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arxiv: 2510.17243 · v2 · submitted 2025-10-20 · ✦ hep-ph

Deep Neural Network extraction of Unpolarized Transverse Momentum Distributions

I. P. Fernando , D. Keller This is my paper

Pith reviewed 2026-05-18 06:41 UTC · model grok-4.3

classification ✦ hep-ph
keywords deep neural networksunpolarized TMDsDrell-Yan processtransverse momentum distributionsstructure kernelQCD phenomenologyFermilab experimentsmomentum space extraction
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The pith

Deep neural networks extract unpolarized TMDs directly from Drell-Yan data in momentum space.

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

This paper develops a physics-informed deep learning framework to extract unpolarized transverse momentum dependent parton distributions from fixed-target Drell-Yan measurements at Fermilab. The method learns a structure kernel in transverse momentum space whose convolution reproduces the observed cross sections, then reconstructs the underlying TMD integrand using a differentiable quadrature layer. It avoids factorized functional forms and Fourier transforms to impact-parameter space while propagating uncertainties from data, PDFs, algorithm, and method. A sympathetic reader would care because the approach yields x- and Q-dependent TMDs that broaden with increasing Q, offering a minimally biased route to access three-dimensional hadron structure in QCD.

Core claim

The extraction proceeds in two stages. Stage I regresses the cross section onto a structure kernel S(qT, x1, x2; QM) after factoring out known kinematic prefactors and charge-weighted PDF combinations, with experimental and PDF uncertainties propagated via Monte Carlo replicas. Stage II reconstructs the normalized integrand s(x, k; Q) from this kernel through an end-to-end differentiable k quadrature layer. Applied to E288 and E605 data, the procedure reproduces the measured qT spectra across Q values and produces TMDs whose width increases with Q, with uncertainty bands that consistently include experimental, PDF, algorithmic, and methodological contributions.

What carries the argument

The structure kernel S(qT, x1, x2; QM) learned by regression in stage I, from which the normalized integrand s(x, k; Q) is recovered via end-to-end differentiable k quadrature in stage II. This kernel and reconstruction step carry the argument by enabling direct TMD extraction while remaining entirely in momentum space.

If this is right

  • The extracted TMDs depend on both x and Q and broaden as Q increases.
  • The method reproduces the measured qT spectra across the Q range covered by the input data.
  • Uncertainty bands on the TMDs incorporate experimental, PDF, algorithmic, and methodological components in a unified way.
  • The same two-stage framework supplies a transferable template for polarized TMD extraction and other QCD inverse problems.

Where Pith is reading between the lines

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

  • The same momentum-space reconstruction could be applied to semi-inclusive deep inelastic scattering data for cross-checks of the extracted TMD evolution.
  • Confirmation of the observed Q-broadening in independent analyses would help constrain non-perturbative models of TMD scale dependence.
  • A data-driven method of this type might eventually reduce reliance on parametric assumptions in global TMD fits.

Load-bearing premise

The normalized integrand s(x, k; Q) can be reconstructed from the learned structure kernel via differentiable quadrature without the network architecture or training procedure introducing significant bias that distorts the TMD shape or its Q dependence.

What would settle it

Independent Drell-Yan measurements at comparable kinematics that the reproduced qT spectra fail to match, or extracted TMDs whose Q dependence contradicts direct width measurements from a separate experiment or facility.

Figures

Figures reproduced from arXiv: 2510.17243 by D. Keller, I. P. Fernando.

Figure 2
Figure 2. Figure 2: A. cross-sections For the initial controlled test we adopt a simple generator with no explicit x– or Q–dependence for simplicity, strue(k) = 1 π e −k 2 , (18) which implies the analytic auto–convolution Strue(qT ) = 1 2π exp − q 2 T 2  . (19) Pseudo–data are produced at E288–like kine￾matics (with the Υ veto) using Eq. (2) together with modern collinear inputs (e.g. NNPDF4.0). We then carry out the end-t… view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Cross-section closure with pseudo-data at [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Closure for [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Representative cross-section comparison after the joint [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Representative cross-section comparison after the joint [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Unpolarized TMDs reconstructed at [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Transverse-momentum–dependent parton distributions [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Transverse-momentum–dependent parton distributions [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Transverse-momentum–dependent parton [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The root-mean-square (RMS) transverse [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
read the original abstract

Building on the first-ever application of neural networks in TMD phenomenology: "Extraction of the Sivers function with deep neural networks", we now present a momentum space, physics-informed deep learning framework for the direct extraction of unpolarized transverse momentum dependent parton distributions (TMDs) from fixed target Drell-Yan data (E288, E605). Rather than transforming to impact-parameter space, we remain in k and embed a normalized integrand s(x, k; Q) whose auto-convolution produces the observed qT spectra. The extraction proceeds in two steps. Stage I learns the structure kernel S(qT , x1, x2; QM ) by regressing the cross-section with known kinematic prefactors and charge-weighted PDF combinations factored out; experimental and PDF uncertainties are propagated with Monte Carlo replicas. Stage II reconstructs s(x, k; Q) with an end-to-end differentiable k quadrature layer. Applied to Fermilab cross-section data from experiments E288 and E605, the method reproduces the measured qT spectra across Q and yields x and Q dependent TMDs that broaden with Q, with uncertainty bands that consistently propagate experimental, PDF, algorithmic and methodological components. The approach is minimally biased (no factorized Ansatze and no bT transform) and provides a transferable template for polarized TMDs and related QCD inverse problems.

