Ultraviolet divergences in non-renormalizable supersymmetric theories

TL;DR

This paper reviews the ultraviolet divergence structure of non-renormalizable supersymmetric theories, highlighting the role of on-shell invariance of counterterms and the full effective Lagrangian's symmetry.

Contribution

It provides a pedagogical overview of the UV behavior and counterterm structure in extended supersymmetric theories, emphasizing on-shell invariance and the full effective Lagrangian.

Findings

Counterterms may not be off-shell invariant under supersymmetry.

The full effective Lagrangian maintains the original symmetry.

Higher-dimensional counterterms contribute to scattering amplitudes.

Abstract

We present a pedagogical review of our current understanding of the ultraviolet structure of N = (1,1) 6D supersymmetric Yang-Mills theory and of N = 8 4D supergravity. These theories are not renormalizable, they involve power ultraviolet divergences and, in all probability, an infinite set of higher-dimensional counterterms that contribute to on-mass-shell scattering amplitudes. A specific feature of supersymmetric theories (especially, of extended supersymmetric theories) is that these counterterms may not be invariant off shell under the full set of supersymmetry transformations. The lowest-dimensional nontrivial counterterm is supersymmetric on shell. Still higher counterterms may lose even the on-shell invariance. On the other hand, the full effective Lagrangian, generating the amplitudes and representing an infinite sum of counterterms, still enjoys the complete symmetry of original theory. We also discuss simple supersymmetric quantum-mechanical models that exhibit the same behaviour.