Addressing $γ_5$ in Nondimensional Regularizations: A Case Study on the Bumblebee Model
We examine the subtleties of regularization schemes in four-dimensional space ($4S$), related in particular to the introduction of the $γ_5$ 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 $γ_5$ 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 $γ_5$ interactions in physical dimension of divergent amplitudes, confirming the approach in [1].