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REVIEW 3 major objections 5 minor 33 references

Comment on "LaMET's Asymptotic Extrapolation vs. Inverse Problem"

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This comment argues that LaMET's reconstruction of parton distributions from noisy lattice data is an ill-posed inverse problem whose uncertainty is controlled by the assumed smoothness of the PDF, not by the exponential decay of the…

desk verdict A clear, well-argued reply-comment that restates the authors' earlier critique of rigid LaMET fits; the only new element is a toy model whose lesson is explicitly conditional on PDF smoothness. read the letter →

arxiv 2506.24037 v1 pith:YPLMG7DT submitted 2025-06-30 hep-lat hep-ph

Hervé Dutrieux , Joe Karpie , Christopher J. Monahan , Kostas Orginos , Anatoly Radyushkin , David Richards , Savvas Zafeiropoulos This is my paper · ORCID
classification hep-lathep-ph PACS 12.38.Gc
keywords LaMETpartondistributionfunctionsinverseproblemregularizationdependenceBayesianpriorsGaussianprocesslatticeQCDuncertaintyquantification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This comment responds to a rebuttal paper that defends rigid few-parameter extrapolation of noisy lattice data in the large momentum effective theory (LaMET). The authors argue that recovering a continuous quasi-PDF from a limited, noisy set of Fourier harmonics is an ill-posed inverse problem, and its tell-tale sign is strong sensitivity to the choice of regularization, not whether the integral relation can be formally inverted. They maintain that the exact exponential asymptotic behavior of space-like matrix elements has little effect on the moderate-x region; the real regulator is the assumed smoothness of parton distributions. If they are right, uncertainty bands from rigid fits are systematically too small, and the advertised point-by-point reconstruction of light-cone PDFs is instead smeared at a width of order the effective mass over the hadron momentum. The paper is a correction of what its authors call misunderstandings of their earlier proposal, not a new computation.

What carries the argument

The machinery is the truncated Fourier relation $f(y,P_z)=P_z\int_{-\infty}^{\infty}\frac{dz}{2\pi}e^{iyP_z z}h(z,P_z)$ between the quasi-PDF and the space-like matrix elements, together with a diagnostic that defines ill-posedness by sensitivity to regularization rather than by formal invertibility. The second ingredient is a scaling estimate: exponential suppression of high harmonics with a decay length $P_z/m_{\rm eff}$ acts like a convolution in $x$-space with width of order $m_{\rm eff}/P_z$, so any feature narrower than that width is unrecoverable. The paper's proposed alternative regularization is a Bayesian prior combining smoothness and correlation structure in $x$-space with a Fourier-space prior that enforces exponential decay only beyond the largest measured separation, with hyperparameters chosen per dataset.

What would settle it

Run a closure test with a synthetic PDF that has a narrow bump of width much smaller than $m_{\rm eff}/P_z$ at, say, $x=0.5$: generate its Fourier harmonics, impose the expected exponential decay, add noise at the level of current lattice data, and reconstruct with the paper's Bayesian procedure. If the bump survives at the claimed point-by-point resolution, the smearing argument is wrong; if the bump is erased, the comment's central limitation is confirmed.

Watch

Extended reading notes

Core claim

The central claim is that LaMET and pseudo-PDF approaches face the same ill-posed inverse problem: both reconstruct a continuous function in $x$ from a limited, noisy set of Fourier-space data. The authors' diagnostic is empirical: the problem is ill-posed when different physically reasonable regularizations of that data produce markedly different reconstructions and uncertainty estimates. They argue that the exponential decay of large-$z$ matrix elements is not the controlling regulator: many current datasets end before one decay length, and varying the decay form barely changes the moderate-$x$ reconstruction. The quantity that actually controls the answer is the prior assumption that usual parton distributions are smooth on scales larger than $m_{\rm eff}/P_z$, with sharp features erased by the Fourier inversion. From this it follows that the advertised point-by-point reconstruction of light-cone PDFs is dubious, because the reconstruction is intrinsically smeared at that width.

