Classifying irreducible fixed points of five scalar fields in perturbation theory

Classifying perturbative fixed points near upper critical dimensions plays an important role in understanding the space of conformal field theories and critical phases of matter. In this work, we consider perturbative fixed points of $N=5$ scalar bosons coupled with quartic interactions preserving an arbitrary subgroup $G\subset {\rm O}(5)$. We perform an exhaustive algorithmic search over the symmetry groups $G$ which are irreducible and satisfy the Landau condition, so that the fixed point can be reached by fine-tuning a single mass term and there is no need to tune the cubic couplings. We also impose stability of the RG flow in the space of quartic couplings, and reality. We thus prove that there exist no new stable fixed points in $d=4-\epsilon$ dimensions beyond the two known ones: namely the ${\rm O}(5)$ invariant fixed point and the Cubic(5) fixed point. This work is a continuation of the classification of such fixed points with $N=4$ scalars by Toledano, Michel, Toledano, and Br\'ezin in 1985.