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Unpolarized gluon PDF of the nucleon from lattice QCD at physical point in the continuum limit

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Lattice QCD, extrapolated to the continuum and infinite-momentum limits, reproduces the nucleon's unpolarized gluon PDF within global-fit uncertainty.

desk verdict 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. read the letter →

arxiv 2510.26425 v2 pith:RCUSQAJQ submitted 2025-10-30 hep-lat

Chen Chen , Chunhua Zeng , Hongxin Dong , Liuming Liu , Xiaomin Shen , Peng Sun , Xiaonu Xiong , Yi-Bo Yang
show 3 more authors
Fei Yao Jian-Hui Zhang Shiyi Zhong
This is my paper · ORCID
classification hep-lat MSC 81V0581T25 PACS 12.38.Gc11.15.Ha
keywords gluonPDFlatticeQCDLaMETlarge-momentumeffectivetheorynucleonstructurehybridrenormalizationcontinuumlimitinfinitemomentum
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

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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).
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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

4 major / 4 minor

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.

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 (4)
  1. [Light-cone PDF, Eq. (7), Eq. (14), Table IV] 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.
  2. [Eq. (8), Table I, Fig. 3] 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.
  3. [Title, abstract, Table I] 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.
  4. [Results, after Eq. (8)] 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.
minor comments (4)
  1. [Throughout] 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].
  2. [Fig. 12 caption] 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.
  3. [Fig. 4] 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.
  4. [Conclusion] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CT18/NNPDF/JAM24 comparisons are external anchors, and the renormalization/tail fits are to lattice data rather than to the target PDF.

full rationale

The claimed prediction is the continuum/infinite-momentum xg0(x) from Eq. (8), obtained by Fourier transforming measured renormalized matrix elements (Eq. 14), NLO matching (Eq. 11), and a joint extrapolation. The global-fit PDFs (CT18, NNPDF, JAM24) enter only in the final comparison (Fig. 4); nothing is fitted to them, so the agreement is not forced. The self-renormalization parameters (k, Lambda_QCD, m0, d, g(z)) are fitted to zero-momentum bare matrix elements via Eq. (6) and then used in Eq. (3) to define renormalized long-distance matrix elements for all momenta; this is a standard renormalization condition, and the target xg0 is obtained from independently measured nonzero-momentum matrix elements, not from the fit function. The large-lambda tail (Eq. 7) is an explicit extrapolation ansatz fitted to the measured h_R in a restricted range (Table IV) and varied for systematics; while it is a genuine model-form risk (the functional form is never varied, and the Fourier integral extends beyond the measured lambda), the result is not equivalent to the tail model by construction - it is a data-driven extrapolation with a stated assumption. The citations to Refs. [13,40] for the hybrid scheme and tail form are published methodology with stated assumptions, not a self-citation uniqueness theorem. No step exhibits a parameter fitted to the target quantity and then renamed as a prediction.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central claim rests on three nested layers of fitted quantities: (i) hybrid-renormalization parameters k, Lambda_QCD, m0, d and the free function g(z), fitted to zero-momentum matrix elements (Eq. 6, Table II) and used to define the renormalized correlator everywhere; (ii) per-(ensemble, momentum) tail parameters l1, a1, lambda0 (Eq. 7) completing the Fourier transform; (iii) per-x functions f, h, d in the joint continuum/momentum extrapolation (Eq. 8) that produce the headline curve. Boundary choices z0, z1, zs, zmax and the scale mu = 2 GeV are additional hand-set inputs. No free parameters are fitted to the global-fit PDFs used for comparison, which keeps the circularity burden moderate-low rather than high. No invented physical entities.

