The Process $gg\to h^0 Z^{*}$ in the Inverted Hierarchy Scenario of the 2HDM Type-I at the LHC

While searching at the Large Hadron Collider (LHC) for the production and decay of the CP-odd scalar ($A^0$) in the 2-Higgs-Doublet Model (2HDM) with Natural Flavour Conservation (NFC) via the channels $gg\to A^0$ (through one-loop triangle diagrams) and $A^0\to h^0 Z^*$ (with $m_{h^0} =125$ GeV or $m_{h^0} < 125$ GeV, with $Z$ off-shell), respectively, a factorisation of the two processes is normally performed, with the $A^0$ state being on-shell. While this approach is gauge-invariant, it is not capturing the presence of either of the following two channels: $gg\to Z^*\to h^0Z^*$ (through one-loop triangle diagrams) or $gg\to h^0Z^*$ (through one-loop box diagrams). As the resolution of the $A^0$ mass cannot be infinitely precise, we affirm that all such contributions should be computed simultaneously, whichever the $h^0$($Z^{*}$) decay(splitting) products, thereby including all possible interferences amongst themselves. The cross section of the ensuing complete process is significantly different from that obtained in the factorisation case, being of the order up to ten percent in either direction at the integrated level and larger (including changes in the shape of kinematical observables) at the differential level. We thus suggest that the complete calculation ought to be performed while searching for $A^0$ in this channel. We illustrate this need for the case of a 2HDM of Type-I in the inverted hierarchy scenario with $m_{h^0}<125$ GeV.