{"id":"62c23b78-ea92-4e5b-94ed-040d75a7a434","arxiv_id":"2505.00583","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A DSE rainbow-ladder calculation produces quark and gluon gravitational form factors, mass radii, and D-terms for nucleon and pion, finding gluon distributions generated purely by quark dressing.","lead":"This paper uses a quark model to estimate how much of a proton's and pion's mass is carried by gluons, the force particles that hold quarks together. The result suggests gluon content and mass radii can emerge from quark dressing alone, matching recent lattice calculations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The constant Landau-gauge compensation in §4 is the load-bearing assumption: if its 7%/27% factors are inaccurate or scale-dependent, the fitted gluon strength, <x>_g(2 GeV), and R_g^E shift, leaving the 'realistic gluon' claim unsupported.","rationale":"The reader’s weakest-assumption analysis identifies the same point I would choose: the constant one-loop Landau-gauge compensation factors in §4 are the least secure link between the model’s gauge-dependent DSE-RL matrix elements and the physical light-cone quantities compared with global data and lattice QCD. The paper is candid about this, stating that the 7%/27% corrections are '1-loop estimated' and are needed 'to maintain the fit to data at higher scales.' Because the gluon-in-quark kernel strength Kg is fit using these compensated values, the gluon fraction and all derived gluon form-factor shapes inherit the compensation. If the compensation is scale-dependent — note the two different model scales μ0^π=0.9 GeV and μ0^N=0.64 GeV — the pion calibration would not transfer to the nucleon, and the close agreement of R_g^E with LQCD would be accidental. The claim 'quark dressing by itself can produce realistic gluon contributions' is weakened to 'quark dressing plus a hand-tuned gauge correction can reproduce the data.' This does not invalidate the exploratory calculation but does justify the conditional verdict: the paper should test the robustness of its outputs to the compensation choice and, ideally, provide an independent gauge-invariant check. I therefore do not change the reader’s conditional-accept verdict.","tokens_in":10533,"tokens_out":5006,"duration_ms":49579,"concrete_test":"Recompute the model-scale gluon and quark momentum fractions without the compensation factors (set them to zero) and with two bounding sets (e.g., 5%/15% and 10%/40% corrections) in the fit of Kg in §4; then evolve to 2 GeV and re-extract A_g(0), R_g^E, and R_g/R_q for the nucleon from Table 1. If R_g^E moves by more than ~0.05 fm away from the LQCD value 0.81 fm, or if <x>_g at 2 GeV shifts by more than the global-fit uncertainty quoted in [25], the claimed realism of the gluon part is an artifact of the compensation and the central claim fails. A complementary check is to compare the uncompensated model-scale <x>_g with the value obtained if the Wilson line were included in the operator definition (e.g., from a light-front or Euclidean lattice calculation at the same scale), which would directly test the need for the 27% correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing point is the constant Landau-gauge compensation introduced in §4: 'we employ the 1-loop estimated compensation [13] of a 7% decrease of <x>_q(mu0) and a 27% increase of <x>_g(mu0) to maintain the fit to data at higher scales.' The gluon-in-quark kernel Kg is calibrated by requiring the compensated pion fractions to match global fits after evolution, and this same kernel is then applied unchanged to the nucleon. Every quantitative output in the paper — the gluon momentum fraction at 2 GeV, the energy/mass radius R_g^E = 0.765 fm, the ratio R_g/R_q, and the comparison with LQCD — inherits these two numbers. If the 1-loop estimate is inaccurate, or if the required compensation depends on the model scale (μ0^π = 0.9 GeV vs μ0^N = 0.64 GeV) or on the hadron, the 'realistic' gluon