Branching fractions and CP asymmetries of the quasi-two-body decays in $$B_{s} \rightarrow K^0({\overline{K}}^0)K^\pm \pi ^\mp $$Bs→K0(K¯0)K±π∓ within PQCD approach

Motivated by the first untagged decay-time-integrated amplitude analysis of $$B_s \rightarrow K_SK^{\mp }\pi ^{\pm }$$ decays performed by LHCb collaboration, where the decay amplitudes are modeled to contain the resonant contributions from intermediate resonances $$K^*(892)$$ , $$K_0^*(1430)$$ and $$K_2^*(1430)$$ , we comprehensively investigate the quasi-two-body decays in the three-body processes , and calculate the branching fractions and the time-dependent CP asymmetries within the perturbative QCD approach based on the $$k_T$$ factorization. In the quasi-two-body space region the calculated branching fractions with the considered intermediate resonances are in good agreement with the experimental results of LHCb by adopting proper $$K\pi $$ pair wave function, describing the interaction between the kaon and pion in the $$K\pi $$ pair. Furthermore, within the obtained branching fractions of the quasi-two-body decays, we also calculate the branching fractions of the corresponding two-body decays, and the results consist with the LHCb measurements and the earlier studies within errors. For these considered decays, since the final states are not flavour-specific, the time-dependent CP could be measured. We then calculate six CP-violation observables, which can be tested in the ongoing LHCb experiment.