**31**: 215-226

https://doi.org/10.1140/epjc/s2003-01332-0

theoretical physics

## Direct *CP* violation in $B \rightarrow \pi^{ + }\pi^{-}\pi$: determination of $\alpha$ without discrete ambiguity

^{1}
Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, 5005, Adelaide, Australia

^{2}
Laboratoire de Physique Corpusculaire, Université Blaise Pascal, CNRS/IN2P3, 24 avenue des Landais, 63177, Aubiére Cedex, France

^{*} e-mail: oleitner@physics.adelaide.edu.au

Direct *CP* violation in the hadronic decays $\Bar{B}^{0} \rightarrow \pi^{ + }\pi^{-} \pi^{0}$ is investigated near the peak of the $\rho^{0}$, taking into account the effect of $\rho $-$ \omega$ mixing. The branching ratios for the processes $B^{\pm,0} \rightarrow \rho^{\pm,0}\pi^{\pm,0}$ and $B^{-} \rightarrow \omega \pi^{-}$ 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*
_{
CP
}, has a maximum of order $-40\%$ to $- 70\%$ for $\Bar{B}^{0} \rightarrow \rho^{0}(\omega) \pi^{0}$ 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 the decays $\Bar{B}^{0} \rightarrow \pi^{ + }\pi^{-} \pi^{0}$ would remove the $\mod(\pi)$ ambiguity in the determination of the *CP* violating phase angle $\alpha$.

*© Springer-Verlag, 2003*