Pith. sign in

REVIEW 4 major objections 5 minor 69 references

Transverse single-spin asymmetries in $\gamma$SIDIS as a direct probe of quark-gluon-quark longitudinal momentum structure

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Transverse single-spin asymmetries in photon-tagged deep-inelastic scattering can map the proton's quark-gluon-quark correlators point-by-point across their full momentum-fraction support, with the paper's numerical estimates reaching 10…

desk verdict A careful, transparent first numerical look at a new twist-3 observable, worth refereeing, but the 10% headline panels sit at Q² values where the paper's own factorization condition is not comfortably met. read the letter →

arxiv 2505.02711 v2 pith:767VGLKI submitted 2025-05-05 hep-ph hep-ex

Michael Harris , Jacob Marsh , Daniel Pitonyak , Alexei Prokudin , Jack Putnam , Daniel Rein , Marc Schlegel This is my paper · ORCID
classification hep-phhep-ex
keywords transversesingle-spinasymmetriesquark-gluon-quarkcorrelatorstwist-3factorizationsemi-inclusivedeep-inelasticscatteringisolatedphotonproductionElectron-IonColliderSiversfunctiond2matrixelement
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 argues that a transverse single-spin asymmetry measured in the semi-inclusive deep-inelastic production of isolated photons (γSIDIS) would give the first direct, point-by-point view of the proton's quark-gluon-quark correlations, the twist-3 functions $F_{FT}(x,x')$ and $G_{FT}(x,x')$. Earlier observables only saw these functions under integrals over $x$ or $x'$, or along the diagonal $x=x'$. The authors rewrite the analytic cross section in a compact form in which the polarized numerator depends on $F_{FT}$ and $G_{FT}$ evaluated at the distinct momentum fractions $(x_B,\tilde x_B)$ and $(x_B,0)$. Using models anchored to the Sivers function and to a lattice-QCD value for the $d_2$ matrix element, they scan the Electron-Ion Collider phase space and find the asymmetry can reach 10% or more at $\sqrt{s}=29$ GeV. If the measurement works, it would supply information about multi-parton correlations in the nucleon that no earlier observable could provide.

What carries the argument

The object that carries the argument is the pair of twist-3 quark-gluon-quark correlators $F_{FT}(x,x')$ and $G_{FT}(x,x')$, which are light-cone matrix elements of a quark-antiquark pair joined by a gluon field strength, encoding the longitudinal momentum sharing among two quarks and a gluon in the polarized proton. The essential mechanism is the kinematic structure of Eq. (6): the polarized cross section is built from the linear combinations $F_\pm = F_{FT} \pm G_{FT}$ evaluated at exactly two points, the off-diagonal hard-pole point $(x_B,\tilde x_B)$ and the soft-fermion-pole point $(x_B,0)$, rather than integrated over a momentum fraction. A second piece of machinery is the model that makes numerics possible: a Fourier expansion of $F_{FT}$ and $G_{FT}$ in polar coordinates adjusted to the support region, normalized by the first transverse moment of the Sivers function, with one Fourier coefficient fixed by the lattice-QCD value of the $d_2$ matrix element.

What would settle it

Measure $A_{UT}^{\gamma\mathrm{SIDIS}}$ at the Electron-Ion Collider at $\sqrt{s}=29$ GeV in the phase-space region identified as largest (mid or backward electron rapidity, forward photon rapidity, electron $p_T$ below about 3 GeV, photon $p_T$ above about 3 GeV, and $\phi'=\phi_\gamma=0$); if the asymmetry is consistent with zero at few-percent precision across that region, the claim that this observable is a practical pointwise probe of $F_{FT}$ and $G_{FT}$ would be falsified for these models. A second check is to test whether the asymmetry in that region follows the $Q^2$ dependence implied by factorization; strong violations at low $Q^2$ would indicate power corrections dominate.

