# $F_4$ symmetric $\phi^3$ theory at four loops

## TL;DR

This paper computes four-loop renormalization group functions for six-dimensional $F_4$ symmetric $\

## Contribution

It provides the first four-loop calculations of RG functions for $F_4$ symmetric $\

## Key findings

- Anomalous dimensions and beta functions are computed at four loops.
- The $eta$-function and anomalous dimensions match conformal bootstrap results.
- Critical exponents are determined up to $O(\e^4)$.

## Abstract

The renormalization group functions for six dimensional scalar $\phi^3$ theory with an $F_4$ symmetry are provided at four loops in the modified minimal subtraction (MSbar) scheme. Aside from the anomalous dimension of $\phi$ and the $\beta$-function this includes the mass operator and a $\phi^2$-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 $\Delta_\phi$ 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 $E_6$.