Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 3 / e / Solution

The renormalization-group fixed points are the zeros of the beta function:For $N>2$ and $ϵ>0$, the positive fixed point $g_c^{\ast }$ separates the low-temperature ordered flow toward $g=0$ from the high-temperature strong-coupling flow. At $ϵ=0$ the two perturbative fixed points merge at zero.