The nested SU(N) off-shell Bethe Ansatz and exact form factors

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1 The nested SUN off-shell Bethe Ansatz and exact form factors Hratchia M. Babujian, Angela Foerster, and Michael Karowski Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 4, 495 Berlin, Germany November, 2006 This work is dedicated to the 75th anniversary of H. Bethe s foundational work on the Heisenberg chain Abstract The form factor equations are solved for an SUN invariant S-matrix under the assumption that the anti-particle is identified with the bound state of N particles. The solution is obtained explicitly in terms of the nested off-shell Bethe ansatz where the contribution from each level is written in terms of multiple contour integrals. PACS:.0.-z;.0.Kk;.55.Ds Keywords: Integrable quantum field theory, Form factors Introduction The Bethe ansatz [], was initially formulated by Bethe 75 years ago to solve the eigenvalue problem for the isotropic Heisenberg model. The approach has found applications in the context of several integrable systems in different areas, such as Statistical Mechanics, Quantum Field Theory, Condensed Matter Physics, Atom and Molecular Physics, among others. The original techniques have been refined into several directions: Lieb and Lininger [2] solved the one-dimensional bose gas problem with δ-function potential using the Bethe ansatz. The 6- vertex model was solved by Lieb [3, 4] with the same technique. C.N. Yang and C.P. Yang [5] proved Bethe s hypothesis for the ground state of the anisotropic Heisenberg spin chain. Due to Yang [7] and Baxter [8] we have Address: Alikhanian Brothers 2, Yerevan, Armenia babujian@physik.fu-berlin.de Address: Instituto de Física da UFRGS, Av. Bento Gonçalves 9500, Porto Alegre, RS - Brazil angela@if.ufrgs.br karowski@physik.fu-berlin.de Yang and Yang decided to honor Bethe s insight by calling his assumption Bethe s hypothesis [6], now usually called Bethe ansatz.

2 INTRODUCTION 2 the fundamental Yang-Baxter equation for the two-particle S-matrix or for the matrix of the Boltzmann weights, which guarantees exact integrability of the system. Subsequently, Faddeev and collaborators [9, 0] formulated these ideas in an elegant algebraic way, known as the algebraic Bethe ansatz. Yang [7] and Sutherland [] generalized the technique of the Bethe ansatz for those cases where the underlying symmetry group is larger than SU2. This method is now called the nested Bethe ansatz. This technique was applied in [2] to derive the spectrum of the chiral SUN Gross-Neveu model [3]. The algebraic nested Bethe ansatz was formulated in [4] for the SUN and in [5] for the O2N symmetric case, respectively. Another generalization of the Bethe ansatz is the off-shell Bethe ansatz 2, which was originally formulated by one of the authors H.B. [6, 7, 8, 9] to calculate correlation function in WZNW models see also [20, 2]. This version of the Bethe ansatz paves the way to an analysis of off-shell quantities and opens up the intriguing possibility to merge the Bethe ansatz and the form factor approach. In this context we point out that recently the form factor program has received renewed interest in connection with condensed matter physics [22, 23, 24] and atomic physics [25]. In particular, applications to Mott insulators and carbon nanotubes [26, 22] doped two-leg ladders [27] and in the field of Bose-Einstein condensates of ultracold atomic and molecular gases [28, 25] have been discussed and in some instances correlation functions have been computed. 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 [29] and Chew [30]. Remarkably, this approach works very well for integrable quantum field theories in + dimensions [3, 32, 33, 34, 35, 36]. One of the present authors M.K. et al. formulated the on-shell program [32] i.e. the exact determination of the scattering matrix using the Yang-Baxter equations and the off-shell program [35] i.e. the exact determination of form factors which are matrix elements of local operators. This approach was developed further and studied in the context of several explicit models by Smirnov [37] who proposed the form factor equations i v see below as extensions of similar formulae in the original article [35]. The formulae were proven by two of the authors H.B. and M.K. et al. [38]. In this article the techniques of the off-shell Bethe ansatz was used to determine the form factors for the sine-gordon model. There, however, the underlying group structure is simple and there was no need to use a nested version of the off-shell Bethe ansatz. In the present article we will focus on the determination of the form factors for an SUN model. The procedure is similar as for the scaling ZN Ising and affine AN Toda models [39, 40] because the bound state structures of these models are similar. However, the algebraic structure of the form factors for the SUN model is more complicated, because the S-matrix possesses backward scattering. Therefore we have to apply a nontrivial algebraic off-shell Bethe ansatz. For N > 2 we have to develop the nested version of this technique see also [4]. 2 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 eigen state of a hamiltonian, but they are integrated over.

3 INTRODUCTION 3 It is expected that the results of this paper apply to the chiral SUN Gross-Neveu model [3, 42, 43, 44]. In a separate article [45] we will investigate these physical applications and compare our exact results with two different /N-expansions of the chiral Gross-Neveu model [42] and [44]. We note that SUN form factors were also calculated in [37, 46, 47] using other techniques, see also the related paper [48].. The SUN S-matrix The general solutions of the Yang-Baxter equations, unitarity and crossing relations for a UN invariant S-matrix have been obtained in [49]. The S-matrix for the scattering of two particles belonging to the vector representation of SUN can be written as S δγ αβ θ = δ αγδ βδ bθ + δ αδ δ βγ cθ. Unitarity reads as S i θs i θ = for the S-matrix eigenvalues S + θ = bθ + cθ, S θ = bθ cθ. The amplitude S + θ = aθ is the highest weight w = 2, 0,, 0 S-matrix eigenvalue for the two particle scattering. It will be essential for the Bethe ansatz below. As usual in this context we use in the notation v...n V...n = V V n for a vector in a tensor product space. The vector components are denoted by v α = v α...α n. Below we will also use co-vectors v...n V...n the dual of V...n with components v α. A linear operator connecting two such spaces with matrix elements A α...α n α...α n is denoted by A...n...n : V...n V...n where we omit the upper indices if they are obvious. All vector spaces V i are isomorphic to a space V whose basis vectors label all kinds of particles e.g. V = C N for the vector representation of SUN. The vector spaces V i is associated to a rapidity variable θ i. An S-matrix such as S ij θ ij = S ji ij θ i θ j acts nontrivially only on the factors V i V j and exchanges these factors. Using this notation, the Yang-Baxter relation writes as S 2 θ 2 S 3 θ 3 S 23 θ 23 = S 23 θ 23 S 3 θ 3 S 2 θ 2 2 and implies here the relation between the amplitudes [49] cθ = iη θ bθ, η = 2π N. A solution [49, 42, 50] of all these equations writes as aθ = bθ + cθ = Γ θ 2πi Γ Γ + θ 2πi N + θ 2πi Γ N θ 2πi. 3

