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 $bb\bar{b}\bar{c}$ and $cc\bar{c}\bar{b}$ 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 ${\mathcal{M}}_{\mathrm{c}}\to {\eta }_c{B}_c^{+}$ and $J\mathrm{/}\psi B_c^{\ast +}$ 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 $b\bar{b}$ and $c\bar{c}$ 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 $m=(15728\pm 90)\mathrm{M}\mathrm{e}\mathrm{V}$ and width $\mathrm{\Gamma }[{\mathcal{M}}_b]=(93\pm 17)\mathrm{M}\mathrm{e}\mathrm{V}$ of the molecule ${\mathcal{M}}_b$ , and $\tilde{m}=(9712\pm 72)\mathrm{M}\mathrm{e}\mathrm{V}$ and $\mathrm{\Gamma }[{\mathcal{M}}_c]=(70\pm 10)\mathrm{M}\mathrm{e}\mathrm{V}$ in the case of ${\mathcal{M}}_c$ offer valuable guidance for experimental searches at existing facilities.