Testing neutrino mass hierarchy under type-II seesaw scenario in $U(1)_X$ from colliders

The origin of tiny neutrino mass is a long standing unsolved puzzle of the Standard Model (SM), which allows us to consider scenarios beyond the Standard Model (BSM) in a variety of ways. One of them being a gauge extension of the SM may be realized as in the form of an anomaly free, general $U(1)_X$ extension of the SM, where an $SU(2)_L$ triplet scalar with a $U(1)_X$ charge is introduced to have Dirac Yukawa couplings with the SM lepton doublets. Once the triplet scalar developes a Vacuum Expectation Value (VEV), light neutrinos acquire their tiny Majorana masses. Hence, the decay modes of the triplet scalar has a direct connection to the neutrino oscillation data for different neutrino mass hierarchies. After the breaking of the $U(1)_X$ gauge symmetry, a neutral $U(1)_X$ gauge boson $(Z^\prime)$ acquires mass, which interacts differently with the left and right handed SM fermions. Satisfying the recent LHC bounds on the triplet scalar and $Z^\prime$ boson productions, we study the pair production of the triplet scalar at LHC, 100 TeV proton proton collider FCC, $e^-e^+$ and $\mu^-\mu^+$ colliders followed by its decay into dominant dilepton modes whose flavor structure depend on the neutrino mass hierarchy. Generating the SM backgrounds, we study the possible signal significance of four lepton final states from the triplet scalar pair production. We also compare our results with the purely SM gauge mediated triplet scalar pair production followed by four lepton final states, which could be significant only in $\mu^- \mu^+$ collider.