Scalar molecules $\eta _{b}B_{c}^{-}$ and $\eta _{c}B_{c}^{+} $ with asymmetric quark contents

The hadronic scalar molecules $\mathcal{M}_{b}$ and $\mathcal{M}_{c}$ with asymmetric quark contents $bb \overline{b}\overline{c}$ and $cc \overline{c} \overline{b}$ are explored by means of the QCD sum rule method. Their masses and current couplings are calculated using the two-point sum rule approach. The obtained results show that they are strong-interaction unstable particles and transform to ordinary mesons' pairs. The molecule $\mathcal{M} _{b}$ dissociates through the process $\mathcal{M}_{\mathrm{b}}\to \eta _{b}B_{c}^{-}$. The decays $\mathcal{M}_{\mathrm{c}}\rightarrow \eta _{c}B_{c}^{+}$ and $J/\psi B_{c}^{\ast +}$ are dominant modes for the molecule $\mathcal{M}_{c}$. The full decay widths of the molecules $\mathcal{ \ \ M}_{b}$ and $\mathcal{M}_{c}$ are estimated using these decay channels, as well as ones generated by the annihilation of $b\overline{b}$ and $c \overline{c}$ quarks in $\mathcal{M}_{b}$ and $\mathcal{M}_{c}$, respectively. The QCD three-point sum rule method is employed to find partial widths all of these channels. This approach is required to evaluate the strong couplings at the molecule-meson-meson vertices under consideration. The mass $m=(15728 \pm 90)~\mathrm{MeV}$ and width $\Gamma[ \mathcal{M}_b] =(93 \pm 17)~ \mathrm{MeV}$ of the molecule $\mathcal{M}_{b}$ , and $\widetilde{m}=(9712 \pm 72)~\mathrm{MeV}$ and $\Gamma[\mathcal{M}_c] =(70 \pm 10)~ \mathrm{MeV}$ in the case of $\mathcal{M}_{c} $ offer valuable guidance for experimental searches at existing facilities.