On O(1) contributions to the free energy in Bethe Ansatz systems: the exact g-function

TL;DR

This paper demonstrates that the Thermodynamic Bethe Ansatz can accurately compute the O(1) contributions to the free energy in Bethe Ansatz models, linking these to the exact g-function, and extends previous results to massless theories.

Contribution

It provides an all-orders proof of the O(1) free energy contributions related to the g-function and introduces a new formula for massless theories with diagonal scattering.

Findings

Confirmed the TBA's capability to compute O(1) free energy terms.

Established the relation between O(1) contributions and the g-function.

Derived a new formula for the g-function in massless models.

Abstract

We investigate the sub-leading contributions to the free energy of Bethe Ansatz solvable (continuum) models with different boundary conditions. We show that the Thermodynamic Bethe Ansatz approach is capable of providing the O(1) pieces if both the density of states in rapidity space and the quadratic fluctuations around the saddle point solution to the TBA are properly taken into account. In relativistic boundary QFT the O(1) contributions are directly related to the exact g-function. In this paper we provide an all-orders proof of the previous results of P. Dorey et al. on the g-function in both massive and massless models. In addition, we derive a new result for the g-function which applies to massless theories with arbitrary diagonal scattering in the bulk.