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

May 28, 2008

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. The general solution is illustrated for some operators, such as the energy momentum tensor, the fields and the current.

PACS: 11.10.-z; 11.10.Kk; 11.55.Ds Keywords: Integrable quantum field theory, Form factors

Contents

1 Introduction 2

2 General setting 3 2.1 The SU(N) S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Generalized Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Nested “off-shell” Bethe ansatz for SU(N) . . . . . . . . . . . . . . . . . . . . . . 6

3 SU(N) form factors 9 3.1 The form factor formula and SU(N) weights . . . . . . . . . . . . . . . . . . . . 10 4 Examples 15

∗Address: Yerevan Physics Institute, Alikhanian Brothers 2, Yerevan, 0036 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 The energy momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 The fields ψ_{α}^{(±)}(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 The current J_{α(ρ)}^{μ}(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Summary 18

A The SU(N) bound state S-matrix 18

B Proof of Theorem 1 20

C Proof of lemma 2 25

1 Introduction

Thirty-five years after the discovery of asymptotic freedom [1, 2], Quantum Chromodynamics (QCD), the theory of the strong interactions between quarks and gluons, remains a challenge. One of the most important trends in theoretical physics in the last several decades is the development of exact methods which are completely different from the perturbation theory since the perturbation theory is not applicable in the strong coupling regime. Resolution of the strong coupling problem would give us a full understanding of the structure of the interactions in non-Abelian gauge theory and in particular would shed light on the mysterious confinement phenomena in QCD. One promising possibility to overcome the limitations of the perturbation theory is the application of exact integrability. This is one of the reasons why investigation of two dimensional exactly solvable lattice statistical systems, 1-dimensional spin chains and quantum field theory models in 1 + 1 dimensions in the framework of the Bethe ansatz method is still quite actual [3]. Some two dimensional exactly integrable field theories, such as nonlinear σ-model or Gross-Neveu model exhibit asymptotic freedom, a phenomenon initially observed in non-Abelian gauge theories. From this point of view the two-dimensional integrable quantum field theories are in a sense a laboratory for the investigations of those properties of quantum field theories,which can not be described via perturbation theory. One may hope that some of these properties, at least qualitatively, could be extended to the higher dimensions too.

The Bethe ansatz method [4], was first introduced by Bethe to solve the isotropic Heisenberg model. Yang [5] and Sutherland [6] 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. A great impulse in the theory of integrable models was given by Faddeev and collaborators [7] with the development of the algebraic Bethe ansatz method. Another modification of the Bethe ansatz is the off-shell Bethe ansatz, which was originally formulated in [8] to calculate correlation function in WZNW models (see also [9, 10]). 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.

The exact determination of form factors, which are matrix elements of local operators, was formulated in the seventies [11]. This approach was developed further and studied in the context of several explicit models by Smirnov [12] . Subsequently [13] the techniques of the off-shell Bethe ansatz were 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. Recently the form factor program has received renewed interest in different areas, such as condensed matter physics [14, 15, 16] and atomic physics [17]. In particular, applications to Mott insulators and carbon nanotubes [18, 14] doped two-leg ladders [19] and in the field of Bose-Einstein condensates of ultracold atomic and molecular gases [20, 17] have been discussed. In addition, form factors have been also employed to compute corrections to the entanglement, an essential property in the field of Quantum Computation [21].

In the present article we will focus on the determination of the form factors for an SU(N) invariant S-matrix. We have to apply the nested off-shell Bethe ansatz to get the vectorial part of the form factors. This solves the missing link of Smirnov’s [12] formula for the SU(N) form factors, where the vectors were given by an “indirect definition” characterized by necessary properties but not provided explicitly. The procedure is similar as for the scaling Z(N) Ising and affine A(N −1) Toda models [22, 23] 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 develop a nontrivial algebraic nested off-shell Bethe ansatz. We note that SU(N) form factors were also calculated in [12, 24, 25] using other techniques, see also the related paper [26, 27]. We believe that our integral representation may shed some light for a better understanding of the correlation functions of SU(N) invariant models, which are also interesting for applications [28].

The paper is organized as follows: In Section 2 we recall some known formulae and derive some simple results, which will be used in the following. In Section 3 we construct the general form factor formula. In Section 4 we discuss form factors of some operators such as the energy momentum tensor, the Dirac field and the SU(N) current. A summary of our results can be found in 5. In appendix A we present some basic properties of the SU(N) bound state S-matrix. In appendix B we provide the proofs of the main theorem and in appendix C we prove the lemma dedicated to the higher level Bethe ansatz.

2 General setting

2.1 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 [29]. The S-matrix for the scattering of two particles of rank 1 (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)

The amplitudes which we use in the following are given by

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})}  (2)

and

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

The amplitude a(θ) is the highest weight S-matrix eigenvalue for the two particle scattering. It will be essential for the Bethe ansatz below.

The S-matrix eigenvalue b(θ) − c(θ) has a pole at θ = iη 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 [30, 31] picture that the antiparticle of a particle of rank r is to be identified with the particle of rank N − r (see also [22, 23]).

