Integrability and renormalizability for the fully anisotropic ${\rm SU}(2)$ principal chiral field and its deformations

TL;DR

This paper explores the link between classical integrability and one-loop renormalizability in anisotropic SU(2) principal chiral models, discovering a new integrable four-parameter family via Poisson-Lie deformation with explicit RG invariants.

Contribution

It introduces a new classically integrable four-parameter family of sigma models derived from the SU(2) PCF through Poisson-Lie deformation, demonstrating their one-loop renormalizability.

Findings

Discovery of a new integrable four-parameter sigma model family.

Explicit RG flow equations for the couplings.

Analytical expressions for the system's RG invariants.

Abstract

For the class of $1+1$ dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic ${\rm SU}(2)$ Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic ${\rm SU}(2)$ PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants).