Bethe Ansatz and exact form factors of the O(N) Gross Neveu-model

We apply previous results on the O(N) Bethe Ansatz [1 to 3] to construct a general form factor formula for the O(N) Gross-Neveu model. We examine this formula for several operators, such as the energy momentum, the spin-field and the current. We also compare these results with the 1/N expansion of this model and obtain full agreement. We discuss bound state form factors, in particular for the three particle form factor of the field. In addition for the two particle case we prove a recursion relation for the K-functions of the higher level Bethe Ansatz.


Introduction
The O(N ) σ-and Gross-Neveu (GN) models are integrable and asymptotically free quantum field theories in 1+1 dimension. The S-matrices of these two models correspond to two solutions of the Yang-Baxter equation [4,5]. In previous articles we constructed the O(N ) nested off-shell Bethe Ansatz [1,2] and applied this technique to construct the exact form factors for the O(N ) σ model [3]. Here we extend this work and construct the form factors for the O(N ) Gross-Neveu model for arbitrary number of fundamental particles (for the two-particle case see [6]). The model exhibits a very rich bound state structure and kinks (see e.g. [7]), turning this study even more challenging.
Before we recall the S-matrix and all other details of this model we should mention that the integrable structure present in 1+1 dimension is now becoming relevant and actual in higher dimensional gauge theories under specific circumstances. Remarkably, in the articles [8][9][10][11] (see also references therein) a non-perturbative formulation of planar scattering in the N = 4 Supersymmetric Yang-Mills theory (SYM) with the so called polygonal Wilson loops was proposed and a new decomposition of the Wilson loops in terms of the fundamental building blocks-Pentagon transitions was introduced. These transitions are directly related to the dynamics of the Gubser-Klebanov-Polyakov flux-tube [12], which can be computed exactly by exploring the integrability. In addition, three axioms about the transitions that single particles must satisfy were postulated and, interestingly, it is possible to verify that these axioms correspond to some deformations of the form factor equations in 1 + 1-dimensional integrable quantum field theories. Such exact and constructive developments in the N = 4 SYM theory opens, indeed, large perspectives in the view of using the exact integrability and the full machinery of the form factor program to get physical insights, specially in the case of non-trivial symmetry groups, such as SU (N ) and O(N ).
In this article we consider the O(N )-Gross-Neveu model for N = even. We do not use any Lagrangian to construct the model, nevertheless, we give the following motivation. (1.1) It is known from semi-classical calculations [14] that there are bound states of two fundamental fermions in the scalar and the anti-symmetric tensor channel. Furthermore there are kinks such that the fundamental fermions are kink-kink bound states. The bootstrap program does not use the Lagrangian, but we are looking for an factorizing S-matrix of an O(N )-isovector N -plett of self conjugate fundamental fermions. However, now we assume bound states in the scalar and anti-symmetric tensor channel of two of them.
In this article we use the techniques of [1,3] to construct the form factors of the O(N )-Gross-Neveu model. We apply the general results to compute exact form factors for the energy-momentum, the spin-field and the current. The exact results are compared with the ones obtained in perturbation theory using the 1/N expansion. The final aim of the form factor program is to obtain explicit results for the correlation functions or Wightman functions in the framework of 2-dimensional integrable QFTs. In [6,15] the concept of generalized form factors was introduced and developed further by Smirnov [16]. We call the matrix elements of fields with many particle states: "generalized form factors". Matrix difference equations (the generalized Watson's equations) are solved by using the "off-shell Bethe Ansatz" [1,[17][18][19], which was introduced in [20] to solve the Knizhnik-Zamolodchikov equations. Other approaches to form factors in integrable quantum field theories can be found in [21][22][23][24][25][26][27][28][29]. For articles considering the form factor program for Bethe Ansatz solvable models with nesting see also [30][31][32][33].
The general form factor formula in terms of an integral representation is the main result of this paper. It solves the form factors equations. 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 α (θ). Here we restrict α to the fundamental particles of the model, which form an isovector N -plett of O(N ). Following [6] we write where F (θ) is the minimal form factor function. For the K-function we propose the same Ansatz as for the σ-model in [3] in terms of a nested 'off-shell' Bethe Ansatz Hereh(θ, z) is a scalar function which depends only on the S-matrix. The scalar p-function p O (θ, z) which is in general a simple function of e θ i and e z j depends on the specific operator O(x). This Ansatz transforms the complicated form factor matrix equations (see (3.1)-(3.3) below) to simple equations for the scalar function p O (θ, z) (see also [19]). The integration contour C θ will be specified in section 4. The stateΨ α in (1.4) is a linear combination of the basic Bethe Ansatz co-vectors (see [3] and (4.1)) The nested off-shell Bethe Ansatz is obtained by making for Lβ(z) an Ansatz like (1.4) and iterating this procedure. In the present paper we mainly consider the case where α correspond to the fundamental fermions of the O(N )-Gross-Neveu model Lagrangian (1.1). In forthcoming publications we will consider the kinks [34] and we will discuss, in particular, the O(6)-Gross-Neveu model in more detail [35]. The 'off-shell' Bethe Ansatz states are highest weight states if they satisfy certain matrix difference equations (see for instance [1]). For n particle states the O(N ) weights are (w 1 , . . . , w N/2 ) = n − n 1 , . . . , n N/2−2 − n − − n + , n − − n + where n 1 = m, n 2 , . . . are the numbers of integrations in (1.4) and the higher levels of the nesting. In particular n ± are the numbers of positive/negative chirality spinors. The various levels of the nested Bethe Ansatz correspond to the nodes of the Dynkin diagram of the corresponding Lie algebra (see for instance [36][37][38] and references therein). Here we have D N/2 for N = even (see Fig. 2). In [3] we used for the O(N ) σ-model the group isomorphy O(4) ≃ SU (2) ⊗ SU (2) to start the nesting procedure with form factors of the SU (2) chiral Gross-Neveu model [39]. For the O(N ) Gross-Neveu model it is also possible to use the group isomorphy O(6) ≃ SU (4) to start the nesting with form factors of the SU (4) chiral Gross-Neveu model [39]. This will be performed in detail in a separate paper [35]. For the on-shell Bethe Ansatz for N even see also [40]. Section 2 provides some known results and the notation for the O(N ) Gross-Neveu S-matrix, the bound states, etc. In Section 3 we recall the general form factor equations and obtain the minimal form factor function. In Section 4 we present the general exact form factors formula for the O(N )-Gross-Neveu model and discuss the higher levels of the nested off-shell Bethe Ansatz. In section 5 the general results are applied to some examples. The more complicated proofs and calculations are delegated to the appendices. We consider the fundamental particles of the Lagrangian (1.1) which are fermions and transform as the vector representation of O(N ). The structure of the S-matrix is the same as that of the nonlinear σ-model [3] however, here we are looking for a solution of the O(N )-Yang-Baxter equations with a bound state pole in the physical strip 0 < Im θ < π. Therefore, here "minimality" implies that the S-matrix for the scattering of two fundamental particles is of the form This S-matrix was given by Zamolodchikov-Zamolodchikov [5]. The first factor in (2.1) is the sine-Gordon breather-breather [41] amplitude and S min is the minimal O(N ) S-matrix which is the one of the nonlinear σ-model (see e.g. [3]). The position of the pole is dictated by the condition [42] that the pole has to be cancelled by a zero in the amplitude S min + . This condition 2 fixes the pole and therefore the bound state mass spectrum m k = 2m sin 1 2 kπν (k = 1, 2, . . . , N/2 − 2) .

