$Z_2$ Non-Restoration and Composite Higgs: Singlet-Assisted Baryogenesis w/o Topological Defects

Simple scalar-singlet extensions of the Standard Model with a (spontaneously broken) $Z_2$ symmetry allow for a strong first order electroweak phase transition, as sought in order to realize electroweak baryogenesis. However they generically also lead to the emergence of phenomenologically problematic domain walls. Here we present a framework with a real scalar singlet that features a different thermal history that avoids this problem by never restoring the $Z_2$ symmetry in the early universe. This is accomplished by considering $D>4$ operators that emerge on general grounds, understanding the model as the low energy tail of a more complete theory, like for example in composite Higgs scenarios. Sticking to the latter framework, we present a concrete $SO(6)/SO(5)$ composite realization of the idea. To this end, we additionally provide a complete classification of the structure of the Higgs potential (and the Yukawa couplings) in $SO(6)/SO(5)$ models with fermions in the ${\bf 1, 6, 15}$ or ${\bf 20^\prime}$ of $SO(6)$.