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We propose a $mu-tau$ reflection symmetric Littlest Seesaw ($mutau$-LSS) model. In this model the two mass parameters of the LSS model are fixed to be in a special ratio by symmetry, so that the resulting neutrino mass matrix in the flavour basis (after the seesaw mechanism has been applied) satisfies $mu-tau$ reflection symmetry and has only one free adjustable parameter, namely an overall free mass scale. However the physical low energy predictions of the neutrino masses and lepton mixing angles and CP phases are subject to renormalisation group (RG) corrections, which introduces further parameters. Although the high energy model is rather complicated, involving $(S_4times U(1))^2$ and supersymmetry, with many flavons and driving fields, the low energy neutrino mass matrix has ultimate simplicity.
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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 $( u_e, u_{mu}, u_{tau})$ proportional to $(0,1,1)$, while the subdominant right-handed neutrino, mainly responsible for the solar neutrino mass, has couplings to $( u_e, u_{mu}, u_{tau})$ 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 $ngeq 1$ 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 $S_4times Z_3times Z_3$, which fixes $n=3$ and the leptogenesis phase $eta = 2pi/3$, 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 $delta_{CP}=-87^{circ}$.
We embed $mu-tau$ reflection symmetry into the minimal seesaw formalism, where two right-handed neutrinos are added to the Standard Model of particle physics. Assuming that both the left- and right-handed neutrino fields transform under $mu-tau$ reflection symmetry, we obtain the required forms of the neutrino Dirac mass matrix and the Majorana mass matrix for the right-handed neutrinos. To investigate the neutrino phenomenology at low energies, we first consider the breaking of $mu-tau$ reflection symmetry due to the renormalization group running, and then systematically study various breaking schemes by introducing explicit breaking terms at high energies.
We propose a two Higgs doublet Type III seesaw model with $mu$-$tau$ flavor symmetry. We add an additional SU(2) Higgs doublet and three SU(2) fermion triplets in our model. The presence of two Higgs doublets allows for natural explanation of small neutrino masses with triplet fermions in the 100 GeV mass range, without fine tuning of the Yukawa couplings to extremely small values. The triplet fermions couple to the gauge bosons and can be thus produced at the LHC. We study in detail the effective cross-sections for the production and subsequent decays of these heavy exotic fermions. We show for the first time that the $mu$-$tau$ flavor symmetry in the low energy neutrino mass matrix results in mixing matrices for the neutral and charged heavy fermions that are not unity and which carry the flavor symmetry pattern. This flavor structure can be observed in the decays of the heavy fermions at LHC. The large Yukawa couplings in our model result in the decay of the heavy fermions into lighter leptons and Higgs with a decay rate which is about $10^{11}$ times larger than what is expected for the one Higgs Type III seesaw model with 100 GeV triplet fermions. The smallness of neutrino masses constrains the neutral Higgs mixing angle $sinalpha$ in our model in such a way that the heavy fermions decay into the lighter neutral CP even Higgs $h^0$, CP odd Higgs $A^0$ and the charged Higgs $H^pm$, but almost never to the heavier neutral CP even Higgs $H^0$. The small value for $sinalpha$ also results in a very long lifetime for $h^0$. This displaced decay vertex should be visible at LHC. We provide an exhaustive list of collider signature channels for our model and identify those that have very large effective cross-sections at LHC and almost no standard model background.
In the framework of the seesaw models with triplets of fermions, we evaluate the decay rates of $mu to e gamma$ and $tau to l gamma$ transitions. We show that although, due to neutrino mass constraints, those rates are in general expected to be well under the present experimental limits, this is not necessarily always the case. Interestingly enough, the observation of one of those decays in planned experiments would nevertheless contradict bounds stemming from present experimental limits on the $mu to eee$ and $tau to 3 l$ decay rates. Such detection of radiative decays would therefore imply that there exist sources of lepton flavour violation not associated to triplet fermions.
The Littlest Seesaw (LS) model involves two right-handed neutrinos and a very constrained Dirac neutrino mass matrix, involving one texture zero and two independent Dirac masses, leading to a highly predictive scheme in which all neutrino masses and the entire PMNS matrix is successfully predicted in terms of just two real parameters. We calculate the renormalisation group (RG) corrections to the LS predictions, with and without supersymmetry, including also the threshold effects induced by the decoupling of heavy Majorana neutrinos both analytically and numerically. We find that the predictions for neutrino mixing angles and mass ratios are rather stable under RG corrections. For example we find that the LS model with RG corrections predicts close to maximal atmospheric mixing, $theta_{23}=45^circ pm 1^circ$, in most considered cases, in tension with the latest NOvA results. The techniques used here apply to other seesaw models with a strong normal mass hierarchy.
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