Bottomonium-like tetraquarks in a chiral quark model

The low-lying bottomonium-like tetraquarks $b\bar{b}q\bar{q}$ $(q=u,\,d,\,s)$ with spin-parity $J^P=0^+$, $1^+$ and $2^+$, and isospin $I=0,\,1$ or $\frac{1}{2}$, are systematically investigated within the theoretical framework of real- and complex-scaling range of a chiral quark model, which has already been successfully applied in analysis of several multiquark systems. A complete four-body $S$-wave function, which includes meson-meson, diquark-antidiquark and K-type arrangements of quarks, along with all possible color configurations are considered. In the $b\bar{b}q\bar{q}$ $(q=u,\,d)$ tetraquark system, we found resonance states of $B^{(*)} \bar{B}^{(*)}$, $\Upsilon\omega$, $\Upsilon\rho$ and $\eta_b \rho$ nature with all possible $I(J^P)$ quantum numbers. Their masses are generally located in the energy range $11.0-11.3$ GeV and their widths are less than $10$ MeV. In addition, extremely narrow resonances, with two-meson strong decay widths less than $1.5$ MeV, are obtained in both $b\bar{b}u\bar{s}$ and $b\bar{b}s\bar{s}$ tetraquark systems. Particularly, four radial excitations of $\Upsilon K^*$ and $\eta_b K^*$ are found at $\sim11.1$ GeV in $J^P=0^+$, $1^+$ and $2^+$ channels of $b\bar{b}u\bar{s}$ system. One $\Upsilon(1S)\phi(2S)$ resonance state is obtained at $11.28$ GeV in the $J^P=1^+$ sector.