pith. sign in

arxiv: 2512.06121 · v2 · pith:RH63JXLZnew · submitted 2025-12-05 · ✦ hep-lat · hep-ex· hep-ph· hep-th

Pion and Kaon PDFs from Lattice QCD via Large Momentum Effective Theory and Short-Distance Factorization

Joshua Miller , Joseph Torsiello , Isaac Anderson , Krzysztof Cichy , Martha Constantinou , Joseph Delmar , Sarah Lampreich This is my paper

Pith reviewed 2026-05-21 17:30 UTC · model grok-4.3

classification ✦ hep-lat hep-exhep-phhep-th
keywords pion PDFkaon PDFlattice QCDLaMETparton distribution functionsquark distributionsmeson structureSU(3) breaking
0
0 comments X

The pith

Lattice QCD produces the unpolarized quark distributions inside the pion and kaon from finite-momentum matrix elements.

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

The paper presents a first-principles lattice calculation of the unpolarized quark PDFs for the pion and kaon. It computes matrix elements for boosted mesons on a 32^3 x 64 ensemble with pion mass 260 MeV and applies both LaMET and short-distance factorization to convert the results into light-cone distributions. A sympathetic reader would care because these are the simplest hadrons, so their PDFs test basic QCD dynamics and provide benchmarks free of phenomenological assumptions. The work also examines valence-only distributions and SU(3) flavor-symmetry breaking while parametrizing the momentum dependence to reach the infinite-momentum limit.

Core claim

We present a first-principles lattice-QCD calculation of the unpolarized quark PDF for the pion and the kaon. The lattice data rely on matrix elements calculated for boosted mesons coupled to non-local operators containing a Wilson line on an N_f=2+1+1 ensemble with pion mass 260 MeV and boosts up to 2.07 GeV. To match to light-cone PDFs we employ LaMET and SDF in parallel, present results for standard and valence distributions, explore SU(3) breaking, and parametrize the momentum dependence for the infinite-momentum limit.

What carries the argument

Large-momentum effective theory (LaMET) and short-distance factorization (SDF) that convert finite-momentum lattice matrix elements into light-cone parton distribution functions.

If this is right

  • The calculation supplies both full and valence quark distributions for the pion and kaon.
  • It quantifies SU(3) flavor-symmetry-breaking effects between the two mesons.
  • Parametrization of the boost dependence yields the infinite-momentum PDFs within the LaMET framework.
  • Parallel use of LaMET and SDF allows direct assessment of method-dependent systematics in the same dataset.

Where Pith is reading between the lines

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

  • The same boosted-meson technique could be applied to other light mesons to trace how quark distributions evolve with flavor and mass.
  • Validation against upcoming experimental extractions from Drell-Yan or prompt-photon processes at facilities such as JLab or COMPASS would test the lattice results outside the current momentum range.
  • Extension to polarized or transverse-momentum-dependent distributions would follow naturally once the unpolarized baseline is established.

Load-bearing premise

The lattice momenta up to 2.07 GeV together with the specific matching procedures in LaMET and SDF are sufficient to control higher-order corrections and produce reliable light-cone PDFs after extrapolation.

What would settle it

Direct numerical comparison of the extracted PDFs with independent global-fit results from experimental data or with new lattice calculations performed at substantially higher boosts would show whether the matching and extrapolation are accurate.

Figures

Figures reproduced from arXiv: 2512.06121 by Isaac Anderson, Joseph Delmar, Joseph Torsiello, Joshua Miller, Krzysztof Cichy, Martha Constantinou, Sarah Lampreich.

