On ${\varphi }^3$ Theory Above Six Dimensions

We study the scalar ${\varphi }^3$ theory above six dimensions. The beta function $\beta (g)=-ϵg-\frac{3}{4}{g}^3$ in $d=6-2ϵ$ dimensions has a UV fixed point when $ϵ<0$. Like the $O(N)$ vector models above four dimensions, such a fixed point observed perturbatively in fact corresponds to a pair of complex CFTs separated by a branch cut. Using both the numerical bootstrap method and Gliozzi’s fusion rule truncation method, we argue that the fixed points of the ${\varphi }^3$ theory above six dimensions exist.