The ultraviolet Behaviour of Integrable Quantum Field Theories, Affine Toda Field Theory

TL;DR

This paper analyzes the ultraviolet behavior of affine Toda field theories using thermodynamic Bethe ansatz equations, providing analytical approximations, numerical comparisons, and exploring solution properties for simply laced Lie algebras.

Contribution

It offers the first analytical approximations for TBA solutions in affine Toda theories and studies their existence, uniqueness, and convergence properties.

Findings

Derived approximate analytical solutions for TBA equations

Compared analytical and numerical solutions for explicit models

Established existence, uniqueness, and convergence rates of solutions

Abstract

We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda field theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for theories in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against the numerical solutions. We address the question of existence and uniqueness of the solutions of the TBA-equations, derive precise error estimates and determine the rate of convergence for the applied numerical procedure. A general expression for the Fourier transformed kernels of the TBA-equations allows to derive the related Y-systems and a reformulation of the equations into a universal form.