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

TL;DR

The paper constructs a UV-complete cubic scalar theory in dimensions near six and shows it flows to a real, IR-stable fixed point for sufficiently large N, reproducing the large-N expansion of the conventional critical O(N) theory continued to d=6-ε. This provides evidence for interacting, unitary O(N) CFTs in five dimensions and suggests a holographic duality with Vasiliev higher-spin theory in AdS$_6$ under alternate boundary conditions. It analyzes fixed points at finite N, computes operator mixing and anomalous dimensions, and uses these results to test the 5d F-theorem while presenting a counterexample to a C_T monotonicity statement. The work also connects to broader topics by comparing with GNY/GN models to test the 3d F-theorem and by outlining the implications for higher-dimensional conformal data and holography.

Abstract

We revisit the classic $O(N)$ symmetric scalar field theories in $d$ dimensions with interaction $(φ^i φ^i)^2$. 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 $σφ^i φ^i$ and $σ^3$. 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.

Figures (8)

• Figure 1: Interacting unitary $O(N)$ symmetric scalar CFT's exist for dimensions $2<d<6$, with $d=4$ excluded. In $6-\epsilon$ and $4-\epsilon$ dimensions they may be described as weakly coupled IR fixed points of the cubic and quartic scalar theories, respectively. In $4+\epsilon$ and $2+\epsilon$ dimensions they are weakly coupled UV fixed points of the quartic theory and of the $O(N)$ Non-linear $\sigma$ Model, respectively.
• Figure 2: The $1/N$ anomalous dimension of $\phi^i$ and the coefficient of the two-point function of $\sigma$ in the large $N$ critical $O(N)$ theory for $2<d<6$.
• Figure 3: Feynman rules of the theory in Euclidean space.
• Figure 4: Diagrams contributing to the 1-loop $\beta$-functions.
• Figure 5: (a) zeros of $\beta_1$ at $N=2000$. (b) zeros of $\beta_2$ at $N=2000$. (c) RG flow directions at $N=2000$. (d) only two non-trivial real solutions at $N=500$.