Molecular states $J/\psi B_{c}^{+}$ and $\eta_{c}B_{c}^{\ast +} $

Hadronic molecules $\mathfrak{M}=J/\psi B_{c}^{+}$ and $\widetilde{\mathfrak{ \ M}}=\eta _{c}B_{c}^{\ast +}$ are investigated in the framework of QCD sum rule method. These particles with spin-parities $J^{\mathrm{P}}=1^+$ have the quark contents $cc \overline{c}\overline{b}$. We compute their masses and current couplings and find that they are numerically very close to each other coinciding within accuracy of the sum rule method. Therefore, we concentrate on the molecule $J/\psi B_{c}^{+}$ and explore features of this state in a detailed form. Our prediction $m=(9740 \pm 70)~\mathrm{MeV}$ for its mass means that $\mathfrak{M}$ easily decays to pairs of ordinary mesons through strong interactions. There are two mechanisms responsible for transformations of $\mathfrak{M}$ to conventional mesons. The fall-apart mechanism generates the dominant decay channels $\mathfrak{M} \to J/\psi B_{c}^{+}$ and $\mathfrak{M} \to \eta _{c}B_{c}^{\ast +}$. Annihilation of $\overline{c}c$ quarks triggers subdominant processes with various final-state $B$ and $D$ mesons: Six of such channels are considered in this work. The partial widths all of decays are computed using the three-point sum rule approach. The width $\Gamma[ \mathfrak{M}]=(121 \pm 17)~ \mathrm{MeV }$ of the hadronic axial-vector molecule $\mathfrak{M}$, as well as its mass provide valuable information for running and future experiments.