# Normal form for renormalization groups

## TL;DR

This paper introduces a normal form approach to renormalization groups, unifying various singular behaviors near critical points into universality classes, and improving the understanding of scaling functions and singularities.

## Contribution

It applies normal form theory to renormalization groups, systematically classifying singularities and scaling functions into universality families, enhancing analysis of critical phenomena.

## Key findings

- Unified treatment of power law, logarithmic, and exponential singularities
- Prediction of nonlinear universal scaling functions
- Improved handling of critical singularities in classic cases

## Abstract

The results of the renormalization group are commonly advertised as the existence of power law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis. We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables. We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions. We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.