Multipoint Green's functions in 1+1 dimensional integrable quantum field theories

We calculate the multipoint Green functions in 1+1 dimensional integrable quantum field theories. We use the crossing formula for general models and calculate the 3 and 4 point functions taking in to account only the lower nontrivial intermediate states contributions. Then we apply the general results to the examples of the scaling $Z_{2}$ Ising model, sinh-Gordon model and $Z_{3}$ scaling Ising model. We demonstrate this calculations explicitly. The results can be applied to physical phenomena as for example to the Raman scattering.


Introduction
A complete set of dynamical correlation functions contains the entire information about a given system.Unfortunately, in practice only few such functions can be measured by available experimental techniques.Usually experiments, such as neutron scattering measurements, probe two point functions.However, there are exceptions and there are several experimental techniques such as resonance Raman and resonance X-ray scattering which measure four-point functions or even something more complicated [1,2,3,4].These higher order correlation functions carry information about the nonlinear dynamics which is especially important and interesting in strongly correlated models.It is also an interesting theoretical problem since such models usually require some special nonperturbative approaches.The latter fact brings us to (1+1)-dimensional models where such approaches are available.
The problem becomes especially interesting for massive quantum field theories where almost nothing is known about multipoint correlation functions.Meanwhile, as will be demonstrated in this paper, it is possible to calculate them by using the results for 1 INTRODUCTION 2 matrix elements or form factors of various operators.In the present paper we will obtain three-and four-point functions for massive integrable models in (1+1)-dimensions.For a low particle intermediate state approximation we apply the general results to three models.We calculate correlation functions of the order parameter fields for the offcritical Z 2 Ising model and the Z 3 Potts model perturbed by the thermal operator and for the fundamental field in the sinh-Gordon model.The models are chosen in a sequence of increasing complexity: the Ising model is equivalent to the model of non-interacting massive Majorana fermions with a trivial S-matrix, the sinh-Gordon model is very similar to the Ising one, but has the simplest possible nontrivial S-matrix (a diagonal one without poles), and the Z 3 model takes the complexity one step further having a diagonal S-matrix with one pole on the physical sheet corresponding to a bound state of the fundamental particles.In this article we will explore the crossing formula [5,6] in order to start calculation of the multipoint Green functions or Wightman functions in 1+1 dimensional integrable quantum field theories.It is well known that the n-particle form factor of the local field φ(x) 0|φ(0)|θ 1 , ...θ n is an analytic function of the variables θ 1 , .., θ n .More general form factors as θ 1 , ..., θ n |ϕ(0)|θ n+1 , ..., θ n+k already are not functions but distributions or generalized functions [5,6,7].In fact the crossing formula is defining the generalized form factors in the language of simple form factors.For example, in the case of the 3-particle form factor we have1 θ 1 |ϕ(0)|θ 2 , θ 3 = 0|ϕ(0)|θ 1 + iπ − i , θ 2 , θ 3 + δ θ1θ2 0|ϕ(0)|θ 3 + δ θ1θ3 0|ϕ(0)|θ 2 S(θ 23 ) The -prescription and the δ-functions makes the left hand side a distribution.In more complicated cases we can define the generalized form factors as explained in [5,7].In this article we will consider 3 and 4 point Green functions.Using this definition we will evaluate the multipoint correlators or Green functions defined as time order products of operators: 0|T ϕ 1 (x 1 )ϕ 2 (x 2 )...ϕ n (x n )|0 We will transform these correlators into sums of products of matrix elements inserting between the fields the identity |θ 1 , ...θ n θ n , ...θ 1 | = 1 and then using the crossing formula we will step by step calculate the Wightman and Green's functions.
The results can be applied to physical phenomena as for example to Raman scattering and nonlinear susceptibility [8].

Green's functions
Below in this Section we will do our calculations in the most general form valid for all integrable models.In the next sections we will apply the results to several concrete examples.We will concentrate on the most difficult case of the four-point function, the calculations of the three-point one are comparatively straightforward.
The Green's functions are time ordered n-point functions, written as a sum over all permutations of the fields ϕ i and variables x i where w πϕ (πx) = 0 | ϕ π1 (x π1 ) . . .ϕ πn (x πn )| 0 is the Wightman function and Θ(πt) = Θ(t π1 − t π2 ) . . .Θ(t π(n−1) − t πn ).The Fourier transform is the Green's function in momentum space where we have used translation invariance and split off the energy momentum δ-function defining Ξ(k).The full Green's function may be decomposed into the connected ones

The Green's functions in low particle approximation
Inserting sets of intermediate states . . .
and n k = the number of particles of type k in the state |p (j) .For explicit calculation it is convenient to take the limit k 1 i → 0, then the δ-functions in (4) simplify to 2πδ p (1) j

S-matrix and form factors
For integrable quantum field theories the n-particle S-matrix factorizes into n(n−1)/2 two-particle ones where the product on the right hand side has to be taken in a specific order (see e.g.[9]).
The numbers θ ij are the rapidity differences θ ij = θ i − θ j , which are related to the momenta of the particles by p = m (cosh θ, sinh θ).To simplify the calculations we will consider only theories with diagonal scattering and only one type of particles.The generalization to more types of particles is straightforward and will be used for the Z 3model.

