Stop comparing resummation methods

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.

11 pages

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• Non-perturbative determination of the sphaleron rate for first-order phase transitions
• Scale dependence improvement of the quartic scalar field thermal effective potential in the optimized perturbation theory
• A Catalog of First-Order Electroweak Phase Transitions in the Standard Model Effective Field Theory
• Hot news on the phase structure of the SMEFT