM scheme. Proposition 2.2. (i) For a collection of mass distributions M = {µ1, . . . , µj} let φ : Yd,k → (R2k) and τ : Xd,k → Uk × (R2k) be the corresponding test maps. If (0, . . . , 0) ∈ φ (Yd,k) or (0, . . . , 0) ∈ τ (Xd,k) then there exists an arrangement of k hyperplanes H in Rd equiparting the collection M. (ii) If there is no Wk-equivariant map with respect to the actions defined above, Yd,k → (R2k) \{(0, . . . , 0)}, or Yd,k → S (R2k) ≈ Sj(2 k−1)−1, or Xd,k → Uk × (R2k) \{(0, . . . , 0)}, or Xd,k → S Uk × (R2k) ≈ Sj(2 k−1)+k−2, then the triple (d, j, k) is admissible. (iii) Specifically, for k = 2, if there is no D8 ∼= W2 equivariant map, with the already defined actions, Yd,2 → (R4) \{(0, . . . , 0)}, or Yd,2 → S ≈ S3j−1, or Xd,2 → U2 × (R4) \{(0, . . . , 0)}, or S2d+1 ≈ Xd,2 → S U2 × (R4) ≈ S3j , then the triple (d, j, 2) is admissible. Remark 2.3. The action of Wk on the sphere S(U2 × (R4) j) is fixed point free, but not free. For k = 2, the action of the unique Z4 subgroup of W2 = D8 on the sphere S(U2 × (R4) j) is fixed point free. The necessary condition for the non-existence of an equivariant Wk-map Xd,k → S(Uk × (R2k) implied by the equivariant Kuratowski–Dugundji theorem [4, Theorem 1.3, page 25] is dk + k − 1 > j(2k − 1) + k − 2 ⇐⇒ d ≥ 2 j . (7) For k = 2 the condition (7) becomes d ≥ ⌈ 3 j⌉. (8) 3 The Fadell–Husseini index theory 3.1 Equivariant cohomology Let X be a G -space and X → EG×GX → BG the associated universal bundle, with X as a typical fibre. EG is a contractible cellular space on which G acts freely, and BG := EG/G. The space EG ×G X = (EG×X) /G is called the Borel construction of X with respect to the action of G. The equivariant cohomology of X is the ordinary cohomology of the Borel construction EG×G X , H∗G(X) := H ∗(EG×G X). The equivariant cohomology is a module over the ring H∗G(pt) = H ∗(BG). When X is a free G-space the homotopy equivalence EG×G X ≃ X/G induces a natural isomorphism H∗G(X) ∗(X/G). The universal bundle X → EG ×G X → BG, for coefficients in the ring R, induces a Serre spectral sequence converging to the graded group Gr(H∗G(X,R)) associated with H G(X,R) appropriately filtered. In this paper “ring” means commutative ring with a unit element. The E2-term is given by p(BG,Hq(X,R)), (9) where Hq(X,R) is a system of local coefficients. For a discrete group G, the E2-term of the spectral sequence can be interpreted as the cohomology of the groupG with coefficients in the G-moduleH∗(X,R), ∼= Hp(G,Hq(X,R)). (10) 3.2 IndexG,R and Index Let X be a G-space, R a ring and π∗X the ring homomorphism in cohomology π∗X : H ∗(BG,R)→ H∗(EG×G X,R) induced by the projection EG×G X → EG×G pt ≈ BG. The Fadell–Husseini (ideal-valued) index of a G-space X is the kernel ideal of π∗X , IndexG,RX := kerπ X ⊆ H ∗(BG,R). The Serre spectral sequence (9) yields a representation of the homomorphism π∗X as the composition H∗(BG,R)→ E 2 → E 3 → E 4 → · · · → E ∞ ⊆ H ∗(EG×G X,R). The k-th Fadell–Husseini index is defined by IndexkG,RX = ker H∗(BG,R)→ E , k ≥ 2, Index1G,RX = {0}. From the definitions the following properties of indexes can be derived. Proposition 3.1. Let X, Y be G-spaces. (1) IndexkG,RX ⊆ H ∗(BG,R) is an ideal, for every k ∈ N; (2) Index1G,RX ⊆ Index G,RX ⊆ Index G,RX ⊆ · · · ⊆ IndexG,RX; k∈N Index G,RX = IndexG,RX. Proposition 3.2. Let X and Y be G-spaces and f : X → Y a G-map. Then IndexG,R(X) ⊇ IndexG,R(Y ) and for every k ∈ N IndexkG,R(X) ⊇ Index G,R(Y ). Proof. Functoriality of all constructions implies that the following diagrams commute: f ✲ Y EG×G X f̂ ✲ EG×G Y and consequently applying cohomology functor H∗(EG×G X,R) ✛ H∗(EG×G Y,R) H∗(BG,R) πX = πY ◦ f̂ and π X = f ∗ ◦ π∗Y . Thus kerπ X ⊇ kerπ Example 3.3. Sn is a Z2-space with the antipodal action. The action is free and therefore EZ2 ×Z2 S n ≃ Sn/Z2 ≈ RP n ⇒ H∗ (Sn, R) ∼= H ∗(RPn, R). 1. R = F2: The cohomology ring H ∗(BZ2,F2) = H ∗(RP∞,F2) is the polynomial ring F2[t] where deg(t) = 1. The Z2-index of S n is the principal ideal generated by tn+1: IndexZ2,F2S n = Indexn+2 Z2,F2 Sn = 〈tn+1〉 ⊆ F2[t]. 2. R = Z: The cohomology ring H∗(BZ2,Z) = H ∗(RP∞,Z) is the quotient polynomial ring Z[τ ]/〈2τ〉 where deg(τ) = 2. The Z2-index of S n is the principal ideal IndexZ2,ZS n = Indexn+2 2 〉, for n odd, 2 〉, for n even. Example 3.4. Let G be a finite group and H a subgroup of index 2. Then H ⊳G and G/H ∼= Z2. Let V be the 1-dimensional real representation of G defined for v ∈ V by g · v = v, for g ∈ H, −v, for g /∈ H. There is a G-homeomorphism S(V ) ≈ Z2. Therefore by [21, last equation on page 34]: EG×G S(V ) ≈ EG×G (G/H) ≈ (EG×G G) /H ≈ EG/H ≈ BH IndexG,RS(V ) = ker resGH : H ∗(G,R)→ H∗(H,R) . (11) 3.3 The restriction map and the index Let X be a G-space and K ⊆ G a subgroup. Then there is a commutative diagram of fibrations [12, pages 179-180]: EG×G X ✛ EG×K X BG = EG/G ✛Bi EG/K = BK induced by inclusion i : K ⊂ G. Here EG in the lower right corner is understood as a K-space and consequently a model for EK. The map Bi is a map between classifying spaces induced by inclusion i. Now with coefficients in the ring R we define resGK := H ∗(f) : H∗(EG×G X,R)→ H ∗(EG×K X,R). If G is a finite group, then the induced map on the cohomology of the classifying spaces resGK = (Bi) ∗ : H∗(BG,R)→ H∗(BK,R) coincides with the restriction homomorphism between group cohomologies resGK : H ∗(G,R)→ H∗(K,R). Proposition 3.5. Let X be a G-space, and K and L subgroups of G. (A) The morphism of fibrations (12) provides the following commutative diagram in cohomology: H∗(EG×G X,R) resGK✲ H∗(EG×K X,R) H∗(BG,R) resGK ✲ H∗(BK,R) (B) For every x ∈ H∗(BG,R) and y ∈ H∗(EG×G X,R), resGK(x · y) = res K(x) · res K(y). (C) L ⊂ K ⊂ G ⇒ resGL = res L ◦ res (D) The map of fibrations (12) induces a morphism of Serre spectral sequences i : E i (EG×G X,R)→ E i (EK ×K X,R) such that (1) Γ∗,∗∞ = res K : H ∗+∗(EG×G X,R)→ H ∗+∗(EG×K X,R), (2) Γ 2 = res K : H ∗(BG,R)→ H∗(BK,R). (E) Let R and S be commutative rings and φ : R→ S a ring homomorphism. There are morphisms: (1) in equivariant cohomology Φ∗ : H∗(EG×G X,R)→ H ∗(EG×G X,S), (2) in group cohomology Φ∗ : H∗(G,R)→ H∗(G,S), and (3) between Serre spectral sequences Φ i : E i (EG×G X,R)→ E i (EG×G X,S), induced by φ such that the following diagram commutes: H∗(EG×G X,R) ✲ H ∗(EG×K X,R) H∗(EG×G X,S) ✲ H∗(EG×K X,S) H∗(BG,R) ✲ H∗(BK,R) H∗(BG,S) H∗(BK,S) Remark 3.6. By a morphism of spectral sequences in properties (D) and (E) we mean that i ◦ ∂i = ∂i ◦ Γ i and Φ i ◦ ∂i = ∂i ◦ Φ These relations are applied in the situations where the right hand side is 6= 0 for a particular element x, to imply that the left hand side Γ i ◦ ∂i(x) or Φ i ◦ ∂i(x) is also 6= 0. In particular, then ∂i(x) 6= 0. Figure 1: Illustration of Proposition 3.5 (D) and (E) Proposition 3.7. Let X be a G-space and K a subgroup of G. Let R and S be rings and φ : R → S a ring homomorphism. Then (1) resGK (IndexG,RX) ⊆ IndexK,RX, (2) resGK IndexrG,RX ⊆ IndexrK,RX for every r ∈ N, (3) Φ∗(IndexG,RX) ⊆ IndexG,SX, (4) Φ∗(IndexrG,RX) ⊆ Index G,SX. Proof. The assertions about the IndexG,R follow from diagrams (13) and (14). The commutative diagrams E∗,0r (EG×G X,R) Γ∗,0r✲ E∗,0r (EK ×K X,R) H∗(BG,R) resGK ✲ H∗(BK,R) E∗,0r (EG×G X,R) Φ∗,0r✲ E∗,0r (EG×G X,S) H∗(BG,R) Φ∗ ✲ H∗(BG,S) imply the partial index assertions. 3.4 Basic calculations of the index 3.4.1 The index of a product Let X be a G-space and Y an H-space. Then X × Y has the natural structure of a G×H-space. What is the relation between the three indexes IndexG×H(X × Y ), IndexG(X), and IndexH(Y )? Using the Künneth formula one can prove the following proposition [14, Corollary 3.2], [32, Proposition 2.7] when the coefficient ring is a field. Proposition 3.8. Let X be a G-space and Y an H-space and H∗(BG, k) ∼= k[x1, . . . , xn], H ∗(BH, k) ∼= k[y1, . . . , ym] the cohomology rings of the associated classifying spaces with coefficients in the field k. If IndexG,kX = 〈f1, . . . , fi〉 and IndexH,k(Y ) = 〈g1, . . . , gj〉, IndexG×H,kX = 〈f1, . . . , fi, g1, . . . , gj〉 ⊆ k[x1, . . . , xn, y1, . . . , ym]. The (Z2) k-index of a product of spheres can be computed using this proposition and Example 3.3. Corollary 3.9. Let Sn1 × · · · × Snk be a (Z2) k-space with the product action. Then Index(Z2)k,F2S n1 × · · · × Snk = 〈tn1+11 , . . . , t k 〉 ⊆ F2[t1, . . . , tk]. Unfortunately when the coefficient ring is not a field the claim of Proposition 3.8 does not hold. Example 3.10. Let Sn × Sn be a (Z2) 2-space with the product action. From the previous corollary Index(Z2)2,F2S n × Sn = 〈tn+11 , t 2 〉 ⊆ F2[t1, t2] = H ∗((Z2) 2,F2). (15) To determine the Z-index we proceed in two steps. Cohomology ring H∗((Z2) 2,Z): Following [24, Section 4.1, page 508] the short exact sequence of coefficients → F2 → 0 (16) induces a long exact sequence in group cohomology [8, Proposition 6.1, page 71] which in this case reduces to a sequence of short exact sequences for k > 0, 0→ Hk((Z2) → Hk((Z2) 2,F2)→ H k+1((Z2) 2,Z)→ 0. (17) Therefore, as in [24, Proposition 4.1, page 508], H∗((Z2) 2,Z) ∼= (Z[τ1, τ2]⊗ Z[µ]) /I (18) where deg τ1 = deg τ2 = 2, deg µ = 3 and the ideal I is generated by the relations 2τ1 = 2τ2 = 2µ = 0 and µ 2 = τ1τ2(τ1 + τ2). The ring morphism c : Z → F2 in the coefficient exact sequence (16) induces a morphism in group cohomology c∗ : H ∗((Z2) 2,Z)→ H∗((Z2) 2,F2) given by: τ1 7−→ t 1, τ2 7−→ t 2, µ 7−→ t1t2(t1 + t2). (19) The arguments used in the computation of the cohomology with integer coefficients come from the Bockstein spectral sequence [7], [10, pages 104-110] associated with the exact couple H∗((Z2) 2,Z) ✛ H∗((Z2) H∗((Z2) 2,F2) where deg(p) = deg(q) = 0 and deg(δ) = 1. The first differential d1 = q ◦ δ coincides with the first Steenrod square Sq1 : H∗((Z2) 2,F2)→ H ∗+1((Z2) 2,F2) and therefore is given by 1 7→ 0, t1 7→ t 1, t2 7→ t Consequently, t1t2 7→ t 1t2 + t1t 2. The spectral sequence stabilizes at the second step since the derived couple is where F2 is in dimension 0. 0 ✛ 0 Index(Z2)2,ZS n × Sn: The (Z2) 2-action on Sn × Sn, as a product of antipodal actions, is free and therefore E(Z2) 2 ×(Z2)2 (S n × Sn) ≃ (Sn × Sn) /(Z2) 2 ≈ RPn × RPn. Using equality (15), Proposition 3.5.E.3 on the coefficient morphism c : Z→ F2, the isomorphism H∗(Z2)2(S n × Sn,Z) ∼= H∗(RPn × RPn,Z) and the existence of the (Z2) 2-inclusions Sn−1 × Sn−1 ⊂ Sn × Sn ⊂ Sn+1 × Sn+1, it can be concluded that Index(Z2)2,ZS n × Sn = 1 , τ 2 〉, for n odd 1 , τ 2 , τ 1 µ, τ 2 µ〉, for n even ⊆ H∗((Z2) 2,Z). (20) 3.4.2 The index of a sphere We need to know how to compute the index of a sphere admitting an action of a finite group different from the antipodal Z2-action. The following three propositions will be of some help [14, Proposition 3.13], [32, Proposition 2.9]. Proposition 3.11. Let G be a finite group and V an n-dimensional complex representation of G. Then IndexG,ZS(V ) = 〈cn(VG)〉 ⊂ H ∗(G,Z) where cn(VG) is the n-th Chern class of the bundle V → EG×G V → BG. Proof. If the group G acts on H∗(S(V ),Z) trivially, then from the Serre spectral sequence of the sphere bundle S(V )→ EG×G S(V )→ BG it follows that IndexG,ZS(V ) = 〈e(VG)〉 ⊂ H ∗(G,Z), where e(V ) is the Euler class of the bundle V → EG×G V → BG. Now V is a complex G-representation, therefore the group G acts trivially on H∗(S(V ),Z). From [22, Exercise 3, page 261] it follows that e(VG) = cn(VG) and the statement is proved. Proposition 3.12. Let U , V be two G-representations and let S(U), S(V ) be the associated G-spheres. Let R be a ring and assume that H∗(S(U), R), H∗(S(V ), R) are trivial G-modules. If IndexG,R(S(U)) = 〈f〉 ⊆ H∗(BG,R) and IndexG,R(S(V )) = 〈g〉 ⊆ H ∗(BG,R), then IndexG,RS(U ⊕ V ) = 〈f · g〉 ⊆ H ∗(BG,R). Proposition 3.13. (A) Let V be the 1-dimensional (Z2) k-representation with the associated ±1 vector (α1, . . . , αk) ∈ (Z2) k (as defined in Section 2). Then Index(Z2)k,F2S(V ) = 〈ᾱ1t1 + · · ·+ ᾱktk〉 ⊆ F2[t1, . . . , tk], where ᾱi = 0 if αi = 1, and ᾱi = 1 if αi = −1. (B) Let U be an n-dimensional (Z2) k-representation with a decomposition U ∼= V1 ⊕ · · · ⊕ Vn into 1- dimensional (Z2) k-representations V1, . . . , Vn. If (α1i, . . . , αki) ∈ (Z2) k is the associated ±1 vector of Vi, then Index(Z2)k,F2S(U) = (ᾱ1it1 + · · ·+ ᾱkitk) ⊆ F2[t1, . . . , tk]. Example 3.14. Let V−+, V+− and V−− be 1-dimensional real (Z2) 2-representations introduced in Section 2.2. Then by Proposition 3.13 Index(Z2)2,F2S(V−+) = 〈t1〉, Index(Z2)2,F2S(V+−) = 〈t2〉, Index(Z2)2,F2S(V−−) = 〈t1 + t2〉. On the other hand, Example 3.4 and the restriction diagram (42) imply that Index(Z2)2,ZS(V−+) = 〈τ1, µ〉, Index(Z2)2,ZS(V+−) = 〈τ2, µ〉, Index(Z2)2,ZS(V−−) = 〈τ1 + τ2, µ〉. 4 The cohomology of D8 and the restriction diagram The dihedral group W2 = D8 = (Z2) 2 ⋊ Z2 = (〈ε1〉 × 〈ε2〉)⋊ 〈σ〉 can be presented by D8 = 〈ε1, σ | ε 1 = σ 2 = (ε1σ) = 1〉. Then 〈ε1σ〉 ∼= Z4 and ε2 = σε1σ. 4.1 The poset of subgroups of D8 The poset Sub(G) denotes the collection of all nontrivial subgroups of a given group G ordered by inclusion. The poset Sub(G) can be interpreted as a small category G in the usual way: • Ob(G) = Sub(G), • for every two objects H and K, subgroups of G, there is a unique morphism fH,K : H → K if H ⊇ K, and no morphism if H + K, i.e. Mor(H,K) = {fH,K } , H ⊇ K, ∅ , H + K. The Hasse diagram of the poset Sub(D8) is presented in the following diagram. 〈ε1, ε2〉 Z2 × Z2 〈ε1σ〉 〈ε1ε2, σ〉 Z2 × Z2 〈ε1ε2〉 〈ε1ε2σ〉 4.2 The cohomology ring H∗(D8,F2) The dihedral group D8 is an example of a wreath product. Therefore the associated classifying space can, as in [1, page 117], be written explicitly as BD8 = B(Z2) 2 ×Z2 EZ2 ≈ (B(Z2) 2)×Z2 EZ2, where Z2 = 〈σ〉 acts on (BZ2) 2 by interchanging coordinates. Presented in this way BD8 is the Borel construction of the Z2-space (BZ2) 2. Thus BD8 fits into a fibration B(Z2) 2 → (B(Z2) 2)×Z2 EZ2 → BZ2. (21) There is an associated Serre spectral sequence with E2-term Hp(BZ2,H B(Z2) Hp(Z2, H Hp+q(BD8,F2) Hp+q(D8,F2) which converges to the cohomology of the group D8 with F2-coefficients. This spectral sequence is also the Lyndon-Hochschild-Serre (LHS) spectral sequence [1, Section IV.1, page 116] associated with the group extension sequence: 1→ (Z2) 2 → D8 → D8/(Z2) 2 → 1. In [1, Theorem 1.7, page 117] it is proved that the spectral sequence (22) collapses at the E2-term. Therefore, to compute the cohomology of D8 we only need to read the E2-term. Lemma 4.1. (i) H∗ ∼=ring F2[a, a+ b], where deg(a) = deg(a + b) = 1 and the Z2-action induced by σ is given by σ · a = a+ b. (ii) H∗ )Z2 ∼=ring F2[b, a(a+ b)]. (iii) Hi ∼=Z2-module F2[Z2] si,1 ⊕ F 2 , where si,1 ≥ 0, si,2 ≥ 0 and F2[Z2] denotes a free Z2-module and F2 a trivial one. (iv) E 2 = H ∗(Z2, H ) ∼=ring H ∗(Z2,F2) ⊕si,2 ⊕F 2 , where F 2 denotes a ring concentrated in dimension 0. Proof. (i) The statement follows from the observation that B(Z2) 2 ≈ (B(Z2)) , and consequently ∼=ring H ∗ (Z2,F2)⊗H ∗ (Z2,F2) ∼=ring F2[a]⊗ F2[a+ b]. The Z2-action interchanges copies on the left hand side. Generators on the right hand side are chosen such that the Z2-action coming from the isomorphism swaps a and a+ b. (ii) With the induced Z2-action b = a+(a+ b) and a(a+ b) are invariant polynomials. They generate the ring of all invariant polynomials. (iii) The cohomology Hi is a Z2-module and therefore a direct sum of irreducible Z2- modules. There are only two irreducible Z2-modules over F2: the free one F2[Z2] and the trivial one F2. (iv) The isomorphism follows from (iii) and the following two properties of group cohomology [20, Exercise 2.2, page 190] and [8, Corollary 6.6, page 73]. Let M and N be G-modules of a finite group G. (a) H∗(G,M ⊕N) ∼= H∗(G,M)⊕H∗(G,N) (b) M is a free G-module ⇒ H∗(G,M) = H0(G,M) ∼= MG. Applied in our case, this yields 2 =ring H ∗(Z2, H ∼=ring H ∗(Z2,F2[Z2] si,1 ⊕ F ∼=ring H ∗(Z2,F2[Z2]) ⊕si,1 ⊕H∗(Z2,F2) ⊕si,2 ∼=ring H 0(Z2,F2[Z2]) ⊕si,1 ⊕H∗(Z2,F2) ⊕si,2 ∼=ring (F2[Z2] Z2)⊕si,1 ⊕H∗(Z2,F2) ⊕si,2 ∼=ring F2 ⊕si,1 ⊕H∗(Z2,F2) ⊕si,2 Let the cohomology of the base space of the fibration (21) be denoted by H∗(Z2,F2) = F2[x]. The E2-term (22) can be pictured as in Figure 2. The cohomology of D8 can be read from the picture. If we denote y := b, w := a(a+ b) (23) and keep x as we introduced above, then H∗(D8,F2) = F2[x, y, w]/〈xy〉. Also, the restriction homomorphism resD8H1 : H ∗(D8,F2) = F2[x, y, w]/〈xy〉 → H ∗(H1,F2) = F2[a, a+ b] (24) can be read off since it is induced by the inclusion of the fibre in the fibration (21). On generators, resD8H1(x) = 0, res (y) = b, resD8H1(w) = a(a+ b). (25) b4,b2aÝa + bÞ a2Ýa + bÞ2 å 1 a2Ýa + bÞ2 å x a2Ýa + bÞ2 å x a2Ýa + bÞ2 å x a2Ýa + bÞ2 å x 3 b3,baÝa + bÞ 0 0 0 0 aÝa + bÞ å 1 aÝa + bÞ å x aÝa + bÞ å x2 aÝa + bÞ å x3 aÝa + bÞ å x4 1 b 0 0 0 0 0 1 x x2 x3 x4 0 1 2 3 4 Figure 2: E2-term 4.3 The cohomology diagram of subgroups with coefficients in F2 Let G be a finite group and R an arbitrary ring. Then the diagram Res(R) : G→ Ring (covariant functor) defined by Ob(G) ∋ H 7−→ H∗(H,R) ( H ⊇ K) 7−→ resHK : H ∗(H,R)→ H∗(K,R) is the cohomology diagram of subgroups of G with coefficients in the ring R. In this section we assume that R = F2. 4.3.1 The Z2 × Z2-diagram The cohomology of any elementary abelian 2-group Z2 × Z2 is a polynomial ring F2 [x, y], deg(x) = deg(y) = 1. The restrictions to the three subgroups of order 2 are given by all possible projections F2 [x, y]→F2 [t], deg(t) = 1: (x 7→ t, y 7→ 0) or (x 7→ 0, y 7→ t) or (x 7→ t, y 7→ t) . Thus the cohomology diagram of the subgroups of Z2 × Z2 is Z2 × Z2 F2 [x, y] F2[t1] F2[t2] x 7→ t2 y 7→ 0 F2[t3] 4.3.2 The D8-diagram For the dihedral group D8, from [9] and (24), the two top levels of the diagram can be presented by: F2 [x, y;w]/〈xy 〉 deg : 1, 1, 2 F2 [a, b] deg : 1, 1 F2 [e, u]/〈e deg : 1, 2 x, y 7→ e w 7→ u F2 [c, d] deg : 1, 1 Let H∗(Ki,F2) = F2[ti], deg(ti) = 1. From [1, Corollary II.5.7, page 69] the restriction resH2K3 : H∗(H2,F2) = F2[e, u] /〈e −→ (H∗(K3,F2) = F2[t3]) is given by e 7→ 0, u 7→ t23. Thus, the restriction res is given by x 7→ 0, y 7→ 0, w 7→ t23. Using diagrams (26), (27) with the property (C) from Proposition 3.5 we almost completely reveal the cohomology diagram of subgroups of D8. The equalities resD8K3 = res ◦ resD8H2 = res ◦ resD8H1 = res ◦ resD8H3 imply that • resH1K3 : (H ∗(H1,F2) = F2[a, b]) −→ (H ∗(K3,F2) = F2[t3]) is given by a 7→ t3, b 7→ 0, • resH3K3 : (H ∗(H3,F2) = F2[c, d]) −→ (H ∗(K3,F2) = F2[t3]) is given by c 7→ t3, d 7→ 0. F2 [a, b] deg : 1, 1 F2 [e, u]/〈e deg : 1, 2 F2 [c, d] deg : 1, 1 F2[t3] deg : 1 u 7→ t23 e 7→ 0❄ The cohomology diagram (26) of subgroups of Z2 × Z2 and the part (28) of the D8 diagram imply that • resH1K1 : F2[a, b] −→ F2[t1] and res : F2[a, b] −→ F2[t2] are given by (a 7→ t1, b 7→ t1 and a 7→ 0, b 7→ t2) or (a 7→ 0, b 7→ t1 and a 7→ t2, b 7→ t2) , • resH3K4 : F2[c, d] −→ F2[t4] and res : F2[a, b] −→ F2[t5] are given by (c 7→ t4, d 7→ t4 and c 7→ 0, d 7→ t5) or (c 7→ 0, d 7→ t4 and c 7→ t5, d 7→ t5) . Proposition 4.2. For all i 6= 3, resD8Ki (w) = 0, while res (w) 6= 0. Proof. The result follows from the diagram (27) in the following way: (a) For i ∈ {1, 2}: resD8Ki (w) = res ◦ resD8H1(w) = res (a(a+ b)) = 0 since either a 7→ ti, b 7→ ti or a 7→ 0, b 7→ ti. (b) For i ∈ {4, 5}: resD8Ki (w) = res ◦ resD8H3(w) = res (c(c+ d)) = 0 since either c 7→ ti, d 7→ ti or c 7→ 0, d 7→ ti. Corollary 4.3. The cohomology of the dihedral group D8 is H∗(D8,F2) = F2[x, y, w]/〈xy〉 where (a) x ∈ H1(D8,F2) and res (x) = 0, (b) y ∈ H1(D8,F2) and res (y) = 0, (c) w ∈ H1(D8,F2) and res (w) = resD8K2(w) = res (w) = resD8K5(w) = 0 and res (w) 6= 0. Assumption Without lose of generality we can assume that resH1K1(a) = t1, res (b) = t1, res (a) = 0, resH1K2(b) = t2. (29) 4.4 The cohomology ring H∗(D8,Z) In this section we present the cohomology H∗(D8,Z) based on: A. Evens’ approach [13, Section 5, pages 191-192], where the concrete generators in H∗(D8,Z) are identified using the Chern classes of appropriate irreducible complex D8-representations. We also consider LHS spectral sequences associated with following two extensions 1→ H1 → D8 → D8/(Z2) 2 → 1 and 1→ H2 → D8 → D8/Z4 → 1. (30) Unfortunately, the ring structure on E∞ -terms of these LHS spectral sequences does not coincide with the ring structure on H∗(D8,Z). B. The Bockstein spectral sequence of the exact couple H∗(D8,Z) ×2 ✲ H∗(D8,Z) H∗(D8,F2) where d1 = c ◦ δ = Sq 1 : H∗(D8,F2) → H ∗+1(D8,F2) is given by d1 (x) = x 2, d1 (y) = y 2 and d1 (w) = (x + y)w [1, Theorem 2.7. page 127]. This approach allows determination of the ring structure on H∗(D8,Z). 4.4.1 Evens’ view Let V C+− ⊕ V −+ = C⊕ C, V −− = C and U 2 = C be the complex D8-representations given by A. For (u, v) ∈ V C+− ⊕ V ε1 · (u, v) = (u,−v), ε2 · (u, v) = (−u, v), σ · (u, v) = (v, u). B. For u ∈ V C−−: ε1 · u = −u, ε2 · u = −u, σ · u = u. C. For u ∈ UC2 : ε1 · u = u, ε2 · u = u, σ · u = −u. There are isomorphisms of real D8-representations V C+− ⊕ V ∼= (V+− ⊕ V−+) , V C−− ∼= (V−−) , UC2 = (U2) Let χ1, ξ ∈ H ∗(D8,Z) be 1-dimensional complex D8-representations given by character (here we assume the identification c1 : Hom(G,U (1))→ H 2(G,Z), [3, page 286]): χ1(ε1) = 1, χ1(ε2) = 1, χ1(σ) = −1, ξ(ε1) = −1, ξ(ε2) = −1, ξ(σ) = −1. Then χ1 = U 2 , ξ = U 2 ⊗ V −− and consequently 2 ) = χ1, and c1(U 2 ) + c1(V −−) = ξ. (31) The cohomology H∗(D8,Z) is given in [13, pages 191-192] by H∗(D8,Z) = Z[ξ, χ1, ζ, χ] (32) where deg ξ = degχ1 = 2, deg ζ = 3, degχ = 4 2ξ = 2χ1 = 2ζ = 4χ = 0, χ 1 = ξ · χ1, ζ 2 = ξ · χ. (33) There are four 1-dimensional irreducible complex representations of D8: 1, ξ = UC2 ⊗ V −−, χ1 = U 2 , ξ ⊗ χ1 = V and one 2-dimensional complex representation which is denoted by ρ in [13, pages 191-192]: ρ = V C+− ⊕ V It is computed in [13, pages 191-192] that c(V C+− ⊕ V −+) = 1 + ξ + χ and c2(V +− ⊕ V −+) = χ. (34) The relations (31) and (34) along with Proposition 3.11 imply the following statement. Proposition 4.4. IndexD8,ZS(V −−) = 〈ξ+χ1〉, IndexD8,ZS(U 2 ) = 〈χ1〉, IndexD8,ZS(V −+) = 〈χ〉. Before proceeding to the Bockstein spectral sequence approach we give descriptions of the E2-terms of two LHS spectral sequences. Even though it is not an easy consequence, it can be proved that both spectral sequences stabilize and that E2 = E∞. LHS spectral sequences of the extension 1 → H1 → D8 → D8/H1 → 1. The LHS spectral sequence of this extension (22) allows computation of the cohomology ringH∗(D8,F2) with F2 coefficients. If we now consider Z coefficients, then the E2-term has the form 2 = H p(D8/H1, H q (H1,Z)) ∼= H p(Z2, H ). (35) The spectral sequence converges to the graded group Gr (Hp+q(D8,Z)) associated with H p+q(D8,Z) appropriately filtered. To present the E2-term we choose generators of H ∗ (H1,Z) consistent with the choices made in Lemma 4.1. Let c : Z → F2 be reduction mod 2 and c∗ : H ∗(D8,Z) → H ∗(D8,F2) the induced map in cohomology. Consider the following presentation of the H1 cohomology ring: H∗ (H1,Z) = Z[α, α+ β]⊗ Z[µ] (36) where A. deg(α) = deg(β) = 2, deg(µ) = 3; B. 2α = 2β = 2µ = 0 and µ2 = αβ(α + β); C. σ action on H∗ (H1,Z) is given by σ · α = α+ β and σ · µ = µ; D. c∗(α) = a 2, c∗ (β) = b 2, c∗ (µ) = ab(a+ b). å3 0 0 1 0 0 0 0 0 0 1 x2 0 1 2 3 4 0 0 0 0 m xåm åmm x2 a(a+b), b a(a+b)åxå0 0 åx 2a(a+b)2 Figure 3: E2-term of extension 1 → H1 → D8 → D8/H1 → 1. Now the E2-term (Figure 3) is given by ∼= Hp(Z2, H Hp(Z2,Z), q = 0 0, q = 1 Hp(Z2,F2[Z2]), q = 2 Hp(Z2,F2), q = 3 . . . , q > 3. The morphism of LHS spectral sequences of the extension 1→ H1 → D8 → D8/H1 → 1 induced by the mod 2 reduction c : Z → F2 (Proposition 3.5 E.3) gives a proof that E2 = E∞ for Z coefficients. The ring structures on E∞ and H ∗(D8,Z) do not coincide. Moreover there is no element in E∞ of exponent 4. One thing is clear: the element µ in the E2 = E∞-term coincides with the element ζ in the Evens’ presentation (32) of H∗(D8,Z). LHS spectral sequences of the extension 1 → H2 → D8 → D8/H2 → 1. The E2-term has the form: 2 = H p(D8/H2, H q (H2,Z)) ∼= H p(Z2, H q (Z4,Z)) ∼= Hp(Z2,Z), q = 0 0, q odd Hp(Z2,Z4), q even and 4 ∤ q Hp(Z2,Z4), q > 0 even and 4|q, where Z4 ∼= Z4 is a non-trivial Z2-module. Using [8, Example 2, pages 58-59] the E2-term has the shape given in the Figure 4. This diagram provides just two hints: there might be elements of exponent 4 6 ¨2 ¨2 ¨2 ¨2 ¨2 ¨2 ¨2 5 0 0 0 0 0 0 0 4 ¨4 ¨2 ¨2 ¨2 ¨2 ¨2 ¨2 3 0 0 0 0 0 0 0 2 ¨2 ¨2 Q ¨2 ¨2 ¨2 ¨2 ¨2 1 0 0 0 0 0 0 0 0 ¨ 0 ¨2 0 ¨2 0 ¨2 0 1 2 3 4 5 6 Figure 4: E2-term of extension 1 → H2 → D8 → D8/H2 → 1. in the cohomology H∗(D8,Z) and definitely there is only one element ζ of degree 3 from the Evens’ presentation. Conclusion. The LHS spectral sequences of different extensions gives an incomplete picture of the cohomology ring with integer coefficients, H∗(D8,Z). Therefore, for the purposes of the computations with Z coefficients we use the Bockstein spectral sequence utilizing results obtained from the LHS spectral sequence with F2 coefficients. Presentations of these two spectral sequences will be used in the description of the restriction diagram in Section 4.5. 4.4.2 The Bockstein spectral sequence view Let G be a finite group. The exact sequence 0 → Z → Z → F2 → 0 induces a long exact sequence in group cohomology, or an exact couple H∗(G,Z) ×2 ✲ H∗(G,Z) H∗(G,F2). The spectral sequence of this exact couple is the Bockstein spectral sequence. It converges to (H∗(G,Z)/torsion)⊗ F2 which in the case of a finite group G is just F2 in dimension 0. Here “torsion” means Z-torsion. The first differential d1 = c ◦ δ is the Bockstein homomorphism and in this case coincides with the first Steenrod square Sq1 : H∗(G,F2)→ H ∗+1(G,F2). Let H be a subgroup of G. The restriction resGH commutes with the maps in the exact couples associated to the groups G and H and therefore induces a morphism of Bockstein spectral sequences [10, page 109 before 5.7.6]. Consider two Bockstein spectral sequences associated with D8 and its subgroup H2 ∼= Z4. A. Group D8. The exact couple is H∗(D8,Z) ×2 ✲ H∗(D8,Z) H∗(D8,F2). and d1 = c ◦ δ = Sq 1 is given by d1 (x) = x 2, d1 (y) = y 2 and d1 (w) = (x + y)w, [1, Theorem 2.7. page 127]. The derived couple is Then by [10, Remark 5.7.4, page 108] there are elements X ,Y ∈ 2 ·H∗(D8,Z) ×2 ✲ 2 ·H∗(D8,Z) 〈x2, y2, xw, yw,w2〉/〈x2, y2, xw + yw〉. H2(D8,Z), M ∈ H 3(D8,Z) of exponent 2 such that c∗(X ) = x 2, c∗(Y) = y 2, c∗(M) = (x + y)w and XY = 0. B. Group Z4. The exact couple is H∗(Z4,Z) ×2 ✲ H∗(Z4,Z) H∗(Z4,F2). Since H∗(Z4,Z) = Z[U ]/〈4U〉, degU = 2 and H ∗(Z4,F2) = F2 [e, u]/〈e 〉, deg e = 1, deg u = 2, the unrolling of the exact couple to a long exact sequence [8, Proposition 6.1, page 71] H0(Z4,Z) Z , 1 ×2✲ H 0(Z4,Z) Z , 1 0(Z4,F2) F2 , 1 1(Z4,Z) ×2✲ H 1(Z4,Z) 1(Z4,F2) F2 , e 2(Z4,Z) Z4 , U ×2✲ H 2(Z4,Z) Z4 , U 2(Z4,F2) F2 , u 3(Z4,Z) ×2✲ H 3(Z4,Z) 3(Z4,F2) F2 , eu 4(Z4,Z) . . . allows us to show that for j ≥ 0 : δ(ui) = 0 and δ(eui) = 2U i+1. Thus d1 = 0 and the derived couple is 2 ·H∗(Z4,Z) ×2 ✲ 2 ·H∗(Z4,Z) H∗(Z4,F2). Moreover, by definition of the differential of a derived couple we have that i) = 0 and d2(eu i) = ui+1. The restriction map resD8H2 : H ∗(D8,F2) → H ∗(H2,F2) is determined by the restriction diagram (27). Therefore, the morphism between spectral sequences induced by the restriction resD8H2 implies that: resD8H2 (d2[xw]) = d2 resD8H2 [xw] = d2(eu) = u and consequently d2[xw] = [w Here [ · ] denotes the class in the quotient 〈x2, y2, xw, yw,w2〉/〈x2, y2, xw + yw〉. Thus, by [10, Remark 5.7.4, page 108] there is an element W ∈ H4(D8,Z) of exponent 4 such that c∗(W) = w 2 and M2 = W(X + Y). The second derived couple of (37) stabilizes. Thus the cohomology ring H∗(D8,Z) and the map c∗ : H ∗(D8,Z) −→ H ∗(D8,F2) are described. Theorem 4.5. The cohomology ring H∗(D8,Z) can be presented by H∗(D8,Z) = Z[X ,Y,M,W ]/I where degX = degY = 2, degM = 3, degW = 4 and the ideal I is generated by the equations 2X = 2Y = 2M = 4W = 0, XY = 0,M2 =W(X + Y). (39) The map c∗ : H ∗(D8,Z) −→ H ∗(D8,F2), induced by the reduction of coefficients Z→ F2, is given by X 7→x2 , Y 7→y2 , M 7→w(x + y) , W 7→w2. (40) Remark 4.6. The correspondence between the Evens’ and Bockstein spectral sequence view is given by X ↔ χ1 , Y ↔ ξ + χ1 , M↔ ζ , W ↔ χ (41) 4.5 The D8-diagram with coefficients in Z Let G be a finite group and R and S rings. A ring homomorphism φ : R → S induces a morphism of diagrams (natural transformation of covariant functors) Φ : Res(R) →Res(S). The morphism Φ on each object H ∈ Ob(G) is defined by the coefficient reduction Φ(H) : H∗(H,R) → H∗(H,S) induced by φ. Particularly in this section, as a tool for the reconstruction of the diagram Res(Z), we use the diagram morphism C : Res(Z) → Res(F2) induced by the coefficient reduction homomorphism c : Z→ F2. 4.5.1 The Z2 × Z2-diagram The cohomology restriction diagram Res(F2) of the elementary abelian 2-group Z2 × Z2 is given in the diagram (26). Using the presentation of cohomology H∗(Z2 × Z2,Z) and the homomorphism H ∗(Z2 × Z2,Z)→ H ∗(Z2 × Z2,F2) given in Example 3.10 we can reconstruct the restriction diagram Res(Z): Z2 × Z2 Z[τ1, τ2]⊗ Z[µ] deg τ1 = deg τ2 = 2, deg µ = 3 2τ1 = 2τ2 = 2µ = 0, µ2 = τ1τ2(τ1 + τ2) 3 , τ Z2 Z[θ1] deg θ1 = 2 2θ1 = 0 Z2 Z[θ2] deg θ2 = 2 2θ2 = 0 τ1 7→θ2, τ2 7→ 0, µ 7→ 0 Z2 Z[θ3] deg θ3 = 2 2θ3 = 0 4.5.2 The D8-diagram In a similar fashion, using: • the D8 restriction diagram (27) and (28) with F2 coefficients, • the Z2 × Z2 restriction diagrams (42) with Z coefficients, • the presentation of cohomology H∗(H1,Z) given in (36), • the Bockstein presentation of H∗(D8,Z) given in Theorem 4.5, • a glance at the restriction maps resD8H1 and res obtained from the E2 = E∞ terms of the LHS spectral sequences Figure 3 and Figure 4, and • the homomorphism c∗ : H ∗(D8,Z)→ H ∗(D8,F2) described in (40), we can reconstruct the restriction diagram of D8 with Z coefficients. D8 Z[X ,Y,M,W ] deg : 2, 2, 3, 4 2X = 2Y = 2M = 4W = 0, XY = 0,M2 =W(X + Y) H1 Z[α, α + β, µ] deg : 2, 2, 3, 2α = 2β = 2µ = 0, µ2 = αβ(α + β) H2 Z[U ] deg : 2 4U = 0, Y 7→ 2U W7→ U2 X 7→2U H3 Z[γ, γ + δ, η] deg : 2, 2, 3, 2γ = 2δ = 2η = 0, η2 = γδ(γ + δ) K3 Z[θ3] deg θ3 = 2 U 7→θ3 Now the determination of the diagram morphism C :Res(Z) →Res(F2) induced by the coefficient reduction homomorphism c : Z→ F2 is just a routine exercise. 5 IndexD8,F2S(R In this section we show the following equality: IndexD8,F2S(R 4 ) = Index D8,F2 4 ) = 〈w jyj〉. The D8-representation R 4 can be decomposed into a sum of irreducibles in the following way R4 = (V−+ ⊕ V+−)⊕ V−− ⇒ R 4 = (V−+ ⊕ V+−) where V−+ ⊕ V+− is a 2-dimensional irreducible D8-representation. Since in this section F2 coefficients are assumed, Proposition 3.12 implies that computing the indexes of the spheres S(V−+ ⊕ V+−) and S(V−−) suffices. The strategy employed uses Proposition 3.7 and the following particular facts. A. Let X = S(T ) for some D8-representation T . Then the E2-term of the Serre spectral sequence associated to ED8 ×D8 X is 2 = H p(D8,F2)⊗H q(X,F2). (44) The local coefficients are trivial since X is a sphere and the coefficients are F2. Since only ∂dimT,F2 may be 6= 0, from the multiplicative property of the spectral sequence it follows that IndexD8,F2X = 〈∂ 0,dimV −1 dimV,F2 (1⊗ l)〉 where l ∈ HdimV −1(X,F2) is the generator. Therefore, IndexD8,F2(X) = Index dimV+1 D8,F2 B. For any subgroup H of D8, with some abuse of notation, dimV,0 dimV ◦ ∂ 0,dimV−1 dimV,F2 (1⊗ l) = ∂ 0,dimV −1 dimV,F2 0,dimV −1 dimV (1⊗ l), (45) where Γ denotes the restriction morphism of Serre spectral sequences introduced in Proposition 3.5(D). Therefore, for every subgroup H of D8 we get IndexD8,F2X = 〈a〉, IndexH,F2X = 〈aH〉 =⇒ res K(a) = aH . In particular, if aH 6= 0 then a 6= 0. Our computation of IndexD8,F2X for X = S(V−+ ⊕ V+−) and X = S(V−−) has two steps: • compute IndexH,F2X = 〈aH〉 for all proper subgroups H of D8, • search for an element a ∈ H∗(D8,F2) such that for every computed aH resGK(a) = aH . 5.1 IndexD8,F2S(V−+ ⊕ V+−) = 〈w〉 Proposition 3.13 and the properties of the action of D8 on V−+⊕V+− provide the following information: IndexH1,F2S(V−+ ⊕ V+−) = 〈a(a+ b)〉 or 〈b(a+ b)〉 or 〈ab〉. Since initially we do not know which of the possible generators a, b, a + b of F2[a, b] correspond to the generators ε1, ε2, ε1ε2, we have to take all three possibilities into account. Similarly: IndexH3,F2S(V−+ ⊕ V+−) = 〈c(c+ d)〉 or 〈d(c+ d)〉 or 〈cd〉. Furthermore, ε1 acts trivially on V+− ⇒ IndexK1,F2S(V−+ ⊕ V+−) = 0 ε2 acts trivially on V−+ ⇒ IndexK2,F2S(V−+ ⊕ V+−) = 0 σ acts trivially on {(x, x) ∈ V−+ ⊕ V+−} ⇒ IndexK4,F2S(V−+ ⊕ V+−) = 0 ε1ε2σ acts trivially on {(x,−x) ∈ V−+ ⊕ V+−} ⇒ IndexK5,F2S(V−+ ⊕ V+−) = 0. The only nonzero element of H2(D8,F2) satisfying all requirements of commutativity with restrictions is w. Hence, IndexD8,F2S(V−+ ⊕ V+−) = 〈w〉. (46) Remark 5.1. The side information coming from this computation is that generators ε1 and ε2 of the group H1 correspond to generators a and a+ b in the cohomology ring H ∗(H1,F2). This correspondence suggested the choice of generators in Lemma 4.1(i). 5.2 IndexD8,F2S(V−−) = 〈y〉 Again, V−− is a concrete D8-representation, and from Proposition 3.13: IndexH1,F2S(V−−) = 〈a+ b〉, or 〈a+ (a+ b)〉, or 〈b+ (a+ b)〉. Again, we allow all three possibilities since we do not know the correspondence between generators of H1 and the chosen generators of H∗(Hq,F2). Furthermore, since K1 and K2 act nontrivially on V−−, IndexK1,F2S(V−−) = 〈t1〉, IndexK2,F2S(V−−) = 〈t2〉 . On the other hand, H3 acts trivially on S(V−−) and so IndexH3,F2S(V−−) = 0. By commutativity of the restriction diagram, or since the groups K3, K4 and K5 act trivially on V(1,1), it follows that IndexK3,F2S(V−−) = IndexK4,F2S(V−−) = IndexK5,F2S(V−−) = 0. The only element satisfying the commutativity requirements is y ∈ H1(D8,F2), so IndexD8,F2S(V−−) = 〈y〉. (47) Remark 5.2. From the previous remark the fact that IndexH1,F2S(V−−) = 〈b〉 = 〈a + (a + b)〉 follows directly. Alternatively, equation (47) is a consequence of (11) and (27). 5.3 IndexD8,F2S(R 4 ) = 〈y From Proposition 3.12 we get that IndexD8,F2S(R 4 ) = IndexD8,F2S((V−+ ⊕ V+−) ⊕j ⊕ V −−) = 〈y jwj〉. Remark 5.3. In the same way we can compute that IndexD8,F2(U2) = 〈x〉. (48) Therefore IndexD8,F2(U2 ⊕ R 4 ) = 0. This means that on the join CS/TM scheme the Fadell–Husseini index theory with F2 coefficients yields no obstruction to the existence of the equivariant map in question. 6 IndexD8,ZS(R In this section we show that IndexD8,ZS(R 4 ) = Index 4 ) = 2 〉, for j even 2 M,Y 2 〉, for j odd. 6.1 The case when j is even We give two proofs of the equation (49) in the case when j is even. Method 1: According to definition of the complex D8 -representations V +− ⊕ V −+ and V −−, in Section 4.4.1, we have an isomorphism of real D8-representations 4 = (V−+ ⊕ V+−) V C+− ⊕ V V C−− Thus by Propositions 3.11 and 3.12, properties of Chern classes [3, (5) page 286] and equations (31) and (34) we have that IndexD8,ZS(R 4 ) = 〈c 3j V C+− ⊕ V V C−− 〉 = 〈c2 V C+− ⊕ V 2 · c1 V C−− 2 (ξ + χ1) The correspondence between Evens’ and Bockstein spectral sequence views implies the statement. Method 2: The group D8 acts trivially on the cohomologyH ∗(S(R 4 ),Z). Then the E2-term of the Serre spectral sequence associated to ED8 ×D8 S(R 4 ) is a tensor product 2 = H p(D8,Z)⊗H q(S(R 4 ),Z). Since only ∂3j,Z may be 6= 0, the multiplicative property of the spectral sequence implies that IndexD8,ZS(R 4 ) = Index dimV +1 4 ) = 〈∂ 0,3j−1 3j,Z (1 ⊗ l)〉 where l ∈ H3j−1(S(R 4 ),Z) is a generator. The coefficient reduction morphism c : Z → F2 induces a morphism of Serre spectral sequences (Proposition 3.5. E. 3) associated with the Borel construction of the sphere S(R 4 ). Thus, 0,3j−1 3j,Z (1 ⊗ l) 0,3j−1 3j,F2 (s∗(1 ⊗ l)) ∈ H 3j(D8,F2) and according to the result of the previous section 0,3j−1 3j,Z (1⊗ l) = yjwj . Now, from the description of the map c∗ : H ∗(D8,Z) −→ H ∗(D8,F2) in (40) follows the statement for j even. 6.2 The case when j is odd The group D8 acts nontrivially on the cohomology H ∗(S(R 4 ),Z). Precisely, the D8-module Z = H3j−1(S(R 4 ),Z) is a nontrivial D8-module and for z ∈ Z: ε1 · z = z, ε2 · z = z, σ · z = −z. Then the E2-term of the Serre spectral sequence associated to ED8 ×D8 S(R 4 ) is not a tensor product 2 = H p(D8, H q(S(R 4 ),Z)) = Hp(D8,Z) , q = 0 Hp(D8,Z) , q = 3j − 1 0 , q 6= 0, 3j − 1. To compute the index in this case we have to study the H∗(D8,Z)-module structure of H ∗(D8,Z). Since the use of LHS-spectral sequence, as in the case of field coefficients (Proposition 7.4), cannot be of significant help we apply the Bockstein spectral sequence associated with the following exact sequence of D8-modules: → Z → F2 → 0. (51) Proposition 6.1. (A) 2 ·H∗(D8,Z) = 0 (B) H∗(D8,Z) is generated as a H ∗(D8,Z)-module by three elements ρ1, ρ2, ρ3 of degree 1, 2, 3 such ρ1 · Y = 0, ρ2 · X = 0, ρ3 · X = 0 c(ρ1) = x, c(ρ2) = y 2, c(ρ3) = yw where c is the map induced by the map Z → F2 from the exact sequence (51). Proof. The strategy of the proof is to consider four exact couples induced by the exact sequence (51): H∗(D8,Z) ×2 ✲ H∗(D8,Z) H ∗(H1,Z) ×2 ✲ H∗(H1,Z) H∗(D8,F2) H∗(H1,F2) H∗(H2,Z) ×2 ✲ H∗(H2,Z) H ∗(K4,Z) ×2 ✲ H∗(K4,Z) H∗(H2,F2) H∗(K4,F2) and the corresponding morphisms induced by resD8H1 , res and resD8K4 . Our notation is as in the restriction diagram (27). 1. The module Z as a H1-module is a trivial module. Therefore in the H1 exact couple d1 is the usual Bockstein homomorphism and so d1(a) = Sq 1(a) = a2, d1(b) = Sq 1(b) = b2. Thus from the restriction homomorphism resD8H1 we have: resD8H1 (d1(1)) = d1 resD8H1 (1) = d1(1) = 0 ⇒ d1(1) ∈ ker resD8H1 resD8H1 (d1(x)) = d1 resD8H1 (x) = d1(0) = 0 ⇒ d1(x) ∈ ker resD8H1 resD8H1 (d1(y)) = d1 resD8H1 (y) = d1(b) = b 2 ⇒ d1(y) ∈ y 2 + ker resD8H1 resD8H1 (d1(w)) = d1 resD8H1 (w) = ba(a+ b) ⇒ d1(w) ∈ yw + ker resD8H1 2. The module Z as a H2-module is a non-trivial module. The H2 ∼= Z4 exact couple unrolls into a long exact sequence [8, Proposition 6.1, page 71] H0(Z4,Z) ×2✲ H 0(Z4,Z) 0(Z4,F2) F2 , 1 1(Z4,Z) F2 , λ ×2✲ H 1(Z4,Z) F2 , λ 1(Z4,F2) F2 , e 2(Z4,Z) ×2✲ H 2(Z4,Z) 2(Z4,F2) F2 , u 3(Z4,Z) F2 , λU ×2✲ H 3(Z4,Z) F2 , λU 3(Z4,F2) F2 , eu 4(Z4,Z) . . . Here we have used the facts that Hi(Z4,Z) = F2 , i odd 0 , i even and that multiplication by U ∈ H2(Z4,Z) in H ∗(Z4,Z) is an isomorphism [11, Section XII. 7. pages 250-253]. The long exact sequence describes the boundary operator: δ(1) = λ, δ(e) = 0, δ(u) = λU and consequently the first differential: d1(1) = e, d1(e) = 0, d1(u) = eu. The restriction homomorphism resD8H2 implies that: resD8H2 (d1(1)) = d1 resD8H1 (1) = d1(1) = e ⇒ d1(1) ∈ x+ ker resD8H2 resD8H1 (d1(x)) = d1 resD8H1 (x) = d1(e) = 0 ⇒ d1(x) ∈ ker resD8H2 resD8H1 (d1(y)) = d1 resD8H1 (y) = d1(e) = 0 ⇒ d1(y) ∈ ker resD8H2 resD8H1 (d1(w)) = d1 resD8H1 (w) = d1(u) = eu ⇒ d1(w) ∈ yw + ker resD8H2 3. The module Z as a K4-module is a non-trivial module. Then the K4 ∼= Z2 exact couple unrolls into H0(Z2,Z) ×2✲ H 0(Z2,Z) 0(Z2,F2) F2 , 1 1(Z2,Z) F2 , ϕ ×2✲ H 1(Z2,Z) F2 , ϕ 1(Z2,F2) F2 , t4 2(Z2,Z) ×2✲ H 2(Z2,Z) 2(Z2,F2) F2 , t 3(Z2,Z) F2 , ϕT ×2✲ H 3(Z2,Z) F2 , ϕT 3(Z2,F2) F2 , t 4(Z2,Z) . . . Similarly, Hi(Z2,Z) = F2 , i odd 0 , i even and multiplication by T ∈ H2(Z2,Z) in H ∗(Z2,Z) is an isomorphism [11, Section XII. 7. pages 250-253]. Then d1(1) = t4, d1(t 4 ) = 0, d1(t 4 ) = t for i ≥ 0. This implies that resD8K4 (d1(w)) = d1 resD8H1 (w) = d1(0) = 0 (55) resD8K4 (d1(y)) = d1 resD8H1 (y) = d1(0) = 0. (56) From (52), (53) and the restriction diagram (27) follows: d1(1) = x, d1(x) = 0, d1(w) ∈ {yw, yw + x 3} and d1(y) ∈ {y 2, y2 + x2}. (57) Since resD8K4 (yw) = 0, res yw + x3 = t34 6= 0 and res = 0, resD8K4 y2 + x2 = t24 6= 0, then the equations (55) and (56) resolve the final dilemmas (57). Thus d1(w) = yw. According to [10, Remark 5.7.4, page 108] there are elements ρ1, ρ2, ρ3 of degree 1, 2, 3 and of exponent 2 in H ∗(D8,Z) satisfying property (B) of this proposition. The property (A) follows from the properties of Bockstein spectral sequence and the fact that the derived couple of the D8 exact couple is: 0 ✲ 0 where F2 appears in dimension 0. Remark 6.2. The proposition does not describes the completeH∗(D8,Z)-modulo structure onH ∗(D8,Z). It gives only the necessary information for the computation of IndexD8,ZS(R 4 ). The complete result can be found in [18, Theorem 5.11.(a)]. Thus, the index is given by IndexD8,ZS(R 4 ) = 〈∂ 1,3j−1 3j,Z (ρ1), ∂ 2,3j−1 3j,Z (ρ2), ∂ 3,3j−1 3j,Z (ρ3)〉. The morphism C from spectral sequence (50) to spectral sequence (44) induced by the reduction homo- morphism Z→ F2 implies that: 1,3j−1 3j,Z (ρ1)) = ∂ 1,3j−1 3j,F2 (c∗(ρ1)) = ∂ 1,3j−1 3j,F2 (x) = 0 2,3j−1 3j,Z (ρ2)) = ∂ 2,3j−1 3j,F2 (c∗(ρ2)) = ∂ 2,3j−1 3j,F2 (y2) = yj+2wj = yj+1wj−1(y + x)w 3,3j−1 3j,Z (ρ3)) = ∂ 3,3j−1 3j,F2 (c∗(ρ3)) = ∂ 3,3j−1 3j,F2 (yw) = yj+1wj+1 The sequence of D8 inclusion maps ⊕(j−1) 4 ) ⊂ S(R 4 ) ⊂ S(R ⊕(j+1) provides (Proposition 3.2) a sequence of inclusions: 2 〉 = IndexD8,ZS(R ⊕(j−1) 4 ) ⊇ IndexD8,ZS(R 4 ) ⊇ IndexD8,ZS(R ⊕(j+1) 4 ) = 〈Y 2 〉. (59) The relations (58), (59) and (40), along with Proposition 6.1 imply that for j odd: IndexD8,ZS(R 4 ) = 〈Y 2 M,Y Remark 6.3. The index IndexD8,ZS(Uk × R 4 ) appearing in the join test map scheme can now be computed. From Example 3.4 and the restriction diagram (43) it follows that IndexD8,ZS(Uk) = IndexD8,ZD8/H1 = ker resD8H1 : H ∗(D8,Z)→ H ∗(H1,Z) = 〈X 〉. The inclusions IndexD8,ZS(Uk ×R 4 ) ⊆ IndexD8,ZS(R 4 ) and IndexD8,ZS(Uk ×R 4 ) ⊆ IndexD8,ZS(Uk) imply that IndexD8,ZS(Uk ×R 4 ) ⊆ IndexD8,ZS(R 4 ) ∩ IndexD8,ZS(Uk) = {0}. Thus, as in the case of F2 coefficients, the Fadell–Husseini index theory with Z coefficients on the join CS/TM scheme does not lead to any obstruction to the existence of the equivariant map in question. 7 IndexD8,F2S This section is devoted to the proof of the equality IndexD8,F2S d × Sd = 〈πd+1, πd+2, w d+1〉. (60) The index will be determined by the explicit computation of the Serre spectral sequence associated with the Borel construction Sd × Sd → ED8 ×D8 Sd × Sd → BD8. The group D8 acts nontrivially on the cohomology of the fibre, and therefore the spectral sequence has nontrivial local coefficients. The E2-term is given by 2 = H p(BD8,H q(Sd × Sd,F2)) = H p(D8, H q(Sd × Sd,F2)) Hp(D8,F2) , q = 0, 2d Hp(D8,F2[D8/H1]) , q = d 0 , q 6= 0, d, 2d. The nontriviality of local coefficients appears in the d-th row of the spectral sequence. In Section 7.4 there is a sketch of an alternative proof of the fact (60) suggested by a referee for an earlier, F2-coefficient, version of the paper. 7.1 The d-th row as an H∗(D8,F2)-module Since the spectral sequence is an H∗(D8,F2)-module and the differentials are module maps we need to understand the H∗(D8,F2)-module structure of the E2-term. Proposition 7.1. H∗(D8,F2[D8/H1]) ∼=ring H ∗(H1,F2). Proof. Here H1 = 〈ε1, ε2〉 ∼= Z2 × Z2 is a maximal (normal) subgroup of index 2 in D8. Method 1: The statement follows from Shapiro’s lemma [8, Proposition 6.2, page 73] and the fact that when [G : H ] <∞, then there is an isomorphism of G-modules CoindGHM ∼= IndGHM . Method 2: There is an exact sequence of groups 1→ H1 → D8 → D8/H1 → 1. The associated LHS spectral sequence [1, Corollary 1.2, page 116] has the E2-term: 2 = H p(D8/H1, H q(H1,F2[D8/H1])) p(Z2, H q((Z2) 2,F2 ⊕ F2)) ∼= Hp(Z2;H q((Z2) 2,F2)⊕H q((Z2) 2,F2)). The action of the group D8/H1 ∼= Z2 on the sum is given by the conjugation action of G on the pair (H1, H q(H1,F2[D8/H1])) [8, Corollary 8.4, page 80]. Since F2[Z2] is a free Z2-module H0(Z2;F2[Z2]) = (F2[Z2]) Z2 = F2 and Hp(Z2;F2[Z2]) = 0 for p > 0. Thus ∼= Hp(D8/H1;H q((Z2) 2,F2)⊕H q((Z2) 2,F2)) p(D8/H1;F2[Z2] ∼= Hp(D8/H1;F2[Z2]) q+1 ∼= Hp(Z2;F2[Z2]) )Z2 ∼= F 2 , p = 0 0 , p > 0. Thus the E2-term has the shape as in Figure 5 (concentrated in the 0-column) and collapses. The first information about the H∗(D8,F2)-module structure on H ∗(D8,F2[D8/H1]), as well as the method for revealing the complete structure, comes from the following proposition. ÝHDÝH1,F2Þ ã H DÝH1,F2ÞÞ D8/H1 0 1 Figure 5: The A2-term of the LHS spectral sequence Proposition 7.2. We have x ·H∗(D8,F2[D8/H1]) = 0 for the nonzero element x ∈ H 1(D8,F2) that is characterized by resD8H1(x) = 0. Proof. Method 1: The isomorphismH∗(D8,F2[D8/H1]) ∼=ring H ∗(H1,F2) induced by Shapiro’s lemma [8, Propo- sition 6.2, page 73] carries the H∗(D8,F2)-module structure to H ∗(H1,F2) via the restriction homomor- phism resD8H1 : H ∗(D8,F2)→ H ∗(H1,F2). In this way the complete H ∗(D8,F2)-module structure is given on H∗(D8,F2[D8/H1]). In particular, since res (x) = 0, the proposition is proved. Method 2: The exact sequence of groups 1→ H1 → D8 → D8/H1 → 1 induces two LHS spectral sequences 2 = H p (D8/H1, H q (H1,F2[D8/H1])) =⇒ H p+q(D8,F2[D8/H1]), (62) 2 = H p (D8/H1, H q (H1,F2)) =⇒ H p+q(D8,F2). (63) The spectral sequence (63) acts on the spectral sequence (62) t → A u+r,v+s In the E∞-term this action becomes an action of H ∗(D8,F2) on H ∗(D8,F2[D8/H1]). Since we already discussed both spectral sequences we know that 2 = A ∞ and B 2 = B From Figures 2 and 5 it is apparent that x ∈ B 2 = B ∞ acts by x · A 2 = 0 for every p and q. Corollary 7.3. Indexd+2D8,F2S d × Sd = im ∂d+1 : E d+1 → E ∗+d+1,0 ⊆ y ·H∗(D8,F2). Proof. Let α ∈ E d+1 and ∂d+1(α) /∈ y ·H ∗(D8,F2). Then x · ∂d+1(α) 6= 0. Since ∂d+1 is a H ∗(D8,F2)- module map and x acts trivially on H∗(D8,F2[D8/H1]), as indicated by Proposition 7.2, there is a contradiction 0 = ∂d+1(x · α) = x · ∂d+1(α) 6= 0. Proposition 7.4. H∗(D8,F2[D8/H1]) is generated as an H ∗(D8,F2)-module by H0(D8,F2[D8/H1]) and H 1(D8,F2[D8/H1]). Proof. Method 1: We already observed that Shapiro’s lemma H∗(D8,F2[D8/H1]) ∼=ring H ∗(H1,F2) carries the H∗(D8,F2)-module structure to H ∗(H1,F2) via the restriction homomorphism res : H∗(D8,F2) → H∗(H1,F2). Thus H ∗(H1,F2) as an H ∗(D8,F2)-module is generated by 1 ∈ H 0(H1,F2) together with a ∈ H1(H1,F2). Method 2: There is the exact sequence of D8-modules 0→ F2 → F2[D8/H1]→ F2 → 0, (64) where the left and right modules F2 are trivial D8-modules. The first map is a diagonal inclusion while the second one is a quotient map. The sequence (64) induces a long exact sequence on group cohomology [8, Proposition 6.1, page 71], 0→ H0 (D8,F2) → H0 (D8,F2[D8/H1]) → H0 (D8,F2) H1 (D8,F2) → H1 (D8,F2[D8/H1]) → H1 (D8,F2) → . . . From the exact sequence (65), compatibility of the cup product [8, page 110, (3.3)] and Proposition 7.2 one can deduce that δ0(1) = x. Then by chasing along sequence (65) with compatibility of the cup product [8, page 110,(3.3)] as a tool it can be proved that H∗(D8,F2[D8/H1]) is generated as a H ∗(D8,F2)-module by I = i0(1) and A ∈ q 1 ({y}). 7.2 Indexd+2D8,F2S d × Sd = 〈πd+1, πd+2〉 The index by definition is Indexd+2D8,F2S d × Sd = im ∂d+1 : E d+1 → E ∗+d+1,0 ∂d+1 : H ∗ (D8,F2[D8/H1])→ H ∗+d+1(D8,F2 From Proposition 7.4 this image is generated as a module by the ∂d+1-images of H 0 (D8,F2[D8/H1]) and of H1 (D8,F2[D8/H1]). The ∂d+1 image is computed by applying restriction properties given in Proposition 3.5 to the subgroup H1. With the identification of H ∗ (D8,F2[D8/H1]) given by Shapiro’s lemma the morphism of spectral sequences of Borel constructions induced by restriction is specified in Figure 6. Also, Indexd+2D8,F2S d × Sd = 〈∂D8d+1(1), ∂ d+1(a), ∂ d+1(b), ∂ d+1(a+ b)〉. a + b 0 ^d+1 y^d+1 0 1 d + 1 d + 2 2d 11612 d 11ã12 a ã Ýa +bÞ Ýa +bÞã a b ã b 0 ad+1 + Ýa +bÞd+1 ad+2 + Ýa +bÞd+2 aÝa +bÞÝad + Ýa +bÞd Þ bÝad+1 + Ýa +bÞd+1 Þ 0 1 2 3 4 d + 1 d + 2 Ed+1 term of the Borel construction Sd × Sd ¸ ED8 ×D8 ÝS d × Sd Þ ¸ BD8 term of the Borel construction Sd × Sd ¸ EH1 ×H1 ÝS d × Sd Þ ¸ BH1 Figure 6: The morphism of spectral sequences To simplify notation let ρd := a d + (a+ b)d+1. Then from 7−→ 11 ⊕ 12 7−→ ρd+1 {a, a+ b, b} a⊕ (a+ b) (a+ b)⊕ a 7−→ {ρd+2, a(a+ b)ρd, bρd+1} it follows that resD8H1 ∂D8d+1(1), ∂ d+1(a), ∂ d+1(b) , ∂ d+1(a+ b) = {ρd+2, a(a+ b)ρd, bρd+1}. The formula ρd+2 = a d+2 + (a+ b)d+2 = (a+ a+ b) ρd+1 + a(a+ b) ai(a+ b)d−1−i = bρd+1 + a(a+ b)(a+ a+ b) ai(a+ b)d−1−i = bρd+1 + a(a+ b)ρd together with Remark 1.3 and the knowledge of the restriction resD8H1 implies that resD8H1(πd) = ρd. Therefore, there exist xα, xβ, xγ, xδ ∈ ker(resD8H1) such that ∂D8d+1(1) = πd+1 + xα ∂D8d+1(a), ∂ d+1(b), ∂ d+1(a+ b) = {πd+2 + xβ, yπd+1 + xγ, wπd + xδ} . Since y divides πd, Proposition 7.2 implies that α = β = γ = δ = 0, and Indexd+2D8,F2S d × Sd = 〈∂D8d+1(1), ∂ d+1(a), ∂ d+1(b), ∂ d+1(a+ b)〉 = 〈πd+1, πd+2, yπd+1, wπd〉 = 〈πd+1, πd+2〉. Remark 7.5. The property that the concretely described homomorphism resD8H1 : H ∗(D8,F2[D8/H1])→ H ∗(H1,F2[D8/H1]) is injective holds more generally [13, Lemma on page 187]. 7.3 IndexD8,F2S d × Sd = 〈πd+1, πd+2, w In the previous section we described the differential ∂D8d+1 of the Serre spectral sequence associated with the Borel construction Sd × Sd → ED8 ×D8 Sd × Sd → BD8. The only remaining, possibly non-trivial, differential is ∂D82d+1. The following proposition describing E 2d+1 can be obtained from Figure 6. Proposition 7.6. E 2d+1 = ker ∂D8d+1 : E d+1 → E ∗+d+1,d = x ·H∗(D8,F2) Proof. The restriction property from Proposition 3.5(D), applied to the element 1 ∈ E d+1 = H ∗(D8,F2) implies that ∂D8d+1(1) 6= 0. Proposition 7.2, together with the fact that multiplication by y and by w in H∗(D8,F2[D8/H1]) is injective, implies that ker ∂D8d+1 : E d+1 → E ∗+d+1,d = xH∗(D8,F2). The description of the differential ∂D82d+1 : E 2d+1 → E ∗+2d+1,0 2d+1 comes in an indirect way. There is a D8-equivariant map Sd × Sd → Sd ∗ Sd ≈ S((V+− ⊕ V−+) ⊕(d+1) given by Sd × Sd ∋ (t1, t2) 7→ t2 ∈ S d ∗ Sd. The result of Section 5.1 and the basic property of the index (Proposition 3.2) imply that IndexD8,F2S d × Sd ⊇ IndexD8,F2S((V+− ⊕ V−+) ⊕(d+1) ) = 〈wd+1〉. Thus wd+1 ∈ IndexD8,F2S d × Sd. Since by Corollary 7.3 wd+1 /∈ Indexd+1D8,F2S d × Sd it follows that wd+1 ∈ im ∂D82d+1 : E 2d+1 → E 2d+2,0 But the only nonzero element in E 2d+1 is x, therefore ∂D82d+1 (x) = w This concludes the proof of equation (60). 7.4 An alternative proof, sketch The objective of our index calculation is to find the kernel of the map (cf. Section 3) H∗(ED8 ×D8 Sd × Sd ,F2) = H (Sd × Sd,F2)← H (pt,F2) = H ∗(ED8 ×D8 pt,F2). (66) This map is induced by the map of spaces ED8 ×D8 (S d × Sd)→ ED8 ×D8 pt. (67) From the definition of the product ×D8 the map (67) is induced by ED8 × (S d × Sd) → ED8 × pt, i.e. by (Sd × Sd)→ pt. The map (67), again by definition of product ×D8 is ED8 × (S d × Sd) /D8 → (ED8 × pt) /D8. (68) Let S2 ∼= Z2 denotes the quotient group D8/H1. There is a natural homeomorphisms [23, Proposition 1.59, page 40] ED8 × (S d × Sd) /S2 → ((ED8 × pt) /H1) /S2 (69) which is induced by the map ED8 × (S d × Sd) /H1 → (ED8 × pt) /H1 (70) Since ED8 is also a model for EH1, the map (70) is a projection map in the Borel construction of S with respect to the group H1: Sd × Sd ✲ ED8 × (S d × Sd) The group D8 acts freely on ED8 × (S d × Sd) and on ED8 × pt. Therefore the S2 actions on the spaces( ED8 × (S d × Sd) /H1 and (ED8 × pt) /H1 are also free. There are natural homotopy equivalences ED8 × (S d × Sd) /S2 ≃ ES2 ×S2 ED8 × (S d × Sd) ((ED8 × pt) /H1) /S2 ≃ ES2 ×S2 ((ED8 × pt) /H1) which transform the map (69) into a map of Borel constructions ES2 ×S2 ED8 × (S d × Sd) → ES2 ×S2 ((ED8 × pt) /H1) (72) induced by the map (70) on the fibres. The map between Borel constructions (72) induces a map of associated Serre spectral sequences which on the E2-term looks like 2 = H p(S2, H q((ED8 × Sd × Sd )/H1,F2))← H p(S2, H q((ED8 × pt) /H1,F2)) = H 2 . (73) The spectral sequence H 2 is the one studied in section 4.2. It converges to H ∗(D8,F2) and H 2 = H Lemma 7.7. E 2 = E Proof. The action of H1 on S d × Sd is free. Therefore ED8 × Sd × Sd /H1 ≃ Sd × Sd /H1 = RP d × RP d (74) where the induced action of S2 from ED8 × Sd × Sd /H1 onto RP d × RP d interchanges the copies of RP d × RP d. The S2-homotopy equivalence (74) induces an isomorphism of induced Serre spectral sequences of Borel constructions 2 = H p(S2, H ED8 × Sd × Sd /H1,F2)) ∼= H p(S2, H q(RP d × RP d,F2)) = G Since for the spectral sequence G 2 , by [1, Theorem 1.7, page 118], we know that G 2 = G ∞ , the same must hold for the spectral sequence E We have obtained the following presentation of the map (66) and the related map of the fibres (70). Proposition 7.8. (A) The map H∗D8(pt,F2) → H (Sd × Sd,F2) gives rise to a map of spectral sequences of S2-Borel constructions 2 = H p(S2, H q((ED8 × pt) /H1,F2))→ H p(S2, H q((ED8 × Sd × Sd )/H1,F2)) = E 2 (75) which is induced by the map on fibres ED8 × (S d × Sd) /H1 → (ED8 × pt) /H1. (B) The map on the fibres is the projection map in the H1-Borel construction Sd × Sd → ED8 × (S d × Sd) /H1 → BH1 . It is completely determined in F2 cohomology by its kernel: H∗(H1,F2)→ H ED8 × (S d × Sd) /H1,F2) = IndexH1,F2S d × Sd = 〈ad+1, (a+ b)d+1〉. The E 2 = E ∞ andH 2 = H ∞ are described by [1, Lemma 1.4, page 117]. Therefore, IndexD8,F2S or the kernel of the map of spectral sequences (75) is completely determined by the kernel of the map of S2-invariants F2[a, a+ b] F2[a, a+ b]/〈a d+1, (a+ b)d+1〉 H∗(H1,F2) S2 → H∗( ED8 × (S d × Sd) /H1,F2) where S2 action is given by a 7−→ a+ b. The equation (60) IndexD8,F2S d × Sd = 〈πd+1, πd+2, w d+1〉. is a consequence of the previous discussion, identification of elements (23) in the spectral sequence (22) and the following proposition about symmetric polynomials. Proposition 7.9. (A) A symmetric polynomial aik(a+ b)jk ∈ F2[a, a+ b] S2 is in the kernel of the map (76) if and only if for every monomial ad+1 | aik(a+ b)jk or (a+ b)d+1 | aik(a+ b)jk . (B) The kernel of the map (76), as an ideal in F2[a, a+ b] S2 is generated by ad+1 + (a+ b)d+1, ad+2 + (a+ b)d+2, ad+1(a+ b)d+1. The approach presented here, with all its advantages, has two disadvantages: (1) The carrier of the combinatorial lower bound for the mass partition problem, the partial index Indexd+2D8,F2S d × Sd, cannot be obtained without extra effort. (2) It cannot be used for computation of the index Indexd+2D8,ZS d × Sd; the spectral sequence H 2 , if considered with Z coefficients, is the sequence (35) whose E∞-term has a ring structure that differs from H∗(D8,Z). These were our reasons for presenting this idea just as a sketch. 8 IndexD8,ZS Let Π0 = 0, Π1 = Y and Πn+2 = YΠn+1 +WΠn, for n ≥ 0, be a sequence of polynomials in H ∗(D8,Z). This section is devoted to the proof of the equality Indexd+2D8,ZS d × Sd = 〈Π d+2 ,Π d+4 ,MΠ d 〉 , for d even 〈Π d+1 ,Π d+3 〉 , for d odd. The index is determined by the explicit computation of the Ed+2-term of the Serre spectral sequence associated with the Borel construction Sd × Sd → ED8 ×D8 Sd × Sd → BD8. As in the previous section, the group D8 acts nontrivially on the cohomology of the fibre and thus the coefficients in the spectral sequence are local. The E2-term is given by 2 = H p(BD8,H q(Sd × Sd,Z)) = Hp(D8, H q(Sd × Sd,Z)) Hp(D8,Z) , q = 0, 2d Hp(D8, H d(Sd × Sd,Z)) , q = d 0 , q 6= 0, d, 2d. The local coefficients are nontrivial in the d-th row of the spectral sequence. 8.1 The d-th row as an H∗(D8,Z)-module The D8-module M := H d(Sd × Sd,Z), as an abelian group, is isomorphic to Z× Z. Since the action of D8 on M depends on d we distinguish two cases. 8.1.1 The case when d is odd The action on M is given by ε1 · (x, y) = (x, y), ε2 · (x, y) = (x, y), σ · (x, y) = (y, x). Thus, there is an isomorphism of D8-modules M ∼= Z[D8/H1]. The situation resembles the one in Section 7.1, and therefore the following propositions hold. Proposition 8.1. H∗(D8,Z[D8/H1]) ∼=ring H ∗(H1,Z). Proof. The claim follows from Shapiro’s lemma [8, Proposition 6.2, page 73] and the fact that when [G : H ] <∞ there is an isomorphism of G-modules CoindGHM ∼= IndGHM . Proposition 8.2. Let T ∈H∗(D8,Z) and P ∈ H ∗(H1,Z) ∼= H ∗(D8,Z[D8/H1]). (A) The action of H∗(D8,Z) on H ∗(D8,Z[D8/H1]) is given by T ·P := resD8H1 (T ) ·P . Here P on the right hand side is an element of H∗(H1,Z) and on the left hand side is its isomorphic image under the isomorphism from the previous proposition. In particular, X·H∗(D8,Z[D8/H1]) = 0. (B) H∗(D8,Z)-module H ∗(D8,Z[D8/H1]) is generated by the two elements 1, α ∈ H∗(H1,Z) ∼= H ∗(D8,Z[D8/H1]) of degree 0 and 2. (C) The map H∗(D8,Z[D8/H1])→ H ∗(D8,F2[D8/H1]), induced by the coefficient map Z→ F2, is given by 1, α 7−→ 1, a2. Proof. The isomorphism H∗(D8,Z[D8/H1]) ∼=ring H ∗(H1,Z) induced by Shapiro’s lemma [8, Propo- sition 6.2, page 73] carries the H∗(D8,Z)-module structure to H ∗(H1,Z) via res : H∗(D8,Z) → H∗(H1,Z). In this way the complete H ∗(D8,Z)-module structure is given on H ∗(D8,Z[D8/H1]). The claim (B) follows from the restriction diagram (43). The morphism of restriction diagrams induced by the coefficient reduction homomorphism c : Z→ F2 implies the last statement. 8.1.2 The case when d is even The action on M is given by ε1 · (x, y) = (−x, y), ε2 · (x, y) = (x,−y), σ · (x, y) = (y, x). In this case we are forced to analyze the Bockstein spectral sequence associated with the exact sequence of D8-modules →M → F2[D8/H1]→ 0, (79) i.e. with the exact couple H∗(D8,M) ×2 ✲ H∗(D8,M) H∗(D8,F2[D8/H1]). First we study the Bockstein spectral sequence H∗(H1,M) ×2 ✲ H∗(H1,M) H∗(H1,F2[D8/H1]). As in Section 7.2, we have that H∗(H1,F2[D8/H1]) = F2[a, a + b] ⊕ F2[a, a + b]. The module M as an H1-module can be decomposed into the sum of two H1-modules Z1 and Z2. The modules Z1 ∼=Ab Z and Z2 ∼=Ab Z are given by ε1 · x = −x, ε2 · x = x and ε1 · y = y, ε2 · y = −y for x ∈ Z1 and y ∈ Z2. This decomposition also induces a decomposition of H1-modules F2[D8/H1] ∼= F2 ⊕ F2. Thus, the exact couple (81) decomposes into the direct sum of two exact couples H∗(H1, Z1) ×2 ✲ H∗(H1, Z1) H ∗(H1, Z2) ×2 ✲ H∗(H1, Z2) H∗(H1,F2) H∗(H1,F2) Since all the maps in these exact couples are H∗(H1,Z)-module maps, the following proposition com- pletely determines both exact couples. Proposition 8.3. In the exact couples (82) differentials d1 = c ◦ δ are determined, respectively, by d1(1) = a, d1(b) = b(b+ a) and d1(1) = a+ b, d1(a) = d1(b) = ab. (83) Proof. In both claims we use the following diagram of exact couples induced by restrictions, where i ∈ {1, 2}: (H1, Zi) ×2 ✲ H∗(H1, Zi) (H1,F2) (K1, Zi) ×2 ✲ H∗(K1, Zi) (K1,F2) (K2, Zi) ×2 ✲ H∗(K2, Zi) (K2,F2) (K3, Zi) ×2 ✲ H∗(K3, Zi) (K3,F2) The first exact couple. The module Z1 is a non-trivial K1 and K3-module, but a trivial K2-module. Therefore by the long exact sequences (54), properties of Steenrod squares and the assumption at the end of the Section 4.3.2: (A) K1-exact couple: d1(1) = t1 and d1(t1) = 0; (B) K2-exact couple: d1(1) = 0 and d1(t2) = t (C) K3-exact couple: d1(1) = t3 and d1(t3) = 0. resH1K1(d1(1)) = t1 resH1K2(d1(1)) = 0 resH1K3(d1(1)) = t3 ⇒ d1(1) = a resH1K1(d1(b)) = 0 resH1K2(d1(b)) = t resH1K3(d1(b)) = 0 ⇒ d1(b) = b(b+ a). The second exact couple. The module Z2 is a non-trivial K2 and K3-module, while it is a trivial K1- module. Therefore by the long exact (54), properties of Steenrod squares and the assumption at the end of the Section 4.3.2: (A) K1-exact couple: d1(1) = 0 and d1(t1) = t (B) K2-exact couple: d1(1) = t2 and d1(t2) = 0; (C) K3-exact couple: d1(1) = t3 and d1(t3) = 0. resH1K1(d1(1)) = 0 resH1K2(d1(1)) = t2 resH1K3(d1(1)) = t3 ⇒ d1(1) = a+ b resH1K1(d1(b)) = t resH1K2(d1(b)) = 0 resH1K3(d1(b)) = 0 ⇒ d1(b) = ab. Remark 8.4. The result of the previous proposition can be seen as a key step in an alternative proof of the equation (20). Proposition 8.5. In the exact couple (80), with identification H∗(D8,F2[D8/H1]) = F2[a, a + b], the differential d1 = s ◦ δ satisfies d1(1) = a, d1(a+ b) = d1(b) = b(b+ a), d1(a 2) = a3. (84) (This determines d1 completely since c and δ are H ∗(D8,Z)-module maps.) Proof. Recall from the Remark 7.5 that the restriction map resD8H1 : H ∗(D8,F2[D8/H1])→ H ∗(H1,F2[D8/H1]) is injective. Then the equations (84) are obtained by filling the empty places in the following diagrams d1 ✲ a+ b d1 ✲ a2 ❄ d1✲ a⊕ (a+ b) (a+ b)⊕ a ❄ d1✲ b(b+ a)⊕ ab a2 ⊕ (a+ b)2 d1✲ a3 ⊕ (a+ b)3 where all vertical maps are resD8H1 . Corollary 8.6. H∗(D8,M) is generated as a H ∗(D8,Z)-module by three elements ζ1, ζ2, ζ3 of degree 1, 2, 3 such that c(ζ1) = a, c(ζ2) = b(a+ b), c(ζ3) = a where c is the map H∗(D8,M)→ H ∗(D8,F2[D8/H1]) from the exact couple (80). 8.2 Indexd+2D8,ZS d × Sd The relation between the sequences of polynomials πd ∈ H ∗(D8,F2) and Πd ∈ H ∗(D8,Z) is described by the following lemma. Lemma 8.7. Let c∗ : H ∗(D8,Z)→ H ∗(D8,F2) be the map induced by the coefficient morphism Z→ F2 (explicitly given by (40)). Then for every d ≥ 0, c∗(Πd) = π2d. Proof. Induction on d ≥ 0. For d = 0 and d = 1 the claim is obvious. Let d ≥ 2 and let us assume that claim holds for every d ≤ k + 1. Then c∗(Πk+2) = c∗(YΠk+1 +WΠk) hypo. = y2π2k+2 + w 2π2k = y 2π2k+2 + ywπ2d+1 + ywπ2d+1 + w = y(yπ2k+2 + wπ2d+1) + w(yπ2d+1 + wπ2k) = yπ2k+3 + wπ2k+2 = π2k+4. There is a sequence of D8-inclusions S1 × S1 ⊂ S2 × S2 ⊂ · · · ⊂ Sd−1 × Sd−1 ⊂ Sd × Sd ⊂ Sd+1 × Sd+1 ⊂ · · · implying a sequence of ideal inclusions Index S1 × S1 ⊇ Index4D8,ZS 2 × S2 ⊇ · · · ⊇ Indexd+1D8,ZS d−1 × Sd−1 ⊇ Indexd+2D8,ZS d × Sd ⊇ · · · (85) 8.2.1 The case when d is odd In this section we prove that Index Sd × Sd = 〈Π d+1 ,Π d+3 〉. (86) The proof can be conducted as in the case of F2 coefficients (Section 7.2). The results of Section 7.2 can also be used to simplify the proof of equation (86). The morphism c∗ : H ∗(D8,Z) → H ∗(D8,F2) induced by the coefficient morphism Z → F2 is a part of the morphism C of Serre spectral sequences (78) and (61). Thus, for 1 ∈ E d+1 = H 0(D8, H d(Sd × Sd,Z)), 1̂ ∈ E0,dd+1 = H 0(D8, H d(Sd × Sd,F2)), α ∈ E d+1 = H 2(D8, H d(Sd × Sd,Z)) and a ∈ E1,dd+1 = H 1(D8, H d(Sd × Sd,Z)), C(∂d+1(1)) = ∂d+1(C(1)) = ∂d+1(1̂) = πd+1 = C Π d+1 C(∂d+1(α)) = ∂d+1(C(α)) = ∂d+1(a 2) = ∂d+1(w · 1̂ + y · a) = wπd+1 + yπd+2 = πd+3 = C Π d+3 From Proposition 8.2 and the sequence of inclusions (85) it follows that ∂d+1(1) = Π d+1 and ∂d+1(α) = Π d+3 Finally, the statement (B) of Proposition 8.2 implies equation (86). 8.2.2 The case when d is even In this section we prove that Indexd+2D8,ZS d × Sd = 〈Π d+2 ,Π d+4 ,MΠ d 〉. (87) The previous section implies that ,Π d+2 〉 ⊇ Indexd+2D8,ZS d × Sd ⊇ 〈Π d+2 ,Π d+4 〉. (88) From Corollary 8.6 we know that Indexd+2D8,ZS d × Sd is generated by three elements ∂d+1(ζ1), ∂d+1(ζ2), ∂d+1(ζ3) of degrees d + 2, d + 3, d + 4. Thus, ∂d+1(ζ1) = Π d+2 and ∂d+1(ζ2) = MΠ d . Since Π d+4 〈Π d+2 ,MΠ d 〉, then ∂d+1(ζ3) = Π d+4 . The proof of the equation (87) is concluded. 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Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76. Princeton Univer- sity Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974, vii+331 pp. [30] E. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996), 147-167. [31] R. T. Živaljević, Topological methods, Chap. 14 in CRC Handbook on Discrete and Computational Geom- etry, J. E. Goodman and J. O’Rourke, eds., 2nd edition 2004, CRC Press, Boca Raton FL, pp. 305-329. [32] R. Živaljević, User’s guide to equivariant methods in combinatorics II, Publ. Inst. Math. Belgrade, 64(78), 1998, 107-132. http://arxiv.org/abs/1004.0746 1 Introduction 1.1 The hyperplane mass partition problem 1.2 Statement of the main result (k=2) 1.3 Proof overview 1.4 Evaluation of the index bounds 1.4.1 F2-evaluation 1.4.2 Z-evaluation 2 Configuration space/Test map scheme 2.1 Configuration space 2.2 Test map 2.3 The test space 3 The Fadell–Husseini index theory 3.1 Equivariant cohomology 3.2 IndexG,R and IndexG,Rk 3.3 The restriction map and the index 3.4 Basic calculations of the index 3.4.1 The index of a product 3.4.2 The index of a sphere 4 The cohomology of D8 and the restriction diagram 4.1 The poset of subgroups of D8 4.2 The cohomology ring H(D8,F2) 4.3 The cohomology diagram of subgroups with coefficients in F2 4.3.1 The Z2Z2-diagram 4.3.2 The D8-diagram 4.4 The cohomology ring H(D8,Z) 4.4.1 Evens' view 4.4.2 The Bockstein spectral sequence view 4.5 The D8-diagram with coefficients in Z 4.5.1 The Z2Z2-diagram 4.5.2 The D8-diagram 5 IndexD8,F2S(R4j) 5.1 IndexD8,F2S(V-+V+-)="426830A w"526930B 5.2 IndexD8,F2S(V–)="426830A y"526930B 5.3 IndexD8,F2S(R4j)="426830A yjwj"526930B 6 IndexD8,ZS(R4j) 6.1 The case when j is even 6.2 The case when j is odd 7 IndexD8,F2SdSd 7.1 The d-th row as an H(D8,F2)-module 7.2 IndexD8,F2d+2SdSd="426830A d+1,d+2"526930B 7.3 IndexD8,F2SdSd="426830A d+1,d+2,wd+1"526930B 7.4 An alternative proof, sketch 8 IndexD8,Z SdSd 8.1 The d-th row as an H(D8,Z )-module 8.1.1 The case when d is odd 8.1.2 The case when d is even 8.2 IndexD8,Zd+2SdSd 8.2.1 The case when d is odd 8.2.2 The case when d is even

0704.1944

A new method of alpha ray measurement using a Quadrupole Mass Spectrometer

A new method of alpha ray measurement using a Quadrupole Mass Spectrometer Y. Iwata a,∗, Y. Inoue b, M. Minowa a aDepartment of Physics, School of Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan bInternational Center for Elementary Particle Physics(ICEPP), University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Abstract We propose a new method of alpha(α)-ray measurement that detects helium atoms with a Quadrupole Mass Spectrometer(QMS). A demonstration is undertaken with a plastic-covered 241Am α-emitting source to detect α-rays stopped in the capsule. We successfully detect helium atoms that diffuse out of the capsule by accumulating them for one to 20 hours in a closed chamber. The detected amount is found to be proportional to the accumulation time. Our method is applicable to probe α-emitting radioactivity in bulk material. Key words: Quadrupole Mass Spectrometer, Helium atoms, Counting 1 Introduction There are many ways to detect α rays such as gas-flow counters and solid- state detectors[1], but all detect the corresponding ionization signal by the incident α ray, instead of 4He itself. However, because α particles travel only a few centimeters in the air and can be easily stopped by a piece of thin foil or paper, it is harder to detect their rays than those of beta or gamma radiation. In this work, we suggest a new method of α ray measurement that aims to detect 4He neutral atoms. If one wants to measure α-emitting radioactivity in bulk material with ordinary detectors, one has to rely on α rays emitted ∗ Corresponding author. Tel.: +81 3 5841 7622; fax: +81 3 5841 4186. Email address: [email protected] (Y. Iwata). Preprint submitted to Elsevier 13 November 2018 http://arxiv.org/abs/0704.1944v1 from the thin surface of the material because of their short range. However, many materials diffuse stopped α particles out of their surface in the form of neutral helium atoms. The amount of the released helium atoms can then be measured by a Quadrupole Mass Spectrometer (QMS) in terms of the mass number A = 4. Therefore, there is the advantage of being able to measure α- emitting radioactivity in bulk materials. It can be seen that a higher sensitivity for alphas can be expected. 2 Experimental setup and method To examine our method of α-ray measurement, we used an Amersham X.825 disc type 241Am α source (Fig. 1) and a vacuum system including a QMS. The 241Am source, an α-emitter with a half life of T1/2 = 432.2 y, is covered with epoxy resin 25mm in diameter and 3mm in thickness. This source emits 3.25 × 105 alphas per second, which eventually stop in the resin and diffuse out as 4He neutral atoms. The time needed for diffusion in resin is negligible 1 compared to the time elapsed of about 10 years or more after the production of this α source. Therefore, the helium production rate of the source is thought to be the same as that of alpha (3.25×105 s−1). A Quadrupole Mass Spectrometer (QMS) is often used for trace element analysis because of its high isotopic selectivity and efficiency[3]. The QMS we used is a Pfeiffer Vacuum QMS200 with a channeltron detector. The mass range is A = 1–100 and the detection limit is 1× 10−12 Pa [4]. The experimental setup is shown in Figs. 2 and 3. A liquid nitrogen trap was introduced to capture unwanted out-gas especially from the resin. We controlled the valves, V1 and V2 (see Fig. 3), to measure the integrated QMS channeltron current QHe of helium as follows: (a) Draw a vacuum to ∼ 10−6Pa with both valves (V1, V2) open. (b) Keep valve V1 closed for Tac =1h,4h,10h and 20h to accumulate helium atoms from the source. The number of helium atoms expected was NHe = 1.17× 109, 4.68× 109, 1.17× 1010 and 2.34× 1010, respectively. (c) After closing valve V2, open valve V1 to introduce helium atoms to the QMS through an orifice for a couple of seconds. (d) Open valve V2 to introduce most of the remaining atoms at one time for helium detection. In our QMS system, the bypass valve V2 and the orifice(φ0.3mm) are placed for a high pressure sample gas. Here, we utilized them in step (c) to avoid a 1 Diffusion time of helium through various samples of epoxy resin adhesives is found in ref [2]. drastic change in the channeltron current measured by the QMS, which was observed with just V1 open after step (b). Considering the conductance of the orifice, the amount of helium atoms lost in step (c) is estimated to be less than 2%, so we measured the integration of the channeltron currents for 5 seconds after opening valve V2 in step (d). The integration time of 5s was determined after considering the time needed for opening valve V2 by hand (∼ 1s) and the evacuation time T0 ∼ 0.3s (≪5s) expected by the pumping speed of our vacuum system. To estimate the amount of spurious signal caused by the out-gas from the resin, we also measured channeltron currents against a 57Co source as a control covered with the same epoxy capsule as the 241Am source. Assuming that both sources are emitting out-gas of similar composition and amount, the net integrated channeltron current QHe can be defined as the difference between the integrated current QAm and QCo with the 241Am and the 57Co source, respectively, under the same accumulation time Tac. In terms of NHe described above, QHe can be written as QHe = QAm −QCo = eGRiNHe, (1) where e is the elementary charge, G ∼ 7 × 103 is the amplification factor of the channeltron, and Ri is the detection efficiency of the QMS, i.e., the ratio of the number of helium atoms detected by the QMS compared to the initial quantity NHe. Note that NHe and QHe are proportional to the accumulation time Tac. 3 Result and analysis We measured a set of QAm and QCo twice for each Tac =1h,4h,10h and 20h. Fig. 4 shows a few examples of the experimental result for Tac = 20h, 10h and 4h. A clear difference between two sources can be seen for the data of Tac = 20h and 10h. The difference can also be seen in Fig. 4-(C) (Tac = 4h), though a drastic change in the channeltron current makes it unclear. Fig. 5 shows the relationship between the accumulation time Tac and the net amount of helium QHe = QAm − QCo with the best fit under the assumption of linearity. Here, because of the difficulty in evaluating the exact error of QHe, we estimated it at QCo for each data point, which may be a conservative overestimation. The unclarity for the data of Tac = 4h described above is reflected in the error bars in this figure. The best-fit proportionality coefficient (1.9±0.2)× 10−12[C/h] corresponds to Ri ∼ 2 × 10 −6 according to Eq.(1). We can see certain degree of linearity between Tac and QHe, but the obtained detection efficiency Ri was rather small, probably because most of the helium atoms were evacuated by the vacuum pump before they got ionized. To estimate the ultimate detection limit under this system apart from the influence of the out-gas coming from the epoxy capsule, we measured the distribution of QBG, the background channeltron current without any sources integrated for 5 seconds, as shown in Fig. 6. The best fit to the data by the normal distribution is also shown in the figure. In this way, the standard deviation σ = 9.3 × 10−14 [C] was obtained. We defined the detection limit Qlimit as Qlimit = (1.645σ)× 2 = 2.2× 10−13 [C] (2) at 95% confidence level. Here, the factor 2 is introduced to take into account the two sets of the data, the 241Am and the 57Co source, that were used for detecting the helium signal. The obtained Qlimit corresponds to ∼ 10 8 helium atoms under our system. Although we did not try to obtain higher detection efficiency Ri with this mea- surement, since the valves were controlled manually, there are some possible solutions to improve Ri: • Atom buncher [5] An atom buncher is a cold trap to capture target gas atoms. It consists of a metal surface that often is cooled by liquid helium. Some kinds of laser can be used to temporarily heat a spot on the surface to evaporate gas atoms. The detection efficiency Ri would improve if the heated area is placed close to the ionization chamber of the QMS. This device can be applied to the detection of helium by cooling the surface to a temperature low enough to trap the helium atoms. • Pulsed Supersonic Valve [6] The Pulsed Supersonic Valve(PSV) is an electromagnetic device to generate supersonic free gas jet. This valve consists of two parallel metallic plates as a gate for the sample gas. The gas is only allowed to be temporarily introduced when the gate is opened by electromagnetic repulsion between these two plates. The jet of sample gas is then injected into the ionization chamber. Similar to the atom buncher described above, higher Ri can be achieved by locating the PSV near the ionization region of the QMS. Attaching devices such as an atom buncher and a PSV to improve detection efficiency and so obtain a higher Ri, much smaller quantities of helium atoms could be detected than under our present system. Also, some QMS are said to be able to count target atoms one by one[4], so our system could theoretically be improved to detect a single helium atom; that is, each α ray regardless of its energy if the detection efficiency Ri gets close to one. Our method can be applied to the accurate estimations of α-induced soft errors in very-large-scale integrated circuit (VLSI). Soft errors are caused by α rays from a minute amount of radioactive substance in LSI packages, and have become a serious problem in VLSI circuits[7]. Using mass analysis in vacuum may enable us to reduce the detection limit of α-emitting radioactivity in the packages compared to the conventional method with a gas flow proportional counter. 4 Conclusion We proposed a new method for alpha(α) ray measurement by detecting helium atoms with a QMS. An 241Am α source and a 57Co source as a control were used to examine our method. The result showed that we could successfully detect helium atoms, but the detection efficiency was only 2× 10−6 under our system. Additional devices such as an atom buncher and a PSV to improve detection efficiency may allow us to reduce the detection limit. This detector cannot measure the energy of the α particles, but that feature is reasonably common to many other conventional α detectors and is not a shortcoming in most applications. Our method may become practicable to choose low α- active material for LSI packages, which is essential to reduce α-induced soft error rates in VLSI circuits. References [1] Kai Siegbahn, Alpha-, beta- and gamma-ray spectroscopy, Amsterdam, 1965. [2] A. Gerlach, W. Keller, J. Schulz and K. Schumacher, Microsystem Technologies 7 (2001) 17. [3] N. Trautmann, G. Passler and K. Wendt, Anal. Bioanal. Chem. 378 (2004) [4] Pfeiffer Vacuum catalog: Mass spectrometer, 2005. [5] G. S. Hurst, et al., J. Appl. Phys. 55 (1984) 1278. [6] H. Pauly, Atom, Molecule, and Cluster Beams I, Springer, 2000. [7] Y. Tosaka, et al., Jpn. J. Appl. Phys. 45(4B) (2006) 3185. Fig. 1. Plastic-covered 241Am α source. The central “point” is 241Am. Fig. 2. Schematic view of our detection method. Fig. 3. Photograph of the experimental setup. 0 3 6 9 12 15 Time [s] V1 open V2 open 241Am α source 20h 57Co source (control) 20h (A) Tac = 20h. 0 3 6 9 12 15 Time [s] V1 open V2 open 241Am α source 10h 57Co source (control) 10h (B) Tac = 10h. 0 3 6 9 12 15 Time [s] V1 open V2 open 241Am α source 4h 57Co source (control) 4h (C) Tac = 4h. Fig. 4. A few examples of the experimental result. 0 4 8 12 16 20 Tac [h] f(Tac)=1.9*10 -12Tac fit f(Tac) Fig. 5. Linearity between Tac and QHe (Tac = 1h, 4h, 10h, 20h). 2.4 2.6 2.8 3.0 3.2 BG integral (5s) [10-12C] σ = 9.3*10-14 [C] data for fitting Fig. 6. Distribution of QBG. Introduction Experimental setup and method Result and analysis Conclusion References

0704.1945

Magnetorotational Collapse of Population III Stars

arXiv:0704.1945v1 [astro-ph] 16 Apr 2007 Magnetorotational Collapse of Population III Stars Yudai Suwa1, Tomoya Takiwaki1, Kei Kotake2, and, Katsuhiko Sato1,3 1Department of Physics, School of Science, The University of Tokyo, Tokyo 113-0033 2National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588,Japan 3Research Center for the Early Universe, School of Science, the University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan [email protected] (Received 2006 October 12; accepted 2007 April 11) Abstract We perform a series of two-dimensional magnetorotational core-collapse simula- tions of Pop III stars. Changing the initial distributions of rotation and magnetic fields prior to collapse in a parametric manner, we compute 19 models. By so do- ing, we systematically investigate how rotation and magnetic fields affect the collapse dynamics and explore how the properties of the black-hole formations and neutrino emissions could be affected. As for the microphysics, we employ a realistic equation of state and approximate the neutrino transfer by a multiflavour leakage scheme. With these computations, we find that the jet-like explosions are obtained by the magneto- driven shock waves if the initial magnetic field is as large as 1012G. We point out that the black-hole masses at the formation decrease with the initial field strength, on the other hand, increase with the initial rotation rates. As for the neutrino properties, we point out that the degree of the differential rotation plays an important role to determine which species of the neutrino luminosity is more dominant than the others. Furthermore, we find that the stronger magnetic fields make the peak neutrino lumi- nosities smaller, because the magnetic pressure acts to halt the collapse in the central regions, leading to the suppression of the releasable gravitational binding energies. Key words: stars: supernovae: general — black hole physics — neutrinos — methods: numerical — magnetohydrodynamics: MHD 1. INTRODUCTION Great attention has been paid to Population III, the first stars to form in the universe, because they are related to many unsettled problems in cosmology and the stellar physics. The Population III (Pop III) stars ionize and enrich the metallicity of the intergalactic medium and thus provide important clues to the subsequent star formation history (for reviews, see e.g., http://arxiv.org/abs/0704.1945v1 Barkana & Loeb 2001; Bromm & Larson 2004; Glover 2005). Pop III stars are also important for the understanding of the chemical evolution history. Recent discovery of hyper metal poor stars such as HE 0107-5240 (Christlieb et al. 2002) and HE 1327-2326 (Frebel et al. 2005) has given us good opportunities to investigate the nucleosynthesis in Pop III stars (Heger & Woosley 2002; Umeda & Nomoto 2002; Umeda & Nomoto 2003; Daigne et al. 2004; Iwamoto et al. 2005). Gamma-ray bursts at very high redshift are pointed out to be accompanied by the gravitational collapse of Pop III stars (Schneider et al. 2002; Bromm & Loeb 2006). The Swift satellite, which is now running 1, is expected to directly detect the Pop III stars accompanied by the high-z gamma-ray bursts. The evolutions of Pop III stars have been also studied for long. From their studies, Pop III stars are predicted to have been predominantly very massive with M >∼ 100M⊙ (Nakamura & Umemura 2001; Abel et al. 2002; Bromm et al. 2002, see references therein). Massive stars in the range of 100M⊙ <∼M ∼ 260M⊙ encounter the electron-positron pair instability during their evolution. This instability sets off explosive oxygen burning, and if the burning provides enough energy to reverse the collapse, the stars are thought to become pair instability supernovae (Bond et al. 1984; Fryer et al. 2001; Heger & Woosley 2002), whose detectability has been recently reported (Scannapieco et al. 2005; Weinmann & Lilly 2005). More massive stars, which also encounter pair-instability, are so tightly bound and the fusion of oxygen is unable to reverse infall. Such stars are thought to collapse to black holes (BHs) finally (Bond et al. 1984; Fryer et al. 2001), which we pay attention to in this paper. So far there have been a few hydrodynamic simulations studying the gravitational- collapse of the BH forming Pop III stars. In the two-dimensional, gray neutrino transport simulations by Fryer et al. (2001), they investigated the collapse of a rotating Pop III star of 300M⊙, leading to the BH formation. They discussed the effects of rotation on the emitted neutrino luminosities, gravitational waves, and furthermore, the possibility of such stars to be the gamma-ray bursts. In their Newtonian study, the central BH was excised and treated as an absorbing boundary after the formation. Although such simplification is not easy to be validated, they followed the dynamics long after the formation of the BH and obtained many findings. More recently, Nakazato et al. (2006) performed one-dimensional, but, general rel- ativistic simulations in the range of 100 ∼ 10000M⊙, in which the state-of-the-art neutrino physics are taken into account. Their detailed calculations revealed the properties of the emer- gent neutrino spectrum, and based on that, they discussed the detectability of such neutrinos as the supernova relic neutrino background (see also Ando & Sato 2004; Iocco et al. 2005). They successfully saw the formation of the apparent horizon, however, the dynamics in the later phases was not referred. In this paper we study the magnetorotational collapse of Pop III stars by performing the two-dimensional magnetohydrodynamic (MHD) simulations (see, also, Akiyama et al. 2003; 1 See http://swift.gsfc.nasa.gov Kotake et al. 2004a; Kotake et al. 2004b; Takiwaki et al. 2004; Yamada & Sawai 2004; Ardeljan et al. 2005; Sawai et al. 2005; Obergaulinger et al. 2006, for MHD computations of core-collapse supernovae, and Kotake et al. 2006 for a review). As for the microphysics, we employ a realistic equation of state based on the relativistic mean field theory and take into account the neutrino cooling by a multiflavor leakage scheme, in which state-of-the-art reactions of neutrinos are included. In our Newtonian simulations, the formation of the BHs is ascribed to a certain condition, and after the formation, the central region is excised and treated as an absorbing boundary in order to follow the dynamics later on. Since the distributions of rotation and magnetic fields in the progenitors of Pop III stars are highly uncertain, we change them in a parametric manner and systematically investigate how rotation and magnetic fields affect the dynamics. We also explore how the natures of explosions, the properties of the BHs and neutrino luminosities could be affected due to the incursion of the rotation and magnetic fields. This paper is organized as follows. In §2, we describe the numerical methods and the initial conditions. In §3, we present the results. We give a summary and discussion in §4. 2. METHOD 2.1. Basic Equations The basic evolution equations are written as follows, + ρ∇ ·v = 0, (1) =−∇P − ρ∇Φ+ (∇×B)×B, (2) =−P∇ ·v−Lν , (3) =∇× (v×B) , (4) △Φ= 4πGρ, (5) where ρ,P,v,e,Φ,B,Lν , , are the mass density, the gas pressure including the radiation pres- sure from neutrino’s, the fluid velocity, the internal energy density, the gravitational potential, the magnetic field, the neutrino cooling rate, and Lagrange derivative, respectively. In our 2D calculations, axial symmetry and reflection symmetry across the equatorial plane are assumed. Spherical coordinates (r, θ) are employed with logarithmic zoning in the radial direction and regular zoning in θ. One quadrant of the meridian section is covered with 300 (r)× 30 (θ) mesh points. The minimum and maximum mesh spacings are 2 km and 60 km, respectively. We also calculated some models with 60 angular mesh points, however, any significant difference was obtained. Therefore, we will report in the following the results obtained from the models with 30 angular mesh points. We employed the ZEUS-2D code (Stone & Norman 1992) as a base and added major changes to include the microphysics. First we added an equation for elec- tron fraction to treat electron captures and neutrino transport by the so-called leakage scheme (Kotake et al. 2003). Furthermore, we extend the scheme to include all 6 species of neutrino (νe, ν̄e, νX), which is indispensable for the computations of the Pop III stars. Here νX means νµ, ν̄µ, ντ and ν̄τ . As for the reactions of νX , pair, photo, and plasma processes are included using the rates by Itoh et al. (1989). The Lν , in Eq. (3) is the cooling rate of the relevant neutrino reactions (see Takiwaki et al. 2007, for details). As for the equation of state, we have incorporated the tabulated one based on relativistic mean field theory instead of the ideal gas EOS assumed in the original code (Shen et al. 1998). 2.2. Initial Models and Boundary Condition In this paper, we set the mass of the Pop III star to be 300M⊙. This is consistent with the recent simulations of the star-formation phenomena in a metal free environment, providing an initial mass function peaked at masses 100−300M⊙ (see, e.g., Nakamura & Umemura 2001). We choose the value because we do not treat the nuclear-powered pair instability supernovae (M<∼260M⊙) and, for convenience, for the comparison with the previous study, which employed the same stellar mass (Fryer et al. 2001). We start the collapse simulations of 180M⊙ core of the 300M⊙ star. The core, which is the initial condition of our simulations, is produced in the following way. According to the prescription in Bond et al. (1984), we set the polytropic index of the core to n= 3 and assume that the core is isentropic of ∼ 10kB per nucleon (Fryer et al. 2001) with the constant electron fraction of Ye = 0.5. We adjust central density to 5× 10 6 g cm−3, by which the temperature of the central regions become high enough to photodisintegrate the iron (∼ 5× 109K), thus initiating the collapse. Given central density, the distribution of electron fraction, and entropy, we construct numerically the hydrostatic structures of the core. Since we know little of the angular momentum distributions in the cores of Pop III stars (see, however, Fryer et al. 2001), we add the following rotation profiles in a parametric manner to the non-rotating core mentioned above. We assume the cylindrical rotation of the core and change the degree of differential rotation in the following two ways. 1. As for the differential rotation models, we assume the following distribution of the initial angular velocity, Ω(X,Z) = Ω0 X2+X20 Z4+Z40 , (6) where Ω is the angular velocity and Ω0 is the model constant. X and Z denote distance from rotational axis and the equatorial plane, respectively. We adopt the value of param- eters, X0 and Z0, as 2×10 8cm,2×109cm, respectively. Since the radius of the outer edge of the core is taken to be as large as 3.5× 109cm, the above profile represents that the cores rotate strongly differentially. 2. As for the rigid rotation models, the initial angular velocity is given by, Ω(X,Z) = Ω0. (7) As for the initial configuration of the magnetic fields, we assume that the field is nearly uniform and parallel to the rotational axis in the core and dipolar outside (see Figure 1). For the purpose, we consider the following effective vector potential, Ar = Aθ = 0, (8) r3+ r30 r sinθ, (9) where Ar,θ,φ is the vector potential in the r,θ,φ direction, respectively, r is the radius, r0 is the radius of the core, and B0 is the model constant. In this study, we adopt the value of r0 as 3.5× 109 cm. This vector potential can produce the uniform magnetic fields when r is small compared with r0, and the dipole magnetic fields for vice versa. We set the outflow boundary conditions for the magnetic fields at the outer boundary of the calculated regions. It is noted that this is a far better way than the loop current method for constructing the dipole magnetic fields (LeBlanc & Wilson 1970), because our method produces no divergence of the magnetic fields near the loop current. Fig. 1. The configuration of the initial magnetic fields. Note that B0 = 10 12G for this figure. The arrows represent the vector of the poloidal magnetic fields. The contour shows the logarithm of the magnetic pressure (: B2/8π). Changing the initial rotational and magnetic energies by varying the values of Ω0 and B0, we compute 19 models in this paper, namely, one spherical and 18 magnetorotational models. In Table 1, we summarize the differences of the initial models. Note that the models are named after this combination, with the first letters, B12, B11, B10, indicating the strength of initial magnetic field, the following letter, TW1, TW2, TW4 indicating the initial T/|W | and final capital letter D or R representing the initial rotational law (D: Differential rotation, R: Rigid rotation). Note that T/|W | represents the ratio of the rotational to the gravitational energy. Table 1. Models and Parameters.∗ T/|W | B0 1% 2% 4% 1010G B10TW1{D,R} B10TW2{D,R} B10TW4{D,R} 1011G B11TW1{D,R} B11TW2{D,R} B11TW4{D,R} 1012G B12TW1{D,R} B12TW2{D,R} B12TW4{D,R} ∗ This table shows the name of the models. In the table they are labeled by the strength of the initial magnetic field and rotation. T/|W | represents the ratio of the rotational to the gravitational energy. B0 represents the strength of the initial magnetic field. In this paper we assume a BH is formed when the condition 6Gm(r) >r is satisfied, where c,G,m(r) are the speed of light, the gravitational constant and the mass coordinate, respectively. This condition means that we assume that fluids cannot escape from the inner region below the radius of the marginally stable orbit of a Schwarzschild BH. When this condition is satisfied, we excise the region inside the radius calculated and then treat it as an absorbing boundary. Afterwards, we enlarge the boundary of the excised region to take into account the growth of the mass infalling into the central region. Although it is not accurate at all to refer the central region as the BH, we cling to the simplification in this paper in order to follow and see the dynamics later on. 3. RESULT 3.1. Spherical Collapse First of all, we briefly describe the hydrodynamic features of spherical collapse as a baseline for the MHD models mentioned later. Note in the following that by “massive stars”, we mean the stars of ≈ O(10)M⊙ with the initial composition of the solar metallicity, which are considered to explode as supernovae at their ends of the evolution (Heger et al. 2003). As in the case of massive stars, the gravitational collapse is triggered by the electron- capture reactions and the photodisintegration of iron nuclei. On the other hand, the gravita- tional contraction is stopped not by the nuclear forces as in the case of massive stars but by the (gradient of) thermal pressure. This is because the progenitor of Pop III stars has high entropy, i.e. high temperature. We call this bounce as “thermal bounce” for convenience. The evolution of density, temperature, entropy and radial velocity around the thermal bounce are shown in Figure 2. Unlike the case of massive stars, no outgoing shock propagates outward after the thermal bounce. At the bounce, the size of the inner core, which is 200 km in radius and 6M⊙ in the masscoordinate, grows gradually due to the mass accretion. As seen from the figure, the materials in the accreting shock regions obtain higher entropy and temperature than 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] -1×1010 -8×109 -6×109 -4×109 -2×109 2×109 0 500 1000 1500 2000 Radius [km] -1×1010 -8×109 -6×109 -4×109 -2×109 2×109 0 500 1000 1500 2000 Radius [km] -1×1010 -8×109 -6×109 -4×109 -2×109 2×109 0 500 1000 1500 2000 Radius [km] -1×1010 -8×109 -6×109 -4×109 -2×109 2×109 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] 0 500 1000 1500 2000 Radius [km] Fig. 2. The evolutions of density, temperature, entropy and radial velocities in the spherical model. Solid line is for -37 ms from bounce, dashed line is for -1 ms, dotted line is for 19 ms and dashed dotted line is for 32 ms, respectively. the ones in the case of massive stars. The higher temperatures are good for producing a large amount of µ- and τ - neutrinos through the pair annihilation of electrons and positrons. This also makes different features of the neutrino emissions from the case of massive stars, in which the electron-neutrino (νe) luminosity dominates over those of the other species near the epoch of core bounce. As shown in the top panel of Figure 3, the total luminosity of νX (νµ, ντ , ν̄µ and ν̄τ ) begins to dominate over the total luminosity of electron neutrinos and anti-electron neutrinos at 25 msec after bounce (see the first intersection of the lines in the figure). At 87 msec after the bounce, the core is so heavy that it promptly collapses to a BH. In Figure 4, we show the evolution of the BH calculated by the procedure described in §2.2. The mass of the BH is initially 20M⊙, rapidly increases to 35 M⊙ because the rest of the dense inner core falls into the BH soon after the formation. The growth rate of the mass is slowed down afterwards when the quasi-steady accretion flow to the BH is established (Fig 4). The rapid decrease of the neutrino luminosity ∼100 msec after bounce (top panel of Figure 3) corresponds to the epoch when the neutrinospheres are swallowed into the BH. Note in the bottom panel, -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time After Bounce [s] -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time After Bounce [s] Fig. 3. Upper panel: Time evolutions of neutrino luminosity in the spherical model. The time is measured from the thermal bounce. Solid line represents the total luminosity of electron neutrinos and anti-electron neutrinos. Dashed line represents the total luminosity of νX (νµ, ντ , ν̄µ and ν̄τ ) neutrino. Before the thermal bounce, the luminosity of electron + anti-electron neutrinos dominate that of νX luminosity, though, after the bounce, it reverses. At ∼ 0.1 second after the bounce, the luminosities drastically decrease due to the BH formation. Lower panel: Time-integrated neutrino luminosities. Solid line and Dashed line are total energy emitted by electron neutrinos and anti-electron neutrinos and X neutrinos, respectively. -0.2 0 0.2 0.4 0.6 0.8 Time [s] Spherical Fig. 4. Evolution of the BH mass in the spherical model. The time is measured from the BH formation. The black filled circle indicates the epoch of the black-hole formation. the total emitted energy are calculated by dtLν , which represents the energy carried out from the core by neutrinos. Again from the quantity, it is shown that νX deprives dominantly of the gravitational energy of the core than νe and ν̄e. 3.2. Rotational Collapse 3.2.1. Effect of Differential Rotation Now we move on to discuss the features in the rotational core-collapse. The deviation of the dynamics from the spherical collapse comes from the initial rotation rates and the degree of the differential rotation initially imposed. To see the effects of the differential rotation on the collapse-dynamics, we first take models of B10TW1D (differential rotation) and B10TW1R (rigid rotation) as examples and mention the difference of them. The effects of the initial rotation rates are discussed later in §3.2.2. We first describe the collapse of model B10TW1D. As in the case of spherical collapse, the rotating core experiences the collapse due to the neutrino emission and the photodisintegration, but the difference appears at the time of the thermal bounce. Due to the pressure support supplied by the centrifugal force, model B10TW1D bounces at the pole at the epoch 17 msec later than that of the spherical collapse. The time evolutions after bounce is presented in Figure 5. It is shown that the materials of the inner core oscillate about 20 msec after bounce (see from the top left down to the bottom), and then the shock wave begins to propagate along the rotational axis (see from the top right down to the bottom). This jet-like shock wave finally stalls at Z ∼ 2×108 cm, where Z is the distance from the center along the rotational axis. It is noted that the shock wave formed at bounce does not stall in the strongly magnetized models as discussed in §3.3. In this weakly magnetized model, the stellar mantle just collapses to the central region after the shock-stall, and then leading to the formation of the BH. In this model, we follow the hydrodynamics until more than 99 % of the materials outside collapse to the BH (typically 2 sec after bounce). Fig. 5. Entropy profiles of differential rotation model of B10TW1D 50 (left top), 63 (left middle), 73 (left bottom), 87 (right top), 113 (right middle), and 127 (right bottom) ms after bounce, respectively. The color coded contour shows the logarithm of entropy (kB) per nucleon and arrows represent the velocity fields. Model B10TW1R thermally bounces rather isotropically in the center, not like model B10TW1D. This is because the central regions have less angular momentum in comparison with the differentially rotating model of B10TW1D. In Figure 6, the time evolutions of entropy after bounce are shown. Unlike B10TW1D (Figure 5), B10TW1R directly collapses to form the BH without producing the outgoing shock waves. This is because the central part has less pressure support from the centrifugal force due to the uniform rotation profile initially imposed. On the other hand, the model B10TW1R has more angular momentum than that of model B10TW1D in the outer part of the core. This leads to the suppression of the accretion rates of the infalling matter to the inner core. As a result, the core of model B10TW1R oscillates in a longer period than that of model B10TW1D because the dynamical timescale, which is proportional to ρ−1/2, becomes longer due to the smaller density there. Fig. 6. Same as Figure 5 but for the rigidly rotating model of B10TW1R at 56 (left top), 76 (left bottom), 106 (right top), and 134 (right bottom) ms after bounce, respectively. Color coded contour shows the logarithm of entropy (kB) per nucleon and arrows represent the velocity fields. Next we compare the masses of the BH at the formation and the subsequent growth between the two models (see Figure 7). The initial mass of BH of models B10TW1R and B10TW1D are 40 and 70 M⊙, respectively (see the black filled circles in the Figure). Both of them are larger than that of the spherical collapse model (∼ 20 M⊙). As mentioned, the reason that the earlier formation of less massive BH of model B10TW1R is that the model has smaller centrifugal forces in the central regions than model B10TW1D. On the other hand, reflecting the smaller mass accretion rates to the BH, the growth rate of BH’s mass of the model B10TW1R is smaller than that of the model B10TW1D (compare the slopes of the lines in the Figure after the BH formation). The luminosity of neutrinos and the total leaked energy of the model B10TW1D are shown in the left panel of Figure 8. It is found that the luminosity of µ and τ neutrinos (νX) do not overwhelm that of the electron neutrinos even after the bounce unlike the spherical collapse and that most energy are emitted by electron neutrinos (bottom panel). This is because the rotation suppresses the compression of the core, which lowers the temperature in the central regions than that of the spherical model. It should be noted that the energy production rates by the pair annihilation processes sharply depend on the temperature. The neutrino features of B10TW1R are found to be intermediate between the model B10TW1D and the spherical model (see right panels). If the initial rotation rates of the above two models become larger, the bounce occurs -0.2 0 0.2 0.4 0.6 0.8 Time[s] B10TW1D B10TW1R Fig. 7. Evolution of the masses of BH for the rotating models. Solid and dashed lines are for models B10TW1D and B10TW1R, respectively. The black circle indicates the epoch of the black-hole formation. Note that the time is measured from the epoch of the BH formation in the spherical model. more later due to the stronger centrifugal forces. In addition, the interval of the core oscilla- tions becomes longer. Except for such differences, the hydrodynamic features before the BH formation are mainly determined by the degree of the differential rotation as mentioned above and no qualitative changes are found as the initial rotation rates become larger. 3.2.2. Effects of Rotation on the BH mass and Neutrino emission In this section, we proceed to describe how the initial rotation rate and the degree of the differential rotation affect the growth of the BH masses and the neutrino emissions. The effects of rotation on the initial masses of the BHs for the almost purely rotating models, labeled by B10, are shown in Figure 9. As seen, larger the initial rotation rate becomes, the heavier BH is found to be produced. This tendency is independent of the degree of the differential rotation. This is simply because rapid rotation tends to halt the infall of the matter to the center, thus heavier masses are required to fulfill the condition of BH formation. It is furthermore found that the initial mass is larger for the differential rotation models than the rigid rotation models. This is regardless of the initial rotation rates. This is due to the smaller angular momentum of the rigid rotation models in the central regions than that of the differential rotation models as mentioned. In Figure 10, the growth of the BH mass for the corresponding models is shown. It is found that the epoch of the formation is delayed as the initial rotation rates become larger regardless of the degree of the differential rotation. As for the growth rates of BH’s mass, it is found that they are almost the same for the differential rotation models regardless of the initial rotation rates (see the left panel of Figure 10). This is because the outer part of the core has little angular momentum due to the strong differential rotation imposed, and thus falls to the center in the similar way. On the other hand, the initial rotation rates affect the evolution of BHs in the rigid rotation models (see the right panel of Figure 10). As the initial rotation -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time After Thermal Bounce [s] -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time After Thermal Bounce [s] -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time After Thermal Bounce [s] -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time After Thermal Bounce [s] Fig. 8. Same as Figure 3 but for models B10TW1D (left) and B10TW1R (right), respectively. 0 1 2 3 4 5 T/|W| [%] Differential Rotation Rigid Rotation Fig. 9. Effects of initial rotation rate and the degree of differential rotation on the initial mass of the BH. In this figure, the sequence of the models labeled by “B10”, which are almost purely rotating model, is chosen. Note that model B10TW4R is absent because this model does not produce the BH during the simulation time. rates become larger, the growth rates of the BHs become smaller due to the larger angular momentum imposed initially. -0.2 0 0.2 0.4 0.6 0.8 Time[s] B10TW1D B10TW2D B10TW4D Spherical -0.2 0 0.2 0.4 0.6 0.8 Time[s] B10TW1R B10TW2R Spherical Fig. 10. Time evolution of the BH mass for the almost purely rotating models labeled by B10. Left panel: Solid, dashed, dotted, and dashed-dotted lines, are for models B10TW1D, B10TW2D, B10TW4D, and the spherical model, respectively. Right panel: Solid, dashed, dotted lines are for models B10TW1R, B10TW2R, and the spherical model, respectively. The time is measured from the thermal bounce of each model. The black filled circles of each panel represent the epoch of the BH formation. BH’s mass at the formation affects the total energy emitted by the neutrinos because the neutrinos in the region of BH cannot escape to the outside of the core afterwards. The total energy emitted by neutrinos are shown in Figure 11. One can see the general trend in the figure that the emitted energy rapidly rises and then becomes constant. The transition to the constant phase corresponds to the formation of the BH. Also in this case, differential rotation models have similar features after the formation of the BHs (solid line of right panel of Figure 11). It is interesting that the model B10TW2R (dashed line of right panel of Figure 11) by contrast has different behaviors of total emitted energy. This is because this model produces the stable accretion disk around the central BH. As a result, the materials of the disk accretes only slowly to the BH, and thus can emit neutrinos for a longer time. The small accretion rate of this model is also prominent as seen in the Figure It is interesting to note that only about 10% of the gravitational energy of the core can be carried away by neutrinos even in the most rapidly rotating model considered here (B10TW4D). On the other hand, it is well known that neutrinos carry away 99 % of the gravitational energy of the protoneutron stars in case of the massive stars. The discrepancy stems obviously from the fact that most part of the inner core is absorbed to the BH in case of the Pop III stars. 3.3. Magnetorotational Collapse In this section, we present the results of the MHD models. First of all, we mention the magnetohydrodynamic (MHD) features in section 3.3.1, then discuss the MHD effects on the 2×1054 4×1054 6×1054 8×1054 1×1055 -0.4 -0.2 0 0.2 0.4 Time After Bounce [s] B10TW1D B10TW2D B10TW4D Spherical 2×1054 4×1054 6×1054 8×1054 1×1055 -0.4 -0.2 0 0.2 0.4 Time After Bounce [s] B10TW1R B10TW2R Spherical Fig. 11. Time-integrated neutrino leakage energy. Left panel: Solid line means B10TW1D, dashed line means B10TW2D, dotted line means B10TW4D and dash-dotted line means Spherical model. Right panel: Solid line means B10TW1R, dashed line means B10TW2R and dotted line means Spherical model. The time is measured from the thermal bounce of each model. BH mass and neutrino emissions in section 3.3.2. 3.3.1. MHD Feature Amongst the computed models, we find that the models with the strongest magnetic field (B = 1012G) can only produce the jet-like shock waves along the rotational axis, which can propagate outside of the core without shock-stall. First of all, we mention the properties of such models taking model B12TW1D as an example. The collapse dynamics before bounce is almost the same as the corresponding weak magnetic field model of B10TW1D. This is because the amplified magnetic fields by the com- pression and the field-wrapping are, of course larger than the weaker field model, but still much smaller than the matter pressure in the central regions. After the bounce, the toroidal mag- netic fields produced by the wrapping, provide the additional pressure support, thus acting to push the infalling matter as jetlike outflow rather than rotates along the magnetic field. The jet is launched when the magnetic pressure overcomes the local ram pressure of the accreting matter. This feature is different from another jet driving mechanism, the magneto-centrifugal acceleration (Blandford & Payne 1982). The MHD features of model B12TW1D after bounce are presented in Figure 12. From the right panels, it is shown that the regions behind the jet-like shock wave (Z ≥ 1.5×109 cm) become dilute with the density of ρ ∼ 105 g cm−3 and have very high entropy s ∼ 102kB. The bottom panel shows the jet is driven by magnetic pressure because the plasma beta (≡ gas pressure / magnetic pressure) of the region of jet is much smaller than unity. As the jet propagates in the core, a newborn BH is produced (see the white circles of right panels of Figure 12). The mass of the BH is initially 57.9M⊙, which is smaller than the one of B10TW1D (70.4M⊙). The reason of the difference is mentioned in §3.3.2. The properties of jet of model B12TW1D are shown in Figure 13. There are profiles of Fig. 12. Time evolution of shock waves of the strongest magnetized model of B12TW1D. The top panel of the figure shows the logarithm of entropy (kB) per nucleon, the middle panel of the figure shows logarithm of density (g cm−3), and bottom panel shows the logarithm of plasma beta. All left figures are at 119 ms from bounce, and the right figures are at 305 ms. The white circles of the right panels represent the BHs. density, radial velocity, magnetic field, and pressure at 104 ms after bounce. The density of matter in the jet region is ∼ 107 g cm−3. The speed of shock front is as large as 40 % of speed of light, which is mildly relativistic. It is easily seen that the toroidal magnetic field overwhelms the poloidal component behind the shock front. In the inner region, the poloidal magnetic field is larger due to compression. The magnetic pressure overwhelms gas pressure throughout the jet region as already depicted in Figure 12. Now we move on to discuss how rotation affects the dynamics while fixing the initial field strength. Figure 14 shows the properties of models B12TW1{D,R} and B12TW4{D,R} when the jet-like shock wave reaches to 1×109 cm. As clearly seen, the main difference between T/|W | = 1 % and 4 % is the degree of the collimation of the shock wave. As the initial rotation rates become large in the differential rotation models, the compression of the magnetic fields is hindered, thus leading to the suppression of the hoop stress of magnetic fields in the central regions (Takiwaki et al. 2004). As a result, the collimation of the shock wave becomes less 0 1×109 2×109 Radius [cm] -4×1010 -2×1010 2×1010 0 1×109 2×109 Radius [cm] 0 1×109 2×109 Radius [cm] poloidal toroidal 0 1×109 2×109 Radius [cm] magnetic pressure gas pressure Fig. 13. Various physical quantities around the rotational axis at 104 ms after bounce for model B12TW1D. Density (left top), radial velocity (right top), absolute value of magnetic field (left bottom), and pressure are shown. In the left bottom panel, the solid line and dashed line represent poloidal com- ponent and toroidal component, respectively. In the right bottom panel, the dashed line represents gas pressure and solid line represents magnetic pressure. (compare the top panels). In contrast, the difference of the degree of collimation between models B12TW1R and B12TW4R is smaller than B12TW1D and B12TW4D. This is because the materials in the inner region of rigid rotating models rotate more slowly than the differential models. Thus the degree of the collimation of the shocks depends weekly on the initial T/|W |. Models with the weaker initial magnetic fields do not produce the jet-like explosion except for model B11TW4R. These models collapse to BHs before formation of jets because of weak magnetic pressure. After forming a BH, especially differentially rotating model, rest parts of star rotate slowly so that the magnetic pressure does not grow up. Thus, when BH is formed before jet rises, rest of core only collapse to BH and entire the core are absorbed by 3.3.2. MHD Effects on the BH Mass and Neutrino Emission The MHD effects on the initial masses of BHs are shown in Tables 2 and 3. As seen, the initial BH mass gets smaller when the initial magnetic field becomes stronger. The angular Fig. 14. Profiles of the shock propagation for models B12TW1D (top left), B12TW4D (top right), B12TW1R (bottom left), and B12TW4R (bottom right), respectively. They show the color coded contour plots of logarithm of entropy (kB) per nucleon. Various profiles are found by changing the strength of the initial magnetic field and rotation. momentum transport by the magnetic fields is an important agent to affect the BH mass. This feature is seen in Figure 15, which represents the distribution of mean specific angular momentum of models B12TW1D and B10TW1D. The central region of B12TW1D has smaller angular momentum than B10TW1D due to angular momentum transport by magnetic fields. The peak of model B12TW1D represents the position of shock front on the equatorial plane. The transport of the angular momentum makes centrifugal force of central region smaller and enhances the collapse. This leads the BH mass smaller. This tendency is more prominent for the rigid rotation models (compare Tables 2 and 3) because the rotation of the central region is slower than differential rotation models and contraction of core is more significant, which leads the amplification of magnetic field and larger angular momentum transport. Figure 16 shows the relation between BH mass and angular momentum at BH formation. The angular momentum of BH gets larger with its mass at the time of BH formation. This is because the matter with large angular momentum cannot collapse due to centrifugal force and requires large Table 2. Initial Mass of Black Holes for Differentially Rotating Models [M⊙]. T/|W | B0 1% 2% 4% 1010G 70.4 87.3 106.6 1011G 70.4 87.3 106.6 1012G 57.9 75.8 96.6 Table 3. Initial Mass of Black Holes for Rigidly Rotating Models [M⊙]. T/|W | B0 1% 2% 4% 1010G 40.5 75.8 — 1011G 38.6 38.6 — 1012G 15.3 15.1 15.1 0 20 40 60 80 100 120 Mass Coordinate [M B12TW1D B10TW1D Fig. 15. Mean specific angular momentum over the shells as a function of the mass coordinate just before BH formation. The solid line and dotted line represent model B12TW1D and B10TW1D, respectively. amount of accreting matter to collapse BH, as already mentioned in §3.2. Next, we discuss the MHD effects of neutrino emissions. Figure 17 shows the peak neutrino luminosities as a function of the initial rotation rates. It is shown that the magnetic fields make the peak luminosities smaller, when fixing the initial degree of the differential rotation (compare B10D and B12D, and B10R and B12R). It is noted that the Pop III stars have gentle slope of the density prior to core-collapse, so that materials of the outer region have a great deal of the total gravitational energy of the iron core. Thus the stronger magnetic pressures, which prevent the accretion, make the liberating gravitational energy of the accreting matter smaller, and thus, results in the suppression of the peak luminosities. In case of the rigid rotation, the stronger centrifugal forces in the outer regions, lead to the stronger suppression of the releasable gravitational energy than in the case of the differential rotation. As a result, 10 20 30 40 50 60 70 80 90 100 110 BH Mass [M Fig. 16. Relation between BH mass and angular momentum at the BH formation. the peak luminosities for the rigidly rotating models decreases more steeply with the initial rotation rates than the ones for the differentially rotating models (compare B10R and B10D). 2×1055 4×1055 6×1055 0 1 2 3 4 5 T/|W| [%] Fig. 17. Effects of rotation and magnetic fields on the peak luminosity of neutrinos. 4. SUMMARY AND DISCUSSION We studied the magnetorotational core-collapse of Pop III stars by performing the two- dimensional magnetohydrodynamic simulations. Since the distributions of rotation and mag- netic fields in the progenitors of Pop III stars are highly uncertain, we changed them in a parametric manner and systematically investigated how rotation and magnetic fields affect the dynamics of Pop III stars. In addition, we explored how rotation and magnetic fields affect the formation of the BHs and the neutrino emissions. In the current Newtonian simulations, the BH formation was ascribed to a certain condition, and after the formation, the central region was excised and treated as an absorbing boundary. As for the microphysics, we took into account the neutrino cooling of 6 species by a leakage scheme with a realistic equation of state. With these computations, we have obtained the following results, 1. In the spherical model, the gravitational contraction is stopped by the gradient of the ther- mal pressure, not by the nuclear forces as in the case of the massive stars ≈O(10)M⊙ with the initial composition of the solar metallicity, because the progenitor of Pop III stars has high entropy, i.e. high temperature initially. Such high temperature also makes different features of the neutrino emissions from the case of the massive stars. The luminosity of µ- and τ - neutrinos dominates over that of the electron neutrinos after core bounce. Thus the gravitational energy of the core is carried away dominantly by µ- and τ - neutrinos. 2. As the initial rotation rates of the core become larger, it is found that the epoch of the BH formation is delayed later and that the initial masses of the BHs become larger. Fixing the initial rotational energy, the BH masses at the formation become larger as the degree of the differential rotation becomes stronger. As the initial degree of the differential rotation becomes larger, the electron neutrino luminosity is found to be more dominant over that of µ and τ neutrinos after core bounce, because the pair creation processes of µ and τ , which sharply depend on the temperature, are more suppressed. 3. We find that the jet-like explosions can be produced even in Pop III stars if the magnetic field is as large as 1012G prior to core-collapse. This jet-like shock wave is completely magneto-driven. 4. Jet-like shocks in the stronger magnetic field models are themselves found to make the initial mass of the BH smaller. The angular momentum transport by magnetic fields is found to be an important agent to make the initial mass of BHs smaller because the transport of the angular momentum enhances the collapse of the central regions. As a result, it is found that the initial BH masses for the most strongly magnetized models are found to become smallest when fixing the initial rotation rates. As for the neutrino luminosities, we point out that the stronger magnetic fields make the peak luminosities smaller, because they can halt the collapse of the materials. Here we shall refer to the limitations of this study. First of all, we mimicked the neutrino transfer by the leakage scheme. Although the scheme is a radical simplification, we checked that we could reproduce, at least, the qualitative features of the neutrino luminosities. The supremacy of νX neutrinos’ luminosity in the spherical collapse of Pop III stars obtained in this study is consistent with the foregoing studies by Fryer et al. (2001) and Nakazato et al. (2006), in which the more elaborate neutrino transport schemes were employed. Furthermore, the effects of rotation on the emergent neutrino luminosities are consistent with Fryer & Heger (2000), in which one model of the rotational collapse of the massive stars was investigated. Secondly, the simulations were done with the Newtonian approximation and we defined the BH formation by the marginally stable orbit of a Schwarzschild BH. This treatment is totally inaccurate because the core rotates so rapidly that fully general relativistic (MHD) simulations with the appropriate implementations of the microphysics are necessary, however, are still too computationally prohibitive and beyond our scope of this paper. Remembering these caveats, this calculation is nothing but a demonstration showing how the combinations of rotation and magnetic fields could produce the variety of the dynamics, and the important consequences in the properties of the neutrino emissions, and these outcomes, of course, should be re-examined by the more sophisticated simulations. In this study, we followed the dynamics till ∼ 1 sec after the formation of the BHs and saw the shock break-out from the cores in the strongly magnetized models. But if we were to follow the dynamics in the much later phase in the weaker magnetized models, the magneto-driven outflows might be produced, due to the long-term field-wrapping and/or the development of the so-called magnetorotational instability, as demonstrated in the study of collapsar (see, e.g., Proga et al. 2003; Fujimoto et al. 2006). But it should be noted that the dynamical phases considered here and the other ones are apparently different (and thus they are complimentary). In the latter studies, the central BHs with a rotationally supported disk around are treated as an initial condition for the computations. Our core-collapse simulations presented here showed that the outer region rotates much slowly than the Keplerian one and most of them directly collapses to BH. Thus the amplification of the magnetic field in the disks, which needs rapid rotation, might not be so efficient as previously demonstrated. To clarify it, we are now preparing for the long-term simulations, in which the final states obtained here are taken as an initial condition. Then we discuss the validity of the initial strength of the magnetic fields assumed in this study. For the purpose, we estimate the strength of the magnetic field just before collapse with Eq. (13) of Maki & Susa (2004), in which the thermal history of the primordial collapsing clouds was calculated in order to investigate the coupling of the magnetic field with the primordial gas. For example, Bini∼10 −7 G and nH,ini∼10 3 cm−3, which are the values they employed, lead B ∼ 1011 G if the magnetic flux is conserved during the contraction and the clouds collapse to 106 g cm−3. Although the above parameters chosen are slightly optimistic, the magnetic fields assumed in this study may not be so unrealistic. We pointed out that the total neutrino energy emitted from rotation models increases several times than the one of the spherical collapse model. However, the detection of such neutrinos as the diffusive backgrounds might be difficult because the Pop III stars are too distant (see Iocco et al. 2005). Alternatively, the detection of gravitational waves from Pop III stars as the backgrounds seems more likely (Buonanno et al. 2005; Sandick et al. 2006) by the currently planning air-borne laser interferometers such as LISA2, DECIGO (Seto et al. 2001) and BBO (Ungarelli et al. 2005), and needs further investigation. We found that the Pop III stars are able to produce jet-like explosions with mass ejections when the central cores are strongly magnetized. This may be important with respect to its relevance to the nucleosynthesis in such objects (Ohkubo et al. 2006). 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0704.1946

Hyperfine Quenching of the $4s4p ^{3}P_{0}$ Level in Zn-like Ions

EPJ manuscript No. (will be inserted by the editor) Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions J. P. Marques1, F. Parente2, and P. Indelicato3 1 Centro de F́ısica Atómica e Departamento F́ısica, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Ed. C8, 1749-016 Lisboa, Portugal, e-mail: [email protected] 2 Centro de F́ısica Atómica da Universidade de Lisboa e Departamento F́ısica da Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Monte da Caparica, 2825-114 Caparica, Portugal, e-mail: [email protected] 3 Laboratoire Kastler Brossel, École Normale Supérieure; CNRS; Université P. et M. Curie - Paris 6 Case 74; 4, place Jussieu, 75252 Paris CEDEX 05, France, e-mail: [email protected] Received: October 24, 2018/ Revised version: date Abstract. In this paper, we used the multiconfiguration Dirac-Fock method to compute with high precision the influence of the hyperfine interaction on the [Ar]3d104s4p 3P0 level lifetime in Zn-like ions for stable and some quasi-stable isotopes of nonzero nuclear spin between Z = 30 and Z = 92. The influence of this interaction on the [Ar]3d104s4p 3P1 − [Ar]3d 104s4p 3P0 separation energy is also calculated for the same ions. PACS. 31.30.Gs – 31.30.Jv – 32.70.Cs 1 Introduction It has been found before that the hyperfine interaction plays a fundamental role in the lifetimes and energy sep- arations of the 3P0 and 3P1 levels of the configurations 1s2p in He-like [1,2,3,4,5,6,7,8,9], [He]2s2p in Be-like [10, 11,12], and [Ne]3s3p in Mg-like [11,13] ions, and also in 3d4J = 4 level in the Ti-like ions [14]. In the He-like ions, in the region Z ≈ 45, these two lev- els undergo a level crossing [15] and are nearly degenerate due to the electron-electron magnetic interaction, which leads to a strong influence of the hyperfine interaction on the energy splitting and on the 3P0 lifetime for isotopes with nonzero nuclear spin. In Be-like, Mg-like and Zn-like ions a level crossing of the 3P0 and 3P1 levels has not been found [16], but hyperfine interaction still has strong influ- ence on the lifetime of the 3P0 metastable level and on the energy splitting, for isotopes with nonzero nuclear spin. Until recently, laboratorymeasurements of atomic mes- tastable states hyperfine quenching have been performed only for He-like systems, for Z = 28 to Z = 79 [3,4,5,8, 17,18,19,20]. Hyperfine-induced transition lines in Be-like systems have been found in the planetary nebula NGC3918 [12]. Measured values of transition probabilities were found to agree with computed values [11,21]. Divalent atoms are being investigated, both theoreti- cally and experimentally, in order to investigate the possi- ble use of the hyperfine quenched 3P0 metastable state for ultraprecise optical clocks and trapping experiments [22]. Recently, dielectronic recombination rate coefficients were measured for three isotopes of Zn-like Pt48+ in the Send offprint requests to: J. P. Marques Heidelberg heavy-ion storage ring TSR [23]. It was sug- gested that hyperfine quenching of the 4s4p 3P0 in isotopes with non-zero nuclear spin could explain the differences detected in the observed spectra. In this paper we extend our previous calculations [2, 10,13] to the influence of the hyperfine interaction on the 1s22s22p63s23p63d104s4p levels in Zn-like ions. In Fig. 1 we show the energy level scheme, not to scale, of these ions. The 3P0 level is a metastable level; one- photon transitions from this level to the ground state are forbidden, and multiphoton transitions have been found to be negligible in similar systems, so the same behavior can be expected for Zn-like ions. Therefore, in first approxima- tion, we will consider the lifetime of this level as infinite. The energy separation between the 4 3P0 and 4 3P1 lev- els is small for Z values around the neutral Zn atom and increases very rapidly with Z. Hyperfine interaction is ex- pected to have a strong influence on the energy splitting and on the 4 3P0 lifetime for isotopes with nonzero nuclear spin. The different steps of this calculation are described in Refs. [2,24]. Here we will emphasize only the fundamental topics of the theory and the characteristic features of Zn- like systems. In this work we used the multi-configuration Dirac- Fock code of Desclaux and Indelicato [25,26,27] to evalu- ate, completely ab initio the 4s4p fine structure energies and transition probabilities. Several terms, such as the nonrelativistic (J indepen- dent) contribution to the correlation, are the same for all levels. In this calculation Breit interaction is included, as well as radiative corrections, using the method described, http://arxiv.org/abs/0704.1946v1 2 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions Fig. 1. Energy level and transition scheme for Zn-like ions (not to scale). for instance, in Ref. [28] and references therein. The MCDF method, in principle, allows for precise calculations be- cause it can include most of the correlation relatively eas- ily, i. e., with a small number of configurations. Here, cor- relation is important in the determination of transition energies to the ground state, which are used in the calcu- lation of transition probabilities. We found that the largest effect is obtained by using [Ar]3d104s2 and 4p2 as the con- figuration set for the ground state. For the excited states we included all configuration state functions (CSF) origi- nated from [Ar]3d104s4p, 4p4d, and 4d4f (which is usually defined as intrashell correlation), because in second-order perturbation theory the dominant energy difference de- nominators correspond to these configurations. Correla- tion originating from interaction with the Ar-like core has been neglected. In particular we included, in a test cal- culation, spin-polarization from s subshells and found a negligible influence in both energies and transition proba- bilities. We note that the fine structure energy separation, ∆E0;fs = E3P1 −E3P0 , which is the important parameter in the calculation of the hyperfine quenching, is not very sensitive to correlation, similarly to what we found for sys- tems with smaller number of electrons [10,13]. The same set of CSF have been used for energy, transition proba- bilities, and for the calculation of the hyperfine matrix elements. All energy calculations are done in the Coulomb gauge for the retarded part of the electron-electron interaction, to avoid spurious contributions (see for example Refs. [29, 31]). The lifetime calculations are all done using exact relativistic formulas. The length gauge has been used for all transition probabilities. 2 Relativistic calculation of hyperfine contribution to fine structure splitting and to transition probabilities In the case of a nucleus with nonzero spin, the hyperfine interaction between the nucleus and the electrons must be taken into account. The correspondent Hamiltonian can be written as Hhfs = (k)·T(k), (1) where M(k)and T(k) are spherical tensors of rank k, rep- resenting, respectively, the nuclear and the atomic parts of the interaction. As in the case of He-like, Be-like and Mg-like ions, the only sizable contribution from Eq. (1) is the magnetic dipole term (k = 1). The hyperfine inter- action mixes states with the same F = J + I values. In our case, we are interested in the 3P0 level, so J = 0 and F = I. The contribution of this interaction for the total energy has been evaluated through the diagonalization of the matrix: Htot = iΓ0 +W0,0 W0,1 W0,2 W1,0 E1 + iΓ1 +W1,1 W1,2 W2,0 W2,1 E2 + iΓ2 +W2,2 Here, Ef is the unperturbed level energy and Γf is the radiative width of the unperturbed level (f = 0, 1, 2 stands, respectively, for 3P0, 3 P1, 1 P1). In reality there is a fourth level, 3P2, which we included in all calculations but was found unnecessary, because the influence of this level in the 3P0 lifetime is negligible. This can be ex- plained by the large energy separation between the 3P2 and 3P0 levels and also because the probability of the allowed M2 transition 4s4p 3P2 → 4s 2 1S0 is many or- ders of magnitude smaller than those of the E1 transitions 4s4p 3P1 → 4s 2 1S0 and 4s4p 1P1 → 4s 2 1S0. Also, the magnetic dipole hyperfine matrix element between the 3P2 and 3P0 levels is very small. This also leads to a negligi- ble influence of the 3P2 level on the 3P1 − 3 P0 separation energy. This is consistent with the results of Plante e John- son [32], who found that the magnetic quadrupole term of the hyperfine interaction affects the 3P2 level only at high Z. The influence of the 1P1 level, however, must be taken into account, specially for light nuclei, because the large energy separation between 1P1 and 3P0 levels is compen- sated by the much shorter lifetime of the 1P1 level. The J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 3 hyperfine matrix element Wf,f ′ = Wf ′,f = 〈[Ar] 3d 104s4p f |Hhfs|[Ar] 3d 104s4p f ′〉 may be written as WJ1,J2 = 〈I, J1, F,MF |M (1)·T(1)|I, J2, F,MF 〉, (3) where I is the nuclear spin and F the total angular mo- mentum of the atom, and may be put in the form: WJ1,J2 = (−1) I+J1+F I J1 F J2 I 1 × 〈I||M(1)||I〉〈J1||T (1)||J2〉. (4) The 6j symbol leads to W0,0 = 0. Also the nuclear mag- netic moment µI in units of the nuclear magneton µN may be defined by µIµN = 〈I||M (1)||I〉 I 1 I −I 0 I , (5) with µN = eh/2πmpc. The electronic matrix elements were evaluated on the basis set |3P0〉, | 3P1〉, | 1P1〉 with all intrashell correlation included. The final result is then obtained by a diagonalization of the 3× 3 matrix in Eq. (2), the real part of each eigen- value being the energy of the correspondent level and the imaginary part its lifetime. 3 Results and discussion The MCDF method has been used to evaluate the influ- ence of the hyperfine interaction on the [Ar]3d104s4p 3P0, 3P1 and 1P1 levels for all Z values between 30 and 92 and for all stable and some quasi-stable isotopes of nonzero nuclear spin. Contributions from other levels have been found to be negligible. A detailed list of the contributions to the the theoretical 3P0, 3P1 and 1P1 level energies is presented in Table 1, for Z = 36, 54, and 82. In Table 2 we present, for all possible values of the nuclear spin I, Z, and the mass number A, the diagonal and off-diagonal hyperfine matrix elements Wi,j , and the nuclear magnetic moment, µI [37], in nuclear magneton units. The indexes 0, 1, and 2 in the hyperfine matrix elements stand for 3P0, 3P1, and 1P1, respectively. In Table 3 are presented, for all possible values of I, Z, and A, the unperturbed separation energies∆E0 = E3P1− E3P0 , the hyperfine-affected separation energy ∆Ehf, the 4s4p 3P1 and 4s4p 1P1 levels lifetime values, τ1 and τ2, respectively, which are not affected, within the precision shown, by the hyperfine interaction, and finally the per- turbed lifetime τ0 of the 4s4p 3P0 level. As we pointed out before, the unperturbed value of τ0 is assumed to be infinite. In Fig. 2 is plotted the difference E = ∆Ehf −∆E0 as a function of Z for the different possible values of nuclear spin. The influence of the hyperfine interaction on this energy is shown to increase slowly with Z, the increase becoming more rapid for Z ' 60. In Fig. 3 is plotted the perturbed 4s4p 3P0 level life- time τ0, for the different nuclear spin values I, as a func- tion of Z. One can easily conclude that the opening of a new channel for the decay of the 4s4p 3P0 level has a dramatic effect on its lifetime. After submitting this paper, it was brought to our at- tention the work by Liu et al [38], which contains inde- pendent calculations of the hyperfine quenched lifetime of the 3P0 level in several Zn-like ions. Our results for this lifetime are three times higher than the values found by Liu et al for Z = 30 and 1.5 higher at Z = 47. To look for the origin of this discrepancy we used the 3P1 → 1 S0 and 1P1 → 1 S0 transition energies and probabilities of Liu et al to diagonalize the matrix in eq. 2 and obtained results very close to our own calculations. As we do not know the values of the hyperfine matrix elements calculated by Liu et al, the reasons for this discrepancy remain unknown. One of the most interesting practical implication of these calculations comes from the relationship between the 4s4p 3P0 − 4s4p 3P1 levels energy separation and the 4s4p 3P0 level lifetime (Eq. 2). As referred in [13] this energy separation can be estimated from a measurement of the hyperfine-quenched 4s4p 3P0 lifetime of Zn-like ions with nuclear spin I 6= 0. This method has been demon- strated for heliumlike Ni26+ [4], Ag45+ [3], Gd62+ [17] and Au77+ [8]. In the Zn-like ions case, as in the correspond- ing Be-like and Mg-like ions, the situations is different, be- cause, even for the highest Z values, the lifetimes involved are much longer than in heavy heliumlike ions. However, measurements of Be-like hyperfine quenched transition rates have been performed from astrophysical sources [12] and the hyperfine quenching of the Zn-like 195Pt48+ ion was observed in the TSR heavy-ion storage-ring [23]. The continuous progress in storage rings, ion sources and ion traps leads us to believe that lifetimes between 0.1 s and 10 µs could be measured, with some accuracy, by directly looking at the light emitted by the ions as a function of time, after the trap has been loaded. Very good vacuum inside the trap is needed, of course, if long life- times are to be measured. It remains to be demonstrated that this method is experimentally feasible. Such experi- ments would be able to provide, for different isotopes, the unperturbed energy separation, because nuclear magnetic moments are well known. This would be an interesting way to test our relativistic calculations. Acknowledgments This research was partially supported by the FCT projects POCTI/FAT/50356/2002 and POCTI/0303/2003 (Portu- gal), financed by the European Community Fund FEDER, and by the French-Portuguese collaboration (PESSOAPro- gram, Contract n◦ 10721NF). 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Phys 77 1 (2005) 37. P. Raghavan, At. Data Nucl. Data Tables 42, 189 (1989) 38. Y. Liu, R. Hutton, Y. Zou, M. Andersson, T. Brage, J. Phys. B 39 3147 (2006) http://dirac.spectro.jussieu.fr/mcdf J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 5 Table 1. Contribution to the energy of the 4s4p3P0, 3 P1 and 1P1 levels (in eV). Z = 36 Coulomb† −75616.613 −75616.265 −75609.264 Magnetic† 42.785 42.779 42.765 Retardation (order ω2)† −4.095 −4.095 −4.094 Retardation (> ω2) −0.197 −0.197 −0.197 Self-energy (SE) 31.358 31.363 31.366 Self-energy screening −2.735 −2.740 −2.743 VP [ α(Zα)]correction to e-e interaction 0.033 0.033 0.033 Vacuum Polarization α(Zα)3 + α2(Zα) 0.010 0.010 0.010 2nd order (SE-SE + SE-VP + S-VP-E)‡ −0.031 −0.031 −0.031 Recoil −0.003 −0.003 −0.003 Relativistic Recoil♯ 0.009 0.009 0.009 Total energy −75549.478 −75549.137 −75542.148 Z = 54 Coulomb† −195348.063 −195344.974 −195317.667 Magnetic† 171.169 171.142 170.943 Retardation (order ω2)† −17.418 −17.418 −17.417 Retardation (> ω2) −1.711 −1.713 −1.731 Self-energy (SE) 127.610 127.615 127.669 Self-energy screening −9.166 −9.170 −9.205 VP [ α(Zα)]correction to e-el interaction 0.124 0.124 0.124 Vacuum Polarization α(Zα)3 + α2(Zα) 0.242 0.242 0.242 2nd order (SE-SE + SE-VP + S-VP-E)‡ −0.253 −0.253 −0.253 Recoil −0.012 −0.012 −0.012 Relativistic Recoil♯ 0.042 0.042 0.042 Total energy −195077.437 −195074.376 −195047.265 Z = 82 Coulomb† −513070.812 −513061.827 −512876.005 Magnetic† 710.586 710.511 708.897 Retardation (order ω2)† −72.655 −72.654 −72.655 Retardation (> ω2) −14.666 −14.668 −14.965 Self-energy (SE) 600.177 600.173 600.249 Self-energy screening −38.198 −38.203 −38.200 VP [ α(Zα)]correction to e-e interaction 0.572 0.572 0.570 Vacuum Polarization α(Zα)3 + α2(Zα) 4.192 4.192 4.189 2nd order (SE-SE + SE-VP + S-VP-E)‡ −2.975 −2.975 −2.974 Recoil −0.051 −0.051 −0.051 Relativistic Recoil♯ 0.191 0.191 0.191 Total energy −511883.638 −511874.738 −511690.755 † Contains the Uheling potential contribution to all order and all order Breit interaction. ‡ Calculated using the results of Ref. [33,34,35]. ♯ The formulas and definitions used to evaluate this term are Appendix A of Ref. [36] 6 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions -0.005 0.005 0.015 0.025 25 35 45 55 65 75 85 95 I=1/2 I=3/2 I=5/2 I=7/2 I=9/2 Fig. 2. Influence of the hyperfine interaction on the 4s4p 3P1 − 3P0 energy separation, as a function of the nuclear spin I and the atomic number Z. The quantity E = ∆Ehf −∆E0 is the contribution of the hyperfine interaction to the fine structure splitting ∆E0. The symbols represent values for the differente nuclear spins; some elements have several isotopes with identical spins but different µI values. 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 25 35 45 55 65 75 85 95 I=1/2 I=3/2 I=5/2 I=7/2 I=9/2 Fig. 3. Influence of the hyperfine interaction on the lifetime of the 4s4p 3P0 level, as a function of the nuclear spin I and the atomic number Z. The unperturbed lifetime is, to a very good approximation, infinite. J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 7 Table 2. Hyperfine Matrix elements Wi,j in eV. The indexes 0, 1, and 2 stand for 3P1, and 1P1 respectively. I is the nuclear spin and µ the nuclear magnetic moment in nuclear magneton units. I Z A W1,1 W2,2 W0,1 W0,2 W1,2 µI 1/2 34 77 −3.312×10−05 −3.245×10−06 −2.382×10−05 1.708×10−05 3.268×10−05 0.535042 39 89 2.697×10−05 4.139×10−07 1.670×10−05 −1.055×10−05 −2.199×10−05 −0.137415 45 103 4.648×10−05 −4.491×10−06 2.564×10−05 −1.321×10−05 −2.923×10−05 −0.0884 47 107 7.878×10−05 −1.013×10−05 4.218×10−05 −2.011×10−05 −4.507×10−05 −0.11368 47 109 9.057×10−05 −1.164×10−05 4.849×10−05 −2.311×10−05 −5.182×10−05 −0.130691 48 111 4.696×10−04 −6.718×10−05 2.480×10−04 −1.136×10−04 −2.560×10−04 −0.594886 48 113 4.912×10−04 −7.027×10−05 2.594×10−04 −1.188×10−04 −2.678×10−04 −0.622301 50 115 9.289×10−04 −1.573×10−04 4.788×10−04 −2.018×10−04 −4.591×10−04 −0.91883 50 117 1.012×10−03 −1.714×10−04 5.216×10−04 −2.199×10−04 −5.002×10−04 −1.00104 50 119 1.059×10−03 −1.793×10−04 5.457×10−04 −2.301×10−04 −5.233×10−04 −1.04728 52 123 9.385×10−04 −1.801×10−04 4.734×10−04 −1.834×10−04 −4.199×10−04 −0.736948 52 125 1.132×10−03 −2.172×10−04 5.708×10−04 −2.212×10−04 −5.063×10−04 −0.888505 54 129 1.232×10−03 −2.602×10−04 6.102×10−04 −2.170×10−04 −4.984×10−04 −0.777976 69 169 1.391×10−03 −3.983×10−04 6.364×10−04 −1.217×10−04 −2.710×10−04 −0.231 70 171 −3.210×10−03 9.272×10−04 −1.463×10−03 2.694×10−04 5.974×10−04 0.49367 74 183 −1.035×10−03 3.081×10−04 −4.659×10−04 7.391×10−05 1.610×10−04 0.117785 76 187 −6.581×10−04 1.982×10−04 −2.942×10−04 4.344×10−05 9.378×10−05 0.064652 78 195 −7.176×10−03 2.186×10−03 −3.189×10−03 4.387×10−04 9.383×10−04 0.60952 80 199 −6.867×10−03 2.112×10−03 −3.030×10−03 3.892×10−04 8.250×10−04 0.505885 81 203 −2.364×10−02 7.305×10−03 −1.040×10−02 1.290×10−03 2.724×10−03 1.622258 81 205 −2.387×10−02 7.377×10−03 −1.050×10−02 1.303×10−03 2.751×10−03 1.638215 82 207 −9.267×10−03 2.876×10−03 −4.060×10−03 4.873×10−04 1.024×10−03 0.592583 3/2 31 69 −1.363×10−05 −1.123×10−06 −2.652×10−05 2.024×10−05 1.535×10−05 2.01659 31 71 −1.732×10−05 −1.427×10−06 −3.370×10−05 2.572×10−05 1.950×10−05 2.56227 33 75 −2.179×10−05 −2.286×10−06 −3.670×10−05 2.687×10−05 2.235×10−05 1.43948 35 79 −5.715×10−05 −4.904×10−06 −8.849×10−05 6.206×10−05 5.431×10−05 2.1064 35 81 −6.161×10−05 −5.286×10−06 −9.538×10−05 6.690×10−05 5.855×10−05 2.270562 37 87 −1.198×10−04 −6.370×10−06 −1.745×10−04 1.165×10−04 1.056×10−04 2.75131 54 131 −3.649×10−04 7.708×10−05 −4.042×10−04 1.438×10−04 1.477×10−04 0.6915 56 135 −5.429×10−04 1.235×10−04 −5.920×10−04 1.932×10−04 1.987×10−04 0.837953 56 137 −6.073×10−04 1.382×10−04 −6.622×10−04 2.161×10−04 2.222×10−04 0.937365 64 155 3.449×10−04 −9.328×10−05 3.595×10−04 −8.381×10−05 −8.504×10−05 −0.2572 64 157 4.556×10−04 −1.232×10−04 4.750×10−04 −1.107×10−04 −1.123×10−04 −0.3398 65 159 −2.936×10−03 8.052×10−04 −3.049×10−03 6.825×10−04 6.901×10−04 2.014 76 189 −2.239×10−03 6.745×10−04 −2.238×10−03 3.305×10−04 3.191×10−04 0.659933 77 191 −5.500×10−04 1.666×10−04 −5.481×10−04 7.811×10−05 7.506×10−05 0.1507 77 193 −5.974×10−04 1.810×10−04 −5.954×10−04 8.484×10−05 8.154×10−05 0.1637 79 197 −6.243×10−04 1.911×10−04 −6.182×10−04 8.215×10−05 7.824×10−05 0.148158 80 201 2.535×10−03 −7.798×10−04 2.501×10−03 −3.212×10−04 −3.045×10−04 −0.560226 91 231 −1.960×10−02 6.265×10−03 −1.845×10−02 1.656×10−03 1.506×10−03 2.01 5/2 30 67 −1.975×10−06 −3.436×10−08 −6.734×10−06 5.115×10−06 2.303×10−06 0.875479 37 85 −3.537×10−05 −1.880×10−06 −7.865×10−05 5.252×10−05 3.118×10−05 1.353352 40 91 6.158×10−05 −2.554×10−07 1.273×10−04 −7.804×10−05 −4.821×10−05 −1.30362 42 95 6.081×10−05 −2.597×10−06 1.207×10−04 −6.931×10−05 −4.375×10−05 −0.9142 42 97 6.209×10−05 −2.652×10−06 1.233×10−04 −7.078×10−05 −4.467×10−05 −0.9335 44 99 5.826×10−05 −4.623×10−06 1.116×10−04 −5.968×10−05 −3.835×10−05 −0.6413 44 101 6.530×10−05 −5.182×10−06 1.251×10−04 −6.689×10−05 −4.298×10−05 −0.7188 46 105 7.773×10−05 −8.788×10−06 1.442×10−04 −7.151×10−05 −4.664×10−05 −0.642 51 121 −7.648×10−04 1.385×10−04 −1.332×10−03 5.382×10−04 3.597×10−04 3.3634 53 127 −8.004×10−04 1.617×10−04 −1.366×10−03 5.072×10−04 3.406×10−04 2.813273 59 141 −2.218×10−03 5.482×10−04 −3.622×10−03 1.040×10−03 6.985×10−04 4.2754 63 151 −2.567×10−03 6.842×10−04 −4.106×10−03 9.970×10−04 6.642×10−04 3.4717 63 153 −1.133×10−03 3.020×10−04 −1.812×10−03 4.401×10−04 2.932×10−04 1.5324 66 161 4.556×10−04 −1.264×10−04 7.196×10−04 −1.548×10−04 −1.021×10−04 −0.4803 66 163 −6.384×10−04 1.771×10−04 −1.008×10−03 2.169×10−04 1.431×10−04 0.673 70 173 8.428×10−04 −2.434×10−04 1.312×10−03 −2.416×10−04 −1.568×10−04 −0.648 75 185 −6.034×10−03 1.807×10−03 −9.246×10−03 1.415×10−03 8.981×10−04 3.1871 75 187 −6.096×10−03 1.826×10−03 −9.341×10−03 1.429×10−03 9.073×10−04 3.2197 82 205 −2.226×10−03 6.908×10−04 −3.331×10−03 3.998×10−04 2.460×10−04 0.7117 92 233 1.702×10−03 −5.451×10−04 3.263×10−03 −2.839×10−04 −1.256×10−04 0.59 8 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions Table 2. Continued I Z A W1,1 W2,2 W0,1 W0,2 W1,2 µI 7/2 34 79 9.003×10−06 8.821×10−07 2.967×10−05 −2.128×10−05 −8.883×10−06 −1.018 51 123 −4.141×10−04 7.500×10−05 −9.673×10−04 3.910×10−04 1.948×10−04 2.5498 53 129 −5.326×10−04 1.076×10−04 −1.220×10−03 4.528×10−04 2.267×10−04 2.621 55 133 −6.480×10−04 1.424×10−04 −1.459×10−03 4.970×10−04 2.494×10−04 2.582025 55 135 −6.857×10−04 1.507×10−04 −1.544×10−03 5.260×10−04 2.639×10−04 2.7324 57 139 −8.527×10−04 2.001×10−04 −1.892×10−03 5.915×10−04 2.968×10−04 2.783046 60 143 4.324×10−04 −1.093×10−04 9.424×10−04 −2.593×10−04 −1.296×10−04 −1.065 60 145 2.664×10−04 −6.731×10−05 5.805×10−04 −1.597×10−04 −7.984×10−05 −0.656 62 147 3.935×10−04 −1.032×10−04 8.483×10−04 −2.147×10−04 −1.069×10−04 −0.812 62 149 3.235×10−04 −8.486×10−05 6.975×10−04 −1.765×10−04 −8.790×10−05 −0.6677 67 163 −3.107×10−03 8.721×10−04 −6.561×10−03 1.356×10−03 6.644×10−04 4.23 67 165 −3.063×10−03 8.598×10−04 −6.467×10−03 1.337×10−03 6.550×10−04 4.17 68 167 4.484×10−04 −1.271×10−04 9.431×10−04 −1.875×10−04 −9.150×10−05 −0.56385 71 175 −2.238×10−03 6.514×10−04 −4.657×10−03 8.257×10−04 3.979×10−04 2.2323 72 177 −8.583×10−04 2.519×10−04 −1.781×10−03 3.041×10−04 1.459×10−04 0.7935 73 181 −2.765×10−03 8.176×10−04 −5.722×10−03 9.414×10−04 4.495×10−04 2.3705 92 235 −3.699×10−03 1.185×10−03 −5.287×10−03 4.599×10−04 2.731×10−04 −0.38 9/2 32 73 3.089×10−06 3.176×10−07 1.419×10−05 −1.060×10−05 −3.306×10−06 −0.879468 36 83 1.124×10−05 7.920×10−07 4.326×10−05 −2.963×10−05 −1.029×10−05 −0.970669 38 87 1.959×10−05 6.773×10−07 7.138×10−05 −4.640×10−05 −1.662×10−05 −1.093603 41 93 −1.930×10−04 4.552×10−06 −6.573×10−04 3.901×10−04 1.449×10−04 6.1705 43 99 −2.464×10−04 1.512×10−05 −8.076×10−04 4.478×10−04 1.696×10−04 5.6847 49 113 −5.500×10−04 8.617×10−05 −1.648×10−03 7.241×10−04 2.856×10−04 5.5289 49 115 −5.512×10−04 8.636×10−05 −1.651×10−03 7.257×10−04 2.862×10−04 5.5408 72 179 5.392×10−04 −1.582×10−04 1.403×10−03 −2.395×10−04 −9.166×10−05 −0.6409 83 209 −7.663×10−03 2.388×10−03 −1.922×10−02 2.230×10−03 8.126×10−04 4.1103 5 57 138 −7.965×10−04 1.869×10−04 −2.439×10−03 7.625×10−04 2.772×10−04 3.713646 83 208 −7.773×10−03 2.422×10−03 −2.146×10−02 2.491×10−03 8.243×10−04 4.633 7 67 166 −1.322×10−03 3.711×10−04 −5.264×10−03 1.088×10−03 2.827×10−04 3.6 71 176 −1.588×10−03 4.624×10−04 −6.233×10−03 1.105×10−03 2.824×10−04 3.169 9 73 180 −2.189×10−03 6.472×10−04 −1.083×10−02 1.781×10−03 3.558×10−04 4.825 83 210 −2.545×10−03 7.929×10−04 −1.217×10−02 1.412×10−03 2.699×10−04 2.73 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions 9 Table 3. Influence of the hyperfine interaction on the 4s4p 3P1 − 4s4p 3P0 energy separation and on the lifetime of the level, as a function of the nuclear spin I and the atomic number Z. ∆E0 is the unperturbed energy separation (in eV), and ∆Ehf is the perturbed energy (in eV) when the hyperfine interaction is taken into account (the 5 digits do not necessarily represent the accuracy of the calculation - they are intended to show the effect at low Z). τ0, τ1 and τ2 represent the perturbed lifetimes (in s) of 4s4p 3P0, P1 and P1 levels respectively. I Z A ∆E0 ∆Ehf τ1 τ2 τ0 1/2 34 77 0.20461 0.20458 2.139×10−07 1.340×10−10 7.847×10+00 39 89 0.61806 0.61809 1.393×10−08 5.025×10−11 1.317×10+01 45 103 1.41748 1.41753 1.976×10−09 2.375×10−11 5.111×10+00 47 107 1.74706 1.74714 1.221×10−09 1.923×10−11 1.847×10+00 47 109 1.74706 1.74715 1.221×10−09 1.923×10−11 1.397×10+00 48 111 1.92132 1.92179 9.804×10−10 1.739×10−11 5.286×10−02 48 113 1.92132 1.92181 9.804×10−10 1.739×10−11 4.831×10−02 50 115 2.28581 2.28674 6.623×10−10 1.433×10−11 1.391×10−02 50 117 2.28581 2.28682 6.623×10−10 1.433×10−11 1.172×10−02 50 119 2.28581 2.28687 6.623×10−10 1.433×10−11 1.071×10−02 52 123 2.66778 2.66872 4.673×10−10 1.185×10−11 1.397×10−02 52 125 2.66778 2.66891 4.673×10−10 1.185×10−11 9.609×10−03 54 129 3.06295 3.06418 3.425×10−10 9.804×10−12 8.274×10−03 69 169 6.17717 6.17856 7.752×10−11 2.398×10−12 7.234×10−03 70 171 6.38647 6.38326 7.246×10−11 2.174×10−12 1.367×10−03 74 183 7.22323 7.22219 5.650×10−11 1.464×10−12 1.353×10−02 76 187 7.64172 7.64106 5.051×10−11 1.196×10−12 3.400×10−02 78 195 8.06074 8.05357 4.545×10−11 9.804×10−13 2.895×10−04 80 199 8.48051 8.47364 4.115×10−11 7.937×10−13 3.215×10−04 81 203 8.69084 8.66722 3.922×10−11 7.194×10−13 2.725×10−05 81 205 8.69084 8.66699 3.922×10−11 7.194×10−13 2.672×10−05 82 207 8.90152 8.89226 3.745×10−11 6.494×10−13 1.794×10−04 3/2 31 69 0.06865 0.06864 4.446×10−06 4.528×10−10 8.711×10+00 31 71 0.06865 0.06863 4.446×10−06 4.528×10−10 5.395×10+00 33 75 0.14992 0.14990 4.831×10−07 1.808×10−10 3.524×10+00 35 79 0.26849 0.26843 1.071×10−07 1.044×10−10 5.357×10−01 35 81 0.26849 0.26843 1.071×10−07 1.044×10−10 4.610×10−01 37 87 0.42429 0.42417 3.448×10−08 6.944×10−11 1.270×10−01 54 131 3.06295 3.06259 3.425×10−10 9.804×10−12 1.883×10−02 56 135 3.46760 3.46706 2.611×10−10 8.197×10−12 8.660×10−03 56 137 3.46760 3.46699 2.611×10−10 8.197×10−12 6.920×10−03 64 155 5.12988 5.13022 1.130×10−10 3.876×10−12 2.277×10−02 64 157 5.12988 5.13034 1.130×10−10 3.876×10−12 1.304×10−02 65 159 5.33938 5.33645 1.041×10−10 3.534×10−12 3.155×10−04 76 189 7.64172 7.63948 5.051×10−11 1.196×10−12 5.872×10−04 77 191 7.85115 7.85060 4.785×10−11 1.081×10−12 9.806×10−03 77 193 7.85115 7.85055 4.785×10−11 1.081×10−12 8.310×10−03 79 197 8.27049 8.26987 4.329×10−11 8.850×10−13 7.729×10−03 80 201 8.48051 8.48305 4.115×10−11 7.937×10−13 4.730×10−04 91 231 10.82070 10.80116 2.591×10−11 2.538×10−13 8.872×10−06 5/2 30 67 0.04546 0.04546 1.834×10−05 1.371×10−09 1.771×10+02 37 85 0.42429 0.42425 3.448×10−08 6.944×10−11 6.251×10−01 40 91 0.72914 0.72920 9.434×10−09 4.348×10−11 2.221×10−01 42 95 0.97889 0.97895 4.695×10−09 3.344×10−11 2.395×10−01 42 97 0.97889 0.97895 4.695×10−09 3.344×10−11 2.297×10−01 44 99 1.26339 1.26345 2.584×10−09 2.646×10−11 2.732×10−01 44 101 1.26339 1.26346 2.584×10−09 2.646×10−11 2.175×10−01 46 105 1.57892 1.57900 1.541×10−09 2.132×10−11 1.597×10−01 51 121 2.47489 2.47413 5.525×10−10 1.302×10−11 1.779×10−03 53 127 2.86396 2.86316 3.984×10−10 1.079×10−11 1.661×10−03 59 141 4.08585 4.08364 1.828×10−10 6.211×10−12 2.274×10−04 63 151 4.92049 4.91793 1.233×10−10 4.274×10−12 1.747×10−04 63 153 4.92049 4.91936 1.233×10−10 4.274×10−12 8.977×10−04 66 161 5.54887 5.54933 9.615×10−11 3.205×10−12 5.667×10−03 66 163 5.54887 5.54823 9.615×10−11 3.205×10−12 2.885×10−03 70 173 6.38647 6.38731 7.246×10−11 2.174×10−12 1.702×10−03 10 J. P. Marques et al.: Hyperfine Quenching of the 4s4p 3P0 Level in Zn-like Ions Table 3. Continued I Z A ∆E0 ∆Ehf τ1 τ2 τ0 5/2 75 185 7.43246 7.42645 5.348×10−11 1.325×10−12 3.433×10−05 75 187 7.43246 7.42639 5.348×10−11 1.325×10−12 3.364×10−05 82 205 8.90152 8.89930 3.745×10−11 6.494×10−13 2.670×10−04 92 233 11.03725 11.03356 2.506×10−11 2.288×10−13 1.090×10−04 7/2 34 79 0.20461 0.20462 2.139×10−07 1.340×10−10 5.060×10+00 51 123 2.47489 2.47448 5.525×10−10 1.302×10−11 3.371×10−03 53 129 2.86396 2.86343 3.984×10−10 1.079×10−11 2.084×10−03 55 133 3.26430 3.26365 2.976×10−10 8.929×10−12 1.435×10−03 55 135 3.26430 3.26362 2.976×10−10 8.929×10−12 1.281×10−03 57 139 3.67249 3.67164 2.304×10−10 7.463×10−12 8.424×10−04 60 143 4.29381 4.29424 1.645×10−10 5.650×10−12 3.350×10−03 60 145 4.29381 4.29408 1.645×10−10 5.650×10−12 8.830×10−03 62 147 4.71127 4.71166 1.351×10−10 4.695×10−12 4.108×10−03 62 149 4.71127 4.71159 1.351×10−10 4.695×10−12 6.075×10−03 67 163 5.75836 5.75527 8.929×10−11 2.915×10−12 6.802×10−05 67 165 5.75836 5.75531 8.929×10−11 2.915×10−12 6.999×10−05 68 167 5.96779 5.96824 8.264×10−11 2.646×10−12 3.294×10−03 71 175 6.59569 6.59346 6.757×10−11 1.972×10−12 1.350×10−04 72 177 6.80490 6.80404 6.369×10−11 1.786×10−12 9.241×10−04 73 181 7.01409 7.01133 5.988×10−11 1.618×10−12 8.953×10−05 92 235 11.03725 11.03895 2.506×10−11 2.288×10−13 2.864×10−04 9/2 32 73 0.10442 0.10442 1.298×10−06 2.644×10−10 2.655×10+01 36 83 0.34167 0.34168 5.882×10−08 8.403×10−11 2.142×10+00 38 87 0.51641 0.51643 2.146×10−08 5.882×10−11 7.376×10−01 41 93 0.84950 0.84931 6.536×10−09 3.802×10−11 8.194×10−03 43 99 1.11697 1.11672 3.448×10−09 2.967×10−11 5.277×10−03 49 113 2.10110 2.10055 8.000×10−10 1.577×10−11 1.184×10−03 49 115 2.10110 2.10055 8.000×10−10 1.577×10−11 1.179×10−03 72 179 6.80490 6.80544 6.369×10−11 1.786×10−12 1.491×10−03 83 209 9.11260 9.10502 3.584×10−11 5.848×10−13 8.025×10−06 5 57 138 3.67249 3.67170 2.304×10−10 7.463×10−12 5.069×10−04 83 208 9.11260 9.10493 3.584×10−11 5.848×10−13 6.433×10−06 7 67 166 5.75836 5.75705 8.929×10−11 2.915×10−12 1.057×10−04 71 176 6.59569 6.59411 6.757×10−11 1.972×10−12 7.540×10−05 9 73 180 7.01409 7.01193 5.988×10−11 1.618×10−12 2.501×10−05 83 210 9.11260 9.11009 3.584×10−11 5.848×10−13 2.003×10−05 Introduction Relativistic calculation of hyperfine contribution to fine structure splitting and to transition probabilities Results and discussion

0704.1947

R-matrices in Rime

CPT-P49-2006 R-MATRICES IN RIME Oleg Ogievetsky∗ Centre de Physique Théorique†, Luminy, 13288 Marseille, France Todor Popov Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, BG-1784, Bulgaria Abstract We replace the ice Ansatz on matrix solutions of the Yang–Baxter equation by a weaker condition which we call rime. Rime solutions include the standard Drinfeld–Jimbo R-matrix. Solutions of the Yang–Baxter equation within the rime Ansatz which are maximally different from the standard one we call strict rime. A strict rime non-unitary solution is parameterized by a projective vector ~φ. We show that in the finite dimension this solution transforms to the Cremmer–Gervais R-matrix by a change of basis with a matrix containing symmetric functions in the components of ~φ. A strict unitary solution (the rime Ansatz is well adapted for taking a unitary limit) in the finite dimension is shown to be equivalent to a quantization of a classical ”boundary” r-matrix of Gerstenhaber and Giaquinto. We analyze the structure of the elementary rime blocks and find, as a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly described in a rime form. We discuss then connections of the classical rime solutions with the Bézout operators. The Bézout operators satisfy the (non-)homogeneous associative classical Yang–Baxter equation which is related to the Rota–Baxter operators. We calculate the Rota–Baxter operators corresponding to the Bézout operators. We classify the rime Poisson brackets: they form a 3-dimensional pencil. A normal form of each individual member of the pencil depends on the discriminant of a certain quadratic polynomial. We also classify orderable quadratic rime associative algebras For the standard Drinfeld–Jimbo solution, there is a choice of the multiparameters, for which it can be non-trivially rimed. However, not every Belavin–Drinfeld triple admits a choice of the multiparameters for which it can be rimed. We give a minimal example. ∗On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, Leninsky prospekt 53, 119991 Moscow, Russia †Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud Toulon – Var; laboratoire affilié à la FRUMAM (FR 2291) http://arxiv.org/abs/0704.1947v3 Contents 1 From ice to rime 3 2 Rime Yang–Baxter solutions 6 2.1 Non-unitary rime R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Unitary rime R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Rime and Cremmer–Gervais R-matrices 9 4 Classical rime r-matrices 12 4.1 Non-skew-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 BD triples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Skew-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 Bézout operators 20 5.1 Non-homogeneous associative classical Yang–Baxter equation . . . . . . . . . . . . . . 21 5.2 Linear quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Algebraic meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.4 Rota–Baxter operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.5 ∗-multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Rime Poisson brackets 30 6.1 Rime pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7 Orderable quadratic rime associative algebras 38 Appendix A. Equations 40 Appendix B. Blocks 41 B.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 B.2 GL(2) and GL(1|1) R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 B.3 Riming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Appendix C. Rimeless triple 47 References 48 1 From ice to rime A well known class of solutions R̂ ∈End (V ⊗ V ), V is a vector space, of the Yang–Baxter equation YB(R̂) = 0, where YB(R̂) := (R̂⊗ 11)(11 ⊗ R̂)(R̂ ⊗ 11)− (11⊗ R̂)(R̂ ⊗ 11)(11 ⊗ R̂) , (1) is characterized by the so called ice condition (see lectures [21] for details) which says that R̂ can be different from zero only if the set of the upper and the set of the lower indices coincide, kl 6= 0 ⇒ {i, j} ≡ {k, l}. (2) We introduce the ”rime” Ansatz, relaxing the ice condition: the entry R̂ kl can be different from zero if the set of the lower indices is a subset of the set of the upper indices, 6= 0 ⇒ {k, l} ⊂ {i, j} . (3) Matrices for which it holds will be referred to as “rime” matrices. Figuratively, in the rime, in contrast to the ice, situation, putting an apple and a banana in a fridge, there is a non-zero amplitude to find next morning two apples instead (but never an apple and an orange). The Yang–Baxter equation for a matrix R̂ is equivalent to the equality of two different reorderings of xiyjzk, using xiyj = R̂ kxl, xizj = R̂ kxl and yizj = R̂ kyl, to the form z•y•x•. One of advantages of the rime Ansatz is that only indices i, j and k appear in the latter expression. Another advantage is that for fixed values of i and j, the elements x• and y• with these values of indices form a subsystem. A rime R-matrix has the following structure = αijδ + βijδ + γijδ l + γ (no summation) . (4) To avoid redundancy, fix βii = 0, γii = 0 = γ ii. We denote by αi the diagonal elements R̂ ii, αi = αii. Throughout the text we shall assume that the matrix R̂ is invertible which, in particular, implies that αi 6= 0 for all i. The order of growth of the number of unknowns in the Yang–Baxter system for a rime matrix is n2, where n =dimV . Arbitrary permutations and rescalings of coordinates preserve the rime condition. The ice and rime matrices are made of 4× 4 elementary building blocks, respectively, R̂ice = α1 0 0 0 0 β12 α12 0 0 α21 β21 0 0 0 0 α2 and R̂rime = α1 0 0 0 γ12 β12 α12 γ γ′21 α21 β21 γ21 0 0 0 α2 . (5) In appendix B we analyze the structure of the 4× 4 rime blocks. We call a rime matrix strict if αijγij 6= 0 ∀ i and j, i 6= j. Note that strict rime matrices are necessarily not ice. Proposition 1. Let R̂ be a rime matrix (4). Then R̂ is a solution of the Yang–Baxter equation if it is of the form = (1− βji)δilδ + βijδ + γijδ l − γjiδ , (6) where βij and γij satisfy the system βijβji = γjiγij , (7) βij + βji = βjk + βkj =: β , (8) βijβjk = (βjk − βji)βik = (βij − βkj)βik , (9) γijγjk = (βji − βjk)γik = (βkj − βij)γik . (10) Proof. The Yang–Baxter system of equations YB(R̂) abc = 0 for a rime matrix is given in the appendix A. The subset (257) - (259) together with its image under the involution (256) reads ij(γij + γ ji) = 0 = αijγij(γji + γ ij) , (11) αij(βijβji + γijγ ij) = 0 = αij(βijβji − γijγji) , (12) ij(αij + βji − αi) = 0 = αijγij(αij + βji − αj) , (13) ij(αji + βij − αi) = 0 = αijγij(αji + βij − αj) . (14) These equations are implied by (and, in the strict rime situation, are equivalent to) the following system γ′ij = −γji , αij + βji = αi , αji + βij = αi , (15) βijβji = γjiγij . (16) One checks that other equations YB(R̂) abc = 0, for which two indices among {i, j, k} are different, follow from (15) and (16). The last two equations from (15) imply αi = αj for all i and j. As an overall rescaling of a solution of the Yang–Baxter equation by a constant is again a solution of the Yang–Baxter equation, we can, without loss of generality, set it to one, αi = 1 . (17) Eqs. (15) and (17) yield the form (6) of the matrix R̂ and eq. (7). Using (15), we rewrite the subset (266) - (268) together with its image under the involution (256) in the form (βij + βji − βik − βki)γijγik = 0 , (18) αij(βijβjk + βikβji − βikβjk) = 0 = αji(βjiβkj + βkiβij − βkiβkj) , (19) αij(γijγjk + γik(βjk − βji)) = 0 = αji(γjiγkj + γki(βkj − βij)) . (20) These equations are implied by (and, in the strict rime situation, are equivalent to) eqs. (8), (9) and (10). One checks that other equations YB(R̂) abc = 0 with three different indices {i, j, k} follow from the system (7)–(10). The proof is finished. � Lemma 1. The rime Yang–Baxter solution R (6) is of Hecke type, R̂2 = βR̂+ (1− β)11⊗ 11 . (21) Moreover, when β 6= 2, R is of GL-type: it has two eigenvalues 1 and β − 1 with multiplicities n(n+1) n(n−1) , respectively. When β = 2 the matrix R̂ has a nontrivial Jordanian structure. Proof. In view of the block structure of rime matrices it is enough to check the Hecke relation (21) for one elementary (4 × 4) block which follows from (7) and (8). When β 6= 2 the multiplicities m1 and mβ−1 are solutions of the system m1 +mβ−1 = n 2 , m1 + (β − 1)mβ−1 = n+ n(n− 1) β (≡ TrR̂) . (22) When β = 2 the matrix R̂ has only one eigenvalue 1 but R̂ 6= 11⊗ 11 due to (7) and (8). � Unitary solutions, R̂2 = 11⊗ 11, are singled out by the value of the parameter β = 0. Lemma 2. A strict rime Yang–Baxter solution R (6) can be brought to a rime matrix = (1 − βji)δilδ + βijδ − βijδikδil + βjiδ , (23) that is, to a solution (6) with γij = −βij , by a change of basis. Proof. The strict rime condition αijγij 6= 0 implies βijβji 6= 0 in view of (7). Thus for a strict rime R-matrix all βij and γij are nonvanishing. The ratio of eqs. (9) and (10) is well-defined and it follows from eqs. (7) and (8) that γijγjk βijβjk (βji − βjk)γik (βji − βjk)βik = −γik γijγji βijβji = 1 , (24) ξijξjk = ξik ξijξji = 1 , (25) where ξij = − . Eq. (25) is solved by ξij = with di 6= 0, i = 1, . . . , n, hence β’s and γ’s are related by γij = − βij . (26) A change of basis with a matrix D, R̂ 7−→ (D ⊗D) R̂ (D−1 ⊗D−1) , (27) where Dij = djδ j , transforms R to the form (23). � Under the strict rime condition, the Yang–Baxter system of equations (see appendix A) reduces to eqs. (8) and (9). However, the matrix (23), where the parameters βij are subject to eqs. (8) and (9), is a solution of the Yang–Baxter equation without a strict rime assumption. Remark. Right and left even quantum spaces are defined by, respectively, kxl = xixj , xjxiR kl = xlxk ; (28) right and left odd quantum spaces are defined by, respectively, kξl = (β − 1)ξiξj , ξjξiRijkl = (β − 1)ξlξk . (29) Assume that β 6= 2. The left even space is classical1 as well as the right odd space [xi, xj ] = 0 , [ξ i, ξj ] = 0 , (30) where [ , ] and [ , ]+ stand for the commutator and the anti-commutator. The relations for the right even space are [xi, xj ] + (βijx i + βjix j)(xi − xj) = 0 ; (31) the relations for the left odd space read (2− β)ξ2i + ξiρ+ (1− β)ρξi = 0 , (32) [ξi, ξj ]+ − βijξiξj − βjiξjξi = 0 , i 6= j , (33) where ρ = j ξj. 2 Rime Yang–Baxter solutions In this section we solve eqs. (8) and (9) thus obtaining explicitly rime Yang–Baxter solutions. 2.1 Non-unitary rime R-matrices Proposition 2. The non-unitary strict rime Yang–Baxter solutions (23) with a parameter β = βji+ βij 6= 0 are parameterized by a point φ ∈ PCn in a projective space, φ = (φ1 : φ2 : . . . : φn), such that φi 6= 0 for all i and φi 6= φj for all i and j, i 6= j. These solutions are given by βij = φi − φj . (34) Proof. Taking the ratio of the following pairs of equations from (9) βijβjk = (βjk − βji)βik , βkjβji = (βji − βjk)βki (35) we find that quantities ηij = −βij/βji verify equations ηijηjk = ηik , ηijηji = 1 , (36) 1Let R̂ be a rime R-matrix (not necessarily strict). When β 6= 2, the following statement holds. If (i) the left even space is classical (which implies that γ′ij = −γji, αij + βji = 1 and αi = 1 in our normalization) and (ii) the R-matrix is Hecke (which implies that βij + βji = β) then the system of equations from the appendix A again reduces to (7), (9) and (10) as in the strict rime situation. whose solutions are ηij = φi/φj for some constants φi 6= 0, i = 1, . . . , n. Substituting the relation βji = − βij into β = βij + βji, we obtain βij − βij = β which establishes (34). � Remark. There is a different parameterization, βij = − φi − φj , of strict rime solutions; it is related to the parameterization (34) by φi 7−→ (φi)−1. A direct check shows that the condition φi 6= 0 is not necessary: the formula (34) with φi 6= φj for all i and j, i 6= j, gives a rime solution of the Yang–Baxter equation. However when one of φi is 0, the matrix (23) is no more strict. 2.2 Unitary rime R-matrices For a unitary strict rime Yang–Baxter solution (23), R̂2 = 11, we have β = 0, so βij = −βji. Proposition 3. The unitary strict rime Yang–Baxter solutions (23) are parameterized by a vector (µ1, . . . , µn) such that µi 6= µj, βij = µi − µj . (37) Proof. Since βij = −βji we can rewrite βijβjk = (βjk − βji)βik as βijβjk = (βij + βjk)βik or . (38) These equations are solved by = µi − µj , (39) which is equivalent to (37). � Remark. The unitary R-matrices of Proposition 3 can be obtained as a limit β → 0 of the non- unitary R-matrices of Proposition 2. Indeed, for the following expansion of the parameters φi in the “small” parameter β, φi = 1 + βµi + o(β) , (40) the expression (34) has a limit (37), βij = β(1 + βµi + o(β)) βµi − βµj + o(β) β→0−→ βij = µi − µj . (41) 2.3 Properties 1. Denote the R-matrix (23) with βij as in (34) by R̂(~φ ). Let R̂21 = PR̂12P , where P is the permutation operator. Then the following holds: R̂21(~φ ) = F −1 ⊗ F−1 R12(~φ−1) F ⊗ F , (42) where F = diag(φ1, φ2, . . . , φn) and ~φ −1 is a vector with components φ−1i . Denote the R-matrix (23) with βij as in (37) by R̂(~µ ). Then the following holds: R̂21(~µ) = R̂12(−~µ) . (43) 2. The R-matrix (23) is skew invertible in the sense that there exists an operator Ψ̂R, which satisfies (see, e.g. [21]) Tr2(R̂12(Ψ̂R)23) = P13 . (44) The matrices of the left and right quantum traces (that is, the left and right traces of the skew inverse Ψ̂R), (QR)1 = Tr2((Ψ̂R)12) and (Q̃R)2 = Tr1((Ψ̂R)12), are given by the formulas j = −βjk l: l 6=k (1− βjl) , k 6= j , and (QR)jj = (1− βjl) ; (45) (Q̃R) j = βjk l: l 6=k (1− βlj) , k 6= j , and (Q̃R)jj = (1− βlj) . (46) The matrices QR and Q̃R satisfy QRQ̃R = (1− β)n−111. For (34), one has Spec QR = Spec Q̃R = {(1 − β)a, a = 0, . . . , n − 1}. The eigenvector wa(~φ) of the matrix QR with the eigenvalue (1 − β)n−1−a coincides with the eigenvector of the matrix Q̃R with the eigenvalue (1−β)a. One has (wa(~φ))j = eĵa(~φ), where eĵi (~φ) is the i-th elementary symmetric function of (φ1, φ2, . . . , φn) with φj omitted. For (37), the Jordanian form of the matrix QR, as well as of Q̃R, is non trivial: it is a single block. In the basis {wi(~µ)}, i = 0, 1, 2, . . . , n− 1, where (wi(~µ))j = eĵi (~µ), one has QR wi(~µ) = n− 1− s ws(~µ) . (47) 3. For an R-matrix R̂, the group of invertible matrices Y satisfying R̂12Y1Y2 = Y1Y2R̂12 (48) form the invariance group GR of R̂. The matrices QR and Q̃R belong to the invariance group as well as the matrices proportional to the identity matrix. One can write down formulas for the group GR for a rime R-matrix (23) uniformly in terms of βij as in (45) and (46) but the properties are different in the non-unitary and unitary cases and we describe them separately. 3a. The invariance group G R(~φ ) for the R-matrix R̂(~φ ) is 2-parametric. It consists of matrices Y (u, v), u, v 6= 0, where Y (u, v) l:l 6=j uφj − vφl φj − φl and Y (u, v)ij = (u− v)φj φj − φi l:l 6=i,j uφj − vφl φj − φl , i 6= j . (49) One has R(~φ ) = Y (1− β, 1) , Q̃ R(~φ ) = Y (1, 1 − β) . (50) The composition law is the component-wise multiplication of the parameters {u, v}, Y (u1, v1)Y (u2, v2) = Y (u1u2, v1v2) . (51) The point u = v = 1 corresponds to the identity matrix, Y (1, 1) = 11; the determinant of Y (u, v) is (uv)n(n−1)/2; u = v corresponds to global rescalings; the connected component of unity of the subgroup SG R(~φ ) consisting of matrices with determinant 1 is uv = 1; the generator η of the connected component of unity of the subgroup SG R(~φ ) is traceless and reads ηij = φj − φi , i 6= j , and ηjj = − l:l 6=j φj − φl . (52) 3b. For the R-matrix R̂(~µ ), the group SGR(~µ ), consisting of matrices with determinant 1 is 1-parametric as well. It is formed by matrices Y (0)(a), where Y (0)(a) l:l 6=j µj − µl ) and Y (0)(a)ij = µj − µi l:l 6=i,j µj − µl ) , i 6= j . (53) The expression (53) can be obtained by taking a limit of (49), similarly to (41) and letting additionally u = 1 + aβ/2 + o(β) and v = 1− aβ/2 + o(β). One has QR(~µ ) = Y (0)(−1) , Q̃R(~µ ) = Y (0)(1) . (54) The composition law is Y (0)(a1)Y (0)(a2) = Y (0)(a1 + a2). The point a = 0 in (53) corresponds to the identity matrix, Y (0)(0) = 11; the generator η(0) of the invariance group SGR(~µ ) is (η(0))ij = µj − µi , i 6= j , and (η(0))jj = l:l 6=j µj − µl . (55) 3 Rime and Cremmer–Gervais R-matrices The Cremmer–Gervais R-matrix arises in the exchange relations of the chiral vertex operators in the non-linearly W -extended Virasoro algebra [6]. The Cremmer–Gervais solution [6] of the Yang– Baxter equation in its general two-parametric form reads (see, e.g. [17]; we use a rescaled matrix with eigenvalues 1 and −q−2) (R̂CG,p) kl = q −2θijpi−jδilδ k + (1− q s: i≤s<j pi−sδskδ i+j−s l − (1− q s: j<s<i pi−sδskδ i+j−s l , (56) where θij is the step function (θij = 1 when i > j and θij = 0 when i ≤ j). The parameter value p = q2/n specifies the SL(n) Cremmer–Gervais R-matrix (its diagonal twist being the GL(n) solution (56)). The Cremmer–Gervais solution is a non-diagonal twist of the standard Drinfeld–Jimbo solution [18, 9]. Let R̂CG := R̂CG,1, that is, the solution (56) with p = 1. The matrix D(p)ij = δ i−1 (57) with arbitrary p satisfies (R̂CG)12D(p)1D(p)2 = D(p)1D(p)2(R̂CG)12. It was observed in [10] that if R̂12D1D2 = D1D2R̂12 for some R-matrix R̂ and operator D then D1R̂12D 1 is again an R-matrix (this operation was also used in [15] to partially change the statistics of ghosts in the super-symmetric situation). The two-parametric matrix R̂CG,p (56) can be obtained from the Cremmer–Gervais matrix R̂CG by this operation as well, (R̂CG,p)12 = D(p)1(R̂CG)12D(p) 1 . (58) Let R̂ be the non-unitary rime matrix from Proposition 2 with φi 6= φj. Proposition 4. The matrix R̂ transforms into the Cremmer–Gervais solution R̂CG R̂ = (X ⊗X) R̂CG (X−1 ⊗X−1) (59) by a change of basis with the invertible matrix Xkj = ej−1 (φ1, . . . , φ̂k, . . . , φn) =: e j−1 (60) whose inverse is (X−1)ji = (−1)j−1φn−ji k:k 6=i (φi − φk) . (61) Here the hat over φj means that this entry is omitted in the expression and ei are the elementary symmetric polynomials ei(x1, . . . , xN ) = s1<...<si xs1xs2 . . . xsi . The projective parameters (φ1 : φ2 : . . . : φn) are removed by the transformation X and the only essential parameter β in R̂ is related to the parameter q in R̂CG by q−2 = 1− β . (62) Proof. Due to the Lagrange interpolation formula, the matrix, inverse to the Vandermonde matrix ||V jk || j,k=1 = φ k is (V −1)kj = (−1)j−1ek̂j−1 l:l 6=k (φk − φl) . (63) The matrix X (60) has the form X = DV −1 d, where Dmk = δ l:l 6=k(φk − φl) and dij = (−1)j−1δij are diagonal n× n matrices. Thus, its inverse is X−1 = d−1 V D−1, which establishes (61). We now prove the matrix identity (59) in the form R̂(X ⊗X) = (X ⊗X) R̂CG . (64) The substitution of the explicit form of the rime matrix R̂ (23) with βij = βφi/(φi − φj) and R̂CG (56) reduces (64) to a set of relations between the symmetrical polynomials eâk−1 l−1 = eîa−1e b−1(R̂CG) kl . (65) There are two subcases: i) i = j and ii) i 6= j. i) The left hand side of eq. (65) with i = j is just eîk−1e l−1 due to the rime condition. Eq. (65) is satisfied because of the symmetry relation (R̂CG) kl = δ l + δ k − (R̂CG)bakl . ii) For i 6= j eq. (65), where q−2 = 1− β, reduces, after some algebraic manipulations, to φi − φj k−1 − φje k−1)(e l−1 − e s: s≥max(1,k−l+2) (eîl+s−2e k−s − e l+s−2e k−s) , 1 ≤ i, j, k, l ≤ n . In fact, the sum in the right hand side goes till s = min(k, n + 1 − l) since eĵr = 0 when r ≥ n − 1; moreover we can start the summation from s = 1 because when 1 < k−l+2 the sum for 1 ≤ s ≤ k−l+1 is anti-symmetric under s←→ k − l + 2− s and thus vanishes. To prove (66) we write er = e r + φie r−1; therefore e r = e r + φje r−1 and e r = e r + φie r−1 and eq. (66) becomes − (φi − φj)eîĵk−1e l−2 = (φi − φj) l+s−2e k−s−1 − e l+s−3e k−s) . (67) The sum in the right hand side telescopes to the value of (−eîĵl+s−3e k−s) at s = 1, that is, to (−e l−2). The proof is complete. � It should be noted that the matrix X = X(~φ ) does not depend on q. The change of the basis with the matrix X(~φ′ )X(~φ )−1 transforms the R-matrix R̂(~φ ) to R̂(~φ′ ). We have (X(~φ′ )X(~φ )−1)ij = φj − φ′i (φj − φ′k) l:l 6=j (φj − φl) The structure of the matrices X andX−1 shows that when the dimension is infinite, the R-matrices R̂CG,1 and R̂(~φ ) (as well as the R-matrices R̂(~φ ) and R̂(~φ′ ) for different φ and φ ′) are in general not equivalent. The right even quantum plane for the Cremmer–Gervais matrix R̂CG,1 is defined by the following equations yiyj = q2yjyi + (q2 − 1)(yi+1yj−1 + . . .+ yj−1yi+1), i < j . (69) If i+ 1 < j − 1, one uses the formula (69) recursively to get the ordering relations. The change of basis with the matrix X, eîj−1y j , (70) transformes the quantum plane (69) into the rime quantum plane (31) exhibiting coordinate two- dimensional subplanes. The change of basis (70) can be written in terms of a ”generating function”: ej(φ1, . . . , φn) y j . (71) . (72) Remark. The standard Drinfeld–Jimbo R-matrix admits, for a certain choice of multi-parameters, a different rime form. The relations uivj = (R̂c) kul for this choice are uivi = viui , uivj = vjui + (1− q−2)viuj , i < j , uivj = q−2 vjui , i > j . The left even space for this R-matrix is classical. The change of variables with the matrix X̃ij = 1− θji, U i := u1 + u2 + · · ·+ ui , V i := v1 + v2 + · · ·+ vi , (74) transforms the relations (73) into U iV i = V iU i , U iV j = V jU i + (1− q−2)V iU j − (1− q−2)V iU i , i < j , U iV j = q−2 V jU i + (1− q−2)V jU j , i > j . The matrix X, defined by eq. (60), degenerates if φi = φj for some i and j. Interestingly, the R-matrix X ⊗XR̂c X−1 ⊗X−1 admits limits limφσ(2)→0 limφσ(3)→0 . . . limφσ(n)→0 for an arbitrary permutation σ ∈ Sn and the result is always rime. In particular, X̃ ⊗ X̃R̂c X̃−1 ⊗ X̃−1 = lim . . . lim X ⊗XR̂c X−1 ⊗X−1 . (76) 4 Classical rime r-matrices The classical limit of an R-matrix is a classical r-matrix, a solution of the classical Yang–Baxter (cYB) equation [r12, r13] + [r12, r23] + [r13, r23] = 0 . (77) We are going to show that the classical limits of the rime R-matrices from Section 2 are equivalent to the Cremmer-Gervais r-matrices in the non-skew-symmetric case and to the ”boundary” r-matrix of Gerstenhaber and Giaquinto [14] (see also [4]; this r-matrix is attributed to A. G. Elashvili there) in the skew-symmetric case. Similar equivalences appeared in the study of the gauge transformations of the dynamical r-matrices in the Calogero-Moser model [12, 13] 2. In the sequel we use the following conventions. An R-matrix acts in a space V ⊗ V . A basis of V is {ei} (labeled by a lower index); an operator A in V has matrix coefficients Aji , A(ei) = A i ej , so for a vector ~v = viei one has (A~v) i = Aij~v j ; the matrix units are eij , e j(ek) = δ kej , so the multiplication rule is eije l = δ j ; eαi are the sl(n) simple positive root elements, eαi = e i ; P is the permutation operator, P (ei ⊗ ej) = ej ⊗ ei, so P (eij ⊗ ekl ) = eil ⊗ ekj and (PB)klij = Blkij for an operator B in V ⊗ V having matrix coefficients Bklij , B(ei ⊗ ej) = Bklij ek ⊗ el. 4.1 Non-skew-symmetric case Proposition 5. The non-unitary rime R-matrix (Proposition 2) is a quantization of the non-skew- symmetric r-matrix i,j:i 6=j φi − φj (eij ⊗ e i − e i ⊗ e j + e i ∧ eij) , (78) where x ∧ y := x ⊗ y − y ⊗ x. The change of basis with the matrix Xjk = ek−1 (φ1, . . . , φ̂j , . . . , φn) transforms r into the parameter-free cYB solution rCG rCG = i,j:i<j (ei+s−1j ⊗ e j−s+1 i − e i+s−1 i ⊗ e j−s+1 j ) . (79) Proof. The coefficients βij (34) are linear in the deformation parameter β (β = 0 is the classical point). Hence R = 11⊗ 11 + βr , (80) where R = PR̂ and r is given by (78). The matrix RCG−11⊗11, where RCG = PR̂CG, is linear with respect to the parameter β = 1−q−2 as well, RCG = 11⊗ 11 + β rCG (81) thus the formula (59) implies r = (X ⊗X) rCG (X−1 ⊗X−1). � We mentioned two ways of obtaining the numerical two-parametric R-matrix (R̂CG,p) from the R- matrix (R̂CG,1): by a diagonal twist and by the operation (58). There is one more way which consists of changing the representation. We shall illustrate it on the example of the classical GL r-matrix (79). A change of representation of the Lie algebra GL, eij 7→ eij + c δij11 , (82) 2We thank László Fehér for drawing our attention to the references [12, 13]. where c is a constant, produces the following effect on the r-matrix (79): rCG 7→ rCG + c η ⊗ 11− 11⊗ η − (n− 1)11 ⊗ 11 , (83) where n =dimV and η = − n(n+ 1) j , tr η = 0 . (84) The classical version of the operation (58) is as follows. Let η be an arbitrary generator of the invariance group of an r-matrix r, [r, η1 + η2] = 0 . (85) Then the operator r(c) = r + c(η1 − η2) , (86) where c is a constant, is again a classical r-matrix (a solution of the cYBe). The operator η in (84) is, up to a scale, the unique traceless generator of the invariance group (see (57)) of the r-matrix (79). Thus, the representation change and the operation (86) give the same family of r-matrices (up to an addition of a multiple of the identity operator, which does not violate the cYBe). 4.2 BD triples. Each block in the strict rime classical r-matrix (78) looks even more ”rimed”, 0 0 0 0 β′12 −β′12 β′21 −β′21 −β′12 β′12 −β′21 β′21 0 0 0 0 , (87) where β′ij = βij/β = φi/(φi − φj). The multiplication from the left by P acts on each block as a permutation of the second and third lines, so the rime r-matrix (87) enjoys the symmetry Pr = −r. We shall now discuss this symmetry property in the context of Belavin–Drinfeld triples. In [3] Belavin and Drinfeld gave, for a simple Lie algebra g, a description of non-unitary (non- skew-symmetric) cYB solutions r ∈ g⊗ g, satisfying r12 + r21 = t, where t ∈ g ⊗ g is the g-invariant element. The non-unitary solutions are put into correspondence with combinatorial objects called Belavin–Drinfeld triples (BD-triples for short). The Belavin–Drinfeld triple (Π1,Π2, τ) for a simple Lie algebra g consists of the following data: Π1,Π2 are subsets of the set of simple positive roots Π of the algebra g and τ is an invertible mapping: Π1 → Π2 such that 〈τ(ρ), τ(ρ′)〉 = 〈ρ, ρ′〉 for any ρ, ρ′ ∈ Π1 and τk(ρ) 6= ρ for any ρ ∈ Π1 and any natural k for which τk(ρ) is defined. The r-matrix for a triple (Π1,Π2, τ) has the form r = r0 + e−α ⊗ eα + α,β∈∆+:α<β e−α ∧ eβ , (88) where < is a partial order on the set of positive roots ∆+ defined by the rule: α < β for α, β ∈ ∆+ if there exists a natural k such that τk(α) = β. The part r0 belongs to h⊗ h, where h is the Cartan subalgebra of g; r0 contains continuous ”multiparameters”, which satisfy (τ(α) ⊗ id + id⊗ α)(r0) = 0 for all α ∈ Π1 . (89) We are dealing with matrix solutions r of the cYB equation, r ∈ gl(n)⊗ gl(n), so r12 + r21 can be a linear combination of P and 11⊗ 11. Let Π = {α1, . . . , αn−1} be the set of the positive simple roots for the Lie algebra sl(n). There are two Cremmer–Gervais BD triples, T+ and T−. For the Cremmer–Gervais triple BD- triple T+, Π1 = {α1, α2, . . . , αn−2}, Π2 = {α2, α3, . . . , αn−1} and τ(αi) = αi+1. The data (Π1,Π2, τ) is encoded in the graph . . . • • • • • . . . • • The triple T− can be obtained from the triple T+ either by setting Π 1 = Π2, Π 2 = Π1 and τ ′ = τ−1 or by applying the outer automorphism of the underlying An−1 Dynkin diagram; the graph corresponding to the triple T− is . . . • • • • . . . • • • The r-matrix (79) corresponds to the triple (90) for a certain choice of the multiparameters. Here is the r-matrix r′ corresponding to the triple (91) r′CG = i,j:i<j (eij−s+1 ⊗ e i+s−1 − e j−s+1 ⊗ e i+s−1) (92) for a certain choice of the multiparameters, for which it satisfies r′P = −r′. For the r-matrices (79) and (92), one has r12+ r21 = P −11⊗11. The Cartan part of the r-matrices (79) and (92) are r0 = − i,j:i<j eii ⊗ e j , r 0 = − i,j:i<j j ⊗ e i . (93) The following lemma shows that a classical r-matrix r for a triple T can have a symmetry with respect to the multiplication by P from one side if and only if all segments (connected components) of Π1 are mapped by τ according to either (90) or (91). Lemma 3. A non-skew-symmetric classical r-matrix with a Belavin–Drinfeld data (Π1,Π2, τ) can satisfy Pr = −r (respectively, rP = −r) for a certain choice of the multiparameters if and only if τ(αi) = αi+1 (respectively, τ(αi) = αi−1) for all i ∈ Π1. Proof. Assume that τ(αm) = αm+k for some natural k, k ≥ 1. Then r contains the term em+km+k+1∧e with the coefficient 1. Such r-matrix cannot satisfy rP = −r for if rP = −r then r contains the term m+k+1 ∧ em+km with the coefficient (−1) but the coefficient in e−α ∧ eβ is 1 in the formula (88). If Pr = −r then r should contain also the term em+1m+k+1 ∧ e m . It then follows that (i) the Lie subalgebra generated by Π1 contains e m therefore the interval [αm, αm+1, . . . , αm+k−1] is contained in Π1; (ii) the Lie subalgebra generated by Π2 contains e m+k+1 therefore the interval [αm+1, αm+2, . . . , αm+k] is contained in Π2; (iii) the image of the interval [αm, αm+1, . . . , αm+k−1] under τ is the interval [αm+1, αm+2, . . . , αm+k]. This implies that the interval [αm+1, αm+2, . . . , αm+k−1] is τ -invariant (since τ(αm) = αm+k) which contradicts to the nilpotency of τ unless this interval is empty, that is, k = 1. Similarly, rP = −r is possible only if τ(αi) = αi−1 for all i ∈ Π1. It is left to show that when τ(αi) = αi+1 (respectively, τ(αi) = αi−1) for all i ∈ Π1 the multipa- rameters can indeed be adjusted to fulfill Pr = −r (respectively, rP = −r). We leave it as an exercise for the reader to check that with the assignment (93) for r (respectively, for r′) the compatibility condition (89) is verified. The proof is finished. � Remark. Two extreme BD triples can be rimed, the empty (Drinfeld–Jimbo) one and the “maximal” Cremmer–Gervais one. However, not every triple can be rimed: already the triple O • • • • • provides a counterexample. We outline a computer-aided proof in appendix C. 4.3 Skew-symmetric case A skew-symmetric classical r-matrix r ∈ g ∧ g is canonically associated with a quasi-Frobenius Lie subalgebra (f, ω) of g (see, e.g., [24]). A Lie algebra f which admits a non-degenerate 2-cocycle ω is called quasi-Frobenius; it is Frobenius if ω is a coboundary, i.e., ω(X,Y ) = λ([X,Y ]) for some λ ∈ f∗. We describe now the skew-symmetric r-matrix arising in the classical limit of the unitary rime R-matrix from Proposition 3. Proposition 6. The unitary rime R-matrix (Proposition 3) is a quantization of the skew-symmetric r-matrix i,j:i<j µi − µj (eij − e j) ∧ (e i − e i) ∈ gl(n) ∧ gl(n) . (95) This skew-symmetric classical r-matrix corresponds to a Frobenius Lie algebra (g0(n), δλn) spanned by the generators Zij := e j − e j , i 6= j, with the Frobenius structure determined by the coboundary of the 1-cochain λn = − i,j:i 6=j j , where {zij}, i 6= j, is the basis in g∗0(n), dual to the basis {Zij} in g0(n), zij(Z l ) = δ Proof. An artificial introduction of a small parameter c by a rescaling µi 7→ c−1µi in the formula for the R-matrix R̂ in Proposition 3 gives R = 11⊗ 11 + c r , (96) where r is given by (95). The n(n− 1) matrices Zij := eij − e j , i 6= j, form an associative subalgebra of the matrix algebra, l = (δ − δil )(Zki − Z li) (97) (we set Zii = 0 for all i); with respect to the commutators these matrices form a Lie subalgebra g0(n) of the Lie algebra gl(n), g0(n) ⊂ gl(n): [Zij , Z i ] = Z i − Z j , [Z i , Z i ] = Z i − Z i , [Z j , Z ] = Z − Zik , i 6= j 6= k 6= i , (98) all other brackets vanish. The skew-symmetric solution (95) of the cYB equation, i,j:i<j Zij ∧ Z µi − µj , (99) is non-degenerate on the carrier subalgebra g0(n). The carrier subalgebra g0(n) is necessarily quasi- Frobenius, having a 2-cocycle ω given by the inverse of the r-matrix, that is, ω(ZA, ZB) = rAB , where r ABrBC = δ C , r = rABZA ∧ ZB . (100) We have ω(Zij , Z l ) = −(µi − µj)δliδ . (101) It is easy to check that the 2-cycle ω is a coboundary, ω(Zij, Z l ) = λn([Z j , Z l ]) , λn = − i,j:i 6=j j ∈ g∗0(n) , (102) thus the subalgebra g0(n) is Frobenius. � The ”Frobenius” r-matrix (95) (and its quantization) was considered in the work [2]. Proposition 7. The skew-symmetric rime classical r-matrix (95), r = i<j(µi−µj)−1Zij∧Z i , where µ = (µ1, µ2, . . . , µn) is an arbitrary vector such that µi 6= µj, belongs to the orbit of the parameter-free classical r-matrix i,j:i<j ei+ki ∧ e j−k+1 j . (103) More precisely, r = AdXµ ⊗AdXµ(b) , (104) where the element Xµ ∈ GL(n) is defined by (Xµ)jk = ek−1 (µ1, . . . , µ̂j , . . . , µn). 3This matrix is the same X as in Proposition 4 but depending on variables µi. Proof. The equality r = AdXµ ⊗ AdXµ(b) is equivalent to a set of relations for the elementary symmetric functions ei, (Xµ ⊗Xµ) b = r (Xµ ⊗Xµ) ⇔ eîr−1e s−1 b l−1 , (105) where j−k+1 δi+ka − δi−k+1a δ and r (δiaδ b + δ − δiaδ − δjaδib)/(µi − µj) , i 6= j , 0 , i = j . Both operators b ab and r ab are symmetric in the lower indices and anti-symmetric in the upper indices, that is, Pb = −b , bP = b and Pr = −r , rP = r . (106) Eqs. (105) have the following form (eîb+s−2e a−s−1 − e b+s−2e a−s−1) = µi − µj (eîa−1 − e a−1)(e b−1 − e b−1) . (107) Due to (66), the left hand side of (107) equals µi − µj a−2 − µje a−2)(e b−1 − e b−1) . (108) The right hand side of (107) equals (108) as well because eîa−1 = ea−1 − µieîa−2. � As in the non-skew-symmetric case, in the infinite dimension the operators b and r are in general not equivalent. The sl(n) cYB solution. Let I = i=1 e i be the central element of gl(n). The generators Z̃ij = Z I ∈ sl(n) satisfy the same relations (98) as Zij thus they form a subalgebra g̃0(n) of the Lie algebra sl(n) which is isomorphic to g0(n), g̃0(n) ≃ g0(n). This isomorphism gives rise to another solution r̃ ∈ sl(n) ∧ sl(n) of the cYB equation, i,j:i<j Z̃ij ∧ Z̃ µi − µj ∈ sl(n) ∧ sl(n) . (109) We have the following lemma about the carrier Lie algebra of r̃ (the Lie subalgebra of sl(n) spanned by the generators Z̃ij). Lemma 4. The subalgebra g̃0(n) ⊂ sl(n) of dimension dim g̃0(n) = n(n − 1) is isomorphic to the maximal parabolic subalgebra p of sl(n) obtained by deleting the first negative root. Proof. The vector v = i=1 ei is an eigenvector for all elements Z̃ Z̃ij(v) = v for all i and j , i 6= j . (110) In a basis in which the first vector is v, the linear span of the generators Z̃ij is ∗ ∗ . . . ∗ 0 ∗ ∗ 0 ∗ . . . ∗ , (111) with the traceless condition. The comparison of dimensions finishes the proof. � Gerstenhaber and Giaquinto [14] found a classical r-matrix bCG which they called “boundary” because it lies in the closure of the solution space of the YB equation. The cYB solution bCG corre- sponds to a Frobenius subalgebra (p,Ω), where p is the parabolic subalgebra of sl(n) as above and the 2-cocycle Ω is a coboundary, Ω = δλbCG , λbCG = (eii+1) ∗ ∈ p∗ . (112) The r-matrix bCG is a twist of b (see [8]). Since the carriers of r̃ and bCG are isomorphic, the r-matrices are equivalent. We shall now prove that the same matrix Xµ transforms bCG into r̃. Proposition 8. The boundary classical r-matrix bCG ∈ sl(n) ∧ sl(n), bCG = ) eii ∧ e i,j:i<j ei+ki ∧ e j−k+1 j , (113) transforms into the cYB solution r̃ ∈ sl(n) ∧ sl(n), i,j:i<j Z̃ij ∧ Z̃ µi − µj , where Z̃ij = e j − e eii , (114) by a change of basis with the matrix Xµ ∈ GL(n), r̃ = AdXµ ⊗AdXµ(bCG) . (115) Proof. Due to Proposition 7 we have r = AdXµ ⊗ AdXµ(b). The cYB solution bCG is the sum of b and other terms, bCG = b + i,j(1 − ) eii ∧ e j . Therefore it is enough to show that r̃ − r = AdXµ ⊗AdXµ(bCG − b). One has r̃ − r = 1 i,j:i 6=j µi − µj , bCG − b = I ∧ (1− j j . (116) Thus we have to show that (1− j i,j:i 6=j µi − µj Xµ , (117) which amounts to the following identities for the elementary symmetric functions: (1− b− 1 )eîb−2 = j:j 6=i b−1 − e µi − µj . (118) Replacing, in the right hand side, e b−1 by e b−1 + µie b−2, e b−1 by e b−1 + µje b−2 and noticing that c = (n − c)ec, c = 1, 2, . . . , n (for the elementary symmetric functions in n variables) finishes the proof. � The passage to the sl(n) solution is another instance of the representation change. The general representation change (82) produces the following effect on the numerical r-matrix (103): b 7→ b− cη(0) ∧ 11 , (119) where η(0) is the generator of the invariance group of the r-matrix (103), η(0) = (n− j)ej+1j . (120) The representation change and the operation (86) produce the same 1-parametric family (119) of skew-symmetric r-matrices. The choice c = −1/n corresponds to the r-matrix bCG. 5 Bézout operators The Bézout operator [5] is the following endomorphism b(0) of the space P of polynomials of two variables x and y: b(0)f(x, y) = f(x, y)− f(y, x) or b(0) = (I − P ) , (121) where I is the identity operator and P is a permutation, Pf(x, y) = f(y, x). For any natural n, the subspace Pn of polynomials of degree less than n in x and less than n in y is invariant with respect to the operator b(0). The matrix of the restriction of b(0) onto Pn, written in the basis {xayb} of powers (in the decreasing order) coincides with the operator (103). The non-skew-symmetric matrix (79) is the matrix of the operator (I − P ) (122) in this basis. The rime bases are formed by the non-normalized Lagrange polynomials {li(x)lj(y)}, li(t) = s:s 6=i (t− φs), at points {φi}, i = 1, 2, . . . , n. We shall call the operators b(0) and b Bézout r-matrices. The Bézout r-matrices were rediscovered in several different contexts related to the Yang–Baxter equation (except the fact that they are the Cremmer–Gervais r-matrices, they appear, for instance, in [7] and [19]). The standard r-matrix r(s), for the choice of the multi-parameters for which it can be non-trivially rimed (see the remark at the end of section 3), has the following form in terms of polynomials r(s) : xiyj 7→ θ(i− j)xiyj − θ(j − i)xjyi . (123) The subspaces Pn are invariant with respect to r The properties of the Bézout r-matrices b(0) and b (and of the operator r(s)) become more trans- parent when they are viewed as operators on polynomials. In particular, (b(0))2 = 0 , b(0)P = −b(0) , Pb(0) = b(0) , b(0) + b(0)21 = 0 , (124) b2 = b , bP = −b , b+ b21 = I − P , (125) (r(s))2 = r(s) , r(s)P = −r(s) , r(s) + r(s)21 = I − P . (126) The description of the invariance groups of the operators b(0) and b is especially transparent when these operators are viewed as operators on the spaces of polynomials. Let ∂x and ∂y be the derivatives in x and y. We have (∂x + ∂y) = 0 which implies that ∂x is the generator of the invariance group of b(0); the group is formed by translations. Similarly, (x∂x + y∂y) = 0 which implies that x∂x is the generator of the invariance group of b; the group is formed by dilatations. The operation (86) implies that the operators b(0) + c(∂x − ∂y) , b+ c(x∂x − y∂y) (127) are solutions of the cYBe (the quantum version is easy as well) for an arbitrary constant c. 5.1 Non-homogeneous associative classical Yang–Baxter equation The operators b(0), b and r(s) satisfy an equation stronger than the cYBe. For an endomorphism r of V ⊗ V , define r ◦ r := r12r13 + r13r23 − r23r12 , r ◦′ r := r13r12 + r23r13 − r12r23 . (128) The equation r ◦ r = 0 (as well as r ◦′ r = 0) is called associative classical Yang–Baxter equation (acYBe) [1, 20]. We introduce a non-homogeneous associative classical Yang–Baxter equation (nhacYBe): r ◦ r = cr13 , (129) where c is a constant. Let F be the space of polynomials in one variable. For the space F ⊗ F of polynomials in two variables, we denote by x (respectively, y) the generator of the first (respectively, second) copy of F . For F ⊗ F ⊗ F , the generators are denoted by x, y and z. Lemma 5. 1. Let M be an operator on the space F ⊗ F . Assume that M(xf) = f + yM(f) , (130) M(yf) = −f + xM(f) (131) for an arbitrary f ∈ F ⊗ F . Then4 M ◦M(xF ) = zM ◦M(F ) , M ◦M(yF ) = xM ◦M(F ) , M ◦M(zF ) = yM ◦M(F ) (132) for an arbitrary F ∈ F ⊗ F ⊗ F . 2. The operator M = b(0) verifies (130) and (131). 3. Moreover, the unique solution of eqs. (130) and (131) (for the operator M on the space F ⊗ F) together with the ”initial” condition M(1) = 0 is M = b(0). Proof. A direct calculation. � Proposition 9. 1. The Bézout operator b(0) satisfies the acYBe. 2. The Bézout operator b and the operator r(s) satisfy the nhacYBe with c = 1. Proof. A direct calculation for b(0). Another way is to notice that the relations (132) for M = b(0) reduce the verification of b(0) ◦ b(0)(F ) = 0 for a monomial F ∈ F ⊗ F ⊗ F to the case F = 1, which is trivial. For the Bézout operator b ≡ xb(0) (x here is the operator of multiplication by x), we have, for an arbitrary F ∈ F ⊗ F ⊗ F , b ◦ b (F ) = xb(0)12 (xb 13 (F )) + xb 13 (yb 23 (F ))− yb 23 (xb 12 (F )) 13 (F ) + yb 13 (F ) + xyb 23 (F )− xyb 12 (F ) 13 (F ) + xyb (0) ◦ b(0)(F ) = b13(F ) . (133) We used eq. (130) for b(0) in the second equality. For the operator r(s), the identity θ(i− k)θ(i− j) + θ(i− k)θ(j − k)− θ(i− j)θ(j − k) = θ(i− k) (134) for the step function θ is helpful. � In each of cases (124-126), the operator r satisfies a quadratic equation r2 = u1r+u2I, the relation r + r21 = αP + βI with some constants α and β and the nhacYBe with some constant c. Several general comments about relations between the constants appearing in these equations are in order here. 4Eq. M ◦M(xF ) = zM ◦M(F ) follows from (130) alone. 1. Assume that an r-matrix (a solution of the cYBe) satisfies r ◦r = cr13. Then r ◦′ r = cr13. Taking the combinations (r ◦ r− cr13)−P23(r ◦′ r− cr13)P23 and (r ◦ r− cr13)−P12(r ◦′ r− cr13)P12, we find r13(Sr)23 − (Sr)23r12 = c(r13 − r12) , (Sr)12r13 − r23(Sr)12 = c(r13 − r23) , (135) where (Sr)12 := r12 + r21. If (Sr)12 = αP12 + βI with some constants α and β, as in (124-126), then it follows from (135) that (β − c)(r13 − r12) = 0 thus c = β . (136) This explains the value of the constant c in lemma 9. 2. For an endomorphism r of V ⊗ V , assume that r ◦ r = βr13 and (Sr)12 = αP12 + βI. Then P23(r ◦ r − βr13)P23 = r13r12 + r12r32 − r32r13 − βr12 = r13r12 + r12(αP23 + βI − r23)− (αP23 + βI − r23)r13 − βr12 = r ◦′ r − βr13 . (137) Thus, if (Sr)12 = αP12 + βI then r ◦ r = βr13 implies r ◦′ r = βr13. 3. Assume that r ◦ r = cr13 for an endomorphism r of V ⊗ V . Then for r̃ = r+ aI + bP , a and b are constants, we have r̃ ◦ r̃ = (c+ 2a)r̃13 + bP13(Sr)23 − a(a+ c)I − bcP13 + b2P23P12 . (138) If, in addition, (Sr)12 = αP12 + βI, then r̃ ◦ r̃ = (c+ 2a)r̃13 − a(c+ a)I + b(β − c)P13 + b(α+ b)P23P12 . (139) This shows that the equation r ◦ r = c1r13 + c2I + c3P13 + c4P23P12, c1, c2, c3 and c4 are constants, reduces to r ◦ r = c̃1r13 + c̃3P13 by a shift r 7→ r + aI + bP . If r ◦ r = βr13 and (Sr)12 = αP12 + βI then r̃ ◦ r̃ = (β + 2a)r̃13 − a(β + a)I + b(α+ b)P23P12 . (140) The combination P23P12 does not appear for b = 0 or b = −α. The choice b = −α corresponds, modulo a shift of r by a multiple of I, to r 7→ r21, so we consider only b = 0. Then, with the choice a = −β we find that the operator r̃ = r − βI satisfies the nhacYBe (and (Sr)12 = αP12 − βI). For the choice a = −β/2 we find that the operator r̃ = r − I satisfies r̃ ◦ r̃ = β , (Sr̃)12 = αP12 . (141) In particular, the operator 2(x− y) P (142) satisfies (141) with β = 1 and α = −1. Also, b̃2 = 1 4. Assume that r2 = ur + v and r12 + r21 = αP12 + βI for an endomorphism r of V ⊗ V . Squaring the relation r12 − βI = αP12 − r21 and using the same relation again, we obtain (u− β)(2r12 − βI − αP12) = 0 . (143) Thus, if r is not a linear combination of I and P then u = β . (144) 5. Assume that r ◦ r = cr13 and rP = −r for an endomorphism r of V ⊗ V . The nhacYBe has the following equivalent form: [r13, r23] = (r12 − cI)r13P23 . (145) Indeed, r13r23 − r23r13 = (−r13r23 + r23r12)P23 = (r12 − cI)r13P23 . (146) Here in the first equality we used r23P23 = −r23 and moved P23 to the right; in the second equality we used the nhacYBe r ◦ r = cr13. 5.2 Linear quantization Consider an algebra with three generators r12, r13 and r23 and relations r13r23 = r23r12 − r12r13 + βr13 , r13r12 = r12r23 − r23r13 + βr13 , r212 = βr12 + v , r 13 = βr13 + v , r 23 = βr23 + v . (147) Choose an order, say, r13 > r23 > r12. Consider (147) as ordering relations. The overlaps in (147) lead to exactly one more relation: r23r12r23 = r12r23r12 . (148) Thus the algebra in question is 12-dimensional (it follows from (147) and (148) that a general element of the algebra is a product AB of an element A of the Hecke algebra generated by r12 and r23 and a polynomial B, of degree less than 2, in r13). We conclude that the nhacYBe together with the quadratic equation for r imply the YBe. Note that the other form of the YBe also follows: r23r13r12 − r12r13r23 = (r12r23 − r13r12 + βr13)r12 − r12(r23r12 − r12r13 + βr13)r23 = −r13(βr12 + v) + βr13r12 + (βr12 + v)r13 − βr12r13 = 0. (149) Here in the first equality both nhacYBe for r were used; the quadratic relation for r was used in the second equality. Therefore, the quantization of such r-matrix is ”linear”5: a combination R = I + λr , (150) where λ is an arbitrary constant, satisfies the YBe R12R13R23 = R23R13R12. 5It was noted in [8] that the operator b(0) satisfies both forms of the YBe, squares to zero and that its quantization has the simple form (150). 5.3 Algebraic meaning We shall clarify the algebraic meaning of the non-homogeneous associative classical Yang–Baxter equation in the general context of associative algebras. Let A be an algebra. Let r ∈ A⊗ A. The operation δ(0) : A→ A⊗A , δ(0)(u) = (u⊗ 1) r − r (1⊗ u) (151) (the algebra A does not need to be unital, (u⊗1)(a⊗ b) stands for ua⊗ b and (a⊗ b)(u⊗1) for au⊗ b) is coassociative if and only if [1] (u⊗ 1⊗ 1) (r ◦′ r) = (r ◦′ r) (1⊗ 1⊗ u) ∀ u ∈ A . (152) In particular, δ(0) is coassociative if (r ◦′ r) = 0. Assume now that the algebra A is unital. Define the operations δ and δ̃ : A→ A⊗ A, δ(u) := (u⊗ 1) r − r (1⊗ u)− c (u⊗ 1) , (153) δ̃(u) := (u⊗ 1) r − r (1⊗ u) + c (1⊗ u) , (154) where c is a constant. Proposition 10. The coassociativity of each of the operations δ and δ̃ is equivalent to (u⊗ 1⊗ 1) (r ◦′ r − c r13) = (r ◦′ r − c r13) (1⊗ 1⊗ u) ∀ u ∈ A . (155) Proof. A straightforward calculation. � In particular, the operations δ and δ̃ are coassociative if r ◦′ r = c r13. The map (151) has the following property: δ(0)(uv) = (u⊗ 1) δ(0)(v) + δ(0)(u) (1 ⊗ v) ; (156) that is, δ(0) is a derivation with respect to the standard structure of A ⊗ A as a bi-module over A, uU := (u⊗ 1)U and Uu := U(1⊗ u) for u ∈ A and U ∈ A⊗ A. For the operations δ and δ̃, the analogue of the property (156) reads δ(uv) = (u⊗ 1) δ(v) + δ(u) (1 ⊗ v) + c (u⊗ v) , (157) δ̃(uv) = (u⊗ 1) δ̃(v) + δ̃(u) (1⊗ v)− c (u⊗ v) . (158) 5.4 Rota–Baxter operators Let A be an algebra. An operator r : A → A is called Rota–Baxter operator of weight α if r(A)r(B) + αr(AB) = r r(A)B +Ar(B) (159) for arbitrary A,B ∈ A (α is a constant). We refer to [22] for further information about the Rota– Baxter operators. The Rota–Baxter operators of weight zero are closely related to the acYBe [23]. It turns out that the Rota–Baxter operators of non-zero weight are related to the nhacYBe. We shall discuss this relation and calculate the Rota–Baxter operators corresponding to the Bézout operators. It is surprising that the Bézout operators, which rather have the sense of derivatives, become, being interpreted as operators on matrix algebras, the Rota–Baxter operators which are designed to axiomatize the properties of indefinite integrations and summations. 1. For an endomorphism r of V ⊗ V , define two endomorphisms, r and r′, of the matrix algebra Mat(V ): r(A)1 := Tr2(r12A2) , r ′(A)2 := Tr1(r12A1) , A ∈ Mat(V ) , (160) where Tri is the trace in the copy number i of the space V . Assume that r satisfies the nhacYBe (129). Multiplying (129) by A2B3, A,B ∈ Mat(V ), and taking traces in the spaces 2 and 3, we find r(A)r(B) + r r′(A)B Ar(B) = cTr(A)r(B) . (161) Assume, in addition, that r12 + r21 = αP12 + βI. Then r(A) + r′(A) = αA+ βTr(A) 11 . (162) If c = β then expressing r′(A) by (162) and substituting into (161), we find that the term with Tr(A) drops out and r is the Rota–Baxter operator of weight α on the algebra of matrices. Similarly, r′ is the Rota–Baxter operator of weight α as well. 2. We shall calculate the Rota–Baxter operators corresponding to the Bézout operators in the poly- nomial basis. The action of the operator b(0) on monomials xkyl reads b(0)(xkyl) = −(xl−1yk + xl−2yk+1 + · · ·+ xkyl−1) , k < l , 0 , k = l , xk−1yl + xk−2yl+1 + · · · + xlyk−1 , k > l . (163) The action of the operator b on monomials xkyl reads b(xkyl) = −(xlyk + xl−1yk+1 + · · ·+ xk+1yl−1) , k < l , 0 , k = l , xkyl + xk−1yl+1 + · · ·+ xl+1yk−1 , k > l . (164) Shortly, b(0)(xkyl) = θ(k − l) k−l−1 xl+syk−s−1 − θ(l − k) l−k−1 xk+syl−s−1 , (165) b(xkyl) = θ(k − l) xl+syk−s − θ(l − k) xk+syl−s . (166) We list several useful matrix forms of the operators b(0) and b in the basis formed by monomials, ea ⊗ eb := xayb; for the operator b(0): b(0) = i,j,a,b θ(j − a) θ(j − b) δi+j a+b+1 eja ∧ eib i,j,a,b θ(j − a) θ(j − b)− θ(i− b) θ(i− a) a+b+1 e a ⊗ eib i,j:i<j i+a−1 ∧ e (167) and for the operator b: i,j,a,b θ(j + 1− a) θ(a− i) δi+ja+b (e a ⊗ eib − eia ⊗ e i,j,a,b θ(j + 1− a) θ(a− i)− θ(i+ 1− a) θ(a− j) a+b e a ⊗ eib i,j:i<j i+a ⊗ e j−a − eii+a ⊗ e i,j:i<j j−i−1 i+a ∧ e j−a + e j ⊗ e i − eij ⊗ e (168) where x ∧ y = x⊗ y − y ⊗ x. The Rota–Baxter operator r corresponding to b(0) reads (A)ij = θ(j − i) Ai−sj−s−1 − θ(i+ 1− j) Ai+s+1j+s . (169) In the right hand side of (169), the summations are over those s ≥ 0 for which the corresponding matrix element in the sum makes sense; that is, the range of s in the first sum is s = 0, 1, . . . , i − 1 and, in the second sum, s = 0, 1, . . . n− i− 1; The Rota–Baxter operator rb corresponding to b reads (with the same convention about the sum- mation ranges) rb(A) j = θ(j + 1− i) Ai−s−1j−s−1 − θ(i− j) Ai+sj+s . (170) Its weight is -1. For the operator r(s), given by eq. (123), the corresponding Rota–Baxter operator r(s) is r(s)(A)ij = −θ(j − i)Aij , i 6= j , s:s<i Ass , i = j . (171) Its weight is -1. We shall give also the Rota–Baxter operator for the Bézout r-matrix b in the rime basis, that is, for the r-matrix (78); it has weight 1 (since r12+r21 = P −I for r in (78)). The Rota–Baxter operator has the form r(A)ij = φj − φi (Aij −A j) , i 6= j , s:s 6=i φi − φs (Ais −Ass) , i = j . (172) 5.5 ∗-multiplication 1. Let r : A→ A be a Rota–Baxter operator of weight α (see eq.(159)) on an algebra A. It is known that the operation A ∗B := r(A)B +Ar(B)− αAB , A,B ∈ A , (173) defines an associative product on the space A. This product is closely related to the coproducts (153) and (154) by duality. We shall illustrate it in the context of the matrix algebras. Define an operation ∗̃ by 〈δ̃(u), B ⊗A〉 = 〈u,A∗̃B〉 , (174) where δ̃ is given by (154). We have then 〈δ̃(u), B ⊗A〉 = Tr12 u1rB1A2 − ru2B1A2 + c u2B1A2 = Tr1 u1 Tr2(rA2)B1 − Tr1 u1A1 Tr2(B2 r21) c u1 Tr(B)A1 r(A)B −A r′(B) + cATr(B) (175) A∗̃B = r(A)B −A r′(B) + cATr(B) . (176) In eq. (175), xi stands for the copy of an element x in the space number i in A ⊗ A; the operators r and r′ are given by (160); to obtain the second and the third terms in the second line of (175) we renumbered 1↔ 2 and then moved r cyclically under the trace in the second term. Assume, as before, that r12+ r21 = αP12 +βI and c = β. Then, expressing r ′(A) by (162), we find that the term with Tr(B) drops out and it follows that A∗̃B = A ∗B . (177) 2. We shall describe the ∗-multiplication in the simplest example of the Rota–Baxter operators 169) and (170) corresponding to the Bézout operators for the the polynomials of degree less than 2 (that is, for 2× 2 matrices A = a11 a a21 a ≡ aije For the operator b(0) = e21 ∧ e11, we have (A) = −a21 a11 (178) and the ∗-multiplication reads A ∗o à ≡ Ar (Ã) + r (A)à = −a21ã11 −a21ã12 + a11(ã11 + ã22) −a21ã21 a21ã11 . (179) This algebra is isomorphic to the algebra of 3× 3 matrices of the form ∗ ∗ ∗ 0 0 ∗ 0 0 0 with the identification e11 7→ 0 1 0 0 0 1 0 0 0  , e12 7→ −1 0 0 0 0 0 0 0 0  , e21 7→ 0 0 1 0 0 0 0 0 0  , e22 7→ 0 0 0 0 0 1 0 0 0  . (180) For the operator b = e22 ⊗ e11 − e12 ⊗ e21, we have rb(A) = −a21 a11 (181) and the ∗-multiplication reads A ∗ à ≡ Arb(Ã) + rb(A)Ã+Aà = a11ã 2 + a 1 + ã a11ã 2 + a 1 + ã . (182) This algebra is isomorphic to the algebra of 3× 3 matrices of the form ∗ ∗ ∗ 0 ∗ 0 0 0 0 with the identification e11 7→ 1 0 0 0 1 0 0 0 0  , e12 7→ 0 0 1 0 0 0 0 0 0  , e21 7→ 0 1 0 0 0 0 0 0 0  , e22 7→ 0 0 0 0 1 0 0 0 0  . (183) 6 Rime Poisson brackets The Poisson brackets having the form {xi, xj} = fij(xi, xj) , i, j = 1, 2, . . . , n , (184) with some functions fij of two variables, we shall call rime. In this section we study quadratic rime Poisson brackets, {xi, xj} = aij(xi)2 − aji(xj)2 + 2νijxixj , i, j = 1, 2, . . . , n . (185) We show that there is a three-dimensional pencil of such Poisson brackets and then find the invariance group and the normal form of each individual member of the pencil. 6.1 Rime pencil In this subsection we establish that the quadratic rime Poisson brackets form a three-dimensional Poisson pencil. The left hand side of (185) contains a matrix aij with zeros on the diagonal, aii = 0, and an anti-symmetric matrix νij, νij = −νji. The Jacobi identity constraints these matrices to satisfy aijajk + aik(νij + νjk) = 0 , i 6= j 6= k 6= i . (186) We shall describe a general solution of eq. (186) in the strict situation, that is, when all aij and νij are different from zero for i 6= j. The left hand side of νij + νjk = −aijajk/aik is anti-symmetric with respect to (i, k), that is ΥijΥjkΥki = 1 for Υij = −aij/aji, which readily implies the existence of a vector φi such that Υij = φ j . Therefore, aik = φicikφ k , (187) where the matrix cij is anti-symmetric, cij = −cji. Next, 2νki = −(νij +νjk)+ (νjk+νki)+ (νki+νij); using (186) to express each bracket in the right hand side, we find νki = cijcki cjkcki cijcjk . (188) The right hand side of eq. (188) does not depend on j which imposes further restrictions on the matrix cij when n > 3. Writing the sum νij + νjk + νkl + νli in two ways, as (νij + νjk) + (νkl + νli) and as (νjk + νkl) + (νli + νij), and using (186) to express each bracket in terms of the matrix c, we obtain cijcjk − cilclk cjkckl − cjicil . (189) Replacing j by m in (188) gives the condition on the matrix c: cijcki cjkcki cijcjk cimcki cmkcki cimcmk . (190) Using eq. (189) to rewrite the combination cijcjk cimcmk , we find (cjkckm − cjicim)Ψijkm = 0 , where Ψijkm = cjkcim cijckm ckicjm . (191) The quantity Ψijkm is totally anti-symmetric with respect to its indices. Therefore, if Ψijkm 6= 0 then the combinations (cjkckm − cjicim) vanish for all permutations of indices. This is however impossible: the system of three linear equations cijcjk − cimcmk = 0 , cikckm − cijcjm = 0 , cimcmj − cikckj = 0 (192) for unknowns {cjk, ckm, cmj} has, by definition, a non-zero solution but the determinant of the system is different from zero. Thus the Pfaffian Ψijkm vanishes for each quadruple {i, j, k,m}; in other words, the coefficients of the matrix 1/cij satisfy the Plücker relations; therefore the form 1/cij is decomposable, c−1ij = sitj−sjti, for some vectors ~s and ~t. For each i, at least one of si or ti is different from zero. Making, if necessary, a change of basis in the two dimensional plane spanned by ~s and ~t, we can therefore always assume that all components of, say, the vector ~s are different from zero, si 6= 0 ∀ i. We represent the bivector 1/cij in the form (u−1i = si and ψi = −ti/si) = u−1i u j (ψi − ψj) . (193) Substituting (193) into (188) we obtain νki + u2k + u ψk − ψi u2i − u2j ψi − ψj u2k − u2j ψk − ψj . (194) Replacing j by m in the right hand side and equating the resulting expressions, we find that the independency of the right hand side on j implies: Eijkm := (ψi − ψj)(ψi − ψk)(ψi − ψm) (ψj − ψi)(ψj − ψk)(ψj − ψm) (ψk − ψi)(ψk − ψj)(ψk − ψm) (ψm − ψi)(ψm − ψj)(ψm − ψk) = 0 . (195) for every quadruple {i, j, k,m}. The quantity Eijkm is totally symmetric. Selecting three values of the index, say, 1,2 and 3, we can form the quadruple {i, 1, 2, 3} for each i. Solving Ei123 = 0, we obtain the following expression for u2i : u2i = A1 (ψi −M2)(ψi −M3) (M1 −M2)(M1 −M3) (ψi −M1)(ψi −M3) (M2 −M1)(M2 −M3) (ψi −M1)(ψi −M2) (M3 −M1)(M3 −M2) (196) for some constants A1, A2, A3,M1,M2 and M3. The right hand side is the value, at the point ψi, of a quadratic polynomial which equals to Aa at the points Ma, a = 1, 2, 3. Since A1, A2, A3,M1,M2 and M3 are arbitrary, we can simply write u2i = aψ i + bψi + c . (197) With the expressions (197) for u2i , the equalities (195) are identically satisfied which shows that (197) is the general solution. Upon rescaling xi 7→ φiuixi with φi from (187), the Poisson brackets (185) simplify. The following statement is established (for n = 2 or 3, (197) does not impose a restriction on the anti-symmetric matrix cij with all off-diagonal entries different from zero). Proposition 11. Up to a rescaling of variables, the general strict quadratic rime Poisson brackets have the form {xi, xj} = ̺(ψj)(x i)2 + ̺(ψi)(x ψi − ψj (ψi − ψj) a− ̺(ψi) + ̺(ψj) ψi − ψj xixj , (198) where ~ψ is an arbitrary vector with pairwise distinct components and ̺(t) = at2+bt+c is an arbitrary quadratic polynomial 6. Thus the strict quadratic rime Poisson brackets form the three-dimensional pencil (parameterized by the polynomial ̺). The Poisson brackets (198) can be rewritten in the following forms: {xi, xj} = ψi − ψj ̺(ψj)x i − ̺(ψi)xj (xi − xj) + a (ψi − ψj)xixj , (199) {xi, xj} = au2ij + buijvij + cv ψi − ψj ψi − ψj , (200) where uij = ψjx i − ψixj and vij = xi − xj. Remark 1. For ̺(t) = bt (respectively, ̺(t) = c) these Poisson brackets appear in the classical limit of the commutation relations (31) in the non-unitary (respectively, unitary) case (with the parameterization βij = − ψi − ψj in the non-unitary case). Remark 2. The strict rime linear Poisson brackets {xi, xj} = aijxi − ajixj , aij 6= 0 for all i, j = 1, 2, . . . , n : i 6= j (201) (or strict rime Lie algebras) are less interesting. The Jacobi identity is aikakj = aijajk for all i 6= j 6= k 6= i . (202) 6To have nonvanishing coefficients in the formula (198) one has to impose certain inequalities for the components of the vector ~ψ and the coefficients of the polynomial ̺; however, the formula (198) defines Poisson brackets without these inequalities. Rescale variables x2, x3, . . . , xn to have a1i = 1, i = 2, . . . , n. Then the condition (202) with one of i, j, k equal 1 implies aij = aji and aij = ai1/aj1, i, j = 2, . . . , n; it follows that a i1 = a j1, i, j = 2, . . . , n. For n > 3, the condition (202) with i, j, k > 1 forces ai1 = aj1, i, j = 2, . . . , n. Denote by ν this common value, ai1 = ν, i, j = 2, . . . , n. After a rescaling x 1 7→ νx1 we find a unique strict rime Lie algebra, [xi, xj ] = xi−xj for all i and k, which is almost trivial: [xi, xk −xl] = −(xk −xl) for all i, k and l and [xi − xj , xk − xl] = 0 for all i, j, k, l. For n = 3, there is one more possibility: a31 = −a21. After a rescaling x1 7→ a21x1, the solution reads [x1, x2] = x1 − x2 , [x1, x3] = x1 + x3 , [x2, x3] = −x2 + x3 . (203) This Lie algebra is isomorphic to sl(2); the isomorphism is given, for example, by h 7→ x1 − x3, e 7→ x1+x3 and f 7→ x2− (x1+x3)/4 (here h, e and f are the standard generators of sl(2), [h, e] = 2e, [h, f ] = −2f and [e, f ] = h). 6.2 Invariance In this subsection we analyze the invariance group of each individual member of the Poisson pencil from the proposition 11. We find that the Poisson brackets (198), with arbitrary (non-vanishing) ̺, admit a non-trivial 1-parametric invariance group. The transformation law of Poisson brackets {xi, xj} = f ij(x) under an infinitesimal change of variables, x̃i = xi + ǫ ϕi(x), ǫ2 = 0, is {x̃i, x̃j} = f ij(x̃) + ǫ δxf ij, where δxf ij = {ϕi, xj}+ {xi, ϕj} + ϕk∂kf ij. For a linear infinitesimal transformation, ϕi(x) = Aijx j, we have ij = Aik{xk, xj}+A i, xk} − xlAkl ∂k{xi, xj} . (204) Specializing to the Poisson brackets (198), we find ij = Uji − Uij (205) Uij := +Ajs(ψija− ̺i + ̺j s:s 6=i (xs)2 + (xi)2 + (ψsia− ̺s + ̺i )xixs (206) where ψij = ψi − ψj and ̺s = ̺(ψs). The Poisson brackets (198) remain rime under the infinitesimal linear transformation with the matrix A if the coefficients in (xs)2, xsxi and xsxj, s 6= i, j, in (205) vanish which gives the following system: (xs)2 , s 6= i, j ⇒ Ajs = 0 , (207) xixs , s 6= i, j ⇒ 2Ais ψsja− ̺i + ̺j − ̺s + ̺i = 0 . (208) Eq. (207) implies that Alk = νk̺l/ψlk, l 6= k, with arbitrary constants νk. For a quadratic polynomial ̺, this solves eq. (208) as well. The coefficient in xjxs vanishes due to the anti-symmetry. The Poisson brackets (198) are invariant under the infinitesimal linear transformation with the matrix A if, in addition to (207) and (208), the coefficients in (xi)2, (xj)2 and xixj in (205) vanish which gives: xixj ⇒ ̺jAij + ̺iA i = 0 , (209) (xi)2 ⇒ Aii ψija− ̺i + ̺j s:s 6=i = 0 . (210) Eq. (209) implies that νk are equal, νk = ν. The matrix A is defined up to a multiplicative factor, so we can set ν to 1. Since the Poisson brackets (198) are quadratic, a global rescaling leaves them invariant, so we can add to A a matrix, proportional to the identity matrix and make A traceless. The traceless condition, together with eq. (210) determines the diagonal entries, Aii = a(n − 1)ψi + b+ ̺i s:s 6=i . The coefficient in (xj)2 vanishes due to the anti-symmetry. We summarize the obtained results. Proposition 12. (i) The infinitesimal linear transformation with the matrix A leaves the Poisson brackets (198) rime if and only if Alk = , l 6= k , (211) with arbitrary constants νk. (ii) Up to a global rescaling of coordinates, the invariance group of the Poisson brackets (198) is 1-parametric, with a generator A, A(̺)ij = , i 6= j , and A(̺)ii = ̺′i + ̺iξi , ξi := s:s 6=i , (212) where ̺′i is the value of the derivative of the polynomial ̺ at the point ψi. Since the Poisson brackets transformed with the matrix (211) are still rime, it follows from the proposition 11 that they can be written, after an appropriate rescalings of coordinates, in the form (198). In other words, the variation δx can be compensated by a variation of ψ’s and ̺ and a diagonal transformation of the coordinates. We have − δxf ij = δ(1) + δ(2) , (213) where δ(1) = ̺i̺j(x i − xj)2 (νi − νj) + a νj̺j(x i)2 − νi̺i(xj)2 (214) δ(2) = (Ãii − à i)2 − ̺i(xj)2 , Ãii := A i − ̺′iνi − s:s 6=i . (215) Choose Aii to set à i to 0; this is a diagonal transformation of the coordinates. Then δ (2) vanishes and the variation of f ij is reduced to δ(1). On the other hand, under a variation of ψ′s, ψi 7→ ψi + δψi, the Poisson brackets (198) transform in the following way: (xi − xj)2 (̺iδψj − ̺jδψi) + a (xj)2δψi − (xi)2δψj (216) and we conclude that with the choice δψi = ǫ̺iνi (217) the variation δ(1) is compensated by the variation δψ. The coefficients of the polynomial ̺ stay the same. In the next subsection we will study relations between the variation of ψ’s and the polynomial Remark. With ξi as in (212), define three operators, (B−)ij = , i 6= j , and (B−)ii = −ξi , (218) (B0)ij = , i 6= j , and (B0)ii = −( + ψiξi) , (219) (B+)ij = , i 6= j , and (B+)ii = −((n− 1)ψi + ψ2i ξi) . (220) The operators B+, B0 and B− generate an action of the Lie algebra sl(2), [B0, B−] = −B− , [B0, B+] = B+ , [B+, B−] = −2B0 (221) (to obtain the usual commutation relations for the generators of sl(2), change the sign of B+). This is the usual projective action of sl(2) on polynomials f(t) of degree less than n, B− : f(t) 7→ f ′(t) , B0 : f(t) 7→ tf ′(t)− f(t) , B+ : f(t) 7→ t2f ′(t)− (n− 1)t f(t) , (222) written in the basis of the non-normalized Lagrange polynomials, li(t) = s:s 6=i (t − ψs), at points {ψi}, i = 1, 2, . . . , n. Indeed, in the basis {li(t)}, a polynomial f(t), deg(f) ≤ n − 1, takes the form f ili, where f i = li(ψi) −1f(ψi). We have l′i(t) = a:a6=i b:b6=a,i (t− ψb) , so l′i(ψk) = b:b6=k,i ψkb = lk(ψk) , k 6= i . (223) Also, l′i(t) = li(t) s:s 6=i t− ψs , so l′i(ψi) = −li(ψi)ξi . (224) Therefore, l′i(t) = −ξili(t) + k:k 6=i lk(t), which is exactly (218). For functions on the set of points {ψi}, the operator of multiplication by t acts as a diagonal matrix Diag(ψ1, ψ2, . . . , ψn) and (219)-(220) follow. Define an involution ̟ on the space of matrices7, ̟(Y )ij = Y j , i 6= j and ̟(Y )ii = −Y ii , Y ∈ Matn . (226) B(̺) = aB+ + bB0 + cB− , B(̺) : f 7→ ̺(t)f ′(t)− n− 1 ̺′(t)f(t) . (227) In the basis {li(t)} for B, the generator (212) of the invariance group is A(̺) = ̟(B(̺)) . (228) Note that the operators ̟(B−), ̟(B0) and ̟(B+) do not form a Lie algebra. 6.3 Normal form In this subsection we derive a normal form of each individual member of the Poisson pencil from the proposition 11. It depends only on the discriminant of the polynomial ̺. When the discriminant of ̺ is different from zero, the Poisson brackets (198) are equivalent to the Poisson brackets defined by the r-matrix (79). When the polynomial ̺ is different from zero but its discriminant is zero, the Poisson brackets (198) are equivalent to the Poisson brackets defined by the r-matrix (103). Under a variation of the polynomial ̺, ̺(t) 7→ (a + δa)t2 + (b + δb)t + (c + δc), we have for the Poisson brackets (200): u2ijδa+ uijvijδb+ v ψi − ψj . (229) The variation of ̺ can be compensated by a variation (216) of ψ’s if the coefficients in (xj)2, xixj and (xi)2 in the combination (δψ + δ̺)f ij vanish, which gives the following system: (xj)2 ⇒ ̺iδψj − ̺jδψi + aδψi + ψ2i δa+ ψiδb+ δc = 0 , (230) xixj ⇒ − 2(̺iδψj − ̺jδψi) 2ψiψjδa+ (ψi + ψj)δb + 2δc = 0 . (231) A combination 2×(230)+(231) gives 2aδψi + 2ψiδa+ δb = 0 . (232) 7The involution ̟ is the difference of two complementary projectors. The involution ̟ satisfies ̟(Y1)̟(Y2) +̟(Y1Y2) = ̟(̟(Y1)̟(Y2)) + Y1Y2 , ̟(̟(Y1)Y2) + Y1̟(Y2) , ̟(Y1̟(Y2)) +̟(Y1)Y2 (225) for arbitrary Y1, Y2 ∈ Matn. All other linear dependencies between Y1Y2, ̟(Y1)Y2, Y1̟(Y2), ̟(Y1)̟(Y2), ̟(Y1Y2), ̟(̟(Y1)Y2), ̟(Y1̟(Y2)) and ̟(̟(Y1)̟(Y2)) are consequences of the three identities (225). Substituting the expression (232) for δψ’s into (230) gives δD(̺) = 0 , where D(̺) = b2 − 4ac . (233) The coefficient in (xi)2 in (δψ + δ̺)f ij vanishes due to the anti-symmetry. Therefore, a necessary condition for a variation of ̺ to be compensated by a variation of ψ’s is that the discriminant D(̺) does not vary. We shall now see that the discriminant is the unique invariant. Explicitly, under a shift ψj 7→ ψj + ζ, we have uij 7→ uij + ζvij and vij 7→ vij (in the notation (200)), which produces the following effect on the coefficients of the polynomial ̺: a 7→ a , b 7→ b+ 2ζa , c 7→ c+ ζb+ ζ2a . (234) A dilatation ψj 7→ λψj produces the following effect on the coefficients of ̺: a 7→ λa , b 7→ b , c 7→ λ−1c . (235) The inversion ψj 7→ ψ−1j accompanied by a change of variables x̃i = ψ i produces the following effect on the coefficients of ̺: a 7→ −c , b 7→ −b , c 7→ −a . (236) The set of operators (234) and (235) generates the action of the affine group on the space of the polynomials ̺. The affine group, together with the inversion (236) generates an action8 of so(3) (the spin 1 representation of sl(2)) on the space of the polynomials ̺ and the classification reduces to that of orbits. The orbits (in the complex situation) of non-zero polynomials are of two types: ”massive”, D(̺) 6= 0, or ”light-like”, D(̺) = 0. Particular representatives of both types appear in the Poisson brackets, corresponding to the rime r-matrices (see the remark 1 after the proposition 11) and thus to the r-matrices studied in subsections 4.1 and 4.3. We obtain the following statement. Proposition 13. Let ̺(t) be a non-zero quadratic polynomial. If the discriminant of ̺ is different from zero, D(̺) 6= 0, then there exists a change of the param- eters ψi in the Poisson brackets (198) which sets ̺(t) to bt, ̺(t) 7→ bt; these are the Poisson brackets corresponding to the r-matrix rCG (subsection 4.1). If the discriminant of ̺ is zero, D(̺) = 0, then there exists a change of the parameters ψi in the Poisson brackets (198) which sets ̺(t) to c, ̺(t) 7→ c; these are the Poisson brackets corresponding to the r-matrix bCG (subsection 4.3). The generator A(̺) of the invariance group can be easily described in both cases, D(̺) 6= 0 and D(̺) = 0, in the parameter-free basis (that is, for the r-matrices rCG and bCG; in the rime basis the generators are given by (52) and (55), respectively). For D(̺) 6= 0 (respectively, D(̺) = 0), it coincides with the matrix of the operator B0 (respectively, B−), as in the remark in subsection 6.2, in the basis {ti} of powers of the variable t. This implies somewhat unexpectedly that for an arbitrary polynomial ̺(t) the matrices A(̺) and ̟(A(̺)) are related by a similarity transformation. Note that in the basis {ti} of powers, the operators aB++bB0+cB− and ̟(aB++bB0+cB−) are also related by a similarity transformation for arbitrary a, b and c but here it is obvious: ̟(aB++bB0+cB−) = aB+−bB0+cB−, so the operator ̟(aB+ + bB0 + cB−) belongs to sl(2) and moreover lies on the same (complex) orbit as aB+ + bB0 + cB− with respect to the adjoint action. 8Let e+ be the generator of the 1-parametric group (234) and h the generator of the 1-parametric group (235). Denote by I the inversion (236). The remaining generator e− is Ie+I. 7 Orderable quadratic rime associative algebras Consider an associative algebra A defined by quadratic relations giving a lexicographical order. This means that xjxk for j < k is a linear combination of terms xaxb with a ≥ b and either a > j or a = j and b > k. We shall say that such algebra A is rime if {a, b} ⊂ {j, k}. In other words, the relations in the algebra are xjxk = fjkx kxj + gjkx kxk , j < k . (237) We shall classify the strict rime algebras A (that is, the algebras for which all coefficients fij and gij are different from zero for i < j). The only possible overlaps for the set of relations (237) are of the form (xjxk)xl = xj(xkxl), j < k < l. The ordered form of the expression (xjxk)xl is (xjxk)xl = fjkfjlfkl x lxkxj + fjkfjlgkl x lxlxj + f2klgjk x lxkxk + (fklgjkgkl + f kl(fjkgjl + gjkgkl))x lxlxk + (fjkgjlgkl + gjkg kl + fklgkl(fjkgjl + gjkgkl))x lxlxl . (238) The ordered form of the expression xj(xkxl) is xj(xkxl) = fjkfjlfkl x lxkxj + f2jlgkl x lxlxj + fklfjlgjk x lxkxk + fklgjl x lxlxk + (gklgjl + fjlgklgjl)x lxlxl . (239) Equating coefficients, we find xlxlxj : fjkfjlgkl = f jlgkl , (240) xlxkxk : f2klgjk = fklfjlgjk , (241) xlxlxk : fklgjkgkl + f kl(fjkgjl + gjkgkl) = fklgjl , (242) xlxlxl : fjkgjlgkl + gjkg kl + fklgkl(fjkgjl + gjkgkl) = gklgjl + fjlgklgjl . (243) In the strict situation, eqs. (240) and (241) simplify, respectively, to fjk = fjl , for j < k and j < l , (244) fkl = fjl , for j < l and k < l . (245) Eqs. (244) and (245) imply that fjk’s are all equal, fjk =: f . (246) The substitution of (246) into (242) gives (in the strict situation) (f + 1) gjkgkl + gjl(f − 1) = 0 for j < k < l . (247) Eq. (243) follows from (246) and (247). We have thus two cases: (i) f = −1 and no extra conditions on gjk’s; (ii) f 6= −1 and gjkgkl = (1− f) gjl for j < k < l ; (248) 1− f 6= 0 since gjk 6= 0 and gkl 6= 0. In the case (ii), make an appropriate rescaling of generators, xi 7→ dixi to achieve gi,i+1 = 1− f for all i = 1, . . . , n− 1 . (249) It then follows from eq. (248) that gij = 1− f for all i < j . (250) We summarize the obtained results. Proposition 14. Up to a rescaling of variables, the general orderable quadratic strict rime algebra has relations (i) either of the form xjxk = −xkxj + gjkxkxk , j < k , (251) with no conditions on the coefficients gjk; (ii) or of the form xjxk = fxkxj + (1− f)xkxk , j < k , (252) with arbitrary f (it is strict when f 6= 0, 1). By construction, the algebras of types (i) and (ii) possess a basis formed by ordered monomials and thus have the Poincaré series of the algebra of commuting variables. The algebra with defining relations (252) is the quantum space for the R-matrix (75). The relations (252) can be written in the form (xj − xk)xk = fxk(xj − xk) , j < k ; (253) this is a quantization of the Poisson brackets {xj , xk} = xk(xj − xk) , j < k . (254) It would be interesting to know if the algebra with the defining relations (251) admits an R-matrix description. Acknowledgements It is our pleasure to thank László Fehér, Alexei Isaev and Milen Yakimov for enlightening discussions. The work was partially supported by the ANR project GIMP No.ANR-05-BLAN-0029-01. The second author (T. Popov) was also partially supported by the Program “Bourses d’échanges scientifiques pour les pays de l’Est européen” and by the Bulgarian National Council for Scientific Research project PH- 1406. Appendix A. Equations Here we give the list of the equations YB(R̂) abc = 0 for the rime matrix kl = αijδ k + βijδ l + γijδ l + γ l , (255) with a convention αi = αii and βii = γii = γ ii = 0. The rime Ansatz implies that YB(R̂) abc can be different from zero only if the set of lower indices is contained in the set of upper indices. Therefore, the equations split into two lists: the first one with two different indices among {i, j, k} and the second one with three different indices. The full set of equations YB(R̂) abc = 0 is invariant under the involution ι, ι : αi ↔ αi , αij ↔ αji , βij ↔ βji , γij ↔ γ′ji , (256) for if R̂ is a solution of the YBe then R̂21 = PR̂P is a solution of the YBe as well. We shall write only the necessary part of the equations, the rest can be obtained by the involution ι. The equations YB(R̂) abc = 0 with two different indices are: αijγij(γji + γ ij) = 0 , (257) αij(βijβji + γijγ ij) = 0 = αij(βijβji − γijγji) , (258) αijγij(αij + βji − αj) = 0 = αijγij(αji + βij − αj) , (259) βij(α i − αijαji − αiβij) + (αi − βij)γijγ′ij = 0 , (260) (αi − αj)γ2ij + αijγij(γij + γ′ji) = 0 , (261) αijβijγ ji + (αiβij + γ ijγij)γij = 0 , (262) (αij − αji − βij + βji)γijγ′ji = 0 = (αij − αji − βij + βji)βijβji , (263) ji(αj − αij) + βjiγij(αi − βji) + γij(βijβji + γjiγ′ji) = 0 , (264) (α2i − αi(αji + βji) + βijβji − γijγji)γij = (α2i − αi(αij + βij) + βijβji − γ′ijγ′ji)γ′ji . (265) The equations with three different indices {i, j, k} are: (αij − αki − βij + βki)γijγ′ki = 0 , (266) αij(βijβjk + βikβji − βikβjk) = 0 , (267) αij(γijγjk + γik(βjk − βji)) = αij(γijγ′kj + γik(βkj − βij)) = 0 , (268) (αijαji − αjkαkj)βik + βijβjk(βij − βjk) = 0 , (269) (αi + βik − βji)βjiγik + γikγjiγ′ji + αik(γjkγ′ji + βjkγ′ki) = 0 , (270) (αi + αij − αkj − βkj)γijγik − γ2ikγkj + γij(αikγ′ki − γijγ′kj) = 0 , (271) (αi − βkj)βijγik + (βikβkj + γijγ′ij)γik + αikβijγ′ki − (βij − βik)γijγ′kj = 0 , (272) αij(γijγjk + γik(αjk − αji))=αji(γijγjk + γik(αjk − αji))=αij(γijγ′kj + γik(αkj − αij))=0. (273) Appendix B. Blocks We analyze here the structure of 4×4 blocks of an invertible and skew-invertible rime R-matrix cor- responding to two-dimensional coordinate planes. We denote the matrix elements as in (5). The skew-invertibility of a rime R-matrix imposes restrictions on its entries: in the line R̂i∗j∗ only two entries can be non-zero, R̂ ji and R̂ jj; in the line R̂ ∗i only two entries can be non-zero, R̂ ji and ii . Therefore, αij = 0 ⇒ γijγ′ij 6= 0 and γijγ′ij = 0 ⇒ αij 6= 0 . (274) Dealing with a single block, this becomes especially clear: to skew invert a 4×4 block is the same as to invert the matrix α1 0 γ12 β12 0 0 α12 γ γ′21 α21 0 0 β21 γ21 0 α2 , (275) whose determinant is (α12β12 − γ12γ′12)(α21β21 − γ21γ′21)− α1α2α12α21 . (276) B.1 Solutions Here we classify solutions which are neither ice nor strict rime. For an ice R-matrix, α12 6= 0 and α21 6= 0. For a rime R-matrix, αij might vanish and we consider the cases according to the number of αij ’s which can be zero. 1. Both α12 and α21 do not vanish, α12α21 6= 0. If γ12γ21 6= 0 then by (257), γ′12γ′21 6= 0. This is strict rime. If both γ12 = 0 and γ21 = 0 then eq. (259) implies (αji + βij − αj)γ′ji = 0; eq. (261) implies (αi − αj + αji)γ′ji = 0 and eq. (262) implies βijγ′ji = 0. Combining these, we find γ′ij = 0, this is ice. It is left to analyze the situation when only one of γ’s is different from zero, say γ12 6= 0 and γ21 = 0. We have the following chain of implications: (257) ⇒ γ′12 = 0 , (277) (259) ⇒ β12 = α2 − α21 , β21 = α2 − α12 , (278) (258) ⇒ (α2 − α12)(α2 − α21) = 0 , (279) (260) ⇒ (α1 − α2)(α2 − α21)(α1 + α21) = 0 , (α1 − α2)(α2 − α12)(α1 + α12) = 0 , (280) (261)&(265) ⇒ (α1 − α2 + α12)γ12 + α12γ′21 = 0 , (α1 − α2 + α21)γ′21 + α21γ12 = 0 . (281) Eqs. (262), (263) and (264) are satisfied. By the second line in (281), γ′21 6= 0. Now the system of inequalities and equations is invariant under R̂ ↔ R̂21, so up to this trans- formation we can solve eq. (279) by setting α21 = α2. Then, by (281), γ 21 = −γ12α2/α1, β’s are expressed in terms of α’s by (278) and the remaining system for α’s reduces to a single equation (α1 − α2)(α1 + α12) = 0. We obtain two solutions: 1a. α2 = α1; α1, α12 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set α1α12 = 1 and denote q = α1, γ = γ12: R̂(q;γ) = α1 0 0 0 γ12 0 α12 0 −γ12 α1 α1 − α12 0 0 0 0 α1 q 0 0 0 γ 0 q−1 0 −γ q q − q−1 0 0 0 0 q . (282) The R-matrix (282) is semi-simple if (and only if) q+q−1 6= 0 and it is then an R-matrix of GL(2)-type, Spec(R̂) = {q, q, q,−q−1}. This solution is a specialization of (15)-(16). 1b. α12 = −α1; α1, α2 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set α1α2 = −1 and denote q = α1, γ = γ12/q: R̂(q;γ) = α1 0 0 0 γ12 0 −α1 0 −γ12α2/α1 α2 α1 + α2 0 0 0 0 α2 q 0 0 0 qγ 0 −q 0 q−1γ −q−1 q − q−1 0 0 0 0 −q−1 . (283) The R-matrix (283) is semi-simple if (and only if) q+ q−1 6= 0 and it is then an R-matrix of GL(1|1)- type, Spec(R̂) = {q, q,−q−1,−q−1}. 2. Assume that α12 = 0. By the invertibility, β12β21 6= 0; by the skew-invertibility, γ12γ′12 6= 0; now eqs. (257) and (258) imply β12β21 = γ12γ21, γ 12 = −γ21 and γ′21 = −γ12. Eq. (259) implies α2 = α1, β12 = α1 − α21 and β21 = α1. The rest is satisfied and we obtain a solution, in which α1, β12 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set α1β12 = −1 and denote q = α1, γ = γ12: R̂(q;γ) = α1 0 0 0 γ12 β12 0 −α1β12/γ12 −γ12 α1 − β12 α1 α1β12/γ12 0 0 0 α1 q 0 0 0 γ −q−1 0 1/γ −γ q + q−1 q −1/γ 0 0 0 q . (284) The R-matrix (284) is semi-simple if (and only if) q+q−1 6= 0 and it is then an R-matrix of GL(2)-type, Spec(R̂) = {q, q, q,−q−1}. This solution is a specialization of (15)-(16). 3. Finally, assume that α12 = α21 = 0. By the invertibility, β12β21 6= 0; by the skew-invertibility, γ12γ′12γ21γ′21 6= 0; now eq. (261) implies α2 = α1, eq. (263) implies β21 = β12; eq. (262) implies γ12γ 12 = γ21γ 21 = −α1β12; eq. (265) implies that γ12γ21 can take three values: α 12 or (−α1β12). The rest is satisfied and we obtain a solution, in which α1, β12 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set α1β12 = −1 and denote q = α1, γ = γ12: R̂(q,ω;γ) = α1 0 0 0 γ12 β12 0 −α1β12/γ12 −α1β12γ12/ω 0 β12 ω/γ12 0 0 0 α1 q 0 0 0 γ −q−1 0 1/γ γ/ω 0 −q−1 ω/γ 0 0 0 q , (285) where ω = q2, 1, q−2. The R-matrix (285) is semi-simple if (and only if) q + q−1 6= 0 and it is then an R-matrix of GL(1|1)-type, Spec(R̂) = {q, q,−q−1,−q−1}. It follows from the analysis above that if γij 6= 0 in an invertible and skew-invertible rime R-matrix then γ′ji 6= 0. In each of the cases (282)-(285), the parameter γ 6= 0 can be set to an arbitrary (non-zero) value by a diagonal change of basis. The R-matrices (282)-(285) are skew-invertible. B.2 GL(2) and GL(1|1) R-matrices 1. In dimension 2, except the standard R-matrices of GL-type, GL(2) (q,p) q 0 0 0 0 0 p 0 0 p−1 q − q−1 0 0 0 0 q GL(1|1) (q,p) q 0 0 0 0 0 p 0 0 p−1 q − q−1 0 0 0 0 −q−1 , (286) there are two non-standard one-parametric families of non-unitary R-matrices of the type GL(1|1): the eight-vertex one, R̂I(q) = q − q−1 + 2 0 0 q − q−1 0 q − q−1 q + q−1 0 0 q + q−1 q − q−1 0 q − q−1 0 0 q − q−1 − 2 , (287) and the matrix R̂(II) for which the matrix R = PR̂ can be given an upper-triangular form, R̂II(q,ε) = q 0 0 q + q−1 0 0 εq−1 0 0 εq q − q−1 0 0 0 0 −q−1 , (288) where ε = ±1. The R-matrices (286), (287) and (288) are semi-simple if (and only if) q + q−1 6= 0. Up to the transformations R̂↔ R̂21 and R̂↔ R̂t (the transposition), basis changes and rescalings R̂ 7→ c R̂ (where c is a constant), the complete list of semi-simple invertible and skew-invertible R-matrices includes (see [16] for a description of all solutions of the Yang–Baxter equation in two dimensions and [11] for the classification of GL(2)-type R-matrices), in addition to (286)-(288), the one-parametric family of Jordanian solutions R̂ (h1:h2) (h1:h2) 1 h1 −h1 h1h2 0 0 1 −h2 0 1 0 h2 0 0 0 1 (289) (the Jordanian R-matrix is of GL(2)-type; it is unitary; the essential parameter is the projective vector (h1 : h2)), as well as the permutation-like solution R̂ (a,b,c) and one more solution R̂ (a,b,c) 1 0 0 0 0 0 a 0 0 b 0 0 0 0 0 c 0 0 0 a 0 1 0 0 0 0 1 0 a 0 0 0 . (290) The R-matrix R̂ (a,b,c) is Hecke when ab = 1 and c = ±1 and it is then standard (and unitary). The R-matrix R̂ is Hecke when a2 = 1; it is then unitary and related to the standard R-matrix by a change of basis with the matrix Without the demand of semi-simplicity, the full list of invertible and skew-invertible R-matrices contains two more solutions, (h1:h2: 1 h1 h2 h3 0 0 1 h1 0 1 0 h2 0 0 0 1 , R̂( ′′′) = 1 0 0 1 0 0 −1 0 0 −1 0 0 0 0 0 1 . (291) The essential parameter for the R-matrix R̂ (h1:h2: is the projective vector (h1 : h2 : h3). The R-matrix R̂ (h1:h2: is semi-simple if and only if h2 = −h1 and h3 = −h21; it then belongs to the family (289) of Jordanian R-matrices. 2. For the R-matrices from the list above, the transformations R̂ ↔ R̂21, R̂ ↔ R̂t and R̂ ↔ R̂−1 partly overlap or reduce to parameter or basis changes. We shall write formulas for the Hecke R- matrices only. For the standard R-matrix R̂(q,p) := R̂ GL(2) (q,p) R̂t(q,p) = R̂(q,p−1) , (R̂(q,p))21 = (π ⊗ π)R̂(q,p)(π −1 ⊗ π−1) , R̂−1 (q,p) = (R̂(q−1,p−1))21 , (292) where π = For the standard R-matrix R̂(q,p) := R̂ GL(1|1) (q,p) R̂t(q,p) = R̂(q,p−1) , (R̂(q,p))21 = (π ⊗ π)R̂(−q−1,p)(π −1 ⊗ π−1) , R̂−1 (q,p) = (R̂(q−1,p−1))21 . (293) For the non-standard GL(1|1) R-matrix R̂(q) := R̂I(q), R̂t(q) = R̂(q) , (R̂(q))21 = R̂(q) , R̂ = (D ⊗D)R̂(q−1)(D ⊗D)−1 , (294) where D = For the non-standard GL(1|1) R-matrix R̂(q,ε) := R̂II(q,ε), R̂t(q,ε) = (π̃ ⊗ π̃)(R̂(−q−1,−ε))21(π̃ −1 ⊗ π̃−1) , R̂−1 (q,ε) = (R̂(q−1,ε))21 , (295) where π̃ = For the Jordanian R-matrix R̂(h1:h2) := R̂ (h1:h2) R̂t(h1:h2) = (π ⊗ π)R̂(h2:h1)(π −1 ⊗ π−1) , (R̂(h1:h2))21 = R̂(−h1:−h2) , R̂ (h1:h2) = R̂(h1:h2) . (296) B.3 Riming We shall now identify the rime R-matrices (282)-(285). 1. GL(2) The R-matrices (282) and (284) are related by a change of basis (the number in brackets refers to the corresponding equation), (282) (q;γ) T ⊗ T = T ⊗ T R̂(284) (q;γ) , T = q −1/γ . (297) In turn, the R-matrix (282) is related to the standard R-matrix R̂ GL(2) (q,q−1) by a change of basis, (282) (q;γ) T ⊗ T = T ⊗ T R̂GL(2) (q,q−1) , T = q − q−1 0 . (298) In the unitary situation (that is, q − q−1 = 0), the R-matrix R̂(282) (q;γ) belongs to the family of Jordanian R-matrices. Note that for the R-matrices (282) and (284), the left even quantum spaces are classical. 2. GL(1|1) The R-matrix (283) is related to the R-matrix (285) with the choice ω = β212, (283) (q;γ) T ⊗ T = T ⊗ T R̂(285) (−q−1,q2;1) , T = . (299) We have (285) (q,1;γ) T ⊗ T = T ⊗ T R̂I , T = γ −γτ , where τ2 = q − 1 q + 1 , (300) (285) (q,q2;γ) T ⊗ T = T ⊗ T R̂II (q,1) , T = γq−1 −γq−1 , (301) (285) (q,q−2;γ) T ⊗ T = T ⊗ T (R̂II (q,1) )21 , T = γq −γq . (302) In the unitary situation (that is, for q = ±1) only eq. (300) changes; but now different choices for ω coincide. 3. Since the standard R-matrices are rime as well, we conclude that in dimension 2, all non-unitary Hecke R-matrices fit into the rime Ansatz. When h1 = 0, the Jordanian R-matrix R̂ (0:h2) is rime as well. However, when h1 6= 0, the Jordanian R-matrix R̂ (h1:h2) cannot be rimed. Indeed, assume that h1 6= 0 and let A = (T ⊗ T )R̂ (h1:h2) (T ⊗ T )−1 with some invertible matrix T . Then (Det(T ))2A1112 = h1 (T 2 (Det(T )− h2 T 11 T 21 ) , (Det(T ))2A1121 = −h1 (T 11 )2 (Det(T ) + h2 T 11 T 21 ) , (Det(T ))2A2212 = h1 (T 2 (Det(T )− h2 T 11 T 21 ) , (Det(T ))2A1121 = −h1 (T 21 )2 (Det(T ) + h2 T 11 T 21 ) . (303) For an invertible T , the non-rime entries (303) of A cannot vanish simultaneously. 4. We remark also that all non-standard R-matrices of GL(1|1)-type are uniformly described by the formula (285). The right quantum spaces for the R-matrix R̂ (285) (q,ω;γ) , with γ = 1, read (R̂− q11⊗ 11)ijkl x kxl = 0 : (q + q−1)xy = x2 + y2 , (q + q−1)xy = ω−1x2 + ωy2 ; (304) (R̂ + q−111⊗ 11)ij xkxl = 0 : x2 = 0 , y2 = 0 . (305) Using the diamond lemma, it is straightforward to verify that the Poincaré series of the quantum space (304) is of GL(1|1)-type if and only if ω = q−2, 1 or q2. Appendix C. Rimeless triple We sketch here a proof that the triple (94) cannot be rimed. Relations xiyj = R̂ kxl, where R̂ is the R-matrix for the triple (94) with arbitrary multiparameters, read xiyi = yixi , i = 1, 2, 3, 4 (306) x1y2 = y2x1 , x1y3 = y3x1 , x1y4 = y4x1 − rs y3x2 , x2y1 = y1x2 + (1− q−2)y2x1 , x2y3 = y3x2 , x2y4 = y4x2 , (307) x3y1 = y1x3 + (1− q−2)y3x1 , x3y2 = y2x3 + (1− q−2)y3x2 , x3y4 = y4x3 , x4y1 = y1x4 + (1− q−2)y4x1 + 1 y2x3 , x4y2 = y2x4 + (1− q−2)y4x2 , x4y3 = y3x4 + (1− q−2)y4x3 . (308) The parameter q enters the characteristic equation for R̂, R̂2 = (1− q−2)R̂+ q−211⊗ 11; p, r and s are the multiparameters. The only needed restriction is q2 6= 1. Denote by 〈l(1), l(2)〉 a two-dimensional plane spanned by l(1) and l(2). We say that two linear forms l(1) and l(2) (in four variables) form a rime pair if, for the ordering relations (306) and (307)-(308), each product l(α)(x)l(β)(y), α = 1, 2, β = 1, 2, is a linear combination of l(1)(y)l(1)(x), l(1)(y)l(2)(x), l(2)(y)l(1)(x) and l(2)(y)l(2)(x). If, in addition, l(α)(x)l(α)(y) is proportional to l(α)(y)l(α)(x) for α = 1 and 2, we say that l(1) and l(2) form a rime basis in the plane 〈l(1), l(2)〉. We call a plane rime if it admits a rime basis. Fork Lemma. Assume that l(1)(x) = x1 + a2x 2 + a3x 3 and l(4)(x) = b2x 2 + b3x 3 + x4 form a rime pair for some a2, a3, b2 and b3. Then either a3b2 6= 0 and a2 = b3 = 0 or a2b3 6= 0 and a3 = b2 = 0. If a3b2 6= 0 then r = s = 1 , l(1)(x) = x1 + wx3 and l(4)(x) = x4 + q − q−1 x2 , w 6= 0 is arbitrary . (309) If a2b3 6= 0 then , r = s , l(1)(x) = x1 + wx2 and l(4)(x) = x4 + q − q−1 x3 , w 6= 0 is arbitrary . (310) Moreover, if r = s = 1 and p 6= q−1 then the rime plane 〈l(1), l(4)〉 admits a unique, up to rescalings, rime basis {l(1), l(4)}; if p = q−1 and r = s 6= 1 then the rime plane 〈l(1), l(4)〉 admits a unique, up to rescalings, rime basis {l(1), l(4)}; if p = q−1 and r = s = 1 then any two independent linear combinations of l(1) and l(4) form a rime basis in the plane 〈l(1), l(4)〉. Proof. A straightforward calculation. � Assume that a rime basis {x̃i} for the triple (94) exists, x̃i = Aijxj , the matrix Aij is invertible. Rename the rime variables x̃i in such a way that the minor A11 A A41 A is non-zero and A11A 4 6= 0; normalize the variables x̃1 and x̃4 to have A11 = A 4 = 1. The plane 〈x̃1, x̃4〉 is, by definition, rime, with a rime basis {x̃1, x̃4}. Suppose that r = s = 1 and p 6= q−1 or p = q−1 and r = s 6= 1. Then, by Fork Lemma, the rime basis in the plane 〈x̃1, x̃4〉 is, up to proportionality, unique, so we know the variables x̃1 and x̃4. The variables x̃1 and x̃2 form a rime plane. Therefore, if the variable x̃2 contains x4 with a non-zero coefficient then, by Fork Lemma, x̃2 must be proportional to x̃4, contradicting to the linear independence of the variables x̃2 and x̃4. Similarly, the variable x̃2 cannot contain x1 with a non-zero coefficient (the plane 〈x̃2, x̃4〉 is rime). Thus, x̃2 is a linear combination of x2 and x3. Same for x̃3: it is a linear combination of x2 and x3. One of the variables, x̃2 or x̃3, say, x̃2, contains x2 with a non-zero coefficient. Writing rime equations for the plane 〈x̃1, x̃2〉 in the case r = s = 1 and p 6= q−1 (for the plane 〈x̃2, x̃4〉 in the case p = q−1 and r = s 6= 1) quickly leads to a contradiction. Therefore, if the relations (306) and (307)-(307) can be rimed then p = q−1 and r = s = 1. It follows from Fork Lemma that x̃4 = (q − q−1)c2c3x1 + c2x2 + c3x3 + x4 for some c2 and c3. The planes 〈x̃a, x̃4〉, a = 1, 2, 3, are rime. Subtracting from the variables x̃a the variable x̃4 with appropriate coefficients, we find three linearly independent combinations l(x) = d1x 1 + d2x 2 + d3x 3 , (311) each forming a rime pair with x̃4. We must have: l(x)l(y) is a linear combination of l(y)l(x), l(y)x̃4, ỹ4l(x) and ỹ4x̃4. It follows, after a straightforward calculation, that d2d3 = 0. Moreover, d2 = d3 = 0 is excluded by Fork Lemma. In the case d2 6= 0 and d3 = 0 (respectively, d3 6= 0 and d2 = 0), the rime condition implies that d1 = (q − q−1)c2d3 (respectively, d1 = (q − q−1)c3d2). Thus, only two linearly independent combinations (311) can form a rime pair with x̃4, the final contradiction. References [1] M. Aguiar, Infinitesimal Hopf algebras; Contemp. Math. 267 (2000) 1–30. [2] G. E. 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Demazure, Une nouvelle formule des caractères; Bull. Sci. Math. (2) 98 no. 3 (1974), 163–172. [8] R. Endelman and T. Hodges, Generalized Jordanian R-matrix of Cremmer–Gervais type; Lett. Math. Phys. 52 (2000), 225–237. ArXiv: math.QA/0003066. [9] P. Etingof, T. Schedler and O. Schiffmann, Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras; J. Amer. Math. Soc. 13 no. 3 (2000), 595–609. ArXiv: math.QA/9912009. [10] H. Ewen and O. Ogievetsky, Classification of the GL(3) Quantum Matrix Groups. ArXiv: q-alg/9412009. [11] H. Ewen, O. Ogievetsky and J. Wess, Quantum Matrices in two Dimensions; Lett. Math. Phys. 22 (1991), 297–305. [12] L. Fehér and B. G. Pusztai, On the classical R-matrix of the degenerate Calogero-Moser models; Czech. J. Phys. 50 (2000), 59–64. ArXiv: math-ph/9912021. [13] L. Fehér and B. G. Pusztai, The non-dynamical r-matrices of the degenerate Calogero-Moser models; J. Phys. A33 (2000), 7739–7759. ArXiv: math-ph/0005021. [14] M. 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Mudrov, Associative triples and Yang–Baxter equation; Israel J. Math. 139 (2004), 11–28. ArXiv: math.QA/0003050 http://arxiv.org/abs/math/0003066 http://arxiv.org/abs/math/9912009 http://arxiv.org/abs/q-alg/9412009 http://arxiv.org/abs/math-ph/9912021 http://arxiv.org/abs/math-ph/0005021 http://arxiv.org/abs/q-alg/9609014 http://arxiv.org/abs/q-alg/9712036 http://arxiv.org/abs/math/0010190 http://arxiv.org/abs/math/0003050 [21] O. Ogievetsky, Uses of quantum spaces; in: Quantum symmetries in theoretical physics and mathematics, Contemp. Math. 294, Amer. Math. Soc., Providence, RI (2002), 161–232. [22] G. C. Rota, Baxter operators, an introduction; in: Gian-Carlo Rota on combinatorics, Contemp. Mathematicians, Birkhäuser Boston, Boston, MA (1995), 504–512. [23] M. A. Semenov-Tyan-Shanskii, What is a classical r-matrix; Funktsional. Analiz i Prilozhen. 17 no. 4 (1983), 17–33. [24] A. Stolin, On the rational solutions of the classical Yang–Baxter equation; Ph. D. Thesis, Stock- holm 1991. From ice to rime Rime Yang–Baxter solutions.25cm Non-unitary rime R-matrices Unitary rime R-matrices Properties Rime and Cremmer–Gervais R-matrices Classical rime r-matrices.25cm Non-skew-symmetric case BD triples. Skew-symmetric case Bézout operators.25cm Non-homogeneous associative classical Yang–Baxter equation Linear quantization Algebraic meaning Rota–Baxter operators *-multiplication Rime Poisson brackets.25cm Rime pencil Invariance Normal form Orderable quadratic rime associative algebras.1cm Appendix A. Equations Appendix B. Blocks.25cm B.1 Solutions B.2 GL(2) and GL(1|1) R-matrices B.3 Riming Appendix C. Rimeless triple References

