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REVIEW 4 major objections 5 minor 69 references

This paper proposes that azimuthal modulations in dihadron fragmentation at electron-positron colliders can measure light-quark Yukawa couplings linearly, reaching limits of order 10^-4.

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2026-08-03 15:30 UTC pith:TMNFCULS

load-bearing objection The linear-yukawa interference observable is a genuinely new idea and the flavor tagging is clever, but the O(10^-4) limits rest on unquantified isospin and FF assumptions, so the paper is a promising proposal rather than a completed measurement prediction. the 4 major comments →

arxiv 2512.16492 v2 pith:TMNFCULS submitted 2025-12-18 hep-ph hep-exnucl-exnucl-th

Unveiling Light-Quark Yukawa Flavor Structure via Dihadron Fragmentation at Lepton Colliders

Qing-Hong Cao , Xin-Kai Wen , Bin Yan , Shu-Tao Zhang This is my paper
classification hep-ph hep-exnucl-exnucl-th
keywords light-quark Yukawa couplingsdihadron fragmentationtransverse spin asymmetriesazimuthal modulationsHiggs flavor structuree+e- collisionsCP violation in Higgs sector
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.

Light-quark Yukawa couplings — the Higgs interactions with up and down quarks — are among the least constrained parts of the Standard Model, with current bounds allowing several orders of magnitude variation. This paper argues that the interference between the Higgs signal and the Standard Model continuum in e^-e^+ -> q qbar Z produces a transverse spin on the produced quarks, and that this spin becomes visible as a cos/sin phi_R azimuthal modulation in dihadron fragmentation. Because the modulation is linear in the Yukawa couplings y_u and y_d, it evades the quadratic suppression of rate-based measurements. By tagging the recoiling quark with a pi±, K±, or proton, the up- and down-quark contributions can be separated, yielding projected 68% limits of order 10^-4 at 250 GeV with 50 ab^-1. If correct, this turns fragmentation dynamics into a direct probe of Higgs flavor structure.

Core claim

At leading order, the interference between the Higgs-mediated amplitude for e^-e^+ -> q qbar Z and the Standard Model continuum generates a single helicity flip of the light quark, leaving it transversely polarized. The paper shows that this polarization is encoded in the quark density matrix through the linear relations s_x C = omega y_q + tildomega tilde y_q and s_y C = tildomega y_q - omega tilde y_q, so the resulting cos phi_R and sin phi_R modulations in the dihadron fragmentation are directly proportional to the Yukawa couplings rather than their squares. Using the pi+ pi- interference fragmentation functions — whose isospin and charge-conjugation relations select only up and down quar

What carries the argument

The key objects are the interference dihadron fragmentation functions H_q^{h1 h2}(z,M_h), which quantify the probability that a transversely polarized quark fragments into a collimated pair of hadrons, together with the unpolarized dihadron fragmentation function D_q^{h1 h2}. Their ratio acts as the spin analyzing power: it sets the amplitude of the sin phi_R and cos phi_R azimuthal modulations in the cross section. The paper exploits the charge-conjugation and isospin relations for pi+ pi- pairs, H_u = -H_d and H_s = H_c = H_b = 0, so only u and d quarks contribute to the modulation, while the recoiling hadron h' acts as a flavor tag through the single-hadron fragmentation functions. The li

Load-bearing premise

That the pi+pi- interference fragmentation functions satisfy exact isospin and charge-conjugation relations (D_u = D_d, H_u = -H_d, H_s,c,b = 0) and that current global fits of fragmentation functions are accurate enough that their uncertainties do not shift the projected 10^-4 limits.

What would settle it

A 250 GeV e+e- run with 50 ab^-1 measuring the sin phi_R and cos phi_R asymmetries in the pi±, K±, and p/pbar channels; if the three 68% confidence contours do not intersect at a common y_u,y_d consistent with the predicted slopes — or if all are null at the level of a few times 10^-4 — the linear-interference mechanism and the isospin relations behind it are falsified.

