{"id":"f488630e-6e4a-4f51-a9cb-16248c928c7b","arxiv_id":"2505.06726","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Using a gauge-invariant chiral quark model with DGLAP evolution, the author predicts pion and kaon gluon distributions and claims lattice QCD agreement, but the model's momentum fractions exceed one.","lead":"This paper computes gluon distributions inside pions and kaons by starting from quark distributions in a chiral quark model and evolving them with QCD equations. It claims agreement with recent lattice data, but the reported momentum fractions add up to more than the total momentum, which undermines the result.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reported pion/kaon valence plus gluon momentum fractions at Q = 5.2 GeV sum to ~1.03–1.04 before sea quarks are included, violating the DGLAP momentum sum rule and undermining the central comparisons.","rationale":"The reader's verdict is REJECT, and the stress-test identifies the same load-bearing internal inconsistency: the reported valence and gluon momentum fractions at Q = 5.2 GeV sum to more than 1 before sea quarks are included. This violates the momentum sum rule that any QCD-evolved parton set must satisfy. The reader's weakest_assumption focuses on the no-intrinsic-gluon/sea input and the fitted value of Q0; while that circularity is a genuine concern, the momentum sum rule violation is more definitive because it does not depend on assumptions about the initial scale. If the DGLAP evolution conserves momentum, the total momentum fraction at Q = 5.2 must be unity; since the reported valence and gluon contributions alone already exceed unity, the evolution is internally inconsistent. The same evolution is used to produce the Q = 2 GeV gluon distributions that are compared with lattice QCD and JAM, so the central claim loses its support. The proposed concrete test uses an independent, standard DGLAP code to check whether the sum rule is preserved; such a check would settle whether the inconsistency is in the reported tables, in the parametrizations, or in the evolution code itself. Given the clear sum-rule violation, the reader's rejection remains appropriate, and the verdict is UNCHANGED.","tokens_in":14638,"tokens_out":5434,"duration_ms":53892,"concrete_test":"Rerun the NLO DGLAP evolution from Q0 = 0.42 GeV using a standard public code (e.g., HOPPET) with the same valence input, and compute the first moments of the valence, sea, and gluon distributions at Q = 2 GeV and Q = 5.2 GeV. If ∫dx x[q_S(x)+g(x)] is not 1, or equivalently if valence+gluon alone exceeds 1 with a positive sea, the paper's evolution is internally inconsistent; if it is exactly 1, then the tabulated moments in Tables I and II are not mutually consistent and the parametrizations in Eqs. (21)-(35) should be rechecked against each other.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III, Table I reports the pion up-valence first moment ⟨x⟩ = 0.21, which the text multiplies by 2 to give a total valence momentum fraction of 0.41, and Table I also reports the gluon first moment ⟨x⟩_g = 0.62. The sum 0.41 + 0.62 = 1.03 already exceeds 1. The same happens for the kaon in Table II: up 0.19 + antistrange 0.23 = 0.42, plus gluon 0.60, summing to 1.02. Because NLO DGLAP evolution from a valence-only input necessarily generates a positive sea-quark distribution, the total momentum fraction ∫dx x[q_S(x)+g(x)] must be exactly 1 at every scale. The reported numbers leave no room for the sea and therefore violate the momentum sum rule. This is not a matter of convention or of an extra fitted parameter: it indicates that the coupled singlet/gluon evolution is not implemented consistently, or that the valence and gluon distributions are not mutually consistent. Since the same evolution is used to produce the Q = 2 GeV gluon distributions that are the paper's main agreement claims, the inconsistency casts doubt on those results as well. Additionally, the initial scale Q0 = 0.42 GeV is fixed by fitting to the pion valence data used later for validation ('it was found that Q0 = 0.42 GeV fits the experimental data'), so the valence agreement at Q = 5.2 is partly circular; the paper does not show an independent determination of Q0.