The Form Factor Program: a Review and New Results - the Nested SU(N) Off-Shell Bethe Ansatz

The purpose of the ''bootstrap program'' for integrable quantum field theories in 1+1 dimensions is to construct explicitly a model in terms of its Wightman functions. In this article, this program is mainly illustrated in terms of the sinh-Gordon model and the SU(N) Gross-Neveu model. The nested off-shell Bethe ansatz for an SU(N) factorizing S-matrix is constructed. We review some previous results on sinh-Gordon form factors and the quantum operator field equation. The problem of how to sum over intermediate states is considered in the short distance limit of the two point Wightman function for the sinh-Gordon model.


Introduction
The bootstrap program to formulate particle physics in terms of the scattering data, i.e. in terms of the S-matrix goes back to Heisenberg [1] and Chew [2]. Remarkably, this approach works very well for integrable quantum field theories in 1 + 1 dimensions [3,4,5,6,8,7]. The program does not start with any classical Lagrangian. Rather it classifies integrable quantum field theoretic models and in addition provides their explicit exact solutions in term of all Wightman functions. We achieve contact with the classical models only, when at the end we compare our exact results with Feynman graph (or other) expansions which are usually based on Lagrangians. However, there is no reason that the resulting quantum field theory is related to a classical Lagrangian.
One of the authors (M.K.) et al. [4] formulated the on-shell program i.e. the exact determination of the scattering matrix using the Yang-Baxter equations. The concept of generalized form factors was introduced by one of the authors (M.K.) et al. [8]. In this article consistency equations were formulated which are expected to be satisfied by these quantities. Thereafter this approach was developed further and studied in the context of several explicit models by Smirnov [9] who proposed the form factor equations (i)-(v) (see below) as extensions of similar formulae in the original article [8]. These formulae were then proven by two of the authors et al. [10]. In the present article we apply the form factor program for an SU (N ) invariant S-matrix (see [11]). The procedure is similar to that for the Z(N ) and A(N − 1) cases [12,13] because the bound state fusions are similar in these three models. However, the algebraic structure of the form factors for the SU (N ) model is more intricate, because the S-matrix also describes backward scattering. We have to apply the nested "off-shell" 1 Bethe ansatz, which was originally formulated by one of the authors (H.B.) [14,15,16] to calculate correlation function in WZNW models (see also [17,18]).
Finally the Wightman functions are obtained by taking integrals and sums over intermediate states. The explicit evaluation of all these integrals and sums remains an open challenge for almost all models, except the Ising model [19,20,21,7,22]. In this article we discuss this problem for the examples of the sinh-Gordon model (see [23]). We investigate the short distance behavior of the two-point Wightman function of the exponentiated field.

The "bootstrap program"
The 'bootstrap program' for integrable quantum field theories in 1 + 1-dimensions provides the solution of a model in term of all its Wightman functions. The result is obtained in three steps: 1. The S-matrix is calculated by means of general properties such as unitarity and crossing, the Yang-Baxter equations (which are a consequence of integrability) and the additional assumption of 'maximal analyticity'. This means that the two-particle S-matrix is an analytic function in the physical plane (of the Mandelstam variable (p 1 + p 2 ) 2 ) and possesses only those poles there which are of physical origin. The only input which depends on the model is the assumption of a particle spectrum with an underlining symmetry. Typically there is a correspondence of fundamental representations with multiplets of particles.
A classification of all S-matrices obeying the given properties is obtained.
2. Generalized form factors which are matrix elements of local operators out p ′ m , . . . , p ′ 1 |O(x)| p 1 , . . . , p n in are calculated by means of the S-matrix. More precisely, the equations (i)-(v) as listed in Section 3 are solved. These equations follow from LSZ-assumptions and again the additional assumption of 'maximal analyticity' [10].
3. The Wightman functions are obtained by inserting a complete set of intermediate states.
In particular the two point function for a Hermitian operator O(x) reads Up to now a direct proof that these sums converge exists only for the scaling Ising model [19,20,21] and the non-unitary 'Yang-Lee' model [24].
Recently, Lechner [25] has shown that models with factorizing S-matrices exist within the framework of algebraic quantum field theory. For the algebraic approach to quantum field theories with factorizing S-matrices see the works of Schroer and Schroer-Wiesbrock [26,27,28]. A determinant representation for correlation functions of integrable models has been obtained by Korepin et al. [29,30].

