Direct CP Violation in $B \to \pi^{+} \pi^{-} \pi$: Determination of   $\alpha$ without discrete ambiguity
  Direct CP violation in the hadronic decays $\Bar{B}^{0} \to \pi^{+}\pi^{-} \pi^{0}$ is investigated near the peak of the $\rho^{0}$ 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 $\Bar{B}^{0} \to \rho^{0}(\omega) \pi^{0}$ when the invariant mass of the $\pi^{+}\pi^{-}$ 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 $\Bar{B}^{0} \to \pi^{+}\pi^{-} \pi^{0}$ would remove the mod$(\pi)$ ambiguity in the determination of the CP violating phase angle $\alpha$. 
O. Leitner, X.-H. Guo, A.W. Thomas