Dark matter and muon $(g-2)$ in local $U(1)_{L_mu-L_tau}$-extended Ma Model

We consider right-handed neutrino dark matter $N_1$ in local $U(1)_{L_mu-L_tau}$-extended Ma model. With the light $U(1)_{mu-tau}$ gauge boson ($m_{Z} sim {cal O}(100)$ MeV) and small $U(1)_{mu-tau}$ gauge coupling ($g_{Z}sim 10^{-4}-10^{-3}$) which can accommodate the muon $(g-2)$ anomaly and is still allowed by other experimental constraints, we show that we can get correct relic density of dark matter for wide range of dark matter mass ($M_1 sim 10-100$ GeV), although the gauge coupling constant $g_{Z}$ is small. This is due to the fact that the annihilation cross section of dark matter pair is enhanced by $M_1^4/m_{Z}^4$ in the processes $N_1 N_1 to Z Z$ or $N_1 N_1 to Z H_2$. We also consider the constraints from direct detection, collider searches.