0704.1948

Quadratic centers defining elliptic surfaces

Quadratic centers defining Elliptic Surfaces Sébastien GAUTIER August 26, 2021 Abstract Let X be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one. To each X we associate an elliptic surface (a smooth complex compact surface which is a genus one fibration). We give the list of all such vector fields and determine the corresponding elliptic surfaces. 1 Introduction The second part of the 16th Hilbert problem asks for an upper bound to the number of limit cycles of a plane polynomial vector field of degree less or equal to n. Even in the case of quadratic systems (n = 2) the problem remains open. An infinitesimal version of the 16th Hilbert problem can be formulated as follows: Find an upper bound Z(f, n) to the number of limit cycles of a polynomial vector field of degree n, close to a polynomial vector field with a first integral f . The associated foliation on the plane is defined by R−1df + εω = 0 (1) where R−1df = Pdx + Qdy is a given polynomial one-form, degP, degQ ≤ n, R−1 is an integrating factor, and ω is a polynomial one-form of degree n with coefficients depending analytically on the small parameter ε. A progress in solving the infinitesimal 16th Hilbert problem is achieved mainly in the case when f is a polynomial of degree three, or F = y2 + P (x) where P is a polynomial of degree four see [I02, P90, G01]. A key point is that the generic leaves Γc = {f = c} ⊂ C2 of the polynomial foliation R−1df = 0 are elliptic curves. We expect that the perturbations of more general polynomial foliations with elliptic leaves (which we call ”elliptic foliations”) can be studied along the same lines. This leads naturally to the following (open) problem. For a given n > 1 determine, up to an affine equivalence, the elliptic polynomial foliations Pdx+ Qdy = 0, degP, degQ ≤ n. The present paper adresses the above problem in the quadratic case, n = 2. In view of applications to the 16th Hilbert problem most important is the case when the non-perturbed foliation is real and possesses a center. Such foliations are well-known since Dulac (1908) and Kapteyn (1912). Moreover, when the leaves of the foliation (the orbits of the quadratic vector field) are algebraic curves, there is a (rational) first integral f [J79]. Reminding the classification of quadratic vector fields with a center, a rational first integral of the foliation induced is thus of four different kinds: f = P3(x, y) with P3 ∈ R3[x, y] (Hamiltonian case) (2) f = xλ(y2 + P2(x)) with λ ∈ Q P2 ∈ R2[x, y] (reversible case) (3) f = xλyµ(ax+ by + c) with λ, µ ∈ Q P2 ∈ R2[x, y] a,b,c real numbers (Lotka-Volterra case) (4) http://arxiv.org/abs/0704.1948v2 f = P2(x, y) −3P3(x, y) 2 with P3 ∈ R3[x, y] P2 ∈ R2[x, y] (codimension 4 case) (5) In section 2 we give the classification, up to an affine equivalence, of all elliptic foliations with a first integral of the form (3) or (4). The Hamiltonian cases obviously induce an elliptic foliation and have already been studied. Remarks concerning the codimension 4 case can be found in [GGI, G07]. In our classification, the base field is supposed to be C, so all parameters a, b, c, λ, ... are complex. We get a finite list of such foliations with a center as well several infinite series of degenerated foliations which can not have a center (when the base field is R). Most of these elliptic foliations were not previously studied in the context of the 16th Hilbert problem (but see [CLLL06, YL02, ILY05]). The section 3 deals with the topology of the singular surface induced. An elliptic foliation in C2 (or more generally a foliation with an algebraic first integral f) gives rise canonically to an elliptic surface as follows. Suppose that f is chosen in such a way that the generic fiber of the map f : C2 → C is an irreducible algebraic curve. The induced rational map f : P2 99K P1 have a finite number of points of indetermination. After a finite number of blow-ups of P2 at these points we get (by the Hironaka’s desingularisation theorem ) an induced analytic map 2 ⊂ K f→ P1 where K is a smooth complex surface. We may further suppose that K is minimal in the sense that the fibers do not contain exceptional curves of first kind. The pair (K, f) is then the elliptic surface associated to the elliptic foliation R−1df = 0. It is unique up to a fiber preserving isomorphism. In this last section, we compute the singular fibers of the elliptic surfaces obtained. The singular fibers of an elliptic surface are classified by Kodaira [Ko63]. Such computations are 2-folds. First of all, it permits to identifiate isomorphic elliptic surfaces of non affine equivalent foliations, wich on its own is of interest. But the most important is that it immediately gives the local monodromy of the singular fibers. Here, the number of singular fibers (except in the Hamiltonian case) do not exceed 4 so the local monodromy of the singular fibers gives a good description of the (global) monodromy group of the associated Picard-Fuchs equation (or equivalently, the homological invariant of the surface [Ko63]), which on its turn is necessary when studying zeros of Abelian integrals (or limit cycles of the perturbed foliation (1)), see [G01, P90, I02, GGI, G07] for details. 2 Quadratic centers which define elliptic foliations Let F = F(ω) be a foliation on the plane C2 defined by a differential form ω = Pdx +Qdy. We say that F(df) is elliptic provided that its generic leaves are elliptic curves. As stated in the Introduction, in the present paper we suppose that F has, eventually after an affine change of the variables in C2, a first integral of the form (3) or (4). Such a foliation will be called reversible (having an integral of the form (3) but not (4)), of Lotka-Voltera type (having an integral of the form (4) but not (3)), or of reversible Lotka-Voltera type. 2.1 The reversible case An elliptic foliation of Lotka-Voltera type has three invariant lines. From this we deduce that a re- versible Lotka-Voltera foliation has always a first integral f = xλ(y2+P2(x)) where P2 is a polynomial of degree at most two, and the bi-variate polynomial y2+P2(x) is irreducible. In this section we prove the following: Theorem 1 The reversible foliation F(df) is elliptic if and only if, after an affine change of the variables, it has a first integral of the form: (rv1) f = x−3(y2 + ax2 + bx+ c) (rv2) f = x(y2 + cx2 + bx+ a) (rv3) f = x−3/2(y2 + ax2 + bx+ c) (rv4) f = x−1/2(y2 + cx2 + bx+ a) (rv5) f = x−4(y2 + ax2 + bx+ c) (rv6) f = x2(y2 + cx2 + bx+ a) (rv7) f = x−4/3(y2 + bx+ c) (rv8) f = x−2/3(y2 + cx2 + bx) (rv9) f = x−4/3(y2 + ax2 + bx) (rv10) f = x−2/3(y2 + bx+ a) (rv11) f = x−5/3(y2 + ax2 + bx) (rv12) f = x−1/3(y2 + bx+ a) (rv13) f = x−5/4(y2 + ax2 + bx) (rv14) f = x−3/4(y2 + bx+ a) (rv15) f = x−7/4(y2 + ax2 + bx) (rv16) f = x−1/4(y2 + bx+ a) (rv17) f = x−5/2(y2 + ax2 + bx) (rv18) f = x1/2(y2 + bx+ a). (i)f = x−1+ k (y2 + x), k ∈ Z∗ \ 2Z, (ii)f = x−1+ k (y2 + x), k ∈ Z∗ \ 3Z. Remark 1 We shall assume moreover that c 6= 0 for (rv3), (rv4). Proof. Let Γt be the set of (x, y) ∈ C2 such that for some determination of the multi-valued function xλ holds f(x, y) = t. If the connected components of Γt for all t are algebraic curves, then λ ∈ Q and we put λ = p (p ∈ Z, q ∈ N∗ and gcd(p, q) = 1) with P2(x) = ax2 + bx+ c ∈ C2[x]. As the foliation is reversible we may suppose that y2 + ax2 + bx + c is irreducible, or simply b2−4ac 6= 0. We shall suppose first that a 6= 0, c 6= 0, that is to say the quadric {y2+ax2+bx+c = 0} is not tangent to the line at infinity in P2 and to the line {x = 0}. 2.1.1 The case a 6= 0, c 6= 0, b2 − 4ac 6= 0. After a scaling of t and an affine transformation we may suppose that f = xλ(y2 + x2 + bx+ c). (6) By abuse of notation we put Γt = {xp/q(y2 + x2 + bx+ c) = t} and in a similar way we define Γ̃t = {Xp(Y 2 +X2q + bXq + c) = t}. (7) Lemma 1 The map ϕ : C2 → C2 : (X,Y ) → (x, y) = (Xq, Y ) induces an isomorphism of Γt and Γ̃t. Indeed, it is straightforward to check that ϕ : Γ̃t → Γt is a bijection and therefore is a bi-holomorphic To compute the genus of Γ̃t or Γt we distinguish two cases: 1. The case when p < 0. We obtain the hyper-elliptic curve {y2 = −x2q − bxq + tx−p − c}. The roots of the polyno- mial −x2q − bxq+ tx−p− c are different and non zeros since t is generic. Consequently, its genus is one if and only if the degree of the polynomial is 3 or 4 and thus we get: (a) f = x−3(y2 + x2 + bx+ c) (b) f = x−4(y2 + x2 + bx+ c) (c) f = x− 2 (y2 + x2 + bx+ c) (d) f = x− 2 (y2 + x2 + bx+ c) 2. Suppose now p ≥ 0. We easily have: y2xp = t − x2q+p − bxq+p − cxp. Thus after a birational transformation, we obtain: y2 = xp(t− x2q+p − bxq+p − cxp). Since t is generic, all the roots of t− x2q+p − bxq+p − cxp are different and do not vanish. • If p is even we have: )2 = t− x2q+p − bxq+p − cxp. Consequently, it is elliptic when 2q+ p either equal 3 or 4. This gives the following curves: (a) x(y2 + x2 + bx+ c) = t (b) x2(y2 + x2 + bx+ c) = t. • If p is odd, then we have: )2 = x(t− x2q+p − bxq+p − cxp). Since all the roots of t−x2q+p−bxq+p−cxp are different and non zeros, the curve is elliptic if and only 2q + p either equals 2 or 3, which gives the solution (b) above. This we have obtained the cases (rv1)-(rv6) in Theorem 1. 2.1.2 The case a 6= 0, c = 0, b2 − 4ac 6= 0. This means that the quadric {y2+ax2+bx = 0} is tangent to the line {x = 0} and is transversal to the line at infinity, see Figure 7. After an affine transformation and scaling of t we get P2(x) = ax 2 + bx with a 6= 0. Therefore we need to compute the genus of {xp(y2 + x2q + bxq) = t} for generic t. If p ≥ 0 the same computations as case a 6= 0, c 6= 0 give the same solutions of the problem. Let p < 0 and suppose that p = 2a is even. We have (xay)2 = −x2q+p − bxq+p + t, so if −p ≤ q, it has genus one if and only 2q + p = 3 or 4, hence q ≤ 4 and (p, q) = (−2, 3). If −p ≥ 2q the curve above is birational to the curve y2 = −x−p−q − bx−p−2q + t and so it has genus one if −p− q = 3 or 4 which leads to (p, q) = (−4, 1). If q < −p < 2q, because p is even and q is odd, it is equivalent to calculate the genus of {y2 = x(tx−(q+p) − xq − b)}. Therefore (p, q) = (−4, 3). Now suppose that p = 2a + 1 is odd. The curve is birationally equivalent to {y2 = −x(x2q+p + bxq+p − t)}. As above we get: (p, q) = (−1, 2), (−3, 1),(−5, 2), (−5, 3), (−5, 4), (−7, 4). To resume, we proved Proposition 1 The foliation F(xλ(y2 + x2 + bx)) is elliptic if and only it has under affine transfor- mation a first integral of the kind : f = x(y2 + x2 + bx) f = x2(y2 + x2 + bx) f = x−3(y2 + x2 + bx) f = x−4(y2 + x2 + bx) f = x− 2 (y2 + x2 + bx) f = x− 2 (y2 + x2 + bx) f = x− 3 (y2 + x2 + bx) f = x− 3 (y2 + x2 + bx) f = x− 4 (y2 + x2 + bx) f = x− 4 (y2 + x2 + bx) f = x− 3 (y2 + x2 + bx) Here we get (rv1)− (rv6) except (rv3) according to Remark 1, (rv8) and the end of the left column of Theorem 1. 2.1.3 a = 0, c 6= 0, b2 − 4ac 6= 0. This means that the quadric {y2 + bx+ c = 0} is tangent to the line at infinity and is transversal to the line {x = 0}. The birational change of variables x → 1/x, y → y/x shows that this is equivalent to the case a 6= 0, c = 0 and we get: Proposition 2 The foliation F(xλ(y2 + x+ c)) is elliptic if and only it has under affine transforma- tion a first integral of the kind : f = x(y2 + x+ c) f = x2(y2 + x+ c) f = x−3(y2 + x+ c) f = x−4(y2 + x+ c) f = x− 2 (y2 + x+ c) f = x 2 (y2 + x+ c) f = x− 3 (y2 + x+ c) f = x− 3 (y2 + x+ c) f = x− 4 (y2 + x+ c) f = x− 4 (y2 + x+ c) f = x− 3 (y2 + x+ c) Here we get (rv1)− (rv6) except (rv4) according to Remark 1, (rv7) and the end of the right column of Theorem 1. 2.1.4 a = c = 0, b 6= 0 This means that the quadric {y2 + ax2 + bx+ c = 0} is tangent to the line at infinity and to the line {x = 0}. Up to affine change of re-scalings we may suppose f = xp/q(y2 + x). If p is even the curve Γ̃t is birational to y 2 = −xq+p + t. If p is odd the curve Γ̃t is birational to y 2 = −x(xq+p − t). For p ≥ 0 this curve is elliptic if and only: (p, q) = (1, 1), (2, 1) and (1, 2). Now, if p ≤ 0 and q + p ≥ 0 the conditions are q + p = 2 , 3 or 4 with p even. The case q + p ≤ 0 gives similarly−q − p = 2 , 3 or 4 with p even. Notice that we must have q prime with the integers 2 or 3 or 4 when considering all cases. This gives the following: Proposition 3 The foliation F(xλ(y2 + bx)) with b 6= 0 is elliptic if and only if it has a first integral of the kind : f = x−1+ k (y2 + x), k ∈ Z∗ \ 2Z f = x−1+ 3k (y2 + x), k ∈ Z∗ \ 3Z Finally, Theorem 1 is proved. 2.2 The Lotka-Volterra case Theorem 2 The Lotka-Volterra foliation F(df) is elliptic if and only if, after an affine change of the variables, it has a first integral of the form: (lv1) f = x2y3(1− x− y) (lv2) f = x−6y2(1− x− y) (lv3) f = x−6y3(1 − x− y) (lv4) f = x−4y2(1− x− y) (lv5) f = x−6y3(1− x− y)2 (iii)f = x k (1 + y), (iv)f = x−1+ k (x+ y) with k ∈ Z∗ \ 3Z and l − k ∈ 3Z, (v)f = x k (1 + y), (vi)f = x−1+ k (x+ y) with k ∈ Z∗ \ 2Z and l − k ∈ 4Z (vii)f = x k (1 + y), (viii)f = x−1+ k (x+ y) with k ∈ Z∗, l ∈ 2Z and kl − 2 ∈ 6Z (ix)f = x k (1 + y) (x)f = x−1+ k (1 + y) with k, l ∈ Z∗ \ 2Z (xi)f = x k (1 + y) (xii)f = x−1+ k (1 + y) with k ∈ Z∗ \ 3Z and l ∈ Z∗ \ 2Z and moreover gcd(k, l) = 1. Proof. An algebraic first integral is given by f = xp1yp2(ax+ by + c)r, p1, p2 ∈ Z, r ∈ N∗. This defines a divisor in P2: D = p1L1 + p2L2 + rL3 where Li, i = 1..3 are projective lines. As in previous section, the study below will depend on the geometry of the reduced divisor D̃ (i.e without multiplicities) associated to D. First we will consider the generic case, that is the projective lines Li, i = 1..3 have normal crossings toward each other. 2.2.1 The case a 6= 0, b 6= 0, c 6= 0. First we may suppose under affine transformation b = c = −a = 1. The expression of D̃ invites us to divide the study in 4: p1 > 0, p2 > 0 (8) p1 < 0, p2 > 0, p1 + p2 + q > 0 (9) p1 < 0, p2 > 0, p1 + p2 + q < 0 (10) p1 < 0, p2 > 0, p1 + p2 + q = 0. (11) In the shape of (10) the generic leaf is birational to the algebraic curve Xp1Y p2 = t which is rational. Hence the generic case will be an obvious consequence of the 3 following propositions: Proposition 4 . The algebraic curve xpyq(1 − x− y)r = 1 with 0 ≤ p ≤ q ≤ r, gcd(p, q, r) = 1 is of genus one if and only if (p, q, r) = (1, 1, 1) or (1, 1, 2) or (1, 2, 3). Proposition 5 The algebraic curve yq(1−x−y)r = xp with p, q, r > 0 , −p+q+r < 0 gcd(p, q, r) = 1 is of genus one if and only if (p, q, r) = (1, 2, 2) or (3, 2, 2). Proposition 6 The algebraic curve yq(1 − x − y)r = xp with p > 0, q > r > 0 , −p + q + r > 0 gcd(p, q, r) = 1 is elliptic if and only if (p, q, r) = (3, 1, 1), (4, 1, 1), (4, 2, 1), (6, 2, 1), (6, 3, 1) or (6, 3, 2). Proof of Proposition 4. Let ω be a one form on a compact Riemann surface S. We write ω = i aiPi with Pi points of S. This sum is finite and we define the degree of ω : deg(ω) = i ai. According to the Poincaré-Hopf formula (see [GH78]), any 1-form ω on S satisfies: deg(ω) = 2g − 2. (12) Now, we use this formula with the riemann surface C̃ obtained after desingularisation of the irre- ducible algebraic curve C defined by the equation xpyq(1 − x − y)r = 1. Let π : C̃ → C be such a desingularisation map. We compute below the degree of the one-form π∗ω where (by abuse of notation): ω = − x[q − qx− (q + r)y] = y[p− py − (p+ r)x] . The 1-form above has been chosen such that it has nor zeros nor poles outside the singular locus of C. Yet, C is only singular in the three singular points meeting the line at infinity: [1 : 0 : 0], [0 : 1 : 0], [1 : −1 : 0]. First we investigate the local behavior of ω near [0 : 1 : 0]. We get local coordinates near [1 : 0 : 0] as follows: Write x = 1 with u → 0. After this change of coordinates, the equation becomes: yq(u− 1− yu)r = up+r. Since u → 0, we have the m = gcd(q, p + r) different parametrisations of the m local branches near this point: u = t y = −e 2ikπm t m (1 + o(t m )). k = 0...m− 1. For each branch, locally, ω = − q −1(1 + o(t −1))dt. Finally, for π∗ω we get after a finite number of blowing-ups m points where our 1-form has a zero of order The study is completely similar for the remaining singular points: near [0 : 1 : 0] we obtain n = gcd(p, q + r) points where the 1-form π∗ω has a zero of order p − 1 and near [1 : −1 : 0], we have l = gcd(r, p+ q) points where π∗ω has a zero of order r Finally, the numbers involved satisfy the following relation: p+ q + r −m− n− l = 2g − 2 (13) and consequently, this curve is elliptic when: p+ q + r = m+ n+ l. (14) Now we have to resolve this diophantine equation: We always have: m ≤ q, n ≤ p and l ≤ r. Hence (14) is true if and only if: gcd(q, r + p) = q; gcd(p, q + r) = p; gcd(r, p+ q) = r. Let α, β, γ ∈ N∗ such that: r + p = qα; (a) r + q = pβ; (b) p+ q = rγ. (c) Using (a) and (b), we obtain (α+ 1)q = (β + 1)p. Using (b) and (c), we obtain (γβ − 1)q = (γ + 1)p. Hence we have: β + 1 γβ − 1 γ + 1 which gives the following equation: αβγ = 2 + α+ β + γ. (15) The solutions of this equation are under symmetry (2, 2, 2), (3, 3, 1) and (5, 2, 1) which gives at last the solutions (p, q, r) of Proposition 4. Proof of Proposition 5. The proof is similar. We still use (12) with ω a 1− forme that without both zeros and poles outside the singular locus of the algebraic curve C defined by yq(1− x− y)r = xp: ω = − x[q − qx− (q + r)y] y[−p+ py − (r − p)x] Here the singular locus is no longer as before. It has two singular points at infinity: [1 : 0 : 0] et [1 : −1 : 0] and moreover (0, 0) et (0, 1) in the affine chart. Local considerations as above naturally leads us to the following Lemma 2 The irreducible algebraic curve above has genus one if and only p, q and r satisfy the equation: q + r = gcd(p, q) + gcd(q, r) + gcd(r, r + q − p) + gcd(q, r + q − p). (16) Writing m = gcd(p, q), n = gcd(q, r), l = gcd(r, r + q − p) and s = gcd(q, r + q − p), then there exists integers α, β, γ, δ such that r = nα = lβ and q = sγ = mδ so that (16) is equivalent to: q + r = γ + δ r (17) Hence we have γ + δ = γδ and α + β = αβ and therefore α = β = γ = δ = 2.Hence m = s and as m divides p and q then m divides r and finally m = 1. Similarly, we get n = 1 and consequently q = r = 2. Now, remind that p < r + q = 4 so that p either equals 1 or 3 (2 is excluded as gcd(p, q, r) = 1). Now we easily verify that (1, 2, 2) and (3, 2, 2) are the solutions to the equation 16) above which proves Proposition 5. Proof of Proposition 6. After the birational change of variable: x → 1 and y → y , the genus (which is a birationnal invariant for curves) is the same as the genus of the algebraic curve: xp−r−qyq(1 − x− y)r = 1. Then this is an immediate consequence of Proposition 4 above. Consequently we found (lv1− 5) of Theorem 2. 2.2.2 The case a = 0. Under affine transformation we may suppose b = c = 1. Geometrically {y = 0} and {y = 1} both intersect at infinity. Notice first the following: xλyµ(1 + y) = (xyn)λyµ−nλ(1 + y) for n ∈ Z, hence xλyµ(1 + y) = t is birational to xλyµ−nλ(1 + y) = t so that we only need to study when λ and µ are strictly positives. This naturally leads to the following: Proposition 7 The algebraic curve: C = {(x, y) ∈ C2, xpyq(1 + y)r = 1} with 0 ≤ r ≤ q and 0 ≤ p, where gcd(p, q, r) = 1. is elliptic if and only, under permutations of {y = 0} and {y + 1 = 0} it is in the following list: x3y1+3u(1 + y)1+3v = 1; x3y2+3u(1 + y)2+3v = 1 x4y1+4u(1 + y)1+4v = 1; x4y3+4u(1 + y)3+4v = 1; x4y2(1+2u)(1 + y)r = 1, r ∈ Z∗ \ 2Z x6y2+6u(1 + y)1+6v = 1; x6y5+6u(1 + y)4+6v = 1; x6y3(2u+1)(1 + y)r = 1, r ∈ Z∗ \ 3Z. Proof of Proposition 7. We still use (12) with a judiciously chosen ω without zeros nor poles in its regular locus: x(q + (q + r))y = − dy py(1 + y) This curve has two points at infinity, namely [1 : 0 : 0] and [0 : 1 : 0], where C is singular (C is regular in the affine chart) Near [1 : 0 : 0], we have two branches where a local equation of each is respectively: Y q = up (1 + Y )r = up where x = 1 . Thus, writing: m = pgcd(p, q) and n = pgcd(p, r), we obtain the parametrisations: Y = t u = t Y = t u = t Both give a pole of order 1 for π∗ω, hence we obtain, adding up the different possible parametrisations, −m− n in the Poincaré-Hopf formula. A similar calculus near the other point at infinity gives a zero of order p − 1 where l = (p, q+ r) with l different parametrisations. Thus we finally obtain the equality: p = m+ n+ l. (18) We want to resolve this equation. Consider: p = nγ p = mβ p = lα with α, β, γ, δ ∈ Z∗. Then (18) is equivalent to the following well-known equation: αβγ = αβ + αγ + βγ. (19) The solutions are up to permutation: (3, 3, 3) (2, 4, 4) (2, 3, 6). The solution (3, 3, 3) implies m = n = l. As gcd(p, q, r) = 1 we have: m = n = l = 1 and so p = 3, 1 = (r, 3), 1 = (q, 3). Hence r = 1 , 2 mod (3) and so does q. Finally, reminding l = gcd(p, q + r), we conclude that (p, q, r) = (3, 1 + 3u, 1 + 3v), (3, 2 + 3u, 2 + 3v). The solution (2, 4, 4) implies l = 2n and m = n = 1 and the same argument shows that (p, q, r) = (4, 1 + 4u, 1+ 4v) or (4, 3 + 4u, 3+ 4v). There are 2 other solutions (permutations of (2, 4, 4)). Under permutations of the two lines {y = 0} and {y + 1 = 0}, we only need to study (4, 2, 4). A similar resolution thus gives (p, q, r) = (4, 2(1 + 2u), 1 + 2v). The solution (2, 3, 6) implies m = 2, n = 1, l = 3, so (p, q, r) = (6, 2+6u, 1+6v) or (6, 5+ 6u, 4+6v). As above, we need to take under consideration the solutions (3, 2, 6) and (6, 2, 3) wich respectively gives (p, q, r) = (6, 3(2u+ 1), r) with gcd(r, 3) = 1 and (6, 2(3u+ 1), 6v + 1) or (6, 2(3u+ 2), 6v + 5). Finally the proposition is proved. This we obtain the last cases of the left column of Theorem 2. 2.2.3 The case c = 0. Under affine transformation we may suppose a = b = 1. Here the three lines {x = 0}, {y = 0} and {x+ y = 0} intersect themselves at the origine. Now, the algebraic curve xλyµ(x+ y) = t is obviously birational to xλ+µ+1yµ(1+ y) = t , so this case falls from the preceding results and we get the last cases of the right column of Theorem 2. We have investigated in fact all the possible first integrals. Indeed, if our foliation admits a first integral: f = x−αy−β(ax+ by+ c) with α, β real positive numbers, then after affine transformation it has a first integral: g = X− α (AX +BY + C). Hence Theorem 2 is proved. 2.3 The reversible Lotka-Voltera case Theorem 3 The reversible Lotka-Voltera foliation F(df) is elliptic if and only if, after an affine change of the variables, it has a first integral of the form (rlv1) f = xy(1− x− y) (rlv2) f = x−3y(1− x− y) (rlv3) f = x2y(1− x− y) (rlv4) f = x−4y(1− x− y) (rlv5) f = x−3y2(1 − x− y)2 (rlv6) f = x−1y2(1− x− y)2 (xiii)f = x k (y + 1)(y − 1), k ∈ Z∗ \ 3Z (x1v)f = x−2+ 3k (x+ y)(x− y), k ∈ Z∗ \ 3Z (xv)f = x k (y + 1)(y − 1), k ∈ Z∗ \ 2Z (xvi)f = x−2+ 4k (x+ y)(x− y), k ∈ Z∗ \ 2Z Proof. Without loss of generality we may suppose that the foliation has a first integral of the form f = xλ(y2 + a(x− b )2) if a 6= 0 and f = xλ(y2 + c) otherwise (we deal about quadratic foliations so that c necessary does not vanish). For a = 0 this a consequence of Proposition 7. Now look at a 6= 0 : if b = 0, the curve xλ(y2 + ax2) = t is birational to xλ+2(y2 + 1) = t thus the conditions are p+ 2q = 3, 4 or −2q − p = 3, 4. if b 6= 0 the quadric is a reducible polynomial so that this case is a straightforward consequence of Propositions 4, 5 and 6. We notice that the last cases of reversible Lotka-Volterra are exactly the last cases of Lotka-Volterra under the condition l = k when it is possible (For (vii) and (viii) of Theorem 2 we can’t have k = l). This gives Theorem 3. Notice that the case b = 0 is also a consequence of the degenerate Lotka-Voltera case with the three invariant lines involved intersecting themselves, but the calculus is here so easy that we proved it directly and is useful to test our preceeding calculus. 3 Topology of the singular fibers and Kodaira’s classification Now we focus on the singular fibers of the induced elliptic surfaces. First of all, Recall that two bira- tional elliptic surfaces have the same minimal model (see [Ka75, M89]). Some of our previous elliptic surfaces are obviously birationnals and therefore have the same singular fibers under permutation. First we investigate such mappings. Then we give some examples of computation of the singular fibers to illustrate the way we obtained Tables 2, 3, 4, 5, 6, 7, 8. 3.1 The reversible case 3.1.1 Birational mappings The first integrals are given by the algebraic equation: xλ(y2 + ax2 + bx+ c) = t with a, b, c complex numbers satisfying some conditions and λ a rational number. We have an easy birational mapping (we already used it, see Section 2.1.3): X = 1 , Y = y which leads to x−2−λ(y2 + cx2 + bx+ a) = t. When considering this mapping in P2 with homogeneous coordinates [x : y : z] this last permutes in fact the projactive lines {x = 0} and {z = 0}. Thus for each line of Table 1 we only need to study either the right or the left element. For degenerate cases, notice the change of variables (X,Y ) = (xy, y) birationnally leads (i) (resp. (ii)) to (x) (resp. (xii)) Lotka-Volterra elliptic case with l = 1. Consequently, such cases will be a consequence of the calculus of the singular fibers of the Lotka-Volterra cases (see below). Be careful that the geometry of the divisors appearing in the first integrals (including the line at infinity) is of importance as we shall blow-up the indetermination points. Birationally, the different geometrical description of the divisors in the reversible case are the following: (1) The divisors are in general position (see Figure 1). This concerns (rv2), (rv4), (rv6) with a, b, c 6= 0. (2) {Q = 0} and {x = 0} are in general position and {Q = 0} and {z = 0} have only one tan- gent double point (see Figure 7). This concerns (rv2) with c = 0, (rv3) with a = 0, (rv6) with c = 0, (rv7), (rv10), (rv12), (rv14), (rv16). (3) Both projective lines {x = 0} and {z = 0} have a double tangent point with the quadric . This concerns (i) and (ii). λ < −1 λ > −1 (rv1) x−3(y2 + ax2 + bx+ c) = t (rv2) x(y2 + cx2 + bx+ a) = t (rv3) x− 2 (y2 + ax2 + bx+ c) = t, c 6= 0 (rv4) x− 12 (y2 + cx2 + bx+ a) = t, c 6= 0 (rv5) x−4(y2 + ax2 + bx+ c) = t (rv6) x2(y2 + cx2 + bx+ a) = t (rv7) x− 3 (y2 + bx+ c) = t (rv8) x− 3 (y2 + cx2 + bx) = t (rv9) x− 3 (y2 + ax2 + bx) = t (rv10) x− 3 (y2 + bx+ a) = t (rv11) x− 3 (y2 + ax2 + bx) = t (rv12) x− 3 (y2 + bx+ a) = t (rv13) x− 4 (y2 + ax2 + bx) = t (rv14) x− 4 (y2 + bx+ a) = t (rv15) x− 4 (y2 + ax2 + bx) = t (rv16) x− 4 (y2 + bx+ a) = t (rv17) x− 2 (y2 + ax2 + bx) = t (rv18) x 2 (y2 + bx+ a) = t (i) x−1+ k (y2 + x) = t k ∈ Z∗ \ 2Z (ii) x−1+ k (y2 + x) = t k ∈ Z∗ \ 3Z Table 1: The elliptic reversible cases. 3.1.2 The singular fibers t = 0 and t = ∞ Here we illustrate the results with examples: EXAMPLE 1: The singular fibers of (rv4): Embedding our first integral in P2, one have: (y2 + ax2 + bxz + cz2)2 Geometrically, there are two lines: {x = 0} and {z = 0} with multiplicities respectively−1 and−3 that intersect in [0 : 1 : 0], and a conic that intersects both lines in four points, namely A1 = [0 : −c : 1], A2 = [0 : − −c : 1], B1 = [ −a : 1 : 0], B2 = [− −a : 1 : 0] with normal crossing each time (see Figure 1). {x = 0} B1 B2 {z = 0} Figure 1: Geometrical situation of (rv4). The rational function is not defined in these four points, thus we need to blow-up them. • Near A1 In local coordinates the rational function becomes: Y We need two blowing-ups to define the rational function near this point: → separation of both local branches Remind that blowing a point of P2 that belongs to a divisor D decreases the self-intersection of D by one (see [GH78] for example). Writing the self-intersection and the multiplicities (the self-intersection numbers are inside the () ) we get the situation of Figure 2. We study A2 along the same lines. See Figure 3. • Near B1. Locally the rational function becomes Y The successive blowing-ups give the following local equations until separation: → separation of the branches (1) −1 (3) 2 (0) −1 (3) 2 −1(−1) Figure 2: The successive blowing-ups of A1. Figure 3: Summary ot the situation after blowing-up A1 and A2. (2) (1) (−3)(0) (0) (0)−1 Figure 4: The successive blowing-ups of B1. see Figure 4. The situation is still the same near B2. We obtain 2 singular fibers. See Figure 5. 2 (−2) −1 (−3)−1 (−3)−1 Figure 5: The fibers t = 0 and t = ∞ of (rv4). We now have to recognize these singular fibers in Kodaira’s classification (see [Ko63]). The sin- gular fiber t = 0 is I∗0 , but we don’t recognize the other. This is because we have branches with self-intersection −1. Recall that Kodaira’s classification involves minimal elliptic surfaces i.e no fiber contains an exceptionnal curve of the first kind. Finally we get IV (the singular fiber at infinity). See Figure 6. −1 (−3) −1(−3)−1(−3) (−2) −1 −1 Figure 6: Contraction of the fiber t = ∞ of (rv4). {x = 0} {z = 0} Figure 7: The divisor associated to (rv12). EXAMPLE 2 : The singular fibers of (rv12) : −4 (−2) −4 (−2) 3 (1) −2 (−1) 3 (0) −1 (−2) −5 −2 2 (−2) −1 (−3) −5 (−1) (−2) 3 (2) −2 (−2) −4(4) Figure 8: Successive blowing-ups of (rv12) near A. The rational funcion here is: (y2 + bxz + az2)3 The geometrical situation is explained in Figure 7. • Study near A: Locally the rational function becomes: Y . To begin with, we have: (Y 2 + Z)3 → (Y + Z) Z5Y 2 → (Z + 1) Z5Y 4 → separation of both local branches Next we have to blow-up the point with local coordinate: (Y = 0, Z = −1), what gives locally : → separation of both local branches. the geometrical explanations are given in Figure 8. • Study near B: Locally the rational function becomes: Y . Such a calculus has already been done in Example We finally obtain two fibers (see Figure 9). For t = 0 we recognize IV ∗. For t = ∞, one have to contract divisors with self-intersection −1 as in Figure 10. We finally get III of Kodaira’s classification. −4 (−2) Figure 9: The fibers at t = 0 and t = ∞ of (rv12). −4 (−2) −1 (−5) −2 (−2) −1 −1 (−4) (−4) Figure 10: Contraction of divisors with self-intersection −1 for the fiber at infinity of (rv12). 3.1.3 Other(s) singular fiber(s) Be careful that we do not have the full list of singular fibers: we also have to consider the singular points of our foliation whiches do not intersect both lines and the conic curve above. Here, the results concern the whole class of reversible systems inducing elliptic fibrations: Writing: f = xλ(y2 + ax2 + bx+ c), (x, y) is a singular point if and only : xλy = 0 xλ−1(λy2 + (λ+ 2)ax2 + (λ+ 1)bx+ λc) = 0. Consequently (x0, y0) is a singular point which does not intersect both lines and the conic curve above if and only: y0 = 0 x0 is a zero of the polynomial : P = (λ+ 2)ax 2 + (λ+ 1)bx+ λc and x0 6= 0. Let δ be the discriminant of P . We resume the general happening below: If a 6= 0, reminding P (x0) = 0, we get: (x0, 0) = x ′(x0) (x0, 0) = x 0 (2(λ− 1)P ′(x0) + 2a(λ+ 2)x0). As λ 6= 0, −1 in the lists obtained concerning the whole reversible case above, different critical points have different critical values and P ′(x0) = 0 ⇔ δ = 0 ⇔ (λ + 1)b2 = 4λac, we have the following (remind that for the reversible case b2 − 4ac 6= 0): Lemma 3 For a 6= 0 and c 6= 0, if δ 6= 0 we obtain two different singular curves with a normal crossing, that is I1 in Kodaira’s classification and if δ = 0, we obtain one singular fiber with a cusp, that is II. If a 6= 0 and c = 0, or a = 0 and c 6= 0, we obtain one singular fiber with normal crossing, that is I1. Otherwise, there are no more singular fibers. Finally, we are now able to compute all the singular fibers. The results are given in Tables 2, 3 and 4. Fibration {t = ∞} {t = 0} t1 t2 (rv1) IV ∗ I2 I1 I1 (rv2) IV ∗ I2 I1 I1 (rv3) IV I0∗ I1 I1 (rv4) IV I0∗ I1 I1 (rv5) III∗ I1 I1 I1 (rv6) III∗ I1 I1 I1 Table 2: The elliptic reversible case with 4 singular fibers. Fibration {t = ∞} {t = 0} t1 (rv1) a, b, c 6= 0, δ = 0 IV ∗ I2 II a 6= 0 ,c = 0 III∗ I2 I1 a = 0, c 6= 0 IV ∗ III I1 (rv2) a, b, c 6= 0, δ = 0 IV ∗ I2 II a 6= 0 ,c = 0 III∗ I2 I1 a = 0, c 6= 0 IV ∗ III I1 (rv3) a, b, c 6= 0, δ = 0 IV I0∗ II a = 0 IV I1∗ I1 (rv4), a, b, c 6= 0, δ = 0 IV I0∗ II a = 0 IV I1∗ I1 (rv5) a, b, c 6= 0, δ = 0 III∗ I1 II a 6= 0 ,c = 0 II∗ I1 I1 a = 0, c 6= 0 III∗ II I1 (rv6) a, b, c 6= 0, δ = 0 III∗ I1 II a 6= 0 ,c = 0 II∗ I1 I1 a = 0, c 6= 0 III∗ II I1 (rv7) IV ∗ III I1 (rv8) IV ∗ III I1 (rv9) I1∗ IV I1 (rv10) I1∗ IV I1 (rv11) III IV ∗ I1 (rv12) III IV ∗ I1 (rv13) IV ∗ III I1 (rv14) IV ∗ III I1 (rv15) III∗ II I1 (rv16) III∗ II I1 (rv17) II∗ I1 I1. (rv18) II∗ I1 I1. Table 3: The elliptic reversible case with 3 singular fibers. Fibration {t = ∞} {t = 0} (i), k = 1 mod (4) III∗ III k = 3 mod (4) III III∗ (ii) k = 1, 2 mod (6) II∗ II k = 3, 5 mod (6) II II∗ Table 4: The reversible case with 2 singular fibers. 3.2 The Lotka-Volterra case 3.2.1 Birational mappings The same birational mapping: X = 1 and Y = y , leads: xλyµ(ax+ by + c) = t birationally to: x−λ−µ−1yµ(cx+ by + a) = t. Using this, one can immediately verify that (lv1), (lv2), (lv3) and (lv5) are birationals and (lv4) is birational to (rlv3) and (rlv4). For the last Lotka-Volterra cases, there are another obvious birational mappings: X = xyu(1 + y)v , Y = y Y = xyu(1 + y)v , X = x. with u, v ∈ Z judiciously chosen. 3.2.2 The fibers t = 0 and t = ∞ EXAMPLE 3:The singular fibers of (lv1) Here the rational function is: x2y3(z − x− y) The geometrical situation is explained in Figure 11. The intersection points with opposite multiplicity need to be blown-up. Here there are 3 points: A1 = [0 : 1 : 0], A2 = [1 : 0 : 0], A3 = [1 : −1 : 0]. Near A1, locally the rational function becomes such that we only need three blowing-ups to separate local branches. This situation is well-known like near A2 and A3, where we need respectively 2 and 6 blowing-ups. See Figures 12 and 13. The fiber at infinity is II∗. For the fiber t = 0, we need two contractions as explained in Figure 14 and we finally get I1. EXAMPLE 4: The singular fibers of (vii) with l = 4 mod (6). Under birational equivalence, the rational function we need to consider is: y4(z + y)5 We have to blow up 3 points: A = [0 : 0 : 1], B = [1 : 0 : 0] and C = [0 : 1 : −1]. For A and C we have normal crossings and the situation is similar to precedent ones. We need to pay little more attention for the blowing-up of B: Locally the rational function becomes: Y 4(Y +Z)5 . Here the first blowing- up separates the three branches. Now we need to blow-up the intersection point of the branch with multiplicity 6 and the branch with multiplicity −3. Locally the rational function is: Y and we get in a well-known situation. We obtain II∗ for t = 0. For t = ∞ we need three contractions to finally obtain II (see Figure 16). {z = 0} {x = 0} {1− x− y = 0} {y = 0} Figure 11: The geometrical situation of (lv1). 2 3 4 Figure 12: The fiber at infinity for (lv1). (−5)1(−2)2 Figure 13: The fiber t = 0 for (lv1). 3.2.3 Other singular fiber We write: f = xλyµ(ax+ by + c). This is here elementary linear algebra and it immediately gives the following: Lemma 4 Under the assumptions λ, µ 6= 0, λ 6= −1, µ 6= −1 and λ + µ + 1 6= 0, the function above gives rise to another singular fiber if and only if a, b, c 6= 0 and the corresponding singular fiber is I1. (−5)1(−2)2 1 (−3) 2 (−1) 1 (−4) Figure 14: Contraction of the fiber t = 0 for (lv1). {x = 0} {y = 0} {z = 0} {z + y = 0} Figure 15: Geometrical situation of (vii). Remark 2 Such assumptions hold for our Lotka-Volterra and reversible Lotka-Volterra systems in- ducing elliptic fibrations. −6 (−1) Figure 16: Contractions of t = 0 for (vii). Fibration Fiber {t = ∞} Fiber t = 0 Other singular fiber (lv1) II∗ I1 I1 (lv2) II∗ I1 I1 (lv3) II∗ I1 I1 (lv5) II∗ I1 I1 (lv4) III∗ I2 I1 Table 5: The Lotka-Volterra case with 3 singular fibers. 3.3 The reversible Lotka-Volterra case The calculus are similar and are left to the reader. The results are contained in Tables 7 and 8. Fibration {t = ∞} t = 0 (iii), k = l = 1 mod (3) IV ∗ IV k = l = 2 mod (3) IV IV ∗ (iv), k = l = 1 mod (3) IV ∗ IV k = l = 2 mod (3) IV IV ∗ (v), k = l = 1 mod (4) III∗ III k = l = 3 mod (4) III III∗ (vi), k = l = 1 mod (4) III∗ III k = l = 3 mod (4) III III∗ (vii), l = 2 mod (6) II∗ II l = 4 mod (6) II II∗ (viii), l = 2 mod (6) II∗ II l = 4 mod (6) II II∗ (ix), k = 1 mod (4) III∗ III k = 3 mod (4) III III∗ (x), k = 1 mod (4) III∗ III k = 3 mod (4) III III∗ (xi), k = 1, 2 mod (6) II∗ II k = 4, 5 mod (6) II II∗ (xii), k = 1, 2 mod (6) II∗ II k = 4, 5 mod (6) II II∗ Table 6: The elliptic Lotka-Volterra case with 2 singular fibers. Fibration Fiber {t = ∞} Fiber t = 0 Other singular fiber (rlv1) IV ∗ I3 I1 (rlv2) IV ∗ I3 I1 (rlv3) III∗ I2 I1 (rlv4) III∗ I2 I1 (rlv5) IV I1∗ I1 (rlv6) IV I1∗ I1 Table 7: The elliptic reversible Lotka-Volterra case with 3 singular fibers. Fibration Fiber {t = ∞} Fiber t = 0 (ix), k = 1 mod (3) IV ∗ IV k = 2 mod (3) IV IV ∗ (x), k = 1 mod (3) IV ∗ IV k = 2 mod (3) IV IV ∗ (xi), k = 1 mod (4) III∗ III k = 3 mod (4) III III∗ (xii), k = 1 mod (4) III∗ III k = 3 mod (4) III III∗ Table 8: The elliptic reversible Lotka-Volterra case with 2 singular fibers. Acknowledgements: The author would like to thank L. Gavrilov and I. D. Iliev and for their attention to the paper, many useful comments and stimulating remarks. References [BPV84] W. Barth , C. Peters, A. Van de Ven: Compact complex surfaces, Erg. Math., Springer- Verlag (1984). [CGLPR] J. Chavarriga, B. Garciá, J. Llibre, J. S Pérez Del Rı́o and J. A. Rodŕıguez: Polynomial first integrals of quadratic vector fields, Journal of Differential Equations Volume 230, Issue 2, 15 November 2006, Pages 393-421. [CLLL06] G. Chen, C. Li, C. Liu, J. Llibre: The cyclicity of period annuli of some classes of reversible quadratic systems, Discrete Contin. Dyn. Syst. 16 (2006), no. 1, 157-177. [G07] S. Gautier: Feuilletages elliptiques quadratiques plans et leurs pertubations, P.H.D Thesis, 7/12/2007. [GGI] S. Gautier, L. Gavrilov, I. D. Iliev: Pertubations of quadratic centers of genus one, http://arxiv.org/abs/0705.1609. [G01] L. Gavrilov: The infinitesimal 16th Hilbert problem in the quadratic case, Invent. math 143, 449-497 (2001). [GH78] P. Griffiths and J. Harris: Principles of algebraic geometry, John Wiley and Sons, New York (1978). [I98] I. D. Iliev: Pertubations of quadratic centers, Bull. Sci. math. 22 (1998). 107-161. [ILY05] I. D. Iliev, C.Li, J.Yu: Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), no. 1, 305–330. [I02] Y. Ilyashenko: Centennial history of Hilbert’s 16th problem, Amer. Math. So., 39 (2002), no 3, 301-354. [J79] JC. Jouanolou: Equations de Pfaff algébriques, Lec. Notes in Math., 708 (1979). [Ka75] K. Kas: Weirstrass normal forms and invariants of elliptic surfaces, Trans. of the A.M.S, 225, (1977). [Ko60] K. Kodaira: On compact analytic surfaces, I, Annals of Math., 71 (1960),111-152. [Ko63] K. Kodaira: On compact analytic surfaces, II, Annals of Math., 77 (1963),563-626. [M89] R. Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica. [Doc- torate in Mathematical Research], ETS Editrice, Pisa, 1989. [P90] G. S. Petrov: Nonoscillation of elliptic integrals, Funct. Anal. Appl., 3 (1990), 45-50 [YL02] J. Yu, C. Li: Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one homoclinic loop, J. Math. Anal. Appl., 269 (2002), no. 1, 227–243. http://arxiv.org/abs/0705.1609 Introduction Quadratic centers which define elliptic foliations The reversible case The case a=0, c=0, b2-4ac=0. The case a=0, c= 0, b2-4ac=0. a=0, c=0, b2-4ac=0. a=c=0, b=0 The Lotka-Volterra case The case a =0, b =0, c =0. The case a=0. The case c=0. The reversible Lotka-Voltera case Topology of the singular fibers and Kodaira's classification The reversible case Birational mappings The singular fibers t=0 and t= Other(s) singular fiber(s) The Lotka-Volterra case Birational mappings The fibers t=0 and t= Other singular fiber The reversible Lotka-Volterra case

0704.1949

Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane

arXiv:0704.1949v1 [cond-mat.stat-mech] 16 Apr 2007 typeset using JPSJ.sty <ver.1.0b> Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane Kouji Ueda1), Roman Krcmar2), Andrej Gendiar2), and Tomotoshi Nishino1) 1Department of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan 2Institute of Electrical Engineering, Slovak Academy of Sciences, SK-841 04 Bratislava, Slovakia (Received ) Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed. KEYWORDS: CTMRG, Ising Model, Scaling, Curvature §1. Introduction Baxter’s method of corner transfer matrix (CTM) has been known as one of the representative tool for analyti- cal stydy of statistical models in two dimension (2D).1–3) The method is also of use for numerical calculations of one point functions, such as the local energy and the magnetization.3) This numerical application is a kind of numerical renormalization group (RG) method, where the block spin transformation is obtained from the di- agonalization of CTMs. Such a RG scheme has many aspects in common with the density matrix renormal- ization group (DMRG) method,4–7) expecially when the method is applied to 2D classical lattice models.8) Introducing the flexibility in the system extension pro- cess of the DMRG method to the Baxter’s method of CTM, the authors developed the corner transfer matrix renormalization group (CTMRG) method.9–12) In this article we report a modification of the CTMRG method, for the purpose of applying the method to classical lat- tice models on the hyperbolic plane. Using the recursive structure of the lattice, we obtain one point functions at the center of sufficiently large finite size systems. Quite recently Hasegawa, Sakaniwa, and Shima re- ported deviations of critical indices of the Ising model on the hyperbolic plane from the well known Ising univer- sality classes in two dimension.14, 15) They predicted that phase transition of such systems would be mean-field like. To confirm their prediction, in the next section we con- sider the Ising model on a lattice, which is represented as the tiling of pentagons. The necessary modification of the CTMRG method on this lattice is explained in §3. We calculate the nearest neighbor spin correlation func- tion and the spontaneous magnetization at the center of large scale finite size systems. Critical indices for these one point functions are studied in §4, and we confirme the mean-field like properties of the phase transition. Con- clusions are summarized in the last section. We discuss a possible deformation of corner Hamiltonian in the hy- perbolic plane. §2. Ising Model on the Tiling of Pentagons Let us consider the hyperbolic plane, which is the two dimensional surface with constant negative curvature. Figure 1 shows a part of a sufficiently large regular lat- tice on the plane, where the lattice is constructed as the tiling of pentagons.13) All the arcs are geodesics, which divides the lattice into two parts of similar structure. Each lattice point represented by an open circle is the crossing points of two geodesics. Fig. 1. Ising model on a regular lattice on the hyperbolic plane. Open circles represent Ising spins on the lattice point, and ) is the local Boltzmann weight defined in Eq. (2.2). Two geodesics drawn by thick arcs divide the system into four quadrants, which are called as the corner. We investigate the ferromagnetic Ising model on this lattice. The Hamiltonian of the system is defined as the sum of nearest neighbor Ising interactions H = −J 〈i,j〉 σiσj , (2.1) where 〈i, j〉 represents pair of neighboring sites, and σi = ±1 and σj = ±1 are the Ising spins on the lattice http://arxiv.org/abs/0704.1949v1 2 Kouji Ueda, Roman Krcmar, Andrej Gendiar, and Tomotoshi Nishino points. Throughout this article we assume the absence of external magnetic field. For the latter conveniences, we represent the system as the ‘interaction round a face (IRF)’ model. The local Boltzmann weight for each face of pentagonal shape — the IRF weight — is given by W (σa, σb, σc, σd, σe) (2.2) = exp (σaσb + σbσc + σcσd + σdσe + σeσa) where β = 1/kBT is the inverse temperature, and σa, σb, σc, σd, and σe are the spin variables around the face as shwon in Fig. 1. The partition function of a finite size system (with suf- ficiently large diameter) is formally written as the con- figuration sum of the Boltzmann weight of the whole system all the spins all the faces W , (2.3) where we are interested in the thermodynamic limit of this system. Note that it is rather hard to investigate the system by use of the Monte Carlo simulations, since the number of sites contained in a cluster blows up exponen- tially with respect to its diameter. As a complemental numerical tool, we employ the CTMRG method. §3. Corner Transfer Matrix Renormalization Group Method The two geodesics shown by thick arcs in Fig. 1 divide the system into four parts, which are called as corners.3) Figure 2 shows the structure of a corner. We label the spins on a cut as {σ1, σ2, σ3, . . .}, and those on another cut as {σ′1, σ 3, . . .}, where σ1 is equivalent to σ 1. The corner transfer matrix is the Boltzmann weight with re- spect to a corner, which is calculated as a partial sum of the product of IRF weights in the corner C(σ′1, σ 3, . . . |σ1, σ2, σ3, . . .) spins inside the quadrant faces in the quadrant W . (3.1) The configuration sum is taken over spins ‘inside’ the corner, leaving those spins on the cuts. Con- ventionally the matrix C is interpreted as block di- agonal with respect to σ1 and σ 1, and the element C(σ′1, σ 3, . . . |σ1, σ2, σ3, . . .) for those cases σ1 6= σ is set to zero. The CTM thus defined is symmetric in the case of the pentagonal lattice under consideration. As shown in Fig. 2 a corner has the structure where three parts labeled by P̄ and two parts labeled by C̄ are joined to a face W . The ‘fusion’ relation can be represented by a formal equation9, 10) C = W · P̄ C̄P̄ C̄P̄ C̄ . (3.2) Note that C̄ is a corner of smaller size. For convenience, let us observe the structure of the part of the system shown in Fig. 3, where two P̄ , and C̄ are joined to W . Labeling the shown part by P , we can Fig. 2. Recursive structure of a corner C. formally write the fusion relation in the same manner P = W · P̄ C̄P̄ . (3.3) For a conventional reason we call P as the ‘half-row’, although P is not a row on the hyperbolic plane. It is easily understood that P̄ is a half-row of smaller size. We have thus obtained recursive structure of the corner C and the half-row P . Fig. 3. Recursive structure of the ‘half-raw’ P . As we have defined CTM for a corner, let us express the Boltzmann weight with respect to the half-row P (σ′1, σ 3, . . . |σ1, σ2, σ3, . . .) spins inside the half-row faces in the half-row W (3.4) in the matrix form, where the positions of spins {σ1, σ2, σ3, . . .} and {σ 3, . . .} are shown in Fig. 3. We call the weight in the matrix form as the half-row transfer matrix (HRTM). The 4-th power of the CTM ρ = C4 (3.5) is a kind of density matrix, since its trace gives the par- tition function Z = Tr ρ = TrC4 (3.6) of a finite size cluster that consists of four corners. The CTMRG Method Applied to the Ising Model on the Hyperbolic Plane 3 matrix dimension of CTM, and also that of the HRTM, increases exponentially with respect to the system size. In order to obtain Z numerically up to sufficiently large systems, we introduce the block spin transformation that is created from the diagonalization of the density matrix ρ.4, 5) Instead of directly diagonalizing ρ in Eq. (3.5), we first create its contraction ρ′(σ′2, σ 3 . . . |σ2, σ3, . . .) ρ(σ′1, σ 3 . . . |σ1, σ2, σ3, . . .) (3.7) and then diagonalize it ρ′(σ′2, σ 3 . . . |σ2, σ3, . . .) A(σ′2, σ 3 . . . | ξ)λξ A(σ2, σ3, . . . | ξ) , (3.8) where the eigenvalue λξ is non-negative. Following the convention in DMRG, we assume the decreasing order for λ . The orthogonal matrix A(σ2, σ3, . . . | ξ) repre- sents the block spin transformation from the ‘row-spin’ {σ2, σ3, . . .} to the effective spin variable ξ. We keep m numbers of representative states, which correspond to major eigenvalues, for the block spin variable ξ. Apply- ing the matrix A to the CTM and the HRTM, we obtain ‘renormalized matrices’ of 2m-dimension C(σ′1, σ 2, . . . |σ1, σ2, . . .) → C(σ ′, ξ′ |σ, ξ) P (σ′1, σ 2, . . . |σ1, σ2, . . .) → P (σ ′, ξ′ |σ, ξ) , (3.9) where we have dropped the indices from σ1 and σ 9, 10) Combining the recursive structures in Eqs. (3.2) and (3.3), and the renormalization scheme in Eqs. (3.7)-(3.9), we can obtain the CTM and HRTM in the renormalized form for arbitrary system size by way of successive ex- tension of the system. Fig. 4. Extension pocess of (a) CTM and (b) HRTM in the renor- malized expression. Suppose that we have C(σ′, ξ′ |σ, ξ) and P (σ′, ξ′ |σ, ξ) for a finite size cluster. In order to explain the extension process, let us rewrite these matrices as C̄(s′ , ζ′| s, ζ) and P̄ (s′ , ζ′| s, ζ). (1) Substitute C̄(s′ , ζ′| s, ζ) and P̄ (s′ , ζ′| s, ζ) into the fusion process in Eqs. (3.2) and (3.3). Figure 4 shows these fusion processes among W , C̄, and P̄ , where rectangles correspond to the block spin vari- ables. The spin variables that are contracted out are shown by black marks. As a result, we obtain the extended CTM C(σ′ , s′ , ζ′ |σ, s, ζ) and the extended HRTM P (σ′ , s′ , ζ′|σ, s, ζ). (2) From the extended CTM C(σ′ , s′ , ζ′|σ, s, ζ) obtain the density matrix ρ(σ′ , s′ , ζ′|σ, s, ζ) by Eq. (3.5). Contracting out the spin at the center, obtain ρ′ (s′ , ζ′| s, ζ) as Eq. (3.7), and diagonalizing it to ob- tain the block spin transformation matrix A(s, ζ| ξ) from Eq. (3.8). (3) Applying A(s, ζ| ξ) to both C(σ′ , s′ , ζ′|σ, s, ζ) and P (σ′ , s′ , ζ′|σ, s, ζ), obtain the extended CTM C(σ′ , ξ′ |σ, ξ) in the initial form, and the same for HRTM to obtain P (σ′ , ξ′ |σ, ξ). (4) return to the first step. The system size, which is the length of the longest geodesics in the system, increases by 2 for each itera- tion.16) In order to start the above extension process, we set the initial condition C̄(σ′ |σ) = P̄ (σ′ |σ) = δ(σ′ | 1) δ(σ| 1) (3.10) that represents ferromagnetic boundary, where δ(a| b) = δa,b is the Cronecker’s delta. During the iteration we can obtain one point functions at the center of the system. For example, the sponta- neous magnetization is calculated as 〈σ〉 = σ,s,ζ σρ(σ, s, ζ|σ, s, ζ) σ,s,ζ ρ(σ, s, ζ|σ, s, ζ) . (3.11) In the same manner we obtain the nearest neighbor spin correlation function 〈σs〉. It should be noted that one point functions thus calculated at the center do not al- ways represent the averaged property of the whole sys- tem even in the thermodynamic limit, since the area near the boundary has non-negligible weight in the hyperbolic plane. §4. Numerical Result Compared With the Bethe Approximation Let us calculate the spontaneous magnetization 〈σ〉, and spin correlation function 〈σs〉 for the nearest spin pair. We regard the Ising interaction strength J as the energy unit, and use the temperature where the Boltz- mann constant kB is equal to unity. Most of the numer- ical calculations are performed keeping m = 40 states. The dumping of the density matrix eigenvalues is very fast, and actually the calculated results with m = 10 do not differ from those obtained with m = 40 even at the critical temperature TC. The iteration number required for the numerical convergence is at most 400000 for the calculated data points. Figure 5 shows the calculated results. The square of the spontaneous magnetization 〈σ〉2 is a linear function of temperature in the neighborhood of TC. From the behavior we estimate the transition temperature TC = 2.799. The nearest neighbor spin correlation function 4 Kouji Ueda, Roman Krcmar, Andrej Gendiar, and Tomotoshi Nishino 2.775 2.780 2.785 2.790 2.795 2.800 2.805 2.775 2.780 2.785 2.790 2.795 2.800 2.805 Fig. 5. Square of the spontaneous magnetization 〈σ〉2 (upper) and the nearest neighbor spin correlation function 〈σs〉 (lower) with respect to the temperature T . 〈σs〉 has a kink at TC, and is linear in T around there. These calculated results support the existence of mean- field like transition, that is subject to the critical indices β = 1/2 and α = 0. We thus confirmed the prediction by Hasegawa, Sakaniwa, and Shima.14, 15) Compared with the transition temperature of the square lattice Ising model T Square C = 2.269, the calcu- lated TC is fairly higher and is close to the transition temperature calculated from the Bethe approximation TBetheC = 2.885. 17, 18) The result suggest that neglection of the ‘loop back effect’ is not so conspicuous in the hy- perbolic plane. §5. Conclusion and discussion We have calculated the spontaneous magnetization and the nearest neighbor spin correlation function of the Ising model on a pentagonal lattice on the hyperbolic plane. The numerical algorithm of the CTMRG method is modified for this purpose. The calculated critical tem- perature is TC = 2.799, and we observe the mean-field like phase transition. The modified CTMRG method we have developped is applicable to regular lattices that consists of geodesics on the hyperbolic plane. For those lattices that does not contain geodesics, one has to either treating asym- metric density matrix or to draw geodesics by use of transformation such as duality transformation and the star-triangle relation. Generalization of the modified CTMRG method to the vertex model is straight for- ward. It may be interesting to classify ordered states of eight-vertex model on a variety of regular lattices in the hyperbolic plane. An interest is in the eigenvalue structure of the den- sity matrix at the transition temperature. Its analytic form is not well defined in the thermodynamic limit of classical lattice models on the flat 2D plane.19, 20) The rapid eigenvalue dumping observed on the hyperbolic plane suggests that there would be a way of regulariz- ing the CTM at the criticality. The classical-quantum correspondence from such a view point is worth consid- ering. Formally speaking the corner transfer matrix C can be written as the exponential of the corner Hamil- tonian HC. When the lattice is on the flat plane, the simplest example of the corner Hamiltonian is written in the sum of local operators HC = h(σ1, σ2) + 2h(σ2, σ3) + 3h(σ3, σ4) + . . . = h1 + 2h2 + 3h3 + 4h4 + . . . , (5.1) where hi = h(σi , σi+1) is the local hamiltonian that acs between neighboring sites. A possible one parameter de- formation of the corner Hamiltonian to the hyperbolic geometry may given by HC(Λ) = h1 + sinh 2Λ sinhΛ sinh 3Λ sinhΛ h3 + . . . , (5.2) which is reduced to HC in Eq. (5.1) in the limit Λ → 0. The deformed Hamiltonian satisfies the recursive struc- HC(Λ) = coshΛ sinh 2Λ sinhΛ sinh 3Λ sinhΛ h4 + . . . + h1 + coshΛ h2 + cosh 2Λ h3 + . . . (5.3) discussed in Okunishi’s RG scheme on the corner Hamil- tonian.21) Such a deformation has similar regularization effect proposed by Okunishi quite recently.22) T. N. and A. G is partially supported by a Grant- in-Aid for Scientific Research from the Ministry of Ed- ucation, Science, Sports and Culture. T. N. thank to Okunishi for valuable discussions. 1) R.J. Baxter, J. Math. Phys. 9 (1968) 650. 2) R.J. Baxter, J. Stat. Phys. 19 (1978) 461. 3) R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982) p. 363. 4) S. R. White: Phys. Rev. Lett. 69 (1992) 2863. 5) S. R. White: Phys. Rev. B 48 (1993) 10345. 6) Density-Matrix Renormalization - A new numerical method in physics -, eds. I. Peschel, X. Wang, M. Kaulke and K. Hall- berg, (Springer Berlin, 1999), and references there in. 7) U. Schollwöck: Rev. Mod. Phys. 77 (2005) 259 and references there in. 8) T. Nishino, J. Phys. Soc. Jpn. 64 (1995) 3598. 9) T. Nishino, J. Phys. Soc. Jpn. 65 (1996) 891. 10) T. Nishino, J. Phys. Soc. Jpn. 66 (1997) 3040. 11) A. Gendiar and T. Nishino, Phys. Rev. E 65 (2002) 046702. 12) K. Ueda and T. Nishino, K. Ueda, R. Otani, Y. Nishio, A. Gendiar, and T. Nishino, J. Phys. Soc. Jpn. 74 (2005) supplement p.111. 13) A number of examples of the regular lattices on the hyperbolic plane is listed in the following URL; http://www2u.biglobe.ne.jp/˜hsaka/mandara/index.html. 14) H. Shima and Y. Sakaniwa: cond-mat/0511539. 15) I. Hasegawa, Y. Sakaniwa, and H. Shima: cond-mat/0612509. 16) Precisely speaking, such a way of system extension does not guarantee the cluster shape where every spins at the boundary is equally distant from the center. 17) H.A. Bethe, Proc. Roy. Soc. London A 150 (1935) 552-75. 18) Since the configuration number of the lattice is z = 4, the mean-field approximation applied to the Ising model on the pentagonal lattice under consideration gives the same transi- tion temperature as the approximation applied to the square lattice Ising model. Such an equivalence holds up to the Bethe approximation. 19) I. Peschel, J. Stat. Mech. (2004) P06004 and references there CTMRG Method Applied to the Ising Model on the Hyperbolic Plane 5 20) K. Okunishi, Y. Hieida, Y. Akutsu, Phys. Rev. E 59 (1999) R6227. 21) k. Okunishi, J. Phys. Soc. Jpn. 74 (2005) 3186. 22) K. Okunishi, to appear in J. Phys. Soc. Jpn, preprint arXiv:cond-mat/0702581.

0704.1950

Inferring periodic orbits from spectra of simple shaped micro-lasers

Inferring periodic orbits from spectra of simple shaped micro-lasers M. Lebental1,2, N. Djellali1, C. Arnaud1, J.-S. Lauret1, J. Zyss1 R. Dubertrand2, C. Schmit2 , and E. Bogomolny2∗ CNRS, Ecole Normale Supérieure de Cachan, UMR 8537, Laboratoire de Photonique Quantique et Moléculaire, 94235 Cachan, France CNRS, Université Paris Sud, UMR 8626, Laboratoire de Physique Théorique et Modèles Statistiques, 91405 Orsay, France (Dated: October 26, 2018) Dielectric micro-cavities are widely used as laser resonators and characterizations of their spectra are of interest for various applications. We experimentally investigate micro-lasers of simple shapes (Fabry-Perot, square, pentagon, and disk). Their lasing spectra consist mainly of almost equidistant peaks and the distance between peaks reveals the length of a quantized periodic orbit. To measure this length with a good precision, it is necessary to take into account different sources of refractive index dispersion. Our experimental and numerical results agree with the superscar model describing the formation of long-lived states in polygonal cavities. The limitations of the two-dimensional approximation are briefly discussed in connection with micro-disks. PACS numbers: 42.55.Sa, 05.45.Mt, 03.65.Sq I. INTRODUCTION Two-dimensional micro-resonators and micro-lasers are being developed as building blocks for optical telecommunications [1, 2]. Furthermore they are of in- terest as sensors for chemical or biological applications [2, 3, 4] as well as billiard toy models for quantum chaos [5, 6]. Towards fundamental and applied considerations, their spectrum is one of the main features. It was used, for instance, to experimentally recover some information about the refractive index [7] or geometrical parameters In this paper we focus on cavities much larger than the wavelength and propose to account for spectra in terms of periodic orbit families. Cavities of the simplest and most currently used shapes were investigated: the Fabry-Perot resonator, polygonal cavities such as square and pentagon, and circular cavities. Our experiments are based on quasi two-dimensional organic micro-lasers [9]. The relatively straightfor- ward fabrication process ensures good quality and repro- ducibility as well as versatility in shapes and sizes (see Fig. 1). The experimental and theoretical approaches developed in this paper can be easily extended to more complicated boundary shapes. Moreover this method is useful towards other kinds of micro-resonators, as it de- pends only on cavity shape and refractive index. The paper is organized as follows. In Section II a de- scription of the two-dimensional model is provided to- gether with its advantages and limitations. In Section III micro-lasers in the form of a long stripe are investigated as Fabry-Perot resonators to test the method and eval- ∗Electronic address: [email protected] FIG. 1: Optical microscope photographs of some organic micro-lasers: stripe (partial view, used as Fabry-Perot res- onator), square, pentagon, disk, quasi-stadium, and cardioid. Typical dimension: 100 µm. uate its experimental precision. This protocol is then further applied to polygonal cavities. In Section IV the case of square cavities is discussed whereas in Section V dielectric pentagonal cavities are investigated. The the- oretical predictions based on a superscar model are com- pared to experiments as well as numerical simulations and a good agreement is found. Finally, in Section VI the case of several coexisting orbits is briefly dealt with on the example of circular cavities. II. PRELIMINARIES Dielectric micro-cavities are quasi two-dimensional ob- jects whose thickness is of the order of the wavelength but with much bigger plane dimensions (see Fig. 1). Al- http://arxiv.org/abs/0704.1950v2 mailto:[email protected] though such cavities have been investigated for a long time both with and without lasing, their theoretical de- scription is not quite satisfactory. In particular, the au- thors are not aware of true three-dimensional studies of high-excited electromagnetic fields even for passive cavi- ties. Usually one uses a two-dimensional approximation but its validity is not under control. Within such approximation fields inside the cavity and close to its two-dimensional boundary are treated differ- ently. In the bulk, one considers electromagnetic fields as propagating inside an infinite dielectric slab (gain layer) with refractive index ngl surrounded by medias with re- fractive indices n1 and n2 smaller than ngl. In our ex- periments, the gain layer is made of a polymer (PMMA) doped with a laser dye (DCM) and sandwiched between the air and a polymer (SOG) layer (see Fig. 2 (a) and [9]). It is well known (see e.g. [10] or [11]) that in such ge- ometry there exist a finite number of propagating modes confined inside the slab by total internal reflection. The allowed values of transverse momentum inside the slab, q, are determined from the standard relation e2ihqr1r2 = 1 (1) where h is the slab thickness and r1,2 are the Fresnel re- flection coefficients on the two horizontal interfaces. For total internal reflection ri = exp(−2iδi) (2) where δi = arctan sin2 θ − n2i ngl cos θ  (3) Here θ is the angle between the direction of wave prop- agation inside the slab and the normal to the interface. The νi parameter is 1 (resp. (ngl/ni) 2) when the mag- netic field (resp. the electric field) is perpendicular to the slab plane. The first and second cases correspond respectively to TE and TM polarizations. Denoting the longitudinal momentum, p = nglk sin θ, as p = neffk, the effective refractive index, neff , is de- termined from the following dispersion relation = arctan − n21 + arctan − n22 + lπ , l ∈ N . (4) This equation has only a finite number of propagating solutions which can easily be obtained numerically. Fig. 3 presents possible propagating modes for our experimental setting n1 = 1 (air), n2 = 1.42 (SOG) [28] and ngl = 1.54 deduced from ellipsometric measurements (see Fig. 2 (b)) in the observation range. 0.800.700.600.50 Wavelength (µm) FIG. 2: (a) Notations for refractive indexes and propagation wavenumbers. From top to bottom, the layers of our samples [9] are the air (n1 = 1), a polymer (PMMA) doped with a laser dye (DCM) (ngl = 1.54), and another polymer (SOG) (n2 = 1.42) or silica (n2 = 1.45). (b) Refractive index of the gain layer versus the wavelength inferred from ellipsometric measurements. FIG. 3: Effective refractive index versus the thickness over wavelength variable calculated from Eq. (4). The refractive indices are assumed to be constant: 1 for air, 1.42 for SOG, and 1.54 for the gain layer (horizontal black lines). The TE polarization is plotted with solid blue lines and TM polariza- tion with dotted red lines. Integer l (see (4)) increases from left to right starting from l = 0. The Maxwell equations for waves propagating inside the slab are thus reduced to the two-dimensional scalar Helmholtz equation: ∆+ n2effk Ψin(x, y) = 0 . (5) Ψ represents the field perpendicular to the slab, i.e. the electric field for TM and the magnetic field for TE po- larization [29]. This equation adequately describes the wave propa- gation inside the cavity. But when one of these propa- gating modes hits the cavity boundary, it can partially escape from the cavity and partially be reflected inside it. To describe correctly different components of electro- magnetic fields near the boundary, the full solution of the three dimensional vectorial Maxwell equations is re- quired, which to the authors knowledge has not yet been adressed in this context. Even the much simpler case of scalar scattering by a half-plane plate with a small but finite thickness is reduced only to numerical solution of the Wiener-Hopf type equation [12]. To avoid these complications, one usually considers that the fields can be separated into TE and TM po- larization and obey the scalar Helmholtz equations (5) ∆+ n2in,outk Ψin,out(x, y) = 0 . (6) with nin is the neff effective index inferred from Eq. (4) and nout the refractive index of the surrounding media, usually air so nout = 1. This system of two-dimensional equations is closed by imposing the following boundary conditions Ψin|B = Ψout|B , νin |B = νout ∂Ψout |B . (7) Here ~τ indicates the direction normal to the boundary and ν depends on the polarization. When the electric (resp. magnetic) field is perpendicular to the cavity plane, called TM polarization (resp. TE polarization), νin,out = 1 (resp. νin,out = 1/n in,out). Notice that these definitions of ν are not the same for horizontal and ver- tical interfaces. We consider this standard two-dimensional approach keeping in mind that waves propagating close to the boundary (whispering gallery modes) may deviate significantly from two-dimensional predictions. In particular leakage through the third dimension could modify the life-time estimation of quasi-stationary states. Our polymer cavities are doped with a laser dye and uniformly pumped one by one from above [9], so that the pumping process induces no mode selection. The complete description of such lasing cavities requires the solution of the non-linear Maxwell-Bloch equations (see e.g. [13, 14, 15] and references therein). For clarity, we accept here a simplified point of view (see e.g. [1] Sect. 24) according to which true lasing modes can be represented as a linear combination of the passive modes which may lase (i.e. for which gain exceeds losses) Ψlasing = CmΨm . (8) From physical considerations, it is natural to consider the Ψm modes as the quasi-stationary states of the passive cavity. Though this choice leads to well known difficulties (see e.g. [1]) it is widely noticed and accepted at least for modes with small losses (cf. [14, 15, 16]). For each individual lasing mode, the Cm coefficients could be determined only after the solution of the full Maxwell-Bloch equations. But due to the statistical na- ture of fluorescence the lasing effect starts randomly and independently during each pump pulse. So it is quite nat- ural to average over many pump pulses. Then the mean spectrum exhibits peaks at frequencies of all possible las- ing modes. The experimental data studied in this paper are recorded after integration over 30 pump pulses and agree with this simple statistical model. More refined verifications are in progress. III. FABRY-PEROT RESONATOR The Fabry-Perot configuration is useful for the calibra- tion control of further spectral experiments due to the non ambiguous single periodic orbit family which sus- tains the laser effect. A long stripe can be considered to a good approxima- tion as a Fabry-Perot resonator. In fact the pumping area is very small compared to the length (see Fig. 4 (a)) and the material is slightly absorbing, so that reflec- tions at far extremities can be neglected. Moreover the pumping area is larger than the width of the stripe, thus the gain is uniformly distributed over the section. For a Fabry-Perot cavity, the emission is expected along both θ = 0 and θ = π directions (see Fig. 4 (a) for notations). Fig. 4 (b) shows that this directional emission is observed experimentally which confirms the validity of our set-up. Long stripe Detector -10 -5 0 5 10 θ (degrees) FIG. 4: (a) Diagram summarizing the main features of the Fabry-Perot experiment. (b) Detected intensity versus θ angle for a Fabry-Perot experiment. The experimental spectrum averaged over 30 pump pulses is made up of almost regularly spaced peaks (see Fig. 5 (a)) which is typically expected for a Fabry- Perot resonator. In fact, due to coherent effects, the k wavenumbers of quasi-bound states of a passive Fabry- Perot cavity are determined from the quantization con- dition along the only periodic orbit of L = 2W length as for (1): r2ei L k neff (k) = 1 (9) where r is the Fresnel reflection coefficient and neff is the effective refractive index (4). The solutions of this equation are complex numbers: the imaginary part cor- responds to the width of the resonance and the real part (called km afterwards) gives the position of a peak in the spectrum and verify L km neff (km) = 2π m , m ∈ N . (10) With δkm = km+1−km assumed to be small, the distance between adjacent peaks is constrained by δkm[neff (km) + km ∂neff (km)] L = 2π . (11) We call nfull = neff (km) + km ∂neff (km) (12) the full effective refractive index. It is a sum over two terms: one corresponding to the phase velocity, neff (km), and the other one to the group velocity, ∂neff (km). If nfull is considered as a constant over the observation range, which is true with a good accu- racy, δk can be retrieved from the experimental spec- trum. For instance, the Fourier transform of the spec- trum (intensity versus k) is made up of regularly spaced peaks (Fig. 5 (b) inset), with the first one (indicated with an arrow) centered at the optical length (L nfull) and the others at its harmonics. So the geometrical length of the periodic orbit can be experimentally inferred from the knowledge of nfull which is independently determined as described below. For the Fabry-Perot resonator, the ge- ometrical length is known to be 2W , thus allowing to check the experimental precision. The relative statistical errors on the W width is estimated to be less than 3 %. The error bars in Fig. 5 (b) are related to the first peak width of the Fourier transform and are less than 5 % of the optical length. The full effective refractive index, nfull, is indepen- dently inferred from ellipsometric measurement (Fig. 2 (b)) and standard effective index derivation described in the previous Section. Depending on the parameter h/λ (thickness over wavelength), one or several modes are al- lowed to propagate. Our samples are designed such as only one TE and TM modes exist with neff effective refractive index according to Eq. (4). In Fig. 3 the refractive index of the gain layer, ngl, is assumed to be constant: ngl = 1.54 in the middle of the experimental window, λ varying from 0.58 to 0.65 µm. From Eq. (4) a neff = 1.50 is obtained in the ob- servation range with a h = 0.6 µm thickness, and corre- sponds to the phase velocity term. The group velocity term km ∂neff (km) is made up of two dispersion contri- butions: one from the effective index (about 4 %) and the other from the gain medium (about 7 %). The de- pendance of ngl with the wavelength is determined with the GES 5 SOPRA ellipsometer from a regression with the Winelli II software (correlation coefficient: 0.9988) and plotted on Fig. 2 (b). Taking into account all contri- butions (that means calculating the effective refractive index with a dispersed ngl), the nfull full effective re- fractive index is evaluated to be 1.645 ± 0.008 in the observation range. So the group velocity term made up 625620615610605 Wavelength (nm) 22020018016014012010080 Width (µm) 2.0x10 1.00.0 Optical length (nm) FIG. 5: (a) Experimental spectrum of a Fabry-Perot res- onator with W = 150 µm. (b) Optical length versus Fabry- Perot width W . The experiments (red points) are linearly fitted by the solid red line. The dashed blue line corresponds to the theoretical prediction without any adjusted parameter. Inset: Normalized Fourier transform of the spectrum in (a) expressed as intensity versus wavenumber. of the two types of refractive index dispersion contribute for 10 % to the full effective index, which is significant compared to our experimental precision. The nfull in- dex depends only smoothly on polarization (TE or TM), and on the h thickness, which is measured with a surface profilometer Veeco (Dektak3ST). Thus, the samples are designed with thickness 0.6 µm and the precision is re- ported on the full effective index which is assumed to be 1.64 with a relative precision of about 1 % throughout this work. Considering all of these parameters, we obtain a satis- factory agreement between measured and calculated op- tical lengths, which further improves when taking into ac- count several Fabry-Perot cavities with different widths as shown on Fig. 5 (b). The excellent reproducibil- ity (time to time and sample to sample) is an addi- tional confirmation of accuracy and validity. With these Fabry-Perot resonators, we have demonstrated a spectral method to recover the geometrical length of a periodic 615610605600595 Wavelength (nm) 1401201008060 Width (µm) 2.0x10 1.00.0 Optical length (nm) FIG. 6: (a) Experimental spectrum of a square-shaped micro- laser of 135 µm side width. (b) Optical length versus a square side width. The experiments (red points) are linearly fitted by the solid red line. The dashed blue line corresponds to the theoretical prediction (diamond periodic orbit) without any adjusted parameter. Top inset: Two representations of the diamond periodic orbit. Bottom inset: Normalized Fourier transform of the spectrum in (a) expressed as intensity versus wavenumber. orbit which can now be confidently applied to different shapes of micro-cavities. IV. SQUARE MICRO-CAVITY In the context of this paper square-shaped micro- cavities present a double advantage. Firstly, they are increasingly used in optical telecommunications [2, 17]. Secondly, the precision and validity of the parameters used above can be tested independently since there is only one totally confined periodic orbit family. In fact the refractive index is quite low (about 1.5), so the di- amond (see Fig. 6 (b), top inset) is the only short- period orbit without refraction loss (i.e. all reflection angles at the boundary are larger than the critical angle χc = arcsin(1/n) ≈ 42◦.) In a square-shaped cavity light escapes mainly at the corners due to diffraction. Thus the quality design of cor- ners is critical for the directionality of emission but not for the spectrum. Indeed for reasonably well designed squared micro-cavity (see Fig. 1), no displacement of the spectrum peaks is detectable by changing the θ obser- vation angle. The spectra used in this paper are thus recorded in the direction of maximal intensity. Fig. 6 (a) presents a typical spectrum of a square- shaped micro-cavity. The peaks are narrower than in the Fabry-Perot resonator spectrum, indicating a better confinement, as well as regularly spaced, revealing a sin- gle periodic orbit. Data processing is performed exactly as presented in the previous Section: for each cavity the Fourier transform of the spectrum is calculated (Fig. 6 (b), bottom inset) and the position of its first peak is located at the optical length. Fig. 6 (b) summarizes the results for about twenty different micro-squares, namely: the optical length inferred from the Fourier transform versus the a square side width. These experimental results are fitted by the solid red line. The dotted blue line corresponds to an a priori slope given by nfull (1.64) times the geometrical length of the diamond periodic orbit (L = 2 2a). The excellent agreement confirms that the diamond periodic orbit family provides a dominant contribution to the quantization of dielectric square resonator. This result is far from obvious as square dielectric cav- ities are not integrable. At first glance the observed dom- inance of one short-period orbit can be understood from general considerations based on trace formulae which are a standard tool in semiclassical quantization of closed multi-dimensional systems (see e.g. [18, 19] and refer- ences therein). In general trace formulae express the density of states (and other quantities as well) as a sum over classical periodic orbits. For two-dimensional closed cavities d(k) ≡ δ(k − kn) ≈ ikLp−iµp + c.c. (13) where k is the wavenumber and kn are the eigenvalues of a closed cavity. The summation on the right part is performed over all periodic orbits labeled by p. Lp is the length of the p periodic orbit, µp is a certain phase accu- mulated from reflection on boundaries and caustics, and amplitude cp can be computed from classical mechanics. In general for integrable and pseudo-integrable systems (e.g. polygonal billiards) classical periodic orbits form continuous periodic orbit families and in two dimensions where Ap is the geometrical area covered by a periodic or- bit family (see the example of circular cavities in Section Non-classical contributions from diffractive orbits and different types of creeping waves (in particular, lateral waves [19]) are individually smaller by a certain power of 1/k and are negligible in semiclassical limit k → ∞ compared to periodic orbits. There exist no true bound states for open systems. One can only compute the spectrum of complex eigen- frequencies of quasi-stationary states. The real parts of such eigenvalues give the positions of resonances and their imaginary part measure the losses due to the leak- age from the cavity. For such systems it is quite natural to assume that the density of quasi-stationary states d(k) ≡ 1 Im(kn) (k − Re(kn))2 + Im(kn)2 can be written in a form similar to (13) but the contri- bution of each periodic orbit has to be multiplied by the product of all reflection coefficients along this orbit (as it was done in a slightly different problem in [19]) d(k) ≈ r(j)p  eikLp−iµp + c.c. . (16) Here Np is the number of reflections at the boundary and p is the value of reflection coefficient corresponding to the jth reflection for the p periodic orbit. When the incident angle is larger than the critical angle the modulus of the reflection coefficient equals 1 (see Eq. (2)), but if a periodic orbit hits a piece of boundary with angle smaller than the critical angle, then |rp| < 1 thus reducing the contribution of this orbit. Therefore, the dominant contribution to the trace for- mula for open dielectric cavities is given by short-period orbits (cp ∝ 1/ Lp) which are confined by total internal reflection. For a square cavity with n = 1.5 the diamond orbit is the only confined short-period orbit which ex- plains our experimentally observation of its dominance. Nevertheless, this reasoning is incomplete because the summation of contributions of one periodic orbit and its repetitions in polygonal cavities does not produce a com- plex pole which is the characteristics of quasi-stationary states. In order to better understand the situation, we have performed numerical simulations for passive square cav- ities in a two-dimensional approximation with TM po- larization (see Section II and [20]). Due to symmetries, the quasi-stationary eigenstates can be classified accord- ing to different parities with respect to the square diago- nals. In Fig. 7 (a), the imaginary parts of wavenumbers are plotted versus their real part for states antisymmet- ric according to the diagonals (that means obeying the Dirichlet boundary conditions along the diagonals) and called here (− −) states. These quasi-stationary states are clearly organized in families. This effect is more pronounced when wave func- tions corresponding to each family are calculated. For in- stance, wave functions for the three lowest families with 30 50 70 90 Re(a k) −0.35 30 80 130 180 230 Re (a k) −0.14 −0.07 10 30 50 70 FIG. 7: (a) Imaginary parts versus real parts of the wavenum- bers of quasi-stationary states with (− −) symmetry for a dielectric square resonator with neff = 1.5 surrounded by air with n = 1. (b) The same as in (a) but for the states with the smallest modulus of the imaginary part (the most confined states). Inset. Empty triangles: the difference (20) between the real part of these wavenumbers and the asymptotic ex- pression. Filled circles: the same but when the correction term (21) is taken into account. (− −) symmetry are presented in Fig. 8. The other mem- bers of these families have similar patterns. The existence of such families was firstly noted in [21] for hexagonal di- electric cavities, then further detailed in [20]. One can argue that the origin of such families is anal- ogous to the formation of superscar states in pseudo- integrable billiards discussed in [22] and observed ex- perimentally in microwave experiments in [23]. In gen- eral, periodic orbits of polygonal cavities form continuous families which can be considered as propagating inside (a) (b) (c) FIG. 8: Squared modulus of wave functions with − − symmetry calculated with numerical simulations. (a) ak = 68.74− .026 i, (b) ak = 68.84 − .16 i, (c) ak = 69.18 − .33 i. (a) (b) (c) FIG. 9: Squared modulus of wave functions calculated within the superscar model (17) and corresponding to the parameters of Fig. 8. FIG. 10: Unfolding of the diamond periodic orbit. Thick lines indicate the initial triangle. straight channels obtained by unfolding classical motion (see Fig. 10). These channels (hatched area Fig. 10) are restricted by straight lines passing through cavity cor- ners. In [22] it was demonstrated that strong quantum mechanical diffraction on these singular corners forces wave functions in the semiclassical limit to obey simple boundary conditions on these (fictitious) channel bound- aries. More precisely it was shown that for billiard prob- lems Ψ on these boundaries take values of the order of k) → 0 when k → ∞. This result was obtained by using the exact solution for the scattering on peri- odic array of half-planes. No such results are known for dielectric problems. Nevertheless, it seems natural from semiclassical considerations that a similar phenomenon should appear for dielectric polygonal cavities as well. Within such framework, a superscar state can be con- structed explicitly as follows. After unfolding (see Fig. 10), a periodic orbit channel has the form of a rectan- gle. Its length equals the periodic orbit length and its width is determined by the positions of the closest sin- gular corners. The unfolded superscar state corresponds to a simple plane wave propagating inside the rectangle taking into account all phase changes. It cancels at the fictitious boundaries parallel to the x direction and is periodic along this direction with a periodicity imposed by the chosen symmetry class. This procedure sets the wavenumber of the state and the true wavefunction is obtained by folding back this superscar state. Superscar wave functions with (− −) symmetry asso- ciated with the diamond orbit (see Fig. 10) are expressed as follows: Ψ(− −)m,p (x, y) = sin κ(−)m x + sin κ(−)m x ′ − 2δ where x′ and y′ are coordinates symmetric with respect to square side. In coordinates as in Fig. 10 x′ = y , y′ = x In (17) m and p are integers with p = 1, 2, . . . , and m ≫ 1. l = 2a is the half of the diamond periodic orbit length [30], δ is the phase of the reflection coeffi- cient defined by r = exp(−2iδ). For simplicity, we ig- nore slight changes of the reflection coefficient for differ- ent plane waves in the functions above. So δ is given by (3) with ν = 1 for TM polarization and θ = π/4. And m is the momentum defined by κ(−)m l− 4δ = 2πm . (18) This construction conducts to the following expression for the real part of the wavenumbers [31] neff lRe(km,p) = 2π δ)2 + p2 = 2π(m+ δ) +O( 1 ) . (19) To check the accuracy of the above formulae we plot in Fig. 9 scar wave functions (17) with the same parameters as those in Fig. 8. The latter were computed numerically by direct solving the Helmholtz equations (6) but the former looks very similar which supports the validity of the superscar model. The real part of the wavenumbers is tested too. In Fig. 7 (b) the lowest loss states (with the smallest mod- ulus of the imaginary part) with (− −) symmetry are presented over a larger interval than in Fig. 7 (a). The real parts of these states are compared to superscar pre- dictions (19) with p = 1, leading to a good agreement. To detect small deviations from the theoretical formula, we plot in the inset of Fig. 7 (b) the difference between a quantity inferred from numerical simulations and its superscar prediction from (19). δ)2 + p2 . (20) From this curve it follows that this difference tends to zero with m increasing, thus confirming the existence of the term proportional to p2. By fitting this difference with the simplest expression (a) (b) FIG. 11: Simplest whispering gallery periodic orbit family for a pentagonal cavity. (a) Solid line indicates the inscribed pentagon which is an isolated periodic orbit. A periodic orbit in its vicinity is plotted with dashed line. It belongs to the family of the five-pointed star periodic orbit. The fundamen- tal domain is indicated in grey. (b) Boundary of the family of the five-pointed star periodic orbit. we find that c ≃ −6.9. By subtracting this correction term from the difference (20), one gets the curve indi- cated with filled circles in inset of Fig. 7 (b). The result is one order of magnitude smaller than the difference it- self. All these calculations confirm that the real parts of resonance wavenumbers for square dielectric cavities are well reproduced in the semiclassical limit by the above superscar formula (19) and our experimental results can be considered as an implicit experimental confirmation of this statement. V. PENTAGONAL MICRO-CAVITY The trace formula and superscar model arguments can be generalized to all polygonal cavities. The pentagonal resonator provides a new interesting test. In fact, due to the odd number of sides, the inscribed pentagonal orbit (indicated by solid line in Fig. 11 (a)) is isolated. The shortest confined periodic orbit family is twice longer. It is represented with a dashed line in Fig. 11 (a) and can be mapped onto the five-pointed star orbit drawn in Fig. 11 (b) by continuous deformation. In this Section we compare the predictions of the superscar model for this periodic orbit family with numerical simulations and experiments. Due to the C5v symmetry, pentagonal cavities sustain 10 symmetry classes corresponding to the rotations by 2π/5 and the inversion with respect to one of the sym- metry axis. We have studied numerically one symmetry class in which wave functions obey the Dirichlet bound- ary conditions along two sides of a right triangle with angle π/5 (see Fig. 11 (a) in grey). The results of these computations are presented in Fig. 12. As for the square cavity, lowest loss states are orga- nized in families. The wave functions of the three lowest loss families are plotted in Fig. 13 and their superscar structure is obvious. The computation of pure superscar states can be per- 40 60 80 100 Re (a k) 8 12 16 20 −0.25 FIG. 12: (a) Wave numbers for a pentagonal cavity. a is the side length of the cavity. The three most confined families are indicated by solid, dashed and long-dashed lines. (b) The difference (28) between the real part of quasi-energies and superscar expression (25) for the three indicated families in formed as in the previous Section. The five-pointed star periodic orbit channel is shown in Fig. 15. In this case boundary conditions along horizontal boundaries of pe- riodic orbit channel are not known. By analogy with su- perscar formation in polygonal billiards [22], we impose that wave functions tend to zero along these boundaries when k → ∞. Therefore, a superscar wave function propagating in- side this channel takes the form Ψscar(x, y) = exp(iκx) sin( py)Θ(y)Θ(w − y) , (22) where w is the width of the channel (for the five-pointed star orbit w = a sin(π/5) where a is the length of the pen- tagon side). Θ(x) is the Heavyside function introduced here to stress that superscar functions are zero (or small) outside the periodic orbit channel. The quantized values of the longitudinal momentum, κ, are obtained by imposing that the function (22) is periodic along the channel when all phases due to the reflection with the cavity boundaries are taken into ac- count κL = 2π . (23) Here M is an integer and L is the total periodic orbit length. For the five-pointed star orbit (see Fig. 11) L = 10a cos( ) , (24) and δ is the phase of the reflection coefficient given by (3) with ν = 1 (for TM polarization) and θ = 3π/10. For these states the real part of the wavenumber is the following nLRek = 2π +O( 1 ) . (25) Wave function inside the cavity are obtained by folding back the scar function (22) and choosing the correct rep- resentative of the chosen symmetry class. When Dirich- let boundary conditions are imposed along two sides of a right triangle passing through the center of the pentagon (see Fig. 15), M must be written as M = 5(2m) if p is odd and M = 5(2m − 1) if p is even. Then the wave function inside the triangle is the sum of two terms Ψm,p(x, y) = sin(κmx) sin( py)Θ(y)Θ(w − y) + + sin(κmx ′ − 2δ) sin( π py′)Θ(y′)Θ(w − y′) (26) where the longitudinal momentum is = 2π(m+ δ − ξ) (27) with ξ = 0 for odd p and ξ = 1/2 for even p. x′ and y′ in (26) are coordinates of the point symmetric of (x, y) with respect to the inversion on the edge of the pentagon. In the coordinate system when the pentagon edge passes through the origin (as in Fig. 15) x′ = x cos 2φ+ y sin 2φ , y′ = x sin 2φ− y cos 2φ and φ = π/5 is the inclination angle of the pentagon side with respect to the abscissa axis. Wavefunctions ob- tained with this construction are presented in Fig. 14. They correspond to the first, second, and third perpen- dicular excitations of the five-star periodic orbit family (p = 1, 2, and 3). To check the agreement between numerically computed real parts of the wavenumbers and the superscar predic- tion (25) and (27), we plot in Fig. 12 (b) the following (a) (b) (c) FIG. 13: Squared modulus of wave functions for pentagonal cavity with (− −) symmetry calculated with numerical simulations. (a) ak = 104.7 − 0.017 i, (b) ak = 102.2 − 0.05 i, (c) ak = 105.0 − 0.12 i. (a) (b) (c) FIG. 14: Squared modulus of wave functions calculated within the superscar model and corresponding to the parameters of Fig. 13. FIG. 15: Unfolding of the five-star periodic orbit for a pen- tagonal cavity. Thick lines indicate the initial triangle. difference n cosφ Rekanum δ − ξ 2 tanφ . (28) For pure scar states ζ = p2. As our numerical sim- ulations have not reached the semiclassical limit (see scales in Figs. 7 and 12), we found it convenient to fit numerically the ζ constant. The best fit gives ζ ≈ 0.44, 2.33, and 5.51 for the three most confined families (for pure scar functions this constant is 1, 4, 9 respectively). The agreement is quite good with a relative accuracy of the order of 10−4 (see Fig. 12 (b)). Irrespective of precise value of ζ the total optical length, nL, is given by (25) and leads to an experimental prediction twice longer than the optical length of the inscribed pentagon, which is an isolated periodic orbit and thus can not base superscar wavefunctions. Comparison with experiments confirms the superscar nature of the most confined states for pentagonal res- onators. In fact, the spectrum and its Fourier transform in Fig. 16 correspond to a pentagonal micro-laser with side a = 80 µm, and show a periodic orbit with optical length 1040 ± 30 µm to be compared with the five-star optical length nfull10a cos(π/5) = 1061 µm. The agree- ment is better than 2%. This result is reproducible for cavities with the same size. Other sizes have been tested as well. For smaller cavities, the five-pointed star orbit is not identifiable due to lack of gain, whereas for bigger ones it is visible but mixed with non confined periodic orbits. This effect, not spe- cific to pentagons, can be assigned to the contribution of different periodic orbit families which become important when the lasing gain exceeds the refractive losses. We will describe this phenomenon in a future publication [24]. The good agreement of numerical simulations and ex- periments with superscar predictions gives an additional credit to the validity of this approach even for non-trivial configurations. 620615610605600 Wavelength (nm) 3.0x10 2.01.00.0 Optical length (nm) FIG. 16: Experimental spectrum of a pentagonal micro-laser of 80 µm side length. Inset: Normalized Fourier transform of the spectrum expressed as intensity versus wavenumber. VI. MICRO-DISKS Micro-disk cavities are the simplest and most widely used micro-resonators. In the context of this work, they are of interest because of the coexistence of several pe- riodic orbit families with close lengths. For low index cavities (n ∼ 1.5) each regular polygon trajectory with more than four sides is confined by total internal reflec- tion. In the two-dimensional approximation passive circu- lar cavities are integrable and the spectrum of quasi- stationary states can be computed from an explicit quan- tization condition J ′m(nkR) Jm(nkR) m (kR) m (kR) . (29) Here R is the radius of the disk, n the refractive index of the cavity, and ν = 1 (resp. ν = n2) for the TM (resp. TE) polarization. For each angular quantum number m, an infinite sequence of solutions, km,q, is deduced from (29). They are labeled by the q radial quantum number. (a) (b) FIG. 17: (a) Two examples of periodic orbits: the square and the pentagon. (b) Two representations of the square periodic orbit and the caustic of this family in red. For large |k| the km,l wavenumbers are obtained from a semiclassical expression (see e.g. [25]) and the density of quasi-stationary states (15) can be proved to be rewrit- ten as a sum over periodic orbit families. The derivation of this trace formula assumes only the semi-classical ap- proximation (|k|R ≫ 1) and can be done in a way similar to that of the billiard case (see e.g. [27]), leading to an expression closed to (16) d(k) ∝ |rp|Np cos(nLpk−Np (2δp + Here the p index specifies a periodic orbit family. This formula depends on periodic orbit parame- ters: the number of bounces on the boundary, Np, the incident angle on the boundary, χp, the length, Lp = 2NpR cos(χp), and the area covered by periodic orbit family, Ap = πR 2 cos2(χp), which is the area included between the caustic and the boundary (see Fig. 17 (b)). 2δp is the phase of the reflection coefficient at each bounce on the boundary (see Eq. (3)) and |rp| is its modulus. For orbits confined by total internal reflection δp does not depend on kR in the semi-classical limit, and rp is exponentially close to 1 [25, 26]. From (30) it follows that each periodic orbit is singled out by a weighing coefficient cp = |rp|Np . Considering the experi- mental values |k|R ∼ 1000, |rp| can be approximated to unity with a good accuracy for confined periodic orbits, and thus cp = depends only on geometrical quantities. Fig. 18 shows the evolution of cp for polygons when the number of sides is increasing. As the critical angle is close to 45◦, the diameter and triangle periodic orbits are not confined and the dominating contribution comes from the square periodic orbit. So we can reasonably conclude that the spectrum (15) of a passive two-dimensional micro-disk is dominated by the square periodic orbit. The experimental method described in the previous Sections has been applied to disk-shaped micro-cavities. A typical experimental spectrum is shown on Fig. 19 (a). The first peak of its Fourier transform (see Fig. 19 (b) 5.0 5.5 6.0 6.5 Geometrical length FIG. 18: Vertical red sticks: cp coefficient for polygons con- fined by total internal reflection (square, pentagon, hexagon, etc...). The dotted blue line indicates the position of the perimeter. inset) has a finite width coming from the experimental conditions (discretization, noise, etc...) and the contribu- tions of several periodic orbits. This width is represented as error bars on graph 19 (b). The continuous red line fitting the experimental data is surrounded by the dashed green line and the dotted blue line corresponding to the optical length of the square and hexagon respectively, calculated with nfull = 1.64 as in the previous Sections. The perimeter (continuous black line) overlaps with a large part of the error bars which evidences its contribu- tion to the spectrum, but it is not close to experimental data. These experimental results seem in good agreement with the above theoretical predictions. But actu- ally these resonances, usually called whispering gallery modes, are living close to the boundary. Thus both roughness and three-dimensional effects must be taken into account. At this stage it is difficult to evaluate and to measure correctly such contributions for each periodic orbit. For micro-disks with a small thickness (about 0.4 µm) and designed with lower roughness, the results are more or less similar to those presented in Fig. 19 (b). VII. CONCLUSION We demonstrate experimentally that the length of the dominant periodic orbit can be recovered from the spec- tra of micro-lasers with simple shapes. Taking into ac- count different dispersion corrections to the effective re- fractive index, a good agreement with theoretical predic- tions has been evidenced first for the Fabry-Perot res- onator. Then we have tested polygonal cavities both with experiments and numerical simulations, and a good agreement for the real parts of wavenumbers has been obtained even for the non trivial configuration of the pen- tagonal cavity. The observed dominance of confined short-period orbits is, in general, a consequence of the trace formula and the 2000I 630620610600 Wavelength (nm) 7060504030 Radius (µm) 2.0x10 1.00.0 Optical length (nm) FIG. 19: (a) Experimental spectrum of a micro-disk of 30 µm radius. (b) Optical length versus radius. The experiments (red points) are linearly fitted by the solid red line. The other lines correspond to theoretical predictions without any adjusted parameters: the dashed green line to the square, the dotted blue line to the hexagon, and the solid black line to the perimeter. Inset: Normalized Fourier transform of the spectrum in (a) expressed as intensity versus wavenumber. formation of long-lived states in polygonal cavities is re- lated to strong diffraction on cavity corners. Finally, the study of micro-disks highlights the case of several orbits and the influence of roughness and three- dimensional effect. Our study opens the way to a systematic exploration of spectral properties by varying the shape of the boundary. In increasing the experimental precision even tiny details of trace formulae will be accessible. The improvement of the etching quality will suppress the leakage due to sur- face roughness and lead to a measure of the diffractive mode losses which should depend on symmetry classes. From the point of view of technology, it will allow a better prediction of the resonator design depending on the ap- plications. From a more fundamental physics viewpoint, it may contribute to a better understanding of open di- electric billiards. VIII. ACKNOWLEDGMENTS The authors are grateful to S. Brasselet, R. Hierle, J. Lautru, C. T. Nguyen, and J.-J. Vachon for experimental and technological support and to C.-M. Kim, O. Bohi- gas, N. Sandeau, J. Szeftel, and E. Richalot for fruitful discussions. [1] A.E. Siegman, Lasers, (University Science Books, Mill Valley, California, 1986). [2] K. 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Bogomolny and C. Schmit, Phys. Rev. Lett. 92, 244102 (2004). [23] E. Bogomolny, B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, F. Schäfer, and C. Schmit, Phys. Rev. Lett. 97, 254102 (2006). [24] M. Lebental, N. Djellali, J.-S. Lauret, J. Zyss, R. Duber- trand, C. Schmit, and E. Bogomolny, in preparation. [25] J. Nöckel, PhD dissertation (1997). http://www.eng.yale.edu/stonegroup/publications.html [26] H. Schomerus and M. Hentschel, Phys. Rev. Lett. 96, 243903 (2006). [27] M. Brack, R.K. Bhaduri, Semiclassical physics, (Addison-Wesley Publishing Company, 1997). [28] For some samples, the underlying layer is silica with re- fractive index n2 = 1.45, so neff is slightly different. [29] This definition is consistent all over the paper. In the literature, these names are sometimes permutated. [30] For a given symmetry class, the length entering the quan- tization condition may be a part of the total periodic orbit length. [31] The estimation of the imaginary parts of these states as well as the field distribution outside the cavity is beyond the scope of this paper and will be discussed elsewhere. http://arxiv.org/abs/cond-mat/0610229 http://www.eng.yale.edu/stonegroup/publications.html

0704.1951

Zeta function and cryptographic exponent of supersingular curves of genus 2

ZETA FUNCTION AND CRYPTOGRAPHIC EXPONENT OF SUPERSINGULAR CURVES OF GENUS 2 GABRIEL CARDONA AND ENRIC NART Abstract. We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to k-isomorphism and we exhibit equations and the zeta function of all their k/k-twists. As an application we obtain a direct computation of the cryptographic exponent of the Jacobians of these curves. Introduction One-round tripartite Diffie-Hellman, identity based encryption, and short digital signatures are some problems for which good solutions have recently been found, making critical use of pairings on supersingular abelian varieties over a finite field k. The cryptographic exponent cA of a supersingular abelian variety A is a half- integer that measures the security against an attack on the DL problem based on the Weil or the Tate pairings. Also, it is relevant to determine when pairings can be efficiently computed. Rubin and Silverberg showed in [RS04] that this invariant is determined by the zeta function of A. In this paper we give a direct, non-algorithmic procedure to compute the zeta function of a supersingular curve of genus 2, providing thus a direct computation of the cryptographic exponent of its Jacobian. This is achieved in Sect. 1. For even characteristic the results are based on [MN06] and are summarized in Table 2; for odd characteristic we use results of Xing and Zhu on the structure of the group of k-rational points of a supersingular abelian surface and we almost determine the zeta function in terms of the Galois structure of the set of Weierstrass points of the curve (Tables 3, 4). In the rest of the paper we obtain a complete answer in the case of curves with many automorphisms. In Sect. 2 we study the extra information provided by these automorphisms and we show how to obtain the relevant data to compute the zeta funtion of a twisted curve in terms of data of the original curve and the 1-cocycle defining the twist. In Sect. 3 we select a family of representatives of these curves up to k-isomorphism and we apply the techniques of the previous section to deal with each curve and all its k/k-twists. The results are displayed in an Appendix in the form of tables. In what cryptographic applications of pairings concerns, curves with many au- tomorphisms are interesting too because they are natural candidates to provide The authors acknowledge support from the projects MTM2006-15038-C02-01 and MTM2006- 11391 from the Spanish MEC. http://arxiv.org/abs/0704.1951v1 distortion maps on the Jacobian. In this regard the computation of the zeta func- tion is a necessary step to study the structure of the endomorphism ring of the Jacobian (cf. [GPRS06]). 1. Zeta Function and Cryptographic Exponent Let p be a prime number and let k = Fq be a finite field of characteristic p. We denote by kn the extension of degree n of k in a fixed algebraic closure k, Gk := Gal(k/k) is the absolute Galois group of k, and σ ∈ Gk the Frobenius automorphism. Let C be a projective, smooth, geometrically irreducible, supersingular curve of genus 2 defined over k. The Jacobian J of C is a supersingular abelian surface over k (the p-torsion subgroup of J(k) is trivial). Let us recall how supersingularity is reflected in a model of the curve C: Theorem 1.1. If p is odd, any curve of genus 2 defined over k admits an affine Weierstrass model y2 = f(x), with f(x) a separable polynomial in k[x] of degree 5 or 6. The curve is supersingular if and only if M (p)M = 0, where M , M (p) are the matrices: cp−1 cp−2 c2p−1 c2p−2 , M (p) = p−1 c 2p−1 c , f(x)(p−1)/2 = If p = 2 a curve of genus 2 defined over k is supersingular if and only if it admits an affine Artin-Schreier model y2 + y = f(x), with f(x) an arbitrary polynomial in k[x] of degree 5. For the first statement see [Yui78] or [IKO86], for the second see [VV92]. For any simple supersingular abelian variety A defined over k, Rubin and Silver- berg computed in [RS04] the cryptographic exponent cA, defined as the half-integer such that qcA is the size of the smallest field F such that every cyclic subgroup of A(k) can be embedded in F ∗. This invariant refines the concept of embedding de- gree, formerly introduced as a measure of the security of the abelian variety against the attacks to the DLP by using the Weil pairing [MOV93] or the Tate pairing [FR94] (see for instance [Gal01]). Let us recall the result of Rubin-Silverberg, adapted to the dimension two case. After classical results of Tate and Honda, the isogeny class of A is determined by the Weil polynomial of A, fA(x) = x 4 + rx3 + sx2 + qrx + q2 ∈ Z[x], which is the characteristic polynomial of the Frobenius endomorphism of the surface. For A supersingular the roots of fA(x) in Q are of the form q ζ, where q is the positive square root of q and ζ is a primitive m-th root of unity. Theorem 1.2. Suppose A is a simple supersingular abelian surface over Fq and let ℓ > 5 be any prime number dividing |A(Fq)|. Then, the smallest half-integer cA such that qcA − 1 is an integer divisible by ℓ is given by m/2, if q is a square, m/(2,m), if q is not a square . In particular, the cryptographic exponent cA is an invariant of the isogeny class of A. The complete list of simple supersingular isogeny classes of abelian surfaces can be found in [MN02, Thm. 2.9]. It is straightforward to find out the m-th root of unity in each case. We display the computation of cA in Table 1. Table 1. Cryptographic exponent cA of the simple supersingular abelian surface A with Weil polynomial fA(x) = x 4 + rx3 + sx2 + qrx + q2 (r, s) conditions on p and q cA (0,−2q) q nonsquare 1 (0, 2q) q square, p ≡ 1 (mod 4) 2 q, 3q) q square, p ≡ 1 (mod 3) 3/2 (−2√q, 3q) q square, p ≡ 1 (mod 3) 3 (0, 0) (q nonsquare, p 6= 2) or (q square, p 6≡ 1 (mod 8)) 4 (0, q) q nonsquare 3 (0,−q) (q nonsquare, p 6= 3) or (q square, p 6≡ 1 (mod 12)) 6 q, q) q square, p 6≡ 1 (mod 5) 5/2 (−√q, q) q square, p 6≡ 1 (mod 5) 5 5q, 3q) q nonsquare, p = 5 5 2q, q) q nonsquare, p = 2 12 Therefore, the computation of the cryptographic exponent of the Jacobian J of a supersingular curve C amounts to the computation of the Weil polynomial of J , which is related in a well-known way to the zeta function of C. We shall call fJ(x) the Weil polynomial of C too. The computation of fJ(x) has deserved a lot of attention because for the crypto- graphic applications one needs to know the cardinality |J(Fq)| = fJ(1) of the group of rational points of the Jacobian. However, in the supersingular case the current “counting points” algorithms are not necessary because there are more direct ways to compute the polynomial fJ(x). The aim of this section is to present these explicit methods, which take a different form for p odd or even. For p = 2 the computation of fJ(x) is an immediate consequence of the methods of [MN06], based on ideas of van der Geer-van der Vlugt; for p > 2 we derive our results from the group structure of J(Fq), determined in [Xin96], [Zhu00], and from the exact knowledge of what isogeny classes of abelian surfaces do contain Jacobians [HNR06]. In both cases we shall show that fJ (x) is almost determined by the structure as a Galois set of a finite subset of k, easy to compute from the defining equation of C. 1.1. Computation of the Zeta Function when p = 2. We denote simply by tr the absolute trace trk/F2 . Recall that ker(tr) = {x + x2 | x ∈ k} is an F2-linear subspace of k of codimension 1. Every projective smooth geometrically irreducible supersingular curve C of genus 2 defined over k admits an affine Artin-Schreier model of the type: C : y2 + y = ax5 + bx3 + cx+ d, a ∈ k∗, b, c, d ∈ k, which has only one point at infinity [VV92]. The change of variables y = y + u, u ∈ k, allows us to suppose that d = 0 or d = d0, with d0 ∈ k\ker(tr) fixed. Twisting C by the hyperelliptic twist consists in adding d0 to the defining equation. If we denote by J ′ the Jacobian of the twisted curve we have fJ′(x) = fJ(−x). Thus, for the computation of fJ(x) we can assume that d = 0. The structure as a Gk-set of the set of roots in k of the polynomial P (x) = a2x5 + b2x+ a ∈ k[x] almost determines the zeta function of C [MN06, Sect.3]. Table 2. Weil polynomial x4 + rx3 + sx2 + qrx+ q2 of the curve y2+ y = ax5+ bx3+ cx, for q nonsquare (left) and q square (right) P (x) N, M (r, s) (1)(4) N = 0 (± 2q, 2q) N = 1 (0, 0) (2)(3) M = 0 (± 2q, q) M = 1 (0, q) N = 0 (±2 2q, 4q) (1)3(2) N = 1 (0, 2q) N = 2 (0, 0) N = 3 (0,−2q) P (x) N, M (r, s) (5) (±√q, q) N = 0 (0,−q) (1)2(3) N = 1 (0, q) N = 2 (±2√q, 3q) M = 0 (±2√q, 2q) (1)(2)2 M = 1 (0, 0) M = 2 (0, 2q) N = 1 (0,−2q) (1)5 N = 3 (0, 2q) N = 5 (±4√q, 6q) In Table 2 we write P (x) = (n1) r1(n2) r2 · · · (nm)rm to indicate that ri of the irreducible factors of P (x) have degree ni. Also, we consider the linear operator T (x) := tr((c+ b2a−1)x) and we define N := number of roots z ∈ k of P (x) s.t. T (z) = 0, M := number of irred. quadratic factors x2 + vx+ w of P (x) s.t. T (v) = 0 . The ambiguity of the sign of r can be solved by computing nD in the Jacobian, where n is one of the presumed values of |J(Fq)| and D is a random rational divisor of degree 0. 1.2. Computation of the Zeta Function when p is odd. Let A be a super- singular abelian surface over k and let rk2(A) := dimF2(A[2](k)). The structure of A(k) as an abelian group was studied in [Xin96], [Zhu00], where it is proven that it is almost determined by the isogeny class of A. In fact, if Fi(x) are the different irreducible factors of fA(x) in Z[x]: fA(x) = Fi(x) ei , 1 ≤ s ≤ 2 =⇒ A(k) ≃ ⊕si=1 (Z/Fi(1)Z) except for the following cases: (a) p ≡ 3 (mod 4), q is not a square and fA(x) = (x2 + q)2, (b) p ≡ 1 (mod 4), q is not a square and fA(x) = (x2 − q)2. (c) q is a square and fA(x) = (x 2 − q)2. The possible structure of A(k) in cases (a) and (b) is: A(k) ≃ (Z/F (1)Z)m ⊕ (Z/(F (1)/2)Z⊕ Z/2Z)n , where F (x) denotes respectively x2 + q, x2 − q, and m, n are non-negative integers such that m+ n = 2 [Zhu00, Thm. 1.1]. In case (c) we have either: A(k) ≃ (Z/((q − 1)/2)Z)2 ⊕ (Z/2Z)2 , or A(k) ≃ (Z/((q − 1)/2m)Z)⊕ (Z/((q − 1)/2n)Z)⊕ (Z/2m+nZ) , where 0 ≤ m, n ≤ v2(q − 1) [Xin96, Thm. 3]. In this last case we have rk2(A) > 1; in fact, v2(1 − q) + v2(1 + q) = v2(1 − q) = (1/2)v2(F (1)) and we can apply [Xin96, Lem. 4] to conclude that A(k) has a subgroup isomorphic to (Z/2Z)2. Table 3. Weil polynomial x4 + rx3 + sx2 + qrx+ q2 of the curve C when q is nonsquare. The sign ǫ is the Legendre symbol (−1/p) W p rk2(J) (r, s) (1)6 or (1)4(2) 4, 3 (0,−2ǫq) (1)2(2)2 or (2)3 2 (0,±2q) (1)3(3) 2 not possible (1)(2)(3) p > 3 1 not possible p = 3 (± 3q, 2q) (1)2(4) or (2)(4) 1 (0, 0) (1)(5) p 6= 5 0 not possible p = 5 (± 5q, 3q) p ≡ 1 (mod 3) (0, q) (3)2 p ≡ −1 (mod 3) 0 (0, ǫq) p = 3 not possible (6) p ≡ −1 (mod 3) 0 (0,±q) p 6≡ −1 (mod 3) (0, q) Consider now a supersingular curve C of genus 2 defined over k, given by a Weierstrass equation y2 = f(x), for some separable polynomial f(x) ∈ k[x] of degree 5 or 6. Let J be its Jacobian variety, W = {P0, P1, P2, P3, P4, P5} ⊆ C(k) the set of Weierstrass points of C, andW (k) ⊆W the subset of k-rational Weierstrass points. Our aim is to show that the structure ofW as a Gk-set contains enough information on the 2-adic value of |C(k)| and |J(k)| to almost determine the polynomial fJ(x) = x4 + rx3 + sx2 + qrx+ q2. From the fundamental identities |C(k)| = q + 1 + r, |J(k)| = fJ(1) = (q2 + 1) + (q + 1)r + s, and the free action of the hyperelliptic involution on C(k) \W (k) we get (1) r ≡ |W (k)| (mod 2), s ≡ |J(k)| (mod 2) . On the other hand, J [2] is represented by the classes of the 15 divisors: Pi − P0, 1 ≤ i ≤ 5, and Pi + Pj − 2P0, 1 ≤ i < j ≤ 5, together with the trivial class. Lemma 1.3. Let D = Pi − Pj, with i 6= j, or D = Pi + Pj − 2P0, with 0, i, j pairwise different. Then, the class of the divisor D is k-rational if and only if Pi, Pj are both k-rational or quadratic conjugate. Hence, the Galois structure of W determines rk2(J) and this limits the possible values of the zeta function of C. Our final results are given in Tables 3, 4, where we write W = (n1) r1(n2) r2 · · · (nm)rm to indicate that there are ri Gk-orbits of length ni of Weierstrass points. If f(x) has degree 6 this Galois structure mimics the decomposition f(x) = (n1) r1(n2) r2 · · · (nm)rm (same notation as in Sect. 1.1) of f(x) into a product of irreducible polynomials k[x]. If f(x) has degree 5 then W = (1)f(x), because in these models the point at infinity is a k-rational Weierstrass point. The proof of the content of Tables 3 and 4 is elementary, but long. Instead of giving all details we only sketch the main ideas: Table 4. Weil polynomial x4 + rx3 + sx2 + qrx+ q2 of the curve C when q is a square W p rk2(J) (r, s) (1)6 4 (0,−2q) or (±4√q, 6q) (1)4(2) 3 (0,−2q) (1)2(2)2 or (2)3 2 (0,±2q) (1)3(3) p > 3 2 not possible p = 3 (±√q, 0) (1)(2)(3) 1 not possible (1)2(4) or (2)(4) p ≡ 1 (mod 8) 1 not possible p 6≡ 1 (mod 8) (0, 0) (1)(5) p ≡ 1 (mod 5) 0 not possible p 6≡ 1 (mod 5) (±√q, q) (3)2 (0, q) or (±2√q, 3q) (6) p ≡ 5 (mod 12) 0 (0,±q) p 6≡ 5 (mod 12) (0, q) (I) Waterhouse determined all possible isogeny classes of supersingular elliptic curves [Wat69]. Thus, it is possible to write down all isogeny classes of supersingular abelian surfaces by adding to the simple classes given in Table 1 the split isogeny classes. By [HNR06] we know exactly what isogeny classes of abelian surfaces do not contain Jacobians and they can be dropped from the list. By the results of Xing and Zhu we can distribute the remaining isogeny classes according to the possible values of rk2. (II) Each structure of W as a Gk-set determines the value of rk2 and, after (I), it has a reduced number of possibilities for the isogeny classes. By using (1) and looking for some incoherence in the behaviour under scalar extension to k2 or k3 of both, the Galois structure of W and the possible associated isogeny classes, we can still discard some of these possibilities. In practice, among the few possibilities left in Tables 3 and 4 we can single out the isogeny class of the Jacobian of any given supersingular curve by computing iterates of random divisors of degree zero. However, if C has many automorphisms they provide enough extra information to completely determine the zeta function. This will be carried out in the rest of the paper. In the Appendix we display equations of the supersingular curves with many automorphisms and their Weil polynomial. 2. Zeta Function of Twists In this section we review some basic facts about twists and we show how to compute different properties of a twisted curve in terms of the defining 1-cocycle. From now on the ground field k will have odd characteristic. Let C be a supersingular curve of genus 2 defined over k and let W ⊆ C(k) be the set of Weierstrass points of C. We denote by Aut(C) the k-automorphism group of C and by Autk(C) the full automorphism group of C. Let φ : C −→ P1 be a fixed k-morphism of degree 2 and consider the group of reduced geometric automorphisms of C: (C) := {u′ ∈ Autk(P 1) | u′(φ(W )) = φ(W )} . We denote by Aut′(C) the subgroup of reduced automorphisms defined over k. Any automorphism u of C fits into a commutative diagram: // P1 for certain uniquely determined reduced automorphism u′. The map u 7→ u′ is a group homomorphism (depending on φ) and we have a central exact sequence of groups compatible with Galois action: 1 −→ {1, ι} −→ Autk(C) φ−→ Aut′ (C) −→ 1, where ι is the hyperelliptic involution. This leads to a long exact sequence of Galois cohomology sets: (2) 1 → {1, ι} → Aut(C) φ→ Aut′(C) δ→ H1(Gk, {1, ι}) → H1(Gk,Autk(C)) → → H1(Gk,Aut′k(C)) → H 2(Gk, {1, ι}) ≃ Br2(k) = 0 . The k/k-twists of C are parameterized by the pointed set H1(Gk,Autk(C)) and, since k is a finite field, a 1-cocycle is determined just by the choice of an automorphism v ∈ Autk(C). The twisted curve Cv associated to v is defined over k and is determined, up to k-isomorphism, by the existence of a k-isomorphism f : C −→ Cv, such that f−1fσ = v. For instance, the choice v = ι corresponds to the hyperelliptic twist C′; if C is given by a Weierstrass equation y2 = f(x) then C′ admits the model y2 = tf(x), for t ∈ k∗\(k∗)2. We say that C is self-dual if it is k-isomorphic to its hyperelliptc twist. If fJ(x) is the Weil polynomial of C, the Weil polynomial of C ′ is fJ′(x) = fJ(−x); in particular, for a self-dual curve one has fJ(x) = x 4+ sx2+ q2 for some integer s. It is easy to deduce from (2) the following criterion for self-duality: Lemma 2.1. The curve C is self-dual if and only if |Aut′(C)| = |Aut(C)|. One can easily compute the data Aut(Cv), Aut ′(Cv) of the twisted curve Cv, in terms of Autk(C), Aut (C) and the 1-cocycle v. Let f : C −→ Cv be a geometric isomorphism such that f−1fσ = v. We have Autk(Cv) = f Autk(C)f −1, and the k-automorphism group is (3) Aut(Cv) = {fuf−1 | u ∈ Autk(C), u v = v u Once we fix any k-morphism of degree two, φv : Cv −→ P1, it determines a unique geometric automorphism f ′ of P1 such that φvf = f ′φ. The reduced group of k-automorphisms of Cv is (4) Aut′(Cv) = {f ′u′(f ′)−1 | u′ ∈ Aut′k(C), u ′ v′ = v′(u′)σ} . In order to compute the zeta function of Cv we consider the geometric isomor- phism f : J −→ Jv induced by f . We still have f−1fσ = v∗, where v∗ is the automorphism of J induced by v. Clearly, πvf = f σπ, where π, πv are the respec- tive q-power Frobenius endomorphisms of J, Jv. Hence, f−1πvf = f −1fσ(fσ)−1πvf = v∗π . In particular, πv has the same characteristic polynomial than v∗π. From this fact one can deduce two crucial results (cf. [HNR06, Props.13.1,13.4]). Proposition 2.2. Suppose q is a square. Let C be a supersingular genus 2 curve over k with Weil polynomial (x + q)4 and let v be a geometric automorphism of C, v 6= 1, ι. Then, the Weil polynomial x4+rx3+sx2+rx+q2 of Cv is determined as follows in terms of v (in the column v6 = 1 we suppose v2 6= 1, v3 6= 1, ι): v v2= 1 v2= ι v3= 1 v3= ι v4= ι v5= 1 v5= ι v6= 1 v6= ι (r, s) (0,−2q) (0, 2q) (−2√q, 3q) (2√q, 3q) (0, 0) (−√q, q) (√q, q) (0, q) (0,−q) Proposition 2.3. Suppose q is nonsquare. Let C be a supersingular genus 2 curve over k with Weil polynomial (x2 + ǫq)2, ǫ ∈ {1,−1}, and let v be a geometric automorphism of C. Then, the Weil polynomial x4 + rx3 + sx2 + rx + q2 of Cv is determined as follows in terms of the order n of the automorphism vvσ: n 1 2 3 4 6 (r, s) (0, 2ǫq) (0,−2ǫq) (0,−ǫq) (0, 0) (0, ǫq) In applying these results the transitivity property of twists can be helpful. Lemma 2.4. Let u, v be automorphisms of C and let f : C → Cv be a geometric isomorphism with f−1fσ = v. Then the curve Cu is the twist of Cv associated to the automorphism fuv−1f−1 of Cv. For a curve with a large k-automorphism group the following remark, together with Tables 3 and 4, determines in some cases the zeta function: Lemma 2.5. Let F ⊆ C(k) be the subset of k-rational points of C that are fixed by some non-trivial k-automorphism of C. Then, |C(k)| ≡ |F| (mod |Aut(C)|) . Proof. The group Aut(C) acts freely on C(k) \ F . � Note that F contains the set W (k) of k-rational Weierstrass points, all of them fixed by the hyperelliptic involution ι of C. In order to apply this result to the twisted curve Cv we need to compute the Gk-set structure of Wv and |Fv| solely in terms of v. Lemma 2.6. (1) For any P ∈ W the length of the Gk-orbit of f(P ) ∈ Wv is the minimum positive integer n such that v vσ · · · vσn−1(P σn) = P . In particular, |Wv(k)| = |{P ∈ W | v(P σ) = P}|. (2) The map f−1 stablishes a bijection between Fv and the set {P ∈ C(k) | v(P σ) = P = u(P ) for some 1 6= u ∈ Autk(C), s.t. u v = v uσ}. 3. Supersingular curves with many automorphisms For several cryptographic applications of the Tate pairing the use of distor- tion maps is essential. A distortion map is an endomorphism ψ of the Jacobian J of C that provides an input for which the value of the pairing is non-trivial: eℓ(D1, ψ(D2)) 6= 1 for some fixed ℓ-torsion divisors D1, D2. The existence of such a map is guaranteed, but in practice it is hard to find it in an efficient way. Usually, one can start with a nice curve C with many automorphisms, consider a concrete automorphism u 6= 1, u 6= ι, and look for a distortion map ψ in the subring Z[π, u∗] ⊆ End(J), where π is the Frobenius endomorphism of J and u∗ is the automorphism of the Jacobian induced by u. If Z[π, u∗] = End(J) it is highly probable that a distortion map is found. If Z[π, u∗] 6= End(J) it can be a hard problem to prove that some nice candidate is a distortion map, but at least one is able most of the time to find a “denominator” m such that mψ lies in the subring Z[π, u∗]; in this case, if ℓ ∤ m one can use mψ as a distortion map on divisors of order ℓ. Several examples are discussed in [GPRS06]. The aim of this section is to exhibit all supersingular curves of genus 2 with many automorphisms, describe their automorphisms, and compute the character- istic polynomial of π, which is always a necessary ingredient in order to analyze the structure of the ring Z[π, u∗]. Recall that a curve C is said to have many auto- morphisms if it has some geometric automorphism other than the identity and the hyperelliptic involution; in other words, if |Autk(C)| > 2. Igusa found equations for all geometric curves of genus 2 with many automor- phisms, and he grouped these curves in six families according to the possible struc- ture of the automorphism group [Igu60], [IKO86]. Cardona and Quer found a faith- ful and complete system of representatives of all these curves up to k-isomorphism and they gave conditions to ensure the exact structure of the automorphism group of each concrete model [Car03], [CQ06]. The following theorem sums up these results. Theorem 3.1. Any curve of genus 2 with many automorphisms is geometrically isomorphic to one and only one of the curves in these six families: Equation of C Aut′ (C) Autk(C) y2 = x6 + ax4 + bx2 + 1 a, b satisfy (5) C2 C2 × C2 y2 = x5 + x3 + ax a 6= 0, 1/4, 9/100 C2 × C2 D8 y2 = x6 + x3 + a p 6= 3, a 6= 0, 1/4, −1/50 S3 D12 y2 = ax6 + x4 + x2 + 1 p = 3, a 6= 0 S3 D12 y2 = x6 − 1 p 6= 3, 5 D12 2D12 y2 = x5 − x p 6= 5 S4 S̃4 p = 5 PGL2(F5) S̃5 y2 = x5 − 1 p 6= 5 C5 C10 (5) (4c3−d2)(c2−4d+18c−27)(c2−4d−110c+1125) 6= 0, c := ab, d := a3+ b3. Ibukiyama-Katsura-Oort determined, using Theorem 1.1, when the last three curves are supersingular [IKO86, Props. 1.11, 1.12, 1.13]: y2 = x6 − 1 is supersingular iff p ≡ −1 (mod 3) y2 = x5 − x is supersingular iff p ≡ 5, 7 (mod 8) y2 = x5 − 1 is supersingular iff p ≡ 2, 3, 4 (mod 5) It is immediate to check that y2 = ax6 + x4 + x2 + 1 is never supersingular if p = 3. One can apply Theorem 1.1 to the other curves in the first three families to distinguish the supersingular ones. Theorem 3.2. Suppose q is a square and let C be a supersingular curve belonging to one of the first five families of Theorem 3.1. Then there a twist of C with Weil polynomial (x+ q)4, and this twist is unique. Proof. Let E be a supersingular elliptic curve defined over Fp. By [IKO86, Prop. 1.3] the Jacobian J of C is geometrically isomorphic to the product of two supersin- gular elliptic curves, which is in turn isomorphic to E×E by a well-known theorem of Deligne. The principally polarized surface (J,Θ) is thus geometrically isomorphic to (E × E, λ) for some principal polarization λ. Since E has all endomorphisms defined over Fp2 , (E × E, λ) is defined over Fp2 and by a classical result of Weil it is Fp2 -isomorphic to the canonically polarized Jacobian of a curve C0 defined over Fp2 . By Torelli, C0 is a twist of C. The Weil polynomial of C0 is (x ± because the Frobenius polynomial of E is x2 + p. The fact that C0 and C 0 are the unique twists of C0 with Weil polynomial (x ± q)4 is consequence of Proposition 2.2. � � Corollary 3.3. Under the same assumptions: (1) The Weil polynomial of C is (x±√q)4 if and only if W = (1)6. (2) If C belongs to one of the first three families of Theorem 3.1, then it admits no twist with Weil polynomial x4 ± qx2 + q2 or x4 + q2. (3) If any of the curves y2 = x5 + x3 + ax, y2 = x6 + x3 + a is supersingular then a ∈ Fp2 . Proof. (1) By Table 4, the set W0 of Weierstrass points of C0 has Gk-structure W0 = (1) 6 and Lemma 2.6 shows that for all automorphisms v 6= 1, ι one has Wv 6= (1)6; thus, only the twists C0 and C′0 have W = (1)6. (2) The geometric automorphisms v of C0 satisfy neither v 6 = 1, v2 6= 1, v3 6= 1, ι, nor v6 = ι, nor v4 = ι; thus, by Proposition 2.2 the Weil polynomial of a twist of C0 is neither x 4 ± qx2 + q2 nor x4 + q2. (3) The Igusa invariants of C0 take values in Fp2 and a can be expressed in terms of these invariants [CQ05]. � � In a series of papers Cardona and Quer studied the possible structures of the pointed sets H1(Gk,Autk(C)) and found representatives v ∈ Autk(C) (identified to 1-cocycles of H1(Gk,Autk(C))) of the twists of all curves with many automor- phisms [Car03], [CQ05], [Car06], [CQ06]. In the next subsections we compute the zeta function and the number of k-automorphisms of these curves when they are supersingular. A general strategy that works in most of the cases is to apply the techniques of Sect. 2 to find a twist of C with Weil polynomial (x ± √q)4 (for q square) or (x2±q)2 (for q nonsquare) and apply then Propositions 2.2, 2.3 to obtain the zeta function of all other twists of C. The results are displayed in the Appendix in the form of Tables, where we exhibit moreover an equation of each curve. 3.1. Twists of the curve C : y2 = x5 − 1, for p 6≡ 0, 1 (mod 5). We have φ(W ) = {∞} ∪ µ5 and Aut′k(C) ≃ µ5. The zeta function of C can be computed from Tables 3,4 and Lemma 2.5 applied to C⊗k2. If q 6≡ 1 (mod 5) the only twists are C, C′. If q ≡ 1 (mod 5) there are ten twists and their zeta function can be deduced from Proposition 2.2. Table 5 summarizes all computations. 3.2. Twists of the curve C : y2 = x5 − x, for p ≡ 5, 7 (mod 8). Now φ(W ) = {∞, 0, ±1, ±i}. If p = 5 we have Aut′ (C) = Aut(P1). If p 6= 5 the group Aut′ is isomorphic to S4 and it is generated by the transformations T (x) = ix, S(x) = , with relations S3 = 1 = T 4, ST 3 = TS2. For q nonsquare the zeta function of C is determined by Table 3; since the curve is defined over Fp we obtain the zeta function of C over k by scalar extension. In all cases we can apply Propositions 2.2 and 2.3 to determine the zeta function of the twists of C. Tables 6, 7, 8 summarize all computations. 3.3. Twists of the curve C : y2 = x6 − 1, for p ≡ −1 (mod 3), p 6= 5. We have φ(W ) = µ6 and Aut (C) = {±x, ±ηx,±η2x,± 1 }, where η ∈ Fp2 is a primitive third root of unity. The zeta function of C can be computed from Tables 3,4 and Lemma 2.5 applied to C and C⊗k2. The zeta function of all twists can be determined by Propositions 2.2, 2.3. Tables 9, 10 summarize all computations. 3.4. Twists of the supersingular curve C : y2 = x6 +x3 + a, for p > 3. Recall that a is a special value making the curve C supersingular and a 6= 0, 1/4, −1/50. We have now φ(W ) = {θ, ηθ, η2θ, A }, Aut′ (C) = {x, ηx, η2x, A where A, z, θ ∈ k satisfy A3 = a, z2 + z + a = 0, θ3 = z. The Galois action on W and on Aut′ (C) depends on z and a/z being cubes or not in their minimum field of definition k∗ or (k2) ∗. This is determined by the fact that a is a cube or not. Lemma 3.4. If a is a cube in k∗ then z, a/z are both cubes in k∗ or in (k2) according to 1− 4a ∈ (k∗)2 being a square or not. If a is not a cube in k∗ then z, a/z are both noncubes in k∗ or in (k2) ∗, according to 1− 4a ∈ (k∗)2 being a square or not. Proof. Let us check that all situations excluded by the statement lead to W = (1)3(3) or Wv = (1) 3(3) for some twist, in contradiction with Tables 3, 4. Suppose q ≡ −1 (mod 3). If 1− 4a is a square then a, z, a/z are all cubes in k∗. If 1 − 4a is not a square then a is a cube and if z, zσ are not cubes in k2 we have θσ = ω(A/θ), with ω3 = 1, ω 6= 1, and the twist by v = (ω−1(A/x), ay/x3) has Wv = (1) 3(3) by Lemma 2.6. Suppose q ≡ 1 (mod 3). If 1−4a is not a square we have z(q2−1)/3 = a(q−1)/3, so that a is a cube in k∗ if and only if z, zσ are cubes in k∗2 . Suppose now that 1−4a is a square. If exactly one of the two elements z, a/z is a cube we have W = (1)3(3); thus z, a/z are both cubes or noncubes in k∗. In particular, if a is not a cube then z, a/z are necessarily both noncubes. Finally, if a is a cube and z, a/z are noncubes in k∗, Lemma 2.6 shows that Wv = (1) 3(3) for the twist corresponding to v = (ηx, y). � � For the computation of the zeta functions of the twists it is useful to detect that some of the combinations a square/nonsquare and 1−4a square/nonsquare are not possible. Lemma 3.5. Suppose q ≡ 1 (mod 3). (1) If q ≡ −1 (mod 4) then 1− 4a is not a square. (2) If q is nonsquare then a is not a square. (3) If q is a square then a and 1− 4a are both squares. Proof. Let Cv be the twist of C corresponding to v(x, y) = (ηx, y). (1) Supose 1− 4a is a square. If a is a cube we have W = (1)6 and if a is not a cube we have Wv = (1) 6; by Table 3 we get (r, s) = (0,−2 q) in both cases. On the other hand, Lemmas 2.5 and 2.6 applied to C⊗k k2 show in both cases that s ≡ 1 (mod 3); thus, p ≡ 1 (mod 4). (2) Suppose a is a square. If a is a cube (respectively a is not a cube) we have W = (1)6 or W = (2)3 (respectively Wv = (1) 6 or Wv = (2) 3), according to 1 − 4a being a square or not. In all cases we have (r, s) = (0,±2q) by Table 3, and a straightforward application of Lemma 2.5 and (2) of Lemma 2.6 leads to r ≡ −1 (mod 3), which is a contradiction. (3) In all cases in which a or 1 − 4a are nonsquares we get (r, s) = (0, q) either for the curve C or for the curve Cv. This contradicts Corollary 3.3. � � After these results one can apply the general strategy. The results are displayed in Tables 11, 12, 13. 3.5. Twists of the supersingular curve C : y2 = x5+x3+ax. Recall that a is a special value making C supersingular and a 6= 0, 1/4, 9/100. Given z ∈ k satisfying z2 + z + a = 0 we have φ(W ) = {0, ∞, ± a/z}, Autk(C) = (ω2 x, ω y) | ω4 = 1 | w4 = a Lemma 3.6. If q ≡ 1 (mod 4) then a and 1−4a are both squares or both nonsquares in k∗. If q is a square then necessarily a and 1− 4a are both squares. Proof. If a 6∈ (k∗)2, 1−4a ∈ (k∗)2, thenW = (1)4(2) and (r, s) = (0,−2q) by Tables 3,4; this contradicts Lemma 2.5 because |Aut(C)| = |F| = 4 and r ≡ 2 (mod 4). Suppose now a ∈ (k∗)2, 1 − 4a 6∈ (k∗)2. If a ∈ (k∗)4 then W = (1)2(2)2 and (r, s) = (0,±2q); this contradicts Lemma 2.5 because |Aut(C)| = 8, |F| = 6 if q ≡ 1 (mod 8) and |F| = 2 or 10 if q ≡ 5 (mod 8), so that r ≡ 4 (mod 8) in both cases. If a 6∈ (k∗)4 we get a similar contradiction for the curve Cv for v(x, y) = (−x, iy). If a, 1 − 4a are nonsquares, then W = (1)2(4) and the Weil polynomial of C is x4 + q2 by Tables 3,4. If q is a square this contradicts Corollary 3.3. � � Lemma 3.7. If q is a square then a ∈ (k∗)4 if and only if z ∈ (k∗)2. Proof. Suppose a ∈ (k∗)4, z 6∈ (k∗)2 and let us look for a contradiction. Consider the k-automorphisms u(x, y) = (−x, iy), v(x, y) = (w y) of C, where w4 = a. By Lemma 2.6, Wu = (1) 6 and Cu has Weil polynomial (x ± q)4 by Corollary 3.3; since u2 = ι, the Weil polynomial of C is (x2 + q)2 by Proposition 2.2 and Lemma 2.4. The quotient E := C/v is an elliptic curve defined over k and the Frobenius endomorphism π of E must satisfy π2 = −q. Since q is a square, E has four automorphisms and its j invariant is necessarily jE = 1728. Now, E has a Weierstrass equation: Y 2 = (X + 2w)(X2 + 1 − 2w2), where X = (x2 + w2)x−1, Y = y(x + w)x−2 are invariant under the action of v. The condition jE = 1728 is equivalent to a = 0 (which was excluded from the beginning) or a = (9/14)2; in this latter case z is a square in Fp2 and we get a contradiction. Suppose now a 6∈ (k∗)4, z ∈ (k∗)2. We haveW = (1)6 and C has Weil polynomial (x ± √q)4 by Corollary 3.3. By Proposition 2.2, the Weil polynomial of Cu is (x2+q)2. For any choice of w = 4 a, the morphism f(x, y) = (x+w x−w , 1+2w2 (x−w)3 ) sets a k2-isomorphism between C and the model: Cu : y 2 = (x2 − 1)(x4 + bx2 + 1), b = (12 a− 2)/(2 a+ 1), of Cu. The quotient of this curve by the automorphism (−x, y) is the elliptic curve E : Y 2 = (X − 1)(X2 + bX + 1). Arguing as above, E has j-invariant 1728, and this leads to a = 0 (excluded from the beginning) or a = (9/14)2, which is a contradiction since a would be a fourth power in Fp2 . � � After these results one is able to determine the zeta function of all twists of C when q is a square; the results are displayed in Table 14. In the cases where the Weil polynomial is (x − ǫ√q)4, ǫ = ±1, the methods of section 2 are not sufficient to determine ǫ; our computation of this sign follows from a study of the 4-torsion of an elliptic quotient of the corresponding curve. In order to deal with the case q nonsquare we need to discard more cases. Lemma 3.8. Suppose q nonsquare. If q ≡ −1 (mod 4) then a and 1 − 4a cannot be both nonsquares. If q ≡ 1 (mod 4) and a ∈ (k∗)2 then a ∈ (k∗)4 if and only if z 6∈ (k∗)2. Proof. If a, 1 − 4a are both nonsquares the polynomial x4 + x2 + a is irreducible and the Weil polynomial of C is x4 + q2 by Table 3; hence, the Weil polynomial of C ⊗k k2 is (x2 + q2)2. If q ≡ −1 (mod 4) we have a ∈ k∗ ⊆ (k∗2)4 and this contradicts Table 14. Suppose q ≡ 1 (mod 4) and a ∈ (k∗)2; by Lemma 3.6, 1−4a is also a square and z ∈ k∗. If a ∈ (k∗)4 and z ∈ (k∗)2 we get W = (1)6, and (r, s) = (0,−2q) by Table 3; we get a contradiction because the Jacobian J of C is simple ([MN02, Thm. 2.9]) and C has elliptic quotients over k because the automorphisms (w2/x, (w3y)/x3) are defined over k. If a 6∈ (k∗)4 and z 6∈ (k∗)2 we get an analogous contradiction for the curve Cu twisted by u(x, y) = (−x, iy). � � The results for the case q nonsquare follow now by the usual arguments and they are displayed in Tables 15, 16. 3.6. Twists of the supersingular curve C : y2 = x6+ax4+bx2+1. Recall that a, b ∈ k are special values satisfying (5) and making C supersingular; in particular p > 3. The curve C has four twists because Autk(C) = Aut(C) = {(±x,±y)} is commutative and has trivial Galois action. The Jacobian of C is k-isogenous to the product E1 × E2 of the elliptic curves with Weierstrass equations y2 = x3 + ax2 + bx + 1, y2 = x3 + bx2 + ax + 1, obtained as the quotient of C by the respective automorphisms v = (−x, y), ιv = (−x,−y). For q nonsquare, these elliptic curves have necessarily Weil polynomial x2 + q and the Weil polynomial of C is (x2 + q)2. Lemma 3.9. If q is a square C has Weil polynomial (x±√q)4. Proof. By Theorem 3.2 and Proposition 2.2 C has Weil polynomial (x ± √q)4 or (x2 − q)2. In both cases the elliptic curves E1, E2 have Weil polynomial (x± and we claim that they are isogenous. Since E(k) ≃ (Z/(1±√q)Z)2 as an abelian group, our elliptic curves have four rational 2-torsion points and the polynomial Table 5. Twists of the curve y2 = x5 − 1 for p ≡ 2, 3, 4 (mod 5). The sign ǫ = ±1 is determined by √q ≡ ǫ (mod 5). The last row provides eight inequivalent twists corresponding to the four nontrivial values of t ∈ k∗/(k∗)5 Cv v (r, s) s.d. |Aut(Cv)| y2 = x5 − 1 (x, y) q ≡ ±2 (mod 5) q ≡ −1 (mod 5) q ≡ 1 (mod 5) (0, 0) (0, 2q) (−4ǫ√q, 6q) y2 = tx5 − 1, t 6∈ (k∗)5 (t 5 x, y) q ≡ 1 (mod 5) (ǫ√q, q) no 10 Table 6. Twists of the curve y2 = x5 − x when q ≡ −1 (mod 8) Cv v (r, s) s.d. |Aut(Cv)| y2 = x5 − x (x, y) (0, 2q) yes 8 y2 = x5 + x (ix, 1+i√ y) (0, 2q) yes 4 y2 = (x2 + 1)(x2 − 2tx − 1)(x2 + 2 x − 1) t2 + 1 6∈ (k∗)2 (− ) (0,−2q) yes 24 y2 = (x2 + 1)(x4 − 4tx3 − 6x2 + 4tx + 1), t2 + 1 6∈ (k∗)2 , i−1√ (0, 0) yes 4 y2 = x6 − (t + 3)x5 + 5( 2+t−s )x4 + 5(s − 1)x3 +5( 2−t−s )x2 + (t − 3)x + 1 irred., s2 + t2 = −2 2(1−i)y (x+i)3 (0, q) no 6 x3 + ax2 + bx + 1 has three roots e1, e2, e3 ∈ k. Since e1e2e3 = 1, either one or three of these roots are squares. If only one root is a square we have W = (1)2(2)2, Wv = (1) 4(2) and C, Cv have both Weil polynomial (x 2 ± q)2, in contradiction with Theorem 3.2. Hence, the three roots are squares, W = (1)6, and C has Weil polynomial (x±√q)4 by Corollary 3.3. � � The zeta function of the twists of C is obtained from Propositions 2.2 and 2.3. The results are displayed in Table 17. For q square the sign of (x ± √q)4 can be determined by analyzing the 4-torsion of the elliptic curve y2 = x3 + ax2 + bx+ 1. Finally, there are special curves over k whose geometric model y2 = x6 + ax4 + bx2+1 is not defined over k (cf. [Car03, Sect.1]). It is straightforward to apply the techniques of this paper to determine their zeta function too. 4. Appendix In this appendix we display in several tables the computation of the zeta func- tion of the supersingular curves of genus 2 with many automorphisms. For each curve Cv, we exhibit the number of k-automorphisms and the pair of integers (r, s) determining the Weil polynomial fJv(x) = x 4 + rx3 + sx2 + qrx+ q2 of Cv. In the column labelled “s.d” we indicate if C is self-dual. For the non-self-dual curves we exhibit only one curve from the pair Cv, C We denote by η, i ∈ k a primitive third, fourth root of unity. For n a positive integer and x ∈ k∗ we define νn(x) = 1 if x ∈ (k∗)n, νn(x) = −1 otherwise . In all tables the parameters s, t take values in k∗. Conclusion. We show that the zeta function of a supersingular curve of genus two is almost determined by the Galois structure of a finite set easy to describe in terms of a defining equation. For curves with many automorphisms this result is refined to obtain a direct (non-algoritmic) computation of the zeta function in all cases. As Table 7. Twists of the curve y2 = x5 − x when q ≡ 5 (mod 8) Cv v p (r, s) s.d. |Aut(Cv)| y2 = x5 − x (x, y) p > 5 p = 5 (0,−2q) yes 24 y2 = x5 − 4x (−x, iy) (0, 2q) yes 8 y2 = x5 − 2x (ix, 1+i√ y) (0, 0) yes 4 y2 = (x2 + 2)(x4 − 12x2 + 4) , i−1√ p > 5 p = 5 (0, 2q) yes y2 = f(t, x)f( 18+(5i−3)t (5i+3)−2t , x) f(t, x) = x3 − tx2 + (t − 3)x+ 1 irred. 2(1−i)y (x+i)3 p > 5 p = 5 (0, q) y2 = x5 − x − t, trk/F5 (t) = 1 (x + 1, y) p = 5 ( 5q, 3q) no 10 y2 = x6 + tx5 + (1 − t)x + 2, irred. ( 3 x−1 , (x+1)3 ) p = 5 (0,−q) yes 6 Table 8. Twists of the curve y2 = x5 − x when p ≡ 5, 7 (mod 8) and q is a square. Here ǫ = (−1/√q) and ǫ′ = (−3/√q) Cv v p (r, s) s.d. |Aut(Cv)| y2 = x5 − x (x, y) p > 5 p = 5 (−4ǫ√q, 6q) no 48 y2 = x5 − t2x, t 6∈ (k∗)2 (−x, iy) (0, 2q) yes 8 y2 = x5 − tx, t 6∈ (k∗)2 (ix, 1+i√ y) (0, 0) no 8 y2 = (x2 − t)(x4 + 6tx2 + t2), t 6∈ (k∗)2 , i−1√ p > 5 p = 5 (0,−2q) yes 4 y2 = (x3 − t)(x3 − (15 3 − 26)t), t 6∈ (k∗)3 2(1−i)y (x+i)3 p > 5 p = 5 q, 3q) no y2 = x5 − x − t, trk/F5(t) = 1 (x+ 1, y) p = 5 ( q, q) no 10 y2 = x6 + tx5 + (1 − t)x+ 2, irred. ( 3 x−1 , (x+1)3 ) p = 5 (0, q) no 12 Table 9. Twists of the curve y2 = x6 − 1 when q ≡ −1 (mod 3), p 6= 5. Here ǫ = (−1/p) Cv v (r, s) s.d. |Aut(Cv)| y2 = x6 − 1 (x, y) (0, 2q) iff ǫ = −1 6 + 2ǫ y2 = x6 − t, t 6∈ (k∗)2 (−x,−y) (0, 2q) iff ǫ = 1 6 − 2ǫ y2 = x(x2 − 1)(x2 − 9) ( 1 ) (0,−2ǫq) yes 12 y2 = (x4 − 2stx3 + (7s + 1)x2 + 2tsx + 1)· ·(x2 − 4 x− 1), t2 + 4 ∈ k∗ \ (k∗)2, s−1 = t2 + 3 (− ) (0, 2ǫq) yes 12 y2 = x6 + 6tx5 + 15sx4 + 20tsx3 + 15s2x2+ +6ts2x + s3, s = t2 − 4 6∈ (k∗)2, gcd(x(q+1)/3 − 1, x2 − tx + 1) = 1 ) (0, ǫq) yes 6 y2 = x6 + 6x5 + 15sx4 + 20sx3 + 15s2x2+ +6s2x + s3, s = t2/(t2 + 4) 6∈ (k∗)2, gcd(x(q+1)/3 + 1, x2 − tx − 1) = 1 ) (0,−ǫq) yes 6 Table 10. Twists of the curve y2 = x6 − 1 when p ≡ −1 (mod 3), p 6= 5 and q is a square. Here ǫ = (−3/√q) Cv v (r, s) s.d. |Aut(Cv)| y2 = x6 − 1 (x, y) (−4ǫ√q, 6q) no 24 y2 = x6 − t3, t 6∈ (k∗)2 (−x, y) (0,−2q) yes 12 y2 = x6 − t2, t 6∈ (k∗)3 (ηx, y) (2ǫ√q, 3q) no 12 y2 = x6 − t, t 6∈ ((k∗)2 ∪ (k∗)3) (−ηx,−y) (0, q) no 12 y2 = x(x2 + 3t)(x2 + t ), t 6∈ (k∗)2 ( 1 ) (0, 2q) yes 4 y2 = x6 + 15tx4 + 15t2x2 + t3, t 6∈ (k∗)2 (− 1 ) (0,−2q) yes 4 Table 11. Twists of the supersingular curve y2 = x6+x3+a, a 6= 0, 1/4,−1/50, when q ≡ −1 (mod 3). Here ǫ = ν2(a) and A is the cubic root of a in k Cv v (r, s) s.d. |Aut(Cv)| y2 = x6 + x3 + a (x, y) (0, 2q) iff ǫ = −1 3 + ǫ y2 = θ−3(x − θ)6 − g(x)3 + aθ3(x − θσ)6 g(x) min. polyn. of θ ∈ k2 \ k, Nk2/k(θ) = A y) (0, 2ǫq) iff ǫ = −1 9 + 3ǫ y2 = θ(x− η)6 − g(x)3 + aθ−1(x − η2)6 g(x) = x2 + x+ 1, θ ∈ k2 \ (k∗2) 3, Nk2/k(θ) = a y) (0,−ǫq) no 6 Table 12. Twists of the supersingular curve y2 = x6+x3+a, a 6= 0, 1/4,−1/50, when q ≡ 1 (mod 3) and q is nonsquare. Here A is a cubic root of a in k and n = 3, if a ∈ (k∗)3, whereas A = a, n = 1, if a 6∈ (k∗)3 Cv v ν3(a) (r, s) s.d. |Aut(Cv)| y2 = x6 + x3 + a (x, y) (0,−2q) (0, q) y2 = x6 + tx3 + t2a, t 6∈ (k∗)3 y2 = x6 + ax3 + a3 3 x, y) (0, q) (0,−2q) y2 = θ−n(x − θ)6 − g(x)3 + aθn(x− θσ)6 g(x) min. polyn. of θ ∈ k2 \ k, Nk2/k(θ) = A y) (0, 2q) yes 2 Table 13. Twists of the supersingular curve y2 = x6+x3+a, a 6= 0, 1/4,−1/50, when q is a square. Here ǫ = (−3/√q). Also, A is a cubic root of a in k and n = 3, if a ∈ (k∗)3, whereas A = a, n = 1, if a 6∈ (k∗)3 Cv v ν3(a) (r, s) s.d. |Aut(Cv)| y2 = x6 + x3 + a (x, y) (−4ǫ√q, 6q) q, 3q) y2 = x6 + tx3 + t2a, t 6∈ (k∗)3 y2 = x6 + ax3 + a3 3 x, y) q, 3q) (−4ǫ√q, 6q) no y2 = θ−n(x − θ)6 − g(x)3 + aθn(x− θσ)6 g(x) min. polyn. of θ ∈ k2 \ k, Nk2/k(θ) = A y) (0,−2q) no 4 Table 14. Twists of the supersingular curve y2 = x5 + x3 + ax, a 6= 0, 1/4, 9/100, when q is a square. The last row provides two inequivalent twists according to the two values of a. Here ǫ = −(−1/√q)ν4(z) and ǫ′ = −(−1/√q)ν4(tz), where z2 + z + a = 0 Cv v ν4(a) (r, s) s.d. |Aut(Cv)| y2 = x5 + x3 + ax (x, y) q, 6q) (0, 2q) y2 = x5 + tx3 + at2x, t 6∈ (k∗)2 (−x, t (0, 2q) q, 6q) y2 = g(x) θ2(x− θσ)4 + g(x)2+ +aθ−2(x− θ)4 , Nk2/k(θ) = g(x) min. polyn. of θ ∈ k2 \ k y) (0,−2q) yes 4 Table 15. Twists of the supersingular curve y2 = x5 + x3 + ax, a 6= 0, 1/4, 9/100, when q is nonsquare and a 6∈ (k∗)2 Cv v (−1/p) (r, s) s.d. |Aut(Cv)| y2 = x5 + x3 + ax (x, y) (0, 0) (0, 2q) y2 = (x2 − a) a)4 + (x2 − a)2+ +aθ−1(x + , θ ∈ k2, Nk2/k(θ) = a (0, 2q) (0, 0) Table 16. Twists of the supersingular curve y2 = x5 + x3 + ax, a 6= 0, 1/4, 9/100, when q is nonsquare and a ∈ (k∗)2. Here ǫ = (−1/p). If p ≡ −1 (mod 4) we assume that a belongs to (k∗)2 Cv v ν4(a) (r, s) s.d. |Aut(Cv)| y2 = x5 + x3 + ax (x, y) (0, 2q) (0,−2q) iff ǫ = −1 6 + 2ǫ y2 = x5 + tx3 + at2x, t 6∈ (k∗)2 (−x, t (0,−2ǫq) (0, 2q) y2 = g(x) θ2(x− θσ)4 + g(x)2+ +aθ−2(x − θ)4 , Nk2/k(θ) = g(x) min. polyn. of θ ∈ k2 \ k y) (0, 2q) iff ǫ = 1 6 − 2ǫ y2 = g(x) θ2(x− θσ)4 + g(x)2+ +aθ−2(x − θ)4 , Nk2/k(θ) = − g(x) min. polyn. of θ ∈ k2 \ k y) (0, 2ǫq) yes 4 Table 17. Twists of the supersingular curve y2 = x6 + ax4 + bx2 + 1 satisfying (5) Cv v (r, s) s.d. |Aut(Cv)| y2 = x6 + ax4 + bx2 + 1 (x, y) q nonsq. q square (0, 2q) (±4√q, 6q) no 4 y2 = x6 + atx4 + bt2x2 + t3 t 6∈ (k∗)2 (−x,−y) q nonsq. q square (0, 2q) (0,−2q) no 4 an application one gets a direct computation of the cryptographic exponent of the Jacobian of these curves. Also, the computation of the zeta function is necessary to determine the structure of the endomorphism ring of the Jacobian and to compute distortion maps for the Weil and Tate pairings. Acknowledgement. It is a pleasure to thank Christophe Ritzenthaler for his help in finding some of the equations of the twisted curves. References [Car03] G. Cardona, On the number of curves of genus 2 over a finite field, Finite Fields and Their Applications 9 (2003), 505-526. [CQ05] G. Cardona, J. Quer, Field of moduli and field of definition for curves of genus 2, in Computational aspects of algebraic curves (T. Shaska, ed.) pp. 71-83., Lecture Notes Series on Computing 13 (World Scientific). [Car06] G. Cardona, Representations of Gk-groups and the genus 2 curve y 2 = x5 − x, Journal of Algebra 303 (2006), 707-721. [CQ06] G. Cardona, J. Quer, Curves of genus 2 with group of automorphisms isomorphic to D8 or D12, Trans. Amer. Math. Soc. to appear. [FR94] G. Frey, H.-G. Rück, A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves, Mathematics of Computation 62 (1994), 865-874. [Gal01] S. D. 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Nart, Zeta functions of supersingular curves of genus 2, Canadian Journal of Mathematics 59 (2007), 372-392. [MOV93] A.J. Menezes, T. Okamoto, S.A. Vanstone, Reducing Elliptic Curve Logarithms to Log- arithms in a Finite Field, IEEE Trans. on Information Theory 39 (1993), 1639-1646. [RS04] K. Rubin, A. Siverberg, Supersingular abelian varieties in cryptology, In Advances in Cryptology-Crypto’2002, volume 2442 of Lecture Notes in Computer Science, 336-353. Springer-Verlag 2004. [VV92] G. van der Geer, M. van der Vlugt, Supersingular curves of genus 2 over finite fields of characteristic 2, Math. Nachrichten 159 (1992), 73-81. [Wat69] W.C. Waterhouse, Abelian varieties over finite fields, Annales Scientifiques de l’École Normale Supérieure (4) 2 (1969), 521-560. [Xin96] C.P. Xing, On supersingular abelian varieties of dimension two over finite fields, Finite Fields and Their Applications 2 (1996), 407-421. [Yui78] N. Yui, On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2, Journal of Algebra 52 (1978), 378-410. [Zhu00] H.J. Zhu, Group Structures of Elementary Supersingular Abelian Varieties over Finite Fields, Journal of Number Theory 81 (2000), 292-309. Dept. Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122, Palma de Mallorca, Spain E-mail address: [email protected] Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain E-mail address: [email protected] Introduction 1. Zeta Function and Cryptographic Exponent 1.1. Computation of the Zeta Function when p=2 1.2. Computation of the Zeta Function when p is odd 2. Zeta Function of Twists 3. Supersingular curves with many automorphisms 3.1. Twists of the curve C2mu-:6muplus1muy2=x5-1, for p0,1 (mod 5) 3.2. Twists of the curve C2mu-:6muplus1muy2=x5-x, for p5,7 (mod 8) 3.3. Twists of the curve C2mu-:6muplus1muy2=x6-1, for p-1 (mod 3), p=5 3.4. Twists of the supersingular curve C2mu-:6muplus1muy2=x6+x3+a, for p>3 3.5. Twists of the supersingular curve C2mu-:6muplus1muy2=x5+x3+ax 3.6. Twists of the supersingular curve C2mu-:6muplus1muy2=x6+ax4+bx2+1 4. Appendix References

0704.1952

Dynamic Effects Increasing Network Vulnerability to Cascading Failures

Dynamic Effects Increasing Network Vulnerability to Cascading Failures Ingve Simonsen,1, 2, ∗ Lubos Buzna,1, 3 Karsten Peters,1 Stefan Bornholdt,4 and Dirk Helbing1 1Dresden University of Technology, Andreas-Schubert-Straße 23, D-01086 Dresden, GERMANY 2Department of Physics, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, NORWAY 3University of Zilina, Univerzitna 8215/5, SK-01026 Zilina, SLOVAKIA 4Institute for Theoretical Physics, University of Bremen, Otto Hahn Allee 1, D-28334 Bremen, GERMANY (Dated: June 10, 2021) We study cascading failures in networks using a dynamical flow model based on simple conserva- tion and distribution laws to investigate the impact of transient dynamics caused by the rebalancing of loads after an initial network failure (triggering event). It is found that considering the flow dy- namics may imply reduced network robustness compared to previous static overload failure models. This is due to the transient oscillations or overshooting in the loads, when the flow dynamics adjusts to the new (remaining) network structure. We obtain upper and lower limits to network robustness, and it is shown that two time scales τ and τ0, defined by the network dynamics, are important to consider prior to accurately addressing network robustness or vulnerability. The robustness of networks showing cascading failures is generally determined by a complex interplay between the network topology and flow dynamics, where the ratio χ = τ/τ0 determines the relative role of the two of them. PACS numbers: 89.75.-k; 89.20.-a; 75.40.Gb; 89.65.-s Societies rely on the stable operation and high perfor- mance of complex infrastructure networks, which are crit- ical for their optimal functioning. Examples are electri- cal power grids, telecommunication networks, water, gas and oil distribution pipelines, or road, railway and airline transportation networks. Their failure can have serious economic and social consequences, as various large-scale blackouts and other incidents all over the world have recently shown. It is therefore a key question how to better protect such critical systems against failures and random or deliberate attacks [1, 2, 3]. Issues of net- work robustness and vulnerability have not only been addressed by engineers [4, 5], but also by the physics community [2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In the initial studies of this kind [2, 6, 7, 8, 11], the pri- mary concern was dedicated to what can be termed struc- tural robustness; the study of different classes of network topologies and how they were affected by the removal of a finite number of links and/or nodes (e.g. how the av- erage network diameter changed). It was concluded that the more heterogeneous a network is in terms of, e.g., degree distribution, the more robust it is to random fail- ures, while, at the same time, it appears more vulnerable to deliberate attacks on highly connected nodes [7, 11]. Later on, the concepts of network loads, capacities, and overload failures were introduced [9, 10, 12, 14, 15, 16]. For networks supporting the flow of a physical quantity, the removal of a node/link will cause the flow to redis- tribute with the risk that some other nodes/links may be overloaded and failure prone. Hence a triggering event can cause a whole sequence of failures due to overload, and may even threaten the global stability of the net- work. Such behavior has been termed cascading fail- ures. A seminal work in this respect is the paper by Motter and Lai [9]. These authors defined the load of a node by its betweenness centrality [3, 9]. Subsequent studies introduced alternative measures for the network loads [12] as well as more realistic redistribution mecha- nisms [12, 14, 15, 16]. In all studies cited above, the redistribution of loads is treated time-independent or static. We will refer to them collectively as static overload failure models. The load redistributions in such models are instantaneously and discontinuously switched to the stationary loads of the new (perturbed) network, i.e. the transient dynam- ical adjustment towards the new stationary loads of the perturbed network is neglected. The aim of this Letter is to compare robustness es- timates of complex networks against cascading failures where the dynamical flow properties are taken into ac- count relative to those where they are not (static case). This work does not intend to target a specific system (or network); instead we aim at being as generic as pos- sible in the choice of dynamical model with the conse- quence that particular details and features of a specific system have to be neglected, i.e. we work with a minimal model as often favored in physics. Nevertheless, the con- ceptually simple dynamical phenomenological flow model that we propose, incorporates flow conservation, network topology, as well as load redistribution features that are shared by real-life systems. On this background, it is expected (cf. Fig. 1) that the model results will reflect some important properties of real-life systems. For matters of illustration and to facilitate compari- son with previous results [9, 10, 12, 14, 15, 16], we have worked with topologies of power transmission networks. Although our model seems to capture stylized features of electrical networks (see Fig. 1), we stress that our goal is not a realistic representation of those, nor is our model re- stricted to such systems. Within the proposed model, we http://arxiv.org/abs/0704.1952v2 0 10 20 30 40 time [s] 0 10 20 30 40 50 time [s] 0 10 20 30 40 time [steps] 0 10 20 30 40 50 time [steps] (a) (b) (c) (d) FIG. 1: (Color online) Comparison of the time-dependent link loads after an triggering event (taking place at t = 0) as predicted by state-of-the-art power simulators (Figs. 1(a) [18] and (b) [19]), and the simple “flow-conserving” model de- scribed and used in the present work (Figs. 1(c) and (d)). want to demonstrate that time-dependent adjustments can play a crucial role. In particularly, we will show that static overload failure models give the lower limit of the vulnerability of flow networks to failures and attacks, and hence of the probability of cascading failures. In order to study this, in the very tradition of physics, we use a simple flow model with few parameters, which however considers the network topology, flow conserva- tion, and the distribution of loads over the neighboring links of a node [20, 22]. We assume a network consist- ing of N nodes and represent it by a matrix W , whose entries Wij ≥ 0 (with i, j = 1, 2, . . . ,N ) shall reflect the weight of the (directed) link from node j to i (with Wij = 0 indicating no link present). The relative weights Tij = Wij/wi shall define the elements of the transfer ma- trix T , where wj = i=1 Wij is the total outgoing weight of node j [22]. These elements describe the distribution of the overall flow (per unit weight) cj(t) reaching node j at time t over the neighboring links i. When the flow is assumed to reach the neighboring nodes i at time step t+1, we obtain ci(t+1) = j=1 Tijcj(t)+j i [20, 22, 23], where we have added possible source terms (j±i > 0) or sink terms (j±i < 0). In vectorial notation, the network flow equation reads c(t+ 1) = Tc(t) + j± , (1) resembling Kirchhoff’s first law from circuit theory. If j± = 0, the stationary solution to Eq. (1) is a con- stant vector with components c i (∞) ∼ 1/ N , while with a source term present (j± 6= 0), it can be expressed as c(∞) = c(0)(∞) + (1− T )+ j± with (1− T )+ denot- ing the so-called generalized inverse [24] of the singu- lar matrix 1 − T . Hence, the total directed current on link j → i at time t becomes Cij(t) = Wijcj(t), from which also the (undirected) load Lij(t) of this link can be defined via Lij(t) = Cij(t) + Cji(t) [28]. Closed- form expressions for the flow dynamics at single nodes have been derived in Ref. [23]. These allow one to study the wave-like spreading and dissipation of perturbations in the network while propagating via neighboring links, second-next, etc. (see Fig. 2). Such perturbations may result from the redistribution of flows after the failure of an overloaded link. In the seminal work of Motter and Lai [9], failure of a node was based on the long-term overload, i.e. a node was assumed to fail whenever the stationary load in the perturbed network (considering previously broken nodes) exceeded the node capacity. The (node) capacities were defined as 1 + α times the (stationary) loads of the orig- inal network with α ≥ 0 being a global tolerance factor (i.e. a relative excess capacity or safety margin). In other words, the evaluation of overloading was previously done after the system relaxed, without considering the time- history of how it got to this state (static overload failure models) [9, 10, 12, 14, 15, 16]. In this Letter, we generalize this approach towards a dynamical overload failure model. Specifically, in our computer simulations we assumed a link from node j to i to be overloaded (and to fail) whenever the time- dependent load Lij(t) exceeded the link capacity Cij for at least a time period τ , the overload exposure time. The link capacities were defined analogously to Motter and Lai [9] as Cij = (1 + α)Lij , (2) where Lij denote the stationary loads of the original net- work. In the following, we study the transient dynamical ef- fects and overload situations that may occur before the stationary state is reached. While for τ = 0, a failure results immediately after a first-time overload, τ > 0 implies that the system will have to be overloaded for a certain time period in order to cause a failure. The static overload failure model corresponds to τ → ∞, or in prac- tice, τ ≫ τ0, where τ0 denotes the transient time of the system (the inverse of the smallest non-zero eigenvalue of T ). Therefore, the ratio χ = τ/τ0 can be used to inter- polate between the static (χ → ∞) and (instantaneous) dynamical overload failure (χ = 0) models. While the static overload failure model describes the upper limit of network robustness (the best case), the dynamic overload failure model with τ = 0 gives the lower limit (the worst case) due to an overshooting flow dynamics (Fig. 2). Re- alistic cases are expected to lie between these two limiting cases, corresponding to a finite value of χ. Apart from network robustness to overload failures, the value of χ also determines the dynamics of failure cascades. In the dynamic case with χ = 0, close-by links are more likely to be overloaded and to fail than in the static case (χ → ∞). Therefore, in the dynamic scenario 0 200 400 600 800 1000 time [steps] 0.980 0.990 1.000 1.010 1.020 1.030 Link A Link B 0 20 40 60 80 100 time [steps] Link A 0 200 400 600 800 1000 time [steps] 1.000 1.005 1.010 1.015 Link C Link D Link E 0 10 20 30 40 50 60 time [steps] Link F FIG. 2: (Color online) (a) Illustration of the dynamics of our network flow model assuming the topology of the UK high-voltage power transmission grid (300–400 kV) consist- ing of 120 geographically correctly placed nodes (generators, utilities, and transmission stations) and 165 links (transmis- sion lines). The network was treated as unweighted and undirected. In our simulations, twenty of the existing net- work nodes were chosen randomly to play the role of gener- ator (source) and utility (sink) nodes ( ˛n±i /N ˛ = 2.5 · 10−4), ten of each kind. In Fig. 2(a) the location of these nodes are indicated by filled red squares and filled green diamonds respectively. At time t = 0, before which the network loads were in the stationary state (Lij(t < 0)), the network was perturbed by removing a transmission line in Scotland (the red dashed link marked by 0 in Fig.2(a)). The resulting nor- malized transient link loads, Lij(t)/Lij(t < 0), are depicted in Figs. 2(b) and (c) for some selected links of the UK trans- mission grid, as indicated in Fig. 2(a). The horizontal dash- dotted lines correspond to the normalized stationary loads of the links. one tends to have a pronounced “failure wave” sweeping over the network. In order to further illustrate the difference between the static and dynamic cascading failure models, as well as getting a quantitative measure of the level of overesti- mation of robustness, we investigate one of the networks already studied by Motter and Lai [9] — the Northwest- ern American power transmission network obtained from Ref. [26] (see also Refs. [11, 27]). To evaluate the effect of an initial network perturbation and the following cascade (if any), we study the fraction of nodes and links, GN (α) and GL(α), respectively, remaining in the giant compo- nent of the network after potential cascading failures have ceased, which have been initiated by the random failure of a link [9]. Both quantities behave similarly [29]. They are displayed in Fig. 3 for the US power transmission net- work as functions of the tolerance parameter α. It has been checked and found that our static overload failure model well reproduces the general behavior previously reported in Ref. [9]. Namely, global cascading failure will occur under random attacks (or failures) mainly for heterogeneous networks. According to Fig. 3, there is a pronounced difference between the static and (instantaneous) dynamic overload failure model, corresponding to upper and lower esti- mates to the network robustness. As is shown in the inset to Fig. 3, it can be as significant as 80%, and for more homogeneous link weights we have found differences even higher than 95%. Only for quite significant toler- ance factors (α ≥ 50%), the discrepancy between the two estimates becomes insignificant. Moreover, it has also been found that the static model tends to be more sensitive to the location of sources (and sinks). Thus, our results show that the role of the dynamical process taking place on the network can be important when esti- mating the robustness of networks to failures and random attacks. It is not only the topology of the network that matters, but also the properties of the network dynamics as measured by χ = τ/τ0. The change in one or both of them will require a new robustness estimate. In conclusion, we have simulated a simple network flow model considering, besides network topology, a flow- conserving dynamics and distribution of loads. Within this framework, we have studied the role of the tran- sient dynamics of the redistribution of loads towards the steady state after the failure of network links. This tran- sient dynamics is often characterized by overshootings and/or oscillations in the loads, which may result in char- acteristic “failure waves” spreading over the network. We have furthermore found that, considering only the loads in the steady state (the static overload model), gives a best case estimate (upper limit) of the robustness. The worst case (lower limit) of robustness can be determined by the instantaneous dynamic overload failure model and may differ considerably. Our simple dynamical approach provides additional in- sights into systems in which network topology is com- bined with flow, conservation and distribution laws. These are potentially useful to understand, better de- sign and protect critical infrastructures against failures. For instance, overloads related to high electrical currents cause (through over-heating of wires) a slow spreading 0 0.2 0.4 0.6 0.8 Static Dynamic (τ = 0) Dynamic (τ = 1) Dynamic (τ = 10) 0 0.2 0.4 0.6 FIG. 3: (Color online) The robustness of the Northwestern US power transmission grid [26], consisting of 4941 nodes, with an average node degree of 2.67 (cf. also Refs. [11, 15, 27]). The average fraction of links (or nodes), G(α), remaining in the giant component of this network (after cascading) is de- picted as function of the tolerance parameter α, using the static and dynamic overload failure models described in the text. To obtain these results, the links were assigned weights, drawn from a uniform distribution on the interval [1, 10], and 200 generator and utility nodes of strength ˛ = 10−8 were assigned randomly (100 of each type). The results were obtained by averaging over all possible triggering events (sin- gle link removals). The inset shows the difference, ∆G(α), between the static and dynamic overload failure models. of failures as compared to the adjustment dynamics of the currents. This corresponds to χ ≫ 1. In contrast, within the validity limits of Ohm’s law, one may also use our model to mimic effects of overloads related to over- voltages. In this case, we have χ ≪ 1, and link failures reflect the anticipatory disconnection of lines to prevent damages of the network and its components. Other ex- amples, besides electrical power grids, are traffic systems, where overloaded streets cause unreasonably long travel times along links, which may be interpreted as effective link failures. The resulting choice of alternative routes corresponds to a rebalancing of loads and is expected to cause transient effects, with finite values of χ. As the model allows for effective simulations, it could also be useful for close to real-time planning and opti- mization of network topologies and load sharing, partic- ularly for large networks. Fully realistic state-of-the-art simulation tools for, say, electrical power grids that in- clude network capacities, inductors, power generation, etc., are computationally expensive and therefore not so well suited for real-time simulation of large networks or their topological optimization. Hence, simpler models could quickly and efficiently give a useful overview that could serve as the starting point for more detailed off-line simulations using classical power network simulators. The authors gratefully acknowledge the support from the EU Integrated Project IRRIIS (027568) and ESF COST Action P10 “Physics of Risk” and comments by J.L. Maŕın and A. Diu. ∗ Electronic address: [email protected] [1] A. Kaufmann, D. Grouchko, and R. Cruon, Mathemat- ical Models for the Study of the Reliability of Systems (Academic Press, 1977). [2] R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002). [3] M. E. J. Newman, SIAM Rev. 45, 167 (2003). [4] R. Billinton and W. Li, Reliability Assessment of Elec- tric Power Systems Using Monte Carlo Methods (Plenum Press, NY, 1994). [5] Y. Dai, J. McCalley, N. Samra, and V. Vittal, IEEE T. Power Syst. 16, 4 (2001). [6] R. Albert, A. L. H. Jeong, and A.-L. Barabási, Nature 406, 378 (2000). [7] D. J. Watts, Proc. Natl. Acad. Sci. USA 99, 5766 (2002). [8] P. Holme, B. J. Kim, C. N. Yoon, and S. K. Han, Phys. Rev. E 65, 056109 (2002). [9] A. E. Motter and Y.-C. Lai, Phys. Rev. E 66, 065102(R) (2002). [10] A. E. Motter, Phys. Rev. Lett. 93, 098701 (2004). [11] R. Albert, I. Albert, and G. L. Nakarado, Phys. Rev. E 69, 025103(R) (2004). [12] P. Crucitti, V. Latora, and M. Marchiori, Phys. Rev. E 69, 045104(R) (2004). [13] A. Scirè, I. Tuval, and V. M. Egúıluz, Europhys. Lett. 71, 318 (2005). [14] L. Huang, L. Yang, and K. Yang, Phys. Rev. E 73, 036102 (2006). [15] J. Bakke, A. Hansen, and J. Kertész, Europhys. Lett. 76, 717 (2006). [16] L. Dall’Asta, A. Barrat, M. Barthélemy, and A. Vespig- nani, J. Stat. Mech. P04006 (2006). [17] P. Kaluza, M. Ipsen, M. Vingron, and A. S. Mikhailov, Phys. Rev. E 75, 015101(R) (2007). [18] R. Sadikovic, Ph.D. thesis, ETH Zurich, Switzerland (2006). [19] The EUROSTAG power simulation package (http://wwweurostag.epfl.ch/). [20] K. A. Eriksen, I. Simonsen, S. Maslov, and K. Sneppen, Phys. Rev. Lett. 90, 148701 (2003). [21] I. Simonsen, K. A. Eriksen, S. Maslov, and K. Sneppen, Physica A 336, 167 (2003). [22] I. Simonsen, Physica A 357, 317 (2005). [23] D. Helbing and R. Molini, Phys. Lett. A 212, 130 (1996). [24] A. Ben-Israel and T. Greville, Generalized Inverses (Springer-Verlag, Berlin, 2003), 2nd ed. [25] The UK power transmission grid data were digitized from a map of the European transmission grid purchased from Glückauf Verlag. [26] D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998). [27] L. A. N. Amaral, A. Scala, M. Barthelemy, and H. E. Stanley, Proc. Natl. Acad. Sci. USA 97, 11149 (2000). [28] Alternatively one could have defined directed loads by Lij(t) = Cij(t), but this possibility will not be considered herein. [29] This is a consequence of the network being either al- most unaffected by an initial link removal, or experienc- ing global failure where the whole network collapses. mailto:[email protected]

0704.1953

Ramsey fringes formation during excitation of topological modes in a Bose-Einstein condensate

Ramsey fringes formation during excitation of topological modes in a Bose-Einstein condensate E. R. F. Ramos a, L. Sanz a, V. I. Yukalov b and V. S. Bagnato a aInstituto de F́ısica de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970, São Carlos-SP Brazil bBogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research Dubna 141980, Russia Abstract The Ramsey fringes formation during the excitation of topological coherent modes of a Bose-Einstein condensate by an external modulating field is considered. The Ramsey fringes appear when a series of pulses of the excitation field is applied. In both Rabi and Ramsey interrogations, there is a shift of the population maximum transfer due to the strong non-linearity present in the system. It is found that the Ramsey pattern itself retains information about the accumulated relative phase between both ground and excited coherent modes. Key words: Bose-Einstein condensation, Ramsey fringes, Coherence, Non-linear dynamics PACS: 03.75.Kk, 39.20.+q, 31.15.Gy, 42.25.Kb 1 Introduction The creation of Bose-Einstein condensates (BEC) in non-ground states, as originally proposed by Yukalov, Yukalova, and Bagnato [1,2] is nowadays a topic of wide interest. This method allows for the direct formation of frag- mented nonequilibrium condensates (see the review article by Leggett [3]). Email address: [email protected] (E. R. F. Ramos). Preprint submitted to Elsevier 11 August 2021 http://arxiv.org/abs/0704.1953v1 One of important applications of coupling between different coherent modes of BEC is the possibility to produce various transverse modes in the atom laser [5,6,7,8]. Many experiments are devoted to the study of properties of trapped BECs by coupling their collective states [9,10]. Starting with a sample of atoms Bose-condensed in the ground state of a con- fining potential, it is possible to promote atoms from one trap level to another by using resonance excitation [1,11]. This is done by applying an additional weak external field, with a fixed spatial distribution, and oscillating in time with a frequency near the transition frequency between the ground and an ex- cited state. Previous calculations demonstrated the possibility of macroscopic transfer between the levels, confirming the feasibility of this procedure, and analyzed several applications [1,2,11]. In the present work, we investigate the formation of the Ramsey-like fringes, due to the interference between the ground and a non-ground states of BEC, excited by means of a near resonant field. We keep in mind a time-domain ver- sion of the separated oscillatory-field method, as developed by Ramsey [13,14], consisting of a sequence of two Rabi π/2-pulses, which are equivalent to the oscillatory fields of the Ramsey method. Previously, the formation of Ramsey fringes in double Bose-Einstein condensates [15] was studied. But this is for the first time that the Ramsey patterns are obtained, when BECs states with different quantum numbers, associated with the trap potential, are coupled. The eventual measurement of those fringes would quantify the coherence of the process. An experimental setup for the observation of the topological coherent modes, through spatial distribution observation, is presently in progress in our research group [16,17]. This paper is structured as follows. In Sec. 2, we briefly review the dynamics of the coherent modes, based on the Gross-Pitaevskii equation for BEC [4]. Also, we recall the idea of the resonant excitation, which makes it possible to couple the ground and non-ground states. Section 3 is devoted to the formation of the Ramsey-like fringes and to the possibility of their experimental observation. In Section 4, we summarize our results. 2 Gross-Pitaevskii equation and coherent modes At low temperatures, dilute Bose gas, as is known, is well described by the Gross-Pitaevskii equation. This equation describes the coherent states of the Bose system with the Hamiltonian Ψ̂† (r, t) + Vho (r, t) Ψ̂ (r, t) dr+ drdr′Ψ̂† (r, t) Ψ̂† (r′, t)V (r− r′) Ψ̂ (r′, t) Ψ̂ (r, t) , (1) with N bosons assumed to be confined by a harmonic trap potential, Vho (r, t). The Ψ̂ (r, t) (Ψ̂† (r, t)) are the boson field operators that annihilate (create) an atom at position r and V (r− r′) is a two-body interaction due to atomic collisions. In a dilute cold gas, the most relevant scattering process is associated with elastic binary collisions at low energy [3,4], giving the effective interaction potential V (r− r′) = Asδ (r− r ′) = (N − 1) 4π~2as δ (r− r′) . (2) Here, as is the “zero-energy” s-wave scattering length. For an effective attrac- tive (repulsive) interaction as is negative (positive). The confining harmonic potential is written as Vho (r, t) = 2 + ω2yy 2 + ω2zz where m0 is the atomic mass and ωk (k = x, y, z) are the trap oscillation frequencies along each axis. The topological coherent modes are the solutions to the stationary Gross- Pitaevskii equation. The nonlinear coherent modes form an overcomplete ba- sis and are normalized (|ϕj| = 1). The eigenvalue problem for stationary functions is given by [11] Ĥ [ϕj]ϕj = Ejϕj (4) where Ĥ [ϕj ] = − + Vho (r, t) + As |ϕj| follows from the Hamiltonian (1). The resonant pumping makes it possible a coherent transfer of condensed atoms between the collective levels of atoms in the harmonic trap, as is de- scribed in Ref. [11]. At initial time, N condensed atoms are assumed to be in the ground state of the trap with a frequency ω0. The aim is to transfer the atoms to a non-ground state (labelled as p). The energy difference of the level- p, relative to the ground state, is ~ (ωp − ω0). Following Yukalov et al. [11], this transfer can be obtained through the action of an external oscillatory field given by Vp (r, t) = V (r) cosωt . (6) The corresponding non-linear Schrödinger equation, associated with the Hamil- tonian (5), writes as ∂φ (r, t) Ĥ + Vp (r, t) φ (r, t) . (7) The modulating field is called external in the sense that it is not a part of the harmonic trap setup, but rather the field Vp (r, t) is a perturbation in Eq.(7). The solution φ (r, t), can be expressed as a sum over the coherent modes: φ (r, t) = n cn (t)ϕn (r, t). Although one could consider a more general form of the external field for transferring atoms between an arbitrary pair of collec- tive levels [11], it is sufficient to work with the potential (6) in order to create the Ramsey interference fringes. The functions c0 (t), associated with ground state, and cp (t), ascribed to an excited state, determine the behavior of the populations in each mode n0 (t) = |c0 (t)| , np (t) = |cp (t)| . (8) There are some conditions on the involved physical parameters in order to observe a macroscopic population of a non-ground state (np 6= 0): First, it is required that the resonance condition between the external field of frequency ω and the frequency associated with the level transition, ωp0 = ωp−ω0, be ful- filled. Also, the detuning ∆ω = ω−ωp0 should be small enough, |∆ω/ωp0| ≪ 1. At the same time, one should remember that not solely the external coupling filed, but also the interatomic collisions cause atomic transitions between the modes. In order to quantify both effects, it is convenient to define the inter- action intensity α, associated with interatomic collisions, and the transition amplitude β of the coupling field as αm,k = |ϕm (r)| 2 |ϕk (r)| − |ϕm (r)| (r)Vp (r)ϕp (r) dr. (9) For simplicity, we set α ≡ α0,p = αp,0 for the folowing analysis. These quan- tities have dimensions of frequency (Hz). Their values are to be smaller than the transition frequency ωp0, so that ≪ 1 , ≪ 1. (10) For the time dependent process, with the potential Vp, the solution of Eq.(7) can be represented as φ (r, t) = cn (t)ϕn (r) exp Ent . (11) Another condition is that the coefficients cn (t) evolve slowly in time, when compared to the oscillatory terms in the equation (11), so that ≪ 1. (12) It is straightforward to obtain a system of coupled differential equations for the coefficients c0 and cp, whose detailed derivation can be found in Ref. [1], =αnpc0 + βei∆ωtcp, =αn0cp + β∗e−i∆ωtc0. (13) Let us mention that the form of these coupled equations resembles the equa- tions of the Rabi two-level problem [18]. It is expected that, if we choose the appropriate physical parameters, Rabi oscillations between the topolog- ical modes could be observed. The solution to this system provides us with all necessary information about the dynamics of the populations, including both ground and non-ground states. An analytical solution of Eqs.(13) can be obtained when |β| ≪ |α|, as in Ref. [2]. Then the population fractions are given by n0 =1− , (14) with the effective collective frequency Ω defined as + [α (n0 − np)−∆ω] . (15) It is worth emphasizing that this collective frequency is not a constant, but a function of the instant populations n0 and np, and that the analytical solution is available under the condition that the populations of the modes have small changes. Thus, the population difference ∆n = |np − n0| is almost a constant. When we are interested in a macroscopic population of an excited state, it is necessary to solve the coupled equations (13) using a numerical procedure based on the fourth-order Runge-Kutta method. Thus, we can deal with α and β values not restricted to the condition |β| ≪ |α|. In order to find the time dependent solutions for coefficients c0 (t) and cp (t), we consider as initial conditions c0 (0) = 1 and cp (0) = 0 (all N atoms ini- tially condensed in the ground state). In figure 1, we plot the results for the fractional populations, associated with the non-ground and excited states and 0 10 20 30 40 50 0 10 20 30 40 50 Fig. 1. Evolution of the density of populations (n0 in dotted line, np in grey line) and the population unbalance ∆n = |np − n0| (black line) as function of dimensionless quantity αt for ∆ω ≈ 0 and different values of coupling parameter β: (a) β = 0.40α; (b) β = 0.46α; (c) β ≈ 0.50α and (d) β = 0.60α. the population difference as functions of the dimensionless parameter αt. Grey lines show the population of the non-ground state (np = 0 at initial time) and dotted lines show n0 (t). The evolution of the population difference, ∆n (t), is plotted in black bold lines. For all cases, the detuning is fixed so that ∆ω ≈ 0, and we simulate a slight change of the external field Vp, which is characterized by the coupling parameter β defined in Eq.(9). From the previ- ous works [19,11,2], we know that there are critical effects associated with the coupled system given by Eqs.(13), which, however, we shall not consider here. Since it is possible to transfer populations between two different topological states, under well defined conditions, prescribing when this process is efficient, we are in a position to analyze what happens if we manipulate the excitation time domain in order to detect Ramsey-like interference fringes as the oscilla- tions of the fractional populations. 3 π/2 and π- pulses and the formation of the Ramsey-like fringes In this section, we demonstrate the formation of the Ramsey-like fringes for the case of two topological modes coupled by an external resonant field. Given a certain value of β, we define the maximum of the atom population that is transferred from the ground to an excited state. It is also possible to estimate the necessary time for the coupling to be switched on. When a π pulse is applied, the time of the coupling, t1, is sufficient for the performance of a complete Rabi oscillation. If the coupling is switched off at the time, when the half of the maximum population is transferred, the applied pulse is the so- called Rabi π/2-pulse. We show below that Ramsey fringes are obtained when two Rabi π/2-pulses are applied with a time interval τ between them. The latter procedure is similar to that one accomplished for the coupled hyperfine levels of Rubidium [20], and the formation of the fringes confirms the existence of a relative phase between both topological modes. In figure 2, we plot our results for the fractional population density of the ground state, n0, considering three pulse configurations, mentioned above, as functions of the detuning ∆ω. The normalized quantity n0 gives us information about the atomic population in the mode and an indirect quantification of the visibility for the imaging process. The total time of the simulations is equal in all three cases. The procedure of obtaining the patterns implies the solution of system (13) for a fixed value of β, which means a fixed value of the spatial amplitude function in Vp, allowing for the variation of the detuning ∆ω. In figure 2, black lines show the expected behavior of n0, when equivalent Rabi pulses are applied. Both, π (bold line) and π/2 (thick line), pulses exhibit common features. First, we observe a sole peak, with a maximum at ∆ω 6= 0. The shift of the maximum of the resonant condition is associated with the contribution from the nonlinear terms of the Hamiltonian (5). Second, the maximual value of n0 depends on the number of atoms transferred between the coupled states, which, in turn, depends on the coupling time t1. This is related to the halfwidth Γ ≈ 1 . If we compare both black lines in figure 2, we note that Γπ ∼ 1.44 and Γπ/2 ∼ 2.76 ≈ 2Γπ. The possibility of the formation of Ramsey fringes was first suggested in Ref. [11]. Our numerical calculations demonstrate that the fringes are really obtained when two π/2 pulses are applied separately. In this way, given a fixed -6 -4 -2 0 2 4 6 Fig. 2. The fractional population of a non-ground state, np, as a function of detuning ∆ω for three different kinds of pulse configurations. For all cases β = 0.4α. Grey line: Ramsey fringes due to the application of two π/2 pulses with an interval τ = 8t1; Bold black line: a π Rabi pulse; Thick black line: π/2 Rabi pulse. number of condensed atoms N , approximately half of the population is trans- ferred from the ground to an excited state during the first π/2 pulse, evolving freely when the coupling is switched off, and then, a second pulse concludes the excitation. The fractional population after this process, as a function of the detuning ∆ω, is plotted in figure 2 with the grey line. The process simu- lates the effect of the application of two π/2 Rabi pulses separated by τ = 8t1. The maximal value for np is found at the same frequency as in the configu- ration of a single pulse, which confirms our previous conclusion that the shift in the frequency is a hallmark of the nonlinear effects due to elastic two-body collisions. In figure 3, we plot the behavior of the fractional population as a function of β, considering the same pulse configuration as above, with an initial π/2 pulse, an interval τ = 4t1, and a second π/2 pulse. If the coupling amplitude is weak, the Ramsey pattern loses its visibility. Increasing β improves visibility, but at the same time causes power shifts, represented by the fringes displacement. The number of the Ramsey fringes, contained within the spectral width, is strongly dependent on the time interval τ between the applied π/2- pulses. This feature is shown in figure 4, where four different situations are considered. We observe that the absolute maximal values of np and its position, as a function of ∆ω, are not connected with the changes of τ . What does depend -4 -2 0 2 4 Fig. 3. Ramsey fringes of the fractional populatuion np as a function of ∆ω, after the application of two π/2- pulses separated by τ = 4t1, with t1 being the time needed for the excitation of the half of the population from the ground to a non-ground state. The lines correspond to the increasing values of β. Light grey line: β = 0.1α; Grey line: β = 0.2α; Black line: β = 0.3α on τ is the number of fringes, which increases as τ increases. If τ is equal to the coupling time t1, two additional peaks appear at the both sides of the main peak. For τ = 2t1, we obtain two peaks at each side, and so on, as it can be seen comparing the four plots in figure 4. The appearance of these auxiliary peaks is the hallmark for the accumulation of a relative phase between both topological states. During the time τ , when the coupling is switched off, the dynamics of the whole system (ground + non-ground state) is associated with the non-linear term in the Hamiltonian (5). This free dynamics determines the accumulated relative phase. -4 -2 0 2 4-4 -2 0 2 4 = 2t = t = 3t = 8t Fig. 4. Ramsey fringes of the fractional population np as a function of ∆ω, after the application of the Ramsey pulse configuration for different τ values. The gray line in all plots corresponds to the Rabi π/2-pulse. 4 Conclusions In this work, we use numerical calculations for solving the system of coupled equations describing the resonant excitation of two coherent non-linear modes of BEC. Our results, obtained by means of the fourth-order Runge-Kutta method, give us an insight into the behavior of the fractional mode popula- tions of non-ground collective atomic states in a harmonic trap. The principal novelty of the present paper is the investigation of the system response to different coupling configurations and the demonstration of the appearance of the Ramsey-like fringes. The formation of the Ramsey-like fringes is a signature of the actual coherent character of topological modes in the studied nonlinear system. In both, Rabi and Ramsey pulse configurations, there is a shift of the maximum population transfer due to the strong effect of non-linearity of the system. The Ramsey pattern itself contains information on the accumulated relative phase, and the number of secondary peaks is proportional to the time τ , as defined in Section 3. As possible extension of this work, it would be interesting to consider the influence on the Ramsay fringes of the trap geometry and of different external pumping fields Vp (r, t). Other possibilities could be related to the manipilation of the scattering length via the Feshbach resonance techniques and to the effects of changing the total number of atoms, which affects the interatomic intensity α. Acknowledgments The authors wish to thank E. P. Yukalova for her important contribution at the early stage of this work. Special thanks are to E. A. L. Henn and K. M. F. Mag- alhães for helpful discussions. This work was supported by Fapesp (Fundação de Amparo à Pesquisa do Estado de São Paulo), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Ńıvel Superior), and by CNPq (Conselho Na- cional de Desenvolvimento Cient́ıfico e Tecnológico). References [1] V. I. Yukalov, E. P. Yukalova, and V. S. Bagnato, Phys. Rev. A 56, 4845 (1997). [2] P. W. Courteille, V. S. Bagnato, and V. I. Yukalov, Laser Phys. 11, 659 (2001). [3] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). [4] F. Dalfovo, S. Giorgini, and L. P. Pitaevskii, Rev. Mod. Phys. 71, 463 (1999). [5] M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, Phys. Rev. Lett. 78, 582 (1997). [6] E. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, and W. D. Phillips, Science 283, 1706 (1999). [7] I. Bloch, T. Hansch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999). [8] B. Anderson and M. Kasevich, Science 282, 1686 (1998). [9] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 78, 586 (1997). [10] M. R. Matthews, D. S. Hall, D. S. Jin, J. R. Ensher, C. E. Wieman, E. A. Cornell, F. Dalfovo, C. Minniti, and S. Stringari, Phys. Rev. Lett. 81, 243 (1998). [11] V. I. Yukalov, E. P. Yukalova, and V. S. Bagnato, Phys. Rev. A 66, 043602 (2002). [12] V. I. Yukalov, E. P. Yukalova, and V. S. Bagnato, Proc. SPIE 4243, 150 (2001). [13] N. F. Ramsey, Phys. Rev. 76, 996 (1949). [14] N. F. Ramsey, Phys. Rev. 78, 695 (1950). [15] A. Eschmann, R. J. Ballagh, and B. M. Caradoc-Davies, J. Opt. B:Quantum Semiclass. Opt. 1, 383 (1999). [16] K. M. F. Magalhães, S. R. Muniz, E. A. L. Henn, R. R. Silva, L. G. Marcassa, and V. S. Bagnato, Laser. Phys. Lett. 2, 214-219 (2005). [17] E. A. L. Henn, K. M. F. Magalhães, G. B. Seco, and V. S. Bagnato, Abstract for the XXIX Encontro nacional da matéria condensada (National meeting on condensed matter physics). May 2006, São Lourenço-MG, Brazil. [18] I. Rabi, Phys. Rep. 51, 652 (1937). [19] V. I. Yukalov, E. P. Yukalova, and V. S. Bagnato, Laser Phys. 13, 861 (2003). [20] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 81, 1543 (1998). Introduction Gross-Pitaevskii equation and coherent modes /2 and - pulses and the formation of the Ramsey-like fringes Conclusions References

0704.1954

The Allen-Cahn Action functional in higher dimensions

THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS LUCA MUGNAI AND MATTHIAS RÖGER Abstract. The Allen–Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen–Cahn equation. Formal calculations suggest a reduced action functional in the sharp interface limit. We prove the corresponding lower bound in two and three space dimensions. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a general- ized velocity and generalized reduced action functional in a class of evolving measures. 1. Introduction In this paper we study the (renormalized) Allen–Cahn action functional Sε(u) := ε∂tu+ − ε∆u+ W ′(u) dx dt. (1.1) This functional arises in the analysis of the stochastically perturbed Allen–Cahn equation [2, 21, 13, 30, 8, 10, 12] and is related to the probability of rare events such as switching between deterministically stable states. Compared to the purely deterministic setting, stochastic perturbations add new features to the theory of phase separations, and the analysis of action functionals has drawn attention [8, 13, 18, 19, 26]. Kohn et alii [18] considered the sharp-interface limit ε → 0 of Sε and identified a reduced action functional that is more easily accessible for a qualitative analysis. The sharp interface limit reveals a connection between minimizers of Sε and mean curvature flow. The reduced action functional in [18] is defined for phase indicator functions u : (0, T ) × Ω → {−1, 1} with the additional properties that the measure of the phase {u(t, ·) = 1} is continuous and the common boundary of the two phases {u = 1} and {u = −1} is, apart from a countable set of singular times, given as union of smoothly evolving hypersurfaces Σ := ∪t∈(0,T ){t} × Σt. The reduced action functional is then defined as S0(u) := c0 ∣v(t, x)−H(t, x) dHn−1(x)dt + 4S0nuc(u), (1.2) S0nuc(u) := 2c0 Hn−1(Σi), (1.3) Date: November 12, 2021. 2000 Mathematics Subject Classification. Primary 49J45; Secondary 35R60, 60F10, 53C44. Key words and phrases. Allen-Cahn equation, stochastic partial differential equations, large deviation theory, sharp interface limits, motion by mean curvature. http://arxiv.org/abs/0704.1954v2 2 LUCA MUGNAI AND MATTHIAS RÖGER where Σi denotes the i th component of Σ at the time of creation, where v denotes the normal velocity of the evolution (Σt)t∈(0,T ), where H(t, ·) denotes the mean curvature vector of Σt, and where the constant c0 is determined by W , c0 := 2W (s) ds. (1.4) (See Section 9 for a more rigourous definition of S0). Several arguments suggest that S0 describes the Gamma-limit of Sε: • The upper bound necessary for the Gamma-convergence was formally proved [18] by the construction of good ‘recovery sequences’. • The lower bound was proved in [18] for sequences (uε)ε>0 such that the associated ‘energy-measures’ have equipartitioned energy and single multi- plicity as ε→ 0. • In one space-dimension Reznikoff and Tonegawa [26] proved that Sε Gamma-converges to an appropriate relaxation of the one-dimensional ver- sion of S0. The approach used in [18] is based on the evolution of the phases and is sensible to cancellations of phase boundaries in the sharp interface limit. Therefore in [18] a sharp lower bound is achieved only under a single-multiplicity assumption for the limit of the diffuse interfaces. As a consequence, it could not be excluded that creating multiple interfaces reduces the action. In the present paper we prove a sharp lower-bound of the functional Sε in space dimensions n = 2, 3 without any additional restrictions on the approximate se- quences. To circumvent problems with cancellations of interfaces we analyze the evolution of the (diffuse) surface-area measures, which makes information available that is lost in the limit of phase fields. With this aim we generalize the functional S0 to a suitable class of evolving energy measures and introduce a generalized formulation of velocity, similar to Brakke’s generalization of Mean Curvature Flow [5]. Let us informally describe our approach and main results. Comparing the two functionals Sε and S0 the first and second term of the sum in the integrand (1.1) describe a ‘diffuse velocity’ and ‘diffuse mean curvature’ respectively. We will make this statement precise in (6.13) and (7.1). The mean curvature is given by the first variation of the area functional, and a lower estimate for the square of the diffuse mean curvature is available in a time-independent situation [28]. The velocity of the evolution of the phase boundaries is determined by the time-derivative of the surface-area measures and the nucleation term in the functional S0 in fact describes a singular part of this time derivative. Our first main result is a compactness result: the diffuse surface-area measures converge to an evolution of measures with a square integrable generalized mean cur- vature and a square integrable generalized velocity . In the class of such evolutions of measures we provide a generalized formulation of the reduced action functional. We prove a lower estimate that counts the propagation cost with the multiplicity of the interface. This shows that it is more expensive to move phase boundaries with higher multiplicity. Finally we prove two statements on the Gamma-convergence (with respect to L1(ΩT )) of the action functional. The first result is for evolu- tions in the domain of S0 that have nucleations only at the initial time. This is in particular desirable since minimizers of S0 are supposed to be in this class. The THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 3 second result proves the Gamma convergence in L1(ΩT ) under an assumption on the structure of the set of measures arising as sharp interface limits of sequences with uniformly bounded action. We give a precise statement of our main results in Section 4. In the remainder of this introduction we describe some background and motivation. 1.1. Deterministic phase field models and sharp interface limits. Most diffuse interface models are based on the Van der Waals–Cahn–Hilliard energy Eε(u) := |∇u|2 + 1 W (u) dx. (1.5) The energy Eε favors a decomposition of Ω into two regions (phases) where u ≈ −1 and u ≈ 1, separated by a transition layer (diffuse interface) with a thickness of order ε. Modica and Mortola [23, 22] proved that Eε Gamma-converges (with respect to L1-convergence) to a constant multiple of the perimeter functional P , restricted to phase indicator functions, Eε → c0P , P(u) := d|∇u| if u ∈ BV (Ω, {−1, 1}), ∞ otherwise. P measures the surface-area of the phase boundary ∂∗{u = 1} ∩ Ω. In this sense Eε describes a diffuse approximation of the surface-area functional. Various tighter connections between the functionals Eε and P have been proved. We mention here just two that are important for our analysis. The (accelerated) L2-gradient flow of Eε is given by the Allen–Cahn equation ε∂tu = ε∆u− W ′(u) (1.6) for phase fields in the time-space cylinder (0, T ) × Ω. It is proved in different formulations [7, 9, 17] that (1.6) converges to the Mean Curvature Flow H(t, ·) = v(t, ·) (1.7) for the evolution of phase boundaries. Another connection between the first variations of Eε and P is expressed in a (modified) conjecture of De Giorgi [6]: Considering Wε(u) := − ε∆u+ 1 W ′(u) dx (1.8) the sum Eε +Wε Gamma-converges up to the constant factor c0 to the sum of the Perimeter functional and the Willmore functional W , Eε +Wε → c0P + c0W , W(u) = H2 dHn−1, (1.9) where Γ denotes the phase boundary ∂∗{u = 1} ∩ Ω. This statement was recently proved by Röger and Schätzle [28] in space dimensions n = 2, 3 and is one essential ingredient to obtain the lower bound for the action functional. 4 LUCA MUGNAI AND MATTHIAS RÖGER 1.2. Stochastic interpretation of the action functional. Phenomena such as the nucleation of a new phase or the switching between two (local) energy minima require an energy barrier crossing and are out of the scope of deterministic models that are energy dissipative. If thermal fluctuations are taken into account such an energy barrier crossing becomes possible. In [18] ‘thermally activated switching’ was considered for the stochastically perturbed Allen–Cahn equation ε∂tu = ε∆u− W ′(u) + 2γηλ (1.10) Here γ > 0 is a parameter that represents the temperature of the system, η is a time-space white noise, and ηλ is a spatial regularization with ηλ → η as λ → 0. This regularization is necessary for n ≥ 2 since the white noise is too singular to ensure well-posedness of (1.10) in higher space-dimensions. Large deviation theory and (extensions of) results by Wentzell and Freidlin [15, 14] yield an estimate on the probability distribution of solutions of stochas- tic ODEs and PDEs in the small-noise limit. This estimate is expressed in terms of a (deterministic) action functional. For instance, thermally activated switching within a time T > 0 is described by the set of paths u(0, ·) = −1, ‖u(t, ·)− 1‖L∞(Ω) ≤ δ for some t ≤ T , (1.11) where δ > 0 is a fixed constant. The probability of switching for solutions of (1.10) then satisfies γ lnProb(B) = − inf S(λ)ε (u). (1.12) Here S(λ)ε is the action functional associated to (1.10) and converges (formally) to the action functional Sε as λ → 0 [18]. Large deviation theory not only estimates the probability of rare events but also identifies the ‘most-likely switching path’ as the minimizer u in (1.12). We focus here on the sharp interface limit ε → 0 of the action functional Sε. The small parameter ε > 0 corresponds to a specific diffusive scaling of the time- and space domains. This choice was identified [8, 18] as particularly interesting, exhibiting a competition between nucleation versus propagation to achieve the op- timal switching. Depending on the value of |Ω|1/d/ T a cascade of more and more complex spatial patterns is observed [8, 18, 19]. The interest in the sharp interface limit is motivated by an interest in applications where the switching time is small compared to the deterministic time-scale, see for instance [20]. 1.3. Organization. We fix some notation and assumptions in the next section. In Section 3 we introduce the concept of L2-flows and generalized velocity. Our main results are stated in Section 4 and proved in the Sections 5-8. We discuss some implications for the Gamma-convergence of the action functional in Section 9. Finally, in the Appendix we collect some definitions from Geometric Measure Theory. Acknowledgment. We wish to thank Maria Reznikoff, Yoshihiro Tonegawa, and Stephan Luckhaus for several stimulating discussions. The first author thanks the Eindhoven University of Technology for its hospitality during his stay in summer 2006. The first author was partially supported by the Schwerpunktprogramm DFG SPP 1095 ‘Multiscale Problems’ and DFG Forschergruppe 718. THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 5 2. Notation and Assumptions Throughout the paper we will adopt the following notation: Ω is an open bounded subset of Rn with Lipschitz boundary; T > 0 is a real number and ΩT := (0, T ) × Ω; x ∈ Ω and t ∈ (0, T ) denote the space- and time-variables respectively; ∇ and ∆ denote the spatial gradient and Laplacian and ∇′ the full gradient in R× Rn. We choose W to be the standard quartic double-well potential W (r) = (1− r2)2. For a family of measures (µt)t∈(0,T ) we denote by L1 ⊗ µt the product measure defined by L1 ⊗ µt (η) := µt(η(t, ·)) dt for any η ∈ C0c (ΩT ). We next state our main assumptions. Assumption 2.1. Let n = 2, 3 and let a sequence (uε)ε>0 of smooth functions be given that satisfies for all ε > 0 Sε(uε) ≤ Λ1, (A1) |∇uε|2 + W (uε) (0, x) dx ≤ Λ2, (A2) where the constants Λ1,Λ2 are independent of ε > 0. Moreover we prescribe that ∇uε · νΩ = 0 on [0, T ]× ∂Ω. (A3) Remark 2.2. It follows from (A3) that for any 0 ≤ t0 ≤ T ε∂tuε + − ε∆uε + W ′(uε) ε(∂tuε) − ε∆uε + W ′(uε) |∇uε|2 + W (uε) (t0, x) dx − 2 |∇uε|2 + W (uε) (0, x) dx. By the uniform bounds (A1), (A2) this implies that ε(∂tuε) − ε∆uε + W ′(uε) dxdt ≤ Λ3, (2.1) 0≤t≤T |∇uε|2 + W (uε) (t, x) dx ≤ Λ4, (2.2) where Λ3 := Λ1 + 2Λ2, Λ4 := Λ1 + Λ2. Remark 2.3. Our arguments would also work for any boundary conditions for which ∂tu∇u · νΩ vanishes on ∂Ω, in particular for time-independent Dirichlet con- ditions or periodic boundary conditions. 6 LUCA MUGNAI AND MATTHIAS RÖGER We set wε := −ε∆uε + W ′(uε) (2.3) and define for ε > 0, t ∈ (0, T ) a Radon measure µtε on Ω, µtε := |∇uε|2(t, ·) + W (uε(t, ·)) Ln, (2.4) and for ε > 0 measures µε, αε on ΩT , µε := |∇uε|2 + W (uε) Ln+1, (2.5) αε := ε1/2∂tuε + ε −1/2wε )2Ln+1. (2.6) Eventually restricting ourselves to a subsequence ε→ 0 we may assume that µε → µ as Radon-measures on ΩT , (2.7) αε → α as Radon-measures on ΩT , (2.8) for two Radon measures µ, α on ΩT , and that α(ΩT ) = lim inf αε(ΩT ). (2.9) 3. L2-flows We will show that the uniform bound on the action implies the existence of a square-integrable weak mean curvature and the existence of a square-integrable generalized velocity. The formulation of weak mean curvature is standard in Geo- metric Measure Theory [1, 31]. Our definition of L2-flow and generalized velocity is similar to Brakke’s formulation of mean curvature flow [5]. Definition 3.1. Let (µt)t∈(0,T ) be any family of integer rectifiable Radon measures such that µ := L1⊗µt defines a Radon measure on ΩT and such that µt has a weak mean curvature H(t, ·) ∈ L2(µt) for almost all t ∈ (0, T ). If there exists a positive constant C and a vector field v ∈ L2(µ,Rn) such that v(t, x) ⊥ Txµt for µ-almost all (t, x) ∈ ΩT , (3.1) ∂tη +∇η · v dµtdt ≤ C‖η‖C0(ΩT ) (3.2) for all η ∈ C1c ((0, T ) × Ω), then we call the evolution (µt)t∈(0,T ) an L2-flow. A function v ∈ L2(µ,Rn) satisfying (3.1), (3.2) is called a generalized velocity vector. This definition is based on the observation that for a smooth evolution (Mt)t∈(0,T ) with mean curvature H(t, ·) and normal velocity vector V (t, ·) η(t, x) dHn−1(x) − ∂tη(t, x) dHn−1(x) − ∇η(t, x) · V (t, x) dHn−1(x) H(t, x) · V (t, x)η(t, x) dHn−1(x). Integrating this equality in time implies (3.2) for any evolution with square- integrable velocity and mean curvature. THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 7 Remark 3.2. Choosing η(t, x) = ζ(t)ψ(x) with ζ ∈ C1c (0, T ), ψ ∈ C1(Ω), we deduce from (3.2) that t 7→ µt(ψ) belongs to BV (0, T ). Choosing a countable dense subset (ψi)i∈N ⊂ C0(Ω) this implies that there exists a countable set S ⊂ (0, T ) of singular times such that any good representative of t 7→ µt(ψ) is continuous in (0, T ) \ S for all ψ ∈ C1(Ω). Any generalized velocity is (in a set of good points) uniquely determined by the evolution (µt)t∈(0,T ). Proposition 3.3. Let (µt)t∈(0,T ) be an L 2-flow and set µ := L1⊗µt. Let v ∈ L2(µ) be a generalized velocity field in the sense of Definition 3.1. Then v(t0, x0) ∈ T(t0,x0)µ (3.3) holds in µ-almost all points (t0, x0) ∈ ΩT where the tangential plane of µ exists. The evolution (µt)t∈(0,T ) uniquely determines v in all points (t0, x0) ∈ ΩT where both tangential planes T(t0,x0)µ and Tx0µ t0 exist. We postpone the proof to Section 8. In the set of points where a tangential plane of µ exists, the generalized velocity field v coincides with the normal velocity introduced in [4]. We turn now to the statement of a lower bound for sequences (uε)ε>0 satisfying Assumption 2.1. As ε→ 0 we will obtain a phase indicator function u as the limit of the sequence (uε)ε>0 and an L 2-flow (µt)t∈(0,T ) as the limit of the measures (µε)ε>0. We will show that in Hn-almost all points of the phase boundary ∂∗{u = 1} ∩Ω a tangential plane of µ exists. This implies the existence of a unique normal velocity field of the phase boundary. 4. Lower bound for the action functional In several steps we state a lower bound for the functionals Sε. We postpone all proofs to Sections 5-8. 4.1. Lower estimate for the mean curvature. We start with an application of the well-known results of Modica and Mortola [23, 22]. Proposition 4.1. There exists u ∈ BV (ΩT , {−1, 1})∩L∞(0, T ;BV (Ω)) such that for a subsequence ε→ 0 uε → u in L1(ΩT ), (4.1) uε(t, ·) → u(t, ·) in L1(Ω) for almost all t ∈ (0, T ). (4.2) Moreover d|∇′u| ≤ Λ3 + TΛ4, d|∇u(t, ·)| ≤ Λ4 (4.3) holds, where c0 was defined in (1.4). The next proposition basically repeats the arguments in [19, Theorem 1.1]. Proposition 4.2. There exists a countable set S ⊂ (0, T ), a subsequence ε → 0 and Radon measures µt, t ∈ [0, T ] \ S, such that for all t ∈ [0, T ] \ S µtε → µt as Radon measures on Ω, (4.4) 8 LUCA MUGNAI AND MATTHIAS RÖGER such that µ = L1 ⊗ µt, (4.5) and such that for all ψ ∈ C1(Ω) the function t 7→ µt(ψ) is of bounded variation in (0, T ) (4.6) and has no jumps in (0, T ) \ S. Exploiting the lower bound [28] for the diffuse approximation of the Willmore functional (1.8) we obtain that the measures µt are up to a constant integer- rectifiable with a weak mean curvature satisfying an appropriate lower estimate. Theorem 4.3. For almost all t ∈ (0, T ) µt is an integral (n− 1)-varifold, µt has weak mean curvature H(t, ·) ∈ L2(µt), and the estimate |H |2 dµ ≤ lim inf w2ε dxdt (4.7) holds. 4.2. Lower estimate for the generalized velocity. Theorem 4.4. Let (µt)t∈(0,T ) be the limit measures obtained in Proposition 4.2. Then there exists a generalized velocity v ∈ L2(µ,Rn) of (µt)t∈(0,T ). Moreover the estimate |v|2 dµ ≤ lim inf ε(∂tuε) 2 dxdt (4.8) is satisfied. In particular, ( 1 µt)t∈(0,T ) is an L 2-flow. We obtain v as a limit of suitably defined approximate velocities, see Lemma 6.2. On the phase boundary v coincides with the (standard) distributional velocity of the bulk-phase {u(t, ·) = 1}. However, our definition extends the velocity also to ‘hid- den boundaries’, which seems necessary in order to prove the Gamma-convergence of the action functional; see the discussion in Section 9. Proposition 4.5. Define the generalized normal velocity V in direction of the inner normal of {u = 1} by V (t, x) := v(t, x) · (t, x), for (t, x) ∈ ∂∗{u = 1}. Then V ∈ L1(|∇u|) holds and V |∂∗{u=1} is the unique vector field that satisfies for all η ∈ C1c (ΩT ) V (t, x)η(t, x) d|∇u(t, ·)|(x)dt = − u∂tη dxdt. (4.9) THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 9 4.3. Lower estimate of the action functional. As our main result we obtain the following lower estimate for Sε. Theorem 4.6. Let Assumption 2.1 hold, and let µ, (µt)t∈[0,T ], and S be the mea- sures and the countable set of singular times that we obtained in Proposition 4.2. Define the nucleation cost Snuc(µ) by Snuc(µ) := µt(ψ)− lim µt(ψ) + sup µt(ψ)− µ0(ψ) + sup µT (ψ)− lim µt(ψ) (4.10) where the sup is taken over all ψ ∈ C1(Ω) with 0 ≤ ψ ≤ 1. Then lim inf Sε(uε) ≥ |v −H |2 dµ+ 4Snuc(µ). (4.11) In the previous definition of nucleation cost we have tacitly chosen good represen- tatives of µt(ψ) (see [3]). With this choice the jump parts in (4.10) are well-defined. Eventually let us remark that, in view of Theorem 4.3, we can conclude that Snuc does indeed measure only (n− 1)-dimensional jumps. Theorem 4.6 improves [18] in the higher-multiplicity case. We will discuss our main results in Section 9. 4.4. Convergence of the Allen–Cahn equation to Mean curvature flow. Let n = 2, 3 and consider solutions (uε)ε>0 of the Allen–Cahn equation (1.6) sat- isfying (A2) and (A3). Then Sε(uε) = 0 and the results of Sections 4.1-4.3 apply: There exists a subsequence ε → 0 such that the phase functions uε converge to a phase indicator function u, such that the energy measures µtε converge an L 2-flow (µt)t∈(0,T ), and such that µ-almost everywhere H = v (4.12) holds, where H(t, ·) denotes the weak mean curvature of µt and where v denotes the generalized velocity of (µt)t∈(0,T ) in the sense of Definition 3.1. Moreover Snuc(µ) = 0, which shows that for any nonnegative ψ ∈ C1(Ω) the function t 7→ µt(ψ) cannot jump upwards. From (1.6) and (5.3) below one obtains that for any ψ ∈ C1(Ω) and all ζ ∈ C1c (0, T ) ∂tζ µ ε(ψ) dt = − ψ(x)w2ε (t, x) +∇ψ(x) · ∇uεwε(t, x) dxdt. (4.13) We will show that suitably defined ‘diffuse mean curvatures’ converge as ε→ 0, see (7.1). Using this result we can pass to the limit in (4.13) and we obtain for any nonnegative functions ψ ∈ C1(Ω), ζ ∈ C1c (0, T ) that t(ψ) dt ≤ − H2(t, x) +∇ψ(x) ·H(t, x) dµt(x)dt, which is an time-integrated version of Brakke’s inequality. 10 LUCA MUGNAI AND MATTHIAS RÖGER 5. Proofs of Propositions 4.1, 4.2 and Theorem 4.3 Proof of Proposition 4.1. By (2.1), (2.2) we obtain that |∇′uε|2 + W (uε) dxdt ≤ Λ3 + TΛ4. This implies by [22] the existence of a subsequence ε → 0 and of a function u ∈ BV (ΩT ; {−1, 1}) such that uε → u in L1(ΩT ) d|∇′u| ≤ lim inf |∇′uε|2 + W (uε) dxdt ≤ Λ3 + TΛ4 After possibly taking another subsequence, for almost all t ∈ (0, T ) uε(t, ·) → u(t, ·) in L1(Ω) (5.1) holds. Using (2.2) and applying [22] for a fixed t ∈ (0, T ) with (5.1) we get that d|∇u|(t, ·) ≤ lim inf µtε(Ω) ≤ Λ4. Before proving Proposition 4.2 we show that the time-derivative of the energy- densities µtε is controlled. Lemma 5.1. There exists C = C(Λ1,Λ3,Λ4) such that for all ψ ∈ C1(Ω) |∂tµtε(ψ)| dt ≤ C‖ψ‖C1(Ω). (5.2) Proof. Using (A3) we compute that ε(ψ) = ε∂tuε + (t, x)ψ(x) dx − ε(∂tuε) (t, x)ψ(x) dx ε∇ψ(x) · ∂tuε(t, x)∇uε(t, x) dx. (5.3) By (2.1), (2.2) we estimate ε∇ψ · ∂tuε∇uε dxdt ε(∂tuε) 2 + ε|∇uε|2 ≤ (Λ3 + TΛ4)‖∇ψ‖C0(Ω) (5.4) and deduce from (A1), (2.1), (5.3) that |∂tµtε(ψ)| dt ≤ (Λ1 + Λ3)‖ψ‖C0(Ω) + C(Λ3, TΛ4)‖∇ψ‖C0(Ω), which proves (5.2). � Proof of Proposition 4.2. By (2.7) µε → µ as Radon-measures on ΩT . Choose now a countable family (ψi)i∈N ⊂ C1(Ω) which is dense in C0(Ω). By Lemma 5.1 THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 11 and a diagonal-sequence argument there exists a subsequence ε→ 0 and functions mi ∈ BV (0, T ), i ∈ N, such that for all i ∈ N µtε(ψi) → mi(t) for almost-all t ∈ (0, T ), (5.5) ε(ψi) → m′i as Radon measures on (0, T ). (5.6) Let S denote the countable set of times t ∈ (0, T ) where for some i ∈ N the measure m′i has an atomic part in t. We claim that (5.5) holds on (0, T ) \ S. To see this we choose a point t ∈ (0, T ) \ S and a sequence of points (tj)j∈N in (0, T ) \ S, such that tj ր t and (5.5) holds for all tj . We then obtain m′i([tj , t]) = 0 for all i ∈ N, (5.7) ε(ψi)([tj , t]) = m i([tj , t]) for all i, j ∈ N. (5.8) Moreover |mi(t)− µtε(ψi)| ≤ |mi(t)−mi(tj)|+ |mi(tj)− µtjε (ψi)|+ |µtjε (ψi)− µtε(ψi)| ≤ |m′i([tj , t])|+ |mi(tj)− µtjε (ψi)|+ |∂tµtε(ψi)([tj , t])| Taking first ε → 0 and then tj ր t we deduce by (5.7), (5.8) that (5.5) holds for all i ∈ N and all t ∈ (0, T ) \ S. Taking now an arbitrary t ∈ (0, T ) such that (5.5) holds, by (2.2) there exists a subsequence ε→ 0 such that µtε → µt as Radon-measures on Ω. (5.9) We deduce that µt(ψi) = mi(t) and since (ψi)i∈N is dense in C 0(Ω) we can identify any limits of (µtε)ε>0 and obtain (5.9) for the whole sequence selected in (5.5), (5.6) and for all t ∈ (0, T ), for which (5.5) holds. Moreover for any ψ ∈ C0(Ω) the map t 7→ µtε(ψ) has no jumps in (0, T ) \ S. After possibly taking another subsequence we can also ensure that as ε→ 0 µ0ε → µ0, µTε → µT as Radon measures on Ω. This proves (4.4). By the Dominated Convergence Theorem we conclude that for any η ∈ C0(ΩT ) η dµ = lim η dµε = lim η(t, x) dµtε(x) dt = η(t, x) dµt(x) dt, which implies (4.5). By (5.2), the L1(0, T )-compactness of sequences that are uniformly bounded in BV (0, T ), the lower-semicontinuity of the BV -norm under L1-convergence, and (4.4) we conclude that (4.6) holds. � Proof of Theorem 4.3. For almost all t ∈ (0, T ) we obtain from Fatou’s Lemma and (2.1), (2.2) that lim inf µtε(Ω) + w2ε(t, x) dx < ∞. (5.10) 12 LUCA MUGNAI AND MATTHIAS RÖGER Let S ⊂ (0, T ) be as in Proposition 4.2 and fix a t ∈ (0, T ) \ S such that (5.10) holds. Then we deduce from [28, Theorem 4.1, Theorem 5.1] and (4.4) that µt is an integral (n− 1)-varifold, µt ≥ c0 |∇u(t, ·)|, and that µt has weak mean curvature H(t, ·) satisfying |H(t, x)|2 dµt(x) ≤ lim inf wε(t, x) 2 dx. (5.11) By (5.11) and Fatou’s Lemma we obtain that |H(t, x)|2 dµt(x) dt ≤ lim inf wε(t, x) ≤ lim inf w2ε dxdt, which proves (4.7). For later use we also associate general varifolds to µtε and consider their conver- gence as ε→ 0. Let νε(t, ·) : Ω → Sn−11 (0) be an extension of∇uε(t, ·)/|∇uε(t, ·)| to the set {∇uε(t, ·) = 0}. Define the projections Pε(t, x) := Id−νε(t, x)⊗νε(t, x) and consider the general varifolds V tε and the integer rectifiable varifold c t defined V tε (f) := f(x, Pε(t, x)) dµ ε(x), (5.12) V t(f) := f(x, P (t, x)) dµt(x) (5.13) for f ∈ C0c (Ω × Rn×n), where P (t, x) ∈ Rn×n denotes the projection onto the tangential plane Txµ t. Then we deduce from the proof of [28, Theorem 4.1] that V tε → V t as ε→ 0 (5.14) in the sense of varifolds. � 6. Proof of Theorem 4.4 6.1. Equipartition of energy. We start with a preliminary result, showing the important equipartition of energy: the discrepancy measure ξε := |∇uε|2 − W (uε) Ln+1 (6.1) vanishes in the limit ε→ 0. To prove this we combine results from [28] with a refined version of Lebesgue’s dominated convergence Theorem [25], see also [27, Lemma 4.2]. Proposition 6.1. For a subsequence ε→ 0 we obtain that |ξε| → 0 as Radon measures on ΩT . (6.2) THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 13 Proof. Let us define the measures ξtε := |∇uε|2 − W (uε) (t, ·)Ln on Ω. For ε > 0, k ∈ N, we define the sets Bε,k := {t ∈ (0, T ) : w2ε(t, x) dx > k}. (6.3) We then obtain from (2.1) that w2ε(t, x) dxdt ≥ |Bε,k|k. (6.4) Next we define the (signed) Radon-measures ξtε,k by ξtε,k := ξtε for t ∈ (0, T ) \ Bε,k, 0 for t ∈ Bε,k. (6.5) By [28, Proposition 4.9], we have |ξtεj | → 0 (j → ∞) as Radon measures on Ω (6.6) for any subsequence εj → 0 (j → ∞) such that lim sup w2εj (t, x) dx < ∞. By (2.2), (6.5) we deduce that for any η ∈ C0(ΩT ,R+0 ), k ∈ N, and almost all t ∈ (0, T ) |ξtε,k|(η(t, ·)) → 0 as ε→ 0 (6.7) and that |ξtε,k|(η(t, ·)) = 1−XBε,k(t) |ξtε| η(t, ·) ≤ Λ4‖η‖C0(ΩT ). (6.8) By the Dominated Convergence Theorem, (6.7) and (6.8) imply that |ξtε,k|(η(t, ·)) dt → 0 as ε→ 0. (6.9) Further we obtain that |ξtε|(η(t, ·)) dt ≤ |ξtε,k|(η(t, ·)) dt + |ξtε|(η(t, ·)) dt |ξtε,k|(η(t, ·)) dt + µtε(η(t, ·)) dt. (6.10) For k ∈ N fixed we deduce from (2.2), (6.4), (6.10) that lim sup |ξtε|(η(t, ·)) dt ≤ lim |ξtε,k|(η(t, ·)) dt + ‖η‖C0(ΩT )Λ4 . (6.11) By (6.9) and since k ∈ N was arbitrary this proves the Proposition. � 14 LUCA MUGNAI AND MATTHIAS RÖGER 6.2. Convergence of approximate velocities. In the next step in the proof of Theorem 4.4 we define approximate velocity vectors and show their convergence as ε→ 0. Lemma 6.2. Define vε : ΩT → Rn by vε := − ∂tuε |∇uε| |∇uε| if |∇uε| 6= 0, 0 otherwise. (6.12) Then there exists a function v ∈ L2(µ,Rn) such that (ε|∇uε|2 Ln+1, vε) → (µ, v) as ε→ 0 (6.13) in the sense of measure function pair convergence (see the Appendix B) and such that (4.8) is satisfied. Proof. We define Radon measures µ̃ε := ε|∇uε|2 Ln+1 = µε + ξε. (6.14) From (2.7), (6.2) we deduce that µ̃ε → µ as Radon measures on ΩT . (6.15) Next we observe that (µ̃ε, vε) is a function-measure pair in the sense of [16] (see also Definition B.1 in Appendix B) and that by (2.1) |vε|2 dµ̃ε ≤ ε(∂tuε) 2 dxdt ≤ Λ3. (6.16) By Theorem B.3 we therefore deduce that there exists a subsequence ε → 0 and a function v ∈ L2(µ,Rn) such that (6.13) and (4.8) hold. � Lemma 6.3. For µ-almost all (t, x) ∈ ΩT v(t, x) ⊥ Txµt. (6.17) Proof. We follow [24, Proposition 3.2]. Let νε : ΩT → Sn−11 (0) be an extension of ∇uε/|∇uε| to the set {∇uε = 0} and define projection-valued maps Pε : ΩT → Pε := Id− νε ⊗ νε. Consider next the general varifolds Ṽε, V defined by Ṽε(f) := f(t, x, Pε(t, x)) dµ̃ε(t, x), (6.18) V (f) := f(t, x, P (t, x)) dµt(x) (6.19) for f ∈ C0c (ΩT × Rn×n), where P (t, x) ∈ Rn×n denotes the projection onto the tangential plane Txµ From (5.14), Proposition 6.1, and Lebesgue’s Dominated Convergence Theorem we deduce that Ṽε = V (6.20) as Radon-measures on ΩT × Rn×n. THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 15 Next we define functions v̂ε on ΩT × Rn×n by v̂ε(t, x, Y ) = vε(t, x) for all (t, x) ∈ ΩT , Y ∈ Rn×n. We then observe that ΩT×Rn×n v̂2ε dVε = v2ε dµ̃ε ≤ Λ3 and deduce from (6.20) and Theorem B.3 the existence of v̂ ∈ L2(V,Rn) such that (Vε, v̂ε) converge to (V, v̂) as measure-function pairs on ΩT × Rn×n with values in We consider now h ∈ C0c (Rn×n) such that h(Y ) = 1 for all projections Y . We deduce that for any η ∈ C0c (ΩT ,Rn) η · v dµ = lim ΩT×Rn×n η(t, x) · h(Y )v̂ε(t, x, Y ) dVε(t, x, Y ) η(t, x) · v̂(t, x, P (t, x)) dµ(t, x), which shows that for µ-almost all (t, x) ∈ ΩT v̂(t, x, P (t, x)) = v(t, x). (6.21) Finally we observe that for h, η as above η(t, x) · P (t, x)v(t, x) dµ(t, x) ΩT×Rn×n η(t, x)h(Y ) · Y v̂(t, x, Y ) dV (t, x, Y ) = lim ΩT×Rn×n η(t, x)h(Y ) · Y v̂ε(t, x, Y ) dVε(t, x, Y ) = lim η(t, x) · Pε(t, x)vε(t, x) dµ̃ε(t, x) = 0 since Pεvε = 0. This shows that P (t, x)v(t, x) = 0 for µ-almost all (t, x) ∈ ΩT . � Proof of Theorem 4.4. By (2.1) there exists a subsequence ε → 0 and a Radon measure β on ΩT such that ε(∂tuε) Ln+1 → β, β(ΩT ) ≤ Λ3. (6.22) Using (A3) we compute that for any η ∈ C1c ((0, T )× Ω) η dαε = ε(∂tuε) dxdt − 2 ∂tη dµε ε∇η · ∂tuε∇uε dxdt . (6.23) As ε tends to zero the term on the left-hand side and the first two terms on the right-hand-side converge by (2.7), (2.8) and (6.22). For the third term on the 16 LUCA MUGNAI AND MATTHIAS RÖGER right-hand side of (6.23) we obtain from (6.13) that ∇η · ε∂tuε∇uε dxdt = − lim ∇η · vε ε|∇uε|2 dxdt ∇η · v dµ. Therefore, taking ε→ 0 in (6.23) we deduce that η dα = η dβ − 2 ∂tη dµ− 2 ∇η · v dµ holds for all η ∈ C1c ((0, T )× Ω). This yields that ∂tη +∇η · v dµ ≤ ‖η‖C0(ΩT ) α(ΩT ) + β(ΩT ) which shows together with (6.17) that v is a generalized velocity vector for (µt)t∈(0,T ) in the sense of Definition 3.1. The estimate (4.8) was already proved in Lemma 6.2. � 7. Proof of Theorem 4.6 We start with the convergence of a ‘diffuse mean curvature term’. Lemma 7.1. Define Hε := |∇uε|2 let µ̃ε = ε|∇uε|2 Ln+1, and let vε, v be as in (6.12), (6.13). Then (µ̃ε, Hε) → (µ,H), (7.1) (µ̃ε, vε −Hε) → (µ, v −H) (7.2) as ε→ 0 in the sense of measure function pair convergence. In particular η|v −H |2 dµ ≤ α(η) (7.3) holds for all η ∈ C0(ΩT ,R+0 ). Proof. We use similar arguments as in the proof of Proposition 6.1. For ε > 0, k ∈ N, we define sets Bε,k := {t ∈ (0, T ) : wε(t, x) 2 dx > k}. (7.4) We then obtain from (2.1) that w2ε dxdt ≥ |Bε,k|k. (7.5) Next we define functionals T tε,k ∈ C0c (Ω,Rn)∗ by T tε,k(ψ) := ψ(x) · wε(t, x)∇uε(t, x) dx for t ∈ (0, T ) \ Bε,k, ψ(x) ·H(t, x) dµt(x) for t ∈ Bε,k. (7.6) THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 17 Considering the general (n− 1)-varifolds V tε , V t defined in (5.12), (5.13) we obtain from [28, Proposition 4.10] and (5.14) that ψ · wεj (t, x)∇uεj (t, x) dx = − lim δV tεj (ψ) = − δµt(ψ) = ψ ·H(t, x) dµt(x) (7.7) for any subsequence εj → 0 (j → ∞) such that lim sup w2εj dxdt < ∞. Therefore we deduce from (7.6), (7.7) that for all η ∈ C0c (ΩT ,Rn), k ∈ N, and almost all t ∈ (0, T ) T tε,k(η(t, ·)) → η(t, x) ·H(t, x) dµt(x) as ε→ 0 (7.8) and that ∣T tε,k(η(t, ·)) 1−XBε,k(t) η(t, x) · wε(t, x)∇uε(t, x) dx + XBε,k(t) η(t, x) ·H(t, x) dµt(x) ≤‖η‖C0(ΩT ) 1−XBε,k(t) wε(t, x) )1/2( |∇uε(t, x)|2 dx |η(t, x)||H(t, x)| dµt(x) ≤‖η‖C0(ΩT ) |η(t, x)||H(t, x)| dµt(x), (7.9) where the right-hand side is bounded in L1(0, T ), uniformly with respect to ε > 0. By the Dominated Convergence Theorem, (7.8) and (7.9) imply that T tε,k(η(t, ·)) dt → η ·H dµ as ε→ 0. (7.10) Further we obtain that η · wε∇uε dxdt− η ·H dµ T tε,k(η(t, ·)) dt− η ·H dµ η(t, x) ·H(t, x) dµt(x)dt η · wε∇uε dx dt (7.11) 18 LUCA MUGNAI AND MATTHIAS RÖGER The last term on the right-hand side we further estimate by η(t, x) · wε(t, x)∇uε(t, x) dx dt ≤‖η‖C0(ΩT ) w2ε dxdt |Bε,k|1/2 ≤‖η‖C0(ΩT )Λ3 Λ4, (7.12) where we have used (2.2) and (7.5). For the second term on the right-hand side of (7.11) we obtain η(t, x) ·H(t, x) dµt(x)dt ≤ |Bε,k|1/2‖η‖1/2C0(ΩT ) supp(η) H2 dµ ‖η‖1/2 C0(ΩT ) Λ3, (7.13) where we have used (4.7) and (2.1). Finally, for k ∈ N fixed, by (7.10) we deduce T tε,k(η(t, ·)) dt − η ·H dµ = 0. (7.14) Taking ε→ 0 in (7.11) we obtain by (7.12)-(7.14) that η · wε∇uε dxdt− η ·H dµ ≤ Λ3√ ‖η‖C0(ΩT ) Λ3 (7.15) for any k ∈ N, which proves (7.1). Using (6.13) this implies (7.2). Finally we fix an arbitrary nonnegative η ∈ C0(ΩT ) and deduce that the measure-function pair (µ̃ε, η(vε − Hε)) converges to (µ, η(v − H)). The estimate (7.3) then follows from Theorem B.3. � Let Π : [0, T ]× Ω → [0, T ] denote the projection onto the first component and Π# the pushforward of measures by Π. For ψ ∈ C0(Ω) we consider the measures αψ := Π# on [0, T ], that means αψ(ζ) := ζ(t)ψ(x) dα(t, x), for ζ ∈ C0([0, T ]), and set αΩ := Π#α. We then can estimate the atomic part of αΩ in terms of the nucleation cost. Lemma 7.2. Let Snuc(µ) be the nucleation cost defined in (4.10). Then (αΩ)atomic[0, T ] ≥ 4Snuc(µ). (7.16) THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 19 Proof. Let η ∈ C1(ΩT ,R+0 ) be nonnegative. We compute that ηdαε = ε(∂tuε) w2ε + 2∂tuεwε η∂tuεwε dxdt = − 4 ∂tη dµε + 4 ∇η · ε∂tuε∇uε dxdt + 4µTε (η(T, ·))− 4µ0ε(η(0, ·)). (7.17) Passing to the limit ε→ 0 we obtain from (2.7), (4.4), (6.13) that ηdα ≥ − 4 ∂tη dµ− 4 ∇η · v dµ+ 4µT (η(T, ·))− 4µ0(η(0, ·)). (7.18) We now choose η(t, x) = ζ(t)ψ(x) where ζ ∈ C1([0, T ],R+0 ), ψ ∈ C1(Ω,R 0 ) in (7.18) and deduce that ζdαψ ≥ − 4 t(ψ) dt+ 4 ∇ψ · v(t, x) dµt(x) dt + 4ζ(T )µT (ψ)− 4ζ(0)µ0(ψ). (7.19) This shows that αψ ≥ 4∂t(µt(ψ)) + 4 ∇ψ(x) · v(t, x) dµt(x) µT (ψ)− lim µt(ψ) δT + 4 µt(ψ)− µ0(ψ) δ0. (7.20) Evaluating the atomic parts we obtain that for any 0 < t0 < T αψ({t0}) ≥ 4∂t(µt(ψ))({t0}), which implies that αΩ({t0}) ≥ 4 sup t(ψ))({t0}). (7.21) where the supremum is taken over all ψ ∈ C1(Ω) with 0 ≤ ψ ≤ 1. Moreover we deduce from (7.20) αΩ({0}) ≥ 4 sup µt(ψ)− µ0(ψ) , (7.22) αΩ({T }) ≥ 4 sup µT (ψ) − lim µt(ψ) , (7.23) where the supremum is taken over ψ ∈ C(Ω) with 0 ≤ ψ ≤ 1. By (7.21)-(7.23) we conclude that (7.16) holds. � Proof of Theorem 4.6. By (7.3) we deduce that α ≥ |v −H |2µ. Since µ = L1 ⊗ µt we deduce from the Radon-Nikodym Theorem that (αΩ)ac[0, T ] ≥ |v −H |2 dµ, (7.24) and from (7.16) that (αΩ)atomic[0, T ] ≥ 4Snuc(µ), (7.25) 20 LUCA MUGNAI AND MATTHIAS RÖGER where (αΩ)ac and (αΩ)atomic denote the absolutely continuous and atomic part with respect to L1 of the measure αΩ. Adding the two estimates and recalling (2.9) we obtain (4.11). � 8. Proofs of Proposition 3.3 and Proposition 4.5 Define for r > 0, (t0, x0) ∈ ΩT the cylinders Qr(t0, x0) := (t0 − r, t0 + r) ×Bnr (x0). Proof of Proposition 3.3. Define Σn(µ) := (t, x) ∈ ΩT : the tangential plane of µ in (t, x) exists (8.1) and choose (t0, x0) ∈ Σn(µ) such that v is approximately continuous with respect to µ in (t0, x0). (8.2) Since v ∈ L2(µ) we deduce from [11, Theorem 2.9.13] that (8.2) holds µ-almost everywhere. Let P0 := T(t0,x0)µ, θ0 > 0 (8.3) denote the tangential plane and multiplicity at (t0, x0) respectively, and define for any ϕ ∈ C0c (Q1(0)) the scaled functions ϕ̺ ∈ C0c (Q̺(t0, x0)), ϕ̺(t, x) := ̺ ̺−1(t− t0), ̺−1(x− x0) We then obtain from (8.3) that ϕ̺ dµ → θ0 ϕdHn as ̺ց 0. (8.4) From (3.2), the Hahn–Banach Theorem, and the Riesz Theorem we deduce that ϑ ∈ C1c (ΩT )∗, ϑ(η) := ∇′η · dµ (8.5) can be extended to a (signed) Radon-measure on ΩT . Since by the Radon-Nikodym Theorem Dµ|ϑ| exists and is finite µ-almost everywhere we may assume without loss of generality that Dµ|ϑ|(t0, x0) < ∞. (8.6) We next fix η ∈ C1c (Q1(0)) and compute that ϑ(̺η̺) = dµ. (8.7) From (8.2), (8.4) we deduce that the right-hand side converges in the limit ̺→ 0, dµ = θ0 v(t0, x0) ∇′η dµ. (8.8) For the left-hand side of (8.7) we deduce that lim inf |ϑ(̺η̺)| ≤ ‖η‖C0c (Q1(0)) lim inf̺ց0 ̺ −n+1|ϑ|(Q̺(t0, x0)) (8.9) THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 21 and observe that (8.6) implies ∞ > lim |ϑ|(Q̺(t0, x0)) µ(Q̺(t0, x0)) ≥ lim inf ̺−n|ϑ|(Q̺(t0, x0)) lim sup ̺−nµ(Q̺(t0, x0)) ≥ c lim inf ̺−n|ϑ|(Q̺(t0, x0)), (8.10) since by (8.4) for any ϕ ∈ C0c (Q2(0),R+0 ) with ϕ ≥ 1 on Q1(0) lim sup ̺−nµ(Q̺(t0, x0)) ≤ lim sup ϕ̺ dµ ≤ C(ϕ). Therefore (8.7)-(8.10) yield v(t0, x0) ∇′η dµ = 0. (8.11) Now we observe that the integral over the projection of ∇′η onto P0 vanishes. This shows that ∇′η dHn ∈ P⊥0 . (8.12) Since η can be chosen such that the integral in (8.12) takes an arbitrary direction normal to P0 we obtain from (8.11) that v(t0, x0) satisfies (3.3). If Tx0µ t0 exists T(t0,x0)µ = {0} × Tx0µt0 ⊕ span v(x0) and we obtain that v is uniquely determined. � To prepare the proof of Proposition 4.5 we first show that µ is absolutely con- tinuous with respect to Hn. Proposition 8.1. For any D ⊂⊂ Ω there exists C(D) such that for all x0 ∈ D and almost all t0 ∈ (0, T ) lim sup Qr(t0, x0) ≤ C(D)Λ4 + lim inf w2ε(t0, x) dx. (8.13) In particular, lim sup µ(Bρ(t0, x0)) < ∞ for µ− almost every (t0, x0) (8.14) and µ is absolutely continuous with respect to Hn, µ << Hn. (8.15) Proof. Let r0 := min dist(D, ∂Ω), |t0|, |T − t0| Then we obtain for all r < r0, x0 ∈ D, from (6.2) and [28, Proposition 4.5] that ∫ t0+r r1−nµt Bnr (x0) ∫ t0+r r1−n0 µ Bnr0(x0) 4(n− 1)2 ∫ t0+r lim inf w2ε(t, x) dx (8.16) 22 LUCA MUGNAI AND MATTHIAS RÖGER By Fatou’s Lemma and (2.1) t 7→ lim inf w2ε(t, x) dx is in L 1(0, T ) (8.17) and by (2.2) we deduce for almost all t0 ∈ (0, T ) that lim sup ∫ t0+r r1−nµt Bnr (x0) ≤ 2r1−n0 Λ4 + 2(n− 1)2 lim inf w2ε(t0, x) dx. Since r0 depends only on D,Ω the inequality (8.13) follows. By (8.17) the right-hand side in (8.13) is finite for L1-almost all t0 ∈ (0, T ) and θ∗n(µ, (t, x)) is bounded for almost all t ∈ (0, T ) and all x ∈ Ω. By (2.2) we deduce that for any I ⊂ (0, T ) with |I| = 0 µ(I × Ω) ≤ Λ4|I| = 0 which implies (8.14). To prove the final statement let B ⊂ ΩT be given with Hn(B) = 0. (8.18) Consider the family of sets (Dk)k∈N, Dk := {z ∈ ΩT : θ∗n(µ, z) ≤ k}. By (8.14), [31, Theorem 3.2], and (8.18) we obtain that for all k ∈ N µ(B ∩Dk) ≤ 2nkHn(B ∩Dk) = 0. (8.19) Moreover we have that µ(B \ Dk) = 0 (8.20) by (8.14). By (8.19), (8.20) we conclude that µ(B) = 0, which proves (8.15). � To prove Proposition 4.5 we need that Hn-almost everywhere on ∂∗{u = 1} the generalized tangent plane of µ exists. We first obtain the following relation between the measures µ and |∇′u|. Proposition 8.2. There exists a nonnegative function g ∈ L2(µ,R+0 ) such that g µ ≥ c0 |∇′u|. (8.21) In particular, |∇′u| is absolutely continuous with respect to µ, |∇′u| << µ. (8.22) Proof. Let G(r) = 2W (s) ds. (8.23) THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 23 On the set {|∇uε| 6= 0} we have |∇G(uε)| = |∇G(uε)| |∇′G(uε)| |∇′G(uε)| |∇G(uε)| ∂tG(uε)2 + |∇G(uε)|2 |∇′G(uε)| 1 + |v2ε | |∇′G(uε)|. (8.24) Letting µ̃ε as in (6.14) we get from (6.16), (2.2), and Theorem B.3 the existence of a function g ∈ L2(µ) such that (up to a subsequence) (µ̃ε, 1 + |vε|2) = (µ, g) (8.25) as measure-function pairs on ΩT with values in R. Let η ∈ C0c (ΩT ). Then 1 + |vε|2 |∇G(uε)| dxdt − 1 + |vε|2 dµ̃ε 1 + |vε|2 2W (uε) ε|∇uε| ε|∇uε| dxdt η2(1 + |vε|2)ε|∇uε|2 dxdt )1/2∥ 2W (uε) ε|∇uε| L2(ΩT ) ≤‖η‖L∞(2TΛ4 + Λ3)1/2(2|ξε|(ΩT ))1/2. (8.26) Thanks to (8.25), (8.26) and (6.2) we conclude that (|∇G(uε)| Ln+1, 1 + |vε|2) = (µ, g) (8.27) as measure-function pairs on ΩT with values in R. Again by (2.1) we have {0=|∇uε|<W (uε)} |∇′G(uε)| dxdt {0=|∇uε|<W (uε)} |∂tuε| 2W (uε) dxdt ε(∂tuε) 2 dxdt )1/2( {0=|∇uε|<W (uε)} W (uε) 2Λ3(|ξε|(ΩT ))1/2, which vanishes by (6.2) as ε→ 0. This implies together with (8.24) and (8.27) that η g dµ = lim 1 + |vε|2|∇G(uε)| dxdt = lim η|∇′G(uε)| dxdt ≥ η d|∇′u|, where in the last line we used that η d|∇′u| = η d|∇′G(u)| ≤ lim inf η|∇′G(uε)| dxdt. 24 LUCA MUGNAI AND MATTHIAS RÖGER Considering now a set B ⊂ ∂∗{u = 1} with µ(B) = 0 we conclude that |∇′u|(B) ≤ 2 g dµ = 0, since g ∈ L2(µ). � Proposition 8.3. In Hn-almost-all points in ∂∗{u = 1} the tangential-plane of µ exists. Proof. From the Radon-Nikodym Theorem we obtain that the derivative f(z) := D|∇′u|µ(z) := lim µ(Bn+1r (z)) |∇′u|(Bn+1r (z)) (8.28) exists for |∇′u|-almost-all z ∈ ΩT and that f ∈ L1(|∇′u|). By (8.15) we deduce µ⌊∂∗{u = 1} = f |∇′u|. (8.29) Similarly we obtain that = Dµ|∇′u|(z) is finite for µ-almost all z ∈ ∂∗{u = 1}. By (8.22) this implies that f > 0 |∇′u|-almost everywhere in ΩT . (8.30) Since |∇′u| is rectifiable and f measurable with respect to |∇′u| we obtain from (8.29), (8.30) and [31, Remark 11.5] that µ⌊∂∗{u = 1} is rectifiable. (8.31) Moreover Hn-almost-all z ∈ ∂∗{u = 1} satisfy that µ(Bn+1r (z) \ ∂∗{u = 1}) µ(Bn+1r (z)) = 0, (8.32) lim sup µ(Bn+1r (z)) < ∞. (8.33) In fact, (8.32) follows from [11, Theorem 2.9.11] and (8.22), and (8.33) from Propo- sition 8.1 and (8.22). Let now z0 ∈ ∂∗{u = 1} satisfy (8.32), (8.33). For an arbitrary η ∈ C0c (Bn+11 (0)) we then deduce that lim sup ΩT \∂∗{u=1} r−1(z − z0) r−n dµ(z) ≤‖η‖C0c (Bn+11 (0)) lim supr→0 Bn+1r (z0) \ ∂∗{u = 1} Bn+1r (z0) ) lim sup Bn+1r (z0) by (8.32), (8.33). Therefore r−1(z − z0) r−n dµ(z) = lim ∂∗{u=1} r−1(z − z0) r−n dµ(z) if the latter limit exists. By (8.31) we therefore conclude that in Hn-almost-all points of ∂∗{u = 1} the tangent-plane of µ exists and coincides with the tangent plane of µ⌊∂∗{u = 1}. � THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 25 Proof of Proposition 4.5. Since u ∈ BV (ΩT ) and u(t, ·) ∈ BV (Ω) for almost all t ∈ (0, T ) we obtain that ∂tu,∇u are Radon measures on ΩT and that ∇u(t, ·) is a Radon measure on Ω for almost all t ∈ (0, T ). Moreover we observe that v ∈ L1(|∇u|) since |v| d|∇u| ≤ |v| d|∇′u| ≤ 2 g|v| dµ ≤ 2 ‖g‖L2(µ)‖v‖L2(µ) < ∞ by Theorem 4.4 and Proposition 8.2. From (3.3) and Proposition 8.3 we deduce that for any η ∈ C1c (ΩT ) η d∂tu = ηv d∇u = d|∇u| = ηV d|∇u(t, ·)| dt, which proves (4.9). � 9. Conclusions Theorem 4.6 suggests to define a generalized action functional S in the class of L2-flows by S(µ) := inf |v −H |2 dµ+ 4Snuc(µ), (9.1) where the infimum is taken over all generalized velocities v for the evolution (µt)t∈(0,T ). In the class of n-rectifiable L 2-flows we have S(µ) = |v −H |2 dµ+ 4Snuc(µ), (9.2) where v is the unique normal velocity of (µt)t∈(0,T ) (see Proposition 3.3). In the present section we compare the functional S with the functional S0 defined in [18] (see (1.2)) and discuss the implications of Theorem 4.6 on a full Gamma convergence result for the action functional. For the ease of the exposition we focus in this section on the switching scenario. Assumption 9.1. Let a sequence (uε)ε>0 of smooth functions uε : ΩT → R be given with uniformly bounded action (A1), zero Neumann boundary data (A3), and assume for the initial- and final states that for all ε > 0 uε(0, ·) = −1, uε(T, ·) = 1 in Ω. (9.3) Following [18] we define the reduced action functional on the set M ⊂ BV (ΩT , {−1, 1})∩ L∞(0, T, BV (Ω)) such that • for every ψ ∈ C0c (Ω) the function u(t, ·)ψ dx is absolutely continuous on [0, T ]; • (∂∗{u(t, ·) = 1})t∈(0,T ) is up to countably many times given as a smooth evolution of hypersurfaces. 26 LUCA MUGNAI AND MATTHIAS RÖGER By Assumption 9.1 the functional S0nuc can be rewritten as S0(u) := c0 ∣v(t, x) −H(t, x) dHn−1(x)dt + 4S0nuc(u), (9.4) S0nuc(u) := |∇u(t, ·)|(ψ)− lim |∇u(t, ·)|(ψ) + sup |∇u(t, ·)|(ψ) (9.5) where the sup is taken over all ψ ∈ C1(Ω) with 0 ≤ ψ ≤ 1. In [18, Proposition 2.2] a (formal) proof of the limsup- estimate was given for a subclass of ‘nice’ functions in M. Following the ideas of that proof, using the one- dimensional construction [18, Proposition 3.1], and a density argument we expect that the limsup-estimate can be extended to the whole set M. We do not give a rigorous proof here but rather assume the limsup-estimate in the following. Assumption 9.2. For all u ∈ M there exists a sequence (uε)ε>0 that satisfies Assumption 9.1 such that u = lim uε, S0(u) ≥ lim sup Sε(uε). (9.6) The natural candidate for the Gamma-limit of Sε with respect to L1(ΩT ) is the L1(ΩT )-lower semicontinuous envelope of S0, S(u) := inf lim inf S0(uk) : (uk)k∈N ⊂ M, uk → u in L1(ΩT ) . (9.7) 9.1. Comparison of S and S0. If we associate with a function u ∈ M the measure |∇u| on ΩT we can compare S0(u) and S( c02 |∇u|). Proposition 9.3. Let u ∈ M and let µ = L1 ⊗ µt be an L2-flow of measures. Assume that for almost all t ∈ (0, T ) µt ≥ c0 |∇u(t, ·)| (9.8) and that the nucleation cost S0nuc(u) is not larger than the nucleation cost Snuc(µ). S0(u) ≤ S(µ) (9.9) holds. For µ = c0 |∇u| we obtain that S0(u) = S(c0 |∇u|). (9.10) Proof. The locality of the mean curvature [29] shows that the weak mean curvature of µt and the (classical) mean curvature coincide on ∂{u(t, ·) = 1}. By Proposition 4.5 any generalized velocity v and the (classical) normal velocity V are equal on the phase boundary. This shows that the integral part of S0(u) is not larger than the integral part of S(µ), with equality if µt = c0 |∇u(t, ·)| for almost all t ∈ (0, T ). This proves (9.9). For the measure c0 |∇u| we observe that the nucleation cost Snuc( c02 µ) equals the nucleation cost S nuc(u) and we obtain (9.10). � If higher multiplicities occur for the measure µ, the nucleation costs of µ and u may differ and the value of S0(u) might be larger than S(µ) as the following example shows. Let Ω = (0, L), let {u = 1} be the shaded regions in Figure 1, and THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 27 x1 x2 Figure 1. The phases {u = 1} x1 x2 Figure 2. The measure µ let µ be the measure supported on the phase boundary and with double density on a hidden boundary connecting the upper and lower part of the phase {u = 1}, see Figure 2. At time t2 a new phase is nucleated but this time is not singular with respect to the evolution (µt)t∈(0,T ). On the other hand, no propagation cost occurs for the evolution (u(t, ·))t∈(t1,t0) whereas there is a propagation cost for (µt)t∈(t1,t2). The difference in action is given by S0(u)− S(µ) = 8c0 − 2c0 (x2 − x1)2 t2 − t1 where x1 is the annihilation point at time t1 and x2 the nucleation point at time t2, see Figure 1. This shows that as soon as (x2 − x1) < 4 t2 − t1 we have S(µ) < S0(u). x1 x2 Figure 3. Phases {uk = 1} 1x x2= Figure 4. The limit 28 LUCA MUGNAI AND MATTHIAS RÖGER The same example with x2 = x1 shows that S0 is not lower-semicontinuous and that a relaxation is necessary in order to obtain the Gamma-limit of Sε. In fact consider a sequence (uk)k∈N with phases {uk = 1} given by the shaded region in Figure 3. Assume that the neck connecting the upper and lower part of the shaded region disappears with k → ∞ and that uk converges to the phase indicator function u with phase {u = 1} indicated by the shaded regions in Figure 4. Then a nucleation cost at time t2 appears for u. For the approximations uk however there is no nucleation cost for t > 0 and the approximation can be made such that the propagation cost in (t1, t2) is arbitrarily small, which shows that S0(u) > lim inf S0(uk). The situation in higher space dimensions is even more involved than in the one- dimensional examples discussed above. For instance one could create a circle with double density (no new phase is created) at a time t1 and let this double-density circle grow until a time t2 > t1 where the double-density circle splits and two circles evolve in different directions, one of them shrinking and the other one growing. In this way a new phase is created at time t2. In this example S counts the creation of a double-density circle at time t1 and the cost of propagating the double-density circle between the times t1, t2. In contrast S0 counts the nucleation cost of the new phase at time t2, which is larger as the nucleation cost Snuc at times t1, but no propagation cost between the times t1, t2. The analysis in [18] suggests that minimizers of the action functional exhibit nucleation and annihilation of phases only at the initial- and final time. This class is therefore particularly interesting. Theorem 9.4. Let (uε)ε>0 satisfy Assumption 9.1 and suppose that Assumption 9.2 holds. Suppose that uε → u in L1(ΩT ), u ∈ M, and that u exhibits nucleation and annihilation of phases only at the final and initial time. Then S(u) = S0(u) ≤ lim inf Sε(uε) (9.11) holds. In particular, Sε Gamma-converges to S0 for those evolutions in M that have nucleations only at the initial time. Proof. From the definition of the functional S we deduce that S(u) ≤ S0(u) (9.12) and that there exists a sequence (uk)k∈N ⊂ M such that u = lim uk, S(u) = lim S0(uk). Assumption 9.2 implies that for all k ∈ N there exists a sequence (uε,k)ε>0 such uk = lim uε,k, S0(uk) ≥ lim sup Sε(uε,k). (9.13) Therefore we can choose a diagonal-sequence (uε(k),k)k∈N such that S(u) ≥ lim sup Sε(k)(uε(k),k). (9.14) By Proposition 4.1, 4.2 there exists a a subsequence k → ∞ such that uε(k),k → u, µε(k),k → µ, µ ≥ |∇u|, (9.15) THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 29 where the last inequality follows from η d|∇u(t, ·)| ≤ lim inf η|∇G(uε)| dx ≤ lim inf ηdµtε = ηdµt, with G as in (8.23). By Theorem 4.6 we further deduce that lim inf Sε(k)(uε(k),k) ≥ S(µ). This implies by (9.14) that S(u) ≥ S(µ). (9.16) Since µ0 = 0 and µt ≥ c0 |∇u(t, ·)| the nucleation cost of µ at t = 0 is not lower than the nucleation cost for u. Since by assumption there are no more nucleation times we can apply Proposition 9.3 and obtain that S0(u) ≤ S(µ). By (9.12), (9.16) we conclude that S0(u) = S(u) = S(µ). Applying Proposition 4.1 and Theorem 4.6 to the sequence (uε)ε>0 we deduce that there exists a subsequence ε→ 0 such that µε → µ̃, µ̃ ≥ |∇u| (9.17) and such that lim inf Sε(uε) ≥ S(µ̃). Repeating the arguments above we deduce from Proposition 9.3 that S0(u) ≤ S(µ̃) S0(u) ≤ lim inf Sε(uε). Combining the upper bound (9.6) with (9.11) proves the Gamma convergence of Sε in u. � 9.2. Gamma convergence under an additional assumption. Using Theorem 4.6 we can prove the Gamma convergence of Sε under an additional assumption on the structure of the set of those measures that arise as limit of sequences with uniformly bounded action. Assumption 9.5. Consider any sequence (uε)ε>0 with uε → u in L1(ΩT ) that satisfies Assumption 9.1. Define the energy measures µε according to (2.5) and let µ be any Radon measure such that for a subsequence ε→ 0 µ = lim µε. (9.18) Then we assume that there exists a sequence (uk)k∈N ⊂ M such that u = lim uk, S(µ) ≥ lim S0(uk). (9.19) For any u ∈ M that exhibits nucleation and annihilation only at initial and final time the Assumption 9.5 is always satisfied: The proof of Theorem 9.4 and our results in Section 4 show that for any limit µ as in (9.18) we can apply Proposition 9.3. Therefore S0(u) ≤ S(µ) and the constant sequence u satisfies (9.19). However, a characterization of those u ∈ M such that Assumption 9.5 holds is open. Theorem 9.6. Suppose that the Assumptions 9.1, 9.2, and 9.5 hold. Then Sε → S as ε→ 0 (9.20) in the sense of Gamma-convergence with respect to L1(ΩT ). 30 LUCA MUGNAI AND MATTHIAS RÖGER Proof. We first prove the limsup-estimate for Sε,S. In fact, fix an arbitrary u ∈ L1(ΩT , {−1, 1}) with S(u) < ∞. We deduce that there exists a sequence (uk)k∈N as in (9.7) such that S(u) = lim S0(uk). (9.21) By (9.6) for all k ∈ N there exists a sequence (uε,k)ε>0 such that uε,k = uk, S0(uk) ≥ lim sup Sε(uε,k). Choosing a suitable diagonal sequence uε(k),k we deduce that S(u) ≥ lim Sε(k)(uε(k),k), (9.22) which proves the limsup-estimate. We next prove the liminf -estimate. Consider an arbitrary sequence (uε)ε>0 that satisfies the Assumption 9.1. By Theorem 4.6 there exists u ∈ BV (ΩT , {−1, 1}) and a measure µ on ΩT such that uε → u in L1(ΩT ), µε → µ (9.23) for a subsequence ε→ 0, and such that lim inf Sε(uε) ≥ S(µ). (9.24) By Assumption 9.5 there exists a sequence (uk)k∈N ⊂ M such that (9.19) holds. By (9.24) and the definition of S this yields that lim inf Sε(uε) ≥ S(µ) ≥ lim S0(uk) ≥ S(u) (9.25) and proves the liminf -estimate. � Appendix A. Rectifiable measures and weak mean curvature We briefly summarize some definitions from Geometric Measure Theory. We always restrict ourselves to the hypersurface case, that is ‘tangential-plane’ and ‘rectifiability’ of a measure in Rd means ‘(d− 1)-dimensional tangential-plane’ and ‘(d− 1)-rectifiable’. Definition A.1. Let µ be a Radon-measure in Rd, d ∈ N. (1) We say that µ has a (generalized) tangent plane in z ∈ Rd if there exist a number Θ > 0 and a (d− 1)-dimensional linear subspace T ⊂ Rd such that r−d+1 y − z dµ(y) = Θ η dHd−1, for every η ∈ C0c (Rd). (A.1) We then set Tzµ := T and call Θ the multiplicity of µ in z. (2) If for µ-almost all z ∈ Rd a tangential plane exists then we call µ rectifiable. If in addition the multiplicity is integer-valued µ-almost everywhere we say that µ is integer-rectifiable. (3) The first variation δµ : C1c (R d,Rd) of a rectifiable Radon-measure µ on Rd is defined by δµ(η) := divTzµ η dµ. THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 31 If there exists a function H ∈ L1loc(µ) such that δµ(η) = − H · η dµ we call H the weak mean-curvature vector of µ. Appendix B. Measure-function pairs We recall some basic facts about the notion of measure function pairs introduced by Hutchinson in [16]. Definition B.1. Let E ⊂ Rd be an open subset. Let µ be a positive Radon- measure on E. Suppose f : E → Rm is well defined µ-almost everywhere, and f ∈ L1(µ,Rm). Then we say (µ, f) is a measure-function pair over E (with values in Rm). Next we define two notions of convergence for a sequence of measure-function pairs on E with values in Rm. Definition B.2. Suppose {(µk, fk)}k and (µ, f) are measure-function pairs over E with values in Rm. Suppose µk = µ, as Radon-measures on E. Then we say (µk, fk) converges to (µ, f) in the weak sense (in E) and write (µk, fk) → (µ, f), if µk⌊fk → µ⌊f in the sense of vector-valued measures, that means fk · η dµk = f · η dµ, for all η ∈ C0c (E,Rm). The following result is a slightly less general version of [16, Theorem 4.4.2], however this is enough for our aims. Theorem B.3. Let F : Rm → [0,+∞) be a continuous, convex function with super-linear growth at infinity, that is: |y|→∞ F (y) = +∞. Suppose {(µk, fk)}k are measure-function pairs over E ⊂ Rd with values in Rm. Suppose µ is Radon-measure on E and µk → µ as k → ∞. Then the following are true: (1) if F (fk) dµk < +∞, (B.1) then some subsequence of {(µk, fk)} converges in the weak sense to some measure function (µ, f) for some f . (2) if (B.1) holds and (µk, fk) → (µ, f) then lim inf F (fk) dµk ≥ F (f) dµ. (B.2) 32 LUCA MUGNAI AND MATTHIAS RÖGER References [1] William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417–491, 1972. [2] H. 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Freidlin and Alexander D. Wentzell. Fluktuatsii v dinamicheskikh sistemakh pod deistviem malykh sluchainykh vozmushchenii. “Nauka”, Moscow, 1979. Teoriya Veroyatnostei i Matematicheskaya Statistika. [Probability Theory and Mathematical Statistics]. [15] Mark I. Freidlin and Alexander D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, second edition, 1998. Translated from the 1979 Russian original by Joseph Szücs. [16] John E. Hutchinson. Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J., 35(1):45–71, 1986. [17] Tom Ilmanen. Convergence of the Allen-Cahn equation to Brakke’s motion by mean curva- ture. J. Differential Geom., 38(2):417–461, 1993. [18] Robert Kohn, Felix Otto, Maria G. Reznikoff, and Eric Vanden-Eijnden. Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation. Comm. Pure Appl. Math., 60(3):393–438, 2007. [19] Robert V. Kohn, Maria G. Reznikoff, and Yoshihiro Tonegawa. Sharp-interface limit of the Allen-Cahn action functional in one space dimension. Calc. Var. Partial Differential Equa- tions, 25(4):503–534, 2006. [20] Robert V. Kohn, Maria G. Reznikoff, and Eric Vanden-Eijnden. Magnetic elements at finite temperature and large deviation theory. J. Nonlinear Sci., 15(4):223–253, 2005. [21] Georgios T. Kossioris Markos A. Katsoulakis and Omar Lakkis. Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem. Interfaces Free Bound., 9(1):1–30, 2007. [22] Luciano Modica. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal., 98(2):123–142, 1987. [23] Luciano Modica and Stefano Mortola. Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. B (5), 14(1):285–299, 1977. [24] Roger Moser. A generalization of Rellich’s theorem and regularity of varifolds minimizing curvature. THE ALLEN–CAHN ACTION FUNCTIONAL IN HIGHER DIMENSIONS 33 [25] Pavel I. Plotnikov and Victor N. Starovŏıtov. The Stefan problem with surface tension as a limit of the phase field model. Differentsial′nye Uravneniya, 29(3):461–471, 550, 1993. [26] Maria G. Reznikoff and Yoshihiro Tonegawa. Higher multiplicity in the one-dimensional Allen- Cahn action functional. preprint, 2007. [27] Matthias Röger. Existence of weak solutions for the Mullins-Sekerka flow. SIAM J. Math. Anal., 37(1):291–301, 2005. [28] Matthias Röger and Reiner Schätzle. On a modified conjecture of De Giorgi. Mathematische Zeitschrift, 254(4):675–714, 2006. [29] Reiner Schätzle. Lower semicontinuity of the Willmore functional for currents. preprint, 2007. [30] Tony Shardlow. Stochastic perturbations of the Allen-Cahn equation. Electron. J. Differential Equations, pages No. 47, 19 pp. (electronic), 2000. [31] Leon Simon. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University Centre for Mathematical Analysis, Canberra, 1983. Luca Mugnai, Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig Matthias Röger, Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig E-mail address: [email protected], [email protected] 1. Introduction 1.1. Deterministic phase field models and sharp interface limits 1.2. Stochastic interpretation of the action functional 1.3. Organization Acknowledgment 2. Notation and Assumptions 3. L2-flows 4. Lower bound for the action functional 4.1. Lower estimate for the mean curvature 4.2. Lower estimate for the generalized velocity 4.3. Lower estimate of the action functional 4.4. Convergence of the Allen–Cahn equation to Mean curvature flow 5. Proofs of Propositions ??, ?? and Theorem ?? 6. Proof of Theorem ?? 6.1. Equipartition of energy 6.2. Convergence of approximate velocities 7. Proof of Theorem ?? 8. Proofs of Proposition ?? and Proposition ?? 9. Conclusions 9.1. Comparison of S and S0 9.2. Gamma convergence under an additional assumption Appendix A. Rectifiable measures and weak mean curvature Appendix B. Measure-function pairs References

