Scalar dark matter and Muon $g-2$ in a $U(1)_{L_{\mu}-L_{\tau}}$ model

We consider a simple scalar dark matter model within the frame of gauged $L_{\mu}-L_{\tau}$ symmetry. A gauge boson $Z'$ as well as two scalar fields $S$ and $\Phi$ are introduced to the Standard Model (SM). $S$ and $\Phi$ are SM singlet but both with $U(1)_{L_{\mu}-L_{\tau}}$ charge. The real component and imaginary component of $S$ can acquire different masses after spontaneously symmetry breaking, and the lighter one can play the role of dark matter which is stabilized by the residual $Z_2$ symmetry. A viable parameter space is considered to discuss the possibility of light dark matter as well as co-annihilation case, and we present current $(g-2)_{\mu}$ anomaly, Higgs invisible decay, dark matter relic density as well as direct detection constriants on the parameter space.