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REVIEW 1 major objections 106 references

Leading-twist to higher-twist generalized parton distributions of the pseudoscalar mesons at non-zero skewness

T0 review · 1 major / 0 minor · reviewed 2026-07-02 · grok-4.3

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Pith's one-line read Higher-twist GPDs of the pion and kaon show SU(3) breaking via shifted strange-quark localizations and mass suppression.

desk verdict LFQM calculation fills in twist-4 GPDs for pion and kaon but model artifacts are unquantified. read the letter →

arxiv 2607.00503 v1 pith:T5DHIYCE submitted 2026-07-01 hep-ph

Abi Jebarson A , Navpreet Kaur , Harleen Dahiya This is my paper · ORCID
classification hep-ph
keywords generalizedpartondistributionspionkaonhigher-twistSU(3)flavorsymmetrybreakinglight-frontquarkmodelimpactparameternon-zeroskewness
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

The paper evaluates the complete set of eight generalized parton distributions up to twist-4 for the pion and kaon at non-zero skewness inside the DGLAP region. It employs the light-front quark model with the Brodsky-Huang-Lepage prescription to generate the momentum-space distributions, then performs Fourier transforms to obtain impact-parameter dependent parton distributions in the transverse plane and diffraction patterns along the longitudinal direction. The resulting numbers indicate that the strange quark in the kaon moves spatial localizations relative to the lighter quarks, while higher-twist amplitudes remain large in the pion yet become heavily suppressed once the larger kaon mass is accounted for. A reader would care because these distributions supply a three-dimensional tomographic image of how flavor symmetry breaking and mass scales shape the partonic content of the lightest mesons.

What carries the argument

Light-front quark model with the Brodsky-Huang-Lepage prescription, which supplies the momentum-space GPDs that are Fourier-transformed into impact-parameter and longitudinal distributions.

What would settle it

A lattice QCD computation or future scattering measurement that finds identical transverse localizations for up and strange quarks inside the kaon, or comparable higher-twist amplitudes in the kaon and pion despite the mass difference, would falsify the reported numerical patterns.

Watch

Extended reading notes

Core claim

Utilizing the light-front quark model with the Brodsky-Huang-Lepage prescription, the complete set of eight GPDs up to twist-4 are computed for spin-0 mesons in the non-zero skewness domain within the DGLAP region. The numerical results reveal the consequences of SU(3) flavor symmetry breaking, as the strange quark in the kaon dynamically shifts spatial localizations compared to the lighter up quarks. We also observe that while higher-twist correlations exhibit massive amplitude scaling in the pion, they are heavily suppressed by the larger macroscopic mass of the kaon.

Load-bearing premise

The light-front quark model with the Brodsky-Huang-Lepage prescription provides an accurate description of the higher-twist GPDs of the pion and kaon in the non-zero skewness DGLAP region without significant higher-order corrections or model artifacts.

Editorial extensions

If this is right

  • The strange quark in the kaon produces a dynamic shift in parton spatial localizations relative to the pion.
  • Higher-twist correlations display large amplitudes in the pion but become heavily suppressed once the kaon mass is introduced.
  • Fourier transformation of the GPDs yields impact-parameter dependent parton distributions in the transverse plane.
  • Corresponding diffraction patterns appear in the longitudinal coordinate space.
  • The set of distributions assembles a three-dimensional tomographic picture of the mesons' partonic structure.

Reading between the lines

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

  • The observed mass-driven suppression suggests that parton correlation strengths may correlate directly with the overall meson mass scale across other light mesons.
  • The flavor-dependent shift in transverse localization could be compared against transverse-momentum dependent distributions extracted from different processes.
  • Extension of the same model setup to vector mesons would test whether the reported SU(3) patterns persist when spin-1 degrees of freedom are added.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript computes the complete set of eight GPDs up to twist-4 for the pion and kaon using the light-front quark model with the Brodsky-Huang-Lepage prescription, restricted to the DGLAP region at nonzero skewness. Fourier transforms yield impact-parameter dependent PDFs in the transverse plane and longitudinal diffraction patterns; the numerical results are presented as exhibiting SU(3) flavor-symmetry breaking via shifts in strange-quark spatial localization relative to up quarks, together with strong mass-driven suppression of higher-twist amplitudes in the kaon relative to the pion.

