# Fixed points and the spontaneous breaking of scale invariance

## TL;DR

This paper analyzes three-dimensional scalar field theories at large N, revealing fixed points, phase structure, and the spontaneous breaking of scale invariance, with implications for understanding vacuum stability and asymptotic safety.

## Contribution

It provides an analytical solution of the renormalization group flows in the large N limit, clarifying the phase diagram and the Bardeen-Moshe-Bander phenomenon.

## Key findings

- Identification of Wilson-Fisher and UV fixed points
- Line of asymptotically safe fixed points with marginal sextic coupling
- Resolution of the unboundedness puzzle of the effective potential

## Abstract

We investigate critical $N$-component scalar field theories and the spontaneous breaking of scale invariance in three dimensions using functional renormalisation. Global and local renormalisation group flows are solved analytically in the infinite $N$ limit to establish the phase diagram including the Wilson-Fisher fixed point and a line of asymptotically safe UV fixed points characterised by an exactly marginal sextic coupling. We also study the Bardeen-Moshe-Bander phenomenon of spontaneously broken scale invariance and the stability of the vacuum for general regularisation. Our findings clarify a long-standing puzzle about the apparent unboundedness of the effective potential. Implications for other theories are indicated.