  The fully heavy axial-vector diquark-antidiquark structures $bb\overline{c}
\overline{c}$ are explored by means of the QCD sum rule method. They are
modeled as four-quark mesons $T_{\mathrm{1}}$ and $T_{\mathrm{2}}$ composed of
$b^{T}C\sigma _{\mu \nu }\gamma _{5}b$, $\overline{c}\gamma ^{\nu }C
\overline{c}^{T}$ and $b^{T}C\gamma _{\mu }\gamma _{5}b$, $\overline{c}C
\overline{c}^{T}$ diquarks, respectively. The spectroscopic parameters of the
tetraquarks $T_{\mathrm{1}}$ and $T_{\mathrm{2}}$ are determined in the context
of the QCD two-point sum rule method. Results obtained for masses of these
states $m_{1} =(12715\pm 86)~\mathrm{MeV}$ and $m_{2}=(13383\pm 92)~
\mathrm{MeV}$ are used to fix their strong decay channels. The full width $
\Gamma (T_{\mathrm{1}})$ of the diquark-antidiquark state $T_{\mathrm{1}}$ is
estimated by considering the processes $T_{\mathrm{1}} \to B_{c}^{-}B_{c}^{\ast
-}$ and $T_{\mathrm{1}} \to B_{c}^{\ast -}B_{c}^{\ast -} $. The decays to
mesons $B_{c}^{-}B_{c}^{\ast -}$, $B_{c}^{-}(2S)B_{c}^{ \ast -}$ and
$B_{c}^{\ast -}B_{c}^{\ast -}$ are employed to evaluate $\Gamma
(T_{\mathrm{2}})$. Results obtained for the widths $\Gamma (T_{\mathrm{1}
})=(44.3\pm 8.8)~\mathrm{MeV}$ and $\Gamma (T_{\mathrm{2}})=(82.5\pm 13.7)~
\mathrm{MeV}$ of these tetraquarks in conjunction with their masses are useful
for future experimental studies of fully heavy resonances.
