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REVIEW 5 major objections 5 minor 54 references

Probing Gluon Linear Polarization with Dihadron Fragmentation in $\chi_b$ Decays

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper argues that $\chi_{b0}$ decays give a first direct probe of the linearly polarized gluon dihadron fragmentation function, through an Artru–Collins-type angular correlation between the two pion pairs.

desk verdict A genuinely new observable for the linearly polarized gluon dihadron fragmentation function, with an honest but model-dependent benchmark and a kinematic-region inconsistency in the numerics that should be fixed before publication. read the letter →

arxiv 2608.10570 v1 pith:OALU45ZF submitted 2026-08-11 hep-ph hep-exnucl-exnucl-th

Zhi-Guo He , Guanghui Li , Yang Liu , Yu-Jie Tian , Xin-Kai Wen , Bin Yan This is my paper · ORCID
classification hep-phhep-exnucl-exnucl-th
keywords gluonlinearpolarizationdihadronfragmentationfunctionArtru-Collinsasymmetrychi_b0decayNRQCDfactorizationbottomoniumspectatormodelunpolarized
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

This paper proposes a way to measure, for the first time, the dihadron fragmentation function of a linearly polarized gluon. The idea is to use the two-gluon decay of the $P$-wave bottomonium state $\chi_{b0}$: at leading order the two gluons carry correlated linear polarizations, and when each fragments into a $\pi^+\pi^-$ pair, the two pairs inherit an angular correlation of the Artru–Collins type, specifically a $\cos(2\phi_1-2\phi_2)$ modulation. If the prediction holds, measuring that correlation directly accesses the linearly polarized gluon dihadron fragmentation function $H_1^{\angle,g}$, while the azimuthally averaged rate constrains the unpolarized gluon dihadron fragmentation function $D_1^g$. A spectator-model estimate puts the asymmetry at the percent level, within reach of existing lepton-collider data, with much better precision at a future high-luminosity run. This would open a new window on the spin-dependent hadronization of gluons, which has so far remained experimentally unconstrained.

What carries the argument

The load-bearing object is the linearly polarized gluon dihadron fragmentation function $H_1^{\angle,g}$, the gluon analogue of the quark interference dihadron fragmentation function that appears in Collins-type asymmetries. It enters the decay distribution through the product $H_1^{\angle,g}(z_1,M_1)H_1^{\angle,g}(z_2,M_2)$ multiplying $\cos(2\phi_1-2\phi_2)$, the Artru–Collins-type angular correlation between the two dihadron planes. The derivation combines NRQCD factorization, which separates the $\chi_{b0}$ annihilation into short-distance coefficients and long-distance matrix elements, with collinear factorization for the fragmentation of each gluon into a $\pi^+\pi^-$ pair; the hard two-gluon production from the color-singlet channel is what supplies the correlated linear polarizations. To make a numerical prediction, the gluon dihadron fragmentation functions are modeled with a scalar-spectator model for $g\to\pi^+\pi^-X$, with parameters fitted against the global estimate of $D_1^g$ and constrained by the $\rho$ and $\omega$ resonance structure in the pion-pair invariant mass.

What would settle it

A measurement of the $\cos(2\phi_1-2\phi_2)$ asymmetry in $\chi_{b0}\to\pi^+\pi^-\pi^+\pi^-X$ over the same $z$ and $M$ bins as Fig. 4, restricted to high-thrust events where the collinear condition $|P_i|\gg M_i$ holds; if the modulation is statistically consistent with zero where the spectator-model benchmark predicts percent-level values, while the azimuthally averaged rate matches the predicted $D_1^g$, the central claim is falsified. A second check is the sign and the $z_1$-dependence: the model predicts a specific growing trend with $z_1$, so a flat or opposite trend would rule out the proposed linear-polarization mechanism.

