Are 2HDMs with a gauged $U(1)$ symmetry alive?

We investigate the phenomenology of 2 Higgs doublet models (2HDMs) with a new $U(1)$ gauge symmetry, $U(1)_X$, by which flavor changing neutral currents are forbidden at tree level. As an important consequence of the spontaneous breaking of both the $U(1)_X$ and electroweak symmetries by electroweak vacuum expectation values, upper limits appear on masses of an additional gauge boson $Z'$ and extra Higgs bosons which are less than the TeV scale. In addition, the standard model (SM) like Higgs boson $h$ and a heavier Higgs boson $H$ mainly decay into a pair of $Z'$ which induces four lepton final states. These new decay modes cannot be suppressed by taking no $Z$-$Z'$ mixing and/or the Higgs alignment limit. We find that the minimum setup of these 2HDMs has been excluded by current data for four lepton searches at LHC. Such severe constraints can, however, be avoided by introducing a pair of vector-like fermions $\chi$ which are singlet under the SM symmetry but charged under $U(1)_X$, and can be a candidate of dark matter. Thanks to the existence of $\chi$, $Z'$ can mainly decay into $\chi\bar{\chi}$ instead of SM leptons. As benchmark models, we consider the $U(1)_H$ and $U(1)_R$ models realized by fixing specific $U(1)_X$ charges, and find regions of parameter space allowed by theoretical and current experimental constraints. We clarify that $m_H \in [160, 220]$ GeV and $\tan \beta \in [3, 4.4]$ are allowed in the $U(1)_H$ model, while $m_H \in [160, 380]$ GeV and $\tan \beta \in [1.6, 4.4]$ are allowed in the $U(1)_R$ model. In both the models, the $Z'$ mass is constrained to be $100~\text{GeV} \lesssim m_{Z'} \lesssim 110$ GeV. Such a quite limited parameter space can further be explored at future collider experiments, e.g., High-Luminosity LHC and lepton colliders.