Transverse-momentum-dependent wave functions with Glauber gluons in $B\to \pi \pi$, $\rho \rho$ decays

Abstract

We investigate the Glauber-gluon effect on the $B\to \pi \pi$ and $\rho \rho$ decays, which is introduced via a convolution of a universal Glauber phase factor with transverse-momentum-dependent(TMD) meson wave functions in the $k_T$ factorization theorem. For an appropriate parametrization of the Glauber phase, it is observed that a TMD wave function for the pion ($\rho$ meson) with a weak (strong) falloff in parton transverse momentum $k_T$ leads to significant (moderate) modification of the $B^0\to {\pi }^0{\pi }^0$ ($B^0\to {\rho }^0{\rho }^0$) branching ratio: the former (latter) is enhanced (reduced) by about a factor of 2 (15\%). This observation is consistent with the dual role of the pion as a massless Nambu-Goldstone boson and as a $q\bar{q}$ bound state, which requires a tighter spatial distribution of its leading Fock state relative to higher Fock states. The agreement between the theoretical predictions and the data for all the $B\to \pi \pi$ and ${\rho }^0{\rho }^0$ branching ratios is then improved simultaneously, and it is possible to resolve the $B\to \pi \pi$ puzzle.

Comments: RevTex file, 11 pages, 4 figures, and 1 table, contents improved and references added; matching the published version