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 A_CP=Br₋/Br₊ 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_e or ν-bar _e); if N is Majorana, both LC and lepton number violating (LV) decays contribute (LV: ν= ν-bar _μ or ν = ν_μ). The results show that the CP asymmetry A_CP 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, ΔM_N not >> Γ_N. 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. (paper)