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Extraction of the Collins-Soper Kernel from a Joint Analysis of Experimental and Lattice Data

Artur Avkhadiev1,2,*, Valerio Bertone3,†, Chiara Bissolotti2,‡, Matteo Cerutti3,§, Yang Fu1,∥, Simone Rodini4,5,¶, Phiala Shanahan1,**, Michael Wagman6,††, and Yong Zhao2,‡‡

  • *Contact author: aavkhadi@anl.gov
  • †Contact author: valerio.bertone@cea.fr
  • ‡Contact author: cbissolotti@anl.gov
  • §Contact author: matteo.cerutti@cea.fr
  • ∥Contact author: yangfu@mit.edu
  • Contact author: simone.rodini@unipv.it
  • **Contact author: phiala@mit.edu
  • ††Contact author: mwagman@fnal.gov
  • ‡‡Contact author: yong.zhao@anl.gov
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Phys. Rev. Lett. 136, 171902 – Published 29 April, 2026

DOI: https://doi.org/10.1103/zphz-4k8q

Abstract

We present a first joint extraction of the Collins-Soper kernel (CSK) combining experimental and lattice QCD data in the context of an analysis of transverse-momentum-dependent distributions (TMDs). Based on a neural-network parametrization, we perform a Bayesian reweighting of an existing fit of TMDs using lattice data, as well as a joint TMD fit to lattice and experimental data. We consistently find that the inclusion of lattice information shifts the central value of the CSK by approximately 10% and reduces its uncertainty by 40%–50%, highlighting the potential of lattice inputs to improve TMD extractions.

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References (59)

