For the first time, we implement the deep-neural-network-based variational Monte Carlo approach for the multiquark bound states, whose complexity surpasses that of electron or nucleon systems due to strong SU(3) color interactions. We design a novel and high-efficiency architecture, DeepQuark, to address the unique challenges in multiquark systems such as stronger correlations, extra discrete quantum numbers, and intractable confinement interaction. Our method demonstrates competitive performance with state-of-the-art approaches, including diffusion Monte Carlo and Gaussian expansion method, in the nucleon, doubly heavy tetraquark, and fully heavy tetraquark systems. Notably, it outperforms existing calculations for pentaquarks, exemplified by the triply heavy pentaquark. For the nucleon, we successfully incorporate three-body flux-tube confinement interactions without additional computational costs. In tetraquark systems, we consistently describe hadronic molecule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></mrow></math> and compact tetraquark <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi><mi>b</mi></mrow></msub></mrow></math> with an unbiased form of wave function ansatz. In the pentaquark sector, we obtain weakly bound <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow><mrow><mo>*</mo></mrow></msubsup></mrow></math> molecule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo stretchy="false">(</mo><mn>5715</mn><mo stretchy="false">)</mo></mrow></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi><mo>=</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math> and its bottom partner <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>b</mi><mi>b</mi><mover accent="true"><mrow><mi>b</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo stretchy="false">(</mo><mn>15569</mn><mo stretchy="false">)</mo></mrow></math>. They can be viewed as the analogs of the molecular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></mrow></math>. We recommend experimental search of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo stretchy="false">(</mo><mn>5715</mn><mo stretchy="false">)</mo></mrow></math> in the D-wave <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><msub><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math> channel. DeepQuark holds great promise for extension to larger multiquark systems, overcoming the computational barriers in conventional methods. It also serves as a powerful framework for exploring confining mechanism beyond two-body interactions in multiquark states, which may offer valuable insights into nonperturbative QCD and general many-body physics.

China

Wei-Lin Wu

Germany

Lu Meng

Shi-Lin Zhu

DeepQuark: A Deep-Neural-Network Approach to Multiquark Bound States

176Lu is the key nucleus for understanding the evolution of planetary bodies in the solar system, and the temperature and nucleosynthesis of the <italic>s</italic> process, due to its very long terrestrial half-life <italic>T</italic> <inf>1/2</inf>&#8239;&#8776;&#8239;3.7&#8239;&#215;&#8239;10 10 years. The very long half-life is caused by two anomalous first-forbidden transitions from the <math altimg='si2.svg'><msup><mn>7</mn><mo>&#8722;</mo></msup></math> state with extremely large comparative half-life log <italic>ft</italic>&#8239;&#8776;&#8239;19 which have never been appeared in other cases of the existing nuclear databases. We analyze the underlying mechanism and reason for the anomaly in first-forbidden transitions of 176Lu for the first time, which is based on the projected shell model. It is found that the possible <italic>K</italic>-forbidden nature is indispensable for describing the two extremely weak first-forbidden transitions, and the transition strengths are very sensitive to the detailed configuration mixing and <italic>K</italic> mixing in the nuclear wave functions. The half-life of the <math altimg='si2.svg'><msup><mn>7</mn><mo>&#8722;</mo></msup></math> state is calculated to be <italic>T</italic> <inf>1/2</inf>&#8239;&#8776;&#8239;1.95&#8239;&#215;&#8239;10 10 years. The sensitive dependence of the stellar <math altimg='si1.svg'><msup><mi>&#946;</mi><mo>&#8722;</mo></msup></math>-decay rate of 176Lu on <italic>s</italic>-process temperature is discussed as well.

