Glauber gluons in spectator amplitudes forB→πMdecays

Abstract

We extract the Glauber divergences from the spectator amplitudes for two-body hadronic decays $B\ensuremath{\rightarrow}{M}_{1}{M}_{2}$ 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 simultaneous roles as both a $q\overline{q}$ 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 $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$, $\ensuremath{\pi}\ensuremath{\rho}$, $\ensuremath{\pi}\ensuremath{\omega}$, and $\ensuremath{\pi}K$ data, and resolve the long-standing $\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}K$ puzzles. The nontrivial success of this modified pQCD formalism is elaborated.