- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Dynamical gluon effects in twist-3 generalized parton distributions of the proton

Ziqi Zhang1,2,3,*, Chandan Mondal1,2,3,†, Siqi Xu1,2,3,‡, Xingbo Zhao1,2,3,§, and James P. Vary4,∥ (BLFQ Collaboration)

  • *Contact author: zhangziqi@impcas.ac.cn
  • †Contact author: mondal@impcas.ac.cn
  • ‡Contact author: xsq234@impcas.ac.cn
  • §Contact author: xbzhao@impcas.ac.cn
  • ∥Contact author: jvary@iastate.edu
PDF

Phys. Rev. D 113, 094031 – Published 22 May, 2026

DOI: https://doi.org/10.1103/1yjm-sqck

Abstract

Within the basis light-front quantization framework, we systematically investigate the subleading-twist (twist-3) generalized parton distributions (GPDs) of the proton’s valence quarks beyond the Wandzura-Wilczek (WW) approximation. The twist-3 GPDs are not independent; through the equations of motion they decompose into a nongenuine contribution and a genuine twist-3 term. The latter encodes quark–quark–gluon correlations and involves interference between the light-front Fock sectors |qqq⟩ and |qqqg⟩, which are typically neglected in the WW approximation. Using light-front wave functions obtained from diagonalizing the proton light-front Hamiltonian for its |qqq⟩ and |qqqg⟩ Fock components, we compute these GPDs via their overlap representations. To further explore their physical implications, we also evaluate several twist-3–related quantities, including the quark orbital angular momentum, the total quark spin contribution, and the quark spin-orbit correlation. Our results provide new nonperturbative input on higher-twist dynamics particularly multiparton interference effects relevant for future measurements at the EicC and the EIC.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (95)

  1. Stefan Diehl, Experimental exploration of the 3D nucleon structure, Prog. Part. Nucl. Phys. 133, 104069 (2023).
  2. Cédric Lorcé, Andreas Metz, Barbara Pasquini, and Peter Schweitzer, Parton distribution functions and their generalizations, arXiv:2507.12664.
  3. F. X. Girod, R. A. Niyazov, H. Avakian, J. Ball, I. Bedlinskiy, V. D. Burkert, R. De Masi, L. Elouadrhiri, M. Garçon, M. Guidal et al., Measurement of deeply virtual Compton scattering beam-spin asymmetries, Phys. Rev. Lett. 100, 162002 (2008).
  4. S. Pisano, A. Biselli, S. Niccolai, E. Seder, M. Guidal, M. Mirazita, K. P. Adhikari, D. Adikaram, M. J. Amaryan, M. D. Anderson et al., Single and double spin asymmetries for deeply virtual Compton scattering measured with clas and a longitudinally polarized proton target, Phys. Rev. D 91, 052014 (2015).
  5. Hyon-Suk Jo, F. X. Girod, H. Avakian, V. D. Burkert, M. Garçon, M. Guidal, V. Kubarovsky, S. Niccolai, P. Stoler, K. P. Adhikari et al., Cross sections for the exclusive photon electroproduction on the proton and generalized parton distributions, Phys. Rev. Lett. 115, 212003 (2015).
  6. M. Hattawy, N. A. Baltzell, R. Dupré, S. Bültmann, R. De Vita, A. El Alaoui, L. El Fassi, H. Egiyan, F. X. Girod, M. Guidal et al., Exploring the structure of the bound proton with deeply virtual Compton scattering, Phys. Rev. Lett. 123, 032502 (2019).
  7. C. Munoz Camacho, Alexandre Camsonne, Malek Mazouz, Catherine Ferdi, Gagik Gavalian, Elena Kuchina, M. Amarian, K. A. Aniol, Matthieu Beaumel, Hachemi Benaoum et al., Scaling tests of the cross section for deeply virtual Compton scattering, Phys. Rev. Lett. 97, 262002 (2006).
