On γ5γ5 in higher-order QCD calculations and the NNLO evolution of the polarized valence distribution

We discuss the prescription for the Dirac matrix γ5γ5 in dimensional regularization used in most second- and third-order QCD calculations of collider cross sections. We provide an alternative implementation of this approach that avoids the use of an explicit form of γ5γ5 and of its (anti-)commutation relations in the most important case of no more than one γ5γ5 in each fermion trace. This treatment is checked by computing the third-order corrections to the structure functions F2F2 and g1g1 in charged-current deep-inelastic scattering with axial-vector couplings to the W  -bosons. We derive the so far unknown third-order helicity-difference splitting function ΔPns(2)s that contributes to the next-to-next-to-leading order (NNLO) evolution of the polarized valence quark distribution of the nucleon. This function is negligible at momentum fractions x≳0.3x≳0.3 but relevant at x≪1x≪1.