Pith. sign in

REVIEW 3 major objections 5 minor 4 cited by

Pionic gluons from global QCD analysis of experimental and lattice data

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read First global pion fit to combine experiment and lattice QCD finds that lattice data cut gluon uncertainty by more than half at $x \gtrsim 0.5$ and reveal a pion gluon density above the proton's at large $x$.

desk verdict First pion PDF global fit to include lattice gluonic data; the uncertainty reduction is real, but the claimed large-x pion-proton enhancement is not yet nailed down because it rests on a truncated, single-spacing lattice systematics model. read the letter →

arxiv 2507.22730 v1 pith:PSRHK2ZX submitted 2025-07-30 hep-ph hep-exhep-latnucl-th

William Good , Patrick C. Barry , Huey-Wen Lin , W. Melnitchouk , Alex NieMiera , Nobuo Sato This is my paper
classification hep-phhep-exhep-latnucl-th
keywords pionpartondistributionfunctionsgluonPDFlatticeQCDpseudo–Ioffe-timedistributionsDrell-Yanleadingneutronelectroproductionglobalanalysisshort-distancefactorization
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

Pions decay too quickly to be used as fixed targets, so their internal quark and gluon structure has had to be inferred from indirect experiments. This paper combines Drell-Yan and leading-neutron electroproduction data with lattice-QCD calculations of gluonic reduced pseudo–Ioffe-time distributions in a single global fit. The lattice input reduces the uncertainty on the pion's gluon distribution by more than half for momentum fractions $x \gtrsim 0.5$ and indicates that the pion carries more large-$x$ gluons than the proton does. If the result survives, it gives the first real constraint on pion gluons where experiment is blind and strengthens the case that gluons contribute about equally to pion and proton mass.

What carries the argument

The carrying object is the reduced pseudo–Ioffe-time distribution (RpITD), a double ratio of pion matrix elements of gluonic correlators, $M(\nu,z^2) = [M(z,P_z)/M(z,0)]/[M(0,P_z)/M(0,0)]$, which a short-distance factorization relates to the gluon and quark-singlet PDFs. At leading order the RpITD is essentially a Fourier transform of $xg(x)/\langle xg\rangle$, giving direct sensitivity to the shape of the gluon PDF; NLO matching kernels handle perturbative corrections and quark-gluon mixing. The extra terms $z^2 B_1(\nu)$ and $(1/z)P_1(\nu)$ absorb higher-twist and discretization effects, truncated at $N=2$ in this analysis. This machinery allows 35 lattice points from one lattice spacing and pion mass to enter a Bayesian global fit alongside experimental data, with the systematic corrections themselves optimized during the fit.

What would settle it

Recompute the pion gluon PDF after including lattice RpITD data at one or more smaller lattice spacings, or after adding the excluded $z=a$ and $z=2a$ points with a larger systematic model. If the inferred large-$x$ gluon enhancement over the proton changes sign or disappears, the central claim is refuted; if it is stable across spacings, the claim is supported.

Watch

Extended reading notes

Core claim

The central claim is that gluonic lattice-QCD data, in the form of reduced pseudo–Ioffe-time distributions (RpITDs), can be fitted simultaneously with experimental Drell-Yan and leading-neutron electroproduction data in a global QCD analysis, and that doing so sharply improves knowledge of the pion's gluon PDF. With systematic corrections for higher-twist and discretization effects included, the lattice data are described with a reduced $\chi^2_{\rm lat}$ of 1.25 (a Z-score of 1.05), while the fit to experimental data is essentially unchanged ($\chi^2_{\rm exp}=0.86$). The resulting gluon distribution is slightly larger at high $x$ than in the experiment-only fit, with uncertainties reduced by over 50% at $x \gtrsim 0.5$. The paper reports that the pion gluon density is enhanced relative to the proton for $x \gtrsim 0.2$ and suppressed at small $x$, with a pion gluon momentum fraction $\langle x\rangle_g^{\pi} = 0.43(4)$ at $\mu = 2$ GeV close to the proton's range 0.40–0.42; since gluons carry three quarters of $\langle x\rangle_g$ of the hadron mass, both hadrons have roughly 30% of their mass from gluons.

