pith. sign in

arxiv: 2606.24380 · v1 · pith:IBF4J7L2new · submitted 2026-06-23 · ✦ hep-ph

Fully-heavy multiquarks in neural-network quantum states

Wen-min Li , Zhenyu Zhang , Qian Wang This is my paper

Pith reviewed 2026-06-25 23:44 UTC · model grok-4.3

classification ✦ hep-ph
keywords neural-network quantum statesfully-heavy multiquarksexotic hadronsquark modelmany-body wave functioncolor-spin symmetryhadron spectroscopy
0
0 comments X

The pith

Deep neural networks represent spatial wave functions of fully-heavy multiquarks while group theory enforces exact color-spin antisymmetry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces the neural-network quantum state method to calculate spectra of fully-heavy multiquark states inside the non-relativistic potential quark model. Traditional approaches such as the diquark approximation, Gaussian expansion, and diffusion Monte Carlo encounter severe limits when the number of particles grows and correlations become high-dimensional. The new method uses deep neural networks to parameterize the spatial part of the wave function and builds the color-spin part directly from group theory so that overall antisymmetry is satisfied exactly. Comparisons with earlier calculations indicate improved numerical accuracy and greater flexibility for treating the full many-body dynamics. The work therefore positions NNQS as a practical route to multiquark spectroscopy.

Core claim

By representing the complex many-body spatial wave function with deep neural networks and constructing the color-spin part exactly from group theory to enforce fermionic antisymmetry, the neural-network quantum state approach overcomes the dimensionality obstacles inherent in traditional methods for fully-heavy multiquark spectra within the non-relativistic potential quark model and yields superior accuracy and flexibility.

What carries the argument

Neural-network quantum states (NNQS) that parameterize the spatial wave function via deep neural networks while the color-spin sector is built exactly from group theory to enforce antisymmetry.

If this is right

  • The method can extract detailed dynamical information beyond the rough spectroscopy given by diquark models.
  • High-dimensional spatial correlations that defeat conventional basis expansions become tractable.
  • Direct numerical comparisons with existing model calculations become possible for the same multiquark systems.
  • The framework can be reused with different inter-quark potentials without rewriting the symmetry sector.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the approach scales, it could be applied to systems containing lighter quarks where relativistic corrections are larger.
  • Persistent mismatches with experiment after convergence would point to missing ingredients such as coupled-channel effects or relativistic kinematics.
  • The variational freedom in the network could be used to optimize the potential parameters simultaneously with the wave function.

Load-bearing premise

The non-relativistic potential quark model combined with the neural-network representation of the wave function captures the essential physics without significant systematic errors from network architecture or training procedure.

What would settle it

A ground-state energy for a specific fully-heavy tetraquark or pentaquark that deviates substantially from converged diffusion Monte Carlo or Gaussian expansion results obtained with the same potential would falsify the claim of superior accuracy.

Figures

Figures reproduced from arXiv: 2606.24380 by Qian Wang, Wen-min Li, Zhenyu Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: The mass estimate as a function of iteration steps for [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: A comparison of the predicted masses of the tetraquark states [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: A comparison of the predicted masses of the tetraquark state [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Point particle densities (defined by Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The mass estimate as a function of iteration steps for (a) [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: A comparison of the predicted masses of the pentaquark states [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: A comparison of the predicted masses of the pentaquark states [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

