On the two-loop BSM corrections to $h⟶\gamma \gamma$ in a triplet extension of the SM

We compute the two-loop BSM contributions to the $h⟶\gamma \gamma$ decay width in the SM extended with a real triplet of $SU(2)$. We consider scenarios in which the neutral components of doublet and triplet do not mix, so that the lighter neutral scalar $h$ has (at least approximately) SM-like couplings to fermions and gauge bosons. We focus on the two-loop corrections controlled by the quartic scalar couplings, and obtain explicit and compact formulas for the $h\gamma \gamma$ amplitude by means of a low-energy theorem that connects it to the derivative of the photon self-energy w.r.t. the Higgs field. We briefly discuss the numerical impact of the newly-computed contributions, showing that they may be required for a precise determination of $\mathrm{\Gamma }[h\to \gamma \gamma ]$ in scenarios where the quartic scalar couplings are large.