Functional Equations of Form Factors for Diagonal Scattering Theories

TL;DR

This paper applies the form factor bootstrap method to diagonal scattering theories, deriving functional equations and analyzing singularities for ADE and affine Toda theories, including explicit form factors and operator identification.

Contribution

It determines the functional equations and singularity parameterizations for form factors in ADE and affine Toda theories, extending to multiple particles and identifying operators.

Findings

Derived functional equations for minimal two-particle form factors.

Explicit form factors for $A^{(1)}_{2}$ affine Toda theory up to four particles.

Identified the operator structure and pole contributions in the form factors.

Abstract

Form factor bootstrap approach is applied for diagonal scattering theories. We consider the ADE theories and determine the functional equations satisfied by the minimal two-particle form factors. We also determine the parameterization of the singularities in two particle form factors. For $A^{(1)}_{2}$ Affine Toda field theory which is the simplest non-self conjugate theory, form factors are derived up to four-body and identification of operator is done. Generalizing this identification to the $A^{(1)}_N$ Affine Toda cases, we fix the two particle form factors. We also determine the additional pole structure of form factors which comes from the double pole of the $S$-matrices of the $A^{(1)}_N$ theory. For $A_N$ theories, existence of the conserved ${\bf Z}_{N+1}$ charge leads to the division of the set of form factors into $N+1$ decoupled sectors.