Dynamical study of $S$-wave $\bar{Q}Q\bar{q}q$ system

We perform the energy spectra calculating for the $S$-wave $\bar{Q}Q\bar{q}q$ (where $Q=c,b$ and $q=u,d,s$) system within two constituent quark models. The bound states of $B^{\ast }{\bar{B}}^{\ast }$ with $I(J^{PC})=1(0^{++}),1(1^{+-}),0(2^{++})$ and $B{\bar{B}}^{\ast }$ with isospin $I=1$, or $0$ are obtained in color-singlet-singlet channel. If considering the coupling of color channels, apart from the deep bound states appear in $[b\bar{q}{]}^{(\ast )}[q\bar{b}{]}^{(\ast )}$ scenario, a bound state $[c\bar{q}{]}^{\ast }[q\bar{c}{]}^{\ast }$ with $I(J^{PC})=1(0^{++})$ is also formed. The $B{\bar{B}}^{\ast }$ and the $B^{\ast }{\bar{B}}^{\ast }$ with quantum number $1(1^{+-})$ are well candidates for $Z_b^{\pm }(10610)$ and $Z_b^{\pm }(10650)$ reported by Belle collaboration respectively, while the $B{\bar{B}}^{\ast }$ with isospin $0$ can be interpret as a candidate for $Z_b^0(10610)$. A bound state $[c\bar{q}{]}^{\ast }[q\bar{c}{]}^{\ast }$ with $I(J^{PC})=1(0^{++})$ is comparable with $Z_c^{+}(4025)$ announced by BES.