Since the LHCb collaboration successively reported the hidden-charm exotic states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub></math>, these exotic states have rightfully attracted significant attention from theoretical perspectives. Motivated by these exotic states, this study systematically investigates the hidden-charm double-strange pentaquark system with the quark configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi><mi>s</mi><mi>s</mi><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></math> using the resonating group method within the framework of the quark delocalization color screening model. The dynamical estimations, incorporating channel coupling, identify five resonance states in the hidden-charm double-strange systems, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">Ω</mi><mo>*</mo></msup><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, through scattering phase shifts processes of various channels. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, with mass range and decay widths of (4684 MeV–4688 MeV, 6.4 MeV–24.2 MeV) and (4751 MeV–4756 MeV, 3.2 MeV–18.7 MeV), can be derived from the scattering processes of channels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Ξ</mi><msub><mi>η</mi><mi>c</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Ξ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi></msub></math>, it should be noted that resonance state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math>, in addition to the channels mentioned above, can also be observed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">Ξ</mi><mo>*</mo></msup><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math>. For the quantum number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">Ω</mi><mo>*</mo></msup><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> can be identified in channels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Ξ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">Ξ</mi><mo>*</mo></msup><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="normal">Ξ</mi><mo>*</mo></msup><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math>, respectively, the estimated masses and decay widths of these resonance states range from approximately (4749 MeV–4755 MeV, 16.4 MeV) and (4771 MeV-4772 MeV, 18.8 MeV). However, the residual resonance state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi></msub></math>, for which the estimated mass and decay width are about 4600 MeV and 21.7 MeV, is expected to be found only in channel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math>. Given the current theoretical estimations, experiments are anticipated to further explore hidden-charm double-strange pentaquark states through the possible decay channels, thereby offering a strong theoretical foundation for their experimental investigation.

China

Xuejie Liu

Yue Tan

Xiaoyun Chen

Dianyong Chen

Hongxia Huang

Jialun Ping

Study on the properties of hidden-charm pentaquarks with double strangeness

Recently, the BESIII Collaboration updated the data for the exotic state, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></math>, with a mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2318.3</mn><mo>±</mo><mn>1.2</mn><mo>±</mo><mn>1.2</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> from the positronium annihilation process. Inspired by the experiment, we systematically investigated the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover></math> meson family spectrum from the perspectives of two-quark, four-quark, and mixed two-four-quark structures, including <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>*</mo></msubsup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2460</mn><mo stretchy="false">)</mo></math>, with the help of the Gaussian expansion method, which is a very accurate calculation method for few-body systems. We found that with appropriate spin-orbit coupling parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mi>s</mi></msub></math>, the chiral quark model can effectively describe both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2460</mn><mo stretchy="false">)</mo></math>. The four-quark calculation shows that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>K</mi></math> state does not form a bound state, but the effective potential calculation shows an attractive interaction between them. Finally, we performed mixed two-four-quark calculations, and the results, with the help of the unquenched effect, show that the energies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2460</mn><mo stretchy="false">)</mo></math> are in better agreement with experimental values. Therefore, we conclude that these two particles are good candidates for mixed two-four-quark states.

Yuheng Wu

Youchang Yang

Systematic study of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></mrow></math> in quark model

In this work, we systematically study <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1440</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo></math> in both the quenched three-quark and five-quark frameworks using the Gaussian expansion method within the chiral quark model. Our calculations show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo></math> can be reproduced as a three-quark state [<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1</mn><mi>P</mi><mo stretchy="false">)</mo></math>], while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1440</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo></math> cannot be accommodated as the three-quark candidates, [<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>2</mn><mi>S</mi><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1</mn><mi>P</mi><mo stretchy="false">)</mo></math>], respectively. In the five-quark framework, we find that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mi>K</mi></math> state for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo></math> cannot form a bound state, while in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> channel there will <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo></math> form a shallow bound state. Based on the complex-scaling method, we performed complete coupled-channel calculations and obtained six resonance states with energies ranging from 1.8 to 2.2 GeV, in addition to one bound state located around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Σ</mi><mi>π</mi></math> channel. However, neither molecular candidates in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mi>K</mi></math> channel for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo></math> nor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo></math> are included in these states. This is because the strong coupling between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Σ</mi><mi>π</mi></math> will make the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> unbound, while the weak coupling between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mi>K</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Σ</mi><mi>K</mi></math> cannot help form a stable structure around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mi>K</mi></math> threshold. Thus, under the quenched quark model, our results support <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo></math> as a three-quark state, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1440</mn><mo stretchy="false">)</mo></math> is neither a three-quark nor a five-quark state. In addition, we find that although <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo></math> can be primarily a five-quark state, it requires a mixture of three- and five-quark components for stability. In the future, an exploration on the mixing effects between bare baryons with these relevant two-body hadronic channel components will be carried out to further test our conclusions.

