Phase transition in scalar $\phi^4$-theory beyond the super daisy resummations

TL;DR

This paper investigates the nature of the temperature-induced phase transition in scalar $\

Contribution

It introduces a beyond super daisy resummation approach using the second Legendre transform to analyze the phase transition.

Findings

The phase transition remains first order beyond super daisy resummations.

No imaginary parts in free energy indicate physical stability.

Loss of smallness in the expansion parameter suggests limitations of the resummation.

Abstract

The temperature phase transition in scalar $\phi^4(x)$ field theory with spontaneous symmetry breaking is investigated in a partly resummed perturbative approach. The second Legendre transform is used and the resulting gap equation is considered in the extrema of the free energy functional. It is found that the phase transition is of first order in the super daisy as well as in a certain beyond super daisy resummations. No unwanted imaginary parts in the free energy are found but a loss of the smallness of the effective expansion parameter near the phase transition temperature is found in both cases. This means an insufficiency of the resummations or a deficit of the perturbative approach.