Resolution to the B -> pi K puzzle
  We calculate the important next-to-leading-order contributions to the B -> pi K, pi pi decays from the vertex corrections, the quark loops, and the magnetic penguins in the perturbative QCD approach. It is found that the latter two reduce the leading-order penguin amplitudes by about 10%, and modify only the B -> pi K branching ratios. The main effect of the vertex corrections is to increase the small color-suppressed tree amplitude by a factor of 3, which then resolves the large difference between the direct CP asymmetries of the B^0 -> pi^\mp K^\pm and B^\pm -> pi^0 K^\pm modes. The puzzle from the large B^0 -> pi^0 pi^0 branching ratio still remains. 
Hsiang-nan Li, Satoshi Mishima, A.I. Sanda