Hard thermal contributions to phase transition observables at NNLO

TL;DR

This paper develops a high-order effective field theory for gauge-Higgs models at high temperature, quantifies the impact of higher-order corrections on phase transition observables, and derives three-loop masses and couplings relevant for gravitational-wave signals.

Contribution

It provides the first three-loop scalar and Debye mass calculations for gauge-Higgs models and analyzes the significance of higher-dimensional operators and loop corrections on thermodynamic parameters.

Findings

One-loop dimension-six effects often dominate over higher-loop corrections.

Derived three-loop scalar and Debye masses for U(1) and SU(N) gauge-Higgs models.

Demonstrated gauge independence of physical parameters and identified missing contributions in master integrals.

Abstract

To construct the high-temperature effective field theory of gauge-Higgs models up to $\mathcal{O}(g^6)$ in the gauge coupling, we integrate out hard modes to three-loop level and use the next-to-next-to-leading order effective potential. For the Abelian Higgs model, we quantify the impact of both higher-dimensional operators and higher-loop corrections on thermodynamic parameters relevant for gravitational-wave observables, finding that one-loop dimension-six effects typically dominate over two- and three-loop corrections to super-renormalizable parameters for the strongest transitions. We derive the three-loop scalar and Debye masses for the ${\rm U(1)}$ and ${\rm SU}(N)$ gauge-Higgs models, as well as the two-loop quartic couplings for the Abelian case, show gauge independence of physical parameters, and demonstrate that no new master integrals are required for the matching, while consistency of 4d and 3d renormalizability points to previously missing contributions in these master integrals.