Addressing $\gamma_5$ in Nondimensional Regularizations: A Case Study on the Bumblebee Model

TL;DR

This paper investigates how different regularization schemes, especially those involving the $oldsymbol{ extgamma_5}$ matrix, affect solutions in a Lorentz-violating Bumblebee model, proposing an improved method for consistent calculations.

Contribution

It introduces an extended Implicit Regularization (IREG) method in quasi-four-dimensional space to resolve ambiguities with $oldsymbol{ extgamma_5}$, comparing it with other schemes in a Lorentz-violating context.

Findings

The enhanced IREG method effectively resolves $oldsymbol{ extgamma_5}$ ambiguities.

Results are consistent with previous approaches in physical dimensions.

The scheme provides a reliable way to compute $oldsymbol{ extgamma_5}$ interactions in divergent amplitudes.

Abstract

We examine the subtleties of regularization schemes in four-dimensional space ($4S$), related in particular to the introduction of the $\gamma_5$ matrix. To illustrate we use a "Bumblebee" model featuring dynamically induced Lorentz symmetry violation. The analysis centers on how different regularization methods affect the solutions to the gap equation in this model. We highlight the resolution of ambiguities associated with the $\gamma_5$ matrix in ultraviolet divergent integrals by employing an enhanced Implicit Regularization (IREG) method. This method extends IREG to a quasi-four-dimensional space, $Q4S = 4S \oplus X$, drawing parallels with the consistent approach of Dimensional Reduction (DRED). Comparative analysis is conducted against results from the 't Hooft-Veltman regularization scheme, conventional IREG in strict $4S$, and sharp momentum cutoff techniques. Our results illustrate a scheme to compute $\gamma_5$ interactions in physical dimension of divergent amplitudes, confirming the approach in [1].