Parisi-Sourlas Supersymmetry in Random Field Models

By the Parisi-Sourlas conjecture, the critical point of a theory with random field (RF) disorder is described by a supersymmeric (SUSY) conformal field theory (CFT), related to a d-2 dimensional CFT without SUSY. Numerical studies indicate that this is true for the RF φ3 model but not for the RF φ4 model in d<5 dimensions. Here we argue that the SUSY fixed point is not reached because of new relevant SUSY-breaking interactions. We use a perturbative renormalization group in a judiciously chosen field basis, allowing systematic exploration of the space of interactions. Our computations agree with the numerical results for both cubic and quartic potential.