  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].
