$QQ\bar{q}\bar{q}$ in a chiral constituent quark model

Inspired by $\Xi_{cc}$ reported by LHCb Collaboration and $X(5568)$ reported by $D0$ Collaboration, the $QQ\bar{q}\bar{q}$ ($Q=c,b,s$, $q=u,d$) tetraquark states, are studied in the present work. With the help of gaussian expansion method, two structures, diquark-antiquark and meson-meson, with all possible color configurations are investigated systematically in a chiral quark model to search for the possible stable states. The results show that there is no bound state in the isovector $QQ\bar{q}\bar{q}$ system, while there are rather deep bound states in the isoscalar $bb\bar{q}\bar{q}$, $cc\bar{q}\bar{q}$ and $bc\bar{q}\bar{q}$ systems. There are also several shallow bound states in $QQ^{\prime}$ system. Mixing two structures of diquark-antidiquark and meson-meson can introduce more attractions and convert some unbound isoscalar states into shallow bound states. The large mass of the heavy quark is beneficial to the formation of the bound state. The separations between quarks are calculated to unravel the spacial structure of the system.