Higgs decay into a lepton pair and a photon: a roadmap to $\mathbf{H\to Z\gamma}$ discovery and probes of new physics

The decay $H\to \ell^+\ell^- \gamma$, $\ell=e,\mu$, receives contributions from $H\to Z[\to \ell^+\ell^-] \gamma$ and a non-resonant contribution, both of which are loop-induced. We describe how one can separate these sub-processes in a gauge-independent way, define the decay rate $\Gamma(H\to Z\gamma)$, and extract the latter from differential $H\to \ell^+\ell^- \gamma$ branching ratios. For $\ell=\mu$ also the tree decay rate, which is driven by the muon Yukawa coupling, is important. We propose kinematic cuts optimized to separate the three contributions, paving the way to the milestones (i) discovery of $H\to Z \gamma$, (ii) discovery of $H\to \mu^+\mu^- \left. \gamma\right|_{\rm tree}$, and (iii) quantification of new physics in both the effective $H$-$Z$-$\gamma$ and non-resonant $H$-$\ell^+$-$\ell^-$-$\gamma$ couplings.