Load-bearing premise

The result hinges on the assumption that the 35 included lattice RpITD points, all at one lattice spacing and a pion mass near 310 MeV, are fully described by leading-twist matching plus two truncated systematic terms, and that only the excluded $z=a$ and $z=2a$ points are unreliable.

Editorial extensions

If this is right

  • Uncertainties on the pion gluon PDF shrink by more than 50% for $x \gtrsim 0.5$, turning a part of the PDF that was essentially unconstrained into a quantity that can be compared with proton results.
  • The pion's gluon distribution lies above the proton's at large $x$ and below it at small $x$, while total gluon momentum fractions remain close ($0.43(4)$ versus about 0.40–0.42).
  • Gluons contribute roughly 30% of both the pion and proton mass through the relation $m_{\rm glue} = \frac{3}{4}\langle x\rangle_g$, a common gluonic structure across hadrons of very different masses.
  • Adding lattice data does not degrade the fit to experimental data, so the two sources of information are complementary rather than conflicting.

Reading between the lines

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

  • A direct stress test would be to repeat the lattice calculation at smaller lattice spacings and lighter pion masses; if the large-$x$ gluon enhancement persists, it is physical, and if it fades, it is likely a discretization artifact.
  • The same simultaneous-fit strategy could be applied to the kaon or to flavor-singlet quark distributions, where lattice RpITDs could fill kinematic regions experiments cannot reach.
  • If confirmed, the near-equality of pion and proton gluon momentum fractions would sharpen the puzzle of why a Goldstone boson and the proton share the same gluonic mass fraction despite very different masses.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents the first global QCD analysis of pion parton distribution functions that simultaneously fits experimental Drell-Yan and leading-neutron electroproduction data with lattice-QCD data on the gluonic reduced pseudo-Ioffe-time distribution (RpITD) from a single MILC ensemble at a valence pion mass of about 310 MeV and a single lattice spacing a = 0.1207 fm. The central result is that including the lattice data, with parametrized higher-twist and discretization corrections B1 and P1 truncated at N=2, reduces the uncertainty on the pion gluon PDF at x≳0.2 by more than 50% and yields a gluon density that is claimed to be clearly enhanced at large x relative to the proton. The experimental fit quality is unchanged (χ2_exp = 0.86) before and after the inclusion of lattice data, and the lattice fit improves from χ2_lat = 1.86 (Z = 2.96) to 1.25 (Z = 1.05) when the systematic terms are included. The authors also obtain ⟨x⟩π_g = 0.43(4) at µ = 2 GeV, which is similar to the proton values, and they interpret the balance between low-x suppression and high-x enhancement as pointing to a common gluonic mass fraction in the pion and proton.

Significance. If the result holds, the paper demonstrates that lattice RpITD data can constrain the pion gluon distribution in the large-x region where experiment is essentially insensitive, and the claimed large-x enhancement over the proton would be a novel hadron-structure observation with implications for the decomposition of the pion mass. The analysis is methodologically careful in several respects: it uses Bayesian posterior sampling with data resampling, it accounts for the strong correlations of the lattice data through the covariance-matrix eigendecomposition, and it verifies that the experimental fit is not distorted by the lattice input. The comparison of the pion gluon with three independent proton PDF sets and the extraction of the gluon momentum fraction are useful. However, the physical conclusion rests on the adequacy of the truncated systematic model and on a single lattice ensemble, so the result should be viewed as a strong hint rather than a definitively established finding at this stage.