(2.2)
2 An additional pole in S GN + would contradict positivity in the Hilbert space (for details see [42]).
For each "principal" quantum number k there exist particles b (r) k which are anti-symmetric tensors of rank r = k, k − 2, · · · ≥ 0, i.e. they transform according to the r-th fundamental representation of O(N ). These particles are bosons/fermions for k even/odd. In addition Note the intimate connection between the spectrum of the GN-model, figure 1, and the Dynkin diagram figure 2. There exist exclusively such one-particle states which transform For the Bethe Ansatz it is convenient as in [3] to use instead of the real basis |α r , (α = 1, 2, . . . , N ) the complex basis |1 , |2 , . . . , |2 , |1 , α = 1, 2, . . . , N/2 .
Then the S-matrix writes in terms of the components as with the rapidity difference θ of the particles and the "charge conjugation matrices" The Yang-Baxter-, crossing-and unitarity-relation write as in [3]. The highest weight . For later convenience we introducẽ We will also needS(z) the S-matrix for O(N − 2) where ν is replaced byν = 2/(N − 4).
(ii) Crossing equation with the charge conjugation matrix C1 1 and the statistics factor σ O 1 of the operator O with respect to the particle 1.
The statistics factors in (ii) and (iii) are not arbitrary, but consistency and crossing implies that both are the same and that the for anti-particle σ O 1 σ Ō 1 = 1 holds (see also [19]). In [18,43] was shown that the form factor equations follow from general LSZ assumptions and "maximal analyticity".

