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arxiv: 2603.28604 · v2 · submitted 2026-03-30 · ✦ hep-lat · hep-ph· nucl-th

Recognition: no theorem link

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

Constantia Alexandrou (University of Cyprus & The Cyprus Institute)

Pith reviewed 2026-05-14 00:35 UTC · model grok-4.3

classification ✦ hep-lat hep-phnucl-th
keywords lattice QCDhadron structureElectron-Ion Collidergeneralized parton distributionstransverse-momentum-dependent distributionsnucleon form factorspion structure
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The pith

Lattice QCD now computes precise charges, form factors, GPDs, and TMDs for nucleons and mesons to inform EIC experiments.

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

The paper reviews substantial recent progress in lattice QCD computations of hadron structure. It shows that charges, electromagnetic form factors, and lower Mellin moments can be obtained to high precision, that generalized parton distributions are now accessible either directly or through moment reconstruction, and that transverse-momentum-dependent distributions can be calculated directly on the lattice. These quantities together give complementary non-perturbative information on the internal structure of the pion, kaon, and nucleon. The review positions these results as directly relevant to the experimental program planned for the Electron-Ion Collider.

Core claim

Hadron structure calculations using lattice QCD have advanced to the point where charges, form factors, and lower Mellin moments are available at high precision, generalized parton distributions can be computed directly or reconstructed from moments, and transverse-momentum-dependent distributions are accessible through direct lattice methods. Together these provide detailed complementary insights into hadron internal structure that inform the scientific agenda of the Electron-Ion Collider, with emphasis on the pion, kaon, and nucleon.

What carries the argument

Lattice QCD matrix-element calculations that extract charges, form factors, Mellin moments, generalized parton distributions, and transverse-momentum-dependent distributions from Euclidean correlation functions.

Load-bearing premise

The summarized lattice QCD results are mature, free of uncontrolled systematics, and can be used at EIC kinematics without further model-dependent corrections.

What would settle it

An EIC measurement of a nucleon or meson observable that differs from the corresponding lattice prediction by an amount larger than the combined stated uncertainties.

Figures

Figures reproduced from arXiv: 2603.28604 by Constantia Alexandrou (University of Cyprus & The Cyprus Institute).

