Lepton flavor physics at $\mu^+ \mu^+$ colliders

We discuss sensitivities to lepton flavor violating (and conserving) interactions at future muon colliders, especially at $\mu^+\mu^+$ colliders. Compared with the searches for rare decays of $\mu$ and $\tau$, we find that the TeV-scale future colliders have better sensitivities depending on the pattern of hierarchy in the flavor mixings. As an example, we study the case with the type-II seesaw model, where the flavor mixing parameters have direct relation to the neutrino mass matrix. At a $\mu^+ \mu^+$ collider, the number of events of the $\mu^+ \mu^+ \to \mu^+ \tau^+$ process can be larger than $\mathcal{O}(100)$ with the center of mass energy $\sqrt s = 2$ TeV, and with an integrated luminosity ${\cal L} = 1$ ab$^{-1}$, while satisfying bounds from rare decays of $\mu$ and $\tau$. We discuss impacts of the overall mass scale of neutrinos as well as CP violating phases to the number of expected events.