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 $Z^{\mathrm{\prime }}$ and a new scalar boson $\phi$ in a spontaneously broken local $U(1{)}_{L_{\mu }-L_{\tau }}$ model. We consider the $Z^{\mathrm{\prime }}$ associated production from the process ${\mu }^{+}{\mu }^{-}\to \phi Z^{\mathrm{\prime }}$ in addition to a muonphilic $Z^{\mathrm{\prime }}$ 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 $Z^{\mathrm{\prime }}{Z}^{\mathrm{\prime }}$ 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.