Dimensional regularization vs methods in fixed dimension with and without $\gamma_5$

TL;DR

This paper compares fixed-dimension regularization methods with dimensional regularization, focusing on challenges involving the $oldsymbol{\gamma_5}$ matrix and proposes a consistent procedure akin to dimensional reduction.

Contribution

It analyzes the algebraic complexities of fixed-dimension regularization methods and introduces a consistent procedure to address issues similar to those in dimensional regularization.

Findings

Fixed-dimension methods face similar difficulties as dimensional regularization.

A proposed consistent procedure improves handling of $oldsymbol{\gamma_5}$ in fixed dimensions.

The approach aligns fixed-dimension methods with dimensional reduction techniques.

Abstract

We study the Lorentz and Dirac algebra, including antisymmetric $\epsilon$ 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.