Pith. sign in

REVIEW 5 major objections 4 minor 2 cited by

A neural network trained on charmonium and bottomonium data reconstructs a non-quadratic dilaton that simultaneously matches the mass spectra and the monotonic drop of leptonic decay constants.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 08:04 UTC pith:B4YIHZSN

load-bearing objection A reasonable method demonstration whose headline 'resolution' overstates an in-sample fit. the 5 major comments →

arxiv 2601.18133 v4 pith:B4YIHZSN submitted 2026-01-26 hep-ph

Heavy Quarkonium Spectrum and Decay Constants from a Neural-Network-Based Holographic Model

Yu Zhang , Xun Chen , Miguel Angel Martin Contreras This is my paper
classification hep-ph
keywords dilatonAdS/QCDholographic modelheavy quarkoniumcharmoniumbottomoniumdecay constantsmachine learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to show that the heavy-quarkonium spectrum and its decay constants can be described by a single, smooth dilaton field learned directly from experimental data, rather than assumed from an analytic ansatz. Using a multilayer perceptron to represent the derivative of the dilaton, the authors build a holographic potential, solve the resulting Schrödinger-like equation, and train on PDG masses and leptonic decay constants for charmonium and bottomonium. The trained profile is non-quadratic, and the combined fit achieves RMS deviations of 1.26% for charmonium and 3.32% for bottomonium. If correct, this resolves a longstanding tension in bottom-up AdS/QCD models, which previously could reproduce either masses or decay constants but not both.

Core claim

The central claim is that a data-driven inverse construction of the dilaton field, learned by an MLP with the UV boundary condition Phi(0)=0, can simultaneously reproduce the radial excitation masses and the monotonically decreasing leptonic decay constants of heavy vector quarkonia. The reconstructed dilaton is non-quadratic: near the UV it deviates from the soft-wall quadratic form, lifting the degeneracy of decay constants, while in the IR it grows rapidly, encoding confinement. From this single profile, the model yields masses and decay constants for charmonium and bottomonium in close agreement with experiment, including the trend that higher excited states have progressively smaller de

What carries the argument

The central object is the dilaton field Phi(z), parametrized through its derivative Phi'(z) by a multilayer perceptron with automatic differentiation. The dilaton and its derivatives generate the holographic potential U(z) = 3/(4z^2) + Phi'(z)/(2z) + Phi'(z)^2/4 - Phi''(z)/2, which enters a Schrödinger-like equation whose eigenvalues give meson masses. Decay constants are then extracted from the near-boundary UV limit of the bulk-to-boundary modes via f_n = (1/g5 M_n) lim_{z->0} e^{-B(z)} partial_z v_n(z). The non-quadratic shape of Phi(z) is what breaks the degeneracy of decay constants and allows a simultaneous fit.

Load-bearing premise

The argument stands on the assumption that fitting the very masses and decay constants that define the loss yields a dilaton profile that is the physical background, even though the coupling g5 that sets the absolute scale of all decay constants is never specified and the network is trained on the same data it is evaluated against.

What would settle it

Pin down g5 using an independent normalization, for example from the vector current correlator or lattice QCD, and recompute the absolute decay constants; if the monotonic suppression disappears or the PDG values are missed, the claimed resolution is an artifact of an unspecified scale. Alternatively, hold out the highest radial excitations (e.g., bottomonium n=5,6) during training and check whether the learned dilaton predicts them.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the claim is correct, a single smooth dilaton profile, not an ad hoc quadratic ansatz, encodes the heavy-quark effects that make heavy-quarkonium Regge trajectories nonlinear and decay constants fall with excitation level.
  • The reconstruction method can be applied to other hadronic channels, providing a data-driven route to holographic backgrounds without presuming a specific analytic form.
  • The framework can be extended to finite temperature, where the learned dilaton may be used to compute spectral functions and quarkonium melting.
  • The non-quadratic UV behavior of the dilaton becomes a testable prediction for how the vector decay constants decrease, connecting the holographic background to the Van Royen-Weisskopf suppression.
  • The MLP approach offers a template for incorporating additional constraints, such as lattice QCD data, into holographic model building.

