Cobimaximal lepton mixing from soft symmetry breaking

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.