Quasi-two-body decays Bc → D(s)[ρ(770),ρ(1450),ρ(1700)→]ππ in the perturbative QCD factorization approach

In this paper, we studied the quasi-two-body Bc→D(s)[ρ(770),ρ(1450),ρ(1700)→]ππ decays by employing the perturbative QCD (PQCD) factorization approach. The two-pion distribution amplitudes Φππ are applied to include the final-state interactions between the pion pair, while the time-like form factors Fπ(w2) 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 Bc→D(s)[ρ(770),ρ(1450),ρ(1700)→]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 R1a,1b,1c and R2a,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 Bc→D(s)[ρ(770)→]ππ and Bc→D(s)ρ(770) decays are confirmed by our numerical results; (c) taking currently known B(ρ(1450)→ππ) and B(ρ(1700)→ππ) as input, we extracted the theoretical predictions for B(Bc→Dρ(1450)) and B(Bc→Dρ(1700)) from the PQCD predictions for the decay rates of the quasi-two-body decays Bc→D[ρ(1450),ρ(1700)→]ππ. All the PQCD predictions will be tested in the future experiments.