The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz

The purpose of the ''bootstrap program'' for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU(N) Gross-Neveu...

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Дата:	2006
Автори:	Babujian, H.M., Foerster, A., Karowski, M.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2006
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz / H.M. Babujian, A. Foerster, M. Karowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 54 назв. — англ.

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spelling	irk-123456789-1460852019-02-08T01:23:13Z The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz Babujian, H.M. Foerster, A. Karowski, M. The purpose of the ''bootstrap program'' for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU(N) Gross-Neveu model. The nested off-shell Bethe ansatz for an SU(N) factorizing S-matrix is constructed. We review some previous results on sinh-Gordon form factors and the quantum operator field equation. The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model. 2006 Article The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz / H.M. Babujian, A. Foerster, M. Karowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 54 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81T08; 81T10; 81T40 http://dspace.nbuv.gov.ua/handle/123456789/146085 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The purpose of the ''bootstrap program'' for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU(N) Gross-Neveu model. The nested off-shell Bethe ansatz for an SU(N) factorizing S-matrix is constructed. We review some previous results on sinh-Gordon form factors and the quantum operator field equation. The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model.
format	Article
author	Babujian, H.M. Foerster, A. Karowski, M.
spellingShingle	Babujian, H.M. Foerster, A. Karowski, M. The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Babujian, H.M. Foerster, A. Karowski, M.
author_sort	Babujian, H.M.
title	The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz
title_short	The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz
title_full	The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz
title_fullStr	The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz
title_full_unstemmed	The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz
title_sort	form factor program: a review and new results − the nested su(n) off-shell bethe ansatz
publisher	Інститут математики НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/146085
citation_txt	The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz / H.M. Babujian, A. Foerster, M. Karowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 54 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 082, 16 pages The Form Factor Program: a Review and New Results – the Nested SU(N) Off-Shell Bethe Ansatz? Hratchya M. BABUJIAN †, Angela FOERSTER ‡ and Michael KAROWSKI § † Yerevan Physics Institute, Alikhanian Brothers 2, Yerevan, 375036, Armenia E-mail: babujian@physik.fu-berlin.de ‡ Instituto de F́ısica da UFRGS, Av. Bento Gonçalves 9500, Porto Alegre, RS - Brazil E-mail: angela@if.ufrgs.br § Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany E-mail: karowski@physik.fu-berlin.de URL: http://www.physik.fu-berlin.de/~karowski/ Received September 29, 2006, in final form November 16, 2006; Published online November 23, 2006 Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper082/ Abstract. The purpose of the “bootstrap program” for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU(N) Gross–Neveu model. The nested off-shell Bethe ansatz for an SU(N) factorizing S-matrix is constructed. We review some previous results on sinh-Gordon form factors and the quantum operator field equation. The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model. Key words: integrable quantum field theory; form factors 2000 Mathematics Subject Classification: 81T08; 81T10; 81T40 1 Introduction The bootstrap program to formulate particle physics in terms of the scattering data, i.e. in terms of the S-matrix goes back