Spontaneous breaking of global symmetries in supersymmetric theories

The analogue of Goldstone’s theorem in supersymmetric theories is proven by examining the properties of the superpotential. The method is extended to the case in which a global U(1) symmetry is broken at a scale M and supersymmetry is spontaneously broken at a scale Λ≪M. We find that the fermion and scalar superpartners of the Goldstone boson acquire masses which can be of order ${\Lambda }^2$/M and Λ, respectively, but not larger. Under certain conditions, for example, if the D terms are zero or if the Goldstone boson is a gauge singlet, the scalar mass also is of order ${\Lambda }^2$/M at most. The results are generalized to the case in which the global U(1) is an R symmetry and the case of explicit, softly broken supersymmetry. Simple models that illustrate some of these results are examined and other related issues are considered.