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In this work, we propose a model by combining the parameterization method andthe QCD factorization to study the $B\to\pi K$ puzzle. The parameterizationintroduces twelve parameters, the weak angle $\gamma$ and the other elevenhadronic parameters. These hadronic parameters are calculated by the QCDfactorization (QCDF). The calculational accuracy of the QCDF is improved byincluding the twist-3 three parton tree and one loop corrections. Threeadditional nonperturbative strong phases from the gluonic penguin, colorsuppressed, and color favored tree diagrams are required to account for thedirect CP asymmetries. The weak angle, three nonperturbative strong phases, andfour scale variables are assumed as fitting parameters. These eight parametersare determined by a least squares fit to the eight measurements. The fit showsthat the hadronic parameters need to be process-dependent. Theprocess-dependent hadronic parameters break the isospin symmetry `dynamically'.A large negative relative phase associated with the internal up quark gluonicpenguin diagram is observed. The weak angle $\gamma$ is found to be$72.1\pm5.8$ degree, being consistent with the world averaged value,$72.1\pm5.8$ degree. By a least squares fit to the mixing induced asymmetry$S(\pi^{0}K_{S})=0.58\pm0.06$, the weak angle $\beta$ is determined to be$22.9\pm2.1$ degree, which is in a well agreement with the world averagedvalue, $22.5\pm4.4\pm0.6$ degree. The model predicts the ratio$C^{\prime}/T^{\prime}$ (color suppressed/color favored) to be $0.59\pm0.08$.The above evidences indicate that the model can solve the $B\to\pi K$ puzzleconsistently.

Minimal QCDF Model for $B\to\pi K$ Puzzle