A model realizing the Harrison-Perkins-Scott lepton mixing matrix

We present a supersymmetric model in which the lepton mixing matrix $U$ obeys, at the seesaw scale, the Harrison--Perkins--Scott \textit{Ansatz}--vanishing $U_{e3}$, maximal atmospheric neutrino mixing, and $\sin^2{\theta_\odot} = 1/3$ ($\theta_\odot$ is the solar mixing angle). The model features a permutation symmetry $S_3$ among the three lepton multiplets of each type--left-handed doublets, right-handed charged leptons, and right-handed neutrinos--and among three Higgs doublets and three zero-hypercharge scalar singlets; a fourth right-handed neutrino, a fourth Higgs doublet, and a fourth scalar singlet are invariant under $S_3$. In addition, the model has seven $\mathbbm{Z}_2$ symmetries, out of which six do not commute with $S_3$. Supersymmetry is needed in order to eliminate some quartic terms from the scalar potential, quartic terms which would make impossible to obtain the required vacuum expectation values of the three Higgs doublets and three scalar singlets. The Yukawa couplings to the charged leptons are flavour diagonal, so that flavour-changing neutral Yukawa interactions only arise at loop level.