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arxiv: 2604.21476 · v1 · submitted 2026-04-23 · ✦ hep-lat · hep-ph

Recognition: unknown

Reconstructing the full kinematic dependence of GPDs from pseudo-distributions

Anatoly Radyushkin, C\'edric Mezrag, Christopher Monahan, David Richards, Eloy Romero, Herv\'e Dutrieux, Joe Karpie, Kostas Orginos, Robert G. Edwards, Savvas Zafeiropoulos

Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:59 UTC · model grok-4.3

classification ✦ hep-lat hep-ph
keywords generalized parton distributionspseudo-distributionslattice QCDdouble distributionsGaussian process regressionnucleon structureisovector GPDsproton GPDs
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0 comments X

The pith

Lattice QCD now reconstructs the full (x, ξ, t) dependence of proton GPDs by extracting double distributions directly from pseudo-distribution data.

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

The paper develops a procedure to obtain the complete dependence of the unpolarized isovector proton GPDs H^{u-d} and E^{u-d} on momentum fraction x, skewness ξ, and momentum transfer t from lattice QCD data computed in the pseudo-distribution formalism. For the first time the reconstruction extracts the underlying double distributions straight from the lattice results, thereby enforcing a Lorentz-invariance property that GPDs must satisfy. Multidimensional Gaussian process regression regularizes the ill-posed inverse problem that converts the lattice observables into the GPDs while allowing an assessment of model dependence. The demonstration uses an ensemble with 358 MeV pions, 0.094 fm spacing, and hadron momenta reaching 2.7 GeV, yielding additional skewness-dependent moments beyond earlier work.

Core claim

Starting from lattice pseudo-distributions at several hadron momenta, the full kinematic dependence of the GPDs is recovered through an intermediate step that extracts the corresponding double distributions directly; this step automatically incorporates the polynomiality property required by Lorentz symmetry, while the multidimensional Gaussian process supplies the regularization needed to control the inverse problem.

What carries the argument

Multidimensional Gaussian process regression applied to recover double distributions from pseudo-distribution lattice data.

If this is right

  • GPDs become available over the full kinematic domain without additional functional parametrizations.
  • The extracted double distributions guarantee that the GPDs obey polynomiality in skewness.
  • Skewness-dependent moments of the GPDs are obtained directly from the lattice data.
  • Systematic uncertainty from regularization can be quantified by varying the Gaussian process hyperparameters.

Where Pith is reading between the lines

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

  • The same regularization framework could be applied to polarized GPDs once corresponding lattice data become available.
  • At the physical pion mass the method would supply tighter three-dimensional constraints on nucleon structure.
  • Direct comparison of the reconstructed GPDs against phenomenological extractions from deeply virtual Compton scattering would test the reconstruction independently.
  • The approach provides a route toward fully symmetry-constrained, parameter-light determinations of GPDs.

Load-bearing premise

The Gaussian process regression introduces only controllable model dependence and the chosen lattice ensemble at 358 MeV pion mass with its available kinematic points is sufficient to constrain the entire (x, ξ, t) dependence.

What would settle it

Repeating the reconstruction on a finer lattice spacing or lighter pion mass and finding that the resulting double distributions violate support properties or produce GPDs inconsistent with the forward limit would falsify the method.

Figures

Figures reproduced from arXiv: 2604.21476 by Anatoly Radyushkin, C\'edric Mezrag, Christopher Monahan, David Richards, Eloy Romero, Herv\'e Dutrieux, Joe Karpie, Kostas Orginos, Robert G. Edwards, Savvas Zafeiropoulos.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the support in ( view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (top) Prior and posterior in view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of the reconstruction in view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The discretization of the DD we use, with 30 view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The kinematic coverage of this study, consisting of view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Ratio view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (top) Dispersion relation view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Elastic form factors of the proton for the isovec view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Applying the pseudo-inverse of the kinematic matrix view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Demonstration of the effect of increasing the prior view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Baseline reconstruction of the real part of the Ioffe-time distribution corresponding to the unmatched view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Baseline reconstruction of the unmatched DD corre view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Baseline reconstruction of the unmatched view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. (top) Baseline reconstruction of the unmatched view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Same as Fig view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Effect of varying hyperparameters to assess the model dependence. The solid bands represent the baseline reconstruc view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Matching vs pure evolution of a toy model from view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Matched GPDs at view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Final extraction of isovector unpolarized GPDs at view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Comparison of our extraction (colored bands) with the Goloskokov-Kroll (GK) model (dashed lines) at view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The colored bands represent the left-hand side of view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23 view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Result of dipole fits on the GPR extraction of generalized form factors. For view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. (left) An ECS in one dimension, at large distance from the boundaries of the domain. (right) Effect of boundaries view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. The absolute value of an ECS in 1D (left) and 2D (right), at large distance from the boundaries of the domain. On the view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. (top) When the double integrals of ECSs are re view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28. (top) Acceleration of the computation time on view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. Result from running the exemplary code of the view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30. The Gaussian windows of view at source ↗
read the original abstract

