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 Percacci and Vacca [Phys. Rev. D 90, 107702 (2014)].
