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Nucleon Electric Dipole Moments in Paramagnetic Molecules through Effective Field Theory

Wouter Dekens1, Jordy de Vries2,3, Lemonia Gialidi2,3,*, Javier Menéndez4,5, Heleen Mulder3,6,†, and Beatriz Romeo7

  • *Contact author: lgialidi@nikhef.nl
  • †Contact author: hmulder@nikhef.nl
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Phys. Rev. Lett. 136, 201803 – Published 19 May, 2026

DOI: https://doi.org/10.1103/5rpp-f6h4

Abstract

Electric dipole moment (EDM) measurements using paramagnetic molecules have significantly advanced over the last decade. Traditionally, these experiments have been analyzed in terms of the electron EDM. However, paramagnetic molecules are also sensitive to hadronic sources of charge-parity (CP) violation, highlighting the need for a new framework to interpret the experimental results. In this Letter, we introduce an effective field theory framework to relate molecular EDMs to the EDMs of neutrons and protons. We identify the dominant contributions through power counting and pinpoint the necessary nuclear matrix elements. As a practical application, we employ the nuclear shell model to calculate these nuclear matrix elements for the polar molecule BaF. Finally, we estimate the limits on the nucleon EDMs set by current molecular EDM experiments.

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References (73)

  1. M. Pospelov and A. Ritz, Ann. Phys. (Amsterdam) 318, 119 (2005).
  2. J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Prog. Part. Nucl. Phys. 71, 21 (2013).
  3. J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, Nature (London) 473, 493 (2011).
  4. J. Baron et al. (ACME Collaboration), Science 343, 269 (2014).
  5. W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, and E. A. Cornell, Phys. Rev. Lett. 119, 153001 (2017).
  6. ACME Collaboration, Nature (London) 562, 355 (2018).
  7. T. S. Roussy et al., Science 381, adg4084 (2023).
  8. P. Aggarwal et al. (NL-eEDM Collaboration), Eur. Phys. J. D 72, 197 (2018).
  9. A. C. Vutha, M. Horbatsch, and E. A. Hessels, Phys. Rev. A 98, 032513 (2018).
  10. C. J. Ho, S. C. Wright, B. E. Sauer, and M. R. Tarbutt, Phys. Rev. Res. 5, 043233 (2023).
  11. R. Alarcon et al., in Snowmass 2021 (2022), arXiv:2203.08103.
  12. G. Arrowsmith-Kron et al., Rep. Prog. Phys. 87, 084301 (2024).
  13. M. Athanasakis-Kaklamanakis, M. Au, R. Berger, S. Degenkolb, J. D. Vries, S. Hoekstra, A. Keshavarzi, N. Neri, D. Ries, P. Schmidt-Wellenburg et al., arXiv:2505.22281.
  14. C. Abel et al., Phys. Rev. Lett. 124, 081803 (2020).
  15. V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, Phys. Rev. D 102, 035001 (2020).
  16. V. V. Flambaum, I. B. Samsonov, and H. B. Tran Tan, J. High Energy Phys. 10 (2020) 077.
  17. H. Mulder, R. Timmermans, and J. de Vries, J. High Energy Phys. 07 (2025) 232.
  18. L. I. Schiff, Phys. Rev. 132, 2194 (1963).
  19. V. V. Flambaum, I. B. Khriplovich, and O. P. Sushkov, Sov. Phys. JETP 60, 873 (1984), http://jetp.ras.ru/cgi-bin/dn/e_060_05_0873.pdf.
  20. J. H. de Jesus and J. Engel, Phys. Rev. C 72, 045503 (2005).
  21. Y. Ema, T. Gao, and M. Pospelov, Phys. Rev. Lett. 129, 231801 (2022).
  22. J. de Vries, E. Mereghetti, R. G. E. Timmermans, and U. van Kolck, Ann. Phys. (Amsterdam) 338, 50 (2013).
  23. W. Dekens and J. de Vries, J. High Energy Phys. 05 (2013) 149.
  24. J. Kley, T. Theil, E. Venturini, and A. Weiler, Eur. Phys. J. C 82, 926 (2022).
  25. J. Kumar and E. Mereghetti, J. High Energy Phys. 09 (2024) 028.
  26. S. Weinberg, Phys. Rev. Lett. 63, 2333 (1989).
  27. R. J. Crewther, P. Di Vecchia, G. Veneziano, and E. Witten, Phys. Lett. 88B, 123 (1979); 91B, 487(E) (1980).
  28. K. Ottnad, B. Kubis, U. G. Meissner, and F. K. Guo, Phys. Lett. B 687, 42 (2010).
  29. E. Mereghetti, J. de Vries, W. H. Hockings, C. M. Maekawa, and U. van Kolck, Phys. Lett. B 696, 97 (2011).

  30. We define 2Ef2Ei⟨hfe|Seff|hie⟩=(2π)4δ4(pf+pe′−pi−pe)A, with the nuclear states satisfying ⟨p|q⟩=(2π)3δ3(p−q).

  31. Strictly speaking, the momentum space potential is infrared divergent. However, performing the Fourier transform in dimensional regularization leads to a potential in coordinate space that is IR finite. It is also possible to deal with the potential in momentum space by defining a subtraction procedure, see Appendix C of Ref. [32] where a similar potential was encountered.

