Broken phase scalar effective potential and Phi-derivable approximations

TL;DR

This paper analyzes the temperature-dependent effective potential of a scalar phi^4 theory within the Hartree Phi-derivable approximation, demonstrating the absence of second order phase transitions and discussing renormalization issues.

Contribution

It provides an analytical proof that no second order phase transition occurs in the Hartree approximation and introduces new computational techniques for Phi-derivable approximations.

Findings

No second order phase transition in the approximation

Renormalization issues at finite temperature addressed

New computational techniques introduced

Abstract

We study the effective potential of a real scalar phi^4 theory as a function of the temperature T within the simplest Phi-derivable approximation, namely the Hartree approximation. We apply renormalization at a "high" temperature T* where the theory is required to be in its symmetric phase and study how the effective potential evolves as the temperature is lowered down to T=0. In particular, we prove analytically that no second order phase transition can occur in this particular approximation of the theory, in agreement with earlier studies based on the numerical evaluation or the high temperature expansion of the effective potential. This work is also an opportunity to illustrate certain issues on the renormalization of Phi-derivable approximations at finite temperature and non-vanishing field expectation value, and to introduce new computational techniques which might also prove useful when dealing with higher order approximations.