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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 Glauberphase factor with transverse-momentum-dependent(TMD) meson wave functions inthe $k_T$ factorization theorem. For an appropriate parametrization of theGlauber phase, it is observed that a TMD wave function for the pion ($\rho$meson) with a weak (strong) falloff in parton transverse momentum $k_T$ leadsto 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) byabout a factor of 2 (15\%). This observation is consistent with the dual roleof the pion as a massless Nambu-Goldstone boson and as a $q \bar q$ boundstate, which requires a tighter spatial distribution of its leading Fock staterelative to higher Fock states. The agreement between the theoreticalpredictions 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 resolvethe $B \to \pi \pi$ puzzle.

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