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Hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon at the physical pion mass

Zoltan Fodor1,2,3,4, Antoine Gérardin5,*, Laurent Lellouch5, Kalman K. Szabo2, Balint C. Toth1,2, and Christian Zimmermann5,†

  • *Contact author: antoine.gerardin@cpt.univ-mrs.fr
  • †Contact author: christian.zimmermann@univ-amu.fr
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Phys. Rev. D 111, 114509 – Published 16 June, 2025

DOI: https://doi.org/10.1103/wdrk-7nrt

Abstract

We present a lattice quantum chromodynamics (QCD) calculation of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon using Nf=2+1+1 flavors of staggered quarks with masses tuned to their physical values. Our final result, in the continuum limit, reads aμhlbl=125.5(11.5)stat(1.9)syst×10−11 where the first error is statistical and the second is systematic. Light, strange, and charm-quark contributions are considered. In addition to the connected and leading disconnected contributions, we also include an estimate of the subleading disconnected diagrams. Our result is compatible with previous lattice QCD and data-driven dispersive determinations.

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References (62)

  1. B. Abi et al. (Muon g-2 Collaboration), Measurement of the positive muon anomalous magnetic moment to 0.46 ppm, Phys. Rev. Lett. 126, 141801 (2021).
  2. D. P. Aguillard et al. (Muon g-2 Collaboration), Measurement of the positive muon anomalous magnetic moment to 0.20 ppm, Phys. Rev. Lett. 131, 161802 (2023).
  3. G. W. Bennett et al. (Muon g-2 Collaboration), Final report of the muon E821 anomalous magnetic moment measurement at BNL, Phys. Rev. D 73, 072003 (2006).
  4. M. Abe et al., A new approach for measuring the muon anomalous magnetic moment and electric dipole moment, Prog. Theor. Exp. Phys. 2019, 053C02 (2019).
  5. T. Aoyama et al., The anomalous magnetic moment of the muon in the standard model, Phys. Rep. 887, 1 (2020).
  6. C. Aubin, T. Blum, C. Tu, M. Golterman, C. Jung, and S. Peris, Light quark vacuum polarization at the physical point and contribution to the muon g−2, Phys. Rev. D 101, 014503 (2020).
  7. M. Cè et al., Window observable for the hadronic vacuum polarization contribution to the muon g−2 from lattice QCD, Phys. Rev. D 106, 114502 (2022).
  8. 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).
  9. A. Bazavov et al. (Fermilab Lattice, HPQCD, MILC Collaborations), Light-quark connected intermediate-window contributions to the muon g−2 hadronic vacuum polarization from lattice QCD, Phys. Rev. D 107, 114514 (2023).
  10. T. Blum et al. (RBC, UKQCD Collaborations), Update of Euclidean windows of the hadronic vacuum polarization, Phys. Rev. D 108, 054507 (2023).
  11. A. Boccaletti et al., High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly, arXiv:2407.10913.
  12. D. Djukanovic, G. von Hippel, S. Kuberski, H. B. Meyer, N. Miller, K. Ottnad, J. Parrino, A. Risch, and H. Wittig, The hadronic vacuum polarization contribution to the muon g−2 at long distances, J. High Energy Phys. 04 (2025) 098.
  13. K. Melnikov and A. Vainshtein, Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited, Phys. Rev. D 70, 113006 (2004).
  14. P. Masjuan and P. Sanchez-Puertas, Pseudoscalar-pole contribution to the (gμ−2): A rational approach, Phys. Rev. D 95, 054026 (2017).
  15. G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Dispersion relation for hadronic light-by-light scattering: Two-pion contributions, J. High Energy Phys. 04 (2017) 161.
  16. M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Dispersion relation for hadronic light-by-light scattering: Pion pole, J. High Energy Phys. 10 (2018) 141.
  17. A. Gérardin, H. B. Meyer, and A. Nyffeler, Lattice calculation of the pion transition form factor with Nf=2+1 Wilson quarks, Phys. Rev. D 100, 034520 (2019).
  18. J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment, Phys. Lett. B 798, 134994 (2019).
