$CP$-violating observables of four-body B₍ₛ₎ → (ππ)(KK̄) decays in perturbative QCD

In this work, we investigate six helicity amplitudes of the four-body B₍ₛ₎ → (ππ)(KK̄) decays in the perturbative QCD (PQCD) approach. The ππ invariant mass spectrum is dominated by the vector resonance ρ(770) together with scalar resonance f₀(980), while the vector resonance φ(1020) and scalar resonance f₀(980) are expected to contribute in the KK̄ invariant mass range. We extract the two-body branching ratios B(B₍ₛ₎→ ρφ) from the corresponding four-body decays B₍ₛ₎→ ρφ→ (ππ)(K K). The predicted B(B⁰ₛ→ ρφ) agrees well with the current experimental data within errors. The longitudinal polarization fractions of the B₍ₛ₎→ ρφ 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 Bₛ⁰→ ρ⁰φ→(π⁺π⁻)(K⁺K⁻) decay is induced by both tree and penguin operators, the values of the ACPdir and A¹T-true are calculated to be (21.8+2.7-3.3)% and (-10.23+1.73-1.56)% respectively. While for pure penguin decays B⁰→ ρ⁰φ→(π⁺π⁻)(K⁺K⁻) and B⁺→ ρ⁺φ→(π⁺π⁰)(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" A¹T-fake=(-20.92+6.26-2.80)% of the B⁰→ ρ⁰φ→(π⁺π⁻)(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.