Convexity, gauge-dependence and tunneling rates

TL;DR

The paper resolves the long-standing issue of gauge dependence in tunneling rates by formulating decay as a false-vacuum problem governed by a non-convex, complex effective action $Γ^T_F$. Nielsen identities guarantee that the value of $Γ^T_F$ at its quantum extrema—and hence the tunneling rate encoded in its imaginary part—is gauge-independent, independent of the chosen gauge fixing, provided the Faddeev–Popov operator is invertible. By contrast, the true vacuum effective action $Γ$ is convex and cannot directly yield tunneling rates; tunneling emerges from a multi-saddle, false-vacuum formalism that naturally incorporates quantum corrections via a generalized bounce, including derivative terms. The approach unifies and extends prior semiclassical results (Callan–Coleman, Garbrecht) and has important implications for evaluating the stability of the Standard Model vacuum and cosmological phase transitions, where gauge-invariant, nonperturbative treatment of tunneling is essential.

Abstract

We clarify issues of convexity, gauge-dependence and radiative corrections in relation to tunneling rates. Despite the gauge dependence of the effective action at zero and finite temperature, it is shown that tunneling and nucleation rates remain independent of the choice of gauge-fixing. Taking as a starting point the functional that defines the transition amplitude from a false vacuum onto itself, it is shown that decay rates are exactly determined by a non-convex, false vacuum effective action evaluated at an extremum. The latter can be viewed as a generalized bounce configuration, and gauge-independence follows from the appropriate Nielsen identities. This holds for any election of gauge-fixing that leads to an invertible Faddeev-Popov matrix.