Anomalous magnetic dipole moment

Short description: Value in quantum electrodynamics

In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. The magnetic moment, also called magnetic dipole moment, is a measure of the strength of a magnetic source.

The "Dirac" magnetic moment, corresponding to tree-level Feynman diagrams (which can be thought of as the classical result), can be calculated from the Dirac equation. It is usually expressed in terms of the g-factor; the Dirac equation predicts g=−2. For particles such as the electron, this classical result differs from the observed value by a small fraction of a percent. The difference is the anomalous magnetic moment, denoted a and defined as a=|g|−22 (the magnitude of g is used in the calculation since the g-factor is, in fact, negative).

## Contents

* 1 Electron
* 2 Muon
* 3 Tau
* 4 Composite particles
* 5 See also
* 6 References
* 7 Bibliography
* 8 External links

## Electron

One-loop correction to a fermion's magnetic dipole moment.

The one-loop contribution to the anomalous magnetic moment—corresponding to the first and largest quantum mechanical correction—of the electron is found by calculating the vertex function shown in the adjacent diagram. The calculation is relatively straightforward [1] and the one-loop result is: ae=α2π≈0.0011614, where α is the fine-structure constant. This result was first found by Julian Schwinger in 1948 [2] and is engraved on his tombstone. As of 2016, the coefficients of the QED formula for the anomalous magnetic moment of the electron are known analytically up to α3 [3] and have been calculated up to order α5:[4][5][6] ae=0.001159652181643(764)

The QED prediction agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron one of the most accurately verified predictions in the history of physics. (See Precision tests of QED for details.)

The current experimental value and uncertainty is:[7] ae=0.00115965218059(13) According to this value, ae is known to an accuracy of around 1 part in 10 billion (1010). This required measuring g to an accuracy of around 1 part in 10 trillion (1013).

## Muon

Proposed Minimal Supersymmetric Standard Model one-loop corrections to the muon g−2 involving particles beyond the standard model: a neutralino and a smuon, and a chargino and a muon sneutrino respectively.

The anomalous magnetic moment of the muon is calculated in a similar way to the electron. The prediction for the value of the muon anomalous magnetic moment includes three parts:[8] aμSM=aμQED+aμEW+aμhadron.

Of the first two components, aμQED represents the photon and lepton loops, and aμEW the W boson, Higgs boson and Z boson loops; both can be calculated precisely from first principles. The third term, aμhadron, represents hadron loops; it cannot be calculated accurately from theory alone. It is either estimated from experimental measurements of the ratio of hadronic to muonic cross sections (R) in electron–antielectron (e−–e+) collisions, or else calculated from scratch in lattice gauge theory.

The E821 Experiment at Brookhaven National Laboratory (BNL) studied the precession of muon and antimuon in a constant external magnetic field as they circulated in a confining storage ring.[9] The E821 Experiment reported the following average value[8] aμ=0.0011659209(6).

In 2024, the Fermilab collaboration "Muon g−2" doubled the accuracy of this value over the group's previous measurements from the 2018 data set. The data for the experiment were collected during the 2019–2020 runs. The independent value came in at aμ=0.00116592057(25), which, combined with measurements from Brookhaven National Laboratory, yields a world average of aμ=0.00116592059(22).[10] In June 2025, the "Muon g−2" team published their final results for the magnetic moment as an eprint,[11] at aμ=0.001165920705(148), bringing the world average to aμ=0.001165920715(145).

For a long time, experimental measurements and theoretical predictions of this quantity disagreed, making the muon's anomalous magnetic moment a long-standing discrepancy between the Standard Model and experiment. As of July 2017, for instance, the measurement disagreed with the Standard Model by 3.5 standard deviations.[12] This disagreement was seen as suggesting physics beyond the Standard Model may be having an effect, or else that the theoretical/experimental errors were not completely under control.