contribution is not actually produced by quark dressing alone; it is partly inserted by an external scale choice. The paper itself flags this as a limitation, but the central claim depends on it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript presents an exploratory application of the Dyson-Schwinger/Bethe-Salpeter rainbow-ladder truncation to the quark and gluon components of the hadron energy-momentum tensor (EMT) for the pion and the nucleon. The author derives a formal connection between EMT matrix elements and generalized momentum-fraction moments of GPDs, then computes the gravitational form factors A_p(Q^2), the energy distributions E_p(Q^2), the angular-momentum form factor J_p(Q^2), and the D-term, together with associated radii. The gluon-in-quark kernel is calibrated using the pion momentum fractions with a 7%/27% Landau-gauge compensation and is then applied unchanged to the nucleon, with model scales μ_0^π = 0.9 GeV and μ_0^N = 0.64 GeV. The main numerical results include a nucleon gluon mass radius R_g^E = 0.765 fm, close to the LQCD value 0.81 fm, and quark/gluon radius ratios for the pion. The paper argues that the results support the idea that quark dressing alone can generate realistic gluon contributions to hadron structure.","tokens_in":10900,"tokens_out":7896,"duration_ms":77177,"significance":"If correct, the central claim is significant because it would provide a single non-perturbative mechanism, quark dressing in rainbow-ladder truncation, that accounts simultaneously for gluon momentum fractions, gravitational form factors, mass/energy radii, and the D-term for both the nucleon and the pion. The formal connection in Eqs. (7)-(9) is clearly laid out, the nucleon results are not directly fit to nucleon gluon data, and the model generates a wide range of observables (A, E, J, D, radii) within one framework. The comparison between the predicted R_g^E = 0.765 fm and the LQCD result 0.81 fm is a useful benchmark. However, the calibration of the gluon-in-quark kernel via the pion and the use of an external 7%/27% compensation make the pion agreement partly circular; the independent support comes mainly from the nucleon and from the Q^2-dependent quantities, so the significance will be fully established only after the sensitivity issues described in the major comments are addressed.","major_comments":[{"comment":"The constant Landau-gauge compensation is load-bearing. The gluon-in-quark kernel strength is set so that the compensated pion model-scale momentum fractions match global PDF data after evolution, and that same kernel is then applied unchanged to the nucleon; consequently the nucleon gluon momentum fraction at 2 GeV, the mass radius R_g^E = 0.765 fm quoted in Table 1, and the comparison with the LQCD value 0.81 fm all inherit the 7% and 27% factors. The paper does not quantify the sensitivity of these outputs to the magnitude of the compensation or to its possible scale dependence between μ0^π = 0.9 GeV and μ0^N = 0.64 GeV, nor does it assess whether a constant rescaling can correct a Q^2-dependent gauge or Wilson-line omission that would feed the radii extracted from slopes. Without such a sensitivity analysis, the central claim that quark dressing alone produces realistic gluon contributions is not fully established.","section":"§4, gluon-in-quark kernel and compensation"},{"comment":"The values of \\bar{C}^{N}_{q/g}(0) and \\bar{C}^{π}_{q/g}(0) in Eq. (17) are introduced as an 'estimate' with no derivation, formula, or uncertainty. These constants directly control the difference between the parton energy fractions E_p(0)/M and A_p(0), and through Eq. (3) and the subsequent radius relations they affect the E_r radii in Table 1 and the D(0) values quoted in §5. Please either derive these estimates from the model or provide a sensitivity study showing that the quoted radii and D-terms are robust to their variation.","section":"§5, Eq. (17)"},{"comment":"No evolution scheme is specified for