Muon collider signatures for a $Z'$ with a maximal $\mu-\tau$ coupling in $U(1)_{L_\mu-L_\tau}$

The gauged $U(1)_{L_\mu - L_\tau}$ model is a candidate model for explaining the muon g-2 anomaly because the $Z'$ in the model has a natural normal coupling to muon. Due to other experimental data constraints the viable mass range for the usual $Z'$ is constrained to be lower than a few hundred MeV. It has been shown that if the $Z'$ has a maximal off-diagonal mixing, $(\bar \mu \gamma^\mu \tau + \bar \tau \gamma^\mu \mu) Z'_\mu$, a large mass for $Z'$ is possible. This class of models has a very interesting signature for detection, such as $\mu^-\mu^+ \to \tau^- \tau^+$ pair, $\mu^- \mu^+ \to \mu^\pm\mu^\pm \tau^\mp \tau^\mp$ at a muon collider. In this work we study in detail these processes. We find that the in the parameter space solving the muon g-2 anomaly, t-channel $\tau^- \tau^+$ pair production can easily be distinguished at more than 5$\sigma$ level from the s-channel production as that predicted in the standard model. The smoking gun signature of doubly same sign $\mu^\pm \mu^\pm + \tau^\mp\tau^\mp$ pairs production can have a 5$\sigma$ sensitivity, at a muon collider of 3 TeV with $\mathcal{O}$($fb^{-1}$) luminosity.