Zi-Xuan Ma

Xiaohuang Hu

Qi Huang

Investigating the nature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo></math> in a quenched chiral quark model

General one-loop formulas for loop-induced processes $gg/\gamma \gamma \rightarrow \phi_i\phi_j$ with $\phi_i\phi_j = hh,~hH,~HH$ are presented in this paper. The analytic expressions evaluated in this study are valid for a class of Higgs Extensions of the Standard Model, $\textit{e.g.,}$ Inert Doublet Higgs, Two Higgs Doublet, Zee-Babu, and Triplet Higgs Models$\textit{.}$ Analytic expressions for one-loop form factors are expressesd in terms of the basic scalar one-loop two-, three-, and four-point functions following the output format of the packages $ {\tt LoopTools}$ and $ {\tt Collier}$ . Hence, physical results can be evaluated numerically using one of these packages. The analytic results are tested using several checks such as the ultraviolet finiteness and infrared finiteness of the one-loop amplitudes. Furthermore, the amplitudes obey the ward identity due to massless gauge bosons in the initial states. This identity is also verified numerically. Regarding applications, we present the phenomenological results for the investigated processes in the Zee-Babu model as a typical example. In particular, production cross-sections for the processes $\gamma \gamma\rightarrow hh$ are scanned over the parameter space of the Zee-Babu Model.

Vietnam

Khiem Hong Phan

Dzung Tri Tran

Thanh Huy Nguyen

One-loop analytical expressions for $\textit{gg}$/$\textit{γγ}$ → ϕ$_{i}$ϕ$_{j}$ in Higgs extensions of the standard model and their applications * This research is supported by Vietnam National Foundation for Science and Technology Development (103.01-2023.16). Khiem Hong Phan and Dzung Tri Tran express their gratitude to all the valuable support from Duy Tan University, for the 30th anniversary of establishment (Nov. 11, 1994 - Nov. 11, 2024) towards ''Integral, Sustainable and Stable Development''

Recently, a charmonium <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3960</mn><mo stretchy="false">)</mo></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> decays in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><msubsup><mi>D</mi><mi>s</mi><mo>−</mo></msubsup></math> invariant-mass spectrum is discovered by the LHCb collaboration with the quantum number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>. Motivated by the discovery, in this work, we systematically investigated the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> tetraquark states with the quantum numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup><mo>,</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup><mo>,</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup><mo>,</mo><msup><mn>2</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> in the framework of the chiral quark model. In our calculations, we considered the meson-meson structure of the tetraquark states and the diquark-antidiquark structure, as well as the channel coupling of all channels of these two configurations are considered in this work. For example, all color structures including color singlet, hidden color channel, and the mixing of them are also taken into account. The numerical results indicate that no bound states were found in our model. There exist several resonant states by using the stabilization method, the so-called real scaling method. Among these states, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> resonant state with mass 3927 MeV matches very well with the energy of the newly discovered exotic state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3960</mn><mo stretchy="false">)</mo></math> reported by the LHCb collaboration. As a result, our calculations suggest that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3960</mn><mo stretchy="false">)</mo></math> can be interpreted as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> tetraquark state with quantum number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>. Apart from that, we also find several resonance states with mass 4179 and 4376 MeV with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>, there is likely one resonance state in the energy range of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>4310</mn><mi>–</mi><mn>4336</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, along with two resonance states at the energy of 4395 and 4687 MeV, respectively. Besides, two resonance states at 4300 and 4355 MeV for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup></math>, as well as one state at 4788 MeV for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>, are found, which are likely to be new exotic states. More experimental data is needed to confirm the existence of these resonance states.

