  We investigate the renormalization group flows of multicomponent scalar
theories with $U(1)$ gauge symmetry using the functional renormalization group
method. The scalar sector is built up from traces of matrix fields that belong
to simple, compact Lie algebras. We find that in general these theories are
non-renormalizable even at zero gauge coupling, but if we add a $U(1)$ factor
to the Lie algebra structure, then they are consistent. In accordance with our
earlier findings, fluctuations introduce anomalous, regulator dependent gauge
contributions, which are only consistent with the flow equation for a given set
of gauge fixing parameters. We establish connections between regularization
procedures in the standard covariant and the $R_\xi$ gauges arguing that one is
not tied by introducing regulators at the level of the functional integral, and
it is allowed to switch between schemes at different levels of the
calculations. We calculate $\beta$ functions, classify fixed points, and
clarify compatibility of the flow equation and the Ward-Takahashi identity
between the scalar wavefunction renormalization and the charge rescaling
factor.
