Radiative Corrections

Beyond tree level, Feynman diagrams contain loops — integrals over internal momenta that often diverge in the ultraviolet. These radiative corrections are of order α ≈ 1/137 and represent genuine quantum effects with measurable consequences: the Lamb shift, the running of α, and most famously the anomalous magnetic moment aₑ = (g−2)/2, which QED predicts to twelve decimal places.

Key Concepts

Electron self-energy Σ(p̸): one-loop correction to propagator, shifts mass and field normalization
Vacuum polarization Π(q²): one-loop photon self-energy, makes α run with momentum transfer
Vertex correction Λμ: one-loop correction to electron-photon vertex, anomalous magnetic moment
Ward-Takahashi identity: kμΛμ(p,p+k) = Σ(p+k) − Σ(p) — exact relation tying vertex to self-energy
Lamb shift: vacuum polarization + vertex correction split 2S₁/₂ and 2P₁/₂ levels in hydrogen (1057 MHz)
Anomalous magnetic moment: aₑ = α/2π − 0.328(α/π)² + ... ≈ 0.001159652 — tested to 12 digits

Key Equations

Full electron propagator

S_F(p)=\frac{i}{\not\!p-m-\Sigma(p)+i\varepsilon}

Vacuum polarization

\Pi^{\mu\nu}(q)=(q^2g^{\mu\nu}-q^\mu q^\nu)\Pi(q^2)

Running coupling

\alpha(q^2)\approx\frac{\alpha}{1-\frac{\alpha}{3\pi}\ln(q^2/m_e^2)}

Anomalous magnetic moment

a_e=\frac{g-2}{2}=\frac{\alpha}{2\pi}-0.32848\Bigl(\frac{\alpha}{\pi}\Bigr)^2+\cdots\approx0.00115965

Worked Example

Example Problem

Problem

Using aₑ ≈ α/2π at one loop, compute aₑ numerically with α = 1/137.036.

Solution

aₑ = (1/137.036)/(2π) = 1/861.0 = 0.001161. The full QED prediction to 5 loops gives 0.0011596521...; the measured value is 0.0011596521811... Agreement to 12 digits — the most precise test of any theory.

Key Takeaways

Loop diagrams generate corrections of order α/π ≈ 1/432 — small but measurable to extraordinary precision
Vacuum polarization makes α run: from 1/137 at q²→0 to ~1/128 at q² = M²_Z (vacuum screens charge at long distances)
The Lamb shift (hydrogen 2S−2P splitting of 1057 MHz) was the first experimental evidence for quantum field theory beyond tree level
The anomalous magnetic moment aₑ = (g−2)/2 is computed to 5 loops in QED and agrees with experiment to 12 significant figures