Exponential Hierarchies from Anderson Localization in Theory Space

We present a new mechanism for generating exponential hierarchies in four-dimensional field theories inspired by Anderson localization in one dimension, exploiting an analogy between the localization of electron energy eigenstates along a one-dimensional disordered wire and the localization of mass eigenstates along a local "theory space" with random mass parameters. Mass eigenstates are localized even at arbitrarily weak disorder, with exponentially suppressed couplings to sites in the theory space. The mechanism is quite general and may be used to exponentially localize fields of any spin. We apply the localization mechanism to two hierarchies in standard model parameters-the smallness of neutrino masses and the ordering of quark masses-and comment on the possible relevance to the electroweak hierarchy problem. This raises the compelling possibility that some of the large hierarchies observed in and beyond the standard model may result from disorder, rather than order.