Renormalization Group Flow

Wilsonian RG — Coupling Space Fixed Points & Flow Lines

The Wilsonian RG integrates out high-energy modes, generating effective couplings at lower scales. The beta function β(g) = μ dg/dμ determines how couplings run with energy. Fixed points g* where β(g*)=0 govern UV (asymptotic freedom) and IR (confinement) behavior. In QCD: β(g) = −b₀g³ − b₁g⁵ −… with b₀=(11Nc−2Nf)/(16π²) > 0 giving asymptotic freedom.

b₀ = 1.50
b₁ = 0.50
ε = 0.10
7
UV fixed pt g*_UV:
IR fixed pt g*_IR:
β(g) at g=0.5:
Regime: