Stop comparing resummation methods

TL;DR

This paper clarifies that resummation methods are consistent when based on a proper power counting from scale hierarchies, resolving common issues in thermal field theory studies.

Contribution

It demonstrates that establishing a power counting scheme ensures the consistency of resummation methods and addresses multiple longstanding problems in thermal field theory.

Findings

Resummation consistency depends on power counting from scale hierarchies.

Adopting a strict perturbative expansion resolves gauge dependence and IR issues.

Proper resummation can eliminate artifacts like the Goldstone boson catastrophe.

Abstract

I argue that the consistency of any resummation method can be established if the method follows a power counting derived from a hierarchy of scales. I.e., whether it encodes a top-down effective field theory. This resolves much confusion over which resummation method to use once an approximation scheme is settled on. And if no hierarchy of scales exists, you should be wary about resumming. I give evidence from the study of phase transitions in thermal field theory, where adopting a consistent power-counting scheme and performing a strict perturbative expansion dissolves many common problems of such studies: gauge dependence, strong renormalization scale dependence, the Goldstone boson catastrophe, IR divergences, imaginary potentials, mirages (illusory barriers), perturbative breakdown, and linear terms.