# Leading exponential finite size corrections for non-diagonal form factors

## TL;DR

This paper derives leading exponential finite volume corrections for non-diagonal form factors in 2D integrable models, expressing them through infinite volume form factors and scattering matrices, with numerical validation.

## Contribution

It provides explicit formulas for exponential finite-size corrections in non-diagonal form factors, including bound state effects, in integrable quantum field theories.

## Key findings

- Formulas relate finite volume corrections to infinite volume data.
- Corrections include μ-terms for bound states and F-terms for virtual particles.
- Numerical checks confirm the theoretical predictions in specific models.

## Abstract

We derive the leading exponential finite volume corrections in two dimensional integrable models for non-diagonal form factors in diagonally scattering theories. These formulas are expressed in terms of the infinite volume form factors and scattering matrices. If the particles are bound states then the leading exponential finite-size corrections ($\mu$-terms) are related to virtual processes in which the particles disintegrate into their constituents. For non-bound state particles the leading exponential finite-size corrections (F-terms) come from virtual particles traveling around the finite world. In these F-terms a specifically regulated infinite volume form factor is integrated for the momenta of the virtual particles. The F-term is also present for bound states and the $\mu$-term can be obtained by taking an appropriate residue of the F-term integral. We check our results numerically in the Lee-Yang and sinh-Gordon models based on newly developed Hamiltonian truncations.