Asymptotic factorization of n-particle SU(N) form factors

We investigate the high energy behavior of the SU(N) chiral Gross-Neveu model in 1 + 1 dimensions. The model is integrable and matrix elements of several local operators (form factors) are known exactly. The form factors show rapidity space clustering, which means factorization, if a group of rapidities is shifted to infinity. We analyze this phenomenon for the SU(N) model. For several operators the factorization formulas are presented explicitly.

F Exponential behavior 37 1 Introduction The Bjorken scattering or inelastic lepton-hadron scattering at high energies has been a very important and crucial stage in the development of modern QCD [1][2][3]. This well known experimental investigation in high energy physics is very actual and has now a modern continuation, being part of lepton-hadron experimental research [4,5]. The essential point in these studies is the behavior of the structure functions of the hadrons [3]. They describe the parton (quark) structure of the hadrons and the nature of the interaction between the quarks inside of the hadrons. The amplitude of the lepton-hadron interaction consists of two parts, where the lepton part is well known. The hadron part, whose invariant decomposition provides the hadron form factors or structure functions [3], is not known.
In QCD the calculation of the structure function for all values of the Bjorken variable x is still an open problem.
On the other side, the existence of exact integrable models in 1+1 dimensional asymptotically free theories may be relevant, providing valuable insights into this discussion. Remarkably, due to integrability, it is possible to obtain exact form factors of local operators [6][7][8][9] 1 . In the remarkable papers [19][20][21] Balog and Weisz define analogs of the structure functions in two-dimensional integrable quantum field theories. In particular, they consider form factors of the current operator (related to the structure-function) of the O(3) sigma model, which are accurately computed over the whole x range; in addition, the structure functions and some moments are compared with renormalized perturbation theory. They also calculate structure functions in the O(N ) sigma model using 1/N expansion and make some conjectures on possible universal formulae in 4 dimensional QCD for small x. Interestingly, in [21] the authors employ the so called cluster behavior of the form factors to calculate the same structure functions. Here we mention that in all of the previously cited papers the authors use only 2,3 and 4 particle form factors in O(3) or in O(N ) sigma models.
In this article we will start an investigation of the above mentioned problems in an opposite order: we will analyze the cluster behavior of the SU (N ) chiral Gross-Neveu model 2 , which is an asymptotically free theory. For this, we do not only use the 2,3 and 4 particle form factors, but also the general n -particle form factors. We should point out that the first investigation of the cluster behavior of the exact form factors was performed by Smirnov [7] in the case of the sine-Gordon, the SU (2) Thirring model and the O(3) sigma model. He also applied these results to the current algebra [7]. For the sinh-Gordon model the cluster property of form factors was investigated in [22]. Here we will consider the high energy behavior of the exact form factors in 1+1 dimensional asymptotically 1 Other approaches to form factors in integrable quantum field theories can be found in [10][11][12][13][14][15][16][17][18]. 2 For N = 2 also called SU (2) Thirring model. free quantum field theories [7,23], with connection to the factorization property and the Bjorken scattering.
The paper is organized as follows: In Section 2 we recall some known formulae, which will be used in the following. In particular we present the SU (N ) S-matrix and construct the form factors which are n-particle matrix elements of local operators. In Section 3 we investigate the "rapidity space clustering" of form factors, which describes the behavior of form factors, if a group of rapitities is shifted to infinity. Several examples of operators are considered, as the Noether current, the energy-momentum tensor, the fundamental field of the SU (N ) chiral Gross-Neveu model, etc. In Section 4 we present the proofs. Some more technical details are delegated to the Appendices.

