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REVIEW 4 major objections 4 minor 45 references

Relations of meson and nucleon electromagnetic and gravitational radii with quarks and gluons contributions

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A GPD-based calculation claims the quark gravitational form factor of the nucleon is a dipole with radius ~0.54 fm, below the proton's electromagnetic radius, while the gluon form factor is a tripole with radius comparable to the charge rad

desk verdict Plausible quark/gluon radius separation, but the gluon tripole form is a finite-range fit with a borrowed t-slope, not an independent result. read the letter →

arxiv 2509.11009 v1 pith:TZFQWMQ7 submitted 2025-09-13 hep-ph hep-ex

O.V. Selyugin , O.V. Teryaev This is my paper
classification hep-phhep-ex
keywords generalizedpartondistributionsgravitationalformfactorsmassradiusquark-gluonmomentumfractionselectromagneticnucleonstructurepomeroncouplingdipoleandtripolefits
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 uses the momentum-transfer dependence of generalized parton distributions (GPDs)—already fitted to nucleon electromagnetic form factors—to compute the quark and gluon parts of the nucleon's gravitational form factor. It claims the quark gravitational form factor is a dipole with Λ_q^2 ≈ 1.6 GeV^2, giving a quark mass radius around 0.54 fm, below the proton's electromagnetic radius of 0.8409 fm. Carrying the same t-dependence into six gluon parton distributions, it claims the gluon form factor is a tripole with Λ_g^2 ≈ 0.9 GeV^2, making the gluon gravitational radius comparable to the proton's charge radius. The result is a picture in which the nucleon's matter is more concentrated than its charge, and the gluon contribution stretches the mass distribution outward to roughly the electromagnetic size.

What carries the argument

The central object is the GPD H(x,t) with a factorized t-dependence H_q(x,t)=q(x) exp(α t f(x)), where f(x) is fixed from fits to proton and neutron electromagnetic form factors. The gravitational form factor A(t) is the second x-moment (one extra power of x) of this GPD at zero skewness; the mass radius follows from the slope of A(t) at t=0. For quarks, the paper integrates x-weighted valence-quark GPDs; for gluons, it uses six gluon PDFs with the same exp(α t f(x)) t-dependence. Power-law fits A(t) = A(0) Λ^2/(Λ^2 - t)^n then convert into radii and large-t falloff.

What would settle it

Measure A_g(t) out to |t| ≈ 2 GeV^2, either in exclusive quarkonium photoproduction or in a lattice calculation of the gluonic energy-momentum form factor, and fit Λ_g^2 and n. If the best fit is a dipole with Λ_g^2 > 1.6 GeV^2 rather than a tripole with Λ_g^2 ≈ 0.9 GeV^2, then the gluon radius would fall below the quark radius and the paper's ordering would fail.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the quark and gluon gravitational form factors have different power-law shapes in momentum transfer. The quark part A_q(t) follows a dipole (n=2) with Λ_q^2 = 1.6 ± 0.1 GeV^2, while the gluon part A_g(t) follows a tripole (n=3) with Λ_g^2 = 0.9 ± 0.2 GeV^2. Since the mass radius is the slope of A(t) at t=0, the quark radius (~0.54 fm) is smaller than the proton's electromagnetic radius, and the gluon radius is comparable to it. The gluon form factor therefore falls faster with |t| than the quark form factor, and adding gluons increases the nucleon's total mass radius. The sum A_q(0)+A_g(0) ≈ 1, with the quark carrying about 55% and gluons about 45% of th

Load-bearing premise

The gluon GPD is assumed to have the same t-dependence as the quark GPD—carried over from fits to electromagnetic form factors—with no gluon-specific constraint; the paper itself notes the picture may hold only for |t| ≤ 2, so a different gluon slope would undo the tripole exponent and the claimed radius equality.