0704.1955

Metrical characterization of super-reflexivity and linear type of Banach spaces

METRICAL CHARACTERIZATION OF SUPER-REFLEXIVITY AND LINEAR TYPE OF BANACH SPACES FLORENT BAUDIER† Abstract. We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain’s result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with the metrics dp induced by the ℓp norms. 1. Introduction and Notation We fix some notation and recall basic results. Let (M,d) and (N, δ) be two metric spaces and an injective map f : M → N . Following [11], we define the distortion of f to be dist(f) := ‖f‖Lip‖f −1‖Lip = sup x 6=y∈M δ(f(x), f(y)) d(x, y) . sup x 6=y∈M d(x, y) δ(f(x), f(y)) If dist(f) is finite, we say that f is a metric embedding, or simply an embedding of M into And if there exists an embedding f from M into N , with dist(f) ≤ C, we use the notation →֒ N . Denote Ω0 = {∅}, the root of the tree. Let Ωn = {−1, 1} n, Tn = i=0 Ωi and n=0 Tn. Thus Tn is the finite tree with n levels and T the infinite tree. For ε, ε′ ∈ T , we note ε ≤ ε′ if ε′ is an extension of ε. Denote |ε| the length of ε; i.e the numbers of nodes of ε. We define the hyperbolic dis- tance between ε and ε′ by ρ(ε, ε′) = |ε|+ |ε′|−2|δ|, where δ is the greatest common ancestor of ε and ε′. The metric on Tn, is the restriction of ρ. For a Banach space X , we denote BX its closed unit ball, and X ∗ its dual space. T embeds isometrically into ℓ1(N) in a trivial way. Actually, let (eε)ε∈T be the canonical basis of ℓ1(T ) (T is countable), then the embedding is given by ε 7→ s≤ε es. Laboratoire de Mathématiques, UMR 6623 Université de Franche-Comté, 25030 Besançon, cedex - France †[email protected] 2000 Mathematics Subject Classification. (46B20) (51F99) http://arxiv.org/abs/0704.1955v1 2 FLORENT BAUDIER† Aharoni proved in [1] that every separable metric space embeds into c0, so T does. The main result of this article is an improvement of Bourgain’s metrical characterization of super-reflexivity. Bourgain proved in [2] that X is not super-reflexive if and only if the finite trees Tn uniformly embed into X (i.e with embedding constants independent of n). Obviously if T embeds into X then the T ′ns embed uniformly into X and X is not super- reflexive, but if X is not super-reflexive we did not know whether the infinite tree T embeds into X . In this paper, we prove that it is indeed the case : Theorem 1.1. Let X be a non super-reflexive Banach space, then (T, ρ) embeds into X. The proof of the direct part of Bourgain’s Theorem essentially uses James’ characteriza- tion of super-reflexivity (see [7]) and an enumeration of the finite trees Tn. We recall James’ Theorem : Theorem 1.2 (James). Let 0 < θ < 1 and X a non super-reflexive Banach space, then : ∀ n ∈ N, ∃ x1, x2, . . . , xn ∈ BX , ∃ x 2, . . . , x n ∈ BX∗ s.t : x∗k(xj) = θ ∀k < j x∗k(xj) = 0 ∀k ≥ j 2. Metrical characterization of super-reflexivity The main obstruction to the embedding of T into any non-super-reflexive Banach space X is the finiteness of the sequences in James’ characterization. How, with a sequence of Bourgain’s type embedding, can we construct a single embedding from T into X ? In [13], Ribe shows in particular, that 2 lpn and ( 2 lpn) l1 are uniformly homeo- morphic, where (pn)n is a sequence of numbers such that pn > 1, and pn tends to 1. But T embeds into l1, hence via the uniform homeomorphism T embeds into 2 lpn . However T does not embed into any lpn(they are super-reflexive). The problem solved in the next theorem, inspired in part by Ribe’s proof, is to construct a subspace with a Schauder decomposition Fn where T2n+1 embeds into Fn and to repast properly the embeddings in order to obtain the desired embedding. Proof of Theorem 1.1 : Let (εi)i≥0, a sequence of positive real numbers such that∏ i≥0(1 + εi) ≤ 2, and fix 0 < θ < 1. Let kn = 2 2n+1+1 − 1. First we construct inductively a sequence (Fn)n≥0 of subspaces of X , which is a Schauder finite dimensional decomposition of a subspace of X s.t the projection from i=0 Fi onto⊕p i=0 Fi, with kernel i=p+1 Fi (with p < q) is of norm at most i=p (1+εi), and sequences xn,1, xn,2, . . . , xn,kn ∈ BFn x∗n,1, x n,2, . . . , x ∈ BX∗ s.t : x∗n,k(xn,j) = θ ∀k < j x∗n,k(xn,j) = 0 ∀k ≥ j. Metrical characterization of super-reflexivity and linear type of Banach spaces 3 Denote Φn : Tn → {1, 2, . . . , 2 n+1− 1} the enumeration of Tn following the lexicographic order. It is an enumeration of Tn such that any pair of segments in Tn starting at incompa- rable nodes (with respect to the tree ordering ≤) are mapped inside disjoint intervals. Let Ψn = Φ2n+1 and Γn = T2n+1. X is non super-reflexive, hence from James’ Theorem : ∃ x0,1, x0,2, . . . , x0,7 ∈ BX , ∃ x 0,1, x 0,2, . . . , x 0,7 ∈ BX∗ s.t : x∗0,k(x0,j) = θ ∀k < j x∗0,k(x0,j) = 0 ∀k ≥ j. Γ0 = T2 embeds into X via the embedding f0(ε) = s≤ε x0,Ψ0(s) (see [2]). Let F0 = Span{x0,1, . . . , x0,7}, then dim(F0) < ∞. Suppose that F0, . . . , Fp, and xp,1, xp,2, . . . , xp,kp ∈ BFp x∗p,1, x p,2, . . . , x ∈ BX∗ verifying the required conditions, are constructed for all p ≤ n. We apply Mazur’s Lemma (see [9] page 4) to the finite dimensional subspace i=0 Fi of X . Thus there exists Yn ⊂ X such that dim(X/Yn) < ∞ and : ‖x‖ ≤ (1 + εn)‖x+ y‖, ∀(x, y) ∈ Fi × Yn But Yn is of finite codimension in X , hence is not super-reflexive. From James’ Theorem and Hahn-Banach Theorem: ∃ xn+1,1, xn+1,2, . . . , xn+1,kn+1 ∈ BYn , ∃ x∗n+1,1, x n+1,2, . . . , x n+1,kn+1 ∈ BX∗ , s.t : x∗n+1,k(xn+1,j) = θ ∀k < j x∗n+1,k(xn+1,j) = 0 ∀k ≥ j. Γn+1 embeds into Yn via the embedding fn+1(ε) = s≤ε xn+1,Ψn+1(s) . Let Fn+1 = Span{xn+1,j ; 1 ≤ j ≤ kn+1}, then dim(Fn+1) < ∞, which achieves the induc- tion. Now define the following projections : Let, Pn the projection from Span( i=0 Fi) onto F0 · · · Fn with kernel Span( i=n+1 Fi). It is easy to show that ‖Pn‖ ≤ i=n(1 + εi) ≤ 2. We denote now Π0 = P0 and Πn = Pn − Pn−1 for n ≥ 1. We have that ‖Πn‖ ≤ 4. 4 FLORENT BAUDIER† From Bourgain’s construction, for all n : ρ(ε, ε′) ≤ ‖fn(ε)− fn(ε ′)‖ ≤ ρ(ε, ε′), where fn denotes the Bourgain’s type embedding from Γn in Fn, i.e fn(ε) = s≤ε xn,Ψn(s). Note that : ∀ n, ∀ ε ∈ Γn ‖fn(ε)‖ ≤ |ε|. Now we define our embedding. f : T → Y = Span( i=0 Fi) ⊂ X ε 7→ λfn(ε) + (1− λ)fn+1(ε) , if 2 n ≤ |ε| ≤ 2n+1 where, 2n+1 − |ε| f(∅) = 0. We will prove that : (2) ∀ε, ε′ ∈ T ρ(ε, ε′) ≤ ‖f(ε)− f(ε′)‖ ≤ 9ρ(ε, ε′). Remark 2.1 We have θ |ε| ≤ ‖f(ε)‖ ≤ |ε|. First of all, we show that f is 9−Lipschitz. We can suppose that 0 < |ε| ≤ |ε′| w.r.t remark 2.1. If |ε| ≤ 1 |ε′| then : ρ(ε, ε′) ≥ |ε′| − |ε| ≥ |ε|+ |ε′| Hence, ‖f(ε)− f(ε′)‖ ≤ 3ρ(ε, ε′). |ε′| < |ε| ≤ |ε′|, we have two different cases to consider. 1) if 2n ≤ |ε| ≤ |ε′| < 2n+1. Then, let 2n+1 − |ε| and λ′ = 2n+1 − |ε′| ‖f(ε)− f(ε′)‖ = ‖λfn(ε)− λ ′fn(ε ′) + (1 − λ)fn+1(ε)− (1− λ ′)fn+1(ε ≤ λ‖fn(ε)− fn(ε ′)‖+ |λ− λ′|(‖fn(ε ′)‖+ ‖fn+1(ε ′)‖) + (1 − λ)‖fn+1(ε)− fn+1(ε ≤ ρ(ε, ε′) + 2ρ(ε, ε′) + 2ρ(ε, ε′) ≤ 5ρ(ε, ε′), Metrical characterization of super-reflexivity and linear type of Banach spaces 5 because ‖fn(ε ′)‖ < 2n+1, ‖fn+1(ε ′)‖ < 2n+1 and, |λ− λ′| = |ε′| − |ε| ρ(ε, ε′) 2) if 2n ≤ |ε| ≤ 2n+1 ≤ |ε′| < 2n+2. Then, let 2n+1 − |ε| and λ′ = 2n+2 − |ε′| ‖f(ε)− f(ε′)‖ = ‖λfn(ε) + (1− λ)fn+1(ε)− λ ′fn+1(ε ′)− (1− λ′)fn+2(ε ≤ λ(‖fn(ε)‖ + ‖fn+1(ε)‖) + (1− λ ′)(‖fn+1(ε ′)‖+ ‖fn+2(ε ′)‖) + ‖fn+1(ε)− fn+1(ε ≤ ρ(ε, ε′) + 2λ|ε|+ 2(1− λ′)|ε′| ≤ 9ρ(ε, ε′), because, ρ(ε, ε′) , so λ|ε| ≤ 2ρ(ε, ε′). Similarly 1− λ′ = |ε′| − 2n+1 ρ(ε, ε′) and (1− λ′)|ε′| ≤ 2ρ(ε, ε′). Finally, f is 9-Lipschitz. Now we deal with the minoration. In our next discussion, whenever |ε| (respectively |ε′|) will belong to [2n, 2n+1), for some integer n, we shall denote 2n+1 − |ε| (respectively λ′ = 2n+1 − |ε′| We can suppose that ε is smaller than ε′ in the lexicographic order. Denote δ the greatest common ancestor of ε and ε′. And let d = |ε| − |δ| (respectively d′ = |ε′| − |δ|). 1) if 2n ≤ |ε|, |ε′| ≤ 2n+1. We have, x∗n,Ψn(δ)Πn(f(ε)− f(ε ′)) = θ(λd− λ′d′) x∗n+1,Ψn+1(δ)Πn+1(f(ε)− f(ε ′)) = θ((1 − λ)d− (1 − λ′)d′). Hence, ‖f(ε)− f(ε′)‖ ≥ θ(d− d′) −x∗n,Ψn(ε)Πn(f(ε)− f(ε ′)) = θλ′d′ −x∗n+1,Ψn+1(ε)Πn+1(f(ε)− f(ε ′)) = θ(1− λ′)d′. 6 FLORENT BAUDIER† ‖f(ε)− f(ε′)‖ ≥ Finally if we distinguish the cases d ≤ d′, and d′ < d we obtain : ‖f(ε)− f(ε′)‖ ≥ θ(d + d′) ρ(ε, ε′). 2) if 2n ≤ |ε| ≤ 2n+1 ≤ 2q+1 ≤ |ε′| ≤ 2q+2, or 2n ≤ |ε′| ≤ 2n+1 ≤ 2q+1 ≤ |ε| ≤ 2q+2. If n < q, |x∗q+1,Ψq+1(δ)Πq+1(f(ε)− f(ε ′)) + x∗q+2,Ψq+2(δ)Πq+2(f(ε)− f(ε ′))| = θMax(d, d′) Hence, ‖f(ε)− f(ε′)‖ ≥ ρ(ε, ε′). If n = q and |ε| ≤ |ε′|, |x∗n+1,Ψn+1(ε)Πn+1(f(ε)− f(ε ′)) + x∗n+2,Ψn+2(δ)Πn+2(f(ε)− f(ε ′))| ≥ θd′. ‖f(ε)− f(ε′)‖ ≥ ρ(ε, ε′). If n = q and |ε′| < |ε|, x∗n+1,Ψn+1(δ)Πn+1(f(ε)−f(ε ′))−x∗n+1,Ψn+1(ε)Πn+1(f(ε)−f(ε ′))+x∗n+2,Ψn+2(δ)Πn+2(f(ε)−f(ε ′)) = θd. Hence, ‖f(ε)− f(ε′)‖ ≥ ρ(ε, ε′). Finally T →֒ X . Corollary 2.2. X is non super-reflexive if and only if (T, ρ) embeds into X. Proof : It follows clearly from Bourgain’s result [2] and Theorem 1.1. � Metrical characterization of super-reflexivity and linear type of Banach spaces 7 3. Metric characterization of the linear type First we identify canonicaly {−1, 1}n with Kn = {−1, 1} k>n{0}. Let p ∈ [1,∞). Then we define an other metric on T = Kn as follows : ∀ ε, ε′ ∈ T , dp(ε, ε ′) = ( |εi − ε The length of ε ∈ T can be viewed as |ε| = (dp(ε, 0)) The norm ‖.‖p on ℓp coincides with dp for the elements of T . We recall now two classical definitions : Let X and Y be two Banach spaces. If X and Y are linearly isomorphic, the Banach- Mazur distance between X and Y , denoted by dBM (X,Y ), is the infimum of ‖T ‖ ‖T over all linear isomorphisms T from X onto Y . For p ∈ [1,∞], we say that a Banach space X uniformly contains the ℓnp ’s if there is a constant C ≥ 1 such that for every integer n, X admits an n-dimensional subspace Y so that dBM (ℓ p , Y ) ≤ C. We state and prove now the following result. Theorem 3.1. Let p ∈ [1,∞). If X uniformly contains the ℓnp ’s then (T, dp) embeds into X. Proof : We first recall a fundamental result due to Krivine (for 1 < p < ∞ in [8]) and James (for p = 1 and ∞ in [7]). Theorem 3.2 (James-Krivine). Let p ∈ [1,∞] and X be a Banach space uniformly con- taining the ℓnp ’s. Then, for any finite codimensional subspace Y of X, any ǫ > 0 and any n ∈ N, there exists a subspace F of Y such that dBM (ℓ p , F ) < 1 + ǫ. Using Theorem 3.2 together with the fact that each ℓnp is finite dimensional, we can build inductively finite dimensional subspaces (Fn) n=0 of X and (Rn) n=0 so that for every n ≥ 0, Rn is a linear isomorphism from ℓ p onto Fn satisfying ∀u ∈ ℓnp ‖u‖ ≤ ‖Rnu‖ ≤ ‖u‖ and also such that (Fn) n=0 is a Schauder finite dimensional decomposition of its closed linear span Z. More precisely, if Pn is the projection from Z onto F0 ⊕ ...⊕ Fn with kernel Span ( i=n+1 Fi), we will assume as we may, that ‖Pn‖ ≤ 2. We denote now Π0 = P0 and Πn = Pn − Pn−1 for n ≥ 1. We have that ‖Πn‖ ≤ 4. We now consider ϕn : Tn → ℓ p defined by ∀ε ∈ Tn, ϕn(ε) = εiei, where (ei) is the canonical basis of ℓ p . The map ϕn is clearly an isometric embedding of Tn into ℓnp . 8 FLORENT BAUDIER† Then we set : ∀ε ∈ Tn, fn(ε) = Rn(ϕn(ε)) ∈ Fn. Finally we construct a map f : T → X as follows : f : T → X ε 7→ λfm(ε) + (1− λ)fm+1(ε) , if 2 m ≤ |ε| < 2m+1, where, 2m+1 − |ε| Remark 3.3 We have 1 p ≤ ‖f(ε)‖ ≤ |ε| Like in the proof of Theorem 1.1 ,we prove that f is 9-Lipschitz using exactly the same computations. We shall now prove that f−1 is Lipschitz. We consider ε, ε′ ∈ T and assume again that 0 < |ε| ≤ |ε′|. We need to study two different cases. Again, whenever |ε| (respectively |ε′|) will belong to [2m, 2m+1), for some integer m, we shall denote 2m+1 − |ε| (respectively λ′ = 2m+1 − |ε′| 1) if 2m ≤ |ε|, |ε′| < 2m+1. dp(ε, ε ′) ≤ ‖λ i=1 εiei − λ ∑|ε′| i=1 ε iei‖p + ‖(1− λ) i=1 εiei − (1− λ ∑|ε′| i=1 ε iei‖p ≤ 2‖Πm(f(ε)− f(ε ′))‖ + 2‖Πm+1(f(ε)− f(ε ≤ 16‖f(ε)− f(ε′)‖. 2) if 2m ≤ |ε| ≤ 2m+1 ≤ 2q+1 ≤ |ε′| < 2q+2. if m < q, dp(ε, ε ′) ≤ 2dp(ε ′, 0) ≤ 2((1− λ′)dp(ε ′, 0) + λ′dp(ε ′, 0)) ≤ 2(2‖Πq+2(f(ε)− f(ε ′))‖+ 2‖Πm+1(f(ε)− f(ε ′))‖) ≤ 32‖f(ε)− f(ε′)‖. Metrical characterization of super-reflexivity and linear type of Banach spaces 9 if m = q, dp(ε, ε ′) ≤ λdp(ε, 0) + ‖(1− λ) i=1 εiei − λ ∑|ε′| i=1 ε iei‖p + (1 − λ ′)dp(ε ′, 0) ≤ 2‖Πm(f(ε)− f(ε ′))‖+ 2‖Πm+1(f(ε)− f(ε ′))‖ + 2‖Πm+2(f(ε)− f(ε ≤ 24‖f(ε)− f(ε′)‖. Finally we obtain that f−1 is 32-Lipschitz, and T →֒ X . In the sequel a Banach space X is said to have type p > 0 if there exists a constant T < ∞ such that for every n and every x1, . . . , xn ∈ X , εjxj‖ ≤ T p where the expectation Eε is with respect to a uniform choice of signs ε1, . . . , εn ∈ {−1, 1} The set of p’s for which X contains ℓnp ’s uniformly is closely related to the type of X according to the following result due to Maurey, Pisier [10] and Krivine [8], which clarifies the meaning of these notions. Theorem 3.4 (Maurey-Pisier-Krivine). Let X be an infinite-dimensional Banach space. pX = sup{p ; X is of type p}, Then X contains ℓnp ’s uniformly for p = pX. Equivalently, we have pX = inf{p ; X contains ℓ p’s uniformly}. We deduce from Theorem 3.1 two corollaries. Corollary 3.5. Let X a Banach space and 1 ≤ p < 2. The following assertions are equivalent : i) pX ≤ p. ii) X uniformly contains the ℓnp ’s. iii) the (Tn, dp)’s uniformly embed into X . iv) (T, dp) embeds into X. Proof : ii) implies i) is obvious. i) implies ii) is due to Theorem 3.2 and the work of Bretagnolle, Dacunha-Castelle and Krivine [4]. For the equivalence between ii) and iii) see the work of Bourgain, Milman and Wolfson [3] and Krivine [8]. iv) implies iii) is obvious. And ii) implies iv) is Theorem 3.1. 10 FLORENT BAUDIER† Corollary 3.6. Let X be an infinite dimensional Banach space, then (T, d2) embeds into Proof : This corollary is a consequence of the Dvoretsky’s Theorem [6] and Theorem 3.1. References [1] I. Aharoni, Every separable metric space is Lipschitz equivalent to a subset of c . Israel J. Math. 19 (1974), 284–291. [2] J. Bourgain, The metrical interpretation of super-reflexivity in Banach spaces. Israel J. Math. 56 (1986), 221-230. [3] J. Bourgain, V. Milman, H. Wolfson, On type of metric spaces. Trans. Amer. Math. Soc. volume 294, number 1, march 1986, 295-317. [4] J. Bretagnolle, D. Dacunha-Castelle, J.L. Krivine, Lois stables et espaces Lp. Ann. Instit. H. Poincaré, 2 (1966), 231-259. [5] J. Diestel, Sequences and Series in Banach Spaces. Springer-Verlag (1984). [6] A. Dvoretzky, Some results on convex bodies and Banach spaces. Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) 123–160. [7] R. C. James, Super-reflexive spaces with bases. Pacific J. Math. 41 (1972), 409-419. [8] J. L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976), 1-29. [9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer Berlin 1977. [10] B. Maurey, G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58(1) 1976, 45-90. [11] M. Mendel, A. Naor, Metric cotype, arXiv:math.FA/0506201 v3 29 Apr 2006 [12] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces. CBMS Regional Conference Series in Mathematics, 60. [13] M. Ribe, Existence of separable uniformly homeomorphic non isomorphic Banach spaces. Israel J. Math. 48 (1984), no. 2-3, 139-147. http://arxiv.org/abs/math/0506201 1. Introduction and Notation 2. Metrical characterization of super-reflexivity 3. Metric characterization of the linear type References

0704.1956

Search for exclusive events using the dijet mass fraction at the Tevatron

Search for exclusive events using the dijet mass fraction at the Tevatron O. Kepka∗ DAPNIA/Service de physique des particules, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France IPNP, Faculty of Mathematics and Physics, Charles University, Prague and Center for Particle Physics, Institute of Physics, Academy of Science, Prague C. Royon† DAPNIA/Service de physique des particules, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France In this paper, we discuss the observation of exclusive events using the dijet mass fraction as measured by the CDF collaboration at the Tevatron. We compare the data to pomeron exchange inspired models as well as Soft color interaction ones. We also provide the prediction on dijet mass fraction at the LHC using both exclusive and inclusive diffractive events. I. INTRODUCTION Double pomeron exchange (DPE) processes are expected to extend the physics program at the LHC not only due to the possible Higgs boson detection but also because of the possibility to study broader range of QCD physics and diffraction [1, 2, 3, 4, 5, 6, 7, 8]. The processes are theoreticaly characterized by large rapidity gap regions devoid of particles between centrally produced heavy object and the scattered hadrons which leave the interaction intact. This is attributed to the exchange of a colorless object, the pomeron (or the reggeon). In the LHC environment, however, the rapidity gap signature will not appear because of the high number of multiple interactions occuring at the same time, and the diffractive events will be identified tagging the escaping protons in the beam pipe. One generally considers two classes of DPE processes, namely, exclusive DPE events if the central object is produced alone carrying away the total available diffractive energy and inclusive events when the total energy is used to produce the central object and in addition the pomeron remnants. Exclusive events allow a precise reconstruction of the mass and kinematical properties of the central object using the central detector or even more precisely using very forward detectors installed far downstream from the interaction point. The most appealing exclusive process to be studied at the LHC is the Higgs boson production but since it cannot be observed at the Tevatron due to the low production cross section, one should find other ways to look for exclusive events at the Tevatron, for example in dijet, diphoton channels. It is needed to mention that until recently, there was not a decisive measurement that would provide enough evidence for the existence of exclusive production. Although exclusive production yields kinematically well constrained final state objects, their experimental detection is non-trivial due to the overlap with the inclusive DPE events. In those events, the colliding pomerons are usualy viewed as an object with partonic sub-structure. A parton emitted from the pomeron takes part in the hard interaction and pomeron remnants accompanying the central object are distributed uniformly in rapidity. Exclusive events usually appear as a small deviation from the inclusive model predictions which need to be studied precisely before accepting a new kind of production. In particular, the structure of the pomeron as obtained from HERA is not precisely known at high momentum fraction, and specifically, the gluon in the pomeron is not well constrained. It is not clear if such uncertainty could not lead to mis-identifying observed processes as exclusive. This would for instance preclude the spin analysis of the produced object. In this paper, we aim to investigate the observation of exclusive production at the Tevatron. Indeed, we use the dijet mass fraction distribution measured by the CDF collaboration and show that even taking into account uncertainties associated with the pomeron structure, one is unable to give a satisfactory description of the data without the existence of exclusive events. We also include other approach to explain diffraction in our study, the so called Soft color interaction model (the properties of all the models are discussed later). As an outlook, we apply current models for the DPE production for LHC energies and demonstrate the possible appereance of exclusive events through the dijet mass fraction. The paper is organized as follows: in the second section we give a brief description of the inclusive, exclusive, and Soft color interaction models. The third section discusses how well the various models can explain the preliminary ∗Electronic address: [email protected] †Electronic address: [email protected] http://arxiv.org/abs/0704.1956v1 mailto:[email protected] mailto:[email protected] Tevatron dijet mass fraction data and the constraints implied by data on the current models. In the fourth part, we foreshadow an application of the dijet mass fraction distribution as a tool to observe exclusive events at LHC energies. Finally, we discuss issues concerning the dijet mass fraction reconstruction and fast detector simulation in the Appendix. II. THEORETICAL MODELS Inclusive and exclusive DPE models used in this paper are implemented in the Monte Carlo program DPEMC [9]. The Soft color interaction model is embeded in the PYTHIA program [18]. The survey of the different models follows. A. Inclusive Models The first inclusive model to be mentioned is the so called “Factorized model”. It is an Ingelman-Schlein type of model [10] describing the diffractive double pomeron process as a scattering of two pomerons emitted from the proton, assuming a factorization of the cross section into a regge flux convoluted with the pomeron structure functions. For ep single diffraction, it is necessary to introduce secondary reggeon trajectory to describe the observed single diffractive non-factorable cross section. In the case of the Tevatron, the pomeron trajectory alone is sufficient to describe present data and the cross section is factorable as it was advocated in [11]. Factorization breaking between HERA and Tevatron comes only through the survival probability factor, denoting the probability that there is no additional soft interaction which would destroy the diffractively scattered protons. In other word, the probability to destroy the rapidity gap does not depend on the hard interaction. At Tevatron energies, the factor was measured to be approximately 0.1, and calculation suggested the value of 0.03 for the LHC. Pomeron structure functions, reggeon and pomeron fluxes are determined from the DIS ep collisions fitting the diffractive structure function FD at HERA. For one of the most recent published diffractive structure function analysis see e.g [12]. On the other hand, the Bialas-Landshoff (BL) inclusive model [8], is a purely non-perturbative calculation utilising only the shape of the pomeron structure function and leaving the overall normalization to be determined from the experiment; one can for example confront the prediction of DPE cross section with the observed rate at the Tevatron [11] and obtain the missing normalization factor 1. Both models use the pomeron structure measured at HERA which is gluon dominated. In this paper, we use the results of the QCD fits to the most recent Pomeron structure function data measured by the H1 collaboration [12]. The new gluon density in the Pomeron is found to be slightly smaller than the previous ones, and it is interesting to see the effect of the new PDFs with respect to the Tevatron measurements. However, the gluon density at high β, where β denotes the fraction of the particular parton in the pomeron, is not well constrained from the QCD fits performed at HERA. To study this uncertainty, we multiply the gluon distribution by the factor (1 − β)ν as shown in Fig. 1. QCD fits to the H1 data lead to the uncertainty on the ν parameter ν = 0.0± 0.5 [12]. We will see in the following how this parameter influences the results on dijet mass fraction as measured at the Tevatron. B. Exclusive Models Bialas-Landsoff exclusive model [13] is based on an exchange of two “non-perturbative” gluons between a pair of colliding hadrons which connect to the hard subprocess. Reggeization is employed in order to recover the pomeron parameters which successfully described soft diffractive phenomena, e.g. total cross section at low energies. A calculation of qq̄ and gg production and more details can be found in [13] and [14], respectively. On the contrary, the Khoze, Martin, Ryskin (KMR) [15] model is purely a perturbative approach. The interaction is obtained by an exchange of two gluons directly coupled to the colliding hadrons (no pomeron picture is introduced). While one gluon takes part in the creation of the central object, the other serves to screen the color flow across the rapidity gap. If the outgoing protons remain intact and scatter at small angles, the exchanged di-gluon system, in 1 One more remark is in order. In the BL inclusive model, the partonic content of the pomeron is expressed in terms of the distribution functions as fi/P(βi) ≡ βiGi/P(βi), where the Gi/P(βi) are the true parton densities as measured by the HERA collaboration, and βi denotes the momentum fraction of the parton i in the pomeron. The integral of fi/P(βi) is normalized to 1, so that in the limit fi/P(βi) → δ(βi) the exclusive cross section is recovered [9]. FIG. 1: Uncertainty of the gluon density at high β (here β ≡ z). The gluon density is multiplied by the factor (1− β)ν where ν=-1., -0.5, 0.5, 1. The default value ν = 0 is the gluon density in the pomeron determined directly by a fit to the H1 FD2 data with an uncertainty of about 0.5. both models, must obey the selection rules JZ = 0, C-even, P-even. Such constrains are also applied to the hard subprocesses for the production of the central object. The two models show a completely different pT dependence of the DPE cross section. The energy dependence of the BL model is found to be weaker since the Pomeron is assumed to be soft whereas it is not the case for the KMR model. C. Soft Color Interaction Model The Soft color interaction model (SCI) [17, 18] assumes that diffraction is not due to a colorless exchange at the hard vertex but rather to a string rearrangement in the final state during hadronisation. This model gives a probability (to be determined by the experiment) that there is no string connection, and so no color exchange, between the partons in the proton and the scattered quark produced during the hard interaction. Since the model does not imply the existence of a pomeron, there is no need of a concept like survival probability and a correct normalisation is found between single diffraction Tevatron and HERA data without any new parameter, which is one of the big successes of this model. III. DIJET MASS FRACTION AT THE TEVATRON Dijet mass fraction (DMF) turns out to be a very appropriate observable for identifying the exclusive production. It is defined as a ratio RJJ = MJJ/MX of the dijet system invariant mass MJJ to the total mass of the final state system MX (excluding the intact beam (anti)protons). If the jet algorithm has such properties that the outside-cone effects are small, the presence of an exclusive production would manifest itself as an excess of the events towards RJJ ∼ 1; for exclusive events, the dijet mass is essentially equal to the mass of the central system because no pomeron remnant is present. The advantage of DMF is that one can focus on the shape of the distribution; the observation of exclusive events does not rely on the overall normalization which might be strongly dependent on the detector simulation and acceptance of the roman pot detector. In the following analysis, we closely follow the measurement performed by the CDF Collaboration. One can find more information about the measurement and the detector setup in a note discussing preliminary results [16]. In this paragraph, we will mention only the different cuts which are relevant for our analysis. To simulate the CDF detector, we use a fast simulation interface [19], which performs a smearing of the deposited cell energy above a 0.5GeV threshold and reconstructs jets using a cone algorithm. Properties of the event such as the rapidity gap size were evaluated at the generator particle level. CDF uses a roman pot detector to tag the antiprotons on one side (corresponding to ηp̄ < 0). For the DMF reconstruction, we require the antiprotons to have the longitudinal momentum loss in the range 0.01 < ξp̄ < 0.12 and we apply the roman pot acceptance obtained from the CDF Collaboration (the real acceptance is greater than 0.5 for 0.035 < ξp̄ < 0.095). On the proton side, where no such device is present, a rapidity gap of the size 3.6 < ηgap < 5.9 is required. In the analysis, further cuts are applied: two leading jets with a transverse momentum above the threshold pjet1,jet2 > 10GeV or p jet1,jet2 T > 25GeV in the central region |ηjet1,jet2 | < 2.5, a third jet veto cut (p T < 5GeV) as well as an additional gap on the antiproton side of the size −5.9 < ηgap < −3.6. For the sake of brevity, the threshold for the transverse momentum of the two leading jets will be in the following denoted as pminT , if needed. The dijet mass is computed using the jet momenta for all events passing the above mentioned cuts. In order to follow as much as possible the method used by the CDF collaboration, the mass of the diffractive system MX is calculated from the longitudinal antiproton momentum loss ξp̄ within the roman pot acceptance, and the longitudinal momentum loss of the proton ξpartp is determined from the particles in the central detector (−4 <∼ ηpart <∼ 4), such that: sξp̄ξ p , (1) ξpartp = particles pT exp η, (2) summing over the particles with energies higher than 0.5GeV in the final state at generator level. To reconstruct the diffractive mass, ξpartp was multiplied by a factor 1.1, obtained by fitting the correlation plot between the momentum loss of the proton at generator level ξp and ξ p at particle level with a straight line. The DMF reconstruction is deeply dependent on the accuracy of the detector simulation. Since we are unable to employ the complete simulation in our analysis, we discuss possible effects due to the various definitions of DMF on the generator and the particle level in the Appendix. A. Inclusive model prediction We present first the dijet mass fraction calculated with FM and BL inclusive models. As stated in a previous section, we want to explore the impact of the high β gluon uncertainty in the pomeron. To do this, we multiply the gluon density by a factor (1− β)ν , for diverse values of ν = −1,−0.5, 0, 0.5, 1. The impact of the parameter is shown in Fig. 2 and Fig. 3 for jets with pT > 10GeV and pT > 25GeV, respectively. The computed distributions were normalized in shape, since there was no luminosity determination, implying no cross section estimation, in the CDF measurement. The interesting possible exclusive region at high RJJ is enhanced for ν = −1, however, not in such extent that would lead to a fair description of the observed distributions. As a consequence, the tail of the measured dijet mass fraction at high RJJ cannot be explained by enhancing the gluon distribution at high β, and an another contribution such as exclusive events is required. A particular property seems to disfavour the BL inclusive model at the Tevatron. Indeed, the dijet mass fraction is dumped at low values of RJJ , especially for jets pT > 10GeV. Since the cross section is obtained as a convolution of the hard matrix element and the distribution functions, the dumping effect is a direct consequence of the use of a multiplicative factor β in the parton density functions in the pomeron (see footnote 1). We will come back on this point when we discuss the possibility of a revised version of the BL inclusive model in the following. x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary =0.5ν =-0.5ν FM INC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary =0.5ν =-0.5ν BL INC FIG. 2: Dijet mass fraction for jets pT > 10GeV. FM (left) and BL (right) models, inclusive contribution. The uncertainty of the gluon density at high β is obtained by multiplying the gluon distribution by (1− β)ν for different values of ν (non-solid lines). As we have seen, inclusive models are not sufficient to describe well the measured CDF distributions. Thus, it opens an area to introduce different types of proceses/models which give a significant contribution at high RJJ . B. Exclusive models predictions In this section, we will study the enhancement of the dijet mass distribution using exclusive DPE processes, with the aim to describe the CDF dijet mass fraction data. We examine three possibilities of the interplay of inclusive plus exclusive contributions, specifically: 1. FM + KMR 2. FM + BL exclusive 3. BL inclusive + BL exclusive The full contribution is obtained by fitting the inclusive and exclusive distribution to the CDF data, leaving the overall normalization N and the relative normalization between the two contributions rEXC/INC free. More precisely, the DMF distribution is obtained with the fit as N(σINC(RJJ )+ r EXC/INCσEXC(RJJ )). The fit was done for jets with pminT = 10GeV and p T = 25GeV, separately. The overall normalization factor cannot be studied since the CDF collaboration did not determine the luminosity for the measurement. On the other hand, the relative normalization between the inclusive and exclusive production is a useful information. The relative normalization allows to make predictions for higher pT jets or for LHC energies for instance. For this sake, the relative normalizations rEXC/INC should not vary much between the two pminT mea- surements. Results are summarized in Table I. We give the inclusive and σINC and the exclusive cross sections σEXC, obtained directly from the models, and the relative scale factor needed to describe the CDF data to be applied to the exclusive contribution only. Whereas the relative normalization changes as a function pminT by an order of magnitude for the exclusive BL model, it tends to be rather stable for the KMR model (the uncertainty on the factor 2.5 might be relatively large since we do not have a full simulation interface and the simulation effects tend to be higher at low jet transverse momentum). Finally, in Fig. 4 and 5, the fitted distributions are depicted for pminT = 10, 25GeV jets, respectively. The Tevatron data are well described by the combination of FM and KMR model. We attribute the departure from the smooth distribution of the data to the imperfection of our fast simulation interface. On the contrary, the x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary =0.5ν =-0.5ν FM INC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary =0.5ν =-0.5ν BL INC FIG. 3: Dijet mass fraction for jets pT > 25GeV. FM (left) and BL (right) models, inclusive contribution. The uncertainty of the gluon density at high β is obtained by multiplying the gluon distribution by (1− β)ν for different values of ν (non-solid lines). contributions rEXC/INC(10) σINC(10)[pb] σEXC(10)[pb] rEXC/INC(25) σINC(25)[pb] σEXC(25)[pb] FM + KMR 2.50 1249 238 1.0 7.39 3.95 FM + BL exc 0.35 1249 1950 0.038 7.39 108 BL inc + BL exc 0.46 2000 1950 0.017 40.6 108 TABLE I: Cross sections for inclusive diffractive production σINC, exclusive cross section σEXC to be rescaled with a relative additional normalization between inclusive and exclusive events rEXC/INC for pT > 10GeV and pT > 25GeV jets and for different models (see text). Note that the fit to the data is parametriezed as N(σINC(RJJ ) + r EXC/INC EXC(RJJ )). BL inclusive model is disfavoured because it fails to describe the low RJJ region. It is due to the βi factor in the parton density fi/P(βi) used by the BL inclusive model (see footnote 1 where the variables are defined) that the RJJ distribution is shifted towards higher values. This factor was introduced to maintain the correspondence between the inclusive and exclusive model in the limit fi/P(xi) → δ(xi). On the contrary, this assumption leads to properties in contradiction with CDF data. Using the BL inclusive model without this additional normalization factor leads to a DMF which is in fair agreement with data. Indeed, we show in Fig. 6 the predictions of the “modified” model (i.e. defined as fi/P(βi) ≡ Gi/P(βi)) for pT > 10GeV and pT > 25GeV jets. We see that the low RJJ region is described well and that fitting the prediction of the exclusive KMR model with the BL inclusive model yields roughly the same amount of exclusive events as using the factorable models. The BL inclusive model will be revised to take these effects into account. We will not mention further this ”modified” version of the BL inclusive model since it gives similar results as the factorable models. The exclusive BL model leads to a quite reasonable description of the DMF shape for both pminT cuts in combination with FM, however, it fails to grasp the shape of the exclusive cross section measured as a function of the jet minimal transverse momentum pminT . To illustrate this, we present the CDF data for exclusive cross section corrected for detector effects compared with the predictions of both exclusive models after applying the same cuts as in the CDF measurement, namely: p jet1,2 T > p T , |ηjet1,2| < 2.5, 3.6 < ηgap < 5.9, 0.03 < ξp̄ < 0.08. The BL exclusive model x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 preliminary DPE data (stat.) exclusive contribution FM INC + KMR EXC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution BL INC + BL EXC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution FM INC + BL EXC FIG. 4: Dijet mass fraction for jets pT > 10GeV. FM + KMR (left), BL + BL (right), FM + BL (bottom) models. We notice that the exclusive contribution allows to describe the tails at high RJJ . shows a much weaker pT dependence than the KMR model and is in disagreement with data. 2 Let us note that the cross section of exclusive events measured by the CDF collaboration is an indirect measurement since it was obtained by subtracting the inclusive contribution using an older version of the gluon density in the pomeron measured at HERA. In that sense, the contribution of exclusive events using the newest gluon density from HERA might change those results. However, as we noticed, modifying the gluon density even greatly at high β by multiplying the gluon distribution by (1 − β)ν does not change the amount of exclusive events by a large factor, and thus does not modify the indirect measurement performed by the CDF collaboration much. x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution FM INC + KMR EXC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution BL INC + BL EXC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution FM INC + BL EXC FIG. 5: Dijet mass fraction for jets pT > 25GeV. FM + KMR (left), BL + BL (right), FM + BL (bottom) models. We note that the exclusive contribution allows to describe the tails at high RJJ . To finish the discussion about the pomeron like models, it is worth mentioning that these results assume that the survival probability has no strong dependence on β and ξ. If this is not the case, we cannot assume that the shape of the gluon distribution as measured at HERA could be used to make predictions at the Tevatron. However, this is a reasonable assumption since the survival probability is related to soft phenomena occuring during hadronisation effects which occur at a much longer time scale than the hard interaction. In other words, it is natural to suppose that the soft phenomenon will not be influenced by the hard interaction. x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution BL INC modified + KMR EXC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution + KMR EXC BL INC modified FIG. 6: Dijet mass distribution at the Tevatron calculated with the ”modified” parton densities (see text) for 10GeV (left) and 25 GeV (right) jets, KMR exclusive model included. (GeV)min 10 15 20 25 30 35 Bialas-Landshoff CDF preliminary FIG. 7: Exclusive cross section as a function of the minimal transverse jet momentum pminT measured by the CDF collaboration and compared to the prediction of the KMR and BL exclusive models. We note that the BL model overshoots the CDF measurement while the KMR model is in good agreement. C. Prospects of future measurements at the Tevatron In this section, we list some examples of observables which could be used to identify better the exclusive contribution in DMF measurements at the Tevatron. We present the prediction as a function of the minimal transverse momentum of the two leading jets pminT . Since the BL inclusive model does not describe the DMF at low RJJ , we choose to show x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 -1Luminosity 200pb >10GeV >50GeV exclusive contribution IS INC+ KMR EXC FIG. 8: Dijet mass fraction for two values of minimal transverse jet momentum pminT . We note that the relative exclusive contribution is higher at high pminT . (GeV) 0 10 20 30 40 50 60 70 80 = 0.5ν =-0.5ν IS + BL IS + KMR -1Number of jet events for 200pb (GeV) 0 10 20 30 40 50 60 70 80 = 0.5ν =-0.5ν IS + KMR Mean value of dijet mass fraction FIG. 9: Number of jet events and mean of the dijet mass fraction as a function of the minimal jet pminT . We note that the ideal value of pminT to enhance the exclusive contribution is of the order of 30-40 GeV which leads to a high enough production cross section as well as a large effect of the exclusive contribution on the dijet mass fraction. only the FM prediction in combination with both, KMR and BL exclusive models. The same roman pot acceptance and restriction cuts as in the CDF measurement were used, specifically, 0.01 < ξp̄ < 0.12, p jet1,2 T > p T , |ηjet1,2| < 2.5, 3.6 < |ηgap| < 5.9. Moreover, we adopted a normalization between inclusive and exclusive events as obtained for the pT > 25GeV analysis in the previous section because we are less sensitive to the imperfections of the fast simulation interface for higher pT jets. Fig. 8 illustrates the appearance of DMF for two separate values of minimum jet pminT . The character of the distribution is clearly governed by exclusive events at x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 preliminary DPE data (stat.only) Soft color interaction -10 -5 0 5 10 FIG. 10: Dijet mass fraction at the Tevatron for jets pT > 10GeV (left) and the η distribution of produced particles (right) for the Soft color interaction model. high pminT . Fig. 9 shows the rate of DPE events. In addition to the curves denoting inclusive contribution with the varied gluon density for ν = −0.5, 0, 0.5, the full contribution for both exclusive models is shown. For the FM model which is in better consistency with accessible data, the measurement of the DPE rate does not provide an evident separation of exclusive contribution from the effects due to the pomeron uncertainty since the noticable difference appears when the cross sections are too low to be observable. It is possible, however, to examine the mean of the DMF distribution. As seen in Fig. 9, this observable disentangles well the exclusive production with the highest effect between 30 and 40GeV. It needs to be stressed that even though we obtain a hint in understanding the exclusive production phenomena at the Tevatron, the final picture cannot be drawn before precisely measuring the structure of the pomeron. For this purpose the DMF or the DPE rate are not suitable at the Tevatron. In the former, there is no sensitivity to the high β gluon variation, whereas in the latter, the gluon variation and the exclusive contribution cannot be easily separated. The way out is to perform QCD fits of the pomeron structure in gluon and quark for data at low RJJ where the exclusive contribution is negligible. Another possibility is to perform silmutaneously the global fits of pomeron structure functions using DGLAP evolution and of the exclusive production. A final important remark is that this study was assuming pomeron like models for inclusive diffraction. It is worth studying other models like Soft color interaction processes and find out if they also lead to the same conclusion concerning the existence of exclusive events. D. Soft color interaction model The Soft color interaction model uses different approach to explain diffractive events. In this model, diffraction is due to special color rearrangement in the final state as we mentionned earlier. It is worth noticing that in this model, the CDF data are dominated by events with tagged antiproton on the p̄ (ηp̄ < 0) side and a rapidity gap on the p side. In other words, in most of the events, there is only one single antiproton in the final state accompanied by a bunch of particles (mainly pions) flowing into the beam pipe. This is illustrated in Fig. 10 right which shows the rapidity distribution of produced particles and we notice the tail of the distribution at high rapidity. We should not omit to mention that on the other hand, the probability to get two protons intact (which is important for the double tagged events) is in SCI model extremly small. After applying all CDF cuts mentioned above, the comparison between SCI and CDF data on RJJ is shown in Figs. 10 (left) and 12. Whereas it is not possible to describe the full dijet mass fraction for a jet with pT > 10GeV, Jet1η -4 -2 0 2 4 Jet2η -4 -2 0 2 4 FIG. 11: Rapidity distribution of a leading jet (left) and a second leading jet (right) in the SCI model when calculating dijet mass fraction. x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 preliminary DPE data (stat.only) Soft color interaction exclusive contribution SCI + KMR EXC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 preliminary DPE data (stat.only) Soft color interaction FIG. 12: Dijet mass fraction at the Tevatron for jets pT > 10GeV for the SCI model and KMR exclusive model (left), and for jets pT > 25GeV for the SCI model only (right). it is noticeable that the exclusive contribution is found to be lower than in the case of the pomeron inspired models. Indeed, performing the same independent fit of SCI and KMR exlusive contribution one finds that only 70% of the exclusive contribution needed in case of pomeron inspired models is necessary to describe the data. For jets with pT > 25GeV, no additional exclusive contribution is needed to describe the measurement which can be seen in Fig. 12. Since most events are asymmetric in the sense that only the antiproton is strictly intact and on the other side, there is a flow of particles in the beam pipe, it is worth studying the rapidity distribution of jets for this model. The results are shown in Fig. 11. We note that the rapidity distribution is boosted towards high values of rapidity and not centered around zero like for pomeron inspired models and CDF data. Moreover, the cross section for pT > 10GeV jets is in the SCI model σSCI = 167 pb, about only 13% of the cross section predicted by the pomeron inspired models which however give a correct prediction of a large range of observables including DPE cross sections. Therefore, such properties disfavour the SCI model. However, it would be worth studying and modifying the SCI model since the probability to observe two protons in the final state (and/or two gaps) should be higher than the square probability of observing one proton (and/or one gap) only (single diffraction) as it was seen by the CDF collaboration [21]. The model needs to be adjusted to take this into account and than it would be interesting to see the impact on the dijet mass fraction and the existence of exclusive events. IV. DIJET MASS FRACTION AT THE LHC It was suggested that exclusive production at the LHC could be used to study the properties of a specific class of centrally produced objects like Higgs bosons. However, it relies on many subtleties such as a good understanding of the inclusive production. The perturbative nature of the diffractive processes results in the factorization of the cross section to a regge flux and pomeron structure functions, while factorization breaking appears via the survival probability only. The gluon density in the pomeron is of most important matter, since its value at high momentum fraction will control the background to exclusive DPE, and the pomeron flux and the survival probability factor will have to be measured at the LHC to make reliable predictions. The flux depends on the pomeron intercept αP whose impact on the DMF distribution for LHC energies is shown in Fig. 13. The pomeron intercept is parametrized as αP = 1 + ǫ and the prediction is made for four values of ǫ = 0.5, 0.2, 0.12, 0.08. The updated HERA pomeron structure function analysis [12] suggests that the “hard pomeron” intercept value is close to αP = 1.12. Nevertheless, new QCD fits using single diffractive or double pomeron exchange data will have to be performed to fully constrain the parton densities and the pomeron flux at the LHC. FIG. 13: Sensitivity of the dijet mass fraction to different values of the pomeron intercept αP = 1 + ǫ. We also give the dependence of the DMF on jet pT at the LHC. DPE events in this analysis were selected ap- plying the roman pot acceptance on both sides from the interaction point, and using a fast simulation of the CMS detector [20] (the results would be similar using the ATLAS simulation) and asking two leading jets with pT >= 100, 200, 300, 400GeV. We have disfavored the predictions of the BL exclusive model at the Tevatron. The BL exclusive shows weak pT dependence which makes the model unphysical for LHC energies since it predicts cross sections even higher than the inclusive ones. We therefore focus on the predictions of FM and KMR models, only. As in the previous sections, we also include a study of the uncertainty on the gluon density enhancing the high β gluon with a factor (1− β)ν . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 FIG. 14: Dijet mass fraction at the LHC as a function of jet minimal transverse momentum pminT , FM inclusive model. FIG. 15: Dijet mass fraction at the LHC for jets pT > 200GeV and pT > 400GeV, respectively, FM inclusive + KMR exclusive models. 100 150 200 250 300 350 400 FIG. 16: Number of DPE events at the LHC as a function of minimal transverse momentum pminT of two leading jets. FM inclusive + KMR exclusive models. The gluon variation is displayed for different ν values. 100 150 200 250 300 350 400 FIG. 17: Average value of the dijet mass fraction as a function of minimal transverse momentum pminT of the leading jets. Exclusive contribution and different values of ν are shown. FM + KMR models. The dijet mass fraction as a function of different pT is visible in Fig. 14. The exclusive contribution manifests itself as an increase in the tail of the distribution which can be seen for 200GeV jets (left) and 400GeV jets (right), respectively in Fig. 15. Exclusive production slowly turns on with the increase of the jet pT which is demonstrated in Fig. 16 where the number of expected DPE events is shown. However, with respect to the uncertainty on the gluon density this appearance is almost negligiable. One can use the average position of the DMF as a function of the minimal jet transverse momentum pminT to study the presence of the exclusive contribution, see Fig. 17. This is true especially for high pT jets. The exclusive production at the LHC plays a minor role for low pT jets. Therefore, measurements e.g for pT < 200GeV where the inclusive production is dominant could be used to constrain the gluon density in the pomeron. Afterwards, one can look in the high pT jet region to extract the exclusive contribution from the tail of the DMF. V. CONCLUSION The aim of this paper was to investigate whether we can explain the excess of events at the high dijet mass fraction measured at the Tevatron without the exclusive production. The result is actually two fold. Concerning the pomeron induced models (”Factorized model” and Bialas-Landshoff inclusive models) we found that the uncertainty on the high β gluon density in the Pomeron has a small impact at high RJJ . Therefore, an additional contribution is needed to describe the CDF data with these models. We examined the exclusive KMR model and Bialas-Landshoff exclusive model predictions for the role of the additional contribution and found that the best descriprion of data is achieved by the combination of the Factorized inclusive model (or the modified inclusive Bialas- Landshoff one) and the KMR exclusive model. The exclusive contribution at the Tevatron can be magnified requesting higher pT jets and studying specific observables like a mean of the dijet mass fraction, for example. Though, one of the limitations of using high pT jets is due to the rate of DPE events which falls logarithmicaly allowing measurements for jets up to approximately 40GeV. The Bialas-Landshoff exclusive model seems to be disfavoured by Tevatron data since it shows a softer jet pT dependance and predicts unphysically large DPE rates at LHC energies. In the case of the Soft color interaction model which is not based on pomeron exchanges, the need to introduce an additional exclusive production is less obvious. For low pT jets the amount of exclusive events to describe the data is smaller than in case of Factorized model, but for high pT jets no additional contribution is necessary. This draws a new question: whether the double pomeron exchange events could be explained by a special rearrangement of color only? The CDF data are in this model dominated by single diffractive events. The probability of tagging two protons in the final state within this model is very small, contradicting the CDF observation. So even though the SCI model is not applicable for DPE events in the current state it would be worth adjusting this model to correctly predict the rate of double tagged events and study the model prediction of dijet mass fraction and other DPE induced processes. Dijet mass fraction at the LHC could be used to select the exclusive events. Indeed, it is possible to study jets with pT > 200GeV for instance, and to focus on events with DMF above 0.8 which is dominated by exclusive production (see Fig. 15). However, as it was advocated earlier, a complete QCD analysis consisting of measuring the gluon density in the Pomeron (especially at high β) and study the QCD evolution of exclusive events as a function of jet pT is needed to fully understand the observables, and make predictions for diffractive Higgs production and its background at the LHC as an example. Acknowledgments The authors want to thank M. Boonekamp, R. Enberg, D. Goulianos, G. Ingelman, R. Pechanski and K. Tereashi for useful discussions and for providing them the CDF data and roman pot acceptance. VI. APPENDIX Throughout the paper, we have purpously omitted a discussion of imperfections concerning the dijet mass fraction reconstruction within our framework, postponing it to this section. In this appendix, all calculations are done for jets with pT > 10GeV. • In our analysis, we define the dijet mass fraction as a ratio of the two leading jet invariant mass MJJ to the central diffractive mass MX . The latter was determined using the momentum loss ξp̄ measured in a roman pot on the antiproton side and the ξpartp obtained from particles on the generator level, such as MX = (sξp̄ξ In this case, we must ensure that all of the produced diffractive energy MX is deposited into the central detector. If this is not the case, our MX at generator level might be sensibly larger than the one measured by the CDF collaboration. The energy flow of the particles on the generator level as a function of rapidity is shown in Fig. 18, upper plot. The middle plot shows the energy flow weighted by the transverse momentum of the particle ET . We see that most of the energy is deposited in the calorimeter region, i.e. for |η| < 4. In p̄ tagged events, protons most frequently loose a smaller momentum fraction (roughly ξp ∼ 0.025) than the tagged antiproton for which the acceptance turns on for ξp̄ > 0.035. This can be seen from the ξp population plot in the bottom of Fig. 18. Thus, a collision of more energetic pomeron from the antiproton side with a pomeron from the proton side is boosted towards the p̄ as it is seen on the energy flow distributions. • A comparison between the proton momentum loss obtained from particles ξpartp calculated using formula (2) and the proton momentum loss at generator level ξp leads to the factor 1.1 mentionned in a previous section. The dependance is displayed in Fig. 19. • The size of the rapidity gap runs as a function of the momentum loss ξ like ∆η ∼ log 1/ξ. The size of the gap which increases with decreasing ξ for inclusive models can be seen in Fig. 20. Regions of high rapidity show the p̄ hits whereas the low rapidity region is due to the produced particles detected in the central detector; they are well separated by a rapidity gap. For exclusive events, the size of a rapidity gap is larger and does not show such a strong ξ dependance as for inclusive models. • The simulation interface plays a significant role in the determination of the exclusive contribution. As previously stated, we cannot profit from having access to the full simulation interface and having under control all the effects of the detector. In order to eliminate some effects of the simulation we plot the dijet mass distribution RJJ using the information from the generator and check whether the need of exclusive events to describe the data is still valid. Specifically, we require the same cuts as in Section III but the diffractive mass MRP is evaluted using the true (anti)protons momentum loss (ξp̄)ξp at generator level MRPX = sξp̄ξp. (3) The dijet mass fraction calculated with MRP is shown in Fig. 21. We see that the distribution is shifted to lower values of RJJ , requesting slightly more exclusive events to describe the CDF data. The description of the data is also quite good. • The role of the simulation interface to reconstruct jets can be illustrated by comparing the above distributins to DMF calculated at generator level defined as RJJ = sξp̄ξpβ1β2 sξp̄ξp β1β2, (4) where β1, β2 denote the fraction of the pomeron carried by the interacting parton. As can be seen, in Fig. 21 (right), the DMF distribution at pure generator level shows a completely different shape not compatible with CDF data and shows the importance of jet reconstruction. [1] M. Boonekamp, R. Peschanski and C. Royon, Phys. Lett. B598 (2004) 243. [2] C. Royon, Mod. Phys. Lett. A18 (2003) 2169. [3] M. Boonekamp, A. De Roeck, R. Peschanski and C. Royon, Acta Phys. Polon. B33 (2002) 3485, Phys. Lett. B550 (2002) 93. [4] M. Boonekamp, J. Cammin, S. Lavignac, R. Peschanski and C. Royon, Phys. Rev. D73 (2006) 115011. [5] B. Cox, J. Forshaw, B. Heinemann, Phys. Lett. B540 (2002) 263. [6] V. A. Khoze, A. D. Martin and M. G. Ryskin, arXiv:hep-ph/0702213. [7] V. A. Khoze, A. D. Martin and M. G. Ryskin, Eur. Phys. J. C48 (2006) 467. [8] M. Boonekamp, R. Peschanski, C. Royon, Phys. Rev. Lett. 87 (2001) 251806; M. Boonekamp, R. Peschanski, C. Royon, Nucl. Phys. B669 (2003) 277, Err-ibid B676 (2004) 493; [9] M. Boonekamp and T. Kucs, Comput. Phys. Commun. 167 (2005) 217. [10] G. Ingelman, P.E.Schlein, Phys.Lett. B152 (1985) 256. [11] A. A. Affolder et al. [CDF Collaboration], Phys. Rev. Lett. 84 (2000) 5043. [12] C. Royon, L. Schoeffel, S. Sapeta, R. Peschanski and E. Sauvan, arXiv:hep-ph/0609291. [13] A. Bialas, P. V. Landshoff, Phys. Lett. B256 (1990) 540; A. Bialas, W. Szeremeta, Phys. Lett. B296 (1992) 191; A. Bialas, R. Janik, Zeit. für. Phys. C62 (1994) 487. [14] A. Bzdak, Acta Phys. Polon. B35 (2004) 1733. [15] V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C19 (2001) 477, Err-ibid C20 (2001) 599; V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C23 (2002) 311; V.A. Khoze, A.D. Martin, M.G. Ryskin, Eur. Phys. J. C24 (2002) 581. [16] The CDF Collaboration, CDF note 8493, (2006) [17] R. Enberg, G. Ingelman, A. Kissavos, N. Timneanu, Phys. Rev. Lett. 89 (2002) 081801. [18] We used the Pythia SCI Monte Carlo program described in http://www.isv.uu.se/thep/MC/scigal/. [19] Fast simulation of the CDF and DØ detectors, SHW package. [20] CMSIM, fast simulation of the CMS detector, CMS Collab., Technical Design Report (1997); TOTEM Coll., Technical Design Report, CERN/LHCC/99-7; ATLFAST, fast simulation of the ATLAS detector, ATLAS Coll., Technical Design Report, CERN/LHC C/99-14. [21] CDF Coll., Phys. Rev. Lett. 91 (2003) 011802. http://arxiv.org/abs/hep-ph/0702213 http://arxiv.org/abs/hep-ph/0609291 http://www.isv.uu.se/thep/MC/scigal/ -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 FIG. 18: Upper and medium plots: Rapidity and ET weighted rapidity distributions of all particles produced (except the protons); Lower plot: momentum loss of the proton in double pomeron exchange events ξp for FM (left) and BL (right) inclusive models. 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 FIG. 19: Comparision of the proton momentum loss ξpartp calculated with formula (2) and the proton momentum loss ξp at generator level. 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. 20: Rapidity of particles on the p̄ side vs. p̄ momentum loss: inclusive models (top) for FM (left) and BL (right); exlusive models (bottom) for KMR (left) and BL (right). Hits of scattered p̄ are included. x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary exclusive contribution FM INC + KMR EXC x/Mjj=MjjR 0 0.2 0.4 0.6 0.8 1 DPE data (stat.) preliminary FM INC FIG. 21: Dijet mass fraction for jets pT > 10GeV: FM + KMR (left), and at generator level calculated according to (4). Introduction Theoretical Models Inclusive Models Exclusive Models Soft Color Interaction Model Dijet mass fraction at the Tevatron Inclusive model prediction Exclusive models predictions Prospects of future measurements at the Tevatron Soft color interaction model Dijet mass fraction at the LHC Conclusion Acknowledgments Appendix References

0704.1957

Entanglement Cost for Sequences of Arbitrary Quantum States

Entanglement Cost for Sequences of Arbitrary Quantum States Garry Bowen ∗and Nilanjana Datta† January 10, 2019 Abstract The entanglement cost of arbitrary sequences of bipartite states is shown to be expressible as the minimization of a conditional spectral entropy rate over sequences of separable extensions of the states in the sequence. The expression is shown to reduce to the regularized entanglement of formation when the nth state in the sequence consists of n copies of a single bipartite state. 1 Introduction A fundamental problem in entanglement theory is to determine how to op- timally convert entanglement, shared between two distant parties Alice and Bob, from one form to another. Entanglement manipulation is the process by which Alice and Bob convert an initial bipartite state ρAB which they share, to a required target state σAB using local operations and classical communication (LOCC). If the target state σAB is a maximally entangled state, then the protocol is called entanglement distillation, whereas if the initial state ρAB is a maximally entangled state, then the protocol is called entanglement dilution. Optimal rates of these protocols were originally eval- uated under the assumption that the entanglement resource accessible to Alice and Bob consist of multiple copies, i.e., tensor products ρ⊗nAB , of the initial bipartite state ρAB, and the requirement that the final state of the protocol is equal to n copies of the desired target state σ⊗nAB with asymptot- ically vanishing error in the limit n→ ∞. The distillable entanglement and entanglement cost computed in this manner are two asymptotic measures of entanglement of the state ρAB . Moreover, in the case in which ρAB is pure, these two measures of entanglement coincide and are equal to the von ∗Centre for Quantum Computation, DAMTP, University of Cambridge, Cambridge CB3 0WA, UK †Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail:[email protected]) http://arxiv.org/abs/0704.1957v3 Neumann entropy of the reduced state on any one of the subsystems, A or In this paper we focus on entanglement dilution, which, as mentioned earlier, is the entanglement manipulation process by which two distant par- ties, say Alice and Bob, create a desired bipartite target state from a maxi- mally entangled state which they initially share, using LOCC. In [6, 7] the optimal rate of entanglement dilution, namely, the entanglement cost, was evaluated in the case in which Alice and Bob created multiple copies of a desired target state ρAB , with asymptotically vanishing error, from a shared resource of singlets, using local operations and classical communication. In particular, for the case of a pure target state ρAB , the entanglement cost was shown [7] to be equal to the entropy of entanglement of the state, i.e., the von Neumann entropy of the reduced state on any one of the two sub- systems A and B. Moreover, in [8] it was shown that for an arbitrary mixed state ρAB, the entanglement cost is equal to the regularized entanglement of formation of the state (see (10) for its definition). The practical ability to transform entanglement from one form to another is useful for many applications in quantum information theory. However, it is not always justified to assume that the entanglement resource available consists of states which are multiple copies (and hence tensor products) of a given entangled state, or to require that the final state of the protocol is of the tensor product form. More generally an entanglement resource is characterized by an arbitrary sequence of bipartite states which are not necessarily of the tensor product form. Sequences of bipartite states on AB are considered to exist on Hilbert spaces H⊗nA ⊗H B for n ∈ {1, 2, 3 . . .}. A useful tool for the study of entanglement manipulation in this general scenario is provided by the Information Spectrum method. The information spectrum method, introduced in classical information theory by Verdu & Han [1, 2], has been extended into quantum information theory by Hayashi, Ogawa and Nagaoka [3, 4, 5]. The power of the information spectrum ap- proach comes from the fact that it does not depend on the specific structure of sources, channels or entanglement resources employed in information the- oretical protocols. In this paper we evaluate the optimal asymptotic rate of entanglement dilution for an arbitrary sequence of bipartite states. The case of an arbitrary sequence of pure bipartite state was studied in [9]. The paper is organized as follows. In Section 2 we introduce the necessary notations and definitions. Section 3 contains the statement and proof of the main result, stated as Theorem 1. Note that if we consider the sequence of bipartite states to consist of tensor products of a given bipartite state, then our main result reduces to the known results obtained in [6, 8] (see the discussion after Theorem 1 of Section 3). Finally, in Section 4 we show how Theorem 1 yields an alternative proof of the equivalence of the asymptotic entanglement cost and the regularised entanglement of formation [8]. 2 Notations and definitions LetB(H) denote the algebra of linear operators acting on a finite–dimensional Hilbert space H of dimension d and let D(H) denote the set of states (or density operators, i. e. positive operators of unit trace) acting on H. Fur- ther, let H(n) denote the Hilbert space H⊗n. For any state ρ ∈ D(H), the von Neumann entropy is defined as S(ρ) := −Tr (ρ log ρ). Let Λn be a quantum operation used for the transformation of an initial bipartite state ωn to a bipartite state ρn, with ωn, ρn ∈ D ((HA ⊗HB)⊗n). For the entanglement manipulation processes considered in this paper, Λn either consists of local operations (LO) alone or LO with one-way or two- way classical communication. We define the efficacy of any entanglement manipulation process in terms of the fidelity Fn := Tr ρnΛn(ωn) between the output state Λn(ωn) and the target state ρn. An entanglement manipulation process is said to be reliable if the asymptotic fidelity F := lim infn→∞ Fn = 1. For given orthonormal bases {|χiA〉} i=1 and {|χiB〉} i=1 in Hilbert spaces A andH B , of dimensions d A and d B respectively, we define the canonical maximally entangled state of Schmidt rank Mn ≤ min{dnA, dnB} to be |ΨMnAB〉 = |χiA〉 ⊗ |χiB〉. (1) In fact, in the following, we consider HA ≃ HB , for simplicity, so that dnA = d B . Here and henceforth, the explicit n-dependence of the basis states |χiA〉 and |χiB〉 has been suppressed for notational simplicity. The quantum information spectrum approach requires the extensive use of spectral projections. Any self-adjoint operator A acting on a finite di- mensional Hilbert space may be written in its spectral decomposition A = i λiπi, where πi denotes the operator which projects onto the eigenspace corresponding to the eigenvalue λi. We define the positive spectral projec- tion on A as {A ≥ 0} = πi, i.e., the projector onto the eigenspace of positive eigenvalues of A. For two operators A and B, we can then define {A ≥ B} as {A−B ≥ 0}. The following key lemmas are used repeatedly in the paper. For their proofs see [3, 4]. Lemma 1. For self-adjoint operators A, B and any positive operator 0 ≤ P ≤ I the inequality P (A−B) A ≥ B (A−B) holds. Lemma 2. Given a state ρn ∈ D(H⊗n) and a self-adjoint operator ωn ∈ B(H⊗n), we have {ρn ≥ enγωn}ωn ≤ e−nγ . (3) for any real number γ. In the quantum information spectrum approach one defines spectral di- vergence rates, which can be viewed as generalizations of the quantum rel- ative entropy. The spectral generalizations of the von Neumann entropy, the conditional entropy and the mutual information can all be expressed as spectral divergence rates. Definition 1. Given a sequence of states ρ̂ = {ρn}∞n=1, with ρn ∈ D(H⊗n), and a sequence of positive operators ω̂ = {ωn}∞n=1, with ωn ∈ B(H⊗n), the quantum spectral sup-(inf-) divergence rates are defined in terms of the difference operators Πn(γ) = ρn − enγωn, for any arbitrary real number γ, D(ρ̂‖ω̂) = inf γ : lim sup {Πn(γ) ≥ 0}Πn(γ) D(ρ̂‖ω̂) = sup γ : lim inf {Πn(γ) ≥ 0}Πn(γ) respectively. The spectral entropy rates and the conditional spectral entropy rates can be expressed as divergence rates with appropriate substitutions for the sequence of operators ω̂ = {ωn}∞n=1. These are S(ρ̂) = −D(ρ̂‖Î) and S(ρ̂) = −D(ρ̂‖Î), where Î = {In}∞n=1, with In being the identity operator acting on the Hilbert space H⊗n. Further, for sequences of bipartite states ρ̂AB = {ρnAB}∞n=1, S(A|B) = −D(ρ̂AB‖ÎA ⊗ ρ̂B) (4) S(A|B) = −D(ρ̂AB‖ÎA ⊗ ρ̂B) . (5) In the above, ÎA = {InA}∞n=1 and ρ̂B = {ρnB}∞n=1, with InA being the identity operator in B(H A ) and ρ B = TrAρ AB, the partial trace being taken on the Hilbert space H A . Various properties of these quantities, and relationships between them, are explored in [10]. For sequences of states ρ̂ = {ρ⊗n} and ω̂ = {ω⊗n}, with ρ, ω ∈ D(H), it has been proved [5] that D(ρ̂‖ω̂) = D(ρ̂‖ω̂) = S(ρ‖ω), (6) where S(ρ‖ω) := Tr ρ log ρ− Tr ρ logω, is the quantum relative entropy. Two parties, Alice and Bob, share a sequence of maximally entangled states {|ΨMnAB〉} n=1, and wish to convert them into a sequence of given bi- partite states {ρnAB}∞n=1, with ρn ∈ D(H⊗n) and |Ψ AB〉 ∈ H B . The protocol used for this conversion is known as entanglement dilution. The concept of reliable entanglement manipulation may then be used to define an asymptotic entanglement measure, namely the entanglement cost. Definition 2. A real-valued number R is said to be an achievable dilu- tion rate for a sequence of states ρ̂AB = {ρnAB}, with ρnAB ∈ D((HA ⊗ ⊗n), if ∀ε > 0, ∃N such that ∀n ≥ N a transformation exists that takes |ΨMnAB〉〈Ψ AB | → ρnAB with fidelity F 2n ≥ 1− ε and logMn ≤ R. Definition 3. The entanglement cost of the sequence ρ̂AB is the infimum of all achievable dilution rates: EC(ρ̂AB) = inf R (7) To simplify the expressions representing the entanglement cost, we define the following sets of sequences of states. Firstly, given a sequence of target states ρ̂AB = {ρnAB}∞n=1, define the set Dcq(ρ̂AB) as the set of sequences of tripartite states ˆ̺RAB = {̺nRAB}∞n=1 such that each ̺nRAB is a classical- quantum state (cq-state) of the form ̺nRAB = R〉〈inR| ⊗ |φ AB〉〈φ AB |, (8) where ρnAB = AB〉〈φ AB | and the set of pure states {|i R〉} form an orthonormal basis of H⊗nR . We refer to the state ̺ RAB as a cq-extension of the bipartite state ρnAB. Let D AB) denote the set of all possible cq- extensions of ρnAB . The entanglement of formation of the bipartite state ρnAB ∈ D((HA ⊗ ⊗n) is defined as EF (ρ AB) := min i S(ρ where ρ A = TrB |φ AB〉〈φ AB |, the partial trace being taken over the Hilbert space H⊗nB , and the minimization is over all possible ensemble decomposi- tions of the state ρnAB. Alternatively, the entanglement of formation of the state ρnAB can be expressed as EF (ρ AB) = min Dncq(ρ S(A|R)̺n , (9) where S(A|R)̺n denotes the conditional entropy S(A|R)̺n = S(̺nRA)− S(̺nR), with ̺nRA = TrB ̺ RAB and ̺ R = TrA ̺ RA, the state ̺ RAB being a cq- extension of the state ρnAB. The regularized entanglement of formation of a bipartite state ρAB ∈ D(HA ⊗HB) is defined as E∞F (ρAB) := lim EF (ρ AB) (10) The sup-conditional entropy rate S(A|R) of the sequence ˆ̺RA := {̺nRA}∞n=1, defined as S(A|R) = −D(ˆ̺RA‖ÎA ⊗ ˆ̺R), (11) where ˆ̺R := {̺nR}∞n=1, will be of particular significance in this paper. Note: For notational simplicity, the explicit n-dependence of quantities are suppressed in the rest of the paper, wherever there is no scope of any ambi- guity. 3 Entanglement Dilution for Mixed States The asymptotic optimization over entanglement dilution protocols leads to the following theorem. Theorem 1. The entanglement cost of a sequence of bipartite target states ρ̂AB = {ρnAB}∞n=1, is given by EC(ρ̂AB) = min Dcq(ρ̂AB) S(A|R), (12) or equivalently minDcq(ρ̂AB) S(B|R), where Dcq(ρ̂AB) is the set of sequences of tripartite states ˆ̺RAB = {̺nRAB}∞n=1 defined above. The proof of Theorem 1 is contained in the following two lemmas. How- ever, before going over to the proof, we would first like to point out that previously known results on entanglement dilution [7, 8] can be recovered from the above theorem. In [8] it was proved that the entanglement cost of an arbitrary (mixed) bipartite state ρAB , evaluated in the case in which Alice and Bob create multiple copies (i.e., tensor products) of ρAB (with asymp- totically vanishing error, from a shared resource of singlets, using LOCC) is given by the regularized entanglement of formation E∞F (ρAB) (10). In Section 4 we prove how this result can be recovered from Theorem 1. As regards the entanglement cost of pure states, in [9] we obtained an expres- sion for the entanglement cost of an arbitrary sequence of pure states and we proved that this expression reduced to the entropy of entanglement of a given pure state (say, |ψAB〉), if the sequence consisted of tensor products of this state – thus recovering the result first proved in [6]. Lemma 3. (Coding) For any sequence ρ̂AB = {ρnAB}∞n=1 and δ > 0, the dilution rate R = S(A|R) + δ, (13) where S(A|R) is the sup-conditional spectral rate given by (11), is achievable. Proof. Let the target bipartite state ρnAB have a decomposition given by ρnAB = pi|φiAB〉〈φiAB |, (14) where the Schmidt decomposition of |φiAB〉 is given by |φiAB〉 = λi,k|ψi,kA 〉|ψ B 〉, (15) with the Schmidt coefficients λik being arranged in non-increasing order, i.e., λi1 ≥ λi2 . . . ≥ λidn , for dn = dimH Alice locally prepares the classical-quantum state (cq-state) ρnRAA′ = i pi|iR〉〈iR| ⊗ |φiAA′〉〈φiAA′ | ∈ D ((HR ⊗HA ⊗HA′)⊗n). She then does a unitary operation on the system RAA′ given by InA ⊗ΘnRA′ ρnRAA′ InA ⊗ΘnRA′ where ΘnRA′ := |jR〉〈jR| ⊗ |χlA′〉〈ψ |, (16) with {|χlA′〉} l=1 being a fixed orthonormal basis in H A′ . This results in the state pi|iR〉〈iR| ⊗ λi,k λi,k ′ |ψi,kA 〉〈ψ A | ⊗ |χ A′〉〈χk A′ |, (17) where, once again, the explicit n-dependence of the terms has been sup- pressed for notational simplicity. Note that Alice’s operation amounts to a coherent implimentation of a projective measurement on R with rank one projections |jR〉〈jR|, followed by a unitary Uj = l |χlA′〉〈ψ A′ | on A ′, condi- tional on the outcome j. Alice teleports the A′ state to Bob. The resultant shared state is νnRAB := pi|iR〉〈iR|⊗ k,k′=1 λi,k λi,k ′|ψi,kA 〉〈ψ A |⊗|χ B〉〈χk B |+σnRAB (18) where σnRAB is an unnormalized error state. Note that the sum over the index k is truncated to Mn. This truncation occurs due to the so-called quantum scissors effect [11], i.e., if the quantum state to be teleported lives in a space of dimension higher than the rank Mn of the shared entangled state (used by the two parties for teleportation), then all higher-dimensional terms in the expansion of the original state are cut off. Moreover the system A′ is now referred to as B, since it is now in Bob’s possession. Alice also sends the “classical” state R to Bob through a classical chan- nel. Bob then acts on the system RB, which is now in his possession, with the unitary operator (ΘnRB) †. The final shared state can therefore be ex- pressed as InA ⊗ (ΘnRB)† νnRAB InA ⊗ΘnRB = ωnRAB + σ̃ pi|iR〉〈iR| ⊗ |φ̂iAB〉〈φ̂iAB |+ σ̃nRAB , where |φ̂iAB〉 := (Q A ⊗ I B)|φiAB〉 , (20) with Q A being the orthogonal projector onto span of the Schmidt vectors corresponding to theMn largest Schmidt coefficients of |φiAB〉, and σ̃nRAB := (ΘnRB) †σnRABΘ By Uhlmann’s theorem (see [12]) it follows that F (ωnAB + σ̃ AB , ρ AB) ≥ F (ωnAB, ρ 1 and the fidelity between the state ωnAB of the entanglement dilution protocol and the target state ρnAB is bounded below by Fn ≥ max ∣〈ρnABC |ωnABC〉 ∣, (21) where |ωnABC〉 is any fixed purification of the final state ωnAB and the maxi- mization is taken over all purifications of ρnAB. By choosing purifications |ωnCAB〉 = pi|iC〉|φ̂iAB〉 and |ρnCAB〉 = pi|iC〉|φiAB〉, we obtain the following lower bound to F 2n(ωnAB, ρnAB). Let QnRA := i |iR〉〈iR| ⊗ Q A and ρ RA := i pi|iR〉〈iR| ⊗ ρ A , where A = TrB |φ AB〉〈φiAB |. Then F 2n ≥ ∣〈ωnCAB|ρnCAB〉 QnRAρ Explicitly examining the projection operator PnRA := { i pi|iR〉〈iR|⊗ρ e−nαρnR⊗InA}, where α is a real number, we can express it in the form PnRA = i pi|iR〉〈iR|⊗ A − e −nαInA ≥ 0} = i |iR〉〈iR|⊗ {ρ A ≥ e −nαIA}. The rank of each of the projectors {ρn,iA ≥ e−nαIA} is then bounded by Tr[{ρ e−nαIA}] ≤ enα by Lemma 2, and hence by comparing PnRA with QnRA we can see that Mn = ⌈enα⌉ implies that Tr[QnRAρnRA] ≥ Tr[PnRAρnRA]. For any δ > 0 we can always choose a positive integer N such that for all n ≥ N there is an integer Mn satisfying S(A|R) + δ ≥ 1n logMn > S(A|R). Thus, using a sequence of maximally entangled states {|ΨMnAB〉} n=1 of Schmidt rank Mn, from the definition of S(A|R) it follows that 2 ≥ lim QnRAρ ≥ lim PnRAρ = 1. (23) and entanglement dilution at the rate R = S(A|R) + δ is achievable. Lemma 4. (Weak Converse) For any arbitrary sequence of states ρ̂AB, any entanglement dilution protocol with a rate ∗ < min S(A|R), (24) 1Take a purification |ω〉 such that F (ω, ρ) = 〈ω|ρ〉. Then utilize purifications |ω〉|0〉, |σ〉|1〉, and |ρ〉|0〉, which along with Uhlmann’s theorem implies F (ω + σ, ρ) ≥ 〈ω|ρ〉 = F (ω, ρ). where S(A|R) is the sup-conditional spectral rate given by (11), is not reli- able. Proof. Let TnAB denote any LOCC operation used for transforming the max- imally entangled state |ΨMnAB〉 ∈ H B to the target state ρ AB in this Hilbert space, such that F |ΨMnAB〉〈Ψ , ρnAB → 1 as n → ∞. Em- ploying the Lo & Popescu theorem [13], the final state of the protocol is expressible as ωnAB := T |ΨMnAB〉〈Ψ A ⊗ U B )|Ψ AB〉〈Ψ AB |(K A ⊗ U † (25) A = I A, and U B is unitary. Let |ωnCAB〉 := k |knC〉 ⊗ (K A ⊗ U B )|Ψ AB〉, denote a purification of the final state, ωnAB, of the entanglement dilution protocol, with C denoting a reference system, and {|knC〉} denoting an orthonormal basis in its Hilbert Space H⊗nC . By Uhlmann’s theorem, for this fixed purification |ω CAB〉, the fidelity is given by AB , ω AB) = max |〈ρnCAB |ωnCAB〉|, (26) where the maximization is over all purifications |ρnCAB〉 of the target state ρnAB. However, this maximization is equivalent to a maximization over all possible unitary transformations acting on the reference system C. This in turn corresponds to a particular decomposition of the purification of the target state ρnAB with respect to a fixed reference system [14]. Explicitly we then have |ρnCAB〉 = |knC〉|φ AB〉, where |φn,kAB〉〈φ AB | is the given decomposition of ρnAB obtained from the maximization. AB , ω AB) = |〈ρnCAB |ωnCAB〉| = | pk〈φn,kAB |K A ⊗ U AB〉|, (27) Note that AB〉 = |χjA〉U |χjA〉U B〉, (28) where Nn = dimH A and P k=1 |χkA〉〈χkA|. For simplicity, let us consider HA ≃ HB ≃ H, and let the state |φn,kAB〉 ∈ (HA⊗HB)⊗n ≃ H⊗2n have a Schmidt decomposition |φn,kAB〉 = A,i 〉⊗ |ψn,kB,i〉. Further, let W kA and W kB be unitary operators in B(H⊗n) such that W k|ψn,kA,j〉 = |χ A〉 and Wk|ψ B,j〉 = |χ B〉. Then from (28) it follows that AB〉 = W kA|ψ A,j〉U B,j〉, A,j〉|ψ B,j〉 (29) where V A := (U TW kA. Here we have used the relation j |j〉 ⊗ U |j〉 = T |j〉 ⊗ |j〉 for U unitary and {|j〉} an orthonormal basis in 〈φn,kAB |K A ⊗ U AB〉 = Tr where ρ A = TrB |φ AB〉〈φ AB | = A,i 〉〈ψ A,i |. Then from (27), using the Cauchy Schwarz inequality we then obtain pn,kρ A · P pn,kρ PMnA (K pn,kρ pn,kTr where P A = (V †PMnA V A . The third inequality follows by the following argument. Express the third line as qkπk with qk = PMnA (K and πk = Tr A pkρ ≥ 0. From the properties Tr[PMnA ] = Mn and A = I A it follows that k qk = 1 and qk ≥ 0 for all k. Then using the concavity of the map x 7→ x, we have that k : qk>0 k : qk>0 πk, (31) yielding the inequality in the last line of (30). Defining the projection operator PnRA := |jnR〉〈jnR| ⊗ P and the state ρnRA := pn,k|knR〉〈knR| ⊗ ρ the square of the fidelity can then be bounded by F 2n ≤ Tr PnRAρ pn,kTr , (33) whereQ A is the orthogonal projector onto the span of the Schmidt vectors corresponding to the Mn largest Schmidt coefficients of |φn,kAB〉. Note that eqs. (22) and (33) yields an alternative proof of the following lemma stated in [15]: Lemma 5. The entanglement dilution fidelity for a given bipartite state ρnAB := i pi|φiAB〉〈φiAB |, under an LOCC transformation Λn is given by F 2(Λn(ΨMnAB), ρ AB) = λij , (34) where λij , j = 1, . . . ,Mn denote the Mn largest Schmidt coefficients of |φiAB〉. From (32), using Lemma 1, with Πn(γ) := ρnRA − e−nγρnR ⊗ InA, PnRAρ PnRAΠ + e−nγ pkTr[P {Πn(γ) ≥ 0}Πn(γ) since Tr[Pn,k] = Tr[PMn ] =Mn. Hence for Mn ≤ enR we have F 2n ≤ Tr {Πn(γ) ≥ 0}Πn(γ) + e−n(γ−R). (35) Choosing a number γ and δ > 0 such that R+ δ = γ < S(A|R), the second term on RHS of (35) tends to zero as n → ∞. However, since γ < S(A|R) the first term on RHS of (35) does not converge to 1 as n→ ∞. Hence, the asymptotic fidelity F is not equal to 1. It is then straightfoward to show that the particular choice of decom- position of each ρnAB imposed by the fidelity criterion gives a minimization over possible cq-sequences. Suppose there exists a cq-sequence σ̂RAB with Sσ(A|R) = Sρ(A|R) − ε for some ε > 0. It then follows from the cod- ing theorem that the rate R = Sσ(A|R) + ε/2 is asymptotically attainable. However, if we take F ′n = |〈σnRR′AB |ωnRR′AB〉| then this is less than the max- imization over all possible purifications, bounding the asymptotic fidelity below 1, giving a contradiction. 4 The regularized entanglement of formation The application of the main result to the case of multiple copies of a single bipartite state provides a new proof of the equivalence [8] between the reg- ularized entanglement of formation E∞F (ρAB) (10), of a bipartite state ρAB, and its entanglement cost EC(ρAB). First note that as the entanglement of formation is a bounded non- increasing function of n we have infn EF (ρ AB) = limn→∞ EF (ρ AB) = E∞F (ρAB). Consider a sequence ρ̂AB = {ρnAB}∞n=1 of a bipartite states. For any state ρnAB in the sequence, let S(An|Bn)ρnAB denote the conditional en- tropy: S(An|Bn)ρn = S(ρnAB)− S(ρnB). From results in [5] it can be shown that the conditional entropy rate of the sequence is bounded above by the sup-conditional spectral entropy rate: E∞F (ρAB) = lim sup S(An|Bn) ≤ S(A|B) (36) Thus, for any sequence of cq-states ˆ̺RAB = {̺nRAB}∞n=1 on RAB, which reduce to product sequences ρ̂AB = {̺⊗n}∞n=1 on AB, we have from (9) and EF (̺ ⊗n) ≤ lim inf S(An|Rn)̺n ≤ S(A|R), where S(A|R) denotes the sup-conditional spectral entropy rate defined in (11). For the reverse inequality we simply construct states of block size m on RAB such that ωmn = ( i |iR〉〈iR|⊗|φmi 〉〈φmi |AB)⊗⌊n/m⌋⊗σRAB , where σ is an asymptotically irrelevant buffer state whenever m does not divide n. Using the chain rule [16] S(A|R) ≤ S(RA)−S(R), the definitions of S(RA) and S(R), and (6), we obtain S(A|R) ≤ S(ωmmRA )− S(ωmmR ) i S(ω for ωmA,i = TrB |φmi 〉〈φmi |AB . Taking the infimum over both m and decompo- sitions then implies EC(ρAB) = E F (ρAB) (37) for product sequences, and hence the regularized entanglement of formation for a bipartite state is equal to its entanglement cost. Acknowledgments This work is part of the QIP-IRC supported by the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 213681. References [1] S. Verdu and T. S. Han, IEEE Trans. Inform. Theory 40, 1147 (1994). [2] T. S. Han, Information-Spectrum Methods in Information Theory Springer-Verlag, (2002). [3] T. Ogawa and H. Nagaoka, IEEE Trans. Inform. Theory 46, 2428 (2000). [4] H. Nagaoka and M. Hayashi, quant-ph/0206185. [5] M. Hayashi and H. Nagaoka, IEEE Trans. Inform. Theory 49,1753 (2003). [6] C.H.Bennett, H.J.Bernstein, S.Popescu, and B.Schumacher, Phys. Rev. A, vol. 53, 2046, 1996. [7] C.H.Bennett, D.P.DiVincenzo, J.A.Smolin, and W.K.Wootters, Phys. Rev. A, vol. 54, 3824, 1996. [8] P. M.Hayden, M. Horodecki and B. M.Terhal, J. Phys. A 34, 6891(2001). [9] G. Bowen and N. Datta, IEEE Trans. Inform. Theory 54,3677 (2008). [10] G. Bowen and N. Datta, Proceedings 2006 IEEE International Sympo- sium on Information Theory p.451 (2006), quant-ph/0604013. [11] D. T. Pegg, L. S. Phillips and S. M. Barnett, Phys. Rev. Lett., 81, p. 1604, 1998. [12] M. A.Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press,Cambridge, (2000). [13] H.K.Lo and S.Popescu, Phys. Rev. A, vol. 63, 022301, (2001) [14] L.Hughston, R.Jozsa,and W.Wootters, Phys. Lett. A 183,14, (1993). [15] M. Hayashi, Quantum Information: an Introduction (Springer-Verlag, Berlin, Heidelberg, 2006). [16] G.Bowen and N.Datta, quant-ph/0610003. 1 Introduction 2 Notations and definitions 3 Entanglement Dilution for Mixed States 4 The regularized entanglement of formation

0704.1958

Line and continuum variability of two intermediate-redshift, high-luminosity quasars

Astronomy & Astrophysics manuscript no. echopap c© ESO 2021 November 8, 2021 Line and continuum variability of two intermediate-redshift, high-luminosity quasars D.Trevese1, D.Paris1, G. M. Stirpe2, F.Vagnetti3, and V. Zitelli2 1 Dipartimento di Fisica, Università di Roma “La Sapienza”, P.le A.Moro 2, I-00185 Roma(Italy) e-mail: [email protected] 2 INAF - Osservatorio Astronomico di Bologna, via Ranzani, 1 - 40127 Bologna (Italy) 3 Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, I-00133 Roma (Italy) ABSTRACT Context. It has been shown that the luminosity of active galactic nuclei and the size of their broad line region obey a simple relation of the type RBLR = aLγ, from faint Seyfert nuclei to bright quasars, allowing single-epoch determination of the central black hole mass MBH = bLγ∆2Hβ from their luminosity L and width of Hβ emission line. Adopting this mass determination for cosmological studies requires the extrapolation to high redshift and luminosity of a relation whose calibration, relies so far on reverberation mapping mea- surements performed for L <∼1046erg s−1 and redshift z <∼ 0.4. Aims. We initiated a campaign for the spectrophotometric monitoring of a few luminous, intermediate redshift quasars whose apparent magnitude, V < 15.7, allows observations with a 1.8 m telescope, aimed at proving that emission lines vary and respond to continuum variations even for luminosities >∼1047erg s−1, and determining eventually their MBH from reverberation mapping. Methods. We have repeatedly performed simultaneous spectrophotometric observations of quasars and reference stars to determine relative variability of continuum and emission lines. We describe the observations and methods of analysis. Results. For the quasars PG 1634+706 and PG 1247+268 we obtain light-curves respectively for CIII](λλ1909Å), MgII(λλ2798Å) and for CIV(λλ1549Å), CIII](λλ1909Å) emission lines with the relevant continua. During 3.2 years of observation, in the former case no continuum variability has been detected and the evidence for line variability is marginal, while in the latter case both continuum and line variability are detected with high significance and the line variations appear correlated with continuum variations. Conclusions. The detection of the emission line variability in a quasar with L ∼1047erg s−1 encourages the prosecution of the moni- toring campaign which should provide a black hole mass estimate in other 5-6 years, constraining the mass-luminosity relation in a poorly explored range of luminosity. Key words. galaxies:active – quasars: emission lines – quasars: general – quasar:individual:PG 1634+706, PG 1247+268 1. Introduction Supermassive black holes (SMBHs) are believed to inhabit most, if not all, the bulges of present-epoch galaxies (Kormendy & Richstone, 1995), and strong evidences exist of a correlation be- tween the black hole mass and either the mass Mbulge and lu- minosity (Marconi et al., 2003, and refs. therein) or the veloc- ity dispersion σ∗ of the host bulge (Ferrarese & Merrit, 2000; Tremaine et al., 2002). This strongly suggests that the formation and growth of SMBHs and galaxies are physically related pro- cesses and provides a basis for a theory of cosmic structure for- mation, incorporating the feedback from Active Galactic Nuclei (AGNs) (Silk & Rees, 1998; Vittorini, Shankar, & Cavaliere, 2005, and refs.therein). Black hole masses determinations based on stellar or gas kinematics are intrinsically limited, by angu- lar resolution, to relatively nearby objects and cannot be applied to bright AGNs where the nuclear light prevails over the galac- tic component, just in the central region where the galactic gas or star motion is dominated by the black hole gravitational field. The reverberation-mapping technique does not suffer of this lim- itation and represents the only mean to measure the mass of SMBH in bright AGNs. Emission lines, in the optical-UV region, are interpreted as recombination of a gas which is photoionized by the continuum radiation emitted by the inner region of the nucleus, presumably an accretion disk surrounding the black hole. Emission lines re- spond to variation of the ionizing continuum. Although the phys- ical origin of these variations is poorly known (e.g. Trevese & Vagnetti, 2002; Vanden Berk et al., 2004; De Vries et al., 2005) it is possible to use the response of lines to continuum varia- tions to investigate the structure of the line emitting region. This requires long campaigns of accurate spectrophotomeric monitor- ing of AGNs, which have led in the past to major progresses to- wards understanding the physics of the “atmosphere” of Seyfert 1 galaxies. A summary of these results is given in Peterson (1993). Line widths, e.g. the FWHM ∆Hβ of the Hβ emission line, correspond to r.m.s. velocities of the emitting gas clouds. A cross-correlation analysis of continuum and emission-line light- curves, evidencing a time delay τ of line respect to continuum variations, allows to estimate the size R = τ/c of the region where the line photons are generated. If the gas motion in the Broad Line Region (BLR) is dominated by gravitation (Peterson & Wandel, 2000), the size estimate RBLR can be combined with the line width to yield a primary estimate of the virial mass of the black hole MBH ∝ ∆2Hβ/GRBLR and the relevant Eddington ratio. For Seyfert 1 galaxies typical BLR sizes are of the order of light-days to light-weeks. Similar studies are more difficult for quasars (QSOs) which require a longer monitoring. 2 D. Trevese et al.: Line and continuum variability A long term campaign for a subsample of 28 QSOs was started in 1991 with the Wise 1.0m and the Steward 2.3m tele- scopes (Maoz et al., 1994). As a result, nine years later Kaspi et al. (2000) provided mass estimates for the entire sample. The new data, combined with previous results on Seyfert 1 galax- ies, thus spanning a much wider range of intrinsic luminosity, allowed to establish an average relation between the intrinsic lu- minosity and the size, RBLR = aLγ, with γ ' 0.7, which allows a secondary estimate of the black hole mass based on single-epoch observations of luminosity and line width: MBH = bLγ∆2Hβ , where both the constant b and γ are determined statistically from the available echo-mapping data. Recent studies show that γ is in the range 0.5-0.7, depending on how luminosity is defined, which lines are selected for the echo-mapping and the fitting pro- cedure adopted (Kaspi et al., 2005; Bentz et al., 2006). The extreme importance of secondary mass estimates relies on the fact that on the basis of single epoch observations it is possible to study the evolution in cosmic time of the mass distri- bution of QSOs/AGNs, and to extend the studies of the relation existing between QSOs and host bulges properties. However, the above correlations with primary masses, based on echo- mapping, were established for relatively close and faint AGNs with z ≤ 0.4 and [λLλ(5100Å) <∼ 1046 erg s−1], thus it is presently unknown whether they can be extrapolated to higher luminosities and/or redshifts. For instance, the extrapolation of the MBH − L relation (Kaspi et al., 2000), together with the assumption that the known MBH − Mbulge − σ∗ relations holds (Tremaine et al., 2002) , leads to predict the existence of galax- ies with Mbulge ∼ 1013.1 − 1013.4M� and σ∗ exceeding 800 km s−1. Such galaxies have never been observed, and their exis- tence would put important constraints on galaxy formation mod- els (Netzer, 2003). Therefore it is essential to extend the primary mass measures to higher redshifts and luminosities. On the other hand, for high QSO luminosities a large size of the broad line region is expected. This would cause both a smoothing of the line light-curve and larger time delays with respect to contin- uum variations (Wilhite et al., 2005), thus the very detectability and the amplitude of line variations are open questions. A sample of objects with redshifts in the range 2 < z < 3.4 and apparent magnitude as faint as mV ∼18 is being monitored by Kaspi et al. (2004) with the 9m Hobby-Eberly Telescope (HET; Ramsey et al., 1998) and new results have been presented recently (Kaspi et al., 2006). During their 6-year monitoring of 6 QSOs, significant continuum and emission-line variations were detected in all targets and a preliminary black hole mass estima- tion is given for one of them. In the present paper we describe a new monitoring campaign limited to objects with V < 15.7 and 1 < z < 4 which, thanks to their apparent brightness, can be observed with the medium- small 1.82 m Copernicus Telescope at Cima Ekar (Asiago, Italy), through a service mode scheduling of a long term monitoring, and allow to investigate whether: i) echo-mapping is feasible for objects as bright as λLλ(5100Å) ∼ 1047 erg s−1 and ii) the RBLR-luminosity correlation can be extrapolated to such bright- ness. The paper is organized as follows. Section 2 describes the sample, observations and the data reduction procedure. Section 3 describes the results for two quasars of the sample. Section 4 summarizes the results and discuss future prospects. In the fol- lowing we derive λLλ(5100Å) from the flux in the Johnson V band, extrapolating the flux density to the rest-frame λ = 5100Å with a power law fν ∼ ν−0.5, and assuming a standard cosmology Ho = 70 km s−1 Mpc−1, ΩM = 0.3, and ΩΛ = 0.7. Table 1. The quasars monitored Object z V log[λLλ(5100Å)] [erg s−1] APM 08279+5255 3.911 15.20 47.7 PG 1247+268 2.042 15.60 47.0 PG 1634+706 1.337 15.27 46.7 HS 2154+2228 1.290 15.30 46.7 4000 5000 6000 7000 Fig. 1. Average spectra of PG 1634+706 (upper panel) and PG 1247+268 (lower panel). Spectral ranges for continuum deter- mination (short ticks) and line flux evaluation (long ticks), as reported in Table 2, are shown. 2. Observations 2.1. Object selection The sample has been extracted from the Veron-Cetty & Veron (2003) (11th ed.) catalog with the condition δ > 0, V < 15.7 mag and z > 1 in order to select objects of bright enough intrinsic luminosity to investigate the bright end extension of the RBLR vs. λLλ(5100Å) relation (Kaspi et al., 2000, 2005). These conditions identify 12 objects, only four of which were monitored, owing to the limits on observing time. These four objects are listed in Table 1. Observations at intermediate redshift allow to sample the variability of MgII λ2798, CIII] λ1909, CIV λ1559 lines, instead of Hα, Hβ, Hγ observed in the low redshift study of Kaspi et al. (2000). This allows to study either BLR at smaller distance from the center, or regions of the same sizes of those producing the Balmer lines, but using lines which respond to different part of the continuum spectrum. The main emission lines falling within the observed wavelength interval are indicated for two objects in Table 2. Figure 1 shows the average spectra of PG 1634+706 and PG 1247+268. With respect to other QSO monitoring programmes, ours is the first which includes the MgII line in part of the ob- served sources. This line is particularly significant, because i) past Seyfert 1 monitoring campaigns conducted with IUE have D. Trevese et al.: Line and continuum variability 3 shown its lag to be similar to that of Hβ (Clavel et al., 1991; Reichert et al., 1994); ii) this line is most often used to derive es- timates of black-hole mass from single-epoch spectra of high-z QSOs (McLure & Jarvis, 2002), because its width is tightly cor- related to that of Hβ. Deriving a lag for the MgII line would therefore allow to estimate the black-hole mass most consis- tently with respect to the results of Seyferts, which are mostly based on the monitoring of Hβ. APM 08279+5255 is one of the most luminous known QSOs if its emission is considered isotropic. However it has been shown to be lensed by a foreground galaxy (Irwin et al., 1998). Three components, separated by a few tenths of an arcsec have been detected in near-infrared images obtained with Keck tele- scope and different models of the lensing field predict a few days delay between photometric variations of the components (Ledoux et al., 1998; Ibata et al., 1999; Egami et al., 2000): a short time compared with the expected time scale of intrin- sic variations. APM 08279+5255 is also a Broad Absorption Line QSO (Irwin et al., 1998), which makes more difficult to define regions free from either emission or absorption features to measure continuum variations. The analysis of this object is deferred to a forthcoming paper. HS 2154+2228 has been ob- served so far only 5 times and the analysis requires further mon- itoring. The other two objects PG 1247+268 and PG1634+706 are analyzed in this paper with the aim of : i) verifying the ad- equacy of the observational data and reduction procedures, un- der the assumption that variability amplitudes and characteristic time scales can be extrapolated from the properties of fainter objects; ii) possibly detecting line variations in objects as lumi- nous as λLλ(5100Å) ∼ 1047 erg s−1, and compare their amplitude with continuum variations. For both objects a star of comparable magnitude, as close as possible to the QSO, has been selected for the relative spectrophotometric calibration, described in the next section. 2.2. Spectrophotometric Observations and Data Reduction Observations were carried out at Asiago 1.82 m telescope equipped with the Faint Object Spectrograph & Camera AFOSC which is a focal reducer with reduction factor of 0.58, designed to allow a quick switching between spectroscopic and imaging modes. The scale at the focal plane is 21.7 ′′/mm. The detector is a 1024x1024 thinned CCD array TEK1024 with 22x22 µm2 pixels corresponding to a scale of 0.473 arcsec pixel−1 and a FOV of 8.14x8.14 arcmin2. We adopted a 8”.44-wide slit and a grism with a dispersion of 4.99 Å pixel−1, providing a typ- ical resolution of ∼ 15 Å in the spectral range 3500-8450 Å. Spectrophotometric exposures are performed after orienting the slit to include both the QSO and the reference star of comparable magnitude, located at (12:50:11.5 +26:33:32) and (16:34:57.4 +70:32:49) (J2000) for PG 1247+268 and PG 1647+706 respec- tively. The reference stars are included as internal calibrators for QSO spectra, as described by Maoz et al. (1990) and Netzer et al. (1990). The wide slit is necessary to avoid different fractional losses of the QSO and star light due to possible non perfect slit alignment and differential refraction, which could cause spuri- ous variation of the flux ratios. Lamp flats and Hg-Cd ark spectra were also taken for wavelength calibration. At each epoch, typi- cal science observations consist of two consecutive exposures of 1800 s. QSO and star spectra,Q(λ) and S (λ) were extracted with the standard IRAF procedures. The QSO/star ratio as a function of wavelength is computed for each exposure k = 1, 2 µ(k)(λ) = Q(k)(λ)/S (k)(λ). (1) This quantity is independent of extinction changes during the night. This allows a check of consistency between the two expo- sures and the rejection of the data if inconsistencies occur (under the assumption that QSO variations are negligible on 1 h time scale). In fact, typical spectra of two consecutive exposures have a ratio µ(1)(λ)/µ(2)(λ) of order unity, with deviations smaller than 0.02 when averaged over 500 Å, at least in the 4000Å − 7000Å range. When discrepancies are larger than 0.04 both exposures are rejected. It is important to notice that, because of the large slit width, there are small changes in the λ scale due to changes of the ob- ject position within the slit (which are in general negligible in the case of pairs of consecutive exposures). This suggested to proceed as follows. As a first step we register the zero point of the λ scale. However, since the changes of position within the slit do not correspond exactly to rigid shifts of the λ scale, we select portions of the spectra in wavelength intervals |∆λ| <∼ 1000Å around each of the QSO emission lines considered (see Table 2), and we determine the shifts for each portion. Once the wavelength scales are registered, the QSO and star spectra taken in the two exposures are co-added and the ratio µi(λ) = (Q i + Q i )/(S i + S i ) (2) is computed, at each epoch i. The reference star is flux calibrated at a reference epoch. Since our aim is to compute relative flux variations, we reduce all quasar spectra to this reference epoch, by multiplying all the µi(λ) by the flux calibrated star spectrum f S (λ): f Qi (λ) ≡ µi(λ) f S (λ). We stress that we are not interested in the absolute flux calibration, whose accuracy is of the order of 20% and which is applied for the sole purpose reporting the QSO spectra in physical units. The star spectrum f S (λ) adopted is the same for all epochs, thus it does not affect relative flux changes we want to measure. The QSO spectrum f Q(λ) = c(λ) + l(λ) can be decom- posed in a line l(λ) and continuum c(λ) spectra. Two values of the continuum, cshort and clong at shorter and longer wave- length with respect to the most prominent QSO emission lines, are evaluated in regions as free as possible of other emission features, defined by the wavelengths ranges (λshort,1,λshort,2) and (λlong,1,λlong,2) respectively. Table 2 reports the observer-frame wavelengths defining the continuum regions and the intervals for line integration. These were chosen on the basis of the analysis of Clavel et al. (1991); Reichert et al. (1994) of UV spectra of low redshift AGNs observed with the International Ultraviolet Explorer (IUE), with slight modifications to maximize the S/N ratio (clong of MgII(λ2798 Å) falls outside the wavelength range covered by the IUE spectrograph). Line fluxes are computed as: [F(Q)(λ) − cint(λ)]dλ, (3) where cint(λ) is the linear interpolation through cshort and clong. The extremes λ1 and λ2, also listed in Table 2, not necessarily coincide with λshort ≡ (λshort,1 + λshort,2)/2 and λlong ≡ (λlong,1 + λlong,2)/2, and are chosen to optimize the fl signal to noise ratio. Direct B and R images in 15x15 arcmin fields centered on the QSOs were taken to check possible variability of the reference 4 D. Trevese et al.: Line and continuum variability Table 2. Wavelength intervals for lines and continua [Å] Object z line λshort,1 λshort,2 λ1 λ2 λlong,1 λlong,2 PG1634+706 1.337 CIII]λ1909 4252 4298 4382 4590 4630 4674 PG1634+706 1.337 MgIIλ2798 6215 6285 6251 6777 6755 6815 PG1247+268 2.042 CIVλ1549 4380 4502 4598 4801 5202 5262 PG1247+268 2.042 CIII]λ1909 5535 5595 5770 5900 6025 6085 CIII] 2800 3200 3600 4000 t (JD-2450000) 2800 3200 3600 4000 t (JD-2450000) Fig. 2. Light-curves of PG 1634+706 in the observer’s frame, as relative flux variations ∆ f / f . Upper panels: emission lines, middle panel: shorter wavelength continuum cshort, lower panel: longer wavelength continuum clong, left: CIII]/λ1909 Å, right: MgII λ2798 Å. star. Comparison of the reference star with the brightest star in the field showed a r.m.s. fractional flux variation of 0.02 in both cases. Thus we can assume that the reference star is not variable, at this level of accuracy. Relative-flux calibration 1-σ errors, reported in figures 2 and 3, are estimated by adding in quadrature the r.m.s. uncertainty on the flux of the reference star to the r.m.s. fractional uncertain- ties of line or continuum fluxes, which, in turn, are computed by comparing pairs of exposures which are added to form quasar spectra of individual epochs. 3. Results and discussion Figures 2 and 3 show the light-curves in the observer’s frame, for emission lines and continua listed in Table 2, for PG1634+706 and PG1247+268 respectively. The light curves are expressed in terms of the relative flux variations ∆ f / f , with respect to the flux f at the reference epoch where the absolute calibration was performed (see Section 2.2). The total time base is 3.3 years in the observer’s frame, corresponding in the rest-frame to 1.4 yr for the former QSOs and 1.1 yr for the latter. In the case of PG1634+706 no significant variations are detected: neither in CIII] line, which has the lowest S/N ratio due to its intrinsic faintness, nor in the MgII line. The relevant continua are also 2800 3200 3600 4000 t (JD-2450000) CIII] 2800 3200 3600 4000 t (JD-2450000) Fig. 3. Light-curves of PG 1247+268 in the observer’s frame, as relative flux variations ∆ f / f . Upper panels: emission lines, middle panel: shorter wavelength continuum cshort, lower panel: longer wavelength continuum clong, left: CIV/λ1549 Å, right: CIII] λ1909 Å. consistent with no variability at a level of ∆ f / f ∼ 0.02-0.03 r.m.s.. The result is different in the case of PG 1247+268. The faintest line CIII] λ1909 Å shows a marginal evidence of vari- ability and the stronger line CIV λ1549 Å seems to decrease steadily during the observing period. An almost steady decrease does not allow to derive possi- ble line-continuum time delays from cross-correlation analysis. However we can establish quantitatively the evidence of contin- uum and, most important, of line variability. For this purpose we define the unbinned discrete structure function UDS F(τi j) = |y(ti) − y(t j)|, (4) where y(t) represents any of the line or continuum light-curves considered, ti and t j are two observation epochs and τi j = ti − t j is the time delay. The (binned) structure function can be defined in bins of time delay, centered at τ: S (τ) = ∑ UDS F(τi j)  , (5) where the sum is extended to all the M values of UDSF belong- ing to a given bin of τ. We adopt the average of UDSF, instead D. Trevese et al.: Line and continuum variability 5 0 200 400 600 800 1000 0 200 400 600 800 1000 Fig. 4. Structure functions for PG 1634+706 in the observer’s frame. Upper panels: emission lines, middle panel: shorter wave- length continuum cshort, lower panel: longer wavelength contin- uum clong, left: CIII]/λ1909 Å, right: MgII λ2798 Å. of UDSF2, since it is more stable, i.e. less sensitive to deviant points; the factor 2 in equation (4) makes S (τ) equal to the standard deviation in the case of a Gaussian distribution (see Di Clemente et al., 1996). Structure functions for lines and continua 0 200 400 600 800 1000 0 200 400 600 800 1000 Fig. 5. Structure functions for PG 1247+268.Upper panels: emission lines, middle panel: shorter wavelength continuum cshort, lower panel: longer wavelength continuum clong, left: CIV/λ1549 Å, right: CIII] λ1909 Å. of the two quasars PG 1634+706 and PG 1247+268 are reported in Figures 4 and 5 respectively, as a function of the time delay in the observer’s frame. The error bars reported in the figures represent simply the standard deviation of the UDSF in each bin and not the uncertainty on their average value. The latter would be 1/(M − 1) times the standard deviation, for M uncorrelated UDSF values. In the case of PG 1634+706 the structure function analy- sis simply confirms what already appears from the light-curves, namely no significant variations of lines and continua are de- tected. The r.m.s. noise level is about 0.02-0.03 for continua and 0.05 for lines. On the contrary, in the case of PG 1247+268, there is a clear increase of variability for long time lags. To establish the signifi- cance of this variability it is necessary to evaluate the probability of the null hypothesis that a value of S (τ) is produced by pure noise. For this purpose we generated mock noise light-curves , n(ti), i=1,Nepo, where Nepo is the number of observing epochs, by extracting random numbers with a Gaussian distribution of standard deviation σ, which represents pure noise, under the as- sumption that the photometric noise is not correlated on the time scale of our minimum sampling interval (namely about 10 days). Then, the relevant structure function S mock(τ) has been computed with the same binning adopted for real data. The simulation was iterated for 107 times to generate, for each bin, the statistical dis- tribution of the S mock(τ) values. The value adopted for σ has been estimated, conservatively, from the observed structure functions of figure 4 and 5 them- selves, as the value of S (τ) in the first bin. This is, in fact, an overestimate of noise, since it includes short-time-scale varia- tions of the QSO, which however should be negligible on the basis of previous studies of (fainter) QSOs (Giveon et al., 1999; Kaspi et al., 2000). The almost constant value of the structure functions in the case of PG 1634+706 confirms the adopted hy- pothesis that noise is not correlated on long time scale. For the CIII] line the r.m.s. noise (∼ 0.05) is larger than in the case of the CIV line (∼ 0.02). This depends on the intrinsic faintness of the CIII] line. According to the above simulations the probability of the null hypothesis, for the second and fourth bins, is P(> S ) ∼ 4 × 10−4. The local maximum of S (τ), for CIII] at τ ∼400 days, is due to an “oscillation” of the light-curve on a time scale comparable with the total time base of the ob- servations. This indicates that a longer time base is needed to properly average individual oscillations and to obtain a “station- ary” structure function. Further sampling, trying to avoid yearly periodicity due to periods of optimal observability, is needed to obtain the resolution required for time delay measurements. The structure function of CIV line shows a much more sig- nificant variability: the probability of the null hypothesis in the case of the last two bins, where S (τ) is 0.13 and 0.21 respec- tively, is less than 10−6. The systematic decrease of CIV line luminosity and the relevant continua might seem to rely on the value of the light-curves at the first epoch. Thus we have recom- puted the S (τ) after removing the first epoch. Indeed, in the case of continua the systematic decrease of S (τ) is no longer evident. On the contrary, for the line it remains practically unchanged (except that the last bin is no longer present). Thus the fact that the probability of the null hypothesis is smaller than 10−6 does not rely on that particular point in the light-curve. The main goal of new echo-mapping campaigns is to obtain a direct measure of the central black hole mass and to establish whether the scaling of QSO broad line region size and masses re- main the same at higher quasar luminosity. The very detectabil- ity of line variation in the most luminous QSOs was, until re- 6 D. Trevese et al.: Line and continuum variability cently, an open question, since 3C273 with (λLλ(5100Å) <∼ 1046 erg s−1) was the brightest QSO with detected line variability (Kaspi et al., 2005). An evidence of line variability in brighter objects has been reported more recently by Kaspi et al. (2006). The present detection of line variability in one of the two quasars considered strengthens the notion that quasar emission lines re- spond to continuum changes in high luminosity QSOs like in lower luminosity ones and indicates that echo-mapping cam- paigns are feasible for λLλ(5100Å) >∼ 1047 erg s−1. Our results are consistent with those of Kaspi et al. (2006), who have monitored sources of comparable luminosities and redshifts. We have also detected variability, of comparable am- plitude, in the continuum and CIII] and CIV emission lines in one of our sources. Considering the shorter time baseline of our observations, the fact that variability has not been found in PG 1634+706 is consistent with the segments without variations which can be seen in the Kaspi et al. (2006) light curves on this time scale. To estimate the further monitoring needed to measure line- continuum delays, we should take into account that, on aver- age, the amplitude of variability increases with the rest-frame time lag. However this dependency is weak: log S (τrest) ∼ 0.15 log τrest, according to previous statistical studies on large quasar samples (Vanden Berk et al., 2004; De Vries et al., 2005). In the case of our quasars PG 1634+706 and PG 1247+268, the observer’s frame τobs = τrest(1+z) is dilated by a factor 2.3 and 3 respectively. The size of the broad line region, based on Balmer line reverberation-mapping, is expected to scale with luminosity as RBLR ∝ Lγ with γ=0.5-0.7, depending on how L is measured and the RBLR vs. L fit is obtained (Kaspi et al., 2000, 2005; Bentz et al., 2006; Kaspi et al., 2006). A Balmer line BLR of about 3 light years in the rest frame is expected for the relevant luminosi- ties. On the other the hand CIV line corresponds to a BLR size 2 times smaller than Balmer lines (Peterson & Wandel, 2000). Considering that in general a time baseline of about twice the light-crossing time is required in order to obtain a reliable lag from the light curves, we estimate that reverberation mapping should be feasible after 5-6 more years of monitoring. 4. Summary – We have initiated a campaign for the monitoring of 4 high luminosity QSOs and we present the result for two of them on the basis of 3.3 years of observation with the 1.82 m tele- scope of the Asiago Observatory. – We discuss the data reduction, the procedures adopted and the level of accuracy attained in relative spectrophotometric variability measurements. – We perform a structure function analysis of the light-curve and numerical simulations to establish the confidence level of line variability detection. – We detect line and continuum variability in one of the two QSOs, PG 1247+268 of λLλ(5100Å) = 1047 erg s−1, with a probability of less than 10−6 of the null hypothesis that the observed structure function is produced by pure noise. – This detection supports the notion that emission lines re- spond to continuum variations as in substantially less lumi- nous QSOs. The results encourage the prosecution of the campaign which should provide time delay, and black hole mass, estimates in 5-6 years. Acknowledgements. We acknowledge support from the Asiago Observatory team, in particular Hripsime Navasardian, for observations. References Clavel,J., Reichert, G. A., Alloin,D. et al., 1991, ApJ, 366, 64 Bentz, M. C., Peterson, B. M., Pogge, R. W., Vestergaard, M., & Onken, C. A., 2006, ApJ, 644,133 De Vries, W.H., Becker, R. H., White, R.L., & Loomis, C., 2005, ApJ, 129, 615 di Clemente, A., Giallongo, E., Natali, G., Trevese, D., & Vagnetti, F., 1996, ApJ, 463, 466 Egami,E., Matthews, K., Ressler, M., et al., 2000, ApJ, 535, 561 Ferrarese, L., & Merrit, D. 2000, ApJ, 539, L9 Giveon, Uriel, Maoz, Dan, Kaspi, Shai, Netzer, Hagai, Smith, Paul S., 1999, MNRAS, 306, 637 Ibata, R. A., Lewis, M.J., Irwin, M. J., Lehar, J., & Totten, E. J., 1999, ApJ, 118,1922 Irwin, M. J., Ibata, R. A., Lewis, G. F., & Totten, E. J., 1998, ApJ, 505, 529 Kaspi, S., Smith, P.S., Meters, H., Maoz, D., Jannuzi, B.T. &Giveon, U., 2000, ApJ, 533, 631 Kaspi, S., Netzer, H., Maoz, D., Shemmer, O., Brandt, W. N.,& Schneider, D. P., 2004, in Coevolution of Black Holes and Galaxies, Carnegie Obs. Astroph. Ser. Vol. 1, L. C. Ho ed., http://www.ociw.edu/ociw/symposia/series/ Kaspi, S., Maoz, D., Netzer, H., Peterson, B. M., Vestergaard, M., & Jannuzi, B. T., 2005, ApJ, 629, 61 Kaspi, S., Bradt, W. N., Maoz, D., Netzer, H., Schneider, D. P., & Shemmer, O., 2006, ApJ (in press), astro-ph/0612722 Kormendy, J., & Richstone, D. 1995, ARA&A,33, 581 Ledoux, C., Theodore, B., Petitjean, P., et al., 1998, A&A,339,L77 Maoz, D., Netzer, H., Leibowitz, E., et al. 1990, ApJ, 351, 75 Maoz, D., Smith, P. S., Jannuzi, B. T., Kaspi, S., & Netzer, H., 1994, ApJ, 421, Marconi, A., Hunt, L. K., 2003, ApJ, 589L, 21 McLure, R. J., & Jarvis, M. J., 2002, MNRAS, 337, 109 Netzer, H., Maoz, D., Laor, A. et al., 1990, ApJ, 353, 108 Netzer, H., 2003, ApJ, 583, L5 Peterson, B. M., 1993, PASP, 105, 247 Peterson, B. M., & Wandel, A., 2000, ApJ, 540, L13 Silk, J. & Rees, M.J. 1998, A&A, 331, L1 Ramsey, L. W., et al. 1998, SPIE, 3352, 34 Reichert, G. A., Rodriguez-Pascual, P. M., Alloin, D. et al., 1994, ApJ, 425, 582 Tremaine, S. et al., 2002, ApJ, 574, 740 Trevese, D., & Vagnetti, F., 2002, ApJ, 564, 624 Vanden Berk, D, 2004, ApJ, 601, 692 Veron-Cetty M.P., Veron P., 2003, A&A, 412, 399 Vittorini, V., Shankar, F., & Cavaliere, A., 2005, MNRAS, 363, 1376 Wilhite, B. C.,Vanden Berk, D. E., Kron, R. G. et al., 2005, ApJ, 633, 638 http://www.ociw.edu/ociw/symposia/series/ http://arxiv.org/abs/astro-ph/0612722 Introduction Observations Object selection Spectrophotometric Observations and Data Reduction Results and discussion Summary

0704.1959

Adiabatic passage in a three-state system with non-Markovian relaxation: The role of excited-state absorption and two-exciton processes

Adiabatic passage in a three-state system with non-Markovian relaxation: The role of excited-state absorption and two-exciton processes. B. D. Fainberg1,2∗and V. A.Gorbunov1 1Faculty of Sciences, Physics Department, Holon Institute of Technology, 52 Golomb St., Holon 58102, Israel 2Raymond and Beverly Sackler Faculty of Exact Sciences, School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel August 8, 2021 Abstract The influence of excited-state absorption (ESA) and two-exciton processes on a co- herent population transfer with intense ultrashort chirped pulses in molecular systems in solution has been studied. An unified treatment of adiabatic rapid passage (ARP) in such systems has been developed using a three-state electronic system with relax- ation treated as a diffusion on electronic potential energy surfaces. We have shown that ESA has a profound effect on coherent population transfer in large molecules that ∗Corresponding author. E-mail: [email protected] http://arxiv.org/abs/0704.1959v1 necessitates a more accurate interpretation of experimental data. A simple and phys- ically clear model for ARP in molecules with three electronic states in solution has been developed by extending the Landau-Zener calculations putting in a third level to random crossing of levels. A method for quantum control of two-exciton states in molecular complexes has been proposed. 1 Introduction. The possibility of the optical control of molecular dynamics using properly tailored pulses has been the subject of intensive studies in the last few years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Chirped pulses can selectively excite coherent wave packet motion either on the ground electronic potential energy surface of a molecule or on the excited electronic potential energy surface due to the intrapulse pump-dump process [1, 5, 11, 12]. In addition, they are very efficient for achieving optical population transfer between molecular electronic states. Total electronic population inversion can be achieved using coherent light-matter interactions like adiabatic rapid passage (ARP) in a two- or three- state system [24, 25], which is based on sweeping the pulse frequency through a resonance. Since the overwhelming majority of chemical reactions are carried out in liquid solution, adiabatic passage in molecules in solution was studied for two-state electronic system (ARP) in Refs.[26, 27, 28], and for stimulated Raman adiabatic passage (STIRAP) configuration in Refs.[29, 30]. It has been shown in Ref.[26] that relaxation does not hinder a coherent population transfer for positive chirped pulses and moderate detuning of the central pulse frequency with respect to the frequency of Franck-Condon transition. However, a two electronic state model for molecular systems is of limited utility. Indeed, excited-state absorption (ESA) occurs for majority of complex organic molecules [31, 23]. Even a molecular dimer consisting from two-level chromophores has an additional excited state corresponding to two-exciton excitation. An unified treatment of ARP in such sys- tems can be developed using three-state electronic system interacting with reservoir (the vibrational subsystems of a molecule (chromophores) and a solvent). More often than not ESA in complex organic molecules corresponds to a transition from the first excited singlet state S1 to a higher singlet state Sn (n > 1), which relaxes back to S1 very fast [32, 33, 34, 31]. Therefore, it would look as if ESA does not influence on population transfer S0 → S1 from the ground state S0. However, in the presence of ESA an exciting pulse interacts with both S0 → S1 and S1 → Sn transitions. It is well known that coherent optical interactions occurring in adjacent optical transitions in a three-state system markedly affect each other. The examples are STIRAP, lasing without inversion, coherent trapping, electromagnetically induced transparency and others. (For textbook treatments of these effects see, for example, [35]). Therefore, one would expect an appreciable change of a population transfer S0 → S1 with chirped pulses in the presence of excited state absorp- tion in the coherent regime when the chirp rate in the frequency domain is not large and, consequently, the pulse is rather short. Our objective is to answer the following questions: “How do ESA and two-exciton pro- cesses influence on a coherent population transfer in molecular systems in solution? What is the potential of chirped pulses for selective excitation of the single and two-exciton states and their selective spectroscopy?” In addition, the three-state system under discussion enables us to consider STIRAP as well. Therefore, we shall also briefly concern slowing down the pure dephasing on STIRAP in intense fields when relaxation is non-Markovian. The outline of the paper is as follows. In Sec.2 we present equations for the density matrix of a three-state molecular system under the action of shaped pulses when the interaction with a dissipative environment can be described as the Gaussian-Markovian modulation (so called the total model). In Sec.3 we formulate a number of approaches to this model that enables us, first, to clarify the underlying physics and, second, to understand the validity of the results obtained by the total model. The ESA effects on ARP in complex molecules are considered in Sec.4. In Sec.5 we study population transfer in molecular dimers with taking into account two-exciton processes. In Sec.6 we consider slowing down the pure dephasing on STIRAP in strong fields when the system-bath interaction is not weak (non-Markovian relaxation). We summarize our results in Sec.7. In the Appendix we extend calculations of two-photon excitation of a quantum ladder system by a chirped pulse [36] to non-zero two-photon detuning. 2 Basic equations Let us consider a molecular system with three electronic states n = 1, 2 and 3 in a solvent described by the Hamiltonian |n〉 [En +Wn(Q)] 〈n| (1) where E3 > E2 > E1, En is the energy of state n,Wn(Q) is the adiabatic Hamiltonian of reservoir R (the vibrational subsystems of a molecular system and a solvent interacting with the three-level electron system under consideration in state n). The molecular system is affected by two shaped pulses of carrier frequencies ω1 and ω2 E(t) = i=1,2 Ei(t) + c.c. = i=1,2 ~Ei (t) exp[−iωit+ iϕi (t)] + c.c. (2) which are resonant to optical transitions 1 → 2 and 2 → 3, respectively (ladder configura- tion). Here Ei (t) and ϕi (t) describe the change of the pulse amplitude and phase, respec- tively, in a time t. The instantaneous pulse frequencies are ωi (t) = ωi − dϕidt . The influence of the vibrational subsystems of a solute and a solvent on the electronic transition can be described as a modulation of this transition by low frequency (LF) vibra- tions {ωs} [37, 38]. In accordance with the Franck-Condon principle, an electronic tran- sition takes place at a fixed nuclear configuration. Therefore, for example, the quantity u(Q) = W2(Q) −W1(Q) − 〈W2(Q) −W1(Q)〉1 is the disturbance of nuclear motion under electronic transition 1 → 2. Here 〈〉n ≡ TrR (...ρRn) denotes the trace operation over the reservoir variables in the electronic state n, ρRn = exp (−βWn) /TrR exp (−βWn) , β = 1/kBT . The relaxation of electronic transition 1 → 2 stimulated by LF vibrations is described by the correlation function K(t) = 〈u(0)u(t)〉 of the corresponding vibrational disturbance with characteristic attenuation time τs [12, 38]. We suppose that ~ωs ≪ kBT . Thus {ωs} is an al- most classical system and operators Wn are assumed to be stochastic functions of time in the Heisenberg representation. The quantity u can be considered as a stochastic Gaussian vari- able. We consider the Gaussian-Markovian process when K(t)/K(0) ≡ S(t) = exp(−|t|/τs). The corresponding Fokker-Planck operator Lj = τ + (q − dj) ∂∂q + 1 describes the diffusion in the effective parabolic potential Uj (q) = Ej + ω̃2 (q − dj)2 (3) of electronic state j where τ−1s = D̃nβω̃ 2 and D̃ is the diffusion coefficient. Going to a di- mensionless generalized coordinate x = qω̃ β, one can obtain the equations for the elements of the density matrix ρij(x, t) by the generalization of the equations of Ref.[26]. Switching to the system that rotates with instantaneous frequency ρ̃12(x, t) = ρ12(x, t) exp[−i(ω1t− ϕ1(t))], ρ̃23(x, t) = ρ23(x, t) exp[−i(ω2t− ϕ2(t))], ρ̃13(x, t) = ρ13(x, t) exp{−i[(ω1 + ω2)t− (ϕ1(t) + ϕ2(t))], (4) we get ρ11(x, t) = Im[Ω1ρ̃12(x, t)] + L1ρ11(x, t) ρ22(x, t) = − Im[Ω1ρ̃12(x, t) + Ω∗2ρ̃32(x, t)] + L2ρ22(x, t) + 2Γ32ρ33(x, t) ρ33(x, t) = − Im[Ω2ρ̃23(x, t)] + (L3 − 2Γ32)ρ33(x, t) (5) ρ̃12(x, t) = i ω21 − ω1(t)− (~β)−1x2x ρ̃12(x, t) + Ω∗1[ρ22(x, t)− ρ11(x, t)]− Ω2ρ̃13(x, t) + L12ρ̃12(x, t) (6) ρ̃13(x, t) = i ω31 − ω1(t)− ω2(t)− (~β)−1x3x ρ̃13(x, t) + Ω∗1ρ̃23(x, t)− Ω∗2ρ̃12(x, t) + (L13 − Γ32)ρ̃13(x, t) (7) ρ̃23(x, t) = i (ω31 − ω21)− ω2(t)− (~β)−1 (x3 − x2)x ρ̃23(x, t) + Ω∗2 (t) [ρ33(x, t)− ρ22(x, t)]+ Ω1ρ̃13(x, t) + (L23 − Γ32)ρ̃23(x, t) (8) where Ω1 = D21E1/~ and Ω2 = D32E2/~ are the Rabi frequencies for transitions 1 → 2 and 2 → 3, respectively. Here ωi1 = ωeli1 + x2i /(2~β) is the frequency of Franck-Condon transition 1 → i, ωelij = (Ei − Ej)/~ is the frequency of purely electronic transition j → i, Dij are matrix elements of the dipole moment operator, 2Γ32 is a probability of nonradiative transition 3 → 2 for the excited state absorption problem (see below); |xj | = (~βω1jst )1/2 is a dimensionless shift between the potential surfaces of states 1 and j (x1 = 0), which is related to the corresponding Stokes shift ω st of the equilibrium absorption and luminescence spectra for transition 1 → j. The last magnitude can be written as ω1jst = ~βσ 2s where σ 2s denotes the LF vibration contribution to a second central moment of an absorption spectrum for transition 1 → j. The terms Lj = τ + (x− xj) on the right-hand side of Eqs.(5) describe the diffusion in the corresponding effective parabolic potential Uj(x) = Ej + (x− xj)2 (j = 1, 2, 3), (10) Lij = (Li + Lj)/2. The partial density matrix of the system ρ̃ij (x, t) describes the system distribution with a given value of x at time t. The complete density matrix averaged over the stochastic process which modulates the system energy levels, is obtained by integration of ρ̃ij (x, t) over the generalized coordinate x: 〈ρ̃〉ij (t) = ρ̃ij (x, t) dx (11) where diagonal quantities 〈ρ〉jj (t) are nothing more nor less than the populations of the electronic states: 〈ρ〉jj (t) ≡ nj , n1 + n2 + n3 = 1. We solve coupled Eqs.(5)-(8), using a basis set expansion with eigenfunctions of diffusion operator L13, similar to Ref. [26]. The solutions, corresponding to the procedure described in this section, are termed the total model for short, bearing in mind that they take into account all the relaxations (diffu- sions) related to populations and electronic coherences between all the electronic states. 3 Approximate models In this section we describe a number of approaches to the total model (Eqs.(5)-(8)). 3.1 System with frozen nuclear motion For pulses much shorter than τs one can ignore all the terms ∼ Li, Lij on the right-hand sides of Eqs.(5)-(8). It means that our system can be described as an ensemble of inde- pendent three-level systems with different transition frequencies corresponding to a pure inhomogeneously broadened electronic transitions. In this case the density matrix equations can be integrated independently for each x. After this the result must be averaged over x. Solutions of the undamped equations for the density matrix are interesting from the point of view of evaluation of the greatest possible population of excited states due to coherent effects, because these solutions ignore all the irreversible relaxations destructing coherence. In addition, a comparison between the latter solutions and calculations for the total model enables us to clarify the role of relaxation in the chirp dependence of population transfer (see Sec.4 below). The approach under discussion in this section is termed ”relaxation-free” model for short. 3.2 Semiclassical (Lax) approximation For broad electronic transitions satisfying the ”slow modulation” limit, we have σ s ≫ 1, where σ 2s is the LF vibration contribution to a second central moment of an absorption spectrum for transition i → j. In the last case electronic dephasing is fast, and one can use a semiclassical (short time) approximation [39]. This limit is also known as the case of appreciable Stokes losses because the perturbation of the nuclear system under electronic excitation i → j (a quantity Wj−Wi) is large. Then one can ignore the last term Lij ρ̃ij(x, t) on the right-hand side of the corresponding equation for the nondiagonal element of the density matrix [26, 40, 12, 41] that describes relaxation (diffusion) of ρ̃ij(x, t) (Eqs.(6) and (8)). The solutions, which correspond to disregarding terms Lij ρ̃ij(x, t) for broad electronic transitions i → j are termed ”partial relaxation” model for short [26]. It is worthy to note that the ”partial relaxation” model offers a particular advantage over the total model. The point is that the first can be derived not assuming the standard adiabatic elimination of the momentum p for the non-diagonal density matrix [41], which is incorrect in the ”slow modulation” limit [42]. This issue is quite important in the light of the limits imposed on Eqs.(6) and (8) for nondiagonal elements of the density matrix [43, 44]. Indeed, in the Wigner representation [45, 46, 47] equation for ρ̃12 may be written in the rotating frame as (see Eq.(6)) ρ̃W12(q, p, t) = i[(U2 (q)− U1 (q))/~− ω1(t)]ρ̃W12(q, p, t)− Ω2ρ̃W13(q, p, t)+ Ω∗1[ρW22(q, p, t)− ρW11(q, p, t)] + LFP12ρ̃W12(q, p, t) (12) Eq.(12) has been derived for harmonic potentials, Eq.(3), by generalization of equations of Refs.[48, 49, 42, 41] where LFP12 = −p + γp+ (U1 (q) + U2 (q)) is the Fokker-Planck operator for overdamped Brownian oscillator with attenuation constant In the case of appreciable Stokes losses when the perturbation of the nuclear system under electronic excitation 1 → 2 (a quantity (U2 (q)−U1 (q))/~−ωel21) is large, the quantity ρ̃W12(q, p, t) oscillates fast due to the first term on the right-hand side of Eq.(12) (see also Ref.[42]). Therefore, to the first approximation, on can neglect changes of ρ̃W12(q, p, t) due to the last term on the right-hand side of Eq.(12). Neglecting this term, integrating both side of Eq.(12) over momentum, and bearing in mind that ρ̃ij(q, t) = ρ̃Wij(q, p, t)dp (13) and x = qω̃ β, we get ρ̃12(x, t) = i[ω21−ω1(t)−(~β)−1x2x]ρ̃12(x, t)+ Ω∗1[ρ22(x, t)−ρ11(x, t)]− Ω2ρ̃13(x, t) (14) that is nothing more nor less Eq.(6) without the last term L12ρ̃12(x, t) on the right-hand side. As a matter of fact, a derivation of Eq.(14) does not involve the assumption that the momentum is instantly equilibrated. The same can be done with Eq.(8) for ρ̃23. 4 Adiabatic population transfer in the presence of excited- state absorption We shall study the ESA effects on ARP in complex molecules by the example of Coumarin 153 in liquid solution [31]. In the frequency domain, the electric field can be written as |E(ω̃)| exp[iΦ(ω̃)] and the phase term Φ(ω̃) can be expanded in a Taylor series Φ(ω̃) = Φ(ω) + (1/2)Φ′′(ω)(ω̃ − ω)2 + ... We shall consider linear chirped pulses of the form E(t) = E0 exp[− (δ2 − iµ)(t− t0)2] (15) where the parameters δ and µ are determined by the formulae [11, 12]: δ2 = 2{τ 2p0 + [2Φ′′ (ω) /τp0] 2}−1, µ = −4Φ′′ (ω) τ 4p0 + 4Φ ′′2 (ω) , (16) τp0 = tp0/ 2 ln 2, tp0 is the pulse duration of the corresponding transform-limited pulse. Fig.1 shows populations of electronic states after the completion of the one pulse action as functions -5 -4 -3 -2 -1 0 1 2 3 4 5 �"(�) (x10 Figure 1: Populations of electronic states after the completion of the pulse action as functions of Φ′′(ν) in a three-state system. Calculations without decay of the upper state 3 into state 2: n1 (dotted line), n2 (solid line), n3 (dashed line). Line with hollow circles - n2 in the model with fast decay 3 → 2 Γ32=10 ps−1. For comparison we also show n2 for a two-state system (line with squares). Total relaxation model with diffusion of all matrix elements. of the chirp rate in the frequency domain Φ′′(ν) = 4π2Φ′′ (ω). For the molecule under consideration a two-photon resonance occurs at the doubled frequency of the Franck-Condon transition 1 → 2. Absorption spectrum corresponding to transition 1 → 3 is rather narrow that means x3 = 0. The values of parameters for Fig.1 were as follows: the pulse duration of the transform-limited (non-chirped) pulse tp0 = 10 fs, ω st = 2686 cm −1, D12 = D32 = 6 D [31], τs = 70 fs, the saturation parameter, which is proportional to the pulse energy [26], Q′ ≡ π|D12Emax|2tp/(2~2 2σ122s ) = 5; the one-photon resonance for Franck-Condon transition 1 → 2 occurs at the pulse maximum, i.e. ω = ω21. Fig.2 contrasts calculations using the total model (Fig.1) with those of the partial relax- ation model. The latter includes both diffusion of all the diagonal elements of the density matrix and one off-diagonal element ρ13. The point is that transition 1 → 3 occurs with- out changing the state of vibrational subsystems of a molecule and a solvent, and therefore -5 -4 -3 -2 -1 0 1 2 3 4 5 �"(�) (x10 Figure 2: Populations of electronic states n1 (dotted lines), n2 (solid lines) and n3 (dashed lines) after the completion of the pulse action calculated without decay of the upper state 3 into state 2 as functions of Φ′′(ν). The partial relaxation and the total models - lines with and without hollow circles, respectively. All the parameters are identical to those of Fig.1. can not be described in a semiclassical (short time) approximation. Fig.2 shows a good agreement between calculation results for the models under consideration. One can see from Fig.1, first, that population n2 for a molecule with a fast decay 3 → 2, which closely resembles experimental data [11] for LD6901, is distinctly different from that of a two-state system for |Φ′′(ν)| < 15 · 103 fs2 when the excited pulse is rather short. This means that the excited state absorption has a profound effect on coherent population transfer in complex molecules. Second, n3 strongly decreases when |Φ′′(ν)| increases. To understand these results, we will consider first two transitions separately. One can obtain the following criterion for the adiabaticity of one transition in the absence of relax- ation: Q′ >> 1 where Q′ is the saturation parameter. It conforms to the value of Q′ = 5 used in our calculations. The condition Q′ >> 1 follows from the adiabatic criterion for a two-level system: 1According to Ref.[23], LD690 shows ESA. -5 -4 -3 -2 -1 0 1 2 3 4 5 �”(�� �x104 fs2) Figure 3: Populations of electronic states n1 (dotted line), n2 (solid line), n3 (dashed line) and n2 + n3 (line with hollow circles) after the completion of the pulse action as functions of Φ′′(ν) for the relaxation-free model τs → ∞. Other parameters are identical to those of Fig.1. In the case under consideration the combined population n2 + n3 does not depend on Φ′′(ν). dω(t) ≪ |Ω1,2(t)|2 (17) where Ω1,2(t) = |D21,32E(t)|/~ are the Rabi frequencies for transitions 1 → 2 and 2 → 3, respectively. Adiabatic criterion Eq.(17) was fulfilled in our simulations for both transitions 1 → 2 and 2 → 3 at any Φ′′(ν). However, Fig.1 shows that n3 strongly decreases when |Φ′′(ν)| increases. To clarify the reasons for strong decreasing n3 it is instructive to carry out the corresponding calculations for the relaxation-free model of Sec.3.1 shown in Fig.3. In this case excitation of state 3 with a transform-limited pulse is slightly more effective as compared to a strongly chirped pulse of the same energy. The point is that a two-photon resonance occurs for a number of spectral components of a transform-limited pulse and only at the maximum of a strongly chirped pulse. However, Fig.3 does not show strong decreasing the population of state 3 when Φ′′(ν) increases. This means that relaxation is responsible for strong decreasing n3 as a function of Φ ′′(ν) in spite of the fact that relaxation does not destroy ARP when the Rabi frequencies exceed the reciprocal irreversible dephasing time (T ′)−1 [27] Ω1,2 >> 1/T ′ (18) The last condition was fulfilled in our simulations at least for |Φ′′(ν)| . 104 fs2. To clarify this issue, we shall consider a population transfer between randomly fluctuating levels. 4.1 Population transfer between randomly fluctuating levels The picture of randomly fluctuating levels [27] offers a simple and physically clear explanation of numerical results [26] obtained for population transfer in a two-state system. Here we shall generalize the Landau-Zener (LZ) calculations putting in a third level [50] to random crossing of levels. Let us write the Schrödinger equations for the amplitudes of states a1,2,3 for the system under consideration. Switching to new variables ãk: ak = ãk exp , (19) we obtain in the rotating wave approximation (U1 − U2)/~+ ω1(t) −Ω1/2 0 −Ω1/2 0 −Ω2/2 0 −Ω2/2 (U3 − U2)/~− ω2(t) Throughout this section effective parabolic potentials (10) are considered as functions of gen- eralized coordinate α = x σ122s−ω12st : Uj(α) = Ej+ ~2ω12st {α+ ω12st [ ω12st+(−1)sgn(xj) st ]}2. (U3 − U2)/~− ω2(t) = [(ωel32 + ω12st /2)− ω2] + α + µ2t (21) for x3 = x1 (that corresponds to Coumarin 153), and (U1 − U2)/~+ ω1(t) = [ω1 − (ωel21 − ω12st /2)] + α− µ1t, (22) for linear chirped pulses ω1,2 (t) = ω1,2 − µ1,2t. Let us define instantaneous crossings of state 2 with photonic repetitions 1′ and 3′ of states 1 and 3, respectively. They are determined by the conditions that quantities Eqs.(21) and (22) are equal to zero: α12(t) = (ω 21 − ω12st /2)− ω1 + µ1t ≡ α12(0) + µ1t (23) α23(t) = ω2 − (ωel32 + ω12st /2)− µ2t ≡ α23(0)− µ2t Near the intersection points one can consider α as a linear function of time. For small t, α(t) ≈ α12(0)+ α̇t. Let α12(0) = α23(0), i.e. states 2, 1′ and 3′ cross at the same point when t = 0. This means ωel21 + ω 32 = ω1 + ω2 (24) i.e. the two-photon resonance occurs for t = 0. Then Eqs.(20) take the following form (α̇− µ1)t −Ω1/2 0 −Ω1/2 0 −Ω2/2 0 −Ω2/2 (α̇ + µ2)t that can be reduced to Eqs.(2) of Ref.[50]. Using the solution obtained in [50] and consid- ering identical chirps when µ1 = µ2 ≡ µ, we get for the initial condition |a1(−∞)|2 = 1, |a2,3(−∞)|2 = 0 |a3(∞)|2 = (1− P )(1−Q) for − |µ| < α̇ < |µ| P (1− P )(1−Q) for both α̇ > −µ when µ < 0, and α̇ < −µ when µ > 0 Q(1− P )(1−Q) for both α̇ < µ when µ < 0, and α̇ > µ when µ > 0 where P = exp 4|α̇− µ| , Q = exp 4|α̇ + µ| Similar to Ref.[27], we consider α as a stochastic Gaussian variable. Consequently, we must average Eqs.(26) over random crossing of levels described by Gaussian random noise induced by intra- and intermolecular fluctuations. It can be easily done for a differentiable (non-Markovian) Gaussian process [27], bearing in mind an independence of α and α̇ from each other for such processes. Therefore, we shall consider in this section a differentiable (non-Markovian) Gaussian noise, as opposed to previuous sections. In addition, we consider a slow modulation limit when σ122sτ s >> 1. Averaging Eqs.(26), we obtain the following expression for the population of state 3 when µ > 0 ∫ −|µ| P (1− P )(1−Q)f(α, α̇)dα̇ + ∫ |µ| (1− P )(1−Q)f(α, α̇)dα̇ Q(1 − P )(1−Q)f(α, α̇)dα̇] (28) Here f(α, α̇) is the joint probability density for α and its derivative α̇: f(α, α̇) = σ122s (−k̈(0)) 2k̈(0) , (29) k̈(0) is the second derivative of the correlation function k(t) =< α(0)α(t) >= σ122s exp(−|t|/τs) of the energetic fluctuations evaluated at zero. Eq.(28) is written for µ > 0 (negatively chirped pulse). One can easily show that n3 is symmetrical with respect to the chirp sign. The point is that a simple stochastic model of this Section misses any chromophore’s effects on bath, in particular the dynamical Stokes shift (see Ref.[51] for details). This is opposite to the models of previous sections, which do describe the dynamical Stokes by the drift term (the second term on the right-hand side of Eq.(9)). Integrating Eq.(28) with respect to α and entering a dimensionless variable y = α̇/ |µ| , we get P (1− P )(1−Q) exp (1− P )(1−Q)] exp Q(1− P )(1−Q) exp dy] (30) where P = exp 2|y − 1| and Q = exp 2|y + 1| , (31) 2 |µ| > 0, ξ = − k̈(0) > 0 (32) are dimensionless parameters. When adiabatic criterion Eq.(17) is satisfied, parameter κ is much larger than 1 since |dω(t)/dt| = |µ| for a linear chirped pulse. Then the integrals on the right-hand side of Eq.(30) can be evaluated by the method of Laplace, similar to Ref.[27]. The result is espe- cially simple for strong interaction, Eq.(18), where the irreversible dephasing time of transi- tions 1 → 2 and 2 → 3 is given by [27] T ′ = 1/[−k̈(0)]1/4. Then, as one can see also from Eqs.(30) and (31), the main contribution to n3 is given by dy = erf |µ|T ′2√ Since erf(1.5) = 0.966, we obtain that relaxation does not hinder a population transfer to state 3 when |µ|T ′2 ≥ 2 (34) For strongly chirped pulses [52], µ|T ′2/ 2 ≈ 2 2π2T ′2/|Φ′′(ν)|. Eq.(34) expresses an extra criterion for coherent population transfer to those we have obtained before for a two-level system [27], Eqs.(17) and (18). New criterion (34) implies conservation of the “counter-movement” of the “photonic repetitions” of states 1 and 3, in spite of random crossing of levels. Condition (34) is exemplified by Fig.4. In addition, Fig.4 shows an excellent agreement of simple formula (33) with numerical calculations. It is worthy to note that condition (18) was fulfilled in our simulations, though in the last case T ′ = (τs/σ2s) 1/3 is determined independently of k̈(0) [53], which does not exist for the Gaussian-Markovian process. 4.2 Influence of excited-state absorption when detuning from two- photon resonance occurs For Coumarin 153 in liquid solution considered above a two-photon resonance occurs at the doubled frequency of the Franck-Condon transition 1 → 2. In this section we consider populations of electronic states when the condition for two-photon resonance is violated. Figs. 5 and 6 show populations of electronic states for the total model after the completion 15 20 25 30 T', fs Figure 4: Population of state 3 as a function of the irreversible dephasing time T ′ for Φ′′(ν) = 104 fs2 calculated by Eq.(33) (solid line with circles) and numerical solution of Eqs.(5)-(8) (dashed line with squares). n3,no relaxation ≡ n3(T ′ → ∞). Other parameters are identical to those of Fig.1. -5 -4 -3 -2 -1 0 1 2 3 4 5 �"( ) (x10 Figure 5: Populations of electronic states after the completion of the pulse action as functions of Φ′′(ν) in a three-state system. The frequency of purely electronic transition 3 → 2, ωel32, decreases by ω12st /4 with the conservation of x3 = 0. Calculations without decay of the upper state 3 into state 2: n1 (dotted line), n2 (solid line), n3 (dashed line). The corresponding populations in the model with fast decay 3 → 2 Γ32 = 10 ps−1 are shown by the same lines with hollow circles. -5 -4 -3 -2 -1 0 1 2 3 4 5 "(�) (x10 Figure 6: Populations of electronic states after the completion of the pulse action as functions of Φ′′(ν) in a three-state system. Equilibrium position of state 3 is offset to the right by x3 = x2/2 and down so that frequencies of Franck-Condon transitions 1 → 2 and 2 → 3 are equal: ω21 = ω32. Calculations without decay of the upper state 3 into state 2: n1 (dotted line), n2 (solid line), n3 (dashed line). The corresponding populations in the model with fast decay 3 → 2 Γ32 = 10 ps−1 are shown by the same lines with hollow circles. of the pulse action as functions of Φ′′(ν) for the same values of parameters as for Fig.1 with the only difference concerning the position of state 3. The frequency of purely electronic transition 3 → 2 ωel32 decreases by ω12st /4 with the conservation of x3 = 0 for Fig. 5. Equilib- rium position of state 3 is offset to the right by x3 = x2/2 and down so that frequencies of Franck-Condon transitions 1 → 2 and 2 → 3 are equal: ω21 = ω32 for Fig. 6. One can see from Figs.1, 5 and 6, first, that population n1 and, as a consequence, n2+n3 depend only slightly on the occurrence of fast decay 3 → 2. Second, populations n2 and n3 in the absence of fast decay 3 → 2 are very sensitive to the violation of the two-photon resonance condition. However, a behavior of n2, when fast decay 3 → 2 occurs, and n1 as functions of Φ′′(ν) is very similar for the figures under discussion, regadless of the two- photon resonance condition. Experimental measurements commonly correspond to n2 and are carried out under the fast decay 3 → 2 conditions. Thus the behavior of n2 for fast decay 3 → 2 shown in Figs.1, 5 and 6 is rather versatile. 5 Population transfer in the presence of two-exciton processes. Selective excitation of single and two- exciton states with chirped pulses Consider a dimer of chromophores each with two electronic states described by the Frenkel exciton Hamiltonian [54, 55, 56] and excited with electromagnetic field Eq.(2). The Hamil- tonian of the dimer is given by m=1,2 ~Ω̄mB mBm + ~J(B 1 B2 +B 2 B1) +Hbath +Heb − m=1,2 Dm ·E(t)(B+m +Bm) (35) where B+m = |m〉〈0| (Bm = |0〉〈m|) are exciton creation (annihilation) operators associated with the chromophore m, which satisfy the commutation rules [Bn, B m] = δnm(1−2B+mBm), δnm is the Kroenecker delta; |0〉 and |m〉 denote the ground state and a state corresponding to the excitation of chromophore m, respectively. Dm is the transition dipole moment of molecule m, Hbath represents a bath and Heb its coupling with the exciton system. We assume that the bath is harmonic and that the coupling is linear in the nuclear coordinates Heb = −~ mBn (36) where αmn represent collective bath coordinates. ~Ω̄1(~Ω̄2) and ~J are the exciton energy of 1 (2) chromophore and their coupling energy at the equilibrium nuclear coordinate of the ground electronic state. One can consider αmn as diagonal: αmn = αmδnm on the assumption that the electronic coupling constant fluctuation amplitude is negligibly smaller than the site energy fluctuation amplitude [55]. Diagonalizing the electronic Hamiltonian m=1,2 ~Ω̄mB mBm + ~J(B 1 B2 +B 2 B1) (37) by unitary transformation [57] U−1 = cos θ sin θ − sin θ cos θ where tan 2θ = Ω̄1 − Ω̄2 , 0 < θ < π/2, (39) one can get the eigenstates for the one-exciton states |ei〉 and the transition dipole moments Dei (i = 1, 2) corresponding to the transitions between the ground and single-exciton states = U−1 A1 cos θ + A2 sin θ −A1 sin θ + A2 cos θ Here aei = |ei〉, Dei and Ai = B+i |0〉, Di; D1 and D2 are the site transition moments. The two one-exciton energies are given by ~Ω̄e1 = ~Ω̄1 cos 2 θ + ~Ω̄2 sin 2 θ + ~J sin 2θ, (41) ~Ω̄e2 = ~Ω̄1 sin 2 θ + ~Ω̄2 cos 2 θ − ~J sin 2θ, The two-exciton state wavefunction and its energy are as following |e3〉 = B+1 B+2 |0〉 ≡ B+e3 |0〉 (42) ~Ω̄e3 = ~Ω̄1 + ~Ω̄2 (43) The transition dipole moments between the single-exciton and two-exciton states are given De1e3 = D1 sin θ +D2 cos θ, De2e3 = D1 cos θ −D2 sin θ (44) However, the transition between the ground and two-exciton states is not allowed. In the eigenstate representation, the Hamiltonian of Eq.(35) is rewritten as i=1,2,3 ~(Ω̄ei − αei)B+eiBei − ~ i,j=1,2 i 6=j αeiejB Bej +Hbath− i=1,2 [Dei(B +Bei) +Deie3(B Be3 +B Bei)] · E(t) (45) Here the interaction with the bath is given by αe1 αe1e2 αe2e1 αe2 = U−1HebU = ~ α1 cos 2 θ + α2 sin 2 θ 1 (α2 − α1) sin 2θ (α2 − α1) sin 2θ α1 sin2 θ + α2 cos2 θ αe3 = α1 + α2, (47) for the single-exciton and two-exciton states, respectively. Eqs.(46) and (47) define the fluctuating parts of the single-exciton and two-exciton state transition frequencies. Consider various correlation functions. Assuming that baths acting on different chro- mophores are uncorrelated 〈αm(t)αn(0)〉 = 0 for m 6= n (48) and that the site energy fluctuation correlation functions are identical for the two monomers [55, 56], we get 〈αe1(t)αe1(0)〉 = 〈αe2(t)αe2(0)〉 = ~−2K(t)(cos4 θ + sin4 θ), 〈αe3(t)αe3(0)〉 = 2~−2K(t) (49) where K(t) = ~−2〈α1(t)α1(0)〉 = ~−2〈α2(t)α2(0)〉 ≡ ~−2〈ᾱ(t)ᾱ(0)〉. The further calculations simplify considerably if the off-diagonal part of the interaction with the bath in the exciton representation αe1e2 = αe2e1 in Eq.(46) can be neglected. This approximation is discussed in Refs.[55, 58]. The correlation function K(t) can be represented as the Fourier transform of the power spectrum Φ(ω) of ~α1(= ~α2) [59] K(t) = dωΦ(ω) exp(iωt) where Φ(−ω) = Φ(ω) exp(−β~ω) (50) Using Eq.(50), the real and imaginary parts of K(t) = K ′(t) + iK ′′(t) can be written as K ′(t) = dωΦ(ω)[1 + exp(−β~ω)] cosωt K ′′(t) = dωΦ(ω)[1− exp(−β~ω)] sinωt In the high temperature limit one get K ′(t) = 2 dωΦ(ω) cosωt K ′′(t) = ~β dωΦ(ω)ω sinωt where K(0) = K ′(0) = 2 dωΦ(ω) = ~2σ2 = ~ωStβ −1; σ2 and ωSt are a second cen- tral moment and the Stokes shift of the equilibrium absorption and luminescence spectra, respectively, for each monomer. Similar to Sec.2, we will consider ᾱ = −u/~ as a stochastic Gaussian variable with the correlation function 〈ᾱ(t)ᾱ(0)〉 = σ2 exp(−|t|/τs) corresponding to the Gaussian-Markovian process. In this case the Fokker–Planck operators for the excited state of each monomer has the following form Lm = τ + (x− xm) where x = qω̃ β = ᾱ/ σ2 is a dimensionless generalized coordinate. Bearing in mind Eqs.(49), the Fokker–Planck operators for the eigenstates |j〉 = |0〉,|ei〉 of the exciton Hamil- tonian can be written by Eq.(9) where x0 = 0, xe1 = xe2 = xm(cos 4 θ+sin4 θ) and xe3 = 2xm. The corresponding transition frequencies at the equilibrium nuclear coordinate of the ground electronic state are defined by Eqs.(41) and (43). Consider a homodimer complex consisting of identical molecules with Ω̄1 = Ω̄2 ≡ Ω̄ and D1 = D2 ≡ D. For this case, using Eqs.(39), (40), (41), (43) and (44), we obtain θ = π/4, -5 -4 -3 -2 -1 0 1 2 3 4 5 �"( ) (x104 fs2) Figure 7: Populations of the ground (dotted line), single- (solid line) and two-exciton (dashed line) states of a homodimer complex after the completion of the pulse action as functions of Φ”(ν) for J = −300 cm−1 (J < 0 - J-aggregate), Q′ = 2.9, tp0 = 10 fs, τs = 100 fs. The partial relaxation and the total models - lines with and without hollow circles, respectively. ~Ω̄e1,2 = ~(Ω̄± J), ~Ω̄e3 = 2~Ω̄ (52) De1 = De1e3 = 2D, De2 = De2e3 = 0 We thus need to consider only three states: |0〉, |e1〉 and |e3〉, since state |e2〉 is not excited with light. Letting |1〉, |2〉 and |3〉 represent |0〉, |e1〉 and |e3〉, respectively, we arrive at a three-state system considered above where ω21 = Ω̄ + J, ω31 = 2Ω̄, D21 = D32 = x1 = 0, x2 = xm, x3 = 2xm. Fig.7 shows populations of single and two-exciton states after the excitation with a linear chirped pulse, Eqs.(15) and (16), as functions of Φ′′(ν). Here the one-photon resonance for Franck-Condon transition 1 → 2 occurs at the pulse maximum, i.e. ω = ω21 = Ω̄ + J , and the Stokes shift of the equilibrium absorption and luminescence spectra for each monomer is equal to ωmonst = 400 cm −1. Fig.7 also contrasts calculations using the total model (lines without hollow circles) with those of the partial relaxation model when only diagonal matrix elements of the density matrix undergo diffusion (lines with hollow circles). Fig.7 shows a good agreement between the calculation results for both models. Furthermore, one can see strong suppressing the population of the two-exciton state for negatively chirped (NC) pulse excitation. As a matter of fact, one can suppress or enhance two-exciton processes using positively or NC pulses. Our calculations (see table below) show twofold benefits of NC pulse excitation (Φ′′ = −104 fs2) with respect to the transform limited pulse (Φ′′ = 0) of the same duration (tp = 71 fs) and energy tuned to one-exciton transition: the population transfer to the single exciton state is larger, and that to the two-exciton state is smaller. Populations after the completion of pulse action Transform limited pulse (Φ′′ = 0, tp = 71fs) NC pulse (Φ′′ = −104fs2, tp = 71fs) n2 0.317 0.573 n3 0.208 0.057 It is worthy to note good selective properties of chirped pulses, bearing in mind strong overlapping Franck-Condon transitions 1 → 2, ω21, and 2 → 3, ω32. Really, the corresponding frequencies differ by ω32 − ω21 = −2J − 34ω st for the model under consideration that comes to ω32 − ω21 = 300 cm−1 for the used values of parameters. On the other hand, the bandwidth of the absorption spectrum at half maximum for transition 2 → 3 comes to ∆ω = 2 2 ln 2σ232s ≈ 1024 cm−1 that is larger than ω32 − ω21. Here σ232s = (~β)−1ω23st is the LF vibration contribution to a second central moment of an absorption spectrum for transition 2 → 3 and ω23st = (~β)−1(x3 − x2)2 = 94ω st is the corresponding Stokes shift. This issue can be understandable in terms of the competition between sequential and direct paths in a two-photon transition [36]. Consider a three-level atomic ladder system in the absence of relaxation with close transition frequencies ω21 ≈ ω32 where ω21 can be associated with one-exciton excitation and frequency ω31 - with two-exciton excitation.The system is affected by one phase modulated pulse of carrier frequency ω, Eqs.(2), (15) and (16). In the Appendix we have calculated the excited-state amplitude a3 due to two-photon transition 1 → 3 involving a nearly resonant intermediate level 2 for such system. Amplitude a3 = aTP +aS consists of two contributions. The first one aTP corresponds exactly to that of the nonresonant two-photon transition. This contribution aTP ∼ 1/|Φ′′(ω)|, and it is small for strongly chirped pulses [52] 2|Φ′′(ω)| ≫ τ 2p0 (53) This result has a clear physical meaning. The point is that the phase structure (chirp) of the pulse determines the temporal ordering of its different frequency components. For a strongly chirped pulse when a pulse duration is much larger than that of the corresponding transform- limited one, one can ascribe to different instants of time the corresponding frequencies [52]. As a matter of fact, in the case under consideration different frequency components of the field are determined via values of the instantaneous pulse frequency ω(t) for different instants of time. Therefore, only a small part of the whole pulse spectrum directly excites the two- photon resonance. The second contribution is given by [36] aS = − D32D21π E(ω21)E(ω32){1− sgn[(ω21 − ω32)Φ′′ (ω)]} (54) where E(ω̃) is the Fourier transform of the positive frequency components of the field ampli- tude E (t) exp[iϕi (t)]. The consideration of the Appendix enables us to extend the results of Ref.[36] to non-zero two-photon detuning Ω2 = ω31 − 2ω 6= 0. Eq.(54) describes a sequential process, the contribution of which is a steplike function. This process can be suppressed when the pulse frequencies arrive in counter-intuitive order (ω32 before ω21) that occurs in our simulations of a J-aggregate for NC excitation. Fig.7 and the table above show that the selective properties of chirped pulses under discussion are conserved on strong field excita- tion and for broad transitions. The selective excitation of single and two-exciton states can be used for preparation of initial states for nonlinear spectroscopy based on pulse shaping [60, 61]. 6 Strong interaction and STIRAP The three-state system under discussion enables us to consider STIRAP as well. STIRAP in molecules in solution was studied in Refs.[29], where the solvent fluctuations were represented as a Gaussian random process, and in Ref.[30], where the system-bath coupling was taken to be weak in the sense that the relaxation times were long in comparison to the bath correlation time, τc. Intense fields were shown in Ref.[30] to effectively slow down the dephasing when the energetic distance between the dressed (adiabatic) states exceeds 1/τc. The point of the last paper is that in contrast to usual undressed states, which intersect, the dressed (adiabatic) states do not intersect. Therefore, the spectral density of the relaxation induced noise, which has a maximum at zero frequency, strongly diminishes for frequencies corresponding to the light-induced gap between dressed states, resulting in suppressing pure dephasing between the dressed states. In this section we show that this conclusion holds also for non- Markovian relaxation when the system-bath interaction is not weak and, therefore, can not be characterized only by τc. In the rotating wave approximation the Schrödinger equations for STIRAP in Λ-configuration can be written as follows U ′1 −~Ω1/2 0 −~Ω1/2 U2 −~Ω2/2 0 −~Ω2/2 U ′3 where U ′1 = U1 + ~ω1 and U 3 = U3 + ~ω2 are ”photonic replications” of effective parabolic potentials U1(x) and U3(x) (Eq.(10)), respectively. We consider the two-photon resonance condition when ω1 − ω2 = (E3 −E1)/~ and x1 = x3 = 0 that would appear reasonable when |1〉 and |3〉 are different vibrational levels of the same electronic state. Then U ′1 = U ′3. Adiabatic states Uad corresponding to Eq.(55) can be found by equation U ′1 − Uad −~Ω1/2 0 −~Ω1/2 U2 − Uad −~Ω2/2 0 −~Ω2/2 U ′3 − Uad This gives the following adiabatic states Uad0 = U 1 = U Uad± = (U2 + U (U2 − U ′1)2 + ~2(Ω21 + Ω22) (56) One can see that initial U ′1 and final U 3 diabatic states coincide with one of adiabatis states Uad0 . For strong interaction the last will be well separated from other adiabatic states U ± due to avoided crossing. Therefore, during STIRAP the system will remain in the same adiabatic state Uad0 , which is U 1 for t = −∞ and U ′3 for t = +∞. Its evolution due to relaxation stimulated by LF vibrations can be described by the corresponding Fokker-Planck operator Lad0 = L1,3 = τ + x ∂ describing diffusion in adiabatic potential Uad0 = U 1 = U This means that during transition 1 → 3 the system motion along a generalized coordinate x does not change. In other words, such a transition will not be accompanied by pure dephasing. This conclusion is a generalization of the previous result [30] relative to slowing down the dephasing in strong fields, which was obtained for weak system-bath interaction, to non-Markovian relaxation. 7 Conclusion In this work we have studied the influence of ESA and two-exciton processes on a coherent population transfer with intense ultrashort chirped pulses in molecular systems in solution. An unified treatment of ARP in such systems has been developed using a three-state elec- tronic system with relaxation treated as a diffusion on electronic potential energy surfaces. We believe that such a simple model properly describes the main relaxation processes related to overdamped motions occurring in large molecules in solutions. Our calculations show that even with fast relaxation of a higher singlet state Sn (n > 1) back to S1, ESA has a profound effect on coherent population transfer in complex molecules that necessitates a more accurate interpretation of the corresponding experimental data. In the absence of Sn → S1 relaxation, the population of state |3〉, n3, strongly decreases when the chirp rate in the frequency domain |Φ′′(ν)| increases. In order to appreciate the physical mechanism for such behavior, an approach to the total model - the relaxation-free model - was invoked. A comparison between the total model behavior and that of the relaxation-free model has shown that relaxation is responsible for strong decreasing n3 as a function of Φ′′(ν) in spite of meeting adiabatic criteria for both transitions 1 → 2 and 2 → 3 separately. By this means usual criteria for ARP in a two-state system must be revised for a three-state system. To clarify this issue, we have developed a simple and physically clear model for ARP with a linear chirped pulse in molecules with three electronic states in solution. The relaxation effects were considered in the framework of the LZ calculations putting in a third level generalized for random crossing of levels. The model has enabled us to obtain a simple formula for n3, Eq.(33), which is in excellent agreement with numerical calculations. In addition, the model gives us an extra criterion for coherent population transfer to those we have obtained before for a two-state system [27]. New criterion, Eq.(34), implies conservation of the “counter-movement” of the “photonic repetitions” of states 1 and 3, in spite of random crossing of levels. Furthermore, we also applied our model to a molecular dimer consisting from two-level chromophores. A strong suppressing of two-exciton state population for NC pulse excitation of a J-aggregate has been demonstrated. We have shown that one can suppress or enhance two-exciton processes using positively or NC pulses. As a matter of fact, a method for quantum control of two-exciton states has been proposed. Our calculations show good selective properties of chirped pulses in spite of strong overlapping transitions related to the excitation of single- and two-exciton states. In the light of the limits [43, 44] imposed on Eqs.(6) and (8) for nondiagonal elements of the density matrix for the total model, we used a semiclassical (Lax) approximation (Eq.(14)) (the partial relaxation model). The latter offers a particular advantage over the total model. The point is that the partial relaxation model can be derived not assuming the standard adiabatic elimination of the momentum for the non-diagonal density matrix, which is incorrect in the ”slow modulation” limit [42]. A good agreement between calcu- lation results for the partial relaxation and the total models in the slow modulation limit (see Figs.2 and 7) shows that a specific form of the relaxation term in the equations for nondiagonal elements of the density matrix ρ̃12(x, t) and ρ̃23(x, t) is unimportant. By this means the limits imposed on the last equation [43, 44] are of no practical importance for the problem under consideration in the slow modulation limit. This issue can be explained as follows. Our previous simulations [26] show that in spite of a quite different behavior of the coherences (nondiagonal density matrix elements) for the partial relaxation and the total models, their population wave packets ρjj (x, t) behave much like. Since we are interested in the populations of the electronic states nj = ρjj (x, t) dx only, which are integrals of ρjj (x, t) over x, the distinctions between the two models under discussion become minimal. In conclusion, we have also demonstrated slowing down the pure dephasing on STIRAP in strong fields when the system-bath interaction is not weak (non-Markovian relaxation). Acknowledgement This work was supported by the Ministry of absorption of Israel. Appendix Consider a three-level system E1 < E2 < E3 with close transition frequencies ω21 ≈ ω32 where ω21 can be associated with a single-exciton excitation and frequency ω31 - with two-exciton excitation. The system is affected by one phase modulated pulse of carrier frequency ω, Eq.(2). The excited-state amplitude for a two-photon transition involving a nearly resonant intermediate level, can be written as [62, 36] a3 = − D32D21 E(ω21)E(ω32) + E(Ω + ω)E(Ω2 − Ω + ω) Ω− (ω21 − ω) where E(ω̃) is the Fourier transform of the positive frequency components of the field am- plitude E (t) exp[iϕi (t)], Ω = ω̃ − ω, P is the principal Cauchy value, Ω2 = ω31 − 2ω is the two-photon detuning. For linear chirped excitation, Eqs.(15) and (16), E(ω̃) is given by E(ω̃) = πE0τp0 exp{− Ω2[τ 2p0/2− iΦ′′(ω)]} (58) Using Eq.(58) and introducing a new variable z = Ω− Ω2/2, Eq.(57) can be written as a3 = − D32D21π 2(E0τp0)2 {exp[−1 (δ2 + (Ω2 − δ)2)(τ 2p0/2− iΦ′′(ω))] + exp[− Ω22(τ p0/2− iΦ′′(ω))] exp[−z2(τ 2p0/2− iΦ′′(ω))] z − (δ − Ω2/2) } (59) where δ = ω21−ω is one-photon detuning. The integral on the right-hand side of Eq.(59) can be evaluated for strongly chirped pulses [52], Eq.(53), when a pulse duration is much larger than that of the corresponding transform-limited one. In this case two frequency ranges give main contributions to the integral. The first one results from the method of stationary phase [63], and it is localized near the two-photon resonance z = ω − ω31/2 = 0 in the small range ∆ω ∼ 1/ |Φ′′(ω)|. 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Basic equations Approximate models System with frozen nuclear motion Semiclassical (Lax) approximation Adiabatic population transfer in the presence of excited-state absorption Population transfer between randomly fluctuating levels Influence of excited-state absorption when detuning from two-photon resonance occurs Population transfer in the presence of two-exciton processes. Selective excitation of single and two-exciton states with chirped pulses Strong interaction and STIRAP Conclusion

0704.1960

Harmonic bilocal fields generated by globally conformal invariant scalar fields

Harmonic bilocal fields generated by globally conformal invariant scalar fields Nikolay M. Nikolov1,3, Karl-Henning Rehren2, Ivan Todorov1,3 September 23, 2021 1 Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria 2 Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany 3 Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I–34014 Trieste, Italy Abstract The twist two contribution in the operator product expansion of φ1(x1) φ2(x2) for a pair of globally conformal invariant, scalar fields of equal scal- ing dimension d in four space–time dimensions is a field V1(x1, x2) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of V1 can be equivalently characterized by a “single–pole property” concern- ing the pole structure of the (rational) correlation functions involving the product φ1(x1) φ2(x2). This property is established for the dimension d = 2 of φ1, φ2. As an application we prove that any system of GCI scalar fields of conformal dimension 2 (in four space–time dimensions) can be presented as a (possibly infinite) superposition of products of free massless fields. Subject classification: PACS 2003: 11.10.-z. 03.70.+k, MSC 2000: 81T10 1 Introduction Global Conformal Invariance (GCI) of Minkowski space Wightman fields yields rationality of correlation functions [14]. This result opens the way for a non- perturbative construction and analysis of GCI models for higher dimensional Quantum Field Theory (QFT), by exploring further implications of the Wight- man axioms. By choosing the axiomatic approach, we avoid any bias about the possible origin of the model, because we aim at a broadest possible perspective. On the other hand, the assumption of GCI limits the analysis to a class of theories http://arxiv.org/abs/0704.1960v4 Harmonic Bilocal Fields 2 that can be parameterized by its (generating) field content and finitely many coefficients for each correlation function (see Sect. 2). As anomalous dimensions under the assumption of GCI are forced to be integral, there is no perturbative approach within this setting, but it is conceivable that a theory with a contin- uous coupling parameter may exhibit GCI at discrete values (that appear as renormalization group fixed points). An example of this type is provided by the Thirring model: it is locally conformal invariant for any value of the coupling constant g and becomes GCI for positive integer g2 [5]. Previous axiomatic treatments of conformal QFT were focussed on the rep- resentation theory and harmonic analysis of the conformal group [6, 10] as tools for the Operator Product Expansion (OPE). The general projective realization of conformal symmetry in QFT was already emphasized in [15, 16] and found to constitute a (partial) organization of the OPE. GCI is complementary in that it assumes true representations (trivial covering projection). A necessary condition for this highly symmetric situation is the presence of infinitely many conserved tensor currents (as we shall see in Sect. 3.3). The first cases studied under the assumption of GCI were theories generated by a scalar field φ(x) of (low) integral dimension d > 1. (The case d = 1 corresponds to a free massless field with a vanishing truncated 4-point function wtr4 .) The cases 2 6 d 6 4, which give rise to non-zero w 4 were considered in [12, 13, 11].1 The main purpose in these papers was to study the constraints for the 4- point correlation (=Wightman) functions coming from theWightman (= Hilbert space) positivity. This was achieved by using the conformal partial wave expan- sion. An important technical tool in this expansion is the splitting of the OPE into different twist contributions (see (2.10)). Each partial wave gives a non- rational contribution to the complete rational 4-point function. It is therefore remarkable that the sum of the leading, twist two, conformal partial waves (cor- responding to the contributions of all conserved symmetric traceless tensors in the OPE of basic fields) can be proven in certain cases to be a rational function. This means that the twist two part in the OPE of two fields φ is convergent in such cases to a bilocal field, V1(x1, x2), which is our first main result in the present paper. Throughout, “bilocal” means Huygens (= space–like and time–like) lo- cality with respect to both arguments. Proving bilocality exploits the bounds on the poles due to Wightman positivity, and the conservation laws for twist two tensors which imply that the bilocal fields are harmonic in both arguments. Trivial examples of harmonic bilocal fields are given by bilinear free field constructions of the form : ϕ(x1)ϕ(x2) :, : ψ̄(x1)γµ(x1 − x2) µψ(x2) :, or (x1 − µ(x1 − x2) ν : Fµσ(x1)F ν (x2) :. A major purpose of this paper is to explore whether harmonic twist two fields can exist which are not of this form, and whether they can be bilocal. Moreover, we show that the presence of a bilocal 1The last two references are chiefly concerned with the case d = 4 (in D = 4 space-time dimensions) which appears to be of particular interest as corresponding to a (gauge invariant) Lagrangian density. The intermediate case d = 3 is briefly surveyed in [18]. Harmonic Bilocal Fields 3 field V1 completely determines the structure of the theory in the case of a scaling dimension d = 2. The first step towards the classification of d = 2 GCI fields was made in [12] where the case of a unique scalar field was considered. Here we extend our study to the most general case of a theory generated by an arbitrary (countable) set of d = 2 scalar fields. Our second main result states that such fields are always combinations of Wick products of free fields (and generalized free fields). The paper is organized as follows. Section 2 contains a review of relevant results concerning the theory of GCI scalar fields. In Sect. 3 we study conditions for the existence of the harmonic bilocal field V1(x1, x2). We prove that Huygens bilocality of V1(x1, x2) is equivalent to the sin- gle pole property (SPP), Definition 3.1, which is a condition on the pole structure of the leading singularities of the truncated correlation functions of φ1(x1)φ2(x2) whose twist expansion starts with V1(x1, x2). This nontrivial condition qualifies a premature announcement in [2] that Huygens bilocality is automatic. Indeed, the SPP is trivially satisfied for all correlations of free field construc- tions of harmonic fields with other (products of) free fields, due to the bilinear structure of V1. Thus any violation of the SPP is a clear signal for a nontriv- ial field content of the model. Moreover, the SPP will be proven from general principles for an arbitrary system of d = 2 scalar fields (the case studied in [2]). Yet, although the pole structure of U(x1, x2) turns out to be highly constrained in general by the conservation laws of twist two tensor currents, the SPP does not follow for fields of higher dimensions, as illustrated by a counter-example of a 6-point function of d = 4 scalar fields involving double poles (Sect. 3.5). The existence of V1(x1, x2) as a Huygens bilocal field in a theory of dimension d = 2 fields allows to determine the truncated correlation functions up to a single parameter in each of them. This is exploited in Sect. 4, where an associative algebra structure of the OPE of d = 2 scalar fields and harmonic bilocal fields is revealed. The free-field representation of these fields is inferred by solving an associated moment problem. 2 Properties of GCI scalar fields 2.1 Structure of correlation functions and pole bounds We assume throughout the validity of the Wightman axioms for a QFT on the D = 4 flat Minkowski space–time M (except for asymptotic completeness) – see [17]. Our results can be, in fact, generalized in a straightforward way to any even space–time dimension D. The condition of GCI in the Minkowski space is an additional symmetry condition on the correlation functions of the theory [14]. In the case of a scalar field φ(x), it asserts that the correlation functions of φ(x) Harmonic Bilocal Fields 4 are invariant under the substitution φ(x) 7→ det , (2.1) where x 7→ g(x) is any conformal transformation of the Minkowski space, ∂g its Jacobi matrix and d > 0 is the scaling dimension of φ. An important point is that the invariance of Wightman functions ∣∣φ(x1) · · · φ(xn) under the transformation (2.1) should be valid for all xk ∈ M in the domain of definition of g (in the sense of distributions). It follows that d must be an integer in order to ensure the singlevaluedness of the prefactor in (2.1). Thus, GCI implies that only integral anomalous dimensions can occur. The most important consequences of GCI in the case of scalar fields φk(x) of dimensions dk are summarized as follows. (a) Huygens Locality ([14, Theorem 4.1]). Fields commute for non light–like separations. This has an algebraic version: (x1 − x2) φ1(x1), φ2(x2) = 0 (2.2) for a sufficiently large integer N . (b) Rationality of Correlation Functions (cf. [14, Theorem 3.1]). The general form of Wightman functions is: ∣∣φ1(x1) · · ·φn(xn) {µjk} C{µjk} (ρjk) µjk , (2.3) where here and in what follows we set ρjk := (xjk − i 0 e0) 2 = (xjk) 2 + i 0 x0jk , xjk := xj − xk ; (2.4) the sum in Eq. (2.3) is over all configurations of integral powers {µjk = µkj} subject to the following conditions: j (6=k) µjk = −dk, (2.5) and pole bounds µjk > − dj + dk δdjdk − 1 . Equation (2.5) follows from the conformal invariance under (2.1); the pole bounds express the absence of non-unitary representations in the OPE of two fields [14, Lemma 4.3]. Under these conditions the sum in (2.3) is always finite and there are a finite number of free parameters for every n-point correlation function. We shall refer to the form (2.3) as a Laurent polynomial in the variables ρjk. 2Writing correlation functions in terms of the conformally invariant cross ratios is particu- larly useful to parameterize 4-point functions. A basis of cross ratios for an n-point function is used in the proof of Lemma 3.6. The general systematics of the pole structure, however, is more transparent in terms of the present variables. Harmonic Bilocal Fields 5 (c) The truncated Wightman functions ∣∣φ1(x1) · · · φn(xn) are of the same form like (2.3) but with pole degrees µtr bounded by µtrjk > − dj+dk (2.6) (cf. [14, Corollary 4.4]). The cluster condition, expressing the uniqueness of the vacuum, requires that if a non-empty proper subset of points xk among all xi (i = 1, . . . , n) is shifted by t · a (a2 6= 0), then the truncated function must vanish in the limit t → ∞. For the two-point clusters {xj , xk}, this condition is ensured by (2.6) in combination with with (2.5). For higher clusters, it puts further constraints on the admissible linear combinations of terms of the form (2.3). Note however, that because of possible cancellations the individual terms need not vanish in the cluster limit. The cluster condition will be used in establishing the single pole property for d = 2. 2.2 Twist expansion of the OPE and bi–harmonicity of twist two contribution The most powerful tool provided by GCI is the explicit construction of the OPE of local fields in the general (axiomatic) framework. Let φ1(x) and φ2(x) be two GCI scalar fields of the same scaling dimension d and consider the operator distribution U(x1, x2) = (ρ12) φ1(x1)φ2(x2)− 〈0|φ1(x1)φ2(x2)|0〉 . (2.7) As a consequence of the pole bounds (2.6), U(x1, x2) is smooth in the difference x12. This is to be understood in a weak sense for matrix elements of U between bounded energy states. Obviously, U(x1, x2) is a Huygens bilocal field in the sense that [ (x1 − x) 2(x2 − x) U(x1, x2), ψ(x) = 0 (2.8) for every field ψ(x) that is Huygens local with respect to φk(x). Then, one introduces the OPE of φ1(x1)φ2(x2) by the Taylor expansion of U in x12 U(x1, x2) = µ1,...,µn=0 12 · · · x µ1...µn (x2) , (2.9) where Xnµ1...µn(x2) are Huygens local fields. We can consider the series (2.9) as a formal power series, or as a convergent series in terms of the analytically continued correlation functions of U(x1, x2). We will consider at this point the series (2.9) just as a formal series. (See also [1] for the general case of constructing OPE via multilocal fields in the context of vertex algebras in higher dimensions.) Since the prefactor in (2.7) transforms as a scalar density of conformal weight (1−d, 1−d) then U(x1, x2) transforms as a conformal bilocal field of weight (1, 1). Harmonic Bilocal Fields 6 Hence, the local fields Xnµ1...µn in (2.9) have scaling dimensions n+2 but are not, in general, quasiprimary.3 One can pass to an expansion in quasiprimary fields by subtracting from Xnµ1...µn derivatives of lower dimensional fields X µ1...µn′ . The resulting quasiprimary fields Okµ1...µℓ are traceless tensor fields of rank ℓ and dimension k. The difference k − ℓ (“dimension − rank”) (2.10) is called twist of the tensor field Okµ1...µℓ . Unitarity implies that the twist is non-negative [10], and by GCI, it should be an even integer. In this way one can reorganize the OPE (2.9) as follows U(x1, x2) = V1(x1, x2) + ρ12 V2(x1, x2) + (ρ12) 2 V3(x1, x2) + · · · , (2.11) where Vκ(x1, x2) is the part of the OPE (2.9) containing only twist 2κ contri- butions. Note that Eq. (2.11) contains also the information that the twist 2κ contributions contain a factor (ρ12) κ−1 (i.e. Vκ are “regular” at x1 = x2), which is a nontrivial feature of this OPE (obtained by considering 3-point functions). Thus, the expansion in twists can be viewed as a light-cone expansion of the Since the twist decomposition of the fields is conformally invariant then each Vκ will be behave, at least infinitesimally, as a scalar (κ, κ) density under con- formal transformations. Every Vκ is a complicated (formal) series in twist 2κ fields and their deriva- tives: Vκ(x1, x2) = Kµ1...µℓκ (x12, ∂x2)O µ1...µℓ (x2) , (2.12) where K µ1...µℓ κ (x12, ∂x2) are infinite formal power series in x12 with coefficients that are differential operators in x2 acting on the quasiprimary fields O. The important point here is that the series K µ1...µℓ κ (x12, ∂x2) can be fixed universally for any (even generally) conformal QFT. This is due to the universality of con- formal 3-point functions. The explicit form of K µ1...µℓ κ (x12, ∂x2) can be found in [6, 7] (see also [13]). Thus, we can at this point consider Vκ(x1, x2) only as generating series for the twist 2κ contributions to the OPE of φ(x1)φ(x2) but we still do not know whether these series would be convergent and even if they were, it would not be evident whether they would give bilocal fields. In the next section we will see that this is true for the leading, twist two part under certain conditions, which are automatically fulfilled for d = 2. The higher twist parts Vκ (κ > 1) are certainly not convergent to Huygens bilocal fields, since their 4-point functions, computed in [13], are not rational. 3Quasiprimary fields transform irreducibly under conformal transformations. Harmonic Bilocal Fields 7 The major difference between the twist two tensor fields and the higher twist fields is that the former satisfy conservation laws: ∂xµ1O µ1...µℓ (x) = 0 (ℓ > 1) . (2.13) This is a well known consequence of the conformal invariance of the 2-point func- tion and the Reeh–Schlieder theorem. It includes, in particular, the conservation laws of the currents and the stress–energy tensor. It turns out that V1(x1, x2) encodes in a simple way this infinite system of equations. Theorem 2.1. ([13]) The system of differential equations (2.13) is equivalent to the harmonicity of V1(x1, x2) in both arguments (bi–harmonicity) as a formal series, i.e., �x1V1(x1, x2) = 0 = �x2V1(x1, x2). The proof is based on the explicit knowledge of the K series in (2.12) and it is valid even if the theory is invariant under infinitesimal conformal transformations only. The separation of the twist two part in (2.11) amounts to a splitting of U of the form U(x1, x2) = V1(x1, x2) + ρ12 Ũ(x1, x2) . (2.14) This splitting can be thought in terms of matrix elements of U(x1, x2) expanded as a formal power series according to (2.9). It is unique by virtue of Theorem 2.1, due to the following classical Lemma: Lemma 2.2. ([3, 1]) Let u(x) be a formal power series in x ∈ C4 (or, CD) with coefficients in a vector space V . Then there exist unique formal power series v(x) and ũ(x) with coefficients in V such that u(x) = v(x) + x2 ũ(x) (2.15) and v(x) is harmonic in x (i.e., �x v(x) = 0). (2.15) is called the harmonic decomposition of u(x) (in the variable x around x = 0), and the formal power series v(x) is said to be the harmonic part of u(x). 3 Bilocality of twist two contribution to the OPE Let us sketch our strategy for studying bilocality of V1(x1, x2). The existence of the field V1(x1, x2) can be established by constructing its cor- relation functions. On the other hand, every correlation function4 ·V1(x1, x2) · 4This short-hand notation stands for ˛φ3(x3) · · ·φk(xk) V1(x1, x2) φk+1(xk+1) · · · φn(xn) , here and in the sequel. Harmonic Bilocal Fields 8 of V1 is obtained (originally, as a formal power series in x12) under the split- ting (2.14). It thus appears as a harmonic decomposition of the corresponding correlation function · U(x1, x2) · of U : · U(x1, x2) · · V1(x1, x2) · + ρ12 · Ũ(x1, x2) · . (3.1) Note that we should initially treat the left hand side of (3.1) also as a formal power series in x12 in order to make the equality meaningful. It is important that this series is always convergent as a Taylor expansion of a rational function in a certain domain around x1 = x2 in M , for the complexified Minkowski space MC = M + iM , according to the standard analytic properties of Wightman functions. We shall show in Sect. 3.1 that this implies the separate convergence of both terms in the right hand side of (3.1). Hence, the key tool in constructing V1 are the harmonic decompositions F (x1, x2) = H(x1, x2) + ρ12 F̃ (x1, x2) (3.2) of functions F (x1, x2) that are analytic in certain neighbourhoods of the diagonal {x1 = x2}. Recall that H in (3.2) is uniquely fixed as the harmonic part of F in x1 around x2, due to Lemma 2.2. This is equivalent to the harmonicity �x1 H(x1, x2) = 0. On the other hand, according to Theorem 2.1 we have to consider also the second harmonicity condition on H, �x2 H(x1, x2) = 0, i.e., H is the harmonic part in x2 around x1. This leads to some “integrability” conditions for the initial function F (x1, x2), which we study in Sect. 3.2. Next, to characterize the Huygens bilocality of V1, we should have rationality of its correlation functions · V1(x1, x2) · , which is due to a straightforward extension of the arguments of [14, Theorem 3.1]. But we have started with the correlation functions of U , which are certainly rational. Hence, we should study another condition on U , namely that its correlation functions have a rational harmonic decomposition. We show in Sect. 3.3 that this is equivalent to a simple condition on the correlation functions of U , which we call “Single Pole Property” (SPP). In this way we establish in Sect. 3.4 that V1 always exists as a Huygens bilocal field in the case of scalar fields of dimension d = 2. However, for higher scaling dimensions one cannot anymore expect that V1 is Huygens bilocal in general. This is illustrated by a counter-example, involving the 6-point function of a system of d = 4 fields, given at the end of Sect. 3.5. 3.1 Convergence of harmonic decompositions To analyze the existence of the harmonic decomposition of a convergent Taylor series we use the complex integration techniques introduced in [1]. Let MC =M + iM be the complexification of Minkowski space, which in this subsection is assumed to be D–dimensional, and E = x : (i x0, x1, . . . , xD−1) its Euclidean real submanifold, and SD−1 ⊂ E the unit sphere in E. We Harmonic Bilocal Fields 9 denote by ‖·‖ the Hilbert norm related to the fixed coordinates in MC: ‖x‖ |x0|2 + · · · + |xD−1|2. Let us also introduce for any r > 0 a real compact submanifold Mr of MC: ζ ∈MC : ζ = r e iθw, ϑ ∈ [0, π], w ∈ SD−1 (3.3) (note that ϑ ∈ [π, 2π] gives another parameterization of Mr). Then there is an integral representation for the harmonic part of a convergent Taylor series. Lemma 3.1. (cf. [1, Sect. 3.3 and Appendix A]) Let u(x) be a complex formal power series that is absolutely convergent in the ball ‖x‖ < r, for some r > 0, to an analytic function U(x). Then the harmonic part v(x) of u(x) (around x = 0), which is provided by Lemma 2.2, is absolutely convergent for |x2|+ 2 r ‖x‖ < r2. (3.4) The analytic function V (x) that is the sum of the formal power series v(x) has the following integral representation: V (x) = (z− x)2 U(z) , V1 = = iπ|SD−1|, (3.5) where r′ < r, |x2| + 2 r′ ‖x‖ < r′2, and the (complex) integration measure is obtained by the restriction of the complex volume form dDz (= dz0∧ · · · ∧dzD−1) on MC (∼= CD) to the real D–dimensional submanifold Mr′ (3.3), r′ > 0. Proof. Consider the Taylor expansion in x of the function 1 − x (z − x)2 and write it in the form (cf. [1, Sect. 3.3]) (z− x)2 (z2)− −ℓHℓ(z, x), Hℓ(z, x) = hℓµ(z)hℓµ(x), (3.6) where {hℓµ(u)} is an orthonormal basis of harmonic homogeneous polynomials of degree ℓ on the sphere SD−1. This expansion is convergent for ∣∣+ 2 ∣∣z · x ∣∣ (3.7) since its left–hand side is related to the generating function for Hℓ: 1− λ2 x2 y2 (1− 2λ x · y + λ2 x2 y2) λℓHℓ(x, y) , (3.8) the expansion (3.8) being convergent for λ 6 1 if |x2y2|+2|x ·y| < 1. Then if we fix r′ < r and z varies on Mr′ , a sufficient condition for (3.7) is |x 2| + 2 r′ ‖x‖ < r′2 (since sup w∈ SD−1 |w · x| = ‖x‖). Harmonic Bilocal Fields 10 On the other hand, writing u(z) = k=0 uk(z), where uk are homogeneous polynomials of degree k, we get by the absolute convergence of u(z) the relation (valid for |x2| + 2 r′ ‖x‖ < r′2) (z− x)2 U(z) = k,ℓ=0 (z2)− −ℓHℓ(x, z)uk(z) . (3.9) Noting next that in the parameterization (3.3) of Mr′ we have d = i r′D eiD ϑ dϑ ∧ dσ(w), where dσ(w) is the volume form on the unit sphere, we obtain for the right hand side of (3.9): k,ℓ=0 eiϑ(k−ℓ) dσ(w) |SD−1| Hℓ(x,w)uk(w) . Now if we write, according to Lemma 2.2, uk(z) = ck,j,µ′ (z 2)j hk−2j,µ′(z) then we get by the orthonormality of hℓ,µ(w) k,ℓ=0 δℓ,k−2j eiϑ(k−ℓ) ck,j,µ hk−2j,µ(x) ck,0,µ hk,µ(x) = v(x) . The latter proves both: the convergence of v(x) in the domain (3.4) (since r′ < r was arbitrary) and the integral representation (3.5). � As an application of this result we will prove now Proposition 3.2. For all n and k, and for all local fields φj (j = 3, . . . , n) the Taylor series ∣∣φ3(x3) · · · φk(xk) V1(x1, x2) φk+1(xk+1) · · · φn(xn) (3.10) in x12 converge absolutely in the domain ‖x12‖+ ‖x12‖2 + ∣∣x212 ‖x2j‖+ ‖x2j‖2 + ∣∣x22j ∣∣x22j ∣∣ ∀ j (3.11) (j = 3, . . . , n). They all are real analytic and independent of k for mutually nonisotropic points. Proof. Let Fk(x12, x23, . . . , x2n) ∣∣φ3(x3) · · ·φk(xk) U(x1, x2) φk+1(xk+1) · · · φn(xn) (3.12) Harmonic Bilocal Fields 11 be the correlation functions, analytically continued in x12. As Fk, which is a rational function, depends on x := x12 via a sum of products of powers (x− x2j) it has a convergent expansion in x for ∣∣+ 2 ∣∣x · x2j ∣∣x22j ∣∣ . (3.13) If we want Fk to have a convergent Taylor expansion for ‖x‖ < r we get the following sufficient condition ∣∣x22j ∣∣− 2 r ‖x2j‖. (3.14) By Lemma 3.1 we conclude that the series (3.10) is convergent for |x212|+ 2 r ‖x12‖ < r 2. (3.15) Combining both (sufficient) conditions (3.14) and (3.15) for r we find that they are compatible if ‖x12‖ + ‖x12‖2 + ∣∣x212 ‖x2j‖2 + ∣∣x22j ∣∣ − ‖x2j‖, which is equivalent to (3.11). � Note that one can also prove a similar convergence property for the correla- tion functions of several V1. Remark 3.1. The domain of convergence of (3.10) should be Lorentz invariant. Hence, (3.10) are convergent in the smallest Lorentz invariant set containing the domain (3.11). Such a set is determined by the values of the invariants x212, x and x12 · x2j and it turns out to be the set ∣∣x212 ∣∣ 12 ∣∣x22j ∣∣ 12 6 ∣∣x12 · x2j (∣∣x22j ∣∣ 12 − ∣∣x212 ∣∣ 12 )2 or equivalently √∣∣x212 ∣∣∣∣x22j ∣∣x12 · x2j ∣∣2 < (∣∣x22j ∣∣ 12 − ∣∣x212 ∣∣ 12 )2 . (3.16) Outside the domain of convergence (3.16), the correlations of V1(x1, x2) have to be defined by analytic continuation. When the correlations are rational, V1 is Huygens bilocal, but the counter-example presented in Sect. 3.5 shows that rationality is not automatic. Then, it is not even obvious that the continuations are single–valued within the tube of analyticity required by the spectrum con- dition, i.e., that V1 exists as a distribution in all of M × M . Nontrivial case studies, however, show that at least for xk space–like to both x1 and x2, the continuation is single–valued and preserves the independence on the position k in (3.10) where V1(x1, x2) is inserted. This leads us to conjecture Conjecture 3.3. The twist two field V1(x1, x2), whose correlations are defined as the analytic continuations of the harmonic parts of those of U(x1, x2), exists and is bilocal in the ordinary sense, i.e., it commutes with φ(x) and V1(x, x ′) if x and x′ are space–like to x1 and x2. We hope to return to this conjecture elsewhere (see also the Note added in proof). Note that the argument that locality implies Huygens locality [14] does not pass to bilocal fields. Harmonic Bilocal Fields 12 3.2 Consequences of bi–harmonicity Now our objective is to find the harmonic decomposition of the rational functions F (x1, x2) that depend on x1 and x2 through the intervals ρik = (xi−xk) 2, i = 1, 2, k = 3, . . . , n, for some additional points x3, . . . , xn. The F ’s, as correlation functions of U(x1, x2), have the form F (x1, x2) = (ρ12) q Fq(x1, x2) ≡ (ρ12) {ρik}{i,k}6={1,2} , (3.17) Fq(x1, x2) = {µ1i},{µ2i} Cq,{µ1j},{µ2j} (ρ1j) (ρ2i) µ2j , (3.18) where M ∈ N and µ1j , µ2j (j = 3, . . . , n) are integers > −d such that j>3 µ1j j>3 µ2j = −1 − q, and the coefficients Cq,{µ1j},{µ2j} may depend on ρjk (j, k > 3). IfH is the harmonic part of F in x12, then the leading part F0 (of order (ρ12) is also the leading part of H. We shall now proceed to show that bi–harmonicity of H (Theorem 2.1), together with the first principles of QFT including GCI, implies strong constraints on F0. Proposition 3.4. Let F0(x1, x2) be as in (3.18), and let H(x1, x2) be its har- monic part with respect to x1 around x2. Then H is also harmonic with respect to x2, if and only if F0 satisfies the differential equation (E1D2 − E2D1)F0 = 0, (3.19) where E1 = i=3 ρ2i∂1i (with ∂jk = ∂kj = ), D1 = 36j<k6n ρjk∂1j∂1k, and similarly for E2 and D2, exchanging 1 ↔ 2. Proof. By Proposition 3.2 (see also Remark 3.1) we can consider H as a function in the 2n − 3 variables ρ1i, ρ2i (i > 3) and ρ12, analytic in some domain that includes ρ12 = 0. Expanding H = q(ρ12) qHq/q!, the functions Hq are homogeneous of degree −1 − q in both sets of variables ρ1i and ρ2i, and H0 = F0. To impose the harmonicity with respect to the variable x1, we use the identity [11, App. C] �x1F = −4 26i<j6n ρij ∂1i∂1j F ρij =(xi−xj)2 , (3.20) valid for homogeneous functions of ρ1i of degree −1, to express the wave operator �x1 as a differential operator with respect to the set of variables ρ1i (i > 2). This yields the recursive system of differential equations E1Hq+1 = −D1Hq. (3.21) Performing the same steps with respect to the variable x2, one obtains E2Hq+1 = −D2Hq. (3.22) Harmonic Bilocal Fields 13 Eq. (3.19) then arises as the integrability condition for the pair of inhomogeneous differential equations for H1 (putting q = 0), observing that E2E1 − E1E2 =∑ ρ1i∂1i − ρ2i∂2i vanishes on H1 by homogeneity. Conversely, if (3.19) is fulfilled, thenH1 exists and satisfies (D1E2−D2E1)H1 = −(D1D2 − D2D1)H0 = 0 because D1 and D2 commute. But this is equivalent to (D2E1 − D1E2)H1 = 0, which is in turn the integrability condition for the existence of H2, and so on. It follows that bi–harmonicity imposes no further conditions on the leading function H0 = F0. � The differential equation (3.19) imposes the following constraints on the lead- ing part F0 of the rational correlation function F (3.17): Corollary 3.5. Assume that the function F0 as in (3.18) satisfies the differential equation (3.19). Then (i) If F0 contains a “double pole” of the form (ρ1i) µ1i(ρ1j) µ1j with i 6= j and µ1i and µ1j both negative, then its coefficients must be regular in ρ2k (k 6= i, j). (ii) F0 cannot contain a “triple pole” of the form (ρ1i) µ1i(ρ1j) µ1j (ρ1k) µ1k with i, j, k all different and µ1i, µ1j , µ1k all negative. The same hold true, exchanging 1 ↔ 2. Proof. Pick any variable, say ρ2k, and decompose F0 = r>−p(ρ2k) rfr as a Lau- rent polynomial in ρ2k. The differential equation (3.19) turns into the recursive system ρij∂1i∂1j − i,j 6=k ρ2iρkj∂1i∂2j  r · fr = Xrfr−1 + Y fr of differential equations for the functions fr which are Laurent polynomials in the remaining variables. The precise form of the polynomial differential operators Xr and Y does not matter. Assume the lowest power −p of ρ2k to be negative. For r = −p, the right-hand-side vanishes. Because the term ρij∂1i∂1j on the left-hand-side would produce a singularity that cannot be cancelled by any other term, f−p cannot have a “double pole” in any pair of variables ρ1i, ρ1j with i 6= j and i, j 6= k. This property passes recursively to all fr with r < 0, because also the right-hand-side never can contain such a pole. This implies that a double pole in a pair of variables ρ1i, ρ1j with i 6= j cannot multiply a term that is singular in ρ2k unless k = i or k = j, proving (i). If the coefficient of the double pole were singular in ρ1k, k 6= i, j, then the resulting double pole in the pair ρ1i, ρ1k resp. ρ1j , ρ1k would imply regularity also in ρ2j resp. ρ2i. Hence the coefficient of a triple pole must be regular in all ρ2m, which contradicts the total homogeneity −1 of F0 in these variables. This proves the statement (ii). � 3.3 A necessary and sufficient condition for Huygens bilocality Definition 3.1. (“Single Pole Property”, SPP) Let f(x1, . . . , xn) be a Laurent polynomial in the variables ρij , i.e., regarded as a function of x1 only, it is a Harmonic Bilocal Fields 14 finite linear combination of functions of the form (ρ1j) µ1j ≡ (x1 − xj) , (3.23) where µ1j (j > 2) are integers and the coefficients may depend on the parameters ρjk (j, k > 2). Then f is said to satisfy the single pole property with respect to x1 if it contains no terms for which there are j 6= k (j, k > 2) such that both µ1,j and µ1,k are negative. The significance of SPP stems from the fact that the harmonic parts H of F0, i.e., the correlation functions of V1, are again Laurent polynomials if and only if F0 satisfies the SPP. Namely, if H is a harmonic Laurent polynomial, the same argument as in [11, Lemma C.1] (using the representation (3.20) of the wave operator) shows that H fulfils the SPP with respect to x1, and so does F0, because it is the leading part of order (ρ12) 0 of H. The converse is an immediate consequence of Lemma 3.6 (allowing for a relabelling and multiple counting of the points x3, . . . , xn, which are not required to be distinct). Lemma 3.6. Let n > 4. Every finite linear combination of monomials of the gn(x1) = i=4 ρ1i (ρ13) i=4(x1 − xi) [(x1 − x3)2]n−2 (3.24) has a rational harmonic decomposition in x1 around x2 gn(x1) = hn(x1) + (x1 − x2) 2 · g̃n(x1) (3.25) i.e., hn is harmonic with respect to x1 and g̃n is regular at x1 = x2, and both hn and g̃n are rational. More precisely, (ρ13) n−2(ρ23) n−3hn is a homogeneous polynomial of total degree 2(n − 3) in the variables {ρij : 1 6 i < j}, which is separately homogoneous of degree n− 3 in the variables {ρ1i : i > 2} and in the variables {ρ12, ρ2i : i > 3}. Proof. It is convenient to introduce the variables ρ1iρ23 ρ13ρ2i , si = ρ12ρ3i ρ13ρ2i , uij = ρ12ρ23ρij ρ13ρ2iρ2j (4 6 i < j 6 n). (3.26) We claim that hn(x1) is of the form hn(x1) = fn(ti, si, uij) , (3.27) where fn are polynomials of degree n − 3 such that fn(ti, si = 0, uij = 0) =∏n i=4 ti. Because all si and uij contain a factor ρ12, these properties ensure that g̃n given by (gn − hn)/ρ12 is regular in ρ12. Harmonic Bilocal Fields 15 Using again the identity (3.20) for the wave operator, and transforming this into a differential operator with respect to the set of variables (3.26), we find �x1 hn(x1) = −4 (ρ13) ·Dfn(ti, si, uij), (3.28) where D is the differential operator D = (1 + t∂t + s∂s + u∂u)(s∂t + s∂s + u∂u)− (s∂s + u∂u)∂t − u∂t∂t (3.29) with shorthand notations for degree-preserving operators t∂t = ti∂ti , s∂t = si∂ti , s∂s = si∂si , u∂u = 46i<j6n uij∂uij and degree-lowering operators ∂ti , u∂t∂t = 46i<j6n uij∂ti∂tj . To solve the condition Dfn = 0 for harmonicity, we make an ansatz fn(ti, si, uij) = (sk, ukl) · i∈N\K (ti − si), where N ≡ {4, . . . , n}, g are polynomials in the variables sk, ukl (k, l ∈ K) only, and g = 1. Then the harmonicity condition Dfn = 0 is equivalent to the recursive system (n− 2− |K|+∆)∆ g K\{k} k,l∈K,k<l (ukl − sk − sl) g K\{k,l} where |K| is the number of elements of the set K and the differential operator ∆ = s∂s + u∂u measures the total polynomial degree r in sk and ukl. Since one can divide by (n − 2− |K|+ r)r if r > 0, there is a unique polynomial solution such that g (sk = 0, ukl = 0) = 0 (K 6= ∅), and g is of order 6 |K|. So fn is of order n − 3. (Explicitly, the first three functions are f3 = 1, f4 = t4 − s4 and f5 = (t4 − s4)(t5 − s5) + (u45 − s4 − s5).) An inspection of the recursion also shows that all possible factors ρ2i in the denominators of the arguments of fn cancel with the factors in the prefactor in (3.27), thus hn can have poles only in ρ13 and ρ23 of the specified maximal degree. This proves the Lemma. � The upshot of the previous discussion is a necessary and sufficient condition for the Huygens bilocality of V1 which directly refers to the local correlation functions of the theory: Harmonic Bilocal Fields 16 Theorem 3.7. The field V1(x1, x2) weakly converges on bounded energy states to a Huygens bilocal field which is conformal of weight (1, 1), if and only if the leading parts F0 of the Laurent polynomials F (3.17) satisfy the “single pole property” (Def. 3.1) with respect to both x1 and x2. In this case, the formal series H converge to Laurent polynomials in (xi − xj) 2 subject to the same pole bounds, specified in Theorem 2.1, as F . Proof. We know already that if V1 is a Huygens bilocal field, then its correlation functions H are Laurent polynomials of the form (2.3), and that this implies the SPP for F0 with respect to x1 and x2. Conversely, if the SPP holds for F0 with respect to x1 and x2, then H are Laurent polynomials by Lemma 3.6, and hence V1 is relatively Huygens bilocal with respect to the fields φi. Since the general argument [4] that relative locality implies local commutativity of a field with itself refers only to local fields, we want to give an explicit argument for the case at hand. All the previous remains true when in (3.10) or (3.17) a product of fields φk(xk)k+1φ(xk+1) is replaced by U(xk, xk+1). By assumption, and because U is bilocal, the contributions of order (ρk,k+1) 0 to the correlation functions of U(xk, xk+1) fulfil the SPP with respect to xk and xk+1. By Lemma 3.6, this property is preserved upon the passage to the harmonic parts with respect to x1 and x2. One may therefore continue in the same way with xk, xk+1, and eventually find that all mixed correlation functions of φ’s and V1’s converge to rational functions. By this convergence we conclude that all products of φ’s and V1’s converge on the vacuum, and this then defines V1 as a Huygens bilocal field, since its matrix elements will satisfy Huygens locality. The conformal properties of V1 follow from the preservation of the homo- geneity and the pole degrees in the harmonic decomposition, as guaranteed by Lemma 3.6. � For n = 4 points, the SPP is trivially satisfied because of homogeneity. Hence the 4-point function ∣∣V ∗1 V1 is always rational. It follows that its expansion in (transcendental) partial waves [11] cannot terminate. This means that (unless V1 = 0 in which case there is not even a stress-energy tensor) a GCI QFT necessarily contains infinitely many conserved tensor fields of arbitrarily high spin. 3.4 The case of dimension 2 Let us consider now the case of scalar fields φk of dimension 2. We claim that in this case, Corollary 3.5 in combination with the cluster condition is sufficient to establish the SPP, Definition 3.1. Hence we conclude by Theorem 3.7 that the twist two harmonic fields V1(x1, x2) are indeed Huygens bilocal fields. To prove our claim, we use that by (2.6), µij > −1, hence the SPP is equiv- alent to the statement that there can be no term contributing to ∣∣φ1(x1) · · · φn(xn) , for which there is i with more than two µij negative (j 6= i). Thus assume that there is a term with, say, µ12 = µ13 = µ14 = −1. It constitutes a Harmonic Bilocal Fields 17 double pole for each of the three harmonic fields V1(x1, xj) (j = 2, 3, 4). Then by homogeneity (2.5), there must be more poles in xj (j = 2, 3, 4), but these cannot be of the form ρjk with k > 4 by Corollary 3.5. Hence (up to permutations of 2, 3, 4) µ23 = µ24 = −1, µ34 = 0. Again by homogeneity (2.5), the dependence on x1, . . . , x4 must be given by a linear combination of terms ρ1kρ4ℓ ρ12ρ13ρ14ρ23ρ24 (3.30) with k, ℓ > 4. Applying the cluster limit (Sect. 2.1) to the points x1, x2, x3, x4 in (3.30), the limit diverges ∼ t4. This behavior is tamed to ∼ t2 by anti– symmetrization in k, ℓ, but it cannot be cancelled by any other terms. Hence the assumption leads to a contradiction. This proves the SPP if the generating scalar fields have dimension d = 2. 3.5 A d = 4 6-point function violating the SPP We proceed with an example of 6-point function violating the SPP in the case of two d = 4 GCI scalar fields Li(x) such that the bilocal field U(x1, x2) obtained from L1(x1)L2(x2) has a non-zero skew–symmetric part. Let L be any linear combination of L1 and L2. The following admissible contribution to the truncated part of the 6-point function 〈0|U(x1, x2)L(x3)L(x4)U(x5, x6)|0〉 clearly violates the SPP: F0(x1, x2) = A12A56 ρ15ρ26ρ34 − 2ρ15ρ23ρ46 − 2ρ15ρ24ρ36 ρ13ρ14ρ23ρ24 · ρ34 · ρ35ρ45ρ36ρ46 , (3.31) where Aij stands for the antisymmetrization in the arguments xi, xj . It is ad- missible as a truncated 6-point structure because (ρ12ρ56) −3F0 obeys all the pole bounds of Sect. 2 for a correlation 〈0|L1(x1)L2(x2)L(x3)L(x4)L1(x5) L2(x6)|0〉 of six fields of dimension d = 4. On the other hand, F0 satisfies the differential equation (E1D2 − E2D1)F0(x1, x2) = 0 (3.32) (and similar in the variables x5 and x6), ensuring that F0 is the leading part of a bi–harmonic function, analytic in a neighborhood of x1 = x2 and x5 = x6, repre- senting a contribution to the twist two 6-point function 〈0|V1(x1, x2)L(x3)L(x4) V1(x5, x6)|0〉, of which F0 is the leading part. This function cannot be a Laurent polynomial in the ρij by our general argument that the leading part of a bi– harmonic Laurent polynomial cannot satisfy the SPP. Hence the twist two field V1(x1, x2) cannot be Huygens bilocal. The resulting contribution to the conserved local current 4-point function 〈0|Jµ(x1)L(x3)L(x4)Jν(x5)|0〉 tr is obtained through Jµ(x) = i(∂xµ−∂yµ) V1(x, y)|x=y. It also satisfies the pertinent pole bounds. This structure is rational as it should, because only the leading part F0 contributes. In fact, while the 6-point structure involving the harmonic field cannot be reproduced by free fields due to its double Harmonic Bilocal Fields 18 pole, the resulting 4-point structure does arise as one of the three independent connected structures contributing to 4-point functions involving two Dirac cur- rents : ψ̄aγ µψb : and two Yukawa scalars ϕ : ψ̄cψd : (allowing for internal flavours a, b, . . . ). 4 The theory of GCI scalar fields of scaling dimension d = 2 The scaling dimension d = 2 is the minimal dimension of a GCI scalar field for which one could expect the existence of nonfree models. It turns out however, that in this case the fields can be constructed as composite fields of free, or generalized free, fields. Namely, we will establish the following result. Theorem 4.1. Let {Φm(x)} m=1 be a system of real GCI scalar fields of scaling dimension d = 2. Then it can be realized by a system of generalized free fields {ψm(x)} and a system of independent real massless free fields {ϕm(x)}, acting on a possibly larger Hilbert space, as follows: Φm(x) = αm,j ψj(x) + j,k=1 βm,j,k :ϕj(x)ϕk(x) : , (4.1) where αm,j and βm,j,k = βm,k,j are real constants such that α2m,j < ∞ and j,k=1 m,j,k < ∞. Here, we assume the normalizations 〈0|ϕj(x1) ϕk(x2) |0〉 = δjk (ρ12) −1, 〈0|ψj(x1) ψk(x2) |0〉 = δjk (ρ12) The proof of Theorem 4.1 is given at the end of Sect. 4.2. The main reason for this result is the fact that in the d = 2 case the harmonic bilocal fields exist and furthermore, they are Lie fields. This was originally recognized in [12], [2] under the assumption that there is a unique field φ of dimension 2. We are extending here the result to an arbitrary system of d = 2 GCI scalar fields. If we assume the existence of a stress-energy tensor as a Wightman field5, the generalized free fields must be absent in (4.1), and the number of free fields must be finite. In this case, the iterated OPE generates in particular the bilocal field 1 i :ϕi(x)ϕi(y):. As this field has no other positive-energy representation than those occurring in the Fock space [2], nontrivial possibilities for correlations between non-free fields and the fields (4.1) are strongly limited. 4.1 Structure of the correlation functions We consider a GCI QFT generated by a set of hermitean (real) scalar fields. We denote by F the real vector space of all GCI real scalar fields of scaling 5A stress-energy tensor always exists as a quadratic form between states generated by the fields Φm from the vacuum [8]. Harmonic Bilocal Fields 19 dimension 2 in the theory. (Note that the space F may be larger than the linear span of the original system of d = 2 fields of Theorem 4.1.) We shall find in this section the explicit form of the correlation functions of the fields from F . Theorem 4.2. Let φ1(x), . . . , φn(x) ∈ F then their truncated n-point functions have the form 〈0|φ1(x1) · · · φn(xn)|0〉 c(n)(φσ1 , . . . , φσn) ρσ1σ2 · · · ρσnσ1 (4.2) where c(n) are multilinear functionals c(n) : F⊗n → R with the inversion and cyclic symmetries c(n)(φ1, . . . , φn) = c (n)(φn, . . . , φ1) = c (n)(φn, φ1, . . . , φn−1). Before we prove the theorem, let us first illustrate it on the example of the free field realization (4.1). In this case one finds Φm1 ,Φm2 αm1,jαm2,j + j,k=1 βm1,j,k βm2,j,k αm1,jαm2,j + Tr βm1βm2 , Φm1 , . . . ,Φmn = Tr βm1 · · · βmn for n > 2 (4.3) where βm = βm,j,k Proof of Theorem 4.2. We first recall the general form (2.3) of the truncated correlation function with pole bounds (2.6) that read in this case: µtrjk > −1. The argument in Sect. 3.4 shows that the nonzero contributing terms in Eq. (2.3) have for every j = 1, . . . , n exactly two negative µtrjk or µ kj for some k = k1, k2 different from j. The nonzero terms are therefore products of “disjoint cyclic products of prop- agators” of the form 1/ρk1k2ρk2k3 · · · ρkr−1krρkrk1 . But cycles of length r < n are in conflict with the cluster condition (Sect. 2). We conclude that 〈0|φ1(x1) · · · φn(xn)|0〉 tr is a linear combination of terms like those in (4.2) with some co- efficients cσ(φ1, . . . , φn) depending on the permutations σ ∈ Sn and on the fields φj (multilinearly). Locality, i.e. 〈0|φ1(x1) · · · φn(xn)|0〉 tr = 〈0|φσ′1(xσ ) · · · φσ′n(xσ′n)|0〉 tr, then implies cσ′σ(φ1, . . . , φn) = cσ(φσ′1 , . . . , φσ′n) (σ, σ ′ ∈ Sn), so that cσ(φ1, . . . , φn) = c (n)(φσ1 , . . . , φσn) for some c (n) : F⊗n → R. The equali- ties c(n)(φ1, . . . , φn) = c (n)(φn, . . . , φ1) = c (n)(φn, φ1, . . . , φn−1) are again due to locality. � As we already know by the general results of the previous section, the har- monic bilocal field exist in the case of fields of dimension d = 2. Moreover, the knowledge of the correlation functions of the d = 2 fields allows us to find the form of the correlation functions of the resulting bilocal fields. This yields an algebraic structure in the space of real (local and bilocal) scalar fields, which we proceed to display. Harmonic Bilocal Fields 20 Let us introduce together with the space F of d = 2 fields also the real vector space V of all real harmonic bilocal fields. We shall consider F and V as built starting from our original system of d = 2 fields {Φm} of Theorem 4.1, by the following constructions. (a) If φ1(x), φ2(x) ∈ F then introducing the bilocal (1, 1)–field U(x1, x2) = φ1(x1)φ2(x2)− 〈0|φ1(x1)φ2(x2)|0〉 in accord with Eq. (2.7), we consider its harmonic decomposition U(x, y) = V1(x, y) + (x−y) 2 Ũ(x, y). We denote V1(x, y) by φ1 ∗ φ2; this defines a bilinear map F ⊗ F (b) If now v(x, y) ∈ V then vt(x, y) := v(y, x) also belongs to V and γ(v) is a field from F . (c) If v(x, y), v′(x, y) ∈ V then there is a harmonic bilocal field (v∗v′) := w-lim x′ → y′ x′−y′ x, x′ y′, y −〈0|v x, x′ y′, y . (4.4) The existence of the above weak limit (i.e., a limit within correlation functions) will be established below together with the independence of x′ = y′ and the regularity of the resulting field for (x− y)2 = 0. (d) If v(x, y) ∈ V and φ(x) ∈ F then we can construct the following bilocal field belonging to V: (v ∗ φ) := w-lim x′ → y x′ − y x, x′ − 〈0|v x, x′ , (4.5) where again the existence of the limit and the regularity for (x− y)2 = 0 will be established later. One can define similarly a product φ ∗ v ∈ V, but it would then be expressed as: (vt ∗ φ)t. To summarize, we have three bilinear maps: F ⊗ F →V, V ⊗ V →V, V ⊗ →V, and two linear ones: V →V, V →F . Applying these maps we construct F and V inductively, starting from our original system of d = 2 fields, given in Theorem 4.1, and at each step of this inductive procedure, we establish the existence of the above limits in (c) and (d). In fact, we shall establish this together with the structure of the truncated correlation functions for the fields in F and V.6 Before we state the inductive result it is convenient to introduce the vector space  = F × V (4.6) 6Since we shall use the notion of truncated correlation functions also for bilocal fields let us briefly recall it. If B1, . . . , Bn are some smeared (multi)local fields then their truncated correlation functions are recursively defined by: 〈0|B1 · · ·Bn|0〉 = ∪̇P = {1,...,n} {j1,...,jk} ∈P 〈0|Bj1 · · ·Bjk |0〉 tr (the sum being over all partitions P of {1, . . . , n}) Harmonic Bilocal Fields 21 and endow it with the following bilinear operation (φ1, v1) ∗ (φ2, v2) := 0, φ1 ∗ φ2 + v1 ∗ v2 + v1 ∗ φ2 + (v 2 ∗ φ1) , (4.7) and with the transposition (φ, v)t := (φ, vt) . (4.8) The spaces F and V will be considered as subspaces in Â. Thus, the new operation ∗ in  combines the above listed three operations. We shall see later that  is actually an associative algebra under the product (4.7). We note that the transposition t (4.8) is an antiinvolution with respect to the product: (q1 ∗ q2) t = qt2 ∗ q 1, for every q1, q2 ∈ Â. Proposition 4.3. There exist multilinear functionals c(N) : Â⊗N → R (4.9) such that if we take elements q1, . . . , qn+m ∈  : qk := vk xk[0], xk[1] ∈ V, where [ε] stands for a Z/2Z–value and k = 1, . . . , n, and qk := φk−n ∈ F for k = n+ 1, . . . , n +m, then the truncated correlation functions can be written in the following form: 〈0|v1 x1[0], x1[1] · · · vn xn[0], xn[1] · · ·φm |0〉tr 2(n+m) σ∈Sn+m (ε1,...,εn)∈ (Z/2Z) Kσ,ε Tσ,ε x1[0], . . . , xn[1], xn+1, . . . , xn+m . (4.10) Here: Kσ,ε are coefficients given by Kσ,ε := c (n+m) [εσ1 ] σ1 , . . . , q [εσn+m ] , where we set εn+1 = · · · = εn+m = 0, and q [0] := q, q[1] := qt (for q ∈ Â); the terms Tσ,ε are the following cyclic products of intervals Tσ,ε = xσn+m − xσ1[ε1] xσk[1+εk] − xσk+1[εk+1] xσn[1+εn] − xσn+1 xσn+k − xσn+k+1 . (4.11) It follows by Eq. (4.10) that the limits in the steps (c) and (d) above are well defined. Before the proof let us make some remarks. First, we used the same notation c(n) as in Theorem 4.2 since the above multilinear functionals are obviously an extension of the previous, i.e., Eq. (4.10) reduces to Eq. (4.2) for m = 0. Let us also give an example for Eq. (4.10) with n = m = 1: 〈0|v(x1, x2)φ(x3)|0〉 = c(2)(v, φ) ρ23 ρ31 + c(2)(vt, φ) ρ13 ρ32 + c(2)(φ, v) ρ31 ρ23 + c(2)(φ, vt) ρ32 ρ13 . (4.12) Harmonic Bilocal Fields 22 As one can see, c(n) (as well as c(n) of Theorem 4.2) possess a cyclic and an inversion symmetry: q1, . . . , qn = c(n) qn, q1 . . . , qn−1 = c(n) qtn, . . . , q . (4.13) This is the reason for choosing the prefactors in Eqs. (4.2) and (4.10) (the inverse of the orders of the symmetry groups). Proof of Proposition 4.3. According to our preliminary remarks it is enough to prove that Eq. (4.10) is consistent with the operations F ⊗ F →V, V ⊗ V V ⊗ F →V and V Starting with F ⊗ F →V one should prove that any truncated correlation function ·φ1(x1)φ2(x2)· given by Eq. (4.10) yields a harmonic decomposition: ·φ1(x1)φ2(x2) · ·(φ1∗φ2) x1, x2 +ρ12R(x1, x2), with a correlation function · (φ1 ∗ φ2) x1, x2 given by Eq. (4.10) and a rational function R regular at ρ12 = 0. This gives us relations of the type c(n+2)(q1, . . . , φ1, φ2, . . . , qn) = c (n+1)(q1, . . . , φ1 ∗ φ2, . . . , qn) . (4.14) Next, having correlation functions of type · v1(x1, x2)v2(x3, x4) · v(x1, x2)φ(x3) · of the form (4.10), one verifies that the limits (4.4) and (4.5) exist within these correlation functions, and they yield expressions for · (v1 ∗ x1, x4 · (v ∗ φ) x1, x3 consistent with (4.10). As a result we obtain again relations between the c’s: c(n+2)(q1, . . . , v1, v2, . . . , qn) = c (n+1)(q1, . . . , v1 ∗ v2, . . . , qn) , c(n+2)(q1, . . . , v, φ, . . . , qn) = c (n+1)(q1, . . . , v ∗ φ, . . . , qn) . (4.15) Finally, one verifies that setting x1 = x2 in · v(x1, x2) · we obtain the correlation functions · γ(v) with the relation c(n+1)(q1, . . . , (v + v t), . . . , qn) = 2 c (n+1)(q1, . . . , γ(v), . . . , qn) . (4.16) This completes the proof of Proposition 4.3 as well as the proof that the products V ⊗ V →V and V ⊗ F →V are well defined. � 4.2 Associative algebra structure of the OPE Note that Eqs. (4.14), (4.15) read (under (4.7)) q1, . . . , qk, qk+1, . . . , qn = c(n−1) q1, . . . , qk ∗ qk+1, . . . , qn . (4.17) This implies that the bilinear operation ∗ on  is an associative product. Indeed, consider the element q := q1 ∗q2 ∗q3−q1∗ q2 ∗q3 for q1, q2, q3 ∈ Â. By (4.7) q is a bilocal field. Equation (4.17) implies that all c’s in which q enters vanish and hence, by Eq. (4.10) q has zero correlation functions with all other Harmonic Bilocal Fields 23 fields, including itself. But then this (bilocal) field is zero by the Reeh–Schlieder theorem, since its action on the vacuum will be identically zero. Thus, introducing the cartesian product  (4.6) was not only convenient for combining three types of bilinear operations in one but also as a compact expres- sion for the associativity (Eqs. (4.14), (4.15)). However,  carries a redundant information due to the following relation: −γ(v), (v + vt) ∗ q = 0 = q ∗ −γ(v), (v + vt) (4.18) for every v ∈ V and q ∈ Â. To prove (4.18) we point out first that it is equivalent to the identities v ∗ φ = γ(v) ∗ φ and v′ ∗ v = v′ ∗ γ(v) for v = vt ∈ V and any φ ∈ F , v′ ∈ V. These identities can be established again first for the c’s, and then proceeding by using the Reeh–Schlieder theorem, as in the above proof of associativity. Hence, the redundancy in  is because we can identify symmetric bilocal fields v = vt ∈ V with their restrictions to the diagonal, γ(v) ∈ F , and this is compatible with the product ∗. Let us point out that the restriction of the map γ to the t–invariant subspace Vs := {v ∈ V : v = v t} is an injection into F . The latter follows from a simple analysis of the 4-point functions of v and the Reeh–Schlieder theorem: if v(x, y) = v(y, x) and 〈0|v(x, x)v(y, y)|0〉 = 0 then 〈0|v(x, x′)v(y, y′)|0〉 = 0. In this way we see that we can identify in  the symmetric harmonic bilocal fields v = vt with their restriction on the diagonal γ(v) ∈ F . Formally, the above considerations can be summarized in the following ab- stract way. Let us introduce the quotient A :=  −γ(v), (v + vt) : v ∈ V . (4.19) It is an associative algebra according to Eq. (4.18). The involution t :  →  can be transferred to an involution on the quotient (4.19) and we denote it by t as well. The spaces F and V are mapped into A by the natural compositions F →  → A and V →  → A. The injectivity of γ on Vs implies that the maps F → A and V → A so defined are actually injections. Hence, we shall treat F and V also as subspaces of A. Furthermore, A becomes a direct sum of vector spaces A = F ⊕ Va , (4.20) q ∈ A : qt = q = F ⊇ Vs v ∈ V : vt = v q ∈ A : qt = −q = Va := v ∈ V : vt = −v Hence, the t–symmetric elements of A are identified with the d = 2 local fields, while the t–antisymmetric elements of A, with the antisymmetric, harmonic bilocal (1, 1) fields. (Neither F nor Va are subalgebras of A.) To summarize, the associative algebra A is obtained from  by identifying the space Vs of symmetric bilocal fields with its image γ ⊆ F . Harmonic Bilocal Fields 24 For simplicity we will denote the equivalence class in A of an element q ∈  again by q. Also note that the c’s can be transferred as well, to multilinear functionals on A, since the kernel of the quotient (4.19) is contained in the kernel of each c(n) by (4.16). We shall use the same notation c(n) also for the multilinear functional c(n) on A. Example 4.1. Let us illustrate the above algebraic structures on the simplest example of a QFT generated by a pair of d = 2 GCI fields Φ1 and Φ2 given by normal a pair of two mutually commuting free massless fields ϕj : Φ1(x) = : ϕ21(x) : − : ϕ 2(x) : and Φ2(x) = ϕ1(x)ϕ2(x). Their OPE algebra involves a set of four independent harmonic bilocal fields Vjk(x1, x2) := :ϕj(x1)ϕk(x2) : (j, k = 1, 2), which satisfy Vjk(x1, x2) = Vkj(x1, x2) = Vjk(x2, x1). For in- stance, we have Φ1 ∗ Φ2 = V12 − V21. 7 Also note that Φ1 = γ(V1) for V1(x1, x2) = :ϕ1(x1)ϕ1(x2) : − :ϕ2(x1)ϕ2(x2) :, etc. By the associativity and Eq. (4.17) we have q1, . . . , qn = c(2) q1 ∗ · · · ∗ qn−1, qn (4.21) for q1, . . . , qn ∈ A. Let us consider now c (2) and define the following symmetric bilinear form on A: 〈 q1, q2 := c(2) qt1, q2 . (4.22) First note that F and Va are orthogonal with respect to this bilinear form: this is due to the fact that there is no nonzero three point conformally invariant scalar function of weights (2, 1, 1), which is antisymmetric in the second and third arguments. Next, we claim that (4.22) is strictly positive definite. This is a straightforward consequence of the Wightman positivity and the Reeh–Schlieder theorem (one should consider separately the positivity on F and Va). In partic- ular, (4.22) is nondegenerate. By Eqs. (4.13) and (4.17) we have: q1 ∗ q2, q3 q2, q 1 ∗ q3 (4.23) for all q1, q2, q3 ∈ A. Let us introduce now an additional splitting of F . Denote by F0 the kernel of the product, i.e., F0 := ψ ∈ F : ψ ∗ q = 0 ∀q ∈ A ψ ∈ F : q ∗ ψ = 0 ∀q ∈ A (4.24) (the second equality is due to the identity φ ∗ q = (qt ∗ φ)t). Let F1 be the orthogonal complement in F of F0 with respect to the scalar product (4.22): F1 := φ ∈ F : = 0 ∀ψ ∈ F0 . (4.25) The meaning of fields belonging to F0 becomes immediately clear if we note that c(n) for n > 3 are zero if one of the arguments belongs to F0 (this is due to 7i.e., in the OPE Φ1(x1)Φ2(x2) there appears the antisymmetric bilocal field V12(x1, x2) − V21(x1, x2) that involves only odd rank conserved tensor currents in its expansion in local fields Harmonic Bilocal Fields 25 Eq. (4.21)). Hence, all their truncated functions higher than two point are zero, i.e., the fields belonging to F0 are generalized free d = 2 fields. Furthermore, these fields commute with all other fields from F1 and Va ≡ A (1): this is because of the vanishing of c(2)(ψ, q) if ψ ∈ F0 and q ∈ F1 ⊕ Va, as well as of all c(n+1)(ψ, q1, . . . , qn) for n > 2 if ψ ∈ F0 and q1, . . . , qn ∈ A (by (4.21) and (4.24)). Clearly, F1⊕Va is a subalgebra of A: this follows from Eq. (4.23) with q3 ∈ F0 along with the definitions (4.24) and (4.25). Let us denote it by B := F1 ⊕ Va . (4.26) We are now ready to state the main step towards the proof of Theorem 4.1. Proposition 4.4. There is a homomorphism ι from the associative algebra B into the algebra of Hilbert–Schmidt operators over some real separable Hilbert space, such that q1, . . . , qn · · · ι , (4.27) and ι are symmetric operators while ι are antisymmetric. We shall give the proof of this proposition in the subsequent subsection. The main reason leading to it is that B becomes a real Hilbert algebra with an integral trace on it. Here we proceed to show how Theorem 4.1 can be proven by using the above results. Proof of Theorem 4.1. Let Φm = Φ m + Φ m be the decomposition of each field Φm according to the splitting F = F0 ⊕ F1. Take an orthonormal basis ψm in F0 and let Φ αm,j ψj , and βm = βm,j,k be the symmetric matrix corresponding to the Hilbert–Schmidt operator ι (m = 1, 2, . . . ). Then Eqs. (4.3) and (4.27) show that the constants αm,j and βm,j,k so defined satisfy the conditions of Theorem 4.1. � Remark 4.1. In general, we have F1 % Vs. This is because the elements of F1 correspond, by Proposition 4.4, to Hilbert–Schmidt symmetric operators and on the other hand, the elements of V are obtained, according to the inductive construction of Sect. 4.1, as products of elements of F and will, hence, correspond to trace class operators. 4.3 Completion of the proofs It remains to prove Proposition 4.4. We start with an inequality of Cauchy– Schwartz type. Lemma 4.5. Let q1, q2 ∈ A be such that each of them belongs either to F or to Va. Then we have q1 ∗ q2, q1 ∗ q2 q1 ∗ q1, q1 ∗ q1 q2 ∗ q2, q2 ∗ q2 . (4.28) Harmonic Bilocal Fields 26 Proof. Consider q1∗q1+λ q2∗q2, q1∗q1+λ q2∗q2 > 0 and use that q1∗q1, q2∗q2 q1 ∗ q2, q1 ∗ q2 if each of q1, q2 belongs either to F or to Va. � The space B (4.26) is a real pre–Hilbert space with a scalar product given by (4.22). It is also invariant under the action of t (actually the eigenspaces of t are F1 and Va). The left action of B on itself gives us an algebra homomorphism ι : B → LinR B (4.29) of B into the algebra of all operators over B. Moreover, the elements of F are mapped into symmetric operators and the elements of Va, into antisymmetric (this is due to (4.23)). Lemma 4.6. Every element of B is mapped into a Hilbert–Schmidt operator. Proof. Since B is generated by F1 (according to the inductive construction of F and V in Sect. 4.1) it is enough to show this for the elements of F1. Let φ ∈ F1 and consider the commutative subalgebra Bφ of B generated by φ. The algebra Bφ is freely generated by φ, i.e., is isomorphic to the algebra λR[λ] of polynomials in a single variable λ (↔ φ), since φ belongs to the orthog- onal complement of F0 (4.24). For a p(λ) ∈ λR[λ] we shall denote by φ[p] the corresponding element of Bφ. In particular, φ[p1] ∗ φ[p2] = φ[p1p2]. (4.30) Setting φ∗(n+1) := φ∗n ∗ φ, c := c(2) φ∗n, φ φ∗n, φ (4.31) (φ∗1 := φ, n > 1) we obtain a positive definite functional over the algebra λ2R[λ] ∼= φ ∗ Bφ (due to Eq. (4.23) and the positivity of (4.22)). Then, by the Hamburger theorem about the classical moment problem ([9, Chap. 12, Sect. 8]) we conclude that there exists a bounded positive Borel mea- sure dµ on R, such that λ2 p (λ) dµ(λ) (4.32) for every p(λ) ∈ R[λ]. Using this we can extend the fields φ[p](x) to φ[f ](x) for Borel measurable functions f having compact support with respect to µ in R\{0}. The latter can be done in the following way. Fix ε ∈ (0, 1) and let g1, . . . , gn be Schwartz test functions on M . By Theorem 4.2 the correla- tors 〈0|φ[p1][g1] · · ·φ [pn][gn]|0〉 depend polynomially on c φ[pk1 ], . . . , φ [pkj ] pk1 · · · pkj for all {k1, . . . , kj} ⊆ {1, . . . , n}. But for every ε ∈ (0, 1) there exists a norm ‖q‖ε = Aε sup |λ|6ε qk(λ) ∣∣∣ + Bε R \ (−ε, ε) ∣∣qk(λ) ∣∣ dµ(λ) (4.33) Harmonic Bilocal Fields 27 on λ2R[λ] ∋ q(λ), where Aε and Bε are some positive constants, such that for every q1, . . . , qm ∈ λ 2R[λ] ∣∣∣ c q1(λ) · · · qm(λ) ]∣∣∣ 6 ∣∣qk(λ) dµ(λ) ‖qk‖ε . Hence, ∣∣〈0|φ[p1][g1] · · ·φ[pn][gn]|0〉 ∣∣ 6 C ‖pk‖ε ‖gk‖S for some constant C and Schwartz norm ‖·‖S (not depending on pk and gk). Since for every ε ∈ (0, 1) the Banach space L1 R\{(−ε, ε)}, µ is contained in the completion of λ2R[λ] with respect to the norms (4.33), we can extend the linear functional c[p(λ)] as well as the correlators 〈0|φ[p1][g1] · · · φ [pn][gn]|0〉 to a functional c[f(λ)] and correlators 〈0|φ[f1][g1] · · ·φ [fn][gn]|0〉 defined for Borel functions f, f1, . . . , fn compactly supported with respect to µ in R\{0}. Thus, we can extend the fields φ[p] by extending their correlators. By the continuity we also have for arbitrary Borel functions f, fk, compactly supported in R\{0}: φ[f1] ∗ φ[f2] = φ[f1f2], c(n) φ[f1], . . . , φ[fn] f1 · · · fn f ] = dµ(λ) (4.34) (cp. (4.32)), and c(n) determine the correlation functions of φ[fk] as in Theo- rem 4.2. In particular, for every characteristic function χS of a compact subset S ⊂ R\{0} we have φ[χS ]∗φ[χS ] = φ[χS ]. Hence, for such a d = 2 field we will have that all its truncated correlation functions are given by (4.2) with all normalization constants c(n) equal to one and the same value c(2) φ[χs], φ[χs] . Then, as shown in [12, Theorem 5.1], Wightman positivity requires this value to be a non-negative integer, i.e., φ[χS ], φ[χS ] dµ(λ) ∈ {0, 1, 2, . . . } (4.35) (it is zero iff φ[χS ] = 0). Hence, the restriction of the measure dµ(λ)/λ2 to R\{0} is a (possibly infinite) sum of atom measures of integral masses, each supported at some γk ∈ R\{0} for k = 1, . . . , N (and N could be infinity). In particular, the measure µ is supported in a bounded subset of R. By Lemma 4.5 we can define ι(φ[f ]) as a closable operator on B if f is a Borel measurable function with compact support in R\{0}. It follows then that the projectors ι(φ[χS ]), for a compact S ⊆ R\{0}, provide a spectral decomposition for ι(φ) (in fact, ι(φ[f ]) = f ). Thus, ι(φ) has discrete spectrum with eigen- values γk (k ∈ N), each of a multiplicity given by the integer c χ{γk} , φ χ{γk} Then ι(φ) is a Hilbert–Schmidt operator since γ2k c χ{γk} , φ χ{γk} dµ(λ) R\{0} dµ(λ) <∞ Harmonic Bilocal Fields 28 (µ being a bounded measure). � The completion of the proof of Proposition 4.4 is provided now by the fol- lowing corollary. Corollary 4.7. For every q1, q2 ∈ B one has c q1, q2 ι(q1)ι(q2) Proof. If q1 = q2 ∈ F1 this follows from the proof of Lemma 4.6 and hence, by a polarization, for any q1, q2 ∈ F1. The general case can be obtained by using the facts that B is generated by F1 and c (2) has the symmetry c(2)(q1 ∗ q2, q3) = c(2)(q1, q2 ∗ q3). � 5 Discussion. Open problems The main result of Sect. 4, the (generalized) free field representation of a system {φa} of GCI scalar fields of conformal dimension d = 2 (Theorem 4.1), is obtained by revealing and exploiting a rich algebraic structure in the space F × V of all d = 2 real scalar fields and of all harmonic bilocal fields of dimension (1, 1). However, this structure is mainly due to the fact that we are in the case of lower scaling dimension: there is only one possible singular structure in the OPE (after truncating the vacuum part). One can try to establish such a result in spaces of spin–tensor bilocal fields (of dimension ) satisfying linear (first order) conformally invariant differential equations (that again imply harmonicity). If these equations together with the corresponding pole bounds imply such singularities in the OPE, which can be “split” one would be able to prove the validity of free field realizations in such more general theories, too. One may also attempt to study models, say in a theory of a system of scalar fields of dimension d = 4, without leaving the realm of scalar bilocal harmonic fields V1 (of dimension (1, 1)). In [11] there have been found examples of 6–point functions of harmonic bilocal fields, which do not have free field realizations. However, our experience with the d = 2 case shows that in order to complete the model (including the check of Wightman positivity for all correlation functions) it is crucial to describe the OPE in terms of some simple algebraic structure (e.g., associative, or Lie algebras). On the other hand going beyond bilocal V1’s is a true signal of nontriviality of a GCI model. Our analysis of Sect. 3 shows that this can be characterized by a simple property of the correlation functions: the violation of the single pole property (of Sect. 3.3). From this point of view a further exploration of the example of Sect. 3.5 within a QFT involving currents appears particularly attractive. Note added in proof. In [19], we have determined the biharmonic function whose leading part is given by Eq. (3.31). It involves dilogarothmic functions, whose arguments are algebraic functions of conformal cross ratios. This exem- plifies the violation of Huygens bilocality for the biharmonic fields, Theorem 3.7. Harmonic Bilocal Fields 29 Yet, in support of Conjecture 3.3, it is shown that the structure of the cuts is in a nontrivial manner consistent with ordinary bilocality. Acknowledgements. We thank Yassen Stanev for an enlightening discussion. This work was started while N.N. and I.T. were visiting the Institut für Theoretische Physik der Universität Göttingen as an Alexander von Humboldt research fellow and an AvH awardee, respectively. It was continued during the stay of N.N. at the Albert Einstein Institute for Gravitational Physics in Potsdam and of I.T. at the Theory Group of the Physics Department of CERN. The paper was completed during the visit of N.N. and I.T. to the High Energy Section of the I.C.T.P. in Trieste, and of K.-H.R. at the Erwin Schrödinger Institute in Vienna. We thank all these institutions for their hospitality and support. N.N. and I.T. were partially supported by the Research Training Network of the European Commission under contract MRTN-CT-2004-00514 and by the Bulgarian National Council for Scientific Research under contract PH-1406. References [1] B. Bakalov, N.M. Nikolov, Jacobi identity for vertex algebras in higher dimensions, J. Math. Phys. 47 (2006) 053505; math-ph/0604069. [2] B. Bakalov, N.M. Nikolov, K.–H. Rehren, I. Todorov, Unitary positive-energy representations of scalar bilocal quantum fields, Commun. Math. Phys. 271 (2007) 223–246; math-ph/0604069. [3] V. Bargmann, I.T. Todorov, Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(N), J. Math. Phys. 18 (1977) 1141–1148. [4] H.–J. Borchers, Über die Mannigfaltigkeit der interpolierenden Felder zu einer interpolierenden S-Matrix, N. Cim. 15 (1960) 784–794. [5] D. Buchholz, G. Mack, I.T. Todorov, The current algebra on the circle as a germ of local field theories, Nucl. Phys. B (Proc. Suppl.) 5B (1988) 20–56. [6] V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova, I.T. Todorov, Harmonic Analysis of the n-Dimensional Lorentz Group and Its Applications to Conformal Quantum Field Theory, Springer, Berlin et al. 1977. [7] F.A. Dolan, H. Osborn, Conformal four point functions and operator product expansion, Nucl. Phys. B 599 (2001) 459–496; hep-th/0011040. [8] M. Dütsch, K.–H. Rehren, Generalized free fields and the AdS-CFT correspondence, Ann. H. Poincaré 4 (2003) 613–635; math-ph/0209035. http://arxiv.org/abs/math-ph/0604069 http://arxiv.org/abs/math-ph/0604069 http://arxiv.org/abs/hep-th/0011040 http://arxiv.org/abs/math-ph/0209035 Harmonic Bilocal Fields 30 [9] N. Dunford, J. Schwartz, Linear Operators, Part 2. Spectral Theory. Self Adjoint Operators in Hilbert Space, Interscience Publishers, N.Y., Lon- don, 1963. [10] G. Mack, All unitary representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1–28. [11] N.M. Nikolov, K.–H. Rehren, I.T. Todorov, Partial wave expan- sion and Wightman positivity in conformal field theory, Nucl. Phys. B 722 (2005) 266–296; hep-th/0504146. [12] N.M. Nikolov, Ya.S. Stanev, I.T. Todorov, Four dimensional CFT models with rational correlation functions, J. Phys. A 35 (2002) 2985–3007; hep-th/0110230. [13] N.M. Nikolov, Ya.S. Stanev, I.T. Todorov, Globally conformal in- variant gauge field theory with rational correlation functions, Nucl. Phys. B 670 (2003) 373–400; hep-th/0305200. [14] N.M. Nikolov, I.T. Todorov, Rationality of conformally invariant lo- cal correlation functions on compactified Minkowsi space, Commun. Math. Phys. 218 (2001) 417–436; hep-th/0009004. [15] B. Schroer, J.A. Swieca, Conformal transformations of quantized fields, Phys. Rev. D 10 (1974) 480–485. [16] B. Schroer, J.A. Swieca, A.H. Völkel, Global operator expansions in conformally invariant relativistic quantum field theory, Phys. Rev. D 11 (1975) 1509–1520. [17] R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That, Benjamin, 1964; Princeton Univ. Press, Princeton, N.J., 2000. [18] I. Todorov, Vertex algebras and conformal field theory models in four dimensions, Fortschr. Phys. 54 (2006) 496–504. [19] N.M. Nikolov, K.–H. Rehren, I.T. Todorov, Pole structure and bihar- monic fields in conformal QFT in four dimensions. e-print arXiv:0711.0628, to appear in: “LT7: Lie Theory and its Applications in Physics”, Proceed- ings Varna 2007, ed. V. Dobrev (Heron Press, Sofia). http://arxiv.org/abs/hep-th/0504146 http://arxiv.org/abs/hep-th/0110230 http://arxiv.org/abs/hep-th/0305200 http://arxiv.org/abs/hep-th/0009004 http://arxiv.org/abs/0711.0628 Introduction Properties of GCI scalar fields Structure of correlation functions and pole bounds Twist expansion of the OPE and bi–harmonicity of twist two contribution Bilocality of twist two contribution to the OPE Convergence of harmonic decompositions Consequences of bi–harmonicity A necessary and sufficient condition for Huygens bilocality The case of dimension 2 A d=4 6-point function violating the SPP The theory of GCI scalar fields of scaling dimension d=2 Structure of the correlation functions Associative algebra structure of the OPE Completion of the proofs Discussion. Open problems

0704.1962

A simple test of quantumness for a single system

A simple test of quantumness for a single system Robert Ali ki Institute of Theoreti al Physi s and Astrophysi s, University of Gda«sk, Wita Stwosza 57, PL 80-952 Gda«sk, Poland Ni holas Van Ryn S hool Of Physi s, Quantum Resear h Group, University of KwaZulu-Natal, Westville Campus, Private Bag x54001, Durban, South Afri a. November 17, 2018 Abstra t We propose a simple test of quantumness whi h an de ide whether for the given set of a essible experimental data the lassi al model is insu� ient. Take two observables A,B su h that for any state ψ their mean values satisfy 0 ≤ 〈ψ|A|ψ〉 ≤ 〈ψ|B|ψ〉 ≤ 1. If there exists a state φ su h that the se ond moments ful�ll the inequality 〈φ|A2|φ〉 > 〈φ|B2|φ〉 then the system annot be des ribed by the lassi al probabilisti s heme. An example of an optimal triple (A,B, φ) in the ase of a qubit is given. Although we are on�dent that the proper theory des ribing all physi al phenomena is the quantum theory, there are many situations where the lassi al des ription in terms of fun tions and probability distributions over a suitable "phase-spa e" is su� ient. In parti ular, the systems onsisting of a large number of parti les and/or emerging in quantum states hara terized by large quantum numbers are supposed to behave lassi ally. The standard explanation of this fa t refers to the stability properties of quantum states with respe t to the intera tion with an environment. For large quantum systems the intera tion with the environment is so strong that most quantum states rapidly de ay (de ohere) and the remaining manifold of experimentally a essible states an be des ribed by lassi al models. However, the a tual border between quantum and lassi al worlds is still a topi of theoreti al debate and experimental e�orts [1℄. This question is parti ularly important in the �eld of quantum information. Useful large s ale quantum omputations would demand preservation of some quantum properties for rather large physi al systems, say at least for 103 qubits. Moreover, some promising implementations of a qubit are based on mesos opi systems whi h lie on the aforementioned border. In parti ular, the so- alled super ondu ting qubits are systems omposed of 108− 109 parti les (Cooper pairs). Therefore, it is important to �nd a simple operational test of quantumness whi h ould be applied to a single system implementing a qubit. It is generally believed that the simplest operational and model independent test of quantumness (or stri tly speaking non- lassi ality) is based on Bell inequalities whi h involve systems omposed of at least two parts [2℄. However, this test is problemati for systems whi h annot be spatially well-separated ("lo ality loophole"). We propose here a mu h simpler operational test whi h an be applied to a single system and formulated in terms of inequalities between mean values of properly hosen observables. In ontrast to Bell inequalities this test does not refer to the notion of lo ality, asso iated with the spa e-time, but involves only the most fundamental di�eren es between lassi al and quantum observables. It is based on the observation that for any two real fun tions f, g satisfying 0 ≤ f(x) ≤ g(x) (1) and any probability distribution ρ(x) the following inequality holds 〈f2〉ρ ≡ 2(x)ρ(x)dx ≤ 2(x)ρ(x)dx ≡ 〈g2〉ρ . (2) On the other hand, as illustrated below for the ase of a single qubit, for quantum systems we an always �nd a pair of observables A,B su h that for all states ψ their mean values satisfy 0 ≤ 〈ψ|A|ψ〉 ≤ 〈ψ|B|ψ〉 but there exists a state φ su h that for the se ond moments 〈φ|A2|φ〉 > 〈φ|B2|φ〉. The above statement follows from the interesting and nontrivial mathemati al result in the theory of C∗-algebras [3℄, whi h is presented below for ompleteness. Consider an abstra t formalism where (bounded) observables are elements of a ertain C∗-algebra A, and states are positive normalized fun tionals A ∋ A 7→ 〈A〉ρ with 〈A〉ρ denoting the mean value of an observable A in a state http://arxiv.org/abs/0704.1962v3 ρ. For all pra ti al purposes we an restri t ourselves to two extreme ases: the �rst being a lassi al model where A is an algebra of fun tions on a ertain "phase-spa e" with 〈A〉ρ = A(x)ρ(x)dx, where ρ(x) is some probability distribution, and the se ond being a �nite quantum model in whi h A is an algebra of matri es and 〈A〉ρ = Tr(ρA), where ρ is some density matrix. For any pair of observables A,B ∈ A, the order relation A ≤ B means that 〈A〉ρ ≤ 〈B〉ρ for all states ρ (in fa t it is enough to take all pure states). Now we an formulate the following: Theorem. If the following impli ation 0 ≤ A ≤ B =⇒ A2 ≤ B2 (3) always holds then the algebra A is ommutative, i.e. isomorphi to the algebra of ontinuous fun tions on a ertain ompa t spa e. As a onsequen e of the above theorem, for any quantum system there exists a pair of observables (identi�ed with matri es) (A,B) su h that the eigenvalues of A,B and B−A are nonnegative but the matrix B2−A2 possesses at least one negative eigenvalue. In order to apply our test of quantumness in an experiment, one should �rst guess a pair of observables A,B with nonnegative values of possible out omes and perform a statisti al test of the inequality 〈A〉ρ ≤ 〈B〉ρ with as large as possible number of di�erent, generally mixed, initial states ρ. These states should be as pure as possible, otherwise the quantumness ould be not dete ted. Then one should sear h amongst these for any states σ satisfying the relation for the se ond moments 〈A2〉σ > 〈B2〉σ. If the violation of the lassi al relation (2) holds, it means that the system exhibits some quantum hara teristi s. Although there are always in�nitely many triples (A,B, σ), the e�e ts of external noise a ting on the system and measuring apparatus an easily wash out the deviations from " lassi ality". Therefore, instead of a random guess it is useful to �nd the examples of su h triples whi h maximally violate lassi ality and an be used to optimally design the experimental setting. This an easily be a hieved in the ase of a qubit whi h is the most important example for quantum information. We sear h for a pair of 2× 2 matri es A,B and a pure state φ whi h satisfy 0 ≤ A ≤ B ≤ I , 〈A2〉φ ≡ 〈φ|A2|φ〉 > 〈B2〉φ ≡ 〈φ|B2|φ〉 , (4) where the observables A and B are normalized in su h a way that their upper bound is the identity. Su h a triple is given as an example as ξ∗ a2 , B = , φ = . (5) The matrix B is hosen to be diagonal, with the identity as its upper bound, and this hoi e of basis an be made sin e the solution to this problem is unique up to unitary equivalen e. From the upper bound it an be seen that both eigenvalues of B should be at most 1, and to maximize the violation of (3) one of the eigenvalues is hosen to be �xed at 1. The usual ondition |α|2 + |β|2 = 1 applies for the parameters of the state φ. The positivity of the observables A, B and (B − A) is ensured by the requirement that both their diagonal elements and determinants are positive. These onditions are expressed below as; 0 ≤ b ≤ 1, (6) 0 ≤ a1 ≤ 1, (7) 0 ≤ a1a2 − |ξ|2, (8) 0 ≤ (1 − a1)(b− a2)− |ξ|2. (9) In order for this triple to satisfy equation (4), it is su� ient that one of the eigenvalues of (B2 −A2) is found to be negative while remaining within the onstraints listed above. The eigenvalues of the matrix (B2 −A2) are found to be b2 + 1− a2 − 2|ξ|2 ± (b2 − 1 + a2 )2 + 4(a1 + a2)2|ξ|2 . (10) In order to �nd the most negative value whi h one of the eigenvalues an attain, one need only onsider one of the two. Sin e the square root is always positive, one eigenvalue always remains greater than the other. A numeri al te hnique is used to al ulate the maximal violation of the inequality (3) sin e we are unable to �nd an exa t solution to this optimisation problem. In �nding the optimal set of parameters a1, a2, b and real ξ, it an be seen that the maximal violation of the inequality (3) arises when the onditions (8) and (9) are equalities rather than inequalities. Using a1a2 = |ξ|2, (11) (1− a1)(b− a2) = |ξ|2 (12) redu es the problem from four unknowns to two, and the triplet an then be written as a1a2√ a1a2 a2 , B = , φ = . (13) The parameters whi h result in one of the eigenvalues of (B2 −A2) attaining it's most negative value, and thus maximally violating inequality (3), while still remaining within the onstraints give us the triplet 0.724 0.249 0.249 0.0854 , B = 0 0.309 , φ = 0.391 0.920 . (14) In this example, the value of 〈φ|B|φ〉 − 〈φ|A|φ〉 is 0.0528, a positive value, whereas it an be seen that 〈φ|B2|φ〉 − 〈φ|A2|φ〉 = −0.0590 whi h learly demonstrates the quantum nature of this example. The eigenve tors and orre- sponding eigenvalues of A using these parameters is al ulated to be 0.946 0.325 = 0.809 0.946 0.325 −0.325 0.946 . (16) To give a on rete example, one an apply these results to the polarization of a single photon. Choosing a polarization basis as |H〉, |V 〉 and attributing the values 1 to |H〉 and 0.309 to |V 〉 we obtain the observable B. The observable A orresponds to a rotated polarization basis |H ′〉 = cos(19◦)|H〉 + sin(19◦)|V 〉 , |V ′〉 = − sin(19◦)|H〉 + cos(19◦)|V 〉 with the eigenvalues 0.809 and 0, respe tively. The maximal violation of lassi ality should be observed in the neighborhood of the state |φ〉 = cos(67◦)|H〉+ sin(67◦)|V 〉. In prin iple, the proposed test with the parameters obtained above ould be used to support the quantum pi ture for di�erent implementations of a qubit in luding, for example, "super ondu ting qubits" with a still questionable quantum hara ter [4℄. A knowledgements. The authors thank P. Badzi�ag, M. Horode ki, R. Horode ki, W.A. Majewski and M. �ukowski for dis ussions. Finan ial support by the POLAND/SA COLLABORATION PROGRAMME of the National Resear h Foundation of South Afri a and the Polish Ministry of S ien e and Higher Edu ation and by the European Union through the Integrated Proje t SCALA is a knowledged. Referen es [1℄ E. Joos, H.D. Zeh, C. Kiefer, D. Giulini, J. Kups h, and I,-O. Stamates u, De oheren e and the Appearan e of a Classi al World in Quantum Theory, 2nd ed. Springer,Berlin 2003. [2℄ J.S. Bell, Rev. Mod. Phys. 38, 447 (1966) [3℄ R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory, A ademi Press, New York ,1983 [4℄ R. Ali ki, quant-ph/0610008 http://arxiv.org/abs/quant-ph/0610008

0704.1963

SPITZER: Accretion in Low Mass Stars and Brown Dwarfs in the Lambda Orionis Cluster

