Pith. sign in

REVIEW 3 major objections 7 minor 123 references

Lattice QCD gives the first nonperturbative constraints on the gluon Collins-Soper kernel for transverse momenta from 300 MeV to 1.3 GeV.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-31 11:09 UTC pith:4XHXAADR

load-bearing objection First lattice numbers on the gluon CS kernel; real advance, but the Casimir-scaling 1–2σ claim is statistical only and the small-b_T points are uncontrolled. the 3 major comments →

arxiv 2607.24587 v1 pith:4XHXAADR submitted 2026-07-27 hep-lat hep-phnucl-th

First constraints on the nonperturbative gluon Collins-Soper kernel

Artur Avkhadiev , Yang Fu , Phiala E. Shanahan , Michael L. Wagman , Yong Zhao This is my paper
classification hep-lat hep-phnucl-th
keywords gluon Collins-Soper kernellattice QCDtransverse-momentum-dependent distributionsLarge-Momentum Effective Theoryquasi-TMD beam functionsCasimir scalingnonperturbative QCD
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The gluon Collins-Soper kernel governs how gluon transverse-momentum distributions change with rapidity. Until now it was known only from perturbation theory, where it is assumed to follow Casimir scaling from the quark kernel. This work extracts the first lattice-QCD constraints on that kernel in the nonperturbative regime, using boosted-pion matrix elements of staple-shaped gluon operators, Large-Momentum Effective Theory matching at next-to-next-to-leading logarithmic accuracy, and a single ensemble near the physical pion mass. The results sit within one-to-two standard deviations of Casimir-scaled quark-kernel values and open a path to first-principles input for gluon TMD phenomenology at present and future colliders.

Core claim

The nonperturbative gluon Collins-Soper kernel γ_g(b_T, μ = 2 GeV) is constrained for the first time over q_T ∈ [300 MeV, 1.3 GeV] from lattice QCD at M_π = 172(3) MeV and a = 0.15 fm, with uNNLL LaMET matching of quasi-TMD beam functions; the constraints are compatible within 1–2σ with Casimir scaling of the continuum-extrapolated quark kernel.

What carries the argument

The Pz derivative of the logarithm of the renormalized quasi-TMD gluon beam function after subtraction of the uNNLL LaMET matching kernel H_g, which isolates the Collins-Soper kernel once power corrections in 1/(x Pz) are controlled.

Load-bearing premise

Discretization artifacts and small-b_T power corrections remain uncontrolled on the single lattice spacing a = 0.15 fm and are expected to be large below roughly 0.3 fm.

What would settle it

A multi-spacing continuum extrapolation of the same quasi-TMD matrix elements that moves the extracted γ_g outside the present 1–2σ band around Casimir-scaled quark results, or a statistically significant deviation once full convolutional b_T-dependent matching is included.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Phenomenological fits of unpolarized gluon TMDs can replace pure Casimir-scaling assumptions with first lattice anchors in the nonperturbative region.
  • Future higher-precision lattice determinations can test whether Casimir scaling of the CS kernels survives beyond perturbation theory.
  • The same framework supplies nonperturbative input for gluon-sensitive observables at the EIC, LHC quarkonium, and Higgs q_T spectra.
  • Coulomb-gauge or flow-based formulations are identified as practical routes to the precision already achieved for the quark kernel.

Where Pith is reading between the lines

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

  • If the mild tension with pure perturbation theory at the smallest b_T persists after continuum extrapolation, it would signal genuine nonperturbative breaking of Casimir scaling that global TMD analyses must absorb.
  • The computational cost scaling noted for adjoint Wilson lines implies that gluon CS-kernel campaigns will likely migrate to Coulomb-gauge quasi-TMDs sooner than the corresponding quark program did.
  • Joint experimental-plus-lattice fits already performed for the quark kernel can be repeated for gluons once a second lattice spacing appears.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This Letter reports the first lattice-QCD constraint on the nonperturbative gluon Collins-Soper (CS) kernel γ_g(b_T, μ=2 GeV), the rapidity-evolution anomalous dimension of gluon TMDs. The extraction uses quasi-TMD beam functions built from staple-shaped gluon field-strength operators with adjoint Wilson-line divergence subtraction, computed on a single MILC HISQ ensemble (a=0.15 fm, L≈4.8 fm, 1105 configurations) with Wilson-clover valence quarks at M_π=172(3) MeV, boosted pion states at P_z ≈ 1.0–2.1 GeV, and Large-Momentum Effective Theory matching at (b_T-unexpanded) NNLL accuracy. The kernel is extracted from the ln P_z slope of matched quasi-beam functions, with x-windows selected by a perturbative-coupling bound and a χ² criterion, yielding constraints for transverse scales q_T ∈ [300 MeV, 1.3 GeV] (b_T = 0.15–0.60 fm). The results are stated to be compatible within 1–2σ with Casimir scaling of the continuum-extrapolated quark CS kernel, while the paper itself notes that O(a) artifacts and small-b_T power corrections are uncontrolled and comparable in size to the gluon–Casimir difference.

