J. L. Ping

Lepton scattering is an established ideal tool for studying inner structure of small particles such as nucleons as well as nuclei. As a future high energy nuclear physics project, an Electron-ion collider in China (EicC) has been proposed. It will be constructed based on an upgraded heavy-ion accelerator, High Intensity heavy-ion Accelerator Facility (HIAF) which is currently under construction, together with a new electron ring. The proposed collider will provide highly polarized electrons (with a polarization of $\sim$80%) and protons (with a polarization of $\sim$70%) with variable center of mass energies from 15 to 20 GeV and the luminosity of (2-3) $\times$ 10$^{33}$ cm$^{-2}$ s$^{-1}$. Polarized deuterons and Helium-3, as well as unpolarized ion beams from Carbon to Uranium, will be also available at the EicC. The main foci of the EicC will be precision measurements of the structure of the nucleon in the sea quark region, including 3D tomography of nucleon; the partonic structure of nuclei and the parton interaction with the nuclear environment; the exotic states, especially those with heavy flavor quark contents. In addition, issues fundamental to understanding the origin of mass could be addressed by measurements of heavy quarkonia near-threshold production at the EicC. In order to achieve the above-mentioned physics goals, a hermetical detector system will be constructed with cutting-edge technologies. This document is the result of collective contributions and valuable inputs from experts across the globe. The EicC physics program complements the ongoing scientific programs at the Jefferson Laboratory and the future EIC project in the United States. The success of this project will also advance both nuclear and particle physics as well as accelerator and detector technology in China.

Inspired by the discovery of two pentaquarks $P_{c}(4380)$ and $P_{c}(4450)$ at the LHCb detector, we study possible hidden-charm molecular pentaquarks in the framework of quark delocalization color screening model. Our results suggest that both $N\eta_{c}$ with $IJ^{P}=\frac{1}{2}\frac{1}{2}^{-}$ and $NJ/\psi$ with $IJ^{P}=\frac{1}{2}\frac{3}{2}^{-}$ are bounded by channels coupling. However, $NJ/\psi$ with $IJ^{P}=\frac{1}{2}\frac{3}{2}^{-}$ may be a resonance state in the $D-$wave $N\eta_{c}$ scattering process. Moreover, $P_{c}(4380)$ can be explained as the molecular pentaquark of $\Sigma^{*}_{c}D$ with quantum numbers $IJ^{P}=\frac{1}{2}\frac{3}{2}^{-}$. The state $\Sigma^{*}_{c}D^{*}$ with $IJ^{P}=\frac{1}{2}\frac{5}{2}^{-}$ is a resonance, it may not be a good candidate of the observed $P_{c}(4450)$ because of the opposite parity of the state to $P_{c}(4380)$, although the mass of the state is not far from the experimental value. In addition, the calculation is extended to the hidden-bottom pentaquarks, the similar properties as that of hidden-charm pentaquarks system are obtained.

With the development of high energy physics experiments, a large amount of exotic states in the hadronic sector have been observed. In order to shed some insights on the nature of the tetraquark and pentaquark candidates, a constituent quark model, along with the gaussian expansion method, has been employed systematically in the real- and complex-range investigations. We review herein the double- and full-heavy tetraquarks, but also the hidden-charm, -bottom and doubly charmed pentaquarks. Several experimentally observed exotic hadrons are well reproduced within our approach; moreover, their possible compositeness are suggested and other properties analyzed accordingly such as their decay widths and general patterns in the spectrum. Besides, we report also some predictions not yet seen by experiment within the studied tetraquark and pentaquark sectors.

Abstract The full-heavy tetraquarks $$bb{\bar{b}}{\bar{b}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mi>b</mml:mi> <mml:mover> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mover> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> and $$cc{\bar{c}}{\bar{c}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>c</mml:mi> <mml:mover> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mover> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> are systematically investigated within the chiral quark model and the quark delocalization color screening model. Two structures, meson–meson and diquark–antidiquark, are considered. For the full-beauty $$bb{\bar{b}}{\bar{b}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mi>b</mml:mi> <mml:mover> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mover> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> systems, there is no any bound state or resonance state in two structures in the chiral quark model, while the wide resonances with masses around $$19.1-19.4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>19.1</mml:mn> <mml:mo>-</mml:mo> <mml:mn>19.4</mml:mn> </mml:mrow> </mml:math> GeV and the quantum numbers $$J^{P}=0^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mi>P</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> , $$1^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:math> , and $$2^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:math> are possible in the quark delocalization color screening model. For the full-charm $$cc{\bar{c}}{\bar{c}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>c</mml:mi> <mml:mover> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mover> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> systems, the results are qualitative consistent in two quark models. No bound state can be found in the meson–meson configuration, while in the diquark–antidiquark configuration there may exist the resonance states, with masses range between 6.2 to 7.4 GeV, and the quantum numbers $$J^{P}=0^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mi>P</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> , $$1^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:math> , and $$2^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:math> . And the separation between the diquark and the antidiquark indicates that these states may be the compact resonance states. The reported state X (6900) is possible to be explained as a compact resonance state with $$IJ^{P}=00^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msup> <mml:mi>J</mml:mi> <mml:mi>P</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>00</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> in present calculation. All these full-charm resonance states are worth searching in the experiments further.

In the framework of the chiral quark model along with complex scaling range, we perform a dynamical study on the low-lying $S$-wave doubly-heavy tetraquark states ($QQ\bar{q}\bar{q}$, $Q=c, b$ and $q=u, d$) with an accurate computing approach, Gaussian expansion method. The meson-meson and diquark-antidiquark configurations within all possible color structures for spin-parity quantum numbers $J^P=0^+$, $1^+$ and $2^+$, and in the $0$ and $1$ isospin sectors are considered. Possible tightly bound and narrow resonance states are obtained for doubly-charm and doubly-bottom tetraquarks with $IJ^P=01^+$, and these exotic states are also obtained in charm-bottom tetraquarks with $00^+$ and $01^+$ quantum numbers. Only loosely bound state is found in charm-bottom tetraquarks of $02^+$ states. All of these bound states within meson-meson configurations are loosely bound whether in color-singlet channels or coupling to hidden-color ones. However compact structures are available in diquark-antidiquark channels except for charm-bottom tetraquarks in $02^+$ states.

We use a color-magnetic interaction model (CMIM), a traditional constituent quark model (CQM) and a multiquark color flux-tube model (MCFTM) to systematically investigate the properties of the states $[Q_1Q_2][\bar{Q}_3\bar{Q}_4]$ ($Q=c,b$). The dynamical investigation indicates that the CMIM can not completely absorb QCD dynamical effects through the effective constituent quark mass and overestimates the color-magnetic interaction in the states under the assumption of the same spatial configurations. The Coulomb interaction plays a critical role in the dynamical model calculations on the heavy hadrons, which induces the fact that none of bound states $[Q_1Q_2][\bar{Q}_3\bar{Q}_4]$ can be found in the dynamical models. The color configuration $\left[[Q_1Q_2]_{\mathbf{6}_c}[\bar{Q}_3\bar{Q}_4]_{\bar{\mathbf{6}}_c}\right]_{\mathbf{1}}$ should be taken seriously in the ground states due to the strong Coulomb attraction between the $[Q_1Q_2]_{\mathbf{6}_c}$ and $[\bar{Q}_3\bar{Q}_4]_{\bar{\mathbf{6}}_c}$. The color configuration $\left[[Q_1Q_2]_{\bar{\mathbf{3}}_c}[\bar{Q}_2\bar{Q}_4]_{\mathbf{3}_c}\right]_{\mathbf{1}}$ is absolutely dominant in the excited states because of the strong Coulomb attraction within the $[Q_1Q_2]_{\bar{\mathbf{3}}_c}$ and $[\bar{Q}_2\bar{Q}_4]_{\mathbf{3}_c}$. The $J/\Psi$-pair resonances recently observed by LHCb are difficult to be accommodated in the CMIM. The broad structure ranging from 6.2 to 6.8 GeV can be described as the ground tetraquark state $[cc][\bar{c}\bar{c}]$ in the various dynamical models. The narrow structure $X(6900)$ can be identified as the excited state $[cc][\bar{c}\bar{c}]$ with $L=1$ ($L=2$) in the CQM (MCFTM).

We systematically investigate the stability of the doubly heavy tetraquark states $$[Q_1Q_2][\bar{q}_3\bar{q}_4]$$ ($$Q=c$$ and b, $$q=u$$, d and s) within the framework of the alternative color flux-tube model involving a multi-body confinement potential, $$\sigma $$-exchange, one-gluon exchange and one-Goldstone-boson-exchange interactions. Our numerical analysis indicates that the states $$[bb][\bar{u}\bar{d}]$$ with $$01^+$$ and $$[bb][\bar{u}\bar{s}]$$ with $$1^+$$ are the most promising stable states against the strong interaction. The states $$[cc][\bar{u}\bar{d}]$$ with $$01^+$$, $$[bc][\bar{u}\bar{d}]$$ with $$00^+$$ and $$01^+$$, and $$[bb][\bar{u}\bar{d}]$$ with $$01^-$$ as stable states are also predicted in the model. The dynamical mechanism producing those stable states with tetrahedral structure are discussed in the model. The states $$[bb][\bar{u}\bar{d}]$$ and $$[bc][\bar{u}\bar{d}]$$ with $$IJ^P=12^+$$ would not be stable although they are, respectively, below the $$\bar{B}^*\bar{B}^*$$ and $$D^*\bar{B}^*$$ thresholds because they can decay into D-wave $$\bar{B}\bar{B}$$ and $$D\bar{B}$$ through the strong interaction. The states should have narrow widths if they really exist as our model prediction.

Recently, a new hidden-charm pentaquark state $P_{cs}(4459)$ was reported by the LHCb Collaboration. Stimulated by the fact that all hidden-charm pentaquark states in $S=0$ systems were successfully predicted by our chiral quark model, we extended this study to the $S=-1$ systems. All possible quantum numbers $IJ^P=0(\frac{1}{2})^-$, $0(\frac{3}{2})^-$, $0(\frac{5}{2})^-$, $1(\frac{1}{2})^-$, $1(\frac{3}{2})^-$ and $1(\frac{5}{2})^-$ have been investigated. The calculation results shows that the newly observed state $P_{cs}(4459)$ can be explained as $\Xi_c \bar{D}^*$ molecular state and the quantum number is $0(\frac{1}{2})^-$. In addition, we also find other molecular states $\Xi_c \bar{D}$, $\Xi_c^* \bar{D}$ and $\Xi_c' \bar{D}^*$. It is worth mentioning that $\Xi_c \bar{D}^*$ can form a two-peak structure from states in system $0(\frac{1}{2})^-$ and $0(\frac{3}{2})^-$. The decay width of all molecular states is given with the help of real scaling method. These hidden-charm pentaquark states is expected to be further verified in future experiments.

The potential of hyperspectral imaging and X-ray techniques for the non-destructive determination of the ginsenosides Rg1 + Re and Rb1 in ginseng was investigated. The random forest (RF) models were established using spectral information extracted from hyperspectral data to predict ginsenosides content. The RF model was optimized by data pre-processing methods and feature screening methods. Multiple feature screening methods combined with partial least squares regression models were used to find hyperspectral image feature information (color information and texture information) related to ginsenosides. A significant positive correlation between density extracted from X-ray images and the ginsenosides content was found by building the univariate linear regression models. Finally, the prediction performance of the integrated learning model based on the three data blocks was better than the model constructed by single data blocks (Rg1 + Re: R2p = 0.8691, RMSEP = 0.0439%; Rb1: R2p = 0.8291, RMSEP = 0.0803%). The results indicate that the developed method is highly feasible for non-destructive evaluation of ginseng quality.

Many proposals have been put forward to explore four-quark states $QQ\overline{q}\overline{q}$ ($Q=s$, $c$, $b$; $q=u$, $d$) by experiment, so a systematic study of $QQ\overline{q}\overline{q}$ spectrum with different constituent quark models by a high precision, few-body method, the Gaussian expression method, is useful. Three quark models: the Bhaduri, Cohler, Nogami quark model, the chiral quark model (ChQM), and the quark delocalization color screening model are all employed for a systematic calculation of the $S$-wave $QQ\overline{q}\overline{q}$ spectrum with different color structures, using the Gaussian expression method. The results show that only the $bb\overline{q}\overline{q}$ state with $(I,J)=(0,1)$ is bound in different color structures within the different quark models. The binding energy varies from several MeV for a di-meson structure to over 100 MeV for a diquark-antidiquark structure. For the $cc\overline{q}\overline{q}$ system, the state with $(I,J)=(0,1)$ is bound in a di-meson structure, and also bound in a diquark-antidiquark structure if pseudoscalar meson exchanges are accounted for. All are weakly bound states. The mixture of diquark-antidiquark and molecular structures is discussed in the framework of quark models for the first time; $cc\overline{q}\overline{q}$ with $(I,J)=(0,1)$ is below the threshold in addition to $bb\overline{q}\overline{q}$ in both the ChQM and the Bhaduri, Cohler, Nogami quark model. In the same channel, $ss\overline{q}\overline{q}$ is also a possible bound state with mass around 1.4 GeV in ChQM.