Integrability
Integrability in (quantum) field theories means that there exist infinitely many local conservation laws A consequence of such conservation laws in 1+1 dimensions is that there is no particle production and the n-particle S-matrix is a product of 2-particle S-matrices Bound states: Let γ be a bound state of particles α and β with mass m γ = m 2 α + m 2 β + 2m α m β cos η , (0 < η < π).
Then the 2 particle S-matrix has a pole such that where η is called the fusion angle and Γ γ αβ is the 'bound state intertwiner' [31,32]. The bound state S-matrix, that is the scattering matrix of the bound state (12) with a particle 3, is obtained by the "bootstrap equation" [31] where we use the usual short hand notation of matrices acting in the spaces corresponding to the particles 1, 2, 3 and (12).
Examples of integrable models in 1+1-dimensions (which we will consider in this review) are the sinh-Gordon model defined by the classical field equation where the Fermi fields form an SU (N ) multiplet. Further integrable quantum field theories are: scaling Z N -Ising models, nonlinear σ-models, O(N )Gross-Neveu models, Toda models etc.

The S-matrix
The sinh-Gordon S-matrix is given by the analytic continuation β → iβ of the sin-Gordon breather S-matrix [5] The model has the self-dual point at The SU (N ) S-matrix: All solutions of U (N )-invariant S-matrix satisfying unitarity, crossing and the Yang-Baxter equation have been obtained in [33]. Following [34,35], we adopt the view that in the SU (N ) Gross-Neveu model, the anti-particles are bound states of N − 1 particles. This implies that there is no particle anti-particle backward scattering and that the SU (N ) S-matrix should be given by solution II of [33]. The scattering of the fundamental particles which form a multiplet corresponding to the vector representation of SU (N ) is (see [36,37,38] and [39] for N = 2) where due to Yang-Baxter c(θ) = − 2πi N θ b(θ) holds and the highest weight amplitude is given as There is a bound state pole at θ = iη = 2πi/N in the antisymmetric tensor sector which agrees with Swieca's [34] picture that the bound state of N − 1 particles is to be identified with the anti-particle. Similar as in the scaling Z(N )-Ising and A(N − 1)-Toda models [12,13] this will be used to construct the form factors in SU (N ) model [11].

Form factors
For a local operator O(x) the generalized form factors [8] are defined as for θ 1 > · · · > θ n . For other orders of the rapidities they are defined by analytic continuation. The index α i denotes the type of the particle with momentum p i . We also use the short notations For the SU (N ) Gross-Neveu model α denotes the types of particles belonging to all fundamental representations of SU (N ) with dimension N r , r = 1, . . . , N − 1. In most formulae we restrict α = 1, . . . , N to the multiplet of the vector representation. Similar as for the S-matrix, 'maximal analyticity' for generalized form factors means again that they are meromorphic and all poles in the 'physical strips' 0 ≤ Im θ i ≤ π have a physical interpretation. Together with the usual LSZ-assumptions [40] of local quantum field theory the following form factor equations can be derived: (i) The Watson's equations describe the symmetry property under the permutation of both, the variables θ i , θ j and the spaces i, j = i + 1 at the same time for all possible arrangements of the θ's.
(ii) The crossing relation which implies a periodicity property under the cyclic permutation of the rapidity variables and spaces where σ O α takes into account the statistics of the particle α with respect to O (e.g., σ O α = −1 if α and O are both fermionic, these numbers can be more general for anyonic or order and disorder fields, see [13]). For the charge conjugation matrix C1 1 we refer to (3).
(iii) There are poles determined by one-particle states in each sub-channel given by a subset of particles of the state in (8).
In particular the function F O α (θ) has a pole at θ 12 = iπ such that (iv) If there are also bound states in the model the function F O α (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 where the bound state intertwiner Γ (12) 12 is defined by (4) (see [31,32]). (v) Naturally, since we are dealing with relativistic quantum field theories we finally have if the local operator transforms under Lorentz transformations as The properties (i)-(iv) may be depicted as These equations have been proposed by Smirnov [9] as generalizations of equations derived in the original articles [8,7,41]. They have been proven [10] by means of the LSZ-assumptions and 'maximal analyticity'.
We will now provide a constructive and systematic way of how to solve the equations (i)-(v) for the co-vector valued function F O 1...n once the scattering matrix is given.

