Adding CP to flavour symmetries

I De Medeiros Varzielas

doi:10.1088/1742-6596/631/1/012020

Abstract

I propose the use of CP-odd invariants, which are independent of basis and valid for any choice of CP transformation, as a powerful approach to study CP in the presence of flavour symmetries. As examples of the approach I focus on Lagrangians invariant under Δ(27). I comment on the consequences of adding a specific CP symmetry to a Lagrangian and distinguish cases where several Δ(27) singlets are present depending on how they couple to the triplets. One of the examples included is a very simple toy model with explicit CP violation with calculable phases, which is referred to as explicit geometrical CP violation by comparison with previously known cases of (spontaneous) geometrical CP violation.

Highlights

This contribution to the proceedings of DISCRETE 2014 follows closely the layout of seminar I presented in the conference
The main conclusion to be drawn is that the invariant approach is a powerful method to study the CP properties of specific Lagrangians, in the presence of flavour symmetries
One can insert into the relevant CP-odd invariants the couplings that respect the flavour symmetry, and obtain a basis independent answer if CP is violated by those couplings

Summary

Introduction

This contribution to the proceedings of DISCRETE 2014 follows closely the layout of seminar I presented in the conference. I include here an expanded discussion of situations with ∆(27) singlets, including cases with explicit geometrical CP violation (first identified recently, in [1]). The invariant approach I refer to the Invariant Approach (IA) to CP [2] as an approach that starts by splitting the Lagrangian into LCP , a part that automatically conserves CP (e.g. kinetic terms, gauge interactions) and the remaining part Lrem.:. Impose the most general CP transformations (that leave LCP invariant). Them and see if it restricts Lrem. If the most general CP transformations restrict the shape of Lrem. An example of this type of restrictions is if the most general CP transformations force some coefficient to be real. Gets results just from the Lagrangian. Independent of basis. Shows relevant quantities for physical processes

The invariant approach for Standard Model leptons

Explicit geometrical CP violation

Conclusions