  19. G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, Longitudinal short-distance constraints for the hadronic light-by-light contribution to (g−2)μ with large-Nc Regge models, J. High Energy Phys. 03 (2020) 101.
  20. V. Pauk and M. Vanderhaeghen, Single meson contributions to the muon‘s anomalous magnetic moment, Eur. Phys. J. C 74, 3008 (2014).
  21. I. Danilkin and M. Vanderhaeghen, Light-by-light scattering sum rules in light of new data, Phys. Rev. D 95, 014019 (2017).
  22. F. Jegerlehner, The anomalous magnetic moment of the muon, Springer Tracts Mod. Phys. 274, 1 (2017).
  23. M. Knecht, S. Narison, A. Rabemananjara, and D. Rabetiarivony, Scalar meson contributions to aμ from hadronic light-by-light scattering, Phys. Lett. B 787, 111 (2018).
  24. G. Eichmann, C. S. Fischer, and R. Williams, Kaon-box contribution to the anomalous magnetic moment of the muon, Phys. Rev. D 101, 054015 (2020).
  25. P. Roig and P. Sánchez-Puertas, Axial-vector exchange contribution to the hadronic light-by-light piece of the muon anomalous magnetic moment, Phys. Rev. D 101, 074019 (2020).
  26. A. Gérardin, H. B. Meyer, and A. Nyffeler, Lattice calculation of the pion transition form factor π0→γ*γ*, Phys. Rev. D 94, 074507 (2016).
  27. C. Alexandrou et al. (Extended Twisted Mass Collaboration), Pion transition form factor from twisted-mass lattice QCD and the hadronic light-by-light π0-pole contribution to the muon g−2, Phys. Rev. D 108, 094514 (2023).
  28. A. Gérardin, W. E. A. Verplanke, G. Wang, Z. Fodor, J. N. Guenther, L. Lellouch, K. K. Szabo, and L. Varnhorst, Lattice calculation of the π0, η and η′ transition form factors and the hadronic light-by-light contribution to the muon g−2, Phys. Rev. D 111, 054511 (2025).
  29. T. Lin, M. Bruno, X. Feng, L.-C. Jin, C. Lehner, C. Liu, and Q.-Y. Luo, Lattice QCD calculation of the π0-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon, arXiv:2411.06349.
  30. T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, C. Lehner, and C. Tu, Hadronic light-by-light contribution to the muon anomaly from lattice QCD with infinite volume QED at physical pion mass, Phys. Rev. D 111, 014501 (2025).
  31. T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, Using infinite volume, continuum QED and lattice QCD for the hadronic light-by-light contribution to the muon anomalous magnetic moment, Phys. Rev. D 96, 034515 (2017).
  32. E.-H. Chao, R. J. Hudspith, A. Gérardin, J. R. Green, and H. B. Meyer, The charm-quark contribution to light-by-light scattering in the muon (g−2) from lattice QCD, Eur. Phys. J. C 82, 664 (2022).
  33. E.-H. Chao, R. J. Hudspith, A. Gérardin, J. R. Green, H. B. Meyer, and K. Ottnad, Hadronic light-by-light contribution to (g−2)μ from lattice QCD: A complete calculation, Eur. Phys. J. C 81, 651 (2021).
  34. E.-H. Chao, A. Gérardin, J. R. Green, R. J. Hudspith, and H. B. Meyer, Hadronic light-by-light contribution to (g−2)μ from lattice QCD with SU(3) flavor symmetry, Eur. Phys. J. C 80, 869 (2020).
  35. T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner, Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment from lattice QCD, Phys. Rev. Lett. 124, 132002 (2020).
  36. N. Asmussen, E.-H. Chao, A. Gérardin, J. R. Green, R. J. Hudspith, H. B. Meyer, and A. Nyffeler, Hadronic light-by-light scattering contribution to the muon g−2 from lattice QCD: semi-analytical calculation of the QED kernel, J. High Energy Phys. 04 (2023) 040.
  37. E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong (HPQCD, UKQCD Collaborations), Highly improved staggered quarks on the lattice, with applications to charm physics, Phys. Rev. D 75, 054502 (2007).
  38. W. E. A. Verplanke, Z. Fodor, A. Gerardin, J. N. Guenther, L. Lellouch, K. K. Szabo, B. C. Toth, and L. Varnhorst, Lattice QCD calculation of the η and η′ meson masses at the physical point using rooted staggered fermions, Phys. Rev. D 111, 094506 (2025).
  39. S. Borsanyi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature (London) 593, 51 (2021).