Significance. If the model extension proves reliable, the work supplies a three-dimensional tomographic picture of pseudoscalar-meson partonic structure and quantifies flavor-breaking and mass effects on higher-twist GPDs, which could serve as input for phenomenological analyses. The explicit computation of the full twist-4 set in a single consistent framework is a positive feature.

major comments (1)
  1. [Abstract] Abstract: the central claims of SU(3) breaking in spatial localizations and massive suppression of twist-4 amplitudes in the kaon are obtained from the LFQM+BHL wave functions; however, the BHL prescription is tuned to leading-twist observables, and the manuscript provides no quantification of artifacts introduced when the same ansatz is applied to twist-4 operators (which involve additional transverse derivatives or quark-gluon correlations) nor validation against known limits in the DGLAP region at ξ eq 0. This directly affects the load-bearing numerical conclusions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive critique. The concern regarding the applicability of the BHL prescription beyond leading twist is valid and directly impacts the interpretation of our numerical results. We address it below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims of SU(3) breaking in spatial localizations and massive suppression of twist-4 amplitudes in the kaon are obtained from the LFQM+BHL wave functions; however, the BHL prescription is tuned to leading-twist observables, and the manuscript provides no quantification of artifacts introduced when the same ansatz is applied to twist-4 operators (which involve additional transverse derivatives or quark-gluon correlations) nor validation against known limits in the DGLAP region at ξ eq 0. This directly affects the load-bearing numerical conclusions.

    Authors: We agree that the BHL prescription originates from leading-twist phenomenology and that extending the identical ansatz to twist-4 operators introduces uncontrolled systematic effects not quantified in the present work. The manuscript treats the LFQM+BHL framework as a consistent but approximate model across twists without additional validation at ξ=0 or explicit error estimates for the higher-twist operators. We will revise the abstract to tone down the definitiveness of the claims, add an explicit limitations paragraph in the introduction, and include a brief discussion of the model’s leading-twist heritage together with a statement that the twist-4 results should be viewed as exploratory within this single framework. No new numerical validation at ξ=0 will be added at this stage. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: direct model evaluation of GPDs

full rationale

The paper applies the established LFQM+BHL light-front wave function to evaluate the eight GPDs (leading through twist-4) in the DGLAP region at nonzero skewness. The numerical outputs (SU(3) breaking in IPDPDFs, mass-driven suppression of higher-twist amplitudes) are obtained by direct integration over the model wave functions with distinct up- and strange-quark masses; they do not reduce to the input meson masses or decay constants by algebraic identity or by renaming a fit. No self-citation chain, uniqueness theorem, or ansatz smuggling is invoked to force the central claims. The calculation is therefore a standard phenomenological prediction whose validity can be tested against external data or lattice results.

Assumptions & free parameters 1 free parameters · 1 assumptions · 0 invented entities

The calculation rests on the light-front quark model whose parameters are not derived from first principles within the paper.

free parameters (1)
  • LFQM parameters (quark masses, harmonic oscillator scale)
    BHL prescription in LFQM requires parameters fitted to meson properties to define the light-front wave functions.
assumptions (1)
  • domain assumption Light-front quark model with BHL prescription accurately captures twist-4 GPDs in the DGLAP region
    All numerical results are generated inside this model framework.

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Cite this review

Pith. "Pith review of Leading-twist to higher-twist generalized parton distributions of the pseudoscalar mesons at non-zero skewness." pith.science (2026). https://pith.science/paper/T5DHIYCE

@misc{pith2026260700503,
  author       = {Pith},
  title        = {Pith review of: Leading-twist to higher-twist generalized parton distributions of the pseudoscalar mesons at non-zero skewness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T5DHIYCE}},
  note         = {Machine review of arXiv:2607.00503}
}
abstract