to Heisenberg [1] and Chew [2]. Remarkably, this approach works very well for integrable quantum field theories in 1+1 dimensions [3, 4, 5, 6, 8, 7]. The program does not start with any classical Lagrangian. Rather it classifies integrable quantum field theoretic models and in addition provides their explicit exact solutions in term of all Wightman functions. We achieve contact with the classical models only, when at the end we compare our exact results with Feynman graph (or other) expansions which are usually based on Lagrangians. However, there is no reason that the resulting quantum field theory is related to a classical Lagrangian. One of the authors (M.K.) et al. [4] formulated the on-shell program i.e. the exact determi- nation of the scattering matrix using the Yang–Baxter equations. The concept of generalized form factors was introduced by one of the authors (M.K.) et al. [8]. In this article consistency equations were formulated which are expected to be satisfied by these quantities. Thereafter this approach was developed further and studied in the context of several explicit models by Smirnov [9] who proposed the form factor equations (i)–(v) (see below) as extensions of similar ?This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The full collection is available at http://www.emis.de/journals/SIGMA/LOR2006.html mailto:babujian@physik.fu-berlin.de mailto:angela@if.ufrgs.br mailto:karowski@physik.fu-berlin.de http://www.physik.fu-berlin.de/~karowski/ http://www.emis.de/journals/SIGMA/2006/Paper082/ http://www.emis.de/journals/SIGMA/LOR2006.html 2 H.M. Babujian, A. Foerster and M. Karowski formulae in the original article [8]. These formulae were then proven by two of the authors et al. [10]. In the present article we apply the form factor program for an SU(N) invariant S-matrix (see [11]). The procedure is similar to that for the Z(N) and A(N−1) cases [12, 13] because the bound state fusions are similar in these three models. However, the algebraic structure of the form factors for the SU(N) model is more intricate, because the S-matrix also describes back- ward scattering. We have to apply the nested “off-shell”1 Bethe ansatz, which was originally formulated by one of the authors (H.B.) [14, 15, 16] to calculate correlation function in WZNW models (see also [17, 18]). Finally the Wightman functions are obtained by taking integrals and sums over intermediate states. The explicit evaluation of all these integrals and sums remains an open challenge for almost all models, except the Ising model [19, 20, 21, 7, 22]. In this article we discuss this problem for the examples of the sinh-Gordon model (see [23]). We investigate the short distance behavior of the two-point Wightman function of the exponentiated field. 2 The “bootstrap program” The ‘bootstrap program’ for integrable quantum field theories in 1 + 1-dimensions provides the solution of a model in term of all its Wightman functions. The result is obtained in three steps: 1. The S-matrix is calculated by means of general properties such as unitarity and crossing, the Yang–Baxter equations (which are a consequence of integrability) and the additional assumption of ‘maximal analyticity’. This means that the two-particle S-matrix is an an- alytic function in the physical plane (of the Mandelstam variable (p1 +p2)2) and possesses only those poles there which are of physical origin. The only input which depends on the model is the assumption of a particle spectrum with an underlining symmetry. Typi- cally there is a correspondence of fundamental representations with multiplets of particles. A classification of all S-matrices obeying the given properties is obtained. 2. Generalized form factors which are matrix elements of local operators out 〈 p′m, . . . , p ′ 1 \|O(x)\| p1, . . . , pn 〉in are calculated by means of the S-matrix. More precisely, the equations (i)–(v) as listed in Section 3 are solved. These equations follow from LSZ-assumptions and again the additional assumption of ‘maximal analyticity’ [10]. 