Kähler Corrections for the Volume Modulus of Flux Compactifications

No-scale models arise in many compactifications of string theory and supergravity, the most prominent recent example being type IIB flux compactifications. Focussing on the case where the no-scale field is a single unstabilized volume modulus (radion), we analyse the general form of supergravity loop corrections that affect the no-scale structure of the Kähler potential. These corrections contribute to the 4d scalar potential of the radion in a way that is similar to the Casimir effect. We discuss the interplay of this loop effect with string-theoretic α corrections and its possible role in the stabilization of the radion. In flux compactifications of type IIB supergravity, all complex structure moduli and the dilaton are generically fixed by the non-trivial superpotential induced by the 3form field strength [1, 2]. However, this superpotential is independent of the Kähler moduli. Even if supersymmetry is broken by the non-zero vacuum expectation value of the superpotential W , one of the flat directions associated with the Kähler moduli survives. The resulting 4d model is of no-scale type and the no-scale field T is the Kähler modulus related to an overall rescaling of the compact volume. Perturbative corrections generically renormalize the Kähler potential, destroy the no-scale structure and lift the flat directions. We will be interested in loop corrections to the no-scale Kähler potential of the volume modulus T (the radion). In the large-volume limit, such corrections should be calculable within the low-energy effective field theory. They are potentially relevant for the stabilization of the radion and the uplifting to a metastable de Sitter vacuum [3,4]. To understand the supergravity 1-loop corrections, we first focus on a situation where W = 0 and supersymmetry is unbroken. We consider the corrections to the radion kinetic term and to the Einstein-Hilbert term of the 4d effective theory. Before Weyl rescaling, these corrections are independent of the 10d Planck mass and their form can therefore be inferred from dimensional arguments. It is then straightforward to derive the corresponding Kähler corrections, which are of the form 1/(T + T̄ ), with the compact volume scaling as V ∼ (ReT ). After a small non-zero W has been introduced as a perturbation, they induce a potentially important contribution to the radion scalar potential. For the purpose of our technical discussion, we first adopt a slightly more general perspective. Consider a d-dimensional supergravity theory, compactified to 4d on a kdimensional manifold (d = 4 + k) for which its total volume V corresponds to a flat direction. We write the metric as ds = gμνdx dx +R(x)g̃mndy dy , (1) where Greek and Latin indices run over 0...3 and 5...d respectively and the decomposition gmn = R g̃mn is defined in such a way that the volume of the compact space measured with the metric g̃mn is 1. The physical volume is V = R . In spite of its various interesting physical effects [2], we neglect for simplicity the possible warp factor, i.e., we assume that gμν does not depend on y. This is justified in the large volume limit. Assuming that the fundamental d-dimensional Einstein-Hilbert term has coefficient M/2, the 4d action reads