Predicting the singlet vector channel in a partially Higgsed gauge theory

We study a toy version of a grand-unified theory on the lattice: An $SU(3)$ gauge theory, which experiences a Brout-Englert-Higgs effect due to a single Higgs field in the fundamental representation. This yields a perturbative breaking pattern $SU(3) \rightarrow SU(2)$. We investigate the singlet vector channel, finding an apparently non-degenerate and massive ground state. This is in contradistinction to the perturbative prediction of three massless and five massive vector states, even though the correlation functions of the gauge bosons exhibit a weak-coupling behavior, being almost tree-level-like. However, a combination of perturbation theory with the Fr\"ohlich-Morchio-Strocchi mechanism allows to predict the physical spectrum in this channel.