We systematically investigate the S-wave singly heavy tetraquark systems containing two or three strange quarks, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>n</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>n</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mi>Q</mi><mo>=</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>n</mi><mo>=</mo><mi>u</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></math>, within the constituent quark potential model. We solve the four-body Schrödinger equation using the Gaussian expansion method and identify resonances via the complex scaling method. There are no bound states below the lowest two-meson thresholds. We obtain several compact resonances with <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><mo>,</mo><msup><mn>2</mn><mo>+</mo></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math>, and <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>n</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>n</mi><mo stretchy="false">¯</mo></mover></math>. The pole positions are mainly distributed around 7.0–7.2 GeV (bottom) and 3.7–3.9 GeV (charm), with widths from a few to several tens of MeV. These resonances decay into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub><msup><mi>η</mi><mo>′</mo></msup><mo>,</mo><msubsup><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mo>*</mo></msubsup><mi>ϕ</mi><mo>,</mo><msup><msub><mi>D</mi><mi>s</mi></msub><mo>*</mo></msup><msup><mi>K</mi><mo>*</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>*</mo></msubsup><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> (and their bottom counterparts), providing targets for future experimental searches.

China

Xin-He Zheng

Yao Ma

Shi-Lin Zhu

Singly heavy tetraquark resonant states with multiple strange quarks

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.

Wei-Lin Wu

Germany

Lu Meng

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

The decay of the Higgs boson and the nature of dark matter remain fundamental challenges in particle physics. We investigate the 95 GeV diphoton excess and dark matter within the framework of the triplet-extended minimal supersymmetric standard model (TMSSM). In this model, an additional hypercharge $ Y=0 $ , $ S U(2)_L $ triplet superfield is introduced. Mixing between the triplet and doublet Higgs states enhances the diphoton signal strength of the 95 GeV Higgs boson, resulting in $ \mu_{\gamma\gamma}^{{\rm{CMS+ATLAS}}} = 0.24_{-0.08}^{+0.09} $ , which is consistent with experimental observations. This enhancement arises primarily from charged Higgs and chargino loop contributions, together with an LEP excess in the $ Zb\bar{b} $ channel around the same mass within the $ 2\sigma $ range. Additionally, the model accommodates viable dark matter candidates in the form of a bino-dominated neutralino. The relic density is reduced to the observed value through resonance-enhanced annihilation via the Higgs portal or co-annihilation with the triplino or higgsino. This reduction remains consistent with constraints from direct and indirect detection experiments. A comprehensive parameter scan demonstrates that the TMSSM can simultaneously explain the 95 GeV diphoton excess, observed 125 GeV Higgs mass, and dark matter relic density, establishing a compelling and theoretically consistent framework.

Zetian Li

Ning Liu

Bin Zhu

Interplay of 95 GeV diphoton excess and dark matter in supersymmetric triplet model * Supported by the National Natural Science Foundation of China (NNSFC) (12275232, 12335005) and Bin Zhu is also supported by the Natural Science Foundation of Shandong Province (ZR2025QA20)

We explore resonant production of sterile neutrino dark matter via the Shi-Fuller (SF) mechanism, revisiting its cosmological viability in light of recent results demonstrating that lepton-number asymmetries <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>α</mi></msub><mo>≳</mo><mn>1</mn></math> at temperatures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">T</mi><mo>&gt;</mo><mn>20</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> are consistent with big bang nucleosynthesis (BBN). Using a quasiclassical Boltzmann transport calculation of the dark matter production, we compute the nonthermal phase space distributions of sterile neutrinos across a broad range of particle mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>s</mi></msub></math> and mixing angle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>sin</mi><mn>2</mn></msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo></mrow></math> parameter space. We then evolve the resulting distributions through linear structure formation using CLASS and fit the resulting matter power spectra to thermal warm dark matter (WDM) transfer functions, enabling a direct mapping between SF models and equivalent thermal WDM particle masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>th</mi></msub></math>. This allows us to reinterpret existing structure formation limits and Lyman-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi></math> forest preferences in the context of SF production. We find that lepton asymmetries <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>L</mi><mo>≳</mo><mn>0.5</mn></math> at high temperatures open significant viable parameter space in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>s</mi></msub><mo>≳</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>keV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>sin</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>2</mn><mi>θ</mi><mo stretchy="false">)</mo><mo>≲</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>14</mn></mrow></msup></math> regime, compatible with both x-ray constraints from NuSTAR and INTEGRAL/SPI and recent Lyman-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>α</mi></math> inferences of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>th</mi></msub><mo>≈</mo><mn>4.1</mn><mtext> </mtext><mtext> </mtext><mi>keV</mi></math>. Following lepton number evolution below 20 MeV, we also specifically show that this lepton asymmetry parameter space is compatible with BBN and cosmic microwave background constraints. We present updated constraints, a refined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>th</mi></msub></math> fitting function, and power-law approximations for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>L</mi></math> across the parameter space. Our results motivate future x-ray observations targeting the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>20</mn><mtext> </mtext><mtext> </mtext><mi>keV</mi></math> photon regime and testing of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>th</mi></msub><mo>≳</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>keV</mi></math> WDM region.

United States

Cannon M. Vogel

Helena García Escudero

Julien Froustey

Kevork N. Abazajian

Return of the lepton number: Sterile neutrino dark matter production and the revival of the Shi-Fuller mechanism

This paper presents a systematic study of hyperon non-leptonic two-body decays induced by light quark transitions, particularly the $ s \rightarrow u\bar{u}d $ process, within the framework of $\textit{SU}$(3) flavor symmetry. The effective weak Hamiltonian is decomposed into irreducible $\textit{SU}$(3) representations, including the 27-plet and octet components, and applied to analyze decays of octet and decuplet baryons and charmed baryons. Both the irreducible representation amplitude (IRA) approach and the topological diagrammatic analysis (TDA) are employed to construct decay amplitudes and constrain the parameter space. $\textit{SU}$(3) symmetry-breaking effects arising from the strange quark mass are incorporated systematically. A global fit to current experimental data allows us to extract form factors and predict branching ratios and asymmetry parameters for several decay channels. Our results demonstrate the predictive power of $\textit{SU}$(3) flavor symmetry while highlighting significant symmetry-breaking effects, especially in amplitudes related to the 27-plet. Notably, the $ \Sigma^+ \rightarrow p\pi^0 $ decay channel exhibits a deviation exceeding $ 1\sigma $ from experimental measurements, suggesting the possible presence of new decay mechanisms or contributions beyond the Standard Model. Besides, we also evaluate the size of symmetry breaking. However, the large uncertainty of the experimental data makes it difficult to precisely determine the value of symmetry breaking. We strongly recommend that future experimental efforts aim to reduce the measurement uncertainties, especially for the processes $ \Lambda^0\to p\pi^- $ and $ \Lambda^0\to n\pi^0 $ , which have larger experimental errors compared to other data and play an important role in determining the symmetry breaking parameter. This study provides a systematic framework for future tests of the Standard Model and the search for new physics in hyperon decays.

Xin Wu

Qi Chen

Ye Xing

Zhi-Peng Xing

Ruilin Zhu

$\textit{SU}$(3) flavor symmetry analysis of hyperon non-leptonic two body decays * Ruilin Zhu is supported by NSFC (12322503, 12075124) and the Natural Science Foundation of Jiangsu (BK20211267). Zhi-Peng Xing is supported by NSFC (12375088, 12335003, 12405113). Ye Xing is supported by NSFC (12005294)

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.

<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)