  The standard model (SM) one-loop contributions to the most general
$H^*Z^*Z^*$ coupling are obtained via the background field method in terms of
Passarino-Veltman scalar functions, from which the contributions to the $H^*ZZ$
and $HZZ^*$ couplings are obtained in terms of two $CP$-conserving $h_{1,2}^V$
and one $CP$-violating $h_3^V$ form factors ($V=H, Z$). The current CMS
constraints on the $HZZ$ coupling ratios are then used to obtain bounds on the
real and absorptive parts of the anomalous $HZZ$ couplings. The former are up
to two orders of magnitude tighter than previous ones, whereas the latter are
the first one of this kind. The effects of the absorptive parts of the $HZZ$
anomalous couplings, which have been overlooked in the past, are analyzed via
the partial decay width $\Gamma_{H^\ast\rightarrow ZZ}$, and a significant
deviation from the SM tree-level contribution is observed at low energies,
though it becomes negligible at high energies. We also explore the possibility
that polarized $Z$ gauge bosons are used for the study of non-SM $HZZ$
contributions via a new left-right asymmetry $\mathcal{A}_{LR}$, which is
sensitive to $CP$-violating complex form factors and can be as large as the
unity at most, though in a more conservative scenario it is four to five orders
of magnitude larger than the SM prediction arising up to the three-loop level.
The partial decay widths $\Gamma_{H^\ast\rightarrow Z_LZ_L}$ and
$\Gamma_{H^\ast\rightarrow Z_RZ_R}$ are also studied in several scenarios and
it is observed that the deviations from the SM can be large at high energies
and increases as the energy increases. Thus, the use of polarized $Z$ gauge
bosons could give hints of $CP$ violation. The Mathematica code for our
analytical results and the numerical evaluation is available in our GitLab
site.