Figure 1
Figure 1. Figure 1: Gauge ensembles used for hadron structure simulated for 𝑚𝜋 < 145 MeV. Near-term goals in LQCD for EIC physics include extending form factors to larger momentum trans￾fers, computing higher Mellin moments, reduc￾ing discretization uncer￾tainties, and reaching higher boosts (𝑃𝑧 ≳ 3 GeV) in the computations of GPDs, as well as incorporating QED effects for quantities where percent precision can now be reached… view at source ↗
Figure 2
Figure 2. Figure 2: Pion vector form factor from: i) Left panel: ETMC results extrapolated to the chiral limit for low 𝑄 2 compared to data from CERN [21]; ii) Middle panel: BNL/JLab results compared to experimental data and phenomenology [22]; iii) Right panel: 𝜒QCD for 𝑚𝜋 = 252, 174 and 137 MeV for intermediate 𝑄 2 [23]. 𝑄 2 -range, as well as increasing the accuracy for the r.m.s radius will still be important. LQCD result… view at source ↗
Figure 3
Figure 3. Figure 3: LQCD results on the pion charge radius compared to the PDG value (gray band). Matrix elements of first derivative operators (𝑛 = 2) or of the energy and momentum tensor for quarks and glu￾ons give access to the unpolarized and tensor generalized form factors (GFFs), 𝐴 𝑞,𝑔 20 (𝑄 2 ), 𝐶𝑞,𝑔 20 (𝑄 2 ), 𝐵𝑞,𝑔 𝑇20 (𝑄 2 ). Most of LQCD computations focus on the momentum fraction, 𝐴 𝑞,𝑔 20 (0) = ⟨𝑥⟩𝑞,𝑔. Recently, E… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Continuum extrapolation of the momentum fraction of the pion and kaon using ETMC ensembles; Middle: Contributions of quarks and gluons to the pion and kaon momentum faction; Right: Comparison of the pion and kaon momentum fractions with other recent LQCD and phenomenology results. Figures are from Ref. [24]. The pion comparison figure includes the recent MIT group values [25]. 0.050 0.075 0.100 0.125… view at source ↗
Figure 5
Figure 5. Figure 5: Results on the third and fourth unpolarized moments for the pion (left), and the u- (right top) and s- (right bottom) quark in the kaon. LQCD data are compared to analyses of experimental data and other theoretical determinations. The gray band denotes the spread of experimental analyses. All results are in the MS scheme at 2 GeV. 0.06 0.08 0.10 0.12 hx 2iK u ETMC 2025 JAM + LQCD 2025 Han et al. 2021 Bedna… view at source ↗
Figure 6
Figure 6. Figure 6: Results on the unpolarized valence PDF for 𝑥𝑞 𝜋 𝑢 (𝑥) (left top), 𝑥𝑞𝐾 𝑢 (𝑥) (right top) and 𝑥𝑞𝐾 𝑠 (𝑥) (left bottom) constructed using ETMC data (hatched orange band) compared to phenomenological results: for 𝑥𝑞 𝜋 𝑢 (𝑥) from JAM (blue band) [32], FANTO (green band) [33], and xFitter (red band) [34]; for 𝑥𝑞𝐾 𝑢 (𝑥) and 𝑥𝑞𝐾 𝑠 from Ref. [29] (green band) and JAM with LQCD input (red band) [30]. The right bottom… view at source ↗
Figure 7
Figure 7. Figure 7: Recent result on the nucleon isovector charges. The gray bands show the FLAG 2024 average of then published data, see Ref. [42] and references within. The dotted line is the experimental value of 𝑔 𝑢−𝑑 𝐴 . 0.75 0.85 0.95 QCD18 Mainz19 PNDME25 ETMC26 g u A 0.5 0.4 0.3 g d A 0.08 0.04 0.00 g s A 0.65 0.75 0.85 Mainz19 PNDME25 ETMC26 g u T 0.3 0.2 0.1 g d T 0.01 0.00 0.01 g s T [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 8
Figure 8. Figure 8: Results on the axial and tensor charges for the u-, d- and s-quarks (left panel) from ETMC [44], PNDME [45], Mainz [46] and 𝜒QCD [47]. The right panel shows results on the u- and d-quark transversity without and with LQCD input on 𝑔 𝑢,𝑑 𝑇 (figure taken from Ref. [48]). Nucleon electromagnetic (EM) form factors have been computed over many years. However, it is only recently that LQCD results have included … view at source ↗
Figure 9
Figure 9. Figure 9: ETMC results on the EM form factors. Left: proton (top) and neutron (bottom) electric; Middle: proton (top) and neutron (bottom) magnetic [49]; Right: strange electric (top) and magnetic (bottom) [50– 52]. Red bands show fits to the z-expansion while the blue one is a fit to the Galster form. Bottom: LQCD results on the proton and neutron radii and magnetic moments (four plots on the left) and proton stran… view at source ↗
Figure 10
Figure 10. Figure 10: LQCD results on EM radii and magnetic moments for the proton and neutron (6 panels on the left) and strange radii and magnetic moment (3 panels on the right) [50]. extrapolated to the continuum limit. In [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Preliminary continuum extrapolated results at physical pion mass by ETMC in the MS at 2 GeV. Left two panels show the continuum extrapolation of the momentum fraction and angular momentum for quarks and gluons; Right two panels show the contribution of quarks and gluons to the momentum fraction and spin of the nucleon. 0.0 0.5 1.0 1.5 2.0 t [GeV2 ] 5 4 3 2 1 0 Di (t) total g u+d+s 0.0 0.5 1.0 1.5 2.0 0.0 … view at source ↗
