# Global Wilson-Fisher fixed points

## TL;DR

This paper investigates the Wilson-Fisher fixed point in three dimensions for $O(N)$ models using combined analytical and numerical methods, providing detailed insights into fixed point solutions, critical exponents, and convergence properties.

## Contribution

It introduces a hybrid analytical-numerical approach to determine global fixed point solutions for real and imaginary fields, including convergence analysis and singularity relations.

## Key findings

- Computed universal and non-universal quantities for all N
- Analyzed convergence and divergence of finite-N results
- Established links between singularities in effective actions

## Abstract

The Wilson-Fisher fixed point with $O(N)$ universality in three dimensions is studied using the renormalisation group. It is shown how a combination of analytical and numerical techniques determine global fixed point solutions to leading order in the derivative expansion for real or purely imaginary fields with moderate numerical effort. Universal and non-universal quantitites such as scaling exponents and mass ratios are computed, for all $N$, together with local fixed point coordinates, radii of convergence, and parameters which control the asymptotic behaviour of the effective action. We also explain when and why finite-$N$ results do not converge pointwise towards the exact infinite-$N$ limit. In the regime of purely imaginary fields, a new link between singularities of fixed point effective actions and singularities of their counterparts by Polchinski are established. Implications for other theories are indicated.