# Anomalous magnetic dipole moment

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

$a = \frac{g-2}{2}$

## Anomalous magnetic moment of the electron

One-loop correction to the 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 diagram on the right. The calculation is relatively straightforward[1] and the one-loop result is:

$a = \frac{\alpha}{2 \pi} \approx 0.0011614$

where $\alpha$ is the fine structure constant. This result was first found by Schwinger in 1948.[2] As of 2009, the coefficients of the QED formula for the anomalous magnetic moment of the electron have been calculated through order $\alpha^4$,[3] and are known analytically up to $\alpha^3$.[4] The QED prediction agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron the most accurately verified prediction in the history of physics. (See precision tests of QED for details.)

The current experimental value and uncertainty is:[5]

$a = 0.00115965218073 (28)$

According to this value, a is known to an accuracy of around 1 part in 1 billion (109). This required measuring g to an accuracy of around 1 part in 1 trillion (1012).

## Anomalous magnetic moment of the muon

One-loop MSSM corrections to the muon g-2 involving 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; its measurement provides a precision test of the Standard Model. The prediction for the value of the muon anomalous magnetic moment includes three parts: αμSM = αμQED + αμEW + αμhad. The first two components represent the photon and lepton loops, and the W boson and Z boson loops, respectively, and can be calculated precisely from first principles. The third term represents hadron loops, and cannot be calculated accurately from theory alone. It is estimated from experimental measurements of the ratio of hadronic to muonic cross sections (R) in e+e- collisions. As of November 2006, the measurement disagrees with the Standard Model by 3.4 standard deviations,[6] suggesting beyond the Standard Model physics may be having an effect (or theoretical/experimental errors not completely under control).

The E821 experiment at Brookhaven National Laboratory (BNL) studied the precession of muon and anti-muon in a constant external magnetic field as they circulated in a confining storage ring. The E821 Experiment reported the following average value (from the July 2009 review by Particle Data Group)

$a = \frac{g-2}{2} = 0.00116592089(54)(33)$

where the first errors are statistical and the second systematic.

## Anomalous magnetic moment of composite particles

Composite particles often have a huge anomalous magnetic moment. This is true for the proton, which is made up of charged quarks, and the neutron, which has a magnetic moment even though it is electrically neutral.

## Notes

1. ^ See section 6.3 of Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, 1995.
2. ^ J. Schwinger, Phys. Rev. 73, 416L (1948)
3. ^ Aoyama et al.; Hayakawa, M.; Kinoshita, T.; Nio, M. (2008). "Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron". Physical Review D 77 (5): 053012. Bibcode:2008PhRvD..77e3012A. doi:10.1103/PhysRevD.77.053012.
4. ^ Laporta and Remiddi; Remiddi, E (1996). "The analytical value of the electron (g − 2) at order α3 in QED". Physics Letters B 379: 283–291. arXiv:hep-ph/9602417. Bibcode:1996PhLB..379..283L. doi:10.1016/0370-2693(96)00439-X.
5. ^ Hanneke, D.; S. Fogwell Hoogerheide, G. Gabrielse (24 September 2010). Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment. arXiv:[physics.atom-ph arXiv:1009.4831v1 [physics.atom-ph]].
6. ^ Hagiwara, K.; A. D. Martin; Daisuke Nomura; T. Teubner (31 May 2007). "Improved predictions for g−2 of the muon and $\alpha_\mathrm{QED}(M_Z^2)$". Physics Letters B 649 (2–3): 173–179. arXiv:hep-ph/0611102. Bibcode:2007PhLB..649..173H. doi:10.1016/j.physletb.2007.04.012.