Figure 1
Figure 1. Figure 1: FIG. 1. The ground state energies for the pion (left) and kaon (right) at the various values of the momentum boost, [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The ratio of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The ratio of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Pion bare matrix element, [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Bare matrix elements for the kaon up flavor for momentum boost [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Bare matrix elements for the kaon strange flavor for momentum boost [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Reduced-ITD for the pion as a function of the Ioffe time [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Reduced-ITD for the kaon up-quark as a function of the Ioffe time [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Reduced-ITD for the kaon strange-quark as a function of the Ioffe time [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Interpolation of Re[ [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Interpolation of Im[ [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Interpolation of Re[ [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Interpolation of Re[ [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Interpolation of Im[ [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Interpolation of Im[ [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Reduced (blue), evolved (red), and matched (green) ITDs for [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Parametrization of the real (left) and imaginary (right) parts of [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p019_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p020_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Similar as Fig [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. SU(3) symmetry breaking from the ratios of [PITH_FULL_IMAGE:figures/full_fig_p021_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26 [PITH_FULL_IMAGE:figures/full_fig_p022_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27 [PITH_FULL_IMAGE:figures/full_fig_p022_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28 [PITH_FULL_IMAGE:figures/full_fig_p022_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29 [PITH_FULL_IMAGE:figures/full_fig_p023_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30 [PITH_FULL_IMAGE:figures/full_fig_p023_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: FIG. 31. The real (blue) and imaginary (red) renormalized matrix element, [PITH_FULL_IMAGE:figures/full_fig_p024_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: FIG. 32. Light-cone pion PDF, [PITH_FULL_IMAGE:figures/full_fig_p024_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: FIG. 33. Light-cone kaon PDFs, [PITH_FULL_IMAGE:figures/full_fig_p024_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: FIG. 34. Light-cone pion PDFs, [PITH_FULL_IMAGE:figures/full_fig_p025_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: FIG. 35. Light-cone up-quark kaon PDFs, [PITH_FULL_IMAGE:figures/full_fig_p025_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: FIG. 36. Light-cone strange-quark kaon PDFs, [PITH_FULL_IMAGE:figures/full_fig_p025_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: FIG. 37. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p026_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: FIG. 38. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p026_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: FIG. 39. Fits on the pion PDF, [PITH_FULL_IMAGE:figures/full_fig_p027_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: FIG. 40. Fits on the kaon up-quark PDF, [PITH_FULL_IMAGE:figures/full_fig_p027_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: FIG. 41. Fits on the kaon strange-quar PDF, [PITH_FULL_IMAGE:figures/full_fig_p027_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: FIG. 42. Comparison of pion (blue), kaon up-quark (red), and kaon strange-quark (green) [PITH_FULL_IMAGE:figures/full_fig_p028_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: FIG. 43. Comparison of pion (blue), kaon up-quark (red), and kaon strange-quark (green) [PITH_FULL_IMAGE:figures/full_fig_p028_43.png] view at source ↗
read the original abstract

In this work, we present a first-principles lattice-QCD calculation of the unpolarized quark PDF for the pion and the kaon. The lattice data rely on matrix elements calculated for boosted mesons coupled to non-local operators containing a Wilson line. The calculations on this lattice ensemble correspond to two degenerate light, a strange, and a charm quark ($N_f=2+1+1$), using maximally twisted mass fermions with a clover term. The lattice volume is $32^3\times 64$, with a lattice spacing of 0.0934 fm, and a pion mass of 260 MeV. Matrix elements are calculated for hadron boosts of $|P_3| = 0,~0.41,~0.83,~1.25,~1.66,$ and 2.07 GeV. To match lattice QCD results to their light-cone counterparts, we employ two complementary frameworks: the large-momentum effective theory (LaMET) and the short-distance factorization (SDF). Using these approaches in parallel, we also test the lattice data to identify methodology-driven systematics. Results are presented for the standard quark PDFs, as well as the valence sector. Beyond obtaining the PDFs, we also explore the possibility of extracting information on SU(3) flavor-symmetry-breaking effects. For LaMET, we also parametrize the momentum dependence to obtain the infinite-momentum PDFs.

