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arxiv: 2604.05154 · v1 · submitted 2026-04-06 · ✦ hep-ph

Probing Gluon TMD Models with Drell--Yan Structure Functions

Jan Ferdyan This is my paper

Pith reviewed 2026-05-10 18:54 UTC · model grok-4.3

classification ✦ hep-ph
keywords gluon TMDDrell-Yan structure functionshigh-energy factorizationWeizsacker-Williams modelATLAS datatransverse momentum distributionsparity violating structure functions
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The pith

A modified Weizsäcker-Williams gluon TMD model provides the closest match to ATLAS Drell-Yan structure functions at 8 TeV.

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

The paper computes both parity-conserving and parity-violating structure functions for the Drell-Yan process in proton-proton collisions at 8 TeV center-of-mass energy. It applies the high-energy factorization approach with tree-level matrix elements from two partonic channels and tests four classes of gluon transverse-momentum-dependent models, including Gaussian, Weizsäcker-Williams, Kimber-Martin-Ryskin, and Jung-Hautmann forms together with their phenomenological adjustments. Direct comparison to 2016 ATLAS measurements shows clear differences in fit quality across the full dataset and for individual structure functions. The lowest chi-squared per degree of freedom occurs for one of the adjusted Weizsäcker-Williams variants. This comparison isolates which model ingredients improve agreement and therefore deserve attention in future parametrizations.

Core claim

Tree-level high-energy factorization applied to the channels q_val g* to q V* and g* g* to q qbar V* yields structure functions whose agreement with ATLAS data at sqrt(S) = 8 TeV is best for a phenomenologically modified Weizsäcker-Williams gluon TMD model, as measured by chi-squared per degree of freedom, while the unadjusted versions of Gaussian, KMR, and JH models perform noticeably worse.

What carries the argument

High-energy factorization formalism that convolves tree-level hard-scattering matrix elements with different gluon TMD distributions to generate Drell-Yan structure functions for comparison with data.

Load-bearing premise

The high-energy factorization formula with only tree-level matrix elements for the two specified channels reproduces the Drell-Yan process at 8 TeV without higher-order corrections or other contributions.

What would settle it

A next-to-leading-order calculation of the same structure functions that reverses the ranking of the TMD models or produces a substantially lower chi-squared for a different model would undermine the preference for the modified Weizsäcker-Williams form.

Figures

Figures reproduced from arXiv: 2604.05154 by Jan Ferdyan.

Figure 1
Figure 1. Figure 1: General form of the diagrams for the DY process. Big circles represent all possible QCD LO subdiagrams. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Drell–Yan total cross section for γ ∗ exchange in pp collision at the LHC at √ S = 7 TeV. We compared the ATLAS data [9] to the cross sections calculated with different gluon TMD models: (a) the KMR-based TMD models, (b) the WW-based TMD models, (c) the CCFM-driven TMD models, (d) KMR(3) , WW(3), JH2013, and Gaussian models as the models with the best overall data description (including the structure f… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the structure functions: (a) A0, (b) A2, (c) ALT , (d) A1, (e) A3, (f) A4 calculated with different modifications of the KMR TMD model with the ATLAS data [10]. The bands around the curves represent the estimated Monte Carlo integration uncertainties. Here, the WW-based models are closest to the data at large qT , and there is no significant difference between them for this observable. All th… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the structure functions: (a) A0, (b) A2, (c) ALT , (d) A1, (e) A3, (f) A4 calculated with different modifications of the WW TMD model with the ATLAS data [10]. The bands around the curves represent the estimated Monte Carlo integration uncertainties. originates from a large discrepancy in A2 at high qT values. In the case of function A1, all models show a clear decline at very large qT , cont… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the structure functions: (a) A0, (b) A2, (c) ALT , (d) A1, (e) A3, (f) A4 calculated with the JH TMD models with the ATLAS data [10]. The bands around the curves represent the estimated Monte Carlo integration uncertainties. more gentle and monotonic increase in this region. The x rescaling has a big impact on this observable. We also notice a large spread of predictions for the A4 function. … view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the structure functions: (a) A0, (b) A2, (c) ALT , (d) A1, (e) A3, (f) A4 calculated with different TMD models with ATLAS data. For this purpose, we used KMR(3), WW(3), the JH2013 model, and the Gaussian model as representatives of each TMD class that provides the best description of the data within each class. The bands around the curves represent the estimated Monte Carlo integration uncert… view at source ↗
Figure 7
Figure 7. Figure 7: The ratios of the qvalg ∗ channel contributions to the total cross section Rqvalg∗ . the sensitivity to the x dependence in the kT factorization approach and its distinction from the collinear approach. An important note is that the ATLAS data used for comparison with the predictions are much denser in the central part of the qT spectrum than at its edges. Additionally, the high qT region is subject to gre… view at source ↗
Figure 8
Figure 8. Figure 8: The ratio of electroweak couplings c Z f for leptons and quarks. The dashed vertical line corresponds to the measured value of sin2 θW . References [1] S. D. Drell and Tung-Mow Yan. Massive Lepton Pair Production in Hadron-Hadron Collisions at High-Energies. Phys. Rev. Lett., 25:316–320, 1970. [Erratum: Phys.Rev.Lett. 25, 902 (1970)]. [2] John C. Collins and Davison E. Soper. Angular Distribution of Dilept… view at source ↗
read the original abstract