major comments (3)
  1. [Lattice data, Eqs. (2)-(3); Fit results] The central claim of a large-x gluon enhancement depends on the N=2 truncation of the B1 and P1 systematic terms and on the exclusion of the z=a and z=2a data, but no stability analysis is presented. Because all lattice data come from a single lattice spacing a=0.1207 fm, an O(a^2) discretization artifact cannot be separated from the 1/z higher-twist term; the text itself states that the 1/z term 'cannot accurately account' for the discretization effects at the two smallest z values, which indicates that the model is known to be incomplete in exactly the short-distance region where this correction is largest. The reduction of χ2_lat from 1.86 to 1.25 is a global diagnostic and, given the high correlations among the lattice data, does not by itself validate the specific x>0.2 region that drives the enhancement. I ask the authors to repeat the fit with N=3 and N=4 and with zmin=2a and zmin=4a, and to report the resulting gluon PDFs. If the large-x enhancement is not robust to these variations, the claim should be downgraded from a 'clear enhancement' to an indication.
  2. [Lattice data (lattice ensemble description)] The lattice RpITD data are obtained at a valence pion mass of approximately 310 MeV, far from the physical pion mass, at one lattice spacing and one lattice volume. The comparison with proton PDFs at the physical point therefore assumes that the pion-mass dependence of the gluon PDF is negligible, but no assessment of this dependence is given. This is a systematic uncertainty that is not included in the quoted PDF bands. Please either estimate the pion-mass effect using available lattice data at other masses or a model-based correction, or explicitly state that the enhancement is established only at Mπ≈310 MeV and that a chiral extrapolation is required before a physical-pion comparison with the proton is made.
  3. [Fit results and Fig. 4] The phrase 'clear enhancement of gπ compared with gp at large x' is not supported by a quantitative significance statement. The figure shows the pion and proton gluon bands relative to a common reference, but no ratio with combined uncertainties is quoted, and the proton PDF sets themselves carry uncertainties. Since the experiment-only pion gluon PDF is essentially unconstrained at x>0.2, the apparent enhancement is driven entirely by the lattice data, so the statistical significance of the enhancement (for example, the posterior probability that the ratio gπ/gp exceeds 1 at x=0.5, including both pion and proton uncertainties) should be computed. Without such a statement, the abstract's claim of a 'clear enhancement' is overstated.
minor comments (5)
  1. [Lattice data (notation)] In Eq. (1), the definition M(ν,z^2) ≡ M(z,Pz)/M(z,0) divided by M(0,Pz)/M(0,0) is confusing because the left-hand side uses both ν and z^2 while the right-hand side uses z and Pz; please clarify the argument notation.
  2. [Lattice data (text)] 'approximately 1.3 M two-point correlators' should spell out that M denotes million.
  3. [Fig. 2 caption] The inset labeled 'rel. error' should be defined in the caption; I assume it is the 68% uncertainty relative to the median PDF, but this should be stated.
  4. [Fig. 4 caption] The caption should state explicitly that the proton bands are also divided by the same reference pion gluon PDF gref, and that the pion and proton uncertainties are shown separately; this is not immediately clear from the figure.
  5. [Global analysis (reproducibility)] The best-fit values (or posterior ranges) of the lattice systematic parameters {a, b, bn, pn} and the priors used for them are not reported; providing these in a supplemental table would improve reproducibility and allow readers to judge the size of the fitted corrections.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the pion gluon extraction is data-driven from external lattice and experimental inputs; only minor, non-load-bearing self-citations appear.

full rationale

The derivation chain is self-contained. The lattice RpITD data of Ref. [35] are raw reduced correlator ratios, not a previously fitted gluon PDF, and the experimental DY and LN data are external. The PDF parameters in Eq. (4) and the systematic coefficients {b_n, p_n, a} are optimized against the combined likelihood, and the reported large-x gluon enhancement is a fitted output, not an input. The N=2 truncation in Eq. (2) and the z >= 3a cut are explicit modeling choices; the paper candidly states that the z=a and 2a points cannot be accounted for by the 1/z P1(nu) term, which is a robustness limitation rather than a circular step. Self-citations to Refs. [34] and [35] involve overlapping authors, but they are not load-bearing in the logical sense: the lattice data are externally published raw data, and the systematic ansatz is stated in the text rather than imported as an unexamined theorem. No equation reduces the output to an input by construction.

