Fully covering the MSSM Higgs sector at the LHC

In the context of the Minimal Supersymmetric extension of the Standard Model (MSSM), we reanalyze the search for the heavier CP-even $H$ and CP-odd $A$ neutral Higgs bosons at the LHC in their production in the gluon-fusion mechanism and their decays into gauge and lighter $h$ bosons and into top quark pairs. We show that only when considering these processes, that one can fully cover the entire parameter space of the Higgs sector of the model. Indeed, they are sensitive to the low $ aneta$ and high Higgs mass ranges, complementing the traditional searches for high mass resonances decaying into $ au$-lepton pairs which are instead sensitive to the large and moderate $ aneta$ regions. The complementarity of the various channels in the probing of the complete $[aneta, M_A]$ MSSM parameter space at the previous and upcoming phases of the LHC is illustrated in a recently proposed simple and model independent approach for the Higgs sector, the $h$MSSM, that we also refine in this paper.