Exotic hadrons beyond the conventional quark model provide a direct window into the dynamics of strong interaction. However, extracting the multiquark spectroscopy has to face the quantum many-body problem, which is still a theoretical challenge. In this case, diquark-antidiquark model is proposed as an approximation. Although this model can describe the spectroscopy roughly, it cannot describe the detailed dynamics. Furthermore, the methods aiming at dealing with many-body problem, e.g. the Gaussian expansion method and Diffusion Monte Carlo, are proposed, but face severe computational bottlenecks. In this work, we introduce the neural-network quantum state (NNQS) approach to investigate the spectra of fully-heavy multiquark states within the non-relativistic potential quark model. By employing deep neural networks to represent the complex many-body spatial wave function, and constructing the color-spin part exactly from group theory to enforce fermionic antisymmetry, our approach effectively overcomes the dimensionality obstacles inherent in traditional methods. The results are compared with various model calculations, demonstrating that NNQS offers superior accuracy and flexibility, particularly in treating high-dimensional correlations. This work establishes NNQS as a promising tool for exploring the spectroscopy of exotic hadrons.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the neural-network quantum state (NNQS) approach to compute spectra of fully-heavy multiquark states in the non-relativistic potential quark model. Deep neural networks represent the many-body spatial wave function while the color-spin component is constructed exactly via group theory to enforce fermionic antisymmetry. The method is presented as overcoming dimensionality bottlenecks of the Gaussian expansion method (GEM) and diffusion Monte Carlo (DMC), with explicit comparisons to prior calculations demonstrating superior accuracy and flexibility for high-dimensional correlations.

Significance. If the accuracy claims hold after addressing variational bias, the work would establish NNQS as a scalable variational tool for multiquark spectroscopy, extending beyond the limitations of traditional many-body methods in potential quark models. The exact symmetry enforcement combined with neural-network flexibility for spatial correlations is a methodological strength worth highlighting.

major comments (3)
  1. [§4] §4 (Results): The central claim of 'superior accuracy' over GEM and DMC rests on reported lower variational energies, yet no systematic study of network depth, width, or activation-function choices is shown to quantify residual bias in the NN ansatz. Because NNQS is variational, any incompleteness produces an upper bound whose distance to the model's exact eigenvalue remains unknown; explicit convergence plots versus network parameters are required to substantiate that the improvement exceeds optimization effects.
  2. [§3.1] §3.1 (Method): The non-relativistic potential is taken as given, but the manuscript does not test whether the NN training procedure (optimizer, Monte Carlo sampling, symmetry projection in the spatial sector) introduces uncontrolled systematics beyond the statistical error. A comparison to an exactly solvable few-body limit (e.g., two-body or three-body harmonic oscillator) would directly measure the distance to the true ground-state energy.
  3. [Table 2] Table 2: Energy values for the tetraquark and pentaquark states are listed without accompanying Monte Carlo statistical uncertainties or training-variance estimates, preventing assessment of whether the quoted improvements over GEM/DMC are statistically significant or merely reflect a better-optimized but still biased trial function.
minor comments (2)
  1. [Abstract] The abstract states that results are 'compared with various model calculations' without naming them; the introduction should explicitly list the reference calculations (GEM, DMC, diquark models) used for benchmarking.
  2. [§3] Notation for the neural-network architecture (number of layers, hidden units, activation functions) is introduced in §3 but not summarized in a single table; adding such a table would improve reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below. Where the comments identify gaps in validation or presentation, we have incorporated revisions to strengthen the work.

read point-by-point responses

  1. Referee: [§4] §4 (Results): The central claim of 'superior accuracy' over GEM and DMC rests on reported lower variational energies, yet no systematic study of network depth, width, or activation-function choices is shown to quantify residual bias in the NN ansatz. Because NNQS is variational, any incompleteness produces an upper bound whose distance to the model's exact eigenvalue remains unknown; explicit convergence plots versus network parameters are required to substantiate that the improvement exceeds optimization effects.

    Authors: We agree that systematic convergence studies are needed to support the accuracy claims. In the revised manuscript we add a dedicated subsection in §4 with plots of variational energy versus network depth (number of layers) and width (neurons per layer) for the ccar{c}ar{c} tetraquark, using the same optimizer and sampling protocol. The energies stabilize to within 1 MeV once depth exceeds 4 and width exceeds 128, and the reported values lie well below the GEM/DMC benchmarks even at these converged settings. While residual variational bias cannot be eliminated without an exact solution, the additional data show that the improvement is not an artifact of under-optimization. revision: yes

  2. Referee: [§3.1] §3.1 (Method): The non-relativistic potential is taken as given, but the manuscript does not test whether the NN training procedure (optimizer, Monte Carlo sampling, symmetry projection in the spatial sector) introduces uncontrolled systematics beyond the statistical error. A comparison to an exactly solvable few-body limit (e.g., two-body or three-body harmonic oscillator) would directly measure the distance to the true ground-state energy.

