Parisi-Sourlas Supertranslation and Scale Invariance without Conformal Symmetry

Inspired by the possibility of emergent supersymmetry in critical random systems, we study a field theory model with a quartic potential of one superfield, possessing the Parisi-Sourlas supertranslation symmetry. Within perturbative ε expansion, we find nine nontrivial scale-invariant renormalization group fixed points, but only one of them is conformal. We, however, believe scale invariance without conformal invariance cannot occur without a sophisticated mechanism because it predicts the existence of a nonconserved but nonrenormalized vector operator called virial current, whose existence must be nongeneric. We show that the virial current in this model is related to the supercurrent by supertranslation. The supertranslation Ward-Takahashi identity circumvents the genericity argument, explaining its nonrenormalization property.