\begin{document}$ U(1)_{L_\mu-L_\tau} $\end{document} at a muon collider. Charged under the \begin{document}$ U(1)_{L_\mu-L_\tau} $\end{document} symmetry, the heavy neutral leptons can be pair produced via the new gauge boson \begin{document}$ Z' $\end{document} at the muon collider as \begin{document}$ \mu^+\mu^-\to Z^{\prime *}\to NN $\end{document} and \begin{document}$ \mu^+\mu^-\to Z^{\prime (*)} \gamma\to NN\gamma $\end{document}. We then performed a detailed analysis on the lepton number violation signature \begin{document}$ \mu^+\mu^-\to NN\to \mu^\pm\mu^\pm W^\mp W^\mp $\end{document} and \begin{document}$ \mu^+\mu^-\to NN \gamma\to \mu^\pm\mu^\pm W^\mp W^\mp \gamma $\end{document} at the 3 TeV muon collider, where the hadronic decays of W boson are treated as fat-jets J. These lepton number violation signatures have quite clean backgrounds at the muon collider. Our simulation shows that a wide range of viable parameter space is within the reach of the 3 TeV muon collider. For instance, with new gauge coupling \begin{document}$ g'=0.6 $\end{document} and an integrated luminosity of 1000 fb\begin{document}$ ^{-1} $\end{document}, the \begin{document}$ \mu^\pm\mu^\pm JJ $\end{document} signal could probe \begin{document}$ m_{Z'}\lesssim 13 $\end{document} TeV. Meanwhile, if the gauge boson mass satisfies \begin{document}$ 2 m_N<m_{Z'}<\sqrt{s} $\end{document}, the \begin{document}$ \mu^\pm\mu^\pm JJ\gamma $\end{document} signature would be more promising than the \begin{document}$ \mu^\pm\mu^\pm JJ $\end{document} signature."> Heavy neutral leptons in gauged <i>U</i>(1)<sub><i>L</i><sub><i>µ</i></sub>−<i>L</i><sub><i>τ</i></sub></sub> at muon collider -
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