Significance. If the results hold, this is a genuine first: no nonperturbative constraint on the gluon Collins-Soper kernel exists to date, and the quantity enters every global TMD analysis, the recent gluon-TMD extraction from Higgs data [28], and EIC physics. The manuscript ships several concrete strengths: (i) a fully documented extraction chain with internal cross-checks — Lanczos matrix elements validated against summed-ratio/statistical fits (SM B, Figs. 4–6), ℓ→∞ extrapolations checked against ℓ-dependent ansätze (SM D), DFT truncation checked against analytic-tail transforms and b_z^cut variations (SM E), and consistency across four operator definitions including differently trace-subtracted ones (SM I); (ii) external, non-circular benchmarks — the kernel is obtained from the P_z-dependence of lattice matrix elements after perturbative matching and compared against N⁴LL perturbation theory and the continuum-extrapolated quark result of Ref. [42], not fit to the data it aims to inform; (iii) an unusually candid accounting of which systematics are controlled and which are not; and (iv) a clear, falsifiable path to improvement (Coulomb-gauge and flow-based observables, NNLO and convolution-

major comments (3)
  1. [Intro; 'Statistical and systematic uncertainties'; Fig. 3] Intro (4th paragraph) and Summary vs. 'Statistical and systematic uncertainties' section: the headline statement that the results are 'compatible within 1–2σ with the Casimir scaling of the quark CS kernel' uses bands that are statistical (plus checked analysis variations) only. The body of the paper states that O(a) discretization artifacts and small-b_T power corrections 'cannot be fully quantified from the present results and are expected to be significant at the current precision level,' with magnitude 'comparable to the difference between the final results and ... the perturbatively-expected Casimir scaling of the quark kernel.' The uncertainty class that decides whether 1–2σ agreement is meaningful is therefore exactly the one absent from the bands. Moreover, the comparison is between a finite-a gluon result (a=0.15 fm, O(a)-unimproved valence action) and the continuum-extrapolated
  2. [Abstract; Fig. 3 caption] Abstract and main text: the quoted upper end of the range, q_T = 1.3 GeV, is Fourier-conjugate to b_T = 0.15 fm = 1a — precisely the point the Fig. 3 caption flags as uncontrolled ('expected to be significant for b_T ≲ 0.30 fm (b_T/a ≲ 2)'). Two of the four b_T points entering Fig. 3 lie in or at the boundary of the flagged region. As written, the abstract's range [300 MeV, 1.3 GeV] rests on the b_T/a=1 point without qualification. The authors should either restrict the quoted range or state explicitly in the abstract that the upper end corresponds to b_T/a=1 where discretization artifacts and power corrections are uncontrolled; at minimum the b_T=0.15 fm point in Fig. 3 should be visually distinguished from the controlled points.
  3. [Eq. (42); SM Sec. G, Eqs. (43)–(48); Figs. 13–14] Eq. (42) and SM Sec. G: the uNNLL matching kernel is constructed by taking the b_T-dependent correction δC^{NkLO}(b_T, μ, xP_z) derived for quark matching in Ref. [41] and rescaling it by C_A/C_F. This is a perturbative Casimir-scaling assumption built into the matching, while the paper's phenomenological motivation is precisely to test Casimir scaling nonperturbatively. The assumption is defensible at the relevant perturbative order, but it is load-bearing because the matching correction is as large as the signal: Fig. 14b,c show δγ shifts of roughly −1 to −3 in the retained x-range, of the same order as the extracted γ_g(b_T), and the uNNLL–NNLL difference exceeds statistical errors at small b_T (Figs. 2, 13). The text should state explicitly at what order the b_T-dependent piece is known to Casimir-scale, what residual uncertainty this construction leaves, and how the uNNLL–NNLL sprea
minor comments (7)
  1. [Eq. (18); SM Sec. F, Fig. 12b] The final x-interval selection uses an asymmetric, χ²-driven window (Eq. (18), δ_χ²=0.5). SM Sec. F reports that symmetric intervals shift the final constraints negatively by up to 1σ at small b_T. Since this shift is of the same size as other effects discussed, it would be cleaner to report it as an explicit analysis systematic rather than absorbing it into the window choice; a sentence of justification for the a posteriori asymmetry would also help.
  2. ['Extraction of the gluon CS kernel'] The final-value construction uses weights given by the ratio of a bootstrap fit p-value to the bootstrap variance at each x. This is unconventional; a sentence motivating this weight choice, and a check against a simple inverse-variance average, would strengthen it.
  3. ['Computational setup'; SM Sec. B] The mixed-action setup (HISQ sea, Wilson-clover valence with c_sw=1.0) yields continuum-dispersion deviations growing to 0.14(1) at n_z=8 (SM Sec. B, Fig. 5 discussion). The insensitivity check on N_{g/π} is reassuring, but a brief comment on expected mixed-action and O(a) effects in the matrix elements themselves would be useful context for Major Comment 1.
  4. [SM Secs. D–E; Eq. (15)] Finite-volume effects for nonlocal operators (Ref. [121]) are invoked at SM Sec. D/E to explain behavior at ℓ, |b_z| ∼ L, and the alternate operator in Eq. (52) is noted to have 'higher sensitivity to finite-volume effects at large ℓ.' A quantitative statement (e.g., M_π L, and the b_z^cut=1.35 fm truncation relative to L/2=2.4 fm) would clarify how close the analysis sits to finite-volume systematics.
  5. [SM Sec. E, Fig. 10] Fig. 10 is labeled 'LO Fourier-transformed quasi-TMD beam functions'; 'LO' here appears to refer to the tree-level Fourier implementation rather than matching order — please clarify to avoid confusion with the LO matching curves of Fig. 13.
  6. [Eq. (17); SM Sec. F] Eq. (17) uses α_s at NLO with α_s(2 GeV) from the 2009 world average (Ref. [123]); the value is reasonable, but a one-line sensitivity statement (the bound δ_p.c.=0.5 is itself a free analysis parameter) would be appropriate.
  7. [Abstract and throughout] The arXiv text shows missing spaces in several places ('scalesq_T', 'massM_π', 'dependenceB^ℓ', 'functionsdonotaccount'); presumably extraction artifacts, but please check the proofs.

