φ⁴ Theory & Renormalization Group Flow

The φ⁴ scalar field theory in d=3 has a Wilson-Fisher fixed point governing the Ising universality class. RG beta functions β(λ) drive coupling constants from bare to renormalized values. Visualize the RG flow in (m², λ) space.

Bare mass m² m² = 0.50 Coupling λ λ = 1.00 Dimension d d = 3.0

β(λ) = (d-4)λ + 3λ²/(16π²) + ... | γ(λ) = λ²/(16π²) + ... | WF fixed point: λ* = (4-d)·16π²/3