Supersymmetry counterterms revisited

TL;DR

This paper revisits the onset of ultraviolet divergences in non-renormalisable supersymmetric theories and revises the standard power‑counting using harmonic superspace. By allowing larger off‑shell supersymmetry (M) to be manifest, it derives higher‑derivative counterterms, predicting later divergence onset and achieving agreement with unitarity calculations for maximally supersymmetric theories. Specifically, it shows $D=5$ maximal SYM diverges at six loops and argues that $D=4$ $N=8$ supergravity could diverge no earlier than five to six loops, depending on the off‑shell SUSY that can be maintained. The discussion highlights the potential existence of off‑shell formulations with $M=6$–$7$ and the broader implications for quantum gravity and M‑theory.

Abstract

Superspace power-counting rules give estimates for the loop order at which divergences can first appear in non-renormalisable supersymmetric field theories. In some cases these estimates can be improved if harmonic superspace, rather than ordinary superspace, is used. The new estimates are in agreement with recent results derived from unitarity calculations for maximally supersymmetric Yang-Mills theories in five and six dimensions. For N=8 supergravity in four dimensions, we speculate that the onset of divergences may correspondingly occur at the six loop order.