Scalar Potentials and Accidental Symmetries in Supersymmetric U(1)' Models

We address two closely related problems associated with the singlet scalars' potential that are often present in supersymmetric U(1)' models, especially those which maintain the gauge unification of the MSSM in a simple way. The first is the possibility of an accidental global symmetry which results in a light Goldstone boson. The second is the problem of generating a vacuum expectation value for more than one field without reintroducing the $\mu$ problem. We give sufficient conditions for addressing both issues and provide a concrete example to generate them.