4 INTRODUCTION 4 This S-matrix possesses a bound state pole in S θ i.e. in the anti-symmetric tensor channel. It is consistent with Swieca s [5, 50, 44] picture that the antiparticle is a bound state of N particles see also [39, 40]. For later convenience and in order to simplify the formulae we introduce Sθ = Sθ aθ = θ Piη θ iη 4 where is the unit, P the permutation operator. We depict this matrix as S δγ αβ θ 2 = and the amplitudes are explicitly δ γ = δ αγ δ βδ bθ2 + δ αδ δ βγ cθ 2 θ α θ 2 β bθ =.2 Generalized Form factors θ iη, cθ = θ iη θ iη. For a state of n particles of kind α i with rapidities θ i and a local operator Ox we define the form factor functions F O α...α n θ,, θ n, or using a short hand notation F O α θ, by 0 Ox θ,, θ n in α = e ixp + +p n F O α θ, for θ > > θ n. 5 where α = α,, α n and θ = θ,, θ n. For all other arrangements of the rapidities the functions Fα O θ are given by analytic continuation. Note that the physical value of the form factor, i.e. the left hand side of 5, is given for ordered rapidities as indicated above and the statistics of the particles. The Fα O θ are considered as the components of a co-vector valued function F O...n θ V...n = V...n which may be depicted as F O...nθ = O θ. 6 θ n Now we formulate the main properties of form factors in terms of the functions F...n O. They follow from general LSZ-assumptions and maximal analyticity, which means that F...n O θ is a meromorphic function with respect to all θ s and in the physical strips 0 < Im θ ij < π θ ij = θ i θ j i < j there are only poles of physical origin as for example bound state poles. The generalized form factor functions satisfy the following Form factor equations: The co-vector valued auxiliary function F...n O θ is meromorphic in all variables θ,, θ n and satisfies the following relations:

5 INTRODUCTION 5 i The Watson s equations describe the symmetry property under the permutation of both, the variables θ i, θ j and the spaces i, j = i + at the same time F O...ij..., θ i, θ j, = F O...ji..., θ j, θ i, S ij θ ij 7 for all possible arrangements of the θ s. ii The crossing relation implies a periodicity property under the cyclic permutation of the rapidity variables and spaces out, p O0 p 2,, p n in,conn. 2...n = F O...nθ + iπ, θ 2,, θ n σ O C = F O 2...nθ 2,, θ n, θ iπc 8 where σα O takes into account the statistics of the particle α with respect to O. The charge conjugation matrix C will be discussed below. iii There are poles determined by one-particle states in each sub-channel given by a subset of particles of the state in 5. In particular the function F O α θ has a pole at θ 2 = iπ such that Res θ 2 =iπ F O...nθ,, θ n = 2i C 2 F O 3...nθ 3,, θ n σ O 2 S 2n S iv If there are also bound states in the model the function F O α θ has additional poles. If for instance the particles and 2 form a bound state 2, there is a pole at θ 2 = iη, 0 < η < π such that Res F 2...nθ O, θ 2,, θ n = F2...n O θ 2,, θ n 2Γ θ 2 =iη where the bound state intertwiner Γ 2 2 and the values of θ, θ 2, θ 2 and η are given in [52, 53]. v Naturally, since we are dealing with relativistic quantum field theories we finally have F O...nθ + µ,, θ n + µ = e sµ F O...nθ,, θ n if the local operator transforms under Lorentz transformations as O e sµ O where s is the spin of O.

6 INTRODUCTION 6 The property i - iv may be depicted as i ii iii iv O conn. = 2i Res θ 2 =iπ 2 Res θ 2 =iη O O = O = O O = O O Ọ O.. = where denotes the statistics factor σ O. As was shown in [38] the properties i iii follow from general LSZ-assumptions and maximal analyticity. We will now provide a constructive and systematic way of how to solve the form factor equations i v for the co-vector valued function F...n O, once the scattering matrix is given. Minimal form factor: The solutions of Watson s and the crossing equations i and ii for two particles with no poles in the physical strip 0 Im θ π and at most a simple zero at θ = 0 are the minimal form factors. In particular those for highest weight states are essential for the construction of the off-shell Bethe ansatz. One easily finds the minimal solution of using 3 as F θ = c exp 0 F θ = a θ F θ = F 2πi θ dt t sinh 2 t e t N sinh t cosh t θ. 2 N iπ It belongs to the highest weight w = 2, 0,, 0. We define the corresponding Jost-function as for the ZN models [39, 40] by the equation N 2 k=0 N φ θ + kiη k=0 F θ + kiη =, η = 2π N 3 which is typical for models where the bound state of N particles is the anti-particle [40]. The solution is θ φθ = Γ Γ 2πi N θ 4 2πi

7 INTRODUCTION 7 and satisfies the relations φθ = φ θa θ = φn iη θ = aθ 2πi φ2πi θ = φθ 2πi. 5 bθ bθ Notice that the equations 4 and 3 also determine the normalization constant c in 2 as c = Γ 2 /N 2 exp e sinh N t N t N 2N t sinh 2 dt. t t sinh t 0 Generalized form factors: The co-vector valued function 6 for n- particles can be written as [35] F...nθ O = K...nθ O F θ ij 6 i<j n where F θ is the minimal form factor function 2. The K-function K O...n θ contains the entire pole structure and its symmetry is determined by the form factor equations i and ii where the S-matrix is replaced by Sθ = Sθ/aθ K Ọ..ij..., θ i, θ j, = K Ọ..ji..., θ j, θ i, S ij θ ij 7 K O...nθ + iπ, θ 2,, θ n σ O C = K O 2...nθ 2,, θ n, θ iπc for all possible arrangements of the θ s..3 Nested off-shell Bethe ansatz for SUN 8 We consider a state with n particles and define as usual in the context of the algebraic Bethe ansatz [9, 0] the monodromy matrix T...n,0 θ, θ 0 = S 0 θ 0 S n0 θ n0 = n 0. 9 It is a matrix acting in the tensor product of the quantum space V...n = V 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 = C N as the space of the vector representation of SUN. The Yang-Baxter algebra relation for the S-matrix 2 yields T...n,a θ, θ a T...n,b θ, θ b S ab θ a θ b = S ab θ a θ b T...n,b θ, θ b T...n,a θ, θ a 20 a a b b b = b a a n n

8 INTRODUCTION 8 which implies the basic algebraic properties of the sub-matrices A, B, C, D with respect to the auxiliary space defined by Ã...n θ, z B...n,β θ, z T...n,0 θ, z C β...n θ, z Dβ, 2 β, β N. 2...n,β θ, z We propose the following ansatz for the general form factor F...n O θ or the K- function defined by 6 in terms of a nested off-shell Bethe ansatz and written as a multiple contour integral Kα O θ = N n dz m! C θ R dz m C θ R hθ, z p O θ, z Ψ α θ, z 22 where hθ, z is a scalar function which depends only on the S-matrix and not on the specific operator Ox hθ, z = τz = n i= j= m φθ i z j i<j m τz i z j 23 φθφ θ, φθ = φθaθ = φ θ. 24 For the SUN S-matrix the function φθ is given by 3 with the solution 4. The integration contour C θ is depicted in Fig.. The constant R is defined by R = θ dz φθ z where the integration contour is a small circle around z = θ as part of C θ. θ n + 2πi N θ 2 + 2πi N θ + 2πi N θ n θ n 2πi N θ 2 θ 2 2πi N θ θ 2πi N θ n 2πi θ 2 2πi θ 2πi Figure : The integration contour C θ. The bullets refer to poles of the integrand resulting from aθ i z j φθ i z j and the small open circles refer to poles originating from bθ i z j and cθ i z j. The dependence on the specific operator Ox is encoded in the scalar p- function p O θ, z which is in general a simple function of e θ i and e z j see below.

