Constrained Sequential Dominance Research Articles

Modular S4×SU(5) GUT

Modular symmetry offers the possibility to provide an origin of discrete flavour symmetry and to break it along particular symmetry preserving directions without introducing flavons or driving fields. It is also possible to use a weighton field to account for charged fermion mass hierarchies rather than a Froggatt-Nielsen mechanism. Such an approach can be applied to flavoured Grand Unified Theories (GUTs) which can be greatly simplified using modular forms. As an example, we consider a modular version of a previously proposed $S_4\times SU(5)$ GUT, with Gatto-Sartori-Tonin and Georgi-Jarlskog relations, in which all flavons and driving fields are removed, with their effect replaced by modular forms with moduli assumed to be at various fixed points, rendering the theory much simpler. In the neutrino sector there are two right-handed neutrinos constituting a Littlest Seesaw model satisfying Constrained Sequential Dominance (CSD) where the two columns of the Dirac neutrino mass matrix are proportional to $(0,1, -1)$ and $(1, n, 2-n)$ respectively, and $n=1+\sqrt{6}\approx 3.45$ is prescribed by the modular symmetry, with predictions subject to charged lepton mixing corrections. We perform a numerical analysis, showing quark and lepton mass and mixing correlations around the best fit points.

Neutrino mixing by modifying the Yukawa coupling structure of constrained sequential dominance

In the constrained sequential dominance (CSD), tri-bimaximal mixing (TBM) pattern in the neutrino sector has been explained, by proposing a certain Yukawa coupling structure for the right-handed neutrinos of the model. However, from the current experimental data it is known that the values of neutrino mixing angles are deviated from the TBM values. In order to explain this neutrino mixing, we first propose a phenomenological model where we consider Yukawa couplings which are modified from that of CSD. Essentially, we add small complex parameters to the Yukawa couplings of CSD. Using these modified Yukawa couplings, we demonstrate that neutrino mixing angles can deviate from their TBM values. We also construct a model, based on a flavor symmetry, in order to justify the modified form of Yukawa couplings of our work.

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 $\delta_{CP}=-87^{\circ}$.

Testing constrained sequential dominance models of neutrinos

Constrained sequential dominance (CSD) is a natural framework for implementing the see-saw mechanism of neutrino masses which allows the mixing angles and phases to be accurately predicted in terms of relatively few input parameters. We analyze a class of CSD(n) models where, in the flavour basis, two right-handed neutrinos are dominantly responsible for the ‘atmospheric’ and ‘solar’ neutrino masses with Yukawa couplings to proportional to and respectively, where n is a positive integer. These coupling patterns may arise in indirect family symmetry models based on A4. With two right-handed neutrinos, using a χ2 test, we find a good agreement with data for CSD(3) and CSD(4) where the entire Pontecorvo–Maki–Nakagawa–Sakata mixing matrix is controlled by a single phase η, which takes simple values, leading to accurate predictions for mixing angles and the magnitude of the oscillation phase We carefully study the perturbing effect of a third ‘decoupled’ right-handed neutrino, leading to a bound on the lightest physical neutrino mass meV for the viable cases, corresponding to a normal neutrino mass hierarchy. We also discuss a direct link between the oscillation phase and leptogenesis in CSD(n) due to the same see-saw phase η appearing in both the neutrino mass matrix and leptogenesis.

Neutrino mixing angles in sequential dominance to NLO and NNLO

Neutrinos with hierarchical masses and two large mixing angles may naturally originate from sequential dominance (SD). Within this framework we present analytic expressions for the neutrino mixing angles including the next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) corrections arising from the second lightest and lightest neutrino masses. The analytic results for neutrino mixing angles in SD presented here, including the NLO and NNLO corrections, are applicable to a wide class of models and may provide useful insights when confronting the models with data from high precision neutrino experiments. We also point out that for special cases of SD corresponding to form dominance (FD) the NLO and NNLO corrections both vanish. For example we study tri-bimaximal (TB) mixing via constrained sequential dominance (CSD) which involves only a NNLO correction and tri-bimaximal-reactor (TBR) mixing via partially constrained sequential dominance (PCSD) which involves a NLO correction suppressed by the small reactor angle and show that the analytic results have good agreement with the numerical results for these cases.

On the origin of neutrino flavour symmetry

We study classes of models which are based on some discrete family symmetry which is completely broken such that the observed neutrino flavour symmetry emerges indirectly as an accidental symmetry. For such models we discuss the D-term flavon vacuum alignments which are required for such an accidental flavour symmetry consistent with tri-bimaximal lepton mixing to emerge. We identify large classes of suitable discrete family symmetries, namely the Δ(3n 2 ) and Δ(6n 2 ) groups, together with other examples such as Z 7 x Z 3 . In such indirect models the implementation of the type I see-saw mechanism is straightforward using constrained sequential dominance. However the accidental neutrino flavour symmetry may be easily violated, for example leading to a large reactor angle, while maintaining accurately the tri-bimaximal solar and atmospheric predictions.