The general bound state S-matrix formula [32, 33] for the scattering of a bound state with another particle can be used to calculate iteratively the scattering of a particle of rank N − 1 with a particle of rank 1 (for details see appendix A and [34]). The result is

S_{(ρ)α}^{γ(σ)}(θ) = (−1)^{N−1} (δ_{(ρ)}^{(σ)}δ_{α}^{γ} b(πi − θ) + C^{γ(σ)}C_{(ρ)α} c(πi − θ))  (3)

where (σ) = (σ_{1} … σ_{N−1}) and (ρ) = (ρ_{1} … ρ_{N−1}) denote the bound states. We have introduced the charge conjugation matrices

C_{(α_{1}...α_{N−1})α_{N}} = C_{α_{1}(α_{2}...α_{N})} = ϵ_{α_{1}...α_{N}}
C^{α_{1(}α_{2}...α_{N})} = C^{(α_{1}...α_{N−1})α_{N}} = (−1)^{N−1}ϵ^{α_{1}α_{2}...α_{N}}  (4)

with ϵ_{α_{1}...α_{N}} and ϵ^{α_{1}α_{2}...α_{N}} total anti-symmetric and ϵ_{1...N} = ϵ^{1...N} = 1. The charge conjugation matrices satisfy

C_{α(ρ)}C^{(ρ)β} = δ_{α}^{β} , C_{α(ρ)}A_{β}^{α}C^{β(ρ)} = (−1)^{N−1} tr A
[figure: •; •; =; ,; A; •; •; =; (−1)^{N−1} tr A .] (5)

Formula (3) may be interpreted as an unusual (because of the sign (−1)^{N−1}) crossing relation

S_{(ρ)α}^{γ(σ)}(θ) = (−1)^{N−1}C_{(ρ)δ}S_{αβ}^{δγ}(πi − θ)C^{β(σ)}

supporting Swieca’s picture that the antiparticle of a particle of rank 1 is to be identified with the bound state of rank N − 1.

In order to simplify the formulae we extract the factor a(θ) from the S-matrix (1)

S_{αβ}^{δγ}(θ) = a(θ)S̃_{αβ}^{δγ}(θ)

such that

S̃_{αβ}^{δγ}(θ) = δ_{α}^{γ}δ_{β}^{δ} b̃(θ) + δ_{α}^{δ}δ_{β}^{γ} c̃(θ)  (6)

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

2.2 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}.  (7)

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 (7), 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}; …].  (8)

The general form factor can be written as [11]

F_{1...n}^{O}(\underline{θ}) = K_{1...n}^{O}(\underline{θ}) ∏_{1≤i<j≤n} F (θ_{ij})  (9)

where F (θ) is the minimal form factor function (10) and the K-function K_{1...n}^{O}(\underline{θ}) is given below. It satisfies the form factor equations (i) – (v) (see below and [11, 12, 13, 33, 23]). For bosons and fermions they follow from general LSZ-assumptions and “maximal analyticity” [13]. For anyons there are additional statistics factors [35, 23, 36]. As we will discuss below in section 3 the statistics factors are modified due to the unusual crossing relation of the S-matrix (see [31]). We will also 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} for the SU(N) S-matrix.

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 (2) 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} )) .  (10)

It belongs to the highest weight vector w = (2, 0, …, 0) of the two particle state. We define the corresponding ‘Jost-function’ as for the Z(N) models [22, 23] by the equation

∏_{k=0}^{N−2} ϕ (θ + kiη) ∏_{k=0}^{N−1} F (θ + kiη) = 1 , η = \frac{2π}{N}  (11)

which is typical for models where the bound state of N − 1 particles is the anti-particle [23]. The solution is

ϕ(θ) = Γ (\frac{θ}{2πi}) Γ (1 − \frac{1}{N} − \frac{θ}{2πi})  (12)

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) . (13)

The equations (12) and (11) also determine the normalization constant c in (10) (for details see appendix A and [34]).

2.3 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 [7, 37] the monodromy matrix

T̃_{1...n,0}(\underline{θ}, θ_{0}) = S̃_{10}(θ_{10}) ⋯ S̃_{n0}(θ_{n0}) = 1  n 0 …  .  (14)

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 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})  (15)

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

2.3 Nested “off-shell” Bethe ansatz for SU(N) 7

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 .  (16)

An ‘off-shell’ Bethe ansatz for a co-vector valued function K\underline{α}(\underline{θ}) is given by the multiple contour integral

K\underline{α}(\underline{θ}) = \frac{1}{m!} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ⋯ ∫_{C\underline{θ}} \frac{dz_{m}}{R} h̃(\underline{θ}, \underline{z}) p(\underline{θ}, \underline{z}) Ψ̃\underline{α}(\underline{θ}, \underline{z})  (17)

with the scalar functions

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})  (18)

τ(z) = \frac{1}{ϕ(θ)ϕ(−θ)} , ϕ̃(θ) = ϕ(θ)a(θ) = ϕ(−θ) .  (19)

where the function ϕ(θ) is defined by the equation (11) with the solution (12). Below we will use the co-vector valued function K\underline{α}(\underline{θ}) to construct form factors for operators O(x). The scalar p-function p(\underline{θ}, \underline{z}) which is in general a simple function of e^{θ_{i}} and e^{z_{j}} (see below) will depend on the specific operator O(x). It turns out that by means of the integral representation we transform the complicated form factor equations (24) – (28) (which are in general matrix equations) to simple equations for the p-function (see section 3). The integration contour C\underline{θ} (see Fig. 1) can be characterized as follows: there is a finite number of complex numbers a_{i}(\underline{θ}), b_{j}(\underline{θ}) such that the positions of all poles of the integrand in (17) are of the form

(1) : a_{i}(\underline{θ}) + 2πik , k ∈ ℕ
(2) : b_{j}(\underline{θ}) − 2πil , l ∈ ℕ

then C\underline{θ} runs from −∞ to +∞ such that all poles (1) are above and all poles (2) are below the contour (this is usually not a straight line). This contour is just the same as that used for the definition of Meijer’s G-function. 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 K-function is in general a linear combination of the fundamental building blocks [38, 22, 23] given by (17) - (19). We consider here cases where the sum consists only of one term.

