A predictive A4 model, charged lepton hierarchy and tri-bimaximal sum rule

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Abstract

We propose a novel A4 model in which the Tri-Bimaximal (TB) neutrino mixing and the charged lepton mass hierarchy are reproduced simultaneously. At leading order, the residual symmetry of the neutrino sector is Z2×Z2 which guarantees the TB mixing without adjusting ad hoc free parameters. In the charged lepton sector, one of the previous Z2 is maximally broken and the resulting mass matrix is nearly diagonal and hierarchical. A natural mechanism for the required vacuum alignment is given with the help of the supersymmetry and an Abelian symmetry factor. In our model, subleading effects which could lead to appreciable deviations from TB mixing are very restrictive giving rise to possible next-to-leading predictions. From an explicit example, we show that our “constrained” A4 model is a natural framework, based on symmetry principle, to incorporate the TB sum rule: sin2θ12=1/3+22(cosδsinθ13)/3.

Introduction

Nowadays continuous improvement on the knowledge of neutrino oscillation parameters makes desirable a neutrino model building going beyond the mere fitting procedure. In particular the leptonic mixing pattern, so different from the one in the quark sector, provides a non-trivial theoretical challenge. The present data [1], at 1σ:θ12=(34.5±1.4)°,θ23=(42.33.3+5.1)°,θ13=(0.00.0+7.9)°, are fully compatible with the so-called Tri-Bimaximal (TB) mixing matrix:UTB=(2/31/301/61/31/21/61/3+1/2), which corresponds tosin2θ12=13(θ12=35.3°),sin2θ23=12,sin2θ13=0. Several interesting ideas leading to a nearly TB mixing have been suggested in the last years [2]. The TB mixing has the advantage of correctly describing the solar mixing angle, which, at present, is the most precisely known. Indeed, its 1σ error, 1.4 degrees corresponds to less than λc2 radians, where λc0.22 denotes the Cabibbo angle.

TB pattern belongs to the class of mixing textures which are independent on the mass eigenstates. Mass-independent mixing textures usually exhibit an underlying discrete symmetry nature [3]. It has been realized that the TB mixing matrix of Eq. (2) can naturally arise as the result of a particular vacuum alignment of scalars that break spontaneously certain discrete flavour symmetries. A class of very promising models are based on A4 flavour symmetry [4], [5] and subsequently extended to the group T [6] to cover a reasonable description also for quarks. Despite of the success, the original A4 models proposed by Altarelli and Feruglio (AF) in [5] require improvement for various reasons. First of all, the leading order results of AF are affected by a large number of subleading corrections. Even though these corrections are hopefully under control, they are totally independent and the model looses the possibility to go beyond the leading prediction. Furthermore the mass eigenvalues are completely unspecified by A4 and the charged lepton mass hierarchy can be explained only by an extra Froggatt–Nielsen (FN) [7] U(1)FN factor. Finally, A4 alone seems unfavorable to accommodate quark masses and eventually be embedded into a GUT theory, despite some recent attempts in this direction: Pati–Salam [8], SU(5) [9], [10], SO(10) [11]. Indeed, it is a quite non-trivial task to reproduce all charged fermion hierarchies compatible with a natural mechanism for the vacuum alignment. Particular efforts have been made in this direction. For example, in [12], the problem of fermion hierarchy is partially solved by embedding A4 into a continuous left–right symmetry. However the question of a natural vacuum alignment in their model remains open. In the recent proposal of A4 model in a 5D SUSY SU(5) GUT [10], the fermion mass hierarchies and mixing are a result of the interplay of three different sources: a discrete flavour group based on A4, wave-function suppressions of bulk fields and an additional U(1)FN.

In the literature, there are many mechanisms based on different discrete groups that can successfully describe TB mixing at leading order. Another important issue for the model building is if there were some criteria to distinguish various constructions. One possibility is to go beyond the leading order prediction. There is a remarkable sum rule in the lepton mixing sector first considered in [13], [14]:sin2θ12=sin2θ12ν+sin22θ12νcosδsinθ13, where δ is the Dirac CP violation phase. This sum rule can be derived model-independently when maximal θ23 and θ13=0 come from the neutrino sector at leading order and a non-vanishing θ13 arises only when one includes small charged lepton mixing [14], [16], [17] (under certain assumptions such as θ12e or θ13e dominance [16] as will be explained in Section 5.1). However, it is important to point out that only in flavour models where sin2θ12ν=1/3 holds almost exactly, this sum rule would be a precise test. Unfortunately, in the dynamical realization of TB mixing pattern based on broken discrete flavour symmetries, Eq. (4) is not in general precisely predicted. For example, in the context of the AF models, higher order corrections generically contribute to neutrino masses as well as to the charged lepton masses, all of relative order λc2, being λc the Cabibbo angle. Then one could only expect that sin2θ12ν=1/3+O(λc2) and sin2θ13=O(λc2), if the model is, in some sense, “unconstrained”.