SPITZER: Accretion in Low Mass Stars and Brown Dwarfs in the Lambda Orionis Cluster David Barrado y Navascués Laboratorio de Astrof́ısica Espacial y F́ısica Fundamental, LAEFF-INTA, P.O. Box 50727, E-28080 Madrid, SPAIN [email protected] John R. Stauffer Spitzer Science Center, California Institute of Technology, Pasadena, CA 91125 Maŕıa Morales-Calderón, Amelia Bayo Laboratorio de Astrof́ısica Espacial y F́ısica Fundamental, LAEFF-INTA, P.O. Box 50727, E-28080 Madrid, SPAIN Giovanni Fazzio, Tom Megeath, Lori Allen Harvard Smithsonian Center for Astrophysics, Cambridge, MA 02138 Lee W. Hartmann, Nuria Calvet Department of Astronomy, University of Michigan ABSTRACT We present multi-wavelength optical and infrared photometry of 170 previously known low mass stars and brown dwarfs of the 5 Myr Collinder 69 cluster (Lambda Orionis). The new photometry supports cluster membership for most of them, with less than 15% of the previous candidates identified as probable non-members. The near infrared photometry allows us to identify stars with IR excesses, and we find that the Class II population is very large, around 25% for stars (in the spectral range M0 - M6.5) and 40% for brown dwarfs, down to 0.04 M⊙, despite the fact that the Hα equivalent width is low for a significant fraction of them. In addition, there are a number of substellar objects, classified as Class III, that have optically thin disks. The Class II members are distributed in an inhomogeneous way, lying preferentially in a filament running toward the south-east. The IR excesses for the Collinder 69 members range from pure Class II (flat or nearly flat spectra longward of 1 µm), to transition disks with no near-IR excess but excesses beginning within the IRAC wavelength range, to two stars with excess only detected at 24 µm. Collinder 69 thus appears to be at an age where it provides a natural laboratory for the study of primordial disks and their dissipation. Subject headings: open clusters and associations: individual (Lambda Orionis Star Forming region) – stars: low-mass, brown dwarfs – stars: pre-main-sequence http://arxiv.org/abs/0704.1963v1 1. Introduction The star-formation process appears to operate successfully over a wide range of initial conditions. In regions like Taurus, groups of a few stars to a few tens of stars are the norm. The molecular gas in Taurus is arranged in a number of nearly paral- lel filaments, possibly aligned with the local mag- netic field, and with the small stellar groups sited near end-points of the filaments (Hartmann 2004). No high mass stars have been formed in the Taurus groups, and the Initial Mass Function (hereafter, IMF) also appears to be relatively deficient in brown dwarfs (Briceño et al. 2003; Luhman 2004) –but see also (Guieu et al. 2006) for an alternate view. The Taurus groups are not gravitation- ally bound, and will disperse into the field on short timescales. At the other end of the mass spectrum, regions like the Trapezium cluster and its surrounding Orion Nebula cluster (ONC) have produced hundreds of stars. The ONC includes several O stars, with the earliest having spectral type O6 and an estimated mass of order 35 M⊙. The very high stellar density in the ONC (10,000 stars/pc3 at its center (McCaughrean & Stauffer 1994)) suggests that star-formation in the ONC was gravity dominated rather than magnetic field dominated. It is uncertain whether the ONC is currently gravitationally bound or not, but it is presumably at least regions like the ONC that are the progenitors of long-lived open clusters like the Pleiades. UV photoionization and ablation from O star winds likely acts to truncate the circum- stellar disks of low mass stars in the ONC, with potential consequences for giant planet formation. An interesting intermediate scale of star- formation is represented by the Lambda Ori association. The central cluster in the associ- ation –normally designated as Collinder 69 or the Lambda Orionis cluster– includes at least one O star, the eponymous λ Ori, with spec- tral type O8III. However, a number of lines of evidence suggests that one of the Coll 69 1Based on observations collected Spitzer Space Tele- scope; at the German-Spanish Astronomical Center of Calar Alto jointly operated by the Max-Planck-Institut für Astronomie Heidelberg and the Instituto de Astrof́ısica de Andalućıa (CSIC); and at the WHT operated on the island of La Palma by the Isaac Newton Group in the Spanish Ob- servatorio del Roque de los Muchachos of the Instituto de Astrofsica de Canarias stars has already passed through its post-main sequence evolution and become a supernova, and hence indicating it was more massive than Lambda Ori (see the complete Initial Mass Func- tion in Barrado y Navascués, Stauffer, & Bouvier (2005)). A census of the stars in Coll 69 by Dolan & Mathieu (2001) –hereafter, DM– indi- cates that the cluster is now strongly unbound. DM argue that this is due to rapid removal of molecular gas from the region that occurred about 1 Myr ago when the supernova exploded. They interpreted the color-magnitude diagram of Coll 69 as indicating a significant age spread with a maximum age of order 6 Myr; an alternative in- terpretation is that the cluster has negligible age spread (with age ∼6 Myr) and a significant num- ber of binary stars. While DM identified a large population of low mass stars in Coll 69, only four of 72 for which they obtained spectra are clas- sical T Tauri stars (based on their Hα emission equivalent widths). Much younger stars, includ- ing classical T Tauri stars, are present elsewhere in the Lambda Ori SFR, which DM attribute to star-formation triggered by the supernova rem- nant shock wave impacting pre-existing molecular cores in the region (the Barnard 30 and Barnard 35 dark clouds, in particular). We have obtained Spitzer space telescope IRAC and MIPS imaging of a ∼one square degree region centered on the star λ Ori in order to (a) search for circumstellar disks of members of the Coll 69 cluster and (b) attempt to identify new, very low mass members of the cluster in order to determine better the cluster IMF (in a forthcoming paper). In §2, we describe the new observations we have obtained; and in §3 we use those data to recon- sider cluster membership. In §4 we use the new candidate member list and the IR photometry to determine the fraction of cluster members with cir- cumstellar disks in both the stellar and substellar domain, and we sort the stars with disks according to their spectral slope from 1 to 24 µm. 2. The data 2.1. Optical and Near Infrared photome- The optical and the near IR data for the bright candidate members come from Barrado y Navascués et al. (2004) –hereafter, Paper I. The RI –Cousins system– data were collected with the CFHT in 1999, whereas the JHKs come from the 2MASS All Sky Survey (Cutri et al. 2003). For cluster members, the completeness limit is located at I(complete,cluster) ∼ 20.2 mag, whereas 2MASS provides near infrared data down to a limiting magnitude of J=16.8, H=16.5, and Ks=15.7 mag. In some cases, low resolution spectroscopy in the optical, which provides spectral types and Hα equivalent widths, is also available. Twenty- five objects out of the 170 CFHT1999 candidate members are in common with Dolan & Mathieu (1999, 2001). Those 25 stars also have Hα and lithium equivalent widths, and radial velocities. 2.1.1. New deep Near Infrared photometry For the objects with large error in 2MASS JHKs, or without this type of data due to their intrinsic faintness, we have obtained additional measurements with the WHT (La Palma Observa- tory, Spain) and INGRID (4.1×4.1 arcmin FOV) in November 2002 and February 2003, and with the Calar Alto 3.5m telescope (Almeria, Spain) and Omega2000 in October 2005 (15.36×15.36 ar- cmin FOV). In all cases, for each position, we took five individual exposures of 60 seconds each, with small offsets of a few arcseconds, thus totalling 5 minutes. In the case of the campaigns with IN- GRID, we observed the area around the star λ Orionis creating a grid. Essentially, we have ob- served about 2/3 of the CFHT 1999 optical survey region in J (in the area around the star and west of it), with some coverage in H and K. On the other hand, the Omega2000 observations, taken under a Director’s Discretionary Time program, were fo- cused on the faint candidate members. Except for one object (LOri154), we collected observa- tions in the J, H and Ks filters. The conditions of the first observing run with INGRID were photo- metric, and we calibrated the data using standard stars from Hunt et al. (1998) observed throughout the nights of the run. The average seeing was 0.9 arcsec. We had cloud cover during the second run with INGRID, and the data were calibrated using the 2MASS catalog and the stars present in each individual image. The dispersion of this calibra- tion is σ=0.05 magnitudes in each filter, with a seeing of about 1.0 arcsec. Finally, no standard stars were observed during the DDT observations at Calar Alto. The seeing in this case was 1.2 arc- sec. The faint Lambda Orionis candidate mem- bers were calibrated using also 2MASS data. In this last case, the dispersion is somewhat higher, probably due to the worse seeing and the larger angular pixel scale of the detector, with σ=0.1 mag. Note that this is dispersion not the error in the calibration. These values correspond to the FWHM of the gaussian distribution of the val- ues zeropoint(i)=magraw(i)-mag2MASS(i), for any star i, which also includes the photometric errors in the 2MASS photometry and any contribution due to the cluster stars being photometrically vari- able. Since there is a large number of stars per field (up to 1,000 in the Omega2000 images), the peak of this distribution can be easily identified and the zero points derived. A better estimate of the error in the calibration is based on the distance between mean, median and mode values, which are smaller than half of the FWHM (in the case of the mean and the median, almost identical to the hundredth of magnitude). Therefore, the errors in the calibration can be estimated as 0.025 and 0.05 magnitudes for the INGRID and the Omega2000 datasets, respectively. All the data were processed and analyzed with IRAF2, using aperture photometry. These measurements, for 166 candidate members, are listed in Table 1 (WHT/INGRID) and Table 3 (CAHA/Omega2000). Note that the errors listed in the table correspond to the values produced by the phot task with the digi.apphot package and does not include the errors in the calibrations. 2.2. Spitzer imaging Our Spitzer data were collected during March 15 (MIPS) and October 11 (IRAC), 2004, as part of a GTO program. The InfraRed Array Camera (IRAC, Fazio et al. (2004)) is a four channel cam- era which takes images at 3.6, 4.5, 5.8, and 8.0 µm with a field of view that covers ∼5.2×5.2 arcmin. IRAC imaging was performed in mapping mode with individual exposures of 12 seconds “fram- etime” (corresponding to 10.4 second exposure times) and three dithers at each map step. In or- der to keep the total observation time for a given 2IRAF is distributed by National Optical Astronomy Obser- vatories, which is operated by the Association of Universi- ties for Research in Astronomy, Inc., under contract to the National Science Foundation, USA map under three hours, the Lambda Ori map was broken into two segments, each of size 28.75×61.5 arcmin - one offset west of the star λ Ori and the other offset to the east, with the combined image covering an area of 57×61.5 arcmin, leav- ing the star λ Orionis approximately at the cen- ter. Each of the IRAC images from the Spitzer Science Center pipeline were corrected for instru- mental artifacts using an IDL routine developed by S. Carey and then combined into the mosaics at each of the four bandpasses using the MOPEX package (Makovoz & Khan 2005). Note that the IRAC images do not cover exactly the same FOV in all bands, providing a slice north of the star with data at 3.6 and 5.8 micron, and another slice south of it with photometry at 4.5 and 8.0 microns. The size of these strips are about 57×6.7 arcmin in both cases. The Multiband Imaging Photometer for Spitzer (MIPS, Rieke et al. (2004)) was used to map the cluster with a medium rate scan mode and 12 legs separated by 302 arcsec in the cross scan direction. The total effective integration time per point on the sky at 24 µm for most points in the map was 40 seconds, and the mosaic covered an area of 60.5×98.75 arcmin centered around the star λ Orionis. Since there were no visible arti- facts in the pipeline mosaics for MIPS 24 µm we used them as our starting point to extract the pho- tometry. We obtained MIPS 70 µm and 160 µm imaging of the λ Ori region, but very few point sources were detected and we do not report those data in this paper. The analysis of the data was done with IRAF. First, we detected objects in each image using the “starfind” command. Since the images in the [3.6] and [4.5] bands are deeper than those in the [5.8] and [8.0] bands, and since the fluxes of most ob- jects are brighter at those wavelengths, the num- ber of detections are much larger at the IRAC short wavelengths than at the longer ones. Only a relatively few objects have been detected at 24 µm with MIPS. As a summary, 164 objects were de- tected at at 3.6 and 4.5 micron, 145 at 5.8 micron, 139 at 8.0 micron and 13 at 24 micron. We have performed aperture photometry to de- rive fluxes for C69 cluster members. For the IRAC mosaics we used an aperture of 4 pixels radius, and the sky was computed using a circular annu- lus 4 pixels wide, starting at a radius 4 pixels away from the center. It is necessary to apply an aper- ture correction to our 4-pixel aperture photometry in order to estimate the flux for a 10-pixel aper- ture, because the latter is the aperture size used to determine the IRAC flux calibration. In some cases, due to the presence of nearby stars, hot pix- els, or because of their faintness, a 2 pixel aperture and the appropriate aperture correction were used (see notes to Table 3). For the MIPS photome- try at 24 µm, we used a 5.31 pixels (13 arcsec) aperture and a sky annulus from 8.16 pixels (20 arcsec) to 13.06 pixels (32 arcsec). An aperture correction was also applied. Table 2 provides the zero points, aperture corrections and conversion factors between magnitudes and Jansky, as pro- vided by the Spitzer Science Center website. 2.2.1. Data cross-correlation The coverage on the sky of our Spitzer/IRAC data is an approximate square of about 1 sq.deg, centered on the star λ Orionis. The optical data taken with the CFHT in 1999 covers an area of 42×28 arcmin, again leaving the star in the center of this rectangle. Therefore, the optical survey is completely included in the Spitzer mapping, and we have been able to look for the counterpart of the cluster candidates presented in Paper I. The analysis of the area covered by Spitzer but without optical imaging in the CFHT1999 sur- vey will be discussed in a forthcoming paper. We have not been able to obtain reliable Spitzer pho- tometry for some candidate members from Paper I, especially at the faint end of the cluster se- quence. The faintest detected object, LOri167, depending on the isochrone and the model, would have a mass of ∼0.017 M⊙, if it is a member (Barrado y Navascués, Huélamo, & Morales Calderón (2005)). The results are listed in Table 3, where non- members and members are included, respectively (see next section for the discussion about the mem- bership). In both cases, we include data corre- sponding to the bands R and I –from CFHT–, J, H and Ks –from 2MASS and CAHA–, [3.6], [4.5], [5.8], and [8.0] –from IRAC– and [24] –from MIPS. Additional near IR photometry from WHT can be found in Table 1. 3. Color-Color and Color-Magnitude Dia- grams and new membership assignment Before discussing membership of the Paper I stars based on all of the new optical and IR data, we have made an initial selection based on the IRAC colors. Figure 1 (see further discussion in the next section) displays a Color-Color Diagram with the four IRAC bands. We have found that 31 objects fall in the area defined by Allen et al. (2004) and Megeath et al. (2004) as Class II ob- jects (ie, TTauri stars). Another two candidate members are located in the region correspond- ing to Class I/II objects. We consider all these 33 objects as bona-fide members of the C69 clus- ter. Harvey et al. (2006) have discussed the con- fusion by extragalactic and other sources when analysing Spitzer data (in their case, they used Serpens, a cloud having a large extinction). We believe that this contribution is negligible for our Lambda Ori data, since those Class II objects de- tected at 24 micron are in the TTauri area de- fined by Sicilia-Aguilar et al. (2005), as displayed in her figure 5. There can be a higher level of contamination among the objects classiefied as Class III. All of the objects in Figure 1 had previ- ously been identified as cluster candidate members based on optical CMDs –it is unlikely that a signif- icant number of AGN would have passed both our optical and our IR criteria (and also have been unresolved in our optical CFHT images). More- over, prior to our Spitzer data, only 25% candidate members which had optical, near-IR data and op- tical low-resolution spectroscopy turned out to be non-members (Paper I). After adding the Spitzer photometry, we are quite confident in the mem- bership of the selection. Figure 2 and Figure 3 display several color- magnitude diagrams (CMD) using the data listed in Tables 1 and 3. In the case of the panels of the first figure, we present optical and IR, including Spitzer/IRAC data; whereas in the second set of figures only IR data are plotted. For the sake of simplicity, we have also removed the non-members from Figure 3. Based on these diagrams and on the spectro- scopic information included in Paper I, we have reclassified the candidate members as belonging or not to the cluster. In color-magnitude diagrams, C69 members lie in a fairly well-defined locus, with a lower bound that coincides approximately with the 20 Myr isochrone in this particular set of the- oretical models (Baraffe et al. 1998). Stars that fall well below (or blueward) of that locus are likely non-members; stars that fall above or redward of that locus are retained because they could have IR excesses or above average reddening. We com- bine the “votes” from several CMD’s to yield a qualitative membership determination, essentially yes, no or maybe. In total, out of 170 candi- dates, 19 are probable non-members, four have du- bious membership and the rest (147 objects) seem to be bona-fide members of the cluster. There- fore, the ratio of non-members to initial candidate members is 13.5 %. In any case, only additional spectroscopy (particularly medium and high res- olution) can be used to establish the real status. Proper motion might be helpful, but as shown by Bouy et al. (2007), some bona-fide member can appear to have discrepant proper motions when compared with the average values of the associa- tion. Table 4 shows the results for each candidate in the different tests used to determine its mem- bership, the membership as in Paper I, and the final membership based on the new photometry. The second and last columns show the spectro- scopic information. Note that the degree of confi- dence in the new membership classification varies depending on the available information and in any event it is always a matter of probability. As Table 3 shows, the Spitzer/IRAC data does not match completely the limiting magnitudes of either our optical survey with CFHT or the 2MASS JHKs data. In the case of the band [3.6], essentially all the Lambda Orionis candidate mem- bers should have been detected (except perhaps the faintest ones, at about I=22 mag). Some ob- jects in the faint end have 3.6 micron data, but lack 2MASS NIR, although in most cases we have supplied it with our own deep NIR survey. In the case of the Spitzer data at 4.5 micron, some ad- ditional candidate members fainter than I=20.9 mag were not found, due to the limiting magni- tude of about [4.5]lim=16.3 mag. The data at 8.0 micron only reach [8.0]lim=14.0 mag, which means that only cluster members with about I=18.6 mag –orKs=14.9– can be detected at that wavelength. This is important when discussing both the mem- bership status based on color-magnitude diagrams and the presence of infrared excesses by examin- ing color-color diagrams. Note, however, that ob- jects with IR excesses have fainter optical/near-IR counterparts than predicted in the table. Figure 4 presents another CMD with the op- tical magnitudes from the CFHT survey (R and I), where we display the 170 candidate members using different symbols to distinguish their ac- tual membership status. Small dots correspond to non-members based on the previous discussion, whereas plus symbols, crosses and circles denote probable members. In the first case –in most cases due to their faintness– they do not have a com- plete set of IRAC magnitudes, although they can have either a measurement at 3.6 and 4.5, or even at 5.8 microns. In the case of the objects rep- resented by crosses, they have been classified as Class III objects (Weak-line TTauri stars and sub- stellar analogs if they indeed belong to the clus- ter) based on an IRAC color-color diagram (see next section and Figure 1). Finally, big circles correspond to Class II sources. The pollution rate seems to be negligible in the magnitude range I=12-16 (1.2–0.17 M⊙ approx, equivalent to M0 and M5, respectively), where our initial selection based on the optical and the near infrared (2MASS data) has worked nicely. However, for fainter can- didates, the number of spurious members is very large and the pollution rate amounts to about 15% for objects with 16 < I < 19, and about 45% for I≥19 (approximately the magnitude beyond the reach of the 2MASS survey). At a distance of 400 pc and for an age of 5 Myr, and according to the models by Baraffe et al. (1998), the substellar borderline is located at about I=17.5 mag. Table 5 lists other values for different ages, as well other bands –J , Ks and L′– discussed in this paper. Among our 170 CFHT candidate members, there are 25 objects fainter than that magnitude, and which pass all of our membership criteria which are probable brown dwarfs. Out of these 25 objects, 12 have low-resolution spectroscopy and seem to be bona- fide members and, therefore, brown dwarfs. The other 13 objects are waiting for spectral confir- mation of their status. Assuming an age of 3 or 8 Myr would increase or decrease the number of brown dwarf by seven in each case. In the first case (3 Myr), five out of the seven possible BDs have spectroscopic membership, whereas in the second case only three were observed in Paper I. As a summary, we have found between 18 and 32 good brown dwarfs candidates (depending on the final age) in the Lambda Orionis cluster, and between 17 and 9 have their nature confirmed via low-resolution spectroscopy. Note that even this technique does not preclude the possibility that a few among them would actually be non-members. Finally, the planetary mass domain starts at about Ic=21.5, using a 5 Myr isochrone (DUSTY models from Baraffe et al. (2002)). In that re- gion, there is only one promising planetary mass candidate, LOri167 (Barrado y Navascués et al . (2007)). 4. Discussion 4.1. The Color-Color diagrams, the diag- nostic of IR excess and the disk ratio The Spitzer/IRAC colors are a powerful tool to reveal the dust and, therefore, the population of Class I and II sources in a stellar association. Fig- ure 1 (after Allen et al. (2004) and Megeath et al. (2004)) displays the colors derived from the mea- surements at 3.6 minus 4.5 microns, versus those obtained at 5.8 minus 8.0 microns. This diagram produces an excellent diagnostic, allowing an easy discrimination between objects with and without disks. Note that due to the limiting magnitudes of the IRAC bands (see the discussion in previous section), objects fainter than about I=18.6 mag cannot have a complete set of IRAC colors and therefore cannot be plotted in the diagram. This fact imposes a limit on our ability to discover mid- IR excesses at the faint end of the cluster sequence. For Lambda Orionis cluster members, assuming a distance of 400 pc and an age of 5 Myr (and ac- cording to the models by Baraffe et al. (1998)), this limit is located at 0.040 M⊙. Figure 1 con- tains a substantial number of objects in the region corresponding to the Class II sources. In total, there are 31 objects located within the solid rect- angle out of 134 Lambda Orionis members with data in the four IRAC bands. Among them, three (LOri045, LOri082 and LOri092) possibly have rel- atively large photometric errors in their 5.8 µm flux, because inspection of their SEDs indicates they are likely diskless. Two additional objects, LOri038 and LOri063, have IRAC colors indicat- ing Class I/II (actually, LOri038 is very close to the Class II region). The SED (see below) in- dicates that both are Class II stars. Therefore, the fraction of cluster members that are Class II PMS stars based on their IR excesses is ∼22–25%, for the spectral range M0–M6.5. This is differ- ent from what was inferred by Dolan & Mathieu (1999, 2001) and by us (Paper I), based on the distribution of the Hα emission and near-infrared color-color diagrams. The Spitzer/IRAC data clearly demonstrate that Lambda Orionis cluster does contain a significant number of stars with dusty circumstellar disks. No embedded objects (Class I) seem to be present, in agreement with the age range for the association (3-8 Myr or even slightly larger). Note that our different IR excess frequency compared to Dolan & Mathieu may re- sult from their sample being primarily of higher mass stars than ours. Figure 5 is a blow-up of the region in Fig- ure 1 corresponding to the Class II sources. We have also added big minus and plus symbols, and large squares, to indicate those objects with mea- sured Hα equivalent widths (in low- and medium- resolution spectrum). We have used the satura- tion criterion by Barrado y Navascués & Mart́ın (2003) to distinguish between objects with high W(Hα) –plus symbols– and normal W(Hα) – minus symbols. In principal, an object with a W(Hα) value above the saturation criterion is ei- ther accreting or is undergoing a flare episode. There are two low mass stars (LOri050 and LOri063) with an Hα line broader than 200 km/s (Muzerolle et al. 2003), another independent indi- cation of accretion (based on Natta et al. (2004), they should have very large accretion rates ∼10−9 M⊙/yr). The theoretical disk models used to in- terpret IRAC Color-Color Diagram by Allen et al. (2004) suggest that the accretion rates increase from the bottom-left to the top-right of the figure. This is in agreement with our results, since most of the accreting objects (assuming that strong Hα is a good indicator of accretion) lie in the area of the figure with the larger excesses (top-right). A couple of objects with very low Hα emission are located near the edge of the Class II area (bottom-left), a fact that suggest that they may have a relatively thin disk, with small or negligible accretion. Actually these objects are surrounded by thin disks instead of thick primordial disks (see next section). Regarding the brown dwarfs in the cluster, several probable members (LOri126, LOri129, LOri131, LOri132, LOri139 and LOri140) are lo- cated within the precinct of Classical TTauri stars. They are just at the border between stars and substellar objects, with magnitudes in the range I=17.52–18.21 and J=15.38–16.16 (the boundary is located at I=17.55 and J=15.36 for 5 Myr, see Table 5). In Paper I we presented low-resolution spectroscopy of LOri126, LOri139 and LOri140, which suggests they are cluster members (the spectral types are M6.5, M6.0 and M7.0 with a Hα equivalent width of 26.2, 19.7 and 72.8 Å, respec- tively). In addition we have confirmed the mem- bership of LOri129 via medium-resolution spec- troscopy (spectral type, M6.0 with a Hα equiva- lent width of 12.1 Å). In total, there are 15 brown dwarf candidates with a complete set of IRAC colors, six of which fall in the Class II region, thus making the fraction of brown dwarfs with IR colors indicative of cir- cumstellar disks close to 40% (down to 0.04 M⊙), similar to the 50% obtained by (Bouy et al. 2007) in Upper Sco brown dwarfs, using mid-IR photom- etry or the 50% derived by Guieu et al. (2006) in Taurus brown dwarfs with Spitzer. 4.2. The Spectral Energy Distribution We have plotted the SEDs of our Lambda Ori- onis candidate members in Figures 6-8. There is clearly a range from approximately flat spectrum, to black-body in the near-IR but starting to show excesses at IRAC wavelengths, to only showing ex- cess at 24 micron. A way to study the presence of a circumstellar disk around an object is to analyze the shape of the SED. After Lada et al. (2006) we have used the 3.6–8.0 µm slope for each source de- tected in at least three IRAC bands to distinguish between objects with optically thick, primordial disks, objects surrounded by optically thin or ane- mic disks and objects without disks. The results of this test are presented in Table 4. In Figures 6-8 the SEDs are sorted in agreement with their IRAC slope classification: diskless objects (slope index or α<−2.56) in Figure 6, thick disks (α>−1.8) in Figure 7, and objects surrounded by thin disks (−1.8<α<−2.56) in Figure 8. In this last figure we also include two low mass stars which present an excess only at 24 micron, due to a transition disk (see below). According to the IRAC slope the fraction of cluster members detected in at least three of the IRAC bands with optically thick disks is 14%, while the total disk fraction is found to be 31% (similar to the 25% derived with the IRAC CCD). This fraction is lower than the 50% found by Lada et al. (2006) in IC348 (1-3 Myr) as ex- pected due to the different age of the clusters. Figure 9 illustrates the evolution of the disk fraction with the age for several stellar associa- tions (assuming that the infrared excess serves as a proxy of the presence of a circumstellar disk). The ratios for the different associations come from IRAC data (Hartmann et al. 2005; Lada et al. 2006; Sicilia-Aguilar et al. 2006) in order to avoid different results depending on the technique used (Bouy et al. 2007). The ratio for the Lambda Ori- onis cluster (Collinder 69) is about 30% and, as stated before, for the objects below the substellar borderline, the fraction of Classical TTauri objects seems to be larger. According to its older age, the thick disk fraction in Collinder 69 is lower than that of IC348 (this fraction is represented by open squares in Figure 9). Among the objects classified as Class III sources from Figure 1, only two (LOri043 and LOri065) have a measurement at [24] with an unambiguous detection. These two stars do not have excesses at 3.6 or 4.5 micron. Therefore, they can be clas- sified as transition objects, the evolutionary link between the primordial disks and debris disks. A third of the Class II sources (11 out of 33) have measurements in the [24] band, all of them with clear excess, as expected from their Class II status. The lack of IR excesses at shorter wave- lengths for LOri043 and LOri065 probably stems from an inner disk hole or at least less inner dust than for the Class II sources. Models of simi- lar 24 µm-only excess sources and a discussion of their disk-evolutionary significance can be found in Sicilia-Aguilar et al. (2006); Muzerolle et al. (2006); D’Alessio et al. (2006). Figure 10 shows the same diagram as in Figure 1 but the MIPS 24 µm information is included as dashed squares. The small circles stand for objects having optically thin (dashed) or optically thick (solid) disks based on their IRAC slope. The diagram shows a smooth transition between the three types of objects: disk- less, thin, and primordial disks. LOri103 has a thin disk based on its 3.6–5.8 µm slope. It has been classified as Class III due to its magnitude at 8.0 µm but we believe that it is actually a Class II source and the faint magnitude at this bandpass is probably due to the presence of a nebulosity. There are some objects classified as Class III sources by the IRAC CCD (they are outside, but close to the Class II area in the diagram), but have disks based on their IRAC slope. All these objects are brown dwarfs according to the models by Baraffe et al. (1998) (5 Myr) which pass all our membership criteria and thus the ratio of substel- lar objects bearing disks increases to 50 % (note that we need detections in at least three IRAC bands to calculate the IRAC slope). None of our brown dwarf candidates have been detected at 24 micron. This is probably due to the detection limits for this band. As a summary, of the 170 objects presented in Paper I, 167 are discussed here (the other three are spurious detections or the Spitzer photometry is not reliable). Excluding the sources classified as non-members, there are 22 which cannot be classi- fied due to the incompleteness of the IRAC data, 95 have been classified as diskless, another two have transition disks, 25 thin disks and 20 thick disks. All this information has been listed in Ta- ble 4. Note that there are nine objects classified as Class III from color-color diagrams but which have thin disks according to the SED slopes, and an- other one (LOri156, a very low mass brown dwarf candidate with a very intense Hα) which has a thick disk based on the slope of the IRAC data. 4.3. The Spatial distribution of the mem- We have plotted the spatial distribution of our good candidate members in Figure 11. Four-point stars represent B stars and λ Orionis (O8III). The Class III members (crosses) are approximately randomly distributed across the survey region. Both the Class II sources and the B stars give the impression of being concentrated into linear grouping - with most of the B stars being aligned vertically near RA = 83.8, and a large number of the Class II sources being aligned in the East- Southeast direction (plus some less well-organized alignments running more or less north-south). It is possible the spatial distributions are reflective of the birth processes in C69 - with the youngest objects (the Class II sources and B stars) tracing the (former?) presence of dense molecular gas, whereas the Class III sources have had time to mix dynamically and they are no longer near the locations where they were born. Figure 12 shows three different views of the cen- tral portion of the Spitzer mosaic at 3.6 microns for the C69 region. Figures 12a and 12b (with 12b being a blow-up of the center of 12a) emphasize the distribution of Class II sources relative to the cluster center; Figure 12c shows the distribution of our brown dwarf candidates. The star λ Orionis is at the center of each of these figures. The object located south of the star λ Orionis is BD+09 879 C (or HD36861 C, a F8 V star), with an angular distance of about 30 arcsec from the close binary λ Orionis AB (the projected distance, if BD+09 879 C is a cluster member, would be 12,000 AU from the AB pair). The apparent relative lack of cluster candidate members within about 75 arcsec from the star λ Orionis may be illusory, as this re- gion was “burned out” in the optical images of the CFHT1999 survey and is also adversely affected in our IRAC images. There are a number of Class II sources at about 75-90 arcsec from λ Ori, corre- sponding to a projected separation of order 30,000 AU, so at least at that distance circumstellar disks can survive despite the presence of a nearby O star. Regarding the distribution of brown dwarfs, a significant number of them (30 %) are within the the inner circle with a diameter of 9 arcmin (our original optical survey covered an area of 42×28 arcmin). However, there are substellar members at any distance from the star λ Ori (Figure 12c), and there is no substantial evidence that the clus- ter brown dwarfs tend to be close to the massive central star. We have estimated the correlation in spatial distribution of different sets of data: Class II vs. Class III candidates, objects with any kind of disk (thin, thick and transition) vs. diskless objects, and stellar vs. substellar objects (following the substellar frontier given in Table 5 for different ages and bands). We have computed the two-sided Kuiper statistic (invariant Kolmogorov-Smirnov test), and its associated probability that any of the previously mentioned pairs of stellar groups were drawn from the same distribution. We have calcu- lated the two dimensional density function of each sample considering a 4.5×3 arcmin grid-binning in a 45×30 arcmin region centered at 05:35:08.31, +09:56:03.6 (the star Lambda Orionis). The test reveals that in the first case, the cumulative dis- tribution function of Class II candidates is sig- nificantly different from that of Class III candi- dates, with a probability for these data sets be- ing drawn from the same distribution of ∼ 1%. This situation changes when comparing the set of objects harbouring any kind of disk with that of diskless objects, finding a probability of ∼ 50% in this case (and hence no conclusion can be drawn, other than that there is no strong correlation). On the other hand, regarding a correlation with age, the test points out a trend in the relationship be- tween the spatial distribution of stellar and sub- stellar objects depending on the assumed age. The value of the probability of these two populations sharing a common spatial distribution decreases from a ∼ 30% when assuming an age of 3 Myr, to ∼ 0.001% for an age of 8 Myr. The value assuming an age of 5 Myr is ∼ 1%. The spatial distribution of objects detected at 24 micron can be seen in Figure 13. The nebulosity immediately south of the star λ Orionis (close to BD +09 879 C) corresponds to the HII region LBN 194.69-12.42 (see the detail in Figure 12b in the band [3.6]). Most of the detected members are lo- cated within the inner 9 arcmin circle, with an ap- parent concentration in a “filament” running ap- proximately north-south (i.e. aligned with the B stars as illustrated in Figure 11b). Out of the clus- ter members discovered by Dolan & Mathieu, 11 are within the MIPS [24] image (see Figure 13) and have fluxes above the detection level. The clos- est member to λ Orionis is D&M#33 (LOri034), about 2 arcmin east from the central star. The MIPS image at 24 micron suggest that there are two bubbles centered around the λ Ori- onis multiple star (actually, the center might be the C component or the HII region LBN 194.69- 12.42). The first one is about 25 arcmin away, and it is located along the North-East/South-West axis. More conspicuous is the smaller front lo- cated at a distance of 10.75 arcmin, again cen- tered on the HII region and not in λ Orionis AB. In this case, it is most visible located in the direc- tion West/North-West, opposite to the alignment of Class II objects and low mass members with ex- cess at 24 micron. Similar structures can be found at larger scales in the IRAS images of this region, at 110 and 190 arcmin. The star 37 Ori, a B0III, is located at the center of the cocoon at the bottom of the image. The source IRAS 05320+0927 is very close and it is probably the same. Note that while BD +09 879 C would appear to be the source of a strong stellar wind and/or large UV photon flux, it is not obvious that the visible star is the UV emitter because the spectral type for BD +09 879 C is given as F8V (Lindroos 1985). It would be useful to examine this star more closely in order to try to resolve this mystery. 5. Conclusions We have obtained Spitzer IRAC and MIPS data of an area about one sq.deg around the star λ Ori- onis, the central star of the 5 Myr Lambda Orio- nis open cluster (Collinder 69). These data were combined with our previous optical and near in- frared photometry (from 2MASS). In addition, we have obtained deep near infrared imaging. The samples have been used to assess the member- ship of the 170 candidate members, selected from Barrado y Navascués et al. (2004). By using the Spitzer/IRAC data and the crite- ria developed by Allen et al. (2004) and Hartmann et al. (2005), we have found 33 objects which can be classified as Classical TTauri stars and substel- lar analogs (Class II objects). This means that the fraction of members with disks is 25% and 40%, for the stellar (in the spectral range M0 - M6.5) and substellar population (down to 0.04 M⊙). How- ever, by combining this information with Hα emis- sion (only a fraction of them have spectroscopy), we find that some do not seem to be accreting. Moreover, as expected from models, we see a correlation in the [3.6] - [4.5] vs. [5.8] - [8.0] dia- gram for objects with redder colors (more IR ex- cess) to have stronger Hα emission. In addition, following Lada et al. (2006) and the classification based on the slope of the IRAC data, we found that the ratio of substellar members bearing disks (optically thin or thick) is∼ 50%, whereas is about 31% for the complete sample (14% with thick disks). This result suggests that the timescale for primordial disks to dissipate is longer for lower mass stars, as suggested in Barrado y Navascués & Mart́ın (2003). We have also found that the distribution of Collinder members is very inhomogeneous, specif- ically for the Class II objects. Most of them are located in a filament which goes from the central star λ Orionis to the south-east, more or less to- wards the dark cloud Barnard 35. In addition, there are several Class II stars close to the central stars. If the (previously) highest mass member of C69 has already evolved off the main sequence and become a supernova, either the disks of these Class II stars survived that episode or they formed subsequent to the supernova. We have also derived the fluxes at 24 micron from Spitzer/MIPS imaging. Only a handful – 13– of the low mass stars were detected (no brown dwarfs). Most of them are Class II objects. 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Name I error J error H error Ks error LOri006 12.752 11.67 0.01 10.90 0.01 10.94 0.01 LOri007 12.779 11.65 0.01 – – – – LOri008 12.789 11.53 0.01 10.85 0.01 10.62 0.01 LOri009 12.953 11.79 0.01 – – – – LOri011 13.006 11.59 0.01 – – – – LOri015 13.045 11.90 0.01 – – – – LOri016 13.181 12.00 0.01 11.41 0.01 11.27 0.01 LOri020 13.313 11.95 0.01 – – – – LOri021 13.376 12.26 0.01 – – – – LOri022 13.382 12.20 0.01 11.44 0.01 11.22 0.01 LOri024 13.451 12.35 0.01 – – – – LOri026 13.472 12.00 0.01 – – – – LOri027 13.498 12.51 0.01 – – – – LOri030 13.742 12.44 0.01 11.81 0.01 11.64 0.01 LOri031 13.750 12.34 0.01 – – – – LOri034 13.973 12.43 0.01 – – – – LOri035 13.974 12.56 0.01 – – – – LOri036 13.985 12.53 0.01 – – – – LOri037 13.988 13.43 0.01 – – – – LOri048 14.409 12.78 0.01 12.17 0.01 12.00 0.01 LOri049 14.501 13.13 0.01 – – – – LOri050 14.541 13.17 0.01 – – – – LOri053 14.716 13.17 0.01 – – – – LOri055 14.763 13.24 0.01 – – – – LOri056 14.870 13.33 0.01 – – – – LOri057 15.044 13.43 0.01 – – – – LOri060 15.144 13.60 0.01 – – – – LOri061 15.146 13.38 0.01 12.74 0.01 12.54 0.01 LOri062 15.163 13.60 0.01 – – – – LOri063 15.340 13.72 0.01 13.02 0.01 12.69 0.01 LOri065 15.366 13.66 0.01 13.04 0.01 12.85 0.01 LOri068 15.200 13.73 0.01 – – – – LOri069 15.203 13.28 0.01 – – – – LOri071 15.449 13.72 0.01 – – – – LOri073 15.277 13.68 0.01 – – – – LOri076 15.812 14.12 0.01 13.51 0.01 13.28 0.01 LOri077 15.891 14.11 0.01 – – – – LOri082 16.022 14.18 0.01 13.64 0.01 13.35 0.01 LOri083 16.025 14.22 0.01 13.63 0.01 13.32 0.01 LOri085 16.043 14.21 0.01 13.58 0.01 13.26 0.01 LOri087 16.091 14.44 0.01 – – – – LOri088 16.100 14.14 0.01 – – – – LOri089 16.146 14.43 0.01 – – – – LOri093 16.207 14.47 0.01 – – – – LOri094 16.282 14.37 0.01 – – – – LOri096 16.366 14.59 0.01 13.98 0.01 13.72 0.01 LOri099 16.416 14.62 0.01 – – – – LOri100 16.426 14.83 0.01 – – – – LOri102 16.505 14.57 0.01 14.05 0.01 13.78 0.01 LOri104 16.710 14.90 0.01 – – – – LOri105 16.745 14.84 0.01 – – – – LOri107 16.776 14.91 0.01 14.35 0.01 14.05 0.01 LOri115 17.077 15.35 0.01 – – – – LOri116 17.165 15.31 0.01 – – – – LOri120 17.339 15.36 0.01 – – – – LOri130 17.634 15.76 0.01 – – – – LOri131 17.783 15.29 0.01 14.83 0.01 14.41 0.01 LOri132 17.822 15.77 0.01 – – – – LOri134 17.902 15.72 0.01 15.17 0.01 14.82 0.01 LOri135 17.904 15.63 0.01 15.14 0.01 14.79 0.01 LOri136 17.924 15.53 0.01 – – – – Table 2 Zero points, aperture corrections and conversion factors between the magnitudes and the fluxes in Jansky. Channel Ap. correction ap=4px (mag) Ap. correction ap=2px (mag) Zero Point (mag)a Flux mag=0 (Jy) [3.6] 0.090 0.210 17.26 280.9b [4.5] 0.102 0.228 16.78 179.7b [5.8] 0.101 0.349 16.29 115.0b [8.0] 0.121 0.499 15.62 64.1b [24] 0.168d 11.76 7.14c aZero Points for aperture photometry performed with IRAF on the BCD data. bReach et al. (2005) cEngelbracht et al. (2006) dThe aperture used for MIPS [24] was always 5.31 pixels. Table 3 Candidate members of the Lambda Orionis cluster (Collinder 69) Name R error I error J error H error Ks error [3.6] error [4.5] error [5.8] error [8.0] error [24] error Mem1 LOri001 13.21 0.00 12.52 0.00 11.297 0.022 10.595 0.022 10.426 0.021 10.228 0.003 10.255 0.004 10.214 0.009 10.206 0.010 – – Y LOri002 13.44 0.00 12.64 0.00 11.230 0.024 10.329 0.023 10.088 0.019 9.935 0.003 10.042 0.003 9.930 0.009 9.880 0.008 – – Y LOri003 13.39 0.00 12.65 0.00 11.416 0.023 10.725 0.022 10.524 0.023 10.262 0.003 10.318 0.004 10.239 0.010 10.171 0.010 – – Y LOri004 13.71 0.00 12.65 0.00 11.359 0.022 10.780 0.023 10.548 0.021 10.287 0.003 10.249 0.004 10.185 0.009 10.127 0.009 – – Y LOri005 13.38 0.00 12.67 0.00 11.378 0.022 10.549 0.022 10.354 0.023 10.204 0.003 10.321 0.004 10.218 0.009 10.158 0.009 – – Y LOri006 13.55 0.00 12.75 0.00 11.542 0.026 10.859 0.026 10.648 0.021 10.454 0.003 10.454 0.004 10.399 0.011 10.319 0.010 – – Y LOri007 13.72 0.00 12.78 0.00 11.698 0.027 11.101 0.024 10.895 0.030 10.668 0.004 10.636 0.004 10.615 0.012 10.482 0.013 – – Y LOri008 13.60 0.00 12.79 0.00 11.548 0.029 10.859 0.023 10.651 0.024 10.498 0.003 10.495 0.004 10.440 0.011 10.256 0.012 – – Y LOri009 13.70 0.00 12.95 0.00 11.843 0.024 11.109 0.024 10.923 0.023 10.834 0.004 10.873 0.005 10.788 0.012 10.743 0.014 – – Y LOri010 13.70 0.00 12.96 0.00 11.880 0.026 11.219 0.026 11.041 0.023 10.916 0.004 10.953 0.005 10.733 0.012 10.839 0.016 – – Y LOri011 13.84 0.00 13.01 0.00 11.604 0.026 10.784 0.024 10.554 0.024 10.378 0.003 10.521 0.004 10.444 0.011 10.326 0.011 – – Y LOri012 13.80 0.00 13.03 0.00 11.816 0.026 10.971 0.024 10.795 0.023 10.619 0.003 10.758 0.005 10.627 0.012 10.543 0.012 – – Y LOri013 14.21 0.00 13.03 0.00 11.656 0.022 10.918 0.022 10.719 0.023 10.511 0.003 10.480 0.004 10.467 0.011 10.344 0.012 – – Y LOri014 13.84 0.00 13.03 0.00 11.941 0.024 11.278 0.027 11.092 0.023 10.902 0.004 10.904 0.005 10.839 0.014 10.797 0.014 – – Y LOri015 13.83 0.00 13.05 0.00 11.870 0.024 11.127 0.024 10.912 0.019 10.808 0.004 10.886 0.005 10.824 0.013 10.882 0.015 – – Y LOri016 14.07 0.00 13.18 0.00 11.958 0.024 11.284 0.027 11.053 0.024 10.833 0.004 10.817 0.006 10.378 0.011 10.700 0.014 – – Y LOri017 13.99 0.00 13.19 0.00 12.188 0.024 11.482 0.023 11.323 0.021 11.165 0.005 11.206 0.006 11.173 0.017 11.072 0.019 – – Y LOri018 14.21 0.00 13.26 0.00 11.991 0.024 11.284 0.022 11.090 0.023 10.804 0.004 10.798 0.005 10.722 0.012 10.636 0.014 – – Y LOri019 14.33 0.00 13.31 0.00 12.019 0.026 11.316 0.024 11.067 0.021 10.880 0.004 10.866 0.005 10.767 0.013 10.788 0.018 – – Y LOri020 14.65 0.00 13.31 0.00 11.856 0.028 11.214 0.026 11.025 0.027 10.676 0.003 10.609 0.004 10.573 0.012 10.485 0.012 – – Y LOri021 14.26 0.00 13.38 0.00 12.258 0.027 11.560 0.026 11.296 0.021 11.129 0.004 11.107 0.005 11.081 0.016 11.065 0.019 – – Y LOri022 14.41 0.00 13.38 0.00 12.102 0.023 11.411 0.022 11.156 0.019 11.010 0.004 10.985 0.005 10.895 0.014 10.683 0.014 – – Y LOri023 14.43 0.00 13.44 0.00 12.221 0.027 11.471 0.022 11.290 0.024 11.090 0.004 11.114 0.005 11.071 0.015 10.928 0.018 – – Y LOri024 14.43 0.00 13.45 0.00 12.139 0.030 11.446 0.026 11.223 0.028 11.018 0.004 11.019 0.005 10.972 0.015 10.877 0.016 – – Y LOri025 14.36 0.00 13.45 0.00 12.163 0.044 11.409 0.051 11.090 0.033 10.668 0.003 10.674 0.004 10.613 0.012 10.576 0.012 – – Y LOri026 14.57 0.00 13.47 0.00 12.046 0.028 11.324 0.024 11.092 0.025 10.882 0.004 10.833 0.005 10.811 0.014 10.742 0.013 – – Y LOri027 14.49 0.00 13.50 0.00 12.378 0.026 11.718 0.023 11.503 0.021 11.305 0.005 11.306 0.006 11.237 0.016 11.179 0.025 – – Y LOri028 14.86 0.00 13.65 0.00 12.488 0.024 11.872 0.022 11.687 0.021 11.439 0.005 11.417 0.006 11.348 0.017 11.297 0.021 – – Y LOri029 14.89 0.00 13.69 0.00 12.210 0.026 11.460 0.027 11.071 0.019 10.259 0.003 9.830 0.003 9.321 0.006 8.416 0.003 5.684 0.007 Y LOri030 14.95 0.00 13.74 0.00 12.427 0.027 11.686 0.026 11.428 0.021 11.208 0.007 11.157 0.007 11.119 0.019 10.997 0.023 – – Y LOri031 14.90 0.00 13.75 0.00 12.412 0.028 11.654 0.023 11.442 0.028 11.206 0.004 11.188 0.006 11.150 0.015 11.079 0.016 – – Y LOri032 15.04 0.00 13.80 0.00 12.410 0.029 11.714 0.023 11.493 0.021 11.252 0.004 11.215 0.006 11.178 0.016 11.080 0.019 – – Y LOri033 14.82 0.00 13.81 0.00 12.455 0.033 11.800 0.042 11.502 0.027 11.146 0.004 11.149 0.005 11.060 0.015 11.020 0.019 – – Y LOri034 15.10 0.00 13.97 0.00 12.442 0.026 11.639 0.026 11.184 0.023 10.068 0.003 9.734 0.003 9.314 0.007 8.325 0.003 5.738 0.007 Y LOri035 15.25 0.00 13.97 0.00 12.546 0.024 11.842 0.027 11.609 0.019 11.371 0.005 11.349 0.006 11.283 0.017 11.259 0.021 – – Y LOri036 15.47 0.00 13.98 0.00 12.576 0.024 11.936 0.023 11.706 0.021 11.395 0.005 11.378 0.006 11.287 0.018 11.260 0.019 – – Y LOri037 15.17 0.00 13.99 0.00 12.459 0.024 11.727 0.026 11.492 0.021 11.302 0.005 11.309 0.006 11.198 0.016 11.180 0.018 – – Y LOri038 15.10 0.00 14.01 0.00 12.684 0.030 11.954 0.029 – – 11.455 0.005 11.320 0.006 10.970 0.014 9.857 0.008 6.211 0.010 Y LOri039 15.25 0.00 14.02 0.00 12.755 0.030 12.004 0.023 11.775 0.023 11.523 0.005 11.534 0.007 11.434 0.018 11.373 0.025 – – Y LOri040 15.38 0.00 14.06 0.00 12.553 0.024 11.877 0.022 11.594 0.024 11.364 0.005 11.319 0.006 11.231 0.017 11.218 0.025 – – Y LOri041 15.55 0.00 14.10 0.00 12.500 0.024 11.856 0.023 11.587 0.027 11.255 0.004 11.187 0.006 11.131 0.015 11.123 0.021 – – Y LOri042 15.31 0.00 14.14 0.00 12.813 0.027 12.099 0.026 11.853 0.023 11.604 0.005 11.633 0.007 11.546 0.019 11.479 0.025 – – Y LOri043 15.46 0.00 14.16 0.00 12.707 0.024 12.021 0.026 11.741 0.024 11.512 0.005 11.496 0.007 11.408 0.019 11.393 0.024 8.479 0.102 Y LOri044 15.39 0.00 14.17 0.00 12.924 0.024 12.318 0.024 12.065 0.023 11.837 0.006 11.804 0.007 11.751 0.023 11.674 0.024 – – Y LOri045 15.56 0.00 14.23 0.00 12.768 0.023 12.102 0.026 11.844 0.023 11.602 0.005 11.596 0.007 12.023 0.039 11.474 0.026 – – Y LOri046 15.64 0.00 14.36 0.00 13.033 0.023 12.478 0.026 12.252 0.026 11.906 0.006 11.852 0.008 11.787 0.024 11.763 0.032 – – Y LOri047 15.91 0.00 14.38 0.00 12.732 0.026 12.097 0.031 11.827 0.026 11.474 0.005 11.400 0.006 11.303 0.017 11.342 0.025 – – Y LOri048 15.78 0.00 14.41 0.00 12.887 0.027 12.196 0.029 11.932 0.026 11.612 0.006 11.521 0.008 11.448 0.020 10.920 0.018 8.119 0.087 Y Table 3—Continued Name R error I error J error H error Ks error [3.6] error [4.5] error [5.8] error [8.0] error [24] error Mem1 LOri049 15.77 0.00 14.50 0.00 13.173 0.027 12.592 0.029 12.253 0.023 12.004 0.006 11.992 0.009 12.043 0.027 12.427 0.057 – – Y LOri050 15.90 0.00 14.54 0.00 12.877 0.027 12.236 0.027 11.955 0.031 11.471 0.005 11.089 0.005 10.529 0.011 9.537 0.007 7.268 0.029 Y LOri051 15.91 0.00 14.60 0.00 13.266 0.024 12.559 0.022 12.285 0.021 12.017 0.006 11.995 0.008 11.969 0.024 11.919 0.035 – – Y LOri052 15.93 0.00 14.63 0.00 13.117 0.023 12.454 0.024 12.192 0.019 11.917 0.006 11.863 0.008 11.764 0.022 11.791 0.029 – – Y LOri053 16.08 0.00 14.72 0.00 13.173 0.032 12.521 0.023 12.278 0.027 11.995 0.006 11.954 0.008 11.886 0.022 11.862 0.034 – – Y LOri054 16.19 0.00 14.73 0.00 13.189 0.024 12.509 0.022 12.271 0.027 11.974 0.006 11.948 0.009 11.805 0.025 11.862 0.038 – – Y LOri055 16.12 0.00 14.76 0.00 13.184 0.026 12.477 0.026 12.253 0.026 12.044 0.006 12.038 0.009 12.015 0.029 11.902 0.038 – – Y LOri056 16.43 0.00 14.87 0.00 13.211 0.029 12.567 0.026 12.267 0.029 12.011 0.004211.906 0.005211.913 0.019211.853 0.0322– – Y LOri057 16.63 0.00 15.04 0.00 13.412 0.024 12.773 0.023 12.487 0.030 12.177 0.007 12.078 0.009 11.988 0.030 11.992 0.033 – – Y LOri058 16.57 0.00 15.06 0.00 13.521 0.024 12.935 0.022 12.643 0.027 12.332 0.007 12.269 0.010 12.172 0.032 12.637 0.072 – – Y LOri059 16.57 0.00 15.10 0.00 13.574 0.026 12.884 0.026 12.682 0.032 12.317 0.007 12.270 0.009 12.218 0.030 12.679 0.066 – – Y LOri060 16.56 0.00 15.14 0.00 13.598 0.030 12.961 0.030 12.663 0.029 12.423 0.008 12.418 0.011 12.408 0.041 12.377 0.051 – – Y LOri061 16.58 0.00 15.15 0.00 13.533 0.023 12.833 0.026 12.525 0.027 12.052 0.006 11.851 0.008 11.519 0.019 10.730 0.015 8.047 0.046 Y LOri062 16.62 0.00 15.16 0.00 13.634 0.029 13.005 0.030 12.725 0.027 12.370 0.007 12.246 0.009 12.153 0.030 11.306 0.021 7.834 0.035 Y LOri063 16.80 0.01 15.34 0.00 13.756 0.029 13.066 0.029 12.663 0.030 11.666 0.006 11.368 0.007 11.768 0.028 10.397 0.014 6.055 0.010 Y LOri064 16.78 0.01 15.34 0.00 13.782 0.026 13.098 0.025 12.846 0.029 12.486 0.008 12.489 0.011 12.378 0.034 12.245 0.053 – – Y LOri065 16.89 0.01 15.37 0.00 13.820 0.024 13.123 0.029 12.843 0.027 12.526 0.008 12.504 0.011 12.494 0.032 12.641 0.063 8.424 0.075 Y LOri066 17.12 0.01 15.40 0.00 13.506 0.024 12.901 0.026 12.654 0.029 12.221 0.007 12.170 0.010 12.196 0.039 12.578 0.072 – – Y LOri067 17.05 0.01 15.53 0.00 14.000 0.033 13.356 0.027 13.102 0.036 12.794 0.010 12.727 0.014 12.702 0.046 12.786 0.071 – – Y LOri068 16.76 0.01 15.20 0.00 13.521 0.027 12.902 0.026 12.628 0.027 12.348 0.005212.246 0.006212.029 0.016212.145 0.0362– – Y LOri069 16.89 0.01 15.20 0.00 13.384 0.027 12.774 0.027 12.425 0.027 12.089 0.006 12.015 0.008 12.034 0.026 11.903 0.042 – – Y LOri070 17.18 0.01 15.61 0.00 14.042 0.032 13.405 0.029 13.067 0.031 12.809 0.009 12.779 0.013 12.799 0.041 12.559 0.060 – – Y LOri071 17.13 0.00 15.63 0.00 13.749 0.030 13.129 0.024 12.839 0.031 12.470 0.008 12.382 0.010 12.276 0.031 12.250 0.044 – – Y LOri072 17.00 0.00 15.35 0.00 13.554 0.026 12.944 0.032 12.631 0.027 11.993 0.006 11.860 0.008 11.836 0.026 11.718 0.037 – – Y LOri073 16.84 0.01 15.28 0.00 13.644 0.028 12.992 0.023 12.715 0.027 12.376 0.007 12.274 0.009 12.187 0.029 12.162 0.031 – – Y LOri074 17.03 0.01 15.39 0.00 13.663 0.026 13.088 0.025 12.720 0.024 12.312 0.007 12.310 0.010 12.290 0.030 12.259 0.046 – – Y LOri075 16.95 0.01 15.23 0.00 13.396 0.026 12.794 0.026 12.526 0.024 12.089 0.006 11.990 0.008 11.924 0.026 11.936 0.038 – – Y LOri076 17.39 0.01 15.81 0.00 14.216 0.027 13.527 0.027 13.201 0.032 12.916 0.010 12.843 0.014 12.669 0.048 12.754 0.072 – – Y LOri077 17.45 0.00 15.89 0.00 14.031 0.027 13.416 0.027 13.109 0.035 12.761 0.009 12.717 0.012 12.700 0.046 12.650 0.073 – – Y LOri078 17.35 0.00 15.92 0.00 14.227 0.041 13.593 0.053 13.286 0.040 12.766 0.009 12.844 0.014 12.789 0.046 12.554 0.069 – – Y LOri079 17.51 0.00 16.00 0.00 14.221 0.032 13.536 0.032 13.338 0.039 13.002 0.006212.970 0.008212.876 0.078212.724 0.0482– – Y LOri080 17.51 0.00 16.01 0.00 13.804 0.023 13.196 0.022 12.891 0.033 12.504 0.008 12.424 0.010 12.597 0.041 12.190 0.031 – – Y LOri081 17.61 0.00 16.02 0.00 14.669 0.032 13.692 0.032 13.209 0.037 12.620 0.008 12.360 0.010 12.050 0.030 11.632 0.028 8.062 0.056 Y LOri082 17.57 0.00 16.02 0.00 14.200 0.033 13.570 0.025 13.281 0.033 13.008 0.011 12.954 0.015 13.586 0.100 12.830 0.100 – – Y LOri083 17.56 0.00 16.02 0.00 14.265 0.030 13.638 0.035 13.375 0.040 13.012 0.010 12.946 0.013 12.947 0.057 13.017 0.074 – – Y LOri084 17.48 0.00 16.03 0.00 14.077 0.024 13.448 0.027 13.188 0.034 12.888 0.010 12.800 0.013 12.863 0.061 12.745 0.067 – – Y LOri085 17.65 0.01 16.04 0.00 14.189 0.026 13.622 0.037 13.233 0.027 12.584 0.008 12.315 0.010 12.090 0.028 11.510 0.029 8.089 0.056 Y LOri086 17.59 0.00 16.09 0.00 14.482 0.032 13.867 0.032 13.503 0.040 13.251 0.011 13.205 0.016 13.160 0.057 13.277 0.098 – – Y LOri087 17.54 0.00 16.09 0.00 14.186 0.039 13.601 0.030 13.279 0.035 12.978 0.010 12.894 0.013 13.002 0.064 12.669 0.060 – – Y LOri088 17.78 0.00 16.10 0.00 14.140 0.031 13.543 0.037 13.228 0.039 12.923 0.009 12.853 0.013 12.865 0.040 12.676 0.051 – – Y LOri089 17.79 0.00 16.15 0.00 14.380 0.032 13.839 0.035 13.512 0.039 13.156 0.011 13.123 0.015 13.682 0.086 12.877 0.081 – – Y LOri090 17.77 0.00 16.17 0.00 14.515 0.041 13.881 0.023 13.651 0.051 13.226 0.011 13.116 0.015 12.930 0.047 13.126 0.100 – – Y LOri091 18.01 0.00 16.18 0.00 14.184 0.032 13.556 0.032 13.289 0.031 12.868 0.009 12.803 0.013 13.087 0.062 12.462 0.054 – – Y LOri092 17.84 0.00 16.19 0.00 14.441 0.030 13.841 0.038 13.537 0.040 13.158 0.011 13.053 0.014 13.517 0.087 12.992 0.089 – – Y LOri093 17.82 0.00 16.21 0.00 14.462 0.030 13.836 0.039 13.604 0.052 13.169 0.011 13.104 0.015 13.256 0.073 12.982 0.098 – – Y LOri094 18.03 0.00 16.28 0.00 14.404 0.034 13.802 0.030 13.425 0.038 12.955 0.009 12.994 0.014 13.184 0.058 12.894 0.085 – – Y LOri095 17.96 0.00 16.35 0.00 14.564 0.033 13.913 0.029 13.613 0.048 13.247 0.012 13.278 0.017 13.176 0.067 13.340 0.138 – – Y LOri096 18.02 0.02 16.37 0.00 14.627 0.038 13.965 0.037 13.638 0.047 13.039 0.010 12.732 0.013 12.527 0.045 12.029 0.039 – – Y Table 3—Continued Name R error I error J error H error Ks error [3.6] error [4.5] error [5.8] error [8.0] error [24] error Mem1 LOri098 18.12 0.00 16.40 0.00 14.647 0.037 13.985 0.045 13.682 0.039 13.393 0.012 13.301 0.016 13.284 0.075 13.182 0.115 – – Y LOri099 18.14 0.00 16.42 0.00 14.709 0.034 14.074 0.035 13.676 0.043 13.421 0.013 13.335 0.018 13.211 0.069 13.352 0.124 – – Y LOri100 18.08 0.00 16.43 0.00 14.768 0.044 14.044 0.042 13.821 0.044 13.446 0.012 13.325 0.017 13.163 0.066 13.318 0.123 – – Y LOri101 18.14 0.00 16.48 0.00 15.019 0.038 14.372 0.044 14.110 0.066 13.763 0.015 13.627 0.021 13.860 0.105 13.475 0.153 – – N LOri102 18.24 0.00 16.50 0.00 14.634 0.047 14.083 0.050 13.809 0.057 13.296 0.012 13.213 0.015 13.275 0.073 13.101 0.108 – – Y LOri103 18.30 0.00 16.55 0.00 14.643 0.029 14.126 0.029 13.833 0.055 13.425 0.013 13.387 0.018 13.140 0.067 13.628 0.151 – – Y LOri104 18.48 0.03 16.71 0.01 14.667 0.030 14.136 0.036 13.721 0.042 13.143 0.011 12.877 0.014 12.694 0.055 11.762 0.035 – – Y LOri105 18.58 0.00 16.75 0.00 14.922 0.040 14.340 0.052 13.993 0.053 13.621 0.016 13.593 0.021 13.601 0.093 13.536 0.159 – – Y LOri106 18.48 0.00 16.76 0.00 14.776 0.043 14.161 0.057 13.743 0.045 13.295 0.012 12.967 0.014 12.558 0.045 11.832 0.034 8.849 0.151 Y LOri107 18.85 0.00 16.78 0.00 14.656 0.036 13.987 0.035 13.621 0.052 13.213 0.017 13.152 0.019 13.215 0.082 13.046 0.160 – – Y LOri108 18.64 0.00 16.80 0.00 14.840 0.033 14.256 0.048 13.918 0.050 13.498 0.013 13.464 0.018 13.387 0.073 13.303 0.124 – – Y LOri109 18.67 0.00 16.81 0.00 14.96 0.01 14.47 0.01 14.18 0.01 13.699 0.015 13.654 0.022 13.519 0.077 13.708 0.190 – – Y LOri110 18.54 0.00 16.82 0.00 15.043 0.051 14.475 0.056 14.144 0.060 13.798 0.016 13.796 0.026 14.017 0.130 13.454 0.177 – – Y LOri111 18.88 0.00 16.86 0.00 14.801 0.038 14.165 0.043 13.786 0.051 13.419 0.012 13.330 0.017 13.601 0.082 13.664 0.185 – – Y LOri112 18.72 0.00 16.87 0.00 14.991 0.042 14.358 0.048 14.148 0.062 13.412 0.013 13.335 0.017 13.334 0.079 13.299 0.145 – – Y LOri113 18.71 0.00 16.99 0.00 15.18 0.01 14.62 0.01 14.30 0.01 13.723 0.017 13.579 0.021 13.263 0.070 12.448 0.054 – – Y LOri114 18.99 0.00 17.06 0.00 15.092 0.044 14.389 0.053 14.006 0.064 13.502 0.015 13.414 0.020 13.525 0.096 13.051 0.108 – – Y LOri115 18.80 0.00 17.08 0.00 15.449 0.047 14.821 0.068 14.594 0.104 14.083 0.017 14.012 0.030 13.942 0.119 13.346 0.131 – – Y LOri116 19.05 0.01 17.17 0.00 15.343 0.057 14.573 0.055 14.411 0.082 13.977 0.017 13.847 0.024 14.340 0.190 13.614 0.141 – – Y LOri117 19.24 0.01 17.21 0.00 15.10 0.01 14.36 0.01 14.17 0.01 13.418 0.018 13.102 0.024 13.063 0.105 – – – – Y LOri118 19.10 0.01 17.23 0.00 15.269 0.044 14.686 0.064 14.181 0.057 13.430 0.013 13.251 0.016 12.844 0.045 12.178 0.043 – – Y LOri119 19.11 0.00 17.30 0.00 15.26 0.01 14.74 0.01 14.41 0.01 13.568 0.014 13.492 0.019 13.408 0.088 13.590 0.170 – – Y LOri120 19.23 0.00 17.34 0.00 15.335 0.050 14.770 0.059 14.337 0.087 13.878 0.015 13.688 0.020 13.458 0.086 12.783 0.070 – – Y LOri121 19.12 0.00 17.37 0.00 15.533 0.060 15.093 0.086 14.748 0.099 14.336 0.019 14.310 0.031 14.053 0.144 – – – – Y LOri122 19.31 0.00 17.38 0.00 15.428 0.066 14.852 0.060 14.462 0.080 14.096 0.018 13.973 0.027 14.315 0.190 13.659 0.174 – – Y LOri124 19.30 0.00 17.45 0.00 15.661 0.073 15.059 0.082 14.778 0.112 14.353 0.012214.235 0.018214.303 0.086213.976 0.1262– – Y LOri125 19.29 0.04 17.51 0.01 15.661 0.073 15.059 0.082 14.778 0.112 14.353 0.012214.235 0.018214.303 0.086213.976 0.1262– – Y LOri126 19.52 0.01 17.52 0.00 15.62 0.01 15.04 0.01 14.67 0.01 13.709 0.014 13.577 0.021 13.118 0.063 12.352 0.049 – – Y LOri127 19.87 0.10 17.53 0.01 13.016 0.023 12.606 0.027 12.468 0.024 12.401 0.007 12.348 0.010 12.301 0.033 12.352 0.037 – – N LOri128 19.53 0.01 17.58 0.00 15.624 0.077 15.099 0.087 14.769 0.109 14.150 0.021 14.115 0.031 14.327 0.254 13.956 0.209 – – Y LOri129 19.51 0.01 17.59 0.00 15.383 0.056 14.816 0.072 14.526 0.102 13.625 0.014 13.317 0.016 13.194 0.058 12.598 0.057 – – Y LOri130 19.44 0.01 17.63 0.00 15.731 0.059 15.265 0.092 14.735 0.110 14.408 0.013214.382 0.020214.041 0.077214.348 0.2562– – Y LOri131 19.79 0.01 17.78 0.00 15.429 0.054 14.900 0.063 14.380 0.090 13.991 0.017 13.909 0.025 13.865 0.119 13.344 0.106 – – Y LOri132 19.99 0.01 17.82 0.00 15.583 0.067 14.962 0.078 14.913 0.145 14.173 0.019 14.087 0.025 14.076 0.140 13.630 0.129 – – Y LOri133 19.68 0.01 17.83 0.00 16.290 0.101 15.900 0.167 15.378 0.203 15.066 0.032 14.941 0.041 15.077 0.312 – – – – N LOri134 19.91 0.01 17.90 0.00 15.543 0.057 14.937 0.074 14.666 0.107 14.321 0.020 14.071 0.027 13.880 0.113 13.884 0.154 – – Y LOri135 19.91 0.01 17.90 0.00 15.671 0.072 15.082 0.087 14.908 0.138 14.334 0.014214.166 0.018214.171 0.113213.871 0.1082– – Y LOri136 20.06 0.12 17.92 0.01 15.560 0.085 14.828 0.090 14.576 0.108 14.139 0.016214.224 0.023213.948 0.083213.607 0.1072– – Y LOri137 19.89 0.08 17.96 0.09 – – – – – – 16.454 0.073 16.789 0.270 – – – – – – N LOri138 20.01 0.01 17.96 0.00 15.821 0.078 15.204 0.083 14.971 0.133 14.527 0.022 14.469 0.035 14.150 0.123 – – – – Y LOri139 20.04 0.01 18.16 0.00 16.16 0.01 15.53 0.01 15.06 0.01 14.054 0.017 13.658 0.019 13.151 0.056 12.663 0.062 – – Y LOri140 20.34 0.01 18.21 0.00 15.981 0.078 15.224 0.089 14.750 0.113 14.030 0.017 13.704 0.023 13.299 0.066 12.786 0.078 – – Y LOri141 20.44 0.01 18.25 0.00 16.61 0.01 15.89 0.01 15.68 0.02 15.100 0.034 15.668 0.089 – – – – – – N LOri142 20.34 0.01 18.27 0.00 16.25 0.01 15.58 0.01 15.26 0.01 14.705 0.028 14.674 0.041 – – – – – – Y LOri143 20.32 0.01 18.30 0.00 16.11 0.01 15.61 0.02 15.23 0.01 14.835 0.015 14.896 0.023 14.765 0.089 14.553 0.126 – – Y LOri144 20.24 0.11 18.30 0.11 17.69 0.02 16.90 0.02 16.55 0.03 16.476 0.094 16.424 0.184 – – – – – – N? LOri146 20.88 0.26 18.60 0.02 16.230 0.107 15.470 0.110 14.936 0.128 14.404 0.022214.199 0.035213.836 0.103213.614 0.1752– – Y LOri147 20.54 0.01 18.60 0.00 16.58 0.02 15.93 0.02 15.62 0.02 15.348 0.023215.675 0.048215.128 0.2242– – – – Y? Table 3—Continued Name R error I error J error H error Ks error [3.6] error [4.5] error [5.8] error [8.0] error [24] error Mem1 LOri148 20.77 0.02 18.62 0.00 16.39 0.01 16.12 0.01 15.98 0.02 14.869 0.030 14.989 0.051 14.530 0.210 – – – – Y? LOri149 21.07 0.02 18.95 0.00 99.99 0.00 88.88 0.00 16.97 0.02 17.135 0.132216.829 0.0992– – – – – – N LOri150 21.29 0.03 19.00 0.00 16.656 0.152 16.134 0.197 15.560 0.214 15.015 0.032 15.133 0.070 14.942 0.390 – – – – Y LOri151 20.98 0.02 19.00 0.00 17.40 0.02 16.76 0.02 16.52 0.04 15.801 0.056 15.716 0.103 – – – – – – N LOri152 21.43 0.04 19.05 0.00 16.773 0.173 16.657 0.295 15.870 0.285 16.313 0.086 16.316 0.158 – – – – – – N? LOri153 21.30 0.03 19.17 0.00 17.09 0.01 16.37 0.01 16.09 0.03 15.223 0.036 15.139 0.072 – – – – – – Y LOri154 21.79 0.05 19.31 0.00 16.804 0.169 16.143 0.192 15.513 0.219 15.071 0.035 15.953 0.141 – – – – – – Y LOri155 21.87 0.06 19.36 0.00 16.97 0.01 16.30 0.01 15.84 0.02 15.085 0.019215.412 0.045214.878 0.163214.517 0.1832– – Y LOri156 22.05 0.06 19.59 0.01 17.06 0.02 16.34 0.02 15.89 0.02 14.942 0.029 14.688 0.038 14.127 0.148 13.870 0.146 – – Y LOri157 22.09 0.06 19.63 0.01 18.08 0.03 17.42 0.03 17.00 0.04 16.907 0.103 16.719 0.161 – – – – – – N LOri158 22.07 0.05 19.67 0.01 18.59 0.03 17.86 0.05 17.61 0.08 17.627 0.123217.565 0.2452– – – – – – N? LOri159 22.25 0.06 20.01 0.01 18.21 0.02 17.62 0.05 17.47 0.09 16.422 0.089 16.627 0.207 – – – – – – N? LOri160 22.82 0.13 20.29 0.02 18.11 0.03 17.14 0.02 16.38 0.03 15.669 0.052 15.384 0.079 – – – – – – Y LOri161 23.09 0.19 20.34 0.01 17.71 0.02 16.90 0.02 16.51 0.03 16.361 0.122 16.451 0.249 – – – – – – Y LOri162 23.22 0.51 20.42 0.02 17.64 0.03 16.90 0.09 16.52 0.04 15.675 0.062 15.733 0.112 – – – – – – Y LOri163 22.96 0.24 20.42 0.02 17.86 0.04 17.02 0.08 16.76 0.05 15.666 0.062 15.904 0.119 – – – – – – Y LOri164 23.11 0.17 20.44 0.01 18.75 0.04 18.17 0.05 18.31 0.13 – – – – – – – – – – N LOri165 23.12 0.22 20.73 0.02 18.77 0.08 18.11 0.16 17.90 0.09 16.377 0.112 16.130 0.189 – – – – – – N? LOri166 23.33 0.18 20.75 0.02 18.26 0.03 88.88 0.00 17.38 0.04 16.655 0.055217.232 0.2372– – – – – – Y? LOri167 23.86 0.64 20.90 0.02 18.01 0.03 17.17 0.07 16.83 0.09 15.935 0.063 16.060 0.129 – – – – – – Y LOri168 24.15 0.62 21.54 0.04 19.39 0.09 18.58 0.08 18.70 0.25 – – – – – – – – – – N? LOri169 24.83 1.10 21.88 0.05 20.10 0.10 19.47 0.15 018.93 0.42 – – – – – – – – – – Y? LOri170 25.41 2.61 22.06 0.07 20.35 0.20 19.20 0.20 019.39 0.46 – – – – – – – – – – N? 1Final membership. 22-pixel aperture radius used for the photometry due to the presence of nearby objects or hot pixels. ∗LOri097 and LOri145 are artifacts. LOri123 is a non member and has uncertainties in its photometry. Table 4 Candidate members of the Lambda Orionis cluster (Collinder 69) Name SpT Phot.Mem1 Mem2 Mem3 IRAC classification4 SED slope5 Disk type Comment6 LOri001 – Y Y Y Y Y Y Y Mem Y III -2.8 Diskless Ha- WHa(DM)=2.51 DM#01 LOri002 – Y Y Y Y Y Y Y NM- Y III -2.74 Diskless – LOri003 – Y Y Y Y Y Y Y Mem Y III -2.71 Diskless Ha- WHa(DM)=3.35 DM#46 LOri004 – Y Y Y Y Y Y Y Mem Y III -2.65 Diskless – LOri005 – Y Y Y Y Y Y Y NM- Y III -2.75 Diskless – LOri006 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless – LOri007 – Y Y Y Y Y Y Y Mem Y III -2.63 Diskless – LOri008 – Y Y Y Y Y Y Y Mem Y III -2.56 Diskless Ha- WHa(DM)=1.65 DM#51 LOri009 – Y Y Y Y Y Y Y Mem Y III -2.71 Diskless – LOri010 – Y Y Y Y Y Y Y Mem Y III -2.69 Diskless – LOri011 – Y Y Y Y Y Y Y NM- Y III -2.70 Diskless – LOri012 – Y Y Y Y Y Y Y NM- Y III -2.70 Diskless – LOri013 – Y Y Y Y Y Y Y Mem Y III -2.66 Diskless Ha- WHa(DM)=4.41 DM#04 LOri014 – Y Y Y Y Y Y Y Mem Y III -2.71 Diskless Ha- WHa(DM)=1.45 DM#58 LOri015 – Y Y Y Y Y Y Y Mem Y III -2.89 Diskless – LOri016 – Y Y Y Y Y Y Y Mem Y III -2.58 Diskless – LOri017 – Y Y Y Y Y Y Y Mem Y III -2.72 Diskless Ha- WHa(DM)=0.80 DM#60 LOri018 – Y Y Y Y Y Y Y Mem Y III -2.63 Diskless Ha- WHa(DM)=2.02 DM#56 LOri019 – Y Y Y Y Y Y Y Mem Y III -2.71 Diskless – LOri020 – Y Y Y Y Y Y Y Mem Y III -2.63 Diskless – LOri021 – Y Y Y Y Y Y Y Mem Y III -2.76 Diskless Ha- WHa(DM)=1.47 DM#25 LOri022 – Y Y Y Y Y Y Y Mem Y III -2.65 Diskless Ha- WHa(DM)=4.39 DM#44 LOri023 – Y Y Y Y Y Y Y Mem Y III -2.65 Diskless Ha- WHa(DM)=1.95 DM#50 LOri024 – Y Y Y Y Y Y Y Mem Y III -2.67 Diskless – LOri025 – Y Y Y Y Y Y Y Mem? Y III -2.72 Diskless Ha- WHa(DM)=3.95 DM#59 LOri026 – Y Y Y Y Y Y Y Mem Y III -2.69 Diskless Ha- WHa(DM)=6.07 DM#12 LOri027 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless – LOri028 – Y Y Y Y Y Y Y Mem? Y III -2.67 Diskless – LOri029 – Y Y Y Y Y Y Y NM- Y II -0.72 Thick Ha+ WHa(DM)=30.00 DM#36 LOri030 – Y Y Y Y Y Y Y Mem Y III -2.60 Diskless – LOri031 M4.0 Y Y Y Y Y Y Y Mem Y III -2.69 Diskless Ha- WHa=3.8 DM#20 WHa(DM)= 3.45 LOri032 – Y Y Y Y Y Y Y Mem Y III -2.64 Diskless Ha- WHa(DM)=6.83 DM#55 LOri033 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless Ha- WHa(DM)=3.14 DM#39 LOri034 – Y Y Y Y Y Y Y NM- Y II -0.85 Thick Ha+ WHa(DM)=10.92 DM#33 LOri035 – Y Y Y Y Y Y Y Mem Y III -2.70 Diskless Ha- WHa(DM)=4.13 DM#29 LOri036 – Y Y Y Y Y Y Y Mem Y III -2.67 Diskless – LOri037 – Y Y Y Y Y Y Y Mem Y III -2.67 Diskless Ha- WHa(DM)=3.63 DM#11 LOri038 – Y – Y Y Y – – Mem Y I/II -1.00 Thick Ha+ WHa(DM)=24.95 DM#02 LOri039 – Y Y Y Y Y Y Y Mem Y III -2.65 Diskless Ha- WHa(DM)=3.59 DM#49 LOri040 – Y Y Y Y Y Y Y Mem Y III -2.66 Diskless Ha- WHa(DM)=3.90 DM#41 LOri041 – Y Y Y Y Y Y Y Mem Y III -2.69 Diskless Ha- WHa(DM)=8.20 DM#38 LOri042 M4.0 Y Y Y Y Y Y Y Mem Y III -2.67 Diskless Ha- WHa=4.3 DM#54 WHa(DM)= 4.22 LOri043 – Y Y Y Y Y Y Y Mem Y III -2.69 Transition – LOri044 – Y Y Y Y Y Y Y Mem Y III -2.65 Diskless – LOri045 – Y Y Y Y Y Y Y Mem Y II7 -2.68 Diskless – LOri046 – Y Y Y Y Y Y Y Mem? Y III -2.67 Diskless – LOri047 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless Ha- WHa(DM)=8.65 DM#47 LOri048 – Y Y Y Y Y Y Y Mem Y II -2.07 Thin – LOri049 – Y Y Y Y Y Y Y Mem Y III -3.32 Diskless – LOri050 M4.5 Y Y Y Y Y Y Y Mem Y II -0.60 Thick 200km WHa=15.6 LOri051 – Y Y Y Y Y Y Y Mem Y III -2.73 Diskless – LOri052 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless – LOri053 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless – LOri054 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless – LOri055 M4.5: Y Y Y Y Y Y Y Mem Y III -2.68 Diskless Ha- WHa=8.2 LOri056 M4.5: Y Y Y Y Y Y Y Mem Y III -2.68 Diskless Ha- WHa=7.2 LOri057 M5.5 Y Y Y Y Y Y Y Mem Y III -2.62 Diskless Ha- WHa=8.4 LOri058 M4.5: Y Y Y Y Y Y Y Mem Y III -3.16 Diskless Ha- WHa=7.3 LOri059 M4.5 Y Y Y Y Y Y Y Mem Y III -3.23 Diskless Ha- WHa=8.7 LOri060 M4.5: Y Y Y Y Y Y Y Mem Y III -2.79 Diskless Ha- WHa=4.1 LOri061 – Y Y Y Y Y Y Y Mem Y II -1.32 Thick – LOri062 – Y Y Y Y Y Y Y Mem Y II -1.66 Thick – LOri063 M4.5: Y Y Y Y Y Y Y Mem? Y I/II -1.58 Thick Ha+FL WHa=12.8 LOri064 – Y Y Y Y Y Y Y Mem Y III -2.54 Thin – Table 4—Continued Name SpT Phot.Mem1 Mem2 Mem3 IRAC classification4 SED slope5 Disk type Comment6 LOri065 – Y Y Y Y Y Y Y Mem Y III -2.97 Transition – LOri066 – Y Y Y Y Y Y Y Mem Y III -3.25 Diskless – LOri067 – Y Y Y Y Y Y Y Mem Y III -2.83 Diskless – LOri068 M5.0 Y Y Y Y Y Y Y Mem? Y III -2.57 Diskless Ha+ WHa=16.6 LOri069 – Y Y Y Y Y Y Y Mem? Y III -2.65 Diskless – LOri070 – Y Y Y Y Y Y Y Mem Y III -2.57 Diskless – LOri071 M5.0 Y Y Y Y Y Y Y Mem Y III -2.58 Diskless Ha- WHa=8.0 LOri072 – Y Y Y Y Y Y Y Mem? Y III -2.55 Thin – LOri073 M5.0 Y Y Y Y Y Y Y Mem? Y III -2.59 Diskless Ha+? WHa=12.0 LOri074 – Y Y Y Y Y Y Y Mem? Y III -2.78 Diskless – LOri075 M5.5 Y Y Y Y Y Y Y Mem? Y III -2.67 Diskless Ha- WHa=9.4 WHa=9.4 LOri076 – Y Y Y Y Y Y Y Mem Y III -2.62 Diskless – LOri077 M5.0 Y Y Y Y Y Y Y Mem Y III -2.72 Diskless Ha- WHa=8.8 LOri078 – Y Y Y Y Y Y Y Mem Y III -2.58 Diskless – LOri079 – Y Y Y Y Y Y Y Mem? Y III -2.51 Thin – LOri080 M5.5 Y Y Y Y Y Y Y Mem Y II -2.55 Thin Ha+? WHa=14.3 LOri081 M5.5 N Y Y Y Y Y Y Mem+ Y II -1.70 Thick Ha- WHa=4.2 LOri082 M4.5 Y Y Y Y Y Y Y Mem+ Y II7 -2.82 Diskless Ha- WHa=8.6 LOri083 – Y Y Y Y Y Y Y Mem Y III -2.85 Diskless – LOri084 – Y Y Y Y Y Y Y Mem Y III -2.71 Diskless – LOri085 – Y Y Y Y Y Y Y Mem Y II -1.63 Thick – LOri086 – Y Y Y Y Y Y Y Mem Y III -2.86 Diskless – LOri087 M4.5 Y Y Y Y Y Y Y Mem+ Y III -2.54 Thin Ha- WHa=6.7 LOri088 – Y Y Y Y Y Y Y Mem Y III -2.58 Diskless – LOri089 M5.0 Y Y Y Y Y Y Y Mem Y II -2.50 Thin Ha- WHa=5.1 LOri090 – Y Y Y Y Y Y Y Mem Y III -2.69 Diskless – LOri091 M5.5 Y Y Y Y Y Y Y Mem Y II -2.48 Thin Ha+? WHa=14.7 LOri092 – Y Y Y Y Y Y Y Mem Y II -2.79 Diskless – LOri093 – Y Y Y Y Y Y Y Mem Y III -2.68 Diskless – LOri094 M5.5 Y Y Y Y Y Y Y Mem Y III -2.82 Diskless Ha- WHa=10.4 LOri095 M6.0 Y Y Y Y Y Y Y Mem+ Y III -2.91 Diskless Ha- WHa=7.3 LOri096 – Y Y Y Y Y Y Y Mem Y II -1.71 Thick – LOri098 M5.0 Y Y Y Y Y Y Y Mem+ Y III -2.61 Diskless Ha- WHa=12.9 LOri099 M5.25 Y Y Y Y Y Y Y Mem Y III -2.74 Diskless Ha- WHa=6.6 LOri100 M5.5 Y Y Y Y Y Y Y Mem Y III -2.67 Diskless Ha+? WHa=13.1 LOri101 – N N ? ? Y Y Y Mem? N III -2.6 Diskless – LOri102 – Y Y Y Y Y Y Y Mem? Y III -2.65 Diskless – LOri103 – Y Y Y Y Y Y Y Mem? Y III8 -2.31 Thin – LOri104 – Y Y Y Y Y Y Y Mem Y II -1.30 Thick – LOri105 – Y Y Y Y Y Y Y Mem Y III -2.75 Diskless – LOri106 M5.5 Y Y Y Y Y Y Y Mem Y II -1.16 Thick Ha+ WHa=54.0 LOri107 M6.0 Y Y Y Y Y Y Y Mem+ Y III -2.68 Diskless Ha- WHa=11.7 LOri108 – Y Y Y Y Y Y Y Mem Y III -2.61 Diskless – LOri109 M5.5 Y Y Y Y Y Y Y Mem Y III -2.82 Diskless Ha- WHa=10.1 LOri110 M5.5 Y Y Y Y Y Y Y Mem Y II -2.52 Thin Ha- WHa=9.1 LOri111 – Y Y Y Y Y Y Y Mem Y III -3.19 Diskless – LOri112 – Y Y Y Y Y Y Y NM- Y III -2.72 Diskless – LOri113 M5.5 Y Y Y Y Y Y Y Mem Y II -1.37 Thick Ha+ WHa=22.0 LOri114 M6.5 Y Y Y Y Y Y Y Mem+ Y II -2.38 Thin Ha- WHa=10.9 LOri115 M5.0 Y Y Y Y Y Y Y NM+ Y II -2.02 Thin Ha- WHa=8.5 LOri116 M5.5 Y Y Y Y Y Y Y Mem+ Y II -2.43 Thin Ha- WHa=11.1 LOri117 M6.0 Y Y Y Y Y Y Y Mem Y – -2.20 Thin Ha+? WHa=22.9 LOri118 M5.5 Y Y Y Y Y Y Y Mem+ Y II -1.37 Thick Ha- WHa=10.1 LOri119 M5.5 Y Y Y Y Y Y Y NM? Y III -2.85 Diskless Ha+? WHa=12.7 LOri120 M5.5 Y Y Y Y Y Y Y Mem+ Y II -1.59 Thick Ha- WHa=7.4 LOri121 – Y Y Y Y ? Y Y NM- Y – -2.31 Thin – LOri122 – Y Y Y Y Y Y Y Mem Y II -2.46 Thin – LOri124 M5.5 Y Y Y Y Y Y Y Mem? Y III -2.56 Diskless Ha- WHa=8.4 LOri125 – Y Y Y Y Y Y Y NM- Y III -2.56 Diskless – LOri126 M6.5 Y Y Y Y Y Y Y Mem+ Y II -1.24 Thick Ha+? WHa=26.2 LOri127 – N N N N N N N NM- N III -2.78 Diskless – LOri128 – Y Y Y Y Y Y Y Mem? Y III -2.69 Diskless – LOri129 M6.0 Y Y Y Y Y Y Y Mem? Y II -1.71 Thick Ha+? WHa=12.1 LOri130 M5.5 Y Y Y Y Y ? Y Mem+ Y III -2.69 Diskless Ha- WHa=8.7 Table 4—Continued Name SpT Phot.Mem1 Mem2 Mem3 IRAC classification4 SED slope5 Disk type Comment6 LOri131 – Y Y Y Y Y Y Y Mem? Y II -2.12 Thin – LOri132 – Y Y Y Y Y Y N NM- Y II -2.25 Thin – LOri133 M4.5 N N N ? ? N Y NM+ N – -2.91 Diskless – LOri134 M5.0 Y Y Y Y ? ? Y NM+ Y III -2.34 Thin – LOri135 M7.0 Y Y Y Y Y Y ? Mem? Y III -2.56 Diskless Ha- WHa=15.5 LOri136 – Y Y Y Y Y Y Y Mem? Y III -2.17 Thin – LOri137 – – – N N – – – ? N – – – – LOri138 – Y Y Y Y Y Y Y NM- Y – -2.13 Thin – LOri139 M6.0 Y Y Y Y Y Y Y Mem+ Y II -1.22 Thick Ha+? WHa=19.7 LOri140 M7.0 Y Y Y Y Y Y Y Mem+ Y II -1.40 Thick Ha+ WHa=72.8 LOri141 M4.5 N N ? N Y Y Y NM+ N – -5.35 Diskless – LOri142 – Y Y Y Y Y Y Y Mem? Y – – – – LOri143 M6.5 Y Y Y Y Y Y Y Mem+ Y III -2.59 Diskless Ha+ WHa=35.7 LOri144 – N N N N N N Y ? N? – – – – LOri146 – Y Y Y Y Y Y Y Mem Y III -1.90 Thin – LOri147 M5.5 Y Y ? N ? N Y NM+ Y? – -2.41 Thin – LOri148 – Y N Y Y Y Y N NM- Y? – -2.19 Thin – LOri149 – – N N N – N – ? N – – – – LOri150 M8.0 Y Y Y Y Y Y Y Mem+ Y – -2.72 Diskless Ha- WHa=15.6 LOri151 M5.5 N N N ? Y Y Y NM? N – – – – LOri152 – Y Y N N N N Y NM- N? – – – – LOri153 – ? Y Y Y Y Y Y ? Y – – – – LOri154 M8.0 Y Y Y ? Y Y Y Mem+ Y – – – Ha- WHa=16.9 LOri155 M8.0 Y Y Y Y Y Y Y Mem+ Y III -2.03 Thin Ha+? WHa=38.0 LOri156 M8.0 Y Y Y Y Y Y Y Mem+ Y III -1.54 Thick Ha+ WHa=101.7 LOri157 – N N N N N N Y ? N – – – – LOri158 – N N N N N N Y ? N? – – – – LOri159 – N N N N Y Y N ? N? – – – – LOri160 – N Y Y Y Y Y Y ? Y – – – – LOri161 M8.5 Y Y Y Y Y Y Y Mem+ Y – – – Ha+ WHa=123 LOri162 – Y Y Y Y Y Y Y ? Y – – – – LOri163 – Y Y Y Y Y Y Y ? Y – – – – LOri164 – N N – – – – N ? N – – – – LOri165 M7.5 N N Y Y Y Y ? NM? N? – – – – LOri166 – ? ? Y N Y Y ? ? Y? – – – – LOri167 – Y Y Y Y Y Y Y ? Y – – – – LOri168 – N N – – – – Y ? N? – – – – LOri169 – N N – – – – – ? Y? – – – – LOri170 – N N – – – – – ? N? – – – – 1Membership is Ivs(I-J); Ivs(I-K); Ivs(I-3.6); Ivs(I-4.5); Jvs(J-3.6);Kvs(K-3.6); Jvs(J-K). 2Membership as in Paper I. 3Final membership. 4Classification as measured in the IRAC CCD –[3.6]-[4.5] versus [5.8]-[8.0]. Class III stands for diskless members and Class II are Classical TTauri stars or substellar analogs. 5IRAC slope. Lada et al. (2006) classified the objects according to their IRAC slope: α <-2.56 for a diskless object, -2.56< α <-1.80 for a transition object, and α >-1.80 for objects bearing optically thick disks 6Ha+ = W(Halpha) above the saturation criterion. Ha- = W(Halpha) below the saturation criterion. 200km = width of Halpha equal or larger than this value. WHa(DM) = from Dolan & Mathieu Probably diskless objects. The different results on IRAC CCD and IRAC slope are probably due to an uncertain measure at 5.8 µm. Probably a class II source with an uncertain measure at 8.0 µm. Table 5 Location of the substellar frontier, using models by Baraffe et al. (1998) and a distance of 400 pc. Values such as 340 or 450 pc would modify the listed magnitudes by −0.35 and +0.26, respectively. We have included an interstellar reddening of E(B − V )=0.12, equivalent to AI=0.223, AJ=0.106, AK=0.042, AL=0.022 Age (Myr) Ic J Ks L′ 1 16.72 14.35 13.32 12.88 3 17.18 14.87 13.84 13.40 5 17.55 15.36 14.35 13.92 8 17.92 15.80 14.80 14.36 10 18.13 16.01 15.01 14.57 16 18.52 16.40 15.40 14.96 20 18.71 16.59 15.60 15.15 Fig. 1.— Spitzer/IRAC CCD. Class I/II (big empty circles, magenta), Class II (big empty circles, red) and Class III –or not members– (crosses) have been classified using this diagram (After Allen et al. (2004) and Hartmann et al. (2005)). Fig. 2.— Optical/IR Color-Magnitude Diagram. Non-members appear as dots. Class II sources (Classical TTauri stars and substellar analogs) have been included as big (red) circles, whereas Class III (Weak-line TTauri) objects appear as crosses, and other Lambda Orionis members lacking the complete set of IRAC photometry are displayed with the plus symbol. The figure includes 1, 3, 5, 10, 20, 50, and 100 Myr isocrones from Baraffe et al. (1998) as solid lines, as well as 5 Myr isochrones corresponding to dusty and COND models (Chabrier et al. 2000; Baraffe et al. 2002), as dotted and dashed lines. Fig. 3.— Near-IR and Spitzer Color-Magnitude Diagram. Class II sources (Classical TTauri stars and substellar analogs) have been included as big (red) circles, whereas Class III (Weak-line TTauri) objects appear as crosses, and other Lambda Orionis members lacking the complete set of IRAC photometry are displayed with the plus symbol. The figure includes 1, 5, 10, 20, and 100 Myr isochrones from Baraffe et al. (1998) as solid lines, as well as 5 Myr isochrones corresponding to dusty and COND models (Chabrier et al. 2000; Baraffe et al. 2002), as dotted and dashed lines. Note that in the last panel we have the the L and M data for the NextGen models, since Spitzer photometry has not been computed for this set of models. Fig. 4.— Optical Color-Magnitude Diagram with the CFHT magnitudes and our new membership classifi- cation. Symbols as in previous figures. Fig. 5.— Spitzer/IRAC CCD for Class II objects. We have included information regarding the Hα emission. Fig. 6.— Spectral Energy Distributions for some stellar members of the Lambda Orionis cluster sorted according to their IRAC slope: simple photosphere spectra. Objects lacking IRAC slope or being in the boundary between two types have been classified after visual inspection. Fig. 7.— Spectral Energy distribution for some stellar members of the Lambda Orionis cluster sorted according to their IRAC slope: flat, or sloping IR spectra with the excess starting in the near-IR (thick disks). Fig. 8.— Spectral Energy distribution for some stellar members of the Lambda Orionis cluster sorted according to their IRAC slope: spectra with excesses begining in the IRAC or MIPS range (thin disks and transition objects). LOri043 and LOri065 were classified as diskless objects but have been sorted as objects bearing thin disks due to their excess at MIPS [24]. Fig. 9.— The fraction of Class II stars and massive brown dwarfs in several SFRs and young clusters (filled squares). Open squares stand for thick disk fractions of IC348 and C69. Fig. 10.— Spitzer/IRAC CCD. We show with different symbols (see key) cluster members with different types of disks. Fig. 11.— a) Spatial distribution of our sample. IRAS contour levels at 100 microns also have have been included as solid (magenta) lines. The big, thick-line rectangle corresponds to the CFHT survey (Paper I). Class II sources (Classical TTauri stars and substellar analogs) have been included as big (red) circles, whereas Class III (Weak-line TTauri) objects appear as crosses, and other Lambda Orionis members lacking the complete set of IRAC photometry are displayed with the plus symbol. b) Spatial distribution of the low mass stars from Dolan & Mathieu (1999, 2001). OB stars appear as four-point (blue) stars, with size related to magnitude (the bigger, the brighter). The overplotted thick triangles indicates those stars whose Hα equivalent width is larger than the saturation criterion defined by Barrado y Navascués & Mart́ın (2003), thus suggesting the presence of active accretion. Based on Hα alone, the fraction of accreting stars would be 11%. Fig. 12.— Spitzer/IRAC image at 3.6 micron centered around the star λ Orionis. a) The size is about 9×9 arcmin, equivalent to 192,000 AU. The double circle indicates the presence of a Class II object, whereas squares indicate the location of cluster members from Dolan & Mathieu (1999;2001). The intensity of the image is in logarithmic scale. b) Detail around the star λ Orionis. The size is about 3.3×3.3 arcmin, equivalent to 80,000 AU. c) Distribution of bona-fide brown dwarfs. The size of the image is 45×30 arcmin. North is up, East is left. Fig. 13.— Spitzer/MIPS image at 24 microns which includes the members of the Lambda Orionis cluster visible at this wavelength, including those cluster members by Dolan & Mathieu (1999, 2001) as big circles and CFHT member as small circles detected at this wavelength. The size is about 60.5×60.5 arcmin. North is up, East is left. The figure is centered on the star λ Ori AB. Introduction The data Optical and Near Infrared photometry New deep Near Infrared photometry Spitzer imaging Data cross-correlation Color-Color and Color-Magnitude Diagrams and new membership assignment Discussion The Color-Color diagrams, the diagnostic of IR excess and the disk ratio The Spectral Energy Distribution The Spatial distribution of the members Conclusions

0704.1964

A data-analysis driven comparison of analytic and numerical coalescing binary waveforms: nonspinning case

A data-analysis driven comparison of analytic and numerical coalescing binary waveforms: nonspinning case Yi Pan,1 Alessandra Buonanno,1 John G. Baker,2 Joan Centrella,2 Bernard J. Kelly,2 Sean T. McWilliams,1 Frans Pretorius,3 and James R. van Meter2, 4 Department of Physics, University of Maryland, College Park, MD 20742 Gravitational Astrophysics Laboratory, NASA Goddard Space Flight Center, 8800 Greenbelt Rd., Greenbelt, MD 20771 Department of Physics, Princeton University, Princeton, NJ 08544 Center for Space Science & Technology, University of Maryland Baltimore County, Physics Department, 1000 Hilltop Circle, Baltimore, MD 21250 (Dated: October 22, 2018) We compare waveforms obtained by numerically evolving nonspinning binary black holes to post- Newtonian (PN) template families currently used in the search for gravitational waves by ground- based detectors. We find that the time-domain 3.5PN template family, which includes the inspiral phase, has fitting factors (FFs) ≥ 0.96 for binary systems with total massM = 10–20M⊙. The time- domain 3.5PN effective-one-body template family, which includes the inspiral, merger and ring-down phases, gives satisfactory signal-matching performance with FFs ≥ 0.96 for binary systems with total mass M = 10–120M⊙. If we introduce a cutoff frequency properly adjusted to the final black-hole ring-down frequency, we find that the frequency-domain stationary-phase-approximated template family at 3.5PN order has FFs ≥ 0.96 for binary systems with total mass M = 10–20M⊙. However, to obtain high matching performances for larger binary masses, we need to either extend this family to unphysical regions of the parameter space or introduce a 4PN order coefficient in the frequency- domain GW phase. Finally, we find that the phenomenological Buonanno-Chen-Vallisneri family has FFs ≥ 0.97 with total mass M = 10–120M⊙. The main analyses use the noise spectral-density of LIGO, but several tests are extended to VIRGO and advanced LIGO noise-spectral densities. PACS numbers: 04.25.Dm, 04.30.Db, 04.70.Bw, x04.25.Nx, 04.30.-w, 04.80.Nn, 95.55.Ym I. INTRODUCTION The search for gravitational-waves (GWs) from coa- lescing binary systems with laser interferometer GW de- tectors [1, 2, 3, 4, 5] is based on the matched-filtering technique, which requires accurate knowledge of the waveform of the incoming signal. In the last couple of years there have been several breakthroughs in numeri- cal relativity (NR) [6, 7, 8], and now independent groups are able to simulate the inspiral, merger and ring-down phases of generic black- hole (BH) merger scenarios, in- cluding different spin orientations and mass ratios [9]. However, the high computational cost of running such simulations makes it difficult to generate sufficiently long inspiral waveforms that cover the parameter space of as- trophysical interest. References [10, 11] found good agreement between ana- lytic (based on the post-Newtonian (PN) expansion) and numerical waveforms emitted during the inspiral, merger and ring-down phases of equal-mass, nonspinning binary BHs. Notably, the best agreement is obtained with 3PN or 3.5PN adiabatic waveforms [12] (henceforth denoted as Taylor PN waveforms) and 3.5PN effective-one-body (EOB) waveforms [13, 14, 15, 16, 17, 18, 19]. In addi- tion to the inspiral phase the latter waveforms include the merger and ring-down phases. Those comparisons suggested that it should be possible to design hybrid nu- merical/analytic templates, or even purely analytic tem- plates with the full numerics used to guide the patching together of the inspiral and ring-down waveforms. This is an important avenue to template construction as eventu- ally thousands of waveform templates may be needed to extract the signal from the noise, an impossible demand for NR alone. Once available, those templates could be used by ground-based laser interferometer GW detectors, such as LIGO, VIRGO, GEO and TAMA, and in the fu- ture by the laser interferometer space antenna (LISA) for detecting GWs emitted by solar mass and supermassive binary BHs, respectively. This paper presents a first attempt at investigating the closeness of the template families currently used in GW inspiral searches to waveforms generated by NR simula- tions. Based on this investigation, we shall propose ad- justments to the templates so that they include merger and ring-down phases. In contrast, Ref. [21] examined the use of numerical waveforms in inspiral searches, and compared numerical waveforms to the ring-down tem- plates currently used in burst searches. Similar to the methodology presented here, fitting factors (FFs) [see Eq. (2) below] are used in Ref. [21] to quantify the ac- curacy of numerical waveforms for the purpose of de- tection, as well as the overlap of burst templates with the waveforms. Reference [21] found that by computing FFs between numerical waveforms from different reso- lution simulations of a given event, one can recast the numerical error as a maximum FF that the numerical waveform can resolve. In other words, any other tem- plate or putative signal convolved with the highest reso- lution numerical simulation that gives a FF equal to or larger than this maximum FF is, for the purpose of de- http://arxiv.org/abs/0704.1964v2 tection, indistinguishable from the numerical waveform. We will explore this aspect of the problem briefly. The primary conclusions we will draw from the analysis do not crucially depend on the exactness of the numerical waveforms. What counts here is that the templates can capture the dominant spectral characteristics of the true waveform. For our analysis we shall focus on two nonspinning equal-mass binary simulation waveforms which differ in length, initial conditions, and the evolution codes used to compute them: Cook-Pfeiffer quasi-equilibrium ini- tial data built on Refs. [22, 23, 24, 25, 26] evolved with Pretorius’ generalized harmonic code [6], and Brandt- Brügmann puncture data [27] evolved using the Goddard group’s moving-puncture code [8]. We also consider two nonspinning unequal-mass binary simulations with mass ratios m2/m1 = 1.5 and m2/m1 = 2 produced by the Goddard group. The paper is organized as follows. In Sec. II we dis- cuss the phase differences between PN inspiraling tem- plates. In Sec. III we build hybrid waveforms by stitch- ing together PN and NR waveforms. We try to under- stand how many NR cycles are needed to obtain good agreement between NR and PN waveforms, to offer a guide for how long PN waveforms can be used as accu- rate templates. In Sec. IV we compute the FFs between several PN template families and NR waveforms. We first focus on low-mass binary systems with total mass M = 10–30M⊙, then high-mass binary systems with to- tal mass M = 30–120M⊙. Finally, Sec. V contains our main conclusions. In Appendix A we comment on how different representations of the energy-balance equations give GW frequencies closer to or farther from the NR ones. II. PHASE DIFFERENCES IN POST-NEWTONIAN INSPIRALING MODELS Starting from Ref. [28], which pointed out the impor- tance of predicting GW phasing with the highest possi- ble accuracy when building GW templates, many subse- quent studies [14, 18, 19, 20, 29, 31, 32, 33] (those ref- erences are restricted to the nonspinning case) focused on this issue and thoroughly tested the accuracy of those templates, proposing improved representations of them. These questions were motivated by the observation that comparable-mass binary systems with total mass higher than 30M⊙ merge in-band with the highest signal-to- noise ratio (SNR) for LIGO detectors, It follows that the corresponding templates demand an improved analysis. In the absence of NR results and under the urgency of providing templates to search for comparable-mass bi- nary BHs, the analytic PN community pushed PN cal- culations to higher PN orders, notably 3.5PN order [12], and also proposed ways of resumming the PN expan- sion, either for conservative dynamics (the EOB ap- proach [13, 16, 17]), radiation-reaction effects (the Padé resummation [19]), or both [14, 18]. Those results lead to several conclusions: (i) 3PN terms improve the compar- ison between analytic and (numerical) quasi-equilibrium predictions [23, 26, 34, 35]; (ii) Taylor expanded and re- summed PN predictions for equal-mass binary systems are much closer at 3.5PN order than at previous PN orders, indicating a convergence between the different schemes [18, 20, 31]; (iii) the two-body motion is quasi- circular until the end of a rather blurred plunge [14], (iv) the transition to ring-down can be described by an ex- tremely short merger phase [14, 18]. Today, with the NR results we are in a position to sharpen the above con- clusions, and to start to assess the closeness of analytic templates to numerical waveforms. Henceforth, we restrict the analysis to the three time- domain physical template families which are closest to NR results [10, 11]: the adiabatic Taylor PN model (Tpn) [see, e.g., Eqs. (1), (10), and (11)–(13) in Ref. [30]] computed at 3PN and 3.5PN order, and the nonadia- batic EOB model (Epn) [see e.g., Eqs. (3.41)–(3.44) in Ref. [14]] computed at 3.5 PN order. We shall denote our models as Tpn(n) and Epn(n), n being the PN or- der. The Tpn model is obtained by solving a particular representation of the balance equation. In Appendix A we briefly discuss how time-domain PN models based on different representations of the energy-balance equation would compare with NR results. The waveforms we use are always derived in the so- called restricted approximation which uses the amplitude at Newtonian order and the phase at the highest PN order available. They are computed by solving PN dy- namical equations providing the instantaneous frequency ω(t) and phase φ(t) = φ0 + ω(t′)dt′, thus h(t) = Aω(t)2/3 cos[2φ(t)] , (1) where t0 and φ0 are the initial time and phase, respec- tively, and A is a constant amplitude, irrelevant to our discussion. The inclusion of higher-order PN corrections to the amplitude can be rather important for certain unequal-mass binary systems, and will be the subject of a future study. When measuring the differences between waveforms we weight them by the power spectral-density (PSDs) of the detector, and compute the widely used fitting factor (FF) (i.e., the ambiguity function or normalized over- lap), or equivalently the mismatch defined as 1-FF. Fol- lowing the standard formalism of matched-filtering [see, e.g., Refs. [19, 31, 36]], we define the FF as the overlap 〈h1(t), h2(t)〉 between the waveforms h1(t) and h2(t): 〈h1(t), h2(t)〉 ≡ 4Re h̃1(f)h̃ Sh(f) FF ≡ max t0,φ0,λi 〈h1, h2(t0, φ0, λ 〈h1, h1〉〈h2(t0, φ0, λi), h2(t0, φ0, λi)〉 where h̃i(f) is the Fourier transform of hi(t), and Sh(f) is the detector’s PSD. Thus, the FF is the normalized overlap between a target waveform h1(t) and a set of tem- plate waveforms h2(t0, φ0, λ i) maximized over the initial time t0, initial phase φ0, and other parameters λ i. Some- times we are interested in FFs that are optimized only over t0 and φ0; we shall denote these as FF0. For data- analysis purposes, the FF has more direct meaning than the phase evolution of the waveforms, since it takes into account the PSDs and is proportional to the SNR of the filtered signal. Since the event rate is proportional to the cube of the SNR, and thus to the cube of the FF, a FF= 0.97 corresponds to a loss of event rates of ∼ 10%. A template waveform is considered a satisfactory repre- sentation of the target waveform when the FF is larger than 0.97. When comparing two families of waveforms, the FF is optimized over the initial phase of the template wave- form, and we also need to specify the initial phase of the target waveform. Since there is no preferred initial phase of the target, two options are usually adopted: (i) the ini- tial phase maximizes the FF or (ii) it minimizes the FF. The resulting FFs are referred to as the best and mini- max FFs [29]. All FFs we present in this paper are min- imax FFs. Although the FF of two waveform families is generally asymmetric under interchange of the template family [31], the best and the minimax FF0s are symmet- ric (see Appendix B of Ref. [29] for details). Henceforth, when comparing two waveform families using FF0, we do not need to specify which family is the target. We shall consider three interferometric GW detectors: LIGO, advanced LIGO and VIRGO. The latter two have better low-frequency sensitivity and broader bandwidth. For LIGO, we use the analytic fit to the LIGO de- sign PSD given in Ref. [20]; for advanced LIGO we use the broadband configuration PSD given in Ref. [37]; for VIRGO we use the PSD given in Ref. [20]. In Fig. 1, we show the FF0s as functions of the ac- cumulated difference in the number of GW cycles be- tween waveforms generated with different inspiraling PN models and for binary systems with different component masses. We first generate two waveforms by evolving two PN models, say, “PN1” and “PN2” which start at the same GW frequency fGW = 30Hz and have the same initial phase. The two waveforms are terminated at the same ending frequency fGW = fend up to a maximum fend,max = min(fend,PN1 , fend,PN2), where fend,PN is the frequency at which the PN inspiraling model ends. (For Tpn models this is the frequency at which the PN energy has a minimum; for Epn models it is the EOB light-ring frequency.) Then, we compute the difference in phase and number of GW cycles accumulated until the ending frequency ∆NGW = [φPN1(fend)− φPN2(fend)] . (3) By varying fend (up to fend,max) ∆NGW changes, though not necessarily monotonically. Although there seems to be a loose correlation between the FF0s and ∆NGW, it is hard to quantify it as a one-to-one correspondence. 0 0.2 0.4 0.6 0.8 1.0 1.2 Tpn(3.5) & Epn(3.5), (3+3) Tpn(3.5) & Epn(3.5), (15+3) Tpn(3.5) & Epn(3.5), (15+15) Tpn(3) & Tpn(3.5), (3+3) Tpn(3) & Tpn(3.5), (15+3) Tpn(3) & Tpn(3.5), (15+15) FIG. 1: We show FF0s between waveforms generated from the three PN models Tpn(3), Tpn(3.5) and Epn(3.5) versus ∆NGW [see Eq. (3)]. The FF0s are evaluated with LIGO’s PSD. Note that for Tpn(3.5) and Epn(3.5) and a (15+3)M⊙ binary, the lowest FF0 is 0.78 and the difference in the number of GW cycles ∆NGW ≃ 2. In the limit ∆NGW → 0, the FF0 goes to unity. For example, a phase difference of about half a GW cy- cle (∆NGW ≃ 0.5) is usually thought to be significant. However, here we find relatively high FF0s between 0.97 and > 0.99, depending on the masses of the binary and the specific PN model used. This happens because the FF between two waveforms is not determined by the to- tal phase difference accumulated, but rather by how the phase difference accumulates across the detector’s most sensitive frequency band. The relation between FFs and phase differences is also blurred by the maximization over the initial time and phase: shifting the phase by half a cycle from the most sensitive band to a less sen- sitive band can increase the matching significantly. We conclude that with LIGO’s PSD, after maximizing only on initial phase and time, Epn(3.5) and Tpn(3.5) tem- plates are close to each other for comparable-mass bi- nary systems M = 6–30M⊙ with FF0>∼ 0.97, but they can be different for mass ratios m2/m1 ≃ 0.3 with FF0 as low as ≃ 0.8. Tpn(3) and Tpn(3.5) templates have FF0>∼ 0.97 for the binary masses considered. Note that for m2/m1 = 1 [≃ 0.3] binary systems, Tpn(3.5) is closer to Epn(3.5) [Tpn(3)] than to Tpn(3) [Epn(3.5)]. Note also that when maximizing on binary masses the FFs can increase significantly, for instance, for a (15 + 3)M⊙ bi- nary, the FF between Tpn(3.5) and Epn(3.5) waveforms becomes > 0.995, whereas FF0 ≃ 0.8. III. BUILDING AND COMPARING HYBRID WAVEFORMS Recent comparisons [10, 11] between analytic and nu- merical inspiraling waveforms of nonspinning, equal-mass binary systems have shown that numerical waveforms are in good agreement with Epn(3.5), Tpn(3) and Tpn(3.5) waveforms. Those results were assessed using eight and sixteen numerical inspiral GW cycles. Can we conclude from these analyses that Epn(3.5), Tpn(3.5) and Tpn(3) can safely be used to build a template bank for detecting inspiraling GW signals? A way to address this question is to compute the mismatch between hybrid waveforms built by attaching either Epn or Tpn waveforms to the same numerical waveform, and varying the time when the attachment is made. This is equivalent to varying the number of numerical GW cycles n in the hybrid tem- plate. The larger n the smaller the mismatch, as we are using the same numerical segment in both waveforms. For a desired maximum mismatch, say 3%, we can then find the smallest number n of numerical cycles that is required in the hybrid waveform. This number will, of course, depend on the binary mass and the PSD of each detector. A. Hybrid waveforms We build hybrid waveforms by connecting PN wave- forms to NR waveforms at a chosen point in the late inspiral stage. As mentioned before, we use NR wave- forms generated with Pretorius’ [10] code and the God- dard group’s [38] code. Pretorius’ waveform is from an equal-mass binary with total mass M , and equal, co- rotating spins (a = 0.06). The simulation lasts ≃ 671M , and the waveform has ≃ 8 cycles before the formation of the common apparent horizon. The Goddard waveform refers to an equal-mass nonspinning binary. The simula- tion lasts about ≃ 1516M , and the waveform has ≃ 16 cycles before merger. Since we will present results from these two waveforms it is useful to first compare them by computing the FF0. Although the binary parameters considered by Pretorius and Goddard are slightly different, we expect the wave- forms, especially around the merger stage, to be fairly close. Comparisons between (shorter) waveforms com- puted with moving punctures and generalized-harmonic gauge were reported in Ref. [39], where the authors dis- cussed the different initial conditions, wave extraction techniques, and compared the phase, amplitude and fre- quency evolutions. Since the two simulations use dif- ferent initial conditions and last for different amounts of time we cut the longer Goddard waveform at roughly the frequency where the Pretorius waveform starts. In this way we compare waveforms that have the same length between the initial time and the time at which the wave amplitude reaches its maximum. In Fig. 2, we show the FF0 as function of the total binary mass. Despite dif- ferences in the two simulations the FF0s are rather high. The waveforms differ more significantly at lower frequen- cies. Indeed, as the total mass decreases the FF0s also decrease as these early parts of the waveform contribute more to the signal power given LIGO’s PSD. Any waveform extracted from a numerical simulation will inherit truncation errors, affecting both the wave- form’s amplitude and phase [10, 21, 38]. To check whether those differences would change the results of the comparisons between NR and PN waveforms, we plot in Fig. 2 the FF0s versus total binary mass between two Goddard waveforms generated from a high and a medium resolution run [38]. The FF0s are extremely high (> 0.995). Based on the comparisons between high and medium resolution waveforms, we can estimate the FFs between high resolution and exact waveforms. If we have several simulations with different resolutions, specified by the mesh-spacings xi, and xi are sufficiently small, we can assume that the waveforms hi are given by hi = h0 + x i hd , (4) where n is the convergence factor of the waveform, h0 is the exact waveform generated from the infinite resolution run (x0 → 0), and hd is the leading order truncation error contribution to the waveform and is independent of the mesh spacing xi. We find that the mismatch between the waveforms hi and hj, 1− FFij , scales as 1− FFij ∝ (x i − x 2 . (5) In the Goddard simulations, the high and medium reso- lution runs have mesh-spacing ratio xh/xm = 5/6, and the waveform convergence rate is n = 4 [38]. The FF between the high resolution and exact waveforms hh and h0 is given by FF0h = 1− 0.87(1− FFhm) , (6) where FFhm is the FF between the high and medium resolution waveforms hh and hm. That is to say, the mismatch between hh and h0 is slightly smaller than that between hh and hm, where the latter can be derived from the FFs shown in Fig. 2. Henceforth, we shall always use high-resolution waveforms. A similar calculation for Pre- torius’ waveform gives FF0h = 1−0.64(1−FFhm), though here xh/xm = 2/3 and n = 2. See Fig. 6 of Ref. [21] for a plot of FFhm calculated from the evolution of the Cook- Pfeiffer initial data 1; there FF0 ranges from ≈ 0.97 for M/Ms = 30 to ≈ 0.99 for M/Ms = 100. In other words, the mismatch between Goddard’s and Pretorius’ wave- form shown in Fig.2 is less than the estimated mismatch from numerical error in the latter waveform. We build hybrid waveforms by stitching together the PN and NR waveforms computed for binary systems with the same parameters. At the point where we connect the 1 The plot in Ref. [21] is for “d=16” corotating Cook-Pfeiffer initial data, whereas the results presented here are from “d=19” initial data. However, the resolutions used for both sets were the same, and thus the mismatches should be similar, in particular in the higher mass range. 30 40 50 60 70 80 90 100 Total mass of the binary ( ) 0.980 0.985 0.990 0.995 1.000 Goddard high and medium resolutions Pretorius and Goddard waveforms FIG. 2: FF0 between NR waveforms as a function of the binary total-mass M . The solid curve are generated for wave- forms from Pretorius and the Goddard group. The longer Goddard waveform is shortened such that both waveforms last ≃ 671M and contain ≃ 8 cycles. The dashed curve is gen- erated for waveforms from the high-resolution and medium resolution simulations of the Goddard group. All FFs are evaluated using LIGO’s PSD. two waveforms, we tune the initial time t0 so that the frequency of the PN waveform is almost the same as the frequency of the NR waveform (there is a subtlety trying to match exactly the frequencies that is discussed at the end of this section). The initial phase φ0 is then chosen so that the strain of the hybrid waveform is continuous at the connecting point. In Fig. 3, we show two examples of hybrid waveforms of an equal-mass binary. We stitch the waveforms at points where effects due to the initial-data transient pulse are negligible. We find an amplitude difference on the order of ∼ 10% between the Goddard waveform and the re- stricted PN waveform. This difference is also present in Pretorius’ waveform, but it is somewhat compensated for by amplitude modulations caused by eccentricity in the initial data. In Ref. [38] it was shown that PN waveforms with 2.5PN amplitude corrections give better agreement (see e.g., Fig. 12 in Ref. [38]). However, the maximum amplitude errors in the waveforms are also on the or- der of 10% [10, 38]. Since neither 2PN nor other lower PN order corrections to the amplitudes are closer to the 2.5PN order, we cannot conclude that 2.5PN amplitude corrections best approximate the numerical waves. Thus, we decide to use two sets of hybrid waveforms: one con- structed with restricted PN waveforms, and the other with restricted PN waveforms rescaled by a single ampli- tude factor, which eliminates amplitude differences with the NR waveforms. We shall see that the difference be- tween these two cases is small for the purpose of our tests. The amplitude difference between PN and NR wave- forms is computed at the same connecting-point GW fre- -500-1000 -1000-1500 -1500-2000 -2000-2500 -2500 Time (M) Time (M) FIG. 3: We show two examples of hybrid waveforms, start- ing from 40Hz. The PN waveforms are generated with the Tpn(3.5) model, and the NR waveforms in the upper and lower panels are generated from Pretorius’ and Goddard’s simulations, respectively. We mark with a dot the point where we connect the PN and NR waveforms. quency. There is another effect which causes a jump in the hybrid-waveform amplitude. This is a small fre- quency difference between PN and NR waveforms at the connecting point. All our NR waveforms contain small eccentricities [10, 38]. As a consequence, the frequency evolution ω(t) oscillates. To reduce this effect we fol- low what is done in Ref. [38] and fit the frequency to a monotonic quartic function. When building the hybrid waveform, we adjust the PN frequency to match the quar- tic fitted frequency (instead of the oscillatory, numerical frequency) at the connecting point. Since the restricted PN amplitude is proportional to ω2/3(t) [see Eq. (1)], this slight difference between ωs at the connecting point creates another difference between the NR and PN am- plitudes. Nevertheless, this difference is usually smaller (for Goddard’s waveform) or comparable (for Pretorius’) to the amplitude difference discussed above. -25 -20 -15 -10 -5 0 Time (100M) Original Whitened -25 -20 -15 -10 -5 0 Time (100M) -800 -600 -400 -200 0 Time (M) -800 -600 -400 -200 0 Time (M) -200 -150 -100 -50 0 50 Time (M) -200 -150 -100 -50 0 50 Time (M) (5+5) (10+10) (20+20) (15+15) (30+30) (50+50) FIG. 4: Distribution of GW signal power. In each panel, we plot a hybrid waveform (a Tpn waveform stitched to the Goddard waveform) in both its original form (blue curve) and its “whitened” form (red curve) [44]. We show waveforms from six binary systems with total masses 10M⊙ 20M⊙, 30M⊙, 40M⊙, 60M⊙ and 100M⊙. The vertical lines divide the waveforms into segments, where each segment contributes 10% of the total signal power. B. Distribution of signal power in gravitational waveforms To better understand the results of the FFs between hybrid waveforms, we want to compute how many sig- nificant GW cycles are in the LIGO frequency band. By significant GW cycles we mean the cycles that contribute most to the signal power, or to the SNR of the filtered signal. Since GW frequencies are scaled by the total bi- nary mass, the answer to this question depends on both the PSD and the total mass of a binary. In Fig. 4, we show the effect of the LIGO PSD on the distribution of signal power for several waves emitted by coalescing binary systems with different total masses. In each panel, we plot a hybrid waveform (a Tpn waveform stitched to the Goddard waveform) in both its original form and its “whitened” form [44]. The whitened wave- form is generated by Fourier-transforming the original waveform into the frequency domain, rescaling it by a fac- tor of 1/ Sh(f), and then inverse-Fourier-transforming it back to the time domain. The reference time t = 0 is the peak in the amplitude of the unwhitened waveform. The amplitude of a segment of the whitened waveform indicates the relative contribution of that segment to the signal power and takes into account LIGO’s PSD. Both waveforms are plotted with arbitrary amplitudes, and the unwhitened one always has the larger amplitude. The absolute amplitude of a waveform, or equivalently the distance of the binary, is not relevant in these figures un- less the redshift z becomes significant. In this case the mass of the binary is the redshifted mass (1+ z)M . Ver- tical lines in each figure divide a waveform into segments, where each segment contributes 10% of the total signal power. In each plot, except for the 10M⊙-binary one, we show all 9 vertical lines that divide the waveforms into 10 segments. In the 10M⊙-binary plot we omit the early part of the inspiral phase that accounts for 50% of the signal power, as it would be too long to show. The absolute time-scale of a waveform increases lin- early with total mass M ; equivalently the waveform is shifted toward lower frequency bands. For a M = 10M⊙ binary, the long inspiral stage generates GWs with fre- quencies spanning the most sensitive part of the LIGO band, around 150Hz, while for an M = 100M⊙ binary, only the merger signal contributes in this band. Thus, for low-mass binary systems, most of the contribution to the signal power comes from the long inspiral stage of the waveform, while for high-mass binary systems most of the contribution comes from the late inspiral, merger, and ring-down stages. Understanding quantitatively the distribution of signal power will let us deduce how many, and which, GW cycles are significant for the purpose of data analysis. We need accurate waveforms from either PN models or NR simulations for at least those signifi- cant cycles. From Fig. 4 we conclude that: • For an M = 10M⊙ binary, the last 25 inspiral cy- cles, plus the merger and ring-down stages of the waveform contribute only 50% of the signal power, and we need 80 cycles (not shown in the figure) of accurate inspiral waveforms to recover 90% of the signal power. For an M = 20M⊙ binary, the last 23 cycles, plus the merger and ring-down stages of the waveform contribute> 90% of the signal power, and current NR simulations can produce waveforms of such length; • For an M = 30M⊙ binary, the last 11 inspiral cy- cles, plus the merger and ring-down stages of the waveform contribute > 90% of the signal power, which means that, for binary systems with to- tal masses higher than 30M⊙, current NR simula- tions, e.g., the sixteen cycles obtained in Ref. [38], can provide long enough waveforms for a matched- filter search of binary coalescence, as also found in Ref. [21]; • For an M = 100M⊙ binary, > 90% of the signal power comes from the last inspiral cycle, merger and ring-down stages of the waveform, with two cycles dominating the signal power. It is thus pos- sible to identify this waveform as a “burst” signal. Similar analyses can be also done for advanced LIGO and VIRGO. C. Comparing hybrid waveforms We shall now compute FF0s between hybrid wave- forms. We fix the total mass of the equal-mass binary in each comparison, i.e., we do not optimize over mass parameters, but only on phase and time. We use the mismatch, defined as 1− FF0, to measure the difference between waveforms and we compute them for LIGO, ad- vanced LIGO, and VIRGO. Note that by using FF0, we test the closeness among hybrid waveforms that are gen- erated from binary systems with the same physical pa- rameters; in other words, we test whether the waveforms are accurate enough for the purpose of parameter estima- tion, rather than for the sole purpose of detecting GWs. In the language of Ref. [19] we are studying the faithful- ness of the PN templates 2. Since at late inspiral stages PN waveforms are partly replaced by NR waveforms, differences between hybrid waveforms from two PN models are smaller than those between pure PN waveforms. In general, the more NR cycles we use to generate hybrid waveforms, the less the 2 Following Ref. [19], faithful templates are templates that have large overlaps, say >∼ 96.5%, with the expected signal maximiz- ing only over the initial phase and time of arrival. By contrast when the maximization is done also on the binary masses, the templates are called effectual. difference is expected to be between these hybrid wave- forms. This is evident in Figs. 5, 6 where we show mis- matches between hybrid waveforms for binary systems with different total masses as a function of the number of NR cycles n. Specifically, the mismatches are taken between two hybrid waveforms generated from the same NR waveform (from the Goddard group, taking the last n cycles, plus merger and ring-down) and two different PN waveforms generated with the same masses. The mismatches are lower for binary systems with higher total masses, since most of their signal power is concentrated in the late cycles close to merger (see Fig. 4). Comparing results between LIGO, advanced LIGO and VIRGO, we see that for the same waveforms the mismatches are lowest when evaluated with the LIGO PSD, and highest when evaluated with the VIRGO PSD. This is due to the much broader bandwidth of VIRGO, especially at low frequency: the absolute sensitivity is not relevant; only the shape of the PSD matters. In VIRGO, the inspiral part of a hybrid waveform has higher weight- ing in its contribution to the signal power. As already observed at the end of Sec. II, we can see also that the difference between the Epn(3.5) and Tpn(3.5) models is smaller than that between the Tpn(3) and Tpn(3.5) mod- Figures 5, 6 show good agreement among hybrid wave- forms. In Sec. IV, as a further confirmation of what was found in Refs. [10, 11], we shall see that PN waveforms from Tpn and Epn models have good agreement with the inspiral phase of the NR waveforms. Therefore, we argue that hybrid waveforms are likely to have high accuracy. In fact, for the late evolution of a compact binary, where NR waveforms are available, the PN waveforms are close to the NR waveforms, while for the early evolution of the binary, where we expect the PN approximations to work better, the PN waveforms (from Tpn and Epn models) are close to each other. Based on these observations, we draw the following conclusions for LIGO, advanced LIGO, and VIRGO data-analysis: • For binary systems with total mass higher than 30M⊙, the current NR simulations of equal-mass binary systems (16 cycles) are long enough to re- duce mismatches between hybrid waveforms gen- erated from the three PN models to below 0.5%. Since these FFs are achieved without optimizing the binary parameters, we conclude that for these high-mass binary systems, the small difference be- tween hybrid waveforms indicates low systematic error in parameter estimation, i.e., hybrid wave- forms are faithful [19]. • For binary systems with total mass around 10–20M⊙, 16 cycles of NR waveforms can reduce the mismatch to below 3%, which is usually set as the maximum tolerance for data-analysis purpose (corresponding to ∼ 10% loss in event rate). By a crude extrapolation of our results, we estimate that with 30 NR waveform cycles, the mismatch might 4 6 8 10 12 14 Number of NR waveform cycles Restricted PN Rescaled PN 4 6 8 10 12 14 Number of NR waveform cycles Quartic-fit frequency Oscillatory frequency FIG. 5: We show the mismatch between hybrid waveforms as a function of the number of NR waveform cycles used to generate the hybrid waveforms. The LIGO PSD is used to evaluate the mismatches. In the left panel, we compare the Epn(3.5) and Tpn(3.5) models. In the right panel, we compare the Tpn(3) and Tpn(3.5) models. From top to bottom, the four curves correspond to four equal-mass binary systems, with total masses 10M⊙, 20M⊙, 30M⊙, and 40M⊙. The dots show mismatches taken between hybrid waveforms that are generated with different methods. In the left panel, we adjust the amplitude of restricted PN waveforms, such that they connect smoothly in amplitude to NR waveforms. In the right panel, to set the frequency of PN waveforms at the joining point, we use the original orbital frequency, instead of the quartic fitted one. (See Sec. IIIA for the discussion on amplitude scaling and frequency fitting). 4 6 8 10 12 14 Number of NR waveform cycles Advanced LIGO PSD VIRGO PSD 4 6 8 10 12 14 Number of NR waveform cycles Advanced LIGO PSD VIRGO PSD FIG. 6: Mismatch between hybrid waveforms as a function of the number of NR waveform cycles used to generate the hybrid waveforms. Following the settings of Fig. 5, we show comparisons between Epn(3.5) and Tpn(3.5), and Tpn(3) and Tpn(3.5) models in the left and the right panels, respectively. The solid and dashed sets of curves are generated using the PSDs of advanced LIGO and VIRGO. In each set, from top to bottom, the three curves correspond to three equal-mass binary systems, with total masses 20M⊙, 30M⊙, and 40M⊙. be reduced to below 1%. • For binary systems with total mass lower than 10M⊙, the difference between the Tpn(3) and Tpn(3.5) models is substantial for Advanced LIGO and VIRGO. Their mismatch can be > 4% and > 6% respectively (not shown in the figure). In this mass range, pursuing more NR waveform cycles in late inspiral phase does not help much, since the signal power is accumulated slowly over hundreds of GW cycles across the detector band. Neverthe- less, here we give mismatches for FF0s which are not optimized over binary masses. For the purpose of detection only, optimization over binary param- eters leads to low enough mismatches (see also the end of Sec. II). In the language of Ref. [19] hybrid waveforms for total mass lower than 10M⊙ are ef- fectual but not faithful. IV. MATCHING NUMERICAL WAVEFORMS WITH POST-NEWTONIAN TEMPLATES In this section, we compare the complete inspiral, merger and ring-down waveforms of coalescing compact binary systems generated from NR simulations with their best-match PN template waveforms. We also compare hybrid waveforms with PN template waveforms for lower total masses, focusing on the late inspiral phase pro- vided by the NR waveforms. We test seven families of PN templates that either have been used in searches for GWs in LIGO (see e.g., Refs. [40, 41]), or are promising candidates for ongoing and future searches with ground-based detectors. We evaluate the perfor- mance of PN templates by computing the FFs maxi- mized on phase, time and binary parameters. As we shall see, for the hybrid waveforms of binary systems with total mass M ≤ 30M⊙, both the time-domain fam- ilies Tpn(3.5) and Epn(3.5), which includes a superposi- tion of three ring-down modes, perform well, confirming what found in Refs. [10, 11]. The standard stationary- phase-approximated (SPA) template family in the fre- quency domain has high FFs only for binary systems with M < 20M⊙. After investigating in detail the GW phase in frequency domain, and having understood why it hap- pens (see Sec. IVB2), we introduce two modified SPA template families (defined in Sec. IVB 2) for binary sys- tems with total mass M ≥ 30M⊙. Overall, for masses M ≥ 30M⊙, the Epn(3.5) template family in the time do- main and the two modified SPA template families in the frequency domain exhibit the best-match performances. A. Numerical waveforms and post-Newtonian templates For binary systems with total mass M ≥ 30M⊙, the last 8–16 cycles contribute more than 80–90% of the sig- nal power, thus in this case we use only the NR wave- forms. By contrast, for binary systems with total mass 10 ≤ M ≤ 30M⊙, for which the merger and ring-down phases of the waveforms contribute only ∼ 1–10%, we use the hybrid waveforms, generated by stitching Tpn waveforms to the Goddard NR waveforms. We want to emphasize that FFs computed for differ- ent target numerical waveforms can not directly be com- pared with each other. For instance, the Goddard wave- form is longer than the Pretorius waveform, and the FFs are sometime slightly lower using the Goddard waveform. This is a completely artificial effect, due to the fact that it is much easier to tune the template parameters and obtain a large FF with a shorter target waveform than a longer one. We consider seven PN template families. The two time-domain families introduced in Sec. II are: • Tpn(3.5) [30, 31]: The inspiral Taylor model. • Epn(3.5) [10, 13, 14, 16, 19]: The EOB model which includes a superposition of three quasi-normal modes (QNMs) of the final BH. These are labeled by three in- tegers (l,m, n) [42]: the least damped QNM (2, 2, 0) and two overtones (2, 2, 1) and (2, 2, 2). The ring-down wave- form is given as: hQNM(t) = −(t−tend)/τ22n cos [ω22n(t− tend) + φn] , where ωlmn and τlmn are the frequency and decay time of the QNM (l,m, n), determined by the mass Mf and spin af of the final BH. The quantities An and φn in Eq. (7) are the amplitude and phase of the QNM (2, 2, n). They are obtained by imposing the continuity of h+ and h×, and their first and second time derivatives, at the time of matching tmatch. Besides the mass parameters, our Epn model contains three other physical parameters: ǫt, ǫM and ǫJ . The parameter ǫt takes into account possible dif- ferences between the time tend at which the EOB models end and the time tmatch at which the matching to ring- down is done. More explicitly, we set tmatch = (1+ǫt)tend, and if ǫt > 0, we extrapolate the EOB evolution, and set an upper limit for the ǫt search where the extrapolation fails. The parameters ǫM and ǫJ describe possible differ- ences between the values of the mass Mend ≡ Eend and angular momentum âend ≡ Jend/M end at the end of the EOB inspiral and the final BH mass and angular mo- mentum. (The end of the EOB inspiral occurs around the EOB light-ring.) The differences are due to the fact that the system has yet to release energy and angular momentum during the merger and ring-down phase be- fore settling down to the stationary BH solution. If the total binary mass and angular momentum at the end of the EOB inspiral are Mend and Jend, we set the total mass and angular momentum of the final stationary BH to be Mf = (1 − ǫM )Mend and Jf = (1 − ǫJ)Jend, and use af ≡ Jf/Mf to compute ωlmn and τlmn. We consider the current Epn model with three parameters ǫt, ǫM and ǫJ , as a first attempt to build a physical EOB model for matching coherently the inspiral, merger and ring-down phases. Since the ǫ-parameters are related to physical quantities, e.g., the loss of energy during ring-down, they are functions of the initial physical parameters of the bi- nary, such as masses, spins, etc. In the near future we expect to be able to fix the ǫ-values by comparing NR and (improved) EOB waveforms for a large range of binary parameters. We also consider five frequency-domain models, in which two (modified SPA models) are introduced later in Sec. IVB 2, and three are introduced here: • SPAc(3.5) [33]: SPAc PN model with an appropriate cutoff frequency fcut [30, 31]; (5 + 5)M⊙ (10 + 10)M⊙ (15 + 15)M⊙ Signal Power (%) (30, 0.2) (80, 2) (85, 10) 〈hNR−hybr, hTpn(3.5)〉 0.9875 0.9527 0.8975 (M/M⊙, η) (10.18, 0.2422) (19.97, 0.2500) (29.60, 0.2499) Mωorb 0.1262 0.1287 0.1287 〈hNR−hybr, hEpn(3.5)〉 0.9836 0.9522 0.9618 (M/M⊙, η) (10.15, 0.2435) (19.90, 0.2500) (29.49,0.2488) (ǫt, ǫM , ǫJ )(%) (-0.02, 12.19, 30.87) (-0.02, 75.03, 95.00) (0.05, 2.38, 92.06) Mωorb 0.1346 0.1345 0.1345 〈hNR−hybr, hSPAc(3.5)〉 0.9690 0.9290 0.8355 (M/M⊙, η) (10.16, 0.2432) (19.93, 0.2498) (29.08, 0.2500) (fcut/Hz) 1566.8 263.9 529.6 TABLE I: FFs between hybrid waveforms [Tpn(3.5) waveform stitched to the Goddard waveform] and PN templates. In the first row, the two numbers in parentheses are the percentages of the signal-power contribution from the 16 inspiraling NR cycles and the NR merger/ring-down cycles. (The separation between inspiral and merger/ring-down is obtained using the EOB approach as a guide, i.e., we match the Epn(3.5) model and use the EOB light-ring position as the beginning of the merger phase.) In the PN-template rows, the first number in each block is the FF, and the numbers in parentheses are template parameters that achieve this FF. The last number in each block of the Tpn(3.5) and Epn(3.5) models is the ending orbital frequency of the best-match template. For the Epn model, the ending frequency is computed at the point of matching with the ring-down phase, around the EOB light ring. • BCV [31]: BCV model with an amplitude correction term (1−αf2/3) and an appropriate cutoff frequency fcut. • BCVimpr [31]: Improved BCV model with an am- plitude correction term (1 − αf1/2) and an appropri- ate cutoff frequency fcut. We include this improved BCV model because Ref. [10] found a deviation of the Fourier-transform amplitude from the Newtonian predic- tion f−7/6 during the merger and ring-down phases (see Fig. 22 of Ref. [10]). Here we shall assume n = −2/3 in the fn power law to get the (1 − αf1/2) form of the amplitude correction. While it was found [10] that the value of n is close to −2/3 for the l = 2,m = 2 wave- form, this value varies slightly if other multiple moments are included and if binary systems with different mass ra- tios are considered. Finally, the α parameter is expected to be negative, but in our actual search it can take both positive and negative values. B. Discussion of fitting-factor results In Table I, we list the FFs for hybrid target waveforms and three PN template families: Tpn(3.5), Epn(3.5), and SPAc(3.5), together with the template parameters at which the best match is obtained. As shown in the first row, in this relatively low-mass range, i.e. 10M⊙ < M < 30M⊙, the merger/ring-down phases of the waveforms contribute only a small fraction of the total signal power, while the last 16 inspiraling cycles of the NR waveform contribute a significant fraction. Therefore, confirming recent claims by Refs. [10, 11], we can conclude that the PN template families Tpn(3.5) and Epn(3.5) have good agreement with the inspiraling NR waveforms. The Tpn(3.5) model gives a low FF for M = 30M⊙ because 0.0 0.2 0.4 0.6 0.8 1.0 af /Mf FIG. 7: Frequencies and decay times of the least damped QNM 220, and two overtones 221 and 222. The scales of the frequency and the decay time are listed on the left and right sides of the plot, respectively. for these higher masses the merger/ring-down phases, which the Tpn model does not include, start contribut- ing to the signal power. Note that both time-domain templates give fairly good estimates of the mass parame- ters. The SPAc(3.5) template family gives FFs that drop substantially when the total binary mass increases from 10M⊙ to 30M⊙, indicating that this template family can only match the early, less relativistic inspiral phase of the hybrid waveforms. Nevertheless, it turns out that by slightly modifying the SPA waveform we can match the NR waveforms with high FFs (see Sec. IVB2). In Table II, we list the FFs for full NR waveforms and five PN template families: Epn(3.5), SPAextc (3.5), SPAYc (4), BCV, and BCVimpr, together with the tem- (15 + 15)M⊙ (20 + 20)M⊙ (30 + 30)M⊙ (50 + 50)M⊙ 〈hNR−Pretorius, hEpn(3.5)〉 0.9616 0.9599 0.9602 0.9787 (M/M⊙, η) (27.93, 0.2384) (35.77, 0.2426) (52.27, 0.2370) (96.60, 0.2386) (ǫt, ǫM , ǫJ )(%) (-0.08, 0.63, 99.70) (-0.03, 0.48, 94.38) (-0.12, 0.00, 64.14) (0.04, 0.01, 73.01) 〈hNR−Pretorius, hSPA c (3.5)〉 0.9712 0.9802 0.9821 0.9722 (M/M⊙, η) (19.14, 08037) (24.92, 0.9097) (36.75, 0.9933) (58.06, 0.9986) (fcut/Hz) (589.6) (476.9) (318.9) (195.9) 〈hNR−Pretorius, hSPA c (4)〉 0.9736 0.9824 0.9874 0.9851 (M/M⊙, η) (29.08, 0.2460) (38.63, 0.2461) (57.58, 0.2441) (96.55, 0.2457) (fcut/Hz) (666.5) (501.2) (332.5) (199.4) 〈hNR−Pretorius, hBCV〉 0.9726 0.9807 0.9788 0.9662 (ψ0/10 4, ψ1/10 2) (2.101, 1.655) (1.178, 1.744) (0.342, 2.385) (-0.092, 3.129) (102α, fcut/Hz) (-1.081, 605.5) (-0.834, 461.7) (0.162, 320.4) (1.438, 204.3) 〈hNR−Pretorius, hBCVimpr〉 0.9727 0.9807 0.9820 0.9803 (ψ0/10 4, ψ1/10 2) (2.377, 0.930) (1.167, 1.762) (0.431, 2.077) (-0.109, 3.158) (102α, fcut/Hz) (-3.398, 571.9) (-2.648, 458.3) (-1.196, 319.1) (-3.233, 196.0) (15 + 15)M⊙ (20 + 20)M⊙ (30 + 30)M⊙ (50 + 50)M⊙ 〈hNR−Goddard, hEpn(3.5)〉 0.9805 0.9720 0.9692 0.9671 (M/M⊙, η) (29.25, 0.2435) (38.27, 0.2422) (56.66, 0.2381) (83.52, 0.2233) (ǫt, ǫM , ǫJ )(%) (0.05, 0.03, 99.90) (0.05, 0.27, 99.17) (0.09, 0.01, 54.56) (0.10, 1.71, 79.75) 〈hNR−Goddard, hSPA (3.5)〉 0.9794 0.9785 0.9778 0.9693 (M/M⊙, η) (21.41, 0.5708) (27.27, 0.6695) (37.67, 0.9911) (60.90, 0.9947) (fcut/Hz) (552.7) (444.4) (318.5) (191.7) 〈hNR−Goddard, hSPA (4)〉 0.9898 0.9905 0.9885 0.9835 (M/M⊙, η) (30.28, 0.2456) (40.23, 0.2477) (60.54, 0.2455) (100.00, 0.2462) (fcut/Hz) (674.6) (506.6) (330.5) (195.0) 〈hNR−Goddard, hBCV〉 0.9707 0.9710 0.9722 0.9692 (ψ0/10 4, ψ1/10 2) (3.056, -1.385) (1.650, -0.091) (0.561, 1.404) (-0.113, 3.113) (102α, fcut/Hz) (0.805, 458.3) (0.559, 412.6) (0.218, 309.2) (1.063, 198.7) 〈hNR−Goddard, hBCVimpr〉 0.9763 0.9768 0.9782 0.9803 (ψ0/10 4, ψ1/10 2) (2.867, -0.600) (1.514, 0.448) (0.555, 1.425) (-0.165, 3.373) (102α, fcut/Hz) (0.193, 578.0) (-1.797, 441.1) (-4.472, 308.1) (-4.467, 193.4) TABLE II: FFs between NR waveforms and PN templates which include merger and ring-down phases. The upper table uses Pretorius’ waveform, and the lower table uses Goddard’s high-resolution long waveform. The first number in each block is the FF, and numbers in parentheses are template parameters that achieve this FF. plate parameters at which the best match is obtained. The SPAextc (3.5) and SPA c (4) families are modified ver- sions of the SPA family, defined in Sec. IVB 2. We shall investigate these results in more detail in the following sections. 1. Effective-one-body template performances The Epn model is the only available time-domain model that explicitly includes ring-down waveforms. It achieves high FFs ≥ 0.96 for all target waveforms, con- firming the necessity of including ring-down modes and proving that the inclusion of three QNMs with three tuning parameters ǫt, ǫM and ǫJ is sufficient for detec- tion. As we see in Table II, the values of the tuning parameters ǫM and ǫJ , where the FFs are achieved, are different from their physical values. For reference, the Goddard numerical simulation predicts Mf ≃ 0.95M and âf ≡ Jf/M f ≃ 0.7 [38], and Epn(3.5) predicts Mend = 0.967 and âend ≡ Jend/M end = 0.796, so the two tuning parameters should be ǫM ≃ 1.75% and ǫJ ≃ 11%. In our search, e.g., for M = 30M⊙, ǫJ tends to be tuned to its lowest possible value and ǫt tends to take its high- est possible value, indicating that pushing the end of the Epn(3.5) inspiral to a later time gives higher FFs. Since the parameters ǫM and ǫJ depend on the QNM frequency and decay time, we show in Fig. 7 how ωlmn and τlmn vary as functions of af [42] for the three modes used in the Epn(3.5) model. The frequencies ωlmn of the three modes are not really different, and grow monotoni- cally with increasing af . The decay times τlmn, although different for the three modes, also grow monotonically with increasing af . Thus, the huge loss of angular mo- mentum ǫJ , or equivalently the small final BH spin re- quired in the Epn(3.5) model to achieve high FFs, indi- cates that low ring-down frequencies and/or short decay times are needed for this model to match the numerical 0.10 0.12 0.14 0.16 0.18 Time (s) -0.02 NR frequency & waveform Epn frequency & waveform (15+15) 0.40 0.45 0.50 0.55 0.60 Time (s) (50+50) FIG. 8: Frequency evolution of waveforms from the Epn(3.5) model, and the NR simulations of the Goddard group. In the left and right panels, we show frequency evolutions for two equal-mass binary systems with total mass 30M⊙ and 100M⊙. In each panel, there are two nearly monotonic curves and two oscillatory curves, where the former are frequency evolutions and the latter are binary coalescence waveforms. The solid curves (blue) are from the NR simulations, while the dashed curves (red) are from the Epn(3.5) model. The vertical line in each plot shows the position where the three-QNM ring-down waveform is attached to the EOB waveform. merger and ring-down waveforms. In Fig. 8, we show Goddard NR and Epn(3.5) wave- forms, as well as their frequency evolutions, for two equal-mass binary systems with total masses 30M⊙ and 100M⊙. In the low-mass case, i.e., M = 30M⊙, since the inspiral part contributes most of the SNR, the Epn(3.5) model fits the frequency and phase evolution of the NR inspiral well, with the drawback that at the joining point the EOB frequency is substantially higher than that of the NR waveform. Then, in order to fit the early ring- down waveform which has higher amplitude, the tuning parameters have to take values in Table II such that the ring-down frequency is small enough to get close to the NR frequency during early ring-down stage, as indicated in Fig. 8. The late ring-down waveform does not con- tribute much to the SNR, and thus it is not too sur- prising that waveform optimizing the FF does not ade- quately represent this part of the NR waveform. In the higher mass case,M = 100M⊙, the Epn(3.5) model gives a much better, though not perfect, match to the merger and ring-down phases of the NR waveform, at the ex- pense of misrepresenting the early inspiral part. Again, this is not unexpected considering that in this mass range the merger and ring-down waveforms dominate the con- tribution to the SNR. Comparing the two cases discussed above, we can see that with the current procedure of matching the inspiral and ring-down waveforms in the EOB approach it is not possible to obtain a perfect match with the entire NR waveform. However, due to the limited detector sensi- tivity bandwidth, the FFs are high enough for detection. The large systematic error in estimating the physical pa- rameters will be overcome by improving the EOB match- ing procedure during the inspiral part, and also by fixing the ǫ-parameters to physical values obtained by compar- ison with numerical simulations. Finally, in Figs. 9, 10 we show the frequency-domain amplitude and phase of the NR and EOB waveforms. Quite interestingly, we notice that the inclusion of three ring-down modes reproduce rather well the bump in the NR frequency-domain amplitude. The EOB frequency- domain phase also matches the NR one very well. 2. Stationary-phase-approximated template performances Figures 9, 10 also show the frequency-domain phases and amplitudes for the best-match SPAc(3.5) waveforms. We see that at high frequency the NR and SPAc(3.5) phases rise with different slopes 3. Based on this obser- vation we introduce two modified SPA models: • SPAextc (3.5): SPAc PN model with unphysical values of η and an appropriate cutoff frequency fcut. The range of the symmetric mass-ratio η = m1m2/(m1+m2) 2 is ex- tended from its physical range 0 ∼ 0.25 to the unphysical 3 By looking in detail at the PN terms in the SPAc(3.5) phase, we find that the difference in slope is largely due to the logarithmic term at 2.5PN order. 100 200 500 NR phase SPAc(3.5) ext(3.5) Epn(3.5) 100 200 500 Frequency (Hz) NR amplitude Epn(3.5) FIG. 9: For M = 30M⊙ equal-mass binary systems, we com- pare the phase and amplitude of the frequency-domain wave- forms from the SPAc models and NR simulation (Goddard group). We also show the amplitude of the waveform from the Epn(3.5) model. range 0 ∼ 1. • SPAYc (4): SPAc PN model with an ad hoc 4PN order term in the phase, and an appropriate cutoff frequency fcut. The phase of the SPA model is known up to the 3.5PN order (see, e.g., Eq. (3.3) of Ref. [33]): ψ(f) = 2πft0 − φ0 − 128ηv5 k , (8) where v = (πMf)1/3. The PN coefficients αks, k = 0, . . . , N , (with N = 7 at 3.5PN order) are given by Eqs. (3.4a), (3.4h) of Ref. [33]. We add the following term at 4PN order: α8 = Y log v , (9) where Y is a parameter which we fix by imposing high matching performances with NR waveforms. Note that a constant term in α8 only adds a 4PN order term that is linear in f , which can be absorbed into the 2πft0 term. Thus, to obtain a nontrivial effect, we need to introduce a logarithmic term. The coefficient Y could in princi- ple depend on η. We determine Y by optimizing the FFs of equal and unequal masses. We find that in the equal-mass case Y does not depend significantly on the binary total mass and is given by Y = 3923. The latter is also close to the best match value obtained for unequal masses. More specifically, it is within 4.5% for binary sys- tems of mass ratiom2/m1 = 2. To further explore the de- pendence of Y on η, we need a larger sample of waveforms 50 100 200 NR phase SPAc(3.5) ext(3.5) Epn(3.5) M=100 50 100 200 Frequency (Hz) NR amplitude Epn(3.5) FIG. 10: For M = 100M⊙ equal-mass binary systems, we compare the phase and amplitude of the frequency-domain waveforms from the SPAc models and NR simulation (God- dard group). We also show the amplitude of the waveform from the Epn(3.5) model. for unequal-mass binary systems 4. As seen in Table II, the two modified SPAc template families have FF> 0.97 (except for one 0.9693) for all target waveforms, even though no explicit merger or ring-down phases are in- cluded in the waveform. The SPAYc (4) model provides also a really good estimation of parameters. In Fig. 11 we plot Goddard NR and SPAYc (4) wave- forms for two equal-mass binary systems with total massesM = 30M⊙ andM = 100M⊙. We can clearly see tail-like ring-down waveforms at the end of the SPAYc (4) waveforms, which result from the inverse Fourier trans- form of frequency domain waveforms that have been cut at f = fcut. This well-known feature is called the Gibbs phenomenon. At first glance, it may appear surpris- ing that the often inconvenient Gibbs phenomenon [44] can provide reasonable ring-down waveforms in the time domain. However, by looking at the spectra of these waveforms in the frequency domain (see the amplitudes in Figs. 9 & 10), we see that the SPAYc (4) cuts off at the frequency fcut (obtained from the optimized FF) where the NR spectra also start to drop. Thus, even though the frequency-domain SPAc waveforms are dis- continuous, while the frequency-domain NR waveforms are continuous (being combinations of Lorentzians), the SPAc time-domain waveforms contain tails with frequen- cies and decay rates similar to the NR ring-down modes. We expect that the values of the cutoff frequency fcut at which the FFs are maximized are well-determined by 4 Note that the auxiliary phase introduced in Eq. (239) of Ref. [43] also gives rise to a term in the SPA phase of the kind f log v, except an order of magnitude smaller than Y . the highest frequency of the NR waveforms, i.e. by the frequency of the fundamental QNM. In the next section, we shall show quantitative results to confirm this guess. 3. Buonanno-Chen-Vallisneri template performances In Table II we see that the BCV and BCVimpr fam- ilies give almost the same FFs for relatively low-mass binary systems (M = 30, 40M⊙), while the BCVimpr family gives slightly better FFs for higher mass binary systems (M = 60, 100M⊙). For higher-mass binary sys- tems, we find that the α parameter takes negative values with reasonable magnitude. This is because the ampli- tude of the NR waveforms in the frequency domain de- viates from the f−7/6 power law only near the merger, which lasts for about one GW cycle. This merger cycle is important only when the total mass of the binary is high enough (see Fig. 4). [See also Ref. [46] where similar tests have been done.] The BCV and BCVimpr template families give FFs nearly as high as those given by the SPAYc (4) family, but the latter has the advantage of being parametrized directly in terms of the physical binary parameters, and it gives fairly small systematic errors. C. Frequency-domain templates for inspiral, merger and ring-down In this section, we extend our comparisons between the SPAc families and NR waveforms to higher total- mass binary systems (40M⊙ to 120M⊙) and to unequal- mass binary systems with mass-ratios m2/m1 = 1.5 and 2. The numerical simulations for unequal-mass binary systems are from the Goddard group. They last for ≃ 373M and ≃ 430M , respectively, and the NR waveforms have ≃ 4 cycles before the merger. In Figs. 12 and 13 we show the FFs for SPAextc (3.5) and SPAYc (4) templates, and the values of fcut that achieved these FFs 5. For all mass combinations (ex- cept for M = 40M⊙ for artificial reasons) the FFs of SPAc(3.5) templates are higher than 0.96, and the FFs of SPAYc (4) templates are higher than 0.97, confirming that both families of templates can be used to search for GWs from coalescing binary systems with equal-masses as large as 120M⊙ and mass ratios m2/m1 = 2 and 1.5. Figure 13 shows that all the fcut values from our searches 5 Note that because of the short NR waveforms for unequal-mass binary systems, we need to search over the starting frequency of templates with a coarse grid, and this causes some oscillations in our results. The oscillations are artificial and will be smoothed out in real searches. For instance, the drop of FFs at 40M⊙ for unequal-mass binary systems happens because the NR wave- forms are too short and begin right in the most sensitive band of LIGO. are within 10% larger than the frequency of the funda- mental QNM ω220 of an equal-mass binary. We have checked that if we fix fcut = 1.07ω220/2π, the FFs drop by less than 1%. In Fig. 15, we show the same information as in Fig. 7, except that here we draw ωlnm and τlnm as functions of the mass-ratio η of a nonspinning binary. We compute the spin of the final BH in units of the mass of the fi- nal BH using the quadratic fit given by Eq. (3.17a) of Ref. [47]: ≃ 3.352η − 2.461η2 . (10) As Fig. 15 shows, ω220 does not change much, confirming the insensitivity of the fcut on η. However, in real searches we might request that the template family have some deviations from the wave- forms predicted by NR. For example, a conservative tem- plate bank might cover a region of fcut ranging from the Schwarzschild innermost stable circular orbit (ISCO) fre- quency, or the innermost circular orbit (ICO) frequency determined by the 3PN conservative dynamics, up to a value slightly higher than the frequency of the fundamen- tal QNM. The number of templates required to cover the fcut dimension depends on the binary masses. We find that to cover the fcut dimension from the 3PN ICO frequency to the fundamental QNM frequency with an SPAextc (3.5) template bank, imposing a mismatch < 0.03 between neighboring templates, we need only two (∼20) templates if M = 30M⊙ (M = 100M⊙) and η = 0.25. In the latter case, the match between templates is more sensitive to fcut since most signal power comes from the last two cycles, sweeping through a large frequency range, right in LIGO’s most sensitive band. The num- ber of templates directly affects the computational power needed, and the false-alarm rate. Further investigations are needed in order to determine the most efficient way to search over the fcut dimension. For the purpose of parameter estimation, Fig. 14 shows that the SPAYc (4) templates are rather faithful, giving reasonable estimates of the chirp mass: systematic errors less than about 8% in absolute value for binary systems with M = 40M⊙ up to M = 120M⊙. A difference of ≃ 8% may seem large, but the SPAYc (4) templates are not exactly physical, and more importantly, for large- mass binary systems, most of the information on the chirp mass comes only from the last cycle of inspiral. We notice that when the total binary mass is higher than 120M⊙, the FFs are relatively high (from 0.93 to 0.97), and the estimates of the chirp mass are still good (within 10%). However, for binary systems with such high to- tal masses, the ring-down waveform dominates the SNR, and the SPAYc (4) template family becomes purely phe- nomenological. A direct ring-down search might be more efficient. All results for unequal-mass binary systems are ob- tained using the C22 component of Ψ4 [10], which is the leading order quadrupole term contributing to the 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Time (s) NR waveform (15+15) 0.45 0.50 0.55 0.60 Times (s) (50+50) FIG. 11: Binary coalescence waveforms from the SPAYc (4) model, and the NR simulations of the Goddard group. In the left and right panels we show waveforms for two equal-mass binary systems with total mass 30M⊙ and 100M⊙. The solid lines show the waveforms from the NR simulation, and the dashed lines give the best-matching waveforms from the SPAYc (4) model. 40 60 80 100 120 Total mass ( ) ext(3.5): equal-mass ext(3.5): mass-ratio 2:1 ext(3.5): mass-ratio 3:2 (4): equal-mass (4): mass-ratio 2:1 (4): mass-ratio 3:2 FIG. 12: FFs as functions of the total binary mass. The FFs are computed between either the SPAextc (3.5) or the SPA c (4) templates and the NR waveforms for equal-mass and unequal- mass binary systems. GW radiation. For unequal-mass binary systems, higher- order multipoles can also be important, and we need to test the performance of the template family directly using Ψ4. For Ψ4 extracted in the direction perpendicular to the binary orbit, we verified that higher-order multipoles do not appreciably change the FFs. A natural way of improving the SPAc models would be to replace the discontinuous frequency cut with a linear combination of Lorentzians. We show here a first attempt 40 60 80 100 120 Total mass ( ) ext(3.5): equal-mass ext(3.5): mass-ratio 2:1 ext(3.5): mass-ratio 3:2 (4): equal-mass (4): mass-ratio 2:1 (4): mass-ratio 3:2 Frequency of the fundamental QNM FIG. 13: Cutoff frequencies as functions of the total binary mass. We show the best-match fcut for SPA c (3.5) and SPAYc (4) templates of Fig. 12. The solid black curve is the fundamental QNM frequency ω220/2π. The frequencies are in units of Hz. at doing so. The Lorentzian L is obtained as a Fourier transform of a damped sinusoid, e.g., for the fundamental QNM we have ei2πft e±iω220t−|t|/τ220 2/τ220 1/τ2220 + (2πf ± ω220) ≡ 2L±220(f) (11) 40 60 80 100 120 Total mass ( ) -0.08 -0.06 -0.04 -0.02 (4): equal-mass (4): mass-ratio 2:1 (4): mass-ratio 3:2 FIG. 14: Systematic errors of the chirp mass as functions of the total binary mass when SPAYc (4) templates are used. We show errors of the chirp masses that optimize the FFs of Fig. 12. 0.00 0.05 0.10 0.15 0.20 0.25 FIG. 15: Frequencies and decay times of the least damped QNM 220, and two overtones 221 and 222. The scales of the frequency and the decay time are listed on the left and right sides of the plot, respectively. and the (inverse) Fourier transform of Eq. (7) reads h̃QNM(f) = An L+22n(f) e iφn + L−22n(f) e Restricting to positive frequencies we only keep the L−22n(f) terms. In the frequency domain we attach the fundamental mode continuously to the SPAYc (4) wave- form at the ring-down frequency ω220 by tuning the am- plitude and phase A0 and φ0. We denote this model SPAL1 (note that we also need to introduce the mass- parameter of the final BH as a scale for ω220 and τ220). Similarly, we define the SPAL3 model where all three QNMs are combined. With the three amplitudes and phases as parameters, this model is similar to the spin- BCV template family [30] and we can optimize automat- ically over the 6 parameters. As an example, we compute the FFs between the SPAL1(4) or SPAL3(4) and the NR waveform of an equal-mass M = 100M⊙ binary. Using the LIGO PSD, we obtain 0.9703 and 0.9817, respec- tively. Those FFs are comparable to the FFs obtained with the simpler SPAc model, shown in Fig. 12. It is known that adding more parameters increases the FFs but also increases the false-alarm probability. By fur- ther investigation and comparison with NR waveforms our goal is to express the phase and amplitude parame- ters of the Lorentzian in terms of the physical binary pa- rameters, relating them to the amplitudes and phases of the QNMs and the physics of the merger. Those parame- ters are somewhat similar to the ǫ-parameters introduced above for the EOB model when modeling the merger and ring-down phases. We wish to emphasize that the results we presented in this section are preliminary, in the sense that we consid- ered only a few mass combinations and the NR waveforms of unequal-mass binary systems are quite short. Never- theless, these results are interesting enough to propose a systematic study of the efficiency of these template fam- ilies through Monte Carlo simulations in real data. V. CONCLUSIONS In this paper we compared NR and analytic waveforms emitted by nonspinning binary systems, trying to under- stand the performance of PN template families developed during the last ten years and currently used for the search for GWs with ground-based detectors, suggesting possi- ble improvements. We first computed FF0s (maximized only on time and phase) between PN template families which best match NR waveforms [10, 11], i.e., Tpn(3), Tpn(3.5) and Epn(3.5). We showed how the drop in FF0s is not simply determined by the accumulated phase difference between waveforms, but also depends on the detector’s PSD and the binary mass. Thus, waveforms which differ even by one GW cycle can have FF0 ∼ 0.97, depending on the binary masses (see Fig. 1). We then showed that the NR waveforms from the high-resolution and medium-resolution simulations of the Goddard group are close to each other (FF0 around 0.99, see Fig. 2). We also estimated that the FF0 between high-resolution and exact NR waveforms is even higher, based on the numerical convergence rates of the Goddard simulations. Second, by stitching PN waveforms to NR waveforms we built hybrid waveforms, and computed FF0s (max- imized only on time and phase) between hybrid wave- forms constructed with different PN models, notably Tpn(3), Tpn(3.5) and Epn(3.5) models. We found that for LIGO’s detectors and equal-mass binary systems with total mass M > 30M⊙, the last 11 GW cycles plus merger and ring-down phases contribute > 90% of the signal power. This information can be used to set the length of NR simulations. The FF0s between hybrid waveforms are summarized in Figs. 5, 6. We found that for LIGO’s detectors and binary systems with total mass higher than 10M⊙, the current NR simulations for equal-mass binary systems are long enough to reduce the differences between hybrid waveforms built with the PN models Tpn(3), Tpn(3.5) and Epn(3.5) to the level of < 3% mismatch. For GW detectors with broader bandwidth like advanced LIGO and VIRGO, longer NR simulations will be needed if the total binary masses M < 10M⊙. With the current avail- able length of numerical simulations, it is hard to esti- mate from the FFs between hybrid waveforms how long the simulations should be. Nevertheless, from our study of the distribution of signal power, we estimate that for M < 10M⊙ binary systems, at least ∼ 80 NR inspiraling cycles before merger are needed. Finally, we evaluated FFs (maximized on binary masses, initial time and phase) between full NR (or hy- brid waveforms, depending on the total binary mass) and several time and frequency domain PN template families. For time-domain PN templates and binary masses 10M⊙ < M < 20M⊙, for which the merger/ring- down phases do not contribute significantly to the to- tal detector signal power, we confirm results obtained in Refs. [10, 11], notably that Tpn(3.5) and Epn(3.5) mod- els have high FFs with good parameter estimation, i.e., they are faithful. We found that the frequency-domain SPA family has high FFs only for binary systems with M < 20M⊙, for which most of the signal power comes from the early stages of inspiral. Furthermore, we found that it is possible to improve the SPA family by either extending it to unphysical regions of the parameter space (as done with BCV templates) or by introducing an ad hoc 4PN-order constant coefficient in the phase. Both modified SPA families achieve high FFs for high-mass binary systems with total masses 30M⊙ < M < 120M⊙. For time-domain PN templates and binary masses M >∼ 30M⊙, we found that if a superposition of ring- down modes is attached to the inspiral waveform, as nat- urally done in the EOB model, the FFs can increase from ∼ 0.8 to > 0.9. We tested the current Epn(3.5) template family obtained by attaching to the inspiral waveform three QNMs [10] around the EOB light-ring. In order to properly take into account the energy and angular- momentum released during the merger/ring-down phases we introduced [10] two physical parameters, ǫM and ǫJ , whose dependence on the binary masses and spins will be determined by future comparisons between EOB and NR waveforms computed for different mass ratios and spins. We found high FFs >∼0.96. Due to small differ- ences between EOB and NR waveforms during the fi- Total mass ( ) 50 100 150 200 Enhanced LIGO Template: NR Template: SPAc ext(3.5) Template: Epn(3.5) FIG. 16: The sky-average SNR for LIGO and Enhanced or mid LIGO detector versus total mass for an equal-mass binary at 100Mpc. nal cycles of the evolution, the best-matches are reached at the cost of large systematic error in the merger–ring- down binary parameters. Thus, the Epn(3.5) template family can be used for detection, but for parameter es- timation it needs to be improved when matching to the ring-down, and also during the inspiral phase. The re- finements can be achieved (i) by introducing deviations from circular motion, (ii) adding higher-order PN terms in the EOB dynamics, (iii) using in the EOB radiation- reaction equations a GW energy flux closer the the NR flux, (iv) designing a better match to ring-down modes, etc.. The goal would be to achieve dephasing between EOB and NR waveforms of less than a few percent in the comparable-mass case, as obtained in Ref. [48] in the extreme mass-ratio limit. Indeed, with more accurate nu- merical simulations, especially those using spectral meth- ods [49], it will be possible to improve the inspiraling templates by introducing higher-order PN terms in the analytic waveforms computed by direct comparison with NR waveforms. Frequency-domain PN templates with an appropriate cutoff frequency fcut provide high FFs (> 0.97), even for large masses. This is due to oscillating tails (Gibbs phenomenon) produced when cutting the signal in the frequency domain. We tested the SPAextc (3.5) and the SPAYc (4) template families for total masses up to 120M⊙, and three mass ratios m2/m1 = 1, 1.5, and 2. We always get FFs > 0.96, even when using a fixed cutoff frequency, fcut = 1.07ω220/2π. Because of its high efficiency, faith- fulness, i.e., low systematic error in parameter estima- tion, and simple implementation, the SPAYc (4) template family (or variants of it which include Lorentzians) is, to- gether with the EOB model, a good candidate for search- ing coherently for GWs from binary systems with total masses up to 120M⊙. In Fig. 16, we show the sky averaged SNRs of a sin- gle LIGO and Enhanced or mid LIGO [50] detector, for an equal-mass binary at 100Mpc. The SNR peaks at the total binary mass M ≃ 150M⊙ and shows the importance of pushing current searches for coalescing binary systems to M > 100M⊙. In the mass range 30M⊙ < M < 120M⊙, the SNR drops only slightly if we filter the GW signal with SPAextc (3.5) or Epn(3.5) instead of using NR waveforms. The difference be- tween Epn(3.5) and SPAextc (3.5) is almost indistinguish- able. When M > 120M⊙, although the SPA c (3.5) and Epn(3.5) template families give fairly good SNRs, it is maybe not a good choice to use them as the number of cycles reduces to a few. The key problem in detecting such GWs is how to veto triggers from non-Gaussian, nonstationary noise, instead of matching the effectively short signal. This is a general problem in searches for short signals in ground-based detectors. Acknowledgments A.B. and Y.P. acknowledge support from NSF grant PHY-0603762, and A.B. also from the Alfred Sloan Foun- dation. The work at Goddard was supported in part by NASA grants O5-BEFS-05-0044 and 06-BEFS06-19. B.K. was supported by the NASA Postdoctoral Program at the Oak Ridge Associated Universities. S.T.M. was supported in part by the Leon A. Herreid Graduate Fel- lowship. Some of the comparisons with PN and EOB models were obtained building on Mathematica codes de- veloped in Refs. [14, 30, 31, 45] APPENDIX A: COMMENT ON WAVEFORMS OBTAINED FROM THE ENERGY-BALANCE EQUATION In adiabatic PN models, like the Tpn model used in this paper, waveforms are computed under the assump- tion that the binary evolves through an adiabatic se- quence of quasi-circular orbits. More specifically, one sets ṙ = 0 and computes the orbital frequency ω from the energy-balance equation dE(ω)/dt = F(ω), where E(ω) is the total energy of the binary system and F(ω) is the GW energy flux. Both E(ω) and F(ω) are computed for circular orbits and expressed as a Taylor expansion in ω. The adiabatic evolution ends in principle at the innermost circular orbit (ICO) [35], or minimum energy circular orbit (MECO) [30], where (dE/dω) = 0. By rewriting the energy-balance equation, ω(t) can be integrated directly as ω̇(t) = dE(ω)/dω . (A1) The RHS of Eq. (A1) can be expressed as an expansion in powers of ω. The expanded version is widely used in generating adiabatic PN waveforms [20, 30, 31, 45], it is used to generate the so-called Tpn template family. It turns out that Tpn(3) and Tpn(3.5) are quite close to the NR inspiraling waveforms [10, 11]. We wonder whether using the energy-balance in the form of Eq. (A1), i.e., without expanding it, might give PN waveforms closer to or farther from NR waveforms. In principle the adiabatic sequence of circular orbits described by Eq. (A1) ends at the ICO, so the adiabatic model should work better until the ICO and start deviating (with ω going to infinity) from the exact result beyond it. In Fig. 17 we show the NR orbital frequency ω(t) to- gether with the PN orbital frequency obtained by solv- ing the unexpanded and expanded form of the energy- balance equation. The frequency evolution in these two cases is rather different, with the orbital-frequency com- puted from the expanded energy-balance equation agree- ing much better with the NR one. 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0704.1965

Quantum entanglement of decohered two-mode squeezed states in absorbing and amplifying environment

Quantum entanglement of decohered two-mode squeezed states in absorbing and amplifying environment Phoenix S. Y. Poon and C. K. Law Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China (Dated: November 5, 2018) We investigate the properties of quantum entanglement of two-mode squeezed states interacting with linear baths with general gain and loss parameters. By explicitly solving for ρ from the master equation, we determine analytical expressions of eigenvalues and eigenvectors of ρTA (the partial transposition of density matrix ρ). In Fock space, ρTA is shown to maintain a block diagonal structure as the system evolves. In addition, we discover that the decoherence induced by the baths would break the degeneracy of ρTA , and leads to a novel set of eigenvectors for the construction of entanglement witness operators. Such eigenvectors are shown to be time-independent, which is a signature of robust entanglement of two-mode squeezed states in the presence of noise. PACS numbers: 03.67.Mn, 03.65.Yz, 42.50.Dv, 42.50.Lc I. INTRODUCTION Optical two-mode squeezed vacuum (TMSV) has been a major source of continuous-variable entanglement for quantum communication [1]. In recent years, intriguing applications such as quantum teleportation [2, 3, 4, 5, 6] and quantum dense coding [7, 8] have been demon- strated experimentally with TMSV. Theoretically, it is also known that TMSV maximizes the EPR correlation when a fixed amount of entanglement is given [9]. In order to exploit fully the non-classical properties of such entangled light fields, it is important to understand deco- herence effects as they propagate through noisy environ- ments [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. This belongs to a more subtle topic involving the characterization and quantification of mixed state entanglement in general. For bipartite systems, Peres and Horodecki have de- veloped a powerful criterion of entanglement, which is known as the PPT (positive partial transposition) crite- rion [20, 21, 22]. If the partial transposition of a den- sity matrix (denoted by ρTA) has one or more negative eigenvalues, then the state is an entangled state. Phys- ically, the partial transposition for separable states can be considered as a time-reversal operation, and one can construct a variety of uncertainty relations serving as in- dicators of entanglement [23, 24]. For two-mode Gaus- sian states such as TMSV, PPT provides a necessary and sufficient condition of separability [25, 26]. The dynamics of disentanglement of TMSV in various noisy situations has been addressed by several authors recently [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. The fact that an amplitude damped TMSV remains Gaus- sian enables an elegant description of entanglement based on the properties of covariance matrix associated with the density operators [27]. In particular, from the time- dependent solution of Wigner function [15, 16, 17] or the corresponding characteristic function [19], one can quantify the degradation of entanglement by calculating the negativity [28] and relative entropy [29]. It is now known that for an initial TMSV at a non-zero thermal bath, quantum entanglement vanishes completely in a fi- nite time [27]. However, we notice that there are much less investiga- tions addressing the structure of ρTA directly, and yet ρTA is what the PPT criterion originally based upon. Since ρTA could manifest differently in various basis, the study of ρTA in Fock space, for example, could reveal entangle- ment properties not easily found by the Wigner function method [30]. An example we notice is the construction of entanglement witness operators via the projectors formed by the eigenvectors of ρTA with negative eigenvalues [31]. Such entanglement witness operators, which correspond a variety of observables for the detection of entanglement, are determined by ρTA . The main purpose of this paper to indicate some key features of decohered entanglement as revealed by eigen- values and eigenvectors of ρTA . Our analysis will con- centrate on the structures of eigenvectors in Fock space, which is also where interesting non-local correlations of continuous-variable systems can be observed [32]. For a TMSV under the influence of amplitude damping (or gaining in an amplifier), we solve for the time evolution of ρ and determine the exact eigenvectors and eigenvalues of ρTA analytically. These eigenvectors are shown to have a strong correlation in photon numbers, and hence ρTA is a block diagonal matrix in Fock space. Therefore witness operators associated with each block involve only a finite number of Fock vectors, which implies that the detection of entanglement can only require a small portion of the Hilbert space. This is in contrast to entanglement detec- tion based on uncertainty relations in which the entire Hilbert space is usually involved [23, 24, 25, 26]. In this sense the eigenvectors of ρTA access the entanglement sig- natures ‘locally’, which is a complement to ‘global’ char- acterization (of Gaussian states) using covariance matri- ces. As we shall see below, as long as the initial state is a TMSV, the corresponding eigenvectors do not change with time, indicating that the entanglement carried by TMSV is robust against amplitude damping. http://arxiv.org/abs/0704.1965v1 FIG. 1: The sub-matrix structure of ρTA , with initial TMSV, in the Fock basis. II. MASTER EQUATION AND SOLUTION To begin with, we consider the time evolution of an initial TMSV, each coupled with a separate phase- insensitive linear bath. In terms of the annihilation op- erators a and b of the two modes, the master equation governing the dynamical process is [33]: ρ̇ = G(2a†ρa− aa†ρ− ρaa† + 2b†ρb− bb†ρ− ρbb†) + L(2aρa† − a†aρ− ρa†a+ 2bρb† − b†bρ− ρb†b), where G and L are the gain and loss parameters respec- tively, both having a dimension of time−1. Depending on the values of G and L, the master equation describes amplifying or damping effects due to the coupling with the baths. For dissipation in thermal baths, each of temperature T , we have the parameters G = γ nth and L = γ (nth +1), where nth = exp(hω/kT )−1 is the average number of photons in each of the modes (with frequency ω) at thermal equilibrium, and γ/2 is the decay rate of the mode amplitudes. In this paper we focus on the ini- tial TMSV with the squeezing parameter r > 0: |ψ(0)〉 = exp[r(a†b† − ab)] |00〉 = 1− λ2 λn |nn〉 ,(2) where λ ≡ tanh r and |00〉 is the two-mode vacuum state. A. Block structures of ρ TA in Fock space To investigate the entanglement properties of the den- sity matrix ρ, we study its partial transposition ρTA . The ρTA is of infinite dimension, however, by examining the master equation in the Fock basis, block structures of ρTA can be identified. Let us denote the matrix elements of ρTA by ρTAn,m,p,q = nm|ρTA |pq = 〈pm|ρ|nq〉 , (3) which is governed by the following differential equation: ρ̇TAn,m,p,q = G[2 npρTAn−1,m,p−1,q mqρTAn,m−1,p,q−1 −(n+m+ p+ q + 4)ρTAn,m,p,q] (n+ 1)(p+ 1)ρTAn+1,m,p+1,q (m+ 1)(q + 1)ρTAn,m+1,p,q+1 −(n+m+ p+ q)ρTAn,m,p,q]. (4) ρTAn,m,p,q(t = 0) = δpmδnq(1− λ2)λm+n. (5) corresponds to the initial condition (2). It can be seen from Eq. (4) that each el- ement ρTAn,m,p,q(t) is coupled with elements ρTAn+l,m+k,p+l,q+k(0) only, for integers l and k. Therefore ρTAn+l,m+k,m+l,n+k (t) are the only non-zero elements at any time t > 0 because of the initial con- dition. By noting that the sum of the first two indices equal to that of the last two, we can group all non-zero elements ρTAn+l,m+k,m+l,n+k (t) into sub-matrices MS according to the sum index S = n + l + m + k, i.e., ρTAn+l,m+k,m+l,n+k (t) is contained in Mn+l+m+k. We can therefore express ρTA in a direct sum of MS as follows: ρTA(t) = MS(t), (6) where the sub-matrix MS has a dimension of S+1, since elements inMS have its first two indices as {0, S}, {1, S− 1}, ... , {S, 0}. Fig. 1 shows the sub-matrix structure of ρTA . Note that characteristic sum S is equal to the total number of photons that the two modes contain. From Eq. (4) we observe that probabilistic flow occurs between elements in neighboring sub-matrices, with emission or absorption of one photon in one of the modes at one time. The time evolution of a typical sub-matrix of ρTA is illustrated schematically in Fig. 2, which will be dis- cussed in detail in the later part of the paper. Ini- tially, only opposite-diagonal elements are present, hav- ing the magnitude as λS . As time increases, element flows from neighboring sub-matrices, and disentangle- ment of the sub-matrices occurs at a critical time t = tc (Section IIIA). In the case of thermal bath, ρ evolves into a diagonal form in the long time limit, settling as the thermal equilibrium state ρTAn,m,p,q = δnpδmq (nth+1)2 ( nth nth+1 )n+m. B. Analytic solution of ρ in position space To analyze the properties of ρTA , it is more conve- nient to first determine ρ in position space and then FIG. 2: (Color online) Schematic diagram showing the evo- lution of the distribution of elements in a sub-matrix of ρTA , assuming the baths are thermal baths. The tc is the critical time for disentanglement. make the transformation to Fock space. The position space method of finding ρ was previously employed in Ref. [11, 34] in studying entanglement in various oscil- lator systems. In this subsection, we present an explicit solution of master equation (1) with an initial TMSV. We remark that our method is different from that given in [11], as the latter involves a Fourier transform of the density matrix, i.e., the momentum space. Here we solve the density matrix entirely in position space (Appendix A). This turns out to be more convenient for the real symmetric Gaussian states considered here, since fewer differential equations are involved. In addition, the re- sultant solution is more transparent for further analysis of eigenvectors in the next section. Let us denote the ‘position’ operators as x = 1√ (a+a†) and y = 1√ (b + b†), and define ρ(x1, y1;x2, y2; t) ≡ 〈x1, y1| ρ(t) |x2, y2〉 , (7) then the master equation (1) becomes, ρ̇ = −1 [L(x21 + x 2 − 2x1x2 + y21 + y22 − 2y1y2 − 4 −∂2x1 − ∂ − ∂2y1 − ∂ − 2∂x1∂x2 − 2∂y1∂y2 −2x1∂x2 − 2x2∂x1 − 2y1∂y2 − 2y2∂y1) +G(x21 + x 2 − 2x1x2 + y21 + y22 − 2y1y2 + 4 −∂2x1 − ∂ − ∂2y1 − ∂ − 2∂x1∂x2 − 2∂y1∂y2 +2x1∂x2 + 2x2∂x1 + 2y1∂y2 + 2y2∂y1)]ρ. (8) For an initial state (2), ρ(x1, y1;x2, y2; t) takes a Gaussian form at any time t, ρ(x1, y1;x2, y2; t) = Ξ(t) exp[−A(t)(x21 + x22 + y21 + y22) +B(t)(x1y1 + x2y2) +C(t)(x1x2 + y1y2) +D(t)(x1y2 + x2y1)], (9) where A(t), B(t), C(t) and D(t) are real time-dependent coefficients, and the normalization factor is: Ξ(t) = [2A(t)− C(t)]2 − [B(t) +D(t)]2. (10) By substituting Eq. (9) into the master equation, the coefficients are found to obey a set of coupled equations that can be solved analytically (Appendix A). For the TMSV considered here, we have, A(t) = [2A0η + G− L(η − 1) + 2(〈x2〉2t − 〈xy〉 B(t) = [B0η + 〈xy〉t 2(〈x2〉2t − 〈xy〉 C(t) = [2A0η + G− L(η − 1)− 2(〈x2〉2t − 〈xy〉 D(t) = [−B0η + 〈xy〉t 2(〈x2〉2t − 〈xy〉 ]. (11) Here A0 = cosh 2r, B0 = sinh 2r and η(t) = exp[2(G− L)t] are defined, and the expectation values are given by, = A0η + 2(G− L) (η − 1), 〈xy〉t = η. (12) III. PROPERTIES OF ρ According to PPT criterion, the appearance of nega- tive eigenvalues of ρTA is a signature of entanglement. In this section, we solve the eigenvectors and eigenvalues of ρTA as the system evolves. Then we discuss how deco- herence affects the entanglement properties of ρTA . The eigenvalues and eigenvectors of ρTA are defined by: ρTA(x1, y1;x2, y2; t)ϕn,m(x2, y2; t)dx2dy2 = ξn,m(t)ϕn,m(x1, y1; t). (13) Our main technique of solving the eigen-problem is the use of Mehler formula which expands a double Gaussian function into a series of orthogonal functions. After some calculations (see Appendix B), we obtain the expression of eigenvalues, ξn,m(t) = Ξ(t)π β1)n+1 β2)m+1 and the corresponding eigenvectors, ϕn,m(x1, y1) = 2n+mn!m!π x1 − y1√ x1 + y1√ × exp[−1 (x21 + y 1)] (15) where Hn are the Hermite polynomials. In writing Eq. (14), we have defined α1(t) = [2A(t)−B(t) + C(t) +D(t)], β1(t) = [2A(t) +B(t)− C(t) +D(t)], α2(t) = [2A(t) +B(t) + C(t)−D(t)], β2(t) = [2A(t)−B(t)− C(t)−D(t)] (16) and the solution of A, B, C and D are given by Eq. (11). Note that in writing Eq. (15) from (B8), we have used the fact that for TMSV, α1β1 = α2β2 = for all time t ≥ 0. We now transform the eigenvectors from the position space to the Fock space. Note that ρTA is a basis de- pendent operation. The eigenvectors of ρTA defined in two different basis sets do not transform directly. An exception is the case when the two sets of basis vectors transform by a real unitary matrix [35], which is the case here. This allows us to write down the eigenvector |ϕn,m〉 in Fock space from Eq. (15): |ϕn,m〉 = [ a† − b†√ a† + b†√ )m] |00〉 , (17) which can be connected to the sub-matrices of ρTA (Fig. 1) via the photon number sum S ≡ m + n to label the eigenket, so that |ϕn,S−n〉 = n!(S − n)! ΓS,n,j j!(S − j)! |j, S − j〉 . (18) Here we have used the abbreviation ΓS,n,j ≡ min(j,n) (−1)n−kCS−nj−k C k , (19) with CSr ≡ S!r!(S−r)! . In this way |ϕn,S−n〉 and ξn,S−n (n = 0, 1, ..., S) are the nth eigenvector and eigenvalue of the block with characteristic sum S. A. Evolution of negative eigenvalues By inspecting Eq. (14), we find that all the negative eigenvalues in each block share the same value at t = 0 (Fig. 3). The same is true also for positive eigenval- ues. However, such a strong degeneracy is broken by coupling with the baths. This is illustrated in Fig. 3 where the time-dependence of individual eigenvalues in various block indices S is shown. Except at t = 0 and at the critical time tc, we see that the negative eigenvalues possessing different values. It is important to observe that the eigenvalues are neg- ative for odd n, when β1, or in other words, B(t) > C(t). All eigenvalues turn zero at the same time, when we have B(t) = C(t), except for the only eigenvalue with n = 0 in each block. Such a critical time tc is given 2 (L−G) log G+ Lλ G (1 + λ) which is always positive finite as long as G 6= 0. For the case G → 0, we have tc → ∞. We remark that the FIG. 3: (Color online) The eigenvalues of sub-matrices of ρTA of an amplifier system with G = 1.5γ, L = 0.5γ and initial squeezing factor r = tanh−1 0.2, with characteristic sum (a) S = 1, (b) S = 4, (c) S = 7, and (d) S = 10. Red line indicates the n = 0 eigenvalue which does not turn zero at tc, while blue line indicates the most negative eigenvalue with n = 1. disentanglement time tc was previous obtained in Ref. [26] for thermal baths, here we obtained a general ex- pression (20) that applies to linear amplifiers as well. In particular, in the case when gain and loss parameters are equal, i.e., G = L, the critical time can be reduced to We point out that at the time of disentanglement t = tc, there is only one non-zero eigenvalue (with the index n = 0) in each sub-matrix (Fig. 3). Therefore ρTA at the critical time is highly degenerate, and the corresponding symmetry property of ρTA(t = tc) is indicated in the relation: ρTAj,S−j,j,S−j = ρ j,S−j,S−j,j = ρTAS−j,j,j,S−j = ρ S−j,j,S−j,j. (21) This results in the symmetric distribution of elements as shown in Fig. 2 schematically. As a further remark, it is interesting that negative eigenvalues may not necessarily be monotones over time. This can be seen by differentiating Eq. (14) and looking at the initial rate: ξ̇n,S−n = (−1)n tanhS r csch rsech3r ×{(G− L)(S − 2n) +(G+ L)(S − 2n) cosh2r −[LS +G(2 + S)] sinh 2r}, (22) which can result in a negative value for certain param- eters, i.e., some eigenvalues of odd n can become more negative over time (Fig. 3d). An exceptional case is when G = 0, where we find that for odd n, the derivative at t = 0 must be positive by inspecting Eq. (22). FIG. 4: (Color online) The negativity N of ρTA with different G, fixing the parameters L = γ and initial squeezing factor r = tanh−1 0.2. B. Negativity and sub-negativity Negativity N serves as a computable measure of en- tanglement defined by the trace norm of ρTA minus 1 divided by 2 [28]. For separable states, ρTA is still a den- sity matrix with trace 1 and hence N = 0. However, for non-separable states with negative ρTA , we have N > 0. Specifically, N equals the sum of the absolute value of negative eigenvalues of ρTA [28, 36]. For the system con- sidered in this paper, N reads, N = πΞ(t) ). (23) Alternatively,N can be derived from the symplectic spec- trum of the covariance matrix associated with the den- sity operator [27]. In Fig. 4 we show the time evolu- tion of negativity N for initial TMSV with different G parameters. An example for the G = 0 case is the zero- temperature bath dissipation scenario. Fixing L, we ob- serve that a larger gain G leads to a shorter tc. We can also calculate the negativity in a sub-matrix MS , which measures the contribution of entanglement from the corresponding block that builds up ρTA . Specif- ically, the sub-matrix negativity NS is defined the same way as negativity N but restricted to the sub-matrix of ρTA with the characteristic sum S. To our knowledge, such a sub-matrix negativity, which requires the calcu- lations of individual eigenvalues, has not been discussed before. Explicitly, NS takes the form: NS = −πΞ(t) β1)2n+2 S−2n−1 β2)S−2n where P is defined as the integral part of S−1 . In Fig. 5 the behavior of NS of some blocks is illustrated. It is surprising that that for higher sub-matrices the corre- sponding negativity NS may increase over time. Physi- cally, the increase of NS is due to the probability flow in FIG. 5: The negativity of sub-matrices NS of ρ TA , for an am- plifier system with G = 1.5γ, L = 0.5γ and initial squeezing factor r = tanh−1 0.2, with characteristic sum (a) S = 1, (b) S = 4, (c) S = 7, and (d) S = 10. Fock space arising from damping or amplifying mecha- nisms. Since each sub-matrix of ρTA does not necessarily conserve probability (i.e., the trace of MS is not a con- stant), it is possible that some blocks could have their negativity increasing with time. However, the increase of NS does not violate the fact that the overall negativity N of ρTA is an entanglement monotone that does not increase under LOCC (the mas- ter equation corresponds to local operations). Eq. (14) reveals that eigenvalues of higher blocks are of smaller order of magnitude. As we see in Fig. 5, although nega- tivity of individual higher sub-matrices may increase over time, their contribution for negativity is smaller by sev- eral orders than the sub-matrices with lower S, and there- fore the overall negativity is still monotonic decreasing. C. Robust structure of entanglement witness An entanglement witness operator W is designed for the detection of entanglement such that Tr(ρW) < 0 for some non-separable states ρ, but Tr(ρsepW) ≥ 0 for all separable states ρsep [21]. For each eigenvec- tor |φ〉 of ρTA with a negative eigenvalue, one can con- struct an W by W = |φ〉〈φ|TA [31] meeting the criteria above. In our system, we can construct a family of W from eigenvectors |ϕn,S−n〉 with odd n accordingly, i.e., WS,n = |ϕn,S−n〉 〈ϕn,S−n|TA . From Eq. (18), the explicit form of WS,n reads, WS,n = 2Sn!(S − n)!ΓS,n,jΓS,n,l j!(S − j)!l!(S − l)!} |l, S − j〉 〈j, S − l| , FIG. 6: Example of entanglement witness W2,1. which shows that WS,n operates in only a finite dimen- sion in the Fock space. Note that Tr(ρWS,n) = ξn,S−n, and ξ1,S−1 is the most negative eigenvalue in each sub- matrix, therefore WS,1 provides the most significant en- tanglement detection among all witnesses constructed from vectors lying within the sub-matrix MS . Some ex- amples for entanglement witnesses with n = 1 are shown in Fig. 6 and Fig. 7. We observe that W also has a block diagonal structure in which non-zero elements are: 〈j, S − l|WS,n |l, S − j〉. This allows us to divide W into sub-blocks ΛK , with each sub-block characterized by a difference K ∈ [−S, S]: WS,n = ΛK , (26) where the element 〈j, S − l|WS,n |l, S − j〉 lies in the sub- block with K = S − j − l. The explicit form of ΛK is min(S,S−K) l=max(0,−K) 2Sn!(S − n)!ΓS,n,S−K−lΓS,n,l (S −K − l)!(K + l)!l!(S − l)!} |l,K + l〉 〈S −K − l, S − l| , (27) having a dimension of S − |K|+ 1. We remark that the simplest 2×2 entanglement witnessW1,1 was constructed in [37] using a different approach. Here our general WS,n applies to all S and n. Finally, let us emphasize the robustness feature of TMSV against decoherence. We have seen from Eq. (18) explicitly that the eigenvectors |ϕn,S−n〉 remain un- changed with time. The time-independent |ϕn,S−n〉 sug- gests that TMSV is robust against noise, in the sense that structure of entanglement witnessWS,n is preserved. The degradation of entanglement would only affect the eigen- values. We stress that such a time-independent property of eigenvectors is specific to initial TMSV, and does not hold for arbitrary initial states in general. In Appendix B, we derive the eigenvectors of ρTA evolving from an ini- tial two-mode symmetric Gaussian states with arbitrary FIG. 7: Part of the entanglement witness W3,1, showing the blocks with characteristic difference from -3 to 0. W3,1 is symmetric about the top-right to bottom-left diagonal. real coefficients A0, B0, C0 and D0. We find that the time dependence of the eigenvectors arises solely from the evolution of the factors α1β1 and α2β2. Because of the system-bath interactions, these factors are generally dependent on time. However, the state evolving from TMSV is an exception in which one can show that the corresponding α1β1 and α2β2 equal the constant 1/16 at all times. IV. CONCLUSION To summarize, by deriving an exact analytic solution for ρTA and examining its eigenvectors, we discover sev- eral important features about the loss of entanglement of a TMSV suffered from decoherence. Both amplitude damping and amplification effects have been included in our analysis. Throughout the decoherence process, the block diagonal structure of ρTA is shown to be main- tained in Fock basis. As each block spans only a finite portion of the Fock space, the existence of negative eigen- values in a block implies that entanglement information can ‘survive’ in the corresponding photon-number sub- space. If quantum entanglement of the system is to be destroyed completely, then all the blocks have to be made positive. In other words, by simply mixing the system with another state involving finite Fock space would not destroy the entanglement. For the decoherence process considered in this paper, all the blocks turn positive at a critical time tc, which agree with the previous analy- sis based on the covariance matrix. At t < tc, we de- rived an explicit expression of the negativity as well as the negativity of sub-matrices in order to characterize the time-dependence of entanglement. Interestingly, the negativity of some sub-matrices could increase with time when G is non-zero, although the effect is weak accord- ing to our calculations. The signature of entanglement in photon number subspace can be detected by the witness operatorsWS,n, and we have constructed WS,n explicitly in this paper. These witness operators are also block di- agonal, and most remarkably, they are time-independent even though the system is under the influence of noise. We interpret such a property as a kind of robust entan- glement structure inherited in TMSV. Acknowledgments This work is supported by the Research Grants Council of the Hong Kong SAR, China (Project No. 401305). APPENDIX A: TIME EVOLUTION OF GENERAL REAL SYMMETRIC TWO-MODE GAUSSIAN DENSITY OPERATOR In this Appendix, we consider the time evolution of real symmetric two-mode Gaussian states, each coupling linearly with a separate bath with general gain and loss parameters. The density matrix is represented in position space, obeying the master equation as in Eq. (8). For the solution of the two-mode real symmetric Gaussian state presented in Eq. (9), a direct substitution leads to the following coupled differential equations: Ȧ = (G− L)C + 1 (G+ L)[1− (2A− C)2 − (B +D)2], Ḃ = −2(G− L)D − 2(G+ L)(2A− C)(B +D), Ċ = 4(G− L)A+ (G+ L)[1 + (2A− C)2 + (B +D)2], Ḋ = −2(G− L)B − 2(G+ L)(2A− C)(B +D). (A1) Without loss of generality, we consider the G 6= L case. From Eqs. (A1), B − D and 2A + C have the simple solution, B(t)−D(t) = (B0 −D0)η(t), 2A(t) + C(t) = (2A0 + C0)η(t) + G− L [η(t)− 1], where η(t) ≡ exp[2(G− L)t] and the zero subscripts de- note the values at t = 0. The other two combinations, B+D and 2A−C, can be found by noting that they are related to the second moments 〈x2〉 and 〈xy〉 by: 2A(t)− C(t) = 2[〈x2〉2t − 〈xy〉 B(t) +D(t) = 〈xy〉t 2[〈x2〉2t − 〈xy〉 . (A3) with the subscript t denoting the value at time t. Under the condition: = 0 (A4) which applies to TMSV, and from the Heisenberg equa- tions of motions of a and b, the time-dependence of the second moments are given by: η(t) + 2(G− L) [η(t)− 1], 〈xy〉t = 〈xy〉0 η(t). (A5) Here the initial second moments are given by: 2A0 − C0 2[(2A0 − C0)2 − (B0 +D0)2] 〈xy〉0 = B0 +D0 2[(2A0 − C0)2 − (B0 +D0)2] . (A6) Thus the time evolution of the coefficients of the two- mode Gaussian state solution are as follows: A(t) = [(2A0 + C0)η + G− L (η − 1) 2(〈x2〉2t − 〈xy〉 B(t) = [(B0 −D0)η + 〈xy〉t 2(〈x2〉2t − 〈xy〉 C(t) = [(2A0 + C0)η + G− L (η − 1) 2(〈x2〉2t − 〈xy〉 D(t) = [−(B0 −D0)η + 〈xy〉t 2(〈x2〉2t − 〈xy〉 ], (A7) where the expectation values are given in Eq. (A5). In the case of TMSV, the solution is reduced to Eq. (12) as the initial coefficients satisfy: C0 = D0 = 0 (A8) 4A20 −B20 = 1. (A9) In particular, by noting that A0 = 1−λ2 , we have A0 = cosh 2r. APPENDIX B: DERIVATION OF EIGENVECTORS AND EIGENVALUES OF ρ In position space, ρTA(x1, y1;x2, y2; t) = ρ(x2, y1;x1, y2; t). The eigenvectors ϕnm and eigenvalues ξnm of ρ TA are defined by Eq. (13). Applying the trans- formation xj = (uj + vj)/ 2 and yj = (−uj + vj)/ (j = 1, 2), ρTA becomes a neat product of two double Gaussians as follows: ρTA(u1, v1;u2, v2; t) = Ξ(t) exp[−α1(t)(u1 − u2)2 − β1(t)(u1 + u2)2] × exp[−α2(t)(v1 − v2)2 − β2(t)(v1 + v2)2], (B1) where α1 = (2A−B+C+D), β1 = 14 (2A+B−C+D), (2A+B +C −D) and β2 = 14 (2A−B −C −D). We apply Mehler’s Formula twice, one for the u1, u2 double Gaussian, and one for the v1, v2 double Gaussian function. This would lead to the Schmidt decomposition on ρTA : ρTA(u1, v1;u2, v2; t) = Ξ(t) α1β1α2β2 λnfn(u1)fn(u2) λ̃mf̃m(v1)f̃m(v2), (B2) where the Schmidt modes fn(u) and f̃m(u) are: fn(u) = 2n−1n! 4Hn[2(α1β1) exp(−2 α1β1u f̃m(u) = 2m−1m! 4Hm[2(α2β2) exp(−2 α2β2u 2), (B3) and the coefficients λn(t) and λ̃m(t) λn(t) = 2(α1β1) β1)n+1 λ̃m(t) = 2(α2β2) β2)m+1 , (B4) where Hn(u) are the Hermite polynomials. Rearranging terms, Eq. (B2) gives: ρTA(x1, y1;x2, y2; t) ≡ ξn,m(t)ϕn,m(x1, y1; t) ×ϕn,m(x2, y2; t), (B5) where the eigenvectors ϕn,m are ϕn,m(x1, y1; t) 2n−12m−1n!m! α1β1α2β2 ×Hn[2(α1β1) x1 − y1√ )]Hm[2(α2β2) x1 + y1√ × exp[− α1β1(x1 − y1)2] exp[− α2β2(x1 + y1) and the eigenvalues ξn,m of ρ TA are, ξn,m(t) = Ξ(t)π β1)n+1 β2)m+1 These expressions of eigenvectors and eigenvalues are for any time-dependent coefficients A(t), B(t), C(t) and D(t), i.e., applicable to states evolving from arbitrary initial values A0, B0, C0 and D0. The special case with initial TMSV is given in Eq. (14) and Eq. 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0704.1966

Some new observations on interpolation in the spectral unit ball

SOME NEW OBSERVATIONS ON INTERPOLATION IN THE SPECTRAL UNIT BALL GAUTAM BHARALI Abstract. We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ωn, n ≥ 2. We begin by showing that a known necessary condition for the existence of a O(D; Ωn)-interpolant (D here being the unit disc in C), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem – one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of Ωn, n ≥ 2. 1. Introduction and Statement of Results The interpolation problem referred to in the title, and which links the assorted results of this paper, is the following (D here will denote the open unit disc centered at 0 ∈ C): (*) Given M distinct points ζ1, . . . , ζM ∈ D and matrices W1, . . . ,WM in the spec- tral unit ball Ωn := {W ∈Mn(C) : r(W ) < 1}, find conditions on {ζ1, . . . , ζM} and {W1, . . . ,WM} such that there exists a holomorphic map F : D −→ Ωn satisfying F (ζj) =Wj, j = 1, . . . ,M . In the above statement, r(W ) denotes the spectral radius of the n × n matrix W . Under a very slight simplification – i.e. that the interpolant F in (*) is required to satisfy supζ∈D r(F (ζ)) < 1 – the paper [5] provides a characterisation of the interpo- lation data ((ζ1,W1), . . . , (ζM ,WM )) that admit an interpolant of the type described. However, this characterisation involves a non-trivial search over a region in Cn Thus, there is interest in finding alternative characterisations that either: a) circum- vent the need to perform a search; or b) reduce the dimension of the search-region. In this regard, a new idea idea was introduced by Agler & Young in the paper [1]. This idea was further developed over several works – notably in [2], in the papers [7] and [8] by Costara, and in David Ogle’s thesis [13]. It can be summarised in two steps as follows: • If the matrices W1, . . . ,WM are all non-derogatory, then (*) is equivalent to an interpolation problem in the symmetrized polydisc Gn, n ≥ 2, which is defined as Gn := (s1, . . . , sn) ∈ C n : all the roots of zn + (−1)jsjz n−j = 0 lie in D 1991 Mathematics Subject Classification. Primary: 30E05, 47A56; Secondary: 32F45. Key words and phrases. Complex geometry, Carathéodory metric, minimial polynomial, Schwarz lemma, spectral radius, spectral unit ball. This work is supported in part by a grant from the UGC under DSA-SAP, Phase IV. To appear in Integral Eqns. Operator Theory. http://arxiv.org/abs/0704.1966v2 2 GAUTAM BHARALI • The Gn-interpolation problem is shown to share certain aspects of the clas- sical Nevanlinna-Pick problems, either by establishing conditions for a von Neumann inequality for Gn – note that Gn is compact – or through function theory. It would be useful, at this stage, to recall the following Definition 1.1. A matrix A ∈ Mn(C) is said to be non-derogatory if the geometric multiplicity of each eigenvalue of A is 1 (regardless of its algebraic multiplicity). The matrix A being non-derogatory is equivalent to A being similar to the companion matrix of its characteristic polynomial – i.e., if zn + j=1 sjz n−j is the characteristic polynomial then A is non-derogatory ⇐⇒ A is similar to 0 −sn 1 0 −sn−1 . . . . . . 0 1 −s1 The Agler-Young papers treat the case n = 2, while the last two works cited above consider the higher-dimensional problem. The reader is referred to [2] for a proof of the equivalence of (*), given non-derogatory matricial data, and the appropriate Gn-interpolation problem. The similarity condition given in Definition 1.1 is central to establishing this equivalence. Before presenting the first result of this paper, we need to examine what is known about (*) from the perspective of the Gn-interpolation problem. Since we would like to focus on the matricial interpolation problem, we will paraphrase the results from [13] and [8] in the language of non-derogatory matrices. Given an n × n complex matrix W , let its characteristic polynomial χW (z) = zn+ j=1(−1) jsj(W )z n−j , and define the rational function f(z;W ) := j=1 jsj(W )(−1) jzj−1 j=0 (n− j)sj(W )(−1) Then, the most general statement that is known about (*) is: Result 1.2 (paraphrased from [13] and [8]). Let ζ1, . . . , ζM be M distinct points in D and let W1, . . . ,WM ∈ Ωn be non-derogatory matrices. If there exists a map F ∈ O(D,Ωn) such that F (ζj) =Wj , j = 1, . . . ,M , then the matrices (1.1) 1− f(z;Wj)f(z;Wk) 1− ζjζk j,k=1 ≥ 0 for each z ∈ D. Here, and elsewhere in this paper, given two complex domains X and Y , O(X;Y ) will denote the class of all holomorphic maps from X into Y . Remark 1.3. The matrices in (1.1) may appear different from those in [13, Corollary 5.2.2], but the latter are, in fact, ∗-congruent to the matrices above. Even though Result 1.2 provides only a necessary condition, (1.1) is more tractable for small values of M than the Bercovici-Foias-Tannenbaum condition. Its viability as a sufficient condition, at least for small M , has been discussed in both [13] and [8]. This is reasonable because the latter condition is sufficient when n = 2 and M = 2 (and the given matrices are, of course, non-derogatory); see [4]. Given all these developments, it seems appropriate to begin with the following: INTERPOLATION IN THE SPECTRAL UNIT BALL 3 Observation 1.4. When n ≥ 3, the condition (1.1) is not sufficient for the existence of a O(D; Ωn)-interpolant for the prescribed data ((ζ1,W1), . . . , (ζM ,WM )), where each Wj ∈ Ωn, j = 1, . . . ,M , is non-derogatory. The above observation relies on ideas from complex geometry; specifically – esti- mates for invariant metrics on the symmetrized polydisc Gn, n ≥ 3. Our argument follows from a recent study [11] of the Carathéodory metric on Gn, n ≥ 3. This argument is presented in the next section. Observation 1.4 takes us back to the drawing board when it comes to realising goals of the type (a) or (b) (as in the opening paragraph) to determine whether a O(D; Ωn)- interpolant exists for a given data-set. Thus, new conditions that are inequivalent to (1.1) are desirable for the same reasons as those offered in [2] and [3]. To wit: all extant approaches to implementing the Bercovici-Foias-Tannenbaum solution of (*) are computational, and rely upon various search algorithms. Rigorous analytical results, even if they only indicate when a data-set ((ζ1,W1), . . . , (ζM ,WM )) does not admit an O(D; Ωn) interpolant – i.e. necessary conditions – provide tests of existing algorithms/software and illustrate the complexities of (*). We will say more about this; but first – notations for our next result. Given z1, z2 ∈ D, the pseudohyperbolic distance between these points, written MD(z1, z2), is defined as: MD(z1, z2) := z1 − z2 1− z2z1 ∣∣∣∣ ∀z1, z2 ∈ D We can now state our next result. Theorem 1.5. Let F ∈ O(D; Ωn), n ≥ 2, and let ζ1, ζ2 ∈ D. Write Wj = F (ζj), and σ(Wj) := the set of eigenvalues of Wj , j = 1, 2 (i.e. elements of σ(Wj) are not repeated according to multiplicity). If λ ∈ σ(Wj), then let m(λ) denote the multiplicity of λ as a zero of the minimal polynomial of Wj. Then: (1.2)  maxµ∈σ(W2) λ∈σ(W1) MD(µ, λ) m(λ), max λ∈σ(W1) µ∈σ(W2) MD(λ, µ) ζ1 − ζ2 1− ζ2ζ1 ∣∣∣∣ . Referring back to our previous paragraph: one could ask whether Theorem 1.5 is able to highlight any complexities of (*) that Result 1.2 misses. There are two parts to the answer: 1) The Jordan structure of the data-set ((ζ1,W1), (ζ2,W2)): Several well-known examples from [6] and [2] reveal that the existence of a O(D; Ωn)-interpolant, n ≥ 2, is sensitive to the Jordan structure of the matrices W1, . . . ,WM . How- ever, to the best of our knowledge, there are no results in the literature to date that incorporate information on the Jordan structures or the minimal polynomials of W1, . . . ,WM . In contrast, the following example shows that information on minimal polynomials is vital – i.e. that with the correct infor- mation about the minimal polynomials of F (ζ1) and F (ζ2), condition (1.2) is sharp. 4 GAUTAM BHARALI Example 1.6. For n ≥ 3 and d = 2, . . . , n − 1, define the holomorphic map Fd : D −→ Ωn by Fd(ζ) :=  1 0 0 . . . . . . ... 0 0 ζIn−d  , ζ ∈ D, where In−d denotes the identity matrix of dimension n− d for 1 < d < n. Let ζ1 = 0 and ζ2 = ζ. One easily computes – in the notation of Theorem 1.5 – that: µ∈σ(W2) λ∈σ(W1) MD(µ, λ) m(λ) = |ζ|, λ∈σ(W1) µ∈σ(W2) MD(λ, µ) m(µ) = |ζ|2, where the first equality holds because W1 is nilpotent of order d. So, (1.2) is satisfied as an equality for the given choice of ζ1 and ζ2 – which is what was meant above by saying that (1.2) is sharp. � 2) Comparison with (1.1): Theorem 1.5 would not be effective in testing any of the existing algorithms used in the implementation of the Bercovici-Foias- Tannenbaum solution to (*) if (1.1) were a universally stronger necessary condition than (1.2). However, (1.1) is devised with non-derogatory data in mind, whereas no simple interpolation condition was hitherto known for pairs of arbitrary matrices in Ωn. Hence, by choosing any one of W1 and W2 to be derogatory, one would like to examine how (1.1) and (1.2) compare. This leads to our next observation. Observation 1.7. For each n ≥ 3, we can find a data-set ((ζ1,W1), (ζ2,W2)) for which (1.2) implies that it cannot admit any O(D; Ωn)-interpolant, whereas (1.1) pro- vides no information. An example pertinent to this observation is presented at the end of Section 3. As for Theorem 1.5, it may be viewed as a Schwarz lemma for mappings between D and the spectral unit ball. Note that the inequality (1.2) is preserved under automorphisms of D and under the “obvious” automorphisms of Ωn (the full automorphism group Aut(Ωn), n ≥ 2, is not known). The proof of Theorem 1.5 is presented in Section 3. The key new idea in the proof of Theorem 1.5 – i.e. to focus on the minimal polynomial of certain crucial matrices that lie in the range of F – pays off in obtaining a result that is somewhat removed from the our main theme. The result in question is a generalisation of the following theorem of Ransford and White [14, Theorem 2]: (1.3) G ∈ O(Ωn; Ωn) and G(0) = 0 =⇒ r(G(X)) ≤ r(X) ∀X ∈ Ωn. One would like to generalise (1.3) in the way the Schwarz-Pick lemma generalises the Schwarz lemma for D – i.e. by formulating an inequality that is valid without assuming that the holomorphic mapping in question has a fixed point. This generalisation is as follows: INTERPOLATION IN THE SPECTRAL UNIT BALL 5 Theorem 1.8. Let G ∈ O(Ωn; Ωn), n ≥ 2, and define dG := the degree of the minimal polynomial of G(0). Then: (1.4) r(G(X)) ≤ r(X)1/dG + r(G(0)) 1 + r(G(0))r(X)1/dG ∀X ∈ Ωn. Furthermore, the inequality (1.4) is sharp in the sense that there exists a non-empty set Sn ⊂ Ωn such that given any A ∈ Sn and d = 1, . . . , n, we can find a G A,d ∈ O(Ωn; Ωn) such that dGA,d = d, and r(GA,d(A)) = r(A)1/d + r(GA,d(0)) 1 + r(GA,d(0))r(A)1/d .(1.5) 2. A Discussion of Observation 1.4 We begin this discussion with a couple of definitions from complex geometry. Given a domain Ω ⊂ Cn, the Carathéodory pseudodistance between two points z1, z2 ∈ Ω is defined as cΩ(z1, z2) := sup {pD(f(z1), f(z2)) : f ∈ O(Ω;D)} , where pD is the Poincaré distance on D (and pD is given by pD(ζ1, ζ2) = tanh −1(MD(ζ1, ζ2)) for ζ1, ζ2 ∈ D). In the same setting, the Lempert functional on Ω× Ω, is defined as (2.1) κ̃Ω(z1, z2) := inf {pD(ζ1, ζ2) : ∃ψ ∈ O(D; Ω) and ζ1, ζ2 ∈ D such that ψ(ζj) = zj, j = 1, 2.} . It is not hard to show that the set on the right-hand side above is non-empty. The reader is referred to Chapter III of [10] for details. Next, we examine a few techni- cal objects. For the remainder of this section, S = (s1, . . . , sn) will denote a point in Cn, n ≥ 2. For z ∈ D define the rational map fn(z;S) := (s̃1(z;S), . . . , s̃n−1(z;S)), n ≥ 2, by s̃j(z;S) := (n− j)sj − z(j + 1)sj+1 n− zs1 , S ∈ Cn s.t. n− zs1 6= 0, j = 1, . . . , (n− 1). Next, define F (Z; ·) := f2(z1; ·) ◦ · · · ◦ fn(zn−1; ·) ∀Z = (z1, . . . , zn−1) ∈ D where the second argument varies through that region in Cn where the right-hand side above is defined. The connection of these objects with our earlier discussions is established via f(z;S) := j=1 jsj(−1) jzj−1 j=0 (n− j)sj(−1) , z ∈ D, and S varies through that region in Cn where the right-hand side above is defined. Note the resemblance of f(z;S) to f(z;W ) defined earlier. From Theorem 3.5 of [8], we excerpt: Result 2.1. Let S = (s1, . . . , sn) denote a point in C n. Then: 1) f(z; ·) = F (z, . . . , z; ·) ∀z ∈ D, wherever defined. 2) S ∈ Gn if and only if supz∈D |f(z;S)| < 1, n ≥ 2. 6 GAUTAM BHARALI 3) If S ∈ Gn, n ≥ 2, then |f(z;S)| = sup |F (Z;S)|. For convenience, let us refer to the Carathéodory pseudodistance on Gn, n ≥ 2, by cn. Next, define – here we refer to Section 2 of [11] – the following distance function on Gn (2.2) pn(S, T ) := max Z∈(∂D)n−1 pD(F (Z;S), F (Z;T )) ∀S, T ∈ Gn. This is the distance function – whose properties have been studied in [11] – we shall exploit to support Observation 1.4. The well-definedness of the right-hand side above follows from parts (2) and (3) of Result 2.1 above. Furthermore, since F (Z;S), F (Z;T ) ∈ D for each Z ∈ D whenever S, T ∈ Gn, n ≥ 2, it follows simply from the definition that (2.3) cn(S, T ) ≥ pn(S, T ) ∀S, T ∈ Gn. Since we have now adopted certain notations from [11], we must make the following Note. We have opted to rely on the notation of [8]. This leads to a slight dis- crepancy between our definition of pn in (2.2) and that in [11]. This discrepancy is easily reconciled by the observation that F (·;S) used here and in [8] will have to be read as F (·;−s1, s2, . . . , (−1) nsn) in [11]. This is harmless because S ∈ Gn ⇐⇒ (−s1, s2, . . . , (−1) nsn) ∈ Gn. Let us now refer back to the condition (1.1) with M = 2. An easy calculation involving 2× 2 matrices reveals that When M = 2, (1.1) ⇐⇒ sup ∣∣∣∣∣ f(z;W1)− f(z;W2) 1− f(z;W2)f(z;W1) ∣∣∣∣∣ ≤ ζ1 − ζ2 1− ζ2ζ1 ∣∣∣∣ . If W1 is nilpotent of order n (recall that all matrices occuring in (1.1) are non- derogatory), then f(·;W1) ≡ 0. Of course,W2 ∈ Ωn implies that (s1(W2), . . . , sn(W2)) ∈ Gn. By part (2) of Result 2.1, f(z;W2) ∈ D ∀z ∈ D. This leads to the following key fact: (2.4) When M = 2 and W1 is nilpotent of order n, (1.1) ⇐⇒ sup tanh−1|f(z;W2)| = sup pD(0, f(z;W2)) ≤ pD(ζ1, ζ2). We now appeal to Proposition 2 in [11], i.e. pn(0, ·) 6= cn(0, ·) for each n ≥ 3. Let us now fix n ≥ 3. Let S0 ∈ Gn \ {0} be such that cn(0, S0) > pn(0, S0). Let ε0 > 0 be such that cn(0, S0) = pn(0, S0) + 2ε0. Let us write S0 = (s0,1, . . . , s0,n) and choose two matrices W1,W2 ∈ Ωn as follows: W1 = a nilpotent of order n, W2 = 0 (−1)n−1s0,n 1 0 (−1)n−2s0,n−1 . . . . . . 0 1 s0,1 INTERPOLATION IN THE SPECTRAL UNIT BALL 7 i.e. W2 is the companion matrix of the polynomial z j=1(−1) js0,jz n−j . We emphasize the following facts that follow from this choice of W1 and W2 f(·,W1) = f(·; 0, . . . , 0) ≡ 0, f(·,W2) = f(·;S0),(2.5) W1 and W2 are, by construction, non-derogatory. The relations in (2.5) are cases of a general correspondence between matrices in Ωn and points in Gn , given by the surjective, holomorphic map Πn : Ωn −→ Gn, where Πn(W ) := (s1(W ), . . . , sn(W )), and sj(W ), j = 1, . . . n, are as defined in the beginning of this article. Let us pick two distinct points ζ1, ζ2 ∈ D such that (2.6) pD(ζ1, ζ2)− ε0 < pn(0, S0) ≤ pD(ζ1, ζ2). Assume, now, that (1.1) is a sufficient condition for the existence of a O(D; Ωn)- interpolant. Then, in view of the choices of W1,W2, the second inequality in (2.6), and (2.5) we get (2.7) sup tanh−1|f(z;W2)| = sup tanh−1|f(z;W2)| = pn(0, S0) ≤ pD(ζ1, ζ2). The first equality in (2.7) is a consequence of part (2) of Result 2.1: since S0 ∈ Gn, the rational function f(·;W2) = f(·;S0) ∈ O(D) C(D), whence the equality follows from the Maximum Modulus Theorem. But now, owing to the equivalence (2.4), the estimate (2.7) implies, by assumption, that there exists an interpolant F ∈ O(D; Ωn) such that F (ζj) =Wj, j = 1, 2. Then, Πn ◦F : D −→ Gn satisfies Πn ◦F (ζ1) = 0 and Πn ◦ F (ζ2) = S0. Then, by the definition of the Lempert functional (for convenience, we denote the Lempert functional of Gn by κ̃n) κ̃n(0, S0) ≤ pD(ζ1, ζ2) < pn(0, S0) + ε0 (from (2.6), 1st part) < cn(0, S0). (by definition of ε0) But, for any domain Ω, the Carathéodory pseudodistance and the Lempert function always satisfy cΩ ≤ κ̃Ω. Hence, we have just obtained a contradiction. Hence our assumption that (1.1) is sufficient for the existence of an O(D,Ωn)-interpolation, for n ≥ 3, must be false. 3. The Proof of Theorem 1.5 The proofs in this section depend crucially on a theorem by Vesentini. The result is as follows: Result 3.1 (Vesentini, [15]). Let A be a complex, unital Banach algebra and let r(x) denote the spectral radius of any element x ∈ A. Let f ∈ O(D;A). Then, the function ζ 7−→ r(f(ζ)) is subharmonic on D. The following result is the key lemma of this section. The proof of Theorem 1.5 is reduced to a simple application of this lemma. The structure of this proof is reminiscent of [12, Theorem 1.1]. This stems from the manner in which Vesentini’s theorem is used. The essence of the trick below goes back to Globevnik [9]. The reader will notice that Theorem 1.5 specialises to Globevnik’s Schwarz lemma when W1 = 0. 8 GAUTAM BHARALI Lemma 3.2. Let F ∈ O(D; Ωn). For each λ ∈ σ(F (0)), define m(λ) :=the multiplic- ity of λ as a zero of the minimal polynomial of F (0). Define the Blaschke product B(ζ) := λ∈σ(F (0)) ζ − λ 1− λζ )m(λ) , ζ ∈ D. Then |B(µ)| ≤ |ζ| ∀µ ∈ σ(F (ζ)). Proof. The Blaschke product B induces a matrix function B̃ on Ωn: for any matrix A ∈ Ωn, we set B̃(A) := λ∈σ(F (0)) (I− λA)−m(λ)(A− λI)m(λ), which is well-defined on Ωn because whenever λ 6= 0, (I− λA) = λ(I/λ−A) ∈ GL(n,C). Furthermore, since ζ 7−→ (ζ − λ)/(1− λζ), |λ| < 1, has a power-series expansion that converges uniformly on compact subsets of D, it follows from standard arguments (3.1) σ(B̃(A)) = {B(µ) : µ ∈ σ(A)} for any A ∈ Ωn. By the definition of the minimal polynomial, B̃ ◦F (0) = 0. Since B̃ ◦F (0) = 0, there exists a holomorphic map Φ ∈ O(D;Mn(C)) such that B̃ ◦ F (ζ) = ζΦ(ζ). Note that (3.2) σ(B̃ ◦ F (ζ)) = σ(ζΦ(ζ)) = ζσ(Φ(ζ)) ∀ζ ∈ D. Since σ(B̃ ◦ F (ζ)) ⊂ D, the above equations give us: (3.3) r(Φ(ζ)) < 1/R ∀ζ : |ζ| = R, R ∈ (0, 1). Taking A =Mn(C) in Vesentini’s theorem, we see that ζ 7−→ r(Φ(ζ)) is subharmonic on the unit disc. Applying the Maximum Principle to (3.3) and taking limits as R −→ 1−, we get (3.4) r(Φ(ζ)) ≤ 1 ∀ζ ∈ D. In view of (3.1), (3.2) and (3.4), we get |B(µ)| ≤ |ζ|r(Φ(ζ)) ≤ |ζ| ∀µ ∈ σ(F (ζ)). We are now in a position to provide 3.3. The proof of Theorem 1.5. Define the disc automorphisms Mj(ζ) := ζ − ζj 1− ζjζ , j = 1, 2, and write Φj = F ◦M j , j = 1, 2. Note that Φ1(0) = W1. For λ ∈ σ(W1), let m(λ) be as stated in the theorem. Define the Blaschke product B1(ζ) := λ∈σ(W1) ζ − λ 1− λζ )m(λ) , ζ ∈ D. INTERPOLATION IN THE SPECTRAL UNIT BALL 9 Applying Lemma 3.2, we get ζ1 − ζ2 1− ζ2ζ1 ∣∣∣∣ = |M1(ζ2)| ≥ λ∈σ(W1) 1− λµ λ∈σ(W1) MD(µ, λ) m(λ) ∀µ ∈ σ(Φ1(M1(ζ2))) = σ(W2).(3.5) Now, swapping the roles of ζ1 and ζ2 and applying the same argument to B2(ζ) := µ∈σ(W2) ζ − µ 1− µζ )m(µ) , ζ ∈ D, we get (3.6) ζ1 − ζ2 1− ζ2ζ1 ∣∣∣∣ ≥ µ∈σ(W2) MD(λ, µ) m(µ) ∀λ ∈ σ(W1). Combining (3.5) and (3.6), we get  maxµ∈σ(W2) λ∈σ(W1) MD(µ, λ) m(λ), max λ∈σ(W1) µ∈σ(W2) MD(λ, µ) ζ1 − ζ2 1− ζ2ζ1 ∣∣∣∣ . We conclude this section with an example. Example 3.4. An illustration of Observation 1.7 We begin by pointing out that the phenomenon below is expected for n = 2. We want to consider n > 2 and show that there is no interpolant for the following data, but that this cannot be inferred from (1.1). First the matricial data: let n = 2m, m ≥ 2, and let W1 = any block-diagonal matrix with two m×m-blocks that are each nilpotent of order m.(3.7) Next, for an α ∈ D, α 6= 0, let W2 = the companion matrix of the polynomial (z 2m − αzm). Note that, by construction, W2 is non-derogatory. We have the characteristic poly- nomials χW1(z) = zm and χW2(z) = z2m − αzm. Hence f(·;W1) ≡ 0, f(z;W2) = −mαzm−1 2m−mαzm We recall, from Section 2, the following equivalent form of (1.1): (3.8) When M = 2, (1.1) ⇐⇒ sup ∣∣∣∣∣ f(z;W1)− f(z;W2) 1− f(z;W2)f(z;W1) ∣∣∣∣∣ ≤ ζ1 − ζ2 1− ζ2ζ1 ∣∣∣∣ . Since, clearly, f(·;W2) ∈ O(D) C(D), by the Maximum Modulus Theorem ∣∣∣∣∣ f(z;W1)− f(z;W2) 1− f(z;W2)f(z;W1) ∣∣∣∣∣ = supz∈∂D |2m−mαzm| 2m−m|α| < |α|.(3.9) 10 GAUTAM BHARALI Observe that σ(W1) = {0} and σ(W2) = {0, |α| 1/mei(2πj+Arg(α))/m, j = 1, . . . ,m}. Therefore, µ∈σ(W2) λ∈σ(W1) MD(µ, λ) m(λ) = |α|, λ∈σ(W1) µ∈σ(W2) MD(λ, µ) m(µ) = 0. We set ζ1 = 0 and pick ζ2 ∈ D in such a way that (3.10) 2m−m|α| < |ζ2| = ζ1 − ζ2 1− ζ2ζ1 ∣∣∣∣ < |α|. Such a choice of ζ2 is made possible by the inequality (3.9). In view of the last calculation above, we see that the data-set ((W1, ζ1), (W2, ζ2)) constructed violates the inequality (1.2). Thus, there is no O(D,Ω2m)-interpolant for this data-set. In contrast, since the equivalent form (3.8) of (1.1) is satisfied, the latter does not yield any information about the existence of a O(D,Ω2m)-interpolant. � 4. The Proof of Theorem 1.8 In order to prove Theorem 1.8, we shall need the following elementary Lemma 4.1. Given a fractional-linear transformation T (z) := (az + b)/(cz + d), if T (∂D) ⋐ C, then T (∂D) is a circle with centre(T (∂D)) = bd− ac |d|2 − |c|2 , radius(T (∂D)) = |ad− bc| ||d|2 − |c|2| We are now in a position to present 4.2. The proof of Theorem 1.8. Let G ∈ O(Ωn; Ωn) and let λ1, . . . , λs be the distinct eigenvalues of G(0). Define m(j) :=the multiplicity of the factor (λ− λj) in the minimal polynomial of G(0). Define the Blaschke product BG(ζ) := ζ − λj 1− λjζ )m(j) , ζ ∈ D. BG induces the following matrix function which, by a mild abuse of notation, we shall also denote as BG BG(Y ) := (I− λjY ) −m(j)(Y − λjI) m(j) ∀Y ∈ Ωn, which is well-defined on Ωn precisely as explained in the proof of Lemma 3.2. Once again, owing to the analyticity of BG on Ωn, σ(BG(Y )) = {BG(λ) : λ ∈ σ(Y )} ∀Y ∈ Ωn, whence BG : Ωn −→ Ωn. Therefore, if we define H(X) := BG ◦G(X) ∀X ∈ Ωn, INTERPOLATION IN THE SPECTRAL UNIT BALL 11 then H ∈ O(Ωn; Ωn) and, by construction, H(0) = 0. By the Ransford-White result, r(H(X)) ≤ r(X), or, more precisely µ∈σ(G(X)) µ− λj 1− λjµ  ≤ r(X) ∀X ∈ Ωn. In particular: µ∈σ(G(X)) distM(µ;σ(G(0))) ≤ r(X) ∀X ∈ Ωn, where, for any compactK D and µ ∈ D, we define distM(µ;K) := minζ∈K ∣∣(µ− ζ)(1− ζµ)−1 For the moment, let us fixX ∈ Ωn. For each µ ∈ σ(G(X)), let λ (µ) be an eigenvalue of G(0) such that ∣∣∣(µ− λ(µ))(1 − λ(µ)µ)−1 ∣∣∣ = distM(µ;σ(G(0))). Now fix µ ∈ σ(G(X)). The above inequality leads to (4.1) ∣∣∣∣∣ µ− λ(µ) 1− λ(µ)µ ∣∣∣∣∣ ≤ r(X) 1/dG . Applying Lemma 4.1 to the Möbius transformation T (z) = |µ|z − λ(µ) 1− λ(µ)|µ|z we deduce that ∣∣∣∣∣ ζ − λ(µ) 1− λ(µ)ζ ∣∣∣∣∣ ≥ ||µ| − |λ(µ)|| 1− |µ||λ(µ)| ∀ζ : |ζ| = |µ|. Applying the above fact to (4.1), we get |µ| − |λ(µ)| 1− |µ||λ(µ)| ≤ r(X)1/dG ⇒ |µ| ≤ r(X)1/dG + |λ(µ)| 1 + |λ(µ)|r(X)1/dG , µ ∈ σ(G(X)).(4.2) Note that the function t 7−→ r(X)1/dG + t 1 + r(X)1/dGt , t ≥ 0, is an increasing function on [0,∞). Combining this fact with (4.2), we get |µ| ≤ r(X)1/dG + r(G(0)) 1 + r(G(0))r(X)1/dG which holds ∀µ ∈ σ(G(X)), while the right-hand side is independent of µ. Since this is true for any arbitrary X ∈ Ωn, we conclude that r(G(X)) ≤ r(X)1/dG + r(G(0)) 1 + r(G(0))r(X)1/dG ∀X ∈ Ωn. In order to prove the sharpness of (1.2), let us fix an n ≥ 2, and define Sn := {A ∈ Ωn : A has a single eigenvalue of multiplicity n}. 12 GAUTAM BHARALI Pick any d = 1, . . . , n, and define Md(X) :=   [tr(X)/n], if d = 1,  0 tr(X)/n 1 0 0 . . . . . .  , if d ≥ 2, and, for the chosen d, define G(d) by the following block-diagonal matrix (d)(Y ) := Md(X) tr(X) ∀X ∈ Ωn. For our purposes GA,d = G(d) for each A ∈ Sn; i.e., the equality (1.5) will will hold with the same function for each A ∈ Sn. To see this, note that • r(G(d)(X)) = |tr(X)/n|1/d; and • G(d)(0) is nilpotent of degree d, whence dG(d) = d. Therefore, r(A)1/d + r(G(d)(0)) 1 + r(G(d)(0))r(A)1/d = r(A)1/d = r(G(d)(A)) ∀A ∈ Sn, which establishes (1.5) � References [1] J. Agler and N.J. Young, A commutant lifting theorem for a domain in C2 and spectral interpo- lation, J. Funct. Anal. 161 (1999), 452-477. [2] J. Agler and N.J. Young, The two-point spectral Nevanlinna-Pick problem, Integral Equations Operator Theory 37 (2000), 375-385. [3] J. Agler and N.J. Young, The two-by-two spectral Nevanlinna-Pick problem, Trans. Amer. Math. Soc. 356 (2004), 573-585. [4] J.Agler, N.J. Young, The hyperbolic geometry of the symmetrized bidisc, J. Geom. Anal. 14 (2004), 375-403. [5] H. Bercovici, C. Foias and A. Tannenbaum, Spectral variants of the Nevanlinna-Pick interpo- lation problem in Signal Processing, Scattering and Operator Theory, and Numerical Methods (Amsterdam, 1989), 23-45, Progr. Systems Control Theory 5, Birkhuser Boston, Boston, MA, 1990. [6] H. Bercovici, C. Foias, A. Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991), 741-763. [7] C. Costara, The 2 × 2 spectral Nevanlinna-Pick problem, J. London Math. Soc. (2) 71 (2005), 684-702. [8] C. Costara, On the spectral Nevanlinna-Pick problem, Studia Math. 170 (2005), 23-55. [9] J. Globevnik, Schwarz’s lemma for the spectral radius, Rev. Roumaine Math. Pures Appl. 19 (1974), 1009-1012. [10] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter Expositions in Mathematics no. 9, Walter de Gruyter & Co., Berlin, 1993. [11] N. Nikolov, P. Pflug, P.J. Thomas and W. Zwonek, Estimates of the Carathéodory metric on the symmetrized polydisc, arXiv preprint arXiv:math.CV/0608496. [12] A. Nokrane and T. Ransford, Schwarz’s lemma for algebroid multifunctions, Complex Variables Theory Appl. 45 (2001), 183–196. [13] D.Ogle, Operator and Function Theory of the Symmetrized Polydisc, Thesis (1999), http://www.maths.leeds.ac.uk/ nicholas/. [14] T.J. Ransford and M.C. White, Holomorphic self-maps of the spectral unit ball, Bull. London Math. Soc. 23 (1991), 256-262. INTERPOLATION IN THE SPECTRAL UNIT BALL 13 [15] E. Vesentini, On the subharmonicity of the spectral radius, Boll. Un. Mat. Ital. (4) 1 1968, 427-429. Department of Mathematics, Indian Institute of Science, Bangalore – 560 012 E-mail address: [email protected] 1. Introduction and Statement of Results 2. A Discussion of Observation ?? 3. The Proof of Theorem ?? 4. The Proof of Theorem ?? References

0704.1967

The microcanonical ensemble of the ideal relativistic quantum gas

arXiv:0704.1967v2 [nucl-th] 9 Oct 2007 The microcanonical ensemble of the ideal relativistic quantum gas F. Becattini, L. Ferroni Università di Firenze and INFN Sezione di Firenze Abstract We derive the microcanonical partition function of the ideal relativistic quantum gas of spinless bosons in a quantum field framework as an expansion over fixed mul- tiplicities. Our calculation generalizes well known expressions in literature in that it does not introduce any large volume approximation and it is valid at any volume. We discuss the issues concerned with the definition of the microcanonical ensemble for a free quantum field at volumes comparable with the Compton wavelength and provide a consistent prescription of calculating the microcanonical partition func- tion, which is finite at finite volume and yielding the correct thermodynamic limit. Besides an immaterial overall factor, the obtained expression turns out to be the same as in the non-relativistic multi-particle approach. This work is introductory to derive the most general expression of the microcanonical partition function fixing the maximal set of observables of the Poincaré group. Key words: PACS: 1 Introduction The microcanonical ensemble of the relativistic gas is a subject which has not received much attention in the past. The reason of scarce interest in this problem is the peculiarity of physical applications, which have been essentially confined within statistical approaches to hadron production and the bag model [1]; these are indeed the only cases where involved volumes and particle num- bers are so small that microcanonical corrections to average quantities become relevant. Otherwise, the involved energies or volumes are so large that canoni- cal and grand-canonical ensembles are appropriate for most practical purposes (e.g. in relativistic heavy ion collisions). Recently [2], it has been pointed out that the equivalence, in the thermodynamic limit, between grand-canonical, canonical and microcanonical ensembles does not apply to fluctuations, more http://arxiv.org/abs/0704.1967v2 in general to moments of multiplicity distributions of order > 1. Indeed, effects of the difference between statistical ensembles might be unveiled in studying multiplicity distributions in relativistic nuclear collisions, [3]. In view of these phenomenological applications, it would be then desirable to have an in-depth analysis of the microcanonical ensemble of a relativistic quantum gas. The main difficulty in tackling this problem stems from the need of imposing a finite volume. This is necessary to have a correct thermodynamic limit because, being the energy E finite by construction, also V must be finite if the limit with E/V fixed is to be taken. Strange as it may seem, a full and rigorous treatment of the relativistic microcanonical ensemble of an ideal gas at finite volume is still missing. In all previous works on the subject, at some point, the large volume approximation is introduced; this is tacitly done, for instance, considering the single-particle level density as continuous, namely replacing sums over discrete quantum states with momentum space integrations [4]: d3p . (1) In a previous work [5], we have derived an expression of the microcanoni- cal partition function of an ideal relativistic quantum gas with explicit finite- volume corrections (see eqs. (23),(24)). However, that expression was obtained in an essentially multi-particle first-quantization framework, which, as pointed out in ref. [6] should be expected to become inadequate at very low volumes, comparable with the Compton wavelength of particles. In this regime, under- lying quantum field effects should become important and pair creation due to localization an unavoidable effect [7]. Indeed, there are several studies of the microcanonical ensemble of a free quantum field [8], but all of them, again, at some point invoke a large-volume approximation. In the limit of large volumes one obtains the same expressions of the microcanonical partition function and, consequently, of statistical averages as in the first-quantization multi-particle approach followed in ref. [5]. The aim of this work is to derive a general expression of the microcanonical partition function in a full relativistic quantum field framework, valid for any finite volume, generalizing the results obtained in [5]. We will do this for the simplest case of an ideal gas of spinless bosons and postpone the treatment of particles with spin to a forthcoming publication [9]. We will show that the expression of the microcanonical partition function obtained in ref. [5] in a non-relativistic multi-particle approach holds provided that a consistent prescription of subtracting terms arising from field degrees of freedom outside the considered volume is introduced. The paper is organized as follows: in Sects. 2 and 3 we will argue about general features of the microcanonical ensemble for a relativistic system and discuss several issues concerning a proper definition of the microcanonical partition function. In Sect. 4 we will cope with the further issues related to the definition of a microcanonical ensemble for a quantum field at finite volume. Sects. 5 and 6 include the main body of this work, where the microcanonical partition function is worked out in a quantum field theoretical framework. In Sect. 7 we will summarize and discuss the results. 2 On the definition of the microcanonical partition function It is well known that the fundamental tool to calculate statistical averages in any ensemble is the partition function. For the microcanonical ensemble one has to calculate the microcanonical partition function (MPF) which is usually defined as the number of states with a definite value E of total energy: states δ(E − Estate) . (2) For a quantum system, the MPF is the trace of the operator δ(E − Ĥ): 1 Ω = trδ(E − Ĥ) (3) with proper normalization of the basis states. For instance, for one non- relativistic free particle, one has to calculate the trace summing over plane waves normalized so as to 〈p|p′〉 = δ3(p− p′): Ω = tr(E − Ĥ) = E − p 〈p|p〉 = 1 (2π)3 d3p δ E − p Thereby, one recovers the well known classical expression implying that the MPF is the number of phase space cells with size h3 and given energy E. This number is infinite as the volume is unbounded and it is thus impossible to calculate a meaningful thermodynamic limit at finite energy density. Hence, one usually considers a system confined within a finite region by modifying the hamiltonian Ĥ with the addition of infinite potential walls, i.e. setting Ĥ ′ = Ĥ+ V̂ where Ĥ is the actual internal hamiltonian and V̂ an external potential implementing infinite walls. Classically, this leads to a finite Ω, namely: (2π)3 d3p δ E − p 1 Throughout this work quantum operators will be distinguished from ordinary numbers by a symbol “ˆ”. where V is the volume of the region encompassed by the potential walls. 2 Also the corresponding quantum problem can be easily solved and one has: Ω = tr(E − Ĥ ′) ≡ E − k where the sum runs over all wave vectors k which, for a rectangular box with side Li and periodic boundary conditions, are labelled by three integers (n1, n2, n3) such that ki = niπ/Li. The difficulty of the quantum expression (6) with respect to the classical one (5) is that, for a given energy E, a set of integers fulfilling the constraint imposed by the Dirac’s δ: in general does not exist. Therefore, the MPF vanishes except for a discrete set of total energies, for which it is divergent. One has a finite result only for the integral number of states 0 dE Ω(E), that is the number of states with an energy less than a given E ′, but this is clearly a stepwise and non-differentiable function of E ′. This holds for an ideal gas of any finite number of particles: strictly speaking, the MPF cannot be defined at finite energy and volume as a continous func- tion. Only in the thermodynamic limit E → ∞ and V → ∞ an expression like (6) becomes meaningful because it is then possible to replace the sum over discrete levels with a phase space integration: cells (2π)3 d3p . (7) Therefore, for a truly finite quantum system, one needs a better definition of microcanonical partition function. A definition which does not suffer from previous drawbacks is the following: Ω = trV δ(E − Ĥ) ≡ 〈hV |δ(E − Ĥ)|hV 〉 . (8) where Ĥ is the internal hamiltonian, without external confining potential, and the |hV 〉’s form a complete set of normalized localized states, i.e. a complete set of states for the wavefunctions vanishing out of the region V . It should be stressed that these states are not a basis of the full Hilbert space because wavefunctions which do not vanish out of V cannot be expanded onto this basis; thence the notation trV instead of tr meaning that the trace in eq. (8) is not a proper one. The difference between (8) and a definition like (3) is that |hV 〉 are not eigenstates of the hamiltonian Ĥ and the right hand side of 2 We will use the same symbol V to denote both the finite region and its volume. (8) does not reduce to a discrete sum of δ’s. In fact, this is crucial to have a continuous function of E, unlike (6). As an example, let us work out the definition (8) for the single free particle confined in a rectangular box by infinite potential walls and compare it to (4). A complete set of states for this problem is: |k〉 = exp(ik · x) if x ∈ V k = πnx/Lxπ̂i + ny/Ly ĵπnz/Lzk̂ 0 if x /∈ V Therefore, the MPF definition (8) implies: 〈k|δ(E − Ĥ)|k〉 = d3p |〈k|p〉|2δ E − p where we have inserted a resolution of the identity by using a complete set of states for the full Hilbert space. The sum in (10) can be calculated and yields: |〈k|p〉|2 = V (2π)3 d3x exp[i(k− p) · x] d3x′ exp[ik · (x− x′)] exp[−ip · (x− x′)] (2π)3 d3x′ δ3(x− x′)] exp[−ip · (x− x′)] = V (2π)3 where the completeness relation in V : exp[ik · (x− x′)] = δ3(x− x′) (12) has been used. Thus, by using (11), the eq. (10) turns into: (2π)3 d3p δ E − p that is the same expression (5) as in the classical case. The MPF (13) is now manifestly a continuous function of E and, remarkably, its thermodynamic limit V → ∞ is the same as the thermodynamic limit of the “pure” quantum expression (6) (because of (7)). Since the only strict re- quirement for a well defined MPF is to reproduce the correct thermodynamic limit, for a gas one can choose a definition like (8) instead of Ω = trδ(E− Ĥ ′) in (6). We emphasize again that in the passage from (6) to (8) the hamilto- nian embodying an external confining potential is replaced with the internal hamiltonian while, at the same time, the localized eigenstates of the former hamiltonian are used to calculate the trace. 3 The microcanonical partition function of a relativistic system In special relativity, the microcanonical ensemble must include momentum conservation beside energy’s to fulfill Lorentz invariance. This means that the MPF definition (8) should be generalized to [5]: 〈hV |δ4(P − P̂ )|hV 〉 , (14) P being the four-momentum of the system and P̂ the four-momentum op- erator. The MPF now being a number of states per four-momentum cell, it is a Lorentz-invariant quantity. The calculation of the MPF is easiest in the rest-frame of the system, where P = (M, 0) and the four-volume V u, u being the four-velocity and V the proper volume of the system, reduces to (V,0), according to the usual formulation of statistical relativistic thermodynamics [10,11]. The eq. (14) can be further generalized by enforcing the conservation of not only energy-momentum but of the maximal set of conserved quantities, i.e. a maximal set of commuting observables built with the generators of the Poincaré group. To achieve this, one has to replace δ4(P − P̂ ) in (14) with a generic projector Pi over an irreducible state of the representation of the Poincaré group [5,6], i.e: 〈hV |Pi|hV 〉 . (15) This ensemble is still generally defined as microcanonical ensemble and (15) microcanonical partition function. In this work, we will confine to a microcanonical ensemble where only energy- momentum is fixed, i.e. our projector Pi in eq. (15) will be: Pi = δ 4(P − P̂ ) (16) and to an ideal quantum gas, i.e. with P̂ being the free four-momentum op- erator. In fact, it should be stressed that δ4(P − P̂ ) is not a proper projector, because P2 = aP where a is a divergent constant. This is owing to the fact that normalized projectors onto irreducible representations cannot be defined for non-compact groups, such as the space-time translation group T(4). Nev- ertheless, we will maintain this naming even for non-idempotent operators, relaxing mathematical rigor, because it will be favorable to adopt the projec- tion formalism. It is worth pointing out that the definition (8) involving only the internal (free) hamiltonian, is much more fit than (6) for a relativistic generalization. Besides the advantage of restoring continuity in E, discussed in previous section, this formulation can be easily extended to the full set of conservation laws with- out major conceptual difficulty. Conversely, had one tried to generalize (6), one should have defined a finite region and afterwards sought the observables commuting with the hamiltonian supplemented with an external confining po- tential. This would have not been an easy task, and, moreover, a maximal set of observables commuting with the modified Ĥ would not, in general, define a Poincaré algebra. This is a well known problem in the static bag model where the translational invariance is manifestly broken and momentum is thus not conserved. On the other hand, in the definition (15), we deal with the original Poincaré algebra of unmodified (free) operators and enforce the localization through the projector onto confined states. The eq. (15) can be recast as a full trace by inserting a complete set of states |f〉 into (15): 〈hV | |f〉〈f |Pi|hV 〉 〈f |Pi |hV 〉〈hV |f〉 ≡ 〈f |PiPV |f〉 = tr [PiPV ] (17) where |hV 〉〈hV | , (18) is, by definition, the projector onto the Hilbert subspace HV of confined states (i.e. of wavefunctions vanishing out of V ). The formula (17) is the starting point to carry out a calculation of the MPF at finite volume. The first thing to do is to expand (17) as a sum of partition functions at fixed multiplicities, i.e.: ΩN (19) for a single species gas and: Ω{Nj} (20) for a multi-species gas, where {Nj} = (N1, . . . , NK) is a set of particle mul- tiplicities for each species j = 1, . . . , K, defining a channel. ΩN or Ω{Nj} are obtained by summing over all possible values of kinematical variables with fixed multiplicities. So, if |f〉 ≡ |N, {p}〉 where {p} labels the set of kinemati- cal variables of particles in the state |f〉, ΩN reads: 〈N, {p}|PiPV |N, {p}〉 . (21) Likewise, for a multi-species gas, the microcanonical partition function is ex- pressed as an expansion over all possible channels: Ω{Nj} = 〈{Nj}, {p}|PiPV |{Nj}, {p}〉 . (22) and Ω{Nj} is defined as the microcanonical channel weight. The microcanonical channel weights (22) have been calculated in ref. [5] with energy-momentum conservation (i.e. using (16) in a multi-particle, first quan- tization framework. For a single species ideal spinless gas: d3p1 . . .d 3pN δ ρ∈ SN FV (pρ(n) − pn) (23) while for a multi-species gas of K spinless bosons: Ω{Nj} = d3pnj ρj∈ SNj FV (pρj(nj) − pnj ) (24) being N = j Nj . In eqs. (23),(24) ρj labels permutation belonging to the permutation group SNj and FV are Fourier integrals over the region V : FV (p− p′) ≡ (2π)3 d3x ei(p−p ′)·x (25) If the volume is large enough so as to allow the approximation: FV (p− p′) = (2π)3 d3x ei(p−p ′)·x ≃ δ3(p− p′) (26) the microcanonical channel weights (24) can be resummed explicitely into the microcanonical partition function according to (20) and one obtains [5]: limε→0 (2π)4 ∫ +∞−iε −∞−iε d3y eiP ·y exp (2π)3 d3p log(1− e−ip·y)−1 A full analytical calculation of eq. (27) is possible only for the limiting case of vanishing masses (e.g. microcanonical black body). For the massive case, four-dimensional integrations cannot be worked out analytically and one has to resort to numerical computation. The eq. (27) was implicitely obtained in ref. [4] where the first expression of the microcanonical partition function of a multi-species ideal relativistic quantum gas was derived as an expansion (20) over channels, by using the large-volume approximation (7) from the very beginning. This shows that the approximation (26) is indeed equivalent to the (7), as also demonstrated in ref. [5]. Noticeably, the MPF definition eq. (14) without any large volume ap- proximation involves the appearance of Fourier integrals accounting for Bose- Einstein and Fermi-Dirac correlations in the quantum gas, which do not show up in the large-volume approximation enforced in ref. [4]. This approach also allows to investigate further generalizations when the volume is so small that relativistic quantum field effects must be taken into account. 4 Microcanonical ensemble and field theory The calculation of the partition function (14) in a quantum field framework brings in new difficulties with respect to the first-quantization scheme. This problem has been approached in literature with a functional approach, in- spired of the usual grand-canonical thermal field theory [8]. However, these calculations aim at the limit of large volumes and are therefore insensitive to the difficulties pertaining to the strict requirement of finite volume discussed in detail in Sect. 2. As a result, for a free field, the derived expressions are equivalent to the formula (27). Instead of starting with a functional integration from the very beginning, we calculate the microcanonical partition function of a free field by first expanding it at fixed multiplicities like in eqs. (19),(21) (or channels, for multi-species gas like in eq. (20)), where |N, {p}〉 are Fock space states with definite particle multiplicity and kinematical variables {p}. To carry out this calculation, we first need to find an expression of the microcanonical state weight: ω ≡ 〈N, {p}|PiPV |N, {p}〉 . (28) By using (16) and choosing |N, {p}〉 as an eigenstate of the four-momentum operator with eigenvalue Pf = i pi, the eq. (28) becomes: ω = δ4(P − Pf) 〈N, {p}|PV |N, {p}〉 , (29) To calculate ω and, by a further integration, ΩN we need to know the projector PV . Since PV is defined as the projector onto the Hilbert subspace of localized states, it can be easily written down in a multi-particle non-relativistic quan- tum mechanical (NRQM) framework. As an example, for a non-relativistic spinless single particle, it reads (see also Sect. 2): |kV 〉〈kV | (30) where |kV 〉 is a normalized state of the particle confined in a region V , with a corresponding wavefunction ψkV (x) vanishing out of V . The symbol kV stands for a set of three numbers labelling the kinematical modes of the confined states (e.g. discrete wavevectors, or energy and angular momenta) and the set |kV 〉 form a complete set of states for the wavefunctions vanishing out of V . The projector (30) can be easily extended to the many-body case and: Ñ ,{k} |Ñ, {kV }〉〈Ñ, {kV }| . (31) where the symbol {kV } denotes a multiple set of kinematical modes of the confined states while Ñ is the integrated occupation number, i.e. the sum of occupation numbers over all single-particle kinematical modes. In the NRQM approach these numbers are simply particles multiplicities, implying: 〈N, {p}|Ñ, {kV }〉 6= 0 iff N = Ñ . (32) To calculate the microcanonical state weight, thence the MPF, the products 〈Ñ, {kV }|N, {p}〉 can be worked out on the basis of (32) similarly to what has been done in Sect. 2 for a single particle, yielding, for a scalar boson [5]: 〈N, {p}|PV |N, {p}〉 = ρ∈ SN FV (pρ(n) − pn) (33) where ρ is a permutation of the integers 1, . . . , N and FV are Fourier inte- grals (25) over the system region V . From the above equation the expression of the microcanonical channel weight in eq. (23) follows. We will refer to the expression (33) as the NRQM one, meaning that is has been obtained in this first-quantized multi-particle NRQM framework, where, for instance, particles and antiparticles are simply considered as different species and their contri- butions factorize. One could envisage that a projector like (31), written in terms of Fock space states, could be simply carried over to the relativistic quantum field case, where |Ñ , {kV }〉 are states of the localized problem, obtained by solving the free field equations in a box with suitable boundary conditions. Yet, some difficulties soon arise. First of all, whilst in NRQM the single-particle local- ized wavefunction |kV 〉 and the free plane wave state |p〉 live in the same Hilbert space, in quantum field theory the localized and the non-localized problem are associated with distinct Hilbert spaces. Thus, unlike in NRQM, it is not clear how to calculate a product like 〈Ñ, {kV }|N, {p}〉. Secondly, even if there was a definite prescription for it, it should be expected that the inte- grated occupation numbers of the localized problem do not coincide with actual particle multiplicities unless the volume is infinite. To understand this point, one should keep in mind that properly called particles arise from solutions of the free field equations over the whole space and that the hamiltonian eigen- states of the localized problem are conceptually different. Consequently, the integrated particle number operators in the whole space should differ from in- tegrated number operators within the finite region. Hence, unlike in NRQM, a state with definite integrated occupation numbers Ñ (we purposely refrain from calling them particle numbers) should be expected to have non vanish- ing components on all free states with different numbers of actual particles, namely: |Ñ〉V = α0,Ñ |0〉+ α1,Ñ |1〉+ . . .+ αÑ ,Ñ |Ñ〉+ . . . (34) where α are non-trivial complex coefficients, and (32) no longer holds. Only in the large volume limit one expects that the integrated occupation numbers coincide with actual multiplicities and eq. (32) applies. This kind of effect is pointed out in the introduction of Landau’s book on quantum field theory [7]: when trying to localize an electron, an electron-positron pair unavoidably appears, meaning that the localized single “particle” is indeed a superposition of many true, asymptotic particle states. Another relevant manifestation of this difference which is probably more familiar, is the Casimir effect, which is related to the difference between the true vacuum state |0〉 and the localized vacuum state |0〉V . Hence, all formulae derived under the approximation (32) are asymptotic ones, valid in the limit V → ∞ but not at strictly finite volume. Thus, one should expect significant finite-volume corrections to the eqs. (33) and the ensuing MPF (23) in a quantum field treatment. If we want to give an expression like 〈Ñ , {kV }|N, {p}〉 a precise meaning in a quantum field framework, we first need to map the Hilbert space HV of the localized problem into the Hilbert space H of the free field over the whole space. This can be done by mapping the field eigenstates and operators of H into HV in a natural way as: Ψ(x)HV −→ Ψ(x)H |ψ(x)〉HV −→ |ψ(x)〉H (35) This allows writing linear, non-bijective, Bogoliubov relations expressing the annihilation and creation operators of the finite region problem as a function of those of the whole (real scalar) field (see Appendix A for derivation): d3pF (k,p) εk + εp ap + F (k,−p) εk − εp a†p (36) where k are triplets of numbers labelling kinematical modes, just like the aforementioned kV , εk is the associated energy, εp = p2 +m2 and: F (k,p) = (2π)3 d3xu∗k(x) e ip·x (37) uk being a complete set of orthonormal wavefunctions for the finite region. A remarkable feature of relativistic quantum fields is that, unlike in NRQM, the localized annihilation operators have non-vanishing components onto the creation operators in the whole space, as shown by (36). This confirms our ex- pectation that a localized state with definite integrated occupation numbers is a non-trivial linear combination of states with different particle multiplicities. Expectedly, as the volume increases, these components become smaller and in the infinite volume limit one recovers ak = ap (see Appendix A). Starting from the Bogoliubov relations (36), it should be possible, in principle, to calculate the coefficients in eq. (34), thence the microcanonical state weight (29) by using the expansion (31). In fact, we do not really need to do that. It is more advantageous, as pointed out in ref. [6], to write the projector PV in terms of field states rather than occupation numbers of field modes within the finite region. Indeed, in the general definition in eq. (17): |hV 〉〈hV | (38) the states |hV 〉 are a complete set of states of the Hilbert space of the localized problem HV , where the degrees of freedom are values of the field in each point of the region V , i.e. {ψ(x)} |x ∈ V . Therefore, the above projector is a resolution of the identity of the localized problem and can be written as (for a real scalar field): Dψ|ψ〉〈ψ| (39) where |ψ〉 ≡ ⊗x|ψ(x)〉 and Dψ is the functional measure; the index V means that the functional integration must be performed over the field degrees of freedom in the region V , that is Dψ = ∏x∈V dψ(x). The normalization of the states is chosen so as to 〈ψ(x)|ψ′(x)〉 = δ(ψ(x) − ψ′(x)) to ensure the idempotency of PV . If we now want to give expressions like: 〈{Nj}, {p}|PV |{Nj}, {p}〉 (40) a clear meaning, we should find a way of completing the tensor product in the projector (39) with the field states outside the region V such a way the scalar product can be performed unambiguously. Unfortunately the answer to this question is not unique and the projector can be extended to H in infinitely many ways. What is important is that the result of the calculation is independent of how the projector has been extended. Thus, at the end of the calculation, one has to check whether spurious terms appear, possibly divergent, depending explicitely on the chosen state of the field outside V and these terms must be subtracted away. In general, all terms depending on the degrees of freedom of the field out of V must be dropped from the final result. In this work, we will extend the projector with eigenstates of the field, where the field function ψ(x) is some arbitrary function outside the region V . Thus, the projector PV (39) is mapped to: Dψ|ψ〉〈ψ| |ψ〉 ≡ ⊗x∈V |ψ(x)〉 ⊗x/∈V |ψ(x)〉 (41) where the index V still implies Dψ = ∏x∈V dψ(x). We will see that, with the definition (41), spurious terms depending on the degrees of freedom outside V do arise indeed, but that they can be subtracted “by hand” in a consistent 5 Single particle channel We will start calculating the expectation value 〈p|PV |p〉 of a single parti- cle channel in the simple cases of neutral and charged scalar fields. This is preparatory to the general case of multiparticle states in Sect. 6. 5.1 Neutral scalar field We consider a gas made of one type of spinless boson described by the free real scalar field 3 (in Schrödinger representation): Ψ(x) = a(p) eip·x+a†(p) e−ip·x where ε ≡ p2 +m2 is the energy, p is the modulus of the three-momentum and the normalization has been chosen so as to have the following commutation rule between annihilation and creation operators: [a(p), a†(p′)] = δ3(p− p′) . (43) We start writing the one-particle Fock state |p〉 in terms of creation and an- nihilation operators acting on the vacuum: 〈p|PV |p〉 = 〈0|a(p) PV a†(p)|0〉 . (44) 3 Henceforth, the capital letter Ψ will denote field operators while for field functions we will use the small letter ψ. Since PV is defined, according to (41) as a functional integral of eigenvectors of the field operator Ψ, it is convenient to express creation and annihilation operators in terms of the field operators. We shall use following expressions which are the most appropriate for our task: 〈0|a(p)= 〈0| 1 d3x e−ip·x 2ε Ψ(x) a†(p)|0〉= 1 d3x eip·x 2ε Ψ(x) |0〉 . (45) These expressions can be easily checked by plugging, on the right hand side, the field operators in (42). By using eq. (45) in (44): 〈0|a(p)PV a†(p)|0〉 = (2π)3 d3x′ eip·(x−x ′) 2ε 〈0|Ψ(x′)PVΨ(x)|0〉 (46) It can be easily checked now that the rightmost factor in the above equation turns out to be (by using the definition (41)): 〈0|Ψ(x′)PVΨ(x)|0〉 = Dψ |〈ψ|0〉|2ψ(x′)ψ(x) . (47) where ψ(x) and ψ(x′) are field functions or the eigenvalues of the field operator relevant to the state |ψ〉, that is: Ψ(x)|ψ〉 = |ψ〉ψ(x) . (48) It is possible to find a solution of the functional integral (47) by first consid- ering the infinite volume limit, when the projector PV reduces to the identity. In this limiting case, the functional integral in (41) is now performed over all possible field functions and eq. (41) becomes a resolution of the identity; 〈0|Ψ(x′)Ψ(x)|0〉 is just the two-point correlation function that we write, ac- cording to (47): 〈0|Ψ(x′)Ψ(x)|0〉 = Dψ |〈ψ|0〉|2ψ(x′)ψ(x) (49) The product 〈ψ|0〉 is known as the vacuum functional and reads [12], for a scalar neutral field: 〈ψ|0〉 = N exp d3x2 ψ(x1)K(x1 − x2)ψ(x2) where N is a field-independent normalization factor, which is irrelevant for our purposes. The function K(x′ −x) is called kernel and fulfills the equation [12]: ∫ d3x′ e−ip·x K(x′ − x) = 2ε e−ip·x (51) whose solution is: K(x′ − x) = 1 (2π)3 d3p e−ip·(x ′−x) 2ε . (52) The functional integral (49) is therefore a gaussian integral and can be solved by using the known formulae for multiple gaussian integrals of real vari- ables [12]: I2N = ψ(ξi) exp d3x2 ψ(x1)K(x1 − x2)ψ(x2) pairings of ξ1,...,ξ2N pairs K−1(paired variables) where paired variables means couples (ξi, ξj) whose difference ξi−ξj (or, what is the same, ξj − ξi as K−1 is symmetric) is the argument of K−1. The factor I0 is just the normalization of the vacuum state I0 = 〈0|0〉 which is set to 1. The inverse kernel K−1 can be found from its definition: d3x′K(y − x′)K−1(x′ − x) = δ3(x− y) ∀ x,y (54) leading to: K−1(x′ − x) = 1 (2π)3 e−ip·(x ′−x) = 〈0|Ψ(x′)Ψ(x)|0〉 (55) The last equality comes from (53) and (49) in the special case N = 2 or can be proved directly from field Fourier expansion (42). We are now in a position to solve the functional integral (47) at finite volume. First, the functional integration variables are separated from those which are not integrated, i.e. the field values out of V (V̄ denotes the complementary of |〈ψ|0〉|2= |N |2 exp d3x2 ψ(x1)K(x1 − x2)ψ(x2) = |N |2 exp d3x2 ψ(x1)K(x1 − x2)ψ(x2) × exp d3x2 ψ(x1)K(x1 − x2)ψ(x2) × exp d3x2 ψ(x1)K(x1 − x2)ψ(x2) where we have taken advantage of the symmetry of the kernel K. The (56) is a gaussian functional, with a general quadratic form in the field values in the region V ; it can be integrated in (47) according to standard rules [12], yielding: Dψ |〈0|ψ〉|2ψ(x′)ψ(x) = K−1V (x′,x)|N |2 det ]−1/2 exp [ d3x1d d3x2d 3x′2K V (x1,x 1)K(x1 − x2)K(x′1 − x′2)ψ(x′2)ψ(x2) d3x2 ψ(x1)K(x1 − x2)ψ(x2) = K−1V (x Dψ |〈0|ψ〉|2 = K−1V (x′,x)〈0|PV |0〉 (57) The function K−1V is the inverse of K over the region V , namely the inverse K(x′ − x)ΘV (x′)ΘV (x) ≡ KV (x′ − x) , (58) the function ΘV (x) being the Heaviside function: ΘV (x) = 1 if x ∈ V 0 otherwise. The inverse kernel K−1V fulfills, by definition, the integral equation: d3x′KV (y − x′)K−1V (x′,x) = δ3(x− y) ∀ x,y ∈ V (60) Note that, because of the finite region of integration, the inverse kernel may now depend on both the space variables instead of just their difference. Also note that K−1V is real and symmetric, being KV real and symmetric. Therefore, the result of the functional integration yields the simple formula: 〈0|Ψ(x′)PVΨ(x)|0〉 = K−1V (x′,x)〈0|PV |0〉 (61) where the factor 〈0|PV |0〉 is a positive constant which we will leave unex- panded. Altogether, the presence of the projector PV in the eq. (61) modifies the two- point correlation function by introducing a constant factor 〈0|PV |0〉 and re- placing the inverse kernel K−1 with a different one K−1V . It can be easily proved, by using the general formulae of gaussian integrals, that this holds true in the more general case of many-points correlation function. In fact, the (53) holds for general quadratic forms in the field ψ and so the eq. (61) can be generalized to: Ψ(xn)PV n=N+1 Ψ(xn)|0〉 = 〈0|PV |0〉 pairings of x1,...,x2N pairs K−1V (paired var.) . The remaining task is to calculate the inverse kernel K−1V by means of (60). In fact, we will look for a solution of the more general equation: d3x′K(y− x′)K−1V (x′,x) = δ3(x− y) ∀ x ∈ V,y (63) with unbounded y. It is clear that a solution K−1V of equation (63) is also a solution of (60) because KV equals K when y ∈ V . Setting y unbounded allows us to find an implicit form for K−1V . In fact (63) implies: (2π)3 d3x′ eip·x 2ε K−1V (x ′,x) = eip·x (2π)3 which is obtained multiplying both sides of (63) by eip·y /(2π)3 and integrating over the whole space in d3y. We are now in a position to accomplish our task of calculating 〈p|PV |p〉. By plugging (61) into (46) we get: 〈0|a(p)PV a†(p)|0〉 = (2π)3 d3x′ eip·(x−x ′) 2ε K−1V (x ′,x)〈0|PV |0〉 . The integration domain in (65) can be split into the region V and the com- plementary V̄ for both variables. The inverse kernel K−1V is not defined out of V and can then be set to an arbitrary value, e.g. zero. Otherwise, even if one chose a non vanishing prolongation of K−1V , an integration outside the domain V would involve the degrees of freedom of the field out of V and, according to the general discussion at the end of Sect. 4, the contributing term should be dropped. Therefore, retaining only the physically meaningful term, the (65) turns into: 〈0|a(p)PV a†(p)|0〉 = (2π)3 d3x′ eip·(x−x ′) 2ε K−1V (x ′,x)〈0|PV |0〉 . In the above equation one can easily recognize the complex conjugate of the left hand side of (64). Hence, replacing it with the complex conjugate of the right hand side, one gets: 〈0|a(p)PV a†(p)|0〉 = (2π)3 d3x 〈0|PV |0〉 = (2π)3 〈0|PV |0〉 (67) which is the same result of NRQM in (33) in the simple case N = 1, times a factor 〈0|PV |0〉. This factor still contains a dependence on the field degrees of freedom out of V , according to the projector expression in eq. (41), which should disappear at some point. However, we will see that this factor appears at any multiplicity and therefore becomes irrelevant for the calculations of the statistical averages. 5.2 Charged scalar field The calculation done for a neutral scalar field can be easily extended to a charged scalar field. The 2-component charged scalar field in Schrödinger rep- resentation reads: Ψ(x) = a(p) eip·x +b†(p) e−ip·x Ψ†(x) = b(p) eip·x +a†(p) e−ip·x where a, a† and b, b† are annihilation and creation operators of particles and antiparticles respectively. They satisfy commutation relations: [a(p), a†(p′)] = [b(p), b†(p′)] = δ3(p− p′) (69) [a(p), b(p′)] = [a†(p), b(p′)] = 0 Likewise, the fields obey the commutation relations: [Ψ(x),Ψ†(y)] = 0 . (70) and it is then possible to construct field states |ψ, ψ†〉. The projector PV can be written as: D(ψ†, ψ)|ψ, ψ†〉〈ψ, ψ†| (71) with suitable state normalization and arbitrary field functions ψ(x) out of the region V 4 Similarly to eq. (45), one can write: 4 The functional measure in the equation (71) reads dψ(x)dψ∗(x)/iπ. Anyhow, its explicit form is not important for our purposes. 〈0|a(p)= 〈0| 1 d3x e−ip·x 2ε Ψ(x) (72) a†(p)|0〉= 1 d3x eip·x 2ε Ψ†(x) |0〉 〈0|b(p)= 〈0| 1 d3x e−ip·x 2ε Ψ†(x) b†(p)|0〉= 1 d3x eip·x 2ε Ψ(x) |0〉 The chain of arguments of the previous subsection can be repeated and the functional integral: Ψ(xn)PV Ψ†(x′n)|0〉 = D(ψ†, ψ) |〈0|ψ, ψ†〉|2 ψ(xn)ψ †(x′n) found to be a multiple gaussian integral. Letting ρ be a permutation of the integers 1, . . . , N , the integration on the right hand side of eq. (73) yields: Ψ(xn)PV Ψ†(x′n)|0〉 = 〈0|PV |0〉 K−1V (xn,x ρ(n)) (74) which differs from the corresponding expression for the real scalar field because now the field is complex and ψ can only be coupled to ψ† [13]: D(ψ†, ψ) ψ(ξn)ψ †(ξ′n) d3x2 ψ †(x1)K(x1 − x2)ψ(x2) K−1(ξn − ξ′ρ(n)) (75) However, the functional integral involving only two fields ψ and ψ† yields the same result as for neutral particles. Thus, the kernel K is still the same and so is the integral equation (64) defining K−1V . The expectation value of PV in a state with only one particle (or antiparticle) will also be the same as in eq. (67), that is: 〈0|a(p)PV a†(p)|0〉 = 〈0|b(p)PV b†(p)|0〉 = (2π)3 〈0|PV |0〉 . (76) 6 Multiparticle channels We have seen in the previous section that the expectation value 〈p|PV |p〉 for a single spinless particle is the same obtained in a NRQM approach [5] times an overall immaterial factor 〈0|PV |0〉. In this section, we will tackle the calcula- tion of the general multiparticle state. We will see that, by using the projector definitions in eqs. (41),(71) and subtracting the spurious contributions stem- ming from external field degrees of freedom, the final result is still the same as in the NRQM calculation times the factor 〈0|PV |0〉. We will first address the case of N charged particles. 6.1 Identical charged particles We will consider a state with N identical charged particles; for N antiparticles the result is trivially the same. We want to calculate: 〈N, {p}|PV |N, {p}〉 = 〈0| a(pn)PV a†(pn)|0〉 . (77) Since: [a†,Ψ†] = [a,Ψ] = [b,Ψ†] = [b†,Ψ] = 0 (78) one can replace creation and annihilation operators with their expressions in (72) disregarding the position of the operators with respect to the vacuum state. In formula: 〈N, {p}|PV |N, {p}〉= (2π)3 d3x′n e −ipn·(xn−x′n) 2εn Ψ(xn)PV Ψ†(x′n)|0〉 (2π)3 d3x′n e −ipn·(xn−x′n) 2εn K−1V (xn,x ρ(n))〈0|PV |0〉 , (80) where SN is the permutation group of rank N . Now, like for the single particle case, we restrict the integration domain to V in (79) in order to get rid of external degrees of freedom and, by repeatedly using eq. (64), we are left with: 〈N, {p}|PV |N, {p}〉 〈0|PV |0〉 (2π)3 d3x′n e −ipn·(x′ρ(n)−x (2π)3 d3x′n e ρ−1(n) −pn)·x′n Hence, using the definition (25): 〈N, {p}|PV |N, {p}〉 = FV (pρ(n) − pn)〈0|PV |0〉 (82) which is exactly the expression (33) obtained in NRQM for N identical bosons, times the factor 〈0|PV |0〉. 6.2 Identical neutral particles The case of N identical neutral particles is more complicated because of the possibility of particle pairs creation. This is reflected in the formalism in the occurrence of many additional terms in working out the eq. (77). We start with the calculation for a state with two neutral particles of four-momenta p1 and p2 respectively, generalizing to N particles thereafter. In terms of creation and annihilation operators: 〈p1, p2|PV |p1, p2〉 = 〈0|a(p1)a(p2) PV a†(p2)a†(p1)|0〉 (83) which we can rewrite using eq. (45) for a(p1) and a †(p1) as: 〈p1, p2|PV |p1, p2〉= (2π)3 d3x′1 e −ip1·(x1−x1′) × 2ε1〈0|Ψ(x1)a(p2) PV a†(p2)Ψ(x′1)|0〉 . (84) In order to use the expression (45) for annihilation and creation a(p2) and a†(p2) operators we have to get them acting on the vacuum state, hence they should be moved from their position in eq. (84) outwards. This can be done by taking advantage of the following commutation rules: [Ψ(x), a(p)] = − e−ip·x [a(p)†,Ψ(x)] = − eip·x Using (85) and then plugging (45) in eq. (84) we get: 〈p1, p2|PV |p1, p2〉 = ε2(2π)6 d3x′1 e −ip1·(x1−x1′) e−ip2·(x1−x 1)〈0|PV |0〉 − 2ε1 (2π)6 d3x′1 d3x2 e −ip1·(x1−x1′) e−ip2·(x2−x 1) 〈0|Ψ(x2)PVΨ(x1)|0〉 − 2ε1 (2π)6 d3x′1 d3x′2 e −ip1·(x1−x1′) e−ip2·(x1−x 2) 〈0|Ψ(x′1)PVΨ(x′2)|0〉 4ε1ε2 (2π)6 d3x′1 d3x′2 e −ip1·(x1−x1′) e−ip2·(x2−x2 ×〈0|Ψ(x2)Ψ(x1)PV Ψ(x′1)Ψ(x′2)|0〉 (86) where advantage has been taken of the fact that [PV ,Ψ(x)] = 0. The eq. (86) has four different terms, among which only the last was present in the case of 2 charged particles in Subsect. 6.1. Instead, the first three terms arise from contractions of the annihilation and creation operators with field operators on the same side with respect to PV . By using (62) and replacing all unbounded integrations with integrations over V to eliminate spurious degrees of freedom of the field, the eq. (86) yields: 〈p1, p2|PV |p1, p2〉 〈0|PV |0〉 ε2(2π)6 d3x′1 e −ip1·(x1−x1′) e−ip2·(x1−x − 2ε1 (2π)6 d3x′1 e −ip1·(x1−x1′) e−ip2·(x2−x 1) K−1V (x2,x1) − 2ε1 (2π)6 d3x′1 d3x′2 e −ip1·(x1−x1′) e−ip2·(x1−x 2) K−1V (x 4ε1ε2 (2π)6 d3x′1 d3x′2 e −ip1·(x1−x1′) e−ip2·(x2−x2 K−1V (x1,x V (x2,x 2) +K V (x1,x V (x2,x +K−1V (x1,x2)K . (87) We can now use (64) to integrate away the inverse kernel in eq. (87). For the second term on the right hand side, we choose to integrate exp(−ip2 · V (x2,x1) in x2 and for the third term exp(ip1 · x′1)K−1V (x′1,x′2) in x′1. At the same time, we perform the integration of the last term in an arbitrary couple of variables with indices 1 and 2, obtaining: 〈p1, p2|PV |p1, p2〉 〈0|PV |0〉 (2π)6 d3x′1 e −ip1·(x1−x1′) e−ip2·(x1−x (2π)6 d3x′1 e −ip1·(x1−x1′) 2ε1 e−ip2·(x1−x (2π)6 d3x′2 e −ip1·(x1−x2′) e−ip2·(x1−x (2π)6 (2π)6 d3x2 e −i(p1−p2)·(x1−x2) (2π)6 d3x′2 e −ip1·(x1−x2′) e−ip2·(x1−x 2) (88) Four terms in the above sum cancel out and we are left with: 〈p1, p2|PV |p1, p2〉 〈0|PV |0〉 (2π)6 + | FV (p1 − p2) |2= FV (pρ(n) − pn) (89) This is the same result one would obtain for two identical bosons in a NRQM framework [5]. The second term on the right hand side accounts for the well known phenomenon of Bose-Einstein correlation in the emission of identical boson pairs. Looking back at the whole derivation, we find that the terms arising from pairings of field variables on the same side with respect to PV have cancelled with the terms stemming from commutation of annihilation and creation op- erators with field operators; the only surviving terms are N ! pairings of field variables on different sides of PV , just like in the case of charged particles. This cancellation property holds and so, the formula (82) applies to the case of N neutral particles as well. A proof based on the form of the thermodynamic limit V → ∞ is given in Appendix B. 6.3 Particle-antiparticle case For a state with one particle and one antiparticle, the expectation value of PV reads: 〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉 (90) implying, in view of (72): 〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉 (2π)3 d3x′1 e −ip1·(x1−x1′) 2ε1〈0|Ψ(x1)b(p2)PV b†(p2)Ψ†(x′1)|0〉 .(91) Like for neutral particles, the b and b† operators are moved outwards to get them acting on the vacuum state by using the commutators: [Ψ(x), b(p)] = e−ip·x [b(p)†,Ψ†(x)] = eip·x By using the eq. (72) and, like in the previous subsection, restricting the integration to the region V to eliminate external degrees of freedom, we obtain: 〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉 ε2(2π)6 d3x′1 e −ip1·(x1−x1′) e−ip2·(x1−x 1)〈0|PV |0〉 − 2ε1 (2π)6 d3x′1 d3x2 e −ip1·(x1−x1′) e−ip2·(x2−x 1) 〈0|Ψ†(x2)PVΨ(x1)|0〉 − 2ε1 (2π)6 d3x′1 d3x′2 e −ip1·(x1−x1′) e−ip2·(x1−x 2) 〈0|Ψ†(x′1)PVΨ(x′2)|0〉 4ε1ε2 (2π)6 d3x′1 d3x′2 e −ip1·(x1−x1′) e−ip2·(x2−x2 ×〈0|Ψ†(x2)Ψ(x1)PV Ψ†(x′1)Ψ(x′2)|0〉 (93) whose result is (87) except the (last-1)th term, because now: 〈0|Ψ†(x2)Ψ(x1) PV Ψ†(x′1)Ψ(x′2)|0〉 〈0|PV |0〉 = K−1V (x1,x V (x2,x 2) +K V (x1,x2)K which is a consequence of the general expression (74). Again, four out of five terms cancel out in eq. (93) and we get: 〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉 〈0|PV |0〉 (2π)6 where the missing term with respect to the neutral particle case (89) is the one involving permutations; this is a natural result because particles and an- tiparticles are, of course, not identical. Hence, the expectation value of PV on a particle-antiparticle pair shows a remarkable factorization property, that is: 〈0|a(p1)b(p2)PV b†(p2)a†(p1)|0〉 = 〈0|a(p1)PV a†(p1)|0〉〈0|b(p2)PV b†(p2)|0〉 where the = sign holds provided that the integrations are restricted to the region V . Extrapolating to the most general case of N+ particles and N− antiparticles, it can be argued by using the limit V → ∞ that the factorization of the microcanoncal state weight holds (see Appendix B) at any multiplicity, i.e. particles and antiparticles behave like two different species: 〈{N+}, {p+}, {N−}, {p−}|PV |{N+}, {p+}, {N−}, {p−}〉 ρ+∈SN+ FV (pρ+(n+) − pn+) ρ−∈SN FV (pρ−(n−) − pn−)〈0|PV |0〉 .(97) 7 Summary and discussion On the basis of eqs. (19), (21), (16) and (82), which applies to charged as well as neutral particles, and taking into account that the states |N, {p}〉 have been chosen to be eigenvectors of four-momentum, we can finally write down the full expression of the microcanonical partition function of a relativistic quantum gas of neutral spinless bosons as: Ω= 〈0|PV |0〉 d3x exp[ix · (pρ(n) − pn)] (98) with P = (M, 0) and the factor 1/N ! has been introduced in order to avoid multiple counting when integrating over particle momenta. Similarly, on the basis of eqs. (20) (22) and (97), the microcanonical partition function of a relativistic quantum gas of charged spinless bosons can be written as: Ω= 〈0|PV |0〉 N+,N−=0 N+!N−! N++N−∏ N++N−∑ ρ+∈SN+ d3x exp[ix · (pρ+(n+) − pn+)] ρ−∈SN d3x exp[ix · (pρ−(n−) − pn−)] . (99) The generalization to a multi-species gas of spinless bosons is then easily achieved: Ω= 〈0|PV |0〉 d3pnj ρj∈ SNj d3x exp[ix · (pρj(nj) − pnj )] . (100) where N = j Nj. The formulae (98), (99) and (100) are our final result. The finite volume Fourier integrals in the above expressions nicely account for quantum statistics correlations known as Bose-Einstein and Fermi-Dirac correlations. We stress once more that for a charged quantum gas, particles and antiparticles can be handled as belonging to distinct species and they correspond to different labels j in the multi-species generalization of (100). The expression of the MPF (98) is the same as obtained in a NRQM calculation in ref. [5], quoted in this work in (24), times an overall factor 〈0|PV |0〉 which is immaterial for the calculation of statistical averages in the microcanonical en- semble. More specifically, the expectation value of PV on a free multi-particle state (see eqs. (82) and (97)) is the same as in the NRQM calculation (33) times 〈0|PV |0〉. This result has been achieved enforcing a subtraction prescrip- tion, namely that all terms depending on the degrees of freedom outside the system region V must be subtracted “by hand” in all terms at fixed multi- plicities. The factor 〈0|PV |0〉 is still dependent on those spurious degrees of freedom, according to the PV definition in eq. (41), but this does not affect any statistical average because it always cancels out. In the thermodynamic limit, this factor tends to 1 as PV → I and the large volume limit result known in literature [4] (i.e. eq. (27)) is recovered. This result looks surprising in a sense because one would have expected, a pri- ori that quantum relativistic effects would affect the statistical averages bring- ing in a dependence on the ratio between the Compton wavelength and the linear size of the region, getting negligible at small values, i.e. when V → ∞. This is because the condition (32) from which the MPF expressions (23,24) ensue, no longer holds in quantum field theory at finite volume and only ap- plies with good approximation at very large volumes, as discussed in Sect. 4. However, in the calculation of statistical averages, only the projector PV enters and this implies the summation over all states at different integrated occupa- tion numbers |Ñ〉V , according to (31). So, even though the coefficients in the expansion (34) are different from zero and do depend on the volume, it turns out that summing all of them at fixed N , one gets the same result as in the NRQM approximation (32). In formula, by using (31): 〈N, {p}|PV |N{p}〉 = Ñ,{kV } |〈N, {p}|Ñ, {kV }〉|2 Ñ,{kV } |〈0|Ñ, {kV }〉|2 {kV } |〈N, {p}|N, {kV }〉|2NR (101) where the index NR means that one should make a non-relativistic quantum mechanical calculation. This calculation can be extended to the most general case of the microcanon- ical ensemble of an ideal relativistic quantum gas by fixing the maximal set of observables of the Poincaré algebra, i.e. spin, third component and par- ity besides the energy-momentum four-vector. This will be the subject of a forthcoming publication. Acknowledgments We are grateful to F. Colomo and L. Lusanna for stimulating discussions. A Bogoliubov relations for a real scalar field They can be derived by first expressing the localized annihilation operator as a function of the field Ψ and its conjugated moment Ψ̇. For the localized problem we have: Ψ(x) = akuk(x) e −iεkt+c.c. (A.1) where k is a vector of three numbers labelling the modes in the region V , εk is the associated energy and uk is a complete set of orthogonal wavefunctions over the region V : u∗k(x)uk(x ′) = δ3(x− x′) d3xu∗k(x)uk′(x) = δk,k′ (A.2) and vanishing out of V . Inverting the (A.1) we have: d3xu∗k(x) e Ψ(x) (A.3) which are valid at any time. We now enforce the mapping (35) and replace the localized field operators at t = 0, i.e. in the Schrödinger representation, with those in the full Hilbert space. In other words, we replace in (A.3): Ψ(x) → 1 (2π)3/2 eip·x ap + c.c. Ψ̇(x) → 1 (2π)3/2 −i√εp√ eip·x ap + c.c. (A.4) where εp = p2 +m2, obtaining: d3pF (k,p) εk + εp ap + F (k,−p) εk − εp a†p (A.5) where: F (k,p) = (2π)3/2 d3xu∗k(x) e ip·x (A.6) that is the Bogoliubov relations (36). We observe that the localized annihila- tion operator is a non-trivial combination of annihilation and creation opera- tors in the whole space. However, the term in (A.5) depending on the creation operator a†p vanishes in the thermodynamic limit, as expected. This is most easily shown for a rectangular region, where one has uk(x) = exp[ik · x]/ k is given by the eq. (9) and εk = k2 +m2. Hence: F (k,−p) ∝ lim d3x e−i(p+k)·x ∝ δ3(p+ k) (A.7) and, consequently εk−εp → 0, which make the second term in (A.5) vanishing. We can use the eq. (A.5) to work out linear relations between Fock space states of the localized problem and asymptotic Fock space states. These can be obtained enforcing the destruction of the localized vacuum state: ak|0〉V = 0 and writing: |0〉V = α0|0〉+ d3pα1(p)|p〉+ d3p1d 3p2 α2(p1,p2)|p1,p2〉+ . . . (A.8) Working out such relations is beyond the scope of this paper. B Multiparticle state with N particles In order to prove eq. (82) for neutral particles, the first step is to realize that a general expansion of: 〈0|a1 . . . aNPV a†N . . . a 1|0〉 (B.1) (ai is a shorthand for a(pi)) must yield, on the basis of what shown for the case of 2 particles, a sum of terms like these: d3x1 . . . n,m=1 e±ipn·xm f(ε1, . . . , εN)〈0|PV |0〉 (B.2) where f is a generic function involving a sum of ratios or product of any number of particle energies and ± stands for either a + or a − sign. That (B.2) ought to be the final expression can be envisaged on the basis of a repeated application of the equations (45),(85), (62),(64) in turn. The eq. (B.2) can also be written as: d3x1 . . . d3xN e pn·x1 . . . ei pn·xn f(ε1, . . . , εN)〈0|PV |0〉 (B.3) where ∑′ stands for an algebraic sum with terms having either sign. If we now take the thermodynamic limit V → ∞ of (B.1), one has PV → I, thence: 〈0|a1 . . . aNPV a†N . . . a 1|0〉 = 〈0|a1 . . . aNa†N . . . a δ3(pn − pρ(n)) (B.4) This tells us that the function f ’s in each term (B.2) must reduce to a triv- ial factor 1 because they would otherwise survive in the thermodynamic limit, being a factor depending only on the pn’s. Moreover, since the thermodynamic limit involves only Dirac’s δ’s with differences of two momenta as argument, there can be only difference of two momenta as argument of exponential func- tions in (B.3). Finally, by comparing the eq. (B.3) with the (B.4), we conclude that the only possible expression at finite V is: 〈0|a1 . . . aNPV a†N . . . a 1|0〉 = 〈0|PV |0〉 (2π)3 d3xn e −i(pn−pρ(n))·xn (B.5) which is precisely (82). For a state with N+ particles and N− antiparticles, the validity of eq. (97) can be argued with a similar argument, i.e. by constraining the form of general terms like (B.2) taking advantage of the limit V → ∞. In the case of particles and antiparticles, the thermodynamic limit tells us that: 〈0|a1 . . . aN+b1 . . . bN−PV a . . . a . . . b = 〈0|a1 . . . aN+b1 . . . bN−a . . . a . . . b ρ+∈SN+ δ3(pn+ − pρ+(n+)) ρ−∈SN δ3(pn− − pρ−(n−)) (B.6) and this determines the form of the expression: 〈0|a1 . . . aN+b1 . . . bN−PV a . . . a . . . b at finite V to be (97). The absence of integrals mixing particles with antipar- ticles momenta such as: ∫ d3xi e i(pn+−pn−)·xi is owing to the absence of such differences as arguments of δ’s in eq. (B.6). References [1] A. Chodos et al, Phys. Rev. D 9 (1974) 3471. [2] V. V. Begun, M. Gazdzicki, M. I. Gorenstein and O. S. Zozulya, Phys. Rev. C 70 (2004) 034901. [3] V. V. Begun, M. Gazdzicki, M. I. Gorenstein, M. Hauer, V. P. Konchakovski and B. Lungwitz, arXiv:nucl-th/0611075. [4] M. Chaichian, R. Hagedorn, M. Hayashi, Nucl. Phys. B 92 (1975) 445. [5] F. Becattini, L. Ferroni, Eur. Phys. J. C 35 (2004) 243. [6] F. Becattini, J. Phys. Conf. Ser. 5 175 (2005). [7] L. Landau and E. Lifshitz, Quantum Electrodynamics. [8] A. E. Strominger, Annals Phys. 146 (1983) 419 A. Iwazaki, Phys. Lett. B 141 (1984) 342 J. D. Brown, J. W. York, Phys. Rev. D 47, (1993) 1420. [9] F. Becattini, L. Ferroni, The microcanonical ensemble of the ideal relativistic quantum gas with angular momentum conservation, in preparation. [10] B. Touschek, N. Cim. 58 B (1968) 295. [11] D. E. Miller and F. Karsch, Phys. Rev. D 24 (1981) 2564. [12] S. Weinberg, The Quantum Theory Of Fields Vol. 1 Cambridge University Press. [13] M. Stone The Physics of Quantum Fields Springer, New York.

