- 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

Double parton distributions of the proton from basis light-front quantization

Tian-Cai Peng1,2,3,4,*, Zhi Hu5,†, Sreeraj Nair6,7,8,‡, Siqi Xu6,7,8,§, Xiang Liu1,2,3,4,9,∥, Chandan Mondal6,7,8,¶, Xingbo Zhao6,7,8,10,**, and James P. Vary11,†† (BLFQ Collaboration)

  • *Contact author: pengtc20@lzu.edu.cn
  • †Contact author: huzhi0826@gmail.com
  • ‡Contact author: sreeraj@impcas.ac.cn
  • §Contact author: xsq234@impcas.ac.cn
  • ∥Contact author: xiangliu@lzu.edu.cn
  • Contact author: mondal@impcas.ac.cn
  • **Contact author: xbzhao@impcas.ac.cn
  • ††Contact author: jvary@iastate.edu
PDF

Phys. Rev. D 112, 014018 – Published 8 July, 2025

DOI: https://doi.org/10.1103/k1yx-2h5b

Abstract

Within the basis light-front quantization framework, we systematically investigate the unpolarized and longitudinally polarized double parton distributions (DPDs) of quarks inside the proton. We utilize the light-front wave functions of the proton derived in the valence sector from a Hamiltonian quantized on the light front. The interaction terms of the Hamiltonian consist of a one-gluon exchange interaction at fixed coupling and a three-dimensional confinement potential. Our current analysis yields significant correlations of the quarks’ longitudinal momenta with their transverse separation. We also demonstrate that our calculations do not support the commonly used x−k→⊥ factorization of the DPDs in x and k⊥. Our results are qualitatively consistent with those of other phenomenological models.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (111)

