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arxiv: 2604.06510 · v1 · submitted 2026-04-07 · ✦ hep-ph · hep-ex· hep-lat· nucl-ex· nucl-th

Distribution amplitudes and functions of ground-state scalar and pseudoscalar charmonia

X.-Y. Zeng , Y.-Y. Xiao , Z.-N. Xu , C. D. Roberts , J. Rodr\'iguez-Quintero This is my paper

Pith reviewed 2026-05-10 18:18 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-latnucl-exnucl-th
keywords charmoniadistribution amplitudesdistribution functionsSchwinger function methodsorbital angular momentumQCD symmetriesglue momentum fractionheavy quarkonia
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0 comments X

The pith

Ground-state charmonia like χ_c0 and η_c are not simple hydrogen-like systems but have mixed orbital angular momentum and nontrivial distribution amplitudes with symmetry-driven negative support.

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

The paper uses continuum Schwinger function methods to test whether charmonia behave like simple atomic systems and concludes they do not. The χ_c0 is found to include more than pure P-wave orbital angular momentum while the η_c wave function contains components beyond S-wave. Their distribution amplitudes and functions are complex, with the χ_c0 amplitude showing balanced positive and negative domains enforced by QCD symmetries. The light-front momentum fraction carried by glue is the same in both states and ten percent lower than the corresponding fraction in the pion, although differences in the distribution functions shrink under scale evolution.

Core claim

Continuum Schwinger function methods applied to ground-state scalar and pseudoscalar charmonia show that the χ_c0 is not a simple P-wave system and the η_c wave function contains more than S-wave contributions. The associated distribution amplitudes are nontrivial, with the χ_c0 DA exhibiting domains of positive and negative support due to QCD symmetries; the same sign structure appears in the χ_c0 distribution function. Differences between the χ_c0 and η_c distribution functions diminish under evolution, and both states carry an identical glue momentum fraction that is ten percent smaller than the glue fraction in the pion.

What carries the argument

Continuum Schwinger function methods that compute wave functions, distribution amplitudes, and distribution functions, exposing orbital angular momentum mixing and QCD symmetry constraints on the sign of the χ_c0 distribution amplitude.

If this is right

  • Charmonium models must include mixed orbital angular momentum rather than assuming pure S- or P-wave states.
  • Distribution amplitudes of scalar charmonia contain negative domains required by QCD symmetries.
  • The glue light-front momentum fraction is identical in χ_c0 and η_c and ten percent lower than the pion value.
  • Differences between χ_c0 and η_c distribution functions shrink with increasing resolution scale.

Where Pith is reading between the lines

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

  • Comparable angular-momentum mixing and sign structures may appear in bottomonium states.
  • The results supply concrete benchmarks for lattice QCD or potential-model studies of heavy-meson structure.
  • Predictions for charmonium production rates or decay distributions in high-energy collisions may need adjustment for the reduced glue fraction.

Load-bearing premise

The chosen interaction kernel and truncation in the continuum Schwinger function approach reproduce the non-perturbative dynamics of charmonia without artifacts that change the reported wave-function mixing or sign structure of the distribution amplitudes.

What would settle it

An independent lattice QCD calculation of the χ_c0 distribution amplitude that finds it strictly positive definite everywhere would show that the negative-support domains are a method-specific artifact.

Figures

Figures reproduced from arXiv: 2604.06510 by C. D. Roberts, J. Rodr\'iguez-Quintero, X.-Y. Zeng, Y.-Y. Xiao, Z.-N. Xu.

