- 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

Proton and neutron electromagnetic form factors in the continuum limit using lattice QCD ensembles with physical pion masses

Constantia Alexandrou1,2, Simone Bacchio1, Giannis Koutsou1, Bhavna Prasad1, and Gregoris Spanoudes2

PDF

Phys. Rev. D 113, 114524 – Published 29 June, 2026

DOI: https://doi.org/10.1103/tt39-n1df

Abstract

We compute the electromagnetic form factors of the proton and neutron using lattice QCD. We employ Nf=2+1+1 twisted mass clover-improved fermions with quark masses tuned to their physical values. Three ensembles with lattice spacings of a=0.080  fm, 0.068 fm, and 0.057 fm and approximately the same physical volume allow us to obtain the continuum limit directly at the physical pion mass. For each ensemble, we use several values of the sink-source time separation, ranging from 0.5 to 1.5 fm, to allow for a thorough analysis of excited-state effects via multistate fits. The disconnected contributions are also analyzed using high statistics combined with techniques to mitigate stochastic noise in the estimation of the fermion loop. These techniques include low-mode deflation, dilution in the color and spin components, and hierarchical probing. We study the momentum-transfer dependence of the form factors using the z-expansion and dipole Ansätze, thereby enabling the extraction of the electric and magnetic radii and the magnetic moments, as well as the Zemach and Friar radii in the continuum limit. Results for the proton and neutron electric and magnetic mean square radii are ⟨rE2⟩p=0.860(38)(23)  fm, ⟨rE2⟩n=−0.147(48)  fm2, ⟨rM2⟩p=0.870(53)(15)  fm, and ⟨rM2⟩n=0.913(67)(19)  fm, and for the proton and neutron magnetic moments, μp=2.849(92)(52) and μn=−1.819(76)(29), respectively. In all cases, the first error is statistical, and the second is systematic, where the latter includes an estimate of the error from the fits to the momentum dependence of the form factors and from the continuum extrapolation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (76)

