Littlest Seesaw

We propose the Littlest Seesaw (LS) model consisting of just two right-handed neutrinos, where one of them, dominantly responsible for the atmospheric neutrino mass, has couplings to $(\nu_e,\nu_{\mu},\nu_{\tau})$ proportional to $(0,1,1)$, while the subdominant right-handed neutrino, mainly responsible for the solar neutrino mass, has couplings to $(\nu_e,\nu_{\mu},\nu_{\tau})$ proportional to $(1,n,n-2)$. This constrained sequential dominance (CSD) model preserves the first column of the tri-bimaximal (TB) mixing matrix (TM1) and has a reactor angle $\theta_{13} \sim (n-1) \frac{\sqrt{2}}{3} \frac{m_2}{m_3}$. This is a generalisation of CSD ($n=1$) which led to TB mixing and arises almost as easily if $n\geq 1$ is a real number. We derive exact analytic formulas for the neutrino masses, lepton mixing angles and CP phases in terms of the four input parameters and discuss exact sum rules. We show how CSD ($n=3$) may arise from vacuum alignment due to residual symmetries of $S_4$. We propose a benchmark model based on $S_4\times Z_3\times Z'_3$, which fixes $n=3$ and the leptogenesis phase $\eta = 2\pi/3$, leaving only two inputs $m_a$ and $m_b=m_{ee}$ describing $\Delta m^2_{31}$, $\Delta m^2_{21}$ and $U_{PMNS}$. The LS model predicts a normal mass hierarchy with a massless neutrino $m_1=0$ and TM1 atmospheric sum rules. The benchmark LS model additionally predicts: solar angle $\theta_{12}=34^\circ$, reactor angle $\theta_{13}=8.7^\circ$, atmospheric angle $\theta_{23}=46^\circ$, and Dirac phase $\delta_{CP}=-87^{\circ}$.