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

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 new scalar 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 carry out numerical simulation analysis for signal and background processes to estimate a discovery significance for different benchmark points. It is shown that our signal can be observed with integrated luminosity less than $\mathcal{O}(100)$ fb$^{-1}$ for both $\mu^+ \mu^-$ and $\mu^+ \mu^+$ colliders with more than 5-$\sigma$ significance.