9 INTRODUCTION 9 By means of the ansatz 6 and 22 we have transformed the complicated form factor equations i - v which are in general matrix equations into much simpler scalar equations for the p-function see below. The K-function is in general a linear combination of the fundamental building blocks [54, 39, 40] given by We consider here cases where the sum consists only of one term. If in 2 the range of β s is non-trivial, i.e. if N > 2 the Bethe ansatz co-vectors are of the form Ψ α θ, z = L β z Φ β αθ, z 25 where summation over all β = β,, β m with β i > is assumed. The basic Bethe ansatz co-vectors Φ β...n V...n are defined as Φ β...n θ, z = Ω...n Φ β αθ, z = β C βm...n θ, z m C β...n θ, z β m z m z θ θ n., 2 β i N α i N. 26 α α n Here the pseudo-vacuum is the highest weight co-vector with weight w = n, 0,, 0 Ω...n = e e where the unit vectors eα α =,, N correspond to the particle of type α which belong to the vector representation of SUN. The pseudo-vacuum vector satisfies Ω...n Bβ...n θ, z = 0 Ω...n Ã...n θ, z = Ω...n Ω...n Dβ...n,β θ, z = δβ β n bθi zω...n. i= 27 The amplitudes of the scattering matrices are given by eqs. and 3. The technique of the nested Bethe ansatz means that one makes for the coefficients L β z in 25 the analogous construction as for K α θ where now the indices β take only the values 2 β i N. This nesting is repeated until the space of the coefficients becomes one dimensional. In this article we will focus on the determination of the form factors for an SUN S-matrix. The paper is organized as follows: In Section 2 we construct the general form factor formula for the simplest SU2 case. In Section 3 we construct the general form factor formula for the SU3 case, which is more complex due to the presence of the nesting procedure more explicitly, we have here two levels. We extend these results in Section 4, where the general form factors for SUN are constructed and discussed in detail.

10 2 SU2 FORM FACTORS 0 2 SU2 form factors In this section we start with the simplest case. We perform the form factor program for the SU2 S-matrix. The results should apply to the well-known SU2 Gross-Neveu model [3], investigated by Bethe ansatz methods in [55, 56]. From the technical point of view the calculation of the form factors is very similar to the one done for the sine-gordon model in [38, 53]. S-matrix: The SU2 S matrix 3 is given by and 3 for N = 2. It turns out that the amplitudes satisfy the relations bθ = aiπ θ and cθ = ciπ θ which may be written as S δγ αβ δ θ = αδ γ β δ biπ θ + ɛδγ ɛ αβ ciπ θ. 28 We have introduced ɛ αβ = α β, α β ɛαβ = which are antisymmetric and ɛ 2 = ɛ 2 = such that ɛ αβ ɛ βγ = δ γ α : = and ɛ αβ A α γ ɛ γβ = A = tr A 29 for a matrix A. Formula 28 may be understood as an unusual crossing relation c.f. [5] S δγ αβ θ = C αα Sα δ βγ iπ θcγ γ = 30 if we define the charge conjugation matrices as C αβ = ɛ αβ and C αβ = ɛ αβ. This means that particle is the anti-particle of 2 and vice versa. For θ 0 and iπ the S-matrix yield the permutation matrix and the annihilation-creation operator, respectively, or in terms of S = S/a S δγ αβ 0 = δδ αδ γ β = δ α γ β, Res S δγ θ=iπ αβ θ = iπɛδγ ɛ αβ = iπ δ γ. α β ansatz for form factors: in components as Because of 29 the crossing equation 8 writes K O α...α n θ, θ 2,, θ n σ O = K O α 2...α nα θ 2,, θ n, θ 2πi 3 3 It is related to the sine-gordon S-matrix for β 2 = 8π by a, b, c SU2 = a, b, c S.G. see e.g. [38].

11 2 SU2 FORM FACTORS The form factor equation iii here reads as Res F...nθ O,, θ n = 2i ɛ 2 F3...nθ O 3,, θ n σ O 2 S 2n S θ 2 =iπ Using the crossing relation 30 it turns out that the statistics factor in 8 and 32 has to satisfy σ O 2 = n with the solutions σ O = i Q O for Q O = n mod We define the SU2 Jost-function φθ and τz by the same equations as for the sine-gordon [38] and the Z2 models [39, 40] φθf θf θ + iπ = 34 τzφzφ z = 35 with the solution for φθ given by 4. The functions φ z and τz are for SU2 explicitly given as φz = φ z z = Γ Γ 2πi 2 z, τz = z sinh z 2πi 4π 3. The scalar function hθ, z in 22 encodes only data from the scattering matrix. The p-function p O θ, z on the other hand depends on the explicit nature of the local operator O. It is analytic in all variables and in order that the form factors satisfy i, ii and iii of eqs. 7-9 p nm θ, z has to satisfy i 2 p nm θ, z is symmetric under θ i θ j and z i z j ii 2 { σpnm θ + 2πi, θ 2,, z = m p nm θ, θ 2,, z p nm θ, z + 2πi, z 2, = n p nm θ, z, z 2, iii 2 if θ 2 = iπ : p nm θ, z z =θ = m σp nm θ, z z =θ 2 = σp n 2m θ 3,, θ n, z 2,, z m + p 36 where σ is the statistics factor of 3 and 32. In order to simplify the notation we have suppressed the dependence of the p-function p O and the statistics factor σ O on the operator Ox. By means of the ansatz 6 and we have transformed the complicated form factor equations i - iii to the simple ones i 2 - iii 2 for the p-function. Theorem The co-vector valued function F O α θ given by the ansatz 6 and 22 satisfies the form factor equations i, ii and iii see 7-9 if the p-function p O θ, z satisfies the equations i 2, ii 2, iii 2 of 36 and the normalization constants satisfies the recursion relation iπ φiπf iπn n = 2iN n 2. The proof of this theorem for SU2 is similar to that for the sine-gordon model in [38] and may easily obtained from the ones for SU3 or SUN established below.

12 3 SU3 FORM FACTORS 2 3 SU3 form factors In this section we construct the form factors for the SU3 model, which corresponds to the simplest example where the nested Bethe ansatz technique has to be applied together with the off-shell Bethe ansatz. In comparison to the previous SU2 case, there are additional properties to be obeyed by the second-level Bethe ansatz function See lemma 3 below for details. 3. S-matrix The SU3 S-matrix is given by and 3 for N = 3. The eigenvalue S has a pole at θ = iη = 2 3iπ which means that there exist bound states of two fundamental particles α + β ρσ with ρ < σ 3 which transform as the anti-symmetric SU3 tensor representation. The general bound state S- matrix formula [52, 53] for the scattering of a bound state with another particle reads in particular for the SU3 case as S γ ρ σ ρσγ θ 23 Γ ρσ αβ = σ Γρ α β S γ α αγ θ 3 S γ β βγ θ θ2 =iη = where θ 2 is the bound state rapidity and η the bound state fusion angle. The bound state fusion intertwiner Γ ρσ αβ is defined by i Res α θ=iη Sβ αβ β α θ = Γ β α ρσ Γρσ αβ = ρσ. ρ<σ α β With a convenient choice of an undetermined phase factor one obtains Γ ρσ αβ δ = Γ αδ ρ β σ δσ αδ ρ β, Γ β α ρσ = Γ δρ β δσ α δσ β δρ α 38 where Γ = i iaiηiη is a number. external particles we calculate Choosing in 37 special cases for the which may be written as S θ = bθ + 3 iπbθ 3iπ = aiπ θ S θ = bθ + 3 iπaθ 3iπ = bπi θ S θ = bθ + 3 iπcθ 3iπ = cπi θ S γ α β αβγ θ = δ γ γ δ α β αβ bπi θ + ɛγ α β ɛ αβγ cπi θ

