On the addition of an $SU(2)$ quadruplet of scalars to the Standard Model

TL;DR

This paper analyzes an extension of the Standard Model with an $SU(2)$ quadruplet of scalars, deriving exact conditions for potential stability and significantly reducing computational effort in stability analysis.

Contribution

It provides exact analytical equations for phase space boundaries and efficient procedures to determine bounded-from-below conditions, streamlining stability checks.

Findings

Derived exact analytical equations for phase space boundaries.

Developed procedures to efficiently check potential boundedness.

Reduced computational time by three orders of magnitude.

Abstract

We consider the extension of the Standard electroweak Model through an $SU(2)$ quadruplet of scalars with hypercharge either $3/2$ or $1/2$ (with an additional reflection symmetry in the latter case). We establish, through $\textit{exact analytical equations}$, the boundaries of the phase spaces of the gauge-invariant terms that appear in the (renormalizable) scalar potentials. We devise procedures for the determination of necessary and sufficient bounded-from-below conditions on those potentials; we emphasize that one mostly needs to scan the scalar potential over a few $\textit{lines}$, instead of $\textit{surfaces}$, in order to establish the boundedness-from-below; this fact allows one $\textit{to reduce by three orders of magnitude the computational time}$ devoted to that establishment.