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.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Standard lattice QCD inputs plus domain assumptions about effective-theory matching.

free parameters (2)
  • lattice spacing a
    Fixed at 0.0934 fm from prior calibration of the ensemble.
  • pion mass
    Set to 260 MeV; not the physical value.
axioms (2)
  • domain assumption LaMET and SDF provide controlled matching at the simulated momenta
    Invoked when converting lattice matrix elements to light-cone PDFs.
  • domain assumption Finite-volume and discretization effects are manageable after extrapolation
    Required for the infinite-momentum limit.

pith-pipeline@v0.9.0 · 5810 in / 1207 out tokens · 102231 ms · 2026-05-21T17:30:50.389002+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 3 Pith papers

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

  1. Pion Parton Distribution Functions in the Light-Cone Quark Model and Experimental Constraints

    hep-ph 2026-04 unverdicted novelty 4.0

    Pion valence quark PDFs from the light-cone quark model are evolved with DGLAP and shown to match experimental data, enabling first NLO F2 predictions and Drell-Yan cross-section estimates.

  2. An Analysis on the Parton Distribution Functions of Heavy Mesons

    hep-ph 2026-05 unverdicted novelty 3.0

    Light-cone quark model PDFs for kaon and heavy mesons are evolved via NLO DGLAP to predict EIC structure functions and COMPASS Drell-Yan cross sections while showing heavy constituents dominate momentum fractions.

  3. Hadron Structure from lattice QCD in the context of the Electron-Ion Collider

    hep-lat 2026-03 unverdicted novelty 2.0

    Lattice QCD now delivers high-precision results on hadron internal structure that directly support the scientific program of the Electron-Ion Collider.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages · cited by 3 Pith papers · 37 internal anchors

  1. [1]

    In this approach, we calculate the matrix elements for each individual momentum boost,P 3, and multiple values ofz

    quasi-distributions approach The first approach we use to reconstruct thex-dependence of the spin-0 unpolarized PDFs,q f M, is the quasi- distribution method, which utilizes large-momentum effective theory (LaMET). In this approach, we calculate the matrix elements for each individual momentum boost,P 3, and multiple values ofz. The LaMET formalism states...

    work page

  2. [2]

    Pion and Kaon Form Factors using Lattice QCD

    pseudo-distributions approach An alternative approach to analyzing the lattice data and reconstructing thex-dependence of the pion and kaon PDFs is the Ioffe-time pseudo-distribution method. It is useful to express the matrix elements of Eq. (1) as a function of the Lorentz invariant Ioffe time,ν=z·P, wherez µ = (0,0,0, z) andP µ = (P 0,0,0, P 3). This en...

    work page 2021

  3. [3]

    A. W. Thomas, W. Melnitchouk, and F. M. Steffens, Phys. Rev. Lett.85, 2892 (2000), arXiv:hep-ph/0005043 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  4. [4]

    Large-N_c Quark Distributions in the Delta and Chiral Logarithms in Quark Distributions of the Nucleon

    J.-W. Chen and X.-d. Ji, Phys. Lett.B523, 73 (2001), arXiv:hep-ph/0105296 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  5. [5]

    Leading Chiral Contributions to the Spin Structure of the Proton

    J.-W. Chen and X.-d. Ji, Phys. Rev. Lett.88, 052003 (2002), arXiv:hep-ph/0111048 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  6. [6]

    $\bar d - \bar u$ asymmetry in the proton in chiral effective theory

    Y. Salamu, C.-R. Ji, W. Melnitchouk, and P. Wang, Phys. Rev. Lett.114, 122001 (2015), arXiv:1409.5885 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  7. [7]

    P. T. P. Hutauruk, I. C. Cloet, and A. W. Thomas, Phys. Rev. C94, 035201 (2016), arXiv:1604.02853 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  8. [8]

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

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    P. C. Barry, C.-R. Ji, N. Sato, and W. Melnitchouk (Jefferson Lab Angular Momentum (JAM)), Phys. Rev. Lett.127, 232001 (2021), arXiv:2108.05822 [hep-ph]. 1 This manuscript was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes...

    work page arXiv 2021

  10. [10]

