Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Gluon Unpolarized, Polarized, and Transversity GPDs from Lattice QCD: Lorentz-Covariant Parametrization (Part I)

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read New projections make all eight gluon GPDs computable on the lattice

desk verdict A genuinely needed gluon-GPD formalism, with a solid spin-0 part and a spin-1/2 core that rests on an unshown computer-algebra calculation; referee it, but require the algebra to be made checkable. read the letter →

arxiv 2412.14110 v2 pith:WEZCJ4OU submitted 2024-12-18 hep-lat hep-phnucl-th

Jakob Schoenleber , Raza Sabbir Sufian , Taku Izubuchi , Yi-Bo Yang This is my paper · ORCID
classification hep-lathep-phnucl-th
keywords gluonGPDslatticeQCDLorentz-covariantparametrizationgeneralizedpartondistributionstransversityquasi-PDFLaMETpseudo-PDF
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 works out the Lorentz-covariant decomposition of the off-forward gluon matrix element that lattice QCD can compute, and derives the specific projections that pick out each of the eight leading-twist gluon generalized parton distributions (GPDs) for spin-0 and spin-1/2 hadrons. The authors show that the tensor structures entering the decomposition are linearly dependent, and that a Gram-Schmidt procedure over an indefinite inner product yields a minimal basis: 18 independent structures for spin-0 and 77 for spin-1/2. For this basis they provide explicit projection formulas that, at zero separation squared, reproduce the light-cone combinations defining $H_g$, $E_g$, $\tilde H_g$, $\tilde E_g$, $H_g^T$, $E_g^T$, $\tilde H_g^T$, and $\tilde E_g^T$ without contaminations. If correct, this removes the main formal obstacle to first-principles lattice calculations of the gluon GPDs, which are needed to understand the nucleon's mass, spin, and mechanical structure.

What carries the argument

The central object is the Lorentz-covariant decomposition $M^{\mu\nu;\alpha\beta}_{s's} = \frac{1}{2m}\sum_\ell M_\ell T_\ell^{\mu\nu;\alpha\beta}_{s's}$ of the off-forward gluon matrix element, with $T_\ell$ built from the metric, the vectors $P^\mu$, $\Delta^\mu$, $z^\mu$, and the spinor structures $\bar u(p')u(p)$ and $\bar u(p')i\sigma^{\mu\nu}u(p)$. The mechanism that carries the argument is the Gram-Schmidt procedure with respect to the contraction inner product $(u,v)=u_{\mu\nu;\alpha\beta}v^{\mu\nu;\alpha\beta}$, which systematically exposes all linear relations among the candidate tensors and produces a linearly independent basis. On that basis the authors invert the system to find projection tensors $P[F]$ such that $(P[F],M)=\sum_\ell f_\ell^{F}M_\ell|_{z^2=0}$ reproduces each GPD's generalized Ioffe-time distribution; the explicit basis and projections for the spin-1/2 case are given in Appendix B.

What would settle it

Independently re-run the tensor enumeration and Gram-Schmidt reduction with a different ordering of the initial spin-1/2 structures (or a different computer algebra system) and check whether the resulting basis has 77 elements and yields the same 25 relations; in addition, substitute a set of random momenta and numerical Dirac matrices into Eqs. (B3)-(B10) and verify that each right-hand side equals the corresponding GPD definition at $z^2=0$. Any mismatch would disprove the central claim.

Watch

Extended reading notes

Core claim

The central claim is that all eight leading-twist gluon GPDs for spin-0 and spin-1/2 hadrons can be isolated from Euclidean off-forward matrix elements of the bilocal gluon operator by solving the 'projection problem' in a carefully chosen Lorentz-covariant basis. The paper demonstrates that the set of allowed tensor structures is overcomplete, with one linear relation for the spin-0 case and 25 for spin-1/2, and that eliminating the redundant structures defines a basis in which the generalized Ioffe-time distributions become explicit linear combinations of invariant amplitudes at $z^2=0$. The authors then list the projection tensors that realize these combinations, and stress that the projections are exact in the sense that, for the chosen basis, no residual $O(z^2)$ contaminations couple into the physical light-cone quantities; changing the basis shifts the result only by $O(z^2)$.

Load-bearing premise

The spin-1/2 result rests on the assumption that the initial list of 102 tensor structures is complete and that the Gram-Schmidt reduction correctly finds exactly 25 linear relations; if any structure is missing or any relation is misidentified, the published projectors would no longer isolate the claimed GPD combinations.

Editorial extensions

If this is right

  • A lattice calculation of the specific trace combinations listed in Appendix B can, for the first time, numerically determine all eight gluon GPDs from first principles rather than from model assumptions.
  • The exact $z^2=0$ projections separate each physical light-cone combination from contaminating amplitudes, letting the $z^2$ dependence of each invariant amplitude be studied directly on the lattice.
  • The spin-1/2 projections give access to the gluon total angular momentum $J_g$ and gluon orbital angular momentum through the $H_g$ and $E_g$ combinations, and to the gluon transversity distributions that are essentially unconstrained experimentally.
  • The same projection method applied to the pion (spin-0) case provides a path to the gluon GPD of the pion, which is otherwise nearly unknown.
  • The explicit forward-case result in Appendix C extends the strategy back to gluon PDFs and shows that the polarized forward case contains a single linear relation among the 21 tensor structures.

