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 non-trivial 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 non-conserved but non-renormalized vector operator called virial current, whose existence must be non-generic. 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 non-renormalization property.

6+2 pages, v2: appendix added, reference added