Light $Z^{\mathrm{\prime }}$ and Dark Matter from U(1)${}_X$ Gauge Symmetry

We consider a U(1)${}_X$ gauge symmetry extension of the Standard Model (SM) with a $Z^{\mathrm{\prime }}$-portal Majorana fermion dark matter that allows for a relatively light gauge boson $Z^{\mathrm{\prime }}$ with mass of 10 MeV$-$ a few GeV and a much heavier dark matter through the freeze-in mechanism. In a second scenario the roles are reversed, and the dark matter mass, in the keV range or so, lies well below the $Z^{\mathrm{\prime }}$ mass, say, $\sim 1$ GeV. We outline the parameter space that can be explored for these two scenarios at the future Lifetime Frontier experiments including Belle-II, FASER, LDMX and SHiP.