  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.
