One-Loop Effective Potential in Scherk–Schwarz Compactifications of Pure d=5 Supergravities

We perform a systematic analysis of the one-loop effective potential of pure (Formula presented.) supergravities, with supersymmetry fully broken by a Scherk–Schwarz compactification on the circle, as a function of the radial modulus. We discuss the precise correspondence between the effective potential (Formula presented.) in the full compactified theory and its counterpart (Formula presented.) in the reduced theory. We confirm that (Formula presented.) is finite for any (Formula presented.), in contrast to (Formula presented.). We find that for broken (Formula presented.) supergravity (Formula presented.) is negative definite even after accounting for the Kaluza–Klein states. We outline a program for future work where the study of a different kind of Scherk-Schwarz compactifications, still at the field theory level but with at least three extra dimensions, could lead to qualitatively new results.