Reading between the lines

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

  • Because the projections depend on the chosen basis only through $O(z^2)$ differences, future lattice implementations may use alternative bases (e.g., eliminating a different tensor) without changing physics; these differences can serve as a systematic check of lattice artefacts.
  • The same Lorentz-covariant projection machinery could be applied to other nonlocal operators, such as quark-gluon mixed operators or higher-spin targets, where the tensor enumerations and Gordon identities are analogous.
  • If the 25 linear relations for spin-1/2 are verified independently, the basis could also be used to classify power corrections and to design optimized lattice momentum setups that minimize the number of independent matrix elements needed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript develops a Lorentz-covariant parameterization of the off-forward gluon matrix element (5) for spin-0 and spin-1/2 hadrons, with the goal of constructing projections that, at z^2=0, reduce exactly to the eight leading-twist gluon GPDs H_g, E_g, tilde-H_g, tilde-E_g, H_g^T, E_g^T, tilde-H_g^T, and tilde-E_g^T. For the spin-0 case the calculation is shown in detail: a 19-element tensor list is reduced via one linear relation to 18 invariant amplitudes, and the resulting projection reproduces the known forward unpolarized gluon PDF projector in the limit Delta -> 0. For the spin-1/2 case the authors report an initial set of 102 structures, 25 linear relations, and a final 77-element basis listed in Appendix B, with the corresponding GITD combinations and projection formulas given in Eqs. (B3)-(B10) and (B13)-(B21). The abstract claims that this is the first derivation of such exact projections for all eight gluon GPDs, enabling their separation in lattice QCD.

Significance. If the spin-1/2 basis and the projection coefficients are correct, the paper fills a genuine gap: lattice QCD calculations exist for quark GPDs and for gluon PDFs, but a complete set of projection operators for the off-forward gluon matrix element has not appeared in the literature. The spin-0 section is a worked, internally consistent example, and the forward-limit check against Balitsky-Morris-Radyushkin (Ref. [102]) provides one solid external benchmark. The paper is also commendably explicit about the non-uniqueness of the projections and about the meaning of 'exactness' in the presence of O(z^2) contaminations. However, the advertised significance rests on the 102-to-77 reduction in Section V, which is currently asserted rather than demonstrated. Because the inner product in Eq. (42) is indefinite and the Gram-Schmidt output is order-dependent, the central claim is not independently verifiable from the text as it stands.

major comments (2)
  1. [Sec. V, Appendix B, Eq. (42)] The load-bearing step for the spin-1/2 result is the reduction from 102 tensor structures to a basis of 77 independent tensors, yet the manuscript gives neither the initial enumeration of the 102 structures, the ordering used in the Gram-Schmidt procedure, the 25 linear relations, nor the computer-algebra code that produced them. The inner product in Eq. (42) is indefinite, so the outcome depends on the order and on algebraic simplifications; an undetected error in this hidden reduction would silently alter the coefficients in Eqs. (B13)-(B21) and invalidate the claim that these projections isolate the eight gluon GPDs. Because Eqs. (B13)-(B21) are the only concrete deliverables for the spin-1/2 case, this is a load-bearing gap. The authors should provide the full reduction data or a machine-readable notebook, together with an independent verification that (P[F],M) = sum_l f_l^(F) M_l holds for random numerical tensor components.
  2. [Sec. V, Eqs. (47)-(54), Appendix C] The forward-limit check in Appendix C exercises only a 20-tensor subspace and therefore does not constrain the 77-tensor off-forward basis or the long projector coefficients. The claimed exactness of the projections should be cross-checked in an additional nontrivial limit. For example, one can require that as xi -> 0, or in the Delta_perp -> 0 limit where the kinematics permit, the combinations in Eqs. (B3)-(B10) reduce to known forward PDF projectors, and that the pole terms in f_l^{(tilde-E_g)} (noted after Eq. (B12)) cancel when combined with the physical matrix elements. Without such a check, a single coefficient typo in the long expressions (B13)-(B21) would be undetectable from the text and would break the advertised separation of the eight gluon GPDs.
minor comments (4)
  1. [Sec. IV, Eqs. (37)-(38)] The symbol p0 appears without definition in the forward-limit expressions; please define it as the hadron energy in the forward frame so that the limit is unambiguous.
  2. [Sec. V, after Eq. (41)] The sentence 'the f's are tensors that can be built out of the available vectors P^mu, Delta^mu, z^mu and the metric tensor g^munu' is imprecise; please specify the index structure of the f's and how the spinor indices are contracted in Eq. (41).
  3. [Appendix B] The projection formulas (B13)-(B21) are very long and contain repeated combinations such as M_{02;12}+M_{12;02}; providing these expressions in a machine-readable format would substantially aid reproducibility and ease checking for transcription errors.
  4. [Sec. III, Eq. (16)] The relation xi = eta / sqrt(1 - m^2 z^2/omega^2 + t z^2/(4 omega^2)) is stated without derivation; a brief derivation or an explicit reference would help the reader understand the O(P^{-2}) corrections claimed after Eq. (16).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the projection derivation is a self-contained linear-algebra construction, with the spin-0 forward limit checked against the independent Balitsky–Morris–Radyushkin result.