We perform a dynamical calculation of the $\ensuremath{\Delta}\ensuremath{\Delta}$ dibaryon candidates with $I{J}^{P}={03}^{+}$ and $I{J}^{P}={30}^{+}$ in the framework of two constituent quark models: the quark delocalization color screening model and the chiral quark model. Our results show that the dibaryon resonances with $I{J}^{P}={03}^{+}$ and $I{J}^{P}={30}^{+}$ can be formed in both models. The mass and width of $I{J}^{P}={03}^{+}$ state are smaller than that of $I{J}^{P}={30}^{+}$ state due to the one-gluon-exchange interaction between quarks. The resonance mass and decay width of $I{J}^{P}={03}^{+}$ state in both models agree with that of the recently observed resonance in the reaction $pn\ensuremath{\rightarrow}d{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$. The $I{J}^{P}={30}^{+}$ $\ensuremath{\Delta}\ensuremath{\Delta}$ is another dibaryon candidate with smaller binding energy and larger width. The hidden-color channel coupling is added to the chiral quark model, and we find it can lower the mass of the dibaryons by 10$\ensuremath{-}$20 MeV.

The recent experimental results of the LHCb Collaboration suggested the existence of pentaquark states with a charmonium. To understand the structure of the states, a dynamical calculation of 5-quark systems with quantum numbers $I{J}^{P}=\frac{1}{2}(\frac{1}{2}{)}^{\ifmmode\pm\else\textpm\fi{}}$, $\frac{1}{2}(\frac{3}{2}{)}^{\ifmmode\pm\else\textpm\fi{}}$ and $\frac{1}{2}(\frac{5}{2}{)}^{\ifmmode\pm\else\textpm\fi{}}$ is performed in the framework of the chiral quark model with the help of the Gaussian expansion method. The results show that there are several negative parity resonance states while all of the positive parity states are the scattering states. The ${P}_{c}(4380)$ state is suggested to be the pentaquark state of ${\mathrm{\ensuremath{\Sigma}}}_{c}^{*}\overline{D}$. Although the energy of ${\mathrm{\ensuremath{\Sigma}}}_{c}{\overline{D}}^{*}$ is very close to the mass of ${P}_{c}(4450)$, the inconsistent parity prevents the assignment. The calculated distances between quarks confirm the molecular nature of the states.

Inspired by $$\varXi _{cc}$$ 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–antidiquark 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 iso-vector $$QQ{\bar{q}}{\bar{q}}$$ system, while there are rather deep bound states in the iso-scalar $$bb{\bar{q}}{\bar{q}}$$ , $$cc{\bar{q}}{\bar{q}}$$ 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 iso-scalar 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. We also compare our result with that of other approaches.

Abstract The constituent quark model is used to compute the ground and excited state masses of QQQ baryons containing either c or b quarks. The quark model parameters previously used to describe the properties of charmonium and bottomonium states were used in this analysis. The non-relativistic three-body bound state problem is solved by means of the Gaussian expansion method which provides sufficient accuracy and simplifies the subsequent evaluation of the matrix elements. Several low-lying states with quantum numbers <?CDATA $ J^P=\frac{1}{2}^\pm, \frac{3}{2}^\pm, \frac{5}{2}^\pm$?> and <?CDATA $ \frac{7}{2}^+$?> are reported. We compare the results with those obtained by the other theoretical formalisms. There is a general agreement for the mass of the ground state in each sector of triply heavy baryons. However, the situation is more puzzling for the excited states, and appropriate comments about the most relevant features of our comparison are given.

Abstract A contact interaction is used to calculate an array of pion twist-two, -three and -four generalised transverse light-front momentum dependent parton distribution functions (GTMDs). Despite the interaction’s simplicity, many of the results are physically relevant, amongst them a statement that GTMD size and shape are largely prescribed by the scale of emergent hadronic mass. Moreover, proceeding from GTMDs to generalised parton distributions, it is found that the pion’s mass distribution form factor is harder than its electromagnetic form factor, which is harder than the gravitational pressure distribution form factor; the pressure in the neighbourhood of the pion’s core is commensurate with that at the centre of a neutron star; the shear pressure is maximal when confinement forces become dominant within the pion; and the spatial distribution of transversely polarised quarks within the pion is asymmetric. Regarding transverse momentum dependent distribution functions, their magnitude and domain of material support decrease with increasing twist. The simplest Wigner distribution associated with the pion’s twist-two dressed-quark GTMD is sharply peaked on the kinematic domain associated with valence-quark dominance; has a domain of negative support; and broadens as the transverse position variable increases in magnitude.

Abstract Motivated by the recent observation of the $$\Lambda _c(2910)^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>2910</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math> state in the $$\Sigma _{c}(2455)^{0,++}\pi ^{\pm }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Σ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>2455</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:mrow></mml:math> spectrum by the Belle Collaboration, we investigate the explanation of $$\Lambda _{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Λ</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math> states within the pentaquark framework using the quark delocalization color screening model (QDCSM). To check for bound states and resonance states, we utilize the real-scaling method. Additionally, we calculate the root mean square of cluster spacing to study the structure of the states and further estimate whether a state is a resonance state or not. Our numerical results show that $$\Lambda _{c}(2910)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>2910</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> cannot be interpreted as a molecular state, and $$\Sigma _{c}(2800)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Σ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>2800</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> cannot be explained as the ND molecular state with $$J^P=1/2^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mo>-</mml:mo></mml:msup></mml:mrow></mml:math> . However, we find that $$\Lambda _{c}(2595)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>2595</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> can be interpreted as a molecular state with $$J^P=\frac{1}{2}^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>-</mml:mo></mml:msup></mml:mrow></mml:math> , where the main component is $$\Sigma _{c}\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Σ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mi>π</mml:mi></mml:mrow></mml:math> . Similarly, we interpret $$\Lambda _{c}(2625)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>2625</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> as a molecular state with $$J^P=\frac{3}{2}^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>-</mml:mo></mml:msup></mml:mrow></mml:math> , where the main component is $$\Sigma _{c}^{*}\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>Σ</mml:mi><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mrow /><mml:mo>∗</mml:mo></mml:mrow></mml:msubsup><mml:mi>π</mml:mi></mml:mrow></mml:math> . We also suggest that $$\Lambda _{c}(2940)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>2940</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is likely to be interpreted as a molecular state with $$J^P=3/2^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mo>-</mml:mo></mml:msup></mml:mrow></mml:math> , where the main component is $$ND^{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mrow /><mml:mo>∗</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math> . In addition, we predict two new states: a $$J^P=3/2^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mo>-</mml:mo></mml:msup></mml:mrow></mml:math> $$\Sigma _{c}\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Σ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mi>ρ</mml:mi></mml:mrow></mml:math> resonance state with a mass of 3140–3142 MeV, and a $$J^P=\frac{5}{2}^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mfrac><mml:mn>5</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>-</mml:mo></mml:msup></mml:mrow></mml:math> $$\Sigma _{c}^*\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>Σ</mml:mi><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup><mml:mi>ρ</mml:mi></mml:mrow></mml:math> state with a mass about 3187–3188 MeV.

According to the classification of the quark model, the hadrons going beyond three-quark baryon and quark-antiquark meson pictures are called exotic hadrons. Many new hadrons have been observed since 2003, some of which exhibit exotic behaviors. There are a lot of excellent review articles on exotic hadrons available so far; the present article tries to focus on the recent experimental and theoretical progress on the exotic states from the perspective of the quark model. Although lattice quantum chromodynamics may give the final answer of the problem, the phenomenological models are still powerful tools to explore the exotic states and to provide insight on the phenomenology of hadrons. The spatial and color structures of multiquark states and the channel coupling calculation are emphasized through reviewing some bound states, molecular and color structure resonances. Finally, the unquench effects of some exotic states are reviewed. With the accumulation of experimental data on multiquark states and inspiration of underlying theory developments, more reasonable phenomenological models incorporating multi-body interactions and high Fock components to unify the description of normal hadrons and exotic hadrons are expectable.

Inspired by the observation of hidden-charm pentaquark ${P}_{c}$ and ${P}_{cs}$ states by the LHCb Collaboration, we explore the $qqc\overline{c}c$ ($q=u$ or $d$) pentaquark systems in the quark delocalization color screening model. The interaction between baryons and mesons and the influence of channel coupling are studied in this work. Three compact $qqc\overline{c}c$ pentaquark states are obtained, whose masses are 5259 MeV with $I({J}^{P})=0(1/{2}^{\ensuremath{-}})$, 5396 MeV with $I({J}^{P})=1(1/{2}^{\ensuremath{-}})$, and 5465 MeV with $I({J}^{P})=1(3/{2}^{\ensuremath{-}})$. Two molecular states are obtained, which are $I({J}^{P})=0(1/{2}^{\ensuremath{-}}){\mathrm{\ensuremath{\Lambda}}}_{c}J/\ensuremath{\psi}$ with 5367 MeV and $I({J}^{P})=0(5/{2}^{\ensuremath{-}})\text{ }\text{ }{\mathrm{\ensuremath{\Xi}}}_{cc}^{*}{\overline{D}}^{*}$ with 5690 MeV. These predicted states may provide important information for future experimental search.

The quark delocalization and color screening model, a quark potential model, is used for a systematic search of dibaryon candidates in the u, d, and s three flavor world. Color screening, which appears in unquenched lattice gauge calculations, and quark delocalization (which is similar to electron delocalization in molecular physics) are both included. Flavor symmetry breaking and channel coupling effects are studied. The model is constrained not only by baryon ground state properties but also by the N-N scattering phase shifts. The deuteron and zero energy dinucleon resonance are both reproduced qualitatively. The model predicts two extreme types of dibaryonic systems: ``molecular'' like the deuteron, and highly delocalized six-quark systems among which only a few narrow dibaryon resonances occur in the u, d, and s three flavor world. Possible high spin dibaryon resonances are emphasized.

We look for $\ensuremath{\Delta}\ensuremath{\Delta}$ and $N\ensuremath{\Delta}$ resonances by calculating $\mathit{NN}$ scattering phase shifts of two interacting baryon clusters of quarks with explicit coupling to these dibaryon channels. Two phenomenological nonrelativistic chiral quark models giving similar low-energy $\mathit{NN}$ properties are found to give significantly different dibaryon resonance structures. In the chiral quark model (ChQM), the dibaryon system does not resonate in the $\mathit{NNS}$ waves, in agreement with the experimental SP07 $\mathit{NN}$ partial-wave scattering amplitudes. In the quark delocalization and color screening model (QDCSM), the $S$-wave $\mathit{NN}$ resonances disappear when the nucleon size $b$ falls below 0.53 fm. Both quark models give an ${\mathit{IJ}}^{P}={03}^{+}\ensuremath{\Delta}\ensuremath{\Delta}$ resonance. At $b=0.52$ fm, the value favored by the baryon spectrum, the resonance mass is 2390 (2420) MeV for the ChQM with quadratic (linear) confinement, and 2360 MeV for the QDCSM. Accessible from the ${}^{3}{D}_{3}^{\mathit{NN}}$ channel, this resonance is a promising candidate for the known isoscalar ABC structure seen more clearly in the $\mathit{pn}\ensuremath{\rightarrow}d\ensuremath{\pi}\ensuremath{\pi}$ production cross section at 2410 MeV in the recent preliminary data reported by the CELSIUS-WASA Collaboration. In the isovector dibaryon sector, our quark models give a bound or almost bound ${}^{5}{S}_{2}^{\ensuremath{\Delta}\ensuremath{\Delta}}$ state that can give rise to a ${}^{1}{D}_{2}^{\mathit{NN}}$ resonance. None of the quark models used have bound $N\ensuremath{\Delta}P$ states that might generate odd-parity resonances.

In the framework of the color flux-tube model with a four-body confinement potential, the lowest charged tetraquark states $[Qq][{\overline{Q}}^{\ensuremath{'}}{\overline{q}}^{\ensuremath{'}}](Q=c,b,q=u,d,s)$ are studied by using the variational method, the Gaussian expansion method. The results indicate that some compact resonance states with three-dimensional spatial structures can be formed. These states cannot decay into two color singlet mesons $Q{\overline{q}}^{\ensuremath{'}}$ and ${\overline{Q}}^{\ensuremath{'}}q$ through the breakdown and recombination of color flux tubes but into $Q{\overline{Q}}^{\ensuremath{'}}$ and $q{\overline{q}}^{\ensuremath{'}}$. The four-body confinement potential is a crucial dynamical mechanism for the formation of these compact resonance states. The decay process is similar to that of a compound nucleus but due to the multibody color confinement. The newly observed charged states ${Z}_{c}(3900)$ and ${Z}_{c}(4025)/{Z}_{c}(4020)$ can be interpreted as the $S$-wave tetraquark states $[cu][\overline{c}\overline{d}]$ with quantum numbers $I{J}^{P}=1{1}^{+}$ and $1{2}^{+}$, respectively.

