{"id":"a90faa24-3586-42e5-aec0-42a71f878b70","arxiv_id":"2511.22947","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Pion valence PDFs and vector, tensor, and scalar form factors computed from NJL-model GPDs match experiment and lattice QCD after evolution, giving charge radii 0.56, 0.63, and 0.83 fm.","lead":"This paper computes pion generalized parton distributions in the covariant Nambu–Jona-Lasinio model and derives parton distributions, generalized form factors, and charge radii from them. The resulting pion PDFs and dressed form factors are compared with E615 data, JAM analyses, and lattice QCD, with radii r_S=0.56 fm, r_V=0.63 fm, r_T=0.83 fm.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PDF 'excellent agreement' is conditional on the untested DGLAP input scale: μ0²=0.18 GeV² is not fixed by mπ/fπ, so the E615/JAM comparison may reflect tuning rather than prediction.","rationale":"The paper is a serious NJL calculation, and the PDF comparison with E615/JAM is the most concrete quantitative claim. In good faith, the model does have parameters fixed by physical inputs, and the GPD framework is coherent. However, the claim 'this result validates our approach' (Sec. 4.2) is only valid if the agreement survives variation of the model scale and regulator. The current manuscript does not demonstrate this: μ0²=0.18 GeV² is introduced without justification, no alternative regulators are tested, and the comparison is entirely post-evolution. The reader's weakest assumption identifies exactly this point, and my independent reading agrees. The scalar-form-factor mismatch is acknowledged and, while a limitation, does not by itself overturn the central PDF claim. Because the concern is about underdetermination rather than internal inconsistency, a conditional verdict remains appropriate; no change from the reader's conditional verdict is needed.","tokens_in":12011,"tokens_out":6992,"duration_ms":81088,"concrete_test":"Vary the DGLAP initial scale over μ0² = 0.10, 0.18, 0.30 GeV² (and, if feasible, replace the proper-time regulator with Pauli-Villars), keeping mπ, fπ, Mu, and Λ_IR fixed, and recompute the evolved x u_v(x) at μ²=4 and 27 GeV². Compare to E615 and JAM with a binned χ² or KS statistic. If the agreement persists across this range with a shallow χ² valley near 0.18, the PDF claim is robust; if the agreement degrades sharply, the 'excellent agreement' is an artifact of the chosen input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that the pion PDFs derived from the NJL GPDs agree 'excellent' with E615 and JAM (Sec. 4.2, Fig. 4)—rests on an input that is not fixed by the model. The NJL parameters Mu=400 MeV, Gπ, and Λ_UV are fixed by mπ and fπ, but the DGLAP initial scale μ0²=0.18 GeV² is chosen ad hoc, and the model-scale PDF is assumed to be valence-only with H_u(x,0,0)=1 (or at least no sea). Because the comparison is made only after DGLAP evolution from 0.18 to 4 and 27 GeV², the final shape is largely controlled by μ0 and by the regulator that defines the input PDF. The paper reports no sensitivity study, no χ², and no error bands, so 'excellent agreement' cannot be distinguished from a tuned input. The scalar-form-factor discrepancy (bottom center of Fig. 5) is acknowledged and is less central; the PDF comparison is the load-bearing evidence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates pion generalized parton distributions (GPDs) at zero skewness in the covariant Nambu–Jona-Lasinio (NJL) model with proper-time regularization. Starting from explicit one-loop expressions for the vector, tensor, and twist-3 scalar pion GPDs, the authors extract the valence pion PDF from the forward limit and the scalar, vector, and tensor generalized form factors from the relevant Mellin moments. The PDFs are evolved via DGLAP from a model scale μ0²=0.18 GeV² to μ²=27 and 4 GeV² and compared with E615 data and the JAM global analysis. The form factors are compared with lattice QCD and experimental data, with 'dressed' results obtained by inserting an external quark form factor F1_Qbar(Q²) from Ref. 12. The authors report excellent agreement for the PDFs and dressed vector/tensor form factors, and obtain charge radii r_Sπ=0.56 fm, r_Vπ=0.63 fm, r_Tπ=0.83 fm.","tokens_in":1399,"tokens_out":1481,"duration_ms":49929,"significance":"If robust, the paper provides a transparent, relatively simple model calculation of all three pion GPDs at zero skewness and shows that the