Dihedral Flavor Symmetry from Dimensional Deconstruction

Extra dimension deconstructed on a closed chain has naturally the symmetry of a regular polygon, the dihedral symmetry D_N. We assume that the fields are irreducible representations of the binary dihedral group Q_2N, which is the covering group of D_N. It is found that although the orbifold boundary conditions break the dihedral invariance explicitly, the Q_2N symmetry appears as an intact internal, global flavor symmetry at low energies. A concrete predictive model based on Q_6N$ with an odd N is given.


I. INTRODUCTION
The Yukawa sector of the standard model (SM) contains a large number of redundant parameters. The presence of the redundant parameters is not related to a symmetry in the SM. That is, they will appear in higher orders in perturbation theory even if they are set equal to zero at the level. These redundant parameters may become physical parameters when going beyond the SM, and, moreover, they can induce flavor changing neutral currents (FCNCs) and CP violating phenomena that are absent or strongly suppressed in the SM. One of the most well known examples is the case of the minimal supersymmetric model (MSSM). Since the SM can not control the redundant parameters, the size of the new FCNCs and CP violating phases may be unacceptably large unless there is some symmetry, or one fine tunes their values 1 .
A natural guidance to constrain the Yukawa sector and to reduce the redundancy of this sector is a flavor symmetry. It has been recently realized that nonabelian discrete flavor symmetries, especially dihedral symmetries, can not only reduce the redundancy, but also partly explain the large mixing of neutrinos 2 . When supersymmetrized, it has been found that the same flavor symmetries can suppress FCNCs that are caused by soft supersymmetry breaking terms [17,18](see also [19]- [23]).
In this letter we address the question of the origin of dihedral flavor symmetries. We will find that dimensional deconstruction [24,25] is a possible origin of dihedral flavor symmetries.

II. DIHEDRAL INVARIANCE IN AN EXTRA DIMENSIONAL SPACE
Consider an extra dimension which is compactified on a closed one-dimensional lattice with N sites. We assume that the lattice has the form of a regular polygon with N edges as it is illustrated in Fig. 1. 1 For recent reviews, see, for instance, [1] and references therein. 2 Models based on dihedral flavor symmetries, ranging from D3(≃ S 3 ) to Q 6 and D 7 , have been recently discussed in [2]- [16].
The regular polygon is invariant under the symmetry operations of the dihedral group D N . The D N operations are 2N discrete rotations, where N of 2N rotations are combined with a parity transformation. Clearly, a discrete polygon rotation of n × θ N , n ∈ {1, . . . , N} corresponds to a discrete translation of the lattice sites of n × a, where a is the lattice spacing and The coordinate of the extra dimension is denoted by y, and the N sites are located at y = y 0 , y 1 , . . . , y N −1 . (y N +i is identified with y i .) Under a D N transformation, the set of coordinates (y 0 , y 1 , . . . , y N −1 ) changes to (y ′ 0 , y ′ 1 , . . . , y ′ N −1 ), which we express in terms of a N × N real matrix. The matrix for the fundamental rotation (i.e., a rotation of θ N ) is given by and that for the parity transformation is Then the 2N group elements of D N are: Using the properties, P 2 There exist two-dimensional representations forR N andP D [2,12]: which are useful representations in finding irreducible representations (irreps) of D N (θ N is given in (1)). It follows that D N is a subgroup of SO(3), which one sees if one embedsR N andP D into 3 × 3 matrices Therefore, D N has only real representations. SU(2) is the universal covering group of SO(3), and has pseudo real and real irreps. Q 2N is a finite subgroup of SU (2). It can be interpreted as the covering group of D N in the sense that the defining matricesR 2N andP Q for Q 2N satisfy whereR The set of 4N elements of Q 2N is given by There exist only one-and two-dimensional irreps for D N and Q 2N . For Q 2N , there are N − 1 different two-dimensional irreps, which we denote by 2 ℓ with odd ℓ is a pseudo real representation, while 2 ℓ with even ℓ is a real representation, where 2 ℓ with even ℓ is exactly 2 ℓ/2 of D N . Under the fundamental rotation (i.e., a rotation of θ N which is defined in (1)), 2 ℓ transforms with the matrix It is straightforward to calculate the Clebsch-Gordan coefficients for tensor products of irreps [12]. There exist four different one-dimensional irreps of Q 2N . Because of the relation (7), each of them has a definite Z 4 charge. Further, under the fundamental rotation, they either remain unchanged or change their sign. Therefore, one-dimensional irreps can be characterized according to Z 2 × Z 4 charge: where the 1 +,0 is the true singlet of Q 2N , and only 1 −,1 and 1 −,3 are complex irreps. Note that all the real representations of Q 2N are exactly those of D N , which is one of the reasons why we would like to call Q 2N as the covering group of D N .

