Pith. sign in

REVIEW 4 major objections 4 minor 38 references

This review argues that spatially nonlocal lattice QCD matrix elements, factorized perturbatively, now give access to the x-dependence of parton distribution functions.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-04 00:53 UTC pith:5CB3LYLZ

load-bearing objection Competent, well-organized encyclopedia review of lattice PDF methods; no new results, but a reliable map of the field whose central claim rests more on cited papers than on evidence presented here. the 4 major comments →

arxiv 2608.00251 v1 pith:5CB3LYLZ submitted 2026-07-31 hep-lat hep-exhep-phhep-th

Parton distribution functions from lattice QCD

Martha Constantinou , Krzysztof Cichy This is my paper
classification hep-lat hep-exhep-phhep-th PACS 12.38.Gc
keywords lattice QCDparton distribution functionsquasi-PDFspseudo-PDFsIoffe-time distributionshadron structurenonlocal operatorsshort-distance factorization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This review chapter argues that a decade of method development has removed the main obstacle to computing parton distribution functions (PDFs) on the lattice: PDFs are defined on the light cone, while lattice QCD lives in Euclidean spacetime. The authors show that spatially nonlocal matrix elements at finite hadron momentum can be factorized into light-cone PDFs through perturbative matching, in two complementary frameworks—large-momentum effective theory (quasi-PDFs) and short-distance factorization (pseudo-PDFs/Ioffe-time distributions). If this is right, lattice QCD is no longer limited to a few low Mellin moments; it can produce x-dependent quark and gluon distributions, for the proton, pion, kaon, and beyond leading twist. A sympathetic reader would care because these first-principles results can fill gaps in experimental and global-QCD knowledge of hadron structure.

Core claim

The central claim is that the Euclidean–light-cone obstruction is overcome by a controlled chain: compute boosted-hadron matrix elements of gauge-invariant Wilson-line operators at spatial separation z, renormalize them nonperturbatively, then apply a perturbative matching kernel—in momentum space in LaMET, in coordinate space in SDF—to obtain light-cone PDFs. The same matrix elements also feed a short-distance operator product expansion that yields Mellin moments. The authors survey the state of the field, including physical-quark-mass and continuum-extrapolated results for proton, pion, kaon, gluon, and twist-3 distributions, and conclude that lattice QCD has moved from proof-of-principle

What carries the argument

The load-bearing object is the spatially nonlocal Euclidean matrix element M(z,P3) = <P|ψ̄(0)Γ W(0,z) ψ(z)|P>, a quark bilinear separated by a straight Wilson line along the boost direction. In LaMET, its Fourier transform defines a quasi-PDF that equals the light-cone PDF up to a perturbative kernel and power corrections of order Λ²_QCD/P3². In SDF, the same matrix element is written as a function of Ioffe time ν = zP3 and matched at short distance z² with corrections z²Λ²_QCD. These two routes, plus the short-distance OPE for moments, are what convert Euclidean data into partonic structure.

Load-bearing premise

The whole program rests on the factorization relations holding with small power corrections at the finite hadron momenta and Wilson-line separations currently simulated—if Λ²_QCD/P3² and z²Λ²_QCD are not negligible at accessible boosts, the extracted x-dependence may still carry uncontrolled distortions.