Circularity Check

0 steps flagged

No significant circularity: gluon CS kernel is extracted from new lattice quasi-TMD matrix elements via LaMET, not forced by definition or by fitting the phenomenological targets it aims to inform.

full rationale

The load-bearing chain is: (i) compute bare gluon quasi-TMD beam-function matrix elements on a single MILC ensemble; (ii) subtract divergences, extrapolate staple length, Fourier-transform; (iii) remove perturbative LaMET matching H_g and fit the residual P_z slope to obtain γ_g(b_T, μ). Equation (5) and the fit form (16) define an extraction from Euclidean data, not a tautology: the lattice correlators are independent inputs, and the matching kernels are taken from fixed-order/resummed QCD (adapted from external Refs. [71,72] plus the authors’ prior multiplicative uNNLL prescription). Comparison to Casimir-scaled continuum quark results and to N4LL perturbation theory is an external benchmark, not a fitted constraint. Self-citations supply reusable methodology (Lanczos spectroscopy, uNNLL form, quark-kernel continuum band) but do not close a definitional loop that would make the reported γ_g values equal to their inputs by construction. Uncontrolled O(a) and small-b_T systematics affect correctness/robustness of the 1–2σ Casimir claim; they are not circularity. Score 0 with empty steps is the honest finding.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The claim rests on standard lattice QCD and LaMET machinery plus several analysis cuts and the uncontrolled continuum limit. No new physical entities are postulated. The largest external loads are the validity of uNNLL multiplicative matching for gluons, the neglect of O(a) artifacts on one spacing, and hand-chosen power-correction and χ² thresholds that define the x-windows entering the final average.

free parameters (7)
  • δ_p.c. = 0.5 = 0.5
    Threshold on α_s²(2x̄P_z) that sets the b_T-independent x∈[0.27,0.73] window used for the final kernel average.
  • δ_χ² = 0.5 = 0.5
    Goodness-of-fit cut relative to χ²_min that further restricts asymmetric x-intervals at each b_T.
  • b_z^cut = 1.35 fm (9a) = 1.35 fm
    Truncation point of the discrete Fourier transform; varied in [5a,10a] but fixed for finals.
  • ℓ_min = 1.20 fm (8a) = 1.20 fm
    Lower edge of the constant fit used for the ℓ o∞ extrapolation of staple matrix elements.
  • μ = 2 GeV (MS-bar) = 2 GeV
    Final renormalization/rapidity scale at which γ_g is quoted; matching logs are resummed from μ_0=2xP_z to this value.
  • gradient-flow times t/a² = 2.0 (gluons) and 1.0 (2pt) = 2.0 / 1.0
    Smearing parameters chosen to improve signal-to-noise; not varied systematically in the final error.
  • Gaussian momentum smearing K_z = 1.16 GeV, 32 iterations, ε=0.2 = K_z=1.16 GeV
    Interpolating-operator parameters fixed for all boosts.
axioms (6)
  • domain assumption Large-Momentum Effective Theory quasi-TMD factorization: γ_g equals d ln B~/d ln P_z minus the perturbative matching kernel H_g up to power corrections in Λ/(xP_z) and 1/(b_T x P_z) that vanish as P_z o∞.
    Eq. (5) and the Theoretical framework section; underpins the entire extraction.
  • domain assumption Multiplicative, P_z-independent renormalizability of the chosen trace-subtracted gluon quasi-TMD operator after Wilson-loop subtraction, so renormalization constants drop out of the P_z derivative.
    Stated after Eq. (5); relies on Refs. [73,74] and the operator definition (6).
  • domain assumption uNNLL (b_T-unexpanded NNLL) matching kernel constructed from NLO fixed-order coefficients plus NNLO resummation and a CA/CF-rescaled b_T-dependent correction adapted from the quark case is adequate for the quoted precision.
    Perturbative matching section and SM Sec. G; full convolutional b_T-dependent NNLO matching is unavailable.
  • ad hoc to paper O(a) discretization effects and finite-volume corrections are smaller than or comparable to the reported statistical errors for the b_T range emphasized, or at least do not reverse the 1–2σ Casimir-scaling compatibility.
    Single a=0.15 fm, L=3.2 fm ensemble; paper acknowledges artifacts are expected to be significant for b_T≲0.30 fm but still presents those points in Fig. 3.
  • domain assumption Boosted pion matrix elements with the mixed-action clover valence setup at M_π=172 MeV adequately represent the universal CS kernel (process- and hadron-independent).
    Universality of the CS kernel is standard TMD factorization; pion is used as the external state throughout.
  • domain assumption Nested bootstrap median estimators with ZCW-filtered Lanczos Ritz values yield unbiased matrix elements and energies for the noise levels encountered.
    SM Sec. B; method validated against ratio fits but is relatively new.

pith-pipeline@v1.2.0-grok45-kimik3 · 36308 in / 4036 out tokens · 68898 ms · 2026-07-31T11:09:27.972855+00:00 · methodology

0 comments
read the original abstract

The gluon Collins-Soper kernel, which encodes the rapidity evolution of transverse-momentum-dependent gluon distributions, is constrained for the first time in the nonperturbative regime, for transverse momentum scales $q_{T} \in [ 300\text{ MeV}, 1.3\text{ GeV}]$. The constraints are determined in lattice QCD at a close-to-physical pion mass $M_\pi = 172(3)\text{ MeV}$, a single lattice spacing $a=0.15\text{ fm}$, and next-to-next-to-leading logarithmic matching in Large-Momentum Effective Theory. These results represent the first step toward a controlled determination of the gluon Collins-Soper kernel in QCD, with eventual phenomenological import and relevance to present and future experiments sensitive to the gluon structure of hadronic matter.

