  The phase diagrams and the nature of the phase transitions in multicomponent
gauge theories with an Abelian gauge field are important topics with various
physical applications. While an early renormalization-group-based study
indicated that the direct transition from a fully ordered to a fully disordered
state is continuous for $N = 1$ and $N > 183$, recently it was demonstrated
that the transition is discontinuous for $N = 2$. We quantitatively study the
dependence on $N$ of the degree of discontinuity of this transition. Our
results suggest that the transition is discontinuous at least up to $N = 7$.
Furthermore, we demonstrate that, at increased coupling strength, the phase
transitions of the neutral and charged sectors of the model split, which for $N
> 2$ yields a new phase with composite order. The transition from the
composite-order phase to the fully disordered phase is then also discontinuous,
at least for $N = 3$ and $N = 4$. Via a duality argument, this indicates that
van der Waals-type interaction between directed loops may be responsible for
the discontinuous phase transitions in these models.