3. The Wightman functions are obtained by inserting a complete set of intermediate states. In particular the two point function for a Hermitian operator O(x) reads 〈 0 \|O(x)O(0)\| 0 〉 = ∞∑ n=0 1 n! ∫ · · · ∫ dp1 · · · dpn (2π)n2ω1 · · · 2ωn × ∣∣∣〈 0 \|O(0)\| p1, . . . , pn 〉in ∣∣∣2 e−ix ∑ pi . Up to now a direct proof that these sums converge exists only for the scaling Ising model [19, 20, 21] and the non-unitary ‘Yang–Lee’ model [24]. Recently, Lechner [25] has shown that models with factorizing S-matrices exist within the framework of algebraic quantum field theory. For the algebraic approach to quantum field theories with factorizing S-matrices see the works of Schroer and Schroer–Wiesbrock [26, 27, 28]. A determinant representation for correlation functions of integrable models has been obtained by Korepin et al. [29, 30]. 1“Off-shell” in the context of the Bethe ansatz means that the spectral parameters in the algebraic Bethe ansatz state are not fixed by Bethe ansatz equations in order to get an eigenstate of a Hamiltonian, but they are integrated over. The Form Factor Program 3 Integrability Integrability in (quantum) field theories means that there exist infinitely many local conservation laws ∂µJ µ L(t, x) = 0 (L = ±1,±3, . . . ). A consequence of such conservation laws in 1+1 dimensions is that there is no particle production and the n-particle S-matrix is a product of 2-particle S-matrices S(n)(p1, . . . , pn) = ∏ i<j Sij(pi, pj). If backward scattering occurs the 2-particle S-matrices will not commute and one has to specify the order. In particular for the 3-particle S-matrix there are two possibilities S(3) = S12S13S23 = S23S13S12 � � � �� @ @ @ @@ • 1 2 3 = � � � �� @ @ @ @@ = � � � �� @ @ @ @@ 1 2 3 1 2 3 which yield the “Yang–Baxter Equation”. The two particle S-matrix is of the form Sβ′α′ α β (θ12) = � ��• @ @@ α β α′β′ where α, β etc denote the type of the particles and the rapidity difference θ12 = θ1 − θ2 > 0 is defined by pi = mi(cosh θi, sinh θi). We also use the short hand notation S12(θ12). It satisfies unitarity S21(θ21)S12(θ12) = 1 : � � @ @ � � @ @ = 1 2 1 2 (1) and crossing S12(θ1 − θ2) = C22̄ S2̄1(θ2 + iπ − θ1)C2̄2 = C11̄ S21̄(θ2 − (θ1 − iπ))C1̄1 (2) � � �� @ @ @@ 1 2 = � � �� A A AA � � 1 2 = � � �� @ @ @@ � 1 2 where C11̄ and C11̄ are charge conjugation matrices. We have introduced the following graphical rule, that a line changing the “time direction” also interchanges particles and anti-particles and changes the rapidity as θ → θ ± iπ Cαβ̄ = �� θ θ − iπ α β̄ , Cαβ̄ = ��θ θ + iπ α β̄ . (3) 4 H.M. Babujian, A. Foerster and M. Karowski Bound states: Let γ be a bound state of particles α and β with mass mγ = √ m2 α +m2 β + 2mαmβ cos η , (0 < η < π). Then the 2 particle S-matrix has a pole such that iRes θ=iη Sβ′α′ αβ (θ) = Γβ′α′ γ Γγ αβ (4) iRes θ=iη � � ��• @ @ @@ α β α′β′ = �� • @@ •��@@ α β γ α′β′ where η is called the fusion angle and Γγ αβ is the ‘bound state intertwiner’ [31, 32]. The bound state S-matrix, that is the scattering matrix of the bound state (12) with a particle 3, is obtained by the “bootstrap equation” [31] S(12)3(θ(12)3) Γ(12) 12 = Γ(12) 12 S13(θ13)S23(θ23) � � @ @ @ �� 1 2 3 (12) • = �� @ @ @@�� 1 2 3 (12) • where we use the usual short hand notation of matrices acting in the spaces corresponding to the particles 1, 2, 3 and (12). Examples of integrable models in 1+1-dimensions (which we will consider in this review) are the sinh-Gordon model defined by the classical field equation ϕ̈(t, x)− ϕ′′(t, x) + α β sinhβϕ(t, x) = 0 and the SU(N) Gross–Neveu model described by the Lagrangian L = ψ̄ iγ∂ ψ + g2 2 ( (ψ̄ψ)2 − (ψ̄γ5ψ)2 ) , where the Fermi fields form an SU(N) multiplet. Further integrable quantum field theories are: scaling ZN -Ising models, nonlinear σ-models, O(N)Gross–Neveu models, Toda models etc. The S-matrix The sinh-Gordon S-matrix is given by the analytic continuation β → iβ of the sin-Gordon breather S-matrix [5] S(θ) = sinh θ + i sinπν sinh θ − i sinπν with − 1 ≤ ν = −β2 8π + β2 ≤ 0. (5) The model has the self-dual point at ν = − 1 2 or β2 = 4π. The Form Factor Program 5 The SU(N) S-matrix: All solutions of U(N)-invariant S-matrix satisfying unitarity, crossing and the Yang–Baxter