0704.1968

The spin-flip phenomenon in supermassive black hole binary mergers

The spin-flip phenomenon in supermassive black hole binary mergers László Árpád Gergely1,2,3⋆ and Peter L. Biermann4,5,6,7,8‡ 1Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, Szeged 6720, Hungary 2Department of Experimental Physics, University of Szeged, Dóm tér 9, Szeged 6720, Hungary 3Department of Applied Science, London South Bank University, 103 Borough Road, London SE1 0AA, UK 4Max Planck Institute for Radioastronomy, Bonn, Germany 5Department of Physics and Astronomy, University of Bonn, Germany 6Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL, USA 7Department of Physics, University of Alabama at Huntsville, AL, USA 8FZ Karlsruhe and Physics Department, University of Karlsruhe, Germany ⋆ E-mail: [email protected] ‡ E-mail: [email protected] ABSTRACT Massive merging black holes will be the primary sources of powerful gravi- tational waves at low frequency, and will permit to test general relativity with candidate galaxies close to a binary black hole merger. In this paper we identify the typical mass ratio of the two black holes but then show that the distance where gravitational radiation becomes the dominant dissipative effect (over dy- namical friction) does not depend on the mass ratio; however the dynamical evolution in the gravitational wave emission regime does. For the typical range of mass ratios the final stage of the merger is preceded by a rapid precession and a subsequent spin-flip of the main black hole. This already occurs in the inspiral phase, therefore can be described analytically by post-Newtonian techniques. We then identify the radio galaxies with a superdisk as those in which the rapidly precessing jet produces effectively a powerful wind, entraining the environmental gas to produce the appearance of a thick disk. These specific galaxies are thus candidates for a merger of two black holes to happen in the astronomically near future. Subject headings: compact binaries, gravitational radiation, radio galaxies,jets http://arxiv.org/abs/0704.1968v4 – 2 – 1. Introduction The most energetic phenomenon that involves general relativity in the observable uni- verse is the merger of two supermassive black holes (SMBHs). Therefore the study of these mergers may provide one of the most stringent tests of general relativity even before the dis- covery and precise measurement of the corresponding gravitational waves (see, e.g., Schäfer 2005). Most galaxies have a central massive black hole (Kormendy & Richstone 1995, Sanders & Mirabel 1996, Faber et al. 1997), and after their initial growth (for one possible example how this might happen, see Munyaneza & Biermann, 2005, 2006), their evolution is governed by mergers. Therefore, the two central black holes also merge (see Zier & Biermann 2001, 2002; Biermann et al. 2000; Merritt & Ekers 2002; Merritt 2003; Gopal-Krishna et al. 2003, 2004, 2006; Gopal-Krishna & Wiita 2000, 2006; Zier 2005, 2006, 2007). Before the two black holes get close, the galaxies begin to round each other, distorting the shape of a radio galaxy fed by one or both of the two black holes; thence the Z-shaped radio galaxies (Gopal-Krishna et al. 2003). When they merge, under specific circumstances to be clarified in this paper, a spin-flip may occur. For a black hole nurturing activity around it, the spin axis defines the axis of a relativistic jet, and therefore a spin-flip results in a new jet direction: thence the X-shaped radio galaxies (Rottmann 2001, Chirvasa 2001, Biermann et al. 2000, Merritt and Ekers 2002). In fact, observations suggest that all activity around a black hole may result in a relativistic jet even for radio-weak quasar activity (Falcke et al. 1996, Chini et al. 1989a, b). A famous color picture showing the past spin-flip of the M87 black hole (Owen et al. 2000) clearly shows a weak radio counter-jet, misaligned with the modern active jet by about 30◦. The feature of the X-shaped radio galaxy jets is so common and yet very short-lived that all radio galaxies may have been through this merger (Rottmann 2001), and thus should have undergone a spin-flip. This can be also deduced from the observation that many compact steep spectrum sources show a misaligned double radio structure, where an inner pair of hot spots is misaligned with an outer pair of hot spots (Marecki et al. 2003). We conclude that theoretical arguments and observations consistently suggest that black holes merge and result in a spin-flip. From these and some other data we deduce a few basic tenets that the theory needs to explain: 1. In the X-shaped radio galaxies the angles between two pairs of jets in projection are typically less than 30 degrees. The real angles can be even about 45◦. The jets are believed to signify the spin axis of the more active (therefore presumably the more massive) black hole before the merger and the spin axis of the merged black hole. Therefore, a substantial spin-flip should have occurred. – 3 – 2. In the X-shaped radio galaxies one pair of jets has a steep radio spectrum. This implies that it has not recently been resupplied energetically, it is an old pair of jets; and its synchrotron age is typically a few 107 years. The other pair of jets has a relatively flat radio spectrum (this is the new jet; Rottmann 2001). Radio continuum spectroscopy thus supports the spin-flip model. 3. Again, as Rottmann (2001) shows, the statistics of X-shaped radio galaxies are such, that every radio galaxy may have passed through this stage during its evolution. This matches with arguments based on far-infrared observations that central activity in galaxies such as starbursts and feeding of the activity of a central black hole, is often, maybe always, preceded by a merger of galaxies (Sanders & Mirabel 1996). 4. There is another critical observation of the spectrum of radio galaxies. For many of them the radio spectrum has a low-frequency cutoff, suggesting a cutoff in the energy distribution of the electrons at approximately the pion mass (the electrons / positrons are decay products from pions, produced in hadronic collisions; Falcke et al. 1995, Biermann et al. 1995, Falcke and Biermann 1995a, 1995b, 1999, Gopal-Krishna et al. 2004). Hadronic collisions with ensuing pion production at the foot ring of the radio jet occur naturally and thermally in the case that the rotation parameter of the black hole is larger than 0.95, and if the foot ring is an advection-dominated accretion flow (ADAF) or radiatively inefficient accretion flow (RIAF; Donea & Biermann 1996, Mahadevan 1998, Gopal-Krishna et al. 2004). If this is true for all radio galaxies, the spin of the black hole both before and after the spin-flip must be more than 95 % of the maximally allowed value. This is a major constraint on the process of the spin-flip. If we assume that this holds for all radio galaxies, then a fortiori it also holds for those which have just undergone a binary black hole merger, and so their spin ought to be high as well. 5. When two black holes merge, the emission of strong gravitational waves is certain (Peters & Mathews 1963, Peters 1964, Thorne 1979). Compact binaries are driven by grav- itational radiation through a post-Newtonian (PN) regime (the inspiral), a plunge and a ring-down phase toward the final state. It is commonly believed than the spin-flip phe- nomenon is likely to be caused by the gravitational radiation escaping the merging system (Rottmann 2001, Biermann et al. 2000, Merritt and Ekers 2002). Recent numerical work on the final stages of the coalescence supports this (see Brügmann 2007; Campanelli et al. 2007a, b; Gonzalez et al. 2007a, b). Therefore, it is mandatory to investigate what happens when the two black holes get close to each other, and this we propose to treat in this paper. We present here a model which allows to have a merger transition going from a high-spin stage to another high-spin stage, using mostly physical insight from outside of the innermost stable orbit (ISO). In contrast – 4 – with available numerical simulation, our method, limited to a certain typical range of mass ratios of the two black holes, has the advantage that the evolution of the compact binary can be treated in the framework of an analytical PN expansion with two small parameters. In Section 2, we review the current state of observations on the masses of supermassive galactic black holes, which roughly scale with the bulge masses of their host galaxies. The observations suggest that the most massive black holes have about 3 × 109 solar masses (M⊙) and the most reliable determination of the low-mass central black hole (in our galaxy) is about 3 × 106 M⊙ (Ghez et al. 2005, Schödel & Eckart 2005). There is some evidence for central massive black holes of slightly lower mass (Barth et al. 2005), but the error bars are very large. This implies that the maximum mass ratio is about 103. We carefully analyze the statistics and argue in Section 2 that mass ratios in the range 3 : 1 to 30 : 1 cover most of the plausible range in mergers of galactic central black holes. Roughly speaking, this means that typically one mass is dominant by a factor of order 10. Therefore, we find that neither the much discussed case of equal masses nor that of the extreme mass ratios (test particles falling into a black hole) describes typical central galactic SMBH mergers. In Section 3, we study the relative magnitudes of the spin of the dominant black hole and of the orbital angular momentum of the system. Their ratio depends on two factors: the mass ratio and the separation of the binary components (the inverse of which scales with the post-Newtonian parameter). We show that for the typical mass ratio interval the orbital momentum left when the system is reaching ISO is much smaller than the dominant spin. So in the typical mass range case whatever happens during the plunge and ring-down phases of the merger, in which the remaining orbital momentum is dissipated, it cannot change essentially the direction of the spin. By contrast, for equal mass mergers the orbital angular momentum dominates until the end of the inspiral, while for extreme mass ratio mergers the larger spin dominates from the beginning of the gravitational wave driven merger phase. In Section 4, we discuss the transition from the dynamical friction dominated regime to the gravitational radiation dominated regime, in order to establish the initial data for the PN treatment. The interaction of the black holes with the already merged stellar environment generates a dynamical friction when the separation of the black holes is between a few parsecs (pc) and one hundredth of a pc. Gravitational radiation has a small effect in this regime. Due to the dynamical friction, some of the orbital angular momentum of the binary black hole system is transferred to the stellar environment, such that the stellar population at the poles of the system tends to be ejected and a torus is formed (Zier & Biermann 2001, 2002, Zier 2006). This connects to the ubiquitous torus around active galactic nuclei (AGNs), detected first in X-ray absorption (Lawrence & Elvis 1982, Mushotzky 1982), and later confirmed by optical polarization of emission lines (Antonucci & Miller 1985). Dynamical – 5 – friction is enhanced as in a merger the phase-space distribution is strongly disturbed by large fluctuations of the mass distribution (Lynden-Bell 1967, Toomre & Toomre 1972, Barnes & Hernquist 1992, Barnes 2001). There had been a major worry that the two black holes stall in their approach to each other (Valtonen 1996, Yu 2003, Merritt 2005, Milosavljević & Merritt 2003a, b, Makino & Funato 2004, Berczik et al. 2005, 2006, Matsubashi et al. 2007) before they get to the emission of gravitational waves; that the loss-cone mechanism for feeding stars into orbits that intersect the binary black holes is too slow. However, Zier (2006) has demonstrated that direct interaction with the surrounding stars slightly further outside speeds up the process, and so very likely no stalling occurs. Relaxation processes due to cloud/star-star interactions are rather strong, as shown by Alexander (2007), using observations of our galaxy. These interactions repopulate the stellar orbits in the center of the galaxy. New work by Merritt et al. (2007) is consistent with Zier (2006) and Alexander (2007). Also in a series of papers Sesana, Haardt, and Madau have recently shown that even in the absence of two-body relaxation or gas dynamical processes, unequal mass and/or eccentric binaries with the mass larger than 105 M⊙ can shrink to the gravitational wave emission regime in less than a Hubble time due to the binary orbital decay by three-body interactions in the gravitationally bound stellar cusps (Sesana et al. 2006, 2007a, 2007b). Finally, Hayasaki (2008) has considered the ”last parsec problem” under the assumption of the existence of three accretion disks: one around each black hole and a third one, which is circumbinary. The circumbinary disk removes orbital angular momentum from the binary via the binary-disk resonant interaction, however, the mass transfer to each individual black hole adds orbital angular momentum to the binary. The critical parameter of the mass transfer rate is such that for SMBH binaries, it becomes larger than the Eddington limit, thus these binaries will merge within a Hubble time by this mechanism. The angular momentum transfer from orbit to disk was already considered as a key physical concept in binary stars by Biermann & Hall (1973). All these recent works suggest that by one mechanism or another the SMBHs will approach each other to distances smaller than approximately one hundredth pc, when the gravitational radiation becomes the dominant dissipative effect. In Section 4, we analyze the characteristic timescales of the dynamical friction and gravitational radiation as function of the total mass, stellar distribution radius and mass ratio of the compact binary and we establish the values of the transition radius and PN parameter, for which the gravitational radiation is overtaking dynamical friction. In Section 5, we discuss the post-Newtonian evolution of the compact binaries, following Apostolatos et al. (1994) and Kidder (1995). The new element is the emphasis of the role of the mass ratio as a second small parameter in the formalism. The leading order conservative effect contributing to the change in the orientation of spins is the spin-orbit (SO) coupling. The backreaction of the gravitational radiation, which is the leading order dissipative effect – 6 – below the transition radius, appears at one PN order higher. We show here that for the characteristic range of mass ratios the spin-flip occurs during the gravitational radiation dominated inspiral regime, outside ISO. In the process we evaluate the timescales for the change of the spin tilt as compared to the timescales of precessional motion and gravitational radiation driven inspiral. As a by-product, we are able to show that for the typical mass range the so-called transitional precession occurs quite rarely. We interpret and discuss the resulting model in Section 6. Here, we give a tentative outline of the time sequence of the activity of two merging galaxies, leading to an AGN episode of the primary black hole. A recent review of the generic aspects of these galaxy nuclei as sources for ultra high energy cosmic rays is in (Biermann et al. 2008). Finally, we summarize our findings in the concluding remarks. Following our arguments about the phase just barely before the merger we propose there that the superwinds in radio galaxies (Gopal-Krishna et al. 2007, Gopal-Krishna & Wiita 2006) are in this stage, as the rapidly precessing jet acts just like a powerful wind. The primary goal of our paper is to put the derived physics into observational context, so as to allow tests to be done in radio and other wavelengths. 2. The relevant mass ratio range In Lauer et al. (2007), the mass distribution of galactic central black holes is described, confirming earlier work, and also consistent with a local analysis (Roman & Biermann 2006). Arguments based on Häring & Rix (2004), Gott & Turner (1977), Hickson (1982), and Press & Schechter (1974) reasoning lead to a similar result, as does a recent observational survey (Ferrarese at al. 2006b). Wilson & Colbert (1995) also find a broken power law. The probability for a specific mass ratio is an integral over the black hole mass distribution, folded with the rate to actually merge (proportional to the capture cross-section and the relative velocity for two galaxies), e.g., isomorphic to the discussion in Silk & Takahashi (1979) for the merger of clumps of different masses. The black hole mass distribution ΦBH(MBH), the number of massive central black holes in galaxies per unit volume, and black hole mass interval, can be described as a broken power law, from about ma ≃ 3 × 10 6 M⊙ to about mb ≃ 3 × 10 9 M⊙, with a break near m⋆ ≃ 10 8 M⊙. The lower masses have been discussed in some detail by Barth et al. (2005). The values of ma, mb and m∗ imply that we have two mass ranges of a factor of 30 each. The masses above 108 M⊙ are rapidly becoming rare with higher mass, so that the lower mass range is statistically more important. That ratio range is then 1 : 1 to 30 : 1; while in the higher mass range the maximal range of the masses is also 30 : 1. – 7 – The mass of the central massive black hole scales with the mass of the spheroidal com- ponent, as with the total mass of a galaxy (the dark matter), see Benson et al. (2007). The rate of black hole mergers is some fraction of all mergers of massive galaxies. If, as argued by Zier (2006) the approach of the two black holes does not stall, then each merger of two massive galaxies will inevitably lead to the merger of the two central black holes. This is supported by the statistical arguments of Rottmann (2001), using radio observations, that all strong central activity in galaxies may involve a merger of two black holes. Therefore, observational evidence suggests that black holes do merge, and do so on the rather short timescales of AGNs. The interactions and mergers of galaxies clearly depend on the three angular momenta: the two intrinsic spins, and the relative orbital angular momentum, as well as on the initial distance and relative velocity of the two galaxies. Once all these parameters are given, the evolution is quite deterministic. The observations of Gilmore et al. (2007) strongly suggest, that the initial seed galaxies are today’s dwarf elliptical galaxies, all of which are consistent with a lower bound to a common total mass of 5 × 107 M⊙. This implies that all galaxies, and a fortiori all central black holes, have undergone very many mergers. The observations of Bouwens & Illingworth (2006) and Iye et al. (2006) strongly suggest that much of this merger history happened earlier than redshift 6, perhaps mostly between redshifts 9 and 6. Each individual merger runs along a well-defined evolutionary track, but all of these mergers are completely uncorrelated with each other. Therefore, the ensemble of very many mergers can be treated statistically, and this is what we proceed to do, using the constant mass ratio between the spheroidal component of galaxies and their central black holes. We thus use the merger rate of galaxies as closely equivalent to the merger rate of the central black holes. The statistics of the mergers is given by the integral for the number of mergers N(q) per volume and time for a given mass ratio q, defined to be larger than unity. This merger rate is the product of the distribution of the first black hole with the distribution of the second black hole multiplied by a rate F . The latter in principle depends on both the cross-section and relative velocity of the two galaxies, the velocities however are not very different, as the universe is not old enough for mass segregation. The cross-section in turn depends on the two masses, thus F = F (q,m). If we integrate for all cases, in which the first black hole is less massive than the second black hole, we undercount by a factor of 2, and we have to correct for this factor. The general relationship is N(q) = 2 ∫ mb/q ΦBH(m)ΦBH(qm)F (q,m)dm (1) – 8 – It is likely that the more massive black hole, and so the more massive host galaxy, will dominate the merger rate F , so that it can be approximated as a function of qm alone, and a power law behavior with F ∼ qξ with ξ > 0 should be adequate for a first approximation. To estimate ξ roughly we just observe, that dwarf spheroidals have a core radius of a few hundred pc (Gilmore et al. 2007), while our Galaxy has a core radius of about 3 kpc (Klypin et al. 2002), so a factor of 10 in radius (102 in cross-section) for a factor of about 104 in mass, thus the exponent is likely to be approximately 1/2; therefore a reasonable first estimate for any cross-section is ξ = 1/2. In this instance we use the approximate equivalence of galaxy mergers with black hole mergers. As the black hole mass distribution has a break at q∗ = 30, we use ΦBH(m) ∼ m −α̃ for the first mass range, and ΦBH(m) ∼ m −β̃ for the second. For the range q from 1 to 30 we have as a dominant contribution N(q) ∼ ∫ m⋆/q )−α̃( )−α̃( )−α̃( )−β̃ ( ∫ mb/q )−β̃ ( )−β̃ ( dm (2) and for the case of q above 30 we have the contribution N(q) ∼ ∫ mb/q )−α̃( )−β̃ ( dm . (3) The various models shown in Lauer et al.(2007) show that a range of values of α̃ and β̃ is possible, with α̃ ranging between approximately 1 and 2, and β̃ from 3 to larger values. Benson et al.(2007) propose α̃ ≈ 0.65. We adopt here the approximate values for α̃ and β̃ of 1 and 3, to be cautious, and for ξ we adopt 1/2. With these values the above integrands are monotonically decreasing functions and the integrals are dominated by the lower limits. Thus, the four terms scale with q as qξ−α̃, q−1+α̃, qξ−β̃, and again qξ−β̃. Let us consider the four terms: the first term is small galaxies merging with small galaxies, and so not very interesting, as the cross-section is low. However, for this distribution the number of mergers in the mass ratio range 30 : 1 to 3 : 1 versus 3 : 1 to 1 : 1 is about 5. The more extreme mass ratios are more common. For the second term this ratio of mergers in the two mass ratio ranges is about 14. As this is massive galaxies merging with smaller – 9 – galaxies (above and below the break m⋆), this is the most interesting case, and also quite common. The third term is almost negligible, and the fourth term adds cases to the second term with more extreme mass ratios, beyond 30 : 1, and so emphasizes the large mass ratio range. So, among the relevant cases the rate of mergers of mass ratio of more than 3 : 1 to those with a smaller mass ratio is in the range of 5 : 1 to 14 : 1, about an order of magnitude. Focussing on those cases where one black hole is at 108 M⊙ or larger, the ratio is larger than 14 : 1. Speculating that the exponent ξ could be larger would enhance all these effects; enlarging α̃ would weaken them. Therefore we will deal in the following with this much more common extended mass ratio range 30 : 1 to 3 : 1, which as will be shown, allows to use analytical methods. 3. The spin and orbital angular momentum in the PN regime We assume the compact binary system to be composed of two masses mi, i = 1, 2 , each having the spin Si. By definition, the characteristic radius Ri of compact objects is of the same order of magnitude that the gravitational radius RG = Gmi/c 2 (where c is the velocity of light and G is the gravitational constant). Therefore, the magnitude of the spin vector can be approximated as Si ≈ miRiVi ≈ Gm iVi/c 2, where Vi is the characteristic rotation velocity of the ith compact object As black holes rotate fast due to accretion, Vi/c is of order unity. Equivalently we can introduce Si = (G/c)m iχi, with χi being the dimensionless spin parameter. Then maximal rotation implies χi = 1. The PN expansion is done in terms of the small parameter , (4) where m = m1+m2 is the total mass and v is the orbital velocity of the reduced mass particle µ = m1m2/m, which is in orbit about the fixed mass m (according to the one-centre problem in celestial mechanics). The two expressions for ε are of the same order of magnitude due to the virial theorem. As in certain expressions odd powers of v/c may occur, it is common to have half-integer orders in the post-Newtonian treatment of the inspiral of a compact binary system. Whenever the masses of the two compact objects are comparable, either of Gmi/c also represent one post-Newtonian order. However, as we have argued before, for colliding galactic black holes it is typical that their masses differ by 1 order of magnitude, so that we have a second small parameter in the formalism. By choosing m2 as the smaller mass, we – 10 – can also define the mass ratio ∈ (0, 1) . (5) In the literature the symmetric mass ratio is also frequently employed. The two mass ratios are related as (1 + ν) , (7) and for small ν we have η = ν − 2ν2 + O (ν3). For the typical mass ratio range of SMBH binaries either η or ν can be chosen as the second small parameter in the formalism. However, while these stay constant, the PN parameter ε evolves during the inspiral toward higher values. Indeed, the separation of the components of the binary with m = 108M⊙ evolves as = 4. 781 3× 10−6 , (8) where rS represents the Schwarzschild radius. The interaction of the galactic black holes with the stellar environment begins when the black holes are a few kpc away from each other (then ε ≈ 10−8). The dynamical friction becomes subdominant at about 0.005 pc (Zier & Biermann 2001, Zier 2006), when the gravitational radiation becomes the leading dissipative effect. Thus, ε = ε∗ ≈ 10−3 is the value of the PN parameter for which the gravitational radiation is driving the dissipation of energy and orbital angular momentum. Then follows the inspiral stage of the evolution of compact binaries, which continues until the domain of validity of the post-Newtonian approach is reached, at few gravitational radii, at ISO. Further away a numerical treatment is necessary in order to describe the plunge, which is finally followed by the ring-down. The PN formalism can be considered valid until ε ≈ 10−1. Theoretically, it is possible for a small ν, that at certain stage of the inspiral, the increasing ε1/2 becomes of the same order of magnitude as ν and later on it even exceeds ν. Such a situation would shift the numerical value of several contributions to various physical quantities into the range of higher or lower PN orders, depending of the involved power of The spin ratio (for similar rotation velocities V1 ≈ V2) can be expressed as = ν2 . (9) – 11 – Table 1: The evolution of the ratio S1/L ≈ ε 1/2ν−1 in the range ε = 10−3÷10−1 for various values of the mass ratio ν. S1/L = ε 1/2ν−1 ε ≈ 10−3 ε ≈ 10−1 ν = 1 0.03 (S1 ≪ L) 0.3 (S1 < L) ν = 1/3 0.1 (S1 < L) 1 (S1 ≈ L) ν = 1/30 1 (S1 ≈ L) 10 (S1 > L) ν = 1/900 30 (S1 ≫ L) 300 (S1 ≫ L) The ratio of the spins to the orbital angular momentum becomes Gm22V2/c ≈ ε1/2ν , (10) ≈ ε1/2ν−1 . (11) We note that the approximations in the above formulae (9)-(11) are related only to the fact that we have assumed maximal rotation (thus Vi/c . 1). First, we note that the above ratios involving the spins of the compact objects already contain ε1/2. Thus, the counting of the inverse powers of c2 is not equivalent with the PN order, when compact objects are involved. Further, while the ratio S2/L is shifted toward higher orders by a small ν (therefore S2 ≪ L during all stages of the inspiral), the order of the ratio of the spin of the dominant black hole to the magnitude of the orbital angular momentum is not fixed. Indeed, it is determined by the relative magnitude of the small parameters ε and ν. As ε increases during the inspiral, whenever ν falls in the range of ε1/2, the initial epoch with S1 < L is followed by S1 ≈ L and S1 > L epochs (Table 1). We have concluded in the previous section that the range of mass ratios q between 3 : 1 and 30 : 1 is the most common. For such binaries the sequence of the three epochs S1 < L, S1 ≈ L, and S1 > L is fairly representative. We call this intermediate mass ratio mergers, which has to be contrasted with the case of equal mass mergers, where the orbital angular momentum dominates throughout the inspiral; and with the case of extreme mass ratio mergers (which we define as having mass ratios larger than 30 : 1), where the larger spin dominates from the beginning of the inspiral to the end of the PN phase, as can be seen from our Table 1. – 12 – 4. The timescales The value ε ≈ 10−3 from which the PN analysis with the gravitational radiation as the leading dissipative effect can be applied was adopted in the previous section for a compact binary with total mass m = 108M⊙ and mass ratio ν = 10 −1. This was based on the analysis in Zier & Biermann (2001) and Zier (2006), where it was shown that at around 5 × 10−3 pc gravitational radiation takes over from dynamical friction in the interaction with stars in the angular momentum loss of the black hole binary. Further arguments for the binary to reach the gravitational wave emission regime were presented by Alexander (2007), Sesana et al. (2006, 2007a, 2007b), and Hayasaki (2008). In this section we raise the question, whether the value of the transition radius (and the corresponding value of the PN parameter) depends on m and ν. In order to answer this, we compare the characteristic timescales of the gravitational radiation and dynamical friction. The timescale of gravitational radiation (as will be derived in Section 5) is ε4η . (12) The timescale for the secondary black hole to lose angular momentum by gravitational interaction with the surrounding stellar distribution is (Binney & Tremaine 1987) tfr = 2πG2m2ρdistrΛ . (13) With the maximally allowable change in the velocity ∆v/v = 1 we find the relevant full timescale. Here, Λ = ln(bmax/bmin) is the logarithm of the ratio of the maximal distance within the system, divided by the typical distance between objects. The latter is large for clouds, so the ratio is low and Λ of the order unity, while for stars or dark matter particles Λ can be taken as 10÷ 20. For the merger, the estimate based on clouds is more appropriate, thus, following the reasoning of Binney & Tremaine (1987) we adopt Λ = 3. The compact stellar distribution with density ρdistr, radius rdistr, and mass mdistr is of the same order in mass as the black hole binary (Zier & Biermann 2001, Ferrarese et al. 2006a) with a scale rdistr of a few pc, and so under the assumption of a spherically symmetric distribution we ρdistr = 4πr3distr . (14) By employing the definitions of the PN parameter and mass ratio, we get r3distr ε−3/2η (1 + ν) . (15) – 13 – This gives the full timescale of the dynamical friction. The two timescales become comparable for a PN parameter: ε∗ = K (ν) c2rdistr )6/11 , (16) with K (ν) being a factor of order unity, defined as: K (ν) = (1 + ν) ]2/11 ∈ (0.938, 1. 064) . (17) corresponding to the distance r∗ r∗ = K−1 (ν) )5/11 distr . (18) Notably, the dependence on the mass ratio is rather weak and in practice it can be neglected. Inserting then for rdistr = 5 pc and using as a reference value for the mass m = 10 8 M⊙, we obtain ε∗ ≈ 10−3 and r∗ ≈ 0.005 pc in agreement with the discussion in Zier & Biermann (2001). We also note that in fact any other reasonable value for Λ and ∆v/v will give a factor of order unity in Eq. (14), as this number arises by taking the power 2/11. The weak dependence on ν through K is due to the same reason. The value of the PN parameter thus scales with m6/11 and radius r −6/11 distr as ε∗ ≈ 10−3 108 M⊙ )6/11 ( rdistr )6/11 , (19) while the transition radius scales with m5/11 and r distr as r∗ ≈ 0.005 pc 108M⊙ )5/11 ( rdistr )6/11 . (20) For m = 109M⊙ and for the same radius of stellar distribution then r ≈ 0.01 pc. We conclude that both the dependence of the transition radius and the corresponding PN parameter on the total mass and on the stellar distribution radius are weak, while there is practically no dependence on the mass ratio. 5. The inspiral of spinning compact binaries in the gravitational radiation dominated regime In the first subsection of this section we present the conservative dynamics of an isolated compact binary in a post-Newtonian treatment, emphasizing the role of the second small – 14 – parameter, as a new element. Then in the second subsection we take into account the effect of gravitational radiation, deriving how the spin-flip occurs for the typical mass ratio range. The limits of validity of our results obtained by using these two small parameters will be considered below in subsection 5.3. 5.1. Conservative dynamics below the transition radius The interchange in the dominance of either L or S1 has a drastic consequence on the dynamics of the compact binary. To see this, let us summarize first the conservative dynam- ics, valid up to the second post-Newtonian order. The constants of the motion are the total energy E and the total angular momentum vector J = L+ S1 + S2 (Kidder et al. 1993). The angular momenta L, S are not conserved separately. The spins obey a precessional motion (Barker & O’Connell 1975, Barker & O’Connell 1979): Ṡi = Ωi × Si , (21) with the angular velocities given as a sum of the spin-orbit, spin-spin, and quadrupole- monopole contributions. The latter come from regarding one of the binary components as a mass monopole moving in the quadrupolar field of the other component. The leading order contribution due to the SO interaction (discussed in Kidder et al. 1993, Apostolatos et al. 1994, Kidder 1995, Ryan 1996, Rieth & Schäfer 1997, Gergely at al. 1998a, 1998b, 1998c, O’Connell 2004), cause the spin axes to tumble and precess. The spin-spin (Kidder 1995, Apostolatos 1995, Apostolatos 1996, Gergely 2000a, 2000b), mass quadrupolar (Poisson 1998, Gergely & Keresztes 2003, Flanagan & Hinderer 2007, Racine 2008), magnetic dipolar (Ioka & Taniguchi 2000, Vasúth et al. 2003), self-spin (Mikóczi et al. 2005) and higher order spin-orbit effects (Faye et al. 2006, Blanchet et al. 2006) slightly modulate this process. The SO precession occurs with the angular velocities G (4 + 3ν) 2c2r3 LN , (22) G (4 + 3ν−1) 2c2r3 LN , (23) where LN = µr × v is the Newtonian part of the orbital angular momentum. The total orbital angular momentum L also contains a contribution LSO (Kidder 1995), which for compact binaries is of the order of ε3/2LN . Due to the conservation of J, the orbital angular momentum evolves as – 15 – 2c2r3 (4 + 3ν)S1 + 4 + 3ν−1 ×L . (24) (By adding a correction term of order ε3/2 relative to the leading order terms, we have changed LN into L on the right-hand side of the above equation.) To leading order in ν we obtain: Ṡ1 = L× S1 , (25) S1 × L . (26) (Again, an LSO term was added to LN on the right-hand side of Eq. (25), in order to have a pure precession of S1.) Thus, the leading order conservative dynamics gives the following picture: the dominant spin S1 undergoes a pure precession about L, while L does the same about S1. Despite the precession (23), the spin S2 can be ignored to leading order, as its magnitude is ν 2 times smaller than S1, e.g., Eq. (9). By adding the vanishing terms (2G/c 2r3)S1 × S1 and (2G/c2r3)L× L to the right-hand sides of Eqs. (25) and (26), respectively, we obtain Ṡ1 = J× S1 , (27) J× L . (28) Thus, the precessions can also be imagined to happen about J, which represents an invariant direction in the conservative dynamics up to 2PN. Higher order contributions to the conservative dynamics slightly modulate this pre- cessional motion. In fact, for both the spin-spin and quadrupole-monopole perturbations an angular average L̄ can be introduced, which is conserved up to the 2PN order (Gergely 2000a). As L̄ differs from L just by terms of order 2PN, and L̄ is conserved, the real evolution of L differs from a pure precession only slightly. Finally, we note that as the SO precessions are 1.5 PN effects and the gravitational ra- diation appears at 2.5 PN, at the transition radius the SO precession timescale is ε−1 times shorter than the timescale of dynamical friction. The modifications induced by the preces- sions in the angular momentum transfer toward the stellar environment will be discussed elsewhere (Zier et al. 2009, in preparation). – 16 – Fig. 1.— The old jet points in the direction of the original spin S1. When the two black holes approach each other due to the motion of their host galaxies, a slow precessional motion of both the spin and of the orbital angular momentum L begins (left figure) about the direction of the total angular momentum J, which is due to the spin-orbit interaction. Gravitational radiation carries away both energy and angular momentum from the system, such that the direction of J stays unchanged. As a consequence the precessional orbit slowly shrinks and the magnitude of L decreases. This is accompanied by a continuous increase in the angle α and a decrease in β. When the magnitudes of L and S become comparable (middle figure), the precessional motions are much faster (for typical values see Table 2). In the typical mass ratio range ν = 1/3 ÷ 1/30 the magnitude of L becomes small as compared to the magnitude of the spin, which is unchanged by gravitational radiation (except the upper boundary of the range at ν = 1/3 when L and S are still comparable). Before reaching the innermost stable orbit, the spin becomes almost aligned to the (original direction of the) total angular momentum, and a new jet can form along this direction. Therefore, for the typical mass ratio range the spin-flip phenomenon has occurred in the inspiral phase and not much orbital angular momentum is left to modify the direction of the spin during the plunge and ring-down. In the regime in between the initial and final states the precessing jet acts as a superwind, sweeping away the environment of the jets. – 17 – 5.2. Dissipative dynamics below the transition radius Dynamics becomes dissipative from 2.5 PN orders. Then gravitational quadrupolar ra- diation carries away both energy and angular momentum. Orbital eccentricity is dissipated faster, than the rate of orbital inspiral (Peters 1964), thus the orbit will circularize.1 Con- sidering circular orbits and averaging over one orbit gives the following dissipative change in gw = − 32Gµ2 L̂ , (29) where L̂ represents the direction of the orbital angular momentum. Then the total change in L is given by the sum of Eqs. (26) and (29). The spin-induced quasi-precessional effects both modulate the dynamics and they have an important effect on gravitational wave detection (see Lang & Hughes 2006, 2008, Racine 2008, Gergely & Mikóczi 2008). The dissipative dynamics, with the inclusion of the leading order SO precessions and the dissipative effects due to gravitational radiation, averaged over circular orbits, was discussed in detail in Apostolatos et al. (1994) for the one-spin case S2 = 0 and equal masses ν = 1. For the typical mass ratio ν ∈ (1/30, 1/3), and keeping only the leading order contributions in the ν-expansion also gives S2 = 0 (the leading order contributions in S2 are of order ν2). In this subsection we will analyze in depth the angular evolutions and the timescales involved. As for any vector X with magnitude X and direction X̂ one has Ẋ = X X + ẊX̂, the change in the direction can be expressed as Ẋ− ẊX̂ /X . Also the identity X2 = X2 gives Ẋ = X̂ · Ẋ. Then Eqs. (27)-(29) imply Ṡ1 = 0 , J× Ŝ1 , L̇ = − 32Gµ2 J× L̂ . (30) 1As we focus here on spin-flips, we will not dwell on possible recoil as a result of the momentum loss due to gravitational radiation in the merger of the two black holes (see for example Brügmann 2008, Gonzalez et al. 2007a, b). The accuracy of determining the distance between two separate and independent active black holes (Marcaide & Shapiro 1983, Brunthaler et al. 2005) is reaching a precision, which may soon allow for recoil to be measurable; no such evidence has been detected yet. – 18 – The total angular momentum J is also changed by the emitted gravitational radiation. As no other change occurs up the 2PN orders, J̇ =L̇L̂ and J̇ = L̇ L̂ · Ĵ L̂ · Ĵ . (31) Note that from the second Eq. (31) it is immediate that the direction of J changes violently, whenever J is small compared to L̇. To leading order in ν , the vectors L, S1, and J form a parallelogram, characterized by the angles α = cos−1 L̂ · Ĵ and β = cos−1 Ŝ1·Ĵ . From Eqs.(30) and (31), we obtain α̇ = − sinα > 0 , (32) sinα < 0 . (33) In the latter equation, we have used that Ŝ1·L̂ = cos (α + β). Thus, we have found the following picture for the inspiral of the compact binary after the transition radius. By disregarding gravitational radiation, the SO precessions (25) and (26) assure that the vectors L and S1 are precessing about J (a fixed direction), but also about each other (then the respective axes of precession evolve in time). Gravitational radiation slightly perturbs this picture. The angle α+ β between the orbital angular momentum and the dominant spin stays constant during the inspiral, even with the gravitational radiation taken into account. By contrast, the angle between J and L continuously increases, while the angle between J and S1 decreases with the same rate. This also means that due to gravitational radiation, the vectors L and S1 do not precess about J any more in an exact sense. They keep precessing about each other, however. The change in the total angular momentum J̇ =L̇L̂ is about the orbital angular momen- tum, which in turn basically (disregarding gravitational radiation) undergoes a precessional motion about J. This shows that the averaged change in J is along J. This conclusion, however, depends strongly on whether the precessional angular frequency Ωp is much higher than the change in the angles α and β. Indeed, if these are comparable, the component perpendicular to J in the change J̇ =L̇L̂ will not average out during one precessional cycle, as due to the increase of α it can significantly differ at the beginning and at the end of the same precessional cycle, see Fig 1. The regime with Ωp ≫ α̇ can be well approximated by a precessional motion of both L and S1 about a fixed Ĵ, with the magnitudes of L and J slowly shrinking, the angle – 19 – α slowly increasing and β slowly decreasing. As a result, during the inspiral, the orbital angular momentum slowly turns away from J, while S1 slowly approaches the direction of J. This regime is characteristic for the majority of cases, and it was called simple precession in Apostolatos et al. (1994). Whenever Ωp ≈ α̇, the conclusion of having J̇ in the direction of Ĵ does not hold for the average over one precession. This results in a change in the direction of J in each precessional cycle. The evolution becomes much more complicated (in fact no approximate analytical solution is known), and it was called transitional precession in Apostolatos et al. (1994). Let us see now when the two types of evolution typically occur. For this, we note that the inspiral rate L̇/L is of the order ε4η , (34) while the precessional angular velocity Ωp = 2GJ/c 2r3 gives the estimate ε5/2η . (35) Finally, the tilt velocity of L is of the order ε7/2ην sin (α + β) ε9/2ην−1 sin (α + β) (36) We have used sinα = sin (α+ β) ≈ ε1/2ν−1 sin (α+ β) (37) for the first and second expressions of α̇ , respectively, together with Eq. (11). For compar- ison, these are all represented in Table 2 for all three regimes characterized by S1/L ≈ 0.3, S1 ≈ L and S1/L ≈ 3, respectively. The numbers from the second line of Table 2 demonstrate that for the chosen typical example the precession timescale can get as short as a day, going from 3000 years to three years to a day in the three columns above. This last stage is obviously quite close to the plunge. From the first line we can infer upper limits of how close the merger actually is, so 30 million years in Column 1, 300 years in Column 2, and a few months in Column 3. As the inspiral rate increases in time, rather than being a constant, these numbers are higher than the real values. The accuracy of the third estimate is further obstructed by the fact that after ISO the plunge follows, but as this comprises only a few orbits, the PN prediction can – 20 – Table 2: Order of magnitude estimates for the inspiral rate L̇/L, angular pre- cessional velocity Ωp and tilt velocity α̇ of the vectors L and S1 with respect to J, represented for the three regimes with L > S1, L ≈ S1 and L < S1, charac- teristic in the domain of mass ratios ν = 0.3÷ 0.03. The numbers in brackets represent inverse time scales in seconds−1, calculated for the typical mass ratio ν = 10−1, post-Newtonian parameter 10−3, 10−2 and 10−1, respectively and m = 108M⊙ (then c 3/Gm = 2× 10−3 s−1). L > S1 L ≈ S1 L < S1 −L̇/L 32c ε4η (≈ 10−15) 32c ε4η (≈ 10−11) 32c ε4η (≈ 10−7) ε5/2η (≈ 10−11) 2c ε5/2η J ≈ 10−8 J ε3 (≈ 10−5) sin(α+β) ε9/2 η (≈ 10−16) 32c ε9/2 η ≈ 10−11L ε7/2ην (≈ 10−8) be considered relevant as an order of magnitude estimate. By multiplying the numbers in the third line with the precession timescales Ω−1p we actually get the relevant tilt angle, varying from 2 arcsec during one precession (6× 10−4 arcseconds per year) at large separations to 3 arcmin per precession (per day) close to the ISO. We see, that the rate of precession and the tilt velocity become comparable in the S1 ≈ L epoch (in which ε1/2ν−1 ≈ 1) for sin (α + β) ]−1/3 ε2ν−1 ≈ ε1/2 ≈ ν , (38) that is, for the chosen numerical example ν = 10−1 and for the square bracket of order unity, this gives J/L ≈ 10−1 and the rate of α̇ ≈ Ωp ≈ 10 −9 (this is still 100 times faster than the rate of inspiral). The total angular momentum J can be that small only if L and S1 are almost perfectly anti-aligned, thus α + β ≈ π − δ, δ ≪ 1, and L ≈ S1. What does the condition for transitional precession (38) mean in terms of their angle, how close should that be to the perfect anti-alignment? To answer the question we note that L cosα + S1 cos β ≈ δ sinα , (39) Then, comparing with the estimates (37) and (38) we conclude, that transitional precession can occur only if the deviation from the perfect anti-alignment is of the order of ν3/2. This is a highly untypical case in galactic black hole binaries. – 21 – 5.3. The limits of validity One might seriously question whether pushing the values of the parameters beyond the range for which we use them would actually demonstrate that our approximations cannot possibly be correct. We would like to address the following two concerns specifically. a) Is the high orbital angular momentum limit L >> S1, obtained by a sufficient increase of the separation of the two black holes a correct limit? b) Is the extreme mass limit ν → 0, for which the treatment of Section 5 may seem increasingly accurate, correct? Concerning the limit (a): in any expansion with a small parameter one condition always holds: the parameter must be small, and in any PN expansion we can always reach a stage, for which the physics basis fails as inadequate, as other additional physical processes become dominant, or for which there is no observational support, despite the mathematical validity of the expansion. From among the effects affecting the directions of the spins the dynamics discussed in Section 5 takes into account (1) the leading order conservative effect, given by the precession due to spin-orbit coupling and (2) the leading order dissipative effect due to gravitational radiation. Other conservative and dissipative effects are neglected, being weaker. Meaningful results can be traced from this model only when these assumptions hold. This implies that the post-Newtonian parameter ε varies between 10−3 and 10−1, corresponding to orbital separations of 500 Schwarzschild radii = 0.005 pc to 5 Schwarzschild radii, given a 108 M⊙ black hole. This is emphasized in the paragraph following Eq. (8). The ”initial” and ”final” phases in the dynamics described above therefore refer to a well-defined range of orbital separation, they are not arbitrary. The choice for the initial state is further justified by the discussion of Section 4. One cannot apply the dynamics discussed above at arbitrarily large distances, where the orbital momentum indeed would dominate, simply because the dynamics is no longer valid there. At larger separations the leading order dissipative effect is due to dynamical friction, thus the discussion of the previous two subsections does not apply. Concerning (b): according to the summary presented in Table 1, there are three possi- bilities in the PN regime, where the dynamics discussed above holds: (i) from mass ratios ν from 1 to 1/3 the orbital angular momentum dominates throughout the whole range of separations between 0.005 pc and 5 Schwarzschild radii, as noted; (ii) from mass ratios ν from 1/3 to 1/30 initially the orbital angular momentum dominates over the spin, but their ratio is reversed at the final separation; (iii) from mass ratios ν smaller than 1/30 the spin – 22 – is dominant throughout the process. Our claim is that the spin-flip is produced only if the total angular momentum (whose direction stays unchanged) initially is dominated by the orbital angular momentum, finally by the spin, thus only for the case (ii). In the dynamics presented in the previous two subsections we have neglected the second spin. As even for the highest mass ratio ν = 1/3 in the regime (ii) the second spin is 1 order of magnitude smaller than the leading spin, we consider this assumption justified. With decreasing mass ratios it becomes increasingly accurate to neglect the second spin, as according to Eq. (7) the ratio of the spins goes with ν2. However, not all results of the previous subsection become increasingly accurate with a decreasing mass ratio. We emphasize, what is different in case (iii) as compared to (ii). The difference is in the initial conditions, which allow to obtain a spin-flip in case (ii), but not in (iii). Mathematically, the difference between these two cases can be seen from Eq. (33), showing that the angular tilt velocity of the dominant spin scales with ν2. For the extreme mass ratios ν < 1/30, e.g., Table 1, the spin dominates over the orbital angular momentum throughout the whole PN regime. Therefore, the ratio S1/J is of order unity. With decreasing ν however the change in the direction of the spin, represented by α̇ goes fast to zero, thus no spin-flip is produced in the PN regime for extreme mass ratios. At the end of this subsection we derive an analytical expression relating α to the con- served α + β, the evolving PN parameter ε and the mass ratio ν. From the Figure 1 J = L cosα+ S1 cos β. By introducing the angle α + β and employing the estimate (11) we obtain 1 + ε1/2ν−1 cos (α + β) cosα + ε1/2ν−1 sin (α + β) sinα . (40) Inserting this into the second expression (37) and rearranging we find sin 2α 1 + cos 2α sin (α + β) ε−1/2ν + cos (α + β) . (41) For an initial configuration of 0.005 pc (such that ε ≡ ε∗ = 10−3) and mass ratio ν = 10−1, the initial misalignment between L and J is αinitial ≈ 18 ◦, 10◦, 0◦ for the dominant spin in the plane of orbit, spanning 45◦ with the plane of orbit and perpendicular to the plane of orbit (such that α+β = 90◦, 45◦, 0◦), respectively. Then βinitial = 72 ◦, 35◦, 0◦. For the same mass ratio and relative configurations, the angle α at the end of the PN epoch (at ε = 10−1) becomes αfinal ≈ 73 ◦, 35◦, 0◦, respectively. This can be translated into a misalignment between S1 and J of βfinal = 17 ◦, 10◦, 0◦, and a spin-flip of ∆β = 55◦, 25◦, 0◦, respectively. – 23 – 5.4. Summary In the typical range of mass ratios ν = 0.03÷ 0.3 the initial condition L > S1 is always transformed into S1 < L, but the transition is very rarely accompanied by the so-called transitional precession. In all other cases the precession is simple. As the precession angle of the dominant spin is decreasing in time from the given initial value to a small value, the precessional cone becomes narrower in time. At the end of the inspiral phase the dominant spin S1 will point roughly along J. This means that a spin-flip has occurred during the post- Newtonian evolution, already in the inspiral phase of the merger. On the other hand, as the inspiral phase ends with L < S1, irrespective of what happens in the next phases, during the plunge and ring-down, L is not high enough to cause additional significant spin-flip. For smaller mass ratios (for extreme mass ratio mergers) the larger spin already domi- nates the total angular momentum from the beginning of the inspiral, thus no spin-flip will occur by the mechanism presented here. Alternatively, from the second expression (36) one can see that the rate of tilt of the spin decreases with ν2, thus it goes fast to zero in the extreme mass ratio case. However, as we argued in Section 2, such mass ratios are less typical for galactic central SMBH mergers. This also shows that an infalling particle will not change the spin of the supermassive black hole. For the (again untypical) equal mass SMBH mergers the orbital angular momentum stays dominant until the end of the inspiral phase. In this case, however the possibility remains open to have a spin-flip later on, during the plunge. 6. Discussion The considerations from this paper lead to the following time sequence for the transient feeding of a SMBH including a merger with another SMBH. First: Two galaxies with central black holes approach each other to within a distance where dynamical friction keeps them bound, spiraling into each other. If there is cool gas in either one, it can begin to form stars rapidly, along tidal arms. The galactic central supermassive SMBH binary influences gas dynamics and star formation activity also in the nuclear gas disk, due to various resonances between gas motion and SMBH binary motion (Matsui et al. 2006), creating some characteristic structures, such as filament structures, formation of gaseous spiral arms, and small gas disks around SMBHs. If either galaxy happens to have radio jets, then due to the orbital motion, these jets get distorted and form the Z-shape (Gopal-Krishna et al. 2003, Zier 2005). – 24 – Second: The central regions in each galaxy begin to act as one unit, in a sea of stars and dark matter of the other galaxy. During this phase, as the cool gas from the other partner typically has low angular momentum with respect to the receiving galaxy, the central region can be stirred up, and produce a nuclear starburst (Toomre and Toomre 1972). The central black hole can get started to be fed at a high rate, but its emission will be submerged in all the far-infrared emission from the gas and dust heated by the massive stars produced in the starburst. In this case, there is dynamical friction, which can act so as to select certain symmetries, such as corotation, counter rotation, or rotation at 90◦ (as in NGC2685, a polar ring galaxy; Richter et al. 1994). Third: The black holes begin to lose orbital angular momentum due to the interaction with the nearby stars (Zier and Biermann 2001, 2002). Other mechanisms for angular momentum loss are also known (Sesana et al. 2006, 2007a,b, Alexander 2007, Hayasaki 2008). The two black holes approach each other to that critical distance where the interaction with the stars and the gravitational radiation remove equivalent fractions of the orbital angular momentum. Then, as shown in this paper, the spin axes tumble and precess. This phase can be identified with the apparent superdisk, as the rapidly precessing jet produces the hydrodynamic equivalent of a powerful wind, by entraining the ambient hot gas, pushing the two radio lobes apart and so giving rise the a broad separation (Gopal-Krishna et al. 2003, 2007, Gopal-Krishna & Wiita 2006). Gopal-Krishna & Wiita (2006) emphasize the apparent asymmetry, which we propose to attribute to line-of-sight effects and the distortion due to the recent merger. The base of the radio structure is so broad and so asymmetric, that the central AGN will appear to be offset from the projected center of gap. The recent arguments of Worrall et al. (2007) seem to be consistent with this point of view. The spin direction of the combined two black holes is preserved, although the strength of the spin decreases. As during simple precession the total angular momentum shrinks considerably, but its direction is conserved, on the other side the magnitude of the spin stays constant, this means that the orbital angular momentum shrinks. For comparable mass binaries it will be still higher than the spin at ISO (therefore the dynamics below ISO, which can be analyzed only numerically, should be responsible for any spin-flip in the comparable masses case). For extreme mass ratio binaries the result of the shrinkage of the orbital angular momentum is L < S at ISO. Therefore, the spin at ISO should be roughly aligned with the direction of J = L+S, which (as initially L was dominant), is close to the direction of the initial L. In certain cases, especially for equal masses of the two black holes, a strong recoil has been found (González et al. 2007a, b). However, as we noted earlier, the equal mass ratio is untypical. Fourth: The two black holes actually merge, and the merged black hole keeps the – 25 – spin axis from the orbital angular momentum of the previously existing binary, whenever the mass ratio is relatively large. In the case that the mass ratio is between 1 : 1 and 1 : 3, then even at the innermost stable orbit a substantial fraction of the orbital angular momentum can survive, possibly leading to a spin-flip later on. This very short phase should be accompanied by extreme emission of low-frequency gravitational waves. The final stage in this merger leads to a rapid increase in the frequency of the waves, called “chirping”, but this chirping will depend on the angles involved. The angle between the orbital spin of the combined two black holes, and intrinsic spin of the more massive black hole influences the highest frequency of the chirp; for a large angle this frequency will be lower than for a small angle between the two spins. Whether there is another observable feature, such as the induced decay of heavy dark matter particles, from the merger of the two black holes at that event such as speculated by Biermann & Frampton (2006) is not clear at this time. Fifth: Now the newly oriented more massive merged black hole starts its accretion disk and jet anew, boring a new hole for the jets through its environment. This stage can be identified perhaps with giga hertz peaked radio sources (GPS). If the new jet points at the observer, then 3C147 may be one example (Junor et al. 1999). Sixth: The newly oriented jets begin to show up over some kpc, and this corresponds to the X-shaped radio galaxies, while the old jets are fading but still visible. This also explains many of the compact steep spectrum sources, with disjoint directions for the inner and outer jets. Seventh: The old jets have faded, and are at most visible in the low radio frequency bubbly structures, such as seen for the Virgo cluster region around M87 (Owen et al. 2000). The feeding is slowing down, and there is no longer an observable accretion disk, but probably only an advection-dominated disk. However, a powerful jet is still there, although below or even far below the maximal power. The feeding is still from the residual material stemming from the merger. Eighth: The feeding of the black hole is down to catching some gas out of a common red giant star wind as presumably is happening in our Galactic center. This stage seems to exist for all black holes, even at very low levels of activity (e.g., Perez-Fournon & Biermann 1984, Elvis et al. 1984, Nagar et al. 2000). If this concept described here is true, then the superdisk radio galaxies should have large outer distortions in their radio images, that may be detectable at very high sensitivity, as they should correspond to recently active Z-shaped sources. Also, the superdisk should be visible in X-rays, although if the cooling is efficient the temperature may be relatively low. Table 2 suggests that the merger is imminent, if the precession of the jet is measurable within – 26 – a few years, and the opening angle of the precession is much narrower than the wind cone, reflecting the earlier longer time precession (see Gopal-Krishna et al. 2007). Therefore, with very sensitive radio interferometry it might be possible to detect the underlying jet despite its rapid precession, although immediately before the actual merger the feeding of the jet will be turned off. As more and more pieces of evidence suggest that AGNs are the sources of ultra-high energy cosmic rays (Biermann & Strittmatter 1987, Biermann et al. 2007) we need to ask what we could learn next. Clearly, after a spin-flip, the new relativistic jet bores through a new environment, with lots of gas, and so suffers a strong decelerating shock. In such a shock particles are accelerated to maximal energies, and at the same time, as they leave the shock region interact with all that interstellar gas. Therefore, such sites are primary sources for any new particles, such as high energy neutrinos (Becker et al. 2007). Such discoveries may well be possible long before we detect the low frequency gravitational waves from the black hole merger. As at such high energy neutrinos travel straight across the universe, and suffer little loss other than from the adiabatic expansion of the universe, the black holes resulting from a merger of two black holes, with subsequent spin-flip, will be primary targets for searches for ultra-high energy neutrinos, and perhaps other photos and particles at extreme energies. 7. Concluding Remarks Whereas it has been questioned in the past whether the central SMBHs of merging galaxies will be able to actually merge or their approach will stall (due to the process of loss-cone depletion) at a distance where dissipation through gravitational radiation is not yet efficient (for a review of these considerations see Merritt & Miloslavljević 2005), the role of the dynamical friction as bringing close the SMBHs to the transition radius, from where gravitational radiation undertakes the control of the dissipative process has been recently confirmed (Zier 2006) and also complementary mechanisms were proposed (Alexander 2007, Sesana et al. 2006, 2007a,b, Hayasaki 2008). The space mission LISA is predicted to detect the merger of SMBHs. The statistical arguments of Rottmann (2001), using radio observations, suggest that all strong central activity in galaxies may involve a merger of two black holes. Therefore, we have assumed in this paper that whenever galaxies merge, so will do their central SMBHs. Even if there would be exceptions under this rule, this would reflect only in the inclusion of an overall factor . 1 in the number of mergers of SMBHs as compared to the number of mergers of galaxies, derived in Section 2, which would not affect the mass ratio estimates of our paper. Guided by reasonable and simple assumptions we have shown that binary systems of – 27 – SMBH binaries formed by galaxy mergers typically have a mass ratio range between 1/3 and 1/30. Following this, we have proven that for the typical mass ranges a combination of the SO precession and gravitational radiation driven dissipation produces the spin-flip of the dominant black hole already in the inspiral phase, except for the particular configuration of the spin perpendicular to the orbital plane. During this process the magnitude of the spin is unchanged, therefore the merger of a high spin (and high rotation parameter) black hole with the smaller black hole results in a similar high spin state at the end of the inspiral phase. These are the main results of our paper. There is a related discussion undergoing in the literature, whether the high spin of SMBHs is produced by prolonged accretion phases or by frequent mergers. Even a scenario, where the SMBHs have typically low spin (King & Pringle 2006) was advanced, based on the assumption of short periods of small accretion from random directions. Hughes & Blandford (2003), extrapolating results from ν = q−1 ≪ 1 binaries to comparable masses, have shown that mergers spin-down black holes. Volonteri et al. (2005) have studied the distribution of SMBH spins under the combined action of accretion and mergers, and found that the dominant spin-up effect is by gas accretion. Recently, Berti & Volonteri (2008) have consid- ered the problem of mergers by taking into account improvements in the numerical general relativistic methods (Pretorius 2007), and a recent semianalytical formula, which gives the final spin in terms of the initial dimensionless spins, mass ratio, and relative angles of orbital angular momentum and spins (Rezzolla et al. 2008a,b,c, Barausse & Rezzolla 2009). They have found that mergers can result in a high spin end state only if the dominant spin is aligned with the orbital angular momentum of the system (thus the smaller mass orbits in the equatorial plane of the larger). Their considerations extend from comparable masses to mass ratios of 1/10. However, Berti and Volonteri (2008) neglected the angular momentum exchange and transport between black hole, jet, and inner accretion disk by magnetic fields (see, e.g., Blandford 1976); this may modify or even sharpen the conclusions. We can add three remarks to this discussion. First, we have shown by analytical means, that for the typical mass ratio range the inspiral phase ends with a considerably lower value of the orbital angular momentum compared to the spin (see the last picture in Figure 1). A heuristic argument then shows that such a small angular momentum could not significantly change the direction of the spin during the next phases of the merger. Apart from this small orbital angular momentum, the problem being axially symmetric, we do not expect significant further spin-flip due to gravitational radiation in the last stages of the inspiral. Second, the configuration of orbital angular momentum aligned with the dominant spin is not a preferred one in the gravitational radiation dominated post-Newtonian regime. It is not clear yet whether such an alignment could be the by-product of previous phases of the – 28 – inspiral, when dynamical friction (Zier & Biermann 2001), three-body interactions (Sesana et al. 2006, 2007a,b), relaxation processes due to cloud-star interactions (Alexander 2007), three disk model accretion (Hayasaki 2008), and other possible mechanisms occur. Since the stellar system is often slightly flattened, differential dynamical friction could produce the near alignment necessary to allow very high spin after a merger. Third, themagnitude of the spin is practically unchanged in the inspiral phase, discussed here. This is because the loss in the spin vector by gravitational radiation, a second PN order effect, calculated from the Burke-Thorne potential (Burke 1971), is perpendicular to the spins, yielding another precessional effect (Gergely et al. 1998c). Below ISO this estimate should break down, as indicated by numerical simulations reporting on various fractions of the spin radiated away. In this context we want to emphasize the unchanged magnitude of the spin during the inspiral, as important initial data for the numerical evolution during the plunge and ring-down. We also mention here the results of the numerical relativity community showing a con- siderable recoil of the merged SMBH in particular cases, mostly for equal masses and peculiar configurations of the angular momenta (Brügmann 2008, Gonzalez et al. 2007a, b, Koppitz et al. 2007). It has also been shown that the recoil regulates the SMBH mass growth, as the SMBH wanders through the host galaxy for 106 ÷ 108 years (Blecha and Loeb 2008). According to the empirical formula of Campanelli et al. (2007a, see also Lousto & Zlochower 2009) the recoil velocity scales with q−2/ (1 + q−1) (1 + q−1), which for q−1 = ν ≪ 1 reduces to a scaling with q−2. Therefore, we do not expect significant recoil effects in the typical mass ratio range of the SMBH mergers. We suggest that the precessional phase of the merger of two black holes, occurring prior to the spin-flip, is visible as a superdisk in radio galaxies (Gopal-Krishna et al. 2007). The precessing jet appears as a superwind separating the two radio lobes in the final stages of the merger. According to our model such radio galaxies are candidates for subsequent SMBH mergers. Further observations and theoretical work may be capable of identifying such candidates likely to merge, and determine the timescale for this to happen. The restart of powering a relativistic jet (after the spin-flip and merging) will produce ultra-high energy hadrons, neutrinos and other particles. Based on the estimates given in Table 2 for the precessional and inspiral timescales, we can say the following. If we were to observe a precession timescale of three years in a superdisk radio galaxy, we could confidently predict a plunge in about 300 years, which should be observable. Faster precession timescales would take some effort to identify. However, if we were able to even identify a precession timescale of days to weeks, then the plunge would be predicted to happen a few months to a few years thence: powerful gravitational waves at – 29 – very low frequency would then be emitted. The picture developed here differs from that in Wilson & Colbert (1995) in that we do not identify just the rare mergers of two massive black holes of about equal masses with radio galaxies and radio quasars. We intend to revisit the interactions with the stars (Zier et al. 2009, in preparation), discuss the spin of the black holes in another work (Kovács et al. 2009, in preparation) developed from Duţan & Biermann (2005), finally to work out quantitatively the relation of the merger of black holes and the statistics of radio galaxies (Gopal-Krishna et al. 2009, in preparation). 8. Acknowledgements We are grateful for discussions with Gopal-Krishna and C. Zier. P.L.B. acknowledge further discussions with J. Barnes, B. Brügmann, and G. Schäfer. L.Á.G. was successively supported by OTKA grants 46939, 69036, the János Bolyai Grant of the Hungarian Academy of Sciences, the London South Bank University Research Opportunities Fund and the Polányi Program of the Hungarian National Office for Research and Technology (NKTH). Sup- port for P.L.B. was from the AUGER membership and theory grant 05 CU 5PD 1/2 via DESY/BMBF and VIHKOS. 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0704.1969

On the Young-Fibonacci insertion algorithm

ON THE YOUNG-FIBONACCI INSERTION ALGORITHM JANVIER NZEUTCHAP Abstract. This work is concerned with some properties of the Young-Fibonacci insertion algo- rithm and its relation with Fomin’s growth diagrams. It also investigates a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada’s algebra associated to the Young-Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are conveniently inter- preted as chains in the Young-Fibonacci lattice. A property of Killpatrick’s evacuation is given a simpler proof, but this evacuation is no longer needed in making Roby’s and Fomin’s constructions coincide. We provide the set of Young-Fibonacci tableaux of size n with a structure of graded poset, induced by the weak order on permutations of the symmetric group, and realized by transitive clo- sure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients in the transition matrix from the analogue of complete symmet- ric functions to analogue of the Schur functions in Okada’s algebra. We end with a quite similar observation for four posets on Young-tableaux studied by Taskin. Contents 1. Introduction 1 1.1. The Young-Fibonacci lattice 2 2. Young-Fibonacci tableaux and Young-Fibonacci insertion algorithm 3 2.1. Converting a chain in YFL into a standard Young-Fibonacci tableau 3 2.2. Redefining the Young-Fibonacci Insertion Algorithm 4 3. Young-Fibonacci insertion and growth in differential posets 6 3.1. Equivalence between Roby’s and Fomin’s constructions 7 3.2. Another viewpoint of Killpatrick’s evacuation for Young-Fibonacci tableaux 8 4. Fibonacci numbers and a statistic on Young-Fibonacci tableaux 9 5. A weak order on Young-Fibonacci tableaux 10 6. A connection with Okada’s algebra associated to the Young-Fibonacci lattice 15 7. Kostka numbers, the Littlewood Richardson rule, and four posets on Young tableaux 16 Concluding remarks and perspectives 18 Acknowledgements 18 References 19 1. Introduction The Young lattice (YL) is defined on the set of partitions of positive integers, with covering relations given by the natural inclusion order. The differential poset nature of this graph was generalized by Fomin who introduced graph duality [13]. With this extension he introduced [15] a generalization of the classical Robinson-Schensted-Knuth [1, 2] algorithm, giving a general scheme for establishing bijective correspondences between couples of saturated chains in dual graded graphs, both starting at a vertex of rank 0 and having a common end point of rank n, on the one hand, and permutations of the symmetric group Sn on the other hand. This approach naturally leads to the Robinson-Schensted insertion algorithm. 2000 Mathematics Subject Classification. Primary 05-06; Secondary 05E99. Key words and phrases. Schensted-Fomin, insertion algorithm, Young-Fibonacci, lattice, tableaux, evacuation, poset, Okada’s algebra, Kostka numbers. http://arxiv.org/abs/0704.1969v1 2 J. NZEUTCHAP Roby [17] gave an insertion algorithm analogous to the Schensted correspondence, which maps a permutation σ onto a couple made of a Young-Fibonacci tableau P (σ) and a path tableau Q(σ). Roby’s path tableau Q(σ) is canonically interpreted as a saturated chain in the Fibonacci lattice Z(1) introduced by Stanley [11] and also by Fomin [14]. Roby also showed that Fomin’s approach is partially equivalent to his construction. Indeed in Roby’s construction, only the saturated chain Q̂ obtained from Fomin’s growth diagram has an interpretation as a representation of the path tableau Q(σ), while there seems to be no way to translate the Young-Fibonacci tableau P (σ) into its equivalent chain P̂ . Contrarily to the approach of Killpatrick [8] who has used an evacuation to relate the two constructions of Roby and Fomin, we show that with a suitable mechanism for converting a saturated chain in the Young-Fibonacci lattice into a Young-Fibonacci tableau, Roby’s construction naturally coincides with Fomin’s one. The paper is organized as follows. In Section 1.1 we recall the definition of the Young-Fibonacci lattice, then in Section 2 we define a mechanism for converting a saturated chain in this lattice into a standard Young-Fibonacci tableau. In the same section, we also introduce a modification in Roby’s algorithm, in such a way that both the insertion and recording tableaux of any permutation will have an interpretation in terms of saturated chain in the Young-Fibonacci lattice. In Section 3.1 we relate Roby’s algorithm with Fomin’s construction using growth diagrams and we compare it to Killpatrick’s work. In Section 4, we define an analogue of Kostka numbers for Young-Fibonacci tableaux, and we point out one of their relation with usual Fibonacci numbers. In Section 5 we define and we study some properties of a poset on Young-Fibonacci tableaux. This poset turns out to be a model for the interpretation as well as the computation of another analogue of Kostka numbers, introduced by Okada [16] in an analogue of the algebra of symmetric functions, associated to the Young-Fibonacci lattice. We prove this result is Section 6, and in the last section of the paper we prove a similar result relating usual Kostka numbers with four posets on Young-tableaux studied by Taskin [9]. 1.1. The Young-Fibonacci lattice. A Fibonacci diagram or snakeshape of size n is a column by column graphical representation of a composition of an integer n, with parts equal to 1 or 2. The number of such compositions is the nth Fibonacci number. A partial order is defined on the set of all snakeshapes, in such a way to obtain an analogue of the Young lattice of partitions of integers (YL). This lattice is called the Young-Fibonacci lattice (YFL) and it was introduced by Stanley [11] and also by Fomin [14]. As we will see in the sequel, there is a considerable similarity between the two lattices, as well as the combinatorics of tableaux their induce. The covering relations in YFL are given below, for any snakeshape u. (1) u is covered by the snakeshape obtained by attaching a single box just in front ; (2) u is covered by the snakeshape obtained by adding a single box on top of its first single-boxed column, reading u from left to right. (3) if u starts with a series of two-boxed columns, then it is covered by all snakeshapes obtained by inserting a single-boxed column just after any of those columns. The rank |u| of a snakeshape u is the sum of digits of the corresponding Fibonacci word. Its length will be denoted ℓ(u). Let u and v be two snakeshapes such that v covers u in YFL, the cell added to u to obtain v is an inner corner of v, it is also called an outer corner of u. Remark 1.1. Young-Fibonacci tableaux (YFT) will naturally appear as numberings of snake- shapes, satisfying certain conditions described in the sequel, the same way as Young tableaux are numberings of partitions of integers with prescribed numbering conditions. The numbering conditions of Young-Fibonacci tableaux are deduced from the description of the Young-Fibonacci insertion algorithm (Section 2.2). ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 3 Below is a pictorial representation of a finite realization of YFL, from rank 0 up to rank n = 4, with black cells representing inner corners. Figure 1. The Young-Fibonacci lattice. Now let us look at the problem of converting a saturated chain in YFL into a standard YFT. 2. Young-Fibonacci tableaux and Young-Fibonacci insertion algorithm In YL, any saturated chain starting at the empty partition can be canonically converted into a standard Young tableau, and this representation is convenient in many ways. It consists in labeling the boxes as their occur in the chain. As already observed by Roby [17], one question which presents itself is to do the same in YFL for any saturated chain starting at the empty snakeshape ∅. The need of such a conversion mechanism will appear in section 3.1 in the interpretation of two saturated chains in a growth diagram. One may also use the canonical labeling to convert a saturated chain of YFL into a tableau, but Roby had already pointed out that one major problem with this canonical labeling is that except for the trivial rule that each element in the top row must be greater than the one below it, no other obvious rules govern what numberings are allowed for a given shape. We suggest that one first defines simple rules governing what numberings are allowed for a given shape, so that it be easy to decide if a numbering of a snakeshape is a legitimate Young-Fibonacci tableau or not. The convention we use is described in the next section. 2.1. Converting a chain in YFL into a standard Young-Fibonacci tableau. Since we do not use the same conventions as Roby [17] and Fomin [13], let us give the following definition of Young-Fibonacci tableaux. Definition 2.1. A numbering of a snakeshape with distinct nonnegative integers is a standard Young-Fibonacci tableau (SYFT) under the following conditions. (1) entries are strictly increasing in columns ; (2) any entry on top in any column has no entry greater than itself on its right. To convert a chain in YFL into a standard YFT, one will follow the canonical approach as far as the new box added to the chain lies in the first column. Example with the chain Q̂ = (∅, 1, 2, 12, 22, 221, 2211, 21211) ; the sub-chain (∅, 1, 2, 12, 22) is converted as follows. ∅ → 1 → 4 J. NZEUTCHAP Now moving from the shape 22 to the shape 221 in YFL, one inserts a box just after a two-boxed column of the previous shape. In such a situation, one will move the entry on top in that column into the newly created box, and then shift the other entries of the top row to the right. Finally, if n is the largest entry in the partial tableau obtained, then label the box on top in the first column with (n + 1). The conversion started above keeps on as follows, xk means that writing or moving the label x is the kth action performed during the current step. 22 → 221 : 3 1 21 3 1 2 3 1 2 221 → 2211 : 3 1 2 3 1 2 3 1 41 2 3 1 4 2 3 1 4 2 2211 → 21211 : 3 1 4 2 3 1 4 2 3 61 1 4 2 3 6 1 4 2 It easily follows from the description above that this mechanism produces only legitimate YFT (Definition 2.1), and that the conversion is reversible. Now another question which presents itself is how to count standard YFT of a given shape u 6= ∅, we denote this number by Fu. Let us first recall the formula counting linear extensions of a binary tree poset P. (2.1) |Ext(P)| = d1d2 · · · dn where for the ith node vi, di is the number of nodes v ≤P vi. This formula is due to Knuth [3], and since any snakeshape u can be canonically assimilated to a poset Pu, then we have the following. Proposition 2.2. Standard Young-Fibonacci tableaux of a given shape are counted by the hook- length formula for binary trees. To apply the formula to a snakeshape u, count it cells from right to left and from bottom to top, labeling the first box and each box appearing in the bottom row of any two-boxed column. The number of standard YFT of the given shape is the product of all the labels obtained. Example 2.3. Let us consider u = 2212. a snakeshape u = 2212 its poset Pu hook lengths 6 4 1 F2212 = 2×3×5×7 = 6× 4× 1 2.2. Redefining the Young-Fibonacci Insertion Algorithm. We refer the reader to [17, 8] for a description of the original algorithm ; below is the one we consider. ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 5 Definition 2.4. The Young-Fibonacci insertion algorithm maps a permutation σ onto a couple of standard YFT built as follows. The insertion tableau P (σ) is built by reading σ from right to left, matching any of the letters encountered (and not yet matched) with the maximal one (not yet matched) on its left if any, provided that the latter be greater than the first. The recording tableau Q(σ) records the positions of the letters, in the reverse order of the one in which they are matched. Example 2.5. For σ = 2715643, we have the following. 2 7 1 5 6 4 3 1 2 3 4 5 6 7 • P (σ) = 7 6 2 3 4 5 1 and Q(σ) = 7 6 3 2 5 4 1 Remark 2.6. That both P (σ) and Q(σ) are standard Young-Fibonacci tableaux (Definition 2.1) is clear from the description of the algorithm. This is not the case in the original algorithm where P (σ) and Q(σ) are not of the same type. Indeed, with the original insertion algorithm, the insertion tableau is the same as the tableau P (σ) above, but the recording tableau Q(σ) which follows does not satisfy Definition 2.1. QRoby(σ) = 3 7 4 2 6 5 1 The definition of Q(σ) we adopt is inspired from the hypoplactic [4] and sylvester [5] insertion algorithms, where Q(σ) also records the positions in σ of the labels of P (σ). With this definition, some essential properties of the Young-Fibonacci correspondence have a much easier combinatorial proof, which is not always the case in [17]. For example, let us recall the involution property. Theorem 2.7. [17] For any permutation σ, P (σ−1) = Q(σ). Proof. Consider the geometric construction by Killpatrick [8], and recall that P (σ) corresponds to reading vertical coordinates of the rightmost and uppermost x in that order, for any broken line. As for Q(σ), we have defined it in such a way that it corresponds to reading horizontal coordinates of the uppermost and rightmost x in that order. The construction for σ−1 is obtained by transposing the one for σ. � Another fundamental property of Roby’s algorithm which is easily proved using Definition 2.4 follows. Theorem 2.8. [17] Let σ be an involution of the symmetric group, then the cycle decomposition of σ is the column reading of its insertion tableau P (σ). We give two other canonical words associated with a tableau t ; so if we let YFC(t) denotes the equivalence class made of permutations having t as insertion tableau, then YFC(t) has at least three canonical elements. The first canonical element is its canonical involution, that is the only involution the cycles of which coincide with the columns of t, as stated in Theorem 2.8. The two other canonical elements are the maximal (resp. minimal) element for the lexicographical order. We will make use of these elements in Section 5. Lemma 2.9. Let t be a Young-Fibonacci tableau, w1 the left-to-right reading of its top row and w2 the right-to-left reading of its bottom row, then w1.w2 (where . denotes the usual concatenation of words) is the maximal element (for the lexicographical order) of YFC(t), it is denoted wtmax. Lemma 2.10. The word consisting of the right-to-left and up-down column reading of t is the minimal element (for the lexicographical order) of YFC(t), it is denoted wtmin. Proof. Clear from the description of the Young-Fibonacci insertion algorithm. � 6 J. NZEUTCHAP An example is given with the tableau t below ; its canonical involution is (13)(26)(48)(5)(7) = 36185274, the maximal canonical element is 86315274, and the minimal one is 31562784. 8 6 3 4 7 2 5 1 We will see (Theorem 5.12) that YFC(t) is the set of linear extensions of a given poset, and additionally, YFC(t) is an interval of the weak order on the symmetric group (Theorem 5.9). 3. Young-Fibonacci insertion and growth in differential posets In this section we show that with the modification we have introduced in Roby’s original insertion algorithm, together with the conversion mechanism discussed in section 2.1, the Young-Fibonacci insertion algorithm naturally coincides with Fomin’s approach using growth diagrams. So we claim that Killpatrick’s evacuation [8] is no longer needed in making the two constructions coincide. We give a simplification of Killpatrick’s theorem relating Roby’s original algorithm to Fomin’s one through an evacuation process, and we will later need this evacuation in the proof of Theorem 6.2 giving a combinatorial interpretation of Okada’s analogue of Kostka numbers. Let us recall that Fomin’s construction with growth diagrams consists in using some local rules in filling a diagram giving rise to a pair of saturated chains in YL. For any permutation σ, the growth diagram d(σ) is build the following way. First draw the permutation matrix of σ ; next fill the left and lower boundary of d(σ) with the empty snakeshape ∅. The rest of the construction is iterative ; d(σ) is filled from the lower left corner to the upper right corner, following the diagonal. At each step and for any configuration as pictured below, z is obtained by applying the local rules to the vertices t, x, y and the permutation matrix element α. We refer the reader to [15] for more details on this construction. a1 α b1 Figure 2. A square in a growth diagram. Algorithm 1 : local rules for YFL 1: if x 6= y and y 6= t then 2: z := t, with a two-boxed column added in front 3: else 4: if x = y = t and α = 1 then 5: z := t, with a single-boxed column added in front 6: else 7: z is defined in such a way that the edge bi is degenerated whenever ai is degenerated 8: end if 9: end if ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 7 3.1. Equivalence between Roby’s and Fomin’s constructions. Let us build Fomin’s growth diagram for the permutation σ = 2715643. ∅ � �� ��� ∅ � � �� ∅ � � �� ∅ � � �� ∅ � � �� ∅ � � � ∅ ∅ ∅ � � � � � ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Figure 3. Example of growth diagram for the Young-Fibonacci insertion. We get the paths Q̂ = (∅, 1, 2, 12, 22, 221, 2211, 21211) and P̂ = (∅, 1, 11, 21, 211, 1211, 2211, 21211) on the upper and right boundary respectively. Now let us convert them into Young-Fibonacci tableaux, using the mechanism discussed in section 2.1. ∅ → 1 → 2 1 → 2 4 1 2 5 4 1 7 6 3 2 5 4 1 = Q̂(σ) = Q(σ) ∅ → 1 → 3 4 1 6 5 2 3 4 1 7 6 2 3 4 5 1 = P̂ (σ) = P (σ) So as we can see on this example, the two constructions naturally coincide. Remark 3.1. Let us mention that because Roby used the canonical labeling to convert a chain into a tableau, there seemed to be no way to convert the chain P̂ into its equivalent tableau P (σ). Killpatrick’s algorithm was then an approach to relate P̂ with P (σ). Our own approach consists in the introduction of a modification of the original algorithm, and a new labeling process. Theorem 3.2. Let (P̂ (σ), Q̂(σ)) be the pair of Young-Fibonacci tableaux obtained from the permu- tation σ by using Fomin’s growth diagram and let (P (σ), Q(σ)) be the Young-Fibonacci insertion and recording tableaux using Roby’s insertion modified (Definition 2.4), then P̂ (σ) = P (σ) and Q̂(σ) = Q(σ). Proof. The equality P̂ (σ) = P (σ) follows from that any snakeshape P̂k appearing in P̂ is the shape of the tableau P (σ/[1..k]) where σ/[1..k] is the restriction of σ to the interval [1..k]. Indeed, the path P̂ is obtained applying to P (σ) the reverse process of the one described in section 2.1. In so doing, the cell added to P̂k to get P̂k+1 lies in the first column when either σ/[1..k+1] ends with the letter k + 1 or σ/[1..k+1] does not end with the letter k + 1 but σ/[1..k] ends with the letter k. A quite similar reasoning is used to prove the equality Q̂(σ) = Q(σ). � 8 J. NZEUTCHAP 3.2. Another viewpoint of Killpatrick’s evacuation for Young-Fibonacci tableaux. For a tableau t, this operation is defined only for top entries of the columns of t. Let a0 be such an entry, the tableau resulting from the evacuation of a0 is denoted ev(t, a0) and is built as follows. (1) if a0 is a single-boxed column, then just delete this column and, if this is necessary, shift one component of the remaining tableau to connect it with the other one (e.g of line 3 in the table below) ; (2) otherwise, the box containing a0 is emptied and one compares the entry a1 that was just below a0 with the entry a2 on top of the column just to the right if any. If a2 < a1 then this terminates the evacuation process (e.g of line 4 in the table below). Otherwise, move a2 on top of a1, creating a new empty box in the tableau. If the new empty box is a single-boxed column, then this terminates the evacuation process (e.g of line 7, step 4, in the table below), otherwise, iteratively repeat the process with the entries just below and to the right of this new empty box. Let t be a tableau of size n and shape u. If one successively evacuates the entries n, (n−1), · · · , 1 from t, labeling the boxes of u according to the positions of the empty cells at the end of the evacuation of entries, one gets a path tableau denoted ev(t). Recall that a path tableau is the canonical labeling of a saturated chain. Remark 3.3. ev(t) is the same tableau as the one described by Killpatrick [8], with Young- Fibonacci tableaux defined as in Definition 2.1. 7 t = 7 6 2 3 4 5 1 • 6 2 3 4 5 1 6 • 2 3 4 5 1 6 5 2 3 4 • 1 7 6 5 2 3 4 1 • 5 2 3 4 1 5 • 2 3 4 1 3 4 1 3 4 1 3 • 1 3 5 7 4 6 2 3 5 7 1 1 → • ev(t) = 4 6 2 3 5 7 1 Table 1. Evacuation on Young-Fibonacci tableaux. Lemma 3.4. Let w be a word with no letter repeated, let a0 be one of its letters appearing as a top element in a column of P (w), and let w0 be the word obtained from w by deleting the only occurrence of a0, then ev(P (w), a0) = P (w0). ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 9 Proof. Easily from the description of the evacuation and the description of the Young-Fibonacci insertion algorithm (Definition 2.4). � We give a simpler proof of the following theorem by Killpatrick, relating ev(P (σ)) with P̂ . Indeed, using the canonical labeling, Roby has converted the path P̂ into a path tableau P̂ (σ) and Theorem 3.5. [8] ev(P (σ)) = P̂ (σ). Proof. Follows from Lemma 3.4 and the remark that any snakeshape P̂k appearing in P̂ is the shape of the tableau P (σ/[1..k]) where σ/[1..k] is the restriction of σ to the interval [1..k]. � 4. Fibonacci numbers and a statistic on Young-Fibonacci tableaux In this section we point out a property of Young-Fibonacci numbers defined as an analogue of Kostka numbers. Recall that the usual Kostka numbers Kλ, µ are defined for two partitions λ and µ of the same integer n and they appear when expressing Schur functions sλ in terms of the monomial symmetric functions mµ, and in the expression of the complete symmetric functions hµ in terms of Schur functions sλ. (4.1) sλ = Kλ, µmµ ; hµ = Kλ, µ sλ We will not focus on the algebraic interpretation of the Kλ, µ but rather on their combinatorial interpretation in terms of tableaux. Indeed, Kλ, µ counts the number of distinct semi-standard Young-tableaux of shape λ and valuation µ, that is to say with µi entries i for i = 1 .. ℓ(µ). It is then natural to introduce the same definition with Young-Fibonacci tableaux. Definition 4.1. A semi-standard Young-Fibonacci tableau is a numbering of a snakeshape with nonnegative integers, not necessarily distinct, preserving the conditions stated in Definition 2.1. Definition 4.2. Let u and v be two snakeshapes of size n, the Young-Fibonacci number associated to u and v, denoted Nu, v is the number of distinct semi-standard Young-Fibonacci tableaux of shape u and valuation v, that is to say with vi entries i for i = 1 .. ℓ(v). For example, for u = 221 and v = 1211, there are 4 distinct semi-standard Young-Fibonacci tableaux of shape u and valuation v. So N221, 1211 = 4. 1 2 2 2 1 2 2 2 1 3 1 2 Proposition 4.3. Young-Fibonacci numbers are generated by the recurrence formulas below, where both u and v are snakeshapes. (4.2) N∅, ∅ = 1 ; N2, 2 = 0 N1u, v1 = Nu, v ; N1u, v2 = Nu, v1 N2u, v1 = w∈ v1 − Nu,w ; N2u, v2 = w∈ v1 − Nu, w1 where v1 denotes the multiset of snakeshapes obtained from v either by deleting a single occurrence of 1, or by decreasing a single entry not equal to 1, for example 21121 = [1112, 212, 212, 2111]. Proof. Easily from the definition of Young-Fibonacci tableaux and Young-Fibonacci numbers. � 10 J. NZEUTCHAP v = 222 2211 2121 2112 21111 1221 1212 12111 1122 11211 11121 11112 111111 u = 222 2 3 4 5 6 4 5 6 5 7 8 12 15 2211 4 5 5 7 9 5 7 9 7 9 9 12 15 2121 2 3 4 4 5 4 4 5 4 6 8 8 10 2112 1 1 1 1 1 2 2 3 3 4 4 4 5 21111 2 3 3 3 4 3 3 4 3 4 4 4 5 1221 2 2 3 4 4 3 4 4 4 4 6 8 8 1212 1 1 1 1 1 1 2 2 3 3 3 4 4 12111 2 2 2 3 3 2 3 3 3 3 3 4 4 1122 1 1 1 1 1 1 1 1 2 2 3 3 3 11211 1 1 2 2 2 2 2 2 2 2 3 3 3 11121 1 1 1 1 1 1 1 1 2 2 2 2 2 11112 0 0 0 0 0 1 1 1 1 1 1 1 1 111111 1 1 1 1 1 1 1 1 1 1 1 1 1 Table 2. Matrix of Young-Fibonacci numbers for n = 6. Theorem 4.4. Let n ≥ 2 be a positive integer, then the number of couples (u, v) of snakeshapes of size n such that Nu, v = 0 is the (n− 2) th Fibonacci number. Proof. The proof is done by induction on n. Indeed using equation (4.2) it is easy to see that Nu, v 6= 0 whenever u 6= 1 n−22. So the problem is equivalent to counting the number of snakeshapes v such that N = 0. But N = 0 if and only if there exists a snakeshape w such that v = 2w. Then the problem is finally equivalent to counting the snakeshapes of size (n − 2), and hence the result. � 5. A weak order on Young-Fibonacci tableaux In what follows, we introduce a partial and graded order denoted � on the set YFTn of Young- Fibonacci tableaux of size n. We will see (Theorem 5.8) that this partial ordering on YFTn is such that the map from the weak order on the symmetric group Sn which sends each permutation σ onto its Young-Fibonacci insertion tableau P (σ) is order-preserving. More particularly, standard Young-Fibonacci classes on Sn are intervals of the weak order on Sn. Recall that the weak or- der on permutations of Sn is the transitive closure of the relation σ ≤p τ if τ = σδi for some i, where δi is the adjacent transposition (i i+1). An inversion of a permutation σ is a couple (j, i), 1 ≤ i < j ≤ n such that σ−1(i) > σ−1(j), that is to say j appears on the left of i in σ. Note that this is not the definition commonly used . The set of inversions of a permutation σ will be denoted inv(σ), and the number of inversions denoted #inv(σ). We will be making use of an analogous notion of non-inversion of a permutation σ which is a couple (i, j), 1 ≤ i < j ≤ n such that σ−1(i) < σ−1(j), that is to say i appears on the left of j in σ. The set of non-inversions of a permutation σ will be denoted ord(σ). Definition 5.1. To introduce �, we define the operation of shifting an entry in a tableau t as follows. (1) the bottom entry a of any column of t may move and bump up the entry c on its left if c is a single-boxed column of t. In the example below, the letter 1 is the one being shifted. 2 4 3 1 shift the entry 1 −−−−−−−−−−−−−−−−→ 2 4 1 ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 11 (2) In the case a was the bottom entry in a two-boxed column, the top entry b will just fall down. In the two examples below, the letter 2 (resp. 3) is the one being shifted. 5 2 3 1 shift 2 −−−−−−−−→ 2 4 3 1 5 3 1 shift 3 −−−−−−−−→ 3 4 1 (3) In the case the column just to the left of a is two-boxed, with bottom entry c and a < c, then a may replace c which on its turn is shifted to the right in such a way that if c < b then c will just replace a ; otherwise c is placed as a new single-boxed column between a and b, and b just falls down. In the two examples below, the letter 1 (resp. 2) is shifted. 2 1 3 shift 1 −−−−−−−−→ 1 2 3 4 2 1 shift 2 −−−−−−−−→ 2 4 3 1 Remark 5.2. It easily follows from the definition that shifting an entry in a tableau always produces a legitimate tableau of the same size. In an analogous way, given a tableau t, one defines the reverse operation of finding all the tableaux t′ such that shifting an entry in t′ gives back t. For example, one will check that 3 4 1 is obtained from 5 3 1 4 3 1 3 4 2 1 by shifting 3 or 1. Finally it is clear that this operation is antisymmetric, that is to say if t′ is obtained from t by shifting a given entry, then t cannot be obtained from t′ by shifting an entry. The latter observation is enforced by the following lemma which also defines the graduation of the poset (YFTn,�) we will soon introduce. Lemma 5.3. Let t2 be a tableau obtained by shifting an entry in a tableau t1, and let σ1 (resp. σ2) be the minimal permutation canonically associated to t1 (resp. t2) as stated in Lemma 2.10, then the inversions sets of σ1 and σ2 are related by the relation #inv(σ2) = #inv(σ1) + 1. Proof. The proof takes into account all the situations one can encounter in shifting an entry in t1. (1) t1 = T2 c a T1 and t2 = T2 a ⋆T1, where T1 and T2 are partial YFT having minimal canonical words w1 and w2 (see Lemma 2.10 for the definition), and ⋆ means any entry preserving the conditions of Definition 2.1, and possibly no entry. The minimal permutations associated to t1 and t2 are σ1 = w1⋆acw2 and σ2 = w1⋆caw2 respectively, and clearly σ2 has one more inversion than σ1. (2) t1 = T2 c a T1 and t2 = T2 a c ⋆ T1, with a < c < d ; one has σ1 = w1 ⋆adcw2 and σ2 = w1⋆cdaw2. The inversion (dc) appears in σ1 but not in σ2, whereas the inversions (da) and (ca) appear in σ2 but not in σ1 ; so σ2 has one more inversion. (3) t1 = T2 c a T1 and t2 = T2 a c T1, with a < c < b < d ; one has σ1 = w1badcw2 and σ2 = w1bcdaw2. The inversion (dc) appears in σ1 but not in σ2, whereas the inversions (da) and (ca) appear in σ2 but not in σ1 ; so σ2 has one more inversion. � We are now in position to provide YFTn with a structure of poset. Definition 5.4 (weak order on YFTn). Let t and t ′ be two tableaux of size n, then t is said smaller than t′ and we write t � t′ if one can find a sequence t0 = t, t1, · · · , tk = t ′ of tableaux of size n such that ti+1 be obtained from ti by shifting an entry, for i from 0 to k − 1. Proposition 5.5. (YFTn,�) is a graded poset, the rank of a Young-Fibonacci tableau being the number of inversions of its minimal canonical permutation. 12 J. NZEUTCHAP Proof. Follows from Lemma 5.3. � Remark 5.6. Note that this remarkable property of graduation of the poset of standard Young- Fibonacci tableaux of size n does not apply to the similar poset YTn of standard Young tableaux of size n. The reader interested may refer to [9] where Taskin studied many nice properties of four partial orders on YTn. ρ = 6 ρ = 5 ρ = 4 ρ = 3 ρ = 2 ρ = 1 54321 ρ = 0 Figure 4. The graded weak order on Young-Fibonacci tableaux of size 5. Remark 5.7. As one will easily check it on the figure above, (YFTn,�) is not a lattice for n = 5 for example. Indeed let a = 5421 and b = 3421, then a and b do not have a least upper bound. Theorem 5.8. Let t1 and t2 be two tableaux, then t1 � t2 if and only if one can find two permu- tations τ1 and τ2 such that P (τ1) = t1, P (τ2) = t2 and τ1 ≤p τ2. Proof. It is enough to prove this statement for the case t2 is obtained by shifting an entry in t1, and the proof is carried out as a parallel process of the proof of Lemma 5.3. So go back to the latter proof and (1) take τi = σi ; (2) take τ1 = w1⋆dacw2 and τ2 = w1⋆dcaw2 ; (3) take τ1 = w1bdacw2 and τ2 = w1bdcaw2. This shows that one can find two permutations τ1 and τ2 such that P (τ1) = t1, P (τ2) = t2 and τ2 = τ1δi for some i, whenever t1 � t2. Reciprocally let τ1 and τ2 be two permutations such that ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 13 P (τ1) = t1 and P (τ2) = t2 and τ2 = τ1δi for some i. Then t2 is obtained from t1 by shifting the entry i in t1. � We now look at the structure of the Young-Fibonacci classes ; below are two pictures of the poset (YFT4,�). On the picture on the left, vertices are Young-Fibonacci classes corresponding to Young-Fibonacci tableaux in the picture on the right. Recall that the rank of a class is the number of inversions of its minimal element in the lexicographical order. The unique involution of any class is enclosed in a rectangle. A double edge means that there are two couples (τ1, τ2) and (τ satisfying the conditions of Theorem 5.8. 3241 ρ=4 3142 ρ=3 1342 ρ=2 2134 1324 1243 ρ=1 1234 ρ=0 1 3 2 4 1 2 2 3 1 4 3 1 4 2 1 3 2 1 4 3 2 1 Figure 5. The graded weak order on Young-Fibonacci classes of size 4. It is easy to check that each class appearing as a vertex of the poset (YFT4,�) is an interval of the weak order (S4,≤p), and this is a general observation. Theorem 5.9. Let t be a standard Young-Fibonacci tableau of size n, then YFC(t) is an interval of the weak order (Sn,≤p), more over YFC(t) = [w min, w max]. To prove this statement, we will first relate YFC(t) with linear extensions of a poset canonically associated to t, and then we will prove that the set of linear extensions of this poset is an interval of the weak order. Definition 5.10. Let t be a standard Young-Fibonacci tableau of size n, its canonical poset Pt is the poset defined on the set {1, 2, ..., n} with the covering relations below. (1) the right-to-left reading of the bottom row of t forms a chain in the poset ; (2) each entry on top in a two-boxed column of t is covered by the corresponding entry on bottom row. Note 5.11. A permutation σ is a toset (totally ordered set) with covering relations defined by σ(i) ≤σ σ(j) whenever i < j, that is to say x ≤σ y if x appears to the left of y in σ. Let P be a poset and σ a permutation, σ is said to be a linear extension of P if its relations preserve the relations in P, that is to say if x ≤P y then x ≤σ y. The set of linear extensions of a poset P will be denoted Ext(P). 14 J. NZEUTCHAP Theorem 5.12. Let t be a standard Young-Fibonacci tableau, then YFC(t) = Ext(Pt). Proof. That any permutation σ having t as insertion tableau is a linear extension of Pt is clear from Definitions 2.4 and 5.10. Conversely, if σ is a linear extension of Pt, then t is naturally built reading σ from right to left following the description given in Definition 2.4. At each new step the first letter one reads is the maximal one (for ≤Pt) not yet read in the chain described in rule (1) of Definition 5.10. � Theorem 5.13. Let t be a standard YFT of size n, then Ext(Pt) is the interval [w min, w max] in (Sn,≤p). To prove this statement we make use of the following well known lemma. Lemma 5.14. Let σ and τ be two permutations of Sn, then the three properties below are equivalent. (1) σ ≤p τ ; (2) ord(τ) ⊆ ord(σ) ; (3) inv(σ) ⊆ inv(τ). Proof. (of Theorem 5.13) It easily follows from the definition that Pt can be partitioned into an antichain A = (y1, y2, · · · , yℓ) and a chain C = (x1 <Pt x2 <Pt · · · <Pt xk) such that for i = 1..ℓ there exists j(i) ≤ k such that yi <Pt xj(i), and additionally for i1 < i2 one has yi1 < yi2 and xj(i1) <Pt xj(i2). For illustrations, we use the following example. A = (3, 6, 7) C = (2 <Pt 5 <Pt 1 <Pt 4) 7 6 3 4 1 5 2 a tableau t of shape u = 2212 its canonical posetPt The set I is made of the inver- sions below. (3, 2), (6, 1), (7, 4) (3, 1), (6, 4) (2, 1), (5, 1), (5, 4). The set O is made of the or- dered pairs below. (2, 5), (2, 4), (1, 4) (3, 5), (3, 4). For σ ∈ Ext(P), inv(σ) includes at least the set (yi, xj(i)), i = 1..ℓ (yi, xr) / xj(i) > xr and xj(i) <Pt xr (xi, xj) / xi > xj and xi <Pt xj which is nothing but inv(wtmin) ; so by [Lemma 5.14 - (3)], w min ≤p σ. Moreover, ord(σ) includes at least the set O = { (yi, xr) / xj(i) <Pt xr } ∪ { (xi, xj) / xi < xj and xi <Pt xj } which is nothing but ord(wtmax) ; so by [Lemma 5.14 - (2)], σ ≤p w max and hence σ ∈ [w min, w max]. Conversely, for σ ∈ [wtmin, w max], applying Lemma 5.14 to w min, σ and w max it appears that σ has the inversions yi ≤σ xj(i) for i = 1..ℓ, and the relations x1 <σ x2 <σ · · · <σ xk. So P (σ) = t and hence σ ∈ Ext(Pt). � Proof. (of Theorem 5.9) Follows from Theorem 5.12 and Theorem 5.13. � Definition 5.15. Let u be a snakeshape of size n, the row canonical tableau rTu is the one such ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 15 (1) top cells of rTu are labeled with entries n, n− 1, · · · from left to right ; (2) bottom cells in two-boxed columns are labeled with entries 1, 2, · · · from left to right. The column canonical tableau cTu is built by labeling the cells of u from right to left and bottom to top. Lemma 5.16. Let u be a snakeshape of size n, then cTu (resp. rTu) is the unique tableau of shape u having minimal rank ρumin (resp. maximal rank ρ max) in the poset (YFTn,�). For any snakeshape u, ρumin is the number of double-boxed columns of u and ρ max is obtained as follows. Label each bottom cell with the number of double-boxed columns on its left and do the same but add 1 for each top cell of double-boxed columns of u. ρumax is the sum of labels obtained. ρ = 12 1̂ ρ = 11 • 7 6 4 1 2 5 3 • row canonical tableau • • • • • • ρ = 3 • • 7 5 2 6 4 3 1 column canonical tableau • • • ρ = 0 0̂ Figure 6. Row canonical and column canonical tableaux of shape 2212. Proof. (of Lemma 5.16) Easily from the definitions. � We will now relate (YFTn,�) to a transition matrix in Okada’s algebra associated to YFL. 6. A connection with Okada’s algebra associated to the Young-Fibonacci lattice A Young-Fibonacci analogue of the ring of symmetric functions [6] was given and studied by S. Okada [16], with a Young-Fibonacci analogue of Kostka numbers, appearing when expressing the analogue of a complete symmetric function hv in terms of the analogue of Schur functions su. (6.1) hv = Ku, v su Young-Fibonacci analogue of Kostka numbers are generated by the recurrence formulas below [16], where Ka, b is defined for two snakeshapes of the same weight and ✄ denotes the covering relation in YFL. (6.2) K1u, 1v = Ku, v (r1) K2u, 2v = Ku, v (r2) K1u, 2v = 0 (r3) K2u, 1v = w✄uKw, v (r4) As it is stated below, the hook-length formula for binary trees illustrated in Example 2.3 is an alternative formula for computingKu, 1n = Fu which is the dimension of a representation in Okada’s algebra. Proposition 6.1. Let u be a snakeshape of size n, then Fu is the dimension of the module Vu corresponding to u in the nth homogenous component of Okada’s algebra associated to YFL. 16 J. NZEUTCHAP Proof. dim(Vu) is the number of saturated chains from ∅ to u in YFL, hence the result. � Here is a more general statement giving a combinatorial interpretation of Ku, v using (YFTn,�). Theorem 6.2. Let u and v be two snakeshapes of size n, and let 1̂ be the maximal tableau in (YFTn,�), then Ku, v is the number of tableaux t of shape u in the interval [rTv, 1̂]. 1 2 3 1 4 2 1 3 2 5 1 2 1 4 3 2 5 1 3 2 4 1 2 5 4 1 2 Example 6.3. In the matrix below, the number Ku, 1121 counts the number of standard Young- Fibonacci tableaux of shape u in the interval [rT1121, 1̂]. 221 212 2111 122 1211 1121 1112 15 221 1 1 2 1 2 3 4 8 212 . 1 1 1 1 1 3 4 2111 . . 1 . 1 1 1 4 122 . . . 1 1 1 2 3 1211 . . . . 1 1 1 3 1121 . . . . . 1 1 2 1112 . . . . . . 1 1 15 . . . . . . . 1 Iterating this for each snakeshape v of size n, one builds the transition matrix for expressing the analogue of complete symmetric function hv in terms of the analogue of Schur functions su. Figure 7. (YFT5,�) and Okada’s analogue of Kostka matrix for n = 5. Proof. (of Theorem 6.2) A proof consists in showing that for any couple (a, b) of snakeshapes appearing in the left hand side of equation (6.2), there is a one-to-one correspondence between tableaux satisfying the conditions of the theorem for (a, b) and those satisfying the conditions of the theorem for the couples of snakeshapes in the corresponding right hand side of the relation. For (r1), given a tableau t of shape u such that rTv � t, t is mapped onto the tableau t ′ of shape 1u obtained from t by attaching a cell labeled n+1 to its left, and rT1v � t ′. For (r2), one attaches a two-boxed column to the left of t, with 1 as bottom entry and n+2 as top entry, in addition one standardizes t by increasing all its entries. Then t′ is of shape 2u and rT2v � t ′. For (r3) it easily follows from the definition of the operation of shifting an entry in a tableau that there is no tableaux t1 and t2 of shape 1u and 2v respectively, such that t2 � t1. For (r4), let t be a tableau of shape 2u such that rT1v � t, then t is mapped onto the tableau t ′ = ev(t, n), that is the tableau obtained from t by evacuating its maximal letter (the evacuation process originally due to Killpatrick [8] is described in Section 3.2). Indeed, let w be the shape of t′, then w ✄ u and rTv � t 7. Kostka numbers, the Littlewood Richardson rule, and four posets on Young tableaux The poset (YFTn,�) of Young-Fibonacci tableaux we defined in Section 5 is an analogue of one among four partial orders on the set YTn of standard Young tableaux of size n [9]. The weak order (YTn,�weak) is defined as in Theorem 5.8 with P (σ) denoting the Schensted insertion tableau of σ. Let λ and µ be two partitions of lengths ℓ(λ) and ℓ(µ), λ is said greater than µ in ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 17 the dominance order and one writes λ ≥dom µ if for each 1 ≤ i ≤ min(ℓ(λ), ℓ(µ)), the inequality λ1 + λ2 + · · · + λi ≥ µ1 + µ2 + · · · + µi holds. Let t be a standard Young tableau of size n, and 1 ≤ i ≤ j ≤ n. We denote λ(t/i,j) the shape of the tableau obtained from t by first restricting t to the segment [i, j], then lowering all entries by i − 1, and finally sliding the skew tableau obtained into normal shape by jeu-de-taquin. The chain order �chain on standard Young tableaux is defined as follows. Definition 7.1. [9] Let t and t′ be two standard Young tableaux of size n, then t �chain t ′ if and only if for each 1 ≤ i ≤ j ≤ n, λ(t/i,j) ≥dom λ(t The reader interested may refer to [9] for the definition of the two other orders, as well as for the properties of those posets. The four posets happen to coincide up to rank n = 5. 12345 Figure 8. Partial order on Young tableaux of size 5. Below is a Young tableaux analogue of Theorem 6.2. Theorem 7.2. Let λ, µ be two partitions of size n, let rTµ be the row canonical standard Young tableau of shape µ, that is to say rTµ has shape µ and is increasingly filled from let to right and bottom to top. And let 0̂ be the minimal tableau in the poset of standard Young tableaux of size n. Then Kλ, µ is the number of standard Young tableaux of shape λ in the interval [0̂, rTµ], for any one of the posets studied in [9]. 18 J. NZEUTCHAP 1 2 4 1 2 5 1 2 4 1 2 4 5 1 2 3 4 0̂ = 1 2 3 4 5 Example 7.3. In the matrix below, the num- ber Kλ, 221 counts the number of standard Young tableaux of shape λ in the interval [0̂, rT221]. µ = 5 41 32 311 221 2111 11111 λ = 5 1 1 1 1 1 1 1 41 . 1 1 2 2 3 4 32 . . 1 1 2 3 5 311 . . . 1 1 3 6 221 . . . . 1 2 5 2111 . . . . . 1 4 11111 . . . . . . 1 Iterating this for each partition µ of size n, one builds the transition matrix for expressing the complete symmetric function hµ in terms of the Schur functions sλ. Figure 9. Poset of Young tableaux and Kostka matrix for n = 5. Proof. (of Theorem 7.2) For a given partition µ, let nscrt(µ) be the row canonical semi-standard Young tableau of shape µ, that is the tableau filled with 1’s on its first line, 2’s on its second line and so on. Let nsclt(n) be the semi-standard Young tableau of shape n and having µi entries i for i = 1..ℓ(µ). Consider the extension of Definition 7.1 to the set Tab(µ) of semi-standard Young tableaux having µi entries i for i = 1..ℓ(µ). Then for each t ∈ Tab(µ), one has nsclt(n) �chain t �chain nscrt(µ). There is a canonical bijection mapping (Tab(µ),�chain) onto ([0̂, rTµ],�chain) and this map is order preserving. So Theorem 7.2 holds for the partial order �chain. From ([9], Theorem 1.1) and the remark that [0̂, rTµ] = rTµ1 ∗ rTµ2 ∗ · · · ∗ rTµℓ(µ) , it follows that the set of tableaux in [0̂, rTµ] does not depend on the choice of the partial order. � Concluding remarks and perspectives There are quite many similarities between the Robinson-Schensted algorithm and the Young- Fibonacci insertion algorithm. As well as between the combinatorics of Young tableaux and the combinatorics of Young-Fibonacci tableaux. One of the questions we have not explored in this paper is the one of the existence of an algebra of Young-Fibonacci tableaux, which would be an analogue of the Poirier-Reutenauer Hopf algebra of Young tableaux [12]. Such an algebra would certainly help in giving a combinatorial description (in terms of tableaux) of the product of Schur functions in Okada’s algebra associated to the Young-Fibonacci lattice. We are currently looking for a suitable definition of this algebra. Acknowledgements The author is grateful to F. Hivert for helpful comments and suggestions throughout this work. ON THE YOUNG-FIBONACCI INSERTION ALGORITHM 19 References [1] C. Schensted, Longest increasing and decreasing subsequences. Canad. J. Math., vol. 13, 1961, pp. 179-191. [2] D. E. Knuth, Permutations, matrices and generalized Young tableaux, Pacific J. Math. 34(1970), 709–727. [3] —, The art of computer programming, vol.3: Searching and sorting (Addison-Wesley, 1973). [4] D. Krob and J.-Y. Thibon, Noncommutative symmetric function IV: Quantum linear groups and Hecke algebras at q=0, J. Alg. Comb. 6 (1997), 339-376. [5] F. Hivert, J. C. Novelli, and J.-Y. Thibon, The Algebra of Binary Search Trees, Theo. Comp. Science 339(2005), 129-165. [6] I. G. MacDonald, Symmetric functions and Hall Polynomials, 2nd ed, Clarendon Press, Oxford Sce Publications, 139(1995). [7] J. Nzeutchap, On the Young-Fibonacci Insertion Algorithm, to appear in the Proceedings of FPSAC’07. [8] K. Killpatrick, Evacuation and a Geometric Consturction for Fibonacci Tableaux, J. Comb. Th, Ser A, 110 (2005), 337-351. [9] M. Taskin, Properties of four partial orders on standard Young tableaux, J. Comb. Theory, Ser A, 113(2006), 1092-1119. [10] R. P. Stanley, Differential Posets, J. Amer. Math. Soc. 1 (1998), 919-961. [11] —, The Fibonacci lattice, Fibonacci Quarterly 13(1998), 215-232. [12] S. Poirier and C. Reutenauer, Algèbre de Hopf des tableaux, Ann. Sci. Math. Qébec 19 (1995), 79-90. [13] S. V. Fomin, Duality of Graded Graphs, J. Alg. Comb. 3(1994), 357-404. [14] —, Generalized Robinson-Schensted-Knuth correspondence, Zapiski Nauchn. Sem. LOMI. 155 (1986), 156-175. [15] —, Schensted Algorithms for Dual Graded Graphs, J. Alg. Comb. 4(1995), 5-45. [16] S. Okada, Algebras associated to the Young-Fibonacci lattice, Trans AMS 346(1994), 549-568. [17] T. Roby, Applications and extensions of Fomin’s generalization of the Robinson-Schensted correspondence to differential posets, Ph.D. thesis, MIT, 1991. LITIS EA 4051 (Laboratoire d’Informatique, de Traitement de l’Information et des Systèmes), Avenue de l’Université, 76800 Saint Etienne du Rouvray, France E-mail address: [email protected] URL: http://monge.univ-mlv.fr/∼nzeutcha 1. Introduction 1.1. The Young-Fibonacci lattice 2. Young-Fibonacci tableaux and Young-Fibonacci insertion algorithm 2.1. Converting a chain in YFL into a standard Young-Fibonacci tableau 2.2. Redefining the Young-Fibonacci Insertion Algorithm 3. Young-Fibonacci insertion and growth in differential posets 3.1. Equivalence between Roby's and Fomin's constructions 3.2. Another viewpoint of Killpatrick's evacuation for Young-Fibonacci tableaux 4. Fibonacci numbers and a statistic on Young-Fibonacci tableaux 5. A weak order on Young-Fibonacci tableaux 6. A connection with Okada's algebra associated to the Young-Fibonacci lattice 7. Kostka numbers, the Littlewood Richardson rule, and four posets on Young tableaux Concluding remarks and perspectives Acknowledgements References

0704.1970

Thermoelectric response near a quantum critical point: the case of CeCoIn5

Thermoelectric response near a quantum critical point: the case of CeCoIn5 K. Izawa1,2,3, K. Behnia4, Y. Matsuda3,5, H. Shishido5,6, R.Settai6, Y. Onuki6 and J. Flouquet2 1Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo, 152-8551 Japan 2DRFMC/SPSMS, Commissariat à l’Energie Atomique, F-38042 Grenoble, France 3Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 4Laboratoire Photons Et Matière(CNRS), ESPCI, 75231 Paris, France 5Department of Physics, Kyoto University, Kyoto 606-8502, Japan and 6Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan We present a study of thermoelectric coefficients in CeCoIn5 down to 0.1 K and up to 16 T in order to probe the thermoelectric signatures of quantum criticality. In the vicinity of the field-induced quantum critical point, the Nernst coefficient ν exhibits a dramatic enhancement without saturation down to lowest measured temperature. The dimensionless ratio of Seebeck coefficient to electronic specific heat shows a minimum at a temperature close to threshold of the quasiparticle formation. Close to Tc(H), in the vortex-liquid state, the Nernst coefficient behaves anomalously in puzzling contrast with other superconductors and standard vortex dynamics. PACS numbers: 74.70.Tx, 72.15.Jf, 71.27.+a CeCoIn5 is an unconventional superconductor with an intriguing normal state[1]. Its behavior is peculiar near the upper critical field, where the energy scale governing various electronic properties is vanishingly small and in- creases with increasing magnetic field[2, 3], a behavior ex- pected in presence of a Quantum Critical Point(QCP)[4]. The proximity of this QCP to the upper critical field in CeCoIn5 is puzzling[5, 6, 7]. The possible existence of a FFLO state[8] and/or an elusive magnetic order are a subject of recent intense research. On the other hand, even in the absence of magnetic field, the normal state presents strong deviation from the standard Fermi-liquid behavior[1, 9]. The application of pressure leads to the destruction of superconductivity and the restoration of the Fermi liquid[6, 10, 11]. The link between the field- induced and the pressure-induced routes to the Fermi liquid is yet to be clarified. During the last three years, the anomalous properties of CeCoIn5 near the field-induced QCP have been re- ported thanks to measurements of specific heat[3], elec- tric resistivity[2], thermal transport[7] and Hall effect[13]. In this paper, new insight on the quantum criticality is given via thermoelectric response down to 0.1K. As far as we know, this is the first experimental investigation of the thermoelectric tensor in the vicinity of a QCP, a subject of several theoretical studies[16, 17, 18]. Single crystals were grown by self-flux method. Thermoelec- tric coefficients were measured with one heater and two RuO2 thermometers in magnetic field along c-axis. The heat current was applied along the basal plane. Previous studies of thermoelectricity in CeCoIn5 detected a large Nernst coefficient and a field-dependent Seebeck coeffi- cient in the non-Fermi liquid regime above Tc[14] and an additional field scale at 23 T[15]. Here, we find that the most spectacular thermoelectric signature of quantum criticality is a drastic enhancement of the Nernst coeffi- cient, ν. The vanishingly small Fermi energy, which was previously detected by a nearly diverging enhancement of the A coefficient of resistivity (ρ = ρ0 +AT 2)[2] and the Sommerfeld coefficient of specific heat (γ = Cel/T )[3], leads also to an apparently diverging ν/T . These results show two distinct anomalies close to Hc2(0) and Tc(0) which are different in the origin. This conclusion cannot be derived from other probes mentioned above. We also find a milder enhancement of the Seebeck coefficient near the QCP. Moreover, the ratio of thermopower to elec- tronic specific heat, expressed in appropriate units[19], remains close to unity even in the vicinity of the QCP. The temperature dependence of this ratio presents a min- imum at a temperature roughly marking the formation of well-defined quasi-particles[7]. Figure 1 presents the data obtained by measuring the Nernst and the Seebeck coefficients at various magnetic fields. Since the thermoelectric response of Fermions is expected to be T -linear well below their Fermi temper- ature, what is plotted in the figure is the temperature dependence of the two coefficients divided by temper- ature. As seen in fig. 1(a), the Seebeck coefficient, S vanishes in the superconducting state. In the normal state, S/T increases with decreasing temperature for all fields. For fields exceeding 5.4 T, the normal state ex- tends down to zero temperature and a finite S/T in the zero-temperature limit can be extracted. For a field of 16 T (which is well above the quantum critical region) S/T saturates to a value of about 13 µVK−2. For fields between 5.4 T and 16 T, S/T presents a non-monotonous temperature dependence. An upturn below 0.15 K is vis- ible for µ0H ≃ 5.5 T (i.e. in the vicinity of the QCP) curves. Note that this upturn leads to a moderate en- hancement of S/T . The overall change in the magnitude of S/T is about 70 %. On the other hand, the tem- perature dependence of the Nernst coefficient divided by temperature ν/T reveals a more dramatic signature of Quantum criticality. As seen in figure 1(b), for µ0H = 5.5 T and µ0H = 6 T, below 1 K, ν/T is steadily increas- ing with decreasing temperature. No such enhancement occurs for µ0H =16 T, far above QCP. At the lowest measured temperature (∼ 0.1 K), ν/T is five-fold en- http://arxiv.org/abs/0704.1970v2 sample 2 3 4 5 6 7 2 3 4 T ( K ) 5.45T 5.2T 5T 4.5T 2 3 4 5 6 7 2 3 4 T ( K ) 0.5 T 5.3 T 5.2 T 5 T 4.5 T 4 T 5.4 T 16 T FIG. 1: (a) The Seebeck coefficient divided by temperature as a function of temperature for different magnetic fields in a semi-log plot. Note the upturn near the QCP.(b) Temper- ature dependence of the Nernst coefficient divided by tem- perature ν/T . Close to the QCP, this quantity never satu- rates. The inset defines the convention used for the sign of the Nernst coefficient(see text). hanced near the QCP (∼6 T) compared to its 16 T value. Since the thermal Hall conductivity κxy in CeCoIn5 be- comes large at low temperatures due to enhancement of the mean-free-path of the electrons [12], the transverse thermal gradient ∇yT could generate a finite transverse electric field Ey. Therefore, the (measured) adiabatic and the (theoretical) isothermal Nernst coefficients are not identical in CeCoIn5. However, using the value of |∇yT |/|∇xT | ∼ 0.1 at 5.2 T reported in Ref. [12], the dif- ference between these two is estimated to be about 10 %, indicating that the observed enhancement is not due to a finite ∇yT . We will argue below that this enhancement reflects a concomitant decrease in the magnitude of the normalized Fermi energy as previously documented by specific heat and resistivity measurements. The thermoelectric response of CeCoIn5 in the vicin- ity of QCP can be better understood by complementing our data with the information extracted by other experi- mental probes[2, 3], which originally detected a quantum critical behavior near Hc2. In particular, an interesting issue to address is the fate of the correlation observed between thermopower and specific heat of many Fermi liquids in the zero-temperature limit In a wide range of systems, the dimensionless ratio linking these two is of the order of unity (q = SNAe ≃ ±1, with γ = Cel/T , NA the Avogadro number and e the charge of electron)[19]. What happens to such a correlation at a quantum criti- cal point? Combining the specific heat data reported by Bianchi et al.[3] with our thermopower results allows us to address these questions. Fig. 2(a) presents q computed in this way as a function of temperature. The first feature to remark is that q remains of the order of unity even in the quantum critical region. Note that, theoretically, this correlation arises because S/T and γ are both inversely proportional to the normalized Fermi energy and thus q is expected to be of the order of (and not rigorously equal to) unity[17]. According to our result (q ≃ 0.9 at 6 T and 0.1 K), this correlation holds even when the normal- 15105 m0H ( T ) 1/2 ( µΩ n/T (0.1 K) A1/2 g (0.1 K) (a.u.) 2 3 4 5 6 7 8 T ( K ) FIG. 2: (a) The temperature dependence of q, the dimen- sionless ratio of thermopower to electronic specific heat at three magnetic fields. The temperature marked by the arrow designs the threshold of the quasi-particle formation accord- ing to the temperature dependence of the Lorenz number as reported by Paglione et al.[7] (b) A comparison of the field dependence of ν/T , γ (as reported in ref. [3]) and A1/2 (taken from ref. [2]). ized Fermi Energy becomes vanishingly small. The sec- ond feature of interest in figure 2(a) is the temperature dependence of q, which presents a minimum. For both fields, the temperature at which this minimum occurs is close to the one where the Lorenz number(L = κ linking thermal, κ, and electric,σ, conductivities present also a minimum. Paglione and co-workers, who report this latter feature, argue that this temperature marks the formation of well-defined quasi-particles[7]. This is a temperature below which both thermal and electric re- sistivities display a T 3/2 temperature dependence. Re- markably, Miyake and Kohno, who provided a theoretical framework in a periodic Anderson model for the corre- lation between thermopower and specific heat, predicted that q should deviate downward from unity in presence of an antiferromagnetic (AF) QCP leading to hot lines on the Fermi surface[17]. We now turn to the Nernst coefficient. In a simple picture, it is proportional to the energy derivative of the relaxation time at the Fermi energy[20]. In a first ap- proximation, it tracks a magnitude set by the cyclotron frequency, the scattering time and the Fermi energy[21]. Since it scales inversely with the Fermi Energy, there is no surprise that it becomes large in heavy-fermion metals[14, 15] and in particular in heavy-Fermion semi- metals[22, 23], where both the heavy mass of electrons and the smallness of the Fermi wave-vector contribute to its enhancement(ν/T ∝ 1/(kF ǫF )). Now, since the Fermi energy (broadly defined as the characteristic energy scale of the system) becomes very small near a QCP, one would expect a large Nernst coefficient in agreement with the experimental observation reported here. With these phenomenological considerations in mind let us compare the behavior of the Nernst coefficient with specific heat and resistivity. Both γ and A, the T 2 term of the resistivity (ρ = ρ0 +AT 2) inversely scale with the Fermi Energy, ǫF . Therefore, both are enhanced when the Fermi energy is small. Since these two quantities are linked by the Kadowaki-Woods relation (γ2 ∝ A), the enhancement is more pronounced in A than in γ. Fig- ure 2(b) compares the field-dependence of A1/2, γ and ν/T . In a naive picture, the enhancement of the three quantities are comparable in magnitude. This quanti- tative correlation suggests that the main reason for the enhancement of ν/T near QCP is due to a small ǫF . It is instructive to trace a contour plot of this quantity in the temperature-field plane. This is done in Fig. 3 with a logarithmic color scale in order to enhance the contrast. Note that contrary to the other probes, there is no need to subtract an offset from the Nernst data. In the case of specific heat, one should subtract the Schottky contribu- tion at low temperature[3] and high-field, and the phonon contribution at high temperature. In the case of resistiv- ity the T 2 behavior is interrupted at low temperature and high-fields by an upturn due to the temperature- dependent magnitude of ωcτ [2]. As seen in Fig. 3, ν/T becomes very large near the QCP, which constitutes the main hearth of the figure. However, there is a second one at zero field just above Tc, which was identified by previ- ous measurements[14]. This zero-field hot region corre- sponds to a purely linear resistivity and anomalously en- hanced Hall coefficient[9] due to strong anisotropic scat- tering by AF fluctuations[11], which can also enhance the Nernst coefficient[24]. On the other hand, close to the QCP, the magnitude of Hall coefficient[13] is compara- ble to its value at room-temperature or in LaCoIn5[11]. Therefore, there appears to be two distinct sources for the enhancement of the Nernst coefficient. In the zero- field regime just above Tc, it is enhanced mostly because of strong inelastic scattering associated with AF fluctu- ations, but in the zero-temperature regime just above Hc2, it becomes large because of the smallness of the Fermi energy. The occurrence of superconductivity im- pedes to explore the route linking together these two hot regions of the (B,T) plane. The inset of the figure com- pares the evolution of energy scales detected by different experimental probes near the QCP. We now turn to the puzzling behavior of the Nernst coefficient in the vicinity of the superconducting transi- tion. Deep into the superconducting state, there is no measurable Nernst signal, as illustrated by the existence of the black area in Fig. 3. On the other hand, close to Hc2(T) (or alternatively, near Tc(H)), vortices can move and an additional contribution to the Nernst sig- nal is expected. In the entire range of our study, the Nernst coefficient keeps the same sign which is presented in the inset of Fig. 1. Such a Nernst coefficient is neg- ative according to a textbook convention on the sign of the thermoelectric coefficients[25]. However, the liter- ature on the vortex Nernst effect[26] usually takes for positive the Nernst signal generated by vortices moving from hot to cold, which leads to an opposite convention. The sign of the Nernst effect in CeCoIn5 is negative ac- cording to the textbook convention[25], but positive ac- cording to the vortex one[26, 27]. Indeed, contrary to quasi-particles, the Nernst signal produced by vortices 0 1 2 3 4 0.2 0.4 0.6 T(K) 0.01000 0.02185 0.04775 0.1044 0.2280 0.4983 1.089 2.380 5.200 FIG. 3: Contour plot of ν/T in the (B,T) plane. The color scale is logarithmic. Note the presence of two hot regions close to Hc2 and Tc. The inset is a zoom on the region near Hc2. The variation of three temperature scales, the onset of T 2 resistivity(solid squares), the minimum in L/L0(open squares) the and minimum in q(open circles) with magnetic field is also shown. should have a fixed sign. A thermal gradient ∇xT gen- erates a force on a vortex because its core has an excess of entropy. The direction of this force is thus thermo- dynamically determined; vortices move along the ther- mal gradient from hot to cold region. The orientation of electric field is also unambiguously set by the direction of the vortex movement and the vortex Nernst signal is not expected to have an arbitrary sign. In order to sepa- rate the vortex and the quasi-particle contributions to the Nernst signal, we put under careful scrutiny the effect of superconducting transition on three coefficients : ρ(T ), S(T ) and N(T ) . As illustrated in fig. 4(a) and 4(b), with the onset of superconductivity, the Nernst signal, N , collapses faster than both resistivity and the Seebeck coefficient. This robust feature was observed for all mag- netic fields. On the other hand, the collapse in ρ(T ) and S(T ) closely track each other. This latter feature, which was also observed in cuprates[27], suggests that the See- beck response is essentially generated by quasi-particles. Therefore, the most natural assumption regarding their contribution to the Nernst signal in the vortex liquid regime is that Nqp(T ) also follows ρ(T ) and S(T ) and the vortex contribution to the Nernst signal can be ob- tained by subtracting the normalized Seebeck coefficient off the normalized Nernst one. Fig. 4(c) and 4(d) show that this procedure clearly resolves a signal of opposite sign. Thus, the most straightforward interpretation of the faster collapse of N(T ) implies an additional source of Nernst signal in the vortex liquid regime with a sign opposite to the predominant one and also to the one ex- pected for vortices moving along the heat flow. This result appears incompatible with the standard picture of vortex dynamics driven by a thermal gradient. However, one shall not forget that additional forces on vortices besides thermal force may be present. CeCoIn5 1.51.00.5 T ( K ) 2.42.22.01.8 T ( K ) S/Sn N/Nn r/rn FIG. 4: (a)(b) Normalized magnitudes r of r = N/Nn(open circles), S/Sn(solid circles) and ρ/ρn(solid line) in the vicinity of superconducting transition for 1 T and 5 T. For all fields, the onset of superconductivity leads to a faster collapse of the Nernst signal. Note also the small shoulder at 1 T. (c)(d) The additional contribution to transverse thermoelectricity in the vortex liquid regime obtained by subtracting the normalized Seebeck coefficient off the normalized Nernst signal. is distinguished from other superconductors by the possi- ble occurrence of an anti-ferromagnetic state in the nor- mal core of its vortices. This feature could decrease the entropy excess of the vortices and reduce the intensity of the thermal force, which can therefore be vanquished by another source of vortex movement. As first noted by Ginzburg[28], in a superconductor subject to a ther- mal gradient, a quasi-particle current (which carry heat) and a supercurrent (which does not) counterflow in or- der to keep the charge current zero[27]. In ordinary con- ditions, this counterflow generates a transverse Magnus force on vortices[29]. Its role in the context of superclean CeCoIn5[12] needs an adequate theoretical treatment. Another remarkable feature of Fig. 4 is the presence of a small shoulder in the temperature dependence of the Nernst effect at the end of the transition. The shoulder is present in an extended range of magnetic fields and only disappears in the proximity of Hc2. There seems to be a narrow temperature window, where a thermal gradient can create a transverse electric field, but a current does not produce any electric field. The simplest explanation for such a discrepancy would imply a threshold force to depin vortices, fdp attained by the applied temperature gradient, but not by the applied current. However, this feature was found to be robust and no change was de- tected by modifying the magnitude of the applied ther- mal gradient. Clearly, the sign and the fine structure of the Nernst effect in the vortex liquid regime of CeCoIn5 need further investigation. We thank J-P. Brison, H. Kontani, N. Kopnin, K. Maki and K. Miyake for helpful discussions and specially N. P. Ong for his illuminating input on the sign of the Nernst effect. K.I. acknowledges a European Union Marie Curie fellowship. This work was supported by the Agence Na- tionale de la Recherche through the ICENET project. [1] C. Petrovic et al., J. Phys. Condens. Matter 13, L337 (2001). [2] J. Paglione et al.,Phys. Rev. Lett. 91, 246405 (2003). [3] A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003). [4] G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001). [5] E. D. Bauer et al., Phys. Rev. Lett. 94, 047001 (2005). [6] F. Ronning et al., Phys. Rev. B 73, 064519 (2006). [7] J. Paglione et al.,Phys. Rev. Lett. 97, 106606 (2006). [8] Y. Matsuda and H. Shimahara, J. Phys. Soc. Jpn 76, 051005 (2007). [9] Y. Nakajima et al., J. Phys. Soc. Jpn. 73, 5 (2004). [10] V. A. Sidorov et al., Phys. Rev. Lett. 89, 157004 (2002). [11] Y. Nakajima et al., J. Phys. Soc. Jpn. 76, 024703 (2007). [12] Y.Kasahara et al., Phys. Rev. B 72, 214515 (2005). [13] S. Singh et al., Phys. Rev. Lett. 98, 057001 (2007). [14] R. Bel et al., Phys. Rev. Lett. 92, 217002 (2004). [15] I. Sheikin et al., Phys. Rev. Lett. 96, 077207 (2006). [16] I. Paul and G. Kotliar, Phys. Rev. B 64, 184414 (2001). [17] K. Miyake and Kohno, J. Phys. Soc. Jpn. 74, 254 (2005). [18] D. Podolsky et al., Phys. Rev. B 75, 014520 (2007). [19] K. Behnia et al., J. Phys.: Condens. Matter 16, 5187 (2004). [20] E. H. Sondheimer, Proc. R. Soc. London, Ser. A 193, 484(1948). [21] K. Behnia et al., Phys. Rev. Lett. 98, 076603 (2007). [22] R. Bel et al., Phys. Rev. B 70, 220501(R) (2004). [23] A. Pourret et al. Phys. Rev. Lett. 96, 176402 (2006). [24] H. Kontani, Phys. Rev. Lett. 89, 237003 (2002). [25] G. S. Nolas et al., Thermoelectrics, Springer (2001). [26] Y. Wang et al., Phys. Rev. B 73, 024510 (2006). [27] R. P. Huebener, Supercond. Sci. Technol. 8, 189 (1995). [28] V. L. Ginzburg, Sov. Phys. Usp. 34, 101(1991). [29] H. -C. Ri et al., Phys. Rev. B 47, 12312 (1993).

0704.1971

Dissipative dynamics of superfluid vortices at non-zero temperatures

Dissipative dynamics of superfluid vortices at non-zero temperatures Natalia G. Berloff and Anthony J. Youd Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA We consider the evolution and dissipation of vortex rings in a condensate at non-zero temperature, in the context of the classical field approximation, based on the defocusing nonlinear Schrödinger equation. The temperature in such a system is fully determined by the total number density and the number density of the condensate. A vortex ring is introduced into a condensate in a state of thermal equilibrium, and interacts with non-condensed particles. These interactions lead to a gradual decrease in the vortex line density, until the vortex ring completely disappears. We show that the square of the vortex line length changes linearly with time, and obtain the corresponding universal decay law. We relate this to mutual friction coefficients in the fundamental equation of vortex motion in superfluids. PACS numbers: 03.65.Sq, 03.75.Kk, 05.65.+b, 67.40.Vs, 67.57.De The processes of self-organization, formation of large- scale coherent localized structures and interactions of these structures with small-scale fluctuations are at the heart of nonlinear sciences, ranging from classical tur- bulence, superfluids, ultracold gases and Bose–Einstein Condensates (BECs), to the formation of the early Uni- verse. Turbulence is characterised by the co-existence of motions with many length and time scales described by many degrees of freedom. The key to our understand- ing of turbulence is to elucidate physics of interactions between large scales (eg. large eddies) and small scales (eg. turbulent fluctuations), and to develop mathemat- ical models that account for the effects of small scales without actually solving for them. According to the Lan- dau description, superfluid 4He consists of the ground state and the excitations — quasiparticles drifting on top of the ground state. In the language of relativistic quan- tum fields this corresponds to vacuum and matter (eg. gravity waves interacting with a vacuum). Indeed, there is a close relationship between superfluid hydrodynamics and quantum gravity, so that at some level of hierarchy of parameters the interactions of the quantum vacuum and matter can be described by the defocusing nonlinear Schrödinger (NLS) equation [1]. The dynamics of Bose condensates depends on the energy exchange between the condensed and non-condensed parts of the gas. Again, the NLS equation (reformulated as the Gross–Pitaevskii (GP) equation [2]) describes equilibrium and dynamical properties of BEC as well as the formation of BEC from a strongly degenerate gas of weakly interacting bosons [3, 4]. The formation of the large-scale coherent localized ground state (condensate) from a non-equilibrium initial state has been studied in a number of papers addressing different stages of the formation. Weak turbulence the- ory has been used to predict the self-similar evolution of the field in the regime of random phases of Fourier ampli- tudes [5, 6], the transition from the regime of weak tur- bulence to superfluid turbulence via states of strong tur- bulence in the long-wavelength region of energy space [7], and the final stage, resulting in the formation of a genuine condensate [8, 9]. The related question about the effect of finite temperature on the BEC dynamics has also been addressed recently [10]. For instance, it was shown that the presence of the thermal cloud in a trapped conden- sate creates an effective dissipation that forces a single vortex to move away from the centre and disappear [11]. The problem of the vortex tangle interacting with the normal fluid (thermal cloud) is the key question in su- perfluid turbulence. The Landau two-fluid theory of su- perfluidity pre-dated the discovery of quantised vortex lines and therefore omitted significant dynamical effects. This was remedied— in the limit in which the mean spac- ing between the vortex lines is small compared with any other length scale of interest — by HVBK theory [12, 13]. In this limit, the superfluid vorticity is treated as a con- tinuum, but the discrete nature of the vorticity gives rise to an extra force on the superfluid component, arising from the tension in the vortex lines. This term is absent from the classical Euler equation of motion for an inviscid fluid. The vortex lines also create a force of mutual fric- tion between superfluid and normal fluid in addition to the mutual friction included by Landau in his equations, and represents the effects of collisions of the quasiparti- cles with the vortex cores. Such forces were introduced into the Landau model in an ad hoc way. This Letter is the first attempt to study the effect of these collisions quantitatively: we shall find the vortex line decay law at non-zero temperature in the context of the defocus- ing NLS equation. The NLS equation is a good starting point, as the non-dissipative Landau two-fluid model can be obtained from the equations of conservation of mass and momentum for a one-component barotropic fluid us- ing a general expression for the internal energy functional of the density [14]. Through the Madelung transforma- tion the NLS equation can be written in that form. Anal- ogously, the transport coefficients in the Landau model http://arxiv.org/abs/0704.1971v2 have been obtained directly from the NLS equation by following the Chapman–Engskog expansion [15]. Note that the separation of scales needed to carry out the derivation of the Landau two-fluid model from the NLS equation does not allow the inclusion of vortices as part of the ground state. It is natural, therefore, to attempt to derive the corresponding effects of the interactions of vortices with the quasiparticles directly from the NLS equation. We consider the normalised defocusing NLS equation for the complex function ψ [2]: i∂tψ = −∇2ψ + |ψ|2ψ. (1) The dynamics conserves the total number of parti- cles N = |ψ|2dx, and the total energy E =∫ ( |∇ψ|2 + 1 dx. We consider the uniform discrete system of volume V = N 3, which is a periodic box on a computational grid with 1283 discrete points. Our goal is to determine the universal decay law for the vortex line density in the entire range of tempera- tures from 0 to the critical temperature of condensation, Tλ. Our approach consists of three essential steps. We aimed to: (1) achieve the thermal equilibrium state for the given number of particles and given energy, starting from a non-equilibrium stochastic initial condition for the wavefunction ψ; (2) introduce a vortex ring into this state and follow its decay via interactions with non-condensed quasiparticles; (3) relate the decay rate to the tempera- ture at equilibrium, where we derive the expression for the relative temperature, T/Tλ, as a function of the total number density, ρ = N/V , and the number density of the condensate, ρ0. We performed large scale numerical simulations of Eq. (1) starting from a strongly non-equilibrium initial condition[7], where the phases of the complex Fourier am- plitudes ak(t) = ψ(x, t)e−ip·x dx are distributed ran- domly at t = 0. Here the momentum p takes quantised values p = (2π/N )n with n = (0, 0, 0), (±1, 0, 0), · · ·. This initial state describes a weakly interacting Bose gas that is so rapidly cooled below the critical BEC tem- perature that the particles remain in a strongly non- equilibrium state. The kinetics of the initial weak tur- bulent state has been analysed in [5, 6, 16] discovering a quasiparticle cascade from high energies to low energies in the wave number space. The ordering of the system and the violation of the assumptions of weak turbulence occurs very rapidly in a low-energy part of the spectrum, with the formation of a quasi-condensate consisting of a tangle of quantised vortices. The vortex tangle decays as the system reaches a state of thermal equilibrium with some portion (ρ0 ≡ |a0|2/V ) of particles occupying the zero momentum state (genuine condensate) and the rest of the non-condensed particles being distributed accord- ing to the Rayleigh–Jeans equilibrium distribution [17], modified by the presence of nonlinear interactions with the condensate [9]: p 6=0 |2 = T ωB(p) , (2) where T is the temperature and ωB(p) is the Bogoli- ubov dispersion relation (see below). An ultraviolet cut- off for this distribution appears naturally through the spatial discretization of the NLS equation. The numer- ical scheme consists of fourth-order finite difference dis- cretization in space and fourth-order Runge–Kutta in time, so it is globally fourth-order accurate. This scheme corresponds to the Hamiltonian system in the discrete variables ψjkn, such that iψ̇jkn = , j, k, n = 1, ...,N , (3) where ψ∗jkn[ Ψ2 − 43Ψ1 + ψjkn] + |ψjkn|4 with Ψ2 = ψj+2,k,n + ψj−2,k,n + ψj,k+2,n + ψj,k−2,n + ψj,k,n+2 + ψj,k,n−2 and Ψ1 = ψj+1,k,n + ψj−1,k,n + ψj,k+1,n + ψj,k−1,n + ψj,k,n+1 + ψj,k,n−1. The thermodynamic description of the condensation process has been obtained in [9] by adapting the Bogoli- ubov theory of a weakly interacting Bose gas [18] to the classical system (1). We follow the same basic idea to de- rive expressions for the energy and non-condensed part of the discretised energy (4) written in terms of the Fourier amplitudes ap as K2(p)a pap + p1,p2,p3,p4 a∗p1a ap3ap4δp1+p2−p3−p4 , where δp is the Kronecker delta symbol and K2(p) = sin2(pi/2)(7− cos(pi)). (6) The Bogoliubov transformation bp = upap − vpa∗−p, such that up = 1/ 1−Q2p and vp = Qp/ 1−Q2p with Qp = [−K2−2ρ0+ωB(p)]/ρ0 diagonalises the term in (5), which is quadratic in a0, to ωB(p) b pbp, where excludes the p = 0 mode. Here ωB(p) = K22 + 2ρ0K2 is the Bogoliubov-type dispersion relation. Using the equilibrium distribution of the uncondensed particles (2) the non-condensed number density can then be expressed in terms of the basis used in this diagonal- isation as ρ− ρ0 = ′K2(p) + ρ0 . (7) The discretised energy density H/V in the new basis takes the form ρ2 + (ρ− ρ0)2 1. (8) The Eqs. (7)–(8) are analogous to Eqs. (8)–(9) of [9] but modified for the discrete Hamiltonian discretization (4). Given the energy density, H/V , and the total number density, ρ, one can determine the temperature, T , at equilibrium and the number density of the condensed particles, ρ0, from Eqs. (7) and (8). The condensate fraction ρ0/ρ as a function of the energy density H/V is shown in FIG.1. This figure can be compared with FIG.2 of [9] for the spectral representation of the total energy. The analytical formulae (7)–(8) predict the sub- critical behaviour of condensation, whereas the numerics does not support this conclusion, as shown in the insert of FIG.1. We use a linear approximation for small ρ0 to determine the critical maximum energy for condensation as shown in the insert. This energy is then used to deter- mine the critical temperature for condensation Tλ (= T for minH/V for which ρ0 = 0) from (7)–(8). We found a phenomenological formula that determines T/Tλ as a function of ρ0 and ρ as 1− α√ρ − α√ρ , (9) where α is the only fitting parameter that we found as α = 0.227538. The insert in FIG.1 shows the graph of T/Tλ as a function of ρ0/ρ for ρ = 1/2. Eq. (9) gives an excellent fit to the values computed from (7)–(8) across all the values of ρ0 and ρ. In order to analyze the decay of the vortex line length at non-zero temperatures, we insert a vortex ring into a state of thermal equilibrium and follow its decay due to the interactions with the non-condensed particles. The condensate healing length, which determines the size of the vortex core, is calculated based on the density of the condensate, and in our non-dimensional units is ξ = 1/ ρ0. In healing lengths, the radius of the ring is set to R0 = 10. The new initial state is ψv(t = 0) = ψeq ∗ ψvortex, where ψeq is the equilibrium state and ψvortex is a wavefunction of the vortex ring [19]. The vortex line length, L, is calculated as a function of time with high frequencies being filtered out from the field ψ, according to ãp = ap ∗ max( 1− p2/p2c, 0), where the cut-off wavenumber is chosen as pc = 10(2π/N ) [20]. The first important conclusion of our numerical simula- tions is that at all temperatures, the square of the vortex line length decays linearly with time, = −γ(ρ, T/Tλ), (10) where γ does not depend on t. FIG. 2 shows this depen- dence for various temperatures. The actual isosurfaces of the decaying vortex line are shown in the inserts. FIG. 1: (colour online) Condensate fraction, ρ0/ρ, as a func- tion of the energy density as obtained from the numerical simulations (points) and from the analytical expressions (7)– (8) (solid line). The inserts show (a) the plot of T/Tλ as a function of ρ0/ρ, obtained using Eqs. (7)–(8) or Eq. (9) and (b) subcritical condensation predicted by Eqs. (7)–(8) (black line), the linear approximation used to obtain the crit- ical temperature of condensation (gray (red) line), and nu- merical calculations (blue dots). The total number density is ρ = 1/2. 0.5 1 1.5 2 H/V 2.55 2.56 2.57 2.58 2.59 H/V 0 0.2 0.4 0.6 0.8 FIG. 2: (colour online) The decay of the square of the vortex line length as a function of time for various T indicated next to the graphs. The fit to the linear function is shown by the gray (red) lines. The inserts show isosurface plots of the vortex line (for filtered fields ψ; see text) for T = 0.52Tλ at time=130 (left) and time=1300 (right); between these two times the vortex line length is reduced by a factor of 2. The perturbations to the vortex line due to collisions with non- condensed particles are clearly seen on the left insert. These collisions generate Kelvin waves that also radiate energy to sound. The total number density is ρ = 1/2. 100 200 300 400 500 600 0.44Tλ 0.27Tλ 0.63Tλ This result agrees with predictions of the HVBK theory for superfluid helium [12] according to which the funda- mental equation of the motion of a vortex line, vL, is given by (see also page 90, Eq. (3.17) of [21]) vL = vsl+αs ′×(vn − vsl)−α′s′×[s′×(vn − vsl)], (11) where vsl is the local superfluid velocity that consists of the ambient superfluid flow velocity and the self-induced vortex velocity ui, vn is the normal fluid velocity, s is a position vector of a point on the vortex and s′ is the unit tangent at that point. Mutual friction parameters α and α′ are ad hoc coefficients in the HVBK theory that are functions of ρn, ρ, and T only. Eq. (11) is a gen- eral and universal equation used to follow the evolution of three-dimensional vortex motion in an arbitrary flow. When formulated for a single vortex ring Eq. 11 reads dR/dt = −αui, where ui = κ[log(8R/ξ) − δ + 1]/(4πR) and δ is the vortex core parameter. For the GP vortices δ ≈ 0.38 [2]. In dimensionless units used in our paper ui = [log(8R) − δ + 1]/R. After integration of the equa- tion for Ṙ we get αt = (R20 − R2)/[2(log(8R̂) + δ − 1)], where R̂ is the mean radius of the ring. When this is com- pared with (10) we get the following relationship between γ and α: γ = 8π2(log(8R̂) + δ − 1)α. From our numer- ics we obtained a general result valid across all ranges of temperatures and total densities: γ ≈ Kρ(T/Tλ)2, where K ≈ 68. Note that for a GP condensate T/Tλ ≈ ρn/ρ to the first order (see insert (a) of FIG.1), so alterna- tively, we can write γ ≈ K1ρn(T/Tλ) FIG. 3 shows the comparison of the numerically calculated γ/ρ and the quadratic fit K(T/Tλ) 2. Thus, we found that the mutual friction coefficient in condensate superfluids is given by α ≈ K2ρn(T/Tλ). The existence of the transverse force on superfluid vor- tices which is parametrised by the parameter α′ has been a subject of much debate in mid-1990s, when calcula- tions of the classical Magnus force applied to superfluid vortices have been offered and argued about [22]. The criticism is based on the observation that the classical hydrodynamic equations are inapplicable in the vortex core. Whether or not the details of the non-classical vor- tex dynamics are crucial to the existence of the trans- verse force is still an open question. The estimate of α′ can be obtained from our numerical procedure as follow- ing. Eq. (11) written for a distance travelled by a single vortex ring takes form (see Eq. (3.53) on page 107 of [21]) dz/dt = (1−α′)ui. We compared the distances travelled by a vortex ring at various temperatures obtained nu- merically with the distances travelled by a vortex ring in the absence of the transverse force according to the ana- lytical formula dz/dt = ui, where ui = ui(R(t)) and R(t) varies with time according to (10). The insert of FIG.3 shows these distances for T/Tλ = 0.27. Our calculations fail to detect any significant presence of the transverse force for any temperature considered: the deviation from the analytical curve is insignificant within the accuracy of (10). We plan to perform a more thorough analyti- cal and numerical study of transverse force from a single FIG. 3: (Color online) Values of γ/ρ as a function of tempera- ture T/Tλ for various values of the total number density ρ de- picted in various shades of gray (in various colours): ρ = 1/2 (dark (red)), ρ = 1/4 (light (green)) and ρ = 3/4 (medium (blue)). The plot of the quadratic fit γ/ρ = 68(T/Tλ) given by the dashed line. The relative temperature is calcu- lated using Eq. (9). The result is not sensitive to whether we use the values of ρ and ρ0 that correspond to the state of thermodynamical equilibrium before the introduction of the vortex ring or after the vortex ring disappears and the sys- tem equilibrates. The insert shows the distance travelled by a vortex ring as a function of time for T/Tλ = 0.27 (red dots – distances calculated using dz/dt = ui, black line using nu- merics). Curves depart when the vortex ring becomes small in radius and the analytical formula is no longer accurate ap- proximation of the vortex velocity. 0 0.2 0.4 0.6 0.8 0 200 400 600 800 phonon acting on a single vortex in context of the GP model in future. In summary, we considered the effect of temperature on the decay of vortex line density via interactions with non-condensed particles in the context of the defocus- ing NLS equation. We obtained a simple expression for the temperature at equilibrium as a function of the to- tal number density and the number density of the con- densed particles. Depending on these two parameters, a vortex ring introduced into the condensate shows dif- ferent decay rates with time. We identified this decay law as linear for the square of the vortex line length and showed the universal dependence of the decay rate on temperature and total density. It has been suggested that the emission of sound by vortex reconnections and vortex motion is the only active dissipation mechanism responsible for the decay of superfluid turbulence. The decay of superfluid turbulence via Kelvin wave radiation and vortex reconnections was studied in the framework of the GP equation [23] at near zero temperature, via collision of two vortex rings, and confirmed that in the Kelvin wave cascade, where energy is transferred to much shorter wavelengths with a cut-off below a critical wave- length, the vortex line density can be described by the famous Vinen equation [24] d(L/V )/dt = −χ(L/V )2. It has also been shown [25] that the presence of localized finite amplitude sound waves greatly enhances the dissi- pation of the vortex tangle, essentially changing the de- cay law to exponential decay. This Letter complements the existing Kelvin wave cascade scenario by consider- ing an opposite limit when there are no reconnections, and the decay mechanism depends only on the energy exchange with non-condensed particles. This mechanism exceeds the energy transfer via the Kelvin wave cascade. Finally, we related our results about a single vortex ring to the mutual friction coefficients in the general equation of vortex motion. NGB acknowledges the support from EPSRC-UK. She is also very grateful to Professor Joe Vinen for several illuminating discussions about superfluid turbulence and his suggestion to look into the decay of vortex rings in the context of the NLS equation. [1] G. Volovik, gr-qc/0612134. [2] V. L. Ginzburg and L. P. Pitaevskii, Sov. Phys. JETP 7, 858 (1958); L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961); E. P. Gross J. Math. Phys. 4, 195 (1963). [3] E. Levich and V. Yakhot, J. Phys. A: Math. Gen. 11, 2237 (1978). [4] Yu. Kagan and B.V. Svistunov, Phys. Rev. Lett. 79, 3331 (1997). [5] V.E.Zakharov, S.L. Musher, and A.M.Rubenchik, Phys. Rep. 129, 285 (1985) and S. Dyachenko, A.C. Newell, A. Pushkarev and V.E.Zakharov, Physica D 57, 96 (1992). [6] B.V. Svistunov, J. Moscow Phys. Soc. 1, 373 (1991); Yu. Kagan and B.V. Svistunov, Zh. Eksp. Theor. Fiz. 105, 353 (1994) [Sov. Phys. JETP 78, 187 (1994)]. [7] N. G. Berloff and B. V. Svistunov, Phys. Rev. A 66, 013603 (2002) [8] M.J.Davis, S.A. Morgan, and K. Burnett, Phys. Rev. Lett 87, 160402 (2001) and Phys. Rev. A 66, 053618 (2002) [9] C. Connaughton et al, Phys. Rev. Lett. 95, 26901 (2005). [10] M.Brewczyk et al J. Phys. B: At. Mol. Opt. Phys. 40, R1-R37 (2007) and reference within. [11] H. Schmidt et al J.Opt.B: Quantum Simiclass. Opt. 5 S96 (2003). [12] H.E. Hall and W. F. Vinen, Proc. R. Soc. Lond., A238, 215 (1956); I.L. Bekharevich and I.M. Khalatnikov Soviet Phys., JETP, 13, 643 (1961). [13] R.N. Hills, and P.H. Roberts, Archiv. Rat. Mech. & Anal., 66, 43 (1977a); Int. J. Eng. Sci., 15, 305 (1977b); J. Low Temp. Phys., 30, 709 (1978a); J. Phys. C11, 4485 (1978b). [14] S.J. Putterman and P.H.Roberts, Physica, 117A, 369 (1983). [15] T.R.Kirkpatrick and J. R. Dorfman J. Low Temp. Phys. 58, 301 (1985); 399 (1985). [16] Yu. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Zh. Eksp. Teor. Fiz. 101, 528 (1992) [Sov. Phys. JETP 75, 387 (1992)]; Yu. Kagan and B.V. Svistunov, Zh. Eksp. Theor. Fiz. 105, 353 (1994) [Sov. Phys. JETP 78, 187 (1994)]. [17] V. E. Zakharov, V. S. L’vov and G. Falkovich, Kol- mogorov Spectra of Turbulence I (Springer, Berlin, 1992); A. C. Newell, S. Nazarenko and L. Biven, Physica D 152, 520 (2001). [18] N. N. Bogoliubov, Journal of Physics 11, 23 (1947). [19] N.G. Berloff J. Phys. A: Math. and Gen., 37(5), 1617 (2004). [20] Various choices of pc taken from the interval (5, 15) × (2π/N ) give different L, but similar vortex line decay rates. A more accurate, but more computationally inten- sive, way to calculate the vortex line length in a quasi- condensate (a condensate with vortices) is through a time averaging of the field that removes the high frequencies, see the discussion in [7]. [21] R.J.Donnelly “Quantized Vortices in Helium II”, Cam- bridge University Press, Cambridge 1991. [22] S.V.Iordanskii, Sov. Phys. JETP 22, 160 (1966); E. B. Sonin, Sov. Phys. JETP 42 469 (1975); Phys. Rev. B 55, 485 (1997), Ao and Thouless, Phys. Rev. Lett. 70, 2158 (1993). [23] M. Leadbeater, D.C. Samuels, C.F. Barenghi and C.S. Adams, Phys. Rev. A 67 015601 (2003). [24] W.F. Vinen, Proc. R. Soc. London. Ser. A 242, 493 (1957) [25] N.G. Berloff Phys. Rev A, 69 053601 (2004) http://arxiv.org/abs/gr-qc/0612134

0704.1972

Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painleve transcendent

Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlevé transcendent A.R. Its Department of Mathematical Sciences Indiana University – Purdue University Indianapolis Indianapolis IN 46202-3216, U.S.A. [email protected] A.B.J. Kuijlaars and J. Östensson Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200B 3001 Leuven, Belgium [email protected] [email protected] June 3, 2018 Abstract We describe a new universality class for unitary invariant random matrix ensem- bles. It arises in the double scaling limit of ensembles of random n × n Hermitian matrices Z−1n,N |detM |2αe−N Tr V (M)dM with α > −1/2, where the factor |detM |2α induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n,N → ∞ such that n2/3(n/N − 1) = O(1). We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|2αe−NV (x). Our main attention is on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution of the Painlevé XXXIV equation. This solution is related to a particular solution of the Painlevé II equation, which however is different from the usual Hastings-McLeod solution. 2000 Mathematics Subject Classification: 15A52, 33E17, 34M55 http://arxiv.org/abs/0704.1972v1 2 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON 1 Introduction and statement of results 1.1 Unitary random matrix models For n ∈ N, N > 0, and α > −1/2, we consider the unitary random matrix ensemble Z−1n,N | detM |2αe−N TrV (M) dM, (1.1) on the space M(n) of n× n Hermitian matrices M , where V is real analytic and satisfies V (x) log(x2 + 1) = +∞. (1.2) This is a unitary random matrix ensemble in the sense that it is invariant under conju- gation, M 7→ UMU−1, by unitary matrices U . As is well-known [11, 38], it induces the following probability density on the n eigenvalues x1, . . . , xn of M P (n,N)(x1, . . . , xn) = Ẑ |xj|2αe−NV (xj) |xi − xj |2. (1.3) The eigenvalue distribution is determinantal with kernel Kn,N built out of the polynomials pj,N(x) = κj,N x j + · · · , κj,N > 0, orthonormal with respect to the weight |x|2αe−NV (x) on R. Indeed, as shown by Dyson, Gaudin, and Mehta, see e.g. [11, 21, 38], for any m = 1, . . . , n− 1, the m-point correlation function R(n,N)m (x1, . . . , xm) ≡ (n−m)! · · · P (n,N)(x1, . . . , xn) dxm+1 · · · dxn (1.4) is given by R(n,N)m (x1, . . . , xm) = det (Kn,N(xi, xj))1≤i,j≤m , (1.5) where Kn,N(x, y) = |x|α|y|αe− N(V (x)+V (y)) pj,N(x) pj,N(y). (1.6) In the limit n,N → ∞, n/N → 1, the global eigenvalue regime is determined by V as follows. The equilibrium measure µV for V is the unique minimizer of IV (µ) = |x− y|dµ(x)dµ(y) + V (x)dµ(x) (1.7) taken over all Borel probability measures µ on R. Since V is real analytic we have that µV is supported on a finite union of disjoint intervals [13], and it has a density ρV such n,N→∞,n/N→1 Kn,N(x, x) = ρV (x), x 6= 0. The limiting mean eigenvalue density is independent of α. CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 3 The factor | detM |2α changes the local eigenvalue behavior near 0. This is reflected in the local scaling limits of Kn,N around 0 that do depend on α. If 0 is in the bulk of the spectrum and ρV (0) > 0, then instead of the usual sine kernel we get a Bessel kernel depending on α [37]. If 0 is in the bulk and ρV (0) = ρ V (0) = 0, ρ V (0) > 0, then the local scaling limits of the kernel near 0 are associated with the Hastings-McLeod solution of the Painlevé II equation q′′ = sq + 2q3 − α [9]. In this paper we study the effect of α in case 0 is an endpoint of the spectrum which is such that the density ρV vanishes like a square root at 0. For α = 0 the scaling limit is the well-known Airy kernel, see the papers [6, 26, 39, 42] and also [3, 12], and so we are asking the question: What is the α-generalization of the Airy kernel? For α > −1/2, we have found a new one-parameter family of limiting kernels as stated in Theorem 1.1 below. In Theorem 1.1 we also assume that the eigenvalue density ρV is regular, which means the following. • The function x 7→ 2 log |x − s|ρV (s)ds − V (x) defined for x ∈ R, assumes its maximum value only on the support of ρV . • The density ρV is positive on the interior of its support. • The density ρV vanishes like a square root at each of the endpoints of its support. Theorem 1.1 For every α > −1/2, there exists a one-parameter family of kernelsKedgeα (x, y; s) such that the following holds. Let V be a real analytic external field on R such that its mean limiting eigenvalue density ρV is regular. Suppose that 0 is a right endpoint of the support of ρV so that for some constant c1 = c1,V > 0 ρV (x) ∼ |x|1/2 as x→ 0− . (1.8) Then there exists a second constant c2 = c2,V > 0 such that n,N→∞ (c1n)2/3 (c1n)2/3 (c1n)2/3 = Kedgeα (x, y; s) (1.9) whenever n,N → ∞ such that = L ∈ R (1.10) and s = −c2,V L. For α = 0, the limiting kernels reduce to the kernel 0 (x, y; s) = Ai(x+ s)Ai′(y + s)− Ai′(x+ s)Ai(y + s) x− y , (1.11) which is the (shifted) Airy kernel from random matrix theory mentioned above, see also Subsection 4.1 below. For α 6= 0, a new type of special functions is needed to describe the limiting kernel Kedgeα (x, y; s). This description is given in the next subsections. 4 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Figure 1: Contour for the model RH problem. 1.2 The model RH problem We describe Kedgeα (x, y; s) through the solution of a special Riemann-Hilbert (RH) prob- lem, that we will refer to as the model RH problem. The model RH problem is posed on a contour Σ in an auxiliary ζ-plane, consisting of four rays Σ1 = {arg ζ = 0}, Σ2 = {arg ζ = 2π/3}, Σ3 = {arg ζ = π}, and Σ4 = {arg ζ = −2π/3} with orientation as shown in Figure 1. As usual in RH problems, the orientation defines a + and a − side on each part of the contour, where the +-side is on the left when traversing the contour according to its orientation. For a function f on C \ Σ, we use f± to denote its limiting values on Σ taken from the ±-side, provided such limiting values exist. The contour Σ divides the complex plane into four sectors Ωj also shown in the figure. The model RH problem reads as follows. Riemann-Hilbert problem for Ψα (a) Ψα : C \ Σ → C2×2 is analytic. (b) Ψα,+(ζ) = Ψα,−(ζ) , for ζ ∈ Σ1, Ψα,+(ζ) = Ψα,−(ζ) e2απi 1 , for ζ ∈ Σ2, Ψα,+(ζ) = Ψα,−(ζ) , for ζ ∈ Σ3, Ψα,+(ζ) = Ψα,−(ζ) e−2απi 1 , for ζ ∈ Σ4. (c) Ψα(ζ) = ζ −σ3/4 1√ (I + O(1/ζ1/2))e−( ζ3/2+sζ1/2)σ3 as ζ → ∞. Here σ3 = ( 1 00 −1 ) is the third Pauli matrix. (d) Ψα(ζ) = O |ζ |α |ζ |α |ζ |α |ζ |α as ζ → 0, if α < 0; and CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 5 Ψα(ζ) = |ζ |α |ζ |−α |ζ |α |ζ |−α as ζ → 0 with ζ ∈ Ω1 ∪ Ω4, |ζ |−α |ζ |−α |ζ |−α |ζ |−α as ζ → 0 with ζ ∈ Ω2 ∪ Ω3, if α ≥ 0. Here, and in what follows, the O-terms are taken entrywise. Note that the RH problem depends on a parameter s through the asymptotic condition at infinity. If we want to emphasize the dependence on s we will write Ψα(ζ ; s) instead of Ψα(ζ). The model RH problem is not uniquely solvable. Indeed, if Ψα is a solution, then( Ψα is also a solution for any η = η(s), and it turns out that this is the only freedom we have (see Proposition 2.1). Theorem 1.2 The model RH problem is solvable for every s ∈ R. Let Ψα be a solution of the model RH problem and put ψ1(x; s) ψ2(x; s) Ψα,+(x; s) , for x > 0, Ψα,+(x; s)e −απiσ3 , for x < 0. (1.12) Then the limiting kernel Kedgeα (x, y; s) can be written in the “integrable form” Kedgeα (x, y; s) = ψ2(x; s)ψ1(y; s)− ψ1(x; s)ψ2(y; s) 2πi(x− y) . (1.13) The function ψ2 depends on the particular choice of solution Ψα to the model RH problem. Indeed, for any η we have that the mapping Ψα 7→ Ψα leaves ψ1 invariant and changes ψ2 to ψ2 + ηψ1. However, this does not change the expression (1.13) for the kernel Kedgeα (x, y; s). It follows from (1.12) and part (c) of the model RH problem that ψ1 and ψ2 have the asymptotic behavior ψ1(x; s) = 2x1/4 x3/2−sx1/2(1 +O(x−1/2)), (1.14) ψ2(x; s) = ix1/4√ x3/2−sx1/2(1 +O(x−1/2)), (1.15) as x→ +∞, and ψ1(x; s) = 2|x|−1/4 cos |x|3/2 − s|x|1/2 − απ − π/4 +O(x−3/4), (1.16) ψ2(x; s) = −i 2|x|1/4 sin |x|3/2 − s|x|1/2 − απ − π/4 +O(x−1/4), (1.17) as x→ −∞. Remark 1.3 The kernel Kedgeα (x, y; s) describes an edge effect for the random matrix ensemble (1.1). If we assume that 0 is the rightmost point in the support of ρV , and if given M we let λmax(M) denote its largest eigenvalue, then it follows under the assumptions of Theorem 1.1, in particular the limit assumption (1.10), that n,N→∞ (c1n) 2/3λmax ≤ t = det 1− Kα,s|(t,∞) , (1.18) 6 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON where Kα,s|(t,∞) is the trace class operator in L2(t,∞) with kernel Kedgeα (x, y; s). To prove (1.18) one must show that the operator with kernel 1 (c1n)2/3 (c1n)2/3 (c1n)2/3 converges in the trace class norm on L2(t,∞) to the operator with kernel Kedgeα (x, y; s). This requires good estimates on the rate of convergence in (1.9), which can be established as in [12]. For α = 0, the kernel is the (shifted) Airy kernel, and the Fredholm determinant (1.18) has an equivalent expression in terms of a special solution of the Painlevé II equation. The resulting distribution is the famous Tracy-Widom distribution [42, 43]. It would be very interesting to find an analogous expression for general α. The connection to the model RH problem given in Theorem 1.2 can be used in obtaining such an expression, following the approach of [5] and [27]. We are planning to address this question in a future publication. 1.3 Connection with the Painlevé XXXIV equation The model RH problem is related to a special solution of the equation number XXXIV from the list of Painlevé and Gambier [29], u′′ = 4u2 + 2su+ (u′)2 − (2α)2 . (1.19) All solutions of (1.19) are meromorphic in the complex plane. Theorem 1.4 Let Ψα(ζ ; s) be a solution of the model RH problem. Then u(s) = −s − i d Ψα(ζ ; s)e ( 23 ζ 3/2+sζ1/2)σ3 1√ ζσ3/4 (1.20) exists and satisfies (1.19). The function (1.20) is a global solution of (1.19) (i.e., it does not have poles on the real line), and it does not depend on the particular solution of the model RH problem. The connection with the Painlevé XXXIV equation leads to the following characteri- zation of ψ1 and ψ2. Theorem 1.5 Let u be the solution of Painlevé XXXIV given by (1.20). Then there exists a solution Ψα of the model RH problem so that the functions ψ1 and ψ2 defined by (1.12) satisfy the following system of linear differential equations ψ1(x; s) ψ2(x; s) u′/(2x) i− iu/x −i(x+ s+ u+ ((u′)2 − (2α)2)/(4ux)) −u′/(2x) ψ1(x; s) ψ2(x; s) (1.21) and have asymptotics (1.14)–(1.17). In fact we will prove that for ζ ∈ C \ Σ, Ψα(ζ ; s) = u′/(2ζ) i− iu/ζ −i(ζ + s+ u+ ((u′)2 − (2α)2)/(4uζ)) −u′/(2ζ) Ψα(ζ ; s) (1.22) CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 7 from which (1.21) readily follows in view of (1.12). We emphasize that (1.21) and (1.22) hold for one particular solution of the model RH problem. Any other solution( Ψα(ζ ; s) also satisfies a system of linear differential equations, but with matrix u′/(2ζ) i− iu/ζ −i(ζ + s+ u+ ((u′)2 − (2α)2)/(4uζ)) −u′/(2ζ) . (1.23) In order to make Theorem 1.5 a genuine, i.e., independent of the Ψα RH problem, characterization of ψ1 and ψ2, we need an independent of formula (1.20) characterization of the solution u(s) of equation (1.19). This can be achieved by indicating the asymptotic behavior of u(s) as s → ∞, cf. the characterization of the Hastings-McLeod solution of Painlevé II equation [28]. We discuss this issue in detail in the last section of the paper, see in particular Proposition 4.1 and the end of Remark 4.2 where the possible asymptotic characterizations of the solution u(s) are given. 1.4 Overview of the rest of the paper In Section 2 we give the proofs of Theorem 1.1 and Theorem 1.2. We start by presenting the RH problem for orthogonal polynomials on the line [23]. The eigenvalue correlation kernel Kn,N can be explicitly expressed in terms of the solution of this RH problem [11, 15]. As in earlier papers, see e.g. [3, 4, 8, 9, 14, 15], we apply the Deift-Zhou steepest descent method for RH problems, see [16]. For the local analysis near 0, we need the model RH problem for Ψα(ζ ; s) as introduced in Subsection 1.2. We show, following the methodology of [25], that the model RH problem has a solution for every s ∈ R. Then we follow the usual steps in the steepest descent analysis for RH problems, which lead us to the proofs of Theorem 1.1 and 1.2. Section 3 is devoted to the proofs of Theorem 1.4 and Theorem 1.5. We start by discussing the RH problem, in the form due to Flaschka and Newell, associated with the Painlevé II equation q′′ = sq+2q3− ν. Following [2], we show that for a special choice of monodromy data the Flaschka-Newell RH problem is related to the model RH problem. The parameters in the Painlevé equations are related by ν = 2α + 1/2. The monodromy data corresponds to a solution of Painlevé II which is different from the Hastings-McLeod solution that has appeared more often in random matrix theory [4, 8, 9, 28, 42]. The known results (asymptotics, Lax pair etc.) for the RH problem for Painlevé II are then transferred to the model RH problem, and then used to complete the proofs of Theorems 1.4 and 1.5. In particular it gives rise to the special solution u of the Painlevé XXXIV equation defined by (1.20). In Section 4 we make some concluding remarks. For the important special cases α = 0 and α = 1, we show how the model RH problem can be explicitly solved in terms of Airy functions, and how the limiting kernel Kedgeα (x, y; s) as well as the special Painlevé XXXIV solution u can be explicitly computed in both cases. Our final remarks concern the characterization of u, in the case of general α, through its asymptotic behavior at infinity. 8 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON 2 Proof of Theorem 1.1 and 1.2 2.1 The Riemann-Hilbert problem for orthogonal polynomials The RH problem for orthogonal polynomials on the line, for our particular weight, is the following (cf. [23]). Riemann-Hilbert problem for Y • Y : C \ R → C2×2 is analytic. • Y+(x) = Y−(x) 1 |x|2αe−NV (x) for x ∈ R\{0}, with R oriented from left to right. • Y (z) = (I +O(1/z)) 0 z−n as z → ∞. • If α < 0, then Y (z) = O 1 |z|2α 1 |z|2α as z → 0. If α ≥ 0, then Y (z) = O ( 1 11 1 ) as z → 0. The RH problem has the unique solution Y (z) = pn,N(z) 2πiκn,N pn,N(s)|s|2αe−NV (s) s− z ds −2πi κn−1,N pn−1,N(z) −κn−1,N pn−1,N(s)|s|2αe−NV (s) s− z ds  , (2.1) where pj,N(x) = κj,N x j + · · · is the orthonormal polynomial with respect to the weight |x|2αe−NV (x). By (1.6) and the Christoffel-Darboux formula for orthogonal polynomials, we have Kn,N(x, y) = |x|α|y|αe− N(V (x)+V (y))κn−1,N pn,N(x) pn−1,N(y)− pn−1,N(x) pn,N(y) x− y . (2.2) Thus, using (2.1) and the fact that det Y ≡ 1, we may express the eigenvalue correlation kernel directly in terms of Y : Kn,N(x, y) = 2πi(x− y) |x| α|y|αe− 12N(V (x)+V (y)) Y −1+ (y)Y+(x) . (2.3) The main idea for the proof of Theorems 1.1 and 1.2 is to apply the powerful steepest descent analysis for RH problems of Deift and Zhou [16] to the RH problem satisfied by Y . In the case at hand it consists of constructing a sequence of invertible transformations Y 7→ T 7→ S 7→ R, where the matrix-valued function R is close to the identity. By unfolding the above transformations asymptotics for Y and thus, in view of (2.3), for Kn,N in various regimes may be derived. Our main attention will be devoted to the local behavior of Y near 0. Around 0 we construct a local parametrix with the help of the model RH problem, which we next discuss in more detail. CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 9 2.2 The model RH problem The model RH problem is not uniquely solvable. Proposition 2.1 Let Ψα be a solution of the model RH problem. Then the following hold. (a) detΨα ≡ 1. (b) For any η ∈ R (which may depend on s), we have that Ψα also solves the model RH problem. (c) Any two solutions are related as in part (b), i.e., if Ψ α and Ψ α are two solutions of the model RH problem, then Ψ α for some η = η(s). Proof. (a) We have that detΨα is analytic in C \ {0}, since all jump matrices have determinant one. In case α < 0 we get from condition (d) of the RH problem that detΨα(ζ) = O(|ζ |2α) as ζ → 0. Since 2α > −1 it follows that the singularity at the origin is removable. In case α ≥ 0 we find from condition (d) of the RH problem that detΨα(ζ) = O(1) as ζ → 0 in Ω1 ∪ Ω4. Thus the singularity at the origin cannot be a pole. Since detΨα = O(|ζ |−2α) as ζ → 0, it cannot be an essential singularity either and therefore the singularity at the origin is removable also in this case. Hence detΨα is entire. From condition (c) of the RH problem it follows that detΨα(ζ) → 1 as ζ → ∞, and so part (a) of the proposition follows from Liouville’s theorem. (b) It is clear that Ψα satisfies the conditions (a), (b), and (d) of the model RH problem. To establish (c) it is enough to observe that ζ−σ3/4 = ζ−σ3/4 2ζ1/2 = ζ−σ3/4 (I +O(1/ζ1/2)) as ζ → ∞. (c) In view of part (a) we know that Ψ α is invertible. Then Ψ −1 is analytic in C \ {0} and, by arguments similar to those in the proof of part (a), the singularity at the origin is removable. As ζ → ∞ we get from condition (c) of the model RH problem Ψ(2)α (ζ)(Ψ α (ζ)) −1 = ζ−σ3/4 (I +O(ζ−1/2)) ζσ3/4 = I +O ζ−1/2 ζ−1 1 ζ−1/2 The statement now follows from Liouville’s theorem. ✷ In the following we will need more information about the behavior at the origin of functions satisfying properties (a), (b), and (d) of the model RH problem. The following result is similar to Proposition 2.3 in [9]. Proposition 2.2 Let Ψ satisfy conditions (a), (b), and (d) of the RH problem for Ψα. Then, with all branches being principal, the following hold. 10 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON • If α− 1 /∈ N0, there exist an analytic matrix-valued function E and constant matrices Aj such that Ψ(ζ) = E(ζ) ζασ3 Aj, for ζ ∈ Ωj . (2.4) Letting vj denote the jump matrix for Ψ on Σj, we have A1 = A4 v1, A1 = A2 v2, A3 = A4 v4, (2.5) 2 cosαπ 2 cosαπ −eαπi e−απi  . (2.6) • If α− 1 ∈ N0, then Ψ has logarithmic behavior at the origin: There exist an analytic matrix-valued function E and constant matrices Aj such that Ψ(ζ) = E(ζ) ζα log ζ 0 ζ−α Aj, for ζ ∈ Ωj . (2.7) Letting vj denote the jump matrix for Ψ on Σj, we now have A1 = A4 v1, A1 = A2 v2, A3 = A4 v4, (2.8) 0 e3πi/4 eπi/4 eπi/4  . (2.9) • In all cases it holds that detAj = 1 and (A1)21 = (A4)21 = 0. (2.10) Proof. The statement (2.10) is an immediate consequence of the explicit formulas for the Aj ’s. Consider the case α− 1 /∈ N0. Define E by (2.4), i.e., let E(ζ) = Ψ(ζ)A−1j ζ −ασ3, for ζ ∈ Ωj , (2.11) with Aj as in (2.5), (2.6). Then E is analytic in C \ Σ. We now show that E is indeed entire. The relations (2.5) and the condition (b) of the model RH problem imply that E is analytic also on Σ1 ∪ Σ2 ∪ Σ4. Moreover, on Σ3 E−1− (ζ)E+(ζ) = ζ − A3 v3A Now, by (2.5), (2.6), and straightforward computation A3 v3A 2 = A2 v2 v 1 v4 v3A 2 = e 2απiσ3 = ζ−ασ3− ζ + . (2.12) CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 11 Thus, E is analytic also on Σ3, and therefore in C \ {0}. We next show that the singularity at 0 is removable. If α < 0, we see from (2.11) and the condition (d) of the model RH problem, that as ζ → 0 E(ζ) = O |ζ |α |ζ |α |ζ |α |ζ |α |ζ |−α 0 0 |ζ |α 1 |ζ |2α 1 |ζ |2α so (since 2α > −1) the isolated singularity at 0 is indeed removable. If α ≥ 0 and ζ → 0 in Ω1 we find in the same way (also using (A1)21 = 0) that E(ζ) = O |ζ |α |ζ |−α |ζ |α |ζ |−α |ζ |−α 0 0 |ζ |α so that E is bounded near 0 in Ω1 and thus 0 cannot be a pole. Since 0 cannot be an essential singularity either, we conclude that the singularity is indeed removable. In case α − 1 ∈ N0 the proof is almost identical, only now the equation (2.12) is replaced by A3 v3A 2 = A2 v2 v 1 v4 v3A −1 −2i ζ−α − 1 ζα log ζ ζα log ζ 0 ζ−α . (2.13) 2.3 Existence of solution to the model RH problem We will need that for s ∈ R the model RH problem indeed has a solution. To prove existence of a solution to the model RH problem it suffices to prove existence of a (unique) solution Ψ (spec) α to the special RH problem obtained when the asymptotics (c) at infinity is replaced by the following stronger condition Ψ(spec)α (ζ) = (I +O(1/ζ))ζ −σ3/4 1√ ζ3/2+sζ1/2)σ3 , (2.14) as ζ → ∞. A key element in the proof of unique solvability of the RH problem for Ψ (spec) α is the following vanishing lemma (cf. [25]). Proposition 2.3 (vanishing lemma) Let α > −1/2, s ∈ R, and put θ(ζ) = θ(ζ ; s) = ζ3/2+sζ1/2. Suppose that Fα satisfies the conditions (a), (b), and (d) in the RH problem for Ψα but, instead of condition (c), has the following behavior at infinity: Fα(ζ) = O(1/ζ)ζ −σ3/4 1√ e−θ(ζ)σ3 , (2.15) as ζ → ∞. Then Fα ≡ 0. 12 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Proof. The ideas of the proof are similar in spirit to those in Deift et al. [14]. Let Gα be defined as follows: Gα(ζ) =   Fα(ζ)e θ(ζ)σ3 , for ζ ∈ Ω1, Fα(ζ)e θ(ζ)σ3 e2απie2θ(ζ) 1 , for ζ ∈ Ω2, Fα(ζ)e θ(ζ)σ3 −e−2απie2θ(ζ) 1 , for ζ ∈ Ω3, Fα(ζ)e θ(ζ)σ3 , for ζ ∈ Ω4. (2.16) Then Gα satisfies the following RH problem. Riemann-Hilbert problem for Gα (a) Gα : C \ R → C2×2 is analytic. (b) Gα,+(ζ) = Gα,−(ζ)vGα(ζ) for ζ ∈ R \ {0}, where vGα(ζ) = e−2θ(ζ) −1 , for ζ > 0, 1 −e2απie2θ+(ζ) e−2απie2θ−(ζ) 0 , for ζ < 0. (2.17) (c) Gα(ζ) = O(ζ −3/4) as ζ → ∞. (d) Gα has the following behavior near the origin: If α < 0, Gα(ζ) = O |ζ |α |ζ |α |ζ |α |ζ |α , as ζ → 0, (2.18) and if α ≥ 0, Gα(ζ) = |ζ |−α |ζ |α |ζ |−α |ζ |α as ζ → 0, Im ζ > 0, |ζ |α |ζ |−α |ζ |α |ζ |−α as ζ → 0, Im ζ < 0. (2.19) The jumps in (b) follow from straightforward computations which uses that θ+(ζ) + θ−(ζ) = 0 for ζ < 0. The behavior (c) of Gα at infinity (uniformly in each sector) follows directly from (2.15), (2.16), and the fact that Re θ(ζ) < 0 for ζ ∈ Ω2 ∪ Ω3. The behavior (2.18) at the origin is immediate from the condition (d) of the RH problem for Fα, and so is the behavior (2.19) if ζ → 0 with ζ ∈ Ω1 ∪ Ω4. To prove (2.19) if ζ → 0 with ζ ∈ Ω2 ∪ Ω3, we need Proposition 2.2. Consider first the case α − 12 /∈ N0 and ζ ∈ Ω2. CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 13 Then we have, using (2.16), (2.4), (2.5), and (2.10) Gα(ζ) = Fα(ζ)e θ(ζ)σ3 e2απie2θ(ζ) 1 = E(ζ)ζασ3A2 e2απi 1 eθ(ζ)σ3 = E(ζ)ζασ3A2 e2απi 1 e−θ(ζ)σ3 = E(ζ)ζασ3A1 e−θ(ζ)σ3 = E(ζ)ζασ3 e−θ(ζ)σ3 , where ∗ denotes an unspecified constant. Using the boundedness of E and θ at the origin, we find (2.19) as ζ → 0 in the sector Ω2. The case ζ ∈ Ω3 is treated similarly. Using (2.7), (2.8) instead of (2.4), (2.5), the same argument works in case α − 1 ∈ N0. Note that in spite of the logarithmic entry in (2.8), there are no logarithmic entries in (2.19). Introduce the auxiliary matrix-valued function Hα(ζ) = Gα(ζ) (Gα(ζ̄)) ∗, ζ ∈ C \ R. (2.20) Then Hα is analytic and Hα(ζ) = O(ζ −3/2), as ζ → ∞. (2.21) From the condition (d) in the RH problem for Gα it follows that Hα has the following behavior near the origin: Hα(ζ) = |ζ |2α |ζ |2α |ζ |2α |ζ |2α as ζ → 0, in case α < 0, as ζ → 0, in case α ≥ 0. (2.22) Since α > −1/2, we see from (2.21) and (2.22) that each entry of Hα,+ is integrable over the real line, and by Cauchy’s theorem and (2.21) Hα,+(ζ) dζ = 0. (2.23) That is, by (2.20), Gα,+(ζ) (Gα,−(ζ)) ∗ dζ = 0. (2.24) Adding (2.24) to its Hermitian conjugate and using (2.17) we obtain Gα,−(ζ) [vGα(ζ) + (vGα(ζ)) ∗] (Gα,−(ζ)) Gα,−(ζ) (Gα,−(ζ)) ∗ dζ + Gα,−(ζ) 2e−2θ(ζ) 0 (Gα,−(ζ)) ∗ dζ. (2.25) 14 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Here we also used that θ+(ζ) = −θ−(ζ) ∈ iR for ζ < 0, which holds because s is real. The identity (2.25) implies that the first column of Gα,− vanishes identically on R. Thus, in view of the form of the jump matrix in (2.17), the second column of Gα,+ vanishes identically on R as well. It follows that the first column of Gα vanishes identically in the lower half-plane, and the second column vanishes identically in the upper half-plane. To prove that the full matrix Gα vanishes identically in both half-planes, we shall use a Phragmen-Lindelöf type theorem due to Carlson [7, 41]. Define for j = 1, 2, gj(ζ) = (Gα)j1(ζ), for Im ζ > 0, (Gα)j2(ζ), for Im ζ < 0. (2.26) The conditions of the RH problem for Gα yield that both g1 and g2 have analytic contin- uation across (0,∞) and that they are both solutions of the following scalar RH problem. Riemann-Hilbert problem for g • g : C \ (−∞, 0] → C is analytic with jump g+(ζ) = g−(ζ) e −2απie2θ−(ζ), for ζ ∈ (−∞, 0). (2.27) • g(ζ) = O(ζ−3/4) as ζ → ∞. • g(ζ) = O(|ζ |−|α|) as ζ → 0. We are going to prove that this RH problem has only the trivial solution. Let g be any solution and define ĝ by ĝ(ζ) = g(ζ2), if Re ζ > 0, g(ζ2)e−2απie−2( ζ3+sζ), if Re ζ < 0, Im ζ > 0, g(ζ2)e2απie−2( ζ3+sζ), if Re ζ < 0, Im ζ < 0. (2.28) The jump property (2.27) ensures that ĝ is analytic across the imaginary axis. Now define h(ζ) = 1 + ζ ĝ(ζ4/3), for Re ζ ≥ 0, (2.29) with (as usual) the principal branches of the fractional powers. Then it can be checked that h is analytic in Re ζ > 0, bounded for Re ζ ≥ 0, and satisfies |h(ζ)| ≤ Ce−c|ζ|4, if ζ ∈ iR, for some positive constants c and C. Hence, by Carlson’s theorem, h ≡ 0 in Re ζ ≥ 0. Therefore g ≡ 0, and so g1 and g2 are both identically zero. It follows that the full matrix Gα vanishes identically in both half-planes. Thus Fα ≡ 0 by (2.16), and this completes the proof of the proposition. ✷ We now show how (unique) solvability of the RH problem for Ψ (spec) α can be deduced from the above vanishing lemma. CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 15 Proposition 2.4 The RH problem for Ψ (spec) α has a unique solution for every s ∈ R. Proof. The idea of the proof is this: Given a solution Ψ (spec) α to the above RH prob- lem, we show how to construct a solution mα to a certain normalized RH problem (i.e., mα(ζ) → I as ζ → ∞) characterized by a jump matrix v on a contour Σ̃, and con- versely. To prove the proposition it therefore suffices to prove (unique) solvability of the normalized RH problem. This can be done by utilizing the basic relationship between normalized RH problems and singular integral equations. We recall briefly, in our setting, some standard facts regarding this relationship. For further details, and proofs, the reader is referred to the papers [17, 18, 19, 44], and to the appendix of [32]. Let C denote the Cauchy operator Ch(ζ) = s− ζ ds, h ∈ L 2(Σ̃), ζ ∈ C \ Σ̃, (2.30) and denote by C±h(ζ), ζ ∈ Σ̃, the limits of Ch(ζ ′) as ζ ′ → ζ from the (±)-side of Σ̃. The operators C± are bounded on L 2(Σ̃). Let v(ζ) = (v−(ζ)) −1v+(ζ), ζ ∈ Σ̃, (2.31) be a pointwise factorization of v(ζ) with v±(ζ) ∈ GL(2,C), and define ω± through v±(ζ) = I ± ω±(ζ), ζ ∈ Σ̃. (2.32) Our choice of factorization will imply that ω± ∈ L2(Σ̃) ∩ L∞(Σ̃). The singular integral operator Cω : L 2(Σ̃) → L2(Σ̃), defined by Cωh = C+(hω−) + C−(hω+), h ∈ L2(Σ̃), (2.33) is then bounded on L2(Σ̃). Moreover, it makes sense to study the singular integral equation (1− Cω)µ = I (2.34) for µ ∈ I + L2(Σ̃). For if we write µ = I + h, then (2.34) takes the form (1− Cω)h = CωI ∈ L2(Σ̃). (2.35) Suppose that µ ∈ I + L2(Σ̃) is a solution of (2.34). Then, indeed mα(ζ) = I + C(µ(ω+ + ω−))(ζ), ζ ∈ C \ Σ̃, (2.36) solves the normalized RH problem. Thus, if we can prove that the operator 1 − Cω is a bijection in L2(Σ̃), then solvability of the RH problem for mα, and hence of that for (spec) α , has been established. Bijectivity of 1 − Cω in L2(Σ̃) is proved in two steps. We first show that, for an appropriate choice of ω = (ω−, ω+) in the above factorization, 1− Cω is Fredholm in L2(Σ̃) with index 0. Second, we show that the kernel of 1− Cω is trivial. Now, it is a standard fact that ker (1 − Cω) = {0} if and only if the associated homogeneous RH problem (for say m0α) has only the trivial solution. But the explicit relation between Ψ (spec) α and mα also establishes a relation between solutions Fα and m 16 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Figure 2: Contour for the RH problem for mα. of the associated homogeneous RH problems. In view of Proposition 2.3, which states that Fα ≡ 0, the second step has thus already been accomplished. We now establish the above mentioned relation between Ψ (spec) α and mα, derive the RH problem satisfied by mα, and finally show that a factorization of v may be chosen so that 1− Cω is Fredholm with index 0, cf. [25]. Let D = {ζ ∈ C | |ζ | < 1}. Set θ(ζ) = 2 ζ3/2 + sζ1/2 and mα(ζ) = (spec) α (ζ)A ζ−α −κα ζα log ζ , for ζ ∈ Ωj ∩ D, (spec) α (ζ)e θ(ζ)σ3 1√ ζσ3/4, for ζ ∈ Ωj ∩ D (2.37) with {Aj}4j=1 being the matrices in Proposition 2.2, and where κα = 1 if α−1/2 ∈ N0 and 0 otherwise. By Proposition 2.2 it follows that mα is analytic in D. Let Σ̃ = Σ ∪ ∂D and orient the components of Σ̃ as in Figure 2. This makes Σ̃ a complete contour, meaning that C\Σ̃ can be expressed as the union of two disjoint sets, C\Σ̃ = Ω+∪Ω−, Ω+∩Ω− = ∅, such that Σ̃ is the positively oriented boundary of Ω+ and the negatively oriented boundary of Ω−. Let Σ̃j = Ωj ∩ ∂D. Computations show that mα satisfies the following normalized RH problem. As in Proposition 2.2 we use vj to denote the jump matrix on Σj in the model RH problem. Riemann-Hilbert problem for mα • mα : C \ Σ̃ → C2×2 is analytic. • mα,+(ζ) = mα,−(ζ)v(ζ) for ζ ∈ Σ̃, where CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 17 v(ζ) =   I, for ζ ∈ Σ̃ ∩ D, ζ−σ3/4 1√ e−θσ3vje θσ3 1√ ζσ3/4, for ζ ∈ Σj ∩ D , j ∈ {1, 2, 4}, I, for ζ ∈ Σ3 ∩ D ζα κα ζα log ζ 0 ζ−α θσ3 1√ ζσ3/4, for ζ ∈ Σ̃j , j ∈ {1, 3}, ζ−σ3/4 1√ e−θσ3A−1j ζ−α −κα ζα log ζ , for ζ ∈ Σ̃j , j ∈ {2, 4}. • mα(ζ) = I +O(1/ζ) as ζ → ∞. The analyticity of mα on Σ3 ∩ D follows since θ+(ζ) + θ−(ζ) = 0 for ζ < 0. It is important to note that v(ζ)− I decays exponentially as ζ → ∞ along Σ̃. Next observe that, at any of the points 0, A, B, C,D of self-intersection of Σ̃ (see Figure 2), precisely four contours come together. At a fixed point of self-intersection, say P , order the contours that meet at P counterclockwise, starting from any contour that is oriented outwards from P . Denoting the limiting value of the jump matrices over the jth contour at P by v(j)(P ), we then have the cyclic relation v(1)(P ) v(2)(P ) v(3)(P ) v(4)(P ) = I. (2.38) This is trivial in case P = 0, and follows by direct computation in the other cases. We remark that the cyclic relation (2.38) at C is a consequence of the relation (2.12) in the case α − 1/2 6∈ N0, and of (2.13) in the case α − 1/2 ∈ N0 (see the proof of Proposition 2.2). Outside small neighborhoods of the points of self-intersection we choose the trivial factorization v+ = v, v− = I in (2.31), so that ω+ = v − I, ω− = 0 by (2.32). Using the cyclic relations (2.38), we are then able to choose a factorization of v in the remaining neighborhoods in such a way that ω+ is continuous along the boundary of each connected component of Ω+, and similarly, ω− is continuous along the boundary of each connected component of Ω−. The exponential decay of v(ζ)− I as ζ → ∞ ensures that ω± ∈ L2(Σ̃)∩L∞(Σ̃). From this it follows that 1− Cω is Fredholm in L2(Σ̃). Indeed, set ω̃− = I − v−1− , ω̃+ = v−1+ − I. (2.39) The choice of ω̃ = (ω̃−, ω̃+) is motivated by the relations ω̃− ω− = ω̃− + ω−, ω̃+ ω+ = −(ω̃+ + ω+). (2.40) A direct calculation, using C+ − C− = 1 and (2.40), shows that (1− Cω)(1− Ceω) = 1 + T, (2.41) where Tf = C+((C−[f(ω̃+ + ω̃−)])ω−) + C−((C+[f(ω̃+ + ω̃−)])ω+) (2.42) 18 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON for f ∈ L2(Σ̃). Standard computations, using continuity of the functions ω+ resp. ω− along the boundary of each connected component of Ω+ resp. Ω−, show that T is compact in L2(Σ̃). Similar computations show that (1− Ceω)(1− Cω) = 1 + S, with S compact in L2(Σ̃). So 1− Ceω is a pseudoinverse for 1− Cω, which is therefore Fredholm in L2(Σ̃). It follows from general theory that the index of the operator 1−Cω equals the winding number of det v along Σ̃, the latter being defined in the natural way. Now, since det v ≡ 1, this is trivially zero. This completes the proof of Proposition 2.4. ✷ Remark 2.5 The RH problem for Ψ (spec) α is indeed solvable for all s ∈ C \D, where D is a discrete set in C (disjoint from R according to Proposition 2.4), and the solution Ψ (spec) is meromorphic in s with poles in D. To see this, we first observe that the factorization (2.31), (2.32) can be done so that ω± are both analytic in s. It follows that s 7→ 1−Cω is an analytic map taking values in the Fredholm operators on L2(Σ̃). Since we know that 1−Cω is invertible for s ∈ R, we then get, by a version of the analytic Fredholm theorem [44], that µ defined by (2.34) is meromorphic. Thus mα and hence Ψ (spec) α is meromorphic in s. 2.4 Some preliminaries on equilibrium measures Before we embark on the steepest descent analysis for the RH problem of Subsection 2.1, we recall certain properties of equilibrium measures, see [11, 40]. We use the following notation: , Vt(x) = V (x). (2.43) As explained in the Introduction, we are interested in the case where n/N → 1 as n,N → ∞, which means that we are interested in t close to 1. For every t we consider the energy functional IVt(µ) as in (1.7), and its minimizer µt. The equilibrium measure dµt = ρt dx is characterized by the following Euler-Lagrange variational conditions: There is a constant lt ∈ R such that log |x− s|ρt(s) ds− Vt(x) + lt = 0, x ∈ suppµt, (2.44) log |x− s|ρt(s) ds− Vt(x) + lt ≤ 0, x ∈ R \ supp µt. (2.45) For t = 1, we have that the support of µV consists of a finite union of disjoint intervals, see [13], say supp µV = [aj , bj ] with a1 < b1 < a2 < · · · < ak < bk. Due to the assumption that the density ρV of µV is regular, we have the following proposition. Proposition 2.6 For every t in an interval around 1, we have that the density ρt of µt is regular, and that supp µt consists of k intervals, say supp µt = [aj(t), bj(t)] CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 19 with a1(t) < b1(t) < a2(t) < · · · < ak(t) < bk(t). In this interval around 1, the functions t 7→ aj(t) and t 7→ bj(t) are real analytic with a′j(t) < 0 and b′j(t) > 0. Proof. See Theorem 1.3 (iii) and Lemma 8.1 of [36]. ✷ For the rest of the proof of Theorem 1.1 we shall assume that suppµV consists of one interval. In the general case (when suppµV consists of k ≥ 2 intervals) one proceeds analogously, but the parametrix away from the end points given in Subsection 2.5.4 must then instead be constructed with the help of the θ-function of B-periods for the two- sheeted Riemann surface y2 = Πkj=1[(z − aj)(z − bj)] obtained by gluing together two copies of the slit plane C \ j=1[aj , bj] in the standard way [14, 37]. Since the formulas will be more complicated in the multi-interval case, but do not contribute to the main issue of the present paper, we chose to give the proof in full for the one-interval case only. 2.5 Steepest descent analysis 2.5.1 Preliminaries We assume from now on that k = 1, so that supp µV consists of one interval which we take as supp(µV ) = [a, 0], a < 0. Then there is δ1 > 0 such that µt is supported on one interval [at, bt] for every t ∈ (1− δ1, 1 + δ1), and its density ρt is regular. Hence ρt is positive on (at, bt) and vanishes like a square root at the end points, and it takes the form [14] ρt(x) = (bt − x)(x− at)ht(x), for x ∈ [at, bt], (2.46) where ht is positive on [at, bt], and analytic in the domain of analyticity of V . In addition, ht depends analytically on t ∈ (1− δ1, 1 + δ1). We are going to use the equilibrium measure µt in the first transformation of the RH problem. We remark that in [8, 9, 10, 20] a modified equilibrium measure was used in the steepest descent analysis of a RH problem at a critical point. It is likely that we could have modified the equilibrium measure in the present situation as well, but the approach with the unmodified µt also works, as we will see, and we chose to use it in this paper. In the one-interval case one can show by explicit computation that at = − t(bt − at)ht(at) t(bt − at)ht(bt) , (2.47) which indeed shows that d at < 0 and bt > 0. It follows that bt > 0 for t ∈ (1, 1 + δ1) and bt < 0 for t ∈ (1− δ1, 1). In both cases we have at < 0. We introduce two functions ϕt and ϕ̃t as follows. For z ∈ C \ (−∞, bt] lying in the domain of analyticity of V (which we may restrict to be simply connected, without loss of generality), we put ϕt(z) = ((s− bt)(s− at))1/2ht(s) ds, (2.48) 20 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON and for z ∈ C \ [at,∞) also in the domain of analyticity of V , ϕ̃t(z) = ((s− bt)(s− at))1/2ht(s) ds. (2.49) It follows from (2.48) that ϕt(z) = −at ht(bt)(z − bt)3/2χt(z), (2.50) where χt is analytic in a neighborhood of bt and χt(bt) = 1. Taking ft(z) = ϕt(z) −at ht(bt) (z − bt)χ2/3t (z), (2.51) we see that ft is analytic in a neighborhood of bt with ft(bt) = 0, f ′t(bt) = −at ht(bt) 6= 0, (2.52) and ft(z) real for real values of z. Hence, in particular, ft(0) > 0, if t < 1, f1(0) = 0, and ft(0) < 0, if t > 1. (2.53) Moreover, ft → f1 as t → 1, uniformly in a neighborhood of 0. We choose a small disc U (0) around 0 and δ2 > 0 sufficiently small, so that ft is a conformal map from U (0) onto a convex neighborhood of 0 for every t ∈ (1− δ2, 1 + δ2). Similarly, there exists a disc U (a) centered at a < 0, and a δ3 > 0, so that f̃t(z) = ϕ̃t(z) (2.54) is a conformal map from U (a) onto a convex neighborhood of 0 for every t ∈ (1−δ3, 1+δ3). We let δ0 = min(δ1, δ2, δ3) and we fix t ∈ (1− δ0, 1 + δ0). In what follows we also take the neighborhoods U (0) and U (a) as above. 2.5.2 First transformation Y 7→ T We introduce the so-called g-function: gt(z) = log(z − s) dµt(s) = log(z − s) ρt(s) ds, z ∈ C \ (−∞, bt], (2.55) where log denotes the principal branch. Then gt is analytic in C \ (−∞, bt]. Define T by T (z) = e nltσ3 Y (z) e− nltσ3 e−ngt(z)σ3 , z ∈ C \ R, (2.56) where lt is the constant from (2.44)–(2.45). By a straightforward calculation it then follows that T has the following jump matrix vT on R (oriented from left to right): vT (x) = e−n(gt,+(x)−gt,−(x)) |x|2α en(gt,+(x)+gt,−(x)−Vt(x)+lt) 0 en(gt,+(x)−gt,−(x)) . (2.57) Because of the identities, see [11, 14], gt,+(x) + gt,−(x)− Vt(x) + lt = −2ϕt(x), for x > bt, (2.58) gt,+(x) + gt,−(x)− Vt(x) + lt = −2ϕ̃t(x), for x < at, (2.59) we see that the RH problem for T is the following. CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 21 Figure 3: Opening of a lens around [at, 0]. Riemann-Hilbert problem for T • T : C \ R → C2×2 is analytic. • T+(x) = T−(x) vT (x) for x ∈ R, with vT (x) =   1 |x|2α e−2nϕ̃t(x) , for x < at, e2nϕt,+(x) |x|2α 0 e2nϕt,−(x) , for x ∈ (at, bt), 1 |x|2α e−2nϕt(x) , for x > bt. • T (z) = I +O(1/z) as z → ∞. • If α < 0, then T (z) = O 1 |z|2α 1 |z|2α as z → 0. If α ≥ 0, then T (z) = O ( 1 11 1 ) as z → 0. 2.5.3 Second transformation T 7→ S The opening of lenses is based on the following factorization of vT on (at, bt): vT (x) = e2nϕt,+(x) |x|2α 0 e2nϕt,−(x) |x|−2α e2nϕt,−(x) 1 0 |x|2α −|x|−2α 0 |x|−2α e2nϕt,+(x) 1 Introduce a lens around the segment [at, 0] as in Figure 3 (recall that at < 0). In the disc U (0) around 0 we take the lens such that z 7→ ζ = ft(z)− ft(0), see (2.51), maps the parts of the upper and lower lips of the lens that are in U (0) into the rays arg ζ = 2π/3 and arg ζ = −2π/3, respectively. Similarly, in the disc U (a) we choose the lens so that z 7→ ζ = f̃t(z), see (2.54), maps the parts of the upper and lower lips of the lens that are in U (a) into the rays arg ζ = π/3, and arg ζ = −π/3, respectively. The remaining parts of the lips of the lens are arbitrary. However, they should be contained in the domain of analyticity of V , and we take them so that Reϕt(z) < −c < 0 for z on the lips of the lens outside U (0) and U (a), with c > 0 independent of t. It is important to note that the lens is around [at, 0], and not around [at, bt]. 22 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Define S by S(z) =   T (z), for z outside the lens, T (z) (−z)−2α e2nϕt(z) 1 , for z in the upper part of the lens, T (z) (−z)−2α e2nϕt(z) 1 , for z in the lower part of the lens. (2.60) Here the map z 7→ (−z)−2α is defined with a cut along the positive real axis. Then, from (2.60) and the RH problem for T , we find that S is the unique solution of the following RH problem. Riemann-Hilbert problem for S • S : C \ΣS → C2×2 is analytic, where ΣS consists of the real line and the upper and lower lips of the lens, with orientation as in Figure 3. • S+(z) = S−(z) vS(z) for z ∈ ΣS, where vS is given as follows. For t < 1, so that bt < 0, we have vS(z) =   1 |z|2α e−2nϕ̃t(z) , for z ∈ (−∞, at), 0 |z|2α −|z|−2α 0 , for z ∈ (at, bt), 0 |z|2α e−2nϕt(z) −|z|−2α e2nϕt(z) 0 , for z ∈ (bt, 0), 1 |z|2α e−2nϕt(z) , for z ∈ (0,∞), (−z)−2α e2nϕt(z) 1 , for z on both lips of the lens, while, for t ≥ 1, so that bt ≥ 0, we have vS(z) =   1 |z|2α e−2nϕ̃t(z) , for z ∈ (−∞, at), 0 |z|2α −|z|−2α 0 , for z ∈ (at, 0), e2nϕt,+(z) |z|2α 0 e2nϕt,−(z) , for z ∈ (0, bt), 1 |z|2α e−2nϕt(z) , for z ∈ (bt,∞), (−z)−2α e2nϕt(z) 1 , for z on both lips of the lens. • S(z) = I +O(1/z) as z → ∞. • If α < 0, then S(z) = O 1 |z|2α 1 |z|2α as z → 0. If α ≥ 0, then S(z) = O ( 1 11 1 ) as z → 0 from outside the lens and S(z) = O |z|−2α 1 |z|−2α 1 as z → 0 from inside the lens. CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 23 The next step is to approximate S by a parametrix P , consisting of three parts P (∞), P (a), and P (0): P (z) = P (0)(z), for z ∈ U (0) \ ΣS, P (a)(z), for z ∈ U (a) \ ΣS , P (∞)(z), for z ∈ C \ (U (0) ∪ U (a) ∪ (at, 0)), (2.61) where U (a) and U (0) are small discs centered at a and 0, respectively, that have been introduced before. The parametrices P (∞), P (a) and P (0) are constructed below. 2.5.4 The parametrix P (∞) The parametrix P (∞) is a solution of the following RH problem. Riemann-Hilbert problem for P (∞) • P (∞) : C \ [at, 0] → C2×2 is analytic. • P (∞)+ (x) = P − (x) 0 |x|2α −|x|−2α 0 for x ∈ (at, 0), oriented from left to right. • P (∞)(z) = I +O(1/z) as z → ∞. The RH problem for P (∞) can be explicitly solved as in [9]. Take D(z) = zα φ 2z − at , for z ∈ C \ [at, 0], (2.62) where φ(z) = z + (z − 1)1/2 (z + 1)1/2 is the conformal map from C \ [−1, 1] onto the exterior of the unit circle. Then D+(x)D−(x) = |x|2α for x ∈ (at, 0). It follows that D(∞)−σ3 P (∞)(z)D(z)σ3 satisfies the normalized RH problem with jump matrix ( 0 1−1 0 ) on (at, 0) (oriented from left to right), whose solution is well-known, see e.g. [11, 14], and it leads to P (∞)(z) = D(∞)σ3 (βt(z) + βt(z) −1) 1 (βt(z)− βt(z)−1) (βt(z)− βt(z)−1) 12 (βt(z) + βt(z) D(z)−σ3 , (2.63) for z ∈ C \ [at, 0], where βt(z) = z − at , for z ∈ C \ [at, 0]. (2.64) 2.5.5 The parametrix P (a) The parametrix P (a) is defined in the disc U (a) around a, where P (a) satisfies the following RH problem. 24 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Riemann-Hilbert problem for P (a) • P (a) : U (a) \ ΣS → C2×2 is analytic. • P (a)+ (z) = P − (z) vS(z) for z ∈ U (a) ∩ ΣS . • P (a)(z) P (∞)(z) = I +O(n−1), as n→ ∞, uniformly for z ∈ ∂U (a) \ ΣS. We seek P (a) in the form P (a)(z) = P̂ (a)(z) enϕ̃t(z)σ3 (−z)−ασ3 , for z ∈ U (a) \ ΣS, where (−z)−α is defined with a branch cut along [0,∞). Then P̂ (a) satisfies a RH problem with constant jumps and can be constructed in terms of the Airy function in a standard way; for more details see the presentation in [11]. 2.5.6 The parametrix P (0) The parametrix P (0), defined in the disk U (0) around 0, should satisfy the following RH problem. Riemann-Hilbert problem for P (0) • P (0) : U (0) \ ΣS → C2×2 is continuous and analytic on U (0) \ ΣS . • P (0)+ (z) = P − (z) vS(z) for z ∈ ΣS ∩ U (0) (with the same orientation as ΣS). • P (0)(z) P (∞)(z) = I+O(n−1/3), as n→ ∞, t→ 1 such that n2/3(t−1) = O(1), uniformly for z ∈ ∂U (0) \ ΣS. • P (0) has the same behavior near 0 as S has (see the RH problem for S). A parametrix P (0) with these properties can be constructed using a solution Ψα of the model RH problem of Subsection 1.2. The construction is done in three steps. Step 1: Transformation to constant jumps. We seek P (0) in the form P (0)(z) = P̂ (0)(z) enϕt(z)σ3 z−ασ3 , for z ∈ U (0) \ ΣS, (2.65) where as usual z−α denotes the principal branch. It then follows from the RH problem for P (0) that P̂ (0) should satisfy the following RH problem. Riemann-Hilbert problem for P̂ (0) • P̂ (0) : U (0) \ ΣS → C2×2 is continuous and analytic on U (0) \ ΣS . CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 25 • For z ∈ ΣS ∩ U (0), we have + (z) = P̂ − (z)×   , for z ∈ (0,∞) ∩ U (0), e2απi 1 , for z in U (0) on the upper lip of the lens, , for z ∈ (−∞, 0) ∩ U (0), e−2απi 1 , for z in U (0) on the lower lip of the lens. uniformly for z ∈ ∂U (0) \ ΣS. • If α < 0, then P̂ (0)(z) = O |z|α |z|α |z|α |z|α as z → 0, while if α ≥ 0 we have that P̂ (0)(z) = O |z|α |z|−α |z|α |z|−α as z → 0 from outside the lens, and P̂ (0)(z) = O |z|−α |z|−α |z|−α |z|−α as z → 0 from inside the lens. Note that the jump matrices of P̂ (0) do not depend on t. The reader may note the similarities between the above RH problem for P̂ (0) and the RH problem for Ψα from Subsection 1.2. In the next step we show how we can use Ψα to construct a solution of the RH problem for P̂ (0). Step 2: The construction of P̂ (0) in terms of Ψα. Recall that ΣS in U (0) was taken such that z 7→ ft(z)− ft(0) maps ΣS ∩U (0) onto a subset of Σ, where Σ is the contour in the RH problem for Ψα, see Subsection 1.2. We choose any solution Ψα of the model RH problem and we define P̂ (0) by P̂ (0)(z) = E(z) Ψα 3 (ft(z)− ft(0));n 3 ft(0) , for z ∈ U (0) \ ΣS, (2.66) where E = En,N is analytic in U (0). Taking P (0) as in (2.65) with P̂ (0) as in (2.66) we find that all the conditions of the RH problem for P (0) are satisfied, except for the matching condition P (0)(z) P (∞)(z) = I +O(n−1/3), (2.67) as n→ ∞, t→ 1 such that n2/3(t− 1) = O(1). 26 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Step 3: Matching condition. To be able to satisfy (2.67) we have to take E in the following way E(z) = P (∞)(z) zασ3 3 (ft(z)− ft(0)) )σ3/4 , for z ∈ U (0) \ [at, 0], (2.68) where both branches are taken as principal. Clearly then E is analytic in U (0) \ [at, 0]. It turns out that E has analytic continuation to U (0). This follows by direct calculation, but it relies on the fact that we chose [at, 0] as the jump contour for P With the choice (2.68) for E, we now show that (2.67) is satisfied as well. By (2.65), (2.66), we have for z ∈ ∂U (0) \ ΣS, P (0)(z) = E(z) Ψα 3 (ft(z)− ft(0));n 3 ft(0) enϕt σ3 z−ασ3 and we are interested in the behavior as n→ ∞, t→ 1 such that n2/3(t− 1) = O(1). We show first that n2/3ft(0) remains bounded. Lemma 2.7 Suppose n → ∞, t → 1 such that n2/3(t − 1) = O(1). Then n2/3ft(0) remains bounded. More precisely, if n2/3(t− 1) → L ∈ R, then n2/3ft(0) → −c2,V L = s, (2.69) where c2,V = (c1,V ) 2/3dbt (2.70) and c1,V is the constant in (1.8). Proof. It follows from (2.51), that ft(0) = −atht(bt) (−bt)χ2/3t (0) −ah1(0) (t− 1)dbt +O((t− 1)2) as t→ 1. By (1.8) and (2.46), we have c1,V = −ah1(0), (2.71) so that (2.69)–(2.70) indeed follows if n2/3(t− 1) → L. ✷ If we use the formula (2.47) for the t-derivative of bt at t = 1, then we find from (2.70) c2,V = 2(−a)−1/2 c−1/31,V . (2.72) Now we continue with the proof of (2.67). CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 27 Lemma 2.8 Suppose that n → ∞, t → 1 such that n2/3(t − 1) = O(1). Then (2.67) holds. Proof. In the proof all O-terms are for n→ ∞, t→ 1 such that n2/3(t− 1) is bounded. By Lemma 2.7 the values n2/3ft(0) remain bounded. Since the asymptotic condition (c) in the RH problem for Ψα is valid uniformly for s in bounded subsets of R, we find by (2.65), (2.66), and (2.68) P (0)(z) = E(z) 3 (ft(z)− ft(0)) )−σ3/4 1√ I +O(n−1/3) × exp −θ(n2/3(ft(z)− ft(0));n2/3ft(0))σ3 enϕtσ3z−ασ3 = P (∞)(z)(I +O(n−1/3)) exp θ(n2/3(ft(z)− ft(0));n2/3ft(0))− nϕt (2.73) uniformly for z ∈ ∂U (0). As before we denote θ(ζ ; s) = 2 ζ3/2 + sζ1/2. The next step is to evaluate the expression in the exponential factor. We have θ(n2/3(ft(z)− ft(0));n2/3ft(0))− nϕt (ft(z)− ft(0))3/2 − (ft(z))3/2 + nft(0)(ft(z)− ft(0))1/2. We will show that this is O(n−1/3) uniformly for z ∈ ∂U (0). To that end, it is enough to show that F (t, z) := (ft(z)− ft(0))3/2 − (ft(z))3/2 + ft(0) (ft(z)− ft(0))1/2 = O((t− 1)2) as t→ 1, (2.74) uniformly for z ∈ ∂U (0). By (2.53), we have F (1, z) = 0. (2.75) Moreover, F (t, z) = (ft(z)− ft(0)) (ft(z)− ft(0))− (ft(z)) ft(z) ft(0) (ft(z)− ft(0)) ft(0) (ft(z)− ft(0))− (ft(z)− ft(0)). Let t = 1 and again use (2.53) and (2.51). Due to cancellations one finds F (1, z) = 0. (2.76) Since, in addition, F (t, z) is analytic in both variables and bounded with respect to z in ∂U (0), it follows from a Taylor expansion that F (t, z) = O((t− 1)2), as claimed in (2.74). θ(n2/3(ft(z)− ft(0));n2/3ft(0))− nϕt = O(n−1/3), 28 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Figure 4: Contour for the RH problem for R. so that (2.73) leads to P (0)(z) = P (∞)(z) I +O(n−1/3) uniformly for z ∈ ∂U (0). Then (2.67) follows since P (∞)(z) and its inverse are bounded in n and t, uniformly for z ∈ ∂U (0). ✷ This completes the construction of the parametrix P (0). Remark 2.9 The local parametrix P (0) is constructed with the help of a solution Ψα of the model RH problem. Since the solution Ψα is not unique (see Proposition 2.1), the local parametrix is not unique. In what follows we can take any P (0) and it will not affect the final results (Theorems 1.1 and 1.2). 2.5.7 Third transformation S 7→ R Having P (∞), P (a), and P (0), we take P as in (2.61), and then we define R(z) = S(z)P−1(z), for z ∈ C \ (∂U (0) ∪ ∂U (a) ∪ ΣS). (2.77) Since S and P have the same jump matrices on U (0)∩ΣS, U (a)∩ΣS and (a, 0)\(U (0)∪U (a)), we have that R is analytic across these contours. What remains are jumps for R on the contour ΣR shown in Figure 4 with orientation that is also shown in the figure. Then, R satisfies the following RH problem. Riemann-Hilbert problem for R • R : C \ ΣR → C2×2 is analytic. • R+(z) = R−(z) vR(z) for z ∈ ΣR, where P (∞) (P (0))−1, on ∂U (0), P (∞) (P (a))−1, on ∂U (a), P (∞) vS (P (∞))−1, on ΣR \ (∂U (0) ∪ ∂U (a)). (2.78) • R(z) = I +O(1/z) as z → ∞. Now let n → ∞, t → 1 such that n2/3(t − 1) = O(1). It then follows from the construction of the parametrices (see in particular the RH problems for P (0) and P (a)) I +O(n−1/3), on ∂U (0), I +O(n−1), on ∂U (a). (2.79) CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 29 Furthermore, by regularity of the eigenvalue density, there is a constant c > 0 such that Reϕt(z) > c > 0, for z ∈ ΣR ∩ (0,∞), Re ϕ̃t(z) > c > 0, for z ∈ ΣR ∩ (−∞, a), Reϕt(z) < −c < 0, for z ∈ ΣR \ (∂U (0) ∪ ∂U (a) ∪ R). This implies (see the RH problem for S) that vS = I+O(e −cn) uniformly on ΣR \ (∂U (0)∪ ∂U (a)), so that by (2.78) vR = I +O(e −2cn) on ΣR \ (∂U (0) ∪ ∂U (a)). (2.80) The O-terms in (2.79) and (2.80) are uniform on the indicated contours. In addition, it follows from (2.58), (2.59), (2.55), and the growth condition (1.2) on V that for any C > 0 there exists r = r(C) > 1 such that ϕt(x) ≥ C log x for x ≥ r, and ϕ̃t(x) ≥ C log |x| for x ≤ −r. Combined with (2.80) this implies that ||vR − I||L2(ΣR\(∂U (0)∪∂U (a))) = O(e −2cn), as n→ ∞. (2.81) Thus, by (2.79)–(2.81), as n → ∞ and t → 1 such that n2/3(t − 1) = O(1), the jump matrix for R is close to I in both L2 and L∞ norm on ΣR, indeed ||vR − I||L2(ΣR)∩L∞(ΣR) = O(n−1/3). (2.82) Standard estimates using L2-boundedness of the operators C± on L 2(ΣR) together with the correspondence between RH problems and singular integral equations now imply that R(z) = I +O(n−1/3), uniformly for z ∈ C \ ΣR, (2.83) as n → ∞, t → 1 such that n2/3(t − 1) = O(1). To get the uniform bound (2.83) up to the contour one needs a contour deformation argument. Again, see the presentation in [11] for more details. This completes the steepest descent analysis of the RH problem for Y . 2.6 Completion of the proofs of Theorem 1.1 and 1.2 Having completed the steepest descent analysis we are now ready for the proofs of Theorem 1.1 and 1.2. We start by rewriting the kernel (2.3) for x, y ∈ U (0) ∩ R according to the transformations Y 7→ T 7→ S 7→ R that we did in the steepest descent analysis. To state the result it is convenient to introduce B = Bn,N as B(z) = R(z)E(z), for z ∈ U (0), (2.84) where E and R are defined in (2.68) and (2.77). We also define for x, s ∈ R the column vector ~ψα(x; s) = ψ1(x; s) ψ2(x; s) Ψα,+(x; s) , for x > 0, Ψα,+(x; s)e −απiσ3 , for x < 0, (2.85) cf. (1.12). We then have the following result. 30 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Lemma 2.10 Let x, y ∈ U (0) ∩ R. Then, Kn,N(x, y) = 2πi(x− y) 3 (ft(y)− ft(0));n 3 ft(0) ))T ( 0 1 × B−1(y)B(x) ~ψα 3 (ft(x)− ft(0));n 3 ft(0) . (2.86) Proof. We start from the formula (2.3) for the eigenvalue correlation kernel. Using (2.56) we obtain, for any x, y ∈ R, Kn,N(x, y) = |x|α e n(2gt,+(x)−Vt(x)+lt) |y|α e 12n(2gt,+(y)−Vt(y)+lt) 2πi(x− y) T−1+ (y) T+(x) . (2.87) Using (2.58) and the fact that gt,+ = gt,− on (bt,∞), it follows that 2gt − Vt + lt = −2ϕt on (bt,∞). Then, by analytic continuation, 2gt,+−Vt+ lt = −2ϕt,+ on all of R. Therefore we can rewrite (2.87) as Kn,N(x, y) = |x|α e−nϕt,+(x) |y|α e−nϕt,+(y) 2πi(x− y) T−1+ (y) T+(x) (2.88) Now we analyze the effect of the transformations T 7→ S 7→ R on the expression |x|αe−nϕt,+(x)T+(x) ( 10 ) in case x ∈ U (0) ∩ R. The result is that for x ∈ U (0) ∩ R, |x|αe−nϕt,+(x)T+(x) = B(x)Ψα,+ 3 (ft(x)− ft(0));n 3 ft(0) (2.89) in case x > 0, and |x|αe−nϕt,+(x)T+(x) = B(x)Ψα,+ 3 (ft(x)− ft(0));n 3 ft(0) e−απiσ3 (2.90) in case x < 0. Since the calculations for (2.89) are easier, we will only show how to obtain (2.90). If x ∈ U (0) ∩ R and x < 0, then it follows from (2.60) that |x|αe−nϕt,+(x)T+(x) = |x|αe−nϕt,+(x)S+(x) |x|−2αe2nϕt,+(x) = S+(x) |x|αe−nϕt,+(x) . (2.91) From (2.77), (2.61), (2.65), (2.68), and (2.84), we find that S+(x) = B(x)Ψα,+ 3 (ft(x)− ft(0));n 3ft(0) enϕt(x)x−α Inserting this into (2.91) and noting that x−α+ |x|α = e−απi we indeed obtain (2.90). In a similar way, we find for y ∈ U (0) ∩ R, |y|αe−nϕt,+(y) T−1+ (y) = Ψ−1α,+ 3 (ft(y)− ft(0));n 3ft(0) B−1(y), (2.92) CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 31 in case y > 0, and |y|αe−nϕt,+(y) T−1+ (y) = eαπiσ3Ψ−1α,+ 3 (ft(y)− ft(0));n 3ft(0) B−1(y), (2.93) in case y < 0. To rewrite (2.92) and (2.93) we use the following fact, which is easy to check. If A is an invertible 2× 2 matrix having determinant 1, then A−1 = . (2.94) If we apply (2.94) to Ψα,+ in (2.92) and (2.93), then we get |y|αe−nϕt,+(y) T−1+ (y) = ΨTα,+ 3 (ft(y)− ft(0));n 3ft(0) B−1(y), (2.95) in case y > 0, and |y|αe−nϕt,+(y) T−1+ (y) = e−απiσ3ΨTα,+ 3 (ft(y)− ft(0));n 3 ft(0) B−1(y), (2.96) in case y < 0. Then (2.86) follows if we insert (2.89), (2.90), (2.95), and (2.96) into (2.88) and use the definition (2.85). ✷ As in Theorem 1.1 we now fix x, y ∈ R. We define (c1n)2/3 , and yn = (c1n)2/3 (2.97) where c1 is the constant from (1.8). In order to take the limit of (c1n) −2/3Kn,N(xn, yn) we need one more lemma. Recall that B = RE is defined in (2.84). Lemma 2.11 Let n→ ∞, t→ 1 such that n2/3(t− 1) → L. Let x, y ∈ R and let xn and yn defined as in (2.97), Then the following hold. (a) n2/3ft(0) → s, (b) n2/3(ft(xn)− ft(0)) → x and n2/3(ft(yn)− ft(0)) → y, (c) B−1(yn)B(xn) = I +O where the implied constant in the O-term is uniform with respect to x and y. Proof. (a) This follows from Lemma 2.7. (b) By (1.8) and (2.46) we have c1 = −ah1(0), so that f ′1(0) = c 1 by (2.52). Taking note of the definitions (2.97), we then obtain part (b), since ft → f1 uniformly in U (0). 32 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON (c) We have R−1(yn)R(xn) = I +R −1(yn)(R(xn)− R(yn)) = I +R−1(yn) (xn − yn) R′(txn + (1− t)yn)dt. (2.98) Recall that R is analytic in U (0), and that R(z) = I + O(n−1/3) by (2.83), uniformly in U (0). Since detR ≡ 1, we find that R−1(yn) remains bounded as n → ∞. It also follows from (2.83) and Cauchy’s theorem, that R′(z) = O(n−1/3) for z in a neighborhood of the origin. By (2.98) we then obtain R−1(yn)R(xn) = I +O . (2.99) Using analyticity of E in a neighborhood of the origin with E(z) = O(n1/6), see (2.68), and the fact that detE ≡ 1, we obtain in the same way E−1(yn)E(xn) = I +O . (2.100) The implied constants in (2.99) and (2.100) are independent of x and y. Using (2.99), (2.100), and the fact that E(xn) = O(n 1/6) and E−1(yn) = O(n 1/6), we obtain from (2.84) B−1(yn)B(xn) = E −1(yn) E(xn) = E−1(yn)E(xn) +O(n 1/6)O O(n1/6) = I +O This completes the proof of part (c). ✷ Proof of Theorems 1.1 and 1.2. We let n,N → ∞, t = n/N → 1, in such a way that n2/3(t− 1) → L. Then by parts (a) and (b) of Lemma 2.11, we have ~ψα(n 2/3(ft(xn)− ft(0));n2/3ft(0)) → ~ψα(x; s) and similarly if we replace xn by yn. The existence of the limit (1.9) then follows easily from Lemma 2.10 and part (c) of Lemma 2.11, which proves Theorem 1.1. We also find that the limiting kernel Kedgeα (x, y; s) is given by Kedgeα (x, y; s) = 2πi(x− y) ~ψα(y; s) ~ψα(x; s) and so (1.13) follows because of (2.85). The model RH problem is solvable for every s ∈ R by Proposition 2.4 and so we have also proved Theorem 1.2. ✷ CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 33 S5 S6❄ Figure 5: Contour for the RH problem for ΨFNν . 3 Proof of Theorems 1.4 and 1.5 We prove Theorem 1.4 and Theorem 1.5 by first establishing, with the help of [2], a connection between the model RH problem and the RH problem for Painlevé II in the form due to Flaschka and Newell [22]. We can then use known properties of the RH problem for Painlevé II to prove the theorems. 3.1 The Painlevé II RH problem We review the RH problem for the Painlevé II equation q′′(s) = sq+2q3−ν, as first given by Flaschka and Newell [22], see also [24] and [25]. We will assume that ν > −1/2. The RH problem involves three complex constants a1, a2, a3 satisfying a1 + a2 + a3 + a1a2a3 = −2i sin νπ, (3.1) and certain connection matrices Ej . Let Sj = {w ∈ C | 2j−36 π < argw < π} for j = 1, . . . , 6, and let ΣFN = C \ j Sj . Then ΣFN consists of six rays ΣFNj for j = 1, . . . , 6, all chosen oriented towards infinity as in Figure 5. The RH problem is the following. Riemann-Hilbert problem for ΨFNν • ΨFNν : C \ ΣFN → C2×2 is analytic, • ΨFNν,+ = ΨFNν,− on ΣFN1 , ΨFNν,+ = Ψ on ΣFN2 , ΨFNν,+ = Ψ on ΣFN3 , 34 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON ΨFNν,+ = Ψ on ΣFN4 , ΨFNν,+ = Ψ on ΣFN5 , ΨFNν,+ = Ψ on ΣFN6 . • ΨFNν (w) = (I +O(1/w))e−i( w3+sw)σ3 as w → ∞. • If ν − 1 6∈ N0, then ΨFNν (w) = B(w) 0 w−ν Ej , for w ∈ Sj , (3.2) where B is analytic. If ν ∈ 1 + N0, then there exists a constant κ such that ΨFNν (w) = B(w) wν κwν logw 0 w−ν Ej , for w ∈ Sj, (3.3) where B is analytic. The connection matrix E1 is given explicitly in [24, Chapter 5]. It is determined (up to inessential left diagonal or upper triangular factors) by ν and the Stokes multipliers a1, a2, and a3, except in the special case + n, a1 = a2 = a3 = i(−1)n+1, n ∈ Z, (3.4) where an additional parameter c ∈ C ∪ {∞} is needed. For example, for ν 6∈ 1 + N0 and 1 + a1a2 6= 0, we have 0 d−1 e−νπi − a2 1 + a1a2 −1 + a1a2 2 cos νπ eνπi + a2 2 cos νπ  , (3.5) where d 6= 0 is arbitrary. In the special case (3.4), when E1 depends on the additional parameter c ∈ C ∪ {∞}, by [24, Chapter 5, (5.0.21)] we may take E1 as , if c ∈ C, while E1 = if c = ∞. (3.6) Assuming that the branch cuts for the functions in (3.2) and (3.3) are chosen along argw = −π/6, we obtain the other connection matrices from E1 through the formula Ej+1 = Ejv j , j = 1, . . . , 5, (3.7) where vFNj is the jump matrix on Σ j . We shall refer to the Stokes multipliers a1, a2, and a3, and in the special case (3.4) also to the additional parameter c, as the monodromy data for Painlevé II. We note that in the special case (3.4) we have κ = 0 in (3.3). CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 35 The special case (3.4) has geometric interpretation. Indeed, (3.4) describes the singular point of the algebraic variety (3.1), that is, the point at which the (complex) gradient of the left-hand side of (3.1) vanishes. The singularity may be removed by attaching a copy of the Riemann sphere (see also [30]). The monodromy data does not depend on s. The RH problem is uniquely solvable, except for a discrete set of s-values, and its solution ΨFNν depends on s through the asymp- totic condition at infinity. We write ΨFNν (w; s) if we want to emphasize its dependence on s. If we take q(s) = 2i lim ΨFNν (w; s) w3+sw)σ3, (3.8) then q satisfies the Painlevé II equation q′′ = sq + 2q3 − ν. In addition ΨFNν satisfies the Lax pair for Painlevé II Ψ = LΨ, L = (−4iw2 − i(s + 2q2) 4wq + 2ir + ν 4wq − 2ir + ν 4iw2 + i(s+ 2q2) , (3.9) Ψ = PΨ, P = −iw q , (3.10) where q = q(s) and r = r(s) = q′(s). In this way there is a one-to-one correspondence between monodromy data and solutions of Painlevé II. We also need the more precise asymptotic behavior ΨFNν (w; s) = H(s) q(s) −q(s) −H(s) +O(1/w2) w3+sw)σ3 (3.11) as w → ∞, where H(s) = (q′(s))2 − sq2(s)− q4(s) + 2νq(s) (3.12) is the Hamiltonian for Painlevé II. Note that H ′ = −q2. We finally note that ΨFNν satisfies the symmetry property ΨFNν (w; s) = σ1Ψ ν (−w; s)σ1, (3.13) where σ1 = ( 1 0 ). Indeed, by a straightforward calculation (see also [24, Chapter 5]) we check that the function σ1Ψ ν (−w; s)σ1 solves exactly the same RH problem as the function ΨFNν (w; s). Unique solvability of the RH problem yields equation (3.13). 3.2 Connection with Ψα The Hastings-McLeod solution of Painlevé II corresponds to the Stokes multipliers a1 = −eνπi, a2 = 0, and a3 = e−νπi. This is not the solution that interests us here. We use instead the solution corresponding to a1 = e −νπi, a2 = −i, a3 = −eνπi. (3.14) For these Stokes multiplies (3.14) we obtain from (3.5) the following connection matrix E1 in case ν 6∈ 12 + N0 (where we take d = (e νπi − i)/(2 cos νπ)) eνπi − i 2 cos νπ ieνπi + 1 2 cos νπ −e−νπi 1  . (3.15) 36 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON For ν ∈ 3 + 2N0, it follows from (3.14) and the formulas in [24, Chapter 5, (5.0.18)] that we can take . (3.16) If ν ∈ 1 + 2N0, then we are in the special case (3.4). We then choose c = i, so that for ν ∈ 1 + 2N0 we have monodromy data a1 = e −νπi = −i, a2 = −i, a3 = −eνπi = −i, c = i. (3.17) Lemma 3.1 For any ν > −1/2, we have that (E2)21 = (E3)21 = 0. (3.18) Proof. In all cases we may check from (3.14), (3.15), (3.16), (3.17), and (3.6) that the second row of E1 is given by −a1 1 . So by (3.7) we have that E2 = E1 ( a1 1 ) is upper triangular. Then also E3 = E2 ( 0 1 ) is upper triangular and therefore (3.18) holds. ✷ The following proposition holds for more general monodromy data, and it was estab- lished in [2], see also [35]. For the reader’s convenience we present a detailed proof for our particular case. Proposition 3.2 ([2]) For α > −1/2, let ΨFN2α+1/2 be the unique solution of the RH prob- lem for Painlevé II with parameter ν = 2α + 1/2 and monodromy data (3.14) in case α 6∈ N0 (so that ν 6∈ 12 + 2N0), and monodromy data (3.17) in the special case α ∈ N0. Then, for any η = η(s), we have that Ψα(ζ ; s) = η(s) 1 ζ−σ3/4 eπiσ3/4ΨFN2α+1/2(w;−21/3s)e−πiσ3/4 (3.19) where w = eπi/22−1/3ζ1/2 with Imw > 0, is a solution of the model RH problem for Ψα given in Subsection 1.2. Proof. Because of Proposition (2.1) we may take η(s) = 0 without loss of generality. Clearly Ψα is analytic on C\Σ. The correct asymptotics as ζ → ∞ follows immediately, as well as the correct jumps across Σ1, Σ2, and Σ4. A little bit more work is needed to check the jump across Σ3 = (−∞, 0) and the behavior at z = 0. In order to analyze the jump across Σ3, we suppose that ζ ∈ Σ3. Then we have that w+ ≡ eπi/22−1/3ζ1/2+ = −w− ≡ −eπi/22−1/3ζ − (< 0), CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 37 and hence by (3.19) and the symmetry property (3.13), Ψα,+(ζ ; s) = ζ −σ3/4 eπiσ3/4ΨFN2α+1/2(w+;−21/3s)e−πiσ3/4 −σ3/4 eπiσ3/4ΨFN2α+1/2(−w−;−21/3s)e−πiσ3/4 −σ3/4 eπiσ3/4σ1Ψ 2α+1/2(w−;−21/3s)σ1e−πiσ3/4 −σ3/4 eπiσ3/4ΨFN2α+1/2(w−;−21/3s)e−πiσ3/4 −σ3/4 eπiσ3/4ΨFN2α+1/2(w−;−21/3s)e−πiσ3/4 = Ψα,−(ζ ; s) This shows that Ψα has the correct jump across Σ3, and it follows that Ψα satisfies the parts (a), (b), and (c) of the model RH problem. Consider now a neighborhood of the point ζ = 0. We recall that (3.2) or (3.3) holds with B(w) analytic at 0. A corollary of the symmetry property (3.13) is the equation B(w) = σ1B(−w)σ3, if ν 6∈ 12 + N0, σ1B(−w) 1 O(w2ν) as w → 0, if ν ∈ 1 + N0, which yields the formula (cf. [24, Chapter 5]) B(0)σ3 = σ1B(0). The last relation, together with detB(0) = 1, in turn implies that B(0) can be represented in the form B(0) = 0 b−1 , b 6= 0. If ζ ∈ Ωj then w ∈ Sπ(j), j = 1, 2, 3, 4, where π denotes the permutation 1 2 3 4 3 4 1 2 Therefore, for the function Ψα(ζ ; s) defined by equation (3.19) with η(s) = 0, we find that (assuming that α 6∈ 1 Ψα(ζ ; s) = ζ −σ3/4 1√ eπiσ3/4B(0)e−πiσ3/4 I +O(ζ1/2) ζσ3/4ζασ3Ẽπ(j) = ζ−σ3/4 0 b−1 I +O(ζ1/2) ζσ3/4ζασ3Ẽπ(j) = ζ−σ3/4 I +O(ζ1/2) ζσ3/4ζασ3 0 b−1 Ẽπ(j) = O(1)ζασ3 0 b−1 Ẽπ(j), as ζ → 0 in Ωj , (3.20) 38 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON where we have introduced the notation Ẽj ≡ eπiσ3/4 eπi/22−1/3 )(2α+1/2)σ3 −πiσ3/4. (3.21) From (3.20) it immediately follows that Ψα(ζ ; s) = O(ζ −|α|) as ζ → 0, which is the required behavior in the model RH problem if α < 0, or if α ≥ 0 and j ∈ {2, 3}. If α ≥ 0 and j ∈ {1, 4}, then π(j) ∈ {2, 3}, and it follows from Lemma 3.1 and (3.21) that Ẽπ(j) Eπ(j) Then (3.20) also yields the required behavior of Ψα(ζ ; s) as ζ → 0 in Ω1 ∪ Ω4. The calculation leading to (3.20) is valid for ν 6∈ 1 +N0, or α 6∈ 12N0. In fact it is also valid if α ∈ N0, since then we are in the special case (3.4) where κ = 0 in (3.3) and so no logarithmic terms appear. Logarithmic terms only appear if α ∈ 1 + N0, and then a similar calculation leads to Ψα(ζ ; s) = O(1)ζ 0 b−1 1 O(log ζ) Ẽπ(j), with Ẽj again given by (3.21). Since α > 0, the required behavior as ζ → 0 then follows in a similar way. This completes the proof of the proposition. ✷ 3.3 Differential equation Recall that ΨFNν has the Lax pair (3.9) and (3.10). Then Ψα defined by (3.19) also satisfies a system of differential equations. It will involve the solution q of the Painlevé II equation with parameter ν = 2α + 1/2 and monodromy data (3.14) or (3.17). We put r = q′ and U(s) = q2(s) + r(s) + , (3.22) V (s) = q2(s)− r(s) + s . (3.23) The functions U and V both satisfy the Painlevé XXXIV equation in a form similar to (1.19), namely (cf. [24, Chapter 5]): U ′′(s) = (U ′(s))2 2U(s) + 2U2(s)− sU(s)− (2α) 2U(s) , (3.24) V ′′(s) = (V ′(s))2 2V (s) + 2V 2(s)− sV (s)− (2α + 1) 2V (s) . (3.25) Then we obtain the following differential equations for Ψα. Lemma 3.3 Let Ψα be given by (3.19). (a) If η ≡ 0, then Ψα satisfies Ψα(ζ ; s) = AΨα(ζ ; s), (3.26) Ψα(ζ ; s) = BΨα(ζ ; s), (3.27) CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 39 where −21/3q(−21/3s) + α i− i2−1/3U(−21/3s)1 −iζ + i2−1/3V (−21/3s) 21/3q(−21/3s)− α , (3.28) −21/3q(−21/3s) i −iζ 21/3q(−21/3s) . (3.29) (b) For general η we have that Ψα satisfies Ψα(ζ ; s) = η(s) 1 −η(s) 1 Ψα(ζ ; s), (3.30) with A given by (3.28). Proof. This follows by straightforward calculations from (3.9), (3.10), and (3.19). ✷ The Lax pair (3.26)–(3.27), after the replacement ζ 7→ ζ − s, becomes the Lax pair from [2, 35]. Equations (3.24)–(3.25) can also be derived directly from the compatibility conditions of the Lax pair (3.26)–(3.27) in a usual way. It is a fact [33], that the solution q of the Painlevé II equation (with parameter ν = 2α+1/2 and monodromy data (3.14) or (3.17)) has an infinite number of poles on the positive real line, see also (4.29) below. If −21/3s is such a pole then ΨFN2α+1/2(·,−21/3s) does not exist. So to be precise, if we assume that η is analytic on R, then (3.19) does not define Ψα for values of s ∈ R which belong to the discrete set of values s where q(−21/3s) has poles. The relation (3.19) defines Ψα for all s ∈ R only if we are able to choose η so that all the poles on the real line of the right-hand side of (3.19) cancel out. Such a choice of η would require η itself to have poles at the poles of q(−21/3s). We will describe two special choices for η. The first choice is such that (3.19) is equal to the special solution Ψ (spec) α , which is characterized by the asymptotic condition (2.14). From Proposition 2.4 we know that Ψ (spec) α exists for all s ∈ R, so that we can already conclude that the special choice η = η(spec) will have poles at the poles of q(−21/3s), and that the real poles of the right-hand side of (3.19) will indeed cancel out. The second choice of η is made so that the differential equation (3.30) takes a nice form. It will lead to the differential equation (1.21) for ψ1 and ψ2. This η is denoted η0, and it is defined by the simple formula η0(s) = i2 1/3q(−21/3s), (3.31) from which it is already clear that it has poles at the poles of q(−21/3s). For the choice (3.31) we can already check that the differential equation (3.30) leads to Ψα(ζ ; s) = A0Ψα(ζ ; s) (3.32) where −η0 1 (α + iuη0)/ζ i− iu/ζ −iζ + i(v + η20) + η0(2α + iuη0)/ζ −(α + iuη0)/ζ , (3.33) 40 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON u(s) = 2−1/3U(−21/3s), (3.34) v(s) = 2−1/3V (−21/3s). (3.35) 3.4 Special choice η(spec) Lemma 3.4 Let H be the Hamiltonian for Painlevé II as in (3.12), with parameter ν = 2α + 1/2, and let η(spec)(s) = i2−2/3 q(−21/3s) +H(−21/3s) . (3.36) Then the choice η = η(spec) in (3.19) leads to the special solution Ψ (spec) α of the model RH problem characterized by (2.14). Proof. It follows from (3.11) and (3.19) by straightforward computation, that Ψα(ζ ; s) = ζ−σ3/4 eπiσ3/4ΨFN2α+1/2(w;−21/3s)e−πiσ3/4 −η(spec) 1 0 i2−2/3(H − q)(−21/3s) + ζ−σ3/4 eπiσ3/4O(1/ζ)e−πiσ3/4 ζσ3/4 × ζ−σ3/4 1√ ζ3/2+sζ1/2)σ3 (3.37) as ζ → ∞. From (3.37) it is clear that we need to take η = η(spec) in order to be able to obtain (2.14). Thus the lemma follows. ✷ From the calculation (3.37) we also note that for any solution Ψα of the model RH problem we have Ψα(ζ ; s)e (ζ3/2+sζ1/2)σ3 ζσ3/4 i2−2/3(H − q)(−21/3s) +O(ζ−3/2) (3.38) as ζ → ∞. This property will be used later in the proof of Theorem 1.4. Since the left-hand side of (3.38) is analytic in s for s ∈ R, it also follows from (3.38) that H − q does not have poles on the real line. This and similar properties are collected in the following lemma. Recall that U is given by (3.22). Lemma 3.5 The following hold. (a) H − q has no poles on the real line. (b) U has no poles on the real line. (c) U has a zero at each of the real poles of q and Uq has no poles on the real line. (d) Uq takes the value ν − 1/2 at each of the real poles of q. CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 41 Proof. (a) We noted already that part (a) follows from (3.38). (b) Since H ′ = −q2, we have that U(s) = q2(s) + q′(s) + s/2 = −(H − q)′(s) + s/2, (3.39) and so it follows from part (a) that U has no poles on the real line either. (c) Differentiating (3.22), we obtain U ′ = 2qq′ + q′′ + = 2qq′ + sq + 2q3 − ν + 1 = 2Uq − ν + 1 . (3.40) Thus also Uq has no poles on the real line, which means that U has a zero at each of the real poles of q. (d) Using (3.40), we get (Uq − ν + 1 )q = (U ′ − Uq)q = (Uq)′ − U(q2 + q′) = (Uq)′ − U(U − s/2). (3.41) Since the right-hand side of (3.41) is analytic on the real line by parts (b) and (c), we conclude that Uq−ν+ 1 has a zero at each of the real poles of q. This proves part (d). ✷ It is well-known and easy to check that each pole of q is simple and has residue +1 or −1. Indeed, the Laurent series for q at a pole s0 has the form q(s) = s− s0 + q1(s− s0) + · · · , where q−1 ∈ {−1, 1}. Using this, one easily verifies that either q2+ q′ or q2− q is analytic at s0 (depending on the sign of the residue q−1). Our result that U = q 2 + q′ + s/2 is analytic on R can then also be stated as follows. Corollary 3.6 The solution q of the Painlevé II equation with parameter ν = 2α + 1/2 and monodromy data (3.14) or (3.17) has only simple poles on the real line, with residue 3.5 Special choice η0 As already announced we will also use the special choice η = η0 given by (3.31). By (3.31) and (3.36) we have that η0(s)− η(spec)(s) = i2−2/3 q(−21/3s)−H(−21/3s) and so it follows from part (a) of Lemma 3.5 that η0 − η(spec) is analytic on the real line. Since Ψ (spec) α exists for all s ∈ R, it follows that the solution of the model RH problem associated with η0 exists for all s ∈ R as well, and it is analytic in s. The differential equation for Ψα with η = η0 is given by (3.32) with A0 as in (3.33). It then follows that A0 is analytic on the real line, and we will explicitly verify this by rewriting its entries in terms of the function u from (3.34) u(s) = 2−1/3U(−21/3s). 42 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON The analyticity of u is immediate from (3.34) and part (b) of Lemma 3.5. The analyticity of uη0 follows from (3.34), (3.31) and part (c) of Lemma 3.5. Using also (3.40) we get u′ = 2iuη0 + ν − 1/2 = 2iuη0 + 2α. (3.42) Next, it follows from (3.22), (3.23), (3.34), (3.35), and (3.31) that v(s) + η0(s) 2 = −u(s)− s. (3.43) We can use (3.42) and (3.43) to eliminate η0 and v from the entries in A0, and we get from (3.33) that u′/(2ζ) i− iu/ζ −iζ − i(u+ s)− i((u′)2 − (2α)2)/(4uζ) −u′/(2ζ) . (3.44) 3.6 Proof of Theorem 1.4 and 1.5 After these preparations the proofs of Theorems 1.4 and 1.5 are short. Proof of Theorem 1.4. From (3.24) and (3.34) it follows that u satisfies the Painlevé XXXIV equation in the form (1.19). From (3.38) it follows that Ψα(ζ ; s)e (ζ3/2+sζ1/2)σ3 ζσ3/4 = i2−2/3(H − q)(−21/3s) which in view of (3.39) and (3.34) leads to (1.20). This proves Theorem 1.4. ✷ Proof of Theorem 1.5. Let Ψα be the solution of the model RH problem given by (3.19) with η = η0 as in (3.31). Then Ψα(ζ ; s) = A0Ψα(ζ ; s), (3.45) with A0 given by (3.44). The differential equation (3.45) is valid for ζ ∈ C \ Σ. We can take the limit ζ → x with x ∈ R\{0} to obtain a differential equation for Ψα,+(x; s), with the same matrix A0 (but with ζ replaced by x). Using (1.12), we obtain the differential equation (1.21) for ψ1 and ψ2. This completes the proof of Theorem 1.5. ✷ 4 Concluding remarks 4.1 The case α = 0 The case α = 0 is classical and well understood. We know that K 0 (x, y; s) is the (shifted) Airy kernel, see (1.11). We will show here how this follows from the calculations from the previous section. In the special case α = 0, we have ν = 1/2, and then the Painlevé II equation has special solutions built out of Airy functions. To be precise if Ai and Bi are the standard Airy functions, then for any C1 and C2, not both zero, we have that q(s) = C1Ai(−2−1/3s) + C2Bi(−2−1/3s) (4.1) CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 43 is a solution of q′′ = sq + 2q3 − 1 . These are exactly the solutions that correspond to the special Stokes multipliers a1 = a2 = a3 = −i. The corresponding solutions to the RH problem were given by Flaschka and Newell [22, Section 3F(iv)]. For example, for w in sector S1 we have (see also [24, Chapter 11]) ΨFN1/2 (w; s) = 1− iq(s)/w −2−1/3i/w 1 + iq(s)/w 2−1/3i/w Ai(z) Bi(z) Ai′(z) Bi′(z) (4.2) with z = −22/3w2 − 2−1/3s and α0 = 21/6 πeiπ/4. The expressions for ΨFNν (w; s) in the other sectors follow by multiplying (4.2) by the appropriate jump matrices. It follows from (4.1) and (4.2) that the extra parameter c in the monodromy data for (4.1) is iC1 − C2 C1 − iC2 . (4.3) So if we take c = i as in (3.17) then C2 = 0 and the corresponding solution (4.1) is q(s) = log Ai(−2−1/3s) = −2−1/3Ai ′(−2−1/3s) Ai(−2−1/3s) . (4.4) Note that the solution (4.4) is special among all solutions (4.1) in its behavior for s→ −∞. Indeed, from the asymptotic behavior for the Airy functions it follows that for (4.4) we q(s) ∼ 1 2(−s)1/2 as s→ −∞, while for the other solutions (4.1) we have q(s) ∼ −1 2(−s)1/2 as s→ −∞. So according to Proposition 3.2 we should be using q given by (4.4) and then define Ψ0 as in (3.19). If ζ is in sector Ω3, then w = e iπ/22−1/3ζ1/2 is in sector S1, so that by (4.2) ΨFN1/2 (w;−21/3s) = 1 + iη0(s)ζ −1/2 −ζ−1/2 1− iη0(s)ζ−1/2 ζ−1/2 Ai(ζ + s) Bi(ζ + s) Ai′(ζ + s) Bi′(ζ + s) where η0(s) = i2 1/3q(−21/3) as in (3.31). Then (3.19) with η = η0 yields for ζ ∈ Ω3, Ψ0(ζ ; s) = η0(s) 1 0 ζ1/2 eπiσ3/4 1 + iη0(s)ζ −1/2 −ζ−1/2 1− iη0(s)ζ−1/2 ζ−1/2 Ai(ζ + s) Bi(ζ + s) Ai′(ζ + s) Bi′(ζ + s) e−πiσ3/4 eπiσ3/4 Ai(ζ + s) Bi(ζ + s) Ai′(ζ + s) Bi′(ζ + s) e−πiσ3/4 Ai(ζ + s) + iBi(ζ + s) −(Ai(ζ + s)− iBi(ζ + s)) −i(Ai′(ζ + s) + iBi′(ζ + s)) i(Ai′(ζ + s)− iBi′(ζ + s)) 44 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON Since (see e.g. formula (10.4.9) in [1]) Ai(z)± iBi(z) = 2e±πi/3Ai(e∓2πi/3z) we can write Ψ0 in the more familiar form Ψ0(ζ ; s) = eπi/3Ai(e−2πi/3(ζ + s)) −e−πi/3Ai(e2πi/3(ζ + s)) −ie−πi/3Ai′(e−2πi/3(ζ + s)) ieπi/3Ai′(e2πi/3(ζ + s)) , for ζ ∈ Ω3. (4.5) For ζ ∈ Ω1 we find in a similar way (or by multiplying (4.5) on the right by ( 1 1−1 0 )), that Ψ0(ζ ; s) = Ai(ζ + s) eπi/3Ai(e−2πi/3(ζ + s)) −iAi′(ζ + s) −ie−πi/3Ai′(e−2πi/3(ζ + s)) , for ζ ∈ Ω1. (4.6) Then it follows from (1.12) and (4.6) that for x > 0, ψ1(x; s) = 2πAi(x+ s), ψ2(x; s) = − 2πiAi′(x+ s), (4.7) and a similar calculation shows that (4.7) also holds for x < 0. Therefore, by (1.13) 0 (x, y; s) = ψ2(x; s)ψ1(y; s)− ψ1(x; s)ψ2(y; s) 2πi(x− y) Ai(x+ s)Ai′(y + s)− Ai′(x+ s)Ai(y + s) x− y , (4.8) which is indeed the (shifted) Airy kernel. 4.2 The case α = 1 The case α = 1 can be solved explicitly in terms of Airy functions as well. Let Ψ0 be a solution of the model RH problem with parameter α = 0. Then for any matrix X = X(s), it is easy to check that Ψ1(ζ ; s) = (I − X(s))Ψ0(ζ ; s) (4.9) satisfies the conditions (a), (b), and (c) of the model RH problem for α = 1. For a special choice of X we will have that the condition (d) is also satisfied. Let’s take Ψ0 given by (4.6) for ζ ∈ Ω1. Then the condition (d) of the model RH problem yields the following condition on X (I − 1 X(s)) Ai(ζ + s) −iAi′(ζ + s) = O(ζ), as ζ → 0. (4.10) The condition (4.10) is satisfied if and only if we take X(s) = Ai′(s)2 − sAi(s)2 Ai(s) −iAi′(s) Ai′(s) −iAi(s) . (4.11) Note that the denominator in (4.11) cannot be zero for s ∈ R. Indeed, its derivative is −Ai(s)2, so that it is decreasing for s ∈ R, and since the limit for s → +∞ is equal to CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 45 0, it follows that Ai′(s) − sAi(s)2 > 0 for all s ∈ R. Note also that if we take the limit x, y → 0 in (4.8), then 0 (0, 0; s) = Ai ′(s)2 − sAi(s)2. (4.12) Using (1.12), (4.6), (4.9), (4.11), and (4.12), we obtain that ψ1(x; s) = 2πAi(x+ s)− Ai(x+ s)Ai′(s)− Ai(s)Ai′(x+ s) x(Ai′(s)2 − sAi(s)2) Ai(s) Ai(x+ s)− K 0 (x, 0; s) 0 (0, 0; s) Ai(s) ψ2(x; s) = − 2πiAi′(x+ s) + Ai(x+ s)Ai′(s)−Ai(s)Ai′(x+ s) x(Ai′(s)2 − sAi(s)2) Ai Ai′(x+ s)− K 0 (x, 0; s) 0 (0, 0; s) Ai′(s) 1 (x, y; s) = ψ2(x; s)ψ1(y; s)− ψ1(x; s)ψ2(y; s) 2πi(x− y) 0 (x, y; s)− 0 (x, 0; s)K 0 (y, 0; s) 0 (0, 0; s) . (4.13) To compute the relevant solution u of the Painlevé XXXIV equation for α = 1, we may assume that we have taken Ψ 0 in (4.9), and then use (1.20), (4.9), (2.14), and the fact that u ≡ 0 for α = 0, to obtain that u(s) = iX ′12(s), which by (4.11) leads to u(s) = Ai(s)2 Ai′(s)2 − sAi(s)2 = − d 0 (0, 0; s). (4.14) Its graph is shown in Figure 6. One can verify from the explicitly known asymptotic formulas for Ai that u(s) = +O(s−7/2) as s→ +∞. (4.15) On the negative real axis, u has an infinite number of zeros. These are the zeros of the Airy function Ai, and an infinite number of additional zeros that interlace with the zeros of Ai. Equations (4.9) and (4.14) constitute the Schlesinger and (induced by it) Bäcklund transformations, respectively, for the case of Painlevé XXXIV and applied to its zero vacuum solution (for the general theory of Schlesinger transformations see [31]; see also [24, Chapter 6]). 4.3 Asymptotic characterization of the Painlevé function u(s) We finally want to characterize the solution u of the Painlevé XXXIV equation by its asymptotic properties. Recall that u is connected to the solution of the Painlevé II equation q′′ = sq + 2q3 − ν with ν = 2α+ 1/2 by the formulas u(s) = 2−1/3U(−21/3s), U(s) = q2(s) + q′(s) + s . (4.16) 46 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON –10 –8 –6 –4 –2 2 4 6 8 10 Figure 6: The solution of the Painlevé XXXIV equation for α = 1. Assume that ν > −1/2. It is shown in [34] (see also [24, Chapters 5, 11]) that the solution q(s) of the Painlevé II equation corresponding to the Stokes multipliers (3.14) exhibits the following asymptotic behavior in the sector arg s ∈ q(s) = [ν+1/2]∑ bn(−s)−3n/2 +O s−3[ν+1/2]/2−1 + c+(−s)− (−s)3/2(1 +O(s−1/4) as s→ ∞, arg s ∈ , arg(−s) ∈ , (4.17) q(s) = [ν+1/2]∑ bn(−s)−3n/2 +O s−3[ν+1/2]/2 + c−(−s)− (−s)3/2(1 +O(s−1/4) as s→ ∞, arg s ∈ , arg(−s) ∈ , (4.18) where we have used the notation [r] for the integer part of the positive number r, i.e. [r] ∈ N0, [r] ≤ r < [r] + 1. The coefficients c+ and c− of the exponential terms, which oscillate on the respective boundaries of the sector , are given by the formulae c+ = − eπ(ν+ )i + 1 + ν), (4.19) c− = − e−π(ν+ )i + 1 + ν), (4.20) CRITICAL EDGE BEHAVIOR IN RANDOM MATRIX ENSEMBLES 47 where Γ denotes the Gamma function. Moreover, either the relations (4.17), (4.19) or the relations (4.18), (4.20) can be taken as a characterization of the solution q(s). Alternatively, the solution q(s) can be characterized by its comparison to the Boutroux tri-tronquée solution q(tri−tronq)(s) of the Painlevée II equation, which is defined as the unique solution satisfying the asymptotic condition q(tri−tronq)(s) ∼ bn(−s)−3n/2, as s→ ∞, arg s ≡ π + arg(−s) ∈ . (4.21) The solution q(s) we are working with is the one whose asymptotic behavior as s→ −∞ is given by the equation q(s)− q(tri−tronq)(s) = −e −π(ν+ 1 )i + 1 × |s|− 32 ν− 14 e− 2 |s|3/2(1 +O(s−1/4) , as s→ −∞. (4.22) The coefficients bn of the asymptotic series in (4.17), (4.18), and (4.21) are determined by substitution into the Painlevé II equation. Indeed, the following recurrence relation takes place b0 = 1, b1 = bn+2 = 9n2 − 1 bmbn+2−m − n+2−l∑ blbmbn+2−l−m. (4.23) Using relation (4.16) between the Painlevé II and Painlevé XXXIV functions we arrive at the asymptotic characterization of the function u(s) of Theorem 1.5. Proposition 4.1 The solution u(s) of the Painlevé XXXIV equation which appears in Theorem 1.5 is uniquely characterized by one of the following asymptotic conditions u(s) = [2α+1]∑ − 3n+1 s−3[2α+1]/2−1 + d+s −3α+ 1 1 +O(s−1/4) as s→ ∞, arg s ∈ , (4.24) u(s) = [2α+1]∑ − 3n+1 s−3[2α+1]/2−1 + d−s −3α+ 1 1 +O(s−1/4) as s→ ∞, arg s ∈ , (4.25) 48 A.R. ITS, A.B.J. KUIJLAARS, and J. ÖSTENSSON where e±2απi − 1 2−6α− 3Γ(1 + 2α). (4.26) Alternatively, the solution u(s) can be characterized by the asymptotic relation u(s)− u(tri−tronq)(s) = −e −2απi − 1 2−6α− 3Γ(1 + 2α) × s−3α+ 12 e− 43s3/2 1 +O(s−1/4) , as s→ +∞. (4.27) The Painlevé XXXIV tri-tronquée solution u(tri−tronq)(s) is determined by the asymptotic condition u(tri−tronq)(s) ∼ α√ − 3n+1 2 , as s→ ∞, arg s ∈ −π, π . (4.28) Finally, the coefficients an of the asymptotic series above can be expressed in terms of the coefficients bn defined in (4.23), with ν replaced by 2α + 1/2: 2 an = bn+1 − 3n− 2 k,m≥1;k+m=n+1 bkbm. Remark 4.2 The leading asymptotics of the Painlevé II function q(s) as s → +∞ is known (see [33]; see also [24, Chapter 10]). Unfortunately, the leading term is not enough to derive the corresponding asymptotics as s → −∞ of the Painlevé XXXIV function u(s). Indeed, the leading asymptotics of q(s) as s→ +∞ is of the form q(s) ∼ s3/2 + χ , (4.29) (the phase χ is known) and it cancels out in the right-hand side of equation (3.22). The better way to study the large negative s asymptotics of the function u(s) is via the direct analysis of the model RH problem for Ψα. The case α = 1 shows that we might expect oscillating behavior as s → −∞ (see Figure 6) and indeed, assuming that α− 1/2 6∈ N0, we are able to show that u(s) = (−s)3/2 − απ +O(1/s2), as s→ −∞. (4.30) The proof of (4.30) will be given in a future publication. Moreover, we conjecture that asymptotics (4.30) determines the solution u(s) uniquely. Acknowledgements Alexander Its was supported in part by NSF grant #DMS-0401009. 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Anal. 20 (1989), 966–986. http://arxiv.org/abs/nlin/0410009 Introduction and statement of results Unitary random matrix models The model RH problem Connection with the Painlevé XXXIV equation Overview of the rest of the paper Proof of Theorem 1.1 and 1.2 The Riemann-Hilbert problem for orthogonal polynomials The model RH problem Existence of solution to the model RH problem Some preliminaries on equilibrium measures Steepest descent analysis Preliminaries First transformation Y T Second transformation T S The parametrix P() The parametrix P(a) The parametrix P(0) Third transformation S R Completion of the proofs of Theorem 1.1 and 1.2 Proof of Theorems 1.4 and 1.5 The Painlevé II RH problem Connection with Differential equation Special choice (spec) Special choice 0 Proof of Theorem 1.4 and 1.5 Concluding remarks The case = 0 The case = 1 Asymptotic characterization of the Painlevé function u(s)

0704.1973

Vibrational effects on low-temperature properties of molecular conductors

Vibrational effects on low-temperature properties of molecular conductors Jernej Mravlje a,∗, Anton Ramšak b,a, Rok Žitko a aJožef Stefan Institute, Jamova 39, Si-1000, Ljubljana, Slovenija bFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Si-1000, Ljubljana, Slovenija Abstract We calculate characteristic correlation functions for the Anderson model with additional phonon-assisted coupling to the odd conduction channel. This model describes, for example, the behavior of a molecule embedded between two electrodes in linear transport experiments where the position of the molecule with respect to the leads affects the tunneling amplitudes. We use variational projection-operator method and numerical renormalization group (NRG) method. The spin is Kondo screened either by even or odd conduction channel depending on the gate voltage and electron-phonon coupling. However, in all regimes the gate- voltage dependence of the zero temperature conductance is found to be qualitatively the same as in the model with no coupling to the vibrational mode. Key words: Kondo effect, molecular conductors, dynamic symmetry breaking PACS: 72.15.Qm,73.23.-b,73.22.-f In recent years the studies of quantum impurity sys- tems have undergone a considerable revival due to improve- ments in experimental techniques for measuring the elec- tron transport through quantum dots and single molecules, as well as due to the development of the DMFT technique which maps interacting lattice problems to quantum impu- rity problems with an additional self-consistency condition. The prototype model for this class of problems is the An- derson model for a single impurity in a metallic host with Himp = Un↑n↓ + ǫn, where n = n↑ + n↓ is the number of electrons occupying the impurity orbital with energy ǫ relative to the chemical potential, and with the Coulomb repulsion U due to the double occupancy of the impurity orbital. An important class of quantum impurity models include coupling to the bosonic degrees of freedom describing the vibrational modes of the molecule or phonons in the bulk. There are two basic types of the electron-phonon coupling, (i) the Holstein coupling of form nx, where n is the elec- tron density and x the oscillator displacement, and (ii) the coupling to hopping term of form vx, where v is the hop- ping (hybridization) operator that couples the impurity to the conduction band. While type (i) is more relevant when the oscillation is related to the change in volume to which the electron is confined (breathing modes), type (ii) is rel- ∗ Corresponding author. E-mail address: [email protected], Fax: +38661 477-3724. evant whenever the displacement modulates the hopping probability. The addition of the Holstein term to the AndersonHamil- tonian effectively reduces the Coulomb repulsion and the hybridization Γ. The effect of the electrons on the phonon propagator is also interesting: when effective U changes sign, a peak in the phonon propagator at reduced frequen- cies (the ’soft mode’) emerges [1,2]. The soft mode is re- lated to the charge susceptibility, which is increased in this regime [3]. Very recently, similar behavior was found also in the case where the electron-phonon coupling term is of the form Hel−ph = gxvodd, where vodd describes the hopping from impurity orbital to the odd conduction channel (antisym- metric combination of the orbitals of the noninteracting part of the Hamiltonian) [4]. The model without phonons consist of veven, which couples the impurity only to the even conduction channel (symmetric combination of orbitals of the noninteracting part of Hamiltonian). The same model (but for finite U instead of U → ∞ treatment of Ref. [4]) was analyzed also with the varia- tional projection-operator method [5]. In this method the ground state is expressed in terms of the ground state of an auxiliary noninteracting Hamiltonian [6]. Several vari- ants were tested, with parameters chosen so as to allow for coupling, (i) only to even channel, (ii) only to odd channel, (iii) a combination of both. Variational method applied to Preprint submitted to Elsevier October 26, 2018 http://arxiv.org/abs/0704.1973v1 0 0.5 1 (iii) 0 0.25 0.5 (c) (d) x (iii) Figure 1. Variational results [U/Γ = 5,Γ/D = 0.04,Ω = 2.5Γ; Ω is the phonon frequency, D the bandwidth. δ = 0 (left pannels), g/U = 0.36 (right pannels)]: (a, b) Variational ground state energy. (c) Displacement fluctuations and displacement for (iii). In (i) and (ii) the parity symmetry is retained, hence the displacement vanishes. (d) The conductance. variant (iii) leads in certain parameter regimes to a ground state of broken parity symmetry (see Fig. 1(a,b)), marked by non-vanishing expectation value of displacement (Fig. 1(c), thin-dotted) and consequently [5] considerably re- duced conductance through a molecule ( plotted as a func- tion of departure from particle-hole symmetric point δ = ǫ+U/2 in Fig. 1(d)). As discussed below, the ground state should have a well defined parity, therefore only the vari- ants (i) and (ii) of the variational procedure correspond to the ground state of correct symmetry. While it would be in- structive to implement the variational method in a manner which correctly took into account the tunneling between the classically degenerate minima of the oscillator poten- tial [4], it currently appears that the implementation would require calculating matrix elements between two distinct Hartree-Fock vacua, which precludes the use of Wick’s the- orem upon which our current implementation of the varia- tional procedure is based [6]. We thus present preliminary results obtained with the NRG method, which does not suffer from this problem. The parity in NRG results is not broken; the expectation values of the displacement and of the hopping term vodd thus vanish. In Fig. 2(a) we compare the fluctuations of hopping to even and odd channels as a measure of the ’ac- tivity’ of corresponding channels. For g large enough the latter are larger, corresponding to increased fluctuations of the displacement and the emergence of the soft mode. The ground state of the system, as seen from the NRG renor- malization flow (not shown here) corresponds to the Fermi liquid ground state of the single-channel Kondo problem [7] with the characteristic quasi-particle scattering phase shift δq.p ∼ π/2 in the even or odd channel, depending on whether the effective phonon mediated coupling to the odd channel is smaller or larger than the direct coupling to the even channel. When the couplings match (marked in Fig. 2(b) by vertical lines), an unstable fixed-point of the two- channel Kondo model type is found. 0 0.5 1 1.5 0 0.5 1 1.5 Ω/U=0.5 Ω/U=0.75 Ω/U=1 Figure 2. NRG results [U/Γ = 25, D = U ]: (a) fluctuations of hop- ping to even (full) and odd (dashed) channel, (b) fluctuations of displacement. 0 0.2 0.4 0.6 0.8 1 even Figure 3. NRG results [U/Γ = 25, D = U,Ω/U = 1, g/Ω = 1.3]: Zero temperature quasiparticle scattering phase shifts in even (circles) and odd (squares) channel plotted versus departure from p-h symmetry calculated using NRG. Corresponding conductance is also plotted (dashed). Note that the phase shifts are defined modulo π. In Fig. 3 the scattering phase-shifts are plotted as a function of δ. The coupling to the phonon-assisted channel is chosen so that in the ground state the impurity spin is screened by the odd channel for small δ, while for larger δ it is screened by the even channel, as seen from δq.p.;even,odd ∼ π/2, respectively. Further away from the symmetric point the model is tuned into the valence fluctuating regime where the Kondo effect is suppressed. When the model is used to describe a molecule (or a quantum dot) embedded between two leads, the scattering phase shifts directly de- termine the differential conductance (Fig. 3 dashed). The conductance curve shown is qualitatively equal to that of the generic one-electron transistor in the single-channel Kondo regime, despite the fact that the Kondo effects occurs in different channels as the gate voltage is swept. The work was supported by SRA under grant Pl-0044. References [1] A. C. Hewson and D. Meyer, J. Phys.: Condens. Matter 14, 427 (2002). [2] G. S. Jeon, T. Park, and H. Choi, Phys. Rev. B 68, 045106 (2003). [3] J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 72, 121403(R) (2005). [4] C. A. Balseiro, P. S. Cornaglia, and D. R. Grempel, Phys. Rev. B 74, 235409 (2006). [5] J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 74, 205320 (2006). [6] T. Rejec and A. Ramšak, Phys. Rev. B 68, 033306 (2003). [7] For a review of the Kondo physics see: A. C. Hewson, The Kondo Problem to Heavy Fermions, University Press, Cambridge, 1993. References

0704.1974

Agile low phase noise radio-frequency sine wave generator applied to experiments on ultracold atoms

