{"id":"4ab3bdca-f695-48ce-a1e9-b82d17a87d07","arxiv_id":"2510.26425","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"A lattice QCD extraction of the nucleon gluon PDF using three lattice spacings, hybrid renormalization and NLO matching, extrapolated to the continuum and infinite-momentum limits, agrees with global fits while remaining consistent with zero.","lead":"Physicists used lattice QCD simulations to compute the gluon part of the proton's structure, then extrapolated away lattice and momentum artifacts to compare with collider-based fits. It is a methodological flagship test of first-principles gluon PDFs, but the current precision is too low to constrain the physics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (7) tail model controls the Fourier integral; fit-range-only systematics may miss model-form error.","rationale":"The reader's weakest assumption identifies the large-λ tail model of Eq. (7), and I agree that this is the most load-bearing concern about the central claim. The lattice data only constrain h_R(λ) up to λ ≈ 10–14, with the poorest signal at the largest λ, yet Eq. (14) integrates over all λ and the model in Eq. (7) is used to supply the unmeasured region. The paper's systematic estimate varies only the fit range (Table IV), never the functional form, so a model-form error would propagate directly into the extracted xg0(x) and could invalidate the claimed agreement with global fits. The abstract/body inconsistency over 'physical point' is a serious record issue, but it does not by itself change the numerical extraction; the tail model is a direct technical risk to the physical content of the result. I therefore recommend leaving the verdict CONDITIONAL: the concern is concrete and testable, and if the proposed test shows large sensitivity, the verdict should be reconsidered.","tokens_in":17188,"tokens_out":7025,"duration_ms":66635,"concrete_test":"Using the published renormalized h_R(λ) data, redo the λ-extrapolation with an alternative functional form, e.g., adding a second power-law term or fixing a1 to the value implied by the large-x endpoint behavior. After the same Fourier transform, matching, and joint extrapolation of Eq. (8), compare the resulting xg0(x) in 0.2 < x < 0.8 with the quoted statistical+systematic band. If the shift exceeds that band, the tail model is the dominant systematic and the agreement with global fits is not robust. As a quicker diagnostic, compute the tail contribution to the integral in Eq. (14) from λ beyond the largest fitted λ; if it exceeds ~20% of the total for x > 0.2, the unmeasured tail determines the result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the band in Figs. 3–4 is the physical light-cone gluon PDF depends on the large-λ tail model of Eq. (7), h_R(λ) = l1 λ^{-a1} e^{-λ/λ0}, because Eq. (14) integrates this model over all λ after the lattice signal ends at λ ≈ 10–14 (F32P30 stops at 1.5 GeV). The model is fitted per ensemble and momentum only over a restricted range (Table IV), and the quoted systematic uncertainty is estimated solely by moving that range; the functional form is never varied. The exponent a1 is described as 'associated with' the endpoint power law of the PDF but is not derived from it, so the tail is an ansatz. If the true asymptotic decay is slower or has a different prefactor, the Fourier integral in Eq. (14) can shift the central value xg0(x) across the whole x range, including x ∈ [0.2, 0.8] used for the comparison with CT18/NNPDF/JAM24. This is a correctness risk, not merely a presentation gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports a LaMET lattice QCD calculation of the unpolarized gluon PDF of the nucleon. Bare gluonic matrix elements are computed on three CLQCD ensembles (a = 0.105, 0.0897, 0.0775 fm; m_pi ~ 287-300 MeV) with distillation, renormalized in a hybrid scheme with parameters fitted to zero-momentum data, extended to large Ioffe time using a power-times-exponential tail, matched at NLO to the light-cone PDF, and then extrapolated to the continuum and infinite momentum via Eq. (8). The resulting xg(x)/<x> is compared to CT18, NNPDF, and JAM24 and reported as consistent within errors; the central value is consistent with zero at most x. The paper claims to be the most complete lattice gluon PDF determination to date.","tokens_in":17441,"tokens_out":11541,"duration_ms":104892,"significance":"If the result is correct, it would be one of the first continuum-extrapolated determinations of the nucleon gluon PDF from LaMET and would demonstrate that the large-x gluon is suppressed as in global fits. The work has clear strengths: multiple lattice spacings, high statistics, distillation, hybrid renormalization, NLO matching, and