equation have been obtained in [33]. Following [34, 35], we adopt the view that in the SU(N) Gross–Neveu model, the anti-particles are bound states of N − 1 particles. This implies that there is no particle anti-particle backward scattering and that the SU(N) S-matrix should be given by solution II of [33]. The scattering of the fundamental particles which form a multiplet corresponding to the vector representation of SU(N) is (see [36, 37, 38] and [39] for N = 2) Sδγ αβ(θ) = � � �� @ @ @@ @@I @@I α β γδ p1 p2 p3p4 = δαγδβδ b(θ) + δαδδβγ c(θ) (6) where due to Yang–Baxter c(θ) = −2πi Nθ b(θ) holds and the highest weight amplitude is given as a(θ) = b(θ) + c(θ) = − Γ ( 1− θ 2πi ) Γ ( 1− 1 N + θ 2πi ) Γ ( 1 + θ 2πi ) Γ ( 1− 1 N − θ 2πi ) . (7) There is a bound state pole at θ = iη = 2πi/N in the antisymmetric tensor sector which agrees with Swieca’s [34] picture that the bound state of N − 1 particles is to be identified with the anti-particle. Similar as in the scaling Z(N)-Ising and A(N − 1)-Toda models [12, 13] this will be used to construct the form factors in SU(N) model [11]. 3 Form factors For a local operator O(x) the generalized form factors [8] are defined as FO α1...αn (θ1, . . . , θn) = 〈 0 \| O(0) \| p1, . . . , pn 〉inα1...αn (8) for θ1 > · · · > θn. For other orders of the rapidities they are defined by analytic continuation. The index αi denotes the type of the particle with momentum pi. We also use the short notations FO α (θ) or FO 1...n(θ)2. For the SU(N) Gross–Neveu model α denotes the types of particles belonging to all funda- mental representations of SU(N) with dimension ( N r ) , r = 1, . . . , N − 1. In most formulae we restrict α = 1, . . . , N to the multiplet of the vector representation. Similar as for the S-matrix, ‘maximal analyticity’ for generalized form factors means again that they are meromorphic and all poles in the ‘physical strips’ 0 ≤ Im θi ≤ π have a physical interpretation. Together with the usual LSZ-assumptions [40] of local quantum field theory the following form factor equations can be derived: (i) The Watson’s equations describe the symmetry property under the permutation of both, the variables θi, θj and the spaces i, j = i+ 1 at the same time FO ...ij...(. . . , θi, θj , . . . ) = FO ...ji...(. . . , θj , θi, . . . )Sij(θij) (9) for all possible arrangements of the θ’s. (ii) The crossing relation which implies a periodicity property under the cyclic permutation of the rapidity variables and spaces out,1̄〈 p1 \| O(0) \| p2, . . . , pn 〉in,conn. 2...n = C1̄1σO1 F O 1...n(θ1 + iπ, θ2, . . . , θn) = FO 2...n1(θ2, . . . , θn, θ1 − iπ)C11̄ (10) 2The later means the co-vector in a tensor product space with the components FO α . 6 H.M. Babujian, A. Foerster and M. Karowski where σOα takes into account the statistics of the particle α with respect toO (e.g., σOα = −1 if α and O are both fermionic, these numbers can be more general for anyonic or order and disorder fields, see [13]). For the charge conjugation matrix C1̄1 we refer to (3). (iii) There are poles determined by one-particle states in each sub-channel given by a subset of particles of the state in (8). In particular the function FO α (θ) has a pole at θ12 = iπ such that Res θ12=iπ FO 1...n(θ1, . . . , θn) = 2iC12 F O 3...n(θ3, . . . , θn) ( 1− σO2 S2n · · ·S23 ) . (11) (iv) If there are also bound states in the model the function FO α (θ) has additional poles. If for instance the particles 1 and 2 form a bound state (12), there is a pole at θ12 = iη (0 < η < π) such that Res θ12=η FO 12...n(θ1, θ2, . . . , θn) = FO (12)...n(θ(12), . . . , θn) √ 2Γ(12) 12 (12) where the bound state intertwiner Γ(12) 12 is defined by (4) (see [31, 32]). (v) Naturally, since we are dealing with relativistic quantum field theories we finally have FO 1...n(θ1 + µ, . . . , θn + µ) = esµFO 1...n(θ1, . . . , θn) (13) if the local operator transforms under Lorentz transformations as FO → esµFO where s is the “spin” of O. The properties (i)–(iv) may be depicted as (i) �� O . . . . . . = �� O � � A A. . . . . . (ii) �� O conn. . . . = σO � �� O . . . = � �� O . . . (iii) 1 2i Res θ12=iπ �� O . . . = �� O . . . − σO # !