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

Abstract

The $B\to \pi$ 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 $B\to \pi$ and $B\to K$ depending on the CKM matrix element parameters $\rho$ and $\eta$. From this analysis, the form factors required to reproduce the experimental data for branching ratios are $F^{B\to \pi }=0.31\pm 0.12$ and $F^{B\to K}=0.37\pm 0.13$. We calculate the direct CP violating asymmetry parameter, $a_{CP}$, for $B\to {\pi }^{+}{\pi }^{-}\pi$ and $B\to {\pi }^{+}{\pi }^{-}K$ decays, in the case where $\rho -\omega$ mixing effects are taken into account. Based on these results, 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 $\rho -\omega$ 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 $b\to s$.

Comments: 74 pages, 15 figures, 8 tables. A few misprints corrected, two references added