In this work, we will give a realization of the TB pattern solving the charged lepton hierarchy problem and discuss a possible subleading prediction according to Eq. (4). The model is supersymmetric and based only on a discrete symmetry A4×G where G is an Abelian factor. The mixing matrix of Eq. (2) is obtained in the neutrino sector by the spontaneous A4 breaking. Supersymmetry (SUSY) is introduced to simplify the discussion of the vacuum alignment. At the lowest order of the expansion parameters |φ/Λ|1, only the tau mass is generated together with the TB mixing. The muon and electron masses are subsequently generated by higher orders of the same expansion, similar to what happens in the recently proposed S3 model [18]. This is one of the distinguished features of our model: the charged lepton hierarchy is also controlled by the spontaneous breaking of A4 without introducing an extra U(1)FN factor. The Abelian factor G is given by Z3×Z3.1 The presence of an Abelian factor G is essential for our construction. First of all, G guarantees the misalignment in flavour space between the neutrino and the charged lepton mass eigenstates, responsible for both TB mixing and charged lepton hierarchy. Furthermore, G plays an important role in suppressing subleading contributions in order to keep the model predictive. From this point of view, the present A4 model is more constrained than the other A4 models and we will refer it as a constrained A4 model. In particular, subleading corrections can affect only the charged lepton sector and TB mixing in the neutrino sector remains almost exactly. Then the neutrino sum rule of Eq. (4) can be realized in the constrained A4 model without adjusting ad hoc small perturbations and offer a possible precision test of our model.

In Section 2 we outline the main features of our model focusing on the symmetry breaking pattern. We then move to solve explicitly the vacuum alignment problem in a SUSY context in Section 3 finding a new type of minimum of the scalar potential. In Section 4 we construct a simple model of leptons with the symmetry breaking pattern according to the vacuum alignment. In Section 5, we will discuss possible predictive deviations from the TB pattern, in particular those related to the sum rule of Eq. (4). Finally, in Section 6, we comment on other aspects of our model and then conclude.

Section snippets

The ingredients for an alternative realization of A4

The group A4 has 12 elements and four non-equivalent irreducible representations: one triplet and three independent singlets 1, 1′ and 1″. Elements of A4 are generated by the two generators S and T obeying the relations:S2=(ST)3=T3=1. We will consider the following unitary representations of T and S:for 1S=1,T=1,for 1S=1,T=ei4π/3ω2,for 1S=1,T=ei2π/3ω, and for the triplet representationT=(1000ω2000ω),S=13(122212221). The tensor product of two triplets is given by 3×3=1+1+1+3S+3A. From

Vacuum alignment

In this section we will discuss the minimization of the scalar potential. To achieve the desired alignment in a simple way, we work with a supersymmetric model, with N=1 SUSY, eventually broken by small soft breaking terms. In association with φT we introduce another scalar field ξ1 under A4. In general, it is not easy to realize non-trivial minima for flavon fields preserving different subgroups of A4. The key point is that the Abelian part of the discrete symmetry G should forbid unwanted

A A4×Z3×Z3 model for leptons

In this section we propose a very simple SUSY model for leptons based on the following pattern of symmetry breaking of A4×Z3×Z3φT=(0,vT,0),φS=(vS,vS,vS),ξ=u,ξ˜=0,ξ=u. In the charged lepton sector the flavour symmetry A4×Z3 is broken by (φT,ξ) down to GT+ (with Z3 unbroken) at leading order where only the tau mass is generated. The muon and electro masses are generated by higher order contributions. In the neutrino sector A4×Z3 is broken by (φS,ξ) down to GS (with Z3 unbroken)

Deviations from TB mixing

The results of the previous sections hold almost exactly predicting very precisely the TB mixing in the lepton sector. However, from the phenomenological point of view, it is also interesting to explore a natural mechanism to generate sizable deviations from TB mixing like a non-vanishing θ13. In this sector we shall demonstrate that the previous construction can be completed by adding an additional singlet of A4. This singlet should be responsible for a non-vanishing θ13 without destroying the

Further discussions and conclusion

Both charged fermion mass hierarchies and large lepton mixings can be potentially achieved via spontaneous breaking of the flavour symmetry. However, in most cases, the flavour group is of the type D×U(1)FN where D is a discrete component that controls the mixing angles and U(1)FN is an Abelian continuous symmetry that describes the mass hierarchies. It would be a very attractive task to construct economical and constrained models where the same flavon fields producing the mixing pattern by VEV

Acknowledgements

We thank Ferruccio Feruglio for useful suggestions and for his encouragement in our work. We thank also Claudia Hagedorn and Luca Merlo for useful discussions and for reading the preliminary manuscript of the work. We recognize that this work has been partly supported by the European Commission under contracts MRTN-CT-2004-503369 and MRTN-CT-2006-035505.

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