D-term Uplifts in Non-Supersymmetric Heterotic String Models

Recently, we proposed that the one-loop tadpole diagram in perturbative non-supersymmetric heterotic string vacua that contain an anomalous $U(1)$ symmetry, leads to an analog of the Fayet-Iliopoulos $D$-term in $\mathcal{N}=1$ supersymmetric models, and may uplift the vacuum energy from negative to positive value. In this paper, we extend this analysis to new types of vacua, including those with Stringy Scherk-Schwarz (SSS) spontaneous supersymmetry breaking versus those with explicit breaking. We develop a criteria that facilitates the extraction of vacua with Scherk-Schwarz breaking. We develop systematic tools to analyse the T-duality property of some of the vacua and demonstrate them in several examples. The extraction of the anomalous $U(1)$ $D$-terms is obtained in two ways. The first utilises the calculation of the $U(1)$-charges from the partition function, whereas the second utilises the free fermionic classification methodology to classify large spaces of vacua and analyse the properties of the massless spectrum. The systematic classification method also ensures that the models are free from physical tachyons. We provide a systematic tool to relate the free fermionic basis vectors and one-loop Generalised GSO phases that define the string models, to the one-loop partition function in the orbifold representation. We argue that a $D$-term uplift, while rare, is possible for both the SSS class of models, as well as in those with explicit breaking. We discuss the steps needed to further develop the arguments presented here.