Glauber gluons in spectator amplitudes for $B \to \pi M$ decays

We extract the Glauber divergences from the spectator amplitudes for two-body hadronic decays $B \to M_1 M_2$ 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 $M_1 M_2$ 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 special role as a $q \bar q$ bound state and as a pseudo Nambu-Goldstone boson simultaneously. Treating the Glauber phases as additional inputs in the perturbative QCD (PQCD) approach, we find a good fit to all the $B \to \pi\pi$, $\pi\rho$, $\pi\omega$, 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.