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Quasi-two-body decays $B_{(s)}\to K^*γ\to Kπγ$ in perturbative QCD approach

In this work we study the quasi-two-body decays $B\to K^*γ\to Kπγ$ in the perturbative QCD (PQCD) approach. The two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the $Kπ$ pair, which involve the time-like form factors and the Gegenbauer polynomials. We calculate the CP averaged branching ratios for the decays $B_{(s)}\to K^*γ\to Kπγ$. Our results are in agreement with the new update data measured by Belle II, which suggests these quasi-two-body decays are more appropriate to be analyzed in three-body framework than in the two-body one. We also predict the direct CP-violation asymmetries for the considered decay modes and find that $A_{CP}(B_{u,d}\to K^*γ\to Kπγ)$ is small and less than $1\%$ in magnitude, while $A_{CP}(B_{s}\to K^*γ\to Kπγ)$ is larger and can arrive at a few percent. Our predictions can be tested by the future B meson experiments.