Muonium to antimuonium conversion and the decay ${\mathrm{\mu }}^{+}$→${\mathit{e}}^{+}$ν${\mathrm{¯}}_{\mathit{e}}$${\nu }_{\mathrm{\mu }}$ in left-right symmetric models

We show that in the minimal left-right symmetric model with triplet Higgs bosons the range of the muon neutrino mass for which ${\nu }_{\mathrm{\mu }}$ is required by cosmological considerations to be unstable can be probed by muonium to antimuonium conversion (M→M¯) and/or by the exotic muon decay ${\mathrm{\mu }}^{+}$→${\mathit{e}}^{+}$ ν${\mathrm{¯}}_{\mathit{e}}$ ${\nu }_{\mathrm{\mu }}$. We point out that if all the leptonic mixing matrices are hierarchical and the Dirac masses of the neutrinos are equal to the masses of the corresponding charged leptons, there is a lower bound in this range for the rates of both of these processes. We find ‖${\mathit{G}}_{\mathit{M}\mathit{M}\mathrm{¯}}$‖≳7×${10}^{\mathrm{-}5}$ ${\mathit{G}}_{\mathit{F}}$ and ‖${\mathit{G}}_{\mathrm{\mu }}^{\left(\mathit{e}\right)}$‖≳2×${10}^{\mathrm{-}4}$ ${\mathit{G}}_{\mathit{F}}$ for the strength of the M→M¯ and ${\mathrm{\mu }}^{+}$→${\mathit{e}}^{+}$ ν${\mathrm{¯}}_{\mathit{e}}$ ${\nu }_{\mathrm{\mu }}$ interactions.