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We analyze the quasi-two-body decays $B\to K\rho\to K\pi\pi$ in theperturbative QCD (PQCD) approach, in which final-state interactions between thepions in the resonant regions associated with the $P$-wave states $\rho(770)$and $\rho^\prime(1450)$ are factorized into two-pion distribution amplitudes.Adopting experimental inputs for the time-like pion form factors involved intwo-pion distribution amplitudes, we calculate branching ratios and direct $CP$asymmetries of the $B\to K\rho(770),K\rho^\prime(1450)\to K\pi\pi$ modes. It isshown that agreement of theoretical results with data can be achieved, throughwhich Gegenbauer moments of the $P$-wave two-pion distribution amplitudes aredetermined. The consistency between the three-body and two-body analyses of the$B\to K\rho(770)\to K\pi\pi$ decays supports the PQCD factorization frameworkfor exclusive hadronic $B$ meson decays.

Quasi-two-body decays $B\to K\rho\to K\pi\pi$ in perturbative QCD approach