Dimensional regularization vs methods in fixed dimension with and without ${\gamma }_5$

Abstract

We study the Lorentz and Dirac algebra, including antisymmetric $ϵ$ tensors and the ${\gamma }_5$ matrix, in implicit gauge-invariant regularization/renormalization methods defined in fixed integer dimensions. They include constrained differential, implicit and four-dimensional renormalization. We find that these fixed-dimension methods face the same difficulties as the different versions of dimensional regularization. We propose a consistent procedure in these methods, similar to the consistent version of regularization by dimensional reduction.