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

The central extraction depends on the QCD factorization framework, the single-pion-exchange dominance assumption for the neutron data, and the parametrized model for lattice systematic effects. The fitted PDF shape and systematic coefficients are free parameters; no fully independent entity is introduced.

free parameters (5)
  • Valence quark PDF shape parameters (N_v, alpha_v, beta_v, gamma_v) = Not reported in text
    Parameters of the template f(x, mu0) used for the pion valence distribution; constrained by charge and baryon number conservation and fitted to experimental and lattice data.
  • Sea quark PDF shape parameters (N_s, alpha_s, beta_s; gamma_s=0) = Not reported in text
    Template parameters for the flavor-symmetric sea; gamma set to zero due to weak sensitivity.
  • Gluon PDF shape parameters (N_g, alpha_g, beta_g; gamma_g=0) = Not reported in text
    Template parameters for the pion gluon distribution; the main target of the lattice constraints.
  • Lattice systematic parameters (a, b, b1, b2, p1, p2) = Not reported in text
    Coefficients of the N=2 expansions of the higher-twist B1(nu) and discretization P1(nu) terms in Eq. (2); optimized in the global fit. Parametrization of the systematic terms.
  • Prior width on Jacobi parameter a = 0.75
    Width of the normal prior centered at 0 used for the parameter a in Eq. (3) to improve convergence.
assumptions (5)
  • domain assumption Leading-twist short-distance factorization and NLO matching in Eq. (1) relate the lattice RpITD to the pion gluon and quark-singlet PDFs.
    The pseudo-PDF framework is assumed to be valid at the accessed distances; incorrect matching would bias the extracted PDFs.
  • domain assumption H1 and ZEUS leading neutron data are dominated by single-pion exchange.
    Standard Sullivan-process assumption; if pion exchange is not dominant, the constraints on the pion PDF are invalid.
  • domain assumption The template parametrization in Eq. (4) is flexible enough to describe the true pion PDFs at the input scale.
    A too-rigid functional form would bias shape and uncertainty estimates.
  • domain assumption The z=a and z=2a lattice points can be excluded because the discretization model cannot describe them.
    The paper states discretization effects at small z 'cannot be accurately accounted for' by the systematic term; this is a modeling assumption, not a proven fact.
  • standard math Pion PDFs satisfy DGLAP evolution and the momentum/valence sum rules.
    Standard QCD evolution and sum rules are assumed in the global fit.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Pionic gluons from global QCD analysis of experimental and lattice data." pith.science (2026). https://pith.science/paper/PSRHK2ZX

@misc{pith2026250722730,
  author       = {Pith},
  title        = {Pith review of: Pionic gluons from global QCD analysis of experimental and lattice data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSRHK2ZX}},
  note         = {Machine review of arXiv:2507.22730}
}
abstract

We perform the first global QCD analysis of parton distribution functions (PDFs) in the pion, with lattice-QCD data on gluonic pseudo--Ioffe-time distributions fitted simultaneously with experimental Drell-Yan and leading neutron electroproduction data. Inclusion of the lattice results with parametrized systematic corrections significantly reduces the uncertainties on the gluon PDF at parton momentum fractions $x \gtrsim 0.2$, revealing a higher gluon density in the pion at large $x$ than in the proton. The similar gluon momentum fractions in the pion and proton further suggests a relative suppression of the pion gluon density at small $x$.

Figures

Figures reproduced from arXiv: 2507.22730 by the authors.

Figure 1
Figure 1. FIG. 1. ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Gluon PDFs [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. FIG. 3. 68% credibility interval bands for the valence (blue) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. First simultaneous global QCD analysis of kaon and pion parton distributions with lattice QCD constraints

    hep-ph 2025-10 conditional novelty 6.0 of 10

    First simultaneous pion+kaon PDF fit with lattice constraints finds kaon valence anti-up softer than pion's and strange valence more peaked at high x.

  2. New insights from the flavor dependence of quark transverse momentum distributions in the pion

    hep-ph 2025-09 conditional novelty 6.0 of 10

    First flavor-dependent extraction of unpolarized quark TMDs in the pion, finding a wider transverse momentum tail for valence d quarks than for sea quarks, with large uncertainties.

  3. Impact of parity-violating deep-inelastic scattering on the weak mixing angle and high-$x$ parton distributions

    hep-ph 2026-06 unverdicted novelty 4.0 of 10

    Simulations of 11 and 22 GeV PVDIS pseudodata indicate potential constraints on low-Q² sin²θ_W and high-x strange quark and d/u PDFs, with strong correlations requiring simultaneous QCD-electroweak analysis.