Jing-Wen Ran

Long-Jun Wang

Anomaly in first-forbidden transitions of 176Lu

We employ Heavy Baryon Chiral Perturbation Theory (HB $$\chi $$ PT), a non-relativistic effective field theory that treats baryons as heavy static sources, to calculate the electromagnetic polarizabilities of spin-3/2 baryons in two sectors: the light-flavor decuplet baryons and singly heavy sextet baryons. We derive the analytical expressions up to $$\mathcal {O}\left( p^3\right) $$ . Our results indicate that the long-range chiral corrections provide substantial contributions to the polarizabilities. In addition, magnetic dipole (M1) transitions of the baryons can significantly affect the magnetic polarizabilities and may reverse their signs. For the decuplet baryons, the $$\Delta ^+$$ and $$\Delta ^0$$ exhibit the largest electric polarizabilities. Their values, $$\alpha _E(\Delta ^+) = (17.5 \pm 9.5)\times 10^{-4} \, \textrm{fm}^3$$ and $$\alpha _E(\Delta ^0) = (17.0 \pm 9.3)\times 10^{-4} \, \textrm{fm}^3$$ , significantly exceed those typically observed for nucleons. Meanwhile, the electric polarizabilities of spin-3/2 singly heavy baryons are comparable to those of their spin-1/2 partners.

Liang-Zhen Wen

Yan-Ke Chen

Electromagnetic polarizabilities of the spin- $$\frac{3}{2}$$ baryons in heavy baryon chiral perturbation theory

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%.

Jun-Zhang Wang

Zi-Yang Lin

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.

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

The hadronic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Σ</mi><mi>c</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><msup><mi>D</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> interactions are revisited, with a focus on their short-range parts, motivated by a tension between the interpretations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><mn>4312</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><mn>4440</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><mn>4457</mn><mo stretchy="false">)</mo></math> in effective field theory (EFT) frameworks and the one-boson-exchange (OBE) model. While the three states can be interpreted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Σ</mi><mi>c</mi></msub><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> molecular states within EFT frameworks, this is not feasible in the single-channel OBE model with consistent cutoff. In this work, the possibility to reconcile OBE model with EFTs by refitting the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi></math>-, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>-exchange interaction is explored and ruled out. It is pointed out that the problem in OBE arises from the strong short-range spin-dependent one-pion-exchange (OPE) interaction and the fixed signs of other short-range interactions in OBE model also prevent the cancellation. To address this issue, the short-range subtraction strategies within the OBE model are revisited. Two subtraction schemes are explored: removing the delta-function from all interactions and eliminating it only from the pseudoscalar-meson-exchange component. These schemes favor different spin assignments for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><mn>4440</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><mn>4457</mn><mo stretchy="false">)</mo></math>. Though solving the problem, there is no clear dynamical picture to support the subtraction schemes. We propose a new quark-exchange mechanism motivated by the Pauli principle. Different from the two subtraction schemes in OBE, the quark-exchange mechanism offers an explanation grounded in microscopic dynamics. It is shown that the spin-dependent quark-exchange interaction cancels those from OPE. The differences in the predictions for the spin, isospin, and open-charm partner states of the experimental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow></msub></math> states offer a way to distinguish between the subtracted OBE model and the OBE model with quark-exchange contributions.

Ru Xu

Toward modeling the short-range interactions of hidden and open charm pentaquark molecular states

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 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

Experimentally, the doubly charm tetraquark channel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mi>c</mi><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover></math> with <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> features an exotic hadron, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math>, with isospin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>0</mn></math> near the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> threshold, while no peak was observed for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>1</mn></math>. We present a lattice QCD study of this channel with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>1</mn></math>, <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub><mo>≃</mo><mn>280</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. Finite-volume energies calculated across five charm quark masses consistently feature a positive energy shift with respect to noninteracting energies, indicating repulsive interaction at energies near the threshold. These energies are used to compute the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> scattering amplitude using both the standard Lüscher method and the recently proposed effective field theory-based approach in the plane wave basis, which incorporates the long-range interactions and the left-hand cut. Both analyses render a small negative scattering length and a scattering amplitude that does not feature any poles in the energy region near the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> threshold, in line with the LHCb results. We identify that the Wick contraction, resembling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi></math>-channel isovector-vector meson exchanges between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup></math>, plays a key role in distinguishing between the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>1</mn></math> channels, leading to repulsion in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>1</mn></math> and attraction in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>0</mn></math> channel.