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes pion and kaon valence-quark distributions in a gauge-invariant nonlocal chiral quark model with momentum-dependent effective quark mass, then evolves them from an initial scale Q0 = 0.42 GeV by NLO DGLAP evolution. The gluon distributions are generated dynamically by the evolution's singlet/gluon splitting functions. The author compares the pion gluon distribution at Q = 2 GeV with lattice QCD and JAM global analysis results, the kaon gluon distribution at Q = 2 GeV with lattice QCD, and the pion up-valence distribution at Q = 5.2 GeV with the Aicher et al. reanalysis data. The central claim is that the model reproduces the gluonic structure of pseudoscalar mesons and the reanalysis valence data.","tokens_in":14975,"tokens_out":5256,"duration_ms":53358,"significance":"If the central claims were correct, the paper would offer a single-model description of pion and kaon gluon and valence distributions at scales relevant to future EIC, EicC, and COMPASS++/AMBER experiments, with explicit parameterizations that could be used in other analyses. The construction of the gauge-invariant nonlocal model and the derivation of the PDF expression are nontrivial and, in principle, valuable. However, the numerical results fail a basic consistency check: the reported momentum fractions at Q = 5.2 GeV violate the DGLAP momentum sum rule before sea quarks are even included. The same evolution is used for the Q = 2 GeV comparisons that form the paper's main agreement claims, so these results cannot be considered trustworthy. In addition, the initial scale Q0 is fit to the same experimental data that is later used as validation, making the valence agreement partly circular.","major_comments":[{"comment":"The reported momentum fractions at Q = 5.2 GeV violate the momentum sum rule. For the pion, Table I gives 2 <x>_u = 0.41 for the valence quarks and <x>_g = 0.62 for the gluon, which already sum to 1.03. For the kaon, Table II gives 0.19 + 0.23 = 0.42 for valence and <x>_g = 0.60, summing to 1.02. Because the initial scale Q0 = 0.42 GeV is assumed to contain only valence quarks with no gluon or sea, NLO DGLAP evolution must generate a positive sea-quark distribution at higher scales, and the total momentum fraction (valence + sea + gluon) is exactly 1 at every scale for a momentum-conserving evolution. The reported numbers leave no room for the sea and therefore indicate that the DGLAP implementation or the input normalization is not self-consistent. Since the same evolution produces the Q = 2 GeV gluon distributions that are compared with lattice QCD in Figure 3, the central agreement claims are not supported.","section":"Section III, Tables I and II"},{"comment":"The initial scale Q0 = 0.42 GeV is not determined independently. The text states, with reference to Ref. [17], that 'it was found that Q0 = 0.42 GeV fits the experimental data.' The same experimental data, specifically the Aicher et al. reanalysis, are then used as the validation target at Q = 5.2 GeV in Figure 1. Consequently, the claimed agreement with the reanalysis valence distribution is partly a consequence of fitting Q0, not an independent prediction. The paper should either determine Q0 from other observables or explicitly present the valence comparison as a post-fit reproduction rather than as validation.","section":"Section III, Q0 determination"},{"comment":"There is an unexplained inconsistency in the local/nonlocal decomposition of the gluon moments. In Table I, the total pion gluon first moment is <x>_g = 0.62, while the local and nonlocal contributions are listed as 0.52 and 0.34, respectively, whose sum 0.86 exceeds the total. Similarly, Table II gives the kaon gluon total as 0.60, with local 0.50 and nonlocal 0.34, summing to 0.84. The text describes these as the total, local, and nonlocal contributions to the same quantity. If the total is not the sum of the local and nonlocal parts, the definition of these contributions should be stated; otherwise this indicates a numerical error in the reported gluon moments.","section":"Tables I and II, gluon rows"}],"minor_comments":[{"comment":"The manuscript contains many typographical errors and garbled equations, which make it difficult to verify the derivation. Examples include 'NC χQM' in Section III (likely 'NLχQM'), the repeated '⟨x^n⟩π NL' labels in Table II for kaon rows, and unclear subscripts such as 'D2a' and 'D2b' in Eq. (10). The equations should be carefully re-typeset.","section":"Throughout"},{"comment":"The notation for the momentum-dependent mass and the nonlocal derivative terms is hard to follow because half-arrows and square-root symbols are missing or misplaced in the rendered text. The authors should provide a cleaner presentation of the vertex factors and the nonlocal contributions.","section":"Section II, Eqs. (6) and (8)"},{"comment":"The comparison with lattice QCD and JAM is purely