Cobimaximal lepton mixing from soft symmetry breaking

Cobimaximal lepton mixing, i.e. ${\theta }_{23}={45}^{\circ }$ and $\delta = \pm 90^\circ$intheleptonmixingmatrix$V$,arisesasaconsequenceof$S V = V^\ast $\mathcal{P}$$,where$S$ is the permutation matrix that interchanges the second and third rows of $V$ and $\mathcal{P}$ is a diagonal matrix of phase factors. We prove that any such $V$ may be written in the form $V=URP$, where $U$ is any predefined unitary matrix satisfying $SU=U^{\ast }$, $R$ is an orthogonal, i.e. real, matrix, and $P$ is a diagonal matrix satisfying $P^2 = $\mathcal{P}$$. 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.