\begin{document}$ H^\ast\rightarrow ZZ \rightarrow \overline{\ell}_1\ell_1\overline{\ell}_2\ell_2 $\end{document} decay width is presented for both unpolarized and polarized \begin{document}$ Z $\end{document} gauge bosons in the scenario with the most general \begin{document}$ H^*ZZ $\end{document} vertex function, which is expressed in terms of two \begin{document}$\rm CP$\end{document}-even (\begin{document}$ \hat b_Z $\end{document} and \begin{document}$ \hat c_Z $\end{document}) and one \begin{document}$\rm CP$\end{document}-odd (\begin{document}$ \tilde b_Z $\end{document}) anomalous couplings. The SM contributions to the \begin{document}$ H^*ZZ $\end{document} coupling up to the one-loop level are also included. Explicit analytic results for the unpolarized and polarized \begin{document}$ H^\ast\rightarrow ZZ \rightarrow \overline{\ell}_1\ell_1\overline{\ell}_2\ell_2 $\end{document} square amplitudes and the four-body phase space are presented, out of which several observable quantities can be obtained straightforwardly. Regarding numerical analysis, a cross-check was performed via \begin{document}$\mathrm{MadGraph5\_aMC@NLO} $\end{document}, where our model was implemented with the aid of FeynRules. We then considered the most stringent bounds on anomalous complex \begin{document}$ H^*ZZ $\end{document} couplings and analyzed the effects of the polarizations of the \begin{document}$ Z $\end{document} gauge bosons through the polarized \begin{document}$ H^\ast\rightarrow ZZ \rightarrow \overline{\ell}_1\ell_1\overline{\ell}_2\ell_2 $\end{document} decay width as well as left-right and forward-backward asymmetries, which were found to be sensitive to new-physics effects. Particular focus was put on the effects of the absorptive parts of the anomalous \begin{document}$ H^*ZZ $\end{document} couplings, which have been largely overlooked up to now in LHC analyses. It was found that the studied observable quantities, particularly the left-right asymmetries, can be helpful to search for effects of \begin{document}$\rm CP$\end{document}-violation in the \begin{document}$ H^*ZZ $\end{document} coupling and set bounds on the absorptive parts. For completeness, we also analyzed the case of unpolarized \begin{document}$ Z $\end{document} gauge bosons."> Polarized and unpolarized off-shell <inline-formula><tex-math id="M1">\begin{document}${ \boldsymbol H^\ast{\bf\to} \boldsymbol Z\boldsymbol Z{\bf\rightarrow 4}\boldsymbol\ell}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/d5a84c49-222f-45cd-99bb-ead7d4945df0/CPC-2024-0317_M1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/d5a84c49-222f-45cd-99bb-ead7d4945df0/CPC-2024-0317_M1.png"/></alternatives></inline-formula> decay above the <inline-formula><tex-math id="M2">\begin{document}${{\bf 2}\boldsymbol m_Z} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/d5a84c49-222f-45cd-99bb-ead7d4945df0/CPC-2024-0317_M2.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/d5a84c49-222f-45cd-99bb-ead7d4945df0/CPC-2024-0317_M2.png"/></alternatives></inline-formula> threshold -
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