Watch

Extended reading notes

Core claim

The central claim is that $A_{UT}^{\gamma\mathrm{SIDIS}}$ is a direct probe of the dynamical twist-3 quark-gluon-quark correlators $F_{FT}$ and $G_{FT}$ in their full two-dimensional support. The numerator of the asymmetry reduces, after combining the two correlators into $F_\pm = F_{FT} \pm G_{FT}$ and compressing four azimuthal spin structures into two, to a sum over Compton and interference channels of hard coefficients times $F_\pm(x_B,\tilde x_B)$ plus soft-fermion-pole coefficients times $F_\pm(x_B,0)$. The hard-pole term samples the off-diagonal point $(x_B,\tilde x_B)$ and the soft-fermion-pole term samples $(x_B,0)$; the Bethe-Heitler and soft-gluon-pole contributions cancel. With a model built from the first transverse moment of the Sivers function and a lattice constraint for $d_2$, the authors find $|A_{UT}|$ around 3–5% in a minimal scenario and 10% or more in a fuller scenario, concentrated at low electron $p_T$, high photon $p_T$, mid or backward electron rapidity, mid or forward photon rapidity, and aligned azimuthal angles at $\sqrt{s}=29$ GeV. They conclude the asymmetry will likely be measurable at the EIC and would provide unprecedented information on $F_{FT}$ and $G_{FT}$ across their full support.

Load-bearing premise

The load-bearing premise is that twist-3 collinear factorization applies at the kinematics scanned, where the cuts only impose $Q^2 > 1$ GeV$^2$ and $\tilde Q^2 > 1$ GeV$^2$ with $M^2 \approx 0.88$ GeV$^2$; the large-asymmetry region, with low electron transverse momentum and mid or backward electron rapidity, can sit at $Q^2$ of only a few GeV$^2$ where higher-twist corrections may be sizable.

Editorial extensions

If this is right

  • If $A_{UT}^{\gamma\mathrm{SIDIS}}$ is measured, $F_{FT}(x,x')$ and $G_{FT}(x,x')$ can be extracted point-by-point over their entire support, replacing earlier observables that only sense integrals or the diagonal $x=x'$.
  • In most of the phase space where the asymmetry is large, the soft-fermion-pole terms $F_\pm(x_B,0)$ are comparable to or larger than the hard-pole terms, so the same measurement also constrains the previously unmeasured functions $F_{FT}(x,0)$ and $G_{FT}(x,0)$.
  • At $\sqrt{s}=29$ GeV with the electron at mid or backward rapidity, the photon at mid to forward rapidity, small electron transverse momentum, large photon transverse momentum, and azimuthal angles aligned, $|A_{UT}|$ is predicted to be 10% or more under the full model.
  • Raising the center-of-mass energy suppresses the asymmetry strongly: at $\sqrt{s}=63$ GeV most of the phase space drops to near zero and at $\sqrt{s}=141$ GeV only extreme forward kinematics survive.
  • Comparing electron and positron beams produces a charge asymmetry that isolates the interference channel, giving access to valence-type $q-\bar q$ combinations of the correlators.

Reading between the lines

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

  • Because the numerator is a sum of hard-pole and soft-fermion-pole terms, a single asymmetry measurement cannot separate $F_\pm(x_B,\tilde x_B)$ from $F_\pm(x_B,0)$; combining electron and positron beam measurements may help disentangle the two kinematic slices by weighting charge combinations differently.
  • If the predicted 10% asymmetries are confirmed, the same experiment would provide an indirect check of the relation between $F_{FT}(x,x)$ and the Sivers first moment, and of the lattice $d_2$ constraint, at momentum fractions not accessible before.
  • The strong energy dependence suggests that EIC running at the lowest collision energy gives the best discovery window; a dedicated low-energy run may be worth more than high-luminosity high-energy running for this observable.
  • A null result in the predicted high-asymmetry region would be informative either way: it would either rule out the Sivers-normalized model of the correlators or signal that twist-3 collinear factorization needs higher-twist corrections at $Q^2$ of a few GeV$^2$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. Summary: This paper proposes the transverse single-spin asymmetry A_UT^{\gamma SIDIS} in semi-inclusive deep-inelastic production of isolated photons as an observable that accesses the quark-gluon-quark correlators F_FT(x,x') and G_FT(x,x') point-by-point over their full support. Building on the analytic cross section of Ref. [34], the authors rewrite the polarized cross section in a compact form using two azimuthal spin structures and the combinations F_n\pm = F_FT \pm G_FT (Eqs. (5)-(6)), and tabulate all 16 hard-scattering coefficients in Appendix A. They construct models for F_FT and G_FT from the JAM3D-22 Sivers first moment f_1T^{\perp(1)} and the lattice d2 matrix element, using a Fourier expansion in polar coordinates (Eqs. (12)-(14)) with two parameter scenarios. Numerical results at \sqrt{s} = 29, 63, and 141 GeV are presented as heat maps, with the finding that |A_UT| can reach 10% or more for \sqrt{s} = 29 GeV at mid-to-backward electron rapidity, mid-to-forward photon rapidity, low electron p_T, high photon p_T, and aligned azimuthal angles. The paper also discusses the relative importance of hard-pole and soft-fermion-pole contributions and the possibility of a beam-charge asymmetry to separate the Compton and Interference channels.

