On critical models with $Nleq 4$ scalars in $d=4-epsilon$

We adopt a combination of analytical and numerical methods to study the renormalization group flow of the most general field theory with quartic interaction in $d=4-epsilon$ with $N=3$ and $N=4$ scalars. For $N=3$, we find that it admits only three nondecomposable critical points: the Wilson-Fisher with $O(3)$ symmetry, the cubic with $H_3=(mathbb{Z}_2)^3rtimes S_3$ symmetry, and the biconical with $O(2)times mathbb{Z}_2$. For $N=4$, our analysis reveals the existence of new nontrivial solutions with discrete symmetries and with up to three distinct field anomalous dimensions.