Fermion Masses and Flavor Mixings in a Model with Flavor Symmetry
GuiJun Ding ^{1}^{1}1email address: dinggj@ustc.edu.cn
Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
We present a supersymmetric model of quark and lepton based on flavor symmetry. The symmetry is broken down to Klein four and subgroups in the neutrino and the charged lepton sectors respectively. TriBimaximal mixing and the charged lepton mass hierarchies are reproduced simultaneously at leading order. Moreover, a realistic pattern of quark masses and mixing angles is generated with the exception of the mixing angle between the first two generations, which requires a small accidental enhancement. It is remarkable that the mass hierarchies are controlled by the spontaneous breaking of flavor symmetry in our model. The next to leading order contributions are studied, all the fermion masses and mixing angles receive corrections of relative order with respect to the leading order results. The phenomenological consequences of the model are analyzed, the neutrino mass spectrum can be normal hierarchy or inverted hierarchy, and the combined measurement of the decay effective mass and the lightest neutrino mass can distinguish the normal hierarchy from the inverted hierarchy.
1 Introduction
Neutrino has provided us a good window to the new physics beyond the Standard Model. Neutrino oscillation experiments have provided solid evidence that neutrinos have small but nonzero masses. Global data fit to the current neutrino oscillation data demonstrates that the mixing pattern in the leptonic sector is so different from the one in the quark sector. Two independent fits for the mixing angles and the mass squared differences are listed in Table 1.
Ref. [1]  Ref. [2]  
parameter  best fit  3 interval  best fit  3 interval  
7.05–8.34  7.07–8.34  
(2.07–2.75) 



0.25–0.37  0.26–0.40  
0.36–0.67  0.33–0.64  
0.056  0.049 
As is obvious, the current neutrino oscillation data is remarkably compatible with the so called TriBimaximal (TB) mixing pattern [3], which suggests the following mixing pattern
(1) 
These values lie in the range of global data analysis shown in Table 1^{2}^{2}2 is exactly within the range of the second global data fit, whereas it slightly above the up limit of the first fit.. Correspondingly, the leptonic PontecorvoMakiNakagawaSakata (PMNS) mixing matrix is given by
(2) 
where and are the Majorana CP violating phases, and is given by
(3) 
The mixing in the quark sector is described by the famous CKM matrix [4], and there is large mass hierarchies within the quarks and charged leptons sectors respectively [5]. The origin of the observed fermion mass hierarchies and flavor mixings is a great puzzle in particle physics. Nowadays promising candidates for understanding such issue are the models based on spontaneously breaking flavor symmetry, various models based on discrete or continuous flavor symmetry have been proposed so far [6, 7]. Recently it was found that flavor symmetry based on discrete group is particularly suitable to reproduce specific mixing pattern at leading order [8]. The models are especially attractive, it has received considerable interest in the recent past [9, 10, 11, 12, 13, 14]. So far various flavor models have been proposed, and their phenomenological consequences were analyzed [15, 16, 17, 18]. These models assumed that symmetry is realized at a high energy scale, the lepton fields transform nontrivially under the symmetry group, and the flavor symmetry is spontaneously broken by a set of flavons with the vacuum expectation value (VEV) along a specific direction. The misalignment in the flavor space between the charged lepton and the neutrino sectors results in the TB lepton mixing.
If extend the symmetry to the quark sector, the quark mixing matrix turns out to be unity matrix at leading order [11], However, the subleading contributions of the higher dimensional operators are too small to provide large enough deviations of from the identity matrix. The possible ways of resolving this issue are to consider new sources of symmetry breaking or enlarge the symmetry group. Two discrete groups [19, 20, 21, 22, 23, 24, 25, 26] and [27, 28, 29, 30, 31, 32] are found to be promising, both groups have two dimensional irreducible representation, which is very useful to describing the quark sector. The symmetry is particularly interesting, as a horizontal symmetry group has been proposed long ago [33], and some models with different purposes have been built [34]. Recently it was claimed to be minimal flavor group capable of yielding the TB mixing without fine tuning [35, 36, 37]. However, Grimus et al. were against this point [38].
In this work, we build a SUSY model based on flavor group, the neutrino mass is generated via the conventional type I SeeSaw mechanism [39]. Our model naturally produces the TB mixing and the charged lepton mass hierarchy at leading order. Furthermore, we extend the model to the quark sector, the realistic patterns of quark masses and mixing angles are generated. In our model the mass hierarchies are controlled by the spontaneous breaking of the flavor symmetry instead of the FroggattNielsen (FN) mechanism [40].
This article is organized as follows. Section 2 is the group theory of group, where the subgroup, the equivalent class, and the representation of are presented. In section 3 we justify the vacuum alignment of our model in the supersymmetric limit. In section 4 we present our model in both the lepton and quark sectors, its basic features and theoretical predictions are discussed. In section 5 we analyze the phenomenological implications of the model in detail, which include the mass spectrum, neutrinoless double beta decay and the Majorana CP violating phases etc. The corrections induced by the next to leading order terms are studied in section 6. Finally we summarize our results in the conclusion section.
2 The discrete group
is the permutation group of 4 objects. The group has 24 distinct elements, and it can be generated by two elements and obeying the relations
(4) 
Without loss of generality, we could choose
(5) 
where the cycle (1234) denotes the permutation , and (123) means . The 24 elements belong to 5 conjugate classes and are generated from and as follows
The structure of the group is rather rich, it has thirty proper subgroups of orders 1, 2, 3, 4, 6, 8, 12 or 24. Concretely, the subgroups of are as follows

