Critical $O(N)$ Models in $6-\epsilon$ 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 $6-\epsilon$ 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 $d=6-\epsilon$. 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.

40 pages, 8 figures. Minor corrections, references added

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• Entanglement entropy across a deformed sphere
• Critical exponents of O(N) models in fractional dimensions
• TASI Lectures on the Higher Spin - CFT duality
• A challenge to the $a$-theorem in six dimensions
• $\epsilon$-Expansion in Critical ${\varphi }^3$-Theory on Real Projective Space from Conformal Field Theory
• Large $N_f$ quantum field theory
• Conformal Bootstrap Approach to O(N) Fixed Points in Five Dimensions
• Are there scaling solutions in the O(N)-models for large N in d>4?
• ${\varphi }^3$ theory with $F_4$ flavor symmetry in $6-2\epsilon$ dimensions: 3-loop renormalization and conformal bootstrap
• More on the cubic versus quartic interaction equivalence in the $O(N)$ model
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• Pure anti-de Sitter supergravity and the conformal bootstrap
• Renormalization-Group Behavior of ${\varphi }^3$ Theories in $d=6$ Dimensions
• Eight dimensional QCD at one loop
• Renormalization group improvement of the effective potential in six dimensions
• AdS/CFT in Fractional Dimension and Higher Spin Gravity at One Loop
• Conformal correlators in the critical $O(N)$ vector model
• Long-Range Vector Models at Large N
• Critical long-range vector model in the UV
• A Finite Energy-Momentum Tensor for the ${\varphi }^3$ theory in $6$ dimensions
• beta-functions in higher dimensional field theories
• Renormalization of supersymmetric chiral theories in rational spacetime dimensions