Minimal form factors: 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 [6] F min is essential. We take the solution 3

The fundamental Theorem
Following [6] we write the general form factor F O 1...n (θ) for n-fundamental particles as (1.3) where F (θ) is the minimal form factor function (3.11). The K-function K O 1...n (θ) is determined by the form factor equations (i) -(v). We propose the K-function in terms of a nested 'off-shell' Bethe Ansatz (1.4) as a multiple contour integral.
The basic Bethe Ansatz co-vectors in (1.5) are defined as (for more details see [1,3]) The Bethe Ansatz co-vectors (4.1) are generalizations of vectors introduced by Tarasov [45] for the Korepin-Izergin model. Below we will use the following relations for special components of Π (for more details see [1][2][3]) The scalar functionh(θ, z) in (1.4) depends only on the S-matrix and not on the specific (4.4) The functionsφ and τ satisfy the shift equations which are related to the form factor equation The form factor equation (iii) or (3.3) (as will be discussed in appendix A) requires that The function (4.8) satisfies this relation. Notice that the equations (4.8) and (4.9) also determine the normalization constant c in (3.11) and (3.12). Similar as in [3] the integration contours C
The proof of this theorem can be found in appendix A.

Higher level off-shell Bethe Ansatz
For this discussion it is convenient to introduce the variables u, v defined by and define in addition we have here the bound state relation (iv) (k) Res 12...n (u 1 , u 2 , . . . , u n ) = F (k) (12)...n (u (12) , . . . , u n ) √ 2Γ 12 . (4.16) The form factor equations (i) -(iv) of (3.1)-(3.4) for O(N − 2k) are similar to these higher level equations. There are, however, two differences: There is only one term on the right hand side in (iii) (k) .
We assume that the p-function p (k) (u, v) satisfies the equations α (u) of (4.12) satisfy the equations (i) (k) , (ii) (k) and (iii) (k) , if the corresponding relations are satisfied for K determine the numbers m k = n k+1 for a given number of particles n = n 0 The proof of this lemma can be found in appendix C.1.

Examples
In this section, to illustrate our general results we present some simple examples.

Current
The O(N ) Noether current 7 J αβ µ =ψ α γ µ ψ β transforms as the antisymmetric tensor representation of O(N ). This operator has therefore the weights w J = (w 1 , . . . , w N/2 ) = (1, 1, 0, . . . , 0) (see [1,2]), which implies with (4.18) that where n i are the numbers of integrations in the various levels of the off-shell Bethe Ansatz. The existence of a pseudo-potential J αβ (x) follows from the conservation law ∂ µ J αβ µ = 0 For the form factors of both operators we have the relation Because the Bethe Ansatz yields highest weight states we obtain the matrix elements of the highest weight component of J αβ which means in the complex basis J(x) = J 12 (x). We propose the form factors of the operator J(x) (for n = m + 1 = n 1 + 1 = n 2 + 2 even) In the real basis.
ExpressingΨ α (θ, z) in terms of all higher level Bethe Ansatzes there appears the product of all level p-functions p J (θ, z). For the example of the current it depends on the θ i and the second level z which satisfies (4.10) with charge Q J = 0 weight vector w J = (1, 1, 0, . . . , 0) statistics factor σ J = 1 spin s J = 0, s Jµ = 1 .
Bound state form factor of ψ: We discuss the bound state fusion of 2 fundamental fermions f + f → b 2 , a boson of mass m 2 (see (2.2)). Writing (5.5) as we apply the form factor equation (iv), i.e. (3.4) Res The result may be written as where the functions f ij may be calculated in terms of hypergeometric functions 3 F 2 (for more details see appendix D). For example f 13 is plotted for N = 12 in Fig. 4.

