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

We analyze the quasi-two-body decays $B\to K\rho \to K\pi \pi$ in the perturbative QCD (PQCD) approach, in which final-state interactions between the pions in the resonant regions associated with the $P$-wave states $\rho (770)$ and ${\rho }^{\mathrm{\prime }}(1450)$ are factorized into two-pion distribution amplitudes. Adopting experimental inputs for the time-like pion form factors involved in two-pion distribution amplitudes, we calculate branching ratios and direct $CP$ asymmetries of the $B\to K\rho (770),K{\rho }^{\mathrm{\prime }}(1450)\to K\pi \pi$ modes. It is shown that agreement of theoretical results with data can be achieved, through which Gegenbauer moments of the $P$-wave two-pion distribution amplitudes are determined. 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 framework for exclusive hadronic $B$ meson decays.