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

If this is right

  • A 250 GeV electron-positron collider with 50 ab^-1 of data could constrain the up- and down-quark Yukawa couplings to the 10^-4 level, far beyond projected hadron-collider bounds, with the up and down couplings separated rather than lumped together.
  • Because the asymmetry is a cross-section ratio, major systematic uncertainties cancel, so the projected reach is statistics-dominated and scales with luminosity and with future fragmentation-function fits.
  • The sign of the sin phi_R modulation flips across the Higgs-boson threshold in the Z-boson energy; binning the Z energy exploits this flip and enhances the CP-even sensitivity.
  • The same linear observable also carries sensitivity to CP-odd Yukawa components; comparing h+ and h- tags can isolate the CP-odd contributions, extending the method to light-quark CP violation.
  • If the predicted flavor-slope pattern holds, the method gives a direct test of the Standard-Model relation y_f = sqrt(2)m_f/v for the first generation, so any deviation would be a signal of new physics in the Higgs sector.

Where Pith is reading between the lines

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

  • The reliance on exact isospin and charge-conjugation relations for the pi+pi- interference fragmentation functions means that even a small SU(2)-breaking correction, or nonzero strange/charm/bottom interference functions, would mix the extracted y_u and y_d; a dedicated estimate of these effects is a natural next step.
  • If interference fragmentation functions for other dihadron pairs (such as K+K- or proton-antiproton) were extracted from future data, the flavor tagging could move from the recoiling hadron into the dihadron pair itself, potentially sharpening the separation.
  • An explicit treatment of the z dependence of the interference fragmentation functions is expected to improve the reach, since the spin analyzing power grows at large z; the present Letter integrates over z, so a differential analysis is a straightforward extension.
  • The same interference mechanism — a single helicity flip projected onto a transverse spin — may also be exploitable in other processes with a measurable recoil hadron, such as tagged Z or W associated production at lepton colliders.

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

4 major / 5 minor

Summary. The manuscript proposes e+e−→qq̅Z with one quark fragmenting into a π+π− pair and the recoiling antiquark into a tagged single hadron h′ (π±, K±, p/p̅) as a probe of the light-quark Yukawa couplings y_u and y_d. The central idea is that interference between the SM continuum amplitude and the Higgs-mediated amplitude produces an azimuthal modulation in the dihadron fragmentation angle φ_R that is linear in y_q, and that the flavor of the fragmenting quark is tagged by h′. Using the JAM π+π− interference DiFFs and NNPDF LO single-hadron FFs, the authors project 68% confidence-level limits of order 10^-4 on y_u and y_d at √s=250 GeV and 50 ab^-1, with K± tags mainly constraining y_u and π±/p tags providing complementary slopes that together break the u/d degeneracy.

Significance. If the proposed mechanism and the numerical projections are correct, this would be a genuinely new avenue to first-generation Yukawa couplings, exploiting the linear-in-y_q interference rather than the quadratic rate suppression. The use of transverse spin in fragmentation is clever and connects Higgs physics to an active QCD program on dihadron fragmentation. The paper also has the merit of using publicly available JAM and NNPDF fragmentation functions and of presenting the observable in a factorized form. However, the quantitative claims rest on several currently unvalidated pillars: the hard coefficients are not shown, the isospin/charge-conjugation relations are assumed exact, the FF uncertainties are not propagated, and no detector-level study is provided. The significance is therefore conditional: the idea is promising and worth publishing as a proposal, but the claimed O(10^-4) reach is not yet established.

