Feynman Rules for the Rational Part of the Standard Model One-loop Amplitudes in the 't Hooft-Veltman ${\gamma }_5$ Scheme

We study Feynman rules for the rational part $R$ of the Standard Model amplitudes at one-loop level in the 't Hooft-Veltman ${\gamma }_5$ scheme. Comparing our results for quantum chromodynamics and electroweak 1-loop amplitudes with that obtained based on the Kreimer-Korner-Schilcher (KKS) ${\gamma }_5$ scheme, we find the latter result can be recovered when our ${\gamma }_5$ scheme becomes identical (by setting $g5s=1$ in our expressions) with the KKS scheme. As an independent check, we also calculate Feynman rules obtained in the KKS scheme, finding our results in complete agreement with formulae presented in the literature. Our results, which are studied in two different ${\gamma }_5$ schemes, may be useful for clarifying the ${\gamma }_5$ problem in dimensional regularization. They are helpful to eliminate or find ambiguities arising from different dimensional regularization schemes.