Abelian Higgs gauge theories with multicomponent scalar fields and multiparameter scalar potentials

We consider multicomponent Abelian Higgs (AH) gauge theories with multiparameter scalar quartic potentials that are extensions, with a smaller global symmetry group, of $SU(N)$-invariant AH theories. In particular, we consider an AH model with a two-parameter scalar potential and $SO(N)$ global symmetry. We discuss the renormalization-group flow of the $SO(N)$-invariant AH field theory and the phase diagram and critical behavior of a corresponding three-dimensional (3D) noncompact lattice AH model. We argue that the phase diagram of 3D noncompact $SO(N)$- and $SU(N)$-symmetric lattice AH models are qualitatively similar. In both cases there are three phases: the high-temperature Coulomb phase, and the low-temperature molecular and Higgs phases that differ for the topological properties of the gauge correlations. However, the main features of the low-temperature ordered phases, and in particular of the Higgs phase, differ significantly in $SO(N)$ and $SU(N)$ models. In particular, in $SO(N)$ models they depend on the sign of the self-interaction parameter $v$ that controls the symmetry breaking from $SU(N)$ to $SO(N)$. As a consequence, also the universal features of the transitions related with the spontaneous breaking of the global symmetry (those between the high-temperature Coulomb phase and the low-temperature molecular and Higgs phases) depend on the sign of $v$.