Discriminating between $CP$ and family transformations in the bilinear space of NHDM

TL;DR

This paper develops a new method to distinguish between generalized CP transformations and Higgs family transformations in the bilinear space of NHDM, extending previous criteria to any number of Higgs doublets.

Contribution

It introduces a novel quantity that discriminates GCPs from Higgs family transformations for any N, and establishes conditions for orthogonal transformations to correspond to valid NHDM transformations.

Findings

New discriminant for GCP vs. Higgs transformations in bilinear space

Criteria applicable for any number of Higgs doublets

Conditions for orthogonal transformations to represent NHDM symmetries

Abstract

The scalar potential of the N-Higgs-doublet model (NHDM) is best analyzed not in the space of N complex doublets $\phi_a$ but in the $N^2$-dimensional space of real-valued bilinears constructed of $\phi_a^\dagger \phi_b$. In particular, many insights have been gained into CP violation in the 2HDM and 3HDM by studying how generalized CP transformations (GCPs) act in this bilinear space. These insights relied on the fact that GCPs, which involved an odd number of mirror reflection, could be clearly distinguished from Higgs family transformations by the sign of the determinant of the transformation matrix. It was recently pointed out that this criterion fails starting from 4HDM, where the reflection/rotation dichotomy does not exist anymore. In this paper, we restore intuition by finding a different quantity which faithfully discriminates between GCPs and Higgs family transformations in the bilinear space for any number of Higgs doublets. We also establish the necessary and sufficient conditions for an orthogonal transformation in the bilinear space to represent a viable transformation back in the space of N doublets, which is helpful if one prefers to build an NHDM directly in the bilinear space.