Direct CP Violation in $B \to π^{+} π^{-} π$: Determination of $α$ without discrete ambiguity

Direct CP violation in the hadronic decays $\Bar{B}^{0} \to π^{+}π^{-} π^{0}$ is investigated near the peak of the $ρ^{0}$ taking into account the effect of $ρ- ω$ mixing. Branching ratios for processes $B^{\pm,0} \to ρ^{\pm,0}π^{\pm,0}$ and $B^{-} \to ωπ^{-}$ are calculated as well. We find that the CP violating asymmetry is strongly dependent on the CKM matrix elements. For a fixed $N_{c}^{eff}$, the CP violating asymmetry, $a$, has a maximum of order -40% to -70% for $\Bar{B}^{0} \to ρ^{0}(ω) π^{0}$ when the invariant mass of the $π^{+}π^{-}$ pair is in the vicinity of the $ω$ resonance. The sensitivity of the asymmetry to $N_{c}^{eff}$ is small in that case. Moreover, we find that in the range of $N_{c}^{eff}$ which is allowed by the most recent experimental branching ratios from the BABAR, BELLE and CLEO Collaborations, the sign of $\sin δ$ is always positive. Thus, a measurement of direct CP violation in decays $\Bar{B}^{0} \to π^{+}π^{-} π^{0}$ would remove the mod$(π)$ ambiguity in the determination of the CP violating phase angle $α$.

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