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Gluonic Energy Momentum Tensor Form Factors of the Proton

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Pith's one-line read Near-threshold J/psi photoproduction fixes the proton's gluon mass radius at 0.755 ± 0.067 fm

desk verdict A clear proceedings summary of an already-published gluon form factor extraction; the tripole-based lattice comparison is a useful but shape-dependent addition, and the numbers should not be treated as new. read the letter →

arxiv 2505.05671 v1 pith:PAVVFHA2 submitted 2025-05-08 nucl-ex

Zein-Eddine Meziani This is my paper · ORCID
classification nucl-ex
keywords gluonicgravitationalformfactorsenergy-momentumtensornear-thresholdJ/psiphotoproductiongluonmassradiusholographicQCDpressureandsheardistributionsJ/psi-007experimentlattice
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports that near-threshold photoproduction of the $J/\psi$ on the proton, measured in a Hall C experiment, can be used to extract the gluonic gravitational form factors $A_g(t)$ and $C_g(t)$ of the proton. Using a non-perturbative holographic QCD model and a two-dimensional fit of the differential cross section in photon energy and momentum transfer $t$, the paper obtains a gluon mass radius of $\sqrt{\langle r^2_m\rangle_g}=0.755\pm0.067$ fm. This value agrees with the lattice QCD result of $0.7464\pm0.055$ fm. The extracted form factors are then Fourier-transformed in the Breit frame to give the gluon energy, pressure, and shear force densities. The reader should care because gluons dominate the proton's mass, and these densities are currently the main experimental window into how that mass is spatially distributed.

What carries the argument

The machinery is the set of gluonic gravitational form factors defined through the nucleon matrix element of the QCD energy-momentum tensor, together with the holographic QCD formula that connects them to the $J/\psi$ photoproduction cross section. The load-bearing piece is the tripole ansatz $A_g(t)=A_g(0)(1-t/m_A^2)^{-3}$ and $C_g(t)=C_g(0)(1-t/m_C^2)^{-3}$, chosen so the fit can be compared directly with lattice QCD. Fourier transforms of $A_g$ and $D_g=4C_g$ in the Breit frame then produce the spatial mass, pressure, and shear distributions.

What would settle it

Fit the same $J/\psi$-007 differential cross sections with the fully calculated $A_g(t)$ and $D_g(t)$ shapes from the holographic model instead of the tripole ansatz, or with a different functional form such as the dipole-dipole form used in newer lattice calculations; if the resulting gluon mass radius moves outside the quoted $0.755\pm0.067$ fm uncertainty, the tripole-based extraction is not robust.

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Extended reading notes

Core claim

The paper's central claim is that the $J/\psi$-007 near-threshold photoproduction cross sections constrain the gluonic gravitational form factors through the holographic QCD cross-section formula. Fitting $A_g(t)$ and $C_g(t)$ as tripoles, the paper reports $m_A=1.575\pm0.059$ GeV, $m_C=1.12\pm0.21$ GeV, and $C_g(0)=-0.45\pm0.132$, with $A_g(0)=0.414\pm0.008$ fixed by a global QCD analysis. These parameters give a gluon mass radius of $0.755\pm0.067$ fm, consistent with lattice QCD. The same form factors, with $B_g\simeq0$ and $\bar C_g$ ignored, produce Breit-frame densities for gluon energy, pressure, and shear. The paper presents this as evidence that gluonic gravitational form factors can be measured at the threshold and used to map the proton's gluon structure.

Load-bearing premise

The load-bearing premise is that the true $t$-dependence of $A_g(t)$ and $C_g(t)$ is the tripole form of Eq. (3), which the paper adopts for consistency with lattice QCD rather than deriving from the holographic model.

