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Direct CP Violation, Branching Ratios and Form Factors $B \to π$, $B \to K$ in $B$ Decays

The $B \to π$ and $B \to K$ transitions involved in hadronic B decays are investigated in a phenomenological way through the framework of QCD factorization. By comparing our results with experimental branching ratios from the BELLE, BABAR and CLEO Collaborations for all the B decays including either a pion or a kaon, we propose boundaries for the transition form factors $B \to π$ and $B \to K$ depending on the CKM matrix element parameters $ρ$ and $η$. From this analysis, the form factors required to reproduce the experimental data for branching ratios are $F^{B \to π}= 0.31 \pm 0.12$ and $F^{B \to K}= 0.37\pm 0.13$. 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 $ρ-ω$ mixing effects are taken into account. Based on these results, 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 $ρ-ω$ 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 $b \to s$.