Massless N=1 Super Sinh-Gordon: Form Factors approach

The N=1 Super Sinh-Gordon model with spontaneously broken supersymmetry is considered. Explicit expressions for form-factors of the trace of the stress energy tensor Theta, the energy operator epsilon, as well as the order and disorder operators sigma and mu are proposed.


Introduction
The SShG model can be considered as a perturbed super Liouville field theory, which lagrangian density is given by with the background charge Q = b + 1/b. This model is a CFT with central charge The super Sinh-Gordon model is 1+1 dimensional integrable quantum field theory with N = 1 supersymmetry. We consider the Lagrangian (ψ∂ψ + ψ∂ψ) + 2iµb 2 ψψ sinh bφ + 2πµ 2 b 2 cosh 2 bφ.
In this model the supersymmetry is spontaneously broken [1]: the bosonic field becomes massive, but the Majorana fermion stays massless and plays the role of Goldstino. In the IR limit, the effective theory for the Goldstino is to the lowest order the Volkov-Akulov lagrangian [2] L IR = (ψ∂ψ + ψ∂ψ) − 4 M 2 (ψ∂ψ)(ψ∂ψ) + · · · (1) where supersymmetry is realized non linearly. The irrelevant operator along which the Super Liouville theory flows into Ising is the product of stress-energy tensor TT = (ψ∂ψ)(ψ∂ψ), which is the lowest dimension non derivative operator allowed by the symmetries. The dots include higher dimensional irrelevant operators. The scattering in the left-left and right-right subchannels is trivial, but not in the right-left channel. The following scattering matrices were proposed in [1] For the right and left movers, the energy momentum is parametrized in terms of the rapidity variables θ and θ ′ by p 0 = p 1 = M 2 e θ (and p 0 = −p 1 = M 2 e −θ ′ ). The mass scale of the theory M −2 is equal to 2 sin πν. The form factors 2 F r,l (θ 1 , θ 2 , . . . , θ r ; θ ′ 1 , θ ′ 2 , . . . , θ ′ l ) are defined to be matrix elements of an operator between the vacuum and a set of asymptotics states. The form factor bootstrap approach [3,4,5] (developed originally for massive theories, but that turned out to be also an effective tool for massless theories [6,7]) leads to a system of linear functional relations for the matrix elements F r,l ; let us introduce the minimal form factors which have neither poles nor zeros in the strip 0 < ℑmθ < π and which are solutions of the equations Then the general form factor is parametrized as follows: and the function Q r,l depends on the operator considered. The RR and LL scattering formally behave as in the massive case, so annihilation poles occur in the RR and LL subchannel. This leads to the residue formula and a similar expression in the LL subchannel. It is important to note that these equations do not refer to any specific operator.

Expression for form factors
The minimal form factors read explicitly: The latter form factor has asymptotic behaviour when θ → −∞ where . The logarithmic contributions come from resonances. The residue condition (2) written in terms of the function Q r,l reads Let us introduce now the functions as well as We assign odd Z 2 -parity to both right and left-movers (ψ R → −ψ R , ψ L → −ψ L , φ → φ) and even (odd) parity to right (left) movers under duality transformations (ψ R → ψ R , ψ L → −ψ L , φ → −φ).

Neveu-Schwarz sector: trace of the stress-energy tensor
The operator Θ has non zero matrix elements on (even,even) number of particles. The first form factor is determined by using the Lagrangian: Q 2,2 = −4πM 2 . We introduce the sets S = (1, . . . , 2r) and S ′ = (1, . . . , 2l), and propose where T,T are respectively subsets of S andS, the notation '#' stands for 'number of elements', and by definitionT = S\T ,T ′ = S ′ \T ′ . Let us show that this representation does indeed satisfy the residue condition (4): only two cases will contribute to this computation, namely when 1 ∈ T, 2 ∈T and 2 ∈ T, 1 ∈T . It amounts to evaluate the residue at θ 12 = iπ of the quantity: The last line is nothing but Q 2r−2,2l (θ 3 , θ 4 , . . . θ 2r ; θ ′ 1 , θ ′ 2 , . . . , θ ′ 2l ); the evaluation of the residue at θ 12 = iπ of the term into brackets gives explicitly and equation (4) is satisfied. As a remark, we would like to note that the leading infrared behaviour of F 2,2 is given by TT , which defines the direction of the flow in the IR region. To determine the subleading IR terms that appear in the expansion (1), one uses the asymptotic development for f RL given by equation (3). For example (up to the logarithmic terms): The terms into brackets give where L −1 = e θ 1 + e θ 2 and L −2 = e 2θ 1 + e 2θ 2 . So the next irrelevant operator appearing in (1) is T 2T 2 (up to derivatives).

Disorder operator µ.
It has non vanishing matrix elements when the sum of left and right movers is even. As it is explained in [9], there is an additional minus sign in front of the product of S matrices in the residue condition (2).
• The expressions for the form factors of σ and µ give the expected leading IR behaviour [4,9,10]: where r + l is odd for σ and even for µ.

Concluding remarks
We understand it is important to check the UV properties of the form factors proposed in this letter; we hope to present numerical checks in a future publication. As far as the operators Θ, σ, µ are concerned, we expect our representation to be the correct answer to the problem: indeed, the form factors of the operators σ and µ have the expected leading IR behaviour; moreover we recover immediately the form factors of the operators Θ, σ, µ in the Tricritical Ising model perturbed by the subenergy that defines a massless flow to the Ising model [12,13], simply by replacing in our formulae S RL and f RL by their corresponding values that can be found in [14,6]. We checked for a low number of particles that they correctly reproduce the results of [6] where the first form factors of the operators Θ, σ, µ are computed in terms of symmetric polynomials 3 . We also obtained agreement (again for a low number of particles) with [7], where an expression quite similar to ours for the form factors of the operator Θ is proposed (with an arbitrary number of intermediate particles). The case of the energy operator ǫ could be slightly more tricky: although it is evoked in [6], only its lowest form factor with one left mover and one right mover is explicitly given there. Finally, the representations we provide for the functions Q r,l are in principle general enough 4 to provide results for other massless models flowing to the Ising model, but where the S-matrix has a more complicated structure of resonance poles [14].