  1. D. E. Soper, Parton distribution functions, Nucl. Phys. B, Proc. Suppl. 53, 69 (1997).
  2. Multiple Parton Interactions at the LHC, edited by P. Bartalini and J. R. Gaunt (World Scientific Publishing, Singapore, 2019), Vol. 29.
  3. A. Tumasyan et al., Observation of same-sign WW production from double parton scattering in proton-proton collisions at s=13  TeV, Phys. Rev. Lett. 131, 091803 (2023).
  4. S. Leontsinis, Observation of triple J/ψ meson production in proton-proton collisions at s=13  TeV, Rev. Mex. Fis. Suppl. 3, 0308046 (2022).
  5. A. Tumasyan et al., Measurement of double-parton scattering in inclusive production of four jets with low transverse momentum in proton-proton collisions at s=13  TeV, J. High Energy Phys. 01 (2022) 177.
  6. R. Aaij et al., Observation of enhanced double parton scattering in proton-lead collisions at sNN=8.16  TeV, Phys. Rev. Lett. 125, 212001 (2020).
  7. A. M. Sirunyan et al., Evidence for WW production from double-parton interactions in proton–proton collisions at s=13  TeV, Eur. Phys. J. C 80, 41 (2020).
  8. I. Belyaev and D. Savrina, Study of double parton scattering processes with heavy quarks, Adv. Ser. Dir. High Energy Phys. 29, 141 (2018).
  9. A. M. Sirunyan et al., Constraints on the double-parton scattering cross section from same-sign W boson pair production in proton-proton collisions at s=8  TeV, J. High Energy Phys. 02 (2018) 032.
  10. L. An, Measurements of soft QCD and double parton scattering at LHCb, in 5th Large Hadron Collider Physics Conference (2017), arXiv:1708.09773.
  11. S. Koshkarev and S. Groote, Double J/ψ production as a test of parton momentum correlations in double parton scattering, Phys. Lett. B 847, 138296 (2023).
  12. R. Kirschner, Generalized Lipatov-Altarelli-Parisi equations and jet calculus rules, Phys. Lett. 84B, 266 (1979).
  13. V. Shelest, A. Snigirev, and G. Zinovjev, Gazing into the multiparton distribution equations in QCD, Phys. Lett. 113B, 325 (1982).
  14. G. m. Zinovev, A. m. Snigirev, and V. p. Shelest, Equations for many parton distributions in quantum chromodynamics, Theor. Math. Phys. 51, 523 (1982).
  15. A. M. Snigirev, QCD status of factorization ansatz for double parton distributions, Phys. Rev. D 68, 114012 (2003).
  16. V. L. Korotkikh and A. M. Snigirev, Double parton correlations versus factorized distributions, Phys. Lett. B 594, 171 (2004).
  17. F. A. Ceccopieri, An update on the evolution of double parton distributions, Phys. Lett. B 697, 482 (2011).
  18. F. A. Ceccopieri, A second update on double parton distributions, Phys. Lett. B 734, 79 (2014).
  19. M. Mekhfi, Correlations in color and spin in multiparton processes, Phys. Rev. D 32, 2380 (1985).
  20. M. Mekhfi and X. Artru, Sudakov suppression of color correlations in multiparton scattering, Phys. Rev. D 37, 2618 (1988).
  21. A. V. Manohar and W. J. Waalewijn, A QCD analysis of double parton scattering: Color correlations, interference effects and evolution, Phys. Rev. D 85, 114009 (2012).
  22. M. Diehl, F. Fabry, and A. Vladimirov, Two-loop evolution kernels for colour dependent double parton distributions, J. High Energy Phys. 05 (2023) 067.
  23. F. Abe et al., Double parton scattering in p¯p collisions at s=1.8  TeV, Phys. Rev. D 56, 3811 (1997).
  24. V. M. Abazov et al., Study of double parton interactions in diphoton+dijet events in pp¯ collisions at s=1.96  TeV, Phys. Rev. D 93, 052008 (2016).
  25. R. Aaij et al., Measurement of the J/ψ pair production cross-section in pp collisions at s=13  TeV, J. High Energy Phys. 06 (2017) 047; 10 (2017) 068(E).
  26. M. Aaboud et al., Measurement of J/ψ production in association with a W± boson with pp data at 8 TeV, J. High Energy Phys. 01 (2020) 095.
  27. R. Abdul Khalek et al., Science requirements and detector concepts for the electron-ion collider: EIC yellow report, Nucl. Phys. A1026, 122447 (2022).
  28. D. P. Anderle et al., Electron-ion collider in China, Front. Phys. (Beijing) 16, 64701 (2021).
  29. J. Katich et al., Measurement of the target-normal single-spin asymmetry in deep-inelastic scattering from the reaction He3↑(e,e′)X, Phys. Rev. Lett. 113, 022502 (2014).
  30. A. Metz, D. Pitonyak, A. Schafer, M. Schlegel, W. Vogelsang, and J. Zhou, Single-spin asymmetries in inclusive deep inelastic scattering and multiparton correlations in the nucleon, Phys. Rev. D 86, 094039 (2012).
