Relationship between electron $g$ value and fine structure constant from QED

Question

The wiki article on the fine structure constant says that 'The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant $\alpha$' but doesn't give the mathematical expression of this relationship. What is this relationship, how is it derived -- roughly speaking -- from QED and where can I find it written down?

Answer 1

According to Wikipedia - Gyromagnetic ratio the $g$-factor of the electron in terms of the fine structure constant $\alpha$ is $$g_e=2\left(1+\frac{\alpha}{2\pi}+ \cdots \right).$$ (The $\cdots$ here stand for higher powers of $\alpha$.)

This formula can be derived from quantum electrodynamics. Considering more and more complex Feynman diagrams in the calculation yields the terms proportional to $\alpha$, $\alpha^2$, $\alpha^3$, ....

_{(figures from "Theory of the Anomalous Magnetic Moment of the Electron" which in turn took the figures from "Quantum electrodynamics calculation of lepton anomalous magnetic moments: Numerical approach to the perturbation theory of QED")}