Z and Higgs boson decays with doubly-charged scalars at one-loop: current constraints, future sensitivities, and application to lepton-triality models

We analyse the $Z$ and Higgs boson decays $Z\to \ell^+ \ell^-$ ($\ell = e, \mu,\tau$), $H\to \gamma\gamma$ and $H\to Z\gamma$ that are induced at one-loop level in models with a doubly-charged isosinglet scalar. After discussing current constraints, we derive the parameter space that will be probed by the HL-LHC and the possible future colliders the ILC, CEPC and FCC. We then apply those constraints to lepton triality models which are based on a discrete $Z_3$ family symmetry and were recently studied in the context of charged-lepton flavour-violating processes at Belle II and the proposed $\mu^+ \mu^+$ and $\mu^+ e^-$ collider known as $\mu$TRISTAN. We find that the future constraints that can be imposed by $Z \to \ell^+ \ell^-$ on the lepton flavour conserving couplings of the triality models reduce the viable parameter space to probe lepton flavour violating processes. The constraints from Higgs boson decays are the first on the Higgs portal sector of the triality models.