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

Abstract

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.

22 pages, 14 figures. v2: minor corrections, version to appear in Phys. Rev. D

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