Effective potentials for supersymmetric three-scale hierarchies

We consider the effective potential in models in which supersymmetry breaks at a scale μ but the Goldstone fermion couples only to fields of mass M>>μ. We show that all large perturbative logarithms are removed by taking the renormalization point to be O(M). This makes it possible to calculate the effective potential at large X in those inverted-hierarchy models where the Goldstone fermion couples only to superheavy fields. A general formula for the one-loop logarithm in these models is given. We illustrate the results with an SU(n) example in which the direction as well as the magnitude of the gauge symmetry breaking is undetermined at the tree level. For this example a large perturbative hierarchy does not form and the unbroken subgroup is always SU(n-1) x U(1). In an appendix we show that O'Raifeartaigh models with just one undetermined scalar field always have a decoupled Goldstone fermion when the undetermined field is large, but that this need not be true in more general inverted-hierarchy models