  1. J. C. Bernauer et al. (A1 Collaboration), Electric and magnetic form factors of the proton, Phys. Rev. C 90, 015206 (2014).
  2. V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, The structure of the nucleon: Elastic electromagnetic form factors, Eur. Phys. J. A 51, 79 (2015).
  3. R. Pohl et al., The size of the proton, Nature (London) 466, 213 (2010).
  4. J. Golak, G. Ziemer, H. Kamada, H. Witala, and W. Gloeckle, Extraction of electromagnetic neutron form-factors through inclusive and exclusive polarized electron scattering on polarized He3 target, Phys. Rev. C 63, 034006 (2001).
  5. W. Xiong et al., A small proton charge radius from an electron–proton scattering experiment, Nature (London) 575, 147 (2019).
  6. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Electromagnetic form factors of the nucleon from Nf=2+1 lattice QCD, Phys. Rev. D 109, 094510 (2024).
  7. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Precision calculation of the electromagnetic radii of the proton and neutron from lattice QCD, Phys. Rev. Lett. 132, 211901 (2024).
  8. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. Vaquero Aviles-Casco, Proton and neutron electromagnetic form factors from lattice QCD, Phys. Rev. D 100, 014509 (2019).
  9. R. Tsuji, Y. Aoki, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, K. Sato, E. Shintani, H. Watanabe, and T. Yamazaki (PACS Collaboration), Nucleon form factors in Nf=2+1 lattice QCD at the physical point: Finite lattice spacing effect on the root-mean-square radii, Phys. Rev. D 109, 094505 (2024).
  10. Y.-C. Jang, R. Gupta, H.-W. Lin, B. Yoon, and T. Bhattacharya, Nucleon electromagnetic form factors in the continuum limit from (2+1+1)-flavor lattice QCD, Phys. Rev. D 101, 014507 (2020).
  11. R. Frezzotti and G. C. Rossi, Chirally improving Wilson fermions. 1. O(a) improvement, J. High Energy Phys. 08 (2004) 007.
  12. R. Frezzotti, P. A. Grassi, S. Sint, and P. Weisz (Alpha Collaboration), Lattice QCD with a chirally twisted mass term, J. High Energy Phys. 08 (2001) 058.
  13. M. Constantinou et al. (ETM Collaboration), Non-perturbative renormalization of quark bilinear operators with Nf=2 (tmQCD) Wilson fermions and the tree-level improved gauge action, J. High Energy Phys. 08 (2010) 068.
  14. C. Alexandrou et al., Simulating twisted mass fermions at physical light, strange and charm quark masses, Phys. Rev. D 98, 054518 (2018).
  15. J. Finkenrath et al., Twisted mass gauge ensembles at physical values of the light, strange and charm quark masses, Proc. Sci., LATTICE2021 (2022) 284 [arXiv:2201.02551].
  16. C. Alexandrou et al. (Extended Twisted Mass Collaboration), Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions, Phys. Rev. D 107, 074506 (2023).
  17. C. Alexandrou et al. (Extended Twisted Mass Collaboration), Quark masses using twisted-mass fermion gauge ensembles, Phys. Rev. D 104, 074515 (2021).
  18. S. Gusken, A study of smearing techniques for hadron correlation functions, Nucl. Phys. B, Proc. Suppl. 17, 361 (1990).
  19. C. Alexandrou, S. Gusken, F. Jegerlehner, K. Schilling, and R. Sommer, The static approximation of heavy—light quark systems: A systematic lattice study, Nucl. Phys. B414, 815 (1994).
  20. C. Alexandrou et al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys. Rev. D 101, 034519 (2020).
  21. M. Albanese et al. (APE Collaboration), Glueball masses and string tension in lattice QCD, Phys. Lett. B 192, 163 (1987).
  22. C. Alexandrou et al., Moments of the nucleon transverse quark spin densities using lattice QCD, Phys. Rev. D 107, 054504 (2023).
  23. C. McNeile and C. Michael (UKQCD Collaboration), Decay width of light quark hybrid meson from the lattice, Phys. Rev. D 73, 074506 (2006).
  24. C. Alexandrou, M. Constantinou, V. Drach, K. Hadjiyiannakou, K. Jansen, G. Koutsou, A. Strelchenko, and A. Vaquero, Evaluation of disconnected quark loops for hadron structure using GPUs, Comput. Phys. Commun. 185, 1370 (2014).
  25. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco, Nucleon axial form factors using Nf=2 twisted mass fermions with a physical value of the pion mass, Phys. Rev. D 96, 054507 (2017).
