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$CP$-violating observables in four-body $B\rightarrow ϕ(\rightarrow K\bar K)K^*(\rightarrow Kπ)$ decays

We analyse the four-body $B\rightarrow ϕ(\rightarrow K\bar K)K^*(\rightarrow Kπ)$ decays in the perturbative QCD approach,where the invariant mass of $K\bar K$($Kπ$) system is limited in a window of $\pm 15$ MeV ($\pm150$ MeV) around the $ϕ(K^*(892))$ mass.In addition to the P wave resonances,two important S wave backgrounds in the selected invariant mass region are also accounted for. Angular momentum conservation allows six helicity amplitudes to contribute,including three P waves, two single S waves,and one double S wave. We calculated the branching ratio for each component and found sizable S wave contributions,coincide with the experimental observation.The obtained branching ratios of $B^{0(+)}\rightarrow ϕK^{*0(+)}$ are comparable with the previous predictions and support the measurements, whereas the predicted $\mathcal{B}(B^0_s\rightarrow ϕ\bar K^{*0})$ is smaller than the world average. The longitudinal polarizations are predicted to be around 0.7,consistent with previous PQCD results but larger than the data. Aside from the direct CP asymmetries,the true and fake triple product asymmetries(TPAs) are calculated in this work. In the case of neutral modes, both direct CP asymmetries and true TPAs are expected to be zero due to the vanishing weak phase difference. The direct CP asymmetries for the $B^+$ mode are predicted to be tiny,since the tree contributions are suppressed with respect to the penguin ones. The true asymmetries have shown no significant deviations from zero.In contrast,large fake asymmetries are observed in these decays,indicating the presence of significant final state interactions.We give the predictions of the S wave induced TPAs for the first time,which is consistent with LHCb data and would be checked with future measurements from Belle and BABAR experiments if the S wave components can be properly taken into account in angular analysis.