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

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.