  We explore relativistic freeze-in production of scalar dark matter in gauged
$B-L$ model, where we focus on the production of dark matter from the decay and
annihilation of Standard Model (SM) and $B-L$ Higgs bosons. We consider the
Bose-Einstein (BE) and Fermi-Dirac (FD) statistics, along with the thermal mass
correction of the SM Higgs boson in our analysis. We show that in addition to
the SM Higgs boson, the annihilation and decay of the $B-L$ scalar can also
contribute substantially to the dark matter relic density. Potential effects of
electroweak symmetry breaking (EWSB) and thermal mass correction in BE
framework enhance the dark matter relic substantially as it freezes-in near
EWSB temperature via scalar annihilation. However, such effects are not so
prominent when the dark matter freezes-in at a later epoch than EWSB,
dominantly by decay of scalars. The results of this analysis are rather
generic, and applicable to other similar scenarios.
