An effective description of the landscape - I

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 = W-0 (H) + is an element of W-1 (H, L), with is an element of << 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 H-0, 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 W-0 is required to be of order is an element of 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 W-0 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 W-0 is small enough.