Figures

Figures reproduced from arXiv: 2607.24587 by Artur Avkhadiev, Michael L. Wagman, Phiala E. Shanahan, Yang Fu, Yong Zhao.

Figure 1
Figure 1. Figure 1: FIG. 1: Position-space functions [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Estimates of the gluon CS kernel [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Effective energy functions [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Continuum dispersion relation [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Lanczos and statistical-fit extractions of matrix elements [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Numerical results for the subtraction factor [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: A comparison of fit results with two functional [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Constant-fit [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: A complete set of LO Fourier-transformed quasi-TMD beam functions [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Effects of several modeling choices in implementing the Fourier transformation (FT) of Eq. (2) on the final [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Constraints on [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: A comparison of CS kernel constraints as a [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Additive LaMET matching corrections to the CS kernel according to Eq. [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Intermediate-stage results for transverse separations [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: A comparison of CS kernel constraints as function of [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

123 extracted references · 91 linked inside Pith

  1. [1]

    Aschenaueret al.,The RHIC SPIN Pro- gram: Achievements and Future Opportunities, (2015), arXiv:1501.01220 [nucl-ex]

    E.-C. Aschenaueret al.,The RHIC SPIN Pro- gram: Achievements and Future Opportunities, (2015), arXiv:1501.01220 [nucl-ex]

    Pith/arXiv arXiv 2015

  2. [2]

    Aschenaueret al.(RHIC SPIN),The RHIC Cold QCD Program, (2023), arXiv:2302.00605 [nucl-ex]

    E.-C. Aschenaueret al.(RHIC SPIN),The RHIC Cold QCD Program, (2023), arXiv:2302.00605 [nucl-ex]

    Pith/arXiv arXiv 2023

  3. [3]

    Dudeket al.,Physics Opportunities with the 12 GeV Upgrade at Jefferson Lab, Eur

    J. Dudeket al.,Physics Opportunities with the 12 GeV Upgrade at Jefferson Lab, Eur. Phys. J. A48, 187 (2012), arXiv:1208.1244 [hep-ex]

    Pith/arXiv arXiv 2012

  4. [4]

    Diehlet al.(CLAS),Multidimensional, High Preci- sion Measurements of Beam Single Spin Asymmetries in Semi-inclusiveπ+ Electroproduction off Protons in the Valence Region, Phys

    S. Diehlet al.(CLAS),Multidimensional, High Preci- sion Measurements of Beam Single Spin Asymmetries in Semi-inclusiveπ+ Electroproduction off Protons in the Valence Region, Phys. Rev. Lett.128, 062005 (2022), arXiv:2101.03544 [hep-ex]

    Pith/arXiv arXiv 2022

  5. [5]

    Accardiet al.,Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all, Eur

    A. Accardiet al.,Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all, Eur. Phys. J. A52, 268 (2016), arXiv:1212.1701 [nucl-ex]

    Pith/arXiv arXiv 2016

  6. [6]

    Abdul Khaleket al.,Snowmass 2021 White Paper: Electron Ion Collider for High Energy Physics, (2022), arXiv:2203.13199 [hep-ph]

    R. Abdul Khaleket al.,Snowmass 2021 White Paper: Electron Ion Collider for High Energy Physics, (2022), arXiv:2203.13199 [hep-ph]

    Pith/arXiv arXiv 2021

  7. [7]

    Agostiniet al.(LHeC, FCC-he Study Group),The Large Hadron–Electron Collider at the HL-LHC, J

    P. Agostiniet al.(LHeC, FCC-he Study Group),The Large Hadron–Electron Collider at the HL-LHC, J. Phys. G48, 110501 (2021), arXiv:2007.14491 [hep-ex]

    arXiv 2021

  8. [8]

    D. P. Anderleet al.,Electron-ion collider in China, Front. Phys. (Beijing)16, 64701 (2021), arXiv:2102.09222 [nucl- ex]

    Pith/arXiv arXiv 2021

  9. [9]

    P. J. Mulders and J. Rodrigues,Transverse momen- tum dependence in gluon distribution and fragmentation functions, Phys. Rev. D63, 094021 (2001), arXiv:hep- ph/0009343

    arXiv 2001

  10. [10]

    Diehl,Introduction to GPDs and TMDs, Eur

    M. Diehl,Introduction to GPDs and TMDs, Eur. Phys. J. A52, 149 (2016), arXiv:1512.01328 [hep-ph]

    Pith/arXiv arXiv 2016

  11. [11]

    Boussarieet al.,TMD Handbook, (2023), arXiv:2304.03302 [hep-ph]

    R. Boussarieet al.,TMD Handbook, (2023), arXiv:2304.03302 [hep-ph]

    Pith/arXiv arXiv 2023

  12. [12]

