- 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

Higher order electroweak radiative corrections in lepton-proton scattering using covariant approach

A. Aleksejevs*, S. Barkanova†, and M. Ghaffar‡

  • *Contact author: aaleksejevs@mun.ca
  • †Contact author: sbarkanova@mun.ca
  • ‡Contact author: mghaffar@mun.ca
PDF

Phys. Rev. D 113, 113002 – Published 8 June, 2026

DOI: https://doi.org/10.1103/y66m-rzl3

Abstract

We perform detailed calculations of electroweak radiative corrections to parity violating lepton scattering with a proton target (ep and μp) up to quadratic and reducible two-loop level using a covariant approach. Our numerical results are presented at energies relevant for a variety of existing and proposed experimental programs such as Qweak, P2, Measurement of a Lepton-Lepton Electroweak Reaction (MOLLER), Muon Scattering Experiment, and experiments at the Electron-Ion Collider. Analysis shows that such corrections at the next-to-next-to-leading order are quite significant and have to be included in searches of physics beyond the standard model, matching the increasing precision of the future experimental programs at low-energy scales.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. T. D. Lee and C. N. Yang, Question of parity conservation in weak interactions, Phys. Rev. 104, 254 (1956).
  2. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Experimental test of parity conservation in beta decay, Phys. Rev. 105, 1413 (1957).
  3. D. Androic et al. (Qweak Collaboration), First determination of the weak charge of the proton, Phys. Rev. Lett. 111, 141803 (2013).
  4. T. Allison et al. (Qweak Collaboration), The Qweak experimental apparatus, Nucl. Instrum. Methods Phys. Res., Sect. A 781, 105 (2015).
  5. D. Androić et al. (Qweak Collaboration), Precision measurement of the weak charge of the proton, Nature (London) 557, 207 (2018).
  6. D. Becker et al., The P2 experiment—A future high-precision measurement of the weak mixing angle at low momentum transfer, Eur. Phys. J. A 54, 208 (2018).
  7. A. Aleksejevs, S. Barkanova, A. Ilyichev, and V. Zykunov, Electroweak radiative corrections for polarized Moller scattering at future 11 GeV JLab experiment, Phys. Rev. D 82, 093013 (2010).
  8. A. Accardi et al., Electron-Ion Collider: The next QCD frontier—understanding the glue that binds us all, Eur. Phys. J. A 52, 268 (2016).
  9. R. Gilman et al. (MUSE Collaboration), Technical Design Report for the Paul Scherrer Institute Experiment R-12-01.1: Studying the proton “Radius” puzzle with μp elastic scattering, arXiv:1709.09753.
  10. M. Kohl (MUSE Collaboration), The muon scattering experiment (MUSE) at PSI and the proton radius puzzle, EPJ Web Conf. 81, 02008 (2014).
  11. M. J. Musolf and T. W. Donnelly, The interpretation of parity-violating electron-scattering experiments, Nucl. Phys. A 546, 509 (1992).
  12. M. J. Musolf and B. R. Holstein, Electroweak corrections to parity-violating neutral current scattering, Phys. Lett. B 242, 461 (1990).
  13. M. J. Musolf and B. R. Holstein, Observability of the anapole moment and neutrino charge radius, Phys. Rev. D 43, 2956 (1991).
  14. W. J. Marciano and A. Sirlin, Some general properties of the O(α) corrections to parity violation in atoms, Phys. Rev. D 29, 75 (1984).
  15. A. V. Afanasev and N. P. Merenkov, Large logarithms in the beam normal spin asymmetry of elastic electron-proton scattering, Phys. Rev. D 70, 073002 (2004).
  16. I. Akushevich, A. Ilyichev, and N. M. Shumeiko, QED radiative effects in the processes of exclusive photon electroproduction from polarized protons with the next-to-leading accuracy, Phys. Rev. D 90, 033001 (2014).
  17. A. Czarnecki and W. J. Marciano, Electroweak radiative corrections to polarized Moller scattering asymmetries, Phys. Rev. D 53, 1066 (1996).
  18. A. Aleksejevs, S. Barkanova, A. Ilyichev, Y. Kolomensky, and V. Zykunov, One-loop electroweak corrections for polarized Moller scattering at different renormalization schemes and conditions, Phys. Part. Nucl. 44, 161 (2013).
  19. A. G. Aleksejevs, S. G. Barkanova, and V. A. Zykunov, Precise calculations of observables of polarized Moller scattering: From JLAB to ILC energies, Phys. At. Nucl. 75, 209 (2012).
  20. A. Aleksejevs, S. Barkanova, Y. Kolomensky, E. Kuraev, and V. Zykunov, Quadratic electroweak corrections for polarized Møller scattering, Phys. Rev. D 85, 013007 (2012).
  21. A. G. Aleksejevs, S. G. Barkanova, Y. M. Bystritskiy, E. A. Kuraev, A. N. Ilyichev, and V. A. Zykunov, Parity violating Moller scattering asymmetry up to the two-loop level, arXiv:1202.0378.
  22. A. Aleksejevs, S. Barkanova, and V. Zykunov, Two-Loop effects in low-energy electroweak measurements, Nucl. Part. Phys. Proc. 273–275, 2259 (2016).
  23. D. Yu. Bardin and N. M. Shumeiko, On an exact calculation of the lowest-order electromagnetic correction to the point particle elastic scattering, Nucl. Phys. B127, 242 (1977).
