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

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.

36 pages, 9 figures, Sec IV H and Refs added

References in corpus (13)

• Exact evolution equation for the effective potential
• Asymptotic safety guaranteed
• Asymptotic safety in the f(R) approximation
• Ising exponents from the functional renormalisation group
• The local potential approximation in quantum gravity
• Fixed-Functionals of three-dimensional Quantum Einstein Gravity
• Critical exponents of O(N) models in fractional dimensions
• High-accuracy scaling exponents in the local potential approximation
• Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings
• Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation
• Are there scaling solutions in the O(N)-models for large N in d>4?
• Truncation Effects in the Functional Renormalization Group Study of Spontaneous Symmetry Breaking
• The two dimensional N=(2,2) Wess-Zumino Model in the Functional Renormalization Group Approach