We compute structure functions for the Drell--Yan process in proton-proton collisions at the center of mass energy $\sqrt{S} = 8 \mathrm{TeV}$ both parity conserving and parity breaking. For this calculation, we use the high-energy factorization formalism. The hard scattering matrix elements used in our derivation consist of two channels -- $q_\mathrm{val} g^* \to q V^*$ and $g^* g^* \to q \overline{q} V^*$, both at the tree level. We consider four types of gluon TMD models: Gaussian, Weizs\"{a}cker--Williams (WW), Kimber--Martin--Ryskin (KMR), and Jung--Hautmann (JH). We also consider the models with phenomenological adjustments to improve the data description. We derive and compare the structure functions calculated for different gluon TMD models with the ATLAS 2016 data. Based on this comparison, we calculate $\chi^2$ per number of degrees of freedom for each of the predictions. This assessment shows clear differences between the predictions obtained with different TMD models, both in the description of the full data set and in the case of individual structure functions. The best description of the structure functions data is obtained with one of the modified WW models. Our analysis can serve to identify the features of the TMD model that should be considered in future gluon TMD fits.

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

3 major / 3 minor

Summary. The paper computes parity-conserving and parity-violating Drell-Yan structure functions in pp collisions at √S=8 TeV within high-energy factorization, employing tree-level matrix elements for the channels q_val g* → q V* and g* g* → q qbar V*. Predictions from Gaussian, WW, KMR, and JH gluon TMD models (plus phenomenological modifications) are compared to ATLAS 2016 data via χ² per degree of freedom, with the conclusion that a modified WW model yields the best overall description and that the exercise identifies useful features for future TMD fits.

Significance. If the tree-level HEF approximation is reliable, the quantitative χ² ranking of TMD models against existing data provides a practical diagnostic for model features (e.g., the form of the WW modification) that should be retained in global fits. The explicit comparison of multiple models on the same dataset and the separation into individual structure functions are strengths that go beyond qualitative statements.

major comments (3)
  1. [§2] §2 (High-energy factorization formalism): The central claim that differences in χ² reflect genuine TMD-model features rests on the assumption that the tree-level matrix elements for only two channels accurately capture the process at √S=8 TeV. No estimate is given for the size of omitted NLO hard corrections, additional partonic channels, or matching to collinear factorization; any of these can shift the predicted structure functions in a model-dependent manner and thereby alter the χ² ranking.
  2. [§3.2 and §4.3] §3.2 (Phenomenological adjustments to TMD models) and §4.3 (χ² results): The modified WW model is reported to give the lowest χ², but the adjustments appear to be introduced to improve data description. Without an a-priori justification or a demonstration that the same modification improves other observables, the superiority may be post-hoc rather than an independent test of the underlying TMD ansatz.
  3. [Table 1] Table 1 (or equivalent χ² table): The reported χ²/dof values for the full dataset and for individual structure functions show clear numerical differences, yet no uncertainty is attached to these values arising from the neglected higher-order terms. This makes it impossible to judge whether the preference for the modified WW model is statistically robust.
minor comments (3)
  1. [Introduction] The definition of the structure functions (F1, F2, etc.) and their relation to the TMDs should be stated explicitly in the introduction or §2 rather than assumed from prior literature.
  2. [Figure captions] Figure captions should specify the exact kinematic cuts and binning used for the ATLAS data comparison to allow direct reproduction.
  3. [§4] A reference to the original ATLAS 2016 Drell-Yan measurement paper is missing in the data-comparison section.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the scope and limitations of our analysis. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [§2] §2 (High-energy factorization formalism): The central claim that differences in χ² reflect genuine TMD-model features rests on the assumption that the tree-level matrix elements for only two channels accurately capture the process at √S=8 TeV. No estimate is given for the size of omitted NLO hard corrections, additional partonic channels, or matching to collinear factorization; any of these can shift the predicted structure functions in a model-dependent manner and thereby alter the χ² ranking.

