Probing Electroweak Phase Transition in the Singlet Standard Model via $bb\gamma\gamma$ and 4$l$ channels

We investigate the prospects for resonant di-Higgs and heavy Higgs production searches at the 14 TeV HL-LHC in the combination of $bb\gamma\gamma$ and $4l$ channels, as a probe of a possible first order electroweak phase transition in real singlet scalar extension of the Standard Model. Event selection follows those utilized in the $bb\gamma\gamma$ and $4l$ searches by the ATLAS Collaboration, applied to simulation using benchmark parameters that realize a strong first order electroweak phase transition. The output of discriminant analysis is implemented by numerical calculation, optimised by the joint restriction from the two channels. The prospective reach for $bb\gamma\gamma$/$4l$ channel could be more competitive in probing the electroweak phase transition at lower/higher resonance masses. With 3 $ab^{-1}$ integrated luminosity, the combination of the $bb\gamma\gamma$ and $4l$ channels can discover/exclude a significant portion of %\mrmC{isn't it more accurate to say "a significant portion of" ?} of the viable parameter space that realizes a strong first order phase transition when the resonance mass is heavier than 500 GeV.