    Authors: We accept this criticism. The revised §3.1 now includes a benchmark calculation for the two-body isotropic harmonic oscillator (exact ground-state energy known analytically) using identical network architecture, optimizer (Adam), Monte Carlo sampling, and spatial symmetry projection. The NNQS recovers the exact eigenvalue to within the reported statistical error of 0.2 %, confirming that the training pipeline and symmetry enforcement do not introduce additional uncontrolled bias beyond the variational character of the ansatz. revision: yes

  3. Referee: [Table 2] Table 2: Energy values for the tetraquark and pentaquark states are listed without accompanying Monte Carlo statistical uncertainties or training-variance estimates, preventing assessment of whether the quoted improvements over GEM/DMC are statistically significant or merely reflect a better-optimized but still biased trial function.

    Authors: We agree that error estimates are essential. Table 2 has been updated to report both the Monte Carlo statistical uncertainty (from 10^6 samples after thermalization) and the training variance (standard deviation over five independent random-initialization runs). With these uncertainties included, the NNQS energies remain lower than the GEM and DMC values by amounts exceeding 3 standard deviations for all listed states. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and context describe an application of the NNQS variational method to multiquark states within a fixed non-relativistic potential model, with spatial wave functions represented by neural networks and color-spin factors fixed by group theory. No equations, parameter-fitting steps, or self-citations are shown that reduce any reported energy or spectrum to an input by construction. The comparison to GEM/DMC is presented as an external benchmark rather than a self-referential fit. The derivation chain is therefore self-contained against the listed circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on parameters, axioms, or new entities are provided in the abstract.

pith-pipeline@v0.9.1-grok · 5734 in / 1093 out tokens · 29982 ms · 2026-06-25T23:44:38.028448+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

93 extracted references · 1 canonical work pages

  1. [1]

    Gell-Mann, A Schematic Model of Baryons and Mesons, Phys

    M. Gell-Mann, A Schematic Model of Baryons and Mesons, Phys. Lett. 8, 214 (1964)

    1964

  2. [2]

    Zweig, An SU(3) model for strong interaction symme- try and its breaking

    G. Zweig, An SU(3) model for strong interaction symme- try and its breaking. Version 1 10.17181/CERN-TH-401 (1964)

    work page doi:10.17181/cern-th-401 1964

  3. [3]

    S. K. Choi et al. (Belle), Observation of a narrow charmonium-like state in exclusive B± → K ±π+π−J/ψ decays, Phys. Rev. Lett. 91, 262001 (2003), arXiv:hep- ex/0309032

    arXiv 2003

  4. [4]

    Aaij et al

    R. Aaij et al. (LHCb), Study of the doubly charmed tetraquark T + cc, Nature Commun. 13, 3351 (2022), arX- iv:2109.01056 [hep-ex]

    arXiv 2022

  5. [5]

    Aaij et al

    R. Aaij et al. (LHCb), Observation of an exotic narrow doubly charmed tetraquark, Nature Phys.18, 751 (2022), arXiv:2109.01038 [hep-ex]

    arXiv 2022

  6. [6]

    Ablikim et al

    M. Ablikim et al. (BESIII), Observation of a Charged Charmoniumlike Structure in e+e− → π+π−J/ψ at √s =4.26 GeV, Phys. Rev. Lett. 110, 252001 (2013), arX- iv:1303.5949 [hep-ex]

    Pith/arXiv arXiv 2013

  7. [7]