SU(N) form factors
Minimal form factor function F (θ), φ-and τ -function: To construct the form factors we need the "minimal form factor function F (θ)" for two particles [9,26] where G(z) is Barnes G-function. It is the minimal solution of the equations where a(θ) is the highest weight amplitude of the corresponding channel of the S-matrix (2.1).
The φ-function satisfies [9,26] is the minimal F-function for a particle and an anti-particle satisfyinḡ The τ -function is (2.7) n particle form factors: The matrix element of a local operator O(x) for a state of n particles of kind α i with rapidities θ i defines the generalized form factor F O 1...n (θ), which is a co-vector valued function with components F O α (θ) . The form factors satisfy the form factor equations (i) -(v) (see Appendix D). Solutions of these equations can be written as follows: As usual we split off the minimal part [6] where α = (α 1 , . . . , α n ), θ = (θ 1 , . . . , θ n ) and F (θ) is defined by (2.3). The K-function is given by an 'off-shell' Bethe ansatz in terms of the multiple contour integral The integration contour C θ (see Fig. 1) and the scalar function h(θ, z) depend only on the S-matrix and not on the specific . (2.11) The dependence on the specific operator O(x) is encoded in the scalar p-function p O (θ, z) which is in general a simple function of e θ i and e z j .
. . . Figure 1. The integration contour C θ . The bullets refer to poles of the integrand in (2.10).
Bethe state: The stateΨ α in (2.10) is a linear combination of the basic Bethe ansatz co-vectorsΨ As usual in the context of the algebraic Bethe ansatz [27,28] the basic Bethe ansatz co-vectors are obtained from the monodromy matrix where the S-matrixS i0 is given by (2.2). The reference co-vector is defined as usual by ΩB β = 0 which implies

It is an eigenstates ofÃ andD
where the indices 1 . . . n are suppressed. The basic Bethe ansatz co-vectors in (2.12) are defined asΦ The technique of the 'nested Bethe ansatz' means that for the coefficients L β (z) in (2.12) one makes the analogous construction as for K α (θ) in (2.10), where now the indices β take only the values 2 ≤ β i ≤ N . This nesting is repeated until the space of the coefficients becomes one dimensional. The final result is and the complete Bethe ansatz statẽ n j ) and α N −1 = (N, . . . , N ). It is well known (see [29]) that the 'off-shell' Bethe ansatz states are highest weight states if they satisfy certain matrix difference equations. If there are n particles the SU (N ) weights are [26] where n 1 = m, n 2 , . . . are the numbers of C operators in the various levels of the nesting, w O is the weight vector of the operator O and L = 0, 1, 2, . . . ; note that w = (1, . . . , 1) correspond to the vacuum sector.

Examples of local fields:
In this article we consider the following fields: The SU (N ) Noether current J µ a =ψ β γ µ (T a ) β α ψ α transforms as the adjoint representation with highest weights w J = (2, 1, . . . , 1, 0). The The conservation law ∂ µ J µ a (x) = 0 implies that J µ a (x) may be written in terms of the pseudo potential J a (x) as with the quantum numbers charge Q J = 0 weight vector w J = (2, 1, . . . , 1, 0) statistics factor σ J = 1 spin s J = 0. Due the Swieca et al [30] the bound state of N − 1 particles is to be identified with the anti-particle. This means that the anti-particleᾱ of a fundamental particle α of rank 1 is a bound state of rank N − 1 The charge conjugation matrix is given by with C βᾱ Cᾱ γ = δ γ β . In terms of fields this meansψ β = C β(ρ)ψ (ρ) = C β(ρ) ψ ρ 1 . . . ψ ρ N−1 .
For the Bethe ansatz the formulation of the Noether current given by = 0 is more convenient, which means for the pseudo potentials Because the Bethe ansatz yields highest weight states we obtain the matrix elements of the highest weight component The form factor is given by (2.9) and (2.10) with the p-function for the operator J(x) [9] p J (θ, z) = e iπ 1 In particular the one particle and one anti-particle form factor is [9] F Ja αβ (θ, ω) = (T a ) αβ where (T a ) αβ = C δβ (T a ) δ α andF (θ) defined in (2.5) is the "minimal form factor function" for one particle and one anti-particle.
The fundamental field ψ α (x) of the chiral SU (N ) Gross-Neveu model with the quantum numbers charge The p-function of the highest weight component ψ = ψ 1 for n = 1 mod N is [9] p ψ (θ, z) = e and the 1-particle matrix element is The field χᾱ(x) with the quantum numbers The p-function of the highest weight component χ = χN for n = (N − 1) mod N is (3.20) with n j = (n + 1) (1 − j/N ) − 1 and the 1-anti-particle matrix element is (see [26]) (3.21)