  31. A. Afanasev, M. Strikman, and C. Weiss, Transverse target spin asymmetry in inclusive DIS with two-photon exchange, Phys. Rev. D 77, 014028 (2008).
  32. N. Paver and D. Treleani, Multi—quark scattering and large pT jet production in hadronic collisions, Nuovo Cimento A 70, 215 (1982).
  33. M. Mekhfi, Multiparton processes: An application to the double Drell-Yan mechanism, Phys. Rev. D 32, 2371 (1985).
  34. T. Sjöstrand and M. Van Zijl, Multiple parton-parton interactions in an impact parameter picture, Phys. Lett. B 188, 149 (1987).
  35. B. Blok, Y. Dokshitzer, L. Frankfurt, and M. Strikman, The four jet production at LHC and tevatron in QCD, Phys. Rev. D 83, 071501 (2011).
  36. J. R. Gaunt and W. J. Stirling, Double parton scattering singularity in one-loop integrals, J. High Energy Phys. 06 (2011) 048.
  37. M. G. Ryskin and A. M. Snigirev, A fresh look at double parton scattering, Phys. Rev. D 83, 114047 (2011).
  38. B. Blok, Y. Dokshitser, L. Frankfurt, and M. Strikman, pQCD physics of multiparton interactions, Eur. Phys. J. C 72, 1963 (2012).
  39. M. Diehl, D. Ostermeier, and A. Schafer, Elements of a theory for multiparton interactions in QCD, J. High Energy Phys. 03 (2012) 089; 03 (2016) 001(E).
  40. A. V. Manohar and W. J. Waalewijn, What is double parton scattering?, Phys. Lett. B 713, 196 (2012).
  41. M. G. Ryskin and A. M. Snigirev, Double parton scattering in double logarithm approximation of perturbative QCD, Phys. Rev. D 86, 014018 (2012).
  42. J. R. Gaunt and W. J. Stirling, Double parton distributions incorporating perturbative QCD evolution and momentum and quark number sum rules, J. High Energy Phys. 03 (2010) 005.
  43. K. Golec-Biernat and E. Lewandowska, How to impose initial conditions for QCD evolution of double parton distributions?, Phys. Rev. D 90, 014032 (2014).
  44. K. Golec-Biernat, E. Lewandowska, M. Serino, Z. Snyder, and A. M. Stasto, Constraining the double gluon distribution by the single gluon distribution, Phys. Lett. B 750, 559 (2015).
  45. M. Diehl, J. R. Gaunt, D. M. Lang, P. Plößl, and A. Schäfer, Sum rule improved double parton distributions in position space, Eur. Phys. J. C 80, 468 (2020).
  46. M. Diehl and A. Schafer, Theoretical considerations on multiparton interactions in QCD, Phys. Lett. B 698, 389 (2011).
  47. M. Diehl, J. R. Gaunt, P. Plößl, and A. Schäfer, Two-loop splitting in double parton distributions, SciPost Phys. 7, 017 (2019).
  48. H.-M. Chang, A. V. Manohar, and W. J. Waalewijn, Double parton correlations in the bag model, Phys. Rev. D 87, 034009 (2013).
  49. M. Rinaldi, S. Scopetta, and V. Vento, Double parton correlations in constituent quark models, Phys. Rev. D 87, 114021 (2013).
  50. W. Broniowski and E. Ruiz Arriola, Valence double parton distributions of the nucleon in a simple model, Few-Body Syst. 55, 381 (2014).
  51. M. Rinaldi, S. Scopetta, M. Traini, and V. Vento, Double parton correlations and constituent quark models: A light front approach to the valence sector, J. High Energy Phys. 12 (2014) 028.
  52. W. Broniowski, E. Ruiz Arriola, and K. Golec-Biernat, Generalized Valon model for double parton distributions, Few-Body Syst. 57, 405 (2016).
  53. T. Kasemets and A. Mukherjee, Quark-gluon double parton distributions in the light-front dressed quark model, Phys. Rev. D 94, 074029 (2016).
  54. M. Rinaldi, S. Scopetta, M. C. Traini, and V. Vento, Correlations in double parton distributions: Perturbative and non-perturbative effects, J. High Energy Phys. 10 (2016) 063.
  55. M. Rinaldi and F. A. Ceccopieri, Relativistic effects in model calculations of double parton distribution function, Phys. Rev. D 95, 034040 (2017).
  56. M. Rinaldi, S. Scopetta, M. Traini, and V. Vento, A model calculation of double parton distribution functions of the pion, Eur. Phys. J. C 78, 781 (2018).
  57. A. Courtoy, S. Noguera, and S. Scopetta, Double parton distributions in the pion in the Nambu–Jona-Lasinio model, J. High Energy Phys. 12 (2019) 045.
  58. W. Broniowski and E. Ruiz Arriola, Double parton distribution of valence quarks in the pion in chiral quark models, Phys. Rev. D 101, 014019 (2020).
  59. M. Rinaldi, Double parton correlations in mesons within AdS/QCD soft-wall models: A first comparison with lattice data, Eur. Phys. J. C 80, 678 (2020).