resulting PDFs can match phenomenological extractions after QCD evolution. The explicit formulas and the use of a small number of parameters are strengths. However, the central 'excellent agreement' claim is not quantified and rests on an initial evolution scale that is not fixed by the model, and the dressed form-factor agreement depends on an external fitted input. The paper would be a useful contribution if these dependencies were assessed and reported; in its current form the predictive content is difficult to evaluate.","major_comments":[{"comment":"The PDF comparison is the load-bearing quantitative claim, but it is only asserted as 'excellent agreement' without a chi-square or similar measure. More importantly, the comparison is made after DGLAP evolution from the initial scale μ0²=0.18 GeV², which is not determined by the model inputs (M_q, Λ_IR, G_π, Λ_UV are fixed by m_π and f_π). The model-scale PDF is also assumed valence-only. Since the final shape after evolution depends strongly on μ0² and on the regulator, the reader cannot distinguish a prediction from a tuned input. Please provide a sensitivity scan over μ0² (e.g., 0.1–0.3 GeV²) and the proper-time scales, a quantitative measure of agreement, and a discussion of the valence-only assumption.","section":"§4.2, Fig. 4"},{"comment":"The dressed vector and tensor form factors use the external quark form factor F1_Qbar(Q²) from Ref. 12, which itself was fitted to data. The good agreement of the dressed form factors with lattice/experimental data is therefore not an independent prediction of this work. This should be stated more prominently, and the sensitivity of the dressed results to the parameterization of F1_Qbar should be quantified. The scalar case, where the dressed form factor underestimates lattice data, deserves a brief discussion of what this implies for the twist-3 GPD calculation or the dressing prescription.","section":"§4.3, Fig. 5"},{"comment":"The text states that H^u(x,0,0) 'equals unity' in the valence region x∈[0,1], but Eq. (15) identifies H^u(x,0,0)=u_v^π(x), which is not a constant and must integrate to 1. Please correct this inconsistent statement.","section":"§4.1, Eq. (15)"}],"minor_comments":[{"comment":"The statement of the DGLAP evolution should specify the order (LO/NLO), the number of active flavors, and how gluons and sea quarks are initialized at μ0². Reference 21 is a code description; the actual settings used here are not given.","section":"General"},{"comment":"The definition of the scalar GPD appears garbled: 'Mu/P + H^q_S' likely should be something like (M_u/P^+) H_S^q(x,ξ,t). Please clarify the notation.","section":"Eq. (10c)"},{"comment":"Several exponents are not typeset correctly (e.g., 'β1(1−β1−β t' appears missing a closing parenthesis and possibly a factor). Please proofread the formulas carefully.","section":"Eqs. (22)–(24)"},{"comment":"A parenthesis appears missing in the Lagrangian: after '(ψ̄_q ψ_q)^2' the bracket should be closed; also 'h i' typesetting artifacts should be fixed.","section":"Sec. 2, Eq. (1)"},{"comment":"In the description of the scalar form factor, 'at intermediate values of x' should presumably be 'at intermediate values of Q²'. Please correct.","section":"Sec. 4.3"},{"comment":"The panels are on different vertical scales; it would help to include a small inset or a common scale for at least the vector and tensor panels to facilitate visual comparison.","section":"Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The core issue is whether the PDF agreement is a genuine model prediction or an artifact of choosing μ0²=0.18 GeV². A sensitivity study and a quantitative goodness-of-fit measure are essential before the claim can be accepted. The dressed form-factor comparison is weakened by its reliance on the externally fitted F1_Qbar of Ref. 12. The paper is not fatally flawed, but it needs a substantive revision to establish the robustness of its central claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the two things to know: the paper offers the first NJL evaluation of tensor and scalar pion GPDs at zero skewness with proper-time regularization, and the resulting charge radii are new numbers; but the headline claim of 'excellent agreement' for the PDFs rests on an evolution scale that is an input, not a prediction, and the dressed-form-factor comparisons lean on the same group's earlier fitted quark form factor.