Multiple breaking patterns in the Brout-Englert-Higgs effect beyond perturbation theory

In many BSM theories, especially GUTs, introducing a Brout-Englert-Higgs effect allows for multiple breaking patterns of the gauge symmetry. The possibility to select a particular pattern is usually decisive for the phenomenological viability of a theory. Beyond perturbation theory it is necessary to replace the Brout-Englert-Higgs effect by a manifestly gauge-invariant description. We study the simplest case with multiple breaking patterns, an $\textrm{SU}(3)$ Yang-Mills theory coupled to a single scalar `Higgs' field in the adjoint representation, on the lattice. We find that only one pattern remains at fixed parameters and gauge-fixing strategy, and that the associated quantum effective potential emerges from a non-trivial interplay of many aspects.