Discriminating between CP and family transformations in the bilinear space of the N -Higgs-doublet model

Igor P Ivanov,Celso C Nishi

doi:10.1103/physrevd.101.015023

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.

Highlights

N-Higgs-doublet models (NHDM) are a popular conservative framework of building models beyond the standard model
These insights relied on the fact that generalized CP transformations (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
Despite its complexity in the 3HDM as compared to the 2HDM, one still observes the discriminating role of the determinant: all Higgs family transformations are described in the 1 þ 8-dimensional bilinear space by pure rotations R ∈ SOð8Þ with det R 1⁄4 þ1, while all GCPs induce transformations C, which involve an odd number of reflections, so that det C 1⁄4 −1

Summary

INTRODUCTION

N-Higgs-doublet models (NHDM) are a popular conservative framework of building models beyond the standard model (bSM). Three Higgs doublets allow one to implement a novel type of CP symmetry of order 4 [9,25], and a basis-independent algorithm for detection of CP4 in 3HDM was given in [26] All these results were obtained only with the bilinear space formalism, stressing its superior role in analyzing the scalar sector of NHDMs. Despite its complexity in the 3HDM as compared to the 2HDM, one still observes the discriminating role of the determinant: all Higgs family transformations are described in the 1 þ 8-dimensional bilinear space by pure rotations R ∈ SOð8Þ with det R 1⁄4 þ1, while all GCPs induce transformations C, which involve an odd number of reflections, so that det C 1⁄4 −1.

The problem

A resolution

Which transformations can be used in the bilinear space?

CONCLUSIONS