visual; no uncertainty bands for the model curves are provided, despite the model having several parameters (M0, mu, current quark masses). Propagating these uncertainties would strengthen the claim of agreement.","section":"Section III, Figure 3"}],"recommendation":"reject","confidential_remarks":"The momentum sum rule violation at Q = 5.2 GeV is a load-bearing internal inconsistency that invalidates the main numerical comparisons. The circular determination of Q0 further weakens the valence claim. These issues cannot be fixed with local edits; the evolution code and the numerical input need to be redone, and the conclusions may change substantially. I would encourage the author to address these points and resubmit a corrected manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line up front: this is a direct follow-up to the author's PRD 2024 paper (Ref. [36]) that reuses the same NLχQM machinery, and the new lattice comparisons are undercut by an internal momentum-sum inconsistency. Read before citing.\n\nWhat is actually new: the Q=2 GeV comparisons to recent lattice QCD for the pion and kaon gluon distributions, including the two pion-mass settings, the local/nonlocal breakdown, and analytic parametrizations. The observation that BSE-NJL matches the older Conway data while the momentum-dependent NLχQM matches the Aicher–Schäfer–Vogelsang reanalysis is an interesting comment on the large-x puzzle, even if it is not a proof.\n\nThe soft spot is load-bearing. Table I reports the pion valence first moment 2⟨x⟩ = 0.41 and gluon first moment 0.62 at Q = 5.2 GeV. These already sum to 1.03, with no room for the positive sea-quark distribution that NLO DGLAP evolution must generate. Table II does the same for the kaon: 0.19 + 0.23 + 0.60 = 1.02. Any valid PDF set has total momentum fraction exactly 1 at every scale. This is not a convention or a parameter; it signals that the singlet/gluon evolution or the input normalization is not self-consistent. The same evolution chain produces the Q=2 gluon distributions that are the paper's main agreement claim, so that claim is not credible as presented. The reported gluon fraction of 0.62 at 5.2 GeV is also higher than typical phenomenological values, which is consistent with the problem.\n\nThe Q0 circularity is real but secondary. Q0 = 0.42 GeV is tuned to reproduce the Aicher et al. pion valence data, and the same data are later shown as validation. That is a fit, not an independent prediction. It would be acceptable if acknowledged, but the valence comparison gives no independent confirmation.\n\nThe novelty statement is too weak. Refs. [17] and [36] already present the model, the PDF formulas, and the DGLAP-generation method. The author needs to state explicitly what this paper adds. As written, it looks like an incremental update rather than a new result.\n\nWho should read it: hadron-structure phenomenologists and lattice practitioners comparing meson gluon distributions. The paper deserves a serious referee, but the referee should require a fix of the momentum-sum problem and a clear novelty statement. I would not cite the numerical results until that is done.","headline":"Follow-up NLχQM meson-PDF paper whose momentum-sum-rule violation undermines the lattice comparison; useful model details but needs major revision before the numbers can be trusted.","tokens_in":15547,"tokens_out":4096,"would_cite":false,"duration_ms":40450,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A gauge-invariant chiral quark model with momentum-dependent quark mass predicts pion and kaon gluon distributions that agree with recent lattice QCD.","keywords":["pion gluon distribution","kaon gluon distribution","nonlocal chiral quark model","momentum-dependent quark mass","DGLAP evolution","valence quark distribution","large-x behavior","lattice QCD"],"falsifier":"A future measurement of the pion gluon distribution at $Q=2$ GeV whose $x$-shape disagrees with the model's prediction, or a lattice calculation showing a nonzero gluon or sea distribution at $Q_0=0.42$ GeV, would show that the valence-only initial condition cannot carry the argument.","tokens_in":14389,"feed_emoji":"⚛️","tokens_out":8154,"duration_ms":75231,"temperature":0.7,"pith_summary":"This paper tries to show that the gluon content of the pion and kaon can be obtained from a gauge-invariant chiral quark model in which the effective quark mass depends on quark momentum, starting from a valence-quark-only picture at a low scale. The gluon distributions are not put in by hand; they are produced dynamically by next-to-leading-order