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 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 $B\to \pi \pi$, $\pi \rho$, $\pi \omega$, 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.

Figure 1

LO diagrams for a spectator amplitude.

Figure 2

NLO diagrams for Fig. 1 that are relevant to the factorization of the $M_2$ meson wave function. Figures 2 contribute to the Glauber divergences.

Figure 3

More NLO diagrams for Fig. 1.

Figure 4

NLO diagrams for Fig. 1 that contribute to the Glauber divergences associated with the $M_2$ meson.

Figure 5

(a)–(c) Higher-order corrections to Fig. 1 that contain the Glauber divergences associated with the $M_1$ meson. (d) Higher-order correction to Fig. 1 that does not contain the Glauber divergence associated with the $M_1$ meson.

Figure 6

$S_{e1}$ and $S_{e2}$ dependencies of the amplitudes $C$ and $T$, and their ratio $C/T$ for the $B\to \pi \pi$ decays.

Figure 7

$S_{e1}$ and $S_{e2}$ dependencies of the amplitudes $C$ and $T$, and their ratio $C/T$ for the $B\to \pi \rho$ and $\pi K$ decays.

Figure 8

$S_{e1}$ and $S_{e2}$ dependencies of the $B\to \pi \pi$ branching ratios (in units of $10^{-6}$) and direct $CP$ asymmetries.

Figure 9

$S_{e1}$ and $S_{e2}$ dependencies of the $B\to \pi \pi$ mixing-induced $CP$ asymmetries, and $\mathrm{\Delta }{\chi }^2$.

Figure 10

$S_{e1}$ and $S_{e2}$ dependencies of the $B\to \pi \rho$ branching ratios (in units of $10^{-6}$), direct $CP$ asymmetries, and mixing-induced $CP$ asymmetry.

Figure 11

$S_{e1}$ and $S_{e2}$ dependencies of the $B\to \pi \omega$ branching ratios (in units of $10^{-6}$), direct $CP$ asymmetries, and mixing-induced $CP$ asymmetry.

Figure 12

$S_{e1}$ and $S_{e2}$ dependencies of the $B\to \pi K$ branching ratios (in units of $10^{-6}$), direct $CP$ asymmetries, and mixing-induced $CP$ asymmetry.

Figure 13

$S_{e1}$ and $S_{e2}$ dependencies of $\mathrm{\Delta }{\chi }^2$ for all the $B\to \pi \pi$, $\pi \rho$, $\pi \omega$, and $\pi K$ decays. The difference of $\mathrm{\Delta }{\chi }^2$ for each considered quantity due to the inclusion of the Glauber effects is also displayed.

Figure 14

NLO diagrams for Fig. 1 that are relevant to the factorization of the $M_1$ meson wave function.