Three Loop Analysis of the Critical $O(N)$ Models in $6-\epsilon$ Dimensions

Abstract

We continue the study, initiated in arXiv:1404.1094, of the $O(N)$ symmetric theory of $N+1$ massless scalar fields in $6-\epsilon$ dimensions. This theory has cubic interaction terms $\frac{1}{2}{g}_1\sigma ({\varphi }^i{)}^2+\frac{1}{6}{g}_2{\sigma }^3$. We calculate the 3-loop beta functions for the two couplings and use them to determine certain operator scaling dimensions at the IR stable fixed point up to order ${\epsilon }^3$. We also use the beta functions to determine the corrections to the critical value of $N$ below which there is no fixed point at real couplings. The result suggests a very significant reduction in the critical value as the dimension is decreased to $5$. We also study the theory with $N=1$, which has a $Z_2$ symmetry under $\varphi \to -\varphi$. We show that it possesses an IR stable fixed point at imaginary couplings which can be reached by flow from a nearby fixed point describing a pair of $N=0$ theories. We calculate certain operator scaling dimensions at the IR fixed point of the $N=1$ theory and suggest that, upon continuation to two dimensions, it describes a non-unitary conformal minimal model.

34 pages, 9 figures. Some improvements and references added

References in corpus (6)

• Algorithm FIRE -- Feynman Integral REduction
• Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• Critical exponents of the 3d Ising and related models from Conformal Bootstrap
• Five dimensional $O(N)$-symmetric CFTs from conformal bootstrap
• Microscopic models of interacting Yang-Lee anyons
• Are there scaling solutions in the O(N)-models for large N in d>4?

Cited by in corpus (18)

• Conformal Bootstrap in Mellin Space
• Bootstrapping $O(N)$ Vector Models in $4<d<6$
• TASI Lectures on the Higher Spin - CFT duality
• Large $N_f$ quantum field theory
• Conformal Bootstrap Approach to O(N) Fixed Points in Five Dimensions
• More on the cubic versus quartic interaction equivalence in the $O(N)$ model
• ${\varphi }^3$ theory with $F_4$ flavor symmetry in $6-2\epsilon$ dimensions: 3-loop renormalization and conformal bootstrap
• Multi-critical multi-field models: a CFT approach to the leading order
• Homoclinic RG flows, or when relevant operators become irrelevant
• 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
• Six dimensional Landau-Ginzburg-Wilson theory
• Interpolating between $a$ and $F$
• Conformal correlators in the critical $O(N)$ vector model
• Critical long-range vector model in the UV
• Non-perturbative Aspects of Quantum Field Theories from Holography
• beta-functions in higher dimensional field theories