free parameters (8)
  • k (linear divergence coefficient in Z_R) = 1.92(1.18)
    Fitted via Eq. (6) to zero-momentum bare matrix elements; enters the renormalized matrix element Eq. (3) for all |z| > z_s.
  • Lambda_QCD in Z_R = 0.59(0.15) GeV
    Scheme parameter in the log-resummation terms of Z_R (Eq. 4); fitted, not the physical QCD scale.
  • m0 (finite-mass renormalization ambiguity) = 2.78(1.51) GeV
    Linear-in-z term in Z_R absorbing the renormalization-ambiguity shift; strongly correlated with k and Lambda_QCD.
  • d (subleading-log coefficient) = -0.079(0.403)
    Coefficient in Z_R (Eq. 4); fitted with chi^2/dof = 0.09, indicating over-flexible parametrization.
  • g(z) (long-distance residual function) = free function, not tabulated
    Fitted to zero-momentum matrix elements for z > z1 (Eq. 6); exp[g(z) - m0 z] is presented as the 'a-independent renormalized matrix element' in Fig. 1.
  • l1, a1, lambda0 (lambda-tail extrapolation) = per (ensemble, Pz), not tabulated
    Parameters of Eq. (7). Fitted per momentum; systematic uncertainty estimated only by varying the fit range, never the functional form.
  • f(x), h(x), d(x) in Eq. (8) = per-x functions, not tabulated
    Continuum/infinite-momentum extrapolation ansatz functions; fitted jointly to 8 data sets per x-bin (3+3+2 momenta).
  • Boundaries z0, z1, zs, z_max = 0.15, 0.3, 0.3, 1.0 fm
    Hand-set boundaries of the hybrid scheme and of the Z_R fit; zs is varied down to 0.2 fm for one systematic estimate.
assumptions (7)
  • domain assumption LaMET factorization (Eq. 11): quasi-PDF differs from the light-cone PDF by the NLO kernel plus power corrections Lambda_QCD^2/(xPz)^2 and Lambda_QCD^2/((1-x)Pz)^2.
    The available Pz = 0.86-1.97 GeV makes power corrections sizable except in the mid-x window; the only probe is scale variation 2 -> 4 GeV.
  • domain assumption NLO hybrid matching kernel (Eqs. 12-13) is adequate without RG, threshold, or renormalon resummation at mu = 2 GeV.
    Resummations are listed as future work (Refs. [54,55]); the scale-variation systematic is the sole handle on higher orders.
  • domain assumption Hybrid renormalization: Z_R of Eq. (4) removes all UV and linear divergences, with the residual renormalization ambiguity absorbed by m0 z and g(z).
    Adopted from the authors' own Ref. [40]; the functional form and fit ranges are assumed, not independently verified for this operator and action.
  • ad hoc to paper Large-lambda tail has the form l1 lambda^{-a1} exp(-lambda/lambda0) (Eq. 7).
    No derivation ties a1 to the PDF endpoint power; only fit-range variation is used for systematics, never the functional form.
  • ad hoc to paper Pion-mass dependence is negligible (m_pi = 287-300 MeV vs the physical point).
    The body states 'we do not include pion mass dependence in this work'; the abstract nevertheless claims extrapolation to the physical pion mass.
  • domain assumption Discretization and higher-twist terms have the form a^2 f(x) + a^2 Pz^2 h(x) + d(x)/Pz^2 (Eq. 8).
    Symanzik/OPE expectation; with 3 spacings and 2-3 momenta per ensemble the four per-x functions cannot be resolved by the data.
  • domain assumption The operator combination Eq. (1) is multiplicatively renormalizable and free of quark-operator mixing under this smearing.
    Backed by Refs. [43,44]; taken as input rather than verified in this work.

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Pith. "Pith review of Unpolarized gluon PDF of the nucleon from lattice QCD at physical point in the continuum limit." pith.science (2026). https://pith.science/paper/RCUSQAJQ

@misc{pith2026251026425,
  author       = {Pith},
  title        = {Pith review of: Unpolarized gluon PDF of the nucleon from lattice QCD at physical point in the continuum limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCUSQAJQ}},
  note         = {Machine review of arXiv:2510.26425}
}
read the original abstract

We report a state-of-the-art lattice QCD calculation of the nucleon unpolarized gluon parton distribution function employing large-momentum effective theory. The calculation is carried out on the 2+1 flavor CLQCD ensembles with five lattice spacings a={0.105,0.0897,0.0775, 0.0688, 0.0519} fm and various pion masses ranging from 136 MeV to 317 MeV, covering nulceon momenta up to 3 GeV. Distillation technique is applied to improve the signal of two-point correlators. We then apply the state-of-the-art hybrid renormalization and one-loop perturbative matching, and extrapolate the result to the continuum limit, infinite momentum limit and physical pion mass.

Figures

Figures reproduced from arXiv: 2510.26425 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison of renormalized matrix elements with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The renormalized matrix elements of C24P29 (upper) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. LPC prediction of unpolarized gluon PDF in the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: It can be represented as: C3pt (Pz; z;tsep, t) = X t0 (O (z;t0 + t) − ⟨O (z;t0 + t)⟩) × (C2pt (Pz;t0 + tsep, t0) − ⟨C2pt (Pz;t0 + tsep, t0)⟩). (19) Fµν(z) Fµν(0) U[z, 0] t0 + tsep t0 + t t0 FIG. 5. Illustration of the three-point correlator of gluon. Fitting for bare m…
Figure 6
Figure 6. Figure 6: FIG. 6. The ratio of three-to-two point correlator and the fitting results for bare matrix elements of ensemble C24P29 under [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The ratio of three-to-two point correlator and the fitting results for bare matrix elements of ensemble E32P29 under [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The ratio of three-to-two point correlator and the fitting results for bare matrix elements of ensemble F32P30 under [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The fitting result of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Bare matrix elements of C24P29 (left) , E32P29 (middle), and F32P30(right) obtained by fitting the ratio of three [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The large- [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Estimation of statistical and systematic uncertain [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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  1. Parton distribution functions from lattice QCD

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    A review of lattice-QCD approaches to parton distribution functions concludes that the field is moving from feasibility studies to quantitatively controlled calculations.

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Reviewed August 4, 2026 · model on record in the stance chip above.