Naive dimensional analysis and supersymmetry

Abstract

In strongly coupled theories with no small parameters, there are factors of 4\ensuremath{\pi} that appear when the couplings of the low-energy effective Lagrangian are written in units of the effective cutoff \ensuremath{\Lambda}. These numerical factors can be explained using ``naive dimensional analysis.'' We extend these ideas to supersymmetric theories, and show how to systematically include small parameters and couplings to weakly interacting fields. The basic principle is that, if the fundamental theory is strongly coupled, then the effective theory must also be strongly coupled at the scale \ensuremath{\Lambda}. We use our results to analyze several examples where strong supersymmetric dynamics may be relevant for phenomenology. For models that break supersymmetry through strong dynamics with no small parameters, we show that the Goldstino decay constant $F$ is given by $F\ensuremath{\sim}{\ensuremath{\Lambda}}^{2}/(4\ensuremath{\pi}).$ We also consider theories with standard-model gauge bosons coupled directly to strong supersymmetry-breaking dynamics near the weak scale, smoothly confining theories, and a model that breaks supersymmetry through the mechanism of a deformed moduli space.