{"id":"0994d073-8e81-4ac6-a1ac-02c2f40899ae","arxiv_id":"2505.04033","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Using the Feynman-Hellmann approach in lattice QCD, the authors estimate the nucleon's isovector parton momentum fraction <x>_{u-d}=0.141(8) at mu^2=4 GeV^2 and the physical F2 moment 0.177(22) at Q^2 ~ 5 GeV^2, consistent with the FLAG average.","lead":"Lattice QCD simulations have been used to compute how much of the proton's momentum is carried by its up and down quarks, giving an isovector parton momentum fraction of 0.141(8) at a scale of 2 GeV. The result agrees with world averages and demonstrates a method that separates short-distance parton physics from long-distance corrections.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The <x>_{u-d} extraction is not yet robust: Eq. (25) imposes an untested multiplicative higher-twist ansatz, so the leading-twist part may be biased.","rationale":"The reader's weakest_assumption identifies the same load-bearing point: the separation of leading and higher twist in Eq. (25) is a model assumption, not a derived OPE relation, and it is not tested against alternatives. I agree that this is the main soft spot for the central <x>_{u-d} claim. The physical-point moment Eq. (24) is better supported by the mild quark-mass dependence shown in Fig. 3, although it too carries unquantified finite-volume and discretisation systematics; those are acknowledged as future work and do not themselves undermine the paper's stated 'towards' status. The concrete test proposed here would settle whether the higher-twist model is actually responsible for the quoted 0.141(8): if alternative parameterizations shift v_2 beyond the error, the comparison with FLAG is not yet a robust first-principles determination of the PDF moment. Because the reader already assigned CONDITIONAL, my recommendation is to retain that verdict rather than to change it.","tokens_in":12381,"tokens_out":6328,"duration_ms":72139,"concrete_test":"Refit the Q^2-dependent moments of Sec. 5 with an additive power-correction ansatz, M(Q^2)=C_2(Q^2)v_2(\\mu)+B_2/Q^2+B_4/Q^4, or at least with C_2(Q^2)[v_2+A_2/Q^2+A_4/Q^4], using the same covariance matrix and data. Also repeat the two-parameter fit with Q^2_min raised from 1 GeV^2 to 2 and 3 GeV^2. If v_2 at \\mu^2=4 GeV^2 remains within about 0.01 of 0.141 in all variants, the higher-twist model is not the deciding factor; if it shifts by more than the quoted 0.008 error, the headline momentum fraction is model-dependent and requires a systematic uncertainty or a more restricted claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline claim beyond the raw moment is <x>_{u-d}=0.141(8), obtained in Sec. 5 by fitting M_{2,uu-dd}^{(2)}(Q^2)=C_2(Q^2/\\mu^2)[v_2(\\mu)+A_2(\\mu)/Q^2] over 1 ≲ Q^2 ≲ 7.5 GeV^2. This is not the generic OPE form: a genuine twist-4 contribution would have its own Wilson coefficient and scale dependence, while Eq. (25) forces it to share the leading-twist coefficient C_2. The paper states explicitly that this is 'a multiplicative prescription assuming the higher-twist contribution shares the leading-twist Wilson coefficient.' With only two parameters (v_2, A_2) fitted to data starting at Q^2 = 1 GeV^2, and with the power correction reported to be about 30% of the leading twist at Q^2 = 4 GeV^2, the extracted v_2 is sensitive to this untested model choice. No alternative ansatz, no 1/Q^4 term, and no Q^2_min stability check are reported. If the true twist-4 term has a different Q^2 dependence, or if 1/Q^4 terms matter in this window, the quoted 0.141(8) is biased by an amount not included in the error. This does not invalidate the raw moment Eq. (24), but it is the load-bearing step for the comparison with FLAG.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reports a lattice QCD computation of the lowest even isovector Mellin moment of the F2 structure function using the Feynman-Hellmann approach. The authors compute second-order energy shifts from perturbed nucleon correlators on three 2+1-flavour ensembles with m_pi ≈ 410, 360, and 300 MeV at fixed volume and lattice spacing, extract moments from the omega-dependence of the Compton structure function, and extrapolate the lowest moment to the physical pion mass at Q^2 ~ 5 GeV^2, obtaining M^{(2)}_{2,uu-dd}(Q^2~5 GeV^2) = 0.177(22). They then use the Q^2 dependence of the moments at the SU(3) symmetric point in