  8. M. Mazouz, A. Camsonne, C. Munoz Camacho, C. Ferdi, G. Gavalian, E. Kuchina, M. Amarian, K. A. Aniol, M. Beaumel, H. Benaoum et al., Deeply virtual Compton scattering off the neutron, Phys. Rev. Lett. 99, 242501 (2007).
  9. Catherine Adloff, V. Andreev, B. Andrieu, T. Anthonis, V. Arkadov, A. Astvatsatourov, A. Babaev, J. Bähr, P. Baranov, E. Barrelet et al., Measurement of deeply virtual Compton scattering at HERA, Phys. Lett. B 517, 47 (2001).
  10. Francise D. Aaron, A. Aktas, C. Alexa, V. Andreev, B. Antunovic, S. Aplin, A. Asmone, A. Astvatsatourov, S. Backovic, A. Baghdasaryan et al., Measurement of deeply virtual Compton scattering and its t-dependence at HERA, Phys. Lett. B 659, 796 (2008).
  11. Francise D. Aaron, M. Aldaya Martin, C. Alexa, K. Alimujiang, V. Andreev, B. Antunovic, S. Backovic, A. Baghdasaryan, E. Barrelet, W. Bartel et al., Deeply virtual Compton scattering and its beam charge asymmetry in e±p collisions at hera, Phys. Lett. B 681, 391 (2009).
  12. Zeus Collaboration, A measurement of the Q2, W and t dependences of deeply virtual Compton scattering at HERA, J. High Energy Phys. 05 (2009) 108.
  13. A. Airapetian, N. Akopov, Z. Akopov, E. C. Aschenauer, W. Augustyniak, R. Avakian, A. Avetissian, E. Avetisyan, H. P. Blok, A. Borissov et al., Beam-helicity and beam-charge asymmetries associated with deeply virtual Compton scattering on the unpolarised proton, J. High Energy Phys. 07 (2012) 032.
  14. A. Airapetian, N. Akopov, Z. Akopov, E. C. Aschenauer, W. Augustyniak, R. Avakian, A. Avetissian, E. Avetisyan, S. Belostotski, H. P. Blok et al., Beam-helicity asymmetry arising from deeply virtual Compton scattering measured with kinematically complete event reconstruction, J. High Energy Phys. 10 (2012) 042.
  15. R. Abdul Khalek, A. Accardi, J. Adam, D. Adamiak, W. Akers, M. Albaladejo, A. Al-Bataineh, M. G. Alexeev, F. Ameli, P. Antonioli et al., Science requirements and detector concepts for the electron-ion collider: EIC yellow report, Nucl. Phys. A1026, 122447 (2022).
  16. Daniele P. Anderle, Valerio Bertone, Xu Cao, Lei Chang, Ningbo Chang, Gu Chen, Xurong Chen, Zhuojun Chen, Zhufang Cui, Lingyun Dai et al., Electron-ion collider in china, Front. Phys. 16, 1 (2021).
  17. Xiang-Dong Ji, W. Melnitchouk, and X. Song, A study of off forward parton distributions, Phys. Rev. D 56, 5511 (1997).
  18. I. V. Anikin, D. Binosi, R. Medrano, S. Noguera, and V. Vento, Single spin asymmetry parameter from deeply virtual Compton scattering of hadrons up to twist—three accuracy. 1. Pion case, Eur. Phys. J. A 14, 95 (2002).
  19. K. Goeke, Maxim V. Polyakov, and M. Vanderhaeghen, Hard exclusive reactions and the structure of hadrons, Prog. Part. Nucl. Phys. 47, 401 (2001).
  20. V. Yu. Petrov, P. V. Pobylitsa, Maxim V. Polyakov, I. Bornig, K. Goeke, and C. Weiss, Off—forward quark distributions of the nucleon in the large N(c) limit, Phys. Rev. D 57, 4325 (1998).
