Is our vacuum global in an economical 331 model?

TL;DR

This paper analyzes the scalar sector of the economical $SU(3)_c\times SU(3)_L\times U(1)_X$ model with $β=-1/\sqrt{3}$, three scalar triplets, and a softly broken $\mathbb{Z}_2$ symmetry. Using orbit-space methods and the $P$-matrix formalism, it systematically classifies all potential minima, derives full necessary-and-sufficient conditions for the potential to be bounded from below, and assesses whether the electroweak vacuum is the global minimum or metastable. It also provides a parametrisation of the scalar couplings in terms of physical observables (masses, VEVs, and mixing angles) to map viable regions of parameter space under perturbative unitarity and metastability constraints. The results show that in the large-$v_\chi$/alignment regime the EW vacuum is global in much of the parameter space, while nonzero mixings between the triplets can induce metastability, thereby restricting the phenomenologically allowed regions and guiding future explorations of the model.

Abstract

We consider the economical 331 model, based on $β=-1/\sqrt{3}$, with three $SU(3)$ triplets with a softly broken $\mathbb{Z}_2$ symmetry. The resulting scalar potential is commonly used in phenomenology. We systematically determine all the potential minima and obtain the conditions under which the electroweak vacuum is global with the help of orbit space methods. For the case the electroweak vacuum is not global, we calculate bounds on the scalar couplings from metastability. We find a parametrisation of the potential couplings in terms of physical quantities and use it to show the available parameter space.

Figures (4)

• Figure 1: Three-dimensional sections of the orbit space for fixed values of the orbit parameter $\vartheta_{4}$: $\vartheta_{4} = \pm 1$ gives the origin, $\vartheta_{4} = 0$ the largest bounding surface. The intersections of the orbit space with the non-negative orthant are shaded darker.
• Figure 2: Section of the orbit space on the $\vartheta_{1}\vartheta_{4}$-plane. The orbit space consists of two symmetric halves bounded by the graphs of the functions $\vartheta_{4+}$ and $\vartheta_{4-}$ defined on a common convex domain. Example convex combinations of orbit space points are shown in dark grey.
• Figure 3: Parameter space of the economical 331 model. Regions where unitarity is violated are shown in red, where the potential is not bounded from below in blue, where the electroweak vacuum is not global in yellow (metastable in hatched yellow). Regions in white satisfy all constraints.
• Figure 4: Parameter space of the economical 331 model with $\sin \alpha_{12} = -1/\sqrt{2}$, $\sin \alpha_{13} = 0$, $\sin \alpha_{23} = 0.05$. Regions where unitarity is violated are shown in red, where the potential is not bounded from below in blue, where the electroweak vacuum is not global in yellow (metastable in hatched yellow). Regions in white satisfy all constraints.