Discrete Symmetries Excluded by Quantum Breaking

In this note we show that the cosmological domain wall and the de Sitter quantum breaking problems complement each other in theories with discrete symmetries that are spontaneously broken at low energies. Either the symmetry is exact and there is a domain wall problem, or it is approximate and there exists an inconsistent de Sitter minimum. This leaves no room for many extension of the Standard Model based on such discrete symmetries. We give some examples that include NMSSM, spontaneous CP violation at the weak scale and some versions of the Peccei-Quinn scenario with discrete symmetries.