  21. M. Penttinen, Maxim V. Polyakov, and K. Goeke, Helicity skewed quark distributions of the nucleon and chiral symmetry, Phys. Rev. D 62, 014024 (2000).
  22. Sergio Scopetta and Vicente Vento, Generalized parton distributions and composite constituent quarks, Phys. Rev. D 69, 094004 (2004).
  23. Sergio Scopetta and Vicente Vento, Generalized parton distributions in constituent quark models, Eur. Phys. J. A 16, 527 (2003).
  24. S. Boffi, B. Pasquini, and M. Traini, Linking generalized parton distributions to constituent quark models, Nucl. Phys. B649, 243 (2003).
  25. S. Boffi, B. Pasquini, and M. Traini, Helicity dependent generalized parton distributions in constituent quark models, Nucl. Phys. B680, 147 (2004).
  26. Sergio Scopetta and Vicente Vento, Helicity-dependent generalized parton distributions and composite constituent quarks, Phys. Rev. D 71, 014014 (2005).
  27. Chandan Mondal, Helicity-dependent generalized parton distributions for nonzero skewness, Eur. Phys. J. C 77, 640 (2017).
  28. Tanmay Maji, Chandan Mondal, and Dipankar Chakrabarti, Leading twist generalized parton distributions and spin densities in a proton, Phys. Rev. D 96, 013006 (2017).
  29. Dipankar Chakrabarti and Chandan Mondal, Chiral-odd generalized parton distributions for proton in a light-front quark-diquark model, Phys. Rev. D 92, 074012 (2015).
  30. Tanmay Maji, Chandan Mondal, D. Chakrabarti, and O. V. Teryaev, Relating transverse structure of various parton distributions, J. High Energy Phys. 01 (2016) 165.
  31. Chandan Mondal and Dipankar Chakrabarti, Generalized parton distributions and transverse densities in a light-front quark–diquark model for the nucleons, Eur. Phys. J. C 75, 261 (2015).
  32. B. Pasquini and S. Boffi, Virtual meson cloud of the nucleon and generalized parton distributions, Phys. Rev. D 73, 094001 (2006).
  33. B. Pasquini and S. Boffi, Generalized parton distributions in a meson cloud model, Nucl. Phys. A782, 86 (2007).
  34. Alfredo Vega, Ivan Schmidt, Thomas Gutsche, and Valery E. Lyubovitskij, Generalized parton distributions in AdS/QCD, Phys. Rev. D 83, 036001 (2011).
  35. Dipankar Chakrabarti and Chandan Mondal, Generalized parton distributions for the proton in AdS/QCD, Phys. Rev. D 88, 073006 (2013).
  36. Matteo Rinaldi, GPDs at nonzero skewness in ADS/QCD model, Phys. Lett. B 771, 563 (2017).
  37. Marco Claudio Traini, Generalized parton distributions: Confining potential effects within AdS/QCD, Eur. Phys. J. C 77, 246 (2017).
  38. Guy F. de Teramond, Tianbo Liu, Raza Sabbir Sufian, Hans Günter Dosch, Stanley J. Brodsky, and Alexandre Deur, Universality of generalized parton distributions in light-front holographic QCD, Phys. Rev. Lett. 120, 182001 (2018).
  39. Bheemsehan Gurjar, Chandan Mondal, and Dipankar Chakrabarti, Polarized gluon distribution in the proton from holographic light-front QCD, Phys. Rev. D 107, 054013 (2023).
  40. B. C. Tiburzi and G. A. Miller, Exploring skewed parton distributions with two-body models on the light front. 1. Bimodality, Phys. Rev. C 64, 065204 (2001).
  41. B. C. Tiburzi and G. A. Miller, Exploring skewed parton distributions with two-body models on the light front 2: Covariant Bethe-Salpeter approach, Phys. Rev. D 65, 074009 (2002).
