Mass formulae and natural hierarchy in string effective supergravities

TL;DR

This work seeks a natural explanation for $m_{3/2} \ll M_{\rm P}$ within string-derived effective supergravities by requiring the cancellation of one-loop quadratic divergences, quantified by $Q(z,\bar z)=0$, in tandem with a sliding gravitino mass under a vanishing cosmological constant. It develops general mass formulae in spontaneously broken $N=1$ supergravity and introduces Large Hierarchy Compatible (LHC) models whose Kähler potentials possess approximate scaling from target-space dualities, enabling simple, universal MSSM soft terms driven by scaling weights $\lambda$. The authors apply these ideas to string-theoretic contexts, including string-tree level breaking and non-perturbative breaking, showing how ${\cal G}^I {\cal G}_I =3$ and $Q=0$ can arise in Calabi-Yau, orbifold, and fermionic constructions, with explicit patterns for gaugino and scalar masses. The findings suggest that the hierarchy problem can be addressed within a coherent string-inspired SUGRA framework, linking duality-induced scaling to phenomenologically viable soft terms and offering directions for further investigation of higher-loop effects and threshold corrections.

Abstract

We study some conditions for the hierarchy $m_{3/2} << M_P$ to occur naturally in a generic effective supergravity theory. Absence of fine-tuning and perturbative calculability require that the effective potential has a sliding gravitino mass and vanishing cosmological constant, up to ${\cal O}(m_{3/2}^4)$ corrections. In particular, cancellation of quadratically divergent contributions to the one-loop effective potential should take place, including the `hidden sector' of the theory. We show that these conditions can be met in the effective supergravities derived from four-dimensional superstrings, with supersymmetry broken either at the string tree level via compactification, or by non-perturbative effects such as gaugino condensation. A crucial role is played by some approximate scaling symmetries, which are remnants of discrete target-space dualities in the large moduli limit. We derive explicit formulae for the soft breaking terms arising from this class of `large hierarchy compatible' (LHC) supergravities.