# CP-odd invariants for multi-Higgs models and applications with discrete symmetry

## TL;DR

This paper develops a systematic method to construct CP-odd invariants in multi-Higgs models with discrete symmetries, enabling the analysis of CP violation in complex scalar potentials.

## Contribution

It introduces a systematic approach to build all CP-odd invariants up to a certain order for multi-Higgs models with discrete symmetries, aiding CP violation analysis.

## Key findings

- Constructed basis invariants for scalar sectors.
- Distinguished CP-odd from CP-even invariants.
- Determined CP properties of potentials with multiple Higgs fields.

## Abstract

CP-odd invariants are useful for studying the CP properties of Lagrangians in any basis. We explain how to build basis invariants for the scalar sector, and how to distinguish CP-odd invariants from CP-even invariants. Up to a certain order, we use these methods to systematically build all the CP-odd invariants. The CP-odd invariants signal either explicit or spontaneous violation of CP. Making use of the CP-odd invariants, we determine the CP properties of potentials with 3 and with 6 Higgs fields arranged as triplets of specific discrete symmetries in the $\Delta(3n^2)$ or $\Delta(6n^2)$ series (inlcuding $A_4$, $S_4$, $\Delta(27)$ and $\Delta(54)$ as well as the cases for $n>3$).