Study of four-body decays $B_{(s)} \to (ππ)(ππ)$ in the perturbative QCD approach

Abstract

In this work, we make a systematical study on the four-body $B_{(s)} \to (ππ)(ππ)$ decays in the perturbative QCD approach, where the $ππ$ invariant mass spectra are dominated by the vector resonance $ρ(770)$ 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 $B_{(s)} \to (ππ)(ππ)$ decays. We extract the branching ratios of two-body $B_{(s)} \to ρρ$ from the corresponding four-body decay modes and calculate the relevant polarization fractions. We find that the ${\cal B}(B^0 \to ρ^+ρ^-)$ is consistent with the previous theoretical predictions and data. The leading-order PQCD calculations of the ${\cal B}(B^+\to ρ^+ρ^0)$, ${\cal B}(B^0\to ρ^0ρ^0)$ and the $f_0(B^0\to ρ^0ρ^0)$ 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 $B_{(s)}\to (ππ)(ππ)$ decays are also analyzed. The sizable averaged TPA ${\cal A}_{\text{T-true}}^{1, \text{ave}}=25.26\%$ of the color-suppressed decay $B^0\to ρ^0ρ^0 \to (π^+π^-)(π^+π^-)$ is predicted for the first time, which deviates a lot from the so-called "true" TPA $\mathcal{A}_\text{T-true}^1=7.92\%$ due to the large direct $CP$ violation. A large "fake" TPA $\mathcal{A}_\text{T-fake}^1=24.96\%$ of the decay $B^0\to ρ^0ρ^0 \to (π^+π^-)(π^+π^-)$ 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.