The Unique Horizontal Symmetry of Leptons
Abstract
There is a grouptheoretical connection between fermion mixing matrices and minimal horizontal symmetry groups. Applying this connection to the tribimaximal neutrino mixing matrix, we show that the minimal horizontal symmetry group for leptons is uniquely , the permutation group of four objects.
I Introduction
Much expectation is currently being focused on the discovery of the StandardModel Higgs boson, and on possible experimental indications of what lies beyond. While we await these exciting results, we should not lose sight of the generation problem of more than 70 years, of why there are three generations of fermions and what are the relations between them, for which a certain amount of data is already available to guide us. If (horizontal) symmetry is to play a role in the generation problem just as it does in the Standard Model, in supersymmetry, in grand unified and string theories, then the generation problem may even have a direct impact on the Higgs boson being hunted for. This is so because, like other symmetries, the horizontal symmetry is expected to be spontaneously broken to yield the measured quark and lepton masses and mixings. Additional Higgs bosons introduced to break the horizontal symmetry contribute to the mass matrices and hence the fermion masses, making the fermion coupling of whichever Higgs bosons that will be first discovered no longer proportional to its mass. That means the estimated fusion production cross section of the StandardModel Higgs will be off, and its expected decay branching ratio to different fermion pairs will not be as predicted. If that is indeed observed, it would be a strong support for the idea of a horizontal symmetry.
Many horizontal symmetry groups have been proposed in the literature, among them the groups Z , ZZ , D ; LAM , S3 , S4 , A4 , T , D27 , and SOU3 . These models introduce a number of additional Higgs bosons to break the spontaneous symmetry, each of which has an adjustable vacuum expectation value and adjustable Yukawa coupling constants. It is the tuning of these parameters that allow the same piece of experimental data to be explained by so many different models based on many different symmetries.
This is somewhat unsatisfactory because if Nature possesses a horizontal symmetry, it must be unique, but how should we determine which of these is the correct symmetry? In a previous letter CSL , I suggested what seems to me to be a natural criterion: the correct symmetry group should reveal itself experimentally without the adjustment of any dynamical parameter. It was further argued that neutrino mixing should be the piece of data used to obtain such a symmetry, whereas quark mixing and fermion masses have to be obtained from a modeldependent way from a dynamical model based on this symmetry. With this criterion, it was shown that the tribimaximal neutrino mixing matrix HPS leads to the symmetry group . It was further claimed that this group is unique, in the sense that any other viable symmetry group must contain it as a subgroup. The purpose of this paper is to show this uniqueness, as well as other details not fully discussed in CSL . Dynamical models based on to implement quark mixing and fermion masses are not unique and their discussion will not be included in this article.
Since we want to stay away from the adjustment of parameters, the symmetry must be obtained from group theoretical arguments, not dynamical models. The group theoretical technique needed for this purpose was developed in LAM , used there and in CSL to obtain the horizontal symmetry from the tribimaximal mixing matrix. For completeness this result will be reviewed in Sec. II. The proof of the uniqueness of will be carried out in Sec. III, by eliminating all other finite subgroups of and one by one. Sec. IV is devoted to a brief summary and conclusion.
Ii symmetry and mixing
We wish to derive the horizontal symmetry from mixing, and vice versa. Since the mixing matrix depends only on lefthanded fermions, instead of the chargedlepton mass matrix , which connects lefthanded to righthanded charged leptons, it is simpler to deal with , which connects lefthanded charged leptons on both sides. Assuming the active neutrinos to be Majorana, their mass matrix does not involve righthanded neutrinos and can be used as it is.
If and are unitary matrices making and diagonal, then is the neutrino mixing matrix. Since lepton masses are all different, these unitary matrices are essentially unique up to phases. To see that, suppose and are unitary matrices such that also diagonalizes and also diagonalizes , then and , so and must both be diagonal, with . In other words, is unique up to an overall phase change of each of its three columns, and is unique up to an overall sign change of each of its three columns.
If is a unitary transformation of the lefthanded charged leptons, then this transformation is a horizontal symmetry of the charged leptons iff . Similarly, if is a unitary transformation of the active neutrinos, then it is a horizontal symmetry of the active neutrinos iff . These equalities imply that also diagonalizes and also diagonalizes , consequently they satisfy and . The condition implies . Without loss of generality, we shall confine ourselves to the situation when , so that has one and two eigenvalues. In other words, the columns of are eigenvectors of , and the columns of are eigenvectors of with eigenvalue or . The diagonalization matrices and , and hence the mixing matrix , are thereby intimately related to the symmetry operations and . It is for this reason that the mixing matrix can determine the symmetry, and vice versa.
