  Neutron star (NS) as the dark matter (DM) probe has gained a broad attention
recently, either from heating due to DM annihilation or its stability under the
presence of DM. In this work, we investigate spin-$1/2$ fermionic DM $\chi$
charged under the $U(1)_{X}$ in the dark sector. The massive gauge boson $V$ of
$U(1)_{X}$ gauge group can be produced in NS via DM annihilation. The produced
gauge boson can decay into Standard Model (SM) particles before it exits NS,
despite its tiny couplings to SM particles. Thus, we perform a systematic study
on $\chi\bar{\chi}\to2V\to4{\rm SM}$ as a new heating mechanism for NS in
addition to $\chi\bar{\chi}\to2{\rm SM}$ and kinetic heating from DM-baryon
scattering. The self-trapping due to $\chi V$ scattering is also considered. We
assume the general framework that both kinetic and mass mixing terms between
$V$ and SM gauge bosons are present. This allows both vector and axial-vector
couplings between $V$ and SM fermions even for $m_V\ll m_Z$. Notably, the
contribution from axial-vector coupling is not negligible when particles
scatter relativistically. We point out that the above approaches to DM-induced
NS heating are not yet adopted in recent analyses. Detectabilities of the
aforementioned effects to the NS surface temperature by the future telescopes
are discussed as well.
