Interpolating Gauges,Parameter Differentiability,WT-identities and the epsilon term

TL;DR

This paper investigates the mathematical and physical implications of using interpolating gauges in gauge field theories, revealing issues with gauge parameter differentiability, WT-identity derivations, and the importance of the epsilon-term.

Contribution

It demonstrates the challenges and pitfalls of interpolating gauges, especially regarding gauge parameter differentiability and WT-identity derivations, and highlights the significance of the epsilon-term.

Findings

Differentiability issues hinder gauge continuation between Feynman and Coulomb gauges.

Large path-integral changes occur near specific gauge parameter values.

The epsilon-term plays a crucial role in WT-identity derivations.

Abstract

Evaluation of variation of a Green's function in a gauge field theory with a gauge parameter theta involves field transformations that are (close to) singular. Recently, we had demonstrated {hep-th/0106264}some unusual results that follow from this fact for an interpolating gauge interpolating between the Feynman and the Coulomb gauge (formulated by Doust). We carry out further studies of this model. We study properties of simple loop integrals involved in an interpolating gauge. We find that the proof of continuation of a Green's function from the Feynman gauge to the Coulomb gauge via such a gauge in a gauge-invariant manner seems obstructed by the lack of differentiability of the path-integral with respect to theta (at least at discrete values for a specific Green's function considered) and/or by additional contributions to the WT-identities. We show this by the consideration of simple loop diagrams for a simple scattering process. The lack of differentiability, alternately, produces a large change in the path-integral for a small enough change in theta near some values. We find several applications of these observations in a gauge field theory. We show that the usual procedure followed in the derivation of the WT-identity that leads to the evaluation of a gauge variation of a Green's function involves steps that are not always valid in the context of such interpolating gauges. We further find new results related to the need for keeping the epsilon-term in the in the derivation of the WT-identity and and a nontrivial contribution to gauge variation from it. We also demonstrate how arguments using Wick rotation cannot rid us of these problems. This work brings out the pitfalls in the use of interpolating gauges in a clearer focus.