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
−
2
2
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 and the one-loop result is:

a
e
=
α
2
π
≈
0.001
161
4
where α is the fine structure constant. This result was first found by Julian Schwinger in 1948 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 α^{3} and have been calculated up to order α^{5}:

a
e
=
0.001
159
652
181
643
(
764
)
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:

a
e
=
0.001
159
652
180
73
(
28
)

According to this value, *a*_{e} is known to an accuracy of around 1 part in 1 billion (10^{9}). This required measuring *g* to an accuracy of around 1 part in 1 trillion (10^{12}).

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:

α
μ
S
M
=
α
μ
Q
E
D
+
α
μ
E
W
+
α
μ
H
a
d
r
o
n
=
0.001
165
918
04
(
51
)
The first two components represent the photon and lepton loops, and the W boson, Higgs 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 electron–antielectron (e^{−}e^{+}) collisions. As of November 2006, the measurement disagrees with the Standard Model by 3.4 standard deviations, suggesting physics beyond the Standard Model may be having an effect (or that the theoretical/experimental errors are not completely under control). This is one of the long-standing discrepancies between the Standard Model and experiment.

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. The E821 Experiment reported the following average value

a
μ
=
0.001
165
920
91
(
54
)
(
33
)
,
where the first error is statistical and the second one is systematic.

A new experiment at Fermilab called "Muon g−2" using the E821 magnet will improve the accuracy of this value. Data taking will begin in 2017. A measurement with a precision 4 times better is expected after years of running.

Standard Model prediction for tau's anomalous magnetic dipole moment is:

a
τ
=
0.001
177
21
(
5
)

while best measured bound for *a*_{τ} is:

−
0.007
<
a
τ
<
0.005
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.