Results
As examples of the general formula (3.1) we obtain: 1. Particle number n = 0 mod N and k = 0 mod N 2. Particle number n = 0 mod N and k = 1 mod N 3. Particle number n = 1 mod N and k = 0 mod N 4. Particle number n = 1 mod N and k = 1 mod N (3.28)

Proofs
We use the short notation θ W of Section 3 and in addition z W = (z The choice of the k j integrations out of the n j ones in (2.10) is arbitrary therefore there is a factor of n j k j such that n j The asymptotic behavior of the form factors given by (2.9) and (2.10) with θ = θ W for W → ∞ is obtained from the asymptotic behavior of F (θ W ),h(θ W , z W ),Ψ(θ W , z W ) and the p-functions (see Appendix E). In the following, some equations are written for simplicity up to constant factors. Constant factors in eq. (3.1) are finally obtained by form factor equation (iii).

Theorem 1
Theorem 1 The form factor of the pseudo-potential of the current for particle number n = 0 mod N and k = 0 mod N shows the cluster behavior Proof. We use the short notations of (2.9) . . . (2.17) and investigate (for For these values of k j and l j we obtain, more precisely, with (E.18), (E.7) and (E.22) in leading order the asymptotic behavior (up to a constant factor) Theẑ-integral vanishes because of Lemma 1 and therefore in leading order Order 1 W : we have to apply the asymptotic behavior of the h-function (E.12), (E.13), (E.14) and the Bethe state (E.23) and (E.20).
We present a complete proof of this 1 W -term for SU (2) and for general N the example of Appendix B for one particle and one anti-particle. In addition we show consistency of the general clustering formula with the form factor equation (iii) (see Remark 1).
We have to consider the 2 contributions: A) From the h-function: Note that because of Lemma 1 inh 1 (θ, z) of (E.14) only thê z j -dependent terms contribute. Therefore we get on the rhs of (4.2) fromh 1 for and (up to a constant factor) where and (up to a constant factor) where (A.3) has been used. The final result is . The other components are obtained by SU (2)transformations. The constant factor is calculated below and the minus sign is due to SU (2) invariance. In terms of the components J a (3.6) this can be written as in (3.22) (see (4.11)).
Calculation of the functions c J JJ (k, l, W ) : defined by for general N . Here and in the following we use the short notation We also use the satistics factorσ O α , which is related to the "physical" statistics by 4 We apply the general procedure of Appendix C: Using a(W ) → e −iπ(1− 1 N ) of (E.1) and σ J 1 = 1, Q J = 0 we check (C.4) and (C.7) for this casė Therefore, as proofed in Appendix C, c J JJ (k, l, W ) is independent of k and l, because It was used that (2.1), (B.1), (E.1) including 1/W terms and a(θ)a(−θ) = 1 implẏ 2) Taking first W → ∞ and then the Res means where (3.9) was used. As result we obtain from (4.7) and (4.8) Remark 1 Note that this also proves consistency of the clustering formula (3.22) for general N with the form factor equation (iii).
Equivalence: We prove that is equivalent to We have the general relations [31,32] [ (4.10) By (3.6) and (4.9) we obtain for W → ∞ where the relations (4.10) have been used. This proves the equivalency.

Theorem 2
Theorem 2 The form factor of the field φ(x) for particle number n = 0 mod N and k = 0 mod N shows the cluster behavior Remark 2 Note that this is the typical behavior of an exponential of a bosonic field (see [33]).
Proof. We investigate wherek For these values of k j and l j we obtain, more precisely, with (E.18), (E.9) and (E.22) in leading order the asymptotic behavior (up to a constant factor) The constant factor is again calculated using the form factor equation (iii).

