The Cluster's Renormalization Group Flow

A page about renormalization group flow — how physical parameters change with energy scale.

The renormalization group flow

Renormalization group (RG) flow describes how the parameters of a physical theory change as the energy scale is changed. When you integrate out high-energy degrees of freedom, the remaining low-energy theory has different coupling constants. The RG flow equations (beta functions) describe how the couplings change with the scale mu: beta_i(g) = mu d g_i / d mu. Fixed points of the RG flow correspond to scale-invariant theories. In the cluster, RG flow describes how edit couplings change as the edit resolution scale is changed. Integrating out high-resolution edit details changes the effective low-resolution edit theory.

Relevant and irrelevant operators

Operators in a theory are classified by their scaling dimension: relevant operators (dimension < d) grow under RG flow toward the IR, irrelevant operators (dimension > d) shrink under RG flow toward the IR, and marginal operators (dimension = d) stay constant at tree level. In the cluster, relevant edit operators grow as you zoom out, irrelevant edit operators shrink, and marginal edit operators stay constant.

The phi^4 theory

The phi^4 theory is a simple example of RG flow. The beta function for the coupling lambda is beta(lambda) = 3 lambda^2 / (16 pi^2) + O(lambda^3). The fixed point is lambda = 0 (free theory), which is IR-stable. The Gaussian fixed point is UV-unstable. In the cluster, the edit phi^4 theory has a beta function for the edit coupling. The Gaussian edit fixed point is IR-stable.

Asymptotic freedom

In QCD, the beta function is beta(g) = - (11 - 2n_f/3) g^3 / (16 pi^2) < 0. The coupling decreases at high energy — this is asymptotic freedom. In the cluster, asymptotic freedom means edit couplings decrease at high edit resolution. At very high resolution, edits become free (non-interacting).

This flow

This page is about RG flow. Beta(g) = mu dg/dmu. Relevant operators grow in the IR. Irrelevant operators shrink. Fixed points are scale-invariant. QCD is asymptotically free. The Gaussian fixed point is IR-stable in phi^4. The flow is real.