Editorial extensions

If this is right

  • The nucleon's quark matter radius is ~0.54 fm, about 64% of its 0.8409 fm charge radius, so the quark mass distribution is more concentrated than the charge distribution.
  • The gluon gravitational radius is comparable to the proton's electromagnetic radius, so gluons spread the nucleon's mass out to approximately its charge size.
  • The gluon form factor falls like a tripole (n=3), faster than the quark dipole (n=2) at large momentum transfer, meaning gluon contributions dominate the short-distance tail of the mass distribution.
  • With A_q(0)+A_g(0)≈1 (quarks ≈0.55, gluons ≈0.45), the gluon gravitational form factor offers a candidate description of pomeron coupling to hadrons in high-energy scattering.

Reading between the lines

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

  • The paper's assumption that gluons inherit the quark t-dependence is testable: exclusive quarkonium photoproduction or a separate lattice determination of A_g(t) can measure the gluon slope independently; if it differs, the tripole radius shifts.
  • Because the gluon radius lands near the proton charge radius, precision measurements of the charge radius may indirectly constrain the gluon gravitational distribution, and the quark–gluon radius split could show up in the t-dependence of deeply virtual Compton scattering.
  • Neglected sea and strange quarks mean A_q(0)=0.54 is an upper bound for the valence quark contribution; including them would slightly lower the quark share and could change the 55/45 split.
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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 / 4 minor

Summary. The paper computes quark and gluon gravitational form factors (GFFs) of the proton by inserting parton distribution functions (PDFs) into a factorized GPD ansatz H(x,t)=q(x) e^{-α t f(x)}, with α and f(x) previously determined by author from fits to nucleon electromagnetic form factors. The same t-dependence is then applied to gluon PDFs. The resulting A(t) are fitted to the power form A(t)=A(0)(Λ²/(Λ²-t))^n. The paper reports a quark GFF dipole (n_q=2, Λ_q²≈1.6 GeV²), a quark mass radius of about 0.54 fm, a gluon GFF tripole (n_g≈3, Λ_g²≈0.9 GeV²), and a gluon gravitational radius comparable to the proton electromagnetic radius (0.84 fm). A monopole form is also reported for the pion. The abstract and conclusions assert that the gluon form factor drops faster at large t than the quark one.

Significance. If the conclusions were robust, the paper would provide a simple phenomenological relation between electromagnetic and gravitational radii, which is of current interest for hadron structure and for pomeron–hadron coupling. A strength of the paper is that it uses several modern gluon and quark PDF sets, providing a Table 1 with numerical spreads. However, the central gluon result is obtained by assuming that the gluon GPD has exactly the same momentum-transfer dependence as the quark GPD, without any gluon-specific constraint. The large-t behavior is obtained by extrapolating a fit over |t|≤2. These issues make the main claims model-dependent and not yet established.

