pith. sign in

arxiv: 2605.30193 · v1 · pith:DPKP5JGJnew · submitted 2026-05-28 · ✦ hep-lat · hep-ph· hep-th· nucl-th

Mellin Moments of the Unpolarized Gluon PDF in the Proton from Nonlocal Operators in Lattice QCD

Joseph Delmar , Krzysztof Cichy , Martha Constantinou , Yong Zhao This is my paper

Pith reviewed 2026-06-28 23:27 UTC · model grok-4.3

classification ✦ hep-lat hep-phhep-thnucl-th
keywords lattice QCDgluon PDFMellin momentsIoffe-time distributionnonlocal operatorsparton distributionsproton structure
0
0 comments X

The pith

Lattice QCD determines the ratio of the third to first Mellin moment for the gluon PDF at 2 GeV.

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

This paper uses lattice QCD with nonlocal gluon operators in momentum-boosted proton states to compute ratios of higher Mellin moments of the unpolarized gluon PDF to the first moment. The work is performed on an Nf=2+1+1 ensemble of twisted mass fermions at a pion mass of roughly 260 MeV. Within the short-distance OPE of the reduced gluon Ioffe-time distribution, the authors extract these ratios while examining truncation order, Wilson-line separation choices, and quark-singlet mixing under perturbative matching. They report a value for ⟨x³⟩_g/⟨x⟩_g at 2 GeV that folds in statistical errors and the leading theoretical systematics. A sympathetic reader cares because the result supplies a first-principles anchor for the gluon momentum distribution inside the proton that experiments access only indirectly.

Core claim

Using matrix elements of nonlocal gluon operators coupled to boosted proton states on an Nf=2+1+1 ensemble, the short-distance OPE of the reduced gluon Ioffe-time distribution yields the ratio ⟨x³⟩_g/⟨x⟩_g at a scale of 2 GeV, with uncertainties that account for both statistical and the dominant theoretical systematic uncertainties.

What carries the argument

The short-distance operator product expansion (OPE) of the reduced gluon Ioffe-time distribution, which converts the lattice nonlocal matrix elements into ratios of Mellin moments.

If this is right

  • The ratio supplies an additional constraint on the x-dependence of the gluon PDF beyond the total momentum fraction.
  • DGLAP evolution of the moments permits a direct check of perturbative truncation effects by varying the renormalization scale.
  • The same OPE framework can be applied to extract further moments n>3 once more terms are retained.
  • Quantified dependence on the minimum and maximum Wilson-line separations sets a benchmark for controlling finite-separation systematics in future runs.

Where Pith is reading between the lines

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

  • The lattice ratio can be inserted into global PDF analyses to test whether current parametrizations of the gluon at moderate x are consistent with first-principles input.
  • Repeating the calculation at the physical pion mass would isolate the size of the chiral extrapolation uncertainty present at 260 MeV.
  • A side-by-side comparison with moments inferred from jet or heavy-flavor production data would expose any tension between lattice and phenomenological determinations.

Load-bearing premise

The short-distance operator product expansion of the reduced gluon Ioffe-time distribution remains accurate after truncation at the order used, for the chosen minimum and maximum Wilson-line separations.

What would settle it

If extending the OPE to higher orders or altering the Wilson-line separation window shifts the central value of ⟨x³⟩_g/⟨x⟩_g outside the reported uncertainties, the truncation assumption is falsified.

Figures

Figures reproduced from arXiv: 2605.30193 by Joseph Delmar, Krzysztof Cichy, Martha Constantinou, Yong Zhao.

Figure 1
Figure 1. Figure 1: FIG. 1. Left: Bare matrix elements as a function of the length of the Wilson line, [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fixed-order results for the ratio [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A comparison of ratios [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A comparison of moments extracted for different systematics and evolved from initial scale [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: at κ = 1/ √ 2, 1, √ 2. The band is taken at κ = 1, while the variation under κ → 1/ √ 2, √ 2 provides an estimate of the perturbative uncertainty associated with matching-scale variation, in addition to the statistical uncertainties. 0.8 1.0 1.2 1.4 0.00 0.01 0.02 0.03 0.04 0.05 ­ x 3 ® = ­ x ® nmax = 4; z 2 (0:186 fm; 0:279 fm) FIG. 5. Comparison of ratios ⟨x 3 ⟩g/⟨x⟩g for gluon only at final choice of z-… view at source ↗
Figure 6
Figure 6. Figure 6: , which incorporates scale evolution. In particular, we select the analysis with (zmin, zmax) = (2a, 3a) as our preferred window, and truncation order nmax = 4, which provides an optimal choice between perturbative control, stability of the extracted moments, and statistical precision. The corresponding evolved results accounting for mixing effects are presented as points in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. A comparison of ratios [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