full rationale

The paper's central derivation is not circular. The Lorentz-covariant decomposition and the projector solutions are obtained by solving linear systems defined by the choice of tensor basis (Secs. IV-V and Appendix B), not by fitting or by assuming the target GPDs. The claimed GPD projections are constructed so that, at z^2=0, they reproduce the light-cone definitions given in Eqs. (47)-(54); this is an algebraic construction rather than a prediction of independent content, and the paper explicitly acknowledges the basis-dependence of the projectors and that they do not reduce O(z^2) power corrections. The forward spin-0 limit is benchmarked against the independent result of Ref. [102], and the forward polarized case is treated separately in Appendix C. Self-citations appear only in non-load-bearing contexts: Ref. [106] is mentioned for the functional form of power corrections in ongoing/future work, and Ref. [110] is cited for the expected difficulty of GPD reconstruction, not as a premise of the projection method. The manuscript does not display the initial 102-structure enumeration or the 25 linear relations that reduce it to 77 tensors, which is a reproducibility and verification gap rather than a circularity: an error there would make the projectors incorrect, but would not make the derivation depend on its own conclusion. Therefore no circular step can be quoted, and the score reflects only the minimal presence of non-load-bearing self-citations and the unverified computer-algebra reduction, not any definitional self-support.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No parameters are fitted to data and no new physical entities are introduced. The central burden is the completeness of the tensor basis and the validity of the factorization assumption, which are domain assumptions of the standard LaMET and pseudo-PDF approaches.

assumptions (3)
  • domain assumption Lorentz covariance: the off-forward gluon matrix element is spanned by tensors built from the metric, Levi-Civita tensor, P, Delta, and z.
    Used in Sections IV and V to enumerate the initial tensor basis; standard in QCD but not proved in the paper.
  • domain assumption Parity symmetry eliminates Levi-Civita structures coupled to non-pseudoscalar spinor bilinears, and Gordon identities reduce the spinor basis to scalar and sigma terms.
    Invoked in Section V and Appendix A; standard spinor algebra, but the completeness of the reduced set is asserted after computer algebra rather than shown.
  • domain assumption Short-distance factorization of invariant amplitudes as in Eq. (18), with power corrections that vanish in the z3 to 0 limit.
    The paper explicitly defers renormalization, matching, and power corrections to future work, but the identification of light-cone GPDs with the z2=0 combinations depends on this factorization being valid.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gluon Unpolarized, Polarized, and Transversity GPDs from Lattice QCD: Lorentz-Covariant Parametrization (Part I)." pith.science (2026). https://pith.science/paper/WEZCJ4OU

@misc{pith2026241214110,
  author       = {Pith},
  title        = {Pith review of: Gluon Unpolarized, Polarized, and Transversity GPDs from Lattice QCD: Lorentz-Covariant Parametrization (Part I)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WEZCJ4OU}},
  note         = {Machine review of arXiv:2412.14110}
}
abstract

We identify the matrix elements necessary to determine the leading-twist gluon generalized parton distributions (GPDs) $H_g,~E_g,~\wt{H}_g,~\wt{E}_g,~H^T_g,~E^T_g, \wt{H}^T_g ,~\wt{E}^T_g$ in lattice QCD calculations. We present a method to achieve a Lorentz-covariant parameterization of the matrix elements in terms of a linearly independent basis of tensor structures. This parameterization is crucial for projecting lattice QCD matrix elements onto light cone distributions. For the first time, we determine the corresponding components that project onto the linear combinations of invariant amplitudes, which reduce to the different gluon GPDs in the light cone limit and enable their separation in a lattice QCD calculation for spin-$0$ and spin-$\frac{1}{2}$ hadrons. Hence, this work lays the foundation for the numerical determination of the gluon GPDs from first-principle lattice QCD calculations, directly advancing our understanding of the mass and spin structures and mechanical properties of the nucleon, as well as the physics underlying deeply virtual Compton scattering and deeply virtual meson production in a range of experimental processes.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Resummation for Lattice QCD Calculation of Generalized Parton Distributions at Nonzero Skewness

    hep-ph 2025-01 conditional novelty 7.0 of 10

    A threshold factorization and resummation scheme for quasi-GPD matching in LaMET is derived and shown to be self-consistent on a GPD model.

Reference graph

Works this paper leans on

120 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [102]

    Khan et al

    T. Khan et al. (HadStruc), Phys. Rev. D 104, 094516 (2021) , arXiv:2107.08960 [hep-lat]

    arXiv 2021

  2. [1]

    M¨ uller, D

    D. M¨ uller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Ho ˇ rejˇ si,Fortsch. Phys. 42, 101 (1994) , arXiv:hep-ph/9812448

    arXiv 1994

  3. [2]

    Ji, Phys

    X.-D. Ji, Phys. Rev. Lett. 78, 610 (1997) , arXiv:hep-ph/9603249

    arXiv 1997

  4. [3]

    tr[M12;02] + 2z2 3 ( 1 ω2 + 8ξ2 ∆ 2 1z2 3−4ξ2ω2 ) 3mξ tr[M12;12] + ξ3z2 3 ( 4m2 −t ) ( ω 2 ( 4m2ξ2(ξ −η) +ηt ) +m2ξ3z2 3 ( 4m2 −t )) 4∆ 2 1η3mtω 4 tr[iγ5M01;02 +iγ5M02;01] + ξz3(η −ξ) ( t − 4m2ξ2) 2∆ 1η2mtω tr[iγ5M01;12 −iγ5M12;01] − ξ2 ( z2 3 ( 5m2 − 2t ) + 5ω 2) ( 4ω 2(ξ −η) +ξz2 3 ( 4m2 −t )) 12∆ 2 1η3mω 4 tr[iσ 10M02;12] − 1 12∆ 2 1η3mω 4( 4ξ2ω 2 − ∆ ...

