HL-LHC and ILC sensitivities in the hunt for heavy Higgs bosons

The prediction of additional Higgs bosons is one of the key features of physics beyond the Standard Model (SM) that gives rise to an extended Higgs sector. We assess the sensitivity of the Large Hadron Collider (LHC) in the high luminosity (HL) run alone and in combination with a possible future International Linear Collider (ILC) to probe heavy neutral Higgs bosons. We employ the Minimal Supersymmetric Standard Model (MSSM) as a framework and assume the light CP-even MSSM Higgs boson to be the Higgs boson observed at 125 GeV. We discuss the constraints on the MSSM parameter space arising from the precision measurements of the rates of the detected signal at 125 GeV and from direct searches for new heavy Higgs bosons in the $\tau^+\tau^-$, $b\bar{b}$ and di-Higgs ($hh$) final states. A new benchmark scenario for heavy Higgs searches in the $b\bar{b}$ channel is proposed in this context. For the future Higgs rate measurements at the HL-LHC and ILC two different scenarios are investigated, namely the case where the future rate measurements agree with the SM prediction and the case where the rates agree with the predictions of possible realizations of the MSSM Higgs sector in nature.