Intermediate scales, μ parameter, and fermion masses from string models

We address intermediate scales within a class of string models. The intermediate scales occur due to the SM singlets $S_i$ acquiring non-zero VEVs due to radiative breaking; the mass squared $m_i^2$ of $S_i$ is driven negative at ${\mu }_{\mathrm{rad}}$ due to O(1) Yukawa couplings of $S_i$ to exotic particles (calculable in a class of string models). The actual VEV of $S_i$ depends on the relative magnitude of the non-renormalizable terms of the type $S_i^{K+3\mathrm{}}{/M}^K$ in the superpotential. We mainly consider the case in which the $S_i$ are charged under an additional non-anomalous ${U\left(1\right)\mathrm{}}^{\prime }$ gauge symmetry and the VEVs occur along $F$- and $D$-flat directions. We explore various scenarios in detail, depending on the type of Yukawa couplings to the exotic particles and on the initial boundary values of the soft SUSY breaking parameters. We then address the implications of these scenarios for the μ parameter and the fermionic masses of the standard model.