DGLAP evolution from $Q_0=0.42$ GeV up to the comparison scales. The author reports that the pion gluon distribution at $Q=2$ GeV agrees with recent lattice QCD results and with a recent global QCD analysis, that the kaon gluon distribution is consistent with kaon lattice QCD results, and that the evolved pion up-valence distribution at $Q=5.2$ GeV reproduces the reanalysis of the muon-pair production data. If these comparisons hold, the model offers a single, parameter-light route to the quark and gluon structure of light pseudoscalar mesons and a way to understand the long-standing disagreement over the large-$x$ behavior of the pion.","feed_headline":"Chiral quark model generates meson gluons matching lattice QCD","feed_subtitle":"From valence quarks only at 0.42 GeV, DGLAP evolution produces gluon distributions lattice QCD agrees with.","key_machinery":"The central object is the gauge-invariant nonlocal chiral quark model (NL$\\chi$QM), defined by an effective chiral action with a momentum-dependent quark mass. The nonlocal mass function is $M_f = M_0[\\mu^2/(k^2-\\mu^2+i\\epsilon)]^2$, and taking a three-point functional derivative of the action with respect to two meson fields and one gauge field produces the twist-2 parton distribution expression. The derivative terms involving $\\sqrt{M_f}$ with respect to the gauge field generate the nonlocal contributions that a momentum-independent model lacks. The resulting valence distributions at $Q_0=0.42$ GeV are then evolved with the next-to-leading-order DGLAP equations, whose $P_{qg}$ and $P_{gg}$ splitting functions create the gluon distributions dynamically.","core_discovery":"Stated on the paper's own terms: in the gauge-invariant nonlocal chiral quark model (NL$\\chi$QM), the nonlocal interaction terms that arise when the quark mass depends on momentum are not small corrections, because they are essential for matching the reanalysis of the pion Drell-Yan data at $Q=5.2$ GeV. Starting from valence distributions only at $Q_0=0.42$ GeV and evolving with the NLO DGLAP splitting functions, the model generates gluon distributions whose shapes at $Q=2$ GeV agree with lattice QCD results for the pion and are consistent with those for the kaon. The author also reports that the gluon carries about 62 percent of the pion momentum at $Q=5.2$ GeV, and that the interplay between local and nonlocal terms produces a large-$x$ power behavior that differs from momentum-independent models, which the paper suggests may explain the puzzle of conflicting pion data sets.","pith_inferences":["If the method generalizes, the same valence-only initial condition followed by NLO DGLAP evolution could be applied to other pseudoscalar mesons to produce gluon distributions before dedicated data exist.","The sharp contrast between momentum-dependent and momentum-independent models at large $x$ suggests that precise future Drell-Yan data at $x \\gtrsim 0.6$ could discriminate between the two pictures more cleanly than current data.","A testable consequence of the valence-only initial scale is a specific sea-quark distribution at higher $Q$; future measurements of the pion sea would check this indirect prediction."],"forward_implications":["The gluon distribution of the pion at $Q=2$ GeV can be predicted from valence quarks alone, so nonzero intrinsic gluons at the model scale are not needed to match current lattice data.","The same valence-only initial condition generates a kaon gluon distribution consistent with lattice results, so the mechanism extends from the pion to its heavier strange partner.","At $Q=5.2$ GeV the evolved pion up-valence distribution matches the reanalysis data while differing from the older data set, giving a concrete target for future pion Drell-Yan measurements to settle the large-$x$ conflict.","The pion's gluon carries about 62 percent of its momentum at $Q=5.2$ GeV, a number that can be tested against future lattice or experimental determinations.","The provided parameterizations of the gluon and valence distributions can be used directly in other calculations of meson structure observables."],"supporting_citations":[{"why":"Supplies the pion gluon distribution from lattice QCD that the $Q=2$ GeV comparison is made against.","marker":"[29]"},{"why":"Supplies the kaon gluon distribution from lattice QCD used to test the kaon prediction at $Q=2$ GeV.","marker":"[30]"},{"why":"Provides the reanalysis of the pion Drell-Yan data that the evolved up-valence distribution at $Q=5.2$ GeV is fitted to.","marker":"[43]"},{"why":"Establishes that the initial scale $Q_0=0.42$ GeV fits