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Reconstruction of gravitational form factors using generative machine learning

Herzallah Alharazin and Julia Yu. Panteleeva

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Phys. Rev. D 113, 116007 – Published 2 June, 2026

DOI: https://doi.org/10.1103/g1j2-2ww2

Abstract

We develop a generative framework based on denoising diffusion for the model-independent reconstruction of hadronic form factors from sparse and noisy data. The generative prior is built from a large ensemble of synthetic curves drawn from ten distinct functional classes rooted in different theoretical approaches to hadron structure. Applied to the proton gravitational form factors A(t), J(t), and D(t), the framework yields nonparametric reconstructions consistent with lattice QCD across the full kinematic range 0≤−t≤2  GeV2, remaining robust even when only one or two conditioning points are retained. The densely sampled output enables a direct extraction of the chiral low-energy constants c8=−4.6±0.8  GeV−1 and c9=−0.61±0.19  GeV−1. Using these values at the physical pion mass, we obtain D(0)=−4.3±0.8 for the nucleon D-term.

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References (87)

  1. R. L. Jaffe, Phys. Rev. D 103, 016017 (2021).
  2. J. Y. Panteleeva, E. Epelbaum, J. Gegelia, and U.-G. Meißner, Eur. Phys. J. C 83, 617 (2023).
  3. A. Freese and G. A. Miller, Phys. Rev. D 105, 014003 (2022).
  4. X. Ji and C. Yang, Nucl. Phys. B1024, 117342 (2026).
  5. D. Müller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hořejši, Fortschr. Phys. 42, 101 (1994).
  6. X. D. Ji, Phys. Rev. Lett. 78, 610 (1997).
  7. A. V. Radyushkin, Phys. Lett. B 380, 417 (1996).
  8. K. Goeke, M. V. Polyakov, and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 47, 401 (2001).
  9. A. V. Radyushkin, Phys. Rev. D 56, 5524 (1997).
  10. V. D. Burkert, L. Elouadrhiri, and F. X. Girod, Nature (London) 557, 396 (2018).
  11. V. D. Burkert, L. Elouadrhiri, and F. X. Girod, arXiv:2104.02031.
  12. J. C. Collins, L. Frankfurt, and M. Strikman, Phys. Rev. D 56, 2982 (1997).
  13. H. Alharazin, D. Djukanovic, J. Gegelia, and M. V. Polyakov, Phys. Rev. D 102, 076023 (2020).
  14. H. Alharazin, E. Epelbaum, J. Gegelia, U.-G. Meißner, and B.-D. Sun, Eur. Phys. J. C 82, 907 (2022).
  15. H. Alharazin, Phys. Rev. D 109, 016009 (2024).
  16. P. E. Shanahan and W. Detmold, Phys. Rev. D 99, 014511 (2019).
  17. D. C. Hackett, D. A. Pefkou, and P. E. Shanahan, Phys. Rev. Lett. 132, 251904 (2024).
  18. I. Y. Kobzarev and L. B. Okun, Zh. Eksp. Teor. Fiz. 43, 1904 (1962) [Sov. Phys. JETP 16, 1343 (1963)].
  19. H. Pagels, Phys. Rev. 144, 1250 (1966).
  20. M. V. Polyakov, Phys. Lett. B 555, 57 (2003).
  21. M. V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33, 1830025 (2018).
  22. J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, in Proceedings of the 32nd International Conference on Machine Learning, Vol. 37 (PMLR, 2015), p. 2256.
