Consistency of the Regularization of Gauge Theories by High Covariant Derivatives

TL;DR

This paper analyzes the consistency of regularizing gauge theories with higher covariant derivatives and Pauli-Villars regulators, identifying issues with previous methods and proposing a modified, consistent approach that preserves gauge invariance and correctly computes renormalization group functions.

Contribution

The paper identifies problems in Slavnov's original covariant gauge regularization method and introduces a minor modification that ensures consistency and correct renormalization group results.

Findings

Original Slavnov method has issues with massless modes in regulators.

Modified regularization method is consistent and gauge invariant.

The new approach correctly reproduces standard beta and gamma functions.

Abstract

We show that regularization of gauge theories by higher covariant derivatives and gauge invariant Pauli-Villars regulators is a consistent method if the Pauli-Villars vector fields are considered in a covariant in the regulating Pauli-Villars fields is pathological and the original Slavnov proposal in covariant Landau gauge is not correct because of the appearance of massless modes in the regulators which do not decouple when the ultraviolet regulator is removed. In such a case the method does not correspond to the regularization of a pure gauge theory but that of a gauge theory in interaction with massless ghost fields. This explains the problems pointed out by Martin and Ruiz in covariant Landau gauge. However, a minor modification of Slavnov method provides a consistent regularization even for such a case. The regularization that we introduce also solves the problem of overlapping divergences in a way similar to geometric regularization and yields the standard values of the $\beta$ and $\gamma$ functions of the renormalization group equations.