Probing the top-Higgs boson FCNC couplings via the $h\to \gamma \gamma$ channel at the HE-LHC and FCC-hh

We investigate the sensitivity of future searches for the top-Higgs boson Flavour Changing Neutral Current (FCNC) couplings $tqh$~($q=u,c$) at the proposed High Energy Large Hadron Collider~(HE-LHC) and Future Circular Collider in hadron-hadron mode (FCC-hh). We perform a full simulation for two processes in the $h\to \gamma \gamma$ decay channel (where $h$ is the discovered Higgs state): single top quark FCNC production in association with the Higgs boson (plus a jet) and top quark pair production with FCNC decays $t\to qh$. All the relevant backgrounds are considered in a cut based analysis to obtain the limits on the Branching Ratios (BRs) of $t\to uh$ and $t\to ch$. It is shown that, at the HE-LHC with an integrated luminosity of 15 ab${}^{-1}$ and at the FCC-hh with an integrated luminosity of 30 ab${}^{-1}$, {{the BR($t\to uh$) (BR($t\to ch$)) can be probed, respectively, to $7.0(8.5)\times {10}^{-5}$ and $2.3(3.0)\times {10}^{-5}$ at the 95% Confidence Level (CL) (assuming a 10% systematic uncertainty on the background), which is almost two orders of magnitude better than the current 13 TeV LHC experimental results.