major comments (4)
  1. [§4, Eq. (11) and text near Eq. (8)] The central numerical results depend on the hard coefficients C_q, ω_q, and ω̃_q and their integrated moments ⟨C_q⟩, ⟨ω_q⟩, ⟨ω̃_q⟩, but none of these functions are given. Equation (8) only defines them through helicity amplitudes, and the full hard-coefficient analysis is deferred to Ref. [47], which is in preparation. Without explicit expressions, the reader cannot verify the sign structure, the magnitude of the transverse spin, or the threshold behavior that the authors exploit when splitting the Z-energy bins. This is a load-bearing gap because the projected limits in Fig. 3 scale directly with these coefficients. Please include the essential hard-scattering expressions or at least a numerical validation against a benchmark, and make the content of Ref. [47] available or summarize the required formulas.
  2. [Eq. (9) and Fig. 3] The clean separation of y_u and y_d into the two-dimensional exclusion contours assumes D_u^{π+π-}=D_d^{π+π-}, H_u^{π+π-}=-H_d^{π+π-}, and H_s,c,b=0 exactly at μ=m_H. These relations are not exact in QCD, and the JAM π+π− DiFFs cannot constrain H_s,c,b. If H_s is nonzero at even a few percent of H_u, the same sinφ_R/cosφ_R modulations receive a y_s H_s/(H_u) contamination that is absorbed into the fitted effective y_u and y_d. Likewise, a few-percent isospin breaking in D_u vs D_d changes the weight of the two terms in Eq. (11) and rotates the contours in Fig. 3. The paper notes only that π+π− DiFFs are available; it does not quantify the sensitivity to Eq. (9). Please provide a quantitative test, e.g., allow H_s=εH_u and D_u=(1+δ)D_d with ε,δ at the 1–10% level and show how the limits and the u/d separation degrade.
  3. [Eqs. (14)–(15) and Fig. 3] The χ² analysis includes only statistical uncertainties δA≈1/√(σ_h′L), with FF uncertainties ignored. Since the asymmetry is proportional to H/D, a relative uncertainty δR in the analyzing power translates directly into a relative uncertainty of the same order in the extracted y_q. The current JAM and NNPDF fits have substantial uncertainties in the relevant z, M_h, and scale region, and at y_q~10^-4 a 20% shift in H/D is the size of the entire allowed region. The statement that FF uncertainties 'are expected to improve' does not justify omitting them from the projections. Please propagate the JAM and NNPDF error sets, or at least produce a bracketing estimate by scaling H/D by ±1σ and recomputing the contours.
  4. [Numerical results and discussion, Fig. 3] The projected O(10^-4) limits assume a perfect detector: no background contamination, no hadron misidentification, no luminosity asymmetries, and no systematic uncertainty in the recoil-mass selection. The azimuthal asymmetries are defined as ratios, so some common systematics cancel, but the quoted 10^-4 level is below typical charge- and flavor-dependent detector asymmetries. The paper also does not simulate the dominant e+e− backgrounds (e.g., qq̅γ, WW, ZZ, ττ) or the efficiency of the (π+π−)h′ topology with the required energy window. A full experimental study is beyond a Letter, but the authors should at least state the required detector performance and give a rough background estimate to support the 'cleanly disentangles' claim.
minor comments (5)
  1. [Introduction, first line] The text says 'In contract, the Yukawa couplings...' — presumably 'In contrast'. Please correct.
  2. [Eq. (15)] 'their uns over all independent bins' appears to be an editing artifact; please replace with 'their sum over all independent bins'.
  3. [Abstract and Conclusion] The abstract says 'O(10^-4) level' while the conclusion says 'O(10^-4) level' after the introduction says 'O(10^-4)' and the figure shows 10^-3 scale. For clarity, state the exact definition of the 68% limit and compare with the SM predictions y_u^SM≈1.3×10^-5 and y_d^SM≈2.7×10^-5 so the reader sees the exclusion power relative to the SM values.
  4. [Eq. (9) and notation] The superscript 'π+π−' is introduced only in the text before Eq. (9); the DiFFs in Eq. (4) and Eq. (11) carry a generic h1h2 superscript. Please make the notation consistent, especially since the numerical analysis uses only the π+π− channel.
  5. [Ref. [47]] Ref. [47] is cited as 'in preparation' for the full hard-coefficient and phenomenological analysis. The present Letter should be self-contained regarding the formulas used for the claimed limits; at minimum, the companion paper should be cited with a version number or the essential formulas moved here.

