Quasi-two-body decays $B_c\to D_{(s)}[\rho (770),\rho (1450),\rho (1700)\to ]\pi \pi$ in the perturbative QCD factorization approach

In this paper, we studied the quasi-two-body $B_c \to D_{(s)} [ \rho(770),\rho(1450),\rho(1700) \to ] \pi \pi$ decays by employing the perturbative QCD (PQCD) factorization approach. The two-pion distribution amplitudes ${\mathrm{\Phi }}_{\pi \pi }$ are applied to include the final-state interactions between the pion pair, while the time-like form factors $F_{\pi }(w^2)$ associated with the $P$-wave resonant states $\rho (770)$, $\rho (1450)$ and $\rho (1700)$ are extracted from the experimental data of the $e^{+}{e}^{-}$ annihilation. We found that: (a) the PQCD predictions for the branching ratios of the quasi-two-body $B_c \to D_{(s)} [\rho(770), \rho(1450), \rho(1700) \to ] \pi \pi$decaysareintheorderof$10^{-9}$to$ 10^{-5}$andthedirect$CP$violationsaround$(10-40)%$ in magnitude; (b) the two sets of the large hierarchy $R_{1a,1b,1c}$ and $R_{2a,2b,2c}$ for the ratios of the branching ratios of the considered decays are defined and can be understood in the PQCD factorization approach, while the self-consistency between the quasi-two-body and two-body framework for $B_c\to D_{(s)}[\rho (770)\to ]\pi \pi$ and $B_c\to D_{(s)}\rho (770)$ decays are confirmed by our numerical results; © taking currently known $\mathcal{B}(\rho (1450)\to \pi \pi )$ and ${\cal B}(\rho(1700) \to \pi\pi)$ as input, we extracted the theoretical predictions for $\mathcal{B}(B_c\to D\rho (1450))$ and $\mathcal{B}(B_c\to D\rho (1700))$ from the PQCD predictions for the decay rates of the quasi-two-body decays $B_c\to D[\rho (1450),\rho (1700)\to ]\pi \pi$. All the PQCD predictions will be tested in the future experiments.