The effect of dark-matter (DM) heating in the neutron star (NS) is an intriguing topic to study.
In this talk we present such a study with DM as a spin-$1/2$ Dirac particle, $\chi$, carrying U(1) dark charge. The U(1) gauge boson, $V$, interacts with Standard Model (SM) fermions through
kinetic and mass mixings with photon and neutral $Z$ bosons. These mixings enable the annihilation $\chi\bar{\chi} \to VV$ and the subsequent decay of $V$ into a pair of SM fermions ($f\bar{f}$).
This chain of processes is a new NS heating mechanism in addition to $\chi\bar{\chi} \to f\bar{f}$ and the kinetic heating induced by DM-baryon scattering. The new feature of our work also includes the axial-vector term in $\chi f\bar{f}$ vertex, which is non-negligible when particles scatter relativistically. We discuss constraints on dark sector parameters by NS temperatures and the prospect of
measuring these temperatures by JWST.