  42. A. Mukherjee and M. Vanderhaeghen, Helicity dependent twist two and twist three generalized parton distributions in light front QCD, Phys. Rev. D 67, 085020 (2003).
  43. Bolang Lin, Sreeraj Nair, Siqi Xu, Zhi Hu, Chandan Mondal, Xingbo Zhao, and James P. Vary, Generalized parton distributions of gluon in proton: A light-front quantization approach, Phys. Lett. B 847, 138305 (2023).
  44. Bolang Lin, Sreeraj Nair, Chandan Mondal, Siqi Xu, Zhi Hu, Pengxiang Zhang, Xingbo Zhao, and James P. Vary, Chiral-odd gluon generalized parton distributions in the proton: A light-front quantization approach, Phys. Lett. B 860, 139153 (2025).
  45. Ziqi Zhang, Zhi Hu, Siqi Xu, Chandan Mondal, Xingbo Zhao, and James P. Vary (BLFQ Collaboration), Twist-3 generalized parton distribution for the proton from basis light-front quantization, Phys. Rev. D 109, 034031 (2024).
  46. Xiangdong Ji, Parton physics on a Euclidean lattice, Phys. Rev. Lett. 110, 262002 (2013).
  47. Xiangdong Ji, Yu-Sheng Liu, Yizhuang Liu, Jian-Hui Zhang, and Yong Zhao, Large-momentum effective theory, Rev. Mod. Phys. 93, 035005 (2021).
  48. Huey-Wen Lin, Nucleon helicity generalized parton distribution at physical pion mass from lattice QCD, Phys. Lett. B 824, 136821 (2022).
  49. Huey-Wen Lin, Nucleon tomography and generalized parton distribution at physical pion mass from lattice QCD, Phys. Rev. Lett. 127, 182001 (2021).
  50. Shohini Bhattacharya, Krzysztof Cichy, Martha Constantinou, Jack Dodson, Xiang Gao, Andreas Metz, Swagato Mukherjee, Aurora Scapellato, Fernanda Steffens, and Yong Zhao, Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Unpolarized quarks, Phys. Rev. D 106, 114512 (2022).
  51. Constantia Alexandrou, Krzysztof Cichy, Martha Constantinou, Kyriakos Hadjiyiannakou, Karl Jansen, Aurora Scapellato, and Fernanda Steffens, Transversity GPDs of the proton from lattice QCD, Phys. Rev. D 105, 034501 (2022).
  52. C. Alexandrou et al., Moments of the nucleon transverse quark spin densities using lattice QCD, Phys. Rev. D 107, 054504 (2023).
  53. Yuxun Guo, Xiangdong Ji, and Kyle Shiells, Generalized parton distributions through universal moment parameterization: Zero skewness case, J. High Energy Phys. 09 (2022) 215.
  54. Constantia Alexandrou, Krzysztof Cichy, Martha Constantinou, Kyriakos Hadjiyiannakou, Karl Jansen, Aurora Scapellato, and Fernanda Steffens, Unpolarized and helicity generalized parton distributions of the proton within lattice QCD, Phys. Rev. Lett. 125, 262001 (2020).
  55. M. Gockeler, Ph. Hagler, R. Horsley, D. Pleiter, Paul E. L. Rakow, A. Schafer, G. Schierholz, and J. M. Zanotti, Quark helicity flip generalized parton distributions from two-flavor lattice QCD, Phys. Lett. B 627, 113 (2005).
  56. M. Göckeler, Ph. Hägler, R. Horsley, Y. Nakamura, D. Pleiter, P. E. L, Rakow, A. Schäfer, G. Schierholz, H. Stüben, and J. M. Zanotti, Transverse spin structure of the nucleon from lattice QCD simulations, Phys. Rev. Lett. 98, 222001 (2007).
  57. C. Alexandrou et al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys. Rev. D 101, 034519 (2020).
  58. Chentao Tan and Zhun Lu, Chiral-even twist-3 GPDs for the proton in a spectator diquark model, Phys. Rev. D 110, 074027 (2024).