  40. D. Bernecker and H. B. Meyer, Vector correlators in lattice QCD: Methods and applications, Eur. Phys. J. A 47, 148 (2011).
  41. S. Laporta and E. Remiddi, The analytic value of the light-light vertex graph contributions to the electron (g−2) in QED, Phys. Lett. B 265, 182 (1991).
  42. S. Laporta and E. Remiddi, The analytical value of the electron light-light graphs contribution to the muon (g−2) in QED, Phys. Lett. B 301, 440 (1993).
  43. J. H. Kuhn, A. I. Onishchenko, A. A. Pivovarov, and O. L. Veretin, Heavy mass expansion, light by light scattering and the anomalous magnetic moment of the muon, Phys. Rev. D 68, 033018 (2003).
  44. C. T. H. Davies, C. McNeile, K. Y. Wong, E. Follana, R. Horgan, K. Hornbostel, G. P. Lepage, J. Shigemitsu, and H. Trottier, Precise charm to strange mass ratio and light quark masses from full lattice QCD, Phys. Rev. Lett. 104, 132003 (2010).
  45. M. Knecht and A. Nyffeler, Hadronic light by light corrections to the muon g−2: The pion pole contribution, Phys. Rev. D 65, 073034 (2002).
  46. J. Green, O. Gryniuk, G. von Hippel, H. B. Meyer, and V. Pascalutsa, Lattice QCD calculation of hadronic light-by-light scattering, Phys. Rev. Lett. 115, 222003 (2015).
  47. J. Bijnens and J. Relefors, Pion light-by-light contributions to the muon g−2, J. High Energy Phys. 09 (2016) 113,
  48. G. P. Lepage and S. J. Brodsky, Exclusive processes in quantum chromodynamics: Evolution equations for hadronic wave functions and the form-factors of mesons, Phys. Lett. B 87, 359 (1979).
  49. G. P. Lepage and S. J. Brodsky, Exclusive processes in perturbative quantum chromodynamics, Phys. Rev. D 22, 2157 (1980).
  50. S. J. Brodsky and G. P. Lepage, Large angle two photon exclusive channels in quantum chromodynamics, Phys. Rev. D 24, 1808 (1981).
  51. V. A. Nesterenko and A. V. Radyushkin, Comparison of the QCD sum rule approach and perturbative QCD analysis for γ*γ*→π0 process, Sov. J. Nucl. Phys. 38, 284 (1983).
  52. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, M. B. Voloshin, and V. I. Zakharov, Use and misuse of QCD sum rules, factorization and related topics, Nucl. Phys. B237, 525 (1984).
  53. L. Jin, T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, C. Jung, and C. Lehner, The connected and leading disconnected diagrams of the hadronic light-by-light contribution to muon g−2, Proc. Sci. LATTICE2016 (2016) 181 [arXiv:1611.08685].
  54. A. Gérardin, J. Green, O. Gryniuk, G. von Hippel, H. B. Meyer, V. Pascalutsa, and H. Wittig, Hadronic light-by-light scattering amplitudes from lattice QCD versus dispersive sum rules, Phys. Rev. D 98, 074501 (2018).
  55. R. L. Workman et al. (Particle Data Group), Review of particle physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  56. T. A. DeGrand and S. Schaefer, Improving meson two-point functions by low-mode averaging, Nucl. Phys. B, Proc. Suppl. 140, 296 (2005).
  57. L. Giusti, P. Hernandez, M. Laine, P. Weisz, and H. Wittig, Low-energy couplings of QCD from current correlators near the chiral limit, J. High Energy Phys. 04 (2004) 013.
  58. L. Giusti, T. Harris, A. Nada, and S. Schaefer, Frequency-splitting estimators of single-propagator traces, Eur. Phys. J. C 79, 586 (2019).
  59. F. Jegerlehner, Muon g−2 theory: The hadronic part, EPJ Web Conf. 166, 00022 (2018).
  60. J. Prades, E. de Rafael, and A. Vainshtein, The hadronic light-by-light scattering contribution to the muon and electron anomalous magnetic moments, Adv. Ser. Dir. High Energy Phys. 20, 303 (2009).
  61. A. Nyffeler, Hadronic light-by-light scattering in the muon g−2: A new short-distance constraint on pion-exchange, Phys. Rev. D 79, 073012 (2009).
  62. F. Jegerlehner and A. Nyffeler, The muon g−2, Phys. Rep. 477, 1 (2009).

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Phys. Rev. D 111, 114509– Published 16 June, 2025
  • Received 10 February 2025
  • Accepted 22 May 2025
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