Measurement of $B^0\to D_s^{\left(*\right)+}{D}^{*-}$ branching fractions and $B^0\to D_s^{*+}{D}^{*-}$ polarization with a partial reconstruction technique

We present a study of the decays $B^0\to D_s^{\left(*\right)+}{D}^{*-},$ using $20.8\hspace{0.5em}{\mathrm{fb}}^{-1}$ of $e^{+}{e}^{-}$ annihilation data recorded with the BABAR detector. The analysis is conducted with a partial reconstruction technique, in which only the $D_s^{\left(*\right)+}$ and the soft pion from the $D^{*-}$ decay are reconstructed. We measure the branching fractions $\mathcal{B}{(B}^0\to D_s^{+}{D}^{*-})=(1.03\mathrm{}\pm 0.14\pm 0.13\pm 0.26)\%$ and $\mathcal{B}{(B}^0\to D_s^{*+}{D}^{*-})=(1.97\mathrm{}\pm 0.15\pm 0.30\pm 0.49)\%,$ where the first error is statistical, the second is systematic, and the third is the error due to the $D_s^{+}\to \phi {\pi }^{+}$ branching fraction uncertainty. From the $B^0\to D_s^{*+}{D}^{*-}$ angular distributions, we measure the fraction of longitudinal polarization ${\Gamma }_L/\Gamma =(51.9\mathrm{}\pm 5.0\pm 2.8)\%,$ which is consistent with theoretical predictions based on factorization.