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, which are extensions of the $\text{SU}(N)$-invariant AH theories, with a smaller global symmetry group. In particular, we consider an AH model with a two-parameter scalar potential and $\text{SO}(N)$ global symmetry. We discuss the renormalization-group flow of the $\text{SO}(N)$-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 $\text{SO}(N)$- and $\text{SU}(N)$-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 $\text{SO}(N)$ and $\text{SU}(N)$ models. In particular, in $\text{SO}(N)$ models they depend on the sign of the self-interaction parameter $v$ that controls the symmetry breaking from $\text{SU}(N)$ to $\text{SO}(N)$. As a consequence, 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) also depend on the sign of $v$.