    P. C. Barry et al. (Jefferson Lab Angular Momentum (JAM), HadStruc), Phys. Rev. D105, 114051 (2022), arXiv:2204.00543 [hep-ph]

    work page arXiv 2022

  11. [11]

    P. C. Barry, C.-R. Ji, W. Melnitchouk, N. Sato, and F. Steffens (JAM), (2025), arXiv:2510.11979 [hep-ph]

    work page arXiv 2025

  12. [12]

    C. Chen, L. Chang, C. D. Roberts, S. Wan, and H.-S. Zong, Phys. Rev. D93, 074021 (2016), arXiv:1602.01502 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  13. [13]

    C. Shi, C. Mezrag, and H.-s. Zong, Phys. Rev. D98, 054029 (2018), arXiv:1806.10232 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  14. [14]

    K. D. Bednar, I. C. Clo¨ et, and P. C. Tandy, Phys. Rev. Lett.124, 042002 (2020), arXiv:1811.12310 [nucl-th]

    work page arXiv 2020

  15. [15]

    J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Phys. Rev. D101, 034024 (2020), arXiv:1907.01509 [nucl-th]

    work page arXiv 2020

  16. [16]

    W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D107, 094024 (2023), arXiv:2302.03759 [hep-ph]

    work page arXiv 2023

  17. [17]

    Q. Wu, C. Han, D. Qing, W. Kou, J. Xie, X. Chen, and F. Wang, Nucl. Phys. B994, 116321 (2023), arXiv:2204.06798 [hep-ph]

    work page arXiv 2023

  18. [18]

    Bourrely, W.-C

    C. Bourrely, W.-C. Chang, and J.-C. Peng, Phys. Rev. D105, 076018 (2022), arXiv:2202.12547 [hep-ph]

    work page arXiv 2022

  19. [19]

    A. Kock, Y. Liu, and I. Zahed, Phys. Rev. D102, 014039 (2020), arXiv:2004.01595 [hep-ph]

    work page arXiv 2020

  20. [20]

    Kaon quark distribution functions in the chiral constituent quark model

    A. Watanabe, T. Sawada, and C. W. Kao, Phys. Rev. D97, 074015 (2018), arXiv:1710.09529 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  21. [21]

    J. S. Conway et al., Phys. Rev.D39, 92 (1989)

    work page 1989

  22. [22]

    Soft-Gluon Resummation and the Valence Parton Distribution Function of the Pion

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

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  23. [23]

    Badier et al

    J. Badier et al. (NA3), Z. Phys.C18, 281 (1983)

    work page 1983

  24. [24]

    Brommel et al

    D. Brommel et al. (QCDSF-UKQCD), Proceedings, 25th International Symposium on Lattice field theory , PoSLA T- TICE2007, 140 (2007)

    work page 2007

  25. [25]

    The pion form factor from lattice QCD with two dynamical flavours

    D. Br¨ ommelet al. (QCDSF/UKQCD), Eur. Phys. J. C51, 335 (2007), arXiv:hep-lat/0608021

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  26. [26]

    G. Bali, S. Collins, B. Gl¨ assle, M. G¨ ockeler, N. Javadi-Motaghi, J. Najjar, W. S¨ oldner, and A. Sternbeck, PoSLA T- TICE2013, 447 (2014), arXiv:1311.7639 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  27. [27]

    Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass

    A. Abdel-Rehim et al., Phys. Rev. D92, 114513 (2015), [Erratum: Phys.Rev.D 93, 039904 (2016)], arXiv:1507.04936 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  28. [28]

    The pion vector form factor from Lattice QCD at the physical point

    C. Alexandrou et al. (ETM), Phys. Rev.D97, 014508 (2018), arXiv:1710.10401 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  29. [29]

