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

TL;DR

This paper studies a $^3$ quantum field theory with $F_4$ symmetry in near-six dimensions, calculating its renormalization group flow up to three loops and using conformal bootstrap to analyze its fixed point and operator dimensions.

Contribution

It provides a three-loop beta function calculation for $^3$ theory with $F_4$ symmetry and applies conformal bootstrap to confirm the existence and properties of the IR fixed point.

Findings

Stable IR fixed point identified at three loops.

Conformal bootstrap bounds match perturbative predictions in 5.95 dimensions.

Weak kink observed in bootstrap bounds at 5 dimensions.

Abstract

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 $(\Delta_{\phi},\Delta^{\rm 2nd}_{{\mathbf 26}})$=$(1.6,4)$.