Vacuum Stability and Perturbativity of SU(3) Scalars

TL;DR

This paper investigates the vacuum stability and renormalisation group evolution of scalar multiplets in SU(3), revealing that certain representations exhibit walking couplings and revising earlier scale-generation results.

Contribution

It provides a detailed analysis of vacuum stability conditions and orbit space calculations for higher SU(3) scalar multiplets, including the novel finding of walking quartic couplings.

Findings

Scalar quartic couplings of representations 3 and 8 walk rather than run.

The orbit space convex hull simplifies stability analysis.

Revised earlier results on scale generation with large SU(3) multiplets.

Abstract

We calculate the vacuum stability conditions and renormalisation group equations for the extensions of standard model with a higher colour multiplet scalar up to the representation $\mathbf{15'}$ that leaves the strong interaction asymptotically free. In order to find the vacuum stability conditions, we calculate the orbit spaces for the self-couplings of the higher multiplets, which for the representations $\mathbf{15}$ and $\mathbf{15'}$ of $SU(3)_c$ are highly complicated. However, if the scalar potential is linear in orbit space variables, it is sufficient to know the convex hull of the orbit space. In contrast to the self-couplings of other multiplets, we find that the scalar quartic couplings of the representations $\mathbf{3}$ and $\mathbf{8}$ walk rather than run, remaining nearly constant and perturbative over a vast energy range. We describe the conditions for walking couplings using a schematic model. With these technical results at hand we revise earlier results of generation of new scales with large $SU(3)_c$ scalar multiplets. Our results are easily extendable to models of new physics with additional $SU(3)$ or SU($N$) gauge symmetries.