  60. A. Courtoy, S. Noguera, and S. Scopetta, Two-current correlations in the pion in the Nambu and Jona-Lasinio model, Eur. Phys. J. C 80, 909 (2020).
  61. C. Mondal, A. Mukherjee, and S. Nair, Double parton distributions for a positroniumlike bound state using light-front wave functions, Phys. Rev. D 100, 094002 (2019).
  62. A. M. Sirunyan et al., Search for physics beyond the standard model in events with jets and two same-sign or at least three charged leptons in proton-proton collisions at s=13  TeV, Eur. Phys. J. C 80, 752 (2020).
  63. A. Del Fabbro and D. Treleani, A double parton scattering background to Higgs boson production at the LHC, Phys. Rev. D 61, 077502 (2000).
  64. O. Fedkevych and A. Kulesza, Double parton scattering in four-jet production in proton-proton collisions at the LHC, Phys. Rev. D 104, 054021 (2021).
  65. F. A. Ceccopieri and M. Rinaldi, Enlighting the transverse structure of the proton via double parton scattering in photon-induced interactions, Phys. Rev. D 105, L011501 (2022).
  66. B. Blok and J. Mehl, Perturbative color correlations in double parton scattering, Int. J. Mod. Phys. A 39, 2450111 (2024).
  67. K. Golec-Biernat and A. M. Staśto, Momentum sum rule and factorization of double parton distributions, Phys. Rev. D 107, 054020 (2023).
  68. M. Diehl, F. Fabry, and P. Ploessl, Evolution of colour correlated double parton distributions: A quantitative study, J. High Energy Phys. 02 (2024) 229.
  69. M. Diehl, R. Nagar, and P. Plößl, Quark mass effects in double parton distributions, J. High Energy Phys. 09 (2023) 100.
  70. M. Diehl, R. Nagar, P. Ploessl, and F. J. Tackmann, Evolution and interpolation of double parton distributions using Chebyshev grids, Eur. Phys. J. C 83, 536 (2023).
  71. G. S. Bali, M. Diehl, B. Gläßle, A. Schäfer, and C. Zimmermann, Double parton distributions in the nucleon from lattice QCD, J. High Energy Phys. 09 (2021) 106.
  72. C. Zimmermann and D. Reitinger, Double parton distributions in the nucleon on the lattice: Flavor interference effects, Proc. Sci. LATTICE2022 (2023) 131 [arXiv:2211.14151].
  73. J.-H. Zhang, Double parton distributions from Euclidean lattice, arXiv:2304.12481.
  74. J. P. Vary, H. Honkanen, J. 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).
  75. S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rep. 301, 299 (1998).
  76. H. Honkanen, P. Maris, J. P. Vary, and S. J. Brodsky, Electron in a transverse harmonic cavity, Phys. Rev. Lett. 106, 061603 (2011).
  77. X. Zhao, H. Honkanen, P. Maris, J. P. Vary, and S. J. Brodsky, Electron g-2 in light-front quantization, Phys. Lett. B 737, 65 (2014).
  78. P. Wiecki, Y. Li, X. Zhao, P. Maris, and J. P. Vary, Basis light-front quantization approach to positronium, Phys. Rev. D 91, 105009 (2015).
  79. D. Chakrabarti, X. Zhao, H. Honkanen, R. Manohar, P. Maris, and J. P. Vary, Generalized parton distributions in a light-front nonperturbative approach, Phys. Rev. D 89, 116004 (2014).
  80. Z. Hu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Transverse structure of electron in momentum space in basis light-front quantization, Phys. Rev. D 103, 036005 (2021).
  81. S. Nair, C. Mondal, X. Zhao, A. Mukherjee, and J. P. Vary, Basis light-front quantization approach to photon, Phys. Lett. B 827, 137005 (2022).
  82. S. Nair, C. Mondal, X. Zhao, A. Mukherjee, and J. P. Vary, Massless and massive photons within basis light-front quantization, Phys. Rev. D 108, 116005 (2023).
  83. S. Jia and J. P. Vary, Basis light front quantization for the charged light mesons with color singlet Nambu–Jona-Lasinio interactions, Phys. Rev. C 99, 035206 (2019).
  84. J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Parton distribution functions from a light front Hamiltonian and QCD evolution for light mesons, Phys. Rev. Lett. 122, 172001 (2019).
  85. J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Pion and kaon parton distribution functions from basis light front quantization and QCD evolution, Phys. Rev. D 101, 034024 (2020).
  86. L. Adhikari, C. Mondal, S. Nair, S. Xu, S. Jia, X. Zhao, and J. P. Vary, Generalized parton distributions and spin structures of light mesons from a light-front Hamiltonian approach, Phys. Rev. D 104, 114019 (2021).
  87. J. Lan, K. Fu, C. Mondal, X. Zhao, and J. P. Vary, Light mesons with one dynamical gluon on the light front, Phys. Lett. B 825, 136890 (2022).