What would settle it

Take a single isovector unpolarized PDF and repeat the extraction at two different boosts P3 (say 1.5 and 2.5 GeV) with full renormalization and matching; if the matched results disagree by more than the combined statistical and systematic uncertainties, the power-correction assumption is falsified. A second independent check is comparing LaMET and SDF analyses of the same lattice matrix elements: agreement within errors is required by the common light-cone limit.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • First-principles x-dependent PDFs become available for the proton, including flavor decomposition, helicity, transversity, and the separation of valence and sea.
  • Pion and kaon PDFs can be predicted where experimental data are sparse, directly probing flavor-SU(3) breaking in partonic structure.
  • Gluon PDFs, though noisier, are now being determined with continuum and physical-mass extrapolations, including the gluon helicity.
  • Lattice matrix elements can be included directly in global QCD analyses, providing complementary constraints to collider and fixed-target data.
  • The same nonlocal-operator machinery extends beyond leading twist, making twist-3 distributions such as g_T(x) and h_L(x) accessible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the momentum and distance corrections continue to shrink as boosts increase, lattice QCD could deliver PDFs in kinematic corners—large x, sea quarks, specific flavors—where global fits must currently rely on parametrization assumptions.
  • The complementarity of LaMET and SDF on identical matrix elements offers a built-in cross-check: agreement between the two after matching would provide a methodology-independent systematic budget.
  • The same factorization logic should apply to generalized and transverse-momentum-dependent distributions, potentially turning lattice QCD into a unified probe of one- and three-dimensional hadron structure.
  • A testable near-term milestone is precision comparison between lattice and experimental PDFs in a restricted window of x and scale, where both methods are reliable; agreement there would validate the whole chain.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This review chapter surveys the current status of lattice-QCD calculations of parton distribution functions (PDFs) through spatially nonlocal Euclidean matrix elements. The authors present the light-cone definitions of quark and gluon PDFs, the traditional Mellin-moment route, and the modern LaMET/quasi-PDF and SDF/pseudo-PDF frameworks. They discuss renormalization of Wilson-line operators, perturbative matching, finite-momentum and finite-distance corrections, reconstruction of the x-dependence from finite and noisy lattice data, and the extension of these methods to pion, kaon, gluon, and twist-3 distributions. The central claim, stated in §7, is that lattice QCD has moved from benchmark moments to a framework capable of systematically addressing partonic structure, with results ready to complement experimental data and global analyses.

Significance. If the status claim is correct, this is a useful and timely review: it lays out the multi-stage pipeline (boosted matrix elements, renormalization, matching, reconstruction) and clearly separates the LaMET and SDF methodologies. The authors explicitly name the main systematic uncertainties—excited states, finite-volume effects, discretization, power corrections, renormalon ambiguities, and the inverse problem—and they do not pretend that any single calculation controls all of them. The review is strongest in its conceptual organization and in its description of reconstruction methods. The manuscript is a review, so the absence of new calculations is not itself a flaw. The main weakness is quantitative: the pivotal assertion of ‘systematically improvable’ control is not backed by concrete evidence for the size of power corrections at currently accessible momenta and distances. The chapter contains no original analysis, code, or machine-checked derivations, but that is normal for this format; its value lies in synthesis, and the major gap lies in substantiating the field-maturity claim.

major comments (4)
  1. [Sec. 4, Eqs. (16) and (18)] The central claim of §7 rests on the assertion that the factorization relations (16) and (18) are under control at accessible kinematics. §4 states that “Agreement among different boosts after matching provides evidence that residual power corrections are under control,” but no such comparison is shown, no scaling test is reported, and no estimate of O(Λ_QCD^2/P3^2) or O(z^2 Λ_QCD^2) is given for the P3 ≲ 3 GeV and z up to ~1 fm used in current-state-of-the-art calculations. The renormalon discussion is mentioned only in passing. This is load-bearing: if power corrections are not controlled, the phrase “systematically improvable” is premature. Please add at least one explicit multi-boost or multi-z comparison from the cited literature, or soften the claim and identify the unresolved checks.
  2. [Secs. 6.1, 6.3] The review repeatedly characterizes calculations as “continuum-extrapolated,” “physical-mass,” and “NNLO-matched” without reporting final uncertainties or comparing lattice and phenomenological PDFs. The reader cannot verify the “systematically improvable” claim without quantitative anchors. A compact table for the most mature channels (e.g., proton nonsinglet, pion valence, gluon) listing m_pi, lattice spacing, P3 or zmax, matching order, and the dominant estimated systematic uncertainties would convert the narrative into an evidence-based assessment. This is a review, so the absence of original calculations is acceptable, but the absence of numerical context weakens support for the main conclusion.
  3. [Sec. 4, ‘Power corrections and renormalon effects’] The factorization relations (16) and (18) contain power corrections that are not fixed by the perturbative matching. The chapter acknowledges renormalon and higher-twist ambiguities and cites Braun et al. (2019), but it does not explain why these ambiguities do not spoil current extractions at the quoted momenta and distances. A short critical discussion of what is known from renormalon models or from LaMET-vs-SDF consistency would be needed to distinguish an optimistic projection from an established result. Without it, the statement that residual power corrections are controlled remains an unsubstantiated assertion.
  4. [Sec. 7, Summary and prospects] The conclusion states that lattice-QCD PDF calculations “are approaching the level of precision and systematic control needed for phenomenological impact.” In light of the limitations acknowledged earlier in the chapter—especially the open issues with power corrections, reconstruction, and excited-state contamination—this is stronger than the evidence presented. Either provide a representative uncertainty budget from the cited calculations or rephrase the conclusion as a direction that is ‘showing promise’ rather than one that has already reached the required control.
minor comments (4)
  1. [Abstract] The abstract says “As will be demonstrated,” but the chapter is a review and does not include a demonstration of the central claim; consider rewording to “as will be argued on the basis of the reviewed literature.”
  2. [Sec. 1, Introduction] Typo: “transerse-momentum-dependent distributions” should be “transverse-momentum-dependent distributions.”
  3. [Sec. 3, Eq. (13)] The sentence “The matrix element in Eq. (13) depends on the spatial separation z, the hadron momentum P3” is missing a conjunction; it should read “... depends on the spatial separation z and the hadron momentum P3.”
  4. [Sec. 5, Eqs. (16)–(18)] The matching kernels C_LaMET and C_SDF are not defined with their explicit arguments in the displayed equations. Adding the arguments (x/y, μ/P3) and (y, z^2 μ^2) would improve readability and avoid ambiguity in the subsequent discussion.

