A predictive model, charged lepton hierarchy and tri-bimaximal sum rule
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: are fully compatible with the so-called Tri-Bimaximal (TB) mixing matrix: which corresponds to 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 radians, where 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 flavour symmetry [4], [5] and subsequently extended to the group [6] to cover a reasonable description also for quarks. Despite of the success, the original 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 and the charged lepton mass hierarchy can be explained only by an extra Froggatt–Nielsen (FN) [7] factor. Finally, 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], [9], [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 into a continuous left–right symmetry. However the question of a natural vacuum alignment in their model remains open. In the recent proposal of model in a 5D SUSY GUT [10], the fermion mass hierarchies and mixing are a result of the interplay of three different sources: a discrete flavour group based on , wave-function suppressions of bulk fields and an additional .
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]: where δ is the Dirac CP violation phase. This sum rule can be derived model-independently when maximal and come from the neutrino sector at leading order and a non-vanishing arises only when one includes small charged lepton mixing [14], [16], [17] (under certain assumptions such as or dominance [16] as will be explained in Section 5.1). However, it is important to point out that only in flavour models where 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 , being the Cabibbo angle. Then one could only expect that and , 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 where G is an Abelian factor. The mixing matrix of Eq. (2) is obtained in the neutrino sector by the spontaneous breaking. Supersymmetry (SUSY) is introduced to simplify the discussion of the vacuum alignment. At the lowest order of the expansion parameters , 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 model [18]. This is one of the distinguished features of our model: the charged lepton hierarchy is also controlled by the spontaneous breaking of without introducing an extra factor. The Abelian factor G is given by .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 model is more constrained than the other models and we will refer it as a constrained 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 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
The group has 12 elements and four non-equivalent irreducible representations: one triplet and three independent singlets 1, 1′ and 1″. Elements of are generated by the two generators S and T obeying the relations: We will consider the following unitary representations of T and S: and for the triplet representation The tensor product of two triplets is given by . 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 SUSY, eventually broken by small soft breaking terms. In association with we introduce another scalar field under . In general, it is not easy to realize non-trivial minima for flavon fields preserving different subgroups of . The key point is that the Abelian part of the discrete symmetry G should forbid unwanted
A model for leptons
In this section we propose a very simple SUSY model for leptons based on the following pattern of symmetry breaking of In the charged lepton sector the flavour symmetry is broken by down to (with 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 is broken by down to (with 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 . In this sector we shall demonstrate that the previous construction can be completed by adding an additional singlet of . This singlet should be responsible for a non-vanishing 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 where D is a discrete component that controls the mixing angles and 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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