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 presents a two-stage physics-informed deep neural network framework for direct extraction of unpolarized TMDs in momentum space from fixed-target Drell-Yan cross sections (E288, E605). Stage I regresses a structure kernel S(qT, x1, x2; QM) after factoring out kinematic prefactors and charge-weighted PDFs, propagating experimental and PDF uncertainties via Monte Carlo replicas. Stage II recovers the normalized integrand s(x, k; Q) via an end-to-end differentiable k-quadrature layer. The central claims are that the method reproduces the measured qT spectra across Q, yields x- and Q-dependent TMDs that broaden with Q, and propagates combined uncertainties while remaining minimally biased by avoiding factorized Ansätze and bT transforms.

Significance. If the extraction and Q-broadening claims hold after validation, the work supplies a data-driven template for TMD phenomenology that is transferable to polarized functions and other QCD inverse problems. Explicit propagation of experimental, PDF, algorithmic, and methodological uncertainties, together with the avoidance of traditional transforms, constitutes a concrete methodological advance over existing approaches.

major comments (2)
  1. [Stage II] Stage II (differentiable k-quadrature reconstruction of s(x, k; Q)): the headline result that TMDs broaden with Q rests on this layer recovering the k-dependence without architecture-induced bias. The manuscript provides no independent validation—such as recovery tests on synthetic data with known analytic TMDs, variation of quadrature node count, or comparison of high-k tails against traditional bT methods—to demonstrate that the chosen activations and training do not systematically distort the extracted Q dependence or uncertainty bands.
  2. [Abstract and §4] Abstract and §4 (reproduction of qT spectra): while the abstract states that spectra are reproduced across Q, no quantitative metrics (e.g., χ² per degree of freedom, point-by-point residuals, or baseline comparison to existing TMD fits) are reported for the E288/E605 data sets. This omission is load-bearing for the claim that the two-stage procedure successfully inverts the data.
minor comments (2)
  1. [Methods] Notation for the normalized integrand s(x, k; Q) and structure kernel S should be introduced with an explicit equation in the methods section to avoid ambiguity when the quadrature layer is described.
  2. [Results] Figure captions for the extracted TMDs should state the precise kinematic ranges in x and Q over which the Q-broadening is observed, together with the number of Monte Carlo replicas used for the uncertainty bands.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and will revise the manuscript accordingly to incorporate additional validation and quantitative metrics.

read point-by-point responses

  1. Referee: [Stage II] Stage II (differentiable k-quadrature reconstruction of s(x, k; Q)): the headline result that TMDs broaden with Q rests on this layer recovering the k-dependence without architecture-induced bias. The manuscript provides no independent validation—such as recovery tests on synthetic data with known analytic TMDs, variation of quadrature node count, or comparison of high-k tails against traditional bT methods—to demonstrate that the chosen activations and training do not systematically distort the extracted Q dependence or uncertainty bands.

    Authors: We agree that independent validation of the Stage II layer is important to confirm that the observed Q-broadening is not an artifact of the network architecture or quadrature implementation. The current manuscript applies the method directly to experimental data without these synthetic tests. In the revised version we will add a new subsection with recovery tests on synthetic data generated from known analytic TMD forms, results for varied quadrature node counts, and comparisons of high-k tails to traditional bT-space extractions. These additions will quantify any potential bias in the extracted Q dependence and uncertainty bands. revision: yes

  2. Referee: [Abstract and §4] Abstract and §4 (reproduction of qT spectra): while the abstract states that spectra are reproduced across Q, no quantitative metrics (e.g., χ² per degree of freedom, point-by-point residuals, or baseline comparison to existing TMD fits) are reported for the E288/E605 data sets. This omission is load-bearing for the claim that the two-stage procedure successfully inverts the data.

    Authors: We acknowledge that quantitative metrics would strengthen the claim of successful reproduction of the qT spectra. The present manuscript reports qualitative agreement but does not include χ²/dof, residuals, or direct comparisons to other TMD fits. In the revised manuscript we will add these metrics in §4, including χ² per degree of freedom for each dataset, point-by-point residual plots, and comparisons to existing unpolarized TMD extractions from the literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity in neural network TMD extraction from Drell-Yan data

full rationale

The paper describes a two-stage data-driven procedure: Stage I regresses a structure kernel S directly onto measured cross sections with kinematic factors and PDFs factored out, while Stage II applies an end-to-end differentiable quadrature to recover the normalized integrand s(x,k;Q) and thereby the TMDs. Reproduction of the input qT spectra is the expected outcome of a successful fit rather than an independent prediction, and the extracted x- and Q-dependent TMDs (including their broadening) are the direct parametric output of the trained model. No self-definitional loop, fitted quantity renamed as prediction, or load-bearing self-citation that reduces the central claim to its own inputs appears in the abstract or described chain. The method is presented as a phenomenological extraction template whose validity is judged by fit quality and uncertainty propagation against external data, making the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard TMD factorization and the ability of a neural network to parameterize the structure kernel without introducing uncontrolled bias. No new particles or forces are postulated.

free parameters (1)
  • Neural network weights and hyperparameters
    The deep network parameters are optimized to fit the experimental cross-section data during training.
axioms (1)
  • domain assumption The observed qT spectrum arises from the auto-convolution of the normalized integrand s(x, k; Q) with known kinematic prefactors and charge-weighted PDFs.
    Standard assumption in TMD phenomenology for Drell-Yan processes.

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Forward citations

Cited by 3 Pith papers

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