Theorem 3
Theorem 3 The form factor of the energy momentum potential for particle number n = 0 mod N and k = 0 mod N satisfies More precisely

Conjecture 1
The cluster behavior of form factor of T for k = 0 mod N reads as We have no general proof of this conjecture. The problem is that the expansion for large W of the integrand in the contour integral representation in (2.10) must not be interchanged with the integration, this is only allowed up to the 1/W -term.
However, we have checked consistency with the form factor equation (iii), which also yields the function From the asymptotic behavior of F (θ W )h (θ W , z W ) and p T (θ W , z W ) in (E.18), (E.8) and (F.2) we derive for W → ∞ the exponential behavior For these values of k j and l j we obtain, more precisely, with (E.18), (E.8) and (E.22) in leading order the asymptotic behavior (up to a constant factor) However, this means that in leading order Order 1 W : similarly, as in the proof of Theorem 1 we discuss the contribution fromh 1 (θ, z) of (E.14), however, for k 1 = k (1 − 1/N ) , l 1 = l (1 − 1/N ) there are no theẑ j -dependent terms and and therefore this contribution vanishes by Lemma 1.

Theorem 4
Theorem 4 The cluster behavior of the form factor of the pseudo-potential of the current for particle number n = 0 mod N and k = 1 mod N reads as Proof. We investigate The exponential behavior of the integrand is again given by (4.1). For k = 1 mod N the leading asymptotic behavior e − 1 For these values of k j and l j we obtain, more precisely which implies (up to const.) Calculation of the function c J ψχ (k, l, W ) defined by We apply the procedure of Appendix C: Using a(W ) → e −iπ(1− 1 N ) of (E.1) we check (C.4) and (C.7) with σ J = 1, Therefore c J ψχ (k, l, W ) is independent of k and for k = 1 mod N (see C) The special case c J ψχ (1, N − 1, W ) is calculated by the following example, which implies because l 1 = (l + 1) (1 − 1/N ) − 1.

Conjecture 2
Conjecture 2 The form factor of the energy momentum potential for particle number n = 0 mod N and k = 1 mod N shows the cluster behavior (3.26) We have no general proof of this conjecture. The problem is the same as in Conjecture 1, that the expansion for large W of the integrand in the multiple contour integral representation in (2.10) must not be interchanged with the integration. However, the relation (4.17) implies the cluster relation (3.26) and we have again checked consistency with the form factor equation (iii), which also yields the function c T ψχ (k, l, W ) of (3.26).
Calculation of the function c T ψχ (k, l, W ) : In the same way as above for c J ψχ we prove that c T ψχ (k, l, W ) is independent of k and for k = 1 mod N c T ψχ (k, l, W ) = c T ψχ (k 0 , l 0 , W )(−1) (N −1)(l−l 0 )/N The special case c T ψχ (1, N − 1, W ) is calculated by the following example, which implies Example: The particle anti-particle of (3.13) and asymptotic behavior of the particle antiparticle minimal form factor function (E.5) imply

Theorem 5
Theorem 5 The cluster behavior of form factor of the fundamental field for the number particles n = 1 mod N and k = 0 mod N reads as Proof. We investigate From the asymptotic behavior of F (θ W , ω W )h (θ W , ω W , z W ) and p ψ (θ W , z W ) in (E.18), (E.10) and (F.4) we derive for W → ∞ the exponential behavior For k = 0 mod N and l = 1 mod N the leading asymptotic behavior ∝ e − 1 2 W 0 is obtained fork j = 0 i.e. k j = k (1 − j/N ) and l j = (l − 1) (1 − j/N ) by (3.19), which implies (up to const.) However, this means that in leading order because of Lemma 1. The proof of the 1 W contribution is similar to that one of Theorem 1. Order 1 W : we have to apply the asymptotic behavior of the h-function (E.12) and the Bethe state (E.21).

2) Taking first W → ∞ and then the Res means
As result we obtain c ψ Jψ (k, l, W ) = iη 1 W which proves (3.27).

