The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math>, as the first and the most crucial member in the exotic charmoniumlike <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mi>Y</mi><mi>Z</mi></math> family, has been studied for a long time. However, its dynamical origin, whether stemming from a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> hadronic molecule or the first excited <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave charmonium <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></math>, remains controversial. In this Letter, we demonstrate that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> definitely does not result from the mass shift of the higher bare <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></math> resonance pole in the coupled-channel dynamics involving a short-distance <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></math> core and the long-distance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> channels. Instead, it originates from either the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> molecular pole or the shadow pole associated with the antiresonance of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave charmonium, depending on the weak or strong coupling mode, respectively. To differentiate these origins and fully exploit the nature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math>, we conduct a comprehensive analysis in a couple-channel dynamics framework, including the isospin violation, the three-body <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>π</mi></math> effect, the dynamical width of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup></math>, and nonopen-charm decays of the bare <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2</mn><mi>P</mi><mo stretchy="false">)</mo></math>. Our findings highlight the pivotal role of the coupled-channel dynamics in explaining the disparity between the pole widths of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math>, while also predicting a new resonance with <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>1</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> around 4.0 GeV. By matching the newly observed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>4010</mn><mo stretchy="false">)</mo></math> by the LHCb Collaboration to our predicted resonance, we conclude that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> most likely originates from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> pole with a confidence level exceeding 99.7%.

China

Jun-Zhang Wang

Zi-Yang Lin

Yan-Ke Chen

Germany

Lu Meng

Shi-Lin Zhu

Uncovering the mystery of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> with the coupled-channel dynamics

We investigate the existence of bound states and resonances in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi><mo>,</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi><mi>N</mi><mo>,</mo><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi><mi>N</mi></math> systems using HAL QCD interactions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mo>,</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi></math>. We employ the Gaussian expansion method to solve the complex-scaled Schrödinger equation and find no resonances or bound states in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi><mi>N</mi></math> systems. We estimate the interaction between charmonium and nuclei, concluding that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math> is likely to bind with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi mathvariant="normal">H</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>He</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>He</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>4</mn></mrow></mmultiscripts></mrow></math>, and heavier nuclei. For the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math> system, the lattice QCD <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ϕ</mi><mi>N</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>2</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></math> interaction is absent. We combine the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>p</mi></math> correlation function analysis and HAL QCD results in model A. We assume the spin-spin interactions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi></math> systems are inversely proportional to their masses in model B. Model A predicts a stronger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>2</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></math> interaction and permits a two-body bound state, whereas model B suggests the interaction is attractive but too weak to form a bound state. Both models predict bound states for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>0</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math> systems. In model A, these states are deeply bound with binding energies exceeding 15 MeV and remain existent when considering parameter uncertainties. In contrast, these states are very loosely bound in model B, with binding energies below 1 MeV and an existent probability of about 60% when parameter uncertainties are considered. In both models, there exist very loosely bound <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> three-body states which resemble a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi></math>-d atom with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi></math> meson surrounding the deuteron, but their existence is sensitive to parameter uncertainties. No bound states or resonances are found in the isovector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math> system.

Liang-Zhen Wen

Yao Ma

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi><mi>N</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi><mi>N</mi></math> systems based on HAL QCD interactions

In this study, we propose using the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math> pole position to constrain the existence of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> three-body bound state within the one-boson-exchange (OBE) model. The existence of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> bound state remains uncertain due to significant variations in the OBE interaction, particularly in the strength of scalar-meson-exchange interactions, which can differ by a factor about 20 between two commonly used OBE models. This discrepancy renders the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> system highly model-dependent. To address this issue, we constrain the scalar-meson-exchange interaction using the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math> pole position, where the pseudoscalar-meson coupling is well-determined, and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi></math>-exchange interactions nearly cancel each other out, leaving the coupling constant of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>-exchange as the only unknown parameter. Our results indicate that the isospin-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> bound states exist when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math> is a virtual state of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>D</mi><msup><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><mo>/</mo><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> located within approximately <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>−</mo><mn>15</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> of the threshold. However, the three-body bound state is gone when the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math> virtual state pole is more than 20 MeV away from the threshold. Each experimental progress, either on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> state or the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math>, can shed light on the nature of the other state. Another significant outcome is a refined set of OBE model parameters calibrated using the pole positions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mn>3875</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math>, rigorously addressing the cutoff dependence. These parameters provide a valuable resource for more accurate calculations of systems involving few-body <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup></math>, and their antiparticles. Additionally, we find no evidence of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> three-body resonances after extensive search using a combination of the Gaussian expansion method and the complex scaling method.

