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We investigate the four-body decays $B_{(s)} \rightarrow (K\pi)_{S/P}(K\pi)_{S/P}$ in the $K\pi$-pair invariant mass region around the $K^*(892)$resonance in the perturbative QCD (PQCD) approach, where $(K\pi)_{S/P}$ denotesa $S$- or $P$-wave $K\pi$ configuration. The contributions from the $P$-waveresonance $K^*(892)$ and from possible $S$-wave components are included intothe parametrization of the $K\pi$ two-meson distribution amplitudes, which arethe nonperturbative inputs in this formalism. We calculate the branching ratiofor each resonance and observe sizable $S$-wave contributions to the $B_s$modes, well consistent with recently available LHCb data. We also deduce thepolarization fractions for the $K^*\bar{K}^*$ final states, and compare ourpredictions to those in previous PQCD analyses of the corresponding two-body$B\to K^*\bar{K}^*$ decays. The polarization puzzle associated with the two$U$-spin related channels $B^0\rightarrow K^{*0}\bar{K}^{*0}$ and$B^0_s\rightarrow K^{*0}\bar{K}^{*0}$ still exists. In addition to direct $CP$asymmetries, triple-product asymmetries and $S$-wave induced direct $CP$asymmetries originating from the interference among various helicityamplitudes, are presented. It is shown that true triple-product asymmetries arerather small, while direct $CP$ asymmetries and $S$-wave-induced direct $CP$asymmetries are significant in some decays. Our results will be subject tostringent tests with precise data from $B$ factories in the near future.

Four-body decays $B_{(s)} \rightarrow (K\pi)_{S/P} (K\pi)_{S/P}$ in the perturbative QCD approach