Resolution to theB→πKpuzzle

We calculate the important next-to-leading-order contributions to the $B\to \pi K$, $\pi \pi$ 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 $B\to \pi K$ 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 $B^0\to {\pi }^{\mp }{K}^{\pm }$ and $B^{\pm }\to {\pi }^0{K}^{\pm }$ modes. The puzzle from the large $B^0\to {\pi }^0{\pi }^0$ branching ratio still remains.