Circularity Check

0 steps flagged

No significant circularity: y_q enters as a hard-scattering parameter and the projected limits are a statistical inversion of an independently derived linear interference term.

full rationale

The derivation chain is: (i) parametrize light-quark Yukawa couplings in the effective Lagrangian, Eq. (1); (ii) compute e^-e^+ → q qbar Z helicity amplitudes whose Higgs/SM interference produces quark transverse spin linear in y_q, Eq. (8) — 'This structure reflects the linear dependence of H_q^{+-} on y_q − i ytilde_q'; (iii) factorize the dihadron and single-hadron fragmentation with the standard Collins–Soper–? collinear factorization, Eq. (4); (iv) insert external DiFFs and single-hadron FFs from the JAM fits [57–59] and NNPDF global analysis [62], which are global fits to independent experimental data, not to the process proposed here nor to y_q; (v) project 68% limits by a χ^2 that uses statistical errors, Eq. (15). No step defines the observable in terms of y_q, fits y_q to the same data it claims to predict, or imports a uniqueness theorem/ansatz from the authors' prior work. The isospin/charge-conjugation relations in Eq. (9), including H_{s,c,b}=0, are an input symmetry and fit assumption inherited from the JAM analysis, not a conclusion derived from the paper's own equations; their possible failure would degrade the claimed flavor disentanglement, but that is a model-uncertainty/robustness issue rather than circularity. The self-citations [39]–[47] supply context for transverse-spin techniques and defer technical details to an in-preparation companion paper [47]; they are not load-bearing for the linear-interference argument or for the external FFs. Accordingly, there is no exhibited reduction of a prediction to a fit by construction, and the circularity score is 0.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper introduces no new free parameters in the theory; the numerical projection inherits external fragmentation functions from JAM and NNPDF and applies hand-chosen kinematic selections. The main axioms are the collinear factorization of the transverse-spin cross section, the isospin/charge-conjugation symmetries of the π+π− interference DiFFs, suppression of scale dependence, and neglect of non-qqbar backgrounds. No new particles or entities are posited.

free parameters (3)
  • JAM π+π− interference and unpolarized DiFFs = external global fits (Refs. [57-59])
    The magnitude of the azimuthal asymmetry is proportional to the interference DiFF H_1; its uncertainty is not propagated in the projected limits.
  • NNPDF LO single-hadron FFs D^{h'}_q = external global fit (Ref. [62])
    Flavor-tag differences D_{qbar}-D_q set the y_u/y_d separation; their uncertainties are not propagated.
  • Kinematic selection windows = z∈[0.25,0.9], zbar∈[0.1,0.9], M_h∈[0.3,2.0] GeV, E_Z∈[109.2,111.2] GeV
    Hand-chosen optimized cuts that affect the projected reach; the paper does not scan or show stability under variation.
axioms (5)
  • domain assumption Collinear factorization for dihadron fragmentation and single-hadron fragmentation applies to e+e− → q qbar Z + h1 h2 + h′.
    Invoked through Eq. (4) using Refs. [35,36]; no estimate of power corrections at the chosen invariant masses.
  • domain assumption Isospin and charge-conjugation relations in Eq. (9) hold exactly for π+π− DiFFs, and H for s,c,b quarks vanishes.
    Isolates u and d contributions; violation would mix flavors and degrade the claimed disentanglement.
  • domain assumption Continuum e+e− → q qbar with ISR/FSR is the dominant background; other SM processes are negligible after the Z recoil-mass selection.
    No detector-level or full background simulation is presented.
  • domain assumption The hard-scattering helicity amplitudes and functions ω_q, ω̃_q in Eqs. (6)-(8) are correct as stated.
    The derivation is not shown; the paper defers to companion Ref. [47] 'in preparation'.
  • domain assumption Scale dependence of the FFs is negligible at μ=m_H and light-quark masses are negligible, so A≃0 in Eq. (14).
    The paper states scale dependence is 'suppressed here for simplicity'.