Circularity Check

0 steps flagged

No circular derivation: the review synthesizes published lattice-QCD results, and its central claim, while under-supported quantitatively, does not reduce to its own inputs.

full rationale

This is a review chapter, not a paper that fits parameters and then predicts them. No equation in the manuscript is shown to reduce to its own input by construction: Eqs. (16) and (18) are presented as literature factorization results with explicit power-correction terms, and the chapter does not claim to have derived them from the lattice data shown. The heavy presence of the authors' own collaborations in §6 (e.g., Alexandrou et al., Cichy, Constantinou, and co-workers) is real self-citation, but the narrative also relies on many independent groups (HadStruc, LPC, JAM, CLQCD, QCDSF), and no uniqueness theorem or forced-choice argument is imported from the authors' prior work to select the framework. The main weakness is evidential, not circular: the text states in §4 that 'Agreement among different boosts after matching provides evidence that residual power corrections are under control' without presenting such a comparison or a quantitative scaling test, and §7's claim that lattice QCD is 'approaching the level of precision and systematic control needed for phenomenological impact' is an assessment supported by citations rather than by new analysis. The chapter itself repeatedly acknowledges the remaining systematics (excited states, discretization, renormalon ambiguities, reconstruction ill-posedness, missing quantitative power-correction control). These are support gaps and would raise correctness risk, but they do not make the derivation circular. Consistent with the reader's take, a score of 2 reflects the density of non-load-bearing self-citation in the evidence base, not a construction-level circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

This chapter introduces no new free parameters or invented entities. Its narrative rests on standard QCD factorization and on matching relations imported from the cited literature. Any fitted parameters appear only in the original lattice papers, not in this review.

axioms (4)
  • standard math QCD collinear factorization: physical cross sections factorize into perturbative coefficient functions and universal PDFs, Eq. (1).
    Standard collinear factorization in QCD; motivates the universality of PDFs and is cited to review literature (§1).
  • domain assumption LaMET matching: a quasi-PDF at finite P3 shares the same infrared physics as the light-cone PDF, with power corrections of order Λ_QCD^2/P3^2 and M^2/P3^2, Eq. (16).
    Central to one of the two reviewed methods; imported from cited papers by Ji, Xiong, Izubuchi and others (§3).
  • domain assumption SDF/OPE: short-distance Euclidean Ioffe-time distributions factorize into light-cone Ioffe-time distributions with corrections of order z^2 Λ_QCD^2, Eqs. (17)–(20).
    Central to the pseudo-PDF/short-distance factorization program; imported from cited work by Radyushkin, Orginos and others (§3).
  • domain assumption Multiplicative renormalizability of straight Wilson-line operators after subtraction of the linear power divergence.
    Needed to define renormalized nonlocal matrix elements; presented as established in §4, citing Ji–Zhang, Chen–Ji–Zhang, Ishikawa et al.

pith-pipeline@v1.3.0-alltime-deepseek · 32076 in / 11255 out tokens · 105460 ms · 2026-08-04T00:53:28.552266+00:00 · methodology