the range 1 ≲ Q^2 ≲ 7.5 GeV^2, together with an NNLO Wilson coefficient and an assumed 1/Q^2 higher-twist term, to extract the isovector parton momentum fraction <x>_{u-d} = 0.141(8) at mu^2 = 4 GeV^2, which agrees with the FLAG 2+1-flavour average.","tokens_in":12812,"tokens_out":10779,"duration_ms":108021,"significance":"The method is attractive because it accesses physical structure function moments through the Compton amplitude without operator mixing and the power-divergent renormalisation that complicate quasi-PDF and pseudo-PDF approaches. The isovector combination is constructed from correlated uu and dd samples, so disconnected contributions cancel, and the direct extraction of the lowest F2 moment at omega=0 avoids a polynomial extrapolation in omega for that moment. The weighted fit-window averaging is clearly described, and the agreement with FLAG is a useful cross-check. If the quoted <x>_{u-d} survives a more robust treatment of higher-twist contamination and quark-mass dependence, this would be a valuable first-principles constraint on a nucleon PDF moment. However, the headline comparison with FLAG rests on the Q^2-dependence fit of Eq. (25), whose higher-twist ansatz and quark-mass identification are currently not fully tested; the significance is therefore conditional.","major_comments":[{"comment":"The extraction of <x>_{u-d} is the load-bearing step for the paper's headline comparison with FLAG, but the higher-twist model in Eq. (25) is an untested assumption. The text states that the power correction 'shares the leading-twist Wilson coefficient'; this is not the generic OPE form, where a twist-4 operator has its own Wilson coefficient and scale dependence. The two-parameter fit is performed over 1 ≲ Q^2 ≲ 7.5 GeV^2, and the higher-twist term is reported to be about 30% of the leading-twist term at Q^2 = 4 GeV^2, so the extracted v_2 is not a small correction. No alternative parameterisation, no 1/Q^4 term, and no Q^2_min stability test are reported. The quoted 0.141(8) therefore does not include the model dependence of the twist separation. I would like to see a systematic study of this model dependence, for example by varying Q^2_min, adding a 1/Q^4 term, and using an additive higher-twist form, before this number is presented as a determination.","section":"Sec. 5, Eq. (25)"},{"comment":"The physical-point value in Eq. (24) is quoted as 0.177(22) and described as having '~10% uncertainty', but the text says this quantifies only statistical and one source of systematic uncertainty. Finite-volume and discretisation errors are explicitly left to future work, and the linear (a m_pi)^2 extrapolation from 410-300 MeV to the physical point is not tested against a curvature term or an alternative functional form. Since this value is one of the two main results and is used to argue for a mild quark-mass dependence, the abstract and the result should state exactly which systematics are included in the quoted 22, and the missing contributions should be estimated or clearly flagged as unquantified.","section":"Sec. 5, Eq. (24)"},{"comment":"The value <x>_{u-d} = 0.141(8) in Eq. (27) is extracted at the SU(3) symmetric point (m_pi ≈ 410 MeV) and is then compared directly with the FLAG average at the physical point. The argument that the isovector moment has mild quark-mass dependence is based on Fig. 3, which shows the finite-Q^2 physical moment M^{(2)}_2 at Q^2 ~ 5 GeV^2, not the leading-twist matrix element v_2 at fixed mu. These are different quantities because of the higher-twist term in Eq. (25). Please either extrapolate v_2 to the physical point or present Eq. (27) as a value at m_pi ≈ 410 MeV, with the mass-dependence uncertainty made explicit in the comparison with FLAG.","section":"Sec. 5, Fig. 3"}],"minor_comments":[{"comment":"The statement of '~10% uncertainty' for Eq. (24) is not consistent with 22/177 ≈ 12%; please harmonize the wording.","section":"Abstract and Sec. 5"},{"comment":"Please tabulate the Q^2 values, the extracted moments with their total uncertainties, and the fit parameters v_2 and A_2 with a goodness-of-fit measure; currently this information is only in Ref. [15].","section":"Fig. 4 and Sec. 5"},{"comment":"The choice of the single fit window used for propagation after the weighted averaging should be justified more explicitly, since the final uncertainty appears to depend on that choice.","section":"Sec. 4"},{"comment":"The notation in Eq. (26) should specify the scheme and flavour number for the constants 158.07 and 58411.28, and state how a_s(Q^2) is evolved; this would improve reproducibility.","section":"Eq. (26)"},{"comment":"The omega-dependence fits truncate the series at n=6 with positivity and monotonicity priors; a short sensitivity check of M^{(2)}_2 to the truncation order and prior widths would be useful.","section":"Sec. 4, Fig. 1"},{"comment":"The outer uncertainty band in Fig. 1 is described as barely visible; please increase the contrast or use a different representation.","section":"Fig. 1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: this is a proceedings contribution describing work in progress. The stress-test concern about Eq. (25) is valid and should be addressed before the result is quoted as a robust determination. I am not recommending rejection, but the authors should be asked to add stability tests for the higher-twist separation and to state the quark-mass and lattice-systematic caveats more prominently."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this proceedings paper adds two lighter pion masses to the QCDSF Feynman-Hellmann program and produces the first estimate of the physical-point isovector F2 moment from this method, M^(2)_{2,uu-dd}(Q^2~5 GeV^2)=0.177(22). The companion PDF moment <x>_{u-d}=0.141(8) agrees with FLAG but is not at the same level of rigor.\n\nWhat is genuinely new: the m_pi=360 and 300 MeV ensembles allow a quark-mass extrapolation, and the Q^2-dependence analysis at the SU(3) symmetric point is a new reanalysis of Ref. [15] data with a two-parameter twist separation. The fit-window weighting procedure is described in enough detail to be reproducible, and the error bands look honest. The citation pattern is self-referential but appropriate: the Feynman-Hellmann method is theirs, and Refs. [14,15] are the direct predecessors. External NNLO inputs for the Wilson coefficient and alpha_s are standard; no circularity.\n\nWhere it is soft: the ~10% error on Eq. (24) is not a full error budget – no finite-volume, discretisation, or scale-variation systematics, and the chiral extrapolation uses three points with a linear ansatz. For a proceedings that is acceptable if stated, but the abstract overstates it. The larger concern is the twist separation. Eq. (25) forces the power correction to share the leading-twist Wilson coefficient; that is not the generic OPE form. With the lowest data point at Q^2=1 GeV^2 and a 30% higher-twist contribution at 4 GeV^2, the fitted v2 is sensitive to this choice. The paper reports no alternative ansatz, no 1/Q^4 term, no Q^2_min variation. The FLAG agreement is reassuring but not a substitute for a systematic test. The raw moment Eq. (24) does not rely on this ansatz and should survive.\n\nWho is this for: lattice QCD practitioners working on Compton amplitude methods and nucleon structure. It deserves a serious referee, although for a full paper the twist-separation systematics would need to be addressed before the PDF moment is quoted as a final number. My recommendation: engage with it, cite the raw moment if it is relevant to you, and treat <x> as a preliminary cross-check until the model dependence is tested.","headline":"New Feynman-Hellmann moments at lighter pion masses; the PDF-moment extraction rests on an untested higher-twist ansatz, so the <x> result is provisional.","tokens_in":13326,"tokens_out":3977,"would_cite":true,"duration_ms":39006,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T25","81V05"],"pacs":["12.38.Gc"],"model":"deepseek-v4-flash","headline":"This paper reports a direct lattice QCD determination of the isovector parton momentum fraction, $\\langle x\\rangle_{u-d}=0.141(8)$ at 4 GeV$^2$, obtained from the Feynman-Hellmann Compton amplitude.","keywords":["lattice QCD","Feynman-Hellmann theorem","Compton amplitude","nucleon structure functions","Mellin moments","parton momentum fraction","higher twist","isovector moment"],"falsifier":"Repeat the leading/higher-twist separation at the SU(3)-symmetric point with a different model for the power correction, for example letting $A_2$ run independently or including a $1/Q^4$ term; if the extracted $v_2$ moves by more than its 0.008 uncertainty, the quoted $\\langle x\\rangle_{u-d}=0.141(8)$ is model-dependent rather than determined. A second check is to repeat the full $Q^2$ analysis on the $m_\\pi\\approx300$ MeV ensemble, where the physical-point extrapolation is no longer required.","tokens_in":12190,"feed_emoji":"⚛️","tokens_out":10622,"duration_ms":97193,"temperature":0.7,"pith_summary":"This paper aims to show that the isovector momentum fraction $\\langle x\\rangle_{u-d}$ carried by quarks inside the nucleon can be computed from first principles by evaluating the forward Compton amplitude on a lattice, rather than by the quasi- or pseudo-PDF routes that suffer from operator mixing and power-divergent renormalization. Using the second-order Feynman-Hellmann theorem, the authors extract the lowest even Mellin moment of the $F_2$ structure function at three pion masses, find essentially no quark-mass dependence in that range, and quote $M^{(2)}_{2,uu-dd}(Q^2 \\sim 5\\,\\mathrm{GeV}^2) = 0.177(22)$ at the physical point. From the $Q^2$ dependence of the same moment at the SU(3)-symmetric point they then separate the leading-twist part from a power correction and obtain $\\langle x\\rangle_{u-d} = 0.141(8)$ at $\\mu^2 = 4\\,\\mathrm{GeV}^2$, in agreement with the 2+1-flavour lattice average. If correct, this is a direct, first-principles determination of a nucleon PDF moment that also quantifies the size of higher-twist effects at moderate $Q^2$.","feed_headline":"Lattice QCD extracts quark momentum fraction, 0.141(8)","feed_subtitle":"A direct Compton-amplitude calculation separates the main signal from power corrections at 4 GeV^2, matching the lattice-QCD average.","key_machinery":"The load-bearing object is the second-order Feynman-Hellmann theorem for the forward Compton amplitude: perturbing the fermion action by a background electromagnetic current $S(\\lambda)=S_0 + \\lambda \\int d^4z\\,(e^{iq\\cdot z}+e^{-iq\\cdot z})J_\\mu(z)$ shifts the nucleon energy, and the second derivative of that energy at $\\lambda=0$ is the Compton tensor (Eq. 15). The extraction chain is: form the ratio of perturbed to unperturbed two-point functions to isolate the even-order energy shift; reconstruct $F_2/\\omega$ from the $T_{00}+T_{33}$ combination at $\\omega=0$; fit the $\\omega$-expansion truncated at $n=6$ with Bayesian positivity priors to obtain the Mellin moments; then fit the $Q^2$-dependence through the OPE form $M = C\\,[v_2 + A_2/Q^2]$ to separate leading and higher twist. The method avoids the power-divergent operator mixing that affects quasi- and pseudo-PDF moment calculations.","core_discovery":"The central claim is that the physical, quark-mass-dependent lowest even isovector moment of $F_2$ can be obtained directly from the Compton amplitude, and that its $Q^2$ dependence can be used to disentangle the leading-twist PDF moment from higher-twist power corrections. The calculation, performed on 2+1-flavour ensembles at $m_\\pi \\approx 410, 360, 300$ MeV with a single lattice spacing $a = 0.068(3)$ fm, gives $M^{(2)}_{2,uu-dd}(Q^2\\sim5\\,\\mathrm{GeV}^2)=0.177(22)$ after a linear fit in $(a m_\\pi)^2$. Fitting the $Q^2$ dependence of the moments at the SU(3)-symmetric point to $M = C(Q^2/\\mu^2,g)\\,[v_2(\\mu) + A_2(\\mu)/Q^2]$ with an NNLO Wilson coefficient yields $v_2 = \\langle x\\rangle_{u-d} = 0.141(8)$ at $\\mu^2=4\\,\\mathrm{GeV}^2$. The same fit indicates that the power correction is sizable: about 30% of the leading-twist moment at $Q^2=4$ GeV$^2$ and about 10% at $Q^2=10$ GeV$^2$.","pith_inferences":["The paper does not test alternative parameterizations of the power correction; allowing $A_2$ to run with scale, or adding a $1/Q^4$ term, could shift $\\langle x\\rangle_{u-d}$ and is the most direct check of the quoted value.","If the reported higher-twist size holds, global QCD fits that quote PDFs at $Q^2\\sim4$ GeV$^2$ and neglect power corrections may carry a small but non-negligible bias.","The same analysis machinery could be applied to other $\\omega$ moments and to polarised structure functions, where the paper only reports the lowest $F_2$ isovector moment.","A natural extension is to repeat the $Q^2$-dependence fit on the lighter-mass ensembles rather than only at the SU(3)-symmetric point; agreement would test the quark-mass independence of the higher-twist term."],"forward_implications":["The physical isovector $F_2$ moment is determined to about 10% precision at a fixed scale, $Q^2\\sim5$ GeV$^2$, from lattice QCD alone.","The leading-twist momentum fraction $\\langle x\\rangle_{u-d}=0.141(8)$ at 4 GeV$^2$ is consistent