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

TL;DR

This paper discusses the gauge dependence issues of the effective potential, emphasizing the Vilkovisky-DeWitt formalism as a gauge-invariant alternative, and demonstrates its application in the Abelian-Higgs model at high temperature.

Contribution

It advocates for using the Vilkovisky-DeWitt formalism to achieve gauge-independent effective potentials and extends this approach to high-temperature scenarios in the Abelian-Higgs model.

Findings

Vilkovisky-DeWitt formalism ensures gauge independence of the effective potential.

Additional gauge-fixing conditions are necessary for gauge independence.

High-temperature generalization demonstrated in Abelian-Higgs model.

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.