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Towards nucleon structure function moments and parton momentum fractions from lattice QCD

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.04033 v1 pith:X4WN3N5N submitted 2025-05-07 hep-lat hep-ph

K. U. Can , R. Horsley , P. E. L. Rakow , G. Schierholz , H. Stüben , R. D. Young , J. M. Zanotti This is my paper · ORCID
classification hep-lathep-ph MSC 81T2581V05 PACS 12.38.Gc
keywords latticeQCDFeynman-HellmanntheoremComptonamplitudenucleonstructurefunctionsMellinmomentspartonmomentumfractionhighertwistisovectormoment
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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 $Q^2$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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$^2$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [Sec. 5, Eq. (25)] 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.
  2. [Sec. 5, Eq. (24)] 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.
  3. [Sec. 5, Fig. 3] 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.
minor comments (6)
  1. [Abstract and Sec. 5] The statement of '~10% uncertainty' for Eq. (24) is not consistent with 22/177 ≈ 12%; please harmonize the wording.
  2. [Fig. 4 and Sec. 5] 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].
  3. [Sec. 4] 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.
  4. [Eq. (26)] 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.
  5. [Sec. 4, Fig. 1] 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.
  6. [Fig. 1] The outer uncertainty band in Fig. 1 is described as barely visible; please increase the contrast or use a different representation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported moments and <x> are extracted from lattice data using external NNLO coefficients; the higher-twist model is explicit and not an input-output identity.

full rationale

The paper's central quantities, the physical Compton amplitude moments M^(2)_{2,uu-dd}(Q^2), are obtained from lattice correlation functions via the Feynman-Hellmann relation (Eqs. (15)-(20)) and a Bayesian fit to the omega-dependence (Eq. (23)); neither step reuses the reported final values as inputs. The leading-twist extraction <x>_{u-d}=0.141(8) is a two-parameter fit (v2, A2) to these independently computed moments using Eq. (25), with the NNLO Wilson coefficient C^(2) taken from an external source [31] and alpha_s from the PDG/FLAG value of Lambda_MS; the FLAG <x> average is used only for comparison, not as a constraint. The higher-twist ansatz in Eq. (25) is an explicit modeling assumption stated in Sec. 5, not a hidden circular definition, and any concern about its correctness belongs to systematic uncertainty rather than circularity. Self-citations to Refs. [13-16,28] are methodological (derivation of the FH theorem, previous correlator datasets, and quark-mass dependence) and do not smuggle in the target result: the quoted moment and PDF moment are new outputs from lattice data, not consequences of those citations alone.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim depends on two fitted amplitudes (v2 and A2) plus a set of prior-constrained higher moments and an assumed chiral slope. All are legitimate analysis parameters, but their systematics are not fully propagated into the quoted errors.

free parameters (5)
  • A2 (twist-4 amplitude) = not quoted
    Fitted in Eq. (25) to the Q^2-dependence of the moments at the SU(3) point. It absorbs genuine twist-4 and target-mass corrections; no independent determination is given.
  • c (chiral extrapolation slope) = not quoted
    Slope of the linear fit f(a m_pi^2) = M_phys + c (a m_pi)^2 to three ensembles in Sec. 5.
  • M^(2)_{2n} for n=2..6 (higher moments) = not quoted
    Bayesian fit to F2/omega in Sec. 4 with positivity and monotonicity priors; nuisance parameters for the extraction of the lowest moment.
  • v2 = <x>_{u-d} (PDF moment) = 0.141(8) at mu^2 = 4 GeV^2
    The target parameter fitted in Eq. (25) from the Q^2-dependence of the moments; the central result.
  • Renormalization scale mu^2 = 4 GeV^2
    Chosen by hand as the reference scale; no scale-variation systematic is quoted.
assumptions (6)
  • standard math The Compton structure functions satisfy the dispersion relations (3)-(4) and the moment expansions (5)-(6) via analyticity, crossing symmetry, and the optical theorem.
    Invoked in Sec. 2 as the bridge between lattice Compton amplitudes and structure function moments.
  • domain assumption The second-order Feynman-Hellmann relation (15) with O(lambda^4) terms neglected is valid for the applied perturbation.
    The paper states higher-order terms are heavily suppressed for |lambda| = O(1e-2) [15,16]; no direct check for these ensembles is shown.
  • ad hoc to paper The moment series (6) can be truncated at n=6 with positive, monotonically decreasing uu and dd moments for the fitted omega range.
    No test of truncation at other orders (e.g., n=5 or n=7) or of prior sensitivity is presented in Sec. 4.
  • ad hoc to paper The higher-twist contribution takes the multiplicative form A2/Q^2 times the leading-twist Wilson coefficient.
    Explicitly stated in Sec. 5 as a prescription; it is load-bearing for the extraction of <x>_{u-d}.
  • ad hoc to paper The isovector moment is linear in (a m_pi)^2 over the range 300-410 MeV, and the m=constant trajectory makes the extrapolation to the physical point valid.
    Used in Sec. 5 to obtain M^(2),phys=0.177(22); no chiral perturbation theory curvature is included.
  • domain assumption The SU(3)-symmetric-point value of <x>_{u-d} approximates the physical-quark-mass value, so comparison with the FLAG physical-point average is meaningful.
    Justified heuristically via the momentum sum rule and Ref. [28], not by a controlled chiral extrapolation in this paper.

