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Study of the four-body decays $B^{0}_{S} \rightarrow ππππ$ in the perturbative QCD approach

In this work, we analyse the CP-averaged branching ratios and direct CP-violating asymmetries of the four-body decays B_{S} \rightarrow ππππdecay from the S-wave resonances, f_{0}(980) and f_{0}(500) and P-wave resonances, ρ(770) by introducing the S-wave and P-wave ππdistribution amplitudes within the framework of the perturbative QCD approach. We also calculate branching ratios of the two-body decays B^{0}_{S}\rightarrowρ^{0}ρ^{0}, B^{0}_{S}\rightarrowρ^{+}ρ^{-} from the corresponding quasi-two-body decays models and compare our results with those obtained in previous perturbative QCD approach, QCD factorization approach and FAT approach, it is found that the predictions are consistent with present data within errors. The branching ratios of our calculations for the four-body decays B_{S}\rightarrow ππππare at the order of the 10^{-7}. For the CP-violating asymmetries, we found that CP-violating asymmetry can be enhanced largely by the ρ-ωmixing resonances when ππpairs masses are in the vicinity of ωresonance.