We present a lattice QCD determination of the Mellin moments of the unpolarized gluon parton distribution function in the proton. The analysis is based on matrix elements of nonlocal gluon operators coupled to momentum-boosted proton states. The calculation relies on an $N_f=2+1+1$ ensemble of maximally twisted mass fermions with clover improvement and the Iwasaki-improved gauge action, at a pion mass of approximately 260 MeV. Working within the short-distance operator product expansion (OPE) of the reduced gluon Ioffe-time distribution, we extract ratios of higher-order gluon moments, $\langle x^n\rangle$ with $n>1$, to the gluon momentum fraction, $\langle x\rangle$. We investigate systematic effects associated with the truncation of the order of moment in the OPE, the choice of minimum and maximum Wilson-line separations entering the analysis, and the treatment of mixing with the quark-singlet under perturbative matching. The stability of the extracted moments is further studied under scale evolution using DGLAP equations, allowing us to assess uncertainties related to perturbative truncation by varying the scale. Our work provides a determination of the ratio $\langle x^3\rangle_g/\langle x\rangle_g$ at a scale of 2 GeV, with uncertainties that account for both statistical and the dominant theoretical systematic uncertainties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a lattice QCD determination of ratios of Mellin moments of the unpolarized gluon PDF, specifically ⟨x³⟩_g/⟨x⟩_g at 2 GeV, extracted from matrix elements of nonlocal gluon operators on an N_f=2+1+1 twisted-mass clover ensemble at m_π≈260 MeV. The extraction uses the short-distance OPE applied to the reduced gluon Ioffe-time distribution, with reported investigations of OPE truncation order, Wilson-line separation cuts, quark-singlet mixing under perturbative matching, and stability under DGLAP scale evolution to assess perturbative truncation effects.

Significance. If robust, the result supplies a non-perturbative lattice anchor for a higher gluon moment ratio that can be compared directly to global PDF analyses and used to constrain the gluon contribution to the proton's momentum. The explicit checks on multiple sources of systematic uncertainty and the use of scale variation to probe perturbative truncation represent concrete strengths in an area where such controls are essential.

major comments (2)
  1. [Abstract] Abstract: the statement that the quoted uncertainties 'account for both statistical and the dominant theoretical systematic uncertainties' cannot be verified from the provided text, as no numerical tables, extracted central values, or detailed error budgets are shown; this directly affects the load-bearing claim that the ratio is determined with controlled systematics.
  2. [OPE analysis] OPE analysis (described in the abstract and method): the truncation error on the short-distance OPE of the reduced gluon Ioffe-time distribution at the accessed Wilson-line separations is assessed only via internal consistency checks (order variation, min/max cuts, scale evolution); because the OPE is asymptotic, these checks do not independently constrain the size of omitted higher-order coefficients, leaving open the possibility that the reported systematic band underestimates the true truncation uncertainty at z∼0.2–0.6 fm.
minor comments (1)
  1. [Abstract] The abstract refers to 'the choice of minimum and maximum Wilson-line separations entering the analysis' without quoting the numerical range or the criterion used to select them; adding these values would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive comments on our manuscript. We address each major comment below. Where the comments identify areas for improved clarity or documentation, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the statement that the quoted uncertainties 'account for both statistical and the dominant theoretical systematic uncertainties' cannot be verified from the provided text, as no numerical tables, extracted central values, or detailed error budgets are shown; this directly affects the load-bearing claim that the ratio is determined with controlled systematics.