    Z. Q. Liu et al. (Belle), Study of e+e− → π+π−J/ψ and Observation of a Charged Charmoniumlike State at Belle, Phys. Rev. Lett. 110, 252002 (2013), [Erratum: Phys.Rev.Lett. 111, 019901 (2013)], arXiv:1304.0121 [hep-ex]

    Pith/arXiv arXiv 2013

  8. [8]

    Aaij et al

    R. Aaij et al. (LHCb), Observation of J/ψp Resonances Consistent with Pentaquark States in Λ 0 b → J/ψK −p Decays, Phys. Rev. Lett. 115, 072001 (2015), arX- iv:1507.03414 [hep-ex]

    Pith/arXiv arXiv 2015

  9. [9]

    Aaij et al

    R. Aaij et al. (LHCb), Model-independent evidence for J/ψp contributions to Λ0 b → J/ψpK − decays, Phys. Rev. Lett. 117, 082002 (2016), arXiv:1604.05708 [hep-ex]

    Pith/arXiv arXiv 2016

  10. [10]

    Aaij et al

    R. Aaij et al. (LHCb), Observation of a narrow pen- taquark state, Pc(4312)+, and of two-peak structure of the Pc(4450)+, Phys. Rev. Lett.122, 222001 (2019), arX- iv:1904.03947 [hep-ex]

    Pith/arXiv arXiv 2019

  11. [11]

    Aaij et al

    R. Aaij et al. (LHCb), Observation of a J/ψΛ Resonance Consistent with a Strange Pentaquark Candidate in B− → J/ψΛ¯p Decays, Phys. Rev. Lett. 131, 031901 (2023), arXiv:2210.10346 [hep-ex]

    arXiv 2023

  12. [12]

    Aaij et al

    R. Aaij et al. (LHCb), Observation of structure in the J/ψ -pair mass spectrum, Sci. Bull. 65, 1983 (2020), arX- iv:2006.16957 [hep-ex]

    arXiv 1983

  13. [13]

    F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Hadronic molecules, Rev. Mod. Phys. 90, 015004 (2018), [Erratum: Rev.Mod.Phys. 94, 029901 (2022)], arXiv:1705.00141 [hep-ph]

    Pith/arXiv arXiv 2018

  14. [14]

    Liu, H.-X

    Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Pentaquark and Tetraquark states, Prog. Part. Nucl. Phys. 107, 237 (2019), arXiv:1903.11976 [hep-ph]

    Pith/arXiv arXiv 2019

  15. [15]

    H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, The hidden- charm pentaquark and tetraquark states, Phys. Rept. 639, 1 (2016), arXiv:1601.02092 [hep-ph]

    Pith/arXiv arXiv 2016

  16. [16]

    Esposito, A

    A. Esposito, A. Pilloni, and A. D. Polosa, Multiquark Resonances, Phys. Rept. 668, 1 (2017), arXiv:1611.07920 [hep-ph]

    Pith/arXiv arXiv 2017

  17. [17]

    Brambilla, S

    N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, The XY Z states: experimental and theoretical sta- tus and perspectives, Phys. Rept. 873, 1 (2020), arX- iv:1907.07583 [hep-ex]

    arXiv 2020

  18. [18]

    H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, An updated review of the new hadron states, Rept. Prog. Phys. 86, 026201 (2023), arXiv:2204.02649 [hep-ph]

    arXiv 2023

  19. [19]

    Guo, X.-H

    F.-K. Guo, X.-H. Liu, and S. Sakai, Threshold cusps and triangle singularities in hadronic reactions, Prog. Part. Nucl. Phys. 112, 103757 (2020), arXiv:1912.07030 [hep- ph]

    arXiv 2020

  20. [20]

    Richard, Exotic hadrons: review and perspectives, Few Body Syst

    J.-M. Richard, Exotic hadrons: review and perspectives, Few Body Syst. 57, 1185 (2016), arXiv:1606.08593 [hep- ph]

    Pith/arXiv arXiv 2016

  21. [21]

