Cobimaximal lepton mixing from soft symmetry breaking

TL;DR

This paper explores the mathematical structure behind cobimaximal lepton mixing, demonstrating how it can be derived from symmetry principles and presenting models with softly broken symmetries to realize this mixing pattern.

Contribution

It provides a new theorem linking cobimaximal mixing matrices to specific unitary and orthogonal matrices, and shows the equivalence of different symmetry-based model constructions.

Findings

Proves a theorem relating cobimaximal mixing to unitary and orthogonal matrices.

Demonstrates equivalence of two symmetry-based approaches to model cobimaximal mixing.

Constructs two seesaw models with softly broken symmetries and discusses Higgs accommodation.

Abstract

Cobimaximal lepton mixing, i.e. $\theta_{23} = 45^\circ$ and $\delta = \pm 90^\circ$ in the lepton mixing matrix $V$, arises as a consequence of $S V = V^\ast \mathcal{P}$, where $S$ is the permutation matrix that interchanges the second and third rows of $V$ and $\mathcal{P}$ is a diagonal matrix of phase factors. We prove that any such $V$ may be written in the form $V = U R P$, where $U$ is any predefined unitary matrix satisfying $S U = U^\ast$, $R$ is an orthogonal, i.e. real, matrix, and $P$ is a diagonal matrix satisfying $P^2 = \mathcal{P}$. Using this theorem, we demonstrate the equivalence of two ways of constructing models for cobimaximal mixing---one way that uses a standard $CP$ symmetry and a different way that uses a $CP$ symmetry including $\mu$--$\tau$ interchange. We also present two simple seesaw models to illustrate this equivalence; those models have, in addition to the $CP$ symmetry, flavour symmetries broken softly by the Majorana mass terms of the right-handed neutrino singlets. Since each of the two models needs four scalar doublets, we investigate how to accommodate the Standard Model Higgs particle in them.