\begin{document}$ L_{\mu}-L_{\tau} $\end{document} symmetry. A new gauge boson \begin{document}$ Z' $\end{document}, as well as two scalar fields S and Φ, are introduced to the Standard Model (SM), where S and Φ are SM singlets but both carry \begin{document}$U(1)_{L_{\mu}-L_{\tau}} $\end{document} charge. The real component and imaginary component of S can acquire different masses after spontaneously breaking symmetry, and the lighter one can play the role of dark matter, which is stabilized by an extra \begin{document}$ Z_2 $\end{document} symmetry. We consider the possibility of light dark matter as well as the co-annihilation case in our model and present the current \begin{document}$ (g-2)_{\mu} $\end{document} anomaly, Higgs invisible decay, dark matter relic density, and direct detection constraints on the parameter space. Direct detection gives the most stringent restriction on the model, and we found that although most of the parameter space is excluded, a mass region of a few GeV is still allowed where the co-annihilation process is also permitted in the model."> Scalar dark matter and muon <i>g</i>-2 in a <inline-formula><tex-math id="Z-20220506143955">\begin{document}${\boldsymbol U({\bf 1})_{{\boldsymbol L}_{\boldsymbol\mu} - {\boldsymbol L}_{\boldsymbol\tau}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/9bb5fdae-f7fa-4c0c-bb52-b9581bf6cf78/CPC-2022-0074_Z-20220506143955.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/9bb5fdae-f7fa-4c0c-bb52-b9581bf6cf78/CPC-2022-0074_Z-20220506143955.png"/></alternatives></inline-formula> model -
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