    M. Oehm, C. Alexandrou, M. Constantinou, K. Jansen, G. Koutsou, B. Kostrzewa, F. Steffens, C. Urbach, and S. Zafeiropoulos, Phys. Rev. D99, 014508 (2019), arXiv:1810.09743 [hep-lat]

    work page arXiv 2019

  30. [30]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, I. Cloet, M. Constantinou, K. Hadjiyiannakou, G. Koutsou, and C. Lauer (ETM), Phys. Rev. D103, 014508 (2021), arXiv:2010.03495 [hep-lat]

    work page arXiv 2021

  31. [31]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, I. Clo¨ et, M. Constantinou, K. Hadjiyiannakou, G. Koutsou, and C. Lauer (ETM), Phys. Rev. D104, 054504 (2021), arXiv:2104.02247 [hep-lat]

    work page arXiv 2021

  32. [32]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, I. Cloet, M. Constantinou, J. Delmar, K. Hadjiyiannakou, G. Koutsou, C. Lauer, and A. Vaquero (ETM), Phys. Rev. D105, 054502 (2022), arXiv:2111.08135 [hep-lat]

    work page arXiv 2022

  33. [33]

    Quark and gluon momentum fractions in the pion and in the kaon,

    C. Alexandrou et al. (Extended Twisted Mass), Phys. Rev. Lett.134, 131902 (2025), arXiv:2405.08529 [hep-lat]

    work page arXiv 2025

  34. [34]

    Shindler, Phys

    A. Shindler, Phys. Rev. D110, L051503 (2024), arXiv:2311.18704 [hep-lat]

    work page arXiv 2024

  35. [35]

    Francis et al., (2025), arXiv:2509.02472 [hep-lat]

    A. Francis et al., (2025), arXiv:2509.02472 [hep-lat]

    work page arXiv 2025

  36. [36]

    Parton Physics on Euclidean Lattice

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

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  37. [37]

    Ji, Y.-S

    X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao, Rev. Mod. Phys.93, 035005 (2021), arXiv:2004.03543 [hep-ph]

    work page arXiv 2021

  38. [38]

    A. V. Radyushkin, Phys. Rev. D96, 034025 (2017), arXiv:1705.01488 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  39. [39]

    Radyushkin, Int

    A. Radyushkin, Int. J. Mod. Phys. A35, 2030002 (2020), arXiv:1912.04244 [hep-ph]

    work page arXiv 2020

  40. [40]

    Extracting Parton Distribution Functions from Lattice QCD Calculations

    Y.-Q. Ma and J.-W. Qiu, Phys. Rev.D98, 074021 (2018), arXiv:1404.6860 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  41. [41]

    QCD Factorization and PDFs from Lattice QCD Calculation

    Y.-Q. Ma and J.-W. Qiu, Proceedings, QCD Evolution Workshop (QCD 2014): Santa Fe, USA, May 12-16, 2014, Int. J. Mod. Phys. Conf. Ser.37, 1560041 (2015), arXiv:1412.2688 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  42. [42]

    Exploring hadrons' partonic structure using ab initio lattice QCD calculations

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

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  43. [43]

    Zhang, J.-W

    J.-H. Zhang, J.-W. Chen, L. Jin, H.-W. Lin, A. Sch¨ afer, and Y. Zhao, Phys. Rev. D100, 034505 (2019), arXiv:1804.01483 [hep-lat]

    work page arXiv 2019

  44. [44]

    Izubuchi, L

    T. Izubuchi, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, and S. Syritsyn, Phys. Rev.D100, 034516 (2019), arXiv:1905.06349 [hep-lat]

    work page arXiv 2019

  45. [45]

    X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Petreczky, P. Scior, S. Shi, S. Syritsyn, Y. Zhao, and K. Zhou, Phys. Rev. D106, 114510 (2022), arXiv:2208.02297 [hep-lat]

    work page arXiv 2022

  46. [46]

    Holligan and H.-W

    J. Holligan and H.-W. Lin, J. Phys. G51, 065101 (2024), arXiv:2404.14525 [hep-lat]

    work page arXiv 2024

  47. [47]

    Lin, J.-W

    H.-W. Lin, J.-W. Chen, Z. Fan, J.-H. Zhang, and R. Zhang, Phys. Rev. D103, 014516 (2021), arXiv:2003.14128 [hep-lat]

    work page arXiv 2021

  48. [48]