Watch

Extended reading notes

Core claim

At leading order in the NRQCD velocity expansion, the color-singlet component of the $P$-wave bottomonium state $\chi_{b0}$ annihilates into two gluons; because the $\chi_{b0}$ is a scalar, the two gluons emerge with correlated linear polarizations. In collinear factorization, the fragmentation of those gluons into two dihadron pairs gives a differential decay distribution with an azimuthally independent term proportional to $D_1^g(z_1,M_1)D_1^g(z_2,M_2)$ and a modulation proportional to $H_1^{\angle,g}(z_1,M_1)H_1^{\angle,g}(z_2,M_2)\,\cos(2\phi_1-2\phi_2)$. The paper shows that this modulation is the first direct experimental probe of the linearly polarized gluon dihadron fragmentation function $H_1^{\angle,g}$, and that the same decay rate provides direct access to the poorly constrained unpolarized gluon dihadron fragmentation function $D_1^g$. The color-octet subprocess, which produces a light quark-antiquark pair instead of two gluons, dilutes the asymmetry but cannot generate it, because the scalar decay does not correlate the quark and antiquark transverse spins. Using a spectator model for $g\to\pi^+\pi^-X$ tuned to the current global estimate of $D_1^g$, the asymmetry is predicted at the percent level, suggesting existing data could already be sensitive to this observable.

Load-bearing premise

The prediction stands on the assumption that the collinear factorization formula, which requires each pion-pair momentum to be much larger than the pair's invariant mass, remains valid over the full kinematic ranges integrated in Fig. 4, including regions near $z_i = 2M_i/m_\chi$ where the pair momentum can vanish.

Editorial extensions

If this is right

  • A measurement of the $\cos(2\phi_1-2\phi_2)$ asymmetry in $\chi_{b0}\to\pi^+\pi^-\pi^+\pi^-X$ would give the first direct experimental determination of the linearly polarized gluon dihadron fragmentation function.
  • The azimuthally averaged semi-inclusive decay rate directly constrains the unpolarized gluon dihadron fragmentation function $D_1^g$, which current global analyses constrain only indirectly or not at all.
  • At the benchmark estimate the asymmetry is at the percent level, so existing data collected at the $\Upsilon(2S)$ resonance may already have statistical sensitivity to the signal.
  • A dedicated high-luminosity data set at the same energy would substantially improve the statistical precision, allowing the kinematic dependence of both gluon dihadron fragmentation functions to be mapped in $z$ and invariant mass.
  • The same framework extends to $\eta_b$ decays, where the leading-order color-octet quark channel is absent, providing a cleaner probe of gluon fragmentation once enough data become available.

Reading between the lines

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

  • If the asymmetry is confirmed, the same observable could be used to test the universality of $H_1^{\angle,g}$ by comparing with future measurements in other hard processes, since fragmentation functions are universal by the QCD factorization theorems.
  • The extraction of the LDME ratio $\rho_8$ from different input data feeds into the denominator of the asymmetry; comparing the $z_1$-dependence of $A_{12}$ at large $z_1$, where quark dilution is strongest, could help resolve the current discrepancy between different determinations of $\rho_8$.
  • A null result would not by itself rule out gluon linear polarization, since the spectator-model normalization is a benchmark rather than a rigorous prediction; the decisive test is whether the $\cos(2\phi_1-2\phi_2)$ modulation appears with the predicted sign and kinematic shape, not just its integrated size.
  • The thrust-axis reconstruction is an experimental proxy for the partonic $gg$ axis; a dedicated comparison of the measured modulation at different thrust values would test the factorization assumption underlying the prediction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper proposes a new observable for accessing the linearly polarized gluon dihadron fragmentation function H_1^{\sphericalangle,g}. Specifically, it argues that in the color-singlet leading-order decay \chi_{b0}\to gg, the two gluons carry correlated linear polarizations, and that collinear fragmentation of each gluon into a \pi^+\pi^- pair generates an Artru--Collins-type angular correlation proportional to \cos(2\phi_1-2\phi_2). The corresponding azimuthally averaged rate constrains the unpolarized gluon DiFF D_1^g. A scalar-spectator model fit to the JAM D_1^g is used to estimate the size of the asymmetry, yielding percent-level values that the authors argue could be accessible with existing Belle data and more precisely with Belle II.