Editorial extensions

If this is right

  • A single near-threshold measurement can determine both the gluon mass radius and the pressure profile of the proton's gluon field.
  • The extracted $C_g(0)\approx -0.45$ supplies the gluon contribution to the $D$-term, the quantity that governs the internal forces felt by gluons inside the proton.
  • Consistency with lattice QCD supports holographic QCD as an extraction tool for non-perturbative threshold kinematics.
  • Higher-statistics measurements with a large-acceptance detector and $\Upsilon$ photoproduction at a future electron-ion collider would test whether these form factors are universal across quarkonium species and kinematics.
  • Comparing these gluon densities with quark densities from deeply virtual Compton scattering would complete the two-sector picture of mass and pressure in the proton.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Refitting the same cross sections with the holographic model's own calculated shapes for $A_g$ and $D_g$, rather than the tripole form, would show how much of the quoted $0.755$ fm radius is data-driven and how much is form-driven.
  • Because the lattice comparison also uses a tripole $D(t)$ and sits at $m_\pi=400$ MeV, part of the agreement may reflect a shared fit shape; a physical-mass lattice result would be a sharper test.
  • The paper argues $\bar C_g=-\bar C_q$ is positive (because $\bar C_q$ is negative), so including the ignored $\bar C_g$ term would raise the gluon energy density; the plotted density profile is then a lower bound in that sense.
  • Applying the same analysis to electroproduction data at higher $Q^2$ would separate the $t$-dependence of the cross section from possible non-form-factor backgrounds, testing whether threshold photoproduction is dominated by the gluon GFFs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This proceedings contribution reports an extraction of the gluonic gravitational form factors A_g(t) and C_g(t) of the proton from two-dimensional fits to the Hall C J/psi-007 near-threshold photoproduction cross sections. The analysis uses the Mamo-Zahed holographic QCD formula (Eq. 2) with tripole parametrizations for A_g and C_g (Eq. 3). The resulting parameters (Table 1) give a gluon mass radius sqrt(<r^2_m>)_g = 0.755 +/- 0.067 fm, consistent with the lattice result of Pefkou et al. [19]. The paper also presents Breit-frame gluon energy density, pressure, and shear force profiles (Figs. 2-3) and discusses future SoLID and EIC measurements. The fit results themselves are not tabulated or plotted in the manuscript but are attributed to private communication [29].

Significance. If the extraction is reliable, the result provides an experimentally determined gluon mass radius and mechanical densities in the proton, complementing the published Nature analysis [9] and the lattice calculations [19,26]. The paper's strength is its use of a genuinely nonperturbative holographic model and a direct comparison with lattice QCD. However, the central numerical claims are not self-contained: the fit outputs are cited to a private communication, and the tripole form assumed in Eq. (3) controls the extracted radius and profiles. Disagreement with the lattice alone is not the issue; the concern is that the agreement is reached partly by construction because both the fit and the lattice comparison adopt the same tripole ansatz. With the systematics from functional-form variation and neglected B_g and Cbar_g unquantified, the significance of the 0.755 fm radius at the quoted precision is not yet established.

major comments (4)
  1. [Section 3, Eq. (3), and Section 4] The tripole ansatz is the load-bearing assumption of the extraction. The text states that A_g(t) and D_g(t) shapes are fully calculated in the M-Z model [16], yet Eq. (3) replaces these shapes with a tripole 'for a consistent comparison with lattice QCD.' Since the lattice comparison [19] also uses a tripole for D_g (as stated in the Fig. 3 caption), the agreement in Table 1 is partly by construction. All quantities in Section 4 — Eq. (4) mass radius, Eqs. (5)-(7) energy density, Eq. (8) pressure and shear — are direct functions of the tripole parameters m_A and m_C. The paper provides no sensitivity test with dipole, z-expansion, or the M-Z model's own predicted shapes; without such a test the quoted 0.755 +/- 0.067 fm is not protected against bias from the chosen parametrization.
  2. [Section 3, Table 1, and Ref. [29]] The fit results are not present in the manuscript. Table 1 and Figures 1-3 are based on 'the J/psi-007 experiment [29] through a two-dimensional fit,' but [29] is a private communication and a talk, not a published, citable dataset. The reader cannot verify the chi-square, the t-range used, the number of data points, the treatment of correlated systematic uncertainties, or even the exact cross-section values entering the fit. A proceedings paper making a quantitative claim should either show the fit or cite a peer-reviewed article containing it; as it stands, the numerical claims are not independently checkable.
  3. [Section 3, after Eq. (2), and Section 4, Eq. (5)] The neglect of B_g(t) and Cbar_g(t) is acknowledged but not quantified. In Eq. (4), C_g(0) enters the mass radius with a coefficient -6/M_N^2; with C_g(0) = -0.45 +/- 0.132, the omission of Cbar_g is not obviously small, and the text itself notes (Section 4) that Cbar_g would make the gluon energy contribution larger. No estimate of the bias from B_g = 0 or Cbar_g = 0 is given, and the quoted uncertainties in Table 1 therefore exclude a known source of systematic error. This matters for the central claim of consistency with lattice at the 0.07 fm level.
  4. [Section 4, Figs. 2-3] The comparison with lattice is not performed on equal footing. Figure 2 states that the lattice result uses a dipole-dipole combination of form factors, while Figure 3's caption states the lattice D_g(t) is a tripole, and Section 3 says a tripole was used for all GFFs. This inconsistency in the functional forms used for different comparisons makes the visual agreement in Figs. 2-3 difficult to interpret. A single, self-consistent parametrization, or a presentation of the data points rather than curves only, would be needed to support the claim of agreement.
minor comments (5)
  1. [Eq. (1)] The decomposition of the EMT matrix element is written with inconsistent notation: the B-term appears as 'B_{q,g} i P^{{mu} sigma^{nu} rho} Delta_rho / (2 M_N)', which is not the standard form. Please correct the tensor indices and use a consistent convention.
  2. [Eq. (5)] There is a typo: the sentence says 'D_g^{FT}(r) and D_g^{FT}(r) are Fourier transform of A(t) and D(t)'. The first should be A_g^{FT}(r).
  3. [Figure 3 caption] The caption contains a duplicated phrase: 'the functional form of the D(t) form factor is a tripole is a tripole.'
  4. [Abstract and Section 2] The abstract states the analysis uses data from 'electronic decay channels,' but Section 2 says both e+e- and mu+mu- pair decays were detected; please harmonize the wording.
  5. [References] Reference [21] appears in the reference list but is not cited in the body of the text.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the gluonic form factors are fitted to external J/psi-007 cross-section data and the mass radius is a standard transform of those fitted form factors; the tripole ansatz is an explicit assumption, not a self-referential input.