    Jo´ o, J

    B. Jo´ o, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, R. S. Sufian, and S. Zafeiropoulos, Phys. Rev.D100, 114512 (2019), arXiv:1909.08517 [hep-lat]

    work page arXiv 2019

  49. [49]

    R. S. Sufian, J. Karpie, C. Egerer, K. Orginos, J.-W. Qiu, and D. G. Richards, Phys. Rev.D99, 074507 (2019), arXiv:1901.03921 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  50. [50]

    R. S. Sufian, C. Egerer, J. Karpie, R. G. Edwards, B. Jo´ o, Y.-Q. Ma, K. Orginos, J.-W. Qiu, and D. G. Richards, Phys. Rev. D102, 054508 (2020), arXiv:2001.04960 [hep-lat]

    work page arXiv 2020

  51. [51]

    X. Gao, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, S. Syritsyn, and Y. Zhao, Phys. Rev. D102, 094513 (2020), arXiv:2007.06590 [hep-lat]

    work page arXiv 2020

  52. [52]

    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), arXiv:1811.07248 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  53. [53]

    Constantinou, Eur

    M. Constantinou, Eur. Phys. J. A57, 77 (2021), arXiv:2010.02445 [hep-lat]

    work page arXiv 2021

  54. [54]

    Cichy, PoSLA TTICE2021, 017 (2022), arXiv:2110.07440 [hep-lat]

    K. Cichy, PoSLA TTICE2021, 017 (2022), arXiv:2110.07440 [hep-lat]

    work page arXiv 2022

  55. [55]

    Cichy, EPJ Web Conf.258, 01005 (2022), arXiv:2111.04552 [hep-lat]

    K. Cichy, EPJ Web Conf.258, 01005 (2022), arXiv:2111.04552 [hep-lat]

    work page arXiv 2022

  56. [56]

    Perturbative Renormalization of quasi-PDFs

    M. Constantinou and H. Panagopoulos, Phys. Rev. D96, 054506 (2017), arXiv:1705.11193 [hep-lat]. 31

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  57. [57]

    A complete non-perturbative renormalization prescription for quasi-PDFs

    C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, H. Panagopoulos, and F. Steffens, Nucl. Phys. B923, 394 (2017), arXiv:1706.00265 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  58. [58]

    Systematic uncertainties in parton distribution functions from lattice QCD simulations at the physical point

    C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. D99, 114504 (2019), arXiv:1902.00587 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  59. [59]

    A General Method for Non-Perturbative Renormalization of Lattice Operators

    G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas, Nucl. Phys. B445, 81 (1995), arXiv:hep-lat/9411010

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  60. [60]

    Nonperturbative Renormalisation of Composite Operators in Lattice QCD

    M. Gockeler, R. Horsley, H. Oelrich, H. Perlt, D. Petters, P. E. L. Rakow, A. Schafer, G. Schierholz, and A. Schiller, Nucl. Phys.B544, 699 (1999), arXiv:hep-lat/9807044 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  61. [61]

    Renormalization functions for Nf=2 and Nf=4 Twisted Mass fermions

    C. Alexandrou, M. Constantinou, and H. Panagopoulos (ETM), Phys. Rev.D95, 034505 (2017), arXiv:1509.00213 [hep- lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  62. [62]

    Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to Neural Networks

    J. Karpie, K. Orginos, A. Rothkopf, and S. Zafeiropoulos, JHEP04, 057 (2019), arXiv:1901.05408 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  63. [63]

    Backus and F

    G. Backus and F. Gilbert, Geophysical Journal International16, 169 (1968)

    work page 1968

  64. [64]

    Alexandrou, K

    C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. D 105, 034501 (2022), arXiv:2108.10789 [hep-lat]

    work page arXiv 2022

  65. [65]

