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 M-2 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 M-1 and M-2 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 q (q) over bar 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 -> pi pi, pi rho, pi omega, and pi K data, and resolve the long-standing pi pi and pi K puzzles. The nontrivial success of this modified pQCD formalism is elaborated.