CP violations in π± meson decay

We study the pion decays with intermediate on-shell neutrinos N into two electrons and a muon, pi(+/-) -> e(+/-) N -> e(+/-) e(+/-)mu(-/+)nu. We investigate the branching ratios Br-+/- = [Gamma(pi(-) -> e(-) e(-) mu(+) nu) +/- Gamma(pi(+) -> e(+) e(+) mu(-) nu)]/Gamma(pi(-) -> 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: nu = nu(e) or (nu) overbar(e)); if N is Majorana, both LC and lepton number violating (LV) decays contribute (LV: nu = (nu) overbar(mu) or nu = nu(mu)). The results show that the CP asymmetry A(CP) is in general very small, but increases and becomes similar to 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, Delta MN >> Gamma(N). The observation of CP violation in pion decays would be consistent with the existence of the well-motivated nu MSM model with two almost degenerate heavy neutrinos.