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) + epsilon W_1(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 H_0, if they approximately correspond to supersymmetric solutions along the H directions, independently of the form of the K"ahler potential K for H and L, provided K is sufficiently regular. In supergravity W_0 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 K"ahler 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 {f 68} (2003) 046005 by Kachru et al., for instance, is reliable to a very good accuracy, provided W_0 is small enough.