  4. Precision QCD with the Electron-Ion Collider

    hep-ph 2026-04 unverdicted novelty 1.0 of 10

    A workshop summary report outlines discussion topics in perturbative QCD, nuclear structure, and related techniques for the upcoming Electron-Ion Collider.

Reference graph

Works this paper leans on

59 extracted references · 13 canonical work pages · cited by 4 Pith papers

  1. [1]

    Betev et al

    B. Betev et al. , Z. Phys. C 28, 9 (1985)

    work page 1985

  2. [2]

    J. S. Conway et al. , Phys. Rev. D 39, 92 (1989)

    work page 1989

  3. [3]

    J. F. Owens, Phys. Rev. D 30, 943 (1984)

    work page 1984

  4. [4]

    Aurenche, R

    P. Aurenche, R. Baier, M. Fontannaz, M. N. Kienzle- Focacci, and M. Werlen, Phys. Lett. B 233, 517 (1989)

    work page 1989

  5. [5]

    P. J. Sutton, A. D. Martin, R. G. Roberts, and W. J. Stirling, Phys. Rev. D 45, 2349 (1992)

    work page 1992

  6. [6]

    Gluck, E

    M. Gluck, E. Reya, and A. Vogt, Z. Phys. C 53, 651 (1992)

    work page 1992

  7. [7]

    Gluck, E

    M. Gluck, E. Reya, and I. Schienbein, Eur. Phys. J. C 10, 313 (1999), arXiv:hep-ph/9903288

    arXiv 1999

  8. [8]

    Wijesooriya, P

    K. Wijesooriya, P. E. Reimer, and R. J. Holt, Phys. Rev. C72, 065203 (2005), arXiv:nucl-ex/0509012 [nucl-ex]

    arXiv 2005

Show all 59 references
  1. [9]

    Aicher, A

    M. Aicher, A. Schafer, and W. Vogelsang, Phys. Rev. Lett. 105, 252003 (2010), arXiv:1009.2481 [hep-ph]

    2010 arXiv

  2. [10]

    P. C. Barry, N. Sato, W. Melnitchouk, and C.-R. Ji, Phys. Rev. Lett. 121, 152001 (2018), arXiv:1804.01965 [hep-ph]

    2018 arXiv

  3. [11]

    P. C. Barry, C.-R. Ji, N. Sato, and W. Melnitchouk, Phys. Rev. Lett. 127, 232001 (2021), arXiv:2108.05822 [hep-ph]

    2021 arXiv

  4. [12]

    N. Y. Cao, P. C. Barry, N. Sato, and W. Melnitchouk, Phys. Rev. D 103, 114014 (2021), arXiv:2103.02159 [hep- ph]

    2021 arXiv

  5. [13]

    L. Kotz, A. Courtoy, P. Nadolsky, F. Olness, and M. Ponce-Chavez, Phys. Rev. D 109, 074027 (2024), arXiv:2311.08447 [hep-ph]

    2024 arXiv

  6. [14]

    L. Kotz, A. Courtoy, P. Nadolsky, and M. Ponce-Chavez, arXiv:2505.13594 [hep-ph]

    arXiv

  7. [15]

    F. D. Aaron et al. , Eur. Phys. J. C 68, 381 (2010), arXiv:1001.0532 [hep-ex]

    2010 arXiv

  8. [16]

    Chekanov et al

    S. Chekanov et al. , Nucl. Phys. B637, 3 (2002), arXiv:hep-ex/0205076

    2002 arXiv

  9. [17]

    Ji, Phys

    X. Ji, Phys. Rev. Lett. 110, 262002 (2013), arXiv:1305.1539 [hep-ph]

    2013 arXiv

  10. [18]

    X. Ji, Sci. China Phys. Mech. Astron. 57, 1407 (2014), arXiv:1404.6680 [hep-ph]

    2014 arXiv

  11. [19]

    Ma and J.-W

    Y.-Q. Ma and J.-W. Qiu, Phys. Rev. D 98, 074021 (2018), arXiv:1404.6860 [hep-ph]

    2018 arXiv

  12. [20]

    Ma and J.-W

    Y.-Q. Ma and J.-W. Qiu, Phys. Rev. Lett. 120, 022003 (2018), arXiv:1709.03018 [hep-ph]