0 comments
read the original abstract

Parton distribution functions (PDFs) provide one of the most direct ways to describe the partonic structure of hadrons in QCD. They encode nonperturbative information about quarks, antiquarks, and gluons as functions of the partonic momentum fraction $x$, and they connect this microscopic structure to experimentally measurable high-energy scattering processes through QCD factorization. This makes PDFs interesting to pursue with lattice QCD, which provides a first-principles formulation of the strong interaction. The challenge is that PDFs are defined through light-cone correlations, while lattice QCD is formulated in Euclidean spacetime. This chapter introduces the theoretical foundations and current status of lattice-QCD calculations of PDFs, with emphasis on modern approaches based on spatially nonlocal matrix elements. We first review the light-cone definitions of quark and gluon PDFs, their Mellin moments, and the connection to QCD factorization. We then explain how large-momentum effective theory, short-distance factorization, and the short-distance operator product expansion make it possible to relate Euclidean lattice observables to light-cone partonic structure. Particular attention is given to the elements that have improved over the last five years: renormalization of Wilson-line operators, perturbative matching, finite-momentum and finite-distance effects, reconstruction of the $x$ dependence, and systematic uncertainties. We summarize selected lattice results for proton quark PDFs, pion and kaon PDFs, gluon PDFs, and twist-3 distributions, highlighting both recent progress and remaining challenges. As will be demonstrated, lattice QCD is moving from proof-of-principle calculations toward systematically improvable determinations that can complement experimental data and global QCD analyses in mapping the partonic structure of hadrons.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

38 extracted references · 27 linked inside Pith

  1. [1]

    Model independent determination of the light cone wave functions for exclusive processes.Phys

    Aglietti U, Ciuchini M, Corbo G, Franco E, Martinelli G and Silvestrini L (1998). Model independent determination of the light cone wave functions for exclusive processes.Phys. Lett. B441: 371–375. doi:10.1016/S0370-2693(98)01138-1.hep-ph/9806277. Alexandrou C, Cichy K, Drach V, Garcia-Ramos E, Hadjiyiannakou K, Jansen K, Steffens F and Wiese C (2015). La...

    Pith/arXiv arXiv 1998

  2. [2]

    Good W, Hasan K, Chevis A and Lin HW (2023)

    Nucleon Parton Distribution Functions from Boosted Correlations in the Coulomb gauge2602.11283. Good W, Hasan K, Chevis A and Lin HW (2023). The Gluon Moment and Parton Distribution Function of the Pion fromN f = 2 + 1 + 1Lattice QCD 2310.12034. Good W, Hasan K and Lin HW (2025). Toward the First Gluon Parton Distribution from the LaMET.J. Phys. G52 (3): ...

    arXiv 2023

  3. [3]

    Chen LB, Wang W and Zhu R (2021)

    Lattice Calculation of Parton Distribution Function from LaMET at Physical Pion Mass with Large Nucleon Momentum1803.04393. Chen LB, Wang W and Zhu R (2021). Next-to-Next-to-Leading Order Calculation of Quasiparton Distribution Functions.Phys. Rev. Lett.126 (7): 072002. doi:10.1103/PhysRevLett.126.072002.2006.14825. Chen C, Dong H, Liu L, Sun P, Xiong X, ...

    Pith/arXiv arXiv 2021

  4. [5]

    Detmold W and Lin CJD (2006)

    Mellin Moments of the Unpolarized Gluon PDF in the Proton from Nonlocal Operators in Lattice QCD2605.30193. Detmold W and Lin CJD (2006). Deep-inelastic scattering and the operator product expansion in lattice QCD.Phys. Rev. D73: 014501. doi:10.1103/ PhysRevD.73.014501.hep-lat/0507007. Dotsenko VS and Vergeles SN (1980). Renormalizability of Phase Factors...

    Pith/arXiv arXiv 2006

  5. [6]

    Green J, Jansen K and Steffens F (2018)

    Direct calculation of parton distributions in momentum space from lattice QCD 2606.16877. Green J, Jansen K and Steffens F (2018). Nonperturbative Renormalization of Nonlocal Quark Bilinears for Parton Quasidistribution Functions on the Lattice Using an Auxiliary Field.Phys. Rev. Lett.121 (2): 022004. doi:10.1103/PhysRevLett.121.022004.1707.07152. Green J...

    arXiv 2018

  6. [10]

    Chou CY and Chen JW (2022)

    Unpolarized gluon PDF of the nucleon from lattice QCD in the continuum limit2510.26425. Chou CY and Chen JW (2022). One-loop hybrid renormalization matching kernels for quasiparton distributions.Phys. Rev. D106 (1): 014507. doi: 10.1103/PhysRevD.106.014507.2204.08343. Chowdhury TA, Izubuchi T, Kamruzzaman M, Karthik N, Khan T, Liu T, Paul A, Schoenleber J...