We investigate the multidimensional partonic structure of spin-0 mesons, specifically the pion and the kaon, by evaluating their complete set of eight generalized parton distributions (GPDs) up to twist-4. Utilizing the light-front quark model (LFQM) with the Brodsky-Huang-Lepage (BHL) prescription, we compute these distributions in the kinematically rich non-zero skewness ($\xi \neq 0$) domain, strictly within the DGLAP region, $x \in [\xi, 1]$. To construct a three-dimensional tomographic picture, we perform Fourier transforms of the momentum-space GPDs to obtain the impact parameter dependent parton distribution functions (IPDPDFs) in the transverse plane and the corresponding diffraction patterns in the longitudinal coordinate space. The numerical results explicitly reveal the consequences of $\mathrm{SU}(3)$ flavor symmetry breaking, as the strange quark in the kaon dynamically shifts spatial localizations compared to the lighter up quarks. We also observe that while higher-twist correlations exhibit massive amplitude scaling in the pion, they are heavily suppressed by the larger macroscopic mass of the kaon.

Figures

Figures reproduced from arXiv: 2607.00503 by the authors.

Figure 1
Figure 1. FIG. 1. The eight GPDs of the pion ( [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The eight GPDs of the kaon as a function of longitudinal momentum fraction [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The GPDs of the pion [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The kaon GPDs as functions of longitudinal momentum fraction [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The six IPDPDFs (in fm [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The first six and the next six plots show the IPDPDFs (in fm [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Longitudinal spatial distributions ( [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

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Works this paper leans on

106 extracted references · 106 canonical work pages

  1. [1]

    R. P. Feynman, Phys. Rev. Lett.23, 1415 (1969)

    work page 1969

  2. [2]

    J. D. Bjorken and E. A. Paschos, Phys. Rev.185, 1975 (1969)

    work page 1975

  3. [3]

    J. C. Collins and D. E. Soper, Nucl. Phys. B194, 445 (1982)

    work page 1982

  4. [4]

    A. D. Martin, A. J. T. M. Mathijssen, W. J. Stirling, R. S. Thorne, B. J. A. Watt, and G. Watt, Eur. Phys. J. C73, 2318 (2013)

    work page 2013

  5. [5]

    R. L. Jaffe and X.-D. Ji, Nucl. Phys. B375, 527 (1992)

    work page 1992

  6. [6]

    A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C63, 189 (2009)

    work page 2009

  7. [7]

    Hofstadter, Rev

    R. Hofstadter, Rev. Mod. Phys.28, 214 (1956). 27

    work page 1956

  8. [8]

    C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen, Prog. Part. Nucl. Phys.59, 694 (2007)

    work page 2007

Show all 106 references
  1. [9]

    Diehl, Phys

    M. Diehl, Phys. Rept.388, 41 (2003)

    2003

  2. [10]

    Diehl, T

    M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Eur. Phys. J. C39, 1 (2005)

    2005

  3. [11]

    Khodjamirian, T

    A. Khodjamirian, T. Mannel, and N. Offen, Phys. Rev. D75, 054013 (2007)

    2007

  4. [12]

    M¨ uller, D

    D. M¨ uller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hoˇ rejˇ si, Fortsch. Phys.42, 101 (1994)

    1994

  5. [13]

    A. V. Radyushkin, Phys. Rev. D56, 5524 (1997)

    1997

  6. [14]

    Ji, Phys

    X.-D. Ji, Phys. Rev. D55, 7114 (1997)

    1997

  7. [15]

    Diehl, Eur

    M. Diehl, Eur. Phys. J. A52, 149 (2016)

    2016

  8. [16]

    Boffi and B

    S. Boffi and B. Pasquini, Riv. Nuovo Cim.30, 387 (2007)

    2007

  9. [17]

    A. V. Radyushkin, Phys. Lett. B380, 417 (1996)

    1996

  10. [18]

    A. V. Belitsky, D. Mueller, and A. Kirchner, Nucl. Phys. B629, 323 (2002)

    2002

  11. [19]

    A. V. Belitsky and A. V. Radyushkin, Phys. Rept.418, 1 (2005)

    2005

  12. [20]

    Diehl and P

    M. Diehl and P. Kroll, Eur. Phys. J. C73, 2397 (2013)

    2013

  13. [21]

    G. D. Rochester and C. C. Butler, Nature160, 855 (1947)

    1947

  14. [22]

    C. M. G. Lattes, H. Muirhead, G. P. S. Occhialini, and C. F. Powell, Nature159, 694 (1947)

    1947

  15. [23]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio, Phys. Rev.122, 345 (1961)

    1961

  16. [24]

