= Electron electromagnetic vertex form factors
{title2=$F_1(q^2),F_2(q^2)$}
For external <Electrons> <on shell>, factor out signed <electric charge> in the parity-preserving <interaction vertex>:
$$
\bar u(p')\Gamma^\mu u(p)
=\bar u(p')\left[\gamma^\mu F_1(q^2)
+\frac{i\sigma^{\mu\nu}q_\nu}{2m}F_2(q^2)\right]u(p),
\qquad q=p'-p,\quad p^2=p'^2=m^2,
\quad\sigma^{\mu\nu}=\frac i2[\gamma^\mu,\gamma^\nu].
$$
The $u$ are <Dirac spinors>; $\gamma^\mu$ are <gamma matrices>. With <on-shell renormalization scheme> normalization $F_1(0)=1$, the <Gordon identity> gives $g_s/2=F_1(0)+F_2(0)$ and $a_e=F_2(0)$.<QED Vertex Correction (Vadim Kaplunovsky course notes)>[Eqs. (3), (36)–(38)]{ref}
This explains a subtlety: the <Dirac equation> already predicts a <spin> <magnetic dipole moment> with $F_2=0$. Rewriting its <Dirac electromagnetic current> with the <Gordon identity> produces a spin-dependent term even before any <loop order> correction. $F_2$ measures the additional response after that baseline is included. The limit $q\to0$ is taken after extracting its coefficient; setting $q=0$ in the <interaction vertex> at the outset would hide it. This electromagnetic <anomalous magnetic moment> is not a failure of <gauge invariance> of the kind called a <quantum anomaly>.
\Image[/qed-vertex-charge-and-magnetic-form-factors.png]
{title=Charge normalization and the additional magnetic response}
{description=Original schematic of the <electron electromagnetic vertex form factors>. The <photon> line denotes a weak external probe, not necessarily a real emitted <photon>. The lower bars use the chosen <fine-structure constant> $\alpha=1/137$: their vertical scales differ so the small $F_2(0)$ remains visible. The <Dirac equation> baseline is $g_s=2$, not zero <magnetic dipole moment>; the comparison is theoretical, not measured data.}
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