Glauber gluons in spectator amplitudes for B→πM decays

We extract the Glauber divergences from the spectator amplitudes for two-body hadronic decays B→M₁M₂ in the k_T factorization theorem, where M₂ denotes the meson emitted at the weak vertex. Employing the eikonal approximation, the divergences are factorized into the corresponding Glauber phase factors associated with the M₁ and M₂ mesons. It is observed that the latter factor enhances the spectator contribution to the color-suppressed tree amplitude by modifying the interference pattern between the two involved leading-order diagrams. The first factor rotates the enhanced spectator contribution by a phase, and changes its interference with other tree diagrams. The above Glauber effects are compared with the mechanism in elastic rescattering among various M₁M₂ final states, which has been widely investigated in the literature. We postulate that only the Glauber effect associated with a pion is significant, due to its simultaneous roles as both a qq ¯ bound state and a pseudo-Nambu-Goldstone boson. Treating the Glauber phases as additional inputs in the perturbative QCD (pQCD) approach, we find a good fit to all the B→ππ, πρ, πω, and πK data, and resolve the long-standing ππ and πK puzzles. The nontrivial success of this modified pQCD formalism is elaborated.