Operational Discrete Symmetries and CP-Violation

TL;DR

This paper explores the mathematical structure of discrete symmetries like CP, T, and CPT within the Lorentz group framework, revealing their complex interactions and implications for gauge symmetries, especially in the context of the Standard Model.

Contribution

It extends the understanding of discrete symmetries by defining their operations on spinor fields and Minkowski space, and analyzes their compatibility with gauge groups such as SU(n).

Findings

CP and T are not compatible with SU(n) for n > 2.

The symmetry operations extend SL(2,C) actions to O(1,3).

CP-violation can be distinguished through symmetry analysis.

Abstract

The discrete symmetries of the Lorentz group are on the one hand a `complex' interplay between linear and anti-linear operations on spinor fields and on the other hand simple linear reflections of the Minkowski space. We define operations for T, CP and CPT leading to both kinds of actions. These operations extend the action of SL(2,C), representing the action of the proper orthochronous Lorentz group SO^+(1,3) on the Weyl spinors, to an action of the full group O(1,3). But it is more instructive to reverse the arguments. The action of O(1,3) is the natural way how SL(2,C) together with its conjugation structure acts on Minkowski space. Focusing on the symmetries of these (anti-)linear operations we can for example distinguish between CP-invariant and CP-violating symmetries. This is important if gauge symmetries are included. It turns out that, contrary to the general belief, CP and T are not compatible with SU(n) for n > 2, especially with colour-SU(3) or with the U(3)-Cabibbo-Kobayashi-Maskawa matrix.