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, $F_1^\ell(0)=1$. The calculation below uses a nonzero lepton mass, perturbative on-shell reduction with an infrared prescription, and external-leg residue factors. It does not assume an isolated exact charged-particle pole in massless-photon QED. Within this scope, the static limit of the Pauli form factor connects a renormalized three-point function to the measured moment; a vertex at a convenient nonzero momentum does not suffice.

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 $q=p'-p$. The external spinors obey $p^2=p'^2=m_\ell^2$ with positive energies. Use the perturbative pole mass and a common infrared prescription for the vertex and external legs, as on the Ward-identity page. After LSZ normalization, the parity-even, CP-even photon–lepton vertex between these spinors is

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

where

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

These $F_i$ include the external-lepton residue product $R$: with unit photon pole normalization and the charge defined in the Thomson limit, $F_i=R F_i^{\rm proper}$. Here $R$ is the residue of the renormalized propagator, not the bare-to-renormalized constant $Z_2$. In an on-shell field scheme $R=1$. At one loop the residue correction multiplies the tree-level Dirac vertex, so it changes $F_1$ but not the order-$\alpha$ Pauli coefficient, whose tree value vanishes.

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 photon here is an external-field insertion, not an isolated real photon absorbed by a free massive lepton. For the static magnetic limit one may use the Breit frame, $q^0=0$ and $q^2=-|\mathbf q|^2$, and take $\mathbf q\to0$ after separating the two form factors.

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

$$F_1(0)=1.$$

This is a charge condition together with the residue cancellation enforced by the identity, not a statement that $F_1(q^2)$ is identically one. The slope $F_1'(0)$ of the charged-lepton vertex is generally infrared sensitive; it is not by itself a regulator-independent charge radius. The photon-mass dependence of the Dirac form factor and its zero-transfer subtraction is explicit in Schwartz 2014, § 19.3, pp. 347–348, Eqs. (19.50)–(19.56). A finite-momentum scattering observable requires its own soft-radiation and measurement prescription.

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. To state the measured moment, use $A^{\rm em}=-A$ and $\mathbf B^{\rm em}=\boldsymbol\nabla\times\mathbf A^{\rm em}=-\mathbf B$, as in the action’s charge convention. The Dirac Hamiltonian is $H=\boldsymbol\alpha\cdot(\mathbf p+q_\ell\mathbf A)+\beta m_\ell-q_\ell A_0$; its nonrelativistic spin term is $+q_\ell\boldsymbol\sigma\cdot\mathbf B/(2m_\ell)=-q_\ell\boldsymbol\sigma\cdot\mathbf B^{\rm em}/(2m_\ell)$. Matching the full spatial vertex to

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

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 $+iq_\ell=-ie$. Keeping the site field instead gives the identical energy $H_{\rm mag}=+g_\ell q_\ell\mathbf S\cdot\mathbf B/(2m_\ell)$. 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 on-shell decomposition and Gordon reduction are developed in Schwartz 2014, § 17.1, pp. 316–318, Eqs. (17.6)–(17.12).

The one-loop QED anchor

Write $m=m_\ell$ in this calculation and work in Feynman gauge. A small photon mass $\lambda>0$ provides an auxiliary infrared regulator for the loop; it is not a physical photon mass. Keep it until the Pauli projection has been isolated, while treating the charge and external-leg factors with the same prescription. With photon loop momentum $k$, the three denominators are

$$\begin{aligned} D_\gamma&=k^2-\lambda^2+i0,\\ D_p&=(p-k)^2-m^2+i0,\\ D_{p'}&=(p'-k)^2-m^2+i0. \end{aligned}$$

Removing the overall vertex factor $-ie$, the one-loop correction is $\delta\Gamma^\mu=-ie^2\int\!\mathrm d^4k\,N^\mu/[(2\pi)^4D_\gamma D_pD_{p'}]$, where

$$N^\mu=\gamma^\alpha(p'\!\!\!/-k\!\!\!/+m) \gamma^\mu(p\!\!\!/-k\!\!\!/+m)\gamma_\alpha.$$

Assign Feynman parameters $x,y,z$ to $D_p,D_{p'},D_\gamma$, respectively. Combining denominators contributes a factor of two. The shift $\ell=k-xp-yp'$ gives

$$\Delta_\lambda=m^2(1-z)^2-q^2xy+\lambda^2z, \qquad x+y+z=1,$$

and the denominator is $(\ell^2-\Delta_\lambda+i0)^3$. The four-dimensional identity $\gamma^\alpha a\!\!\!/\gamma^\mu b\!\!\!/\gamma_\alpha=-2b\!\!\!/\gamma^\mu a\!\!\!/$, followed by the on-shell Dirac equations, separates the numerator into the basis $\gamma^\mu$, $P^\mu=(p'+p)^\mu$ and $q^\mu$. After odd loop terms are removed and the remaining loop tensors are angularly averaged, its matrix element has the form

$$\bar u(p')N^\mu u(p) =\bar u(p')\left[ C_\gamma\gamma^\mu +2mz(1-z)P^\mu +2m(x-y)(2-z)q^\mu \right]u(p).$$

The unneeded coefficient $C_\gamma$ contains the ultraviolet-divergent Dirac part. The Pauli part is ultraviolet convergent, so its numerator can be reduced in four dimensions; this does not authorize dropping the ultraviolet regulator from $C_\gamma$. The $q^\mu$ term integrates to zero because it is odd under $x\leftrightarrow y$, while the measure and $\Delta_\lambda$ are even.

