Angular structure of Drell-Yan reaction in the TMD factorization approach

Alexey Vladimirov

REVIEW 3 major objections 4 minor 1 cited by

Angular structure of Drell-Yan reaction in the TMD factorization approach

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read TMD factorization with kinematic power corrections reproduces the Drell-Yan angular coefficients and exposes a first LHC-based signal of the Boer-Mulders function.

desk verdict A useful summary of the TMD-with-KPC program for Drell-Yan angular distributions; the A0 and Lam-Tung predictions are genuine, but the first-LHC Boer-Mulders claim is a same-data fit without significance. read the letter →

arxiv 2505.11064 v1 pith:DEWWIR7P submitted 2025-05-16 hep-ph

Alexey Vladimirov This is my paper · ORCID

classification hep-ph PACS 12.38.-t13.85.Qk

keywords Drell-YanTMDfactorizationkinematicpowercorrectionsangularcoefficientsLam-TungrelationBoer-Muldersfunctiontwist-three

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

The paper claims that an extended TMD factorization, in which kinematic power corrections to the leading-power cross-section are resummed to all orders, reproduces the measured angular distributions of Drell-Yan lepton pairs at hadron colliders. These distributions (the coefficients $A_0$, $A_1$, $A_2$ and the Lam-Tung combination $A_0-A_2$) are power suppressed in standard leading-power TMD factorization, yet they carry distinct physical information. The agreement gives the first LHC-based indication that the Boer-Mulders function is nonzero, seen through the low-transverse-momentum behavior of $A_2$. It also turns the Lam-Tung relation into a test of both power corrections and twist-three spin correlations.

What carries the argument

The central object is the kinematic power correction (KPC), a series of $1/Q$-suppressed terms that restore frame and gauge invariance order by order and share the same nonperturbative content as the leading-power term. In ref. 3 this series is summed to all powers, yielding the TMD-with-KPC factorization theorem used here. The machinery then expresses each angular coefficient through twist-two TMD distributions, $f_1$ and the Boer-Mulders function $h_1^\perp$, and, for $A_1$ and $A_3$, through a $q_T/Q$ correction isolated from a twist-three singularity by the leading-TMD approximation.

What would settle it

A high-precision measurement of $A_2$ and $A_0-A_2$ in the region $q_T<20$ GeV with uncertainties below the current few-percent level would settle the claim: if $A_2$ is consistent with zero at low $q_T$, or if $A_0-A_2$ vanishes despite nonzero KPC predictions, the framework's resummation and the Boer-Mulders interpretation would be ruled out.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the TMD-with-KPC factorization theorem gives a complete, frame- and gauge-invariant description of power-suppressed Drell-Yan angular structure in the region $q_T \ll Q$. Expressed through the unpolarized TMD distribution $f_1$ and the Boer-Mulders function $h_1^\perp$, the coefficients $A_0$, $A_2$, and the parity-violating coefficients follow from kinematic power corrections alone, while $A_1$ requires an additional $q_T/Q$ correction obtained from the leading-TMD approximation of twist-three distributions. The paper reports agreement with hadron-collider measurements: $A_0$ is reproduced including its 4-6% signal, the Lam-Tung combination $A_0-A_2$ agrees up to $q_T\sim 30$ GeV, and $A_2$ shows a low-$q_T$ region dominated by the double-Boer-Mulders contribution, which the paper identifies as the first LHC-based evidence of a nonzero Boer-Mulders function.

Load-bearing premise

The whole analysis assumes that the resummed kinematic-power-correction series is the complete, correct restoration of frame and gauge invariance in TMD factorization, and that the twist-three singularity producing the $q_T/Q$ term can be parametrized entirely through twist-two TMD distributions in the leading-TMD approximation.