Significance. The conceptual proposal is attractive and timely: a new EIC observable that encodes pointwise information on quark-gluon-quark correlations would be genuinely unprecedented, and the compressed analytic form plus the full Appendix A listing of hard coefficients is a service to the community. The accompanying Colab notebook is a constructive resource that allows independent exploration. These are real strengths. However, the status of the numerical estimates is limited by three coupled issues: the 10% result is produced in kinematic regions where the paper's own factorization conditions are not satisfied; the Scenario 1 Fourier coefficients are essentially unconstrained and no sensitivity range is given; and the conclusion that the asymmetry 'will be measurable' is not backed by a statistical significance estimate. If the first issue is resolved by stricter cuts and the model uncertainty is quantified, the paper would provide a useful and falsifiable benchmark. As it stands, the central claim is defensible in outline but requires revision.

major comments (4)
  1. [§2 and §4] Equation (3) is stated to be valid when Q^2 >> M^2, \tilde{Q}^2 >> M^2, and Q^2 - \tilde{Q}^2 >> M^2, while Sec. 4 retains points with only Q^2 > 1 GeV^2, \tilde{Q}^2 > 1 GeV^2, and Q^2 - \tilde{Q}^2 > 1 GeV^2 with M^2 \approx 0.88 GeV^2. The high-asymmetry regions of Figs. 3 and 4 (\eta' \approx 0 or -1, p'_T \lesssim 3 GeV, \eta_\gamma \gtrsim 0, p_{\gamma T} \gtrsim 3 GeV at \sqrt{s} = 29 GeV) can have Q^2 of only a few GeV^2, so M^2/Q^2 is of order 0.2-0.4 and power corrections absent from Eqs. (3)-(6) can change a nominal 10% asymmetry by an order-one factor. Please demonstrate that the 10% result survives under cuts that actually enforce the stated hierarchy (e.g., Q^2, \tilde{Q}^2, |Q^2 - \tilde{Q}^2| > 4 GeV^2 or > 4 M^2), or otherwise quantify the size of target-mass and higher-twist corrections in the published plots; without this, the headline claim is not established in the regime where the calculation is controlled.
  2. [§3, Scenario 1 (Eqs. (12)-(15))] The '10% or larger' numerical result is driven by Scenario 1, whose coefficients a^q_3...a^q_7 and b^q_1...b^q_6 are chosen by hand as 'arbitrary values between -1 and 1' (Sec. 3), with only a^q_2 fixed by the lattice d2 constraint through Eq. (17). Because F_FT and G_FT are otherwise unconstrained, the quoted 10% is an output of a single ad-hoc parameter choice, not a bound or a scan; no uncertainties from JAM3D-22 or the lattice d2 values are propagated. Please add a sensitivity study over the coefficient space (the Colab notebook could serve this purpose) reporting the range of |A_UT| in the highlighted kinematic region, or explicitly downgrade the 10% statement to 'an illustrative model scenario' in the abstract and conclusions.
  3. [§4 and §5] The abstract and Sec. 5 state that A_UT^{\gamma SIDIS} 'will be measurable at the EIC', but the paper provides no estimate of the expected statistical uncertainty: no integrated luminosity, no event rates, no acceptance or binning efficiency are used to convert the 10% asymmetry into a significance. Since the central conclusion is measurability, a rough projection (even order-of-magnitude, using typical EIC luminosities and the cross sections in Eqs. (3) and (5)) is needed to support the claim; otherwise the conclusion should be softened to 'potentially observable'.
  4. [§3, model evolution (paragraph after Eq. (14))] The model evolution is 'inherited' from the Sivers-function DGLAP evolution, and the full twist-3 evolution including mixing with trigluon correlators (available via the code of Ref. [61]) is not used. The argument that such evolution effects cancel in asymmetries is standard for TMD ratio observables, but it is not automatic here because the numerator is a twist-3 collinear cross section involving F_FT and G_FT at several (x,x') pairs, while the denominator uses f_1 at scales Q, \tilde{Q}, and \sqrt{Q\tilde{Q}}. Please quantify the impact of the full twist-3 evolution, or state explicitly why it is negligible at the scales of Figs. 3-6; otherwise the kinematic pattern of the 10% regions carries an unquantified dependence on this approximation.
minor comments (5)
  1. [§2, Eq. (5)] The symbols '\epsilon Pll^\prime S' and '\epsilon PlP_\gamma S' are used in Eq. (5) but defined only later in Eq. (A.3); please define them at first use in Sec. 2.
  2. [§4, Figs. 3-6] The color scale in Figs. 3-6 saturates at |A_UT| = 0.10, so '10% or larger' indicates only the saturation of the scale; please report the actual maximum value of |A_UT| found in each scenario, and whether any points exceed 0.10 after the |A_UT| > 1 rejection.
  3. [§4, after Eq. (7)] The sentence 'We must use caution when large asymmetries arise at the periphery of the subgraphs' is not quantitative; please define what fraction of the phase space is considered periphery and how many points are rejected by the |A_UT| \le 1 cut.
  4. [§3, Eq. (14)] There is a typographical artifact 'su fficient' in the sentence following Eq. (14); please correct the spacing and punctuation.
  5. [§4, Fig. 5 caption] The reflection of points across the x_B = \tilde{x}_B line in Fig. 5 exploits Eq. (8), but the caption should clarify that the plotted asymmetry at reflected points is the same because the observable is evaluated at F_FT(x_B,\tilde{x}_B) and the symmetry (8) relates this to the reflected point; as written, 'experimental coverage only explicitly gives points below the line' is not immediately clear.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: A_UT is computed from externally constrained FFT/GFT models; no fitted parameter is renamed as a prediction.