  5. [4]

    tr[iσ 30M12;12] − 5(η −ξ) 3∆ 2 1ηm tr[iσ 31M02;12] − 5 ( 4ξ2ω 2(η −ξ) + ∆ 2 1z2 3(η +ξ) ) 3∆ 2 1ηm (∆ 2 1z2 3 − 4ξ2ω 2) tr[iσ 31M12;02] + ξ ( tz5 3 ( ξ2 ( t − 4m2) −t ) + 4ξ2tω 2z3 3 ) 8η2mω 3 (4∆ 1ξ2ω 2 − ∆ 3 1z2

  6. [5]

    tr[iσ 31M12;12] − ξ2tz2 3(η −ξ) 4∆ 2 1η3mω 2 tr[iσ 32M01;12] (B15) 18 Projection onto ~Eg : (P[~Eg],M ) = − 4mξz3 ( 4ω 2(ξ −η) +ξz2 3 ( 4m2 −t )) 3∆ 1η2tω 3 tr[M02;12] − 4mξz3 ( 16ξ2ω 4(η −ξ) + 4ω 2z2 3 ( 4ηm2ξ2 +t ( −ηξ2 +η +ξ )) +ξz4 3 ( 4m2 −t ) ( 4m2ξ2 −ξ2t +t )) 3∆ 1η2tω 3 (4ξ2ω 2 − ∆ 2 1z2

  7. [6]

    tr[M12;02] + 8mz2 3 ( 8ξ2 4ξ2ω2−∆ 2 1z2 3 − 1 ω2 ) 3ξt tr[M12;12] + 1 4∆ 2 1η3t2ω 4mξ2 ( − 32ξω 4(η −ξ) ( 4m2ξ2 −ξ2t +t ) − 4ω 2z2 3 ( 32m4ξ3(η − 2ξ) + 4m2ξt ( η ( 4 − 5ξ2) + 9ξ3 − 6ξ ) +t2( η ( 3ξ2 − 4 ) ξ − 5ξ4 + 6ξ2 − 1 )) +z4 3 ( 128m6ξ4 − 16m4ξ2( 7ξ2 − 4 ) t + 4m2( 8ξ4 − 8ξ2 + 1 ) t2 + ( − 3ξ4 + 4ξ2 − 1 ) t3) ) tr[iγ5M01;02] − mξ2 ( z2 3 ( 8m2ξ2 + ( ...

  8. [7]

    Indeed, we find that for generic hadron polarizations, there is a single linear r elation between the possible Lorentz structures that can appear

    tr[iσ 10M12;02] + mξ2z2 3 ( 4ω 2(ξ −η) +ξz2 3 ( 4m2 −t )) 4∆ 2 1η3ω 4 tr[iσ 20M12;01] − mξ2z2 3 ∆ 2 1η2ω 2 tr[iσ 21M01;02 −iσ 21M02;01] − mz2 3 ( z2 3 ( 5m2 − 2t ) + 5ω 2) ( 4ξ2ω 2 + ∆ 2 1z2 3 ) 3η2tω 4 (z2 3 ((ξ2 − 1)t − 4m2ξ2) − 4ξ2ω 2) tr[iσ 30M12;12] + 20m(η −ξ) 3∆ 2 1ηt tr[iσ 31M02;12] + 20m ( 4ξ2ω 2(η −ξ) + ∆ 2 1z2 3(η +ξ) ) 3∆ 2 1ηt (∆ 2 1z2 3 − 4ξ...

Show all 120 references
  1. [8]

    M. V. Polyakov and A. G. Shuvaev, (2002), arXiv:hep-ph/0207153

    2002 arXiv

  2. [9]

    A. V. Radyushkin, Phys. Rev. D 56, 5524 (1997) , arXiv:hep-ph/9704207

    1997 arXiv

  3. [10]

    Ji, Phys

    X.-D. Ji, Phys. Rev. Lett. 74, 1071 (1995) , arXiv:hep-ph/9410274

    1995 arXiv

  4. [11]

    Ji, Phys

    X.-D. Ji, Phys. Rev. D 52, 271 (1995) , arXiv:hep-ph/9502213

    1995 arXiv

  5. [12]

    Ashman et al

    J. Ashman et al. (European Muon), Phys. Lett. B 206, 364 (1988)

    1988

  6. [13]

    Ashman et al

    J. Ashman et al. (European Muon), Nucl. Phys. B 328, 1 (1989)

    1989

  7. [14]

    Burkardt, Phys

    M. Burkardt, Phys. Rev. D 62, 071503 (2000) , [Erratum: Phys.Rev.D 66, 119903 (2002)], arXiv:hep-ph/0005108

    2000 arXiv

  8. [15]

    M. V. Polyakov, Phys. Lett. B 555, 57 (2003) , arXiv:hep-ph/0210165

    2003 arXiv

  9. [16]

    M. V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33, 1830025 (2018) , arXiv:1805.06596 [hep-ph]

    2018 arXiv

  10. [17]

    Lorc´ e, H

    C. Lorc´ e, H. Moutarde, and A. P. Trawi´ nski, Eur. Phys. J. C 79, 89 (2019) , arXiv:1810.09837 [hep-ph]

    2019 arXiv

  11. [18]

    kinematic

    is subject to power corrections that vanish in the limit z3 → 0, an important subject for ongoing and future investigations. In the context of GPDs the re is some knowledge of the functional form from the study of renormalons [ 106]. We remark that, in the GPD case, for generi...