If the Hamiltonian has an unbroken symmetry, this symmetry must be simultaneously a horizontal symmetry for the charged leptons and the active neutrinos, thus . This implies up to inconsequential phases, and hence . In order to have a mixing, is not allowed, so whatever horizontal symmetry that is present at high energies must be broken at the present energy down to the residual symmetries for the charged leptons and for the active neutrinos. With , the minimal horizontal symmetry group at high energy is the (finite) group generated by and . We shall denote this by the notation .
From now on we choose a basis in which is diagonal. That means and . is diagonal in that basis because it commutes with whose eigenvalues are all different. Conversely, if is diagonal and nondegenerate, in the sense that all its eigenvalues are different, then must be diagonal as well. This property will be needed to recover the mixing matrix from and , hence we shall assume to be nondegenerate from now on .
Given that , , it is easy to construct from the neutrino mixing matrix . In fact, there are three solutions , such that , with occupying the position of and occupying the other two diagonal entries. If the mixing matrix is chosen to have the tribimaximal form HPS ,
(1) 
a form which is well within one standard deviation of the experimental mixing angles, then
(2) 
There is no need to consider from now on because .
In summary, the minimal horizontal group given by the neutrino mixing matrix is the finite group generated by , and : . The matrices and are given in (2); the matrix is unitary and diagonal but otherwise arbitrary.
Conversely, given a horizontal group for leptons, whether it can yield the tribimaximal mixing (1) without finetuning depends on whether we can choose three elements , and in to act as the residual symmetries after breaking, so that is nondegenerate, and , are given by (2) in the basis where is diagonal. If this can be done, then the mixing is given by (1) because and is diagonal in that basis as remarked earlier. If no such triplets can be found, then we cannot obtain (1) from without finetuning the dynamical parameters.
Since is assumed to be a finite group, there must be a natural integer such that . We require in order to keep nondegenerate. For , each entry of the diagonal must be a different cube root of unity. Let , then there are six possible ’s, given by and the other five permutations of the three entries. Correspondingly there should be six groups . However, since each is invariant under a simultaneous permutation of its second and third columns and rows, a permutation of the (22) and (33) entries of will not give rise to a new group. In this way we cut down the six possible to three, the other two being generated by and . However, since and , these two must generate the same group, so altogether there are only two distinct groups from the six possible ’s. These groups can be explicitly calculated by repeatedly multiplying the three generators. The result is: , where is the symmetric (permutation) group of four objects with 24 elements, which is also the symmetry group of the cube and of the octahedron, and is a 72element group consisting of , and . Since , the horizontal symmetry group for is uniquely , unique in the sense that any other viable horizontal group must contain it as a subgroup.
It remains to show the uniqueness of for . For a given , there are possible ’s, and there is an infinite number of ’s, thus a straightforward calculation by direct multiplication for every case is clearly impossible. Instead, we will provide a proof in the next section in another way, by enumerating and rejecting the other relevant finite groups.
Iii uniqueness of
In this section we shall show that either contains as a subgroup, or else it must have an infinite order.
To do that, we first assume . Since , and since are unitary matrices, is clearly a finite subgroup of or , or, a finite subgroup of or since covers twice. The desired conclusion is reached by enumerating and considering all the finite subgroups of and , which are known. At the end of the section, we will show that the conclusion remains valid when the unitdeterminant assumption of is dropped.
iii.1 Finite Subgroups of and
The finite subgroups of are in oneone correspondence with the simplylaced Lie algebras MBD ; B ; MK . The two infinite series and correspond respectively to the cyclic group with elements, and the dihedral group with elements. The three exceptional groups correspond to the three symmetry groups of the regular polyhedrons, with the symmetry of the tetrahedron, the symmetry of the cube and the octahedron, and the symmetry of the dodecahedron and the icosahedron. They are also equal to the groups respectively, where is the symmetric group, namely, the permutation group of objects, and is the alternating group, namely, all the even permutations of objects.
The finite subgroups of not in are also known MBD ; B ; FFK ; BLW ; LNR ; LNR2 ; the notations used below are those of FFK , in which the number within the parentheses is the order of the group. There are also two infinite series, and , and six ‘exceptional’ ones: .
We shall show in the subsequent paragraphs that these groups cannot yield the tribimaximal mixing matrix (1) without parameter tuning unless they contain as a subgroup. We shall do that by dividing these groups into different categories.
iii.2 Groups without threedimensional irreducible representations
It was shown in LAM that the horizontal symmetry group has to possess a threedimensional (3D) irreducible representation (IR), or else we can never recover the tribimaximal mixing without a tuning of parameters. This is so because of the particular form of (2), and because the lefthanded fermions must belong to a 3DIR to avoid the presence of tunable parameters.