    Pith/arXiv arXiv 2022

  7. [11]

    20Parton distribution functions from lattice QCD Lin HW, Chen JW, Fan Z, Zhang JH and Zhang R (2021)

    Lattice Nucleon Isovector Unpolarized Parton Distribution in the Physical-Continuum Limit2011.14971. 20Parton distribution functions from lattice QCD Lin HW, Chen JW, Fan Z, Zhang JH and Zhang R (2021). Valence-Quark Distribution of the Kaon and Pion from Lattice QCD.Phys. Rev. D103 (1): 014516. doi:10.1103/PhysRevD.103.014516.2003.14128. Liu KF and Dong ...

    Pith/arXiv arXiv 2021

  8. [16]

    Egerer C, Edwards RG, Kallidonis C, Orginos K, Radyushkin A V, Richards DG, Romero E and Zafeiropoulos S (HadStruc) (2021)

    doi:10.1007/ JHEP03(2023)086.2211.04434. Egerer C, Edwards RG, Kallidonis C, Orginos K, Radyushkin A V, Richards DG, Romero E and Zafeiropoulos S (HadStruc) (2021). Towards high-precision parton distributions from lattice QCD via distillation.JHEP11:

    Pith/arXiv arXiv 2023

  9. [17]

    Cichy K and Constantinou M (2019)

    doi:10.22323/1.396.0017.2110.07440. Cichy K and Constantinou M (2019). A guide to light-cone PDFs from Lattice QCD: an overview of approaches, techniques and results.Adv. High Energy Phys.2019: 3036904. doi:10.1155/2019/3036904.1811.07248. 18Parton distribution functions from lattice QCD Cichy K, Del Debbio L and Giani T (2019). Parton distributions from ...

    arXiv 2019

  10. [21]

    Jo´ o B, Karpie J, Orginos K, Radyushkin A, Richards D and Zafeiropoulos S (2019a)

    Asymptotic Long-Distance Expansion of Euclidean Correlators in Lattice Parton Applications2601.12189. Jo´ o B, Karpie J, Orginos K, Radyushkin A, Richards D and Zafeiropoulos S (2019a). Parton Distribution Functions from Ioffe time pseudo-distributions. JHEP12:

    arXiv

  11. [24]

    Karpie J, Whitehill RM, Melnitchouk W, Monahan C, Orginos K, Qiu JW, Richards DG, Sato N and Zafeiropoulos S (Jefferson Lab Angular Momentum, HadStruc) (2024)

    doi:10.1007/JHEP11(2021)024.2105.13313. Karpie J, Whitehill RM, Melnitchouk W, Monahan C, Orginos K, Qiu JW, Richards DG, Sato N and Zafeiropoulos S (Jefferson Lab Angular Momentum, HadStruc) (2024). Gluon helicity from global analysis of experimental data and lattice QCD Ioffe time distributions.Phys. Rev. D109 (3): 036031. doi:10.1103/PhysRevD.109.03603...

    Pith/arXiv arXiv 2021

  12. [28]

    Ma YQ and Qiu JW (2018a)

    Nucleon Transversity Distribution at the Physical Pion Mass from Lattice QCD1810.05043. Ma YQ and Qiu JW (2018a). Exploring Partonic Structure of Hadrons Using ab initio Lattice QCD Calculations.Phys. Rev. Lett.120 (2): 022003. doi:10.1103/PhysRevLett.120.022003.1709.03018. Ma YQ and Qiu JW (2018b). Extracting Parton Distribution Functions from Lattice QC...