Hai-Xiang Zhu

Ning Li

Wei Chen

Constraining the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> three-body bound state via the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math> pole

In this study, we investigate the $R^2$ corrections to the dissociation of heavy quarkonium in the Gauss-Bonnet gravitational background. We analyze the impact of the Gauss-Bonnet parameter $\lambda_{\rm GB}$ on the spectral function of charmonium and bottomonium and examine the effect of $\lambda_{\rm GB}$ on the dissociation of heavy quarkonium. Our results show that $\lambda_{\rm GB}$ reduces the peak height and increases the peak width of the spectral function, suggesting that $\lambda_{\rm GB}$ enhances the dissociation of heavy quarkonium. We also discuss the variation in the dissociation of heavy quarkonium with the ratio of shear viscosity to entropy density and observe that the dissociation is easier in more perfect plasma. Additionally, we observe that the temperature decreases the peak height and widens the peak, thereby accelerating the dissociation.

Zhou-Run Zhu

Manman Sun

Rui Zhou

Jinzhong Han

$\textit{R}$ $^{2}$ corrections to holographic heavy quarkonium dissociation * Rui Zhou supported by the Science and Technology Development Plan Project of Henan Province, China (242102230085), the National Natural Science Foundation of China (12404470), the High Level Talents Research and Startup Foundation Projects for Doctors of Zhoukou Normal University, China (ZKNUC2023017). Manman Sun supported by the National Natural Science Foundation of China (12305076). Zhou-Run Zhu supported by the High Level Talents Research and Startup Foundation Projects for Doctors of Zhoukou Normal University, China (ZKNUC2023018)

We investigate the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave triply heavy tetraquark systems, including the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><msup><mi>Q</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><msup><mi>Q</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> configurations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mi>c</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>b</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>q</mi><mo>=</mo><mi>u</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>d</mi></mrow></math>) with spin-parity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>0</mn><mo>+</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mo>+</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mo>+</mo></msup></math>, within the framework of the constituent quark model. We construct the wave functions for these systems, incorporating complete color-spin wave functions and spatial wave functions with three different Jacobi coordinate configurations. We use complex scaling and the Gaussian expansion method to solve the complex-scaled four-body Schrödinger equation and obtain possible exotic states. To analyze the spatial properties of the tetraquark states, we compute the root-mean-square (rms) radii, which help distinguish between meson-molecular and compact tetraquark states. We find that there do not exist any bound states in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><msup><mi>Q</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><msup><mi>Q</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>Q</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> systems. However, we identify a possible molecular resonant state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mn>3</mn><mi>c</mi><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>5819</mn><mo stretchy="false">)</mo></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover></math> system with spin-parity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>2</mn><mo>+</mo></msup></math>, which lies slightly above the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mi>D</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>2</mn><mi>S</mi><mo stretchy="false">)</mo></math> threshold and has a large rms radius around 2.2 fm. Furthermore, we obtain a series of compact resonant states, some of which exhibit spatial structure similar to the tritium atom in QED, where the light quark circles around the cluster of three heavy quarks. The lowest resonant state in the triply heavy systems has a mass of 5634 MeV, a width of 16 MeV, and spin-parity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math>. We suggest searching for this state in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>D</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><msup><mi>D</mi><mo>*</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><msup><mi>D</mi><mo>*</mo></msup></math> decay channels.

