  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 $\phi^3$ model but not for RF
$\phi^4$ 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
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.