    2018 arXiv

  13. [21]

    A. V. Radyushkin, Phys. Rev. D 96, 034025 (2017), arXiv:1705.01488 [hep-ph]

    2017 arXiv

  14. [22]

    Fan, Y.-B

    Z.-Y. Fan, Y.-B. Yang, A. Anthony, H.-W. Lin, and K.-F. Liu, Phys. Rev. Lett. 121, 242001 (2018), arXiv:1808.02077 [hep-lat]

    2018 arXiv

  15. [23]

    Z. Fan, R. Zhang, and H.-W. Lin, Int. J. Mod. Phys. A 36, 2150080 (2021), arXiv:2007.16113 [hep-lat]

    2021 arXiv

  16. [24]

    Fan and H.-W

    Z. Fan and H.-W. Lin, Phys. Lett. B 823, 136778 (2021), arXiv:2104.06372 [hep-lat]

    2021 arXiv

  17. [25]

    Khan et al

    T. Khan et al. , Phys. Rev. D 104, 094516 (2021), arXiv:2107.08960 [hep-lat]

    2021 arXiv

  18. [26]

    Salas-Chavira, Z

    A. Salas-Chavira, Z. Fan, and H.-W. Lin, Phys. Rev. D 106, 094510 (2022), arXiv:2112.03124 [hep-lat]

    2022 arXiv

  19. [27]

    Z. Fan, W. Good, and H.-W. Lin, Phys. Rev. D 108, 014508 (2023), arXiv:2210.09985 [hep-lat]

    2023 arXiv

  20. [28]

    Delmar, C

    J. Delmar, C. Alexandrou, K. Cichy, M. Constanti- nou, and K. Hadjiyiannakou, Phys. Rev. D 108, 094515 (2023), arXiv:2310.01389 [hep-lat]

    2023 arXiv

  21. [29]

    W. Good, K. Hasan, A. Chevis, and H.-W. Lin, Phys. Rev. D 109, 114509 (2024), arXiv:2310.12034 [hep-lat]

    2024 arXiv

  22. [30]

    Egerer et al

    C. Egerer et al. , Phys. Rev. D 106, 094511 (2022), arXiv:2207.08733 [hep-lat]

    2022 arXiv

  23. [31]

    T. Khan, T. Liu, and R. S. Sufian, Phys. Rev. D 108, 074502 (2023), arXiv:2211.15587 [hep-lat]

    2023 arXiv

  24. [32]

    Karpie, R

    J. Karpie, R. M. Whitehill, W. Melnitchouk, C. Mona- han, K. Orginos, J. W. Qiu, D. G. Richards, N. Sato, and S. Zafeiropoulos, Phys. Rev. D 109, 036031 (2024), arXiv:2310.18179 [hep-ph]

    2024 arXiv

  25. [33]

    NieMiera, W

    A. NieMiera, W. Good, and H.-W. Lin, (2025), arXiv:2506.03002 [hep-lat]

    2025 arXiv

  26. [34]

    P. C. Barry et al. , Phys. Rev. D 105, 114051 (2022), arXiv:2204.00543 [hep-ph]

    2022 arXiv

  27. [35]

    W. Good, K. Hasan, and H.-W. Lin, J. Phys. G 52, 035105 (2025), arXiv:2409.02750 [hep-lat]

    2025 arXiv

  28. [36]

    Balitsky, W

    I. Balitsky, W. Morris, and A. Radyushkin, Phys. Lett. B 808, 135621 (2020), arXiv:1910.13963 [hep-ph]

    2020 arXiv

  29. [37]

    Balitsky, W

    I. Balitsky, W. Morris, and A. Radyushkin, Phys. Rev. D 105, 014008 (2022), arXiv:2111.06797 [hep-ph]

    2022 arXiv

  30. [38]

    W. H. Morris III, Investigating Gluonic Operators in Co- ordinate Space, Ph.D. thesis, Old Dominion U. (2022)

    2022

  31. [39]

    Orginos, A

    K. Orginos, A. Radyushkin, J. Karpie, and S. Zafeiropoulos, Phys. Rev. D 96, 094503 (2017), arXiv:1706.05373 [hep-ph]

    2017 arXiv

  32. [40]