    Pith/arXiv arXiv 1968

  13. [30]

    Monahan C and Orginos K (2017)

    Mellin Moments of Pion and Kaon Unpolarized PDFs from Nonlocal Operators in Lattice QCD2606.28102. Monahan C and Orginos K (2017). Quasi parton distributions and the gradient flow.JHEP03:

    Pith/arXiv arXiv 2017

  14. [32]

    Nocera ER (2026), 7, Global analyses of helicity-dependent parton distribution functions,2607.24613

    Systematic Study of the Self-Renormalized Nucleon Gluon PDF in Large-Momentum Effective Theory2511.14708. Nocera ER (2026), 7, Global analyses of helicity-dependent parton distribution functions,2607.24613. Orginos K, Radyushkin A, Karpie J and Zafeiropoulos S (2017). Lattice QCD exploration of parton pseudo-distribution functions.Phys. Rev. D96 (9): 0945...

    arXiv 2026

  15. [33]

    Salas-Chavira A, Fan Z and Lin HW (2022)

    Kinematic enhancement for nucleon interpolators2606.02447. Salas-Chavira A, Fan Z and Lin HW (2022). First Glimpse into the Kaon Gluon Parton Distribution Using Lattice QCD.Phys. Rev. D106 (9): 094510. doi:10.1103/PhysRevD.106.094510.2112.03124. Shindler A (2024). Moments of parton distribution functions of any order from lattice QCD.Phys. Rev. D110 (5): ...

    Pith/arXiv arXiv 2022

  16. [35]

    Sufian RS, Karpie J, Egerer C, Orginos K, Qiu JW and Richards DG (2019)

    From the quark parton model to QCD2606.27618. Sufian RS, Karpie J, Egerer C, Orginos K, Qiu JW and Richards DG (2019). Pion Valence Quark Distribution from Matrix Element Calculated in Lattice QCD.Phys. Rev. D99 (7): 074507. doi:10.1103/PhysRevD.99.074507.1901.03921. Sufian RS, Egerer C, Karpie J, Edwards RG, Jo´ o B, Ma YQ, Orginos K, Qiu JW and Richards...

    Pith/arXiv arXiv 2019

  17. [36]

    Yao F and et al

    doi:10.1007/JHEP11(2023)021.2212.14415. Yao F and et al. (Lattice Parton) (2023b). Nucleon Transversity Distribution in the Continuum and Physical Mass Limit from Lattice QCD.Phys. Rev. Lett.131 (26): 261901. doi:10.1103/PhysRevLett.131.261901.2208.08008. Zhang J, Chen JW, Jin L, Lin HW, Liu YS, Sch¨ afer A, Yang YB and Zhao Y (2018a). Direct lattice-QCD ...

    Pith/arXiv arXiv 2023

  18. [38]

    Zimmermann C and Sch¨ afer A (2024)

    Parton Distribution Functions from Large Momentum Expansion of Current-Current Correlators2605.03234. Zimmermann C and Sch¨ afer A (2024). Valence quark PDFs of the proton from two-current correlations in lattice QCD.Phys. Rev. D110 (7): 074503. Parton distribution functions from lattice QCD21 doi:10.1103/PhysRevD.110.074503.2405.07712

    Pith/arXiv arXiv 2024

  19. [57]

    Karpie J, Orginos K, Radyushkin A and Zafeiropoulos S (HadStruc) (2021)

    doi:10.1007/JHEP04(2019)057.1901.05408. Karpie J, Orginos K, Radyushkin A and Zafeiropoulos S (HadStruc) (2021). The continuum and leading twist limits of parton distribution functions in lattice QCD.JHEP11:

    Pith/arXiv arXiv 2019

  20. [61]

    Dutrieux H, Karpie J, Orginos K and Zafeiropoulos S (2025)

    doi:10.1007/JHEP04(2024)061.2310.19926. Dutrieux H, Karpie J, Orginos K and Zafeiropoulos S (2025). Simple nonparametric reconstruction of parton distributions from limited Fourier information. Phys. Rev. D111 (3): 034515. doi:10.1103/PhysRevD.111.034515.2412.05227. Edwards RG and et al. (HadStruc) (2023). Non-singlet quark helicity PDFs of the nucleon fr...

    Pith/arXiv arXiv 2024

  21. [77]

    Constantinou M and Panagopoulos H (2017)

    doi:10.1140/epja/s10050-021-00353-7.2010.02445. Constantinou M and Panagopoulos H (2017). Perturbative renormalization of quasi-parton distribution functions.Phys. Rev. D96 (5): 054506. doi: 10.1103/PhysRevD.96.054506.1705.11193. Constantinou M and Panagopoulos H (2023). Improved renormalization scheme for nonlocal operators.Phys. Rev. D107 (1): 014503. d...

