Critical $O(N)$ Models in $6-ϵ$ Dimensions

Abstract

We revisit the classic $O(N)$ symmetric scalar field theories in $d$ dimensions with interaction $({\varphi }^i{\varphi }^i{)}^2$. For $2<d<4$ these theories flow to the Wilson-Fisher fixed points for any $N$. A standard large $N$ Hubbard-Stratonovich approach also indicates that, for $4<d<6$, these theories possess unitary UV fixed points. We propose their alternate description in terms of a theory of $N+1$ massless scalars with the cubic interactions $\sigma {\varphi }^i{\varphi }^i$ and ${\sigma }^3$. Our one-loop calculation in $6-ϵ$ dimensions shows that this theory has an IR stable fixed point at real values of the coupling constants for $N>1038$. We show that the $1/N$ expansions of various operator scaling dimensions match the known results for the critical $O(N)$ theory continued to $d=6-ϵ$. These results suggest that, for sufficiently large $N$, there are 5-dimensional unitary $O(N)$ symmetric interacting CFT's; they should be dual to the Vasiliev higher-spin theory in AdS${}_6$ with alternate boundary conditions for the bulk scalar. Using these CFT's we provide a new test of the 5-dimensional $F$-theorem, and also find a new counterexample for the $C_T$ theorem.

Comments: 40 pages, 8 figures. Minor corrections, references added