    Karpie, K

    J. Karpie, K. Orginos, A. Radyushkin, and 6 S. Zafeiropoulos, JHEP 11, 024 (2021), arXiv:2105.13313 [hep-lat]

    2021 arXiv

  33. [41]

    Bazavov et al

    A. Bazavov et al. , Phys. Rev. D 87, 054505 (2013), arXiv:1212.4768 [hep-lat]

    2013 arXiv

  34. [42]

    Follana, Q

    E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong, Phys. Rev. D 75, 054502 (2007), arXiv:hep-lat/0610092

    2007 arXiv

  35. [43]

    G. S. Bali, B. Lang, B. U. Musch, and A. Sch¨ afer, Phys. Rev. D93, 094515 (2016), arXiv:1602.05525 [hep-lat]

    2016 arXiv

  36. [44]

    Hasenfratz and F

    A. Hasenfratz and F. Knechtli, Phys. Rev. D 64, 034504 (2001), arXiv:hep-lat/0103029

    2001 arXiv

  37. [45]

    J. D. Sullivan, Phys. Rev. D 5, 1732 (1972)

    1972

  38. [46]

    Melnitchouk and A

    W. Melnitchouk and A. W. Thomas, Z. Phys. A 353, 311 (1995), arXiv:hep-ph/9508205

    1995 arXiv

  39. [47]

    J. R. McKenney, N. Sato, W. Melnitchouk, and C.-R. Ji, Phys. Rev. D 93, 054011 (2016), arXiv:1512.04459 [hep-ph]

    2016 arXiv

  40. [48]

    Cerutti, A

    M. Cerutti, A. Accardi, I. P. Fernando, S. Li, J. F. Owens, and S. Park, Phys. Rev. D 111, 094013 (2025), arXiv:2501.06849 [hep-ph]

    2025 arXiv

  41. [49]

    R. D. Ball et al. (NNPDF), Eur. Phys. J. C 82, 428 (2022), arXiv:2109.02653 [hep-ph]

    2022 arXiv

  42. [50]

    Cocuzza, N

    C. Cocuzza, N. T. Hunt-Smith, W. Melnitchouk, N. Sato, and A. W. Thomas, (2025), arXiv:2506.13616 [hep-ph]

    2025

  43. [51]

    Courtoy and P

    A. Courtoy and P. M. Nadolsky, Phys. Rev. D 103, 054029 (2021), arXiv:2011.10078 [hep-ph]

    2021 arXiv

  44. [52]

    Bailey, T

    S. Bailey, T. Cridge, L. A. Harland-Lang, A. D. Mar- tin, and R. S. Thorne, Eur. Phys. J. C 81, 341 (2021), arXiv:2012.04684 [hep-ph]

    2021 arXiv

  45. [53]

    Hou et al

    T.-J. Hou et al. , Phys. Rev. D 103, 014013 (2021), arXiv:1912.10053 [hep-ph]

    2021 arXiv

  46. [54]

    Ji, Phys

    X.-D. Ji, Phys. Rev. Lett. 74, 1071 (1995), arXiv:hep- ph/9410274

    1995

  47. [55]

    Adams et al

    B. Adams et al. , arXiv:1808.00848 [hep-ex]

    arXiv

  48. [56]

    Measurement of Tagged Deep Inelas- tic Scattering (TDIS),

    C. E. Keppel et al., “Measurement of Tagged Deep Inelas- tic Scattering (TDIS),” (2015), jefferson Lab Experiment PR12-15-006

    2015

  49. [57]

    Achenbach et al

    P. Achenbach et al. , Nucl. Phys. A1047, 122874 (2024), arXiv:2303.02579 [hep-ph]

    2024 arXiv

  50. [58]

    D. P. Anderle et al. , Front. Phys. (Beijing) 16, 64701 (2021), arXiv:2102.09222 [nucl-ex]

    2021 arXiv

  51. [59]

    R. G. Edwards and B. Joo, Nucl. Phys. Proc. Suppl. B140, 832 (2005), arXiv:hep-lat/0409003

    2005 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.