    Y.-S. Liu, W. Wang, J. Xu, Q.-A. Zhang, J.-H. Zhang, S. Zhao, and Y. Zhao, Phys. Rev. D100, 034006 (2019), arXiv:1902.00307 [hep-ph]

    work page arXiv 2019

  66. [66]

    Lattice QCD exploration of pseudo-PDFs

    K. Orginos, A. Radyushkin, J. Karpie, and S. Zafeiropoulos, Phys. Rev.D96, 094503 (2017), arXiv:1706.05373 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  67. [67]

    A. V. Radyushkin, Phys. Lett. B781, 433 (2018), arXiv:1710.08813 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  68. [68]

    Moments of Ioffe time parton distribution functions from non-local matrix elements

    J. Karpie, K. Orginos, and S. Zafeiropoulos, JHEP11, 178 (2018), arXiv:1807.10933 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  69. [69]

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

    C. Alexandrou et al., Phys. Rev. D98, 054518 (2018), arXiv:1807.00495 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  70. [70]

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

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  71. [71]

    Updated Lattice Results for Parton Distributions

    C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, F. Steffens, and C. Wiese, Phys. Rev. D96, 014513 (2017), arXiv:1610.03689 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  72. [72]

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

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

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  73. [73]

    Amendolia, B

    S. Amendolia, B. Badelek, G. Batignani, G. Beck, F. Bedeschi, E. Bellamy, E. Bertolucci, D. Bettoni, H. Bilokon, G. Bologna, L. Bosisio, C. Bradaschia, M. Budinich, A. Codino, M. Counihan, M. Dell’Orso, B. D. Piazzoli, F. Fabbri, F. Fidecaro, L. Fo` a, E. Focardi, S. Frank, A. Giazotto, M. Giorgi, M. Green, J. Harvey, G. Heath, M. Landon, P. Laurelli, F. ...

    work page 1984

  74. [74]

    Amendolia, G

    S. Amendolia, G. Batignani, G. Beck, E. Bellamy, E. Bertolucci, G. Bologna, L. Bosisio, C. Bradaschia, M. Budinich, M. Dell’orso, B. D. Piazzoli, F. Fabbri, F. Fidecaro, L. Foa, E. Focardi, S. Frank, P. Gianetti, A. Giazzotto, M. Giorgi, M. Green, G. Heath, M. Landon, P. Laurelli, F. Liello, G. Mannocchi, P. March, P. Marrocchesi, A. Menzione, E. Meroni, ...

    work page 1986

  75. [75]

    M. Bhat, W. Chomicki, K. Cichy, M. Constantinou, J. R. Green, and A. Scapellato, Phys. Rev. D106, 054504 (2022), arXiv:2205.07585 [hep-lat]

    work page arXiv 2022

  76. [76]

    Badier et al

    J. Badier et al. (Saclay-CERN-College de France-Ecole Poly-Orsay), Phys. Lett. B93, 354 (1980)

    work page 1980

  77. [77]

    Bhattacharya, K

    S. Bhattacharya, K. Cichy, M. Constantinou, X. Gao, A. Metz, J. Miller, S. Mukherjee, P. Petreczky, F. Steffens, and Y. Zhao, Phys. Rev. D108, 014507 (2023), arXiv:2305.11117 [hep-lat]

    work page arXiv 2023

  78. [78]

    Bhattacharya, K

    S. Bhattacharya, K. Cichy, M. Constantinou, X. Gao, A. Metz, J. Miller, S. Mukherjee, P. Petreczky, F. Steffens, and Y. Zhao, JHEP01, 146 (2025), arXiv:2410.03539 [hep-lat]

    work page arXiv 2025

  79. [79]

    M.-H. Chu, K. Cichy, M. Constantinou, P. Sznajder, and J. Wagner, (2025), arXiv:2509.15931 [hep-lat]

    work page arXiv 2025

  80. [80]

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

    A. Frommer, K. Kahl, S. Krieg, B. Leder, and M. Rottmann, SIAM J. Sci. Comput.36, A1581 (2014), arXiv:1303.1377 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

Showing first 80 references.