# Generalised P and CP transformations in the 3-Higgs-doublet model

## TL;DR

This paper investigates generalized P and CP symmetries in the three-Higgs-doublet model, classifying their equivalence classes, conditions for invariance, and extending the analysis to n-Higgs-doublet models with explicit matrix forms.

## Contribution

It classifies generalized P and CP transformations in 3HDM, relates them to potential invariance, and extends the analysis to nHDM with explicit transformation matrices.

## Key findings

- Two equivalence classes of generalized P transformations in 3HDM.
- Only one class of CP transformations in 3HDM.
- Explicit form of standard CP transformation matrices for nHDM.

## Abstract

We study generalised P and CP transformations in the three-Higgs-doublet model (3HDM) with Higgs and gauge fields only. We find that there are two equivalence classes, with respect to flavour transformations, of generalised P transformations and there is only one class of CP transformations. We discuss the conditions the potential has to satisfy in order to be invariant under these transformations. We apply the method of bilinears which we briefly review. We discuss the relation to the conventional basis, where the potential is written in terms of scalar products of the doublet fields. In particular we reproduce the known result that a potential is invariant under CP transformations if and only if there is a conventional basis where all parameters are real. Eventually we study standard P and CP transformations in the $n$-Higgs-doublet model (nHDM). We show that for the bilinears of the nHDM the standard CP transformation corresponds to a diagonal linear transformation with only $\pm 1$ as diagonal elements. We give this matrix explicitly for arbitrary $n$.