Study of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> system in the chiral quark model

Inspired by the recent ATLAS 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 resonances: (6923), (6996), (7060) and (7159). (6923) and (7159) are experimental candidates for (6900) and (7200), whose main decay channel is $$J/\psi J/\psi $$ . It is important to note that the another major decay channel of (7159) is $$\chi _{c0} \chi _{c0}$$ , and the $$\chi _{c0} \chi _{c0}$$ is also the main decay channel of (6996), (7060). Therefore, we propose to search experimentally for these two predicted resonances in the $$\chi _{c0} \chi _{c0}$$ invariant mass spectrum.

Further study of $$c\bar{c}c\bar{c}$$ system within a chiral quark model

Within the framework of the quark delocalization color screening model, a systematic investigation of the anticharmed-strange pentaquark system is performed using the resonance group method. The currently estimations predict three bound states with estimated masses to be 2886 MeV, 3039 MeV, and 3153 MeV, respectively. Additionally, three resonance states are identified in various scattering phase shifts processes. Among them, two resonance states <math><mi>Σ</mi><mover><mi>D</mi><mo>¯</mo></mover></math> and <math><msup><mi>Σ</mi><mo>*</mo></msup><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> with quantum number <math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><msup><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo></mrow></math> are detected in channels <math><mi>N</mi><msubsup><mover><mi>D</mi><mo>¯</mo></mover><mi>s</mi><mo>*</mo></msubsup></math> and <math><mi>N</mi><mover><mi>D</mi><mo>¯</mo></mover></math>, and <math><mi>Σ</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> and <math><mi>Λ</mi><mover><mi>D</mi><mo>¯</mo></mover></math>, with masses and decay widths of (<math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>=</mo><mn>3053</mn><mi>–</mi><mn>3055</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mtext>total</mtext></mrow></msub><mo>=</mo><mn>13.0</mn><mi>–</mi><mn>13.4</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>) and (<math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>=</mo><mn>3389</mn><mi>–</mi><mn>3390</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mtext>total</mtext></mrow></msub><mo>=</mo><mn>10.4</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>), respectively. In the <math><mi>Λ</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> and <math><mi>Σ</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> channels, a resonance state with quantum number <math><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>(</mo><msup><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo></msup><mo>)</mo></math> is discovered, with its mass and decay width being 3250–3252 MeV and 4.4,MeV, respectively. These predicted pentaquark states have <math><mover><mi>c</mi><mo>¯</mo></mover><mi>s</mi><mi>n</mi><mi>n</mi><mi>n</mi></math> quark compositions, allowing them to be recognized as genuine pentaquark states. To validate these predictions, it is expected that upcoming experiments will further explore the predicted resonance and bound states in these possible decay channels.

Probing the nature of the anticharmed-strange pentaquark states: Mass spectra, decays, and magnetic moments

The unquenched quark model for the light quarkonium and charmonium states is explored in this study. The quark-pair creation operator in the   $^3P_0$   model, which combines the two-quark and four-quark components, is modified by considering the effects of the created quark pair's energy. Furthermore, the separation between the created quark pair and valence quark pair is modified. All the wave functions, including those for the mesons and the relative motion between two mesons, are obtained by solving the corresponding Schrödinger equation using the Gaussian expansion method. The aim of this study is to find a new set of parameters that can accurately describe the mass spectrum of low-lying light quarkonium and charmonium states. Moreover, certain exotic states, such as   $^3P_0$  , can be described well in the unquenched quark model.