\n\nWhat's good: The model parameters are fixed by m_pi and f_pi, so the forward-limit PDF at the model scale is a genuine model output. The valence-only PDF, after DGLAP evolution from 0.18 to 27 and 4 GeV^2, does look close to E615 and JAM. That is a real success, and it uses a single fixed mu0^2 for both scales. The tensor and scalar GPD shapes as functions of t are documented, and the ordering r_T > r_V > r_S is consistent with lattice. The scalar discrepancy is acknowledged, which is honest.\n\nSoft spots, in proportion: The stress-test worry is real. mu0^2 = 0.18 GeV^2 is not fixed by the NJL model; the paper gives no sensitivity study, no chi^2, and no error bands. 'Excellent agreement' is a qualitative judgment. Without a scan over mu0 and the regulator, the PDF result could be tuned. The dressed form factors use F1_Qbar from Ref. 12, a previous paper by the same authors, which was fit to data, so those curves are not independent predictions. The scalar form factor misses lattice; the authors flag it but don't explain why.\n\nThe paper ships no code and no data files, so the DGLAP evolution step is not independently checkable from the text. That is a minor issue but relevant given the claim.\n\nWho it's for: people doing chiral quark model phenomenology and planning pion structure measurements at EIC/EicC. The paper is a legitimate extension of an established framework, not a breakthrough. It deserves a serious referee: send it to review, but ask for (a) a mu0-sensitivity study, (b) quantified agreement with data, and (c) a clear statement of what is predicted versus fitted.","headline":"Solid but conditional NJL calculation of pion GPDs; the PDF 'excellent agreement' is undermined by an ad hoc DGLAP initial scale.","tokens_in":12823,"tokens_out":3031,"would_cite":true,"duration_ms":29052,"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 single model reproduces pion quark distributions, form factors, and charge radii from one unified object: the pion GPD at zero skewness.","keywords":["pion","generalized parton distributions","Nambu-Jona-Lasinio model","proper-time regularization","parton distribution functions","generalized form factors","charge radii","DGLAP evolution"],"falsifier":"Compute or measure the pion valence PDF at a third scale, for example mu^2 = 10 GeV^2, after DGLAP evolution from mu0^2 = 0.18 GeV^2; if it disagrees with new data by more than the experimental errors, the central claim collapses. Alternatively, a lattice or experimental determination of the tensor charge radius below about 0.7 fm would contradict the predicted ordering and values.","tokens_in":11892,"feed_emoji":"⚛️","tokens_out":6624,"duration_ms":56278,"temperature":0.7,"pith_summary":"The paper aims to show that pion generalized parton distributions (GPDs) at zero skewness, computed in the covariant Nambu–Jona-Lasinio model with proper-time regularization, form a single source for the pion's quark distributions and its scalar, vector, and tensor form factors. In the forward limit the GPD reduces to the valence quark distribution, which after DGLAP evolution from a low model scale agrees with measured pion Drell-Yan data and a recent global QCD analysis at both 4 and 27 GeV^2. The first Mellin moments of the same GPDs yield the vector and tensor form factors, and the twist-3 scalar GPD yields the scalar form factor; once the quark-photon vertex is dressed, these match recent lattice QCD results. From the slopes of these form factors the paper obtains charge radii r_S = 0.56 fm, r_V = 0.63 fm, and r_T = 0.83 fm, ordered r_T >= r_V >= r_S. A sympathetic reader would take this as evidence that a single low-energy model can organize multiple pion observables into one consistent internal-structure picture.","feed_headline":"A single model reproduces pion quark distributions, form factors, and radii","feed_subtitle":"The same GPD yields quark distributions that match experiment and charge radii that match lattice QCD.","key_machinery":"The central object is the pion GPD trio at zero skewness — vector H^u(x, xi=0, t), tensor E^u(x, xi=0, t), and twist-3 scalar H^u_S(x, xi=0, t) — computed from quark-loop diagrams in the covariant NJL model. The proper-time regularization scheme, with infrared cutoff Lambda_IR = 240 MeV and ultraviolet cutoff Lambda_UV = 645 MeV, removes ultraviolet divergences and mimics confinement. The forward limit t=0, xi=0 turns the vector GPD into the valence PDF; the first Mellin moments turn the vector and tensor GPDs into generalized form factors; and the twist-3 GPD yields the scalar form factor. A dressed quark-photon vertex, rather than the bare gamma^+, is the additional piece needed to bring t","core_discovery":"The central claim is that the zero-skewness pion GPDs from the covariant NJL model are quantitatively reliable across three separate observables. The forward limit H^u(x,0,0) gives the valence pion PDF; evolved from the model scale mu0^2 = 0.18 GeV^2 to mu^2 = 27 and 4 GeV^2, it reproduces the measured Drell-Yan data and a global QCD analysis, including the (1-x) behavior near x to 1. The n=0 Mellin moments of the vector and tensor GPDs give the pion electromagnetic and tensor form factors, and the twist-3 scalar GPD gives the scalar form factor; with a dressed quark-photon vertex these match lattice QCD. The resulting charge radii satisfy r_T >= r_V >= r_S, consistent with lattice ordering.","pith_inferences":["If the derivation holds, extending the same GPDs to nonzero skewness would give access to the pion's transverse spatial tomography and to gravitational form factors, which the zero-skewness limit cannot reach; the paper does not perform this extension.","Because the model scale contains only valence quarks, all sea-quark effects seen at high scales are generated by DGLAP evolution; a precise small-x measurement of the pion's sea would stress-test this input.","A sharper test would be to treat mu0^2 and the regulator parameters as fit parameters constrained simultaneously by PDF, form-factor, and charge-radius data; the current paper fixes them from mass and decay constant and then checks the observables.","The paper's finding that the dressed scalar form factor underestimates lattice data suggests the twist-3 sector may need additional physics beyond the dressed vertex; investigating that mismatch could refine the model."],"forward_implications":["The same GPD that matches PDF data also predicts vector, tensor, and scalar form factors, so a future measurement of pion GPDs at nonzero momentum transfer would test the model's internal consistency directly.","The model's valence PDF falls as (1-x)^1 as x approaches 1, matching the global analysis, whereas some other QCD-inspired models predict (1-x)^2; this difference is observable at large x.","The predicted ordering r_T >= r_V >= r_S for the pion's charge radii is a concrete signature that can be checked against future lattice and experimental determinations.","Agreement at two different renormalization scales from a single initial scale supports the use of mu0^2 = 0.18 GeV^2 as a nonperturbative input for DGLAP evolution."],"fun_headline_variants":["One model reproduces pion PDFs, form factors, and radii","Pion GPDs unify quark distributions, form factors, and charge radii","Zero-skewness pion GPDs match experiment and lattice data","Covariant NJL model yields pion radii and distributions","Quantum distributions and radii from one pion GPD"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the valence-only PDF at the arbitrarily chosen initial scale mu0^2 = 0.18 GeV^2, together with the proper-time regulator parameters, is the correct nonperturbative input; if that scale is chosen differently, the claimed agreement with data may not survive.","fun_headline_variants_meta":{"raw":{"variants":["One model reproduces pion PDFs, form factors, and radii","Pion GPDs unify quark distributions, form factors, and charge radii","Zero-skewness pion GPDs match experiment and lattice data","Covariant NJL model yields pion radii and distributions","Quantum distributions and radii from one pion GPD"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000863,"raw_usage":{"total_tokens":3614,"prompt_tokens":811,"completion_tokens":2803,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":2715}},"tokens_in":555,"tokens_out":2803,"duration_ms":17112,"temperature":1.0,"reasoning_tokens":2715,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:36:03.384294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute or measure the pion valence PDF at a third scale, for example mu^2 = 10 GeV^2, after DGLAP evolution from mu0^2 = 0.18 GeV^2; if it disagrees with new data by more than the experimental errors, the central claim collapses. Alternatively, a lattice or experimental determination of the tensor charge radius below about 0.7 fm would contradict the predicted ordering and values.","supporting_citations":[],"review_version":1}