Editorial extensions

If this is right

Angular coefficients that were previously considered inaccessible in TMD factorization become computable and data-compatible observables.
The low-$q_T$ part of $A_2$ becomes a direct handle on the Boer-Mulders function, enabling its first LHC-driven determination.
The Lam-Tung relation $A_0-A_2$ is reinterpreted as a combined test of kinematic power corrections and double-Boer-Mulders effects rather than a null test.
Existing TMDPDF extractions remain usable but will need updating once KPCs are included, with deviations of about 1% at the $Z$ pole and 5-10% at $Q\sim 10$ GeV.
The leading-TMD treatment of the $q_T/Q$ correction extends the practical range of TMD predictions in $q_T$ for coefficients such as $A_1$.

Reading between the lines

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

If these results hold, the distinction between leading-twist and higher-twist observables becomes less sharp: many power-suppressed cross-section components are actually determined by leading-twist TMD content plus universal KPCs, so they can be used to cross-check TMD fits instead of being discarded.
A dedicated high-statistics measurement of $A_2$ at low $q_T$ in a forward detector, combined with independent SIDIS constraints on the Boer-Mulders function, could confirm the sign and size of the effect without relying on the 8 TeV datasets used here.
The same KPC summation should apply to other processes with a measured angular structure, such as $Z$+jet, $W$ production, or SIDIS, where analogous coefficients are currently treated as higher-twist backgrounds; testing one such process would show whether the KPC series is universal.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

This is a short summary of Vladimirov's own TMD-with-KPC program, applied to Drell-Yan angular coefficients. The genuinely useful content is the demonstration that A0 and the Lam-Tung relation A0-A2, both power-suppressed in standard TMD factorization, are described by the kinematic power corrections resummed to all orders. Those comparisons with ATLAS, CMS, and LHCb data are not fitted; they are predictions of the formalism, and they look reasonably good. That is a real step forward, and it deserves credit.

The soft spot is the headline novelty claim. The paper says the A2 data provide the first LHC-based indication of a nonzero Boer-Mulders function. But as the text makes clear, A2 is described by a sum of a double Boer-Mulders term and an unpolarized f1 f1/Q^2 term, and the BM function is extracted from the same A2 data whose agreement is then quoted. There is no BM=0 null hypothesis test, no significance, no covariance or parameter count. Any flexible nonperturbative shape can absorb a missing background. So the claim is an indication at best, not a measurement. The Lam-Tung first two points likewise depend on the same BM fit, so they are consistency checks, not independent predictions.

The paper also gives no derivations, referring instead to refs 3 and 5; the qT/Q correction used for A1 is from ref 6. For a summary, that is acceptable, but it means this manuscript stands or falls on the underlying papers. The large-rapidity A0 discrepancy between ATLAS and LHCb is noted but left unresolved, which is fine for a report.

On balance, the underlying framework and the predicted angular coefficients are serious physics and deserve peer review. This particular manuscript, however, is a proceedings-style summary with an over-claimed novelty. I would send it to a serious referee if submitted as a letter, with a strong request to either add the fit details or soften the Boer-Mulders claim. For a full journal article, I would prefer the detailed analysis from ref 5 plus the new extraction.

I would cite the detailed papers rather than this summary. But it is worth a reading-group discussion on power corrections in TMD factorization.

Referee Report

3 major / 4 minor

Summary. The manuscript reports a phenomenological study of the Drell-Yan angular coefficients A0, A1, A2 and the Lam-Tung combination A0-A2 within the TMD-with-KPC factorization framework, using the ART23 unpolarized TMDPDFs and a parametrized Boer-Mulders function. It claims good agreement with ATLAS, CMS, and LHCb data, identifies A2 as dominated by the double-Boer-Mulders contribution at low qT, and interprets this as the first LHC-based indication of a nonzero Boer-Mulders function. The paper is a short summary that delegates derivations and fit details to refs. [3,5,6].