    Authors: We agree that the abstract's claim regarding the uncertainties cannot be directly verified without the supporting numerical results and error budget. The full manuscript presents the extracted ratio ⟨x³⟩_g/⟨x⟩_g together with statistical and systematic variations in Section 4 and the associated figures, but a consolidated error-budget table was not included. We will add a new table that lists the central value, statistical uncertainty, and individual contributions from OPE truncation, Wilson-line cuts, quark-singlet matching, and DGLAP scale variation. This revision will make the error accounting explicit and allow readers to assess the claim. revision: yes

  2. Referee: [OPE analysis] OPE analysis (described in the abstract and method): the truncation error on the short-distance OPE of the reduced gluon Ioffe-time distribution at the accessed Wilson-line separations is assessed only via internal consistency checks (order variation, min/max cuts, scale evolution); because the OPE is asymptotic, these checks do not independently constrain the size of omitted higher-order coefficients, leaving open the possibility that the reported systematic band underestimates the true truncation uncertainty at z∼0.2–0.6 fm.

    Authors: We acknowledge that the OPE is asymptotic and that consistency checks based on order variation, z-cuts, and scale evolution provide an estimate rather than a rigorous upper bound on omitted higher-order coefficients. These checks follow the standard methodology used in other lattice PDF studies. To strengthen the presentation, we will expand the relevant section to explicitly discuss the asymptotic nature of the series, report the observed convergence pattern, and add a conservative estimate of the possible size of the next term based on the last included coefficient. We will also state that the quoted band should be viewed as an indicative rather than exhaustive uncertainty. This addresses the concern without altering the central result. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation extracts ratios of gluon Mellin moments from lattice matrix elements of nonlocal operators via the short-distance OPE applied to the reduced Ioffe-time distribution on an external N_f=2+1+1 ensemble. The OPE truncation, separation cuts, perturbative matching, and DGLAP evolution are treated as systematic variations on independent lattice inputs rather than self-definitions or fitted parameters renamed as predictions. No load-bearing step reduces by construction to the target ratio itself, and the central result remains a direct computation from external gauge configurations with standard external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Only abstract available; main unverified inputs are the validity of the truncated OPE at the separations used and the perturbative treatment of quark-singlet mixing. The 260 MeV pion mass is an explicit simulation choice away from the physical point.

free parameters (1)
  • Pion mass
    Fixed at approximately 260 MeV by the chosen Nf=2+1+1 ensemble; not the physical value and affects chiral extrapolation uncertainty.
axioms (2)
  • domain assumption Short-distance OPE of the reduced gluon Ioffe-time distribution is valid and truncatable at the order employed for the Wilson-line separations chosen
    Invoked to convert matrix elements into moment ratios; location: abstract description of the analysis method.
  • domain assumption Perturbative matching to the quark-singlet sector introduces controllable errors
    Required when isolating the pure-gluon contribution; abstract notes investigation of this mixing.

pith-pipeline@v0.9.1-grok · 5785 in / 1483 out tokens · 25460 ms · 2026-06-28T23:27:39.400832+00:00 · methodology

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

71 extracted references · 59 canonical work pages · 34 internal anchors

  1. [1]

    0.8 1.0 1.2 1.4 0.00 0.01 0.02 0.03 0.04 0.05 ⟨ x3⟩ / ⟨ x ⟩ nmax = 4, z ∈ (0.186fm, 0.279fm) FIG

    The band is taken atκ= 1, while the variation underκ→1/ √ 2, √ 2 provides an estimate of the perturbative uncertainty associated with matching-scale variation, in addition to the statistical uncertainties. 0.8 1.0 1.2 1.4 0.00 0.01 0.02 0.03 0.04 0.05 ⟨ x3⟩ / ⟨ x ⟩ nmax = 4, z ∈ (0.186fm, 0.279fm) FIG. 5. Comparison of ratios⟨x 3⟩g/⟨x⟩g for gluon only at ...

  2. [2]

    We include a band covering the uncertainties associated with extracted value atκ= 1 and a lighter band including additional theoretical error determined from varyingκ. E. Impact of Mixing with Quark-Singlet Contribution In the extraction of gluon Mellin moments, the perturbative matching receives contributions from mixing with the quark-singlet operator. ...

  3. [3]

    Pion and Kaon Form Factors using Lattice QCD

    The shaded band corresponds to the final lattice-only determination obtained from the analysis without quark-singlet mixing of Eq. (23). The central value is taken from theκ= 1 analysis, corresponding to the natural choiceµ 0 ∼1/z, while the variation underκ→1/ √ 2 andκ→ √ 2 is used to estimate the associated perturbative systematic uncertainty. Our final...

