Wilson-Fisher Fixed Point

Dimension d = 4−ε 0.50 Flow speed 1.0 Coupling g (initial) 0.50

Gaussian (g*=0)

Wilson-Fisher (g*=ε/6+...)

RG flow lines

The Wilson-Fisher fixed point emerges from the Gaussian (free) theory under RG flow for d < 4. At one loop:

dg/dℓ = εg − 3g² + O(g³)

g* = ε/3 + O(ε²)

The fixed point controls the Ising universality class. Critical exponents are computed in ε = 4−d:

η = ε²/54 + O(ε³)

ν = 1/2 + ε/12 + O(ε²)