Nesting: If in (16) 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})  (20)

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

Figure 1: The integration contour C\underline{θ}. The bullets refer to poles of the integrand in (17).

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}  (21)

Here the “pseudo-vacuum” is the highest weight co-vector (with weight vector 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 belongs 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}  (22)

The amplitudes of the scattering matrices are given by eqs. (1) and (2). The technique of the ‘nested Bethe ansatz’ means that one makes for the coefficients L\underline{β}(\underline{z}) in (20) the analogous construction (17) - (19) 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. It is well known (see [39]) that the ‘off-shell’ Bethe ansatz states are highest weight states if they satisfy a matrix difference equation which is here the form factor equation (ii) (see below). The SU(N) weights are

w = (n_{0} − n_{1}, n_{1} − n_{2}, …, n_{N−2} − n_{N−1}, n_{N−1})  (23)

where n_{0} = n, n_{1} = m, … are the numbers of variables in the various levels of the nesting.

3 SU(N) form factors

In this section we perform the form factor program for the general SU(N) S-matrix. For this purpose, we perform the nested Bethe ansatz method with N −1 levels combined with the off-shell Bethe ansatz. Applications of the results to the SU(N) chiral Gross-Neveu model [40] will be investigated in a separate article [34]. We consider operators O(x) with charge Q^{O} where Q^{O} is the smallest number n of fundamental (rank 1) particles such that ⟨0 | O(x) | θ_{1}, …, θ_{n} ⟩^{in} is non-vanishing. For example for the energy momentum tensor and the SU(N)-current Q^{T} = Q^{J} = 0 and for the fundamental field Q^{ψ} = 1.

SU(N) 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})  (24)

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⟨ θ_{1} | O(0) | θ_{2}, …, θ_{n} ⟩_{2...n}^{in,conn.}

= F_{1...n}^{O}(θ_{1} + iπ, θ_{2}, …, θ_{n}) σ̇_{1}C^{¯11} = F_{2...n1}^{O}(θ_{2}, …, θ_{n}, θ_{1} − iπ)C^{1¯1} (25)

where σ̇_{α}^{O} takes into account the statistics of the particle α with respect to O. The charge conjugation matrix C^{¯11} is defined by (4).

(iii) There are poles determined by one-particle states in each sub-channel given by a subset of particles of the state in (7). 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 − σ̇_{2}S_{2n} … S_{23}) . (26)

(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)}  (27)

where the bound state intertwiner Γ_{12}^{(12)} and the values of θ_{1}, θ_{2}, θ_{(12)} and η are given in [32, 33].

10 3 SU(N) FORM FACTORS

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

have

F_{1...n}^{O}(θ_{1}, …, θ_{n}) = e^{sμ} F_{1...n}^{O}(θ_{1} + μ, …, θ_{n} + μ)  (28)

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

The form factor equations (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 phase factor σ̇_{1} = ρσ_{1}^{O}. Here σ_{1}^{O} is the statistics factor of the operator O(x) with respect to the particle 1 and ρ is a sign factor due to the unusual crossing relation of the S-matrix which is determined as follows. The statistics factors in (ii) and (iii) are not arbitrary, consistency implies that both are the same and

σ̇_{1}^{O}σ̇O ; ¯1 = (σ̇_{1}^{O})^{N} = (−1)^{(N−1)Q^{O}}

with the solution

σ_{1}^{O} = e^{iπ(1−1/N)Q^{O}}  (29)
ρ = (−1)^{(N−1)+(1−1/N)(n−Q^{O})} = ±1  (30)

where Q^{O} = n mod N is the charge of the operator O(x). This solution and the decomposition of σ̇_{1}^{O} = ρσ_{1}^{O} is of course not unique, the choice proposed is consistent with [31] and the usual relation of spin and statistics (for all examples discussed below).

3.1 The form factor formula and SU(N) weights

3.1 The form factor formula and SU(N) weights 11

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. Using then the annihilation residue equation (iii) we obtain with (5)

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

= 2i\sqrt{2}^{N−2}Γ ε_{1...N}F_{N+1...n}^{O}(θ_{N+1}, …, θ_{n}) (1 − σ̇_{N}^{O}S_{Nn} … S_{NN+1}) (31)

where Γ is a constant (see (59) in appendix A).

The properties of the K- function: We proposed above the n-particle form factors of an operator O(x) as given by the formula (9). In this expression the K-function K_{1...n}^{O}(\underline{θ}) contains the entire pole structure and its properties are determined by the form factor equations (i) – (iii) which read in terms of the K-function as

K_{...ij...}(…, θ_{i}, θ_{j}, …) = K_{...ji...}(…, θ_{j}, θ_{i}, …) S̃_{ij}(θ_{ij}) (32)
K_{α_{1}α_{2}...α_{n}}(θ_{1} + iπ, θ_{2}, …, θ_{n}) σ̇_{α_{1}}(−1)^{N−1} = K_{α_{2}...α_{n}α_{1}}(θ_{2}, …, θ_{n}, θ_{1} − iπ) (33)

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}(θ_{N+1}, …, θ_{n})

× (1 − σ̇_{N}S_{Nn} … S_{NN+1}) (34)

with the constant

c_{0} = 2i\sqrt{2}^{N−2}Γ ∏_{j=1}^{N−1} F^{−(N−j)}(jiη) .  (35)

For the second equation (33) the relation (5) for the charge conjugation matrices has been used and the third equation (34) follows from (11) and (31).

We propose the K-function K\underline{α}(\underline{θ}) as the integral representation (17) with (18) and (19). The Bethe ansatz co-vector Ψ̃\underline{α}(\underline{θ}, \underline{z}) is of the form (20) - (21) and for the function L\underline{β}(\underline{z}) one makes again an analogous ansatz as for the K-function (17) where the indices run over sets with one element less. For the SU(N) case we have to iterate this N − 2 times.

The higher level Bethe ansatz: We define for l = 1, …, N − 2, with the short notation m = n_{l} and k = n_{l+1}

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})  (36)

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})
Ψ̃(l) ; \underline{β}(\underline{z}, \underline{u}) = L(l+1) ; \underline{γ}(\underline{u})Φ̃(l−1)\underline{γ} ; \underline{β}(\underline{z}, \underline{u})  (37)

where \underline{z} = z_{1}, …, z_{m} , \underline{u} = u_{1}, …, u_{k}, \underline{β} = β_{1}, …, β_{m} , \underline{γ} = γ_{1}, …, γ_{k}. The indices assume the values β_{i} = l + 1, …, N, γ_{i} = l + 2, …, N. By this procedure one obtains the nested Bethe ansatz^{1}.

In order that the form factor F^{O} (\underline{θ}) given by (9), (17) - (21) and (36) satisfies the form factor equations (i) - (iii) the higher level L-functions L(l) ; \underline{β}(\underline{z}) 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}) ,  (38)

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

L_{β_{1},β_{2},...,β_{m}}^{(l)}(z_{1} + 2πi, z_{2}, …, z_{m}) = L_{β_{2},...,β_{m},β_{1}}^{(l)}(z_{2}, …, z_{m}, z_{1})  (39)

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} = 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(l) ; \underline{β̌}(\underline{ž})  (40)

[figure: L^{(l)}; \underline{β}; …] ≈ c_{l} ∏_{i=N−l+1}^{m} ∏_{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})
\underline{ẑ} = (z_{1}, …, z_{N−l}), \underline{ž} = (z_{N−l+1}, …, z_{m}).  (41)

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 and the final configuration of quantum numbers \underline{β̂} and (N, …, l + 1), respectively.

Theorem 1 We make the following assumptions:

1. the p-function p(\underline{θ}, \underline{\underline{z}}) satisfies the equations

\begin{matrix}(i′) & p_{nm}(\underline{θ}, \underline{z}) is symmetric under θ_{i} ↔ θ_{j} and z_{i} ↔ z_{j} \\ (ii′_{1}) & p_{nm}(\underline{θ}, \underline{z}) = σ̇(−1)^{N−1+m}p_{nm}(θ_{1} + 2πi, θ_{2}, …, \underline{z}) \\ (ii′_{2}) & p_{nm}(\underline{θ}, \underline{z}) = (−1)^{n}p_{nm}(\underline{θ}, z_{1} + 2πi, z_{2}, …) \\ (iii′_{1}) & p_{nm}(\underline{θ}, \underline{ẑ}\underline{ž}) = ν p_{n−Nm−N+1}(\underline{θ̌}, \underline{ž}) \\ (iii′_{2}) & p_{nm}(\underline{θ}, \underline{ẑ}\underline{ž}) = (−1)^{m+N−1}σ̇ ν p_{n−Nm−N+1}(\underline{θ̌}, \underline{ž}) \end{matrix} \}  (42)

1In [39] the nested off-shell Bethe ansatz was formulated in terms of “Jackson-type integrals” instead of contour integrals.

where in (iii′_{1}) : \underline{ẑ} = (θ_{2}, …, θ_{N}) and in (iii′_{2}) : \underline{ẑ} = (θ_{1}, …, θ_{N−1}) with θ_{12} = ⋯ = θ_{N−1N} = iη. In (iii′) the short notations (41) and \underline{θ̌} = (θ_{N+1}, …, θ_{n}) are used.

2. the higher level function L(1) ; \underline{β}(\underline{z}) satisfies (i)^{(1)} - (iii)^{(1)} of (38) – (40) and

3. the normalization constant ν satisfies

ν = \frac{2i\sqrt{2}^{N−2}Γ}{(N − 1)!(iη)^{N−1}} ∏_{l=1}^{N−1}(−1)^{n_{l}+l} (ϕ̃(liη)F (liη))^{l−N}  (43)

then the co-vector valued function F\underline{α}(\underline{θ}) given by the ansatz (9) and the integral representation (17) satisfies the form factor equations (i), (ii) and (iii) in the form of (32) – (34).

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

Lemma 2 The higher level function L(l) ; \underline{β}(\underline{z}) satisfies (i)^{(l)} - (iii)^{(l)}, if the higher level pfunction in the integral representation (36) satisfies

(i′)^{(l)} p^{(l)}(\underline{z}, \underline{u}) is symmetric under z_{i} ↔ z_{j}
(ii′)^{(l)} \{ \begin{matrix}p_{mk}^{(l)}(\underline{z}, \underline{u}) = (−1)^{k}p_{mk}^{(l)}(\underline{z}′, \underline{u}) \\ = (−1)^{m}p_{mk}^{(l)}(\underline{z}, \underline{u}′) \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} = p_{m̌ǩ}^{(l)}(\underline{ž}, \underline{ǔ})

(44)

with m̌ = m − N + l, ǩ = m − N + l − 1 and m = n_{l}, k = n_{l+1}. We use the short notation \underline{z}′ = z_{1} + 2πi, z_{2}, …, z_{m} and \underline{u}′ = u_{1} + 2πi, u_{2}, …, u_{k}. In (iii′)^{(l)} the short notations (41) and \underline{ǔ} = (u_{N−l}, …, u_{k}) are used.