Significance. If the factorization in Eq. (1) is valid, this is a genuinely new and interesting proposal: it would be the first direct experimental access to the linearly polarized gluon dihadron fragmentation function, and it simultaneously provides a new constraint on the poorly known unpolarized gluon DiFF. The paper is honest about the model dependence of its numerical benchmark and about the weak constraints on the JAM gluon DiFF, and the observable is in principle falsifiable at B factories. The central idea deserves serious consideration. However, the numerical results and the experimental-reach claims rest on several load-bearing assumptions that are either not derived or not checked against the stated kinematics, so the present manuscript does not yet fully establish its quantitative conclusions.

major comments (5)
  1. [§4, Fig. 4 and Eq. (1)] The kinematic domain used for the numerical integration violates the stated validity condition of Eq. (1). The factorization is introduced with the requirement |P_i| \gg M_i, but the Fig. 4 integrations run over z_i\in[0.19,0.99] and M_i\in[0.28,2.05] GeV subject only to the threshold constraint z_i\ge 2M_i/m_{\chi}. At the boundary |P_i|=0, and for a substantial part of the allowed region |P_i|<M_i (for example, for M_i=2 GeV, |P_i|<M_i for z_i\lesssim 0.57). In this region the pion pair is not collinear with the fragmenting gluon, so the factored product of two independent DiFFs and the \cos(2\phi_1-2\phi_2) correlation are not defined. The numerical values of A_{12} and the Belle/Belle II statistical projections in Fig. 4 are therefore not predictions of the stated factorization. The authors should impose an explicit collinearity cut, for example |P_i| \ge \kappa M_i with \kappa\gg 1, and demonstrate that the asymmetry remains after restricting to the region where Eq. (1) is valid.
  2. [§3, Eq. (12)] The conversion from the unintegrated DiFFs to the (z,M_h)-dependent functions is incomplete. In Eq. (12), the right-hand sides still depend on |R_T|, but the relation between |R_T| and the integration variables (\xi, M_h) is never stated. Without this relation and the corresponding Jacobian, the definitions of D_1^g(z,M_h) and H_1^{\sphericalangle,g}(z,M_h) are mathematically ill-defined, and the spectator-model results in Figs. 3 and 4 cannot be reproduced. The authors should give the explicit mapping, e.g. M_h^2=4(m_\pi^2+R_T^2)/(1-\xi^2) if that is the intended relation, and the resulting measure.
  3. [§4, Eq. (14) and Fig. 4] The statistical sensitivity estimates are not reproducible because the event number N is never evaluated. The text defines N as the number of selected events after kinematic cuts, but no estimate is provided for the expected \chi_{b0} yield at Belle with \mathcal{L}=24.7\,\text{fb}^{-1}, nor are the relevant branching ratios (\Upsilon(2S)\to\gamma\chi_{b0}, \chi_{b0}\to 2\pi^+2\pi^-+X) or selection efficiencies given. Without these inputs, the plotted statistical bands and the claim that existing Belle data may already be sensitive to the asymmetry are unsupported. The authors should provide a concrete event-count estimate and state the cuts used.
  4. [§2, Eq. (1)] The factorization formula itself is asserted rather than derived. The text states the result with a citation to collinear factorization, but does not show how the NRQCD decay amplitude for \chi_{b0}\to gg is combined with the gluon fragmentation functions, how the thrust axis replaces the partonic axis in the presence of hadron transverse momenta, or how the hard coefficients C^{q,g} in Eq. (2) are obtained. Since Eq. (1) is the foundation of the proposed observable, a more explicit derivation or a reference in which the same double-factorization is established should be provided.
  5. [§4, Fig. 4 and Eq. (3)] The numerical benchmark is model-dependent in a way that is not reflected in the quoted uncertainty bands. The 10-parameter spectator-model fit is anchored to the JAM D_1^g, which the paper itself describes as weakly constrained and obtained under indirect assumptions, and no theory uncertainty from the fit is propagated to A_{12}. The text correctly says the best-fit values are used only as a benchmark, but the abstract and conclusion state 'percent-level asymmetries, potentially within reach of existing Belle data' without the accompanying caveat. The authors should either present a range of benchmark values reflecting the fit uncertainties and model sensitivity, or explicitly reframe the reach claim as conditional on the spectator model.
minor comments (5)
  1. [§3, Eq. (4)] Equation (4) contains a garbled factor '(k -)^2' that appears to be a typo; it should presumably be (k-P_h)^2 or an explicit k^2 factor. Please correct the notation.
  2. [§2, text after Eq. (2)] The statement that the scalar \chi_{b0} decay 'does not induce a transverse-spin correlation between the produced quark and antiquark' is too strong. A J=0 decay can produce a spin-singlet q\bar q correlation, and the color-octet ^3S_1^{[8]} channel is a spin-triplet. The relevant point is that the quark-channel spin correlations do not project onto the particular cos(2\phi_1-2\phi_2) moment, because quark interference DiFFs enter at first order in the azimuthal angle. Please clarify the argument.
  3. [§3, Eq. (13)] The fitted spectator-model parameters are quoted without uncertainties or a fit-quality indicator. Given that the fit is used as the numerical basis for the benchmark, a \chi^2/dof and parameter covariance would be needed for reproducibility.
  4. [Throughout] The notation H_b^1 for the NRQCD matrix element is nonstandard and can be confused with a fragmentation function; consider using the conventional \langle O(^3P_0^{[1]})\rangle notation.
  5. [Introduction and Conclusion] The qualitative discussion of \eta_b and \chi_{b2} is useful, but the title and abstract promise a treatment of '\chi_b decays' in general; the actual quantitative analysis is only for \chi_{b0}. It would help to state explicitly that \eta_b and \chi_{b2} are left for future work.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the Artru–Collins-type correlation follows from NRQCD and collinear factorization with an external spectator-model benchmark; only a minor non-load-bearing self-citation appears.