The trivial group only consisting of the unit element.

Six twoelement subgroups generated by a transposition of the form with
, , , , and

Three twoelement subgroups generated by a double transition of the form with
, ,

Four subgroups of order three, which is spanned by a threecycle
,,,

The fourelement subgroups generated by a fourcycle, they are of the form with any fourcycle
, ,

The fourelement subgroups generated by two disjoint transpositions, which is isomorphic to Klein four group
, ,

The order four subgroup comprising of the identity and three double transitions, which is isomorphic to Klein four group

Four subgroups of order six, which is isomorphic to . They are the permutation groups of any three of the four objects, leaving the fourth invariant
, ,
, 
Three eightelement subgroups, which is isomorphic to
,

The alternating group

The whole group
In particular, and are the invariant subgroups of . Since the number of the unequivalent irreducible representation is equal to the number of class, the group has five irreducible representations: , , 2, and . is the identity representation and is the antisymmetric one. The Young diagram for the two dimensional representations is self associated, and the Young diagrams corresponding to the three dimensional representations and are associated Young diagrams. For the same group element, the representation matrices of and are exactly the same if the element is an even permutation. Whereas the overall signs are opposite if the group element is an odd permutation. It is notable that together with is the smallest group containing one, two and three dimensional representations. The character table of group is shown in Table 2.
classes  

1  6  3  8  6 

1  2  2  3  4 

1  1  1  1  1 

1  1  1  1  1 
2 
2  0  2  1  0 

3  1  1  0  1 

3  1  1  0  1 
From the character table of the group, we can straightforwardly obtain the multiplication rules between the various representations
(6) 
The explicit representation matrices of the generators , and other group elements for the five irreducible representations are listed in Appendix A. From these representation matrices, one can explicitly calculate the ClebschGordan coefficients for the decomposition of the product representations, and the same results as those in Ref.[29] are obtained.
3 Field content and the vacuum alignment
The model is supersymmetric and based on the discrete symmetry . Supersymmetry is introduced in order to simplify the discussion of the vacuum alignment. The component controls the mixing angles, the auxiliary symmetry guarantees the misalignment in flavor space between the neutrino and the charged lepton mass eigenstates, and the component is crucial to eliminating the unwanted couplings and reproducing the observed mass hierarchy. The fields of the model and their classification under the flavor symmetry are shown in Table 3, where two Higgses doublets of the minimal supersymmetric standard model are present. If the flavor symmetry is preserved until the electroweak scale, then all the fermions would be massless. Therefore symmetry should be broken by the suitable flavon fields, which are standard model singlets. Another critical issue of the flavor model building is the vacuum alignment, a global continuous symmetry is exploited to simplify the vacuum alignment problem. This symmetry is broken to the discrete R parity once we include the gaugino mass in the model. The matter fields carry +1 Rcharge, the Higgses and the flavon supermultiplets have Rcharge 0. The spontaneous breaking of symmetry can be implemented by introducing a new set of multiplets, the driving fields carrying 2 unit Rcharge. Consequently the driving fields enter linearly into the superpotential. The suitable driving fields and their transformation properties are shown in Table 4. In the following, we will discuss the minimization of the scalar potential in the supersymmetric limit. At the leading order, the most general superpotential dependent on the driving fields, which is invariant under the flavor symmetry group , is given by