pith-pipeline@v1.3.0-alltime-deepseek · 9347 in / 17000 out tokens · 159429 ms · 2026-08-03T15:30:32.674349+00:00 · methodology

0 comments
read the original abstract

Directly probing light-quark Yukawa couplings and their flavor structure remains a major challenge due to their smallness and overwhelming QCD backgrounds. In this Letter, we propose a theoretical framework to access these couplings at lepton colliders through transverse spin dependent azimuthal modulations in dihadron fragmentation. These modulations arise from the interference between Higgs mediated and standard model amplitudes in $e^-e^+\to q\bar{q}Z$, producing angular structures that are linearly sensitive to the Yukawa couplings $y_q$, in contrast to conventional observables that scale as $y_q^2$. By combining channels with an identified accompanying single hadron, $h^\prime=\pi^\pm,K^\pm$, and $p/\bar{p}$, this approach cleanly disentangles the up- and down-quark Yukawa contributions, yielding typical limits at the $\mathcal{O}(10^{-4}\sim 10^{-3})$ level and establishing fragmentation dynamics as a novel and complementary probe of the Higgs flavor structure.

Figures

Figures reproduced from arXiv: 2512.16492 by Bin Yan, Qing-Hong Cao, Shu-Tao Zhang, Xin-Kai Wen.

Figure 2
Figure 2. Figure 2: FIG. 2. Kinematic configuration of [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Representative diagrams for the interference between [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Expected 68% C.L. limits on light-quark Yukawa cou [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

69 extracted references · 54 linked inside Pith

  1. [1]

    Cao, S.-L

    Q.-H. Cao, S.-L. Chen, and Y. Liu, Phys. Rev. D95, 053004 (2017), 1602.01934

    Pith/arXiv arXiv 2017

  2. [2]

    Cao, S.-L

    Q.-H. Cao, S.-L. Chen, Y. Liu, R. Zhang, and Y. Zhang, Phys. Rev. D99, 113003 (2019), 1901.04567

    Pith/arXiv arXiv 2019

  3. [3]

    Li, L.-X

    G. Li, L.-X. Xu, B. Yan, and C. P. Yuan, Phys. Lett. B 800, 135070 (2020), 1904.12006

    Pith/arXiv arXiv 2020

  4. [4]

    Cao, K.-P

    Q.-H. Cao, K.-P. Xie, H. Zhang, and R. Zhang, Chin. Phys. C45, 023117 (2021), 2008.13442

    Pith/arXiv arXiv 2021

  5. [5]

    Q. Bi, K. Chai, J. Gao, Y. Liu, and H. Zhang, Chin. Phys. C45, 023105 (2021), 2009.02000

    Pith/arXiv arXiv 2021

  6. [6]

    Aad et al

    G. Aad et al. (ATLAS), Nature607, 52 (2022), [Erratum: Nature 612, E24 (2022)], 2207.00092

    Pith/arXiv arXiv 2022

  7. [7]

    Tumasyan et al

    A. Tumasyan et al. (CMS), Nature607, 60 (2022), [Er- ratum: Nature 623, (2023)], 2207.00043

    Pith/arXiv arXiv 2022

  8. [8]

    Tumasyan et al

    A. Tumasyan et al. (CMS), Eur. Phys. J. C83, 562 (2023), 2204.12957

    Pith/arXiv arXiv 2023

  9. [9]

    Aad et al

    G. Aad et al. (ATLAS) (2025), 2510.23755

    Pith/arXiv arXiv 2025

  10. [10]

    Aad et al

    G. Aad et al. (ATLAS), JHEP03, 010 (2025), 2407.16320

    Pith/arXiv arXiv 2025

  11. [11]