Two-particle form factors
For the two-particle form factors the form factor equations are easily understood. The usual assumptions of local quantum field theory yield where the rapidity difference is defined by p 1 p 2 = m 2 cosh θ 12 . For integrable theories one has particle number conservation which implies (for any eigenstate of the two-particle S-matrix) Crossing (10) means where for one-particle states the in-and out-states coincide. Therefore Watson's equations follow For general theories Watson's [42] equations only hold below the particle production thresholds. However, for integrable theories there is no particle production and therefore they hold for all complex values of θ. It has been shown [8] that these equations together with "maximal analyticity" have a unique solution.
As an example we write the sinh-Gordon [8] and the (highest weight) SU (N ) form factor function [11] which are the minimal solution of (14) with S SHG (θ) as given by (5) and S SU (N ) (θ) = a (θ) as given by (7), respectively.

The general form factor formula
As usual [8] we split off the minimal part and write the form factor for n particles as By means of the following "off-shell Bethe ansatz" for the (co-vector valued) K-function we transform the complicated form factor equations (i) − (v) into simple ones for the p-functions which are scalar and polynomials in e ±z i . The "of f-shell Bethe ansatz" state Ψ α 1 ...αn (θ, z) is obtained as a product of S-matrix elements and the integration contour C θ depends on the model (see below for the SU (N )-model). The scalar functions depend on S(θ) only (see (25) below), i.e. on the S-matrix, whereas the p-function p O (θ, z) depends on the operator.

The SU (N ) form factors
The form factors for n fundamental particles (of the vector representation of SU (N )) are given by (17)- (19) where the "nested Bethe ansatz" is needed. This means that Ψ is of the form where the indices α i take the values α = 1, . . . , N and the summations run over β i = 2, . . . , N .
We consider a state with n particles and define the monodromy matrix T 1...n,0 (θ, θ 0 ) = S 10 (θ 10 ) · · · S n0 (θ n0 ) = 1 n 0 . . . as a matrix acting in the tensor product of the "quantum space", a space of n particles (with respect to their quantum numbers) V 1...n = V 1 ⊗ · · · ⊗ V n and the "auxiliary space" V 0 . All vector spaces V i are isomorphic to a space V whose basis vectors label all kinds of particles.
This may be depicted as It means that Φ β α (θ, z) is a product of S-matrix elements as given by the picture where at all crossing points of lines there is an S-matrix (6) and the sum over all indices of internal lines is to be taken. The "pseudo-vacuum" is the highest weight co-vector (with weight w = (n, 0, . . . , 0)) where the unit vectors e(α) (α = 1, . . . , N ) correspond to the particle of type α which belong to the vector representation of SU (N ). The pseudo-vacuum basic vector satisfies Ω 1...n B β 1...n (θ, z) = 0, The amplitudes of the scattering matrices are given by equation (6). The technique of the nested Bethe ansatz means that for the co-vector valued function L β 1 ...βm (z) in (20) one makes the second level Bethe ansatz. This ansatz is of the same form as (18) only that the range of the indices is reduced by 1. Iterating this nesting procedure one finally arrives at a scalar function.
The integration contour C θ depends on the θ and is depicted in Fig. 1.
. . . Swieca's picture is that the bound state of N − 1 fundamental particles is to be identified with the anti-particle and lead together with the form factor recursion relations (iii) + (iv) to the equation for the function φ(z) (see [13,11]) with the solution .
In [11] it is shown that the form factors given by (17) and (18) satisfy the form factor equations (i)-(v) if some simple equations for the p-function are satisfied. We note that form factors of this model were also calculated in [9,43,44] using other techniques.

