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In this work, we investigate six helicity amplitudes of the four-body B (ₒ) () (KK) decays in the perturbative QCD (PQCD) approach. The invariant mass spectrum is dominated by the vector resonance (770) together with scalar resonance f₀ (980), while the vector resonance (1020) and scalar resonance f₀ (980) are expected to contribute in the KK invariant mass range. We extract the two-body branching ratios B (B (ₒ) ) from the corresponding four-body decays B (ₒ) () (K K). The predicted B (B⁰ₒ) agrees well with the current experimental data within errors. The longitudinal polarization fractions of the B (ₒ) 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 ⁰ (^+^-) (K^+K^-) decay is induced by both tree and penguin operators, the values of the A^ CP ₃₈ₑ and A^1ₓ-ₓₑₔ₄ are calculated to be (21. 8^+2. 7-₃. ₃) \% and (-10. 23^+1. 73-₁. ₅₆) \% respectively. While for pure penguin decays B⁰ ⁰ (^+^-) (K^+K^-) and B^+ ^+ (^+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" A^1ₓ-₅₀₊₄= (-20. 92^+6. 26-₂. ₈₀) \% of the B⁰ ⁰ (^+^-) (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.
Yan et al. (Mon,) studied this question.