Fayet-Iliopoulos D-Term in Non-Supersymmetric Heterotic String Orbifolds

The Fayet-Iliopoulos $D$-term is a common feature in $\mathcal{N}=1$ string vacua that contain an anomalous $U(1)$ gauge symmetry, and arises from a one--loop diagram in string perturbation theory. The same diagram is generated in string vacua in which supersymmetry is broken directly at the string scale, either via spontaneous Scherk-Schwarz breaking, in which case the gravitino mass is determined by the radius of the circle used in the Scherk-Schwarz mechanism, or via explicit supersymmetry breaking by the GSO projections. We analyse the resulting would-be Fayet-Illiopoulos $D$-term in the non-supersymmetric string vacua and its contribution to the vacuum energy. A numerical estimate in an explicit tachyon-free string-derived model suggests that the would-be $D$-term contribution may uplift the vacuum energy to a positive value.