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Reexamining $ B\to ππ,πK$ Decays in QCD Factorization Approach

Motivated by the recent experimental data, we have revisited the $B\to πK,ππ$ decays in the framework of QCD factorization, with inclusion of the important strong penguin corrections of order $α_s^2$ 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 $B\to πK$ decay rates, and such an enhancement can improve the consistency between the theoretical predictions and the experimental data significantly, while for the tree-dominated $B\to ππ$ 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 $πK$ 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}.