Possible resolution of theB→ππ,πKpuzzles

We show that there exist uncanceled soft divergences in the $k_T$ factorization for nonfactorizable amplitudes of two-body nonleptonic $B$ meson decays, similar to those identified in hadron hadroproduction. These divergences can be grouped into a soft factor using the eikonal approximation, which is then treated as an additional nonperturbative input in the perturbative QCD formalism. Viewing the special role of the pion as a $q\overline{q}$ bound state and as a pseudo Nambu-Goldstone boson, we postulate that the soft effect associated with it is significant. This soft factor enhances the nonfactorizable color-suppressed tree amplitudes, such that the branching ratios $B({\pi }^0{\pi }^0)$ and $B({\pi }^0{\rho }^0)$ are increased under the constraint of the $B({\rho }^0{\rho }^0)$ data, the difference between the direct $CP$ asymmetries $A_{CP}({\pi }^{\mp }{K}^{\pm })$ and $A_{CP}({\pi }^0{K}^{\pm })$ is enlarged, and the mixing-induced $CP$ asymmetry $S_{{\pi }^0{K}_S}$ is reduced. Namely, the known $\pi \pi$ and $\pi K$ puzzles can be resolved simultaneously. DOI: http://dx.doi.org/10.1103/PhysRevD.83.034023 © 2011 American Physical Society