Locality
Let us discuss a further property of the form factors which is directly related to the nature of the operators, that is locality. In fact, this property touches the very heart and most central concepts of relativistic quantum field theory, like Einstein causality and Poincaré covariance, which are captured in local field equations and commutation relations. So far we have presented a formulation of a quantum field theory, which starts from a particle picture. It is basically possible to obtain the particle picture from the field formulation by means of the LSZ-reduction formalism. The reverse problem, namely of how to reconstruct the entire field content, or at least part of it, from the scattering theory is in general still an outstanding challenge. Besides this classification issue there remains also the general question if the operators which are related to the solutions of (i)-(v) are genuinely local, meaning that they (anti)-commute for space-like separations with themselves. It has been shown that (i)-(v) (see e.g. [9,13]) imply for all matrix elements. Note that commutation rules such as (26) hold if the statistics factors in (i)-(v) are trivial, σ O = 1. However, for more general statistics factors there are also fermionic, anyonic and even order-disorder commutation rules as for example for the scaling Z N -Ising model (see [13]).

Examples of operators and their p-functions for the sinh-Gordon model
Since there is no backward scattering the Bethe ansatz in (18) is trivial, the integrations can be performed [45] and the n-particle K-function may be written as For several cases the correspondence between local operators and their p-functions have been proposed in [45] 3 . Here we provide three examples: 1. The normal ordered exponential of the field (see also [48]) 2. Expanding the last relation with respect to γ one obtains the p-functions for normal ordered powers : ϕ(x) N : in particular for N = 1 : which yields (for n = 1) the 'wave function renormalization constant' (see also [8]).
3. The higher conserved currents J µ L (x) which are typical for integrable quantum field theories Quantum sinh-Gordon f ield operator equation Using (28) and (29) one finds [49,32,45] that the quantum sinh-Gordon field equation ϕ(x) + α β : sinh βϕ : (x) = 0 (30) holds for all matrix elements, if the "bare" mass √ α is related to the renormalized mass by where m is the physical mass of the fundamental boson. The result may be checked in perturbation theory by Feynman graph expansions. In particular in lowest order the relation between the bare and the renormalized mass (31) had already been calculated in the original article [8].
The result is which agrees with the exact formula above. The factor πν sin πν in (31) modifies the classical equation and has to be considered as a quantum correction. The proof of the quantum field equation (30) can be found in [45].

Wightman functions
As the simplest case we consider the two-point function of two local scalar operators O(x) and

Summation over all intermediate states
Inserting a complete set of in-states we may write w(x) = ∞ n=0 1 n! dp 1 2π2ω 1 · · · dp n 2π2ω n e −ix(p 1 +···+pn) We have introduced the functions where crossing has been used. In particular we consider exponentials of a scalar bose field where : · · · : means normal ordering with respect to the physical vacuum and which amounts to the condition 0 | : e iγ (′) ϕ(x) : | 0 = 1 and therefore g 0 = 1 holds.
This follows when we distinguish the variable θ 1 in the relation of the g's and the h's above and reorganize the terms on the right hand side as follows h J g I\J .
In principle the higher particle intermediate state integrals may also be calculated, however, up to now we were not able to derive a general formula. For the scaling Ising model, however, this is possible. The constant C in the approximation of 1-intermediate states is given as C 1 = exp (−2h 1 (γ E − ln 2)) = exp 4 sin πν πF (iπ) (γ E − ln 2) .
We have not calculated the integrals appearing in higher-particle intermediate state contributions.