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arxiv: 2504.17706 · v2 · submitted 2025-04-24 · ✦ hep-lat · hep-ph

Inverse problem in the LaMET framework

Herv\'e Dutrieux , Joe Karpie , Christopher J. Monahan , Kostas Orginos , Anatoly Radyushkin , David Richards , Savvas Zafeiropoulos This is my paper

Pith reviewed 2026-05-22 18:57 UTC · model grok-4.3

classification ✦ hep-lat hep-ph
keywords LaMETparton distribution functionslattice QCDinverse problemFourier transformhadron structure
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0 comments X

The pith

The inverse problem from incomplete lattice harmonics creates large uncertainty in LaMET parton distributions unless strong assumptions are added.

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

The paper studies the inverse problem that arises when LaMET is used to extract parton distributions from lattice QCD. Lattice calculations supply matrix elements only up to moderate Fourier harmonics, beyond which the data are noisy or absent. By examining actual non-perturbative results, the authors show that this truncation produces sizable uncertainties in the reconstructed distributions unless very restrictive assumptions are imposed on the missing terms. They further find that the precise form of the asymptotic tail has little practical effect within the x range where LaMET is currently usable. The work calls for more advanced methods to handle the inverse problem in both LaMET and related short-distance factorization approaches.

Core claim

Using non-perturbative lattice data, the authors show that the uncertainty introduced by this inverse problem in a realistic setup remains significant without very restrictive assumptions, and that the importance of the exact asymptotic behavior is minimal for values of x where the framework is currently applicable. The crux of the inverse problem lies in harmonics of the order of λ = z P_z ∼ 5–15, where the signal in the lattice data is often barely existent or unreliable in current studies.

What carries the argument

The Fourier transform required by LaMET to recover the x-dependence of parton distributions from matrix elements known only over a finite, noisy range of harmonics.

If this is right

  • Uncertainties in parton distributions extracted via LaMET stay large with present-day lattice data quality.
  • The precise asymptotic decay of missing harmonics has minimal impact on results at accessible x.
  • More sophisticated techniques are needed to control the inverse problem in LaMET and in short-distance factorization.
  • LaMET does not currently deliver a direct computation of the full x-dependence, contrary to a common misconception.

Where Pith is reading between the lines

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

  • Future lattice work that strengthens the signal specifically in the λ ∼ 5–15 window could shrink the inverse-problem uncertainty without needing extreme assumptions.
  • The same truncation issue is likely to appear in any factorization scheme that converts limited lattice matrix elements into momentum-space distributions.
  • Cross-checking LaMET results against moment-based extractions on the same ensembles would help quantify how much of the spread comes from the inverse problem.

Load-bearing premise

The non-perturbative lattice data sets employed are representative of typical current calculations and that the signal in the harmonics λ = z P_z ∼ 5–15 is indeed barely existent or unreliable.

What would settle it

A new lattice calculation that delivers reliable data for harmonics well above λ = 15 and shows that enforcing or removing an exponential tail changes the extracted distributions by a large amount at currently accessible x would test the claim.

Figures

Figures reproduced from arXiv: 2504.17706 by Anatoly Radyushkin, Christopher J. Monahan, David Richards, Herv\'e Dutrieux, Joe Karpie, Kostas Orginos, Savvas Zafeiropoulos.

Figure 1
Figure 1. Figure 1: FIG. 1. (top) Various BG reconstructions of the same Fourier [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (top) Reconstructions / extrapolations of the same [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (top) Reconstructions / extrapolations of the same [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Various [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

One proposal to compute parton distributions from first principles is the large momentum effective theory (LaMET), which requires the Fourier transform of matrix elements computed non-perturbatively. Lattice quantum chromodynamics (QCD) provides calculations of these matrix elements over a finite range of Fourier harmonics that are often noisy or unreliable in the largest computed harmonics. It has been suggested that enforcing an exponential decay of the missing harmonics helps alleviate this issue. Using non-perturbative data, we show that the uncertainty introduced by this inverse problem in a realistic setup remains significant without very restrictive assumptions, and that the importance of the exact asymptotic behavior is minimal for values of $x$ where the framework is currently applicable. We show that the crux of the inverse problem lies in harmonics of the order of $\lambda=zP_z \sim 5-15$, where the signal in the lattice data is often barely existent in current studies, and the asymptotic behavior is not firmly established. We stress the need for more sophisticated techniques to account for this inverse problem, whether in the LaMET or related frameworks like the short-distance factorization. We also address a misconception that, with available lattice methods, the LaMET framework allows a "direct" computation of the $x$-dependence, whereas the alternative short-distance factorization only gives access to moments or fits of the $x$-dependence.