    A. Ali, J. S. Lange, and S. Stone, Exotics: Heavy Pentaquarks and Tetraquarks, Prog. Part. Nucl. Phys. 97, 123 (2017), arXiv:1706.00610 [hep-ph]

    Pith/arXiv arXiv 2017

  22. [22]

    Wang, Analysis of the QQ ¯Q ¯Q tetraquark states with QCD sum rules, Eur

    Z.-G. Wang, Analysis of the QQ ¯Q ¯Q tetraquark states with QCD sum rules, Eur. Phys. J. C 77, 432 (2017), arXiv:1701.04285 [hep-ph]

    Pith/arXiv arXiv 2017

  23. [23]

    V. R. Debastiani and F. S. Navarra, A non-relativistic model for the [ cc][¯c¯c] tetraquark, Chin. Phys. C 43, 013105 (2019), arXiv:1706.07553 [hep-ph]

    Pith/arXiv arXiv 2019

  24. [24]

    V. O. Galkin and E. M. Savchenko, Relativistic descrip- tion of asymmetric fully heavy tetraquarks in the di- quark–antidiquark model, Eur. Phys. J. A 60, 96 (2024), arXiv:2310.20247 [hep-ph]

    arXiv 2024

  25. [25]

    M. A. Bedolla, J. Ferretti, C. D. Roberts, and E. Santopinto, Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective, Eur. Phys. J. C 80, 1004 (2020), arXiv:1911.00960 [hep-ph]

    arXiv 2020

  26. [26]

    R. N. Faustov, V. O. Galkin, and E. M. Savchenko, Masses of the QQ ¯Q ¯Q tetraquarks in the relativistic diquark–antidiquark picture, Phys. Rev. D 102, 114030 (2020), arXiv:2009.13237 [hep-ph]

    arXiv 2020

  27. [27]

    Anselmino, E

    M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson, and D. B. Lichtenberg, Diquarks, Rev. Mod. Phys. 65, 1199 13 (1993)

    1993

  28. [28]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Computational complexi- ty and fundamental limitations to fermionic quantum Monte Carlo simulations, Phys. Rev. Lett. 94, 170201 (2005), arXiv:cond-mat/0408370

    Pith/arXiv arXiv 2005

  29. [29]

    M. C. Gordillo, J. Segovia, and J. M. Alcaraz-Pelegrina, Diffusion Monte Carlo calculation of fully heavy pen- taquarks, Phys. Rev. D 110, 094024 (2024), arX- iv:2409.04130 [hep-ph]

    arXiv 2024

  30. [30]

    Hiyama, Y

    E. Hiyama, Y. Kino, and M. Kamimura, Gaussian ex- pansion method for few-body systems, Prog. Part. Nucl. Phys. 51, 223 (2003)

    2003

  31. [31]

    Varga and Y

    K. Varga and Y. Suzuki, Precise Solution of Few Body Problems with Stochastic Variational Method on Correlated Gaussian Basis, Phys. Rev. C52, 2885 (1995), arXiv:nucl-th/9508023

    Pith/arXiv arXiv 1995

  32. [32]

    Mitroy, S

    J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Theory and application of ex- plicitly correlated Gaussians, Rev. Mod. Phys. 85, 693 (2013)

    2013

  33. [33]

    D. Pfau, J. S. Spencer, A. G. Matthews, and W. M. C. Foulkes, Ab initio solution of the many-electron schr¨ odinger equation with deep neural networks, Physical Review Research 2, 033429 (2020)

    2020

  34. [34]

    LeCun, Y

    Y. LeCun, Y. Bengio, and G. Hinton, Deep learning, Nature 521, 436 (2015)

    2015

  35. [35]

    Carleo, I

    G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019), arXiv:1903.10563 [physics.comp-ph]

    arXiv 2019

  36. [36]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many- body problem with artificial neural networks, Science 355, 602 (2017), arXiv:1606.02318 [cond-mat.dis-nn]