� �� O . . . (iv) 1√ 2 Res θ12=η �� O . . . = �� O��. . . These equations have been proposed by Smirnov [9] as generalizations of equations derived in the original articles [8, 7, 41]. They have been proven [10] by means of the LSZ-assumptions and ‘maximal analyticity’. We will now provide a constructive and systematic way of how to solve the equations (i)–(v) for the co-vector valued function FO 1...n once the scattering matrix is given. 3.1 Two-particle form factors For the two-particle form factors the form factor equations are easily understood. The usual assumptions of local quantum field theory yield 〈 0 \| O(0) \| p1, p2〉in/out = F ( (p1 + p2)2 ± iε ) = F (±θ12) The Form Factor Program 7 where the rapidity difference is defined by p1p2 = m2 cosh θ12. For integrable theories one has particle number conservation which implies (for any eigenstate of the two-particle S-matrix) 〈 0 \| O(0) \| p1, p2〉in = 〈 0 \| O(0) \| p2, p1〉outS (θ12) . Crossing (10) means 〈 p1 \| O(0) \| p2〉 = F (iπ − θ12) where for one-particle states the in- and out-states coincide. Therefore Watson’s equations follow F (θ) = F (−θ)S (θ) , F (iπ − θ) = F (iπ + θ) . (14) For general theories Watson’s [42] equations only hold below the particle production thresholds. However, for integrable theories there is no particle production and therefore they hold for all complex values of θ. It has been shown [8] that these equations together with “maximal analyticity” have a unique solution. As an example we write the sinh-Gordon [8] and the (highest weight) SU(N) form factor function [11] FSHG(θ) = exp ∫ ∞ 0 dt t sinh t ( cosh(1 2 + ν)t cosh 1 2 t − 1 ) cosh t ( 1− θ iπ ) , (15) FSU(N) (θ) = c exp ∞∫ 0 dt t sinh2 t e t N sinh t ( 1− 1 N )( 1− cosh t ( 1− θ iπ )) (16) which are the minimal solution of (14) with SSHG (θ) as given by (5) and SSU(N) (θ) = a (θ) as given by (7), respectively. 3.2 The general form factor formula As usual [8] we split off the minimal part and write the form factor for n particles as FO α1...αn (θ1, . . . , θn) = KO α1...αn (θ) ∏ 1≤i<j≤n F (θij). (17) By means of the following “off-shell Bethe ansatz” for the (co-vector valued) K-function KO α1...αn (θ) = ∫ Cθ dz1 · · · ∫ Cθ dzm h(θ, z) pO(θ, z) Ψα1...αn(θ, z) (18) we transform the complicated form factor equations (i)−(v) into simple ones for the p-functions which are scalar and polynomials in e±zi . The “off-shell Bethe ansatz” state Ψα1...αn(θ, z) is obtained as a product of S-matrix elements and the integration contour Cθ depends on the model (see below for the SU(N)-model). The scalar functions h(θ, z) = n∏ i=1 m∏ j=1 φ(θi − zj) ∏ 1≤i<j≤m τ(zi − zj), (19) τ(z) = 1 φ(z)φ(−z) depend on S(θ) only (see (25) below), i.e. on the S-matrix, whereas the p-function pO(θ, z) depends on the operator. 8 H.M. Babujian, A. Foerster and M. Karowski The SU(N) form factors The form factors for n fundamental particles (of the vector representation of SU(N)) are given by (17)–(19) where the “nested Bethe ansatz” is needed. This means that Ψ is of the form Ψα1...αn(θ, z) = Lβ1...βm(z)Φβ1...βm α1...αn (θ, z) (20) where the indices αi take the values α = 1, . . . , N and the summations run over βi = 2, . . . , N . The “off-shell Bethe ansatz” state Φβ1...βm α1...αn (θ, z) is obtained using the techniques of the algebraic Bethe ansatz as follows. We consider a state with n particles and define the monodromy matrix T1...n,0(θ, θ0) = S10(θ10) · · ·Sn0(θn0) = 1 n 0 . . . as a matrix acting in the tensor product of the “quantum space”, a space of n particles (with respect to their quantum numbers) V 1...n = V 1 ⊗ · · · ⊗ V n and the “auxiliary space” V 0. All vector spaces V i are isomorphic to a space V whose basis vectors label all kinds of particles. Here we consider V ∼= CN as the space of the vector representation of SU(N). The Yang–Baxter algebra relation for the S-matrix yields T1...n,a(θ, θa)T1...n,b(θ, θb)Sab(θa − θb) = Sab(θa − θb)T1...n,b(θ, θb)T1...n,a(θ, θa) (21) 1 n b a b a . . . $ � = 1 n b a b a . . .