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

In this paper we perform a systematic study for the three $B \to (π^+π^-,π^+π^0,π^0π^0)$ 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 $Br(B^0\to π^+π^-)$ and $Br(B^+\to π^+π^0)$, the NLO pQCD predictions agree with the data within one standard dviation; (b) for $Br(B^0\to π^0π^0)$, 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; (c) for the CP-violating asymmetries of $B^0\to π^+π^-$ decay, the central values of the NLO PQCD predictions agree with the data; and (d) we also examined the relative strength of the LO and NLO contributions from different sources.