Possible solution of the puzzle for the branching ratio and $CP$ violation in $B\to ππ$ decays with a modified perturbative QCD approach

Abstract: We study $B\to \pi\pi$ decay with a modified perturbative QCD approach. The branching ratios and $CP$ violation are calculated with the transverse momenta of partons considered. Sudakov factor associated with each meson is included to suppress soft contribution in QCD. With the wave function of $B$ meson obtained in QCD-inspired relativistic potential model being used, the suppression of Sudakov factor to the soft contribution is not effective enough. Soft scale cutoff and soft form factors of $B\pi$ transition and $\pi\pi$ production have to be introduced. The main next-to-leading-order contributions of vertex correction, the quark-loop and magnetic penguin are included. To solve the long-standing puzzle in $B\to\pi\pi$ decay, that is the theoretical prediction of the branching ratio of $B^0\to \pi<sup>0\pi<sup>0$ being seriously smaller than experimental data, color-octet matrix element which is of long-distance dynamics is introduced. With parameters taken with reasonable values, all the branching ratios and $CP$ violation are well consistent with experimental data.