Is There Scale Invariance in N=1 Supersymmetric Field Theories ?

TL;DR

This paper investigates whether scale invariance in four-dimensional N=1 supersymmetric theories necessarily implies conformal invariance, revealing conditions under which they are equivalent or distinct.

Contribution

It constructs the SUSY dilatation current using the FZ-multiplet and clarifies when scale invariance implies conformal invariance in 4d N=1 SUSY theories.

Findings

Scale invariance implies conformal invariance if the superfield in the dilatation current is conserved.

Without this conservation, theories are only scale-invariant, not conformally invariant.

The work extends understanding of symmetry relations in supersymmetric quantum field theories.

Abstract

In two dimensions, it is well known that the scale invariance can be considered as conformal invariance. However, there is no solid proof of this equivalence in four or higher dimensions. We address this issue in the context of 4d $\mathcal{N}=1$ SUSY theories. The SUSY version of dilatation current for theories without conserved $R$ symmetry is constructed through the FZ-multiplet. We discover that the scale-invariant SUSY theory is also conformal when the real superfield in the dilatation current multiplet is conserved. Otherwise, it is only scale-invariant, despite of the transformation of improvement.