Probing CP Violation in $h \to Z\gamma$ with Background Interference

We show that the parity of the $h Z\gamma$ vertex can be probed by interference between the gluon fusion Higgs production, $gg \to h \to \gamma Z \to \gamma \ell^+\ell^-$, and the background, $gg \to \gamma Z \to \gamma \ell^+\ell^-$, amplitudes. In the presence of a parity violating $hZ\gamma$ vertex, this interference alters the kinematic distribution of the leptons and photon compared to Standard Model (SM) expectations. For a Higgs with SM-sized width and couplings, we find that the size of the effect enters at most at the $10^{-2}$ level. Such a small effect cannot be seen at the LHC, even with futuristic high luminosities. Should there exist other broader scalar particles with larger production cross-section times branching ratio to $Z\gamma$, then the parity structure of their $Z\gamma$ couplings can be probed with this technique.