  Leading two-loop contributions to the di-photon decay of the Higgs boson are
evaluated for the first time in the Inert Doublet Model (IDM). We employ for
this calculation the Higgs low-energy theorem, meaning that we obtain
corrections to the Higgs decay process by taking Higgs-field derivatives of the
leading two-loop contributions to the photon self-energy. Specifically, we have
included purely scalar corrections involving inert BSM Higgs bosons, as well as
external-leg contributions involving both the inert scalars and fermions. Our
calculation has been performed with a full on-shell renormalization, and in the
gauge-less limit. We investigate our results numerically in two scenarios of
the IDM: one with a light dark matter (DM) candidate (Higgs resonance
scenario), and another with all additional scalars heavy (heavy Higgs
scenario). In both cases, we find that the inclusion of two-loop corrections
qualitatively modifies the behavior of the decay width, compared with the
one-loop ($i.e.$ leading) order, and that they increase the deviation from the
Standard Model. These large deviations can be tested at the High-Luminosity
LHC.