The scaling Z 2 Ising model
In the scaling limit this model may be described by an interacting Bose field σ z n = Cm 1/8 σ(x), where C is a numerical constant and m = h − J.The excitations are noninteracting Majorana fermions with the 2-particle S-matrix S(θ) = −1.The field (x) is defined by σ x = (m/J) 1/2 (x) ∼ ψψ(x), where ψ is a free Majorana spinor field.The n-particle form factors for the order parameter σ(x) were proposed in [11,12] as

The 4-point function
We investigate the Fourier transform of the Green's function for the order parameter ϕ(x) = σ(x).From ( 13) for and For the function g Z2 (x) see Fig. 3 This result can be applied to Raman scattering [8].

The sinh-Gordon model
The classical field equation2 is The sinh-Gordon S-matrix was derived in [9,13] 3 where µ is related to the coupling constant by The sinh-Gordon minimal form factor is [14,15]

The Z 3 -model
The model we consider is the Z N -symmetric CFT perturbed by the thermal operator for a particular value N = 3.Such model appears as the continuum limit of the lattice model describing the integrable anti-ferromagnetic chain of spins S = N/2 in an applied magnetic field [16] where P N (x) is the polynomial of the N -th degree [17], [18].The continuum limit of this model at H = 0 is the SU N (2) Wess-Zumino-Novikov-Witten (WZNW) model perturbed by the irrelevant operator where Φ adj is the primary field in the adjoint representation and J, J are the holomorphic and antiholomorphic currents of the su N (2) Kac-Moody algebra.Whence the magnetic field is applied along the z-axis the z-components of the currents acquire finite expectation values where χ ∼ 1/J is the uniform magnetic susceptibility, and the irrelevant operator becomes relevant [19] J a Jb Φ ab adj → 1 4 (χH) 2 Φ zz adj .
The conformal embedding SU N (2) = U (1) × Z N establishes the equivalence between the diagonal component of the adjoint primary field and the thermal operator and hence the equivalence between the massive sector of model (21) and model (20).The Z 3 CFT and the exact solution of the massive theory (20) for N = 3 suggest that in the disordered phase there are 2 types of particles 1 and 2 and two corresponding fields (order parameters σ 1 , σ 2 = σ * 1 ) with where the indices correspond to the emission of particle 1 and 2 (the latter is a bound state of two 1-particles and simultaneously the anti-particle of particle 1).
The two-particle S-matrix for the Z 3 -Potts model perturbed by the thermal operator has been proposed by Köberle and Swieca [20].It coincides with the one derived from the Bethe ansatz solution of model ( 21) [16].The scattering matrix of two particles of type 1 is This S-matrix is consistent with the picture that the bound state of two particles of type 1 is the particle 2 which is the anti-particle of 1.
The form factors of the Z N -model (20) have been proposed in [21,22,10].The minimal solution of the Watson's and the crossing equations where G(x) is the Barnes G-function [23] with the defining relation The form factor of the order parameter field σ 1 and two particles of type 2 is and for the 3 particles of type 1, 1 and 2 where is the minimal form factor of the particles 1 and 2.

The 3-point function
We consider the Green's function ) | 0 , this 3point function was also investigated in [24].As in (10) we have the simple contribution and as in (8) we calculate for the intermediate states 0|σ where we have used the crossing relation The δ-function terms do not contribute because F σ1 22 (0, 0) = 0. Inserting the form factor functions we get (up to constant factors) For the intermediate states 0|σ and as in ( 12) As expected, there is a threshold singularity at k 0 = 2m.

The 4-point function
Due to (B.4) there are contributions to the four point Green's function from I 1 , I 2 and I 3 .The one from I 3 belongs to the disconnected part and the one from I 2 is trivial as in ( 19) and (B.7).We restrict here to the contribution from I 1 .We consider the Green's function τ σ1σ2σ1σ2 (x) = 0 | T σ 1 (x 1 )σ 2 (x 2 )σ 1 (x 3 )σ 2 (x 4 ) | 0 and as in (13) we obtain (for details see Appendix C.3) Obviously, if σ 3 = σ 1 and σ 4 = σ 2 there are three functions g Z3 π (x) It turns out that g Z3 III (x) = g Z3 I (x).For plots of the functions g Z3 I (x) and g Z3 II (x) see Figs. 7 and 8.

Conclusion
In this paper we develop a technique to calculate multipoint Wightman or Green functions in integrable quantum field theories in 1+1 dimension.We insert intermediate states between the fields and use the crossing formula to write the Wightman function in terms of form factors in a model independent way.We expect good approximations for low number of particles in the intermediate states.In the present article we demonstrate this technique explicitly for 3-and 4-point functions of simple models with no backward scattering: the scaling Z 2 Ising, the scaling Z 3 Potts and the sinh-Gordon model.The results can be applied to physical phenomena, for example to Raman scattering [8].In a forthcoming article we will generalize the technique to models with backward scattering, as the O(N ) σ-and the O(N ) Gross-Neveu model.

