- 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

Sixth order QED radiative corrections to lepton anomalies due to the fourth order vacuum polarization insertions within the Mellin-Barnes representation

L. P. Kaptari*

V. I. Lashkevich†

O. P. Solovtsova‡

  • Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia

  • Bogoliubov Lab. Theor. Phys., JINR, Dubna 141980, Russia and Gomel State Technical University, Gomel 246746, Belarus

  • *Contact author: kaptari@theor.jinr.ru
  • †Contact author: lashkevich@gstu.gomel.by
  • ‡Contact author: solovtsova@gstu.gomel.by; olsol@theor.jinr.ru
PDF

Phys. Rev. D 112, 073008 – Published 27 October, 2025

DOI: https://doi.org/10.1103/nnh8-wnv4

Abstract

The explicit form of the sixth order radiative corrections to the lepton L (L=e, μ, and τ) anomalous magnetic moment from QED Feynman diagrams with insertion of the fourth order polarization operators consisting of either two closed lepton loops or one lepton loop crossed by a photon line is discussed in detail. The approach is based on the consistent application of dispersion relations for vacuum polarization operators and the Mellin-Barnes transform for massive photon propagators. Explicit analytical expressions for corrections to the lepton anomaly are obtained for the first time in the whole interval 0<r<∞ of the ratio r of lepton masses mℓ/mL. Asymptotic expansions in the limit of both small r≪1 and large r≫1 computed from the exact expressions are found to be in perfect agreement with the ones earlier reported in the literature. We argue that in the region where the physical lepton mass ratios are located, the asymptotic expansions hold with an accuracy higher than the experimentally measured anomalies. The two loop diagrams with all three leptons different from each other are computed numerically and compared with the corresponding corrections from the pure two-bubble and one-bubble mixed diagrams. It is shown that there are regions of ratios r where all three types of the fourth order polarization operator contribute equally to the anomaly.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (52)