Load-bearing premise

The load-bearing premise is that physically realized parton distributions are 'usual', i.e. smooth enough that high Fourier harmonics are suppressed; if a real PDF contains a sharp moderate-$x$ feature, the exponential smearing will erase it and the asymptotic behavior of the correlator would become important after all.

Editorial extensions

If this is right

  • Uncertainty bands from rigid few-parameter LaMET fits should be read as lower bounds on model dependence, not as the full error.
  • The distinction between LaMET and pseudo-PDF is not one of inverse problem versus forward problem; both share the same ill-conditioned Fourier inversion and the same sensitivity to regularization.
  • At moderate $x$, the precise functional form chosen for the asymptotic $z$ tail is not the dominant error; the smoothness prior on the PDF is.
  • At finite hadron momentum, LaMET cannot resolve sharp $x$-space features narrower than $m_{\rm eff}/P_z$, so point-by-point claims hold only up to that smearing.
  • For datasets with small noise in the range $\lambda\sim5$–$15$, the concerns are reduced; the criticism is aimed at currently noisy datasets rather than at all LaMET measurements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The smearing argument implies a quantitative resolution scale: comparing $m_{\rm eff}/P_z$ with the narrowest features expected in realistic PDFs would show how much of the advertised point-by-point resolution is practically real at current momenta.
  • The regularization-sensitivity diagnostic could be exported to other hadronic inverse problems, such as global PDF extraction from deep-inelastic structure functions, to test whether quoted uncertainty bands depend on parametrization choice in the same way.
  • A concrete community protocol follows: closure tests with a truth distribution containing a deliberately sharp moderate-$x$ feature. The paper's logic predicts such features are systematically washed out by the exponential tail, and the test would confirm or refute that prediction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This is a reply/comment to Chen et al. (arXiv:2505.14619) defending the authors' earlier critique of rigid parametric extrapolations in LaMET. The paper argues that reconstructing quasi-PDFs from finite, noisy lattice matrix elements is an ill-posed inverse problem whose defining symptom is strong sensitivity to the choice of regularization. It criticizes the rigid fits of Refs. [5,6] for underestimating uncertainty, contends that the exact asymptotic behavior of space-like correlators matters little in the moderate-x region when parton distributions are smooth, and claims that LaMET smears point-like x-space features over a width of order meff/Pz, making point-by-point PDF reconstruction dubious. The paper includes two toy-model figures (Figs. 3 and 4), visual re-examinations of published extrapolations (Figs. 1 and 2), and a correction of what the authors see as misunderstandings in Ref. [2].

Significance. If the central claims are correct, the paper provides a useful caution about the reliability of uncertainty bands from rigid few-parameter fits in current LaMET analyses and argues for more conservative, regularization-aware uncertainty quantification. The paper's strengths include explicit toy-model demonstrations, a clearly stated criterion (regularization sensitivity) for identifying ill-posedness, and a falsifiable expectation for smearing of x-space features. It also openly acknowledges that the Gaussian-process priors depend on the dataset (footnote [13]), which is an important limitation. However, the quantitative support for the main claims largely resides in the authors' previous papers (Refs. [1,8]), and the present manuscript does not deliver a systematic robustness test across plausible PDF shapes or hyperparameter choices. The smoothness assumption on which the 'asymptotic behavior matters little' claim rests is explicit but unquantified.

