  We present a field theoretic renormalization group study for the critical
behaviour of a uniformly driven diffusive system with quenched disorder, which
is modelled by different kinds of potential barriers between sites. Due to
their symmetry properties, these different realizations of the random potential
barriers lead to three different models for the phase transition to transverse
order and to one model for the phase transition to longitudinal order all
belonging to distinct universality classes. In these four models that have
different upper critical dimensions d_{c} we find the critical scaling
behaviour of the vertex functions in spatial dimensions d < d_{c} . Its
deviation from purely diffusive behaviour is characterized by the
anomaly-exponent \eta that we calculate at first and second order, respectively
in $\epsilon = d_{c} -d$. In each model \eta turns out to be positive which
means superdiffusive spread of density fluctuations in the driving force
direction.
