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arxiv: 2606.28442 · v1 · pith:5GHDU4IEnew · submitted 2026-06-26 · ✦ hep-ph

Pseudoscalar charmonium and bottomonium: light-front wave functions, distribution amplitudes and distribution functions

Zhen-Ni Xu , Zhao-Qian Yao , Kh\'epani Raya , Jos\'e Rodr\'iguez-Quintero This is my paper

Pith reviewed 2026-06-30 01:33 UTC · model grok-4.3

classification ✦ hep-ph
keywords light-front wave functionscharmoniumbottomoniumdistribution amplitudesgeneralised parton distributionsDyson-Schwinger equationsBethe-Salpeter equationsquarkonia
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0 comments X

The pith

A modified non-separable light-front wave function ansatz reproduces the properties of heavy pseudoscalar quarkonia and yields their zero-skewness GPDs plus electromagnetic and gravitational form factors.

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

The paper extends earlier light-front wave function calculations from light pseudoscalars to charmonium and bottomonium within the Dyson-Schwinger/Bethe-Salpeter framework. It shows that the longitudinal-transverse separability observed for light mesons breaks down for heavy systems and introduces a revised ansatz to restore agreement with the computed wave functions. This ansatz then supplies distribution amplitudes, parton distribution functions, generalised parton distributions at zero skewness, electromagnetic and gravitational form factors, and transverse charge and mass distributions. A sympathetic reader cares because the work supplies a QCD-connected route to the internal momentum structure of heavy quarkonia without relying on model assumptions disconnected from the underlying theory.

Core claim

Using continuum Schwinger methods, the authors compute light-front wave functions for pseudoscalar charmonium and bottomonium, taking the fictitious π_s meson as benchmark. They find clear deviations from the approximate separability of longitudinal and transverse light-cone momentum dependence that holds for light mesons. Motivated by these deviations, they construct a modified non-separable LFWF ansatz that reproduces the computed wave functions and meson properties, thereby permitting direct evaluation of distribution amplitudes, distribution functions, zero-skewness generalised parton distributions, electromagnetic and gravitational form factors, and the associated transverse charge and

What carries the argument

The modified non-separable LFWF ansatz, which replaces the separable form used for light mesons and directly incorporates the coupled longitudinal-transverse momentum dependence obtained from the Bethe-Salpeter wave functions.

If this is right

  • The ansatz reproduces the computed light-front wave functions and static properties of heavy pseudoscalar quarkonia.
  • Zero-skewness generalised parton distributions become directly accessible from the same wave functions.
  • Electromagnetic and gravitational form factors follow from the same construction.
  • Transverse charge and mass distributions can be extracted without additional model input.
  • Distribution amplitudes and parton distribution functions are obtained for the same states.

Where Pith is reading between the lines

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

  • The same non-separable construction could be tested on vector charmonia or bottomonia to check consistency across spin states.
  • Predictions for the transverse distributions could be confronted with future measurements at electron-ion colliders.
  • The framework might be used to trace how the degree of separability evolves with quark mass between the light and heavy regimes.
  • Independent lattice calculations of the same GPDs or form factors would provide a direct numerical test of the Schwinger-method results.

Load-bearing premise

The Dyson-Schwinger and Bethe-Salpeter equations remain quantitatively reliable when applied to bound states whose mass scale is comparable to the QCD scale.

What would settle it

A lattice QCD calculation or experimental measurement of the electromagnetic form factor or transverse charge distribution of the η_c or η_b that deviates substantially from the values obtained with the modified ansatz would falsify the claim that the ansatz and framework are reliable for these heavy systems.

Figures

Figures reproduced from arXiv: 2606.28442 by Jos\'e Rodr\'iguez-Quintero, Kh\'epani Raya, Zhao-Qian Yao, Zhen-Ni Xu.

Figure 1
Figure 1. Figure 1: FIG. 1. Results for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Zero-skewness GPDs, calculated from Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Left panel: Electromagnetic and mass form factors. Solid curves – the electromagnetic form factors of the fictitious [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

Light-front wave functions play a central role in the program of understanding the structure of hadrons as QCD bound states. Using continuum Schwinger methods, based on Dyson-Schwinger and Bethe-Salpeter equations, they can be computed directly within a framework connected to QCD. For light pseudoscalar mesons, previous studies revealed an approximate separability of longitudinal and transverse lightcone momentum dependences in the LFWFs, leading to a simple relation between distribution functions and amplitudes. In this work, we extend those previous studies to the case of pseudoscalar charmonium and bottomonium, using the fictitious $\pi_s$ meson as a benchmark. Motivated by the observed deviations, we propose a modified non-separable LFWF ansatz that successfully reproduces the properties of heavy pseudoscalar quarkonia and allows the calculation of zero-skewness generalised parton distribution functions, electromagnetic and gravitational form factors, and transverse charge and mass distributions.

