Wilson RG — Momentum Shell Integration

Wilson's renormalization group integrates out a thin shell of high-momentum modes Λ/s < |k| < Λ, generating flow equations for coupling constants. Fixed points classify universality classes; relevant operators drive RG flows away while irrelevant ones decay, explaining why few parameters describe critical phenomena.

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RG Flow

Shell

Phase diagram

Mass (relevant) r 0.50 Interaction u (marginal) 1.00 Momentum rescaling s 1.30 RG steps 15

● Gaussian FP (r=u=0) ● Wilson-Fisher FP ● Unstable

φ⁴ flow equations:
dr/dl = 2r + 4u/(1+r) − ...
du/dl = εu − 12u² + ...
(ε = 4−d, d=3: ε=1)

WF fixed point: r*<0, u*>0
η, ν from linearization.