Significance. If the underlying framework is correct, the paper points to a genuinely novel window: kinematic power corrections in TMD factorization generate nonzero angular coefficients that are zero at leading power, and the Lam-Tung relation violation observed at the LHC could be explained within TMD factorization. The comparison to three independent experiments (ATLAS, CMS, LHCb) is a clear strength, and the framework makes concrete, falsifiable predictions. The paper also honestly identifies the A1 coefficient as requiring an additional qT/Q correction. However, the central novelty—the claim of a nonzero Boer-Mulders function at the LHC—is not quantitatively supported in this manuscript: no fit procedure, no parameterization, no significance, and no null test are given. The Lam-Tung low-qT points and the A2 data used for the extraction are not independent, so the 'indication' is a consistency check rather than an independent verification. The paper also claims to be the first twist-three effect at the LHC, a statement that goes beyond the evidence presented.

major comments (3)

[Main text, paragraph after Eq. (1) beginning 'Arguably the most intriguing distribution is A2'] The claim that the Boer-Mulders function is found to be non-zero at the LHC is not supported by any statistical evidence in this manuscript. No parameterization of h1^perp, no number of fitted parameters, no chi-square or likelihood comparison, and no uncertainty quantification of the extraction are presented. In particular, no test of the null hypothesis h1^perp = 0 is shown. Because the A2 data are the same data used both to constrain the BM term and to demonstrate agreement, the 'indication' is currently a model-dependent consistency statement rather than a falsifiable extraction. Please provide a quantitative significance (e.g., Delta chi^2 against BM=0, or a confidence interval excluding zero) or explicitly label the claim as qualitative.
[Main text, paragraph beginning 'The finest test of KPC'] The agreement of the Lam-Tung relation A0-A2 at low qT is presented as a key success, but the first two points are attributed to the double-Boer-Mulders effect using the BM function extracted from the same A2 data. This is not an independent prediction; it is a consistency check of the fit. To avoid circularity, the authors should either (i) show that the BM function extracted from A2 alone predicts the observed A0-A2 without refitting, or (ii) present a BM extraction from a different observable (e.g., SIDIS or other Drell-Yan data) and test A0-A2 with it. The current text obscures this distinction, and the phrase 'almost perfect agreement' overstates the significance of a check that shares its fitting input with the data being described.
[Main text, final paragraph before Conclusion] The description of A1 relies on a qT/Q correction computed in ref. [6] under the 'leading-TMD approximation' in which a twist-three singularity is isolated and parametrized by twist-two TMD distributions. The manuscript does not specify the functional form, the range of validity, or the uncertainty of this approximation. Since A1 is the only coefficient that requires this correction and the paper claims overall agreement with data, the result depends on an assumption whose verification is not shown here. Please state explicitly what is assumed in the parametrization and quantify the sensitivity of the A1 prediction to variations in the twist-three parametrization, or indicate where in ref. [6] such an analysis can be found.

minor comments (4)

[Main text, paragraph beginning 'The distribution A0 is zero'] The abstract's claim of 'good agreement with experimental data' is too strong for the large-rapidity A0 region, where the authors themselves note a contradiction between ATLAS and LHCb and state that the prediction lies in between. Please qualify the abstract or discuss this discrepancy in the main text.
[Figures 1-3] The figures are reproduced from ref. [5], but no caption or text describes the origin of the uncertainty bands (experimental uncertainty, theory scale variation, or parameter uncertainty). Please add a sentence to each caption or in the text identifying the sources of the bands and the parameter choices used for the central curves.
[Title page] The title on the first page contains a typo: 'Drell-Y an reaction' should be 'Drell-Yan reaction'.
[Main text, paragraph after Eq. (1) beginning 'Arguably the most intriguing distribution is A2'] The statement 'To best of our knowledge it is the first twist-three effect ... at LHC' is stronger than what is shown. The Boer-Mulders function is a twist-two TMD whose collinear limit is a twist-three distribution, and the direct observable is a power-suppressed cross-section ratio; please justify the claim with a literature comparison or soften it.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline Boer-Mulders claim is a fit to the same A2 data it is used to describe, so the 'first LHC indication' is not an independent prediction; the Lam-Tung double-BM points inherit the same fitted input.

fitted input called prediction [A2 discussion, p. 3 (between Fig. 2 and Fig. 3 captions)]
"At LP it is described by the product of Boer-Mulders functions, while the unpolarized contribution is suppressed, i.e. schematically it has a form A2∼ h⊥1h⊥1/M2 +f1f1/Q2. Both these terms are very clearly pronounced in the experimental measurement, with h⊥1h⊥1 being dominating qT < 5GeV part, and f1f1 for the remaining part. It allows the first estimation of Boer-Mulders function using the LHC data. Despite large uncertainties, the Boer-Mulders function is found to be non-zero."