full rationale

The derivation chain is not circular. The cross-section formulas in Eqs. (3), (5), and (6) are analytic results reproduced from Ref. [34] and given explicitly in Appendix A; they are not inferred from the asymmetry data being predicted. The quark-gluon-quark models in Eqs. (12)-(13) are constructed from two external inputs: the first transverse-momentum moment of the Sivers function from the JAM3D-22 extraction [59] and the lattice-QCD d2 values [53]. No parameter of the model is fitted to A_UT^{gamma SIDIS}. The asymmetry is then evaluated by combining these inputs with CT18NLO PDFs; the output is not an input by construction. The self-citations to Refs. [23, 27, 34] are transparent: the Fourier ansatz is stated explicitly with its assumptions (i)-(iii), and the hard-scattering coefficients are tabulated in Appendix A, so the load-bearing argument does not reduce to an unverified self-citation. The phase-space limitations noted in Sec. 4, including the Q^2 > 1 GeV^2 cuts versus the stated factorization conditions Q^2 >> M^2, ~Q^2 >> M^2, and Q^2 - ~Q^2 >> M^2, concern the range of validity of twist-3 factorization; that is a robustness or correctness question, not a circularity. No step was found where a claimed prediction is equivalent to its input by definition.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central estimates are not derived from known quantities. The shape of FFT/GFT is imposed through an ad hoc Fourier ansatz whose coefficients are either set to zero, fixed by d2, or chosen by hand; the overall size is inherited from the JAM3D-22 Sivers extraction. The only external constraints are the diagonal Sivers moment and the lattice d2 integral. This makes the 10 percent estimate an illustration of model sensitivity rather than a first-principles prediction.