major comments (4)
  1. [Gluon GPDs and gravitational radius, Eq. (13)] The gluon GFF is computed as A_g(t)=∫ xg(x) e^{-α t f(x)} dx, using the same t-dependence α f(x) extracted from quark electromagnetic form factors. The text states explicitly: 'Using the t dependence of GPDs, obtained for the quarks contributions, the corresponding gluon gravitational form factors were obtained.' No gluon-specific observable enters. Since r_g²=6α⟨f⟩_g, a factor 2 change in the gluon slope α_g changes r_g from roughly 0.62 fm to 1.25 fm. The claim that the gluon radius is comparable to the proton electromagnetic radius (0.84 fm) is therefore not robust unless quark and gluon slopes are equal. The authors should quantify this sensitivity or test against an independent gluon constraint (e.g., lattice gluonic GFFs or exclusive quarkonium production).
  2. [Gluon GPDs and gravitational radius, Eq. (14)] The tripole value n≈3 is obtained by fitting the model-generated A_g(t) to Eq. (14) over |t|≤2, and the text itself states 'our picture for gluon distributions is valid only for |t|≤2.' Within this finite range the model A_g(t) is essentially an exponential in t; n and Λ² are strongly correlated, and the exponent should be regarded as an effective fitting parameter, not the large-t asymptotic power. The abstract's claim that the gluon form factor 'drops faster ... and corresponds to the triple form' is an extrapolation beyond the fitted range. The authors should either fit over multiple ranges to show stability, or explicitly restrict the claim to |t|≤2 and remove the large-t statement.
  3. [Meson gravitomagnetic form factors and radii] The pion fit is reported with Λ²=1.44 GeV² and n=1.07. Using the same relation as for the nucleon, r²=6n/Λ², this gives r≈0.40 fm, not the quoted 0.67 fm. If a different definition is intended, it must be stated; if not, this is an internal inconsistency that undermines the pion mass-radius comparison with Ref. [36].
  4. [Table 1 and Fig. 1] The parameters a and b, which the text says control the large-x behavior of the PDFs and which Fig. 1 displays, are not defined in the text or captions. The column 'χ2tot' contains values such as 155300 and 1380 without stating what is minimized or how the normalization is defined. Without these definitions, the spread in Table 1 and the claimed dependence of n and Λ² on b cannot be assessed.
minor comments (4)
  1. [Eqs. (16)–(17)] The two displayed formulas for the dipole–dipole and dipole–tripole cases are identical, both reading (1/2)(2/Λ_q²)+(1/2)(3/Λ_g²). This appears to be a typo and obscures the distinction between the cases.
  2. [General text] There are numerous typographical errors, e.g. 'different dip inelastic reactions' should be 'deep-inelastic reactions', 'the the', 'n ucleon', and duplicated 'F1q(t)' in Eq. (3). These should be corrected.
  3. [Pion section] The sentence 'The tensor meson dominance model gives [26] Aπ(t)=m²f2/(m²f2−1)=1+t/m²f2+m²f2=...' appears garbled and should be rewritten.
  4. [Conclusions] The statement 'We do not take into account small additional contributions from the sea quarks and strange quarks' is useful, but the numerical consequence for A(0)q+A(0)g≈1 should be quantified, since Table 1 gives A_g(0) values from 0.33 to 0.46, which with A_q(0)=0.54 do not all sum to 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: gravitational radii are derived from GPD t-slopes fitted to electromagnetic form factors and are not used as inputs; the multipole forms are fit outputs, not self-referential premises.

full rationale

The paper's derivation chain is not circular. The quark gravitational form factor is obtained by integrating the second Mellin moment of GPDs, A_q(t)=∫ x q(x) e^{-α t f(x)} dx, where the t-dependence is taken from [11,14] and was fitted to the nucleon electromagnetic form factors. The resulting quark mass radius (0.54 fm) is therefore a genuine prediction from a different moment of the same GPD, not a re-statement of the electromagnetic radius; the ratio of the gravitational to electromagnetic radius depends on the PDFs and the function f(x) and varies across the 19 PDF sets examined. The gluon form factor A_g(t)=∫ xg(x) e^{-α t f(x)} dx uses the same t-dependence as an ansatz, and the tripole form (n≈3, Λ²≈0.9) is obtained by fitting the model-generated A_g(t) to Eq. (14) over |t|≤2. This means the 'triple form' and large-t drop are fit results and extrapolations, which is a robustness concern, but not circularity: the exponent n and Λ² are outputs of the fit, not inputs. The self-citations [11,14] are load-bearing for the GPD t-slope, but those results are externally anchored to a wide set of electromagnetic form factor data, so the citation does not create a logical loop. No equation is defined in terms of the claimed conclusion, and no fitted parameter is renamed as an independent prediction. Therefore, despite model-dependence and finite-range fitting, the paper does not exhibit circular reasoning.

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

The central outputs inherit many upstream choices. The free parameters are the fitted t-slope from electromagnetic form factors and the Lambda^2 and n values fitted to second-moment curves. The axioms are standard GPD sum rules plus the ad hoc transfer of the quark t-dependence to gluons and to the pion. No new particles or mediators are introduced.

