Study of nonleptonic $B_{q}^{\ast}$ ${\to}$ $D_{q}V$ and $P_{q} D^*$ weak decays

Motivated by the powerful capability of measurement for the $b$-flavored hadron rare decays at LHC and SuperKEKB/Belle-II, the nonleptonic $\bar{B}^{\ast}$ ${\to}$ $D\bar{D}^{\ast}$, $D{\rho^-}$, $DK^{\ast-}$, ${\pi}D^{\ast}$ and $KD^{\ast}$ weak decays are studied in detail. With the amplitudes calculated with factorization approach and the form factors of $B^{\ast}$ transition into pseudoscalar meson evaluated with the BSW model, branching fractions and polarization fractions are firstly presented. Numerically, the CKM-favored $\bar{B}_{q}^{\ast}$ ${\to}$ $D_{q}D_{s}^{{\ast}-}$ and $D_{q}{\rho}^{-}$ decays have large branching fractions, $\sim$ $10^{-8}$, which should be sought for with priority and firstly observed by LHC and Belle-II experiments. The $\bar{B}^{\ast}_q$ ${\to}$ $D_qK^{\ast}$ and $D_q{\rho}$ decays are dominated by the longitudinal polarization states. While, the parallel polarization fractions of $\bar{B}^{\ast}_q$ ${\to}$ $D_q\bar{D}^{\ast}$ decays are comparable with the longitudinal ones, numerically, $f_{\parallel}$ $+$ $f_{L}$ ${\simeq}$ 95\% and $f_{L}:f_{\parallel}$ $\simeq$ $5:4$. Some comparisons between $\bar{B}^{*0}_q$ $\to$ $D_q V$ and their corresponding $\bar{B}^{0}_q$ $\to$ $D^*_q V$ decays are performed, and the relation $f_{L,\parallel}(\bar{B}^{\ast 0}\to D V)\simeq f_{L,\parallel}(\bar{B}^0\to D^{\ast +} V^-)$ is presented. Besides, with the implication of $SU(3)$ flavor symmetry, some useful ratios $R_{\rm du}$ and $R_{\rm ds}$ are discussed in detail, and suggested to be verified experimentally.