Comments on the gauge dependence of the effective potential and the utility of the Vilkovisky-DeWitt formalism

TL;DR

The paper tackles the problem that the conventional effective action and its zero-momentum limit, the effective potential, are generally not gauge-invariant or scalar under field reparametrisations. It clarifies the distinct roles of gauge invariance, gauge-fixing independence, and reparametrisation covariance, and shows that at an extremum the effective potential can be gauge-independent if the gauge-fixing function satisfies a vanishing vacuum expectation value in the absence of sources, via Nielsen identities. To address broader ambiguities, it advocates the Vilkovisky-DeWitt formalism, which encodes field-space geometry to render the effective action gauge-invariant and invariant under field reparametrisations, and demonstrates GFP independence within this framework by promoting the gauge-fixing parameter to a BRST-variant field. The Abelian-Higgs model serves as a concrete test bed, yielding a one-loop, gauge- and parametrisation-independent $V_{eff}$, with a high-temperature generalisation, underscoring the method’s robustness for physical predictions. Overall, the work provides a principled route to extract physically meaningful information about spontaneous symmetry breaking from gauge theories and clarifies the conditions under which such quantities are unambiguous.

Abstract

We provide some additional comments on the long-lived discussions surrounding an effective action and potential plagued by a number of ambiguities. We reinforce the importance of an extra condition on the gauge-fixing function, namely the vanishing of its vacuum expectation value in the absence of external sources, when concluding gauge-independence of the effective action and potential at an extremum. We advocate for the alternative construction of the effective action and potential based on the Vilkovisky-DeWitt approach, and demonstrate its independence from the gauge-fixing parameter. We also exhibit a high-temperature generalisation of this alternative construction in the specific case of the Abelian-Higgs model.