1/N expansion: For N → ∞ we obtain
This result agrees with the one obtained by computing Feynman graphs as was done in [6].

Conclusions:
In this article we have enlarged our O(N ) Bethe Ansatz knowledge of the O(N ) Gross-Neveu model, which exhibits a very rich bound state structure and, consequently, creates a rich form factor hierarchy. We have computed the form factors for the fundamental Fermi field, which transforms as a vector representation of O(N ). Then we have also constructed the form factors for the Noether current and the energy-momentum tensor. In addition for the two particle case we have proved the recursion relation for the higher level Kfunctions. Finally we have checked our results against the usual 1/N expansion and found full agreement. In a forthcoming paper we will investigate the kink form factors, possibly proving a kink field equation. Moreover, we will perform a detailed analysis of the O(6) Gross-Neveu model, a starting point in the nesting procedure.

Appendices
A Proof of the main theorem 1 The identity where the C a encircles the poles of Γ(a − z) anti-clockwise may be used to write the Kfunction K O α (θ) defined by the integral representation (1.4) as a sum of "Jackson-type Integrals" as investigated in [1]. These expressions satisfy symmetry properties and a matrix difference equation which are equivalent to the form factor equations (i) and (ii). We have to prove, that due to the assumptions of theorem 1 in addition the residue relations (iii) and (iv) Res 12)...n (θ (12) , . . . , θ n ) √ 2Γ where (4.9) has been used.
• (iii) the residue of consists of two terms This is because for each z j integration with j even the contours will be "pinched" at two points (see Fig. 3): We prove in appendix C.1 the residue formulas for general level k of the off-shell Bethe Ansatz. In particular for k = 0 the general result implies that the pinching (1) gives (1) Res for a suitable choice of the normalization constants in (A.5). Therefore we have proved (1) Res θ 12 =iπ F 1...n (θ 1 , . . . , θ n ) = 2i C 12 F 3...n (θ 3 , . . . , θ n ) .
The bound state form factor F O (12)...n (θ (12) ,θ) is then obtained from the residue Similar as in the proof of (iii) the residue is obtained from pinching at: Here we will not perform the lengthy calculations and write the complicated result, but in appendix D we will calculate the bound state form factors for the examples of section 5.

B Two-particle current form factor
Derivation of (5.3) and (5.4): Proof. The two-particle K-function of the current is with the p-function (5.2) for n = 2 and m = 1 and the Bethe statẽ Doing the integral we obtain and K J 12 (θ) = −K J 21 (θ).
We use again the variables u = θ/(iπν) and v = z/(iπν), consider the component K J 21 (θ) and calculate the integral Writing the integrals in terms of sums over residues we obtain (see Fig. 3) Using the Gauss formula we get which agrees with (B.2) taking (B.1) into account. Therefore using (3.11) and (3.13) we finally obtain with the normalization constant for the pseudo-potential J αβ (x) the two-particle form factors (5.3) and (5.4) for the current. The normalization is chosen such that the form factor agrees for F − (θ) → F − (iπ) = 1 with the free field expression.

