Wilson RG — Epsilon Expansion

Renormalization group flows in coupling-space: fixed points, critical exponents, universality classes

Model Dimension d 3.0 N (O(N) components) 1 Initial u₀ 0.5 Initial r₀ 0.1

ε = 4-d: --

u* (Wilson-Fisher): --

η: --

ν: --

Fixed point: --

β-function (φ⁴): β(u) = -εu + (N+8)/(16π²) u² − ... [Wilson-Fisher 1972]

Fixed point: u* = 16π²ε/(N+8) + O(ε²) → Wilson-Fisher FP

Critical exponents: η = (N+2)/(2(N+8)²) ε² ν⁻¹ = 2 − (N+2)/(N+8) ε

Universality: all systems with same (d,N,symmetry) share identical critical exponents

Wilson RG Flow — ε Expansion