Lepton Magnetic Moments and the QED Form-Factor Expansion

For an on-shell charged lepton, the anomalous magnetic moment is the Pauli form factor at zero momentum transfer,

$$a_\ell\equiv\frac{g_\ell-2}{2}=F_2^\ell(0),$$

provided the Dirac form factor is normalized by the physical charge, $F_1^\ell(0)=1$. This equality is the bridge from a renormalized three-point function to a precision observable. It fixes what must be calculated, matched, and compared: a static limit of an on-shell vertex, not a vertex at a convenient nonzero momentum.

Required background. The Ward–Takahashi identity and charge renormalization supplies $F_1(0)=1$ and the relation between vertex and external-leg renormalization. Form factors and local operator insertions supplies the general on-shell decomposition.

Helpful background. Validation and theory uncertainties gives the rules for comparing a multi-source prediction with a measured value.

The on-shell electromagnetic vertex

Take the lepton charge to be $q_\ell=-e$, with $e>0$, and define $q=p'-p$. After LSZ normalization, the parity-even, CP-even photon–lepton vertex between on-shell spinors is

$$(-iq_\ell)\,\bar u(p')\Gamma^\mu(p',p)u(p),$$

where

$$\Gamma^\mu(p',p) =F_1(q^2)\gamma^\mu +\frac{i\sigma^{\mu\nu}q_\nu}{2m_\ell}F_2(q^2), \qquad \sigma^{\mu\nu}\equiv\frac{i}{2}[\gamma^\mu,\gamma^\nu].$$

Terms proportional to $q^\mu$ do not contribute to a conserved external electromagnetic current, and the on-shell equations reduce the remaining parity-even structures to the two shown. An electric-dipole form factor is a separate CP-odd structure and is outside this page’s scope.

The Ward–Takahashi identity fixes the charge normalization in an on-shell charge scheme,

$$F_1(0)=1.$$

This is a renormalization condition, not a statement that $F_1(q^2)$ is identically one. Its slope contains physical information together with the infrared qualifications appropriate to a charged particle.

Why the Pauli form factor is the anomaly

The Gordon identity,

$$\bar u(p')\gamma^\mu u(p) =\bar u(p')\left[ \frac{(p'+p)^\mu}{2m_\ell} +\frac{i\sigma^{\mu\nu}q_\nu}{2m_\ell} \right]u(p),$$

shows that both form factors multiply the spin coupling in a slowly varying external field. Matching the spatial vertex to

$$H_{\mathrm{mag}}=-\boldsymbol\mu\cdot\mathbf B, \qquad \boldsymbol\mu=g_\ell\frac{-e}{2m_\ell}\mathbf S,$$

gives

$$\frac{g_\ell}{2}=F_1(0)+F_2(0).$$

With $F_1(0)=1$, the pointlike Dirac contribution is $g_\ell=2$, and every extra Pauli coupling is measured by

$$a_\ell=F_2(0).$$

This derivation fixes two common normalization ambiguities: $F_2$ is dimensionless because $1/(2m_\ell)$ is explicit, and the signed charge is already carried by the overall vertex $-iq_\ell$. Reversing the momentum-flow convention $q=p'-p$ changes the signs of both $q_\nu$ and the corresponding photon momentum; it does not change $a_\ell$.

The one-loop QED anchor

At one loop, combine the three propagators in the vertex graph with $x+y+z=1$, shift the loop momentum, and retain the coefficient of $i\sigma^{\mu\nu}q_\nu/(2m_\ell)$. The on-shell result can be written

$$F_2(q^2) =\frac{\alpha}{\pi} \int_0^1dx\int_0^1dy\int_0^1dz\, \delta(1-x-y-z) \frac{m_\ell^2z(1-z)} {m_\ell^2(1-z)^2-q^2xy}.$$

At $q^2=0$, integrate first over the line $x+y=1-z$:

$$F_2(0) =\frac{\alpha}{\pi}\int_0^1dz\,z =\frac{\alpha}{2\pi}.$$

Thus

$$a_\ell^{(1)}=\frac{\alpha}{2\pi},$$

Schwinger’s leading result Schwinger 1948, pp. 416–417. A detailed derivation and the form-factor projection appear in Schwartz 2014, §§ 17.1–17.2, pp. 315–321.

The Pauli form factor at $q^2=0$ is ultraviolet finite at this order because no independent renormalizable Pauli counterterm exists in QED. That statement does not make the whole vertex graph finite: charge and external-leg renormalization are still required to define $F_1$, and intermediate expressions can contain infrared singularities that must cancel in the properly projected static quantity. A gauge-parameter or ultraviolet-pole residue in the final $F_2(0)$ signals an incomplete on-shell reduction or counterterm treatment.