    Pith/arXiv arXiv 2010

  22. [81]

    Jo´ o B, Karpie J, Orginos K, Radyushkin A V, Richards DG, Sufian RS and Zafeiropoulos S (2019b)

    doi:10.1007/JHEP12(2019)081.1908.09771. Jo´ o B, Karpie J, Orginos K, Radyushkin A V, Richards DG, Sufian RS and Zafeiropoulos S (2019b). Pion valence structure from Ioffe-time parton pseudodistribution functions.Phys. Rev. D100 (11): 114512. doi:10.1103/PhysRevD.100.114512.1909.08517. Jo´ o B, Karpie J, Orginos K, Radyushkin A V, Richards DG and Zafeirop...

    Pith/arXiv arXiv 2019

  23. [86]

    Brice˜ no RA and Monahan CJ (2021)

    doi:10.1007/ JHEP05(2021)086.2103.12105. Brice˜ no RA and Monahan CJ (2021). A model-independent framework for determining finite-volume effects of spatially nonlocal operators.Phys. Rev. D 103 (9): 094521. doi:10.1103/PhysRevD.103.094521.2102.01814. Brice˜ no RA, Hansen MT and Monahan CJ (2017). Role of the Euclidean signature in lattice calculations of ...

    Pith/arXiv arXiv 2021

  24. [87]

    Braun VM, Ji Y and Vladimirov A (2021b)

    doi:10.1007/ JHEP10(2021)087.2108.03065. Braun VM, Ji Y and Vladimirov A (2021b). QCD factorization for twist-three axial-vector parton quasidistributions.JHEP05:

    Pith/arXiv arXiv 2021

  25. [95]

    Alexandrou C, Cichy K, Constantinou M, Jansen K, Scapellato A and Steffens F (2018c)

    doi:10.22323/1.334.0095.1811.01589. Alexandrou C, Cichy K, Constantinou M, Jansen K, Scapellato A and Steffens F (2018c). Light-Cone Parton Distribution Functions from Lattice QCD. Phys. Rev. Lett.121 (11): 112001. doi:10.1103/PhysRevLett.121.112001.1803.02685. Alexandrou C, Cichy K, Constantinou M, Jansen K, Scapellato A and Steffens F (2018d). Transvers...

    arXiv 2019

  26. [108]

    Zhang JH, Chen JW and Monahan C (2018b)

    doi:10.22323/1.334.0108. Zhang JH, Chen JW and Monahan C (2018b). Parton distribution functions from reduced Ioffe-time distributions.Phys. Rev. D97 (7): 074508. doi:10.1103/PhysRevD.97.074508.1801.03023. Zhang JH, Chen JW, Jin L, Lin HW, Sch¨ afer A and Zhao Y (2019). First direct lattice-QCD calculation of thex-dependence of the pion parton distribution...

    arXiv 2019

  27. [110]

    Soper DE (2026),

    doi:10.22323/1.334.0110.1811.06074. Soper DE (2026),

    arXiv 2026

  28. [116]

    NieMiera A, Good W and Lin HW (2025a)

    doi:10.1007/JHEP03(2017)116.1612.01584. NieMiera A, Good W and Lin HW (2025a). Kaon Gluon Parton Distribution and Momentum Fraction from 2+1+1 Lattice-QCD with High Statistics. Phys. Rev. D112 (7): 074504. doi:10.1103/89mq-gx33.2506.03002. NieMiera A, Good W, Lin HW and Yao F (2025b). First Self-Renormalized Gluon PDF of Nucleon from Large-Momentum Effect...

    Pith/arXiv arXiv 2017

  29. [137]

    1907.06037

    doi:10.1007/JHEP10(2019)137. 1907.06037. Constantinou M (2021). The x-dependence of hadronic parton distributions: A review on the progress of lattice QCD.Eur. Phys. J. A57 (2):

    Pith/arXiv arXiv 2019

  30. [138]

    Delmar J, Alexandrou C, Cichy K, Constantinou M and Hadjiyiannakou K (2023)

    doi:10.1007/JHEP02(2021)138.2010.03996. Delmar J, Alexandrou C, Cichy K, Constantinou M and Hadjiyiannakou K (2023). Gluon PDF of the proton using twisted mass fermions.Phys. Rev. D 108 (9): 094515. doi:10.1103/PhysRevD.108.094515.2310.01389. Delmar J, Cichy K, Constantinou M and Zhao Y (2026),

    Pith/arXiv arXiv 2021

  31. [148]

    Egerer C, Jo´ o B, Karpie J, Karthik N, Khan T, Monahan CJ, Morris W, Orginos K, Radyushkin A, Richards DG, Romero E, Sufian RS and Zafeiropoulos S (HadStruc) (2022a)

    doi:10.1007/JHEP11(2021)148.2107.05199. Egerer C, Jo´ o B, Karpie J, Karthik N, Khan T, Monahan CJ, Morris W, Orginos K, Radyushkin A, Richards DG, Romero E, Sufian RS and Zafeiropoulos S (HadStruc) (2022a). Towards the determination of the gluon helicity distribution in the nucleon from lattice quantum chromodynamics.Phys. Rev. D106 (9): 094511. doi:10.1...

