\begin{document}$ g-2 $\end{document} anomaly, we study the \begin{document}$ U(1)_{L_\mu-L_\tau} $\end{document} breaking phase transition, gravitational wave spectra, and direct detection at the LHC in an extra \begin{document}$ U(1)_{L_\mu-L_\tau} $\end{document} gauge symmetry extension of the standard model. The new fields include vector-like leptons (\begin{document}$ E_1,\; E_2,\; N $\end{document}), the \begin{document}$ U(1)_{L_\mu-L_\tau} $\end{document}breaking scalar S, and the gauge boson \begin{document}$ Z' $\end{document}, as well as the dark matter candidate \begin{document}$ X_I $\end{document} and its heavy partner \begin{document}$ X_R $\end{document}. A joint explanation of the dark matter relic density and muon \begin{document}$ g-2 $\end{document} anomaly excludes the region where both \begin{document}$\min(m_{E_1},m_{E_2},m_N,m_{X_R})$\end{document} and \begin{document}$\min(m_{Z'},m_S)$\end{document} are much larger than \begin{document}$ m_{X_I} $\end{document}. In the parameter space accommodating the DM relic density and muon \begin{document}$ g-2 $\end{document} anomaly, the model can achieve a first-order \begin{document}$ U(1)_{L_\mu-L_\tau} $\end{document} breaking phase transition, whose strength is sensitive to the parameters of the Higgs potential. The corresponding gravitational wave spectra can reach the sensitivity of U-DECIGO. In addition, the direct searches at the LHC impose stringent bounds on the mass spectra of the vector-like leptons and dark matter."> <i>U</i>(1)<sub><i>L</i><sub><i>μ</i></sub>-<i>L</i><sub><i>τ</i></sub></sub> breaking phase transition, muon <i>g–</i>2, dark matter, collider, and gravitational wave -
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