Possible resolution of the domain wall problem in the NMSSM

We discuss a possibility that the domain wall problem in the next-to-minimal supersymmetric standard model is alleviated without introducing a small explicit $Z_3$ breaking term by analyzing the evolution of the singlet scalar field within an inflationary paradigm. The singlet scalar field which explains the $\mu$-term tracks a time-varying minimum of the effective potential after inflation and slowly rolls down to its global minimum if there exist sufficiently large negative Hubble-induced corrections on the effective potential for the singlet field, which arise through supergravity. As a consequence, the whole Universe is confined within a single domain during and after inflation, which prevents the formation of domain walls. This will further constrain the history of the early Universe along with the Higgs-singlet coupling.