Gauge independence of the bubble nucleation rate in theories with radiative symmetry breaking

TL;DR

This work shows that in theories with radiative symmetry breaking, the gauge dependence of the effective action does not spoil the gauge invariance of the bubble nucleation rate at leading nontrivial order. By deriving a hierarchy of identities from the Nielsen identity for the derivative-expansion functions $V^{\rm eff}(\phi)$ and $Z(\phi)$, Metaxas and Weinberg prove that the exponent $B=B_0+B_1$ remains gauge-invariant, despite gauge-dependent intermediate quantities. They formalize the proof and corroborate it with explicit calculations in scalar electrodynamics in $R_\xi$ gauges, including a demonstration that the two-loop contributions to $V^{\rm eff}_{e^6}$ sum to a gauge-independent result. The findings support using gauge-dependent effective actions to extract gauge-invariant physical rates and illuminate the role of derivative terms in maintaining gauge consistency.

Abstract

In field theories where a metastable false vacuum state arises as a result of radiative corrections, the calculation of the rate of false vacuum decay by bubble nucleation depends on the effective potential and the other functions that appear in the derivative expansion of the effective action. Beginning with the Nielsen identity, we derive a series of identities that govern the gauge dependence of these functions. Using these, we show, to leading nontrivial order, that even though these functions are individually gauge-dependent, one obtains a gauge-independent result for the bubble nucleation rate. Our formal arguments are complemented by explicit calculations for scalar electrodynamics in a class of $R_ξ$ gauges.