# CP and other Symmetries of Symmetries

## TL;DR

This paper explores the role of outer automorphisms, including CP, P, and T transformations, in relativistic quantum field theories, revealing how they can predict CP violation and lead to emergent symmetries and fixed vacuum expectation values.

## Contribution

It introduces the group theory of outer automorphisms in quantum field theories and demonstrates their implications for CP violation and emergent symmetries, including spontaneous geometrical CP violation.

## Key findings

- Outer automorphisms are special symmetries of symmetries in quantum field theories.
- Certain discrete groups predict fixed, parameter-independent CP violating phases.
- Outer automorphisms can fix Higgs VEVs and induce spontaneous geometrical CP violation.

## Abstract

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. The group theory of outer automorphisms is pedagogically introduced and it is shown that CP transformations are special outer automorphisms of the global, local, and space-time symmetries of a theory. It is shown that certain discrete groups allow for a group theoretical prediction of parameter independent CP violating complex phases with fixed geometrical values. The remainder of this thesis pioneers the study of outer automorphisms which are not related to C, P, or T. It is shown how outer automorphisms, in general, relate symmetry invariants and, in theories with spontaneous symmetry breaking, imply relations between different vacuum expectation values. Thereby, outer automorphisms can give rise to emergent symmetries. An example model with a discrete symmetry and three copies of the Standard Model Higgs field is discussed, in which the rich outer automorphism structure completely fixes the Higgs VEVs in their field space direction, including CP violating relative phases. This underlies the phenomenon of spontaneous geometrical CP violation. The possible relevance of outer automorphisms for a wide field of future studies is highlighted.