The paper estimates h⊥1 from the very same A2(qT) data that it then says are described by the h⊥1h⊥1/M2 term. The low-qT part of A2 is therefore not independently predicted: the fitted Boer-Mulders term is adjusted to those points, so agreement there is built in. Calling the resulting nonzero h⊥1 an 'indication' is a fit conclusion, and the text supplies no BM=0 null test, parameter count, or covariance to demonstrate that the nonzero is required rather than absorbed by the fit. The central novelty of the paper thus reduces to a refit of the data it is supposed to explain.
fitted input called prediction [Lam-Tung paragraph, p. 3]
"The first two points are the contribution of the double-Boer-Mulders effect, while the rest is the contribution of unpolarized part (which is formally 1/Q2 suppressed)."

The first two points of A0−A2 are dominated by the same A2 Boer-Mulders term already fitted to A2 data in the preceding paragraph. Since h⊥1 was extracted from A2, the double-Boer-Mulders component of the Lam-Tung prediction is not an independent computation; it is the same fitted input re-expressed as another observable. Only the unpolarized 1/Q2 part of Lam-Tung is a genuinely independent KPC prediction, so presenting the full Lam-Tung agreement as a 'finest test of KPC' overstates the independent content. The BM portion is a consistency check, not a prediction.

full rationale

Most of the paper's derivation chain is independent of its own claims. The TMD-with-KPC framework is cited from ref. 3 as an explicit all-order computation, and the A0 and A1 comparisons are checked against external ATLAS, CMS, and LHCb data; those parts are not circular. The self-citations to refs. 3, 5, and 6 are not by themselves circular, because they point to concrete computations and the external data provide a benchmark. The significant circularity is confined to the Boer-Mulders claim. The text writes A2 as the sum of a double-Boer-Mulders term and an unpolarized term, says the BM term dominates A2 below 5 GeV, and then states that this allows the first estimation of the Boer-Mulders function from the same LHC data. The agreement of A2 with the model is therefore not a prediction of the BM function; it is a fit result, and without a null hypothesis of h⊥1=0 or a significance statement the nonzero finding is not independently supported. The Lam-Tung relation inherits this problem because its first two points are explicitly attributed to the same double-Boer-Mulders effect. The remaining unpolarized part of Lam-Tung and the A0/A1 comparisons retain independent predictive content, so the circularity is partial rather than total. Overall score 6: one or more 'predictions' reduce by construction, while the KPC framework itself is externally testable.

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

The central phenomenological predictions are inherited from the author's own previous papers (refs 3, 5, and 6) and depend on the ART23 TMDPDF fit from ref 1. The only genuinely new empirical element, the Boer-Mulders indication, is fitted to the same data used to claim agreement, which raises the circularity burden. No new particles or fields are introduced.

free parameters (2)

ART23 unpolarized TMDPDF parameters = As extracted in ref 1
The predictions for A0, A1, A2, and the Lam-Tung relation rely on these fitted TMD distributions. They are not refit here, but the central claim depends on their validity inside the KPC framework.
Boer-Mulders TMDPDF h1perp and its nonperturbative parameters = Not quoted; found nonzero with large uncertainties
The A2 expression includes a h1perp h1perp / M2 term, and the paper estimates h1perp from the same A2 data it compares with. The agreement for that term is therefore a fit rather than an independent prediction.

