Asymptotically free and safe fate of symmetry nonrestoration

We investigate the high temperature fate of four dimensional gauge-Yukawa theories featuring short distance conformality either interacting or non-interacting nature. The latter is known as complete asymptotic freedom and, templates, we consider non-abelian gauge two singlet scalars coupled to gauged fermions via Yukawa interactions with(out) fermions. For with fixed points at distance, asymptotically safe, calculable examples. Exploring landscape safe and free above discover a class for which symmetry breaks arbitrary temperatures. In its minimal form this constituted by theory fundamental each under an independent group.