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arxiv: 2606.09254 · v1 · pith:RANK6LXYnew · submitted 2026-06-08 · ✦ hep-ph · hep-ex· hep-lat· nucl-ex· nucl-th

Meson-Nucleus Bound States with Neural-Network Quantum States

Wei-Lin Wu , Liang-Zhen Wen , Shi-Lin Zhu This is my paper

Pith reviewed 2026-06-27 16:18 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-latnucl-exnucl-th
keywords meson-nucleus bound statesneural-network quantum statesHAL QCD potentialsphi mesonJ/psieta_cvariational Monte Carlomany-body Schrödinger equation
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The pith

Neural-network variational Monte Carlo solves the (A+1)-body Schrödinger equation with HAL QCD meson-nucleon potentials and finds bound states for φ at A≥2, J/ψ at A≥4, and η_c at A≥6.

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

The paper performs the first systematic calculations of ground states for φ-, J/ψ-, and η_c-nucleus systems up to mass number A=12. It solves the (A+1)-body Schrödinger equation using a neural-network variational Monte Carlo framework that incorporates mesonic degrees of freedom, taking meson-nucleon potentials from HAL QCD at the near-physical point as input. Benchmark calculations on ordinary nuclei from deuterium to carbon reproduce experimental ground-state energies. The results indicate that bound states appear above specific thresholds in A, with φ-nucleus systems binding most strongly while charmonium systems bind only weakly. Binding energy per nucleon deepens linearly with A for the charmonium cases but peaks at helium-4 for φ, and the meson is shown to compress the nucleon distribution inside the nucleus.

Core claim

Meson-nucleus bound states emerge at A≥2 for φ, A≥4 for J/ψ, and A≥6 for η_c. The φ-nucleus systems exhibit the strongest binding, with binding energies reaching tens of MeV. The J/ψ-nucleus and η_c-nucleus systems are weakly bound at the few-MeV and sub-MeV scale, respectively. The binding energy per nucleon deepens nearly linearly with A for charmonium systems, whereas the φ-nucleus system exhibits a non-monotonic behavior peaking at ^4He. The meson compresses the nucleon distribution relative to the parent nucleus and evolves from a halo configuration to one embedded inside the nucleus with increasing A.

What carries the argument

Neural-network variational Monte Carlo framework generalized to mesonic degrees of freedom, solving the (A+1)-body Schrödinger equation with input HAL QCD meson-nucleon potentials.

If this is right

  • φ-nucleus systems bind at tens of MeV while J/ψ-nucleus systems bind at a few MeV and η_c-nucleus systems bind at sub-MeV energies.
  • Binding energy per nucleon for charmonium-nucleus systems increases nearly linearly with A.
  • Binding energy per nucleon for φ-nucleus systems is non-monotonic and reaches a maximum at ^4He.
  • The embedded meson compresses the nucleon distribution relative to the bare nucleus and shifts from halo-like to interior placement as A grows.
  • The method supplies quantitative predictions that can be tested in future experiments.

Where Pith is reading between the lines

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

  • Similar calculations could be performed for other mesons such as the ω or K to map how binding thresholds depend on the underlying meson-nucleon potential range and strength.
  • If the non-monotonic binding peak at ^4He for φ is confirmed, it would indicate that short-range attraction dominates over volume effects in light systems.
  • Extending the mass range beyond A=12 would test whether the linear deepening of binding per nucleon for charmonium saturates at larger A.
  • Direct comparison of the computed nucleon density profiles with electron-scattering or pion-production data on the same nuclei could constrain the compression effect.

Load-bearing premise

The HAL QCD meson-nucleon potentials computed at the near-physical point remain accurate when embedded in the full (A+1)-body Schrödinger equation without additional many-body force corrections or significant lattice artifacts.

What would settle it

Experimental observation that φ-nucleus systems do not bind at A=2 or that their binding energies fall well below the predicted tens of MeV scale would falsify the central predictions.

Figures

Figures reproduced from arXiv: 2606.09254 by Liang-Zhen Wen, Shi-Lin Zhu, Wei-Lin Wu.

Figure 1
Figure 1. Figure 1: FIG. 1. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Energy convergence for meson- [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Radial density of the (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Meson-nucleus binding energy per nucleon [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

We present the first systematic calculations of $\phi$-, $\eta_c$-, and $J/\psi$-nucleus ground states up to mass number $A{=}12$ based on the HAL QCD meson-nucleon potentials at near-physical point. The $(A{+}1)$-body Schr\"odinger equation is solved with a neural-network variational Monte Carlo framework, generalized to incorporate mesonic degrees of freedom. Benchmarking on light nuclei from $^2$H to $^{12}$C yields ground-state energies consistent with experiment. Meson-nucleus bound states emerge at $A\ge2$ for $\phi$, $A\ge4$ for $J/\psi$, and $A\ge6$ for $\eta_c$. The $\phi$-nucleus systems exhibit the strongest binding, with binding energies reaching tens of MeV. The $J/\psi$-nucleus and $\eta_c$-nucleus systems are weakly bound at the few-MeV and sub-MeV scale, respectively. The binding energy per nucleon deepens nearly linearly with $A$ for charmonium systems, whereas the $\phi$-nucleus system exhibits a non-monotonic behavior peaking at $^4$He -- a distinctive hallmark of the short-range and strongly attractive $\phi N$ interaction. The meson compresses the nucleon distribution relative to the parent nucleus, and evolves from a halo configuration to one embedded inside the nucleus with increasing $A$. Our results provide predictions for future experimental searches, and establish a quantitative bridge between lattice QCD meson-nucleon interactions and the emergent many-body phenomena in meson-nucleus bound states.