In April 2021, an international group of fourteen physicists reported that by using ab-initio quantum chromodynamics and quantum electrodynamics simulations they were able to obtain a theory-based approximation agreeing more with the experimental value than with the previous theory-based value that relied on the electron–positron annihilation experiments.[13] In May 2025, a large international consortium of scientists reported calculations in which the estimate of the quark and gluon contributions that in earlier calculations had been based on experimental data had been replaced with from-scratch lattice QCD calculations.[14] This yielded aμ=0.00116592033(62), which was compatible both with the then-best experimental data and with the final "Muon g−2" result that was published a few weeks after.

## Tau

The Standard Model prediction for the tau's anomalous magnetic dipole moment is[15] aτ=0.00117721(5), while the best measured bound for aτ is[16] aτ=0.0009−0.0031+0.0032,reported by the CMS experiment at the CERN LHC.

## Composite particles

Composite particles often have large anomalous magnetic moments. The nucleons, protons and neutrons, both composed of quarks, are examples. The nucleon magnetic moments are both large and were unexpected; the proton's magnetic moment is much too large for an elementary particle, while the neutron's magnetic moment was expected to be zero due to its charge being zero.

## See also

* Anomalous electric dipole moment
* Electron magnetic moment
* g-factor
* Gordon decomposition

## References

1. ↑ Peskin, M. E.; Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Addison-Wesley. ISBN 978-0-201-50397-5. https://archive.org/details/introductiontoqu0000pesk.
2. ↑ Schwinger, J. (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron". Physical Review 73 (4): 416–417. doi:10.1103/PhysRev.73.416. Bibcode: 1948PhRv...73..416S. http://web.ihep.su/dbserv/compas/src/schwinger48b/eng.pdf.
3. ↑ Laporta, S.; Remiddi, E. (1996). "The analytical value of the electron (g − 2) at order α3 in QED". Physics Letters B 379 (1–4): 283–291. doi:10.1016/0370-2693(96)00439-X. Bibcode: 1996PhLB..379..283L.''g''+−+2)+at+order+''α''3+in+QED&rft.jtitle=[[Physics:Physics+Letters+B|Physics+Letters+B]]&rft.aulast=Laporta&rft.aufirst=S.&rft.au=Laporta, S.&rft.au=Remiddi, E.&rft.date=1996&rft.volume=379&rft.issue=1–4&rft.pages=283–291&rft_id=info:doi/10.1016/0370-2693(96)00439-X&rft_id=info:bibcode/1996PhLB..379..283L&rfr_id=info:sid/en.wikibooks.org:Physics:Anomalous_magnetic_dipole_moment">
4. ↑ Aoyama, T.; Hayakawa, M.; Kinoshita, T.; Nio, M. (2012). "Tenth-Order QED Contribution to the Electron g−2 and an Improved Value of the Fine Structure Constant". Physical Review Letters 109 (11). doi:10.1103/PhysRevLett.109.111807. PMID 23005618. Bibcode: 2012PhRvL.109k1807A.
5. ↑ Aoyama, Tatsumi; Hayakawa, Masashi; Kinoshita, Toichiro; Nio, Makiko (1 February 2015). "Tenth-Order Electron Anomalous Magnetic Moment – Contribution of Diagrams without Closed Lepton Loops". Physical Review D 91 (3). doi:10.1103/PhysRevD.91.033006. Bibcode: 2015PhRvD..91c3006A.
6. ↑ Nio, Makiko (3 February 2015). "QED tenth-order contribution to the electron anomalous magnetic moment and a new value of the fine-structure constant". Fundamental Constants Meeting 2015. Eltville, Germany. http://www.bipm.org/cc/CODATA-TGFC/Allowed/2015-02/Nio.pdf.