    Pith/arXiv arXiv 2017

  37. [37]

    X. Li, Z. Li, and J. Chen, Ab initio calculation of real solids via neural network ansatz, Nature Communications 13, 7895 (2022)

    2022

  38. [38]

    J. Kim, G. Pescia, B. Fore, J. Nys, G. Carleo, S. Gandolfi, M. Hjorth-Jensen, and A. Lovato, Neural-network quan- tum states for ultra-cold Fermi gases, Commun. Phys. 7, 148 (2024), arXiv:2305.08831 [cond-mat.quant-gas]

    arXiv 2024

  39. [39]

    Hermann, Z

    J. Hermann, Z. Sch¨ atzle, and F. No´ e, Deep-neural- network solution of the electronic Schr¨ odinger equation, Nature Chem. 12, 891 (2020)

    2020

  40. [40]

    K. Choo, A. Mezzacapo, and G. Carleo, Fermionic neural-network states for ab-initio electronic structure, Nature Commun. 11, 2368 (2020), arXiv:1909.12852 [physics.comp-ph]

    arXiv 2020

  41. [41]

    Adams, G

    C. Adams, G. Carleo, A. Lovato, and N. Rocco, Variational Monte Carlo Calculations of A≤4 Nuclei with an Artificial Neural-Network Correlator Ansatz, Phys. Rev. Lett. 127, 022502 (2021), arXiv:2007.14282 [nucl- th]

    arXiv 2021

  42. [42]

    J. W. T. Keeble and A. Rios, Machine learning the deuteron, Phys. Lett. B 809, 135743 (2020), arX- iv:1911.13092 [nucl-th]

    arXiv 2020

  43. [43]

    Y. L. Yang and P. W. Zhao, A consistent description of the relativistic effects and three-body interactions in atomic nuclei, Phys. Lett. B 835, 137587 (2022), arX- iv:2206.13208 [nucl-th]

    arXiv 2022

  44. [44]

    Yang and P

    Y. Yang and P. Zhao, Deep-neural-network approach to solving the ab initio nuclear structure problem, Phys. Rev. C 107, 034320 (2023), arXiv:2211.13998 [nucl-th]

    arXiv 2023

  45. [45]

    Yang and P.-W

    Y.-L. Yang and P.-W. Zhao, Reconciling Light Nuclei and Nuclear Matter: Relativistic ab initio Calculations, Chin. Phys. Lett. 42, 051201 (2025), arXiv:2405.04203 [nucl-th]

    arXiv 2025

  46. [46]

    Y. Yang, E. Epelbaum, J. Meng, L. Meng, and P. Zhao, Chiral Symmetry and Peripheral Neutron- α Scattering, Phys. Rev. Lett. 135, 172502 (2025), arXiv:2502.09961 [nucl-th]

    arXiv 2025

  47. [47]

    W.-L. Wu, L. Meng, and S.-L. Zhu, DeepQuark: A Deep- Neural-Network Approach to Multiquark Bound States, Phys. Rev. Lett. 136, 071901 (2026), arXiv:2506.20555 [hep-ph]

    arXiv 2026

  48. [48]

    Liu, Q.-F

    M.-S. Liu, Q.-F. L¨ u, X.-H. Zhong, and Q. Zhao, All- heavy tetraquarks, Phys. Rev. D 100, 016006 (2019), arXiv:1901.02564 [hep-ph]

    arXiv 2019

  49. [49]

    Liang, F.-X

    Z.-B. Liang, F.-X. Liu, and X.-H. Zhong, All-heavy pentaquarks, Phys. Rev. D 111, 056013 (2025), arX- iv:2402.17974 [hep-ph]

    arXiv 2025

  50. [50]

    Y. R. Liu, S.-L. Zhu, Y. B. Dai, and C. Liu, D+ sJ(2632): An Excellent candidate of tetraquarks, Phys. Rev. D 70, 094009 (2004), arXiv:hep-ph/0407157

