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

Abstract

In this paper, we studied the quasi-two-body $B_c \to D_{(s)} [ ρ(770),ρ(1450),ρ(1700) \to ] ππ$ decays by employing the perturbative QCD (PQCD) factorization approach. The two-pion distribution amplitudes $Φ_{ππ}$ are applied to include the final-state interactions between the pion pair, while the time-like form factors $F_π(w^2)$ associated with the $P$-wave resonant states $ρ(770)$, $ρ(1450)$ and $ρ(1700)$ 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 $B_c \to D_{(s)} [ρ(770), ρ(1450), ρ(1700) \to ] ππ$ decays are in the order of $10^{-9}$ to $ 10^{-5}$ and the direct $CP$ violations around $(10-40)\%$ 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 $B_{c} \to D_{(s)} [ρ(770) \to ] ππ$ and $B_{c} \to D_{(s)} ρ(770)$ decays are confirmed by our numerical results; (c) taking currently known ${\cal B}(ρ(1450)\toππ)$ and ${\cal B}(ρ(1700) \to ππ)$ as input, we extracted the theoretical predictions for ${\cal B}(B_{c} \to D ρ(1450)) $ and ${\cal B}(B_{c} \to D ρ(1700))$ from the PQCD predictions for the decay rates of the quasi-two-body decays $B_{c} \to D [ρ(1450), ρ(1700) \to ] ππ$. All the PQCD predictions will be tested in the future experiments.