We propose a reconstruction of the full $(x, \xi, t)$ dependence of unpolarized isovector proton generalized parton distributions (GPDs) $H^{u-d}$ and $E^{u-d}$ from lattice QCD data in the pseudo-distribution formalism. For the first time, we extract double distributions (DDs) directly from lattice data, enforcing therefore an important property of GPDs linked to Lorentz symmetry. We use the flexible framework of multidimensional Gaussian process regression to regularize the inverse problem and present an assessment of the impact of model dependence on the systematic uncertainty. Our lattice ensemble corresponds to a pion mass $m_\pi = 358$~MeV and a lattice spacing $a = 0.094$~fm. We use larger hadron momenta, up to 2.7~GeV, and kinematic coverage compared to our previous computations and extract additional skewness-dependent moments of the GPD.

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 / 2 minor

Summary. The manuscript proposes reconstructing the full (x, ξ, t) dependence of the unpolarized isovector proton GPDs H^{u-d} and E^{u-d} from lattice QCD pseudo-distribution data. It claims the first direct extraction of double distributions (DDs) F(β, α, t) from lattice data on a single ensemble (m_π = 358 MeV, a = 0.094 fm, |P| up to 2.7 GeV), thereby enforcing the Lorentz-symmetry support property of GPDs. Multidimensional Gaussian process regression is used to regularize the inverse problem, with an assessment of model dependence, plus extraction of additional skewness-dependent moments.

Significance. If the central results hold, the work would mark a notable advance in lattice GPD studies by achieving the first direct DD extraction from lattice pseudo-distributions rather than post-processing GPDs. The GP framework offers a data-driven route to full kinematic coverage with quantified systematics, potentially improving consistency with phenomenological parametrizations and enabling better tests of Lorentz invariance in the extracted distributions.

major comments (2)
  1. [Gaussian process regression and DD extraction sections] The central claim that DDs are extracted directly from lattice data (enforcing Lorentz symmetry) rests on the multidimensional GP regression not dominating the reconstruction. With only one ensemble, m_π = 358 MeV, and a finite set of Ioffe-time points, the paper must demonstrate through explicit hyperparameter/kernel variations that the support property and (x, ξ, t) dependence are data-constrained rather than prior-driven; the current assessment of model dependence appears insufficient to establish this.
  2. [Results and systematic uncertainty assessment] The weakest assumption—that the chosen lattice kinematics and GP regularization sufficiently constrain the full three-dimensional surface—requires quantitative support. Limited coverage up to |P| = 2.7 GeV on a single ensemble risks under-constraining the prior; the manuscript should include a dedicated study (e.g., leave-one-out or synthetic data tests) showing stability of the extracted F(β, α, t) under changes to the GP length scales.
minor comments (2)
  1. [Introduction and formalism] Clarify the precise definition and normalization of the double distribution F(β, α, t) relative to the standard GPD integral representation; this notation appears in the abstract but would benefit from an explicit equation in the methods.
  2. [Abstract and results] The abstract states that 'additional skewness-dependent moments' are extracted; specify which moments (e.g., which powers of ξ) and in which figure or table they are shown, to aid readers comparing with prior lattice work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and for recognizing the potential significance of our work on reconstructing GPDs and extracting double distributions from lattice pseudo-distributions. We address the two major comments point by point below. Where the comments identify areas for strengthening the analysis of model dependence and systematic uncertainties, we agree to incorporate additional quantitative studies in a revised manuscript.

read point-by-point responses

  1. Referee: The central claim that DDs are extracted directly from lattice data (enforcing Lorentz symmetry) rests on the multidimensional GP regression not dominating the reconstruction. With only one ensemble, m_π = 358 MeV, and a finite set of Ioffe-time points, the paper must demonstrate through explicit hyperparameter/kernel variations that the support property and (x, ξ, t) dependence are data-constrained rather than prior-driven; the current assessment of model dependence appears insufficient to establish this.