free parameters (5)
  • JAM3D-22 Sivers first moment f_1T^(1),q(x) for u,d,ubar,dbar = central curves from Ref. [59] at scale mu0^2 = 2 GeV^2
    Overall normalization of FFT and GFT through Eqs. (12), (13); this function is itself a fit to transverse-spin SIDIS data.
  • Lattice d2 central values d_u2, d_d2 = 0.026(4)(13) and -0.0086(26)(146) from RQCD Ref. [53]
    Fix the a_2 Fourier coefficients through Eq. (17); large lattice uncertainties are acknowledged but not propagated into the asymmetry plots.
  • Scenario 1 FFT Fourier coefficients a_u4, a_u6, a_u3, a_u5, a_u7 and a_d counterparts = au=(1.1585,-2/3,-2/3,-1/3,-1,-1/3); ad=(-0.6658,2/3,2/3,1/3,1,1/3)
    Hand-picked between -1 and 1 to switch on all Fourier terms; a2 is then fixed by d2. These choices directly set the size of the 10 percent asymmetry.
  • Scenario 1 GFT Fourier coefficients b_q1 through b_q6 = bu=(-1.1585,1/3,2/3,1,2/3,1/3); bd=(0.6658,-1/3,-2/3,-1,-2/3,-1/3)
    Hand-chosen with b1 = -a1; GFT is unconstrained by data and enters the asymmetry directly.
  • Envelope steepness parameter in e(x,x') = 50 in Eq. (14)
    Arbitrary smoothing scale controlling the falloff near the support boundaries |x|, |x'|, |x-x'| to 1; affects the edges of the heat maps, where |A_UT| > 1 points are then removed.
assumptions (6)
  • standard math Levi-Civita identity reduces the four azimuthal spin structures to two, leading to Eq. (5).
    Used in Sec. 2; standard tensor algebra, not a physics assumption.
  • domain assumption Collinear twist-3 factorization applies at Q^2, Q~^2, and Q^2 - Q~^2 much larger than M^2.
    Load-bearing premise for Eqs. (5) and (6); the numerical cuts admit points with Q^2 > 1 GeV^2, close to M^2.
  • ad hoc to paper FFT/GFT model ansatz: constant Fourier coefficients, truncation at n <= 7, Sivers normalization, and envelope e(x,x') as in Eqs. (12) to (14).
    Model assumptions (i) to (iii) in Sec. 3; they enforce known symmetries and integral constraints but are otherwise arbitrary.
  • ad hoc to paper Strange quark a and b coefficients are set to zero; Scenario 0 sets all b to zero; Scenario 1 sets b1 = -a1.
    Choices in Sec. 3 with no data or theory justification beyond simplicity.
  • ad hoc to paper Evolution of FFT/GFT is inherited from Sivers DGLAP evolution; full twist-3 evolution and tri-gluon mixing are neglected.
    Footnote 4 in Sec. 3; authors argue DGLAP-type evolution effects cancel in asymmetry ratios, but this is an approximation.
  • domain assumption Lattice d2 values and the JAM3D-22 Sivers extraction are reliable inputs.
    Used to normalize and constrain the model; their uncertainties are not propagated into the final plots.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Transverse single-spin asymmetries in $\gamma$SIDIS as a direct probe of quark-gluon-quark longitudinal momentum structure." pith.science (2026). https://pith.science/paper/767VGLKI

@misc{pith2026250502711,
  author       = {Pith},
  title        = {Pith review of: Transverse single-spin asymmetries in $\gamma$SIDIS as a direct probe of quark-gluon-quark longitudinal momentum structure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/767VGLKI}},
  note         = {Machine review of arXiv:2505.02711}
}
abstract

Transverse single-spin asymmetries in the semi-inclusive deep-inelastic production of isolated photons ($\gamma$SIDIS), $A_{UT}^{\gamma {\rm SIDIS}}$, provide an unprecedented opportunity to extract the quark-gluon-quark correlators $F_{FT}(x,x')$ and $G_{FT}(x,x')$ point-by-point in their full support $x,x'$. We utilize realistic models for these functions, based on input from the Sivers transverse momentum dependent parton distribution function and imposing constraints from the $d_2$ matrix element calculated in lattice QCD, in order to provide numerical estimates for $A_{UT}^{\gamma {\rm SIDIS}}$ at the Electron-Ion Collider (EIC). We thoroughly explore the EIC phase space in order to isolate in which regions the asymmetry can be sizable, finding it can be as much as $10\%$ or larger for certain kinematics. Given that $F_{FT}(x,x')$ and $G_{FT}(x,x')$ are basically unknown, $A_{UT}^{\gamma {\rm SIDIS}}$ will be an important future measurement to learn about multi-parton correlations in the nucleon.

Figures

Figures reproduced from arXiv: 2505.02711 by the authors.

Figure 1
Figure 1. FFT (x, x ′ ) vs. (x, x ′ ) at a scale µ 2 = 4 GeV2 for Scenario 0 for the up quark (left) and down quark (right) in a proton. Recall that GFT (x, x ′ ) = 0 for Scenario 0. The polar coordinates have been chosen so that we count ϕ starting from the “diagonal” axis of support (x ′ = x) instead of from the x-axis. This is convenient since at ϕ = 0, one has FFT (x, x) = f ⊥(1) 1T (x)/π and GFT (x, x) = 0 (see Eqs. (8),… view at source ↗
Figure 2
Figure 2. FFT (x, x ′ ) vs. (x, x ′ ) (top row) and GFT (x, x ′ ) vs. (x, x ′ ) (bottom row) at a scale µ 2 = 4 GeV2 for Scenario 1 for the up quark (left) and down quark (right) in a proton. This allows us to relate one of the unknown coefficients in FFT (x, x ′ ), say a2, to a4,6,3,5,7 once they are chosen: a q 2 = (d q 2 − (A q 0 + A q 4 a q 4 + A q 6 a q 6 + A q 3 a q 3 + A q 5 a q 5 + A q 7 a q 7 ))/A q 2 , (17) where A … view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Plot of the kinematic points ˜xB vs. xB that enter the arguments of FFT and GFT corresponding to the setup in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

