Gauge Invariant Effective Potential

TL;DR

This paper addresses the gauge dependence problem of the effective potential by analyzing the multiplicative anomaly and proposing the Heat-Kernel method to compute gauge-independent effective actions, especially in four-dimensional non-compact manifolds.

Contribution

It demonstrates that the multiplicative anomaly cancels gauge-dependent terms and advocates the Heat-Kernel expansion as a straightforward approach for gauge-invariant effective potential calculation.

Findings

Gauge dependence arises from invalid determinant factorization.

Multiplicative anomaly cancels gauge-dependent terms in certain manifolds.

Heat-Kernel method naturally yields gauge-invariant effective actions.

Abstract

We show that the long-standing problem of gauge dependence of the effective potential arises due to the factorisation of the determinant of operators, which is invalid when we take the zeta-regularised trace of the operators. We show by correcting for this assumption by computing the multiplicative anomaly, the gauge-dependent terms of the effective potential cancel. We also show that in two- and odd-dimensional non-compact spacetime manifolds where the multiplicative anomaly term is zero, the standard calculation of one-loop effective potential gives a gauge-independent result. These results are in support of our claim that the multiplicative anomaly may play a crucial role in removing the gauge dependence in the effective potential in the four-dimensional non-compact manifold. Noting the non-trivial aspects of this anomaly computation for a generic scenario, we propose the Heat-Kernel method to compute the effective potential where this anomaly emerges as a total derivative, thus redundant. We explicitly show how one can calculate the gauge independent, effective action and the Coleman-Weinberg effective potential by employing the Heat-Kernel method. Based on this result, we advocate the Heat-Kernel expansion as the most straightforward method, as it naturally deals with the matrix elliptic operator for the calculation of manifestly gauge independent, effective actions compared to other conventional methods.