    D. Boer, S. J. Brodsky, P. J. Mulders, and C. Pisano, Direct Probes of Linearly Polarized Gluons inside Unpo- larized Hadrons, Phys. Rev. Lett.106, 132001 (2011), arXiv:1011.4225 [hep-ph]

    Pith/arXiv arXiv 2011

  13. [13]

    Z.-B. Kang, J. Reiten, D. Y. Shao, and J. Terry,QCD evolution of the gluon Sivers function in heavy flavor dijet production at the Electron-Ion Collider, JHEP05, 286, arXiv:2012.01756 [hep-ph]

    Pith/arXiv arXiv 2012

  14. [14]

    M. G. Echevarria, P. A. G. García, and I. Scimemi,Gluon Sivers function in dijet production at the EIC, (2026), arXiv:2603.00375 [hep-ph]

    arXiv 2026

  15. [15]

    Kishore, A

    R. Kishore, A. Mukherjee, A. Pawar, and M. Siddiqah, cos2ϕt azimuthal asymmetry in back-to-back J/ψ-jet pro- duction in ep→eJ/ψ jet X at the EIC, Phys. Rev. D106, 034009 (2022), arXiv:2203.13516 [hep-ph]

    Pith/arXiv arXiv 2022

  16. [16]

    P. M. Nadolsky, C. Balazs, E. L. Berger, and C. P. Yuan,Gluon-gluon contributions to the production of continuum diphoton pairs at hadron colliders, Phys. Rev. D76, 013008 (2007), arXiv:hep-ph/0702003

    Pith/arXiv arXiv 2007

  17. [17]

    J.-W. Qiu, M. Schlegel, and W. Vogelsang,Probing Glu- onic Spin-Orbit Correlations in Photon Pair Production, Phys. Rev. Lett.107, 062001 (2011), arXiv:1103.3861 [hep-ph]

    Pith/arXiv arXiv 2011

  18. [18]

    Anedda, F

    S. Anedda, F. Murgia, and C. Pisano,First insight into transverse-momentum-dependent fragmentation physics at photon-photon colliders, Phys. Rev. D112, 014013 (2025), arXiv:2504.12802 [hep-ph]

    Pith/arXiv arXiv 2025

  19. [19]

    Boer and C

    D. Boer and C. Pisano,Polarized gluon studies with charmonium and bottomonium at LHCb and AFTER, Phys. Rev. D86, 094007 (2012), arXiv:1208.3642 [hep- ph]

    Pith/arXiv arXiv 2012

  20. [20]

    M. G. Echevarria,Proper TMD factorization for quarko- nia production: pp→η c,b as a study case, JHEP10, 144, arXiv:1907.06494 [hep-ph]

    Pith/arXiv arXiv 1907

  21. [21]

    W. J. den Dunnen, J. P. Lansberg, C. Pisano, and M. Schlegel,Accessing the Transverse Dynamics and Polarization of Gluons inside the Proton at the LHC, Phys. Rev. Lett.112, 212001 (2014), arXiv:1401.7611 [hep-ph]

    Pith/arXiv arXiv 2014

  22. [22]

    Lansberg, C

    J.-P. Lansberg, C. Pisano, and M. Schlegel,Associ- ated production of a dilepton and aΥ( J/ψ)at the LHC as a probe of gluon transverse momentum depen- dent distributions, Nucl. Phys. B920, 192–210 (2017), arXiv:1702.00305 [hep-ph]

    Pith/arXiv arXiv 2017

  23. [23]

    N. Kato, L. Maxia, and C. Pisano,Spin asymme- tries for C-even quarkonium production as a probe of gluon distributions, Phys. Rev. D110, 034038 (2024), arXiv:2403.20017 [hep-ph]

    Pith/arXiv arXiv 2024

  24. [24]

    Scarpa, D

    F. Scarpa, D. Boer, M. G. Echevarria, J.-P. Lansberg, C. Pisano, and M. Schlegel,Studies of gluon TMDs and their evolution using quarkonium-pair production at the LHC, Eur. Phys. J. C80, 87 (2020), arXiv:1909.05769 [hep-ph]

    Pith/arXiv arXiv 2020

  25. [25]

    D.Chakrabarti, R.Kishore, A.Mukherjee,andS.Rajesh, Azimuthal asymmetries in J/ψ-photon production at the EIC, Phys. Rev. D107, 014008 (2023), arXiv:2211.08709 [hep-ph]

    Pith/arXiv arXiv 2023

  26. [26]

    Catani and M

    S. Catani and M. Grazzini,QCD transverse-momentum resummation in gluon fusion processes, Nucl. Phys. B 845, 297–323 (2011), arXiv:1011.3918 [hep-ph]

    Pith/arXiv arXiv 2011

  27. [27]

    M. G. Echevarria, T. Kasemets, P. J. Mulders, and 7 C. Pisano,QCD evolution of (un)polarized gluon TMD- PDFs and the HiggsqT-distribution, JHEP07, 158, [Er- ratum: JHEP 05, 073 (2017)], arXiv:1502.05354 [hep-ph]

    Pith/arXiv arXiv 2017

  28. [28]

    Anedda, V

    S. Anedda, V. Bertone, G. Bozzi, and M. Cerutti,A first extraction of gluon TMDs from Higgs data at the LHC, (2026), arXiv:2605.28216 [hep-ph]

    Pith/arXiv arXiv 2026

  29. [29]

    J. C. Collins and D. E. Soper,Back-To-Back Jets: Fourier Transform from B to K-Transverse, Nucl. Phys. B197, 446–476 (1982)

