The three body decays B+→π+π−π+B+→π+π−π+ in perturbative QCD approach

In this work, we study three body decays B+→π+π−π+B+→π+π−π+ in the perturbative QCD (pQCD) approach. The analysis is simplified into the one for two-body decays by introducing two-meson distribution amplitudes. We calculate both nonresonant and resonant contributions, and show the contribution to the B+→π+π−π+B+→π+π−π+ decays spectrum with respect to the dipion invariant mass M(π+π−)M(π+π−). Besides the usual factorizable diagrams, the non-factorizable and annihilation diagrams are also calculated analytically. The pQCD predictions for the branching ratios of B+→π+π−π+B+→π+π−π+ (nonresonant) are Br=5.7×10−6Br=5.7×10−6, the results of the resonant contributions are Br(B+→π+ρ(770))=8.9×10−6Br(B+→π+ρ(770))=8.9×10−6. The above pQCD predictions agree well with current available data. We also predict another resonant contribution Br(B+→π+f0(980))=7.3×10−6Br(B+→π+f0(980))=7.3×10−6, but with large error. We still do not have a clear understanding about the inner structure of the scalar mesons f0f0. Further improvement in theory and experiments are expected.