Fixed points and the spontaneous breaking of scale invariance

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.