For the remaining $P^\mu$ term, the scalar integral and its vertex coefficient are

$$\begin{aligned} \int\frac{\mathrm d^4\ell}{(2\pi)^4} \frac{1}{(\ell^2-\Delta_\lambda+i0)^3} &=-\frac{i}{32\pi^2(\Delta_\lambda-i0)},\\ C_P&=-\frac{\alpha m}{2\pi} \int_{x,y,z\geq0}\!\mathrm dx\,\mathrm dy\,\mathrm dz\, \delta(1-x-y-z)\frac{z(1-z)}{\Delta_\lambda-i0}. \end{aligned}$$

The Gordon identity gives $P^\mu=2m\gamma^\mu-i\sigma^{\mu\nu}q_\nu$ between these spinors. Comparing $C_P P^\mu$ with the defined Pauli structure therefore yields $F_2=-2mC_P$. This projection must precede setting $q=0$, which would erase the explicit Pauli tensor. The loop reduction is given in Schwartz 2014, § 17.2, pp. 318–320, Eqs. (17.15)–(17.31).

For spacelike $q^2\leq0$, the projected integral has a finite $\lambda\to0$ limit. The one-loop result, with terms of order $\alpha^2$ omitted, is

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

The displayed denominator is understood as $m_\ell^2(1-z)^2-q^2xy-i0$ on analytic continuation; below we use the real spacelike/static limit. Infrared finiteness at zero transfer can be checked before removing the regulator: integrating over $x+y=1-z$ gives

$$F_2(0;\lambda) =\frac{\alpha}{\pi}\int_0^1\!\mathrm dz\, \frac{m^2z(1-z)^2}{m^2(1-z)^2+\lambda^2z}.$$

The integrand is between zero and $z$, and approaches $z$ except at an endpoint of measure zero. Dominated convergence therefore justifies the massless-photon limit. In that limit, 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, p. 416. The lepton mass cancels from this coefficient; it still sets the dimensionful moment and the momentum scale over which the form factor varies.

The Pauli form factor at $q^2=0$ is ultraviolet finite at this order, consistently with the absence of an independent renormalizable Pauli counterterm in QED. The explicit projection also proves its one-loop infrared finiteness. Neither fact makes the whole vertex graph finite: charge and external-leg renormalization are still required to define $F_1$. Beyond this order, residue and counterterm factors multiply lower-order Pauli contributions as well. 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. The decomposition and correlations between different perturbative orders are summarized in Aoyama et al. 2020, § 0 and Table 1, manuscript pp. 7–8, PDF. That dated review supports the organization, not a claim about today’s world average or discrepancy.

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. Correlation assumptions in combining hadronic data are discussed concretely in Aoyama et al. 2020, § 2.3.6, manuscript pp. 51–52, PDF.

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 identify:

• 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. Keep the regulated on-shell external normalization, separate the Pauli coefficient at nonzero transfer, and only then take the static limit. Setting $q=0$ in the unprojected vertex removes the very tensor whose coefficient is wanted. A finite-transfer form factor or an off-shell lepton 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.

Exercises

The first spacelike correction. In the one-loop integral with its infrared regulator removed, put $q^2=-Q^2$ with $Q^2\geq0$ and set $x=(1-z)t$, $y=(1-z)(1-t)$. Reduce the integral to one parameter. Find the first correction for $Q^2\ll m_\ell^2$, and determine whether the one-loop $F_2(-Q^2)$ increases or decreases with $Q^2$.

Solution

After the delta function eliminates $y$, the measure is $\mathrm dx\,\mathrm dz=(1-z)\,\mathrm dt\,\mathrm dz$. The denominator contains $(1-z)^2[m_\ell^2+Q^2t(1-t)]$, so the remaining $z$ integral is $\int_0^1z\,\mathrm dz=1/2$. Hence

$$F_2(-Q^2) =\frac{\alpha}{2\pi}\int_0^1 \frac{\mathrm dt}{1+(Q^2/m_\ell^2)t(1-t)}.$$

Expanding the denominator and using $\int_0^1t(1-t)\,\mathrm dt=1/6$ gives

$$F_2(-Q^2) =\frac{\alpha}{2\pi} \left[1-\frac{Q^2}{6m_\ell^2} +O\!\left(\frac{Q^4}{m_\ell^4}\right)\right].$$

The derivative of the integrand with respect to $Q^2$ is negative for $0<t<1$, so this one-loop form factor decreases from its positive static value. The universal value $F_2(0)=\alpha/(2\pi)$ does not mean the finite-transfer function is independent of the lepton mass. This calculation concerns the Pauli form factor, not an infrared-independent interpretation of the Dirac slope $F_1'(0)$ as a charge radius.

References

• Aoyama, Tatsumi, et al. “The Anomalous Magnetic Moment of the Muon in the Standard Model.” Physics Reports 887 (2020): 1–166. DOI. Open PDF, arXiv:2006.04822v2, 13 November 2020; manuscript pagination.
• 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.