Figure 12
Figure 12. Figure 12: Left: The nucleon 𝐷(𝑡)-term vs −𝑡 = 𝑄 2 (left top) [60]; GFF 𝐴20 (𝑡) (right top) and 𝐷(𝑡)-term (left bottom) and r.m.s mass radius (right bottom) for a scalar glueball compared to the other hadrons [61]. need for further investigation. For the isoscalar, strange and charm Mellin moments there are only a few computations. The ETMC collaboration performed a flavor decomposition of the unpolarized moments us… view at source ↗
Figure 13
Figure 13. Figure 13: Low-𝑥 region covered by EIC. Early exploratory studies were done within the LaMET approach in Refs. [79, 80] for the unpolarized distribu￾tions of the pion, of the pion and kaon in Ref. [27] and of the nucleon [81–83] but also within the pseudo￾distribution approach for the the pion unpolarized PDF [84]. Other approaches were also developed. For example, employing matrix elements of two local, spacelike-s… view at source ↗
Figure 14
Figure 14. Figure 14: Left: Results from an analysis of experimental data by JAM with (blue band) and without (red band) using LQCD input. Figure from Ref.[91]. Right top: Nucleon strange helicity PDF (blue band) compared to results from JAM (green band) and NNPDF (red band) [89]. Right bottom: Strange-antistrange asymmetry for the unpolarized PDF [90]. An early example of an im￾proved computation is the evaluation of the vale… view at source ↗
Figure 15
Figure 15. Figure 15: Left top: Valence pion unpolarized PDF using NNLO matching and the hybrid renormalization scheme (red band) compared to JAM (black band) [92]. The green band shows previous results without the improvements. The gray horizontal bands show the 𝑥-values where the PDF should not be trusted; Right top: Nucleon isovector unpolarized PDF from Ref. [95] (green band) and ETMC (orange band); Left bottom: Isovector … view at source ↗
Figure 16
Figure 16. Figure 16 [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Impact of EIC data on the determination of the twist-3 PDF 𝑔𝑇 (𝑥). Figure from Ref. [3]. The pseudo-distribution ap￾proach was used by the HadStruc and ETM col￾laborations. The first an￾alyzed one HISQ ensem￾ble with 𝑚𝜋 ∼ 360 MeV and 𝑎 ∼ 0.09 fm and the second one TMF ensemble with 𝑚𝜋 = 260 MeV and 𝑎 ∼ 0.09 fm. Their re￾sults are compatible. The quasi-distribution approach was employed by LPC and the MSUL… view at source ↗
Figure 18
Figure 18. Figure 18: Nucleon gluon unpolarized PDF. Left: ETMC (red) [101] compared to HadStruc collabotaion (green) [102] and JAM (blue). Middle: LPC (blue) compared to CT18 (yellow), NNPDF (purple) and JAM (red) [103]. Right: MSULAT (green) compared to JAM (red), CT18 (blue) and CJ22 (yellow) [108]. Important quantities for the EIC physics program include higher twist PDFs. The three twist-3 PDFs, scalar 𝑒(𝑥), helicity 𝑔2 (… view at source ↗
Figure 19
Figure 19. Figure 19: Results by ETMC for the nucleon isovector helicity (left top) [113] and transversity (right top) [114] for 𝑄 2 = −𝑡=0, 0.69, 1.02 GeV2 in the Breit-frame; Nucleon isovector unpolarized GPD in the asymmetric frame at various −𝑡 = 𝑄 2 -values and 𝜉=0 (Left bottom) courtesy of K. Cichy, EINN 2025); Pion isovector unpolarized GPD in the asymmetric frame for several values of −𝑡 [115] (right bottom) [PITH_FUL… view at source ↗
Figure 20
Figure 20. Figure 20: Distribution in impact parameter for the pion (top) and nucleon (bottom) for 𝑥 = 0.3 (left) and 𝑥 = 0.5 (right) [117]. The computation was done using one 𝑁𝑓 = 2 + 1 + 1 TMF ensemble with 𝑚𝜋 = 260 MeV and 𝑎 ∼ 0.09 fm, demonstrating the feasibility of the method. However, computing matrix ele￾ments in the Breit frame is very expensive since for each value of the momentum transfer one needs another three-poi… view at source ↗
Figure 21
Figure 21. Figure 21: Top 4 panels: Nucleon axial GFFs for i) isovector 𝐴˜𝑢−𝑑 10 and 𝐴˜𝑢−𝑑 20 (left); ii) isoscalar 𝐴˜𝑢+𝑑 10 and 𝐴 𝑢+𝑑 20 (right), shown with black points compared to those computed from local oper￾ators (red points) [122]. Bottom 4 panels: i) Electric and mag￾netic form factors (left) extracted from LQCD-determined GPDs (blue) and from the matrix element of the electromagnetic current (red) [123]; ii) Pion GFF… view at source ↗
read the original abstract