13 3 SU3 FORM FACTORS 3 with the total anti-symmetric tensors ɛ αβγ and ɛ αγβ ɛ 23 = ɛ 32 =. These formulae may be again understood as a crossing relation c.f. [5] ɛ α β δs γ α αγ θ + β 2 iηsγ βγ θ 2 iη = ɛ γ αβδ Sδ γδ iπ θ 39 γ = γ α β γ δ α β γ δ or S γ α β αβγ θ = C αβδ S δ γ γδ iπ θcδα β if we write the charge conjugation matrices as C αβγ = C αβγ = ɛ αβγ, C αβγ = C αβγ = ɛ αβγ. 40 Therefore we have the relations c.f. 29 and C αρσ C ρσβ = δ β α, C αρσ A α β Cβρσ = tr A 4 C αρσ Γ ρσ βγ = C ρσγ Γ ρσ αβ = ɛ αβγγ. 42 These results 4 are consistent with the picture that the bound state of particles and 2 is to be identified with the anti-particle of 3. For later convenience we consider the total 3-particle S-matrix in the neighborhood of its poles at θ 2, θ 23 = iη 3.2 Form factors S γ β α αβγ θ, θ 2, θ 3 = S β α α β θ 2 S γ α αγ θ 3 S γ β βγ θ 23 iη iη β 2 α θ 2 iη θ 23 iη ɛγ ɛ αβγ. 43 In order to obtain a recursion relation where only form factors for the fundamental particles of type α =, 2, 3 which transform as the SU3 vector representation are involved, we have to apply the bound state relation iv to get the anti-particle and then the creation annihilation equation iii Res F 23...nθ O, θ 2, θ 3,, θ n = F23...n O θ 2, θ 3,, θ n 2Γ 2 2 θ 2 =iη Res F 23...n O θ 2, θ 3,, θ n = 2iC 23 F4...nθ O 4,, θ n θ 23 =iπ σ O 3 S 3n S 34 where Γ 2 2 is the bound state intertwiner 38 see [52, 53], θ 2 = 2 θ + θ 2 is the bound state rapidity, η = 2 3 π is the bound state fusion angle and C 23 is the charge conjugation matrix 40. We obtain with the short notation ˇθ = θ 4,, θ n Res Res F 23...nθ O = 2iε 23 2ΓF O 4...n ˇθ σ O 3 S 3n S θ 23 =iη θ 2 =iη where 42 has been used. 4 The physical aspects of these facts will we discussed elsewhere [45].

14 3 SU3 FORM FACTORS 4 ansatz for form factors: We propose that the n-particle form factor of an operator Ox is given by the same formula 6 as for SU2 where the form factor equations i and ii for K-function write again as 7 and 8. Consistency of 44, 8 and the crossing relation 39 means that the statistics factors are of the form σα O = σ O r α if the particle of type α belongs to a SU3 representation of rank r α =, 2 and σ O r = e iπ 2 3 rq O for Q O = n mod 3 45 as an extension of 33. As for SU2, we propose again for the K-function the ansatz in form of the integral representation 22 with 23. However, the Bethe ansatz co-vector is here for SU3 of the form Ψ α θ, z = L β z Φ β αθ, z 46 where the basic Bethe states Φ β αθ, z are given by 26 and the indices α i run over α i 3 and the β i over 2 β i 3. For the function L β z one makes an analogous ansatz 22 as for K α θ where the indices run here over a set with one element less. By this procedure one obtains the nested Bethe ansatz. The next level function L β z is assumed to satisfy i L...ij..., z i, z j, = L...ji..., z j, z i, S ij z ij 47 ii C L...m z + iπ, z 2,, z m = L 2...m z 2,, z m, z iπc 48 iii there is a pole at z 2 = iη such that with c = φiη. m Res L βz = c z 2 =iη i=3 φz i2 Res S β 32 z 2 =iη β 2 z 2 L ˇβž 49 These properties of the next level Bethe ansatz function L β z are discussed in lemma 3. The minimal two particle form factor function F θ = c exp 0 dt t sinh 2 t e t 2 3 sinh 3 t cosh t θ iπ 50 belongs to the highest weight w = 2, 0, 0. The SU3 and the Z3 model [39, 40] possess the same bound state structure, namely the anti-particle is to be identified with the bound state of two particles. Therefore we define the SU3 Jost-function φ z by the same equation as for the Z3 model φθφθ + iηf θf θ + iηf θ + 2iη = 5

15 3 SU3 FORM FACTORS 5 with the solution z 2 φz = Γ Γ 2πi 3 z. 2πi These equations determine also the constant c in 50. The function τz is again defined by 24. The function hθ, z is scalar and encodes only data from the scattering matrix. The p-function p O θ, z on the other hand depends on the explicit nature of the local operator O. It is analytic in all variables and, in order that the form factors satisfy i - iii of eqs. 7-9, we assume i 3 pθ, z is symmetric under θ i θ j { ii σpθ + 2πi, θ 2,, z = m pθ, θ 2,, z 3 pθ, z + 2πi, z 2, { = n pθ, z, z 2, pθ; iii θ2, θ 3 for θ 2 = θ 23 = iη 3, ž = m pˇθ, ž pθ; θ, θ 2, ž = σ pˇθ, ž 52 with ˇθ = θ 4,, θ n, ž = z 3,, z m. In ii 3 and iii 3 σ is the statistics factor of the operator O with respect to the fundamental particle belonging to the vector representation of SU3. Theorem 2 Let the co-vector valued function F O α θ be defined by the ansatz 6, 22, 23 and 46. Let the p-function satisfy i 3, ii 3 and iii 3 of 52 and let the function L β z satisfy i, ii and iii of Let the normalization constants satisfy the recursion relation 2iη 2 φ2 iη φ2iηf 2 iηf 2iηN n = 2i 2ΓN n Then the function F O α θ satisfies the form factor equations i, ii and iii see 7-9. In particular 44 is satisfied. Proof. Property i in the form of 7 follows directly from i 3, the Yang-Baxter equations and the action of the S-matrix on the pseudo-ground state Ω Ω...ji... C...ji... θ j, θ i S ij θ ij = Ω...ji... Sij θ ij C...ij... θ i, θ j S = Ω...ij... C...ij...θ i, θ j. because θ = S θ/aθ = and F θ = F θaθ. Using i and 4 the property ii in the form of 8 may be rewritten as a matrix difference equation [57, 58, 4] K...n θ σ = K...n θ Q...n θ 54 where θ = θ + 2πi, θ 2, θ n and σ is a statistics factor. The matrix Qθ = Qθ, is a special case for i = of the trace Q...n θ, i = tr 0 TQ,...n,0 θ, i = θ i θ θ i θ n α α i α n α α i α n