III. FIELD THEORY WITH THE DIHEDRAL INVARIANCE
Let us now discuss how to construct field theory models with a dihedral invariance. We denote the five-dimensional coordinate by The coordinates y i of the lattice sites transform to y ′ i with N × N matrices of D N , which are given in (2) and (3). Then it is natural to assume 3 that the fields defined on the lattice are irreps of Q 2N which is the covering group of D N . That is 4 , In Table 1 explicit expressions of the matrices corresponding to the fundamental rotation and the parity transformation are given, where we assume that the gauge fields belong to the true singlet 1 +,0 . Table 1. Explicit expressions of the matrices corresponding to the fundamental rotation (i.e., a rotation of θ N given in (1)) and the parity transformation.R 2N ,P Q andP D are given in (8) and (5) Given the details of the Q 2N irreps, it is then straightforward to construct an invariant action [24,25,26]. Supersymmetrization can also be straightforwardly done [26].

IV. ORBIFOLD BOUNDARY CONDITIONS AND Q 2N FLAVOR SYMMETRY
In the case of a continuos extra dimension, orbifold boundary conditions are used to suppress unnecessary light fields and also to obtain four-dimensional chiral fields. We shall discuss next how an internal Q 2N flavor symmetry can appear even if orbifold boundary conditions break the dihedral invariance (15). Let φ(x, y) be a generic field which satisfies the periodic boundary condition, φ(x, y) = φ(x, y + Na). Then the field φ(x, y) can be decomposed into the cosine and sine modes: where 3 D N may be understood as a twisted product of Z N and Z 2 . Witten [27] has considered this Z N (the symmetry of the boundary of a deconstructed disc) to solve the triplet-doublet splitting problem in GUTs. 4 Nonabelian discrete family symmetries appearing in extra dimension models of [5,28], for instance, are not directly related to a symmetry of the extra dimension.
φ(x) is the zero mode. As in the continuos case, we can drop the cosine or sine modes by imposing an appropriate boundary condition: Under the parity transformation (3), i.e., the zero mode φ(x) and the cosines modes are even, while the sine modes are odd.
Since the D N transformation mixes the cosine and sine modes, the orbifold boundary conditions break the dihedral invariance explicitly. However, the Q 2N invariant construction of an action discussed in the previous section ensures that the Q 2N invariance remains intact as a global, internal symmetry. This is because there is no derivative with respect to y is used in the construction. So, the theory with orbifold boundary conditions is invariant under the internal transformation which should be compared with (15). The internal symmetry is nothing but a global flavor symmetry based on Q 2N .