The LHCb Collaboration has recently reported strong evidences of the existence of pentaquark states in the hidden-charm baryon sector, the so-called ${P}_{c}(4380{)}^{+}$ and ${P}_{c}(4450{)}^{+}$ signals. Five-quark bound states in the hidden-charm sector were explored by us using, for the quark-quark interaction, a chiral quark model which successfully explains meson and baryon phenomenology, from the light to the heavy quark sector. We extend herein such study into the hidden-bottom pentaquark sector, analyzing possible bound-states with spin-parity quantum numbers ${J}^{P}={\frac{1}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, ${\frac{3}{2}}^{\ifmmode\pm\else\textpm\fi{}}$ and ${\frac{5}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, and in the $\frac{1}{2}$ and $\frac{3}{2}$ isospin sectors. We do not find positive parity hidden-bottom pentaquark states; however, several candidates with negative parity are found with dominant baryon-meson structures ${\mathrm{\ensuremath{\Sigma}}}_{b}^{(*)}{\overline{B}}^{(*)}$. Their inner structures have been also analyzed with the computation of the distance among any pair of quarks within the bound-state. This exercise reflects that molecular-type bound-states are favored when only color-singlet configurations are considered in the coupled-channels calculation whereas some deeply-bound compact pentaquarks can be found when hidden-color configurations are added. Finally, our findings resemble the ones found in the hidden-charm sector but, as expected, we find in the hidden-bottom sector larger binding energies and bigger contributions of the hidden-color configurations.

Inspired by the recent report of the exotic states ${X}_{0}(2900)$ and ${X}_{1}(2900)$ with four different quark flavors in the ${D}^{\ensuremath{-}}{K}^{+}$ invariant-mass distributions of the decay process ${B}^{+}\ensuremath{\rightarrow}{D}^{+}{D}^{\ensuremath{-}}{K}^{+}$ by the LHCb Collaboration, we systematically investigate the tetraquarks composed of $ud\overline{s}\overline{c}$ with meson-meson and diquark-antidiquark structures in the quark delocalization color-screening model. We find that the ${X}_{0}(2900)$ can be interpreted as the molecular state $\overline{D}{\text{ }}^{*}{K}^{*}$ with $I{J}^{P}=0{0}^{+}$. Moreover, two bound states are obtained by the channel coupling calculation, with energies 2341.2 MeV for $I{J}^{P}=0{0}^{+}$ and 2489.7 MeV for $I{J}^{P}=0{1}^{+}$, respectively. We also extend our study to the $uc\overline{d}\overline{s}$ systems and find that there is no $S$-wave bound state, so the ${D}_{s0}(2317)$ cannot be identified as the $DK$ molecular state in the present calculation. Besides, several resonance states with the diquark-antidiquark configuration are possible in both $ud\overline{s}\overline{c}$ and $uc\overline{d}\overline{s}$ systems. The states composed of $ud\overline{s}\overline{b}$ are also possible open bottom tetraquark candidates. All of these open charm and open bottom bound states and resonances are worth investigating in future experiments.

Fully-heavy tetraquark states, i.e. $cc\bar{c}\bar{c}$, $bb\bar{b}\bar{b}$, $bb\bar{c}\bar{c}$ ($cc\bar{b}\bar{b}$), $cb\bar{c}\bar{c}$, $cb\bar{b}\bar{b}$, and $cb\bar{c}\bar{b}$, are systematically investigated by means of a non-relativistic quark model based on lattice-QCD studies of the two-body $Q\bar{Q}$ interaction, which exhibits a spin-independent Cornell potential along with a spin-spin term. The four-body problem is solved using the Gaussian expansion method; additionally, the so-called complex scaling technique is employed so that bound, resonance, and scattering states can be treated on the same footing. Moreover, a complete set of four-body configurations, including meson-meson, diquark-antidiquark, and K-type configurations, as well as their couplings, are considered for spin-parity quantum numbers $J^{P(C)}=0^{+(+)}$, $1^{+(\pm)}$, and $2^{+(+)}$ in the $S$-wave channel. Several narrow resonances, with two-meson strong decay widths less than 30 MeV, are found in all of the tetraquark systems studied. Particularly, the fully-charm resonances recently reported by the LHCb Collaboration, at the energy range between 6.2 and 7.2 GeV in the di-$J/\psi$ invariant spectrum, can be well identified in our calculation. Focusing on the fully-bottom tetraquark spectrum, resonances with masses between 18.9 and 19.6 GeV are found. For the remaining charm-bottom cases, the masses are obtained within a energy region from 9.8 GeV to 16.4 GeV. All these predicted resonances can be further examined in future experiments.

Motivated by the two charged bottomonium-like resonances $Z_b$(10610) and $Z_b$(10650) newly observed by the Belle collaboration, the possible molecular states composed of a pair of heavy mesons, $B\bar{B}, B\bar{B}^*, B^*\bar{B}^*, B_s\bar{B}$, etc (in S-wave), are investigated in the framework of chiral quark models by the Gaussian expansion method. The bound states $B\bar{B}^*$ and $B^*\bar{B}^*$ with quantum numbers $I(J^{PC})=1(1^{+-})$, which are good candidates for the $Z_b(10610)$ and $Z_b(10650)$ respectively, are obtained. Other three bound states $B\bar{B}^*$ with $I(J^{PC})=0(1^{++})$, $B^*\bar{B}^*$ with $I(J^{PC})=1(0^{++}), 0(2^{++})$ are predicted. These states may be observed in open-bottom or hidden-bottom decay channel of highly excited $\Upsilon$. When extending directly the quark model to the hidden color channel of the multi-quark system, more deeply bound states are found. Future experimental search of those states will cast doubt on the validity of applying the chiral constituent quark model to the hidden color channel directly.

Inspired by the five newly observed $\Omega^{0}_{c}$ states by the LHCb detector, we study the $\Omega_{c}^{0}$ states as the $S-$wave molecular pentaquarks with $I=0$, $J^{P}=\frac{1}{2}^{-}$, $\frac{3}{2}^{-}$, and $\frac{5}{2}^{-}$ by solving the RGM equation in the framework of chiral quark model. Both the energies and the decay widths are obtained in this work. Our results suggest that $\Omega_{c}(3119)^{0}$ can be explained as an $S-$wave resonance state of $\Xi D$ with $J^{P}=\frac{1}{2}^{-}$, and the decay channels are the $S-$wave $\Xi_{c} K$ and $\Xi^{'}_{c}K$ . Other reported $\Omega^{0}_{c}$ states cannot be obtained in our present calculation. Another $\Omega_{c}^{0}$ state with much higher mass 3533 MeV with $J^{P}=\frac{5}{2}^{-}$ is also obtained. In addition, the calculation is extended to the $\Omega_{b}^{0}$ states, similar results as that of $\Omega^{0}_{c}$ are obtained.

Recently, the experimental results of the LHCb Collaboration suggested the existence of five new excited states of ${\mathrm{\ensuremath{\Omega}}}_{c}^{0}$: ${\mathrm{\ensuremath{\Omega}}}_{c}(3000{)}^{0}$, ${\mathrm{\ensuremath{\Omega}}}_{c}(3050{)}^{0}$, ${\mathrm{\ensuremath{\Omega}}}_{c}(3066{)}^{0}$, ${\mathrm{\ensuremath{\Omega}}}_{c}(3090{)}^{0}$, and ${\mathrm{\ensuremath{\Omega}}}_{c}(3119{)}^{0}$; however, the quantum numbers of these new particles are not determined now. To understand the nature of these states, a dynamical calculation of ${\mathrm{\ensuremath{\Omega}}}_{c}^{0}$ both in five-quark configuration with quantum numbers $I{J}^{P}=0(\frac{1}{2}{)}^{\ensuremath{-}}$, $0(\frac{3}{2}{)}^{\ensuremath{-}}$, $0(\frac{5}{2}{)}^{\ensuremath{-}}$ and in three-quark configuration with positive parity and negative parity was performed in the framework of the chiral quark model with the help of the Gaussian expansion method. The results show the masses both of the $1P$ and the $2S$ states in $ssc$ systems are comparable to experimental data; Besides, $\mathrm{\ensuremath{\Xi}}\overline{D}$, ${\mathrm{\ensuremath{\Xi}}}_{c}\overline{K}$, and ${\mathrm{\ensuremath{\Xi}}}_{c}^{*}\overline{K}$ are also possible candidates of these new particles if the parity is negative. The distances between quark pairs suggest a compact structure nature.

The $1S$, $2S$, $1P$ and $2P$ states of light baryons are investigated within the chiral quark model, paying particular attention to the well-known order reverse problem of $1P$ and $2S$ states. Besides a nonperturbative linear-screened confining interaction and a perturbative one-gluon exchange between quarks, we incorporate the Goldstone-boson exchanges taking into account not only the full octet of pseudoscalar mesons but also the scalar one. The scalar meson exchange potential simulates the higher order multi-pion exchange terms that appear in the chiral Lagrangian and its omission has been already admitted as a deficiency of the original model in describing, for instance, the $\rho-\omega$ splitting. The numerical approach to the three body bound state problem is the so-called Gau\ss ian expansion method, which is able to get a precision as good as Faddeev calculations. With a set of parameters fixed to different hadron and hadron-hadron observables, we find that the chiral potential could play an important role towards the issue on the mass order reverse problem. We extend our calculation to the $qqQ$ and $qQQ$ sectors (with $q$ representing a light $u$-, $d$-, or $s$-quark and $Q$ denoting the charm quark or the bottom one) in which many new states have been recently observed. Some tentative assignments are done attending to the agreement between theoretical and experimental masses; however, we admit that other sources of information are needed in order to make strong claims about the nature of these states.

In the framework of quark delocalization color screening model, both the hidden-charm and hidden-bottom pentaquark resonances are studied in the hadron-hadron scattering process. A few narrow pentaquark resonances with hidden-charm above $4.2$ GeV, and some narrow pentaquark resonances with hidden-bottom above $11$ GeV are found from corresponding scattering processes. Besides, the states $Nη_{c}$, $NJ/ψ$, $Nη_{b}$ and $NΥ$ with $IJ^{P}=\frac{1}{2}\frac{1}{2}^{-}$, as well as $NJ/ψ$ and $NΥ$ with $IJ^{P}=\frac{1}{2}\frac{3}{2}^{-}$ are all possible to be bound by channel-coupling calculation. All these heavy pentaquarks are worth searching in the future experiments.

Stimulated by the recent observation of the exotic X(5568) state by the D0 Collaboration, we study the four-quark system $$us\bar{b}\bar{d}$$ with quantum numbers $$J^P=0^+$$ in the framework of the chiral quark model. Two structures, diquark–antidiquark and meson–meson, with all possible color configurations are investigated by using the Gaussian expansion method. The results show that the energies of the tetraquark states with diquark–antiquark structure are too high to be candidates of X(5568), and no molecular structure can be formed in our calculations. The calculation is also extended to the four-quark system $$us\bar{c}\bar{d}$$ and the same results as that of $$us\bar{b}\bar{d}$$ are obtained.

The LHCb Collaboration, using its full data set from runs $1$ and $2$, announced in $2019$ a surprising update of the hidden-charm pentaquark states $P_c(4380)^+$ and $P_c(4450)^+$, observed in 2015. A new state, $P_c(4312)^+$, was clearly seen at lower energies; furthermore, the original $P_c(4450)$ resonance was resolved into two individual states, named the $P_c(4440)^+$ and the $P_c(4457)^+$. Motivated by the fact that these new hidden-charm pentaquark states were successfully predicted by our chiral quark model, we extend herein such study to the doubly charmed sector. The analyzed total spin and parity quantum numbers are $J^P=\frac12^-$, $\frac32^-$ and $\frac52^-$, in the $I=\frac12$ and $\frac32$ isospin channels. We find several possible narrow baryon-meson resonances (theoretical masses in parenthesis): $IJ^P = \frac12 \frac12^-$ $\Sigma_c D(4356)$, $\frac12 \frac32^-$ $\Sigma^*_c D(4449)$, $\frac32 \frac12^-$ $\Sigma_c D(4431)$, $\frac32 \frac12^-$ $\Sigma_c D(4446)$, $\frac32 \frac32^-$ $\Sigma_c D^*(4514)$ and $\frac32 \frac52^-$ $\Xi^*_{cc} \rho(4461)$ whose widths are $4.8$, $8.0$, $2.6$, $2.2$, $4.0$ and $3.0\,\text{MeV}$, respectively. Moreover, one shallow bound-state is found, too, with quantum numbers $IJ^P = \frac12 \frac32^-$ $\Xi^*_{cc} \pi(3757)$. These doubly charmed pentaquark states are expected to be identified in future experiments.

