In this paper, we use the Hořava gravity model and EHT observations of supermassive black holes (BHs) to investigate signatures of Lorentz violation in real astrophysical environments. The Lorentz violation in the rotating Hořava BH spacetime are confined to the strong gravitational field region, being induced by the BH’s rotation. Due to the non-separability of the photon motion equations in this spacetime, we employed a numerical backward ray-tracing method to generate shadow images for various BH parameters. Subsequently, we extracted coordinate positions characterizing the shadow shape from high-pixel images to evaluate the parameter space of the BH. When evaluating M87*, Lorentz violation can occur with arbitrary magnitude. However, for Sgr A*, we can impose certain parameter constraints on Lorentz violation. These constraints depend on the BH’s spin. Should future observations at the highest inclination angle of the spin axis determine that the spin parameter of Sgr A* is less than 0.6, general relativity would be challenged, and the current EHT observations will serve as crucial evidence for the existence of Lorentz violation in low-energy environments.

China

Wentao Liu

Hongxia Huang

Di Wu

Jieci Wang

Lorentz violation signatures in the low-energy sector of Hořava gravity from black hole shadow observations

<p>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.</p>

Xuejie Liu

Yue Tan

Xiaoyun Chen

Dianyong Chen

Jialun Ping

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

<p>We perform a systematical investigation of the doubly charmed dibaryon system with quantum numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mi>J</mi><mo>=</mo><mn>01</mn></math> and strangeness numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>−</mo><mn>2</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>−</mo><mn>4</mn></math> in the framework of the chiral quark model. Two resonance states with strangeness numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mo>=</mo><mo>−</mo><mn>2</mn></math> are obtained in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><msub><mi mathvariant="normal">Ω</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> scattering channel: the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>*</mo></msubsup><mi mathvariant="normal">Ξ</mi></math> state with resonance mass 5081 MeV and decay width 0.3 MeV, and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup></math> state with mass 5213 MeV and decay width 19.8 MeV. These two predicted charmed dibaryon candidates are worth searching for experimentally. Besides this, we would like to emphasize that the multichannel coupling calculation is important for confirming the existence of multiquark states. The coupling can shift the energy of the resonance, give width to the resonance, and even destroy the resonance. Therefore, to provide the necessary information for experiments to search for exotic hadron states, the coupling calculation between the bound channels and open channels is indispensable.</p>

Yao Cui

Yuheng Wu

Xinmei Zhu

Search for doubly charmed dibaryons in baryon-baryon scattering

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

We perform a systematical investigation of the singly charmed dibaryon system with strangeness numbers <math><mi>S</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, <math><mo>−</mo><mn>3</mn></math>, and <math><mo>−</mo><mn>5</mn></math> in the framework of the chiral quark model. Two resonance states with strangeness numbers <math><mi>S</mi><mo>=</mo><mo>−</mo><mn>1</mn></math> are obtained in the baryon-baryon scattering process. In the <math><mi>Λ</mi><msub><mi>Λ</mi><mi>c</mi></msub></math> and <math><mi>N</mi><msubsup><mi>Ξ</mi><mi>c</mi><mo>*</mo></msubsup></math> scattering phase shifts, the <math><mi>Σ</mi><msub><mi>Σ</mi><mi>c</mi></msub></math> appears as a resonance state with the mass and width 3591–3614 MeV and 11.75 MeV, respectively. In the <math><mi>N</mi><msub><mi>Ξ</mi><mi>c</mi></msub></math> and <math><mi>N</mi><msubsup><mi>Ξ</mi><mi>c</mi><mo>′</mo></msubsup></math> scattering phase shifts, the <math><mi>Σ</mi><msubsup><mi>Σ</mi><mi>c</mi><mo>*</mo></msubsup></math> exhibits as a resonance state with the mass and width 3621–3624 MeV and 14.9 MeV, respectively. All these heavy-flavor dibaryons are worth searching for in experiments. Besides, we would like to emphasize that the coupling calculation between the bound channels and open channels is indispensable. The study of the scattering process maybe an effective way to look for the genuine resonances.

Search for singly charmed dibaryons in baryon-baryon scattering

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

We have investigated the <math><mi>q</mi><mi>s</mi><mi>s</mi><mover><mi>q</mi><mo>¯</mo></mover><mi>q</mi></math> (<math><mi>q</mi><mo>=</mo><mi>u</mi></math> or <math><mi>d</mi></math>) system to find possible pentaquark explanations for the <math><mi>Ξ</mi></math> 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 <math><mi>Ξ</mi><mo>(</mo><mn>1950</mn><mo>)</mo></math> state can be interpreted as a <math><mi>Λ</mi><msup><mover><mi>K</mi><mo>¯</mo></mover><mo>*</mo></msup></math> state with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>. Three states are identified that match the <math><mi>Ξ</mi><mo>(</mo><mn>2250</mn><mo>)</mo></math>, which are a <math><msup><mi>Σ</mi><mo>*</mo></msup><msup><mover><mi>K</mi><mo>¯</mo></mover><mo>*</mo></msup></math> state with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, a <math><msup><mi>Σ</mi><mo>*</mo></msup><msup><mover><mi>K</mi><mo>¯</mo></mover><mo>*</mo></msup></math> state with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>5</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, and a <math><msup><mi>Ξ</mi><mo>*</mo></msup><mi>ρ</mi></math> state with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>5</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>. This may explain the conflicting experimental values for the width of the <math><mi>Ξ</mi><mo>(</mo><mn>2250</mn><mo>)</mo></math> state. A new <math><mi>Ξ</mi></math> 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 <math><mi>Ξ</mi></math> resonances and to the future search for new <math><mi>Ξ</mi></math> resonances. Moreover, it is meaningful to further investigate the <math><mi>Ξ</mi></math> resonances from an unquenched picture on the basis of pentaquark investigation.

Ye Yan

Qi Huang

Investigating <math><mi>Ξ</mi></math> resonances from a pentaquark perspective

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.

Youchang Yang

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.

Exotic <math><mi>Q</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> states in the chiral quark 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 a resonance state or bound state. In this paper, taking the <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> 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.