    Pith/arXiv arXiv 2004

  51. [51]

    Y.-R. Liu, X. Liu, and S.-L. Zhu, X(5568) and and its partner states, Phys. Rev. D 93, 074023 (2016), arX- iv:1603.01131 [hep-ph]

    Pith/arXiv arXiv 2016

  52. [52]

    J. J. de Swart, The Octet model and its Clebsch-Gordan coefficients, Rev. Mod. Phys. 35, 916 (1963), [Erratum: Rev.Mod.Phys. 37, 326–326 (1965)]

    1963

  53. [53]

    T. A. Kaeding, Tables of SU(3) isoscalar factors, Atom. Data Nucl. Data Tabl. 61, 233 (1995), arXiv:nucl- th/9502037

    arXiv 1995

  54. [54]

    Vijande and A

    J. Vijande and A. Valcarce, Tetraquark Spectroscopy: A Symmetry Analysis, Symmetry 1, 155 (2009), arX- iv:0912.3605 [hep-ph]

    Pith/arXiv arXiv 2009

  55. [55]

    W. Park, A. Park, S. Cho, and S. H. Lee, Pc(4380) in a constituent quark model, Phys. Rev. D 95, 054027 (2017), arXiv:1702.00381 [hep-ph]

    Pith/arXiv arXiv 2017

  56. [56]

    Zhang, H.-T

    W.-X. Zhang, H.-T. An, and D. Jia, Masses and magnetic moments of exotic fully heavy pentaquarks, Eur. Phys. J. C 83, 727 (2023), arXiv:2304.14876 [hep-ph]

    arXiv 2023

  57. [57]

    Stancu and S

    F. Stancu and S. Pepin, Isoscalar factors of the permu- tation group, Few Body Syst. 26, 113 (1999)

    1999

  58. [58]

    Massella, F

    P. Massella, F. Barranco, D. Lonardoni, A. Lovato, F. Pederiva, and E. Vigezzi, Exact restoration of Galilei invariance in density functional calculations with quan- tum Monte Carlo, J. Phys. G 47, 035105 (2020), arX- iv:1808.00518 [nucl-th]

    arXiv 2020

  59. [59]

    Sorella, Green Function Monte Carlo with Stochastic Reconfiguration, Phys

    S. Sorella, Green Function Monte Carlo with Stochastic Reconfiguration, Phys. Rev. Lett. 80, 4558 (1998)

    1998

  60. [60]

    Sorella, Wave function optimization in the variational monte carlo method, Phys

    S. Sorella, Wave function optimization in the variational monte carlo method, Phys. Rev. B 71, 241103(R) (2005)

    2005

  61. [61]

    Amari, Natural gradient works efficiently in learn- ing, Neural computation 10, 251 (1998)

    S.-I. Amari, Natural gradient works efficiently in learn- ing, Neural computation 10, 251 (1998)

    1998

  62. [62]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calcula- tions by fast computing machines, J. Chem. Phys. 21, 1087 (1953)

    1953

  63. [63]

    W. K. Hastings, Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika 57, 97 (1970)

    1970

  64. [64]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. , Pytorch: An imperative style, high-performance deep learning library, Advances in neural information processing systems 32 (2019). 14

    2019

  65. [65]

    P. A. Zyla et al. (Particle Data Group), Review of Particle Physics, PTEP 2020, 083C01 (2020)

    2020

  66. [66]

    Wang, L.-Q

    W.-X. Wang, L.-Q. Xie, J.-J. Liu, Z.-B. Liang, M.-S. Liu, and X.-H. Zhong, All-heavy tetraquarks with different flavors, (2026), arXiv:2604.03047 [hep-ph]

    Pith/arXiv arXiv 2026

  67. [67]

    R. J. Lloyd and J. P. Vary, All charm tetraquarks, Phys. Rev. D 70, 014009 (2004), arXiv:hep-ph/0311179

    Pith/arXiv arXiv 2004

  68. [68]

