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

Abstract

We study the pion decays with intermediate on-shell neutrinos $N$ into two electrons and a muon, ${\pi }^{\pm }\to e^{\pm }N\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu$. We investigate the branching ratios ${\mathrm{Br}}_{\pm }=[\Gamma ({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )\pm \Gamma ({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )]/\Gamma ({\pi }^{-}\to \mathrm{all})$ and the CP asymmetry ratio ${\mathcal{A}}_{\mathrm{CP}}={\mathrm{Br}}_{-}/{\mathrm{Br}}_{+}$ 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 ${\bar{\nu }}_e$); if $N$ is Majorana, both LC and lepton number violating (LV) decays contribute (LV: $\nu ={\bar{\nu }}_{\mu }$ or $\nu ={\nu }_{\mu }$). The results show that the CP asymmetry ${\mathcal{A}}_{\mathrm{CP}}$ 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, $\Delta M_N/\gg {\Gamma }_N$. 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.

Comments: 16 pages, 6 figures, v4: new references ([17,18]) added, and ref.[9] extended; the version to appear in J.Phys.G