Cobimaximal lepton mixing from soft symmetry breaking

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 $SV=V^{\ast }\mathcal{P}$, 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 $V=URP$, where $U$ is any predefined unitary matrix satisfying $SU=U^{\ast }$, $R$ 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.

16 pages, no figures

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