The series of groups , being Abelian, has only 1DIR, and the series of groups has at most 2DIR, so both are unsuitable. The groups , and do not have 3DIR either FFK , so they must be rejected as well.
iii.3
This subgroup of gives automatically trimaximal mixing, but it can give bimaximal mixing as well only by tuning some Yukawa coupling constants MA . To obtain tribimaximal mixing without any tuning, we need the rest of the symmetries contained in .
iii.4 and
These two infinite series are discussed in MBD ; B ; FFK ; BLW ; LNR ; EL , with their 3DIR’s given in BLW ; LNR ; EL . We must show that none of them are allowed unless they contain as a subgroup.
There are many 3DIR’s, but every one of those matrices has only one nonzero element in each row and each column, and these nonzero elements all have absolute value 1. This is the only property that is required to rule out these two series.
There are types of matrices with this property. They are
(3) 
with . The question is whether we can pick from them three members which can be turned into and by a unitary transformation , such that is diagonal, nondegenerate, and having an order . are excluded because we need the three eigenvalues of to be different; is not considered because simply generate . Unless three such members can be picked, there will be no new horizontal symmetry contained in these two series of groups.
and can be simultaneously diagonalized by the tribimaximal mixing matrix of (1) so that
(4) 
are diagonal. To find , we first determine the unitary matrix which simultaneously diagonalizes and into diagonal forms and , then identify with and with . The desired matrix so that is then given by .
Such a matrix exists because and commute, for otherwise and would not have commuted. Let be the unitary matrix which renders diagonal, then must be one of these ’s.
Since , we must require . Since and the three eigenvalues of are required to be different, we also have , and hence . The trace of any matrix in and is zero, thus we know that must all come from for . To have the right trace, we need to have . To have unit determinant which and possess, we also require .
Incidentally, this analysis excludes which has . In that case and its belongs to type or .
The unitary diagonalization matrices of the first four types can be taken to be
(5) 
with . can be taken to be the identity matrix.
It is easy to check from (3) that the vanishing of the commutators between and requires either (i) both of them to belong to the same type , or (ii) one of them belongs to type . For , the matrix element of and is , so we see from (4) that when (i) occurs, both and must belong to type or type . Let us consider the different cases separately.
(ia). both and belong to type
In order for to be diagonal, is required, in which case
(6) 
This can be identified with and in (4) by setting and .
Now that we know , we can compute to see whether it can be identified with , which is a diagonal nondegenerate matrix with order . Since has to be in one of the four types , we can calculate them all. The result is that this can never happen no matter what parameters we choose.
(ib). both and belong to type
In that case is the identity and . Again we can verified that none of for can be the desired .
(ii). one of and belongs to and the other to
We shall deal with the case when is of type . The other case when is in is very similar.
Comparing with (4), we see that . If belongs to type , its diagonalization matrix must diagonalize as well, hence . As in case (ia) above, can then be computed and checked whether it can lead to a desired . The result is the same before, no matter which one of the four types that belongs to, it can never gives rise to the desired .
This concludes the proof that and cannot give rise to a new horizontal symmetry not containing .
iii.5 , ,
The remaining three groups, the icosahedral group , the Klein group LNR2 , and the Hessian group , can all be ruled out by using their class structures and character tables. To explain how this is done let us denote any of these three groups by .
The question is whether an equivalent representation can be chosen to enable three members to be picked out to be equal to , respectively. If so, is capable of giving the tribimaximal mixing. If not, can be ruled out. Such an identification requires to have a finite order , and to have order 2, and be given by the formula (2).
Since the proof relies only on characters and eigenvalues, which are the same for all equivalent representations, we do not have to worry about which equivalent representation we choose.
We need not consider those with because they would never give us anything new. If we can find that are equal to , then must contain as a subgroup, so remains to be the only minimal horizontal group compatible with tribimaximal mixing without tuning. If not, then can never give rise to tribimaximal mixing so it is ruled out.
For , we rely on the following strategy to rule out these groups . It is shown below how class structure and character table can be used to determine the eigenvalues of . This tells us what the diagonal forms of are. If can be found to equal to in this representation of diagonal , then the group generated by must be a subgroup of . However, we shall show that in each case when , the element and/or the element has an order larger than the order of the group , hence this group cannot be a subgroup these groups , there can be no that can be found to be identified with . This is how these groups are ruled out.
iii.5.1
The character table from FFK is given in Table I, where and . The first row of the table names the 5 conjugacy classes, the second row gives the permutation structure of each class in cycle notations of . For example, consists of two 1cycles and one 3cycle. The third row tells us the order of the elements, for example, class consists of elements of order 5 and their fourth powers, and consists of the second and third powers of elements of order 5. is the identity element. The next five rows are the characters of the irreducible representations of each class, with the boldface numerals in the first column giving the dimension of the representation. The last two rows will be explained later.