16 3 SU3 FORM FACTORS 6 of a modified monodromy matrix T Q,...n,0 θ, i = S 0 θ θ i P i0 S n0 θ n θ i where P is the permutation matrix. For the special case i = the matrix Qθ, = Qθ may be written as a trace of the ordinary monodromy matrix 9 over the auxiliary space for the specific value of the spectral parameter θ 0 = θ Q...n θ = tr 0 T...n,0 θ, θ 55 because S0 = P. The difference equation 54 may be depicted as K K σ = where we use the rule that the rapidity of a line changes by 2πi if the line bends by in the positive sense. In the following we will suppress the indices n. The Yang-Baxter relations 20 imply the typical commutation rules for the matrices Ã, C, D defined in eq. 2 C β θ, zãθ, θ = Ãθ, θ bθ C β cθ z θ, z Ãθ, z z bθ C β θ, θ 56 z C β θ, z γ D γ θ, θ = Sγ β bz θ β γ z θ D γ γ θ, θ C β θ, z cz θ bz θ Dβ γ θ, z C γ θ, θ where β, β, γ, γ, γ {2, 3}. In addition there are the Zapletal commutation rules [57, 58, 4] where also the matrices A Q, C Q, D Q defined by ÃQ θ Bβ Q T Q θ = θ C β Q θ β DQ β θ are involved [57] C β θ, zãqθ = Ã bθ Q θ C β θ, z cθ z Ãθ, z z bθ C β Q θ 57 z C β θ, z D γ Q γ θ = S γ β bz θ β γ z θ D γ Q γ θ C β θ, z cz θ D γ bz β θ, z θ γ C Q θ. 58 Note that we assign to the auxiliary space of T Q θ corresponding to the horizontal line the spectral parameter θ on the right hand side and θ = θ + 2πi on the left hand side.

17 3 SU3 FORM FACTORS 7 We are now going to prove 54 in the form 3 β Kθ Ã Q θ + D Q β θ = Kθ Qθ = Kθ σ 59 β=2 where Kθ is a co-vector valued function as given by eq. 22 and the Bethe ansatz state 26. To analyze the left hand side of eq. 59 we proceed as follows: We apply the trace of T Q which is ÃQ + 3 D β β=2 Q β to the co-vector Kθ Ω C βm θ, z m C β θ, z T Q γ γ θ = β β m γ θ θ 2 z m. In the contribution from ÃQθ which means γ = γ = one may use Yang- Baxter relations to observe that only the amplitudes S θ z j = appear in the S-matrices Sθ z j which are constituents of the C-operators. Therefore we may shift all z j -integration contours C θ to C θ without changing the values of the integrals, because there are no singularities inside C θ C θ c.f. Fig.. Using a short notation we have Kθ ÃQθ = dz hθ, zpθ, z Ψθ, z ÃQθ C θ with C θ dz = dz m! C θ R dz m C θ R. We now proceed as usual in the algebraic Bethe ansatz and push the ÃQθ and D Q θ through all the C-operators using the commutation rules 57 and 58 and obtain C βm θ, z m C β θ, z ÃQθ = m j= Ã bθ Q θ z j z θ θ n γ C βm θ, z m C β θ, z + uw A,. C βm θ, z m C β θ, z D Q β β θ = m j= β β T bzj θ β β z, θ D β Q β θ C β mθ, z m C β θ, z + uw D. The wanted terms written out explicitly originate from the first term in the commutations rules 57; all other contributions yield the so-called unwanted terms. The next level monodromy matrix is T β β β β z, θ = S0 z θ S m0 z m θ β β β β

18 3 SU3 FORM FACTORS 8 where the β s and also the internal summation indices take the values 2, 3. If we insert these equations into the representation 22 of Kθ we first find that the wanted contribution from ÃQ already gives the result we are looking for. Secondly the wanted contribution from D Q applied to Ω gives zero. Thirdly the unwanted contributions from ÃQ and D Q cancel after integration over the z j. All these three facts can be seen as follows. We have Ω ÃQθ = Ω, Ω D β Q β θ = δβ β n bθi θ Ω = 0 60 which follow from eq. 27. Therefore the wanted term from ÃQ yields [ ] w m Kθ ÃQθ = dz hθ, zpθ, z C θ bθ Ψθ, z j= z j = σ dz hθ, zpθ, z Ψθ, z C θ = σkθ. It has been used that the shift relation of the function φθ in 5 and ii of 52 for the function p imply that m hθ, zpθ, z = σ hθ bθ, zpθ, z. z j j= The wanted contribution from D Q vanish, since b0 = 0. Therefore it remains to prove that the unwanted terms cancel. The commutation relations 57 and 58 imply that the unwanted terms are proportional to a product of Coperators, where one Cθ, z j is replaced by C Q θ. Because of the symmetry i of L β z it is sufficient to consider only the unwanted terms for j = which are denoted by uwa and uw D. They originate from the second term in the commutation rules 57 when ÃQθ is commuted with Cθ, z. Then the resulting Ãθ, z pushed through the other C s. Taking into account only the first terms in 56 we arrive at uwaz = cθ z m bθ z bz z j Ω Ãθ, z C βm θ, z m C β Q θ j=2 Using 27 and Yang-Baxter relations always taking into account the SUN ice rule which means color conservation and ãθ = one obtains [ Ω Ãθ, z C βm θ, z m C ] β Q θ = β α δβ α ˇΦ α ˇθ, ž β β 2 β m θ 2 α α 2 z z m z 2. = θ θ n α n i= β β 2 β m θ z m z 2. θ 2 θ n α α 2 α n

19 3 SU3 FORM FACTORS 9 β where ˇΦ α ˇθ, ž does not depend on α, β, θ, z. Similarly, we obtain the first unwanted contribution from D β β Q β as uw D = cz θ bz θ m j=2 which may be depicted as bzj z T β β z, z β β Ω D β β θ, z C β mθ, z m C β Q θ T β β β z, z β β Ω D β θ, z C β mθ, z m C β Q θ Again 27 and Yang-Baxter relations yield [Ω β D β θ, z C β mθ, z m C ] β Q θ = α δβ β while assumption ii for L β z means = β β β 2 β m z β z m z 2.. θ θ 2 θ n α α 2 α n n i= L β zq β β z = L β z bθi z δ β α ˇΦ β α ˇθ, ž where analogously to eq. 55 Q β z = T β β β β β z, z is the next level Q- matrix and z = z = z + 2πi,, z m. Therefore we finally obtain [ ] uw Kθ ÃQθ = [ C θ α dz hθ, z pθ, z cθ z bθ z ] uw Kθ D β Q β α = dz hθ, z pθ, z cz θ C θ bz θ m j=2 m j=2 bz z j L βzδ β α ˇΦ β α ˇθ, ž bz z j L βz δ β α ˇΦ β α ˇθ, ž. It has been used that the shift relation of the function φθ in 5 and ii of 52 for the function p imply n m bθi z hθ, zpθ, z = bzj z i= j=2 m j=2 hθ, z bz pθ, z. z j

20 3 SU3 FORM FACTORS 20 For the D Q -term we rewrite the z -integral by replacing z z 2πi such that z z and C θ C θ + 2πi and obtain for the sum of the unwanted term from ÃQ and D Q using cz/ bz = c z/ b z [ Kθ ÃQ θ + D uw Q θ] = dz dz dž α C θ C θ +2πi C θ m cz θ bz θ hθ, z p O θ, zl β z j=2 bz z j δβ α ˇΦ β α θ, z. Taking into account the pole structure of cz θ bz θ hθ, z we may replace for the à Q -term dz = 2πi Res + dz d z C θ z =θ C θ C θ where again Cθ Cθ d z = 0 and for the D Q -term dz d z = 2πi Res + dz d z C θ +2πi C θ z =θ C θ C θ which proves the cancellation of the unwanted terms. The proof of iii is similar to that for the Z3 model in [40]. We use the short notations θ = θ,, θ n, ˆθ = θ, θ 2, θ 3, ˇθ = θ4,, θ n, α = α,, α n, ˆα = α, α 2, α 3, ˇα = α 4,, α n, z = z,, z m, ẑ = z, z 2, ž = z 3,, z m, β = β,, β m, ˆβ = β, β 2, ˇβ = β3,, β m and prove 44 which may be depicted as Res Res O θ 23 =iη θ 2 =iη = 2i 2Γ O O. We will show that the K-function given by the integral representation 22 with 23 satisfies Res θ 23 =iη Res K...nθ = c 0 θ 2 =iη n i=4 j=2 3 φθ ij ε 23 K 4...n ˇθ σ 3 S 3n S 34 6 c 0 = 2i 2ΓF 2 iηf 2iη. This is equivalent to 44 because of the ansatz 6 for F...n θ and the relation of F z and φz given by 5. The residues of K...n θ consists of three terms Res θ 2 =iη Res K...nθ = R θ 23 =iη...n + R2...n + R3...n.

