Parisi-Sourlas Supertranslation and Scale without Conformal symmetry

TL;DR

This paper investigates a supersymmetric field theory with Parisi-Sourlas supertranslation symmetry, revealing multiple scale-invariant fixed points and exploring the conditions under which scale invariance can exist without conformal invariance.

Contribution

It identifies nine non-trivial fixed points in a supersymmetric model and clarifies the role of supertranslation in protecting the virial current from renormalization.

Findings

Nine non-trivial scale-invariant fixed points found.

Only one fixed point exhibits conformal invariance.

Supertranslation Ward-Takahashi identity explains virial current non-renormalization.

Abstract

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 $\epsilon$ 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.