  12. [19]

    V. D. Burkert, L. Elouadrhiri, and F. X. Girod, Nature 557, 396 (2018)

    2018

  13. [20]

    P. E. Shanahan and W. Detmold, Phys. Rev. Lett. 122, 072003 (2019) , arXiv:1810.07589 [nucl-th]

    2019 arXiv

  14. [21]

    J. P. Ralston and B. Pire, Phys. Rev. D 66, 111501 (2002) , arXiv:hep-ph/0110075

    2002 arXiv

  15. [22]

    Diehl, Eur

    M. Diehl, Eur. Phys. J. C 25, 223 (2002) , [Erratum: Eur.Phys.J.C 31, 277–278 (2003)], arXiv:hep-ph/0205208

    2002 arXiv

  16. [23]

    Ji, Phys

    X.-D. Ji, Phys. Rev. D 55, 7114 (1997) , arXiv:hep-ph/9609381

    1997 arXiv

  17. [24]

    A. V. Radyushkin, Phys. Lett. B 385, 333 (1996) , arXiv:hep-ph/9605431

    1996 arXiv

  18. [25]

    J. C. Collins, L. Frankfurt, and M. Strikman, Phys. Rev. D 56, 2982 (1997) , arXiv:hep-ph/9611433

    1997 arXiv

  19. [26]

    Goeke, M

    K. Goeke, M. V. Polyakov, and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 47, 401 (2001) , arXiv:hep-ph/0106012

    2001 arXiv

  20. [27]

    Diehl, Phys

    M. Diehl, Phys. Rept. 388, 41 (2003) , arXiv:hep-ph/0307382

    2003 arXiv

  21. [28]

    Vanderhaeghen, P

    M. Vanderhaeghen, P. A. M. Guichon, and M. Guidal, Phys. Rev. D 60, 094017 (1999) , arXiv:hep-ph/9905372

    1999 arXiv

  22. [29]

    E. R. Berger, M. Diehl, and B. Pire, Eur. Phys. J. C 23, 675 (2002) , arXiv:hep-ph/0110062

    2002 arXiv

  23. [30]

    Guidal and M

    M. Guidal and M. Vanderhaeghen, Phys. Rev. Lett. 90, 012001 (2003) , arXiv:hep-ph/0208275. 21

    2003 arXiv

  24. [31]

    A. V. Belitsky and D. Mueller, Phys. Rev. Lett. 90, 022001 (2003) , arXiv:hep-ph/0210313

    2003 arXiv

  25. [32]

    Grocholski, B

    O. Grocholski, B. Pire, P. Sznajder, L. Szymanowski, an d J. Wagner, Phys. Rev. D 105, 094025 (2022) , arXiv:2204.00396 [hep-ph]

    2022 arXiv

  26. [33]

    Duplanˇ ci´ c, K

    G. Duplanˇ ci´ c, K. Passek-Kumeriˇ cki, B. Pire, L. Szym anowski, and S. Wallon, JHEP 11, 179 (2018) , arXiv:1809.08104 [hep-ph]

    2018 arXiv

  27. [34]

    S. J. Brodsky, L. Frankfurt, J. F. Gunion, A. H. Mueller, and M. Strikman, Phys. Rev. D 50, 3134 (1994) , arXiv:hep-ph/9402283

    1994 arXiv

  28. [35]

    Qiu and Z

    J.-W. Qiu and Z. Yu, Phys. Rev. D 107, 014007 (2023) , arXiv:2210.07995 [hep-ph]

    2023 arXiv

  29. [36]

    Ali et al

    A. Ali et al. (GlueX), Phys. Rev. Lett. 123, 072001 (2019) , arXiv:1905.10811 [nucl-ex]

    2019 arXiv

  30. [37]

    Duran et al

    B. Duran et al. , Nature 615, 813 (2023) , arXiv:2207.05212 [nucl-ex]

    2023 arXiv

  31. [38]

    Adhikari et al

    S. Adhikari et al. (GlueX), Phys. Rev. C 108, 025201 (2023) , arXiv:2304.03845 [nucl-ex]

    2023 arXiv

  32. [39]

    Accardi et al

    A. Accardi et al. , Eur. Phys. J. A 52, 268 (2016) , arXiv:1212.1701 [nucl-ex]

    2016 arXiv

  33. [40]

    Abdul Khalek et al

    R. Abdul Khalek et al. , Nucl. Phys. A 1026, 122447 (2022) , arXiv:2103.05419 [physics.ins-det]

    2022 arXiv

  34. [41]

    D. P. Anderle et al. , Front. Phys. (Beijing) 16, 64701 (2021) , arXiv:2102.09222 [nucl-ex]

    2021 arXiv

  35. [42]

    S. V. Goloskokov and P. Kroll, Eur. Phys. J. C 50, 829 (2007) , arXiv:hep-ph/0611290

    2007 arXiv

  36. [43]

    S. V. Goloskokov and P. Kroll, Eur. Phys. J. C 65, 137 (2010) , arXiv:0906.0460 [hep-ph]

    2010 arXiv

  37. [44]

    S. V. Goloskokov and P. Kroll, Eur. Phys. J. A 47, 112 (2011) , arXiv:1106.4897 [hep-ph]

    2011 arXiv

  38. [45]

