Searching singlet extensions of the supersymmetric standard model in $Z_{6-II}$ orbifold compactification of heterotic string

We search for supersymmetric standard model realizations with extra singlets and extra $U(1)$ using the heterotic string compactification on the $Z_{6-II}$ orbifold with two Wilson lines. We analyze the vacuum restabilization mechanism for three representative Pati-Salam string models obtained in the literature and present detailed results for the effective superpotential compatible with the string selection rules. An automated selection of semi-realistic vacua along flat directions in the non-Abelian singlet modes field space is performed by requiring the presence of massless pairs of electroweak Higgs bosons having trilinear superpotential couplings with massless singlet modes and the decoupling of color triplet exotic modes needed to suppress $B$ and $L$ number violating processes.