We point out that the $B\to \rho \rho$ data have seriously constrained the possibility of resolving the $B\to \pi \pi$ puzzle from the large observed $B^0\to {\pi }^0{\pi }^0$ branching ratio in the available theoretical approaches. The next-to-leading-order (NLO) contributions from the vertex corrections, the quark loops, and the magnetic penguin evaluated in the perturbative QCD (PQCD) approach have saturated the experimental upper bound of the $B^0\to {\rho }^0{\rho }^0$ branching ratio and do not help. The NLO PQCD predictions for the $B^0\to {\rho }^{\mp }{\rho }^{\pm }$ and $B^{\pm }\to {\rho }^{\pm }{\rho }^0$ branching ratios are consistent with the data. The inclusion of the NLO jet function from the soft-collinear effective theory into the QCD-improved factorization approach, though enhancing the $B^0\to {\pi }^0{\pi }^0$ branching ratio sufficiently, overshoots the bound of the $B^0\to {\rho }^0{\rho }^0$ branching ratio and deteriorates the predictions for the $B^{\pm }\to {\pi }^0{K}^{\pm }$ and $B^0\to {\pi }^{\mp }{K}^{\pm }$ direct $CP$ asymmetries.

• Received 5 March 2006

DOI:https://doi.org/10.1103/PhysRevD.73.114014