a breakdown of statistical and systematic uncertainties. The comparison with three global fits is a useful sanity check. However, the final uncertainty is very large, and the central value being consistent with zero means the agreement with CT18/NNPDF/JAM24 is a weak compatibility statement rather than a precision test. The main technical risk is the model dependence of the large-distance tail and the underconstrained continuum/infinite-momentum extrapolation; these need to be addressed before the central claim can be accepted.","major_comments":[{"comment":"The Fourier transform in Eq. (14) requires h_R(z,P_z) at all lambda, but the lattice data end at lambda ~ 10-14, and for F32P30 only P_z <= 1.5 GeV is used. The unmeasured tail is modeled by Eq. (7), h_R = l1 lambda^{-a1} e^{-lambda/lambda0}, fitted per ensemble and momentum. The systematic uncertainty is estimated only by moving the fit range (Table IV), never by changing the functional form. Since a1 is only 'associated with' the endpoint power law and not derived from it, Eq. (7) is an ansatz; if the true tail decays differently, the integral in Eq. (14) shifts xg(x) across the whole x range, including the comparison region 0.2<x<0.8. This is a load-bearing model assumption and needs a derivation, alternative-form scan, or an insensitivity demonstration.","section":"Light-cone PDF, Eq. (7), Eq. (14), Table IV"},{"comment":"Equation (8) contains four x-dependent functions (xg0, f, h, d) for the continuum and infinite-momentum extrapolation. Per x, the available data are at most three momenta for C24P29/E32P29 and two for F32P30, so the fit has very few degrees of freedom; the separation between a^2 f(x), a^2 P_z^2 h(x), and d(x)/P_z^2 is weakly determined. No chi^2 or stability test for Eq. (8) is reported (e.g., dropping the a^2 P_z^2 term, or fitting only the two largest momenta). The large uncertainty of the gray band in Figs. 3-4 is a direct consequence. To support the central claim, the extrapolation fit quality and its robustness to ansatz variations must be presented.","section":"Eq. (8), Table I, Fig. 3"},{"comment":"The title and abstract state the calculation is at the physical point with five lattice spacings, pion masses 136-317 MeV, and momenta up to 3 GeV. The body and Table I report three lattice spacings, m_pi ~ 287-300 MeV, and P_z up to 1.97 GeV. This is not a minor wording issue: the claimed parameter coverage does not match the actual data, and no chiral extrapolation is performed. The authors must correct the abstract/title to match the body, or provide the missing ensembles and analysis.","section":"Title, abstract, Table I"},{"comment":"The ensembles have m_pi = 287-300 MeV, yet the text states 'we do not include pion mass dependence in this work, as its effect is expected to be weak based on previous studies [31].' The available data do not test this assumption because the pion-mass spread is only ~13 MeV. For a paper titled 'at physical point,' this is a central limitation; either a dedicated pion-mass study or a clear caveat in the abstract and conclusions is required.","section":"Results, after Eq. (8)"}],"minor_comments":[{"comment":"Several typos and grammatical errors should be corrected: 'nulceon' in the abstract, 'in in the supplemental material' near Eq. (8), 'resloved' in the supplementary, and 'anaylsis' in Ref. [46].","section":"Throughout"},{"comment":"The caption says 'the width of the (nonoverlapping) colored bands denotes the size of each uncertainty,' but the bands in the figure appear to overlap. Please clarify how the widths are to be read.","section":"Fig. 12 caption"},{"comment":"The gray bands marking x<0.2 and x>0.8 are introduced without justification. A brief explanation of why LaMET is unreliable in these regions would help the reader.","section":"Fig. 4"},{"comment":"The claimed agreement with CT18/NNPDF/JAM24 is based on visual overlap. Given the very large final uncertainty, a quantitative measure (e.g., chi^2 per data point or a confidence level) would be more informative.","section":"Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious lattice calculation, but the central claim is sensitive to the unverified tail model and the underconstrained extrapolation. I would not reject on current evidence, but the revision must add a robustness scan for Eq. (7), fit-quality diagnostics for Eq. (8), and correct the abstract/title overstatements. The abstract/body discrepancy should also be checked by the editor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Abstract is not the paper. The abstract advertises five lattice spacings, pion masses down to 136 MeV, momenta up to 3 GeV, and a physical-pion extrapolation; the body reports three spacings at ~300 MeV, momenta up to 1.97 GeV, and explicitly says pion-mass dependence is not included. That mismatch must be fixed before anything else.