$B\to K\pi$ Decays with $1/m_b$ Corrections in $QCD$ Factorization

It is commonly believed that a careful investigation of the subleading terms is crucial for a better understanding of the $QCD$ factorization in charmless B decays. In this work the penguin-dominated $B\to K\pi$ decays are discussed systematically, including the subleading corrections in $1/m_b$ due to soft and hard gluons, besides the annihilation contributions. Soft-gluon effects for all the relevant 4-quark effective operators are calculated within the framework of the light-cone QCD sum rules (LCSR). Our observation is that such soft and hard corrections are less important than the annihilation effects, enhancing only the branching ratios by a few percent; the resultant increase in the branching ratios due to the overall ${\cal O}(1/m_b)$ effects is between about $(22-27)$ of the QCD factorization results with the ${\cal O}(\alpha_s)$ corrections, as the weak phase $\gamma (=\textmd{Im}V_{ub}^*)$ ranges from $40^0$ to $80^0$. Impacts of the involved uncertainties are discussed in some details.