$H\rightarrow Z\gamma$ decay and $CP$ violation

This study examines the impact of $CP$-violation on the signal strength $\mu^{Z\gamma}$, which was reported as $2.2\pm 0.7$ by the LHC. We obtain constraints on the real and absorptive parts of the $CP$-violating form factor $h_3^{Z\gamma}$ and find that they are less than 1.15 GeV. Additionally, we revisit the leading order Standard Model contributions to the $H\rightarrow Z\gamma$ decay and calculate contributions to $h_3^{Z\gamma}$ from FCNC complex couplings mediated by the $Z$ and $H$ bosons. By using the current bounds on such couplings, we find that the FCNC contribution to $h_3^{Z\gamma}$ with top and charm quarks in the loop is of order $10^{-5}$ GeV. While in a model with new quarks that preserves the SM predictions on Higgs decays, the $CP$-violating form factor $h_3^{Z\gamma}$ can be of order $10^{-1}$ GeV and could explain the excess on the signal strength $\mu^{Z\gamma}$.