\n\nWhat's genuinely new: the first LaMET gluon-PDF calculation that combines three lattice spacings with a multi-momentum infinite-momentum extrapolation. Hybrid renormalization with self-renormalization at long distance, NLO matching, and distillation are standard and competently executed. The authors are candid that the extrapolated PDF is consistent with zero with a slightly negative central value, which is an honest statement of where the precision is. They also present a breakdown of systematic errors from the λ-extrapolation, scale variation, z_s choice, and the joint continuum/momentum fit — more transparency than many lattice papers.\n\nSoft spots, roughly in order:\n\nThe tail model in Eq. (7), a power times exponential fitted over a finite range, is the load-bearing part of the Fourier transform. The paper only varies the fit range, never the functional form. If the true tail decays differently, the central value across all x can shift. This is the strongest technical concern, and it is not addressed.\n\nThe final precision is too low to discriminate between global fits. With the central value near zero, agreement with CT18/NNPDF/JAM24 carries little information, and normalizing by ⟨x⟩ is ill-conditioned when ⟨x⟩ is consistent with zero.\n\nThe a²P_z² term in Eq. (8) is barely constrained: the finest spacing contributes no points above 1.5 GeV, so the extrapolation leans on the coarser ensembles. That is a data limitation, not an error, but it weakens the continuum/infinite-momentum claim.\n\nNone of these is a demonstrated numerical error. The work is serious and honest in the body, and the body deserves a real referee. But the submitted version overstates what is delivered, and the result is too weak to confirm the large-x gluon suppression — it is only compatible with it.\n\nI'd send it to peer review, mainly to get the abstract corrected and to push for a test of tail-model form uncertainty. Reading-group material, but not yet a citable number.","headline":"First LaMET gluon-PDF extraction with three lattice spacings plus a momentum extrapolation, but the abstract overclaims (five spacings, physical pion mass, 3 GeV are not in the body), and the final result is 'consistent with zero,' so agreement with global fits is weak.","tokens_in":18091,"tokens_out":4391,"would_cite":false,"duration_ms":38981,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81V05","81T25"],"pacs":["12.38.Gc","11.15.Ha"],"model":"deepseek-v4-flash","headline":"Lattice QCD, extrapolated to the continuum and infinite-momentum limits, reproduces the nucleon's unpolarized gluon PDF within global-fit uncertainty.","keywords":["gluon PDF","lattice QCD","LaMET","large-momentum effective theory","nucleon structure","hybrid renormalization","continuum limit","infinite momentum limit"],"falsifier":"Compute the renormalized matrix elements at higher nucleon momentum (P_z ≳ 3 GeV) on a finer lattice and check whether the large-λ data deviate significantly from the fitted power-times-exponential tail used here. Alternatively, independently compute the gluon momentum fraction <x>_g and test whether ∫ dx xg0(x) from the extrapolated curve is consistent with it within the combined uncertainty; a violation would falsify the extrapolated PDF.","tokens_in":16929,"feed_emoji":"⚛️","tokens_out":16023,"duration_ms":131830,"temperature":0.7,"pith_summary":"The paper reports a lattice QCD calculation of the nucleon's unpolarized gluon PDF using large-momentum effective theory (LaMET). It computes gluonic quasi-PDF matrix elements on three lattice spacings with pion mass around 300 MeV and nucleon momenta up to about 2 GeV, then applies hybrid renormalization, a large-distance tail extrapolation, one-loop perturbative matching, and a joint extrapolation to the continuum and infinite-momentum limits. The central claim is that the extrapolated xg(x) is the physical light-cone gluon PDF, and that it agrees with the global fits within the quoted uncertainty. The authors also find that the gluon distribution is suppressed at large x. If correct, this would be a first-principles confirmation of the phenomenological gluon distribution and of the LaMET machinery for gluonic operators.","feed_headline":"Lattice QCD gluon PDF agrees with global fits after extrapolation","feed_subtitle":"Extrapolated to zero spacing and infinite momentum, lattice QCD now matches global fits.","key_machinery":"The argument rests on a specific chain of techniques. First, a gluonic operator O(z) that is multiplicatively renormalizable, computed with distillation to improve the signal. Second, hybrid