log(M_Pl/m_3/2)

TL;DR

This paper addresses the challenge of obtaining the weak–gravity hierarchy within flux compactifications by replacing a quantized constant in the superpotential with a dynamical quantity provided by a continuous Chern–Simons term. The gaugino condensate then fixes the SUSY-breaking scale via dimensional transmutation while the continuous Chern–Simons term cancels the resulting vacuum energy, enabling a zero-energy starting point prior to SUSY breaking. The resulting framework yields a little hierarchy with $\langle X\rangle \sim \log(M_{Pl}/m_{3/2}) \approx 4\pi^2$ and a moduli/soft-term pattern $m_{modulus} \sim \langle X\rangle m_{3/2} \sim \langle X\rangle^2 m_{soft}$, characteristic of mirage mediation. These ideas have significant implications for phenomenology and cosmology, offering a distinct route to moduli stabilization and SUSY breaking that avoids fine-tuning of quantized constants in flux compactifications.

Abstract

Flux compactifications of string theory seem to require the presence of a fine-tuned constant in the superpotential. We discuss a scheme where this constant is replaced by a dynamical quantity which we argue to be a `continuous Chern--Simons term'. In such a scheme, the gaugino condensate generates the hierarchically small scale of supersymmetry breakdown rather than adjusting its size to a constant. A crucial ingredient is the appearance of the hierarchically small quantity exp(-<X>) which corresponds to the scale of gaugino condensation. Under rather general circumstances, this leads to a scenario of moduli stabilization, which is endowed with a hierarchy between the mass of the lightest modulus, the gravitino mass and the scale of the soft terms, m_modulus ~ <X> m_3/2 ~ <X>^2 m_soft. The `little hierarchy' <X> is given by the logarithm of the ratio of the Planck scale and the gravitino mass, <X> ~ log(M_Pl/m_3/2) ~ 4pi^2. This exhibits a new mediation scheme of supersymmetry breakdown, called mirage mediation. We highlight the special properties of the scheme, and their consequences for phenomenology and cosmology.