The hidden-charm tetraquarks with strangeness, $c\bar{c}s\bar{q}$ $(q=u,\,d)$, in $J^P=0^+$, $1^+$ and $2^+$ are systematically investigated in the framework of real- and complex-scaling range of a chiral quark model, whose parameters have been fixed in advance describing hadron, hadron-hadron and multiquark phenomenology. Each tetraquark configuration, compatible with the quantum numbers studied, is taken into account; this includes meson-meson, diquark-antidiquark and K-type arrangements of quarks with all possible color wave functions in four-body sector. Among the different numerical techniques to solve the Schr\"odinger-like 4-body bound state equation, we use a variational method in which the trial wave function is expanded in complex-range Gaussian basis functions, which is characterized by its simplicity and flexibility. This theoretical framework has already been used to study different kinds of multiquark systems, such as the hidden-charm pentaquarks, $P^+_c$, and doubly-charmed tetraquarks, $T^+_{cc}$. The recently reported $Z_{cs}$ states by the BESIII and LHCb collaborations are generally compatible with either compact tetraquark or hadronic molecular resonance configurations in our investigation. Moreover, several additional exotic resonances are found in the mass range between 3.8 GeV and 4.6 GeV.

The low-lying $sQ\bar{q}\bar{q}$ $(q=u,\,d,\, Q=c,\,b)$ tetraquark states with $J^P=0^+$, $1^+$ and $2^+$, and in the isoscalar and isovector sectors, are systematically investigated in the framework of real- and complex-scaling range of a chiral quark model, whose parameters have been fixed in advance describing hadron, hadron-hadron and multiquark phenomenology, and thus all results presented here are pure predictions. Each tetraquark configuration, compatible with the quantum numbers studied, is taken into account; this includes meson-meson, diquark-antidiquark and K-type arrangements of quarks with all possible color wave functions in four-body sector. Among the different numerical techniques to solve the Schr\"odinger-like 4-body bound state equation, we use a variational method in which the trial wave function is expanded in complex-range Gaussian basis functions, because its simplicity and flexibility. Several compact bound states and narrow resonances are found in both charm-strange $cs\bar{q}\bar{q}$ and bottom-strange $bs\bar{q}\bar{q}$ tetraquark sectors, most of them as a product of the strong coupling between the different channels. The so-called $X_{0,1}(2900)$ signals, recently found by the LHCb collaboration, are unstable in our formalism with several candidates in the single-channel computations.

The fully heavy pentaquarks $cccc\overline{c}$ and $bbbb\overline{b}$ are systematically investigated within the chiral quark model and quark delocalization color screening model. The results are consistent in both of the models, and the effect of the channel coupling is crucial for forming a bound state of the fully heavy pentaquark system. Three possible fully heavy pentaquarks are obtained and are the $cccc\overline{c}$ state with ${J}^{P}=1/{2}^{\ensuremath{-}}$ and the mass of 7891.9--7892.7 MeV, the $bbbb\overline{b}$ state with ${J}^{P}=1/{2}^{\ensuremath{-}}$ and the mass of 23810.1--23813.8 MeV, and the $bbbb\overline{b}$ state with ${J}^{P}=3/{2}^{\ensuremath{-}}$ and the mass of 23748.2--23752.3 MeV. All these fully heavy pentaquark states are worth searching in future experiments.

Abstract The existence of fully heavy dibaryons $$\varOmega _{ccc}\varOmega _{bbb}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccc</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbb</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , $$\varOmega _{ccb}\varOmega _{bbc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccb</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbc</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , $$\varOmega _{ccc}\varOmega _{ccc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccc</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccc</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , and $$\varOmega _{bbb}\varOmega _{bbb}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbb</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbb</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> with $$J=0,~1,~2,~3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> and $$P=\pm 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> are investigated in the framework of a constituent quark model with the help of the resonating group method. The dibaryon composed of six c or b quarks with $$J^{P}=0^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mi>P</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> is able to be bound, because the requirement for antisymmetrization between the same baryon clusters introduces an attractive interaction between two fully heavy baryons. Although it is difficult for the dibaryon with the color-singlet type $$\varOmega _{ccc}\varOmega _{bbb}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccc</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbb</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> to form any bound state, it is possible for the $$\varOmega _{ccb}\varOmega _{bbc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccb</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbc</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> state to be bound. The channel coupling of all channels of both $$\varOmega _{ccb}\varOmega _{bbc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccb</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbc</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and $$\varOmega _{ccc}\varOmega _{bbb}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>ccc</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>bbb</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> structures leads to the bound conclusion of this fully heavy system composed of three c quarks and three b quarks.

Abstract The lowest-lying charmonium-like tetraquarks $$c{\bar{c}}q{\bar{q}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mover><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover><mml:mi>q</mml:mi><mml:mover><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:math> $$(q=u,\,d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mspace /><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> and $$c{\bar{c}}s{\bar{s}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mover><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover><mml:mi>s</mml:mi><mml:mover><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:math> , with spin-parity $$J^P=0^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mi>P</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mn>0</mml:mn><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math> , $$1^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>1</mml:mn><mml:mo>+</mml:mo></mml:msup></mml:math> and $$2^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn><mml:mo>+</mml:mo></mml:msup></mml:math> , and isospin $$I=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and 1, are systematically investigated within the theoretical framework of complex-scaling range for a chiral quark model that has already been successfully applied in former studies of various tetra- and penta-quark systems. A four-body S -wave configuration which includes meson–meson, diquark–antidiquark and K-type arrangements of quarks, along with all possible color wave functions, is comprehensively considered. Several narrow resonances are obtained in each tetraquark channel when a fully coupled-channel computation is performed. We tentatively assign theoretical states to experimentally reported charmonium-like signals such as X (3872), $$Z_c(3900)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>3900</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> , X (3960), X (4350), X (4685) and X (4700). They can be well identified as hadronic molecules; however, other exotic components which involve, for instance, hidden-color channels or diquark–antidiquark structures play a considerable role. Meanwhile, two resonances are obtained at 4.04 GeV and 4.14 GeV which may be compatible with experimental data in the energy interval 4.0–4.2 GeV. Furthermore, the X (3940) and X (4630) may be identified as color compact tetraquark resonances. Finally, we also find few resonance states in the energy interval from 4.5 to 5.0 GeV, which would be awaiting for discovery in future experiments.

The quark delocalization and color screening model is used for a systematic study of the effective potential between baryons in the u, d and s sector. The model is constrained by the properties of baryons and N-N scattering. The effective potentials for the N-N (IJ = 01, 10, 11, 00) channels and the N-Λ and N-Σ (IJ = 121, 120, 321, 320) channels fit the N-N, N-Λ and N-Σ scattering data reasonably well. This model predicts: there are rather strong effective attractions between some decuplet-baryons; the effective attractions between octet-baryons are usually weak; and the attractions between decuplet- and octet-baryons lie in between.

We apply the quark delocalization and color screening model to nucleon-baryon scattering. A semi-quantitative fit to N-N, N-Lambda and N-Sigma phase shifts and scattering cross sections is obtained without invoking meson exchange. Quarks delocalize reasonably in all of the different flavor channels to induce effective nucleon-baryon interactions with both a repulsive core and with an intermediate range attraction in the cases expected.

The quark-delocalization, color-screening model, extended by inclusion of a one-pion-exchange (OPE) tail, is applied to the study of the deuteron and the ${d}^{*}$ dibaryon. The results show that the properties of the deuteron (an extended object) are well reproduced, greatly improving the agreement with experimental data as compared to our previous study (without OPE). At the same time, the mass and decay width of the ${d}^{*}$ (a compact object) are, as expected, not altered significantly.

In the framework of constituent quark model, the effect of hidden color channels on the nucleon-nucleon ($NN$) interaction is studied. By adjusting the color confinement strength between the hidden color channels and color singlet channels and/or between the hidden color channels and hidden color channels, the experimental data of $S$ to $I$ partial-wave phase shifts of $NN$ scattering can be fitted well. The results show that the hidden color channel coupling might be important in producing the intermediate-range attraction of $NN$ interaction. The deuteron properties and dibaryon candidates have also been studied with this model.

Two nonrelativistic quark models, a chiral quark model and a quark delocalization color screening model, are employed to calculate the baryon-baryon scattering phase shifts to look for dibaryon resonances with strangeness by means of the resonating group method. The two models predict similar-strangeness dibaryon resonance states. No resonance appears in the octet-octet channels, but resonance did exist in the octet-decuplet and decuplet-decuplet channels. These resonances were distributed in the energy region 2400--2800 MeV. The widths are generally smaller than 10 MeV but increased to tens of MeV after taking into account the off-shell widths of decuplet baryons. These resonances all appear in the $D$-wave nucleon-hyperon and hyperon-hyperon scatterings and can be searched for through hyperon-nucleon scatterings with hyperon beams and hyperon-hyperon vertex and masses reconstruction with the data collected by relativistic heavy ion collisions.

Inspired by the present experimental results of charged charmonium-like states ${Z}_{c}^{+}$, we present a systematic study of the tetraquark states $[cu][\overline{c}\overline{d}]$ in a color flux-tube model with a multibody confinement potential. Our investigation indicates that charged charmonium-like states ${Z}_{c}^{+}(3900)$ or ${Z}_{c}^{+}(3885)$, ${Z}_{c}^{+}(3930)$, ${Z}_{c}^{+}(4020)$ or ${Z}_{c}^{+}(4025)$, ${Z}_{1}^{+}(4050)$, ${Z}_{2}^{+}(4250)$, and ${Z}_{c}^{+}(4200)$ can be described as a family of tetraquark $[cu][\overline{c}\overline{d}]$ states with the quantum numbers $n{^{2S+1}L}_{J}$ and ${J}^{P}$ of $1{^{3}S}_{1}$ and ${1}^{+}$, $2{^{3}S}_{1}$ and ${1}^{+}$, $1{^{5}S}_{2}$ and ${2}^{+}$, $1{^{3}P}_{1}$ and ${1}^{\ensuremath{-}}$, $1{^{5}D}_{1}$ and ${1}^{+}$, and $1{^{3}D}_{1}$ and ${1}^{+}$, respectively. The predicted lowest mass charged tetraquark state $[cu][\overline{c}\overline{d}]$ with ${0}^{+}$ and $1{^{1}S}_{0}$ lies at $3780\ifmmode\pm\else\textpm\fi{}10\text{ }\text{ }\mathrm{MeV}/{\mathrm{c}}^{2}$ in the model. These tetraquark states have compact three-dimensional spatial configurations similar to a rugby ball with higher orbital angular momentum $L$ between the diquark $[cu]$ and antidiquark $[\overline{c}\overline{d}]$ corresponding to a more prolate spatial distribution. The multibody color flux tube, a collective degree of freedom, plays an important role in the formation of those charged tetraquark states. However, the two heavier charged states ${Z}_{c}^{+}(4430)$ and ${Z}_{c}^{+}(4475)$ cannot be explained as tetraquark states $[cu][\overline{c}\overline{d}]$ in this model approach.

The low-lying $S$-wave $QQ\bar{s}\bar{s}$ ($Q=c, b$) tetraquark states with $IJ^P=00^+$, $01^+$ and $02^+$ are systematically investigated in the framework of complex scaling range of chiral quark model. Every structure including meson-meson, diquark-antidiquark and K-type configurations, and all possible color channels in four-body sector are considered by means of a commonly extended variational approach, Gaussian expansion method. Several narrow and wide resonance states are obtained for $cc\bar{s}\bar{s}$ and $bb\bar{s}\bar{s}$ tetraquarks with $IJ^P=00^+$ and $02^+$. Meanwhile, narrow resonances for $cb\bar{s}\bar{s}$ tetraquarks are also found in $IJ^P=00^+$, $01^+$ and $02^+$ states. These results confirm the possibility of finding hadronic molecules with masses $\sim\,0.6\,\text{GeV}$ above the noninteracting hadron-hadron thresholds.

Abstract Inspired by the recent observation of $$\chi _{c0}(3930)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3930</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , X (4685) and X (4630) by the LHCb Collaboration and some exotic resonances such as X (4350), X (4500), etc. by several experiment collaborations, the $$cs{\bar{c}}{\bar{s}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>s</mml:mi> <mml:mover> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mover> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> tetraquark systems with $$J^{PC}=0^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>PC</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , $$1^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> , $$1^{+-}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>-</mml:mo> </mml:mrow> </mml:msup> </mml:math> and $$2^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> are systematically investigated in the framework of the quark delocalization color screening model(QDCSM). Two structures, the meson–meson and diquark–antidiquark structures, as well as the channel-coupling of all channels of these two configurations are considered in this work. The numerical results indicate that the molecular bound state $$D^{-}_{s}D_{s}^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> <mml:mo>-</mml:mo> </mml:msubsup> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> </mml:mrow> </mml:math> with $$J^{PC}=00^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>PC</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>00</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> can be supposed to explain the $$\chi _{c0}(3930)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3930</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Besides, by using the stabilization method, several resonant states are obtained. Among these states, X (4350), X (4500) and X (4700) can be explained as the compact tetraquark states with $$J^{PC}=00^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>PC</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>00</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , and the X (4274) is possible to be a candidate of the compact tetraquark state with $$J^{PC}=1^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>PC</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Apart from that, the $$J^{PC}=0^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>PC</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> resonance state with energy range 4028–4033 MeV, the two $$J^{PC}=2^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow> <mml:mi>PC</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> resonance states with energy range of 4394–4448 MeV and 4526–4536 MeV are possible to be new exotic states, which are indeed worthy of attention. More experimental tests are expected to check the existence of all these possible resonance states.

