# On the finite volume expectation values of local operators in the sine-Gordon model

## TL;DR

This paper develops linear integral equations to compute finite volume expectation values of local operators in the sine-Gordon model, providing analytical formulas that test and partially confirm a conjecture for non-diagonal scattering theories.

## Contribution

It introduces a new method of calculating finite volume expectation values in the sine-Gordon model and tests a conjecture related to non-diagonal scattering theories.

## Key findings

- Derived integral equations for expectation values in sine-Gordon
- Confirmed the conjecture up to three-particle states with modifications
- Identified the need for a proof of form-factor finiteness in non-diagonal theories

## Abstract

In this paper we present sets of linear integral equations which make possible to compute the finite volume expectation values of the trace of the stress energy tensor ($\Theta$) and the $U(1)$ current ($J_\mu$) in any eigenstate of the Hamiltonian of the sine-Gordon model. The solution of these equations in the large volume limit allows one to get exact analytical formulas for the expectation values in the Bethe-Yang limit. These analytical formulas are used to test an earlier conjecture for the Bethe-Yang limit of expectation values in non-diagonally scattering theories. The analytical tests have been carried out upto three particle states and gave agreement with the conjectured formula, provided the definition of polarized symmetric diagonal form-factors is modified appropriately. Nevertheless, we point out that our results provide only a partial confirmation of the conjecture and further investigations are necessary to fully determine its validity. The most important missing piece in the confirmation is the mathematical proof of the finiteness of the symmetric diagonal limit of form-factors in a non-diagonally scattering theory.