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 R 1 a , 1 b , 1 c and R 2 a , 2 b , 2 c 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.