full rationale

The derivation chain is self-contained against its stated inputs. Eq. (1) is obtained from NRQCD short-distance coefficients (C_g, C_q), LDMEs, and the operator-defined DiFFs; the cos(2 phi_1 - 2 phi_2) term is generated by the tensor-projected linearly polarized gluon DiFF H_1^{angular,g} squared, with no step in which H_1^{angular,g} is defined in terms of the asymmetry or fitted to it. The numerical benchmark uses a scalar-spectator model whose parameters are fitted to the JAM unpolarized gluon DiFF D_1^g and then used to generate H_1^{angular,g}; this is model/calibration dependence, explicitly labeled as a benchmark, not a statistical reduction of the target observable. The only self-citation ([35], the authors' previous chi_b2 paper) is used to note an independent determination of the LDME ratio rho_8, but the two rho_8 values entering the numerics come from CLEO [47] and lattice NRQCD [49], so the self-citation is not load-bearing. A separate domain concern -- Eq. (1) is stated for |P_i| >> M_i while the Fig. 4 integrations extend to z_i = 2M_i/m_chi where |P_i| = 0 -- is a correctness/validity issue, not circularity. Overall score 2 reflects only the minor, non-load-bearing self-citation.

Assumptions & free parameters 10 free parameters · 6 assumptions · 1 invented entities

The QCD factorization structure is standard, but the quantitative predictions depend on a 10-parameter spectator model fitted to a weakly constrained external DiFF extraction, on a disputed LDME ratio, and on the unverified applicability of collinear factorization over the full integration range.

