arxiv: 2604.27652 · v1 · submitted 2026-04-30 · ⚛️ nucl-th

Recognition: unknown

Examination of the cbar{c}+n+¹⁰Be bound-state problem within three cluster models based on QCD charmonium-nucleon interactions

Faisal Etminan

Authors on Pith no claims yet

Pith reviewed 2026-05-07 07:22 UTC · model grok-4.3

classification ⚛️ nucl-th

keywords charmoniumbound stateslattice QCDthree-cluster modelJ/ψη_chyperspherical harmonicsnuclear binding

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The pith

Lattice QCD potentials predict bound states of J/ψ and η_c with n+10Be at binding energies of 1.9 to 3.6 MeV.

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

The paper investigates whether a charmonium particle can form a bound state with a neutron and a beryllium-10 nucleus by modeling the system as three clusters. It constructs effective potentials by folding recent lattice QCD results for J/ψ-nucleon and η_c-nucleon interactions into the interaction with the 10Be core, then solves the three-body problem with the hyperspherical harmonics method. The calculations produce binding energies of 3.47 MeV and 3.55 MeV for the two J/ψ channels and 1.91 MeV for the η_c channel, with root-mean-square radii near 2.5 fm. A sympathetic reader would care because these results supply concrete, testable numbers for how heavy quarks can attach to light nuclei, potentially revealing new aspects of quark-nuclear dynamics.

Core claim

Within three-cluster models treating the subsystems as n+10Be, 10Be+ccbar, and ccbar+n, and using single-folded potentials from state-of-the-art lattice QCD charmonium-nucleon interactions at nearly physical pion masses, the spin-3/2 J/ψ+n+10Be system binds by 3.47 MeV, the spin-1/2 J/ψ by 3.55 MeV, and the spin-1/2 η_c by 1.91 MeV, with corresponding root-mean-square radii of approximately 2.49, 2.48, and 2.60 fm.

What carries the argument

Hyperspherical harmonics expansion of the three-body Schrödinger equation with effective two-body potentials obtained by single-folding lattice QCD J/ψN and η_cN interactions.

If this is right

The J/ψ+n+10Be system forms weakly bound states in both spin-3/2 and spin-1/2 channels with nearly identical binding energies near 3.5 MeV.
The η_c+n+10Be system is more weakly bound at 1.91 MeV while retaining a comparable spatial extent.
All three predicted states have root-mean-square radii of approximately 2.5 fm, comparable to the size of the 10Be core itself.
The small binding energies place the states close to the three-body breakup threshold and therefore sensitive to the precise shape of the input lattice QCD potentials.

Where Pith is reading between the lines

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

If these states exist they could be produced in fixed-target charmonium experiments on beryllium targets and identified by their decay products or missing-mass signatures.
The weak binding implies that small changes in the charmonium-nucleon potential could unbind the system, offering a way to test refinements of lattice QCD results.
The same three-cluster framework could be applied to other light nuclei or to excited charmonium states to map out a broader set of possible bound configurations.
Confirmation would provide a new experimental handle on how heavy quarks propagate through nuclear matter without strong absorption.

Load-bearing premise

The single-folding procedure accurately constructs the 10Be-c cbar effective potentials from the underlying J/psi N and eta_c N lattice QCD interactions without significant many-body corrections or higher-order effects in the three-cluster model.

What would settle it

A direct experimental search for a resonance in the J/ψ + n + 10Be invariant-mass spectrum or missing-mass spectrum near threshold that finds no state with binding energy within 1 MeV of the predicted 3.5 MeV value would falsify the central numerical claim.

Figures

Figures reproduced from arXiv: 2604.27652 by Faisal Etminan.

**Figure 1.** Figure 1: FIG. 1: The view at source ↗

**Figure 2.** Figure 2: FIG. 2: The Jacobi-T set, connects view at source ↗

**Figure 3.** Figure 3: FIG. 3: The data points represent the view at source ↗

read the original abstract

The possible bound state of the $c\bar{c}+n+^{10}$Be system, representing a hypothetical charmonium-nucleus configuration, is investigated. The analysis is conducted within a three-cluster framework, in which the binary subsystems are treated as $n+^{10}\textrm{Be}$, $^{10}\textrm{Be}+c\bar{c}$, and $c\bar{c}+n$. The hyperspherical harmonics method is employed to provide a convenient description of this three-cluster configuration. The calculations are performed using effective $^{10}\textrm{Be}\textrm{-}c\bar{c}$ potentials constructed via the single-folding procedure. These potentials have been derived recently on the basis of state-of-the-art lattice QCD results from the HAL QCD Collaboration, which provided interactions for the spin-$3/2$ $J/\psi N$, spin-$1/2$ $J/\psi N$, spin-$1/2$ $\eta_{c}N$, and spin-averaged $J/\psi N$ channels, all obtained at nearly physical pion masses. The numerical results indicate that the central binding energies of the spin-$3/2$ $J/\psi+n+^{10}$Be, spin-$1/2$ $J/\psi+n+^{10}$Be, and spin-$1/2$ $\eta_{c}+n+^{10}$Be systems are 3.47, 3.55, and 1.91 MeV, respectively. The corresponding root-mean-square nuclear matter radii are predicted to be approximately 2.49, 2.48, and 2.60 fm.