Theorem 6
Theorem 6 The cluster behavior of form factor of the fundamental field for particle number n = 1 mod N and k = 1 mod N reads as Proof. We investigate and obtain as above the exponential behavior (4.21). The leading behavior ∝ e − 1 and (up to const.) proving (3.28).
Calculation of the function c ψ ψφ (k, l, W ) : defined by We apply the procedure of Appendix C: Using a(W ) if we normalize the field φ(x) by F φ ∅ = 0|φ(x)|0 = 1, this gives the result

Summary
In this article we investigate the rapidity clustering of exact multi-particle form factors of the SU (N ) chiral Gross-Neveu model. For some examples of local fields, in particular, the Noether current, the energy momentum tensor, the fundamental spinor field etc, we explicitly demonstrate the clustering or factorization phenomena. In a forthcoming paper we will consider the form factor of the Noether current in a special form, in order to connect the asymptotic clustering with Bjorken scattering.
A Some lemmata For SU (2) the proof of this lemma is quite analog to that for the Sine-Gordon model in [34]. For general N we present an example (see Proposition 1).

Lemma 2
For SU (2) and m = n/2 2) which is a non-highest weight K-function and For SU (2) the proofs are similar to the one of Lemma 1. For general N see the proofs of Propositions 2 and 3.

B.1.1 Bound state S-matrix
The S-matrix of a particle and an anti-particle (which is a bound state of N − 1 particles where the charge conjugation matrices are given by (3.5).
Proof. The weight formula (B.6) implies that n j = 1 for j = 1, . . . , N − 1 and the L-function of level j is where (B.20) -(B.29) have been used. For j = 0 B.3 Theorem 1 for general N and n =n = 2, k =k = 1 We consider form factors of the pseudo potential J(x) for particles and anti-particles. Formula (B.6) means, generalizing (3.8) and the p-function is [26] p J (θ, ω, z, z (N −1) ) = with the asymptotic behavior p J (θ,ž). (B.10) In particular for n =n = 2 and k =k = 1 we prove the proposition: The form factor of the current for n =n = 2 and k =k = 1 satisfies the clustering formula (3.22) in the form Proof. The exponential behavior (4.1) implies for n =n = 2 and k =k = 1 that k j = 1 and l j = 0 for j = 1, . . . , N − 1. We investigate for J = J 1N (N = bound state (1 . . . N − 1)) We have proved in theorem 1 that in leading order Order 1 W : we have to apply the asymptotic behavior of the h-function (E.12) and the Bethe state (E.21).
The result for the contribution of h 1 is and the result for the contribution of from Φ 1 is with the asymptotic behavior In particular for n =n = 2 and k =k = 1 we prove the proposition: The form factor of the current for n = 2,n = 1 and k =k = 1 satisfies the clustering formula (3.27) in the form Proof. The exponential behavior (4.21) implies for n =n = 2 and k =k = 1 that k j = 1 and l j = 0 for j = 1, . . . , N − 1. We investigate for ψ = ψ 1 We have proved in theorem 5 that in leading order Order 1 W : we have to apply the asymptotic behavior of the h-function (E.12) and the Bethe state (E.21).
The result for the contribution of h 1 is and the result for the contribution of from Φ 1 is δκ + C 11 δ 1 δ δ1 κ = C δκ (see (B.30)) the relation (B.16) is proved.
in particular Proof. We use where C a encircles the poles of Γ(a − z) clockwise.
and iterating this result ⇒ 1.
3. and 4. imply that c Ô OǑ (k, l, W ) depends on l as where (l − l 0 ) = 0 mod N and c Ô OǑ (k 0 , l 0 , W ) is obtained by a simple example.

D Form factor equations
The co-vector valued function F O 1...n (θ) is meromorphic in all variables θ 1 , . . . , θ n and satisfies the following relations [6,7]: , where the statistics factor σ O α is determined by the space-like commutation rule of the operator O and the field which creates the particle α. The charge conjugation matrix C1 1 is given by (3.5).
(iii) There are poles determined by one-particle states in each sub-channel given by a subset of particles of the state. In particular the function F O α (θ) has a pole at θ 12 = iπ such that where the bound state intertwiner Γ (12) 12 and the values of θ 1 , θ 2 , θ (12) are given in general in [34][35][36].
(v) Naturally, since we are dealing with relativistic quantum field theories we finally have There exist bound states of r fundamental particles (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 ).

E Asymptotic behavior for W → ∞
We use the short notations of Section 4.