  We develop a renormalization group for weak Harris-marginal disorder in
otherwise strongly interacting quantum critical theories, focusing on systems
which have emergent conformal invariance. Using conformal perturbation theory,
we argue that previously proposed random lines of fixed points with Lifshitz
scaling in fact flow towards other universal fixed points, and this flow is
captured by a "one-loop" analysis. Our approach appears best controlled in
theories with only a few operators with low scaling dimension. In this regime,
we compare our predictions for the flow of disorder to holographic models, and
find complete agreement.
