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

Hratchia M. Babujian^{∗}^{†}, Angela Foerster^{‡}^{§}, and Michael Karowski^{¶}

Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

November 1, 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 SU(N) invariant S-matrix under the assumption that the anti-particle is identified with the bound state of N − 1 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: 11.10.-z; 11.10.Kk; 11.55.Ds Keywords: Integrable quantum field theory, Form factors

Contents

1 Introduction 2 1.1 The SU(N) S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Generalized Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Nested “off-shell” Bethe Ansatz for SU(N) . . . . . . . . . . . . . . . . 7

2 SU(2) form factors 10

3 SU(3) form factors 12 3.1 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 SU(N) form factors 24

∗Address: Alikhanian Brothers 2, Yerevan, 375036 Armenia †E-mail: babujian@physik.fu-berlin.de ‡Address: Instituto de Física da UFRGS, Av. Bento Gonçalves 9500, Porto Alegre, RS - Brazil

§E-mail: angela@if.ufrgs.br ¶E-mail: karowski@physik.fu-berlin.de

4.1 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 The general form factor formula . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A A Lemma 31

B Proof of Theorem 5 32

C The higher level Bethe Ansatz 35

1 Introduction

The Bethe Ansatz [1], 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 6vertex model was solved by Lieb [3, 4] with the same technique. C.N. Yang and C.P. Yang [5] proved ‘Bethe’s hypothesis’^{1} for the ground state of the anisotropic Heisenberg spin chain. Due to Yang [7] and Baxter [8] we have 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, 10] formulated these ideas in an elegant algebraic way, known as the “algebraic Bethe Ansatz”. Yang [7] and Sutherland [11] generalized the technique of the Bethe Ansatz for those cases where the underlying symmetry group is larger than SU(2). This method is now called the “nested” Bethe Ansatz. This technique was applied in [12] to derive the spectrum of the chiral SU(N) Gross-Neveu model [13]. The algebraic nested Bethe Ansatz was formulated in [14] for the SU(N) and in [15] for the O(2N) symmetric case, respectively. Another generalization of the Bethe Ansatz is the “off-shell” Bethe Ansatz, which was originally formulated by one of the authors (H.B.) [16, 17, 18, 19] to calculate correlation function in WZNW models (see also [20, 21]). 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.

1Yang and Yang decided to honor Bethe’s insight by calling his assumption “Bethe’s hypothesis” [6], now usually called “Bethe Ansatz”.

For integrable quantum field theories in 1+1 dimensions one of the authors (M.K.) et al. formulated the on-shell program [29] i.e. the exact determination of the scattering matrix using the Yang-Baxter equations and the off-shell program [30] 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 [31] who proposed the form factor equations (i) – (v) (see below) as extensions of similar formulae in the original article [30]. The formulae were proven by two of the authors (H.B. and M.K.) et al. [32]. 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 SU(N) model. The procedure is similar as for the scaling Z(N) Ising and affine A(N − 1) Toda models [33, 34] because the bound state structures of these models are similar. However, the algebraic structure of the form factors for the SU(N) 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 [35]).

It is expected that the results of this paper apply to the chiral SU(N) Gross-Neveu model [13, 36, 37, 38]. In a separate article [39] we will investigate these physical applications and compare our exact results with two different 1/N-expansions of the chiral Gross-Neveu model [36] and [38]. We note that SU(N) form factors were also calculated in [31, 40, 41] using other techniques, see also the related paper [42].

1.1 The SU(N) S-matrix

The general solutions of the Yang-Baxter equations, unitarity and crossing relations for a U(N) invariant S-matrix have been obtained in [43]. The S-matrix for the scattering of two particles belonging to the vector representation of SU(N) can be written as

S_{αβ}^{δγ}(θ) = δ_{αγ}δ_{βδ} b(θ) + δ_{αδ}δ_{βγ} c(θ) .  (1)

Unitarity reads as S_{i}(−θ)S_{i}(θ) = 1 for the S-matrix eigenvalues

S_{+}(θ) = b(θ) + c(θ) , S_{−}(θ) = b(θ) − c(θ) .

The amplitude S_{+}(θ) = a(θ) 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^{1...n} ∈ V^{1...n} = V^{1} ⊗ ⋯ ⊗ V^{n}

for a vector in a tensor product space. The vector components are denoted by v\underline{α} = v^{α_{1}...α_{n}}. Below we will also use co-vectors v_{1...n} ∈ (V^{1...n})^{†} (the dual of V^{1...n}) with components v\underline{α}. A linear operator connecting two such spaces with

matrix elements A_{α_{1}...α_{n}}^{α′_{1}...α′_{n′}} is denoted by

A_{1...n}^{1′...n′} : V^{1...n} → V^{1′...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 SU(N)). The vector spaces V^{i} is associated to a rapidity variable θ_{i}. An S-matrix such as S_{ij}(θ_{ij}) = S_{ij}^{ji}(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_{12}(θ_{12})S_{13}(θ_{13})S_{23}(θ_{23}) = S_{23}(θ_{23})S_{13}(θ_{13})S_{12}(θ_{12})  (2)

and implies here the relation between the amplitudes [43]

c(θ) = −\frac{iη}{θ} b(θ) , η = \frac{2π}{N} .

A solution [43, 36, 44] of all these equations writes as

a(θ) = b(θ) + c(θ) = −\frac{Γ (1 − \frac{θ}{2πi}) Γ (1 − \frac{1}{N} + \frac{θ}{2πi})}{Γ (1 + \frac{θ}{2πi}) Γ (1 − \frac{1}{N} − \frac{θ}{2πi})} .  (3)

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 [45, 44, 38] picture that the antiparticle is a bound state of N − 1 particles (see also [33, 34]).

For later convenience and in order to simplify the formulae we introduce

S̃(θ) = \frac{S(θ)}{a(θ)} = \frac{1θ − Piη}{θ − iη}  (4)

where 1 is the unit, P the permutation operator. We depict this matrix as

S̃_{αβ}^{δγ}(θ_{12}) = [figure: α; β; γ; δ; θ_{1}; θ_{2}] = δ_{αγ}δ_{βδ} b̃(θ_{12}) + δ_{αδ}δ_{βγ} c̃(θ_{12})

and the amplitudes are explicitly

b̃(θ) = \frac{θ}{θ − iη} , c̃(θ) = \frac{−iη}{θ − iη} .

1.2 Generalized Form factors

For a state of n particles of kind α_{i} with rapidities θ_{i} and a local operator O(x) we define the form factor functions F_{α_{1}...α_{n}}^{O}(θ_{1}, …, θ_{n}), or using a short hand notation F O ; \underline{α}(\underline{θ}), by

⟨ 0 | O(x) | θ_{1}, …, θ_{n} ⟩in ; \underline{α} = e^{−ix(p_{1}+···+p_{n})}F O ; \underline{α}(\underline{θ}) , for θ_{1} > ⋯ > θ_{n}. (5)

where \underline{α} = (α_{1}, …, α_{n}) and \underline{θ} = (θ_{1}, …, θ_{n}). For all other arrangements of the rapidities the functions F O ; \underline{α}(\underline{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 F O ; \underline{α}(\underline{θ}) are considered as the components of a co-vector valued function F_{1...n}^{O}(\underline{θ}) ∈ V_{1...n} = (V^{1...n})^{†} which may be depicted as

F_{1...n}^{O}(\underline{θ}) = [figure: O; θ_{1}; θ_{n}; …].  (6)

Now we formulate the main properties of form factors in terms of the functions F_{1...n}^{O}. They follow from general LSZ-assumptions and “maximal analyticity”, which means that F_{1...n}^{O}(\underline{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_{1...n}^{O}(\underline{θ}) is meromorphic in all variables θ_{1}, …, θ_{n} and satisfies the following relations:

(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 + 1 at the same time

F_{...ij...}^{O}(…, θ_{i}, θ_{j}, …) = F_{...ji...}^{O}(…, 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,¯1⟨ p_{1} | O(0) | p_{2}, …, p_{n} ⟩_{2...n}^{in,conn.}

= F_{1...n}^{O}(θ_{1} + iπ, θ_{2}, …, θ_{n})σ_{1}^{O}C^{¯11} = F_{2...n1}^{O}(θ_{2}, …, θ_{n}, θ_{1} − iπ)C^{1¯1} (8)

where σ_{α}^{O} takes into account the statistics of the particle α with respect to O. The charge conjugation matrix C^{¯11} will be discussed below.

(iii) There are poles determined by one-particle states in each sub-channel. In particular the function F O ; \underline{α}(\underline{θ}) has a pole at θ_{12} = iπ such that

Res_{θ_{12}=iπ} F_{1...n}^{O}(θ_{1}, …, θ_{n}) = 2i C_{12} F_{3...n}^{O}(θ_{3}, …, θ_{n}) (1 − σ_{2}^{O}S_{2n} … S_{23}) .

(9)

(iv) If there are also bound states in the model the function F O ; \underline{α}(\underline{θ}) has additional poles. If for instance the particles 1 and 2 form a bound state (12), there is a pole at θ_{12} = iη, (0 < η < π) such that

Res_{θ_{12}=iη} F_{12...n}^{O}(θ_{1}, θ_{2}, …, θ_{n}) = F_{(12)...n}^{O}(θ_{(12)}, …, θ_{n}) \sqrt{2}Γ_{12}^{(12)}  (10)

where the bound state intertwiner Γ_{12}^{(12)} and the values of θ_{1}, θ_{2}, θ_{(12)} and η are given in [46, 47].

(v) Naturally, since we are dealing with relativistic quantum field theories we finally have

F_{1...n}^{O}(θ_{1} + μ, …, θ_{n} + μ) = e^{sμ} F_{1...n}^{O}(θ_{1}, …, θ_{n})  (11)

if the local operator transforms under Lorentz transformations as O → e^{sμ}O where s is the “spin” of O.

The property (i) - (iv) may be depicted as

[figure: (i); O; …; …; =; O; …; …]

(ii) [figure: O; conn.; …] = [figure: ×; O; …] = [figure: O; …]

(iii)  \frac{1}{2i} Res_{θ_{12}=iπ} [figure: O; …]= [figure: O; …]− [figure: ×; O; …]

[figure: (iv); \frac{1}{\sqrt{2}}; Res_{θ_{12}=iη}; O; …; =; O; …]

where × denotes the statistics factor σ^{O}. As was shown in [32] 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_{1...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

F (θ) = a (θ) F (−θ) = F (2πi − θ)

using (3) as

F (θ) = c exp ∫_{0}^{∞} \frac{dt}{t sinh^{2} t}e^{\frac{t}{N}} sinh t (1 − \frac{1}{N} ) (1 − cosh t (1 − \frac{θ}{iπ} )) . (12)

It belongs to the highest weight w = (2, 0, …, 0). We define the corresponding ‘Jost-function’ as for the Z(N) models [33, 34] by the equation

∏_{k=0}^{N−2} ϕ (θ + kiη) ∏_{k=0}^{N−1} F (θ + kiη) = 1 , η = \frac{2π}{N}  (13)

which is typical for models where the bound state of N − 1 particles is the anti-particle [34]. The solution is

ϕ(θ) = Γ (\frac{θ}{2πi}) Γ (1 − \frac{1}{N} − \frac{θ}{2πi})  (14)

and satisfies the relations

ϕ(θ) = ϕ(−θ)a (−θ) = ϕ((N − 1)iη − θ)

= \frac{1}{−b(θ)}ϕ(2πi − θ) = \frac{a(θ − 2πi)}{−b(θ)}ϕ(θ − 2πi) . (15)

Notice that the equations (14) and (13) also determine the normalization constant c in (12) as

c = Γ^{−2(1−1/N)} (\frac{1}{2} − \frac{1}{2N} ) exp (− ∫_{0}^{∞} e^{\frac{1}{N} t} (\frac{sinh (1 − \frac{1}{N} ) t}{t sinh^{2} t} − \frac{(1 − \frac{1}{N} )}{t sinh t}) dt) .

Generalized form factors: The co-vector valued function (6) for nparticles can be written as [30]

F_{1...n}^{O}(\underline{θ}) = K_{1...n}^{O}(\underline{θ}) ∏_{1≤i<j≤n} F (θ_{ij})  (16)

where F (θ) is the minimal form factor function (12). The K-function K_{1...n}^{O}(\underline{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 S̃(θ) = S(θ)/a(θ)

K_{...ij...}^{O}(…, θ_{i}, θ_{j}, …) = K_{...ji...}^{O}(…, θ_{j}, θ_{i}, …) S̃_{ij}(θ_{ij})  (17)
K_{1...n}^{O}(θ_{1} + iπ, θ_{2}, …, θ_{n})σ_{1}^{O}C^{¯11} = K_{2...n1}^{O}(θ_{2}, …, θ_{n}, θ_{1} − iπ)C^{1¯1}  (18)

for all possible arrangements of the θ’s.