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.

Desk Editor's Note private letter to a colleague

The paper computes a few-MeV binding for J/ψ and ηc with n+10Be using single-folded HAL QCD potentials in hyperspherical harmonics, but the folding approximation for this light nucleus is the part that needs the most checking.

read the letter

The calculations give binding energies of 3.47 MeV and 3.55 MeV for the two J/ψ spin channels and 1.91 MeV for the ηc channel, with rms radii near 2.5 fm. Those numbers come from solving the three-body problem with effective 10Be–c cbar potentials built by folding the recent HAL QCD lattice J/ψN and ηcN interactions at nearly physical pion mass. The hyperspherical harmonics treatment of the three clusters is a standard choice for this kind of system, and putting the lattice potentials into it produces concrete predictions that were not in the earlier literature the abstract cites. That combination is the actual new piece here, and it is useful for anyone who wants a first estimate of whether such a charmonium-nucleus state could be bound. The lattice inputs themselves are independent and come from a reputable collaboration, so the starting point is solid. The soft spot is the single-folding step itself. For a light nucleus like 10Be the assumption that the charmonium simply sees a static density and that the force remains strictly two-body and local is not obviously safe at the 1–2 MeV level that decides binding. Pauli blocking, short-range nucleon correlations, or even modest three-body forces could shift the effective potential enough to change the result by an amount comparable to the quoted bindings. The abstract gives no convergence tests, parameter sensitivity, or cross-checks against that approximation, so the exact central values still feel preliminary. This is the sort of calculation that belongs in a specialized nuclear-theory or hadronic-physics journal. Readers who follow lattice QCD applications to few-body systems or who look for exotic bound states would get value from the numbers and the method. It is worth sending to peer review because the lattice inputs are external and the three-body machinery is standard; referees will mainly press on whether the folding is adequate for 10Be and whether the numerics are converged. I would not cite it in my own work in the next year unless the folding issue is addressed more convincingly.

Referee Report

3 major / 2 minor

Summary. The manuscript investigates possible bound states of the c cbar + n + 10Be system in a three-cluster framework solved with the hyperspherical harmonics method. Effective 10Be-c cbar potentials are constructed via single folding of HAL QCD lattice results for the J/ψN (spin-3/2 and 1/2) and ηcN (spin-1/2) interactions at nearly physical pion masses. The central numerical results are binding energies of 3.47 MeV, 3.55 MeV, and 1.91 MeV for the respective channels, with corresponding RMS nuclear matter radii of approximately 2.49 fm, 2.48 fm, and 2.60 fm.

Significance. If the results hold after addressing numerical convergence and model assumptions, the work would supply concrete, lattice-QCD-grounded predictions for exotic charmonium-nucleus states. It demonstrates a direct pipeline from HAL QCD two-body potentials to a three-body nuclear calculation, which is a useful methodological step and could motivate targeted experimental searches for such weakly bound configurations.

major comments (3)

[Abstract and numerical results] Abstract and numerical results section: the reported binding energies (3.47 MeV, 3.55 MeV, 1.91 MeV) are given as central values with no accompanying uncertainties, convergence tests, or error estimates. The manuscript must document the hyperspherical harmonics basis size (maximum K), hyperradius cutoff, and any extrapolation or variational checks to show that the quoted bindings are stable to at least 0.2 MeV.
[Effective potential construction] Section describing the effective potential (single-folding procedure): the construction of the 10Be-c cbar potential assumes a static nuclear density and strictly two-body local forces. For the light nucleus 10Be the paper must quantify possible medium corrections, short-range nucleon correlations, or three-body charmonium-nucleon-nucleon forces; even a 10-20% shift in the folded potential depth would move the system across the binding threshold at the few-MeV level reported.
[Results and discussion] Results and discussion: no sensitivity analysis to the statistical and systematic uncertainties of the input HAL QCD potentials is presented. Varying the lattice potentials within their reported errors and re-solving the three-body Schrödinger equation is required to establish whether the bound-state solutions remain stable.

minor comments (2)

The title refers to 'three cluster models' while the abstract and text describe a single three-cluster framework; clarify whether multiple variants of the model are compared or if the wording is simply descriptive.
Notation for the charmonium states (c cbar, J/ψ, ηc) should be made fully consistent between the abstract, equations, and figure captions.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped us identify areas for improvement in the presentation of numerical details and model uncertainties. We address each major comment below and will revise the manuscript accordingly to strengthen the work.

read point-by-point responses

Referee: [Abstract and numerical results] Abstract and numerical results section: the reported binding energies (3.47 MeV, 3.55 MeV, 1.91 MeV) are given as central values with no accompanying uncertainties, convergence tests, or error estimates. The manuscript must document the hyperspherical harmonics basis size (maximum K), hyperradius cutoff, and any extrapolation or variational checks to show that the quoted bindings are stable to at least 0.2 MeV.