C Higher level K-functions
C.1 Proof of lemma 2 Remark 3 If in (4.15) Res is replaced by (1) Res Lemma 2 also holds for k = 0 as explained in appendix A.
For the discussion of the general k-level Bethe Ansatz it is convenient to use the variables u, v defined by θ = iπν k u, z = iπν k v and ν k = 2/(N − 2k − 2) (for the S-matrix see (4.11)). In the proof we will replace p (k) (u, v) by 1 which will not change the results, if the p (k) satisfy the conditions (4.17).
Proof. As above in the proof of theorem 1 the relations (i) (k) and (ii) (k) follow from the results of [1]. The proof of (iii) (k) is the same as the corresponding one in [3], only the functionsψ(u) andχ(u) have to replaced byφ(u) and τ ij (v) by τ (v). As in [3] one finally obtains It has been used that for This can be shown by means of (4.5) and the formulas The final result is that equation (4.15) holds for a suitable choice of the normalization constants in (4.12).
Because the function J(u) satisfies (C.5) it is proportional to K(u, k) (as was shown in general in [6]) if there are no zeroes and exactly one pole 11 at u = 1 in 0 ≤ Re u ≤ 1/ν. Finally we obtain where the constant is calculated by taking the residue at u = 1 on both sides of (C.10). Finally we turn to (C.3). By (C.8) and (C.1) we have which provides the normalization (C.4).
In particular for k = 0 and k = 1 The function L(u) is that of (5.10) and it is used to calculate the 2-particle form factor on the energy momentum (5.14) and also to calculate the 3-particle form factor of the field 5.9. In particular the 2-particle K-function of the scalar operatorψψ is up to a constant equal to K(u). With the normalization in (5.12) we obtain (5.14) Fψ ψ α 1 α 2 (θ) = C α 1 α 2v (θ 1 )u(θ 2 ) F 0 (θ 12 ) which agrees with the result of [6]. The normalization is chosen such that the form factor agrees for θ → iπ with the free field expression.

D Bound state form factors
We discuss the form factor equation (iv) Res θ 12 =iη F O 12...n (θ 1 , θ 2 , . . . , θ n ) = F O (12)...n (θ (12) , . . . , θ n ) √ 2Γ (12) 12 for the examples of section (5). Of course, one may easily calculate the residues for twoparticle form factors for the pseudo-potential J αβ (x) (5.3) andψψ(x) (5.14) directly, however we will check here whether the general pinching procedure of appendix A will give the same result. In addition we obtain the bound state form factor of the three-particle form factor for the field.
Two-particle form factor ofψψ: By the form factor equation (iv) (3.4) the two-particle bound state form factor forψψ is Res 12 .
In appendix C.2 we calculated the two-particle form factor forψψ(x) in terms of the integral In appendix A we remarked that the residue is obtained from pinching at: z = θ 1 − iπν ≈ θ 2 for C (o) and z j = θ 2 ≈ θ 1 − iπν for C (e) , therefore (see (C.9)) Res u 12 =1 J = Res (s 11 (u 1 , u 2 , 0, l 2 ) + s 12 (u 1 , u 2 , 0, l 2 )) . Res v 2 =v e(u j ,l 2 ) It turns out that s 12 gives no contribution and such that again Res π (1 − ν) which means that the pinching procedure gives the same result as the direct calculation.
3-particle form factor of ψ: We discuss the bound state fusion of 2 fundamental fermions f + f → b 2 . We write (5.5) as and apply the form factor equation (3.4) to 12χ Res The component Kχ 111 of the K-function (similar as forψψ in appendix C.2) can be written in terms of In appendix A we remarked that the residue is obtained from pinching at: . Therefore the bound state form factor is obtained from J χ (u) = Res The integrals may be calculated in terms of hypergeometric functions 3 F 2 . We obtain where f 1i (θ 03 ) and f i2 (θ 03 ) are the results from the integrations For example up to a constant (see Fig. 4) where F b (u) is the minimal highest weight form factor function in the b or explicitly in terms of G (z) Barnes G-function with u = θ/(iπν). It satisfies Watson's equation where a b (θ) is the highest weight scattering amplitude in the b which is equivalent to (5.11).
3-particle form factor of the fundamental fermi field: We now calculate the three particle form factor of the fundamental fermi field in 1/N -expansion in lowest nontrivial order. For convenience we multiply the field with the Dirac operator and define γ out p 3 | χ δD (0) | θ 1 , θ 2 in αβ = F η δD γ αβ (θ 3 ; θ 1 , θ 2 ) . Figure 7. The connected part of the three particle form factor of the fundamental fermi field in 1/N -expansion.