    Pich, Rept

    A. Pich, Rept. Prog. Phys.58, 563 (1995)

    1995

  17. [25]

    Kaur and H

    S. Kaur and H. Dahiya, Phys. Rev. D100, 074008 (2019)

    2019

  18. [26]

    S. Kaur, N. Kumar, J. Lan, C. Mondal, and H. Dahiya, Phys. Rev. D102, 014021 (2020)

    2020

  19. [27]

    Kaur and H

    S. Kaur and H. Dahiya, AIP Conf. Proc.2249, 030043 (2020)

    2020

  20. [28]

    Meissner, A

    S. Meissner, A. Metz, M. Schlegel, and K. Goeke, JHEP08, 038

  21. [29]

    A. I. Signal, Nucl. Phys. B497, 415 (1997)

    1997

  22. [30]

    Avakian, A

    H. Avakian, A. V. Efremov, P. Schweitzer, and F. Yuan, Phys. Rev. D81, 074035 (2010)

    2010

  23. [31]

    Lorc´ e, B

    C. Lorc´ e, B. Pasquini, and P. Schweitzer, JHEP01, 103

  24. [32]

    Lu and I

    Z. Lu and I. Schmidt, Phys. Lett. B712, 451 (2012)

    2012

  25. [33]

    Mao and Z

    W. Mao and Z. Lu, Eur. Phys. J. C73, 2557 (2013)

    2013

  26. [34]

    W. Mao, Z. Lu, and B.-Q. Ma, Phys. Rev. D90, 014048 (2014)

    2014

  27. [35]

    X. Liu, W. Mao, X. Wang, and B.-Q. Ma, Phys. Rev. D104, 094043 (2021)

    2021

  28. [36]

    Kumericki, S

    K. Kumericki, S. Liuti, and H. Moutarde, Eur. Phys. J. A52, 157 (2016)

    2016

  29. [37]

    Kroll, EPJ Web Conf.85, 01005 (2015)

    P. Kroll, EPJ Web Conf.85, 01005 (2015). 28

    2015

  30. [38]

    I. V. Anikin, B. Pire, and O. V. Teryaev, Phys. Rev. D62, 071501 (2000)

    2000

  31. [39]

    Mazouzet al.(Jefferson Lab Hall A), Phys

    M. Mazouzet al.(Jefferson Lab Hall A), Phys. Rev. Lett.99, 242501 (2007)

    2007

  32. [40]

    Defurneet al., Nature Commun.8, 1408 (2017)

    M. Defurneet al., Nature Commun.8, 1408 (2017)

    2017

  33. [41]

    Dupre, M

    R. Dupre, M. Guidal, and M. Vanderhaeghen, Phys. Rev. D95, 011501 (2017)

    2017

  34. [42]

    Hadjidakiset al.(CLAS), Phys

    C. Hadjidakiset al.(CLAS), Phys. Lett. B605, 256 (2005)

    2005

  35. [43]

    Chekanovet al.(ZEUS), Nucl

    S. Chekanovet al.(ZEUS), Nucl. Phys. B718, 3 (2005)

    2005

  36. [44]

    Airapetianet al.(HERMES), Phys

    A. Airapetianet al.(HERMES), Phys. Lett. B659, 486 (2008)

    2008

  37. [45]

    Diehl, Eur

    M. Diehl, Eur. Phys. J. C25, 223 (2002), [Erratum: Eur.Phys.J.C 31, 277–278 (2003)]

    2002

  38. [46]

    Burkardt, Int

    M. Burkardt, Int. J. Mod. Phys. A18, 173 (2003)

    2003

  39. [47]

    Burkardt, Phys

    M. Burkardt, Phys. Rev. D62, 071503 (2000), [Erratum: Phys.Rev.D 66, 119903 (2002)]

    2000

  40. [48]

    Burkardt, Phys

    M. Burkardt, Phys. Rev. D62, 094003 (2000)

    2000

  41. [49]

    S. J. Brodsky, D. Chakrabarti, A. Harindranath, A. Mukherjee, and J. P. Vary, Phys. Rev. D75, 014003 (2007)

    2007

  42. [50]

    Collins, Int

    J. Collins, Int. J. Mod. Phys. Conf. Ser.4, 85 (2011)

    2011

  43. [51]