  88. C. Mondal, S. Nair, S. Jia, X. Zhao, and J. P. Vary, Pion to photon transition form factors with basis light-front quantization, Phys. Rev. D 104, 094034 (2021).
  89. Y. Li, P. Maris, X. Zhao, and J. P. Vary, Heavy quarkonium in a holographic basis, Phys. Lett. B 758, 118 (2016).
  90. Y. Li, P. Maris, and J. P. Vary, Quarkonium as a relativistic bound state on the light front, Phys. Rev. D 96, 016022 (2017).
  91. M. Li, Y. Li, P. Maris, and J. P. Vary, Radiative transitions between 0−+ and 1−− heavy quarkonia on the light front, Phys. Rev. D 98, 034024 (2018).
  92. J. Lan, C. Mondal, M. Li, Y. Li, S. Tang, X. Zhao, and J. P. Vary, Parton distribution functions of heavy mesons on the light front, Phys. Rev. D 102, 014020 (2020).
  93. S. Tang, Y. Li, P. Maris, and J. P. Vary, Bc mesons and their properties on the light front, Phys. Rev. D 98, 114038 (2018).
  94. S. Tang, Y. Li, P. Maris, and J. P. Vary, Heavy-light mesons on the light front, Eur. Phys. J. C 80, 522 (2020).
  95. C. Mondal, S. Xu, J. Lan, X. Zhao, Y. Li, D. Chakrabarti, and J. P. Vary, Proton structure from a light-front Hamiltonian, Phys. Rev. D 102, 016008 (2020).
  96. S. Xu, C. Mondal, J. Lan, X. Zhao, Y. Li, and J. P. Vary, Nucleon structure from basis light-front quantization, Phys. Rev. D 104, 094036 (2021).
  97. Y. Liu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Angular momentum and generalized parton distributions for the proton with basis light-front quantization, Phys. Rev. D 105, 094018 (2022).
  98. Z. Hu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Transverse momentum structure of proton within the basis light-front quantization framework, Phys. Lett. B 833, 137360 (2022).
  99. T. Peng, Z. Zhu, S. Xu, X. Liu, C. Mondal, X. Zhao, and J. P. Vary, Basis light-front quantization approach to Λ and Λ  c and their isospin triplet baryons, Phys. Rev. D 106, 114040 (2022).
  100. S. Kaur, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Spatial imaging of proton via leading-twist nonskewed GPDs with basis light-front quantization, Phys. Rev. D 109, 014015 (2024).
  101. Z. Zhu, T. Peng, Z. Hu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Transverse momentum structure of strange and charmed baryons: A light-front Hamiltonian approach, Phys. Rev. D 108, 036009 (2023).
  102. Z. Zhang, Z. Hu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Twist-3 generalized parton distribution for the proton from basis light-front quantization, Phys. Rev. D 109, 034031 (2024).
  103. Y. Liu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Skewed generalized parton distributions of proton from basis light-front quantization, Phys. Lett. B 855, 138809 (2024).
  104. S. Kaur, J. Wu, Z. Hu, J. Lan, C. Mondal, X. Zhao, and J. P. Vary, Quark and gluon distributions in ρ-meson from basis light-front quantization, Phys. Lett. B 851, 138563 (2024).
  105. H. Yu, Z. Hu, S. Xu, C. Mondal, X. Zhao, and J. P. Vary, Transverse-momentum-dependent gluon distributions of proton within basis light-front quantization, Phys. Lett. B 855, 138831 (2024).
  106. S. Xu, C. Mondal, X. Zhao, Y. Li, and J. P. Vary, Quark and gluon spin and orbital angular momentum in the proton, Phys. Rev. D 108, 094002 (2023).
  107. B. Lin, S. Nair, S. Xu, Z. Hu, C. Mondal, X. Zhao, and J. P. Vary, Generalized parton distributions of gluon in proton: A light-front quantization approach, Phys. Lett. B 847, 138305 (2023).
  108. Z. Zhu, Z. Hu, J. Lan, C. Mondal, X. Zhao, and J. P. Vary, Transverse structure of the pion beyond leading twist with basis light-front quantization, Phys. Lett. B 839, 137808 (2023).
  109. A. Harindranath, An introduction to light front dynamics for pedestrians, in International School on Light-Front Quantization and Non-Perturbative QCD (To be followed by the Workshop 3–14 June 1996) (1996), arXiv:hep-ph/9612244.
  110. X. Zhao, A. Ilderton, P. Maris, and J. P. Vary, Scattering in time-dependent basis light-front quantization, Phys. Rev. D 88, 065014 (2013).
  111. S. J. Brodsky, G. F. de Teramond, H. G. Dosch, and J. Erlich, Light-front holographic QCD and emerging confinement, Phys. Rep. 584, 1 (2015).

Outline

Information

Phys. Rev. D 112, 014018– Published 8 July, 2025
  • Received 16 October 2024
  • Accepted 19 June 2025
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