Agile low phase noise radio-frequency sine wave generator applied to experiments on ultracold atoms O. Morizot, J. de Lapeyre de Bellair, F. Wiotte, O. Lopez, P.-E. Pottie, and H. Perrin∗ Laboratoire de physique des lasers, Institut Galilée, Université Paris 13 and CNRS, Avenue J.-B. Clément, F-93430 Villetaneuse, France (Dated: October 31, 2018) We report on the frequency performance of a low cost (∼500 $) radio-frequency sine wave genera- tor, using direct digital synthesis (DDS) and a field-programmable gate array (FPGA). The output frequency of the device may be changed dynamically to any arbitrary value ranging from DC to 10 MHz without any phase slip. Sampling effects are substantially reduced by a high sample rate, up to 1 MHz, and by a large memory length, more than 2 × 105 samples. By using a low noise external oscillator to clock the DDS, we demonstrate a phase noise as low as that of the master clock, that is at the level of −113 dB.rad2/Hz at 1 Hz from the carrier for an output frequency of 3.75 MHz. The device is successfully used to confine an ultracold atomic cloud of rubidium 87 in a RF-based trap, and there is no extra heating from the RF source. PACS numbers: 39.25.+k, 06.30.Ft, 07.57.Hm I. INTRODUCTION Radio-frequency (RF) fields are used in cold atom ex- periments for different purposes: for instance, evapo- rative cooling performed in a magnetic trap relies on RF field coupling between the different atomic magnetic states [1, 2]. This technique led to the first observa- tion of Bose-Einstein condensation (BEC) [3, 4]. Also, RF pulses are used for dissociating ultracold molecules produced from ultracold gases through Feshbach reso- nances [5]. More recently, RF fields have been used to- gether with static magnetic fields for trapping utracold atoms at a temperature of a few µK in unusual geome- tries [6, 7]. There is a growing interest for these “RF- based traps” among atomic physicists, for creating dou- ble well traps on atom chips [8, 9, 10] or proposing new kinds of confining potentials [11, 12, 13]. In both cases, a single frequency RF signal must be frequency swept over some range, often larger than the initial frequency, following a precise time function lasting several seconds. Typically the RF frequency is varied between 1 MHz and a few tens of MHz in 0.1 s to 10 s in the ramping stage, and held at the final frequency for seconds in the plateau stage. For cooling purposes, commercial RF generators fit physicists’ needs reasonably well, even if a better res- olution in arbitrary frequency ramps would be appreci- ated. However, in the case of RF-based trapping, the requirements are stronger. The main difference between these two situations is as follows: in evaporative cooling the cold atomic sample is located away from the region of efficient coupling, whereas in the RF-based trapping scheme the atoms sit exactly at the point where the RF field has the largest effect. The quality of the RF source is then much more important than for evaporative cooling. In fact, the cloud position is directly related to the value ∗Electronic address: [email protected] of the RF field frequency, and the trap restoring force, or equivalently the oscillation frequency νt in the harmonic approximation, is linked to the RF amplitude. As a re- sult, any amplitude noise, frequency noise or phase noise of the RF signal during the ramp or the plateau leads to a heating of the cold atomic cloud. This motivated the construction of a synthesizer fitting our requirements. This paper is organized as follows. In section II we give explicit expressions for the heating of the cold atom sample for frequency and amplitude noise in the case of RF-based trapping. In section III, we describe our RF synthesizer. Finally, section IV is devoted to experimen- tal results on its performance and comparison between the different RF sources tested on the BEC experiment. II. REQUIREMENTS ON THE RF SOURCE FOR RF-BASED TRAPPING In this section, we will focus on the RF-dressed trap that we experimentally produce in the laboratory [7]. The extension of the main conclusions to other RF- dressed trap geometries is straightforward. The trap confines the atoms in all three space di- mensions. The trapping force arises from the inter- action between the linearly polarized RF field B(t) = BRF cos(2πνRFt) and the atoms in the presence of an in- homogeneous magnetic field. This interaction results in a transverse confinement of the atoms to the surface of an ellipsoid. The atoms are free to move along the con- fining surface, resulting in a kind of “bubble trap” [6]. Due to gravity, however, the atoms are concentrated at the bottom of the ellipsoid. Their motion is pendulum- like in the horizontal directions, and imposed by the RF interaction along the vertical z axis. This last direction is thus the most sensitive to the RF field properties (fre- quency νRF, amplitude BRF) and we will concentrate on the vertical motion in the following. Along this direc- tion, heating or atomic losses may arise from frequency http://arxiv.org/abs/0704.1974v1 mailto:[email protected] or amplitude noise, phase hops or sudden frequency hops during the RF ramp. A. Dipolar excitation heating Very generally, for atoms in a one dimensional har- monic trap with a trapping frequency νz, any effect pro- ducing a jitter in the trap position z results in linear heating through dipolar excitation. The average energy of the cold atomic cloud E increases linearly as [14]: Mω4z Sz(νz) (1) where ωz = 2πνz, M is the atomic mass and Sz is the one-sided Power Spectral Density (PSD) of the position fluctuations δz, defined as the Fourier transform of the time correlation function [14] Sz(ν) = 4 dτ cos(2πντ)〈δz(t) δz(t + τ)〉. (2) The time variations of energy, E, and temperature, T , are related by Ṫ = Ė/3kB. The factor 3 arises because only one degree of freedom is responsible for the temperature increase, as is the case in our atom trap. The vertical trap position z is linked to the RF frequency νRF by z = Z(νRF) such that Sz is directly proportional to Sy, the PSD of relative frequency noise of the RF source, through: Sz(ν) = Sy(ν). (3) The function Z depends on the geometry of the static magnetic field. In a quadrupolar field, for instance, Z is linear with νRF and its derivative is simply a constant. From Eqs. (1) and (3), we infer that the linear heating rate is proportional to Sy(νz). To fix orders of magnitude, within the static mag- netic field of our Ioffe-Pritchard trap [7], νz may be ad- justed between 600 and 1500 Hz and the typical tem- perature of the cold rubidium 87 atoms ranges from 0.5 to 5 µK. To maintain a temperature below condensa- tion threshold for a few seconds, a linear temperature increase below 0.1 µK/s is necessary. This rate corre- sponds to Sz(νz) = 0.27 nm/ Hz for an intermediate trap frequency of 1000 Hz and νRF = 3 MHz, which in turn corresponds to a one-sided PSD of relative frequency fluctuations of the RF source Sy(νz) = −118 dB/Hz. B. Parametric heating Fluctuations of the RF field amplitude BRF are re- sponsible for parametric heating in the vertical direction. The trapping frequency νz is inversely proportional to BRF [6]: 2F h̄ MγBRF . (4) Here, γ is the gyromagnetic ratio of the atom and F is the total atomic spin (F = 2 for rubidium 87 in its upper hyperfine state). The atoms are assumed to be polarized in their extreme mF = F substate. The cloud tempera- ture increases exponentially due to amplitude noise with a rate Γ, where Γ = π2ν2zSa(2νz) (5) and Sa is the PSD of the relative RF amplitude noise [14]. In order to perform experiments with the BEC within a time scale of a few seconds, Γ should not exceed 10−2 s−1. Again, for a typical oscillation frequency of 1000 Hz, this corresponds to Sa < −90 dB/Hz. This requirement is rather easy to match and does not limit the choice of the RF source, as -110 dB/Hz is commonly reached. How- ever, particular care must be taken in the choice and installation of the RF amplifier usually used after the source. C. Phase hops Controlling the phase of the RF source is not a cru- cial point for evaporative cooling, but is an issue in the case of RF-based traps, where it is associated with trap losses. In the latter situation, the atomic spin follows an effective magnetic field oscillating at the RF frequency. A phase hop results in a sudden flip of this effective field, the atomic spin being then misaligned with the new di- rection of the field. Some of the atoms end up with a spin oriented incorrectly and escape the trap. For this reason, phase hops should be avoided. This is difficult to achieve with an analog synthesizer over a wide frequency sweep. By contrast, Direct Digital Syn- thesis (DDS) technology is well adapted to this require- ment [16]. D. Frequency steps The drawback of DDS technology is that, although the phase is continuous, the frequency is increased by N successive discrete steps δν. A sudden change in the RF frequency also results in atomic losses, through the same mechanism as for phase hops. The effec- tive magnetic field rotates, at most, by the small angle δθ = 2π δν/(γBRF/2) [15]. For a linear ramp over a fre- quency range ∆ν = Nδν, the fraction of atoms remaining after the full ramp is of order (1− Fδθ2/2)N . Given the expression for δθ, this reads: NγBRF ≃ 1− F . (6) Thus, for the remaining fraction to be larger than 95%, the number of frequency steps should be larger than 10F (4π∆ν/γBRF) 2. For example, for a 2 MHz ramp with a typical RF amplitude of 200 mG, N should be larger than 16,000. In addition to this loss effect, a sudden change in the RF frequency results in a sudden shift of the position of the RF-dressed trap. This may cause dipolar heating of the atoms, especially if this frequency change occurs every trap period. The frequency steps should thus be as small as possible, a few tens Hz to a hundred Hz typically. III. DEVICE DESCRIPTION Our experiment has the following requirements. First, during the ramp the gap between two successive frequen- cies must fulfil the criterion discussed in section IID. Sec- ond, adiabaticity criteria require a controlled, optimized ramp. Third, the ramp duration should be tunable from one experiment to the other on a time scale ranging from 50 ms to 10 s. Finally, frequency and amplitude noise must be small enough, as discussed in previous section. Given the amplitude of the frequency sweep we need to perform in our experiment, DDS technology appears to be an ideal solution. We previously used a com- mercial DDS-based RF generator, the Stanford DS-345. Its memory length is limited to 1,500 frequency points for each waveform with an adjustable step duration of 40 Msample/s/N , with N=1 to 234 − 1. The major in- convenience of this device is that it is unable to hold the final frequency at the end of the ramp. Instead, the frequency sweep is looped indefinitely. It forced us to sacrifice either frequency resolution during the ramp or duration of the plateau. To benefit from both a low noise RF spectrum during the ramp and a very small frequency step, and to improve the possibilities of the RF source, we designed a digital RF synthesizer with a > 200, 000 memory length and great agility, fitted to our experimen- tal requirements. The main features of the RF synthesizer are as fol- lows. It is able to generate 262,144 sine waves in a row in the radio frequency band (DC - 10 MHz), owing to its 1 M-byte fast asynchronous Static Random Access Memory (SRAM). Each frequency is an integer chosen by the user. A key feature of the device is a variable sample frequency over the sequence, as the duration of each generated frequency can be tuned from 1 µs to 1 hour. The general architecture of the device is sketched in Fig. 1. It is made up of one evaluation kit DDS board, and a “starter kit” Field Programmable Gate Array (FPGA) board. The device is managed by a Personal Computer (PC). The DDS is clocked by an ultra-stable external reference signal. The output of the DDS is a sine wave, filtered through a 10 MHz low-pass filter. The frequency ramp synthesis starts when a TTL signal is sent to the device. File *.txt Memory Spartan-3 Clock frequency AD 9851 PC FPGA board DDS board TTL start DDS device filter 10MHz RF output FIG. 1: Layout of the system. The DDS board combines digital parameters and an analog reference clock frequency to generate a sine wave [16]. The heart of the DDS board is a digitally pro- grammable device using DDS technology, the AD9851. It has a 32-bit phase accumulator, a 14-bit digital phase-to- amplitude converter and a 10-bit Digital-to-Analog Con- verter (DAC). Its maximum clock frequency is 180 MHz, and its maximum output frequency is 70 MHz. The phase, relative to the clock signal, is encoded in 5 bits, and is adjustable to any value from 0 to 2π. This results in a rather poor phase resolution of 196 mrad. The FPGA board manages the 1 Mb memory, the time settings and the input/output of the device through se- rial port. The FPGA is a Xilinx spartan-3 XC3S200, providing 200, 000 logic gates. These logic gates are designed with VHDL [17]. A Universal Asynchronous Receiver Transmitter(UART) and a Picoblaze microcon- troller are loaded into the FPGA, in order to commu- nicate through serial port to the PC and to load the on-board memory. We wrote our own VHDL scripts to manage the DDS board and the FPGA in-board memory. The FPGA board is clocked internally at 50 MHz. The output sample rate of the device depends on the num- ber of clock cycles ncycle between frequency data trans- ferred to the DDS board. We set ncycle = 50 so that ncycle/50 MHz = 1 µs, large enough thus ensuring safe operation of the frequency data. Software was developed in C with CVI Labwindows in order to configure the device. The user writes a plain text file ordering all the frequencies of the desired frequency ramp. The set of frequencies is separated into 10 groups of adjustable length, with a given sample rate for each. The group lengths and the corresponding sample rate are each translated by the software into 4 bytes. In addition, each frequency in a given group, an integer written as a decimal number, is translated into 4 bytes (32 bits). The software sends these bytes by serial port to the FPGA board. The clock frequency fc and its phase noise level are the key points for setting the frequency performance of the device. The lower the phase noise of the clock signal, the lower is theminimum phase noise of the output frequency of the device (see section IVA). The clock signal used for the experiment, see next sec- tion, is the 10 MHz clock signal from an ultra-stable Oven Controlled Crystal Oscillator (OCXO) BVA-8600. Its phase noise PSD is −115 dB.rad2/Hz at 1 Hz. As this clock frequency is very close to the desired maxi- mum output frequency (≃ 10 MHz, see next section), and to fulfil Shannon’s theorem [18, 19], we generate a higher clock frequency by using the internal frequency clock multiplier, at ×6, of the DDS board. IV. RESULTS The device presented in section III was first tested for its frequency stability performance, as described in sec- tion IVA and summarized in Table I. It was then in- tegrated into a Bose-Einstein condensation experiment, see section IVB. A. Device frequency performance 1. Quantization error, Phase Accumulator truncation and “magic” frequencies By construction, digitization yields to inaccuracies in frequency synthesis. The output frequency of 32-bit res- olution DDS is given by fRF = fc × where w is a binary 32-bit tuning word. The output fre- quency can thus differ slightly from the desired frequency. As fc = 60 MHz, the maximum digitization error δf is 6× 107/232 = 0.014 Hz. As our software only takes integer frequencies as in- put, a given frequency fRF will be synthesized without sampling error if it may be written exactly as an integer in the form given at Eq. (7). This condition is written w = n× 232−p (8) where n is a positive integer and p is the power of 2 in the prime factorization of the clock frequency fc. In our case, fc = 2 8 × 3 × 57 Hz and p = 8 so that every frequency verifying fRF = n× 234375 Hz (9) will yield to no digitization error. n should be less than 2p/3 for the desired frequency to be in the synthesizer range (fc/3). In addition, when the AD9851 converts the calcu- lated phase to an effective output amplitude, only the first most significant 14 bits are used, even though the AD9851 is a 32-bit synthesizer, in order to handle prac- ticable number of entries in a lookup table. Truncating the phase results in errors in amplitude that are periodic in the time domain. These errors will be seen as spurs in the frequency domain. However, for particular frequen- cies which are exactly encoded by the first 14 bits (The TABLE I: Performance of the device with fc = 60 MHz. The relative frequency noise is computed from the phase noise data, and given for 3.75 MHz (“magic” frequency) and 3 MHz (larger noise value). Parameters Min. Max. Units Dynamic 0 10 MHz Line-width - 30 mHz Digitalization error 0 14 mHz Sample rate adjustable 1 MHz Memory length 1 262,144 pts Phase noise@1Hz -113 -78 dB.rad2/Hz Rel. freq. noise@1Hz -244 -207 dB/Hz Rel. ampl. noise@1Hz - -120 dB/Hz last 18 bits are 0.), the phase is not truncated at all, yielding no spurious effects and the best PSD of phase noise. This occurs for every frequency satisfying fRF = n× fc/214 (10) with n a positive integer. As fc = 60 MHz, we have fRF = n × 3662.109375 Hz. The most stringent condi- tion being the first one, we will denote the frequencies satisfying Eq. (9) as “magic frequencies”. In order to il- lustrate the difference between a “magic frequency” and another one, we performed a set of noise PSD measure- ments for two frequencies: a first set for fRF = 3 MHz which is not a “magic” frequency, and a second set for fRF = 3.75 MHz which is a “magic” frequency. 2. Spectral density of noise We recorded the spectral density of noise of our syn- thesizer at a given frequency fRF by FFT analysis of the beat note at 0 Hz with a second synthesized ref- erence signal. The measurement bench is sketched in Fig. 2. In order to generate a tunable reference signal in the RF range we used an analog synthesizer, Rhode & Schwartz SML-01 (R&S), for synthesizing a signal at a high frequency, and then divided by 100 to give fRF for subsequent mixing. The beat note was recorded and an- alyzed by a digital FFT analyzer HP 3562A sampling at 256 kHz. The R&S, as all the measurement devices, was clocked at 10 MHz by the ultra-stable BVA-8600 quartz oscillator. All the plots shown on Fig. 3 are raw spectra of the beat note. The reference signal (R&S) itself was characterized by making a beat note with a second iden- tical R&S based synthesis. This corresponds to the line labelled as “Reference” in Fig. 3. By tuning the phase difference φ between the RF sig- nal and the reference signal to π/2, we recorded phase noise. At fRF = 3 MHz, the PSD of phase noise is −78 dB.rad2.Hz−1 at 1 Hz, which corresponds to a PSD of relative frequency noise of −207 dB.Hz−1. From the data reported on Fig. 3, and assuming a Lorentzian line- shape for the beat note, the linewidth δf of the RF signal, given by δf = πν2RF df Sy(f) , (11) is as small as 30 mHz over a bandwidth ∆f = 100 kHz for an output frequency of 3 MHz [23]. We found similar results for output frequencies from 1 to 5 MHz. At a “magic” frequency, as for example at 3.75 MHz, where truncation effects cancel out, the results are even better, with a PSD of phase noise as low as −113 dB.rad2.Hz−1 at 1 Hz, only 2 dB higher than the phase noise of the BVA 8600. The observed value corresponds to the ultimate phase noise of the DDS chip specified by the manufacturer. The relative frequency noise is then −244 dB.Hz−1. The frequency noise performance is naturally linked to the quality of the master oscillator. To illustrate this fact, the same measurements were repeated with the im- proved OCXO of a DS-345, the ERC EROS-750-SBR-4, as master oscillator. No significant change in the fre- quency performances were noticed at non magic frequen- cies. At a magic frequency, the PSD of phase noise in- creased to −100 dB.rad2/Hz, which is consistent with the phase noise specifications of this quartz. The PSD of relative amplitude noise Sa was recorded with the same measurement bench, by tuning the phase φ to 0. The reference signal was also delivered by the R&S. The recorded spectrum is very close to the “Reference” line itself at the level of −120 dB.Hz−1 at 1 Hz, close to the input noise of the FFT analyzer. Note that for rele- vant excitation frequencies ν = 2νz (larger than 1.2 kHz), the PSD of amplitude noise is lower than −130 dB. In practice, we use in our BEC experiment a pro- grammable RF attenuator Minicircuit ZAS-3, driven by an analog output channel of a National Instrument PC card PCI-6713, in order to vary the RF amplitude sent to the RF coil. We repeated the measurements described above with this attenuator and found an increase of Sa, starting at −115 dB.Hz−1 at 1 Hz, which corresponds to a noise figure of the programmable attenuator of 5.1 dB. At the output of the attenuator, the RF signal is am- plified by a class-A amplifier Kalmus 505 F. Its gain is ¸100SML-01 -8600 10 MHz ZAS-3 FIG. 2: Noise spectrum measurement bench. The gain in the mixer was fitted to the output signal, depending on the phase between the two beating signals. 1 10 100 1k 10k 100k -180 -160 -140 -120 -100 1 10 100 1k 10k 100k -160 -140 -120 -100 2 . Frequency (Hz) DDS 3.75 MHz DDS 3 MHz Reference BVA 8600 FIG. 3: PSD of relative amplitude noise, upper plot, and phase noise,lower plot. Averaging = 100. FFT’s sampling frequency = 256 kHz. The low pass filter cutting frequency of the mixer was set to 200 kHz. The reference curve corresponds to the beat note of two identical R&S synthesizers. Plotted for comparison is the phase noise specifications of the BVA 8600 quartz oscillator, as given by the manufacturer. 37 dB and its noise figure is typically +10 dB according to the manufacturer specifications. B. Integration into the experiment In this section, we present the main experimental re- sults concerning the heating rate of the atomic sample. As described in section II, the RF signal is used for producing the bubble-like trap, where ultracold rubid- ium atoms are accumulated at the bottom of the surface. The trap is very anisotropic with stronger confinement in the vertical direction [7]. For the RF-dressed trap to be efficiently loaded from the standard Ioffe-Pritchard static magnetic trap, described in [20], the RF frequency fRF has to be ramped up from 1 MHz to a final fixed frequency fendRF ranging from 2 to 10 MHz. The static magnetic field, necessary both for magnetic trapping and RF-induced trapping, is always present. A typical ramp is shown on Fig. 4. The frequency is ramped more slowly around 1.3 MHz, corresponding to the resonant frequency at the center of the magnetic trap where adiabaticity of spin rotation is more difficult to obtain. At the end of this ramp, which may last between 75 ms and 500 ms, the RF frequency is held between 0.1 and 10 s for testing the lifetime and heating rate of the atoms in the RF-based trap. The RF signal is amplified by 37 dB with a single stage amplifier, and the RF field is produced by a small circular antenna. The RF field is linearly polarized and its ampli- tude BRF may be adjusted between 70 and 700 mG. We 0 50 100 150 200 Time (ms) FIG. 4: Typical shape of a radio-frequency ramp applied to the ultracold atomic sample. In the present example fRF is increased from 1 to 3 MHz within 150 ms. At the end of this ramp, the RF frequency is maintained at its final value for some holding time in the RF-based trap, dashed line. record the atomic temperature after the RF ramp as a function of time while the atoms are confined in the RF- based trap. The temperature is deduced from the cloud size along z after a 7 ms of ballistic expansion. The same measurement was repeated with different RF sources. First, we used a Agilent 33250A synthesizer with a RF amplitude larger than 500 mG for both the frequency ramp and the final holding frequency. Such RF ana- log synthesizers operated at fixed frequency exhibit very good relative frequency noise in most cases, typically at the −180 dB/Hz level or better. However, as mentioned by Colombe et al. [7] and confirmed by White et al. [21], the relative frequency noise increases by a few decades if the output frequency is driven with an external analog voltage. The frequency was indeed tuned through an ex- ternal voltage control provided by the PC analog board (NI 6713), such that the modulation depth was ±1 MHz on a central frequency of 2 MHz. We obtained both a short lifetime, typically 400 ms at 1/e, and a strong lin- ear heating, as shown on Fig. 5 full circles. The heating rate is measured to be 5.0 µK/s. This rate, given the RF amplitude, corresponds to a relative frequency noise of Sy = −100 dB/Hz at the trap frequency of 600 Hz. This noise is quite high and is related to voltage noise on the external frequency control of the synthesizer. This effect is strong in our case as the frequency is varied with a large modulation depth (∆f/f = 1). We also tested a two step scheme, with a first ramp performed by a Stanford DS-345 DDS (1,500 frequency points), followed by the R&S maintained at fixed fre- quency for the full holding time. This scheme allows one to benefit from the excellent frequency stability of the second device used at fixed frequency. With this setup, 0 1 2 3 4 5 6 7 8 Agilent Our DDS device holding time in the trap (s) FIG. 5: Comparison of heating of the atomic cloud in the bub- ble trap: Agilent 33250A synthesizer, full circles, or present device, open diamonds, is used for creating the RF ramp and the final radio-frequency fendRF . We observe a heating rate of 5.0 µK/s in the first case and 0.47 µK/s the second one, as given by a linear fit, full lines. heating during the plateau stage seemed to be completely suppressed [7]. However, a large dispersion was observed in the atom number data after the switching, which pre- vented us from characterizing the heating very precisely. This dispersion is due to a random phase hop at the switching time between the two synthesizers, resulting in atomic losses. We studied the effect of the random phase hop on the atom number by recording, for each experi- ment, the phase difference at the switching time with a control oscilloscope. The results are presented on Fig. 6, full circles. For the maximum phase hop, π, 80% of the atoms are lost. This figure depends on the atomic tem- perature, the losses being higher at lower temperature, and is well reproduced by theory, as shown on Fig. 6, black line. The theoretical curve is calculated for an RF amplitude of 470 mG by averaging the loss probability over the positions of the atoms, as deduced from a ther- mal distribution at a temperature of 4 µK. The fact that the trap is able to hold two of the five spin components of the F = 2 hyperfine state is taken into account. Finally, we performed the heating rate and lifetime measurements with our DDS device and an amplitude of 70 mG. We found an increased lifetime, up to 9.6 s. During the first second, the vertical “kinetic tempera- ture”, deduced from the vertical cloud size after time of flight, decreases due to thermalization with the horizon- tal degrees of freedom which initially have a lower energy due to the loading procedure: Fig. 5, open diamonds. After 1 second, a linear heating rate of 0.47 µK.s−1 is clearly observable. No exponential parametric heating is measurable. We believe that the remaining linear heat- ing is due to residual excitation by scattered light, which was not present in the previous case. The same heating -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 6 ) phase hop at switching (degrees) experiment theory FIG. 6: Number of atoms remaining after switching between the two synthesizers as a function of the phase hop. Exper- imental data, full circles, are compared to a calculation, full line, assuming a RF amplitude of 470 mG and a temperature of 4 µK. rate was indeed observed within the static magnetic trap, when the RF source was off. Heating directly linked to the RF source is not observable. V. CONCLUSION We built a low cost DDS RF generator, based on a DDS chip and its evaluation kit. It is extremely agile since we have the ability of using up to 10 different time steps during the sequence, allowing an efficient use of the FPGA on-board memory. In contrast to Stanford DS-345 RF synthesizers, the frequency ramp is not repeated at the end of the sequence. The last frequency is kept until a new TTL pulse is proceeded to end the sequence. Its large memory length, up to 262,144 frequency samples, allows very small frequency steps, thus strongly reduc- ing the atomic losses in our RF-dressed trap experiment. The DDS technology ensures that the phase is continu- ous all over the frequency ramp, which is essential for RF trapping applications. Apart from these high resolution features, the device presents excellent spectral properties. The power spec- tral density of phase noise is as low as−113 dB.rad2.Hz−1 at well chosen “magic frequencies”, and in any case re- mains below than −78 dB.rad2.Hz−1 at worst, dominated by phase truncation. This last value yields a linewidth of 30 mHz over a 100 kHz bandwidth. The corresponding relative frequency noise level is 90 dB below the require- ments of our RF-dressed trapping experiment. The amplitude noise of the DDS synthesizer is mea- sured to be below −130 dB.Hz−1 at frequencies above 1 kHz. Amplitude noise turns out to be the limiting noise present in our RF chain at the present time due to the several dB noise figures of the attenuators and amplifiers in use in our experiment. They actually increase the RF amplitude noise to −115 dB.Hz−1 at 1 kHz, which still remains 25 dB below our experiment requirements. All these features make our DDS apparatus very well- suited for cold atom experiments. It may be used for optimized evaporative cooling implementation, with ar- bitrary frequency ramp profiles including a final “RF- knife”, in a standard BEC experiment. It is of particular importance in the case of atom chip experiments [22], where the cooling ramp is usually done very quickly (1 s typically). It is also ideal for developing new RF-based traps. In particular, the possibility of configuring the time step, that is the duration of each frequency point, is very attractive for our application. This feature is all the more interesting in that the memory can be separated in several zones, each being allocated a chosen number of points and a chosen time step, so we can dilate or compress in time the whole or a part of the ramp, while keeping the same frequency series. For instance, if we want to accomplish a ramp of N frequency values with a small time resolution, say 10 µs, and then keep the final frequency constant over ten seconds, we have the ability to address a first memory zone with the first N−1 points and a 10 µs time step, and a second memory zone with the last frequency value and a 10 s time step. A sin- gle time step over the whole sequence would have forced us to sacrifice either the resolution of the ramp or the duration of the final plateau. The device was successfully implemented in our ultra- cold atom experiment. First, we set boundary specifica- tions on the RF source performances for trapping exper- iments regarding heating rate and lifetime. In the same manner, we set conditions on maximum phase hops and frequency step magnitude. With our home made device, we observed no heating due to the RF source during the plateau step. A remaining heating rate of 0.47µK/s was identical to the one obtained in the magnetic trap and was limited by other noise sources, presumably scattered photons. Despite the very large number of frequency points, heating along the vertical axis is still present dur- ing the loading stage of the RF-dressed trap. We at- tribute this heating to non-adiabatic deformation of the confining potential during the loading process, which we could not make slower due to the other noise sources. While the actual performances of the device are al- ready very good, a few improvements are still possible. First, the memory length of the DDS is basically deter- mined by the FPGA in-board memory size. A larger sample size may easily be implemented by replacing the board, provided that the data transfer rate is improved through the use of an Ethernet port or a USB port. Sec- ond, the sample rate is 1 MHz, limited by the data trans- fer rate from the FPGA board. Higher FPGA clock fre- quency, or smaller clock counts ncycle, would allow higher sample rates. The frequency limit due to the AD-9851 DDS chip itself is 3 MHz. This modification would im- prove the temporal resolution by a factor of 3. Finally, the maximum output frequency of 10 MHz may be very easily increased to 20 MHz by replacing the low-pass fil- ter. By clocking the DDS with a 180 MHz clock, one could even reach output frequencies up to 60 MHz. Note that with a similar DDS chip, the AD 9858, frequency clocks as high as 1 GHz are allowed, which makes possi- ble the synthesis of RF frequencies up to 330 MHz. Acknowledgments We are indebted to R. J. Butcher for a critical read- ing of the manuscript. This work was supported by the Région Ile-de-France (contract number E1213) and by the European Community through the Research Train- ing Network “FASTNet” under contract No. HPRN-CT- 2002-00304 and Marie Curie Training Network “Atom Chips” under contract No. MRTN-CT-2003-505032. Laboratoire de physique des lasers is UMR 7538 of CNRS and Paris 13 University. The LPL is a member of the In- stitut Francilien de Recherche sur les Atomes Froids. [1] H. F. Hess, Phys. Rev. B 34, 3476 (1986). [2] N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, and G. P. Kochanski, Phys. Rev. Lett. 61, 935 (1988). [3] E. A. Cornell and C. E. Wieman, Rev. Mod. Phys. 74, 875 (2002). [4] W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002). [5] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Na- ture 424, 47 (2003). [6] O. Zobay and B. M. Garraway, Phys. Rev. Lett. 86, 1195 (2001). [7] Y. Colombe, E. Knyazchyan, O. Morizot, B. Mercier, V. Lorent, and H. Perrin, Europhys. Lett. 67, 593 (2004). [8] T. Schumm, S. Hofferberth, L. M. Andersson, S. Wilder- muth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, Nature Physics 1, 57 (2005). [9] M. H. T. Extavour, L. J. Le Blanc, T. Schumm, B. Cies- lak, S. Myrskog, A. Stummer, S. Aubin, and J. H. Thy- wissen, Proceedings of the International Conference on Atomic Physics, Atomic Physics 20, 241 (2006). [10] G.-B. Jo, Y. Shin, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, M. Vengalattore, and M. Prentiss, Phys. Rev. Lett. 98, 030407 (2007). [11] I. Lesanovsky, T. Schumm, S. Hofferberth, L. M. Ander- sson, P. Krüger, and J. Schmiedmayer, Phys. Rev. A 73, 033619 (2006). [12] Ph. W. Courteille, B. Deh, J. Fortàgh, A. Günther, S. Kraft, C. Marzok, S. Slama, and C. Zimmermann, J. Phys. B: At. Mol. Opt. Phys. 39, 1055 (2006). [13] O. Morizot, Y. Colombe, V. Lorent, H. Perrin, and B. M. Garraway, Phys. Rev. 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0704.1975

Growth rates for geometric complexities and counting functions in polygonal billiards

GROWTH RATES FOR GEOMETRIC COMPLEXITIES AND COUNTING FUNCTIONS IN POLYGONAL BILLIARDS EUGENE GUTKIN AND MICHAL RAMS Abstract. We introduce a new method for estimating the growth of various quantities arising in dynamical systems. We apply our method to polygonal billiards on surfaces of constant curvature. For instance, we obtain power bounds of degree two plus epsilon for billiard orbits between almost all pairs of points in a planar polygon. Introduction and overview Complexity of a dynamical system is measured with respect to a coding of its orbits. The coding, in turn, is determined by partitioning the phase space of the system into elementary pieces. For dynami- cal systems with singularities, such as polygonal billiards, connected components in the complement to the singular set yield a natural par- tition. Convexity of its atoms with respect to the geodesic structure in the phase space imposed by geometric optics, is crucial in the study of billiard complexity [7]. In the present study, as well as in [7], P is a geodesic polygon in a surface of constant curvature. Let, for concreteness, P be a planar polygon. We denote by fP (n) the number of words of length n gener- ated by coding billiard orbits by visited domains of regularity. When P is simply connected, this coincides with the coding by sides in P . It is known that fP (n) is subexponential in n [3, 6], and for general P no better bound is known. If P is a rational polygon (i. e., its angles are commensurable with π [4]), fP (n) = O(n 3) [1, 7]. The current conjecture is that for any planar polygon fP (n) = O(n d) [5]. In order to advance the understanding of billiard complexity, we introduce the notion of partial complexities. Let Ψ be the phase space, and let P be the defining partition. Iterating the dynamics we obtain an increasing tower Pn of partitions; the full complexity is f(n) = Date: October 31, 2018. Key words and phrases. Geodesic polygon, billiard map, billiard flow, complex- ity, counting functions, unfolding of orbits, covering space, exponential map. http://arxiv.org/abs/0704.1975v1 2 EUGENE GUTKIN AND MICHAL RAMS |Pn|. If R ⊂ Φ, let Pn(R) be the induced tower of its partitions. The partial complexity based on R is fR(n) = |Pn(R)|. Particular partial complexities have been studied earlier. For instance, in [8] we obtained polynomial bounds on direction complexity, which is one of the partial complexities investigated here. In this work we introduce a new general approach to estimating par- tial complexities. The setting is as follows. There is a family of subsets Rθ foliating the phase space. Let fθ(n) be the partial complexity with base Rθ. Let gθ(n) be the counting function for singular billiard or- bits starting from Rθ. Under appropriate assumptions, fθ(n) and gθ(n) have the same growth, as n → ∞. See section 4. Let θ ∈ Θ, the parameter space. Suppose that we bound the average counting function G(n) = gθ(n). Tchebysheff inequality and the zero-one law yield bounds for individual gθ(n) valid for almost all θ ∈ Θ. See section 2. Combined with preceding remarks, these yield estimates on partial complexities for almost all values of the parameter. This is the general scheme for our approach to partial complexities. This work implements this scheme for polygonal billiards. We will now describe the contents of the paper in more detail. In section 1 we investigate counting functions and their averages. We establish the relevant framework in sufficient generality, with the view towards a broad range of geometric-dynamic applications. The main results are Propositions 1 and 2 respectively. These yield geometric formulas for averages of counting functions which are valid under mild assumptions of transversality type. Section 2 is analytic, and also quite general. The setting is as follows. There is a family of positive functions, gθ(p), of positive argument (p ∈ N and p ∈ R+ in the discrete and continuous cases respectively), depending on parameter θ ∈ Θ. Set G(p) = gθ(p)dθ. From upper bounds on G(p) we derive estimates on individual gθ(p); they are valid for almost all θ. Precise formulations depend on the details of the situation. See Propositions 3 and 4. Section 3 sets the stage for applications to billiard dynamics. Our billiard table is a geodesic polygon, P , in a simply connected surface of constant curvature. There are two versions of billiard dynamics: the billiard flow and the billiard map. In our discussion of partial complexities, it is convenient to treat them separately. Accordingly, section 3 consists of several subsections; each subsection deals with a particular partial complexity for a particular kind of billiard dynamics. We use two geometric parameters for partial complexities: the direc- tion and the position. The direction complexity tells us how the set of COMPLEXITY, ETC 3 phase points starting in the same direction splits after bouncing off of the sides of P . The direction complexity is defined for planar polygons. The position complexity tells us about the splitting of beams of billiard orbits emanating from a point of P . It is defined in all cases. In each of the subsections of section 3 we define a counting function and check the assumptions of section 1; then we evaluate the integral over the parameter space, i. e., we compute the average counting func- tions. It turns out that they have geometric meanings. Here is a sample of results from section 3. Let GP (l) be the average position counting function for the billiard flow in a geodesic polygon P . For planar poly- gons we have GP (l) = c0(P )l 2. See Corollary 2 in section 3.2. For polygons in S2 we have GP (l) = c+(P )l + c +(P )f(l) where f is a uni- versal periodic function. See Corollary 3 in section 3.4. For polygons in H2 we have GP (l) = c−(P ) cosh l. See Corollary 4 in section 3.5. The coefficients in these formulas depend on how many corners P has and on the number of obstacles in its interior. Section 4, again, is quite general. In this section we obtain relation- ships between partial complexities with one-dimensional base sets and counting functions. The main result of this section is Proposition 5. It says that if the bases are one-dimensional, then the difference be- tween the partial complexity and the counting function is bounded, as time goes to infinity. Other assumptions on the base have to do with convexity in the phase space. The framework of this section is that of piecewise convex transformations [7]. In section 5 we specialize again to polygonal billiards. Combining the material of preceding sections, we obtain bounds on the position and direction complexities for the billard flow and the billiard map. Here is a sample of our results. Let P be a euclidean polygon. Let θ ∈ S1 (resp. z ∈ P ) be any direction (resp. position). Let fdθ(n) (resp. hz(l)) be the direction complexity for the billiard map (resp. position complexity for the billiard flow). Then for almost all directions θ (resp. for almost all positions z) we have fdθ(n) = O(n 1+ε) (resp. hz(l) = O(l 2+ε)), where ε > 0 is arbitrary. See Corollary 6 and Corollary 8. Let now P be a spherical polygon, and let hz(l) be the position complexity for the billiard flow in P . Then for almost every z ∈ P there is a C = C(z) and arbitrarily large l such that hz(l) ≤ Cl. See Corollary 10. For any ε > 0 and almost every z ∈ P we have hz(l) = O(l 1+ε). See Corollary 11. In the study of polygonal billiards the device of unfolding billiard orbits is indispensable [4]. If P ⊂ M , and β is a billiard orbit in P , its unfolding is a geodesic in M . Several arguments in section 3 use 4 EUGENE GUTKIN AND MICHAL RAMS the technique of lifting billiard orbits to the universal covering space of P .1 This notion was not written up in the billiard literature. In our Appendix section 6 we present the relevant definitions and propositions. Proposition 6 puts forward the main property of the universal covering space of a geodesic polygon. It relates the unfoldings and the liftings of billiard orbits. The proofs in section 3 use Corollary 13 of Proposition 6, which deals with the pullbacks of lebesgue measures under unfoldings. In order to put our results into perspective, we will now briefly sur- vey the literature on billiard complexities. The subexponential growth of (full) billiard complexity for arbitrary euclidean polygons is estab- lished in [3] and [6]. Both proofs are indirect, in that they do not yield explicit subexponential bounds. On the other hand, for rational eu- clidean polygons the complexity is cubic. This is contained in [1] for convex and in [7] for all rational polygons. The arguments in [1] and [7] rely on a theorem in [11]; it says that the number of billiard orbits between any pair of corners in a rational polygon grows quadratically in length. From our viewpoint, this is a statement about the position counting functions gz(l). It says that gz(l) = O(l 2) if P ⊂ R2 is ra- tional and z ∈ P is a corner. By comparison, our Corollary 8 and Proposition 5 yield that gz(l) = O(l 2+ε) for any ε > 0 and almost all z ∈ P where P ⊂ R2 is an arbitrary polygon. The directional com- plexity fdθ(n) has been studied in [8] and [9]. The work [9] concerns the directional complexity for the billiard in a rational, planar polygon P . Assume that P is convex. Then [9] derives an explicit formula for fdθ(n), valid for minimal directions θ. (The set of nonminimal direc- tions is countable.) By this formula, fdθ(n) = O(n). On the other hand, [8] shows that fdθ(n) = O(n d) for any P ⊂ R2 and an arbitrary θ. The degree d in the bound does not depend on θ. Our Corollary 6 estimates the complexity fdθ(n) for an arbitrary polygon P ⊂ R 2. It says that fdθ(n) = O(n 1+ε) for any ε > 0 and almost all directions θ. It is plausible that the bounds like Corollary 8, Corollary 6, etc hold for any point z ∈ P , any direction θ ∈ S1, etc. 1. Averages of counting functions In this section we introduce the framework of counting functions in differentiable dynamics. We will apply it to the billiard dynamics later on. Our phase spaces are “manifolds”. By this we will mean compact manifolds with boundaries, corners, and singular points, in general. Our setting involves i) a foliation of the phase space by closed 1Not to be confused with the concept of universal covering space in topology. COMPLEXITY, ETC 5 submanifolds that are fibers for a projection onto a manifold of smaller dimension; ii) a submanifold in the phase space, transversal to the fibers; iii) a weight function on the product of the phase space and the time. See the details below. The dynamics in question may be discrete or continuous. We will expose the two cases separately. The two subsections that follow are parallel, and the treatments differ in technical details. 1.1. Discrete dynamics. Let T : X → X, T−1 : X → X be piecewise diffeomorphisms with the following data. 1. There is a fibration η : X → Θ whose base is a compact manifold and whose fibers Rθ = η −1(θ) are compact submanifolds, such that dim(Rθ) = dim(X)− dim(Θ). We will use the notation X = ∪θ∈ΘRθ. 2. There is a closed submanifold, Y ⊂ X , dim(Y ) = dim(Θ), such that for k ∈ −N2 the manifolds T k(Y ) are transversal to the fibers Rθ. 3. There is a weight function, i. e., a continuous, non-negative function w(x, t) on X × N. The function w may depend only on time, e. g., w = χn, the indicator function of [0, n− 1]. Remark 1. Condition 2 may be weakened, as follows. 2′. There is a closed submanifold, Y ⊂ X , and a set Θex ⊂ Θ of measure zero such that for k ∈ −N and θ ∈ Θ\Θex the manifolds T k(Y ) and Rθ are transversal. All of our results remain valid if we replace condition 2 by the weaker condition 2′. However, in our applications to polygonal billards, condition 2 may not hold only for polygons in surfaces of positive curvature. See section 3.4. To simplify the exposition, we will assume in what follows that Θex = ∅. In view of condition 2, Γ(θ) = {(x, k) : x ∈ Rθ, k ∈ N, T k(x) ∈ Y } is a countable (at most) set. The sets Γk(θ) = {(x, k) : x ∈ Rθ, T k(x) ∈ Y } are finite for all k ∈ N, and Γ(θ) = ∪Γk(θ). We define the weighted counting function by (1) g(θ;w) = (x,k)∈Γ(θ) w(x, k). The pure counting function gn(θ) corresponds to the weight w = χn. We have (2) gn(θ) = |Γk(θ)|. 2By convention, N = 0, 1, . . . . 6 EUGENE GUTKIN AND MICHAL RAMS Proposition 1. Let dθ, dy be finite, lebesgue-class measures on Θ, Y respectively. Then for k ∈ N there are functions rk(·) ≥ 0 on Y , determined by the data 1) and 2) alone, such that g(θ;w)dθ = w(T−k · y, k)rk(y) Proof. For any k ∈ N set fk = η ◦ T −k : Y → Θ. By conditions 1 and 2, fk is a local diffeomorphism. Therefore f k (dθ) = rk(y) dy. It suffices to establish equation (3) for the special case w(x, i) = 0 if i 6= k. A point x ∈ X contributes to the integral in the left hand side of equation (3) iff T k ·x ∈ Y , or equivalently, η(x) = fk(y), y ∈ Y . The claim follows by a straightforward change of variables. 1.2. Continuous dynamics. Let bt : Ψ → Ψ be a flow of piecewise diffeomorphisms on a phase space Ψ with the following data. 1. There is a fibration q : Ψ → Z with a compact base and fibers q−1(z) = Rz ⊂ Ψ, transversal to the flow. We will use the notation Ψ = ∪z∈ZRz. 2. There is a closed submanifold, M ⊂ Ψ, dim(M) = dim(Z) − 1, transversal to the flow, and such that N = ∪t∈R bt · M is transversal to the fibers Rz. 3. There is a weight function, i. e., a continuous, non-negative function w(x, t) on Ψ × R+. In a special case, w depends only on time, e. g., w = χl, the indicator function of [0, l]. In view of condition 2, G(z) = {(x, t) : x ∈ Rz, 0 ≤ t, b t(x) ∈ M} is a countable (at most) set. The sets Gl(z) = {(x, t) : x ∈ Rz, 0 ≤ t ≤ l, bt(x) ∈ M} are finite for all l ∈ R+, and G(z) = ∪Gl(z). We define the weighted counting function by (4) g(z;w) = (x,t)∈G(z) w(x, t). The pure counting function gl(z) corresponds to the weight w = χl. We have (5) gl(z) = |Gl(z)|. Proposition 2. Let dz, dm be finite, lebesgue-class measures on Z,M respectively; let dt be the lebesgue measure on R. Then there exist a 3Our results remain valid if the set of parameters Z ⊂ Z where the transversal- ity fails has measure zero. See Remark 1. In what follows, by condition 2′ we will mean the weakened condition 2 either in the setting of section 1.2 or section 1.1. COMPLEXITY, ETC 7 positive function r(·) on M × R+, determined by the data 1) and 2), and such that g(z;w)dz = w(b−t ·m, t)r(m, t)dmdt Proof. We define the mapping f : M × R+ → Z by f = q ◦ b By conditions 1 and 2, f has full rank almost everywhere. The pull- back by f of dz is absolutely continuous with respect to dmdt, hence f ∗(dz) = r(m, t)dmdt. For 0 < l set wl(x, t) = w(x, t)χl(t), and let gl(z;w) be the corre- sponding counting function. Set Il(w) = gl(z;w)dz. A point, x ∈ Ψ, contributes to Il(w) iff x ∈ ϕ(M× [0, l]). Under the change of variables dz = d(q ◦ ϕ(m, t)) = r(m, t)dmdt, we have Il(w) = M×[0,l] w(b−t ·m, t)r(m, t)dmdt. In the limit l → ∞, we obtain the claim. 1.3. Special cases. We will discuss a few special cases of Proposition 1 and Proposition 2. First, the discrete version. The function gn(θ) counts the number of visits in Y of points x ∈ Rθ during the first n steps of their journey. Set ρk = rk(y)dy, and Rn = k=0 ρk. Then ρk is the volume of Yk = T −k(Y ) with respect to the measure η∗(dθ). Proposition 1 yields gn(θ)dθ = Rn. In the continuous case the function gl(z) counts the number of visits in M of orbits bt · x, x ∈ Rz, during the period 0 ≤ t ≤ l. Let R(l) be the volume of the manifold Nl ⊂ Ψ with respect to the measure q ∗(dz). Proposition 2 yields gl(z)dz = R(l). 2. Bounds on counting functions In this section we analyze the setting of section 1 from the measure theoretic viewpoint. This allows us to obtain pointwise upper bounds on counting functions in a broad spectrum of situations. Let X, µ be a finite measure space. Let f(x; t) (for t ∈ R+) be a family of nonnegative L1 functions on X . Set (9) F (t) = f(x; t)dµ(x). 8 EUGENE GUTKIN AND MICHAL RAMS Lemma 1. For almost every x ∈ X there exists C = C(x) > 0 such that for arbitrarily large n ∈ N there is t ≥ n satisfying f(x, t) < CF (t). Proof. For 0 < C and n ∈ N let Bn(C) = {x ∈ X : CF (t) < f(x; t) ∀t > n}, and set B(C) = Bn(C). Integrating the inequality above, we obtain µ(Bn(C)) ≤ C −1 for any n. Thus µ(B(C)) ≤ C−1, and hence µ(∩C∈R+B(C)) = 0. But ∩C∈R+B(C) ⊂ X is the complement of the set of points x ∈ X satisfying the hypoth- esis of the lemma. Let the setting be as in Lemma 1. In addition, we suppose that i) the functions f(x; t) are nondecreasing in t and ii) F (t) → ∞. Lemma 2. Let ε > 0 be arbitrary. Then for almost every x ∈ X there exists T = T (x, ε) > 0 such that for all t > T we have (10) f(x; t) ≤ F (t)(1 + log(1 + F (t)))1+ε. Proof. Denote by f(x; t−) (resp. F (t−)) the limits of f(x; s) (resp. F (s)), as s → t from the left. For n ∈ N set tn = inf{t : F (t) ≥ 2 Then F (t(n+1) −) ≤ 2F (tn). Let An ⊂ X be the set of points satisfying the inequality (11) f(x; t−n ) ≤ F (tn −)(1 + log(1 + F (tn −)))1+ǫ. It suffices to prove that the set Ak has full measure. Indeed, for x ∈ An and t ∈ [tn, tn+1) we have f(x; t) ≤ f(x; t−n+1) ≤ F (tn+1 −)(1 + log(1 + F (tn+1 −))1+ε ≤ F (tn)(1 + log(1 + F (tn))) 1+ε ≤ F (t)(1 + log(1 + F (t)))1+ε. Thus, the points x ∈ Ak have the property equation (10). If Bn ⊂ X is any sequence of sets, we set lim supn→∞Bn = Let Bn be the complement of An inX . Then lim supn→∞Bn is the com- plement of Ak. It remains to prove that µ(lim supn→∞Bn) = By Tchebysheff inequality, we have (12) µ(Bn) ≤ 2(1 + log(1 + F (tn −)))−(1+ε). COMPLEXITY, ETC 9 Set µn = µ(Bn). Suppose first that F is a continuous function. Then F (tn −) = F (tn) = 2 n. By equation (12) µn ≤ 2(1 + log(1 + 2 n))−(1+ε), hence the series µn converges. Since µ(lim sup Bn) ≤ for any n0 ∈ N, the claim follows. In general, F need not be continuous. It is thus possible that tn = tn+1 for some n ∈ N, implying Bn = Bn+1. From the series we drop the terms µn such that Bn = Bn−1. By equation (12), the remaining terms satisfy µn ≤ 2(1 + log(1 + 2 n−2))−(1+ε). Now the preceding argument applies. In sections 3, 5 we will apply these results in the billiard setting. In section 3 we will estimate the integrals equation (9), hence the bounds provided by Lemmas 1, 2 will be more specific. The propositions below anticipate these applications. Proposition 3. Let the setting and the assumptions be as in Lemma 2. Let 0 < ε be arbitrary. 1. Let F (t) = O(tp) for 0 < p. Then for almost every x ∈ X we have f(x; t) = O(tp+ε). 2. Let F (t) = O(eat) for 0 < a. Then for almost every x ∈ X we have f(x; t) = O(e(a+ε)t). Proof. The first claim is immediate from Lemma 2 and (log t)1+ε = o(tε). The second claim follows the same way from t1+ε = o(eεt). For applications to the billiard map we need a counterpart of Proposi- tion 3 for integer-valued time. We state it below. Its proof is analogous to the proof of Proposition 3. Moreover, the discrete time case may be formally reduced to the continuous time case. We leave details to the reader. Let X, µ be a finite measure space. Let f(x;n), n ∈ N be a sequence of nonnegative L1 functions on X such that for every x ∈ X the nu- merical sequence f(x;n) is nondecreasing. Set F (n) = f(x;n)dµ. Proposition 4. Let 0 < ε be arbitrary. Then the following claims hold. 10 EUGENE GUTKIN AND MICHAL RAMS 1. Let F (n) = O(np) for 0 < p. Then for almost every x ∈ X we have f(x;n) = O(np+ε). 2. Let F (n) = O(ean) for 0 < a. Then for almost every x ∈ X we have f(x;n) = O(e(a+ε)n). Remark 2. All of the bounds f(·) = O(·) in preceding propositions are equivalent to the formally stronger bounds f(·) = o(·). 3. Counting functions for polygonal billiard We will now apply the preceding material to the billiard dynamics. Our billiard table will be a geodesic polygon either in the euclidean plane R2, or the round sphere S2, or the hyperbolic plane H2. We refer to [4], [7], and section 6 for the background. 3.1. Direction counting functions for billiard maps in euclidean polygons. Let P ⊂ R2 be a euclidean polygon, and let T : X(P ) → X(P ) be the billiard map. Elements of the phase space X = X(P ) are oriented geodesic segments in R2 with endpoints in ∂P . A segment x ∈ X ending in a corner of P is singular; the element Tx is not well defined. A billiard orbit x, Tx, . . . , T k−1x is a singular orbit of length k if T k−1x is the first singular element in the sequence. Assigning to x ∈ X its direction, η(x) ∈ S1, we obtain a fibration η : X → S1 with fibers Rθ ⊂ X . See figure 1. We define the counting function gdθ(n) for singular orbits in direction θ as the number of phase points x ∈ Rθ that yield singular orbits of length k ≤ n. Theorem 1. Let P ⊂ R2 be an arbitrary polygon. Let K(P ) be the set of its corners. Let α(v) be the angle of v ∈ K(P ). Let dθ be the lebesgue measure on S1. Let K ⊂ K(P ). Then gdθ(n; v)dθ = Proof. It suffices to prove the claim for a singleton, K = {v}. Let Y = Y (v) ⊂ X be the set of segments x ∈ X ending at v. Let dy be the angular measure on Y . These data fit into the setting of section 1.1, and gdθ(n; v) is the pure counting function. Let B(z, α) be a conical beam of light with apex angle α emanating from z ∈ R2. After reflecting in ∂P , it splits into a finite number of beams B(zi, αi) satisfying αi = α. The preservation of light volume is due to the flatness of ∂P . COMPLEXITY, ETC 11 By preceding remark, the functions rk(·) of Proposition 1 satisfy rk(·) ≡ 1. The claim now follows from the special case of Proposition 1 considered in section 1.3. Let p, q be the numbers of corners, obstacles in P . Let κ(P ) = p+ 2q − 2. Thus, P is simply connected iff q = 0 iff κ(P ) = p− 2. Corollary 1. Let P ⊂ R2 be an arbitrary polygon. Then gdθ(n)dθ = πκ(P )n. Proof. Follows from Theorem 1 via v∈K(P ) α(v) = (p+ 2q − 2)π. Figure 1. Base sets for billiard counting functions 12 EUGENE GUTKIN AND MICHAL RAMS 3.2. Position counting functions for billiard flows in euclidean polygons. Let P ⊂ R2 be a polygon, and let bt : Ψ → Ψ be the billiard flow. See section 6 for details. For z ∈ P and v ∈ K(P ) let gcz(l; v) be the number of billiard flow orbits that start from z ∈ P and wind up at v by time l. Then gcz(l) = v∈K(P ) gcz(l; v) is the number of singular billiard orbits of length at most l starting from z. This is the position counting function for the billiard flow in P . Theorem 2. Let P ⊂ R2 be a euclidean polygon, and let dz be the lebesgue measure on P . Then for any K ⊂ K(P ) we have (15) 2 gcz(l; v)dz = Proof. It suffices to prove the claim for K = {v}. We view elements of Ψ as pairs z, θ where z ∈ P is the basepoint, and θ is the direction. Let M = {(v, θ) : (v,−θ) ∈ Ψ}. Let q : Ψ → P be the obvious projection. Its fibers Rz are the base sets for the counting functions gcz(l; v). See figure 1. Set w = χl. These data satisfy the assumptions of Proposition 2, and gcz(l; v) is the pure counting function. We set dm to be the angular measure, and compute the function r(m, t) in equation (6). By Corollary 13 in section 6, r = tχl. Propo- sition 2 implies the claim. When K = K(P ), the left hand side in equation (15) is the average of the position counting function. The argument of Corollary 1 yields the following. Corollary 2. Let P ⊂ R2 be an arbitrary polygon. Then (16) 2 gcz(l)dz = πκ(P )l 3.3. Position counting functions for billiard maps in euclidean polygons. We will now discuss two billiard map analogs of the pre- ceding example. Let P ⊂ R2 be a euclidean polygon, and let T : X(P ) → X(P ) be the billiard map. The phase space X = X(P ) consists of pairs (s, α) where s is the arclentgh parameter on ∂P , and 0 < α < π is the outgoing angle. See [4, 7] and section 6 for details. An orbit x, Tx, . . . , T k−1x is singular, of (combinatorial) length k if its last segment ends at a corner of P . Let s ∈ ∂P , v ∈ K(P ). Define GDs(n; v) to be the set of phase points (s, α) ∈ X whose orbits of length less than or equal to n end at COMPLEXITY, ETC 13 v. Set gds(n; v) = |GDs(n; v)|, gods(n; v) = (s,α)∈GDs(n;v) sinα. The expressions gds(n) = v∈K(P ) gds(n; v), gods(n) = v∈K(P ) gods(n; v) are the pure position counting function and the optical position counting function for the billiard map in P . Let z ∈ R2 and let γ ⊂ R2 be an oriented piecewise C1 curve. Denote by dzs the projection of the arclength form ds of γ onto the direction perpendicular to the line from z to s ∈ γ. The integral dzs = |opt(γ, z)| ≤ |γ| is the optical length of γ viewed from z. Let z ∈ P . Unfolding k-segment billiard orbits emanating from z, we obtain a set of linear segments in R2. Let ∂z(P ; k) ⊂ R 2 be the curve traced by their endpoints. We say that ∂z(P ; k) ⊂ R 2 is the outer boundary of P , as viewed from z, after k iterates. Theorem 3. Let P be a euclidean polygon, and let K ⊂ K(P ) be a set of corners. Then gds(n; v)ds = |∂v(P ; k)|; gods(n; v)ds = |opt(∂v(P, k))|. Proof. It suffices to prove the claims for a singleton, K = {v}. Let η : X → ∂P be the natural projection. Using the arclength parametriza- tion, we identify ∂P with the interval [0, |∂P |] ⊂ R. For 0 ≤ s ≤ |∂P | let Rs = η −1(s) ⊂ X be the fiber. Then Rs are the base sets for the counting functions gds(n; v), gods(n; v). See figure 1. Let Y = Y (v) ⊂ X be the set of phase points whose T−1-orbits emanate from v. The assumptions of section 1.1 are satisfied. The weight functions are w(s, α, t) = χn(t) and wo(s, α, t) = sinα ·χn(t) for the two cases at hand. Let ϕ be the angle parameter on Y . The measures on ∂P and Y have densities ds and dϕ respectively. The integrals in the right hand side of equation (3) are over the curves ∂v(P ; k), 0 ≤ k ≤ n− 1. The integrands are ds(ϕ) and sinα · ds(ϕ) = dvs(ϕ) in respective cases. We will need estimates on lengths and optical lengths. 14 EUGENE GUTKIN AND MICHAL RAMS Lemma 3. For any polygon P ⊂ R2 there exist 0 < c1 < c2 < ∞ such that for n sufficiently large |opt(∂v(P, k))| ≤ c2n 2, c1n |(∂v(P, k))|. Proof. There exist positive constants d1, d2 and m0 ∈ N, such that for any orbit γ of the billiard map with m > m0 segments, we have d1|γ| ≤ m ≤ d2|γ| [4]. Let v ∈ K(P ). We will estimate |opt(∂v(P, k))|, as n → ∞. Let θ1 ≤ θ ≤ θ2 be the angular parameter for orbits emanating from v; let r(θ) be the geometric length of the orbit. Suppose that r1 ≤ r(θ) ≤ r2. Then the optical length in question is sandwiched between the lengths of circular arcs of radii r1, r2 of angular size θ2−θ1. By preceding remarks, if k is sufficiently large, the bounds r1, r2 are proportional to k. The total angular size does not depend on k. Hence, for sufficiently large k we have linear upper and lower bounds on v∈K |opt(∂v(P, k))|. The other inequality follows from |opt(∂v(P, k))| ≤ |∂v(P, k)|. 3.4. Position counting functions for billiard flows in spherical polygons. The study is analogous to the planar case discussed in sec- tion 3.2; we will use the same notation whenever this does not lead to confusion. We denote by dz the lebesgue measure on S2, and by α(v) the angle of a corner of P . Set (20) ζ(x) = 1− cosx− Theorem 4. Let P ⊂ S2 be a geodesic polygon, and let K ⊂ K(P ). gcz(l; v)dz = l + ζ(l− π⌊l/π⌋) Proof. It suffices to prove the claim when K = {v}. Let M = M(v) ⊂ Ψ be as in section 3.2, and let dα be the angular measure on it. The assumptions 1, 3 of section 1 are satisfied; the transversality of bt ·M and Rz may fail for at most a countable set of parameters Pex ⊂ P . See Remark 3 in section 6. Hence, condition 2′ is fullfilled, and the results of section 1.2 hold. The function gcz(l; v) is a pure counting function. The claim now follows from Proposition 2 and Corollary 13. Let κ(P ) be as in section 3.1. COMPLEXITY, ETC 15 Corollary 3. Let P ⊂ S2 be an arbitrary polygon. Then gcz(l)dz = (κ(P )π + area(P )) l + ζ(l − π⌊l/π⌋) Proof. For a spherical polygon we have v∈K(P ) α(v) = area(P ) + κ(P )π. Substitute this into equation (21). 3.5. Position counting functions for billiard flows in hyper- bolic polygons. Our treatment and our notation are modelled on section 3.4. We denote by dz the lebesgue measure on H2, and by α(v) the angles of corners. Theorem 5. Let P ⊂ H2 be a geodesic polygon, and let K ⊂ K(P ). gcz(l; v)dz = cosh l. Proof. We repeat verbatim the proof of Theorem 4, and use claim 2 in Corollary 13. Let κ(P ) be as in section 3.1. Corollary 4. Let P ⊂ H2 be a polygon. Then gcz(l)dz = (κ(P )π − area(P )) cosh l. Proof. Repeat the argument of Corollary 3; use the formula v∈K(P ) α(v) = κ(P )π− area(P ) relating the angles and the area of geodesic polygons in H2. 4. Relating partial complexities and counting functions In this section we establish a framework that will allow us to study the complexity of a wide class of dynamical systems. Our motivation comes from the billiard dynamics. In fact, polygonal billiard is the target of applications for our results. The framework is more general, however. The following observations served as our guiding principles. First, natural partitions of the billiard-type systems are geared to the singularities. Second, the billiard dynamics satisfies a certain convexity property that is instrumental in the study of complexity. These princi- ples are manifest in the framework of piecewise convex transformations There are two approaches to the billiard dynamics: The billiard flow and the billiard map. See section 6. The framework of piecewise convex 16 EUGENE GUTKIN AND MICHAL RAMS transformations is geared to the billiard map. We begin by establishing its counterpart for flows. 4.1. Piecewise convex transformations and piecewise convex flows. A piecewise convex transformation is a triple (X,Γ, T ), whereX is a two-dimensional convex cell complex, Γ ⊂ X is the graph formed by the union of one-cells, and T : X → X is an invertible map, regular on the two-cells of the complex, and compatible with the convex structure Let Ψ be a compact manifold, with boundary and corners, in general. Let bt : Ψ → Ψ be a flow, possibly with singularities; let X ⊂ Ψ be a cross-section. We will assume that the singular set of the flow is contained in X . For z ∈ X let τ+(z), τ−(z) be the times when z ∈ Ψ first reaches X under bt, b−t for 0 < t. We assume that for any z ∈ Ψ\X there is 0 < ε = ε(z) such that bt(z) is regular for |t| < ε. A piecewise convex flow is determined by the following data: A flow, bt : Ψ → Ψ, a cross-section, X ⊂ Ψ, and the structure of a convex cell complex on X , compatible with the poincare map. Billiard flows for polygons on surfaces of constant curvature are piecewise convex flows 4.2. Partial complexities for maps and flows. Let (X,Γn, T n) be the iterates of a piecewise convex transformation (X,Γ, T ).4 Let F (Γn) be the finite set of open faces of Γn; these are the continuity regions for T n. The function f(n) = |F (Γn)| is the (full) complexity of (X,Γ, T ). Let R ⊂ X be a closed subset. Set FR(n) = {A ∈ F (Γn) : A ∩R 6= ∅}. Definition 1. The function fR(n) = |FR(n)| is the partial complexity of the piecewise convex transformation (X,Γ, T ) based on the subset Let bt : Ψ → Ψ be a piecewise convex flow, and let R ⊂ Ψ be a closed, convex set transversal to the flow. For 0 < l let OR(l) be the set of regular flow orbits of length l starting from R. Let α0, α1 ∈ OR(l). A homotopy is a continuous family of regular orbits αp ∈ OR(l), 0 ≤ p ≤ 1, interpolating between α0, α1. We will say, for brevity, that the orbits α0, α1 are R-homotopic. We denote by HR(l) the set of R-homotopy classes. Definition 2. The function hR(l) = |HR(l)| is the partial complexity (based on R) of the piecewise convex flow bt : Ψ → Ψ. 4 They are piecewise convex transformations as well [7]. COMPLEXITY, ETC 17 Figure 2. Removing a vertex in a graph 4.3. Partial complexities and counting functions. In what fol- lows we assume that R ⊂ Ψ is a convex graph without isolated vertices. For x ∈ R its valence val(x) is the number of edges of x minus one. In particular, if x is an interior point of an edge, then val(x) = 1. Set val(R) = maxx∈R val(x). We endow R \ {x} with the graph structure where x is replaced by 1+val(x) vertices; each of them is the endpoint of a unique edge. If x, y, z, . . . ∈ R are distinct points, then the induc- tively defined graph structure on R without x, y, z, . . . does not depend on the order of removing these points. We will denote this graph by R \ {x, y, z, . . .}. See figure 2 for an illustration. Let E(R) and V (R) be the sets of edges and vertices, and let c(R) be the number of connected components of the graph. Let hi = hi(R) be the betti numbers of R, and set χ(R) = |V (R)| − |E(R)|. Then c(R) = h0, χ(R) = h0 − h1. Lemma 4. Let R be a finite graph, and let x1, . . . , xp ∈ R be distinct points. Then (25) χ(R) + val(xi) ≤ c(R \ {x1, . . . , xp}) ≤ c(R) + val(xi). If R is a forest, then the bound on the right in equation (25) becomes an equality. 18 EUGENE GUTKIN AND MICHAL RAMS Proof. It suffices to prove the claims when R is connected, and we remove a single vertex, x. Equation (25) becomes (26) χ(R) + val(x) ≤ c(R \ {x}) ≤ c(R) + val(x). We have |V (R \ {x})| = |V (R)| + val(x), |E(R \ {x})| = |E(R)|, and χ(R\{x}) = χ(R)+val(x). Equivalently, we have χ(R\{x}) = h0(R)+ val(x)−h1(R) and h0(R\{x}) = h0(R)+val(x)+(h1(R \ {x})− h1(R)). The former (resp. latter) identity implies the left (resp. right) inequal- ity in equation (26). When R is a tree, we have c(R \ {x}) = c(R) + val(x), and the remaining claim follows. We will introduce counting functions for singular orbits of the billiard map and the billiard flow. By definition, an orbit α = {bt(z), 0 ≤ t ≤ l}, does not pass through singular points in Ψ. It is regular if it does not contain any singu- lar points in Ψ; it is singular if one of its endpoints is singular. The set SR(l) of singular orbits of length at most l, based in R, is finite. The quantities gcR(l) = |SR(l)| and gdR(n) = |R ∩ Γn| are the count- ing functions for singular orbits based in R for the flow and the map respectively. Now we will relate partial complexities and counting functions. We do this for a piecewise convex flow bt : Ψ → Ψ and for a piecewise convex transformation (X,Γ, T ). In both cases the partial complexity is based on a 1-dimensional subset, say R. Recall that gcR(l), gdR(n) are the respective counting functions, and hR(l), fR(n) are the respective complexities. We will refer to these situations as the continuous case and the discrete case respectively. Proposition 5. Let the setting be as above. Then the following state- ments hold. 1. In the continuous case there exist h0 ∈ N and l0 ∈ R+ such that hR(l) = h0 + gcR(l) for l0 ≤ l. 2. In the discrete case there exist f0, n0 ∈ N such that for n0 ≤ n we have fR(n) = f0 + gdR(n). Proof. In both cases the graph R is equipped with a tower of finite sets, say X(l) and Xn respectively. Let X∞ ⊂ R be their union. We will compare the number of connected components of graphs R\X(l), R\Xn with the cardinalities of these sets. We consider the discrete case, leaving the continuous case to the reader. Let m < n be any pair of natural numbers. By (the proof of) COMPLEXITY, ETC 19 Lemma 4, c(R \Xn)− c(R \Xm) = [h1(R \Xn)− h1(R \Xm)] + x∈Xn\Xm val(x). We have h1(R \ Xn) ≤ h1(R \ Xm); the inequality holds iff Xn \ Xm breaks cycles in R \ Xm. Since the sequence h1(R \ Xk) ∈ N is nonincreasing, it stabilizes. Thus, there exists n1 ∈ N such that for n1 ≤ m < n we have h1(R \Xn) = h1(R \Xm). The set of points x ∈ R satisfying 1 < val(x) is finite. Thus, there exists n2 ∈ N such that if n2 ≤ k and x ∈ X∞ \Xk, then val(x) = 1. Set n0 = max(n1, n2). Then for n0 ≤ m < n the above equation yields c(R \Xn)− c(R \Xm) = |Xn \Xm|. Specializing to m = n0, we obtain fR(n) = (fR(n0)− gdR(n0)) + gdR(n). 5. Bounds on partial complexities for the billiard We will use the preceding material to derive bounds on partial com- plexities for the polygonal billiard. 5.1. Direction complexities for billiard maps in euclidean poly- gons. We use the setting and the notation of section 3.1. For a polygon P and a direction θ, we denote by fdθ(n) the partial complexity with base Rθ. This is the complexity in direction θ. Corollary 5. For lebesgue almost all directions θ there is C = C(θ) and there are arbitrarily large n such that fdθ(n) ≤ Cn. Proof. Each Rθ is a convex graph in the phase space [7]. By Lemma 1 and Corollary 1, the counting functions gdθ(n) have the desired proper- ties. By the second claim of Proposition 5, the directional complexities do as well. Corollary 6. For any ε > 0 and almost every direction θ we have fdθ(n) = O(n 1+ε). Proof. The proof goes along the lines of the proof of Corollary 5. In- stead of Lemma 1, we use Proposition 4 (the first claim). 5.2. Position complexities for billiard flows in euclidean poly- gons. Let P be a euclidean polygon, and let z ∈ P be any point. We consider the billiard flow in P , and use the setting of section 3.2. Thus, gcz(l) is the position counting function for orbits emanating from z. We denote by hz(l) the corresponding partial complexity. Corollary 7. For almost every point z there is a positive number C = C(z) such that hz(l) ≤ Cl 2 for arbitrarily large l. 20 EUGENE GUTKIN AND MICHAL RAMS Proof. The sets Rz satisfy the assumptions of section 4. The claim follows from Lemma 1, Corollary 2 and the continuous case in Propo- sition 5. Corollary 8. For any ε > 0 and almost every z ∈ P we have hz(l) = O(l2+ε). Proof. The proof is similar to the preceding argument, and we use the first claim in Proposition 3 instead of Lemma 1. 5.3. Position complexities for billiard maps in euclidean poly- gons. This is the billiard map analog of the preceding example. Let P be a euclidean polygon, and let s ∈ ∂P . We use the setting of sec- tion 3.3. There we have defined the counting functions gds(n), gods(n). Let fs(n) be the partial complexity corresponding to gds(n). This is the position complexity for the billiard map. Corollary 9. Let P ⊂ R2 be a polygon such that k=1 |∂v(P ; k)| has a quadratic upper bound.5 Then for almost all s ∈ ∂P we have fs(n) = O(n 2+ε) for any 0 < ε. Proof. The sets Rs ⊂ X satisfy the assumptions of section 4. We use Theorem 3, Lemma 3, and apply Proposition 5. The estimate of Corollary 9 on fs(n) is conditional, because in gen- eral we have no efficient upper bound on |∂v(P ; k)|. 5.4. Position complexities for billiard flows in spherical poly- gons. We use the setting of section 3.4. For a spherical polygon, P ⊂ S2, and z ∈ P , let hz(l) be the position complexity. Corollary 10. For almost every point z ∈ P there is C = C(z) and there are arbitrarily large l such that hz(l) ≤ Cl. Proof. The sets Rz satisfy the assumptions of section 4. We use Lemma 1, Corollary 3, and Proposition 5. Corollary 11. For any ε > 0 and almost every z ∈ P we have hz(l) = O(l1+ε). Proof. See the proof of Corollary 8. 5This holds if P is a rational polygon [11]. COMPLEXITY, ETC 21 5.5. Position complexities for billiard flows in hyperbolic poly- gons. This material is the hyperbolic plane counterpart of section 3.2, and we use the setting of section 3.5. Corollary 12. Let P ⊂ H2 be a geodesic polygon, let z ∈ P , and let hz(l) be the position complexity. Then for almost every point z ∈ P we have hz(l) = O(e (1+ε)l). Proof. We verify that the sets Rz satisfy the assumptions of section 4, and mimick the proof of Corollary 8; we use Corollary 4, Proposition 3, and the continuous case of Proposition 5. 6. Appendix: Covering spaces for polygonal billiards Let M be a simply connected surface of constant curvature, and let P ⊂ M be a connected geodesic polygon. We normalize the metric so that the curvature is either zero (M = R2), or one (M = S2), or minus one (M = H2). Let A be the set of sides in P . We will denote its elements by a, b, . . . . For a side, say a ∈ A, let sa ∈ Iso(M) be the corresponding geodesic reflection. We associate with P a Coxeter system (G,A) [2]. We denote by σa, σb, · · · ∈ G the elements corresponding to a, b, . . . ∈ A. They generate G. The defining relations are σ2a = 1 and (σaσb) n(a,b) = 1; the latter arise only for the sides a, b with a common corner if the angle, θ(a, b), between them is π-rational. In this case n(a, b) is the denominator of θ(a, b)/π. Otherwise n(a, b) = ∞. To any “generalized polyhedron” P corresponds a topological space C endowed with several structures, and a Coxeter system [2]. Our situation fits into the framework of [2], and we apply its results. First, C is a differentiable surface. Second, C is tiled by subsets Pg, g ∈ G, labelled by elements of the Coxeter group G; we call them the tiles, and identify Pe with P . The group G acts on C properly discontinuously, preserving the tiling: g · Ph = Pgh. Since Pe is identified with P ⊂ M , it inherits from M a riemann- ian structure. The action of G is compatible with this structure, and extends it to all of C. This riemannian structure generally has cone singularities at vertices of the tiling C = ∪g∈GPg. 6 Around other points this riemannian structure is isometric to that of M ; in particular, ex- cept for cone points, C has constant curvature. The group G acts on C by isometries. 6Each vertex, v, corresponds to a corner of P . The metric at v is regular iff the corner angle is π/n, n = 2, 3, . . . . 22 EUGENE GUTKIN AND MICHAL RAMS Definition 3. The space C endowed with the riemannian structure, the isometric action of G and the G-invariant tiling C = ∪g∈GPg is the universal covering space of the geodesic polygon P ⊂ M . If X is a riemannian manifold (with boundary and singularities, in general), we denote by TX = ∪x∈XTxX its unit tangent bundle. The classical construct of geodesic flow, GtX : TX → TX , extends to manifolds with boundaries and singularities. In particular, GtX makes sense when X = M,P , or C. Another classical construct, the expo- nential map, also extends to our situation. For x ∈ X as above, and (v, t) ∈ TxX × R+, we set expX(v, t) ∈ X be the base-point of G X(v). We will use the notation expxX to indicate that we are exponentiating from the point x. If X is nonsingular, then expxX : TxX × R+ → X is a differentiable mapping. For X with singularities, such as our P and C, the maps expxX are defined on proper subsets of TxX × R+; these subsets have full lebesgue measure. Generally, the maps do not extend by continuity to all of TxX × R+. Let X, Y be nonsingular riemannian manifolds of the same dimen- sion; let ϕ : X → Y be a local isometry. It induces a local dif- feomorphism Φ : TX → TY commuting with the geodesic flows: Φ ◦GtX = G Y ◦Φ. The exponential maps commute as well: ϕ ◦ exp Y ◦ dxϕ. These relationships hold, in particular, for coverings of nonsingular riemannian manifolds. Suitably interpreted, they extend to (branched) coverings of riemannian manifolds with boundaries, cor- ners, and singularities. In our case X = C, while Y = M , or Y = P . We will now define the mappings f : C → P, F : TC → TP and ϕ : C → M, Φ : TC → TM . The identification Pe = P defines f, ϕ on Pe. To extend them to all of C, we use the tiling C = ∪g∈GPg and the actions of G on C and M . In order to distinguish between these actions, we will denote them by g · x and g(x) respectively. Then there is a unique x ∈ Pe such that z = g · x. We set f(z) = x ∈ P and ϕ(z) = g(x) ∈ M . By basic properties of Coxeter groups [2], the mappings f, ϕ are well defined. Moreover, f : C → P and ϕ : C → M are the unique G-equivariant mappings which are identical on Pe. 7 By construction, both mappings are continuous; they are diffeomorphisms in the interior of each tile, Pg ⊂ C, and on the interior of the union of any pair of adjacent tiles. The potential locus of non-differentiability for both f and ϕ is the set V of vertices in the tiling C = ∪g∈GPg. We have V = f −1(K(P )) where K(P ) is the set of corners of P . By equivariance, ϕ(V ) = 7The action of G on P is trivial. COMPLEXITY, ETC 23 ∪g∈Gg(K(P )) ⊂ M . 8 There are two kinds of points in V : vertices coming from the corners of P with π-rational and π-irrational angles. Their cone angles are integer multiples of 2π and are infinite respec- tively. Vertices v ∈ V with cone angle 2π are, in fact, regular points in C, and the mappings f, ϕ are both regular there. Around a vertex v with cone angle 2kπ > 2π the mapping ϕ is differentiable, but not a diffeomorphism; it is locally conjugate to z 7→ zk. Near such a vertex, ϕ is a branched covering of degree k. At a vertex with infinite cone angle, ϕ has infinite branching. Remark 3. The set ϕ(V ) ⊂ M is countable. (It is finite iff the group generated by geodesic reflections in the sides of P is a finite Coxeter group. Typically, ϕ(V ) ⊂ M is a dense, countable set.) Let M = S2, and let z 7→ z′ denote the antipodal map. Set F = P ∩ (ϕ(V ) ∪ (ϕ(V ))′). Points of F are exceptional, in the following sense. Let z ∈ P be such that the beam Rz of billiard orbits emanat- ing from z contains a sub-beam focusing at a corner of P . Then z ∈ F . This follows from Proposition 6 below. Thus, F contains all points z ∈ P for which the transversality as- sumption in Condition 2 of section 1.2 fails. Since F is countable, the set of exceptional parameters has measure zero, and Condition 2′ is satisfied. See Remark 1 in section 1. Furthermore, the mappings f and ϕ are local isometries. They are isometries on every tile Pg ⊂ C; we have f(Pg) = P , ϕ(Pg) = g(P ) ⊂ M . Let g · a be a side of Pg, let h = σag and let Ph be the adjacent tile. The maps f : Pg → P, Ph → P and ϕ : Pg → g(P ), Ph → h(P ) are coherent around the common (open) side g · a. The map f is never an isometry on Pg ∪Ph; for ϕ this is the case iff the interiors of g(P ), h(P ) are disjoint in M . The latter generally fails for nonconvex P . By coherence of f and ϕ across the sides separating adjacent tiles, we lift them to the tangent bundles, obtaining the mappings of unit tangent bundles F : TC → TP , Φ : TC → TM , which are also defined on vectors based at the vertices of the tiling C = ∪g∈GPg. Let v be a vertex, and let α be the angle of the corner f(v) ∈ K(P ). Then Φ : TvC → Tϕ(v)M is m-to-1 if α = mπ/n and ∞-to-1 if α is π-irrational. The geodesics γ(t) in C cannot be further extended (generally) once they reach a vertex. All other geodesics in C are defined for −∞ < t < 8 The representation M = ∪g∈Gg(P ) is not a tiling, in general. 24 EUGENE GUTKIN AND MICHAL RAMS Using the inclusion P ⊂ M , we identify TP with the subset of TM consisting of M-tangent vectors with base-points in P , and directed in- ward. Any v ∈ TP defines the billiard orbit in P , β(t) = expP (tv), 0 ≤ t, and the geodesic in M , γ(t) = expM(tv), 0 ≤ t. They are related by the canonical unfolding of billiard orbits. This is an inductive pro- cedure which replaces the consecutive reflections about the sides of P by consecutive reflections of the “latest billiard table” g(P ) about the appropriate side, yielding the next billiard table h(P ), and continuing the geodesic straight across the common side of g(P ) and h(P ). See [4] in the planar case and [7], section 3.1, in the general case. Let x ∈ P and let v ∈ TxP . We denote by βv (resp. γv) the billiard orbit in P (resp. the geodesic in M) that emanates from x in the direction v. The unfolding operator, Ux : βv 7→ γv, preserves the parametrisations. Proposition 6. Let x ∈ P, v ∈ TxP . Identify P and Pe ⊂ C and let x ∈ Pe, v ∈ TxC be the corresponding data. Then for t ∈ R+ we have (27) Ux(expP (v, t)) = ϕ(expC(v, t)). Proof. We will freely use the preceding discussion. As t ∈ R+ goes to infinity, expP (v, t) runs with the unit speed along a billiard orbit in P . The curve exp (v, t) is the geodesic in C defined by the data (x, v), and ϕ(exp (v, t)) is the geodesic in M emanating from x in the direction v. The billiard orbit in P and the geodesic in M are related by the unfolding operator. For x ∈ P let ExP = TxP ×R+ be the full tangent space (or the full tangent cone) at x. If S ⊂ TxP is a segment, let ESxP = S × R+ be the corresponding subcone. We use the analogous notation for x ∈ C or x ∈ M . In polar coordinates (t, θ) in R2 the lebesgue measure on ExP is given by the density tdtdθ. Corollary 13. Let x ∈ P ⊂ M be arbitrary, and let expxP : ExP → P be the exponential mapping. The pull-back by expxP of the lebesgue measure on P to ExP is the smooth measure with the density dν(t, θ). 1. When M = R2, we have dν = tdtdθ. 2. When M = H2, we have dν = sinh tdtdθ. 3. When M = S2, we have dν = | sin t|dtdθ. Proof. By Proposition 6, the measure in question coincides with the pullback to the tangent space ExM of the riemannian measure on M by the exponential map ExM → M . The latter is well known. We point out that the preceding material has a billiard map version. We will briefly discuss it now. Let β(t) = (z(t), θ(t)), t ∈ R, be an orbit COMPLEXITY, ETC 25 of the billiard flow. We obtain the corresponding billiard map orbit βd(k), k ∈ Z, by restricting β(t) to the consecutive times tk such that z(tk) ∈ ∂P . The correspondence β(·) 7→ βd(·) is invertible. This allows us to formulate the billiard map versions of the universal covering space, the lifting of billiard map orbits to the universal covering space, and the relationship between the liftings and the unfoldings, à là Proposition 6. Since we are not directly using this material in the body of the paper, we spare the details. References [1] J. Cassaigne, P. Hubert and S. Troubetzkoy, Complexity and growth for polyg- onal billiards, Ann. Inst. Fourier 52 (2002), 835 – 847. [2] M.W. Davis, Groups generated by reflections and aspherical manifolds not cov- ered by Euclidean space, Ann. Math. 117 (1983), 293 – 324. [3] G. Galperin, T. Krüger and S. Troubetzkoy, Local instability of orbits in polyg- onal and polyhedral billiards, Comm. Math. Phys. 169 (1995), 463 – 473. [4] E. Gutkin, Billiards in polygons, Physica 19 D (1986), 311 – 333. [5] E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Reg. & Chaot. Dyn. 8 (2003), 1 – 13. [6] E. Gutkin and N. Haydn, Topological entropy of polygon exchange transfor- mations and polygonal billiards, Erg. Theo. & Dyn. Syst. 17 (1997), 849 – [7] E. Gutkin and S. Tabachnikov, Complexity of piecewise convex transforma- tions in two dimensions, with applications to polygonal billiards on surfaces of constant curvature, Moscow Math. J. 6 (2006), 673 – 701. [8] E. Gutkin and S. Troubetzkoy, Directional flows and strong recurrence for polygonal billiards, Pitman Res. Not. Math. 362 (1996), 21 – 45. [9] P. Hubert, Complexité de suites définies par des billards rationnels, Bull. Soc. Math. Fr. 123 (1995), 257 – 270. [10] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge 1995. [11] H. Masur, The growth rate of trajectories of a quadratic differential, Erg. Theo. & Dyn. Syst. 10 (1990), 151 – 176. IMPA, Rio de Janeiro, Brasil and UMK, Torun, Poland; IMPAN, Warszawa, Poland E-mail address : [email protected],[email protected];[email protected] Introduction and overview 1. Averages of counting functions 1.1. Discrete dynamics 1.2. Continuous dynamics 1.3. Special cases 2. Bounds on counting functions 3. Counting functions for polygonal billiard 3.1. Direction counting functions for billiard maps in euclidean polygons 3.2. Position counting functions for billiard flows in euclidean polygons 3.3. Position counting functions for billiard maps in euclidean polygons 3.4. Position counting functions for billiard flows in spherical polygons 3.5. Position counting functions for billiard flows in hyperbolic polygons 4. Relating partial complexities and counting functions 4.1. Piecewise convex transformations and piecewise convex flows 4.2. Partial complexities for maps and flows 4.3. Partial complexities and counting functions 5. Bounds on partial complexities for the billiard 5.1. Direction complexities for billiard maps in euclidean polygons 5.2. Position complexities for billiard flows in euclidean polygons 5.3. Position complexities for billiard maps in euclidean polygons 5.4. Position complexities for billiard flows in spherical polygons 5.5. Position complexities for billiard flows in hyperbolic polygons 6. Appendix: Covering spaces for polygonal billiards References

0704.1976

Information-Based Asset Pricing

Information-Based Asset Pricing Dorje C. Brody∗, Lane P. Hughston†, and Andrea Macrina† ∗Blackett Laboratory, Imperial College, London SW7 2BZ, UK †Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, UK Abstract. A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the correspond- ing price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such “X-factor” we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents “noise”. The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black-Scholes-Merton type. We consider both the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, with- out the need for specifying on an ad hoc basis the dynamics of the volatility. Key words: Asset pricing; partial information; stochastic volatility; correlation; div- idend growth; Brownian bridge; nonlinear filtering; market microstructure Working paper. Original version: December 5, 2005. This version: October 22, 2018. Email: [email protected], [email protected], [email protected] I. INTRODUCTION In derivative pricing, the starting point is usually the specification of a model for the price process of the underlying asset. Such models tend to be of an ad hoc nature. For example, in the Black-Scholes-Merton-Merton theory, the underlying asset has a geometric http://arxiv.org/abs/0704.1976v1 Brownian motion as its price process. More generally, the economy is often modelled by a probability space equipped with the filtration generated by a multi-dimensional Brownian motion, and it is assumed that asset prices are adapted to this filtration. This example is of course the “standard” model within which a great deal of financial engineering has been carried out. The basic problem with the standard model (and the same applies to various generalisations thereof) is that the market filtration is fixed, and no comment is offered on the issue of “where it comes from”. In other words, the filtration, which represents the revelation of information to market participants, is modelled first, in an ad hoc manner, and then it is assumed that the asset price processes are adapted to it. But no indication is given about the nature of this “information”, and it is not obvious, a priori, why the Brownian filtration, for example, should be regarded as providing information rather than noise. In a complete market there is a sense in which the Brownian filtration provides no ir- relevant information. That is to say, in a complete market based on a Brownian filtration the asset price movements reflect the information content of the filtration. Nevertheless, the notion that the market filtration should be “prespecified” is an unsatisfactory one in financial modelling. The intuition behind the “prespecified-filtration” approach is that the filtration represents the unfolding in time of a succession of random events that “influence” the markets, causing prices to change. For example, bad weather in South America results in a decrease in the supply of coffee beans and hence an increase in the price of coffee. Or, say, a spate of bad derivative deals causes a drop in confidence in banks, and hence a downgrade in earnings projections, and thus a drop in their prices. The idea is that one “abstractifies” these influences in the form of a prespecified background filtration to which price processes are adapted. What is unsatisfactory about this is that little structure is given to the filtration: price movements behave as though they were spontaneous. In reality, we expect the price-formation process to exhibit more structure. It would be out of place in the present context to attempt an account of the process of price formation. Nevertheless, we can improve on the “prespecified” approach. In that spirit we proceed as follows. We note that price changes arise from two sources. The first is that resulting from changes in agent preferences—that is to say, changes in the pricing kernel. Movements in the pricing kernel are associated with (a) changes in investor attitudes towards risk, and (b) changes in investor “impatience”, the subjective discounting of future cash flows. Equally important are changes in price resulting from the revelation of information about the future cash flows derivable from a given asset. When a market agent decides to buy or sell an asset, the decision is made in accordance with the information available to the agent concerning the likely future cash flows associated with the asset. A change in the information available to the agent about a future cash flow will typically have an effect on the price at which they are willing to buy or sell, even if the agent’s preferences remain unchanged. Consider the situation where one is thinking of purchasing an item at a price that seems attractive. But then, one reads an article pointing out an undesirable feature of the product. After reflection, one decides that the price is too high, given the deficiencies that one is now aware of. As a result, one decides not to buy, not at that price, and eventually—because other individuals will have read the same report—the price drops. The movement of the price of an asset should, therefore, be regarded as an emergent phenomenon. To put the matter another way, the price process of an asset should be viewed as the output of (rather than an input into) the decisions made relating to possible trans- actions in the asset, and these decisions should be understood as being induced primarily by the flow of information to market participants. Taking into account this observation we propose in this paper a new framework for asset pricing based on modelling of the flow of market information. The information is that concerning the values of the future cash flows associated with the given assets. For example, if the asset represents a share in a firm that will make a single distribution at some agreed date, then there is a single cash flow. If the asset is a credit-risky discount bond, then the future cash flow is the payout of the bond at maturity. In each case, based on the information available relating to the likely payouts of the given financial instrument, market participants determine estimates for the value of the right to the impending cash flows. These estimates lead to the decisions concerning transactions that trigger movements in the price. In this paper we present a class of models capturing the essence of the scenario described above. In building the framework we have several criteria in mind that we would like to see satisfied. The first of these is that our model for the flow of market information should be intuitively appealing, and should allow for a reasonably sophisticated account of aggregate investor behaviour. At the same time, the model should be simple enough to allow one to derive explicit expressions for the asset price processes thus induced, in a suitably rich range of examples, as well as for various associated derivative price processes. The framework should also be flexible enough to allow for the modelling of assets having complex cash-flow structures. Furthermore, it should be suitable for practical implementation. The framework should be mathematically sound, and manifestly arbitrage-free. In what follows we show how our modelling framework goes a long way towards satisfying these criteria. The role of information in financial modelling has long been appreciated, particularly in the theory of market microstructure (see, e.g., Back [1], Back and Baruch [2], O’Hara [20], and references cited therein). The present framework is perhaps most closely related to the line of investigation represented, e.g., in Cetin et al. [5], Duffie and Lando [8], Giesecke [10], Giesecke and Goldberg [11], Guo et al. [13], and Jarrow and Protter [14]. The work in this paper, in particular, develops that described in Brody et al. [3] and Macrina [19], where preliminary accounts of some of this material appear (see also Rutkowski and Yu [22]). The paper is organised as follows. In Sections II, III, and IV we illustrate the framework for information-based pricing by considering the scenario in which there is a single random cash flow. An elementary model for market information is presented, based on the specifica- tion of a process composed of two parts: a “signal” component containing true information about the upcoming cash flow, and an independent “noise” component which we model in a specific way. A closed-form expression for the asset price is obtained in terms of the market information available at the time the price is being specified. This result is summarised in Proposition 1. In Section V we show that the resulting asset price process is driven by a Brownian motion, an expression for which can be obtained in terms of the market informa- tion process: this construction indicates in explicit terms the sense in which the price process can be viewed as an “emergent” phenomenon. In Section VI we show that the value of a European-style call option, in the case of an asset with a single cash flow, admits a simple formula analogous to that of the Black-Scholes-Merton model. In Section VII we derive pricing formulae for the situation when the random variable associated with the single cash flow has an exponential distribution or, more generally, a gamma distribution. The extension of the framework to assets associated with multiple cash flows is established in Section VIII. We show that once the relevant cash flows are decomposed in terms of a collection of independent market factors, then a closed-form expression for the asset price associated with a complex cash-flow structure can be obtained. In Section IX we show how the standard geometric Brownian motion model can be derived in an information-based setting. This remarkable result motivates the specific choice of information process given in equation (4). In Section X we present a simple model for dividend growth. In Section XI we show that by allowing distinct assets to share one or more common market factors in the determination of one or more of their respective cash flows we obtain a natural correlation structure for the associated asset price processes. This method for introducing correlation in asset price movements contrasts with the ad hoc approach adopted in most financial modelling. In Section XII we demonstrate that if two or more market factors affect the future cash flows of an asset, then the corresponding price process will exhibit unhedgeable stochastic volatility. This result is noteworthy since even for the class of relatively simple models considered here it is possible to identify a candidate for the origin of stochasticity in price volatility, as well as the form it should take, which is given in Proposition 2. In the remaining sections of the paper we consider the case where the rate at which the information concerning the true value of an impending cash flow is revealed is time dependent. The introduction of a time-dependent information rate adds additional flexibility to the modelling framework, and opens the door to the possibility of calibrating the model to the market prices of options. We consider the single-factor case first, and obtain a closed- form expression for the conditional expectation of the cash flow. The result is stated first in Section XIII as Proposition 3, and the derivation is then given in the two sections that follow. In Section XIV we introduce a new measure appropriate for the consideration of a Brownian bridge with a random drift, which is used in Section XV to obtain an expression for the conditional probability density of the random cash flow. The consistency of the resulting price process is established in Section XVI. We show, in particular, that, for the given information process, if we re-initialise the model at some specified future time, the dynamics of the model moving forward from that time can be represented by a suitably re-initialised information process. The statement of this result is given in Proposition 4. The dynamical equation satisfied by the price process is analysed in Section XVII, where we demonstrate in Proposition 5 that the driving process is a Brownian motion, just as in the constant parameter case. In Section XVIII we derive the price of a European-style call option in the case for which the information rate is time dependent. Our approach is based on the idea that first one models the cash flows, then the in- formation processes, then the filtration, and finally the prices. In Section XIX, we solve the corresponding “inverse” problem. The result is stated in Proposition 6. Starting from the dynamics of the conditional distribution of the impending payoff, which is driven by a Brownian motion adapted to the market filtration, we construct (a) the random variable that represents the relevant market factor, and (b) an independent Brownian bridge repre- senting irrelevant information. These two combine to generate the market filtration. We conclude in Section XX with a general multi-factor extension of the time-dependent setup, for which the dynamics of the resulting price processes are given in Propositions 7 and 8. II. THE MODELLING FRAMEWORK In asset pricing we require three ingredients: (a) the cash flows, (b) the investor prefer- ences, and (c) the flow of information to market participants. Translated into a mathematical language, these ingredients amount to the following: (a′) cash flows are modelled as random variables; (b′) investor preferences are modelled with the determination of a pricing kernel; and (c′) the market information flow is modelled with the specification of a filtration. As we have indicated above, asset pricing theory conventionally attaches more weight to (a) and (b) than to (c). In this paper we emphasise the importance of ingredient (c). Our theory will be based on modelling the flow of information accessible to market par- ticipants concerning the future cash flows associated with a position in a financial asset. We start by introducing the notation and assumptions employed in this paper. We model the financial markets with the specification of a probability space (Ω,F ,Q) on which a fil- tration {Ft}0≤t<∞ will be constructed. The probability measure Q is understood to be the risk-neutral measure, and the filtration {Ft} is understood to be the market filtration. All asset-price processes and other information-providing processes accessible to market partic- ipants will be adapted to {Ft}. Several simplifying assumptions will be made that will enable us to concentrate our efforts on the problems associated with the flow of market information. The first of these assumptions is the use of the risk-neutral measure. The “real” probability measure does not enter into the present investigation. Expectation in the measure Q will be denoted by E[−]. Our second assumption is that we take the default-free system of interest rates to be deterministic. The absence of arbitrage then implies that the corresponding system of discount functions {PtT }0≤t≤T<∞ can be written in the form PtT = P0T/P0t for t ≤ T , where {P0t}0≤t<∞ is the initial discount function, which we take to be part of the initial data of the model. The function {P0t}0≤t<∞ is assumed to be differentiable and strictly decreasing, and to satisfy 0 < P0t ≤ 1 and limt→∞ P0t = 0. We also assume, for simplicity, that cash flows occur at pre-determined dates. Now clearly for some purposes we would like to allow for cash flows occurring effectively at random times—in particular, at stopping times associated with the market filtration. But in the present exposition we want to avoid the idea of a “prespecified” filtration with respect to which stopping times are defined. We take the view that the market filtration is a “derived” notion, generated by information about upcoming cash flows, and by the values of cash flows when they occur. We shall therefore regard a “randomly-timed” cash flow as being a set of random cash flows occurring at various times, and with a joint distribution function that ensures only one of these flows is non-zero. Hence in our view the ontological status of a cash flow is that its timing is definite, only the amount is random—and that cash flows occurring at different times are, by their nature, different cash flows. III. MODELLING THE CASH FLOWS First we consider the case of an asset that provides a single isolated cash flow occurring at time T , represented by a random variable DT . We assume that DT ≥ 0. The value St of the cash flow at any earlier time t in the interval 0 ≤ t < T is then given by the discounted conditional expectation of DT : St = PtTE [DT |Ft] . (1) In this way we model the price process {St}0≤t<T of a limited-liability asset that pays a single dividend DT at time T . The construction of the price process here is carried out in such a way as to guarantee an arbitrage-free market if other assets are priced by the same method (see Davis [7] for a closely related point of view). We shall use the terms “cash flow” and “dividend” more or less interchangeably. If a more specific use of one of these terms is needed, then this will be evident from the context. We adopt the convention that when the dividend is paid the asset price goes “ex-dividend” immediately. Hence in the example above we have limt→T St = DT and ST = 0. In the case where the asset pays a sequence of dividends DTk (k = 1, 2, . . . , n) on the dates Tk the price (for t < T1) is given by PtTkE [DTk |Ft] . (2) More generally, for all t ≥ 0, and taking into account the ex-dividend behaviour, we have 1{t<Tk}PtTkE [DTk |Ft] . (3) It will be useful if we adopt the convention that a discount bond also goes ex-dividend on its maturity date. Thus in the case of a discount bond we assume that the price of the bond is given, for dates earlier than the maturity date, by the product of the principal and the relevant discount factor. But at maturity (when the principal is paid) the value of the bond drops to zero. In the case of a coupon bond, there is likewise a downward jump in the price of the bond at the time a coupon is paid (the value lost may be captured back in the form of an “accrued interest” payment). In this way we obtain a consistent treatment of the “ex-dividend” behaviour of all of the asset price processes under consideration. With this convention it follows that price processes are right continuous with left limits. IV. MODELLING THE INFORMATION FLOW Now we present a simple model for the flow of market information, following Brody et al. [3]. We consider first the case of a single distribution, occurring at time T , and assume that market participants have only partial information about the upcoming cash flow DT . The information available in the market about the cash flow is assumed to be contained in a process {ξt}0≤t≤T defined by: ξt = σtDT + βtT . (4) We call {ξt} the market information process. This process is composed of two parts. The term σtDT contains the “true information” about the dividend, and grows in magnitude as t increases. The process {βtT}0≤t≤T is a standard Brownian bridge over the time interval [0, T ]. Thus {βtT } is Gaussian, β0T = 0, βTT = 0, the random variable βtT has mean zero, and the covariance of βsT and βtT for s ≤ t is s(T − t)/T . We assume that DT and {βtT } are independent. Thus the information contained in the bridge process is “pure noise”. We assume that the market filtration {Ft} is generated by the market information process: Ft = σ({ξs}0≤s≤t). The dividend DT is therefore FT -measurable, but is not Ft-measurable for t < T . Thus the value of DT becomes “known” at time T , but not earlier. The bridge process {βtT } is not adapted to {Ft} and thus is not directly accessible to market participants. This reflects the fact that until the dividend is paid the market participants cannot distinguish the “true information” from the “noise” in the market. The introduction of the Brownian bridge models the fact that market perceptions, whether valid or not, play a role in determining asset prices. Initially, all available information is used to determine the a priori probability distribution for DT . The parameter σ represents the rate at which information about the true value of DT is revealed as time progresses. If σ is low, the value of DT is effectively hidden until very near the time of the dividend payment; whereas if σ is high, then the value of the cash flow is for all practical purposes revealed quickly. In the example under consideration we have made some simplifying assumptions concern- ing the information structure. For instance, we assume that σ is constant. In Section XIII, however, we consider a time-dependent information flow rate. We have also assumed that the random dividend DT enters directly into the structure of the information process, and enters linearly. As we shall indicate later, a more general and natural setup is to let the information process depend on a random variable XT which we call a “market factor”; then the dividend is regarded as a function of the market factor. This arrangement has the ad- vantage that it easily generalises to the situation where a cash flow might depend on several independent market factors, or indeed where cash flows associated with different financial instruments have one or more factors in common. Given the market information structure described above for a single cash flow, we proceed to construct the associated price dynamics. The price process {St} for a share in the firm paying the specified dividend is given by formula (1). It is assumed that the a priori probability distribution of the dividend DT is known. This distribution is regarded as part of the initial data of the problem, which in some cases can be calibrated from knowledge of the initial price of the asset along with other price data. The general problem of how the a priori distribution is obtained is an important one—any asset pricing model has to confront this issue—which we defer for later consideration. The initial distribution is not to be understood as being “absolutely” determined, but rather represents the “best estimate” for the distribution given the data available at that time, in accordance with what one might call a Bayesian point of view. We note the fact that the information process {ξt} is Markovian (see Brody et al. [3], and Rutkowski and Yu [22]). Making use of this property together with the fact that DT is FT -measurable we deduce that St = 1{t<T}PtTE [DT |ξt] . (5) If the random variable DT that represents the payoff has a continuous distribution, then the conditional expectation in (5) can be expressed in the form E [DT |ξt] = xπt(x) dx. (6) Here πt(x) is the conditional probability density for the random variable DT : πt(x) = Q(DT ≤ x|ξt). (7) We implicitly assume appropriate technical conditions on the distribution of the dividend that will suffice to ensure the existence of the expressions under consideration. Also, for con- venience we use a notation appropriate for continuous distributions, though corresponding results can be inferred for discrete distributions, or more general distributions, by slightly modifying the stated assumptions and conclusions. Bearing in mind these points, we note that the conditional probability density process for the dividend can be worked out by use of a form of the Bayes formula: πt(x) = p(x)ρ(ξt|DT = x) p(x)ρ(ξt|DT = x)dx . (8) Here p(x) denotes the a priori probability density for DT , which we assume is known as an initial condition, and ρ(ξt|DT = x) denotes the conditional density for the random variable ξt given that DT = x. Since βtT is a Gaussian random variable with mean zero and variance t(T − t)/T , we deduce that the conditional probability density for ξt is ρ(ξt|DT = x) = 2πt(T − t) −(ξt − σtx) 2t(T − t) . (9) Inserting this expression into the Bayes formula we get πt(x) = p(x) exp T−t(σxξt − σ2x2t) p(x) exp T−t(σxξt − σ2x2t) . (10) We thus obtain the following result for the asset price: Proposition 1. The information-based price process {St}0≤t≤T of a limited-liability asset that pays a single dividend DT at time T with a priori distribution Q(DT ≤ y) = p(x) dx (11) is given by St = 1{t<T}PtT xp(x) exp T−t(σxξt − σ2x2t) p(x) exp T−t(σxξt − σ2x2t) , (12) where ξt = σtDT + βtT is the market information. V. ASSET PRICE DYNAMICS IN THE CASE OF A SINGLE CASH FLOW In order to analyse the properties of the price process deduced above, and to be able to compare it with other models, we need to work out the dynamics of {St}. One of the advantages of the model under consideration is that we have an explicit expression for the price at our disposal. Thus in obtaining the dynamics we need to find the stochastic differential equation of which {St} is the solution. This turns out to be an interesting exercise because it offers some insights into what we mean by the assertion that market price dynamics should be regarded as an “emergent phenomenon”. To obtain the dynamics associated with the price process {St} of a single-dividend paying asset let us write DtT = E[DT |ξt]. (13) Evidently, DtT can be expressed in the form DtT = D(ξt, t), where D(ξ, t) is defined by D(ξ, t) = xp(x) exp T−t(σxξ − σ2x2t) p(x) exp T−t(σxξ − σ2x2t) . (14) A straightforward calculation making use of the Ito rules shows that the dynamical equation for {DtT} is given by dDtT = T − t T − t ξt − σTDtT dt+ dξt . (15) Here Vt is the conditional variance of the dividend: Vt = Et (DT − Et[DT ])2 x2πt(x) dx− xπt(x) dx . (16) Therefore, if we define a new process {Wt}0≤t<T by setting Wt = ξt − T − s σTDtT − ξs ds, (17) we find, after some rearrangement, that dDtT = T − t VtdWt. (18) For the dynamics of the asset price we thus have dSt = rtStdt + ΓtTdWt, (19) where rt = −d lnP0t/dt is the short rate, and the absolute price volatility ΓtT is ΓtT = PtT T − t Vt. (20) A slightly different way of arriving at this result is as follows. We start with the condi- tional probability πt(x). Then, using the notation above, for its dynamics we obtain dπt(x) = T − t (x−DtT )πt(x) dWt. (21) Since according to (5) the asset price is given by St = 1{t<T}PtT xπt(x) dx, (22) we can infer the dynamics of {St} from the dynamics of the conditional probability {πt(x)}, once we take into account formula (16) for the conditional variance. As we shall demonstrate later, the process {Wt} defined in (17) is an {Ft}-Brownian motion. Hence from the point of view of the market it is the process {Wt} that drives the asset price dynamics. In this way our framework resolves the paradoxical point of view usually adopted in financial modelling in which {Wt} is regarded on the one hand as “noise”, and yet on the other hand also generates the market information flow. And thus, instead of hypothesising the existence of a driving process for the dynamics of the markets, we are able from the information-based perspective to deduce the existence of such a process. The information-flow parameter σ determines the overall magnitude of the volatility. In fact, σ plays a role analogous to the similarly-labelled parameter in the Black-Scholes- Merton theory. Thus, we can say that the rate at which information is revealed in the market determines the magnitude of the volatility. Everything else being the same, if we increase the information-flow rate, then the market volatility will increase as well. According to this point of view, those mechanisms that one might have thought were destined to make markets more efficient—e.g., globalisation of the financial markets, reduction of trade barriers, improved communications, a robust regulatory environment, and so on—can have the effect of increasing market volatility, and hence market risk, rather than reducing it. VI. EUROPEAN-STYLE CALL OPTIONS Before we turn to the consideration of more general cash flows and market information structures, let us consider the pricing of a derivative on an asset for which the price process is governed by (19). Specifically, we consider the valuation of a European call option on such an asset, with strike price K, and exercisable at a fixed date t. The option is written on an asset that pays a single dividend DT at time T > t. The initial value of the option is C0 = P0tE (St −K)+ . (23) Inserting the expression for St derived in the previous section into this formula, we obtain C0 = P0t E x πt(x)dx−K . (24) For convenience we write the conditional probability πt(x) in the form πt(x) = pt(x) pt(x)dx , (25) where the “unnormalised” density pt(x) is defined by pt(x) = p(x) exp T − t σxξt − 12σ . (26) Substituting (26) into (24) we find that the value of the option is C0 = P0tE (PtTx−K) pt(x)dx , (27) where pt(x)dx. (28) The random variable 1/Φt can be used to introduce a measure B on (Ω,Ft), which we call the “bridge measure”. The option price can then be written: C0 = P0tE (PtTx−K) pt(x)dx . (29) The special feature of the bridge measure, as we establish in Section XIV in a more general context, is that the random variable ξt is Gaussian under B. In particular, under B we find that {ξt} has mean 0 and variance t(T − t)/T . Since pt(x) can be expressed as a function of ξt, when we calculate the expectation in (29) we obtain a tractable formula for C0. To determine the value of the option we define a constant ξ∗ (the critical value) by the following condition: (PtTx−K) p(x) exp T − t σxξ∗ − 1 σ2x2t dx = 0. (30) Then the expectation in (29) can be performed and we find that the option price is C0 = P0T x p(x)N − z∗ + σx dx− P0tK p(x)N − z∗ + σx dx, (31) where N(x) is the standard normal distribution function, and T − t , z∗ = ξ∗ t(T − t) . (32) We see that a tractable expression of the Black-Scholes-Merton-Merton type is obtained. The option pricing problem, even for general p(x), reduces to an elementary numerical problem. It is interesting to note that although the probability distribution for the price St is not of a “standard” type, nevertheless the option valuation problem remains a solvable one. VII. EXAMPLES OF SPECIFIC DIVIDEND STRUCTURES In this section we consider the dynamics of assets with various dividend structures. First we look at a simple asset for which the cash flow is exponentially distributed. The a priori probability density for DT is thus of the form p(x) = exp (−x/δ) , (33) where δ is a constant. We can regard the idea of an exponentially distributed payout as a model for the situation where little is known about the probability distribution of the dividend, apart from its mean. Then from formula (12) we find that the asset price is: St = 1{t<T}PtT x exp(−x/δ) exp T−t(σxξt − σ2x2t) exp(−x/δ) exp T−t(σxξt − σ2x2t) . (34) We note that S0 = P0T δ, so we can calibrate δ by use of the initial price. The integrals in the numerator and denominator in the expression above can be worked out explicitly. Hence, we obtain a closed-form expression for the price in the case of a simple asset with an exponentially-distributed cash flow: St = 1{t<T}PtT B2t /At 2πAt N(Bt/ , (35) where At = σ 2tT/(T − t) and Bt = σTξt/(T − t)− δ−1. Next we consider the case of an asset for which the single dividend paid at T is gamma- distributed. More specifically, we assume the probability density is of the form p(x) = (n− 1)! xn−1 exp(−δx), (36) where δ is a positive real number and n is a positive integer. This choice for the probability density also leads to a closed-form expression for the share price. We find that St = 1{t<T}PtT t Fk(−Bt/ k−n+1 n−k−1 t Fk(−Bt/ , (37) where At and Bt are as above, and Fk(x) = zk exp dz. (38) A recursion formula can be worked out for the function Fk(x). This is given by (k + 1)Fk(x) = Fk+2(x)− xk+1 exp , (39) from which it follows that F0(x) = 2πN(−x), F1(x) = e− x2 , F2(x) = xe 2πN(−x), F3(x) = (x 2 + 2)e− x2 , and so on. In general, the polynomial parts of {Fk(x)}k=0,1,2,... are related to the Legendre polynomials. VIII. MARKET FACTORS AND MULTIPLE CASH FLOWS In this section we proceed to consider the more general situation where the asset pays multiple dividends. This will allow us to consider a wider range of financial instruments. Let us write DTk (k = 1, . . . , n) for a set of random dividends paid at the pre-designated dates Tk (k = 1, . . . , n). Possession of the asset at time t entitles the bearer to the cash flows occurring at times Tk > t. For simplicity we assume n is finite. For each value of k we introduce a set of independent random variables XαTk (α = 1, . . . , mk), which we call market factors or X-factors. For each value of α we assume that the market factor XαTk is FTk-measurable, where {Ft} is the market filtration. For each value of k, the market factors {XαTj}j≤k represent the independent elements that determine the cash flow occurring at time Tk. Thus for each value of k the cash flow DTk is assumed to have the following structure: DTk = ∆Tk(X , XαT2 , ..., X ), (40) where ∆Tk(X , XαT2 , ..., X ) is a function of j=1mj variables. For each cash flow it is, so to speak, the job of the financial analyst (or actuary) to determine the relevant independent market factors, and the form of the cash-flow function ∆Tk for each cash flow. With each market factor XαTk we associate an information process {ξ }0≤t≤Tk of the form ξαtTk = σ XαTkt+ β . (41) Here σαTk is a parameter, and {β } is a standard Brownian bridge over the interval [0, Tk]. We assume that the X-factors and the Brownian bridge processes are all independent. The parameter σαTk determines the rate at which the market factorX is revealed. The Brownian bridge represents the associated noise. We assume that the market filtration {Ft} is gener- ated by the totality of the independent information processes {ξαtTk}0≤t≤Tk for k = 1, 2, . . . , n and α = 1, 2, . . . , mk. Hence, the price of the asset is given by 1{t<Tk}PtTkEt [DTk ] . (42) IX. GEOMETRIC BROWNIAN MOTION MODEL The simplest application of the X-factor technique arises in the case of geometric Brow- nian motion models. We consider a limited-liability company that makes a single cash distribution ST at time T . We assume that ST has a log-normal distribution under Q, and can be written in the form ST = S0 exp rT + ν TXT − 12ν , (43) where the market factor XT is normally distributed with mean zero and variance one, and r > 0 and ν > 0 are constants. The information process {ξt} is taken to be of the form ξt = σtXT + βtT , (44) where the Brownian bridge {βtT} is independent of XT , and where the information flow rate is of the special form . (45) By use of the Bayes formula we find that the conditional probability density is of the Gaussian form: πt(x) = 2π(T − t) 2(T − t) Tx− ξt , (46) and has the following dynamics dπt(x) = T − t Tx− ξt πt(x)dξt. (47) A short calculation then shows that the value of the asset at time t < T is given by St = e −r(T−t) Et[ST ] = e−r(T−t) Tx− 1 ν2Tπt(x)dx = S0 exp rt+ νξt − 12ν . (48) The surprising fact in this example is that {ξt} itself turns out to be the innovation process. Indeed, it is not too difficult to verify that {ξt} is an {Ft}-Brownian motion. Hence, setting Wt = ξt for 0 ≤ t < T we obtain the standard geometric Brownian motion model: St = S0 exp rt+ νWt − 12ν . (49) We see therefore that starting with an information process of the form (44) we are able to recover the familiar asset price dynamics given by (49). An important point to note here is that the Brownian bridge process {βtT } appears quite naturally in this context. In fact, if we start with (49) then we can make use of the following orthogonal decomposition of the Brownian motion (see, e.g., Yor [24]): . (50) The second term in the right, independent of the first term on the right, is a standard representation for a Brownian bridge process: βtT = Wt − WT . (51) Thus by writing XT = WT/ T and σ = 1/ T we find that the right side of (50) is indeed the market information. In other words, formulated in the information-based framework, the standard Black-Scholes-Merton theory can be expressed in terms of a normally distributed X-factor and an independent Brownian bridge noise process. X. DIVIDEND GROWTH As an elementary example of a multi-dividend structure, we shall look at a simple growth model for dividends in the equity markets. We consider an asset that pays a sequence of dividends DTk , where each dividend date has an associated X-factor. Let {XTk}k=1,...,n be a set of independent, identically-distributed X-factors, each with mean 1 + g. The dividend structure is assumed to be of the form DTk = D0 XTj , (52) where D0 is a constant. The parameter g can be interpreted as the dividend growth factor, and D0 can be understood as representing the most recent dividend before time zero. For the price of the asset we have: St = D0 1{t<Tk}PtTkEt . (53) Since the X-factors are independent, the conditional expectation of the product appearing in this expression factorises into a product of conditional expectations, and each such con- ditional expectation can be written in the form of an expression of the type we have already considered. As a consequence we are led to a tractable family of dividend growth models. XI. ASSETS WITH COMMON FACTORS The multiple-dividend asset pricing model introduced in Section VIII can be extended in a very natural way to the situation where two or more assets are being priced. In this case we consider a collection of N assets with price processes {S(i)t }i=1,2,...,N . With asset number (i) we associate the cash flows {D(i)Tk} paid at the dates {Tk}k=1,2,...,n. We note that the dates {Tk}k=1,2,...,n are not tied to any specific asset, but rather represent the totality of possible cash-flow dates of any of the given assets. If a particular asset has no cash flow on one of the dates, then it is assigned a zero cash-flow for that date. From this point, the theory proceeds exactly as in the single asset case. That is to say, with each value of k we associate a set of X-factors XαTk (α = 1, 2, . . . , mk), and a system of market information processes {ξ }. The X-factors and the information processes are not tied to any particular asset. The cash flow occurring at time Tk for asset number (i) is given by a cash flow function of the form (XαT1, X , ..., XαTk). (54) In other words, for each asset each cash flow can depend on all of the X-factors that have been “activated” at that point. Thus for the general multi-asset model we have the following price process system: 1{t<Tk}PtTkEt . (55) It is possible in general for two or more assets to “share” an X-factor in association with one or more of the cash flows of each of the assets. This in turn implies that the various assets will have at least one Brownian motion in common in the dynamics of their price processes. We thus obtain a natural model for the correlation structures in the prices of these assets. The intuition is that as new information comes in (whether “true” or “bogus”) there will be several different assets all affected by the news, and as a consequence there will be a correlated movement in their prices. XII. ORIGIN OF UNHEDGEABLE STOCHASTIC VOLATILITY Based on the general model introduced in the previous sections, we are now in a position to make an observation concerning the nature of stochastic volatility in the equity markets. In particular, we shall show how a natural framework for stochastic volatility arises in the information-based framework. This is achieved without the need for any ad hoc assumptions concerning the dynamics of the stochastic volatility. In fact, a very specific dynamical model for stochastic volatility is obtained—thus leading to a possible means by which the theory proposed here might be tested. We shall work out the volatility associated with the dynamics of the asset price process {St} given by (42). The result is given in Proposition 2 below. First, as an example, we consider the dynamics of an asset that pays a single dividend DT at T . We assume that the dividend depends on the market factors {XαT }α=1,...,m. For t < T we then have: St = PtTE X1T , . . . , X ∣ ξ1tT , . . . , ξ = PtT · · · ∆T (x 1, . . . , xm) π1tT (x1) · · ·πmtT (xm) dx1 · · ·dxm. (56) Here the various conditional probability density functions παtT (x) for α = 1, . . . , m are παtT (x) = pα(x) exp σα x ξαtT − 12(σ α)2 x2t pα(x) exp σα x ξαtT − 12(σα)2 x2t , (57) where pα(x) denotes the a priori probability density function for the factor XαT . The drift of {St}0≤t<T is given by the short rate. This is because Q is the risk-neutral measure, and no dividend is paid before T . Thus, we are left with the problem of determining the volatility of {St}. We find that for t < T the dynamical equation of {St} assumes the form: dSt = rtStdt + ΓαtTdW t . (58) Here the volatility term associated with factor number α is given by ΓαtT = σ T − t PtT Cov X1T , . . . , X , XαT , (59) and {W αt } denotes the Brownian motion associated with the information process {ξαt }, as defined in (17). The absolute volatility of {St} is of the form (ΓαtT ) . (60) For the dynamics of a multi-factor single-dividend paying asset we can thus write dSt = rtStdt+ΓtdZt, where the {Ft}-Brownian motion {Zt} that drives the asset-price process is ΓαsT dW s . (61) The point to note here is that in the case of a multi-factor model we obtain an unhedgeable stochastic volatility. That is to say, although the asset price is in effect driven by a single Brownian motion, its volatility in general depends on a multiplicity of Brownian motions. This means that in general an option position cannot be hedged with a position in the underlying asset. The components of the volatility vector are given by the covariances of the cash flow and the independent market factors. Unhedgeable stochastic volatility thus emerges from the multiplicity of uncertain elements in the market that affect the value of the future cash flow. As a consequence we see that in this framework we obtain a natural explanation for the origin of stochastic volatility. This result can be contrasted with, say, the Heston model [12], which despite its popularity suffers from the fact that it is ad hoc in nature. Much the same can be said for the various generalisations of the Heston model used in commercial applications. The approach to stochastic volatility proposed in the present paper is thus of a new character. Expression (58) generalises to the case for which the asset pays a set of dividends DTk (k = 1, . . . , n), and for each k the dividend depends on the X-factors {{XαTj} α=1,...,mj j=1,...,k }. The result can be summarised as follows. Proposition 2. The price process of a multi-dividend asset has the following dynamics: dSt = rt St dt + 1{t<Tk} σαkTk Tk − t PtTk Cov DTk , X dW αkt DTkd1{t<Tk}, (62) where DTk = ∆Tk(X , XαT2 , · · · , X ) is the dividend at time Tk (k = 1, 2, . . . , n). XIII. TIME-DEPENDENT INFORMATION FLOW We consider now a generalisation of the foregoing material to the situation in which the information-flow rate varies in time. The time-dependent problem is of relevance to many circumstances. For example, there will typically be more activity in a market during the day than at night—such a consideration is important for short-term investments. Alternatively, it may be that the annual report of a firm is going to be published on a specified day—in this case much more information concerning the future of the firm may be made available on that day than normal. We begin our analysis of the time-dependent case by considering the situation where there is a single cash flow DT occurring at T , and the associated market factor is the cash flow itself. In this way we can focus our attention on mathematical issues arising from the time dependence of the information flow rate. Once these issues have been dealt with, we shall consider more complicated cash-flow structures. For the market information process we propose an expression of the form ξt = DT σsds+ βtT , (63) where the function {σs}o≤s≤T is taken to be deterministic and nonnegative. We assume that σ2sds < ∞. The price process {St} of the asset is then given by St = 1{t<T}PtTE [DT |Ft ] . (64) where the market filtration is, as in the previous sections, assumed to be generated by the information process. Our first task is to work out the conditional expectation in (64). This can be achieved by use of a change-of-measure technique, which will be outlined in Section XIV. It will be useful, however, to state the result first. We define the conditional probability density process {πt(x)} by setting πt(x) = Q (DT ≤ x| Ft) . (65) The following result is obtained: Proposition 3. Let the information process {ξt} be given by (63). Then the conditional probability density process {πt(x)} for the random variable DT is given by πt(x) = p(x) e x( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds p(x) e x( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds . (66) We deduce at once from Proposition 3 that the conditional expectation of the random variable DT is DtT = xp(x) e x( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds p(x) e x( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds . (67) The associated price process {St} is then given by St = 1{t<T}PtTDtT . XIV. CHANGES OF MEASURE FOR BROWNIAN BRIDGES Since the information process is a Brownian bridge with a random drift, we shall require formulae relating a Brownian bridge with drift in one measure to a standard Brownian bridge in another measure to establish Proposition 3 . We proceed as follows. First we recall a well- known integral representation for the Brownian bridge. Let the probability space (Ω,F ,Q) be given, with a filtration {Gt}0≤t<∞, and let {Bt} be a standard {Gt}-Brownian motion. Then the process {βtT}, defined by βtT = (T − t) T − s dBs, (68) for 0 ≤ t < T , and by βtT = 0 for t = T , is a standard Brownian bridge over the interval [0, T ]. Expression (68) converges to zero as t → T ; see, e.g., Karatzas and Shreve [16], Protter [21]). The filtration {Gt} is larger than the market filtration {Ft}. In particular, since {βtT} is adapted to {Gt} we can think of {Gt} as the filtration describing the information available to an “insider” who can distinguish between what is noise and what is not. Let DT be a random variable on (Ω,F ,Q). We assume thatDT is G0-measurable and that DT is independent of {βtT}. Thus the value of DT is known “all along” to the insider, but not to the typical market participant. For simplicity in what follows we assume that DT is bounded; this condition can be relaxed with the introduction of an appropriate Novikov-type condition; but we will not pursue the more general situation here. Define the deterministic nonnegative process {νt}0≤t≤T by νt = σt + T − t σsds, (69) and let {ξt} be defined as in (63). We define the process {Λt}0≤t<T by the relation = exp νsdBs − 12D ν2sds . (70) With these elements in hand, we fix a time horizon U ∈ (0, T ) and introduce a probability measure B on GU by the relation dB = Λ−1U dQ. (71) Then we have the following facts: (i) The process {W ∗t }0≤t<U defined by W ∗t = DT νsds+Bt (72) is a B-Brownian motion. (ii) The process {ξt} defined by (63) is a B-Brownian bridge and is independent of DT . (iii) The random variable DT has the same probability law with respect to B and Q. (iv) The conditional expectation for any integrable function f(DT ) of the random variable DT can be expressed in the form Q[f(DT )|F ξt ] = f(DT )Λt ∣F ξt ∣F ξt ] . (73) We note that the measure B is independent of the specific choice of the time horizon U in the sense that if B is defined on GU ′ for some U ′ > U , then the restriction of that measure to GU agrees with the measure B as already defined. When we say that {ξt} is a B-Brownian bridge what we mean, more precisely, is that ξ0 = 0, that {ξt} is B-Gaussian, that EB[ξt] = 0, and that B[ξsξt] = s(T − t) for 0 ≤ s ≤ t ≤ U . Thus with respect to the measure B the process {ξt}0≤t≤U has the properties of a standard [0, T ]-Brownian bridge that has been truncated at time U . The fact that {ξt} is a B-Brownian bridge can be verified as follows. By (63), (68), and (72) we have ξt = DT σsds+ (T − t) T − s σsds+ (T − t) T − s (dW ∗s −DTνsds) σsds− (T − t) T − s + (T − t) T − s dW ∗s = (T − t) T − s dW ∗s , (75) where in the final step we used the relation T − s νsds = T − t σsds. (76) This formula can be verified explicitly by differentiation, which then gives us (69). In (75) we see that {ξt} has been given the standard integral representation of a Brownian bridge. We remark, incidentally, that (73) can be thought of a variation of the Kallianpur-Striebel formula appearing in the literature of nonlinear filtering (see, for example, Bucy and Joseph [4], Davis and Marcus [7], Fujisaki et al. [9], Kallianpur and Striebel [15], Krishnan [17], and Liptser and Shiryaev [18]). XV. DERIVATION OF THE CONDITIONAL DENSITY We have introduced the idea of measure changes associated with Brownian bridges in order to introduce formula (73), which involves the density process {Λt}, which in (70) is defined in terms of the Q-Brownian motion {Bt}. On the other hand, the expectations in (73) are conditional with respect to the information generated by {ξt}. Therefore, it will be convenient to express {Λt} in terms of {ξt}. To do this we substitute (72) in (70) to obtain Λt = exp s − 12D ν2sds . (77) We then observe, by differentiating (75), that dξt = − T − t dt+ dW ∗t . (78) Substituting this relation in (77) we obtain Λt = exp νsdξs + T − s νsξsds ν2sds . (79) In principle at this point all we need to do is to substitute (69) into (79) to obtain the result for {Λt}. In practice, further simplification can be achieved. To this end, we note that by taking the differential of the coefficient of DT in the exponent of (79) we get νsdξs + T − s νsξsds dξt + T − t T − t dξt + T − t T − t σsds+ σsdξs . (80) Then integrating both sides of (80) we obtain: νsdξs + T − s νsξsds = T − t σsds+ σsdξs. (81) Similarly, by taking the differential of the coefficient of −1 D2T in the exponent of (79) and making use of (69), we find ν2t dt = σ2t + 2 T − t σsds+ (T − t)2 T − t σ2sds . (82) Therefore, by integrating both sides of (82) we obtain an identity for the coefficient of −1 D2T . It follows by virtue of the two identities just obtained that {Λt} can be expressed in terms of {ξt}. More explicitly, we have Λt = exp T−t ξt σsds+ σsdξs σ2sds . (83) Note that by transforming (79) into (83) we have eliminated a term having {ξt} in the integrand, thus achieving a considerable simplification. Proposition 3 can then be deduced if we use equation (76) and the basic relation Q (DT ≤ x| Ft) = EQ 1{DT≤x} . (84) In particular, since DT and {ξt} are independent under the bridge measure, by virtue of (73), (83), and (84) we obtain Q (DT ≤ x| Ft) = p(y) e y( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds p(y) e y( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds , (85) from which we immediately infer Proposition 3 by differentiation with respect to x. We conclude this section by noting that an alternative expression for {πt(x)}, written in terms of {W ∗t }, is given by πt(x) = p(x) exp u − 12x ν2udu p(x) exp νudW ∗u − 12x2 ν2udu . (86) XVI. CONSISTENCY RELATIONS Before we proceed to analyse in detail the dynamics of the price process {St}, first we shall establish a useful dynamical consistency relation satisfied by prices obtained in the information-based framework. By “consistency” we have in mind the following. Suppose that we re-initialise the information process at an intermediate time s ∈ (0, T ) by specifying the value ξs of the information at that time. For the framework to be dynamically consistent, we require that the remainder of the period [s, T ] admits a representation in terms of a suitably “renormalised” information process. Specifically, we have: Proposition 4. Let 0 ≤ s ≤ t ≤ T . The conditional probability πt(x) can be written in terms of the intermediate conditional probability πs(x) in the form πt(x) = πs(x) e x( 1T−t ηt σ̃udu+ σ̃udηu)− 1 σ̃udu) σ̃2udu πs(x) e x( 1T−t ηt σ̃udu+ σ̃udηu)− 1 σ̃udu) σ̃2udu , (87) where σ̃u = σu + T − s σvdv (88) is the re-initialised market information-flow rate, and ηt = ξt − T − t T − s ξs (89) is the re-initialised information process. The fact that {ηt}s≤t≤T represents the updated information process bridging the interval [s, T ] can be seen as follows. First we note that ηs = 0 and that ηT = ξT . Substituting (63) in (89) we find that ηt = DT σ̃udu+ γtT , (90) where σ̃u is as defined in (88), and γtT = βtT − T − t T − s βsT . (91) A calculation making use of the covariance of the Brownian bridge {βtT} shows that the Gaussian process {γtT}s≤t≤T is a standard Brownian bridge over the interval [s, T ]. It thus follows that {ηt} is the information bridge interpolating the interval [s, T ]. To verify (87) we note that (86) can be written in the form πt(x) = πs(x) exp u − 12x ν2udu πs(x) exp νudW ∗u − 12x2 ν2udu . (92) The identity (80) then implies that T − t σudu+ σudξu + T − t T − s T − t σ̃udu+ σ̃udηu, (93) where we have made use of (88) and (89). Similarly, (81) implies that ν2udu = T − t T − s σ2udu T − t σ̃udu σ̃2udu. (94) Substitution of (93) and (94) into (92) establishes (87). In particular, the form of (87) is identical to the original formula (66), modulo the indicated renormalisation of the informa- tion process and the associated information flow rate. XVII. EXPECTED DIVIDEND The goal of sections XIII, XIV, and XV was to obtain an expression for the conditional expectation (13) in the case of a single-dividend asset in the case of a time-dependent information-flow rate. In the analysis of the associated price process it will therefore be useful to work out the dynamics of the conditional expectation of the dividend. In particular, an application of Ito’s rule to (67), after some rearrangement, shows that dDtT = νtVt T − t ξt − νtDtT dt+ νtVtdξt, (95) where {Vt} is the conditional variance of the random variable DT , given by (16). Let us define a new process {Wt} by Wt = ξt + T − s ξsds− νsDsTds. (96) We refer to {Wt} as the “innovation process”. It follows from the definition of {Wt} that dDtT = νtVt dWt. (97) Since {DtT } is an {Ft}-martingale we are thus led to conjecture that {Wt} must also be an {Ft}-martingale. In fact, we have the following result: Proposition 5. The process {Wt} defined by (96) is an {Ft}-Brownian motion under Q. Proof. We need to establish that (i) {Wt} is an {Ft}-martingale, and that (ii) (dWt)2 = dt. Writing Et[−] = EQ[−|Ft] and letting t ≤ u we have Et [Wu] = Et [ξu] + Et T − s νsDsTds . (98) Splitting the second two terms on the right into integrals between 0 and t, and between t and u, we obtain Et [Wu] = Et[ξu] + T − s ξsds− νsDsTds T − s Et[ξs]ds− νsEt[DsT ]ds. (99) The martingale property of the conditional expectation implies that Et[DsT ] = DtT for t ≤ s, which allows us to simplify the last term. To simplify the expression for the expectation Et[ξs] for t ≤ s we use the tower property: Et[βsT ] = Et[E[βsT |HT , βtT ]] = Et[E[βsT |βtT ]]. (100) To calculate the inner expectation E[βsT |βtT ] here we use the fact that the random variable βsT/(T − s)− βtT/(T − t) is independent of βtT and deduce that E[βsT |βtT ] = T − s T − t βtT , (101) from which it follows that Et [βsT ] = T − s T − t Et[βtT ]. (102) As a result we obtain Et[ξs] = DtT σvdv + T − s T − t Et[βtT ]. (103) We also recall the definition of {Wt} given by (96), which implies that T − s ξsds− νsDsTds = Wt − ξt. (104) Therefore, substituting (103) and (104) into (99) we obtain Et [Wu] = DtT σsds+Wt − ξt +DtT T − s ds−DtT +Et[βtT ]. (105) Next we split the first term into an integral from 0 to t and an integral from t to u, and we insert the definition (69) of {νt} into the fifth term. The result is: Et [Wu] = Wt +DtT σsds+ Et[βtT ]− ξt. (106) Finally, if we make use of the fact that ξt = Et[ξt], and hence that ξt = DtT σsds+ Et[βtT ], (107) we see that {Wt} satisfies the martingale condition. On the other hand, by virtue of (96) we have (dWt) 2 = dt. We conclude that {Wt} is an {Ft}-Brownian motion under Q. � XVIII. ASSET PRICES AND DERIVATIVE PRICES We are now in a position to consider in more detail the dynamics of the price process of an asset paying a single dividend DT in the case of a time-dependent information flow. For {St} we have St = 1{t<T}PtTDtT , or equivalently St = 1{t<T}PtT xp(x) e x( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds p(x) e x( 1T−t ξt σsds+ σsdξs)− 1 σsds) σ2sds . (108) A calculation making use of (97) shows that for the dynamics of the price we have dSt = rtStdt + ΓtTdWt, (109) where the asset price volatility is given by ΓtT = νtPtTVt, where Vt is the conditional variance of the dividend, given by (16). It should be evident by virtue of its definition that {Vt} is a supermartingale. More specifically, for the dynamics of {Vt} we obtain dVt = −ν2t V 2t dt+ νtκtdWt, (110) where κt denotes the third conditional moment of DT , given by κt = Et [(DT −DtT )3]. Although we have derived (108) by assuming that the price process is induced by the market information {ξt}, the result to be shown in Section XIX demonstrates that we can regard the dynamical equation (109) for the price process as given, and then deduce the structure of the underlying information. The information-based interpretation of the mod- elling framework, however, is more appealing. According to this interpretation there is a flow of market information, which is available to all market participants and is represented by the filtration generated by {ξt}. Given this information, each participant will “act”, in our interpretation, so as to minimise the risk adjusted future P&L variance associated with the cash flow under consideration. The future P&L is determined by the value of DT , and the estimate of DT that minimises its variance is indeed given by the conditional expectation (13). By discounting this expectation with PtT we recover the price process {St}. We note that {ΓtT } is “infinitely stochastic” in the sense that all of the higher-order volatilities (the volatility of the volatility, and so on) are stochastic. These higher-order volatilities have a natural interpretation: the volatility of the asset price is determined by the variance of the random cash flow; the volatility of the volatility is determined by the skewness of DT ; its volatility is determined by the kurtosis of DT ; and so on. The fact that the asset price in the bridge measure is given by a function of a Gaussian random variable means that the pricing of derivatives is numerically straightforward. We have seen this in the case of a constant information-flow rate, but the result holds in the time-dependent case as well. For example, consider a European-style call option with strike K and maturity t, where t ≤ T , for which the value is (23). If we express the asset price St on the option maturity date in terms of the B-Brownian motion {W ∗t } we find C0 = P0tE (PtTx−K)p(x) exp s − 12x ν2sds , (111) where p(x) exp s − 12x ν2sds dx. (112) To proceed we shall use the factor 1/Φt in (111) to make a change of measure on (Ω,Ft). The idea is as follows. We fix a time horizon u at or beyond the option expiration but before the bond maturity, so t ≤ u < T , and define a process {Φt}0≤t≤u by use of the expression (112), where now we let t vary in the range [0, u]. By an application of Ito calculus on (112) we see that dΦt = νtDtTΦt dW t . On the other hand, it follows from (78) and (96) that the B-Brownian motion {W ∗t } and the Q-Brownian motion {Wt} are related by dW ∗t = dWt + νtDtTdt. (113) Therefore, in terms of {Wt} we have dΦt = ν tTΦtdt + νtDtTΦt dWt, (114) from which it follows that dΦ−1t = −νtDtTΦ−1t dWt. (115) Upon integration we deduce Φ−1t = exp νsDsT dWs − 12 sT ds . (116) Since {νsDsT} is bounded, and s ≤ u < T , we see that {Φ−1s }0≤s≤u is a Q-martingale with EQ[Φ−1t ] = 1, where t is the option maturity date. Therefore, the factor 1/Φt in (111) can be used to effect a change of measure Q → B on (Ω,Ft). We note that while on the space (Ω,Gt) it is the process {Λt} introduced in (70) that defines the measure change from B and Q, on (Ω,Ft) it is Φt = EQ[Λt|Ft] that defines the relevant measure change. As a consequence, by changing the measure in (111) we obtain C0 = P0tE (PtTx−K)p(x) exp s − 12x ν2sds . (117) This result should be compared with (27). We note that in the bridge measure the expression s is a Gaussian random variable with mean zero and variance ω2t = ν2sds = T − t σ2sds. (118) Here we have used the relation (82). Therefore, if we set Y = ω−1t s , (119) it follows that Y is B-Gaussian. For the call price we thus have C0 = P0tE (PtTx−K)p(x)eωtxY− ω2t x , (120) and hence C0 = P0t (PtTx−K)p(x)eωtxy− ω2t x dy. (121) We observe that there exists a critical value y = y∗ such that the argument of the “plus” function vanishes in the expression above. Thus y∗ is given by (PtTx−K)p(x) eωtxy ω2t x dx = 0. (122) As a consequence the call price can be written C0 = P0t (PtTx−K)p(x)eωtxy− ω2t x dy. (123) The integration in the y variable can be performed, and we deduce the following represen- tation for the call price: C0 = P0t (PtTx−K)p(x)N(ωtx− y∗)dx. (124) When the cash flow is represented by a discrete random variable and the information-flow rate is constant, this result reduces to an expression equivalent to the option pricing formula derived in Brody et al. [3]. If the cash flow is a continuous random variable and the information flow rate is constant then we recover the expression (31) given in section IV (see also Rutkowski and Yu [22]). We conclude this section with the remark that the simulation of {St} is straightforward. First, we generate a Brownian trajectory, and form the associated Brownian bridge {βtT (ω)}. We then select a value for DT by a method consistent with the a priori probability density p(x), and substitute these in the formula ξt(ω) = DT (ω) σsds + βtT (ω) for some choice of {σt}. Finally, substitution of {ξt(ω)} in (108) gives us a simulated path {St(ω)}. The statistics of the process {St} are obtained by repeating this procedure, the results of which can be used to price derivatives, or to calibrate the information-flow rate {σt}. XIX. EXISTENCE OF THE INFORMATION PROCESS We consider now what might be called the “inverse problem” for information-based asset pricing. The idea is to begin with the conditional density process {πt(x)} and to construct from it the independent degrees of freedom represented by the X-factor DT and the noise {βtT}. The setup is as follows. On the probability space (Ω,F ,Q) let {Wt} be a Brownian motion and let {Ft} be the filtration generated by {Wt}. Let DT be FT -measurable, and let {πt(x)} denote the associated conditional probability density process. We assume that {πt(x)} satisfies the stochastic differential equation dπt(x) = νt(x−DtT )πt(x) dWt, (125) with the initial condition π0(x) = p(x), where {νt} is given by (69), and where DtT = xπt(x) dx. (126) We define the process {ξt} as follows: ξt = (T − t) T − s dWs + νsDsTds . (127) Then we have the following result: Proposition 6. The random variables DT and βtT = ξt − DT σsds are Q-independent for all t ∈ [0, T ]. Furthermore, the process {βtT} is a Q-Brownian bridge. Proof. To establish the independence of DT and βtT it suffices to verify that Q[exβtT+yDT ] = EQ[exβtT ]EQ[eyDT ] (128) for arbitrary x, y. Using the tower property we have Q[exβtT+yDT ] = EQ exξt E e(y−x σsds)DT , (129) where we have inserted the definition of βtT given in the statement of the Proposition. We consider the inner expectation first. From equation (73) for the conditional expectation of a function of DT we deduce that e(y−x σsds)DT = Φ−1t p(z) e(y−x σsds)z ez u− 12z ν2ududz, (130) where the process {Φt} is defined by (112). We now change the probability measure from Q to B, so that the term Φ−1t appearing in (130) drops out to give us exξt E e(y−x σsds)DT p(z) e(y−x σsds)z ez u− 12 z ν2ududz p(z)EB ex(T−t) dW ∗s +(y−x σsds)z+z s − 12 z ν2sds p(z) e(y−x σsds)z− 1 ν2sds+ α2sds E s − 12 α2sds dz, (131) where αs = x(T − t)/(T − s) + zνs, and therefore Q[exβtT+yDT ] = p(z) e(y−x σsds)z− 1 ν2sds+ α2sdsdz (132) Furthermore, making use of relation (76) we have σsds− 12z ν2sds + α2sds = exp t(T − t) . (133) As a consequence, it follows from (132) that exβtT+yDT p(z) eyzdz t(T − t) . (134) This establishes the independence of {βtT} and DT . The factorisation (134) also shows that the process {βtT } is Q-Gaussian, with mean zero and variance t(T − t)/T . To establish that {βtT} is a Brownian bridge, we must show that for s ≤ t the covariance of βsT and βtT is given by s(T − t)/T . Alternatively, it suffices to analyse the moment generating function E[exβsT+yβtT ]. We proceed as follows. First, using the tower property we have exβsT+yβtT exξs+yξt−(x σudu+y σudu)DT exξs+yξtE σudu+y σudu)DT . (135) Next, by use of formula (73), the inner expectation can be carried out to give exβs+yβtT exξs+yξtΦ−1t p(z) e−(x σudu+y σudu)z ez u− 12z ν2ududz .(136) If we change the probability measure to B the random variable Φt in the denominator drops out, and we have exβs+yβtT p(z) e−(x σudu+y σudu)z− 1 ν2udu E exξs+yξt+z dz. (137) Let us consider the inner expectation first. By defining au = x(T − s)/(T − u) and bu = y(T − t)/(T − u) + zνu we can write exξs+yξt+z . (138) However, since {W ∗t } is a B-Brownian motion, using the properties of Gaussian random variable we find that = exp a2udu+ b2udu+ 2 aubudu . (139) Substituting the definitions of {au} and {bu} into the right side of (139) and combining the result with the remaining terms in the exponent of the right side of (137) we find that the terms involving the integration variable z drop out, and we are left with the integral of the density function p(z), which is unity. Gathering the remaining terms we obtain exβsT+yβtT = exp s(T − s) t(T − t) + 2xy s(T − t) . (140) It follows that the covariance of βsT and βtT for s ≤ t is given by exβsT+yβtT x=y=0 s(T − t) . (141) This establishes the assertion that {βtT} is a Q-Brownian bridge. � The result above shows that, for the class of price processes we are considering, even if at the outset we take the “usual” point of view in financial modelling, and regard the price process of the asset as being adapted to some “prespecified” filtration, nevertheless it is possible to deduce the structure of the underlying information-based model. XX. MULTI-FACTOR MODELS WITH A TIME-DEPENDENT INFORMATION FLOW RATE Let us now turn to consider the case of a single cash flowDT that depends on amultiplicity of market factors {XαTk} α=1,...,mk k=1,...,n , where we have the n pre-designated information dates {Tk}k=1,2,...,n, and where for each value of k we have a set ofmk market factors. For simplicity we set T = Tn. Each market factor X is associated with an information process ξαtTk = X σαsTkds+ β , (142) where XαTk and {β } are independent. It should be evident that although the random variable DT representing the cash flow is FT -measurable, the values of some of the X-factors upon which it depends may be revealed at earlier times. That is to say, the uncertainties arising from some of the economic elements affecting the value of the cash flow at time T may be resolved before that time. Since the X-factors are independent, it follows that for each market factor the associated conditional density process παtTk(x) takes the form given in (66), and the corresponding dynamical equation is given by dπαtTk = ν xαk − EQ XαTk |Ft παtTk dW t . (143) The function ναtTk appearing here is given by an expression of the form (69): ναtTk = σ Tk − t σαsTkds. (144) The innovation process {W αkt } is defined in terms of {ξαtTk} via a relation of the form W αkt = ξ Tk − s ξαsTkds− ναsTkX ds. (145) The conditional expectation EQ[DT |Ft] is thus given by the multi-dimensional integral DtT = · · · ∆T (x 1, . . . , x 1 , . . . , x n, . . . , x ×πt1(x11) · · ·πt1(x 1 ) · · ·πtn(x1n) · · ·πtn(xmnn ) dx11 · · ·dx 1 · · ·dx1n · · ·dxmnn , (146) and the price of the asset for t < T is St = PtTDtT . A straightforward application of Ito’s rule then establishes the following result: Proposition 7. The price process {St}0≤t<T of an asset that pays a single dividend DT at time T (= Tn) depending on the market factors {XαTk} α=1,2,...,mk k=1,2,...n , satisfies the dynamical equation dSt = rtStdt+ ναtTkCovt[DT , X ] dW αkt , (147) where DT = ∆T XαT1 , . . . , X . (148) Here Covt[DT , X ] denotes the covariance between the cash-flow DT and the market factor XαTk , conditional on the information Ft generated by the information processes {ξ }0≤s≤t. In the more general case of an asset that pays multiple dividends (see Section VIII) the price is given by 1{t<Tk}PtTkE {XαTj} α=1,2,...,mj j=1,...,k . (149) Proposition 8. The price process {St} of an asset that pays the random dividends DTk on the dates Tk (k = 1, . . . , n) satisfies the dynamical equation dSt = rtStdt+ 1{t<Tk}ν Covt[DTk , X ] dW αkt +∆Tkd1{t<T}, (150) where DTk = ∆Tk {XαTj} α=1,...,mj j=1,...,k . (151) Here Covt[DTk , X ] denotes the covariance between the dividend DTk and the market factor XαTk , conditional on the market information Ft. We conclude that the multi-factor, multi-dividend situation is fully tractable when the information-flow rates associated with the various market factors are time dependent. A straightforward extension of Proposition 8 allows us to formulate the joint price dynamics of a system of assets, the associated dividend flows of which may depend on common market factors. As a consequence, a specific model for stochastic volatility and correlation emerges for such a system of assets, and it is one of the main conclusions of this paper that such a model can be formulated. The information-based “X-factor” approach presented here thus offers a new insights into the nature of volatility and correlation, and as such may find applications in a number of different areas of financial risk analysis. We have in mind, in particular, applications to equity portfolios, credit portfolios, and insurance, all of which exhibit intertemporal market correlation effects. We also have in mind the problem of firm- wide risk management and optimal capital allocation for banking institutions. Acknowledgements. The authors thank T. Bielecki, T. Björk, I. Buckley, H. Bühlmann, S. Carter, I. Constantinou, M. Davis, J. Dear, A. Elizalde, B. Flesaker, H. Geman, V. Hen- derson, D. Hobson, T. Hurd, M. Jeanblanc, A. Lokka, J. Mao, B. Meister, M. Monoyios, M. Pistorius, M. Rutkowski, D. Taylor, and M. Zervos for stimulating discussions. The authors are also grateful for helpful comments made by seminar participants at various meetings where parts of this work have been presented, including: the Developments in Quantitative Finance conference, July 2005, Isaac Newton Institute, Cambridge; the Math- ematics in Finance conference, August 2005, Kruger National Park, RSA; the School of Computational and Applied Mathematics, University of the Witwatersrand, RSA, August 2005; CEMFI (Centro de Estudios Monetarios y Financieros), Madrid, October 2005; the Department of Actuarial Mathematics and Statistics, Heriot-Watt University, December 2005; the Department of Mathematics, King’s College London, December 2005; the Bank of Japan, Tokyo, December 2005; and Nomura Securities, Tokyo, December 2005. 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Financial Studies 6, 327-343 (1993). [13] X. Guo, R. A. Jarrow and Y. Zeng, “Information reduction in credit risk modelling” working paper (2005). [14] R. A. Jarrow and P. Protter, “Structural versus reduced form models: a new informa- tion based perspective” J. Investment Management 2, 34-43 (2004). [15] G. Kallianpur and C. Striebel, “Estimation of stochastic systems: Arbitrary system process with additive white noise observation errors” Ann. Math. Statist. 39, 785 (1968). [16] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus (Berlin: Springer, 1997). [17] V. Krishnan, Nonlinear Filtering and Smoothing (New York: Dover, 2005). [18] R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes Vols. I and II, 2nd ed. (Berlin: Springer, 2000). [19] A. Macrina, “An information-based framework for asset pricing: X-factor theory and its applications” PhD thesis, King’s College London (2006). [20] M. O’Hara, Market Microstructure Theory (Cambridge, Massachusetts: Blackwell, 1995). [21] P. Protter Stochastic Integration and Differential Equations: A New Approach, 2nd ed. (Berlin: Springer, 2003). [22] M. Rutkowski and N. Yu, “On the Brody-Hughston-Macrina approach to modelling of defaultable term structure”, working paper, School of Mathematics, University of New South Wales, Sydney, downloadable at www.defaultrisk.com (2005). [23] W. M. Wonham, “Some applications of stochastic differential equations to optimal nonlinear filtering” J. SIAM A2, 347 (1965). [24] M. Yor, Some Aspects of Brownian Motion, Part I: Some Special Functionals (Basel: Birkhäuser, 1992). [25] M. Yor, Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems (Basel: Birkhäuser, 1996). Introduction The Modelling framework Modelling the cash flows Modelling the information flow Asset price dynamics in the case of a single cash flow European-style call options Examples of specific dividend structures Market factors and multiple cash flows Geometric Brownian motion model Dividend growth Assets with common factors Origin of unhedgeable stochastic volatility Time-dependent information flow Changes of measure for Brownian bridges Derivation of the conditional density Consistency relations Expected dividend Asset prices and derivative prices Existence of the information process Multi-factor models with a time-dependent information flow rate

0704.1977

The Jumping Phenomenon of Hodge Numbers

7 The Jumping Phenomenon of Hodge Numbers October 24, 2018 XuanmingYe Abstract Let X be a compact complex manifold, consider a small defor- mation φ : X → B of X, the dimension of the Dolbeault cohomology groups Hq(Xt,Ω ) may vary under this defromation. This paper will study such phenomenons by studying the obstructions to deform a class in Hq(X,Ω X) with the parameter t and get the formula for the obstructions. 1 Introduction Let X be a compact complex manifold and φ : X → B be a family of complex manifolds such that φ−1(0) = X . Let Xt = φ −1(t) denote the fibre of φ above the point t ∈ B. We denote by OX and ΩpX the sheaves of germs of X of holomorphic functions and p-forms respectively. Recall hp,q = dimCH q(X,Ω X) and Pm = dimH 0(X, (ΩnX) m) where n = dimCX . S.Iitaka proposed a problem whether all Pm are deformation invariants [1]. This problem was solved by Iku Nakamura in his paper [2], and actually he gave us some examples of small deformations of complex parallelisable manifold (by a complex parallelisable manifold we mean a compact complex manifold with the trivial holomorphic tangent bundle) such that the hodge numbers of the fibre of the family jump in these deformations. http://arxiv.org/abs/0704.1977v1 In this paper, we will study such phenomenons from the viewpoint of obstruction theory. More precisely, for a certain small deformation X of X parametrized by a basis B and a certain class [α] of the Dolbeaut cohomology group Hq(X,Ω X), we will try to find out the obstruction to extending it to an element of the relative Dolbeaut cohomology group Hq(X ,Ωp ). We will call those elements which have non trivial obstruction the obstructed elements. In §2 we will summarize the results of Grauert’s Direct Image Theorems and we will try to explain why we need to consider the obstructed elements. Actually, we will see that these elements will play an important role when we study the jumping phenomenon of Hodge numbers. Because we will see that the existence of the obstructed elements is a necessary and sufficient condition for the variation of the Hodge diamond. In §3 we will get a formula for the obstruction to the extension we men- tioned above. Theorem 3.3 Let π : X → B be a deformation of π−1(0) = X, where X is a compact complex manifold. Let πn : Xn → Bn be the nth order deformation of X. For arbitrary [α] belongs to Hq(X,Ωp), suppose we can extend [α] to order n − 1 in Hq(Xn−1,ΩpXn−1/Bn−1). Denote such element by [αn−1]. The obstruction of the extension of [α] to nth order is given by: on,n−1(α) = dXn−1/Bn−1 ◦ κnx(αn−1) + κnx◦dXn−1/Bn−1(αn−1), where κn is the nth order Kodaira-Spencer class and dXn−1/Bn−1 is the relative differential operator of the n− 1th order deformation. In §4 we will use this formula to study carefully the example given by Iku Nakamura, i.e. the small deformation of the Iwasama manifold and discuss some phenomenons. Acknowledgement. The research was partially supported by China-France- Russian mathematics collaboration grant, No. 34000-3275100, from Sun Yat- sen University. The author would also like to thank ENS, Paris for its hos- pitality during the academic years of 2005–2007. Last but most, the author would like to thank Professor Voisin for her patient helps and valuable sug- gestions. 2 Grauert’s Direct Image Theorems and De- formation theory In this section, let us first review some general results of deformation theory. Let X be a compact complex manifold. The manifold X has an underlying differential structure, but given this fixed underlying structure there may be many different complex structures on X . In particular, there might be a range of complex structures on X varying in an analytic manner. This is the object that we will study. Definition 1.0 A deformation of X consists of a smooth proper morphism φ : X → B, where X and B are connected complex spaces, and an isomor- phism X ∼= φ−1(0), where 0 ∈ B is a distinguished point. We call X → B a family of complex manifolds. Although B is not necessarily a manifold, and can be singular, reducible, or non-reduced, (e.g. B = SpecC[ε]/(ε2)), since the problem we are going to research is the phenomenon of the jumping of the Dolbeaut cohomology, we may assume that X and B are complex manifolds. In order to study the jumping of the Dolbeaut cohomology, we need the following important theorem (one of the Grauert’s Direct Image Theorems). Theorem 1.1 Let X, Y be complex spaces, π : X → Y a proper holomor- phic map. Suppose that Y is Stein, and let F be a coherent analytic sheaf on X. Let Y0 be a relatively compact open set in Y . Then, there is an integer N > 0 such that the following hold. I. There exists a complex E · : ...→ E−1 → E0 → ...→ EN → 0 of finitely generated locally free OY0-modules on Y0 such that for any Stein open set W ⊂ Y0, we have Hq(Γ(W, E ·)) ≃ Γ(W,Rqπ∗(F)) ≃ Hq(π−1(W ),F) ∀q ∈ Z. II. (Base Change Theorem). Assume, in addition, that F is π-flat [i.e. ∀x ∈ X, the stalk F is flat over as a module over OY,π(x)]. Then, there exists a complex E · : 0 → E0 → E1 → ...→ EN → 0 of finitely generated locally free OY0-sheaves Ep with the following property: Let S be a Stein space and f : S → Y a holomorphic map. Let X ′ = X ×Y S and f ′ → X and π′ : X ′ → S be the two projections. Then, if T is an open Stein subset of f−1(Y0), we have, for all q ∈ Z Hq(Γ(T, f ∗(E ·))) ≃ Γ(T,Rqπ′∗(F )) ≃ Hq(π′−1(T ),F ′) where F ′ = (f ′)∗(F). Let X ,Y be complex spaces, π : X → Y a proper map. Let F be a π-flat coherent sheaf on X . For y ∈ Y , denote by My the OY -sheaf of germs of holomorphic functions ”vanishing at y”: the stalk of My at y is the maximal ideal of OY,y; that at t 6= y” is OY,t. We set F(y) = analytic restriction of F to π−1(y) = F (OY /My). Since we just need to study the local properties, we may assume, in view of Theorem 1.0, part II, that there is a complex E · : 0 // OP0Y // OP1Y // ... d // OPNY with the base change property in Theorem 1.0, part II. In particular, if y ∈ Y , we have Hq(π−1(y),F(y)) ≃ Hq(E · ⊗ (OY /My)). Apply what we discussed above to our case φ : X → B, we get the follow- ing. There is a complex of vector bundles on the basis B, whose cohomology groups at the point identifies to the cohomology groups of the fiber Xb with values in the considered vector bundle on X , restricted to Xb. Therefore, for arbitrary p, there exists a complex of vector bundles (E·, d·), such that for arbitrary t ∈ B, Hq(Xt,ΩpXt) = H q(E·t) = Ker(d q)/Im(dq−1). Via a local trivialisation of the bundle Ei, the differential of the complex E· are represented by matrices with holomorphic coefficients, and follows from the lower semicontinuity of the rank of a matrix with variable coefficients , it is easy to check that the function dimCKer(d q) and −dimCIm(dq) are upper semicontinuous onB. Therefore the function dimCH q(E·t) is also upper semicontinuous. It seems that either the increasing of dimCIm(d q−1) or the decreasing of dimCKer(d q) will cause the jumping of dimCH q(E·t), however, because of the following exact sequence: 0 → Ker(dq)t → Eqt → Im(dq)t → 0 ∀t, which means the variation of−dimCIm(dq) is exactly the variation of dimCKer(dq), we just need to consider the variation of dimCKer(d q) for all q. In order to study the variation of dimCKer(d q), we need to consider the following problem. Let α be an element of Ker(dq) at t = 0, we try to find out the obstruction to extending it to an element which belongs to Ker(dq) in a neighborhood of 0. Such kind of extending can be studied order by order. Let E q0 be the stalk of the assocaited sheaf of Eq at 0. Let m0 be the maximal idea of OB,0. For arbitrary positive intergal n, since dq can be represented by matrices with holomorphic coefficients, it is not difficult to check dq(E q0 ⊗OB,0 mn0 ) ⊂ E 0 ⊗OB,0 mn0 . Therefore the complex of the vector bundles (E·, d·) induces the following complex: 0 → E00⊗OB,0OB,0/mn0 d0→ E10⊗OB,0OB,0/mn0 d1→ ... d → EN0 ⊗OB,0OB,0/mn0 dN→ 0. Definition 2.2 Those elements of H ·(E·0) which can not be extended are called the first class obstructed elements. Next, we will show the obstructions of the extending we mentioned above. For simplicity, my may assume that dimCB = 1, suppose α can be extended to an element αn−1 such that j q(αn−1))(t) = 0, then αn−1 can be con- sidered as the n−1 order extension of α. Here jn−10 (dq(αn−1))(t) is the n−1 jet of dq(αn−1) at 0. Define a map oqn : H q(E ·0 ⊗OB,0 OB,0/mn0 ) → Hq+1(E·0) by [αn−1] 7−→ [jn0 (dq(αn−1))(t)/tn]. At first, we need to check oqn is well defined. So we need to show that [jn0 (d q(αn−1))(t)/t n] is dq+1-closed. Via a local trivialization of the bundles Ei, the differentials of the complex E· are represented by matrices with holo- morphic coefficients, and from the lower semi-continuity of the rank of a matrix with variable coefficients, we may assume that there always exists 1 , ..., σ l ) which are sections of E q+1 such that (σ 1 |t=0, ..., σ l |t=0) form a basis of Ker(dq+1 : E 0 → E 0 ) and Ker(d q+1 : Eq+1 → Eq+2) ⊂ Span{σq+1j }. So we can write dq(αn−1) = j fjσ Since jn−10 (d q(αn−1))(t)=0, we have fj = 0 and = 0, i = 1..n− 1. (dq(αn−1))|t=0 = j |t=0+...+ j )|t=0 = j |t=0, therefore dq+1( (dq(α))|t=0) = dq+1( j |t=0) = 0, which means ∂ (dq(αn−1))|t=0 is dq+1-closed. Next we are going to show that the equivalent class of ∂ (dq(αn−1))|t=0 in Hq+1(E·0) depends only on j 0 (αn−1)(t). Let (σ 1, ..., σ k) be a bases of Eq, we only need to show that if jn−10 (αn−1)(t) = 0, then (dq(αn−1))|t=0 belongs to Im(dq : Eq → Eq+1). Indeed, we can write αn−1 = j fjσ j while fj(0) = 0, = 0, i = 1...n− 1, then, (dq(αn−1)) = i )) = i )+...+ (dq(σ i )). Therefore, ∂ (dq(Ω))|t=0 = i )|t=0, which belongs to Im(dq : Eq → Eq+1). At last, we are going to show that the equivalent class of ∂ (dq(Ω))|t=0 in Hq+1(E·0) depends only on the equivalent class of αn−1 in H q(E ·0 ⊗OB,0 OB,0/mn0 ). Actually, we only need to show that if αn−1 belongs to Im(dq−1 : E q−10 ⊗OB,0 OB,0/mn0 → E 0 ⊗OB,0 OB,0/mn0 ), we will have ∂ (dq(αn−1))|t=0 belongs to Im(dq : Eq → Eq+1). In fact, let α′n−1 = dq−1( j fjσ j ) such that jn−10 (α n−1)(t) = j 0 (αn−1)(t). From the discussion above, we have (dq(αn−1))|t=0 = (dq(α n−1))|t=0 = (dq(dq−1( j ))) = 0 in Hq(E·). Remark It seems that jn0 (d q(αn−1))(t)/t n depends on the connection of Eq+1. But, by using an induction argument, it is not difficult to prove that if ji0(d q(αn−1))(t) = 0, ∀i < n, then jn0 (dq(αn−1))(t) is independent of the choice of the connection of Eq+1. There is natural a map ρ i : H q(E·0) → Hq(E ·0 ⊗OB,0 OB,0/m 0 ) given by [σ] 7−→ [tiσ], ∀[σ] ∈ Hq(E·0). Denote the map ρ i ◦oqn : Hq(E ·0⊗OB,0OB,0/mn0 ) → Hq+1(E ·0⊗OB,0OB,0/mi+10 ), ∀i ≤ n by o Next we will show that, for arbitrary i, 0 < i ≤ n, αn−1 can be extended to αn which is the nth order extension of α such that j 0 (αn −αn−1)(t) = 0 if and only if o n,n−i([αn−1]) is trivial. For necessarity, (αn−αn−1)(t)/ti is sup- posed to be the preimage of o n,n−i([αn−1]), so o n,n−i([αn−1]) is trivial. There- fore we just need to check whether it is sufficient. In fact, if o n,n−i([αn−1]) is trivial, then there exists a section β of E ·0 ⊗OB,0 OB,0/mi+10 such that dq(β) = o n,n−i([αn−1]). Then it is not difficult to check that αn−1 − tiβ̃ is an n th order extension of α that we need, where β̃ is an extension of β in the neighborhood of 0. Therefore we have the following proposition. Proposition 2.3 Let αn−1 be an n−1 th order extension of α, for arbitrary i, 0 < i ≤ n, αn−1 can be extended to αn which is the nth order extension of α such that ji−10 (αn − αn−1)(t) = 0 if and only if o n,n−i([αn−1]) = 0. In the following, we will show that the obstructions oqn([αn−1]) also play an important role when we consider about the jumping of dimCIm(d q). Note that dimCIm(d q) jumps if and only if there exist a section β of dimCKer(d q+1), such that β0 is not exact while βt is exact for t 6= 0. Definition 2.4 Those nontrivial elements of H ·(E·0) that can always be extended to a section which is only exact at t 6= 0 are called the second class obstructed elements. Note that if α is exact at t = 0, it can be extended to an element which is exact at every point. So the definition above does not depend on the element of a fixed equivalent class. Proposition 2.5 Let [β] be an nontrivial element of Hq+1(E·0). Then [β] is a second class obstructed element if and only if there exist n ≥ 0 and αn−1 in Hq(E ·0 ⊗OB,0 OB,0/mn0 ) such that oqn([αn−1]) = [β]. Proof. If oqn([αn−1]) = [β], then j q(αn−1))(t)/t n is the extension we need. On the contrary, if [β] is a second class obstructed element. There exist β̃ such that β̃t, t 6= 0 is exact. Then (dq)−1(β̃) is a meromorphic section which has a pole at t = 0. Let n be the degree of (dq)−1(β̃). Then let αn−1 = t n(dq)−1(β̃). It is easy to check that oqn([αn−1]) = [β]. Proposition 2.6 Let αn−1 be an element of H q(E ·0 ⊗OB,0 OB,0/mn0 ) such that oqn([αn−1]) 6= 0. Then there exists n ′ ≤ n and α′ be an element of Hq(E ·0 ⊗OB,0 OB,0/mn 0 ), such that ρ ◦ oqn([αn−1]) = o ]) 6= 0. Proof. If o n,n−1([αn−1]) 6= 0, then n = n and α = αn−1. Otherwise, there exists α 1, such that d 1) = ρ ◦ oqn([αn−1]). Note that o n−1,n−2([α 1]) = n−2 ◦ oqn([αn−1]) = o n,n−2([αn−1]). If we go on step by step as above, we can always get the n and α for there is at least one of the o n,i([αn−1]) is nontrivial. This proposition tells us that althought oqn([αn−1]) 6= 0 does not mean that n,n−1([αn−1]) 6= 0, we can always find α such that oqn([αn−1]) comes from obstuctions like o n,n−1([α ]). Therefore we can get the following corollary immediately from Proposition 2.5 and Proposition 2.6. Corollary 2.7 Let [β] be an nontrivial element of Hq+1(E·0). Then [β] is a second class obstructed element if and only if there exist n ≥ 0 and αn−1 in Hq(E ·0 ⊗OB,0 OB,0/mn0 ) such that o n,n−1([αn−1]) = ρ n−1([β]). Let us come back to our problem, suppose α can be extended to an element αn−1 such that j q(αn−1))(t) = 0, since what we care is whether α can be extended to an element which belongs to Ker(dq) in a neighborhood of 0. So, if we have an nth order extension αn of α, it is not necessary that ji−10 (αn−αn−1)(t) = 0, ∀i, 1 < i < n.What we need is just j00(αn−αn−1)(t) = 0 which means αn is an extension of α. So the “real” obstructions come from n,n−1([αn−1]). Since these obstructions is so important when we consider the problem of variation of hodge numbers, we will try to find out an explicit calculation for such obstructions in next section. 3 The Formula for the Obstructions We are going to prove in this section an explicit formula (Theorem 3.3) for the abstract obstructions described above. Let π : X → B be a deformation of π−1(0) = X , where X is a compact complex manifold. For every integer n ≥ 0, denote by Bn = SpecOB,0/mn+10 the nth order infinitesimal neighbor- hood of the closed point 0 ∈ B of the base B. Let Xn ⊂ X be the complex space over Bn. Let πn : Xn → Bn be the nth order deformation of X . In order to study the jumping phenomenon of Dolbeaut cohomology groups, for arbitrary [α] belongs to Hq(X,Ωp), suppose we can extend [α] to order n−1 in Hq(Xn−1,Ω Xn−1/Bn−1 ). Denote such element by [αn−1]. In the following, we try to find out the obstruction of the extension of [αn−1] to nth order. Denote π∗(m0) by M0. Consider the exact sequence 0 → Mn0/Mn+10 ⊗ Ω X0/B0 Xn/Bn Xn−1/Bn−1 which induces a long exact sequence 0 → H0(X,Mn0/Mn+10 ⊗Ω X0/B0 ) → H0(Xn,ΩpXn/Bn) → H 0(Xn−1,Ω Xn−1/Bn−1 → H1(X,Mn0/Mn+10 ⊗ Ω X0/B0 ) → .... The obstruction for [αn−1] comes from the non trivial image of the connecting homomorphism δ∗ : Hq(Xn−1,Ω Xn−1/Bn−1 ) → Hq+1(X,Mn0/Mn+10 ⊗Ω X0/B0 We will calculate it by C̆ech calculation. Cover X by open sets Ui such that, for arbitrary i, Ui is small enough. More precisely, Ui is stein and the following exact sequence splits 0 → π∗n(ΩBn)(Ui) → ΩXn(Ui) → ΩXn/Bn(Ui) → 0. So we have a map ϕi : ΩXn/Bn(Ui) → ΩXn(Ui), such that, ϕi(ΩXn/Bn(Ui))⊕ π∗n(ΩBn)(Ui) ∼= ΩXn(Ui). Denote by ιi, ι−1i the inclusion from π∗n(ΩBn)(Ui) to ΩXn(Ui) and its inverse. Define d Xn/Bn by ϕi ◦ dXn/Bn ◦ ϕ−1i and diBn by ιi ◦ dBn ◦ ι−1i . Then it determines a local decomposition of the exterior dif- ferentiation dXn in Ω dXn = d + diXn/Bn . Denote the set of alternating q-cochains β with values in F by Cq(U,F), i.e. to each q+1-tuple, i0 < i1... < iq, β assigns a section β(i0, i1, ..., iq) of F over Ui0 ∩ Ui1 ∩ ... ∩ Uiq . Let us still using ϕi denote the following map, ϕi : π ) ∧ Ωp Xn/Bn (Ui) → Ωp+rXn (Ui) ϕi(ωi1 ∧ ... ∧ ωir ∧ βj1 ∧ ... ∧ βjp) = ωi1 ∧ ... ∧ ωir ∧ ϕi(βj1) ∧ ... ∧ ϕi(βjp). Define ϕ : Cq(U, π∗n(ΩrBn) ∧ Ω Xn/Bn ) → Cq(U,Ωp+rXn ) by ϕ(β)(i0, i1, ..., iq) = ϕi0(β(i0, i1, ..., iq)) ∀β ∈ Cq(U, π∗n(ΩrBn) ∧ Ω Xn/Bn where i0 < i1... < iq. Define the total Lie derivative with respect to Bn LBn : Cq(U,Ω ) → Cq(U,Ωp+1Xn ) LBn(β)(i0, i1, ..., iq) = d (β(i0, i1, ..., iq)) ∀β ∈ Cq(U,ΩpXn), where i0 < i1... < iq. Define, for each Ui the total interior product with respect to Bn, I (Ui) → ΩpXn(Ui) by I i(µdg1 ∧ dg2 ∧ ...∧ dgp) = µ dg1 ∧ ...∧ dgj−1 ∧ diBn(gj)∧ dgj+1 ∧ ... ∧ dgp. When p = 0, we put I i = 0. Define λ : Cq(U,ΩpXn) → C q+1(U,Ω (λβ)(i0, ..., iq+1) = (I i0 − I i1)β(i1, ..., iq+1) ∀β ∈ Cq(U,ΩpXn). Lemma 3.0 λ ◦ ϕ ≡ δ ◦ ϕ− ϕ ◦ δ mod. π∗n(Ω ) ∧ Ωp−1Xn . Proof. Define J : Cq(U,Ω Xn/Bn ) → Cq((U),ΩpXn) by (J(β))(i0, ..., iq+1) = (−1)(ϕi0 − ϕi1)(β(i1, ..., iq+1), where i0 < i1 < ... < iq+1. For arbitrary β belongs to C q(U,Ω Xn/Bn (δ ◦ ϕ(β))(i0, ..., iq+1) = (−1)jϕ(β)(i0, ..., îj, ..., iq+1) = ϕi1(β)(i1, ..., iq+1) (−1)jϕi0(β)(i0, ..., îj , ..., iq+1), while (ϕ ◦ δ(β))(i0, ..., iq+1) = ϕ( (−1)j(β)(i0, ..., îj , ..., iq+1)) (−1)jϕi0(β)(i0, ..., îj , ..., iq+1). So we have δ ◦ ϕ− ϕ ◦ δ = J . Fix (i0, ..., iq+1) and let ω = β(i1, ..., iq+1). We must show that (I Ii1)(ϕi1(ω)) = (−1)(ϕi0 −ϕi1)(w) mod π∗n(Ω2Bn)∧Ω . By linearity, we may suppose ϕi1(ω) = µdg1 ∧ ... ∧ dgp. Then ϕi0 = µd Xn/Bn (g1) ∧ ... ∧ di0Xn/Bn(gp) = µ(dg1 − di Xn/Bn (g1)) ∧ ... ∧ (dgp − di0Xn/Bn(gp)) = µdg1 ∧ ... ∧ dgp − µdg1 ∧ ...dgj−1 ∧ di0Bn(gj ∧ dgj+1 ∧ ... ∧ dgp +terms in π∗n(Ω ) ∧ Ωp−1Xn . Thus ϕi0 ≡ ϕi1(ω) − I i0 ◦ ϕi1(ω) mod. π∗n(Ω2Bn) ∧ Ω , and I i1 ◦ ϕi1 = 0. which means λ ◦ ϕ ≡ J mod. π∗n(Ω2Bn) ∧ Ω Now we are ready to calculate the formula for the obstructions. Let α̃ be an element of Cq(U,Ωp Xn/Bn ) such that its quotient image in Cq(U,Ωp Xn−1/Bn−1 is αn−1. Then δ ∗([αn−1])= [δ(α̃)] which is an element ofH q+1(X,Mn0/Mn+10 ⊗ X0/B0 ) ∼= mn0/mn+10 ⊗Hq+1(X,Ω X0/B0 Denote rXn the restriction to the complex space Xn. In order to give the obstructions an explicit calculation, we need to consider the following map ρ : Hq(X,Mn0/Mn+10 ⊗ Ω X0/B0 ) → Hq(Xn−1, π∗n−1(ΩBn|Bn−1) ∧ Ω Xn−1/Bn−1 which is defined by ρ[σ] = [ϕ−1 ◦ rXn−1 ◦ LBn ◦ ϕ(σ)]. Lemma 3.1 The map: ρ : Hq(X,Mn0/Mn+10 ⊗ Ω X0/B0 Hq(Xn−1, π n−1(ΩBn|Bn−1) ∧ Ω Xn−1/Bn−1 ) is well defined. Proof. At first, we need to show that if σ is closed, then ϕ−1 ◦ rXn−1 ◦ LBn ◦ ϕ(σ) is closed, which is equivalent to show that δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ) ≡ 0 mod. π∗n−1(Ω Bn|Bn−1 ) ∧ Ωp−1 Xn|Xn−1 Note that dXn ◦ δ = −δ ◦ dXn . Then δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ) = rXn−1 ◦ δ ◦ LBn ◦ ϕ(σ) = −rXn−1 ◦ (δ ◦ d·Xn/Bn + d Xn/Bn ◦ δ + LBn ◦ δ) ◦ ϕ(σ). Since LBn ◦ δ ◦ ϕ(σ) ≡ LBn ◦ (δ ◦ ϕ− λ ◦ ϕ)(σ) ≡ LBn ◦ ϕ ◦ δ(σ) = 0 rXn−1 ◦ (δ ◦ d·Xn/Bn + d Xn/Bn ◦ δ) ◦ ϕ(σ) = 0, we have δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ) ≡ 0 mod. π∗n−1(Ω2Bn|Bn−1) ∧ Ω Xn|Xn−1 Next we need to show that if σ is belongs to Cq(U,Mn0/Mn+10 ⊗Ω X0/B0 then ϕ−1 ◦ rXn−1 ◦ LBn ◦ ϕ ◦ δ(σ) is exact. In fact, as the calculation above: rXn−1 ◦LBn ◦ϕ ◦ δ(σ) ≡ −rXn−1 ◦ (δ ◦ d·Xn/Bn + d Xn/Bn ◦ δ+ δ ◦LBn) ◦ϕ(σ) = −δ ◦ rXn−1 ◦ LBn ◦ ϕ(σ). In general, the map ρ is not injective. However, as we mentioned at the end of the previous section. The “real” obstructions are o n,n−1([αn−1]), but not oqn([αn−1]). So we don’t need ρ to be injective. In the following, we will explain that ρ([δ(α̃)]) is exactly the “real” obstructions we need. In fact, Hq(Xn−1, π n−1(ΩBn|Bn−1)∧Ω Xn−1/Bn−1 ) = (ΩBn|Bn−1)⊗OBn−1H q(Xn−1,Ω Xn−1/Bn−1 Let m = dimCB, let ti, i = 0...m be the local coordinates of B. Then ρ([δ(α̃)]) can be written as: i=0 dti⊗ α̃i, where α̃i ∈ Hq(Xn−1,Ω Xn−1/Bn−1 For a certain direction ∂ , suppose α̃i 6= 0. Then by a simple calculation, it is not difficult to check that α̃i = constant[δ(α̃)/ti] in H q(Xn−1,Ω Xn−1/Bn−1 While [δ(α̃)/ti] is exactly the obstruction o n,n−1([αn−1]) in the direction of we mentioned in the previous section. Now consider the following exact sequence. The connecting homomor- phism of the associated long exact sequence gives the Kodaira-Spencer class of order n [4 1.3.2], 0 → π∗n−1(ΩBn|Bn−1) → ΩXn|Xn−1 → ΩXn−1/Bn−1 → 0. By wedge the above exact sequence with Ω Xn−1/Bn−1 , we get a new exact sequence. The connecting homomorphism of such exact sequence gives us a map from Hq(Xn−1,Ω Xn−1/Bn−1 ) to Hq+1(Xn−1, π ∗(ΩBn|Bn−1) ∧ Ω Xn−1/Bn−1 Denote such map by κnx, for such map is simply the inner product with the Kodaira-Spencer class of order n. By the definition and simply calculation it is not difficult to proof the following lemma. Lemma 3.2 Let θ be an element of Hq(Xn−1,Ω Xn−1/Bn−1 ), let θ̃ be an element of Cq(U,Ωp Xn/Bn ) such that its quotient image is θ. Then [κnxθ] is equal to [ϕ−1 ◦ rXn−1 ◦ δ ◦ ϕ(θ̃)]. Let us come back to the problem we discussed, we have rXn−1 ◦ LBn ◦ ϕ ◦ δ(α̃) ≡ rXn−1 ◦ LBn ◦ (δ ◦ ϕ− λ ◦ ϕ)(α̃) ≡ rXn−1 ◦ LBn ◦ δ ◦ ϕ(α̃) ≡ −rXn−1 ◦ (d·Xn/Bn ◦ δ + δ ◦ d Xn/Bn + δ ◦ LBn) ◦ ϕ(α̃) ≡ −rXn−1 ◦ (d·Xn/Bn ◦ δ + δ ◦ d Xn/Bn ) ◦ ϕ(α̃) −δ ◦ rXn−1 ◦ LBn) ◦ ϕ(α̃). Therefore [rXn−1 ◦ LBn ◦ ϕ ◦ δ(α̃)] = [−rXn−1 ◦ (d·Xn/Bn ◦ δ + δ ◦ d Xn/Bn ) ◦ ϕ(α̃)] = −[d·Xn−1/Bn−1 ◦ rXn−1δ ◦ ϕ(α̃) + rXn−1 ◦ δ ◦ d Xn/Bn ◦ ϕ(α̃)] = −[d·Xn−1/Bn−1 ◦ ϕ ◦ ϕ −1 ◦ rXn−1δ ◦ ϕ(α̃) +rXn−1 ◦ δ ◦ ϕ ◦ dXn/Bn(α̃)] = −[ϕ ◦ dXn−1/Bn−1 ◦ ϕ−1 ◦ rXn−1δ ◦ ϕ(α̃) +rXn−1 ◦ δ ◦ ϕ ◦ ( ˜dXn−1/Bn−1(αn−1))] = −[dXn−1/Bn−1 ◦ κnxαn−1 + κnx◦dXn−1/Bn−1(αn−1)]. From the discussion above, we get the main theorem of this paper. Theorem 3.3 Let π : X → B be a deformation of π−1(0) = X, where X is a compact complex manifold. Let πn : Xn → Bn be the nth order deformation of X. For arbitrary [α] belongs to Hq(X,Ωp), suppose we can extend [α] to order n − 1 in Hq(Xn−1,ΩpXn−1/Bn−1). Denote such element by [αn−1]. The obstruction of the extension of [α] to nth order is given by: on,n−1(αn−1) = dXn−1/Bn−1 ◦ κnx(αn−1) + κnx◦dXn−1/Bn−1(αn−1), where κn is the nth order Kodaira-Spencer class and dXn−1/Bn−1 is the relative differential operator of the n− 1th order deformation. From the theorem, we can get the following corollary immediately. Corollary 3.4 Let π : X → B be a deformation of π−1(0) = X, where X is a compact complex manifold. Suppose that up to order n, the d1 of the Frölicher spectral sequence vanishes. For arbitrary [α] belongs to Hq(X,Ωp), it can be extended to order n + 1 in Hq(Xn+1,Ω Xn+1/Bn+1 4 An Example In this section, we will use the formula in previous section to study the jumping of the Hodge numbers hp,q of small deformations of Iwasawa man- ifold. It was Kodaira who first calculated small deformations of Iwasawa manifold [2]. In the first part of this section, let us recall his result. 1 z2 z3 0 1 z1 0 0 1  ; zi ∈ C ∼= C3 1 ω2 ω3 0 1 ω1 0 0 1  ;ωi ∈ Z+ Z The multiplication is defined by 1 z2 z3 0 1 z1 0 0 1 1 ω2 ω3 0 1 ω1 0 0 1 1 z2 + ω2 z3 + ω2z1 + ω3 0 1 z1 + ω1 0 0 1 X = G/Γ is called Iwasawa manifold. We may consider X = C3/Γ. g ∈ Γ operates on C3 as follows: z′1 = z1 + ω1, z 2 = z2 + ω2, z 3 = z3 + ω1z2 + ω3 where g = (ω1, ω2, ω3) and z ′ = z · g. There exist holomorphic 1-froms ϕ1, ϕ2, ϕ3 which are linearly independent at every point on X and are given ϕ1 = dz1, ϕ2 = dz2, ϕ3 = dz3 − z1dz2, so that dϕ1 = dϕ2 = 0, dϕ3 = −ϕ1 ∧ ϕ2. On the other hand we have holomorphic vector fields θ1, θ2, θ3 on X given by , θ2 = , θ3 = It is easily seen that [θ1, θ2] = −[θ2, θ1] = θ3, [θ1, θ3] = [θ2, θ3] = 0. in view of Theorem 3 in [2], H1(X,O) is spanned by ϕ1, ϕ2. Since Θ is isomorphic to O3, H1(X, TX) is spanned by θiϕλ, i = 1, 2, 3, λ = 1, 2. The small deformation o f X is given by ψ(t) = tiλθiϕλt− (t11t22 − t21t12)θ3ϕ3t2. We summarize the numerical characters of deformations. The deformations are divided into the following three classes: i) t11 = t12 = t21 = t22 = 0, Xt is a parallelisable manifold. ii) t11t22−t21t12 = 0 and (t11, t12, t21, t22) 6= (0, 0, 0, 0),Xt is not parallelisable. iii) t11t22 − t21t12 6= 0, Xt is not parallelisable. h1,0 h0,1 h2,0 h1,1 h0,2 h3,0 h2,1 h1,2 h3,0 i) 3 2 3 6 2 1 6 6 1 ii) 2 2 2 5 2 1 5 5 1 iii) 2 2 1 5 2 1 4 4 1 Now let us explain the jumping phenomenon of the Hodge number by using the obstruction formula. From Corollary 4.3 in [6], it follows that the Dolbeault cohomology groups are: H0(X,Ω) = Span{[ϕ1], [ϕ2], [ϕ3]}, H1(X,O) = Span{[ϕ1], [ϕ2]}, H0(X,Ω2) = Span{[ϕ1 ∧ ϕ2], [ϕ2 ∧ ϕ3], [ϕ3 ∧ ϕ1]}, H1(X,Ω) = Span{[ϕi ∧ ϕλ]}, i = 1, 2, 3, λ = 1, 2, H2(X,O) = Span{[ϕ2 ∧ ϕ3], [ϕ3 ∧ ϕ1]}, H0(X,Ω3) = Span{[ϕ1 ∧ ϕ2 ∧ ϕ3]}, H1(X,Ω2) = Span{[ϕi ∧ ϕj ∧ ϕλ]}, i, j = 1, 2, 3, i < j, λ = 1, 2, H2(X,Ω1) = Span{[ϕi ∧ ϕ2 ∧ ϕ3], [ϕj ∧ ϕ1 ∧ ϕ3]}, i, j = 1, 2, 3, H3(X,O) = Span{[ϕ1 ∧ ϕ2 ∧ ϕ3]}, For example, let us first consider h2,0, in the ii) class of deformation. The Kodaira-Spencer class of the this deformation is ψ1(t) = λ=1 tiλθiϕλ, with t11t22−t21t12 = 0. It is easy to check that o1(ϕ1∧ϕ2) = ∂(int(ψ1(t))(ϕ1∧ ϕ2) − int(ψ1(t))(∂(ϕ1 ∧ ϕ2)) = 0, o1(t11ϕ2 ∧ ϕ3 − t21ϕ1 ∧ ϕ3) = ∂((t11t22 − t21t12)ϕ3 ∧ ϕ2) = 0, and o1(ϕ2 ∧ ϕ3) = −t21ϕ1 ∧ ϕ2 ∧ ϕ1 − t22ϕ1 ∧ ϕ2 ∧ ϕ2, o1(ϕ1 ∧ ϕ3) = −t11ϕ1 ∧ ϕ2 ∧ ϕ1 − t21ϕ1 ∧ ϕ2 ∧ ϕ2. Therefore, we have shown that for an element of the subspace Span{[ϕ1∧ϕ2], [t11ϕ2∧ϕ3− t21ϕ1∧ϕ3]}, the first order obstruction is trivial, while, since (t11, t12, t21, t22) 6= (0, 0, 0, 0), at least one of the obstruction o1(ϕ2 ∧ ϕ3), o1(ϕ1 ∧ ϕ3) is non trivial which partly explain why the Hodge number h2,0 jumps from 3 to 2. For another example, let us consider h1,2, in the ii) class of deformation. It is easy to check that for an element of the subspace (the dimension of such a subspace is 5) Span{[ϕi∧ϕλ∧ϕ3], [t12ϕ3∧ϕ2∧ϕ3− t11ϕ3∧ϕ1∧ϕ3]}, i = 1, 2, λ = 1, 2, the first order obstruction is trivial, while at least one of the obstruction o1(ϕ3 ∧ ϕ2 ∧ ϕ3), o1(ϕ3 ∧ ϕ1 ∧ ϕ3) is non trivial. Remark 1 It is easy to see that, in the ii) or iii) class of deformation, the first order obstruction for any element in H1(X,Ω) is trivial. The reason of Hodge number h1,1’s jumping from 6 to 5 comes from the existence of the second class obstructed elements o1(ϕ3). After simple calculation, it is not difficult to get the structure equation of Xt, t 6= 0. dϕ1 = 0, dϕ2 = 0, dϕ3 = −ϕ1 ∧ ϕ2 + to1(ϕ3), i = 1, 2, λ = 1, 2, which can be considered an example of proposition 2.5. Remark 2 From the example we discussed above, it is not difficult to find out the following fact. Let X be an non-Kähler nilpotent complex par- allelisable manifold whose dimension is more than 2, and φ : X → B be the versal deformation family of X . Then the Hodge number h1,0 will jump in a neighborhood of 0 ∈ B. In fact, let ϕi, i = 1...n, n = dimC(X) be the linearly independent holomorphic 1-forms of X . By the theorem 3 of [2], H1(X,O) is spanned by a subset of {ϕi}, i = 1..n. So we have ∂ : H1(X,O) → H1(X,Ω) is trivial, which means one term of the first order obstruction of the holomor- phic 1-forms vanishes. Let θi, i = 1...n be the dual of ϕi, which are linearly independent holomorphic vector fields. Since X is non-Kähler, which means X is not a torus, there exists ϕi such that ∂ϕi 6= 0. Since X is nilpotent, there exist ϕj such that ∂ϕj = 0. Assume that ∂ϕi = Aϕk ∧ ϕl + ... with A 6= 0. Consider θkϕj in H 1(X, TX). It is easy to check that o1(∂ϕi, θkϕj) 6= 0. References [1] S. Iitaka, Plurigenera and classification of algebraic varieties, Sugaku 24 (1972), 14-27. [2] Nakamura, I(1975). Complex parallelisable manifolds and their small deformations, J.Differential Geom. 10, 85-112. [3] C. Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge University Press 2002. [4] C. Voisin, Symétrie miroir, Société Mathématique de France, Paris, 1996. [5] Bell S. and Narasimhan R., Proper holomorphic mappings of complex spaces, Encyclopedia of Mathematical Sciences, Several Complex Vari- ables VI, Springer Verlag, pp. 1-38, 1991. [6] Cordero, L. A., Fernández, Gray, A. and Ugate, L.(1999). Frölicher Spectral Sequence of Compact Nilmanifolds with Nilpotent Complex Structure. New developments in differential geometry, Budapest 1996, 77-102, Kluwer Acad. Publ., Dordrecht. Introduction Grauert's Direct Image Theorems and Deformation theory The Formula for the Obstructions An Example

0704.1978

Strange dibaryon resonance in the anti-KNN--piYN system

Strange dibaryon resonance in the K̄NN-πY N system Y. Ikeda and T. Sato∗ Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan (Dated: November 4, 2018) Abstract Three-body resonances in the K̄NN system have been studied within a framework of the K̄NN − πY N coupled-channel Faddeev equation. By solving the three-body equation, the en- ergy dependence of the resonant K̄N amplitude is fully taken into account. The S-matrix pole has been investigated from the eigenvalue of the kernel with the analytic continuation of the scattering amplitude on the unphysical Riemann sheet. The K̄N interaction is constructed from the leading order term of the chiral Lagrangian using relativistic kinematics. The Λ(1405) resonance is dy- namically generated in this model, where the K̄N interaction parameters are fitted to the data of scattering length. As a result we find a three-body resonance of the strange dibaryon system with binding energy, B ∼ 79 MeV, and width, Γ ∼ 74 MeV. The energy of the three-body resonance is found to be sensitive to the model of the I = 0 K̄N interaction. PACS numbers: 11.30.Rd, 11.80.Jy, 13.75.Jz ∗Electronic address: [email protected], [email protected] http://arxiv.org/abs/0704.1978v2 mailto:[email protected], [email protected] I. INTRODUCTION The analysis of the kaonic atoms [1] revealed an attractive K̄-nucleus interaction. Al- though the strength of the attraction depends on the parametrization of the density de- pendence of the optical potential [1] and the theoretical study of the K̄ optical potential suggests a rather shallow potential [2], there has been a great interest in the possibilities of K̄-nucleus bound states in recent years. Akaishi and Yamazaki [3, 4] studied the kaon bound states in light nuclei and found deeply bound kaonic states, for example, B ∼ 100 MeV for 3 H. In their study, the kaonic nuclear states were investigated by using the K̄ optical potential, which is constructed by folding the g matrix with a trial nuclear density. The potential model of K̄N − πΣ interaction is determined to reproduce the Λ(1405) and the scattering length. The kaonic nuclear states are further studied by using a method of antisymmetrised molecular dynamics [5] using the K̄N g matrix. Among the simplest K̄ nucleus state, theK−pp state, which has strangeness S = −1, total angular momentum and parity Jπ = 0−, and isospin I = 1/2 dibaryon state, is expected to have largest component of the I = 0 K̄N . An experimental signal of theK−pp bound state is reported by the FINUDA Collaboration from the analysis of the invariant mass distribution of Λ − p in the K− absorption reaction on nuclei [6]. The reported central value of the binding energy, B, and the width, Γ, are (B,Γ) = (115, 67) MeV, which is below the πΣN threshold energy. This data may be compared with the predicted values (B,Γ) = (48, 61) MeV in Ref. [4]. However it was pointed out that the data can be understood by the two- nucleon absorption of K− in nuclei together with the final state interaction of the outgoing baryons [7]. In the attractive interaction of kaon in nuclei, the resonance Λ(1405) in the s-wave and I = 0 channel K̄N scattering state plays an essential role. The energy of the Λ(1405) is below the K̄N threshold and strongly couples with the πΣ state. Although the kaonic nuclear states have been studied so far by using the K̄N g matrix or optical potential, it might be very important to examine the full dynamical calculation of K̄N−πΣ system by taking into account the energy dependence of the resonance t matrix and the coupling with the K̄N−πΣ channel explicitly. Such a theoretical study may be possible in the simplest kaonic nuclei with baryon number B = 2 system. In this work, we study the strange dibaryon system by taking into account the three-body dynamics using the K̄NN−πΣN−πΛN (K̄NN−πY N) coupled-channel Faddeev equation with relativistic and non-relativistic kinematics. Methods to investigate resonances in the three-body system have been developed in the studies of the three-neutron [8, 9], πNN dibaryon [10, 11] and ΣNN hypernuclei [10, 12, 13]. In this work, we employ a method started by Glöckle [8] and Möller [9] and developed by Matsuyama and Yazaki, Afnan, Pearce and Gibson [10, 12, 13] to find a pole of the S matrix in the unphysical energy plane from the eigenvalue of the kernel of the Faddeev equation. To analytically continue the scattering amplitude into the unphysical sheet, the path of the momentum integral must be carefully deformed in the complex plane to avoid possible singularities. The most important interaction for the study of the strange dibaryon system is for the I = 0 K̄N states. The internal structure of the Λ(1405) has been a long standing is- sue. The chiral Lagrangian [14, 15, 16] approach can describe well the low energy K̄N reaction with the meson-baryon dynamics. A genuine q3 picture of the Λ(1405) coupled with meson-baryon [17] may not yet be excluded. Though previous studies of the K̄NN system used phenomenological models of the K̄N potentials, we use s-wave meson-baryon coupled-channel potentials guided by the lowest order chiral Lagrangian. With this model, the strength of the potentials and the relative strength of the potentials among various meson-baryon channels are not parameters but are determined from the SU(3) structure of the chiral Lagrangian. In this model, the Λ(1405) is an ’unstable bound state’, whose pole on the unphysical sheet will become the bound state of K̄N when the coupling between the K̄N and the πY is turned off. We examine a relativistic model as well as a nonrelativistic model to account for the relativistic energy of pion in the πY N state. We briefly explain our K̄NN − πY N coupled-channel equations and the procedure to search for the three-body resonance in section 2. The model of the two-body interactions used in this work is explained in section 3. We then report our results on the K̄NN dibaryon resonance in section 4. This work is the extension of the early version of our analysis reported in Ref. [18]. Recently Shevchenko et al. [19] performed a similar study of the K̄NN system using Faddeev equation starting form the phenomenological K̄N interaction within a nonrelativistic framework. The comparison of our results with theirs will be discussed in section 4. II. COUPLED CHANNEL FADDEEV EQUATION AND RESONANCE POLE We start from the Alt-Grassberger-Sandhas(AGS) equation [20] for the three-body scat- tering problem. The operators Ui,j of the three-body scattering satisfy the AGS equation Ui,j = (1− δi,j)G−10 + n 6=i tnG0Un,j . (1) Here we label the pair of particles j, k by the spectator particle i = 1, 2, 3. The two-body t matrix ti of particles j, k with the spectator particle i is given by the solution of the Lippmann-Schwinger equation: ti = vi + viG0ti. (2) Here G0 = 1/(W −H0 + iǫ) is the free Green’s function of the three particles, and W is the total energy of the three-body system. When the two-body interactions vi are given in separable form with the vertex form factor |gi > and the coupling constant γi as vi = |gi > γi < gi|, (3) the AGS-equation of Eq. (1) is written in the form Xi,j(~pi, ~pj,W ) = (1− δi,j)Zi,j(~pi, ~pj,W ) + n 6=i d~pnZi,n(~pi, ~pn,W )τn(W )Xn,j(~pn, ~pj,W ). The amplitude Xi,j is defined by the matrix element of Ui,j between state vectors G0|~pi, gi > Xi,j(~pi, ~pj,W ) = < ~pi, gi|G0Ui,jG0|~pj, gj > . (5) The state vector |~pi, gi > represents a plane wave state of the spectator i and the state vector |gi > of the interacting pair. The driving term Zi,j of Eq. (4) shown in Fig. 1(a) is given by the particle exchange mechanism defined as Zi,j(~pi, ~pj,W ) = < ~pi, gi|G0|~pj, gj > (6) g∗(~qi)g(~qj) W −Ei(~pi)−Ej(~pj)−Ek(~pk) . (7) FIG. 1: Graphical representation of (a) one particle exchange interaction Zi,j(~pi, ~pj ,W ) and (b) two-body t-matrix τi(W ). The relative momentum of the interacting particles is given by ~qi for spectator particle i. Here the momentum of the exchanged particle k( 6= i, j) is given as ~pk = −~pi − ~pj and g(~qi) is the vertex form factor of the two-body interaction g(~qi) =< gi|~qi >. The energy Ei(~pi) is given by Ei(~pi) = mi + ~p i /2mi for the nonrelativistic model and Ei(~pi) = m2i + ~p i for the relativistic model. The relative momentum is given by ~qi = (mk~pj −mj~pk)/(mj +mk) for the nonrelativistic model, while we define qi = |~qi| for the relativistic model as W 2i +m j −m2k )2 −m2j , (8) (Ej(~pj) + Ek(~pk))2 − ~p2i . (9) The two-body t matrix can be solved for the separable interaction as ti = |gi > τi(W ) < gi|. (10) Here the ’isobar’ propagator τi, illustrated in Fig. 1(b), is given as τi(W ) = [1/γi − |gi(~qi)|2 W −Ei(~pi)− Ejk(~pi, ~qi) ]−1. (11) The two-body t matrix depends on the energy Ei(~pi) of the spectator particle. Here Ejk is the energy of the interacting pair given as Ejk(~pi, ~qi) = mj +mk + ~p i /(mj +mk) + ~q i /µi for the non-relativistic model, while Ejk(~pi, ~qi) = (Ej(~qi) + Ek(~qi))2 + ~p i for the relativistic model. The reduced mass is defined as µi = mjmk/(mj +mk). Following the standard method of angular momentum expansion [21], the AGS equation reduces to following coupled integral equations by keeping only s-wave states: Xi,j(pi, pj,W ) = Zi,j(pi, pj,W ) + × Ki,n(pi, pn,W )Xn,j(pn, pj ,W ). (12) Here we used a simplified notation for the kernel K = Zτ , which can be written as Ki,n(pi, pn,W ) = 2π d(p̂i · p̂n) g∗(qi)g(qn) W −Ei(pi)− Ej(pn)− Ek(~pi + ~pn) τn(W ). (13) The formulas given above are valid for the spinless and distinguishable particles without channel coupling among the Fock-space vectors. In our K̄NN resonance problem, we have included the following K̄NN and πY N states: |a > = |N1, N2, K̄3 >, (14) |b > = |N1, Y2, π3 >, (15) |c > = |Y1, N2, π3 >, (16) with Yi is Σi or Λi. After anti-symmetrizing the amplitude for identical particles of nucle- ons [11], we obtain the following forms of the coupled AGS equations, XYK ,YK XYπ,YK Xd,YK XN∗,YK ZYK ,YK Zd,YK −ZYK ,YKτYK ,YK −ZYK ,YKτYK ,Yπ 2ZYK ,dτd,d 0 0 0 0 −ZYπ ,N∗τN∗,N∗ Zd,YKτYK ,YK Zd,YKτYK ,Yπ 0 0 −ZN∗,YπτYπ ,YK −ZN∗,YπτYπ,Yπ 0 0 XYK ,YK XYπ,YK Xd,YK XN∗,YK . (17) Here we have suppressed the spin-isospin quantum numbers, the spectator momentum pj and the total energy of the three-body system W in Z, X and τ for simplicity. The concise notation of YK , Yπ, d and N ∗ represents the ’isobars’ and their decay channels. The decay channels of isobars YK , Yπ, d and N ∗ are K̄N(I = 0, 1), πΣ(I = 0, 1) and πΛ(I = 1), NN(I = 1) and πN(I = 1/2, 3/2), respectively. Here I is isobar isospin. Those indices uniquely specify the three-body states of X and Z except N∗ showing ΣN∗ and ΛN∗. Therefore we have a nine-channel coupled equation of Eq. (17) for spin singlet, s-wave three- body system. The explicit form of Eq. (7) when we include spin-isospin is summarized in the Appendix. The dominant Fock space component is expected to be |K̄NN >, and therefore the most important amplitudes are XYK ,YK and Xd,YK . They couple to each other through the kaon exchange ZYK ,YK and nucleon exchange ZYK ,d mechanisms. Notice, however, the πY N component is also implicitly included in τYK ,YK when we solve the two-body K̄N − πY coupled-channel equations. The πY N components, XY π,YK and XN∗,YK , couple with the K̄NN components through the pion exchange mechanism ZN∗,Yπ and the πN and πY ’isobars’ τN∗,N∗ , τYπ,Y . The pion exchange mechanism may play an important role in the width of the resonance. In this work, we have not included weak Y N interaction. It was found in Ref. [19] that the Y N interaction plays rather minor role in this strange dibaryon system. To find the resonance energy of the three-body system using the AGS equation of Eq. (17), we follow the method used in Refs. [8, 9, 10, 12, 13]. The AGS equation of Eq. (13) is a Fredholm-type integral equation with the kernel K = Zτ . Using the eigenvalue ηa(W ) and the eigenfunction |φa(W ) > of the kernel for given energy W , Zτ |φa(W ) > = ηa(W )|φa(W ) >, (18) the scattering amplitude X can be written as |φa(W ) >< φa(W )|Z 1− ηa(W ) . (19) At the energy W = Wp where ηa(Wp) = 1, the amplitude has a pole, and therefore Wp gives the bound state or resonance energy. Since a resonance pole appears on the unphysical energy Riemann sheet, we need analytic continuation of the scattering amplitude. We use here the nonrelativistic model to explain a method of analytic continuation, which is based on Refs. [9, 10]. At first we examine the singularities of the kernel of Eq. (13). Above the threshold energy of the three-body break up W > mi+mj+mk, Z(pi, pn,W ) has logarithmic singularities. The branch points appear FIG. 2: The singularities of the one particle exchange interaction Z(pi, pn,W ) in the complex pn plane at W = E + iǫ and the real pi. at pn = ±pZ1,2 , where pZ1 = − 2µjWth − p2i , (20) pZ2 = + 2µjWth − p2i , (21) mi +mk mj(mi +mk) mi +mj +mk Wth = W −mi −mj −mk. For given pi > 0, the cuts run from pZ1 to pZ2 above the positive real axis of complex pn plane and from −pZ1 to −pZ2 below the negative real axis as shown in Fig. 2, while the integration of momentum pn in Eq. (13) is along the real positive axis. Let us consider the case when W has a negative imaginary part. For given pi > 0, the cut from pZ1 to pZ2 moves into the fourth quadrant across the integration contour of pn. Assuming the integrand of Eq. (12) is an analytic function around real positive pn, one can perform an analytic continuation of the amplitudes by deforming the integration contour along the logarithmic singularity as shown in Fig. 3 and then we obtain amplitudes on the unphysical Riemann sheet. FIG. 3: The integration contour C and the singularity of Z at W = E − iΓ/2 and real value of pi. In principle it might be possible to solve the AGS equation keeping the momentum vari- ables real and taking into account the discontinuity across the cut. The moving logarithmic singularities depending on pi make it difficult to solve the integral equation. To overcome this problem we deform the integration contour of pi, pn, into the fourth quadrant of the complex momentum plane so that we take into account the contribution of the cuts. As an example of our K̄NN − πY N problem, we choose the integration contour of pn as shown in solid line in Fig. 4. Here we take the energy W = 10 − i35 +mπ +mΣ +mN MeV, which is below the mass of K̄NN and above the πY N . The shaded region in Fig. 4 shows the cuts of ’Z’ for the pion exchange mechanism. The cuts become ’forbidden regions’ because the position of the cuts depends on pi, which runs the same integration contour as pn. In our numerical calculation, we studied all the ’forbidden regions’ for π,N and K exchange mechanism and determined the integration contour. With the integration contour C in Fig. 4, we choose the physical sheet of K̄NN . The singularities of the isobar propagator τ(W ) arises from the three-body Green’s func- tion in the integrand of τ . The poles are at qn = ± 2µjWth − µjηj p i . Since qs = (pZ1+pZ2)/2, we can analytically continue it into the same unphysical sheet as the case in Z as long as we keep the same deformed contour as the one used in Z. Another singularity we have to worry about is the singularity due to the two-body resonance. Since our K̄N − πΣ system has the two-body resonance Λ(1405), the cut starts from the two-body resonance energy in the complex energy plane. To examine this, we write the approximate energy dependence FIG. 4: The logarithmic singularities of the π exchange mechanism Z(pi, pn, Z) at Wth = 10− i35 MeV in the complex pn plane. C is integration contour of pn and pi. of the τ as τi(W ) ∼ W − p − EΛ∗ −mN . (22) Here pN and mN are the momentum and mass of the spectator nucleon. The reduced mass of the spectator nucleon with the isobar pair K̄N or πΣ is denoted as ηN and EΛ∗ is the pole energy of Λ(1405). At W = + EΛ∗ +mN with pN on the contour C in Figs. 5 (b), the two-body t-matrix has a singularity, which is plotted as a solid line in Fig. 5(a). We illustrate the typical trajectories of the three-body resonance pole W = Wp as curves A and B in Fig. 5. If the pole trajectories A and B intercept the two-body NΛ(1405) cut, then the analytic continuation to the NΛ(1405) unphysical energy sheet must be examined. The same situation from the pN plane is shown in Fig. 5 (b). The momentum p ∗ corresponding to the energy Wp of the three-body resonance is determined by p∗ = ± 2ηN(Wp − EΛ∗ −mN). (23) If p∗ intercepts the contour C, we have to take care of the analytic continuation of the NΛ(1405) energy sheet. As will be seen in section 4, the trajectories of the three-body resonance in our calculation follow line A of Fig. 5(a) and do not intercept the singularity of the two-body resonance. FIG. 5: The singularities due to the three-body resonance and the Λ(1405) in (a) the complex energy plane and (b) the momentum plane. III. MODEL OF THE TWO-BODY INTERACTIONS We take into account the K̄N interactions in Jπ = 1/2−, I = 0 and I = 1 states, the πN interactions in Jπ = 1/2−, I = 1/2 and 3/2 states and the NN interaction in I = 0,1 S0 state. Our s-wave meson-baryon interaction is guided by the leading order effective chiral Lagrangian for the octet baryon ψB and the pseudoscalar meson φ fields given as Lint = 8F 2π tr(ψ̄Bγ µ[[φ, ∂µφ], ψB]). (24) The meson-baryon potential derived from the chiral Lagrangian can be written as < ~p′, β|VBM |~p, α > = −Cβ,α (2π)38F 2π mβ +mα 4Eβ(~p′)Eα(~p) × gβ(~p′)gα(~p). (25) Here ~p and ~p′ are the momentum of the meson in the initial state α and the final state β. The strength of the potential at zero momentum is not an arbitrary constant but is determined by the pion decay constant Fπ. The relative strength between the meson-baryon states is controlled by the constants Cβ,α which are basically determined by the SU(3) flavor structure of the chiral Lagrangian. The parameter of our model is the cutoff Λ of the phenomenologically introduced vertex function gα(~p) = Λ α/(~p 2 + Λ2α) The most important interaction for the study of the K̄NN system is the I = 0 K̄N interaction. We describe the K̄N interaction by the coupled-channel model of the K̄N and the πΣ states. The constants Cβ,α for this channel are given as CK̄N−K̄N = 6, CK̄N−πΣ = 6 and CπΣ−πΣ = 8. The cutoff Λ is determined by fitting the scattering length a −1.70 + i0.68 fm of Ref. [22]. The values of Λ are around 1 GeV and are given as model (a) in tables I and II for the nonrelativistic and the relativistic models. In general, the form factors of the relativistic models are hard compared with those of the non-relativistic models because of the weak relativistic kinetic energy. We found a resonance pole atW = 1420−i30 MeV for the non-relativistic and the relativistic models. The relativistic kinematics might be important in describing πY channel because of the small pion mass. We choose this model (a) as a standard parameter of the K̄N interaction. The K̄N scattering lengths are not very well constrained from the data. The ranges of the K̄N scattering lengths are studied within the chiral unitary model in Ref. [23]. In this work, we simply examined models with the scattering length aI=0 = (−1.70±0.10)+ i(0.68±0.10) fm in order to examine the sensitivity of the energy of the three-body resonance on the input model of the two-body interaction. The cutoff Λ’s for those models are given as models (b)- (e) of Tables I and II. The values of the resonance energy are about 1415 ∼ 1425 MeV, and the width 50 ∼ 70 MeV, which are close to the values of the chiral model in Ref. [16]. One can notice that there is a correlation between the real (imaginary) part of the pole energy of the Λ(1405) and the imaginary (real) part of the scattering length. Those resonance energies are slightly larger than the pole energy reported in Ref. [24]. Therefore as a last model, model (f) reproduces the deeper resonance energy 1406− i25 MeV of Ref. [24]. The scattering length of this model is −1.72 + i0.44 fm, which is, however, somewhat different from the value −1.54 + i0.74 fm in Ref. [24]. The I = 1 K̄N interaction is described by the K̄N − πΣ − πΛ coupled-channel model. The coupling constants Cβ,α are CK̄N−K̄N = 2, CK̄N−πΣ = −2, CK̄N−πΛ = − 6, CπΣ−πΣ = 4 and CπΣ−πΛ = CπΛ−πΛ = 0. The cutoff Λ’s are determined to fit the imaginary part of the scattering length of Ref. [22], which are given as model (A) in Tables III and IV for the non- relativistic and the relativistic models. The real part of the scattering length of those models is larger than aI=1 = 0.37+ i0.60 fm of Ref. [22]. The K−p scattering length predicted from model (aA) , which is model (a) for I = 0 and model (A) for I = 1 interactions, is between the central values of the two kaonic hydrogen data [25, 26, 27]. To study the sensitivity of the models of I = 1 K̄N interaction to the resonance energy ofK−pp system, we constructed model (B) given in Tables III and IV. A similar model of K̄N interaction is developed to study K−d scattering[28]. The range of the vertex form factor found in in Ref. [28], which is monopole form factor with 880MeV cutoff mass, is comparable to ours. The total cross sections of K−p reactions predicted from our models (aA), (aB) and (fA) are shown in Fig. 6 together with the data [29, 30, 31, 32, 33]. The models (aA) and (aB) describe well the K−p → K−p(Fig. 6a), K−p → π+Σ−(Fig. 6b) and K−p → π−Σ+(Fig. 6c) reactions, where both I = 0 and I = 1 interactions contribute to the cross section. The models of I = 0 (I = 1) can be tested from K−p → π0Σ0(Fig. 6d) (K−p → π0Λ(Fig. 6e)) reactions, where models ’a’ and ’A/B’ describe the cross sections well. The model (fA) tends to give smaller cross sections. It is noticed, however, as we will see, that the resonance energy of the K−pp system is more sensitive to the I = 0 K̄N interaction and less sensitive to I = 1 interactions, while both I = 0 and I = 1 interactions are equally important to describe K−p cross sections and kaonic hydrogen data. K̄N(MeV) πΣ(MeV) Scattering Length(fm) Resonance energy(MeV) (a) 1095 1450 −1.70 + i0.68 1419.8 − i29.4 (b) 1105 1550 −1.60 + i0.68 1422.2 − i33.7 (c) 1085 1350 −1.80 + i0.68 1418.5 − i25.0 (d) 1120 1340 −1.70 + i0.59 1414.6 − i29.4 (e) 1070 1540 −1.70 + i0.78 1424.3 − i28.3 (f) 1160 1100 −1.72 + i0.44 1405.8 − i25.2 TABLE I: The cutoff parameters, scattering length and the resonance pole of the relativistic models of I=0 K̄N − πΣ interaction. The form of the s-wave πN interactions is taken as Eq. (25). The constant Cα,β is 4 for I = 1/2 and −2 for I = 3/2 states. The parameters of the potentials are determined by fitting the scattering length and the low energy phase shifts. For I = 1/2 state, the strength of the potential is modified as λCβ,α by introducing a phenomenological factor λ to describe the data of the scattering length (0.1788±0.0050)m−1π [34] and the phase shifts [35]. The fitted parameters λ and Λ are shown in Table V together with the scattering length K̄N(MeV) πΣ(MeV) Scattering Length(fm) Resonance energy(MeV) (a) 946 988 −1.70 + i0.68 1420.1 − i30.1 (b) 954 1035 −1.60 + i0.68 1422.4 − i34.7 (c) 940 944 −1.80 + i0.68 1418.7 − i26.0 (d) 968 933 −1.70 + i0.58 1414.3 − i30.5 (e) 927 1031 −1.70 + i0.78 1424.7 − i29.0 (f) 1000 800 −1.72 + i0.43 1404.8 − i25.5 TABLE II: The cutoff parameters, scattering length and the resonance pole of the nonrelativistic models of I=0 K̄N − πΣ interaction. K̄N(MeV) πΣ(MeV) πΛ(MeV) Scattering Length(fm) (A) 1100 850 1250 0.68 + i0.60 (B) 950 800 1250 0.65 + i0.46 TABLE III: The cutoff parameters, scattering length of the relativistic models of I=1 K̄N − πY interaction. calculated using the models. The models describe well the S11 phase shifts up to 1.2 GeV as shown in Fig. 7. For the I = 3/2 πN scattering, the πN potential is constructed so as to reproduce the scattering length (−0.0927 ± 0.0093)m−1π [34] and the S31 partial wave phase shifts data. Here we introduced a modified dipole form factor as g(~p) = (~p2 + Λ2)2 × (1 + a~p2). (26) K̄N(MeV) πΣ(MeV) πΛ(MeV) Scattering Length(fm) (A) 920 960 640 0.72 + i0.59 (B) 800 940 660 0.68 + i0.45 TABLE IV: The cutoff parameters, scattering length of the nonrelativistic models of I=1 K̄N−πY interaction. FIG. 6: The total cross section of (a) K−p → K−p, (b) K−p → π+Σ− (c) K−p → π−Σ+ (d) K−p → π0Σ0 and (e) K−p → π0Λ reactions in the relativistic model. The solid (dashed, dotted) curve shows the result using model (aA) ((aB), (fA)). Data are taken from Ref. [29, 30, 31, 32, 33]. λ Λ(MeV) scattering length Relativistic 0.90 800 0.175m−1π Nonrelativistic 0.85 800 0.177m−1π TABLE V: Parameters and scattering length for the relativistic and nonrelativistic model of I=1/2 πN interaction. The parameters of the model are Λ and a for the form factor and the strength parameter λ. The obtained parameters are summarized in Table VI. The relativistic model can describe well the phase shifts up to 1.2 GeV as shown in Fig. 7; however, the non-relativistic model starts to deviate from the data at around 1.1 GeV. We used a Yamaguchi-type separable interaction for the nucleon-nucleon potential. To take into account the long range attractive interaction and the short range repulsion of the two-nucleon interaction, we used a two-term separable potential, < ~p′|VBB|~p >= CRgR(~p′)gR(~p)− CAgA(~p′)gA(~p). (27) λ Λ(MeV ) a(fm)2 scattering length Relativistic 2.7 618 0.50 −0.095m−1π Nonrelativistic 3.0 628 0.30 −0.101m−1π TABLE VI: Parameters and scattering length for the relativistic and nonrelativistic model of I=3/2 πN interaction. FIG. 7: The phase shift of the πN scattering for S11 (a), and S31 (b) partial waves. The solid curve shows the relativistic model and the dashed curve shows the nonrelativistic model. Data are taken from Ref. [35]. Here CR (CA) is the coupling strength of the repulsive (attractive) potential. gR(~p) (gA(~p)) is the form factor, whose form is given as gR(~p) = Λ R/(~p 2 + Λ2R) (gA(~p) = Λ A/(~p 2 + Λ2A)), where Λ is a cutoff of the nucleon-nucleon potential. The adjustable parameters in our nucleon-nucleon potential are determined by fits to the data of the 1S0 phase shifts [36]. The best-fit parameters are summarized in Table VI. The low energy phase shifts of the 1S0 state is shown in Fig. 8. ΛR(MeV ) ΛA(MeV ) CR(MeV fm 3) CA(MeV fm Relativistic 1144 333 5.33 5.61 Nonrelativistic 1215 352 5.05 5.84 TABLE VII: Our parameters of the relativistic and nonrelativistic model for NN scattering. FIG. 8: Phase shifts of the NN scattering for 1S0 state. Solid curve shows the relativistic model and dashed curve shows the nonrelativistic model. The phase shifts calculated from the model of Ref. [36] are shown in triangles. IV. RESULTS AND DISCUSSION The dibaryon resonance with Jπ = 0−, S = −1, I = 1/2 is studied using a formalism of the Faddeev equation as explained in section 2. We assume all the angular momentum to be in an s-wave state and the spin singlet state SBB = 0 for the two baryon states. We have included the dominant K̄NN , πΣN and πΛN Fock-space components, whose isospin wave functions are [K̄⊗ [NN ]I=1]I=1/2, [π⊗ [ΣN ]I=1/2,3/2]I=1/2 and [π⊗ [ΛN ]I=1/2]I=1/2. An approximation within this model is that the weak Y N interaction is not included. Let us start to examine the three-body resonance energy by taking into account only the K̄N interactions v I=0,1 K̄N−K̄N neglecting the πY N Fock space. In this case, the bound state pole is expected to lie on the physical Riemann sheet below mK + 2mN if the K̄N attraction is strong enough. Therefore it is not necessary to use the analytic continuation of the amplitude with the deformed contour discussed in section 2, so we simply use the integral over the momentum pi in the real axis. The results are shown in Fig. 9 marked by a and a for the the ’relativistic’ and ’non-relativistic’ models. Here we use the ’standard’ parameters (aA) of the K̄N interaction with nonrelativistic and relativistic kinematics. The binding energies are about 18 MeV. The K̄N interaction included in τ and Z is strong enough to bind the K̄NN system, where the I = 0 K̄N interaction plays a dominant role. We then take into account the NN interaction. Then the binding energy increased furthermore to 25.1 MeV (22.8 MeV) shown as b (b′) for ’relativistic’ (’nonrelativistic’) model. Notice that FIG. 9: The pole trajectories of the K̄NN − πY N scattering amplitude for the Jπ = 0− and I = 1/2 state. The solid curve and the filled circles (dashed curve and filled triangles) show the results of the relativistic (nonrelativistic) model (aA). Here WKNN = mK + 2mN . if we neglect the repulsive component of the NN interaction, we obtain a much more deeply bound state. In the next step, we gradually include the πY N interactions, while the pion-exchange Z diagram is not yet included. To do this, we multiply by factor x the coupling constants Cα,β of the K̄N − πY and πY − πY interactions as xCα,β . When the parameter is zero, x = 0, the πY is disconnected from K̄N and when it takes the value 1, x = 1, we recover the full model. By varying the parameter x from 0 to 1, we can follow the trajectory of the resonance pole from the bound state pole. Now the K̄NN bound state decays into the πY N channel and the bound state pole moves into the unphysical sheet. Since the K̄NN bound state was found above the πΣN threshold, the resonance pole may be on the πY N unphysical and K̄NN physical Riemann sheet, which we have discussed in section 2. The results of the pole trajectories are shown by the solid and dashed curves in Fig. 9 corresponding to the relativistic and the nonrelativistic models. Increasing the coupling to the πY N channel causes the width as well as the binding energy of the resonance increases. For larger binding energy Re(Wpole − WKNN) < −60 MeV the width starts to decrease because of the decreasing phase space for the decay into the πΣN state. The pole position is at −82− i29 MeV (−91−28i) for the relativistic (nonrelativistic) model shown as c(c′). It is noticed that the numerical method to follow the pole trajectories helps us to find whether FIG. 10: The pole trajectories of the three-body resonance (solid curve) and Λ(1405) (dashed curve) using the nonrelativistic model of (aA). we encounter singularities or not. As an example, the pole of the three-body resonance is shown by the solid line in Fig. 10 for 0 < x < 1. The pole of the Λ(1405) is also shown by the dashed curve. The trajectory of the K̄NN resonance is similar to the the case A in Fig. 5, and the integration contour does not intercept the singularity arising from the two-body resonance Λ(1405). Finally we include the π exchange mechanism in Z and πN two-body scattering terms in τ , which adds another mechanism for the decay of the K̄NN into πY N and is important for the width of the three-body resonance. The final results of the K̄NN−πY N resonance poles are denoted by d and d′ in Fig. 9. This mechanism increases the width of the three-body resonance by about 14 MeV, while the effect on the real part is small. The cancellation between the attractive I = 1/2 πN interaction and the repulsive I = 3/2 πN interaction may lead to the small effects on the real part of the resonance energy. The effects of πΛN channel is small which increases binding energy and half-width at most by 1MeV. The pole position of the three-body resonance is W = M − iΓ/2 = 2mN +mK − 79.3 − i37.1 MeV (2mN +mK − 92.2− i35.4 MeV) for relativistic (nonrelativistic) model shown as d and d′. The model dependence of our results on the three-body resonance is summarized in Tables VIII and IX. The K̄NN − πY N resonance pole is located on the K̄NN physical and πY N unphysical sheet with the binding energy, B ∼ 60 − 95 MeV, and the width, Γ ∼ 45 − 80 MeV, using relativistic models. All of our models predict resonance energies above the πΣN threshold. The relatively large model dependence of our results is due to the uncertainty in the models of I = 0 K̄N − πΣ interaction. Comparing the results of model (A) with Model (A) Model (B) (a) −79.3− i37.1 −79.3− i37.3 (b) −93.3− i27.4 −93.3− i27.6 (c) −57.2− i38.6 −56.9− i38.6 (d) −72.4− i31.7 −72.2− i31.9 (e) −87.1− i40.8 −87.1− i41.0 (f) −63.3− i22.2 −63.2− i22.3 TABLE VIII: The pole energy (Wpole −mK − 2mN ) of the three-body resonance using relativistic models. The listed pole energies in MeV can be related to binding energy B and the width Γ as Wpole −mK − 2mN = −B − iΓ/2. Model (A) Model (B) (a) −92.2 − i35.4 −92.3− i35.6 (b) −101.6 − i20.7 −101.6 − i20.7 (c) −72.7 − i53.9 −72.5− i54.9 (d) −83.0 − i33.3 −83.0− i33.6 (e) −98.1 − i33.2 −98.2− i33.3 (f) −66.5 − i24.4 −66.3− i24.4 TABLE IX: The pole energy of the three-body resonance. The same as Table VIII but for the nonrelativistic models. model (B), we can see the three-body pole position is almost independent of the parameters of the I = 1 K̄N − πY interaction. By varying the real(imaginary) part of the fitted scattering length by ±0.1 fm, the binding energy of the three-body resonance is affected by ∼ ±14(8) MeV. Following another way to construct the model, the parameters of the model (f) are fitted to the pole energy of the Λ(1405). This model predicts the scattering length −1.72+ i0.44 fm. The energy of the three-body resonance is found to be B = 63 MeV with a rather small width, Γ = 44 MeV, compared with the models (a-e), which can already be seen in the small imaginary part of the scattering length in model (f). Let us briefly compare our results with those of the other theoretical studies of the K−pp resonance, which use a nonrelativistic approach. Our resonance has a deeper binding energy and a similar width compared with those in Ref. [4]. However, it is not straightforward to compare with the pole energy of Ref. [4] because of the differences in the method to obtain the three-body resonance energy and on the model for the K̄N interaction. Their K̄N potential is stronger and has a short range than ours. Recently Shevchenko, Gal and Mares [19] studied K−pp system using the nonrelativistic coupled-channel Faddeev equation. Though the details of their method is not described in Ref. [19], it seems their approach is quite similar to our present study. They employed a phenomenological K̄N potential model, and reported B ∼ 55− 70 MeV and Γ ∼ 95− 110 MeV. Their result is consistent with our results of the nonrelativistic model. Specially our result using the model (c) gives a quite similar resonance energy and width. In summary we have studied the existence and properties of a strange dibaryon resonance using the K̄NN − πY N coupled channel Faddeev equation. By solving the three-body equation the energy dependence of the resonant K̄N amplitude is fully taken into account. The resonance pole has been investigated from the eigenvalue of the kernel with the analytic continuation of the scattering amplitude on the unphysical Riemann sheet. The model of the K̄N−πY interaction is constructed from the leading order term of the chiral Lagrangian takes into account the relativistic kinematics. The K̄N interaction parameters are fitted to the scattering length given by Martin. We found a resonance pole at B ∼ 79 MeV and Γ ∼ 74 MeV in the relativistic model (aA). However, as the K̄N interaction is not very well constrained by the data, we studied a possible range of the resonance energies by considering different parameter sets of the K̄N −πY interaction. The binding energy and the full width can be in the range of B ∼ 60 − 95 MeV and Γ ∼ 45 − 80 MeV when computed in the relativistic model. In order to connect the resonance found in this work to the experimental signal, further theoretical studies on the production mechanism and further decay of the resonance especially to the Λ− p channel are necessary. Acknowledgments The authors are grateful to Prof. A. Matsuyama for very useful discussions on the three- body resonance. We also thank Drs. B. Juliá-Dı́az, T.-S. H. Lee and Prof. A. Gal for discussions. This work is supported by a Grant-in-Aid for Scientific Research on Priority Areas(MEXT), Japan with No. 18042003. APPENDIX The spin-isospin recoupling coefficient of the particle exchange interaction Z is briefly explained. The coefficient given in Eq. (1.181) of Ref. [21] can be simplified for s-wave states. The three-body state with total spin and isospin (Stot, Itot), which couples with baryon ’isobar’ with spin and isospin (S, I) and the spectator baryon Bi(Si,Ii), is given as |[[M3(S3,I3) ⊗ Bj(Sj ,Ij)](S,I) ⊗ Bi(Si,Ii)](Stot,Itot) > . (A.1) Here baryon i, j represents particle 1 or 2, and the meson is always assigned as the third particle. The wave function of the three-body state, which couples with dibaryon ’isobar’ and the spectator meson M3, is given as |[[B1(S1,I1) ⊗B2(S2,I2)](S,I) ⊗M3(S3,I3)](Stot,Itot) > . (A.2) Then Eq. (7) is extended to include spin-isospin degrees of freedom. The particle exchange interaction for the spectators l, m, the isobars f ′, f with spin-isospin (S ′, I ′) and (S, I) and the exchanged particle n can be expressed as follows Zl,f ′(S′,I′),m,f(S,I)(pl, pm,W ) = Rl,f ′(S′,I′),m,f(S,I) d(p̂l · p̂m) 2πgf ′(S′,I′)(ql)gf(S,I)(qm) W −El(pl)−Em(pm)−En(~pl + ~pm) (A.3) where f represents isobar YK , Yπ, d and N Rl,f ′(S′,I′),m,f(S,I) is given by the overlap of the initial and final spin-isospin wave functions. For the meson(M3) exchange mechanism, Ri,f ′(S′,I′),j,f(S,I) is given as, Ri,f ′(S′,I′),j,f(S,I) = < [[M3(S3,I3) ⊗ Bj(Sj ,Ij)](S′,I′) ⊗ Bi(Si,Ii)](Stot,Itot)| ×|[[M3(S3,I3) ⊗Bi(Si,Ii)](S,I) ⊗Bj(Sj ,Ij)](Stot,Itot) > = (−1)S+S′−S3−StotW (Si, S3, Stot, Sj;S, S ′) (2S + 1)(2S ′ + 1) ×(−1)I+I′−I3−ItotW (Ii, I3, Itot, Ij ; I, I ′) (2I + 1)(2I ′ + 1). (A.4) For the baryon(Bj) exchange mechanism, Ri,f ′(S′,I′),3,f(S,I) is given as, Ri,f ′(S′,I′),3,f(S,I) = < [[M3(S3,I3) ⊗Bj(Sj ,Ij)](S′,I′) ⊗Bi(Si,Ii)](Stot,Itot)| ×|[[B1(S1,I1) ⊗ B2(S2,I2)](S,I) ⊗M3(S3,I3)](Stot,Itot) > = (−1)S3+S−Stot+I3+I−ItotW (S3, Sj , Stot, Si;S ′, S) (2S ′ + 1)(2S + 1) ×W (I3, Ij, Itot, Ii; I ′, I) (2I ′ + 1)(2I + 1) ×(δi,2δj,1 + δi,1δj,2(−1)Si+Sj−S+Ii+Ij−I). (A.5) When we anti-symmetrize the AGS equation, the last factor in the bracket in Eq. (A.5) projects the anti-symmetric two nucleon states. This can be explicitly seen by comparing the exchange of nucleon 2 Z2,YK(S′,I′),3,d(S,I) and nucleon 1 Z1,YK(S′,I′),3,d(S,I) interactions. Using Eq. (A.5), those interactions are related as R1,YK(S′,I′),3,d(S,I) = (−1)S+IR2,YK(S′,I′),3,d(S,I), (A.6) which leads to Eq. (17) as XYK ,YK = (1− (−1)S+I)ZYK ,dτd,dXd,YK + · · · . (A.7) [1] J. Mares, E. Friedman and A. Gal, Nucl. Phys. A770, 84 (2006). [2] L. Tolós, A. Ramos and E. Oset, Phys. Rev. C 74, 015203 (2006). [3] Y. Akaishi and T. Yamazaki, Phys. Rev. C 65, 044005 (2002). [4] T. Yamazaki and Y. Akaishi, Phys. Lett. B535, 70 (2002). [5] A. Dote, H. Horiuchi, Y. Akaishi and T. Yamazaki, Phys. Rev. C 70, 044313 (2004). [6] M. Agnello et al., Phys. Rev. Lett. 94, 212303 (2005). [7] V.K. Magas, E. Oset, A. Ramos and H. Toki, Phys. Rev. C 74, 025206 (2006). [8] W. Glöckle, Phys. Rev. C 18, 564 (1978). [9] K. Möller, Czech. J. Phys. 32, 291 (1982). [10] A. Matsuyama and K. Yazaki, Nucl. Phys. A534, 620 (1991). A. Matsuyama, Phys. Lett. B408, 25 (1997). [11] I.R. Afnan and A.W. Thomas, Phys. Rev. C 10, 109 (1974). [12] B.C. Pearce and I.R. Afnan, Phys. Rev, C 30, 2022 (1984). [13] I.R. Afnan and B.F. Gibson, Phys. Rev. C 47, 1000 (1993). [14] J.A. Oller and U.-G. Meißner, Phys. Lett. B500, 263 (2001). [15] D. Jido et al., Nucl. Phys. A725, 181 (2003). [16] B. Borasoy, R. Nißler and W. Weise, Eur. Phys. J. A 25, 79 (2005). [17] T. Hamaie, M. Arima and K. Masutani, Nucl. Phys. A591, 675 (1995). [18] Y. Ikeda and T. Sato, arXive:nucl-th/0701001. [19] N.V. Shevchenko, A. Gal and J. Mares, Phys. Rev. Lett. 98, 082301 (2007). [20] E.O. Alt, P. Grassberger and W. Sandhas, Nucl. Phys. B2, 167 (1967). [21] I.R. Afnan and A.W. Thomas, in Modern Three-Hadron Physics, edited by A.W. Thomas (Springer, Berlin, 1977), Chap. 1. [22] A.D. Martin, Nucl. Phys. B179, 33 (1981). [23] B. Borasoy, U.-G. Meißner and R. Nißler, Phys. Rev. C 74, 055201 (2006). [24] R.H. Dalitz and A. Deloff, J. Phys. G 17, 289 (1991). [25] M. 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C 49, 2950 (1994). http://arxiv.org/abs/nucl-th/0701001 Introduction Coupled channel Faddeev equation and resonance pole Model of the two-body interactions Results and Discussion Acknowledgments References

0704.1979

SU(2) and SU(4) Kondo effect in double quantum dots

SU(2) and SU(4) Kondo effect in double quantum Jernej Mravlje∗, Anton Ramšak†,∗ and Tomaž Rejec†,∗ ∗Jožef Stefan Institute, Jamova 39, Ljubljana, Slovenia †Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia Abstract. We investigate serial double quantum dot systems with on-site and inter-site interaction by means of Schönhammer-Gunnarsson projection-operator method. The ground state is established by the competition between extended Kondo phases and localized singlet phases in spin and charge degrees of freedom. We present and discuss different phases, as discerned by characteristic correlation functions. We discuss also how different phases would be seen in linear transport measurements. Keywords: Kondo effect, double quantum dots, conductance PACS: 73.23.-b, 73.63.Kv, 72.15.Qm INTRODUCTION In the last decade the advances in experimental techniques enabled the exploration of intriguing many-body effects occurring in solid-state systems such as the Kondo effect [1] by means of measuring the conductance of nanoscale electrical circuits. Tiny pools of electrons defined by electrodes – quantum dots (QDs) – constitute artificial atoms/molecules. Additional gates enable tuning of the orbital levels as well as the tunneling rates, which makes systematic exploration of various effects experimentally accessible. The Kondo effect is essentially the increased scattering rate (with phase shifts near π/2) at low temperatures due to magnetic impurities in host metals. In transport experiments through quantum dots it is seen in another disguise: it is discerned as the amplification of the conductance towards unitary limit. Interesting way to proceed further is to analyze the consequences of inter-impurity interaction by looking at the transport through double quantum dot (DQD) systems. The characteristic feature of the two-impurity Kondo physics is that the two impurities either form an inter-impurity singlet, which is virtually decoupled from conduction elec- trons or they form a double Kondo state SU(2)×SU(2), in which each spin characterized by the SU(2) symmetry group is screened by the conduction electrons [2] depending on the scales of the energies of the inter-impurity singlet formation J and Kondo state formation TK . When the symmetry of the Hamiltonian is larger the Kondo temperature is enhanced. For double quantum dots, which have the capacitative interaction V tuned near the value of the on-dot interaction U , the SU(4) Kondo effect occurs [3]. Here we report our results on the competition between extended Kondo and localized singlet phases in serial DQD systems with inter-dot interaction in the point of particle- hole symmetry [4] and discuss also the phases which occur outside this point. The SU(4) Kondo phase cannot be explored directly by transport experiment through a DQD as the http://arxiv.org/abs/0704.1979v1 conductance is small irrespective of whether the system is in the SU(4) Kondo state or not. Nevertheless, the scale of the SU(4) condensation energy can be estimated by tuning the system away from the point of SU(4) symmetry until the SU(4) Kondo state collapses. The boundary is easy to discern from the conductance data as the conductance is unity whenever the crossover between the phases takes place. MODEL AND METHOD We model DQDs by the two-impurity Anderson Hamiltonian H = Hd +Hl, where Hd corresponds to the isolated dots Hd = ∑ i=1,2 (εni +Uni↑ni↓)+Vn1n2 − t ∑ (c†1σ c2σ +h.c.), with ni = ni↑+ni↓, niσ = c iσ ciσ . The dots are coupled by a tunneling matrix element t and a capacitive V term. The on-site energies ε and the Hubbard repulsion U are taken equal for both dots. Hl describes the noninteracting left and right tight-binding leads with hopping parameter t0 and the coupling of the leads to the DQD. We denote the characteristic tunneling rate of an isolated electron from the dot to the lead by Γ = t ′2/t0, where t ′ is the parameter characterizing the dot-lead hopping. To calculate the ground state of the system we use the Schönhammer and Gunnarsson projection-operator basis [5, 6] |Ψλλ ′〉 = Pλ1Pλ ′2 , which consists of projectors Pλ i; P0i = 1−ni↑ 1−ni↓ , P1i = ∑σ niσ (1−niσ̄ ), P2i = ni↑ni↓ and additional operators involving the operators in leads. We used up to ∼ 100 additional combinations of operators consisting of, for example, P3i =P0iv̂P1i, where v̂ denotes the tunneling to/from dot i. These operators are applied to the state , which is the ground state of the auxiliary noninteracting DQD Hamiltonian of the same form as H, but with U,V = 0, renormalized parameters ε, t, t ′ → ε̃, t̃, t̃ ′ and additional parameter t̃ ′′ which corresponds to hopping from left dot to right lead and vice versa which although absent in the original Hamiltonian is present in the effective Hamiltonian in some parameter regimes. The conductance is calculated using the sine formula [7], G = G0 sin 2[(E+ − E−)/4t0L], where G0 = 2e 2/h and E± are the ground state energies of a large auxiliary ring consisting of L non-interacting sites and an embedded DQD, with periodic and anti-periodic boundary conditions, respectively. GROUND STATE AND CONDUCTANCE OF DQD WITH INTER-DOT INTERACTION Detached DQDs The starting point towards the understanding of the ground state of DQDs are the filling properties of isolated DQDs (i.e. of the Heitler-London or the two-site Hubbard model). The first electron is added when ε = t, and the second when ε =−t +J+[(U + V )−|U −V |]/2, where J = [−|U −V |+ (U −V )2 +16t2]/2 is the difference between singlet and triplet energies. When n = 2 the ground state is [α(|↑↓〉− |↓↑〉)+β (|20〉− |02〉)]/ 2, where α/β = 4t/(V −U + (U −V )2 +16t2). The range of ε where single occupation is favorable is progressively diminished when V 6=U . For large t or at (and near) V =U the molecular bonding and anti-bonding orbitals are formed as is seen here from α ∼ β . Attached DQDs and conductance As we attach DQDs to the leads the ground state either is or is not reminiscent of the ground state of the isolated system. Here the latter possibility is always due to some kind of the Kondo effect. In the top panels of Fig. 1 the ground state of DQDs are presented with pictograms for V = 0,U on the left and right, respectively. The near vertical dividing lines correspond to values of parameters where the ground state of the isolated system is degenerate due to matching energies of states with different occupancies, for example, the rightmost line corresponds to E(0) = 0= E(1) = ε− t. The horizontal U-shaped line is given by J = 2.2TK, where the scale of the Kondo condensation energy is estimated by TK = UΓ/2exp(−πε(ε +U)/2Γ) for U/Γ = 15. FIGURE 1. – Top panels: phases of serial DQDs for V = 0 (left) and V =U (right). The occupancy of the DQD falls from left to right. Extended Kondo phases (with leads in pictograms) and localized singlet phases (without leads in pictograms) occur. – Bottom panels: Conductance and spin-spin correlation of the DQD for t above (full and dotted lines for V = 0; full, dotted and dashed lines for V =U ) and below the localized singlet formation threshold. Note the approaching of S1 ·S2 towards −3/8 for large t indicating the formation of the orbital singlet. For n= 0,4 interaction between electrons (or holes) is not important, hence the ground state is not interesting. For n = 1,3 the ground-state of the isolated DQD is a free spin in (anti-)bonding orbital, which is, when the leads are attached, at low-temperatures screened by conduction electrons as in ’ordinary’ single impurity Anderson model. The most interesting part of the diagrams corresponds to n ∼ 2. Here the ground state of the isolated system is a non-degenerate singlet but the tunneling to the leads breaks this singlet whenever roughly twice the Kondo condensation energy exceeds the triplet excitation energy J. For V ∼ U the J is enhanced hence the area corresponding to SU(2)×SU(2) Kondo is diminished. Near the symmetric point, however, another kind of the Kondo effect arises for V ∼U as a consequence of larger symmetry of the V =U Hamiltonian, which partially restores the occurrence of the Kondo phase. Symmetries The Kondo effect occurs as the consequence of the degeneracy of states of iso- lated impurities. If one looks at the ground state of two isolated impurities coupled by a capacitative (but not tunneling) term V = U , one sees that the 6 states |σ1σ2〉, |20〉 and |02〉 are degenerate. Indeed, by introducing the pseudospin operator [8] T̃ i = 1/2∑ll′=1,2 ∑σ c ll′cl′σ , where τ i are the Pauli matrices, and the combined spin- pseudospin operators W i j = SiT̃ j, one sees that the Hamiltonian is SU(4) symmetric. As long as the SU(4) symmetry breaking terms are small enough V −U, t . TK[SU(4)], the ground state is an SU(4) ’spin’ screened by the electrons in the leads. Orbital representation A complementary way is to rewrite the Hamiltonian in the basis of orbital operators cb,a = (c1 ± c2)/ Hd = ∑ α=a,b εαnα + nα↑nα↓+nα↑nᾱ↓ naσ nbσ + Cflip−Sflip where notation ā = b, b̄ = a is used. The last term of Hd consists of isospin-flip Cflip = T+a T b +h.c. and spin-flip Sflip = S b +h.c. operators, where S λ = c λ↓cλ↑ = (S † are spin and T−λ = cλ↑cλ↓ = (T † isospin lowering and raising operators for the orbitals λ = b,a (or sites λ = 1,2). The full spin (isospin) algebra is closed with operators Szλ = (nλ↑−nλ↓)/2 and T λ = (nλ −1)/2, respectively. When V = U , the spin- and isospin-flip terms in Hd are absent: the Hamiltonian is mapped exactly to the two-level Hamiltonian with intra- and inter-level interaction U with the bonding and anti-bonding levels coupled to even and odd transmission channels, respectively. When V 6= U this mapping is no longer strictly valid: the electrons try to avoid the inter-level repulsion by occupying aligned spin-states in different orbitals, and the isospin-flip terms induce the fluctuations of charge between orbitals. Both mechanisms prohibit electrons from occupying the well-defined orbital states. FIGURE 2. Phases of DQD in the point of particle-hole symmetry. The boundaries between Kondo and localized singlet phases are given by peaks in conductance and abrupt changes in correlation functions. The boundaries of the orbital spin singlet state are given by S1 ·S2 = −3/16 and ∆n21 = ∆n2b on the upper and lower side, respectively. Note the extension of the Kondo phase behind the line J = 2.2TK (dashed) at V ∼U . Numerical results In the lower panels of Fig. 1 the conductance and inter-dot spin-spin correlations are plotted. Note that the orbital picture is indeed more robust for the V =U case as indicated by the broad plateaus in conductance corresponding to the SU(2) Kondo effect of a spin residing in the (anti-)bonding orbitals. Moreover, J is enhanced when compared to the V = 0 case: absence of singlet phase signalled by no peak with unitary conductance and minor spin-spin correlation for all ε occurs only for smaller t. Note also that conductance is small whenever the ground state is practically geometrically separable into parts. In that case the flux can be transported out of the auxiliary ring through the boundary between the parts, yielding zero conductance in our approach [7]. In Fig. 2 we indicate the phases in the (J/TK,V/U) plane. Details are given in Ref. [4]. ACKNOWLEDGMENTS We acknowledge the support of SRA under grant Pl-0044. REFERENCES 1. A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, 1993. 2. B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125–128 (1988). 3. M. R. Galpin, D. E. Logan, and H. R. Krishnamurthy, Phys. Rev. Lett. 94, 186406 (2005). 4. J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 73, 241305(R) (2006). 5. K. Schönhammer, Phys. Rev. B 13, 4336 (1976). 6. O. Gunnarsson, and K. Schönhammer, Phys. Rev. B 31, 4815 (1985). 7. T. Rejec, and A. Ramšak, Phys. Rev. B 68, 035342 (2003). 8. L. D. Leo, and M. Fabrizio, Phys. Rev. B 69, 245114 (2004). INTRODUCTION MODEL AND METHOD GROUND STATE AND CONDUCTANCE OF DQD WITH INTER-DOT INTERACTION Detached DQDs Attached DQDs and conductance Symmetries Orbital representation Numerical results

0704.1980

V-cycle optimal convergence for DCT-III matrices

7 V-cycle optimal convergence for DCT-III ma- trices C. Tablino Possio Dedicated to Georg Heinig Abstract. The paper analyzes a two-grid and a multigrid method for matrices belonging to the DCT-III algebra and generated by a polynomial symbol. The aim is to prove that the convergence rate of the considered multigrid method (V-cycle) is constant independent of the size of the given matrix. Numerical examples from differential and integral equations are considered to illustrate the claimed convergence properties. Mathematics Subject Classification (2000). Primary 65F10, 65F15, 15A12. Keywords. DCT-III algebra, two-grid and multigrid iterations, multi-iterative methods. 1. Introduction In the last two decades, an intensive work has concerned the numerical solution of structured linear systems of large dimensions [6, 14, 16]. Many problems have been solved mainly by the use of (preconditioned) iterative solvers. However, in the multilevel setting, it has been proved that the most popular matrix algebra preconditioners cannot work in general (see [23, 26, 20] and references therein). On the other hand, the multilevel structures often are the most interesting in practical applications. Therefore, quite recently, more attention has been focused (see [1, 2, 7, 5, 27, 9, 12, 10, 13, 22, 25, 19]) on the multigrid solution of multilevel structured (Toeplitz, circulants, Hartley, sine (τ class) and cosine algebras) linear systems in which the coefficient matrix is banded in a multilevel sense and positive definite. The reason is due to the fact that these techniques are very efficient, the total cost for reaching the solution within a preassigned accuracy being linear as the dimensions of the involved linear systems. The work of the author was partially supported by MIUR, grant number 2006017542. http://arxiv.org/abs/0704.1980v1 2 C. Tablino Possio In this paper we deal with the case of matrices generated by a polynomial symbol and belonging to the DCT-III algebra. This kind of matrices appears in the solution of differential equations and integral equations, see for instance [4, 18, 24]. In particular, they directly arise in certain image restoration problems or can be used as preconditioners for more complicated problems in the same field of application [17, 18]. In [7] a Two-Grid (TGM)/Multi-Grid (MGM) Method has been proposed and the theoretical analysis of the TGM has been performed in terms of the alge- braic multigrid theory developed by Ruge and Stüben [21]. Here, the aim is to provide general conditions under which the proposed MGM results to be optimally convergent with a convergence rate independent of the di- mension and to perform the corresponding theoretical analysis. More precisely, for MGM we mean the simplest (and less expensive) version of the large family of multigrid methods, i.e. the V-cycle procedure. For a brief descrip- tion of the TGM and of the MGM (standard V-cycle) we refer to §2. An extensive treatment can be found in [11], and especially in [28]. In all the considered cases the MGM results to be optimal in the sense of Defi- nition 1.1, i.e. the problem of solving a linear system with coefficient matrix Am is asymptotically of the same cost as the direct problem of multiplying Am by a vector. Definition 1.1. [3] Let {Amxm = bm} be a given sequence of linear systems of increasing dimensions. An iterative method is optimal if 1. the arithmetic cost of each iteration is at most proportional to the complexity of a matrix vector product with matrix Am, 2. the number of iterations for reaching the solution within a fixed accuracy can be bounded from above by a constant independent of m. In fact, the total cost of the proposed MGM will be of O(m) operations since for any coarse level s we can find a projection operator P ss+1 such that • the matrix vector product involving P ss+1 costs O(ms) operations wherems = m/2s; • the coarse grid matrix Ams+1 = P ss+1Ams(P ss+1)T is also a matrix in the DCT III algebra generated by a polynomial symbol and can be formed within O(ms) operations; • the convergence rate of the MGM is independent of m. The paper is organized as follows. In §2 we briefly report the main tools re- garding to the convergence theory of algebraic multigrid methods [21]. In §3 we consider the TGM for matrices belonging to DCT-III algebra with reference to some optimal convergence properties, while §4 is devoted to the convergence anal- ysis of its natural extension as V-cycle. In §5 numerical evidences of the claimed results are discussed and §6 deals with complexity issues and conclusions. V-cycle optimal convergence for DCT-III matrices 3 2. Two-grid and Multi-grid methods In this section we briefly report the main results pertaining to the convergence theory of algebraic multigrid methods. Let us consider the generic linear system Amxm = bm, where Am ∈ Cm×m is a Hermitian positive definite matrix and xm, bm ∈ Cm. Let m0 = m > m1 > . . . > ms > . . . > msmin and let P s+1 ∈ Cms+1×ms be a given full-rank matrix for any s. Lastly, let us denote by Vs a class of iterative methods for linear systems of dimension ms. According to [11], the algebraic Two-Grid Method (TGM) is an iterative method whose generic step is defined as follow. xouts = T GM(s, xins , bs) xpres = V s,pre(x s ) Pre-smoothing iterations rs = Asx s − bs rs+1 = P s+1rs As+1 = P s+1As(P Solve As+1ys+1 = rs+1 x̂s = x s − (P ss+1)Hys+1 Exact Coarse Grid Correction xouts = V νpost s,post(x̂s) Post-smoothing iterations where the dimension ms is denoted in short by the subscript s. In the first and last steps a pre-smoothing iteration and a post-smoothing iteration are respectively applied νpre times and νpost times, according to the chosen iterative method in the class Vs. Moreover, the intermediate steps define the so called exact coarse grid correction operator, that depends on the considered projector operator P ss+1. The global iteration matrix of the TGM is then given by TGMs = V νpost s,postCGCsV s,pre, (2.1) CGCs = Is − (P ss+1)HA−1s+1P s+1As As+1 = P s+1As(P H , (2.2) where Vs,pre and Vs,post respectively denote the pre-smoothing and post-smoothing iteration matrices. 4 C. Tablino Possio By means of a recursive procedure, the TGM gives rise to a Multi-Grid Method (MGM): the standard V-cycle is defined as follows. xouts = MGM(s, xins , bs) if s ≤ smin then Solve Asx s = bs Exact solution xpres = V s,pre(x s ) Pre-smoothing iterations rs = Asx s − bs rs+1 = P s+1rs ys+1 = MGM(s+ 1,0s+1, rs+1) x̂s = x s − (P ss+1)Hys+1 Coarse Grid Correction xouts = V νpost s,post(x̂s) Post-smoothing iterations Notice that in MGM the matrices As+1 = P s+1As(P H are more profitably formed in the so called setup phase in order to reduce the computational costs. The global iteration matrix of the MGM can be recursively defined as MGMsmin = O ∈ Csmin×smin , MGMs = V νpost s,post Is − (P ss+1)H (Is+1 −MGMs+1)A−1s+1P ss+1As s,pre, s = smin − 1, . . . , 0. Some general conditions that ensure the convergence of an algebraic TGM and MGM are due to Ruge and Stüben [21]. Hereafter, by ‖·‖2 we denote the Euclidean norm on Cm and the associated induced matrix norm over Cm×m. If X is positive definite, ‖ · ‖X = ‖X1/2 · ‖2 denotes the Euclidean norm weighted by X on Cm and the associated induced matrix norm. Finally, if X and Y are Hermitian matrices, then the notation X ≤ Y means that Y −X is nonnegative definite. Theorem 2.1 (TGM convergence [21]). Let m0, m1 be integers such that m0 > m1 > 0, let A ∈ Cm0×m0 be a positive definite matrix. Let V0 be a class of iterative methods for linear systems of dimension m0 and let P 1 ∈ Cm1×m0 be a given full- rank matrix. Suppose that there exist αpre > 0 and αpost > 0 independent of m0 V-cycle optimal convergence for DCT-III matrices 5 such that ‖V0,pre x‖2A ≤ ‖x‖2A − αpre‖V0,pre x‖2AD−1A for any x ∈ C m0 (2.3a) ‖V0,post x‖2A ≤ ‖x‖2A − αpost ‖x‖2AD−1A for any x ∈ C m0 (2.3b) (where D denotes the main diagonal of A) and that there exists γ > 0 independent of m0 such that y∈Cm1 ‖x− (P 01 )Hy‖2D ≤ γ‖x‖2A for any x ∈ Cm0 . (2.4) Then, γ ≥ αpost and ‖TGM0‖A ≤ 1− αpost/γ 1 + αpre/γ . (2.5) It is worth stressing that in Theorem 2.1 the matrix D ∈ Cm0×m0 can be substi- tuted by any Hermitian positive definite matrix X : clearly the choice X = I can give rise to valuable simplifications [1]. At first sight, the MGM convergence requirements are more severe since the smoothing and CGC iteration matrices are linked in the same inequalities as stated below. Theorem 2.2 (MGM convergence [21]). Let m0 = m > m1 > m2 > . . . > ms > . . . > msmin and let A ∈ Cm×m be a positive definite matrix. Let P ss+1 ∈ Cms+1×ms be full-rank matrices for any level s. Suppose that there exist δpre > 0 and δpost > 0 such that ‖V νpres,prex‖ ≤ ‖x‖2As − δpre ‖CGCsV s,prex‖ for any x ∈ Cms (2.6a) ‖V νposts,posrx‖ ≤ ‖x‖2As − δpost ‖CGCsx‖ for any x ∈ Cms (2.6b) both for each s = 0, . . . , smin − 1, then δpost ≤ 1 and ‖MGM0‖A 6 1− δpost 1 + δpre < 1. (2.7) By virtue of Theorem 2.2, the sequence {x(k)m }k∈N will converge to the solution of the linear system Amxm = bm and within a constant error reduction not depending on m and smin if at least one between δpre and δpost is independent of m and smin. Nevertheless, as also suggested in [21], the inequalities (2.6a) and (2.6b) can be respectively splitted as ‖V νpres,prex‖ ≤ ‖x‖2As − α ‖V s,prex‖AsD−1s As ‖CGCsx‖2As ≤ γ ‖x‖ δpre = α/γ (2.8) and  ‖V νposts,postx‖ ≤ ‖x‖2As − β ‖x‖ ‖CGCs x‖2As ≤ γ ‖x‖ δpost = β/γ (2.9) 6 C. Tablino Possio where Ds is the diagonal part of As (again, the AD −1A-norm is not compulsory [1] and the A2-norm will be considered in the following) and where, more importantly, the coefficients α, β and γ can differ in each recursion level s since the step from (2.8) to (2.6a) and from (2.9) to (2.6b) are purely algebraic and do not affect the proof of Theorem 2.2. Therefore, in order to prove the V-cycle optimal convergence, it is possible to consider the inequalities ‖V νpres,prex‖ ≤ ‖x‖2As − αs ‖V s,prex‖ for any x ∈ Cms (2.10a) ‖V νposts,postx‖ ≤ ‖x‖2As − βs ‖x‖ for any x ∈ Cms (2.10b) ‖CGCsx‖2As ≤ γs ‖x‖ for any x ∈ Cms . (2.10c) where it is required that αs, βs, γs ≥ 0 for each s = 0, . . . , smin − 1 and δpre = min 0≤s<smin , δpost = min 0≤s<smin . (2.11) We refer to (2.10a) as the pre-smoothing property, (2.10b) as the post-smoothing property and (2.10c) as the approximation property (see [21]). An evident benefit in considering the inequalities (2.10a)-(2.10c) relies on to the fact that the analysis of the smoothing iterations is distinguished from the more difficult analysis of the projector operator. Moreover, the MGM smoothing properties (2.10a) and (2.10b) are nothing more than the TGM smoothing properties (2.3a) and (2.3b) with D substituted by I, in accordance with the previous reasoning (see [1]). 3. Two-grid and Multi-grid methods for DCT III matrices Let Cm = {Cm ∈ Rm×m|Cm = QmDmQTm} the unilevel DCT-III cosine matrix algebra, i.e. the algebra of matrices that are simultaneously diagonalized by the orthogonal transform 2− δj,1 (i − 1)(j − 1/2)π i,j=1 (3.1) with δi,j denoting the Kronecker symbol. Let f be a real-valued even trigonometric polynomial of degree k and period 2π. Then, the DCT III matrix of order m generated by f is defined as Cm(f) = QmDm(f)Q m, Dm(f) = diag1≤j≤m f (j − 1)π Clearly, Cm(f) is a symmetric band matrix of bandwidth 2k+1. In the following, we denote in short with Cs = Cms(gs) the DCT III matrix of size ms generated by the function gs. An algebraic TGM/MGM method for (multilevel) DCT III matrices generated by a real-valued even trigonometric polynomial has been proposed in [7]. Here, we V-cycle optimal convergence for DCT-III matrices 7 briefly report the relevant results with respect to TGM convergence analysis, the aim being to prove in §4 the V-cycle optimal convergence under suitable conditions. Indeed, the projector operator P ss+1 is chosen as P ss+1 = T s+1Cs(ps) where T ss+1 ∈ Rms+1×ms , ms+1 = ms/2, is the cutting operator defined as T ss+1 2 for j ∈ {2i− 1, 2i}, i = 1, . . . ,ms+1, 0 otherwise. (3.2) and Cs(ps) is the DCT-III cosine matrix of size ms generated by a suitable even trigonometric polynomial ps. Here, the scaling by a factor 1/ 2 is introduced in order to normalize the matrix T ss+1 with respect to the Euclidean norm. From the point of view of an algebraic multigrid this is a natural choice, while in a geometric multigrid it is more natural to consider just a scaling by 1/2 in the projector, to obtain an average value. The cutting operator plays a leading role in preserving both the structural and spectral properties of the projected matrix Cs+1: in fact, it ensures a spectral link between the space of the frequencies of size ms and the corresponding space of frequencies of size ms+1, according to the following Lemma. Lemma 3.1. [7] Let Qs ∈ Rms×ms and T ss+1 ∈ Rms+1×ms be given as in (3.1) and (3.2) respectively. Then T ss+1Qs = Qs+1[Φs+1,Θs+1Πs+1], (3.3) where Φs+1 = diagj=1,...,ms+1 (j − 1)π , (3.4a) Θs+1 = diagj=1,...,ms+1 − cos , (3.4b) and Πs+1 ∈ Rms+1×ms+1 is the permutation matrix (1, 2, . . . ,ms+1) 7→ (1,ms+1,ms+1 − 2, . . . , 2). As a consequence, let As = Cs(fs) be the DCT-III matrix generated by fs, then As+1 = P s+1As(P T = Cs+1(fs+1) where fs+1(x) = cos (3.5) + cos2 π − x/2 π − x π − x , x ∈ [0, π]. On the other side, the convergence of proposed TGM at size ms is ensured by choosing the polynomial as follows. 8 C. Tablino Possio Definition 3.2. Let x0 ∈ [0, π) a zero of the generating function fs. The polynomial ps is chosen so that p2s(π − x) fs(x) < +∞, (3.6a) p2s(x) + p s(π − x) > 0. (3.6b) In the special case x0 = π, the requirement (3.6a) is replaced by x→x0=π p2s(π − x) fs(x) < +∞. (3.7a) If fs has more than one zero in [0, π], then ps will be the product of the polynomials satisfying the condition (3.6a) (or (3.7a)) for every single zero and globally the condition (3.6b). It is evident from the quoted definition that the polynomial ps must have zeros of proper order in any mirror point x̂0 = π − x0, where x0 is a zeros of fs. It is worth stressing that conditions (3.6a) and (3.6b) are in perfect agreement with the case of other structures such as τ , symmetric Toeplitz and circulant matrices (see e.g. [22, 25]), while the condition (3.7a) is proper of the DCT III algebra and it corresponds to a worsening of the convergence requirements. Moreover, as just suggested in [7], in the case x0 = 0 the condition (3.6a) can also be weakened as x→x0=0 p2s(π − x) fs(x) < +∞. (3.8a) We note that if fs is a trigonometric polynomial of degree k, then fs can have a zero of order at most 2k. If none of the root of fs are at π, then by (3.6a) the degree of ps has to be less than or equal to ⌈k/2⌉. If π is one of the roots of fs, then the degree of ps is less than or equal to ⌈(k + 1)/2⌉. Notice also that from (3.5), it is easy to obtain the Fourier coefficients of fs+1 and hence the nonzero entries of As+1 = Cs+1(fs+1). In addition, we can obtain the roots of fs+1 and their orders by knowing the roots of fs and their orders. Lemma 3.3. [7] If 0 ≤ x0 ≤ π/2 is a zero of fs, then by (3.6a), ps(π − x0) = 0 and hence by (3.5), fs+1(2x 0) = 0, i.e. y0 = 2x0 is a zero of fs+1. Furthermore, because ps(π − x0) = 0, by (3.6b), ps(x0) > 0 and hence the orders of x0 and y0 are the same. Similarly, π/2 ≤ x0 < π, then y0 = 2(π − x0) is a root of fs+1 with the same order as x0. Finally, if x0 = π, then y0 = 0 with order equals to the order of x0 plus two. In [7] the Richardson method has be considered as the most natural choice for the smoothing iteration, since the corresponding iteration matrix Vm := Im − ωAm ∈ Cm×m belongs to the DCT-III algebra, too. Further remarks about such a type of smoothing iterations and the tuning of the parameter ω are reported in [25, 2]. V-cycle optimal convergence for DCT-III matrices 9 Theorem 3.4. [7] Let Am0 = Cm0(f0) with f0 being a nonnegative trigonometric polynomial and let Vm0 = Im0 − ωAm0 with ω = 2/‖f0‖∞ and ω = 1/‖f0‖∞, respectively for the pre-smoothing and the post-smoothing iteration. Then, under the quoted assumptions and definitions the inequalities (2.3a), (2.3b), and (2.4) hold true and the proposed TGM converges linearly. Here, it could be interesting to come back to some key steps in the proof of the quoted Theorem 3.4 in order to highlight the structure with respect to any point and its mirror point according to the considered notations. By referring to a proof technique developed in [22], the claimed thesis is obtained by proving that the right-hand sides in the inequalities γ ≥ 1 ds(x) π − x p2s(π − x) fs(x) , (3.9a) γ ≥ 1 ds(x) π − x p2s(π − x) fs(x) + cos2 ) p2s(x) fs(π − x) , (3.9b) ds(x) = cos p2s(x) + cos π − x p2s(π − x) (3.9c) are uniformly bounded on the whole domain so that γ is an universal constant. It is evident that (3.9a) is implied by (3.9b). Moreover, both the two terms in (3.9b) and in ds(x) can be exchanged each other, up to the change of variable y = π − x. Therefore, if x0 6= π it is evident that Definition 3.2 ensures the required uniform boundedness since the condition p2s(x) + p s(π − x) > 0 implies ds(x) > 0. In the case x0 = π, the inequality (3.9b) can be rewritten as γ ≥ 1 p2s(x) p2s(π − x) p2s(π − x) fs(x) p2s(x) fs(π − x) (3.10) so motivating the special case in Definition 3.2. 4. V-cycle optimal convergence In this section we propose a suitable modification of Definition 3.2 with respect to the choice of the polynomial involved into the projector, that allows us to prove the V-cycle optimal convergence according to the verification of the inequalities (2.10a)-(2.10c) and the requirement (2.11). It is worth stressing that the MGM smoothing properties do not require a true verification, since (2.10a) and (2.10b) are exactly the TGM smoothing properties (2.3a) and (2.3b) (with D = I). Proposition 4.1. Let As = Cms(fs) for any s = 0, . . . , smin, with fs ≥ 0, and let ωs be such that 0 < ωs ≤ 2/‖fs‖∞. If we choose αs and βs such that αs ≤ 10 C. Tablino Possio ωsmin 2, (2− ωs‖fs‖∞)/(1− ωs‖fs‖∞)2 and βs 6 ωs(2 − ωs‖fs‖∞) then for any x ∈ Cm the inequalities ‖Vs,pre x‖2As ≤ ‖x‖ − αs ‖Vs,pre x‖2As (4.1) ‖Vs,post x‖2As ≤ ‖x‖ − βs ‖x‖2As (4.2) hold true. Notice, for instance, that the best bound to βs is given by 1/‖fs‖∞ and it is obtained by taking ωs = 1/‖fs‖∞ [25, 2]. Concerning the analysis of the approximation condition (2.10c) we consider here the case of a generating function f0 with a single zero at x 0. In such a case, the choice of the polynomial in the projector is more severe with respect to the case of TGM. Definition 4.2. Let x0 ∈ [0, π) a zero of the generating function fs. The polynomial ps is chosen in such a way that ps(π − x) fs(x) < +∞, (4.3a) p2s(x) + p s(π − x) > 0. (4.3b) In the special case x0 = π, the requirement (4.3a) is replaced by x→x0=π ps(π − x) fs(x) < +∞. (4.4a) Notice also that in the special case x0 = 0 the requirement (4.3a) can be weakened x→x0=0 ps(π − x) fs(x) < +∞. (4.5a) Proposition 4.3. Let As = Cms(fs) for any s = 0, . . . , smin, with fs ≥ 0. Let P ss+1 = T s+1Cs(ps), where ps(x) is fulfilling (4.3a) (or (4.4a)) and (4.3b). Then, for any s = 0, . . . , smin − 1, there exists γs > 0 independent of ms such that ‖CGCsx‖2As ≤ γs ‖x‖ for any x ∈ Cms , (4.6) where CGCs is defined as in (2.2). Proof. Since CGCs = Is − (P ss+1)T (P ss+1As(P ss+1)T )−1P ss+1As is an unitary projector, it holds that CGCTs As CGCs = As CGCs. Therefore, the target inequality (4.6) can be simplified and symmetrized, giving rise to the matrix inequality C̃GCs = Is −A1/2s (P ss+1)T (P ss+1As(P ss+1)T )−1P ss+1A1/2s ≤ γsAs. (4.7) V-cycle optimal convergence for DCT-III matrices 11 Hence, by invoking Lemma 3.1, QTs C̃GCsQs can be permuted into a 2 × 2 block diagonal matrix whose jth block, j = 1, . . . ,ms+1, is given by the rank-1 matrix (see [8] for the analogous τ case) c2j + s c2j cjsj cjsj s where cj = cos p2f(x j ) sj = − cos π − x[ms]j p2f(π − x[ms]j ). As in the proof of the TGM convergence, due to the continuity of fs and ps, (4.7) is proven if the right-hand sides in the inequalities d̃s(x) π − x p2sfs(π − x) fs(x) (4.8a) d̃s(x) π − x p2sfs(π − x) fs(x) + cos2 ) p2sfs(x) fs(π − x) (4.8b) d̃s(x) = cos p2sfs(x) + cos π − x p2sf(π − x) (4.8c) are uniformly bounded on the whole domain so that γs are universal constants. Once again, it is evident that (4.8a) is implied by (4.8b). Moreover, both the terms in (4.8b) and in d̃s(x) can be exchanged each other, up to the change of variable y = π − x. Therefore, if x0 6= π, (4.8b) can be rewritten as d̂s(x) π − x p2s(π − x) f2s (x) + cos2 ) p2s(x) f2s (π − x) (4.9) where d̂s(x) = cos ) p2s(x) fs(π − x) + cos2 π − x p2s(π − x) fs(x) so that Definition 4.2 ensures the required uniform boundedness. In the case x0 = π, the inequality (4.8b) can be rewritten as p2s(x) fs(π − x) p2s(π − x) fs(x) p2s(π − x) f2s (x) p2s(x) f2s (π − x) (4.10) so motivating the special case in Definition 4.2. � 12 C. Tablino Possio Remark 4.4. Notice that in the case of pre-smoothing iterations and under the assumption Vs,pre nonsingular, the approximation condition ‖CGCsV νpres,prex‖ ≤ γs ‖V νpres,prex‖ for any x ∈ Cms , (4.11) is equivalent to the condition, in matrix form, C̃GCs ≤ γsAs obtained in Propo- sition 4.3. In Propositions 4.1 and 4.3 we have obtained that for every s (independent of m = m0) the constants αs, βs, and γs are absolute values not depending on m = m0, but only depending on the functions fs and ps. Nevertheless, in order to prove the MGM optimal convergence according to Theorem 2.2, we should verify at least one between the following inf–min conditions [1]: δpre = inf 0≤s≤log2(m0) > 0, δpost = inf 0≤s≤log2(m0) > 0. (4.12) First, we consider the inf-min requirement (4.12) by analyzing the case of a gen- erating function f̃0 with a single zero at x 0 = 0. It is worth stressing that in such a case the DCT-III matrix Ãm0 = Cm0(f̃0) is sin- gular since 0 belongs to the set of grid points x j = (j − 1)π/m0, j = 1, . . . ,m0. Thus, the matrix Ãm0 is replaced by Am0 = Cm0(f0) = Cm0(f̃0) + f̃0 with e = [1, . . . , 1]T ∈ Rm0 and where the rank-1 additional term is known as Strang correction [29]. Equivalently, f̃0 ≥ 0 is replaced by the generating function f0 = f̃0 + f̃0 1 +2πZ > 0, (4.13) where χX is the characteristic function of the set X and w 1 = x 0 = 0. In Lemma 4.5 is reported the law to which the generating functions are subjected at the coarser levels. With respect to this target, it is useful to consider the following factorization result: let f ≥ 0 be a trigonometric polynomial with a single zero at x0 of order 2q. Then, there exists a positive trigonometric polynomial ψ such that f(x) = [1− cos(x− x0)]q ψ(x). (4.14) Notice also that, according to Lemma 3.3, the location of the zero is never shifted at the subsequent levels. Lemma 4.5. Let f0(x) = f̃0(x) + c0χ2πZ(x), with f̃0(x) = [1− cos(x)]qψ0(x), q being a positive integer and ψ0 being a positive trigonometric polynomial and with c0 = f̃0 . Let ps(x) = [1+cos(x)] q for any s = 0, . . . , smin− 1. Then, under the same assumptions of Lemma 3.1, each generating function fs is given by fs(x) = f̃s(x) + csχ2πZ(x), f̃s(x) = [1− cos(x)]qψs(x). V-cycle optimal convergence for DCT-III matrices 13 The sequences {ψs} and {cs} are defined as ψs+1 = Φq,ps(ψs), cs+1 = csp s(0), s = 0, . . . , smin − 1, where Φq,p is an operator such that [Φq,p(ψ)] (x) = (ϕpψ) + (ϕpψ) π − x , (4.15) with ϕ(x) = 1 + cos(x). Moreover, each f̃s is a trigonometric polynomial that vanishes only at 2πZ with the same order 2q as f̃0. Proof. The claim is a direct consequence of Lemma 3.1. Moreover, since the func- tion ψ0 is positive by assumption, the same holds true for each function ψs. � Hereafter, we make use of the following notations: for a given function f , we will write Mf = supx |f |, mf = infx |f | and µ∞(f) =Mf/mf . Now, if x ∈ (0, 2π) we can give an upper bound for the left-hand side R(x) in (4.9), since it holds that R(x) = p2s(x) f2s (π − x) p2s(π − x) f2s (x) p2s(x) fs(π − x) p2s(π − x) fs(x) ψ2s (π − x) ψ2s(x) ps(x) ψs(π − x) ps(π − x) ψs(x) ≤ Mψs ps(x) + cos2 ps(π − x) ≤ Mψs we can consider γs = Mψs/m . In the case x = 0, since ps(0) = 0, it holds R(0) = 1/fs(π), so that we have also to require 1/fs(π) ≤ γs. However, since 1/fs(π) ≤Mψs/m2ψs , we take γ s =Mψs/m as the best value. In (2.9), by choosing ω∗s = ‖fs‖−1∞ , we simply find β∗s = ‖fs‖−1∞ ≥ 1/(2qMψs) and as a consequence, we obtain 2qMψs 2qµ2∞(ψs) . (4.16) A similar relation can be found in the case of a pre-smoothing iteration. Neverthe- less, since it is enough to prove one between the inf-min conditions, we focus our at- tention on condition (4.16). So, to enforce the inf–min condition (4.12), it is enough to prove the existence of an absolute constant L such that µ∞(ψs) 6 L < +∞ uniformly in order to deduce that ‖MGM0‖A0 6 1− (2qL2)−1 < 1. 14 C. Tablino Possio Proposition 4.6. Under the same assumptions of Lemma 4.5, let us define ψs = [Φps,q] s(ψ) for every s ∈ N, where Φp,q is the linear operator defined as in (4.15). Then, there exists a positive polynomial ψ∞ of degree q such that ψs uniformly converges to ψ∞, and moreover there exists a positive real number L such that µ∞(ψs) 6 L for any s ∈ N. Proof. Due to the periodicity and to the cosine expansions of all the involved functions, the operator Φq,p in (4.15) can be rewritten as [Φq,p(ψ)] (x) = (ϕpψ) + (ϕpψ) . (4.17) The representation of Φq,p in the Fourier basis (see Proposition 4.8 in [1]) leads to an operator from Rm(q) to Rm(q), m(q) proper constant depending only on q, which is identical to the irreducible nonnegative matrix Φ̄q in equation (4.14) of [1], with q + 1 in place of q. As a consequence, the claimed thesis follows by referring to the Perron–Frobenius theorem [15, 30] according to the very same proof technique considered in [1]. � Lastly, by taking into account all the previous results, we can claim the optimality of the proposed MGM. Theorem 4.7. Let f̃0 be a even nonnegative trigonometric polynomial vanishing at 0 with order 2q. Let m0 = m > m1 > . . . > ms > . . . > msmin , ms+1 = ms/2. For any s = 0, . . . ,msmin−1, let P ss+1 be as in Proposition 4.3 with ps(x) = [1+cos(x)]q, and let Vs,post = Ims − Ams/‖fs‖∞. If we set Am0 = Cm0(f̃0 + c0χ2πZ) with c0 = f̃0(w 2 ) and we consider b ∈ Cm0 , then the MGM (standard V-cycle) converges to the solution of Am0x = b and is optimal (in the sense of Definition 1.1). Proof. Under the quoted assumptions it holds that f̃0(x) = [1− cos(x)]q ψ0(x) for some positive polynomial ψ0(x). Therefore, it is enough to observe that the optimal convergence of MGM as stated in Theorem 2.2 is implied by the inf–min condition (4.12). Thanks to (4.16), the latter is guaranteed if the quantities µ∞(ψs) are uniformly bounded and this holds true according to Proposition 4.6. � Now, we consider the case of a generating function f0 with a unique zero at x0 = π, this being particularly important in applications since the discretization of certain integral equations leads to matrices belonging to this class. For instance, the signal restoration leads to the case of f0(π) = 0, while for the super-resolution problem and image restoration f0(π, π) = 0 is found [5]. By virtue of Lemma 3.3 we simply have that the generating function f1 related to the first projected matrix uniquely vanishes at 0, i.e. at the first level the MGM projects a discretized integral problem, into another which is spectrally and structurally equivalent to a discretized differential problem. With respect to the optimal convergence, we have that Theorem 2.2 holds true with δ = min{δ0, δ̄} since δ results to be a constant and independent of m0. V-cycle optimal convergence for DCT-III matrices 15 More precisely, δ0 is directly related to the finest level and δ̄ is given by the inf-min condition of the differential problem obtained at the coarser levels. The latter constant value has been previously shown, while the former can be proven as follows: we are dealing with f0(x) = (1+cos(x)) qψ0(x) and according to Definition 4.2 we choose p̃0(x) = p0(x) + d0χ2πZ with p0(x) = (1 + cos(x)) q+1 and d0 = Therefore, an upper bound for the left-hand side R̃(x) in (4.10) is obtained as R̃(x) ≤ i.e. we can consider γ0 = Mψ0/m and so that a value δ0 independent of m0 is found. 5. Numerical experiments Hereafter, we give numerical evidence of the convergence properties claimed in the previous sections, both in the case of proposed TGM and MGM (standard V-cycle), for two types of DCT-III systems with generating functions having zero at 0 (differential like problems) and at π (integral like problems). The projectors P ss+1 are chosen as described in §3 in §4 and the Richardson smooth- ing iterations are used twice in each iteration with ω = 2/‖f‖ and ω = 1/‖f‖ respectively. The iterative procedure is performed until the Euclidean norm of the relative residual at dimension m0 is greater than 10 −7. Moreover, in the V-cycle, the exact solution of the system is found by a direct solver when the coarse grid dimension equals to 16 (162 in the additional two-level tests). 5.1. Case x0 = 0 (differential like problems) First, we consider the case Am = Cm(f0) with f0(x) = [2 − 2 cos(x)]q, i.e. with a unique zero at x0 = 0 of order 2q. As previously outlined, the matrix Cm(f0) is singular, so that the solution of the rank-1 corrected system is considered, whose matrix is given by Cm(f0) + (f0(π/m)/m)ee T , with e = [1, . . . , 1]T . Since the position of the zero x0 = 0 at the coarser levels is never shifted, then the function ps(x) = [2− 2 cos(π − x)]r in the projectors is the same at all the subsequent levels s. To test TGM/MGM linear convergence with rate independent of the size m0 we tried for different r: according to (3.6a), we must choose r at least equal to 1 if q = 1 and at least equal to 2 if q = 2, 3, while according to (4.3a) we must always choose r equal to q. The results are reported in Table 1. By using tensor arguments, the previous results plainly extend to the multilevel case. In Table 2 we consider the case of generating function f0(x, y) = f0(x) + f0(y), that arises in the uniform finite difference discretization of elliptic constant coefficient differential equations on a square with Neumann boundary conditions, see e.g [24]. 16 C. Tablino Possio Table 1. Twogrid/Multigrid - 1D Case: f0(x) = [2 − 2 cos(x)]q and p(x) = [2− 2 cos(π − x)]r . q = 1 q = 2 q = 3 m0 r = 1 r = 1 r = 2 r = 2 r = 3 16 7 15 13 34 32 32 7 16 15 35 34 64 7 16 16 35 35 128 7 16 16 35 35 256 7 16 16 35 35 512 7 16 16 35 35 q = 1 q = 2 q = 3 m0 r = 1 r = 1 r = 2 r = 2 r = 3 16 1 1 1 1 1 32 7 16 15 34 32 64 7 17 16 35 34 128 7 18 16 35 35 256 7 18 16 35 35 512 7 18 16 35 35 Table 2. Twogrid/Multigrid - 2D Case: f0(x, y) = [2 − 2 cos(x)]q + [2 − 2 cos(y)]q and p(x, y) = [2 − 2 cos(π − x)]r + [2− 2 cos(π − y)]r. q = 1 q = 2 q = 3 m0 r = 1 r = 1 r = 2 r = 2 r = 3 162 15 34 30 - - 322 16 36 35 71 67 642 16 36 36 74 73 1282 16 36 36 74 73 2562 16 36 36 74 73 5122 16 36 36 74 73 q = 1 q = 2 q = 3 m0 r = 1 r = 1 r = 2 r = 2 r = 3 162 1 1 1 1 1 322 16 36 35 71 67 642 16 36 36 74 73 1282 16 36 36 74 73 2562 16 37 36 74 73 5122 16 37 36 74 73 5.2. Case x0 = π (integral like problems) DCT III matrices Am0 = Cm0(f0) whose generating function shows a unique zero at x0 = π are encountered in solving integral equations, for instance in image restoration problems with Neumann (reflecting) boundary conditions [18]. According to Lemma 3.3, if x0 = π, then the generating function f1 of the coarser matrix Am1 = Cm1(f1), m1 = m0/2 has a unique zero at 0, whose order equals the order of x0 = π with respect to f0 plus two. It is worth stressing that in such a case the projector at the first level is singular so that its rank-1 Strang correction is considered. This choice gives rise in a natural way to the rank-1 correction considered in §5.1. Moreover, starting from the second coarser level, the new location of the zero is never shifted from 0. In Table 3 are reported the numerical results both in the unilevel and two-level case. 6. Computational costs and conclusions Some remarks about the computational costs are required in order to highlight the optimality of the proposed procedure. V-cycle optimal convergence for DCT-III matrices 17 Table 3. Twogrid/Multigrid - 1D Case: f0(x) = 2+2 cos(x) and p0(x) = 2 − 2 cos(π − x) and 2D Case: f0(x, y) = 4 + 2 cos(x) + 2 cos(y) and p0(x, y) = 4− 2 cos(π − x)− 2 cos(π − y). 1D TGM MGM 16 15 1 32 14 14 64 12 13 128 11 13 256 10 12 512 8 10 2D TGM MGM 162 7 1 322 7 7 642 7 7 1282 7 6 2562 7 6 5122 7 6 Since the matrix Cms(p) appearing in the definition of P s+1 is banded, the cost of a matrix vector product involving P ss+1 is O(ms). Therefore, the first condition in Definition 1.1 is satisfied. In addition, notice that the matrices at every level (except for the coarsest) are never formed since we need only to store the O(1) nonzero Fourier coefficients of the related generating function at each level for matrix-vector multiplications. Thus, the memory requirements are also very low. With respect to the second condition in Definition 1.1 we stress that the repre- sentation of Ams+1 = Cms+1(fs+1) can be obtained formally in O(1) operations by virtue of (3.5). In addition, the roots of fs+1 and their orders are obtained according to Lemma 3.3 by knowing the roots of fs and their orders. Finally, each iteration of TGM costs O(m0) operations as Am0 is banded. In conclusion, each iteration of the proposed TGM requires O(m0) operations. With regard to MGM, optimality is reached since we have proven that there exists δ is independent from both m and smin so that the number of required iterations results uniformly bounded by a constant irrespective of the problem size. In addi- tion, since each iteration has a computational cost proportional to matrix-vector product, Definition 1.1 states that such a kind of MGM is optimal. As a conclusion, we observe that the reported numerical tests in §5 show that the requirements on the order of zero in the projector could be weakened. Future works will deals with this topic and with the extension of the convergence analysis in the case of a general location of the zeros of the generating function. References [1] A. Aricò, M. Donatelli, S. Serra-Capizzano, V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl. 26 (2004), no. 1, 186–214. [2] A. Aricò, M. Donatelli, A V-cycle Multigrid for multilevel matrix algebras: proof of optimality. Numer. Math. 105 (2007), no. 4, 511–547 (DOI 10.1007/s00211-006-0049- 18 C. Tablino Possio [3] O. Axelsson , M. Neytcheva, The algebraic multilevel iteration methods—theory and applications. In Proceedings of the Second International Colloquium on Numerical Analysis (Plovdiv, 1993), 13–23, VSP, 1994. [4] R.H. Chan, T.F. Chan, C. 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Vassalos, Matrix algebra preconditioners for mul- tilevel Toeplitz systems do not insure optimal convergence rate. Theoret. Comput. Sci. 315 (2004), no. 2-3, 557–579. [21] J. Ruge, K. Stüben, Algebraic multigrid. In Frontiers in Applied Mathematics: Multi- grid Methods.SIAM, 1987, 73–130. [22] S. Serra Capizzano, Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences. Numer. Math. 92 (2002), no. 3, 433–465. [23] S. Serra-Capizzano, Matrix algebra preconditioners for multilevel Toeplitz matrices are not superlinear. Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343-344 (2002), 303–319. [24] S. Serra Capizzano, C. Tablino Possio, Spectral and structural analysis of high preci- sion finite difference matrices for elliptic operators. Linear Algebra Appl. 293 (1999), no. 1-3, 85–131. [25] S. Serra-Capizzano, C. Tablino-Possio, Multigrid methods for multilevel circulant matrices. SIAM J. Sci. Comput. 26 (2004), no. 1, 55–85. [26] S. Serra-Capizzano and E. Tyrtyshnikov, How to prove that a preconditioner cannot be superlinear. Math. Comp. 72 (2003), no. 243, 1305–1316. [27] H. Sun, X. Jin, Q. Chang, Convergence of the multigrid method for ill-conditioned block Toeplitz systems. BIT 41 (2001), no. 1, 179–190. [28] U. Trottenberg, C.W. Oosterlee and A. Schüller, Multigrid. With contributions by A. Brandt, P. Oswald and K. Stb̈en. Academic Press, Inc., 2001. [29] E. Tyrtyshnikov, Circulant preconditioners with unbounded inverses. Linear Algebra Appl. 216 (1995), 1–23. [30] R.S. Varga, Matrix Iterative Analysis. Prentice-Hall, Inc., Englewood Cliffs, 1962. C. Tablino Possio Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Cozzi 53 20125 Milano Italy e-mail: [email protected] 1. Introduction 2. Two-grid and Multi-grid methods 3. Two-grid and Multi-grid methods for DCT III matrices 4. V-cycle optimal convergence 5. Numerical experiments 5.1. Case x0=0 (differential like problems) 5.2. Case x0= (integral like problems) 6. Computational costs and conclusions References

0704.1981

Improved Measurement of the Positive Muon Lifetime and Determination of the Fermi Constant

Improved Measurement of the Positive Muon Lifetime and Determination of the Fermi Constant D.B. Chitwood,1 T.I. Banks,2 M.J. Barnes,3 S. Battu,4 R.M. Carey,5 S. Cheekatmalla,4 S.M. Clayton,1 J. Crnkovic,1 K.M. Crowe,2 P.T. Debevec,1 S. Dhamija,4 W. Earle,5 A. Gafarov,5 K. Giovanetti,6 T.P. Gorringe,4 F.E. Gray,1, 2 M. Hance,5 D.W. Hertzog,1 M.F. Hare,5 P. Kammel,1 B. Kiburg,1 J. Kunkle,1 B. Lauss,2 I. Logashenko,5 K.R. Lynch,5 R. McNabb,1 J.P. Miller,5 F. Mulhauser,1 C.J.G. Onderwater,1, 7 C.S. Özben,1 Q. Peng,5 C.C. Polly,1 S. Rath,4 B.L. Roberts,5 V. Tishchenko,4 G.D. Wait,3 J. Wasserman,5 D.M. Webber,1 P. Winter,1 and P.A. Żo lnierczuk4 (MuLan Collaboration) Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Department of Physics, University of California, Berkeley, CA 94720, USA TRIUMF, Vancouver, BC, V6T 2A3, Canada Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Department of Physics, Boston University, Boston, MA 02215, USA Department of Physics, James Madison University, Harrisonburg, VA 22807, USA Kernfysisch Versneller Instituut, Groningen University, NL 9747 AA Groningen, The Netherlands The mean life of the positive muon has been measured to a precision of 11 ppm using a low- energy, pulsed muon beam stopped in a ferromagnetic target, which was surrounded by a scintillator detector array. The result, τµ = 2.197 013(24) µs, is in excellent agreement with the previous world average. The new world average τµ = 2.197 019(21) µs determines the Fermi constant GF = 1.166 371(6)× 10−5 GeV−2 (5 ppm). Additionally, the precision measurement of the positive muon lifetime is needed to determine the nucleon pseudoscalar coupling gP . The predictive power of the standard model relies on precision measurements of its input parameters. Impres- sive examples include the fine-structure constant [1], the Z mass [2], and the Fermi constant [3], having relative precisions of 0.0007, 23, and 9 ppm, respectively. The Fermi constant GF is related [3] to the electroweak gauge coupling g by (1 + ∆r) , (1) where ∆r represents the weak-boson-mediated tree-level and radiative corrections, which have been computed to second order [4]. Comparison of the Fermi constant ex- tracted from various measurements stringently tests the universality of the weak interaction’s strength [5]. The most precise determination of GF is based on the mean life of the positive muon, τµ. It has long been known that in the Fermi theory, the QED radiative cor- rections are finite to first order in GF and to all orders in the electromagnetic coupling constant, α [6]. This pro- vides a framework [3] for extracting GF from τµ, 192π3 (1 + ∆q) , (2) where ∆q is the sum of phase space and both QED and hadronic radiative corrections, which have been known in lowest-order since the 1950s [7]. Relation 2 does not depend on the specifics of the underlying electroweak model. Until recently, the uncertainty on extracting GF from τµ was limited by the uncertainty in higher-order QED corrections, rather than by measurement. In 1999, van Ritbergen and Stuart’s calculation of the second-order QED corrections [3] reduced the relative theoretical un- certainty in the determination of GF to less than 0.3 ppm. The dominant uncertainty is currently from τµ, which motivates this work. While the final goal of our experi- ment is a 1 ppm uncertainty on τµ, we report here a re- sult having a precision of 11 ppm—2.5 times better than any previous measurement [8]—based on data obtained in 2004, the commissioning run period. The experimental design is conceptually simple. Lon- gitudinally polarized muons from the πE3 beamline at the Paul Scherrer Institute are stopped in a thin ferro- magnetic target. A fast-switching kicker imposes a cycle on the continuous beam, consisting of a 5 µs “beam-on” period of stopped-muon accumulation, TA, followed by a 22 µs “beam-off” measurement period, TM . The muon decay positrons are detected in a scintillator array which surrounds the target. A decay time spectrum for a sub- set of the detectors is shown in Fig. 1. The background level is indicative of the “extinction” of the beam during TM , caused by the kicker. The beamline is tuned to transport 28.8 MeV/c muons from pions that decay at rest near the surface of the production target. Two opposing 60◦ bends raise the beam from ground level to the experimental area, where it is directed parallel to the optic axis through an ~E × ~B velocity separator that removes the e+ contamination. The muons continue undeflected through the (uncharged) kicker and are focused on a 1.2-cm tall by 6.5-cm wide aperture. The activated (charged) kicker induces a 36 mrad vertical deflection, which causes the beam to http://arxiv.org/abs/0704.1981v2 Time Relative to Kicker Transition [ns] -5000 0 5000 10000 15000 20000 Background Level Measurement Period Accumulation Period Time [ct] 2000 4000 6000 8000 10000 12000 14000 16000 FIG. 1: Data from a subset of the MuLan detectors, illus- trating the accumulation and measurement periods and the background mainly caused by incomplete extinction. The fit region used for the whole data set is indicated as a thick red line. The residuals divided by their uncertainty (i.e, in stan- dard deviations) are shown in the upper inset panel. be blocked at the aperture. In 2004, the average un- kicked muon rate was limited to 2 MHz; approximately 10 muons were accumulated per cycle, of which 4 re- mained undecayed when TM began. The kicker is described in detail in Ref. [9]. Briefly, it consists of two pairs of electrode plates biased to pro- duce a potential difference of up to VK = 25 kV, with a virtual ground at the midplane. Modulators, using series-connected MOSFETs operating in push-pull mode, charge or discharge the plates. In 2004, a partial system achieved an average beam extinction of ε = 260 with a 60 ns switching time [10]. During TM , VK changed by less than 0.25 V. A time dependence of VK at this level, together with a voltage dependent extinction, gives rise to a 3.5 ppm systematic error on the muon lifetime. The parity-violating correlation between the muon’s spin orientation and the emission direction of its de- cay positron can lead to a systematic shift in the ex- tracted lifetime, for the following reasons: Suppose de- tector A at position (θ, φ) counts positrons at the rate ~Pµ) exp(−t/τµ), where ~Pµ is the polarization of the stopped muons. If ~Pµ varies with time because of relax- ation or spin rotation caused by magnetic fields, so will ~Pµ). A temporal variation that is long compared to τµ will manifest itself as an unobserved distortion to the fitted lifetime of the detector. The spin-related system- atic uncertainty in τµ is minimized through both detector symmetry and target choice: the positron detectors are arranged as a symmetric ball covering a large solid angle, and every detector A at position (θ, φ) is mirrored by an- other detector, Ā at (π − θ, φ + π). To the extent that the detector pairs have the same geometrical acceptance FIG. 2: Diagram of the experiment with several detector el- ements removed. Muons enter through the beampipe vac- uum window and traverse the EMC and a helium bag (not shown) before stopping in the AK-3 target. Decay positrons are recorded by the coincidence of inner and outer scintillators in one triangular segment; the outer scintillators are visible. Two example decay trajectories are shown. and efficiency NA+Ā ≡ NA(~Pµ)+NĀ(~Pµ) is independent of the value of ~Pµ, and therefore also its time variation. Finally, a target possessing a high internal magnetic field is used so that the muon spin precession period is ≪ τµ. As depicted in Fig. 2, the muon beam exits its vac- uum pipe through a 9.3-cm diameter, 76-µm-thick Mylar window, then passes through a thin, high-rate multiwire entrance muon chamber (EMC), which records the time and position of muons entering the detector. Roughly 1 in 104 muons stop in the EMC. Their spins precess in the field of a permanent magnet array, which has a mean transverse field of 11 mT at the EMC center. The field orientation was regularly reversed throughout data tak- ing. The region between the exit of the EMC and the target is spanned by a helium-filled balloon (instead of air) to minimize muon stops and scattering. The stopping target is a 0.5-mm thick, 50-cm diame- ter disk of ArnokromeTM III (AK-3) [11], having an in- ternal magnetic field of approximately 0.4 T, oriented transverse to the muon spin axis. The field direction was reversed at regular intervals. Dedicated µSR measure- ments [12] on an AK-3 sample show an 18-ns oscillation period with a large initial asymmetry that relaxes with a time constant of 14 ns. These times are considerably shorter than the accumulation period, TA. Using the dif- ference spectrum of counts from mirrored detectors ver- sus time; e.g., NA−Ā, the longer-term components are shown to be negligible. The decay positrons are recorded by 170 detector el- ements, each consisting of an inner and outer layer of 3-mm-thick, BC-404 plastic scintillator. Each triangle- shaped scintillator is read edge-on using a lightguide mounted at 90◦, which is coupled to a 29-mm photo- multiplier tube (PMT). The 170 elements are organized in groups of six and five to form the 20 hexagon and 10 pentagon faces of a truncated icosahedron (two pen- tagons are omitted for the beam entry and symmetric exit). The distance from target center to an inner scin- tillator is 40.5 cm. The total acceptance is 64%, taking into account the reduction in the geometrical coverage of 70% from positron range and annihilation in the target and detector materials. A clip line reshapes the natural PMT pulse width to a full-width at 20% maximum of 9 ns. These sig- nals are routed to leading-edge discriminators that have 10 ns output widths. On average, a throughgoing positron gives a signal of 70 photoelectrons, producing a 400-mV pulse height. The data taking was split al- most evenly between periods of 80-mV and 200-mV dis- criminator thresholds. The arrival time of a positron is measured with respect to the kicker transition by a CAEN V767 128-channel, multihit TDC. An Agi- lent E4400B frequency synthesizer, operating at approx- imately 190.2 MHz, serves as the master clock. Its abso- lute frequency is accurate to 10−8 and its central value did not change at this level over the course of the run. A clock step-down and distribution system provides a 23.75 MHz square wave as the input clock for each TDC. The master clock frequency was given a concealed off- set within 250 ppm from 190.2 MHz. The analyzers added a fitting offset to τµ when reporting intermedi- ate results. Only after the analysis was complete was the exact oscillator frequency revealed, and the fitting offset removed, to obtain the lifetime. The raw data consist of individual scintillator hit times for each cycle. The kicker transition defines a common (global) t = 0, using the 1.32 ns resolution provided by the 32 subdivisions of the TDC input oscillator period. To avoid problems with differential nonlinearities in the TDC clock period division circuit, coincidence windows, artificial deadtimes, and decay histogram bin widths were always set at integer multiples of the undivided input clock period (42 ns). Events are missed if a positron passes through a detec- tor during the electronic or software-imposed deadtime following a recorded event in the same detector. With peak rates in individual detectors of 7 kHz, the “pileup” probability for a 42-ns deadtime is < 3 × 10−4. If un- accounted for, this leads to a 67 ppm shift in the fit- ted τµ. Pileup can be accommodated by including an exp(−2t/τµ) term in the fit function, but this doubles the uncertainty on the fitted τµ. Instead, an artificial pileup spectrum, constructed from secondary hits occur- ring in a fixed-width time window that is offset from a primary hit, is added back to the raw spectrum, thus restoring, on average, the missing hits. The procedure is repeated using a wide range of artificial deadtime periods and offsets, and the corrected spectra all give consistent lifetimes. The systematic uncertainty from this proce- dure is 2 ppm. Preliminary fits to the decay time spectra using the function N(t) = N0 exp(−t/τµ) + B showed a common structure in the residuals at early times, independent of experimental condition or detector. The structure is caused by an intrinsic flaw in the TDC, which does not lose events but can shift them in time by ±25 ps. This behavior was characterized in extensive laboratory tests using white-noise and fixed-frequency sources, together with signals that simulate the kicker transitions. For a fit start time of tstart = 1 µs, the TDC response settles into a simple pattern that can be described well by a modification of the decay time spectrum by a factor ℑ(t′) = 1 + A cos(2πt′/T + δ) exp(−t′/τTDC), (3) where t′ = t − tstart, and with typical values of A = 5 × 10−4, τTDC = 600 ns, and T = 370 ns. A spectrum of 1011 white-noise events was fit to a constant function, modified by ℑ(t′), achieving a good χ2 and structureless residuals. The function used to fit the decay time spectra is N(t′) = ℑ(t′) · [N0 exp(−t′/τµ) + B]. (4) Because the clock frequency was blinded during the anal- ysis, T and τTDC in ℑ(t ′) could not be fixed in the fits. With the clock frequency revealed, these parameters are found to be consistent with the laboratory values. In an important test of the appropriateness of Eq. 4, the fitted τµ is found to be independent of the fit start time beyond the minimal tstart = 1.05 µs. The systematic uncertainty for the TDC response is 1 ppm. Subsequent to the 2004 run, waveform digitizers (WFDs) replaced the discriminator and TDC timing sys- tem. The WFDs establish the stability of the PMT gain versus time during TM . A gain change, together with a fixed discriminator threshold (as in 2004), will appear as a time-dependent efficiency. The analysis of the PMT gain stability over a range of instantaneous rates indi- cates a systematic effect of less than 1.8 ppm on τµ. A powerful consistency test is performed by grouping data from detectors having a common azimuthal angle φ (see Fig. 2), fitting each group independently, and sorting the lifetime results by cos θ, where θ is the polar angle. For the AK-3 data, the lifetime distribution is flat over cos θ (χ2/dof = 17.9/19). For data taken with a 20-cm diameter sulfur target, surrounded by a permanent mag- net array, the same distribution is not flat (χ2/dof = 8.0). The cause in the latter case is a higher fraction of muons that miss the target and stop downstream along inner detector walls. The SRIM program [13] finds that 0.55% of the muons miss the smaller sulfur target, while only 0.07% miss the AK-3 target. In both cases, ≈ 0.11% suf- fer large-angle scatters in the EMC or backscatter from the target, stopping in upstream detector walls. Sim- ulated decay spectra for all “errant” muons—using the muon stopping distribution predicted by SRIM and in- cluding detector element acceptance, initial muon po- larization, and relaxation—were used to determine the expected distortion to the lifetime. The procedure was tested against a special data set in which all incoming muons were stopped in a plastic plate placed midway between the EMC and the target. The distribution of lifetimes with cos θ was successfully reproduced. For the AK-3 target, a distortion as large as 1 ppm could be expected. However, a negative shift in the range of 4 − 12 ppm is implied for sulfur, principally owing to the ∼ 8 times larger fraction of downstream muon stops. The uncertainty on a correction of this magnitude could be as large as 100%, exceeding the 15 ppm statistical precision of the sulfur data sample. Therefore, we chose not to use the sulfur data in our reported results—a deci- sion made prior to unblinding the clock frequency—even though such a correction would bring the fitted sulfur life- time into excellent agreement with that found for AK-3. We conservatively assign a 2 ppm systematic uncertainty to this procedure for the AK-3 data set. Other small systematic uncertainties are listed in Ta- ble I. Data integrity checks indicate a small fraction of hits (6 × 10−6) that may be duplicates of earlier hits. When the duplicates are removed, τµ shifts upwards by 2 ppm. Since the status of these hits is uncertain, a correction of +1 ppm is applied to τµ with a systematic uncertainty of 1 ppm. A systematic error from queuing losses in the TDC single-channel buffer is calculated to be less than 0.7 ppm. The systematic error from timing shifts induced by previous hits in a channel during the measurement period is less than 0.8 ppm. The final result is based on a fit using Eq. 4 to the 1.8 × 1010 events in the summed AK-3 spectra, giving τµ(MuLan) = 2.197 013(21)(11) µs (11.0 ppm) with a χ2/dof = 452.5/484. The first error is statistical and the second is the quadrature sum of the systematic uncertainties in Table I. Figure 1 indicates the range of the fitted region and the inset displays the residuals di- vided by their uncertainty in units of standard deviations from the fit. The consistency of τµ was checked against experimental conditions, including detector, threshold, target and magnet orientation, extinction factor, kicker voltage, and run number. Only a sub-group of runs at the beginning of the production period exhibited an anoma- lous lifetime compared to the sum. When all runs, in groups of 10, are fit to a constant, a χ2/dof = 108/102 is obtained, suggestive that the sub-group fluctuation was statistical. Our fitted lifetime is in excellent agreement with the world average, 2.197 03(4) µs, which is based on four pre- TABLE I: Systematic uncertainties. Source Size (ppm) Extinction stability 3.5 Deadtime correction 2.0 TDC response 1.0 Gain stability 1.8 Errant muon stops 2.0 Duplicate words (+1 ppm shift) 1.0 Queuing loss 0.7 Multiple hit timing shifts 0.8 Total 5.2 vious measurements [8]. The improved world average is τµ(W.A.) = 2.197 019(21) µs (9.6 ppm). Assuming the standard model value of the Michel pa- rameter η = 0, and light neutrinos, determines the Fermi constant GF = 1.166 371(6)× 10−5 GeV−2 (5 ppm). In a companion Letter [14], a new determination of the induced pseudoscalar coupling gP is reported. It depends mainly on a comparison of negative and positive muon lifetimes, the latter quantity being reported here. We acknowledge the generous support from PSI and the assistance of its accelerator and detector groups. We thank W. Bertl, J. Blackburn, K. Gabathuler, K. Dieters, J. Doornbos, J. Egger, W.J. Marciano, D. Renker, U. Rohrer, R. Scheuermann, R.G. Stuart and E. Thorsland for discussions and assistance. This work was supported in part by the U.S. Department of Energy, the U.S. National Science Foundation, and the John Simon Guggenheim Foundation (DWH). [1] G.Gabrielse, D.Hanneke, T.Kinoshita, M.Nio, and B.Odom, Phys. Rev. Lett. 97, 030802 (2006). [2] S. Schael et al., Phys. Rept. 427, 257 (2006). [3] T. van Ritbergen and R.G. Stuart, Nucl. Phys. B564, 343 (2000); T. van Ritbergen and R.G. Stuart, Phys. Lett. B437, 201 (1998); T. van Ritbergen and R.G. Stuart, Phys. Rev. Lett. 82, 488 (1999). [4] M.Awramik, M. Czakon, A. Freitas, and G.Weiglein, Phys. Rev. D69, 053006 (2004). [5] W.J. Marciano, Phys. Rev. D60, 093006 (1999). [6] S.M. Berman and A. Sirlin, Ann. Phys. 20, 20 (1962). [7] S.M. Berman, Phys. Rev. 112, 267 (1958); T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959). [8] G. Bardin et al., Phys. Lett. B137, 135 (1984); K. Gio- vanetti et al., Phys. Rev.D29, 343 (1984); M.P. Balandin et al., Sov. Phys. JETP 40, 811 (1974); and J. Duclos et al., Phys. Lett. B47, 491 (1973). [9] M.J. Barnes and G.D.Wait, IEEE Trans. Plasma Sci. 32, 1932 (2004); R.B. Armenta, M.J. Barnes, G.D. Wait, Proc. of 15th IEEE Int. Pulsed Power Conf., June 13- 17 2005, Monterey, USA. [10] Using the full system (four modulators instead of two), ε = 800 is achieved with a 45 ns switching time. [11] ArnokromeTM III (AK-3) is an alloy of ≈ 30% Cr, ≈ 10% Co and ≈ 60% Fe. Arnold Engineering Co., Alnico Prod- ucts Division, 300 N. West Street, Marengo, IL 60152. [12] E. Morenzoni and H. Luetkens, private communication. [13] J.F. Ziegler, J.P. Biersack and U.Littmark, The Stopping and Range of Ions in Matter, Pergamon Press, New York, (2003). [14] MuCap Collaboration: V.A. Andreev et al., This volume.

