Fisher's zeros as boundary of renormalization group flows in complex coupling spaces

TL;DR

This paper introduces new methods to analyze the boundary of renormalization group flows in complex coupling spaces, using Fisher's zeros to identify boundaries of attraction basins and supporting this with numerical evidence from various models.

Contribution

It proposes novel techniques to extend RG transformations into complex spaces and links Fisher's zeros to IR fixed point boundaries, supported by numerical calculations.

Findings

Fisher's zeros mark the boundary of attraction basins in complex coupling space.

Numerical evidence shows Fisher's zeros stabilize away from the real axis as volume increases.

Implications for confinement and IR fixed points in beyond Standard Model theories.

Abstract

We propose new methods to extend the renormalization group transformation to complex coupling spaces. We argue that the Fisher's zeros are located at the boundary of the complex basin of attraction of infra-red fixed points. We support this picture with numerical calculations at finite volume for two-dimensional O(N) models in the large-N limit and the hierarchical Ising model. We present numerical evidence that, as the volume increases, the Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with a Wilson action, stabilize at a distance larger than 0.15 from the real axis in the complex beta=4/g^2 plane. We discuss the implications for proofs of confinement and searches for nontrivial infra-red fixed points in models beyond the standard model.