Vacuum Stability Conditions for New $SU(2)$ Multiplets

TL;DR

This paper analyzes the vacuum stability conditions for scalar $SU(2)$ multiplets of dimensions 1 through 6 added to the Standard Model, identifying the parameter space where the potential remains stable.

Contribution

It provides a comprehensive derivation of bounded-from-below and vacuum stability conditions for new scalar multiplets with arbitrary hypercharge in the Standard Model.

Findings

Stability conditions are derived for multiplets of dimensions 1 to 6.

The phase space of the scalar potential is characterized, noting a concavity for the 6-plet.

Explicit stability bounds are established for each multiplet dimension.

Abstract

We consider the addition to the Standard Model of a scalar $SU(2)$ multiplet $\Delta_n$ with dimension $n$ going from $1$ to $6$. The multiplet $\Delta_n$ is assumed to have null vacuum expectation value and an arbitrary (free) hypercharge. We determine the shape of the phase space for the new terms that appear in the scalar potential (SP); we observe in particular that, in the case of a 6-plet, the phase space is slightly concave along one of its boundaries. We determine the bounded-from-below and vacuum stability conditions on the SP for each value of $n$.