Figure 1
Figure 1. Figure 1: Rest-frame OAM decompositions of χc0, ηc BSWFs as defined via Eqs. (11) - (13). There are both positive (above plane) and negative (below plane) contributions, the total sum of which is unity in each case. wherewith Eq. (3) guarantees 1 = P4 i, j L i j . This matrix pro￾vides a measure of the rest-frame OAM pairing contributions to a meson’s canonical normalisation in a form that is somewhat like the bound… view at source ↗
Figure 2
Figure 2. Figure 2: Leading-twist valence dof DAs: Panel A – χc0 (solid purple) and (2/5)φas (dotted black); Panel B – ηc (solid purple) and φas (dotted black). Our calculated results are listed in Table 2B. For DA recon￾struction, again and for the same reasons, we follow a two-step process, beginning with the following Ansatz – the 0−+ ηc DA is even under x ↔ (1 − x): φ5(x) = [x(1 − x)]α √ πΓ(α + 1) 2 2α+1Γ  α + 3 2  , (2… view at source ↗
Figure 3
Figure 3. Figure 3: Hadron scale valence dof DFs: Panel A – χc0 (solid purple) and zeroth-moment unit-normalised φ 2 0 (dot-dashed red); Panel B – ηc (solid pur￾ple) and zeroth-moment unit-normalised φ 2 5 (dot-dashed red) Repeating the steps now for the ηc, beginning with the Ansatz cˇ 5 (x; ζH ) = [x(1 − x)]β n 5 1 , (25) one finds β = 6.20, which yields the moments in Table 4B - row 2: the mean value of fit/input for the m… view at source ↗
Figure 5
Figure 5. Figure 5: Glue DFs at ζ3 in the χc0 (solid purple), ηc (dashed blue), and π (short-dashed green) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Valence dof DFs in Fig [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Charmonia are often supposed to provide simple hydrogen-like ``atomic'' systems that can be used to obtain insights into heavier-quark QCD. We use continuum Schwinger function methods to analyse this hypothesis in connection with ground-state scalar and pseudoscalar charmonia and find that a more complex picture of these states may be necessary. For instance, considering orbital angular momentum, the $\chi_{c0}$ is not a simple $P$-wave system; similarly, the $\eta_c$ wave function contains more than merely $S$-wave contributions. The distribution amplitudes (DAs) and distribution functions (DFs) of these mesons are also nontrivial. For instance, the $\chi_{c0}$ DA is not positive definite: owing to QCD symmetries, it possesses domains of balanced negative and positive support. This feature is also expressed in the $\chi_{c0}$ DF, but differences between $\chi_{c0}$ and $\eta_c$ DFs diminish under scale evolution. Notably, the light-front momentum fraction carried by glue is the same in both states: it is 10\% less than the in-pion glue momentum fraction. Whilst experimental confirmation of the predictions herein is unlikely, our results should serve as benchmarks for complementary theory attempts to understand local and global structural features of heavier-quark 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

2 major / 1 minor

Summary. The paper employs continuum Schwinger function methods to study the ground-state scalar χ_c0 and pseudoscalar η_c charmonia, arguing that these states deviate from simple quark-model pictures: the χ_c0 is not a pure P-wave and the η_c wave function includes non-S-wave components. It computes their distribution amplitudes (DAs) and distribution functions (DFs), showing that the χ_c0 DA has sign-changing support (negative and positive domains) protected by QCD symmetries, with analogous features in the DF that diminish under evolution; both states carry the same light-front glue momentum fraction, which is 10% lower than in the pion. These findings are offered as benchmarks for other theoretical approaches to heavier-quark hadron structure.

Significance. If the central results hold, the work is significant for hadron physics because it supplies concrete, non-perturbative predictions for charmonium DAs and DFs that challenge the hydrogen-like model and quantify gluon contributions and orbital mixing. The continuum Schwinger-function framework provides access to these quantities in a manner complementary to lattice QCD, and the symmetry-protected negativity of the χ_c0 DA is a robust qualitative feature. The paper explicitly positions its outputs as benchmarks, which is a strength when the underlying truncation is controlled.

