Littlest Seesaw model from S4 x U(1)

We show how a minimal (littlest) seesaw model involving two right-handed neutrinos and a very constrained Dirac mass matrix, with one texture zero and two independent Dirac masses, may arise from $S_4\times U(1)$ symmetry in a semi-direct supersymmetric model. The resulting CSD3 form of neutrino mass matrix only depends on two real mass parameters plus one undetermined phase. We show how the phase may be fixed to be one of the cube roots of unity by extending the $S_4\times U(1)$ symmetry to include a product of $Z_3$ factors together with a CP symmetry, which is spontaneously broken leaving a single residual $Z_3$ in the charged lepton sector and a residual $Z_2$ in the neutrino sector, with suppressed higher order corrections. With the phase chosen from the cube roots of unity to be $-2\pi/3$, the model predicts a normal neutrino mass hierarchy with $m_1=0$, reactor angle $\theta_{13}=8.7^\circ$, solar angle $\theta_{12}=34^\circ$, atmospheric angle $\theta_{23}=44^\circ$, and CP violating oscillation phase $\delta_{\rm CP}=-93^\circ$, depending on the fit of the model to the neutrino masses.