Vanishing beta function curves from the functional renormalization group

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)].