The constants satisfy the recursion relation

c_{l} = c_{l+1}(−1)^{n_{l+1}−N+l+1} ∏_{j=1}^{N−l−1} ϕ̃(jiη)  (45)

with the solution

c_{1} = ϕ̃^{N−2}(iη)ϕ̃^{N−3}(2iη) ⋯ ϕ̃((N − 2)iη) ∏_{l=1}^{N−2}(−1)^{n_{l+1}−N+l+1}  (46)

if c_{N−1} = 1

This lemma is proved in appendix C A compact form for full form factors: We may write the above results as

K\underline{α}(\underline{θ}) = \frac{1}{\underline{n}!} ∫ \underline{\underline{dz}} h̃(\underline{θ}, \underline{\underline{z}}) p(\underline{θ}, \underline{\underline{z}}) Ψ̃\underline{α}(\underline{θ}, \underline{\underline{z}})  (47)
h̃(\underline{θ}, \underline{\underline{z}}) = h̃(\underline{θ}, \underline{z}^{(1)}) ⋯ h̃(\underline{z}^{(N−2)}, \underline{z}^{(N−1)})  (48)
Ψ̃\underline{α}(\underline{θ}, \underline{\underline{z}}) = Φ̃\underline{β}_{(N−1)}(\underline{z}^{(N−2)}, \underline{z}^{(N−1)}) ⋯ Φ̃\underline{β}^{(1)} ; \underline{α}(\underline{θ}, \underline{z}^{(1)})  (49)

with the short notations

\underline{n}! = n_{1}! ⋯ n_{N−1}!; \underline{\underline{z}} = \underline{z}^{(1)}, …, \underline{z}^{(N−1)}; \underline{z}^{(l)} = z_{1}^{(l)}, …, z_{n_{l}}^{(l)}
∫ \underline{\underline{dz}} = ∫ \underline{dz}^{(1)} ⋯ ∫ \underline{dz}^{(N−1)}; ∫ \underline{dz}^{(l)} = ∫_{C_{\underline{z}_{(l−1)}}} \frac{dz_{1}^{(l)}}{R} ⋯ ∫_{C_{\underline{z}_{(l−1)}}} \frac{dz_{n_{l}}^{(l)}}{R} .

The function h̃(\underline{θ}, \underline{z}) is given by (18) and (19).

The p-function and SU(N) weights: In general the function p(\underline{θ}, \underline{\underline{z}}) depends on the rapidities \underline{θ} and all integration variables \underline{z}^{(l)} l = 1, …, N −1. In order that the form factor F^{O} (θ) given by (9) and (47) satisfies the form factor equations (i) - (iii) the p-function p^{O}(\underline{θ}, \underline{\underline{z}}) in (17) which depends on the explicit nature of the local operator O(x) is assumed to satisfy

\begin{matrix}(i′) & p(\underline{θ}, \underline{\underline{z}}) is symmetric under θ_{i} ↔ θ_{j} and z_{i}^{(l)} ↔ z_{j}^{(l)} \\ (ii′_{1}) & p(\underline{θ}, \underline{\underline{z}}) = σ̇(−1)^{n_{1}+N−1}p(θ_{1} + 2πi, θ_{2}, …, \underline{\underline{z}}) \\ (ii′_{2}) & p(\underline{θ}, \underline{\underline{z}}) = (−1)^{w_{l}^{O}+w_{l+1}^{O}}p(\underline{θ}, …, z_{i}^{(l)} + 2πi, …) , l = 1, …, N − 1 \\ (iii′_{1}) & p(\underline{θ}, \underline{ẑ}^{(1)}\underline{ž}^{(1)}, …) = ν p(\underline{θ̌}, \underline{ž}^{(1)}, …) \\ (iii′_{2}) & p(\underline{θ}, \underline{ẑ}^{(1)}\underline{ž}^{(1)}, …) = (−1)^{n_{1}+N−1}σ̇ ν p(\underline{θ̌}, \underline{ž}^{(1)}, …) \\ (iii′_{3}) & p(\underline{θ}, …, \underline{z}^{(l)}, \underline{ẑ}^{(l+1)}\underline{ž}^{(l+1)}, …) = p(\underline{θ}, …, \underline{z}^{(l)}, \underline{ž}^{(l+1)}, …) \\ (v′) & p(\underline{θ}, \underline{\underline{z}}) = e^{sμ} p(\underline{θ} + μ, \underline{\underline{z}} + μ) \end{matrix} \} (50)

where in (iii′_{1}) : \underline{ẑ}^{(1)} = (θ_{2}, …, θ_{N}) in (iii′_{2}) : \underline{ẑ}^{(1)} = (θ_{1}, …, θ_{N−1}) and in (iii′_{3}) : \underline{ẑ}^{(l+1)} = (z_{2}^{(l)}, … z_{N−l}^{(l)}), furthermore in (iii′_{1}) and (iii′_{2}) : θ_{12} = ⋯ = θ_{N−1N} = iη and in (iii′_{3}) : z_{12}^{(l)} = ⋯ = z_{z_{N−l−1,N−l}}^{(l)} = iη. Here we have also included the equation (v′) which implies the behavior of the form factor under Lorentz transformations (v) of (28). The sign factor (ii′_{2}) follows from (42) and (44) as (−1)^{n_{l−1}−n_{l+1}}. This sign does not depend on the specific state of F O ; \underline{α} but only on the weights of the operator O(x). This can be seen as follows: We assume that the operator O(x) transforms under SU(N) as a highest weight representation with highest weight vector

w^{O} = (w_{1}^{O}, …, w_{N}^{O}) .  (51)

Then SU(N) invariance implies that the weight vector of the form factor F\underline{α}(\underline{θ}) as covector is of the form

w = w^{O} + L(1, …, 1) , (L = 0, 1, …)  (52)

because w = (1, …, 1) correspond to the vacuum sector. Then because of (23)

(−1)^{n_{l−1}−n_{l+1}} = (−1)^{w_{l}+w_{l+1}} = (−1)^{w_{l}^{O}+w_{l+1}^{O}+2L} = (−1)^{w_{l}^{O}+w_{l+1}^{O}} .

