  In this paper we analyze the renormalization group (RG) flow of field
theories with quenched disorder, in which the couplings vary randomly in space.
We analyze both classical (Euclidean) disorder and quantum disorder,
emphasizing general properties rather than specific cases. The RG flow of the
disorder-averaged theories takes place in the space of their coupling constants
and also in the space of distributions for the disordered couplings, and the
two mix together. We write down a generalization of the Callan-Symanzik
equation for the flow of disorder-averaged correlation functions. We find that
local operators can mix with the response of the theory to local changes in the
disorder distribution, and that the generalized Callan-Symanzik equation mixes
the disorder averages of several different correlation functions. For classical
disorder we show that this can lead to new types of anomalous dimensions and to
logarithmic behavior at fixed points. For quantum disorder we find that the RG
flow always generates a rescaling of time relative to space, which at a fixed
point generically leads to Lifshitz scaling. The dynamical scaling exponent z
behaves as an anomalous dimension (as in other non-relativistic RG flows), and
we compute it at leading order in perturbation theory in the disorder for a
general theory. Our results agree with a previous perturbative computation by
Boyanovsky and Cardy, and with a holographic disorder computation of Hartnoll
and Santos. We also find in quantum disorder that local operators mix with
non-local (in time) operators under the RG, and that there are critical
exponents associated with the disorder distribution that have not previously
been discussed. In large N theories the disorder averages may be computed
exactly, and we verify that they are consistent with the generalized
Callan-Symanzik equations.