assumptions (4)

domain assumption The TMD-with-KPC factorization theorem from ref 3 correctly resums the kinematic power corrections to the leading-power term at all orders.
All angular-coefficient formulas used in the paper come from this theorem; no derivation is reproduced in this manuscript.
domain assumption The qT/Q correction can be isolated from the twist-three TMD singularity and parametrized through twist-two TMD distributions, the leading-TMD approximation of ref 6.
This separation is essential for the description of A1 and A3; the paper states it but does not prove it.
domain assumption Unpolarized TMDPDFs extracted at leading power (ART23, ref 1) remain valid inside the KPC-extended factorization.
The paper assumes the LP extraction can be reused and estimates deviations at the 1-10% level, without a dedicated validation fit.
domain assumption TMD factorization is applicable for qT << Q in the Drell-Yan process.
This is the standard kinematic limitation accepted by the paper, although some comparisons extend to qT ~ 30 GeV, beyond the nominal range.

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Cite this review

Pith. "Pith review of Angular structure of Drell-Yan reaction in the TMD factorization approach." pith.science (2026). https://pith.science/paper/DEWWIR7P

@misc{pith2026250511064,
  author       = {Pith},
  title        = {Pith review of: Angular structure of Drell-Yan reaction in the TMD factorization approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DEWWIR7P}},
  note         = {Machine review of arXiv:2505.11064}
}

read the original abstract

We study angular distributions in the Drell-Yan process using an extended transverse momentum dependent (TMD) factorization framework that includes kinematic power corrections. This approach allows the description of observables previously considered power-suppressed. The results show good agreement with experimental data and provide the first LHC-based indication of the Boer-Mulders function, highlighting the value of power corrections in TMD phenomenology.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Theoretical Summary: Moriond QCD and High-Energy Interactions 2025
hep-ph 2025-06 unverdicted

A conference summary that compiles the main theory results presented at Moriond QCD 2025, spanning hard scattering, precision QCD, flavour, strong coupling, lattice, heavy-ion, and BSM physics.

Reference graph

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13 extracted references · 9 canonical work pages · cited by 1 Pith paper

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Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [6]

Leading qT/Q correction for Drell-Yan in TMD factorization

Arturo Arroyo-Castro, Ignazio Scimemi, and Alexey Vladimirov. Leading qT/Q correction for Drell-Yan in TMD factorization . 3 2025

work page 2025

[2] [1]

Extraction of unpolarized transverse momentum distributions from the fit of Drell-Yan data at N ^ 4 LL

Valentin Moos, Ignazio Scimemi, Alexey Vladimirov, and Pia Zurita. Extraction of unpolarized transverse momentum distributions from the fit of Drell-Yan data at N ^ 4 LL . JHEP , 05:036, 2024

work page 2024

[3] [2]

Determination of unpolarized TMD distributions from the fit of Drell-Yan and SIDIS data at N ^4 LL

Valentin Moos, Ignazio Scimemi, Alexey Vladimirov, and Pia Zurita. Determination of unpolarized TMD distributions from the fit of Drell-Yan and SIDIS data at N ^4 LL . 3 2025

work page 2025

[4] [3]

Kinematic power corrections in TMD factorization theorem

Alexey Vladimirov. Kinematic power corrections in TMD factorization theorem . JHEP , 12:008, 2023

work page 2023

[5] [4]

Measurement of the angular coefficients in Z -boson events using electron and muon pairs from data taken at s =8 TeV with the ATLAS detector

Georges Aad et al. Measurement of the angular coefficients in Z -boson events using electron and muon pairs from data taken at s =8 TeV with the ATLAS detector . JHEP , 08:159, 2016

work page 2016

[6] [5]

Angular distributions of Drell-Yan leptons in the TMD factorization approach

Sara Piloneta and Alexey Vladimirov. Angular distributions of Drell-Yan leptons in the TMD factorization approach . JHEP , 12:059, 2024

work page 2024

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Ian Balitsky. Drell-Yan angular lepton distributions at small x from TMD factorization. JHEP , 09:022, 2021

work page 2021

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Aaij et al

R. Aaij et al. First Measurement of the Z + - Angular Coefficients in the Forward Region of pp Collisions at s=13\,\,TeV . Phys. Rev. Lett. , 129(9):091801, 2022

work page 2022