In GUT-like scenarios it is necessary to break the gauge group to the SM-group. This is usually achieved by adding scalars to the theory, which may introduce one or more possible breaking patterns to the theory. Additionally, a valid description of gauge-symmetry-breaking needs to be manifestly gauge-invariant, which is often neglected in standard phenomenology.
In our work we therefore study the lattice phase diagram of the simplest theory with two breaking patterns: a $SU(3)$ gauge theory with a Higgs in the adjoint representation. This allows us to analyse possible tensions between perturbative breaking patterns and actual possible ones.