promoting the model-scale results at μ0^π = 0.9 GeV and μ0^N = 0.64 GeV to μ = 2 GeV. For the Q^2-dependent form factors A_p(Q^2), E_p(Q^2), and J_p(Q^2), this is not the same as evolving forward PDFs; the manuscript should state the order of the evolution, the treatment of the non-forward matrix elements, and the matching conventions, or restrict the comparisons to the model scale.","section":"§4-§5, scale evolution"},{"comment":"No uncertainties are quoted for any quantitative result, e.g., R_g^E = 0.765 fm, R_g/R_q = 1.22, D_π(0) = −0.89, and D_N(0) = −1.73. Since the model contains fitted parameters (the kernel strength, model scales, nucleon wavefunction parameters) plus the external 7%/27% compensation, the claimed agreement with LQCD cannot be evaluated without parameter and model uncertainties or at least a sensitivity analysis.","section":"§5, Table 1 and D-term paragraph"},{"comment":"The pion comparison is partly circular: the gluon-in-quark kernel is fixed by requiring the pion's model-scale momentum fractions to evolve to global PDF data, so the pion agreement is imposed rather than predicted. The genuinely independent support for the central claim comes from the nucleon and from the Q^2-dependence, radii, and D-terms, which were not used in the fit. The manuscript should state this limitation explicitly and frame the pion results as a consistency check rather than as independent evidence.","section":"§1, §4, §6"}],"minor_comments":[{"comment":"The notation i\\partial_a(z) and the action of the derivative on the Wilson line are not defined; please spell out the integration-by-parts step leading to Eq. (8).","section":"Eq. (7)"},{"comment":"The definition of t^{\\mu\\nu} uses curly-brace symmetrization; please state the convention used for the symmetrization.","section":"Eq. (5)"},{"comment":"The caption should state explicitly whether the displayed D(Q^2) curves are the total (summed parton) form factors and at which scale; the text says D is scale invariant, but the figure does not indicate this.","section":"Figure 3"},{"comment":"The table of \\bar{C} estimates in Eq. (17) would be easier to read as a numbered table rather than an unnumbered inline matrix.","section":"Eq. (17)"},{"comment":"The paper repeatedly refers to Ref. [13] for kernel parameters and the compensation estimate; a brief recapitulation of these inputs would improve the self-containedness of the proceedings article.","section":"§4, references [13] and [26]"}],"recommendation":"major_revision","confidential_remarks":"This is a compact proceedings contribution, but the missing methodological details (evolution scheme, derivation of the \\bar{C} estimates, parameter uncertainties, and sensitivity to the 7%/27% compensation) are important because quantitative comparisons with LQCD are claimed. The author may also be asked to clarify the relation to Refs. [13] and [26]: the reader should be told explicitly which of the present results (e.g., the Q^2-dependent GFFs, the D-term, and the radii) are new relative to those earlier works."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the genuinely new result is the Q^2 dependence of quark and gluon gravitational form factors, the mass/energy distribution radii, and the D-term from a DSE-RL calculation. The nucleon gluon mass radius comes out 0.765 fm, close to the lattice value 0.81 fm, and that comparison is an independent test because the nucleon was not used to fix the gluon kernel. Second, the central claim—that quark dressing alone can produce realistic gluon contributions—is plausible but softer than the abstract sounds. The pion side is partly fit-driven, and the 7%/27% Landau-gauge compensation factors carry real weight.