21 3 SU3 FORM FACTORS 2 This is because each pair of the z integration contours will be pinched at three points. Due to symmetry it is sufficient to determine the contribution from the z - and the z 2 -integrations and multiply the result by mm. The pinching points are z = θ 2, z 2 = θ 3, 2 z = θ, z 2 = θ 2, 3 z = θ 2 iη, z 2 = θ 3 iη. The contribution of is given by z -and z 2 -integrations θ 2 dz θ 3 dz 2 along small circles around z = θ 2 and z 2 = θ 3 see figure. The S-matrices Sθ 2 z and Sθ 3 z 2 yield the permutation operator S0 = P Ω C βm θ, z m C β 2 θ, θ 3 C β θ, θ 2 = α = S β 2 α α 3 θ 3 S β α α α 2 θ 2 m j=3 β β 2 β 3 β m. α α 2 α 3 α 4 α n bθ z j Ω C βm ˇθ, z m C β 3 ˇθ, z 3 We have used the fact that because of the SUN ice rule only the amplitude b contributes to the S-matrices Sθ z j and only ã = to the S-matrices Sθ 2 z j, Sθ 3 z j, Sθ i z, Sθ i z 2 after having applied Yang-Baxter relations. Further we use that for z 2 θ 23 iη by assumption iii c.f. 49 and that because of 43 Res Res θ 2 =iη θ 23 =iη L β z c m i=3 32 φz i2 S β β 2 z 2 L ˇβž c = φz 2 = φθ 23 = φiη φθ 2 φθ 32 3 S α α 2 α 3 θ, θ 2, θ 3 = φiη φ2iη2 iη 2 ɛ α α 2 α 3. We combine this with the scalar functions h and p and after having performed the remaining z j -integrations we obtain R α = c 0 n i=4 j=2 3 φθ ij ɛˆα Kˇα ˇθ because of bθ z φθ z φθ 2 z φz θ 3 φθ 3 zτθ 2 zτθ 3 z =, the relation iii 3 of 52 for the p-functions pθ, θ 2, θ 3, ž θ2 =θ 23 =iη = m pˇθ, ž ˇα.

22 3 SU3 FORM FACTORS 22 and the relation 53 for the normalization constants N n c 2iη 2 φiη φ2iη = N n 3 c 0. The remaining contribution to 6 is due to R 2...n + R3...n. It is convenient to shift the particle with momentum θ 3 to the right by applying S-matrices using i. Then we have to prove that R 2...n + n 3 R3...n S 43 S n3 + c 0 φθ ij ε 23 K 4...n ˇθσ 3 = 0. αα2 ˇαα3 Note that because of i c.f. 7 K...n θs 43 S n3 = i=4 j=2 n aθ i3 K n3 θ,, θ 4,, θ n, θ 3. i=4 62 Due to Lemma 6 in appendix A it is sufficient to prove equation 62 only for α 3 = since the left hand side of this equation is a highest weight co-vector. This is because Bethe ansatz vectors, in particular also off-shell Bethe ansatz vectors are of highest weight see [57]. Therefore we consider this equation for the components with α 3 =. The contribution of pinching 2 is given by the z -, z 2 -integrations along the small circles around z = θ and z 2 = θ 2 see again figure. Now Sθ z and Sθ 2 z 2 yield permutation operators P and the co-vector part of this contribution for α 3 = is Ω C βm θ, z m C β 2 θ, θ 2 C β θ, θ P 3 α β β 2 β m =. α α 2 α 4 α 3 = = δ β α δ β 2 α 2 Ω C βm ˇθ, z m C β 3 ˇθ, z 3 ˇα δ α 3 63 where P 3 projects onto the components with α 3 =. We have used the fact that because of the SUN ice rule the amplitude a only contributes to the S-matrices Sθ z j, Sθ 2 z j, Sθ 3 z j, Sθ i z after having applied Yang-Baxter relations. One derives the formula Res Res θ 23 =iη θ 2 =iη φθ 3 φθ S α α 2 θ 2 = φiη φ2iη2iη 2 ɛ α α 2 using 43, ɛ 32 =, S 32 α α 2 θ 3 +2πi, θ, θ 2 = bθ 3 +2πibθ 32 +2πi S 32 α α 2 θ 2 because of the SU3 ice rule and φθ = bθ +2πi φθ +2πi see 5. We combine this and 63 with the scalar functions h and p taking into account the

23 3 SU3 FORM FACTORS 23 property iii of 43 and after having performed the remaining z j -integrations we obtain R 2...n S 43 S n3 P 3 = c 0 n i=4 j=2 3 φθ ij ε 23 K 4...n ˇθσ 3 P We have used the following equations aθ i3 φθ i = φθ i3 because of 5, the definition 24 of τz which implies φz θ 2 φθ z φθ 2 z φθ 3 zτθ zτθ 2 z =, the second relation iii 3 of 52 for the p-functions pθ, θ, θ 2, ž θ2 =θ 23 =iη = σ pˇθ, ž and the relation 53 for the normalization constants N n c φiη φ2iη2iη 2 = N n 3 c 0. The contribution of pinching 3 is given by the z -, z 2 -integrations along the small circles around z = θ 2 iη and z 2 = θ 3 iη, see again figure. Now δγ Sθ 3 z 2 yields S αβ θ 3 z 2 Γ δγ ρσ Γρσ αβ and the residue of the Bethe ansatz state β β 2 β m i Res z 2 =θ 3 iη Ω Cθ, z m Cθ, z P n = α. α α 2 α 3 α n = vanishes for α 3 = because Γ ρσ αβ is antisymmetric with respect to α, β. Therefore equation 62 is proved for α 3 = and because of Lemma 6 also in general. The higher level Bethe ansatz Lemma 3 Let the constant R and the contour C z be defined as in the context of 22. Then the higher level function L β z = du k! C z R du k C z R hz, up z, u Ψ β z, u Ψ β Ω z, u = C z, u k C z, u hz, u = m i= j= k φz i u j i<j k satisfies i - iii of the equations if β τu i u j. i ii 3 3 p mk z, u is symmetric under z i z j { p mk z + 2πi, z 2,, u = k p mk z, z 2,, u p mk z, u + 2πi, u 2, = m p iii 3 p mk z, u z 2 =iη,u =z 2 = k p m 2k mk z, u, u 2, ž, ǔ. 65

