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

By employing a framework for the quasi-two-body decays in the perturbative QCD (PQCD) factorization approach, we calculate the branching ratios of the decays $B_{(s)} \to D (\rho(1450),\rho(1700))\to D \pi \pi$ with $D=(D_{(s)}, \bar{D}_{(s)})$. The pion vector form factor $F_\pi$, acquired from a {\it BABAR} Collaboration analysis of $e^+ e^- \to \pi^+ \pi^-(\gamma)$ data, is involved in the two-pion distribution amplitudes $\Phi^{I=1}_{\pi\pi}$. The PQCD predictions for the branching ratios of the considered quasi-two-body decays are in the range of $10^{-10} \sim 10^{-4}$. The PQCD predictions for ${\cal B}(B^0\to \bar{D}^0 (\rho^0(1450),\rho^0(1700)) \to \bar{D}^0 \pi^+\pi^-)$ agree well with the measured values as reported by LHCb if one takes still large theoretical errors into account. Unlike the traditional way of the PQCD approach, one can extract the decay rates for the two-body decays $B_{(s)} \to D (\rho(1450),\rho(1700))$ from the results of the corresponding quasi-two-body decays. The PQCD predictions for ${\cal B}(B_{(s)} \to D \rho(1450))$ and ${\cal B}(B_{(s)} \to D \rho(770))$ are similar in magnitude: an interesting relation to be tested by future experimental measurements.