major comments (3)
  1. [Inverse problem and upper-bound on uncertainty; Fig. 4] The claim that the exact asymptotic behavior of space-like correlators 'matters little' in the moderate-x region is demonstrated only for the smooth family of Eq. (3), whose Fourier harmonics decay as 1/lambda^2. The bump example in the same figure shows that if a PDF contains features narrower than the smearing scale meff/Pz, then exponential damping erases those features and the asymptotic behavior becomes the controlling regulator. The paper calls such bump models 'unphysical' but does not justify the implicit assumption that realistic PDFs are smooth on all scales relevant at moderate x. Please either quantify the class of PDFs for which the claim holds (for example by testing additional shapes with (1-x)^beta for beta <= 2 or with moderate-x oscillatory components), or explicitly state the claim as conditional on an assumed minimum feature width in x.
  2. [Footnote [13] and 'Our proposal to study uncertainty'] The authors acknowledge that the Gaussian-process hyperparameters are set with respect to the dataset at hand and that the priors are designed to enforce a desired behavior on the posterior. This makes the demonstration of 'strong sensitivity to the choice of regularization' in Fig. 3 partly a statement about the hyperparameter-setting rule, rather than a purely intrinsic property of the inverse problem. The closure tests in Ref. [8] mitigate this concern, but those tests are not reproduced or summarized here. Please provide a robustness check in which the hyperparameters are varied over a systematic, physically motivated range for the same dataset, and report how the spread of reconstructions changes. Without such a check, the claim that the bands in Fig. 3 reflect the full model uncertainty remains under-supported.
  3. [Point-by-point reconstruction in the LaMET formalism] The argument that LaMET smears x-space features over a width of order meff/Pz is made with a simple Fourier toy model, not with the actual LaMET matching formalism. The light-cone PDF is obtained from the quasi-PDF through a perturbative matching kernel, and it is not automatic that exponential damping of the hadronic matrix element directly translates into a smearing of the light-cone PDF with no compensating structure. Please either derive the smearing kernel in the matching formalism, showing how the exponential decay of h(z,Pz) limits the resolution in x after matching, or restrict the claim to the quasi-PDF itself rather than to the light-cone PDF. As written, the conclusion that 'the claim that LaMET can reconstruct a point-by-point x-dependence of light-cone PDFs ... seems dubious' is plausible but not established by the evidence presented.
minor comments (5)
  1. [Fig. 4 caption] The caption says 'dotted lines' twice: 'We now add an unphysical bump at x = 0.5 (dotted lines). When the exponential decay is enforced (dotted lines), the bump is erased.' The second occurrence should presumably be 'solid lines' for the enforced-exponential-decay curves; please correct this to avoid ambiguity.
  2. [Eq. (3) and Fig. 4] The parameter a in Eq. (3) is said to be varied, but the text does not specify the range of a used or the criteria for choosing the displayed values. For reproducibility, please state the values of a and the decay parameters (meff, Pz, threshold) used in Fig. 4.
  3. [Footnote [13]] The footnote is long and defensive; the discussion about why dataset-dependent priors are still called 'priors' is not central to the scientific reply and would be better placed in a methods appendix or moved to the main text only if it is essential to the argument.
  4. [Fig. 4, lower panels] The term 'unphysical PDF models' is asserted without definition. If the criterion is simply that the bump is not part of the smooth benchmark family, please say so explicitly, or specify what physical conditions (positivity, support, smoothness) are meant.
  5. [Section 'Inverse problem and upper-bound on uncertainty'] The discussion first states that the bound of Ref. [5] is 'not really an upper bound' because parts of the quantity are ignored, then immediately concedes that 'there must be a value of N of the order of a few units where this broad estimate is reasonable.' This reads as contradictory; please clarify the intended status of the bound (e.g., a heuristic estimate with explicit caveats rather than a rigorous upper bound).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are supported by explicit in-paper sensitivity tests and independent benchmarks, with self-citations not load-bearing.

full rationale

This is a comment and defense of a prior analysis, not a new derivation whose conclusion is fed back into its inputs. The central claims - that LaMET reconstruction is an ill-posed inverse problem, that the exact large-lambda asymptotic behavior matters little in the moderate-x region, and that rigid few-parameter fits can underestimate uncertainty - are supported by demonstrations inside the paper. Fig. 4 takes the model of Eq. (3), whose Fourier harmonics decay as 1/lambda^2, applies an exponential suppression exp(1 - lambda meff/Pz), and directly compares the x-space reconstructions; the observed insensitivity for x > 0.3 is a computed sensitivity result, not an input of the argument. Fig. 3 compares two different x-space prior kernels under the same Fourier-space decay prior and shows different uncertainty bands; this is the paper's own tell-tale sign criterion applied to a concrete example, not a circular reduction of the conclusion to the definition. The repeated citations to the authors' own Ref. [1] are frequent, but they do not carry the argument alone: the ill-posed nature is independently acknowledged by Ref. [7], and the Gaussian-process method was benchmarked through closure tests in Ref. [8]. Footnote [13] openly admits that prior scale hyperparameters are set with the dataset in mind; that is an honest limitation of the uncertainty quantification, but the paper is not claiming to predict a held-out quantity from those fitted values, so it does not fit the fitted-input-called-prediction pattern. The load-bearing smoothness-of-usual-parton-distributions assumption is unquantified and could be challenged by sharp x-space features, but that is a physics and correctness criticism, not a circularity. No equation or claim reduces to its own input by construction, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. The free parameters are the GPR hyperparameters and the toy-model parameters, both chosen by hand. The physical axioms are the exponential decay of matrix elements, the smoothness of PDFs, the Fourier relation, and the representativeness of the toy model.