Hadron structure calculations using lattice Quantum Chromodynamics (QCD) have advanced significantly in recent years. Results for charges, form factors, and lower Mellin moments can be obtained to high precision, generalized parton distributions can now be computed either directly or reconstructed from moments, and transverse-momentum-dependent distributions can be accessed through direct lattice calculations. Together, these quantities provide detailed and complementary insights into the internal structure of hadrons. These theoretical developments are highly relevant to the experimental program of the Electron-Ion Collider (EIC) and of other facilities. We review the most pertinent lattice QCD results for hadron structure that inform the EIC scientific agenda, with particular emphasis on the pion, kaon, and nucleon.

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 / 3 minor

Summary. The manuscript is a review that summarizes recent lattice QCD advances in computing hadron structure quantities—charges, electromagnetic and axial form factors, lower Mellin moments of PDFs, GPDs (both direct and moment-reconstructed), and TMDs—for the pion, kaon, and nucleon. It argues that these results are now sufficiently mature to provide complementary, quantitative input for the EIC physics program, particularly for interpreting parton distributions and form-factor measurements at the kinematics accessible to the collider.

Significance. If the cited lattice results are faithfully represented, the review supplies a timely, single-source overview that experimentalists and phenomenologists can use to assess the current theoretical constraints on nucleon and meson structure. By collating precision lattice numbers with EIC-relevant observables, it helps identify which quantities are already under good control and which still require further lattice or model input before EIC data arrive.

major comments (2)
  1. [§4] §4 (GPDs): the statement that GPDs 'can now be computed either directly or reconstructed from moments' is not accompanied by any quantitative comparison of the systematic uncertainties of the two approaches (e.g., truncation of the moment series versus lattice discretization effects in direct calculations), which is load-bearing for the claim that these results are ready for EIC use.
  2. [§5] §5 (TMDs): the assertion that TMDs 'can be accessed through direct lattice calculations' omits discussion of the renormalization and matching procedures required to connect lattice quasi-TMDs to the continuum MS-bar scheme; without this, the precision quoted for the TMD results cannot be assessed for EIC phenomenology.
minor comments (3)
  1. [Abstract] Abstract: the phrase 'high precision' is used without any numerical benchmark (e.g., percent-level errors on charges or moments); a single sentence referencing the typical error budgets of the cited works would improve clarity.
  2. [Table 1] Table 1 (nucleon charges): the caption does not state whether the quoted uncertainties are statistical only or include systematic contributions from chiral and continuum extrapolations.
  3. [References] References: several 2022–2023 papers on lattice TMD renormalization (e.g., works using the Collins-Soper kernel on the lattice) are absent; their inclusion would strengthen the TMD section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments on §§4 and 5. We address each major comment below and will revise the manuscript to strengthen the discussion of systematic uncertainties and renormalization procedures.

read point-by-point responses

  1. Referee: [§4] §4 (GPDs): the statement that GPDs 'can now be computed either directly or reconstructed from moments' is not accompanied by any quantitative comparison of the systematic uncertainties of the two approaches (e.g., truncation of the moment series versus lattice discretization effects in direct calculations), which is load-bearing for the claim that these results are ready for EIC use.

    Authors: We agree that a quantitative comparison of the systematic uncertainties would improve the utility of the review for EIC phenomenology. In the revised manuscript we will expand the discussion in §4 to include a brief comparison of truncation errors in moment reconstruction versus discretization and other lattice systematics in direct calculations, drawing on the specific results and references already cited in the section. revision: yes

  2. Referee: [§5] §5 (TMDs): the assertion that TMDs 'can be accessed through direct lattice calculations' omits discussion of the renormalization and matching procedures required to connect lattice quasi-TMDs to the continuum MS-bar scheme; without this, the precision quoted for the TMD results cannot be assessed for EIC phenomenology.

    Authors: We acknowledge that the present text in §5 does not explicitly address the renormalization and matching steps. In the revised version we will add a concise paragraph outlining the quasi-TMD renormalization procedure and the matching to the MS-bar scheme, thereby allowing readers to assess the quoted precision in the context of EIC applications. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is a review paper that summarizes published lattice QCD results on hadron structure quantities (charges, form factors, Mellin moments, GPDs, TMDs) and their relevance to EIC kinematics. No new derivations, predictions, parameter fittings, or model-independent extrapolations are performed within the text. All referenced results are drawn from external literature via citations, with no self-citation chains or internal equations that reduce claims to inputs defined by the paper itself. The central content is descriptive collation rather than any load-bearing derivation, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The review rests on the standard lattice QCD framework and the assumption that cited calculations are reliable; no new free parameters, axioms, or invented entities are introduced by this paper itself.

axioms (1)
  • domain assumption Lattice QCD on a discrete Euclidean grid accurately approximates continuum QCD in the limit of vanishing lattice spacing and infinite volume.
    Invoked implicitly when stating that lattice results can be obtained to high precision.

pith-pipeline@v0.9.0 · 5421 in / 1111 out tokens · 52402 ms · 2026-05-14T00:35:00.612209+00:00 · methodology

discussion (0)

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