QCD Factorization Based on Six-Quark Operator Effective Hamiltonian from   Perturbative QCD and Charmless Bottom Meson Decays $B_{(s)}\to \pi\pi,\pi K,   KK$
  The charmless bottom meson decays are systematically investigated based on an approximate six quark operator effective Hamiltonian from perturbative QCD. It is shown that within this framework the naive QCD factorization method provides a simple way to evaluate the hadronic matrix elements of two body mesonic decays. The singularities caused by on mass-shell quark propagator and gluon exchanging interaction are appropriately treated. Such a simple framework allows us to make theoretical predictions for the decay amplitudes with reasonable input parameters. The resulting theoretical predictions for all the branching ratios and CP asymmetries in the charmless $B^0, B^+, B_s\to \pi\pi, \pi K, KK$ decays are found to be consistent with the current experimental data except for a few decay modes. The observed large branching ratio in $B\to \pi^0\pi^0$ decay remains a puzzle though the predicted branching ratio may be significantly improved by considering the large vertex corrections in the effective Wilson coefficients. More precise measurements of charmless bottom meson decays, especially on CP-violations in $B\to K K$ and $B_s\to \pi\pi, \pi K, KK$ decay modes, will provide a useful test and guide us to a better understanding on perturbative and nonperturbative QCD. 
Fang Su, Yue-Liang Wu, Yi-Bo Yang and Ci Zhuang