    2021

  4. [4]

    New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC

    T.-J. Houet al., Phys. Rev. D103, 014013 (2021), 1912.10053

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  5. [5]

    Moffatet al., Phys

    Jefferson Lab Angular Momentum (JAM), E. Moffatet al., Phys. Rev. D104, 016015 (2021), 2101.04664. 16

    work page arXiv 2021

  6. [6]

    R. D. Ballet al., The European Physical Journal C82(2022)

    2022

  7. [7]

    Electron Ion Collider: The Next QCD Frontier - Understanding the glue that binds us all

    A. Accardiet al., Electron ion collider: The next QCD frontier - understanding the glue that binds us all, 2014, 1212.1701

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  8. [8]

    Abdul Khaleket al., Nuclear Physics A1026, 122447 (2022)

    R. Abdul Khaleket al., Nuclear Physics A1026, 122447 (2022)

    2022

  9. [9]

    Parton Physics on Euclidean Lattice

    X. Ji, Phys. Rev. Lett.110, 262002 (2013), 1305.1539

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  10. [10]

    X. Ji, Sci. China Phys. Mech. Astron.57, 1407 (2014), 1404.6680

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  11. [11]

    Ji, Y.-S

    X. Jiet al., Rev. Mod. Phys.93, 035005 (2021), 2004.03543

    work page arXiv 2021

  12. [12]

    A. V. Radyushkin, Phys. Rev. D96, 034025 (2017), 1705.01488

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  13. [13]

    One-loop evolution of parton pseudo-distribution functions on the lattice

    A. Radyushkin, Phys. Rev. D98, 014019 (2018), 1801.02427

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  14. [14]

    Exclusive processes in position space and the pion distribution amplitude

    V. Braun and D. Mueller, Eur. Phys. J.C55, 349 (2008), 0709.1348

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  15. [15]

    Extracting Parton Distribution Functions from Lattice QCD Calculations

    Y.-Q. Ma and J.-W. Qiu, Phys. Rev.D98, 074021 (2018), 1404.6860

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  16. [16]

    Exploring hadrons' partonic structure using ab initio lattice QCD calculations

    Y.-Q. Ma and J.-W. Qiu, Phys. Rev. Lett.120, 022003 (2018), 1709.03018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  17. [17]

    Origin of Difference between $\overline{d}$ and $\overline{u}$ Partons in the Nucleon

    K.-F. Liu and S.-J. Dong, Phys. Rev. Lett.72, 1790 (1994), hep-ph/9306299

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  18. [18]

    Deep-inelastic scattering and the operator product expansion in lattice QCD

    W. Detmold and C. J. D. Lin, Phys. Rev.D73, 014501 (2006), hep-lat/0507007

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  19. [19]

    A. J. Chamberset al., Phys. Rev. Lett.118, 242001 (2017), 1703.01153

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  20. [20]

    Detmoldet al., Phys

    HOPE, W. Detmoldet al., Phys. Rev. D105, 034506 (2022), 2109.15241

    work page arXiv 2022

  21. [21]

    Restoration of Rotational Symmetry in the Continuum Limit of Lattice Field Theories

    Z. Davoudi and M. J. Savage, Phys. Rev. D86, 054505 (2012), 1204.4146

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  22. [22]

    Shindler, Phys

    A. Shindler, Phys. Rev. D110, L051503 (2024), 2311.18704

    work page arXiv 2024

  23. [23]

    A guide to light-cone PDFs from Lattice QCD: an overview of approaches, techniques and results

    K. Cichy and M. Constantinou, Adv. High Energy Phys.2019, 3036904 (2019), 1811.07248

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  24. [24]

    Constantinouet al., Prog

    M. Constantinouet al., Prog. Part. Nucl. Phys.121, 103908 (2021), 2006.08636

    work page arXiv 2021

  25. [25]

    Lin, Prog

    H.-W. Lin, Prog. Part. Nucl. Phys.144, 104177 (2025), 2506.05025. [23]Precision QCD with the Electron-Ion Collider, 2026, 2604.04765

    work page arXiv 2025

  26. [26]