24 4 SUN FORM FACTORS 24 Proof. The proofs of the second level equations i and ii are similar to the ones for the first level. For the proof of iii we observe that for z 2 iη there is a pinching at u j = z 2 which means that in Ψ β z, u we may replace Sz 2 u P and we have to consider for z 2 iη and u = z Ω C z, u k C z, z 2. β 3 u = z 3 2 u 2 z z 2 z 3 z m β β 2 β 3 β m k 32 = S β β 2 z 2 bz u j ˇΨ ˇβž, ǔ j=2 u k with ˇβ = β 3,, β m, ž = z 3,, z m, ǔ = u 2,, u k. Therefore because of bz u φz u φz 2 uτz 2 u = and iii 3 for z 2 iη L β z φz 2 m i=3 which proves the claim iii of SUN form factors 23 φz i2 S β β 2 z 2 L ˇβž In this section we perform the form factor program for the general SUN S- matrix. For this purpose, we extend the procedures of the previous section, i.e., the nested Bethe ansatz method now with N levels combined with the off-shell Bethe ansatz. Applications of the results to the SUN Gross-Neveu model [3] will be investigated in a separate article [45]. 4. S-matrix The SUN S-matrix is given by and 3. Again, the eigenvalue S θ has a pole at θ = iη = 2 N iπ which means that there exist bound states of r fundamental particles α + + α r ρ ρ r with ρ < < ρ r which transform as the anti-symmetric SUN tensor representation of rank r, 0 < r < N. The masses of the bound states satisfy m r = m N r which suggests Swieca s [5, 50, 44] picture that the antiparticle of a particle of rank r is to be identified with the particle of rank N r see also [39, 40]. Iterating N times the general bound state formula one obtains for the scattering of a bound state β β 2 β N with another particle δ analogously

25 4 SUN FORM FACTORS 25 to 37 for the SU3 case S δ γ γ 2...γ N β β 2...β N δ θγβ β 2...β N α α 2...α N = Γ γ γ 2...γ N S δ α α α 2...α α N δ θ + iπ iη S δ N 2α N α N δ θ iπ + iη with the total bound state fusion intertwiner Γ β β 2...β N α α 2...α N = Γ β β 2...β N β β 2...β N 2 α N Γ β β 2 β 3 β β 2 α 3 Γ β β 2 α α 2. Taking special cases for the external particles we obtain S N,2...,N,2...,N N θ = bθ + iπ iη bθ iπ + iη = N aπi θ S N,2...,N,2...,N N θ = bθ + iπ iη aθ iπ + iη = N bπi θ S N,2...,N 2,3...,N θ = bθ + iπ iη cθ iπ + iη = N cπi θ. which may be written as S β N β...β N α...α N α N θ = N δ β...β N α...α N δβ N αn bπi θ + ɛ β N β...β N ɛα...α N α N cπi θ with the total anti-symmetric tensors ɛ α...α N and ɛ α...α N ɛ...n = ɛ N... =. These results may be interpreted as an unusual crossing relation ɛ β β 2...β N β S δ β α δ θ S δ N 2β N α N δ θ N = N ɛ α α 2...α N γ S γδ δβ iπ θ 66 δ γ = N δ δ δ α α 2 α N β α α 2 α N β with θ j = θ + iπ jiη if we write the charge conjugation matrices as C α...α N α N = C α α 2...α N = ɛ α...α N, C α...α N α N = C α α 2...α N = ɛ α...α N. Therefore we have the relations c.f. 29 and 4 C αα...α N C α...α N β = δ β α, C αα...α N A α β Cβα...α N = N tr A and C β...β N γγ β...β N α...α N = ɛ α...α N γγ 67 where the constant Γ is Γ = i 2π NΓN. N

26 4 SUN FORM FACTORS 26 These results are consistent with the picture c.f. [5] that the bound state of N particles of rank is to be identified with the anti-particle of rank. As already remarked, the physical aspects of these facts will be discussed elsewhere [45]. For later convenience we consider as a generalization of 43 the total N- particle S-matrix consisting of NN /2 factors in terms of S S 2...N = S2 S3 S N S23 S 2N S N N 68 in the limit θ jj+ iη j =, N which behaves as S β N...β 2 β iη α α 2...α N N! θ 2 iη iη θ N N iη ɛβ N...β 2 β ɛ α α 2...α N. 69 The algebraic structure of this relation follows because one can use the Yang- Baxter equations to shift in 68 any factor S ii+ P ii+ to the right or the left. Therefore the expression is totally anti-symmetric with respect to the α i and the β i. The factor follows from The general form factor formula In order to obtain a recursion relation where only form factors for the fundamental particles of type α which transform as the SUN vector representation are involved, we have to apply iteratively the bound state relation iv to get the N -bound state which is to be identified with the anti-particle Res θ 2 =iη Res θ N 2N =iη F O...nθ = F O...N N...n θ...n, θ N,, θ n 2 N 2 Γ...N...N and finally the annihilation residue equation iii Res F...N N...n O θ...n, θ N,, θ n θ...n N =iπ = 2iC...N N F O N+...nθ N+,, θ n σ O NS Nn S NN+. Similar as for N = 3 we obtain with 67 Res Res F 23...nθ O,, θ n θ N N =iη θ 2 =iη = 2iε...N 2 N 2 ΓF O N+...n θ N+,, θ n σ O NS Nn S NN+. 70 ansatz for form factors: We propose the n-particle form factors of an operator Ox as given by the same formula 6 as for SU2 and SU3 in terms of the K-function and the minimal form factor function F θ given by 2 which belongs to the highest weight w = 2, 0,, 0. The form factor equations i and ii for K-function write again as 7 and 8. Consistency of 70, 8 and the crossing relation 66 means that the statistics factors are of the form

27 4 SUN FORM FACTORS 27 σα O = σ O r α if the particle of type α belongs to an SUN representation of rank r α =,, N and σ O r = e iπ /NrQ O for Q O = n mod N 7 as an extension of 33 and 45. For the K-function K O α θ we propose again the ansatz in form of the integral representation 22 with 23 and 24. The Bethe ansatz co-vector Ψ α θ, z is again of the form 46 and for the function L β z one makes again an analogous ansatz as for the K-function 22 where the indices run over sets with one element less. For the SUN case we have to iterate this N 2 times Ψ l α θ, z = L l β L l β z = k! hz, u = m i= j= z Φ l β α θ, z du C z R k φθ i z j C z du k R hz, up l z, u i<j k τz i z j l Ψ β z, u 72 for l =,, N 2, α i = l,, N, β i = l +,, N, Ψ 0 = Ψ. By this procedure one obtains the nested Bethe ansatz 5. As for the ZN and AN models [39, 40] the Jost-function φ θ = φθ/a θ = φ θ is a solution of the equation N 2 k=0 N φ θ + kiη k=0 F θ + kiη = 73 which is typical for models where the bound state of N particles is the anti-particle. The solution is z φz = Γ Γ 2πi N z 2πi and again we define τz = /φzφ z. The higher level Bethe ansatz: In order that the form factor F O θ given by 6 and 72 satisfies the form factor equations i - iii the higher level L-functions L l β,β 2,...,β m z, z 2,, z m, l < β i N, m = n l have to satisfy: i l Watson s equations ii l the crossing relation and L l...ij..., z i, z j, = L l...ji..., z j, z i, S ij z ij, L l β,β 2,...,β m z, z 2,, z m = L l β 2,...,β m,β z 2,, z m, z 2πi 5 In [4] the nested off-shell Bethe ansatz was formulated in terms of Jackson-type integrals instead of contour integrals.