    D. Y. Ivanov, A. Schafer, L. Szymanowski, and G. Krasnik ov, Eur. Phys. J. C 34, 297 (2004) , [Erratum: Eur.Phys.J.C 75, 75 (2015)], arXiv:hep-ph/0401131

    2004 arXiv

  39. [46]

    Y. Guo, X. Ji, and Y. Liu, Phys. Rev. D 103, 096010 (2021) , arXiv:2103.11506 [hep-ph]

    2021 arXiv

  40. [47]

    Koempel, P

    J. Koempel, P. Kroll, A. Metz, and J. Zhou, Phys. Rev. D 85, 051502 (2012) , arXiv:1112.1334 [hep-ph]

    2012 arXiv

  41. [48]

    S. V. Goloskokov, Y.-P. Xie, and X. Chen, (2024), arXiv:2408.05800 [hep-ph]

    2024 arXiv

  42. [49]

    Y. Guo, X. Ji, and F. Yuan, Phys. Rev. D 109, 014014 (2024) , arXiv:2308.13006 [hep-ph]

    2024 arXiv

  43. [50]

    Bertone, H

    V. Bertone, H. Dutrieux, C. Mezrag, H. Moutarde, and P. S znajder, Phys. Rev. D 103, 114019 (2021) , arXiv:2104.03836 [hep-ph]

    2021 arXiv

  44. [51]

    Moffat, A

    E. Moffat, A. Freese, I. Clo¨ et, T. Donohoe, L. Gamberg, W . Melnitchouk, A. Metz, A. Prokudin, and N. Sato, Phys. Rev. D 108, 036027 (2023) , arXiv:2303.12006 [hep-ph]

    2023 arXiv

  45. [52]

    Ji, Phys

    X. Ji, Phys. Rev. Lett. 110, 262002 (2013) , arXiv:1305.1539 [hep-ph]

    2013 arXiv

  46. [53]

    Abir et al

    R. Abir et al. , (2023), arXiv:2305.14572 [hep-ph]

    2023

  47. [54]

    X. Ji, Sci. China Phys. Mech. Astron. 57, 1407 (2014) , arXiv:1404.6680 [hep-ph]

    2014 arXiv

  48. [55]

    A. V. Radyushkin, Phys. Rev. D 96, 034025 (2017) , arXiv:1705.01488 [hep-ph]

    2017 arXiv

  49. [56]

    Hoodbhoy and X.-D

    P. Hoodbhoy and X.-D. Ji, Phys. Rev. D 58, 054006 (1998) , arXiv:hep-ph/9801369

    1998 arXiv

  50. [57]

    Diehl, Eur

    M. Diehl, Eur. Phys. J. C 19, 485 (2001) , arXiv:hep-ph/0101335

    2001 arXiv

  51. [58]

    R. L. Jaffe and A. Manohar, Phys. Lett. B 223, 218 (1989)

    1989

  52. [59]

    Artru and M

    X. Artru and M. Mekhfi, Z. Phys. C 45, 669 (1990)

    1990

  53. [60]

    Meissner, A

    S. Meissner, A. Metz, and K. Goeke, Phys. Rev. D 76, 034002 (2007) , arXiv:hep-ph/0703176

    2007 arXiv

  54. [61]

    Aschenauer, S

    E.-C. Aschenauer, S. Fazio, K. Kumericki, and D. Muelle r, JHEP 09, 093 (2013) , arXiv:1304.0077 [hep-ph]

    2013 arXiv

  55. [62]

    Leader and C

    E. Leader and C. Lorc´ e, Phys. Rept. 541, 163 (2014) , arXiv:1309.4235 [hep-ph]

    2014 arXiv

  56. [63]

    Chen, X.-F

    X.-S. Chen, X.-F. Lu, W.-M. Sun, F. Wang, and T. Goldman, Phys. Rev. Lett. 100, 232002 (2008) , arXiv:0806.3166 [hep-ph]

    2008 arXiv

  57. [64]

    Wakamatsu, Phys

    M. Wakamatsu, Phys. Rev. D 81, 114010 (2010) , arXiv:1004.0268 [hep-ph]

    2010 arXiv

  58. [65]

    R. L. Jaffe and A. Manohar, Nucl. Phys. B 337, 509 (1990)

    1990

  59. [66]

    Hatta and S

    Y. Hatta and S. Yoshida, JHEP 10, 080 (2012) , arXiv:1207.5332 [hep-ph]

    2012 arXiv

  60. [67]

    Hatta and J

    Y. Hatta and J. Zhou, Phys. Rev. Lett. 129, 252002 (2022) , arXiv:2207.03378 [hep-ph]

    2022 arXiv

  61. [68]

    Boer et al

    D. Boer et al. , (2011), arXiv:1108.1713 [nucl-th]

    2011 arXiv

  62. [69]

    Achenbach et al

    P. Achenbach et al. , Nucl. Phys. A 1047, 122874 (2024) , arXiv:2303.02579 [hep-ph]

    2024 arXiv

  63. [70]

    I. Y. Kobzarev and L. B. Okun, Zh. Eksp. Teor. Fiz. 43, 1904 (1962)

    1962

  64. [71]

    I. Y. Kobsarev and V. I. Zakharov, Annals Phys. 60, 448 (1970)

    1970

  65. [72]

    S. J. Brodsky, D. S. Hwang, B.-Q. Ma, and I. Schmidt, Nucl. Phys. B 593, 311 (2001) , arXiv:hep-th/0003082

    2001 arXiv

  66. [73]