free parameters (6)
  • GPD t-slope parameter alpha = Not stated explicitly; fit to proton and neutron electromagnetic form factors in Selyugin 2014
    The exponential t-dependence e^{-alpha t f(x)} in Eqs. (1), (2), and (12) is fixed by reproducing electromagnetic form factor data, and all gravitational radii inherit this parameter.
  • Quark gravitational form factor exponent n_q = 2
    The A(t) from the quark GPD integral is approximated by a dipole form; n_q = 2 is chosen, not derived from a first-principles constraint.
  • Quark gravitational form factor scale Lambda_q^2 = 1.6 +/- 0.1 GeV^2 (1.58 +/- 0.04 in the model)
    Fitted to the second-moment GPD integral A(t); this scale sets the quark gravitational radius near 0.54 fm.
  • Gluon gravitational form factor exponent n_g = About 3, ranging from 2.6 to 3.0 in Table 1
    Fitted to A_g(t) computed from six gluon PDF sets; the tripole conclusion is a fit result rather than an independent prediction.
  • Gluon gravitational form factor scale Lambda_g^2 = About 0.9 +/- 0.2 GeV^2, ranging from 0.65 to 1.07 in Table 1
    Fitted to A_g(t); this scale leads to a gluon radius comparable to the proton charge radius.
  • Pion gravitational form factor parameters = Lambda^2 = 1.44, n = 1.07
    Fit to pion PDFs and electromagnetic form factor data; produces a pion mass radius of 0.67 fm.
assumptions (6)
  • domain assumption GPD Mellin-moment sum rules connect first and second moments to electromagnetic and gravitational form factors.
    Invoked in Eqs. (3), (7), and (8); standard in the GPD literature (Ji, Radyushkin).
  • ad hoc to paper The t-dependence of quark GPDs factorizes as q(x) e^{-alpha t f(x)}, and the same functional form is applied to gluon GPDs.
    Used in Section 'Gluon GPDs and gravitational radius' with no gluon-specific t-dependence input.
  • domain assumption Published PDF sets at scale mu^2 = 1 represent the x-dependence of quarks and gluons.
    Used as inputs for the integrals in Eqs. (12) and (13).
  • standard math The quark and gluon decomposition of the Belinfante-improved EMT defines the gravitational form factors.
    Standard QCD energy-momentum tensor decomposition, Eqs. (5) and (6).
  • ad hoc to paper For the pion, the same momentum-transfer dependence of GPDs is assumed as for the nucleon.
    Stated in the Conclusions: 'For the meson (pion) the momentum transfer dependence of GPDs was taken the same as for the nucleon.'
  • ad hoc to paper The gravitational form factor can be represented by the power form (Lambda^2/(Lambda^2 - t))^n.
    This fitting ansatz underlies the dipole and tripole conclusions in Eqs. (9), (14), and Table 1.

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Pith. "Pith review of Relations of meson and nucleon electromagnetic and gravitational radii with quarks and gluons contributions." pith.science (2026). https://pith.science/paper/TZFQWMQ7

@misc{pith2026250911009,
  author       = {Pith},
  title        = {Pith review of: Relations of meson and nucleon electromagnetic and gravitational radii with quarks and gluons contributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZFQWMQ7}},
  note         = {Machine review of arXiv:2509.11009}
}
read the original abstract

The electromagnetic and gravitational form factors of the nucleon determined by quark and gluon contributions are calculated using the momentum transfer dependence of generalized parton distributions with different forms of parton distribution functions obtained by various Collaborations. The power forms of gravitational form factors of quarks and gluons are examined. It is shown that the gluon gravitational radius of the nucleon is comparable to the electromagnetic radius of the proton; however, the quark gravitational radius of the nucleon is less than its electromagnetic radius. It is shown that the gluon gravitational form factor drops faster than the quark gravitational form factor at large transfer momenta and corresponds to the triple form.

Figures

Figures reproduced from arXiv: 2509.11009 by the authors.

Figure 1
Figure 1. a)[left]The dependence of ng of the gluon gravitational form factor on the parameter b; b) [right] The dependence of Λ 2 g of the gluon gravitational form factor on the parameter b the obtained gluon gravitational form factors are described by the multupole form with the fitting parameters Λ and n. Ag(t) = A(0) Λ 2 (Λ2 − t) n (14) The results were presented in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The dependence of ng of gluon gravitational form factor over parameter a. Now let us examine different cases: Case I) (dipole -dipole case) Suppose that in both (quark and gluon) cases the gravitational form factors are described by the dipole form. In this case, nq = ng = 2 and for the radius we obtain < r2 >(q+g) = 1 2 2 Λ2 q + 1 2 3 Λ2 g . (16) There can be three different situations: a) Λ 2 q = Λ2 g This leads t… view at source ↗

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