$ϕ^3$ theory with $F_4$ flavor symmetry in $6-2ε$ dimensions: 3-loop renormalization and conformal bootstrap

Abstract

We consider $ϕ^3$ theory in $6-2ε$ with $F_4$ global symmetry. The beta function is calculated up to 3 loops, and a stable unitary IR fixed point is observed. The anomalous dimensions of operators quadratic or cubic in $ϕ$ are also computed. We then employ conformal bootstrap technique to study the fixed point predicted from the perturbative approach. For each putative scaling dimension of $ϕ$ ($Δ_ϕ)$, we obtain the corresponding upper bound on the scaling dimension of the second lowest scalar primary in the ${\mathbf 26}$ representation $(Δ^{\rm 2nd}_{\mathbf 26})$ which appears in the OPE of $ϕ\timesϕ$. In $D=5.95$, we observe a sharp peak on the upper bound curve located at $Δ_ϕ$ equal to the value predicted by the 3-loop computation. In $D=5$, we observe a weak kink on the upper bound curve at $(Δ_ϕ,Δ^{\rm 2nd}_{\mathbf 26})$=$(1.6,4)$.

22 pages, published version