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In this work, we calculate the $CP$-averaged branching ratios and direct$CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to P\rho^\prime(1450), P\rho^{\prime\prime}(1700)\to P \pi\pi$ by employing theperturbative QCD (PQCD) factorization approach, where $P$ is a lightpseudoscalar meson $K, \pi, \eta$ and $\eta^{\prime}$. The considered decaymodes are studied in the quasi-two-body framework by parameterizing thetwo-pion distribution amplitude $\Phi_{\pi\pi}^{\rm P}$. The $P$-wave time-likeform factor $F_{\pi}$ in the resonant regions associated with the$\rho^\prime(1450)$ and $\rho^{\prime\prime}(1700)$ is estimated based onavailable experimental data. The PQCD predictions for the $CP$-averagedbranching ratios of the decays $B_{(s)}\to P\rho^\prime(1450),P\rho^{\prime\prime}(1700)\to P\pi\pi$ are in the order of $10^{-7} \sim10^{-9}$. The branching ratios of the two-body decays $B_{(s)}\toP\rho^\prime(1450), P\rho^{\prime\prime}(1700)$ are extracted from thecorresponding quasi-two-body decay modes. The whole pattern of the pion formfactor-squared $|F_\pi|^2$ measured by {\it BABAR} Collaboration could also beunderstood based on our studies. Our PQCD predictions will be tested by theprecise data from the LHCb and the future Belle II experiments.

Quasi-two-body decays $B_{(s)}\to P\rho^\prime(1450), P\rho^{\prime\prime}(1700)\to P\pi\pi$ in the perturbative QCD approach