Flat directions in de Sitter space.

Abstract

The one loop level effective potential along the flat direction in global supersymmetric models is evaluated in de Sitter space. In flat space the direction \ensuremath{\varphi} only has a curvature determined by the supersymmetry soft breaking mass term ${\mathit{m}}^{2}$${\mathrm{\ensuremath{\varphi}}}^{2}$ and in the supersymmetric limit m\ensuremath{\rightarrow}0 it receives no radiative corrections because of the nonrenormalization theorem. In this paper we show that in de Sitter space the one loop radiative correction induces a term \ensuremath{\sim}-3${\mathit{g}}^{2}$${\mathit{H}}^{2}$${\mathrm{\ensuremath{\varphi}}}^{2}$ ln(${\mathrm{\ensuremath{\varphi}}}^{2}$)/(16${\mathrm{\ensuremath{\pi}}}^{2}$) on the flat direction, where g is a typical coupling constant and H is the Hubble parameter. If the (renormalized) curvature coupling is nearly zero, \ensuremath{\xi}\ensuremath{\simeq}0, and the soft breaking mass is sufficiently small as ${\mathit{m}}^{2}$\ensuremath{\lesssim}${\mathit{g}}^{2}$${\mathit{H}}^{2}$/${\mathrm{\ensuremath{\pi}}}^{2}$, the correction may qualitatively modify the tree level potential and give an unbounded potential for the flat direction. We argue that this behavior is irrespective of the matter content and the details of the model if the flat direction couples to a scalar multiplet (as realistic supersymmetric models should do). The unbounded potential may give a large vacuum expectation value of \ensuremath{\varphi} in the inflationary expanding universe and be desirable for baryogenesis by the Affleck-Dine mechanism. Including the back reaction effect, the unbounded potential may also cause the adjustment mechanism for the cosmological constant problem.