$\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}^{\prime }$ symmetry breaking in supersymmetric ${\mathrm{E}}_6$ models

We study the electroweak and $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}^{\prime }$ symmetry breaking patterns in models with the particle content of supersymmetric ${\mathrm{E}}_6,$ including standard model singlets $S$ and exotic quarks $D,\hspace{0.5em}\overline{D}.$ Motivated by free fermionic string models, we do not require ${\mathrm{E}}_6$-type relations between Yukawa couplings. In particular, we assume that baryon and lepton numbers are conserved, so that the exotic quarks can be light. Gauge invariance allows Yukawa interactions between $S$ and Higgs doublets, and between $S$ and the exotic quarks, allowing radiative $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}^{\prime }$ symmetry breaking and the generation of an effective $\mu$ parameter at the electroweak scale. For both the ${\mathrm{E}}_6\hspace{0.5em}\psi$ and $\eta$ models, universal soft supersymmetry breaking parameters and Yukawa universality at the high (string) scale do not yield acceptable low energy phenomenology. Relaxing universality, we find solutions with phenomenologically acceptable values of $M_{Z^{\prime }}$ and the $Z-Z^{\prime }$ mixing angle. In addition, by varying the $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}^{\prime }$ charge assignments due to the mixing of $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{\chi }$ and $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{\psi }$ of ${\mathrm{E}}_6,$ it is possible to have acceptable low energy phenomenology with universal boundary conditions.