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

1 major / 1 minor

Summary. The manuscript examines the inverse problem in the LaMET framework for computing parton distribution functions from lattice QCD matrix elements. Using non-perturbative data, the authors argue that the uncertainty arising from the finite range of Fourier harmonics remains significant without very restrictive assumptions on the missing terms, that the precise asymptotic decay has minimal impact for the x values where LaMET is currently applicable, and that the crux lies in the harmonics with λ = z P_z ∼ 5–15 where the lattice signal is typically weak or unreliable. The paper also addresses a misconception that current LaMET methods permit a direct computation of the x-dependence, in contrast to moment-based or fit-based approaches in short-distance factorization.

Significance. If the findings hold, the work would be significant for the lattice QCD community by quantifying practical limitations of the inverse problem in realistic setups and underscoring the need for more sophisticated techniques. The reliance on actual non-perturbative data to illustrate the points adds practical relevance, though the strength of the conclusions depends on the representativeness of the data sets employed.

major comments (1)
  1. [Sections discussing non-perturbative data and harmonics λ ∼ 5–15] The central claim that the inverse-problem uncertainty remains significant without restrictive assumptions, and that this affects current LaMET applications in general, rests on the specific non-perturbative lattice data sets used to demonstrate weak signal at λ ∼ 5–15. The manuscript must explicitly identify the ensembles (including references to the original lattice publications), detail parameters such as source-sink separation, smearing, and fitting procedures, and provide evidence that the observed unreliability in this harmonic range is representative of typical contemporary calculations rather than an artifact of the chosen data. Without this justification, the broader conclusion for the field does not fully follow.
minor comments (1)
  1. [Abstract and introduction] In the abstract and introduction, provide a brief quantitative example or reference to a figure showing the x-range where the framework is currently applicable to make the statement about minimal impact of asymptotic behavior more concrete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback on the presentation of our non-perturbative results. We address the single major comment below and will revise the manuscript to incorporate additional details as requested.

read point-by-point responses

  1. Referee: [Sections discussing non-perturbative data and harmonics λ ∼ 5–15] The central claim that the inverse-problem uncertainty remains significant without restrictive assumptions, and that this affects current LaMET applications in general, rests on the specific non-perturbative lattice data sets used to demonstrate weak signal at λ ∼ 5–15. The manuscript must explicitly identify the ensembles (including references to the original lattice publications), detail parameters such as source-sink separation, smearing, and fitting procedures, and provide evidence that the observed unreliability in this harmonic range is representative of typical contemporary calculations rather than an artifact of the chosen data. Without this justification, the broader conclusion for the field does not fully follow.

    Authors: We agree that explicit identification of the data sets strengthens the manuscript and helps readers evaluate the scope of our conclusions. In the revised version we will add a new subsection (or appendix) that fully specifies the lattice ensembles employed, including complete references to the original publications from which the matrix elements were taken. We will also document the source-sink separations, smearing parameters, and fitting procedures used to obtain the quoted matrix elements. To address representativeness, we will include a brief comparison with other recent LaMET calculations in the literature, noting that the rapid degradation of the signal-to-noise ratio for λ = z P_z ≳ 5–10 is a widely reported feature arising from the exponential fall-off of the correlators and the practical limits on achievable P_z. This discussion will clarify that the behavior we observe is characteristic of current-generation ensembles rather than an isolated artifact. revision: yes

Circularity Check

0 steps flagged

Analysis relies on external lattice data; no internal circularity in derivation

full rationale

The paper's central claim—that inverse-problem uncertainty remains significant without restrictive assumptions—is supported by analysis of existing non-perturbative lattice matrix elements rather than any quantity defined or fitted inside the paper itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or described derivation chain. The work draws on external benchmarks (current lattice studies) whose signal quality at λ ∼ 5-15 is treated as representative input, not generated by the present analysis. This yields a minor score consistent with normal self-citation that is not load-bearing for the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard lattice QCD domain assumptions about finite-range matrix elements and noisy high harmonics; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Lattice QCD provides calculations of matrix elements over a finite range of Fourier harmonics that are often noisy or unreliable in the largest computed harmonics.
    Invoked in the opening description of the LaMET setup and the inverse problem.

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