Lepton flavor violation in muonium decay and muon colliders in models with heavy neutrinos

We study the lepton-flavor-violating reaction ${\mu }^{+}{e}^{-}\to e^{+}{e}^{-}$ within two extensions of the standard model that include heavy neutrinos. The reaction is studied in the low energy limit in the form of muonium decay $M\to e^{+}{e}^{-}$ and in the high energy regime of a muon collider. The two theoretical models we consider are: model I, a typical seesaw model that violates lepton flavor and number by inclusion of extra right-handed neutrinos; and model II, a variant where lepton number is conserved and which includes extra right-handed as well as left-handed neutrinos, singlets under the gauge group. We find for muonium decay into $e^{+}{e}^{-}$ the extremely small result $Br(M\to e^{+}{e}^{-})<{10}^{-19}$ in both scenarios. Alternatively, for ${\mu }^{+}{e}^{-}$ collisions up to $\sqrt{s}\sim 50\mathrm{GeV}$, we find $\sigma ({\mu }^{+}{e}^{-}\to e^{+}{e}^{-})<{10}^{-5}\mathrm{fb}$, while for energies above the $W^{+}{W}^{-}$ threshold we find $\sigma ({\mu }^{+}{e}^{-}\to W^{+}{W}^{-})$ up to $\sim 1\mathrm{fb}$.