Direct CP Violation in $B^{\pm }\to {\rho }^{\pm }{\pi }^{+}{\pi }^{-}$ in the ${\rho }^0$-$\omega$ Interference Region

We study direct CP violation in $B^\pm \to \rho^\pm\rho^0(\omega) \to \rho^\pm\pi^+\pi^-$andfocusspecificallyontherateasymmetryinthe$\rho^0$-$\omega$ interference region. Here the strong phase is dominated by isospin violation, so that it can be essentially determined by $e^+e^- \to \rho^0 (\omega) \to \pi^+\pi^-$ data. We find the CP-violating asymmetry to be of the order of 20% at the $\omega$ invariant mass. Moreover, it is robust with respect to the estimable strong-phase uncertainties, permitting the extraction of $\sin \alpha$ from this channel.