Vacuum structure of the ${\mathbb{Z}}_2$ symmetric Georgi-Machacek model

We discuss the vacuum structure of a version of the Georgi-Machecek model with an exact ${\mathbb{Z}}_2$ symmetry acting on the triplet fields. Besides the usual custodial-symmetric model, with $\rho =1$ at tree-level, a model with a dark matter candidate is also viable. The other phases of the model lead to electric charge breaking, a wrong pattern of electroweak symmetry breaking or to $\rho \mathrm{\ne }1$ at tree-level. We derive conditions to have an absolute minimum in each of the two viable phases, the custodial and the dark matter phases.