Polarized \( \mathrm{q}\overline{\mathrm{q}} \) → Z+Higgs amplitudes at two loops in QCD: the interplay between vector and axial vector form factors and a pitfall in applying a non-anticommuting γ5

We consider QCD corrections to two loops for the polarized amplitudes of $$ q\overline{q} $$ → Z + Higgs boson. First we show how the polarized amplitudes of $$ b\overline{b} $$ → Zh associated with a non-vanishing b-quark Yukawa coupling and a scalar or pseudoscalar Higgs boson h can be built up solely from vector form factors (FF) of properly grouped classes of diagrams, bypassing completely the need of explicitly manipulating γ5 in dimensional regularization (up to a few “anomalous”, i.e., triangle diagrams). We determine the contributions of the triangle diagrams in the heavy top limit. We present the analytic results of the vector FF and the triangle-diagram contributions to the axial vector FF, which are sufficient for deriving the two-loop QCD amplitudes for $$ b\overline{b} $$ → Zh with a CP-even and CP-odd Higgs boson h. We derive the respective Ward identity for these amplitudes, which are subsequently verified to two-loop order in QCD using these FF. In addition, the FF of a class of corrections to $$ q\overline{q} $$ → ZH proportional to the top-Yukawa coupling are obtained analytically to two-loop order in QCD in the heavy-top limit using the Higgs-gluon effective Lagrangian where the top quark is integrated out. We address a pitfall that occurs when applying the non-anticommutating γ5 prescription to this class of contributions that has been overlooked so far in the literature. We attribute this issue to the fact that the absence of certain heavy-mass expanded diagrams in the infinite-mass limit of a scattering amplitude with an axial vector current depends on the particular γ5 prescription in use.