69 extracted references · 32 canonical work pages

  1. [34]

    W. S. Albaltan, A. Prokudin, and M. Schlegel, Phys. Lett. B 804, 135367 (2020), 1910.02883

    work page Pith review arXiv 2020

  2. [27]

    D. Rein, M. Schlegel, P. Tollk ¨uhn, and W. V ogelsang (2025),2503.16119

    arXiv 2025

  3. [61]

    Rodini, L

    S. Rodini, L. Rossi, and A. Vladimirov, Eur. Phys. J. C 84, 732 (2024), 2405.01162

    arXiv 2024

  4. [1]

    Bunce et al., Phys

    G. Bunce et al., Phys. Rev. Lett. 36, 1113 (1976)

    work page 1976

  5. [2]

    R. D. Klem et al., Phys. Rev. Lett. 36, 929 (1976)

    work page 1976

  6. [3]

    G. L. Kane, J. Pumplin, and W. Repko, Phys. Rev. Lett. 41, 1689 (1978)

    1978

  7. [4]

    A. V . Efremov and O. V . Teryaev, Sov. J. Nucl. Phys.36, 140 (1982)

    work page 1982

  8. [5]

    Efremov and O

    A. Efremov and O. Teryaev, Phys.Lett. B150, 383 (1985)

    work page 1985

Show all 69 references
  1. [6]

    Qiu and G

    J.-W. Qiu and G. Sterman, Phys. Rev. Lett. 67, 2264 (1991)

    1991

  2. [7]

    Qiu and G

    J.-W. Qiu and G. Sterman, Nucl. Phys. B378, 52 (1992)

    1992

  3. [8]

    Qiu and G

    J.-W. Qiu and G. Sterman, Phys. Rev. D59, 014004 (1998), hep-ph/9806356

    1998 arXiv

  4. [9]

    Kanazawa and Y

    Y . Kanazawa and Y . Koike, Phys. Lett.B478, 121 (2000), hep-ph/0001021

    2000 arXiv

  5. [10]

    Eguchi, Y

    H. Eguchi, Y . Koike, and K. Tanaka, Nucl. Phys.B752, 1 (2006), hep-ph/0604003

    2006 arXiv

  6. [11]

    Kouvaris, J.-W

    C. Kouvaris, J.-W. Qiu, W. V ogelsang, and F. Yuan, Phys. Rev.D74, 114013 (2006), hep-ph/0609238

    2006 arXiv

  7. [12]

    Eguchi, Y

    H. Eguchi, Y . Koike, and K. Tanaka, Nucl. Phys.B763, 198 (2007), hep-ph/0610314

    2007 arXiv

  8. [13]

    J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D78, 114008 (2008), 0808.3629

    2008 arXiv

  9. [14]

    Koike and T

    Y . Koike and T. Tomita, Phys. Lett.B675, 181 (2009), 0903.1923

    2009 arXiv

  10. [15]

    V ogelsang and F

    W. V ogelsang and F. Yuan, Phys. Rev.D79, 094010 (2009), 0904.0410

    2009 arXiv

  11. [16]

    A. Metz, D. Pitonyak, A. Schaefer, M. Schlegel, W. V ogelsang, et al., Phys. Rev. D86, 094039 (2012), 1209. 3138

    2012

  12. [17]

    Z.-B. Kang, I. Vitev, and H. Xing, Phys. Rev. D87, 034024 (2013), 1212.1221. 12

    2013 arXiv

  13. [18]

    Schlegel, Phys

    M. Schlegel, Phys. Rev. D 87, 034006 (2013), 1211.3579

    2013 arXiv

  14. [19]

    Kanazawa and Y

    K. Kanazawa and Y . Koike, Phys. Rev.D88, 074022 (2013), 1309.1215

    2013 arXiv

  15. [20]

