Rare Decays of Charmed Mesons into Three Charged Leptons and Neutrino

The branching ratios and differential distributions for four-lepton decays $$D_{s}^{ - } \to {{\mu }^{ + }}{{\mu }^{ - }}{{\bar {\nu }}_{e}}{\kern 1pt} {{e}^{ - }}$$ and $${{D}^{ - }} \to {{\mu }^{ + }}{{\mu }^{ - }}{{\bar {\nu }}_{e}}{\kern 1pt} {{e}^{ - }}$$ are obtained in the Standard Model. Our calculations include emission of virtual photons by light ( $$d$$ or $$s$$ ) and heavy $$c$$ quarks of the charmed meson and bremsstrahlung of a virtual photon by a charged lepton in the final state. The emission of a virtual photon by a light quarks is described using the vector meson dominance model.