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 uses Dyson-Schwinger and Bethe-Salpeter equations in the continuum Schwinger framework to compute light-front wave functions (LFWFs) for pseudoscalar charmonium and bottomonium, benchmarked against the fictitious π_s. Observing deviations from the longitudinal-transverse separability found in light pseudoscalars, the authors propose a modified non-separable LFWF ansatz that is asserted to reproduce the properties of heavy pseudoscalar quarkonia and is then employed to compute zero-skewness GPDs, electromagnetic and gravitational form factors, and transverse charge and mass distributions.

Significance. If the underlying LFWFs are quantitatively reliable, the work supplies a QCD-connected route to hadronic structure observables for heavy quarkonia that could inform phenomenology of charmonium and bottomonium. The explicit construction of a non-separable ansatz and its use for GPDs and form factors would constitute a concrete extension of prior light-meson studies.

major comments (3)
  1. [Abstract] Abstract: the assertion that the modified ansatz 'successfully reproduces' the properties of heavy pseudoscalar quarkonia is unsupported by any quantitative measures, error estimates, or tabulated comparisons to lattice QCD decay constants, radii, or form factors.
  2. [§4 (LFWF results and ansatz construction)] The central claim that the computed LFWFs enable reliable GPD and form-factor calculations rests on the quantitative accuracy of the DSE/BSE framework for systems whose mass scale is comparable to Λ_QCD; no independent cross-check (e.g., against lattice results for η_c or η_b decay constants or electromagnetic radii) is supplied to validate the transfer from light to heavy systems.
  3. [§5] §5 (GPD and form-factor calculations): because the ansatz parameters are fixed by the same DSE/BSE solutions whose accuracy for heavy quarks is unverified, it is unclear whether the reported GPDs and transverse distributions constitute independent predictions or are shaped by the same inputs used to define the ansatz.
minor comments (2)
  1. [§2] Define the fictitious π_s meson parameters and its relation to the physical pion more explicitly in the text.
  2. [Figures 3-5] Ensure all figures comparing LFWFs or distribution amplitudes include error bands or sensitivity estimates from the model parameters.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We respond to each major comment below, indicating where we agree revisions are warranted and where we maintain the original framing.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the modified ansatz 'successfully reproduces' the properties of heavy pseudoscalar quarkonia is unsupported by any quantitative measures, error estimates, or tabulated comparisons to lattice QCD decay constants, radii, or form factors.

    Authors: We agree the abstract phrasing is assertive. The manuscript shows reproduction via the ansatz matching the non-separable structure and transverse momentum dependence obtained from the DSE/BSE solutions for the heavy systems, in contrast to the light-meson case. To strengthen the claim, we will revise the abstract to 'captures the essential properties' and add a table in section 4 with quantitative comparisons of decay constants to available lattice results. revision: yes

  2. Referee: [§4 (LFWF results and ansatz construction)] The central claim that the computed LFWFs enable reliable GPD and form-factor calculations rests on the quantitative accuracy of the DSE/BSE framework for systems whose mass scale is comparable to Λ_QCD; no independent cross-check (e.g., against lattice results for η_c or η_b decay constants or electromagnetic radii) is supplied to validate the transfer from light to heavy systems.

    Authors: The rainbow-ladder DSE/BSE setup employed here is the same as in earlier works that successfully describe both light pseudoscalars and heavy quarkonia when parameters are fixed to meson masses. We will expand section 4 with a paragraph discussing the framework's prior validation for heavy systems and explicitly note the lack of new lattice cross-checks for electromagnetic radii as a limitation of the present study. revision: partial

  3. Referee: [§5] §5 (GPD and form-factor calculations): because the ansatz parameters are fixed by the same DSE/BSE solutions whose accuracy for heavy quarks is unverified, it is unclear whether the reported GPDs and transverse distributions constitute independent predictions or are shaped by the same inputs used to define the ansatz.

    Authors: The GPDs, form factors and transverse distributions are obtained by direct use of the DSE/BSE LFWFs through the constructed ansatz; they are therefore not independent predictions but consistent extensions within the same continuum framework. This is the intended strength of the approach, ensuring the observables inherit the QCD-connected features of the computed wave functions. We do not plan revisions on this point. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained model construction

full rationale

The paper computes LFWFs directly via DSE/BSE within the continuum Schwinger framework, observes deviations from light-meson separability using the fictitious π_s benchmark, and proposes a modified non-separable ansatz motivated by those deviations. The ansatz is then used to compute GPDs, form factors and distributions. This is a standard model-building step in which an ansatz is calibrated to one set of computed quantities (LFWFs/properties) to enable analytic access to derived quantities; no equation reduces to its input by construction, no parameter is fitted to a subset and then relabeled as an independent prediction of a closely related observable, and no load-bearing premise rests solely on self-citation. The chain remains externally falsifiable against lattice data or other frameworks and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no explicit list of free parameters or axioms; the central claim rests on the transferability of the DSE/BSE truncation scheme from light to heavy quarks and on the validity of the proposed non-separable ansatz.

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discussion (0)

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