Properties of the tensor state $bc\overline{b}\overline{c}$

Spectroscopic parameters and decays of the exotic tensor meson $T$ with content $bc \overline{b}\overline{c}$ are explored in the context of the diquark-antidiquark model. We treat it as a state built of axial-vector diquark $b^{T}C\gamma _{\mu }c$ and antidiquark $\overline{b}\gamma _{\nu }C \overline{c}^{T}$, where $C$ is the charge conjugation matrix. The mass $m$ and current coupling $\Lambda$ of this tetraquark are extracted from two-point sum rules. Our result for $m=(12.70 \pm 0.09)~\mathrm{GeV}$ proves that $T$ is unstable against strong dissociations to two-meson final states. Its dominant decay channels are processes $T \to J/\psi \Upsilon $, $ \eta_{b}\eta _{c}$, and $B_{c}^{(\ast) +}B_{c}^{(\ast) -}$. Kinematically allowed transformations of $T$ include also decays $T\rightarrow D^{(\ast )+}D^{(\ast )-}$ and $D^{(\ast )0}\overline{D}^{(\ast )0}$, which are generated by $b\overline{b}$ annihilation inside of $T$. The full width of $ T $ is estimated by considering all of these channels. Their partial widths are calculated by invoking methods of three-point sum rule approach which are required to evaluate strong couplings at corresponding tetraquark-meson-meson vertices. Our predictions for the mass and width $ \Gamma_{T}=(117.4 \pm 15.9)~ \mathrm{MeV} $ of the tensor state $T$ provide useful information for experimental studies of fully heavy four-quark exotic structures.