$F_4$ symmetric ${\varphi }^3$ theory at four loops

The renormalization group functions for six dimensional scalar ${\varphi }^3$ theory with an $F_4$ symmetry are provided at four loops in the modified minimal subtraction ($\overline{\mathrm{MS}}$) scheme. Aside from the anomalous dimension of $\varphi$ and the $\beta$-function this includes the mass operator and a ${\varphi }^2$-type operator. The anomalous dimension of the latter is computed explicitly at four loops for the $\mathbf{26}$ and $\mathbf{324}$ representations of $F_4$. The $\epsilon$ expansion of all the related critical exponents are determined to $O({\epsilon }^4)$. For instance the value for ${\mathrm{\Delta }}_{\varphi }$ agrees with recent conformal bootstrap estimates in 5 and 5.95 dimensions. The renormalization group functions are also provided at four loops for the group $E_6$.