---
title: The Anomalous Magnetic Moment
module: Quantum Electrodynamics
moduleNumber: 6
lessonNumber: 4
order: 604
summary: >
  The Dirac equation predicts $g=2$; loops shift it. Schwinger's one-loop vertex
  correction gives the anomaly $a=(g-2)/2=\alpha/2\pi$, and the QED series continues
  to five loops. The electron $a_e$ agrees with theory to better than a part in a
  billion, the most precise confrontation of theory and experiment in physics. The
  muon $a_\mu$, heavier and so more sensitive to virtual heavy states, is dominated
  by hadronic uncertainty and sits at the center of a long-running comparison with
  the Standard Model prediction.
topics: [Quantum Electrodynamics]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 7 (the vertex correction and the Schwinger term)"
  - book: Thomson
    ref: "Ch. 6 (the anomalous magnetic moment)"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §1.4 (magnetic moments from the Dirac equation)"
---

The [Dirac equation](/particle-physics/relativistic-wave-equations/dirac-equation-spinors)
predicts a gyromagnetic ratio $g = 2$ for the electron, with no free parameter — a
triumph in 1928. Precise measurement finds $g$ slightly larger than 2, and the
excess is not a failure of the theory but a prediction of it: the same
[vertex correction](/particle-physics/qed/renormalization-running-coupling) that
needed renormalizing shifts $g$ by a calculable amount. Schwinger computed the
leading shift in 1948, and the agreement between the measured and computed values
of the electron's magnetic moment has since become the most stringent quantitative
test in all of science. The muon repeats the story with heavier virtual particles
in reach, which is why its magnetic moment is a sensitive probe for physics beyond
the Standard Model.

## The anomaly

The magnetic moment of a spin-$\tfrac12$ particle of charge $q$ and mass $m$ is

$$
\vec\mu = g\,\frac{q}{2m}\,\vec S,
$$

with $g$ the **gyromagnetic ratio**. The Dirac value is exactly $g = 2$. The
deviation is measured by the **anomalous magnetic moment**

$$
a \equiv \frac{g - 2}{2},
$$

so $a = 0$ is the tree-level Dirac prediction and any nonzero $a$ is a quantum
correction from loops. The name "anomalous" is historical; the anomaly is the
generic situation, and $g = 2$ is only the leading term.

The loop that generates $a$ is the vertex correction: a virtual photon exchanged
between the incoming and outgoing electron legs while the external photon attaches.
This diagram modifies the effective photon-electron coupling, and its low-momentum
limit splits into a piece that renormalizes the charge and a finite piece
proportional to the spin — the extra magnetic moment. The finite piece is
unambiguous: it is one of the safe, calculable outputs of renormalized QED.

$$
% caption: The Schwinger vertex correction. A virtual photon bridges the two
% electron legs while the external photon attaches at the middle vertex. The
% low-energy limit of this diagram supplies the anomalous magnetic moment, the
% leading correction to the Dirac value g equals two.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % external (measuring) photon into apex vertex from above
  \draw[thick, dashed] (0,2.0) -- (0,0.9);
  \node[black, right] at (0.15,1.6) {external photon};
  % electron in: lower-left, up to left vertex, on to apex
  \draw[thick, ->] (-3.0,-0.9) -- (-2.3,-0.5);
  \draw[thick] (-2.3,-0.5) -- (-1.6,0.0);
  \draw[thick] (-1.6,0.0) -- (0,0.9);
  % apex to right vertex, out to lower-right
  \draw[thick] (0,0.9) -- (1.6,0.0);
  \draw[thick, ->] (1.6,0.0) -- (2.3,-0.5);
  \draw[thick] (2.3,-0.5) -- (3.0,-0.9);
  % virtual photon arc bridging the two legs
  \draw[acc, thick, dashed] (-1.6,0.0) .. controls (-0.9,-0.8) and (0.9,-0.8) .. (1.6,0.0);
  \node[acc, below] at (0,-0.65) {virtual photon};
  \fill (0,0.9) circle (1.8pt);
  \fill (-1.6,0.0) circle (1.6pt);
  \fill (1.6,0.0) circle (1.6pt);
  \node[black, left] at (-3.0,-0.9) {electron in};
  \node[black, right] at (3.0,-0.9) {electron out};
\end{tikzpicture}
$$

## Schwinger's one-loop result

Evaluating the vertex-correction diagram at zero external photon momentum and
extracting the spin-dependent form factor gives the leading anomaly,

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

**Schwinger's term**, independent of the fermion mass and therefore the same for
electron and muon at this order.[^schwinger] It was the first quantitative success
of loop-level QED and the number engraved on Schwinger's headstone. The prediction
$g = 2(1 + \alpha/2\pi) = 2.00232$ matched the anomalous Zeeman measurements of the
late 1940s and settled that the loop corrections of QED are real.

Higher orders continue the series in powers of $\alpha/\pi$,

$$
a = \frac{1}{2}\Big(\frac{\alpha}{\pi}\Big)
  + C_2\Big(\frac{\alpha}{\pi}\Big)^{\!2}
  + C_3\Big(\frac{\alpha}{\pi}\Big)^{\!3}
  + C_4\Big(\frac{\alpha}{\pi}\Big)^{\!4}
  + \cdots,
$$

with $C_2 = -0.328478965\ldots$ known analytically and the coefficients now
computed through five loops. Because $\alpha/\pi \approx 0.0023$, each term is
smaller than the last by a factor of a few thousand, and the series converges fast
enough that theory keeps pace with the most precise measurements.

## The electron: the cleanest test of QED

For the electron the anomaly is almost pure QED. Hadronic and weak contributions
enter only at the level of $10^{-12}$ relative to the leading term, because the
electron is light and couples weakly to the heavy virtual states that carry those
effects. The measured value

$$
a_e \approx 1.159\,652\,18 \times 10^{-3}
$$

agrees with the QED calculation to more than ten significant figures — a fractional
agreement of order $10^{-10}$, the best in physics.[^pdg-ae] The comparison runs
both directions: because the QED series is known so well, the electron anomaly is
inverted to provide one of the most precise determinations of the fine-structure
constant $\alpha$ itself. That a single number computed from Feynman diagrams
matches experiment to this depth is the central evidence that QED is correct.

$$
% caption: The electron anomalous moment as a stack of contributions. QED accounts
% for essentially the entire value; hadronic and electroweak pieces enter only near
% the twelfth digit. Measurement and QED theory agree to about ten significant
% figures, shown as the near-perfect overlap of the two bars.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % two bars, experiment vs theory, essentially identical length
  \draw[acc, very thick, fill=acc!12] (0,1.2) rectangle (6.0,1.9);
  \node[acc, left] at (-0.15,1.55) {measured};
  \draw[black, very thick, fill=black!7] (0,0.1) rectangle (5.98,0.8);
  \node[black, left] at (-0.15,0.45) {QED theory};
  % agreement bracket
  \draw[black] (6.15,0.1) -- (6.45,0.1);
  \draw[black] (6.15,1.9) -- (6.45,1.9);
  \draw[black] (6.45,0.1) -- (6.45,1.9);
  \node[black, right, align=left] at (6.55,1.0) {agree to\\one part in ten billion};
  \node[black, below] at (3.0,-0.15) {electron anomaly};
\end{tikzpicture}
$$

## The muon: a heavier, more sensitive probe

The muon anomaly follows the same QED series, but the muon is about 207 times
heavier than the electron, and a virtual particle of mass $M$ contributes to
$a_\ell$ with a weight scaling as $(m_\ell/M)^2$. The muon's sensitivity to any
heavy virtual state — hadronic, electroweak, or beyond the Standard Model — is
therefore enhanced over the electron's by roughly $(m_\mu/m_e)^2 \approx 4\times
10^4$. This makes $a_\mu$ a genuine window on high-mass physics, at the cost that
the calculation must include contributions the electron can ignore.

The Standard Model prediction for $a_\mu$ is a sum of three kinds of contribution:

| Contribution | Approximate size ($\times 10^{-11}$) | Character |
| --- | --- | --- |
| QED (through five loops) | $116\,584\,719$ | Known to high precision |
| Electroweak ($W$, $Z$, Higgs loops) | $154$ | Small, well controlled |
| Hadronic vacuum polarization | $\sim 6800$ | Dominant uncertainty |
| Hadronic light-by-light | $\sim 90$ | Second hadronic uncertainty |

$$
% caption: The contributions to the muon anomaly on a logarithmic scale, spanning
% eight orders of magnitude. QED dominates completely; the electroweak and hadronic
% light-by-light pieces are far smaller; the hadronic vacuum polarization, though
% only the fourth-largest, carries the bulk of the theoretical uncertainty because
% it cannot be computed in perturbation theory.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % log10 sizes: QED 8.07, HVP 3.83, EW 2.19, LBL 1.95 ; scale 0.62
  \draw[acc, very thick, fill=acc!12] (0,2.7) rectangle (5.00,3.3);
  \node[acc, left] at (-0.15,3.0) {QED};
  \draw[black, very thick, fill=black!7] (0,1.8) rectangle (2.37,2.4);
  \node[black, left] at (-0.15,2.1) {hadronic v.p.};
  \draw[black, very thick, fill=black!7] (0,0.9) rectangle (1.36,1.5);
  \node[black, left] at (-0.15,1.2) {electroweak};
  \draw[black, very thick, fill=black!7] (0,0.0) rectangle (1.21,0.6);
  \node[black, left] at (-0.15,0.3) {hadronic l.b.l.};
  % axis
  \draw[->, black] (0,-0.4) -- (5.6,-0.4) node[right, black] {$\log_{10}$ size};
\end{tikzpicture}
$$

