A Procedure g5anchor to Anchor $γ_5$ in Feynman Diagrams for the Standard Model

Abstract

We present a procedure g5anchor to anchor $\gamma_5$ in the definition of a Dirac trace with $\gamma_5$ in Dimensional Regularization (DR) in Feynman diagrams for the Standard Model, based on a recent revision of the works by Kreimer, Gottlieb and Donohue. For each closed fermion chain with an odd number of primitive (i.e.~not-yet-clearly-defined) $\gamma_5$ in a given Feynman diagram, g5anchor returns a definite set of anchor points for $\gamma_5$, in terms of pairs of ordered fermion propagators; at each of these $\gamma_5$ anchor points a fixed expression in terms of the Levi-Civita tensor and elementary Dirac matrices will be inserted together with a sign determined by anticommutatively shifting all $\gamma_5$ from their original places (dictated by the Feynman rules) to this anchor point. The defining expressions for the cyclic $\gamma_5$-odd Dirac traces in DR associated with closed fermion chains in amplitudes, or more generally squared amplitudes, thus follow from this procedure, where the Levi-Civita tensors are not necessarily treated strictly in 4-dimensions. We propose utilizing this definition in practical perturbative calculations in the Standard Model at least to three-loop orders with the current implementation, and maybe to higher loop orders in absence of Yukawa couplings to Higgs fields. Certain limitations and modifications of the KKS and/or the Kreimer scheme are addressed.