Where Pith is reading between the lines

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

  • The absolute scale of the decay constants is set by the gauge coupling g5, which the paper never fixes; the monotonic decrease could in principle be absorbed by choosing a z-dependent normalization, so the claim of 'resolving' the decay-constant problem is stronger than what the data alone can establish without pinning down g5.
  • Since the network is trained on the very masses and decay constants it then reproduces, the fit is in-sample; the real test of the reconstructed dilaton would be its predictions for states beyond the training set, such as higher radial excitations or other channels.
  • The non-quadratic dilaton learned here could be compared against the WKB-derived dilaton and Bethe-Salpeter-inspired forms; if the MLP profile collapses onto one of those analytic curves, it would suggest the data-driven approach is recovering a universal background rather than merely overfitting.
  • A testable extension would be to train on charmonium alone and predict bottomonium, or vice versa, to see whether the learned dilaton has any flavor-universal component or whether each quarkonium sector requires a separate background.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper presents a bottom-up AdS/QCD reconstruction of the dilaton field for heavy vector quarkonia. Instead of an analytic ansatz, a multilayer perceptron represents Phi'(z), with Phi(0)=0 enforced; the resulting potential U(z) is built from Phi and its automatic derivatives, and the Schr\"odinger equation is diagonalized to obtain masses and decay constants. The model is trained on PDG charmonium and bottomonium masses and decay constants, yielding RMS deviations of 1.26% and 3.32%. The authors claim that the resulting non-quadratic dilaton simultaneously reproduces the heavy-quarkonium spectrum and the monotonically decreasing leptonic decay constants, thereby resolving a longstanding difficulty in holographic QCD.

Significance. If the reconstruction could be shown to generalize and the decay-constant scale anchored, the approach would be a useful data-driven complement to analytic dilaton ansatze, and the public code release is a strength. The holographic derivation from the 5D action to the Schr\"odinger equation is standard. However, the central evidence is currently an in-sample fit by a high-capacity network, and the decay-constant absolute scale depends on an unspecified constant g5. The claim of 'resolving' the decay-constant problem therefore goes beyond what the presented tests establish.

major comments (5)
  1. [Tables I, II and Section III] The quoted RMS deviations are fitting errors on the training dataset; the text itself says the network 'demonstrates good fitting performance on the training dataset.' Since the MLP is trained to minimize the discrepancy between predicted and experimental masses and decay constants, the close agreement in the tables is an optimization result, not independent evidence for the physical dilaton. The abstract's 'resolves' needs support from an out-of-sample test, e.g., hold out one state per sector and retrain, or report leave-one-out cross-validation. Without such a test, the quantitative headline is a consistency check.
  2. [Eq. (18)] The decay constant is proportional to 1/g5, but no value for g5 (or R) is stated anywhere in the text. A common multiplicative factor sets the absolute scale of all decay constants in a sector, so the reported agreement of f_n with experiment does not by itself constrain the dilaton. Please state whether g5 is fixed from the holographic dictionary or fitted; if fitted, include it in the parameter count and report its value. As written, the absolute scale of the decay constants is a free parameter, weakening the claim that the model resolves the decay-constant problem.
  3. [Section III, Fig. 2] The MLP has five hidden layers with 128 neurons, i.e., roughly 10^5 parameters, while the training data consist of 8 charmonium and 12 bottomonium observables. The reconstructed dilaton profile is therefore strongly underdetermined; many different Phi(z) can fit the same data. The physical conclusions drawn from the 'non-quadratic' shape in Fig. 2 require a robustness study (different random seeds, architectures, or regularization) or a demonstration that the reconstruction is stable. Without this, the uniqueness of the extracted dilaton is not established.
  4. [Section III vs. Section IV] The training objective is described inconsistently. Section III first says the loss is SmoothL1Loss between the reconstructed potential and a target potential from a 'pre-trained model', then says the final implementation instead reconstructs Phi(z), while Section IV says the Adam optimizer minimizes the discrepancy between the predicted spectrum and experimental data. These are different statements. A single, coherent description of the loss, the role of any pre-trained model, and the exact training target (potential, dilaton derivative, or spectrum) is needed before the fitting claims can be evaluated.
  5. [Eq. (19), Tables I and II] The RMS definition weights all observables equally and does not use the experimental uncertainties. This is misleading for states with large PDG errors; for example, the f(\Upsilon(11020)) deviation is 11.3%, while the quoted experimental uncertainty is about 11.5%, so that point is consistent within 1 sigma. A weighted RMS or chi^2 per degree of freedom should be reported, along with a statement of which states dominate the residuals. This is important because the overall RMS values are the paper's main quantitative evidence.
minor comments (4)
  1. [Throughout] Several typos and formatting issues: 'using a MPL' should be 'MLP', 'anad hoc' needs a space, and the Schr\"odinger umlaut is inconsistently rendered. Reference [90] appears to duplicate the citation style of [43]; please unify the reference list.
  2. [Fig. 3] The insets are hard to read; the curves and axes labels are too small. Please enlarge the inset text and ensure each curve is clearly identifiable in both the main panel and the inset.
  3. [Section III] Numerical details for reproducibility are incomplete: the z-domain, grid spacing, finite-difference stencil, and the Hamiltonian diagonalization cutoff are not specified. These should be reported, since the eigenvalues and decay constants depend on them.
  4. [Eq. (19)] The definition of N and the weighting of masses versus decay constants are unclear. If N is the total number of observables in both sectors, the charmonium and bottomonium RMS values are computed over different numbers of points; please clarify the aggregation.

