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In this paper, we studied the $B_{(s)} \to (D_{(s)},\bar{D}_{(s)}) \rho \to(D_{(s)}, \bar{D}_{(s)})\pi \pi$ decays by employing a framework for thequasi-two-body decays in the perturbative QCD (PQCD) factorization approach. Weuse the two-pion distribution amplitudes $\Phi_{\pi\pi}$, which contains bothresonant and nonresonant contributions from the pion pair, to describe thefinal state interactions (FSIs) between the pions in the resonant region. Wefound that (a) for all considered decays, the PQCD predictions for theirbranching ratios based on the quasi-two-body and the two-body framework agreewell with each other due to ${\cal B}(\rho \to \pi\pi) \approx 100\%$; For$B^+\to \bar{D^0}\rho^+\to \bar{D^0}\pi^+ \pi^0$ and other four considereddecay modes, the PQCD predictions do agree well with the measured values withinerrors; (b) the great difference between the PQCD predictions for ${\calB}(B\to \bar{D} \rho \to \bar{D} \pi\pi)$ and ${\cal B}(B\to D \rho \to D\pi\pi)$ can be understood by the strong CKM suppression factor $ R_{\rm CKM}\approx 3 \times 10^{-4}$; (c) for the $B_s\to D \rho \to D \pi\pi$ and $B_s\to\bar{D} \rho \to \bar{D} \pi\pi$ decays, however, the PQCD predictions of$R_{\rm s1}\approx 0.13$ and $R_{\rm s2}\approx 0.14$ do agree very well withthe moderate CKM suppression factor $R^s_{\rm CKM}\approx 0.14$; and (d) thePQCD predictions for the ratios $R_{D\rho}$ and the strong phase difference$\cos\delta_{D\rho}$ of the three $B \to \bar{D} \rho$ decay modes agree wellwith the LHCb measurements within one standard deviation.

nuclear physics b

The quasi-two-body decays $B_{(s)} \to (D_{(s)},\bar{D}_{(s)}) \rho \to  (D_{(s)}, \bar{D}_{(s)})\pi \pi$ in the perturbative QCD factorization  approach