Direct CP violation in B to rho⁰(ω) PS

We calculate the direct CP violating asymmetry parameter, $a_{CP}$, for $B \to \pi^{+} \pi^{-} \pi$ and $B \to \pi^{+} \pi^{-} K$ decays, in the case where $\rho^0-\omega$ mixing effects are taken into account. We find that the direct CP asymmetry for $B^{-} \to \pi^{+} \pi^{-} \pi^{-}$, $\bar{B}^{0} \to \pi^{+} \pi^{-}\pi^{0}$, $B^{-} \to \pi^{+} \pi^{-} K^{-}$ and $\bar{B}^{0} \to \pi^{+} \pi^{-}\bar{K}^{0}$, reaches its maximum when the invariant mass $\pi^{+} \pi^{-}$ is in the vicinity of the $\omega$ meson mass. The inclusion of $\rho^0-\omega$ mixing provides an opportunity to erase, without ambiguity, the phase uncertainty mod$(\pi)$ in the determination of the CKM angles $\alpha$ in case of $b\to u$ and $\gamma$ in case of $b \to s$.