    Aad et al

    G. Aad et al. (ATLAS), JHEP04, 075 (2025), 2410.19611

    Pith/arXiv arXiv 2025

  12. [12]

    G. T. Bodwin, F. Petriello, S. Stoynev, and M. Velasco, Phys. Rev. D88, 053003 (2013), 1306.5770

    Pith/arXiv arXiv 2013

  13. [13]

    A. L. Kagan, G. Perez, F. Petriello, Y. Soreq, S. Stoynev, and J. Zupan, Phys. Rev. Lett.114, 101802 (2015), 1406.1722

    Pith/arXiv arXiv 2015

  14. [14]

    K¨ onig and M

    M. K¨ onig and M. Neubert, JHEP08, 012 (2015), 1505.03870

    Pith/arXiv arXiv 2015

  15. [15]

    Brivio, F

    I. Brivio, F. Goertz, and G. Isidori, Phys. Rev. Lett.115, 211801 (2015), 1507.02916

    Pith/arXiv arXiv 2015

  16. [16]

    Aad et al

    G. Aad et al. (ATLAS), JHEP02, 045 (2025), 2407.15550

    Pith/arXiv arXiv 2025

  17. [17]

    Hayrapetyan et al

    A. Hayrapetyan et al. (CMS) (2025), 2508.14988

    Pith/arXiv arXiv 2025

  18. [18]

    Aad et al

    G. Aad et al. (ATLAS), Phys. Rev. Lett.135, 231802 (2025), 2507.03595

    arXiv 2025

  19. [19]

    Hayrapetyan et al

    A. Hayrapetyan et al. (CMS), Phys. Rev. Lett.132, 121901 (2024), 2310.05164

    Pith/arXiv arXiv 2024

  20. [20]

    Chekhovsky et al

    V. Chekhovsky et al. (CMS), Phys. Rev. D112, 112001 (2025), 2502.05665

    arXiv 2025

  21. [21]

    Bar-Shalom and A

    S. Bar-Shalom and A. Soni, Phys. Rev. D98, 055001 (2018), 1804.02400

    Pith/arXiv arXiv 2018

  22. [22]

    B. A. Erdelyi, R. Gr¨ ober, and N. Selimovic, JHEP05, 189 (2025), 2410.08272

    Pith/arXiv arXiv 2025

  23. [23]

    Zhou, Phys

    Y. Zhou, Phys. Rev. D93, 013019 (2016), 1505.06369

    Pith/arXiv arXiv 2016

  24. [24]

    Bishara, U

    F. Bishara, U. Haisch, P. F. Monni, and E. Re, Phys. Rev. Lett.118, 121801 (2017), 1606.09253

    Pith/arXiv arXiv 2017

  25. [25]

    Soreq, H

    Y. Soreq, H. X. Zhu, and J. Zupan, JHEP12, 045 (2016), 1606.09621

    Pith/arXiv arXiv 2016

  26. [26]

    Gao, JHEP01, 038 (2018), 1608.01746

    J. Gao, JHEP01, 038 (2018), 1608.01746

    Pith/arXiv arXiv 2018

  27. [27]

    Yu, JHEP02, 083 (2017), 1609.06592

    F. Yu, JHEP02, 083 (2017), 1609.06592

    Pith/arXiv arXiv 2017

  28. [28]

    Falkowski, S

    A. Falkowski, S. Ganguly, P. Gras, J. M. No, K. To- bioka, N. Vignaroli, and T. You, JHEP04, 023 (2021), 2011.09551

    Pith/arXiv arXiv 2021

  29. [29]

    Vignaroli, Symmetry14, 1183 (2022), 2205.09449

    N. Vignaroli, Symmetry14, 1183 (2022), 2205.09449

    Pith/arXiv arXiv 2022

  30. [30]

    Alasfar, R

    L. Alasfar, R. Gr¨ ober, C. Grojean, A. Paul, and Z. Qian, JHEP11, 045 (2022), 2207.04157

    Pith/arXiv arXiv 2022

  31. [31]