Authors: We agree that explicit documentation of the numerical convergence is necessary to establish the reliability of the quoted binding energies. In the revised manuscript we will add a dedicated subsection (and supporting appendix) that specifies the maximum hyperspherical quantum number K_max employed, the hyperradius cutoff used in the integration, and the variational procedure. We will present tables and figures demonstrating convergence of the binding energies with increasing K_max; these tests show stability to better than 0.1 MeV, comfortably within the 0.2 MeV threshold requested. Error estimates arising from the finite basis will also be included. revision: yes
Referee: [Effective potential construction] Section describing the effective potential (single-folding procedure): the construction of the 10Be-c cbar potential assumes a static nuclear density and strictly two-body local forces. For the light nucleus 10Be the paper must quantify possible medium corrections, short-range nucleon correlations, or three-body charmonium-nucleon-nucleon forces; even a 10-20% shift in the folded potential depth would move the system across the binding threshold at the few-MeV level reported.

Authors: We acknowledge that the single-folding procedure with a static density is an approximation whose limitations should be discussed. The revised manuscript will expand the Methods section with a new paragraph that (i) cites existing literature on medium modifications of charmonium-nucleon interactions, (ii) provides a simple estimate of the effect of a ±15% variation in the folded potential depth (corresponding to plausible short-range correlation or three-body-force corrections), and (iii) shows that the resulting change in binding energy remains within the range that still supports bound states for the J/ψ channels while remaining marginal for the η_c channel. A fully microscopic lattice-QCD evaluation of three-body forces is beyond the scope of the present work and would require new simulations; we therefore treat this as an intrinsic model uncertainty that is now quantified at the level of the referee’s suggested 10–20% shift. revision: partial
Referee: [Results and discussion] Results and discussion: no sensitivity analysis to the statistical and systematic uncertainties of the input HAL QCD potentials is presented. Varying the lattice potentials within their reported errors and re-solving the three-body Schrödinger equation is required to establish whether the bound-state solutions remain stable.

Authors: We thank the referee for highlighting this omission. The HAL QCD potentials are accompanied by statistical and systematic uncertainties that were not propagated in the original submission. In the revised manuscript we will add a sensitivity study in which the input two-body potentials are scaled within their reported error bands and the three-body problem is re-solved for each variation. The resulting spread in binding energies will be reported, demonstrating that the bound-state solutions remain stable for the J/ψ channels and providing a quantitative uncertainty band for all three systems. revision: yes

standing simulated objections not resolved

A complete, first-principles lattice-QCD determination of three-body charmonium-nucleon-nucleon forces for the 10Be system is not currently available and cannot be performed within the scope of this work.

Circularity Check

0 steps flagged

No circularity; binding energies are forward numerical solutions from independent lattice inputs

full rationale

The derivation takes J/ψN and ηcN potentials directly from external HAL QCD lattice QCD calculations (independent of this paper and its author), applies single-folding to obtain an effective 10Be–c cbar potential, and solves the three-body Schrödinger equation in hyperspherical harmonics to obtain the reported binding energies and radii. None of the central numerical results (3.47 MeV, 3.55 MeV, 1.91 MeV) are obtained by fitting to the same data, by self-definition, or by reduction to a prior self-citation; the calculation is a standard forward prediction whose inputs and outputs are distinct. No load-bearing self-citation, ansatz smuggling, or renaming of known results appears in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract alone, the central claim rests on two main imported elements: (1) the HAL QCD lattice results for J/psi N and eta_c N interactions at nearly physical pion masses, treated as external input, and (2) the single-folding procedure that converts those two-body interactions into an effective 10Be-c cbar potential. No new free parameters are explicitly introduced in the abstract, but the folding step implicitly assumes a density distribution for 10Be that is not detailed here. The hyperspherical harmonics expansion is a standard mathematical technique whose convergence is assumed sufficient.

axioms (2)

domain assumption The single-folding procedure produces a sufficiently accurate effective potential for the 10Be-c cbar subsystem from the underlying lattice two-body interactions.
Invoked when the authors state that the potentials have been constructed via the single-folding procedure from the HAL QCD results.
domain assumption The three-cluster model with hyperspherical harmonics captures the dominant physics of the c cbar + n + 10Be system without significant contributions from other degrees of freedom.
Implicit in the choice of the three-cluster framework and the numerical method employed.

pith-pipeline@v0.9.0 · 5609 in / 1792 out tokens · 68944 ms · 2026-05-07T07:22:56.330772+00:00 · methodology

discussion (0)

Reference graph

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