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In this work, we make a systematical study on the four-body $B_{(s)} \to(\pi\pi)(\pi\pi)$ decays in the perturbative QCD approach, where the $\pi\pi$invariant mass spectra are dominated by the vector resonance $\rho(770)$ andthe scalar resonance $f_0(980)$. We improve the Gengenbauer moments for thelongitudinal $P$-wave two-pion distribution amplitudes (DAs) by fitting thePQCD factorization formulas to measured branching ratios of three-body andfour-body $B$ decays. With the fitted Gegenbauer moments, we make predictionsfor the branching ratios and direct $CP$ asymmetries of four-body $B_{(s)} \to(\pi\pi)(\pi\pi)$ decays. We extract the branching ratios of two-body $B_{(s)}\to \rho\rho$ from the corresponding four-body decay modes and calculate therelevant polarization fractions. We find that the ${\cal B}(B^0 \to\rho^+\rho^-)$ is consistent with the previous theoretical predictions anddata. The leading-order PQCD calculations of the ${\cal B}(B^+\to\rho^+\rho^0)$, ${\cal B}(B^0\to \rho^0\rho^0)$ and the $f_0(B^0\to\rho^0\rho^0)$ are a bit lower than the experimental measurements, which shouldbe further examined. In addition, the "true" and "fake" triple-productasymmetries (TPAs) in the $B_{(s)}\to (\pi\pi)(\pi\pi)$ decays are alsoanalyzed. The sizable averaged TPA ${\cal A}_{\text{T-true}}^{1,\text{ave}}=25.26\%$ of the color-suppressed decay $B^0\to \rho^0\rho^0 \to(\pi^+\pi^-)(\pi^+\pi^-)$ is predicted for the first time, which deviates a lotfrom the so-called "true" TPA $\mathcal{A}_\text{T-true}^1=7.92\%$ due to thelarge direct $CP$ violation. A large "fake" TPA$\mathcal{A}_\text{T-fake}^1=24.96\%$ of the decay $B^0\to \rho^0\rho^0 \to(\pi^+\pi^-)(\pi^+\pi^-)$ is also found, which indicates the significance ofthe final-state interactions. The predictions in this work can be tested byLHCb and Belle-II experiments in the near future.

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