Hui-Min Yang

Wei-Lin Wu

Triply heavy tetraquark states with different flavors

This work presents the first prediction of tetralepton resonant states containing muons, extending beyond the simplest tetralepton system, dipositronium (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>Ps</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math>). With the rapid advancements in experimental facilities, the production and study of these intriguing states may be within reach. We perform a comprehensive analysis of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave trilepton and tetralepton systems within the framework of a QED Coulomb potential. We employ the Gaussian expansion method to solve the three- or four-body Schrödinger equation and utilize the complex scaling method to identify resonant states. We uncover a series of bound and resonant states in the trilepton systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>, as well as the tetralepton systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>. The energies of these states range from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>−</mo><mn>30</mn><mtext> </mtext><mtext> </mtext><mi>eV</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>−</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>eV</mi></math> below the total mass of three or four leptons, with their widths varying from less than 0.01 eV to approximately 0.07 eV. Additionally, we calculate the spin configurations and root mean square radii of these states, providing insight into their spatial structures. No bound or resonant states are found in the trilepton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>e</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math> systems, nor in the tetralepton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>e</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> system. A comparison with fully heavy tetraquark systems reveals that the additional color degree of freedom in QCD results in the absence of low-energy bound and resonant states. However, this extra degree of freedom allows for a broader range of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi><mi>C</mi></mrow></msup></mrow></math> quantum numbers to produce resonant states, highlighting the rich complexity of QCD systems.

Trilepton and tetralepton bound and resonant states: The QED counterpart of multiquark states

We study the transverse momentum ( $p_T$ ) differential anisotropic flow and flow fluctuation in Pb+Pb collisions at $\sqrt{s_{_{NN}}}$ =5.02 TeV at the LHC. A (3+1)-dimensional CLVisc hydrodynamics framework with fluctuating TRENTO (or AMPT) initial conditions is utilized to simulate the space-time evolution of the quark-gluon plasma (QGP) medium. The effects of shear viscosity and the sub-nucleon structure on anisotropic flow and flow fluctuation are analyzed. Our result shows that shear viscosity tends to suppress both flow coefficients ( ${v_2\{2}\}$ , ${v_2\{4\}}$ , and ${\langle v_2\rangle}$ ) and flow fluctuation ( ${\sigma_{v_2}}$ ) owing to its smearing effect on local density fluctuation. The flow coefficients appear to be insensitive to the sub-nucleon structure, whereas the flow fluctuation ${\sigma_{v_2}}$ tends to be suppressed by the sub-nucleon structure in central collisions but enhanced in peripheral collisions. After taking into account the sub-nucleon structure effect, our numerical result can qualitatively describe the relative flow fluctuations ( ${v_2\{4\}/v_2\{2\}}$ , $F({v_2})$ ) measured by the ALICE Collaboration at the LHC. We further investigate the effects of shear viscosity, sub-nucleon structure, and initial condition model on the flow angle and flow magnitude decorrelations ( ${A_2^f}$ , ${M_2^f}$ ) using the four-particle correlation method. We find that the flow decorrelation effect is typically stronger in central collisions than in peripheral collisions. The flow angle decorrelation is found to be insensitive to the shear viscosity and sub-nucleon structure, whereas the flow magnitude decorrelation shows quite different behaviors when using the TRENTO and AMPT initial condition models. Our study sheds light on the anisotropic flow, transport properties, and initial structure of the QGP created in high-energy nuclear collisions.

Jie Zhu

Canada

Xiang-Yu Wu

Guang-You Qin

Anisotropic flow, flow fluctuation, and flow decorrelation in relativistic heavy-ion collisions: the roles of sub-nucleon structure and shear viscosity * Supported in part by the Natural Science Foundation of China (12225503, 11890710, 11890711, 11935007). Zhu Jie is supported in part by the China Scholarship Council (202306770009). Wu Xiang-Yu is supported in part by the National Science Foundation within the framework of the JETSCAPE collaboration (OAC-2004571) and in part by the Natural Sciences and Engineering Research Council of Canada. Some of the calculations were performed in the Nuclear Science Computing Center at Central China Normal University (NSC$^{3}$), Wuhan, Hubei, China.

We calculate the electromagnetic polarizabilities of the spin-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> singly heavy baryons in the heavy baryon chiral perturbation theory up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>3</mn></msup><mo stretchy="false">)</mo></math>. We estimate the low-energy constants using the magnetic moments of singly charmed baryons from lattice QCD simulations and the experimental decay widths of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Σ</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Σ</mi><mi>c</mi><mo>*</mo></msubsup></math>. Our results indicate that the long-range chiral corrections make significant contributions to the polarizabilities. Additionally, the magnetic dipole transitions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="script">B</mi><mn>6</mn><mo>*</mo></msubsup><mo stretchy="false">→</mo><msub><mi mathvariant="script">B</mi><mn>6</mn></msub><mo>+</mo><mi>γ</mi></math> also provide large contribution to the magnetic polarizabilities.