the experimental data for the evolved valence distribution.","marker":"[17]"},{"why":"Gives the updated gauge-invariant nonlocal chiral quark model expression for pion and kaon gluon distributions that this work extends.","marker":"[36]"},{"why":"Provides the numerical DGLAP evolution solver used to evolve the PDFs and generate the gluon distributions.","marker":"[35]"},{"why":"Provides the global QCD analysis result with which the pion gluon distribution at $Q=2$ GeV is compared.","marker":"[33]"},{"why":"Gives the momentum-independent BSE-NJL model result used as the comparison that differs in large-$x$ power behavior.","marker":"[19]"},{"why":"Provides the original muon-pair production data set for the pion, the older data set that the momentum-independent model fits.","marker":"[3]"}],"fun_headline_variants":["Nonlocal chiral quark model's gluons match lattice QCD","Chiral model reproduces pion and kaon gluon distributions","Gauge-invariant quark model fits lattice meson gluons","Model gluons from valence quarks agree with lattice data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that at the initial scale $Q_0=0.42$ GeV the pion and kaon contain only valence quarks, with no intrinsic gluons or sea quarks, so every predicted gluon distribution is a pure product of DGLAP evolution from that fitted input.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal chiral quark model's gluons match lattice QCD","Chiral model reproduces pion and kaon gluon distributions","Gauge-invariant quark model fits lattice meson gluons","Model gluons from valence quarks agree with lattice data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1579,"prompt_tokens":933,"completion_tokens":646,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":576}},"tokens_in":549,"tokens_out":646,"duration_ms":6893,"temperature":1.0,"reasoning_tokens":576,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:35:33.466840+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A future measurement of the pion gluon distribution at $Q=2$ GeV whose $x$-shape disagrees with the model's prediction, or a lattice calculation showing a nonzero gluon or sea distribution at $Q_0=0.42$ GeV, would show that the valence-only initial condition cannot carry the argument.","supporting_citations":[{"cited_title":"First glimpse into the kaon gluon parton distribution using lattice QCD","cited_arxiv_id":null,"evidence_quote":"Supplies the kaon gluon distribution from lattice QCD used to test the kaon prediction at $Q=2$ GeV."},{"cited_title":"Gluon parton distribution of the pion from lattice QCD","cited_arxiv_id":null,"evidence_quote":"Supplies the pion gluon distribution from lattice QCD that the $Q=2$ GeV comparison is made against."},{"cited_title":"Soft-gluon resummation and the valence parton distribution function of the pion","cited_arxiv_id":null,"evidence_quote":"Provides the reanalysis of the pion Drell-Yan data that the evolved up-valence distribution at $Q=5.2$ GeV is fitted to."},{"cited_title":"Parton-distribution functions for the pion and kaon in the gauge-invariant nonlocal chiral-quark model","cited_arxiv_id":null,"evidence_quote":"Establishes that the initial scale $Q_0=0.42$ GeV fits the experimental data for the evolved valence distribution."},{"cited_title":"Updated analyses of gluon distribution functions for the pion and kaon from the gauge-invariant nonlocal chiral quark model","cited_arxiv_id":null,"evidence_quote":"Gives the updated gauge-invariant nonlocal chiral quark model expression for pion and kaon gluon distributions that this work extends."},{"cited_title":"Numerical solution of Q2 evolution equations in a brute force method","cited_arxiv_id":null,"evidence_quote":"Provides the numerical DGLAP evolution solver used to evolve the PDFs and generate the gluon distributions."},{"cited_title":"Global QCD Analysis of Pion Parton Distributions with Threshold Resummation","cited_arxiv_id":null,"evidence_quote":"Provides the global QCD analysis result with which the pion gluon distribution at $Q=2$ GeV is compared."},{"cited_title":"Gluon and valence quark distributions for the pion and kaon in nuclear matter","cited_arxiv_id":null,"evidence_quote":"Gives the momentum-independent BSE-NJL model result used as the comparison that differs in large-$x$ power behavior."},{"cited_title":"Experimental Study of Muon Pairs Produced by 252-GeV Pions on Tungsten","cited_arxiv_id":null,"evidence_quote":"Provides the original muon-pair production data set for the pion, the older data set that the momentum-independent model fits."}],"review_version":1}