    Authors: We agree that the tree-level HEF approximation with the two specified channels has inherent limitations at √S=8 TeV, and that NLO corrections or additional channels could in principle introduce model-dependent shifts. However, because the same hard-scattering matrix elements are used for every TMD model, the relative differences in predicted structure functions—and thus in χ²—arise primarily from the TMD parametrizations themselves. The exercise therefore remains a useful diagnostic for identifying which TMD features best describe the data within this consistent framework. We will revise §2 to include an explicit discussion of these limitations, citing relevant literature on the expected size of higher-order effects in high-energy factorization for Drell-Yan processes, and to state that while absolute χ² values may change, the comparative ranking provides guidance for future TMD model development. revision: partial

  2. Referee: [§3.2 and §4.3] §3.2 (Phenomenological adjustments to TMD models) and §4.3 (χ² results): The modified WW model is reported to give the lowest χ², but the adjustments appear to be introduced to improve data description. Without an a-priori justification or a demonstration that the same modification improves other observables, the superiority may be post-hoc rather than an independent test of the underlying TMD ansatz.

    Authors: The modifications applied to the WW model follow forms previously employed in the TMD literature to incorporate small-x saturation and non-perturbative effects. We will strengthen §3.2 by adding explicit references to these earlier works and by articulating the physical motivation for each adjustment prior to the numerical comparison. While the present study does not test the same modifications on independent observables, its purpose—as stated in the abstract—is to isolate promising TMD features that can be retained or refined in future global fits. This comparative approach therefore serves as an independent diagnostic rather than a purely post-hoc optimization. revision: partial

  3. Referee: [Table 1] Table 1 (or equivalent χ² table): The reported χ²/dof values for the full dataset and for individual structure functions show clear numerical differences, yet no uncertainty is attached to these values arising from the neglected higher-order terms. This makes it impossible to judge whether the preference for the modified WW model is statistically robust.

    Authors: We acknowledge that quantitative uncertainties on the χ²/dof values arising from omitted higher-order contributions are not provided. Because a complete NLO calculation lies outside the scope of this work, we cannot attach numerical error estimates. We will add a clarifying paragraph adjacent to the χ² table (and in §4.3) that states this limitation explicitly and offers a qualitative assessment of how such corrections might affect the observed differences, noting that the modified WW model yields the lowest χ² consistently across both the full dataset and the individual structure functions. revision: partial

standing simulated objections not resolved
  • A quantitative estimate of the size of NLO hard corrections, additional partonic channels, and their potential model-dependent impact on the χ² ranking, which would require a separate, more extensive calculation beyond the present tree-level analysis.

Circularity Check

1 steps flagged

Phenomenological adjustments tuned to data then used to declare 'best' model via chi2

specific steps
  1. fitted input called prediction [Abstract]
    "We also consider the models with phenomenological adjustments to improve the data description. ... The best description of the structure functions data is obtained with one of the modified WW models."

    Adjustments are explicitly introduced to fit the ATLAS data better; the subsequent chi2 ranking that crowns the modified WW model as superior is then a direct consequence of that fitting step rather than a prediction tested against held-out or independent observables.

full rationale

The paper computes Drell-Yan structure functions in tree-level high-energy factorization for several gluon TMD models and their variants. The central result—that a modified WW model gives the best description—rests on first introducing 'phenomenological adjustments to improve the data description' and then ranking the models by chi2 to the identical ATLAS dataset. This selection step is forced by the tuning procedure itself rather than constituting an independent test of TMD features. No self-citation chains, definitional loops, or renamed known results appear in the derivation; the circularity is localized to the model-adjustment and ranking step.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The calculation rests on the high-energy factorization formalism and tree-level hard matrix elements; TMD models contain multiple free parameters (widths, normalizations, and phenomenological adjustments) that are fitted or chosen to match data.

free parameters (1)
  • TMD model parameters (e.g., Gaussian width, WW scale, KMR/JH parameters)
    Each of the four TMD models and their modified versions introduces parameters that are adjusted to improve agreement with data.
axioms (1)
  • domain assumption High-energy factorization applies to Drell-Yan structure functions at sqrt(S)=8 TeV with the specified tree-level channels
    Invoked to compute the structure functions from the gluon TMDs and hard scattering.

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