    Gluon Quasi-PDF From Lattice QCD

    Z.-Y. Fanet al., Phys. Rev. Lett.121, 242001 (2018), 1808.02077

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  27. [27]

    Accessing gluon parton distributions in large momentum effective theory

    J.-H. Zhanget al., Phys. Rev. Lett.122, 142001 (2019), 1808.10824

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  28. [28]

    Z. Fan, R. Zhang, and H.-W. Lin, Int. J. Mod. Phys. A36, 2150080 (2021), 2007.16113

    work page arXiv 2021

  29. [29]

    Fan and H.-W

    Z. Fan and H.-W. Lin, Phys. Lett. B823, 136778 (2021), 2104.06372

    work page arXiv 2021

  30. [30]

    Salas-Chavira, Z

    A. Salas-Chavira, Z. Fan, and H.-W. Lin, Phys. Rev. D106, 094510 (2022), 2112.03124

    work page arXiv 2022

  31. [31]

    Khanet al., Phys

    HadStruc, T. Khanet al., Phys. Rev. D104, 094516 (2021), 2107.08960

    work page arXiv 2021

  32. [32]

    Egereret al., Phys

    HadStruc, C. Egereret al., Phys. Rev. D106, 094511 (2022), 2207.08733

    work page arXiv 2022

  33. [33]

    T. Khan, T. Liu, and R. S. Sufian, Phys. Rev. D108, 074502 (2023), 2211.15587

    work page arXiv 2023

  34. [34]

    Z. Fan, W. Good, and H.-W. Lin, Phys. Rev. D108, 014508 (2023), 2210.09985

    work page arXiv 2023

  35. [35]

    T. A. Chowdhuryet al., Phys. Rev. D111, 074509 (2025), 2409.17234

    work page arXiv 2025

  36. [36]

    W. Good, F. Yao, and H.-W. Lin, Physics Letters B872, 140067 (2026)

    2026

  37. [37]

    Chenet al., Unpolarized gluon pdf of the nucleon from lattice qcd in the continuum limit, 2025, 2510.26425

    C. Chenet al., Unpolarized gluon pdf of the nucleon from lattice qcd in the continuum limit, 2025, 2510.26425

    work page arXiv 2025

  38. [38]

    Delmaret al., Phys

    J. Delmaret al., Phys. Rev. D108, 094515 (2023), 2310.01389

    work page arXiv 2023

  39. [39]

    K. G. Wilson, Phys. Rev.179, 1499 (1969)

    1969

  40. [40]

    Factorization Theorem Relating Euclidean and Light-Cone Parton Distributions

    T. Izubuchiet al., Phys. Rev. D98, 056004 (2018), 1801.03917

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  41. [41]

    Balitsky, W

    I. Balitsky, W. Morris, and A. Radyushkin, Phys. Lett. B808, 135621 (2020), 1910.13963

    work page arXiv 2020

  42. [42]

    Balitsky, W

    I. Balitsky, W. Morris, and A. Radyushkin, SciPost Phys. Proc.8, 161 (2022), 2106.01916

    work page arXiv 2022

  43. [43]

    Renormalization in Large Momentum Effective Theory of Parton Physics

    X. Ji, J.-H. Zhang, and Y. Zhao, Phys. Rev. Lett.120, 112001 (2018), 1706.08962

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  44. [44]

    On the Renormalizability of Quasi Parton Distribution Functions

    T. Ishikawaet al., Phys. Rev. D96, 094019 (2017), 1707.03107

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  45. [45]

    Nonperturbative renormalization of nonlocal quark bilinears for quasi-PDFs on the lattice using an auxiliary field

    J. Green, K. Jansen, and F. Steffens, Phys. Rev. Lett.121, 022004 (2018), 1707.07152

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  46. [46]

    Lattice QCD exploration of pseudo-PDFs

    K. Orginoset al., Phys. Rev. D96, 094503 (2017), 1706.05373

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  47. [47]

    The nucleon spin and momentum decomposition using lattice QCD simulations

    C. Alexandrouet al., Phys. Rev. Lett.119, 142002 (2017), 1706.02973

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  48. [48]

