Reexamining $B\to \pi \pi ,\pi K$ Decays in QCD Factorization Approach

Abstract

Motivated by the recent experimental data, we have revisited the $B\to \pi K,\pi \pi$ decays in the framework of QCD factorization, with inclusion of the important strong penguin corrections of order ${\alpha }_s^2$ induced by $b\to D{g}^{\ast }{g}^{\ast }$ ($D=d$ or $s$ and $g^{\ast }$ denotes an off-shell gluon) transitions. We find that these higher order strong penguin contributions can provide $\sim 30$ enhancement to the penguin-dominated $B\to \pi K$ decay rates, and such an enhancement can improve the consistency between the theoretical predictions and the experimental data significantly, while for the tree-dominated $B\to \pi \pi$ decays, these higher order contributions play only a minor role. When these strong penguin contributions are summed, only a small strong phase remains and the direct CP asymmetries get small corrections. We also find patterns of the ratios between the CP-averaged branching fractions remain nearly unaffected even after including these higher order corrections and the $\pi K$ puzzle still persists. Our results may indicate that to resolve the puzzle one would have to resort to new physics contributions in the electroweak penguin sector as found by Buras {\it et al}.

Comments: 43 pages, 19 eps figures. Typos corrected. Final version published in PRD72, 074007(2005)