Resolution to theB→πKpuzzle

Abstract

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