Circularity Check

2 steps flagged

Decay-constant 'resolution' is an in-sample fit: the same masses and f_n used as training targets are reported as predictions, with g5 unspecified so the absolute f_n scale is free.

specific steps
  1. fitted input called prediction [Abstract; Section III (Eq. (19), Tables I-II)]
    "Training on PDG data for charmonium and bottomonium yields a non-quadratic dilaton profile that resolves the longstanding difficulty of simultaneously reproducing both the heavy-quarkonium spectrum and the monotonic suppression of leptonic decay constants with radial excitation."

    The 'training data' are precisely the masses and decay constants appearing in Tables I-II: Section III states the profile is 'directly inferred from experimental data, specifically the mass spectrum and decay constants,' and Eq. (19) defines the reported RMS over an N that 'includes both masses and decay constants.' The excellent agreement in Tables I-II is therefore the minimized training objective, not an independent test. The paper even labels the fitted columns 'Theoretical predictions.' No holdout or out-of-sample validation is reported; the limitation paragraph concedes predictive power for n>6 is 'untested.'

  2. fitted input called prediction [Section II, Eq. (18); Section III, Tables I-II]
    "fn = 1 g5 Mn lim z→0 e−B(z) ∂z vn(z) , (18) where g5 is the five-dimensional gauge coupling constant and B(z) is the background function defined earlier."

    No numerical value for g5 (or R) is given anywhere in the manuscript or tables. Since g5 multiplies every f_n, the absolute scale of all decay constants is a free common factor, so matching the experimental f_n values imposes no constraint on the dilaton. What remains—the monotonic suppression pattern—is encoded in the network weights during the same training that uses those decay constants as targets. The decay-constant part of the claimed 'resolution' is thus adjustable in scale and fitted in shape, rather than derived predictively.

full rationale

This is a data-driven inverse-problem paper rather than a first-principles derivation, and the standard AdS/QCD equations (7)-(18) are not themselves circular. The circularity lies in the epistemic claim: the abstract says training on PDG data 'resolves' the long-standing tension, yet Section III explicitly says the dilaton profile is inferred from the mass spectrum and decay constants, and Eq. (19) computes the RMS over both. Hence the agreement reported in Tables I-II is the minimized training loss. It is not identity-by-construction—an arbitrary Phi would not reproduce the data—but with five 128-neuron hidden layers fitting only ten points, the success is a flexible interpolation. The paper itself states that predictive power for n>6 'remains untested,' confirming that no out-of-sample check is provided. Additionally, g5 is never fixed, so the absolute scale of f_n is free. I did not count the MCVD/WKB self-citations as load-bearing circularity because the MLP result is computed from the standard equations and the cited prior work is mostly used for motivation/comparison. Overall, the central 'resolution' claim reduces substantially to a fitted-input-called-prediction pattern: score 6.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 0 invented entities

No new particles or forces are introduced. The free parameters are the network weights and unspecified normalization constants (g5, grid, hyperparameters). The axioms are mostly standard domain assumptions of bottom-up holography, plus the ad hoc assumption that the inverse problem is well-posed and that separate dilatons per flavor are acceptable.

