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 $\rho \neq 1$ 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.