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CP violations in $π^{\pm}$ Meson Decay

We study the pion decays with intermediate on-shell neutrinos $N$ into two electrons and a muon, $π^{\pm} \to e^{\pm} N \to e^{\pm} e^{\pm} μ^{\mp} ν$. We investigate the branching ratios ${\rm Br}_{\pm} = [Γ(π^- \to e^- e^- μ^+ ν) \pm Γ(π^+ \to e^+e^+μ^-ν)]/Γ(π^- \to {\rm all})$ and the CP asymmetry ratio ${\cal A}_{\rm CP} = {\rm Br}_{-}/{\rm Br}_{+}$ 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 ${\bar ν}_e$); if $N$ is Majorana, both LC and lepton number violating (LV) decays contribute (LV: $ν={\bar ν}_μ$ or $ν=ν_μ$). The results show that the CP asymmetry ${\cal A}_{\rm CP}$ is in general very small, but increases and becomes $\sim 1$ 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, $ΔM_N \not\gg Γ_N$. The observation of CP violation in pion decays would be consistent with the existence of the well-motivated $ν$MSM model with two almost degenerate heavy neutrinos.