Title: $CP$-violating observables of four-body $B_{(s)} \to (ππ)(K\bar{K})$ decays in perturbative QCD

Abstract: In this work, we investigate six helicity amplitudes of the four-body $B_{(s)} \to (\pi\pi)(K\bar{K})$ decays in the perturbative QCD (PQCD) approach. The $\pi\pi$ invariant mass spectrum is dominated by the vector resonance $\rho(770)$ together with scalar resonance $f_0(980)$, while the vector resonance $\phi(1020)$ and scalar resonance $f_0(980)$ are expected to contribute in the $K\bar{K}$ invariant mass range. We extract the two-body branching ratios ${\cal B}(B_{(s)}\to \rho\phi)$ from the corresponding four-body decays $B_{(s)}\to \rho\phi\to (\pi\pi)(K \bar K)$. The predicted ${\cal B}(B^0_{s}\to \rho\phi)$ agrees well with the current experimental data within errors. The longitudinal polarization fractions of the $B_{(s)}\to \rho\phi$ decays are found to be as large as $90\%$, basically consistent with the previous two-body predictions within uncertainties. In addition, the triple-product asymmetries (TPAs) of the considered decays are also presented for the first time. Since the $B_s^0\to \rho^0\phi\to(\pi^+\pi^-)(K^+K^-)$ decay is induced by both tree and penguin operators, the values of the ${\cal A}^{\rm CP}_{\rm dir}$ and ${\cal A}^{1}_{\text{T-true}}$ are calculated to be $(21.8^{+2.7}_{-3.3})\%$ and $(-10.23^{+1.73}_{-1.56})\%$ respectively. While for pure penguin decays $B^0\to \rho^0\phi\to(\pi^+\pi^-)(K^+K^-)$ and $B^+\to \rho^+\phi\to(\pi^+\pi^0)(K^+K^-)$, both the direct $CP$ asymmetries and ``true" TPAs are naturally expected to be zero in the standard model (SM). The ``fake" TPAs requiring no weak phase difference are usually none zero for all considered decay channels. The sizable ``fake" ${\cal A}^{1}_{\text{T-fake}}=(-20.92^{+6.26}_{-2.80})\%$ of the $B^0\to \rho^0\phi\to(\pi^+\pi^-)(K^+K^-)$ decay is predicted in the PQCD approach, which provides valuable information on the final-state interactions.Our predictions can be tested by the future experiments.