Light quarkonium and charmonium mass shifts in an unquenched quark model  

Inspired by the recent report from the LHCb Collaboration on <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math>, which can be interpreted as a molecular <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math>, we investigated two trimeson systems of the <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> partner with <math><mi>I</mi><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>0</mn><msup><mn>1</mn><mo>−</mo></msup></math> in the framework of a chiral quark model. In the first case, because of the attraction between <math><msup><mi>D</mi><mo>*</mo></msup></math> and <math><mover><mi>D</mi><mo>¯</mo></mover></math>, we explored the existence of bound states in the system <math><msup><mi>D</mi><mo>*</mo></msup><mi>D</mi><mover><mi>D</mi><mo>¯</mo></mover></math>, which is obtained by adding <math><mover><mi>D</mi><mo>¯</mo></mover></math> into the molecular bound state <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> (<math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math>). Similarly, in the second case, we explored whether there is a bound state in the <math><msup><mi>D</mi><mo>*</mo></msup><mi>D</mi><mi>K</mi></math> system, which is obtained by adding <math><mi>K</mi></math> into the <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math>, given the attraction between <math><mi>D</mi></math> and <math><mi>K</mi></math>. The results show that both of them are bound states, in which the binding energy of the molecular state <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup><mi>K</mi></math> is relatively small, only <math><mn>0.4</mn><mo>±</mo><mn>0.4</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, while the binding energy of <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup><mover><mi>D</mi><mo>¯</mo></mover></math> is up to <math><mn>1.6</mn><mo>±</mo><mn>0.3</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. According to the calculation results of the root-square-mean distances, the spatial structure of the two systems shows the obvious <math><mrow><mo>(</mo><mi>D</mi><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>)</mo><mtext>−</mtext><mo>(</mo><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>/</mo><mi>K</mi><mo>)</mo></mrow></math> structure, in which <math><mi>D</mi></math> is close to <math><msup><mi>D</mi><mo>*</mo></msup></math> while <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> as a whole is relatively distant from the third hadron (<math><mover><mi>D</mi><mo>¯</mo></mover><mo>/</mo><mi>K</mi></math>), which are similar to the nucleon-electron structure. As a result, we strongly recommend that these bound states <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup><mover><mi>D</mi><mo>¯</mo></mover></math> and <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup><mi>K</mi></math> are searched for experimentally.

Dynamical study of <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mi>D</mi><mi>K</mi></mrow></math> and <math><msup><mi>D</mi><mo>*</mo></msup><mi>D</mi><mover><mi>D</mi><mo>¯</mo></mover></math> systems at quark level

In the framework of the chiral quark model, we investigate the <math><mi>Q</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> (<math><mrow><mi>Q</mi><mo>=</mo><mi>c</mi></mrow></math>, <math><mrow><mi>b</mi></mrow></math> and <math><mrow><mi>q</mi><mo>=</mo><mi>u</mi></mrow></math>, <math><mrow><mi>d</mi></mrow></math>) tetraquark system with two structures: <math><mrow><mi>Q</mi><mover><mrow><mi>q</mi></mrow><mrow><mo>¯</mo></mrow></mover><mtext>−</mtext><mi>q</mi><mover><mrow><mi>q</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> and <math><mi>Q</mi><mi>q</mi><mtext>−</mtext><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math>. The bound-state calculation shows that for the single channel, there is no evidence for any bound state below the minimum threshold in both <math><mi>c</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> and <math><mi>b</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> systems. However, after coupling all channels of two structures, we obtain a bound state below the minimum threshold in the <math><mi>c</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> system with the energy of 1998 MeV, and the quantum number is <math><mi>I</mi><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mn>0</mn><mo>+</mo></msup></math>. Meanwhile, in the <math><mi>b</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> system, two bound states with energies of 5414 MeV and 5456 MeV are obtained, and the quantum numbers are <math><mi>I</mi><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mn>0</mn><mo>+</mo></msup></math> and <math><mi>I</mi><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mn>1</mn><mo>+</mo></msup></math>, respectively. Besides, we also employ the real-scaling method to search for resonance states in the <math><mi>c</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> and <math><mi>b</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> systems. Unfortunately, no genuine resonance states were obtained in both systems. We suggest future experiments to search for these three possible bound states.