  24. A. Afanasev, I. Akushevich, V. Burkert, and K. Joo, QED radiative corrections in processes of exclusive pion electroproduction, Phys. Rev. D 66, 074004 (2002).
  25. T. Hahn, Generating Feynman diagrams and amplitudes with feynarts 3, Comput. Phys. Commun. 140, 418 (2001).
  26. T. Hahn and M. Pérez-Victoria, Automated one-loop calculations in four and D dimensions, Comput. Phys. Commun. 118, 153 (1999).
  27. V. Shtabovenko, R. Mertig, and F. Orellana, feyncalc 9.3: New features and improvements, Comput. Phys. Commun. 256, 107478 (2020).
  28. A. Bacchetta, M. Cerutti, L. Manna, M. Radici, and X. Zheng, Signals of strong parity violation in deep inelastic scattering, Phys. Lett. B 849, 138455 (2024).
  29. M. Gorchtein, Beam normal spin asymmetry in the quasireal Compton scattering approximation, Phys. Rev. C 73, 055201 (2006).
  30. P. G. Blunden, W. Melnitchouk, and J. A. Tjon, Two-Photon exchange and elastic Electron-Proton scattering, Phys. Rev. Lett. 94, 142304 (2003).
  31. M. N. Rosenbluth, High energy elastic scattering of electrons on protons, Phys. Rev. 79, 615 (1950).
  32. O. Gayou et al. (Jefferson Lab Hall A Collaboration), Measurement of GEp/GMp in e→p→ep→ to Q2=5.6  GeV2, Phys. Rev. Lett. 88, 092301 (2002).
  33. I. A. Qattan et al., High precision measurements of the proton elastic electromagnetic form factors and their ratio at Q2=0.50, 2.64, 3.20, and 4.10  GeV2, Phys. Rev. C 112, 035205 (2025).
  34. J. Arrington, Implications of the discrepancy between proton form factor measurements, Phys. Rev. C 69, 022201 (2004).
  35. T. J. Hague, J. Arrington, J. Jayne, G. A. Miller, S. N. Santiesteban, and Z. Ye, Updated analysis of neutron magnetic form factor and the nucleon transverse densities, arXiv:2501.18443.
  36. K. Borah, R. J. Hill, G. Lee, and O. Tomalak, Parametrization and applications of the low-Q2 nucleon vector form factors, Phys. Rev. D 102, 074012 (2020).
  37. A. V. Afanasev and C. E. Carlson, Two-photon-exchange correction to parity-violating elastic electron-proton scattering, Phys. Rev. Lett. 94, 212301 (2005).
  38. D. Altarawneh, R. Höllwieser, and M. Wellenzohn, Neutron beta decay and exact conservation of charged weak hadronic vector current in the standard model, Universe 10, 415 (2024).
  39. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  40. G. Passarino and M. Veltman, One-loop corrections for e+e− annihilation into μ+μ− in the Weinberg model, Nucl. Phys. B160, 151 (1979).
  41. W. F. L. Hollik, Radiative corrections in the standard model and their role for precision tests of the electroweak theory, Fortschr. Phys. 38, 165 (1990).
  42. A. Denner, Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortschr. Phys. 41, 307 (1993).
  43. A. Afanasev and A. Ilyichev, Contribution of hard photon emission to the charge asymmetry in elastic lepton- and antilepton-proton scattering, Phys. Rev. D 105, L011301 (2022).
  44. A. Aleksejevs, S. Barkanova, A. Ilyichev, and V. Zykunov, Electroweak radiative corrections for polarized Møller scattering at the future 11 GeV JLab experiment, Phys. Rev. D 82, 093013 (2010).
  45. G. ’t Hooft and M. Veltman, Scalar one-loop integrals, Nucl. Phys. B 153, 365 (1979).
  46. N. L. Hall, P. G. Blunden, W. Melnitchouk, A. W. Thomas, and R. D. Young, Hadronic γZ box corrections in Møller scattering, Phys. Lett. B 731, 287 (2014); Phys. Lett. B733, 380(E) (2014).
  47. J. Ahmed, P. G. Blunden, and W. Melnitchouk, Normal single-spin asymmetries in electron-proton scattering: Two-photon exchange with intermediate-state resonances, Phys. Rev. C 108, 055202 (2023).
  48. A. Afanasev et al., Radiative corrections: From medium to high energy experiments, Eur. Phys. J. A 60, 1281 (2024).
  49. Y.-S. Tsai, Radiative corrections to electron-proton scattering, Phys. Rev. 122, 1898 (1961).
  50. L. C. Maximon and J. A. Tjon, Radiative corrections to electron-proton scattering, Phys. Rev. C 62, 054320 (2000).
  51. J. Mammei (MOLLER Collaboration), The MOLLER experiment, Nuovo Cimento C 035N04, 203 (2012).
  52. J. Benesch et al. (MOLLER Collaboration), The MOLLER experiment: An ultra-precise measurement of the weak mixing angle using Moller scattering, arXiv:1411.4088.
  53. L. Bellafronte, S. Dawson, P. P. Giardino, and H. Liu, Probing top quark—electron interactions at future colliders, Phys. Rev. Lett. 135, 251801 (2025).
  54. F. Gil-Domínguez, J. M. Alarcón, and C. Weiss, Proton charge radius extraction from muon scattering at MUSE using dispersively improved chiral effective field theory, Phys. Rev. D 108, 074026 (2023).
  55. E. Downie (MUSE Collaboration), MUSE—the muon proton scattering experiment, in APS April Meeting Abstracts (APS, 2023), Vol. 2023, p. M17 002.
  56. A. Naik and A. Afanasev, Contribution of π0 exchange in elastic muon-proton scattering, Phys. Rev. C 110, 035202 (2024).

Outline

Information

Phys. Rev. D 113, 113002– Published 8 June, 2026
  • Received 30 July 2025
  • Accepted 8 April 2026
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