  1. H.-W. Lin, W. Melnitchouk, A. Prokudin, N. Sato, and H. Shows, Phys. Rev. Lett. 120, 152502 (2018).
  2. K. Cichy, L. Del Debbio, and T. Giani, J. High Energy Phys. 10 (2019) 137.
  3. L. Del Debbio, T. Giani, and C. J. Monahan, J. High Energy Phys. 09 (2020) 021.
  4. L. Del Debbio, T. Giani, J. Karpie, K. Orginos, A. Radyushkin, and S. Zafeiropoulos, J. High Energy Phys. 02 (2021) 138.
  5. J. Bringewatt, N. Sato, W. Melnitchouk, J.-W. Qiu, F. Steffens, and M. Constantinou, Phys. Rev. D 103, 016003 (2021).
  6. P. C. Barry et al. (Jefferson Lab Angular Momentum (JAM) and HadStruc Collaborations), Phys. Rev. D 105, 114051 (2022).
  7. T.-J. Hou, H.-W. Lin, M. Yan, and C. P. Yuan, Phys. Rev. D 107, 076018 (2023).
  8. J. Karpie, R. M. Whitehill, W. Melnitchouk, C. Monahan, K. Orginos, J. W. Qiu, D. G. Richards, N. Sato, and S. Zafeiropoulos (Jefferson Lab Angular Momentum and HadStruc Collaborations), Phys. Rev. D 109, 036031 (2024).
  9. A. Ablat et al., Eur. Phys. J. Plus 139, 1063 (2024).
  10. C. Cocuzza, N. T. Hunt-Smith, W. Melnitchouk, N. Sato, and A. W. Thomas, Phys. Rev. D 112, 114017 (2025).
  11. A. Ablat, S. Dulat, T.-J. Hou, H.-W. Lin, K. Xie, and C. P. Yuan, arXiv:2502.10630.
  12. W. Good, P. C. Barry, H.-W. Lin, W. Melnitchouk, A. NieMiera, and N. Sato, arXiv:2507.22730.
  13. P. C. Barry, C.-R. Ji, W. Melnitchouk, N. Sato, and F. Steffens (JAM Collaboration), arXiv:2510.11979.
  14. Y. Guo, X. Ji, M. G. Santiago, K. Shiells, and J. Yang, J. High Energy Phys. 05 (2023) 150.
  15. K. Cichy, M. Constantinou, P. Sznajder, and J. Wagner, Phys. Rev. D 110, 114025 (2024).
  16. Y. Guo, F. P. Aslan, X. Ji, and M. G. Santiago, Phys. Rev. Lett. 135, 261903 (2025).
  17. T. Cridge, G. Marinelli, and F. J. Tackmann, J. High Energy Phys. 12 (2025) 043.
  18. I. Moult, H. X. Zhu, and Y. J. Zhu, J. High Energy Phys. 08 (2022) 280.
  19. C. Duhr, B. Mistlberger, and G. Vita, Phys. Rev. Lett. 129, 162001 (2022).
  20. V. Moos, I. Scimemi, A. Vladimirov, and P. Zurita, J. High Energy Phys. 11 (2025) 134.
  21. F. Aslan, M. Boglione, J. O. Gonzalez-Hernandez, T. Rainaldi, T. C. Rogers, and A. Simonelli, Phys. Rev. D 110, 074016 (2024).
  22. A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, F. Delcarro, F. Piacenza, and M. Radici, J. High Energy Phys. 07 (2020) 117.
  23. A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, M. Cerutti, F. Delcarro, M. Radici, L. Rossi, and A. Signori (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), J. High Energy Phys. 08 (2024) 232.
  24. A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, M. Cerutti, F. Piacenza, M. Radici, and A. Signori (MAP (Multi-dimensional Analyses of Partonic Distributions Collaboration), J. High Energy Phys. 10 (2022) 127.
  25. V. Moos, I. Scimemi, A. Vladimirov, and P. Zurita, J. High Energy Phys. 05 (2024) 036.
  26. S. Camarda, G. Ferrera, and M. Schott, Eur. Phys. J. C 84, 39 (2024).
  27. P. C. Barry et al., arXiv:2510.13771.
  28. A. B. Cuerpo, I. Scimemi, and A. Vladimirov, J. High Energy Phys. 11 (2025) 013.
  29. Z.-B. Kang, J. Penttala, and C. Zhang, arXiv:2410.21435.
  30. D. de Florian and M. Grazzini, Nucl. Phys. B704, 387 (2005).
  31. S. Camarda, G. Ferrera, and L. Rossi, J. High Energy Phys. 01 (2026) 150.
  32. A. Bacchetta, V. Bertone, C. Bissolotti, M. Cerutti, M. Radici, S. Rodini, and L. Rossi (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), Phys. Rev. Lett. 135, 021904 (2025).
  33. X. Ji, P. Sun, X. Xiong, and F. Yuan, Phys. Rev. D 91, 074009 (2015).
  34. M. A. Ebert, I. W. Stewart, and Y. Zhao, Phys. Rev. D 99, 034505 (2019).
  35. M. A. Ebert, I. W. Stewart, and Y. Zhao, J. High Energy Phys. 09 (2019) 037.
  36. X. Ji, Y. Liu, and Y.-S. Liu, Nucl. Phys. B955, 115054 (2020).
  37. X. Ji, Y. Liu, and Y.-S. Liu, Phys. Lett. B 811, 135946 (2020).
  38. X. Ji and Y. Liu, Phys. Rev. D 105, 076014 (2022).
  39. A. A. Vladimirov and A. Schäfer, Phys. Rev. D 101, 074517 (2020).
  40. S. Rodini and A. Vladimirov, J. High Energy Phys. 09 (2023) 117.
  41. Z.-F. Deng, W. Wang, and J. Zeng, J. High Energy Phys. 09 (2022) 046.
  42. Y. Zhao, Phys. Rev. Lett. 133, 241904 (2024).
  43. P. Shanahan, M. Wagman, and Y. Zhao, Phys. Rev. D 102, 014511 (2020).
  44. P. Shanahan, M. Wagman, and Y. Zhao, Phys. Rev. D 104, 114502 (2021).
  45. H.-T. Shu, M. Schlemmer, T. Sizmann, A. Vladimirov, L. Walter, M. Engelhardt, A. Schäfer, and Y.-B. Yang, Phys. Rev. D 108, 074519 (2023).
  46. Q.-A. Zhang et al. (Lattice Parton Collaboration), Phys. Rev. Lett. 125, 192001 (2020).
  47. M.-H. Chu et al. (Lattice Parton (LPC) Collaboration), Phys. Rev. D 106, 034509 (2022).
  48. Y. Li et al., Phys. Rev. Lett. 128, 062002 (2022).
  49. M. Schlemmer, A. Vladimirov, C. Zimmermann, M. Engelhardt, and A. Schäfer, J. High Energy Phys. 08 (2021) 004.
  50. C. Alexandrou, S. Bacchio, K. Cichy, M. Constantinou, A. Sen, G. Spanoudes, F. Steffens, and J. Tarello, Phys. Rev. D 113, 074506 (2026).
  51. D. Bollweg, X. Gao, S. Mukherjee, and Y. Zhao, Phys. Lett. B 852, 138617 (2024).
  52. D. Bollweg, X. Gao, J. He, S. Mukherjee, and Y. Zhao, Phys. Rev. D 112, 034501 (2025).
  53. A. Avkhadiev, P. E. Shanahan, M. L. Wagman, and Y. Zhao, Phys. Rev. D 108, 114505 (2023).
  54. A. Avkhadiev, P. E. Shanahan, M. L. Wagman, and Y. Zhao, Phys. Rev. Lett. 132, 231901 (2024).
  55. R. D. Ball, V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, A. Guffanti, J. I. Latorre, J. Rojo, and M. Ubiali (NNPDF Collaboration), Nucl. Phys. B849, 112 (2011); B854, 926(E) (2012); B855, 927(E) (2012).

  56. Note that k2 was estimated to be negligible in Ref. [54] based on the Akaike information criterion [57]. To generate a MC ensemble, we assigned an uncertainty of 0.1 to it.

  57. H. Akaike, IEEE Trans. Autom. Control 19, 716 (1974).

  58. Different from Ref. [32] where we quoted the reduced χ2’s of the central replica, here we report ⟨χ2⟩ because no central replica can be defined in the case of reweighting. The ⟨χ2⟩’s in the reweighting case are to be understood as weighted averages.

  59. TMD fits, https://mapcollaboration.github.io.

Outline

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Phys. Rev. Lett. 136, 171902– Published 29 April, 2026
  • Received 7 November 2025
  • Revised 1 March 2026
  • Accepted 11 March 2026
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