Inspired by the newly reported $Z_{cs}(3985)^{-}$ by the BESIII Collaboration, we systematically investigate the strange hidden-charm tetraquark systems $cs\bar{c}\bar{u}$ with two structures: meson-meson and diquark-antidiquark. Two quark models: the chiral quark model (ChQM) and the quark delocalization color screening model (QDCSM) are used here. Similar results are obtained in both two quark models. There is no any bound state in either ChQM or QDCSM, which excludes the molecular state explanation ($D_{s}D^{*}/D_{s}^{*}D/D_{s}^{*}D^{*}$) of the reported $Z_{cs}(3985)^{-}$. However, the effective potentials for the diquark-antidiquark $cs\bar{c}\bar{u}$ systems shows the possibility of some resonance states with mass range of $3916.5\sim 3964.6$ MeV for $IJ^{P}=\frac{1}{2} 0^{+}$, $4008.8\sim 4091.2$ MeV for $IJ^{P}=\frac{1}{2} 1^{+}$, $4246.8\sim 4418.1$ MeV for $IJ^{P}=\frac{1}{2} 2^{+}$. So the observed $Z_{cs}(3985)^{-}$ state is possible to be explained as a compact resonance state composed of $cs\bar{c}\bar{u}$ with $IJ^{P}=\frac{1}{2} 0^{+}$ or $IJ^{P}=\frac{1}{2} 1^{+}$. The study of the scattering process of corresponding open channels is under way to check this conclusion.

The quark delocalization and color screening model is used to study the N-N interaction of spin-isospin (00) and (11) channels. The predicted $^{1}$${\mathit{p}}_{1}$ phase shifts qualitatively fit the phase shift analysis of scattering data. The $^{3}$${\mathit{p}}_{\mathit{ls}}$ phase shifts are enhanced by the quark delocalization effect, but the $^{3}$${\mathit{p}}_{\mathit{c}}$ is too strongly repulsive. \textcopyright{} 1996 The American Physical Society.

Through a quantitative comparative study of the properties of deuteron and nucleon-nucleon interaction between the chiral quark model and the quark delocalization color screening model, we show that the \ensuremath{\sigma}-meson exchange used in the chiral quark model can be replaced by the quark delocalization and color screening mechanism.

The structural and dynamical properties of the supercritical CO2 fluid confined in the slit nanopores with the hydroxylated and silylated amorphous silica surfaces have been studied using molecular dynamics (MD) simulation. The amorphous bulk silica was obtained by a melt-quench MD simulation technique and the modified silica surfaces were artificially created by the attachment of hydrogen (−OH model) and trimethysilane (−Si(CH3)3 model) to the nonbridging oxygen atoms on the silica surfaces. The VdW interaction potential between the CO2 molecule and the hydroxylated silica surface was determined based on the ab initio quantum mechanics (QM) computation. The adsorption potential distributions of CO2 on the two modified silica surfaces were examined in order to evaluate the different surface interaction characteristics. The density profiles, the radial distribution functions, as well as the interfacial dynamics properties (self-diffusion coefficients and residence time) for the confined supercritical CO2 fluid have been simulated. It is demonstrated that the hydroxylated silica surface gives a stronger confining effect on the supercritical CO2 fluid as compared with the silylated surface. The remarkable impact on the supercritical CO2 fluid from the hydroxylated silica surface can be attributed to the H-bonding interaction between CO2 molecules and surface silanol groups. The analysis of the vibrational density of states of the confined supercritical CO2 fluid reveals the phenomena of the spectral shifts and the Fermi resonance in compaison with the bands in unconfined supercritical CO2. This spectrum behavior is associated with the enhanced interaction from the functional groups on silica surfaces.

Inspired by the newly observed $X$(4160) and $X$(3915) states, we analyze the mass spectrum of these states in different quark models and calculate their strong decay widths by the $^3P_0$ model. According to the mass spectrum of charmonium states predicted by the potential model, the states $\chi_0(3^3P_0), \chi_1(3^3P_1), \eta_{c2}(2^1D_2), \eta_c(4^1S_0)$ all can be candidates for the $X$(4160). However, only the decay width of the state $\eta_{c2}(2^1D_2)$ in our calculation is in good agreement with the data reported by Belle and the decay of $\eta_{c2}(2^1D_2)\to D\bar{D}$, which is not seen in experiment, is also forbidden. Therefore, it is reasonable to interpret the charmonium state $\eta_{c2}(2^1D_2)$ as the state X(4160). For the state X(3915), although the mass of $\chi_0(2^3P_0)$ is compatible with the experimental value, the calculated strong decay width is much larger than experimental data. Hence, the assignment of X(3915) to charmonium state $\chi_0(2^3P_0)$ is disfavored in our calculation.

The tetraquark states with diquark--anti-diquark configuration have been studied in the flux-tube model, in which the multibody confinement is used. In this model approach, the states $Y(2175)$, ${f}_{0}(600)$, ${f}_{0}(980)$, and $X(1576)$ can be assigned as tetraquark states. They are color confinement resonances with three-dimension structure. This study suggests that the multibody confinement should be employed in the quark model study of multiquark states instead of the additive two-body confinement.

Inspired by the discovery of the dibaryon ${d}^{*}$ and the experimental search of $N\mathrm{\ensuremath{\Omega}}$ dibaryon with the STAR data, we study the strange dibaryon $N\mathrm{\ensuremath{\Omega}}$ further in the framework of the quark delocalization color screening model and the chiral quark model. We have shown $N\mathrm{\ensuremath{\Omega}}$ is a narrow resonance in $\mathrm{\ensuremath{\Lambda}}\mathrm{\ensuremath{\Xi}}$ D-wave scattering before. However, the $\mathrm{\ensuremath{\Lambda}}\ensuremath{-}\mathrm{\ensuremath{\Xi}}$ scattering data analysis is quite complicated. Here we calculate the low-energy $N\mathrm{\ensuremath{\Omega}}$ scattering phase shifts, scattering length, effective range, and binding energy to provide another approach of STAR data analysis. Our results show there exists an $N\mathrm{\ensuremath{\Omega}}$ ``bound'' state, which can be observed by the $N\ensuremath{-}\mathrm{\ensuremath{\Omega}}$ correlation analysis with RHIC and LHC data, or by the new developed automatic scanning system at J-PARC. In addition, we also find that the hidden-color channel coupling is important for the $N\mathrm{\ensuremath{\Omega}}$ system to develop intermediate-range attraction.

Possible $H$-like dibaryon states ${\ensuremath{\Lambda}}_{c}{\ensuremath{\Lambda}}_{c}$ and ${\ensuremath{\Lambda}}_{b}{\ensuremath{\Lambda}}_{b}$ are investigated within the framework of the quark delocalization color screening model. The results show that the interaction between two ${\ensuremath{\Lambda}}_{c}$'s is repulsive, so it cannot be a bound state by itself. However, the strong attraction in ${\ensuremath{\Sigma}}_{c}{\ensuremath{\Sigma}}_{c}$ and ${\ensuremath{\Sigma}}_{c}^{*}{\ensuremath{\Sigma}}_{c}^{*}$ channels and the strong channel coupling, due to the central interaction of one-gluon exchange and one-pion exchange, among ${\ensuremath{\Lambda}}_{c}{\ensuremath{\Lambda}}_{c}$, ${\ensuremath{\Sigma}}_{c}{\ensuremath{\Sigma}}_{c}$ and ${\ensuremath{\Sigma}}_{c}^{*}{\ensuremath{\Sigma}}_{c}^{*}$ push the energy of system below the threshold of ${\ensuremath{\Lambda}}_{c}{\ensuremath{\Lambda}}_{c}$ by 3-20 MeV. The ${\ensuremath{\Lambda}}_{b}{\ensuremath{\Lambda}}_{b}$ system has properties similar to those of the ${\ensuremath{\Lambda}}_{c}{\ensuremath{\Lambda}}_{c}$ system, and a bound state is also possible in the ${\ensuremath{\Lambda}}_{b}{\ensuremath{\Lambda}}_{b}$ system.

Within the framework of the color flux-tube model with a multibody confinement potential, we systematically investigate the hidden charmed states observed in recent years. It can be found that most of them can be described as the compact tetraquark states $[cq][\bar{c}\bar{q}]$ ($q=u,d$ and $s$) in the color flux-tube model. The multibody confinement potential based on the color flux-tube picture is a dynamical mechanism in the formation and decay of the compact tetraquark states.

We study the effect of magnetic field on heavy quark-antiquark pair in both Einstein-Maxwell (EM) and Einstein-Maxwell-dilaton (EMD) model. The interquark distance, free energy, entropy, binding energy, and internal energy of the heavy quarkonium are calculated. It is found that the free energy suppresses and the entropy increases quickly with the increase of the magnetic field $B$. The binding energy vanishes at smaller distance when increasing the magnetic field, which indicates the quark-antiquark pair dissociates at smaller distance. The internal energy which consists of free energy and entropy will increase at large separating distance for a nonvanishing magnetic field. These conclusions are consistent both in the EM and EMD model. Moreover, we also find that the quarkonium will dissociate easier in the parallel direction than that in the transverse direction for EMD model, but the conclusion is opposite in EM model. Lattice results are in favor of EMD model. Besides, a Coulomb-plus-linear potential (Cornell potential) can be realized only in EMD model. Thus, a dilaton field is proved to be important in holographic model. Finally, we also show that the free energy, entropy, and internal energy of a single quark in EMD model with the presence of magnetic field.

We perform a systemical investigation of the low-lying doubly-heavy dibaryon systems with strange $S=0$, isospin $I=0$, $1$, $2$ and the angular momentum $J=0$, $1$, $2$, $3$ in the quark delocalization color screening model. We find the effect of channel-coupling cannot be neglected in the study of the multi-quark systems. Due to the heavy flavor symmetry, the results of the doubly-charm and doubly-bottom dibaryon systems are similar with each other. Both of them have three bound states, the quantum numbers of which are $IJ=00$, $IJ=02$ and $IJ=13$, respectively. The energies are $4554$ MeV, $4741$ MeV, and $4969$ MeV respectively for the doubly-charm systems and $11219$ MeV, $11416$ MeV, and $11633$ MeV respectively for the doubly-bottom dibaryon systems. Besides, six resonance states are obtained, which are $IJ=00$ $N\Xi_{cc}$ and $N\Xi_{bb}$ with resonance mass of $4716$ MeV and $11411$ MeV respectively, $IJ=11$ $N\Xi^*_{cc}$ and $N\Xi^*_{bb}$ with resonance mass of $4757$ MeV and $11432$ MeV respectively, and $IJ=12$ $\Sigma_{c}\Sigma^*_{c}$ and $\Sigma_{b}\Sigma^*_{b}$ with resonance mass of $4949$ MeV and $11626$ MeV respectively. All these heavy dibaryons are worth searching for on experiments, although it will be a challenging work.

The fully-charm and -bottom pentaquarks, \emph{i.e.} $cccc\bar{c}$ and $bbbb\bar{b}$, with spin-parity quantum numbers $J^P=\frac{1}{2}^-$, $\frac{3}{2}^-$ and $\frac{5}{2}^-$, are investigated within a Lattice-QCD inspired quark model, which has already successfully described the recently announced fully-charm tetraquark candidate $X(6900)$, and has also predicted several other fully-heavy tetraquarks. A powerful computational technique, based on the Gaussian expansion method combined with a complex-scaling range approach, is employed to predict, and distinguish, bound, resonance and scattering states of the mentioned five-body system. Both baryon-meson and diquark-diquark-antiquark configurations, along with all of their possible color channels are comprehensively considered. Narrow resonances are obtained in each spin-parity channel for the fully-charm and -bottom systems. Moreover, most of them seems to be compact multiquarks whose wave-functions are dominated by either hidden-color baryon-meson or diquark-diquark-antiquark structure, or by the coupling between them.

Inspired by the recent observation of new ${\mathrm{\ensuremath{\Omega}}}_{c}^{0}$ states by the LHCb Collaboration, we explore the excited ${\mathrm{\ensuremath{\Omega}}}_{c}$ states from the pentaquark perspective in the quark delocalization color screening model. Our results indicate that the ${\mathrm{\ensuremath{\Omega}}}_{c}(3185)$ can be well interpreted as a molecular $\mathrm{\ensuremath{\Xi}}D$ predominated resonance state with ${J}^{P}=1/{2}^{\ensuremath{-}}$. The ${\mathrm{\ensuremath{\Omega}}}_{c}(3120)$ can also be interpreted as a molecular ${\mathrm{\ensuremath{\Xi}}}_{c}^{*}\overline{K}$ state with ${J}^{P}=3/{2}^{\ensuremath{-}}$ and a new molecular state ${\mathrm{\ensuremath{\Xi}}}_{c}^{*}{\overline{K}}^{*}$ with ${J}^{P}=5/{2}^{\ensuremath{-}}$ and a mass of 3526 MeV is predicted, which is worth searching in the future. Other reported ${\mathrm{\ensuremath{\Omega}}}_{c}$ states cannot be well described in the framework of pentaquark systems in the present work. The three-quark excited state or the unquenched picture may be a good explanation, which is worth further exploration.

