Electroweak breaking and the $\mu$ problem in supergravity models with an additional U(1)

We consider electroweak symmetry breaking in supersymmetric models with an extra nonanomalous ${\mathrm{U}\left(1\right)\mathrm{}}^{\prime }$ gauge symmetry and an extra standard-model singlet scalar $S$. For appropriate charges the ${\mathrm{U}\left(1\right)\mathrm{}}^{\prime }$ forbids an elementary $\mu$ term, but an effective $\mu$ is generated by the VEV of $S$, leading to a natural solution to the $\mu$ problem. There are a variety of scenarios leading to acceptably small $Z$-$Z^{\prime }$ mixing and other phenomenological consequences, all of which involve some but not excessive fine-tuning. One class, driven by a large trilinear soft supersymmetry-breaking term, implies small mixing, a light $Z^{\prime }$ (e.g., 200 GeV), and an electroweak phase transition that may be first order at the tree level. In another class, with $m_S^2<0\mathrm{}$ (radiative breaking), the typical scale of dimensional parameters, including $M_{Z^{\prime }}$ and the effective $\mu$, is $\sim 1\mathrm{TeV}$, but the electroweak scale is smaller due to cancellations. We relate the soft supersymmetry-breaking parameters at the electroweak scale to those at the string scale, choosing Yukawa couplings as determined within a class of string models. We find that one does not obtain either scenario for universal soft supersymmetry-breaking mass parameters at the string scale and no exotic multiplets contributing to the renormalization group equations. However, either scenario is possible when the assumption of universal soft breaking is relaxed. Radiative breaking can also be generated by exotics, which are expected in most string models.