    Kharzeev, Proc

    D. Kharzeev, Proc. Int. Sch. Phys. Fermi 130, 105 (1996) , arXiv:nucl-th/9601029

    1996 arXiv

  67. [74]

    Kharzeev, H

    D. Kharzeev, H. Satz, A. Syamtomov, and G. Zinovjev, Eur. Phys. J. C 9, 459 (1999) , arXiv:hep-ph/9901375

    1999 arXiv

  68. [75]

    Y.-B. Yang, J. Liang, Y.-J. Bi, Y. Chen, T. Draper, K.-F. Liu, and Z. Liu, Phys. Rev. Lett. 121, 212001 (2018) , arXiv:1808.08677 [hep-lat]

    2018 arXiv

  69. [76]

    F. He, P. Sun, and Y.-B. Yang ( χ QCD), Phys. Rev. D 104, 074507 (2021) , arXiv:2101.04942 [hep-lat]

    2021 arXiv

  70. [77]

    B. Wang, F. He, G. Wang, T. Draper, J. Liang, K.-F. Liu, an d Y.-B. Yang ( χ QCD), Phys. Rev. D 109, 094504 (2024) , arXiv:2401.05496 [hep-lat]

    2024 arXiv

  71. [78]

    Amrath, M

    D. Amrath, M. Diehl, and J.-P. Lansberg, Eur. Phys. J. C 58, 179 (2008) , arXiv:0807.4474 [hep-ph]

    2008 arXiv

  72. [79]

    J. M. M. Chavez, V. Bertone, F. De Soto Borrero, M. Defurn e, C. Mezrag, H. Moutarde, J. Rodr ´ ıguez-Quintero, and J. Segovia, Phys. Rev. D 105, 094012 (2022) , arXiv:2110.06052 [hep-ph]

    2022 arXiv

  73. [80]

    G. F. de T´ eramond, H. G. Dosch, T. Liu, R. S. Sufian, S. J. B rodsky, and A. Deur (HLFHS), Phys. Rev. D 104, 114005 (2021) , arXiv:2107.01231 [hep-ph]

    2021 arXiv

  74. [81]

    Kriesten, P

    B. Kriesten, P. Velie, E. Yeats, F. Y. Lopez, and S. Liuti , Phys. Rev. D 105, 056022 (2022) , arXiv:2101.01826 [hep-ph]

    2022 arXiv

  75. [82]

    Pire and L

    B. Pire and L. Szymanowski, Phys. Rev. D 96, 114008 (2017) , arXiv:1711.04608 [hep-ph]

    2017 arXiv

  76. [83]

    B. Pire, L. Szymanowski, and J. Wagner, Phys. Rev. D 104, 094002 (2021) , arXiv:2104.04944 [hep-ph] . 22

    2021 arXiv

  77. [84]

    Qiu and Z

    J.-W. Qiu and Z. Yu, Phys. Rev. Lett. 131, 161902 (2023) , arXiv:2305.15397 [hep-ph]

    2023 arXiv

  78. [85]

    Chen, H.-W

    J.-W. Chen, H.-W. Lin, and J.-H. Zhang, Nucl. Phys. B 952, 114940 (2020) , arXiv:1904.12376 [hep-lat]

    2020 arXiv

  79. [86]

    Alexandrou et al

    C. Alexandrou et al. , Phys. Rev. D 101, 034519 (2020) , arXiv:1908.10706 [hep-lat]

    2020 arXiv

  80. [87]

    Alexandrou, K

    C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyian nakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. Lett. 125, 262001 (2020) , arXiv:2008.10573 [hep-lat]

    2020 arXiv

  81. [88]

    Lin, Phys

    H.-W. Lin, Phys. Rev. Lett. 127, 182001 (2021) , arXiv:2008.12474 [hep-ph]

    2021 arXiv

  82. [89]

    Lin, Phys

    H.-W. Lin, Phys. Lett. B 824, 136821 (2022) , arXiv:2112.07519 [hep-lat]

    2022 arXiv

  83. [90]

    Alexandrou, K

    C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyian nakou, K. Jansen, A. Scapellato, and F. Steffens, Phys. Rev. D 105, 034501 (2022) , arXiv:2108.10789 [hep-lat]

    2022 arXiv

  84. [91]

    Bhattacharya, K

    S. Bhattacharya, K. Cichy, M. Constantinou, J. Dodson, X. Gao, A. Metz, S. Mukherjee, A. Scapellato, F. Steffens, and Y. Zhao, Phys. Rev. D 106, 114512 (2022) , arXiv:2209.05373 [hep-lat]

    2022 arXiv

  85. [92]

    Lin, Phys

    H.-W. Lin, Phys. Lett. B 846, 138181 (2023) , arXiv:2310.10579 [hep-lat]

    2023 arXiv

  86. [93]

    Holligan and H.-W

    J. Holligan and H.-W. Lin, Phys. Rev. D 110, 034503 (2024) , arXiv:2312.10829 [hep-lat]

    2024 arXiv

  87. [94]

    Bhattacharya, K

    S. Bhattacharya, K. Cichy, M. Constantinou, J. Dodson, A. Metz, A. Scapellato, and F. Steffens, Phys. Rev. D 108, 054501 (2023) , arXiv:2306.05533 [hep-lat]

    2023 arXiv

  88. [95]

    Bhattacharya, K

    S. Bhattacharya, K. Cichy, M. Constantinou, A. Metz, N. Nurminen, and F. Steffens, Phys. Rev. D 110, 054502 (2024) , arXiv:2405.04414 [hep-lat]

