  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.
