Vanishing Beta Function curves from the Functional Renormalisation 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 $D\leq2$ 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)].