Analysis of three-body decays $B \to D (V \to ) PP $ under the factorization-assisted topological-amplitude approach

Motived by the accumulated experimental results on three-body charmed $B$ decays with resonance contributions in Babar, LHCb and Belle (II), we systematically analyze $B_{(s)} \to D_{(s)} (V \to) P_1 P_2 $ decays with $V$ representing a vector resonance ($\rho, K^*, \omega$ or $\phi$) and $P_{1,2}$ as a light pseudoscalar meson (pion or kaon). The intermediate subprocesses $B_{(s)} \to D_{(s)} V$ are calculated with the factorization-assisted topological-amplitude (FAT) approach and the intermediate resonant states $V$ described by the relativistic Breit-Wigner distribution successively decay to $P_1 P_2$ via strong interaction. Taking all lowest resonance states ($\rho, K^*, \omega, \phi$) into account, we calculate the branching fractions of these decay modes as well as the Breit-Wigner-tail effects for $B_{(s)} \to D_{(s)} (\rho ,\omega \to) KK$. Our results agree with the data by Babar, LHCb and Belle (II). Among the predictions that are still not observed, there are some branching ratios of order $10^{-6}-10^{-4}$ which are hopeful to be measured by LHCb and Belle II. Our approach and the perturbative QCD approach (PQCD) adopt the compatible theme to deal with the resonance contributions. What's more, our data for the intermediate two-body charmed $B$-meson decays in FAT approach are more precise. As a result, our results for branching fractions have smaller uncertainties, especially for color-suppressed emission diagram dominated modes.