  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 $v$.
