Regularization

Loop integrals in QFT diverge at large momenta (UV divergences) and sometimes at small momenta (IR divergences). Regularization renders these divergences finite and controlled while preserving the symmetries of the theory. Dimensional regularization — working in d = 4−2ε dimensions — is the modern standard: it respects gauge invariance, handles both UV and IR divergences, and integrates seamlessly into the renormalization group.

Key Concepts

Superficial degree of divergence: D = 4L − 2I_B − I_F; D=0 log-divergent, D=2 quadratic, D<0 finite
Dimensional regularization: d⁴k/(2π)⁴ → μ^{2ε} d^d k/(2π)^d, treat d as complex variable
UV poles appear as 1/ε in dimensional regularization (where ε = (4−d)/2)
Feynman parameters: 1/(AB) = ∫₀¹ dx/[xA+(1−x)B]² — combines denominators for loop integration
Pauli-Villars: add regulator fields with mass M → ∞; less flexible for non-Abelian theories
Dimensional regularization preserves gauge invariance — a crucial property for renormalizing gauge theories

Key Equations

Dim-reg measure

\int\!\frac{d^4k}{(2\pi)^4}\to\mu^{2\varepsilon}\!\int\!\frac{d^dk}{(2\pi)^d},\;d=4-2\varepsilon

Standard loop integral

\int\!\frac{d^dk}{(2\pi)^d}\frac{1}{(k^2-\Delta)^n}=\frac{i(-1)^n}{(4\pi)^{d/2}}\frac{\Gamma(n-d/2)}{\Gamma(n)}\Delta^{d/2-n}

Feynman parameter

\frac{1}{AB}=\int_0^1\!dx\,\frac{1}{[xA+(1-x)B]^2}

UV pole

\frac{1}{\varepsilon}\equiv\frac{2}{4-d}\xrightarrow{d\to4}\text{log UV divergence}

Worked Example

Example Problem

Problem

A massless one-loop integral in dim-reg gives I = μ^{2ε}∫d^dk/k⁴. Using the master formula with Δ=0, find the pole in ε.

Solution

From the formula with n=2: I ∝ Γ(2−d/2)/Δ^{2−d/2}. At d=4−2ε: Γ(ε) = 1/ε − γ_E + O(ε). The integral has a 1/ε UV pole with coefficient proportional to Γ(ε).

Key Takeaways

Every loop integral must be regularized to expose its UV divergence in a controlled, symmetry-preserving way
Dimensional regularization is the gold standard: it preserves gauge invariance and produces 1/ε poles for each UV divergence
The superficial degree of divergence D determines how many counterterms are needed; a renormalizable theory has only finitely many D ≥ 0 structures
Feynman parameterization + Wick rotation reduces any loop integral to a standard Euclidean form solvable by Gamma-function identities