1.3 Nested “off-shell” Bethe Ansatz for SU(N)

We consider a state with n particles and define as usual in the context of the algebraic Bethe Ansatz [9, 10] the monodromy matrix

T̃_{1...n,0}(\underline{θ}, θ_{0}) = S̃_{10}(θ_{10}) ⋯ S̃_{n0}(θ_{n0}) = 1  n 0 …  .  (19)

It is a matrix acting in the tensor product of the “quantum space” 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 ≅ ℂ^{N} as the space of the vector representation of SU(N). The Yang-Baxter algebra relation for the S-matrix (2) yields

T̃_{1...n,a}(\underline{θ}, θ_{a}) T̃_{1...n,b}(\underline{θ}, θ_{b}) S̃_{ab}(θ_{a} − θ_{b}) = S̃_{ab}(θ_{a} − θ_{b}) T̃_{1...n,b}(\underline{θ}, θ_{b}) T̃_{1...n,a}(\underline{θ}, θ_{a}) (20)

[figure: 1; n; b; a; b; a; …; =; 1; n; b; a; b; a; …]

which implies the basic algebraic properties of the sub-matrices A, B, C, D with respect to the auxiliary space defined by

T̃_{1...n,0}(\underline{θ}, z) ≡ ( \begin{matrix}Ã_{1...n}(\underline{θ}, z) & B̃_{1...n,β}(\underline{θ}, z) \\ C̃_{1...n}^{β}(\underline{θ}, z) & D̃_{1...n,β}^{β′}(\underline{θ}, z) \end{matrix} ) , 2 ≤ β, β′ ≤ N . (21)

We propose the following Ansatz for the general form factor F_{1...n}^{O}(\underline{θ}) or the K-function defined by (16) in terms of a nested ‘off-shell’ Bethe Ansatz and written as a multiple contour integral

KO ; \underline{α}(\underline{θ}) = \frac{N_{n}}{m!} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ⋯ ∫_{C\underline{θ}} \frac{dz_{m}}{R} h̃(\underline{θ}, \underline{z}) p^{O}(\underline{θ}, \underline{z}) Ψ̃\underline{α}(\underline{θ}, \underline{z})  (22)

where h̃(\underline{θ}, \underline{z}) is a scalar function which depends only on the S-matrix and not on the specific operator O(x)

h̃(\underline{θ}, \underline{z}) = ∏_{i=1}^{n} ∏_{j=1}^{m} ϕ̃(θ_{i} − z_{j}) ∏_{1≤i<j≤m} τ(z_{i} − z_{j})  (23)

τ(z) = \frac{1}{ϕ(θ)ϕ(−θ)} , ϕ̃(θ) = ϕ(θ)a(θ) = ϕ(−θ) .  (24)

For the SU(N) S-matrix the function ϕ(θ) is given by (13) with the solution (14). The integration contour C\underline{θ} is depicted in Fig. 1. The constant R is defined by R = ∮_{θ} dzϕ̃(θ − z) where the integration contour is a small circle around z = θ as part of C\underline{θ}.

The dependence on the specific operator O(x) is encoded in the scalar pfunction p^{O}(\underline{θ}, \underline{z}) which is in general a simple function of e^{θ_{i}} and e^{z_{j}} (see below). By means of the Ansatz (16) 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 [48, 33, 34] given by (22) - (24). We consider here cases where the sum consists only of one term.

If in (21) the range of β’s is non-trivial, i.e. if N > 2 the Bethe Ansatz co-vectors are of the form

Ψ̃\underline{α}(\underline{θ}, \underline{z}) = L\underline{β}(\underline{z}) Φ̃\underline{β} ; \underline{α}(\underline{θ}, \underline{z})  (25)

• θ_{n} − 2πi θ_{n} − 2πi\frac{1}{N} • θ_{n} [figure] • θ_{n} + 2πi(1 − \frac{1}{N}) …  • θ_{2} − 2πi θ_{2} − 2πi\frac{1}{N} • θ_{2} [figure] • θ_{2} + 2πi(1 − \frac{1}{N}) • θ_{1} − 2πi θ_{1} − 2πi\frac{1}{N} • θ_{1} [figure] • θ_{1} + 2πi(1 − \frac{1}{N})

Figure 1: The integration contour C\underline{θ}. The bullets refer to poles of the integrand resulting from a(θ_{i} − z_{j}) ϕ(θ_{i} − z_{j}) and the small open circles refer to poles originating from b(θ_{i} − z_{j}) and c(θ_{i} − z_{j}).

where summation over all \underline{β} = (β_{1}, …, β_{m}) with β_{i} > 1 is assumed. The basic

Bethe Ansatz co-vectors Φ̃\underline{β} ; 1...n ∈ (V^{1...n})^{†} are defined as

\begin{aligned}Φ̃\underline{β} ; 1...n(\underline{θ}, \underline{z}) & = & Ω_{1...n}C̃_{1...n}^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃_{1...n}^{β_{1}}(\underline{θ}, z_{1}) \\ Φ̃\underline{β} ; \underline{α}(\underline{θ}, \underline{z}) & = & [figure: β_{1}; β_{m}; α_{1}; α_{n}; 1; 1; 1; 1; θ_{1}; θ_{n}; z_{1}; z_{m}; …; ⋮] , \begin{matrix}2 ≤ β_{i} ≤ N \\ 1 ≤ α_{i} ≤ N . \end{matrix} \end{aligned}  (26)

Here the “pseudo-vacuum” is the highest weight co-vector (with weight w = (n, 0, …, 0))

Ω_{1...n} = e(1) ⊗ ⋯ ⊗ e(1)

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 vector satisfies

\begin{aligned}Ω_{1...n} B̃_{1...n}^{β}(\underline{θ}, z) & = & 0 \\ Ω_{1...n} Ã_{1...n}(\underline{θ}, z) & = & Ω_{1...n} \\ Ω_{1...n} D̃_{1...n,β}^{β′}(\underline{θ}, z) & = & δ_{β}^{β′} ∏_{i=1}^{n} b̃(θ_{i} − z)Ω_{1...n} . \end{aligned}  (27)

The amplitudes of the scattering matrices are given by eqs. (1) and (3). The technique of the ‘nested Bethe Ansatz’ means that one makes for the coefficients L\underline{β}(\underline{z}) in (25) the analogous construction (22) - (24) as for K\underline{α}(\underline{θ}) where now the indices \underline{β} take only the values 2 ≤ β_{i} ≤ N. This nesting is repeated until the space of the coefficients becomes one dimensional.

2 SU(2) FORM FACTORS 10

In this article we will focus on the determination of the form factors for an SU(N) S-matrix. The paper is organized as follows: In Section 2 we construct the general form factor formula for the simplest SU(2) case. In Section 3 we construct the general form factor formula for the SU(3) 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 SU(N) are constructed and discussed in detail.

2 SU(2) form factors

In this section we start with the simplest case. We perform the form factor program for the SU(2) S-matrix. The results should apply to the well-known SU(2) Gross-Neveu model [13], investigated by Bethe Ansatz methods in [49, 50]. 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 [32, 47].

S-matrix: The SU(2) S matrix^{2} is given by (1) and (3) for N = 2. It turns out that the amplitudes satisfy the relations b(θ) = −a(iπ − θ) and c(θ) = c(iπ − θ) which may be written as

S_{αβ}^{δγ}(θ) = − (δ_{α}^{γ}δ_{β}^{δ} b(iπ − θ) + ϵ^{δγ}ϵ_{αβ} c(iπ − θ)) .  (28)

We have introduced

ϵ_{αβ} = [figure: •; α; β] , ϵ^{αβ} = [figure: •; α; β]

which are antisymmetric and ϵ_{12} = ϵ^{21} = 1 such that

ϵ_{αβ}ϵ^{βγ} = δ_{α}^{γ} : [figure: •; •; =]  and ϵ_{αβ}A_{γ}^{α}ϵ^{γβ} = [figure: A; •; •] = − tr A (29)

for a matrix A. Formula (28) may be understood as an unusual crossing relation (c.f. [45])

S_{αβ}^{δγ}(θ) = −C_{αα′}S_{βγ′}^{α′δ}(iπ − θ)C^{γ′γ}  (30)

[figure: = −; •; •]

if we define the “charge conjugation matrices” as C_{αβ} = ϵ_{αβ} and C^{αβ} = ϵ^{αβ}. This means that particle 1 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 S̃ = S/a

S̃_{αβ}^{δγ}(0) = δ_{α}^{δ}δ_{β}^{γ} = [figure: α; β; δ; γ] , Res_{θ=iπ} S̃_{αβ}^{δγ}(θ) = iπϵ^{δγ}ϵ_{αβ} = iπ [figure: •; •; α; β; δ; γ].

2It is related to the sine-Gordon S-matrix for β^{2} = 8π by (a, b, c)^{SU(2)} = (a, −b, c)^{S.G.} (see e.g. [32]).

Ansatz for form factors: Because of (29) the crossing equation (18) writes in components as

K_{α_{1}...α_{n}}^{O}(θ_{1}, θ_{2}, …, θ_{n})σ^{O} = −K_{α_{2}...α_{n}α_{1}}^{O}(θ_{2}, …, θ_{n}, θ_{1} − 2πi)  (31)

The form factor equation (iii) here reads as

Res_{θ_{12}=iπ} F_{1...n}^{O}(θ_{1}, …, θ_{n}) = 2i ϵ_{12} F_{3...n}^{O}(θ_{3}, …, θ_{n}) (1 − σ_{2}^{O}S_{2n} … S_{23}) . (32)

Using the crossing relation (30) it turns out that the statistics factor in (18) and (32) has to satisfy (σ^{O})^{2} = (−1)^{n} with the solutions

σ^{O} = i^{Q_{O}} for Q_{O} = n mod 2 .  (33)

We define the SU(2) ‘Jost-function’ ϕ(θ) and τ(z) by the same equations as for the sine-Gordon [32] and the Z(2) models [33, 34]

ϕ(θ)F (θ)F (θ + iπ) = 1  (34)
τ(z)ϕ(z)ϕ(−z) = 1  (35)

with the solution for ϕ(θ) given by (14). The functions ϕ (z) and τ(z) are for SU(2) explicitly given as

ϕ(z) = ϕ̃(−z) = Γ (\frac{z}{2πi}) Γ (\frac{1}{2} − \frac{z}{2πi}) , τ(z) = \frac{z sinh z}{4π^{3}} .

The scalar function h̃(\underline{θ}, \underline{z}) in (22) encodes only data from the scattering matrix. The p-function p^{O}(\underline{θ}, \underline{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}(\underline{θ}, \underline{z}) has to satisfy

(i′)_{2} p_{nm}(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j} and z_{i} ↔ z_{j}
(ii′)_{2} \{ \begin{matrix}σp_{nm}(θ_{1} + 2πi, θ_{2}, …, \underline{z}) = (−1)^{m−1}p_{nm}(θ_{1}, θ_{2}, …, \underline{z}) \\ p_{nm}(\underline{θ}, z_{1} + 2πi, z_{2}, …) = (−1)^{n}p_{nm}(\underline{θ}, z_{1}, z_{2}, …) \end{matrix}
(iii′)_{2} if θ_{12} = iπ : p_{nm}(\underline{θ}, \underline{z})|_{z_{1}=θ_{1}} = (−1)^{m−1}σp_{nm}(\underline{θ}, \underline{z})|_{z_{1}=θ_{2}}
= σp_{n−2m−1}(θ_{3}, …, θ_{n}, z_{2}, …, z_{m}) + p̃

(36)

where σ is the statistics factor of (31) 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 O(x). By means of the Ansatz (16) and (22) - (24) we have transformed the complicated form factor equations (i) - (iii) to the simple ones (i′)_{2} - (iii′)_{2} for the p-function.

Theorem 1 The co-vector valued function F O ; \underline{α}(\underline{θ}) given by the Ansatz (16) and (22) satisfies the form factor equations (i), (ii) and (iii) (see (7) - (9)) if the p-function p^{O}(\underline{θ}, \underline{z}) satisfies the equations (i′)_{2}, (ii′)_{2}, (iii′)_{2} of (36) and the normalization constants satisfies the recursion relation

iπϕ̃(iπ)F (iπ)N_{n} = 2iN_{n−2} .

The proof of this theorem for SU(2) is similar to that for the sine-Gordon model in [32] and may easily obtained from the ones for SU(3) or SU(N) established below.

3 SU(3) form factors

In this section we construct the form factors for the SU(3) 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 SU(2) case, there are additional properties to be obeyed by the second-level Bethe Ansatz function (See lemma 3 below for details).

3.1 S-matrix

The SU(3) S-matrix is given by (1) and (3) for N = 3. The eigenvalue S_{−} has a pole at θ = iη = \frac{2}{3}iπ which means that there exist bound states of two fundamental particles α + β → (ρσ) (with 1 ≤ ρ < σ ≤ 3) which transform as the anti-symmetric SU(3) tensor representation. The general bound state Smatrix formula [46, 47] for the scattering of a bound state with another particle reads in particular for the SU(3) case as

S_{(ρσ)γ}^{γ′(ρ′σ′)}(θ_{(12)3})Γ_{αβ}^{(ρσ)} = Γ_{α′β′}^{(ρ′σ′)}S_{αγ′′}^{γ′α′}(θ_{13})S_{βγ}^{γ′′β′}(θ_{23}) |_{θ_{12}=iη}  (37)

[figure: 1; 2; 3; •] = [figure: 1; 2; 3; •]

where θ_{(12)} is the bound state rapidity and η the bound state fusion angle. The bound state fusion intertwiner Γ_{αβ}^{(ρσ)} is defined by

i Res_{θ=iη} S_{αβ}^{β′α′}(θ) = ∑_{ρ<σ} Γ_{(ρσ)}^{β′α′}Γ_{αβ}^{(ρσ)} = [figure: •; •; α; β; (ρσ); β′; α′].