free parameters (4)
  • MLP weights and biases (two separate networks per quarkonium sector) = not stated (five hidden layers of 128 neurons each)
    The entire function Phi'(z) is parameterized by the network weights. These are optimized to match the PDG masses and decay constants, so the reconstructed dilaton is a fitted object.
  • 5D gauge coupling g5 = not stated
    Needed in Eq. (18) to set the absolute scale of decay constants. Without an independently fixed g5, the reported f_n values can be rescaled arbitrarily, weakening the claimed simultaneous fit.
  • Holographic grid and IR cutoff = not stated
    The Schrödinger equation is discretized on a finite grid and diagonalized; grid spacing and z_max are not reported, which can affect higher eigenvalues.
  • Training hyperparameters and loss weights = not stated
    Learning rate, clipping threshold, and the relative weighting of masses versus decay constants in the loss are not specified, and the loss target is described inconsistently in §III.
axioms (7)
  • domain assumption The vector meson is dual to a massless 5D gauge field with M5^2 R^2 = 0 (Eqs. 4-5).
    Standard AdS/CFT dictionary for the vector current with scaling dimension 3.
  • domain assumption The soft-wall dilaton e^{-Phi(z)} with Phi(0)=0 produces a confining potential and a discrete spectrum (Eq. 2).
    The whole bottom-up construction relies on this mechanism; no derivation of the dilaton from QCD is attempted.
  • standard math The Sturm-Liouville equation can be transformed to a Schrödinger equation with potential (17) and diagonalized.
    The Bogoliubov transformation and spectral theory are standard.
  • standard math A multilayer perceptron can approximate any continuous Phi'(z) (universal approximation theorem).
    Invoked in the introduction to justify the MLP representation.
  • ad hoc to paper The PDG masses and decay constants for 4 charmonium and 6 bottomonium states are sufficient data to determine the dilaton profile.
    The inverse problem is treated as well-posed, but this is not demonstrated and is doubtful given the number of free network parameters.
  • ad hoc to paper Charmonium and bottomonium can be described by two separate dilaton fields, one per sector.
    A single universal dilaton is not constructed; flavor dependence is imposed by construction rather than derived.
  • domain assumption Open-flavor thresholds and coupled-channel effects do not materially affect the fitted low-lying spectrum.
    The paper itself acknowledges these effects are not incorporated, and they may be relevant for states near or above open-flavor thresholds such as Υ(4S) and Υ(5S).

pith-pipeline@v1.3.0-alltime-deepseek · 12487 in / 13790 out tokens · 155593 ms · 2026-08-03T08:04:58.622011+00:00 · methodology

0 comments
read the original abstract

We present a data-driven inverse construction of the dilaton field in a bottom-up AdS/QCD description of heavy vector quarkonia. Instead of adopting an \emph{ad hoc} analytic ansatz, we use a multilayer perceptron to learn \(\Phi'(z)\) as a smooth function of the holographic coordinate, with \(\Phi(0)=0\) imposed to ensure ultraviolet consistency. The dilaton and its derivatives obtained by automatic differentiation generate the holographic potential \(U(z)\), and the associated Schr\"odinger-like equation is discretized and diagonalized to extract the low-lying eigenmodes. Masses and decay constants are then evaluated from the eigenvalues and the near-boundary behavior of the bulk-to-boundary modes. Training on PDG data for charmonium and bottomonium yields a non-quadratic dilaton profile that resolves the longstanding difficulty of simultaneously reproducing both the heavy-quarkonium spectrum and the monotonic suppression of leptonic decay constants with radial excitation. The combined fit achieves RMS deviations of \(1.26\%\) (charmonium) and \(3.32\%\) (bottomonium). This work establishes neural-network reconstruction as a flexible tool for holographic modeling and provides a basis for future extensions incorporating additional channels, lattice constraints, or finite-temperature backgrounds.

Figures

Figures reproduced from arXiv: 2601.18133 by Miguel Angel Martin Contreras, Xun Chen, Yu Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: A neural network flowchart for reconstructing the dilaton field Φ( [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Dilaton fields Φ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Potentials [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

discussion (0)

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

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Holographic spectral functions of exotic spin-1 mesons

    hep-ph 2026-07 conditional novelty 5.0

    In a soft-wall holographic model with gluon condensate, the π1 hybrid and Zc tetraquark-like ground states both melt below the deconfinement temperature, with Zc melting slightly earlier and a larger condensate delayi...