28 4 SUN FORM FACTORS 28 iii l the function L l β z has to possess simple poles at z 2,, z N l,n l = iη and it has to factorize in the neighborhood of these poles as L l β z c l m N l i=n l+ j=2 φz ij L l mn l N l c l φz ij β i=n l+ j=2 N...l+ S ẑ L ˆβ ˇβž 74 N l+... L S l... ˆβ ˇβ where we have used the short notations ˆβ = β,, β N l, ˇβ = β N l+,, β m and ẑ = z,, z N l, ž = z N l+,, z m. The N...l+ S-matrix S ẑ describes the total scattering of the N l particles ˆβ with rapidities ẑ for the initial quantum numbers ˆβ and the final configuration of quantum numbers N,, l +. The constants satisfy the recursion relation with the solution if c N = see appendix C. N l c l = c l+ j= φjiη 75 c = φ N 2 iη φ N 3 2iη φn 2iη 76 Lemma 4 The higher level function L l β z satisfies il - iii l, if the higher level p-function in the integral representation 72 satisfies i l p l z, u is symmetric under z i z j { ii l p l mk z + 2πi, z 2,, u = k p l mk z, z 2,, u p l mk z, u + 2πi, u 2, = m p l mk z, u, u 2, iii l p l mk z, z 2, z N l, ǔ z2 = =z zn l,n l =iη = k N l+ p l ˇmǩž, ǔ 77 with ˇm = M N + l, ǩ = M N + l. This lemma is proved in appendix C. The p-function: In order that the form factor F O θ given by 6 and satisfies the form factor equations i - iii the p-function p O θ, z in 22 which depends on the explicit nature of the local operator O is assumed

29 4 SUN FORM FACTORS 29 to satisfy i p nm θ, z is symmetric under θ i θ j { ii σpnm θ + 2πi, θ 2,, z = m+n p nm θ, θ 2,, z p nm θ, z + 2πi, z 2, = n p nm θ, z, z 2, iii p nm θ, θ 2,, θ N, ž θ2 = =θ N N =iη = m N+ p n Nm N+ ˇθ, ž p nm θ, θ,, θ N, ž θ2 = =θ N N =iη = σ p n Nm N+ ˇθ, ž 78 where θ = θ,, θ n, ˇθ = θ N+,, θ n, z = z,, z m and ž = z N,, z m. In order to simplify the notation, we have in these equations suppressed the dependence of the p-function p O and the statistics factor σ O on the operator Ox. Theorem 5 The co-vector valued function F α θ given by the ansatz 6 and the integral representation 22 satisfies the form factor equations i, ii and iii of 7 9, in particular 70 if. L β z satisfies the equation i, ii, iii and 76 of lemma 4, 2. p O θ, z satisfies the equation i, ii and iii of 78 and 3. the normalization constants satisfy N N j N!iη N φjiηf jiη N n = 2i 2 N 2 ΓN n N. j= The proof of this theorem can be found in appendix B. 4.3 Examples To illustrate our general results we present some simple examples. 79 The energy momentum tensor: For the local operator Ox = T ρσ x where ρ, σ = ± denote the light cone components the p-function is, as for the sine-gordon model in [53] p T ρσ θ, z = n e ρθ i i= i= m e σz i. For the n = N particle form factor there are n l = N l integrations in the l-th level of the off-shell Bethe ansatz. The SUN weights are see [4] w = n n, n n 2,, n N 2 n N, n N =,,,,. We calculate the form factor of the particle α and the bound state β = β,, β N of N particles. In each level all integrations up to one may be performed iteratively using the bound state relation iv similar as in the

30 4 SUN FORM FACTORS 30 proof of theorem 5. Then all remaining integrations in the higher levels can be done by means of the formula i i Γa + cγa + dγb + cγb + d dsγa+sγb+sγc sγd s = 2πi. Γa + b + c + d The result for the form factor of the particle α and the bound state β writes as F T ρσ αβ θ, θ 2 = K T ρσ K T ρσ αβ θ, θ 2 = N T ρσ 2 αβ θ, θ 2 Gθ 2 e ρθ + e ρθ 2 ɛ Sδ ɛγ δγ αɛθ z S β θ 2 z C θ dz R φθ ze σz Lθ 2 z 80 where the summation is over γ and δ > and Gθ is the minimal form factor function of two particles of rank r = and r = N. The functions Gθ and Lθ are given by Lθ = Γ Giπ θf θφθ = 2 + θ Γ 2πi 2 + N θ. 2πi The remaining integral in80 may be performed similar as in [53] with the result 6 0 T ρσ 0 θ, θ 2 in αβ = 4M 2 ɛ αβ e 2 ρ+σθ +θ 2 +iπ sinh 2 θ 2 iπ Gθ 2. θ 2 iπ Similar as in [53] one can prove the eigenvalue equation n dxt ±0 x θ,, θ n in α = 0 for arbitrary states. i= p ± i The fields ψ α x: Because the Bethe ansatz yields highest weight states we obtain the matrix elements of the spinor field ψx = ψ x. The p-function for the local operator ψ ± x is see also [38] p ψ± θ, z = exp ± m z i n θ i. 2 N i= i= For example the -particle form factor is 0 ψ ± 0 θ α = δ α e 2 N θ. 6 In [58, 4] this result has been obtained using Jackson type integrals.

31 A A LEMMA 3 The last two formulae are consistent with the proposal of Swieca et al. [5, 44] that the statistics of the fundamental particles in the chiral SUN Gross- Neveu model should be σ = exp 2πis, where s = 2 N is the spin see also 7. For the n = N + particle form factor there are again n l = N l integrations in the l-th level of the off-shell Bethe ansatz and the SUN weights are w = 2,,,,. Similar as above one obtains the 2-particle and -bound state form factor F ψ± αβγ θ, θ 2, θ 3 = K ψ± αβγ θ, θ 2, θ 3 F θ 2 Gθ 3 Gθ 23 K ψ± αβγ = N ψ e 2 N θ i dz C θ R φθ z φθ 2 zlθ 3 ze ± 2 z ɛ Sδ ɛ δγ α ɛθ z S α 2 ζ θ ζγ 2 z S β θ 3 z. We were not able to perform this integration. In [45] we will discuss the /N expansion. There we will also discuss the physical interpretation of the results for the chiral Gross-Neveu model. Acknowledgments We thank A. Fring, R. Schrader, B. Schroer, and A. Zapletal for useful discussions. H.B. thanks A. Belavin, A. Nersesyan, and A. Tsvelik for interesting discussions. M.K. thanks E. Seiler and P. Weisz for discussions and hospitality at the Max-Planck Institut für Physik München, where parts of this work have been performed. H.B. thanks the condensed matter group of ICTP Trieste for hospitality, where part of this work was done. H.B. was supported partially by the grant Volkswagenstiftung within in the project Nonperturbative aspects of quantum field theory in various space-time dimensions. A.F. acknowledges support from PRONEX under contract CNPq / and CNPq Conselho Nacional de Desenvolvimento Científico e Tecnológico. This work is also supported by the EU network EUCLID, Integrable models and applications: from strings to condensed matter, HPRN-CT Appendix A A Lemma Lemma 6 Let v...n V...n be a highest weight vector, i.e. E...n v...n = 0. Then v...n vanishes if the components v α α 2...α n for α = vanish. as Proof. By definition E acts on the basis vectors e...n α E...n e...n α = n e α Ee αi e αn i= Ee α = e α where e 0 = 0. = e α e α2 e αn