Branching ratio andCPviolation ofB→ππdecays in the perturbative QCD approach

We calculate the branching ratios and $\mathrm{CP}$ asymmetries for $B^0\to {\pi }^{+}{\pi }^{-},\hspace{0.5em}{B}^{+}\to {\pi }^{+}{\pi }^0,$ and $B^0\to {\pi }^0{\pi }^0$ decays, in a perturbative QCD approach. In this approach, we calculate nonfactorizable and annihilation type contributions, in addition to the usual factorizable contributions. We find that the annihilation diagram contributions are not very small as previously argued. Our result is in agreement with the measured branching ratio of $\overrightarrow{B}{\pi }^{+}{\pi }^{-}$ by the CLEO Collaboration. With a non-negligible contribution from annihilation diagrams and a large strong phase, we predict a large direct $\mathrm{CP}$ asymmetry in $B^0\to {\pi }^{+}{\pi }^{-},$ and ${\pi }^0{\pi }^0,$ which can be tested by the current running B factories.