With a convenient choice of an undetermined phase factor one obtains

Γ_{αβ}^{(ρσ)} = Γ (δ_{α}^{ρ}δ_{β}^{σ} − δ_{α}^{σ}δ_{β}^{ρ}) , Γ_{(ρσ)}^{β′α′} = Γ (δ_{ρ}^{β′}δ_{σ}^{α′} − δ_{σ}^{β′}δ_{ρ}^{α′})  (38)

where Γ = i\sqrt{ia(iη)iη} is a number. Choosing in (37) special cases for the external particles we calculate

S_{(12)3}^{3(12)}(θ) = b(θ + \frac{1}{3}iπ)b(θ − \frac{1}{3}iπ) = a(iπ − θ)
S_{(12)2}^{2(12)}(θ) = b(θ + \frac{1}{3}iπ)a(θ − \frac{1}{3}iπ) = b(πi − θ)
S_{(23)1}^{3(12)}(θ) = −b(θ + \frac{1}{3}iπ)c(θ − \frac{1}{3}iπ) = c(πi − θ)

which may be written as

S_{(αβ)γ}^{γ′(α′β′)}(θ) = δ_{γ}^{γ′}δ_{(αβ)}^{(α′β′)}b(πi − θ) + ϵ^{γ′α′β′}ϵ_{αβγ}c(πi − θ)

with the total anti-symmetric tensors ϵ_{αβγ} and ϵ^{αγβ} (ϵ_{123} = ϵ^{321} = 1). These formulae may be again understood as a crossing relation (c.f. [45])

ϵ_{α′β′δ}S_{αγ′′}^{γ′α′}(θ + \frac{1}{2}iη)S_{βγ}^{γ′′β′}(θ − \frac{1}{2}iη) = ϵ_{αβδ′}S_{γδ}^{δ′γ′}(iπ − θ)  (39)

[figure: •; α; β; γ; γ′; δ; =; •; α; β; γ; γ′; δ]

or S_{(αβ)γ}^{γ′(α′β′)}(θ) = C_{(αβ)δ′}S_{γδ}^{δ′γ′}(iπ − θ)C^{δ(α′β′)} if we write the charge conjugation matrices as

C_{(αβ)γ} = C_{α(βγ)} = ϵ_{αβγ} , C^{α(βγ)} = C^{(αβ)γ} = ϵ^{αβγ} .  (40)

Therefore we have the relations (c.f. (29))

C_{α(ρσ)}C^{(ρσ)β} = δ_{α}^{β} , C_{α(ρσ)}A_{β}^{α}C^{β(ρσ)} = tr A  (41)

and

C_{α(ρσ)}Γ_{βγ}^{(ρσ)} = C_{(ρσ)γ}Γ_{αβ}^{(ρσ)} = ϵ_{αβγ}Γ .  (42)

These results^{3} are consistent with the picture that the bound state of particles 1 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_{12}, θ_{23} = iη

S̃_{αβγ}^{γ′β′α′}(θ_{1}, θ_{2}, θ_{3}) = S̃_{α′′β′′}^{β′α′}(θ_{12})S̃_{αγ′′}^{γ′α′′}(θ_{13})S̃_{βγ}^{γ′′β′′}(θ_{23})
≈ 2\frac{iη}{θ_{12} − iη} \frac{iη}{θ_{23} − iη} ϵ^{γ′β′α′}ϵ_{αβγ} .  (43)

3.2 Form factors

In order to obtain a recursion relation where only form factors for the fundamental particles of type α = 1, 2, 3 (which transform as the SU(3) 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_{θ_{12}=iη} F_{123...n}^{O}(θ_{1}, θ_{2}, θ_{3}, …, θ_{n}) = F_{(12)3...n}^{O}(θ_{(12)}, θ_{3}, …, θ_{n})\sqrt{2}Γ_{12}^{(12)}
Res_{θ_{(12)3}=iπ} F_{(12)3...n}^{O}(θ_{(12)}, θ_{3}, …, θ_{n}) = 2iC_{(12)3}F_{4...n}^{O}(θ_{4}, …, θ_{n})
× (1 − σ_{3}^{O}S_{3n} … S_{34})

where Γ_{12}^{(12)} is the bound state intertwiner (38) (see [46, 47]), θ_{(12)} = \frac{1}{2}(θ_{1} + θ_{2}) is the bound state rapidity, η = \frac{2}{3}π is the bound state fusion angle and C_{(12)3} is the charge conjugation matrix (40). We obtain with the short notation \underline{θ̌} = (θ_{4}, …, θ_{n})

Res_{θ_{23}=iη} Res_{θ_{12}=iη} F_{123...n}^{O}(\underline{θ}) = 2iε_{123}\sqrt{2}ΓF_{4...n}^{O}(\underline{θ̌}) (1 − σ_{3}^{O}S_{3n} … S_{34}) (44)

where (42) has been used.

3The physical aspects of these facts will we discussed elsewhere [39].

Ansatz for form factors: We propose that the n-particle form factor of an operator O(x) is given by the same formula (16) as for SU(2) where the form factor equations (i) and (ii) for K-function write again as (17) and (18). Consistency of (44), (18) and the crossing relation (39) means that the statistics factors are of the form σ_{α}^{O} = σ^{O}(r_{α}) if the particle of type α belongs to a SU(3) representation of rank r_{α} = 1, 2 and

σ^{O}(r) = e^{iπ \frac{2}{3} rQ_{O}} for Q_{O} = n mod 3  (45)

as an extension of (33). As for SU(2), 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 SU(3) of the form

Ψ̃\underline{α}(\underline{θ}, \underline{z}) = L\underline{β}(\underline{z})Φ̃\underline{β} ; \underline{α}(\underline{θ}, \underline{z})  (46)

where the basic Bethe states Φ̃\underline{β} ; \underline{α}(\underline{θ}, \underline{z}) are given by (26) and the indices α_{i} run over 1 ≤ α_{i} ≤ 3 and the β_{i} over 2 ≤ β_{i} ≤ 3. For the function L\underline{β}(\underline{z}) one makes an analogous Ansatz (22) as for K\underline{α}(\underline{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\underline{β}(\underline{z}) is assumed to satisfy

(i)^{(1)}

L_{...ij...}(…, z_{i}, z_{j}, …) = L_{...ji...}(…, z_{j}, z_{i}, …) S̃_{ij}(z_{ij})  (47)

(ii)^{(1)}

C^{¯11}L_{1...m}(z_{1} + iπ, z_{2}, …, z_{m}) = L_{2...m1}(z_{2}, …, z_{m}, z_{1} − iπ)C^{1¯1} (48)

(iii)^{(1)} there is a pole at z_{12} = iη such that

Res_{z_{12}=iη} L\underline{β}(\underline{z}) = c_{1} ∏_{i=3}^{m} ϕ̃(z_{i2}) Res_{z_{12}=iη} S̃_{β_{1}β_{2}}^{32}(z_{12})L\underline{β̌}(\underline{ž})  (49)

with c_{1} = ϕ̃(iη).

These properties of the next level Bethe Ansatz function L\underline{β}(\underline{z}) are discussed in lemma 3.

The minimal two particle form factor function

F (θ) = c exp ∫_{0}^{∞} \frac{dt}{t sinh^{2} t}e^{\frac{t}{3}} sinh \frac{2}{3}t (1 − cosh t (1 − \frac{θ}{iπ} ))  (50)

belongs to the highest weight w = (2, 0, 0). The SU(3) and the Z(3) model [33, 34] 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 SU(3) ‘Jost-function’ ϕ (z) by the same equation as for the Z(3) model

ϕ(θ)ϕ(θ + iη)F (θ)F (θ + iη)F (θ + 2iη) = 1  (51)

with the solution

ϕ(z) = Γ (\frac{z}{2πi}) Γ (\frac{2}{3} − \frac{z}{2πi}) .

These equations determine also the constant c in (50). The function τ(z) is again defined by (24).

The function h̃(\underline{θ}, \underline{z}) is scalar and encodes only data from the scattering matrix. The p-function p^{O}(\underline{θ}, \underline{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} p(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j}
(ii′)_{3} \{ \begin{matrix}σp(θ_{1} + 2πi, θ_{2}, …, \underline{z}) = (−1)^{m}p(θ_{1}, θ_{2}, …, \underline{z}) \\ p(\underline{θ}, z_{1} + 2πi, z_{2}, …) = (−1)^{n}p(\underline{θ}, z_{1}, z_{2}, …) \end{matrix}
(iii′)_{3} for θ_{12} = θ_{23} = iη \{ \begin{matrix}p(\underline{θ}; θ_{2}, θ_{3}, \underline{ž}) = (−1)^{m} p(\underline{θ̌}, \underline{ž}) \\ p(\underline{θ}; θ_{1}, θ_{2}, \underline{ž}) = σ p(\underline{θ̌}, \underline{ž}) \end{matrix}

(52)

with \underline{θ̌} = (θ_{4}, …, θ_{n}), \underline{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 SU(3).

Theorem 2 Let the co-vector valued function F O ; \underline{α}(\underline{θ}) be defined by the Ansatz (16), (22), (23) and (46). Let the p-function satisfy (i′)_{3}, (ii′)_{3} and (iii′)_{3} of (52) and let the function L\underline{β}(\underline{z}) satisfy (i)^{(1)}, (ii)^{(1)} and (iii)^{(1)} of (47) - (49). Let the normalization constants satisfy the recursion relation

2(iη)^{2}ϕ̃^{2}(iη)ϕ̃(2iη)F^{2}(iη)F (2iη)N_{n} = 2i\sqrt{2}ΓN_{n−3} .  (53)

Then the function F O ; \underline{α}(\underline{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 (17) follows directly from (i′)_{3}, the Yang-Baxter equations and the action of the S-matrix on the pseudo-ground state Ω

Ω_{...ji...} C̃_{...ji...}(… θ_{j}, θ_{i} …) S̃_{ij}(θ_{ij}) = Ω_{...ji...} S̃_{ij}(θ_{ij}) C̃_{...ij...}(… θ_{i}, θ_{j} …)
= Ω_{...ij...} C̃_{...ij...}(…θ_{i}, θ_{j}…) .

because S̃_{11}^{11}(θ) = S_{11}^{11}(θ)/a(θ) = 1 and F (θ) = F (−θ)a(θ).

Using (i) and (41) the property (ii) in the form of (18) may be rewritten as a matrix difference equation [51, 52, 35]

K_{1...n}(\underline{θ}′)σ = K_{1...n}(\underline{θ}) Q_{1...n}(\underline{θ})  (54)

where \underline{θ}′ = (θ_{1} + 2πi, θ_{2} …, θ_{n}) and σ is a statistics factor. The matrix Q(\underline{θ}) = Q(\underline{θ}, 1) is a special case (for i = 1) of the trace

Q_{1...n}(\underline{θ}, i) = tr_{0} T̃_{Q,1...n,0}(\underline{θ}, i) = [figure: α_{1}; α′_{1}; α_{i}; α′_{i}; α_{n}; α′_{n}; θ_{1}; θ′_{i}; θ_{i}; θ_{n}; …; …]

of a modified monodromy matrix

T̃_{Q,1...n,0}(\underline{θ}, i) = S̃_{10}(θ_{1} − θ′_{i}) ⋯ P_{i0} ⋯ S̃_{n0}(θ_{n} − θ_{i})

where P is the permutation matrix. For the special case i = 1 the matrix Q(\underline{θ}, 1) = Q(\underline{θ}) may be written as a trace of the ordinary monodromy matrix (19) over the auxiliary space for the specific value of the spectral parameter θ_{0} = θ_{1}

Q_{1...n}(\underline{θ}) = tr_{0} T̃_{1...n,0}(\underline{θ}, θ_{1})  (55)

because S̃(0) = P. The difference equation (54) may be depicted as

[figure: K; …]σ = [figure: K; …]

where we use the rule that the rapidity of a line changes by 2πi if the line bends by 360^{0} in the positive sense.

In the following we will suppress the indices 1 … n. The Yang-Baxter relations (20) imply the typical commutation rules for the matrices Ã, C̃, D̃ defined in eq. (21)

C̃^{β}(\underline{θ}, z)Ã(\underline{θ}, θ) = \frac{1}{b̃(θ − z)}Ã(\underline{θ}, θ)C̃^{β}(\underline{θ}, z) − \frac{c̃(θ − z)}{b̃(θ − z)}Ã(\underline{θ}, z)C̃^{β}(\underline{θ}, θ) (56)
C̃^{β}(\underline{θ}, z)D̃_{γ}^{γ′}(\underline{θ}, θ) = \frac{1}{b̃(z − θ)}S̃_{β′γ′′}^{γ′β}(z − θ)D̃_{γ}^{γ′′}(\underline{θ}, θ)C̃^{β′}(\underline{θ}, z)

− \frac{c̃(z − θ)}{b̃(z − θ)}D̃_{γ}^{β}(\underline{θ}, z)C̃^{γ′}(\underline{θ}, θ)

where β, β′, γ, γ′, γ′′ ∈ {2, 3}. In addition there are the Zapletal commutation rules [51, 52, 35] where also the matrices A_{Q}, C_{Q}, D_{Q} defined by

T̃_{Q}(\underline{θ}) = ( \begin{matrix}Ã_{Q}(\underline{θ}) & B̃_{Q}^{β}(\underline{θ}) \\ C̃_{Q}^{β}(\underline{θ}) & D̃_{Q}_{β}^{β′}(\underline{θ}) \end{matrix} )

are involved [51]

C̃^{β}(\underline{θ}, z)Ã_{Q}(\underline{θ}) = \frac{1}{b̃(θ′_{1} − z)}Ã_{Q}(\underline{θ})C̃^{β}(\underline{θ}′, z) − \frac{c̃(θ′_{1} − z)}{b̃(θ′_{1} − z)}Ã(\underline{θ}, z)C̃_{Q}^{β}(\underline{θ}) (57)
C̃^{β}(\underline{θ}, z)D̃_{Q}_{γ}^{γ′}(\underline{θ}) = \frac{1}{b̃(z − θ′_{1})}S̃_{β′γ′′}^{γ′β}(z − θ′_{1})D̃_{Q}_{γ}^{γ′′}(\underline{θ})C̃^{β′}(\underline{θ}′, z)
− \frac{c̃(z − θ_{1})}{b̃(z − θ_{1})}D̃_{γ}^{β}(\underline{θ}, z)C̃_{Q}^{γ′}(\underline{θ}) .  (58)

Note that we assign to the auxiliary space of T̃_{Q}(\underline{θ}) corresponding to the horizontal line the spectral parameter θ_{1} on the right hand side and θ′_{1} = θ_{1} + 2πi on the left hand side.

We are now going to prove (54) in the form

K(\underline{θ}) (Ã_{Q}(\underline{θ}) + ∑_{β=2}^{3} D̃_{Q}_{β}^{β}(\underline{θ})) = K(\underline{θ}) Q(\underline{θ}) = K(\underline{θ}′)σ  (59)

where K(\underline{θ}) 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 T̃_{Q} which is Ã_{Q} + ∑_{β=2}^{3} D̃_{Q}_{β}^{β} to the co-vector K(\underline{θ})

Ω C̃^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β_{1}}(\underline{θ}, z_{1})T̃_{Q}_{γ}^{γ′} (\underline{θ}) = [figure: β_{1}; β_{m}; 1; 1; 1; γ′; γ; 1; 1; θ_{2}; θ_{n}; z_{1}; z_{m}; …; ⋮; θ_{1}; θ′_{1}] .

In the contribution from Ã_{Q}(\underline{θ}) which means γ′ = γ = 1 one may use Yang-Baxter relations to observe that only the amplitudes S̃_{11}^{11}(θ_{1} − z_{j}) = 1 appear in the S-matrices S̃(θ_{1} − z_{j}) which are constituents of the C-operators. Therefore we may shift all z_{j}-integration contours C\underline{θ} to C\underline{θ}′ without changing the values of the integrals, because there are no singularities inside C\underline{θ} ∪ −C\underline{θ}′ (c.f. Fig. 1). Using a short notation we have

K(\underline{θ}) Ã_{Q}(\underline{θ}) = ∫_{C_{\underline{θ}′}} d\underline{z} h̃(\underline{θ}, \underline{z})p(\underline{θ}, \underline{z}) Ψ̃(\underline{θ}, \underline{z}) Ã_{Q}(\underline{θ})

(with ∫_{C\underline{θ}} d\underline{z} = \frac{1}{m!} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ⋯ ∫_{C\underline{θ}} \frac{dz_{m}}{R}). We now proceed as usual in the algebraic

Bethe Ansatz and push the Ã_{Q}(\underline{θ}) and D̃_{Q}(\underline{θ}) through all the C̃-operators using the commutation rules (57) and (58) and obtain

C̃^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β_{1}}(\underline{θ}, z_{1}) Ã_{Q}(\underline{θ}) = ∏_{j=1}^{m} \frac{1}{b̃(θ′_{1} − z_{j})} Ã_{Q}(\underline{θ})
× C̃^{β_{m}}(\underline{θ}′, z_{m}) ⋯ C̃^{β_{1}}(\underline{θ}′, z_{1}) + ∑ uw_{A} ,

C̃^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β_{1}}(\underline{θ}, z_{1}) D̃_{Q}_{β}^{β′}(\underline{θ}) = ∏_{j=1}^{m} \frac{1}{b̃(z_{j} − θ′_{1})}T̃^{(1)} _{\underline{β} β′′}^{β′\underline{β}′}(\underline{z}, θ′_{1})D̃_{Q}_{β}^{β′′}(\underline{θ})

× C̃^{β′_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β′_{1}}(\underline{θ}, z_{1}) + ∑ 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

T̃^{(1)} _{\underline{β} β}^{β′\underline{β}′}(\underline{z}, θ) = (S̃_{10}(z_{1} − θ) ⋯ S̃_{m0}(z_{m} − θ))_{\underline{β} β}^{β′ \underline{β}′}

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 K(\underline{θ}) we first find that the wanted contribution from Ã_{Q} already gives the result we are looking for. Secondly the wanted contribution from D̃_{Q} applied to Ω gives zero. Thirdly the unwanted contributions from Ã_{Q} and D̃_{Q} cancel after integration over the z_{j}. All these three facts can be seen as follows. We have

Ω Ã_{Q}(\underline{θ}) = Ω , Ω D̃_{Q}_{β}^{β′}(\underline{θ}) = δ_{β}^{β′} ∏_{i=1}^{n} b̃(θ_{i} − θ_{1})Ω = 0  (60)

which follow from eq. (27). Therefore the wanted term from Ã_{Q} yields

[K(\underline{θ}) Ã_{Q}(\underline{θ})]^{w} = ∫_{C_{\underline{θ}′}} d\underline{z} ∏_{j=1}^{m} \frac{1}{b̃(θ′_{1} − z_{j})} h̃(\underline{θ}, \underline{z})p(\underline{θ}, \underline{z})Ψ̃(\underline{θ}′, \underline{z})

= σ ∫_{C_{\underline{θ}′}} d\underline{z} h̃(\underline{θ}′, \underline{z})p(\underline{θ}′, \underline{z})Ψ̃(\underline{θ}′, \underline{z})
= σK(\underline{θ}′) .

It has been used that the ‘shift relation’ of the function ϕ(θ) in (15) and (ii’) of (52) for the function p imply that

∏_{j=1}^{m} \frac{1}{b̃(θ′_{1} − z_{j})} h̃(\underline{θ}, \underline{z})p(\underline{θ}, \underline{z}) = σh̃(\underline{θ}′, \underline{z})p(\underline{θ}′, \underline{z}) .

The wanted contribution from D̃_{Q} vanish, since b̃(0) = 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 C̃- operators, where one C̃(\underline{θ}, z_{j}) is replaced by C̃_{Q}(\underline{θ}). Because of the symmetry (i)^{1} of L\underline{β}(\underline{z}) it is sufficient to consider only the unwanted terms for j = 1 which are denoted by uw_{A}^{1} and uw_{D}^{1}. They originate from the second term in the commutation rules (57) when Ã_{Q}(\underline{θ}) is commuted with C̃(\underline{θ}, z_{1}). Then the resulting Ã(\underline{θ}, z_{1}) pushed through the other C’s. Taking into account only the first terms in (56) we arrive at

uw_{A}^{1}(\underline{z}) = −\frac{c̃(θ′_{1} − z_{1})}{b̃(θ′_{1} − z_{1})} ∏_{j=2}^{m} \frac{1}{b̃(z_{1} − z_{j})} Ω Ã(\underline{θ}, z_{1}) C̃^{β_{m}}(\underline{θ}, z_{m}) … C̃_{Q}^{β_{1}}(\underline{θ})

Using (27) and Yang-Baxter relations (always taking into account the ‘SU(N) ice rule’ which means ‘color conservation’) and ã(θ) = 1 one obtains

[Ω Ã(\underline{θ}, z_{1}) C̃^{β_{m}}(\underline{θ}, z_{m}) … C̃_{Q}^{β_{1}}(\underline{θ})]\underline{α} = δ_{α_{1}}^{β_{1}} Φ̌\underline{β̃} ; \underline{α̃}(\underline{θ̌}, \underline{ž})

[figure: β_{1}; β_{2}; β_{m}; 1; 1; 1; α_{1}; α_{2}; α_{n}; 1; 1; 1; 1; 1; θ_{2}; θ_{n}; z_{2}; z_{1}; z_{m}; …; ⋮; θ_{1}] =[figure: β_{1}; β_{2}; β_{m}; 1; 1; 1; α_{1}; α_{2}; α_{n}; 1; 1; 1; θ_{2}; θ_{n}; z_{2}; z_{m}; …; ⋮; θ_{1}]

3 SU(3) FORM FACTORS 19 where Φ̌\underline{β̃} ; \underline{α̃}(\underline{θ̌}, \underline{ž}) does not depend on α_{1}, β_{1}, θ_{1}, z_{1}. Similarly, we obtain the first unwanted contribution from ∑_{β} D̃_{Q}_{β}^{β} as

uw_{D}^{1} = −\frac{c̃(z_{1} − θ_{1})}{b̃(z_{1} − θ_{1})} ∏_{j=2}^{m} \frac{1}{b̃(z_{j} − z_{1})} (T̃^{(1)} _{\underline{β}′β′′}^{β \underline{β}}(\underline{z}, z_{1}))

× ΩD̃_{β}^{β′′}(\underline{θ}, z_{1})C̃^{β′_{m}}(\underline{θ}, z_{m}) ⋯ C̃_{Q}^{β′_{1}}(\underline{θ})

which may be depicted as (T̃^{(1)}_{\underline{β}′ β′′}^{β \underline{β}}(\underline{z}, z_{1})) ΩD̃_{β}^{β′′}(\underline{θ}, z_{1})C̃^{β′_{m}}(\underline{θ}, z_{m}) ⋯ C̃_{Q}^{β′_{1}}(\underline{θ})

=[figure: β_{1}; β_{2}; β_{m}; 1; 1; 1; α_{1}; α_{2}; α_{n}; 1; 1; 1; β; β; θ_{2}; θ_{n}; z_{2}; z_{1}; z_{m}; …; ⋮; θ_{1}] .

Again (27) and Yang-Baxter relations yield

[Ω D̃_{β}^{β′′}(\underline{θ}, z_{1})C̃^{β′_{m}}(\underline{θ}, z_{m}) ⋯ C̃_{Q}^{β′_{1}}(\underline{θ})]\underline{α} = δ_{β}^{β′′} ∏_{i=1}^{n} b̃(θ_{i} − z_{1}) δ_{α_{1}}^{β′_{1}} Φ̌\underline{β̃}′ ; \underline{α̃}(\underline{θ̌}, \underline{ž})

while assumption (ii)^{(1)} for L\underline{β}(\underline{z}) means

L\underline{β}(\underline{z})Q^{(1)}\underline{β} ; \underline{β}′(\underline{z}) = L\underline{β}(\underline{z}′)

where analogously to eq. (55) Q^{(1)}\underline{β} ; \underline{β}′(\underline{z}) = T̃^{(1)}_{\underline{β}′ β}^{β \underline{β}}(\underline{z}, z_{1}) is the next level Qmatrix and \underline{z}′ = (z′_{1} = z_{1} + 2πi, …, z_{m}). Therefore we finally obtain

[K(\underline{θ}) Ã_{Q}(\underline{θ})]uw ; \underline{α}

= − ∫_{C_{\underline{θ}′}} d\underline{z} h̃(\underline{θ}, \underline{z}) p(\underline{θ}, \underline{z})\frac{c̃(θ′_{1} − z_{1})}{b̃(θ′_{1} − z_{1})} ∏_{j=2}^{m} \frac{1}{b̃(z_{1} − z_{j})} L\underline{β}(\underline{z})δ_{α_{1}}^{β_{1}} Φ̌\underline{β̃} ; \underline{α̃}(\underline{θ̌}, \underline{ž})

[K(\underline{θ}) D̃_{Q}_{β}^{β}]uw ; \underline{α}

= − ∫_{C\underline{θ}} d\underline{z} h̃(\underline{θ}, \underline{z}′) p(\underline{θ}, \underline{z}′)\frac{c̃(z_{1} − θ_{1})}{b̃(z_{1} − θ_{1})} ∏_{j=2}^{m} \frac{1}{b̃(z′_{1} − z_{j})} L\underline{β}(\underline{z}′)δ_{α_{1}}^{β_{1}} Φ̌\underline{β̃} ; \underline{α̃}(\underline{θ̌}, \underline{ž}) .

It has been used that the ‘shift relation’ of the function ϕ(θ) in (15) and (ii’) of (52) for the function p imply

∏_{i=1}^{n} b̃(θ_{i} − z_{1}) ∏_{j=2}^{m} \frac{1}{b̃(z_{j} − z_{1})} h̃(\underline{θ}, \underline{z})p(\underline{θ}, \underline{z}) = ∏_{j=2}^{m} \frac{1}{b̃(z′_{1} − z_{j})} h̃(\underline{θ}, \underline{z}′)p(\underline{θ}, \underline{z}′) .

For the D̃_{Q}-term we rewrite the z_{1}-integral by replacing z_{1} → z_{1} − 2πi (such that z′_{1} → z_{1} and C\underline{θ} → C\underline{θ} + 2πi) and obtain for the sum of the unwanted term from Ã_{Q} and D̃_{Q} (using c̃(z)/b̃(z) = −c̃(−z)/b̃(−z))

[K(\underline{θ}) (Ã_{Q}(\underline{θ}) + D̃_{Q}(\underline{θ}))]uw ; \underline{α} = (∫_{C_{\underline{θ}′}} d\underline{z} − ∫_{C\underline{θ}+2πi} dz_{1} ∫_{C\underline{θ}} d\underline{ž})

× \frac{c̃(z_{1} − θ′_{1})}{b̃(z_{1} − θ′_{1})}h̃(\underline{θ}, \underline{z}) p^{O}(\underline{θ}, \underline{z})L\underline{β}(\underline{z}) ∏_{j=2}^{m} \frac{1}{b̃(z_{1} − z_{j})} δ_{α_{1}}^{β_{1}} Φ̌\underline{β̃} ; \underline{α̃}(\underline{θ̃}, \underline{z̃}) .

Taking into account the pole structure of \frac{c̃(z_{1}−θ′_{1})}{b̃(z_{1}−θ′_{1})} h̃(\underline{θ}, \underline{z}) we may replace for the

Ã_{Q} -term

∫_{C_{\underline{θ}′}} d\underline{z} ⋯ = (−2πi Res_{z_{1}=θ′_{1}} + ∫_{C\underline{θ}} dz_{1}) ∫_{C_{\underline{θ}′}} d\underline{z̃}⋯

where again (∫_{C_{\underline{θ}′}} − ∫_{C\underline{θ}}) d\underline{z̃}⋯ = 0 and for the D̃_{Q}-term

∫_{C\underline{θ}+2πi} dz_{1} ∫_{C\underline{θ}} d\underline{z̃} ⋯ = (−2πi Res_{z_{1}=θ′_{1}} + ∫_{C\underline{θ}} dz_{1}) ∫_{C\underline{θ}} d\underline{z̃}⋯

which proves the cancellation of the unwanted terms.

The proof of (iii) is similar to that for the Z(3) model in [34]. We use the short notations

\underline{θ} = (θ_{1}, …, θ_{n}), \underline{θ̂} = (θ_{1}, θ_{2}, θ_{3}), \underline{θ̌} = (θ_{4}, …, θ_{n}),
\underline{α} = (α_{1}, …, α_{n}), \underline{α̂} = (α_{1}, α_{2}, α_{3}), \underline{α̌} = (α_{4}, …, α_{n}),
\underline{z} = (z_{1}, …, z_{m}), \underline{ẑ} = (z_{1}, z_{2}), \underline{ž} = (z_{3}, …, z_{m}),
\underline{β} = (β_{1}, …, β_{m}), \underline{β̂} = (β_{1}, β_{2}), \underline{β̌} = (β_{3}, …, β_{m})

and prove (44) which may be depicted as

Res_{θ_{23}=iη} Res_{θ_{12}=iη} [figure: O; …]= 2i\sqrt{2}Γ ( [figure: O; …] − [figure: O; …; ×]) .