  2. Thermodynamics of the Isospectral family of holographic vector mesons

    hep-ph 2026-05 unverdicted novelty 4.0

    Isospectral transformations in softwall AdS/QCD allow tuning the rho meson's decay constant to its experimental value of 226 MeV, producing a predicted melting temperature of 157 MeV from spectral function computation...

Reference graph

Works this paper leans on

89 extracted references · 74 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Polchinski and M

    J. Polchinski and M. J. Strassler, Phys. Rev. Lett.88, 031601 (2002), arXiv:hep-th/0109174

    Pith/arXiv arXiv 2002

  2. [3]

    Erlich, E

    J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett.95, 261602 (2005), arXiv:hep-ph/0501128

    Pith/arXiv arXiv 2005

  3. [4]

    S. J. Brodsky and G. F. de Teramond, Phys. Rev. D77, 056007 (2008), arXiv:0707.3859 [hep-ph]

    Pith/arXiv arXiv 2008

  4. [5]

    J. M. Maldacena, Int. J. Theor. Phys.38, 1113 (1999), arXiv:hep-th/9711200

    Pith/arXiv arXiv 1999

  5. [6]

    Aharony, S

    O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Phys. Rept.323, 183 (2000), arXiv:hep-th/9905111

    Pith/arXiv arXiv 2000

  6. [7]

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428, 105 (1998), arXiv:hep- th/9802109

    arXiv 1998

  7. [8]

    Karch and E

    A. Karch and E. Katz, JHEP06, 043 (2002), arXiv:hep-th/0205236

    Pith/arXiv arXiv 2002

  8. [9]

    Sakai and S

    T. Sakai and S. Sugimoto, Prog. Theor. Phys.113, 843 (2005), arXiv:hep-th/0412141

    Pith/arXiv arXiv 2005

  9. [10]

    Sakai and S

    T. Sakai and S. Sugimoto, Prog. Theor. Phys.114, 1083 (2005), arXiv:hep-th/0507073

    Pith/arXiv arXiv 2005

  10. [11]

    Boschi-Filho and N

    H. Boschi-Filho and N. R. F. Braga, Eur. Phys. J. C32, 529 (2004), arXiv:hep-th/0209080

    Pith/arXiv arXiv 2004

  11. [12]

    Karch, E

    A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. D74, 015005 (2006), arXiv:hep-ph/0602229 [hep-ph]

    Pith/arXiv arXiv 2006

  12. [13]

    H. R. Grigoryan and A. V. Radyushkin, Phys. Rev. D76, 095007 (2007), arXiv:0706.1543 [hep-ph]

    Pith/arXiv arXiv 2007

  13. [14]

    Erdmenger, N

    J. Erdmenger, N. Evans, I. Kirsch, and E. Threlfall, Eur. Phys. J. A35, 81 (2008), arXiv:0711.4467 [hep-th]

    Pith/arXiv arXiv 2008

  14. [15]

    Colangelo, F

    P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau, and S. Nicotri, Phys. Rev. D78, 055009 (2008), arXiv:0807.1054 [hep-ph]

    Pith/arXiv arXiv 2008

  15. [16]

    Ballon Bayona, H

    A. Ballon Bayona, H. Boschi-Filho, N. R. F. Braga, and M. A. C. Torres, JHEP01, 052 (2010), arXiv:0911.0023 [hep-th]

    Pith/arXiv arXiv 2010

  16. [17]

    A. L. Cotrone, A. Dymarsky, and S. Kuperstein, JHEP03, 005 (2011), arXiv:1010.1017 [hep-th]. 16

    Pith/arXiv arXiv 2011

  17. [18]

    Li and M

    D. Li and M. Huang, JHEP11, 088 (2013), arXiv:1303.6929 [hep-ph]

    Pith/arXiv arXiv 2013

  18. [19]

    N. Wen, X. Cao, J. Chao, and H. Liu, Phys. Rev. D109, 086021 (2024), arXiv:2402.06239 [hep-th]

    Pith/arXiv arXiv 2024

  19. [20]

    X. Cao, S. Qiu, H. Liu, and D. Li, JHEP08, 005 (2021), arXiv:2102.10946 [hep-ph]

    Pith/arXiv arXiv 2021

  20. [21]

    Thomas Arun and R

    M. Thomas Arun and R. Pal, (2025), arXiv:2512.16450 [hep-ph]

    Pith/arXiv arXiv 2025

  21. [22]

