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Fixed Points, Universality, and Continuum Limits

An RG fixed point is a scale-invariant solution of a flow in dimensionless theory space. Its neighborhood organizes far more than zeros of beta functions: the linearized eigenoperators identify which microscopic data must be tuned, nonlinear flow determines crossover and corrections to scaling, and the basin of attraction decides whether a fixed point can define an infrared universality class or an ultraviolet continuum limit.

Enter this chapter

Section titled “Enter this chapter”

Begin with Fixed Points and Linearized RG Flow to connect the eigenvectors of a linearized flow to departures from scale invariance.

Topics

Section titled “Topics”
  1. Fixed Points and Linearized RG Flow — Finds the stability spectrum near a scale-invariant theory.
  2. Relevant, Marginal, and Irrelevant Directions — Classifies perturbations and identifies when nonlinear terms matter.
  3. Critical Exponents, Scaling Relations, and Hyperscaling Caveats — Derives observable exponents and the assumptions behind hyperscaling.
  4. Critical Surfaces, Crossover, and Corrections to Scaling — Connects tuning, crossover, and finite-size corrections.
  5. Universality Classes and Scaling Functions — Separates universal scaling data from microscopic normalizations.
  6. Gaussian and Wilson–Fisher Fixed Points — Develops the interacting scalar fixed point near four dimensions.
  7. Ultraviolet and Infrared Fixed Points: Criteria and Evidence — Tests whether a candidate flow supports the claimed continuum theory.

How the ideas fit together

Section titled “How the ideas fit together”

Linearized flow connects microscopic tuning to macroscopic scaling: perturbations that grow toward the infrared must be tuned away to reach a critical point. Decaying perturbations still produce corrections to scaling, while marginal directions require nonlinear analysis. Universal exponents can survive a smooth change of coupling coordinates even though the fixed-point coordinates move. This local information does not determine the global basin of attraction, and a beta-function zero found in a truncation is not by itself a continuum theory. Wilson and Kogut 1974, §§ 11–12.

Where to go next

Section titled “Where to go next”

Continue to Effective Field Theory: Construction and Power Counting to define the regime and accuracy that make an effective theory predictive.

References

Section titled “References”
  • Wilson, Kenneth G., and John Kogut. “The Renormalization Group and the Epsilon Expansion.” Physics Reports 12 (1974): 75–200. DOI