φ⁴ theory: integrate out high-momentum modes and watch coupling constants flow.
m² is relevant (d<4)
λ is marginal (d=4)
Current m²: 0.50
Current λ: 1.00
Wilson's RG (1971): Start with action S = ∫[½(∂φ)²+ ½m²φ² + λφ⁴]. Integrate momentum shells Λ/b<|k|<Λ.
The remaining modes are rescaled: x→x/b, φ→b^{(d+2)/2}φ. This induces flows in (m², λ).
φ⁴ in d=4: λ is exactly marginal at tree level; one-loop correction gives β(λ) = 3λ²/(16π²) > 0 — the coupling
is asymptotically free in reverse. The Gaussian fixed point (m²=0, λ=0) controls the UV.
d<4: λ becomes relevant; Wilson-Fisher fixed point appears at finite λ* — controls critical phenomena (Ising universality).