    1982

  30. [30]

    J. C. Collins and D. E. Soper,Back-To-Back Jets in QCD, Nucl. Phys. B193, 381 (1981), [Erratum: Nucl.Phys.B 213, 545 (1983)]

    1981

  31. [31]

    J. C. Collins, D. E. Soper, and G. F. Sterman,Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl. Phys. B250, 199–224 (1985)

    1985

  32. [32]

    Landry, R

    F. Landry, R. Brock, P. M. Nadolsky, and C. P. Yuan, Tevatron Run-1Z boson data and Collins-Soper-Sterman resummation formalism, Phys. Rev. D67, 073016 (2003), arXiv:hep-ph/0212159

    Pith/arXiv arXiv 2003

  33. [33]

    V. Moos, I. Scimemi, A. Vladimirov, and P. Zurita, Determination of unpolarized TMD distributions from the fit of Drell-Yan and SIDIS data at N4LL, JHEP11, 134, arXiv:2503.11201 [hep-ph]

    arXiv

  34. [34]

    Bacchetta, V

    A. Bacchetta, V. Bertone, C. Bissolotti, M. Cerutti, M. Radici, S. Rodini, and L. Rossi (MAP (Multi- dimensional Analyses of Partonic distributions)), Neural-Network Extraction of Unpolarized Transverse- Momentum-Dependent Distributions, Phys. Rev. Lett. 135, 021904 (2025), arXiv:2502.04166 [hep-ph]

    Pith/arXiv arXiv 2025

  35. [35]

    Avkhadiev, V

    A. Avkhadiev, V. Bertone, C. Bissolotti, M. Cerutti, Y. Fu, S. Rodini, P. Shanahan, M. Wagman, and Y. Zhao, Extraction of the Collins-Soper Kernel from a Joint Anal- ysis of Experimental and Lattice Data, Phys. Rev. Lett. 136, 171902 (2026), arXiv:2510.26489 [hep-ph]

    arXiv 2026

  36. [36]

    Z.-B. Kang, J. Penttala, and C. Zhang,Determination of the strong coupling constant and the Collins-Soper kernel from the energy-energy correlator ine+e− collisions, (2024), arXiv:2410.21435 [hep-ph]

    Pith/arXiv arXiv 2024

  37. [37]

    Camarda, G

    S. Camarda, G. Ferrera, and L. Rossi,Drell-Yan lepton pair production at low invariant masses: transverse- momentum resummation and non-perturbative effects in QCD, JHEP01, 150, arXiv:2508.06201 [hep-ph]

    arXiv

  38. [38]

    Mendizabal Morentin, S

    A.B.Martinez, F.Hautmann, L.Keersmaekers, A.Lelek, M. Mendizabal Morentin, S. Taheri Monfared, and A. M. van Kampen,Soft-gluon coupling and the TMD par- ton branching Sudakov form factor, Phys. Lett. B868, 139762 (2025), arXiv:2412.21116 [hep-ph]

    Pith/arXiv arXiv 2025

  39. [39]

    Shanahan, M

    P. Shanahan, M. Wagman, and Y. Zhao,Collins-Soper kernel for TMD evolution from lattice QCD, Phys. Rev. D102, 014511 (2020), arXiv:2003.06063 [hep-lat]

    Pith/arXiv arXiv 2020

  40. [40]

    Shanahan, M

    P. Shanahan, M. Wagman, and Y. Zhao,Lattice QCD calculation of the Collins-Soper kernel from quasi-TMDPDFs, Phys. Rev. D104, 114502 (2021), arXiv:2107.11930 [hep-lat]

    Pith/arXiv arXiv 2021

  41. [41]

    Avkhadiev, P

    A. Avkhadiev, P. Shanahan, M. Wagman, and Y. Zhao, Collins-Soper kernel from lattice QCD at the physical pion mass, (2023), arXiv:2307.12359 [hep-lat]

    Pith/arXiv arXiv 2023

  42. [42]

    Avkhadiev, P

    A. Avkhadiev, P. E. Shanahan, M. L. Wagman, and Y. Zhao,Determination of the Collins-Soper Kernel from Lattice QCD, Phys. Rev. Lett.132, 231901 (2024), arXiv:2402.06725 [hep-lat]

    Pith/arXiv arXiv 2024

  43. [43]

    Zhanget al.(Lattice Parton),Lattice QCD Calcu- lations of Transverse-Momentum-Dependent Soft Func- tion through Large-Momentum Effective Theory, Phys

    Q.-A. Zhanget al.(Lattice Parton),Lattice QCD Calcu- lations of Transverse-Momentum-Dependent Soft Func- tion through Large-Momentum Effective Theory, Phys. Rev. Lett.125, 192001 (2020), arXiv:2005.14572 [hep- lat]

    Pith/arXiv arXiv 2020

  44. [44]

    Chuet al.(LPC),Nonperturbative determina- tion of the Collins-Soper kernel from quasitransverse- momentum-dependent wave functions, Phys

    M.-H. Chuet al.(LPC),Nonperturbative determina- tion of the Collins-Soper kernel from quasitransverse- momentum-dependent wave functions, Phys. Rev. D106, 034509 (2022), arXiv:2204.00200 [hep-lat]

    Pith/arXiv arXiv 2022

  45. [45]

    Chuet al.,Lattice Calculation of the Intrinsic Soft Function and the Collins-Soper Kernel, (2023), arXiv:2306.06488 [hep-lat]