& � which in turn implies the basic algebraic properties of the sub-matrices A, B, C, D with respect to the auxiliary space defined by T1...n,0(θ, z) ≡ ( A1...n(θ, z) B1...n,β(θ, z) Cβ 1...n(θ, z) Dβ′ 1...n,β(θ, z) ) , 2 ≤ β, β′ ≤ N. (22) The basic Bethe ansatz co-vectors Φ of equation (20) are obtained by an application of the operators C to a “pseudo-vacuum” state Φ β 1...n(θ, z) = Ω1...nC βm 1...n(θ, zm) · · ·Cβ1 1...n(θ, z1). (23) This may be depicted as Φ β α(θ, z) = �& α1 αn β1 βm 1 1 1 1 θ1 θn z1 zm. . . . . . ... with 2 ≤ βi ≤ N, 1 ≤ αi ≤ N. It means that Φ β α(θ, z) is a product of S-matrix elements as given by the picture where at all crossing points of lines there is an S-matrix (6) and the sum over all indices of internal lines is to be taken. The “pseudo-vacuum” is the highest weight co-vector (with weight w = (n, 0, . . . , 0)) Ω1...n = e(1)⊗ · · · ⊗ e(1) The Form Factor Program 9 where the unit vectors e(α) (α = 1, . . . , N) correspond to the particle of type α which belong to the vector representation of SU(N). The pseudo-vacuum basic vector satisfies Ω1...nB β 1...n(θ, z) = 0, Ω1...nA1...n(θ, z) = n∏ i=1 a(θi − z)Ω1...n, (24) Ω1...nD β′ 1...n,β(θ, z) = δβ′ β n∏ i=1 b(θi − z)Ω1...n. The amplitudes of the scattering matrices are given by equation (6). The technique of the nested Bethe ansatz means that for the co-vector valued function Lβ1...βm(z) in (20) one makes the second level Bethe ansatz. This ansatz is of the same form as (18) only that the range of the indices is reduced by 1. Iterating this nesting procedure one finally arrives at a scalar function. The integration contour Cθ depends on the θ and is depicted in Fig. 1. • θn − 2πi b θn − 2πi 1 N • θn �� - • θn + 2πi(1− 1 N ) . . . • θ2 − 2πi b θ2 − 2πi 1 N • θ2 �� - • θ2 + 2πi(1− 1 N ) • θ1 − 2πi b θ1 − 2πi 1 N • θ1 �� - • θ1 + 2πi(1− 1 N ) - �� Figure 1. The integration contour Cθ. The bullets belong to poles of the integrand resulting from a(θi − zj)φ(θi − zj) and the small open circles belong to poles originating from b(θi − zj) and c(θi − zj). Swieca’s picture is that the bound state of N − 1 fundamental particles is to be identified with the anti-particle and lead together with the form factor recursion relations (iii) + (iv) to the equation for the function φ(z) (see [13, 11]) N−2∏ k=0 φ (z + kiη) N−1∏ k=0 F (z + kiη) = 1, η = 2π N (25) with the solution φ(θ) = Γ ( θ 2πi ) Γ ( 1− 1 N − θ 2πi ) . In [11] it is shown that the form factors given by (17) and (18) satisfy the form factor equations (i)–(v) if some simple equations for the p-function are satisfied. We note that form factors of this model were also calculated in [9, 43, 44] using other techniques. Locality Let us discuss a further property of the form factors which is directly related to the nature of the operators, that is locality. In fact, this property touches the very heart and most central 10 H.M. Babujian, A. Foerster and M. Karowski concepts of relativistic quantum field theory, like Einstein causality and Poincaré covariance, which are captured in local field equations and commutation relations. So far we have presented a formulation of a quantum field theory, which starts from a particle picture. It is basically possible to obtain the particle picture from the field formulation by means of the LSZ-reduction formalism. The reverse problem, namely of how to reconstruct the entire field content, or at least part of it, from the scattering theory is in general still an outstanding challenge. Besides this classification issue there remains also the general question if the operators which are related to the solutions of (i)–(v) are genuinely local, meaning that they (anti)-commute for space-like separations with themselves. It has been shown that (i)–(v) (see e.g. [9, 13]) imply in〈φ \| [O(x),O(y)] \|ψ 〉in = 0 for (x− y)2 < 0 (26) for all matrix elements. Note that commutation rules such as (26) hold if the statistics factors in (i)–(v) are trivial, σO = 1. However, for more general statistics factors there are also fermionic, anyonic and even order-disorder commutation rules as for example for the scaling ZN -Ising model (see [13]). Examples of operators and their p-functions for the sinh-Gordon model Since there is no