Useful checks on any implementation are

$$\operatorname{dim}_{\mathrm{mass}}F_2=0, \qquad F_2(0)\xrightarrow{\alpha\to0}0, \qquad F_1(0)=1, \qquad \frac{\partial F_2(0)}{\partial\xi}=0.$$

From one loop to a Standard Model prediction

For each lepton species, organize the prediction by physics source,

$$a_\ell^{\mathrm{SM}} =a_\ell^{\mathrm{QED}} +a_\ell^{\mathrm{HVP}} +a_\ell^{\mathrm{HLbL}} +a_\ell^{\mathrm{EW}}.$$

The QED term is a perturbative series with mass-dependent coefficients,

$$a_\ell^{\mathrm{QED}} =\sum_{n\geq1}C_n^\ell \!\left(\frac{m_\ell}{m_e},\frac{m_\ell}{m_\mu},\frac{m_\ell}{m_\tau}\right) \left(\frac{\alpha}{\pi}\right)^n, \qquad C_1^\ell=\frac12.$$

Closed lepton loops make the higher coefficients species dependent. Hadronic vacuum polarization (HVP) enters through a dispersive integral over measured hadronic production or through lattice QCD. Hadronic light-by-light scattering (HLbL) requires a four-current hadronic amplitude, again accessible through complementary data-driven and lattice methods. The electroweak term contains weak-boson and Higgs effects. This decomposition, including the need to keep correlations between inputs, is reviewed in Aoyama et al. 2020, §§ 2–5, pp. 1–121.

The sectors are conceptual partitions, not automatically independent random variables. Common hadronic cross-section data, scale setting, radiative corrections, or values of fundamental constants can correlate nominally separate entries.

Contribution	Main inputs	Uncertainties that must remain visible
QED	$\alpha$, lepton mass ratios, perturbative coefficients	Missing orders, numerical integration, input-parameter covariance
HVP	Hadronic spectral data and/or lattice correlators	Experimental systematics, radiative corrections, lattice continuum/volume limits, data–lattice covariance
HLbL	Hadronic amplitudes, short-distance constraints, lattice correlators	Reconstruction choices, discretization and volume effects, shared hadronic inputs
Electroweak	Weak and Higgs masses, couplings, higher-order matching	Missing electroweak orders and parameter dependence
Experiment	Frequency ratios, magnetic-field calibration, mass and moment ratios	Statistical/systematic covariance and external constants

Comparing prediction and experiment

A meaningful comparison starts from two dated values with compatible definitions, not from a bare difference copied from separate summaries. If $a_{\mathrm{exp}}$ and $a_{\mathrm{th}}$ share an input vector $x$, their difference

$$\Delta a=a_{\mathrm{exp}}-a_{\mathrm{th}}$$

has variance

$$\operatorname{Var}(\Delta a) =\operatorname{Var}(a_{\mathrm{exp}}) +\operatorname{Var}(a_{\mathrm{th}}) -2\operatorname{Cov}(a_{\mathrm{exp}},a_{\mathrm{th}}).$$

The covariance term cannot be set to zero merely because one number is called “experimental” and the other “theoretical.” A constant inferred using the same magnetic-moment measurement must not be fed back into the prediction as though it were independent.

Any current numerical comparison should therefore be attached to a dated evidence record containing:

• the exact experimental release and combination rule;
• the theory-input releases, including the chosen HVP and HLbL determinations;
• the value and provenance of $\alpha$ and every mass ratio;
• the full covariance or a justified approximation to it;
• perturbative orders and numerical-integration errors;
• any corrections, replacements, or withdrawals since publication.

This page intentionally gives the invariant derivation and comparison method rather than mutable world averages. A significance without the versions and covariance above is not reproducible.

Common pitfalls

Reading $g-2$ directly from the coefficient of $\gamma^\mu$. The magnetic spin coupling receives both the Gordon-decomposed Dirac term and the Pauli term. First impose $F_1(0)=1$, then use $a_\ell=F_2(0)$.

Taking the wrong limit. The anomaly is $F_2(q^2)$ after putting both external leptons on shell and then taking $q^2\to0$. A Euclidean form factor at finite momentum or an off-shell vertex is not the measured static moment.

Calling a sector error the total theory error. QED truncation, hadronic inputs, electroweak matching, parameters, and numerical integration have different correlations and update cycles. Preserve the components until the final covariance propagation.

Quoting a current discrepancy without provenance. Experimental combinations and hadronic evaluations can change independently. A durable statement names the releases, inputs, covariance treatment, and date.

References

• Aoyama, Tatsumi, et al. “The Anomalous Magnetic Moment of the Muon in the Standard Model.” Physics Reports 887 (2020): 1–166. DOI.
• Schwinger, Julian. “On Quantum-Electrodynamics and the Magnetic Moment of the Electron.” Physical Review 73 (1948): 416–417. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.