An Effective Description of the Landscape - I

TL;DR

The paper analyzes when heavy moduli in 4D ${\cal N}=1$ theories can be reliably frozen rather than integrated out, focusing on superpotentials of the form $W(H,L)=W_0(H)+\epsilon W_1(H,L)$ with $\epsilon\ll1$. In global SUSY, freezing $H$ to the leading SUSY solution $\partial_H W_0=0$ yields an effective theory for the light fields $L$ that matches the full theory to $O(\epsilon)$, irrespective of the Kähler potential, as long as $K$ is regular. In supergravity, a mass hierarchy and negligible backreaction require either $\langle W_0\rangle=O(\epsilon)$ or an almost factorable $K$, after which the light-sector EFT with $H$ frozen reproduces the full theory to leading order in $\epsilon$. The results justify the standard KKLT and large-volume approaches under the stated conditions and provide a general framework for freezing moduli in string-inspired 4D ${\cal N}=1$ SUGRA, with potential extensions to gauge sectors left for future work.

Abstract

We study under what conditions massive fields can be "frozen" rather than integrated out in certain four dimensional theories with global or local N=1 supersymmetry. We focus on models without gauge fields, admitting a superpotential of the form W = W0(H) + epsilon W1(H,L), with epsilon << 1, where H and L schematically denote the heavy and light chiral superfields. We find that the fields H can always be frozen to constant values H0, if they approximately correspond to supersymmetric solutions along the H directions, independently of the form of the Kahler potential K for H and L, provided K is sufficiently regular. In supergravity W0 is required to be of order epsilon at the vacuum to ensure a mass hierarchy between H and L. The backreaction induced by the breaking of supersymmetry on the heavy fields is always negligible, leading to suppressed F^H--terms. For factorizable Kahler potentials W0 can instead be generic. Our results imply that the common way complex structure and dilaton moduli are stabilized, as in Phys. Rev. D 68 (2003) 046005 by Kachru et al., for instance, is reliable to a very good accuracy, provided W0 is small enough.