\n\nWhat the paper does well: the formal step connecting EMT matrix elements to generalized momentum-fraction moments (Eqs. 7–9) is clearly laid out. Using the light-like n^mu projection for A(Q^2) and the rest-frame e^mu projection for the energy distribution is a sensible way to extract multiple form factors from one framework. The gluon-in-quark kernel is fixed once using the pion and then applied unchanged to the nucleon, which gives the nucleon comparison genuine predictive content. The D-term extraction and the radii pattern (R_A < R_T < R_E) are useful, and the paper does not oversell the Q^2 range.\n\nThe soft spots are in proportion. The pion momentum fractions at the model scale are calibrated to reproduce global PDFs after evolution, so the pion gluon fraction is not an independent prediction. The larger concern is the constant 7% decrease of <x>_q and 27% increase of <x>_g used to compensate for Landau gauge and the missing Wilson line. If those factors are inaccurate or scale-dependent, the fitted gluon strength, <x>_g(2 GeV), and R_g^E all shift. The paper flags this, but it is load-bearing for the 'realistic' claim. Still, this is not fatal: the nucleon side remains a meaningful independent test, and a sensitivity study would show how much the conclusions depend on those two numbers. The absence of any uncertainty estimates is a genuine weakness for quantitative comparison with lattice QCD.\n\nThe citation pattern is fine. It leans on the author's own earlier work because this is a direct extension, and the lattice and global-data references are the right comparisons.\n\nWho this is for: people working on hadron structure models and EIC-era observables. It is a proceedings contribution, so the detail level is modest, but it does contain new predictions. I would not desk-reject it. A serious referee can usefully push for uncertainty estimates and a more critical treatment of the compensation factors. My own bottom line is conditional accept, with the referee report focused on those two points.","headline":"Genuinely new DSE-RL results for the Q^2 dependence of gluon gravitational form factors and radii, with a plausible central claim and one load-bearing assumption—the 7%/27% Landau-gauge compensation—deserves a serious referee.","tokens_in":11391,"tokens_out":3407,"would_cite":true,"duration_ms":35364,"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 Dyson-Schwinger model in rainbow-ladder truncation produces realistic gluon contributions to the nucleon and pion gravitational form factors and mass/energy distributions.","keywords":["gravitational form factors","energy-momentum tensor","gluon distributions","Dyson-Schwinger equations","rainbow-ladder truncation","nucleon mass radius","pion","parton momentum fractions"],"falsifier":"Directly compute the $Q^\\mu T^{\\mu\\nu}_g Q^\\nu/Q^2$ projection in the same DSE-RL framework to obtain $\\bar C_g(Q^2)$; if $\\bar C_g(0)$ comes out far from the estimated $0.08$ (nucleon) or $0.02$ (pion), or if the reconstructed gluon mass radius moves outside the lattice-QCD uncertainty band around $0.81\\,\\mathrm{fm}$, the claim that quark dressing alone reproduces the gluon mass distribution would be falsified.","tokens_in":10331,"feed_emoji":"⚛️","tokens_out":12571,"duration_ms":109950,"temperature":0.7,"pith_summary":"This paper argues that the dressing of quarks --- the self-generated cloud of interactions built up by the Dyson-Schwinger equation in rainbow-ladder truncation --- can by itself account for the gluon shares of the nucleon's and pion's momentum, mass, and energy distributions. The author computes quark and gluon gravitational form factors directly from energy-momentum tensor matrix elements, using the identity that connects these matrix elements to the momentum-fraction moments of generalized parton distributions. The resulting nucleon gluon mass radius, $0.765\\,\\mathrm{fm}$ at $2\\,\\mathrm{GeV}$, lands close to the lattice-QCD value $0.81\\,\\mathrm{fm}$, and the pion's gluon-to-quark radius ratio of $1.22$ is close to the lattice value $1.1$. If this holds, quark dressing is a sufficient source for most gluon parton structure without invoking explicit gluonic degrees of freedom.","feed_headline":"Quark dressing alone yields realistic