0704.1983

Universal 2+1-Dimensional Plane Equations in General Relativity and Evolutions of Disk Nebula

Microsoft Word - EvoluGRG.doc Universal 2+1-Dimensional Plane Equations in General Relativity and Evolutions of Disk Nebula Yi-Fang Chang Department of Physics, Yunnan University, Kunming, 650091, China (e-mail: [email protected]) ABSTRACT The general relativity is the base for any exact evolutionary theory of large scale structures. We calculate the universal 2+1-dimensional plane equations of gravitational field in general relativity. Based on the equations, the evolutions of disk nebula are discussed. A system of nebula can form binary stars or single star for different conditions. While any simplified linear theory forms only a single star system. It is proved that the nonlinear interactions are very general, so the binary stars are also common. Key words: general relativity, evolution, disk nebula, binary stars. PACS: 04.20. q; 97.10. Cv; 98.38. Ly; 97.80. d. At the present the general relativity with a precise formulation of the theory is the best astronomic theory for various large scale structures of space-time [1,2]. Since the general solutions of Einstein ¯s fi el d equati ons are ver y co mpl ex, so far,some exact solutions [1,2,3], including from the Schwarzschild solution, Eddington-Robertson solution, and various Kerr solutions to the gravitational lensing in metric theories of gravity [4], spherically symmetric space-times in massive gravity [5], and all static circularly symmetric perfect fluid solutions of 2+1 gravity [6]. They are mostly some static solutions. On the other hand, the evolutions of various celestial bodies exist widely in the universe. In particular, astronomers observed that binary star systems are very common from 1989 [7,8]. The generality of binary star is proposed. Fact, Spyrou presented the results of a systematic study of relativistic celestial mechanics of binary stars in the post-Newtonian approximation (PNA) of general relativity [9], and proposed star models determined the inertial and rest masses of binary stars [10]. Itoh, et al., discussed the equation of motion for relativistic compact binaries with the strong field [11]. Alvi and Liu studied the dynamics of a cluster of collisionless particles orbiting a non-rotating black hole, which is part of a widely separated circular binary, and found that the most stable orbits are close to the companion's orbital plane and retrograde with respect to the companion's orbit [12]. Gu¨¦ro and Letelier studied binary systems around a black hole [13]. Hansen discussed the motion of the binary system composed of an oscillating and rotating coplanar dusty disk and a point-like object [14]. The most exact celestial evolutionary theory should be based on the general relativity. Wilson and Mathews [15,16] reported preliminary results obtained with a relativistic numerical evolution code. Their dynamical calculations suggest that the neutron stars may collapse to black holes prior to merger. Baumgarte, et al., studied the quasiequilibrium model on binary neutron stars in general relativity [17]. Then they presented a new numerical method for the construction of quasiequilibrium formulation of black hole-neutron star binaries, and solved the constraint equations of general relativity, and solved these coupled equations in the background metric of a Kerr-Schild black hole, which accounts for the neutron star's black hole companion [18]. Taniguchi, et al., presented new sequences of general relativistic, quasiequilibrium black hole-neutron star binaries. They solved the gravitational field equations coupled to the equations of relativistic hydrostatic equilibrium for a perfect fluid [19]. Shibata, et al., presented a new implementation for magnetohydrodynamics (MHD) simulations in full general relativity (involving dynamical space-times), and performed numerical simulations for standard test problems in relativistic MHD, including special relativistic magnetized shocks, general relativistic magnetized Bondi ow in stationary space-time, and a long-term evolution for self-gravitating system composed of a neutron star and a magnetized disk in full general relativity [20]. Faber, et al., calculated the dynamical evolutions of merging black hole-neutron star binaries that construct the combined black hole-neutron star space-time in a general relativistic framework. They treated the metric in the conformal flatness approximation, and assumed that the black hole mass is sufficiently large compared to that of the neutron star so that the black hole remains fixed in space [21]. One of more successful theories of the formation of binary stars is the fragmentation proposed by Boss, et al.[22-24], which supposes that binary stars are born during the protostellar collapse phase under their own gravity. Using computer simulation they obtained that an initial spherical cloud in rapidly rotation collapses and flattens to a disk, which later fragments into a binary system. Based on the basic equations of a rotating disk on the nebula, we use the qualitative analysis theory of nonlinear equation and obtain a nonlinear dynamical model of formation of binary stars [25]. Under certain conditions a pair of singular points results in the course of evolution, which corresponds to the binary stars. Under other conditions these equations give a single central point, which corresponds to a single star. This new method and model may be extended and developed. Steinitz and Farbiash established the correlation between the spins (rotational velocities) in binaries, and show that the degree of spin correlation is independent of the components separation. Such a result might be related for example to the nonlinear model for the formation of binary stars from a nebula [26]. So far, the general relativity is mainly applied to neutron star-neutron star binaries, or black hole-neutron star binaries. Wilson, et al., chose the 3-metric to be conformally flat [15,16]. In Cartesian coordinates the line element can be written [17] ))((4222 dtdxdtdxdtds jjiiij . (1) In more general cases, for a flatten disk nebula we take a universal 2+1-dimensional plane metric into diagonal form with 2+1-paramers 2 ),,( drtfdredxedxdxgds vkiik . (2) where ),,( rtvv and ),,( rt . Denoting by 210 ,, xxx , respectively, the (2+1)-dimensional plane coordinates ct, r, , we have for the nonzero components of the metric tensor the expressions [27]: ),,(,, 221100 rtfgegeg , (3) and 1221100 ,, fgegeg v . (4) With these values the calculation leads to the following Christoffel symbols: 00 vvev (5a) v (5b) 11 fef (5c) f (5d) 11 fef . (5e) Here the prime means differentiation with respect to r, while a dot on a symbol means differentiation with respect to ct. All other components are zero. Therefore, we may derive the components of the field equations: ''''2[ (6a) (6b) ])'(''''2[ vvveveG v (6c) (7a) (7b) (7c) These equations may possess various solutions. The energy-momentum tensor of perfect fluid is [3] abbaab pguupT )( . (8) Assume that b 0 and d 0 , so 1 vdf , (9a) )'''''2('' ''''2[ 2 vvvdvfb , (9b) )(2[] f (9c) Here bbbb 321 and dddd 321 . For the evolutions of disk nebula, we consider a system only with time. Let yffdtfdx //)(ln, , the equations (9a) become xxvxy , (10a) yyvxy , (10b) From dx/dt=0 and dy/dt=0 we derive four solutions: (),0,(),,0(),0,0(),( vvyx . (11) Using a qualitative analysis theory of the nonlinear equations, the characteristic matrix of the equations (10) is 2/2/ 2/ 2/2/2/ yvbxby dxxvdy . (12) Its characteristic equation is 02 DT . (13) While DT 42 . For a singular point (0,0), 0,04/, 1 11 vDvT . (14) It is a critical nodal point. For a singular point (0, v ), 1(,0) T . (15) It is a saddle point. For a singular point ( v ,0), 1(,0) T . (16) It is also a saddle point. For a singular point ) .])()1(4[ )1)(1( 11112 vdbdb . (17) In the usual case it is a nodal point. When 11db >1, i.e., T<0, the point is a stable sink, this system may form the binary stars. When 11db <1, i.e., T>0 and D<0, the point is a saddle point, this system forms a single star for (0,0) singularity. They seem to correspond to thin nebula and dense nebula, respectively. When the singularity is a nodal point, ffyx /, may be represented by )0(, 11 khbeyaex . So 0ln,0 11 ktht ebydtfeaxdt is also a nodal point. The equations (6) are simplified to (18a) 2 aTf (18b) 2 ]2[ aTve v . (18c) Then 04 aTexy , and the equations (18b) and (18c) become aTexxv , (19a) aTeyyv , (19b) From dx/dt=0 and dy/dt=0 we derive solutions: 2,1 XvaTevvx , (20a) 2,1 YvaTevvy . (20b) Using a qualitative analysis theory, the characteristic matrix of the equations (19) is 6 2/ 0 . (21) Its characteristic equation is (13). In this case, for a singular point ),( 11 yx , 111 YXXYDYXT . (22a) It is a nodal point, and is a stable sink. For a singular point ),( 21 yx , 222 YXXYDYXT . (22b) It is a saddle point. For a singular point ),( 12 yx , 333 YXXYDYXT . (22c) It is also a saddle point. For a singular point ),( 22 yx , 444 YXXYDYXT . (22d) It is a nodal point, and is an unstable source. For the linear equations of simplified (10) or (19), only singularity (0,0) is a nodal point, and forms only a single star system. Based on the universal 2+1-dimensional plane equations of gravitational field in general relativity, the evolutions of disk nebula are discussed. A system of nebula can form binary stars or single star for different conditions. Here the nonlinear interactions play an important role. They are necessary conditions of the formation of binary stars, but are not sufficient conditions. Generally, any stable star should be a stable fixed point in the evolutionary process in astrophysics. Assume that an evolutionary equation is y=f(x), which corresponds to an equation of the invariant point x*=f(x*). Therefore, if f(x) is a nonlinear n-order function, there will be possibly multiple (stable or unstable) invariant points. While f(x)=ax+b is a linear function, there will be only an invariant point x*=b/(1-a), namely, it can form only a single star. Since the nonlinear interactions are very general, the binary and multiple stars are also common. References 1.Hawking, S.W. & Ellis, G.F.R., 1973. The Large Scale Structure of Space-Time. Cambridge University Press. 2. Misner, C.W., Thorne. K.S. & Wheeler, J.A., 1973. Gravitation. W.H.Freeman and Company. San Francisco. 3.Kramer, D., Stephani, H., Herlt, E., & MacCallum, M., 1980. Exact Solutions of Einstein ¯s Fi el d Equations. Cambridge University Press. Cambridge. 4.Sereno, M., 2003. Phys.Rev. D67,064007. 5.Damour, T., Kogan, I.I., & Papazoglou, A., 2003. Phys.Rev. D67,064009. 6.Garcia, A.A., & Campuzano, C., 2003. Phys.Rev. D67,064014. 7.Leiner, C., & Haas, M., 1989. ApJ.Lett.342,L39. 8.Duquennoy, M., & Mayor, M., 1991. A.Ap.248,485. 9.Spyrou, N., 1981. Gen.Rel.Grav., 13,473. 10.Spyrou, N., 1981. Gen.Rel.Grav., 13,487. 11.Itoh, Y., Futamase, T., & Asada, S., 2000. Phys.Rev., D62,064002. 12.Alvi, K. & Liu, Y.T., 2002. Gen.Rel.Grav., 34,1067. 13.Gu¨¦ro, E. & Letelier, P.S., 2004. Gen.Rel.Grav., 36,2107. 14.Hansen, D., 2005. Gen.Rel.Grav., 37,1781. 15.Wilson, J. R. & Mathews, G. J., 1995, Phys.Rev.Lett. 75, 4161. 16.Wilson, J. R., Mathews, G. J. & Marronetti, P., 1996, Phys.Rev. D 54, 1317. 17.Baumgarte, T.W., G.B.Cook, M.A.Scheel, et al., 1997, Phys.Rev.Lett., 79,1182. 18.Baumgarte,T.W., Skoge,M.L.& Shapiro,S.L., 2004, Phys.Rev. D 70, 064040. 19.Taniguchi,K., Baumgarte,T.W., Faber,J.A. & Shapiro,S.L., 2005, Phys.Rev. D72, 044008. 20.Shibata,M. & Sekiguchi,Y., 2005, Phy.Rev. D72,044014. 21.Faber,J.A., Baumgarte, T.W., Shapiro,S.L., et al., 2006, Phys.Rev. D73, 024012. 22.Boss, A.P., 1991. Nature, 351,298; 1993. ApJ., 410,157; 1995. ApJ., 439,224. 23.Boss, A.P., & Yorke, H.W., 1993. ApJ.Lett., 411,L99; 1995. ApJ.Lett., 439,L55. 24.Boss, A.P., & Myhill, E.A., 1995. ApJ,439,224;451,218. 25.Zhang, Y.F., 2000. Chinese Aston.Astophys. 24,269. 26.Steinitz, R., & Farbiash, N., 2003. Spectroscopically and Spatially Resolving the Components of close Binary Stars. ASP Conference Series. in Dubrovnik, Croatia, Oct.20-24, 2003. 27. Landau, L.D., & Lifshitz, E.M., 1975. The Classical Theory of Field. Pergamon Press. Oxford.

0704.1985

Electromagnetic Higgs production

arXiv:0704.1985v2 [hep-ph] 13 Nov 2007 Preprint typeset in JHEP style - HYPER VERSION December 8, 2021 Electromagnetic Higgs production J. Miller Department of Particle Physics, School of Physics and Astronomy Raymond and Beverley Sackler Faculty of Exact Science Tel Aviv University, Tel Aviv, 69978, Israel Abstract: The cross section for central diffractive Higgs production is calculated, for the LHC range of energies. The graphs for the possible mechanisms for Higgs production, through pomeron fusion and photon fusions are calculated for all possibilities allowed by the standard model. The cross section for central diffractive Higgs production through pomeron fusion, must be multiplied by a factor for the survival probability, to isolate the Higgs signal and reduce the background. Due to the small value of the survival probability 4 × 10−3 , the cross sections for central diffractive Higgs production, in the two cases for pomeron fusion and photon fusion, are competitive. Keywords: Higgs boson, BFKL pomeron, diffractive production, LHC, electromagnetic Higgs production, photon photon fusion, pomeron pomeron fusion, survival probability, standard model. Email: [email protected]; http://arxiv.org/abs/0704.1985v2 http://jhep.sissa.it/stdsearch http://jhep.sissa.it/stdsearch Contents 1. Introduction 1 2. Double diffractive Higgs production at the LHC 4 3. Electromagnetic Higgs production at the LHC 8 3.1 The fermion triangle subprocess for Higgs production in γγ fusion 9 3.2 Boson loop sub-processes for Higgs production in γγ fusion 12 3.3 The cross section for central exclusive Higgs production through γγ fusion 15 4. Conclusion 15 5. Acknowledgements 15 A. Appendix 16 A-1 Evaluation of the integral over the anomalous dimensions γ1 and γ2 of the momentum Q in the gluon density function 16 A-2 Feynman rules for the standard electro-weak theory 19 A-3 Evaluation of the integral over the momentum in the fermion triangle loop 19 1. Introduction The most promising process for observation of the Higgs boson at the LHC is central diffractive production of the Higgs, with large rapidity gaps ( LRG ) between the Higgs and the two emerging protons, after scattering. Namely, p + p → p + [LRG ] + H + [LRG ] + p (1.1) The large rapidity gaps either side of the Higgs reduces the background, so that the Higgs signal will be easier to isolate in central exclusive production. Hence, this process gives the best experimental signature – 1 – for detecting the Higgs at the LHC, and is very interesting for experiments in the search for the Higgs boson. In this paper, two mechanisms are compared for central exclusive Higgs production; (1) γ γ fusion, namely pp → γ γ → H and (2) pomeron exchange, namely pp → IP IP → H, where IP denotes a pomeron. For γ γ fusion, there is no hard re-scattering of the photons to fill up the rapidity gaps, so that large rapidity gaps are automatically present. The motivation for considering IP IP → H, is the observation that for gluon gluon fusion, namely the process gg → H the colour flow induces many secondary parton showers which fill up the rapidity gaps. Instead the process IP IP → H is a colour singlet exchange, where the colour flow is screened, and the large rapidity gaps are preserved. However in the case of IPIP fusion, there are hard re-scattering corrections, giving additional inelastic scattering, which will give emission filling up the rapidity gaps (see ref. [1]). To guarantee the presence of large rapidity gaps after scattering in central exclusive production, in the case of IPIP fusion, one has to multiply by the survival probability. This is the probability that large rapidity gaps, between the Higgs boson and the emerging protons, will be present after scattering. The motive of this paper, was driven by the potentially small value for the survival probability, namely < |S2| >= 4 × 10−3, calculated in ref. [1] for central exclusive Higgs production via IPIP fusion. This is an order of magnitude less than previous estimates, namely < |S2| >= 0.02 for LHC energies (see ref. [2]). Hence, previous calculations in ref. [2] of the cross section for central exclusive Higgs production via IPIP fusion, which included a factor for the survival probability of < |S2| >= 0.02, gave for the exclusive cross section σexcIP IP ( pp→ p+H + P ) = 3 fb. Since the survival probability is predicted in ref. [1] to be an order of magnitude less, it follows that the cross section σexcIP IP ( pp→ p+H + P ) for central exclusive Higgs production will also be an order of magnitude smaller. It also follows that this cross section will be competitive with the cross section for central exclusive Higgs production via γ γ fusion, which is predicted in ref. [2] to be σexcγγ ( p+ p→ p+H + p) = 0.1fb. To illustrate that for the process of Eq. (1.1), the cross sections will be competitive for γγ fusion and IPIP fusion, it is instructive to consider the diagrams for these processes. The notation used for the couplings in the standard model are the following. – 2 – coupling constant expression value ( GeV / c2 ) 1.17 × 10−5 v gw 80 2λ v2 120 ) g2s(M2H) where the mass of the Higgs boson, is derived from the vacuum expectation value of the Higgs SU (2) weak isodoublet, which is , where v = , and µ and λ are parametres in the Higgs potential, which is introduced into the standard model in the spontaneous symmetry breaking of SUL ( 2) × UY ( 1) → UEM ( 1) , which is responsible for giving rise to the W and Z boson masses. In the case for γγ fusion shown in Fig. 3, there are four vertices proportional to αem coupling the photons to the two protons and the photons either side of the subprocess for γγ → H. A factor of g2w = 4 2GF m should also be included, to account for the weak coupling of the Higgs to the sub-process, depicted by the shaded area in Fig. 3. So it is expected that the cross section will be proportional to σexcγγ ( p+ p→ p+H + P ) ∝ 4 2GF m em = 0.6 fb (1.2) where units are defined as 1GeV−2 = 0.3893 mb . In central exclusive Higgs production for the case for γγ fusion, all the couplings of the photons shown in Fig. 3 are known constants, namely they are proportional to αem. Hence, the cross section for this diagram can be calculated exactly. On the other hand, when considering central exclusive Higgs production for the case for IPIP fusion shown in Fig. 1, the couplings of the gluons are not constants. In Fig. 1 there are four gluon couplings with the protons, giving a contribution to the cross section proportional to α4s , where Q2 is the momentum transferred along the pomeron, and it is assumed that αs ∼ 0.2. There are also two gluon couplings with the subprocess for IPIP → H, giving a contribution to the cross section proportional to α2s . Taking the mass of the Higgs to be MH ∼ 100GeV , then it is expected that αs ∼ 0.12. Also in the case of IPIP fusion, a factor of g2w = 4 2GF m w should also be included, to account for the weak coupling of the Higgs to the sub-process shown in Fig. 1. For IPIP fusion, this subprocess is the quark triangle subprocess shown in Fig. 2. Since the gluon itself couples weakly to the Higgs boson, only the contribution of the quark triangle subprocess is taken into account. The amplitude for the quark triangle subprocess, is – 3 – derived in section 3.1 for the electromagnetic case, where αs replaces αem. After multiplying by a factor for the survival probability, which includes the survival probability < |S2 | >= 4× 10−3 which takes into account hard re-scattering of the pomeron, and < |S2 | >= 5× 10−2 which takes into account soft re-scattering of the pomeron, then one obtains for the exclusive cross section for central exclusive Higgs production, in the case of IPIP fusion the value σexcIP IP ( p+ p→ p+H + p) ∝ 4 2GF m < |S2hard| >< |S soft| >= 0.9 fb (1.3) Comparing the estimates of Eq. (1.2) and Eq. (1.3), it is expected that σexcγγ ( p+ p→ p+H + p) and σexcIP IP ( p+ p→ p+H + p) will be competitive. This is the motivation for this paper which re-examines σexcγγ ( p+ p→ p+H + p) for electromagnetic Higgs production. This paper is organised in the following way. In section 2, the details of the calculation of the cross section for central exclusive Higgs production in the case for IPIP fusion is explained in detail. The cross section which is obtained, is multiplied by the factor for the survival probability in ref. [1], to give the exclusive cross section. In section 3, the cross section for central exclusive Higgs production, in the case of γγ fusion is calculated. The mechanism γγ → H proceeds via the fermion triangle and Boson loop sub-processes illustrated in Fig. 4 and Fig. 5. The total cross section for central exclusive Higgs production, in the case of γγ fusion is calculated by taking the sum over all the contributions for these sub-processes for the γγ → H mechanism. Finally, in the conclusion the results found in section 2 and section 3, are compared. All calculations in this paper, are based on the Feynman rules for the standard electro-weak theory, given in Fig. 6, which is to be found in the appendix. 2. Double diffractive Higgs production at the LHC In this section, an explicit expression is derived for the Born amplitude for double diffractive Higgs pro- duction (see Fig. 1). The Higgs couples weakly to the gluon, so the main contribution comes from the quark triangle subprocess (see Fig. 2), and all six flavours of quarks are taken into account. The second t channel gluon in Fig. 1 is included. This is because of the large rapidity (LRG) gap between the Higgs and the proton, which demands that in the t channel, one has colorless exchange. Indeed, if the LRG wasn’t present between the protons, then the Higgs could simply be produced from gluon gluon fusion in a single channel. However, the colour flow induced by a single channel exchange process, could produce many secondary particles. These secondary particles could fill up the LRG. To screen the colour flow, it is necessary to exchange a second t channel gluon. At lowest order in αs, this gluon couples only to the incoming quark lines. The Born amplitude for double diffractive Higgs production by gluon exchange, is given by the expression [3, 4] – 4 – Bremstrahlung gluon Higgs t−channel screening gluon Q BFKL gluons BFKL gluons Figure 1: Double diffractive Higgs production in the Born approximation A~k1 · ~k2 ~k1T · ~k2T ~k1T · ~k2T (2.1) where ~k1 ·~k2 = has been used. In Eq. (2.1), the Weizsäcker - Williams approach, explained in ref. [5] has been used. The factor A~k1 · ~k2 is the amplitude for the quark triangle subprocess of Fig. 2, where A takes the value, [6, 7, 8, 9]  (2.2) The Born amplitude shown in Fig. 1, is extended to proton scattering instead of just quark scattering here. The typical momentum transferred, t1 = ( q1 −Q) 2 and t2 = ( q2 −Q) 2, are rather small and of the order of 1 , where b is the slope of the gluon - proton form factor, and can be estimated to be b = 5.5GeV 2. Therefore, one takes into account the proton couplings to the gluon ladder (see Fig. 1), by including the proton form factors in the gaussian form bt1 − where t1 = ( q1 −Q)2 and t2 = ( q2 −Q)2 (2.3) – 5 – Higgs Figure 2: Quark triangle subprocess for Higgs production in gluon gluon fusion to describe the dependence on the transferred momentum. If the momentum transfer is small, it can be assumed that k1 ∼ k2 ∼ Q. Therefore, with the definition of Eq. (2.3), t1 → 0 and t2 → 0. Hence, the proton form factors e− bt1 and e− bt2 tend to 1, and can be taken outside the integral, and the Born amplitude behaves as bt1e− (2.4) To consider the exclusive process only, with the condition of the LRG, bremsstrahlung gluons must be suppressed. The bremsstrahlung gluons are shown by the dashed lines in Fig. 1, which are suppressed by multiplying by the Sudakov form factor Fs = e −S(k2 ) (2.5) Fs is the probability not to emit bremsstrahlung gluons. S is the mean multiplicity of Bremsstrahlung gluons given as k2⊥, E (2.6) Secondly, evolution of BFKL ladder gluons between the two channels (see Fig. 1), must be taken into account. For proton scattering instead of quark scattering, the naive coupling of the gluons to the external – 6 – quarks must be replaced by a coupling of the gluons to the external proton lines. To include both of these modifications, the naive gluon density for quarks is replaced by the density for protons by the following substitution (2.7) where f x, k2 is the un-integrated gluon density of the proton. After including the Sudakov form factor of Eq. (2.1), and the gluon density function of Eq. (2.7), the amplitude in Eq. (2.1) becomes MIP IP (p+p → p+H+p) = Aπ3s e−S(k x1, Q x2, Q (2.8) where for the gluon densities f (x1,2) = 2 γ1,2e ω( γ1,2) ln (2.9) where γ1,2 are the anomalous dimensions and the numerical coefficient 2 in Eq. (2.9) can be taken from MRST-2002-NLO parameterizations. Using Eq. (2.9), the integration over Q⊥ and over γ1 and γ2 can be evaluated. It turns out that the integrand of Eq. (2.8) has a saddle point given by ln Q2⊥ = ( γ1 + γ2 − 1) , and the essential values of γ1 and γ2 are close to 12 . Hence, the typical Q⊥ is rather large, and depends on the mass of the Higgs. After integrating over γ1 and γ2 and q⊥, the final result for the amplitude MIP IP ( p+p → p+H+p) is derived in the appendix (see section A-1), and the final result is given in Eq. (A-1-16) as MIP IP (p+p → p+H+p) =− 2Aπ4s ω ”( 12) ln s1 ω ”( 12) ln s1 + ω ” ln s1 ) (2.10) Now that the amplitude is known, σIP IP ( p+p → p+H+p) for central Higgs production can be cal- culated for the case of IPIP fusion. To derive the cross section for exclusive central Higgs production, one has to multiply by a factor which takes into account the survival probability for large rapidity gaps < |S2 | > = 0.004, to suppress hard re-scattering. Therefore, the cross section σexcIP IP ( p+p → p+H+p) , for central exclusive Higgs production, without any further hard re-scattering for the case of IPIP fusion, takes the value < |S2hard| > σIP IP ( p+p → p+H+p) = σ IP IP ( p+p → p+H+p) = 0.47 fb (2.11) – 7 – 3. Electromagnetic Higgs production at the LHC In the case of central exclusive Higgs production for the case of γγ fusion, shown in Fig. 3, there is no hard re-scattering to take into account, and all the couplings are known precisely. The shaded area in Fig. 3 depicts the subprocess for the mechanism γγ → H. The possible mechanisms are illustrated in Fig. 4 and Fig. 5, and their contributions to the amplitude are calculated in this section. Gauge invariance requires that the contribution of a subprocess for the mechanism γγ → H, takes the form photon photon Higgs + qq= Figure 3: Diffractive Higgs production in single channel photon exchange. Aµν = A (3.1) where A is a constant, depending on the particular subprocess. It turns out that for the case of when the subprocess is the fermion triangle shown in Fig. 4, summed over all six flavours of quarks, and all three lepton flavours, then the expression for the amplitude takes the form of Eq. (3.1). However in the case of when the subprocess is one of the Boson loops shown in Fig. 5, the expression for the amplitude is not of the form of Eq. (3.1). The correct statement is that the sum over all the amplitudes for the sub-processes shown in Fig. 5, gives a gauge invariant expression of the form of Eq. (3.1). The amplitude for the diagram of Fig. 3, where the process γγ → H proceeds via the sum over the fermion triangle shown in Fig. 4, and Boson loops shown in Fig. 5, is given by Mγγ ( p+p → p+H+p) = − 4παem Afµν +A (3.2) where A µν = Af denotes the amplitude of the quark/anti-quark triangle subpro- cess, summed over all six quark flavours/anti-quark flavours (q, /q̄ = u/ū, d/d̄, s/s̄, c/c̄, t/t̄, b/b̄) and all – 8 – three lepton/anti-lepton flavours (L± = e±, µ±, τ±), shown in Fig. 4, and Abµν = Ab denotes the amplitude of the sum over all the Boson loop sub-processes shown in Fig. 5. At this point, the Weizsacker - Williams formula is used, as explained in refs. [3, 5, 4]. In this approach, the substitution 2Aµν = − 2SM2 2⊥Aµν is used. In the notation used in this paper, q 1⊥ and q 2⊥ denotes two dimen- sional vectors, in the plane transverse to the direction of the momenta of the two incoming protons p 1 and pν2 . Hence, the amplitude of Eq. (3.2) can be written as Mγγ ( p+p → p+H+p) = − 4παem Afµν +A (3.3) In order to calculate the cross section, one has to integrate the squared amplitude over all the transverse momenta q1⊥ and q2⊥. For central exclusive Higgs production in the case of γγ fusion, it is required that the lower limits of integration are qmin1⊥ qmin2⊥ = m2p , where mp is the proton mass, which is assumed in this paper to be 1 GeV. The upper limits of integration, are taken from the electromagnetic form factors for the proton, namely Gp ” , from which the upper limits of the integration are derived to be ( qmax1⊥ ) = ( qmax1⊥ ) = 0.72. σγγ ( p+p → p+H+p) = (Af +Ab)  (3.4) where y = ln s is the rapidity gap between the two incoming protons in Fig. 3, and m is the proton mass, assumed to be 1 GeV. This calculation is for central exclusive Higgs production at the LHC, where it is expected that s = 14000GeV, which gives for the value of the rapidity gap y = 19. 3.1 The fermion triangle subprocess for Higgs production in γγ fusion Central exclusive Higgs production for the case of γγ fusion, can proceed through the subprocess γγ → fermion triangle → H shown in Fig. 4, where the fermions include the six flavours of quarks and anti quarks (u , d , s , c , t , b) and the three lepton and anti lepton flavours (e , µ , τ ). A derivation of the amplitude for the fermion triangle, can be found in refs. [6, 9, 10] for the case where the mass of the fermion in the triangle is much larger than the Higgs mass. In this section, the amplitude of the fermion triangle is derived by taking the sum over all the fermions which could contribute, including the six quark flavours and the three lepton flavours. The H → f f vertex coupling, is proportional to the mass mf of the fermion at the vertex, so that the sub-process amplitude of the fermion triangle will be proportional to the mass of the fermion in the triangle. The fermion masses are assumed to take the following values listed in the table below. – 9 – fermion mass ( GeV / c2 ) quarks u 3 × 10−3 d 6 × 10−3 s 1.3 c 0.1 t 175 b 4.3 fermions e 5.11 × 10−4 µ 0.106 τ 1.7771 Hence, these values indicate that the most significant contribution to the amplitude will come from the top quark triangle. Higgs pH pH Figure 4: Fermion triangle subprocess for Higgs production through γγ fusion To derive explicitly the amplitude of the subprocess Fig. 4, the labeling of momenta shown in the – 10 – diagram is used, and the following notation is introduced. D1 = l 2 −m2f D2 = (l − q1) 2 −m2f D3 = (l + q2) 2 −m2f (3.1.1) where mf denotes the mass of the fermion which forms the triangle, which could be one of the quark flavours or one of the lepton flavours. Then the amplitude for the fermion triangle subprocess, summed over all possibilities of quark and anti-quark flavours, and lepton and anti-lepton flavours takes the form Aµνq = 4παem D1D2D3 where L= e , µ , τ (3.1.2) where q denotes the sum over all six quark flavours q = u , d , s , c , t , b and L= e , µ , τ denotes the sum over all three lepton flavours L = e , µ , τ . On the RHS of Eq. (3.1.2), d is the space-time dimension, and at the end f the calculation, d → 4 is imposed. The reason for not specifying this in the beginning, is because it will be necessary to use dimensional regularisation to cancel divergences, which requires integration over d + ǫ dimensions, in the limit that d → 4 and ǫ → 0. Using the notation shown in the diagram of Fig. 4 for the flow of momenta in the quark and anti - quark triangles, the trace term is given by lx + q x +mf lσ +mfγ lτ − q1τ +mf −lx + q1x +mf −lσ +mfγµγτ −lτ − q2τ +mf = 8mf 2 + 4l µlν + 2 (lµqν1 − lνq 2 )− g ~q1 · ~q2 + l2 −m2f (3.1.3) where ~q1 and ~q2 denotes four dimensional vectors. The first line on the RHS of Eq. (3.1.3) corresponds to the contribution given by the triangle formed by the fermions, ( i.e. quarks q = (u , d , s , c , t , b ) and negatively charged leptons L− = ( e− , µ− , τ−) , and the second line corresponds to the contribution given by the triangle formed by the anti - fermions ( i.e. anti - quarks q̄ = (ū , d̄ , s̄ , c̄ , t̄ , b̄ ) and posi- tively charged leptons L+ = ( e+ , µ+ , τ+) ). Introducing Feynman parametres to rewrite the quotient (D1D2D3) in a more convenient form, Eq. (3.1.2) simplifies to 4παem dd l̃ ∫ 1−x 8mf I l̃2 −∆f ( x, y) where ∆f = m f − M2H x y and l̃µ = lµ − x qν1 + y q 2 (3.1.4) Note in the last step, it was assumed that q21 ≪ q22 ≪ M2H ,m2f so that these terms can be ignored. From the kinematics shown in the diagram of Fig. 3, and Fig. 4, it is clear that 2~q1 · ~q2 = M2H . The trace – 11 – term I was given in Eq. (3.1.3) in terms of the unknown momentum lµ in the fermion triangle in Fig. 4. In terms of the new variable l̃µ, the trace term takes the form 2 + 4l̃ µ l̃ν − 4qν1q 2x y − g ~q1 · ~q2 (1− 2x y)−m2f + l̃2 (3.1.5) The details of the integration over the momentum l̃ on the RHS of Eq. (3.1.4) are given in section A-3 of the appendix, (see Eq. (A-3-1) - Eq. (A-3-4)). Here, dimensional regularisation is used, a technique where one integrates over d + ǫ dimensions, and afterwards d → 4 and ǫ → 0. This removes non gauge invariant terms, in the numerator of the integrand on the RHS of Eq. (3.1.4). In this way, one obtains the following gauge invariant result for the RHS of Eq. (3.1.4). 2αemG If (3.1.6) where f =u , d , s , c , t , b , e , µ , τ ∫ 1−x 1− 4xy ∆f ( x, y) where ∆f ( x, y) = m f −M2Hxy (3.1.7) The integral If is evaluated in section A-3 of the appendix (see Eq. (A-3-5) - Eq. (A-3-13) ). It turns out that, due to the dependence of the factor for the H → f f vertex coupling on the mass of the fermion mf , that the only fermion triangle which gives a significant contribution to the amplitude is for the case where mf ≫ MH . From the table, this is true only for the top quark mt. Hence, it turns out that the only fermion triangle that is necessary to take into, is the top quark triangle, and the contributions from the rest of the triangle sub-processes formed by the rest of the quarks, and the leptons can be neglected. Using this result, the amplitude for the contribution of the fermion triangle has the expression = Af (q 2 − g µν(~q1 · ~q2)) where Af = − (3.1.8) 3.2 Boson loop sub-processes for Higgs production in γγ fusion For central exclusive Higgs production, the Higgs can also be produced through the subprocess γγ → boson loop → H, where the possible boson loops are shown in Fig. 5 ( taken from ref. [7]). H− in Fig. 5 is an un-physical, charged Higgs boson, and φ± is a Fadeev-Popov ghost. The formalism for calculating the amplitude of each diagram in Fig. 5, is similar to the approach used to calculate the amplitude of the fermion triangle above, in section 3.1. Similarly here, after integration over the unknown momentum l in the loop of each diagram in Fig. 5, and after integration over Feynman parametres, the expression for each diagram takes the general form [7] – 12 – gµν + C (3.2.1) where d is the dimension of space-time. Terms proportional to q21 and q 2 were assumed to vanish, and from the kinematics shown in Fig. 3, it was assumed that ~q1 · ~q2 = . In the limit that d → 4, the term proportional to Γ on the RHS of Eq. (3.2.1) tends to infinity. However, when one sums over the contributions to the amplitude given by all the boson sub-processes shown in Fig. 5, these divergencies cancel exactly (see table below). One requires also, that this sum over all the amplitudes for the boson loop sub-processes shown in Fig. 5, satisfies the condition C = − D (3.2.2) such that the amplitude for the sum is gauge invariant. In ref. [7], this sum was taken and the result was an almost gauge invariant expression, since terms proportional to and higher were neglected. In the calculation which lead to the results in this paper, the result gives an exactly gauge invariant expression, after using dimensional regularisation to remove terms which do not satisfy the gauge invariance condition. graph B C D a + crossed 3 ( d− 1) −4 5 b + crossed −2 ( d− 1) 0 2m c + d + crossed −1 ( d− 1) −4 2 e + crossed 0 0 −m f + crossed 0 0 1 g + h + crossed 1 0 0 i + crossed −2 0 0 2j + crossed −1 sum 1 d − 2 −8 8 Hence, plugging the results shown in the table for the coefficients into Eq. (3.2.1), the result of taking the sum over the amplitudes for the sub-processes shown in Fig. 5 gives a gauge invariant expression, and in the limit that d→ 4, the divergencies cancel exactly, such that the expression of Eq. (3.2.1) reduces to where Ab = 4 (3.2.3) – 13 – Higgs Figure 5: Boson loop subprocess for Higgs production in γγ fusion It should be noted from the results for the amplitude of the subprocess of Fig. 5 (a), the subprocess γγ → W triangle → H, interferes destructively with the subprocess γγ → fermion triangle → H shown in Fig. 4. – 14 – 3.3 The cross section for central exclusive Higgs production through γγ fusion Now that the amplitudes for the sub-processes γγ → fermion triangle → H and γγ → boson loop → H have been calculated, the results can be plugged into Eq. (3.4) to derive the cross section, for central exclusive Higgs production through γγ fusion. The result, taking into account all possible sub-processes shown in Fig. 4 and Fig. 5 is found to be σexcγγ ( p+p → p+H+p) = 0.1 fb (3.3.1) 4. Conclusion The results of this paper are summarized in the table below. σexc is the exclusive cross section, which includes multiplication by a factor for the survival probability, for central exclusive Higgs production. The results are given for the mechanisms pp → γγ → H and pp → IPIP → H. Note that in these results, the cross section for central exclusive Higgs production in the case of γγ fusion, is multiplied by a factor for the survival probability of 1. This is because in the case of photon exchange, there is no hard re-scattering to suppress, and the large rapidity gaps between the Higgs and the two emerging protons are automatically present. process < |S2 | > σexc (fb) IPIP 0.023 2.7 IPIP 0.004 0.47 γγ 1 0.1 The results show that, taking the survival probability to be 0.02, which is the value used in ref. [2], then the result for σexcIP IP for central exclusive Higgs production at the LHC, almost agrees with the prediction of ref. [2], (which was 3 fb). However, if the survival probability is an order of magnitude smaller as predicted in ref. [1], then σexcIP IP will be an order of magnitude smaller and, it becomes competitive with σ γγ for central exclusive Higgs production at the LHC. 5. Acknowledgements This paper is dedicated to the memory of Grandpa Herman, the Pindenjara April 15th 1924 - January 2nd 2007. I would like to thank E. Levin for helpful advice in writing this paper. I would also like to thank E. Gotsman, A. Kormilitzin, A. Prygarin for fruitful discussions on the subject. This research was supported in part by the Israel Science Foundation, founded by the Israeli Academy of Science and Humanities, by a grant from the Israeli ministry of science, culture & sport & the Russian Foundation for Basic research of the Russian Federation, and by the BSF grant 20004019. – 15 – A. Appendix A-1 Evaluation of the integral over the anomalous dimensions γ1 and γ2 of the momentum Q2 in the gluon density function The Born amplitude was calculated in Eq. (2.8), in terms of the gluon density as a function of the anomalous dimensions γ1 and γ2, for the two gluon ladders in Fig. 1. One now needs to integrate over γ1 and γ2, and also over the momentum in the t-channel gluon, namely Q. Altogether the necessary integrations take the MIP IP ( p+p → p+H+p) = Aπ3s dγ1dγ2 e−S(k x1, Q x2, Q (A-1-1) Firstly, the integral over Q⊥ is evaluated using the steepest descent technique. The gluon density is given by Q2, x1,2 )γ1,2 ω(γ1,2) ln s0 (A-1-2) where ln , where s0 ∼ 1GeV . This comes from the BFKL ladder gluon exchange (see Fig. 1), while the coefficient was taken from MRST - NLO - 2002 data (see Ref/[?]). ω (γ1,2) is the BFKL kernel defined as ω (γ1,2) = ᾱsχ (γ1,2) = ᾱs (ψ ( 1) − ψ ( γ1,2) − ψ ( 1− γ1,2) ) (A-1-3) where ψ ( f) is the digamma function and ψ ( f) = dΓ( f) . In Eq. (A-1-2), S k2⊥, E is the Sudakov form factor with the typical value [3, 4] S Q2⊥, E⊥ = 3αs , in the notation that E⊥ = Using this substitution Eq. (A-1-1) then becomes MIP IP ( p+p → p+H+p) = 4Aπ3s dγ1dγ2 × exp ω ( γ1) ln + ω ( γ2) ln (A-1-4) where φ − (γ1 + γ2 − 1) lnQ2⊥ (A-1-5) Differentiating the right hand side of Eq. (A-1-5) with respect to ln Q2⊥, one sees that φ has a saddle point at ln Q2⊥ = ln (γ1 + γ2 − 1). Hence, changing the integration variable to u = ln Q2⊥, and expanding φ around the saddle point, Eq. (A-1-4) can be written as – 16 – MIP IP ( p+p → p+H+p) = 4Aπ3se−φ(u0) dγ1dγ2 (u−u0)2 d2φ(u0) ω( γ1) ln +ω( γ2) ln (A-1-6) where u0 = ln (γ1 + γ2 − 1) (A-1-7) Now the right hand side of Eq. (A-1-6) has reduced to a Gaussian integral over u, which can be evaluated by the steepest descent technique, to give the expression MIP IP ( p+p → p+H+p) =4Aπ4s dγ1dγ2 exp (γ1 + γ2 − 1) (γ1 + γ2 − 1) + ln × exp ω ( γ1) ln + ω ( γ2) ln (A-1-8) The BFKL function ω ( γ) has a saddle point at γ = 1 . Near to this point ω ( γ) can be written as ω (γ1,2) = ω γ1,2 − (A-1-9) Hence using Eq. (A-1-9), Eq. (A-1-8) can be reduced to MIP IP ( p+p → p+H+p) =4Aπ4s dγ1dγ2 exp ( f ( γ1, γ2) ) (A-1-10) where the function f ( γ1, γ2) has the form f (γ1, γ2) =ω + (γ1 + γ2 − 1) (γ1 + γ2 − 1) + ln (A-1-11) This function has a saddle point with respect to γ1 given by (γ2 − 1)− ln ln s1 + ω ” ln s1 ) (A-1-12) Hence, expanding f ( γ1, γ2) around γ 1 , the integration over γ1 is evaluated using the steepest descent technique to give the expression – 17 – MIP IP ( p+p → p+H+p) =2Aπ4s exp ( f (γ 1 , γ2)) + ω ” ln s1 (A-1-13) Now the function f (γ 1 , γ2) has a saddle point with respect to γ2 given by ln s1 ln s1s2 + ω ” ln s1s2 1 + 4π ln s1 (A-1-14) Here in the second line it is assumed that s1 ∼ s2. For large s1 and s2, γsp2 is approximately . Using the same method as above, expanding f ( γ 1 , γ2) around γ 2 , the integration over γ2 is evaluated using the steepest descent technique for Eq. (A-1-13), to give the result MIP IP ( p+p → p+H+p) = Aπ4s 2 −2 exp 1 , γ 2 ∼ 12 + ω ” ln s1 ) (A-1-15) where f ( γ 1 , γ 2 ) = − ln s1 ln s1 (A-1-16) – 18 – A-2 Feynman rules for the standard electro-weak theory wmwgµν w w H νρ µ ρµ ν µν ρ µνgρσ µρgνσ µσgνρ Figure 6: Figure 7: Feynman rules in the standard electroweak theory A-3 Evaluation of the integral over the momentum in the fermion triangle loop The amplitude for the fermion triangle subprocess, summed over all quark flavours q = (u , d , s , c , t , b ) – 19 – and lepton flavours L = ( e , µ , τ ) , for the mechanism γγ fermion triangle → H was found in equation Eq. (3.1.4) to take the form 4παem ∫ 1−x ( 1− 2x y) − 2x y M gµν +m2 l̃2 −∆f (x, y) 4παem ∫ 1−x 4l̃µ l̃ν − l̃2 gµν l̃2 −∆f ( x, y) where ∆f = m f − M2H xy and L= e , µ , τ (A-3-1) where q denotes the sum over all six quark flavours q = u , d , s , c , t , b and L= e , µ , τ denotes the sum over all three lepton flavours L = e , µ , τ . From the numerator of the integrand, on the RHS of Eq. (A-3-1), one can see that there is a gauge invariant term ( 1− 4x y) , and the numerator in the integrand of the second line on the RHS gives a vanishing contribution to the integration over l̃, for d → 4. However one is still left with the terms −2xyM gµν + m2 gµν in the numerator of the integrand, which are certainly not gauge invariant. However, conveniently this non gauge invariant piece is exactly equal to ∆f (x, y) introduced in Eq. (3.1.2). To deal with this non gauge invariant piece, it is useful to use dimensional regularisation, when integrating over the l̃2 term in the numerator of the integrand on the second line. In this approach, one initially integrates over d+ ǫ dimensions in the limit that ǫ→ 0 and d→ 4. In this way the non gauge invariant terms disappear. Hence, evaluating the integral over l̃ on the RHS of Eq. (A-3-1) gives 4παemG (−1)3 Γ 3 − d Γ ( 3) ( 4π) ∫ 1−x ( 1− 4x y) + ∆f (x, y) gµν ∆f (x, y) + 2 lim 4παemG (−1)3+1 Γ 3 − d+ǫ 2Γ ( 3) ( 4π) ∫ 1−x dy 8m2f ( 4g µν − ( d+ ǫ) gµν) (A-3-2) In the limit that ǫ → 0 and d → 4, the gamma function Γ 3− d+ǫ and the RHS of Eq. (A-3-2) reduces to 4παemG 2 ( 4π) ∫ 1−x ( 1− 4x y) + ∆f (x, y) gµν ∆f ( x, y) 4παemG 2 ( 4π) ∫ 1−x dy 8m2f ǫ g µν (A-3-3) – 20 – Thus, after canceling ǫ in the numerator and the denominator in the second line on the RHS of Eq. (A- 3-3), the second line exactly cancels the non gauge invariant part of the integrand on the first line. Hence, one is left with the purely gauge invariant expression Aµνq = −2 If (A-3-4) where If is the only remaining integral to evaluate, which takes the form f =u , d , s , c , t , b ,e , µ , τ ∫ 1−x 1− 4xy ∆f (x, y) where ∆f (x, y) = m f −M2Hxy (A-3-5) To evaluate this integral, this are two cases to consider, namely (1) when m2 ≫ M2H , which is true for the top quark when mf = mt = 175GeV, and (2) when m ≪ M2H , which is true for all the rest of the fermions listed in the table. Therefore f If can be separated into two parts, namely If = It + f 6= t If (A-3-6) For case (1), where mf = mt ≫ MH , one can write It in a more convenient way as ∫ 1−x 1− 4xy ∫ 1−x  4 + 1− 4 m ) ∫ 1 x ( 1− x) ) ∫ 1 ( 1− x) + x ( 1− x)2 + x2 ( 1− x)3 + ..... (A-3-7) where in the last step, the logarithm was expanded in a Taylor series around x = 0. Evaluating the integral over x, and since it is assumed that M2H ≪ m2t , retaining terms no smaller than , the RHS of Eq. (A-3-7) becomes + .... (A-3-8) – 21 – For case (2), where mf ≪ MH which includes all the fermions in the table except for the top quark, there are two possible regions of integration, namely (I) when M2H x y > m and (II) when M2H x y < m In the region where M2H x y > m f , the RHS of Eq. (A-3-5) reduces to f 6= t region (I) f 6= t ∫ 1−x 1− 4xy f 6= t x ( 1− x) f 6= t 1− x − f 6= t polylog ( 2 , x = 1) − f 6= t polylog 2 , x = f 6= t  (A-3-9) In the region (II) where M2H x y < m , the RHS of Eq. (A-3-5) reduces to f 6= t region (II) f 6= t ( 1− 4xy) f 6= t f 6= t (A-3-10) Hence, adding the contributions of Eq. (A-3-9) and Eq. (A-3-10) for the contributions of region (I) and region (II) of the integral, gives the result f 6= t If = − f 6= t f 6= t polylog ( 2 , x = 1) − polylog 2 , x = (A-3-11) f 6= t for mf ≪ MH (A-3-12) From Eq. (A-3-12) and Eq. (A-3-8), the result for the evaluation of the integral If has its main contribution from the top quark triangle, such that – 22 – If ≈ It , = for mt ≫ MH (A-3-13) Plugging this result into Eq. (A-3-4) gives the final expression for the amplitude of the fermion triangle subprocess shown in Fig. 4, for the sum over all quark q = ( u , d , s , c , t , b ) contributions and lepton contributions L = e , µ , τ as where Af = − (A-3-14) References [1] J.S. Miller, ”Survivial probability for Higgs diffractive production in high density QCD” (in press) arxiv: hep-ph/0610427 [2] V.A. Khoze A.D.Martin M.G. Ryskin ”Prospects for new physics observations in diffractive processes at the LHC and Tevatron” Eur. Phys. J. C23 311 - 327 (2002) arxiv: hep-ph/0111078 [3] V. Khoze, A.Martin, M.Ryskin, ”The rapidity gap Higgs signal at the LHC” Phys. Lett. B401 (191997) 330-336 arXiv:hep-ph/9701419 [4] V.Khoze, A.Martin, M.Ryskin, ”Dijet hadroproduction with rapidity gaps and QCD double logarithmic effects” Phys. Rev. D56 (191997) 5867-5874 arXiv:hep-ph/9705258 [5] G.Altarelli, G.Parisi, Nucl. Phys. B126 (191977) 298 [6] Thomas G. Rizzo, ”Gluon final states in Higgs - Boson decay”, Phys. Rev. D22 (191980) 178, Addendum-ibid Phys. Rev. D22 (191980) 1824-1825 [7] J. Ellis et al., ”Higgs boson” Nucl. Phys. B106 326- 331 (1976) [8] J. Ellis et al., ”A phenomenological profile of the Higgs boson” Nucl. Phys. B106 (191976) 326-331 [9] S. Dawson, ”Radiative corrections to Higgs boson production” Nucl. Phys. B359 (191991) 283-300 [10] S. Bentvelsen, E. Laenen, P. Motylinski, ”Higgs production through gluon fusion at leading order” NIKHEF 2005 - 007 – 23 –

0704.1987

Pure inductive limit state and Kolmogorov's property

arXiv:0704.1987v1 [math.OA] 16 Apr 2007 Pure inductive limit state and Kolmogorov’s property Anilesh Mohari S.N.Bose Center for Basic Sciences, JD Block, Sector-3, Calcutta-98 E-mail:[email protected] Abstract Let (B, λt, ψ) be a C ∗-dynamical system where (λt : t ∈ IT+) be a semi- group of injective endomorphism and ψ be an (λt) invariant state on the C subalgebra B and IT+ is either non-negative integers or real numbers. The central aim of this exposition is to find a useful criteria for the inductive limit state B →λt B canonically associated with ψ to be pure. We achieve this by exploring the minimal weak forward and backward Markov processes associ- ated with the Markov semigroup on the corner von-Neumann algebra of the support projection of the state ψ to prove that Kolmogorov’s property [Mo2] of the Markov semigroup is a sufficient condition for the inductive state to be pure. As an application of this criteria we find a sufficient condition for a translation invariant factor state on a one dimensional quantum spin chain to be pure. This criteria in a sense complements criteria obtained in [BJKW,Mo2] as we could go beyond lattice symmetric states. http://arxiv.org/abs/0704.1987v1 1 Introduction: Let τ = (τt, t ≥ 0) be a semigroup of identity preserving completely positive maps [Da1,Da2,BR] on a von-Neumann algebra A0 acting on a Hilbert space H0, where either the parameter t ∈ R+, the set of positive real numbers or t ∈ Z+, the set of positive integers. We assume further that the map τt is normal for each t ≥ 0 and the map t → τt(x) is weak ∗ continuous for each x ∈ A0. We say a projection p ∈ A0 is sub-harmonic and harmonic if τt(p) ≥ p and τt(p) = p for all t ≥ 0 respectively. For a sub-harmonic projection p, we define the reduced quantum dynamical semigroup (τ t ) on the von-Neumann algebra pA0p by τ t (x) = pτt(x)p where t ≥ 0 and x ∈ A 0. 1 is an upper bound for the increasing positive operators τt(p), t ≥ 0. Thus there exists an operator 0 ≤ y ≤ 1 so that y = s.limt→∞τt(p). A normal state φ0 is called invariant for (τt) if φ0τt(x) = φ0(x) for all x ∈ A0 and t ≥ 0. The support p of a normal invariant state is a sub-harmonic projection and φ 0, the restriction of φ0 to A 0 is a faithful normal invariant state for (τ t ). Thus asymptotic properties ( ergodic, mixing ) of the dynamics (A0, τt, φ0) is well determined by the asymptotic properties (ergodic, mixing respectively ) of the reduced dynamics (A t , φ 0) provided y = 1. For more details we refer to [Mo1]. In case φ0 is faithful, normal and invariant for (τt), we recall [Mo1] that G = {x ∈ A0 : τ̃tτt(x) = x, t ≥ 0} is von-Neumann sub-algebra of F = {x ∈ A0 : ∗)τt(x) = τt(x ∗x), τt(x)τt(x ∗) = τt(xx ∗) ∀t ≥ 0} and the equality G = IC is a sufficient condition for φ0 to be strong mixing for (τt). Since the backward process [AM] is related with the forward process via an anti-unitary operator we note that φ0 is strongly mixing for (τt) if and only if same hold for (τ̃t). We can also check this fact by exploring faithfulness of φ0 and the adjoint relation [OP]. Thus IC ⊆ G̃ ⊆ F̃ and equality IC = G̃ is also a sufficient condition for strong mixing where F̃ and G̃ are von-Neumann algebras associated with (τ̃t). Thus we find two competing criteria for strong mixing. However it is straight forward whether F = F̃ or G = G̃. Since given a dynamics it is difficult to describe (τ̃t) explicitly and thus this criterion G = IC is rather non-transparent. We prove in section 2 that G = {x ∈ F : τtσs(x) = σsτt(x), ∀t ≥ 0. s ∈R} where σ = (σs : s ∈R) is the Tomita’s modular auto-morphism group [BR,OP] associated with φ0. So G is the maximal von-Neumann sub-algebra ofA0, where (τt) is an ∗-endomorphism [Ar], invariant by the modular auto-morphism group (σs). Moreover σs(G) = G for all s ∈R and τ̃t(G) = G for all t ≥ 0. Thus by a theorem of Takesaki [OP], there exists a norm one projection IEG from A0 onto G which preserves φ0 i.e. φ0IE = φ0. Exploring the fact that τ̃t(G) = G, we also conclude that the conditional expectation IEG commutes with (τt). This enables us to prove that (A0, τt, φ0) is ergodic (strongly mixing) if and only if (G, τt, φ0) is ergodic (strongly mixing). Though τt(G) ⊆ G for all t ≥ 0, equality may not hold in general. However we have τt(G) = τ̃t(G̃) where G̃ = {x ∈ A0 : τt(τ̃t(x)) = x, t ≥ 0}. G = G̃ holds if and only if τt(G) = G, τ̃t(G̃) = G̃ for all t ≥ 0. Thus G0 = t≥0 τt(G) is the maximal von-Neumann sub-algebra invariant by the modular automorphism so that (G0, τt, φ0) is an ∗−automorphisms with (G0, τ̃t, φ0) as it’s inverse dynamics. Once more there exists a conditional expectation IEG0 : A0 → A0 onto G0 commuting with (τt). This ensures that (A0, τt, φ0) is ergodic (strongly mixing) if and only if (G0, τt, φ0) is ergodic (strongly mixing). It is clear now that G0 = G̃0, thus G0 = IC, a criterion for strong mixing, is symmetric or time- reversible. As an application in classical probability we can find an easy criteria for a stochastically complete Brownian flows [Mo5] on a Riemannian manifold driven by a family of complete vector fields to be strong mixing. Exploring the criterion G0 = IC we also prove that for a type-I factor A0 with center completely atomic, strong mixing is equivalent to ergodicity when the time variable is continuous i.e. R+ (Theorem 3.4). This result in particular extends a result proved by Arveson [Ar] for type-I finite factor. In general, for discreet time dynamics (A0, τ, φ0), ergodicity does not imply strong mixing property (not a surprise fact since we have many classical cases). We also prove that τ on a type-I von-Neumann algebra A0 with completely atomic center is strong mixing if and only if it is ergodic and the point spectrum of τ in the unit circle i.e. {w ∈ S1 : τ(x) = wx for some non zero x ∈ A0} is trivial. The last result in a sense gives a direct proof of a result obtained in section 7 of [BJKW] without being involved with Popescu dilation. In section 3 we consider the unique up to isomorphism minimal forward weak Markov [AM,Mo1,Mo4] stationary process {jt(x), t ∈ IT, x ∈ A0} asso- ciated with (A0, τt, φ0). We set a family of isomorphic von-Neumann algebras {A[t : t ∈ IT} generated by the forward process so that A[t ⊆ A[s whenever s ≤ t. In this framework we construct a unique modulo unitary equivalence minimal dilation (A[0, αt, t ≥ 0, φ), where α = (αt : t ≥ 0) is a semigroup of ∗−endomorphism on a von-Neumann algebra A[0 acting on a Hilbert space H[0 with a normal invariant state φ and a projection P in A[0 so that (a) PA[0P = π(A0) (b) Ω ∈ H[0 is a unit vector so that φ(X) =< Ω, XΩ >; (b) Pαt(X)P = π(τt(PXP )) for t ≥ 0, X ∈ A[0; (c) {αtn(PXnP ).....αt3(PX3P )αt2(PX2P )αt1(PX1P )Ω : 0 ≤ t1 ≤ t2.. ≤ tn, n ≥ 1}, Xi ∈ A[0} is total in H[0, where π is the GNS representation of A0 associated with the state φ0. In case φ0 is also faithful, we consider the backward process (j t ) defined in [AM] associate with the KMS adjoint Markov semigroup and prove that commutant of A[t is equal to A t] = {j s(x) : x ∈ A0, s ≤ t} ′′ for any fix t ∈ IT . As an application of our result on asymptotic behavior of a Markov semi- group, we also study a family of endomorphism (B, λt) on a von-Neumann algebra. Following Powers [Po2] an endomorphism αt : B0 → B0 is called shift t≥0 αt(B) is trivial. In general such a shift may not admit an invariant state [BJP]. Here we assume that λt admits an invariant state ψ and address how the shift property is related with Kolmogorov’s property of the canonical Markov semigroup (A0, τt, ψ) on the support projection on the von-Neumann algebra πψ(B) ′′ of the state vector state in the GNS space (Hπ, π,Ω) associated with (B, ψ). As a first step here we prove that Powers’s shift property is equivalent to Kolmogorov’s property of the adjoint Markov semigroup (τ̃t). However in the last section we show that Kolmogorov’s property of a Markov semigroup need not be equivalent to Kolmogorov’s property of the KMS adjoint Markov semigroup. Thus Powers’s shift property in general is not equivalent to Kol- mogorov’s property of the associated Markov semigroup. Section 4 includes the main mathematical result by proving a criteria for the inductive limit state, associated with an invariant state of an injective endomorphism on a C∗ algebra, to be pure. To that end we explore the minimal weak Markov process associated with the reduced Markov semigroup on the corner algebra of the support projection and prove that the inductive limit state is pure if the Markov semigroup satisfies Kolmogorov’s property. Further for a lattice symmetric factor state, Kolmogorov’s property is also necessary for purity of the inductive limit state. The last section deals with an application of our main results on translation invariant state on quantum spin chain. We give a simple criteria for such a factor state to be pure and find its relation with Kolmogorov’s property. Here we also deal with the unique temperature state i.e. KMS state on Cuntz alge- bra to illustrate that Powers shift property is not equivalent to Kolmogorov’s property of the associated canonical Markov map on the support projection. In fact this shows that Kolmogorov’s property is an appropriate notion to describe purity of the inductive state. 2 Time-reverse Markov semigroup and asymptotic properties: In this section we will deal will a von-Neumann algebra A and a completely positive map τ or a semigroup τ = (τt, t ≥ 0} of such maps on A. We assume further that there exists a normal invariant state φ0 for τ and aim to investigate asymptotic properties of the Markov map. We say (A0, τt, φ0) is ergodic if {x : τt(x) = x, t ≥ 0} = {zI, z ∈ IC} and we say mixing if τt(x) → φ0(x) in the weak ∗ topology as t→ ∞ for all x ∈ A0. For the time being we assume φ0 is faithful and recall following [OP,AM], the unique Markov map τ̃ on A0 which satisfies the following adjoint relation φ0(σ1/2(x)τ(y)) = φ0(τ̃(x)σ−1/2(y)) (2.1) for all x, y ∈ A0 analytic elements for the Tomita’s modular automorphism (σt : t ∈ IR) associated with a faithful normal invariant state for a Markov map τ on A0. For more details we refer to the monograph [OP]. We also quote now [OP, Proposition 8.4 ] the following proposition without a proof. PROPOSITION 2.1: Let τ be an unital completely positive normal maps on a von-Neumann algebra A0 and φ0 be a faithful normal invariant state for τ . Then the following conditions are equivalent for x ∈ A0: (a) τ(x∗x) = τ(x∗)τ(x) and σs(τ(x)) = τ(σs(x)), ∀ s ∈R; (b) τ̃ τ(x) = x. Moreover τ restricted to the sub-algebra {x : τ̃ τ(x) = x} is an isomorphism onto the sub-algebra {x ∈ A0 : τ τ̃ (x) = x} where (σs) be the modular auto- morphism on A0 associated with φ0. In the following we investigate the situation further. PROPOSITION 2.2: Let (A0, τt, φ0) be a quantum dynamical system and φ0 be faithful invariant normal state for (τt). Then the following hold: (a) G = {x ∈ A0 : τt(x ∗x) = τt(x ∗)τt(x), τt(xx ∗) = τt(x)τt(x ∗), σs(τt(x)) = τt(σs(x)), ∀ s ∈R, t ≥ 0} and G is σ = (σs : s ∈R) invariant and commuting with τ = (τt : t ≥ 0) on G. Moreover for all t ≥ 0, τ̃t(G) = G and the conditional expectationEG : A0 → A0 onto G0 commutes with (τt). (b) There exists a unique maximal von-Neumann algebra G0 ⊆ G G̃ so that σt(G0) = G0 for all t ∈R and (G0, τt, φ0) is an automorphism where for any t ≥ 0, τ̃tτt = τtτ̃t = 1 on G0. Moreover the conditional expectation EG0 : A0 → A0 onto G0 commutes with (τt) and (τ̃t). PROOF: The first part of (a) is a trivial consequence of Proposition 2.1 once we note that G is closed under the action x → x∗. For the second part we recall [Mo1] that φ0(x ∗JxJ) − φ0(τt(x ∗)Jτt(x)J) is monotonically increasing with t and thus for any fix t ≥ 0 if τ̃tτt(x) = x then τ̃sτs(x) = x for all 0 ≤ s ≤ t. So the sequence Gt = {x ∈ A0 : τ̃tτt(x) = x} of von-Neumann sub-algebras decreases to G as t increases to ∞ i.e. G = t≥0 Gt. Similarly we also have t≥0 G̃t. Since G̃t monotonically decreases to G̃ as t increases to infinity for any s ≥ 0 we claim that τs(G̃ t≥0 τs(G̃ t ), where we have used the symbol A1 = {x ∈ A : ||x|| = 1} for a von-Neumann algebra A. We will prove the non-trivial inclusion. To that end let x ∈ t≥0 τs(G̃ t ) i.e. for each t ≥ 0 there exists yt ∈ G̃ t so that τs(yt) = x. By weak ∗ compactness of the unit ball of A0, we extract a subsequence tn → ∞ so that ytn → y as tn → ∞ for some y ∈ A0. The von-Neumann algebras G̃t being monotonically decreasing, for each m ≥ 1, ytn ∈ G̃tm for all n ≥ m. G̃tm being a von-Neumann algebra, we get y ∈ G̃tm . As this holds for each m ≥ 1, we get y ∈ G̃. However by normality of the map τs, we also have x = τs(y). Hence x ∈ τs(G̃ Now we verify that s≥r τs(G̃ t≥0 τs+t(G t ) = s≥r τs+t(G t ) = 0≤s≤t τt(G s ), where we have used τt(G t ) = G̃ t being isomorphic. Since Gt are monotonically decreasing with t we also note that 0≤s≤t τt(G s ) = τt(G Hence for any r ≥ 0 τs(G̃ 1) = G̃1 (2.2) From (2.2) with r = 0 we get G̃1 ⊆ τt(G̃ 1) for all t ≥ 0. For any t ≥ 0 we also have τt(G̃ s≥t τs(G̃ 1) = G̃1. Hence we conclude τt(G̃ 1) = G̃1 for any t ≥ 0. Now we can easily remove the restriction to show that τt(G̃) = G̃ for any t ≥ 0 by linearity. By symmetry τ̃t(G) = G for any t ≥ 0. Since G is invariant under the modular automorphism (σs) by a theorem of Takesaki [AC] there exists a norm one projection EG : A → A with range equal to G. We claim thatEG commutes with (τt). To that end we verify for any x ∈ A0 and y ∈ G the following equalities: < JGyJGω0,EG(τt(x))ω0 >=< J0yJ0ω0, τt(x)ω0 > =< J0τ̃t(y)J0ω0, xω0 >=< JG τ̃t(y)JGω0,EG(x)ω0 > =< JGyJGω0, τt(EG(x))ω0) > where we used the fact that τ̃ (G) = G for the third equality and range of IEG is indeed G is used for the last equality. This completes the proof of (a). Now for any s ≥ 0, it is obvious that τ̃s(G̃) ⊆ t≥s τ̃s(G̃t). In the following we prove equality in the above relation. Let x ∈ t≥s τ̃s(G̃t) i.e. there exists elements yt ∈ G̃t so that x = τ̃s(yt) for all t ≥ s. If so then we have τs(x) = yt for all t ≥ s as G̃t ⊆ G̃s. Thus for any t ≥ s, yt = ys ∈ G̃ and x ∈ τ̃s(G̃). Now we verify the following elementary relations: τ̃s(G̃) = t≥s τ̃sτt(Gt) = t≥s τ̃sτs(τt−s(Gt))) = t≥s τt−s(Gt) = t≥0 τt(Gs+t) where we have used the fact that τt−s(Gt) ⊆ Gs. Thus we have s≥0 τs(G) ⊆ s≥0 τ̃s(G̃). By the dual symmetry, we conclude the reverse inclusion and hence τs(G) = τ̃s(G̃) (2.3) We set von-Neumann algebra G0 = s≥0 τs(G). Thus G0 ⊆ G and also G0 ⊆ G̃ by (2.3) and for each t ≥ 0 we have τtτ̃t = τ̃tτt = 1 on G0. Since τs(G) is monotonically decreasing, we also note that τt(G0) = s≥0 τs+t(G) = G0. Similarly τ̃t(G0) = G0 by (2.3). That G0 is invariant by the modular group σ follows since G is invariant by σ = (σt) which is commuting with τ = (τt) on G. Same is also true for (τ̃t) by (2.3). By Takesaki’s theorem [AC] once more we guarantee that there exists a conditional expectationEG0 : A0 → A0 with range equal to G0. Since τ̃t(G0) = G0, once more by repeating the above argument we conclude that EG0τt = τtEG0 on A0. By symmetry of the argument,EG0 is also commuting with τ̃ = (τ̃t) We have the following reduction theorem. THEOREM 2.3: Let (A0, τt, φ0) be as in Proposition 2.2. Then the follow- ing statements are equivalent: (a) (A0, τt, φ0) is mixing ( ergodic ); (b) (G, τt, φ0) is mixing ( ergodic ); (c) (G0, τt, φ0) is mixing ( ergodic ). PROOF: That (a) implies (b) is obvious. By Proposition 2.2. we have EGτt(x) = τtEG(x) for any x ∈ A0 and t ≥ 0. Fix any x ∈ A0. Let x∞ be any weak∗ limit point of the net τt(x) as t→ ∞ which is an element in G [Mo1]. In case (b) is true, we find that x∞ =EG(x∞) = φ0(EG(x)) = φ0(x)1. Thus φ0(x)1 is the unique limit point, hence weak∗ limit of τt(x) as t → ∞ is φ0(x)1. The equivalence statement for ergodicity also follows along the same line since the conditional expectationEI on the the von-Neumann algebra I = {x : τt(x) = x, t ≥ 0} commutes with (τt) and thus satisfies EIEG = EGEI = EI . This completes the proof that (a) and (b) are equivalent. That (b) and (c) are equivalent follows essentially along the same line since once more there exists a conditional expectation from G to G0 commuting with (τt) and any weak ∗ limit point of the net τt(x) as t diverges to infinity belongs to τs(G) for each s ≥ 0, thus in G0. We omit the details. Now we investigate asymptotic behavior for quantum dynamical system dropping the assumption that φ0 is faithful. Let p be the support projection of the normal state φ0 in A0. Thus we have φ0(pτt(1− p)p) = 0 for all t ≥ 0, p being the support projection we have pτt(1− p)p = 0 i.e. p is a sub-harmonic projection in A0 for (τt) i.e. τt(p) ≥ p for all t ≥ 0. Then it is simple to check that (A t , φ 0) is a quantum dynamical semigroup where A 0 = pA0p and τ t (x) = pτt(pxp)p for x ∈ A 0 and φ 0(x) = φ0(pxp) is faithful on A 0. In Theorem 3.6 and Theorem 3.12 in [Mo1] we have explored how ergodicity and strong mixing of the original dynamics (A0, τt, φ0) can be determined by that of the reduced dynamics (A t , φ 0). Here we add one more result in that line of investigation. THEOREM 2.4: Let (A0, τt, φ0) be a quantum dynamical systems with a normal invariant state φ0 and p be a sub-harmonic projection for (τt). If s-limitt→∞τt(p) = 1 then the following statements are equivalent: (a) ||φτt − φ0|| → 0 as t→ ∞ for any normal state on φ on A0. (b) ||φpτ t − φ 0|| → 0 as t→ ∞ for any normal state φ p on A PROOF: That (a) implies (b) is trivial. For the converse we write ||φτt − φ0|| = supx:||x||≤1|φτt(x) − φ0(x)| ≤ sup{x:||x||≤1}|φτt(pxp) − φ0(pxp)| + sup{x:||x||≤1}|φτt(pxp ⊥)| + sup{x:||x||≤1}|φτt(p ⊥xp)| + sup{x:||x||≤1}|φτt(p ⊥xp⊥)|. Since τt((1 − p)x) → 0 in the weak ∗ topology and |φτt(xp ⊥)|2 ≤ |φτt(xx ∗)|φ(τt(p ⊥))| ≤ ||x||2φ(τt(p ⊥) it is good enough if we verify that (a) is equivalent to sup{x:||x||≤1}|φτt(pxp) − φ0(pxp)| → 0 as t → ∞. To that end we first note that limsupt→∞supx:||x||≤1|ψ(τs+t(pxp))− φ0(pxp)| is independent of s ≥ 0 we choose. On the other hand we write τs+t(pxp) = τs(pτt(pxp)p) + τs(pτt(pxp)p ⊥)+τs(p ⊥τt(pxp)p)+τs(p ⊥τt(pxp)p ⊥) and use the fact for any nor- mal state φ we have limsupt→∞supx:||x||≤1|ψ(τs(zτt(pxp)p ⊥)| ≤ ||z|| |ψ(τs(p for all z ∈ A0. Thus by our hypothesis on the support projection we conclude that (a) hold whenever (b) is true. In case the time variable is continuous and the von-Neumann algebra is the set of bounded linear operators on a finite dimensional Hilbert space H0, by exploring Lindblad’s representation [Li], Arveson [Ar] shows that a quantum dynamical semigroup with a faithful normal invariant state is ergodic if and only if the dynamics is mixing. In the following we prove a more general result exploring the criteria that we have obtained in Theorem 2.3. Note at this point that we don’t even need the generator of the Markov semigroup to be a bounded operator for which Lindblad’s representation is not yet understood with full generality [CE]. THEOREM 2.5: Let A0 be type-I with center completely atomic and (τt : t ∈R) admits a normal invariant state φ0. Then (A0, τt, φ0) is strong mixing if and only if (A0, τt, φ0) is ergodic. PROOF: We first assume that φ0 is also faithful. We will verify now the criteria that G0 is trivial when (τt) is ergodic. Since G0 is invariant by the modular auto-morphism group associated with the faithful normal state φ0, by a theorem of Takesaki [Ta] there exists a faithful normal norm one projection from A0 onto G0. Now since A0 is a von-Neumann algebra of type-I with center completely atomic, a result of E. Stormer [So] says that G0 is also type-I with center completely atomic. Let Q be a central projection in G0. Since τt(Q) is also a central projection and τt(Q) → Q as t → 0 we conclude that τt(Q) = Q for all t ≥ 0 (center of G being completely atomic and time variable t is continuous ). Hence by ergodicity we conclude that Q = 0 or 1. Hence G0 can be identified with B(K) for a separable Hilbert space K. Since (τt) on B(K) is an automorphism we find a self-adjoint operator H in K so that τt(x) = e itHxe−itH for any x ∈ B(K). Since it admits an ergodic faithful normal state, by [Fr, Mo1] we conclude that {x ∈ B(K) : xeitH = eitHx, t ∈R} = IC, which holds if and only if K is one dimensional. Hence G0 = IC. Now we deal with the general situation. Let p be the support projection of φ0 in A0 and A0 being a type-I von-Neumann algebra with centre completely atomic, the center of A 0 = pA0p being equal to the corner of the center of A0 i.e. pA0 A′0p, is also a type-I von-Neumann algebra with completely atomic centre. (A0, τt, φ0) being ergodic, we have τt(p) ↑ 1 as t ↑ ∞ in the weak topology and (A t , φ 0) is ergodic. Thus by the first part of the argument, t , φ 0) is strongly mixing. Hence by Theorem 3.12 in [Mo1] we conclude that (A0, τt, φ0) is also strong mixing. This completes the proof. We end this section with another simple application of Theorem 2.3 by proving a result originated in [FNW1,FNW2,BJKW]. THEOREM 2.6: Let A0 be a type-I von-Neumann algebra with center completely atomic and τ be a completely positive map with a faithful normal invariant state φ0. Then the following are equivalent: (a) (A0, τn, φ0) is strong mixing. (b) (A0, τn, φ0) is ergodic and {w ∈ S 1, τ(x) = wx, for some non zerox ∈ A0} = {1}, where S 1 = {w ∈ IC : |w| = 1}. PROOF: That ‘(a) implies (b)’ is rather simple and true in general for any von-Neumann algebra. To that end let τ(x) = wx for some x 6= 0 and |w| = 1. Then τn(x) = wnx and since the sequence wn has a limit point say z, |z| = 1 we conclude by strong mixing that zx = φ0(x)I. Hence x is a scaler and thus x = τ(x), x 6= 0. So w = 1 and x = φ0(x)I. Ergodic property also follows by strong mixing as x = φ0(x)I for any x for which τ(x) = x. Now for the converse we will use our hypothesis that φ0 is faithful and A0 is a type-I von-Neumann algebra with completely atomic. To that end we plan to verify that G0 consists of scalers only and appeal to Theorem 2.3 for strong mixing. Since there exists a conditional expectation fromA0 onto G0, by a The- orem of Stormer [So] G0 is once more a type-I von-Neumann algebra with center completely atomic. Let E be a non-zero atomic projection in the center of G0. τ being an automorphism on G0, each element in the sequence {τk(E) : k ≥ 0} is an atomic projection in the center of G0. If τn(E) τm(E) 6= 0 and n ≥ m we find that τm(τn−m(E) E) 6= 0 and thus by faithful and invariance prop- erty of φ0, we get φ(τn−m(E) E) > 0. Once more by faithfulness we find that τn−m(E) E 6= 0. So by atomic property of E and τn−m(E) we con- clude that τn−m(E) = E. Thus either the elements in the infinite sequence E, τ(E), ...., τn(E).... are all mutually orthogonal or there exists a least pos- itive integer n ≥ 1 so that the projections E, τ(E), .., τn−1(E) are mutually orthogonal and τn(E) = E. However for such an infinite sequence with mutu- ally orthogonal projection we have 1 = φ0(I) ≤ φ0( 0≤n≤m−1 τn(E)) = mφ0(E) for all m ≥ 1. Hence φ0(E) = 0 contradicting that E is non-zero and φ0 is faithful. Thus for any w ∈ S1 with wn = 1, we have τ(x) = wx, where x = 0≤k≤n−1w kτk(E) 6= 0. By (b) we have w = 1. Hence n = 1. In other words we have τ(E) = E for any atomic projection in the center of G0. Now by ergodicity we have E = I. Thus G0 is a type-I factor say isomorphic to B(K) for some Hilbert space K and τ(x) = uxu∗ for some unitary element u in G0. Since (G0, τn, φ0) is ergodic we have {u, u∗} ′′ = B(K), which holds if and only if K is one dimensional ( check for an alternative proof that τ(u) = u, thus u = I by ergodicity and thus τ(x) = x for all x ∈ G0 ). Hence G0 = IC. This complete the proof that (b) implies (a). 3 Minimal endomorphisms and Markov semi- groups : An E0-semigroup (αt) is a weak ∗-continuous one-parameter semigroup of unital ∗-endomorphisms on a von-Neumann algebra A acting on a Hilbert space H. Following [Po1,Po2,Ar] we say (αt) is a shift if t≥0 αt(A) = IC. For each t ≥ 0, αt being an endomorphism, αt(A) is itself a von-Neumann algebra and t≥0 αt(A) is a limit of a sequence of decreasing von-Neumann algebras. Exploring this property Arveson proved that (αt) is pure if and only if ||ψ1αt− ψ2αt|| → 0 as t→ ∞ for any two normal states ψ1, ψ2 on A. These criteria gets further simplified in case (αt) admits a normal invariant state ψ0 for which we have (αt) is a shift (in his terminology it is called pure, here we prefer Powers’s terminology as the last section will illustrate a shift need not be pure in its inductive limit ) if and only if ||ψαt−ψ0|| → 0 as t→ ∞ for any normal state ψ. In such a case ψ0 is the unique normal invariant state. However a shift (αt) in general may not admit a normal invariant state [Po2,BJP] and this issue is itself an interesting problem. One natural question that we wish to address here whether similar result is also true for a Markov semigroup (τt) defined on an arbitrary von-Neumann algebra A0. This issue is already investigated in [Ar] where A0 = B(H) and the semigroup (τt) is assumed to be continuous in the strong operator topology. He explored associated minimal dilation to an E0-semigroups and thus make possible to prove that the associated E0-semigroup is a shift if and only if ||φ1τt − φ2τt|| → 0 as t → ∞ for any two normal states φ1, φ2 on A0. In case (τt) admits a normal invariant state the criteria gets simplified once more. In this section we will investigate this issue further for an arbitrary von-Neumann algebra assuming that (τt) is admits a normal invariant state φ0. To that end, we consider [Mo1] the minimal stationary weak Markov for- ward process (H, Ft], jt,Ω, t ∈ R) and Markov shift (St) associated with (A0, τt, φ0) and set A[t to be the von-Neumann algebra generated by the family of operators {js(x) : t ≤ s < ∞, x ∈ A0}. We recall that js+t(x) = S t js(x)St, t, s ∈R and thus αt(A[0) ⊆ A[0 whenever t ≥ 0. Hence (αt, t ≥ 0) is a E0-semigroup on A[0 with a invariant normal state Ω and js(τt−s(x)) = Fs]αt(jt−s(x))Fs] (3.1) for all x ∈ A0. We consider the GNS Hilbert space (Hπφ0 , πφ0(A0), ω0) as- sociated with (A0, φ0) and define a Markov semigroup (τ t ) on π(A0) by τπt (π(x)) = π(τt(x). Furthermore we now identify Hφ0 as the subspace of H by the prescription πφ0(x)ω0 → j0(x)Ω. In such a case π(x) is identified as j0(x) and aim to verify for any t ≥ 0 that t (PXP ) = Pαt(X)P (3.2) for all X ∈ A[0 where P is the projection from H on the GNS space. We use induction on n ≥ 1. If X = js(x) for some s ≥ 0, (4.2) follows from (4.1). Now we assume that (3.2) is true for any element of the form js1(x1)...jsn(xn) for any s1, s2, ..., sn ≥ 0 and xi ∈ A0 for 1 ≤ i ≤ n. Fix any s1, s2, , sn, sn+1 ≥ 0 and consider X = js1(x1)...jsn+1(xn+1). Thus Pαt(X)P = j0(1)js1+t(x1)...jsn+t(xn+1)j0(1). If sn+1 ≥ sn we use (3.1) to conclude (3.2) by our induction hypothesis. Now suppose sn+1 ≤ sn. In that case if sn−1 ≤ sn we appeal to (3.1) and induction hypothesis to verify (3.2) for X . Thus we are left to consider the case where sn+1 ≤ sn ≤ sn−1 and by repeating this argument we are left to check only the case where sn+1 ≤ sn ≤ sn−1 ≤ .. ≤ s1. But s1 ≥ 0 = s0 thus we can appeal to (3.1) at the end of the string and conclude that our claim is true for all elements in the ∗− algebra generated by these elements of all order. Thus the result follows by von-Neumann density theorem. We also note that P = τπt (1) is a sub-harmonic projection [Mo1] for (αt : t ≥ 0) i.e. αt(P ) ≥ P for all t ≥ 0. PROPOSITION 3.1: Let (A0, τt, φ0) be a quantum dynamical semigroup with a normal invariant state for (τt). Then the GNS space Hπφ0 associated with the normal state φ0 on A0 can be realized as a closed subspace of a unique Hilbert space H[0 up to isomorphism so that the following hold: (a) There exists a von-Neumann algebra A[0 acting on H[0 and a unital ∗- endomorphism (αt, t ≥ 0) on A[0 with a pure vector state φ(X) =< Ω, XΩ >, Ω ∈ H[0 invariant for (αt : t ≥ 0). (b) PAP is isomorphic with π(A0) where P is the projection onto Hπφ0 ; (c) Pαt(X)P = τ t (PXP ) for all t ≥ 0 and X ∈ A[0; (d) The closed span generated by the vectors {αtn(PXnP )....αt1(PX1P )Ω : 0 ≤ t1 ≤ t2 ≤ .. ≤ tk ≤ ....tn, X1, .., Xn ∈ A[0, n ≥ 1} is H[0. PROOF: The uniqueness up to isomorphism follows from the minimality prop- erty (d). Following the literature [Vi,Sa,BhP] on dilation we say (A[0, αt, φ) is the minimal E0-semigroup associated with (A0, τt, φ0). By a theorem [Ar, Proposi- tion 1.1 ] we conclude that t≥0 αt(A[0) = IC if and only if for any normal state ψ on A[0, ||ψαt − ψ0|| → 0 as t→ ∞, where ψ0(X) =< Ω, XΩ > for X ∈ A0]. In the following proposition we explore that fact that P is a sub-harmonic projection for (αt) and by our construction αt(P ) = Ft] ↑ 1 as t→ ∞. PROPOSITION 3.2: ||ψαt − ψ0|| → 0 as t → ∞ for all normal state ψ on A[0 if and only if ||φτt − φ0|| → 0 as t → ∞ for all normal state φ on π(A0) where π is the GNS space associated with (A0, φ0). PROOF: Since Fs] ↑ 1 in strong operator topology by our construction and π(A0) is isomorphic to F0]A[0F0], we get the result by a simple application of Theorem 2.4. THEOREM 3.3: Let τ = (τt, t ≥ 0) be a weak ∗ continuous Markov semi- group on A0 with an invariant normal state φ0. Then there exists a weak continuous E0-semigroup α = (αt, t ≥ 0) on a von-Neumann algebra A[0 acting on a Hilbert space H so that Pαt(X)P = τ t (PXP ), t ≥ 0 for all X ∈ A[0, where P is a sub-harmonic projection for (αt) such that αt(P ) ↑ I. Moreover the following statements are equivalent: t≥0 αt(A[0) =C (b) ||φτπt − φ0|| → 0 as t→ ∞ for any normal state φ on π(A0) PROOF: For convenience of notation we denote π(A0) ′′ as A0 in the following proof. That (a) and (b) are equivalent follows by a Theorem of Arveson [Ar] and Proposition 3.2. Following [AM,Mo1] we say (H, St, Ft],Ω) is a Kolmogorov’s shift if strong limt→−∞Ft] = |Ω >< Ω|. We also recall here that Kolmogorov’s shift property holds if and only if φ0(τt(x)τt(y)) → φ0(x)φ0(y) as t→ ∞ for all x, y ∈ A0. In such a case A = B(H) ( see the paragraph before Theorem 3.9 in [Mo1] ). If φ0 is faithful then A0 and π(A0) are isomorphic, thus t≥0 αt(A[0) =C if and only if ||φτt − φ0|| → 0 as t→ ∞ for any normal state φ on A0. Such a property is often called strong ergodic property. Our next result says that there is a duality between strong ergodicity and Kolmogorov’s shift property. To that end we recall the backward process (H, jbt , F[t,Ω) as defined in [AcM,Mo1] where Ft] be the projection on the subspace generated by the vectors {λ : IR → A0 : support of λ ⊆ (−∞, t]} and for any x ∈ A0, j t (x) is the trivial extension of it’s action on Ft] which takes an typical vector λ to λ ′ where λ′(s) = λ(s) for any s < t and λ′(t) = λ(t)σ i (x). For any analytic element x for the automorphism group, we check first that jbt is indeed an isometry if x is so. Now we extend as analytic elements are weak∗ dense to all isometrics and extend by linearity to all elements of A0. We recall here that we have backward Markov property for the process (jbs) as F[tj s(x)F[t = j t (τ̃t−s(x)) for all t ≥ s where (A0, τ̃t, t ≥ 0, φ0 is the dual Markov semigroup defined in (3.1). As in the forward process we have now F[tA t]F[t = j t (A0) where for each t ∈ IR we set A t] for the von-Neumann algebra {jbs(x) : s ≤ t, x ∈ A0} THEOREM 3.4: Let (A0, τt, φ0) be a Markov semigroup with a faithful normal invariant state φ0. Then the following are equivalent: (a) φ0(τ̃t(x)τ̃t(y)) → φ0(x)φ0(y) as t→ ∞ for any x, y ∈ A0. (b) ||φτt − φ0|| → 0 as t→ ∞ for any normal state φ on A0. PROOF: For each t ∈R let Abt] be the von-Neumann algebra generated by the backward processes {jbs(x) : −∞ < s ≤ t} [Mo1]. Assume (a). By Theorem 3.9 and Theorem 4.1 in [Mo1] we verify that weak∗ closure of t] is B(H). Since for each t ∈ R the commutant of Abt] contains A[t we conclude that t∈RA[t is trivial. Hence (b) follows once we appeal to Theorem 3.3. For the converse, it is enough if we verify that φ0(τ̃t(x)Jτ̃t(y)J) → φ0(x)φ0(y) as t→ ∞ for any x, y ∈ A0 with y ≥ 0 and φ0(y) = 1. To that end we check the following easy steps φ0(τ̃t(x)Jτ̃t(y)J) = φ0(τt(τ̃t(x))JyJ) and for any normal state φ, |φ ◦ τt(τ̃t(x))−φ0(x)| ≤ ||φ ◦ τt−φ0||||τ̃t(x)|| ≤ ||φ ◦ τt−φ0||||x||. Thus the result follows once we note that φ defined by φ(x) = φ0(xJyJ) is a normal state. THEOREM 3.5: Let (A0, τt, φ0) be a Markov semigroup with a normal in- variant state φ0. Consider the following statements: (a) φ0(τt(x)τt(y)) → φ0(x)φ0(y) as t→ ∞ for all x, y ∈ A0. (b) the strong limt→−∞Ft] = |Ω >< Ω|. (c) A = B(H) Then (a) and (b) are equivalent statements and in such a case (c) is also true. If φ0 is also faithful (c) is also equivalent to (a) ( and hence ( b)). PROOF: That (a) and (b) are equivalent is nothing but a restatement of Theorem 3.9 in [Mo1]. That (b) implies (c) is obvious since the projection [A′Ω], where A′ is the commutant of A, is the support of the vector state in A. We will prove now (c) implies (a). In case A = B(H), we have t] =C, thus in particular t≤0 αt(A 0]) = C. Hence by Theorem 3.3 applied for the time-reverse endomorphism we verify that ||φτ̃t− φ0|| → 0 as t→ ∞. Now (a) follows once we appeal to Theorem 3.4 for the adjoint semigroups since ˜̃τ t = τt. THEOREM 3.6: Let (A0, τt, φ0) be as in Theorem 3.1. Then the following hold: (a) If (A0, τt, φ0) is mixing then αt(X) → φ(X) as t→ ∞ for all X ∈ B, where B is the C∗ completion of the ∗ algebra generated by {jt(x) : t ∈ IR, x ∈ A0}. (b) If (A0, τt, φ0) is mixing and A is a type-I factor then A = B(H). PROOF: For (a) we refer to [AM, Mo1]. By our hypothesis A is a type-I von-Neumann factor and thus there exists an irreducible representation π of B in a Hilbert space Hπ quasi equivalent to πφ. There exists a density matrix ρ on Hπ such that φ(X) = tr(π(X)ρ) for all X ∈ B. Thus there exists a unitary representation t→ Ut on Hπ so that Utπ(X)U t = π(αt(X)) for all t ∈ IR and X ∈ B. Since φ = φαt on B we also have U t ρUt = ρ. We claim that ρ is a one dimensional projection. Suppose not and then there exists at least two characteristic unit vectors f1, f2 for ρ so that f1, f2 are character- istic vector for unitary representation Ut. Hence we have < fi, π(X)fi >=< fi, π(αt(X))fi > for all t ∈ IR and i = 1, 2. By taking limit we conclude by (a) that < fi, π(X)fi >= φ(X) < fi, fi >= φ(X) for i = 1, 2 for all X ∈ B. This violets irreducibility of representation π. PROPOSITION 3.7: Let (A0, τt, φ0) be as in Theorem 3.5 with φ0 as faithful. Then the commutant of A[t is A t] for each t ∈ IR. PROOF: It is obvious that A[0 is a subset of the commutant of A 0]. Note also that F[0 is an element in A 0] which commutes with all the elements in A[0. As a first step note that it is good enough if we show that F[0(A ′F[0 = F[0A[0F[0. As for some X ∈ (Ab0]) ′ and Y ∈ A[0 if we have XF[0 = F[0XF[0 = F[0Y F[0 = Y F[0 then we verify that XZf = Y Zf where f is any vector so that F[0f = f and Z ∈ Ab0] and thus as such vectors are total in H we get X = Y ). Thus all that we need to show that F[0(A ′F[0 ⊆ F[0A[0F[0 as inclusion in other direction is obvious. We will explore in following the relation that F0]F[0 = F[0F0] = F{0} i.e. the projection on the fiber at 0 repeatedly. A simple proof follows once we use explicit formulas for F0] and F[0 given in [Mo1]. Now we aim to prove that F[0A [0F[0 ⊆ F[0A 0]F[0. Let X ∈ F[0A [0F[0 and verify that XΩ = XF0]Ω = F0]XF0]Ω = F{0}XF{0}Ω ∈ [j 0(A0) ′′Ω]. On the other-hand we note by Markov property of the backward process (jbt ) that 0]F[0 = j b(A0) ′′. Thus there exists an element Y ∈ Ab0] so that XΩ = Y Ω. Hence XZΩ = Y ZΩ for all Z ∈ A[0 as Z commutes with both X and Y . Since {ZΩ : Z ∈ A[0} spans F[0, we get the required inclusion. Since inclusion in the other direction is trivial as F[0 ∈ A [0 we conclude that F[0A [0F[0 = F[0A 0]F[0. F[0 being a projection in A 0] we verify that F[0(A ′F[0 ⊆ (F[0A 0]F[0) ′ and so we also have F[0(A ′F[0 ⊆ (F[0A [0F[0) ′ as Ab0] ⊆ A [0. Thus it is enough if we prove that [0F[0 = (F[0A[0F[0) We will verify the non-trivial inclusion for the above equality. Let X ∈ (F[0A[0F[0) ′ then XΩ = XF0]Ω = F0]XF0]Ω = F{0}XF{0}Ω ∈ [j 0(A0)Ω]. Hence there exists an element Y ∈ F[0A [0F[0 so that XΩ = Y Ω. Thus for any Z ∈ A[0 we have XZΩ = Y ZΩ and thusXF[0 = Y F[0. Hence X = Y ∈ F[0A [0F[0. Thus we get the required inclusion. Now for any value of t ∈ IR we recall that αt(A[0) = A[t and αt(A[0) [0), αt being an automorphism. This completes the proof as αt(A 0]) = A by our construction. One interesting problem that we raised in [Mo1] whether Kolmogorov’s property is time reversible i.e. whether Ft] → |Ω >< Ω| as t → −∞ if and only if F[t → |Ω >< Ω| as t → ∞. That it is true in classical case follows by Kolmogorov-Sinai-Rohlin theory on dynamical entropy for the associated Markov shift [Pa]. In the present general set up, it is true if A0 is a type-I von- Neumann algebra with centre atomic [Mo1]. It is obviously true if the Markov semigroup is KMS symmetric. But in general it is false. In the last section we will give a class of counter example. This indicates that the quantum counter part of Kolmogorov property is unlikely to be captured by a suitable notion of quantum dynamical entropy with Kolmogorov-Sinai-Rohlin property. 4 Inductive limit state and purity: Let (B0, λt, t ≥ 0, ψ) be a unital ∗− endomorphism with an invariant normal state ψ on a von-Neumann algebra B0 acting on a Hilbert space H. Let P be the support projection for ψ. We set A0 = PBP , a von-Neumann algebra acting on H0, the closed subspace P , and τt(x) = Pλt(PxP )P , for any x ∈ A0 and t ≥ 0. Since λt(P ) ≥ P , it is simple to verify [Mo1] that (A0, τt, ψ0) is a quantum dynamical semigroup with a faithful normal invariant state ψ0, where ψ0(x) = ψ(PxP ) for x ∈ A0. Now we set j0(x) = PxP and jt(x) = λt(j0(x)) for t ≥ 0 and x ∈ A0. A routine verification says that Fs]jt(x)Fs] = js(τt−s(x)) for 0 ≤ s ≤ t, where Fs] = λs(P ), s ≥ 0. Let A[0 be the von-Neumann algebra {jt(x) : t ≥ 0, x ∈ A0} ′′. As in Section 4 we check that Pαt(X)P = τt(PXP ) for all X ∈ A[0. However are these vectors {λtn(PXnP )....λt1(PX1P )f : f ∈ H0, 0 ≤ t1 ≤ t2 ≤ .. ≤ tk ≤ ..tn, X1, .., Xn ∈ B0, n ≥ 1} total in H? As an counter example in discrete time we consider an endomorphism on B(H) [BJP] with a pure mixing state and note that A0 is only scalers. Thus the cyclic space generated by the process (jt) on the pure state is itself. Thus the problem is rather delicate even when the von-Neumann algebra is the algebra of all bounded operators on K. We will not address this problem here. Since λt(P )λtn(PXnP )...λt1(PXP )H0 = λtn(PXnP )...λt1(PXP )Ω for t ≥ tn, limt→∞λt(P ) = 1 is a necessary condition for cyclic property. The same counter example shows that it is not sufficient. In the following we explore the fact the support projection P is indeed an element in the von-Neumann algebra A generated by the process (kt(x) : t ≥ 0, x ∈ A0) and asymptotic limit of the endomorphism (B0, λt, t ≥ 0, ψ) is related with that of minimal endomorphism (A[0, αt, t ≥ 0φ). In the following we consider a little more general situation. Let B0 be a C algebra, (λt : t ≥ 0) be a semigroup of injective endomorphisms and ψ be an invariant state for (λt : t ≥ 0). We extend (λt) to an automorphism on the C algebra B−∞ of the inductive limit λt B0 → λt B0 and extend also the state ψ to B−∞ by requiring (λt) invariance. Thus there exists a directed set ( i.e. indexed by IT , by inclusion B[−s ⊆ B[−t if and only if t ≥ s ) of C∗-subalgebras B[t of B−∞ so that the uniform closure of s∈IT B[s is B[−∞. Moreover there exists an isomorphism i0 : B0 → B[0 ( we refer [Sa] for general facts on inductive limit of C∗-algebras). It is simple to note that it = λt ◦ i0 is an isomorphism of B0 onto B[t and ψ−∞it = ψ on B0. Let (Hπ, π,Ω) be the GNS space associated with (B[−∞, ψ[−∞) and (λt) be the unique normal extension to π(B−∞) ′′. Thus the vector state ψΩ(X) =< Ω, XΩ > is an invariant state for automorphism (λt). As λt(B[0) ⊆ B[0 for all t ≥ 0, (π(B[0) ′′, λt, t ≥ 0, ψΩ) is a quantum dynamics of endomorphisms. Let Ft] be the support projection of the normal vector state Ω in the von-Neumann sub-algebra π(B[t) ′′. Ft] ∈ π(B[t) ′′ ⊆ π(B[−∞) ′′ is a monotonically decreasing sequence of projections as t → −∞. Let projection Q be the limit. Thus Q ≥ [π(B[−∞) ′Ω] ≥ |Ω >< Ω|. So Q = |Ω >< Ω| ensures that ψ on B[−∞ is pure. We aim to investigate when Q is pure i.e. Q = |Ω >< Ω|. To that end we set von-Neumann algebra N0 = F0]π(B[0) ′′F0] and define family {kt : N0 → π(B−∞) ′′, t ∈ IT} of ∗−homomorphisms by kt(x) = λt(F0]xF0]), x ∈ N0 It is a routine work to check that (kt : t ∈ IT ) is the unique up to isomor- phism ( in the cyclic space of the vector Ω generated by the von-Neumann algebra {kt(x) : t ∈ IT, x ∈ N0} ) forward weak Markov process associ- ated with (N0, ηt, ψ0) where ηt(x) = F0]αt(F0]xF0])F0] for all t ≥ 0. It is minimal once restricted to the cyclic space generated by the process. Thus Q = |Ω >< Ω| when restricted to the cyclic subspace of the process if and only if ψ0(ηt(x)ηt(y)) → ψ0(x)ψ0(y) as t→ ∞ for all x, y ∈ N0. PROPOSITION 4.1: Let G0] be the cyclic subspace of the vector Ω gen- erated by π(B[0). (a) G0] ∈ π(B[0) ′ and the map h : π(B[0) ′′ → G0]π(B[0) ′′G0] defined by X → G0]XG0] is an homomorphism and the range is isomorphic to π0(B0) ′′, where (Hπ0, π0) is the GNS space associated with (B0, ψ). (b) Identifying the range of h with π0(B0) ′′ we have h ◦ λt(X) = λt(h(X)) for all X ∈ π(B[0) ′′ and t ≥ 0. (c) Let P be the support projection of the state ψ in von-Neumann algebra π0(B0) ′′ and A0 = Pπ0(B0) ′′P . We set τt(x) = Pλt(PxP )P for all t ≥ 0, x ∈ A0 and ψ0(x) = ψ(PxP ) for x ∈ A0. Then (i) h(F0]) = P and h(N0) = A0; (ii) h(ηt(x)) = τt(h(x)) for all t ≥ 0. PROOF: The map π(X)Ω → π0(X)Ω0 has an unitary extension which in- tertwines the GNS representation (H0, π0) with the sub-representation of B[0 on the cyclic subspace G0]. Thus (a) follows. (b) is a simple consequence as i0 : B0 → B[0 is a C ∗ isomorphism which covariant with respect to (λt) for all t ≥ 0 i.e λti0(x) = i0(λt(x)) for all x ∈ B0. That h(F0]) = P is simple as h is an isomorphism and thus also a normal map taking support projection F0] of the state ψ in π(B[0) ′′ to support projection P of the state ψ in π0(B0) ′′. Now by homomorphism property of the map h and commuting property with (λt) we also check that h(N0) = h(F0]π(B[0) ′′F0]) = Pπ0(B0) ′′P = A0 and h(ηt(x)) = h(F0])λt(h(F0])h(x)h(F0])) = Pλt(Ph(x)P )P = τt(h(x)) for all t ≥ 0. THEOREM 4.2: Q is pure if and only if φ0(τt(x)τt(y)) → φ0(x)ψ0(y) as t→ ∞ for all x, y ∈ A0. PROOF: For any fix t ∈ IT since kt(A0) = Ft]π(B[t) ′′Ft], for any X ∈ B[t we have QXΩ = QFt]XFt]Ω = Qkt(x)Ω for some x ∈ A0. Hence Q = |Ω >< Ω| if and only if Q = |Ω >< Ω| on the cyclic subspace generated by {kt(x), t ∈ IT, x ∈ A0}. Theorem 3.5 says now that Q = |Ω >< Ω| if and only if ψ0(ηt(x)ηt(y)) → ψ0(x)ψ0(y) as t→ ∞ for all x ∈ N0, Since h is an homomor- phism and hηt(x) = τt(h(x)), we also have h(ηt(x))ηt(y)) = τt(h(x))τt(h(x)). Since φ0 ◦ h = ψ0 we complete the proof. COROLLARY 4.3: ψ[−∞ is a pure state if φ0(τt(x)τt(y)) → φ0(x)ψ0(y) as t→ ∞ for all x, y ∈ A0. PROOF: It follows by Theorem 4.2 as Q ≤ [π(B[−∞) ′Ω] ≤ |Ω >< Ω|. Our analysis above put very little light whether the sufficient condition given in Corollary 4.3 is also necessary for purity. We will get to this point in next section where we will deal with a class of examples. 5 Kolmogorov’s property and pure transla- tion invariant states: Let ω be a translation invariant state on UHFd algebra A = ⊗ZZMd and ω be the restriction of ω to UHFd algebra B0 = ⊗INMd. There is a one to one correspondence between a translation invariant state ω and λ (one sided shift ) invariant state ω′ on UHFd algebra ⊗INMd. Powers’s [Po] criteria easily yields that ω is a factor state if and only if ω′ is a factor state. A question that comes naturally here which property of ω′ is related with the purity of ω. A systematic account of this question was initiated in [BJKW] inspired by initial success of [FNW1,FNW2,BJP] and a sufficient condition is obtained. In a recent article [Mo2] this line of investigation was further explored and we obtained a necessary and sufficient condition for a translation invariant lattice symmetric factor state to be pure and the criteria can be described in terms of Popescu elements canonically associated with Cuntz’s representation. That the state is lattice symmetric played an important role in the duality argument used in the proof. Here as an application of our general result, we aim now to find one more useful criteria for a translation invariant factor state ω on a one dimensional quantum spin chain ⊗ZZMd to be pure. We also prove that purity of a lattice symmetric translation invariant state ω is equivalent to Kolmogorov’s property of a Markov semigroup canonically associated with ω. First we recall that the Cuntz algebra Od(d ∈ {2, 3, .., }) is the universal C∗-algebra generated by the elements {s1, s2, ..., sd} subject to the relations: s∗i sj = δ 1≤i≤d i = 1. There is a canonical action of the group U(d) of unitary d× d matrices on Od given by βg(si) = 1≤j≤d for g = ((gij) ∈ U(d). In particular the gauge action is defined by βz(si) = zsi, z ∈ IT = S 1 = {z ∈ IC : |z| = 1}. If UHFd is the fixed point subalgebra under the gauge action, then UHFd is the closure of the linear span of all wick ordered monomials of the form si1...siks ...s∗j1 which is also isomorphic to the UHFd algebra Md∞ = ⊗ so that the isomorphism carries the wick ordered monomial above into the matrix element (1)⊗ ei2j2(2)⊗ ....⊗ e (k)⊗ 1⊗ 1.... and the restriction of βg to UHFd is then carried into action Ad(g)⊗Ad(g)⊗ Ad(g)⊗ .... We also define the canonical endomorphism λ on Od by λ(x) = 1≤i≤d and the isomorphism carries λ restricted to UHFd into the one-sided shift y1 ⊗ y2 ⊗ ...→ 1⊗ y1 ⊗ y2.... on ⊗∞1 Md. Note that λβg = βgλ on UHFd. Let d ∈ {2, 3, .., , ..} and ZZd be a set of d elements. I be the set of finite sequences I = (i1, i2, ..., im) where ik ∈ ZZd and m ≥ 1. We also include empty set ∅ ∈ I and set s∅ = 1 = s ∅, sI = si1......sim ∈ Od and s I = s ...s∗i1 ∈ Od. Let ω be a translation invariant state on A = ⊗ZZMd where Md is (d × d) matrices with complex entries. Identifying ⊗INMd with UHFd we find a one to one relation from a λ invariant state on UHFd with that of an one sided shift invariant state on AR = ⊗INMd. Let ω ′ be an λ-invariant state on the UHFd sub-algebra of Od. Following [BJKW, section 7], we consider the set Kω′ = {ψ : ψ is a state on Od such that ψλ = ψ and ψ|UHFd = ω By taking invariant mean on an extension of ω′ to Od, we verify that Kω′ is non empty and Kω′ is clearly convex and compact in the weak topology. In case ω′ is an ergodic state ( extremal state ) Kω′ is a face in the λ invariant states. Before we recall Proposition 7.4 of [BJKW] in the following proposition. PROPOSITION 5.1: Let ω′ be ergodic. Then ψ ∈ Kω′ is an extremal point in Kω′ if and only if ω̂ is a factor state and moreover any other extremal point in Kω′ have the form ψβz for some z ∈ IT . We fix any ω̂ ∈ Kω′ point and consider the associated Popescu system (K,M, vk,Ω) described as in Proposition 2.4. A simple application of Theorem 3.6 in [Mo2] says that the inductive limit state ω̂−∞ on the inductive limit (Od, ω̂) → λ (Od, ω̂) → λ (Od, ω̂) is pure if φ0(τn(x)τn(y)) → φ0(x)φ0(y) for all x, y ∈ M as n→ ∞. This criteria is of limited use in determining purity of ω unless we have πω̂(UHFd) ′′ = πω̂(Od) ′′. We prove a more powerful criteria in the next section, complementing a necessary and sufficient condition obtained by [Mo2], for a translation invariant factor state ω to be pure. To that end note that the von-Neumann algebra {SIS J : |I| = |J | < ∞} acts on the cyclic subspace of Hπω̂ generated by the vector Ω. This is iso- morphic with the GNS representation associated with (B0, ω ′). The inductive limit (B−∞, ω̂−∞) [Sa] described as in Proposition 3.6 in [Mo2] associated with (B0, λn, n ≥ 0, ω ′) is UHFd algebra ⊗ZZMd and the inductive limit state is ω. Let Q be the support projection of the state ω̂ in π0(B0) ′′ and A0 = Qπ(B0) ′′Q. Since ψΩ(Λ(X)) = ψΩ(X) for all X ∈ πω̂(UHFd) ′′, Λ(Q) ∈ πω̂(UHFd) ′′ and Λ(Q) ≥ Q [Mo1]. Thus QΛ(I − Q)Q = 0 and we have (I − Q)S∗kQ = 0 for all 1 ≤ k ≤ d. The reduced Markov map η : A0 → A0 is defined by η(x) = QΛ(QxQ)Q (5.1) for all x ∈ A0 which admits a faithful normal state φ0 defined by ψ0(x) = ψΩ(QxQ), x ∈ A0 (5.2) In particular, Λn(Q) ↑ I as n → ∞. Hence {SIf : |I| < ∞, Qf = f, f ∈ Hπ} is total in Hπω̂ . We set lk = QSkQ, where lk need not be an element in A0. However J ∈ A0 provided |I| = |J | < ∞. Nevertheless we have QΩ = Ω and thus verify that ω̂(sIs J) =< Ω, SIS < Ω, QSIS JQΩ >=< Ω, lI l for all |I|, |J | <∞. In particular we have ω′(sIs J) = ψ0(lI l for all |I| = |J | <∞. For each n ≥ 1 we note that {SIS J : |I| = |J | ≤ n} Λn(πω̂(UHFd) πω̂(UHFd) ′′ and thus πω̂(UHFd) ′′ ⊆ ( n≥1Λn(πω̂(UHFd) ′′)′. Hence Λn(πω̂(UHFd) ′′) ⊆ πω̂(UHFd) πω̂(UHFd) ′. (5.3) Now by Proposition 1.1 in [Ar, see also Mo2] ||ψΛn − ψΩ|| → 0 as n → ∞ for any normal state ψ on πω̂(UHFd) ′′ if ω′ is a factor state. Thus we have arrived at the following well-known result of R. T. Powers [Pow1,BR]. THEOREM 5.2: Let ω′ be a λ invariant state on UHFd ⊗INMd. Then the following statements are equivalent: (a) ω′ is a factor state; (b) For any normal state ψ on A0, ||ψηn − ψ0|| → 0 as n→ ∞; (c) For any x ∈ UHFd ⊗INMd sup||y||≤1|ω ′(xλn(y))− ω ′(x)ω′(y)| → 0 as n→ ∞; (d) ω′(xλn(y)) → ω ′(x)ω′(y) as n→ ∞ for all x, y ∈ UHFd ⊗INMd; PROOF: For any normal state ψ on A0 we note that ψP (X) = ψ(PXP ) is a normal state on πω̂(UHFd) ′′ and ||ψηn − ψ0|| ≤ ||ψPΛn − ψΩ||. Thus by the above argument (a) implies (b). That (c) implies (d) and (d) implies (a) are obvious. We will prove that (b) implies (c). Note that for (c) it is good enough if we verify for all non-negative x ∈ UHFd with finite support and ω ′(x) = 1. In such a case for large values of n the map πω̂(y) → ω ′(xλn(y)) determines a normal state on πω̂(UHFd) ′′. Hence (c) follows whenever (b) hold. COROLLARY 5.3: Let ω be a translation invariant state on UHFd ⊗ZZMd. Then the following are equivalent: (a) ω is a factor state; (b) ω(xλn(y)) → ω(x)ω(y) as n→ ∞ for all x, y ∈ UHFd ⊗ZZMd; PROOF: First we recall ω is a factor state if and only if ω is an extremal point in the translation invariant state i.e. ω is an ergodic state for the translation map. Since the cluster property (b) implies ergodicity, (a) follows. For the converse note that ω is a ergodic state for the translation map if and only if ω′ is ergodic for λ on UHFd ⊗INMd. Hence by Theorem 3.2 we conclude that statement (b) hold for any local elements x, y ∈ UHFd ⊗ZZMd. Now we use the fact that local elements are dense in the C∗ norm to complete the proof. PROPOSITION 5.4: Let ω be a translation invariant extremal state on A and ψ be an extremal point in Kω. Then following hold: (a) H = {z ∈ S1 : ψβz = ψ} is a closed subgroup of S 1 and π(Od) ′′βH = π(UHFd) ′′. Furthermore we have n(π(Od) ′′) = π(Od) ′′ ⋂ π(UHFd)′; (b) If H = S1 then π(Od) ′′ ⋂ π(UHFd)′ = IC; (c) Let (H, π,Ω) be the GNS representation of (Od, ψ) and P be the support projection of the state ψ in π(Od) ′′. Then P ∈ π(UHFd) ′′ is also the support projection of the state ψ in π(UHFd) PROOF: First part of (a) is noting but a restatement of Proposition 2.5 in [Mo2] modulo the factor property of π(UHFd) ′′. For a proof of the factor property we refer to Lemma 7.11 in [BJKW] modulo a modification described in Proposition 3.2 in [Mo2]. We aim now to show that n≥1 Λ n(π(Od) ′′) = π(Od) ′′ ⋂ π(UHFd)′. It is obvious by Cuntz relation that n(π(Od) ′′) ⊆ π(Od) ′′ ⋂ π(UHFd)′. For the converse letX ∈ π(Od) π(UHFd) ′ and fix any n ≥ 1 and set Yn = S with |I| = n. Since X ∈ π(UHFd) ′ we verify that S∗IXSI = S IXSIS JSJ = S∗ISIS JXSJ = S JXSJ for any |J | = n. Thus Yn is independent of the multi- index that we choose. Once gain as X ∈ π(UHFd) ′ we also check that Λn(Yn) = J :|J |=n SJS IXSIS J = X . Hence X ∈ n≥1 Λ n(π(Od) Now π(UHFd) ′′ being a factor, a general result in [BJKW, Lemma 7.12] says that π(Od) π(OHd ) ′ is a commutative von-Neumann algebra generated by an unitary operator u so that βz(u) = γ(z)u for all z ∈ H and some character γ of H . Furthermore there exists a z0 ∈ H so that βz0(x) = uxu ∗ for all x ∈ π(Od) Thus we also have βz0(u) = u = γ(z0)u. So we have γ(z0) = 1. H being S 1 the character can be written as γ(z) = zk all z ∈ H and for some k ≥ 1. Hence ukx(uk)∗ = βzk (x) = x. π(Od) ′′ being a factor uk is a scaler. By multiplying a proper factor we can choose an unitary u ∈ π(Od) π(UHFd) ′ so that uk = 1. However we also check that for all z ∈ S1 we have βz(u k) = γ(z)kuk i.e. γ(z)k = 1 for all z ∈ S1 as uk = 1. Hence γ(z) = 1 for all z ∈ S1. Thus βz(u) = u for all z ∈ S 1 and u is scaler as u is also an element in π(UHFd) ′′ by the first part. π(UHFd) ′′ being a factor we conclude that u is a scaler. Hence πψ(Od) ′′ ⋂π(UHFd)′ is trivial. This completes the proof of (b). It is obvious that βz(P ) = P for all z ∈ H and thus by (a) P ∈ π(UHFd) and thus also the support projection in π(UHFd) ′′ of the state ψ. (c) is a simple consequence of (a) and Corollary 4.3. THEOREM 5.4: Let ω be a translation invariant state on UHFd ⊗ZZMd and P be the support projection of ψ ∈ Kω′ in π(Od) ′′. Further let A0 be the von-Neumann algebra Pπ(UHFd) ′′P acting on the subspace P and completely positive map τ : A0 → A0 defined by τ(x) = PΛ(PxP )P , i.e. τ(x) = k lkxl be the completely positive map on A0 where lk = Pπ(sk)P for all 1 ≤ k ≤ d. Then the following hold: (a) If φ0(τ n(x)τn(y)) → φ0(x)φ0(y) as n→ ∞ for all x, y ∈ A0 then ω is pure; (b) If H = S1 then ||φτn − φ0|| → 0 as n→ ∞ for any normal state on A0; PROOF: (a) follows by an easy application of Corollary 4.3. For a proof for (b) we appeal to [Ar, Proposition 1.1] and the last statement in Proposition 5.3 (a). By a duality argument, Theorem 3.4 in [Mo2], ||ψηn − ψ0|| → 0 as n→ ∞ for any normal state ψ if and only if |ψ0(η̃n(x)η̃n(y)) → ψ0(x)ψ0(y)| as n → ∞ for any x, y ∈ A0, where (A0, η̃, φ0) the KMS-adjoint Markov semigroup [OP,AcM,Mo1] of (A0, η, φ0). We recall the unique KMS state ψ = ψβ on Od where β = ln(d) is a factor state and ψβ ∈ Kω where ω ′ is the unique trace on UHFd. For a proof that H = S1 for ψβ we refer to [BR]. ω is the unique trace on A and so is a factor state. Hence by Proposition 5.4 (d) we have πψ(Od) ′′ ⋂ πψ(UHFd)′ is trivial. Thus n≥1Λ(πψ(Od) ′′) = IC. In particular n≥1 Λ n(πψ(UHFd) ′′) = IC. On the other hand ψβ being faithful, the support projection is the identity operator and thus canonical Markov semigroup τ is equal to Λ. Λ being an endomorphism and ψβ being faithful, we easily verify that τ does not admit Kolmogorov property. On the other handH = S1 and so by Proposition 5.4 (d) ||φτn−φ0|| → 0 as n→ ∞ for any normal state φ on A0. This example unlike in the classical case shows that Kolmogorov’s property of a non-commutative dynamical system in general is not time reversible. REFERENCES • [AM] Accardi, L., Mohari, A.: Time reflected Markov processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top., vol-2, no-3, 397-425 (1999). • [Ar] Arveson, W.: Pure E0-semigroups and absorbing states, Comm. Math. Phys. 187 , no.1, 19-43, (1997) • [BP] Bhat, B.V.R., Parthasarathy, K.R.: Kolmogorov’s existence the- orem for Markov processes on C∗-algebras, Proc. Indian Acad. Sci. 104,1994, p-253-262. • [BR] Bratteli, Ola., Robinson, D.W. : Operator algebras and quantum statistical mechanics, I,II, Springer 1981. • [BJ] Bratteli, Ola; Jorgensen, Palle E. T. Endomorphism of B(H), II, Finitely correlated states on ON , J. Functional Analysis 145, 323-373 (1997). • [BJP] Bratteli, Ola., Jorgensen, Palle E.T. and Price, G.L.: Endomor- phism of B(H), Quantization, nonlinear partial differential equations, Operator algebras, ( Cambridge, MA, 1994), 93-138, Proc. Sympos. Pure Math 59, Amer. Math. Soc. Providence, RT 1996. • [BJKW] Bratteli, O., Jorgensen, Palle E.T., Kishimoto, Akitaka and Werner Reinhard F.: Pure states on Od, J.Operator Theory 43 (2000), no-1, 97-143. • [Da] Davies, E.B.: Quantum Theory of open systems, Academic press, 1976. • [FNW1] Fannes, M., Nachtergaele,D., Werner,R.: Finitely Correlated States on Quantum Spin Chains, Commun. Math. Phys. 144, 443-490 (1992). • [FNW2] Fannes, M., Nachtergaele,D., Werner,R.: Finitely Correlated pure states, J. Funct. Anal. 120, 511-534 (1994). • [Fr] Frigerio, A.: Stationary states of quantum dynamical semigroups. Commun. Math. Phys. 63, 269-276 (1978). • [Li] Lindblad, G. : On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119-130 (1976). • [Mo1] Mohari, A.: Markov shift in non-commutative probability, Jour. Func. Anal. 199 (2003) 189-209. • [Mo2] Mohari, A.: SU(2) symmetry breaking in quantum spin chain, The preprint is under review in Communication in Mathematical Physics, http://arxiv.org/abs/math-ph/0509049. • [Mo3] Mohari, A.: Quantum detailed balance and split property in quan- tum spin chain, Arxiv: http://arxiv.org/abs/math-ph/0505035. • [Mo4] Mohari, A: Jones index of a Markov semigroup, Preprint 2007. • [Mo5] Mohari, A.: Ergodicity of Homogeneous Brownian flows, Stochas- tic Process. Appl. 105 (1),99-116. • [OP] Ohya, M., Petz, D.: Quantum entropy and its use, Text and mono- graph in physics, Springer-Verlag 1995. • [Po] Powers, Robert T.: An index theory for semigroups of ∗- endomorphisms of B(H) and type II1 factors. Canad. J. Math. 40 (1988), no. 1, 86–114. • [Pa] Parry, W.: Topics in Ergodic Theory, Cambridge University Press, Cambridge, 1981. • [Sak] Sakai, S.: C∗-algebras and W∗-algebras, Springer 1971. • [Sa] Sauvageot, Jean-Luc: Markov quantum semigroups admit covariant Markov C∗-dilations. Comm. Math. Phys. 106 (1986), no. 1, 91103. • [So] Stormer, Erling : On projection maps of von Neumann algebras. Math. Scand. 30 (1972), 46–50. • [Vi] Vincent-Smith, G. F.: Dilation of a dissipative quantum dynamical system to a quantum Markov process. Proc. London Math. Soc. (3) 49 (1984), no. 1, 5872.

0704.1988

Classical nucleation theory in ordering alloys precipitating with L12 structure

Classical nucleation theory in ordering alloys precipitating with L12 structure. Emmanuel Clouet∗ and Maylise Nastar Service de Recherches de Métallurgie Physique, CEA/Saclay, 91191 Gif-sur-Yvette, France (Dated: October 28, 2018) By means of low-temperature expansions (LTEs), the nucleation free energy and the precipi- tate interface free energy are expressed as functions of the solubility limit for alloys which lead to the precipitation of a stoichiometric L12 compound such as Al-Sc or Al-Zr alloys. Classical nu- cleation theory is then used to obtain a simple expression of the nucleation rate whose validity is demonstrated by a comparison with atomic simulations. LTEs also explain why simple mean-field approximation like the Bragg-Williams approximation fails to predict correct nucleation rates in such an ordering alloy. Since its initial formulation in 1927 by Volmer, We- ber and Farkas and its modification in 1935 by Becker and Döring the classical nucleation theory (CNT)1,2,3 has been a suitable tool to model the nucleation stage in phase transformations. The success of this theory re- lies on its simplicity and on the few parameters required to predict the nucleation rate. Recently, the use of com- puter simulations have allowed to assess the applicability of the theory for solid-solid phase transformations4,5,6,7. Thanks to a precise control of simulation conditions, it is possible to get accurate estimations of CNT parameters and thus to make a direct comparison between theory predictions and quantities observed during simulations. One thus gains a deeper understanding of the validity of the different assumptions used by the CNT. Previous studies have shown that the capillary approx- imation, which CNT relies on, gives a precise description of cluster thermodynamics. Within this approximation, the free energy of a nucleus is written as the sum of a volume contribution, the nucleation free energy, and a surface contribution corresponding to the energy cost to create an interface between the nucleus and the solvent. For CNT to agree with atomic simulations, care has to be taken in the way these two energetic contributions are obtained. In particular, we have shown that one has to take into account short range order when calculating the nucleation free energy in an ordering alloy4. Usual thermodynamic approximations, like the ideal solid so- lution or the Bragg-Williams approximation, cannot de- scribe short range order and thus can predict values of the cluster size distribution and of the nucleation rate wrong by several orders of magnitude. This is to contrast with more sophisticated mean-field approximations like the cluster variation method (CVM) which provides good predictions of the nucleation rate4. However, an easy use of CNT and a clear determination of the missing ingre- dients in simple mean-field approximations requires an analytical approach which CVM cannot provide. Such an approach has to lead to accurate expressions of the CNT input parameters so as to make the theory predic- tive without any fitting of its parameters. In this Letter, we use low-temperature expansions (LTE)8,9 to derive an analytical formulation of the nucle- ation free energy and the interface free energy in a binary system like Al-Sc or Al-Zr, i.e. a supersaturated Al-X solid solution leading to the nucleation of a stoichiomet- ric Al3X compound with the L12 structure. This struc- ture corresponds to an ordering of the fcc lattice with so- lute X atoms lying on one of the four cubic sublattices8. LTE are well suited to describe short range order in dilute solid solution and nearly stoichiometric ordered compounds10,11,12,13,14 like Al3X compound. The use of this method in CNT framework allows to obtain a fully analytical modelling whose only material parameters are the solubility limit and the solute diffusion coefficient. To do so, we start from the same atomic diffusion model previously developed for Al-Sc-Zr system4,15. This model relies on a rigid lattice with interactions between first- and second nearest neighbors and uses a thermally activated atom-vacancy exchange mechanism to describe diffusion. Despite its simplicity, it has been shown to lead to predictions in good agreement with experimental data15,16,17. Within this atomic model, atoms are con- strained to lie on a fcc lattice and the configurations of a binary Al-X alloy is fully described by the solute atom occupation number pn with pn = 1 if the site n is occu- pied by a solute atom and pn = 0 otherwise. The energy per atom of a given configuration of the Al-X alloys is then given by E = UAl + (UX − UAl) (1− pn) pm (1− pn) pm (1) where the first and second sums, respectively, run on all first and second nearest-neighbor pairs of sites, Ns is the number of lattice sites, UAl (respectively UX) is the en- ergy per atom when only Al (respectively X) atoms lie on the fcc lattice and ω AlX and ω AlX are the first and second nearest neighbor order energies. Al-Sc and Al- Zr thermodynamics are characterized by the order ten- dency between first nearest neighbors (ω AlX < 0) and the demixing tendency between second nearest neighbors AlX > 0). Eq. 1 is a rewriting for binary alloys of the atomic model developed in Refs.4,15 when one neglects vacancy contributions. http://arxiv.org/abs/0704.1988v1 The nucleation free energy entering CNT is defined by ∆Gnuc(x0X) = µAl(x X )− µAl(x X )− µX(x , (2) where µAl(xX) and µX(xX) are the Al and X component chemical potentials in a solid solution of concentration xX, and x X and x X the concentrations of the equilibrium and supersaturated solid solution. LTE are more easy to handle in semi-grand-canonical ensemble where all quantities are written as functions of the effective potential µ = (µAl − µX) /2. Definition of the nucleation free energy then becomes ∆Gnuc(µ) = A(µeq)−A(µ) + (µeq − µ) , (3) where µeq is the effective potential corresponding to equi- librium between the Al solid solution and the Al3X L12 compound. We have defined in Eq. 3 the solid solution semi-grand-canonical free energy A = (µAl + µX) /2 = F (x) + (1− 2x)µ, F (x) being the usual canonical free energy. A LTE consists in developing the partition function of the system around a reference state, keeping in the series only the excited states of lowest energies. Use of the linked cluster theorem8,9 allows then to express the corresponding semi-grand canonical free energy as A(µ) = A0(µ)− kT gi,n exp (−∆Ei,n(µ)/kT ), (4) where the energy of the ground state is A0(µ) = UAl +µ for the Al solid solution and A0(µ) = 3/4 UAl+1/4 UX+ AlX+µ/2 for the Al3X L12 compound. In the sum ap- pearing in Eq. 4, the excited states have been gathered according to their energy state i and the number n of lattice sites involved. LTE parameters corresponding to the excited states with the lowest energies are given in Tab. I. All excitation energies only involve a set of iso- lated atoms or in second nearest neighbor position since flipping two atoms at nearest neighbor position produces an excited state with a much higher energy. The solute concentration in a given phase is obtained by considering the derivative of the corresponding semi- grand-canonical free energy. For the solid solution, one xX(µ) = ∂A(µ) ngi,n exp (−∆Ei,n(µ)/kT ). (5) The solid solution and the L12 compound are in equi- librium when both phases have the same semi-grand canonical free energy. Considering third order LTE TABLE I: Coefficients entering in the low temperature ex- pansion (Eq. 4). The first seven excited states are considered for the solid solution and the first three excited states for the Al3X L12 compound. The effective potential is written as µ = (UX − UAl)/2 + 6ω AlX + δµ. Solid solution L12 compound i n ∆Ei,n(µ) gi,n ∆Ei,n(µ) gi,n 1 1 6ω AlX − 2δµ 1 6ω AlX + 2δµ 1/4 2 2 10ω AlX − 4δµ 3 10ω AlX + 4δµ 3/4 3 2 12ω AlX − 4δµ −19/2 12ω AlX + 4δµ −7/8 4 3 14ω AlX − 6δµ 15 5 4 16ω AlX − 8δµ 3 5 3 16ω AlX − 6δµ −96 6 4 18ω AlX − 8δµ 83 6 3 18ω AlX − 6δµ −774 7 5 20ω AlX − 10δµ 48 7 4 + 20ω AlX − 8δµ −1569/2 (i = 3), this happens for the effective potential µeq = (UX − UAl)/2 + 6ω exp (−6ω(2)AlX/kT ) + 3 exp (−10ω AlX/kT ) exp (−12ω(2)AlX/kT ) , (6) corresponding to the solubility X = exp −6ω(2)AlX/kT + 6 exp −10ω(2)AlX/kT − 16 exp −12ω(2)AlX/kT . (7) As these expressions have to be consistent with the ex- pansion of A, terms with larger exponential arguments than −12ω(2)AlX are discarded. For equilibrium phases, one does not need to go further in the expansion than the third order. Indeed, thermodynamic properties are al- ready well converged as the solid solution and the L12 compound in equilibrium only slightly deviate from their respective ground states. On the other hand, Fig. 1 show that an expansion beyond the third order of the semi-grand canonical free energy A(µ) of the supersat- urated solid solution and of the corresponding concen- tration xX(µ) improves the convergence of the nucle- ation free energy. When only the first excited state is included in the expansion, LTE leads to the same value of the nucleation free energy as the ideal solid solution model. As more excited state are included in the expansion, the value deduced from LTE converges to the one calculated with CVM in the tetrahedron- octahedron approximation4. This is to contrast to the Bragg-Williams approximation which leads to a worse prediction of the nucleation free energy than the ideal solid solution model. 0 0.5 1 1.5 Ideal solid solution Bragg−Williams LTE : 1st order 3rd order 7th order FIG. 1: Variation with the nominal concentration x0Zr of the nucleation free energy ∆Gnuc at T = 723 K obtained with different thermodynamic approximations: CVM, ideal solid solution, Bragg-Williams and low-temperature expan- sions (LTE) to different orders. So as to understand why Bragg-Williams approxima- tion does so bad, it is worth going back to the canonical ensemble. When considering only the third order LTE, thermodynamic quantities can be expressed as functions of the solid solution nominal concentration x0X. In par- ticular, the nucleation free energy is given by ∆GnucLTE(x X) = kT q (x0X)− q (x + 3kT exp AlX/kT q (x0X) 2 − q (xeqX ) q (x0X) − ln [q (xeqX )] , (8) where we have defined the function q (x) = 1 + 4x 6 exp AlX/kT . (9) This expression developed to first order in the concentra- tions x0X and x X leads to ∆GnucLTE(x kT ln 1− xeqX 1− x0X kT ln 1 + 6e2ω x0X − x . (10) Doing the same development for the nucleation free energy calculated within the Bragg-Williams approximation4, we obtain ∆GnucBW(x kT ln 1− xeqX 1− x0X kT ln AlX + 3ω x0X − x . (11) Comparing Eq. 10 with Eq. 11, we see that these two thermodynamic approximations deviate from the ideal solid solution model by a distinct linear term. In the LTE (Eq. 10), the nucleation free energy is only de- pending on the second nearest neighbor interaction and the coefficient in front of the concentration difference is positive. On the other hand, the Bragg-Williams ap- proximation (Eq. 11) incorporates both first and sec- ond nearest neighbor interactions into a global parameter ωAlX = 6ω AlX + 3ω AlX. This leads to a linear correction with a coefficient which can be negative due to the os- cillating nature of the interactions. In particular, this is the case for both binary Al-Zr and Al-Sc alloys4. Bragg- Williams approximation thus leads to a wrong correction of the ideal model because it does not consider properly short range order. In the case of a L12 ordered com- pound precipitating from a solid solution lying on a fcc lattice, one cannot use such an approximation to calcu- late the nucleation free energy. On the other hand, Eq. 8 is a good approximation and can be used to calculate the nucleation free energy even when the second nearest neighbor interaction ω AlX is not known. Indeed, this pa- rameter can be deduced from the solubility limit x inverting Eq. 7, leading to the relation AlX = − kT ln (x X ) + kT 2/3 − . (12) This relation combined with Eq. 8 provides a powerful way for calculating the nucleation free energy from the solid solubility. LTE can be used too to calculate the plane interface free energy σ100 corresponding to a [100] direction. Due to the inhomogeneity perpendicular to the interface, the main contribution arises from broken bonds and excited states, whose energies are lower than in bulk phases, only bring a small correction. One thus does not need to go further than the second order in the expansion. At 0 K, the isotropic interface free energy σ̄ is obtained by mul- tiplying σ100 with the geometric factor (6/π) corre- sponding to a perfect [100] facetting of the precipitates. For low temperatures, this is a good approximation to assume that the same linear relation holds between both quantities4. The isotropic interface free energy given by LTE is then a2σ̄ = (6/π) AlX − 2kT exp (−4ω AlX/kT ) −kT exp (−6ω(2)AlX/kT ) , (13) where a is the fcc lattice parameter. LTE thus allow to calculate all CNT input parameters from the knowledge of the solubility limit. The nucle- ation rate is then obtained from the equation J st(x0X) = −16Ns ∆Gnuc(x0X)√ kTa2σ̄ (a2σ̄)3 kT [∆Gnuc(x0X)] , (14) 21.510.5 T = 723 K T = 773 K T = 823 K T = 873 K T = 723 K T = 773 K T = 823 K T = 873 K T = 723 K T = 773 K T = 823 K T = 723 K T = 773 K T = 823 K FIG. 2: Variation with nominal concentration and tempera- ture of the steady-state nucleation rate Jst for Al3Zr (top) and Al3Sc (bottom) precipitations. Symbols correspond to kinetic Monte Carlo simulations and lines to classical nucle- ation theory whereDX is the X impurity diffusion coefficient in Al. We thus obtain a fully analytical expression of the nucleation rate. Using the same experimental data, i.e. solubility limits and diffusion coefficients, as the ones used to fit the atomic diffusion model of kinetic Monte Carlo simu- lations, we can compare CNT predictions with nucleation rate observed in simulations4. A good agreement is ob- tained both for Al-Zr and Al-Sc binary alloys (Fig. 2). The combination of LTE with CNT thus allows to build a quantitative modeling of nucleation relying on a very limited number of material parameters. Such a model can be directly applied to aluminum alloys where a L12 compound precipitates from a supersaturated solid so- lution as this is the case with Zr, Sc or other rare earth elements like Er, Tm, Yb and Lu18. Li too precipitates in aluminum with a L12 structure, but this system requires another statistical approximation than LTE. Indeed, this approach based on LTE, requires that the precipitating phase only slightly deviates from its perfect stoichiome- try and that the solute solubility remains low. Provided these conditions are fulfilled, it could be applied to al- loys other than aluminum alloys. More generally, LTE demonstrate that the oscillating nature of the interac- tions in an alloy with an ordering tendency has to be taken into account by the CNT and requires a better statistical description than the Bragg-Williams approxi- mation which treats all interactions on the same footing. The authors would like to thank Y. Le Bouar and A. Finel for helpful discussions on LTE, and B. Legrand, F. Soisson and G. Martin for their invaluable help. ∗ Electronic address: [email protected] 1 G. Martin, in Solid State Phase Transformation in Met- als and Alloys (Les Éditions de Physique, Orsay, France, 1978), pp. 337–406. 2 K. F. Kelton, in Solid State Physics, edited by H. Ehren- reich and D. Turnbull (Academic Press, 1991), vol. 45, pp. 75–177. 3 D. Kashchiev, Nucleation : basic theory with applications (Butterworth Heinemann, Oxford, 2000). 4 E. Clouet, M. Nastar, and C. Sigli, Phys. Rev. B 69, 064109 (2004). 5 V. A. Shneidman, K. A. Jackson, and K. M. Beatty, Phys. Rev. B 59, 3579 (1999). 6 F. Soisson and G. Martin, Phys. Rev. B 62, 203 (2000). 7 F. Berthier, B. Legrand, J. Creuze, and R. Tétot, J. Elec- troanal. Chem. 561, 37 (2004); 562, 127 (2004). 8 F. Ducastelle, Order and Phase Stability in Alloys (North- Holland, Amsterdam, 1991). 9 C. Domb and M. S. Green, eds., Phase Transitions and Critical Phenomena, vol. 3 (Academic Press, London, 1974). 10 C. Woodward, M. Asta, G. Kresse, and J. Hafner, Phys. Rev. B 63, 094103 (2001). 11 A. F. Kohan, P. D. Tepesch, G. Ceder, and C. Wolverton, Comput. Mater. Sci. 9, 389 (1998). 12 M. Asta, S. M. Foiles, and A. A. Quong, Phys. Rev. B 57, 11265 (1998). 13 R. W. Hyland, M. Asta, S. M. Foiles, and C. L. Rohrer, Acta Mater. 46, 3667 (1998). 14 Y. Le Bouar, A. Loiseau, and A. Finel, Phys. Rev. B 68, 224203 (2003). 15 E. Clouet, L. Laé, T. Épicier, W. Lefebvre, M. Nastar, and A. Deschamps, Nat. Mater. 5, 482 (2006). 16 E. Clouet, A. Barbu, L. Laé, and G. Martin, Acta Mater. 53, 2313 (2005). 17 E. Clouet and A. Barbu, Acta Mater. 55, 391 (2007). 18 K. E. Knipling, D. C. Dunand, and D. N. Seidman, Z. Metallkd. 2006, 246 (2006). mailto:[email protected]

0704.1989

Jones index of a quantum dynamical semigroup

arXiv:0704.1989v1 [math.OA] 16 Apr 2007 Jones index of a quantum dynamical semigroup Anilesh Mohari S.N.Bose Center for Basic Sciences, JD Block, Sector-3, Calcutta-98 E-mail:[email protected] Abstract In this paper we consider a completely positive map τ = (τt, t ≥ 0) with a faith- ful normal invariant state φ on a type-II1 factor A0 and propose an index theory. We achieve this via a more general Kolmogorov’s type of construction for station- ary Markov processes which naturally associate a nested isomorphic von-Neumann algebras. In particular this construction generalizes well known Jones construction associated with a sub-factor of type-II1 factor. http://arxiv.org/abs/0704.1989v1 1 Introduction: Let τ = (τt, t ≥ 0) be a semigroup of identity preserving completely positive normal maps [Da,BR] on a von-Neumann algebra A0 acting on a separable Hilbert space H0, where either the parameter t ∈R+, the set of positive real numbers orZ +, the set of positive integers. In case t ∈ R+, i.e. continuous, we assume that for each x ∈ A0 the map t→ τt(x) is continuous in the weak ∗ topology. Thus variable t ∈ IT+ where IT is either IR or IN . We assume further that (τt) admits a normal invariant state φ0, i.e. φ0τt = φ0∀t ≥ 0. As a first step following well known Kolmogorov’s construction of stationary Markov processes, we employ GNS method to construct a Hilbert space H and an increasing tower of isomorphic von-Neumann type−II factors {A[t : t ∈ R or Z} generated by the weak Markov process (H, jt, Ft], t ∈ R or Z,Ω) [BP,AM] where jt : A0 → A[t is an injective homomorphism from A0 into A[0 so that the projection Ft] = jt(I) is the cyclic space of Ω generated by {js(x) : −∞ < s ≤ t, x ∈ A0}. The tower of increasing isomorphic von-Neumann algebras {A[t, t ∈ R or Z} are indeed a type-II∞ factor if and only if τ is not an endomorphism. In any case the projection j0(I) is a finite projection in A[−t for all t ≤ 0. In particular we also find an increasing tower of type-II1 factors {Mt : t ≥ 0} defined by Mt = j0(I)A[−tj0(I). Thus Jones in-dices {[Mt : Ms] : 0 ≤ s ≤ t} are invariance for the Markov semigroup (A0, τt, t ≥ 0, φ0) and further the map (t, s) → [Mt : Ms] is not continuous if the variable (t, s) are continuous i.e. if τ = (τt : t ∈ IR+). In discrete time dynamical system we find an invariance sequence {[Mn+1 : Mn] : n ≥ 0} canonically associated with the canonical conditional expectation on a sub-factor B0 of a type-II1 factor A0 where φ0 is the unique normal trace on A0. However unlike Jones construction we have [Mn+1 : Mn] = d 2 where d = [A0 : B0]. This shows that our construction in a sense generalizes two step Jones construction in discrete time. A detailed study, needs to be done to explore this new invariance, which seems to be an interesting problem! Acknowledgment: The author takes the opportunity to acknowledge Prof. Luigi Accardi for an invitation to visit Centro Vito Volterra, University of Rome, Tor Vergata during the summer 2005. The author further gratefully acknowledge Prof. Roberto Longo and Prof. Francesco Fidaleo for valuable discussion which helped the author to realize that the tower of type-II1 sub-factors indeed generalizing well known Jones construction. 2 Stationary Markov Processes and Markov shift: A family (τt, t ≥ 0) of one parameter completely positive maps on a C ∗ algebra or a von-Neumann sub-algebra A0 is called a quantum dynamical semigroup if τ0 = I, τs ◦ τt = τs+t, s, t ≥ 0 Moreover if τt(I) = I, t ≥ 0 it is called a Markov semigroup. We say a state φ0 on A0 is invariant for (τt) if φ0(τt(x)) = φ0(x) ∀t ≥ 0. We fix a Markov semigroup (A0, τt, t ≥ 0) and also a (τt)−invariant state φ0. In the following we briefly recall [AM] the basic construction of the minimal forward weak Markov processes associated with (A0, τt, t ≥ 0, φ0). The construction goes along the line of Kolmogorov’s construction of stationary Markov processes or Markov shift with a modification [Sa,BP] which takes care of the fact that A0 need not be a commutative algebra. Here we review the construction given in [AM] in order to fix the notations and important properties. We consider the class M of A0 valued functions x : IT → A0 so that xr 6= I for finitely many points and equip with the point-wise multiplication (xy)r = xryr. We define the map L : (M,M) → IC by L(x, y) = φ0(x τrn−1−rn(x (.....x∗r2τr1−r2(x yr1)yr2)...yrn−1)yrn) (2.1) where r = (r1, r2, ..rn) r1 ≤ r2 ≤ .. ≤ rn is the collection of points in IT when either x or y are not equal to I. That this kernel is well defined follows from our hypothesis that τt(I) = I, t ≥ 0 and the invariance of the state φ0 for (τt). The complete positiveness of (τt) implies that the map L is a non-negative definite form on M. Thus there exists a Hilbert space H and a map λ : M → H such that < λ(x), λ(y) >= L(x, y). Often we will omit the symbol λ to simplify our notations unless more then one such maps are involved. We use the symbol Ω for the unique element in H associated with x = (xr = I, r ∈ IR) and φ for the associated vector state φ on B(H) defined by φ(X) =< Ω,XΩ >. For each t ∈ IR we define shift operator St : H → H by the following prescription: (Stx)r = xr+t (2.2) It is simple to note that S = ((St, t ∈ IR)) is a unitary group of operators on H with Ω as an invariant element. For any t ∈ IR we set Mt] = {x ∈ M, xr = I ∀r > t} and Ft] for the projection onto Ht], the closed linear span of {λ(Mt])}. For any x ∈ A0 and t ∈ IT we also set elements it(x),∈ M defined by it(x)r = x, if r = t I, otherwise So the map V+ : H0 → H defined by V+x = i0(x) is an isometry of the GNS space {x :< x, y >φ0= φ0(x ∗y)} into H and a simple computation shows that < y, V ∗+StV+x >φ0=< y, τt(x) >φ0 . Hence P 0t = V +StV+, t ≥ 0 where P 0t x = τt(x) is a contractive semigroup of operators on the GNS space asso- ciated with φ0. We also note that it(x) ∈ Mt] and set ⋆-homomorphisms j 0 : A0 → B(H0]) defined by j00(x)y = i0(x)y for all y ∈ M0]. That it is well defined follows from (2.1) once we verify that it preserves the inner product whenever x is an isometry. For any arbitrary element we extend by linearity. Now we define j 0 : A → B(H) by 0 (x) = j 0(x)F0]. (2.3) Thus j 0 (x) is a realization of A0 at time t = 0 with j 0 (I) = F0]. Now we use the shift (St) to obtain the process j f = (j t : A0 → B(H), t ∈ IR) and forward filtration F = (Ft], t ∈ IR) defined by the following prescription: t (x) = Stj 0 (x)S t Ft] = StF0]S t , t ∈ IR. (2.4) So it follows by our construction that jfr1(y1)j (y2)...j (yn)Ω = y where yr = yri , if r = ri otherwise I, (r1 ≤ r2 ≤ .. ≤ rn). Thus Ω is a cyclic vector for the von-Neumann algebra A generated by {jfr (x), r ∈ IR, x ∈ A0}. From (2.4) we also conclude that StXS t ∈ A whenever X ∈ A and thus we can set a family of automorphism (αt) on A defined by αt(X) = StXS Since Ω is an invariant element for (St), φ is an invariant state for (αt). Now our aim is to show that the reversible system (A, αt, φ) satisfies (1.1) with j0 as defined in (2.4), for a suitable choice of IE0]. To that end, for any element x ∈ M, we verify by the relation < y,Ft]x =< y, x > for all y ∈ Mt] that (Ft]x)r = xr, if r < t; τrk−t(...τrn−1−rn−2(τrn−rn−1(xrn)xrn−1)...xt), if r = t I, if r > t where r1 ≤ .. ≤ rk ≤ t ≤ .. ≤ rn is the support of x. We also claim that t (x)Fs] = j s (τt−s(x)) ∀s ≤ t. (2.5) For that purpose we choose any two elements y, y′ ∈ λ(Ms]) and check the following steps with the aid of (2.2): < y,Fs]j t (x)Fs]y ′ >=< y, it(x)y =< y, is(τt−s(x))y ′) > . Since λ(Ms]) spans Hs] it complete the proof of our claim. We also verify that < z, V ∗+j t (x)V+y >φ0= φ0(z ∗τt(x)y), hence V ∗+j t (x)V+ = τt(x), ∀t ≥ 0. (2.6) For any fix t ∈ IT let A[t be the von-Neumann algebra generated by the family of operators {js(x) : t ≤ s < ∞, x ∈ A0}. We recall that js+t(x) = S t js(x)St, t, s ∈R and thus αt(A[0) ⊆ A[0 whenever t ≥ 0. Hence (αt, t ≥ 0) is a E0-semigroup on A[0 with a invariant normal state Ω and js(τt−s(x)) = Fs]αt(jt−s(x))Fs] (2.7) for all x ∈ A0. We consider the GNS Hilbert space (Hπφ0 , πφ0(A0), ω0) associ- ated with (A0, φ0) and define a Markov semigroup (τ t ) on π(A0) by τ t (π(x)) = π(τt(x). Furthermore we now identify Hφ0 as the subspace of H by the prescription πφ0(x)ω0 → j0(x)Ω. In such a case π(x) is identified as j0(x) and aim to verify for any t ≥ 0 that τπt (PXP ) = Pαt(X)P (2.8) for all X ∈ A[0 where P is the projection from [A[0Ω] onto the GNS space [j 0 (A0)Ω] which identified with the GNS space associated with (A0, φ0). It is enough if we verify for typical elements X = js1(x1)...jsn(xn) for any s1, s2, ..., sn ≥ 0 and xi ∈ A0 for 1 ≤ i ≤ n and n ≥ 1. We use induction on n ≥ 1. If X = js(x) for some s ≥ 0, (2.8) follows from (2.5). Now we assume that (2.8) is true for any element of the form js1(x1)...jsn(xn) for any s1, s2, ..., sn ≥ 0 and xi ∈ A0 for 1 ≤ i ≤ n. Fix any s1, s2, , sn, sn+1 ≥ 0 and consider X = js1(x1)...jsn+1(xn+1). Thus Pαt(X)P = j0(1)js1+t(x1)...jsn+t(xn+1)j0(1). If sn+1 ≥ sn, we use (2.5) to conclude (2.8) by our induction hypothesis. Now suppose sn+1 ≤ sn. In such a case if sn−1 ≤ sn we appeal once more to (2.5) and induction hypothesis to verify (2.8) for X. Thus we are left to consider the case where sn+1 ≤ sn ≤ sn−1 and by repeating this argument we are left to check only the case where sn+1 ≤ sn ≤ sn−1 ≤ .. ≤ s1. But s1 ≥ 0 = s0 thus we can appeal to (2.5) at the end of the string and conclude that our claim is true for any typical element X and hence true for all elements in the ∗− algebra generated by these elements of all order. Thus the result follows by von-Neumann density theorem. We also note that P is a sub-harmonic projection [Mo1] for (αt : t ≥ 0) i.e. αt(P ) ≥ P for all t ≥ 0 and αt(P ) ↑ [A[0Ω] as t ↑ ∞. THEOREM 2.1: Let (A0, τt, φ0) be a Markov semigroup and φ0 be (τt)-invariant state on a C∗ algebra A0. Then the GNS space [π(A0)Ω] associated with φ0 can be realized as a closed subspace of a unique Hilbert space H[0 up to isomorphism so that the following hold: (a) There exists a von-Neumann algebra A[0 acting on H[0 and a unital ∗- endomorphism (αt, t ≥ 0) on A[0 with a vector state φ(X) =< Ω,XΩ >, Ω ∈ H[0 invariant for (αt : t ≥ 0). (b) PA[0P is isomorphic with π(A0) ′′ where P is the projection from [A[0Ω] onto [jf (A0)Ω]; (c) Pαt(X)P = τ t (PXP ) for all t ≥ 0 and X ∈ A[0; (d) The closed span generated by the vectors {αtn(PXnP )....αt1(PX1P )Ω : 0 ≤ t1 ≤ t2 ≤ .. ≤ tk ≤ ....tn,X1, ..,Xn ∈ A[0, n ≥ 1} is H[0. PROOF: The uniqueness up to isomorphism follows from the minimality property Following the literature [Vi,Sa,BhP,Bh] on dilation we say (A[0, αt, φ) is the min- imal E0semigroup associated with (A0, τt, φ0). We have studied extensively asymp- totic behavior of the dynamics (A0, τt, φ0) in [AM] and Kolmogorov’s property of the Markov semigroup introduced in [Mo1] was explored to asymptotic behavior of the dynamics (A[0, αt, φ). In particular this yields a criteria for the inductive limit state canonically associated with (A[0, αt, φ) to be pure. The notion is intimately connected with the notion of a pure E0-semigroup introduced in [Po,Ar]. For more details we refer to [Mo2]. 3 Dual Markov semigroup and Time Reverse Markov processes: Now we are more specific and assume that A0 is a von-Neumann algebra and each Markov map (τt) is normal and for each x ∈ A0 the map t → τt(x) is continuous in the weak∗ topology. We assume further that φ0 is also faithful. Following [AM2], in the following we briefly recall the time reverse process associated with the KMS- adjoint ( or Petz-adjoint ) quantum dynamical semigroup (A, τ̃t, φ0). Let φ0 be a faithful state and without loss of generality let also (A0, φ0) be in the standard form (A0, J,P, ω0) [BR] where ω0 ∈ H0, a cyclic and separating vector for A0, so that φ0(x) =< ω0, xω0 > and the closer of the close-able operator S0 : xω0 → x ∗ω0, S possesses a polar decomposition S = J∆ 1/2 with the self-dual positive cone P as the closure of {JxJxω0 : x ∈ A0} in H0. Tomita’s [BR] theorem says that ∆itA0∆ −it = A0, t ∈ IR and JA0J = A 0, where A 0 is the commutant of A0. We define the modular automorphism group σ = (σt, t ∈ IR) on A0 by σt(x) = ∆ itx∆−it. Furthermore for any normal state ψ on A0 there exists a unique vector ζ ∈ P so that ψ(x) =< ζ, xζ >. Note that J π(x)J π(y)Ω = J π(x)∆ 2π(y∗)Ω = 2π(x)∆ 2π(y∗)Ω = π(y)∆ 2π(x∗)∆− 2Ω. Thus the Tomita’s map x → J π(x)J is an anti-linear ∗-homomorphism representation of A0. This observation leads to a notion called backward weak Markov processes [AM]. To that end we consider the unique Markov semigroup (τ ′t) on the commutant A′0 of A0 so that φ(τt(x)y) = φ(xτ t(y)) for all x ∈ A0 and y ∈ A 0. We define weak continuous Markov semigroup (τ̃t) on A0 by τ̃t(x) = Jτ t(JxJ)J. Thus we have the following adjoint relation φ0(σ1/2(x)τt(y)) = φ0(τ̃t(x)σ−1/2(y)) (3.1) for all x, y ∈ A0, analytic elements for (σt). One can as well describe the adjoint semigroup as Hilbert space adjoint of a one parameter contractive semigroup (Pt) on a Hilbert space defined by Pt : ∆ 1/4xω0 = ∆ 1/4τt(x)ω0. For more details we refer to [Ci]. We also note that it(x) ∈ M[t and set ⋆ anti-homomorphisms j 0 : A0 → B(H[0) defined by jb0(x)y = yi0(σ− i (x∗)) for all y ∈ M[0. That it is well defined follows from (2.1) once we verify by KMS relation that it preserves the inner product whenever x is an isometry. For any arbitrary element we extend by linearity. Now we define jb0 : A → B(H) by jb0(x) = j 0(x)F[0. (3.2) Thus jb0(x) is a realization of A0 at time t = 0 with j 0(I) = F[0. Now we use the shift (St) to obtain the process j b = (jbt : A0 → B(H), t ∈ IR) and forward filtration F = (F[t, t ∈ IR) defined by the following prescription: jbt (x) = Stj 0(x)S t F[t = StF[0S t , t ∈ IR. (3.3) A simple computation shows for −∞ < s ≤ t <∞ that s(x)F[t = j t (τ̃t−s(x)) (3.4) for all x ∈ A0. It also follows by our construction that j (y1)j (y2)...j (yn)Ω = (y) where yr = yri , if r = ri otherwise I, (r1 ≥ r2 ≥ .. ≥ rn). Thus Ω is a cyclic vector for the von-Neumann algebra Ab generated by {jbr(x), r ∈ IR, x ∈ A0} ′′. We also von-Neumann algebra Ab generated by {jbr(x), r ≤ t, x ∈ A0} ′′. The following theorems say that there is a duality between the forward and backward weak Markov processes. THEOREM 3.1: [AM] We consider the weak Markov processes (A,H, Ft], F[t, St, j t , j t t ∈ IR, Ω) associated with (A0, τt, t ≥ 0, φ0) and the weak Markov processes (Ã, H̃, F̃t], F̃[t, S̃t, j̃ t , j̃ t , t ∈ IR, Ω̃) associated with (A0, τ̃t, t ≥ 0, φ0). There exists an unique anti-unitary operator U0 : H → H̃ so (a) U0Ω = Ω̃; (b) U0StU 0 = S̃−t for all t ∈ IR; (c) U0j t (x)U 0 = j̃ −t(x), U0J t (x)U0 = j̃ −t(x) for all t ∈ IR; (d) U0Ft]U 0 = F̃[−t, U0F[tU 0 = F̃−t] for all t ∈ IR; THEOREM 3.2: Let (A0, τt, φ0) be as in Theorem 3.1 with φ0 as faithful. Then the commutant of A[t is A for each t ∈ IR. PROOF: It is obvious that A[0 is a subset of the commutant of A . Note also that F[0 is an element in A which commutes with all the elements in A[0. As a first step note that it is good enough if we show that F[0(A )′F[0 = F[0A[0F[0. As for some X ∈ (Ab )′ and Y ∈ A[0 if we have XF[0 = F[0XF[0 = F[0Y F[0 = Y F[0 then we verify that XZf = Y Zf where f is any vector so that F[0f = f and Z ∈ A and thus as such vectors are total in H we get X = Y ). Thus all that we need to show that F[0(A )′F[0 ⊆ F[0A[0F[0 as inclusion in other direction is obvious. We will explore in following the relation that F0]F[0 = F[0F0] = F{0} i.e. the projection on the fiber at 0 repeatedly. A simple proof follows once we use explicit formulas for F0] and F[0 given in [Mo1]. Now we aim to prove that F[0A F[0 ⊆ F[0A F[0. Let X ∈ F[0A F[0 and verify that XΩ = XF0]Ω = F0]XF0]Ω = F{0}XF{0}Ω ∈ [j 0(A0) ′′Ω]. On the other-hand we note by Markov property of the backward process (jbt ) that F[0A F[0 = j b(A0) Thus there exists an element Y ∈ Ab so that XΩ = YΩ. Hence XZΩ = Y ZΩ for all Z ∈ A[0 as Z commutes with both X and Y . Since {ZΩ : Z ∈ A[0} spans F[0, we get the required inclusion. Since inclusion in the other direction is trivial as F[0 ∈ A we conclude that F[0A F[0 = F[0A F[0 being a projection in A we verify that F[0(A )′F[0 ⊆ (F[0A ′ and so we also have F[0(A )′F[0 ⊆ (F[0A ′ as Ab . Thus it is enough if we prove [0F[0 = (F[0A[0F[0) We will verify the non-trivial inclusion for the above equality. Let X ∈ (F[0A[0F[0) then XΩ = XF0]Ω = F0]XF0]Ω = F{0}XF{0}Ω ∈ [j 0(A0)Ω]. Hence there exists an element Y ∈ F[0A F[0 so that XΩ = Y Ω. Thus for any Z ∈ A[0 we have XZΩ = Y ZΩ and thus XF[0 = Y F[0. Hence X = Y ∈ F[0A F[0. Thus we get the required inclusion. Now for any value of t ∈ IR we recall that αt(A[0) = A[t and αt(A[0) ′ = αt(A αt being an automorphism. This completes the proof as αt(A ) = Ab by our construction. 4 Subfactors: In this section we will investigate further the sequence of von-Neumann algebra {A[t : t ∈ IR} defined in the last section with an additional assumption that φ0 is also faithful and thus we also have in our hand backward von-Neumann algebras : t ∈ IR}. PROPOSITION 4.1: Let (A0, τt, φ0) be a Markov semigroup with a faithful normal invariant state φ0. If A0 is a factor then A[0 is a factor. In such a case the following also hold: (a) A0 is type-I (type-II, type-III) if and only if A[0 is type-I (type -II , type-III) respectively; (b) H is separable if and only if H0 is separable; (c) If H0 is separable then A0 is hyper-finite if and only if A[0 is hyper-finite. PROOF: We first show factor property of A[0. Note that the von-Neumann algebra generated by the backward process {jbs(x) : s ≤ 0, x ∈ A0} is a sub-algebra of A′ , the commutant of A[0. We fix any X ∈ A[0 in the center. Then for any y ∈ A0 we verify that Xj0(y)Ω = XF0]j0(y)Ω = F0]XF0]j0(y)Ω = j0(xy)Ω for some x ∈ A0. Since Xj0(y) = j0(y)X we also have j0(xy)Ω = j0(yx)Ω. By faithfulness of the state φ0 we conclude xy = yx thus x must be a scaler. Thus we have Xj0(y)Ω = cj0(y)Ω for some scaler c ∈ IC. Now we use the property that X commutes with forward process jt(x) : x ∈ A0, t ≥ 0 and as well as the backward processes {jbt (x), t ≤ 0} to conclude that Xλ(t, x) = cλ(t, x). Hence X = c. Thus A[0 is a factor. Now if A0 is a type-I factor, then there exists a non-zero minimal projection p ∈ A0. In such a case we claim that j0(p) is also a minimal projection in A[0. To that end let X be any projection in A[0 so that X ≤ j0(p). Since F0]A[0F0] = j0(A0) we conclude that F0]XF0] = j0(x) for some x ∈ A0. Hence X = j0(p)Xj0(p) = F0]Xj0(p) = j0(xp) = j0(px) Thus by faithfulness of the state φ0 we conclude that px = xp. Hence X = j0(q) where q is a projection smaller then equal to p. Since p is a minimal projection in A0, q = p or q = 0 i.e. X = j0(p) or 0. So j0(p) is also a minimal projection. Hence A[0 is a type-I factor. For the converse statement we trace the argument in the reverse direction. Let p be a non-zero projection in A0 and claim that there exists a minimal projection q ∈ A0 so that 0 < q ≤ p. Now since j0(p) is a non-zero projection in a type-I factor A[0 there exists a non-zero projection X which is minimal in A[0 so that 0 < X ≤ j0(p). Now we repeat the argument to conclude that X = j0(q) for some projection q. Since X 6= 0 and minimal, q 6= 0 and minimal in A0. This completes the proof for type-I case. We will prove now the case for Type-II. Let A[0 be type-II then there exists a finite projection X ≤ F0]. Once more X = F0]XF0] = j0(x) for some projection x ∈ A0. We claim that x is finite. To that end let q be another projection so that q ≤ x and q = uu∗ and u∗u = x. Then j0(q) ≤ j0(x) = X and j0(q) = j0(u)j0(u) ∗ and j0(x) = j0(u) ∗j0(u). Since X is finite in A[0 we conclude that j0(q) = j0(x). By faithfulness of φ0 we conclude that q = x, hence x is a finite projection. Since A0 is not type-I, it is type-II. For the converse let A0 be type-II. So A[0 is either type-II or type-III. We will rule out that the possibility for type-III. Suppose not, i.e. if A[0 is type-III, for every projection p 6= 0, there exists u ∈ A[0 so that j0(p) = uu ∗ and F0] = u ∗u. In such a case j0(p)u = uF0]. Set j0(v) = F0]uF0] for some v ∈ A0. Thus j0(pv) = j0(v). Once more by faithfulness of the normal state φ0, we conclude pv = v. So j0(v) = uF0]. Hence j0(v ∗v) = F0]. Hence v ∗v = 1 by faithfulness of φ0. Since this is true for any non-zero projection p in A0, A0 is type-III, which is a contradiction. Now we are left to show the statement for type-III, which is true since any factor needs to be either of these three types. This completes the proof for (a). (b) is obvious if IT is ZZ. In case IT = IR, we use our hypothesis that the map (t, x) → τt(x) is sequentially jointly continuous with respect to weak ∗ topology. For (c) we first recall from [Co] that hyper-finiteness property, being equivalent to injective property of von-Neumann algebra, is stable under commutant and countable intersection operation when they are acting on a separable Hilbert space. Let A0 be hyper-finite and H0 be separable. We will first prove A[0 is hyper-finite when IT = ZZ, i.e. time variable are integers. In such a case for each n ≥ 0, jn being injective, jn(A0) ′′ = {j0(x) : x ∈ A0} ′′ is a hyper-finite von-Neumann algebra. Thus A[0 = {jn(A0) ′′ : n ≥ 0}′′ is also hyper-finite as they are acting on a separable Hilbert space. In case IT = IR, for each n ≥ 1 we set von-Neumann sub-algebras ⊆ A[0 generated by the elements {jt(A0) ′′ : t = r , 0 ≤ r ≤ n2n}. Thus each An is hyper-finite. Since A′ n≥0(A )′ by weak∗ continuity of the map t → τt(x), we conclude that A[0 is also hyper-finite being generated by a countable family of increasing hyper-finite von-Neumann algebras. For the converse we recall for a factorM acting on a Hilbert spaceH, Tomiyama’s property ( i.e. there exists a norm one projection E : B(H) → M, see [BR1] page- 151 for details ) is equivalent to hyper-finite property. For a hyper-finite factor A[0, j0(A0) is a factor in the GNS space identified with the subspace F0]. Let E be the norm one projection from B(H[0) on A[0 and verify that the completely positive map E0 : B(H0) → A0 defined by E0(X) = F0]E(F0]XF0])F0] is a norm one projection from B(F0]) to A0. This completes the proof for (b). PROPOSITION 4.2: Let (A0, τt, φ0) be a dynamical system as in Proposition 4.1. If A[0 is a type-II1 factor which admits a unique normalize faithful normal tracial state then the following hold: (a) Ft] = I for all t ∈ IR; (b) τ = (τt) is a semigroup of ∗−endomorphisms. (c) A[0 = j0(A0). PROOF: Let tr0 be the unique normalize faithful normal trace on A[0. For any fix t ≥ 0 we set a normal state φt on A[0 by φt(x) = tr0(αt(x)). It is simple to check that it is also a faithful normal trace. Since αt(I) = I, by uniqueness φt = tr0. In particular tr0(F0]) = tr0(αt(F0]) = tr0(Ft]), by faithful property Ft] = F0] for all t ≥ 0. Since Ft] ↑ 1 as t→ ∞ we have F0] = I. Hence Ft] = αt(F0]) = I for all t ∈ IR. This proves (a). For (b) and (c) we recall that F0]jt(x)F0] = j0(τt(x)) for all t ≥ 0 and jt : A0 → A[t is an injective ∗− homomorphism. Since Ft] = F0] = I we have jt(x) = F0]jt(x)F0] = j0(τt(x)). Hence A[0 = j0(A0) and j0(τt(x)τt(y)) = j0(τt(xy)) for all x, y ∈ A0. Now by injective property of j0, we verify (b). This completes the proof. We fix a type-II1 factor A0 which admits a unique normalize faithful normal tracial state. Since A[0 is a type-II factor whenever A0 is so, we conclude that A[0 is a type-II∞ factor whenever τt is not an endomorphism on a such a type-II1 factor. The following proposition says much more. PROPOSITION 4.3: Let A0 be a type-II1 factor with a unique normalize normal trace and (A0, τt, φ0) be a dynamical system as in Proposition 4.1. Then the following hold: (a) j0(I) is a finite projection in A[−t for all t ≥ 0. (b) For each t ≥ 0 Mt = j0(I)A[−tj0(I) is a type-II1 factor and M0 ⊆ Ms... ⊆ Mt ⊆ .., t ≥ s ≥ 0 are acting on Hilbert space F0] where M0 = j0(A0). PROOF: By Proposition 4.1 A[0 is a type-II factor. Thus A[0 is either type-II1 or type-II∞. In case it is type-II1, Proposition 4.2 says that A[−t is j0(A0), hence the statements (a) and (b) are true with Mt = j0(A0). Thus it is good enough if we prove (a) and (b) when A[0 is indeed a type-II∞ factor. To that end for any fix t ≥ 0 we fix a normal faithful trace tr on A[−t and consider the normal map x→ j0(x) and thus a normal trace trace on A0 defined by x→ tr(j0(x)) for x ∈ A0. It is a normal faithful trace on A0 and hence it is a scaler multiple of the unique trace on A0. A0 being a type-II1 factor, j0(I) is a finite projection in A[−t. Now the general theory on von-Neumann algebra [Sa] guarantees that Mt is type-II1 factor and inclusion follows as A−s] ⊆ A−t] whenever t ≥ s. That j0(A0) = j0(I)A[0j0(I) follows from Proposition 4.1. We have now one simple but useful result. COROLLARY 4.4: Let (A0, τt, φ0) be as in Proposition 4.1. Then one of the following statements are false: (a) A = B(H) (b) A0 is a type-II1 factor. PROOF : Suppose both (a) and (b) are true. Let φt be the unique normalized trace on Mt. As they are acting on the same Hilbert space, we note by uniqueness that φt is an extension of φs for t ≥ s. Thus there exists a normal extension of (φt) to weak completion M of t≥0 Mt ( here we can use Lemma 13 page 131 [Sc] ). However if A = B(H), M is equal to B(H0]). H0] being an infinite dimensional Hilbert space we arrive at a contradiction. In case M in Corollary 4.4 is a type-II1 factor, by uniqueness of the tracial state we claim that λ(t)λ(s) = λ(t + s) where λ(t) = tr(F−t]) for all t ≥ 0. The claim follows as von-Neumann algebra M is isomorphic with α−t(M) which is equal to F−t]MF−t]. The map t → λ(t) being continuous we get λ(t) = exp(λt) for some λ ≤ 0. If λ = 0 then λ(t) = 1 so by faithful property of the trace we get F−t] = F0] for all t ≥ 0. Hence we conclude that (τt) is a family of endomorphisms by Proposition 4.2. Now for λ < 0 we have tr(F−t]) → 0 as t → ∞. As F−t] ≥ |Ω >< Ω| for all t ≥ 0, we draw a contradiction. Thus the weak∗ completion of t≥0 Mt is a type-II1 factor if and only if (τt) is a family of endomorphism. In otherwords if (τt) is not a family of endomorphism then the weak∗ completion of t≥0 Mt is not a type-II1 factor and the tracial state though exists on M is not unique. 5 Jones index of a quantum dynamical semigroup on II1 factor: We first recall Jones’s index of a sub-factor originated to understand the structure of inclusions of von Neumann factors of type II1. Let N be a sub-factor of a finite factor M . M acts naturally as left multiplication on L2(M, tr), where tr be the normalize normal trace. The projection E0 = [Nω] ∈ N ′, where ω is the unit trace vector i.e. tr(x) =< ω, xω > for x ∈ M , determines a conditional expectation E(x) = E0xE0 on N . If the commutant N ′ is not a finite factor, we define the index [M : N ] to be infinite. In case N ′ is also a finite factor, acting on L2(M, tr), then the index [M : N ] of sub-factors is defined as tr(E0) −1, which is the Murray-von Neumann coupling constant [MuN] of N in the standard representation L2(M, tr). Clearly index is an invariance for the sub-factors. Jones proved [M : N ] ∈ {4 cos2(π/n) : n = 3, 4, · · ·} ∪ [4,∞] with all values being realized for some inclusion N ⊆M . In this section we continue our investigation in the general framework of section 4 and study the case when A0 is type-II1 which admits a unique normalize faithful normal tracial state and (τt) is not an endomorphism on such a type-II1 factor. By Proposition 4.3 A[0 is a type-II∞ factor and (Mt : t ≥ 0) is a family of increasing type-II1 factor where Mt = j0(I)A[−tj0(I) for all t ≥ 0. Before we prove to discrete time dynamics we here briefly discuss continuous case. Thus the map I : (t, s) → [Mt : Ms], 0 ≤ s ≤ t is an invariance for the Markov semigroup (A0, τt, φ0). By our definition I(t, t) = 1 for all t ≥ 0 and range of values Jones’s index also says that the map (s, t) → I(s, t) is not continuous at (s, s) for all s ≥ 0. Being a discontinuous map we also claim the map (s, t) → I(s, t) is not time homogeneous, i.e. I(s, t) 6= I(0, t − s) for some 0 ≤ s ≤ t. If not we could have I(0, s + t) = I(0, s)I(s, s + t) = I(0, s)I(0, t), i.e. I(0, t) = exp(λt) for some λ, this leads to a contradiction. The non-homogenity suggest that I is far from being simple. We devote rest of the section discussing a much simple example in discrete time dynamics. To that end we review now Jones’s construction [Jo, OhP]. Let A0 be a type- II1 factor and φ0 be the unique normalize normal trace. The algebra A0 acts on L2(A0, φ0) by left multiplication π0(y)x = yx for x ∈ L 2(A0, φ0). Let ω be the cyclic and separating trace vector in L2(A0, φ0). The projection E0 = [B0ω] induces a trace preserving conditional expectation τ : a → E0aE0 of A0 onto B0. Thus E0π0(y)E0 = E0π0(E(y))E0 for all y ∈ A0. Let A1 be the von-Neumann algebra {π0(A0), E0} ′′. A1 is also a type-II1 factor and A0 ⊆ A1, where we have identified π0(A0) with A0. Jones proved that [A1 : A0] = [A0 : B0]. Now by repeating this canonical method we get an increasing tower of type-II1 factors A1 ⊆ A2... so that [Ak+1 : Ak] = [A0 : B0] for all k ≥ 0. Thus the natural question: Is Jones tower A0 ⊆ A1 ⊆ ... ⊆ Ak... related with the tower M0 ⊆ M1...Mk ⊆ Mk+1 defined in Proposition 4.3 associated with the dynamics (A0, τn, φ0)? To that end recall the von-Neumann sub-factors M0 ⊆ M1 and the induced representation of M1 on Hilbert subspace H[−1,0] generated by {j0(x0)j−1(x−1)Ω : x0, x−1 ∈ A0}. Ω is the trace vector for M0 i.e. φ0(x) =< Ω, j0(x)Ω >. However the vector state given by Ω is not the trace vector for M1 as M1 6= M0 ( If so we check by trace property that φ0(τ(x)yτ(z)) = φ(j0(x)j−1(y)j0(z)) = φ0(τ(zx)y) for any x, y, z ∈ A0 and so τ(zx) = τ(z)τ(x) for all z, x ∈ A0. Hence by Proposition 4.2 we have M1 = M0). Nevertheless M1 being a type-II1 factor there exists a unique normalize trace on M1. PROPOSITION 5.1: M1 ≡ A2 and [M1 : M0] = d 2 where d = [A0 : B0]. PROOF: Let φ1 be the unique normalize normal trace on A1 and H1 = L 2(A1, φ1). We consider the left action π1(x) : y → xy of A1 on H1. Thus π0(A0) is also acting on H1. Since E0π0(x)E0 = E0π0(τ(x))E0 = E0π0(τ(x)), for any element X ∈ A1, E0X = E0π0(x) for some x ∈ A0. Thus π1(E0) is the projection on the subspace {E0π0(x) : x ∈ A0}. For any y ∈ A0 we set (a) k−1(y) on the subspace π1(E0) by k−1(y)E0π0(x) = E0π0(yx) for x ∈ A0 and extend it to H1 trivially. That k−1(y) is well defined and an isometry for an isometry y follows from the following identities: φ1((E0π0(yz)) ∗E0π0(yx)) = φ1(π0(z ∗y∗)E0π0(yx)) = φ1(E0π0(yx)π0(z ∗y∗)) by trace property = φ1(E0)φ0(π0(yx)π0(z ∗y∗)) being a trace and φ1(E0π0(x)) = φ1(E0)φ0(π0(x)) = φ1(E0)φ0(π0(z ∗y∗)π0(yx)) = φ1(E0)φ0(π0(z = φ1((E0π0(z)) ∗E0π0(x)) (b) k0(y)x = π0(y)x for x ∈ A1. Thus y → k0(y) is an injective ∗-representation of A0 in L 2(A1, φ1). For y, z ∈ A0 we verify that < E0π0(y), k−1(1)k0(x)k−1(1)E0π0(z) >1=< E0π0(y), E0π0(x)E0π0(z) >1 =< E0π0(y), E0π0(τ(x))E0π0(z) >1 =< E0π0(y), E0π0(τ(x))π0(z) >1 Thus k−1(1)k0(x)k−1(1) = k−1(τ(x)) for all x ∈ A0. Note that k−1(1) = π1(E0) and the identity operator in H1 is a cyclic vector for the von-Neumann algebra {k0(x), k−1(x), x ∈ A0} ′′. We have noted before that the vector Ω need not be the tracial vector for M1 and also verify by a direct computation that the space {k0(y)k−1(x)1 : x, y ∈ A0} is equal to {yE0x : y, x ∈ A0} which is a proper subspace of L2(A1, φ1). Now we claim that the type-II1 factor M1 is isomorphic to the von-Neumann algebra {k−1(x), k0(x), x ∈ A0)} ′′. To that end we define an unitary operator from L2(A1, φ1) to L 2(M1, tr1) by taking an element kt1(x1)..ktn(xn) to jt1(x1)..jtn(xn), where tk are either 0 or −1. That it is an unitary operator follows by the tracial property of the respective states and weak Markov property of the homomorphisms. We leave the details and without lose of generality we identify these two weak Markov processes. Since k−1(1) = π1(E0), we conclude that π1(A1) ⊆ M1. In fact strict inclusion hold unless B0 = A0. However by our construction A1 = π0(A0) ∨ E0 is acting on L 2(A0, φ0) and A2 = π1(A1) ∨ E1 is acting on L 2(A1, φ1) where E1 is the cyclic subspace of 1 generated by π1(π0(A0)) i.e. E1 = [π1(π0(A0))1]. From (a) we also have k−1(y)π1(E0)E1 = π1(E0)k0(y)E1 (5.1) for all y ∈ A0. By Temperley-Lieb relation [Jo] we have π1(E0)E1π1(E0) = π1(E0) and thus post multiplying (5.1) by π1(E0) we have k−1(y)π1(E0) = π1(E0)k0(y)E1π1(E0) (5.2) So it is clear now that k−1(y) ∈ π1(A1) ∨ E1 for all y ∈ A0. Thus π1(A1) ⊆ M1 ⊆ π1(A1) ∨E1 = A2. We claim also that E1 ∈ M1. We will show that any unitary ele- ment u commuting with M1 is also commuting with E1. By (5.2) we have π1(E0)k0(y)(uE1u ∗ − E1)π1(E0) = 0 for all y ∈ A0. By taking adjoint we have π1(E0)(uE1u ∗ − E1)k0(y)π1(E0) = 0 for all y ∈ A0. Since A1 = π0(A0) ∨ E0 we conclude by cylicity of the trace vector that π1(E0)(uE1u ∗ − E1) = 0. So we have E1π1(E0)uE1u ∗ = E1π1(E0)E1 = E1 by Temperley-Lieb rela- tion. So u∗E1uπ1(E0) = u ∗E1π1(E0)uE1u ∗u = u∗E1u By taking adjoint we get π1(E0)u ∗E1u = u ∗E1u. Since same is true for u ∗, we conclude that u∗E1u = E1 for any unitary u ∈ M′1. Hence E1 ∈ M1. Hence M1 = A2. Since [M1 : M0] = [M1 : A1][A1 : M0] and [A1 : A0] = [A0 : B0] = d, we conclude the result. THEOREM 5.2: Mm ≡ A2m for all m ≥ 1. PROOF: Proposition 5.1 gives a proof for m = 1. The proof essentially follows the same steps as in Proposition 5.1. We use induction method for m ≥ 1. Assume it is true for 1, 2, ..m. Now consider the Hilbert space L2(A2m+1, tr2m+1), m ≥ 1 and we set homomorphism k−1, k0 from A2m into B(L 2(A2m+1, tr2m+1)) in the following: (a) k0(x)y = xy for all y ∈ A2m+1 and x ∈ A2m (b) π2m+1(E2m) is the projection on the subspace {E2my : y ∈ A2m} and k−1(x) defined on the subspace E2m by k−1(x)E2my = E2mxy for all x ∈ A2m and y ∈ A2m. That k−1 is an homomorphism follows as in Proposition 5.1. Thus an easy adaptation of Proposition 5.1 says that M = {k0(x), k−1(x) : x ∈ A2m} ′′ is a type-II1 factor and proof will be complete once we show that it is isomorphic to Mm+1. To that end we check as in Proposition 5.1 that k−1(E2m) = k−1(I), k−1(I)k0(x)k−1(I) = k−1(E2mxE2m) for all x ∈ A2m and k−1(x)E2m = E2mk0(x)E2m+1 where E2m+1 is the cyclic space of the trace vector generated by A2m. Thus following Proposition 5.1 we verify now that type-II1 factor M = {k0(x), k−1(I), E2m+1, x ∈ A2m} ′′ is isomorphic to Mm+1 = {J−1(x), J0(x) x ∈ Am} ′′, where we used notation J−1(x) = x for all x ∈ Am, J0(x) = SJ−1(x)S ∗ for all x ∈ A2m where we have identified A2m ≡ Mm with {jk(x),−m − 1 ≤ k ≤ −1, x ∈ A0} ′′ and S is the (right) Markov shift. This completes the proof. REFERENCES • [AM] Accardi, L., Mohari, A.: Time reflected Markov processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top., vol-2, no-3, 397-425 (1999). • [Ar] Arveson, W.: Pure E0-semigroups and absorbing states, Comm. Math. Phys. 187 , no.1, 19-43, (1997) • [Bh] Bhat, B.V.R.: An index theory for quantum dynamical semigroups, Trans. Amer. Maths. Soc. vol-348, no-2 561-583 (1996). • [BP] Bhat, B.V.R., Parthasarathy, K.R.: Kolmogorov’s existence theorem for Markov processes on C∗-algebras, Proc. Indian Acad. Sci. 104,1994, p-253- • [BR] Bratelli, Ola., Robinson, D.W. : Operator algebras and quantum statis- tical mechanics, I,II, Springer 1981. • [Da] Davies, E.B.: Quantum Theory of open systems, Academic press, 1976. • [El] Elliot. G. A.: On approximately finite dimensional von-Neumann algebras I and II, Math. Scand. 39 (1976), 91-101; Canad. Math. Bull. 21 (1978), no. 4, 415–418. • [Jo] Jones, V. F. R.: Index for subfactors. Invent. Math. 72 (1983), no. 1, 1–25. • [Mo1] Mohari, A.: Markov shift in non-commutative probability, Jour. Func. Anal. 199 (2003) 189-209. • [Mo2] Mohari, A.: Pure inductive limit state and Kolmogorov’s property, .... • [Mo3] Mohari, A.: SU(2) symmetry breaking in quantum spin chain, .... • [MuN] Murray, F. J.; von Neumann, J., On rings of operators. (English)[J] Ann. Math., Princeton, (2)37, 116-229. • [OP] Ohya, M., Petz, D.: Quantum entropy and its use, Text and monograph in physics, Springer-Verlag 1995. • [Po] Powers, Robert T.: An index theory for semigroups of ∗-endomorphisms of B(H) and type II1 factors. Canad. J. Math. 40 (1988), no. 1, 86–114. • [Sak] Sakai, S.: C∗-algebras and W∗-algebras, Springer 1971. • [Sa] Sauvageot, Jean-Luc: Markov quantum semigroups admit covariant Markov C∗-dilations. Comm. Math. Phys. 106 (1986), no. 1, 91103. • [Vi] Vincent-Smith, G. F.: Dilation of a dissipative quantum dynamical system to a quantum Markov process. Proc. London Math. Soc. (3) 49 (1984), no. 1, 5872.

0704.1990

Vacuum Polarization by a Magnetic Flux Tube at Finite Temperature in the Cosmic String Spacetime

arXiv:0704.1990v2 [hep-th] 3 Apr 2008 Vacuum Polarization by a Magnetic Flux Tube at Finite Temperature in the Cosmic String Spacetime J. Spinelly1 ∗and E. R. Bezerra de Mello2 † 1.Departamento de F́ısica-CCT Universidade Estadual da Paráıba Juvêncio Arruda S/N, C. Grande, PB 2.Departamento de F́ısica-CCEN Universidade Federal da Paráıba 58.059-970, J. Pessoa, PB C. Postal 5.008 Brazil November 11, 2018 Abstract In this paper we analyse the effect produced by the temperature in the vacuum polarization associated with charged massless scalar field in the presence of magnetic flux tube in the cosmic string spacetime. Three different configurations of magnetic fields are taken into account: (i) a homogeneous field inside the tube, (ii) a field proportional to 1/r and (iii) a cylindrical shell with δ-function. In these three cases, the axis of the infinitely long tube of radius R coincides with the cosmic string. Because the complexity of this analysis in the region inside the tube, we consider the thermal effect in the region outside. In order to develop this analysis, we construct the thermal Green function associated with this system for the three above mentioned situations considering points in the region outside the tube. We explicitly calculate in the high-temperature limit, the thermal average of the field square and the energy-momentum tensor. PACS numbers: 98.80.Cq, 11.27. + d, 04.62. + v 1 Introduction It is well known that different types of topological defects may have been created in the early Universe after the Planck time by the vacuum phase transition [1, 2]. These include domain walls, cosmic strings and monopoles. Among them cosmic string and monopole seem to be the best candidates to be detected. ∗E-mail: [email protected] †E-mail: [email protected] http://arxiv.org/abs/0704.1990v2 Many years ago Nielsen and Olesen proposed a theoretical model comprised by Higgs and gauge fields, that by a spontaneous broken of gauge symmetry produces linear topo- logical defect carrying a magnetic flux named vortex [3]. A few years later, Garfinkle investigated the influence of this topological object on the geometry of the spacetime [4]. Coupling the energy-momentum tensor associated with the system to the Einstein equations, he found static cylindrically symmetric solutions. The author also shown that asymptotically the spacetime around the vortex is a Minkowiski one minus a wedge. The core of the vortex has a non-zero thickness and the magnetic flux inside. Two years later Linet [5] obtained, under some specific condition, exact solutions for the complete set of differential equation. He was able to show that the structure of the respective spacetime corresponds to a conical one, with the conicity parameter being expressed in terms of the energy per unity length of the vortex.1 In recent papers, we have investigated the vacuum polarization effects associated with massless scalar [7, 8] and fermionic [9, 10] fields, in the presence of a magnetic flux tubes of finite radius in the cosmic string spacetime at zero temperature. In these analysis we considered that the magnetic fields are confined into an infinitely long tube of radius R around the cosmic string. Three different configurations of magnetic field, H(r), are taking into account in our analysis: i) H(r) = Θ(R− r) , homogeneous field inside, (1) ii) H(r) = 2παRr Θ(R− r) , field proportional to 1/r inside, (2) iii) H(r) = δ(r − R) , cylindrical shell, (3) where R is the radius extent of the tube, Θ is the Heaviside’s function and Φ is the total magnetic flux.2 The ratio of the flux to the quantum flux Φo, can be expressed by δ = Φ/Φ0 = N + γ, where N is the integer part and 0 < γ < 1. In the framework of the quantum field theory at finite temperature, a fundamental quantity is the thermal Green function, Gβ(x, x ′). For the scalar field it should be periodic in the imaginary time with period β, which is proportional to the inverse of the temper- ature. Because we are interested to obtain the thermal Green function, it is convenient to work in the Euclidean analytic continuation of the Green function performing a Wick rotation on the time coordinate, t → iτ . So, we shall work on the Euclidean version of the idealized cosmic string spacetime, which in cylindrical coordinates, can be described by the line element below: ds2 = dτ 2 + dr2 + α2r2dθ2 + dz2 , (4) where α is a parameter smaller than unity which codify the presence of a conical two- surface (r, θ) .3 1The complete analysis about the behavior of the gauge and matter fields near the cosmic string’s core can only be obtained numerically. Some recent numerical analysis [6] about the structure of supermassive cosmic strings show that two different kind of solutions for the metric tensor exist. 2The configurations for magnetic flux provide the same magnetic flux on the two-surface perpendicular to the z−axis in coordinate system defined in (4). 3For a typical Grand Unified Theory, α = 1−O(10−6). The vacuum polarization effects associated with a charged scalar field due to a mag- netic field confined in a tube of finite radius in Minkowski spacetime has been first analysed by Serebryanyi [11]. A few years later this analysis, for an idealized cosmic string space- time, has been considered by Guimarães and Linet [12]; however the magnetic flux was considered as being a line running through the string. The effect of the temperature on this vacuum polarization was also investigated by Guimarães in [13]. In this context, inspired by our previous work [7, 8], we decided to investigate the effect of the temper- ature on the vacuum polarization effects associated with charged massless scalar field in presence of magnetic flux tube in the cosmic string spacetime, considering the three dif- ferent configurations of magnetic field given before. The standard procedure to develop this analysis is by calculating the respective thermal Euclidean Green function. This can be done for an ultrastatic spacetime4 by knowing the Green function at zero tempera- ture. The analysis of the thermal effects on the vacuum polarization effects associated with massless bosonic and fermionic fields in the global monopole spacetime have been considered in [14, 15, 16] few years ago. This paper is organized as follows: In the section 2 we calculate the thermal Euclidean Green function associated with the system for the three different models of magnetic fields. Using the results obtained, we calculate in the sections 3 and 4, the thermal renormalized vacuum expectation value of the field square and the energy-momentum tensor, respectively. We leave for the section 4 our conclusions. 2 The Euclidean thermal Green function The Green function associated with the charged massless scalar field at zero temperature in the presence of a electromagnetic field, must obey the following non-homogeneous second-order differential equation −ggµνDν G(x, x ) = −δ(4)(x− x′) , (5) where Dµ = ∂µ − ieAµ, Aµ being the four-vector potential. In order to reproduce the configurations of magnetic fields along the z−direction given by (1)-(3), we write the the vector potential by Aµ = (0, 0, A(r), 0), with A(r) = a(r) . (6) For the two first models considered, we can represent the radial function a(r) by: a(r) = f(r)Θ(R− r) + Θ(r −R) , (7) f(r) = r2/R2, for the model (i) and r/R, for the model (ii). For the third model, a(r) = Θ(R− r) . (9) 4An ultrastatic spacetime admits a globally defined coordinate system in which the components of the metric tensor are time independent and the conditions g00 = 1 and g0i = 0 hold As we have mentioned, in this work we shall continue in the same line of investigation started in [7, 8], calculating at this time, the thermal contribution on the vacuum polar- ization effects in the region outside the magnetic tube for the three magnetic fields under consideration. The Euclidean Green functions for points outside the magnetic flux tube at zero- temperature are given below: • For the models (i) and (ii), T=0(x, x eiN∆θ 8π2αrr′ sinh u0 ei∆θ sinh(γu0/α) + sinh[(1− γ)u0/α] cosh(u0/α)− cos∆θ dωωJ0 (∆τ)2 + (∆z)2 ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr ′), j = 1, 2 , (10) where n−N − γ , (11) cosh uo = r2 + r + (∆τ)2 + (∆z)2 Djn(ωR) = H ′j(R)I|ν|(ωR)−Hj(R)I ′|ν|(ωR) Hj(R)K |ν|(ωR)−H ′j(R)K|ν|(ωR) . (13) In the above equations the functions Hj(r) are given by: H1(r) = Mσ1,λ1 , (14) with σ1 = ( − ω2R2α )/2 and λ1 = n/2α, and H2(r) = Mσ2,λ2 (ζr) , (15) with σ2 = (δ2 + ω2R2α2)−1/2, λ2 = n/α and ζ = (δ2 + ω2R2α2)1/2. Moreover, Mσ,λ is the Whittaker function, while I|ν| e K|ν| are the modified Bessel functions [17]. • For the model (iii), we have T=0(x, x 8π2αrr′ sinh u0 sinh(u0/α) cosh(uo/α)− cos∆θ dωωJ0 (∆τ)2 + (∆z)2 ein∆θDn(ωR)K|n/α|(ωr)K|n/α|(ωr ′) . (16) where Dn(ωR) = |ν|(ωR)I|n|/ω(ωR)− I|ν|(ωR)I |n|/α(ωR) I|ν|(ωR)K |n|/α(ωR)− I |ν|(ωR)K|n|/α(ωR) . (17) We can observe that the first term on the right hand side of (10) is, up to a gauge transformation, equivalent to the result presented by Guimarães and Linet [12] for a charged massless scalar field in the presence of a magnetic flux running through the cosmic string; and as to (16), its first term corresponds the results found by Smith [18] and by Linet in [19] for a massless scalar field without charge. However, the seconds terms of booths expressions represent corrections on the respective Green functions due to a non-vanishing radius R attributed to the magnetic flux; off course these corrections vanish when we take R → 0. Following the prescription given in the papers by Braden [20] and Page [21], the thermal Green function, GT (x, x ′), can be expressed in terms of the sum GT (x, x GT=0(x, x ′ − lλβ) , (18) where λ is the ”Euclidean” unitary time-like vector and β = 1/kBT , being kB the Boltz- mann constant and T the absolute temperature. In agreement with the equations (10), (16) and (18), the thermal Green functions associated with the massless scalar field in the cosmic string spacetime and in the presence of magnetics field, are given by: • For the models (i) and (ii), T (x, x ′) = G T (x, x dωωJ0 (∆τ + lβ)2 + (∆z)2 ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr ′) , j = 1, 2 . (19) • For the model (iii), T (x, x ′) = GT (x, x dωωJ0 (∆τ + lβ)2 + (∆z)2 ein∆θDn(ωR)K|n|/α(ωr)K|n|/α(ωr ′) . (20) The thermal Green functions G T (x, x ′) and GT (x, x ′) which appear in (19) and (20) have being obtained a few years ago by Guimarães [13] and Linet [22], respectively. So, we shall not repeat them. In fact what we are really interested in this paper is to analyze the seconds terms in these Green functions, which correspond to the thermal contributions due to the finite thickness admitted for the magnetic flux tube. 3 The Computation of 〈φ̂∗(x)φ̂(x)〉Ren. at Non-zero Tem- perature The main objective of this section is to investigate the effects produced by the temperature in the renormalized vacuum expectation value of the square of the charged massless scalar field, 〈φ̂∗(x)φ̂(x)〉, in the presence of a magnetic flux tube of finite radius. Formally this quantity is given by taking the coincidence limit of the Green function: 〈φ̂∗(x)φ̂(x)〉T = lim GT (x, x ′) . (21) However, this procedure provides a divergent result and the divergence comes exclusively from the first terms of the right hand side of (19) and (20)5. In order to obtain a finite and well defined result, we must apply some renormalization procedure. Here we shall adopt the point-splitting renormalization one. It has been observed that the singular behavior of the Green function has the same structure as given by the Hadamard one, which on the other hand can be written in terms of the square of the geodesic distance between two points. So, here we shall adopt the following prescription: we subtract from the Green function the Hadamard one before applying the coincidence limit as shown below: 〈φ̂∗(x)φ̂(x)〉T,Ren. = lim [GT (x, x ′)−GH(x, x′)] . (22) We can write the result as: 〈φ̂∗(x)φ̂(x)〉T,Ren. = 〈φ̂∗(x)φ̂(x)〉T,Reg. + 〈φ̂∗(x)φ̂(x)〉CT=0 + 〈φ̂∗(x)φ̂(x)〉CT . (23) The first term on the right hand side of the above expression, represents, for the models (i) and (ii), the thermal contribution coming from the interaction between charged massless scalar field with a magnetic flux considered as a line running along the cosmic string. From the Guimarães’s paper [13], this term is: 〈φ̂∗(x)φ̂(x)〉T,Reg. = 16π2αβr r cosh u/2 cosh u/2 F (γ)α (u, 0) du , (24) where F (γ)α (u, 0) = −2 sin [πγ/α] cos [u (1− γ) /α] + sin [u (1− γ) /α] cos [πγ/α] cosh u/α− cosπ/α . (25) For the model (iii) an analogous expression can be obtained from the previous one, by taking γ = 06. (An interesting aspect of these results is that the vacuum polarizations depend only on the fractional part of the ration of the magnetic flux by the quantum one, In the high-temperature limit (β → 0), Guimarães showed that, 〈φ̂∗(x)φ̂(x)〉T,Reg. ≈ M (γ) , (26) where the constant M (γ) is given by M (γ) = 16π2α F (γ)α (u, 0) cosh(u/2) du , (27) 5A special feature of these Green functions is that the correction due to the magnetic tube’s radius is finite in the coincidence limit. 6See paper [22]. and in the zero-temperature limit (β → ∞), 〈φ̂∗(x)φ̂(x)〉T,Reg. ≈ , (28) with ω(γ) being given by ω(γ) = − 1 γ − 1 . (29) The two last terms in (23) are corrections on the vacuum polarization due to fi- nite thickness of the radius of the magnetic tube and non-zero temperature. The term 〈φ̂2(x)〉CT=0, corresponds to correction due to finite thickness of the radius of tube only, it was given in [7, 8]. For the two firsts models the corrections are similar. They are given by the component l = 0 in (19), and read (2π)2α Din(ωR)K |ν|(ωr) . (30) Because we are mostly interested to study the vacuum polarization for points very far from the cosmic string, we shall consider R/r ≪ 1. Comparing the behavior of Din(ωR) with the behavior of K2|ν|(ωr), we have shown that we may approximate the integrand of (30) assuming to the coefficient Djn its first order expansion in ωR , which reads: Djn(ωR) = − Γ(|ν| + 1)Γ(|ν|) wnj − |ν|znj wnj + |ν|znj )2|ν| , (31) where wn1 = δ − n Mλ1,λ2(δ/α) + Mγ1,λ1(δ/α), (32) zn1 = Mλ1,λ2(δ/α), (33) wn2 = δ − n Mλ2,λ1(2δ/α) + Mγ2,λ2(2δ/α) , (34) zn2 = Mλ2,λ1(2δ/α) , (35) being γ1 = (n+2α)/2α, γ2 = (n+α)/α and ν = n−N−γ . The most important contributions in (30) comes from the component n = N in the summation. In this way, considering only the dominant term, the corrections are given by: 〈φ̂∗(x)φ̂(x)〉CT=0 = − (2π)2r2 α(2γ + α) wNj − (γ/α)zNj wNj + (γ/α)z )2γ/α . (36) As we can see the corrections present extra dependence on the radial coordinate, conse- quently they are appreciable only in the region close to the tube. As to the third model there happens a very interesting phenomenon. For n 6= 0, the coefficient Dn(ωR) in (16) vanishes at least as fast as (ωR) 2|n|/α when ωR → 0. On the other hand, D0(ωR) vanishes only with the inverse of the logarithm, so quite slowly. In this way the most relevant contribution to the summation comes from n = 0. For this D0(ωR) = + C − α/δ , (37) where C is the Euler constant. In [8] we have explicitly shown that the correction on the vacuum polarization effect for the third model, consequence of a non-vanishing radius of the magnetic flux, is mainly given by: 〈φ̂∗(x)φ̂(x)〉CT=0 = − 8π2αr2 ln e−C+α/δ ) . (38) We also have shown that for α = 0.99 and δ = 0.2 this term is of the same order of the standard vacuum expectation value of the field square in the absence of magnetic flux, up to the distance r which exceeds the radius of the observable Universe. As we have already mentioned, the last term in (23), 〈φ̂∗(x)φ̂(x)〉CT , is consequence of combined effects of the nonvanishing thickness of the magnetic tube and temperature. This is a new contribution and we shall focus on it. For the first two models, this term can be expressed by 〈φ̂∗(x)φ̂(x)〉CT = 2π2αr2 dvvJ0 (vζl) Djn(vR/r)K |ν|(v) , j = 1, 2 , (39) with ζ = β/r. In the above expression we have introduced a dimensionless variable v = ωr. As in the zero temperature analysis, because we are considering R/r ≪ 1, the most important contribution for the above summation comes from the n = N component. So, on basis what we have already discussed for points very far from the string, equation (39) can be approximate to 〈φ̂∗(x)φ̂(x)〉CT = − (2π)2αr2 Γ(γ/α + 1)Γ(γ/α) wNj − (γ/α)zNj wNj + (γ/α)z )2γ/α dvv1+2γ/αJ0 (vζl)K |γ/α|(v) , j = 1, 2 . (40) From the above expression we can observe that the thermal content of this correction is in the summation S given below: S(ζ) = dvv1+2γ/αJ0 (vζl)K |γ/α|(v) . (41) Unfortunately it is not possible to obtain a closed expression to this term, and provide a complete information about the thermal behavior of (40); on the other hand, it is possible to give its main information. In order to do that we shall divide the integral above in two parts: from [0, 2π/ζ ] and from [2π/ζ,∞). Due the strong exponential decay of the modified Bessel function for large argument, the contribution in the interval [2π/ζ,∞) can be neglected in the high-temperature regime, i.e., for ζ ≪ 1; moreover using the series properties for the Bessel function [17], J0(vζl) = − , 0 < v < 2π/ζ , (42) (41) can be approximated to S(ζ) = −1 ∫ 2π/ζ dvv1+2γ/αK2|γ/α|(v) + ∫ 2π/ζ dvv2γ/αK2|γ/α|(v) . (43) In the high-temperature limit the most relevant contribution is given by the second term. Adopting the same approximation criterion to discard the integral in the inter- val [2π/ζ,∞), we may evaluate the integral by taking its upper limit going to infinity. In this way the most relevant contribution to the thermal vacuum polarization effect is [17]7: S(ζ) = 22−2γ/α Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α) Γ (1 + 2γ/α) . (44) Consequently, in the high-temperature regime the correction term, 〈φ̂∗(x)φ̂(x)〉CT , is dom- inated by: 〈φ̂∗(x)φ̂(x)〉CT = − (γ,α,R) )2γ/α , j = 1, 2 , (45) (γ,α,R) 4π3/2α Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α) Γ (1 + 2γ/α) Γ(1 + γ/α)Γ(γ/α) wNj − (γ/α)zNj wNj + (γ/α)z Hence, the expression for renormalized vacuum expectation value of the square of the charged massless scalar field, in the high-temperature limit, is given by 〈φ̂∗(x)φ̂(x)〉T,Ren. ≈ M (γ) (γ,α,R) )2γ/α . (46) From this expression we can observe that the two sub-dominant contributions are of the same order of magnitude for points near the tube. For the third model, 〈φ̂∗(x)φ̂(x)〉CT , is given by the expression below: 〈φ̂∗(x)φ̂(x)〉CT = 2π2αr2 dvvJ0 (vζl) Dn(vR/r)K |n|/α(v) . (47) Again, as in the zero temperature analysis of the vacuum polarization effect, the most relevant contribution comes from the n = 0 component of the above summation for R/r ≪ 1. The respective coefficient is: D0(vR/r) = ln (v/qr) , (48) where e−C+α/δ . (49) In this way (47) can be written by: 〈φ̂∗(x)φ̂(x)〉CT = 2π2αr2 S̄(ζ) , (50) 7By numerical analysis, in [24] we have reached to S(ζ), the same behavior found here, i.e., S(ζ) ≈ 1/ζ. where S̄(ζ) = dvvJ0 (vζl)D0(vR/r)K 0(v) . (51) Applying the previous summation property to the Bessel functions, in the high- temperature regime, (51) can be given by S̄(ζ) = − ∫ 2π/ζ dvvD0(vR/r)K 0(v) + ∫ 2π/ζ dvD0(vR/r)K 0(v) . (52) For ζ << 1 the most relevant contribution to S̄ is given by second term; moreover in this limit we can obtain an approximated expressions for it by taking the upper limit of the integral going to infinite. In this way (52) can be evaluated by S̄(ζ) ∼= K20 (v) ln(v)− ln(qr) . (53) Unfortunately this expression presents a pole for v = qr. However, this pole is con- sequence of the approximation adopted and it is located in the region where where the approximation is no more valid8. In fact the full expression to D0(vR/r) has no singu- larity. D0 is a slowly increasing function of v. As we have mentioned, the full expression to the coefficient D0 vanishes with the inverse of the logarithm for v ≈ 0, and grows less slower than e2vR/r for large v. The square of the modified Bessel function, K20 , provides an integrable logarithmic divergence for small argument, and decays with e−2v for large argument. Except for very small value of v, where the dominant contribution to D0 is given by 1/ ln(v), D0 can be well approximated by 9 −1/ ln(qr). In figure 1, this argument is also numerically justified. There, it is exhibited the behavior of the exact integrand of S(ζ), and its approximated expression discarding the factor ln(v) in the denominator, for specific values of the parameters. Accepting the above arguments, we can represent the main contribution to (53) by: S̄(ζ) = − 1 ζ ln(qr) dvK20(v) , (54) Consequently we obtain 〈φ̂∗(x)φ̂(x)〉CT = − 8αβr ln e−C+α/δ ] . (55) Now for this model, the expression to the renormalized vacuum polarization effect in high-temperature limit reads 〈φ̂∗(x)φ̂(x)〉T,Ren. ≈ M (γ) 8αβr ln e−C+α/δ ] . (56) 8For γ = 0.2, N = 0 and α = 0.99, qr = 2r eC−α/ζ is of order 105 for r/R = 103, so vR/r = 102 consequently bigger than unity. 9In [23], B. Allen at all adopted a similar procedure to provide an approximate expression to the vacuum expectation value of the field square in the cosmic string spacetime considering a general structure to its core; in [8] this procedure has is also been adopted to calculate the 〈φ∗φ〉. 0.5 1 1.5 2 2.5 3 3.5 4 Figure 1: The dashed curve corresponds the exact integrand of the second term of the right hand side of (52), and the solid line the approximated expression discarding ln(v) in the denominator of (53). In the numerical analysis we have used α = 0.99, γ = 0.2, N = 0 and r/R = 103. On basis of previous discussion about the vacuum polarization effect at zero temper- ature, we can conclude that although being sub-dominant, the thermal correction due to the non vanishing radius of the magnetic flux tube, is so relevant as the standard term pro- portional to 1/βr, for points at very large distance from the cosmic string. Consequently, it can be considered as a long-range effect. 4 The Computation of 〈T̂ νµ (x)〉Ren. at Non-zero Tem- perature The energy-momentum tensor, Tµν(x), is a bilinear function of the fields, so we can evaluate its vacuum expectation value, 〈Tµν(x)〉, by the standard method using the Green function [25]. The thermal vacuum average, consequently, can also be obtained by using the thermal Green function. The renormalized vacuum expectation value of the energy-momentum tensor at non- zero temperature for the system adopted here can be calculated by: 〈T̂ νµ 〉T,Ren. = lim µ(α,φ)GT (x, x ′)−Dν′µ(1,0)GH(x, x′)] , (57) where Dν µ(α,φ) stands a second order non-local differential operator in presence of a mag- netic field and cosmic string, defined by µ(α,φ) = (1− 2ξ)DµD ν′ − ξ(DµDν +Dµ′D 2ξ − 1 δνµDσD , (58) ξ being the non-minimal curvature coupling. In the above expression we consider that Dσ = ∇σ−ieAσ, with∇σ being the covariant gravitational derivative and the bar denoting complex conjugate. Again we shall consider the magnetic field configuration given by (7) and (8) for the two first models, and by (9) for the third model. The respective thermal Green function, GT (x ′, x), are given by (19) and (20), respectively. Due to the form of the Green function, we can express (57) by 〈T̂ νµ (x)〉T,Ren. = 〈T̂ νµ (x)〉T,Reg. + 〈T̂ νµ (x)〉CT=0 + 〈T̂ νµ (x)〉CT . (59) The first and second term of the right hand side of the above expression have already been computed in [13] and [8], respectively. As to the first term, Guimarães has shown 〈T̂ νµ (x)〉T,Reg. = diag(−3, 1, 1, 1) + 〈T̃ νµ (x)〉T , (60) where the second term on the right hand side is given in terms of five integrals. However, the author was able to show that in the high-temperature limit (β → 0), this term is proportional to 1/βr3, and in the zero-temperature limit (β → ∞) 〈T̂ νµ (x)〉T,Ren. is proportional to 1/r4. The second term in (59), 〈T̂ νµ (x)〉CT=0, is consequence of a non- vanishing radius to the magnetic flux. For the two first models, its most important contribution is proportional to (1/r4)(R/r)2γ/α, consequently is relevant only in the region near the magnetic tube; on the other hand, for the third model this term presents a long- range effect, similar to what happens in the vacuum expectation value of the field square. The new contribution, 〈T̂ νµ (x)〉CT , is consequence of a non-vanishing radius attributed to the magnetic flux tube and the temperature. For the two first models this term is given by 〈T̂ νµ 〉CT = lim µ(α,φ)G C,(j) T (x, x ′) , (61) C,(j) T (x, x l 6=0 dωωJ0 (∆τ + lβ)2 + (∆z)2 ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr ′) , j = 1, 2 . (62) Following the same procedure adopted in the last section, we shall consider only the n = N component in the summation. Substituting (62) into (61), and using the approx- imated expression to the coefficient D N , after long calculation we arrive at the following result: 〈T̂ νµ 〉CT = − 4π2αr4 Γ (γ/α) Γ (1 + γ/α) wNj − (γ/α)zNj wNj + (γ/α)z )2γ/α 0 , a 1 , a 2 , a where 0 = (4ξ − 1) 3 + I 1 − 2(γ/α)I 2 + 2 (γ/α) 1 = I 1 − 2 (γ/α + 2ξ) I 2 + 4ξ(γ/α)I 4 − I 2 = 2(γ/α) (γ/α− 2ξ) I 4 + 4ξI (4ξ − 1) 3 + I 1 − 2(γ/α)I 2 + 2 (γ/α) 3 = 2I 5 + (4ξ − 1) 3 + I 1 − 2(γ/α)I 2 + 2 (γ/α) dvv3+2γ/αJ0 (vζl)K γ/α+1(v) , (64) dvv2+2γ/αJ0 (vζl)Kγ/α+1Kγ/α(v) , (65) dvv3+2γ/αJ0 (vζl)K γ/α(v) , (66) dvv1+2γ/αJ0 (vζl)K γ/α+1(v) (67) dvv2+2γ/α J1 (vζl) K2γ/α(v) . (68) In order to provide the most relevant contribution to (64)-(67), we adopt the same pro- cedure adopted in last section: we divide the interval of integration in two parts, from [0, 2π/ζ ] and from [2π/ζ,∞). Again in the high-temperature regime, ζ ≪ 1, the integral in the second interval can be neglected due the exponential decay of the modified Bessel function. Finally using the series properties for the Bessel function [17], and after some intermediate steps, we obtain: 〈T̂ νµ 〉CT = − γ,α,R )2γ/α 0 , b 1 , b 2 , b , j = 1, 2 , (69) where γ,α,R (1 + 2γ/α) 2π3/2α Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α) Γ (3 + 2γ/α) Γ(1 + γ/α)Γ(γ/α) wNj − (γ/α)zNj wNj + (γ/α)z being ξ − 1 1 + 5(γ/α) + 8 (γ/α) + 4 (γ/α) 1 + 4(γ/α)− 8ξ 1 + 3(γ/α) + 2 (γ/α) 2 = − 4 + 5(γ/α)− 4 (γ/α)2 − 8ξ 1 + 4(γ/α) + 5 (γ/α) + 2 (γ/α) 3 = − 1 + 4(γ/α) + 8 (γ/α) + 8 (γ/α) 1 + 5(γ/α) + 8 (γ/α) + 4 (γ/α) Analysing the result we can observe that 〈T̂ νµ 〉CT becomes relevant in the region near the magnetic tube. In this region, it is of the same order of magnitude as the sub-dominant contribution obtained in [13] in the high-temperature limit. Also it is possible to check that for the conformal factor ξ = 1/6, the trace of the correction vanishes, i.e., 〈T̂ µµ 〉CT = 0 For the third model the new contribution is given by: 〈T̂ νµ 〉CT = lim µ(α,φ)G C,(3) T (x, x ′) , (70) where C,(3) T (x, x l 6=0 dωωJ0 (∆τ + lβ)2 + (∆z)2 ein∆θDn(ωR)K|n|/α(ωr)K|n|/α(ωr ′) . (71) Using the same procedure adopted to calculate the thermal average of the field square, we shall consider only the component n = 0 in the thermal Green function above. Substi- tuting (71) into (70), using the approximated expression to the coefficient D0, and taking into account the same considerations to overcome the integral problem found in (53), we obtain: 〈T̂ νµ 〉T,Ren. = − 4π2αr4 ln (q/r) 0 , a 1 , a 2 , a , (72) where the coefficients a(0)µ are given by the expressions found in the precedent analysis by taking γ = 0. Moreover in the high-temperature regime (ζ ≪ 1), we obtain: 〈T̂ νµ 〉T,Ren. = − 8αβr3 ln e−C+α/δ ]diag(b 0 , b 1 , b 2 , b 3 ) . (73) Here we also observe that a long-range effect appears, consequently it is so relevant as the sub-dominant contribution proportional to 1/βr3 for points at large distance from the tube. 5 Concluding Remarks In this paper we have analysed the thermal effects on the renormalized vacuum expectation value of the square of a charged massless scalar field, 〈φ̂∗φ̂〉, and in the energy-momentum tensor, 〈T̂ νµ 〉, in the cosmic string spacetime considering the presence of a magnetic flux of finite radius. Three specific configurations of magnetic filed have been considered. For them we could express these vacuum polarization effects as the sum of three different terms as follows: • The first one represents the thermal contribution coming from the interaction be- tween the charged field with a magnetic flux considered as a line running along the cosmic string. This contribution has been exactly calculated in [13]. • The second term is the zero temperature contribution on the vacuum polarization due to the non vanishing radius of the magnetic flux tube. This contribution has been analysed in detail in [8]. • The third contribution is the new one. It comes from the combination of the non vanishing radius of the magnetic flux tube and temperature. It goes to zero for R → 0 and for T → 0. Unfortunately this new contribution cannot be expressed in terms of any analytical function. In order to provide the most important quantitative information about the ther- mal behavior of this contribution, we adopted an approximated procedure. We consider this term in the high-temperature regime. Doing this it was possible, by using the series property for the Bessel function, to obtain an analytical expression to this correction. Apart from the homogeneous thermal contribution to the vacuum polarizations, there appears in calculations of the thermal average of the field square and energy-momentum tensor, corrections due to the geometry of the spacetime, and due to the non-vanishing magnetic flux. The latter presents two parts: one is given as the magnetic flux be a line running along the string, the standard contribution, and the other consequence of a finite transversal radius. These corrections are sub-dominant and depend on the distance to the string. For the two first models for the magnetic fields, the corrections on the thermal average due to the finite radius, are very similar and only relevant in the regions near the tube; as to the third model, it becomes so important as the standard one for large distance. Although the structure of the magnetic field produced by a U(1)−gauge cosmic string cannot be presented by any analytical function, its influence on the thermal vacuum polarization effects of charged matter fields takes place for sure. So in this case, the geometric and magnetic interactions provide contributions. The relevant physical question is how important they are. Trying to clarify this important question, we adopted specific fields configurations which allow us to develop an analytical procedure: we assumed that the magnetic field extent R is much bigger that the cosmic string’s radius considered here equal to zero.10 So our main conclusion is that: although the structure of the magnetic field cannot be very well understood, its influence on the thermal vacuum polarization effect can be so relevant as the influence of the gravitational filed itself. A real cosmic string would be immersed in a bath of primordial heat radiation. The behavior of particle detectors near straight strings immersed in thermal radiation, in free space or passing through black holes, has been analyzed in [26]. The influence of a magnetic field surrounding the string on this detector, can be evaluated by using the corresponding thermal Green functions calculated in this paper. In addition, some results obtained here may shed light upon the vacuum polarization effects induced by a realistic vortex configuration in early cosmology, where the temperature of the Universe was really high. By these results, we can see that thermal contributions to the vacuum expectation 10In fact for the U(1)−gauge cosmic string the magnetic field extent is approximately δ ≈ 1 while the cosmic string radius ξ ≈ 1 . So we are considering the case where >> 1. value of the field square and the energy-momentum tensor, modify the zero-temperature quantities in that epoch. Because of this, they should be taken into account in, for ex- ample, the theory of structure formation. Acknowledgment One of us (ERBM) wants to thanks Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico (CNPq.) for partial financial support, FAPESQ-PB/CNPq. (PRONEX) and FAPES-ES/CNPq. (PRONEX). References [1] Kibble T W, J. Phys. A 9, 1387 (1976). [2] A. Vilenkin, Phys. Rep., 121, 263 (1985). [3] N. B. Nielsen and P. Olesen, Nucl. Phys. B61, 45 (1973). [4] D. Garfinkle, Phys. Rev. D 32, 1323 (1985). [5] B. Linet, Phys. Lett. B 124, 240 (1987). [6] M. Christensen, A. L. Larsen and Y. Verbin, Phys. Red. D 60, 125012 (1999); Y. Brihaye and M. Lubo, ibid 62, 085005 (2000). [7] J. Spinelly and E. R. Bezerra de Mello, Int. J. Mod. Phys. A 17, 4375 (2002). [8] J. Spinelly and E. R. Bezerra de Mello, Class. Quantum Grav. 20, 873 (2003). [9] J. Spinelly and E. R. Bezerra de Mello, Int. J. Mod. Phys. D 13, 607 (2004). [10] J. Spinelly and E. R. Bezerra de Mello, Nucl Phys. B (Proc. Suppl.) 127, 77 (2004). [11] E. M. Serebryanyi, Theor. Math. Phys. 64, 846 (1985). [12] M. E. X. Gumarães and B. Linet, Commun. Math. Phy. 165, 297 (1994). [13] M.E.X. Guimarães, Class. Quantum Grav. 12, 1705 (1995). [14] F. C. Cabral and E. R. Bezerra de Mello, Class. Quantum Grav. 18, 1637 (2001). [15] F. C. Cabral and E. R. Bezerra de Mello, Class. Quantum Grav. 18, 5455 (2001). [16] E. R. Bezerra de Mello and F. C. Cabral, Int. J. Mod. Phys. A 17 879 (2002). [17] I. S. Gradshteyn e I. M Ryzhik, Table of Integral, Series and Products (Academic Press, New York, 1980). [18] A. G. Smith, in Symposium on the Formation and Evolution of Cosmic String, edited by G. W. Gibbons, S. W. Hawking and T. Vachaspati (Cambridge University Press, Cambridge, England, 1989). [19] B. Linet, Phys. Rev. D, 35, 536 (1987). [20] H. W. Braden, Phys. Rev. D 25, 1028 (1982). [21] D. N. Page, Phys. Rev. D 25, 1499 (1982). [22] B. Linet, Class. Quantum Grav. 9, 2429 (1992). [23] B. Allen, B. S. Kay and A. C. Ottewill, Phys. Rev. D 53, 6829 (1996). [24] J. Spinelly and E. R. Bezerra de Mello, PoS(IC2006), 056 (2006). [25] N. D. Birrell and P. C. W Davis, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England, 1982). [26] P. C. Davies and V. Sahni, Class. Quantum Grav. 5, 1 (1988).