We present a phenomenological study of three models with different effective degrees of freedom: a Goldstone boson exchange model which is based on quark-meson couplings, the quark delocalization, color screening model which is based on quark-gluon couplings with delocalized quark wave functions, and the Fujiwara-Nijmegen mixed model which includes both quark-meson and quark-gluon couplings. We find that for roughly two-thirds of 64 states consisting of pairs of octet and decuplet baryons, the three models predict similar effective baryon-baryon interactions. This suggests that the three very different models, based on different effective degrees of freedom, are nonetheless all compatible with respect to baryon spectra and baryon-baryon interactions. We also discuss the differences between the three models and their separate characteristics.

The mass estimate of the d∗(IJP=03+) dibaryon is improved by a dynamical calculation in the quark delocalization, color screening model. The partial decay width of d∗ into an NN D-wave state is also obtained. The mass obtained is slightly larger than that obtained in adiabatic calculations, due to the anharmonicity of the effective potential between two Δ's. The value of the width obtained due to tensor one-gluon exchange is about 5 MeV, comparable in magnitude to earlier results found using pion exchange.

The baryon-antibaryon spectrum consisting of $u$, $d$, $s$, $c$, and $b$ quarks is studied in the color flux-tube model with a multibody confinement interaction. Numerical results indicate that many low-spin ($S\ensuremath{\le}1$) baryon-antibaryon states can form compact bound states and are stable against decaying into a baryon and an antibaryon. They can be searched for in ${e}^{+}{e}^{\ensuremath{-}}$ annihilation and charmonium or bottomonium decay if they really exist. Multibody confinement interaction as a binding mechanism plays an important role in the formation of the baryon-antibaryon bound states; chromomagnetic interaction also provides a strong attraction in many low-spin baryon-antibaryon states. The newly reported states, $X(1835)$, $X(2370)$, $Y(2175)$, $Y(4360)$, and ${Y}_{b}(10890)$, might be interpreted as $N\overline{N}$, $\ensuremath{\Delta}\overline{\ensuremath{\Delta}}$, $\ensuremath{\Lambda}\overline{\ensuremath{\Lambda}}$, ${\ensuremath{\Lambda}}_{c}{\overline{\ensuremath{\Lambda}}}_{c}$, and ${\ensuremath{\Lambda}}_{b}{\overline{\ensuremath{\Lambda}}}_{b}$ bound states, respectively.

The $N{\ensuremath{\Lambda}}_{c}$ and $N{\ensuremath{\Lambda}}_{b}$ systems with $I=\frac{1}{2}$ and ${J}^{P}={0}^{+}$ or ${1}^{+}$ are investigated within the framework of the quark delocalization color screening model (QDCSM) by solving a resonating group method equation. The results show that the interaction between $N$ and ${\ensuremath{\Lambda}}_{c}$ is attractive, but it is not strong enough to form any $N{\ensuremath{\Lambda}}_{c}$ bound state, even with the help of channel coupling. Whereas the attraction between $N$ and ${\ensuremath{\Sigma}}_{c}$ is strong enough to form a bound state $N{\ensuremath{\Sigma}}_{c}{(}^{3}{S}_{1})$, it becomes a resonance state with the resonance mass of 3389--3404 MeV by coupling to the open $N{\ensuremath{\Lambda}}_{c}$ $D$-wave channel. However it does not show up in the $N{\ensuremath{\Lambda}}_{c}$ $S$-wave channel. The corresponding states $N{\ensuremath{\Lambda}}_{b}$, $N{\ensuremath{\Sigma}}_{b}$ have similar properties as that of states $N{\ensuremath{\Lambda}}_{c}$, $N{\ensuremath{\Sigma}}_{c}$, and there exists a $N{\ensuremath{\Sigma}}_{b}{(}^{3}{S}_{1})$ resonance state with energy of 6733--6743 MeV in the $N{\ensuremath{\Lambda}}_{b}$ $D$-wave scattering phase shifts.

We investigate the hidden strange light baryon-meson system. With the resonating-group method, two bound states, ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{-}N$ and $\ensuremath{\phi}\ensuremath{-}N$, are found in the quark delocalization color screening model. Focusing on the $\ensuremath{\phi}\ensuremath{-}N$ bound state around 1950 MeV, we obtain the total decay width of about 4 MeV by calculating the phase shifts in the resonance scattering processes. To study the feasibility of an experimental search for the $\ensuremath{\phi}\ensuremath{-}N$ bound state, we perform a Monte Carlo simulation of the bound state production with an electron beam and a gold target. In the simulation, we use the CLAS12 detector with the Forward Tagger and the BONUS12 detector in Hall B at Jefferson Lab. Both the signal and the background channels are estimated. We demonstrate that the signal events can be separated from the background with some momentum cuts. Therefore it is feasible to experimentally search for the $\ensuremath{\phi}\ensuremath{-}N$ bound state through the near threshold $\ensuremath{\phi}$ meson production from heavy nuclei.

Encouraged by the observation of the pentaquark states ${P}_{c}^{+}(4380)$ and ${P}_{c}^{+}(4450)$, we propose a novel color flux-tube structure, a pentagonal state, for pentaquark states within the framework of a color flux-tube mode involving a five-body confinement potential. Numerical results on the heavy pentaquark states indicate that the states with three color flux-tube structures, diquark, octet, and pentagonal structures, have the closest masses, which can therefore be called QCD isomers, analogous to isomers in chemistry. The pentagonal structure has the lowest energy. The state ${P}_{c}^{+}(4380)$ can be described as the compact pentaquark state $uudc\overline{c}$ with the pentagonal structure and ${J}^{P}={\frac{3}{2}}^{\ensuremath{-}}$ in the color flux-tube model. The state ${P}_{c}^{+}(4450)$ can not be accommodated into the color flux-tube model. The heavy pentaquark states $uudc\overline{b}$, $uudb\overline{c}$, and $uudb\overline{b}$ are predicted in the color flux-tube model. The five-body confinement potential, based on the color flux-tube picture as a collective degree of freedom, is a dynamical mechanism in the formation of the compact heavy pentaquark states.

Stimulated by the state $Y(4626)$ recently reported by the Belle Collaboration, we utilize a multiquark color flux-tube model with a multibody confinement potential and one-gluon-exchange interaction to make an exhaustive investigation on the diquark-antidiquark state $[cs][\overline{c}\overline{s}]$. Numerical results indicate that the spatial configuration of the states $[cs][\overline{c}\overline{s}]$ like a dumbbell, the larger the orbital excitation $L$, the farther the distance between the diquark $[cs]$ and the antidiquark $[\overline{c}\overline{s}]$, the more clearer the shape. The mixing of the color configurations ${[[cs{]}_{{\overline{\mathbf{3}}}_{c}}[\overline{c}\overline{s}{]}_{{\mathbf{3}}_{c}}]}_{\mathbf{1}}$ and ${[[cs{]}_{{\mathbf{6}}_{c}}[\overline{c}\overline{s}{]}_{{\overline{\mathbf{6}}}_{c}}]}_{\mathbf{1}}$ in the ground states is strong, while the color configuration ${[[cs{]}_{{\overline{\mathbf{3}}}_{c}}[\overline{c}\overline{s}{]}_{{\mathbf{3}}_{c}}]}_{\mathbf{1}}$ is absolutely predominant in the excited states. The states $Y(4626)$, $Y(4630)$, and $Y(4660)$ can be uniformly described as the $P$-wave tetraquark state $[cs][\overline{c}\overline{s}]$ with ${1}^{\ensuremath{-}\ensuremath{-}}$. The states $Y(4626)$ and $Y(4630)$ can be described as the same state consisting of a scalar $[cs]$ and a scalar $[\overline{c}\overline{s}]$, while the state $Y(4660)$ is made of a scalar $[cs]$ ($[\overline{c}\overline{s}]$) and an axial-vector $[\overline{c}\overline{s}]$ ($[cs]$). Their hidden-bottom partner is predicted in the model calculation. The states $X(4140)$, $X(4274)$, $X(4350)$, $X(4500)$, and $X(4700)$ are also discussed.

Using the guage/gravity correspondence, we study the free energy, entropy, binding energy, and internal energy of a heavy quarkonium in the rotating quark-gluon plasma (QGP). First, we extend the static black hole with planar horizon to the spinning case, which is dual to the rotating matter. Subsequently, we use the Wilson loop to get the free energy of heavy quarkonium. Entropy, binding energy, and internal energy are naturally given by the thermodynamic relationship. It is found that angular velocity will suppress the dissociation temperature of heavy quarkonium and change the system from confined to deconfined phase with the increase of angular velocity. Thus, the heavy quarkonium will dissolve at a certain separating distance in the deconfined phase. We also compare these thermodynamic quantities of the quark-antiquark pair in the direction parallel and transverse to the rotation direction. At last, the free energy, entropy, and internal energy of a single quark are discussed. It is found that these quantities are decreasing functions of temperature.

A relativistic quark potential model is used to do a systematic search for quasi-stable dibaryon states in the u, d and s three-flavor world. Flavor symmetry breaking and channel coupling effects are included and an adiabatic method and fractional parentage expansion technique are used in the calculations. The relativistic model predicts dibaryon candidates completely consistent with the nonrelativistic model.

By differentiating the dressed quark propagator with respect to a constant external field, the linear response of the nonperturbative dressed quark propagator in the presence of the constant external field can be obtained. From this general method, taking the vector vacuum susceptibility as an illustration, we one extract a model-independent expression for the vacuum susceptibility in the quantum chromodynamical (QCD) sum rule two-point external field formula. This expression for the vacuum susceptibility is quite different from that given in the previous literature. The numerical values of the vector vacuum susceptibility are calculated within the framework of the rainbow-ladder approximation of the Dyson-Schwinger approach. A comparison with the results of the previous approaches is given.

A modified method for calculating the nonperturbative quark vacuum condensates from the global color symmetry model is derived. Within this approach it is shown that the vacuum condensates are free of ultraviolet divergence which is different from previous studies. As special cases, the two-quark condensate $〈\overline{q}q〉$ and the mixed quark-gluon condensate $g〈\overline{q}{G}_{\ensuremath{\mu}\ensuremath{\nu}}{\ensuremath{\sigma}}^{\ensuremath{\mu}\ensuremath{\nu}}q〉$ are calculated. A comparison with the results of other nonperturbative QCD approaches is given.

A full azimuthal phi-wedge of the ATLAS liquid argon end-cap calorimeter has been exposed to beams of electrons, muons and pions in the energy range 6 GeV &lt;= E &lt;= 200 GeV at the CERN SPS. The angular region studied corresponds to the ATLAS impact position around the pseudorapidity interval 1.6 &lt; |eta| &lt; 1.8. The beam test set-up is described. A detailed study of the performance is given as well as the related intercalibration constants obtained. Following the ATLAS hadronic calibration proposal, a first study of the hadron calibration using a weighting ansatz is presented. The results are compared to predictions from Monte Carlo simulations, based on GEANT 3 and GEANT 4 models.

Analogous to the work of hidden charm molecular pentaquarks, we study possible hidden strange molecular pentaquarks composed of $\mathrm{\ensuremath{\Sigma}}$ (or ${\mathrm{\ensuremath{\Sigma}}}^{*}$) and $K$ (or ${K}^{*}$) in the framework of a quark delocalization color screening model. Our results suggest that the $\mathrm{\ensuremath{\Sigma}}K$, $\mathrm{\ensuremath{\Sigma}}{K}^{*}$, and ${\mathrm{\ensuremath{\Sigma}}}^{*}{K}^{*}$ with $I{J}^{P}=\frac{1}{2}{\frac{1}{2}}^{\ensuremath{-}}$ and $\mathrm{\ensuremath{\Sigma}}{K}^{*}$, ${\mathrm{\ensuremath{\Sigma}}}^{*}K$, and ${\mathrm{\ensuremath{\Sigma}}}^{*}{K}^{*}$ with $I{J}^{P}=\frac{1}{2}{\frac{3}{2}}^{\ensuremath{-}}$ are all resonance states by coupling the open channels. The molecular pentaquark ${\mathrm{\ensuremath{\Sigma}}}^{*}K$ with quantum numbers $I{J}^{P}=\frac{1}{2}{\frac{3}{2}}^{\ensuremath{-}}$ can be seen as a strange partner of the LHCb ${P}_{c}(4380)$ state. The possibility of identifying the resonances as nucleon resonances is proposed.

We conduct a dynamical calculation of pentaquark systems with quark contents $sssu\bar{u}$ in the framework of two quark models: the chiral quark model(ChQM) and quark delocalization color screening model(QDCSM). The effective potentials between baryon and meson clusters are given, and the possible bound states are also investigated. Besides, the study of the scattering process of the open channels is also performed to look for any resonance state. The results show that the $\Omega(2012)$ is not suitable for the interpretation as a $\Xi^{*} \bar{K}$ molecular state in present quark models. Two resonance states: the $\Xi^{*}\bar{K}^{*}$ with $IJ^{P}=0\frac{3}{2}^{-}$ ($M=2328\sim2374$ MeV, $\Gamma=57\sim65.5$ MeV) and $IJ^{P}=1\frac{3}{2}^{-}$ ($M=2341\sim2386$ MeV, $\Gamma=31.5\sim100$ MeV) are obtained in both QDCSM and ChQM, which indicates that both of these two states are more possible to be existed and worthy of being searched by future experiments.