  26. C. Alexandrou et al., Nucleon scalar and tensor charges using lattice QCD simulations at the physical value of the pion mass, Phys. Rev. D 95, 114514 (2017); 96, 099906(E) (2017).
  27. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Avilés-Casco, and C. Wiese, Nucleon spin and momentum decomposition using lattice QCD simulations, Phys. Rev. Lett. 119, 142002 (2017).
  28. A. Stathopoulos, J. Laeuchli, and K. Orginos, Hierarchical probing for estimating the trace of the matrix inverse on toroidal lattices, SIAM J. Sci. Comput. 35, S299 (2013).
  29. A. S. Gambhir, A. Stathopoulos, and K. Orginos, Deflation as a method of variance reduction for estimating the trace of a matrix inverse, SIAM J. Sci. Comput. 39, A532 (2017).
  30. G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas, A general method for nonperturbative renormalization of lattice operators, Nucl. Phys. B445, 81 (1995).
  31. Extended Twisted Mass Collaboration, Non-perturbative renormalisation of quark bilinear operators with Nf=2+1+1 Wilson-clover twisted mass fermions (to be published).
  32. M. Gockeler, R. Horsley, H. Oelrich, H. Perlt, D. Petters, P. E. L. Rakow, A. Schafer, G. Schierholz, and A. Schiller, Nonperturbative renormalization of composite operators in lattice QCD, Nucl. Phys. B544, 699 (1999).
  33. C. Alexandrou, S. Bacchio, J. Finkenrath, C. Iona, G. Koutsou, Y. Li, and G. Spanoudes, Nucleon charges and σ-terms in lattice QCD, Phys. Rev. D 111, 054505 (2025).
  34. C. Alexandrou, M. Constantinou, and H. Panagopoulos (ETM Collaboration), Renormalization functions for Nf=2 and Nf=4 twisted mass fermions, Phys. Rev. D 95, 034505 (2017).
  35. O. Bar and H. Colic, Nπ-state contamination in lattice calculations of the nucleon electromagnetic form factors, Phys. Rev. D 103, 114514 (2021).
  36. W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103, 114502 (2021).
  37. E. T. Neil and J. W. Sitison, Improved information criteria for Bayesian model averaging in lattice field theory, Phys. Rev. D 109, 014510 (2024).
  38. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, R. Frezzotti, B. Kostrzewa, G. Koutsou, G. Spanoudes, and C. Urbach (Extended Twisted Mass Collaboration), Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point, Phys. Rev. D 109, 034503 (2024).
  39. S. Galster, H. Klein, J. Moritz, K. H. Schmidt, D. Wegener, and J. Bleckwenn, Elastic electron-deuteron scattering and the electric neutron form factor at four-momentum transfers 5  fm−2<q2<14  fm−2, Nucl. Phys. B32, 221 (1971).
  40. G. Lee, J. R. Arrington, and R. J. Hill, Extraction of the proton radius from electron-proton scattering data, Phys. Rev. D 92, 013013 (2015).
  41. A. S. Meyer, M. Betancourt, R. Gran, and R. J. Hill, Deuterium target data for precision neutrino-nucleus cross sections, Phys. Rev. D 93, 113015 (2016).
  42. R. J. Hill and G. Paz, Model independent extraction of the proton charge radius from electron scattering, Phys. Rev. D 82, 113005 (2010).
  43. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  44. Z. Ye, J. Arrington, R. J. Hill, and G. Lee, Proton and neutron electromagnetic form factors and uncertainties, Phys. Lett. B 777, 8 (2018).
  45. J. Becker et al., Determination of the neutron electric form-factor from the reaction He3(e,e’ n) at medium momentum transfer, Eur. Phys. J. A 6, 329 (1999).
  46. T. Eden et al., Electric form factor of the neutron from the H2(e→,e′n→)1H reaction at Q2=0.255  (GeV/c)2, Phys. Rev. C 50, R1749 (1994).
  47. M. Meyerhoff et al., First measurement of the electric form-factor of the neutron in the exclusive quasielastic scattering of polarized electrons from polarized He3, Phys. Lett. B 327, 201 (1994).
  48. I. Passchier et al., The Charge form-factor of the neutron from the reaction H→2(e→,e′n)p, Phys. Rev. Lett. 82, 4988 (1999).
  49. G. Warren et al. (Jefferson Lab E93-026 Collaboration), Measurement of the electric form-factor of the neutron at Q2=0.5 and 1.0  GeV2/c2, Phys. Rev. Lett. 92, 042301 (2004).
  50. H. Zhu et al. (E93026 Collaboration), A measurement of the electric form-factor of the neutron through d→(e→,e′n)p at Q2=0.5−(GeV/c)2, Phys. Rev. Lett. 87, 081801 (2001).