    M.-H. Chuet al.,Lattice Calculation of the Intrinsic Soft Function and the Collins-Soper Kernel, (2023), arXiv:2306.06488 [hep-lat]

    Pith/arXiv arXiv 2023

  46. [46]

    Tanet al.,Lattice QCD determination of the Collins-Soper kernel in the continuum and physi- cal mass limits, Phys

    J.-X. Tanet al.,Lattice QCD determination of the Collins-Soper kernel in the continuum and physi- cal mass limits, Phys. Rev. D113, 054505 (2026), arXiv:2511.22547 [hep-lat]

    arXiv 2026

  47. [47]

    Schlemmer, A

    M. Schlemmer, A. Vladimirov, C. Zimmermann, M. En- gelhardt, and A. Schäfer,Determination of the Collins- Soper Kernel from Lattice QCD, JHEP08, 004, arXiv:2103.16991 [hep-lat]

    Pith/arXiv arXiv

  48. [48]

    H.-T. Shu, M. Schlemmer, T. Sizmann, A. Vladimirov, L. Walter, M. Engelhardt, A. Schäfer, and Y.-B. Yang, Universality of the Collins-Soper kernel in lattice calcu- lations, (2023), arXiv:2302.06502 [hep-lat]

    Pith/arXiv arXiv 2023

  49. [49]

    Liet al.,Systematic study of transverse-momentum dependent soft function from lattice QCD, (2021), arXiv:2106.13027 [hep-lat]

    Y. Liet al.,Systematic study of transverse-momentum dependent soft function from lattice QCD, (2021), arXiv:2106.13027 [hep-lat]

    Pith/arXiv arXiv 2021

  50. [50]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, K. Cichy, M. Constantinou, A. Sen, G. Spanoudes, F. Steffens, and J. Tarello,Collins- Soper kernel and reduced soft function in lattice QCD, Phys. Rev. D113, 074506 (2026), arXiv:2509.26316 [hep- lat]

    arXiv 2026

  51. [51]

    Bollweg, X

    D. Bollweg, X. Gao, S. Mukherjee, and Y. Zhao,Non- perturbative Collins-Soper kernel from chiral quarks with physical masses, Phys. Lett. B852, 138617 (2024), arXiv:2403.00664 [hep-lat]

    Pith/arXiv arXiv 2024

  52. [52]

    Bollweg, X

    D. Bollweg, X. Gao, J. He, S. Mukherjee, and Y. Zhao, Transverse-momentum-dependent pion structures from lattice QCD: Collins-Soper kernel, soft factor, TMDWF, and TMDPDF, Phys. Rev. D112, 034501 (2025), arXiv:2504.04625 [hep-lat]

    arXiv 2025

  53. [53]

    Francis, C

    A. Francis, C. J. D. Lin, W. Morris, and Y. Zhao,The Collins-Soper kernel from a vacuum soft function, (2026), arXiv:2606.19221 [hep-lat]

    Pith/arXiv arXiv 2026

  54. [54]

    von Manteuffel, E

    A. von Manteuffel, E. Panzer, and R. M. Schabinger, Cusp and collinear anomalous dimensions in four-loop QCD from form factors, Phys. Rev. Lett.124, 162001 (2020), arXiv:2002.04617 [hep-ph]

    Pith/arXiv arXiv 2020

  55. [55]

    Moult, H

    I. Moult, H. X. Zhu, and Y. J. Zhu,The four loop QCD rapidity anomalous dimension, JHEP08, 280, arXiv:2205.02249 [hep-ph]

    Pith/arXiv arXiv

  56. [56]

    C. Duhr, B. Mistlberger, and G. Vita,Four-Loop Rapid- ity Anomalous Dimension and Event Shapes to Fourth Logarithmic Order, Phys. Rev. Lett.129, 162001 (2022), arXiv:2205.02242 [hep-ph]

    Pith/arXiv arXiv 2022

  57. [57]

    G. S. Bali,Casimir scaling of SU(3) static potentials, Phys. Rev. D62, 114503 (2000), arXiv:hep-lat/0006022

    Pith/arXiv arXiv 2000

  58. [58]

    Moult and H

    I. Moult and H. X. Zhu,Energy Correlators: A Journey From Theory to Experiment, (2025), arXiv:2506.09119 [hep-ph]

    Pith/arXiv arXiv 2025

  59. [59]

    Cridge, G

    T. Cridge, G. Marinelli, and F. J. Tackmann,The- ory uncertainties in the extraction ofαs from Drell- Yan at small transverse momentum, JHEP12, 043, arXiv:2506.13874 [hep-ph]

    arXiv

  60. [60]

    Jaarsma, Y

    M. Jaarsma, Y. Li, I. Moult, W. J. Waalewijn, and H. X. Zhu,From DGLAP to Sudakov: Precision Predictions for Energy-Energy Correlators, (2025), arXiv:2512.11950 8 [hep-ph]

    arXiv 2025

  61. [61]

    Chekhovskyet al.(CMS),High-precision measure- ment of the W boson mass with the CMS experiment, Nature652, 321–327 (2026), arXiv:2412.13872 [hep-ex]

    V. Chekhovskyet al.(CMS),High-precision measure- ment of the W boson mass with the CMS experiment, Nature652, 321–327 (2026), arXiv:2412.13872 [hep-ex]

    Pith/arXiv arXiv 2026

  62. [62]