  59. Sameer Jain, Shubham Sharma, and Harleen Dahiya, Deciphering twist-3 chiral-even GPDs in the light-front quark-diquark model, Phys. Rev. D 110, 094030 (2024).
  60. Alexey Vladimirov, Guillermo Portela, and Simone Rodini, Determination of quark-gluon-quark interference within the proton, arXiv:2511.04294.
  61. Xiangdong Ji, Gauge-invariant decomposition of nucleon spin, Phys. Rev. Lett. 78, 610 (1997).
  62. Yoshitaka Hatta and Shinsuke Yoshida, Twist analysis of the nucleon spin in QCD, J. High Energy Phys. 10 (2012) 080.
  63. R. L. Jaffe and Aneesh Manohar, The g1 problem: Deep inelastic electron scattering and the spin of the proton, Nucl. Phys. B337, 509 (1990).
  64. Matthias Burkardt, Transverse force on quarks in deep-inelastic scattering, Phys. Rev. D 88, 114502 (2013).
  65. Fatma P. Aslan, Matthias Burkardt, and Marc Schlegel, Transverse force tomography, Phys. Rev. D 100, 096021 (2019).
  66. Yuxun Guo, Xiangdong Ji, Brandon Kriesten, and Kyle Shiells, Twist-three cross-sections in deeply virtual Compton scattering, J. High Energy Phys. 06 (2022) 096.
  67. J. P. Vary, H. Honkanen, Jun Li, P. Maris, S. J. Brodsky, A. Harindranath, G. F. de Teramond, P. Sternberg, E. G. Ng, and C. Yang, Hamiltonian light-front field theory in a basis function approach, Phys. Rev. C 81, 035205 (2010).
  68. N. Kivel, Maxim V. Polyakov, A. Schäfer, and O. V. Teryaev, Wandzura–Wilczek approximation for the twist-3 DVCS amplitude, Phys. Lett. B 497, 73 (2001).
  69. A. V. Radyushkin and C. Weiss, Dvcs amplitude with kinematical twist-3 terms, Phys. Lett. B 493, 332 (2000).
  70. N. Kivel and Maxim V. Polyakov, Dvcs on the nucleon to the twist-3 accuracy, Nucl. Phys. B600, 334 (2001).
  71. A. V. Radyushkin and C. Weiss, Kinematical twist-3 effects in deeply virtual Compton scattering as a quark spin rotation, Phys. Rev. D 64, 097504 (2001).
  72. Siqi Xu, Chandan Mondal, Xingbo Zhao, Yang Li, and James P. Vary, Quark and gluon spin and orbital angular momentum in the proton, Phys. Rev. D 108, 094002 (2023).
  73. Zhimin Zhu, Siqi Xu, Jiatong Wu, Hongyao Yu, Zhi Hu, Jiangshan Lan, Chandan Mondal, Xingbo Zhao, and James P. Vary, Transverse structure of the proton beyond leading twist: A light-front Hamiltonian approach, Phys. Lett. B 855, 138829 (2024).
  74. Sreeraj Nair, Chandan Mondal, Siqi Xu, Xingbo Zhao, and James P. Vary (BLFQ Collaboration), Proton gravitational structure and mass decomposition on the light front, Phys. Rev. D 112, 114001 (2025).
  75. Yang Li, Pieter Maris, Xingbo Zhao, and James P. Vary, Heavy quarkonium in a holographic basis, Phys. Lett. B 758, 118 (2016).
  76. Xingbo Zhao, Heli Honkanen, Pieter Maris, James P. Vary, and Stanley J, Brodsky, Electron g-2 in light-front quantization, Phys. Lett. B 737, 65 (2014).
  77. Matthias Burkardt, Dynamical vertex mass generation and chiral symmetry breaking on the light front, Phys. Rev. D 58, 096015 (1998).