free parameters (10)
  • f_s = 2.27e3 GeV^-5
    s-wave form-factor strength; fitted to the JAM D1^g benchmark in the relevant kinematic region.
  • f_rho1 = 8.02e-1 GeV^-3
    rho-resonance p-wave coupling; fitted to the JAM D1^g benchmark.
  • f_omega1 = -9.17e-3 GeV^-3
    omega-resonance p-wave coupling; fitted to the JAM D1^g benchmark.
  • f'_omega1 = -5.97 GeV^-3
    omega three-pion channel coupling; fitted to the JAM D1^g benchmark.
  • f_rho2 = -1.74e1 GeV^-3
    Second rho-resonance Lorentz structure coupling; fitted to the JAM D1^g benchmark.
  • f_omega2 = 1.49 GeV^-3
    Second omega-resonance Lorentz structure coupling; fitted to the JAM D1^g benchmark.
  • f'_omega2 = 1.01e1 GeV^-3
    Second omega three-pion channel coupling; fitted to the JAM D1^g benchmark.
  • Lambda_s = 1.83 GeV
    Suppression scale for the s-wave vertex; fitted to the JAM D1^g benchmark.
  • Lambda_p = 3.68 GeV
    Suppression scale for the p-wave vertices; fitted to the JAM D1^g benchmark.
  • M_s = 2.88 GeV
    Mass of the effective scalar spectator; fitted to the JAM D1^g benchmark.
assumptions (6)
  • domain assumption Collinear factorization applies with |P_i| >> M_i, and the thrust axis approximates the partonic gg axis.
    Required for Eq. (1); the paper does not check whether the integrated phase space satisfies this condition.
  • domain assumption NRQCD factorization separates chi_b0 decay into hard coefficients and LDMEs H_b^1 and H_b^8.
    Standard heavy-quarkonium factorization, invoked in the section 'Gluon Linear Polarization from chi_b0 Decay'.
  • domain assumption At LO the only relevant channels are CS bbar(3P0[1]) -> gg and CO bbar(3S1[8]) -> q qbar.
    Used to write Eq. (1) without other NRQCD channels or higher orders.
  • ad hoc to paper The scalar-spectator model with effective s-wave and p-wave vertices approximates the nonperturbative gluon fragmentation correlator.
    This is the model behind Eq. (4) through Eq. (13); it is not derived from QCD and all its parameters are fitted.
  • domain assumption The JAM extractions of D1^q and D1^g are reliable enough to serve as inputs for the benchmark.
    The paper itself notes that JAM D1^g is weakly constrained and indirectly inferred; the quantitative prediction inherits that uncertainty.
  • domain assumption The LDME ratio rho8 is taken from lattice NRQCD (0.044) or CLEO (0.160).
    The two external determinations disagree; the paper explores the ratio R to quantify the sensitivity, but the predicted A12 depends on this external input.
invented entities (1)
  • Effective scalar spectator state with mass M_s
    purpose: Models the unobserved remnant in g -> pi+pi- X to compute the fragmentation correlator
    This is a modeling device, not a physical particle; M_s is a fitted parameter in Eq. (13), and there is no independent experimental handle.

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Pith. "Pith review of Probing Gluon Linear Polarization with Dihadron Fragmentation in $\chi_b$ Decays." pith.science (2026). https://pith.science/paper/OALU45ZF

@misc{pith2026260810570,
  author       = {Pith},
  title        = {Pith review of: Probing Gluon Linear Polarization with Dihadron Fragmentation in $\chi_b$ Decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OALU45ZF}},
  note         = {Machine review of arXiv:2608.10570}
}
abstract

The dihadron fragmentation function (DiFF) of a linearly polarized gluon has not yet been accessed experimentally, leaving an important aspect of spin-dependent gluon hadronization unexplored. We show that, at leading order, the color-singlet decay channel of the $P$-wave bottomonium state $\chi_{b0}$ produces two energetic gluons with correlated linear polarizations. Within collinear factorization, their fragmentation into separate dihadron pairs generates an Artru--Collins-type angular correlation that provides the first direct probe of the linearly polarized gluon DiFF, while the corresponding semi-inclusive decay rate constrains the unpolarized gluon DiFF. A spectator-model benchmark indicates percent-level asymmetries, potentially within reach of existing Belle data. A dedicated Belle~II data set would substantially improve the statistical precision, enabling more stringent constraints on the kinematic dependence of the linearly polarized gluon DiFF.

Figures

Figures reproduced from arXiv: 2608.10570 by the authors.

Figure 1
Figure 1. FIG. 1. Leading-order kinematic configuration for the pro [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic diagram of the gluon fragmentation corre [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Scalar-spectator model predictions for the gluon [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The Artru–Collins-type asymmetry [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 13 canonical work pages

  1. [1]

    We next assess the sensitivity of the predicted asymme- try to the poorly known LDME ratioρ 8

    Assuming an integrated luminosity ofL= 1 ab −1 at Belle II (orange), the statistical precision would be substantially improved, allowing detailed studies of the kinematic dependence of both gluon DiFFs. We next assess the sensitivity of the predicted asymme- try to the poorly known LDME ratioρ 8. The purple lines in Fig. 4 show the ratioR=A 12(ρ8 = 0.160)...

    work page

  2. [2]

    The framework can also be extended toη b decays and to other polarization-sensitive observables, including TMD spin correlations and en- ergy correlators

    Using a spectator-model benchmark together with global fits of the quark DiFFs, we obtain percent-level asymmetries, indicating that existing Belle data may already be sensi- tive to this observable, with substantially improved pre- cision expected at Belle II. The framework can also be extended toη b decays and to other polarization-sensitive observables...

    work page

  3. [3]

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

    work page 1971

  4. [4]

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

    work page 1978

  5. [5]

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

    work page 1978

  6. [6]

    Metz and A

    A. Metz and A. Vossen, Prog. Part. Nucl. Phys.91, 136 (2016), 1607.02521

    arXiv 2016

  7. [7]

    J. C. Collins, Nucl. Phys. B396, 161 (1993), hep- ph/9208213

    arXiv 1993

  8. [8]

    Z.-B. Kang, A. Prokudin, P. Sun, and F. Yuan, Phys. Rev. D93, 014009 (2016), 1505.05589

    arXiv 2016

Show all 54 references
  1. [9]

    C. Zeng, H. Dong, T. Liu, P. Sun, and Y. Zhao, Phys. Rev. D109, 056002 (2024), 2310.15532

    2024 arXiv

  2. [10]

    Boussarie et al

    R. Boussarie et al. (2023), 2304.03302

    2023 arXiv

  3. [11]

    C. Zeng, H. Dong, T. Liue, P. Sun, and Y. Zhao (2024), 2412.18324

    2024 arXiv

  4. [12]

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

    1994 arXiv

  5. [13]

    R. L. Jaffe, X.-m. Jin, and J. Tang, Phys. Rev. Lett.80, 1166 (1998), hep-ph/9709322

    1998 arXiv

  6. [14]

    R. L. Jaffe, X.-m. Jin, and J.-a. Tang, Phys. Rev. D57, 5920 (1998), hep-ph/9710561

    1998 arXiv

  7. [15]

    Bianconi, S

    A. Bianconi, S. Boffi, R. Jakob, and M. Radici, Phys. Rev. D62, 034008 (2000), hep-ph/9907475

    2000 arXiv

  8. [16]

    Barone, A

    V. Barone, A. Drago, and P. G. Ratcliffe, Phys. Rept. 359, 1 (2002), hep-ph/0104283

    2002 arXiv

  9. [17]

    Bacchetta, A

    A. Bacchetta, A. Courtoy, and M. Radici, Phys. Rev. Lett.107, 012001 (2011), 1104.3855

    2011 arXiv

  10. [18]

    Courtoy, A

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

    2012 arXiv

  11. [19]

    Pitonyak, C

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

    2024 arXiv

  12. [20]

    Cocuzza, A

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

    2024 arXiv

  13. [21]

    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

    2024 arXiv

  14. [22]

    T. C. Rogers, M. Radici, A. Courtoy, and T. Rainaldi, Phys. Rev. D111, 056001 (2025), 2412.12282

    2025 arXiv

  15. [23]

    Mahaut, L

    V. Mahaut, L. Polano, A. Bacchetta, V. Bertone, M. Cerutti, M. Radici, and L. Rossi (MAP (Multi- dimensional Analyses of Partonic distributions)), JHEP 02, 051 (2026), 2509.11855

    2026

  16. [24]

    Artru and J

    X. Artru and J. C. Collins, Z. Phys. C69, 277 (1996), hep-ph/9504220

    1996 arXiv

  17. [25]

    D. Boer, R. Jakob, and P. J. Mulders, Nucl. Phys. B504, 345 (1997), hep-ph/9702281

    1997 arXiv

  18. [26]

    Bacchetta and M

    A. Bacchetta and M. Radici, Phys. Rev. D69, 074026 (2004), hep-ph/0311173

    2004 arXiv

  19. [27]

    D. Boer, R. Jakob, and M. Radici, Phys. Rev. D 67, 094003 (2003), [Erratum: Phys.Rev.D 98, 039902 (2018)], hep-ph/0302232

    2003 arXiv

  20. [28]

    Bacchetta, F

    A. Bacchetta, F. A. Ceccopieri, A. Mukherjee, and M. Radici, Phys. Rev. D79, 034029 (2009), 0812.0611

    2009 arXiv

  21. [29]

    Zhou and A

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

    2011 arXiv

  22. [30]

