Remark on integrating out heavy moduli in flux compactification

TL;DR

The paper analyzes whether heavy moduli $S$ and $U^\alpha$ can be integrated out before stabilizing the Kähler modulus $T$ in type IIB flux compactifications with gaugino condensations. It derives a mass-hierarchy condition, $|W_{ab}| \gg m_{3/2}$ with $m_{3/2}=e^{K/2}|W|$, and a light $T$ mass $m_T \approx a m_{3/2}$, to justify a two-step procedure and discusses flux- and racetrack-based ways to realize this. It identifies scenarios where two-step stabilization is valid and where it fails, requiring simultaneous stabilization of multiple moduli or alternative superpotential designs. The appendix provides explicit minima for the simple $h_{1,1}=h_{2,1}=1$ case, supporting the general argument about heavy-moduli decoupling.

Abstract

We study two steps of moduli stabilization in type IIB flux compactification with gaugino condensations. We consider the condition that one can integrate out heavy moduli first with light moduli remaining. We give appendix, where detail study is carried out for potential minima of the model with a six dimensional compact space with $h_{1,1}=h_{2,1}=1$, including the model, whose respective moduli with $h_{1,1}, h_{2,1} \neq 1$ are identified.