QCD factorization at twist-3: the two parton contributions

In this paper, the twist-3 two parton corrections in charmless $B\to PP$ decays are shown to be factorizable under the QCD factorization approach. The factorizability of the twist-3 two parton corrections is constructed on the following findings. Under the energetic meson limit, the pseudoscalar distribution amplitude for a light pseudoscalar meson is allowed to be non-constant by the equations of motion for the quark. The non-constant pseudoscalar distribution amplitude is then used to regularize the end-point divergences in the hard spectator corrections at twist-3 order. By retaining the momentum fraction variable of the spectator quark of the $B$ meson in the propagators, the end-point divergence in the weak annihilation corrections at twist-3 order is resolved. The factorization of the $O(\alpha_s)$ corrections under the two parton approximation is shown valid up-to $O(1/m_b)$ . The hard scattering kernels of order $O(\alpha_s)$ and $O(\Lambda_{QCD}/m_b)$ are explicitly given and found to be infrared finite. The results are applied for making predictions for the branching ratios of $B\to \pi K $ decays.