On vacuum energies and renormalizability in integrable quantum field theories

TL;DR

This paper calculates vacuum energies in perturbed conformal field theories using the thermodynamic Bethe ansatz, revealing how their sign and divergence depend on the models' scaling dimensions and indicating the necessity of counterterms.

Contribution

It provides a detailed analysis of vacuum energies across various models, highlighting the impact of scaling dimensions on divergences and the need for counterterms in certain regimes.

Findings

Vacuum energies are positive or negative depending on the conformal dimension being less or greater than 1/2.

At transition points, vacuum energies diverge, partly due to free fermions.

Different models exhibit distinct ultraviolet and infrared divergence behaviors.

Abstract

We compute for various perturbed conformal field theories the vacuum energies by means of the thermodynamic Bethe ansatz. Depending on the infrared and ultraviolet divergencies of the models, governed by the scaling dimensions of the underlying perturbed conformal field theory in the ultraviolet, the vacuum energies exhibit different types of characteristics. In particular, for the homogeneous sine-Gordon models we observe that once the conformal dimension of the perturbing scalar field is smaller or greater than 1/2, the vacuum energies are positive or negative, respectively. This behaviour indicates the need for additional ultraviolet counterterms in the latter case. At the transition points we obtain an infinite vacuum energy, which is partly explainable with the presence of several free Fermions in the models studied.