Similarly, also the phase factor in (ii′) and (iii′) of (50) σ̇(−1)^{n_{1}+N−1} with σ̇ = ρσ and

ρ(−1)^{n_{1}+N−1} = (−1)^{(1−1/N)(n_{0}−Q^{O})+n_{1}}  (53)

does not depend on the specific state of F O ; \underline{α} but only on the weights of the operator (see (30)).

4 Examples

To illustrate our general results we present some simple examples of solutions of the equations (50) for the p-functions. Details of the following calculations will be published elsewhere [34].

4.1 The energy momentum tensor

The local operator O(x) = T^{ρσ}(x) (where ρ, σ = ± denote the light cone components) has charge Q^{T} = 0, is bosonic and transforms as a scalar under SU(N) and has the weight vector of (51) w^{T} = (0, …, 0). The p-function (as for the sine-Gordon model in [33])

p^{T^{ρσ}}(\underline{θ}, \underline{z}) = N_{n}^{T^{ρσ}} ∑_{i=1}^{n} e^{ρθ_{i}} ∑_{i=1}^{m} e^{σz_{i}} .

solves (50) with

charge  Q^{T} = 0
weight vector w^{T} = (0, …, 0)
statistics factor σ^{T} = 1
spin  s^{T} = 2

if N_{n}^{T^{ρσ}}/N_{n−N}^{T^{ρσ}} = ν. The sign factor (53) is here ρ(−1)^{n_{1}+N−1} = 1.

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 because (see (23) and (52))

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 1). 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_{N}^{T^{ρσ}} (e^{ρθ_{1}} + e^{ρθ_{2}}) ∫_{C\underline{θ}} \frac{dz}{R} ϕ̃(θ_{1} − z)e^{σz}L(θ_{2} − z)  (54)
× 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 (54) may be performed (similar as in [33]) with the result^{2}

⟨ 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 [33] 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.

4.2 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), where we consider here the + component of the spinor. The weight vector is w^{ψ} = (1, 0, …, 0) and the charge is Q^{ψ} = 1. The p-function for the local operator ψ(x) is (see also [13])

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

It solves (50) with

charge  Q^{ψ} = 1
weight vector w^{ψ} = (1, …, 0)
statistics factor σ^{ψ} = e^{iπ(1− \frac{1}{N})}
spin  s^{ψ} = \frac{1}{2} (1 − \frac{1}{N} )

if N_{n}^{ψ}/N_{n−N}^{ψ} = ν. The last two formulae are consistent with the proposal of Swieca et al. [30, 31] 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 (29)). The sign factor (53) is here again ρ(−1)^{n_{1}+N−1} = 1.

The 1-particle form factor is for example

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

2In [41, 39] this result has been obtained using Jackson type integrals.

4.3 The current J_{α(ρ)}^{μ}(x) 17

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 [34] we will discuss the 1/N expansion of this exact form factor and compare the result with the 1/N expansion for the chiral Gross-Neveu model.

4.3 The current J_{α(ρ)}^{μ}(x)

The SU(N) current transforms as the adjoint representation with highest weights w^{J} = (2, 1, …, 1, 0) and has charge Q_{J} = 0. Again, because the Bethe ansatz yields highest weight states we obtain the matrix elements of the highest weight component

J_{α(ρ)}^{μ} = δ_{α1}ϵ_{(ρ)N}J^{μ} , (ρ) = (ρ_{1}, …, ρ_{N−1})
J^{μ} = ϵ^{μν}∂_{ν}φ

with the pseudo potential φ(x). The p-function for the operator φ(x)

p^{φ}(\underline{θ}, \underline{z}, \underline{u}) = N_{n}^{φ} (∑_{i=1}^{n} exp θ_{i})^{−1} exp \frac{1}{2} (∑_{i=1}^{n_{0}} θ_{i} − ∑_{i=1}^{n_{1}} z_{i}^{(1)} − ∑_{i=1}^{n_{N−1}} z_{i}^{(N−1)})

solves (50) with

charge  Q^{φ} = 0
weight vector w^{φ} = (2, 1, …, 1, 0)
statistics factor σ^{φ} = 1
spin  s^{φ} = 0, s^{J} = 1

if N_{n}^{φ}/N_{n−N}^{φ} = ν. The sign factor (53) is here ρ(−1)^{n_{1}+N−1} = −1.

We calculate the form factor of the particle α and the bound state (λ) = (λ_{1}, …, λ_{N−1}) of N −1 particles with weight vector w = (2, 1, …, 1, 0). In each level the integrations may be performed iteratively using the bound state relation (iv), similar as above, however, here no integration remains. The result is

F_{α(λ)}^{φ}(θ, ω) = K_{α(λ)}^{φ}(θ, ω) G(θ − ω)
K_{α(λ)}^{φ}(θ, ω) = N_{2}^{φ}δ_{α1}ϵ_{(λ)N} \frac{e^{\frac{1}{2} (θ+ω)}}{e^{θ} + e^{ω}}

18 A THE SU(N) BOUND STATE S-MATRIX

The current form factor is therefore

F_{α(λ)}^{J_{β(ρ)}^{±}}(θ, ω) = ⟨ 0 | J_{β(ρ)}^{±}(0) | θ, ω ⟩_{α(λ)}^{in} = ±N_{2}δ_{α}^{β}δ_{(λ)}^{(ρ)} (e^{±θ} + e^{±ω}) \frac{e^{\frac{1}{2} (θ+ω)}}{e^{θ} + e^{ω}} G(θ − ω)
= δ_{α}^{β}δ_{(λ)}^{(ρ)} v̄(ω)γ^{±}u(θ) G(θ − ω)/G(iπ)

as expected.

