  The origin of neutrino mass is a big unsolved problem of the Standard Model
(SM) that motivate us to consider beyond the SM (BSM) scenarios where
SM-singlet right-handed neutrinos (RHNs) are introduced to explain the origin
of the light neutrino masses through the seesaw mechanism. There is a variety
of ways which could lead us to this goal and one of them is a general U$(1)$
extension of the SM. In this scenario, three SM-singlet RHNs are introduced to
cancel the gauge and mixed gauge gravity anomalies. After anomaly cancellation,
we notice that the left- and right-handed charged fermions are differently
charged under the general U$(1)$ gauge group evolving a chiral scenario. After
the breaking of the general U$(1)$ symmetry, a neutral BSM gauge boson
$(Z^\prime)$ acquires mass and it is a free parameter. Such $Z^\prime$, being
lighter than $5$ GeV, could be probed at the intensity and lifetime frontiers
like FASER, FASER2, DUNE, and ILC beam dump experiments. The estimated bounds
are needed to be compared with the existing bounds. We find that existing
constraints from Orsay, Nomad, PS191, KEK, LSND, CHARM experiments, and
cosmological scenario like SN1987A can be compared in our case once estimated
for chiral scenarios. Finally, we compare the parameter spaces showing viable
ones that could be probed by FASER, FASER2, DUNE, and ILC beam dump experiments
and already excluded regions from Orsay, Nomad, PS191, KEK, LSND, CHARM, and
SN1987A for a chiral scenario.
