Pseudo scalar dark matter in a generic U$(1)_X$ model

We consider a $U(1)_X$ extension of the Standard Model~(SM), where the spontaneous breaking of $U(1)_X$ gauge group results in a pseudo scalar particle which is the proposed candidate for dark matter. In the model, we introduce three right-handed neutrinos~(RHNs) $N_R^i$ and two extra scalars $\Phi$, $\chi$, which are SM gauge singlets but charged under $U(1)_X$ gauge group. Right-handed neutrinos are required to have the model anomaly free and explain the neutrino oscillation data. The heaviest scalar breaks the $U(1)_X$ gauge symmetry and the other extra scalar gives a pseudo scalar DM candidate. A pseudo scalar dark matter~(DM) is an interesting candidate as it naturally evades the stringent direct detection bounds due to its coupling structure. We study the phenomenology of this pseudo-scalar DM while considering several theoretical and experimental constraints. We find that in our model, there is a feasible parameter space, which satisfies by the DM lifetime bound, relic and direct detection constraints while respecting the colliders and other bounds.