5 Summary

In this paper we have obtained the form factors for an SU(N) invariant S-matrix. Since the S-matrix also describes backward scattering, the algebraic structure of the form factors was more intricate compared to the Z(N) case [23]. We had to combine the nesting procedure with the techniques of the off-shell Bethe ansatz to capture the vectorial nature of the form factors. The solution was obtained explicitly in terms of multiple contour integrals. We exemplified our general solution for several operators, such as the energy momentum tensor, the fields and the current. We believe that our new integral representation may shed some light for a better comprehension of complicated objects, such as the correlation functions for the SU(N) invariant Gross-Neveu model.

Acknowledgments

We thank R. Schrader, B. Schroer, and A. Zapletal for useful discussions. H.B. was supported partially by the Humboldt Foundation. A.F. acknowledges support from DAAD (Deutscher Akademischer Austausch Dienst) and CAPES (Coordenacao de Aperfeicoamento Pessoal de Nivel Superior). This work is also supported by the EU network EU-CLID, ’Integrable models and applications: from strings to condensed matter’, HPRN-CT-2002-00325.

Appendix

A The SU(N) bound state S-matrix

The eigenvalue S_{−}(θ) has a pole at θ = iη = 2πi/N 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 [30, 42, 31] picture that the antiparticle of a particle of rank r is to be identified with the particle of rank N − r (see also [22, 23]).

The general bound state S-matrix formula [32, 33] for the scattering of a bound state with another particle reads 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}  (55)

[figure: (ρσ); (ρ′σ′); α; β; γ′; γ; •] = [figure: (ρ′σ′); α′; α; β′; β; γ′; γ′′; γ; •]

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′].  (56)

With a convenient choice of an undetermined phase factor one obtains

Γ_{αβ}^{(ρσ)} = Γ_{(ρσ)}^{βα} = i\sqrt{ia(iη)iη} (δ_{α}^{ρ}δ_{β}^{σ} − δ_{α}^{σ}δ_{β}^{ρ}) .  (57)

Iterating the general bound state formula N − 1 times one obtains for the scattering of a bound state (ρ) = (ρ_{1}ρ_{2} … ρ_{N−1}) with another particle δ

S_{(ρ)δ}^{δ′(σ)}(θ)Γ(ρ) ; \underline{α} = Γ_{\underline{α}′}^{(σ)}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(ρ) ; \underline{α} = Γ_{α_{1}α_{2}...α_{N−1}}^{(ρ_{1}ρ_{2}...ρ_{N−1})} = Γ_{(ρ_{1}ρ_{2}...ρ_{N−2})α_{N−1}}^{(ρ_{1}ρ_{2}...ρ_{N−1})} … Γ_{(ρ_{1}ρ_{2})α_{3}}^{(ρ_{1}ρ_{2}ρ_{3})}Γ_{α_{1}α_{2}}^{(ρ_{1}ρ_{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) = (−1)^{N}b(θ + iπ − iη) … c(θ − iπ + iη) = c(πi − θ) .

These results may be interpreted as an unusual crossing relation

C_{(β)β}Γ(β) ; \underline{β} (S(θ_{1}) … S(θ_{N−1}))_{\underline{α}δ}^{δ′\underline{β}} = C_{(β)β}S_{(α)δ}^{δ′(β)}(θ)Γ(α) ; \underline{α}

= (−1)^{N−1}C_{(α)γ}Γ(α) ; \underline{α} S_{δβ}^{γδ′}(iπ − θ) (58)

with θ_{j} = θ + iπ − jiη and the charge conjugation matrix C_{(β)β} defined by (4). The total bound state fusion intertwiner satisfies

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  (59)

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

We consider the total N-particle 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}  (60)

in the limit θ_{jj+1} → iη (j = 1, … N − 1). It 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_{1}...β_{N}} ϵ_{α_{1}...α_{N}} .  (61)

The algebraic structure of this relation follows because one can use the Yang-Baxter equations to shift in (60) 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 (6).

The normalization constant c in (10) is calculated from (12) and (11) 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) > 0 .

B Proof of Theorem 1

Proof. Property (i) in the form of (32) follows directly from (i′) of (42), 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 (5) the property (ii) in the form of (33) may be rewritten as a matrix difference equation [43, 41, 39]

K_{1...n}(\underline{θ})Q_{1...n}(\underline{θ}, i) = (−1)^{N−1}K_{1...n}(\underline{θ}′)σ̇_{1}  (62)

where \underline{θ}′ = (θ_{1}, …, θ_{i} + 2πi, …, θ_{n}) and σ̇ is a statistics factor. The matrix Q(\underline{θ}, i) is the trace with respect to the auxiliary space

Q_{1...n}(\underline{θ}, i) = tr_{0} T̃_{Q,1...n,0}(\underline{θ}, i) = (−1)^{N−1} [figure: •; α_{1}; α′_{1}; α_{i}; α′_{i}; α_{n}; α′_{n}; θ_{1}; θ′_{i}; θ_{i}; θ_{n}; …; …; •] (63)