gluon mass in nucleon, pion","feed_subtitle":"A quark-dressing model reproduces lattice-QCD gluon mass radii and momentum fractions for nucleon and pion.","key_machinery":"The central object is the identification of the energy-momentum tensor matrix element $\\langle P'|T^{aa}_p(0)|P\\rangle$ with the generalized momentum-fraction moment $\\langle k\\cdot a/K\\cdot a\\rangle_p(Q^2)$, evaluated through DSE-RL dressed vertices $\\Gamma^a_p(p,Q)$ that solve a Bethe-Salpeter equation with the rainbow-ladder kernel $K(q^2)=D_{RL}^2 e^{-q^2/\\omega^2}+F(q^2)\\,4\\pi\\bar\\alpha_s(q^2)$. Choosing $a=n$ (light-like) yields the standard light-cone momentum fractions, while $a=e$ (rest-frame) yields the mass/energy density; both come from the same dressed vertices, with the gluon case entering through the gluon-in-quark inhomogeneous term built from the kernel $\\Delta_g(q)^2=K_g(q^2)$. This identity lets the calculation reach gravitational form factors without adding explicit gluon degrees of freedom beyond those generated by quark dressing.","core_discovery":"At the model scale ($\\mu_0^N=0.64\\,\\mathrm{GeV}$, $\\mu_0^\\pi=0.9\\,\\mathrm{GeV}$), the DSE-RL calculation yields quark and gluon components of the gravitational form factors $A_p(Q^2)$, $B_p(Q^2)$, $J_p(Q^2)$, the mass/energy distributions $E_p(Q^2)/M$, and the summed $D(Q^2)$, with the gluon contributions generated entirely by quark dressing. After applying the 1-loop-estimated Landau-gauge compensation factors (a $7\\%$ decrease of $\\langle x\\rangle_q$ and a $27\\%$ increase of $\\langle x\\rangle_g$ at the model scale) and evolution to $2\\,\\mathrm{GeV}$, the gluon momentum fractions agree with a global data analysis, the nucleon gluon mass radius is $R_g^{E}=0.765\\,\\mathrm{fm}$ (compared with the lattice value $0.81\\,\\mathrm{fm}$ and a data-based value $0.778\\,\\mathrm{fm}$), and the pion ratio $R_g/R_q=1.22$ for $A(Q^2)$ is close to the lattice value $1.1$ and algebraic-model value $1.31$. The extracted D-term values are $D_\\pi(0)=-0.89$ and $D_N(0)=-1.73$. The radii hierarchy $R_A<R_T<R_E$ indicates that the form factor $A_p(Q^2)$ alone does not represent the mass/energy distribution.","pith_inferences":["A direct calculation of the $\\bar C_g(Q^2)$ form factor via the $Q^\\mu T^{\\mu\\nu}_g Q^\\nu/Q^2$ projection, which the paper flags as future work, would convert the estimated $\\bar C_g(0)$ values into a testable prediction and could shift the quoted radii.","If the compensation factors are actually scale-dependent rather than constant, the close agreement with lattice values could be accidental; a lattice calculation of quark and gluon momentum fractions at the model scale would settle this.","Because the pion fixes the gluon-in-quark kernel and the nucleon uses it unchanged, extending the same calculation to the $\\rho$ meson would test whether the dressing mechanism remains sufficient outside the pseudoscalar channel.","The joint agreement of two very different hadrons with one shared kernel is a stronger constraint than either alone, so the next natural check is whether higher Mellin moments (beyond the second) preserve the pattern."],"forward_implications":["The nucleon gluon mass radius is $R_g^E=0.765\\,\\mathrm{fm}$ at $2\\,\\mathrm{GeV}$, essentially equal to the quark mass radius ($R_g/R_q=0.998$), so gluonic energy is spread as widely as quark energy.","In the pion, the gluon radius exceeds the quark radius by $22\\%$ for the $A(Q^2)$ form factor, matching the pattern seen in lattice QCD and in algebraic GPD models.","The mass/energy radius $R_E$ exceeds the $A(Q^2)$ radius $R_A$ for both hadrons, dramatically so for the pion ($R_q^E=1.178\\,\\mathrm{fm}$ versus $R_q^A=0.397\\,\\mathrm{fm}$), so $A_p(Q^2)$ alone is not a reliable proxy for mass distribution.","The total D-term is fixed by the radius relations: $D_\\pi(0)=-0.89$ and $D_N(0)=-1.73$, giving scale-invariant targets for lattice and experiment.","Quark dressing already produces more