Past exam of the nuclear and particle physics course of the University of Oxford / 2022 / 2 / g / Solution

The decay is allowed in the Standard Model. Begin with ordinary muon decay, ${\mu }^{-}\to {\nu }_{\mu }{W}^{-\ast }$, $W^{-\ast }\to e^{-}{\bar{\nu }}_e$. The outgoing electron can emit a virtual photon that converts to $e^{-}{e}^{+}$. The bottom panel of the figure in part (b) shows one representative diagram. Radiation from other charged lines and exchange of the identical final electrons complete the decay amplitude.

The final total electric charge is $-1$. Its electron-family lepton number is $1+1-1-1=0$, and its muon-family lepton number is $1$, matching the initial muon. The extra pair therefore violates no relevant Standard Model selection rule. It is also kinematically allowed since $m_{\mu }>3m_e$. The virtual pair costs two electromagnetic vertices: the decay amplitude is order $G_F{e}^2$, and the decay rate is suppressed by roughly two powers of the fine-structure constant, with additional phase space and logarithmic factors.