    Beppu, K

    H. Beppu, K. Kanazawa, Y . Koike, and S. Yoshida, Phys. Rev.D89, 034029 (2014), 1312.6862

    2014 arXiv

  16. [21]

    Dai, Z.-B

    L.-Y . Dai, Z.-B. Kang, A. Prokudin, and I. Vitev, Phys. Rev.D92, 114024 (2015), 1409.5851

    2015 arXiv

  17. [22]

    Gamberg, Z.-B

    L. Gamberg, Z.-B. Kang, A. Metz, D. Pitonyak, and A. Prokudin, Phys. Rev. D90, 074012 (2014), 1407.5078

    2014 arXiv

  18. [23]

    Kanazawa, Y

    K. Kanazawa, Y . Koike, A. Metz, D. Pitonyak, and M. Schlegel, Phys. Rev.D93, 054024 (2016), 1512.07233

    2016 arXiv

  19. [24]

    Yoshida, Phys

    S. Yoshida, Phys. Rev. D93, 054048 (2016), 1601.07737

    2016 arXiv

  20. [25]

    A. P. Chen, J. P. Ma, and G. P. Zhang, Phys. Rev.D95, 074005 (2017), 1607.08676

    2017 arXiv

  21. [26]

    A. P. Chen, J. P. Ma, and G. P. Zhang, Phys. Rev.D97, 054003 (2018), 1708.09091

    2018 arXiv

  22. [28]

    D. Rein, M. Schlegel, P. Tollk ¨uhn, and W. V ogelsang (2025),2503.16097

    2025

  23. [29]

    Z.-B. Kang, A. Metz, J.-W. Qiu, and J. Zhou, Phys. Rev. D84, 034046 (2011), 1106.3514

    2011 arXiv

  24. [30]

    Liang, A

    Z.-T. Liang, A. Metz, D. Pitonyak, A. Schaefer, Y .-K. Song, et al., Phys. Lett.B712, 235 (2012), 1203.3956

    2012 arXiv

  25. [31]

    A. Metz, D. Pitonyak, A. Schaefer, and J. Zhou, Phys. Rev. D86, 114020 (2012), 1210.6555

    2012 arXiv

  26. [32]

    Hatta, K

    Y . Hatta, K. Kanazawa, and S. Yoshida, Phys. Rev.D88, 014037 (2013), 1305.7001

    2013 arXiv

  27. [33]

    Kanazawa, A

    K. Kanazawa, A. Metz, D. Pitonyak, and M. Schlegel, Phys. Lett. B742, 340 (2015), 1411.6459

    2015 arXiv

  28. [35]

    Bauer, D

    B. Bauer, D. Pitonyak, and C. Shay, Phys. Rev. D 107, 014013 (2023), 2210.14334

    2023 arXiv

  29. [36]

    Fitzgibbons, M

    S. Fitzgibbons, M. Malda, J. Marsh, D. Pitonyak, and P. Smith, Phys. Lett. B 852, 138606 (2024), 2401.02391

    2024 arXiv

  30. [37]

    Kang and J.-W

    Z.-B. Kang and J.-W. Qiu, Phys. Rev. D79, 016003 (2009), 0811.3101

    2009 arXiv

  31. [38]

    Braun, A

    V . Braun, A. Manashov, and B. Pirnay, Phys. Rev.D80, 114002 (2009), 0909.3410

    2009 arXiv

  32. [39]

    J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D79, 114022 (2009), 0812.4484

    2009 arXiv

  33. [40]

    Ma and Q

    J. Ma and Q. Wang, Phys. Lett. B715, 157 (2012), 1205.0611

    2012 arXiv

  34. [41]

    S. M. Aybat, J. C. Collins, J.-W. Qiu, and T. C. Rogers, Phys. Rev. D85, 034043 (2012), 1110.6428

    2012 arXiv

  35. [42]

    Scimemi and A

    I. Scimemi and A. Vladimirov, Eur. Phys. J. C 78, 802 (2018), 1804.08148

    2018 arXiv

  36. [43]

    Scimemi, A

    I. Scimemi, A. Tarasov, and A. Vladimirov, JHEP 05, 125 (2019), 1901.04519

    2019 arXiv

  37. [44]

    F. Rein, S. Rodini, A. Sch ¨afer, and A. Vladimirov, JHEP 01, 116 (2023), 2209.00962

    2023 arXiv

  38. [45]

    Harris, J

    M. Harris, J. Marsh, D. Pitonyak, A. Prokudin, J. Putnam, D. Rein, and M. Schlegel, https: //colab.research.google.com/github/pitonyak25/AUT_gamSIDIS_lib/blob/main/AUT_ gamSIDIS_public.ipynb

  39. [46]

    Kanazawa and Y

    K. Kanazawa and Y . Koike, Phys. Rev.D82, 034009 (2010), 1005.1468

    2010 arXiv

  40. [47]

    V . M. Braun, T. Lautenschlager, A. N. Manashov, and B. Pirnay, Phys. Rev. D83, 094023 (2011), 1103.1269

    2011 arXiv

  41. [48]

    S. J. Brodsky, J. F. Gunion, and R. L. Jaffe, Phys. Rev. D 6, 2487 (1972)

    1972

  42. [49]

    Hou et al

    T.-J. Hou et al. (2019), 1908.11394. 13

    2019 arXiv

  43. [50]

    D. Boer, P. J. Mulders, and F. Pijlman, Nucl. Phys. B667, 201 (2003), hep-ph/0303034

    2003 arXiv

  44. [51]

    Gockeler, R

    M. Gockeler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schafer, G. Schierholz, H. Stuben, and J. M. Zanotti, Phys. Rev. D 72, 054507 (2005), hep-lat/0506017

    2005 arXiv

  45. [52]

    J. M. Bickerton, Ph.D. thesis, Adelaide U. (2020)

    2020

  46. [53]

    B ¨urger, T

    S. B ¨urger, T. Wurm, M. L¨offler, M. G¨ockeler, G. Bali, S. Collins, A. Sch ¨afer, and A. Sternbeck (RQCD), Phys. Rev. D 105, 054504 (2022), 2111.08306

    2022 arXiv

  47. [54]

    J. A. Crawford, K. U. Can, R. Horsley, P. E. L. Rakow, G. Schierholz, H. St¨uben, R. D. Young, and J. M. Zanotti (QCDSF), Phys. Rev. Lett. 134, 071901 (2025), 2408.03621

    2025 arXiv

  48. [55]

    Burkardt, Phys

    M. Burkardt, Phys. Rev. D88, 114502 (2013), 0810.3589

    2013 arXiv

  49. [56]

    F. P. Aslan, M. Burkardt, and M. Schlegel, Phys. Rev. D 100, 096021 (2019), 1904.03494

    2019 arXiv

  50. [57]

    E. V . Shuryak and A. I. Vainshtein, Nucl. Phys. B201, 141 (1982)

    1982

  51. [58]

    R. L. Ja ffe, Comments Nucl. Part. Phys. 19, 239 (1990)

    1990

  52. [59]

    Gamberg, M

    L. Gamberg, M. Malda, J. A. Miller, D. Pitonyak, A. Prokudin, and N. Sato (Je fferson Lab Angular Momentum (JAM)), Phys. Rev. D106, 034014 (2022), 2205.00999

    2022 arXiv

  53. [60]

    D. W. Duke and J. F. Owens, Phys. Rev. D30, 49 (1984)

    1984

  54. [62]

    Anselmino, M

    M. Anselmino, M. Boglione, U. D’Alesio, F. Murgia, and A. Prokudin, JHEP 04, 046 (2017), 1612.06413

    2017 arXiv

  55. [63]

    M. G. Echevarria, Z.-B. Kang, and J. Terry, JHEP 01, 126 (2021), 2009.10710

    2021 arXiv

  56. [64]

    Bacchetta, F

    A. Bacchetta, F. Delcarro, C. Pisano, and M. Radici, Phys. Lett. B 827, 136961 (2022), 2004.14278

    2022 arXiv

  57. [65]

    M. Bury, A. Prokudin, and A. Vladimirov, JHEP 05, 151 (2021), 2103.03270

    2021 arXiv

  58. [66]

    Anselmino, M

    M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, et al., Eur. Phys. J. A39, 89 (2009), 0805. 2677

    2009

  59. [67]

    Afanasev et al

    A. Afanasev et al. (2019), 1906.09419

    2019 arXiv

  60. [68]

    Arrington et al., Prog

    J. Arrington et al., Prog. Part. Nucl. Phys. 127, 103985 (2022), 2112.00060

    2022 arXiv

  61. [69]

    Abdul Khalek et al., Nucl

    R. Abdul Khalek et al., Nucl. Phys. A 1026, 122447 (2022), 2103.05419. 14

    2022 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.