Possible solution of the puzzle for the branching ratio and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math> violation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math> decays with a modified perturbative QCD approach

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