Scalar Bound States of $D^{\ast }{\bar{D}}^{\ast }$ and $B^{\ast }{\bar{B}}^{\ast }$ in the Bethe-Salpeter Formalism

We study the scalar bound states of $D^{\ast }{\bar{D}}^{\ast }$ and $B^{\ast }{\bar{B}}^{\ast }$ in the Bethe-Salpeter formalism, with the effective interaction kernel extracted from the chiral perturbative theory and the heavy quark effective theory in the ladder approximation and the covariant instantaneous approximation. The results show that, in the scalar case ($J=0$), there can only exist $I=0$ bound states for parameters in proper range, while there cannot exist the $I=1$ bound states in the whole reasonable parameter range, due to more constraints arising from our definition of the Bethe-Salpeter wavefunction.