We will show that the K-function given by the integral representation (22) with (23) satisfies

Res_{θ_{23}=iη} Res_{θ_{12}=iη} K_{1...n}(\underline{θ}) = c_{0} ∏_{i=4}^{n} ∏_{j=2}^{3} ϕ̃(θ_{ij}) ε_{123} K_{4...n}(\underline{θ̌}) (1 − σ_{3}S_{3n} … S_{34}) (61)
c_{0} = 2i\sqrt{2}ΓF^{−2}(iη)F^{−1}(2iη) .

This is equivalent to (44) because of the Ansatz (16) for F_{1...n}(\underline{θ}) and the relation of F (z) and ϕ(z) given by (51). The residues of K_{1...n}(\underline{θ}) consists of three terms

Res_{θ_{12}=iη} Res_{θ_{23}=iη} K_{1...n}(\underline{θ}) = R_{1...n}^{(1)} + R_{1...n}^{(2)} + R_{1...n}^{(3)} .

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_{1}- and the z_{2}-integrations and multiply the result by m(m − 1). The pinching points are

(1) z_{1} = θ_{2}, z_{2} = θ_{3},

(2) z_{1} = θ_{1}, z_{2} = θ_{2},

(3) z_{1} = θ_{2} − iη, z_{2} = θ_{3} − iη.

The contribution of (1) is given by z_{1}-and z_{2}-integrations ∮_{θ_{2}} dz_{1} ∮_{θ_{3}} dz_{2} ⋯ along small circles around z_{1} = θ_{2} and z_{2} = θ_{3} (see figure 1). The S-matrices S̃(θ_{2} − z_{1}) and S̃(θ_{3} − z_{2}) yield the permutation operator S̃(0) = P

(Ω C̃^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β_{2}}(\underline{θ}, θ_{3})C̃^{β_{1}}(\underline{θ}, θ_{2}))\underline{α} =[figure: β_{1}; β_{2}; β_{3}; β_{m}; 1; 1; 1; 1; 1; α_{1}; α_{2}; α_{3}; α_{4}; α_{n}; 1; 1; 1; 1; …; ⋮]

= S̃_{α′_{1}α_{3}}^{β_{2}1}(θ_{13})S̃_{α_{1}α_{2}}^{β_{1}α′_{1}}(θ_{12}) ∏_{j=3}^{m} b̃(θ_{1} − z_{j}) (ΩC̃^{β_{m}}(\underline{θ̌}, z_{m}) ⋯ C̃^{β_{3}}(\underline{θ̌}, z_{3}))\underline{α̌} .

We have used the fact that because of the SU(N) ice rule only the amplitude b̃ contributes to the S-matrices S̃(θ_{1} − z_{j}) and only ã = 1 to the S-matrices S̃(θ_{2} − z_{j}), S̃(θ_{3} − z_{j}), S̃(θ_{i} − z_{1}), S̃(θ_{i} − z_{2}) after having applied Yang-Baxter relations. Further we use that for z_{12} → θ_{23} → iη by assumption (iii)^{(1)} (c.f. (49))

L\underline{β}(\underline{z}) ≈ c_{1} ∏_{i=3}^{m} ϕ̃(z_{i2})S̃_{β_{1}β_{2}}^{32}(z_{12})L\underline{β̌}(\underline{ž})
c_{1} = ϕ̃(z_{12}) = ϕ̃(θ_{23}) = ϕ̃(iη)

and that because of (43)

Res_{θ_{12}=iη} Res_{θ_{23}=iη} ϕ̃(θ_{12})ϕ̃(θ_{13})S̃_{α_{1}α_{2}α_{3}}^{321}(θ_{1}, θ_{2}, θ_{3}) = ϕ̃(iη)ϕ̃(2iη)2 (iη)^{2} ϵ_{α_{1}α_{2}α_{3}} .

We combine this with the scalar functions h̃ and p and after having performed the remaining z_{j}-integrations we obtain

R(1) ; \underline{α} = c_{0} ∏_{i=4}^{n} ∏_{j=2}^{3} ϕ̃(θ_{ij}) ϵ\underline{α̂}K\underline{α̌}(\underline{θ̌})

because of b̃(θ_{1}−z)ϕ̃(θ_{1}−z)ϕ̃(θ_{2}−z)ϕ̃(z−θ_{3})ϕ̃(θ_{3}−z)τ(θ_{2}−z)τ(θ_{3}−z) = −1, the relation (iii’)_{3} of (52) for the p-functions p(\underline{θ}, θ_{2}, θ_{3}, \underline{ž})|_{θ_{12}=θ_{23}=iη} = (−1)^{m} p(\underline{θ̌}, \underline{ž}) and the relation (53) for the normalization constants N_{n}c_{1}2(iη)^{2}ϕ̃(iη)ϕ̃(2iη) = N_{n−3}c_{0}.

The remaining contribution to (61) is due to R_{1...n}^{(2)} + R_{1...n}^{(3)}. 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_{1...n}^{(2)} + R_{1...n}^{(3)}) S_{43} … S_{n3} + c_{0} ∏_{i=4}^{n} ∏_{j=2}^{3} ϕ̃(θ_{ij}) ε_{123} K_{4...n}(\underline{θ̌})σ_{3} )_{α_{1}α_{2}\underline{α̌}α_{3}} = 0 .

(62)

Note that because of (i) (c.f. (17))

K_{1...n}(\underline{θ})S_{43} … S_{n3} = ∏_{i=4}^{n} a(θ_{i3})K_{1...4...n3}(θ_{1}, …, θ_{4}, …, θ_{n}, θ_{3}) .

Due to Lemma 6 in appendix A it is sufficient to prove equation (62) only for α_{3} = 1 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 [51]). Therefore we consider this equation for the components with α_{3} = 1. The contribution of pinching (2) is given by the z_{1}-, z_{2}-integrations along the small circles around z_{1} = θ_{1} and z_{2} = θ_{2} (see again figure 1). Now S̃(θ_{1} − z_{1}) and S̃(θ_{2} − z_{2}) yield permutation operators P and the co-vector part of this contribution for α_{3} = 1 is

(Ω C̃^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β_{2}}(\underline{θ}, θ_{2})C̃^{β_{1}}(\underline{θ}, θ_{1})P_{3}(1))\underline{α}

=[figure: β_{1}; β_{2}; β_{m}; 1; 1; 1; 1; α_{1}; α_{2}; α_{4}; α_{3} = 1; 1; 1; 1; …; ⋮]

= δ_{α_{1}}^{β_{1}}δ_{α_{2}}^{β_{2}} (ΩC̃^{β_{m}}(\underline{θ̌}, z_{m}) ⋯ C̃^{β_{3}}(\underline{θ̌}, z_{3}))\underline{α̌} δ_{α_{3}}^{1} (63)

where P_{3}(1) projects onto the components with α_{3} = 1. We have used the fact that because of the SU(N) ice rule the amplitude a only contributes to the S-matrices S(θ_{1} − z_{j}), S(θ_{2} − z_{j}), S(θ_{3} − z_{j}), S(θ_{i} − z_{1}) after having applied Yang-Baxter relations. One derives the formula

Res_{θ_{23}=iη} Res_{θ_{12}=iη} ϕ̃(θ_{31})ϕ̃(θ_{32})S̃_{α_{1}α_{2}}^{32}(θ_{12}) = −ϕ̃(iη)ϕ̃(2iη)2(iη)^{2}ϵ_{α_{1}α_{2}1}

using (43), ϵ^{321} = 1, S̃_{1α_{1}α_{2}}^{321}(θ_{3} +2πi, θ_{1}, θ_{2}) = b(θ_{31} +2πi)b(θ_{32} +2πi)S̃_{α_{1}α_{2}}^{32}(θ_{12}) (because of the SU(3) ice rule) and ϕ̃(θ) = −b̃(θ+2πi)ϕ̃(θ+2πi) (see (15)). We combine this and (63) with the scalar functions h̃ and p taking into account the property (iii)^{(1)} of (43) and after having performed the remaining z_{j}-integrations we obtain

R_{1...n}^{(2)}S_{43} … S_{n3}P_{3}(1) = −c_{0} ∏_{i=4}^{n} ∏_{j=2}^{3} ϕ̃(θ_{ij}) ε_{123} K_{4...n}(\underline{θ̌})σ_{3}P_{3}(1) .  (64)

We have used the following equations a(θ_{i3})ϕ̃(θ_{i1}) = ϕ̃(θ_{i3}) (because of (15)), the definition (24) of τ(z) which implies ϕ̃(z − θ_{2})ϕ̃(θ_{1} − z)ϕ̃(θ_{2} − z)ϕ̃(θ_{3} − z)τ(θ_{1} − z)τ(θ_{2} − z) = 1, the second relation (iii’)_{3} of (52) for the p-functions p(\underline{θ}, θ_{1}, θ_{2}, \underline{ž})|_{θ_{12}=θ_{23}=iη} = σ p(\underline{θ̌}, \underline{ž}) and the relation (53) for the normalization constants N_{n}c_{1}ϕ̃(iη)ϕ̃(2iη)2(iη)^{2} = N_{n−3}c_{0}.

The contribution of pinching (3) is given by the z_{1}-, z_{2}-integrations along the small circles around z_{1} = θ_{2} − iη and z_{2} = θ_{3} − iη, (see again figure 1). Now S̃(θ_{3} − z_{2}) yields S̃_{αβ}^{δγ}(θ_{3} − z_{2}) → Γ_{(ρσ)}^{δγ}Γ_{αβ}^{(ρσ)} and the residue of the Bethe Ansatz state

i Res_{z_{2}=θ_{3}−iη} (Ω C̃(\underline{θ}, z_{m}) ⋯ C̃(\underline{θ}, z_{1})P_{n}(1) )\underline{α} = [figure: •; •; β_{1}; β_{2}; β_{m}; 1; 1; 1; 1; α_{1}; α_{2}; α_{3}; α_{n} = 1; 1; 1; 1; …; ⋮]

vanishes for α_{3} = 1 because Γ_{αβ}^{(ρσ)} is antisymmetric with respect to α, β. Therefore equation (62) is proved for α_{3} = 1 and because of Lemma 6 also in general.

The higher level Bethe Ansatz

Lemma 3 Let the constant R and the contour C\underline{z} be defined as in the context of (22). Then the higher level function

L\underline{β}(\underline{z}) = \frac{1}{k!} ∫_{C\underline{z}} \frac{du_{1}}{R} ⋯ ∫_{C\underline{z}} \frac{du_{k}}{R} h̃(\underline{z}, \underline{u})p^{(1)}(\underline{z}, \underline{u}) Ψ̃(1) ; \underline{β}(\underline{z}, \underline{u})
Ψ̃(1) ; \underline{β}(\underline{z}, \underline{u}) = (Ω^{(1)} C̃^{(1)}(\underline{z}, u_{k}) ⋯ C̃^{(1)}(\underline{z}, u_{1}))\underline{β}

h̃(\underline{z}, \underline{u}) = ∏_{i=1}^{m} ∏_{j=1}^{k} ϕ̃(z_{i} − u_{j}) ∏_{1≤i<j≤k} τ(u_{i} − u_{j}) .

satisfies (i)^{(1)} - (iii)^{(1)} of the equations (47) - (49) if

(i′)_{3}^{(1)} p_{mk}^{(1)}(\underline{z}, \underline{u}) is symmetric under z_{i} ↔ z_{j}
(ii′)_{3}^{(1)} \{ \begin{matrix}p_{mk}^{(1)}(z_{1} + 2πi, z_{2}, …, \underline{u}) = (−1)^{k}p_{mk}^{(1)}(z_{1}, z_{2}, …, \underline{u}) \\ p_{mk}^{(1)}(\underline{z}, u_{1} + 2πi, u_{2}, …) = (−1)^{m}p_{mk}^{(1)}(\underline{z}, u_{1}, u_{2}, …) \end{matrix}
(iii′)_{3}^{(1)} p_{mk}^{(1)}(\underline{z}, \underline{u})|_{z_{12}=iη,u_{1}=z_{2}} = (−1)^{k−1}p_{m−2k−1}^{(1)}(\underline{ž}, \underline{ǔ}) .

(65)

4 SU(N) FORM FACTORS 24

Proof. The proofs of the second level equations (i)^{(1)} and (ii)^{(1)} are similar to the ones for the first level. For the proof of (iii)^{(1)} we observe that for z_{12} → iη there is a pinching at u_{j} = z_{2} which means that in Ψ̃(1) ; \underline{β}(\underline{z}, \underline{u}) we may

replace S̃(z_{2} − u_{1}) → P and we have to consider (for z_{12} → iη and u_{1} = z_{2})

(Ω^{(1)} C̃^{(1)}(\underline{z}, u_{k}) ⋯ C̃^{(1)}(\underline{z}, z_{2}))\underline{β} = [figure: z_{1}; z_{2}; z_{3}; z_{m}; β_{1}; β_{2}; β_{3}; β_{m}; u_{2}; u_{k}; u_{1} = z_{2}; …; ⋮; 3; 3; 3; 2; 2; 2; 2; 2; 2; 2]

= S̃_{β_{1}β_{2}}^{32}(z_{12}) ∏_{j=2}^{k} b̃(z_{1} − u_{j})Ψ̌\underline{β̌}(\underline{ž}, \underline{ǔ})

with \underline{β̌} = (β_{3}, …, β_{m}), \underline{ž} = (z_{3}, …, z_{m}), \underline{ǔ} = (u_{2}, …, u_{k}). Therefore because of b̃(z_{1} − u)ϕ̃(z_{1} − u)ϕ̃(z_{2} − u)τ(z_{2} − u) = −1 and (iii′)_{3}^{(1)} for z_{12} → iη

L\underline{β}(\underline{z}) ≈ ϕ̃(z_{12}) ∏_{i=3}^{m} ϕ̃(z_{i2})S̃_{β_{1}β_{2}}^{23}(z_{12})L\underline{β̌}(\underline{ž})

which proves the claim (iii)^{(1)} of (49).