major comments (2)
  1. [Abstract] Abstract and the section presenting the glue momentum fractions: the quantitative claim that both χ_c0 and η_c carry a glue momentum fraction 10% lower than the pion is not protected by symmetry and depends on the specific rainbow-ladder truncation and one-parameter interaction kernel. No variation of the kernel infrared strength or inclusion of beyond-rainbow-ladder diagrams is shown, so the 10% deficit cannot yet be regarded as robust against the truncation artifacts highlighted in the stress-test note.
  2. [Results on wave functions] The results section on Bethe-Salpeter amplitudes and orbital angular momentum decomposition: the claims that χ_c0 is not a simple P-wave and η_c contains non-S-wave contributions rest on the same truncation. Because these mixing fractions are not symmetry-protected, a direct demonstration that they survive reasonable changes to the kernel or truncation scheme is required to establish that the deviations from the quark model are physical rather than artifacts.
minor comments (1)
  1. The notation for the distribution amplitudes and functions could be clarified with an explicit table comparing the support properties of χ_c0 and η_c at the initial and evolved scales.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below, indicating the revisions we will make to qualify our claims appropriately while preserving the work's value as a benchmark within the rainbow-ladder framework.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section presenting the glue momentum fractions: the quantitative claim that both χ_c0 and η_c carry a glue momentum fraction 10% lower than the pion is not protected by symmetry and depends on the specific rainbow-ladder truncation and one-parameter interaction kernel. No variation of the kernel infrared strength or inclusion of beyond-rainbow-ladder diagrams is shown, so the 10% deficit cannot yet be regarded as robust against the truncation artifacts highlighted in the stress-test note.

    Authors: We agree that the glue momentum fraction is a dynamical quantity without symmetry protection and that its precise numerical value is tied to the rainbow-ladder truncation and the chosen kernel. The result is presented as a concrete prediction of this standard approach, which has been validated for charmonium masses and other observables in earlier work. To respond to the concern, we will revise the abstract and the relevant results section to state explicitly that the 10% reduction is obtained within the rainbow-ladder truncation, and we will add a short paragraph referencing the stress-test note and discussing the expected stability of this feature for heavy-quark systems based on prior applications of the method. revision: yes

  2. Referee: [Results on wave functions] The results section on Bethe-Salpeter amplitudes and orbital angular momentum decomposition: the claims that χ_c0 is not a simple P-wave and η_c contains non-S-wave contributions rest on the same truncation. Because these mixing fractions are not symmetry-protected, a direct demonstration that they survive reasonable changes to the kernel or truncation scheme is required to establish that the deviations from the quark model are physical rather than artifacts.

    Authors: The orbital angular momentum content is extracted from the Bethe-Salpeter amplitudes solved in the rainbow-ladder truncation. These deviations from pure S- or P-wave states are a direct consequence of the relativistic bound-state dynamics and are consistent with results obtained for lighter mesons in the same framework. While we cannot perform new explicit kernel variations or beyond-rainbow-ladder calculations in the present study, we will expand the results section with additional discussion of the truncation's reliability for heavy quarks, drawing on existing literature where analogous mixing persists under modest parameter changes. The statements will be qualified to make clear that the reported mixing fractions are predictions of the employed truncation. revision: yes

standing simulated objections not resolved
  • Direct demonstration that the orbital mixing fractions and glue momentum deficit survive explicit variations of the kernel infrared strength or inclusion of beyond-rainbow-ladder diagrams, as this would require substantial new computations outside the scope of the current manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies established continuum Schwinger-function methods (DSE/BSE) to compute charmonium wave functions, DAs and DFs. Central features such as the sign-changing support of the χ_c0 DA are explicitly attributed to QCD symmetries rather than model details, while the reported glue momentum fraction and orbital-angular-momentum mixing are presented as outputs of the calculation. No equations or steps in the provided text reduce by construction to fitted parameters, self-citations, or ansätze that render the claims tautological. The method is self-contained against external benchmarks once the interaction kernel is fixed by independent observables, satisfying the criteria for a non-circular analysis.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the applicability of continuum Schwinger methods to charmonia. This framework typically incorporates a model for the gluon propagator and a truncation of the Dyson-Schwinger equations whose parameters are constrained by lower-mass systems or phenomenology.

free parameters (1)
  • interaction kernel parameters
    Standard in Schwinger-function studies; values are chosen or fitted to reproduce known meson properties before being applied to charmonia.
axioms (1)
  • domain assumption The chosen truncation and kernel form capture the essential non-perturbative QCD dynamics for ground-state charmonia
    Invoked throughout the analysis to justify the computed wave functions and distribution amplitudes.

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