{"id":"50e7af7c-daa9-4bb5-afd5-ea53bdd6493f","arxiv_id":"2509.11009","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Nucleon gravitational radii are computed from GPDs: quark radius about 0.54 fm, gluon radius about 0.88 fm, with quark dipole and gluon tripole form factors.","lead":"The authors use generalized parton distributions with several published quark and gluon PDFs to compute the quark and gluon gravitational form factors of the nucleon. They find the matter radius is about 0.54 fm, smaller than the proton charge radius, while the gluon radius comes out close to the charge radius.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The gluon gravitational form factor and its tripole fit inherit the quark GPD t-slope without a gluon-specific constraint, and the exponent n is fit only over |t|<=2; the large-t and radius claims are not independently established.","rationale":"The paper's central quantitative claims are the quark GFF dipole with Λ_q^2≈1.6 GeV^2, the gluon GFF tripole with n≈3 and Λ_g^2≈0.9 GeV^2, and the resulting radius ordering: quark radius < electromagnetic radius ≈ gluon radius. The reader's CONDITIONAL verdict identifies the two weakest supports: the borrowed quark t-dependence for gluons and the extraction of the large-t exponent from a |t|≤2 fit. My independent reading finds the same two issues and considers them genuinely load-bearing.\n\nThe quark part has reasonable internal grounding: the t-dependence is fit to electromagnetic form factor data, and the quark radius ~0.54 fm agrees with Kharzeev's estimate and is broadly consistent with lattice QCD. The gluon part, however, has no comparable anchor. Eq. (13) simply inserts gluon PDFs into the quark-determined exponential t-dependence. The resulting A_g(t) is then fit to a multipole form over |t|≤2. Since the gluon GPD slope is unknown, both n and Λ_g^2 are conditional; because the radius formula r_g^2 = 6n/Λ_g^2 is proportional to the slope α_g, even a moderate change in the gluon slope changes the radius by tens of percent and can move it away from the electromagnetic radius.\n\nThe paper itself contains a limitation statement that supports this concern: it notes that the picture for gluon distributions is valid only for |t|≤2. Yet the exponent n, which controls large-t behavior, is extracted from that same limited region. Thus the 'triple form' and 'drops faster at large transfer momenta' claims are not predictions from data but artifacts of the model and the fitting range.\n\nThis is not an internal inconsistency or a charge of scientific misconduct; it is a standard model-dependence problem. The direct radii in Table 1 may be less vulnerable, and the qualitative agreement with external estimates provides some support. The proposed concrete test would settle whether the tripole and comparable-radius conclusions survive when the gluon t-slope is varied and when the fit range is changed. Since the reader already recommends CONDITIONAL, my stress-test does not change that verdict.","tokens_in":8040,"tokens_out":9058,"duration_ms":103157,"concrete_test":"Take the same six gluon PDFs and recompute Eq. (13) with an independent gluon GPD slope, e.g. α_g f_g(x) = c α_q f_q(x) for c = 0.5, 1, 2; refit Eq. (14) on |t|≤2 and recompute r_g. If n moves outside roughly 2.5–3.5, or r_g outside roughly 0.7–1.0 fm, the tripole and comparable-radius conclusions are not robust. As a second check, refit the original A_g(t) on |t|≤1, |t|≤2, and |t|≤4: if n changes by more than the quoted spread, the exponent is a fit-range artifact rather than a physical large-t power.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (13) evaluates the gluon GFF as A_g(t)=∫ xg(x) e^{-α t f(x)} dx with the same t-dependence α f(x) extracted from quark electromagnetic form factors. No gluon-specific observable fixes this slope; the authors state explicitly: 'Using the t dependence of GPDs, obtained for the quarks contributions, the corresponding gluon gravitational form factors were obtained.' This matters because r_g^2 = -6 A_g'(0)/A_g(0) = 6α⟨f⟩_g scales linearly with the borrowed slope: a factor 2 in α_g changes r_g from roughly 0.62 fm to 1.25 fm, so 'comparable to the electromagnetic radius (0.84 fm)' is not robust unless the quark and gluon GPD slopes are equal.\n\nA second, compounding problem: the tripole value n≈3 is obtained by fitting the model-generated A_g(t) to Eq. (14) over |t|≤2, while the model itself is an exponential in t and the text acknowledges the picture is 'valid only for |t| ≤ 2'. The exponent n controls the large-t tail, so the abstract's 'drops faster ... triple form' is an extrapolation of a finite-range fit, not a prediction. Within the fitted range, n and Λ^2 can trade against each other, and changing the fit range will shift n. The directly computed radii in Table 1 are less exposed, but the tripole claim and the consequent 'faster drop at large t' are not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8509,"tokens_out":7146,"duration_ms":76225,"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":[{"comment":"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).","section":"Gluon GPDs and gravitational