4 SU(N) form factors

In this section we perform the form factor program for the general SU(N) Smatrix. For this purpose, we extend the procedures of the previous section, i.e., the nested Bethe Ansatz method now with N − 1 levels combined with the off-shell Bethe Ansatz. Applications of the results to the SU(N) Gross-Neveu model [13] will be investigated in a separate article [39].

4.1 S-matrix

The SU(N) S-matrix is given by (1) and (3). Again, the eigenvalue S_{−}(θ) has a pole at θ = iη = \frac{2}{N} iπ which means that there exist bound states of r fundamental particles α_{1} + ⋯ + α_{r} → (ρ_{1} … ρ_{r}) (with ρ_{1} < ⋯ < ρ_{r}) which transform as the anti-symmetric SU(N) tensor representation of rank r, (0 < r < N). The masses of the bound states satisfy m_{r} = m_{N−r} which suggests Swieca’s [45, 44, 38] picture that the antiparticle of a particle of rank r is to be identified with the particle of rank N − r (see also [33, 34]).

Iterating N − 1 times the general bound state formula one obtains for the scattering of a bound state (β_{1}β_{2} … β_{N−1}) with another particle δ (analogously 4 SU(N) FORM FACTORS 25 to (37) for the SU(3) case)

S_{(β_{1}β_{2}...β_{N−1})δ}^{δ′(γ_{1}γ_{2}...γ_{N−1})}(θ)Γ_{α_{1}α_{2}...α_{N−1}}^{(β_{1}β_{2}...β_{N−1})}

= Γ_{α′_{1}α′_{2}...α′_{N−1}}^{(γ_{1}γ_{2}...γ_{N−1})}S_{α_{1}δ_{1}}^{δ′α′_{1}}(θ + iπ − iη) … S_{α_{N−1}δ}^{δ_{N−2}α′_{N−1}}(θ − iπ + iη)

with the total bound state fusion intertwiner

Γ_{α_{1}α_{2}...α_{N−1}}^{(β_{1}β_{2}...β_{N−1})} = Γ_{(β_{1}β_{2}...β_{N−2})α_{N−1}}^{(β_{1}β_{2}...β_{N−1})} … Γ_{(β_{1}β_{2})α_{3}}^{(β_{1}β_{2}β_{3})}Γ_{α_{1}α_{2}}^{(β_{1}β_{2})} .

Taking special cases for the external particles we obtain

S_{(1,2...,N−1)N}^{N(1,2...,N−1)}(θ) = b(θ + iπ − iη) … b(θ − iπ + iη) = (−1)^{N−1}a(πi − θ)
S_{(1,2...,N−1)N−1}^{N−1(1,2...,N−1)}(θ) = b(θ + iπ − iη) … a(θ − iπ + iη) = (−1)^{N−1}b(πi − θ)
S_{(2,3...,N)1}^{N(1,2...,N−1)}(θ) = −b(θ + iπ − iη) … c(θ − iπ + iη) = (−1)^{N−1}c(πi − θ) .

which may be written as S_{(α_{1}...α_{N−1})α_{N}}^{β_{N} (β_{1}...β_{N−1})} (θ) = (−1)^{N−1} (δ_{(α_{1}...α_{N−1})}^{(β_{1}...β_{N−1})}δ_{α_{N}}^{β_{N}} b(πi − θ) + ϵ^{β_{N} β_{1}...β′_{N−1}}ϵ_{α_{1}...α_{N−1}α_{N}} c(πi − θ))

with the total anti-symmetric tensors ϵ_{α_{1}...α_{N}} and ϵ^{α_{1}...α_{N}} (ϵ_{1...N} = ϵ^{N...1} = 1). These results may be interpreted as an unusual crossing relation ϵ_{β_{1}β_{2}...β_{N−1}β} S_{α_{1}δ_{1}}^{δ′β_{1}}(θ_{1}) … S_{α_{N−1}δ}^{δ_{N−2}β_{N−1}}(θ_{N−1}) = (−1)^{N−1}ϵ_{α_{1}α_{2}...α_{N−1}γ} S_{δβ}^{γδ′}(iπ − θ) (66)

[figure: •; •; …; •; •; α_{1}; α_{2}; α_{N−1}; δ′; δ; β; = (−1)^{N−1}; •; •; …; •; •; α_{1}; α_{2}; α_{N−1}; δ′; δ; β; γ]

with θ_{j} = θ + iπ − jiη if we write the charge conjugation matrices as

C_{(α_{1}...α_{N−1})α_{N}} = C_{α_{1(}α_{2}...α_{N})} = ϵ_{α_{1}...α_{N}} ,
C^{(α_{1}...α_{N−1})α_{N}} = C^{α_{1(}α_{2}...α_{N})} = ϵ^{α_{1}...α_{N}} .

Therefore we have the relations (c.f. (29) and (41)) C_{α(α_{1}...α_{N−1})}C^{(α_{1}...α_{N−1})β} = δ_{α}^{β} , C_{α(α_{1}...α_{N−1})}A_{β}^{α}C^{β(α_{1}...α_{N−1})} = (−1)^{N−1} tr A

and C_{(β_{1}...β_{N−1})γ}Γ_{α_{1}...α_{N−1}}^{(β_{1}...β_{N−1})} = ϵ_{α_{1}...α_{N−1}γ}Γ  (67)

where the constant Γ is

Γ = i\sqrt{\frac{1}{2π} NΓ^{N} (1 − \frac{1}{N} )} .

These results are consistent with the picture (c.f. [45]) that the bound state of N − 1 particles of rank 1 is to be identified with the anti-particle of rank 1. As already remarked, the physical aspects of these facts will be discussed elsewhere [39].

For later convenience we consider as a generalization of (43) the total Nparticle S-matrix (consisting of N(N − 1)/2 factors) in terms of S̃

S̃_{12...N} = (S̃_{12}S̃_{13} … S̃_{1N}) (S̃_{23} … S̃_{2N}) … S̃_{N−1N}  (68)

in the limit θ_{jj+1} → iη (j = 1, … N − 1) which behaves as

S̃_{α_{1}α_{2}...α_{N}}^{β_{N} ...β_{2}β_{1}} ≈ (N − 1)!\frac{iη}{θ_{12} − iη} … \frac{iη}{θ_{N−1N} − iη} ϵ^{β_{N} ...β_{2}β_{1}}ϵ_{α_{1}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 S̃_{ii+1} ∼ (1 − P)_{ii+1} 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 (4).

4.2 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 SU(N) vector representation) are involved, we have to apply iteratively the bound state relation (iv) to get the (N − 1)-bound state which is to be identified with the anti-particle

Res_{θ_{12}=iη} … Res_{θ_{N−2N−1}=iη} F_{1...n}^{O}(\underline{θ})

= F_{(1...N−1)N...n}^{O}(θ_{(1...N−1)}, θ_{N}, …, θ_{n})\sqrt{2}^{N−2}Γ_{1...N−1}^{(1...N−1)}

and finally the annihilation residue equation (iii)

Res_{θ_{(1...N−1)N} =iπ} F_{(1...N−1)N...n}^{O}(θ_{(1...N−1)}, θ_{N}, …, θ_{n})

= 2iC_{(1...N−1)N}F_{N+1...n}^{O}(θ_{N+1}, …, θ_{n}) (1 − σ_{N}^{O}S_{Nn} … S_{NN+1}) .

Similar as for N = 3 we obtain with (67)

Res_{θ_{N−1N} =iη} … Res_{θ_{12}=iη} F_{123...n}^{O}(θ_{1}, …, θ_{n})

= 2iε_{1...N} \sqrt{2}^{N−2}ΓF_{N+1...n}^{O}(θ_{N+1}, …, θ_{n}) (1 − σ_{N}^{O}S_{Nn} … S_{NN+1}) . (70)

Ansatz for form factors: We propose the n-particle form factors of an operator O(x) as given by the same formula (16) as for SU(2) and SU(3) in terms of the K-function and the minimal form factor function F (θ) given by (12) which belongs to the highest weight w = (2, 0, …, 0). The form factor equations (i) and (ii) for K-function write again as (17) and (18). Consistency of (70), (18) and the crossing relation (66) means that the statistics factors are of the form σ_{α}^{O} = σ^{O}(r_{α}) if the particle of type α belongs to an SU(N) representation of rank r_{α} = 1, …, N − 1 and

σ^{O}(r) = e^{iπ(1−1/N)rQ_{O}} for Q_{O} = n mod N  (71)

as an extension of (33) and (45). For the K-function KO ; \underline{α}(\underline{θ}) we propose again the Ansatz in form of the integral representation (22) with (23) and (24). The Bethe Ansatz co-vector Ψ̃\underline{α}(\underline{θ}, \underline{z}) is again of the form (46) and for the function L\underline{β}(\underline{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 SU(N) case we have to iterate this N − 2 times

Ψ̃(l−1) ; \underline{α}(\underline{θ}, \underline{z}) = L(l) ; \underline{β}(\underline{z})Φ̃(l−1)\underline{β} ; \underline{α}(\underline{θ}, \underline{z})
L(l) ; \underline{β}(\underline{z}) = \frac{1}{k!} ∫_{C\underline{z}} \frac{du_{1}}{R} ⋯ ∫_{C\underline{z}} \frac{du_{k}}{R} h̃(\underline{z}, \underline{u})p^{(l)}(\underline{z}, \underline{u}) Ψ̃(l) ; \underline{β}(\underline{z}, \underline{u})  (72)

h̃(\underline{z}, \underline{u}) = ∏_{i=1}^{m} ∏_{j=1}^{k} ϕ̃(θ_{i} − z_{j}) ∏_{1≤i<j≤k} τ(z_{i} − z_{j})

for l = 1, …, N − 2, α_{i} = l, …, N, β_{i} = l + 1, …, N, (Ψ̃^{(0)} = Ψ̃). By this procedure one obtains the nested Bethe Ansatz^{4}.

As for the Z(N) and A(N − 1) models [33, 34] the ‘Jost-function’ ϕ (θ) = ϕ̃(θ)/a (θ) = ϕ̃(−θ) is a solution of the equation

∏_{k=0}^{N−2} ϕ (θ + kiη) ∏_{k=0}^{N−1} F (θ + kiη) = 1  (73)

which is typical for models where the bound state of N − 1 particles is the anti-particle. The solution is

ϕ(z) = Γ (\frac{z}{2πi}) Γ (1 − \frac{1}{N} − \frac{z}{2πi})

and again we define τ(z) = 1/(ϕ(z)ϕ(−z)).

The higher level Bethe Ansatz: In order that the form factor F^{O} (\underline{θ}) given by (16) and (72) satisfies the form factor equations (i) - (iii) the higher level L-functions L_{β_{1},β_{2},...,β_{m}}^{(l)}(z_{1}, z_{2}, …, z_{m}), (l < β_{i} ≤ N, m = n_{l}) have to satisfy:

(i)^{(l)} Watson’s equations

L_{...ij...}^{(l)}(…, z_{i}, z_{j}, …) = L_{...ji...}^{(l)}(…, z_{j}, z_{i}, …)S̃_{ij}(z_{ij}) ,

(ii)^{(l)} the crossing relation

L_{β_{1},β_{2},...,β_{m}}^{(l)}(z_{1}, z_{2}, …, z_{m}) = L_{β_{2},...,β_{m},β_{1}}^{(l)}(z_{2}, …, z_{m}, z_{1} − 2πi)

and (iii)^{(l)} the function L(l) ; \underline{β}(\underline{z}) has to possess simple poles at z_{12}, …, z_{N−l−1,N−l} =

4In [35] the nested off-shell Bethe Ansatz was formulated in terms of “Jackson-type integrals” instead of contour integrals.

iη and it has to factorize in the neighborhood of these poles as

L(l) ; \underline{β}(\underline{z}) ≈ c_{l} ∏_{i=N−l+1}^{m} ∏_{j=2}^{N−l} ϕ̃(z_{ij}) S̃N...l+1 ; \underline{β̂}(\underline{ẑ}) L\underline{β̌}(\underline{ž})  (74)

[figure: L^{(l)}; \underline{β}; …] ≈ c_{l} ∏_{i=N−l+1}^{mn_{l}} ∏_{j=2}^{N−l} ϕ̃(z_{ij}) [figure: S̃; \underline{β̂}; N l+1; ...; ...; L^{(l)}; \underline{β̌}; …]

where we have used the short notations \underline{β̂} = (β_{1}, …, β_{N−l}),

\underline{β̌} = (β_{N−l+1}, …, β_{m}) and \underline{ẑ} = (z_{1}, …, z_{N−l}), \underline{ž} = (z_{N−l+1}, …, z_{m}). The S-matrix S̃N...l+1 ; \underline{β̂}(\underline{ẑ}) describes the total scattering of the N − l particles

with rapidities \underline{ẑ} for the initial quantum numbers \underline{β̂} and the final configuration of quantum numbers (N, …, l + 1).

The constants satisfy the recursion relation

c_{l} = c_{l+1} ∏_{j=1}^{N−l−1} ϕ̃(jiη)  (75)

with the solution

c_{1} = ϕ̃^{N−2}(iη)ϕ̃^{N−3}(2iη) ⋯ ϕ̃((N − 2)iη)  (76)

if c_{N−1} = 1 (see appendix C).

