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Phys. J. C"],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We examine the subtleties of regularization schemes in four-dimensional space (4<jats:italic>S<\/jats:italic>), related in particular to the introduction of the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\gamma _5$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u03b3<\/mml:mi>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> matrix. To illustrate we use a \u201cBumblebee\u201d 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 <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\gamma _5$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u03b3<\/mml:mi>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> matrix in ultraviolet divergent integrals by employing an enhanced Implicit Regularization (IREG) method. This method extends IREG to a quasi-four-dimensional space, <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$Q4S = 4S \\oplus X$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>Q<\/mml:mi>\n                    <mml:mn>4<\/mml:mn>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mo>\u2295<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, drawing parallels with the consistent approach of Dimensional Reduction (DRED). Comparative analysis is conducted against results from the \u2019t Hooft\u2013Veltman regularization scheme, conventional IREG in strict 4<jats:italic>S<\/jats:italic>, and sharp momentum cutoff techniques. Our results illustrate a scheme to compute <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\gamma _5$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u03b3<\/mml:mi>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> interactions in physical dimension of divergent amplitudes, confirming the approach in Bruque et al. (JHEP 08:109, 2018).<\/jats:p>","DOI":"10.1140\/epjc\/s10052-025-13756-3","type":"journal-article","created":{"date-parts":[[2025,1,21]],"date-time":"2025-01-21T13:47:26Z","timestamp":1737467246000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Addressing $$\\gamma _5$$ in nondimensional regularizations: a case study on the Bumblebee model"],"prefix":"10.1140","volume":"85","author":[{"given":"Ricardo J. 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