Trinification from ${\mathrm{E}}_6$ symmetry breaking

Abstract

In the context of ${\mathrm{E}}_6$ Grand Unified Theories (GUTs), an intriguing possibility for symmetry breaking to the Standard Model (SM) group involves an intermediate stage characterized by either $\mathrm{SU}(3)\times \mathrm{SU}(3)\times \mathrm{SU}(3)$ (trinification) or $\mathrm{SU}(6)\times \mathrm{SU}(2)$. The more common choices of $\mathrm{SU}(5)$ and $\mathrm{SO}(10)$ GUT symmetry groups do not offer such breaking chains. We argue that the presence of a real (rank $2$ tensor) representation $650$ of ${\mathrm{E}}_6$ in the scalar sector is the minimal and likely only reasonable possibility to obtain one of the novel intermediate stages. We analyze the renormalizable scalar potential of a single copy of the $650$ and find vacuum solutions that support regularly embedded subgroups $\mathrm{SU}(3)\times \mathrm{SU}(3)\times \mathrm{SU}(3)$, $\mathrm{SU}(6)\times \mathrm{SU}(2)$, and $\mathrm{SO}(10)\times \mathrm{U}(1)$, as well as specially embedded subgroups ${\mathrm{F}}_4$ and $\mathrm{SU}(3)\times {\mathrm{G}}_2$ that do not contain the SM gauge symmetry. We show that for a suitable choice of parameters, each of the regular cases can be obtained as the lowest among the analyzed minima in the potential.

arXiv comment29 pages, 15 tables, 3 figures

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