On $ϕ^3$ Theory Above Six Dimensions

We study the scalar $ϕ^3$ theory above six dimensions. The beta function $β(g)=-εg-\frac{3}{4}g^3$ 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 $ϕ^3$ theory above six dimensions exist.

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