    Pitonyak, M

    D. Pitonyak, M. Schlegel, and A. Metz, Phys. Rev. D89, 054032 (2014), 1310.6240

    2014 arXiv

  23. [31]

    X.-K. Wen, B. Yan, Z. Yu, and C. P. Yuan, Sci. China Phys. Mech. Astron.69, 271011 (2026), 2408.07255

    2026 arXiv

  24. [32]

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

    2025

  25. [33]

    Cheng and B

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

    2025 arXiv

  26. [34]

    Q.-H. Cao, G. Li, X.-K. Wen, and B. Yan, Rept. Prog. Phys.89, 080501 (2026), 2509.18276

    2026 arXiv

  27. [35]

    Cao, X.-K

    Q.-H. Cao, X.-K. Wen, B. Yan, and S.-T. Zhang, Phys. Rev. Lett.136, 231901 (2026), 2512.16492

    2026 arXiv

  28. [36]

    Z.-B. Kang, A. Metz, D. Pitonyak, and C. Zhang, Phys. Rev. Lett.136, 081905 (2026), 2507.17444

    2026

  29. [37]

    Z.-G. He, G. Li, Y.-J. Tian, X.-K. Wen, and B. Yan (2026), 2603.18874

    2026

  30. [38]

    Hatta, B.-W

    Y. Hatta, B.-W. Xiao, F. Yuan, and J. Zhou, Phys. Rev. Lett.126, 142001 (2021), 2010.10774

    2021 arXiv

  31. [39]

    Hatta, B.-W

    Y. Hatta, B.-W. Xiao, F. Yuan, and J. Zhou, Phys. Rev. D104, 054037 (2021), 2106.05307

    2021 arXiv

  32. [40]

    Song, S.-Y

    Y.-K. Song, S.-Y. Wei, L. Yang, and J. Zhou, Phys. Rev. Lett.136, 131901 (2026), 2509.14960

    2026

  33. [41]

    Brambilla et al

    N. Brambilla et al. (Quarkonium Working Group) (2004), hep-ph/0412158

    2004 arXiv

  34. [42]

    G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D51, 1125 (1995), [Erratum: Phys.Rev.D 55, 5853 (1997)], hep-ph/9407339

    1995 arXiv

  35. [43]

    Vossen et al

    A. Vossen et al. (Belle), Phys. Rev. Lett.107, 072004 (2011), 1104.2425

    2011 arXiv

  36. [44]

    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. [45]

    Bianconi, S

    A. Bianconi, S. Boffi, R. Jakob, and M. Radici, Phys. Rev. D62, 034009 (2000), hep-ph/9907488

    2000 arXiv

  38. [46]

    Radici, A

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

    2015 arXiv

  39. [47]

    Pitonyak, C

    D. Pitonyak, C. Cocuzza, A. Metz, A. Prokudin, and N. Sato, Phys. Rev. D113, 038901 (2026), 2502.15817

    2026 arXiv

  40. [48]

    T. C. Rogers, T. Rainaldi, M. Radici, and A. Courtoy, Phys. Rev. D113, 038902 (2026), 2602.09211

    2026

  41. [49]

    R. A. Briere et al. (CLEO), Phys. Rev. D78, 092007 (2008), 0807.3757

    2008 arXiv

  42. [50]

    G. T. Bodwin, D. K. Sinclair, and S. Kim, Phys. Rev. D 65, 054504 (2002), hep-lat/0107011

    2002 arXiv

  43. [51]

    G. T. Bodwin, E. Braaten, D. Kang, and J. Lee, Phys. Rev. D76, 054001 (2007), 0704.2599

    2007 arXiv

  44. [52]

    Jakob, P

    R. Jakob, P. J. Mulders, and J. Rodrigues, Nucl. Phys. A626, 937 (1997), hep-ph/9704335

    1997 arXiv

  45. [53]

    Bacchetta and M

    A. Bacchetta and M. Radici, Phys. Rev. D74, 114007 (2006), hep-ph/0608037

    2006 arXiv

  46. [54]

    Xie and Z

    X. Xie and Z. Lu, Phys. Lett. B842, 137973 (2023), 2210.16532

    2023 arXiv

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Reviewed August 12, 2026 · model on record in the stance chip above.