full rationale

The derivation chain begins with the M-Z holographic cross-section formula (Eq. 2), which depends on A_g(t) and D_g(t), and the measured J/psi-007 differential cross sections. The paper fits the free parameters m_A, C_g(0), and m_C in the tripole forms of Eq. (3) to these external data, with A_g(0) fixed by the CT18 global PDF analysis. The mass radius in Eq. (4) and the density profiles in Eqs. (5)-(8) are obtained by differentiating and Fourier transforming the fitted form factors; these are derived quantities from the same fit rather than independent predictions, but that is ordinary inverse-problem inference, not circularity. The comparison with lattice QCD uses the independent calculation of Ref. [19]. The paper explicitly states that the tripole form was chosen 'for a consistent comparison with lattice QCD calculations' and notes in Fig. 1 that 'in all cases the form factors used are of tripole functional form'; this is a transparent parametrization assumption that could bias the extracted values if the true t-dependence is non-tripole, but it is not a reduction of the result to its inputs. Self-citations appear (Refs. [9] and [29] report the same experiment's data and fit results, and Refs. [11,16,22] are the M-Z model), but the central input data are external and the lattice benchmark is independent; no equation in the paper is equal by construction to the claimed output.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a specific non-perturbative model, an assumed tripole shape, and two neglected form factors. The fit itself uses three free parameters plus fixed inputs from CT18 and M-Z. Lattice QCD is used as a benchmark but at unphysical pion mass. These are all stated or cited in the paper, but they are assumptions the reader must accept.

