  The compact tetraquark states with fully heavy quark contents $QQ\bar Q\bar
Q$ are studied as the bound states of the diquark-antidiquark within the
Bethe-Salpeter framework. The (anti)diquark masses and form factors used are
the same as we calculated the doubly heavy baryons in a previous work. Under
the instantaneous approximation, the three-dimensional (Bethe-)Salpeter
equation of the tetraquarks is derived and solved numerically to obtain the
corresponding mass spectra and wave functions of the tetraquarks with
$J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$. Our results show that the three ground
states of $cc\bar c\bar c$ locate in the mass range of $6.4\sim6.5$GeV, and the
$bb\bar b\bar b$ states in mass range of $19.2\sim19.3$GeV. The obtained
relativistic wave functions naturally include the mixing effects from the
possible $D$\,(or $G$) partial waves, and then can be further used to do
precise calculations of the tetraquark decays. Based on the obtained results,
the LHCb's observation $X(6900)$ is less likely to be the ground states of
compact $cc\bar c\bar c$ tetraquarks but might be the first or second excited
states. In addition, a widely used propagator-like form factor is also
investigated and discussed.