    Balzani, R

    E. Balzani, R. Gr¨ ober, and M. Vitti, JHEP10, 027 (2023), 2304.09772

    Pith/arXiv arXiv 2023

  32. [32]

    Yan and C

    B. Yan and C. Lee, JHEP03, 123 (2024), 2311.12556

    Pith/arXiv arXiv 2024

  33. [33]

    J. K. L. Michel (2025), 2508.05914

    Pith/arXiv arXiv 2025

  34. [34]

    de Blas et al., JHEP01, 139 (2020), 1905.03764

    J. de Blas et al., JHEP01, 139 (2020), 1905.03764

    arXiv 2020

  35. [35]

    J. C. Collins, S. F. Heppelmann, and G. A. Ladinsky, Nucl. Phys. B420, 565 (1994), hep-ph/9305309

    Pith/arXiv arXiv 1994

  36. [36]

    Collins,Foundations of Perturbative QCD, vol

    J. Collins,Foundations of Perturbative QCD, vol. 32 (Cambridge University Press, 2011), ISBN 978-1-009- 40184-5, 978-1-009-40183-8, 978-1-009-40182-1

    2011

  37. [37]

    Zhou and A

    J. Zhou and A. Metz, Phys. Rev. Lett.106, 172001 (2011), 1101.3273

    Pith/arXiv arXiv 2011

  38. [38]

    Boughezal, D

    R. Boughezal, D. de Florian, F. Petriello, and W. Vogel- sang, Phys. Rev. D107, 075028 (2023), 2301.02304

    Pith/arXiv arXiv 2023

  39. [39]

    Wang, X.-K

    H.-L. Wang, X.-K. Wen, H. Xing, and B. Yan, Phys. Rev. D109, 095025 (2024), 2401.08419

    Pith/arXiv arXiv 2024

  40. [40]

    X.-K. Wen, B. Yan, Z. Yu, and C. P. Yuan (2024), 2408.07255

    Pith/arXiv arXiv 2024

  41. [41]

    X.-K. Wen, B. Yan, Z. Yu, and C. P. Yuan, Phys. Rev. 6 D112, 053004 (2025), 2411.13845

    arXiv 2025

  42. [42]

    Yang, Y.-K

    L. Yang, Y.-K. Song, and S.-Y. Wei, Phys. Rev. D111, 054035 (2025), 2410.20917

    Pith/arXiv arXiv 2025

  43. [43]

    Cheng and B

    K. Cheng and B. Yan, Phys. Rev. Lett.135, 011902 (2025), 2501.03321

    Pith/arXiv arXiv 2025

  44. [44]

    Huang, X.-B

    Y. Huang, X.-B. Tong, and H.-L. Wang (2025), 2508.08516

    Pith/arXiv arXiv 2025

  45. [45]

    Q.-H. Cao, G. Li, X.-K. Wen, and B. Yan (2025), 2509.18276

    arXiv 2025

  46. [46]

    Qin, Y.-K

    X.-y. Qin, Y.-K. Song, and S.-y. Wei (2025), 2504.00739

    arXiv 2025

  47. [47]

    X.-K. Wen, B. Yan, and S. Zhang (2025), in preparation

    2025

  48. [48]

    Yu, Ph.D

    Z. Yu, Ph.D. thesis (2023), 2308.13080

    Pith/arXiv arXiv 2023

  49. [49]

    S. M. Berman, J. D. Bjorken, and J. B. Kogut, Phys. Rev. D4, 3388 (1971)

    1971

  50. [50]

    R. D. Field and R. P. Feynman, Nucl. Phys. B136, 1 (1978)

    1978

  51. [51]

    R. P. Feynman, R. D. Field, and G. C. Fox, Phys. Rev. D18, 3320 (1978)

    1978

  52. [52]

    Ai et al., Chin

    X. Ai et al., Chin. Phys. C49, 123108 (2025), 2505.24810

    arXiv 2025

  53. [53]

