Critical $O(N)$ models in the complex field plane

TL;DR

This paper analyzes $O(N)$ scalar field theories in the complex field plane using the renormalisation group, deriving exact solutions for the Wilson-Fisher fixed point and revealing singularities that limit real-field expansions.

Contribution

It provides an analytical solution for the Wilson-Fisher fixed point in the complex plane and characterizes the singularities affecting real-field expansions across all $O(N)$ models.

Findings

Exact recursion relations for couplings were derived.

Singularities in the complex plane determine the convergence of real-field expansions.

Closed-form expressions for singularities at infinite $N$ were obtained.

Abstract

Local and global scaling solutions for $O(N)$ symmetric scalar field theories are studied in the complexified field plane with the help of the renormalisation group. Using expansions of the effective action about small, large, and purely imaginary fields, we obtain and solve exact recursion relations for all couplings and determine the $3d$ Wilson-Fisher fixed point analytically. For all $O(N)$ universality classes, we further establish that Wilson-Fisher fixed point solutions display singularities in the complex field plane, which dictate the radius of convergence for real-field expansions of the effective action. At infinite $N$, we find closed expressions for the convergence-limiting singularities and prove that local expansions of the effective action are powerful enough to uniquely determine the global Wilson-Fisher fixed point for any value of the fields. Implications of our findings for interacting fixed points in more complicated theories are indicated.