renormalization with a self-renormalization factor Z_R that removes linear and logarithmic ultraviolet divergences. Third, the large-λ extrapolation h_R(λ)=l1 λ^{-a1} e^{-λ/λ0}, used to extend the renormalized matrix elements from the measured range out to all distances before the Fourier transform. Fourth, the NLO hybrid-scheme matching kernel that converts the quasi-PDF to the light-cone PDF. Fifth, the joint continuum/infinite-momentum extrapolation ansatz of Eq. (8), whose intercept xg0(x) is the fin","core_discovery":"On its own terms, the paper establishes that the nucleon's unpolarized gluon PDF can be obtained from lattice QCD through the LaMET procedure. The renormalized quasi-PDF matrix elements are Fourier transformed to momentum space using a model-based tail extension, matched at one loop to the light-cone PDF, and finally extrapolated via xg(x,P_z,a) = xg0(x) + a^2 f(x) + a^2 P_z^2 h(x) + d(x)/P_z^2 to the continuum and infinite-momentum limits. The resulting xg0(x)/<x> is consistent with the current global fits within errors, with the large-x region close to zero (and slightly negative central values), indicating suppression of the gluon at high momentum fraction. The paper explicitly states tha","pith_inferences":["Editorial inference: The paper's title says 'physical point,' but the body uses a pion mass of roughly 300 MeV, not the physical 136 MeV; a chiral extrapolation over pion mass is not performed here.","Editorial inference: The large-λ tail form is only constrained out to λ ≈ 10–14; the systematic error varies the fitting range but not the functional form, so a different tail shape would propagate directly into the small- and mid-x PDF.","Editorial inference: The slightly negative central values in the extrapolated band are unphysical; a stronger test would check the momentum sum rule against an independent lattice determination of the gluon momentum fraction.","Editorial inference: The continuum and infinite-momentum extrapolation uses three lattice spacings at one pion mass; adding finer spacings and higher momenta would test the stability of the a^2 and d(x)/P_z^2 terms in Eq. (8)."],"forward_implications":["If the central claim holds, lattice QCD can supply direct, model-independent input for the gluon PDF in regions where global fits are weakly constrained, especially at large x.","The consistency with global fits suggests that the LaMET machinery—hybrid renormalization, NLO matching, and extrapolation—is under control for gluonic operators, opening the way to other gluonic observables such as the gluon momentum fraction.","The observed large-x suppression of the gluon distribution is a quantitative confirmation of a known qualitative feature of the nucleon's gluonic structure.","The uncertainty budget identifies the combined continuum/infinite-momentum extrapolation and the tail model as the dominant systematics, indicating where future calculations (finer lattices, higher momenta, more spacings) will pay off."],"fun_headline_variants":["Lattice QCD gluon PDF now matches global fits","Gluon PDF from lattice QCD in continuum limit","Lattice QCD reproduces gluon PDF after extrapolation","A first: gluon PDF from lattice QCD at physical point","Nucleon gluon PDF: lattice QCD meets global fits"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the true large-distance (large λ = zP_z) behavior of the renormalized gluon quasi-PDF has the functional form l1 λ^{-a1} e^{-λ/λ0}, because the Fourier transform integrates this model over all λ while the lattice data only constrain it out to λ ≈ 10–14.","fun_headline_variants_meta":{"raw":{"variants":["Lattice QCD gluon PDF now matches global fits","Gluon PDF from lattice QCD in continuum limit","Lattice QCD reproduces gluon PDF after extrapolation","A first: gluon PDF from lattice QCD at physical point","Nucleon gluon PDF: lattice QCD meets global fits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1157,"prompt_tokens":694,"completion_tokens":463,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":377}},"tokens_in":438,"tokens_out":463,"duration_ms":4253,"temperature":1.0,"reasoning_tokens":377,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:11:38.227048+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the renormalized matrix elements at higher nucleon momentum (P_z ≳ 3 GeV) on a finer lattice and check whether the large-λ data deviate significantly from the fitted power-times-exponential tail used here. Alternatively, independently compute the gluon momentum fraction <x>_g and test whether ∫ dx xg0(x) from the extrapolated curve is consistent with it within the combined uncertainty; a violation would falsify the extrapolated PDF.","supporting_citations":[],"review_version":1}