backward scattering the Bethe ansatz in (18) is trivial, the integrations can be performed [45] and the n-particle K-function may be written as KO n (θ) = 1∑ l1=0 · · · 1∑ ln=0 (−1) ∑ li ∏ i<j ( 1 + (li − lj) i sinπν sinh θij ) pOn (θ, l). (27) For several cases the correspondence between local operators and their p-functions have been proposed in [45]3. Here we provide three examples: 1. The normal ordered exponential of the field (see also [48]) O(x) = :eγϕ(x) : ↔ p(θ, l) = ( 2 F (iπ) sinπν )n 2 n∏ i=1 e πν γ β (−1)li , (28) 2. Expanding the last relation with respect to γ one obtains the p-functions for normal ordered powers :ϕ(x)N : in particular for N = 1 :ϕ(x) : ↔ p(θ, l) = πν β ( 2 F (iπ) sinπν )n 2 n∑ i=1 (−1)li , (29) which yields (for n = 1) the ‘wave function renormalization constant’ Zϕ = 〈 0 \|ϕ(0) \|p 〉2 = 8π2ν2 −FSHG(iπ)β2 sinπν (see also [8]). 3. The higher conserved currents Jµ L(x) which are typical for integrable quantum field theories J±L (x) ↔ ±N (JL) n n∑ i=1 e±θi n∑ i=1 eL(θi− iπ 2 (1−(−1)liν)). 3Sinh-Gordon form factors have been presented before in a different form in [46, 47]. The Form Factor Program 11 Quantum sinh-Gordon field operator equation Using (28) and (29) one finds [49, 32, 45] that the quantum sinh-Gordon field equation �ϕ(x) + α β : sinhβϕ : (x) = 0 (30) holds for all matrix elements, if the “bare” mass √ α is related to the renormalized mass by α = m2 πν sinπν (31) where m is the physical mass of the fundamental boson. The result may be checked in pertur- bation theory by Feynman graph expansions. In particular in lowest order the relation between the bare and the renormalized mass (31) had already been calculated in the original article [8]. The result is m2 = α ( 1− 1 6 ( β2 8 )2 +O(β6) ) which agrees with the exact formula above. The factor πν sin πν in (31) modifies the classical equation and has to be considered as a quantum correction. The proof of the quantum field equation (30) can be found in [45]. 4 Wightman functions As the simplest case we consider the two-point function of two local scalar operators O(x) and O′(x) w(x) = 〈 0 \| O(x)O′(0) \| 0 〉. Summation over all intermediate states Inserting a complete set of in-states we may write w(x) = ∞∑ n=0 1 n! ∫ dp1 2π2ω1 · · · ∫ dpn 2π2ωn e−ix(p1+···+pn) × 〈 0 \| O(0) \|p1, . . . , pn 〉in in〈pn, . . . , p1 \| O′(0) \| 0 〉 = ∞∑ n=0 1 n! ∫ dθ1 · · · ∫ dθne −ix ∑ pign(θ). (32) We have introduced the functions gn(θ) = 1 (4π)n 〈 0 \| O(0) \|p1, . . . , pn 〉in in〈pn, . . . , p1 \| O′(0) \| 0 〉 = 1 (4π)nF O(θ1, . . . , θn)FO′ (θn + iπ, . . . , θ1 + iπ). where crossing has been used. In particular we consider exponentials of a scalar bose field O(′)(x) = :eiγ (′)ϕ(x) : where : · · · : means normal ordering with respect to the physical vacuum and which amounts to the condition 〈 0 \| :eiγ(′)ϕ(x) : \| 0 〉 = 1 and therefore g0 = 1 holds. 12 H.M. Babujian, A. Foerster and M. Karowski The Log of the two-point function For specific operators like exponentials of bose fields it might be convenient (see below) to consider a resummation of the sum in (32). For g0 = 1 we may write (see also [50]) w(x) = 1 + ∞∑ n=1 1 n! ∫ dθ1 · · · ∫ dθne −ix ∑ pign(θ) = exp ∞∑ n=1 1 n! ∫ dθ1 · · · ∫ dθne −ix ∑ pihn(θ). It is well known that the functions gn and hn are related by the cummulant formula gI = ∑ I1∪···∪Ik=I hI1 · · ·hIk , where we use the short hand notation gI = gn(θ1, . . . , θn) with I = {1, . . . , n}. The relations for the g’s and the h’s may be depicted with g = �� and h = as� � � � 1 . . . n = 1 . . . n + n∑ i=1 . . i + · · · 1 . . . n Thus as examples g1 = h1, g12 = h12 + h1h2, g123 = h123 + h12h3 + h13h2 + h23h1 + h1h2h3, . . . . . . . . . . . . . . . . . . . . . Due to Lorentz invariance it is sufficient to consider the value x = (−iτ, 0). Let O(x) and O′(x) be scalar operators. Then the functions hn(θ) depend only on the rapidity differences. We use the formula for the modified Bessel function of the third kind i∆+(x) = 〈 0 \|ϕ(x)ϕ(0) \| 0 〉 = 1 4π ∫ dθe−τm cosh θ = 1 2π K0(mτ) to perform one integration lnw(x) = ∞∑ n=1 1 n! ∫ dθ1 · · · ∫ dθne −τm ∑ cosh θihn(θ) = 2 ∞∑ n=1 1 n! ∫ dθ1 · · · ∫ dθn−1hn(θ1, . . . , θn−1, 0)K0(mτξ) with ξ2 = ( n−1∑ i=1 cosh θi + 1 )2 − ( n−1∑ i=1 sinh θi )2 . The Form Factor Program 13 Short distance behavior x → 0 In order to perform the conformal limit of massive models one investigates the short distance behavior (see e.g. [51, 52, 50]). For small τ we use the expansion of the modified Bessel function of the third kind and obtain lnw(x) = −2 ∞∑ n=1 1 n! ∫ dθ1 · · · ∫ dθn−1hn(θ1, . . . , θn−1, 0) × ( lnmτ + ln ξ + γE − ln 2 +O ( τ2 ln τ )) where γE = 0.5772 . . . is Euler’s or Mascheroni’s constant. Therefore the two-point Wightman function has power-like behavior for short distances w(x) ≈ C (mτ)−4∆ for τ → 0 where the dimension is given by ∆ = 1 2 ∞∑ n=1 1 n! ∫ dθ1 · · · ∫ dθn−1hn(θ1, . . . , θn−1,0) in case the integrals exist. This is true for the exponentials of bose fieldsO =:eγϕ(x) : due to the asymptotic behavior for Re θ1 →∞ FO n (θ1, θ2, . . . ) = FO 1 (θ1)FO n−1(θ2, . . . ) +O(e−θ1), gn(θ1, θ2, . . . , θn) = g1gn−1(θ2, . . . , θn) +O(e−\|θ1\|) (see e.g. [45]). Therefore the functions hn satisfy hn(θ) = O(e−\|θi\|) for Re θi → ±∞. This follows when we distinguish the variable θ1 in the relation of the g’s and the h’s above and reorganize the terms on the right hand side as follows gI = ∑ 1∈J⊆I hJgI\J . The constant C is obtained as C = exp ( −2 ∞∑ n=1 1 n! ∫ dθ1 · · · ∫ dθn−1hn(θ1, . . . , θn−1, 0) ( ln 1 2ξ + γE )) and it should be related to the vacuum expectation value G = 〈 0 \|O(x)\| 0 〉C in the ‘conformal normalization’ C = m4∆G−2. Such vacuum expectation value was calculated in [53] for the sine-Gordon model. Example: the sinh-Gordon model The dimension of the exponential of the field O(x) = : eβϕ(x) : for the sinh-Gordon model may be calculated in the 1- and 1+2-particle intermediate state approximation (see Fig. 2) as ∆1+2 = 1 2 ( h1 + 1 2! ∫ dθ h2(θ, 0) + · · · ) = − sinπν πF (iπ) + ( sinπν πF (iπ) )2 ∫ ∞ −∞ dθ (F (θ)F (−θ)− 1) + · · · . 14 H.M. Babujian, A. Foerster and M. Karowski 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 1-particle 1+2-particle ∆(ν) Figure 2. Dimension of an exponential of the field for the sinh-Gordon model: 1- and 1+2-particle intermediate state contributions. The integral may be calculated exactly with the result [54] −π 2 sinπνF 2(iπ)− π cosπν − 1 sinπν + 2 ( 1− πν cosπν sinπν ) . In principle the higher particle intermediate state integrals may also be calculated, however, up to now we were not able to derive a general formula. For the scaling Ising model, however, this is possible. The constant C in the approximation of 1-intermediate states is given as C1 = exp (−2h1 (γE − ln 2)) = exp ( 4 sinπν πF (iπ) (γE − ln 2) ) . We have not calculated the integrals appearing in higher-particle intermediate state contribu- tions. Acknowledgments H.B. was supported partially by the grant Volkswagenstiftung within in the project “Non- perturbative aspects of quantum field theory in various space-time dimensions”. H.B. also acknowledges to ICTP condensed matter group for hospitality, where part of this work was done. A.F. acknowledges support from PRONEX under contract CNPq 66.2002/1998-99 and CNPq (Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico). This work is also supported by the EU network EUCLID, ’Integrable models and applications: from strings to condensed matter’, HPRN-CT-2002-00325. 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[54] Babujian H., Karowski M., Short distance behavior of integrable quantum field theories and exact Wightman functions – summation over all intermediate states, in preparation. http://arxiv.org/abs/math-ph/0105040 http://arxiv.org/abs/math-ph/0112025 http://arxiv.org/abs/hep-th/0204097 http://arxiv.org/abs/hep-th/9211053 http://arxiv.org/abs/hep-th/9707091 http://arxiv.org/abs/hep-th/9909153 http://arxiv.org/abs/hep-th/9611238 1 Introduction 2 The ``bootstrap program'' 3 Form factors 3.1 Two-particle form factors 3.2 The general form factor formula 4 Wightman functions

The form factor program: a review and new results − the nested SU(N) off-shell Bethe ansatz

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