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

We study two-body $B_{(c)}\to M_c(\pi ,K)$ and semileptonic $B_{c}\to M_c\ell^-\bar \nu_\ell$decayswith$M_c=(J/\psi,X_c^0)$,where$X_c^0\equiv X^0(3872)$isregardedasthetetraquarkstateof$c\bar cu\bar u(d\bar d)$. With the decay constant $f_{X_c^0}=(234\pm 52)$~MeV determined from the data, we predict that $\mathcal{B}(B^{-}\to X_c^0{\pi }^{-})=(11.5\pm 5.7)\times {10}^{-6}$, $\mathcal{B}({\bar{B}}^0\to X_c^0{\bar{K}}^0)=(2.1\pm 1.0)\times {10}^{-4}$, 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 $f_c/f_u=(6.4\pm 1.9)\times 10^{-3}$.