  23. J. Ho, A. Jain, and P. Abbeel, in Advances in Neural Information Processing Systems, Vol. 33 (Curran Associates, Red Hook, NY, 2020), p. 6840.
  24. Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, arxiv:2011.13456.
  25. R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer, in Proc. IEEE/CVF Conf. on Computer Vision and Pattern Recognition (CVPR) (IEEE, Piscataway, NJ, 2022), p. 10684.
  26. A. Q. Nichol and P. Dhariwal, in Proc. 38th Int. Conf. on Machine Learning, Vol. 139 (PMLR, 2021), p. 8162.
  27. L. Wang, G. Aarts, and K. Zhou, J. High Energy Phys. 05 (2024) 060.
  28. Q. Zhu, G. Aarts, W. Wang, K. Zhou, and L. Wang, J. High Energy Phys. 05 (2025) 043.
  29. G. Aarts, D. E. Habibi, A. Ipp, D. I. Müller, T. R. Ranner, L. Wang, W. Wang, and Q. Zhu, arXiv:2601.19552.
  30. H. Alharazin, J. Yu. Panteleeva, and B.-D. Sun, arxiv:2602.09045.
  31. X. H. Cao, F. K. Guo, Q. Z. Li, B. W. Wu, and D. L. Yao, arXiv:2507.05375.
  32. J. Gegelia and M. V. Polyakov, Phys. Lett. B 820, 136572 (2021).
  33. S. Cotogno, C. Lorcé, P. Lowdon, and M. Morales, Phys. Rev. D 101, 056016 (2020).
  34. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017).
  35. P. P. Avelino, Phys. Lett. B 795, 627 (2019).
  36. S. Dubynskiy and M. B. Voloshin, Phys. Lett. B 666, 344 (2008).
  37. M. I. Eides, V. Y. Petrov, and M. V. Polyakov, Phys. Rev. D 93, 054039 (2016).
  38. I. A. Perevalova, M. V. Polyakov, and P. Schweitzer, Phys. Rev. D 94, 054024 (2016).
  39. F. J. Ernst, R. G. Sachs, and K. C. Wali, Phys. Rev. 119, 1105 (1960).
  40. R. G. Sachs, Phys. Rev. 126, 2256 (1962).
  41. G. A. Miller, Phys. Rev. C 99, 035202 (2019).
  42. G. A. Miller, Phys. Rev. Lett. 99, 112001 (2007).
  43. C. Lorcé, Phys. Rev. Lett. 125, 232002 (2020).
  44. E. Epelbaum, J. Gegelia, N. Lange, U. G. Meißner, and M. V. Polyakov, Phys. Rev. Lett. 129, 012001 (2022).
  45. A. Freese and G. A. Miller, Phys. Rev. D 108, 034008 (2023).
  46. J. Y. Panteleeva, E. Epelbaum, J. Gegelia, and U. G. Meißner, Phys. Rev. D 106, 056019 (2022).
  47. C. Lorcé, H. Moutarde, and A. P. Trawiński, Eur. Phys. J. C 79, 89 (2019).
  48. H. Hashamipour, M. Goharipour, K. Azizi, and S. V. Goloskokov, Phys. Rev. D 107, 096005 (2023).
  49. D. A. Pefkou, D. C. Hackett, and P. E. Shanahan, Phys. Rev. D 105, 054509 (2022).
  50. C. Alexandrou et al., Phys. Rev. D 101, 094513 (2020).
  51. G. S. Bali et al. (RQCD Collaboration), Phys. Rev. D 100, 014507 (2019).
  52. R. J. Hill and G. Paz, Phys. Rev. D 82, 113005 (2010).
  53. B. Bhattacharya, R. J. Hill, and G. Paz, Phys. Rev. D 84, 073006 (2011).
  54. G. Lee, J. R. Arrington, and R. J. Hill, Phys. Rev. D 92, 013013 (2015).
  55. W. Broniowski and E. Ruiz Arriola, Phys. Rev. D 111, 074017 (2025).
  56. P. Masjuan and S. Peris, Phys. Lett. B 686, 307 (2010).
  57. R. Bijker and F. Iachello, Phys. Rev. C 69, 068201 (2004).
  58. H. Tong, J.-W. Ma, and F. Yuan, J. High Energy Phys. 10 (2022) 046.
  59. S. J. Brodsky and G. F. de Téramond, Phys. Rev. D 78, 025032 (2008).
