One-loop dimensional reduction of the linear sigma model

TL;DR

This paper performs a one-loop dimensional reduction of the linear sigma model, deriving the effective potential, renormalization constants, and the thermal renormalization group equation, while analyzing vacuum stability at large N.

Contribution

It provides a detailed one-loop dimensional reduction of the linear sigma model, including thermal effects and renormalization group equations, which was not previously fully developed.

Findings

Effective potential derived for the reduced theory

Thermal mass and coupling renormalization constants calculated

Vacuum instability at large N confirmed

Abstract

We perform the dimensional reduction of the linear $\sigma$ model at one-loop level. The effective potential of the reduced theory obtained from the integration over the nonzero Matsubara frequencies is exhibited. Thermal mass and coupling constant renormalization constants are given, as well as the thermal renormalization group equation which controls the dependence of the counterterms on the temperature. We also recover, for the reduced theory, the vacuum unstability of the model for large N.