Nonrestoration of spontaneously broken [Formula presented] and [Formula presented] at high temperature

The possibility of [Formula presented] and [Formula presented] violation at high temperature in models where these symmetries are spontaneously broken is investigated. It is found that in minimal models that include singlet fields, high [Formula presented] nonrestoration is possible for a wide range of parameters of the theory, in particular, in models of [Formula presented] violation with a [Formula presented]-odd Higgs field. The same holds true for the invisible axion version of the Peccei-Quinn mechanism. This can provide both a way out for the domain wall problem in these theories and the [Formula presented] violation required for baryogenesis. In the case of spontaneous [Formula presented] violation it turns out that high [Formula presented] nonrestoration requires going beyond the minimal model. The results are shown to hold true when next-to-leading order effects are considered.