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arxiv: 2606.07073 · v1 · pith:MPX2CQWTnew · submitted 2026-06-05 · ✦ hep-ph · nucl-th

Mechanical distribution of the pseudoscalar charmonium and bottomonium on the light-front

Ashutosh Dwibedi , Satyajit Puhan , Sabyasachi Ghosh This is my paper

Pith reviewed 2026-06-27 22:00 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords light-front quark modelgravitational form factorsmechanical distributionscharmoniumbottomoniumpressure distributionenergy-momentum tensortransverse plane
0
0 comments X

The pith

Pressure in pseudoscalar charmonium and bottomonium changes from repulsive to attractive at larger transverse distances.

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

The paper calculates gravitational form factors A and D from light-front wave functions for pseudoscalar charmonium and bottomonium, then obtains the corresponding mechanical distributions in the transverse plane via Fourier transforms. Two Gaussian forms for the spatial wave function are used to test dependence on the internal quark-antiquark distribution. The resulting pressure shows a node that switches sign from positive to negative with rising distance, while the force stays positive everywhere. A reader would care because these quantities describe the mechanical forces holding the meson together and how they vary with distance inside the bound state.

Core claim

In the light-front quark model the gravitational form factors A and D are computed directly from the light-front wave functions of pseudoscalar charmonium and bottomonium. Their Fourier transforms produce transverse-plane distributions in which pressure changes sign from positive (repulsive) to negative (attractive) with increasing transverse distance, the force density remains positive throughout the plane, and most distributions except shear stress are sensitive to wave-function choice near the center but become insensitive at larger radii.

What carries the argument

Gravitational form factors A and D extracted from light-front wave functions, Fourier-transformed to yield transverse mechanical distributions of pressure, force, shear stress and energy density.

If this is right

  • Force remaining positive throughout the transverse plane supports the stability condition proposed in earlier studies.
  • Pressure changes sign once, becoming attractive beyond a certain transverse radius.
  • Most spatial distributions are sensitive to wave-function choice near the meson center but insensitive at the periphery.
  • Shear stress shows noticeable sensitivity to wave function in the intermediate transverse region.
  • Internal energy density and momentum density follow the same center-sensitive, periphery-insensitive pattern as pressure.

Where Pith is reading between the lines

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

  • The location of the pressure node may set a characteristic transverse size that can be compared with other radius measures such as the charge radius.
  • Similar sign-changing pressure profiles might appear in lighter mesons if the same light-front framework is applied.
  • The insensitivity at large radii suggests that peripheral mechanical properties are largely model-independent within this class of wave functions.

Load-bearing premise

The light-front quark model with the two chosen Gaussian forms for the spatial wave function accurately represents the quark-antiquark distribution inside the pseudoscalar charmonium and bottomonium for computing the energy-momentum tensor and its Fourier transforms.

What would settle it

A computation of the same GFFs with a qualitatively different spatial wave function that produces no sign-changing node in the pressure distribution would falsify the reported feature.

Figures

Figures reproduced from arXiv: 2606.07073 by Ashutosh Dwibedi, Sabyasachi Ghosh, Satyajit Puhan.

Figure 3
Figure 3. Figure 3: FIG. 3: (Color online) Variation of the EMFFs with [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Color online) Variation of the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) Variation of all the form factors [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Dependence of spatial distributions of mechanical properties (momentum and pressure) of mesons on their [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The wavefunction dependence of the mechanical distributions (shear, force, and energy) inside [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

We investigate the energy-momentum tensor of pseudoscalar charmonium and bottomonium within the framework of the light-front quark model. The gravitational form factors (GFFs), namely the $A$ and $D$-terms, are evaluated in terms of the light-front wave functions. The corresponding spatial mechanical distributions in the transverse plane are obtained through the Fourier transform of these GFFs. To examine the sensitivity of the results to the internal quark-antiquark distribution inside the meson, two distinct Gaussian forms are employed for the spatial part of the wave function. We analyze several mechanical properties in the transverse plane, including the momentum density, pressure distribution, shear stress, force density, and internal energy density. The pressure distribution exhibits a node where it changes sign from positive (repulsive) to negative (attractive) with increasing transverse distance. The force distribution remains positive throughout the transverse plane, supporting the stability condition proposed in earlier studies. Most of the spatial distributions, except for the shear stress, are found to be sensitive to the choice of the spatial wave function near the center of the meson, while they become nearly insensitive toward the periphery. In contrast, the shear stress distribution exhibits noticeable sensitivity to the choice of wave function in the intermediate transverse region.