than half of the 2 GeV gluon momentum fraction at the model scale, indicating that perturbative evolution adds to, but does not create, most of the gluon strength."],"supporting_citations":[{"why":"Establishes the decomposition of energy-momentum tensor matrix elements into gravitational form factors and their connection to GPD momentum-fraction moments, which is the identity the calculation exploits.","marker":"[1]"},{"why":"Lattice-QCD calculation of the nucleon gravitational form factors that provides the benchmark gluon mass radius of $0.81\\,\\mathrm{fm}$.","marker":"[6]"},{"why":"Lattice-QCD calculation of pion gravitational form factors providing the benchmark $R_g/R_q\\approx 1.1$ for the $A(Q^2)$ radii.","marker":"[7]"},{"why":"Prior DSE-RL calculation that fixes the gluon-in-quark kernel parameters and introduces the $7\\%$/$27\\%$ Landau-gauge compensation factors used here.","marker":"[13]"},{"why":"Defines the rainbow-ladder interaction kernel used to solve the quark propagator, bound-state vertices, and Bethe-Salpeter amplitudes.","marker":"[22]"},{"why":"Global PDF analysis providing the empirical quark and gluon momentum fractions at higher scales that the model-scale fit must reproduce.","marker":"[25]"},{"why":"Supplies the gluon-in-quark dressed vertex and the $Q=0$ momentum-fraction formalism that this work generalizes to $Q>0$.","marker":"[26]"},{"why":"Data-based extraction of the nucleon gluon mass radius ($0.778\\,\\mathrm{fm}$) used as a second benchmark alongside lattice QCD.","marker":"[32]"}],"fun_headline_variants":["Quark dressing alone explains gluon mass in nucleon and pion","Gluon mass from quark dressing matches lattice QCD","Dressed quarks generate realistic gluon momentum fractions","Model gluon mass radii agree with lattice and data","All gluon mass from quark dressing, no extra glue"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the paper's Section 4 assumption that the fixed 1-loop-estimated Landau-gauge compensation factors --- a $7\\%$ decrease of $\\langle x\\rangle_q(\\mu_0)$ and a $27\\%$ increase of $\\langle x\\rangle_g(\\mu_0)$ --- correctly convert the gauge-dependent DSE-RL model-scale results into the light-cone quantities compared with data and lattice QCD.","fun_headline_variants_meta":{"raw":{"variants":["Quark dressing alone explains gluon mass in nucleon and pion","Gluon mass from quark dressing matches lattice QCD","Dressed quarks generate realistic gluon momentum fractions","Model gluon mass radii agree with lattice and data","All gluon mass from quark dressing, no extra glue"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000669,"raw_usage":{"total_tokens":3066,"prompt_tokens":980,"completion_tokens":2086,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":2005}},"tokens_in":596,"tokens_out":2086,"duration_ms":12650,"temperature":1.0,"reasoning_tokens":2005,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:38:25.782477+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly compute the $Q^\\mu T^{\\mu\\nu}_g Q^\\nu/Q^2$ projection in the same DSE-RL framework to obtain $\\bar C_g(Q^2)$; if $\\bar C_g(0)$ comes out far from the estimated $0.08$ (nucleon) or $0.02$ (pion), or if the reconstructed gluon mass radius moves outside the lattice-QCD uncertainty band around $0.81\\,\\mathrm{fm}$, the claim that quark dressing alone reproduces the gluon mass distribution would be falsified.","supporting_citations":[{"cited_title":"Parton Decomposition of Nucleon Spin and Momentum: Gluons from Dressed Quarks","cited_arxiv_id":"2302.07473","evidence_quote":"Prior DSE-RL calculation that fixes the gluon-in-quark kernel parameters and introduces the $7\\%$/$27\\%$ Landau-gauge compensation factors used here."},{"cited_title":"Gluon PDF from Quark dressing in the Nucleon and Pion","cited_arxiv_id":"2103.05839","evidence_quote":"Supplies the gluon-in-quark dressed vertex and the $Q=0$ momentum-fraction formalism that this work generalizes to $Q>0$."}],"review_version":1}