On Lepton-Number-Violating Searches for Muon to Positron Conversion

There is no guarantee that the violation of lepton number, assuming it exists, will primarily manifest itself in neutrinoless double beta decay ($0\nu\beta\beta$). Lepton-number violation and lepton-flavor violation may be related, and much-needed information regarding the physics that violates lepton number can be learned by exploring observables that violate lepton flavors other than electron flavor. One of the most promising observables is $\mu^- \rightarrow e^+$ conversion, which can be explored by experiments that are specifically designed to search for $\mu^- \rightarrow e^-$ conversion. We survey lepton-number-violating dimension-five, -seven, and -nine effective operators in the standard model and discuss the relationships between Majorana neutrino masses and the rates for $0\nu\beta\beta$ and $\mu^- \rightarrow e^+$ conversion. While $0\nu\beta\beta$ has the greatest sensitivity to new ultraviolet energy scales, its rate might be suppressed by the new physics relationship to lepton flavor, and $\mu^- \rightarrow e^+$ conversion offers a complementary probe of lepton-number-violating physics.