    Meissner, A

    S. Meissner, A. Metz, and K. Goeke, Phys. Rev. D76, 034002 (2007)

    2007

  44. [52]

    X.-D. Ji, W. Melnitchouk, and X. Song, Phys. Rev. D56, 5511 (1997)

    1997

  45. [53]

    Scopetta and V

    S. Scopetta and V. Vento, Eur. Phys. J. A16, 527 (2003)

    2003

  46. [54]

    Scopetta and V

    S. Scopetta and V. Vento, Phys. Rev. D69, 094004 (2004)

    2004

  47. [55]

    Boffi, B

    S. Boffi, B. Pasquini, and M. Traini, Nucl. Phys. B649, 243 (2003)

    2003

  48. [56]

    Boffi, B

    S. Boffi, B. Pasquini, and M. Traini, Nucl. Phys. B680, 147 (2004)

    2004

  49. [57]

    Pasquini and S

    B. Pasquini and S. Boffi, Phys. Rev. D73, 094001 (2006)

    2006

  50. [58]

    Mondal and D

    C. Mondal and D. Chakrabarti, Eur. Phys. J. C75, 261 (2015)

    2015

  51. [59]

    Gutsche, V

    T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, J. Phys. G42, 095005 (2015)

    2015

  52. [60]

    N. Kaur, N. Kumar, C. Mondal, and H. Dahiya, Nucl. Phys. B934, 80 (2018)

    2018

  53. [61]

    Theussl, S

    L. Theussl, S. Noguera, and V. Vento, Eur. Phys. J. A20, 483 (2004)

    2004

  54. [62]

    B. C. Tiburzi and G. A. Miller, Phys. Rev. D67, 054014 (2003)

    2003

  55. [63]

    Ji, Phys

    X. Ji, Phys. Rev. Lett.110, 262002 (2013)

    2013

  56. [64]

    Chen, H.-W

    J.-W. Chen, H.-W. Lin, and J.-H. Zhang, Nucl. Phys. B952, 114940 (2020)

    2020

  57. [65]

    H.-T. Ding, X. Gao, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, and Y. Zhao, JHEP02, 056

  58. [66]

    Son and P

    H.-D. Son and P. T. P. Hutauruk, Phys. Rev. D111, 054007 (2025). 29

    2025

  59. [67]

    Zhang, Chin

    J.-L. Zhang, Chin. Phys. C50, 033109 (2026)

    2026

  60. [68]

    Lorc´ e, B

    C. Lorc´ e, B. Pasquini, and P. Schweitzer, Eur. Phys. J. C76, 415 (2016)

    2016

  61. [69]

    Pasquini and S

    B. Pasquini and S. Rodini, Phys. Lett. B788, 414 (2019)

    2019

  62. [70]

    Mukherjee and M

    A. Mukherjee and M. Vanderhaeghen, Phys. Lett. B542, 245 (2002)

    2002

  63. [71]

    Mukherjee and M

    A. Mukherjee and M. Vanderhaeghen, Phys. Rev. D67, 085020 (2003)

    2003

  64. [72]

    Balla, M

    J. Balla, M. V. Polyakov, and C. Weiss, Nucl. Phys. B510, 327 (1998)

    1998

  65. [73]

    Dressler and M

    B. Dressler and M. V. Polyakov, Phys. Rev. D61, 097501 (2000)

    2000

  66. [74]

    Schweitzer, Phys

    P. Schweitzer, Phys. Rev. D67, 114010 (2003)

    2003

  67. [75]

    Wakamatsu and Y

    M. Wakamatsu and Y. Ohnishi, Phys. Rev. D67, 114011 (2003)

    2003

  68. [76]

    Wakamatsu, Phys

    M. Wakamatsu, Phys. Lett. B653, 398 (2007)

    2007

  69. [77]

    Ohnishi and M

    Y. Ohnishi and M. Wakamatsu, Phys. Rev. D69, 114002 (2004)

    2004

  70. [78]

    Cebulla, J

    C. Cebulla, J. Ossmann, P. Schweitzer, and D. Urbano, Acta Phys. Polon. B39, 609 (2008)

    2008

  71. [79]

