CP and other Symmetries of Symmetries

Outer automorphisms of symmetries (``symmetries of symmetries'') in relativistic quantum field theories are studied, including charge conjugation (C), space-reflection (P) , and time-reversal (T) transformations. It is shown that certain discrete groups can predict parameter independent CP violating complex phases with fixed geometrical values. The study of outer automorphisms which are not related to C, P, or T is pioneered. It is shown how outer automorphisms, in general, relate symmetry invariants and imply relations between distinct vacuum expectation values. Thereby, outer automorphisms can give rise to emergent symmetries. This underlies the prediction of spontaneous geometrical CP violation in a three Higgs doublet model. The possible relevance of outer automorphisms for a wide field of future studies is highlighted.