Lepton number and flavor violation mediated by doubly charged scalars at $μ$TRISTAN $μ^+ μ^+$ collider

Charged lepton flavor violation is an important probe of physics beyond the Standard Model. Although $ΔL_μ= -ΔL_e = \pm 1$ lepton flavor violating (LFV) processes, such as $μ\to eγ$ and $μ\to 3e$, are severely constrained experimentally, there is still plenty of room for LFV processes involving lepton number violation (LNV). In this paper, we investigate such processes at a $μ^+μ^+$ collider that remain even if processes with $ΔL_μ= -ΔL_e = \pm 1$ are forbidden by discrete symmetries. We consider processes induced by doubly charged scalars, which can be either an $SU(2)_L$ singlet or a component of an $SU(2)_L$ triplet. The cross sections of these processes at the collider depend on the representation of the scalar unless the muon beam is unpolarized. In the case of $ΔL_μ= -ΔL_e = 2$, we discuss the combination of the muonium-to-antimuonium transition and the $μ^+μ^+ \to e^+e^+γ/Z$ processes at the collider, and how the model parameters, such as the scalar couplings and mass, can be determined from experimental measurements. Doubly charged scalars are often utilized in models for generating neutrino masses, and they can also be used to construct models with LNV processes at a muon collider. If neutrino masses are generated at tree level, the LNV processes at the collider are suppressed, and therefore, we consider models that do not induce neutrino masses at tree level. LNV processes, such as $μ^+μ^+ \to t\bar b t\bar b$ and $μ^+μ^+ \to μ^+ e^- t\bar b t\bar b$, are possible in these models. Even if LNV is indicated by searches at hadron colliders, it is worthwhile to pursue these processes as clean LNV signals at a lepton collider.