QCD factorization for B ---> P P

In this work, we give a detailed discussion for QCD factorization involved in the complete chirally enhanced power corrections in the heavy quark limit for B decays to two light pseudoscalar mesons, and present some detailed calculations of radiative corrections at the order of ${\ensuremath{\alpha}}_{s}.$ We point out that the infrared finiteness of the vertex corrections in the chirally enhanced power corrections requires twist-3 light-cone distribution amplitudes (LCDAs) of the light pseudoscalar symmetric. However, even in the symmetric condition, there is also a logarithmic divergence from the end points of the twist-3 LCDAs in the hard spectator scattering. We point out that the decay amplitudes of $\stackrel{\ensuremath{\rightarrow}}{B}\mathrm{PP}$ predicted by QCD factorization are really free of the renormalization scale dependence, at least at the order of ${\ensuremath{\alpha}}_{s}.$ At last, we briefly compare the QCD factorization with the generalized factorization and PQCD method.