    Kim, J.-P

    Y. Kim, J.-P. Lee, and S. H. Lee, Phys. Rev. D75, 114008 (2007), arXiv:hep-ph/0703172 [hep-ph]

    Pith/arXiv arXiv 2007

  22. [23]

    H. R. Grigoryan, P. M. Hohler, and M. A. Stephanov, Phys. Rev. D82, 026005 (2010), arXiv:1003.1138 [hep-ph]

    Pith/arXiv arXiv 2010

  23. [24]

    Y. Li, P. Maris, X. Zhao, and J. P. Vary, Phys. Lett. B758, 118 (2016), arXiv:1509.07212 [hep-ph]

    Pith/arXiv arXiv 2016

  24. [25]

    N. R. F. Braga, M. A. Martin Contreras, and S. Diles, Phys. Lett. B763, 203 (2016), arXiv:1507.04708 [hep-ph]

    Pith/arXiv arXiv 2016

  25. [26]

    Chen, Phys

    J.-K. Chen, Phys. Lett. B786, 477 (2018), arXiv:1807.11003 [hep-ph]

    Pith/arXiv arXiv 2018

  26. [27]

    Chen, Eur

    J.-K. Chen, Eur. Phys. J. A57, 238 (2021), arXiv:2102.07993 [hep-ph]

    Pith/arXiv arXiv 2021

  27. [28]

    Chen, Nucl

    J.-K. Chen, Nucl. Phys. A1050, 122927 (2024), arXiv:2302.05926 [hep-ph]

    Pith/arXiv arXiv 2024

  28. [29]

    M. A. Martin Contreras and A. Vega, Phys. Rev. D102, 046007 (2020), arXiv:2004.10286 [hep-ph]

    Pith/arXiv arXiv 2020

  29. [30]

    M. A. Martin Contreras and A. Vega, arXiv preprint (2025), arXiv:2509.04956 [hep-ph]

    Pith/arXiv arXiv 2025

  30. [31]

    Navaset al.(Particle Data Group), Phys

    S. Navaset al.(Particle Data Group), Phys. Rev. D110, 030001 (2024)

    2024

  31. [32]

    N. R. F. Braga, L. F. Ferreira, and R. Da Rocha, Phys. Lett. B787, 16 (2018), arXiv:1808.10499 [hep-ph]

    Pith/arXiv arXiv 2018

  32. [33]

    M. A. Martin Contreras and A. Vega, Phys. Rev. D108, 126024 (2023), arXiv:2309.02905 [hep-ph]

    Pith/arXiv arXiv 2023

  33. [34]

    A. F. Krutov, R. G. Polezhaev, and V. E. Troitsky, Phys. Rev. D93, 036007 (2016), arXiv:1602.00907 [hep-ph]

    Pith/arXiv arXiv 2016

  34. [35]

    T. M. Aliev, A. Ozpineci, and M. Savci, Phys. Lett. B678, 470 (2009), arXiv:0902.4627 [hep-ph]

    Pith/arXiv arXiv 2009

  35. [36]

    J. P. Varyet al., Few Body Syst.59, 56 (2018), arXiv:1804.07865 [nucl-th]

    Pith/arXiv arXiv 2018

  36. [37]

    M. E. Carrillo-Serrano, W. Bentz, I. C. Clo¨ et, and A. W. Thomas, Phys. Rev. C92, 015212 (2015), arXiv:1504.08119 [nucl-th]. 17

    Pith/arXiv arXiv 2015

  37. [38]

    Evans and A

    N. Evans and A. Tedder, Phys. Lett. B642, 546 (2006), arXiv:hep-ph/0609112 [hep-ph]

    Pith/arXiv arXiv 2006

  38. [39]

    S. S. Afonin, Phys. Rev. C83, 048202 (2011), arXiv:1102.0156 [hep-ph]

    Pith/arXiv arXiv 2011

  39. [40]

    S. S. Afonin, Int. J. Mod. Phys. A27, 1250171 (2012), arXiv:1207.2644 [hep-ph]

    Pith/arXiv arXiv 2012

  40. [41]

    N. R. F. Braga and L. F. Ferreira, Phys. Lett. B783, 186 (2018), arXiv:1802.02084 [hep-ph]

    Pith/arXiv arXiv 2018

  41. [42]

