Neutrino masses and self-interacting dark matter with mass mixing $Z-Z^{\mathrm{\prime }}$ gauge portal

New light gauge bosons can affect several low-energy experiments, such as atomic parity violation or colliders. Here, we explore the possibility that a dark sector is charged under a new $U(1)$ gauge symmetry, and the portal to the Standard Model is through a $Z-Z^{\mathrm{\prime }}$ mass mixing. In our approach, breaking the new gauge symmetry is crucial to generate neutrino masses. We investigate the parameter space to reproduce neutrino masses, the correct dark matter relic abundance, and to produce the observed core-like DM distribution in galactic centers.