Classification of nonsupersymmetric Pati-Salam heterotic string models

We extend the classification of fermionic ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ heterotic string orbifolds to nonsupersymmetric Pati-Salam models in two classes of vacua, that we dub $\stackrel{\texttildelow{}}{S}$ models and $S$ models. The first corresponds to compactifications of a tachyonic ten-dimensional vacuum, whereas the second corresponds to compactifications of the ten-dimensional tachyon-free $SO(16)\ifmmode\times\else\texttimes\fi{}SO(16)$ heterotic string. In both cases we develop a systematic method to extract tachyon-free four-dimensional models. We show that tachyon-free configurations arise with probability $\ensuremath{\sim}0.002$ and $\ensuremath{\sim}0.01$ in the first and second case, respectively. We adapt the `fertility methodology' that facilitates the extraction of phenomenological models. We show that Pati-Salam $\stackrel{\texttildelow{}}{S}$ models do not contain heavy Higgs scalar representations that are required to break the PS symmetry to the Standard Model and are therefore not phenomenologically viable. Hence, we argue that in $\stackrel{\texttildelow{}}{S}$ models the $SO(10)$ grand unified theory symmetry must be broken at the string scale to the Standard-like Model subgroup. We extract tachyon--free three generation models in both cases that contain an equal number of massless bosonic and fermionic degrees of freedom i.e., with ${a}_{00}={N}_{b}^{0}\ensuremath{-}{N}_{f}^{0}=0$, and analyze their one-loop partition function.