    M. A. Martin Contreras and A. Vega, Phys. Rev. D101, 046009 (2020), arXiv:1910.10922 [hep-th]

    Pith/arXiv arXiv 2020

  42. [44]

    M. A. Martin Contreras, A. Vega, and S. Diles, Phys. Lett. B854, 138723 (2024), arXiv:2308.16007 [hep-ph]

    Pith/arXiv arXiv 2024

  43. [45]

    Ma, L.-G

    Y.-G. Ma, L.-G. Pang, R. Wang, and K. Zhou, Chin. Phys. Lett.40, 122101 (2023), arXiv:2311.07274 [nucl-th]

    Pith/arXiv arXiv 2023

  44. [46]

    Zhou, L.-G

    K. Zhou, L.-G. Pang, S. Shi, and H. Stoecker, inProceedings of Science, F AIRness2022(2023) p. 064

    2023

  45. [47]

    Pang, Int

    L.-G. Pang, Int. J. Mod. Phys. E33, 2430009 (2024)

    2024

  46. [48]

    Li, C.-Q

    Z.-H. Li, C.-Q. Li, and L.-G. Pang, (2023), arXiv:2309.07397 [gr-qc]

    Pith/arXiv arXiv 2023

  47. [49]

    K.-F. Pu, H. Li, H.-L. Lu, and L.-G. Pang, Chin. Phys. C47, 054104 (2023), arXiv:2303.03934 [nucl-th]

    Pith/arXiv arXiv 2023

  48. [50]

    Pang, Nucl

    L.-G. Pang, Nucl. Phys. A1005, 121972 (2021)

    2021

  49. [51]

    Steinheimer, L.-G

    J. Steinheimer, L.-G. Pang, K. Zhou, V. Koch, J. Randrup, and H. Stoecker, JHEP12, 122 (2019), arXiv:1906.06562 [nucl-th]

    Pith/arXiv arXiv 2019

  50. [52]

    L. Wang, B. M. Hare, K. Zhou, H. Stoecker, and O. Scholten, Chaos, Solitons & Fractals 170, 113346 (2023)

    2023

  51. [53]

    Huang, J

    Y. Huang, J. Chen, J. Jia, L.-M. Liu, Y.-G. Ma, and C. Zhang, (2025), arXiv:2501.01352 [nucl-th]

    Pith/arXiv arXiv 2025

  52. [54]

    He, Y.-G

    W.-B. He, Y.-G. Ma, L.-G. Pang, H. Song, and K. Zhou, Nucl. Sci. Tech.34, 88 (2023), arXiv:2303.06752 [hep-ph]

    Pith/arXiv arXiv 2023

  53. [55]

    He, W.-B

    J. He, W.-B. He, Y.-G. Ma, and S. Zhang, Phys. Rev. C104, 044902 (2021), arXiv:2109.06277 [hep-ph]

    Pith/arXiv arXiv 2021

  54. [56]

    Hornik, M

    K. Hornik, M. Stinchcombe, and H. White, Neural Networks2, 359 (1989). 18

    1989

  55. [57]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, Journal of Computational Physics378, 686 (2019)

    2019

  56. [58]

    S. Soma, L. Wang, S. Shi, H. Stoecker, and K. Zhou, Phys. Rev. D107, 083028 (2023), arXiv:2209.08883 [astro-ph.HE]

    Pith/arXiv arXiv 2023

  57. [59]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Nature Reviews Physics3, 422 (2021)

    2021

  58. [60]

    S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, Phys. Rev. D105, 014017 (2022), arXiv:2105.07862 [hep-ph]

    Pith/arXiv arXiv 2022

  59. [61]

    W. He, Q. Li, Y. Ma, Z. Niu, J. Pei, and Y. Zhang, Sci. China Phys. Mech. Astron.66, 282001 (2023), arXiv:2301.06396 [nucl-th]

    Pith/arXiv arXiv 2023

  60. [62]

    Zhou, L.-G

    K. Zhou, L.-G. Pang, N. Su, H. Petersen, H. Stoecker, and X.-N. Wang, inEPJ Web Conf., Vol. 171 (2018) p. 16005

    2018

  61. [63]

    L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stoecker, and X.-N. Wang, Nucl. Phys. A982, 867 (2019)

    2019

  62. [64]

