CP violations in π^±→ meson decay

We study the pion decays with intermediate on-shell neutrinos N into two electrons and a muon, π^±→ → e ^±→N → e^±→e ^±→μ^∓ν. We investigate the branching ratios Br_±→= [Γ(π^- → e^-e^-μ⁺ν) ±→ Γ(π⁺ → e⁺e⁺μ^-ν)] /Γ(π^- → all) and the CP asymmetry ratio for such decays, in the scenario with two different on-shell neutrinos. If N is Dirac, only the lepton number conserving (LC) decays contribute (LC: ν = ν_eor); if N is Majorana, both LC and lepton number violating (LV) decays contribute (LV: or ν = ν_μ). The results show that the CP asymmetry is in general very small, but increases and becomes ∼1 when the masses of the two intermediate neutrinos get closer to each other, i.e., when their mass difference becomes comparable with their decay width, . The observation of CP violation in pion decays would be consistent with the existence of the well-motivated νMSM model with two almost degenerate heavy neutrinos.