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We discuss the discovery potential of a promising signals, $3 \times \mu^+\mu^-$ at a $\mu^+ \mu^-$ collider and $\mu^+ \mu^+ \mu^+ \mu^+ \mu^- \mu^- $at a $\mu^+ \mu^+$ collider, that are obtained via production of $Z'$ and a newscalar boson $\phi$ in a spontaneously broken local $U(1)_{L_\mu - L_\tau}$model. We consider the $Z'$ associated production from the process $\mu^+ \mu^-\to \phi Z'$ in addition to a muonphilic $Z'$ fusion process $\mu^+ \mu^\pm \to\phi \mu^+ \mu^\pm$. The scalar boson is associated with $U(1)_{L_\mu -L_\tau}$ symmetry breaking and dominantly decays into $Z' Z'$ mode. We carryout numerical simulation analysis for signal and background processes toestimate a discovery significance for different benchmark points. It is shownthat our signal can be observed with integrated luminosity less than$\mathcal{O}(100)$ fb$^{-1}$ for both $\mu^+ \mu^-$ and $\mu^+ \mu^+$ colliderswith more than 5-$\sigma$ significance.

Multi muon/anti-muon signals via productions of gauge and scalar bosons in a $U(1)_{L_\mu-L_\tau}$ model at muonic colliders