Lemma 4 The higher level function L(l) ; \underline{β}(\underline{z}) satisfies (i)^{(l)} - (iii)^{(l)}, if the higher level p-function in the integral representation (72) satisfies

(i′)^{(l)} p^{(l)}(\underline{z}, \underline{u}) is symmetric under z_{i} ↔ z_{j}
(ii′)^{(l)} \{ \begin{matrix}p_{mk}^{(l)}(z_{1} + 2πi, z_{2}, …, \underline{u}) = (−1)^{k}p_{mk}^{(l)}(z_{1}, z_{2}, …, \underline{u}) \\ p_{mk}^{(l)}(\underline{z}, u_{1} + 2πi, u_{2}, …) = (−1)^{m}p_{mk}^{(l)}(\underline{z}, u_{1}, u_{2}, …) \end{matrix}
(iii′)^{(l)} p_{mk}^{(l)}(\underline{z}, z_{2}, … z_{N−l}, \underline{ǔ})|_{z_{12}=···=z_{z_{N−l−1,N−l}}=iη} = (−1)^{k−N−l+1} p_{m̌ǩ}^{(l)}(\underline{ž}, \underline{ǔ})

(77)

with m̌ = M − N + l, ǩ = M − N + l − 1.

This lemma is proved in appendix C.

The p-function: In order that the form factor F^{O} (θ) given by (16) and (22)

- (24) satisfies the form factor equations (i) - (iii) the p-function p^{O}(\underline{θ}, \underline{z}) in (22) which depends on the explicit nature of the local operator O is assumed to satisfy

(i′) p_{nm}(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j}
(ii′) \{ \begin{matrix}σp_{nm}(θ_{1} + 2πi, θ_{2}, …, \underline{z}) = (−1)^{m+N−1}p_{nm}(θ_{1}, θ_{2}, …, \underline{z}) \\ p_{nm}(\underline{θ}, z_{1} + 2πi, z_{2}, …) = (−1)^{n}p_{nm}(\underline{θ}, z_{1}, z_{2}, …) \end{matrix}
(iii′) p_{nm}(\underline{θ}, θ_{2}, …, θ_{N}, \underline{ž})|_{θ_{12}=···=θ_{N−1N} =iη} = (−1)^{m−N+1} p_{n−Nm−N+1}(\underline{θ̌}, \underline{ž})
p_{nm}(\underline{θ}, θ_{1}, …, θ_{N−1}, \underline{ž})|_{θ_{12}=···=θ_{N−1N} =iη} = σ p_{n−Nm−N+1}(\underline{θ̌}, \underline{ž})

(78)

where \underline{θ} = (θ_{1}, …, θ_{n}), \underline{θ̌} = (θ_{N+1}, …, θ_{n}), \underline{z} = (z_{1}, …, z_{m}) and \underline{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 O(x).

Theorem 5 The co-vector valued function F\underline{α}(\underline{θ}) given by the Ansatz (16) and the integral representation (22) satisfies the form factor equations (i), (ii) and (iii) of (7) – (9), in particular (70) if

1. L\underline{β}(\underline{z}) satisfies the equation (i)^{(1)}, (ii)^{(1)}, (iii)^{(1)} and (76) of lemma 4,

2. p^{O}(\underline{θ}, \underline{z}) satisfies the equation (i′), (ii′) and (iii′) of (78) and

3. the normalization constants satisfy

(N − 1)!(iη)^{N−1} ( ∏_{j=1}^{N−1} (ϕ̃(jiη)F (jiη))^{N−j}) N_{n} = 2i\sqrt{2}^{N−2}ΓN_{n−N} .

(79)

The proof of this theorem can be found in appendix B.

4.3 Examples

To illustrate our general results we present some simple examples.

The energy momentum tensor: For the local operator O(x) = T^{ρσ}(x) (where ρ, σ = ± denote the light cone components) the p-function is, as for the sine-Gordon model in [47]

p^{T^{ρσ}}(\underline{θ}, \underline{z}) = ∑_{i=1}^{n} e^{ρθ_{i}} ∑_{i=1}^{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 SU(N) weights are (see [35])

w = (n − n_{1}, n_{1} − n_{2}, …, n_{N−2} − n_{N−1}, n_{N−1}) = (1, 1, …, 1, 1) .

We calculate the form factor of the particle α and the bound state (\underline{β}) = (β_{1}, …, β_{N−1}) of N − 1 particles. In each level all integrations up to one may be performed iteratively using the bound state relation (iv) (similar as in the proof of theorem 5). Then all remaining integrations in the higher levels can be done by means of the formula

∫_{−i∞}^{i∞} dsΓ(a+s)Γ(b+s)Γ(c−s)Γ(d−s) = 2πi\frac{Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)}{Γ(a + b + c + d)} .

The result for the form factor of the particle α and the bound state (\underline{β}) writes as

F_{α(\underline{β})}^{T^{ρσ}}(θ_{1}, θ_{2}) = K_{α(\underline{β})}^{T^{ρσ}}(θ_{1}, θ_{2}) G(θ_{12})
K_{α(\underline{β})}^{T^{ρσ}}(θ_{1}, θ_{2}) = N_{2}^{T^{ρσ}} (e^{ρθ_{1}} + e^{ρθ_{2}}) ∫_{C\underline{θ}} \frac{dz}{R} ϕ̃(θ_{1} − z)e^{σz}L(θ_{2} − z)  (80)
× ϵ_{δ\underline{γ}}S̃_{αϵ}^{δ1}(θ_{1} − z)S̃_{(\underline{β})1}^{ϵ(\underline{γ})}(θ_{2} − z)

where the summation is over \underline{γ} and δ > 1 and G(θ) is the minimal form factor function of two particles of rank r = 1 and r = N − 1. The functions G(θ) and L(θ) are given by

G(iπ − θ)F (θ)ϕ(θ) = 1

L(θ) = Γ (\frac{1}{2} + \frac{θ}{2πi}) Γ (−\frac{1}{2} + \frac{1}{N} − \frac{θ}{2πi}) .

The remaining integral in(80) may be performed (similar as in [47]) with the result^{5}

⟨ 0 | T^{ρσ}(0) | θ_{1}, θ_{2} ⟩_{α(\underline{β})}^{in} = 4M^{2}ϵ_{α\underline{β}}e^{\frac{1}{2} (ρ+σ)(θ_{1}+θ_{2}+iπ)} \frac{sinh \frac{1}{2} (θ_{12} − iπ)}{θ_{12} − iπ}G(θ_{12}) .

Similar as in [47] one can prove the eigenvalue equation

(∫ dxT^{±0}(x) − ∑_{i=1}^{n} p_{i}^{±}) | θ_{1}, …, θ_{n}⟩in ; \underline{α} = 0

for arbitrary states.

The fields ψ_{α}(x): Because the Bethe Ansatz yields highest weight states we obtain the matrix elements of the spinor field ψ(x) = ψ_{1}(x). The p-function for the local operator ψ^{(±)}(x) is (see also [32])

p^{ψ^{(±)}}(\underline{θ}, \underline{z}) = exp ±\frac{1}{2} (∑_{i=1}^{m} z_{i} − (1 − \frac{1}{N} ) ∑_{i=1}^{n} θ_{i}) .

For example the 1-particle form factor is

⟨ 0 | ψ^{(±)}(0) | θ ⟩_{α} = δ_{α1} e^{∓ \frac{1}{2}(1− \frac{1}{N})θ} .

5In [52, 35] this result has been obtained using Jackson type integrals.

The last two formulae are consistent with the proposal of Swieca et al. [45, 38] that the statistics of the fundamental particles in the chiral SU(N) Gross-Neveu model should be σ = exp (2πis), where s = \frac{1}{2} (1 − \frac{1}{N} ) is the spin (see also (71)). For the n = N + 1 particle form factor there are again n_{l} = N − l integrations in the l-th level of the off-shell Bethe Ansatz and the SU(N) weights are w = (2, 1, …, 1, 1). Similar as above one obtains the 2-particle and 1-bound state form factor

F_{αβ(\underline{γ})}^{ψ^{(±)}}(θ_{1}, θ_{2}, θ_{3}) = K_{αβ(\underline{γ})}^{ψ^{(±)}}(θ_{1}, θ_{2}, θ_{3})F (θ_{12})G(θ_{13})G(θ_{23})
K_{αβ(\underline{γ})}^{ψ^{(±)}} = N^{ψ}e^{∓ \frac{1}{2}(1− \frac{1}{N}) ∑ θ_{i}} ∫_{C\underline{θ}} \frac{dz}{R} ϕ̃(θ_{1} − z)ϕ̃(θ_{2} − z)L(θ_{3} − z)e^{± \frac{1}{2} z}
× ϵ_{δ\underline{γ}}S̃_{α_{1}ϵ}^{δ1}(θ_{1} − z)S̃_{α_{2}ζ}^{ϵ1}(θ_{2} − z)S̃_{(\underline{β})1}^{ζ(\underline{γ})}(θ_{3} − z) .

We were not able to perform this integration. In [39] we will discuss the 1/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 66.2002/1998-99 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-2002-00325.

Appendix

A A Lemma

Lemma 6 Let v^{1...n} ∈ V^{1...n} be a highest weight vector, i.e. E_{1...n}v^{1...n} = 0. Then v^{1...n} vanishes if the components v^{α_{1}α_{2}...α_{n}} for α_{1} = 1 vanish.

Proof. By definition E acts on the basis vectors e1...n ; \underline{α} = e_{α_{1}} ⊗e_{α_{2}} ⊗⋯⊗e_{α_{n}} as

E_{1...n}e1...n ; \underline{α} = ∑_{i=1}^{n} e_{α_{1}} ⊗ ⋯ ⊗ Ee_{α_{i}} ⊗ ⋯ ⊗ e_{α_{n}}
Ee_{α} = e_{α−1} where e_{0} = 0 .

Therefore we may write

0 = E_{1...n}v^{1...n} = ∑_{\underline{α},α_{1}>1} v^{α_{1}α_{2}...α_{n}}E_{1...n}e1...n ; \underline{α}

= ∑_{\underline{α},α_{1}>1} v^{α_{1}α_{2}...α_{n}} ∑_{i=1}^{n} e_{α_{1}} ⊗ ⋯ ⊗ e_{α_{i}−1} ⊗ ⋯ ⊗ e_{α_{n}}

= ∑_{\underline{α},α_{1}=2} v^{2α_{2}...α_{n}}e_{1} ⊗ ⋯ ⊗ e_{α_{n}} + ∑_{\underline{α},α_{1}>2} w^{α_{1}α_{2}...α_{n}}e_{α_{1}} ⊗ ⋯ ⊗ e_{α_{n}}

for some vector w^{1...n}. Because the second term w^{1...n} is orthogonal to the first one all components v^{2α_{2}...α_{n}} vanish. Iterating this procedure proves the claim.

B Proof of Theorem 5

The proof of the main theorem of this article for SU(N) is a straightforward extension of the one for SU(3) above.

Proof. The form factor equations (i) and (ii) may be proved quite analogously as for SU(3). The proof of (iii) is similar to that for the Z(N) model in [34]. We use the short hand notations

\underline{θ} = (θ_{1}, …, θ_{n}), \underline{θ̂} = (θ_{1}, …, θ_{N}), \underline{θ̌} = (θ_{N+1}, …, θ_{n}),
\underline{α} = (α_{1}, …, α_{n}), \underline{α̂} = (α_{1}, …, α_{N}), \underline{α̌} = (α_{N+1}, …, α_{n}),
\underline{z} = (z_{1}, …, z_{m}), \underline{ẑ} = (z_{1}, …, z_{N−1}), \underline{ž} = (z_{N}, …, z_{m}),
\underline{β} = (β_{1}, …, β_{m}), \underline{β̂} = (β_{1}, …, β_{N−1}), \underline{β̌} = (β_{N}, …, β_{m}) .

That F_{1...n}^{O}(\underline{θ}) given by (16), (22), (23) and (72) satisfies (iii) in the form of (70) is equivalent to that K_{1...n}(\underline{θ}) satisfies

Res_{θ_{N−1N} =iη} … Res_{θ_{12}=iη} K_{1...n}(\underline{θ}) = c_{0} ∏_{i=N+1}^{n} ∏_{j=2}^{N} ϕ̃(θ_{ij}) ε_{1...N}K_{N+1...n}(\underline{θ̌})

× (1 − σ_{N}S_{Nn} … S_{NN+1}) (81)

c_{0} = 2i\sqrt{2}^{N−2}Γ ∏_{j=1}^{N−1} F^{−(N−j)}(jiη)

where the relation of F (z) and ϕ(z) given by (73) has been used. The residues of K_{1...n}(\underline{θ}) consists again of three terms

Res_{θ_{N−1N} =iη} … Res_{θ_{12}=iη} K_{1...n}(\underline{θ}) = R_{1...n}^{(1)} + R_{1...n}^{(2)} + R_{1...n}^{(3)}

because N − 1 of the z integration contours will be “pinched” at three points. Again due to symmetry it is sufficient to determine the contribution from the z_{1}, …, z_{N−1}-integrations and multiply the result by m … (m − N + 2). The pinching points are (1) z_{1} = θ_{2}, …, z_{N−1} = θ_{N},

(2) z_{1} = θ_{1}, …, z_{N−1} = θ_{N−1},

(3) z_{1} = θ_{2} − iη, …, z_{N−1} = θ_{N} − iη,

The contribution of (1) is given by N − 1 integrations along small circles around z_{1} = θ_{2}, z_{2} = θ_{3}, …, z_{N−1} = θ_{N} (see figure 1). The S-matrices S̃(θ_{2} − z_{1}), …, S̃(θ_{N} − z_{N−1}) yield the permutation operator S̃(0) = P. Therefore for θ_{12}, …, θ_{N−2N−1}, θ_{N−1N} → iη

(Ω C̃^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β_{N}} (\underline{θ}, z_{N})C̃^{β_{N−1}}(\underline{θ}, θ_{N}) ⋯ C̃^{β_{1}}(\underline{θ}, θ_{2}))\underline{α}

= [figure: ⋮; :; …; …; θ_{1}; θ_{N}; θ_{n}; α_{1}; α_{N}; α_{n}; β_{1}; β_{N−1}; β_{m}; z_{1}; z_{N−1}; z_{m}; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1]

= ∏_{j=N}^{m} b̃(θ_{1} − z_{j}) (S̃_{1N} … S̃_{12})\underline{β̂},1 ; \underline{α̂} (ΩC̃^{β_{m}}(\underline{θ̌}, z_{m}) ⋯ C̃^{β_{N}} (\underline{θ̌}, z_{N}))\underline{α̌} . (82)

It has been used that due to the SU(N) ice rule only the amplitude b(·) contributes to the S-matrices S(θ_{1} − z_{j}) and a(·) to the S-matrices S(θ_{2} − z_{j}), …, S(θ_{N} −z_{j}), S(θ_{i}−z_{1}), …, S(θ_{i}−z_{N−2}) after having applied Yang-Baxter relations. One observes that the product of S-matrices in (74) together with the one in (82) yields the total N − 1 S-matrix

S̃N...2 ; \underline{β̂}(\underline{ẑ}) (S̃_{1N} … S̃_{12})\underline{β̂},1 ; \underline{α̂} = S̃N...21 ; \underline{α̂}(\underline{θ̂})

for which the residue formula (69) applies

Res_{θ_{12}=iη} … Res_{θ_{N−1N} =iη} S̃_{α_{1}...α_{N}}^{N...21} = (N − 1)!(iη)^{N−1}ϵ^{N...21}ϵ_{α_{1}...α_{N}} .