0704.1991

Single pion electro-- and neutrinoproduction on heavy targets

arXiv:0704.1991v1 [hep-ph] 16 Apr 2007 Preprint typeset in JHEP style - HYPER VERSION DO-TH 07/05 LPSC 07-29 SMU-HEP 07-07 Single pion electro– and neutrinoproduction on heavy targets E. A. Paschos Theoretische Physik III, University of Dortmund, D-44221 Dortmund, Germany E-mail: [email protected] I. Schienbein Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier/CNRS-IN2P3, 53 Avenue des Martyrs, F-38026 Grenoble, France E-mail: [email protected] J.-Y. Yu Southern Methodist University, Dallas, Texas 75275, USA E-mail: [email protected] Abstract: We present a calculation of single pion electroproduction cross sections on heavy targets in the kinematic region of the ∆(1232) resonance. Final state interactions of the pions are taken into account using the pion multiple scattering model of Adler, Nussinov and Paschos (ANP model). For electroproduction and neutral current reactions we obtain results for carbon, oxygen, argon and iron targets and find a significant reduction of the W -spectra for π0 as compared to the free nucleon case. On the other hand, the charged pion spectra are only little affected by final state interactions. Measurements of such cross sections with the CLAS detector at JLAB could help to improve our understanding of pion rescattering effects and serve as important/valuable input for calculations of single pion neutrinoproduction on heavy targets relevant for current and future long baseline neutrino experiments. Two ratios, in Eq. (3.8) and (3.10), will test important properties of the model. Keywords: single pion production, nuclear effects, long baseline experiments. http://arxiv.org/abs/0704.1991v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] http://jhep.sissa.it/stdsearch Contents 1. Introduction 1 2. Free nucleon cross sections 3 3. Cross sections for heavy targets 5 3.1 Pion rescattering in the ANP model 5 3.2 Results for various targets 7 3.2.1 Neutrinoproduction 7 3.2.2 Electroproduction 7 4. Summary 9 A. Charge exchange matrices in the double averaging approximation 11 B. Charge exchange matrices for various amounts of pion absorption 11 C. Forward- and backward charge exchange matrices 12 1. Introduction Neutrino interactions at low and medium energies are attracting attention because they will be measured accurately in the new generation of experiments [1, 2]. One aim of the experiments is to measure the precise form of the cross sections and their dependence on the input parameters. This way we check their couplings and compare the functional dependence of the form factors, where deviations from the dipole dependence have already been established (see e.g. figure 1 in [3] and references therein). Deviation from the standard model predictions can arise either from properties of the neutrinos or from new couplings of the gauge bosons to the particles in the target. Another aim of the experiments is to establish the properties of neutrinos including their masses, mixings and their fermionic nature (Dirac or Majorana particles). This program requires a good understanding of the cross sections, which motivated a new generation of calculations. Since the experiments use nuclear targets, like C12, O16, Ar40, Fe56, ... it is necessary to understand the modifications brought about by the targets. The very old calculations for quasi-elastic scattering and resonance excitation on free nucleons [4, 5] have been replaced by new results where couplings and form factors are now better determined. For the vector couplings comparisons with electroproduction data have been very useful [3, 6]. Axial couplings are frequently determined by PCAC. There are already improvements and checks of the earlier quark models [7]. Comparisons with – 1 – experimental data are also available even though the experimental results are not always consistent with each other [8, 9, 10] but there are plans for improvements that will resolve the differences [1, 2]. For reactions on nuclear targets there are modifications brought about by the propaga- tion of the produced particles in the nuclear medium. They involve absorption of particles, restrictions from Pauli blocking, Fermi motion and charge-exchange rescatterings. One group of papers uses nuclear potentials for the propagation of the particles [11]. Others use a transport theory of the final particles including channels coupled to each other [12]. These groups gained experience by analyzing reactions with electron beams (electroproduction) and adopted their methods to neutrino reactions [12]. Our group investigated 1-π pion production on medium and heavy targets employing the pion multiple scattering model by Adler, Nussinov and Paschos [13] that was developed in order to understand neutral current neutrino interactions with nuclei. This model was useful in the discovery of neutral currents and has been applied to predict neutrino-induced single pion production on Oxygen, Argon and Iron targets [14, 15, 16] which are used in long baseline(LBL) experiments. Among its characteristics is the importance of charge- exchange reactions that modify the π+ : π0 : π− ratios of the original neutrino-nucleon interaction through their scatterings within the nuclei. The presence of this effect has been confirmed by experiments [17]. We note here that our results are valid for isoscalar targets. For non-isoscalar targets like lead, used in the OPERA experiment, it is possible to extend the ANP model [18], which can be done in the future. In this article we take an inverse route and use our calculation in neutrino reactions to go back to the electroproduction of pions on free nucleons and heavy nuclei. The plan of the paper is as follows. In section 2 we summarize the neutrino production cross sections on free nucleons and in the ∆ resonance region. This topic has been described by several groups in the past few years. We present cross sections differential in several variables Eπ, Q 2 and W . We pay special attention to the spectrum dσ/dEπ, where we correct an error we found in our earlier calculation [14]. Then we obtain the electroproduction cross section by setting the axial coupling equal to zero and rescaling, appropriately, the vector current contribution. The main content of the article appears in section 3 where we describe the salient features and results of the ANP model. This model has the nice property that it can be written in analytic form including charge exchange and absorption of pions. This way we can trace the origin of the effects and formulate quantities which test specific terms and parameters. As we mentioned above several features have been tested already, and we wish to use electroproduction data in order to determine the accuracy of the predictions. We present numerical results for different target materials, and study the quality of the averaging approximation and uncertainties of the ANP model due to pion absorption ef- fects. We discuss how the shape of the pion absorption cross section (per nucleon), an important and almost unconstrained ingredient of the ANP model, can be delineated from a measurement of the total fraction of absorbed pions. Finally, in Sec. 4 we summarize the main results. Averaged rescattering matrices for carbon, oxygen, argon, and iron targets and for different amounts of pion absorption have been collected in the appendices and are – 2 – useful for simple estimates of the rescattering effects. 2. Free nucleon cross sections In the following sections, leptonic pion production on nuclear targets is regraded as a two step process. In the first step, the pions are produced from constituent nucleons in the target with free lepton-nucleon cross sections [13]. In the second step the produced pions undergo a nuclear interaction described by a transport matrix. Of course, the resonances themselves propagate in the nuclear medium before they decay, an effect that we will investigate in the future. The leptonic production of pions in the ∆-resonance region is theoretically available and rather well understood as described in articles for both electro- and neutrino produc- tion, where comparisons with available data are in good agreement [3, 6, 7, 19, 12]. The available data is described accurately with the proposed parameterizations. The vector form factors are modified dipoles [3] which reproduce the helicity amplitudes mea- sured in electroproduction experiments at Jefferson Laboratory [7]. The coupling in the axial form factors are determined by PCAC and data. Their functional dependence in Q2 is determined by fitting the dσ distributions. For the vector form factors the magnetic dipole dominance for CV3 (q 2) and CV4 (q 2) gives an accurate description of the data. How- ever, deviations with a non-zero CV5 (q 2) have also been established [7]. This way a small (5%) isoscalar amplitude is reproduced. For the propose of this article we shall use a scaling relation connecting neutrino- to electroproduction. The weak vector current is in the same isospin multiplied with the electromagnetic current and the two are related as follows: < ∆++|V |p >= 3 < ∆+|JI=1em |p >= 3 < ∆0|JI=1em |n > . Taking into account the isospin Clebsch-Gordan factors for the ∆ → Nπ branchings one finds the following contributions of the ∆-resonance to the cross sections for ep → epπ0, ep → enπ+, en → epπ− and en → enπ0 dσem,I=1 dQ2dW dQ2dW   : pπ0 : nπ+ : pπ− : nπ0 (2.1) where dV dQ2dW denotes the cross section for the vector contribution alone to the reaction νp → µ−pπ+. The free nucleon cross sections in Eq. (2.1) will be used in our numerical analysis. We shall call this the reduced electromagnetic formula. Its accuracy was tested in figure (5) of ref. [3]. Further comparisons can be found in [20]. For studies of the pion angular distributions (or what is the same of the pion energy spectrum in the laboratory frame) we begin with the triple differential cross section for – 3 – neutrino production dQ2dWd cos θ⋆π 16πM2 KiW̃i − KiDi(3 cos 2 θ⋆π − 1) (2.2) with Ki being kinematic factors of W and Q 2 and the structure functions W̃i(Q 2,W ) and 2,W ) representing the dynamics for the process. All of them are found in ref. [5]. The angle θ⋆π is the polar angle of the pion in the CM frame with cos θ⋆π = −γECMSπ + Eπ βγ|~p CMSπ | (2.3) where |~p CMSπ | = (ECMSπ ) 2 −m2π with ECMSπ = W 2 +m2π −M2N (2.4) and the rest of the variables defined as W 2 +Q2 −M2N , γ = ν +MN , βγ = ν2 +Q2 . (2.5) It is now straight-forward to convert the cross section differential in the solid angle to the one differential in the laboratory energy of the pion, Eπ, γβ|~p CMSπ | d cos θ⋆π . (2.6) Having expressed all quantities in (2.2) and (2.5) in terms of W, Q2 and Eπ it is possible to compute the pion energy spectrum ∫ Wmax ∫ Q2max dQ2dWdEπ θ(phys). (2.7) The limits of integration are given as Q2min = 0 , Q max = (S −W 2)(S −M2N ) Wmin = MN +mπ , Wmax ≃ 1.6 GeV (2.8) where S = M2N + 2MNE1 is the center-of-mass energy squared with E1 the energy of the incoming lepton in the LAB system. The θ-function takes care of the constraints from the phase space. We integrated the cross section for Eν = 1 GeV and show the spectrum in figures 1–3. In our earlier publication [14] the spectrum for Eπ was incorrect because we did not impose the phase space constraints correctly. The pion spectrum for charged current reactions is correctly reported in figure (4) in ref. [21]. The discrepancy in ref. [14] has been pointed out for neutral currents in ref [12]. The neutrino–nucleon and electron–nucleon cross sections will be used in the rest of this article in order to compute and test effects of nuclear corrections. We deduce the electroproduction cross sections from neutrino production as in Eq. (2.1). For the triple – 4 – differential cross section we follow the same procedure by setting the axial form factors to zero and using the relation dσem,I=1 dQ2dWdEπ dQ2dWdEπ   : ep → epπ0 : ep → enπ+ : en → epπ− : en → enπ0 (2.9) A small isoscalar part in the electromagnetic cross section is omitted since it does not contribute to the ∆-resonance but only to the background, which for W < 1.3 GeV is small and contributes for 1.3 GeV < W < 1.4 GeV. 3. Cross sections for heavy targets In the following we will deal with single pion resonance production in the scattering of a lepton l off a nuclear target T (6C 12, 8O 16, 18Ar 40, 26Fe 56), i.e., with the reactions l + T → l′ + T ′ + π±,0 (3.1) where l′ is the outgoing lepton and T ′ a final nuclear state. Furthermore, in our analysis of nuclear rescattering effects we will restrict ourselves to the region of the ∆(1232) resonance, 1.1 GeV < W < 1.4 GeV, and to isoscalar targets with equal number of protons and neutrons. 3.1 Pion rescattering in the ANP model According to the ANP model [13, 22] the final cross sections for pions (π+, π0, π−)f can be related to the initial cross sections (π+, π0, π−)i for a free nucleon target in the simple form   dσ(ZT A;π+) dQ2dW dσ(ZT A;π0) dQ2dW dσ(ZT A;π−) dQ2dW  = M [T ;Q2,W ]  dσ(NT ;π dQ2dW dσ(NT ;π dQ2dW dσ(NT ;π dQ2dW  (3.2) dσ(NT ;±0) dQ2dW dσ(p;±0) dQ2dW + (A− Z)dσ(n;±0) dQ2dW (3.3) where the free nucleon cross sections are averaged over the Fermi momentum of the nucle- ons.1 For an isoscalar target the matrix M is described by three independent parameters Ap, d, and c in the following form [13] M = Ap 1− c− d d c d 1− 2d d c d 1− c− d  , (3.4) However, the Fermi motion has a very small effect on the W distribution and we neglect it in our numerical analysis. On the other hand, effects of the Pauli exclusion principle have been absorbed into the matrix M and are taken into account. – 5 – where Ap(Q 2,W ) = g(Q2,W ) × f(1,W ). Here, g(Q2,W ) is the Pauli suppression factor and f(1,W ) is a transport function for equal populations of π+, π0, π− which depends on the absorption cross section of pions in the nucleus. The parameters c and d describe the charge exchange contribution. The final yields of π’s depend on the target material and the final state kinematic variables, i.e., M = M [T ;Q2,W ]. In order to simplify the problem it is helpful to integrate the doubly differential cross sections of Eq. (3.2) over W in the (3, 3) resonance region, say, mp +mπ ≤ W ≤ 1.4 GeV. In this case Eq. (3.2) can be replaced by an equation of identical form  dσ(ZT A;π+) dσ(ZT A;π0) dσ(ZT A;π−)  = M [T ;Q2]  dσ(NT ;π dσ(NT ;π dσ(NT ;π  (3.5) where the matrix M [T ;Q2] can be obtained by averaging the matrix M [T ;Q2,W ] over W with the leading W -dependence coming from the ∆ resonance contribution. Moreover, we expect the matrix M to be a slowly varying function of Q2 (for Q2 & 0.3 GeV2). For this reason we introduce a second averaging over Q2 and define the double averaged matrix M [T ] which is particularly useful for giving a simple description of charge exchange effects in different nuclear targets. In the double-averaging approximation (AV2) the final cross sections including nuclear corrections are expressed as follows:  dσ(ZT A;π+) dQ2dW dσ(ZT A;π0) dQ2dW dσ(ZT A;π−) dQ2dW  = M [T ]  dσ(NT ;π dQ2dW dσ(NT ;π dQ2dW dσ(NT ;π dQ2dW  . (3.6) We note that the cross sections are differential in two variables while the matrix M [T ] is the average over these variables. The above discussion will be used for a phenomenological description of nuclear rescat- tering effects. On the other hand, in Ref. [13] a dynamical model has been developed to calculate the charge exchange matrix M . As an example, for oxygen the resulting matrix in the double-averaging approximation is given by 16) = Ap 0.788 0.158 0.0537 0.158 0.684 0.158 0.0537 0.158 0.788  . (3.7) with Ap = 0.766, which contains the averaged Pauli suppression factor and absorption of pions in the nucleus. There are various absorption models described in the original article. Two of them are distinguished by the energy dependence of the absorption cross section beyond the ∆ region. In model [A] the absorption increases as W increases while in [B] – 6 – it decreases for large W ’s (beyond the ∆ region). A comparison of the two absorption models (A) and (B) can be found in [22]. Since the fraction of absorbed pions is still rather uncertain we provide in the appendices ANP matrices for different amounts of absorption. These matrices are useful to obtain an uncertainty band for the expected nuclear corrections. 3.2 Results for various targets In this section we present numerical results for 1-pion leptoproduction differential cross sections including nuclear corrections using the ANP model outlined in the preceding sec- tion. 3.2.1 Neutrinoproduction We begin with a discussion of the nuclear corrections to the pion energy spectra in neutrino scattering shown in Figs. 1–3, where the curves are neutral current reactions. The dotted lines are the spectra for the free nucleon cross sections. The dashed lines include the effect of the Pauli suppression (in step one of the two step process), whereas the solid line in addition takes into account the pion multiple scattering. These curves correct Figs. 8–16 in Ref. [14]. Similar curves have been obtained recently by Leitner et al. [12] who also noticed the error in [14]. Even though the models differ in the transport matrix, they both include charge exchange effects. For example, they both find that for reactions where the charge of the pions is the same with the charge of the current the pion yield shows a substantial decrease. 3.2.2 Electroproduction We now turn to the electroproduction. To be specific, our analysis will be done under the conditions of the Cebaf Large Acceptance Spectrometer (CLAS) at Jefferson Lab (JLAB). The CLAS detector [23] covers a large fraction of the full solid angle with efficient neutral and charged particle detection. Therefore it is very well suited to perform a high statistics measurement on various light and heavy nuclear targets and to test the ideas of pion multiple scattering models. In the future these measurements can be compared with results in neutrinoproduction from the Minerva experiment [1] using the high intensity Numi neutrino beam. If not stated otherwise we use an electron energy Ee = 2.7 GeV in order to come as close as possible to the relevant low energy range of the LBL experiments. For the momentum transfer we take the values Q2 = 0.4, 0.8 GeV2 in order to avoid the experimentally and theoretically more problematic region at very low Q2. Results for larger Q2 and larger energies, say Ee = 10 GeV, are qualitatively very similar. Figure 4 shows the double differential cross section dσ/dQ2dW for π+ and π0 produc- tion versus W for an oxygen target. The solid lines have been obtained with help of Eq. (3.2) including the nuclear corrections. The dashed lines show the result of the double- averaging approximation according to Eq. (3.6) using the ANP matrix in Eq. (3.7). The dotted line is the free cross section in Eq. (3.3). One sees, the double-averaging approxima- tion and the exact calculation give very similar results such that the former is well-suited for simple estimates to an accuracy of 10% of pion rescattering effects. We observe that – 7 – the cross sections for π0 production are largely reduced by about 40% due to the nuclear corrections. This can be understood since the larger π0 cross sections are reduced by ab- sorption effects and charge exchange effects. On the other hand, the π+ cross sections are even slightly enlarged, because the reduction due to pion absorption is compensated by an increase due to charge exchange. The compensation is substantial since the π0 yields are dominant. In Fig. 5 double differential cross sections per nucleon for different target materials are presented. The electron energy and the momentum transfer have been chosen as Ee = 2.7 GeV and Q 2 = 0.4 GeV2, respectively. The results for the pion rescattering corrections have been obtained within the double-averaging approximation (3.6) which allows for a simple comparison of the dependence on the target material in terms of the matrices M [T ] which can be found in Eq. (3.7) and App. A. For comparison the free nucleon cross section (3.3) (isoscalar ) is also shown. As expected, the nuclear corrections become larger with increasing atomic number from carbon to iron. One of the input quantities for calculating the transport function f(λ) in the ANP model is the pion absorption cross section σabs(W ) describing the probability that the pion is absorbed in a single rescattering process. For σabs(W ) the ANP article reported results for two parameterizations, models A and B, taken from Refs. [24, 25] which have very different W -dependence and normalization. However, the predictions of the ANP model in the double-averaging approximation are primarily sensitive to the normalization of the pion absorption cross section at W ≃ m∆ [22]. Using data by Merenyi et al. [26] for a neon target it was found that about 25% ± 5% of pions are absorbed making possible the determination of the normalization of σabs(W ) with a 20% accuracy. In order to investigate the theoretical uncertainty due to pion absorption effects we show in Fig. 6 double differential cross sections dσ/dQ2dW for π+ and π0 production vs W for different amounts of pion absorption in oxygen: 25% (solid line), 20% (dashed line), 30% (dotted line). The π0 and π+ spectra have been calculated in the double-averaging approximation (3.6) utilizing the matrices in App. B. The three curves represent the theoretical uncertainty due to pion absorption effects. For comparison, the free nucleon cross section (3.3) is shown as well. Although the predictions of the ANP model are mainly sensitive to σabs(W ≃ m∆) it would be interesting to obtain more information on the detailed W -shape. The fraction of absorbed pions can be determined by measuring the inclusive pion production cross sections for a nuclear target divided by the free nucleon cross sections, Abs(Q2,W) = 1− k=0,± dσ(ZT A;πk) dQ2dW j=0,± dσ(NT;π dQ2dW = 1−Ap(Q2,W) , (3.8) where Ap has been introduced in (3.4). This quantity is related to σabs(W ) as can be seen by linearizing the transport function f(λ,W ) [16, 22] Abs(Q2,W) ≃ 1 L̄ρ0 × σabs(W) . (3.9) – 8 – Here L̄ is the effective length of the nucleus averaged over impact parameters and ρ0 the charge density in the center. As an example, for oxygen one finds L̄ ≃ 1.9R with radius R ≃ 1.833 fm and ρ0 = 0.141 fm−3. Therefore, the W -dependence of σabs(W ) can be reconstructed from the fraction of absorbed pions, i.e. Abs(Q2,W). Summing over the three charged pions eliminates charge exchange effects. In order to verify the linearized approximation in Eq. (3.9), we show in Fig. 7 the ANP model prediction for Abs(Q2,W) for oxygen and iron targets with Q2 = 0.3 GeV2. This prediction strongly depends on the shape of the cross section σabs(W ) for which we use model B from Refs. [25]. σabs(W ) multiplied by a free normalization factors for oxygen and iron, respectively, is depicted by the dashed lines. Obviously, Eq. (3.9) is quite well satisfied for oxygen and still reasonably good for iron. Finally, the dotted line shows the result of the averaging approximation. We conclude that σabs(W ) can be extracted with help of Eqs. (3.8) and (3.9). For completeness, we mention that the pion absorption in nuclei is reported in various articles [27]. For comparisons one should be careful because the absorption cross sections in pi-nucleus and in neutrino-nucleus reactions are different, in the former case it is a surface effect while in the latter it occurs everywhere in the nucleus. A useful test of charge exchange effects is provided by the double ratio DR(Q2,W ) = π+ + π− π+ + π− (3.10) where (πi)A represents the doubly differential cross section dσ/dQ 2dW for the production of a pion πi in eA scattering. This observable is expected to be rather robust with respect to radiative corrections and acceptance differences between neutral and charged pions.2 In Fig. 8 we show the double ratio for a carbon target in dependence of W for a fixed Q2 = 0.4 GeV2. The dependence on Q2 is weak and results for other values of Q2 are very similar. The solid line shows the exact result, whereas the dotted lines have been obtained in the double averaging approximation with minimal and maximal amounts of pion absorption. As can be seen, the results are rather insensitive to the exact amount of pion absorption. Without charge exchange effects (and assuming similar absorption of charged and neutral pions) the double ratio would be close to unity. As can be seen, the ANP model predicts a double ratio smaller than 0.6 in the region W ≃ 1.2 GeV. A confirmation of this expectation would be a clear signal of pion charge exchange predominantly governed by isospin symmetry. In this case it would be interesting to go a step further and to study similar ratios for pion angular distributions. 4. Summary Lepton induced reactions on medium and heavy nuclei include the rescattering of produced pions inside the nuclei. This is especially noticeable in the ∆-resonance region, where the produced resonance decays into a nucleon and a pion. In the introduction and section We are grateful to S. Manly for drawing our attention to the double ratio. – 9 – 2 we reviewed the progress that has been made in the calculations of neutrino-induced reactions on free protons and neutrons, because we needed them for following calculations. For several resonances the vector form factors have been recently determined by using electroproduction results in Jefferson Laboratory [7]. For the axial form factors modified dipoles give an accurate description of the data. For the purposes of this article (studies of nuclear corrections) it suffices to deduce the electroproduction cross sections through Eqs. (2.1) and (2.9). The main contribution of this article is contained in section 3, where we describe important features of the ANP model and define single- and double averaged transport matrices. Two important aspects of rescattering are emphasized: (i) the absorption of the pions and (ii) charge exchange occurring in the multiple scattering, where we have shown that special features of the data are attributed to each of them. Finally we propose specific ratios of electroproduction reactions that are sensitive to the absorption cross section and to charge exchange effects. Using the model we calculate the transport matrix for various absorption cross sec- tions and nuclei and present the results in appendix A. We also calculated the pion energy spectra with and without nuclear corrections. The results appear in figures 1–3 and can be compared with other calculations [12]. Comparison of the double averaged approximation with the exact ANP calculation shows small differences (figure 4). As mentioned already, electroproduction data are very useful in testing several aspects of the model and its pre- dictions. For the absorption cross section we propose in Eq. (3.8) a ratio that depends only on Ap(Q 2,W ) = g(Q2,W )f(1,W ). Since we consider isoscalar targets and sum over the charges of the pions, charge exchange terms are eliminated. This leaves over the depen- dence on charge independent effects, like the Pauli factor and the average absorption; this is indeed the average absorption of pions and even includes the absorption of the ∆-resonance itself. Another ratio (DR(Q2,W )) is sensitive to charge exchange effects. In the double ratio the dependence on Ap(Q 2,W ) drops out and the surviving terms are isospin dependent. Our calculation shows that the ratio depends on W with the largest reduction occurring in the region 1.1 < W < 1.25 GeV. Finally, the ∆(1232) is a sharply peaked resonance, where the resonant interaction, takes place over small ranges of the kinematic variables, so that averaging over them gives accurate approximations. This is analogous to a narrow width approximation. Several comparisons in this article confirm the expectation that averaged quantities give rather accurate approximations of more extensive calculations. Acknowledgments We wish to thank W. Brooks and S. Manly for many useful discussions, their interest and encouragement. The work of J. Y. Yu is supported by the Deutsche Forschungsgemeinschaft (DFG) through Grant No. YU 118/1-1. – 10 – Appendix A. Charge exchange matrices in the double averaging approximation Carbon: 12) = Ap 0.826 0.136 0.038 0.136 0.728 0.136 0.038 0.136 0.826  (A.1) with Ap = 0.791 . Argon: M(18Ar 40) = Ap 0.733 0.187 0.080 0.187 0.626 0.187 0.080 0.187 0.733  (A.2) with Ap = 0.657 . Iron: M (26Fe 56) = Ap 0.720 0.194 0.086 0.194 0.613 0.194 0.086 0.194 0.720  (A.3) with Ap = 0.631 . B. Charge exchange matrices for various amounts of pion absorption Carbon: 15% absorption 12) = Ap 0.817 0.141 0.041 0.141 0.718 0.141 0.041 0.141 0.817  (B.1) with Ap = 0.831 . 20% absorption 12) = Ap 0.829 0.134 0.037 0.134 0.731 0.134 0.037 0.134 0.829  (B.2) with Ap = 0.782 . – 11 – 25% absorption 12) = Ap 0.840 0.127 0.032 0.127 0.745 0.127 0.032 0.127 0.840  (B.3) with Ap = 0.734 . Oxygen: 15% absorption 16) = Ap 0.771 0.167 0.062 0.167 0.665 0.167 0.062 0.167 0.771  (B.4) with Ap = 0.833 . 20% absorption 16) = Ap 0.783 0.161 0.056 0.161 0.679 0.161 0.056 0.161 0.783  (B.5) with Ap = 0.784 . 25% absorption 16) = Ap 0.797 0.153 0.050 0.153 0.693 0.153 0.050 0.153 0.797  (B.6) with Ap = 0.735 . 30% absorption 16) = Ap 0.810 0.146 0.044 0.146 0.709 0.146 0.044 0.146 0.810  (B.7) with Ap = 0.687 . C. Forward- and backward charge exchange matrices Carbon: 15% absorption M+(6C 12) = Ap+ 0.870 0.100 0.029 0.100 0.799 0.100 0.029 0.100 0.870  ,M−(6C12) = Ap− 0.675 0.251 0.074 0.251 0.498 0.251 0.074 0.251 0.675  (C.1) – 12 – with Ap+ = 0.606 and Ap− = 0.225. 20% absorption M+(6C 12) = Ap+ 0.880 0.094 0.026 0.094 0.811 0.094 0.026 0.094 0.880  ,M−(6C12) = Ap− 0.685 0.247 0.068 0.247 0.505 0.247 0.068 0.247 0.685  (C.2) with Ap+ = 0.578 and Ap− = 0.204. 25% absorption M+(6C 12) = Ap+ 0.889 0.088 0.022 0.088 0.823 0.088 0.022 0.088 0.889  ,M−(6C12) = Ap− 0.695 0.243 0.062 0.243 0.513 0.243 0.062 0.243 0.695  (C.3) with Ap+ = 0.549 and Ap− = 0.184. 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Yu, talk presented at the Second International Workshop on Neutrino-Nucleus Interactions in the few-GeV Region (NUINT’02), Irvine, California, December 2002. Homepage: http://www.ps.uci.edu/∼nuint/ , hep-ph/0308010. [23] B. A. Mecking et al., CLAS Collaboration, Nucl. Instrum. Meth. A503, 513 (2003). [24] M. M. Sternheim and R. R. Silbar, Phys. Rev. D6, 3117 (1972). [25] R. R. Silbar and M. M. Sternheim, Phys. Rev. C8, 492 (1973). [26] R. Merenyi et al., Phys. Rev. D45, 743 (1992). [27] D. Ashery and J. P. Schiffer, Ann. Rev. Nucl. Part. Sci. 36, 207 (1986). C. H. Q. Ingram, Nucl. Phys. A684, 122 (2001). R. D. Ransome, Nucl. Phys. Proc. Suppl. 139, 208 (2005), and references therein. – 15 – π+ ig Eπ (GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 π0 ig Eπ (GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 π- ig Eπ (GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 1: Differential cross section per nucleon for single pion spectra of π+, π0, π− for oxygen with = 1 GeV in dependence of pion energy E . The curves correspond to neutral current reactions. – 16 – π+ ig Eπ (GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 π0 ig Eπ (GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 π- ig Eπ (GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 2: The same as in fig. 1 for argon. – 17 – π+ ig Eπ(GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 π0 ig Eπ(GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 π- ig Eπ(GeV) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 3: The same as in fig. 1 for iron. – 18 – Q2 = 0.4 GeV2 π0 f (exact) π0 f (av. app.) W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Q2 = 0.4 GeV2 π+ f (exact) π+ f (av. app.) W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Q2 = 0.8 GeV2 π0 f (exact) π0 f (av. app.) W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Q2 = 0.8 GeV2 π+ f (exact) π+ f (av. app.) W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Figure 4: Double differential cross sections for single-pion electroproduction for an oxygen target in dependence of W . Spectra for π0 and π+ production are shown for Q2 = 0.4 GeV2 and Q2 = 0.8 GeV2 using an electron energy E = 2.7 GeV. The solid and dotted lines have been obtained according to (3.2) using the exact ANP matrix M(W,Q2) and (3.6) utilizing the double-averaged ANP matrix M in (3.7), respectively. The dashed lines show the free nucleon cross section (3.3). Q2 = 0.4 GeV2 π0 f (8O π0 f (6C π0 f (18Ar π0 f (26Fe W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Q2 = 0.4 GeV2 π+ f (8O π+ f (6C π+ f (18Ar π+ f (26Fe W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Figure 5: Double differential cross sections per nucleon for single-pion electroproduction for different target materials. W -spectra for π0 and π+ production are shown for Q2 = 0.4 GeV2 using an electron energy E = 2.7 GeV. The pion rescattering corrections have been calculated in the double-averaging approximation (3.6) using the ANP matrices in (3.7) and App. A. For comparison, the free nucleon cross section (3.3) is shown. – 19 – Q2 = 0.8 GeV2 π0 f (Abs. 25%) π0 f (Abs. 20%) π0 f (Abs. 30%) W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Q2 = 0.8 GeV2 π+ f (Abs. 25%) π+ f (Abs. 20%) π+ f (Abs. 30%) W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Figure 6: Double differential cross sections per nucleon for single-pion electroproduction for oxygen with 20% (dashed line), 25% (solid line) and 30% (dotted line) pion absorption. Furthermore, Q2 = 0.8 GeV2 and E = 2.7 GeV. The π0 and π+ spectra have been calculated in the double-averaging approximation (3.6) utilizing the matrices in App. B. For comparison, the free nucleon cross section (3.3) is shown as well. exact Av. Approx. N * sigabs(W) W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 Figure 7: The fraction of absorbed pions, Abs(Q2,W), in dependence of W for oxygen and iron targets for Q2 = 0.3 GeV2. Also shown is the cross section σabs(W ) (model B) multiplied by free normalization factors (dashed lines). The dotted lines are the result for Abs(Q2,W) in the averaging approximation. – 20 – Q2 = 0.4 GeV2 W(GeV) 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Figure 8: Double ratio of single pion electroproduction cross sections in dependence of W for fixed Q2 = 0.4 GeV2 as defined in Eq. (3.10). The dotted lines show results in the double averaging approximation with varying amounts of absorption. – 21 –

0704.1992

Yukawa's Pion, Low-Energy QCD and Nuclear Chiral Dynamics

arXiv:0704.1992v1 [nucl-th] 16 Apr 2007 Yukawa’s Pion, Low-Energy QCD and Nuclear Chiral Dynamics Wolfram Weise Physik-Department, Technische Universität München, D-85747 Garching, Germany A survey is given of the evolution from Yukawa’s early work, via the understanding of the pion as a Nambu-Goldstone boson of spontaneously broken chiral symmetry in QCD, to modern developments in the theory of the nucleus based on the chiral effective field theory representing QCD in its low-energy limit. §1. The Beginnings One of the most remarkable documentations in modern science history is the series of articles which form the first volumes of Supplement of the Progress of Theoretical Physics. These Collected Papers on Meson Theory1) celebrated the twenty years’ anniversary of Yukawa’s pioneering work2) which laid the foundations for our understanding of the strong force between nucleons. In those early days of nuclear physics very little was known about this force. Heisenberg’s ”Platzwechsel” interaction between proton and neutron gave a first intuitive picture, and some unsuccessful attempts had been made to connect Fermi’s theory of beta decay with nuclear interactions. Yukawa’s 1935 article introduced the conceptual framework for a systematic approach to the nucleon-nucleon interaction. His then postulated ”U field” did not yet have the correct quantum numbers of what later became the pion∗), but it had already some of its principal features. It provided the basic mechanism of charge exchange between proton and neutron. From the form of the potential, e−µr/r, and from the estimated range of the nuclear force, the mass of the U particle was predicted to be about 200 times that of the electron. This U particle was erroneously identified with the muon discovered in 1937 almost simultaneously by Anderson and by Nishina and their collaborators. The ”real” π meson was discovered a decade later3) in cosmic rays and then produced for the first time4) at the Berkeley cyclotron. Its pseudoscalar nature was soon established. In 1949 the Nobel prize was awarded to Yukawa. In the footsteps of Yukawa’s original work, the decade thereafter saw an impres- sively coherent effort by the next generation of Japanese theorists. A cornerstone of these developments was the visionary conceptual design by Taketani et al.5) of the inward-bound hierarchy of scales governing the nucleon-nucleon interaction, sketched in Fig.1. The long distance region I is determined by one-pion exchange. It contin- ues inward to the intermediate distance region II dominated by two-pion exchange. The basic idea was to construct the NN potential in regions I and II by explicit calculation of π and 2π exchange processes, whereas the detailed behaviour of the ∗) I am grateful to Professor H. Miyazawa for instructive communications on this point. typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1992v1 2 W. Weise interaction in the short distance region III remains unresolved at the low-energy scales characteristic of nuclear physics. This short distance part is given a suitably parametrized form. The parameters are fixed by comparison with scattering data. 1 2 3 IIIIII r [µ−1] one-pion exchange two-pion exchange short distance one-pion exchange two-pion exchange Fig. 1. Hierarchy of scales governing the nucleon-nucleon potential (adapted from Taketani6)). The distance r is given in units of the pion Compton wavelength µ−1 ≃ 1.4 fm. Taketani’s picture, although not at all universally accepted at the time by the international community of theorists, turned out to be immensely useful. Today this strategy is the one conducted by modern effective field theory approaches. It is amazing how far this program had already been developed by Japanese theorists in the late fifties. One example is the pioneering calculation of the two-pion exchange potential7) using dispersion relation techniques and early knowledge8) of the resonant pion-nucleon amplitude which anticipated the ∆ isobar models of later decades. §2. The Pion in the context of Low-Energy QCD Today’s theory of the strong interaction is Quantum Chromodynamics (QCD). There exist two limiting situations in which QCD is accessible with ”controlled” ap- proximations. At momentum scales exceeding several GeV (corresponding to short distances, r < 0.1 fm), QCD is a theory of weakly interacting quarks and gluons. At low momentum scales considerably smaller than 1 GeV (corresponding to long dis- tances, r > 1 fm), QCD is governed by color confinement and a non-trivial vacuum: the ground state of QCD hosts strong condensates of quark-antiquark pairs and gluons. Confinement implies the dynamical breaking of a global symmetry which is exact in the limit of massless quarks: chiral symmetry. Spontaneous chiral symmetry breaking in turn implies the existence of pseudoscalar Nambu-Goldstone bosons. For two quark flavours (Nf = 2) with (almost) massless u and d quarks, these Goldstone bosons are identified with the isospin triplet of pions. At low energy, Goldstone bosons interact weakly with one another or with any massive particles. Low-energy QCD is thus realized as an effective field theory in which these Goldstone bosons are the active, light degrees of freedom. 2.1. Chiral Symmetry and the NJL Model How does Yukawa’s pion figure in the frame of QCD? Historically, the founda- tions for understanding the pion as a Nambu-Goldstone boson9), 10) of spontaneously Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 3 broken chiral symmetry were initiated in the 1960’s, culminating in the PCAC and current algebra approaches11) of the pre-QCD era. A most inspiring work from this early period is the one by Nambu and Jona-Lasinio10)(NJL). Just as the BCS model provided an understanding of the basic mechanism behind superconductivity, the NJL model helped clarifying the dynamics that drives spontaneous chiral symmetry breaking and the formation of pions as Goldstone bosons. Consider as a starting point the conserved color current of quarks, Jaµ = ψ̄γµt where ta (a = 1, ..., 8) are the generators of the SU(Nc = 3) gauge group and ψ are the quark fields with 4NcNf components representing their spin, color and flavor degrees freedom. This current couples to the gluon fields. Any two quarks interact through multiple exchanges of gluons. Assume that the distance over which color propagates is restricted to a short correlation length lc. Then the interaction expe- rienced by low-momentum quarks can be schematically viewed as a local coupling between their color currents: Lint = −Gc Jaµ(x)Jµa(x) , (2.1) where Gc ∼ ḡ2 l2c is an effective coupling strength of dimension length2 which encodes the QCD coupling, averaged over the relevant distance scales, in combination with the squared correlation length, l2c . Now adopt the local interaction (2.1) and write the following model Lagrangian for the quark fields ψ(x): L = ψ̄(x)(iγµ∂µ −m0)ψ(x) + Lint(ψ̄, ψ) . (2.2) In essence, by ”integrating out” gluon degrees of freedom and absorbing them in the four-fermion interaction Lint, the local SU(Nc) gauge symmetry of QCD is now replaced by a global SU(Nc) symmetry. Confinement is obviously lost, but all other symmetries of QCD are maintained. The mass matrix m0 incorporates small ”bare” quark masses. In the limit m0 → 0, the Lagrangian (2.2) has a chiral symmetry of right- and left-handed quarks, SU(Nf )R×SU(Nf )L, that it shares with the original QCD Lagrangian for Nf massless quark flavors. A Fierz transform of the color current-current interaction (2.1) produces a set of exchange terms acting in quark-antiquark channels. For the Nf = 2 case: Lint → (ψ̄ψ)2 + (ψ̄ iγ5 τ ψ) + ... , (2.3) with the isospin SU(2) Pauli matrices τ = (τ1, τ2, τ3). For brevity we have not shown a series of terms with combinations of vector and axial vector currents, both in color singlet and color octet channels. The constant G is proportional to the color coupling strength Gc. The ratio of these two constants is uniquely determined by Nc and Nf . The steps just outlined can be viewed as a contemporary way of introducing the time-honored NJL model10) . This model has been further developed and ap- plied12), 13) to a variety of problems in hadron physics. The virtue of this schematic model is its simplicity in illustrating the basic mechanism of spontaneous chiral sym- metry breaking, as follows. In the mean-field (Hartree) approximation the equation 4 W. Weise of motion derived from the Lagrangian (2.2) leads to a gap equation M = m0 −G〈ψ̄ψ〉 , (2.4) which links the dynamical generation of a constituent quark mass M to the appear- ance of the chiral quark condensate 〈ψ̄ψ〉 = −Tr lim x→ 0+ 〈T ψ(0)ψ̄(x)〉 = −2iNfNc (2π)4 M θ(Λ2 − ~p 2) p2 −M2 + iε . (2.5) This condensate plays the role of an order parameter of spontaneous chiral symme- try breaking. Starting from m0 = 0 a non-zero quark mass develops dynamically, together with a non-vanishing chiral condensate, once G exceeds a critical value of order Gcrit ∼ 10 GeV−2. The procedure requires a momentum cutoff Λ ≃ 2M be- yond which the interaction is ”turned off”. Note that the strong non-perturbative interactions, by polarizing the vacuum and turning it into a condensate of quark- antiquark pairs, transmute an initially pointlike quark with its small bare mass m0 into a massive quasiparticle with a size of order (2M)−1. 2.2. The Pseudoscalar Meson Spectrum The NJL model demonstrates lucidly the appearance of chiral Nambu-Goldstone bosons. Solving Bethe-Salpeter equations in the color singlet quark-antiquark chan- nels generates the lightest mesons as quark-antiquark excitations of the correlated QCD ground state with its condensate structure. Several such calculations have been performed in the past with Nf = 3 quark flavors 12)–14) . Such a model has an un- wanted U(3)R×U(3)L symmetry to start with, but the axial U(1)A anomaly reduces this symmetry to SU(3)R × SU(3)L × U(1)V . In QCD, instantons are considered responsible for U(1)A breaking. In the NJL model, these instanton driven interac- tions are incorporated in the form of a flavor determinant15) det[ψ̄i(1± γ5)ψj ]. This interaction involves all three flavors u, d, s simultaneously in a genuine three-body term. Fig. 2. Symmetry breaking pattern in the pseudoscalar meson nonet calculated in the three-flavor NJL model14) . The symmetry breaking pattern resulting from such a calculation is apparent in the pseudoscalar meson spectrum of Fig.2. Starting from massless u, d and s Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 5 quarks, the pseudoscalar octet emerges as a set of massless Goldstone bosons of spontaneously broken SU(3) × SU(3), while U(1)A breaking drives the singlet η0 away from the Goldstone boson sector. Finite quark masses shift the Jπ = 0− nonet into its empirically observed position, including η-η′ mixing. The very special nature of the pion as a Nambu-Goldstone boson is manifest in a famous relation16) derived from current algebra and PCAC: m2π f π = − (mu +md)〈ψ̄ψ〉+O(m2u,d). (2.6) It involves the pion decay constant, fπ ≃ 0.09 GeV, defined by the matrix element which connects the pion with the QCD vacuum via the axial vector current, A ψ̄γµγ5 〈0|Aµi (x = 0)|πi(p)〉 = ip µfπ . (2.7) Just like the chiral condensate, the pion decay constant is a measure of spontaneous chiral symmetry breaking expressed in terms of a characteristic scale 4πfπ ∼ 1 GeV. The non-zero pion mass, mπ ∼ 0.14 GeV ≪ 4πfπ, reflects the explicit symmetry breaking by the small quark masses, with m2π ∼ mq. One should note that the quark masses mu,d and the condensate 〈ψ̄ψ〉 are both scale dependent quantities. Only their product is scale independent, i.e. invariant under the renormalization group. At a renormalization scale of about 1 GeV, a typical average quark mass (mu +md) ≃ 7 MeV implies |〈ψ̄ψ〉| ≃ (0.3 GeV)3. 2.3. Scales and Symmetry Breaking Patterns The quark masses are the only parameters that set primary scales in QCD. Their classification into sectors of ”light” and ”heavy” quarks determines very different physics phenomena. While the heavy quarks (i.e. the t, b and - within limits - the c quarks) offer a natural ”small parameter” in terms of their reciprocal masses, such that non-relativistic approximations (expansions of observables in powers of m−1t,b,c) tend to work increasingly well with increasing quark mass, the sector of the light quarks (i.e. the u, d quarks and - to some extent - the s quark) is governed by quite different principles and rules. Evidently, the quark masses themselves are now ”small parameters”, to be compared with a characteristic ”large” scale of dynamical origin. This large scale is visible as a characteristic mass gap of about 1 GeV which separates the QCD vacuum from almost all of its excitations, with the exception of the pseudoscalar meson octet of pions, kaons and the eta meson. This mass gap is in turn comparable to 4πfπ, the scale associated with spontaneous chiral symmetry breaking in QCD. In this context it is also instructive to have a look at the spectroscopic pattern of pseudoscalar and vector mesons, starting from heavy-light quark-antiquark pairs in 1S0 and 3S1 states and following those states downward in the mass of the quark. This is illustrated in Fig.3 where we show the masses of mesons composed of a b, c, s or u quark with an anti-d-quark attached. Bare quark masses are subtracted from the meson masses in this plot in order to directly demonstrate the evolution from perturbative hyperfine splitting in the heavy systems to the non-perturbative mass gap in the light ones. In the B̄ and B̄∗ mesons, the d̄ quark is tightly bound 6 W. Weise Fig. 3. Evolution of the splitting between spin singlet (lower) and triplet (upper) quark-antiquark states (the pseudoscalar (Jπ = 0−) and vector (Jπ = 1−) mesons) with varying mass of one of the quarks. The bare quark masses are subtracted from the physical meson masses for convenience of demonstration. to the heavy b quark at small average distance, within the range where perturbative QCD is applicable. The spin-spin interaction is well approximated by perturbative one-gluon exchange, resulting in a small hyperfine splitting. Moving downward in mass to the D and D∗ systems, with the b quark replaced by a c quark, the hyperfine splitting increases but remains perturbative in magnitude. As this pattern evolves further into the light-quark sector, it undergoes a qualitative change via the large mass difference of K̄ and K̄∗ to the non-perturbative mass gap in the π− ρ system, reflecting the Goldstone boson nature of the pion. 2.4. Chiral Effective Field Theory Low-energy QCD is the physics of systems of light quarks at energy and momen- tum scales smaller than the 1 GeV mass gap observed in the hadron spectrum. This 1 GeV scale set by 4πfπ offers a natural separation between ”light” and ”heavy” (or, correspondingly, ”fast” and ”slow”) degrees of freedom. The basic idea of an effective field theory is to introduce the active light particles as collective degrees of freedom, while the heavy particles are frozen and treated as (almost) static sources. The dynamics is described by an effective Lagrangian which incorporates all rele- vant symmetries of the underlying fundamental theory. In QCD, confinement and spontaneous chiral symmetry breaking implies that the ”fast” degrees of freedom are the Nambu-Goldstone bosons. With Yukawa’s pion in mind, we restrict ourselves to Nf = 2. We first briefly summarize the steps17), 18) required in the pure meson sector (baryon number B = 0) and later for the pion-nucleon sector (B = 1). A chiral field is introduced as U(x) = ei τi πi(x)/fπ ∈ SU(2) , (2.8) with the Goldstone pion fields πi(x) normalized by the pion decay constant fπ taken in the chiral limit (mπ → 0). The QCD Lagrangian is replaced by an effective Lagrangian which involves U(x) and its derivatives: LQCD → Leff (U, ∂µU, ...). (2.9) Goldstone bosons interact only when they carry non-zero momentum, so the low- Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 7 energy expansion of (2.9) is an ordering in powers of ∂µU . Lorentz invariance permits only even numbers of derivatives. One writes Leff = L(2) + L(4) + ... , (2.10) omitting an irrelevant constant. The leading term (the non-linear sigma model) involves two derivatives: L(2) = f Tr[∂µU †∂µU ]. (2.11) At fourth order, the terms permitted by symmetries are (apart from an extra con- tribution from the QCD anomaly, not included here): L(4) = (Tr[∂µU †∂µU ])2 + Tr[∂µU †∂νU ]Tr[∂ µU †∂νU ], (2.12) and so forth. The constants l1, l2 (following canonical notations 18)) must be deter- mined by experiment. The symmetry breaking mass term is small, so that it can be handled pertur- batively, together with the power series in momentum. The leading contribution introduces a term linear in the quark mass matrix m = diag(mu,md): L(2) = f Tr[∂µU †∂µU ] + B0 Tr[m(U + U †)], (2.13) with B0 = −〈ψ̄ψ〉/2fπ The fourth order term L(4) also receives symmetry breaking contributions with additional constants li. To the extent that the effective Lagrangian includes all terms dictated by sym- metries of QCD, the chiral effective field theory is the low-energy equivalent19), 20) of the original QCD Lagrangian. Given the effective Lagrangian, the framework for systematic perturbative calculations of the S-matrix involving Goldstone bosons, Chiral Perturbation Theory (ChPT), is then defined by the following rules: Collect all Feynman diagrams generated by Leff . Classify all terms according to powers of a small quantity Q which stands generically for three-momentum or energy of the Goldstone boson, or for the pion mass mπ. The small expansion parameter is Q/4πfπ. Loops are subject to dimensional regularization and renormalization. 2.5. Pion-Pion Scattering When using the Gell-Mann, Oakes, Renner (GOR) relation (2.6) to leading order in the quark mass, it is tacitly assumed that the chiral condensate is large in magnitude and plays the role of an order parameter for spontaneous chiral symmetry breaking. This basic scenario needs to be confirmed. It has in fact been tested by a detailed quantitative analysis of pion-pion scattering, the process most accurately and extensively studied using ChPT. Consider s-wave ππ scattering at very low energy. The scattering lengths in the isospin I = 0, 2 channels, calculated to leading chiral order, are21) 32πf2π , a2 = − 16πf2π , (2.14) 8 W. Weise showing that the ππ interaction properly vanishes in the chiral limit, mπ → 0. The next-to-leading order introduces one-loop iterations of the leading L(2) part of the effective Lagrangian as well as pieces generated by L(4). At that level enters the renormalized constant l̄3 which also determines the correction to the leading-order GOR relation: m2π = 32π2f2 π + O( π) , (2 where π = − mu +md 〈ψ̄ψ〉 (2.16) involves the quark mass in leading order. Here f is the pion decay constant in the chiral limit. An accurate determination of the I = 0 s-wave ππ scattering length therefore provides a constraint for l̄3 which in turn sets a limit for the next-to-leading order correction to the GOR relation. Such an investigation has been performed22) using low-energy ππ phase shifts extracted from the detailed final state analysis of the K → ππ + eν decay. The result, when translated into a statement about the non-leading term entering (2.15), implies that the difference between m2π and the leading GOR expression (2 .16) is less than 5 percent. Hence the “strong condensate” scenario of spontaneous chiral symmetry breaking in QCD appears to be confirmed∗). 2.6. The Pion in Lattice QCD The leading-order relationship m2π ∼ mq is also observed23) in lattice QCD up to surprisingly large quark masses. A detailed recent analysis24) is shown in Fig.4. Within statistical errors, the data for squared pion mass versus quark mass lie on a straight. The lattice results are remarkably compatible with one-loop chiral perturbation theory up to mπ . 0.5 GeV. Fig. 4. Lattice QCD simulation results24) for the squared pion mass m2π as function of the quark mass m in units of the lattice spacing a. Pion masses converted to physical units are attached to the lattice data points. A linear fit (dashed) is shown in comparison with the next-to-leading order ChPT result (solid curve). ∗) One should note, however, that this conclusion is drawn at the level of QCD with only Nf = 2 flavours. Additional corrections may still arise when strange quarks are taken into account. Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 9 2.7. Pion-Nucleon Effective Lagrangian The prominent role played by the pion as a Goldstone boson of spontaneously broken chiral symmetry has its impact on the low-energy structure and dynamics of nucleons as well.25) When probing the nucleon with long-wavelength electroweak and strong fields, a substantial part of the response comes from the pion cloud, the “soft” surface of the nucleon. The calculational framework for this, baryon chiral perturbation theory26), 27) has been applied quite successfully to a variety of low- energy processes (such as low-energy pion-nucleon scattering, threshold pion photo- and electroproduction and Compton scattering on the nucleon). Consider now the sector with baryon number B = 1 and the physics of the pion- nucleon system. The nucleon is represented by an isospin-1/2 doublet, Dirac spinor field ΨN (x) = (p, n) T of proton and neutron. The free field Lagrangian LN0 = Ψ̄N (iγµ∂µ −M0)ΨN (2.17) includes the nucleon mass in the chiral limit, M0. One should note that the nucleon, unlike the pion, has a large mass of the same order as the chiral symmetry breaking scale 4πfπ, which survives in the limit of vanishing bare quark masses, mu,d → 0. The previous pure meson Lagrangian Leff is now replaced by Leff (U, ∂µU,ΨN , ...) which also includes the nucleon field. The additional term involving the nucleon, de- noted by LNeff , is expanded again in powers of derivatives (external momenta) of the Goldstone boson field and of the quark masses: LNeff = L πN + L πN ... (2 In the leading term, L(1)πN there is a replacement of ∂µ by a chiral covariant deriva- tive which introduces vector current couplings between the pions and the nucleon. Secondly, there is an axial vector coupling. This structure of the πN effective La- grangian is again dictated by chiral symmetry. We have L(1)πN = Ψ̄N [iγµ(∂ µ − iVµ) + γµγ5 Aµ −M0]ΨN , (2.19) with vector and axial vector quantities involving the Goldstone boson (pion) fields in the form ξ = Vµ = i (ξ†∂µξ + ξ∂µξ†) = − 1 εabcτa πb ∂ µπc + ... , (2.20) (ξ†∂µξ − ξ∂µξ†) = − µπa + ... , (2.21) where the last steps result when expanding Vµ and Aµ to leading order in the pion fields. So far, the only parameters that enter are the nucleon mass,M0, and the pion decay constant, fπ, both taken in the chiral limit. The nucleon has its own intrinsic structure which leads to a modification of the axial vector coupling term in (2.19). The analysis of neutron beta decay reveals that the γµγ5 term is to be multiplied by the axial vector coupling constant gA, with the empirical value gA ≃ 1.27. 10 W. Weise At next-to-leading order (L(2)πN ), the symmetry breaking quark mass term enters. It has the effect of shifting the nucleon mass from its value in the chiral limit to the physical one: MN =M0 + σN . (2.22) The sigma term σN = mq = 〈N |mq(ūu+ d̄d)|N〉 (2.23) measures the contribution of the non-vanishing quark mass, mq = (mu+md), to the nucleon mass MN . Its empirical value is in the range σN ≃ (45 − 55) MeV and has been deduced37) by a sophisticated extrapolation of low-energy pion-nucleon data using dispersion relation techniques. Up to this point, the πN effective Lagrangian, expanded to second order in the pion field, has the form LNeff = Ψ̄N (iγµ∂µ −MN )ΨN − Ψ̄Nγµγ5 τ ΨN · ∂µπ (2.24) Ψ̄Nγµ τ ΨN · π × ∂µπ + Ψ̄NΨN π 2 + ... , where we have not shown a series of additional terms of order (∂µπ)2 included in the complete L(2)πN . These terms come with further low-energy constants encoding physics at smaller distances and higher energies. These constants need to be fitted to experimental data, e.g. from pion-nucleon scattering. The “effectiveness” of such an effective field theory relies on the proper identifi- cation of the active low-energy degrees of freedom. Pion-nucleon scattering is known to be dominated by the p-wave ∆(1232) resonance with spin and isospin 3/2. The excitation energy of this resonance, given by the mass difference δM =M∆−MN , is not large, just about twice the pion mass. If the physics of the ∆(1232) is absorbed in low-energy constants of an effective theory that works with pions and nucleons only (as commonly done in heavy-baryon ChPT), the limits of applicabilty of such a theory is clearly narrowed down to an energy-momentum range small compared to δM . The B = 1 chiral effective Lagrangian is therefore often extended28) by incorporating the ∆ isobar as an explicit degree of freedom. §3. Chiral Thermodynamics and Goldstone Bosons in Matter Before turning to chiral dynamics in nuclear many-body systems, it is instructive to make a brief digression and touch upon more general issues of chiral symmetry at finite temperature and non-zero baryon density. 3.1. The Chiral Order Parameter As outlined in the previous sections, the QCD ground state (the vacuum) is characterized by the presence of the strong chiral condensate 〈ψ̄ψ〉. The light hadrons are quasiparticle excitations of this condensed ground state, with Yukawa’s pion playing a very special role as Nambu-Goldstone boson of spontaneously broken chiral symmetry. A key question33) is then the following: how do the basic quantities and Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 11 scales associated with this symmetry breaking pattern (the chiral condensate, the pion mass and decay constant) evolve with changing thermodynamical conditions (temperature, baryon density)? Assume a homogenous hadronic medium in a volume V at temperature T and consider the pressure P (T, V, µ) = lnZ = T ln Tr exp d3x (H − µρ) . (3.1) Here µ denotes the chemical potential, ρ the baryon density. The Hamiltonian density H of QCD is expressed in terms of the relevant degrees of freedom in the hadronic phase, derived from the chiral effective Lagrangian Leff . The Nf = 2 Hamiltonian has a mass term, δH = ψ̄mψ = mu ūu+md d̄d, so that H = H0 + δH, with H0 representing the massless limit. Now take the derivative of the pressure with respect to the quark mass and use the GOR relation (2.6) to derive the condensate 〈ψ̄ψ〉T,ρ at finite T and density ρ = ∂P/∂µ, or rather its ratio with the condensate at T = µ = 0, 〈ψ̄ψ〉T,ρ 〈ψ̄ψ〉0 = 1 + dP (T, µ) f2π dm . (3.2) The T dependence of this condensate, at zero chemical potential, is shown in compar- ison with two-flavor lattice QCD results in Fig.5. Its behaviour reflects a continuous crossover transition at a critical temperature Tc ∼ 0.2 GeV which turns into a sec- ond order phase transition in the chiral limit of massless quarks. Above Tc chiral symmetry is restored and the pion stops being realized as a Nambu-Goldstone mode. The chiral condensate therefore has the features of an order parameter. However, it is not directly observable. A related measurable quantity is the pion decay constant. Its temperature and density dependence is in fact an indicator of tendencies towards chiral symmetry restoration, in the following sense. The GOR relation (2.6) continues to hold in matter at finite temperature T < Tc and density ρ, when reduced to a statement about the time component, A ψ†γ5(τa/2)ψ, of the axial current. We can introduce the in-medium pion decay constant, f∗π(T, ρ) through the thermal matrix element 〈 |A0|π〉T,ρ , the in-medium analogue of eq.(2.7). One finds f∗π(T, ρ) 2m∗π(T, ρ) 2 = − mu +md 〈ψ̄ψ〉T,ρ + ... (3.3) to leading order in the quark mass. The in-medium pion mass m∗π (more precisely: the average of the π+ and π− masses) is protected by the pion’s Goldstone boson nature and not much affected by the thermal environment. The “melting” of the condensate by heat or compression translates primarily to the in-medium change of the pion decay constant. The leading behaviour32), 33) of the pion condensate and, consequently, of the pion decay constant, with increasing temperature and density is: f∗π(T, ρ) 〈ψ̄ψ〉T,ρ 〈ψ̄ψ〉0 = 1− T 8 f2π m2π f ρ+ ... . (3.4) 12 W. Weise Fig. 5. Temperature dependence of the chi- ral condensate at zero chemical potential. The curve results from a calculation30) based on an extended NJL model with inclusion of Polyakov loop dynamics (the PNJL model). The data points areNf = 2 lattice QCD results taken from ref.31) . Fig. 6. Pion decay constant as function of temperature T and baryon density ρ. Cal- culation29) based on the PNJL model. Normal nuclear matter density ρ0 = 0.16 fm−3 is indicated for orientation. A typical result for the in-medium behavior of the pion decay constant is displayed in Fig.6. It should be noted that the dropping of the condensate’s magnitude with density is significantly more pronounced than its temperature dependence. The decreasing “chiral gap” 4πf∗π(T, ρ) with changing thermodynamic conditions should thus imply observable changes in the low-energy dynamics of pions in dense matter. 3.2. Low-Energy Pion-Nucleus Interactions Goldstone’s theorem implies that low-momentum pions interact weakly. This is generally true also for low-momentum pions interacting with nuclear many-body systems. As a starting point, consider homogeneous nuclear matter at zero temper- ature with proton density ρp and neutron density ρn. A pion wave in matter has its energy ω and momentum ~q related by the dispersion equation ω2 − ~q 2 −m2π −Π(ω, ~q ; ρp, ρn) = 0 . (3.5) The polarization function, or pion self-energy Π, summarizes all interactions of the pion with the medium. At low densities, Π(±)(ω, ~q ; ρp, ρn) = −T+(ω, ~q ) ρ± T−(ω, ~q ) δρ , (3.6) in terms of the isospin-even (T+) and isospin-odd (T−) pion-nucleon forward scat- tering amplitudes, with ρ = ρp + ρn and δρ = ρp − ρn. We have now specified the self-energies Π(±) for a π+ or π−, respectively. Applications to finite systems, in particular for low-energy pion-nucleus interac- tions34), 35) relevant to pionic atoms, commonly make use of an energy-independent effective potential. Such an equivalent potential is constructed36) by expanding the polarization function for ω −mπ ≪ mπ and |~q |2 ≪ m2π around the physical thresh- old, ω = mπ and |~q | = 0. By comparison with the Klein-Gordon equation for the Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 13 pion wave function φ(~r ) in coordinate space, ω2 −m2π + ~∇2 − 2mπU(~r ) φ(~r ) = 0 , (3.7) the (energy-independent) potential U(~r ) is identified as follows: 2mπU(~r ) = )−1 [ Π(mπ, 0) − ~∇ ∂~q 2 , (3.8) with all derivatives taken at the threshold point. The wave function renormalization factor (1 − ∂Π/∂ω2)−1 encodes the energy dependence of the polarization function Π(ω, ~q ) in the equivalent energy-independent potential (3.8). This potential is ex- pressed in terms of local density distributions ρp,n(~r ) for protons and neutrons, and the standard prescription ~q 2f(ρ) → −~∇f(ρ(~r ))~∇ is used for the ~q 2- dependent parts. In practical calculations of pionic atoms, the Coulomb potential Vc is in- troduced by replacing ω → ω − Vc(~r ), and corrections of higher order beyond the leading terms (3.6), resulting from double scattering and absorption, are added. 3.3. Deeply Bound States of Pionic Atoms Accurate data on 1s states of a negatively charged pion bound to Pb and Sn isotopes41), 42) have set new standards and constraints for the detailed analysis of s-wave pion interactions with nuclei. Such deeply bound pionic states owe their existence, with relatively long lifetimes, to a subtle balance between the attractive Coulomb force and the repulsive strong π−-nucleus interaction in the bulk of the nucleus. As a consequence, the 1s wave function of the bound pion is pushed toward the edge of the nuclear surface. Its overlap with the nuclear density distribution is small, so that the standard π−pn→ nn absorption mechanism is strongly suppressed. The topic of low-energy, s-wave pion-nucleus interactions has a long34), 35) history. Inspired by the measurements of deeply bound pionic atoms it has recently been re-investigated38) from the point of view of the distinct energy dependence of the pion-nuclear polarization operator in calculations based on systematic in-medium chiral perturbation theory39), 40) . Consider a negatively charged pion interacting with nuclear matter and recall the π− self-energy from Eq.(3.6). In the long-wavelength limit (~q → 0), chiral symmetry (the Tomozawa-Weinberg low-energy theorem) implies T−(ω) = ω/(2f2π) + O(ω3). Together with the observed approximate vanishing of the isospin-even threshold am- plitude T+(ω = mπ), it is clear that 1s states of pions bound to heavy, neutron rich nuclei are a sensitive source of information for in-medium chiral dynamics. Terms of non-leading order in density (double scattering (Pauli) corrections of order ρ4/3, absorption effects of order ρ2 etc.) are important and systematically incorporated. Absorption effects and corresponding dispersive corrections appear at the three-loop level and through short-distance dynamics parametrized by πNN contact terms, not explicitly calculable within the effective low-energy theory. The imaginary parts as- sociated with these terms are well constrained by the systematics of observed widths of pionic atom levels throughout the periodic table. With these ingredients the Klein-Gordon equation for the nuclear pion field has been solved with the explicitly energy dependent pion self-energy just described. As 14 W. Weise an example we show predictions38) for binding energies and widths for pionic 1s states bound to a series of Sn isotopes. These calculations include a careful assessment of uncertainties in neutron distributions. Results are shown in Fig.7 in comparison with experimental data.41) Fig. 7. Binding energies (upper pannel) and widths (lower pannel) of pionic 1s states in Sn iso- topes. The curves show predictions38) based on the explicitly energy dependent pionic s-wave polarization operator calculated in two-loop in-medium chiral perturbation theory. Upper and lower curves give an impression of uncertainties related to the πN sigma term. Data from ref.41) The question has been raised43), 44) whether one can actually ”observe” finger- prints of (partial) chiral symmetry restoration in the high-precision data of deeply bound pionic atoms. Pionic atom calculations are usually done with energy inde- pendent phenomenological optical potentials instead of explicitly energy dependent pionic polarisation functions. The connection is provided by Eq.(3.8). Consider a zero momentum π− in low density matter. Its energy dependent leading-order polarisation operator is Π(ω) = −T+(ω) (ρp + ρn) + T−(ω) (ρn − ρp) , and the in- medium dispersion equation at ~q = 0 is ω2−m2π −Π(ω) = 0 . The chiral low-energy expansion of the off-shell amplitude T+(ω) at ~q = 0 implies leading terms of the form T+(ω) = (σN − β ω2)/f2π , with β ≃ σN/m2π required to yield the empirical T+(ω = mπ) ≃ 0. Using Eq.(3.8) one finds for the effective (energy-independent) s-wave potential ρn − ρp 4 f2π 1− σN ρ m2π f ρn − ρp 4 f∗2π (ρ) , (3.9) with the replacement fπ → f∗π(ρ) of the pion decay constant representing the in- medium wave function renormalization. The expression (3.9) is just the one proposed previously in ref.43) based on the relation (3.4) between the in-medium changes of the chiral condensate and the pion decay constant associated with the time component of the axial current. The explicitly energy dependent chiral dynamics represented by Π(ω) ”knows” about these renormalization effects. Their translation into an equivalent, energy-independent potential implies fπ → f∗π(ρ) as given in eq. (3.9). This heuristic reasoning has recently been underlined by a more profound derivation in ref.45) . The analysis of the deeply bound pionic atom data41) along these lines comes to Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 15 the conclusion that, when extrapolated to nuclear matter density ρ0 = 0.16 fm f∗π(ρ0) ≃ 0.8 fπ , (3.10) which is compatible with the theoretical prediction f∗π(ρ0) 2m2π f ρ0 , (3.11) assuming σN ≃ 50 MeV. It is quite remarkable that an optical potential fit to recent precision measurements of π+ and π− differential cross sections at the lowest possible energy (Tπ = 21.5 MeV) on a variety of nuclei reaches a similar conclusion, 46) namely f∗π(ρ0) ≃ 0.83 fπ , although within a different procedure. With the interpretation (3.9), the tendency towards chiral restoration in a nuclear medium as suggested by Eq.(3.4) appears to be - at least qualitatively - visible in low-energy pion-nucleus interactions. §4. Nuclear Chiral Dynamics We now approach a basic question at the origin of modern nuclear physics: is there a path from QCD via its low-energy representation, chiral effective field theory, to the observed systematics of the nuclear chart? Or equivalently: how does Yukawa’s pion and its realization as a Nambu-Goldstone boson figure in the nuclear many-body problem? Pionic degrees of freedom in nuclei have been in the focus right from the begin- nings. The field of exchange currents in nuclei, with the pion as the prime agent, was started already in the early fifties47) and investigated in great breadth in the seven- ties. An instructive overview of these developments is given in volume II of ref.48) The important role played by one-pion exchange and its strong tensor force in the deuteron49) is a long known and well established fact. The description of radiative np capture (n + p → d + γ) in terms of the magnetic pion exchange current50) was a key to establishing the pion as an observable degree of freedom in the deuteron. For heavier nuclei, on the other hand, the role of the pion is not so directly evident. One-pion exchange does not contribute to the bulk nuclear (Hartree) mean field when averaged over nucleon spins. Fock exchange terms involving one-pion exchange are relatively small. The role of pions in binding the nucleus is manifest primarily in the intermediate range attractive force generated by two-pion exchange processes. For decades, nuclear mean field models preferred to replace the complexity of such processes by a phenomenological scalar-isoscalar “sigma” field, although the more detailed treatment of the two-pion exchange nucleon-nucleon interaction had been known before (see e.g. ref.51) and volume I of ref.48)). Recent developments return to these basics by introducing chiral effective field theory as a systematic framework for the treatment of NN interactions and nuclear systems, following ref.52) For an updated review see ref.53) 16 W. Weise 4.1. In-medium chiral perturbation theory and nuclear matter In nuclear matter the relevant momentum scale is the Fermi momentum kF . Around the empirical saturation point with k F ≃ 0.26 GeV ∼ 2mπ, the Fermi momentum and the pion mass are scales of comparable magnitude. This implies that at the densities of interest in nuclear physics, ρ ∼ ρ0 = 2(k 3/3π2 ≃ 0.16 fm−3 ≃ 0.45m3π , pions must be included as explicit degrees of freedom: their propagation in matter is ”resolved” at the relevant momentum scales around the Fermi momentum. At the same time, kF and mπ are small compared to the characteristic chiral scale, 4πfπ ∼ 1 GeV. Consequently, methods of chiral perturbation theory are ex- pected to be applicable to nuclear matter at least in a certain window around k In that range, the energy density E(kF ) = E(kF ) ρ . (4.1) should then be given as a convergent power series in the Fermi momentum. This is the working hypothesis. More precisely, the energy per particle has an expansion E(kF ) Fn(kF /mπ) knF . (4.2) The expansion coefficients Fn are in general non-trivial functions of kF /mπ, the dimensionless ratio of the two relevant scales. These functions must obviously not be further expanded. Apart from kF and mπ, a third relevant “small” scale is the mass difference δM =M∆ −MN ≃ 0.3 GeV between the ∆(1232) and the nucleon. The strong spin-isospin transition from the nucleon to the ∆ isobar is therefore to be included as an additional important ingredient in nuclear many-body calculations, so that the Fn become functions of both kF /mπ and mπ/δM . In-medium chiral perturbation theory is the framework for treating pion ex- change processes in the presence of a filled Fermi sea of nucleons. The chiral pion- nucleon effective Lagrangian, with its low-energy constants constrained by pion- nucleon scattering observables in vacuum, is used to construct the hierarchy of NN interaction terms as illustrated in Fig.8. One- and two-pion exchange processes (as well as those involving low-energy particle-hole excitations) are treated explicitly. They govern the long-range interactions at distance scales d > 1/kF fm relevant to the nuclear many-body problem, whereas short-range mechanisms, with t-channel spectral functions involving masses far beyond those of two pions, are not resolved in detail at nuclear Fermi momentum scales and can be subsumed in contact interac- tions and derivatives thereof. This “separation of scales” argument makes strategies of chiral effective field theory work even for nuclear problems, with the “small” scales (kF ,mπ, δM) distinct from the “large” ones (4πfπ,MN ). In essence, this is the mod- ern realization of Taketani’s programme mentioned in the beginning. Closely related renormalization group considerations have motivated the construction of a universal low-momentum NN interaction V(low k)54) from phase shift equivalent NN potentials such that the ambiguities associated with unresolved short-distance parts disappear. Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 17 Fig. 8. NN amplitude in chiral effective field theory: (upper:) one-pion exchange, two- pion exchange (including ∆ isobar inter- mediate states) and (lower:) contact terms representing short-distance dynamics. Fig. 9. Energy density from in-medium chi- ral perturbation theory at three-loop or- der. Dashed lines show pions. Each (solid) nucleon line means insertion of the in-medium propagator (4.4). Two- and three-body terms involving contact inter- actions are also shown. The two-pion exchange interaction has as its most prominent pieces the second order tensor force and intermediate ∆(1232) states which reflect the strong spin- isospin polarizability of the nucleon. The latter produces a Van der Waals - like NN interaction. At long and intermediate distances it behaves as55) V2π(r) ∼ − e−2mπr P (mπr) , (4.3) where P is a polynomial in mπr. In the chiral limit (mπ → 0), this V2π approaches the characteristic r−6 dependence of a non-relativistic Van der Waals potential. The two-pion exchange force is the major source of intermediate range attraction that binds nuclei. This is, of course, not a new observation. For example, the important role of the second-order tensor force from iterated pion exchange had been emphasised long ago,56) as well as the close connection of the nuclear force to the strong spin-isospin polarizability of the nucleon.57) The new element that has entered the discussion more recently is the systematics provided by chiral effective field theory in dealing with these phenomena. With these ingredients, in-medium ChPT calculations of nuclear and neutron matter have been performed58), 59) up to three-loop order in the energy density, as illustrated in Fig.9. Each nucleon line in these diagrams stands for the in-medium propagator (γ · p+MN ) p2 −M2N + iε − 2πδ(p2 −M2N )θ(p0)θ(kF − |~p |) . (4.4) The regularization of some divergent loops introduces a scale which is balanced by counter terms (contact interactions) so that the result is independent of this reg- ularization scale.60) A limited, small number of constants in these contact terms 18 W. Weise must be adjusted to empirical information such as the equilibrium density of nuclear matter. Stabilization and saturation of nuclear matter at equilibrium is achieved in a non-trivial and model-independent way: the Pauli principle acting on nucleon intermediate states in two-pion exchange processes produces a repulsive term pro- portional to ρ4/3 in the energy per particle. This partial Pauli blocking counteracts the leading attraction from the term linear in ρ. Three-body forces arise necessarily and naturally in this approach. Their contributions is not large at normal nuclear matter density, indicating a convergent hierarchy of terms in powers of the Fermi momentum as long as the baryon density does not exceed about twice the density of equilibrium nuclear matter. Binding and saturation of nuclear matter can thus be seen, in this approach, as a combination of phenomena and effects which relate to the names Yukawa, Van der Waals and Pauli. It does then perhaps not come as a surprise that the resulting nu- clear matter equation of state, see Fig.10, is reminscent of a Van der Waals equation of state. The nuclear liquid turns into a gas at a critical temperature Tc ≃ 15 MeV, quite close to the commonly accepted empirical range Tc ∼ 16− 18 MeV. Fig. 10. The nuclear matter equation of state: pressure versus baryon density calculated in three- loop in-medium chiral perturbation theory.59) Shown are isothermes with temperatures indi- cated. 4.2. Finite nuclei: density functional strategies A description of finite nuclei over a broad range, from 16O to the very heavy ones, is successfully achieved using a (relativistic) universal energy density functional guided by the nuclear matter results. The energy as a functional of density is written E[ρ] = Ekin + E(0)(ρ) + Eexc(ρ) + Ecoul , (4.5) where Ekin and Ecoul are the kinetic and Coulomb energy contributions. The ba- sic idea61), 62) is to construct Eexc from the in-medium chiral perturbation theory calculations discussed previously, representing the pionic fluctuations built on the non-perturbative QCD vacuum in the presence of baryons. Binding and saturation, Yukawa’s pion, low-energy QCD and nuclear chiral dynamics 19 in nuclear matter as well as in finite nuclei, is driven primarily by two-pion exchange mechanisms in combination with the Pauli principle included in Eexc. At the same time the QCD vacuum is populated by strong condensates. The E(0) part of the energy density incorporates the leading changes of these condensates at finite baryon chemical potential (or density). As discussed in ref.61), 62) and refer- ences therein, QCD sum rules at non-zero baryon density suggest that these density dependent changes of condensates generate strong scalar and vector mean fields with opposite signs: at nuclear bulk densities, several hundred MeV of scalar attraction are compensated by an almost equal amount of vector repulsion, such that the net effect of the condensate mean fields almost vanishes and is hardly visible in infinite, homogeneous nuclear matter. However, In finite nuclei, the coherent effect of the strong scalar and vector mean fields produces the large spin-orbit splitting observed empirically. Calculations along these lines, using the chiral pion-nuclear dynamics framework and constraints from the symmetry breaking pattern of low-energy, have been per- formed in refs61), 62) throughout the nuclear chart. The results for nuclear binding energies and radii are comparable in accuracy with those of the best phenomenolog- ical relativistic mean field models available. Examples are shown in Figs.11, 12 and Fig. 11. Deviations (in %) of calculated binding energies (upper pannel) and r.m.s. charge radii (lower pannel) from measured values for a series of nuclei from A = 16 to A = 210. For details of the computations see ref.62) Of particular interest is the systematics through chains of isotopes of deformed nuclei, increasing the number of neutrons by one unit in each step and changing the deformation pattern along the way. The results are sensitive to the detailed isospin dependence of the nuclear interaction. Chiral pion dynamics and its prediction for the isospin structure of the two-pion exchange NN force in the nuclear medium appears to account successfully for the observed properties of such isotopic chains. §5. Concluding Remarks Yukawa’s original U-field which then became the pion is still, more than seventy years after its first release, a generic starting point for our understanding of nuclear 20 W. Weise Fig. 12. Charge form factor of 48Ca. Calcu- lated curve62) in comparison with experi- mental data. Fig. 13. Deviations (in %) of calculated from measured binding energies for a series of isotopic chains from Nd to Pt. For details of the computations see ref.62) systems and interactions. Its property as a Nambu-Goldstone boson of spontaneously broken chiral symmetry is at the origin of a successful effective field theory which represents QCD in its low energy limit. Indications so far are promising that this framework, constrained by the symmetry breaking pattern of low-energy QCD, can serve as foundation for a modern theory of the nucleus. Generating the nucleon- nucleon interaction itself directly from QCD is still a major challenge. 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0704.1993

Competition between local and nonlocal dissipation effects in two-dimensional quantum Josephson junction arrays

Competition between lo al and nonlo al dissipation e�e ts in two-dimensional quantum Josephson jun tion arrays T. P. Polak Consiglio Nazionale delle Ri er he - Istituto Nazionale per la Fisi a della Materia, Complesso Universitario Monte S. Angelo, 80126 Naples, Italy and Istituto di Ciberneti a �E.Caianiello� del CNR, Via Campi Flegrei 34, I-80078 Pozzuoli, Italy. T. K. Kope¢ Institute for Low Temperatures and Stru ture Resear h, Polish A ademy of S ien es, POB 1410, 50-950 Wro law 2, Poland We dis uss the lo al and nonlo al dissipation e�e ts on the existen e of the global phase oher- en e transitions in two dimensional Josephson- oupled jun tions. The quantum phase transitions are also examined for various latti e geometries: square, triangular and honey omb. The T = 0 super ondu tor-insulator phase transition is analyzed as a fun tion of several ontrol parameters whi h in lude self- apa itan e and jun tion apa itan e and both lo al and nonlo al dissipation e�e ts. We found the riti al value of the nonlo al dissipation parameter α1 depends on a geometry of the latti e. The riti al value of the normal state ondu tan e seems to be di� ult to obtain experimentally if we take into onsideration di�erent damping me hanisms whi h are presented in real physi al systems. PACS numbers: 74.50.+r, 67.40.Db, 73.23.Hk I. INTRODUCTION Ma ros opi quantum e�e ts in two-dimensional Josephson jun tion arrays (JJA's) have been ex- tensively studied both theoreti ally 1,2,3,4,5,6,7 experimentally 8,9,10 during the last years. The quan- tum nature of the phase of a super ondu ting order parameter is re�e ted in phase transitions in JJA's. In nondissipative JJA's the two main energy s ales are set by the Josephson oupling EJ between super ondu ting islands and the ele trostati energy EC arising from lo al deviations from harge neutrality. The ratio EC/EJ determines the relevan e of the quantum �u tuations and when it in reases above a riti al value, the phase order is destroyed and the array turns into the insulator. For large apa itive oupling EC ≫ EJ the system an be modeled by a renormalized lassi al two-dimensional (2D) XY model. In the opposite limit, the energy ost for transferring harges between neighboring islands in the array is so high that harges tend to be lo alized. While the nature of the lassi al 2D XY model is well understood, its quantum generalization still poses unsettled issues. Modern fabri ation te hniques allow one to make ar- rays of ultrasmall super ondu ting islands separated by insulators. In su h systems the important fa tor whi h has a profound impa t on the ground state of the JJA's is dissipation aused by Ohmi resistors shunting the jun tions 11,12,13 or quasiparti le tunneling through the jun tions. 14,15,16 Despite several experiments with 2D JJA's 17,18 and super ondu ting granular �lms exis- ten e of the dissipation driven transition and riti al value of the normal state ondu tan e is at least ques- tionable. Phase diagrams in quantum JJA's with both me ha- nisms of dissipation Ohmi and quasiparti le were stud- ied theoreti ally by Zaikin. Cal ulations done within the framework of the instanton te hnique reveal a zero temperature phase diagram with two dissipative phase transitions. The author laims there are regions on the phase diagram where disordered phase and the lassi- al Josephson e�e t ould take pla e. Cu oli et al. presented an analyti al study based on the e�e tive po- tential approa h. They proposed a model in whi h two di�erent relaxation times lead to the ondu tan e matrix with resistive shunts to the ground and among islands. Despite of these a urate analyti al studies the problem of a theoreti al explanation of phase diagrams in JJA's in dissipative environment is still open. Our previous theoreti al work in whi h the attention was fo used on lo al dissipation e�e ts, predi ted the ex- isten e of the riti al value of the dissipation parameter α = 2 independent on geometry of a latti e and mag- neti �eld. Other theoreti al studies 11,12,23,24 suggest a rather broad range of the riti al values of the dissipation parameter α = 0.5, 0.84, 1, 2 whi h depends on the di- mension of the system and me hanism of the dissipation. It seems that an unambiguous in experimental measure- ment of the riti al value of the normal state ondu tan e is elusive. Several groups 17,18,25,26,27,28,29 using di�erent experimental te hniques obtained various riti al values of α = 0.5, 0.8, 1. To explain these theoreti al and ex- perimental di� ulties we propose a model in whi h lo al ( aused by shunt resistors onne ting the islands to a ground) and nonlo al (shunt resistors in parallel to the jun tions) dissipation e�e ts are onsidered. The purpose of this paper is to investigate phase tran- sitions at zero temperature in two-dimensional apa i- tively oupled super ondu ting arrays with emphasis on http://arxiv.org/abs/0704.1993v1 the ompetition of lo al α0 and nonlo al α1 dissipation e�e ts. The detailed phase boundary ru ially depends on the ratio of mutual to self- apa itan es C1/C0 and spe i� planar geometry of the array. Aware of that fa t we onsider apa itive matrix Cij within the range of parameters C1/C0 whi h an be adjusted to the exper- imental samples. We analyze phase diagrams for three di�erent latti es: square (�), triangular (△) and honey- omb (H). We want to emphasize that our approximation annot be used for analysis of the Berezinski-Kosterlitz- Thouless transitions sin e it is appropriate only for phys- i al systems where long-range order appears. The outline of the rest of the paper is the following: In Se . II we de�ne the model Hamiltonian, followed by its path integral formulation in terms of the dimensionality dependent nonmean-�eld like approa h. In Se III we present the zero-temperature phase diagram results for di�erent JJA's geometries. Finally, in Se . IV we dis uss our results and their relevan e to other theoreti al and experimental works. II. MODEL We onsider a two-dimensional Josephson jun tion ar- ray with latti e sites i, hara terized by super ondu ting phase φi in dissipative environment. The orresponding Eu lidean a tion reads: S = SC + SJ + SD, (1) where 〈i,j〉 dτJij {1− cos [φi (τ) − φj (τ)]} , dτdτ ′αij (τ − τ ′) [φi (τ) − φj (τ ′)] and τ is the Matsubara's imaginary time (0 ≤ τ ≤ 1/kBT ≡ β); T is temperature and kB the Boltzmann onstant (~ = 1). The �rst part of the a tion (2) de- �nes the ele trostati energy where Cij is the apa i- tan e matrix whi h is a geometri property of the array. This matrix is usually approximated as a diagonal (self- apa itan e C0) and a mutual one C1 between nearest neighbors. We an write a general expression for the Cij in the following form: Cij = C0 + zC1 for i = j −C1 for nearest neighbors whi h holds for periodi stru tures in any dimension; z is oordination number of the network. The se ond term is the Josephson energy EJ (Jij ≡ EJ for |i− j| = |d| and zero otherwise). The ve tor d forms a set of z latti e translation ve tors, onne ting a given site to its near- est neighbors. The Fourier transformed wave-ve tor de- pendent Josephson ouplings Jk are di�erent for various latti es. The third part of the a tion SD des ribes the dissipation e�e ts and αij (τ − τ ′) is a dissipation ma- trix. We hoose two independent damping me hanisms, the on-site and the nearest-neighbor, be ause usually, the damping is des ribed in terms of shunt resistors R0 on- ne ting the islands to a ground and shunt resistors in parallel to the jun tions related to R1. We an write dissipation matrix similar to Eq. (3) in a more losed form: αij = (α0 + zα1) δij − α1 δi,j+d (4) with the ve tor d running over nearest neighboring is- lands. The dimensionless parameters , α1 = . (5) des ribe strength of the lo al and nonlo al dissipation respe tively, where RQ = 1/4e is quantum resistan e. A. Method Most of existing analyti al works on quantum JJA's have employed di�erent kinds of mean-�eld-like approxi- mations whi h are not reliable for treatment spatial and temporal quantum phase �u tuations. The model in Eq. 2 en odes the phase �u tuation algebra given by Eu- lidean group E2 de�ned by ommutation relations be- tween parti le Li and phase Pj operators, Pj = e iφj , [Li, Pj ] = −Piδij , Li, P i δij , [Pi, Pj ] = 0, (6) with the onserved quantity (invariant of the E2 algebra) i ≡ P xi + P yi = 1. (7) The proper theoreti al treatment of the quantum JJA's must maintain the onstraint in Eq. 7. A formulation of the problem in terms of the spheri al model initiated by Kope¢ and José leads us to introdu e the auxiliary omplex �eld ψi whi h repla es the original operator Pi. Furthermore, relaxing the original �rigid� onstraint and imposing the weaker spheri al ondition: i = N. (8) where N is the number of latti e sites, allows us to im- plementation the spheri al onstraint: [Dψ] δ |ψi|2 −N e−SJ[ψ] [Dφ] e−SC+D[φ] δ [Reψi − Pxi (φ)] ×δ [Imψi − Pyi (φ)] . (9) where [Dψ] = iDψiDψ∗j and [Dφ] = iDφi. It is on- venient to employ the fun tional Fourier representation of the δ fun tional to enfor e the spheri al onstraint in Eq. (8): δ [x (τ)] = ∫ +i∞ dτλ(τ)x(τ), (10) whi h introdu es the Lagrange multiplier λ (τ) thus adding a quadrati term (in ψ �eld) to the a tion in Eq. (2). The evaluation of the e�e tive a tion in terms of the ψ to se ond order in ψi gives the partition fun tion of the quantum spheri al model (QSM) ZQSM = [Dψ] δ |ψi|2 −N e−S[ψ] (11) where the e�e tive a tion reads: S[ψ] = 〈i,j〉 dτdτ ′ {[Jij (τ) δ (τ − τ ′) + W−1ij (τ, τ ′)− λ (τ) δijδ (τ − τ ′) + Nλ (τ) δ (τ − τ ′)} . (12) Furthermore, Wij (τ, τ ′) = [Dφ] ei[φi(τ)−φj(τ ′)]e−SC+D[φ], (13) is the phase-phase orrelation fun tion with statisti al [Dφ] e−SC+D[φ], (14) where a tion SC+D [φ] is just a sum of ele trostati and dissipative terms in Eq. (2). After introdu ing the Fourier transform of the �eld φi (τ) = φk,ne −i(ωnτ−kri) with ωn = 2πn/β, (n = 0,±1,±2, ...) being the Bose Matsubara frequen ies. From Eq. (13) the phase-phase orrelation fun tion reads: W (τ, τ ′) = exp 1− cos [ωn (τ − τ ′)] ω2n + . (16) The harging energy parameter entering Eq. (16) is π (C0 + 4C1) C0 + 4C1 where K (x) is the ellipti integral of the �rst kind32. Furthermore we introdu e quantities E0 = e 2/2C0 and E1 = e 2/2C1 related to the island and jun tion apa i- tan es. The dissipative parameter α and may be expli itly written as α−1 = lim α0 + zα1 − 2α1Ek where Ek is a dispersion and has di�erent form for various latti es. In the present paper we onsider three di�erent geometries of the latti e: square (�), triangular (△) and honey omb (H): = cos kx + cos ky, = cos kx + 2 cos . (19) with the latti e spa ing set to 1. The results of the sum over wave ve tors in Eq. 18 are pla ed in Appendix B. Finally, for small frequen ies, α0 ≤ 2 and α1 ≤ 1 the inverse of orrelation fun tion (16) be omes: W−1 (ωn) = ω2n + |ωn| for ωn 6= 0 0 otherwise In order to determine the Lagrange multiplayer λ we ob- serve that in the thermodynami limit (N → ∞) the steepest des ent method be omes exa t. The ondition the integrand in Eq. (11) has a saddle point λ (τ) = λ0, leads to an impli it equation for λ0: G (k, ωn) , (21) where G−1 (k, ωn) = λ0 − Jk + ω2n + |ωn| . (22) The emergen e of the riti al point in the model is sig- naled by the ondition G−1 (k = 0, ωn = 0) ≡ λ0 − J0 = 0 (23) whi h �xes the saddle point of the Lagrange multiplier λ0 within the ordered phase λ0 = J0. 0.3 0.6 0.9 1.2 1.5 1.8 2.1 Figure 1: Zero-temperature phase diagram for the total harg- ing energy EJ/EC vs parameter of dissipation αi (i = 1 if α0 = 0 and i = 0 if α1 = 0) for triangular (△; α 1 = 2/3), square (�; αcrit1 = 1) and honey omb (H ; α 1 = 4/3) lat- ti e. Insulating (super ondu ting) state is below (above) the urves. III. PHASE DIAGRAMS A Fourier transform of the Green fun tion in Eq. (22) enables one to write the spheri al onstraint (21) expli - itly as: ρ (ξ) λ− ξEJ + 18EC ω ξ |ωn| . (24) where ρ (ξ) = is the density of states. We an easily see that solu- tion of the model requires the knowledge of the DOS for a spe i� latti e with the superimposition of the self- onsisten y ondition for the riti al line in Eq. (24). A Josephson-jun tion array network is hara terized for di�erent latti es by the nearest-neighbor Josephson ou- pling EJ with the following wave-ve tor dependen e = EJE�k = EJE△k = EJEHk (26) where Ek's are given by Eq. 19. The Fourier transform of the apa itan e (dissipative) matri es for a triangular and honey omb latti e an be also found in Appendix 0.2 0.4 0.6 0.8 1 1.2 a a0 1 0= = a0 2,≃ a1 1≃ PCSdiss PCSdiss E EJ0 / E EJ1 / crit crit Figure 2: Zero-temperature phase diagram for square 2D JJA's with self C0 = e 2/2E0 and mutual C1 = e 2/2E1 a- pa itan e (Eq. 17) for two values of lo al and nonlo al dis- sipation parameter α0 = α1 = 0 and α0 = 2 and α1 = 1 (see Appendix). We an distinguish three areas: phase oher- ent state (PCS) where phases in the islands are well de�ned. Insulating state (IS) whi h ould be driven to the phase o- herent state by e�e ts of the dissipation (PCSdiss). Finally, insulating state, where super ondu ting phase is perturbed by strong zero point quantum �u tuations due to Coulomb blo k- ade that lo alizes harge arries to the islands. However sys- tem an be driven to the phase oherent state (PCScritdiss) but only by riti al values of the dissipation parameters (αcrit0 ≃ 2 and αcrit1 ≃ 1). By substituting the value of λ0 = Jmax where Jmax denotes maximum value of the spe trum Jk, and after performing the summation over Matsubara frequen ies, in T → 0 limit we obtain the following result: ρ (ξ) )2 − Jmax−ξEJ )2 − Jmax−ξEJ )2 − Jmax−ξEJ  . (27) The riti al values of the nonlo al dissipation param- eters have a sour e in low temperature properties of the JJA's orrelation fun tion in dissipative environment (Appendix A). The dependen e of the riti al value α1 depi ted in Fig. 1 is a dire t result of the divergen e this phase-phase orrelator. The Fig. 2 and Fig. 3 point out the big di�eren e in values of the self C0 and mutual C1 apa itan e and ompetition between various dissi- pation me hanisms have a severe impa t on phase dia- grams. In typi al real situations mutual apa itan e an be at least two orders of magnitude larger than the self- apa itan e what indi ates the samples are pla ed very lose to E1/EJ axis in Fig. 2. JJA's devoid of dissipation e�e ts an be in two phases: insulator phase (IS) and phase oherent state (PCS). However oupling system to the environment we are able 0.2 0.4 0.6 0.8 1 1.2 a0 PCSdiss Figure 3: Zero-temperature phase diagram for square 2D JJA's in spa e of lo al α0 and nonlo al α1 dissipation pa- rameters for several values of the ratio C1/C0. From the top C1/C0 = 1.25, 1.67, 2.5, 5, 10, 50. Insulating (super ondu t- ing) state below (above) the urves. to drive arrays into PCS even if lo alization of the harge arriers due to Coulomb blo kade is strong and dominates properties of the system. Furthermore for ea h geometry of the latti e there are riti al values of the dissipation parameters that lead arrays to situation (Fig. 2) where phases in the islands are well de�ned and quantum �u - tuations do not perturb a super ondu ting phase - region des ribe as PCScritdiss. Notwithstanding between these two boundary situations there is a region on the phase dia- gram in Fig. 2 where on rete situation depends on the values of the parameters. In this area PCSdiss system an be driven to the PCS but oupling to the environ- ment does not have to be so strong as in PCScritdiss ase. IV. RESULTS Until now only three papers onsidered e�e ts with both me hanisms of the dissipation. 11,20,21 The most in- teresting is Cu oli's work where authors introdu ed the full ondu tan e matrix for triangular and square lat- ti es. It seems their results improve the quantitative a - ura y; nevertheless problem of the theoreti al explana- tion of the phase diagram of JJA's in dissipative environ- ments is thus open. A model of an ordered array of resistively shunted Josephson jun tions was also onsidered by Chakravarty et. al. and simpli�ed at several points. They assumed that apa itan e matrix is diagonal Cij = Cδij . The au- thors laim the results do not depend sensitively on de- tailed form of Cij . Moreover the matrix αij = h/4e where Rij is the shunting resistan e between grains i and j is redu ed to the form in whi h the information about the geometry of the latti e is not in luded. The obtained zero-temperature phase diagram reveals the fa t that the riti al value of the dissipation exists and is proportional to the inverse of the dimension of the system whi h gives us riti al value α = 1/2 for a square latti e, but espe- ially at low temperatures, variational methods are not pre ise enough to per eive su h a subtle transition. In our model the riti al value of the nonlo al dissipa- tion parameter α1 behaves similarly. It depends on the maximum value of the Jk spe trum. Be ause Jk exhibits di�erent hara ters for various latti es hen e we ould observe phenomenon su h as nonmonotoni dependen e of the riti al value of the nonlo al dissipation parameter for various geometries of the array (see Fig. 1). When we assume diagonal form αij = α0δij then obviously our results will not hange when we hange the geometry of the latti e simply be ause the shunt resistors onne ting the islands to ground are the same for ea h island. On the other hand if we take into onsideration that α0 = 0 and only α1 is present, the situation hanges be ause now values of the dissipative matrix strongly depend on the Jk spe trum whi h indi ates various values of the matrix αij depend on the geometry of the stru ture. This ase is present in arrays with shunt resistors in parallel to the jun tions. A standard way to study models with only diagonal harging energies Cij = Cδij orresponds to a om- plete absen e of s reening by the other islands in the array. Noti e the mutual apa itan e an be at least two orders of magnitude larger than the self- apa itan e C1 ≃ 102C0. We propose a more realisti model in whi h both self and mutual apa itan es are nonzero. To see how signi� ant this onsideration is we shall analyze Fig. 3. If we assume, that C1/C0 ≃ 50 (see the lowermost line in Fig. 3) we an see the riti al values of the dissipation parameters hange dramati ally from lo al α0 = 2 and nonlo al α1 = 1 (Appendix A) obtained for fewer real- isti ases C1/C0 = 1 in whi h both of the apa itan es are omparable. In the limit (C1 6= 0, C0 = 0) the latti e model is equivalent to the Coulomb gas model with rit- i al properties not fully understood at present. However we have to emphasize that the range of the Coulomb ma- trix be omes in�nite when C0 is set equal to zero. The phase boundary in Fig. 3 shift downward with in reasing ratio C1/C0 be ause lower value of the Coulomb inter- a tion between nearest neighbors redu es strong quan- tum phase �u tuations and in onsequen e we observe a growth of long-range phase oheren e. Yagi et al. experimented with the super ondu tor- insulator transition in two-dimensional network of Josephson jun tions in detail by varying the jun tions- area. It was observed the riti al tunneling resistan e exhibited signi� ant jun tion area dependen e. The low-temperature behavior of the total harging energy EJ/EC as a fun tion RQ/Rn, where Rn is the tunnel- ing resistan e exhibits the same behavior as the urve obtained from our theory for square latti es (see Figure 1) in the absen e of the lo al dissipation e�e ts α0 = 0. The observed phase boundary, whi h is bending down- ward, and the riti al value of the nonlo al dissipation parameter αcrit1 = 1 is in ex ellent a ordan e with our results. In order to investigate the e�e ts of quantum �u tu- ations and dissipation in JJA's, another group two-dimensional arrays of small jun tions with various EJ/EC and resistors whi h aused dissipation e�e ts. The value of EJ was ontrolled by varying the tunnel resistan e. Ea h island was onne ted to the neighbor- ing ones by a shunt resistor as well as the tunnel jun - tion. The resistan e of the shunt resistors was tuned by varying their length. Ground states of 2D Josephson ar- ray in EJ/EC -RQ/Rs parameters spa e reveal the same behavior as previous experimental results but there is a di�eren e in riti al value of the RQ/Rs ≃ 0.5 whi h also di�ers from value αcrit1 obtained in this paper. These dis- repan ies between experiments an be explained in the framework of our model by taking into a ount di�er- ent value of the ratio jun tion-to-self apa itan es whi h redu es the riti al value of the RQ/Rs and onsidering that not only nonlo al dissipation me hanism is present α0 6= 0 we an obtain RQ/Rs ≃ 0.5 value (see Figure 3). Real numbers strongly depend on the properties of the jun tions used in experiments. V. SUMMARY We have al ulated quantum phase diagrams of two- dimensional Josephson jun tion arrays using the spher- i al model approximation. The al ulations were per- formed for systems using experimentally attainable ge- ometries for the arrays su h us square, triangular and honey omb. The ground state of the Josephson oupled array with a triangular latti e appears to be most stable against the Coulomb e�e ts. This geometry is also the ase in whi h the global oherent state emerged when the value of the nonlo al dissipation parameter α1 is the lowest. In JJA's we an observe the phase oheren e transition whi h is aused by ele trostati and dissipa- tive e�e ts. The detailed phase diagrams ru ially de- pend on the ratio jun tion-to-self apa itan es, C1/C0 and both dissipation me hanism have a big impa t on phase boundaries. The nondiagonal terms in apa itive and dissipative matri es an hange the phase diagrams of the system drasti ally . It is ne essary to take them into onsiderations when we have di�erent sour es of dis- sipation su h us shunt resistors onne ting the islands to a ground and shunt resistors in parallel to the jun tions. The experimental observation of an universal resistan e threshold for the onset of the global oherent state seems possible, but appears to be di� ult. A knowledgments One of the authors wants to thank Dr. Ettore Sarnelli for arefully reading manus ript and fruitful dis ussions. This work was supported by the TRN �DeQUACS� and some parts of it were done in Max-Plan k-Institut für DOS △ H � Jmax/EJ 3 Table I: Maximum values of the spe trum J (k) for three dif- ferent geometries of the latti es: triangular (△), honey omb (H) and square (�) Physik komplexer Systeme, Nöthnitzer Straÿe 38, 01187 Dresden, Germany. Appendix A: SOME PROPERTIES OF THE CORRELATOR Assume that α0 = 0 we write expression for the phase- phase orrelation fun tion (similar to equation used in a previous al ulations but modi�ed by dissipative ma- trix) in form: W (τ) = exp 1− cos (ωnτ) ω2n + . (A1) It is easy to see the sum over ωn is symmetri when we hange ωn → −ωn. The key to obtain the solution is a al ulation the sum or the integral under the expo- nent in Eq. (A1). Be ause we are going to investigate low-temperature properties of the orrelation fun tion we ould write dω. In that ase (getting rid of abs) for large value τ we write W (τ) = exp 1− cos (τω) ω2 + α1 ≃ exp −2γEJ α1JkEC )2EJ/α1Jk where γ = 0.57721 is the Euler-Mas heroni onstant. Finally, after Fourier transform we see that orrelator W−1 (ωm) ∼ |ωm|2EJ/α1Jmax−1 at zero temperature di- verges for α1 ≥ 2EJ/Jmax. Quantity Jmax/EJ means the maximum value of the Jk whi h di�ers for various latti es (see Table I). Appendix B: DISSIPATION PARAMETER FOR CONSIDERED LATTICES In this appendix we give the expli it formulas for the dissipation parameter dis ussed in Se . II and III. 1. Square latti e π (α0 + 4α1) α0 + 4α1 where K (x) = ∫ π/2 1− x2 sin2 φ , (B2) is the ellipti integral of the �rst kind and the unit step fun tion is de�ned by: Θ(x) = 1 for x > 0 0 for x ≤ 0 . (B3) For small values of the α1 we an write dissipation pa- rameter for square latti e as: α� = α0 + 3α1 − , (B4) for large values values of the α1: 3− 2 2. Triangular latti e α−1△ = K (κ) (B6) where (2t+ 3) 1/2 − 1 ]3/2 [ (2t+ 3) 4 (2t+ 3) (2t+ 3) 1/2 − 1 ]3/2 [ (2t+ 3) with t = (α0 + 6α1) /2α1. 3. Honey omb latti e α−1H = α0 + 3α1 K (κ) (B9) where (2t− 1)3/2 (2t+ 3)1/2 (B10) 41/4 (2t) (2t− 1)3/2 (2t+ 3)1/2 (B11) with t = (α0 + 3α1) /2α1. Appendix C: DOS FOR CONSIDERED LATTICES In this appendix we give the expli it formulas for the density of states dis ussed in Se . II and III. 1. Square latti e ρ� (ξ) = , (C1) 2. Triangular latti e ρ△ (ξ) = −Θ(ξ − 3) where 3 + 2 3 + 2ξ − ξ2 −Θ(ξ + 1) 3 + 2ξ [Θ (ξ + 1)−Θ(ξ − 3)] , (C3) κ1 = 4 3 + 2ξ −Θ(ξ + 1) 3 + 2 3 + 2ξ − ξ2 [Θ (ξ + 1)−Θ(ξ − 3)] .(C4) 3. Honey omb latti e ρH (ξ) = 4 |ξ| ρ△ 3− 4ξ2 . (C5) Ele troni address: polak��si a. ib.na. nr.it Ele troni address: kope �int.pan.wro .pl Simánek, Solid State Commun. 31, 419 (1979). S. 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0704.1994

Substrate-induced bandgap in graphene on hexagonal boron nitride

Substrate-induced bandgap in graphene on hexagonal boron nitride Gianluca Giovannetti1,2, Petr A. Khomyakov2, Geert Brocks2, Paul J. Kelly2 and Jeroen van den Brink1 Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands Faculty of Science and Technology and MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. (Dated: October 22, 2018) We determine the electronic structure of a graphene sheet on top of a lattice-matched hexagonal boron nitride (h-BN) substrate using ab initio density functional calculations. The most stable configuration has one carbon atom on top of a boron atom, the other centered above a BN ring. The resulting inequivalence of the two carbon sites leads to the opening of a gap of 53 meV at the Dirac points of graphene and to finite masses for the Dirac fermions. Alternative orientations of the graphene sheet on the BN substrate generate similar band gaps and masses. The band gap induced by the BN surface can greatly improve room temperature pinch-off characteristics of graphene-based field effect transistors. PACS numbers: 71.20.-b, 73.22.-f, 73.20.-r Introduction Less than 3 years ago it was discovered that graphene – a one-atom-thick carbon sheet – can be deposited on a silicon-oxide surface by micromechnical cleavage of high quality graphite [1]. The graphene flakes are micrometers in size, sufficiently large to have con- tacts attached so as to construct field effect transistors (FETs). Electrical transport measurements made clear that at room temperature graphene has an electron mo- bility of at least 10 000 cm2/Vs, a value ten times higher than the mobility of silicon wafers used in microproces- sors [1, 2, 3]. The high mobility is not much affected by a field-induced excess of electrons or holes. A graphene sheet has a honeycomb structure with two crystallographically equivalent atoms in its primitive unit cell. Two bands with pz character belonging to different irreducible representations cross precisely at the Fermi energy at the K and K ′ points in momentum space. As a result undoped graphene is a zero-gap semiconductor. The linear dispersion of the bands results in quasiparti- cles with zero mass, so-called Dirac fermions. At energies close to the degeneracy point the electronic states form perfect Dirac cones. The absence of a gap, preventing the Dirac fermions from attaining a finite mass and com- plicating the use of graphene in electronic devices [4], is related to the equivalence of the two carbon sublattices of graphene. The relativistic nature of the Dirac fermions gives rise to counterintuitive phenomena. One, known as the Klein paradox, is that relativistic electrons exhibit per- fect transmission through arbitrarily high and wide po- tential barriers. This effect is related to an unwanted characteristic of graphene FETs, namely that pinch-off is far from complete [5]. If one applies a gate voltage so that either holes or electrons are injected into the graphene sheet, the FET is open and its conductivity high. One can then try to block the current by tun- ing the gate voltage to move the graphene layer towards the charge neutrality point where the Fermi energy co- incides with the Dirac points; at this energy the density of states vanishes and nominally there are no carriers present. However, it turns out that in spite of the lack of electronic states the conductivity does not vanish in this case. Rather, it assumes the minimal value σmin = 4e where h is Planck’s constant and e the unit of charge. Thus even when pinched-off to its maximum the FET still supports an appreciable electrical current, which is intrinsic to graphene and related to the fact that the Dirac fermions are massless [2, 3, 4, 5, 6, 7]. Inducing a gap The poor pinch-off can only be reme- died by generating a mass for the Dirac fermions. A number of possibilities exist to do so. One is to use bi-layer graphene which will have a gap if the top and bottom layers are made inequivalent, for instance by ap- plying a bias potential [5]. Another is the use of graphene nanoribbons, where gaps arise from the lateral constric- tion of the electrons in the ribbon. The size of the gap then depends on the detailed structure of the ribbon edges [8, 9, 10]. We investigate an alternative possibility and consider graphene on a substrate that makes the two carbon sublattices inequivalent. This breaks the sublat- tice symmetry directly, generating an intrinsic and robust mass for the Dirac fermions. As a substrate, hexagonal boron nitride (h-BN) is a suitable choice [11]. This wide gap insulator has a layered structure very similar to that of graphene but the two atoms in the unit cell are chemically inequivalent. Placed on top of h-BN the two carbon sublattices of graphene become inequivalent as a result of the interaction with the substrate. Our band structure calculations in the local-density approximation show that a gap of at least 53 meV – an energy roughly twice as large as kBT at room temperature – is induced. This can be compared to graphene on a copper (111) metallic surface where the gap is found to be much smaller and can even vanish, depending on the orientation of the graphene sheet. Stable structure The lattice mismatch of graphene with http://arxiv.org/abs/0704.1994v2 carbonnitrogen boron FIG. 1: (Color online) The three inequivalent orientations of single-layer graphene on a h-BN surface. Left: sideview, right: topview. hexagonal boron nitride is less than 2%. Just as in graphite, the interaction between adjacent BN layers is weak. The h-BN layers have an AA′ stacking: the boron atoms in layer A are directly above the nitrogen atoms in layer A′. Within the local density approximation (LDA) the minimum energy separation of adjacent layers is found to be 3.24 Å, which is reasonably close to the experimental value of 3.33 Å. Because generalized gra- dient approximation (GGA) calculations give essentially no bonding between BN planes and lead to excessively large values of c [12], we opt for electronic structure cal- culations within the LDA. Electronically, h-BN is a wide gap insulator, with experimentally a gap of 5.97 eV [13]. This gap is underestimated by about 33% in LDA. A quasi-particle GW correction on top of the LDA brings it into very close agreement with experiment [14, 15] and reinterprets experiment in terms of an indirect gap. For the composite graphene layer on top of h-BN system, we use the LDA lattice parameter for graphene, a = 2.445 On the basis of this structural information we con- struct a unit cell with 4 layers of h-BN and a graphene 0.25 0.15 0.05 −0.05 3.9 3.7 3.5 3.3 3.1 2.9 2.7 2.5 d ( °A) FIG. 2: (Color online) Total energy E of graphene on h-BN surface for the three configurations (a), (b), and (c) as a func- tion of the distance between the graphene sheet and the top h-BN layer. top layer. We represent the vacuum above graphene with an empty space of 12 to 15 Å. The results to be presented below converge quickly as a function of the number of h- BN layers and the width of the vacuum space, consistent with weak interlayer interactions. No significant differ- ence in the final results were found when 6 layers of h-BN were used. The in-plane periodicity is that of a single graphene sheet with a hexagonal unit cell containing two carbon atoms. We consider three inequivalent orienta- tions of the graphene sheet with respect to the h-BN, see Fig. 1: - the (a) configuration with one carbon over B, the other carbon over N. - the (b) configuration with one carbon over N, the other carbon centered above a h-BN hexagon. - the (c) configuration with one carbon over B, the other carbon centered above a h-BN hexagon. The self-consistent calculations were performed with the Vienna Ab-initio Simulation Package (VASP) [16, 17] using a plane wave basis and a kinetic energy cutoff of 600 eV. The Brillouin Zone (BZ) summations were car- ried out with the tetrahedron method and a 36× 36× 1 grid which included the Γ, K and M points. A dipole correction avoids interactions between periodic images of the slab along the z-direction [18]. The total energies of the three configurations are shown as a function of the distance between the h-BN surface and the graphene sheet in Fig. 2. For all distances, the lowest-energy configuration is (c) with one carbon on top of a boron atom and the other above a h-BN ring. The equilibrium separation of 3.22 Å for configuration (c) is smaller than 3.50 Å for configuration (a) and 3.40 Å for configuration (b). For all three configurations the energy landscape is seen to be very flat around the energy min- imum. Though symmetry does not require inequivalent DOS C B N Bands Γ K M FIG. 3: (Color online) Band structure along the ΓK and KM directions in reciprocal space, total and projected densities of states (DOS) for the relaxed structure (c) of graphene on h- BN. Carbon, Boron and Nitrogen projected DOS are shown, with a projection on the p-states in-plane (red/thick grey lines) and out-of-plane (blue/thin grey lines). The inset is a magnification of the bands around the K point, where the gap opens. carbon atoms to be equidistant from the BN layer, in practice the stiffness of the graphene sheet prevents any significant buckling. Band structure With the stable structures in hand, we compute the corresponding electronic band structures and projected densities of states which are shown in Fig. 3 for configuration (c). For the h-BN derived bands a gap of 4.7 eV at theK-point is found, which is nearly identical to the LDA gap value at this particular point in the Bril- louin zone found for bulk h-BN [15]. Within this boron nitride gap, the bands have entirely carbon character as expected on the basis of the weak interlayer interactions in both bulk h-BN and graphite. On the electron-volt scale of Fig. 3 the Dirac cone around the K-point appears to be preserved. However, zooming in on that point in the BZ (see inset) reveals that a gap of 53 meV is opened and the dispersion around the Dirac points is quadratic. The band gaps for the three different configurations are shown in Fig. 4 as a function of the distance between the graphene sheet and the h-BN surface. Decreasing this distance increases the gap, as expected for a phys- ical picture based upon a symmetry-breaking substrate potential. The band gaps that are opened at the equi- librium geometries of the (a) and (b) configurations are 56 meV and 46 meV, respectively, which are compara- ble to the band gap obtained for configuration (c). The largest gap is found for the (a) configuration with one carbon atom above a boron atom and the other above a nitrogen atom. Again, this is expected for gap open- ing induced by breaking the symmetry of the two carbon FIG. 4: (Color online) The values of the gaps for the three configurations (a), (b), and (c) as a function of the distance between the graphene sheet and the top h-BN layer. The calculated equilibrium separations are indicated by vertical arrows. atoms. Since LDA generally underestimates the gap, the values that we obtain put a lower bound on the induced band gaps, which we thus find to be significantly larger than kBT at room temperature. Although the lattice mismatch between graphene and h-BN constants is less than 2% and can be neglected in a first approximation, in a real system incommensurabil- ity will occur and we expect the strong in-plane bond- ing of both graphene and h-BN to prevail over the weak inter-plane bonding. For graphene on Ir(111) where the lattice mismatch is ∼ 10%, Moiré patterns have been ob- served in STM images [19]. There, first-principles calcu- lations showed that regions could be identified where the graphene was in registry with the underlying substrate in high symmetry configurations analogous to the (a), (b) and (c) configurations discussed above, and transition regions with little or no symmetry [19]. The graphene separation from the substrate varied across the surface leading to bending of the graphene sheet. If we could take the lattice mismatch into account in a large super- cell in a similar fashion, some areas of graphene would be forced into the higher energy (a) and (b) configurations with larger separations to the BN substrate. However, the corresponding band gaps are all of the order of the 50 meV we find for the lowest energy (c) configuration, or higher. It seems reasonable to conclude that the broad- ening resulting from lattice mismatch will not reduce the gap substantially. Cu(111) substrate The situation changes markedly for graphene on a Cu(111) surface. The copper surface layer forms a triangular lattice, matching that of graphene to better than 4%. We consider two configurations of graphene on Cu(111). Either the center of each car- bon hexagon is on top of a Cu atom, which we call the symmetric configuration in the following, or every sec- ond carbon atom is on top of a Cu atom, which we call the asymmetric configuration. For the asymmetric and symmetric configurations, LDA calculations yield equi- librium separations of 3.3 Å and 3.4 Å which are compa- rable to those of graphene on h-BN (Fig. 2). The total energy difference between the two configurations is only about 9 meV. In the asymmetric configuration a small gap of 11 meV is opened in the graphene band structure, whereas in the symmetric configuration the gap remains very close to zero. In both cases we find very little mix- ing between copper and carbon states. The difference between the gaps can be explained by the fact that the symmetric configuration preserves the graphene symme- try in the top Cu surface layer, whereas the symmetry is broken in the asymmetric configuration. The effect of this symmetry breaking is small, however, and the re- sulting band gap is much smaller than that induced by h-BN and comparable to the typical thermal broadening reported in experiments [2, 4]. Taking into account the graphene-Cu lattice mismatch in for instance a super cell calculation [19] will not to change this conclusion. For both configurations of graphene on Cu, a charge rearrangement at the interface is found which moves the Fermi level away from the induced gap [20] by much more than the magnitude of the gap itself. This is in contrast to a h-BN substrate, where the Fermi level re- mains in the induced gap. Around the Fermi level of graphene on Cu the band dispersion is still linear. Con- sequently, the properties characteristic of graphene which result from the linear dispersion should be preserved. In for instance tunneling experiments that require adsorp- tion of graphene on a metallic (Cu) substrate [21] one should still be able to observe the intrinsic linear elec- tronic structure of graphene near the Fermi energy, but no longer at the Dirac points. Conclusions Our density functional calculations show that the carbon atoms of a graphene sheet preferentially orient themselves directly above the boron atoms of a h-BN substrate, with one carbon sublattice above the boron sublattice and the other carbon centered above a h- BN ring. Although graphene interacts only weakly with the h-BN substrate, even when a few angstroms away the presence of h-BN induces a band gap of 53 meV, generating an effective mass for the Dirac fermions of 4.7·10−3 me, whereme is the electron mass. The gap that opens at the Dirac points is considerably larger than the one for graphene on Cu(111). Additional quasi-particle interactions, for instance taken into account within aGW scheme, will increase the value of the gap. The opening of a band gap in graphene on h-BN offers the potential to improve the characteristics of graphene-based FETs, decreasing the minimum conductance by orders of magni- tude. Other interesting features such as the valley degree of freedom, which is related to the degeneracy of the K and K ′ points in the Brillouin zone, remain intact and can still be used to control an electronic device [22]. Also the half-integer quantum Hall effect – a peculiar charac- teristic of graphene – remains unchanged [2, 23, 24]. This work was financially supported by “NanoNed”, a nanotechnology programme of the Dutch Ministry of Economic Affairs and by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)” via the re- search programs of “Chemische Wetenschappen (CW)” and the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”. Part of the calculations were performed with a grant of computer time from the “Stichting Na- tionale Computerfaciliteiten (NCF)”. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). [3] Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005). [4] A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007). [5] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Na- ture Physics 2, 620 (2006). [6] H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. Vandersypen, and A. F. Morpurgo, Nature 446, 56 (2007). [7] J. van den Brink, Nature Nanotechnology, 2, 199 (2007). [8] Y. W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97, 216803 (2006). [9] M. Y. Han, B. Ozyilmaz, Y. Zhang, and P. Kim, cond- mat/0702511. [10] Y. M. L. Z. Chen, R. J. Rooks, and P. Avouris, cond- mat/0701599. [11] K. Suenaga, C. Colliex, N. Demoncy, A. Loiseau, H. Pas- card, and F. Willaime, Science 278, 653 (1997). [12] G. Kern, G. Kresse, and J. Hafner, Phys. Rev. B 59, 8551 (1999). [13] K. Watanabe, T. Taniguchi, and H. Kanda, Nature Ma- terials 3, 404 (2004). [14] X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Phys. Rev. B 51, 6868 (1995). [15] B. Arnaud, S. Lebegue, P. Rabiller, and M. Alouani, Phys. Rev. Lett. 96, 026402 (2006). [16] G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996). [17] G. Kresse and J. Furthmuller, Comp. Mat. Sci. 6, 15 (1996). [18] J. Neugebauer and M. Scheffler, Phys. Rev. B 46, 16067 (1992). [19] A. T. N’Diaye, S. Bleikamp, P. J. Feibelman, and T. Michely, Phys. Rev. Lett. 97, 215501 (2006). [20] G. Giovannetti, P. A. Khomyakov, G. Brocks, J. van den Brink, P. J. Kelly, to be published. [21] C. Oshima and A. Nagashima, J. Phys.: Condens. Mat- ter. 9, 1 (1997). [22] A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, Nature Physics 3, 172 (2007). [23] N.M.R. Peres, F. Guinea and A. H. Castro Neto, Phys. Rev. B 73, 125411 (2006). [24] V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005).

0704.1995

Coherent dynamics of domain formation in the Bose Ferromagnet

Coherent dynamics of domain formation in the Bose Ferromagnet Qiang Gu1,2, Haibo Qiu1,3 Department of Physics, University of Science and Technology Beijing, Beijing 100083, China Institut für Laser-Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China (Dated: November 20, 2018) We present a theory to describe domain formation observed very recently in a quenched 87Rb gas, a typical ferromagnetic spinor Bose system. An overlap factor is introduced to characterize the symmetry breaking of MF = ±1 components for the F = 1 ferromagnetic condensate. We demonstrate that the domain formation is a co-effect of the quantum coherence and the thermal relaxation. A thermally enhanced quantum-oscillation is observed during the dynamical process of the domain formation. And the spatial separation of domains leads to significant decay of the MF = 0 component fraction in an initial MF = 0 condensate. PACS numbers: 05.30.Jp, 03.75.Kk, 03.75.Mn Very recently, the Berkeley group observed sponta- neous symmetry breaking in 87Rb spinor condensates [1]. Ferromagnetic domains and domain walls were clearly shown using an in-situ phase-contrast imaging. This ap- pears the first image of the domain structure in a Bose ferromagnet. Although ferromagnetism has been inten- sively studied in the context of condensed matter physics and is regarded as one of the best understood phenom- ena in nature [2], the description of ferromagnetism is not yet complete. The conventional ferromagnets being considered are usually comprised of either classical parti- cles (insulating ferromagnets) or fermions (itinerant fer- romagnets) while Bose systems are seldom touched [3]. The realization of cold spinor 87Rb gases [4], a typical ferromagnetic Bose system, has provided an opportunity to study Bose ferromagnets and thus opens up a way to a comprehensive understanding of ferromagnetism in all kinds of condensed matters. The ferromagnetic spinor Bose gas has attracted nu- merous theoretical interests [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. On one side, researchers expect that it will show some general properties as conventional ferromagnets do. Ho [5], Ohmi and Machida [6] pointed out that this sys- tem has a spontaneous symmetry-broken ground state and a normal spin-wave excitations spectrum at small wave vector k, ωs = csk 2. On the other side, researchers aim at exploring distinct features of the system. Studies on thermodynamics and phase transitions have revealed that the ferromagnetic spinor Bose gas displays a quite surprising phase diagram. Its Curie point can be larger by magnitudes than the energy scale of the ferromag- netic interaction between bosons, and never below the Bose-Einstein condensation point [7, 8]. It means that once the Bose gas condenses, it is already spontaneously magnetized. A conventional ferromagnet usually has some domain structure below the Curie point, as illustrated in Fig. 1a. But whether it is true for a Bose ferromagnet is still some- what controversial. One even questions whether there exists a Curie point in ferromagntic Bose gases [9, 10], as the cold atomic gas under experimental conditions is usually not in the thermodynamic limit while the phase diagram mentioned above is derived from the thermal- equilibrium grand canonical ensemble [7, 8]. Never- theless, a number of theoretical works have discussed the possibility of the domain formation [3, 13, 14, 15]. Within the mean-field theory, Zhang et al. found out that the ferromagnetic condensate has a dynamical in- stability leading to spontaneous domain formation in an initially magnetized Bose gas [13]. Moreover, Mur-Petit et al. showed that a multi-spin-domain structure mani- fests in 87Rb condensates at finite temperatures[14]. The Berkeley experiment confirms that the Bose ferromagnet can indeed form domain structures at least under certain conditions [1]. Then new questions come, is the process of the domain formation similar to that inside a conven- tional ferromagnet, and how does the domain formation affect spin dynamics? We attempt to answer these ques- tions in the present letter. We start with the Hamiltonian for an F = 1 system in the following form [5] ∇ψ†a · ∇ψa − (µ− U)ψ†aψa ψa′ψa + Fab · Fa′b′ψb′ψb where ψa(r) is the field annihilation operator for an atom in state MF = a at point r, µ is the chemical potential and U is the trapping potential. g0 and g2 are the spin- independent and spin-dependent mean-field interaction, respectively. For macroscopically occupied Bose systems it is common to replace the field annihilation operator for the ath spin component by its expectation value, i.e. ϕa(r, t) ≡ 〈ψa(r, t)〉, which for spinor condensates is con- http://arxiv.org/abs/0704.1995v2 (d)(c) (b)(a) FIG. 1: Schematic domain structure inside a Ferromagnet. (a) The bulk material shows no magnetism because domains are randomly oriented. (b) The multi-domain configuration is often simplified to a two-domain structure with opposite sign of magnetization for theoretical convenience. The gray region denotes the domain wall. (c) Two-domain structure for a fer- romagnetic spinor Bose-Einstein condensate. Dotted, dashed and solid lines represent the normalized particle distributions of MF = 1, MF = −1 and MF = 0 bosons respectively. (d) In a homogeneous system, the normalized particle distribution is taken as a constant in each domain; for MF = 0 bosons it remains a constant through the whole system. veniently expressed as ϕa(r, t) = Na(t)ηa(r, t)e iφa(r,t). (2) Here Na(t) is the number of condensed particles, ηa(r, t) denotes the normalized particle distribution with d3rηa = 1 and φa(r, t) the phase. The multiplier of Na and ηa refers to the condensed particle density of the a-component, and the total particle density is ρ(r, t) = Na(t)ηa(r, t). We suppose that the particle distribution can be different for distinct components in our approach. Neglecting excitations, the Hamiltonian is simplified to H = H0 +Hu +Hs, with Naηa(∇φa)2 (U − µ)Naηa + NaηaNa′ηa′ − − 2N+N−η+η− +2N0η0(N+η+ +N−η−) + 4N0 N+N−η0 η+η−cosθ , (3) where θ = φ+ + φ− − 2φ0 is the relative phase. In case that the ground state of the condensate is symmetry-broken, certain magnetic domain structure is formed spontaneously. Within each domain the atomic magnetic moments are aligned in a preferential direc- tion. A direct consequence of domain formation is that the MF = 1 and MF = −1 components are spa- tially separated, as portrayed in Fig. 1c. The integral d3rη+η− measures the extent of overlap between the two, which is called the overlap factor hereinafter. In general, several “overlap factors” should be introduced to Eqs.(3), e.g., α0± = V d3rη2±, α1 = V d3rη+η−, α2± = V d3rη0η± and α3 = V d3rη0 η+η−, where V is the volume of the system. Treating the relative phase θ as a spatially independent constant as previous theory did [11, 12, 13, 14, 15], the term of Hs in Eqs. (3) is rewritten as + + α0−N − − 2α1N+N− +2N0(α2+N+ + α2−N−) + 4α3N0 N+N−cosθ . (4) According to their definition, these overlap factors are not totally independent from each other and the number of independent ones can be further reduced. For simplic- ity, we consider a homogeneous spinor Bose gas with a two-domain structure, as shown in Fig. 1d, and neglect the domain wall. In this case, we derive, after some in- tegration and algebraic manipulation, that there is only one independent overlap factor and the above equation is reduced to (2− α)(N2+ +N2−)− 2αN+N− + 2N0(N+ +N−) + 4 N+N−cosθ , (5) where the reduced overlap factor is just given by α = d3rη+η−. Our model allows the α-factor to vary from one to zero, corresponding to the case that the two com- ponents are from thoroughly mixed to completely sep- arated. For a homogeneous system, the gradient term H0 can be doped. The spin-irrelevant term Hu remains a constant since the total density distribution is hardly responsive to the evolution of domain structure [1, 15], ρ(r, t) ≈ ρ(r, 0). Thus the dynamics of domain formation is only determined by the Hamiltonian expressed in Eq. The Berkeley experiment considered a pure spinor con- densate initially prepared in the unmagnetized state. It is important to emphasize that the total spin is con- served in an atomic quantum gas under experimental conditions[16, 17, 18]. Therefore the particle numbers of MF = 1 and −1 component are always equal, N+ = N−. Such a system is described by the Hamiltonian Hs = −(1− α)(1 − n0)2 − 2n0(1− n0) αcosθ Here Hs = Hs/(N | g22 |) with total particle number N = N+ + N− + N0; na = Na/N is the fraction of MF = a component. n0 and θ form a pair of conjugate variables 0.0 0.2 0.4 0.6 0.8 1.0 -2 -1 0 1 2 -2 -1 0 1 2 FIG. 2: Contour plot of the energy surface in the θ-n0 plane with the overlap factor α = 0.5 (a) and α = 0.1 (b). (c) shows the fraction of MF = 0 atoms at the minima, n̄0, as a function of α. and their equations of motion are given by n0 = −2 αn0(1− n0)sinθ, (7a) θ = 2(1− α)(1 − n0) −2(1− 2n0) αcosθ . (7b) The population dynamics depicts the process of the do- main formation; and the magnetization of magnetic do- mains is defined as m = N+η+ −N−η−. A number of theoretical works have investigated the spin dynamics of the 87Rb spinor condensate [11, 12, 13]. We notice that previous theories usually treated the three components being mixed as they share the same spatial wave function, which is known as the single-mode approx- imation. Therefore the domain structure is smeared out and thus the ferromagnetic feature was not sufficiently dealt with. Those theories correspond the special case of α = 1 in this letter. The dynamical behaviors of Eqs. (7) can be visual- ized by the phase-space portrait with constant energy lines. Figure 2a and 2b plot the contour lines of en- ergy with the overlap factor α = 0.5 and 0.1, respec- tively. In case of α = 1, there are energy minima along n0 = 1/2 at θ = 2nπ [11, 12, 13] and all the contour lines are closed loops around those minima. As domains 0 30 60 90 120 0 30 60 90 120 0 30 60 90 120 0 30 60 90 1200.0 0 30 60 90 1200.0 0 30 60 90 120 1.0 (f)(e) (d)(c) (b)(a) FIG. 3: Dynamics of domain formation for Bose systems with the overlap factor α = 0.1 and TR = 20, 10, and 2 from top to bottom. The left column shows the particle fractions of MF = 0 (n0, solid lines) and MF = ±1 (n±, dotted lines) components. The right column shows the magnetization of magnetic domains, m. The initial state configuration is (n+ = 0.005, n0 = 0.99, n− = 0.005). build up, the value of α drops down and two apparent changes take place, seen in Fig. 2a and 2b. (i) There exist two distinct regimes in the phase space diagram. The newly appeared regime lies in the upper region of the figures, which consists of a set of open curves. Each line corresponds to a rotation type of solution, in which the relative phase θ is ”running” with the time. The closed orbits, lying in the lower region, represent libra- tion type of solutions, in which θ oscillates around the minimum. (ii) The positions of those minima move to- wards to smaller values of n0. The minima points are connected to the ground state of the system. Figure 2c shows the MF = 0 particle fraction at the minima, n̄0. The less α is, the smaller n̄0 is. It means that the domain formation tends to reduce the number of MF = 0 atoms. This point has also been affirmed by Mur-Petit et al. who obtained a state with equipartition in populations, (n+ ≈ 1/3, n0 ≈ 1/3, n− ≈ 1/3), from a starting state (0.005,0.99,0.005) [14]. We derive that n0 = n+ = n− = 1/3 when α drops to 0.25. In the limit case of α = 0, n0 = 0 and n+ = n− = 0.5. The state of the quenched MF = 0 condensate is viewed as a point lying in the upper region of the phase space diagram (Fig. 2a or 2c), when Eqs. (7) yield a self- trapping solution[19]. This motion reflects the quantum mechanical nature of the Bose-Einstein condensate. The oscillating amplitude of n0 is very small and the resulting magnetization m is so low in magnitude that it is hard to be probed experimentally during this stage[1]. The self- trapping effect prevents the growth of magnetic domains. This case is similar to the classical Larmor precession of a spin around magnetic field: it is rotating all the time, but the spin direction can be never parallel to the field without energy dissipation. Then one has to take into consideration the effect of thermal agitation, which can change the energy of the system and drive the system into thermal equilibrium. Therefore, if n0 departs from its thermal equilibrium value n̄0, it will relaxes to n̄0 exponentially with a charac- teristic time scale TR, called the relaxation time. Assum- ing that the relaxation velocity is proportional to n̄0−n0, ∂n̄0/∂t ∝ n̄0−n0, Eq. (7a) can be replaced with the fol- lowing one [20], n0 = −2 αn0(1− n0)sinθ + n̄0 − n0 . (8) TR scales qualitatively the thermal dissipation rate. Longer TR denotes weaker thermal agitation. Figure 3 displays the population of MF = 0 and ±1 components, as well as the magnetization m. The evo- lution of n0 and m reflects the dynamical process of the domain formation. As shown in Fig. 3a and 3b, n0 de- creases oscillatorily with time driven by the the thermal agitation, and meantime m arises. A very interesting re- sult is that the oscillation amplitude increases as the sys- tem relaxes. Generally, the thermal agitation suppresses the macroscopic quantum coherence, and thus tends to kill oscillations of the population, while here we see that the oscillation is enhanced. Furthermore, the magnetiza- tion persists in oscillating for a very long period of time after the amplitude reaches its maximum. This result is qualitatively consistent with the Berkeley group’s obser- vation of the unstable magnetization mode[1]. If the thermal agitation gets stronger, the oscillation will be enhanced first, and then suppressed, as shown in Fig. 3c and 3d. Correspondingly, we divide the whole process into two periods with respect to the domain for- mation, the growing period and the stabilizing period. In the latter period, n0 andm oscillate around their thermal equilibrium values and the amplitudes decrease gradu- ally, then we have a stable domain structure eventually. If sketching the solution in the phase space diagram, one can find that the growing period is represented by the trajectory in the libration regime, and solutions for the stabilizing period lie in the rotation regime. Based on this understanding, Fig. 3a and 3b show only the grow- ing period. Given the thermal agitation strong enough, the quantum mechanical feature will be smeared out in both periods, as Fig. 3e and 3f indicate. According to the above discussions, the present model can qualitatively describe the dynamic process of the do- main formation in a quenched MF = 0 condensate. Sig- nificantly, we show that the spatial separation of mag- netic domains brings about much nontrivial effects on the spin dynamics of the ferromagnetic condensate. To get a quantitative description, more local details of the particle distribution and the relative phase should be considered. In conclusion, we have investigated the dynamics of do- main formation in a ferromagnetic spinor Bose-Einstein condensate, taking into account of the symmetry- breaking of the MF = 1 and −1 components. Magnetic domains develop with the separation ofMF = ±1 compo- nents. Our results suggest that the MF = 0 component in the condensate can significantly decay to a very small value, far less than 1/2 as previous theories predicted. The domain structure is formed and stabilized with the help of the thermal dissipation. A thermally enhanced quantum-oscillation is observed during the process. This work is supported by the National Natural Sci- ence Foundation of China (Grant No. 10504002), the Fok Yin-Tung Education Foundation, China (Grant No. 101008), and the Ministry of Education of China (NCET-05-0098). Q.G. acknowledges helpful discussions with K. Sengstock, K. Bongs and L. You and support from the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg No. 463. [1] L.E. Sadleret al., Nature 443, 312(2006). [2] P. Mohn, Magnetism in the Solid state: An Introduction (Springer-Verlag, Berlin, 2003). [3] Q. Gu, Chapter 6 in Progress in Ferromagnetism Re- search, Ed. by V.N. Murray, (Nova Science Publishers, Inc., New York, 2006). [4] J. Stenger et al., Nature 396, 345 (1998). [5] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998). [6] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). [7] Q. Gu and R.A. Klemm, Phys. Rev. A 68, 031604(R) (2003); Q. Gu, K. Bongs, and K. Sengstock, Phys. Rev. A 70, 063609 (2004). [8] K. Kis-Szabo, P. Szepfalusy, and G. Szirmai, Phys. Rev. A 72, 023617 (2005); G. Szirmai, K. Kis-Szabo, and P. Szepfalusy, Eur. Phys. J. D 36, 281 (2005). [9] T. Isoshima, T. Ohmi, and K. Machida, J. Phys. Soc. Jpn. 69, 3864 (2000). [10] W. Zhang, S. Yi, and L. You, Phys. Rev. A 70, 043611 (2004). [11] H. Schmaljohann et al., Appl. Phys. B: Lasers Opt. 79, 1001 (2004); D. R. Romano and E. J. V. de Passos, Phys. Rev. A 70, 043614 (2004). [12] M.-S. Chang et al., Nature Physics 1, 111 (2005). [13] W. Zhang et al., Phys. Rev. Lett. 95, 180403 (2005). [14] J. Mur-Petit et al., Phys. Rev. A 73, 013629 (2006). [15] T. Isoshima, K. Machida, and T. Ohmi, Phys. Rev. A 60, 4857 (1999). [16] M.-S. Chang et al., Phys. Rev. Lett. 92, 140403 (2004). [17] J. Kronjäger et al., Phys. Rev. A 72, 063619 (2005). [18] H. Schmaljohann et al., Phys. Rev. Lett. 92, 040402 (2004). [19] A. Smerzi et al. Phys. Rev. Lett. 79, 4950 (1997). [20] F. Bloch, Phys. Rev. 70, 460(1946). References

0704.1996

A Wave-function for Stringy Universes

LPTENS–07/16 April 2007 A Wave-function for Stringy Universes∗ Costas Kounnas,1 Nicolaos Toumbas2 and Jan Troost1 1 Laboratoire de Physique Théorique, Ecole Normale Supérieure,† 24 rue Lhomond, F–75231 Paris Cedex 05, France 2 Department of Physics, University of Cyprus, Nicosia 1678, Cyprus. Abstract We define a wave-function for string theory cosmological backgrounds. We give a prescription for computing its norm following an earlier analysis within general relativity. Under Euclidean continuation, the cosmologies we discuss in this paper are described in terms of compact parafermionic worldsheet systems. To define the wave-function we provide a T-fold description of the parafermionic conformal field theory, and of the corresponding string cosmology. In specific examples, we compute the norm of the wave-function and comment on its behavior as a function of moduli. ∗ Research partially supported by the EU under the contracts MRTN-CT-2004-005104, MRTN- CT-2004-512194 and ANR (CNRS-USAR) contract No 05-BLAN-0079-01 (01/12/05). † Unité mixte du CNRS et de l’Ecole Normale Supérieure associée à l’université Pierre et Marie Curie 6, UMR 8549. http://arxiv.org/abs/0704.1996v3 1 Introduction Our goal in this paper is to embed the Hartle-Hawking no-boundary proposal for a wave- function description of the quantum state of the universe [1][2] in a string theoretic frame- work. The Hartle-Hawking proposal pertains in particular to de Sitter–like universes in general relativity. A partial list of interesting recent work on related topics is [3][4][5] [6][7][8]. One motivation for embedding the Hartle-Hawking proposal into string theory is that it provides us with a calculable quantity in de Sitter-like compactifications of a quantum theory of gravity. These quantities are hard to come by (see e.g. [9][10] and references thereto). Two essential properties which a cosmological background must fulfill in order to admit a wave-function description under the no-boundary proposal are the following. First the cosmology must be spatially closed. More importantly, the cosmology should admit a con- tinuation to a positive definite Euclidean geometry that is compact and has no boundaries or singularities. The most familiar example is the case of n-dimensional de Sitter space, dSn, where these properties are satisfied. In global coordinates, the dSn metric is given by ds2 = R2(−dt2 + cosh2 t dΩ2), (1) where dΩ2 is the metric on a round unit (n − 1)-sphere and R is the radius of curvature. The spatial slices of constant time t are (n− 1)-spheres of radius R cosh t. We can rotate to Euclidean signature by setting t = iτ = i(π/2− θ), upon which we obtain an n-sphere Sn of radius R. The Euclidean continuation is a compact smooth manifold. In a field theoretic framework, the quantum state of a de Sitter cosmology can be ex- pressed as a functional of fields, including both matter fields and metric fluctuations, on a spatial slice of time-reversal symmetry. For the reversal t→ −t, this is the slice t = 0 in the de Sitter space dSn. That slice is also the equator θ = π/2 of the corresponding Euclidean sphere Sn. Imagine cutting de Sitter space along this slice and gluing smoothly one half of it to half a sphere Sn. Under the Hartle-Hawking proposal, we express the wave-function as a Euclidean path integral over half the sphere Sn with the condition that the metric gij and the matter fields, collectively denoted by φ, take specific values (hij , φ0) on the boundary equator θ = π/2: Ψ(hij, φ0) = [dg][dφ]e−SE(g,φ). (2) No other boundary condition needs to be specified due to the compactness of the Euclidean manifold. Here, SE is the Euclidean gravitational action in the presence of matter fields and a positive cosmological constant Λ. The norm of the wave-function is given by the full Euclidean path integral on Sn. It can be computed in the semi-classical approximation by evaluating the Euclidean action for a given solution to the classical equations of motion. One solution is empty de Sitter space of radius R ∼ Λ−1/2. In this approximation, and in the case of four dimensions, the norm is given by [1]: ||ΨHH ||2 ∼ e 3λ , (3) where the dimensionless parameter λ is proportional to the cosmological constant: λ = 2GΛ/9π . The compactness of the Euclidean manifold ensures that the full path integral is free of any infrared divergences. However the field theory in question is non-renormalizable, and to go beyond the semi-classical approximation, we need to impose an ultraviolet cutoff. One way to deal with the ultraviolet ambiguities is to embed the calculation in a string theoretic framework, where we expect the ultraviolet divergences to be absent. Unfortunately there are no known classical de Sitter solutions in string theory to begin with. Therefore, we seek other cosmological backgrounds which are exact solutions to string theory and for which we can generalize the Hartle–Hawking computation. To this end, notice that any tachyon free, compact Euclidean string background provides us with a finite, calculable quantity, namely the string partition function Zstring. Associated to the classical string background is a two-dimensional worldsheet conformal field theory (CFT). At the perturbative level, the string partition function can be computed as usual as a sum of CFT vacuum amplitudes over compact worldsheets of all topologies. Our proposal is that when such Euclidean string backgrounds admit a continuation to a Lorentzian cosmol- ogy, the Hartle-Hawking construction can be generalized with the norm of the wave-function given by ||Ψcosm.||2 = eZstring . (4) We will motivate this formula by working out specific examples in string perturbation theory. As we will explain, the relevant string partition function has to be thermal. Given the discussion above, a first candidate to consider is a Euclidean model for which the two-dimensional CFT is of the form SU(2)k × K, the first factor corresponding to an SU(2) Wess-Zumino-Witten (WZW) model at level k and the second factor K corresponding to a suitable internal compact conformal field theory. The WZW factor is equivalent to a sigma model on a 3-sphere of radius (kα′)1/2 and with k units of NSNS 3-form flux through the sphere. The dilaton field is constant and by choosing this to be small we can apply string perturbation theory. Unfortunately, however, the continuation to Lorentzian signature results in a dS3 cosmology with imaginary flux, and it is not clear whether such a Lorentzian background is physical. (See [11] for an alternative non-compact, time-like Liouville model for which the SU(2) WZW factor describes the internal space.) The only known string theory example which satisfies all the criteria we described so far is based on the parafermionic SU(2)|k|/U(1) coset model [12], which can be realized as a gauged SU(2) WZW model at level |k|. We consider Euclidean backgrounds corresponding to a two- dimensional CFT of the form SU(2)|k|/U(1) × K where K is again an internal compact conformal field theory. Such a Euclidean background admits a Lorentzian continuation to a cosmological background belonging to a class of models studied in [13][14], and which are described by two-dimensional CFTs of the form SL(2, R)−|k|/U(1) × K. To avoid having to deal with the tachyonic instabilities of bosonic string theory, we consider solutions of this form in superstring theory. The total central charge must be ctot = 15 (ĉ = 10) in order for worldsheet (super-)gravitational anomalies to cancel. When we fix the internal conformal field theory K, the level |k| is determined by anomaly cancellation. The non-trivial time-dependence of the cosmology necessarily breaks space-time super- symmetry. As in the de Sitter case, the Euclidean path integral can be interpreted as a thermal ensemble. Thus from the point of view of the Euclidean N = 2 worldsheet super- conformal system, space-time supersymmetry will be broken by specific boundary conditions, analogous to the thermal co-cycles that appear in the partition function of superstring the- ories on flat space at finite temperature [15]. For large level |k|, the effective temperature of the models is of order T ∼ 1/ |k|α′ [14][28]. In this paper, we will explore some low level |k| models. In order for the corresponding cosmological wave-function to be computable in string perturbation theory, the effective temperature must be below the Hagedorn tempera- ture. A Hagedorn temperature would signal a phase transition, as proposed in [15][16][17]. We will construct explicitly low level |k| models for which the effective temperature is below the Hagedorn temperature and so string perturbation theory can be applied. It is well known that the geometric sigma model approach to the parafermionic coset model (and to the corresponding Lorentzian cosmology) leads to a metric with curvature singularities and strong coupling. However, the underlying CFT is perfectly well behaved at these apparently singular regions, and by using T-duality a weakly coupled description of these regions can be obtained [18]. Using this fact, we construct an almost geometrical description of the CFT in terms of a compact, non-singular T-fold [19][20] with a well– defined partition function. These considerations allow us to define the wave-function of the Lorentzian cosmology. Our paper is organized as follows. In section 2, we review properties of the two-dimensional SL(2, R)−|k|/U(1) × K conformal field theory that corresponds to a cosmological back- ground. It is the analogue of the de Sitter universe. In section 3, we describe how to ana- lytically continue the cosmology to a compact Euclidean space-time described at the string level by a two-dimensional parafermionic model of the form SU(2)|k|/U(1) × K. Then, we discuss in section 4 how to obtain an almost geometrical description of these backgrounds in terms of T-folds. We discuss in sections 5 and 6 how to calculate a wave-function and its norm for the cosmology. In section 7 we discuss the thermal nature of the wave-function. In section 8 we apply the definition of the wave-function to some particular compact models and for which perturbation theory can be used to compute its norm. Finally we discuss interpretations of the results in the concluding sections. 2 The cosmological solution In this section, we review in some detail the cosmological solution of string theory which is based on an SL(2, R)/U(1) gauged WZW model at level k [13]. We can define a WZW conformal field theory on the group manifold SL(2, R), at least classically. The worldsheet action is given by d2zTr(g−1∂gg−1∂̄g) + Tr(g−1dg ∧ g−1dg ∧ g−1dg), (5) where Σ is the worldsheet Riemann surface, M is a 3-manifold whose boundary is Σ and g is an element of SL(2, R). For concreteness, we parameterize the SL(2, R) group manifold as follows with ab+ uv = 1. The conformal field theory has an SL(2, R)× SL(2, R) global symmetry. We choose to gauge an axial U(1) subgroup under which g → hgh. In particular, we consider the non-compact U(1) subgroup generated by δg = ǫ g + ǫg . (7) Infinitesimally, we have the transformations δa = 2ǫa, δb = −2ǫb, δu = δv = 0. To gauge this U(1) symmetry, we introduce an Abelian gauge field and render the action invariant. The action is quadratic and non-derivative in the gauge field, and so this can be integrated out in a straightforward way [21]. In the region 1−uv > 0, we can use the gauge freedom to set a = b and integrate out the gauge field. The resulting action is expressed in terms of gauge invariant degrees of freedom only, and it turns out to be S = − k ∂u∂̄v + ∂v∂̄u 1− uv , (8) while a non-trivial coupling to the worldsheet curvature is generated corresponding to a dilaton background [21]. This action can be identified with a non-linear sigma-model action with background metric ds2 = −kα′ dudv 1− uv . (9) The non-trivial dilaton is given by e2Φ = 1− uv . (10) The metric (9) is a Lorentzian metric whose precise causal structure, however, depends on the sign of k. For positive level k, u and v are Kruskal-like null coordinates of a 2- dimensional black hole. In this case, the time-like coordinate is given by u + v, and the metric has space-like singularities in future and past times at uv = 1. For negative level k, one obtains a cosmological solution [13]. It consists of a singularity- free light-cone region, and there are (apparent) time-like singularities in the regions outside the light-cone horizons. Indeed, for negative level k we may set u = −T +X and v = T +X and the metric becomes ds2 = |k|α′−dT 2 + dX2 1 + T 2 −X2 . (11) The surfaces of constant time T intersect the singularities at X = ± 1 + T 2. Even though the singularities follow accelerated trajectories, their proper distance remains finite with respect to the string frame metric L = (|k|α′) 1+T 2 1+T 2 1 + T 2 −X2 = π(|k|α′) 2 . (12) So with respect to stringy probes, the cosmology is spatially closed. The singularity-free light-cone region is the region T 2 − X2 ≥ 0 (or uv ≤ 0). The future part of this region describes an expanding, asymptotically flat geometry with the string coupling vanishing at late times. See e.g. [11][13][14][23][24][25][26][27][28][29] for some discussions of these types of models. To see this, we parameterize the region uv ≤ 0 with coordinates (x, t) such that u = −tex, v = te−x (13) and the metric becomes ds2 = |k|α′−dt 2 + t2dx2 1 + t2 , (14) while the dilaton field becomes e2Φ = 1 + t2 . (15) The scalar curvature is given by |k|α′(1 + t2) . (16) Initially the curvature is set by the level |k| and it is positive. No matter how small the level |k| is, asymptotically the scalar curvature vanishes. An observer in this region never encounters the singularities. These are hidden behind the visible horizons at T = ±X . However signals from the singularities can propagate into the region uv < 0, and therefore influence its future evolution. Thus when |k| is small the early universe region t ∼ 0 is highly curved, with curvature of order the string scale. In this sense, it is similar to a big-bang cosmology. Despite the regions of large curvature, this cosmological background has a well defined CFT description and can be described in a string theoretic framework. The cosmological background can also be realized as a solution of superstring theory by generalizing the worldsheet theory to a superconformal SL(2, R)/U(1) model. The central charge of the superconformal SL(2, R)/U(1) model at negative level k, is given by c = 3− 6 |k|+ 2 , ĉ = 2− 4 |k|+ 2 In superstring theory, we must tensor it with other conformal field theories so as to satisfy the condition ĉtot = 10 for worldsheet gravitational anomalies to cancel. An interesting case considered in [13] is the case where we add two large (however com- pact) free super-coordinates (y, z) together with a compact, superconformal CFT of central charge δĉ = 6 + 4/(|k| + 2). The resulting background is a four dimensional cosmological background whose metric in Einstein frame is given by ds2E = |k|α′(−dt2 + t2dx2) + (1 + t2)(R2ydy2 +R2zdz2). (18) This is an anisotropic cosmology which at late times however, and for large Ry ∼ Rz, asymptotes to an isotropic flat Friedman-Robertson cosmology. The cosmological region t2 = −uv ≥ 0 is non-compact, and when Ry,z are large it has the desired four-dimensional interpretation. This is so irrespective of how small the level k is. In the region uv > 0 (t2 < 0), sigma-model time-like singularities appear at uv = 1 (t2 = −1). As we propose later in this work, these singularities are resolved at the string level, since the structure of the space-time manifold is replaced by a non-singular T -fold. The string partition function depends crucially on the extra ĉ = 6+4/(|k|+2) supercon- formal system, which is taken to be compact. In contrast to the four dimensional part defined by (t, x, y, z), for the internal, Euclidean ĉ = 6+4/(|k|+2) system, the naive six-dimensional interpretation, which is valid for large level |k| with curvature corrections of order 1/(|k|α′), is not valid for small values of |k| [18]. For example, for |k| = 2, the system can be taken to be a seven-dimensional torus. In general, small |k| implies that the generalized curvatures (i.e. including dilaton gradients etcetera) are large and the moduli/radii are small. We remind the reader of the example of the SU(2)k=1 Wess-Zumino-Witten model which is equivalent to a (one-dimensional) compact boson at self dual radius. For large level |k|, however, the sigma model manifold is a large three-dimensional sphere with NSNS 3-form flux. 3 The Euclidean continuation Let us consider the region 1− uv ≥ 0 of the two-dimensional cosmology, and set u = −T + X, v = T +X . We can rotate to Euclidean signature by setting T → −iTE . The Euclidean continuation is a disk of unit coordinate radius parameterized by Z = X+ iTE , Z̄ = X− iTE such that |Z|2 ≤ 1. The metric (9) becomes ds2 = |k|α′ dZdZ̄ 1− ZZ̄ = |k|α′dρ 2 + ρ2dφ2 1− ρ2 and the dilaton e2Φ = 1− ZZ̄ 1− ρ2 , (20) where we have also set Z = ρeiφ with 0 ≤ ρ ≤ 1. The singularity becomes the boundary circle ρ = 1. The radial distance of the center to the boundary of the disk is finite, but the circum- ference of the boundary circle at ρ = 1 is infinite. Geometrically the space looks like a bell. This Euclidean background corresponds to a well defined worldsheet conformal field theory based on an SU(2)/U(1) gauged WZW model at level |k|. From the point of view of the WZW worldsheet theory, the Euclidean continuation can be understood as a double analytic continuation as follows. We parameterize the SL(2, R) group manifold as in equation (6). Let us also set a = X̃ − T̃ , b = X̃ + T̃ so that the group element becomes X̃ − T̃ X − T −X − T X̃ + T̃ X̃2 +X2 − T̃ 2 − T 2 = 1. (22) This parameterization shows that the SL(2, R) group manifold is a 3-dimensional hyper- boloid. Then it is clear that upon the double analytic continuation T → −iTE , T̃ → −iT̃E the group element becomes the following SU(2) matrix −Z̄ W̄ with WW̄ + ZZ̄ = 1. After the analytic continuation we also have that a → W = X̃ + iT̃E , b→ W̄ = X̃ − iT̃E . A useful parameterization of the SU(2) group manifold for our purposes is W = cos θeiχ, Z = sin θeiφ (24) and the metric on S3 in these coordinates becomes ds2 = dθ2 + sin2 θdφ2 + cos2 θdχ2. (25) The ranges of the angles are as follows 0 ≤ θ ≤ π, 0 ≤ χ, φ,≤ 2π. The original global SL(2, R) × SL(2, R) symmetry naturally continues to the SU(2) × SU(2) global symmetry of the resulting SU(2) WZW model. The non-compact U(1) axial symmetry subgroup that we gauge continues to a compact U(1) subgroup generated by δg = iǫ g + iǫg , (26) which amounts to the following infinitesimal transformations δW = 2iǫW , δW̄ = −2iǫW̄ and δZ = δZ̄ = 0. In the parameterization (24), the U(1) symmetry corresponds to shifts of the angle χ. Gauging this symmetry results in the SU(2)/U(1) coset model. In the Euclidean set-up, we take the level |k| to be an integer for the WZW model to be well-defined. After the analytic continuation described, we end up with the action (see e.g. [31] for a review): d2z∂θ∂̄θ + tan2 θ∂φ∂̄φ +cos2 θ(∂χ + tan2 θ∂φ + Az)(∂̄χ− tan2 θ∂̄φ+ Az̄). In the Euclidean theory the gauge freedom can be fixed by setting the imaginary part of W (equivalently the angle χ) to zero. The equations of motion for the gauge field can then be used to integrate the gauge field out. This amounts to setting the last term in (27) to zero and producing a dilaton e2Φ = e2Φ0/ cos2 θ. We end up with a sigma model action with metric ds2 = |k|α′(dθ2 + tan2 θdφ2) (28) which is equivalent to the metric (19) after the coordinate transformation Z = sin θeiφ. The curvature singularity occurs at θ = π/2. The procedure of fixing the gauge χ = 0 and using the equations of motion to integrate the gauge field out is not valid near θ = π/2, since it results into singular field configurations on the worldsheet. However, the full action (27) is perfectly well behaved at θ = π/2. To see this, we expand the Lagrangian in (27) around θ = π/2. Setting θ = π/2− θ̃, we obtain that d2zφFzz̄ +O(θ̃ 2), (29) where we expressed the action in terms of manifestly gauge invariant degrees of freedom. The leading term in this expansion describes a simple topological theory, which shows that an alternative, non-geometric description of the theory can be given including the region near θ = π/2. We return to this point later on. From the form of the action near θ = π/2, we also learn that the U(1) symmetry cor- responding to shifts of the angle φ is quantum mechanically broken to a discrete symmetry Z|k|. This is because compact worldsheets can support gauge field configurations for which Fzz̄ = 2πin, with n an integer, and such configurations must be summed over in the full path integral. It is clear then that the path integral is only invariant under discrete shifts of the angle φ: δφ = 2πm/|k|. This breaking of the classical U(1) symmetry to Z|k| is in accordance with the algebraic description of the SU(2)/U(1) coset in terms of a system of Z|k| parafermionic currents ψ±l(z), l = 0, 1 . . . |k| − 1 [with ψ0 = 1, ψ†l ≡ ψ−l = ψk−l], of conformal weights hl = l(|k| − l)/|k| (see also [32]). These satisfy the OPE relations ψl(z)ψl′(0) = cll′z −2ll′/|k|(ψl+l′(0) + . . . ) ψl(z)ψ l (0) = z −2hl(1 + 2hlz 2T (0)/c+ . . . ) (30) which are invariant under the Z|k| global symmetry: ψl → e2πil/kψl. Here T is the energy momentum tensor of the parafermions, c the central charge (which is the same as the central charge of the coset model) and the coefficients cll′ are the parafermionic fusion constants [12]. In the infinite level |k| limit, the conformal weights of the parafermion fields become integers. In this limit the sigma model metric is flat, and we recover the full rotational invariance of flat space [33]. The system can be also generalized to an N = 2 superconformal system by tensoring the Z|k| parafermions with a free compact boson as described in [36]. Finally we can check that the central charge remains the same after the analytic contin- uation. Indeed, it is the very fact that the central charge of the conformal field theory is smaller than the central charge corresponding to two macroscopic flat dimensions that codes the de Sitter nature of the two-dimensional cosmology. 4 The cosmological T-fold The parafermionic T-fold It is interesting to take a closer look at the geometry that we associate to the parafermionic model SU(2)|k|/U(1). As we already discussed, we describe it in terms of a metric and dilaton profile: ds2 = |k|α′(dθ2 + tan2 θdφ2) cos θ , (31) where φ ∼ φ+2π and θ takes values in the interval [0, π/2]. This description breaks down near θ = π/2. Nevertheless, the parafermionic conformal field theory is perfectly well-behaved, and we can wonder whether there is a more appropriate, almost-geometrical description. We argue that such a description exists in terms of a T-fold. To obtain it, we perform a T-duality along the angular direction φ on the geometry described above: ds2 = |k|α′dθ2 + cot2 θdφ̃2 sin θ . (32) By changing variables θ̃ = π/2− θ, we see that this is equivalent to: ds2 = |k|α′dθ̃2 + tan2 θ̃dφ̃2 cos θ̃ . (33) This description is therefore at weak curvature (apart from an orbifold–like singularity) and weak coupling near θ = π/2. Moreover, we can identify it as a Z|k| orbifold of a vectorially (or axially) gauged SU(2)/U(1) coset. Indeed, it is true for the parafermionic theory that the T-dual and the Z|k| orbifold give two models with identical spectrum due to the coset character identity χj,m = χj,−m (see e.g. [30][31] for reviews). We now use these facts to give an almost geometrical description of the parafermionic theory, in terms of a T-fold [19][20]. We use the description in terms of the first geometry (31) near θ = 0. We cut it just past θ = π/4, where the radius of the circle is |k|α′. We glue it to the T-dual geometry which we consider near θ̃ = 0, or θ = π/2, and which we cut just past θ̃ = π/4, where we have radius α′/|k|. We glue the circles (and their environments) using the T-duality transformation described above. In the gluing process, it is crucial to realize that we glue a patch with a direction of increasing radius to a T-dual patch which in the same direction has decreasing radius. That gives us the parafermionic T-fold. The associated partition function is (see e.g. [31] for a review) : χj,m(τ)χj,m(τ̄ ). (34) One aspect of the model that is rendered manifest by the T-fold description is the breaking of the U(1) rotation symmetry to a discrete Z|k| symmetry, due to the Z|k| orbifolding. This is consistent with our previous discussion of the breaking due to worldsheet instantons. The T- fold yields an almost-geometrical picture of the symmetry breaking. The T-fold description is indeed everywhere regular modulo a benign orbifold singularity. The cosmological T-fold In the case of the two-dimensional cosmology as well, we can obtain a regular T-fold de- scription of the target space of the conformal field theory. We recall that under T-duality (the metric can be obtained by analytically continuing the metric (32) in the direction φ̃), the light-cone and the singularities get interchanged. Consider the cosmology, and cut it at a hyperbola at radius |k|α′, in between the light-cone and the time-like singularities in the Penrose diagram (see the upper part of figure 1). Consider then its T-dual, and cut it along a similar line. Glue the two parts of the T-dual cosmologies along these cuts to obtain the T-fold cosmology. The description we obtain is particularly nice as we no longer need a microscopic origin of a would-be source associated to the time-like singularities, nor do we need to define boundary conditions associated to them. There is no singularity in, nor is there a boundary to the T-fold cosmology. Indeed, the almost-geometrical description is very much like dS2, which we can think of as a hyperboloid embedded in three-dimensional space. The difference is that the T-fold cosmology has two patches glued together via a Figure 1: The analytic continuation of the T-fold. The upper part of the diagram shows the two T-dual descriptions of the cosmology in which the horizon and the (apparent) sin- gularities are exchanged (in bold blue). The (striped black) cut along which they are glued is indicated, as well as the (thin black) line along which the cosmology is cut to obtain a space-like slice (see later). Analytic continuation then gives rise to the lower part of the figure, in which we have sketched the T-fold description of the parafermionic conformal field theory. In bold blue we have the center and the boundary of the disk, and (in black stripes) the T-dual circles along which we glue. T-duality transformation (instead of an ordinary coordinate transformation in the case of two-dimensional de Sitter space). In figure 1 we show how the T-fold description of the parafermions and the two-dimensional cosmology continue into one another after analytic continuation. 5 Defining the wave-function of the universe Later on, we will consider string theory backgrounds which are product models and in which one factor consists of the two-dimensional cosmology discussed in sections 2, 3 and 4. For these models, we wish to define a wave-function of the universe in string theory following ideas of [1] which define a wave-function of dSn universes within a field theoretic context. We consider a time-reversal symmetric space-like slice of the cosmology, within the bound- aries of the (seeming) singularities. See figure 2. This is the slice T = 0. In the past of the space-like slice, we glue half of the target space of an SU(2)/U(1) coset conformal field theory – a half disk. By the analytic continuation discussed in the previous section, this Figure 2: The continuous gluing of the half-disk into the cosmology, when cutting the cos- mology along a space-like slice, and analytically continuing. The figure should be viewed as a simplified version of the previous T-fold picture. gluing is continuous in the backgrounds fields, and moreover in the exact conformal field theory description. A crucial feature of the proposal of [1] for the definition of the wave-function of the universe is that the corresponding Euclidean space is without boundary. In our set-up as well, the Euclidean conformal field theory has a target with no boundary. It is important in this respect that we have obtained an almost-geometric description of the parafermionic conformal field theory1. It is intuitively clear from the T-fold description given in the previous section that the parafermionic theory does not have a boundary2. When we cut the Euclidean T-fold into half, it is clear (from figure 1) that we can glue the boundary of that half-T-fold into the initial surface of the cosmological T-fold. Thus we have determined the precise gluing of T-folds necessary in order to define a wave-function depending on initial data. We now define the wave-function of the universe by performing a “half T-fold” Euclidean path integral over all target space fields with specified values on the boundary: Ψ[h∂ , φ∂, . . . ] = [dg][dφ] . . . e−S(g,φ,... ) , (35) where the path integral is such that the metric, the dilaton and all other space-time fields 1A traditional description of the target space as a disk, which is singular,would lead to the faulty conclusion that the target space has a boundary. 2Since a T-fold is non-geometric, one needs to define the concept of boundary precisely. We believe that a reasonable definition will match our intuition. satisfy g = h∂, φ = φ∂, . . . on the boundary of the half T-fold that we glued into the cosmological solution. The path integral above can in principle be performed off-shell, in a second-quantized string field theory context, where we may also express it as an integral over a single string field Φ. (See e.g. [34] for a concise review). Let us be more specific. The initial space-like slice of the T-fold cosmology has two patches. On each patch, we define a boundary metric h1 and h2, and similarly for other fields. The boundary metrics satisfy the condition that on the overlap of the patches, they match up to a T-duality transformation, symbolically: h1 ∂1∩∂2 = T (h ∂1∩∂2). This is the way in which we can specify boundary data precisely. In the following, we do not emphasize this important part of the definition of the path integral further, not to clutter the formulas. In principle, a T-fold path integral can be computed as follows. Consider again the two patches. Each patch has a non-singular geometric description. Over each patch the path integral reduces to an ordinary field theory path integral, and can be performed in the usual way giving rise to a functional of boundary data. The full path integral can be obtained by integrating the two functionals together over data that belong to the common boundary of the two patches. Since at the common boundary of the patches their fields are related by a T-duality transformation, to do the final integral we would need to perform a T-duality transformation on one of the two functionals. We remark here that this particular feature of definitions of path integrals over T-folds with boundaries is generic. The above description is easily extended to a generic description of T-fold boundary data. Although we do not need a general prescription in this paper, we believe it would be interesting to develop the path integral formalism for T-folds with boundary further. The prescription for the wave-function of the universe we outlined above should have an analogue, via the relation between string oscillators and the target space fields, to a first quantized prescription. Notice that the initial-time data allow multi closed string configura- tions. Summing over histories that lead to them, would allow worldsheets with boundaries (and other topology features), including disconnected ones. The wave-function would take the form Ψ[X∂(σ, τ)] = topologies [dX ]e−S[X(σ,τ)], (36) where the worldsheet path integrals are performed over string configurations X(σ, τ) that satisfy a specified boundary condition at given values of the zero-modes of the string con- figuration, i.e. at a given position of the target space. The equivalence of these descriptions is far from obvious, but it is made plausible by the fact that for two-dimensional string worldsheets, the first quantized description automatically comes with a prescription for the proper weighting of interaction vertices. The initial-time closed string configurations could be specified in terms of macroscopic loop operators discussed for example in [35]. The first quantized prescription considers fluctuations around a given background. A full second quantized prescription also integrates over backgrounds as in general relativity [1]. The wave-function so defined is hard to compute, although it may be obtained presumably for very particular boundary conditions. An example would be boundary conditions that are fixed by taking a Z2 orbifold that folds over the disk onto itself – in that case, one may be able to compute the value of the wave-function for a particular argument. In order to understand better some global properties, we again follow [1] and concentrate on calculating the norm of the wave-function. 6 The norm of the wave-function The norm of the wave-function is easier to compute. It is given by the following calculation: ||Ψ||2 = [dΦ∂] half T−fold [dΦ]e−S(Φ) × conj half T−fold [dΦ]e−S(Φ) T−fold [dΦ]e−S(Φ), (37) where we have expressed it as a string field theory path integral in terms of a string field Φ. The final integral is an integral over all possible string field configurations on the Euclidean T-fold. No boundary conditions need to be specified. We can do this calculation by considering the fluctuations around an on-shell closed string background, in a first quantized formalism: ||Ψ||2 = topologies [dX ]e−S[X(σ,τ)] (38) where X(σ, τ) is any mapping from the string worldsheet into the target space. The sum is over all closed worldsheet topologies, and includes a sum over disconnected diagrams. In fact it is equal to the following exponential of a sum of connected diagrams: ||Ψ||2 = exp(Ztotal), (39) where the function Ztotal is the total string theory partition function, which is defined as a sum over Euclidean worldsheet topologies: Ztotal = ZS2 + ZT 2 + g sZgenus=2 + g2g−2s Zgenus=g. (40) Therefore, to evaluate the norm of the wave-function perturbatively, we need to evaluate the partition function for string theory on the Riemann surfaces of genus 0, 1, 2, . . . and add their contributions with the appropriate power of the string coupling constant. The first contribution is akin to the tree level contribution in ordinary gravity, the second to the one-loop contribution, etc. 7 Thermal nature of the wave-function A natural way to perform the Euclidean path integral in equation (35) over half the space is as follows. The origin X = 0 in one T-fold patch (and similarly for the other), divides the T = 0 slice into two halves: the left half corresponding to X < 0 and the right part corresponding to X > 0. We denote the boundary data on X < 0 by φL and on X > 0 by φR. See figure 3. By dividing the space into angular wedges spanning an overall angle equal to π, we can evaluate the path integral in terms of the generator of angular rotations. This generator is given by the analytic continuation of iHω, where Hω = i∂ω is the Hamiltonian conjugate to “Rindler” time in the region uv > 0 of the Lorentzian cosmology. Indeed in this region, we may set u = ρe−ω, v = ρeω, with the string frame metric and dilaton given ds2 = |k|α′ dρ2 − ρ2dω2 1− ρ2 e2Φ = 1− ρ2 . (41) In this patch, the background metric is static, invariant under time translations, and the dilaton field is space-like. Rotating to Euclidean signature amounts to setting ω = −iφ. Figure 3: The thermal interpretation of the wave-function is obtained by thinking of the path integral as being performed along angular wedges from an initial (right) to a final (left) configuration. So Rindler time translations correspond to angular rotations in the Euclidean space. As we have already discussed, only discrete angular rotations are true symmetries of the string theory background. The boundary data can then be viewed as specifying initial and final conditions for the path integral evolution. This is clearly reviewed for the case of flat Rindler space and black hole spaces in [37]. In particular, the path integral measures the overlap between the data on the right φR, evolved for a Euclidean time π, and the data specified on the left φL (see figure 3), and it can be written as an amplitude Ψ(φL, φR) = 〈φL|e−πHω |φR〉. (42) If we integrate over φL we obtain a thermal density matrix appropriate for the Rindler observer [37] [dφL]Ψ(φL, φR)Ψ ∗(φL, φ R) = 〈φ′R|e−2πHω |φR〉, (43) with dimensionless temperature Tω = 1/2π. The norm of the wave-function is given by the trace ||Ψ||2 = Tre−2πHω (44) and so it can be interpreted as a thermal space-time partition function. The genus-1 string contribution is a thermal one-loop amplitude. In the full Euclidean path integral equation (37), the contributions of the fermionic fields are positive. To perform the full path integral over the whole T-fold, we divide it into angular wedges spanning an angle equal to 2π. Since the space has no non-contractible cycles, the space-time fermionic fields have to be taken anti-periodic in the angular variable and they contribute positively to the path integral. The T-fold patches are glued along the hyperbola ρ = 1/ 2 (see section 4). Near this region, the curvature is low for large enough level |k|. Thus for large level |k|, we may use the metric equation (41) to conclude that observers moving near the region ρ ∼ 1/ 2 measure a proper temperature T ∼ 1/(2π |k|α′). In the cosmological region uv ≤ 0, there is also an effective temperature of the same order as a result of particle production [14][28]. For small level |k|, we need a string calculation to deduce the proper temperature of the system. 8 Specific Examples As we discussed above in order to derive the wave-function of the cosmology, we need to compute the total string partition function for the corresponding Euclidean background. When conformal field theories are compact, the genus-zero contribution to the total string partition function vanishes. This is because the spherical string partition function is divided by the infinite volume of the conformal Killing group. This fact is a first important difference with the calculation in general relativity where the classical contribution is non-zero. In perturbative string theory the leading contribution is the genus-1 amplitude. The Euclidean examples we shall describe here in detail belong to the family of ĉ = 10 superconformal, compact systems. In order for them to admit a Lorentzian continuation to a cosmological (time-dependent) background, space-time supersymmetry must be broken. Moreover, the models must be free of tachyons. The presence of tachyonic modes would indicate that the system undergoes a phase transition. The only known examples with the above properties are of the form SU(2)|k| × K ←→ SL(2, R)−|k| × K , (45) where we indicated the analytic continuation from the Euclidean to the Lorentzian space- time. The level |k| can be taken to be small. As we already discussed, the relevant genus-1 string amplitude has to be thermal. The total superstring model has transverse central charge equal to c = 12 (or ĉ = 8). As a consequence, it has a Hagedorn transition at the fermionic radius RH = 2α′. In order for the genus-1 string amplitude to be finite, the physical temperature of the model has to be below the Hagedorn temperature: T < TH = 1/(2π 2α′). Let us give an argument that this can be realized for any |k| ≥ 2. In writing the norm of the wave-function as a thermal space- time partition function, the role of the Hamiltonian is taken up by the generator of rotations on the disk. For the superconformal SU(2)/U(1) model, the corresponding U(1) current is at level |k| + 2 [36]. Thus we expect the physical temperature of the model to be set by the radius associated to this isometry generator, namely (|k|+ 2)α′. The corresponding temperature is given by (|k|+ 2)α′ and is below the Hagedorn temperature for any positive (integer) level |k|. We will find that this temperature arises naturally in a level |k| = 2 model below. At level |k| = 0, where the minimal model has zero central charge, and consists only of the identity operator (and state), the cosmology disappears. When we reach the Hagedorn transition, the cosmology becomes so highly curved that it is no longer present in the string theory background. 8.1 Compact models In a first class of specific examples that we will discuss in this section, we choose the level |k| = 2, and we take the internal conformal field theory K to be: K = T 2 × i=1,...,7 SU(2)ki where all ki’s are taken equal to 2, so that ĉK = 9 (representing the central charge equivalent of nine flat directions). In the sequel, we set α′ = 1. For this choice, the supersymmetric characters of the whole system are defined in terms of eight level k = 2 parafermionic systems (which are nothing but eight real fermions ψi and eight bosons φi compactified at the self-dual radius R = 1), and also a complex fermion ΨT and a complex boson ΦT for the torus T 2. The N = 2 superconformal operators TF , J are: i=0,1,...7 2φi + iΨT∂ΦT i=0,1,...7 ∂φi +ΨT Ψ̄T . (48) It is convenient to pair the (0, 1), (2, 3), (4, 5), (6, 7) systems respectively in order to obtain four copies of ĉ = 2 systems. For the first copy we define the bosons H0, H1 at radius 2 (or R = 1√ , fermionic T -dual points): (H0 +H1), φ1 = (H0 −H1) (49) and similarly for the others. Then the currents are given by i=0,2,4,6 i(Hi+Hi+1) + ψi+1 e i(Hi−Hi+1) + i∂ΦT e J = i∂H0 + i∂H2 + i∂H4 + i∂H6 + i∂HT , (50) where i∂HT = ΨT Ψ̄T , which is also defined at the fermionic point. Observe that the N = 2 current is given in terms of H0, H2, H4, H6 and HT only and is normalized correctly for a system with ĉ = 10. The N = 2 left-moving characters of a particular ĉ = 2 system (e.g of the one containing H0), are expressed in terms of the usual level-2 Θ-functions 3: η(τ)3 2 ΘH0 α+2H0 β+2G0 γ0−2H0 δ0−2G0 3γ0−4H0 3δ0−4G0 where the arguments (γ0, δ0) and (4H0, 4G0) are integers. The later exemplifies the chiral Zk+2-symmetry of the superconformal parafermionic characters (k + 2 = 4 in our case). Similar expressions are obtained for the other three ĉ = 2 parafermionic systems by replacing (γ0, δ0) with (γi, δi), and (H0, G0) by (Hi, Gi), i = 1, 2, 3. The global existence of the N = 2 superconformal world-sheet symmetry and thus the existence of the left space-time supersymmetry imply (HT , GT ) + i=0,1,2,3 (Hi, Gi) = ǫ mod 2 , (52) 3Our convention for the level-2 Θ function is Θ [ iπτ(n+ )2+2iπ(n+ and similarly for the right supersymmetry. The arguments (ǫ, ǭ) define the chirality of the space-time spinors. A simple choice is to set (HT , GT ) = (0, 0), (Hi, Gi) = (H,G) for i = 0, 1 and (Hi, Gi) = (−H,−G) for i = 2, 3 (and similarly for the right arguments). Then if (ǫ, ǭ) = (1, 1), space-time supersymmetry is broken. For the other choices there is some amount of supersymmetry preserved. In this class of models the only remaining possibility consistent with the global N = 2 super-parameterization consists of shifts on ΦT of the T torus. Z4 orbifold models Using the chiral Z4 symmetry defined above and its subgroups, we can obtain four classes of models: (H,G) = (h, g), where (h, g) = integers, M = 1, 2, 3, 4. (53) In particular, if we orbifold by Z4 (M = 1 or M = 3 ), the ψ0 parafermion decouples from the rest, especially from ψ1. This is clear since in this case, the arguments of Θψ0 and Θψ1 become independent. Initially, the arguments (γi, δi) are taken to be identical for the left- and right-moving characters. In this case the modular invariance of the partition function is manifest. Indeed, using the periodicity property of Θ-functions ∣ = |Θ [γδ ]| , ǫ, ζ integers, (54) and orbifolding by Z4, (M = 1), the genus-1 modular invariant partition function becomes: dτdτ̄ (Imτ)2 η(τ)12η̄(τ̄ )12 h,g=0,1,2,3 γ0,δ0 γ0−h2 δ0− g2 γ1,δ1 γ1−h2 δ1− g2 γ2,δ2 γ3,δ3 (α,β) eiπ(α+β+ǫαβ) Θ (ᾱ,β̄) eiπ(ᾱ+β̄+ǭᾱβ̄) Θ̄ β̄− g . (55) The arguments (α, β) and (ᾱ, β̄) are those associated to the N = 2 left- and right-moving supercurrents. If (ǫ, ǭ) = (1, 1), supersymmetry is broken. For all other choices this partition function is identically zero and there is some amount of supersymmetry preserved. To obtain the above result we have used the fact that the contribution of the superconformal ghosts cancel the oscillator contributions of the T 2 supercoordinates (ΦT ,ΨT ). This is the reason for choosing the Z4 not to act on ΨT , having set (HT , GT ) = (0, 0). The only remnant from the torus contribution is the Γ2,2 lattice, which can be possibly shifted by (Lh/2, Lg/2) with either L = 0 or L = 1, 2, 3 (as we will see later). To proceed we need to identify and insert the thermal co-cycle S q , (α+ ᾱ) p, (β+β̄) associated to the time direction of the cosmology: q, (α+ᾱ) p, (β+β̄) = eiπ( p(α+ᾱ)+ q(β+β̄) ), (56) where p and q are the lattice charges associated to the compactified Euclidean time. Here, Fα = (α + ᾱ) and Fβ = (β + β̄) define the spin of space-time particles: Fα = 1 modulo 2 for fermions and Fα = 0 modulo 2 for bosons. To impose this co-cycle insertion, it is necessary to rewrite the partition function in a form that reveals the charge lattice (p, q) of the Euclidean time direction. To this end, it is convenient to separate the partition function into the “untwisted sector” (h, g) = (0, 0) and “ twisted sectors” (h, g) 6= (0, 0), dτdτ̄ (Imτ)2 Zunt + (h,g)6=(0,0) Ztwist  . (57) To isolate the relevant (p, q) charge lattice, we use the identity |Θ [γδ ]| (m,n) 2 |m+τn| Imτ eiπ(mγ+nδ+mn) , R2 = . (58) Although in the above identity the radius is fixed to the fermionic point R2 = 1/2, we note that the modular transformation properties are the same for any R2, and in particular for the dual-fermionic point with R2 = 2. Using the above identity for the conformal block involving the parafermions ψ0,1 and the H1 field, we obtain (γ,δ) ∣Θψ0,1 [ 2 |ΘH1 [ (γ,δ) (m1,n1),(m2,n2) |m1+τn1| −πR22 |m2+τn2| Imτ eiπ((m1+m2)γ+(n1+n2)δ+m1n1+m2n2) (59) where R21 = R 2 = R 2 = 1/2. Since the arguments (γ, δ) do not appear elsewhere (in equation (55)), summing over them forces (m1+m2) and (n1+n2) to be even integers. This constraint can be solved if we take m1 = p1 + p2, m2 = p1 − p2, n1 = q1 + q2, n2 = q1 − q2 (60) so that, (γ,δ) ∣Θψ0,1 [ 2 |ΘH1 [ = Γ1,1(R+) Γ1,1(R−) = (p1,q1) 2 R+ e −πR2+ |p1+τq1| (p2,q2) 2 R− e −πR2− |p2+τq2|  (61) with R2+ = R − = 2R 2 = 1. Therefore, the partition function of the bosonic part of parafermions factorizes in two Γ1,1 lattices, both of them with twice the initial radius squared. The charge lattices (p1, q1) and (p2, q2) are associated to the ψ0 and ψ1 parafermions. We can see this as follows. Consider the left-charge operators which are well defined in the untwisted sector: Q+ = i dz(ψ0ψ1 + ∂H1), Q− = i dz(ψ0ψ1 − ∂H1) (62) and similarly for the right-moving ones (Q̄±). Then, (Q + Q̄)+ = m1 +m2 = 2p1, (Q− Q̄)+ = n1 + n2 = 2q1 (Q+ Q̄)− = m1 −m2 = 2p2, (Q− Q̄)− = n1 − n2 = 2q2 (63) where we have used the constraint (60). We identify the charges (p1, q1) as the momenta that enter in the thermal co-cycle, and associate the lattice to the Euclidean time direction. Before we proceed further, let us stress the following point. We started with a diagonal modular invariant combination and with initial radii R21 = R 2 = R 2 = 1/2. The anti-diagonal choice implies that the initial values for the radii are at the fermionic T-dual points, namely R21 = R 2 = R 2 = 2. Notice that in all we perform two T-dualities simultaneously so that we remain in the same type II theory. Thus the conformal block, equation (59), can be replaced with the T-dual one with R2 = 2.4 For the anti-diagonal choice, the radii of the corresponding factorized lattices are given by R2+ = R − = 4 instead of unity for the diagonal combination. Thus in the untwisted sector, Zunt has to be replaced with Zunt −→ Zthermalunt = (p1,q1) (α,β),(ᾱ,β̄) Zthermalunt q1, α+ᾱ p1, β+β̄ eiπ(p1(α+ᾱ)+q1(β+β̄)). (64) Performing a similar factorization for the remaining three copies of ĉ = 2 superconformal parafermionic blocks, we can write the thermal untwisted contribution in a compact form: Zthermalunt = ImτΓ9,9 η12η̄12 (α,β),(ᾱ,β̄) Γ1,1(R+) eiπ(α+β+ǫαβ) Θ eiπ(ᾱ+β̄+ǭᾱβ̄) Θ̄ The Γ9,9 lattice factor is composed of a product of lattices: the initial Γ2,2 lattice of the torus T 2 with radii Ry, Rz, the Γ1,1 lattice of the first parafermionic block at radius R−, and the product of three pairs Γ1,1(R +) Γ1,1(R −) for the other three parafermionic blocks. The Γ1,1(R+) lattice is the thermally shifted lattice Γ1,1(R+) (p1,q1) 2 R+ e −πR2+ |p1+τq1| Imτ eiπ((α+ᾱ)p1+(β+β̄)q1). (66) Its coupling with the space-time spin structure breaks space-time supersymmetry so that both bosons and fermions give positive contributions to the thermal partition function. The radius R+ sets the temperature of the system: 2πT = 1/R+. The form of the thermal coupling in equation (65) is similar to the one that appears in the familiar flat type II superstring theories at finite temperature. The difference here is that the temperature is fixed. Since we have succeeded to factorize out the thermal lattice, we can now treat all other radii parameterizing the Γ9,9 lattice as independent moduli. To obtain the four dimensional 4In Gepner’s formalism [36] the anti-diagonal combination corresponds to exchanging (m, m̄)→ (m,−m̄). interpretation we discussed in section 2, we take the radii Ry,z to be large keeping all other ones small. As we already remarked, there are only two choices consistent with the cosmological interpretation of the partition function corresponding to the two values of the radius R+. For the diagonal choice we have a radius R2+ = 1 corresponding to a temperature 2πT = 1/R+ higher than Hagedorn: 2πTH = 1/RH = 1/ 2. This model is tachyonic and so unstable in perturbation theory. For the second anti-diagonal choice R2+ = 4, and the temperature is below Hagedorn: 2πT = 1/2 < 2πTH . This is precisely the temperature that we gathered from general arguments, equation (46) for level |k| = 2. That model gives rise to a well defined, integrable partition function and a finite norm for the wave-function at one loop. The integral is difficult to perform analytically but it can be estimated. We will not carry out this computation here. The remaining part consists of the twisted sectors of the theory, (h, g) 6= 0. Here we shall find new stringy phenomena associated with the fact that we are orbifolding the Euclidean time circle. Since this is twisted, the thermal co-cycle has to be extended consistently. It takes the general form, valid for both the untwisted and the twisted sectors: (q+h), (α+ᾱ) (p+g), (β+β̄) = eiπ( (p+g)(α+ᾱ)+ (q+h)(β+β̄) ). (67) That is, the relevant lattice is augmented by the quantum numbers (g, h) that label the twisted sectors. Again, the thermal co-cycle insures that fermions contribute positively to the partition function. In the twisted sectors, there is no momentum charge and we can set (p, q) = (0, 0). In the twisted sectors, each of the ĉ = 2 superconformal blocks is equivalent to a system described by a free complex boson and a free complex fermion twisted by Z4. This equivalence implies topological identities for each N = 2 twisted superconformal block [38][39]: 2|η|4 (γi,δi) γi−h2 δi− g2 = 22 sin2( πΛ(h, g) where Λ(h, g) = Λ(g, h) depend on the (h, g)-twisted sector. Λ(h, g) = 2 when (h, g) =(0,2), (2,0) and (2,2) while for the remaining 12 twisted sectors Λ(h, g) = 1. Although the above orbifold expressions are derived at the fermionic point, they remain valid for any other point of the untwisted moduli space. Using the above orbifold identity, the “twisted” part of the thermal partition function simplifies to: Zthermaltwist = (α,β,ᾱ,β̄) (h,g)6=(0,0) Imτ Γ2,2 e iπ(α+β+ǫαβ)eiπ(ᾱ+β̄+ǭᾱβ̄)eiπ(α+ᾱ)g+iπ(β+β̄)h × 28 sin8( πΛ(h, g) β̄− g . (69) Furthermore, by using the left- (and right-) Jacobi identities (α,β) eiπ(α+β+ǫαβ)Θ = −|ǫ|Θ , (70) the twisted part of the thermal partition function simplifies further: Zthermaltwist = (h,g)6=(0,0) Imτ Γ2,2 2 8 sin8( πΛ(h, g) ) φ((g, h), (ǫ, ǭ)). (71) The factor φ depends on the initial choice of the left- and right-chirality coefficients ǫ, ǭ φ((g, h), (ǫ, ǭ)) = |ǫ|+ (1− |ǫ|)(1− eiπgh) |ǭ|+ (1− |ǭ|)(1− eiπgh) . (72) It should be noted that expression (71) has a field theoretic interpretation in terms of a momentum lattice only. If the initial non-thermal model was not supersymmetric, |ǫ| = |ǭ| = 1, then the number of massless bosons and fermions will not be equal, nb 6= nf . For all other choices nb = nf . This situation is reflected in the factor φ((g, h), (ǫ, ǭ)), which distinguishes the four different possibilities. In the non supersymmetric case there is a non-vanishing contribution to the partition function even in the absence of the thermal co-cycle. This is equivalent, in field theory, to the one-loop zero temperature contribution to the effective action. This contribution is zero in supersymmetric theories. In the later cases, the corrections are coming from the massive thermal bosons and fermions plus a contribution from massless bosons. We will display here a number of typical examples. The first class is when the Γ2,2 lattice is unshifted by (h, g) and so it factorizes out from the sum over (g, h). Unshifted Γ2,2 lattice In the unshifted case, the sum over (g, h) can be performed easily so that the only remaining dependence is that of ǫ, ǭ. We obtain [40][41], Sthermaltwist = dτdτ̄ (Imτ)2 Zthermaltwist = −C[ǫ, ǭ] log |η(T )|4 |η(U)|4 ImT ImUµ2 where µ2 is an infrared cut-off and T, U parameterize the Kähler and complex structure moduli of the target space torus T 2. The coefficient C[ǫ, ǭ] depends on the initial chiralities of the spinors: C[1, 1] = 240, C[1, 0] = C[0, 1] = 32, C[0, 0] = 64 . (74) Actually, C[ǫ, ǭ] is nothing but the number of the massless bosonic degrees of freedom of the theory. Equation (73) is invariant under the full target space T-duality group acting on the T and U moduli separately. For large volume, ImT ≫ 1, the leading behavior is linear in ImT ∼ RyRz. Assuming iU ∼ Ry/Rz fixed and µ2 ∼ γ/RyRz, we have Sthermaltwist = C[ǫ, ǭ] RyRz − log γ |η(U)|4 ImU . (75) Note that in the large volume limit, the twisted sector contribution to one-loop amplitude depends both on the Kähler and complex structure moduli. Shifted lattice Another illustrating example is when the Z4 action shifts the Γ2,2 lattice simultaneously with the twist we described before. In this case the lattice is replaced by a shifted Γ2,2 lattice. The (Lh/2, Lg/2) shifted lattice, L = 1, 2, 3, is given by 5: (R) = (m,n) 2 |(4m+Lg)+(4n+Lh)τ | Imτ . (76) Here, we will examine in more detail the L = 1 case which corresponds to a 1/4-shifted lattice. 5For brevity we have given the expression for the shifted Γ1,1 lattice but the generalization to the Γ2,2 lattice is straightforward. When (ǫ, ǭ) = (1, 1), the contribution of the twisted sector to the partition function becomes [39][42]: Sthermaltwist = 240 dτdτ̄ (Imτ)2 Γ2,2[T, U ]− Γ2,2[4T, 4U ] . (77) To obtain the above expression we have used the identities (h,g) = Γ2,2(T, U), Γ2,2[ 0 ] = Γ2,2[4T, 4U ] (78) and we subtracted the contribution of the untwisted sector, (g, h) = (0, 0). Integrating over τ we obtain [39] [42] Sthermaltwist = − 60 log |η(T )|16 |η(U)|16 |η(4T )|4 |η(4U)|4 ImT 3 ImU3µ6 . (79) There is no volume factor in the large ImT limit (and this is generic in the case of freely acting orbifolds [39][42]). So for large ImT , (and setting ImTµ2 ∼ γ), we obtain Sthermaltwist = −60 log |η(U)|16 |η(4U)|4 . (80) In the large volume limit, Sthermaltwist only depends on the complex structure modulus of the torus. Comments The total Euclidean one-loop amplitude in both the shifted and unshifted lattice cases is given by: Sthermal = Sthermalunt + S thermal twist (81) The Sthermalunt is nothing but one-quarter of the thermal partition function of type II super- string theory on S1 × T 9 with all nine spatial radii arbitrary, while that of Euclidean time fixed by the temperature: 2πT = 1/R+ = 1/2. By exponentiating S thermal, with the twisted sector contributions in our examples given by equations (75) and (80), we obtain the norms of the corresponding cosmological wave-functions as functions of all moduli. The difference between the shifted and the ordinary model discussed previously can be understood as follows. It is known that the freely acting orbifolds are related to gravitational and gauge field backgrounds with fluxes [43]. This indicates the different interpretations of the two cosmological models with shifted and unshifted Γ2,2. In the shifted model there are non-vanishing magnetic fluxes [43] while in the unshifted case such fluxes are absent. Let us stress here that the thermal Z4-orbifold described involves a twisting that leaves two moduli parameterizing a Γ2,2 lattice, which in the large moduli limit gives us the four dimensional cosmological model discussed in section 2. Many other orbifold-like models can be constructed, which may factorize bigger lattices, admitting a higher dimensional interpretation. In all those cases, the partition function is computable as a function of the moduli. However its analytic form in terms of the moduli in various limits depends crucially on whether the orbifold is freely, or even partially freely acting (or in other words, on the different structure of magnetic fluxes). Models based on asymmetric orbifolds can also be constructed giving rise to a rich family of calculable models. A more detailed analysis would be interesting so as to understand the classification of the low level cosmologies, as well as the characteristic dependence of the norm of the wave-function on the various features of the large class of models. To illustrate the above points, we offer one further simple example based on a Z2 instead of the Z4 orbifold. Z2 orbifold models In the Z2 orbifold models (M = 2 in equation (53)), the factorization of the cosmological CFT factor is not so explicit as it was in the Z4 examples. However the cosmological interpretation remains the same. For the untwisted sector, the genus-1 contribution Sthermalunt is now one half of the thermal partition function of type II theory on S1 × T 9. We proceed to analyze the twisted sector contribution to the genus-1 amplitude. Following similar steps as in the Z4 orbifold case, and now setting (2H, 2G) = (h, g) to be integers defined modulo 2, we obtain for the case (ǫ, ǭ) = (0, 0) Sthermaltwist = dτdτ̄ (Imτ)2 (h,g)6=(0,0) 28 Imτ Γ2,2 . (82) To arrive at the result notice that since the characters (h, g) are defined to be integers modulo 2, the factor of 1/4 in the first line of equation (55) now becomes 1/2; no other modifications in this formula are needed. All the other steps carry through as before. Here also we may classify the models in two classes; in the first class the lattice is taken to be unshifted, while in the second class the Γ2,2 is half-shifted. We will use the definition: (R) = (m,n) 2 |(2m+Lg)+(2n+Lh)τ| Imτ . (83) For the unshifted Γ2,2 Z2-model, and when (ǫ, ǭ) = (0, 0), we obtain: Sthermaltwist = −384 log |η(T )|4 |η(U)|4 ImT ImUµ2 . (84) As in the Z4 models, for large volume, ImT ≫ 1, the leading behavior is linear in ImT ∼ RyRz. Assuming iU ∼ Ry/Rz fixed and µ2 ∼ γ/RyRz we have: Sthermaltwist = 384 RyRz − log γ |η(U)|4 ImU . (85) The Z2-model with half-shifted lattice, Γ2,2[ g ], (g, h = 0, 1), yields [39][42] Sthermaltwist = 384 dτdτ̄ (Imτ)2 Γ2,2[T, U ]− Γ2,2[ 2T, 2U ] . (86) Here also the contribution coming from the untwisted sector (h, g) = (0, 0), is subtracted. To obtain the above expression we have used the identities (h,g) Γ2,2[ g ] = Γ2,2[T, U ], Γ2,2[ 0 ] = Γ2,2[ 2T, 2U ]. (87) Integrating over τ we obtain: Sthermaltwist = −192 log |η(T )|8 |η(U)|8 |η(2T )|4 |η(2U)|4 ImT ImUµ2 . (88) There is no volume factor in the large ImT limit. Thus, for large ImT , (and setting ImTµ2 ∼ γ), we obtain for the half-shifted lattice contribution to Sthermaltwist : Sthermaltwist = −192 log γ |η(U)|8 |η(2U)|4 . (89) As we see, we can obtain explicit expressions for the genus-1 approximation to the norm of the wave-function for these particular Z2-models. The twisted sector contribution is given explicitly by equations (88) and (89). In the above family of models, we have always considered a two-dimensional cosmology at small level. That choice is mainly due to two obstructions that are difficult (but not necessarily impossible) to circumvent. One is associated to the difficulty of continuing from Lorentzian to Euclidean signature in the presence of electric fluxes. The other is that it is difficult to construct compact models with positive central charge deficit (negatively curved Euclidean backgrounds) in string theory, or alternatively, a compact version of linear dilaton type models. We make some further comments on this in the next section. 8.2 Liouville type models Consider string theory cosmological backgrounds based on worldsheet CFTs of the form (see e.g. [13][22][14][23][25][26][27][28][29]): SL(2, R)−|k| SL(2, R)|k|+4 ×K. (90) A nice feature of such models is that the combined central charge of the two SL(2, R)/U(1) factors is independent of |k|, and so this can be taken to be an independent, varying param- eter. Although the partition function of the euclidean cigar background is known [44], we need to deal first with the fact that for such a background the cosmological wave-function is non-normalizable due to the infinite volume of the cigar factor SL(2, R)|k|/U(1). To produce a normalizable wave-function we must face the problem of consistently compactifying this factor, as alluded at the end of the previous section. Moreover, it would be interesting to obtain a compactification scheme which leaves |k| a free parameter. Compactifying the cigar would amount to discretizing its continuous modes keeping at the same time the unitarity and the modular invariance of the torus partition function intact. An interesting aspect of these models is that now the sphere contribution to the string partition function is finite since the volume of the conformal Killing group cancels against the volume of the SL(2, R)|k|/U(1) conformal field theory factor. Nevertheless, we can see that the torus contribution dominates (at any finite string coupling) due to the volume divergence. If a consistent way of cutting off the volume of the cigar is found, we could interpret the torus contribution as a finite thermal correction to the tree-level contribution, realizing a stringy version of the computation in [5][7][45]. A further suggestion for developing our formalism in linear dilaton spaces, is to view the wave-function of the universe as also depending on a boundary condition in the space- like linear dilaton direction of the cigar, in order to obtain a linear dilaton holographic interpretation [46]. Part of the interpretation of the wave-function of the universe would then be as in the Hartle-Hawking picture, and part would be holographic. Finally in these models supersymmetry can be restored asymptotically in the large |k| limit, making contact with linear dilaton models in null directions (see e.g. [47] for recent progress). 9 Discussion We have outlined a framework generalizing the Hartle-Hawking no boundary proposal of the wave-function of the universe to string theory cosmological backgrounds. The class of example cosmologies considered here are described by worldsheet conformal field theories of the general form SL(2, R)−|k|/U(1) × K, where K is an internal, compact CFT. In order to define the analogue of the Hartle-Hawking wave-function, we had to surmount the technical hurdle of realizing that such cosmologies (like the corresponding Euclidean parafermion theories) have an almost geometrical description in terms of a compact non- singular T-fold. We then defined the wave-function of the universe via a Euclidean string field theory path integral (generalizing the no-boundary proposal). For specific examples we computed the norm of the wave-function to leading order in string perturbation theory, as a function of moduli parameters. There are many interesting similar examples to which we can generalize our analysis. In a probabilistic interpretation, with a normalizable wave-function at hand, one can attempt to compute vacuum expectation values for particular physical quantities in various cosmological models, and analyze their properties in various regions of the moduli space. Our purpose in this paper was to provide the framework for such a discussion, which promises to be interesting. In particular, it is an open problem to identify preferred regions in the moduli space in the large class of models to which our analysis applies. More concretely, we believe our construction points out the good use that can be made of T-folds, and generalized geometry, in stringy cosmologies (allowing to evade various no-go theorems in pure geometry). Moreover, we have been able to define a sensible calculation in a de Sitter-like compactification of string theory, after analytically continuing to the Euclidean theory. These calculations are generically hard to come by in de Sitter gravity after quantization, so any well-defined cosmological quantity, like the norm of the wave- function of stringy universes, merits scrutiny. Finally, we calculated a quantity akin to an entanglement entropy in de Sitter space, and showed that it only gets contributions starting at one loop, and we gave its microscopic origin. Acknowledgements We thank Constantin Bachas, James Bedford, Ben Craps, John Iliopoulos, Dieter Luest, Hervé Partouche, Anastasios Petkou, Giuseppe Policastro and Marios Petropoulos for useful discussions. N. T. thanks the Ecole Normale Supérieure and C. K. and J. T. thank the University of Cyprus for hospitality. 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Verlinde, “A matrix big bang,” JHEP 0510, 005 (2005) [arXiv:hep-th/0506180]. http://arxiv.org/abs/hep-th/9808149 http://arxiv.org/abs/hep-th/0506180 Introduction The cosmological solution The Euclidean continuation The cosmological T-fold Defining the wave-function of the universe The norm of the wave-function Thermal nature of the wave-function Specific Examples Compact models Liouville type models Discussion

0704.1997

Query on Negative Temperature, Internal Interactions and Decrease of Entropy

Microsoft Word - negEntr.doc Query on Negative Temperature, Internal Interactions and Decrease of Entropy Yi-Fang Chang Department of Physics, Yunnan University, Kunming, 650091, China (e-mail: [email protected]) Abstract: After negative temperature is restated, we find that it will derive necessarily decrease of entropy. Negative temperature is based on the Kelvin scale and the condition dU>0 and dS<0. Conversely, there is also negative temperature for dU<0 and dS>0. But, negative temperature is contradiction with usual meaning of temperature and with some basic concepts of physics and mathematics. It is a question in nonequilibrium thermodynamics. We proposed a possibility of decrease of entropy due to fluctuation magnified and internal interactions in some isolated systems. From this we discuss some possible examples and theories. Keywords: entropy; negative temperature; nonequilibrium. PACS: 05.70.-a, 05.20.-y 1.Restatement of Negative Temperature In thermodynamics negative temperature is a well-known idea proposed and expounded by Ramsey [1] and Landau [2], et al. Ramsey discussed the thermodynamics and statistical mechanics of negative absolute temperatures in a detailed and fundamental manner [1]. He proved that if the entropy of a thermodynamic system is not a monotonically increasing function of its internal energy, it possesses a negative temperature whenever XUS )/( negative. Negative temperatures are hotter than positive temperature. He pointed out: from a thermodynamic point of view the only requirement for the existence of a negative temperature is that the entropy S should not be restricted to a monotonically increasing function of the internal energy U. In the thermodynamic equation relating TdS and dU, a temperature is )/( XUST . (1) An assumption, entropy S increases monotonically with U, is not necessary in the derivation of many thermodynamic theorems. Negative temperatures are hotter than infinite temperature. Negative temperature is an unfortunate and misleading. If the temperature function had been chosen as 1/T, then the coldest temperature would correspond to - for this function, infinite temperatures on the conventional scale would correspond to 0, and negative temperatures on the conventional scale would correspond to positive values of this function. Ramsey proposed: One of the standard formulations of the second law of thermodynamics must be altered to the following: It is impossible to construct an engine that will operate in a closed cycle and prove no effect other than (a) the extraction of heat from a positive-temperature reservoir with the performance of an equivalent amount of work or (b) the rejection of heat into a negative-temperature reservoir with the corresponding work being done on the engine. A thermodynamic system that is in internal thermodynamic equilibrium, that is otherwise essentially isolated. Negative temperatures are applied to some mutually interacting nuclear spin system. Klein justified Ramsey s criteria for systems capable of negative absolute temperatures [3]. Only premise of Ramsey s statement is Kelvin definition (1). Even he implied that entropy might decrease with U, i.e., in an original definition of temperature T=dU/dS, (2) when dU>0 and dS<0, T<0. Intuitively, the physical meaning of temperature is that it describes whether a body is hot or cold [4]. A definition of an absolute thermodynamic temperature scale is proportional to the quantity of heat. Maxwell s definition is that the temperature of a body is its thermal state considered with reference to its power of communicating heat to other bodies, which was adopted substantially unchanged by Planck and Poincare [5]. In microscopic thermodynamics temperatures are related with the states of molecular motions. Kelvin temperature scale is defined by the relationship (1). It is a little different that Landau proved negative temperature. In Landau s book <Statistical Physics> [2] negative temperature was stated as following: Let us consider some peculiar effects related to the properties of paramagnetic dielectrics. Here the interaction of these moments brings about a new magnetic spectrum, which is superposed on the ordinary spectrum. From this the entropy is ln nnmag EE gNS , (3) where N is the number of atoms, g is the number of possible orientations of an individual moment relative to the lattice, nE are the energy levels of the system of interacting moments, and nE is the average as the ordinary arithmetic mean. This shall regard the atomic magnetic moments fixed at the lattice sites and interacting with one another as a single isolated system. Further, there has the interesting result that the system of interacting moments may have either a positive or a negative temperature. At T=0, the system is in its lower quantum state, and its entropy is zero [2]. In fact, the temperature T=0 (absolute zero) is impossibly achieved. As the temperature increases, the energy and entropy of the system increase monotonically. At T= , the energy is nE and the entropy reaches its maximum value Nlng; these values correspond to a distribution with equal probability over all quantum states of the system, which is the limit of the Gibbs distribution as T . The statement of original negative temperature is based on the two premises: entropy of the system increase monotonically, and the Gibbs distribution holds. From this some strange arguments [2] are obtained: (a). The temperature T=- is physically identical with T= ; the two values give the same distribution and the same values of the thermodynamic quantities for the system. According to the general definition, temperature cannot be infinite, since the quantity of heat or molecular motion all cannot be infinite. Negative temperature, even negative infinite temperature is stranger. In the same book <Statistical Physics>, Landau proved a very important result that the temperature must be positive: T>0 [2]. Moreover, T= =- is excluded by mathematics. (b). A further increase in the energy of the system corresponds to an increase in the temperature from T= , and the entropy decreases monotonically. (c). At T=0- the energy reaches its greatest value and the entropy returns to zero, the system then being in its highest quantum state. This obeys Nernst s theorem, but in which the quantity of heat is zero at T=0, while at T=0- it possesses highest quantum state! I do not know whether T=0=T=0- holds or not. (d). The region of negative temperature lies not below absolute zero but above infinity , i.e., negative temperatures are higher than positive ones . 2.Query on Negative Temperature and Decrease of Entropy In a previous paper [6], we proved that since fluctuations can be magnified due to internal interactions under a certain condition, the equal-probability does not hold. The entropy would be defined as rr tPtPktS )(ln)()( . (4) From this or lnkS in an internal condensed process, possible decrease of entropy is calculated. If various internal complex mechanism and interactions cannot be neglected, a state with smaller entropy (for example, self-organized structures) will be able to appear. In these cases, the statistics and the second law of thermodynamics should be different, in particular, for nonequilibrium thermodynamics [7,8]. Because internal interactions bring about inapplicability of the statistical independence, decrease of entropy due to internal interactions in isolated system is caused possibly. This possibility is researched for attractive process, internal energy, system entropy and nonlinear interactions, etc [6]. In fact, negative temperature derives necessarily decrease of entropy. We think, T= =- is only a finite threshold temperature cT , which corresponds to a value of entropy from increase to decrease, and this entropy is a maximum Nlng. According to the basic equation of thermodynamics, i.e., Euler equation [9], S . (5) If energy is invariance, corresponding temperature should be . (6) This value should be testable and measurable. Such temperature and energy increase continuously, and entropy decreases to a minimum, but cannot be zero. Of course, energy passes necessarily from negative temperature system to positive temperature system. Next, the Gibbs distribution is nAew / . (7) This finds the probability nw of a state of the whole system such that the body concerned is in some definite quantum state (with energy nE ), i.e., a microscopically defined state , and is suitable that the system is assumed to be in equilibrium [2]. So long as assume that the Gibbs distribution hole always, it is necessary for negative temperature. But, in the above example and laser that is another example applied negative temperature, these states are already unstable or metastable nonequilibrium states with higher energy. Bodies of negative temperature are also completely unstable and cannot exist in Nature [2]. The entropy of a body is a function only of its internal energy [2]. In states with negative temperature, the crystal be magnetized in a strong magnetic field, then the direction of the field is reversed so quickly that the spins cannot follow it [2]. This system is in a nonequilibrium state, and its internal energy and entropy are different. Laser should be an ordering process with decrease of entropy. Generally, the Gibbs distribution for a variable number of particles is [2] nNAew /)( , (8) where is the thermodynamic potential. From this the distributions are different for the number N of particles. The number N should be different in magnetic field with reversed direction. The above statement of negative temperature proves just that entropy is able to decrease with internal interactions in an isolated system. The experimental study requires that the spin system be well isolated from the lattice system [1]. This isolation is possible if the ratio of spin-lattice to spin-spin relaxation times is large [2]. This may describes the Figure 1, which is namely Fig.1 [1] and Fig.10 [2], in which the finite threshold value cT corresponds to only a maximum point dS/dE=0. Fig.1 negative temperature According to Eq.(1) or Eq.(2), when the condition dU>0 and dS<0 hold, negative temperature will be obtained. Conversely, if dU<0 and dS>0, we will also derive negative temperature. But, Eq.(2) originates from Clausius entropy dS=dQ/T, so whether above conditions could hold or not? According to Eq.(5), if ii NYX =0, T=U/S. In Fig.1, since U>0 (E>0) and S>0, then T>0. From this case, we cannot obtain negative temperature, and gNET cc ln/ . In another book <Principles of General Thermodynamics> [5], Kelvin temperature (1) of thermodynamic system may be either positive or negative, according to whether, as the system passes through stable states with fixed parameters, the entropy increases or decreases with increasing energy. This is different with Landau s statement. In fact, this statement seems to imply that the negative temperature is unnecessary, so long as the entropy decreases with increasing energy. A normal system can assume only positive Kelvin temperatures. ± A system at a negative Kelvin temperature is in a special state. For if this were not true we could, by definition, do work on the system adiabatically and prove that the system was at a positive Kelvin temperature. A system is capable of attaining negative Kelvin temperatures if for some of its stable states the entropy decreases for increasing energy at fixed values of the parameters. Further, as examples, the entropy for a monatomic gas is given by 'lnln)2/3( SRTRS . (9) Based on this, T and the density cannot be negative. The systems are nuclear spin ones in a pure lithium fluoride (LiF) crystal spin lattice relaxation times were as large as 5 minutes at room temperature while the spin-spin relaxation time was less than seconds. The systems lose internal energy as they gain entropy, and the reversed deflection corresponds to induce radiation. The sudden reversal of the magnetic field produces negative temperature for the Boltzmann distribution [10]. Next, this book [5] discussed heat flow between two systems A and B at unequal temperatures, and derives AdQ . (10) Here let a heat quantity AdQ flow into A from B, so there should be 0AB TT , which is also consistent with inequality (10). Further, according to an efficiency of heat engines . (11) Here if either temperature is negative, the efficiency will be greater than unity. Assume that be possible, this will be testified and infinite meritorious and beneficent deeds. The results arrived at for negative temperatures which are strange to our intuition have no practical significance in the field of energy production. But, °syst e m at negative Kelvin temperatures obey the second law and its many corollaries. Of course, it would be useless to consume work in order to produce a reservoir at a negative temperature which can be used to operate a very efficient heat engine [5]. Therefore, this seems to imply that negative temperature is introduced only in order to obey the second law of thermodynamics. There is the same efficiency of a Carnot engine applied by Ramsey [1]: . (12) Here various results of existence of the machine were discussed in order to be not in contradiction to the principle of increasing entropy. For an equal-temperature process, there is a simple result: dS=(dU+PdV)/T, (13) where U is the internal energy of system. A general case is (dU+PdV)>0, dS>0 for usual temperature T>0; dS<0 if T<0. Further, if T>0 and (dU+PdV)<0, for example, a contractive process is dV<0, dS<0 is possible [6]. In fact, so long as dS<0, the negative-temperature is unnecessary. Otherwise, one of the standard formulations of the second law of thermodynamics must be altered to the following: It is impossible to construct an engine that will operate in a closed cycle and prove no effect other than (1) the extraction of heat from a positive-temperature reservoir with the performance of an equivalent amount of work or (2) the rejection of heat into a negative-temperature reservoir with the corresponding work being done on the engine. The experimental study requires that the spin system be well isolated from the lattice system. This isolation is possible if the ratio of spin-lattice to spin-spin relaxation times is large [1]. The condition in which there are more atomic systems in the upper of two energy levels than in the lower, so stimulated emission will predominate over stimulated absorption. This condition may be described as a negative temperature. In a word, negative temperature is a remarkable question, in particular, for nonequilibrium thermodynamics. Is it a fallacy? From Kelvin scale one obtained infinite temperature and negative temperature, which is inconsistent with other definition of temperature, and with some basic concepts of physics and mathematics. Moreover, negative temperature is confused easily with an absolute zero defined usually by negative 273.16C. 3. Some Possible Examples for Decrease of Entropy For a mixture of the ideal gases, the increased entropy is jj xRndS ln . (14) If the interactions of two mixed gases cannot be neglected, the change of the free energy will be [9]: 7 212211 )lnln( xxxxxxRTGG if , (15) where )/( 2111 nnnx , etc. Then the change of entropy of mixing will be ]/)([)lnln(]/)([ 212211 TxxxxxxRTdGdS . (16) When >0, i.e., the interaction is an attractive force, there is probably dS<0. For instance, for 21 xx 1/2, dS Rln2 )4/( T . (17) When >(4Rln2)T, dS<0 is possible [8]. Many protons mix with electrons to form hydrogen atoms, a pair of positive and negative ions forms an atom, and various neutralization reactions between acids and alkalis form different salts. These far-equilibrium nonlinear processes form some new self-organize structures due to electromagnetic interactions. They should be able to test increase or decrease of entropy in isolated systems. The total rate of production of entropy is [11]: total . (18) If the heat current QJ >0, the total rate will be dS/dt>0 for 21 TT , and dS/dt<0 for 21 TT . We discussed an attractive process based on a potential energy U i , (19) in which entropy decreases [6]. Using a similar method of the theory of dissipative structure in non-equilibrium thermodynamics, we derived a generalized formula, in which entropy may increase or decrease, the total entropy in an isolated system is [6]: ia dSdSdS , (20) in which adS is an additive part of entropy, and idS is an interacting part of entropy. Further, the theory may be developed like the theory of dissipative structure. Barbera discussed the principle of minimal entropy production, whose field equations do not agree with the equations of balance of mass, momentum and energy in two particular cases. The processes considered are: heat conduction in a fluid at rest, and shear flow and heat conduction in an incompressible fluid [13]. Eq.(4) is similar to a generalization of the Boltzmann-Gibbs entropy functional proposed by Tsallis [14], which given by a formula: iq PPkS ln . (21) where iP is the probability of the ith microstate, the parameter q is any real number, )0(),1()1(ln 11 ffqf qq (22) When q 1, it reduces to ii PPkS ln . ( 23) The entropy of the composite system BA verifies )()()1()()()( BSASqBSASBAS qqqqq . (24) Our conclusions are consistent quantitatively with the system theory [6], in which there is [15] )()()( 21 SSS . (25) This corresponds to Tsallis entropy [14]: q , (26) which is nonextensive for Eq.(26) when q>1. Both seem to exhibit decrease of entropy with some internal interactions. For more general case, statistical independence in mathematics corresponds to the independence of probability, i.e., addition of probabilities is ipP . But, addition of dependent probabilities is .)....()1(....)( )()()....( AAApAAAp AApApAAAP (27) Therefore, from independence to interrelation, probability decrease necessarily, whose amount is determined by interaction strength. Correspondingly, entropy on mixture of different systems should decrease. In fact, any internal interaction in a system increases already relativity and orderliness. In present theory, the superfluid helium and its fountain effect must suppose that the helium does not carry entropy, so that the second law of thermodynamics is not violated [9]. It shows that the superfluids possess zero-entropy, but it cannot hold because zero-entropy corresponds to absolute zero according to the third law of thermodynamics. For the liquid or solid 3He the entropy difference [9] is sl SSS >0 (for higher temperature), =0 (for T=0.3K), <0 (for lower temperature). Such a solid state with higher entropy should be disorder than a liquid state in lower temperature! Otherwise, in chemical thermodynamics, the entropy of formation is a variant with pressure. In general, any chemical reaction can take place in either direction. In a word, according to the second law of thermodynamics, all systems in Nature will tend to heat death [16], while product will be impossible. But, world is not pessimistic always. The gravitational interactions produce various ordered stable stars and celestial bodies. The electromagnetic interactions produce various crystals and atoms. The stable atoms are determined by electromagnetic interaction and quantum mechanics. According to the second law of thermodynamics, they should be unstable. The strong interactions produce various stable nuclei and particles. A free proton is stable, which testifies that some quarks may form a structure by strong interaction. Theses stabilities depend mainly on various internal interactions and self-organizations. These attractive interactions correspond to decrease of entropy in our theory [6]. Increase of entropy corresponds to repulsive electromagnetic interactions with the same changes and weak interactions. Any stable objects and their formations from particles to stars are accompanied with internal interactions inside these objects, which have implied a possibility of decrease of entropy. The stability in Nature waits our study and research. References 1.Ramsey, N. Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures. Phys.Rev. 1956, 103,1,20-28. 2.Landau, L.D.; Lifshitz, E.M. Statistical Physics. Pergamon Press. 1980. 3.Klein, M.J. Negative Absolute Temperatures. Phys.Rev. 1956, 104,3,589. 4.Holman, J.P. Thermodynamics. Third Edition. McGraw-Hill. 1980. 5.Hatsopoulos, G.N.; Keenan, J.H. Principles of General Thermodynamics. New York: Robert E. Krieger Publishing Company, Inc. 1981. 6.Chang Yi-Fang. Entropy, Fluctuation Magnified and Internal Interactions. Entropy. 2005, 7,3,190-198. 7.Chang, Yi-Fang. In Entropy, Information and Intersecting Science. Yu C.Z., Ed. Yunnan Univ. Press. 1994. p53-60. 8.Chang Yi-Fang. Possible Decrease of Entropy due to Internal Interactions in Isolated Systems. Apeiron, 1997, 4,4,97-99. 9.Reichl, L.E. A Modern Course in Statistical Physics. Univ.of Texas Press. 1980. 10.Purcell, E.M.; Pound, R.V. A nuclear Spin System at Negative Temperature. Phys.Rev. 1951,81,1,279-280. 11.Lee, J.F., Sears, F.W. and Turcotte, D.L., Statistical Thermodynamics. Addison- Wesley Publishing Company, Inc. 1963. 12.Gioev, D.; Klich, I. Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture. Phys. Rev. Lett. 2006, 96,100503. 13.Barbera,E. On the principle of minimal entropy production for Navier-Stokes- Fourier fluids. Cont.Mecha.Ther. 1999,11,5,327-330. 14.Tsallis, C. Possible Generalization of Boltzmann-Gibbs Statistics. J.Stat.Phys., 1988,52, 1-2,479-487. 15.Wehrl, A. General Properties of Entropy. Rev.Mod.Phys. 1978, 50, 2, 221-260. 16.Rifkin, J.; Toward, T. Entropy----A New World View. New York: Bantam Edition. 1981.

0704.1998

Absence of the Fifth Force Problem in a Model with Spontaneously Broken Dilatation Symmetry

Absence of the Fifth Force Problem in a Model with Spontaneously Broken Dilatation Symmetry E. I. Guendelman and A. B. Kaganovich Physics Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel (Dated: August 3, 2021) Abstract A scale invariant model containing dilaton φ and dust (as a model of matter) is studied where the shift symmetry φ → φ + const. is spontaneously broken at the classical level due to intrinsic features of the model. The dilaton to matter coupling ”constant” f appears to be dependent of the matter density. In normal conditions, i.e. when the matter energy density is many orders of magnitude larger than the dilaton contribution to the dark energy density, f becomes less than the ratio of the ”mass of the vacuum” in the volume occupied by the matter to the Planck mass. The model yields this kind of ”Archimedes law” without any especial (intended for this) choice of the underlying action and without fine tuning of the parameters. The model not only explains why all attempts to discover a scalar force correction to Newtonian gravity were unsuccessful so far but also predicts that in the near future there is no chance to detect such corrections in the astronomical measurements as well as in the specially designed fifth force experiments on intermediate, short (like millimeter) and even ultrashort (a few nanometer) ranges. This prediction is alternative to predictions of other known models. Keywords: Fifth force; Spontaneously broken dilatation symmetry; Coupling depending on the matter density. PACS numbers: 04.50.+h; 04.80.Cc; 95.36.+x; 95.35.+d http://arxiv.org/abs/0704.1998v3 I. INTRODUCTION Possible coupling of the matter to a scalar field can be the origin of a long range force if the mass of the scalar particles is very small. It is well known since the appearance of the Brans- Dicke model[1] that such ”fifth” force could affect the results of tests of General Relativity (GR). In more general cases it may entail a violation of the Einstein’s equivalence principle. A possible existence of light scalar particles interacting to matter could also give rise to testable concequences in an intermediate or submillimeter or even shorter range depending on the scalar mass. Numerous, many years lasting, specially designed experiments, see for example [2]-[10], have not revealed so far any of possible manifestations of the fifth force. This fact, on each stage of the sequence of experiments, is treated as a new, stronger constraint on the parameters (like coupling constant and mass) with hope that the next generation of experiments will be able to discover a scalar force modifying the Newtonian gravity. This is the essence of the fifth force problem in the ”narrow sense”[47]. In this paper we demonstrate that it is quite possible that the fifth force problem in such narrow sense does not exist. Namely we will present a model where the strength of the dilaton to matter coupling measured in experimental attempts to detect a correction to the Newtonian gravity turns out so small that at least near future experiments will not be able to reveal it. On the other hand, if the matter is very diluted then its coupling to the dilaton may be not weak. But the latter is realized under conditions not compatible with the design of the fifth force experiments. The idea of the existence of a light scalar coupled to matter has a well known theo- retical ground, for example in string theory[11] and in models with spontaneously broken dilatation symmetry[12],[13]. The fifth force problem has acquired a special actuality in the last decade when the quintessence[14] and its different modifications, for example coupled quintessence[15], k-essence[16], were recognized as successful models of the dark energy[17]. If the amazing observational fact[18] that the dark energy density is about two times bigger that the (dark) matter density in the present cosmological epoch is not an accidental coin- cidence but rather is a characteristic feature during long enough period of evolution, then the explanation of this phenomenon suggests that there is an exchange of energy between dark matter and dark energy. A number of models have been constructed with the aim to describe this exchange, see for example[15], [19]-[26] and references therein. In the context of scalar field models of the dark energy, the availability of this energy exchange implies the existence of a coupling of the scalar field to dark matter. Then immediately the question arises why similar coupling to the visible matter is very strongly suppressed according to the present astronomical data[10]. Thus the resolution of the fifth force problem in its mod- ern treatment should apparently consist of simultaneous explanations, on the ground of a fundamental theory, of both the very strong suppression of the scalar field coupling to the visible matter and the absence of similar suppression of its coupling to the dark matter. One of the interesting approaches to resolution of the fifth force problem known since 1994 as ”the least coupling principle” based on the idea[27] to use non-perturbative string loop effects to explain why the massless dilaton may decouples from matter. In fact it was shown that under certain assumptions about the structure of the (unknown) dilaton coupling functions in the low energy effective action resulting from taking into account the full non-perturbative string loop expansion, the string dilaton is cosmologically attracted toward values where its effective coupling to matter disappears. The astrophysical effects of the matter density dependence of the dilaton to matter cou- pling was studied in 1989 in the context of a model with spontaneously broken dilatation symmetry in Ref. [13]. However in this model the effect is too weak to be observed now. Another way to describe the influence of the matter density on the fifth force is used in the Chameleon model[28] formulated in 2004. The key point here is the fact that the scalar field effective potential depends on the local matter density ρm if the direct coupling of the scalar field to the metric tensor in the underlying Lagrangian is assumed like in earlier models [29],[30]. Therefore the position of the minimum of the effective potential and the mass of small fluctuations turn out to be ρm-dependent. In space regions of ”high’ matter density such as on the Earth or in other compact objects, the effective mass of the scalar field becomes so big that the scalar field can penetrate only into a thin superficial shell of the compact object. As a result of this, it appears to be possible to realize a situation where in spite of a choice for a scalar to matter coupling of order unity, the violation of the equivalence principle is exponentially suppressed. However, for objects of lower density, the fifth force may be detectable and the corresponding predictions are made. One should note that the model of Ref. [31] with the matter density dependence of the effective dilaton to matter coupling was constructed in 2001 without any specific conjectures in the underlying action intended to solve the fifth force problem[48]. The resolution of the fifth force problem appears as a result which reads: 1) The local effective Yukawa coupling of the dilaton to fermions in normal laboratory conditions equals practically zero automatically, without any fine tuning of the parameters. The term normal laboratory conditions means that the local fermion energy density is many orders of magnitude larger than the dilaton contribution to the dark energy density. 2) Under the same conditions, the Einstein’s GR is reproduced. One of the main ingredients of the model[31] consists in the realization of the idea [33] that the fifth force problem might be resolved if the theory would possess the approximate global shift symmetry of the scalar field φ → φ+ const. (1) In the model [31],[32], the global shift symmetry (1) is spontaneously broken in such a way that the effective potential depends on φ only via M4e−2αφ/Mp where M is an integration constant of the dimensionality of mass that appears as a result of the spontaneous breakdown of the shift symmetry (1). Here α > 0 is a parameter of the order of unity and Mp = (8πG)−1/2. This is a way the model [31],[32] avoids the problem with realization of the global shift symmetry (1) in the context of quintessence type models where the potential is not invariant under the shift of φ. The model with such features was constructed in the framework of the Two Measures Field Theory (TMT) [34]-[41]. In the present paper we show that the main results concerning the decoupling and the restoration of the Einstein’s GR in the model [31],[32] for fermions (which is rather com- plicated), remain also true in a macroscopic description of matter (which is significantly simpler). This should make more clear the way of resolution of the fifth force problem in scale invariant TMT models. Our underlying model involves the coupling of the dilaton φ to dust in such a form that Lagrangians are quite usual, without any exotic term, and the action is invariant under scale transformations accompanied by a corresponding shift (1) of the dilaton. After spontaneous symmetry breaking (SSB), the effective picture in the Einstein frame differs in general very much from the Einstein’s GR. But if the local matter density is many orders of magnitude larger then the vacuum energy density then Einstein’s GR is reproduced, and the dilaton to matter coupling practically disappears without fine tuning of the parameters. II. BASIS OF TWO MEASURES FIELD THEORY AND FORMULATION OF THE SCALE INVARIANT MODEL A. Main ideas of the Two Measures Field Theory TMT is a generally coordinate invariant theory where all the difference from the standard field theory in curved space-time consists only of the following three additional assumptions: 1. The first assumption is the hypothesis that the effective action at the energies below the Planck scale has to be of the form[34]-[41] −gd4x (2) including two Lagrangians L1 and L2 and two measures of integration −g and Φ. One is the usual measure of integration −g in the 4-dimensional space-time manifold equipped with the metric gµν . Another is the new measure of integration Φ in the same 4-dimensional space-time manifold. The measure Φ being a scalar density and a total derivative may be defined for example[49] by means of four scalar fields ϕa (a = 1, 2, 3, 4) Φ = εµναβεabcd∂µϕa∂νϕb∂αϕc∂βϕd. (3) To provide parity conservation one can choose for example one of ϕa’s to be a pseu- doscalar. 2. Generically it is allowed that L1 and L2 will be functions of all matter fields, the dilaton field, the metric, the connection but not of the ”measure fields” ϕa . In such a case, i.e. when the measure fields enter in the theory only via the measure Φ, the action (2) possesses an infinite dimensional symmetry ϕa → ϕa + fa(L1), where fa(L1) are arbitrary functions of L1 (see details in Ref. [35]). One can hope that this symmetry should prevent emergence of a measure fields dependence in L1 and L2 after quantum effects are taken into account. 3. Important feature of TMT that is responsible for many interesting and desirable results of the field theory models studied so far[31],[32],[34]-[41] consists of the assumption that all fields, including also metric, connection and the measure fields ϕa are independent dynamical variables. All the relations between them are results of equations of motion. In particular, the independence of the metric and the connection means that we proceed in the first order formalism and the relation between connection and metric is not a priori according to Riemannian geometry. We want to stress again that except for the listed three assumptions we do not make any changes as compared with principles of the standard field theory in curved space-time. In other words, all the freedom in constructing different models in the framework of TMT consists of the choice of the concrete matter content and the Lagrangians L1 and L2 that is quite similar to the standard field theory. Since Φ is a total derivative, a shift of L1 by a constant, L1 → L1 + const, has no effect on the equations of motion. Similar shift of L2 would lead to the change of the constant part of the Lagrangian coupled to the volume element −gd4x. In the standard GR, this constant term is the cosmological constant. However in TMT the relation between the constant term of L2 and the physical cosmological constant is very non trivial and this makes possible[35],[31],[32],[41] to resolve the cosmological constant problem[50]. Varying the measure fields ϕa, we obtain Bµa∂µL1 = 0 where B a = ε µναβεabcd∂νϕb∂αϕc∂βϕd. (4) Since Det(Bµa ) = Φ3 it follows that if Φ 6= 0, L1 = sM 4 = const (5) where s = ±1 and M is a constant of integration with the dimension of mass. In what follows we make the choice s = 1. One should notice the very important differences of TMT from scalar-tensor theories with nonminimal coupling: a) In general, the Lagrangian density L1 (coupled to the measure Φ) may contain not only the scalar curvature term (or more general gravity term) but also all possible matter fields terms. This means that TMT modifies in general both the gravitational sector and the matter sector; b) If the field Φ were the fundamental (non composite) one then instead of (5), the variation of Φ would result in the equation L1 = 0 and therefore the dimensionfull integration constant M4 would not appear in the theory. Applying the Palatini formalism in TMT one can show (see for example [35]) that in addition to the usual Christoffel coefficients, the resulting relation between metric and con- nection includes also the gradient of the ratio of the two measures ζ ≡ Φ√ which is a scalar field. This means that with the set of variables used in the underlying action (2) (and in particular with the metric gµν) the space-time is not Riemannian. The gravity and matter field equations obtained by means of the first order formalism contain both ζ and its gradient. It turns out that at least at the classical level, the measure fields ϕa affect the theory only through the scalar field ζ . Variation with respect to the metric yields as usual the gravitational equations. But in addition, if L1 involves a scalar curvature term then Eq.(5) provides us with an additional gravitational type equation, independent of the former. Taking trace of the gravitational equations and excluding the scalar curvature from these independent equations we obtain a consistency condition having the form of a constraint which determines ζ(x) as a function of matter fields. It is very important that neither Newton constant nor curvature appear in this constraint which means that the geometrical scalar field ζ(x) is determined by other fields configuration locally and straightforward (that is without gravitational interaction). By an appropriate change of the dynamical variables which includes a redefinition of the metric, one can formulate the theory in a Riemannian space-time. The corresponding frame we call ”the Einstein frame”. The big advantage of TMT is that in a very wide class of models, the gravity and all matter fields equations of motion take canonical GR form in the Einstein frame. All the novelty of TMT in the Einstein frame as compared with the standard GR is revealed only in an unusual structure of the scalar fields effective potential, masses of particles and their interactions with scalar fields as well as in the unusual structure of matter contributions to the energy-momentum tensor: all these quantities appear to be ζ dependent. This is why the scalar field ζ(x) determined by the constraint as a function of matter fields, has a key role in dynamics of TMT models. Note that if we were to assume that for some reasons the gravity effects are negligible and choose to work in the Minkowski space-time from the very beginning, then we would lose the constraint, and the result would be very much different from the one obtained according to the prescriptions of TMT with taking the flat space-time limit at the end. This means that the gravity in TMT plays the more essential role than in the usual (i.e. only with the measure of integration −g) field theory in curved space-time. B. Scale invariant model In the original frame (where the metric is gµν), a matter content of our TMT model represented in the form of the action (2), is a dust and a scalar field (dilaton). The dilaton φ allows to realize a spontaneously broken global scale invariance[36],[37],[31],[32] and together with this it can govern the evolution of the universe on different stages: in the early universe φ plays the role of inflaton and in the late time universe it is transformed into a part of the dark energy (for details see Refs. [31],[32],[41]). We postulate that the theory is invariant under the global scale transformations: gµν → eθgµν , Γµαβ → Γ αβ, φ → φ− θ, ϕa → labϕb (7) where det(lab) = e 2θ and θ = const. Keeping the general structure (2), it is convenient to represent the action in the following form: S = Sg + Sφ + Sm (8) Sg = − (Φ + bg −g)R(Γ, g)eαφ/Mpd4x ; eαφ/Mp (Φ + bφ gµνφ,µφ,ν − ΦV1 + eαφ/Mp d4x ; (Φ + bm −g)Lmd4x , where R(Γ, g) = gµν Γλµν,λ − Γλµλ,ν + ΓλαλΓαµν − ΓλανΓαµλ and the Lagrangian for the matter, as collection of particles, which provides the scale in- variance of Sm reads Lm = −m αφ/Mp δ4(x− xi(λ))√ dλ (9) where λ is an arbitrary parameter. For simplicity we consider the collection of the particles with the same mass parameter m. We assume in addition that xi(λ) do not participate in the scale transformations (7). In the action (8) there are two types of the gravitational terms and of the ”kinetic- like terms” which respect the scale invariance : the terms of the one type coupled to the measure Φ and those of the other type coupled to the measure −g. Using the freedom in normalization of the measure fields ϕa we set the coupling constant of the scalar curvature to the measure Φ to be − 1 . Normalizing all the fields such that their couplings to the measure Φ have no additional factors, we are not able in general to provide the same in terms describing the appropriate couplings to the measure −g. This fact explains the need to introduce the dimensionless real parameters bg, bφ and bm. We will only assume that they are positive, have the same or very close orders of magnitude bg ∼ bφ ∼ bm (10) and besides bm > bg. The real positive parameter α is assumed to be of the order of unity. As usual κ = 16πG and we use Mp = (8πG) −1/2. One should also point out the possibility of introducing two different pre-potentials which are exponential functions of the dilaton φ coupled to the measures Φ and −g with factors V1 and V2. Such φ-dependence provides the scale symmetry (7). We will see below how the dilaton effective potential is generated as the result of SSB of the scale invariance and the transformation to the Einstein frame. According to the general prescriptions of TMT, we have to start from studying the self- consistent system of gravity (metric gµν and connection Γ αβ), the measure Φ degrees of freedom ϕa, the dilaton field φ and the matter particles coordinates x i (λ), proceeding in the first order formalism. For the purpose of this paper we restrict ourselves to a zero temperature gas of particles, i.e. we will assume that d~xi/dλ ≡ 0 for all particles. It is convenient to proceed in the frame where g0l = 0, l = 1, 2, 3. Then the particle density is defined by n(~x) = √−g(3) δ(3)(~x− ~xi(λ)) (11) where g(3) = det(gkl) and Sm = −m d4x(Φ + bm −g)n(~x) e αφ/Mp (12) Following the procedure described in the previous subsection we have to write down all equations of motion, find the consistency condition (the constraint which determines ζ-field as a function of other fields and matter) and make a transformation to the Einstein frame. We will skip most of the intermediate results and in the next subsection present the resulting equations in the Einstein frame. Nevertheless two exclusions we have to make here. The first one concerns the important effect observable when varying Sm with respect to gµν : −g mn(~x) e αφ/Mp g00, (13) Φmn(~x) e αφ/Mp gkl. (14) The latter equation shows that due to the measure Φ, the zero temperature gas generically possesses a pressure. As we will see this pressure disappears automatically together with the fifth force as the matter energy density is many orders of magnitude larger then the dark energy density, which is evidently true in all physical phenomena tested experimentally. The second one is the notion concerning the role of Eq. (5) resulting from variation of the measure fields ϕa. With the action (8), where the Lagrangian L1 is the sum of terms coupled to the measure Φ, Eq. (5) describes a spontaneous breakdown of the global scale symmetry (7). III. EQUATIONS OF MOTION IN THE EINSTEIN FRAME. It turns out that when working with the new metric (φ remains the same) g̃µν = e αφ/Mp(ζ + bg)gµν , (15) which we call the Einstein frame, the connection becomes Riemannian. Since g̃µν is invari- ant under the scale transformations (7), spontaneous breaking of the scale symmetry (by means of Eq.(5)) is reduced in the Einstein frame to the spontaneous breakdown of the shift symmetry (1). Notice that the Goldstone theorem generically is not applicable in this kind of models[37]. The transformation (15) causes the transformation of the particle density ñ(~x) = (ζ + bg) −3/2 e− αφ/Mp n(~x) (16) After the change of variables to the Einstein frame (15) and some simple algebra, the gravitational equations take the standard GR form Gµν(g̃αβ) = T effµν (17) where Gµν(g̃αβ) is the Einstein tensor in the Riemannian space-time with the metric g̃µν . The components of the effective energy-momentum tensor are as follows ζ + bφ ζ + bg φ̇2 − g̃00X + g̃00 Veff(φ; ζ,M)− δ · bg ζ + bg 3ζ + bm + 2bg ζ + bg ζ + bφ ζ + bg (φ,kφ,l − g̃klX) (19) + g̃kl Veff(φ; ζ,M)− δ · bg ζ + bg ζ − bm + 2bg ζ + bg Here the following notations have been used: X ≡ 1 g̃αβφ,αφ,β and δ = bg − bφ and the function Veff(φ; ζ) is defined by Veff(φ; ζ) = M4e−2αφ/Mp + V1 (ζ + bg)2 The dilaton φ field equation in the Einstein frame is as follows ζ + bφ ζ + bg −g̃g̃µν∂νφ (ζ + bg)M 4e−2αφ/Mp − (ζ − bg)V1 − 2V2 − δbg(ζ + bg)X (ζ + bg)2 ζ − bm + 2bg ζ + bg mñ (22) In the above equations, the scalar field ζ is determined as a function ζ(φ,X, ñ) by means of the following constraint (origin of which has been discussed in Sec.2.1): (bg − ζ) M4e−2αφ/Mp + V1 − 2V2 (ζ + bg)2 − δ · bgX ζ + bg ζ − bm + 2bg ζ + bg Applying the constraint (23) to Eq.(22) one can reduce the latter to the form ζ + bφ ζ + bg −g̃g̃µν∂νφ − 2αζ (ζ + bg)2Mp M4e−2αφ/Mp = 0, (24) where ζ is a solution of the constraint (23). One should point out two very important features of the model. First, the φ dependence in all the equations of motion (including the constraint) emerges only in the formM4e−2αφ/Mp where M is the integration constant, i.e. due to the spontaneous breakdown of the scale symmetry (7) (or the shift symmetry (1) in the Einstein frame). Second, the constraint (23) is the fifth degree algebraic equation with respect to ζ + bg and therefore generically ζ is a complicated function of φ, X and ñ. Hence generically each of ζ dependent terms in Eqs. (18)-(22) and (24) describe very nontrivial coupling of the dilaton to the matter. IV. DARK ENERGY IN THE ABSENCE OF MATTER It is worth to start the investigation of the features of our model from the simplest case when the particle density of the dust is zero: ñ(x) ≡ 0. Then the dilaton φ is the only matter which in the early universe plays the role of the inflaton while in the late universe it is the dark energy. The appropriate model in the context of cosmological solutions has been studied in detail in Ref. [41]. Here we present only some of the equations we will need for the purposes of this paper and a list of the main results. In the absence of the matter particles, the scalar ζ = ζ(φ,X) can be easily found from the constraint (23): ζ (ñ=0) = bg − 2 V2 + δ · b2gX M4e−2αφ/Mp + V1 + δ · bgX In the spatially homogeneous case X ≥ 0. Then the effective energy-momentum tensor can be represented in a form of that of a perfect fluid T effµν = (ρ+ p)uµuν − pg̃µν , where uµ = (2X)1/2 with the following energy and pressure densities obtained after inserting (25) into the com- ponents of the energy-momentum tensor (18), (19) where now ñ(x) ≡ 0 ρ(φ,X ;M) ≡ ρ(ñ=0) (27) = X + (M4e−2αφ/Mp + V1) 2 − 2δbg(M4e−2αφ/Mp + V1)X − 3δ2b2gX2 4[bg(M4e−2αφ/Mp + V1)− V2] p(φ,X ;M) ≡ p(ñ=0) = X − M4e−2αφ/Mp + V1 + δbgX 4[bg(M4e−2αφ/Mp + V1)− V2] . (28) Substitution of (25) into Eq. (24) yields the φ-equation with very interesting dynamics. The appearance of the nonlinear X dependence in spite of the absence of such nonlinearity in the underlying action means that our model represents an explicit example of k-essence[16] resulting from first principles. The effective k-essence action has the form Seff = −g̃d4x R(g̃) + p (φ,X ;M) , (29) where p(φ,X ;M) is given by Eq.(28). In the context of spatially flat FRW cosmology, in the absence of the matter particles (i.e ñ(x) ≡ 0), the TMT model under consideration[41] exhibits a number of interesting outputs depending of the choice of regions in the parameter space (but without fine tuning): a) Absence of initial singularity of the curvature while its time derivative is singular. This is a sort of ”sudden” singularities studied by Barrow on purely kinematic grounds[44]. b) Power law inflation in the subsequent stage of evolution. Depending on the region in the parameter space the inflation ends with a graceful exit either into the state with zero cosmological constant (CC) or into the state driven by both a small CC and the field φ with a quintessence-like potential. c) Possibility of resolution of the old CC problem. From the point of view of TMT, it be- comes clear why the old CC problem cannot be solved (without fine tuning) in conventional field theories. d) TMT enables two ways for achieving small CC without fine tuning of dimensionful pa- rameters: either by a seesaw type mechanism or due to a correspondence principle between TMT and conventional field theories (i.e theories with only the measure of integration in the action). e) There is a wide range of the parameters where the dynamics of the scalar field φ, playing the role of the dark energy in the late universe, allows crossing the phantom divide, i.e. the equation-of-state w = p/ρ may be w < −1 and w asymptotically (as t → ∞) approaches −1 from below. One can show that in the original frame used in the underlying action (8), this regime corresponds to the negative sign of the measure of integration Φ+ bφ −g of the dilaton φ kinetic term[51]. This dynamical effect which emerges here instead of putting the wrong sign kinetic term by hand in the phantom model[46], will be discussed in detail in another paper. Taking into account that in the late time universe theX-contribution to ρ(ñ=0) approaches zero, one can see that the dark energy density is positive for any φ provided bgV1 ≥ V2 (30) Then it follows from (25) that |ζ (ñ=0)| ∼ bg. (31) This will be useful in the next section. V. NORMAL CONDITIONS: REPRODUCING EINSTEIN’S GR AND ABSENCE OF THE FIFTH FORCE PROBLEM One should now pay attention to the interesting result that the explicit ñ dependence involving the same form of ζ dependence ζ − bm + 2bg ζ + bg mñ (32) appears simultaneously[52] in the dust contribution to the pressure (through the last term in Eq. (19)), in the effective dilaton to dust coupling (in the r.h.s. of Eq. (22)) and in the r.h.s. of the constraint (23). Let us analyze consequences of this wonderful coincidence in the case when the matter energy density (modeled by dust) is much larger than the dilaton contribution to the dark energy density in the space region occupied by this matter. Evidently this is the condition under which all tests of Einstein’s GR, including the question of the fifth force, are fulfilled. Therefore if this condition is satisfied we will say that the matter is in normal conditions. The existence of the fifth force turns into a problem just in normal conditions. The opposite situation may be realized (see Refs. [31],[32]) if the matter is diluted up to a magnitude of the macroscopic energy density comparable with the dilaton contribution to the dark energy density. In this case we say that the matter is in the state of cosmo-low energy physics (CLEP). It is evident that the fifth force acting on the matter in the CLEP state cannot be detected now and in the near future, and therefore does not appear to be a problem. But effects of the CLEP may be important in cosmology, see Ref. [32]. The last terms in eqs. (18) and (19), being the matter contributions to the energy density (ρm) and the pressure (−pm) respectively, generally speaking have the same order of magnitude. But if the dust is in the normal conditions there is a possibility to provide the desirable feature of the dust in GR: it must be pressureless. This is realized provided that in normal conditions (n.c.) the following equality holds with extremely high accuracy: ζ (n.c.) ≈ bm − 2bg (33) Remind that we have assumed bm > bg. Then ζ (n.c.) + bg > 0, and the transformation (15) and the subsequent equations in the Einstein frame are well defined. Inserting (33) in the last term of Eq. (18) we obtain the effective dust energy density in normal conditions ρ(n.c.)m = 2 bm − bg mñ (34) Substitution of (33) into the rest of the terms of the components of the energy-momentum tensor (18) and (19) gives the dilaton contribution to the energy density and pressure of the dark energy which have the orders of magnitude close to those in the absence of matter case, Eqs. (27) and (28). The latter statement may be easily checked by using Eqs. (25),(31),(33) and (10). Note that Eq. (33) is not just a choice to provide zero dust contribution to the pressure. In fact it is the result of analyzing the equations of motion together with the constraint (23). In the Appendix we present the detailed analysis yielding this result. But in this section we have started from the use of this result in order to make the physical meaning more distinct. Taking into account our assumption (10) and Eq. (31) we infer that ζ (n.c.) and ζ (ñ=0) (in the absence of matter case, Eq. (25)) have close orders of magnitudes. Then it is easy to see (making use the inequality (30)) that the l.h.s. of the constraint (23), as ζ = ζ (n.c.), has the order of magnitude close to that of the dark energy density ρ(ñ=0) in the absence of matter case discussed in Sec. 4. Thus in the case under consideration, the constraint (23) describes a balance between the pressure of the dust in normal conditions on the one hand and the vacuum energy density on the other hand. This balance is realized due to the condition (33). Besides reproducing Einstein equations when the scalar field and dust (in normal con- ditions) are sources of the gravity, the condition (33) automatically provides a practical disappearance of the effective dilaton to matter coupling. Indeed, inserting (33) into the φ-equation written in the form (24) and into Veff(φ; ζ), Eq. (21), one can immediately see that only the force of the strength of the dark energy selfinteraction is present in this case. Note that this force is a total force involving both the selfinteraction of the dilaton and its interaction with dust in normal conditions. Furthermore, in this way one can see explicitly that due to the factor M4e−2αφ/Mp , this total force may obtain an additional, exponential dumping since in the cosmological context shortly discussed in Sec. 4 (see details in Ref. ([41])) a scenario, where in the late time universe φ ≫ Mp, seems to be most appealing. Another way to see the absence of the fifth force problem in the normal conditions is to look at the φ-equation in the form (22) and estimate the Yukawa type coupling constant in the r.h.s. of this equation. In fact, using the constraint (23) and representing the particle density in the form ñ ≈ N/υ where N is the number of particles in a volume υ, one can make the following estimation for the effective dilaton to matter coupling ”constant” f defined by the Yukawa type interaction term fñφ (if we were to invent an effective action whose variation with respect to φ would result in Eq. (22)): f ≡ α ζ − bm + 2bg ζ + bg ζ − bm + 2bg bm − bg ρvacυ Thus we conclude that the effective coupling ”constant” of the dilaton to matter in the normal conditions is of the order of the ratio of the ”mass of the vacuum” in the volume occupied by the matter to the Planck mass taken N times. In some sense this result resembles the Archimedes law. At the same time Eq. (35) gives us an estimation of the exactness of the condition (33). VI. DISCUSSION AND CONCLUSION In the present paper, the idea to construct a model with spontaneously broken dilatation invariance where the dilaton dependence in all equations of motion results only from the SSB of the shift symmetry (1), is implemented from first principles in the framework of Although the dust model studied in this paper is a very crude model of matter, it is quite sufficient for studying the fifth force problem. In fact, all experiments which search for the fifth force deal with macroscopic bodies which, in the zeroth order approximation, can be regarded as collections of noninteracting, point-like motionless particles with very high particle number density ñ(x). Generically the model studied in the present paper is different from Einstein’s GR. For example it allows the long range scalar force and a non-zero pressure of the cold dust. However the magnitude of the particle number density turns out to be the very important factor influencing the strength of the dilaton to matter coupling. This happens due to the constraint (23) which is nothing but the consistency condition of the equations of motion. The analysis of the constraint presented in the Appendix shows that generically it describes a balance between the matter density and dark energy density. It turns out that in the case of a macroscopic body, that is in normal conditions, the constraint allows this balance only in such a way that the dilaton practically decouples from the matter and Einstein’s GR is restored automatically. Thus our model not only explains why all attempts to discover a scalar force correction to Newtonian gravity were unsuccessful so far but also predicts that in the near future there is no chance to detect such corrections in the astronomical measurements as well as in the specially designed fifth force experiments on intermediate, short (like millimeter) and even ultrashort (a few nanometer) ranges. This prediction is alternative to predictions of other known models. Formally one can consider the case of a very diluted matter when the matter energy density is of the order of magnitude comparable with the dark energy density, which is the case opposite to the normal conditions. Only in this case the balance dictated by the constraint implies the existence of a non small dilaton coupling to matter, as well as a possibility of other distinctions from Einstein’s GR. However these effects cannot be detected in fifth force experiments now and in the near future. One should also note here that in the framework of the present model based on the consideration of point particles, the low density limit, strictly speaking, cannot be satisfactory defined. An example of the appropriate low density limit (CLEP state) was realized using a field theory model in Ref. [32] while conclusions for matter in the normal conditions were very similar to results of the present paper. Possible cosmological and astrophysical effects when the normal conditions are not satis- fied may be very interesting. In particular, taking into account that all dark matter known in the present universe has the macroscopic energy density many orders of magnitude smaller than the energy density of visible macroscopic bodies, we hope that the nature of the dark matter can be understood as a state opposite to the normal conditions studied in the present paper. VII. APPENDIX. ζ(x) WHEN THE MATTER IS IN NORMAL CONDITIONS As we mentioned in Sec. 3, solutions ζ = ζ(φ,X, ñ) of the constraint (23) are generi- cally very complicated functions. Nevertheless let us imagine that we solve the constraint, substitute ζ = ζ(φ,X, ñ) into eqs. (18), (19), (22) and solve them with certain boundary or/and initial conditions. Inserting the obtained solutions for φ(x) and X(x) back into ζ = ζ(φ,X, ñ) we will obtain a space-time dependence of the scalar field ζ = ζ(x). Let us analyze possible regimes for the ζ(x) having in mind its possible numerical values. As we have seen at the end of Sec. 4, in the vacuum |ζ (ñ=0)| ∼ bg. One can start asking the following question: what is the effect of inserting dust (into a vacuum) on the magnitude of ζ(x) in comparison with ζ (ñ=0)? One can think of three possible regimes: |ζ(x)| may become significantly larger than bg, may keep the same order of magnitude |ζ(x)| ∼ bg as it was in the vacuum and may become significantly less than bg. Consider each of these possibilities. 1. ζ(x) ≫ bg Let us start from the notion that if formally ζ → ∞ then for any particle density ñ 6= 0, the l.h.s. of the constraint (23) approaches zero while the r.h.s. approaches infinity. Therefore a regime where ζ → ∞ is impossible. Consider now the case ζ(x) ≫ bg with finite ζ . We start from estimations of the order of magnitude of two terms of Veff(φ; ζ), Eq. (21), in the vacuum, i.e. Veff (φ; ζ)|ζ=ζ(ñ=0). Using Eq. (31) we have M4e−2αφ/Mp + V1 (ζ + bg)2 M4e−2αφ/Mp + V1 (ζ + bg)2 In the presence of dust, in the regime ζ(x) ≫ bg we have respectively: M4e−2αφ/Mp + V1 (ζ + bg)2 ñ 6=0 4e−2αφ/Mp + V1 4e−2αφ/Mp + V1 (ζ + bg)2 ñ 6=0 ∼ |V2| ≪ |V2| Therefore generically Veff (φ; ζ)|ñ6=0 ≪ Veff(φ; ζ)|vac (40) where we have ignored possible different values of φ in the vacuum and inside the matter[53]. Further, proceeding in the same manner with the constraint (23) and using the above estimations it is easily to see that in the regime ζ(x) ≫ bg, the absolute value of the l.h.s. of the constraint (23) is much less than the vacuum energy density. But the r.h.s. of the constraint (23) is of the order of the dust contribution to the energy density (see the last term of Eq. (18) in the regime ζ(x) ≫ bg). Therefore in normal conditions (large ñ) the constraint (23) does not allow the regime ζ(x) ≫ bg. 2. |ζ(x)| ∼ bg In this case the l.h.s. of the constraint (23) has the order of the vacuum energy density. Let us start from the assumption that ζ(x), being |ζ(x)| ∼ bg, is different from the value ζ = bm − 2bg. Then the r.h.s. of the constraint (23), being equal to the dust contribution to the pressure (the last term of Eq. (19)), has also the order of magnitude of the dust contribution to the energy density (the last term of Eq. (18)). Therefore in normal conditions (large ñ) the constraint (23) cannot be satisfied if the value ζ is far from bm−2bg. The only way to satisfy the constraint (23) in the regime |ζ(x)| ∼ bg when the dust is in normal conditions is the equality (33). Consequences of this condition are discuused in Sec. 5. 3. |ζ(x)| ≪ bg In this case the l.h.s. of the constraint (23) has again the order of the vacuum energy density. But the r.h.s. of the constraint (23) has generically the same order of magnitude as the dust contribution to the energy density (the last term in Eq. (18)). 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[48] The more detailed description of the model [31] and its results, including new effects as neutrino dark energy which appear when the fermion density is very low, are presented in Ref. [32]. [49] Possible nature of the measure fields ϕa have been discussed in Ref. [40]. It is interesting that the idea of T.D. Lee on the possibility of dynamical coordinates[42] may be related to the measure fields ϕa too. Another possibility consists of the use of a totally antisymmetric three index field [40]. [50] Another way to construct a measure of integration which is a total derivative was recently studied by Comelli in Ref. [43]. Using a vector field (instead of four scalar fields ϕa used in TMT) and proceeding in the second order formalism, it was shown in [43] that it is possible to overcome the cosmological constant problem. [51] Note that by the definition (3) the measure Φ is not positive definite. 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0704.1999

Dark matter caustics and the enhancement of self-annihilation flux

Draft version November 16, 2018 Preprint typeset using LATEX style emulateapj v. 11/26/04 DARK MATTER CAUSTICS AND THE ENHANCEMENT OF SELF-ANNIHILATION FLUX Roya C. Mohayaee , Sergei Shandarin , Joseph Silk Draft version November 16, 2018 ABSTRACT Cold dark matter haloes are populated by caustics, which are yet to be resolved in N-body simulations or observed in the Universe. Secondary infall model provides a paradigm for the study of caustics in typical haloes assuming that they have had no major mergers and have grown only by smooth accretion. This is a particular characteristic of the smallest dark matter haloes of about 10−5 M⊙, which although atypical contain no substructures and could have survived until now with no major mergers. Thus using this model as the first guidline, we evaluate the neutralino self-annihilation flux for these haloes. Our results show that caustics could leave a distinct sawteeth signature on the differential and cumulative fluxes coming from the outer regions of these haloes. The total annihilation signal from the regions away from the centre can be boosted by about forty percents. Subject headings: dark matter haloes, caustics, dark matter detection 1. INTRODUCTION Evidence from the rotation curves of galaxies, gravitational lensing, microwave background radi- ation, peculiar velocity fields, and many other ob- servations indicate that the visible mass, in the form of stars and hot gas, is only a small fraction of the total content of the Universe. The nature of the missing mass, the dark matter, remains unknown but is widely presumed to be Weakly Interacting Massive Particles (WIMPs), such as the lightest su- persymmetric particles, which are yet to be detected in particle accelerators (Jungman, Kamionkowski, & Griest 1996 ; Bertone, Hooper & Silk 2004). Accelerator searches are complemented by the vast experimental efforts to detect these particles in our galaxy and in nearby galaxies which are believed to be embedded in dark matter haloes (Ostriker & Peebles 1973). Such complementary techniques presently involve direct detection in low background laboratory detectors (Goodman & Witten 1985) and indirect detection through observation of en- ergetic neutrinos, gamma rays and other products of self-annihilation of dark matter particles (Silk & Srednicki 1984) . The event rate for self-annihilation depends quadratically on the local dark matter density, which falls off with distance from the center of the halo. The averaged halo density profile obtained in various numerical simulations diverges at the cen- tre but is otherwise smooth and is often fit with a(n asymptotically) double power-law (Navarro, Frenk & White 1996, Moore et al 1998). However, a con- sensus on the precise values of the power exponents, the size of the central core and the resolution of fine high-density structures are yet to be achieved. The fine structures, the caustics, are inevitable outcomes 1 Institut d’astrophysique de Paris, 98 bis boulevard Arago, France 2 Department of Physics and Astronomy, University of Kansas, KS 66045, U.S.A. 3 University of Oxford, Astrophysics, Keble Road, Oxford OX1 3RH, U.K. of the evolution of a collisionless self-gravitating sys- tem described by the Jeans-Vlassov-Poisson equa- tion (for a one-dimensional numerical result see Alard & Colombi 2005). Formally, in three dimen- sions, the most common caustics are surfaces of zero thicknesses over which the density diverges. 4 How- ever, a maximum cut-off to their density is set by the finite non-negligible velocity dispersion of dark matter particles. Their density however remains very high and hence they can be significant for dark matter search experiments (Sikivie & Ipser 1992, Sikivie et al 1997, Natarajan 2007). The effect of velocity dispersion in the smearing of the caustics is expected to dominate over other effects such as par- ticle discreteness which would also smooth the caus- tics but to a far lesser degree. Mergers of haloes can also smear out the caustics substantially and due to this fact we restrict our study to haloes that have grown by slow and smooth accretion. Nevertheless, caustics are robust, in that while they may break up into micro-caustics, they remain in the fine-scale halo substructure and thereby contribute to the gen- eral clumpiness boost of any annihilation signal. Analytic studies of the formation of haloes and caustics have been carried out mainly under vari- ous simplifying assumptions, such as spherical sym- metry, self-similarity, and cold and smooth accre- tion (Gott 1975, Gunn 1977, Fillmore & Goldreich 1984, Bertschinger 1985). In an Einstein-de Sit- ter Universe a spherical overdensity expands and then turns around to collapse. After collapse and at late times, the fluid motion becomes self-similar: its form remains unchanged when its length is re- scaled in terms of the radius, rta, of the shell that is currently at turn-around and is falling onto the 4 The general theory of singularities (Arnol’d, Shandarin& Zel’dovich 1982) also predicts singularities on lines and at points. Despite the greater concentration of mass in these singularities they probably play a less important role in the total annihilation rate because they contain a considerably smaller amount of mass. However, this has not been studied in detail. http://arxiv.org/abs/0704.1999v1 galaxy for the first time. Physically, self-similarity arises because gravity is scale-free and because mass shells outside the initial overdensity are also bound and turn around at successively later times. Self- similar solutions give power-law density profiles on the scale of the halo which provides an explanation for the flattening of the rotation curves of galaxies. However, on smaller scales the density profile con- tains many spikes (i.e. caustics) of infinite density. The position and time of formation of these caus- tics are among the many properties that have been established in the framework of the self-similar in- fall model (Fillmore & Goldreich 1984, Bertschinger 1985). In reality, dark matter has a small velocity dis- persion and haloes do suffer from major mergers and non-sphericity. However, until numerical simu- lations achieve sufficient resolution, the self-similar accretion model provides a useful guideline to haloes which have not undergone major mergers. Here, we use the self-similar model of halo for- mation and a further elaboration which includes the velocity dispersion of dark matter (Mohayaee & Shandarin 2006) as a first guidline to describe the evolution of the smallest haloes which have sur- vived major merger and disruption until now and have grown only by slow accretion. The application of self-similar model to such haloes can be viewed from two contradictory angles. One might assume that minihaloes are expected to be well-represented by this model, since they con- tain no substructures, have not undergone merger and grow very slowly only by smooth mass accre- tion. On the other hand, minihaloes are not typical haloes and selfsimilar accretion model is formulated to describe the evolution of a characteristic halo. Keeping both of these issues in mind, we use self- similar model only as a first guidline for the evolu- tion of minihaloes. A large number of them have been found in simulations (Diemand et al 2005). The simulations estimate the size of these haloes to be of about 0.01 pc (half mass radius) and their mass of 10−6M⊙ at z = 26. Due to resolution prob- lems, these simulations are stopped at this redshift and typical evolution of galactic scale haloes is ex- tended to minihaloes and the conclusion is drawn that about 1015 of these haloes could exist in the halo of MilkWay today. We assume that at least a fraction of these haloes have evolved by slow ac- cretion model from z = 26 until now and use the selfsimilar model to evaluate their radius and mass at z = 0, which are respectively 1 pc and 10−5 M⊙. For these haloes and working self-consistently within our model including the contribution from the caustics, we demonstrate that in the outer re- gions of these haloes caustics can boost the annihi- lation signal by about 40%. 2. SECONDARY-INFALL MODEL WITH VELOCITY DISPERSION The haloes considered in this work grow by smooth and slow accretion. A good example are the earth-mass haloes which were recently resolved (at z=26) in numerical simulations (Diemand et al Fig. 1.— A surface-contour plot of the caustic density. In the self-similar model, caustics form concentric shells of increasing density and decreasing thicknesses and separations as we approach the center of the halo. 2005) and which although small are expected to have clean spherical caustics. We expect that at least a fraction of these haloes have survive disrup- tion and major merger and grow by self-similar ac- cretion model to a virial radius of about 1 pc and a mass of about 10−5 M⊙. To comply with the requirement of slow accretion, we fix the value of the parameter ǫ in the initial den- sity perturbation δ ∼ M−ǫi , where Mi is the initial mass, to unity. We emphasis that the self-similar model aims at describing the evolution of a typi- cal halo. Typical haloes have mass variance (σ(M) which varies as M−(n+3)/6, which sets ǫ = (n+3)/6, where n is the power spectrum index. A typical σ(M) fluctuation grows as t2/(3ǫ). Minihaloes cor- respond to the limit n → −3 part of the spectrum. For this part of the spectrum, there are mass fluc- tuations of comparable amplitude on all scales and consequently adiabatic invariance does not apply for such fluctuations. Hence, we use the self-similar model only as a first guidline for the growth of minihaloes. We as- sume that minihaloes of mass 10−6M⊙ have grown by very small accretion from z = 26 to z = 0. Once again, slow accretion corresponds to the case of ǫ = 1 in the work of Fillmore and Goldreich (1984), hence we shall adopt this value for ǫ. The self-similar density profile is given by (Bertschinger 1985) (−1)jexp where ρ̄ is the critical density and is the dimensionless radius and r is the physical ra- dius and ξi = ln(t/tta) is the dimensionless time given in terms of the turnaround time, tta, of the particle that is at the jth point where λ = λ(ξ) (see Bertschinger 1985 for further explanation). The density (1) is evaluated numerically and plot- ted in Fig. 2 after an appropriate cut-off of the caus- tics which shall be discussed now. In principle the 0.06 0.1 0.2 1e+02 1e+03 1e+04 1e+05 1e+06 2nd caustic 1st caustic Axion Neutralinoρ λ2 λ1 Fig. 2.— The plot is made by numerically solving (1) and cutting the caustics using (3). The density of halos (divided by the critical density ρ̄) can be enhanced significantly at the caustics. This enhancement is far larger for axions (dotted violet spikes) than for neutralinos (continuous black line). The number of the streams increases to the center of the halo which explains the rapid growth of the smooth component of the density profile and the caustic contribution to the halo density diminishes. The dashed-dotted red line marked ρss is the approximate self-similar density profile given by (6). The inset shows a magnified view of the second and third caustics for neutralinos. density at the caustics diverges if the velocity dis- persion of dark matter is zero. In the presence of a small velocity dispersion the maximum density and thickness of the caustic shells and their density pro- files have been evaluated (Mohayaee & Shandarin 2006). The maximum density at the caustics and their profile are given by ρcaustic,k = |∆λk| ρ̄ λk − |∆λk| < λ < λk ρ̄ λ < λk − |∆λk| where λk is the non-dimensional radius of the kth caustic counted inwards and −2λ′′k e−2ξk/3 and the thickness of the caustic shell is given by ∆λk = (3π)2/3e5ξk/9Λk t σ(t) , (5) where t is the age of the Universe, σ is the present- day velocity dispersion of dark matter particles which is that at decoupling re-scaled with the ex- pansion factor. The values of these parameters vary from one caustic to another (see Table 1 of Mo- hayaee & Shandarin 2006 for the first ten caustics). The profile (1) together with appropriate cut-off given by (3) is plotted in Fig. 2. The peaked density profile given by (1) and shown in Fig. 2 has to be evaluated numerically. However, as is evident from Fig. 2 a “self-similar” profile5 is 5 This profile can also be well-fitted by a power-law and an exponential cut-off reached which we fit with ρss = 2.8λ−9/4 (1 + λ3/4)2 ρ̄ , (6) as shown in Fig. 2 by the dashed-dotted red line, marked ρss. The turnaround radius, rta can now be evaluated by considering that at the virial radius the density is about 200 times the background density, and is given by rta ∼ 4rvir, which corresponds to the density profile given by (6). This approximate profile has been shown to be a good fit also to the mass profile (see Mohayaee & Shandarin 2006). In the next section we shall show that using this profile which ignores the caustics would yield an under- estimated value for the flux. Both the extrapolated numerical and the approx- imate density profiles shown in Fig. 2 formally di- verge at the centre. However, due to finite dark matter velocity dispersion, haloes can develop cen- tral cores. Dark matter haloes are expected to have central cores due to the dark matter velocity disper- sion, self-annihilations at the centre, angular mo- mentum, tidal and various other effects. The core could be very small and the minimum scale asso- ciated with a generic dark matter merging history would conserve traces of the original cores in the initial substructure. These should be of order the free-streaming mass as for example computed in Bertschinger (2006). In principle for small core sizes the total flux from the whole of the halo is dominated by the annihi- lation in the centre of the halo and the boost due to caustics is negligible. However, we shall show in the next section that the differential (similarly cu- mulative) flux would be distinctly marked by the caustics and shall have a sawteeth pattern and the contribution to the total flux from the outer region of these haloes by the caustics is significant and can yield a boost factor of about 40%. 3. THE FLUX DUE TO SELF-ANNIHILATION INCLUDING THE EFFECT OF CAUSTICS Caustics if detected would be clear evidence of the existence of dark matter and could rule out al- ternative models of gravity. Two major methods for their detection are through gravitational lensing (see e.g. Gavazzi et al 2006) and the flux of dark matter annihilation product which is expected to be significantly enhanced by the caustics. Here we shall discuss the second method. The flux of the self-annihilation product (e.g. γ- rays) is given by F lux ∼ ρ2(4πr2) dr , (7) where the proportionality coefficient is a function of dark matter particle mass, interaction cross section and the number of photons produced per annihila- tion. The differential and cumulative flux (i.e. the in- tegrand in expression (7) and the integral evaluated from rta inwards) for neutralino (σ = 0.03 cm/s) and a minihalo of rta = 3.24 pc (which corresponds to a virial radius of about 0.8 pc) is shown in Fig. 0.3 0.5 1 r (pc) 1e+02 1e+04 1e+06 approximate [" eq. (6)] numerical [" eq. (1)] 0.14 0.29 0.57 1.1 r (pc) 1e+03 1e+04 1e+05 1e+06 1e+07 for neutralino minihaloes of r ~1 pc ; M~ 10 σ=0.03 cm/s σ=0.0003 cm/s Fig. 3.— Cumulative flux is obtained by summing the flux inwards: i.e. from the first outer caustic towards the most inner (i.e. the integral (7) evaluated inwards). The flux is shown for two different values of the velocity dispersion. The red dashed line shows the cumulative flux obtained by using our approximate analytic expression for the density (6) which neglects the contribution from the caustics and can considerably underestimate the annihilation flux and ignore the distinct sawteeth characteristic of the caustics. The inset shows the differential flux (integrand of expression (7)) using the full density profile (1) as shown by the solid black spiky line and the approximate profile (6) as shown by the dashed- dotted red line. The sawteeth pattern is once again neglected in using the later profile. 3. The fluctuations, due to caustics, become less prominent as we go towards the centre. Decreasing the velocity dispersion would increase both the am- plitude of the peaks in the density profile and the fluctuations in the flux, as shown in Figs. 2 and 3. Using our numerical solution to (1) and approx- imation (6), we can now determine the flux from the neutralino minihaloes (Diemand et al 2005) and its enhancement due to the first twenty caustics. Clearly the total flux from the whole halo is dom- inated by the emission from the centre, where the density of the caustics reaches the background den- sity (see also . However in the outer regions where the first twenty caustics dominate, as shown in Fig. 2 the ratio of the flux using the self-similar density profile given by (6) and the complete density profile (1) gives a boost factor of about Boost = 1.4 (8) Thus, not only we expect a distinct signature on the cumulative and (similarly differential) flux due to caustics as highlighted schematically in Fig. 1 and shown numerically in Fig. 3, we also expect that the total flux from the outer halo region including the first twenty caustics to be boosted by about %40. Quantitative works on the gamma-ray flux is not carried out here, as it requires more realistic model than the self-similar model which can at best ex- plain the growth of a typical halo. Minihaloes are atypical in the sense that they evolve in isolation, accreting almost no mass. In conclusion, we have modeled dark matter haloes by an extended version of secondary infall model to include non-vanishing velocity dispersion. We have shown that the differential and cumula- tive fluxes would have distinct sawteeth pattern due to caustics. We have demonstrated that caustics can boost the total annihilation flux by about 40% percents in the outer regions of smallest haloes of about 10−5 M⊙. As for the prospect of detecting caustics, the nearest minihaloes could be detectable in gamma rays by proper motions observed with GLAST (Koushiappas 2006), and should display a caustic-like substructure. One would expect to find a series of caustics, detectable as arclets. The pre- dicted spacings could be used as a template to dig more deeply into the noisy background. Acknowledgment: S.S. acknowledges the support from LANL T8, CITA and IAP during sabbatical year 2005-06 when most of this study was done. We thank M. Kuhlen and J. Diemand for discussions. 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0704.2000

Search for a Higgs boson produced in association with a Z boson in ppbar collisions

FERMILAB-PUB-07/076-E Search for a Higgs boson produced in association with a Z boson in pp collisions V.M. Abazov,35 B. Abbott,75 M. Abolins,65 B.S. Acharya,28 M. Adams,51 T. Adams,49 E. Aguilo,5 S.H. Ahn,30 M. Ahsan,59 G.D. Alexeev,35 G. Alkhazov,39 A. Alton,64,∗ G. Alverson,63 G.A. Alves,2 M. Anastasoaie,34 L.S. Ancu,34 T. Andeen,53 S. Anderson,45 B. Andrieu,16 M.S. Anzelc,53 Y. Arnoud,13 M. Arov,60 M. Arthaud,17 A. Askew,49 B. Åsman,40 A.C.S. Assis Jesus,3 O. Atramentov,49 C. Autermann,20 C. Avila,7 C. Ay,23 F. Badaud,12 A. Baden,61 L. Bagby,52 B. Baldin,50 D.V. Bandurin,59 P. Banerjee,28 S. Banerjee,28 E. Barberis,63 A.-F. Barfuss,14 P. Bargassa,80 P. Baringer,58 J. Barreto,2 J.F. Bartlett,50 U. Bassler,16 D. Bauer,43 S. Beale,5 A. Bean,58 M. Begalli,3 M. Begel,71 C. Belanger-Champagne,40 L. Bellantoni,50 A. Bellavance,50 J.A. Benitez,65 S.B. Beri,26 G. Bernardi,16 R. Bernhard,22 L. Berntzon,14 I. Bertram,42 M. Besançon,17 R. Beuselinck,43 V.A. Bezzubov,38 P.C. Bhat,50 V. Bhatnagar,26 C. Biscarat,19 G. Blazey,52 F. Blekman,43 S. Blessing,49 D. Bloch,18 K. Bloom,67 A. Boehnlein,50 D. Boline,62 T.A. Bolton,59 G. Borissov,42 K. Bos,33 T. Bose,77 A. Brandt,78 R. Brock,65 G. Brooijmans,70 A. Bross,50 D. Brown,78 N.J. Buchanan,49 D. Buchholz,53 M. Buehler,81 V. Buescher,21 S. Burdin,42,¶ S. Burke,45 T.H. Burnett,82 C.P. Buszello,43 J.M. Butler,62 P. Calfayan,24 S. Calvet,14 J. Cammin,71 S. Caron,33 W. Carvalho,3 B.C.K. Casey,77 N.M. Cason,55 H. Castilla-Valdez,32 S. Chakrabarti,17 D. Chakraborty,52 K. Chan,5 K.M. Chan,55 A. Chandra,48 F. Charles,18 E. Cheu,45 F. Chevallier,13 D.K. Cho,62 S. Choi,31 B. Choudhary,27 L. Christofek,77 T. Christoudias,43 S. Cihangir,50 D. Claes,67 B. Clément,18 C. Clément,40 Y. Coadou,5 M. Cooke,80 W.E. Cooper,50 M. Corcoran,80 F. Couderc,17 M.-C. Cousinou,14 S. Crépé-Renaudin,13 D. Cutts,77 M. Ćwiok,29 H. da Motta,2 A. Das,62 G. Davies,43 K. De,78 P. de Jong,33 S.J. de Jong,34 E. De La Cruz-Burelo,64 C. De Oliveira Martins,3 J.D. Degenhardt,64 F. Déliot,17 M. Demarteau,50 R. Demina,71 D. Denisov,50 S.P. Denisov,38 S. Desai,50 H.T. Diehl,50 M. Diesburg,50 A. Dominguez,67 H. Dong,72 L.V. Dudko,37 L. Duflot,15 S.R. Dugad,28 D. Duggan,49 A. Duperrin,14 J. Dyer,65 A. Dyshkant,52 M. Eads,67 D. Edmunds,65 J. Ellison,48 V.D. Elvira,50 Y. Enari,77 S. Eno,61 P. Ermolov,37 H. Evans,54 A. Evdokimov,73 V.N. Evdokimov,38 A.V. Ferapontov,59 T. Ferbel,71 F. Fiedler,24 F. Filthaut,34 W. Fisher,50 H.E. Fisk,50 M. Ford,44 M. Fortner,52 H. Fox,22 S. Fu,50 S. Fuess,50 T. Gadfort,82 C.F. Galea,34 E. Gallas,50 E. Galyaev,55 C. Garcia,71 A. Garcia-Bellido,82 V. Gavrilov,36 P. Gay,12 W. Geist,18 D. Gelé,18 C.E. Gerber,51 Y. Gershtein,49 D. Gillberg,5 G. Ginther,71 N. Gollub,40 B. Gómez,7 A. Goussiou,55 P.D. Grannis,72 H. Greenlee,50 Z.D. Greenwood,60 E.M. Gregores,4 G. Grenier,19 Ph. Gris,12 J.-F. Grivaz,15 A. Grohsjean,24 S. Grünendahl,50 M.W. Grünewald,29 F. Guo,72 J. Guo,72 G. Gutierrez,50 P. Gutierrez,75 A. Haas,70 N.J. Hadley,61 P. Haefner,24 S. Hagopian,49 J. Haley,68 I. Hall,75 R.E. Hall,47 L. Han,6 K. Hanagaki,50 P. Hansson,40 K. Harder,44 A. Harel,71 R. Harrington,63 J.M. Hauptman,57 R. Hauser,65 J. Hays,43 T. Hebbeker,20 D. Hedin,52 J.G. Hegeman,33 J.M. Heinmiller,51 A.P. Heinson,48 U. Heintz,62 C. Hensel,58 K. Herner,72 G. Hesketh,63 M.D. Hildreth,55 R. Hirosky,81 J.D. Hobbs,72 B. Hoeneisen,11 H. Hoeth,25 M. Hohlfeld,21 S.J. Hong,30 R. Hooper,77 S. Hossain,75 P. Houben,33 Y. Hu,72 Z. Hubacek,9 V. Hynek,8 I. Iashvili,69 R. Illingworth,50 A.S. Ito,50 S. Jabeen,62 M. Jaffré,15 S. Jain,75 K. Jakobs,22 C. Jarvis,61 R. Jesik,43 K. Johns,45 C. Johnson,70 M. Johnson,50 A. Jonckheere,50 P. Jonsson,43 A. Juste,50 D. Käfer,20 S. Kahn,73 E. Kajfasz,14 A.M. Kalinin,35 J.M. Kalk,60 J.R. Kalk,65 S. Kappler,20 D. Karmanov,37 J. Kasper,62 P. Kasper,50 I. Katsanos,70 D. Kau,49 R. Kaur,26 V. Kaushik,78 R. Kehoe,79 S. Kermiche,14 N. Khalatyan,38 A. Khanov,76 A. Kharchilava,69 Y.M. 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Meyer,21 M. Michaut,17 T. Millet,19 J. Mitrevski,70 J. Molina,3 R.K. Mommsen,44 N.K. Mondal,28 R.W. Moore,5 T. Moulik,58 G.S. Muanza,19 M. Mulders,50 M. Mulhearn,70 O. Mundal,21 L. Mundim,3 E. Nagy,14 M. Naimuddin,50 M. Narain,77 N.A. Naumann,34 H.A. Neal,64 J.P. Negret,7 P. Neustroev,39 H. Nilsen,22 http://arxiv.org/abs/0704.2000v1 C. Noeding,22 A. Nomerotski,50 S.F. Novaes,4 T. Nunnemann,24 V. O’Dell,50 D.C. O’Neil,5 G. Obrant,39 C. Ochando,15 D. Onoprienko,59 N. Oshima,50 J. Osta,55 R. Otec,9 G.J. Otero y Garzón,51 M. Owen,44 P. Padley,80 M. Pangilinan,77 N. Parashar,56 S.-J. Park,71 S.K. Park,30 J. Parsons,70 R. Partridge,77 N. Parua,54 A. Patwa,73 G. Pawloski,80 P.M. Perea,48 K. Peters,44 Y. Peters,25 P. Pétroff,15 M. Petteni,43 R. Piegaia,1 J. Piper,65 M.-A. Pleier,21 P.L.M. Podesta-Lerma,32,§ V.M. Podstavkov,50 Y. Pogorelov,55 M.-E. Pol,2 A. Pompoš,75 B.G. Pope,65 A.V. Popov,38 C. Potter,5 W.L. Prado da Silva,3 H.B. Prosper,49 S. Protopopescu,73 J. Qian,64 A. Quadt,21 B. Quinn,66 A. Rakitine,42 M.S. Rangel,2 K.J. Rani,28 K. Ranjan,27 P.N. Ratoff,42 P. Renkel,79 S. Reucroft,63 P. Rich,44 M. Rijssenbeek,72 I. Ripp-Baudot,18 F. Rizatdinova,76 S. Robinson,43 R.F. Rodrigues,3 C. Royon,17 P. Rubinov,50 R. Ruchti,55 G. Safronov,36 G. Sajot,13 A. Sánchez-Hernández,32 M.P. Sanders,16 A. Santoro,3 G. Savage,50 L. Sawyer,60 T. Scanlon,43 D. Schaile,24 R.D. Schamberger,72 Y. Scheglov,39 H. Schellman,53 P. Schieferdecker,24 T. Schliephake,25 C. Schmitt,25 C. Schwanenberger,44 A. Schwartzman,68 R. Schwienhorst,65 J. Sekaric,49 S. Sengupta,49 H. Severini,75 E. Shabalina,51 M. Shamim,59 V. Shary,17 A.A. Shchukin,38 R.K. Shivpuri,27 D. Shpakov,50 V. Siccardi,18 V. Simak,9 V. Sirotenko,50 P. Skubic,75 P. Slattery,71 D. Smirnov,55 R.P. Smith,50 G.R. Snow,67 J. Snow,74 S. Snyder,73 S. Söldner-Rembold,44 L. Sonnenschein,16 A. Sopczak,42 M. Sosebee,78 K. Soustruznik,8 M. Souza,2 B. Spurlock,78 J. Stark,13 J. Steele,60 V. Stolin,36 A. Stone,51 D.A. Stoyanova,38 J. Strandberg,64 S. Strandberg,40 M.A. Strang,69 M. Strauss,75 R. Ströhmer,24 D. Strom,53 M. Strovink,46 L. Stutte,50 S. Sumowidagdo,49 P. Svoisky,55 A. Sznajder,3 M. Talby,14 P. Tamburello,45 A. Tanasijczuk,1 W. Taylor,5 P. Telford,44 J. Temple,45 B. Tiller,24 F. Tissandier,12 M. Titov,17 V.V. Tokmenin,35 M. Tomoto,50 T. Toole,61 I. Torchiani,22 T. Trefzger,23 D. Tsybychev,72 B. Tuchming,17 C. Tully,68 P.M. Tuts,70 R. Unalan,65 L. Uvarov,39 S. Uvarov,39 S. Uzunyan,52 B. Vachon,5 P.J. van den Berg,33 B. van Eijk,35 R. Van Kooten,54 W.M. van Leeuwen,33 N. Varelas,51 E.W. Varnes,45 A. Vartapetian,78 I.A. Vasilyev,38 M. Vaupel,25 P. Verdier,19 L.S. Vertogradov,35 M. Verzocchi,50 F. Villeneuve-Seguier,43 P. Vint,43 E. Von Toerne,59 M. Voutilainen,67,‡ M. Vreeswijk,33 R. Wagner,68 H.D. Wahl,49 L. Wang,61 M.H.L.S Wang,50 J. Warchol,55 G. Watts,82 M. Wayne,55 G. Weber,23 M. Weber,50 H. Weerts,65 A. Wenger,22,# N. Wermes,21 M. Wetstein,61 A. White,78 D. Wicke,25 G.W. Wilson,58 S.J. Wimpenny,48 M. Wobisch,60 D.R. Wood,63 T.R. Wyatt,44 Y. Xie,77 S. Yacoob,53 R. Yamada,50 M. Yan,61 T. Yasuda,50 Y.A. Yatsunenko,35 K. Yip,73 H.D. Yoo,77 S.W. Youn,53 C. Yu,13 J. Yu,78 A. Yurkewicz,72 A. Zatserklyaniy,52 C. Zeitnitz,25 D. Zhang,50 T. Zhao,82 B. Zhou,64 J. Zhu,72 M. Zielinski,71 D. Zieminska,54 A. Zieminski,54 L. Zivkovic,70 V. Zutshi,52 and E.G. Zverev37 (DØ Collaboration) 1Universidad de Buenos Aires, Buenos Aires, Argentina 2LAFEX, Centro Brasileiro de Pesquisas F́ısicas, Rio de Janeiro, Brazil 3Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil 4Instituto de F́ısica Teórica, Universidade Estadual Paulista, São Paulo, Brazil 5University of Alberta, Edmonton, Alberta, Canada, Simon Fraser University, Burnaby, British Columbia, Canada, York University, Toronto, Ontario, Canada, and McGill University, Montreal, Quebec, Canada 6University of Science and Technology of China, Hefei, People’s Republic of China 7Universidad de los Andes, Bogotá, Colombia 8Center for Particle Physics, Charles University, Prague, Czech Republic 9Czech Technical University, Prague, Czech Republic 10Center for Particle Physics, Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic 11Universidad San Francisco de Quito, Quito, Ecuador 12Laboratoire de Physique Corpusculaire, IN2P3-CNRS, Université Blaise Pascal, Clermont-Ferrand, France 13Laboratoire de Physique Subatomique et de Cosmologie, IN2P3-CNRS, Universite de Grenoble 1, Grenoble, France 14CPPM, IN2P3-CNRS, Université de la Méditerranée, Marseille, France 15Laboratoire de l’Accélérateur Linéaire, IN2P3-CNRS et Université Paris-Sud, Orsay, France 16LPNHE, IN2P3-CNRS, Universités Paris VI and VII, Paris, France 17DAPNIA/Service de Physique des Particules, CEA, Saclay, France 18IPHC, Université Louis Pasteur et Université de Haute Alsace, CNRS, IN2P3, Strasbourg, France 19IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France and Université de Lyon, Lyon, France 20III. Physikalisches Institut A, RWTH Aachen, Aachen, Germany 21Physikalisches Institut, Universität Bonn, Bonn, Germany 22Physikalisches Institut, Universität Freiburg, Freiburg, Germany 23Institut für Physik, Universität Mainz, Mainz, Germany 24Ludwig-Maximilians-Universität München, München, Germany 25Fachbereich Physik, University of Wuppertal, Wuppertal, Germany 26Panjab University, Chandigarh, India 27Delhi University, Delhi, India 28Tata Institute of Fundamental Research, Mumbai, India 29University College Dublin, Dublin, Ireland 30Korea Detector Laboratory, Korea University, Seoul, Korea 31SungKyunKwan University, Suwon, Korea 32CINVESTAV, Mexico City, Mexico 33FOM-Institute NIKHEF and University of Amsterdam/NIKHEF, Amsterdam, The Netherlands 34Radboud University Nijmegen/NIKHEF, Nijmegen, The Netherlands 35Joint Institute for Nuclear Research, Dubna, Russia 36Institute for Theoretical and Experimental Physics, Moscow, Russia 37Moscow State University, Moscow, Russia 38Institute for High Energy Physics, Protvino, Russia 39Petersburg Nuclear Physics Institute, St. Petersburg, Russia 40Lund University, Lund, Sweden, Royal Institute of Technology and Stockholm University, Stockholm, Sweden, and Uppsala University, Uppsala, Sweden 41Physik Institut der Universität Zürich, Zürich, Switzerland 42Lancaster University, Lancaster, United Kingdom 43Imperial College, London, United Kingdom 44University of Manchester, Manchester, United Kingdom 45University of Arizona, Tucson, Arizona 85721, USA 46Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 47California State University, Fresno, California 93740, USA 48University of California, Riverside, California 92521, USA 49Florida State University, Tallahassee, Florida 32306, USA 50Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA 51University of Illinois at Chicago, Chicago, Illinois 60607, USA 52Northern Illinois University, DeKalb, Illinois 60115, USA 53Northwestern University, Evanston, Illinois 60208, USA 54Indiana University, Bloomington, Indiana 47405, USA 55University of Notre Dame, Notre Dame, Indiana 46556, USA 56Purdue University Calumet, Hammond, Indiana 46323, USA 57Iowa State University, Ames, Iowa 50011, USA 58University of Kansas, Lawrence, Kansas 66045, USA 59Kansas State University, Manhattan, Kansas 66506, USA 60Louisiana Tech University, Ruston, Louisiana 71272, USA 61University of Maryland, College Park, Maryland 20742, USA 62Boston University, Boston, Massachusetts 02215, USA 63Northeastern University, Boston, Massachusetts 02115, USA 64University of Michigan, Ann Arbor, Michigan 48109, USA 65Michigan State University, East Lansing, Michigan 48824, USA 66University of Mississippi, University, Mississippi 38677, USA 67University of Nebraska, Lincoln, Nebraska 68588, USA 68Princeton University, Princeton, New Jersey 08544, USA 69State University of New York, Buffalo, New York 14260, USA 70Columbia University, New York, New York 10027, USA 71University of Rochester, Rochester, New York 14627, USA 72State University of New York, Stony Brook, New York 11794, USA 73Brookhaven National Laboratory, Upton, New York 11973, USA 74Langston University, Langston, Oklahoma 73050, USA 75University of Oklahoma, Norman, Oklahoma 73019, USA 76Oklahoma State University, Stillwater, Oklahoma 74078, USA 77Brown University, Providence, Rhode Island 02912, USA 78University of Texas, Arlington, Texas 76019, USA 79Southern Methodist University, Dallas, Texas 75275, USA 80Rice University, Houston, Texas 77005, USA 81University of Virginia, Charlottesville, Virginia 22901, USA 82University of Washington, Seattle, Washington 98195, USA (Dated: April 16, 2007) We describe a search for the standard model Higgs boson with a mass of 105 GeV/c2 to 145 GeV/c2 in data corresponding to an integrated luminosity of approximately 450 pb−1 collected with the D0 detector at the Fermilab Tevatron pp collider at a center-of-mass energy of 1.96 TeV. The Higgs boson is required to be produced in association with a Z boson, and the Z boson is required to decay to either electrons or muons with the Higgs boson decaying to a bb pair. The data are well described by the expected background, leading to 95% confidence level cross section upper limits σ(pp → ZH)×B(H → bb) in the range of 3.1 pb to 4.4 pb. PACS numbers: 13.85.Ni, 13.85.Qk, 13.85Rm Over the past two decades, increasingly precise exper- imental results have repeatedly validated the standard model (SM) and the relationship between gauge invari- ance and the embedded coupling strengths. For massive W and Z bosons, gauge invariance of the Lagrangian is preserved through the Higgs mechanism, but the Higgs boson (H) has yet to be observed. The current lower bound on the mass of the Higgs boson from direct ex- perimental searches is MH = 114.4 GeV/c 2 at the 95% confidence level [1]. Searches for pp → WH → e(µ)νbb, pp → WH → WWW ∗, and pp → ZH → ννbb have been recently reported [2, 3, 4]. The CDF collabora- tion has reported results in the pp → WH → ℓν and pp → ZH → ℓ+ℓ−bb (ℓ = e, µ) channels with signifi- cantly smaller data sets [5, 6, 7]. This Letter provides the first results from the D0 experiment of searches for a Higgs boson produced in association with a Z boson, which then decays either to an electron pair or to a muon pair. The Higgs is assumed to decay to a bb pair with a branching fraction given by the SM. The Z(→ ℓ+ℓ−)H channels reported in this letter comprise major compo- nents of the search for a Higgs boson at the Tevatron collider. Z bosons are reconstructed and identified through pairs of isolated, electrons or muons with large momen- tum components transverse to the beam direction (pT ) having invariant mass consistent with that of the Z bo- son. Events are required to have exactly two jets iden- tified as arising from b quarks (b jets). The resulting data are examined for the presence of a (H → bb̄) sig- nal in the b-tagged dijet mass distribution. An efficient b-identification algorithm with low misidentification rate and good dijet mass resolution are essential to enhance signal relative to the backgrounds. The analysis of the di- electron [8] (dimuon [9]) channel is based on 450±27 pb−1 (370± 23 pb−1) of data recorded by the D0 experiment between 2002 and 2004. The D0 detector [10, 11] has a central-tracking sys- tem consisting of a silicon microstrip tracker (SMT) and a central fiber tracker (CFT), both located within a ≈ 2 T superconducting solenoidal magnet, with de- signs optimized for tracking and vertexing covering pseudorapidities |η| < 3 and |η| < 2.5, respectively (η = − ln[tan(θ/2)], with θ the polar angle relative to the direction of the proton beam). Central and for- ward preshower detectors are positioned just outside of the superconducting coil. A liquid-argon and uranium calorimeter has a central section (CC) covering pseudo- rapidities up to |η| ≈ 1.1 and two end calorimeters (EC) that extend coverage to |η| ≈ 4.2, with all three housed in separate cryostats [11]. An outer muon system, cover- ing |η| < 2, consists of a layer of tracking detectors and scintillation trigger counters in front of 1.8 T toroids, followed by two similar layers behind the toroids [12]. Luminosity is measured using plastic scintillator arrays placed in front of the EC cryostats [13]. The primary background to the Higgs signal is the as- sociated production of a Z boson with jets arising from gluon radiation, among which Z+bb̄ production is an irre- ducible background. The other background sources are tt̄ production, diboson (ZZ and WZ) production, and events from multijet production that are misidentified as containing Z bosons. The backgrounds are grouped into two categories with the first category, called physics backgrounds, containing events with Z or W bosons aris- ing from SM processes: inclusive Z + bb̄ production, in- clusive Z + jj production in which j is a jet without b flavor, tt̄, ZZ, and WZ events. This background is estimated from simulation as described below. The sec- ond category, called instrumental background, contains those events from multijet production that have two jets misidentified as isolated electrons or muons which ap- pear to arise from the Z boson decay. This background is modeled using control data samples and the procedure described below. Physics backgrounds are simulated using the leading order alpgen [14] and pythia [15] event generators, with the leading order cteq5l [16] used as parton distri- bution functions. The decay and fragmentation of heavy flavor hadrons is done via evtgen [17]. The simulated events are passed through a detailed D0 detector simula- tion program based on geant [18] and are reconstructed using the same software program used to reconstruct the collider data. The ZH signal, for a range of Higgs masses, is also simulated using pythia with the same processing as applied to data. Determination of the instrumental background and the normalization of the physics back- grounds are discussed below. Candidate Z → ee events are selected using a combi- nation of single-electron triggers. Accepted events must have two isolated electromagnetic (EM) clusters recon- structed offline in the calorimeter. Isolation is defined as I = (E (0.4) total − E (0.2) (0.2) in which E (0.4) total is the total calorimeter energy within ∆R < 0.4 of the electron direc- tion and E (0.2) is the energy in the electromagnetic por- tion of the calorimeter within ∆R < 0.2 of the electron direction. Candidate electrons must satisfy I < 0.15. Each EM cluster must have pT > 20 GeV/c and either |ηdet| < 1.1 or 1.5 < |ηdet| < 2.5, where ηdet is the pseu- dorapidity measured relative to the center of the detector, with at least one cluster satisfying |ηdet| < 1.1. In ad- dition, the lateral and longitudinal shower shape of the energy cluster must be consistent with that expected of electrons. At least one of the two EM clusters is also required to have a reconstructed track matching the po- sition of the EM cluster energy. Events with a dielectron mass of 75 < Mee < 105 GeV/c 2 form the Z boson can- didate sample in the dielectron channel. Candidate Z → µ+µ− events are selected using a set of single-muon triggers. Accepted events must have two isolated muons reconstructed offline. The muons must have opposite charge, pT > 15 GeV/c, and |η| < 2.0 with muon trajectories matched to tracks in the central tracking system (i.e., the SMT and the CFT), where the central track must contain at least one SMT measure- ment. In addition, the central tracks are required to have a distance of closest approach to the interaction vertex in the transverse plane smaller than 0.25 cm. Muon iso- lation is based on the sum of the energy measured in the calorimeter around the muon candidate and the sum of the pT of tracks within ∆R = (∆φ)2 + (∆η)2 = 0.5 of the muon candidate normalized by the muon momentum. The distribution of this variable in background multi- hadron events is converted to a probability distribution such that a low probability corresponds to an isolated muon. The product of the probabilities for both muons in an event is computed, and the event is retained if the product is less than 0.02. Accepted Z boson candidates must have the opening angle of the dimuon system in the transverse plane (azimuth) of ∆φ > 0.4, and invari- ant mass 65 GeV/c2 < Mµµ < 115 GeV/c 2. This mass range differs from that of the dielectrons because of the difference in resolutions of electron energies and muon momenta. After selecting the Z candidate events, we define a Z+dijet sample which, in addition to satisfying the Z candidate selection requirements, has at least two jets in each event. Jets are reconstructed from energy in calorimeter towers using the Run II cone algorithm with ∆R = 0.5 [19] with towers defined as non-overlapping, adjacent regions of the calorimeter ∆η ×∆φ = 0.1× 0.1 in size. The transverse momentum of each jet is cor- rected for multiple pp interactions, calorimeter noise, out- of-cone showering in the calorimeter, and energy response of the calorimeter as determined from the transverse mo- mentum imbalance in photon+jet events [20, 21]. Only jets that pass standard quality requirements and satisfy pT > 20 GeV/c and |η| < 2.5 are used in this analy- sis. The quality requirements are based on the pattern of energy deposition within a jet and consistency with the energy deposition measured by the trigger system. For the Z → ee channel, the normalizations of the smaller tt, WZ and ZZ backgrounds are computed using simulated events and next-to-leading-order (NLO) cross sections. Trigger efficiency, electron identification (ID) efficiency and resolution correction factors are derived from comparisons of data control samples and simulated events. The background contributions from Z+jj, Z+bj and Z + bb processes are normalized to the observed Z+dijet data yield reduced by the expected contribu- tions from the smaller physics and instrumental back- grounds. The relative fractions of the Z + jj, Z + bj and Z+bb backgrounds in the Z+dijet sample are determined from the acceptance and selection efficiencies multiplied by the ratios of the NLO cross sections for these processes computed using the mcfm [22] program and the next-to- leading order cteq6m [23] parton distribution functions. For the Z → µ+µ− channel, all physics backgrounds are determined using simulated events with NLO cross sec- tions applied. Trigger efficiency, muon ID efficiency, and resolution correction factors are derived from comparison of data control samples and the simulated events. Instrumental backgrounds in both channels are deter- mined by fitting the dilepton invariant mass distributions to a sum of non-Z and Z boson contributions. The Z boson lineshape is modeled using a Breit-Wigner distri- bution convoluted with a Gaussian representing detector resolution. The non-Z background, consisting of a sum of events from Drell-Yan production and instrumental background, is modeled using exponentials. The ratio of Z boson to non-resonant Drell-Yan production is fixed by the standard model. The (two) jets arising from Higgs boson decay should contain b-flavored quarks (b jets), whereas background from Z+jets has relatively few events with b jets. To im- prove the signal-to-background ratio, two of the jets in the events from the Z+dijet sample are required to ex- hibit properties consistent with those of jets containing b quarks. The same b-jet identification algorithm [24] is used for the dielectron and dimuon samples. It is based on the finite lifetime of b hadrons giving a low probabil- ity that these tracks appear to arise from the primary vertex and considers all central tracks associated with a jet. A small probability corresponds to jets with tracks with large impact parameter, as expected in b hadron decays. The efficiency for tagging a b jet from Higgs de- cay is approximately 50%, determined as described in the next paragraph. The probability of misidentifying a jet arising from a charm quark as a b jet is roughly 20%. The probability to misidentify a jet arising from a light quark (u, d, s) or gluon as a b jet is roughly 4%. This choice of efficiency and purity optimizes the sensitivity of the analysis. The relatively large per-jet light-flavor misidentification rate can be accommodated because two tagged jets are required in each event. For background yields determined from simulated events, the probability as a function of jet pT and η that a jet of a given flavor would be identified (tagged) as a b jet is applied to each jet in an event. The probability functions are derived from control data samples. For jets in the simulated events, the flavor is determined from a priori knowledge of the parton that gives rise to the jet. The probability of having two b-tagged jets is defined by convoluting the per-jet probabilities assuming there are )2Dijet Mass (GeV/c 0 50 100 150 200 250 Background Signal x10 -1, 370-450 pbOD FIG. 1: The dijet invariant mass distribution in double– tagged Z+dijet events. The Higgs signal corresponds to MH = 115 GeV/c 2. (The uncertainties are statistical only.) no jet-to-jet correlations introduced by the b-tag require- ment. The observed number of Z+2 b-jet events and the predicted background levels are shown in Table I. The invariant mass of the two b jets in the Z + 2 b jet sample is shown in Fig. 1. This distribution is searched for an excess of events. The peak position in the dijet mass spectrum is expected to be at a lower value than the hypothetical Higgs mass because the jet energy is corrected to reflect the energy of particles in the jet cone without a general correction for the lower b jet response compared with light jets. If a muon is within ∆R < 0.5 of the jet, then twice the muon momentum is added to the jet momentum. This is an approximation to the energy of both the muon and the accompanying neutrino. The expected contribution from Higgs boson production shown in Fig. 1 corresponds to MH = 115 GeV/c Systematic uncertainties for signal and background arise from a variety of sources, including uncertainties on the trigger efficiency, on the corrections for differences between data and simulation for lepton reconstruction and identification efficiencies, lepton energy resolution, jet reconstruction efficiencies and energy determination, b-identification efficiency, uncertainties from theory and parton distribution functions for cross sections used for simulated events and uncertainties on the method used for instrumental background estimates. The uncertain- ties from these sources are shown in Table II. These are evaluated by varying each of the corrections by ±1σ, by comparing different methods (for the instrumental back- grounds), and by varying the parton distribution func- tions among the 20 error sets provided as part of the cteq6l library. The variations seen for different pro- cesses for a given uncertainty arise because of differences among the various background processes and because of intrinsic differences in the kinematic spectra from differ- ent Higgs mass hypotheses. The observed yield is consistent with background pre- dictions. Upper limits on the ZH production cross sec- tion are derived at 95% confidence level using the CLs method [25], a modified frequentist procedure, with a log- likelihood ratio classifier. The shapes of dijet invariant- mass spectra of the signal and background are used to produce likelihoods that the data are consistent with the background-only hypothesis or with a background plus signal hypothesis. Systematic uncertainties are folded into the likelihoods via Gaussian distribution, with corre- lations maintained throughout. The data yield, predicted backgrounds and expected and observed limits are shown in Table III for five hypothetical Higgs masses. The limits are also shown in Fig. 2. In summary, we have carried out a search for associated ZH production in events having two high-pT electrons or muons and two jets identified as arising from b quarks. Consistency is found between data and background pre- dictions. A 95% confidence level upper limit on the Higgs boson cross section σ(pp → ZH) × B(H → bb) is set between 4.4 pb and 3.1 pb for Higgs bosons with mass between 105 GeV/c2 and 145 GeV/c2, respectively. )2Higgs Mass (GeV/c 100 110 120 130 140 150 210 -1, 370-450 pbOD 95% C.L. upper limit ( --- expected limit) standard model FIG. 2: The expected and observed cross–section limits are shown as a function of Higgs mass. The cross section based on the SM is shown for comparison. We thank the staffs at Fermilab and collaborating in- stitutions, and acknowledge support from the DOE and NSF (USA); CEA and CNRS/IN2P3 (France); FASI, Rosatom and RFBR (Russia); CAPES, CNPq, FAPERJ, FAPESP and FUNDUNESP (Brazil); DAE and DST (India); Colciencias (Colombia); CONACyT (Mexico); KRF and KOSEF (Korea); CONICET and UBACyT (Argentina); FOM (The Netherlands); PPARC (United Kingdom); MSMT and GACR (Czech Republic); CRC Program, CFI, NSERC and WestGrid Project (Canada); BMBF and DFG (Germany); SFI (Ireland); The Swedish Research Council (Sweden); CAS and CNSF (China); Alexander von Humboldt Foundation; and the Marie TABLE I: Number of observed and expected background events. Z+ ≥ 2 jets 2 b tags Final state Z → ee Z → µ+µ− Z → ℓ+ℓ− Z → ee Z → µ+µ− Z → ℓ+ℓ− Zbb 9.1 8.3 17.4 2.0 1.3 3.3 Zjj 414 437 851 1.2 2.6 3.8 tt̄ 2.7 9.6 12.3 0.80 3.1 3.9 ZZ +WZ 9.2 21.4 30.6 0.32 0.42 0.74 Instrumental 28.0 16.1 44.1 0.18 0.41 0.59 Total background 463 493 956 4.5 7.8 12.3 Observed events 463 545 1008 5 10 15 TABLE II: Systematic uncertainty in background and signal predictions given as the fractional uncertainty on the event totals. The ranges correspond to variations introduced by different processes (background), the dijet mass window re- quirement (background and signal) and intrinsic differences in kinematics arising from different hypothesized Higgs masses (signal). Source Background Signal Lepton ID Efficiencies 11% – 16% 11% – 12% Lepton Resolution 2% 2% Jet ID Efficiency 5% – 11% 8% Jet Energy Reconstruction 10% 7% b–jet ID Efficiency 10% – 12% 9% Cross Sections 6% – 19% 7% Trigger Efficiency 1% 1% Instrumental Background 2% (ee) 12% (µµ) Curie Program. [*] Visitor from Augustana College, Sioux Falls, SD, USA. [¶] Visitor from The University of Liverpool, Liverpool, UK. [§] Visitor from ICN-UNAM, Mexico City, Mexico. [‡] Visitor from Helsinki Institute of Physics, Helsinki, Fin- land. [#] Visitor from Universität Zürich, Zürich, Switzerland. [1] The ALEPH, DELPHI, L3 and OPAL Collaborations, Phys. Lett. B 565, 61 (2003). [2] V.M. Abazov, et al. (D0 Collaboration), Phys. Rev. Lett. 94, 091802 (2005). [3] V.M. Abazov, et al. (D0 Collaboration), Phys. Rev. Lett. 97, 151804 (2006). [4] V.M. Abazov, et al. (D0 Collaboration), Phys. Rev. Lett 97, 161803 (2006). [5] F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 79, 3819 (1997). [6] F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 81, 5748 (1998). [7] D. Acosta et al. (CDF Collaboration), Phys. Rev. Lett. 95, 051801 (2005). [8] J.M. Heinmiller, Ph.D. Dissertation, University of Illinois at Chicago, Fermilab-Thesis-2006-30 (2006). [9] H. Dong, Ph.D. Dissertation, Stony Brook University, in preparation. [10] V.M. Abazov, et al. (D0 Collaboration), Nucl. Instrum. and Methods A 565, 463 (2006). [11] S. Abachi et al. (D0 Collaboration) , Nucl. Instrum. Methods A 338, 185 (1994). [12] V.M. Abazov et al. (D0 Collaboration), Nucl. Instrum. and Methods A 552, 372 (2005). [13] T. Andeen et al., FERMILAB-TM-2365-E (2006), in preparation. [14] M.L. Mangano, M. Moretti, F. Piccinini, R. Pittau, and A. Polosa, J. High Energy Phys. 07, 001 (2003). [15] T. Sjöstrand et al., Comput. Phys. Commun. 135, 238 (2001). [16] H.L. Lai et al., Phys. Rev. D 55, 1280 (1997). [17] D.J. Lange, Nucl. Instrum. and Methods A 462, 152 (2001). [18] R. Brun and F. Carminati, CERN Program Library Long Writeup W5013 (1993). [19] G.C. Blazey et al., hep-ex/0005012. [20] V.M. Abazov et al. (D0 Collaboration), hep-ex/0612040, submitted to Phys. Rev. D. [21] V.M. Abazov et al. (D0 Collaboration), hep-ex/0702018, submitted to Phys. Rev. D. [22] J. Campbell and K. Ellis, http://mcfm.fnal.gov/ [23] J. Pumplin et al., J. High Energy Phys. 07, 12 (2002). [24] S. Greder, Ph.D. dissertation, Université Louis Pasteur, Strasbourg, FERMILAB-THESIS-2004-28. [25] T. Junk, Nucl. Instrum. Methods A 434, 435 (1999), A. Read, proceedings of the “1st workshop on Confidence Limits”, edited by L. Lyons, Y. Perrin and F. James, CERN report 2000-005 (2000). http://arxiv.org/abs/hep-ex/0005012 http://arxiv.org/abs/hep-ex/0612040 http://arxiv.org/abs/hep-ex/0702018 http://mcfm.fnal.gov/ TABLE III: Numbers of predicted background and signal events and the observed yield after all selection requirements, including the addition of a dijet mass window. The mass window is centered on the mean of the reconstructed Higgs mass in simulated ZH events and has a width of ±1.5σ in which σ is the result of a gaussian fit to the reconstructed dijet mass distribution. The upper bounds differ slightly between the Z → ee and Z → µ+µ− events because of different resolutions. The window is applied for illustration, showing the yields in the region of highest predicted signal-to-background ratio. Also shown are the expected and observed upper limits on the cross section for the combined analysis at 95% confidence level computed as described in the text (without the mass window, but weighted by the bin-to-bin signal-to-background ratio). MH = 105 GeV/c MH = 115 GeV/c MH = 125 GeV/c ee µµ ee µµ ee µµ Mass window(GeV/c2) [65, 113] [65, 118] [72, 125] [70, 128] [75, 136] [78, 137] Predicted signal 0.07 0.06 0.05 0.05 0.04 0.03 Background 1.4 3.1 1.3 3.1 1.4 2.8 Data 2 3 1 3 1 4 Expected σ95 4.2 pb 4.1 pb 3.4 pb Observed σ95 4.4 pb 4.0 pb 3.3 pb MH = 135 GeV/c MH = 145 GeV/c ee µµ ee µµ Mass window(GeV/c2) [82, 143] [84, 147] [87, 156] [92, 160] Predicted signal 0.027 0.022 0.015 0.01 Background 1.6 2.9 1.6 2.8 Data 1 5 0 6 Expected σ95 2.8 pb 2.6 pb Observed σ95 3.1 pb 3.4 pb References