  60. Z. Abidin and C. E. Carlson, Phys. Rev. D 79, 115003 (2009).
  61. J. J. Kelly, Phys. Rev. C 70, 068202 (2004),
  62. Z. Ye, J. Arrington, R. J. Hill, and G. Lee, Phys. Lett. B 777, 8 (2018).
  63. L. Schlessinger, Phys. Rev. 167, 1411 (1968).
  64. Z.-F. Cui, D. Binosi, C. D. Roberts, and S. M. Schmidt, Phys. Rev. Lett. 127, 092001 (2021).
  65. X.-H. Cao, F.-K. Guo, Q.-Z. Li, and D.-L. Yao, Nat. Commun. 16, 6979 (2025).
  66. H.-W. Hammer and U.-G. Meißner, Eur. Phys. J. A 20, 469 (2004).
  67. G. Höhler, E. Pietarinen, I. Sabba-Stefanescu, F. Borkowski, G. G. Simon, V. H. Walther, and R. D. Wendling, Nucl. Phys. B114, 505 (1976).
  68. M. A. Belushkin, H.-W. Hammer, and U.-G. Meißner, Phys. Rev. C 75, 035202 (2007).
  69. S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973).
  70. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
  71. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D 9, 3471 (1974).
  72. X. Ji, J. Phys. G 24, 1181 (1998).
  73. M. J. Neubelt, A. Sampino, and P. Schweitzer, Phys. Rev. D 101, 034013 (2020).
  74. T. Tezgin, P. Schweitzer, and N. Kersting, Phys. Rev. D 109, 014010 (2024).
  75. H. Alharazin and J. Yu. Panteleeva, https://github.com/Herzallah15/diffusion-gff-reconstruction (2026); https://github.com/JuliaYu24/GFF_Diffusion_Analysis (2026).
  76. T. Salimans and J. Ho, arXiv:2202.00512.
  77. K. Goeke, J. Grabis, J. Ossmann, P. Schweitzer, A. Silva, and D. Urbano, Phys. Rev. C 75, 055207 (2007).
  78. C. Cebulla, K. Goeke, J. Ossmann, and P. Schweitzer, Nucl. Phys. A794, 87 (2007).
  79. J. Deng and D. Hou, arXiv:2512.17554.
  80. Z. Q. Yao, Y. Z. Xu, D. Binosi, Z. F. Cui, M. Ding, K. Raya, C. D. Roberts, J. Rodríguez-Quintero, and S. M. Schmidt, Eur. Phys. J. A 61, 92 (2025).
  81. Y. Guo, F. Yuan, and W. Zhao, Phys. Rev. Lett. 135, 111902 (2025).
  82. M. Goharipour, H. Hashamipour, H. Fatehi, F. Irani, K. Azizi, and S.  V. Goloskokov (MMGPDs Collaboration), Phys. Rev. D 112, 014016 (2025).
  83. K. Kumerički, Nature (London) 570, E1 (2019).
  84. H. Dutrieux, C. Lorcé, H. Moutarde, P. Sznajder, A. Trawiński, and J. Wagner, Eur. Phys. J. C 81, 300 (2021).
  85. E. Perez, F. Strub, H. de Vries, V. Dumoulin, and A. Courville, FiLM: Visual Reasoning with a General Conditioning Layer in Proceedings of the AAAI Conference on Artificial Intelligence Vol. 32 (2018).
  86. P. Goyal, P. Dollár, R. Girshick, P. Noordhuis, L. Wesolowski, A. Kyrola, A. Tulloch, Y. Jia, and K. He, arXiv:1706.02677.
  87. I. Loshchilov and F. Hutter, Decoupled weight decay regularization, in Proceedings of the 7th International Conference on Learning Representations (ICLR 2019) (2019).

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Phys. Rev. D 113, 116007– Published 2 June, 2026
  • Received 8 March 2026
  • Accepted 28 April 2026
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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.

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