    Sharma, N

    S. Sharma, N. Kumar, and H. Dahiya, Nucl. Phys. B992, 116247 (2023)

    2023

  72. [80]

    Z. Zhu, Z. Hu, J. Lan, C. Mondal, X. Zhao, and J. P. Vary (BLFQ), Phys. Lett. B839, 137808 (2023)

    2023

  73. [81]

    Aslan and M

    F. Aslan and M. Burkardt, Phys. Rev. D101, 016010 (2020)

    2020

  74. [82]

    Sharma and H

    S. Sharma and H. Dahiya, Eur. Phys. J. A59, 235 (2023)

    2023

  75. [83]

    Bhattacharya, K

    S. Bhattacharya, K. Cichy, M. Constantinou, J. Dodson, A. Metz, A. Scapellato, and F. Stef- fens, Phys. Rev. D108, 054501 (2023)

    2023

  76. [84]

    Zhang, Z

    Z. Zhang, Z. Hu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary (BLFQ), Phys. Rev. D109, 034031 (2024)

    2024

  77. [85]

    Sharma and H

    S. Sharma and H. Dahiya, Int. J. Mod. Phys. A37, 2250205 (2022)

    2022

  78. [86]

    F. P. Aslan, M. Burkardt, and M. Schlegel, Phys. Rev. D100, 096021 (2019)

    2019

  79. [87]

    Luan and Z

    X. Luan and Z. Lu, Phys. Rev. D110, 074022 (2024)

    2024

  80. [88]

    Puhan and H

    S. Puhan and H. Dahiya, Phys. Rev. D111, 114039 (2025)

    2025

  81. [89]

    S. J. Brodsky, D. S. Hwang, B.-Q. Ma, and I. Schmidt, Nucl. Phys. B593, 311 (2001)

    2001

  82. [90]

    G. P. Lepage and S. J. Brodsky, Phys. Rev. D22, 2157 (1980)

    1980

  83. [91]

    Bacchetta, F

    A. Bacchetta, F. Conti, and M. Radici, Phys. Rev. D78, 074010 (2008)

    2008

  84. [92]

    Lu and I

    Z. Lu and I. Schmidt, Phys. Rev. D75, 073008 (2007)

    2007

  85. [93]

    Luan and Z

    X. Luan and Z. Lu, Phys. Lett. B833, 137299 (2022)

    2022

  86. [94]

    Ma and Z

    Z.-L. Ma and Z. Lu, Phys. Rev. D98, 054024 (2018). 30

    2018

  87. [95]

    Luan and Z

    X. Luan and Z. Lu, Phys. Rev. D109, 094016 (2024)

    2024

  88. [96]

    Luan and Z

    X. Luan and Z. Lu, Eur. Phys. J. C83, 504 (2023)

    2023

  89. [97]

    S. J. Brodsky, T. Huang, and G. P. Lepage, Springer Tracts Mod. Phys.100, 81 (1982)

    1982

  90. [98]

    Xiao and B.-Q

    B.-W. Xiao and B.-Q. Ma, Phys. Rev. D68, 034020 (2003)

    2003

  91. [99]

    Qian and B.-Q

    W. Qian and B.-Q. Ma, Phys. Rev. D78, 074002 (2008)

    2008

  92. [100]

    Wandzura and F

    S. Wandzura and F. Wilczek, Phys. Lett. B72, 195 (1977)

    1977

  93. [101]

    P. A. M. Dirac, Rev. Mod. Phys.21, 392 (1949)

    1949

  94. [102]

    S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Phys. Rept.301, 299 (1998)

    1998

  95. [103]

    S. J. Brodsky, M. Diehl, and D. S. Hwang, Nucl. Phys. B596, 99 (2001)

    2001

  96. [104]

    Diehl, T

    M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Nucl. Phys. B596, 33 (2001), [Erratum: Nucl.Phys.B 605, 647–647 (2001)]

    2001

  97. [105]

    Sharma and H

    S. Sharma and H. Dahiya, Nucl. Phys. B1001, 116522 (2024)

    2024

  98. [106]

    Burkardt and D

    M. Burkardt and D. S. Hwang, Phys. Rev. D69, 074032 (2004). 31

    2004

Pith tools

Reviewed July 2, 2026 · model on record in the stance chip above.