Dark matter in Inert Doublet Model with one scalar singlet and $U(1)_X$ gauge symmetry

We study Dark Matter (DM) abundance in the framework of the extension of the Standard Model (SM) with an additional $U(1)_X$ gauge symmetry. One complex singlet is included to break the $U(1)_X$ gauge symmetry, meanwhile one of the doublets is considered inert to introduce a DM candidate. The stability of the DM candidate is analyzed with a continuous $U(1)_X$ gauge symmetry as well as discrete $Z_2$ symmetry. We find allowed regions for the free model parameters which are in agreement with the most up-to-date experimental results reported by CMS and ATLAS collaborations, the upper limit on WIMP-nucleon cross section imposed by XENON1T collaboration and the upper limit on the production cross-section of a $Z^{\prime}$ gauge boson times the branching ratio of the $Z^{\prime}$ boson decaying into $\ell^-\ell^+$. We also obtain allowed regions for the DM candidate mass from the relic density reported by the PLANCK collaboration including light, intermediate and heavy masses; depending mainly on two parameters of the scalar potential, $\lambda_{2x}$ and $\lambda_{345}=\lambda_3+\lambda_4+2\lambda_5$. We find that trough $pp\rightarrow \chi\chi \gamma$ production, it may only be possible for a future hadron-hadron Circular Collider (FCC-hh) to be able to detect a DM candidate within the range of masses 10-60 GeV.
Comment: Accepted version to be published in EPJC, typos corrected, cites added