QED supplies essentially the whole value; the electroweak piece is small but
required at current precision; the **hadronic** pieces, in which the virtual photon
turns into strongly interacting quarks and gluons that cannot be computed
perturbatively, carry almost all the theoretical uncertainty.[^pdg-amu] The
hadronic vacuum polarization is estimated two ways — from measured
$e^+e^-\to\text{hadrons}$ cross sections through a dispersion relation, and from
lattice QCD — and the two methods have differed, which is the crux of the current
theory situation.

$$
% caption: The measured muon anomaly compared with the Standard Model prediction,
% each shown as a central value with its uncertainty band. The two have historically
% sat apart by several combined standard deviations, with the theory uncertainty
% dominated by the hadronic contribution; refined lattice and dispersive evaluations
% continue to reshape the comparison.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % horizontal value axis
  \draw[->, black] (0,0) -- (6.6,0) node[right, black] {muon anomaly};
  % experiment marker with error bar (upper)
  \draw[acc, very thick] (4.1,1.5) -- (4.9,1.5);
  \fill[acc] (4.5,1.5) circle (2.2pt);
  \node[acc, right] at (5.0,1.5) {measured};
  % theory marker with error bar (lower)
  \draw[black, very thick] (2.9,0.7) -- (3.7,0.7);
  \fill[black] (3.3,0.7) circle (2.2pt);
  \node[black, right] at (5.0,0.7) {SM prediction};
  % gap indicator
  \draw[black, dashed] (3.3,0.7) -- (3.3,1.5);
  \draw[black, dashed] (4.5,1.5) -- (4.5,0.7);
  \draw[black, <->] (3.3,0.35) -- (4.5,0.35);
  \node[black, below] at (3.9,0.35) {discrepancy};
\end{tikzpicture}
$$

Whatever the final resolution, the structure of the test is fixed: the QED and
electroweak parts are under control, so any persistent gap points at the hadronic
evaluation or at new virtual particles the muon is heavy enough to feel. The
anomalous magnetic moment thus plays two roles — for the electron, the definitive
confirmation that QED is right; for the muon, one of the sharpest low-energy
searches for what lies beyond it.

## Summary

The Dirac prediction $g = 2$ is the tree-level value; loops shift it by the anomaly
$a = (g-2)/2$. Schwinger's one-loop vertex correction gives $a = \alpha/2\pi
\approx 0.00116$, mass-independent, and the QED series in $\alpha/\pi$ continues to
five loops with $C_2 = -0.328\ldots$. For the electron the anomaly is almost pure
QED, and measurement matches theory to about one part in $10^{10}$ — the tightest
test in physics, and a route to determining $\alpha$. For the muon, the enhanced
sensitivity to heavy virtual states of order $(m_\mu/m_e)^2$ makes $a_\mu$ a probe
of new physics, with the hadronic vacuum polarization dominating the theoretical
uncertainty and driving the ongoing comparison with experiment. The precision of
$g - 2$ closes the QED module: a theory with one vertex and one coupling, computed
to five loops, agreeing with nature to ten digits.

[^schwinger]: The vertex-correction diagram and the Schwinger result $a = \alpha/2\pi$ are in Griffiths, Ch. 7, and Thomson, Ch. 6; the magnetic moment from the Dirac equation is in Tong, _The Standard Model_ (Cambridge Part III), §1.4, [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html).

[^pdg-ae]: The measured electron anomaly $a_e = 1.15965218\ldots\times 10^{-3}$, its agreement with the QED calculation to better than a part in $10^{10}$, and its use to extract $\alpha$ are from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).

[^pdg-amu]: The muon anomaly $a_\mu \approx 1.16592\times 10^{-3}$, the breakdown into QED, electroweak, and hadronic contributions, and the experimental value are from the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov), and the Fermilab Muon $g-2$ collaboration measurement. The dispersive-versus-lattice difference in the hadronic vacuum polarization is the dominant issue in the current theory evaluation.