We perform a coupled-channels calculation of the masses of light mesons with the quantum numbers $I{J}^{P=-}$, $(I,J)=0$, 1, by including $q\overline{q}$ and $(q\overline{q}{)}^2$ components in a nonrelativistic chiral quark model. The coupling between two- and four-quark configurations is realized through a ${}^3{P}_0$ quark-pair creation model. With the usual form of this operator, the mass shifts are large and negative, an outcome which raises serious issues of validity for the quenched quark model. Herein, therefore, we introduce some improvements of the ${}^3{P}_0$ operator in order to reduce the size of the mass shifts. By introducing two simple factors, physically well motivated, the coupling between $q\overline{q}$ and $(q\overline{q}{)}^2$ components is weakened, producing mass shifts that are around 10%–20% of hadron bare masses.

In the framework of a nonrelativistic chiral quark model, we continue to study the mass spectra of the fully-heavy $bb\overline{c}\overline{c}$ and $bc\overline{b}\overline{c}$ tetraquarks. In the present calculations, two structures, meson-meson [$\overline{Q}Q$] [$\overline{Q}Q$] and diquark-antidiquark [$QQ$] [$\overline{Q}\overline{Q}$] ($Q=c$ or $b$), and their mixing, along with all possible color, spin configurations are considered. The calculations suggest that no bound state can be formed for $bb\overline{c}\overline{c}$ and $bc\overline{b}\overline{c}$ systems. However, resonances are possible because of the color structure. Several resonances are predicted and their stabilities are checked using the real scaling method.

Inspired by the report of D0 Collaboration on the state $X(5568)$ with four different flavors, a similar state, $ud\overline{s}\overline{b,}$ is investigated in the present work. The advantage of looking for this state over the state $X(5568)$ with quark contents, $bu\overline{d}\overline{s}$ or $bd\overline{u}\overline{s}$, is that the $BK$ threshold is 270 MeV higher than that of $B_s\pi$, and it allows a large mass region for $ud\overline{s}\overline{b}$, which cannot decay to $B_s\pi$, to be stable. The chiral quark model and Gaussian expansion method are employed to do the calculations of four-quark states, $ud\overline{s}\overline{b}$, with quantum numbers $I{J}^P$ ($I=0,1;J=0,1,2;P=+$). Two structures, diquark-antidiquark and meson-meson, with all possible color configurations are considered. The results indicate that energies of the tetraquark with diquark-antiquark configuration are all higher than the threshold of $BK$, but the resonances are still possible because of their structures. For the state of $I{J}^P=00^{+}$ in the meson-meson structure, the energies are just below the corresponding thresholds, where the color channel coupling plays an important role. The bound state is possible. The distances between two objects (quark/antiquark) show that the state is a molecular one.

Stimulated by the newly observed charged hidden-charm state $Z_{cs}(3985{)}^{-}$ by BESIII Collaboration, $Z_{cs}(4000{)}^{+}$, $Z_{cs}(4220{)}^{+}$ and the excited $B_s^0$ states by LHCb Collaboration, a full calculation including masses, decay widths, and the inner structures of the states has emerged in the chiral quark model. For $Z_{cs}$ states, we assign quantum numbers $I(J^P)=\frac{1}{2}(1^{+})$ and quark composition $c\overline{c}s\overline{u}$ according to the experiment. For $B_s^0$ states, systematically, investigations are performed with $I(J^P)=0(0^{+}),0(1^{+}),0(2^{+})$ in both two-body $b\overline{s}$ and four-body $b\overline{s}q\overline{q}$ ($q=u$ or $d$) systems. Each tetraquark calculation takes all structures including meson-meson, diquark-antidiquark, and all possible color configurations into account. Among the numerical techniques to solve the two-body and four-body Schrödinger equation, the spatial wave functions are expanded in series of Gaussian basis functions for high precision, which is the way Gaussian expansion method (GEM) so called. Our results indicate that the low-lying states of the four-quark system are all higher than the corresponding thresholds either for $c\overline{c}s\overline{u}$ or for $b\overline{s}q\overline{q}$ systems. With the help of the real scaling method, we found two molecular resonance states with masses of 4023 and 4042 MeV for the $c\overline{c}s\overline{u}$ system. The state $c\overline{c}s\overline{u}(4042)$ has a close mass and decay width with the recent observed state $Z_{cs}(3985{)}^{-}$. For the $b\overline{s}q\overline{q}$ system with $J=0$, some resonance states are also found. The newly observed excited $B_s^0$ states can be accommodated in the chiral quark model as $2S$ or $1D$ states, and the mixing with four-quark states also needs to be considered.

Recently, some progress has been made in the experiments on double-heavy tetraquarks, such as reported by the LHCb Collaboration and reported by the Belle Collaboration. Coming on the heels of our previous work about and , we present a study on the bound and resonance states of their companions, ( ) tetraquarks with strange flavor in the chiral quark model. Two pictures, meson-meson and diquark-antidiquark ones, and their couplings were considered in our calculations. Isospin violation was neglected herein. Our numerical analysis indicated that the states with and with are the most promising stable states against strong interactions. Besides, we found several resonance states for the double-heavy strange tetraquarks with the real scaling method.