F-term uplifting via consistent D-terms
Z. Lalak, O. J. Eyton-Williams, R. Matyszkiewicz
TL;DR
This work addresses the challenge of obtaining a small cosmological constant while maintaining a hierarchical separation between moduli masses and the gravitino in supergravity with consistent D-terms. It advances a refined F-term uplifting mechanism within a racetrack framework of two condensing sectors, complemented by a spectator charged field, so SUSY breaking and moduli stabilisation are inseparable yet yield $V o 0$ with a small $m_{3/2}$. The key distinction is between cancellable and non-cancellable D-terms: in the former, tuning occurs mainly in the constant superpotential term $W_0$, while in the latter tuning involves $U(1)$ charges and the D-term strength; in both cases, the D-term vanishes at the minimum and the negative supergravity term is cancelled by F-terms. All relevant scales are dynamically generated by field-dependent gauge couplings, with $W=W_0+W(T)$ and $V=V_F+V_W+V_D$, enabling near-vanishing vacuum energy without large hierarchies in input parameters. The results provide a concrete framework for controlled uplift in string-inspired supergravity models and illuminate how D-terms can shape the moduli mass spectrum and vacuum structure.
Abstract
The issue of fine-tuning necessary to achieve satisfactory degree of hierarchy between moduli masses, the gravitino mass and the scale of the cosmological constant has been revisited in the context of supergravities with consistent D-terms. We have studied (extended) racetrack models where supersymmetry breaking and moduli stabilisation cannot be separated from each other. We show that even in such cases the realistic hierarchy can be achieved on the expense of a single fine-tuning. The presence of two condensates changes the role of the constant term in the superpotential, W_0, and solutions with small vacuum energy and large gravitino mass can be found even for very small values of W_0. Models where D-terms are allowed to vanish at finite vevs of moduli fields - denoted `cancellable' D-terms - and the ones where D-terms may vanish only at infinite vevs of some moduli - denoted `non-cancellable' - differ markedly in their properties. It turns out that the tuning with respect to the Planck scale required in the case of cancellable D-terms is much weaker than in the case of non-cancellable ones. We have shown that, against intuition, a vanishing D-term can trigger F-term uplifting of the vacuum energy due to the stringent constraint it imposes on vacuum expectation values of charged fields. Finally we note that our models only rely on two dimensionful parameters: M_P and W_0.