  32. V. Cirigliano, W. Dekens, J. de Vries, S. Gandolfi, M. Hoferichter, and E. Mereghetti, Phys. Rev. C 110, 055502 (2024).
  33. M. Agostini, G. Benato, J. A. Detwiler, J. Menéndez, and F. Vissani, Rev. Mod. Phys. 95, 025002 (2023).
  34. V. Cirigliano, W. Dekens, J. de Vries, S. Gandolfi, M. Hoferichter, and E. Mereghetti, Phys. Rev. Lett. 133, 211801 (2024).
  35. V. Cirigliano et al., J. Phys. G 49, 120502 (2022).
  36. J. Zupan, Eur. Phys. J. C 25, 233 (2002).
  37. R. D. Heil, H. H. Pitz, U. E. P. Berg, U. Kneissl, K. D. Hummel, G. Kilgus, D. Bohle, A. Richter, C. Wesselborg, and P. Von Brentano, Nucl. Phys. A476, 39 (1988).
  38. N. Pietralla et al., Nucl. Phys. A618, 141 (1997).
  39. A. Zilges, P. Von Brentano, C. Wesselborg, R. D. Heil, U. Kneissl, S. Lindenstruth, H. H. Pitz, U. Seemann, and R. Stock, Nucl. Phys. A507, 399 (1990); A519, 848(E) (1990).

  40. We describe the nucleus nonrelativistically, so that the kinetic term takes the form LΨ(0)=Ψi†iv·DΨi. This ensures that ⟨0|Ψi(x)|p⟩=e−ip·x and implies the field has dimension [Ψi]=3/2, so that [C¯SP]=0.

  41. D. J. Broadhurst and A. G. Grozin, Phys. Lett. B 267, 105 (1991).
  42. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5rpp-f6h4 for definitions and detailed results, which includes Refs. [43–45].
  43. S. Stroberg, imsrg++ (2018), https://github.com/ragnarstroberg/imsrg.
  44. W. A. Richter, S. Mkhize, and B. A. Brown, Phys. Rev. C 78, 064302 (2008).
  45. A. Poves, J. Sánchez-Solano, E. Caurier, and F. Nowacki, Nucl. Phys. A694, 157 (2001).
  46. F. Šimkovic, R. Dvornický, D. Stefánik, and A. Faessler, Phys. Rev. C 97, 034315 (2018).
  47. S. e. Morabit, R. Bouabid, V. Cirigliano, J. de Vries, L. Gráf, and E. Mereghetti, J. High Energy Phys. 06 (2025) 082.
  48. B. Romeo, J. Menéndez, and C. Peña Garay, Phys. Lett. B 827, 136965 (2022).
  49. W. Dekens, J. de Vries, D. Castillo, J. Menéndez, E. Mereghetti, V. Plakkot, P. Soriano, and G. Zhou, J. High Energy Phys. 09 (2024) 201.
  50. D. Castillo, L. Jokiniemi, P. Soriano, and J. Menéndez, Phys. Lett. B 860, 139181 (2025).
  51. E. Caurier, F. Nowacki, and A. Poves, Phys. Lett. B 711, 62 (2012).
  52. E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, Rev. Mod. Phys. 77, 427 (2005).
  53. N. Shimizu, arXiv:1310.5431.
  54. T. Miyagi (2020), https://github.com/Takayuki-Miyagi.
  55. E. Caurier, F. Nowacki, A. Poves, and K. Sieja, Phys. Rev. C 82, 064304 (2010).
  56. C. Qi and Z. X. Xu, Phys. Rev. C 86, 044323 (2012).
  57. B. A. Brown, N. J. Stone, J. R. Stone, I. S. Towner, and M. Hjorth-Jensen, Phys. Rev. C 71, 044317 (2005).

  58. We thank J. Engel for discussions on this point.

  59. Such enhancements arise, for example, for sterile neutrino contributions to neutrinoless double beta decay [49].

  60. B. Graner, Y. Chen, E. G. Lindahl, and B. R. Heckel, Phys. Rev. Lett. 116, 161601 (2016).
  61. T. Chupp and M. Ramsey-Musolf, Phys. Rev. C 91, 035502 (2015).
  62. T. Fleig and M. Jung, J. High Energy Phys. 07 (2018) 012.
  63. J. de Vries, E. Mereghetti, R. G. E. Timmermans, and U. van Kolck, Phys. Rev. Lett. 107, 091804 (2011).
  64. J. Bsaisou, C. Hanhart, S. Liebig, U. G. Meissner, A. Nogga, and A. Wirzba, Eur. Phys. J. A 49, 31 (2013).
  65. J. de Vries, E. Mereghetti, and A. Walker-Loud, Phys. Rev. C 92, 045201 (2015).
  66. T. R. Richardson, Phys. Rev. D 112, 095045 (2015).
  67. J. Dragos, T. Luu, A. Shindler, J. de Vries, and A. Yousif, Phys. Rev. C 103, 015202 (2021).
  68. 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).
  69. M. Pospelov and A. Ritz, Phys. Rev. D 89, 056006 (2014).
  70. P. A. B. Haase, D. J. Doeglas, A. Boeschoten, E. Eliav, M. Iliaš, P. Aggarwal, H. L. Bethlem, A. Borschevsky, K. Esajas, Y. Hao et al., J. Chem. Phys. 155, 034309 (2021).
  71. M. Pospelov, Phys. Lett. B 530, 123 (2002).
  72. W. Dekens, J. de Vries, M. Jung, and K. K. Vos, J. High Energy Phys. 01 (2019) 069.
  73. S. Bhattacharya, K. Fuyuto, E. Mereghetti, and T. R. Richardson, Phys. Rev. C 112, 025501 (2025).

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Phys. Rev. Lett. 136, 201803– Published 19 May, 2026
  • Received 10 November 2025
  • Revised 3 March 2026
  • Accepted 20 April 2026
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