$B\to \pi \pi$ decays and effects of the next-to-leading order contributions

In this paper we perform a systematic study for the three $B\to ({\pi }^{+}{\pi }^{-},{\pi }^{+}{\pi }^0,{\pi }^0{\pi }^0)$ decays in the perturbative QCD (pQCD) factorization approach with the inclusion of all currently known next-to-leading order (NLO) contributions from various sources. We found that (a) for the CP-averaged decay rates $Br(B^0\to {\pi }^{+}{\pi }^{-})$ and $Br(B^{+}\to {\pi }^{+}{\pi }^0)$, the NLO pQCD predictions agree with the data within one standard deviation; (b) for $Br(B^0\to {\pi }^0{\pi }^0)$, however, although the NLO contributions can provide a $\sim 100\%$ enhancement to the leading-order (LO) result, it is still not large enough to interpret the data; and (c) for the CP-violating asymmetries of $B^0\to {\pi }^{+}{\pi }^{-}$ decay, the NLO pQCD predictions for the central values of ${\mathcal{A}}_{\pi \pi }$ and ${\mathcal{S}}_{\pi \pi }$ have the same sign as the data but are still smaller in magnitude than the measured values. We also examined the relative strength of the LO and NLO contributions from different sources.

Figure 1

Feynman diagrams that may contribute to the $B\to \pi \pi$ decays in the pQCD approach at leading order.