Quasi-two-body decays $B_{(s)}\to P\rho \to P\pi \pi$ in perturbative QCD approach

In this work, we calculate the $CP$-averaged branching ratios and the direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)} \to P (\rho \to) \pi\pi$byemployingtheperturbativeQCD(PQCD)approach(here$P$ stands for a light pseudoscalar meson $\pi ,K,\eta$ or ${\eta }^{\mathrm{\prime }}$). The vector current time-like form factor $F_{\pi }$, which contains the final state interactions between the pion pair in the resonant region associated with the $P$-wave states $\rho (770)$ along with the two-pion distribution amplitudes, are employed to describe the interactions between the $\rho$ and the pion pair under the hypothesis of the conserved vector current. We found that (a) the PQCD predictions for the branching ratios and the direct $CP$-violating asymmetries for most considered $B_{(s)}\to P(\rho \to )\pi \pi$ decays agree with currently available data within errors; (b) for ${\cal B}(B \to \pi^0 \rho^0 \to \pi^0(\pi^+ \pi^-)$, the PQCD prediction is much smaller than the measured one; and © for $B^{+}\to {\pi }^{+}({\rho }^0\to ){\pi }^{+}{\pi }^{-}$ decay mode, we found a negative $CP$ asymmetry $(-27.5_{-3.7}^{+3.0})\mathrm{\%}$, which agrees with other theoretical predictions but different in sign from those as reported by BaBar and LHCb Collaboration.