7. ↑ Fan, X.; Myers, T. G.; Sukra, B. A. D.; Gabrielse, G. (2023-02-13). "Measurement of the Electron Magnetic Moment". Physical Review Letters 130 (7). doi:10.1103/PhysRevLett.130.071801. PMID 36867820. Bibcode: 2023PhRvL.130g1801F. https://link.aps.org/doi/10.1103/PhysRevLett.130.071801.
8. ↑ 8.0 8.1 Patrignani, C.; Agashe, K. (2016). "Review of Particle Physics". Chinese Physics C (IOP Publishing) 40 (10). doi:10.1088/1674-1137/40/10/100001. ISSN 1674-1137. Bibcode: 2016ChPhC..40j0001P. http://inspirehep.net/record/1489868/files/openaccess_cpc_40_10_100001.pdf.
9. ↑ "The E821 Muon (g−2) Home Page". Brookhaven National Laboratory. http://www.g-2.bnl.gov/.
10. ↑ Aguillard, D. P. et al. (8 August 2024). "Detailed report on the measurement of the positive muon anomalous magnetic moment to 0.20 ppm". Physical Review D (Fermilab) 110 (3). doi:10.1103/PhysRevD.110.032009. Bibcode: 2024PhRvD.110c2009A. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.110.032009. Retrieved 16 March 2015.
11. ↑ The Muon $g-2$ Collaboration et al. (3 June 2025). "Measurement of the Positive Muon Anomalous Magnetic Moment to 127 ppb". Physical Review Letters 135 (10). doi:10.1103/7clf-sm2v. Bibcode: 2025PhRvL.135j1802A.
12. ↑ Giusti, D.; Lubicz, V.; Martinelli, G.; Sanflippo, F.; Simula, S. (2017). "Strange and charm HVP contributions to the muon (g − 2) including QED corrections with twisted-mass fermions". Journal of High Energy Physics 2017 (10): 157. doi:10.1007/JHEP10(2017)157. Bibcode: 2017JHEP...10..157G.
13. ↑ Borsany, Szabolcs (7 April 2021). "Leading hadronic contribution to the muon magnetic moment from lattice QCD". Nature 593 (7857): 51–55. doi:10.1038/s41586-021-03418-1. PMID 33828303. Bibcode: 2021Natur.593...51B.
14. ↑ Aliberti, R (2025). "The anomalous magnetic moment of the muon in the Standard Model: an update". Phys. Rep. 1143: 1–158. doi:10.1016/j.physrep.2025.08.002. Bibcode: 2025PhR..1143....1A.
15. ↑ Eidelman, S.; Passera, M. (2007-01-30). "THEORY OF THE τ LEPTON ANOMALOUS MAGNETIC MOMENT" (in en). Modern Physics Letters A 22 (3): 159–179. doi:10.1142/S0217732307022694. ISSN 0217-7323. Bibcode: 2007MPLA...22..159E.
16. ↑ CMS Collaboration, The (2024-10-01). "Observation of γ γ → τ τ in proton–proton collisions and limits on the anomalous electromagnetic moments of the τ lepton". Reports on Progress in Physics 87 (10): 107801. doi:10.1088/1361-6633/ad6fcb. ISSN 0034-4885. PMID 39146977. Bibcode: 2024RPPh...87j7801C. https://iopscience.iop.org/article/10.1088/1361-6633/ad6fcb.

## Bibliography

* Sergei Vonsovsky (1975). Magnetism of Elementary Particles. Mir Publishers. https://archive.org/details/MagnetismOfElementaryParticles.

## External links

* Overview of the g−2 experiment
* Kusch, P.; Foley, H. M. (1948). "The Magnetic Moment of the Electron". Physical Review 74 (3): 250–263. doi:10.1103/PhysRev.74.250. Bibcode: 1948PhRv...74..250K.
* Aoyama, T. et al. (2020). "The anomalous magnetic moment of the muon in the Standard Model". Physics Reports 887: 1–166. doi:10.1016/j.physrep.2020.07.006. Bibcode: 2020PhR...887....1A.

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Quantum electrodynamics

* Anomalous magnetic dipole moment
* Bhabha scattering
* Bremsstrahlung
* Compton scattering
* Dual photon
* Electron
* Feynman diagram
* Feynman slash notation
* Gupta–Bleuler formalism
* Lamb shift
* Møller scattering
* Path integral formulation
* Photon
* Positron
* Positronium
* Probability amplitude
* Propagator
* QED vacuum
* Self-energy
* Vacuum polarization
* Vertex function
* Virtual particles
* Ward–Takahashi identity
* ξ gauge

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