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 renormalization. Global and local renormalization group flows are solved analytically in the infinite N limit to establish the exact phase diagram of the theory including the Wilson-Fisher fixed point and a line of asymptotically safe UV fixed points. We also study the Bardeen-Moshe-Bander phenomenon of spontaneously broken scale invariance and the stability of the vacuum for general regularization. Our findings clarify a long-standing puzzle about the apparent unboundedness of the effective potential. Implications for other theories are indicated.