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Motivated by the recent experimental data, we have revisited the $B\to \piK,\pi \pi$ decays in the framework of QCD factorization, with inclusion of theimportant strong penguin corrections of order $\alpha_s^2$ induced by $b\to Dg^\ast g^\ast$ ($D=d$ or $s$ and $g^\ast$ denotes an off-shell gluon)transitions. We find that these higher order strong penguin contributions canprovide $\sim 30%$ enhancement to the penguin-dominated $B\to \pi K$ decayrates, and such an enhancement can improve the consistency between thetheoretical predictions and the experimental data significantly, while for thetree-dominated $B\to \pi\pi$ decays, these higher order contributions play onlya minor role. When these strong penguin contributions are summed, only a smallstrong phase remains and the direct CP asymmetries get small corrections. Wealso find patterns of the ratios between the CP-averaged branching fractionsremain nearly unaffected even after including these higher order correctionsand the $\pi K$ puzzle still persists. Our results may indicate that to resolvethe puzzle one would have to resort to new physics contributions in theelectroweak penguin sector as found by Buras {\it et al}.Comment: 43 pages, 19 eps figures. Typos corrected. Final version published in  PRD72, 074007(2005

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Reexamining<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo>→</mml:mo><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>K</mml:mi></mml:math>decays in QCD factorization approach

ReexaminingB→ππ,πKdecays in QCD factorization approach