2  2  

1  1  1  1  1  1  1  1  1  1  1  

1  i  1  i  1  1  1  i  1  1  1  1  1  1  i  i  1  1  1  1 
Fields  


2  

1  1  1  

1  1  1  1  1  1 
(7) 
where the subscript denotes the contraction in , similar rule applies to other subscripts , , and . In the SUSY limit, the vacuum configuration is determined by the vanishing of the derivative of with respect to each component of the driving fields
(8) 
This set of equations admit the solution
(9) 
with
(10) 
From the driving superpotential , we can also derive the equations from which to extract the vacuum expectation values of , and
(11) 
The solution to the above six equations is
(12) 
with the conditions
(13) 
The vacuum expectation values (VEVs) of the flavons can be very large, much larger than the electroweak scale, and we expect that all the VEVs are of a common order of magnitude. This is a very common assumption in the flavor model building, which guarantees the reasonability of the subsequent perturbative expansion in inverse power of the cutoff scale . Acting on the vacuum configurations of Eq.(9) and Eq.(12) with the elements of the flavor symmetry group , we can see that the VEVs of and are invariant under four elements 1, , and , which exactly constitute the Klein four group . On the contrary, the VEVs of and break completely. Under the action of or , the directions of and are invariant except an overall phase. Considering the enlarged group , the vacuum configuration Eq.(9) preserves the subgroup generated by , which is defined as the simultaneous transformation of and . As we shall see later that the flavor symmetry is spontaneously broken down by the VEVs of and in the neutrino sector at the leading order(LO), and it is broken down by the VEVs of and in the charged lepton sector. Whereas both , and , are involved in generating the quark masses. The flavor symmetry is broken into the Klein four symmetry and the symmetry generated by in the neutrino and the charged lepton sector respectively at LO. This symmetry breaking chain is crucial to generating the TB mixing.
4 The model with flavor symmetry
In this section we shall propose a concise supersymmetric (SUSY) model based on flavor symmetry with the vacuum alignment of Eq.(9) and Eq.(12).
4.1 Charged leptons
The charged lepton masses are described by the following superpotential
(14) 
In the above superpotential , for each charged lepton, only the lowest order operators in the expansion in powers of are displayed explicitly. Dots stand for higher dimensional operators. Note that the auxiliary symmetry imposes different powers of and for the electron, mu and tau terms. At LO only the tau mass is generated, the muon and the electron masses are generated by high order contributions. After the flavor symmetry breaking and the electroweak symmetry breaking, the charged leptons acquire masses, and becomes
(15) 
where , and . As a result, the charged lepton mass matrix is diagonal at LO
(16) 
It is obvious that the hermitian matrix is invariant under both and displayed in the Appendix A, i.e.,
(17) 
Conversely, the general matrix invariant under and must be diagonal. Consequently the symmetry is broken to the subgroup in the charged lepton sector. The charged lepton masses can be read out directly as
(18) 
we notice that the charged lepton mass hierarchies are naturally generated by the spontaneous symmetry breaking of symmetry without exploiting the FN mechanism [11]. Using the experimental data on the ratio of the lepton masses, one can estimate the order of magnitude of and . Assuming that the coefficients , and are of , we obtain
(19) 
Obviously the solution to the above equations is
(20) 
we see that the amplitudes of both and are roughly of the same order about , where is the Cabibbo angle.
4.2 Neutrinos
The superpotential contributing to the neutrino mass is as follows
(21) 
where dots denote the higher order contributions, is a constant with dimension of mass, and the factor is a normalization factor for convenience. The first two terms in Eq.(21) determine the neutrino Dirac mass matrix, and the third term is Majorana mass term. After electroweak and symmetry breaking, we obtain the following LO contributions to the neutrino Dirac and Majorana mass matrices
(22) 
where , and . We notice that the Dirac mass matrix is symmetric and it is controlled by two parameters and . The eigenvalues of the Majorana matrix are given by
(23) 
The right handed neutrino masses are exactly degenerate, this is a remarkable feature of our model. Integrating out the heavy degrees of freedom, we get the light neutrino mass matrix, which is given by the famous SeeSaw relation
(24) 
The above light neutrino mass matrix is invariant and it satisfies the magic symmetry . Therefore it is exactly diagonalized by the TB mixing
(25) 
The unitary matrix is written as
(26) 
The phases , and are given by
(27) 
, and in Eq.(25) are the light neutrino masses,
(28) 
Concerning the neutrinos, the symmetry is spontaneously broken by the VEVs of and at the LO. since both and are invariant under the actions of , and , the flavor symmetry is broken down to the Klein four subgroup