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

Abstract

In the heavy quark limit, we establish the Bethe-Salpeter equations for the ground state $Q{Q}^{\prime }\bar{u}\bar{d}$ containing one heavy diquark $Q{Q}^{\prime }$ ($Q,Q^{\prime }=b$ 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 ${\bar{B}}^{\ast 0}{B}^{-}$ or $B^0{B^{\ast }}^{-}$, $B^{-}{D}^{+}$ or ${\bar{B}}^0{D}^0$, and $D^{+}{D^{\ast }}^0$ or ${D^{\ast }}^{+}{D}^0$ mesons, respectively. The ground states $Q{Q}^{\prime }\bar{u}\bar{d}$ may exist and we expect the forthcoming experimental data to confirm them.

Comments: 6 figure, 3 tables