    L. Wang, Y. Jiang, L. He, and K. Zhou, Chin. Phys. Lett.39, 120502 (2022), arXiv:2005.04857 [cond-mat.dis-nn]

    Pith/arXiv arXiv 2022

  63. [65]

    Y.-S. Zhao, L. Wang, K. Zhou, and X.-G. Huang, Phys. Rev. C106, L051901 (2022), arXiv:2105.13761 [hep-ph]

    Pith/arXiv arXiv 2022

  64. [66]

    Y.-L. Du, D. Pablos, and K. Tywoniuk, JHEP2021, 206 (2020), arXiv:2012.07797 [hep-ph]

    Pith/arXiv arXiv 2020

  65. [67]

    Y.-L. Du, D. Pablos, and K. Tywoniuk, Phys. Rev. Lett.128, 012301 (2022), arXiv:2106.11271 [hep-ph]

    Pith/arXiv arXiv 2022

  66. [68]

    Jiang, L

    L. Jiang, L. Wang, and K. Zhou, Phys. Rev. D103, 116023 (2021), arXiv:2103.04090 [nucl-th]

    Pith/arXiv arXiv 2021

  67. [69]

    S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, inPoS LATTICE2021(2022) p. 537

    2022

  68. [70]

    H.-A. Zeng, L. Wang, and M. Huang, (2025), arXiv:2512.06044 [hep-lat]

    Pith/arXiv arXiv 2025

  69. [71]

    Akutagawa, K

    T. Akutagawa, K. Hashimoto, and T. Sumimoto, Phys. Rev. D102, 026020 (2020), arXiv:2005.02636 [hep-th]

    Pith/arXiv arXiv 2020

  70. [72]

    Hashimoto, S

    K. Hashimoto, S. Sugishita, A. Tanaka, and A. Tomiya, Phys. Rev. D98, 106014 (2018), arXiv:1809.10536 [hep-th]

    Pith/arXiv arXiv 2018

  71. [73]

    Yan, S.-F

    Y.-K. Yan, S.-F. Wu, X.-H. Ge, and Y. Tian, Phys. Rev. D102, 101902 (2020), arXiv:2004.12112 [hep-th]. 19

    Pith/arXiv arXiv 2020

  72. [74]

    Hashimoto, K

    K. Hashimoto, K. Ohashi, and T. Sumimoto, Phys. Rev. D105, 106008 (2022), arXiv:2108.08091 [hep-th]

    Pith/arXiv arXiv 2022

  73. [75]

    Song, M.-S

    M. Song, M.-S. H. Oh, Y. Ahn, and K.-Y. Kim, Chin. Phys. C45, 073111 (2021), arXiv:2011.13726 [physics.class-ph]

    Pith/arXiv arXiv 2021

  74. [76]

    Chang and D.-f

    W.-B. Chang and D.-f. Hou, Phys. Rev. D109, 086010 (2024), arXiv:2403.04966 [hep-ph]

    Pith/arXiv arXiv 2024

  75. [77]

    Ahn, H.-S

    B. Ahn, H.-S. Jeong, K.-Y. Kim, and K. Yun, JHEP2024, 141 (2024), arXiv:2401.00939 [hep-th]

    Pith/arXiv arXiv 2024

  76. [78]

    Gu, Y.-K

    Z.-F. Gu, Y.-K. Yan, and S.-F. Wu, (2024), arXiv:2401.09946 [hep-th]

    Pith/arXiv arXiv 2024

  77. [79]

    K. Li, Y. Ling, P. Liu, and M.-H. Wu, Phys. Rev. D107, 066021 (2023), arXiv:2209.05203 [hep-th]

    Pith/arXiv arXiv 2023

  78. [80]

    R.-G. Cai, S. He, L. Li, and H.-A. Zeng, (2024), arXiv:2406.12772 [hep-th]

    arXiv 2024

  79. [81]

    Ahn, H.-S

    B. Ahn, H.-S. Jeong, K.-Y. Kim, and K. Yun, (2024), arXiv:2406.07395 [hep-th]

    Pith/arXiv arXiv 2024

  80. [82]

    Boschi-Filho and N

    H. Boschi-Filho and N. R. F. Braga, JHEP05, 009 (2003), arXiv:hep-th/0212207

    Pith/arXiv arXiv 2003

Showing first 80 references.