Branching fractions of $B_{(c)}$ decays involving $J/\psi$ and $X(3872)$

We study two-body $B_{(c)}\to M_c(\pi,K)$ and semileptonic $B_{c}\to M_c\ell^-\bar \nu_\ell$ decays with $M_c=(J/\psi,X_c^0)$, where $X_c^0\equiv X^0(3872)$ is regarded as the tetraquark state of $c\bar cu\bar u(d\bar d)$. With the decay constant $f_{X_c^0}=(234\pm 52)$~MeV determined from the data, we predict that ${\cal B}(B^-\to X_c^0\pi^-)=(11.5\pm 5.7)\times 10^{-6}$, ${\cal B}(\bar B^0\to X_c^0\bar K^0)=(2.1\pm 1.0)\times 10^{-4}$, and ${\cal B}(\bar B^0_s\to X_c^0\bar K^0)=(11.4\pm 5.6)\times 10^{-6}$. With the form factors in QCD models, we calculate that ${\cal B}(B_c^-\to X_c^0\pi^-,X_c^0 K^-)=(6.0\pm 2.6)\times 10^{-5}$ and $(4.7\pm 2.0)\times 10^{-6}$, and ${\cal B}(B_c^-\to J/\psi \mu^-\bar \nu_\mu, X_c^0 \mu^-\bar \nu_\mu )=(2.3\pm0.6)\times 10^{-2}$ and $(1.35\pm 0.18)\times 10^{-3}$, respectively, and extract the ratio of the fragmentation fractions to be $f_c/f_u=(6.4\pm 1.9)\times 10^{-3}$.