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We study the three-body radiative decays $B\to \phi(\rho) K\gamma$ induced bya flavor-changing neutral current in the perturbative QCD approach.Pseudoscalar-vector ($PV$) distribution amplitudes (DAs) are introduced for thefinal-state $\phi K$ ($\rho K$) pair to capture important infrared dynamics inthe region with a small $PV$-pair invariant mass. The dependence of these $PV$DAs on the parton momentum fraction is parametrized in terms of the Gegenbauerpolynomials, and the dependence on the meson momentum fraction is derivedthrough their normalizations to time-like $PV$ form factors. In addition to thedominant electromagnetic penguin, the subleading chromomagnetic penguin,quark-loop and annihilation diagrams are also calculated. After determining the$PV$ DAs from relevant branching-ratio data, the direct $CP$ asymmetries anddecay spectra in the $PV$-pair invariant mass are predicted for each $B\to\phi(\rho) K\gamma$ mode.

Three-body decays $B \to \phi(\rho) K \gamma$ in perturbative QCD approach