On the gauge dependence of vacuum transitions at finite temperature

TL;DR

The work investigates how gauge fixing affects finite-temperature vacuum transitions in the Abelian Higgs model, using Nielsen identities to separate genuine physical gauge-invariant content from gauge artefacts introduced by perturbative gradient expansions. It computes the finite-temperature effective potential to order $g^3$ and $\lambda$, including daisy resummation and Goldstone/ghost cancellations, and analyzes the gauge dependence of the tunneling action and sphaleron energy within a gradient-expansion framework. The key finding is that, for gauges compatible with perturbation theory (notably around $\xi \sim 1$), the gauge dependence of the tunneling action and sphaleron energy is numerically small in the physically relevant regime $g \ll 1$, $\phi_c/T_c \gtrsim 1$, though the gradient expansion breaks down near the symmetric phase and cannot guarantee complete gauge independence there. The study highlights that fully gauge-invariant results may require moving beyond the gradient expansion to include full momentum dependence, and it provides a tractable pathway to apply these insights to the Standard Model and its extensions.

Abstract

In principle, observables as for example the sphaleron rate or the tunneling rate in a first-order phase transition are gauge-independent. However, in practice a gauge dependence is introduced in explicit perturbative calculations due to the breakdown of the gradient expansion of the effective action in the symmetric phase. We exemplify the situation using the effective potential of the Abelian Higgs model in the general renormalizable gauge. Still, we find that the quantitative dependence on the gauge choice is small for gauges that are consistent with the perturbative expansion.

Figures (14)

• Figure 1: One-loop diagram contributing to the Nielsen coefficient $C_0$.
• Figure 2: The daisy diagrams that are resummed.
• Figure 3: The leading contributions to the self-energy of the gauge bosons.
• Figure 4: The leading contributions to the self-energy of the Goldstone bosons.
• Figure 5: Two-loop contribution to the effective potential. The box denotes the counterterm of the thermal mass.