Gauge Dependence of the High-Temperature 2-Loop Effective Potential for the Higgs Field

TL;DR

This work tackles the gauge dependence of the high-temperature, two-loop effective potential for the Higgs by using a dimensionally reduced 3d theory and comparing a conventional gauge-dependent potential with a gauge-invariant generating functional. It shows that the pressure at the minimum is gauge-independent to order $\hbar^2$ and demonstrates this via explicit IR-divergence cancellation and renormalization in the 3d framework. The paper also analyzes convergence in the broken phase through RG improvement, finding good perturbative behavior for broken-phase observables while cautioning that critical-temperature and latent-heat predictions are not reliable at this order. Overall, the results establish a practical method to extract physical thermodynamic quantities from gauge-dependent finite-temperature calculations and clarify the role of gauge invariance in the equation of state.

Abstract

The high-temperature limit of the 2-loop effective potential for the Higgs field is calculated from an effective 3d theory, in a general covariant gauge. It is shown explicitly that a gauge-independent result can be extracted for the equation of state from the gauge-dependent effective potential. The convergence of perturbation theory is estimated in the broken phase, utilizing the gauge dependence of the effective potential.