Interpolating gauges and the importance of a careful treatment of epsilon term

TL;DR

This paper emphasizes the critical role of the epsilon term in interpolating gauges for gauge theories, showing that improper handling leads to gauge dependence and pathologies, and advocates for a careful, parameter-dependent treatment to ensure consistency.

Contribution

It demonstrates that the epsilon term must be varied with the gauge parameter in interpolating gauges and provides a method to correctly fix this term using finite field-dependent BRS transformations.

Findings

Naive epsilon-term use causes pathologies in the path-integral.

Proper epsilon-term variation ensures gauge independence.

Connecting to Lorentz gauges with correct epsilon-term preserves consistency.

Abstract

We consider the use of interpolating gauges (with a gauge function (F[A;alpha ]) in gauge theories to connect the results in a set of different gauges in the path-integral formulation. We point out that the results for physical observables are very sensitive to the epsilon term that we have to add to deal with singularities and thus cannot be left out of a discussion of gauge-independence generally. We further point out, with reasons, that the fact that we can ignore this term in the discussion of gauge independence while varying of the gauge parameter in Lorentz-type covariant gauges is an exception rather than a rule . We show that generally gauge-independence requires that the epsilon-term has to be varied with alpha. We further show that if we make a naive use of the epsilon term -i\int d^{4}x[{1/2}A^{2}-\bar{c}c]) (that is appropriate for the Feynman gauge) for general interpolating gauges with arbitrary parameter values [i.e.alpha], we cannot preserve gauge independence [except when we happen to be in the infinitesimal neighborhood of the Lorentz-type gauges]. We show with an explicit example that for such a naive use of an epsilon-term, we develop serious pathology in the path-integral as alpha is/are varied. We point out that correct way to fix the epsilon-term in a path-integral in a non-Lorentz gauge is by connecting the path-integral to the Lorentz-gauge path-integral with correct epsilon-term as has been done using the finite field-dependent BRS transformations in recent years.