$\phi^3$ theory with $F_4$ flavor symmetry in $6-2\epsilon$ dimensions: 3-loop renormalization and conformal bootstrap

We consider $\phi^3$ theory in $6-2\epsilon$ 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 $\phi$ 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 $\phi$ ($\Delta_{\phi})$, we obtain the corresponding upper bound on the scaling dimension of the second lowest scalar primary in the ${\mathbf 26}$ representation $(\Delta^{\rm 2nd}_{{\mathbf 26}})$ which appears in the OPE of $\phi\times\phi$. In $D=5.95$, we observe a sharp peak on the upper bound curve located at $\Delta_{\phi}$ equal to the value predicted by the 3-loop computation. In $D=5$, we observe a weak kink on the upper bound curve at $(\Delta_{\phi},\Delta^{\rm 2nd}_{{\mathbf 26}})$=$(1.6,4)$.