Direct CP violation in B \to ρ^0(ω) PS

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

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