radius, Eq. (13)"},{"comment":"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.","section":"Gluon GPDs and gravitational radius, Eq. (14)"},{"comment":"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].","section":"Meson gravitomagnetic form factors and radii"},{"comment":"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.","section":"Table 1 and Fig. 1"}],"minor_comments":[{"comment":"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.","section":"Eqs. (16)–(17)"},{"comment":"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.","section":"General text"},{"comment":"The sentence 'The tensor meson dominance model gives [26] Aπ(t)=m²f2/(m²f2−1)=1+t/m²f2+m²f2=...' appears garbled and should be rewritten.","section":"Pion section"},{"comment":"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.","section":"Conclusions"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis paper gives you a concrete, testable parameterization: the quark gravitational form factor is a dipole with Λ_q^2 ≈ 1.6 GeV^2 and a mass radius about 0.54 fm, while the gluon form factor is a tripole with Λ_g^2 ≈ 0.9 GeV^2 and a radius near the proton charge radius of 0.84 fm. The quark matter radius being smaller than the charge radius, with the gluon field reaching out to roughly the charge radius, is a clean physical picture consistent with Kharzeev's estimate and with lattice hints. That is a useful service.\n\nWhat is new is exactly that power-form separation. The machinery is standard GPD Mellin moments, but the explicit multipole parameters and the radius ordering are not in the cited earlier papers. The authors also do a sensible thing in averaging over many PDF sets and comparing with the pion sector.\n\nThe soft spots are real, though. The gluon form factor is computed with the same t-dependence e^{-α t f(x)} that was fit to quark electromagnetic form factors, with no gluon-specific constraint. Since r_g^2 is linear in that slope, a factor of two in the gluon slope moves r_g from 0.62 to 1.25 fm, so \"comparable to the electromagnetic radius\" is only as good as that borrowed slope. Second, the tripole n≈3 is obtained by fitting Eq. (14) to model-generated A_g(t) over |t|≤2. The text itself says the gluon picture is \"valid only for |t|≤2,\" yet the abstract claims the gluon form factor \"drops faster at large t\" and \"corresponds to the triple form.\" That is a finite-range fit extrapolated to large t, with n and Λ^2 trading against each other. The directly computed radii in Table 1 are less exposed, but the tripole claim is not.\n\nAlso note the χ^2 column in Table 1: some gluon PDFs give enormous values (u4 at 155300), so \"averaging\" over PDFs hides a lot. No errors are propagated through the fit.\n\nWho is this for? Hadron structure people and pomeron phenomenologists. The physical picture is plausible and worth a serious referee, but the paper needs a revision that tests sensitivity to the gluon t-slope, reports fit-range dependence, and stops presenting the tripole as a standalone result. I would send it to review, with a clear request for those checks.\n\nBest,\n[You]","headline":"Plausible quark/gluon radius separation, but the gluon tripole form is a finite-range fit with a borrowed t-slope, not an independent result.","tokens_in":8956,"tokens_out":2267,"would_cite":false,"duration_ms":26419,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["generalized parton distributions","gravitational form factors","mass radius","quark-gluon momentum fractions","electromagnetic form factors","nucleon structure","pomeron coupling","dipole and tripole fits"],"falsifier":"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.","tokens_in":7891,"feed_emoji":"⚛️","tokens_out":7362,"duration_ms":78062,"temperature":0.7,"pith_summary":"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.","feed_headline":"Gluon radius matches proton charge radius; quark radius is 0.54 fm","feed_subtitle":"A GPD calculation puts the quark mass radius at 0.54 fm while gluons extend the nucleon's mass to 0.84 fm.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Gluon radius 0.84 fm, quark radius 0.54 fm: nucleon mass split","Quark mass radius 0.54 fm; gluon radius ties proton charge radius","Dipole quarks, tripole gluons: mass radii differ by 0.30 fm","Gluon gravitational radius equals proton's; quark radius is 0.54 fm","Nucleon mass: quarks to 0.54 fm, gluons to 0.84 fm"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Gluon radius 0.84 fm, quark radius 0.54 fm: nucleon mass split","Quark mass radius 0.54 fm; gluon radius ties proton charge radius","Dipole quarks, tripole gluons: mass radii differ by 0.30 fm","Gluon gravitational radius equals proton's; quark radius is 0.54 fm","Nucleon mass: quarks to 0.54 fm, gluons to 0.84 fm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001095,"raw_usage":{"total_tokens":4391,"prompt_tokens":709,"completion_tokens":3682,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":453,"completion_tokens_details":{"reasoning_tokens":3559}},"tokens_in":453,"tokens_out":3682,"duration_ms":28316,"temperature":1.0,"reasoning_tokens":3559,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T17:14:09.974403+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}