Vacuum Stability and Perturbativity of SU(3) Scalars

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 $15^{\prime }$ 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 $15$ and $15^{\prime }$ 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 $3$ and $8$ 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.

26 pages, 6 figures, 2 tables

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