Interpolating gauges and the importance of a careful treatment of ?-term

Abstract

We consider the use of interpolating gauges (with a gauge function F[A;α]) 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 it 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 preserving gauge-independence as α is varied requires that the e-term has to be varied with α. We further show that if we make a naive use of the (fixed) epsilon term $$-i\in \int d^{4}x[\frac{1}{2}A^2-{\bar c}c]$$ (that is appropriate for the Feynman gauge) for general interpolating gauges with arbitrary parameter values [i.e. α], 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 e-term, we develop serious pathological behavior in the path-integral as α is/are varied. We point out that correct way to fix the e-term in a path-integral in a non-Lorentz gauge is by connecting the path-integral to the Lorentz-gauge path-integral with correct e-term as has been done using the finite field-dependent BRS transformations in recent years.