V. AN EXAMPLE
In what follows, we would like to discuss a concrete model. One of the successful Ansätze for the quark mass matrices is of a nearest neighbor interaction (NNI) type [29,30,31] In [12] it has been proposed to derive the mass matrix (21) solely from a dihedral symmetry, and concluded that two conditions should be met: (i) There should be real as well as pseudo real nonsinglet representations, and (ii) there should be the up and down type Higgs SU(2) L doublets (type II Higgs). The smallest finite group that allows both real and pseudo real nonsinglet representations is Q 6 as we have seen. Further, the Higgs sector of the MSSM fits the desired Higgs structure. Therefore, we assume supersymmetry in four dimensions. The D 3 (S 3 ) model of [6] with a Z 2 symmetry in the leptonic sector is one of the most predictive models for the leptonic sector. However, the Z 2 symmetry in the quark sector is broken, so that the Z 2 symmetry should be seen as an approximate symmetry in that model. It was found, however, that this leptonic sector can be reproduced in a supersymmetric Q 6 model without introducing an additional discrete symmetry into the leptonic sector [12]. In Table  2 we write the Q 6 assignment of the quark and lepton chiral supermultiplets 5 : The same model exists for Q 2N if N is odd and a multiple of 3.
3 . This is an alternative assignment to the one given in the footnote of [12]. The present assignment can more suppress the proton decay [32]. The assignment for the mirror supermultiplets can be simply read off from Table 2.
We impose the following orbifold boundary conditions: All the mirror chiral supermultiplets are odd under the parity transformation (19). Similarly, the N = 1 chiral supermultiplets, which are the N = 2 superpartners of the SU(3) C × SU(2) L × U(1) Y gauge supermultiplets, are also odd. It is then clear that the zero modes of the gauge, matter and Higgs supermultiplets coincide with those of the supersymmetric Q 6 model of [12], and hence it is the low energy effective theory. The low energy Yukawa superpotential W Y is given by and f 122 = f 212 = f 222 = −f 111 = 1. In [12] it has been found that by introducing a certain set of gauge singlet Higgs supermultiplets it is possible to construct a Higgs sector in such a way that CP phases can be spontaneously induced. Therefore, all the parameters appearing in the Lagrangian including the soft supersymmetry breaking (SSB) sector are real. Consequently, no CP violating processes induced by SSB terms are possible in this model, satisfying the most stringent experimental constraint coming from the EDM of the neutron and the electron [35]. Since the Higgs sector is also Q 6 invariant, it is straightforward to derive it from dimensional deconstruction. Consequently, the quark sector contains only 8 real parameters with one independent phase to describe the quark masses and their mixing, and the leptonic sector contains only 6 real parameters with one independent phase to describe 12 independent physical parameters. Predictions in the |V ub | − sin 2φ 1 planes are shown in Fig. 2, while Fig. 3 shows the predictions in the sin 2φ 1 − φ 3 planes. As we can see from Fig. 2 and 3, with accurate measurements of the Cabibbo-Kobayashi-Maskawa matrix elements, the predictions could be tested. The predictions in the leptonic sector are summarized as follows 7 : 1. Inverted neutrino mass spectrum, i.e., m ν 3 < m ν 1 , m ν 2 . 6 The Higgs sector of the model of [12] possesses a permutation symmetry H , which ensures the stability of the VEV < H >. The resulting mass quark matrices are equivalent to (21). The leptonic sector given in [6] can be obtained by the interchange 1 ↔ 2. 7 Large mixing of neutrinos may be obtained in dimensional deconstruction models in a different mechanism.
See, for instance, [5,33,34]. 4. The prediction of < m ee > is shown in Fig. 4. We emphasize that the smallness of sin θ 13 and the almost maximal mixing of the atmospheric neutrinos are consequences of the Q 6 flavor symmetry. sin θ 13 in the present model may be too small to be measured in a laboratory experiment [41], but the tiny deviation from zero (sin 2 θ 13 ≃ m 2 e /2m 2 µ ≃ 10 −5 ) are important in supernova neutrino oscillations [40].

VI. CONCLUSION
In this letter we have looked for a possible origin of dihedral symmetries. It has been recently realized that a flavor symmetry based on a dihedral group can be used to soften the flavor problem of the SM and the MSSM. We have considered an extra dimension compactified on a closed chain, which is assumed to have the form of a regular polygon. Since the symmetry group of the regular polygon is the dihedral group D N , we assumed that the fields are irreps of the covering group of D N , which is the binary dihedral group Q 2N . The construction of an action with the dihedral invariance is straightforward, and moreover we found that the Q 2N symmetry remains as an intact, internal flavor symmetry even if the original dihedral invariance is broken by orbifold boundary conditions. We hope that with our finding we can come closer to a deep understanding of the origin of a flavor symmetry based on a nonabelian finite group.