Gap equation in scalar field theory at finite temperature

TL;DR

This paper studies the two-loop gap equation in scalar field theory at finite temperature, revealing a non-zero imaginary part and providing solutions for the thermal mass up to order g^4.

Contribution

It demonstrates the complex nature of the thermal mass gap equation at two-loop order and offers solutions for the real and imaginary parts of the thermal mass.

Findings

The gap equation has a non-zero finite imaginary part.

The thermal mass solutions are accurate up to g^4 order.

Real and imaginary parts of the thermal mass are explicitly obtained.

Abstract

We investigate the two-loop gap equation for the thermal mass of hot massless $g^2\phi^4$ theory and find that the gap equation itself has a non-zero finite imaginary part. This indicates that it is not possible to find the real thermal mass as a solution of the gap equation beyond $g^2$ order in perturbation theory. We have solved the gap equation and obtain the real and the imaginary part of the thermal mass which are correct up to $g^4$ order in perturbation theory.