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 $\alpha$ gauge with $\alpha \ne 0$ and a given auxiliary preregularization is introduced in order to uniquely define the regularization. The limit $\alpha \to 0$ in the regulating Pauli-Villars fields is pathological and the original Slavnov proposal in the 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. However, a minor modification of the 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.

• Received 4 August 1995

DOI:https://doi.org/10.1103/PhysRevD.54.5290