A virtual photon connects two points on the electron line at the vertex, forming a triangle loop. The UV divergence is related to the self-energy divergence by the Ward-Takahashi identity. The finite piece generates the anomalous magnetic moment.
Julian Schwinger computed F_2(0) = alpha/(2*pi) in 1948, the first successful calculation in renormalized QED. The result agrees with experiment to better than 12 significant figures when extended to higher loop orders.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
A virtual photon connects two points on the electron line at the electromagnetic vertex, forming a triangle loop.
Feynman rules expression
$$\Lambda^\mu(p,p') = \int\frac{d^4k}{(2\pi)^4}\,(-ie\gamma^\nu)\frac{i(\not{p}'-\not{k}+m)}{(p'-k)^2-m^2}(-ie\gamma^\mu)\frac{i(\not{p}-\not{k}+m)}{(p-k)^2-m^2}(-ie\gamma_\nu)\frac{-i}{k^2}$$
Three propagators (two electron, one photon) and three vertices. The loop momentum $k$ is unconstrained and must be integrated over.
Feynman parametrization
$$\frac{1}{ABC} = 2\int_0^1 dx\,dy\,dz\,\delta(x+y+z-1)\frac{1}{[xA+yB+zC]^3}$$
Combine the three propagator denominators into a single denominator using Feynman parameters $x, y, z$. This makes the loop integral tractable by completing the square in $k$.
Loop integration in $d = 4-\epsilon$ dimensions
Shift the loop momentum to complete the square, Wick rotate to Euclidean space, and evaluate the standard integral. The UV divergence appears as a $1/\epsilon$ pole in dimensional regularization.
Gordon decomposition and form factors
$$\bar{u}(p')\Lambda^\mu u(p) = \bar{u}(p')\left[\gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu}q_\nu}{2m}F_2(q^2)\right]u(p)$$
Decompose the vertex into two Lorentz structures: the Dirac form factor $F_1$ (charge) and the Pauli form factor $F_2$ (magnetic moment). The anomalous magnetic moment is $a_e = F_2(0)$.
The Schwinger result
$$F_2(0) = \frac{\alpha}{2\pi} \approx 0.00116$$
At $q^2 = 0$ (static limit), the Pauli form factor evaluates to $\alpha/(2\pi)$. This is the one-loop correction to the electron's anomalous magnetic moment: $g/2 = 1 + \alpha/(2\pi) + \cdots$
Experimental comparison
The electron $g-2$ has been measured to extraordinary precision. Including QED corrections through five loops: $a_e^{\text{theory}} = 0.001\,159\,652\,181\,643(764)$, $a_e^{\text{expt}} = 0.001\,159\,652\,180\,73(28)$. Agreement to better than 1 part in $10^{12}$.
Physical contribution
The Pauli form factor F_2(0) = alpha/(2*pi) = 0.00116... gives the Schwinger correction to the electron anomalous magnetic moment. This is the most celebrated single result in QED.