Prospects for detecting the rare heavy Higgs decay $H\to h\gamma \gamma$ through the $H\to b\bar{b}\gamma \gamma$ channel at the LHC

We study the decay of a heavy CP-even neutral Higgs into an on-shell Standard Model-like Higgs boson and two photons, $H\to h\gamma \gamma$, in the two-Higgs doublet model. We argue that the decay channel $H\to h\gamma \gamma$, followed by the decay of the Standard Model Higgs $h\to b\bar{b}$, could be observed at the 5$\sigma$ level at the High-Luminosity LHC for masses of the heavy Higgs up to 950 GeV for the type-II, 650 GeV for the Lepton Specific and the Flipped 2HDMs, and 350 GeV for the type-I. We also discuss the possible role of the decay $H\to h\gamma \gamma$ in discriminating among different types of 2HDMs and in enhancing the total number of events in the final state $H\to b\bar{b}\gamma \gamma$ compared to the cascade decay $H\to hh$ followed by $h\to \gamma \gamma$ $h\to b\bar{b}$ with identical final state (although with different kinematical distributions).