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

On critical models with $N\leq 4$ scalars in $d=4-\epsilon$