Vacuum Energy of Non-Supersymmetric $\mathbf{\tilde{S}}$ Heterotic String Models

We use the free fermionic formulation of the heterotic-string in four dimensions to study the vacuum structure and energy of non-supersymmetric tachyon free models that correspond to compactifications of tachyonic vacua of the ten dimensional heterotic-string. We explore the class of heterotic $SO(10)$ non-supersymmetric models constructed from the $\tilde{S}$-model in the Free Fermionic Formalism, and investigate the dependence of the potential on the geometric moduli. This paper will explore a sample of $ 10^{9}$ string vacua to find the frequency of viable models, classifying these vacua by the following fertility criteria: tachyon presence; number of spinorial $\boldsymbol{16/\overline{16}}$ representations; vectorial $\boldsymbol{10}$ states; Top Quark Mass Coupling compatibility. Of these we find those that mimic supersymmetric models with equal number of bosons and fermions at the massless level - $a_{00} = 0$. Tachyon free models occur with a frequency of $5.309\times10^{-3}$. Furthermore, models that fulfil the rest of the phenomenological fertility conditions and the additional condition on $a_{00}$ occur with probability $4.0 \times 10^{-9}$ We analyse the partition functions and study the moduli dependence of such models, finding that almost all fertile models have finite, positive potential at the Free Fermionic Point, with $2$ out of $84$ of the fertile cores having negative, finite potential. We demonstratate that the Free Fermionic Point is not necessarily a minimum in the potential. This work provides further evidence that supersymmetry may not be a necessary ingredient of phenomenological models, recreating many of the desirable features of such models without employing supersymmetry.