- 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

Nucleon electric dipole form factor in the QCD instanton vacuum

Wei-Yang Liu* and Ismail Zahed†

  • Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA

  • *Contact author: wei-yang.liu@stonybrook.edu
  • †Contact author: ismail.zahed@stonybrook.edu
PDF

Phys. Rev. D 112, 094048 – Published 24 November, 2025

DOI: https://doi.org/10.1103/d9tl-ycqf

Abstract

In the quantum chromodynamics (QCD) vacuum, the nucleon form factors receive contributions from the underlying ensemble of topological pseudoparticles, which are sensitive to a finite vacuum angle θ. We use this observation to derive a novel relationship between the Pauli and electric dipole form factors, for light quark flavors. This relationship allows for an explicit derivation of the proton and neutron electric dipole moments induced by a small CP violating θ angle, in terms of the vacuum topological susceptibility times pertinent magnetic moments. The results compare well with some recent lattice estimates.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (45)

  1. A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  2. D. Kharzeev, E. Shuryak, and I. Zahed, Phys. Rev. D 102, 073003 (2020).
  3. D. Diakonov, M. V. Polyakov, and C. Weiss, Nucl. Phys. B461, 539 (1996).
  4. T. Schäfer and E. V. Shuryak, Rev. Mod. Phys. 70, 323 (1998).
  5. M. A. Nowak, M. Rho, and I. Zahed, Chiral Nuclear Dynamics (World Scientific Publishing, Singapore, 1996).
  6. W.-Y. Liu, arXiv:2501.07776.
  7. C. Abel et al., Phys. Rev. Lett. 124, 081803 (2020).
  8. S. Syritsyn, T. Izubuchi, and H. Ohki, Proc. Sci., Confinement2018 (2019) 194 [arXiv:1901.05455].
  9. C. Alexandrou, A. Athenodorou, K. Hadjiyiannakou, and A. Todaro, Phys. Rev. D 103, 054501 (2021).
  10. J. Liang, A. Alexandru, T. Draper, K.-F. Liu, B. Wang, G. Wang, and Y.-B. Yang (χQCD Collaboration), Phys. Rev. D 108, 094512 (2023).
  11. D. B. Leinweber, in Workshop on Light-Cone QCD and Nonperturbative Hadron Physics, The University of Adelaide, Adelaide, Australia (1999), pp. 138–143. arXiv:hep-lat/0004025.
  12. C. Michael and P. S. Spencer, Nucl. Phys. B, Proc. Suppl. 42, 261 (1995).
  13. C. Michael and P. S. Spencer, Phys. Rev. D 52, 4691 (1995).
  14. J. C. Biddle, W. Kamleh, and D. B. Leinweber, Proc. Sci., LATTICE2018 (2018) 256 [arXiv:1903.07767].
  15. A. Athenodorou, P. Boucaud, F. De Soto, J. Rodríguez-Quintero, and S. Zafeiropoulos, J. High Energy Phys. 02 (2018) 140.
  16. A. Ringwald and F. Schrempp, Phys. Lett. B 459, 249 (1999).
  17. E. V. Shuryak, Nucl. Phys. B203, 93 (1982).
  18. P. J. Moran and D. B. Leinweber, in QCD Downunder II, Massey University, Auckland, New Zealand (2008). arXiv:0805.4246.
  19. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B191, 301 (1981).
  20. I. Zahed, Phys. Rev. D 104, 054031 (2021).
  21. L. Del Debbio, L. Giusti, and C. Pica, Nucl. Phys. B, Proc. Suppl. 140, 603 (2005).
  22. W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D 107, 094024 (2023).
  23. W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D 109, 074029 (2024).
  24. W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D 110, 054005 (2024).
  25. P. Faccioli and E. V. Shuryak, Phys. Rev. D 64, 114020 (2001).
  26. P. V. Pobylitsa, Phys. Lett. B 226, 387 (1989).
  27. A. I. Vainshtein, V. I. Zakharov, V. A. Novikov, and M. A. Shifman, Sov. Phys. Usp. 25, 195 (1982).
  28. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG) Collaboration), Eur. Phys. J. C 82, 869 (2022).
  29. I. Zahed, Symmetry 14, 932 (2022).
  30. N. I. Kochelev, Phys. Lett. B 565, 131 (2003).
  31. Y. Liu and I. Zahed, arXiv:2102.07248.
  32. C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 59, 694 (2007).
  33. M. A. Belushkin, H. W. Hammer, and U. G. Meissner, Phys. Rev. C 75, 035202 (2007).
  34. P. E. Shanahan, A. W. Thomas, R. D. Young, J. M. Zanotti, R. Horsley, Y. Nakamura, D. Pleiter, P. E. L. Rakow, G. Schierholz, and H. Stüben (CSSM, QCDSF/UKQCD Collaboration), Phys. Rev. D 89, 074511 (2014).
  35. C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40, 100001 (2016).
  36. P. Faccioli, D. Guadagnoli, and S. Simula, Phys. Rev. D 70, 074017 (2004).
  37. E. Mereghetti, J. de Vries, W. H. Hockings, C. M. Maekawa, and U. van Kolck, Phys. Lett. B 696, 97 (2011).
  38. T. Bhattacharya, V. Cirigliano, R. Gupta, E. Mereghetti, and B. Yoon, Phys. Rev. D 103, 114507 (2021).
  39. J. Dragos, T. Luu, A. Shindler, J. de Vries, and A. Yousif, Phys. Rev. C 103, 015202 (2021).
  40. D. Diakonov, Proceedings of the International School of Physics "Enrico Fermi" 130, 397 (1996).
  41. E. V. Shuryak, arXiv:hep-ph/9909458.
  42. M. Hutter, arXiv:hep-ph/0107098.
  43. A. Kock, Y. Liu, and I. Zahed, Phys. Rev. D 102, 014039 (2020).
  44. L. S. Brown, R. D. Carlitz, D. B. Creamer, and C. Lee, Phys. Rev. D 17, 1583 (1978).
  45. A. G. Zubkov, O. V. Dubasov, and B. O. Kerbikov, Int. J. Mod. Phys. A 14, 241 (1999).

Outline

Information

Phys. Rev. D 112, 094048– Published 24 November, 2025
  • Received 10 March 2025
  • Accepted 6 November 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