    Pith/arXiv arXiv 2021

  32. [161]

    Barry PC and et al

    doi:10.21468/ SciPostPhysProc.8.161.2106.01916. Barry PC and et al. (Jefferson Lab Angular Momentum (JAM), HadStruc) (2022). Complementarity of experimental and lattice QCD data on pion parton distributions.Phys. Rev. D105 (11): 114051. doi:10.1103/PhysRevD.105.114051.2204.00543. Bhat M, Cichy K, Constantinou M and Scapellato A (2021). Flavor nonsinglet p...

    Pith/arXiv arXiv 2022

  33. [178]

    Karpie J, Orginos K, Rothkopf A and Zafeiropoulos S (2019)

    doi:10.1007/JHEP11(2018)178.1807.10933. Karpie J, Orginos K, Rothkopf A and Zafeiropoulos S (2019). Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to Neural Networks.JHEP04:

    Pith/arXiv arXiv 2018

  34. [182]

    Miller J, Torsiello J, Anderson I, Cichy K, Constantinou M, Delmar J and Lampreich S (2026a)

    doi:10.1007/JHEP04(2026)182.2510.21041. Miller J, Torsiello J, Anderson I, Cichy K, Constantinou M, Delmar J and Lampreich S (2026a). Pion and kaon PDFs from lattice QCD via large momentum effective theory and short-distance factorization.Phys. Rev. D113 (11): 114506. doi:10.1103/xz1w-8vgf.2512.06121. Miller J, Torsiello J, Cichy K, Constantinou M and Del...

    arXiv 2026

  35. [210]

    Cichy K (2022a)

    doi:10.1007/JHEP05(2026)210.2509.15931. Cichy K (2022a). Overview of lattice calculations of the x-dependence of PDFs, GPDs and TMDs.EPJ Web Conf.258: 01005. doi:10.1051/epjconf/ 202225801005.2111.04552. Cichy K (2022b). Progress inx-dependent partonic distributions from lattice QCD.PoSLATTICE2021:

    Pith/arXiv arXiv 2026

  36. [879]

    Braun VM (2022)

    doi:10.1103/PhysRevD.24.879. Braun VM (2022). Higher Twists.EPJ Web Conf.274: 01012. doi:10.1051/epjconf/202227401012.2212.02887. Braun V and M¨ uller D (2008). Exclusive processes in position space and the pion distribution amplitude.Eur. Phys. J. C55: 349–361. doi:10.1140/ epjc/s10052-008-0608-4.0709.1348. Braun V, Gornicki P and Mankiewicz L (1995). Io...

    arXiv 2022

  37. [1125]

    Hatta Y (2026), 7, Spin Structure of the Nucleon: Overview,2607.20761

    doi:10.1140/epjc/s10052-023-11949-2.2212.11107. Hatta Y (2026), 7, Spin Structure of the Nucleon: Overview,2607.20761. Holligan J and Lin HW (2024a). Nucleon helicity parton distribution function in the continuum limit with self-renormalization.Phys. Lett. B854: 138731. doi:10.1016/j.physletb.2024.138731.2405.18238. Holligan J and Lin HW (2024b). Pion val...

    Pith/arXiv arXiv 2026

  38. [2901]

    Multiplicative Renormalizability of Operators defining Quasiparton Distributions.Phys

    Li ZY, Ma YQ and Qiu JW (2019). Multiplicative Renormalizability of Operators defining Quasiparton Distributions.Phys. Rev. Lett.122 (6): 062002. doi:10.1103/PhysRevLett.122.062002.1809.01836. Li ZY, Ma YQ and Qiu JW (2021). Extraction of Next-to-Next-to-Leading-Order Parton Distribution Functions from Lattice QCD Calculations.Phys. Rev. Lett.126 (7): 072...

    arXiv 2019