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In the heavy quark limit, we establish the Bethe-Salpeter equations for theground state $QQ^{\prime} \bar u \bar d$ containing one heavy diquark$QQ^{\prime}$ ($Q, Q^{\prime}=b$ or $c$) and one light antidiquark $\bar u \bard$. We solve the Bethe-Salpeter equations numerically in the covariantinstantaneous approximations with the kernels containing a scalar confinementterm and a one-gluon-exchange term. Numerical solutions for the Bethe-Salpeterwave 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 thethreshold 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 groundstates $QQ^{\prime} \bar u \bar d$ may exist and we expect the forthcomingexperimental data to confirm them.

$QQ^{\prime} \bar u \bar d$ bound state in the Bethe-Salpeter equation approach