A4see-saw models and form dominance

We introduce the idea of Form Dominance in the (type I) see-saw mechanism, according to which a particular right-handed neutrino mass eigenstate is associated with a particular physical neutrino mass eigenstate, leading to a form diagonalizable effective neutrino mass matrix. Form Dominance, which allows an arbitrary neutrino mass spectrum, may be regarded as a generalization of Constrained Sequential Dominance which only allows strongly hierarchical neutrino masses. We consider alternative implementations of the see-saw mechanism in minimal A4 see-saw models and show that such models satisfy Form Dominance, leading to neutrino mass sum rules which predict closely spaced neutrino masses with a normal or inverted neutrino mass ordering. To avoid the partial cancellations inherent in such models we propose Natural Form Dominance, in which a different flavon is associated with each physical neutrino mass eigenstate.

Lepton flavour violation in the constrained MSSM with constrained sequential dominance

We consider charged lepton flavour violation (LFV) in the constrained minimal supersymmetric Standard Model, extended to include the see-saw mechanism with constrained sequential dominance (CSD), where CSD provides a natural see-saw explanation of tri-bimaximal neutrino mixing. When charged lepton corrections to tri-bimaximal neutrino mixing are included, we discover characteristic correlations among the LFV branching ratios, depending on the mass ordering of the right-handed neutrinos, with a pronounced dependence on the leptonic mixing angle θ13 (and in some cases also on the Dirac CP phase δ).

Invariant see-saw models and sequential dominance

We propose an invariant see-saw (ISS) approach to model building, based on the observation that see-saw models of neutrino mass and mixing fall into basis invariant classes labelled by the Casas–Ibarra R-matrix, which we prove to be invariant not only under basis transformations but also non-unitary right-handed neutrino transformations S. According to the ISS approach, given any see-saw model in some particular basis one may determine the invariant R-matrix and hence the invariant class to which that model belongs. The formulation of see-saw models in terms of invariant classes puts them on a firmer theoretical footing, and allows different see-saw models in the same class to be related more easily, while their relation to the R-matrix makes them more easily identifiable in phenomenological studies. To illustrate the ISS approach we show that sequential dominance (SD) models form basis invariant classes in which the R-matrix is approximately related to a permutation of the unit matrix, and quite accurately so in the case of constrained sequential dominance (CSD) and tri-bimaximal mixing. Using the ISS approach we discuss examples of models in which the mixing naturally arises (at least in part) from the charged lepton or right-handed neutrino sectors and show that they are in the same invariant class as SD models. We also discuss the application of our results to flavour-dependent leptogenesis where we show that the case of a real R-matrix is approximately realized in SD, and accurately realized in CSD.

Predicting neutrino parameters from SO(3) family symmetry and quark-lepton unification

We show how the neutrino mixing angles and oscillation phase can be predicted from tri-bimaximal neutrino mixing, corrected by charged lepton mixing angles which are related to quark mixing angles via quark-lepton unification. The tri-bimaximal neutrino mixing can naturally originate from the see-saw mechanism via constrained sequential dominance (CSD), where CSD can result from the vacuum alignment of a non-Abelian family symmetry such as SO(3). We construct a realistic model of quark and lepton masses and mixings based on SO(3) family symmetry with quark-lepton unification based on the Pati-Salam gauge group. The atmospheric angle is predicted to be approximately maximal $\theta_{23}= 45^\circ$, corrected by the quark mixing angle $\theta_{23}^{\mathrm{CKM}}\approx 2.4^\circ$, with the correction controlled by an undetermined phase in the quark sector. The solar angle is predicted by the tri-bimaximal complementarity relation: $\theta_{12}+ \frac{1}{\sqrt{2}}\frac{\theta_{\mathrm{C}}}{3} \cos (\delta - \pi) \approx 35.26^\circ $, where $\theta_{\mathrm{C}}$ is the Cabibbo angle and $\delta$ is the neutrino oscillation phase. The reactor angle is predicted to be $\theta_{13} \approx \frac{1}{\sqrt{2}}\frac{\theta_{\mathrm{C}}}{3}\approx 3.06^\circ$. The MNS neutrino oscillation phase $\delta$ is predicted in terms of the solar angle to be $\cos (\delta - \pi) \approx (35.26^\circ - \theta_{12}^\circ)/3.06^\circ $. These predictions can all be tested by future high precision neutrino oscillation experiments, thereby probing the nature of high energy quark-lepton unification.