    Dong et al

    M. Dong et al. (CEPC Study Group) (2018), 1811.10545

    Pith/arXiv arXiv 2018

  54. [54]

    Abdallah et al

    W. Abdallah et al. (CEPC Study Group), Radiat. De- tect. Technol. Methods8, 1 (2024), 2312.14363

    arXiv 2024

  55. [56]

    Abada et al

    A. Abada et al. (FCC), Eur. Phys. J. ST228, 261 (2019)

    2019

  56. [57]

    Pitonyak, C

    D. Pitonyak, C. Cocuzza, A. Metz, A. Prokudin, and N. Sato, Phys. Rev. Lett.132, 011902 (2024), 2305.11995

    Pith/arXiv arXiv 2024

  57. [58]

    Cocuzza, A

    C. Cocuzza, A. Metz, D. Pitonyak, A. Prokudin, N. Sato, and R. Seidl (JAM), Phys. Rev. Lett.132, 091901 (2024), 2306.12998

    Pith/arXiv arXiv 2024

  58. [59]

    Cocuzza, A

    C. Cocuzza, A. Metz, D. Pitonyak, A. Prokudin, N. Sato, and R. Seidl (Jefferson Lab Angular Momentum (JAM)), Phys. Rev. D109, 034024 (2024), 2308.14857

    Pith/arXiv arXiv 2024

  59. [60]

    Courtoy, A

    A. Courtoy, A. Bacchetta, M. Radici, and A. Bianconi, Phys. Rev. D85, 114023 (2012), 1202.0323

    Pith/arXiv arXiv 2012

  60. [61]

    Radici, A

    M. Radici, A. Courtoy, A. Bacchetta, and M. Guagnelli, JHEP05, 123 (2015), 1503.03495

    Pith/arXiv arXiv 2015

  61. [62]

    Bertone, S

    V. Bertone, S. Carrazza, N. P. Hartland, E. R. Nocera, and J. Rojo (NNPDF), Eur. Phys. J. C77, 516 (2017), 1706.07049

    Pith/arXiv arXiv 2017

  62. [63]

    N. Sato, C. Andres, J. J. Ethier, and W. Melnitchouk (JAM), Phys. Rev. D101, 074020 (2020), 1905.03788

    Pith/arXiv arXiv 2020

  63. [64]

    Moffat, W

    E. Moffat, W. Melnitchouk, T. C. Rogers, and N. Sato (Jefferson Lab Angular Momentum (JAM)), Phys. Rev. D104, 016015 (2021), 2101.04664

    Pith/arXiv arXiv 2021

  64. [65]

    Cocuzza, W

    C. Cocuzza, W. Melnitchouk, A. Metz, and N. Sato (Jef- ferson Lab Angular Momentum (JAM)), Phys. Rev. D 106, L031502 (2022), 2202.03372

    Pith/arXiv arXiv 2022

  65. [66]

    N. Sato, J. J. Ethier, W. Melnitchouk, M. Hirai, S. Ku- mano, and A. Accardi, Phys. Rev. D94, 114004 (2016), 1609.00899

    Pith/arXiv arXiv 2016

  66. [67]

    J. Gao, C. Liu, X. Shen, H. Xing, and Y. Zhao, Phys. Rev. Lett.132, 261903 (2024), 2401.02781

    Pith/arXiv arXiv 2024

  67. [68]

    J. Gao, C. Liu, X. Shen, H. Xing, and Y. Zhao, Phys. Rev. D110, 114019 (2024), 2407.04422

    Pith/arXiv arXiv 2024

  68. [69]

    Mahaut, L

    V. Mahaut, L. Polano, A. Bacchetta, V. Bertone, M. Cerutti, M. Radici, and L. Rossi (2025), 2509.11855

    arXiv 2025

  69. [70]

    Zhou and J

    B. Zhou and J. Gao, JHEP02, 003 (2025), 2407.10059

    Pith/arXiv arXiv 2025