free parameters (2)
  • Gaussian process hyperparameters (kernel amplitude, correlation length, Fourier-space decay mass) = not given; set per dataset by examination
    The central reconstructions in Fig. 3 and the critique of rigid fits depend on these choices. The authors note that hyperparameters must be set differently for different datasets and do not provide an automated rule, so the uncertainty assessment is partly a product of these hand-chosen values.
  • Toy-model parameters in Fig. 4 (a, meff, Pz, decay threshold) = a varied; meff = 0.3 GeV, Pz = 2 GeV
    Chosen to illustrate the argument, not fitted to data. These choices determine the claimed insensitivity region x > 0.3 and the smearing scale meff/Pz.
assumptions (4)
  • domain assumption Space-like LaMET matrix elements decay exponentially at large separation with a mass scale of order 0.2 to 0.3 GeV.
    Used to set the Fourier-space prior and to argue that features in x are smeared over a width meff/Pz. The paper itself notes that the datasets shown provide no conclusive sign of such decay.
  • domain assumption Parton distributions are smooth, so high Fourier harmonics are suppressed.
    This is the core regulator in the authors' argument that asymptotic behavior matters little. It is introduced in the 'Inverse problem and upper-bound on uncertainty' section as a general expectation, not as a tested hypothesis.
  • standard math The Fourier relation of Eq. (1) between quasi-PDF and matrix elements is valid for all z with appropriate renormalization.
    The inverse problem is defined by inverting this integral relation from finite noisy samples, so this standard Fourier transform identity is a necessary background assumption.
  • ad hoc to paper The toy model f(x) = a(1-x)^3 - (1-x)^4 - (1-x)^5 (Eq. 3) is representative of realistic PDFs.
    Used to draw general conclusions about moderate-x insensitivity to asymptotic behavior and about the smearing of sharp features. No evidence is provided that all PDFs behave this way.

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Cite this review

Pith. "Pith review of Comment on "LaMET's Asymptotic Extrapolation vs. Inverse Problem"." pith.science (2026). https://pith.science/paper/YPLMG7DT

@misc{pith2026250624037,
  author       = {Pith},
  title        = {Pith review of: Comment on "LaMET's Asymptotic Extrapolation vs. Inverse Problem"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPLMG7DT}},
  note         = {Machine review of arXiv:2506.24037}
}
read the original abstract

In arXiv:2504.17706 {Dutrieux:2025jed} we criticized the excessive model-dependence introduced by rigid few-parameter fits to extrapolate lattice data in the large momentum effective theory (LaMET) when the data are noisy and lose signal before an exponential asymptotic behavior of the space-like correlators is established. In reaction, arXiv:2505.14619 {Chen:2025cxr} claims that even when the data is of poor quality, rigid parametrizations are better than attempts at representing the uncertainty using what they call "inverse problem methods". We clarify the fundamental differences in our perspectives regarding how to meaningfully handle noisy lattice matrix elements, especially when they exhibit a strong sensitivity to the choice of regularization in the inverse problem. We additionally correct misunderstandings of {Chen:2025cxr} on our message and methods.

Figures

Figures reproduced from arXiv: 2506.24037 by the authors.

Figure 1
Figure 1. FIG. 1. The set of Fourier data for the pion valence quark [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The set of Fourier data (real and imaginary parts) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (top) Two solutions of the inverse problem in Fourier [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (top) The dotted lines represent the Fourier transform (left) and [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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