    2024 arXiv

  89. [96]

    Hannaford-Gunn, K

    A. Hannaford-Gunn, K. U. Can, J. A. Crawford, R. Horsley , P. E. L. Rakow, G. Schierholz, H. St¨ uben, R. D. Young, and J. M. Zanotti (CSSM/QCDSF/UKQCD), Phys. Rev. D 110, 014509 (2024) , arXiv:2405.06256 [hep-lat]

    2024 arXiv

  90. [97]

    Dutrieux, R

    H. Dutrieux, R. G. Edwards, C. Egerer, J. Karpie, C. Mona han, K. Orginos, A. Radyushkin, D. Richards, E. Romero, and S. Zafeiropoulos (HadStruc), JHEP 08, 162 (2024) , arXiv:2405.10304 [hep-lat]

    2024 arXiv

  91. [98]

    Z. Fan, R. Zhang, and H.-W. Lin, Int. J. Mod. Phys. A 36, 2150080 (2021) , arXiv:2007.16113 [hep-lat]

    2021 arXiv

  92. [99]

    Fan and H.-W

    Z. Fan and H.-W. Lin, Phys. Lett. B 823, 136778 (2021) , arXiv:2104.06372 [hep-lat]

    2021 arXiv

  93. [100]

    Z. Fan, W. Good, and H.-W. Lin, Phys. Rev. D 108, 014508 (2023) , arXiv:2210.09985 [hep-lat]

    2023 arXiv

  94. [101]

    Salas-Chavira, Z

    A. Salas-Chavira, Z. Fan, and H.-W. Lin, Phys. Rev. D 106, 094510 (2022) , arXiv:2112.03124 [hep-lat]

    2022 arXiv

  95. [103]

    Delmar, C

    J. Delmar, C. Alexandrou, K. Cichy, M. Constantinou, an d K. Hadjiyiannakou, Phys. Rev. D 108, 094515 (2023) , arXiv:2310.01389 [hep-lat]

    2023 arXiv

  96. [104]

    W. Good, K. Hasan, A. Chevis, and H.-W. Lin, Phys. Rev. D 109, 114509 (2024) , arXiv:2310.12034 [hep-lat]

    2024 arXiv

  97. [105]

    W. Good, K. Hasan, and H.-W. Lin, (2024), arXiv:2409.02750 [hep-lat]

    2024 arXiv

  98. [106]

    Egerer et al

    C. Egerer et al. (HadStruc), Phys. Rev. D 106, 094511 (2022) , arXiv:2207.08733 [hep-lat]

    2022 arXiv

  99. [107]

    T. Khan, T. Liu, and R. S. Sufian, Phys. Rev. D 108, 074502 (2023) , arXiv:2211.15587 [hep-lat]

    2023 arXiv

  100. [108]

    Balitsky, W

    I. Balitsky, W. Morris, and A. Radyushkin, Phys. Lett. B 808, 135621 (2020) , arXiv:1910.13963 [hep-ph]

    2020 arXiv

  101. [109]

    V. N. Gribov, B. L. Ioffe, and I. Y. Pomeranchuk, Yad. Fiz . 2, 768 (1965)

    1965

  102. [110]

    B. L. Ioffe, Phys. Lett. B 30, 123 (1969)

    1969

  103. [111]

    Braun, P

    V. Braun, P. Gornicki, and L. Mankiewicz, Phys. Rev. D 51, 6036 (1995) , arXiv:hep-ph/9410318

    1995 arXiv

  104. [112]

    V. M. Braun, M. Koller, and J. Schoenleber, Phys. Rev. D 109, 074510 (2024) , arXiv:2401.08012 [hep-ph]

    2024 arXiv

  105. [113]

    F. Yao, Y. Ji, and J.-H. Zhang, JHEP 11, 021 (2023) , arXiv:2212.14415 [hep-ph]

    2023 arXiv

  106. [114]

    X. Ji, Y. Liu, A. Sch¨ afer, W. Wang, Y.-B. Yang, J.-H. Zh ang, and Y. Zhao, Nucl. Phys. B 964, 115311 (2021) , arXiv:2008.03886 [hep-ph]

    2021 arXiv

  107. [115]

    V. M. Braun, JHEP 10, 134 (2023) , arXiv:2308.04270 [hep-ph]

    2023 arXiv

  108. [116]

    T. A. Chowdhury, T. Izubuchi, M. Kamruzzaman, N. Karth ik, T. Khan, T. Liu, A. Paul, J. Schoenleber, and R. S. Sufian, (2024), arXiv:2409.17234 [hep-lat]

    2024 arXiv

  109. [117]

    A. V. Radyushkin, Phys. Rev. D 59, 014030 (1999) , arXiv:hep-ph/9805342

    1999 arXiv

  110. [118]

    A. V. Radyushkin, Phys. Lett. B 449, 81 (1999) , arXiv:hep-ph/9810466

    1999 arXiv

  111. [119]

    A. V. Radyushkin, Phys. Rev. D 109, 014514 (2024) , arXiv:2311.06007 [hep-ph]

    2024 arXiv

  112. [120]

    Chakrabarti, P

    D. Chakrabarti, P. Choudhary, B. Gurjar, T. Maji, C. Mo ndal, and A. Mukherjee, Phys. Rev. D 109, 114040 (2024) , arXiv:2402.16503 [hep-ph]

    2024 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.