$CP$ violation in the $HZZ$ vertex and left-right asymmetries

We investigate new contributions to the $HZZ$ vertex from the Flavor Changing Neutral Current (FCNC) involving the Higgs and $Z$ bosons. Our calculations reveal that the form factors $h_2^V$ and $h_3^V$ ($V=H$, $Z$) can be induced through these couplings, and we present our results in terms of the Passarino-Veltman scalar functions. Using the current limits on $H\overline{t}c$ and $Z\overline{t}c$ couplings, we determine that the new contributions to the $CP$-conserving form factor $h_2^V$ are small compared to the Standard Model (SM) predictions. However, for the $CP$-violating form factor $h_3^V$, the contributions can reach values as high as $10^{-6}$, five orders of magnitude larger than in the SM. Furthermore, we examine how these results influence the left-right asymmetries in the processes $H^\ast\to ZZ$ and $Z^\ast\to ZH$. Our findings suggest that significant deviations from SM predictions may occur when considering FCNC contributions.