  1. P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. A 117, 619 (1928).
  2. J. S. Schwinger, On Quantum electrodynamics and the magnetic moment of the electron, Phys. Rev. 73, 416 (1948); Quantum electrodynamics. III: The electromagnetic properties of the electron: Radiative corrections to scattering, 76, 790 (1949).
  3. X. Fan, T. G. Myers, B. A. D. Sukra, and G. Gabrielse, Measurement of the electron magnetic moment, Phys. Rev. Lett. 130, 071801 (2023).
  4. D. P. Aguillard et al. (Muon g−2 Collaboration), Measurement of the positive muon anomalous magnetic moment to 127 ppb, Phys. Rev. Lett. 135, 101802 (2025).
  5. T. Aoyama, T. Kinoshita, and M. Nio, Theory of the anomalous magnetic moment of the electron, Atoms 7, 28 (2019).
  6. T. Aoyama, N. Asmussen, M. Benayoun et al., The anomalous magnetic moment of the muon in the standard model, Phys. Rep. 887, 1 (2020).
  7. F. Jegerlehner, The anomalous magnetic moment of the muon, Springer Tracts Mod. Phys. 274, 693 (2017).
  8. T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Complete tenth-order QED contribution to the muon g−2, Phys. Rev. Lett. 109, 111808 (2012).
  9. T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Tenth-order QED contribution to the electron g−2 and an improved value of the fine structure constant, Phys. Rev. Lett. 109, 111807 (2012).
  10. T. Aoyama, T. Kinoshita, and M. Nio, Revised and improved value of the QED tenth-order electron anomalous magnetic moment, Phys. Rev. D 97, 036001 (2018).
  11. S. Volkov, Calculation of the total 10th order QED contribution to the electron magnetic moment, Phys. Rev. D 110, 036001 (2024).
  12. T. Aoyama, M. Hayakawa, A. Hirayama, and M. Nio, Verification of the tenth-order QED contribution to the anomalous magnetic moment of the electron from diagrams without fermion loops, Phys. Rev. D 111, L031902 (2025).
  13. R. Aliberti, T. Aoyama, E. Balzani et al., The anomalous magnetic moment of the muon in the standard model: An update, Phys. Rep. 1143, 1 (2025).
  14. G. Gabrielse and G. Venanzoni, Measured lepton magnetic moments, arXiv:2507.11268.
  15. P. Athron, K. Mohling, D. Stockinger, and H. Stockinger-Kim, The muon magnetic moment and physics beyond the standard model, arXiv:2507.09289.
  16. A. Pustyntsev and M. Vanderhaeghen, Implications of recent (g−2)μ measurements for MeV-GeV dark sector searches, arXiv:2506.17750.
  17. S. Laporta, High-precision calculations of multiloop Feynman integrals by difference equations, J. Mod. Phys. A 15, 5087 (2000).
  18. S. Laporta, High-precision calculations of the 4-loop contribution to the electron g−2 in QED, Phys. Lett. B 772, 332 (2017).
  19. D. H. Bailey and D. J. Broadhurst, Parallel integer relation detection: Techniques and applications, Math. Comput. 70, 1719 (2001).
  20. S. Laporta and E. Remiddi, The analytical value of the electron (g-2) at order α3 in QED, Phys. Lett. B 379, 283 (1996).
  21. E. E. Boos and A. I. Davydychev, A method of evaluation massive Feynman diagrams, Theor. Math. Fiz. 89, 56 (1991). [Theor. Math. Phys. 89, 1052 (1991)].
  22. S. Friot, D. Greynat, and E. de Rafael, Asymptotics of Feynman diagrams and the Mellin-Barnes representation, Phys. Lett. B 628, 73 (2005).
  23. S. Friot and D. Greynat, On convergent series representations of Mellin-Barnes integrals, J. Math. Phys. (N.Y.) 53, 023508 (2012).
  24. S. Banik and S. Friot, Multiple Mellin-Barnes integrals and triangulations of point configurations, Phys. Rev. D 110, 036002 (2024).
  25. S. Banik and S. Friot, Multiple Mellin-Barnes integrals with straight contours, Phys. Rev. D 107, 016007 (2023).
  26. J. P. Aguilar, E. de Rafael, and D. Greynat, Muon anomaly from lepton vacuum polarization and the Mellin-Barnes representation, Phys. Rev. D 77, 093010 (2008).
  27. O. P. Solovtsova, V. I. Lashkevich, and L. P. Kaptari, Lepton anomaly from QED diagrams with vacuum polarization insertions within the Mellin-Barnes representation, Eur. Phys. J. Plus 138, 212 (2023).
  28. J. Charles, E. de Rafael, and D. Greynat, Mellin-Barnes approach to hadronic vacuum polarization and gμ−2, Phys. Rev. D 97, 076014 (2018).