We combine (82) with the function L\underline{β}(\underline{z}) with the property (74) for (l = 1) and the scalar functions h̃ and p and after having performed the remaining z_{j}-integrations we obtain

R_{1...n}^{(1)} = c_{0} ∏_{i=N+1}^{n} ∏_{j=2}^{N} ϕ̃(θ_{ij}) ε_{1...N}(\underline{θ̂})K_{N+1...n}(\underline{θ̌}) .

We have used the following equations b̃(θ_{1} − z)ϕ̃(θ_{1} − z) = −ϕ̃(z − θ_{2}) (for θ_{12} = iη) together with the definition (35) of τ(z), the relation (iii’) of (78) for the p-function p(\underline{θ}, θ_{2}, …, θ_{N}, \underline{ž}) = (−1)^{m−N+1} p(\underline{θ̌}, \underline{ž}) (for θ_{12} = ⋯ = θ_{N−1N} = iη), the relation (79) for the normalization constants N_{n}c_{1}(N − 1)!(iη)^{N−1} ∏_{j=1}^{N−1} ϕ̃(jiη) = N_{n−N}c_{0} and the recursion relation (75) for the constants c_{l} with the solution (86).

The remaining contribution to (81) is due to R_{2} and R_{3}

R_{1...n}^{(2)} + R_{1...n}^{(3)} = −c_{0} ∏_{i=N+1}^{n} ∏_{j=2}^{N} ϕ̃(θ_{ij}) ε_{1...N}K_{N+1...n}(\underline{θ̌})σ_{N}S_{Nn} … S_{NN+1}

It is convenient to shift the particle with momentum θ_{N} to the right by applying S-matrices and write

(R_{1...n}^{(2)} + R_{1...n}^{(3)}) S_{N+1N} … S_{nN} + c_{0} ∏_{i=N+1}^{n} ∏_{j=2}^{N} ϕ̃(θ_{ij}) ε_{1...N}K_{N+1...n}(\underline{θ̌})σ_{N} = 0

(83)

where the components of this co-vector are now denoted by v_{α_{1}...α_{N−1N+1}...α_{n}α_{N}}. Note that because of (i) (17)

K_{1...n}(\underline{θ})S_{N+1N} … S_{nN} = ∏_{i=N+1}^{n} a(θ_{iN})K_{1...N+1...nN}(θ_{1}, …, θ_{N+1}, …, θ_{n}, θ_{N}) .

Again due to Lemma 6 in appendix A it is sufficient to prove equation (83) only for α_{N} = 1 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 [51]). Therefore we consider this equation for the components with α_{N} = 1 only. The contribution of R_{1...n}^{(2)} is given by the z_{1}, …, z_{N−1}-integrations along the small circles around z_{1} = θ_{1}, …, z_{N−1} = θ_{N−1} (see again figure 1). Now S̃(θ_{1} − z_{1})_{1}, …, S̃(θ_{N−1} − z_{N−1}) yield permutation operators P and the co-vector part of this contribution for α_{N} = 1 is

(Ω C̃^{β_{m}}(\underline{θ}, z_{m}) ⋯ C̃^{β_{N−1}}(\underline{θ}, θ_{N−1}) ⋯ C̃^{β_{1}}(\underline{θ}, θ_{1})P_{N}(1))_{α_{1}...α_{N−1N+1}...α_{n}α_{N}}

= [figure: ⋮; :; …; …; θ_{1}; θ_{N+1}; θ_{N}; α_{1}; α_{N+1}; α_{N} = 1; β_{1}; β_{N−1}; β_{m}; z_{1}; z_{N−1}; z_{m}; 1; 1; 1; 1; 1; 1; 1; 1; 1]

= δ_{α_{1}}^{β_{1}} ⋯ δ_{α_{N−1}}^{β_{N−1}} (ΩC̃^{β_{m}}(\underline{θ̌}, z_{m}) ⋯ C̃^{β_{N}} (\underline{θ̌}, z_{N}))\underline{α̌} δ_{α_{N}}^{1} (84)

where P_{N}(1) projects onto the components with α_{N} = 1. We have used the fact that because of the SU(N) ice rule the amplitude a(·) only contributes to the S-matrices S(θ_{1} − z_{j}), S(θ_{2} − z_{j}), S(θ_{N} − z_{j}), S(θ_{i} − z_{1}) after having applied Yang-Baxter relations. We use ϕ̃(θ) = −b̃(θ + 2πi)ϕ̃(θ + 2πi) to replace for i = 1, …, N − 1 and β ≠ 1

ϕ̃(θ_{Ni}) = S̃_{11}^{11}(θ_{Ni})ϕ̃(θ_{Ni}) = −S̃_{1β}^{β1}(θ_{Ni} + 2πi)ϕ̃(θ_{Ni} + 2πi)

therefore using again (69) we obtain

Res_{θ_{N−1N} =iη} ⋯ Res_{θ_{12}=iη} ϕ̃(θ_{N1}) ⋯ ϕ̃(θ_{NN−1})S̃_{α_{1}...α_{N−1}}^{N...2}(θ_{1}, ⋯, θ_{N−1})

= (−1)^{N−1}ϕ̃(iη) ⋯ ϕ̃((N − 1)iη)

× Res_{θ_{N−1N} =iη} ⋯ Res_{θ_{12}=iη} S̃_{1α_{1}···α_{N−1}}^{N···21}(θ_{N} + 2πi, θ_{1}, ⋯, θ_{N−1})

= −ϕ̃(iη) ⋯ ϕ̃((N − 1)iη)(N − 1)!(iη)^{N−1}ϵ_{α_{1}···α_{N−1}1} .

We combine (84) with the function L\underline{β}(\underline{z}) with the property (74) (for l = 1) and the scalar functions h̃ and p and after having performed the remaining z_{j}-integrations we obtain

R_{1...n}^{(2)}S_{N+1N} … S_{nN}P_{N}(1) = −c_{0} ∏_{i=N+1}^{n} ∏_{j=2}^{N} ϕ̃(θ_{ij}) ε_{1...N}(\underline{θ̂})K_{N+1...n}(\underline{θ̌})σ_{N}P_{N}(1) .

We have used the following equations: a(θ_{iN})ϕ̃(θ_{i1}) = ϕ̃(θ_{iN}) (because of (15)), the definition (35) of τ(z), the relation (iii’) of (78) for the p-function p(\underline{θ}, θ_{1}, …, θ_{N−1}, \underline{ž}) = σp(\underline{θ̌}, \underline{ž}) (for θ_{12} = ⋯ = θ_{N−1N} = iη), the relation (79) for the normalization constants N_{n}c_{1}(N − 1)!(iη)^{N−1} ∏_{j=1}^{N−1} ϕ̃(jiη) = N_{n−N}c_{0} and the recursion relation (75) for the constants c_{l}.

The contribution of pinching (3) is given by the z_{1}, …, z_{N−1}-integrations along the small circles around z_{1} = θ_{2} − iη, …, z_{N−2} = θ_{N−1} − iη, z_{N−1} = θ_{N} − iη, (see again figure 1). As for N = 3 the S-matrix S̃_{αβ}^{δγ}(θ_{N} − z_{N−1}) yields

Γ_{(ρσ)}^{δγ}Γ_{αβ}^{(ρσ)} and the residue of the Bethe Ansatz state vanishes,

Res_{z_{N−1}=θ_{N} −iη} (Ω C̃(\underline{θ}, z_{m}) ⋯ C̃(\underline{θ}, z_{1})P_{N}(1) )\underline{α} = 0 ,

because Γ_{αβ}^{(ρσ)} is antisymmetric with respect to α, β. Therefore equation (83) is proved for α_{N} = 1 and because of Lemma 6 also in general.

C The higher level Bethe Ansatz

Proof of lemma 3: For the higher level functions L(l) ; \underline{β}(\underline{z}) one may verify the

equations (i)^{(l)} and (ii)^{(l)} quite analogously to the corresponding ones for the main theorem (e.g. for N = 3).

We prove (iii)^{(l)} by induction and assume:

L(l+1) ; \underline{γ}(\underline{u}) ≈ c_{l+1}S̃N...l+2 ; \underline{γ̂}(\underline{û}) ∏_{i=N−l}^{k} ∏_{j=2}^{N−l−1} ϕ̃(u_{ij})L(l+1) ; \underline{γ̌}(\underline{ǔ})  (85)

for u_{12}, …, u_{N−l−2,N−2} → iη. In the integral representation (72) of L(l) ; \underline{β}(\underline{z}) there are pinchings at u_{1} = z_{2}, …, u_{N−l−1} = z_{N−l} if z_{12}, …, z_{N−l−1,N−l} → iη. Therefore in Ψ̃(l) ; \underline{β}(\underline{z}, \underline{u}) the S-matrices S̃(z_{2} − u_{1}), …, S̃(z_{N−l} − u_{N−l−1}) yield the permutation operator P and we have to consider

Φ̃(l)\underline{γ} ; \underline{β}(\underline{z}, \underline{u}) = (Ω^{(l)} C̃^{(l)γ_{k}}(\underline{z}, u_{k}) ⋯ C̃^{(l)γ_{N−l−1}}(\underline{z}, z_{N−l}) ⋯ C̃^{(l)γ_{1}}(\underline{z}, z_{2}))\underline{β}

= [figure: z_{1}; z_{2}; z_{N−l}; z_{m}; u_{N−l−1}; u_{k}; u_{1} = z_{2}; …; …; ⋮; ⋮; γ_{1}; γ_{N−l}; γ_{k}; l+1; l+1; l+1; l+1; l+1; l+1; β_{1}; β_{2}; β_{N−l}; β_{m}]

= (S̃_{12}(z_{12}) … S̃_{1N−l}(z_{1N−l}))_{β_{1}...β_{N−l}}^{γ_{1}...γ_{N−l−1}l+1} ∏_{j=N−l}^{k} b̃(z_{1} − u_{j})Φ̃(l)\underline{γ̌} ; \underline{β̌}(\underline{ž}, \underline{ǔ})

where \underline{β̌} = (β_{N−l+1}, …, β_{m}), \underline{ž} = (z_{N−l+1}, …, z_{m}), \underline{ǔ} = (u_{N−l}, …, u_{k}). We may write for \underline{û} = (u_{1}, …, u_{N−l−1}) = (z_{2}, …, z_{N−l})

S̃N...l+2 ; \underline{γ̂}(\underline{û}) (S̃_{12}(z_{12}) … S̃_{1N−l}(z_{1N−l}))_{β_{1}...β_{N−l}}^{γ_{1}...γ_{N−l−1}l+1} = S̃N...l+1 ; \underline{β̂}(\underline{ẑ})

with the notation \underline{ẑ} = (z_{1}, …, z_{N−l}). Therefore using the assumption (85) we obtain when z_{12}, …, z_{N−l−1,N−l} → iη

L(l) ; \underline{β}(\underline{z}) ≈ \frac{1}{ǩ!} ∮_{z_{2}} \frac{du_{1}}{R} ⋯ ∮_{z_{N−l}} \frac{du_{N−l−1}}{R} ∫_{C\underline{z}} \frac{du_{N−l}}{R} ⋯ ∫_{C\underline{z}} \frac{du_{k}}{R} h̃(\underline{z}, \underline{u})

× L(l+1) ; \underline{γ}(\underline{u}) Φ̃(l)\underline{γ} ; \underline{β}(\underline{z}, \underline{u}) = c_{l}S̃N...l+1 ; \underline{β̂}(\underline{ẑ}) ∏_{i=N−l+1}^{m} ∏_{j=2}^{N−l} ϕ̃(z_{ij})L(l) ; \underline{β̌}(\underline{ž})

where ǩ = k − N + l + 1. The following formulae have been used: b̃(z_{1} − u_{j}) ∏_{i=3}^{N−l} ϕ̃(u_{j} −z_{i}) ∏_{i=1}^{N−l} ϕ̃(z_{i}−u_{j}) ∏_{i=2}^{N−l} τ(z_{i}−u_{j}) = −1, the relation (iii′)^{(l)} of (77) for the p-function p_{mk}^{(l)}(\underline{z}, z_{2}, … z_{N−l}, \underline{ǔ}) = (−1)^{k−N−l+1} p_{m̌ǩ}^{(l)}(\underline{ž}, \underline{ǔ}) (if z_{12} = ⋯ = z_{z_{N−l−1,N−l}} = iη) and the recursion relation c_{l} = c_{l+1} ∏_{j=1}^{N−l−1} ϕ̃(jiη). The solution to this recursion relation with c_{N−1} = 1 is

c_{1} = ϕ̃^{N−2}(iη)ϕ̃^{N−3}(2iη) ⋯ ϕ̃((N − 2)iη)  (86)

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