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

Abstract

Direct CP violation in the hadronic decays $\text{Bar}{B}^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ 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 $\text{Bar}{B}^0\to {\rho }^0(\omega ){\pi }^0$ 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 $\text{Bar}{B}^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ would remove the mod$(\pi )$ ambiguity in the determination of the CP violating phase angle $\alpha$.

Comments: 23 pages, 10 figures, 5 tables. Misprints corrected and two figures updated