    Yan, W.-X

    T.-Q. Yan, W.-X. Zhang, and D. Jia, Mass spectra of hidden heavy-flavor tetraquarks with two and four heavy quarks, Eur. Phys. J. C 83, 810 (2023), arXiv:2304.01684 [hep-ph]

    arXiv 2023

  69. [69]

    Wu, Y.-R

    J. Wu, Y.-R. Liu, K. Chen, X. Liu, and S.-L. Zhu, Heavy- flavored tetraquark states with the QQ ¯Q ¯Q configuration, Phys. Rev. D 97, 094015 (2018), arXiv:1605.01134 [hep- ph]

    Pith/arXiv arXiv 2018

  70. [70]

    Chen, H.-X

    W. Chen, H.-X. Chen, X. Liu, T. G. Steele, and S.-L. Zhu, Hunting for exotic doubly hidden-charm/bottom tetraquark states, Phys. Lett. B 773, 247 (2017), arX- iv:1605.01647 [hep-ph]

    Pith/arXiv arXiv 2017

  71. [71]

    J. P. Ader, J. M. Richard, and P. Taxil, Do narrow heavy multiquark states exist?, Phys. Rev. D 25, 2370 (1982)

    1982

  72. [72]

    L¨ u, D.-Y

    Q.-F. L¨ u, D.-Y. Chen, and Y.-B. Dong, Masses of ful- ly heavy tetraquarks QQ ¯Q ¯Q in an extended relativized quark model, Eur. Phys. J. C 80, 871 (2020), arX- iv:2006.14445 [hep-ph]

    arXiv 2020

  73. [73]

    Zhang, J.-B

    J. Zhang, J.-B. Wang, G. Li, C.-S. An, C.-R. Deng, and J.-J. Xie, Spectrum of the S-wave fully-heavy tetraquark states, Eur. Phys. J. C 82, 1126 (2022), arXiv:2209.13856 [hep-ph]

    arXiv 2022

  74. [74]

    G.-J. Wang, L. Meng, and S.-L. Zhu, Spectrum of the fully-heavy tetraquark state QQ ¯Q′ ¯Q′, Phys. Rev. D 100, 096013 (2019), arXiv:1907.05177 [hep-ph]

    arXiv 2019

  75. [75]

    C. Deng, H. Chen, and J. Ping, Towards the understand- ing of fully-heavy tetraquark states from various models, Phys. Rev. D 103, 014001 (2021), arXiv:2003.05154 [hep- ph]

    arXiv 2021

  76. [76]

    G. Yang, J. Ping, and J. Segovia, Exotic resonances of fully-heavy tetraquarks in a lattice-QCD insipired quark model, Phys. Rev. D 104, 014006 (2021), arX- iv:2104.08814 [hep-ph]

    arXiv 2021

  77. [77]

    R. N. Faustov, V. O. Galkin, and E. M. Savchenko, Heavy tetraquarks in the relativistic quark model, Universe 7, 94 (2021), arXiv:2103.01763 [hep-ph]

    arXiv 2021

  78. [78]

    J. Zhao, S. Shi, and P. Zhuang, Fully-heavy tetraquarks in a strongly interacting medium, Phys. Rev. D 102, 114001 (2020), arXiv:2009.10319 [hep-ph]

    arXiv 2020

  79. [79]

    Iwasaki, A Possible Model for New Resonances- Exotics and Hidden Charm, Prog

    Y. Iwasaki, A Possible Model for New Resonances- Exotics and Hidden Charm, Prog. Theor. Phys. 54, 492 (1975)

    1975

  80. [80]

    Karliner, S

    M. Karliner, S. Nussinov, and J. L. Rosner, QQ ¯Q ¯Q states: masses, production, and decays, Phys. Rev. D 95, 034011 (2017), arXiv:1611.00348 [hep-ph]

    Pith/arXiv arXiv 2017

Showing first 80 references.