Asymptotic safety of scalar field theories

TL;DR

This paper investigates three-dimensional O(N) scalar field theories using Polchinski's renormalisation group, revealing a rich structure of fixed points, phase transitions, and dualities, with implications for asymptotic safety in quantum field theories.

Contribution

It provides an exact solution at infinite N, explores the phase diagram and fixed points, and establishes a duality between different functional renormalisation group formulations.

Findings

Identification of asymptotically safe UV fixed points

Existence of a conformal window for safety

Duality between Polchinski's and Wetterich's RG approaches

Abstract

We study $3d$ $O(N)$ symmetric scalar field theories using Polchinski's renormalisation group. In the infinite $N$ limit the model is solved exactly including at strong coupling. At short distances the theory is described by a line of asymptotically safe ultraviolet fixed points bounded by asymptotic freedom at weak, and the Bardeen-Moshe-Bander phenomenon at strong sextic coupling. The Wilson-Fisher fixed point arises as an isolated low-energy fixed point. Further results include the conformal window for asymptotic safety, convergence-limiting poles in the complex field plane, and the phase diagram with regions of first and second order phase transitions. We substantiate a duality between Polchinski's and Wetterich's versions of the functional renormalisation group, also showing that that eigenperturbations are identical at any fixed point. At a critical sextic coupling, the duality is worked out in detail to explain the spontaneous breaking of scale symmetry responsible for the generation of a light dilaton. Implications for asymptotic safety in other theories are indicated.