Complex RG flows for 2D nonlinear O(N) sigma models

TL;DR

This paper explores the complex renormalization group flows in 2D nonlinear O(N) sigma models, revealing how Fisher's zeros and singular points influence the flow structure and fixed points, with implications for gauge theories and beyond standard model physics.

Contribution

It extends RG analysis to complex coupling spaces in O(N) models, elucidating the role of Fisher's zeros and singularities in the flow dynamics and fixed point structure.

Findings

Fisher's zeros form strings near singular points in the complex plane.

Complex flows are channeled through singular points and end at strong coupling fixed points.

Scheme dependence affects flow behavior when the correlation length exceeds system size.

Abstract

Motivated by recent attempts to find nontrivial infrared fixed points in 4-dimensional lattice gauge theories, we discuss the extension of the renormalization group (RG) transformations to complex coupling spaces for O(N) models on LxL lattices, in the large-N limit. We explain the Riemann sheet structure and singular points of the finite L mappings between the mass gap and the 't Hooft coupling. We argue that the Fisher's zeros appear on "strings" ending approximately near these singular points. We show that for the spherical model at finite N and L, the density of states is stripwise polynomial in the complex energy plane. We compare finite volume complex flows obtained from the rescaling of the ultraviolet cutoff in the gap equation and from the two-lattice matching. In both cases, the flows are channelled through the singular points and end at the strong coupling fixed points, however strong scheme dependence appear when the Compton wavelength of the mass gap is larger than the linear size of the system. We argue that the Fisher's zeros control the global properties of the complex flows. We briefly discuss the implications for perturbation theory, proofs of confinement and searches for nontrivial infrared fixed points in models beyond the standard model.