Study of four-body decays B_(s) ( )( ) in the perturbative QCD approach

In this work, we make a systematical study on the four-body B((s))->(pi pi)(pi pi) decays in the perturbative QCD approach, where the pi pi invariant mass spectra are dominated by the vector resonance rho(770) and the scalar resonance f(0)(980). We improve the Gengenbauer moments for the longitudinal P-wave two-pion distribution amplitudes (DAs) by fitting the PQCD factorization formulas to measured branching ratios of three-body and four-body B decays. With the fitted Gegenbauer moments, we make predictions for the branching ratios and direct CP asymmetries of four-body B(s)->(pi pi)(pi pi) decays. We extract the branching ratios of two-body B-(s)->rho rho from the corresponding four-body decay modes and calculate the relevant polarization fractions. We find that the B(B0 ->rho+rho-) is consistent with the previous theoretical predictions and data. The leading-order PQCD calculations of the B(B+->rho+rho(0)), B(B0 ->rho 0 rho 0) and the f0(B0 ->rho(0)rho 0) are a bit lower than the experimental measurements, which should be further examined. In addition, the "true" and "fake" triple-product asymmetries (TPAs) in the B(s)->(pi pi)(pi pi) decays are also analyzed. The sizable averaged TPA A1,aveT-true=25.26% of the color-suppressed decay B-0 ->rho(0)rho(0)->(pi+pi-)(pi+pi-) is predicted for the first time, which deviates a lot from the so-called "true" TPA A1T-true=7.92% due to the large direct CP violation. A large "fake" TPA A1T-fake=24.96% of the decay B0 ->rho 0 rho 0 ->(pi+pi-)(pi+pi-) is also found, which indicates the significance of the final-state interactions. The predictions in this work can be tested by LHCb and Belle-II experiments in the near future.