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Pith. "Pith review of Towards nucleon structure function moments and parton momentum fractions from lattice QCD." pith.science (2026). https://pith.science/paper/X4WN3N5N

@misc{pith2026250504033,
  author       = {Pith},
  title        = {Pith review of: Towards nucleon structure function moments and parton momentum fractions from lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4WN3N5N}},
  note         = {Machine review of arXiv:2505.04033}
}
abstract

We calculate the lowest even isovector moment of the $F_2$ structure function in $2+1$-flavour lattice QCD with varying quark masses corresponding to $m_\pi \approx [410, 360, 300] \; {\rm MeV}$, at a fixed volume of $V = 48^3 \times 96$ and coupling $\beta = 5.65$ ($a = 0.068(3) \, {\rm fm}$). We directly compute the physical Compton amplitude using the Feynman-Hellmann approach and extract moments of the physical structure function. We report on the quark-mass dependence of the lowest isovector moment and estimate its value at the physical quark-mass point with $\sim 10\%$ uncertainty at fixed $Q^2$. By analysing the $Q^2$ dependence of the moments at the $SU(3)$ symmetric point ($m_\pi \approx 410 \; {\rm MeV}$), we separate the leading- and higher-twist contributions and estimate the parton momentum fraction, $\langle x \rangle_{u-d}$, which agrees with existing results.

Figures

Figures reproduced from arXiv: 2505.04033 by the authors.

Figure 1
Figure 1. The F2 Compton structure function obtained via Eq. (16) for (p, q) = ( (0, 0, 0)  2𝜋 𝐿  , (5, 3, 0)  2𝜋 𝐿  ), where 𝐿 = 48𝑎. The 𝑢𝑢 contribution determined on the 𝑆𝑈(3) symmet￾ric ensemble is shown. 𝑡max = 20𝑎 for this case and we dropped the 𝑡min > 15𝑎 points for clarity due to their vanishing weights. The grey bars associated with the right y-axis denote the weight, Eq. (21), of each point. The final estimate,… view at source ↗
Figure 2
Figure 2. 𝜔 dependence of the Compton structure function F2 at 𝑄 2 ∼ 5 GeV2 . We show the 𝑢𝑢 (blue circles) and 𝑑𝑑 (red diamonds) contributions and the constructed 𝑢𝑢 − 𝑑𝑑 (purple squares) results. Coloured shaded bands show the fits with their 68% credible region of the highest posterior density. We stress that we do not fit to the 𝑢𝑢 − 𝑑𝑑 points but show the constructed points (F 𝑢𝑢 2 (𝜔)/𝜔 − F 𝑑𝑑 2 (𝜔)/𝜔) and fit band Í6 … view at source ↗
Figure 3
Figure 3. The lowest even isovector moment, 𝑀 (2) 2,𝑢𝑢−𝑑𝑑 (𝑄 2 ) as a function of 𝑎𝑚2 𝜋 along the 𝑚 = constant line at fixed 𝑄 2 ∼ 5 GeV2 . Solid line and the shaded band show the fit curve and the 1𝜎 uncertainty respectively. See the main text for the fit function. where the ∼ 10% uncertainty quantifies the statistical and one source of the systematic uncertainties. At 𝑄 2 ∼ 5 GeV2 , the leading contribution to the moment Eq… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The lowest even isovector Mellin moments (LQCD moments) of the 𝐹2 structure function as a function of 𝑄 2 extracted from our lattice F2 Compton structure functions. The solid band (Full fit) shows our fit to the LQCD moments using Eq. (25), the hatched band (QCD evolut…

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