  We consider QCD corrections to two loops for the polarized amplitudes of
$q{\bar q}\to Z +$ Higgs boson. First we show how the polarized amplitudes of
$b \bar{b} \rightarrow Z h$ 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 $\gamma_5$ 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 $b \bar{b} \rightarrow Z h$ 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 $q \bar{q}
\rightarrow ZH$ 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 $\gamma_5$
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 $\gamma_5$
prescription in use.

High Energy Physics - Phenomenology;High Energy Physics - Theory