free parameters (5)
  • A_g(0) (gluon momentum fraction) = 0.414 (fixed from CT18 global fit, +/- 0.008)
    Used as the normalization of A_g(t) in Eq. 3 and in the mass radius formula Eq. 4. If the CT18 input is wrong, the extracted radius shifts.
  • m_A = 1.575 +/- 0.059 GeV
    Tripole mass scale for A_g(t), fit to the two-dimensional cross-section data.
  • m_C = 1.12 +/- 0.21 GeV
    Tripole mass scale for C_g(t), and hence D_g=4C_g, fit to the data; controls the width of the pressure and shear profiles.
  • C_g(0) = -0.45 +/- 0.132
    Normalization of the C_g/D_g form factor at t=0, fit to the data; enters the mass radius through the -6 C_g(0)/M_N^2 term in Eq. 4.
  • N^2 e^2 normalization in Eq. 2 = (7.768)^2 nb/GeV^6
    Overall cross-section normalization taken from the M-Z holographic model [11]; it is not fitted in this paper, but the extracted form factors scale with it.
assumptions (6)
  • domain assumption The Mamo-Zahed holographic QCD cross-section formula (Eq. 2) is the correct description of near-threshold J/psi photoproduction.
    The whole extraction of A_g and C_g goes through Eq. 2; if this model is not the right non-perturbative description, the extracted values are not the true GFFs.
  • domain assumption Tripole functional forms (Eq. 3) accurately represent the t-dependence of A_g and C_g.
    Chosen for consistent comparison with lattice but not derived from QCD; the mass radius and density profiles inherit this shape assumption.
  • domain assumption B_g(t) can be set to zero and Cbar_g(t) ignored.
    Explicitly assumed in Sections 3 and 4. The text notes Cbar_g would increase the gluon energy density, so it is not a negligible correction at the claimed precision.
  • domain assumption CT18 gluon momentum fraction provides the correct A_g(0).
    The central radius formula and normalization use A_g(0)=0.414 from the CT18 global analysis [20].
  • domain assumption Lattice QCD at m_pi=400 MeV is a valid benchmark for the proton.
    The agreement with lattice is used as support; the paper admits the lattice calculation is at an unphysical pion mass.
  • standard math Standard Breit-frame Fourier transform relations (Lorce et al. [25]) connect the form factors to spatial densities.
    Eqs. 4-8 follow from these relations; they are standard but still an input the reader must accept.

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Cite this review

Pith. "Pith review of Gluonic Energy Momentum Tensor Form Factors of the Proton." pith.science (2026). https://pith.science/paper/PAVVFHA2

@misc{pith2026250505671,
  author       = {Pith},
  title        = {Pith review of: Gluonic Energy Momentum Tensor Form Factors of the Proton},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PAVVFHA2}},
  note         = {Machine review of arXiv:2505.05671}
}
abstract

Gravitational form factors (GFFs), defined through the matrix elements of the energy-momentum tensor, provide critical insights into the internal structure of nucleons and nuclei. In particular, their Fourier transforms -- evaluated in the Breit frame -- reveal spatial distributions of mass, pressure, and shear force densities associated with both quark and gluon constituents. This work presents recent measurements of near-threshold $J/\psi$ photoproduction on the proton, performed in Hall C at Jefferson Lab, utilizing data from the electronic decay channels of the $J/\psi$. These results enable the extraction of gluonic gravitational form factors (gGFFs), offering a novel probe of the gluon dynamics within the nucleon. The analysis employs a holographic QCD framework to interpret the threshold behavior of the cross sections and to facilitate the extraction of the gGFFs. The implications of these measurements are discussed in the context of upcoming experimental programs, including the near-threshold electro- and photoproduction studies with SoLID at Jefferson Lab and the $\Upsilon$ production program at the Electron-Ion Collider using the ePIC detector. These future efforts are expected to significantly improve the precision of gGFF determinations and provide essential tests of their universality across different kinematic regimes.

Figures

Figures reproduced from arXiv: 2505.05671 by the authors.

Figure 1
Figure 1. Left panel: The 𝐴𝑔 (𝑘 2 ) form factor (𝑘 2 = |𝑡|) extracted from our two-dimensional cross section data in the holographic QCD approach [11, 22] (orange dash-dot curve), compared to the latest lattice calculation [19] (blue dotted curve). In all cases the form factors used are of tripole functional form. The shaded areas show the corresponding uncertainty bands. Right panel: The extracted 𝐷𝑔 (𝑘 2 ) = 4𝐶𝑔 (𝑘 2 ) form… view at source ↗
Figure 2
Figure 2. The gluon energy density 𝑟 2 𝜖 contribution in the Breit frame, according to [25], with 𝐴 and 𝐷 GFFs extracted using the holographic QCD approach [11, 16, 22] (Green curve). The other bands are the more recent lattice results from [26]. The shaded area shows the corresponding uncertainty band for every contribution. Note that the lattice result uses a dipole-dipole combination of the form factors For the gluons mass… view at source ↗
Figure 3
Figure 3. Left panel: The 𝑟 2 𝑝𝑔 (𝑟) pressure in the Breit frame in the holographic QCD approach [11, 16, 22] (orange dash-dot curve), compared to the lattice calculation [19] (blue dotted curve). In this lattice calculation the functional form of the 𝐷(𝑡) form factor is a tripole is a tripole. The shaded areas show the corresponding uncertainty bands. Right panel: The extracted shear forces density 𝑟 2 𝑠𝑔 (𝑟) of the gluons i… view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.