Prospects of $Z'$ Portal Dark Matter in $U(1)_{L_\mu-L_\tau}$

The gauged $U(1)_{L_\mu-L_\tau}$ model is well-motivated to explain the muon $g-2$ anomaly and dark matter in previous studies. However, the latest NA64$\mu$ experiment has almost excluded all the parameter space for the muon $g-2$, which indicates that the light dark matter benchmark scenarios interpreting muon $g-2$ in previous studies are also not allowed at present. In light of many recent and future experimental results, we revisit the minimal $Z'$ portal dark matter in $U(1)_{L_\mu-L_\tau}$. Focus on the phenomenology of dark matter $\chi$, we first explore the viable parameter space for the light dark matter under various tight constraints. Around the $Z'$ resonance, we find that there is still a large parameter space for $m_\chi\gtrsim10$ MeV via thermal freeze-out. On the other hand, the constraints on $Z'$ above the electroweak scale are quite loose but are less studied for dark matter. We also investigate the heavy dark matter around the TeV scale and corresponding constraints. A large part of the parameter space for dark matter is within the reach of future experiments.