Electromagnetic polarizabilities of the spin-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> singly heavy baryons in heavy baryon chiral perturbation theory

We conduct the first comprehensive P-wave four-body dynamical calculations of the fully charmed tetraquark systems within the quark potential model. We apply the Gaussian expansion method to solve the four-body Schrödinger equation, incorporating both dimeson and diquark-antidiquark spatial configurations. The matrix elements of P-wave states are calculated analytically using the infinitesimally shifted Gaussian basis functions. With the complex scaling method, we obtain several fully charmed P-wave resonant states with compact tetraquark configuration in the mass region of (7.0, 7.2) GeV, including states with exotic quantum numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo><mo>−</mo></mrow></msup><mo>,</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mo>+</mo></mrow></msup></mrow></math>. However, we find no resonant states with the mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi><mo>&lt;</mo><mn>7</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> and width <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>&lt;</mo><mn>200</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. Combining the present investigation with our previous results on S-wave fully charmed tetraquark systems [, <a href="http://dx.doi.org/10.1103/PhysRevD.109.054034">Phys. Rev. D 109, 054034 (2024)</a>], we find no candidates for the experimental states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>6400</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>6600</mn><mo stretchy="false">)</mo></math> in the quark model.

Fully charmed P-wave tetraquark resonant states in the quark model

The BESIII Collaboration recently performed a precise measurement of the <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>D</mi><mover><mi>D</mi><mo>¯</mo></mover></math> Born cross sections, and confirmed the <math><mi>G</mi><mo>(</mo><mn>3900</mn><mo>)</mo></math> structure reported by BABAR and Belle with high significance. We identify the <math><mi>G</mi><mo>(</mo><mn>3900</mn><mo>)</mo></math> as the first <math><mi>P</mi></math>-wave <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> molecular resonance. The experimental and theoretical identification of the <math><mi>P</mi></math>-wave dimeson state holds paramount importance in enhancing our comprehension of the nonperturbative QCD and few-body physics. Its existence is firmly established in a unified meson-exchange model that simultaneously depicts the features of the <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>3872</mn><mo>)</mo></math>, <math><msub><mi>Z</mi><mi>c</mi></msub><mo>(</mo><mn>3900</mn><mo>)</mo></math>, and <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub><mo>(</mo><mn>3875</mn><mo>)</mo></math>. This scenario can be directly examined in the <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> cross section by seeing whether a resonance exists at the threshold. The credibility of the investigations is also ensured by the fact that the <math><mi>P</mi></math>-wave interaction dominantly arises from the well-known long-range pion exchange. Additionally, thanks to the centrifugal barrier, it is easier to form resonances in <math><mi>P</mi></math>-wave than in <math><mi>S</mi></math>-wave. We extensively calculate all systems up to <math><mi>P</mi></math>-wave with various quantum numbers and predict a dense population of the <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> and <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> states, where the <math><mi>S</mi></math>-wave <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> state with <math><msup><mi>I</mi><mi>G</mi></msup><mo>(</mo><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mn>0</mn><mo>−</mo></msup><mo>(</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup><mo>)</mo></math>, <math><mi>P</mi></math>-wave <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> state with <math><msup><mi>I</mi><mi>G</mi></msup><mo>(</mo><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mn>0</mn><mo>+</mo></msup><mo>(</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup><mo>)</mo></math>, and <math><mi>P</mi></math>-wave <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> state with <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><msup><mn>0</mn><mo>−</mo></msup><mo>)</mo></math> are more likely to be observed in experiments.

Jian-Bo Cheng

Identification of the <math><mrow><mi>G</mi><mo>(</mo><mn>3900</mn><mo>)</mo></mrow></math> Structure as the <math><mi>P</mi></math>-Wave <math><mrow><mi>D</mi><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><mo>/</mo><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> Resonance