In the present work, the triply heavy tetraquarks states $QQ\overline{Q}\overline{q}$ with $Q=(c,b)$ and $q=(u,d,s)$ with all possible quantum numbers are systematically investigated in the framework of the chiral quark model with the resonating ground method. Two kinds of structures, including the meson-meson configuration (the color-singlet channels and the hidden-color channels) and the diquark-antidiquark configuration (the color sextet-antisextet and the color triplet-antitriplet), are considered. In the considered system, several bound states are obtained for the $cc\overline{c}\overline{{q}^{\ensuremath{'}}}$, $bb\overline{c}\overline{{q}^{\ensuremath{'}}}$, and $bc\overline{c}\overline{q}$ tetraquarks. From the present estimations, we find that the coupled-channel effect is of great significance for forming below-threshold tetraquark states, which are stable for strong decays.

This study investigates the resonance $\ensuremath{\omega}{\ensuremath{\chi}}_{bJ}$, recently discovered in ${e}^{+}{e}^{\ensuremath{-}}$ collisions with a center-of-mass energy of $\sqrt{s}=10.745\text{ }\text{ }\mathrm{GeV}$ and reported by the Belle II Collaboration, to explore its potential identification with the previously reported $\mathrm{\ensuremath{\Upsilon}}(10753)$. We consider both the traditional $\mathrm{\ensuremath{\Upsilon}}(nS)$ meson and the exotic tetraquark $b\overline{q}q\overline{b}$ state with ${J}^{P}={1}^{\ensuremath{-}}$, employing the chiral quark model to solve the Schr\"odinger equation. Our calculations demonstrate that the mass of $\mathrm{\ensuremath{\Upsilon}}(5S)$, $\mathrm{\ensuremath{\Upsilon}}(4S)$, and $\mathrm{\ensuremath{\Upsilon}}(3D)$, as potential two-quark state candidates, are inconsistent with the experimentally observed $\mathrm{\ensuremath{\Upsilon}}(10753)$, effectively excluding the possibility of $\mathrm{\ensuremath{\Upsilon}}(10753)$ as a two-quark state in our model. Furthermore, we employ the Gaussian expansion method to investigate the tetraquark structure of the resonance, including two molecular structures ($b\overline{b}\text{\ensuremath{-}}q\overline{q}$, $b\overline{q}\text{\ensuremath{-}}q\overline{b}$) and a diquark-antidiquark structure ($\overline{b}\overline{q}\text{\ensuremath{-}}qb$), and perform a full-channel coupling using the real-scaling method. Our investigation yields a total of seven resonant states, with one state having an energy of $10761\ifmmode\pm\else\textpm\fi{}7\text{ }\text{ }\mathrm{MeV}$ and a width of $10.8\ifmmode\pm\else\textpm\fi{}5.2\text{ }\text{ }\mathrm{MeV}$, which is a promising candidate for the experimentally observed $\mathrm{\ensuremath{\Upsilon}}(10753)$. Furthermore, we report no resonances in the energy range below 10.6 GeV, while several resonances were found in the 10.8--10.9 GeV range. These findings suggest the need for future experiments to search for additional resonant states in this energy range, potentially providing further support for our model.

We probe the existence of the $_{{\mathrm{\ensuremath{\Lambda}}}_{c}}^{3}\mathrm{H}$ where the $N{\mathrm{\ensuremath{\Lambda}}}_{c}$ potentials are derived from the quark-delocalization color-screening model (QDCSM). The $N{\mathrm{\ensuremath{\Lambda}}}_{c}$ system is studied and the $N{\mathrm{\ensuremath{\Lambda}}}_{c}$ scattering length as well so the effective ranges are obtained in the QDCSM. We construct effective Gaussian-type $N{\mathrm{\ensuremath{\Lambda}}}_{c}$ potentials which reproduce the $N{\mathrm{\ensuremath{\Lambda}}}_{c}$ scattering data given by the QDCSM. By solving the $NN{\mathrm{\ensuremath{\Lambda}}}_{c}$ three-body Schr\"odinger equation with the Gaussian expansion method, we calculate the energies of the $_{{\mathrm{\ensuremath{\Lambda}}}_{c}}^{3}\mathrm{H}$ with isospin $I=0$, ${J}^{P}=1/{2}^{+}$ and $I=0$, ${J}^{P}=3/{2}^{+}$ under different color screening parameter $\ensuremath{\mu}$. The ${J}^{P}=1/{2}^{+}$ and ${J}^{P}=3/{2}^{+}$ states are both bound when the color screening parameter $\ensuremath{\mu}$ is set to 1.0 ${\mathrm{fm}}^{\ensuremath{-}2}$ or 1.2 ${\mathrm{fm}}^{\ensuremath{-}2}$, where the ${J}^{P}=1/{2}^{+}$ state is bound by 0.08--0.85 MeV and the ${J}^{P}=3/{2}^{+}$ state is bound by 0.15--1.31 MeV with respect to the deuteron-${\mathrm{\ensuremath{\Lambda}}}_{c}$ threshold.

Abstract According to gauge/gravity correspondence, we study the holographic Schwinger effect within an anisotropic background. Firstly, the separate length of the particle–antiparticle pairs is computed within the context of an anisotropic background which is parameterized by dynamical exponent $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> . It is found that the maximum separate length x increases with the increase of dynamical exponent $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> . By analyzing the potential energy, we find that the potential barrier increases with the dynamical exponent $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> at a small separate distance. This observation implies that the Schwinger effect within an anisotropic background is comparatively weaker when contrasted with its manifestation in an isotropic background. Finally, we also find that the Schwinger effect in the transverse direction is weakened compared to the parallel direction in the anisotropic background, which is consistent with the top-down model.

Since the quark model was put forward, theoretical researchers have always attached great importance to the study of hidden color channels (including color octets and diquark structure). Because of the influence of color Van der waals forces, the hidden color channel itself has strong attraction, which provides a dynamic mechanism for the formation of resonance state or bound state. In this paper, taking the $T_{cc}$ system as an example, under the framework of multi-Gaussian expansion method, a set of relatively complete color singlets (that is, the ground state of the color singlet plus its corresponding higher-order component) is used to replace the contribution of the color octet. Similarly, we endeavor to replace the diquark structure with a relatively complete set of molecular states, encompassing both the ground state and excited states. Our results demonstrate that the color octet structure can be effectively replaced by a set of relatively complete color singlet bases, while the diquark structure cannot be entirely substituted by an equivalently comprehensive set of molecular state bases.

Inspired by the detection of ${T}_{cc}$ tetraquark state by LHCb Collaboration, we perform a systemical investigation of the low-lying doubly heavy charm tetraquark states with strangeness in the quark delocalization color screening model in the present work. Two kinds of configurations, the meson-meson configuration and diquark-antidiquark configuration, are considered in the calculation. Our estimations indicate that the coupled channel effects play important role in the multiquark system, and a bound state with ${J}^{P}={1}^{+}$ and a resonance state with ${J}^{P}={0}^{+}$ have been predicted. The mass of the bound state is evaluated to be (3971--3975) MeV, while the mass and width of the resonance are determined to be (4113--4114) MeV and (14.3--16.1) MeV, respectively.

The LHCb collaboration has recently announced the discovery of two hidden-charm pentaquark states with strange quark content, Pcs(4338) and Pcs(4459); its analysis points towards having both hadrons’ isospins equal to zero and spin-parity quantum numbers 12− and 32−, respectively. Herein, we perform a systematical investigation of the qqscc¯(q=u,d) system by means of a chiral quark model, along with a highly accurate computational method, the Gaussian expansion approach combined with the complex scaling technique. baryon-meson configurations in both singlet- and hidden-color channels are considered. The Pcs(4338) and Pcs(4459) signals can be well identified as molecular bound states with dominant components ΛJ/ψ(60%) and ΞcD(23%) for the lowest-energy case and ΞcD∗(72%) for the highest-energy one. In addition, it seems that some narrow resonances can also be found in each allowed I(JP) channel in the energy region of 4.6–5.5 GeV, except for the 1(12−) channel where a shallow bound state with dominant Ξc∗D∗ structure is obtained at 4673 MeV with binding energy EB=−3 MeV. These exotic states are expected to be confirmed in future high-energy experiments.

Inspired by the recent Altas and CMS experiments on the invariant mass spectrum of $J/\psi J/\psi$, we systematically study the $c\bar{c}c\bar{c}$ system of $J^{P}=0^{+}$. In the framework of chiral quark model, we have carried out bound-state calculation and resonance-state calculation respectively by using Real-scaling method. The results of bound-state calculation show that there are no bound states in the $c\bar{c}c\bar{c}$ with $0^{+}$ system. The resonance-state calculation shows that there are four possible stable resonances: $R(6920)$, $R(7000)$, $R(7080)$ and $R(7160)$. $R(6920)$ and $R(7160)$ are experimental candidates for $X(6900)$ and $X(7200)$, whose main decay channel is $J/\psi J/\psi$. It is important to note that the another major decay channel of $R(7160)$ is $\chi_{c0} \chi_{c0} $, and the $\chi_{c0} \chi_{c0} $ is also the main decay channel of $R(7000)$, $R(7080)$. Therefore, we propose to search experimentally for these two predicted resonances in the $\chi_{c0} \chi_{c0}$ invariant mass spectrum.

Abstract The $S$-wave $\bar{q}q\bar{s}Q$ $(q=u,\,d;\,Q=c,\,b)$ tetraquarks, with spin-parity $J^P=0^+$, $1^+$ and $2^+$, in both isoscalar and isovector sectors are systematically studied in a chiral quark model. The meson-meson, diquark-antidiquark and K-type arrangements of quarks, along with all possible color wave functions, are comprehensively considered. The four-body system is solved by means of a highly efficient computational approach, the Gaussian expansion method, along with a complex-scaling formulation of the problem to disentangle bound, resonance and scattering states. This theoretical framework has already been successfully applied in various tetra- and penta-quark systems. In the complete coupled-channel case, and within the complex-range formulation, several narrow resonances of $\bar{q}q\bar{s}c$ and $\bar{q}q\bar{s}b$ systems are obtained, in each allowed $I(J^P)$-channels, within the energy regions $2.4-3.4$ GeV and $5.7-6.7$ GeV, respectively. The predicted exotic states, which are an indication of a richer color structure when going towards multiquark systems beyond mesons and baryons, are expected to be confirmed in future high-energy particle and nuclear experiments.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

Inspired by that Belle\uppercase\expandafter{\romannumeral2} Collaboration recently reported $T_{cc}$, which can be interpreted as a molecular $DD^{*}$, we investigated the trihadron system of $T_{cc}$ partner with $IJ^{P}$=$01^{-}$ in the framework of a chiral quark model. It's widely accepted that the main component of $X(3872)$ contains the molecular $\bar{D}D^{*}$, while the main component of $D_{s0}^{*}(2317)$ is molecular $DK$. Based on these three well-known exotic states, $T_{cc} (DD^{*})$, $X(3872) (\bar{D}D^{*})$ and $D_{s0}^{*}(2317) (DK)$, we dynamically investigate $D^{*}DK$ and $DD^{*}\bar{D}$ systems at quark level to search for possible bound states. The results show that both of them are bound states, in which the binding energy of the molecular state $DD^*K$ is relatively small, only 0.8 MeV, while the binding energy of $DD^*\bar{D}$ is up to 1.9 MeV. According to the calculation results of the Root-square-mean distances, the spatial structure of the two systems shows obvious ($DD^*$)-($\bar{D}$/$K$) structure, in which $D$ is close to $D^*$ while $DD^*$ as a whole is relatively distant from the third hadron ($\bar{D}$/$K$), which are similar to the nucleon-electron structure. As a result, we strongly recommend that these bound states $DD^*\bar{D}$ and $DD^*K$ are searched for experimentally.

We have investigated the $qss\bar{q}q$ ($q = u$ or $d$) system to find possible pentaquark explanations for the $\Xi$ resonances. The bound state calculation is carried out within the framework of the quark delocalization color screening model. The scattering processes are also studied to examine the possible resonance states. The current results indicate that the $\Xi(1950)$ can be interpreted as $\Lambda \bar{K}^*$ state with $J^P = 1/2^-$. Three states are identified that match the $\Xi(2250)$, which are $\Sigma^* \bar{K}^*$ state with $J^P = 3/2^-$, $\Sigma^* \bar{K}^*$ state with $J^P =5/2^-$, and $\Xi^* \rho$ state with $J^P =5/2^-$. This may explain the conflicting experimental values for the width of the $\Xi(2250)$. A new $\Xi$ resonance is predicted, whose mass and width are 2066--2079 MeV and 186--189 MeV, respectively. These results contribute to understanding the nature of the $\Xi$ resonances and to the future search for new $\Xi$ resonances. Moreover, it is meaningful to further investigate the $\Xi$ resonances from an unquenched picture on the basis of pentaquark investigation.

