Towards the spectrum of a GUT from gauge invariance

The description of electroweak physics using perturbation theory is highly successful. Though not obvious, this is due to a subtle field-theoretical effect, the Fr\"ohlich-Morchio-Strocchi mechanism, which links the physical spectrum to that of the elementary particles. This works because of the special structure of the standard model, and it is not a priori clear whether it works for structurally different theories. Candidates for conflicts are, e.g., grand unified theories. We study this situation in a toy model, a $SU(3)$ gauge theory with two Higgs fields and a breaking pattern $SU(3) \rightarrow SU(2) \rightarrow 1$. This mimics the weak-Higgs sector of the standard model. We determine the leading order predictions for the gauge invariant spectrum in this theory, and discuss a setup to test them using lattice gauge theory.