  In the heavy quark limit, we establish the Bethe-Salpeter equations for the
ground state $QQ^{\prime} \bar u \bar d$ containing one heavy diquark
$QQ^{\prime}$ ($Q, Q^{\prime}=b$ or $c$) and one light antidiquark $\bar u \bar
d$. We solve the Bethe-Salpeter equations numerically in the covariant
instantaneous approximations with the kernels containing a scalar confinement
term and a one-gluon-exchange term. Numerical solutions for the Bethe-Salpeter
wave functions are presented. The results show that the masses of $bb \bar u
\bar d$, $bc \bar u \bar d$, and $cc \bar u \bar d$ bound states lie below the
threshold of $\bar{B}^{*0}\,B^-$ or $B^0\,{B^*}^-$, $B^-D^+$ or $\bar{B}^{0}
D^0$, and $D^+\,{D^*}^0$ or ${D^*}^+\,D^0$ mesons, respectively. The ground
states $QQ^{\prime} \bar u \bar d$ may exist and we expect the forthcoming
experimental data to confirm them.
