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The Minimal SUSY $B-L$ Model: Simultaneous Wilson Lines and String Thresholds

In previous work, we presented a statistical scan over the soft supersymmetry breaking parameters of the minimal SUSY $B-L$ model. For specificity of calculation, unification of the gauge parameters was enforced by allowing the two ${\mathbb Z}_{3}\times {\mathbb Z}_{3}$ Wilson lines to have mass scales separated by approximately an order of magnitude. This introduced an additional "left-right" sector below the unification scale. In this paper, for three important reasons, we modify our previous analysis by demanding that the mass scales of the two Wilson lines be simultaneous and equal to an "average unification" mass $\left<M_{U}\right>$. The present analysis is 1) more "natural" than the previous calculations, which were only valid in a very specific region of the Calabi-Yau moduli space, 2) the theory is conceptually simpler in that the left-right sector has been removed and 3) in the present analysis the lack of gauge unification is due to threshold effects--particularly heavy string thresholds, which we calculate statistically in detail. As in our previous work, the theory is renormalization group evolved from $\left<M_{U}\right>$ to the electroweak scale--being subjected, sequentially, to the requirement of radiative $B-L$ and electroweak symmetry breaking, the present experimental lower bounds on the $B-L$ vector boson and sparticle masses, as well as the lightest neutral Higgs mass of $\sim$125 GeV. The subspace of soft supersymmetry breaking masses that satisfies all such constraints is presented and shown to be substantial.