    I. W. Stewart, F. J. Tackmann, and W. J. Waalewijn, Factorization at the LHC: From PDFs to Initial State Jets, Phys. Rev. D81, 094035 (2010), arXiv:0910.0467 [hep-ph]

    Pith/arXiv arXiv 2010

  63. [63]

    A. Jain, M. Procura, and W. J. Waalewijn,Fully- Unintegrated Parton Distribution and Fragmenta- tion Functions at Perturbative kT, JHEP04, 132, arXiv:1110.0839 [hep-ph]

    Pith/arXiv arXiv

  64. [64]

    J.-y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein,The Rapidity Renormalization Group, Phys. Rev. Lett.108, 151601 (2012), arXiv:1104.0881 [hep-ph]

    Pith/arXiv arXiv 2012

  65. [65]

    J.-Y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein,A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP05, 084, arXiv:1202.0814 [hep-ph]

    Pith/arXiv arXiv

  66. [66]

    Ji,Parton Physics on a Euclidean Lattice, Phys

    X. Ji,Parton Physics on a Euclidean Lattice, Phys. Rev. Lett.110, 262002 (2013), arXiv:1305.1539 [hep-ph]

    Pith/arXiv arXiv 2013

  67. [67]

    X. Ji, P. Sun, X. Xiong, and F. Yuan,Soft factor sub- traction and transverse momentum dependent parton distributions on the lattice, Phys. Rev. D91, 074009 (2015), arXiv:1405.7640 [hep-ph]

    Pith/arXiv arXiv 2015

  68. [68]

    M. A. Ebert, I. W. Stewart, and Y. Zhao,Determining the Nonperturbative Collins-Soper Kernel From Lattice QCD, Phys. Rev. D99, 034505 (2019), arXiv:1811.00026 [hep-ph]

    Pith/arXiv arXiv 2019

  69. [69]

    Ji,Parton Physics from Large-Momentum Effective Field Theory, Sci

    X. Ji,Parton Physics from Large-Momentum Effective Field Theory, Sci. China Phys. Mech. Astron.57, 1407– 1412 (2014), arXiv:1404.6680 [hep-ph]

    Pith/arXiv arXiv 2014

  70. [70]

    Ji, Y.-S

    X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao,Large- momentum effective theory, Rev. Mod. Phys.93, 035005 (2021), arXiv:2004.03543 [hep-ph]

    Pith/arXiv arXiv 2021

  71. [71]

    S. T. Schindler, I. W. Stewart, and Y. Zhao,One- loop matching for gluon lattice TMDs, JHEP08, 084, arXiv:2205.12369 [hep-ph]

    Pith/arXiv arXiv

  72. [72]

    R. Zhu, Y. Ji, J.-H. Zhang, and S. Zhao, Gluon transverse-momentum-dependent distributions from large-momentum effective theory, JHEP02, 114, arXiv:2209.05443 [hep-ph]

    Pith/arXiv arXiv

  73. [73]

    Zhang, X

    J.-H. Zhang, X. Ji, A. Schäfer, W. Wang, and S. Zhao, Accessing Gluon Parton Distributions in Large Momen- tum Effective Theory, Phys. Rev. Lett.122, 142001 (2019), arXiv:1808.10824 [hep-ph]

    Pith/arXiv arXiv 2019

  74. [74]

    Li, Y.-Q

    Z.-Y. Li, Y.-Q. Ma, and J.-W. Qiu,Multiplicative Renormalizability of Operators defining Quasiparton Distributions, Phys. Rev. Lett.122, 062002 (2019), arXiv:1809.01836 [hep-ph]

    Pith/arXiv arXiv 2019

  75. [75]

    Ji, L.-C

    X. Ji, L.-C. Jin, F. Yuan, J.-H. Zhang, and Y. Zhao, Transverse momentum dependent parton quasidistribu- tions, Phys. Rev. D99, 114006 (2019), arXiv:1801.05930 [hep-ph]

    Pith/arXiv arXiv 2019

  76. [76]

    Zhang, X

    K. Zhang, X. Ji, Y.-B. Yang, F. Yao, and J.-H. Zhang (Lattice Parton (LPC)),Renormalization of Transverse-Momentum-Dependent Parton Distribution on the Lattice, Phys. Rev. Lett.129, 082002 (2022), arXiv:2205.13402 [hep-lat]

    Pith/arXiv arXiv 2022

  77. [77]

    H. Dorn, D. Robaschik, and E. Wieczorek,Renormal- ization and short distance properties of gauge invariant gluonium and hadron operators, Annalen Phys.40, 166 (1983)

    1983

  78. [78]

    X. Ji, Y. Liu, and Y.-S. Liu,Transverse-momentum- dependent parton distribution functions from large- momentum effective theory, Phys. Lett. B811, 135946 (2020), arXiv:1911.03840 [hep-ph]

    Pith/arXiv arXiv 2020

  79. [79]

    M. A. Ebert, S. T. Schindler, I. W. Stewart, and Y. Zhao, Factorization connecting continuum & lattice TMDs, JHEP04, 178, arXiv:2201.08401 [hep-ph]

    Pith/arXiv arXiv

  80. [80]

    Gaunt, M

    J. Gaunt, M. Stahlhofen, and F. J. Tackmann,The Gluon Beam Function at Two Loops, JHEP08, 020, arXiv:1405.1044 [hep-ph]

    Pith/arXiv arXiv

Showing first 80 references.