Effective Action of Spontaneously Broken Gauge Theories

TL;DR

This paper demonstrates that the gauge invariance of the effective action in Higgs theories is preserved when using the polar basis for Higgs fields, resolving gauge-dependence issues in effective potential calculations.

Contribution

It shows that evaluating the effective action in the polar basis ensures gauge invariance, providing a formal derivation and explicit one-loop calculations for Abelian and non-Abelian Higgs theories.

Findings

Effective action is gauge invariant in the polar basis.

One-loop effective potential is gauge invariant.

Equivalence between covariant gauge in polar basis and unitary gauge.

Abstract

The effective action of a Higgs theory should be gauge-invariant. However, the quantum and/or thermal contributions to the effective potential seem to be gauge-dependent, posing a problem for its physical interpretation. In this paper, we identify the source of the problem and argue that in a Higgs theory, perturbative contributions should be evaluated with the Higgs fields in the polar basis, not in the Cartesian basis. Formally, this observation can be made from the derivation of the Higgs theorem, which we provide. We show explicitly that, properly defined, the effective action for the Abelian Higgs theory is gauge invariant to all orders in perturbation expansion when evaluated in the covariant gauge in the polar basis. In particular, the effective potential is gauge invariant. We also show the equivalence between the calculations in the covariant gauge in the polar basis and the unitary gauge. These points are illustrated explicitly with the one-loop calculations of the effective action. With a field redefinition, we obtain the physical effective potential. The SU(2) non-Abelian case is also discussed.