Appendix A. Crossing
The general crossing formula (31) in [7] . Using the form factor equation (iii) and Lorentz invariance (see e.g.[10]) we can rewrite these equations as ( 6) and (7).And further one derives

Appendix B. Green's function and intermediate states in low particle approximation
Let ϕ(x) a scalar charge-less bosonic field with the normalization 0 Simple examples of Wigthman and Green's functions.
w1: The 2-point Wigthman function in 1-intermediate particle approximation is and the Green's function in this approximation is the free Feynman propagator which implies that also w121: The 4-point Wightman function in 1-2-1-intermediate particle approximation is (with θ = θ1 . . .θ3 ) Using equations (6, 7) and the identity which is using (A.2) and (A.3) equal to where we have introduced From I 3 we calculate which together with (B.1) yields the disconnected part of τ (k) = τdisc (k) + τc (k).This means the connected part of the Green's function is given by I 1 (θ) + I 2 (θ).
General case.We start with (1), insert sets of intermediate states and write . . .
with the notation of (4).We perform the y-integrations and obtain for Ξ defined in ( 3) . . .
which is (13) and F (0; θ, −θ)F (θ, −θ; 0) is given by (see (B.4)) As mentioned in the context of (B.5) the connected part of Ξϕ (k) is obtained by The contribution from I 1 will be calculated for the Z(2)-, the Z(3)-scaling Ising and the sinh-Gordon models.The contribution from I 2 leads to 0/0, therefore the limit k 1 i → 0 has to be taken more carefully.

Contribution of
given by (B.4).Taking first the term with δ κ1θ2 we get (because We write then for small κ i (using S(0) = −1) Similarly we get the other contribution from I 2 and calculate where σ a are the Pauli matrices.This model has numerous condensed matter realizations being one of the most popular models of condensed matter theory.It describes a sequence of coupled two level systems.They may represent spins; then the first term describes an anisotropic exchange interaction.In this case σ z directly couples to external magnetic field: µ B B z n σ z n .States of the two level systems may also correspond to positions of electric charges in a double well potential.Then the first term is the dipole-dipole interaction and the transverse field describes the quantum tunnelling between the wells.Then σ a would be the dipole moment operators.Their interaction with the electric field is given by pE z n σ z n with p being the dipole moment.
Since the dominant interaction is ferromagnetic, the strongest fluctuations take place at zero wave vectors which guarantees a direct coupling to the electromagnetic field creating optimal resonance conditions.The Ising model (C.1) has two phases depending on the sign of m = h − J.The resonance occurs in the paramagnetic phase m > 0 when the ground state average of the order parameter σ z = 0.In that case the electromagnetic field has a nonzero matrix element between the ground state and single magnon state.
In the scaling limit model (C.1) can be described by an interacting Bose field σ z n = Cm 1/8 σ(x), where C is a numerical constant and m = h − J.The excitations are noninteracting Majorana fermions with the 2-particle S-matrix S Z(2) (θ) = −1.The field σ x = (m/J) 1/2 (x) ∼ ψψ(x), where ψ is a free Majorana spinor field.The n-particle form factors for the order parameter σ(x) is given by (14).From (x) ∼ ψψ(x) one has for a free Majorana spinor field (up to a constant)

The 3-point function
We calculate Ξϕϕ (k) in the limit k 1 i → 0. For the various permutations in (4) we obtain: a) For the permutation π = 123 and which is (8).Equations ( 14) and (C.2) imply b) For the permutation π = 321 and c) For the permutation π = 132 and There are 3 contributions from It turns out that all 3 give the same result, therefore For the permutation π = 132 and n 1 = 1, n 2 = 3 we find, similarly Finally with (12) we obtain (15).
From (B.4) and ( 14) we obtain which can be calculated: for Re x < −1 The intrinsic coupling g R [25], defined by Ξ(0 and the field equation The model is super-renormalizable, therefore after introducing normal products in (C.3) there are only two finite renormalization constants.The wave function and the mass renormalization constants are given by [14,26] with [14] Z ϕ = (1 − µ) The S-matrix can be obtained by analytic continuation (from β → iβ) of the sine-Gordon S-matrix which was derived in [9, 13] The minimal sinh-Gordon form factor is [14,15] where the meromorphic function 5 The function E(x) was introduced in [14] and also used in [6] and [7].
has been introduced, for more details see Appendix C.2.2.The 3-particle form factor is [14,15] where the normalization follows from the form factor equation (iii) and (C.4) where g SG (x) = g SG 1 (x) + g SG 2 (x) and From (B.4) and (C.5) we obtain where I Z2 ϕ as defined in (17).We have introduced Therefore as in ( 16) and ( 17) we obtain The functions g SG 1 (x) for µ = 0.3 and µ = 0.5 are plotted in Fig. 4 and 5.The contribution from I 2 follows from (B.7) as The form factor of σ 1 (x) for the 3 particles of type 112 is given by (23) [21,10], where 22 (θ)Γ.The minimal form factor of the particles 1 and 2 is the the S-matrix for the particles 1 and 2.

Figure 6 :
Figure 6: Plot of Re h Z3 (x) (black) and Im h Z3 (x) (red) for the Z 3 model