    Alexandrouet al., Phys

    C. Alexandrouet al., Phys. Rev. D101, 094513 (2020), 2003.08486

    work page arXiv 2020

  49. [49]

    Gaoet al., Phys

    X. Gaoet al., Phys. Rev. D103, 094504 (2021), 2102.01101

    work page arXiv 2021

  50. [50]

    Suet al., Nucl

    Y. Suet al., Nucl. Phys. B991, 116201 (2023), 2209.01236. 17

    work page arXiv 2023

  51. [51]

    S. Moch, J. Vermaseren, and A. Vogt, Nuclear Physics B688, 101–134 (2004)

    2004

  52. [52]

    A. Vogt, S. Moch, and J. Vermaseren, Nuclear Physics B691, 129–181 (2004)

    2004

  53. [53]

    Ledoit and M

    O. Ledoit and M. Wolf, Journal of Multivariate Analysis88, 365 (2004)

    2004

  54. [54]

    Multiplicative renormalizability of quasi-parton operators

    Z.-Y. Li, Y.-Q. Ma, and J.-W. Qiu, Phys. Rev. Lett.122, 062002 (2019), 1809.01836

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  55. [55]

    Simulating twisted mass fermions at physical light, strange and charm quark masses

    C. Alexandrouet al., Phys. Rev. D98, 054518 (2018), 1807.00495

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  56. [56]

    Alexandrouet al., Phys

    Extended Twisted Mass, C. Alexandrouet al., Phys. Rev. D104, 074515 (2021), 2104.13408

    work page arXiv 2021

  57. [57]

    QUDA, M. A. Clarket al., (2016), 1612.07873

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  58. [58]

    Adaptive Aggregation-based Domain Decomposition Multigrid for Twisted Mass Fermions

    C. Alexandrouet al., Phys. Rev. D94, 114509 (2016), 1610.02370

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  59. [59]

    Multigrid accelerated simulations for Twisted Mass fermions

    S. Bacchio, C. Alexandrou, and J. Finkerath, EPJ Web Conf.175, 02002 (2018), 1710.06198

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  60. [60]

    Multigrid approach in shifted linear systems for the non-degenerated twisted mass operator

    C. Alexandrou, S. Bacchio, and J. Finkenrath, Comput. Phys. Commun.236, 51 (2019), 1805.09584

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  61. [61]

    Analytic Smearing of SU(3) Link Variables in Lattice QCD

    C. Morningstar and M. J. Peardon, Phys. Rev. D69, 054501 (2004), hep-lat/0311018

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  62. [62]

    Gluon momentum fraction of the nucleon from lattice QCD

    C. Alexandrouet al., Phys. Rev. D96, 054503 (2017), 1611.06901

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  63. [63]

    G. S. Baliet al., Phys. Rev. D93, 094515 (2016), 1602.05525

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  64. [64]

    Updated Lattice Results for Parton Distributions

    C. Alexandrouet al., Phys. Rev. D96, 014513 (2017), 1610.03689

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  65. [65]

    Gaoet al., Phys

    X. Gaoet al., Phys. Rev. D106, 074505 (2022), 2206.04084

    work page arXiv 2022

  66. [66]

    Gaoet al., Phys

    X. Gaoet al., Phys. Rev. D107, 074509 (2023), 2212.12569

    work page arXiv 2023

  67. [67]

    I. W. Stewart, F. J. Tackmann, and W. J. Waalewijn, Journal of High Energy Physics2010(2010)

    2010

  68. [68]

    Buckleyet al., The European Physical Journal C75(2015)

    A. Buckleyet al., The European Physical Journal C75(2015)

    2015

  69. [69]

    Cocuzzaet al., Phys

    JAM Collaboration (Spin PDF Analysis Group), C. Cocuzzaet al., Phys. Rev. D112, 114017 (2025), 2506.13616

    work page arXiv 2025

  70. [70]

    Stability of parton distributions at high $x$: impact of nuclear and power corrections

    C. Cocuzzaet al., Stability of parton distributions at highx: impact of nuclear and power corrections, 2026, 2605.00666

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  71. [71]

    Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator

    A. Frommeret al., SIAM J. Sci. Comput.36, A1581 (2014), 1303.1377

    work page internal anchor Pith review Pith/arXiv arXiv 2014