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

1 major / 2 minor

Summary. The manuscript computes the gravitational form factors A and D for pseudoscalar charmonium and bottomonium in the light-front quark model, using two distinct Gaussian forms for the spatial wave function. These GFFs are Fourier-transformed to obtain transverse-plane mechanical distributions (momentum density, pressure, shear stress, force density, internal energy density). The central results are a node in the pressure distribution (sign change from positive/repulsive to negative/attractive) and a strictly positive force distribution throughout the plane, interpreted as supporting a stability condition from prior work; most distributions (except shear) are sensitive to wave-function choice near r=0 but insensitive at large r.

Significance. If the model wave functions are shown to be reliable for the relevant EMT matrix elements, the work supplies a concrete light-front illustration of mechanical properties in heavy quarkonia, including the pressure node and the positive force that satisfies the stability criterion. The explicit comparison of two Gaussian ansätze provides a built-in sensitivity test that strengthens the qualitative claim of peripheral insensitivity.

major comments (1)
  1. [Abstract and wave-function section] The location of the pressure node and the sign of the integrated force are direct outputs of the chosen Gaussian radial forms (Abstract and the section describing the two wave functions). No comparison is presented to independent determinations of the D-term (lattice GFFs, Bethe-Salpeter solutions, or potential-model wave functions) at the momentum transfers that control the Fourier transform; without such a benchmark the node position remains an artifact of the ansatz rather than a robust prediction.
minor comments (2)
  1. The manuscript does not report numerical stability checks, convergence tests for the Fourier transforms, or quantitative uncertainties on the extracted node position and force values.
  2. Parameter values for the two Gaussian widths and the precise fitting procedure to meson masses or decay constants are not tabulated, making reproduction of the quoted distributions difficult.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the positive remarks on the significance of our study. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and wave-function section] The location of the pressure node and the sign of the integrated force are direct outputs of the chosen Gaussian radial forms (Abstract and the section describing the two wave functions). No comparison is presented to independent determinations of the D-term (lattice GFFs, Bethe-Salpeter solutions, or potential-model wave functions) at the momentum transfers that control the Fourier transform; without such a benchmark the node position remains an artifact of the ansatz rather than a robust prediction.

    Authors: We agree with the referee that the quantitative location of the pressure node depends on the specific form of the wave function. The manuscript uses two different Gaussian parametrizations precisely to assess this sensitivity, as stated in the abstract and the wave-function section. Key qualitative results, including the existence of the node and the strictly positive force distribution, are reproduced with both choices. We note that independent lattice determinations of the D-term for charmonium and bottomonium are not yet available at the low momentum transfers that dominate the Fourier transform to transverse distributions. Bethe-Salpeter or potential-model calculations could in principle be compared, but would require a separate study. Our work is a model calculation within the light-front quark model, and we will revise the text to more explicitly state the model dependence of the node position while emphasizing the robustness of the positive force. revision: partial

Circularity Check

0 steps flagged

No significant circularity; model computation is self-contained

full rationale

The paper evaluates GFFs A and D directly from light-front wave functions (two Gaussian spatial forms) and obtains mechanical distributions via Fourier transform of those GFFs. Parameters in the wave functions are selected to represent the meson, but the reported pressure node, sign change, and positive force density are explicit outputs of the model's integral expressions rather than inputs or fits that are renamed as predictions. No self-citation chain, uniqueness theorem, or ansatz smuggling is quoted as load-bearing; the derivation chain remains independent of the target mechanical quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on the light-front quark model framework and the choice of Gaussian wave functions whose parameters are adjusted to reproduce meson properties; no new particles or forces are introduced.

free parameters (1)
  • Gaussian width parameter for spatial wave function
    Two distinct values are employed to test sensitivity; each is selected to represent the quark-antiquark distribution inside the meson.
axioms (2)
  • domain assumption The light-front quark model provides a valid description of the energy-momentum tensor for pseudoscalar heavy mesons
    Invoked throughout the calculation of GFFs from light-front wave functions.
  • standard math Fourier transform of GFFs yields the correct transverse-plane mechanical distributions
    Standard relation used to obtain spatial pressure, force, and shear from A and D form factors.

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discussion (0)

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