The anomalous magnetic moment is the departure of a particle’s spin magnetic moment from the value predicted for a pointlike, minimally coupled spin-½ particle by the Dirac equation. It is usually expressed as the dimensionless magnetic anomaly (a=(g-2)/2), where (g) is the particle’s magnetic (g)-factor. For charged leptons, notably the electron and muon, the anomaly arises from quantum corrections and can be both measured and calculated with exceptional precision, making it an important test of the Standard Model. (arxiv.org)
Definition and conventions
For a particle of electric charge (q), mass (m), and spin angular momentum (\mathbf S), the spin magnetic moment is written
[ \boldsymbol{\mu}=g,\frac{q}{2m}\mathbf S. ]
Here the sign of the charge is included in (q), so the electron’s magnetic moment points opposite to its spin when (g) is positive. The Dirac prediction is (g=2); defining
[ a=\frac{g-2}{2} ]
gives (g=2(1+a)). The corresponding anomalous part of the magnetic-moment operator is
[ \delta\boldsymbol{\mu} =\boldsymbol{\mu}-\boldsymbol{\mu}_{\rm Dirac} =a,\frac{q}{m}\mathbf S. ]
Thus, although “anomalous magnetic moment” commonly denotes (a), this quantity is dimensionless rather than a magnetic moment expressed in physical units. The expression (g-2) denotes twice the anomaly, not the anomaly itself. (arxiv.org)
“Anomalous” refers to the deviation from the Dirac value, not necessarily to a disagreement with established theory. A nonzero anomaly is an ordinary prediction of interacting quantum theory. A possible discrepancy between experiment and the Standard Model is instead represented by (a^{\rm exp}-a^{\rm SM}). (pdg.lbl.gov)
Quantum origin
In quantum electrodynamics (QED), interactions modify the coupling between a lepton and an external magnetic field. These modifications are calculated as radiative corrections, represented by loop Feynman diagrams involving photons and other particles. The leading correction is the Schwinger term,
[ a_\ell^{\rm QED} =\frac{\alpha}{2\pi} +O(\alpha^2), ]
where (\alpha) is the fine-structure constant. Its value is approximately (0.00116), explaining why charged-lepton (g)-factors are close to, but slightly greater than, 2. Higher-order contributions include photon corrections, lepton loops, and vacuum polarization. (pdg.lbl.gov)
Within the Standard Model, a charged lepton’s anomaly is conventionally separated into
[ a_\ell^{\rm SM} =a_\ell^{\rm QED} +a_\ell^{\rm electroweak} +a_\ell^{\rm hadronic}. ]
The electroweak contribution includes effects associated with heavy weak-interaction particles. Hadronic contributions involve the strongly interacting sector and are especially important for the muon. They are smaller than the dominant QED contribution but account for most of the uncertainty in the muon’s theoretical prediction. (pdg.lbl.gov)
Electron measurements
The electron anomaly is a stringent test of QED and a means of determining the fine-structure constant. Precision experiments confine a single electron in a Penning trap, using magnetic and electric fields, and infer its magnetic moment from cyclotron and spin-related transition frequencies. (cfp.physics.northwestern.edu)
A measurement published in 2023 reported
[ \frac{g_e}{2} =1.001,159,652,180,59(13), ]
equivalently,
[ a_e=0.001,159,652,180,59(13). ]
Parentheses give the uncertainty in the final quoted digits. The relative uncertainty of (g_e/2) was approximately 0.13 parts per trillion; because (a_e) is much smaller than unity, its relative uncertainty is larger. (cfp.physics.northwestern.edu)
A test of QED requires a prediction based on an independently measured (\alpha). Conversely, assuming the theoretical calculation, the measured electron anomaly can be used to infer (\alpha). These are different uses of the same measurement: a value of (\alpha) extracted from (a_e) cannot independently validate the prediction from which it was obtained. The 2023 study identified disagreement among independent determinations of (\alpha) as a limitation on the comparison. (arxiv.org)
Muon measurements
The muon is approximately 207 times heavier than the electron. In many theories, contributions from particles much heavier than a lepton scale approximately as the square of the lepton mass divided by the heavy-particle mass. Under comparable coupling assumptions, the muon anomaly can therefore be about (4\times10^4) times more sensitive than the electron anomaly to such effects. This enhancement is model-dependent, not a universal rule for every possible new interaction. (arxiv.org)
Muon storage-ring experiments measure the difference between the spin-precession and cyclotron frequencies. A polarized muon beam circulates in a magnetic field, and the arrival times and energies of decay positrons reveal the spin’s precession relative to the particle’s momentum. Nuclear magnetic resonance probes determine the field experienced by the beam. Precision analysis also requires corrections for electric fields, vertical beam motion, and detector effects. (lss.fnal.gov)
The Fermilab Muon (g-2) collaboration announced its final magnetic-anomaly result on June 3, 2025. Combining all its measurement runs, it obtained
[ a_\mu^{\rm FNAL} =0.001,165,920,705(148), ]
with a relative uncertainty of 127 parts per billion. Combining Fermilab’s result with the earlier Brookhaven National Laboratory measurement gave the experimental world average
[ a_\mu^{\rm exp} =0.001,165,920,715(145). ]
These uncertainties refer to the anomaly (a_\mu), not to the full (g)-factor. (news.fnal.gov)
Hadronic contributions and theoretical interpretation
The principal difficulty in predicting the muon anomaly is the low-energy behavior of quantum chromodynamics (QCD). Two important contributions are hadronic vacuum polarization, in which strong-interaction effects modify photon propagation, and hadronic light-by-light scattering, involving a hadronic coupling to multiple photons. These contributions are investigated using experimental scattering data and numerical lattice QCD calculations. (arxiv.org)
The comparison between theory and experiment depends on how these hadronic terms are evaluated. The Muon (g-2) Theory Initiative’s 2025 update adopted a lattice-QCD average for the leading hadronic-vacuum-polarization contribution. It reported
[ a_\mu^{\rm SM} =116,592,033(62)\times10^{-11}. ]
Against the final experimental world average, the difference was approximately
[ a_\mu^{\rm exp}-a_\mu^{\rm SM} =38(63)\times10^{-11}, ]
which was consistent with zero. This changed the interpretation associated with earlier comparisons, rather than overturning the experimentally measured anomaly. (arxiv.org)
The same update documented unresolved tensions among electron–positron collision datasets used in data-driven evaluations of hadronic vacuum polarization. Consequently, claims of a muon (g-2) discrepancy must specify the theoretical prediction and its inputs: disagreement with one evaluation is not automatically evidence of a failure of the Standard Model. (arxiv.org)
References
- Muon g-2: Review of Theory and Experimentarxiv.org
- The Muon Anomalous Magnetic Momentpdg.lbl.gov
- Measurement of the Electron Magnetic Momentcfp.physics.northwestern.edu
- Measurement of the Electron Magnetic Momentarxiv.org
- Muon g-2 announces most precise measurement of the magnetic anomaly of the muonnews.fnal.gov
- The anomalous magnetic moment of the muon in the Standard Modelarxiv.org
- The anomalous magnetic moment of the muon in the Standard Model: an updatearxiv.org