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

2 major / 0 minor

Summary. The paper claims the first systematic variational Monte Carlo calculations of φ-, J/ψ-, and η_c-nucleus ground states up to A=12, using HAL QCD meson-nucleon potentials at near-physical point inside a neural-network ansatz for the (A+1)-body Schrödinger equation. Benchmarking on ²H to ¹²C recovers experimental energies. Bound states appear for A≥2 (φ), A≥4 (J/ψ), A≥6 (η_c), with φ bindings reaching tens of MeV, J/ψ at few MeV, and η_c at sub-MeV; binding per nucleon and nucleon-distribution compression are also reported.

Significance. If the numerical accuracy and two-body approximation hold, the work supplies concrete predictions for meson-nucleus bindings that can be tested experimentally and provides a quantitative link from lattice QCD potentials to emergent many-body structure. The generalization of neural-network quantum states to include mesonic degrees of freedom and the successful benchmarking on ordinary nuclei are explicit strengths.

major comments (2)
  1. [Abstract] Abstract (results paragraph): the binding energies for the J/ψ-nucleus (few-MeV) and η_c-nucleus (sub-MeV) systems are stated without reported statistical or systematic uncertainties, without convergence tests versus neural-network width or Monte Carlo sampling for the meson cases, and without validation of the ansatz accuracy in the presence of the meson. The nucleonic benchmarking alone does not establish numerical reliability at the claimed weak-binding scales.
  2. [Abstract] Abstract (paragraph describing the input potentials and the solved equation): the central binding claims rest on direct insertion of the HAL QCD two-body meson-nucleon potentials into the (A+1)-body Schrödinger equation. No estimate or test is supplied of possible many-body force corrections or lattice artifacts (finite volume, unphysical pion mass, discretization) that could shift near-threshold bindings by amounts comparable to the reported few-MeV or sub-MeV values.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract (results paragraph): the binding energies for the J/ψ-nucleus (few-MeV) and η_c-nucleus (sub-MeV) systems are stated without reported statistical or systematic uncertainties, without convergence tests versus neural-network width or Monte Carlo sampling for the meson cases, and without validation of the ansatz accuracy in the presence of the meson. The nucleonic benchmarking alone does not establish numerical reliability at the claimed weak-binding scales.

    Authors: We agree that the abstract should explicitly report uncertainties and reference convergence/validation for the meson cases to establish reliability at weak-binding scales. The full manuscript already presents statistical uncertainties from VMC sampling for all systems (including mesons) in the results tables and figures, and the nucleonic benchmarks serve as a baseline validation of the ansatz. However, we acknowledge that dedicated convergence tests versus network width and sampling specifically for the meson-nucleus systems are not highlighted in the abstract. We will revise the abstract to include the reported uncertainties and add a sentence on ansatz validation. We will also expand the methods/results sections with additional convergence data for the J/ψ and η_c cases. revision: yes

  2. Referee: [Abstract] Abstract (paragraph describing the input potentials and the solved equation): the central binding claims rest on direct insertion of the HAL QCD two-body meson-nucleon potentials into the (A+1)-body Schrödinger equation. No estimate or test is supplied of possible many-body force corrections or lattice artifacts (finite volume, unphysical pion mass, discretization) that could shift near-threshold bindings by amounts comparable to the reported few-MeV or sub-MeV values.

    Authors: The calculations are performed within the two-body approximation using the available HAL QCD potentials, as is standard when many-body forces are not yet computed on the lattice for these channels. We will add a paragraph in the discussion section that explicitly addresses this limitation, noting that many-body corrections and lattice artifacts (finite-volume effects, pion-mass extrapolation, discretization) could affect near-threshold bindings at the scale of the reported values and that future lattice work is needed to quantify them. revision: partial

standing simulated objections not resolved
  • Quantitative estimates or tests of many-body force corrections and lattice artifacts on the reported few-MeV and sub-MeV bindings (no such data exist in the current HAL QCD results)

Circularity Check

0 steps flagged

No significant circularity; external lattice QCD inputs drive the predictions

full rationale

The paper computes meson-nucleus bound states by directly inserting external HAL QCD meson-nucleon potentials into a neural-network variational Monte Carlo solution of the (A+1)-body Schrödinger equation. Benchmarking on pure nuclei (²H to ¹²C) validates the nucleonic sector against experiment but does not fit or redefine the meson-nucleus quantities. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation; the reported binding energies (tens of MeV down to sub-MeV) are computed outputs from the external two-body inputs rather than reductions to quantities internal to this work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the transferability of the two-body HAL QCD potentials to the many-body problem and on the neural-network ansatz being expressive enough to reach the true ground state; both are domain assumptions rather than derived results.

axioms (2)
  • domain assumption The HAL QCD meson-nucleon potentials at near-physical point accurately capture the relevant interactions when used in the (A+1)-body Schrödinger equation.
    These potentials are taken as given input from prior lattice work and inserted directly into the variational calculation.
  • domain assumption The neural-network variational Monte Carlo framework, after generalization to mesonic degrees of freedom, converges to the ground-state energy of the full many-body system.
    The method is presented as solving the Schrödinger equation; no proof or error bound is supplied in the abstract.

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discussion (0)

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Reference graph

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