Implications of right-handed neutrinos in $B-L$ extended standard model
  with scalar dark matter
  We investigate the Standard Model (SM) with a $U(1)_{B-L}$ gauge extension
where a $B-L$ charged scalar is a viable dark matter (DM) candidate. The
dominant annihilation process, for the DM particle is through the $B-L$
symmetry breaking scalar to right-handed neutrino pair. We exploit the effect
of decay and inverse decay of the right-handed neutrino in thermal relic
abundance of the DM. Depending on the values of the decay rate, the DM relic
density can be significantly different from what is obtained in the standard
calculation assuming the right-handed neutrino is in thermal equilibrium and
there appear different regions of the parameter space satisfying the observed
DM relic density. For a DM mass less than $\mathcal{O}$(TeV), the direct
detection experiments impose a competitive bound on the mass of the
$U(1)_{B-L}$ gauge boson $Z^\prime$ with the collider experiments. Utilizing
the non-observation of the displaced vertices arising from the right-handed
neutrino decays, bound on the mass of $Z^\prime$ has been obtained at present
and higher luminosities at the LHC with 14 TeV centre of mass energy where an
integrated luminosity of 100fb$^{-1}$ is sufficient to probe $m_{Z'} \sim 5.5$
TeV.