  51. R. Madey et al. (E93-038 Collaboration), Measurements of GEn/GMn from the H2(e→,e′n→)H2 reaction to Q2=1.45  (GeV/c)2, Phys. Rev. Lett. 91, 122002 (2003).
  52. D. Rohe et al., Measurement of the neutron electric form-factor Gen at 0.67−(GeV/c)2 via He→3(e→,e′n), Phys. Rev. Lett. 83, 4257 (1999).
  53. J. Bermuth et al., The Neutron charge form-factor and target analyzing powers from He→3(e→,e′n) scattering, Phys. Lett. B 564, 199 (2003).
  54. D. I. Glazier et al., Measurement of the electric form-factor of the neutron at Q2=0.3−(GeV/c)2 to 0.8−(GeV/c)2, Eur. Phys. J. A 24, 101 (2005).
  55. C. Herberg et al., Determination of the neutron electric form-factor in the D(e,e′n)p reaction and the influence of nuclear binding, Eur. Phys. J. A 5, 131 (1999).
  56. R. Schiavilla and I. Sick, Neutron charge form-factor at large q2, Phys. Rev. C 64, 041002 (2001).
  57. M. Ostrick et al., Measurement of the neutron electric form-factor GE,n in the quasifree H2(e→,e′n→)p reaction, Phys. Rev. Lett. 83, 276 (1999).
  58. B. Anderson et al. (Jefferson Lab E95-001 Collaboration), Extraction of the neutron magnetic form factor from quasi-elastic He→3(e→,e′) at Q2=0.1–0.6  (GeV/c)2, Phys. Rev. C 75, 034003 (2007).
  59. H. Gao et al., Measurement of the neutron magnetic form-factor from inclusive quasielastic scattering of polarized electrons from polarized He3, Phys. Rev. C 50, R546 (1994).
  60. H. Anklin et al., Precision measurement of the neutron magnetic form-factor, Phys. Lett. B 336, 313 (1994).
  61. H. Anklin et al., Precise measurements of the neutron magnetic form-factor, Phys. Lett. B 428, 248 (1998).
  62. G. Kubon et al., Precise neutron magnetic form-factors, Phys. Lett. B 524, 26 (2002).
  63. R. Alarcon (BLAST Collaboration), Nucleon form factors and the BLAST experiment, Eur. Phys. J. A 32, 477 (2007).
  64. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco, Nucleon electromagnetic form factors using lattice simulations at the physical point, Phys. Rev. D 96, 034503 (2017).
  65. E. Shintani, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, and T. Yamazaki, Nucleon form factors and root-mean-square radii on a (10.8  fm)4 lattice at the physical point, Phys. Rev. D 99, 014510 (2019); 102, 019902(E) (2020).
  66. A. Antognini et al., Proton structure from the measurement of 2S−2P transition frequencies of muonic hydrogen, Science 339, 417 (2013).
  67. 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).
  68. M. O. Distler, J. C. Bernauer, and T. Walcher, The RMS charge radius of the proton and zemach moments, Phys. Lett. B 696, 343 (2011).
  69. Y.-H. Lin, H.-W. Hammer, and U.-G. Meißner, New insights into the nucleon’s electromagnetic structure, Phys. Rev. Lett. 128, 052002 (2022).
  70. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, M. Salg, and H. Wittig, Zemach and friar radii of the proton and neutron from lattice QCD, Phys. Rev. D 110, L011503 (2024).
  71. K. M. Graczyk and C. Juszczak, Zemach moments of the proton from Bayesian inference, Phys. Rev. C 91, 045205 (2015).
  72. A. V. Volotka, V. M. Shabaev, G. Plunien, and G. Soff, Zemach and magnetic radius of the proton from the hyperfine splitting in hydrogen, Eur. Phys. J. D 33, 23 (2005).
  73. P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATA recommended values of the fundamental physical constants: 2014, Rev. Mod. Phys. 88, 035009 (2016).
  74. C. Alexandrou et al., Large-scale simulations of lattice QCD for nucleon structure using Nf=2+1+1 flavors of twisted mass fermions, Procedia Comput. Sci. 267, 92 (2025).
  75. Jülich Supercomputing Centre, JUWELS: Modular Tier-0/1 supercomputer at the Jülich supercomputing centre, J. Large-Scale Res. Facil. 5, A135 (2019).
  76. Jülich Supercomputing Centre, JUWELS cluster and booster: Exascale pathfinder with modular supercomputing architecture at Juelich supercomputing centre, J. Large-Scale Res. Facil. 7, A183 (2021).

Outline

Information

Phys. Rev. D 113, 114524– Published 29 June, 2026
  • Received 5 August 2025
  • Accepted 27 May 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