Spontaneous Discrete Symmetry Breaking During Inflation and the NMSSM Domain Wall Problem

The Next to Minimal Supersymmetric Standard Model (NMSSM), proposed as a solution of the mu problem of the Minimal Supersymmetric Standard Model, has a discrete Z_{3} symmetry which is spontaneously broken at the electroweak phase transition, resulting in a cosmological domain wall problem. In most cases this domain wall problem cannot be solved by explicit Z_{3} breaking without introducing supergravity tadpole corrections which destabilize the weak scale hierarchy. Here we consider the possibility of solving the domain wall problem of the NMSSM via spontaneous discrete symmetry breaking occuring during inflation. For the case where the discrete symmetry breaking field has renormalizible couplings to the NMSSM fields, we find that the couplings must be less than 10^{-5} if the reheating temperature is larger than 10^{7}GeV, but can be up to 10^{-3} for reheating temperatures of the order of the electroweak phase transition temperature. For the case of non-renormalizible couplings, we present a model which can solve the domain wall problem for large reheating temperatures without requiring any very small coupling constants. In this model the domain walls are eliminated by a pressure coming from their interaction with a coherently oscillating scalar field whose phase is fixed during inflation. This oscillating scalar field typically decays after the electroweak phase transition but before nucleosynthesis, leaving no additional Z_{3} symmetry breaking in the zero-temperature theory.