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.

In this paper we will discuss the derivation of the so-called vanishing beta function curves which can be used to explore the fixed point structure of the theory under consideration. This can be applied to the O(N ) symmetric theories, essentially, for arbitrary dimensions (D) and field component (N ). We will show the restoration of the Mermin-Wagner theorem for theories defined in D ≤ 2 and the presence of the Wilson-Fisher fixed point in 2 < D < 4. Triviality is found in D > 4. Interestingly, one needs to make an excursion to the complex plane to see the triviality of the four-dimensional O(N ) theories. The large -N analysis shows a new fixed point candidate in 4 < D < 6 dimensions which turns out to define an unbounded fixed point potential supporting the recent results by R. Percacci and G. P. Vacca in: "Are there scaling solutions in the O(N )-models for large -N in D > 4?" [Phys. Rev. D 90, 107702 (2014)].