with the 2+1-flavour lattice average, providing an independent cross-check of existing PDF-moment determinations.","Power corrections are not negligible in the studied range: the higher-twist term is roughly 30% of the leading-twist moment at $Q^2=4$ GeV$^2$ and still about 10% at $Q^2=10$ GeV$^2$.","Because the method accesses the physical Compton amplitude directly, it bypasses the renormalization and mixing issues that complicate quasi- and pseudo-PDF calculations of moments.","The extracted moment shows only mild quark-mass dependence between $m_\\pi\\approx300$ and 410 MeV, supporting the linear extrapolation to the physical point used in the analysis."],"supporting_citations":[{"why":"Supplies the second-order Feynman-Hellmann formalism and the moment-extraction procedure used throughout this calculation.","marker":"[14]"},{"why":"Provides the earlier moment and power-correction data whose correlators and $Q^2$ range are reanalysed here.","marker":"[15]"},{"why":"Gives the NNLO Wilson-coefficient expression used in Eq. (26) for running the leading-twist moment.","marker":"[31]"},{"why":"The cited community review supplies the 2+1-flavour average that the result is compared against and the three-flavour QCD scale parameter used for $\\alpha_s$.","marker":"[10]"},{"why":"Supplies the flavour-symmetry analysis of hadron matrix elements that underlies the quark-mass extrapolation of the twist-2 moment.","marker":"[28]"},{"why":"Determines the renormalisation constant $Z_V$ for the local electromagnetic current used in the perturbation.","marker":"[21]"},{"why":"Provides the weighted-averaging fit-window method used to extract energy shifts from the perturbed correlation functions.","marker":"[16]"}],"fun_headline_variants":["Lattice QCD extracts quark momentum fraction: 0.141(8)","Direct Compton amplitude yields <x>_{u-d}=0.141(8) on lattice","Twist-separated lattice PDF moment: <x>_{u-d}=0.141(8)","Quark momentum fraction from lattice QCD: 0.141(8) at 4 GeV²","Feynman-Hellmann approach gives lattice PDF moment 0.141(8)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's headline momentum fraction rests on a specific model of the $Q^2$-dependence of non-leading corrections: one power-suppressed term of the form $A_2/Q^2$ sharing the same short-distance coefficient as the leading term, with no alternative parameterization tested.","fun_headline_variants_meta":{"raw":{"variants":["Lattice QCD extracts quark momentum fraction: 0.141(8)","Direct Compton amplitude yields <x>_{u-d}=0.141(8) on lattice","Twist-separated lattice PDF moment: <x>_{u-d}=0.141(8)","Quark momentum fraction from lattice QCD: 0.141(8) at 4 GeV²","Feynman-Hellmann approach gives lattice PDF moment 0.141(8)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001062,"raw_usage":{"total_tokens":4507,"prompt_tokens":1050,"completion_tokens":3457,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":666,"completion_tokens_details":{"reasoning_tokens":3338}},"tokens_in":666,"tokens_out":3457,"duration_ms":26266,"temperature":1.0,"reasoning_tokens":3338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:39:55.674691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the leading/higher-twist separation at the SU(3)-symmetric point with a different model for the power correction, for example letting $A_2$ run independently or including a $1/Q^4$ term; if the extracted $v_2$ moves by more than its 0.008 uncertainty, the quoted $\\langle x\\rangle_{u-d}=0.141(8)$ is model-dependent rather than determined. A second check is to repeat the full $Q^2$ analysis on the $m_\\pi\\approx300$ MeV ensemble, where the physical-point extrapolation is no longer required.","supporting_citations":[{"cited_title":"Larin, T","cited_arxiv_id":null,"evidence_quote":"Gives the NNLO Wilson-coefficient expression used in Eq. (26) for running the leading-twist moment."},{"cited_title":"Patterns of flavour symmetry breaking in hadron matrix elements involving u, d and s quarks","cited_arxiv_id":"1909.02521","evidence_quote":"Supplies the flavour-symmetry analysis of hadron matrix elements that underlies the quark-mass extrapolation of the twist-2 moment."},{"cited_title":"Gross-Llewellyn Smith sum rule from lattice QCD","cited_arxiv_id":"2502.19704","evidence_quote":"Provides the weighted-averaging fit-window method used to extract energy shifts from the perturbed correlation functions."}],"review_version":1}