Since the quark model was put forward, theoretical researchers have always attached great importance to the study of hidden color channels (including color octets and diquark structure). Because of the influence of color van der Waals forces, the hidden color channel itself has strong attraction, which provides a dynamic mechanism for the formation of a resonance state or bound state. In this paper, taking the ${T}_{cc}$ system as an example, under the framework of the Gaussian expansion method, a set of relatively complete color singlets (that is, the ground state of the color singlet plus its corresponding higher-order component) is used to replace the contribution of the color octet. Similarly, we endeavor to replace the diquark structure with a relatively complete set of molecular states, encompassing both the ground state and excited states. Our results demonstrate that the color-octet structure can be effectively replaced by a set of relatively complete color-singlet bases, while the diquark structure cannot be entirely substituted by an equivalently comprehensive set of molecular state bases.

Dibaryon candidates with strangeness $S=\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}4,\ensuremath{-}5,\ensuremath{-}6$ are studied in terms of the extended quark delocalization and color screening model. The results show that there are only a few promising low lying dibaryon states: The $H$ and di-$\ensuremath{\Omega}$ may be marginally strong interaction stable but model uncertainties are too large to allow any definitive statement. The $SIJ=\ensuremath{-}3,1∕2,2$ $N\ensuremath{\Omega}$ state is $62\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$ lower than the $N\ensuremath{\Omega}$ threshold and $24\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$ lower than the $\ensuremath{\Lambda}\ensuremath{\Xi}\ensuremath{\pi}$ threshold. It might appear as a narrow dibaryon resonance and be detectable in the RHIC detector through the reconstruction of the vertex mass of the $\ensuremath{\Lambda}\ensuremath{\Xi}$ two-body decay. The effects of explicit $K$ and $\ensuremath{\eta}$ meson exchange have been studied and found to be negligible in this model. The mechanisms of effective intermediate range attraction, $\ensuremath{\sigma}$ meson exchange, and kinetic energy reduction due to quark delocalization are discussed.

The pseudorapidity region 2.5<|η|<4.0 in ATLAS is a particularly complex transition zone between the endcap and forward calorimeters. A set-up consisting of 14 resp. 18 of the full azimuthal acceptance of the ATLAS liquid argon endcap and forward calorimeters has been exposed to beams of electrons, pions and muons in the energy range E⩽200GeV at the CERN SPS. Data have been taken in the endcap and forward calorimeter regions as well as in the transition region. This beam test set-up corresponds very closely to the geometry and support structures in ATLAS. A detailed study of the performance in the endcap and forward calorimeter regions is described. The data are compared with MC simulations based on GEANT 4 models.

Nonstrange hexaquark state ${q}^{3}{\overline{q}}^{3}$ spectrum is systematically studied by using the Gaussian expansion method in flux tube models with a six-body confinement potential. All the model parameters are fixed by baryon properties, so the calculation of hexaquark state ${q}^{3}{\overline{q}}^{3}$ is parameter-free. It is found that some ground states of ${q}^{3}{\overline{q}}^{3}$ are stable against disintegrating into a baryon and an anti-baryon. The main components of $X(1835)$ and $X(2370)$, which are observed in the radiative decay of $J/\ensuremath{\psi}$ by BES collaboration, can be described as compact hexaquark states ${N}_{8}{\overline{N}}_{8}$ and ${\ensuremath{\Delta}}_{8}{\overline{\ensuremath{\Delta}}}_{8}$ with quantum numbers ${I}^{G}{J}^{PC}={0}^{+}{0}^{\ensuremath{-}+}$, respectively. These bound states should be color confinement resonances with three-dimensional configurations similar to a rugby ball, however, $X(2120)$ can not be accommodated in this model approach.

We perform a coupled-channels calculation of the masses of light mesons with the quantum numbers $IJ^{P=-}$, $(I,J)=0,1$, by including $q\bar{q}$ and $(q\bar{q})^2$ components in a nonrelativistic chiral quark model. The coupling between two- and four-quark configurations is realized through a $^3P_0$ 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 $^3P_0$ operator in order to reduce the size of the mass shifts. By introducing two simple factors, physically well motivated, the coupling between $q\bar{q}$ and $(q\bar{q})^2$ components is weakened, producing mass shifts that are around 10-20% of hadron bare masses.

Inspired by the report of D0 Collaboration on the state $X(5568)$ with four different flavors, a similar state, $ud\bar{s}\bar{b}$ is investigated in the present work. The advantage of looking for this state over the state $X(5568)$ with quark contents, $bu\bar{d}\bar{s}$ or $bd\bar{u}\bar{s}$, is that the $BK$ threshold is 270 MeV higher than that of $B_s \pi$, and it allows a large mass region for $ud\bar{s}\bar{b}$ to be stable. The chiral quark model and Gaussian expansion method are employed to do the calculations of four-quark states $ud\bar{s}\bar{b}$ with quantum numbers $IJ^{P}$($I=0,1;~ J=0,1,2;~P=+$). 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 for the state of $IJ^P=00^+$ in the meson-meson structure, the energies are below the corresponding thresholds, where the color channel coupling plays an important role. The distances between two objects (quark/antiquark) show that the state is a molecular one.

Recently, the LHCb Collaboration reported their observation of the first two fully open-flavor tetraquark states named $X_0(2900)$ and $X_1(2900)$ with unknown parity. Inspired by the report, we consider all of possible four-quark candidates of $X(2900)$ including molecular structure and diquark structure in a chiral quark model with the help of Gaussian expansion method. Two different structures coupling is also considered. To identify the genuine resonances, real-scaling method (stabilization method) was employed. The results show that no candidate of $X(2900)$ is founded in $00^+$ and $01^+$ $cs\bar{q}\bar{q}$ system below the threshold of $D^*\bar{K}^*$, while there are two states in the $P$-wave excited $cs\bar{q}\bar{q}$ system, $D_1\bar{K}$ and $D_J\bar{K}$, which could be candidates of $X(2900)$. In this way, we assign negative parity to $X_1(2900)$, and $X_0(2900)$ maybe a resonance state above the threshold of $D^*\bar{K}^*$, more calculation is needed.

A six-quark state with the benzene-like color structure based on a color string model is proposed and studied. Calculation with quadratic confinement with multi-string junctions shows that such a state has a ground state energy similar to that of other hidden color six-quark states proposed so far. Its possible effect on NN scattering is discussed.

Inspired by the recent observation of exotic resonances $X(4140)$, $X(4274)$, $X(4350)$, $X(4500)$ and $X(4700)$ reported by several experiment collaborations, we investigated the four-quark system $cs\bar c\bar s$ with quantum numbers $J^{PC}=1^{++}$ and $0^{++}$ in the framework of the chiral quark model. Two configurations, diquark-antidiquark and meson-meson, with all possible color structures are considered. The results show that no molecular state can be formed, but the resonance may exist if the color structure of meson-meson configuration is $8\otimes8$. In the present calculation, the $X(4274)$ can be assigned as the $cs\bar c\bar s$ tetraquark states with $J^{PC}=1^{++}$, but the energy of $X(4140)$ is too low to be regarded as the tetraquark state. $X(4350)$ can be a good candidate of compact tetraquark state with $J^{PC}=0^{++}$. When the radial excitation is taken into account, the $X(4700)$ can be explained as the $2S$ radial excited tetraquark state with $J^{PC}=0^{++}$. As for $X(4500)$, there is no matching state in our calculation.

A new technique for constructing representations of point groups, called symmetrized boson representation is introduced. This method is particularly useful when describing vibrations of large molecules and for high overtones. The method is illustrated by applying it to stretching overtones of octahedral molecules, SF6, WF6, and UF6 with group Oh.

By differentiating the dressed quark propagator with respect to a constant external tensor field, the linear response of the nonperturbative dressed quark propagator in the presence of the constant external tensor field can be obtained. From this we extract a general expression for the vacuum susceptibility which is needed in the QCD sum rule external field method for the coupling of tensor current to hadrons and compare it with the previous approaches. The numerical value of the tensor vacuum susceptibility are calculated within the framework of the rainbow-ladder approximation of the Dyson–Schwinger approach.

We generalize our previous work [Phys. Rev. C 72, 035202 (2005)] on the linear response theory of the dressed quark propagator in the presence of a constant external field to the case of a variable external field in order to make it applicable to a wider class of problems. Using the axial vector vacuum susceptibility as an illustration, we apply this general formalism to extract a new expression for the axial vector vacuum susceptibility in the quantum chromodynamical (QCD) sum rule two-point external field formula. The numerical values of the axial vector vacuum susceptibility are calculated within the framework of the rainbow-ladder approximation of the Dyson-Schwinger approach. A comparison with the results of the previous approaches is given.

The ND and NB systems with I = 0 and 1, $J^{P}=\frac{1}{2}^{\pm}$ , $\frac{3}{2}^{\pm}$ , and $\frac{5}{2}^{\pm}$ are investigated within the framework of the quark delocalization color screening model. The results show that all the positive-parity states are unbound. By coupling to the $ND^{\ast}$ channel, the state ND with I = 0, $ J^{P}=\frac{1}{2}^{-}$ can form a bound state, which can be invoked to explain the observed $\Sigma(2800)$ state. The mass of the $ND^{\ast}$ with I = 0, $J^{P}=\frac{3}{2}^{-}$ is close to that of the reported $ \Lambda_{c}(2940)^{+}$ , which indicates that $ \Lambda_{c}(2940)^{+}$ can be explained as a $ ND^{\ast}$ molecular state in QDCSM. Besides, the $ \Delta D^{\ast}$ with I = 1, $J^{P}=\frac{5}{2}^{-}$ is also a possible resonance state. The results of the bottom case of the NB system are similar to those of the ND system. Searching for these states will be a challenging subject of experiments.

In this paper we calculate the mass and probability fractions of the meson-meson components of $X(3872)$ in an unquenched quark model. Different from most other unquenched quark models, the quark-pair creation operator from $^{3}{P}_{0}$ is modified by considering the effects of the created quark pair's energy and the separation between the created quark pair and the valence quark pair. In the calculation all of the wave functions of the mesons and the relative motion between two mesons are obtained by solving the corresponding Schr\"odinger equation with the help of the Gaussian expansion method. The multichannel couplings of the quark-antiquark state with possible meson-meson states are calculated. The results show that $X(3872)$ can be described as a mixing state of the dominant charmonium state (70%) and meson-meson components (30%).

In the framework of the quark delocalization color screening model, we investigate tetraquarks composed of $$us\bar{d}\bar{b}$$ and $$ud\bar{s}\bar{b}$$ in two structures: meson-meson structure and diquark–antidiquark structure. Neither bound state nor resonance state is found in the system composed of $$us\bar{d}\bar{b}$$ . The reported X(5568) cannot be explained as a molecular state or a diquark–antidiquark resonance of $$us\bar{d}\bar{b}$$ in present calculation. However, two bound states of the diquark–antidiquark structure are obtained in the tetraquarks system composed of $$ud\bar{s}\bar{b}$$ : an $$IJ=00$$ state with the mass of 5701 MeV, and an $$IJ=01$$ state with the mass of 5756 MeV, which maybe the better tetraquark states with four different flavors. Our results indicate that the diquark–antidiquark configuration would be a good choice for the tetraquarks $$ud\bar{s}\bar{b}$$ with quantum numbers $$IJ=00$$ and $$IJ=01$$ . The tetraquarks composed of $$ud\bar{s}\bar{b}$$ is more possible to form bound states than the one composed of $$us\bar{d}\bar{b}$$ . These bound states are worth investigating in future experiments.

The tidal deformabilities and radii of strange quark stars are studied via the quasiparticle model which includes the nonperturbative features of QCD in the low-density region. The results show that the mass constraint of ${M}_{\mathrm{TOV}}&gt;2.0\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ rules out the EOSs which are soft at low densities, while the constraint on the tidal deformability of ${\mathrm{\ensuremath{\Lambda}}}_{1.4}&lt;800$ from GW170817 rules out the EOSs which are too stiff in the low density region. The range for the radius of a $1.4\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ strange quark star is $11.12\text{ }\text{ }\mathrm{km}&lt;{R}_{1.4}&lt;11.98\text{ }\text{ }\mathrm{km}$. ${\mathrm{\ensuremath{\Lambda}}}_{1.4}$ has a strong correlation with ${R}_{1.4}$, and the empirical correlation function is ${\mathrm{\ensuremath{\Lambda}}}_{1.4}=2.86\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}(R/\mathrm{km}{)}^{6.92}$, which is larger than that for neutron stars. The lower bound of ${\mathrm{\ensuremath{\Lambda}}}_{1.4}&gt;513.66$ is also obtained. $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Lambda}}}$ is a monotonically increasing function of the mass ratio $\ensuremath{\eta}$, but the slope is very small. And we conclude that the range of $\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Lambda}}}$ for GW190425 is $184.81&lt;\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Lambda}}}&lt;320.08$.