  78. Hongyao Yu, Zhi Hu, Siqi Xu, Chandan Mondal, Xingbo Zhao, and James P. Vary, Transverse-momentum-dependent gluon distributions of proton within basis light-front quantization, Phys. Lett. B 855, 138831 (2024).
  79. Pengxiang Zhang, Yiping Liu, Siqi Xu, Chandan Mondal, Xingbo Zhao, and James P, Vary, Gluon skewed generalized parton distributions of proton from a light-front Hamiltonian approach, Phys. Lett. B 866, 139584 (2025).
  80. John B. Kogut and Davison E. Soper, Quantum electrodynamics in the infinite-momentum frame, Phys. Rev. D 1, 2901 (1970).
  81. Stephan Meißner, Andreas Metz, and Marc Schlegel, Generalized parton correlation functions for a spin-1/2 hadron, J. High Energy Phys. 08 (2009) 056.
  82. Fatma Aslan, Matthias Burkardt, Cédric Lorcé, Andreas Metz, and Barbara Pasquini, Twist-3 generalized parton distributions in deeply-virtual Compton scattering, Phys. Rev. D 98, 014038 (2018).
  83. Yuxun Guo, Xiangdong Ji, and Kyle Shiells, Novel twist-three transverse-spin sum rule for the proton and related generalized parton distributions, Nucl. Phys. B969, 115440 (2021).
  84. Andrei V. Belitsky, A. Kirchner, Dieter Mueller, and A. Schäfer, Twist-three observables in deeply virtual Compton scattering on the nucleon, Phys. Lett. B 510, 117 (2001).
  85. Andrei V. Belitsky, Dieter Mueller, and A. Kirchner, Theory of deeply virtual Compton scattering on the nucleon, Nucl. Phys. B629, 323 (2002).
  86. Xiangdong Ji, Xiaonu Xiong, and Feng Yuan, Probing parton orbital angular momentum in longitudinally polarized nucleon, Phys. Rev. D 88, 014041 (2013).
  87. Fatma P. Aslan, Matthias Burkardt, and Marc Schlegel, Transverse force tomography, Phys. Rev. D 100, 096021 (2019).
  88. D. V. Kiptily and M. V. Polyakov, Genuine twist-3 contributions to the generalized parton distributionsfrom instantons, Eur. Phys. J. C 37, 105 (2004).
  89. Siqi Xu, Chandan Mondal, Jiangshan Lan, Xingbo Zhao, Yang Li, and James P, Vary, Nucleon structure from basis light-front quantization, Phys. Rev. D 104, 094036 (2021).
  90. Cédric Lorcé, Spin–orbit correlations in the nucleon, Phys. Lett. B 735, 344 (2014).
  91. Hugh Burkhardt and W. N. Cottingham, Sum rules for forward virtual Compton scattering, Ann. Phys. (N.Y.) 56, 453 (1970).
  92. Cedric Lorce and Barbara Pasquini, Quark Wigner distributions and orbital angular momentum, Phys. Rev. D 84, 014015 (2011).
  93. Elliot Leader, Aleksander V. Sidorov, and Dimiter B. Stamenov, Determination of polarized parton densities from a qcd analysis of inclusive¡? format?¿ and semi-inclusive deep inelastic scattering data, Phys. Rev. D 82, 114018 (2010).
  94. Cédric Lorcé, Wilson lines and orbital angular momentum, Phys. Lett. B 719, 185 (2013).
  95. José Manuel Morgado Chávez, Valerio Bertone, Feliciano De Soto Borrero, Maxime Defurne, Cédric Mezrag, Hervé Moutarde, José Rodríguez-Quintero, and Jorge Segovia, Accessing the pion 3D structure at us and china electron-ion colliders, Phys. Rev. Lett. 128, 202501 (2022).

Outline

Information

Phys. Rev. D 113, 094031– Published 22 May, 2026
  • Received 18 January 2026
  • Accepted 6 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation