The quasi-two-body decays B(s)→(D(s),D¯(s))ρ→(D(s),D¯(s))ππ in the perturbative QCD factorization approach

In this paper, we studied the B(s)→(D(s),D¯(s))ρ→(D(s),D¯(s))ππ decays by employing a framework for the quasi-two-body decays in the perturbative QCD (PQCD) factorization approach. We use the two-pion distribution amplitudes Φππ, which contains both resonant and nonresonant contributions from the pion pair, to describe the final state interactions (FSIs) between the pions in the resonant region. We found that (a) for all considered decays, the PQCD predictions for their branching ratios based on the quasi-two-body and the two-body framework agree well with each other due to B(ρ→ππ)≈100%; For B+→D0¯ρ+→D0¯π+π0 and other four considered decay modes, the PQCD predictions do agree well with the measured values within errors; (b) the great difference between the PQCD predictions for B(B→D¯ρ→D¯ππ) and B(B→Dρ→Dππ) can be understood by the strong CKM suppression factor RCKM≈3×10−4; (c) for the Bs→Dρ→Dππ and Bs→D¯ρ→D¯ππ decays, however, the PQCD predictions of Rs1≈0.13 and Rs2≈0.14 do agree very well with the moderate CKM suppression factor RCKMs≈0.14; and (d) the PQCD predictions for the ratios RDρ and the strong phase difference cos⁡δDρ of the three B→D¯ρ decay modes agree well with the LHCb measurements within one standard deviation.