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 = S̃(0) is the permutation matrix. In (63) we use the rule that the rapidity of a line changes by 2πi if the line bends by 360^{0} in the positive sense. Because of (i) it is sufficient to proof the matrix difference equation (62) for i = 1, i.e. for Q(\underline{θ}) = Q(\underline{θ}, 1)

In the following we will suppress the indices 1 … n. The Yang-Baxter relations (15) imply the typical commutation rules for the matrices Ã, C̃, D̃ defined in eq. (16)

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)  (64)
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, …, N}. In addition there are the Zapletal commutation rules [43, 41, 39] 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 [43]

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})  (65)
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}) .  (66)

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 (62) in the form

K(\underline{θ}) Q(\underline{θ}) = K(\underline{θ}) (Ã_{Q}(\underline{θ}) + ∑_{β=2}^{N} D̃_{Q}_{β}^{β}(\underline{θ})) = σ̇(−1)^{N−1}K(\underline{θ}′)  (67)

where K(\underline{θ}) is a co-vector valued function as given by eq. (17) and the Bethe ansatz state (20) and (21). Using a short notation we write

K(\underline{θ}) = ∫_{C\underline{θ}} d\underline{z} h̃(\underline{θ}, \underline{z})p(\underline{θ}, \underline{z}) Ψ̃(\underline{θ}, \underline{z})

(with ∫_{C\underline{θ}} d\underline{z} = \frac{N_{n}}{m!} ∫_{C\underline{θ}} \frac{dz_{1}}{R} ⋯ ∫_{C\underline{θ}} \frac{dz_{m}}{R}). To analyze the left hand side of eq. (67) we proceed

as follows: We apply the trace of T̃_{Q} to the co-vector Ψ̃(\underline{θ}, \underline{z}).

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).

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 (65) and (66) 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 (65); 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, …, N. If we insert these equations into the representation (17) 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}. These three facts follow from the ‘shift relations’ of the ϕ-function (13) and of the higher level L-function (ii)^{(1)} of (39).

The proof of (iii) is similar to that for the Z(N) model in [23]. 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}) .

We prove the form factor equation in the form of (34)

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

× (1 − σ̇_{N}^{O}S_{Nn} … S_{NN+1}) (68) for the K-function. The residues of K_{1...n}(\underline{θ}) consists 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 and z_{1} = θ_{2}, z_{2} = θ_{3}, …, z_{N−1} = θ_{N}

Φ̃\underline{β} ; \underline{α}(\underline{θ}, \underline{z}) → (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}

= ∏_{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} . (69)

It has been used that due to the SU(N) ice rule only the amplitude b(·) (because all β > 1) contributes to the S-matrices S(θ_{1} − z_{j}) (j ≥ N) and a(·) to the S-matrices S(θ_{2}−z_{j}), …, S(θ_{N} −z_{j}) (j ≥ N), S(θ_{i}−z_{1}), …, S(θ_{i}−z_{N−2}) (i ≥ N) after having applied Yang-Baxter relations. One observes that the product of S-matrices in (40) together with the one in (69) yields the total N − 1 particle 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 (61) applies

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

We combine (69) with the function L\underline{β}(\underline{z}) with the property (40) 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})

with c_{0} = c_{1}ν(N − 1)!(iη)^{N−1}(−1)^{m−N+1} ∏_{j=1}^{N−1} ϕ̃(jiη)

We have used the equations (13), (19), the relation (iii’) of (42) for the p-function, the relation (43) for the constant ν and the recursion relation (45) for the constants c_{l} with the solution (73).

The remaining contribution to (68) 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})σ̇_{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 the claim as

(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})σ̇_{N} = 0  (70)

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) (see (32))

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

Because the Bethe ansatz states are of highest weight (see [36]) it is sufficient to prove equation (70) only for α_{N} = 1. 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}}

= 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} (71)

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 (61)

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})

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

We combine (71) with the function L\underline{β}(\underline{z}) with the property (40) (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}σ̇ ∏_{i=N+1}^{n} ∏_{j=2}^{N} ϕ̃(θ_{ij}) ε_{1...N}K_{N+1...n}(\underline{θ̌})P_{N}(1) .

Again we have used the equations (13), (19), the relation (iii’) of (42) for the p-function, the relation (43) for the constant ν and the recursion relation (45) for the constants c_{l} with the solution (73).

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). In C̃β_{N−1} ; \underline{α̃}(\underline{θ}, z_{N−1}) the S-matrix S̃_{α′_{N} 1}^{β′_{N−1}α′′_{N}} (θ_{N} − z_{N−1}) yields Γ_{(ρσ)}^{β′_{N−1}α′′_{N}} Γ_{α′_{N} 1}^{(ρσ)} by

applying Res_{z_{N−1}=θ_{N} −iη}. For α_{N} = 1 this vanishes because Γ_{αβ}^{(ρσ)} is antisymmetric with respect to α, β the SU(N) ice rule implies that also α′_{N} = 1. Therefore equation (70) is proved for α_{N} = 1 and because the left hand side belongs to a highest weight state also in general.

C Proof of lemma 2

Proof. We set n_{l} = m, n_{l+1} = k. 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. 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})  (72)

for u_{12}, …, u_{N−l−2,N−2} → iη. In the integral representation (36) 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}

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

26 REFERENCES

with the notation \underline{ẑ} = (z_{1}, …, z_{N−l}). Therefore using the assumption (72) 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 (44) for the p-function and the recursion relation

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

The solution of this recursion relation with c_{N−1} = 1 is

c_{1} = ϕ̃^{N−2}(iη)ϕ̃^{N−3}(2iη) ⋯ ϕ̃((N − 2)iη) ∏_{l=1}^{N−2}(−1)^{n_{l+1}−N+l+1}  (73)

References

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