Quantum field-theoretic differences of magnetic properties than expected from classical theories
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
$a={\frac {g-2}{2}}$
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:
where $\alpha$ 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 $\alpha ^{3}$^{
[3]} and have been calculated up to order $\alpha ^{5}$:^{
[4]}^{
[5]}^{
[6]}
$a_{\text{e}}=0.001\,159\,652\,181\,643(764)$
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:^{
[7]}
$a_{\text{e}}=0.001\,159\,652\,180\,73(28)$
According to this value, $a_{\text{e}}$ 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:^{
[8]}
Of the first two components, $a_{\mu }^{\mathrm {QED} }$ represents the photon and lepton loops, and $a_{\mu }^{\mathrm {EW} }$ the W boson, Higgs boson and Z boson loops; both can be calculated precisely from first principles. The third term, $a_{\mu }^{\mathrm {hadron} }$, represents hadron loops; it 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 July 2017, the measurement disagrees with the
Standard Model by 3.5
standard deviations,^{
[9]} 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.^{
[10]} The E821 Experiment reported the following average value^{
[8]}
$a_{\mu }=0.001\;165\;920\;9(6).$
A new experiment at
Fermilab called "
Muon g−2" using the E821 magnet will improve the accuracy of this value.^{
[11]} Data taking began in March 2018 and is expected to end in September 2022.^{
[12]} An interim result released on April 7, 2021^{
[13]} yields $a_{\mu }=0.001\,165\,920\,40(54)$ which, in combination with existing measurements, gives a more precise estimate $a_{\mu }=0.001\,165\,920\,61(41)$, exceeding the Standard Model prediction by 4.2 standard deviations. Also, experiment E34 at
J-PARC plans to start its first run in 2024.^{
[14]}
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.^{
[15]}
Tau
The Standard Model prediction for the
tau's anomalous magnetic dipole moment is^{
[16]}
$a_{\tau }=0.001\,177\,21(5),$
while the best measured bound for $a_{\tau }$ is^{
[17]}
$-0.052<a_{\tau }<+0.013.$
Composite particles
Composite particles often have a huge anomalous magnetic moment. 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, which has no charge, was not expected to have a magnetic moment.