  29. B. Ananthanarayan, S. Friot, and S. Ghosh, Three-loop QED contributions to the g−2 of charged leptons with two internal fermion loops and a class of Kampé de Fériet series, Phys. Rev. D 101, 116008 (2020).
  30. O. P. Solovtsova, V. I. Lashkevich, and L. P. Kaptari, Analytical calculations of the tenth order QED radiative corrections to lepton anomalies within the Mellin–Barnes representation, J. Phys. G 51, 055001 (2024).
  31. P. A. Baikov, A. Maier, and P. Marquard, The QED vacuum polarization function at four loops and the anomalous magnetic moment at five loops, Nucl. Phys. B877, 647 (2013).
  32. S. Laporta, The analytical contribution of the sixth order graphs with vacuum polarization insertions to the muon (g−2) in QED, Nuovo Cimento Soc. Ital. Fis. 106A, 675 (1993).
  33. B. Lautrup and E. Rafael, The anomalous magnetic moment of the muon and short-distrance behaviour of quantum electrodynamics, Nucl. Phys. B70, 317 (1974).
  34. R. Barbieri and E. Remiddi, Electron and muon 12(g−2) from vacuum polarision insertions, Nucl. Phys. B90, 233 (1975).
  35. B. E. Lautrup and E. de Rafael, Calculation of the sixth-order contribution from the fourth-order vacuum polarization to the difference of the anomalous magnetic moments of muon and electron, Phys. Rev. 174, 1835 (1968).
  36. V. B. Berestetskii, O. N. Krohnin, and A. K. Khlebnikov, Concerning the radiative corrections to the μ-meson magnetic moment, Zh. Eksp. Teor. Fiz. 30, 788 (1956) [Sov. Phys. JETP 3, 761 (1956)].
  37. S. J. Brodsky and E. de Rafael, Suggested boson-lepton pair coupling and the anomalous magnetic moment of the muon, Phys. Rev. 168, 1620 (1968).
  38. T. L. Trueman and T. Yao, High energy scattering amplitude in perturbation theory, Phys. Rev. 132, 2741 (1963).
  39. I. Dubovyk, J. Gluza, and G. Somogyi, Mellin-Barnes integrals: A primer on particle physics applications, Lect. Notes Phys. 1008, 208 (2022).
  40. T. Kinoshita, B. Nizic, and Y. Okamoto, Eighth order QED contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 41, 593 (1990).
  41. M. L. Laursen and M. A. Samuel, The n-bubble diagram contribution to (g−2) of the electron. Mathematical structure of the analytical expression, Phys. Lett. 91B, 249 (1980).
  42. M. L. Laursen and M. A. Samuel, Borel transform technique and the n bubble diagram contribution to the lepton anomaly, Phys. Rev. D 23, 2478 (1981).
  43. S. Laporta and E. Remiddi, Analytic QED calculations of the anomalous magnetic moment of the electron, Adv. Ser. Dir. High Energy Phys. 20, 119 (2009).
  44. J. A. Mignaco and E. Remiddi, Fourth-order vacuum polarization contribution to the sixth-order electron magnetic moment, Nouvo Cimento A 60, 519 (1969).
  45. A. Czarnecki and M. Skrzypek, The Muon anomalous magnetic moment in QED: Three loop electron and tau contributions, Phys. Lett. B 449, 354 (1999).
  46. M. A. Samuel and G. W. Li, Improved analytic theory of the muon anomalous magnetic moment, Phys. Rev. D 44, 3935 (1991); 48, 1879(E) (1993).
  47. R. Barbieri and E. Remiddi, Infra-red divergences and adiabatic switching. Fourth-order vaccum polarization, Nouvo Cimento Soc. Ital. Fis. A 13, 99 (1973).
  48. G.Källen and A. Sabry, Fourth order vacuum polarization, Dan. Mat. Fys. Medd. 29, 556 (1955).
  49. M. Laursen and M. A. Samuel, Corrections to the sixth order anomalous magnetic moment of the muon, Phys. Rev. D 19, 1281 (1979).
  50. E. Tiesinga, P. J. Mohr, D. B. Nevell, and B. N. Taylor, CODATA recommended values of the fundamental physical constants: 2018, Rev. Mod. Phys. 93, 025010 (2021).
  51. A. V. Kotikov and S. Teber, Multi-loop techniques for massless Feynman diagram calculations, Phys. Part. Nucl. 50, 1 (2019).
  52. Gh. Adam et al., IT-ecosystem of the HybriLIT heterogeneous platform for high-performance computing and training of IT-specialists, Proceedings of the VIII International Conference “Distributed Computing and Grid-technologies in Science and Education” (GRID 2018) (2018) [Report No. CEUR-WS.org/Vol-2267]; https://indico.jinr.ru/event/447/page/439-proceedings-new.

Outline

Information

Phys. Rev. D 112, 073008– Published 27 October, 2025
  • Received 11 August 2025
  • Accepted 26 September 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