    Authors: We agree that explicit demonstration is required to establish that the extracted double distributions and the enforced support property are driven by the lattice data. The manuscript already presents an assessment of model dependence through variations in GP hyperparameters and kernels, with the support property preserved in these tests. To address the referee's concern more rigorously, we will expand this section in the revision with additional explicit hyperparameter and kernel variations, including quantitative metrics (such as variation in the extracted F(β, α, t) surfaces) to show the degree to which the (x, ξ, t) dependence is constrained by the data rather than the prior. revision: yes

  2. Referee: The weakest assumption—that the chosen lattice kinematics and GP regularization sufficiently constrain the full three-dimensional surface—requires quantitative support. Limited coverage up to |P| = 2.7 GeV on a single ensemble risks under-constraining the prior; the manuscript should include a dedicated study (e.g., leave-one-out or synthetic data tests) showing stability of the extracted F(β, α, t) under changes to the GP length scales.

    Authors: We acknowledge that the limited kinematic coverage on a single ensemble requires additional quantitative validation to confirm that the GP regularization does not under-constrain the three-dimensional reconstruction. While the current analysis incorporates some stability checks, we will add a dedicated subsection presenting leave-one-out cross-validation and synthetic data tests (generated from phenomenological GPD models) to demonstrate the stability of the extracted F(β, α, t) under variations in GP length scales. These studies will be performed on the existing dataset and included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in GPD reconstruction from pseudo-distributions

full rationale

The derivation uses lattice pseudo-distribution data as primary input and applies multidimensional Gaussian process regression to regularize the inverse problem while parametrizing via the standard double-distribution representation of GPDs. This representation enforces the Lorentz-invariance support property by construction of the known formalism rather than deriving it from the current fit. No equation or step reduces the extracted DD or full (x, ξ, t) dependence to a tautology, a fitted parameter relabeled as prediction, or a self-citation chain whose validity depends on the present work. The GP kernel and length-scale choices introduce assessed model dependence but do not make the central claim equivalent to the inputs by definition. The lattice ensemble and kinematic coverage supply independent constraints, rendering the reconstruction self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the pseudo-distribution formalism being a valid proxy for GPDs, the applicability of GP regression to the inverse problem, and the physical relevance of the chosen lattice ensemble. No new particles or forces are introduced.

free parameters (1)
  • Gaussian process hyperparameters
    Hyperparameters of the multidimensional GP are optimized or chosen to regularize the reconstruction of the GPDs from discrete lattice points.
axioms (1)
  • domain assumption Lorentz symmetry implies that GPDs can be represented by double distributions
    Invoked when enforcing the DD property during extraction from lattice data.

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Reference graph

Works this paper leans on

144 extracted references · 121 canonical work pages · 2 internal anchors

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    This systematic ef- fect translates in our reconstruction being systematically larger than the GK model forH u−d(+) (upper-right plot of Fig

    gives⟨x⟩ u−d = 0.150(5), the GK model stands at 0.17, and our own extraction at non-physical pion mass at 0.21(1) (see next section). This systematic ef- fect translates in our reconstruction being systematically larger than the GK model forH u−d(+) (upper-right plot of Fig. 21). An interesting aspect of GPD phenomenology isposi- tivity bounds[64, 114]. U...

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    Integrated behavior overα Let us start with the simple casek= 0, where theα- dependence is integrated over. From Eq. (A2), the DD moment is simplyA n,0(t), which in turn is a Mellin mo- ment of the GPD atξ= 0.A n,0(t) behaves with respect to evolution exactly like a PDF moment. In particular, one has: An,0(t, µ2

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    =A n,0(t, µ2 0) αs(µ2 0) αs(µ2 1) γn/(2πβ0) ,(A5) whereγ n are the Mellin moments of the DGLAP evolu- tion operator: γn =C F Z 1 0 dα 1 +α 2 1−α (αn−1 −1) =C F −2ψ(0)(n)−2γ E + 3 2 − 1 n − 1 n+ 1 ,(A6) withψ (0)(n) = d ln Γ(n)/dnthe digamma function and γE =−ψ (0)(1) the Euler-Mascheroni constant.β 0 = (33−2n f)/(12π)≈0.716. In Section 3.3 of [70], we stu...

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    (A11) ensures thatc n,n = 1

    =O n(ξ, t, µ2 0) αs(µ2 0) αs(µ2 1) γn/(2πβ0) .(A12) Let us write the explicit decomposition of Gegenbauer polynomials as: C(3/2) n (x) = nX k=0 k≡n[2] cn,kxk .(A13) The initial factor in the definition of Eq. (A11) ensures thatc n,n = 1. Then the conformal moments write: On(ξ) = [An,0 +A n,2ξ2 +...] +ξ 2cn−1,n−3[An−2,0 +A n−2,2ξ2 +...] +...(A14) 28 Theref...

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    The first term in the right hand side of Eq

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