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

Direct CP violation in the hadronic decays $\Bar{B}^{0} \to \pi^{+}\pi^{-} \pi^{0}$isinvestigatednearthepeakofthe$\rho^{0}$ taking into account the effect of $\rho -\omega$ mixing. Branching ratios for processes $B^{\pm,0} \to \rho^{\pm,0}\pi^{\pm,0}$and$B^{-} \to \omega \pi^{-}$ 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 $\Bar{B}^{0} \to \rho^{0}(\omega) \pi^{0}$whentheinvariantmassofthe$\pi^{+}\pi^{-}$ pair is in the vicinity of the $\omega$ 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 $\sin \delta$ is always positive. Thus, a measurement of direct CP violation in decays $\Bar{B}^{0} \to \pi^{+}\pi^{-} \pi^{0}$wouldremovethemod$(\pi)$ ambiguity in the determination of the CP violating phase angle $\alpha$.