Mixed EW-QCD two-loop amplitudes for $q\bar{q}\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and ${\gamma }_5$ scheme independence of multi-loop corrections

We perform a dedicated study of the $q\bar{q}$-initiated two-loop electroweak-QCD Drell-Yan scattering amplitude in dimensional regularization schemes for vanishing light quark and lepton masses. For the relative order $\alpha$ and ${\alpha }_s$ one-loop Standard Model corrections, details of our comparison to the original literature are given. The infrared pole terms of the mixed two-loop amplitude are governed by a known generalization of the dipole formula and we show explicitly that exactly the same two-loop polarized hard scattering functions are obtained in both the standard 't Hooft-Veltman-Breitenlohner-Maison ${\gamma }_5$ scheme and Kreimer’s anticommuting ${\gamma }_5$ scheme.