Table I: character table for
classes  

perm type  
elem type  ()  
1  1  1  1  1  1 
3  3  0  
3  0  
4  4  1  0  
5  5  1  0  0  

In principle, can be taken to be any of the 60 elements of the group, but to get anything other than , we merely have to concentrate on those elements with order . This leaves elements in class or , both of order . To determine the eigenvalues in each class, we list in the last two rows of the table what class the square and the cube of every belong to. Once this is known, the eigenvalues of every element can be deduced from and , namely, from the character table.
Let us illustrate how the last two rows are obtained. For example, class consists of elements of the type and , hence the square of any element is of the form and , so . Similarly, if , then . We will now proceed to determine the eigenvalues of from the characters.
Let . Then it is well known that
(7) 
Using that for , it will now be shown that , and . To start with, let , and . Then (7) tells us that . Now , and . Thus both and satisfy the quadratic equation , whose two solutions are , so one must be and the other must be . To determine which is which, note that is in the first quadrant of the complex plane and is in the fourth quadrant, so must have a positive real part. Thus and .
Knowing that, it is now easy to verify that if , then its eigenvalues in the 3 representation are . These are also the eigenvalues of in the representation if . Similarly, the eigenvalues for any in the 3 representation and any in the representation, are .
Given three distinct eigenvalues, diagonal ’s can be produced, depending on where the eigenvalues are put. In order to show that or has an order larger than 60, the order of , we compute its three eigenvalues numerically. Let us assume , for otherwise cannot have a finite order. In that case, , and the numerical value for each is approximated by a rational number . Then would have an order larger than 60 if , and that turns out to be true in every case.
iii.5.2
This group is studied in great detail in LNR2 , but for our present purpose, it is sufficient just to use the character table taken from FFK :
Table II: character table for
classes  

perm type  7  
elem type  
1  1  1  1  1  1  1 
3  3  0  
3  0  
6  6  2  0  0  
7  7  1  0  0  
8  8  0  0  1  1  

This table is listed in the same way as the table for , with . From this table we see that candidates for with should come from classes , or . Therefore we need to figure out the eigenvalues of elements in these classes.
For , since the order of its elements is , the three eigenvalues of have to be chosen from the four values . Both the 3 and characters of are 1, and the characters are . Thus if , then and , so the eigenvalues are .
The character of is in 3 and in . It is the reverse for . Let us now prove that and . The proof is similar to the case of . Letting and , it follows that both and satisfy the quadratic equation , whose solutions are . Moreover, should be identified with because their imaginary parts are both positive.
With this relation we can now determine the eigenvalues of in and , both for 3 and . For and 3 or and , and . Hence the eigenvalues of are . Similarly, for and 3 or and , the eigenvalues are . For each set of eigenvalues, diagonal matrices can be produced.
As in the case of , or has an order larger than 168 in every case, thereby ruling out .
iii.5.3
The character table taken from FFK is,
Table III: character table for
classes  

perm type  
elem type  
1  1  1  1  1  1  1  1  1  1  1 
1  1  1  
1  1  1  
2  2  0  1  1  2  
2  0  2  
2  0  2  
3  3  0  0  0  3  3  0  0  
8  0  0  0  0  
8  0  0  0  0  
8  0  0  0  0  

where . To have , must come from , , or . From the 3representation row of the character table, we find that if , then and . The eigenvalues of are then . Since two of the three eigenvalues are identical, we must reject this case. If or , then and . The eigenvalues are then . In spite of having order for elements in , their 3DIR is identical to an with , whose answer is already known. Hence cannot produce anything new.
iii.6 General
We will now relax the condition . Since for some , is an th root of unity, and the unitary matrix can be written , where .
If forms a finite group, then its presentation CM ; WIKI is defined by a number of relations , where are monomials of , and . Three of these relations are , and , but there must be others relating and . If appears times in , then by taking the determinant of the relation on both sides, we see that .
Every group element of can be written as a monomial of : . This monomial is not unique because we can always insert a number of in it. Nevertheless, if , the sum of powers of appearing in and must equal to the power of appearing in , modulo , where is the smallest integer such that . This conclusion can be reached by taking the determinant on both sides of the equation.
Now consider the group generated by and . Since , , hence a relation of is also a relation of . Moreover, the mapping is a homomorphism from to , preserving multiplication relations. The kernel of the mapping is a subgroup of , consisting of all elements for which . If the power of in the monomial is , then it follows that . In other words, is isomorphic to a subgroup of the cyclic group .
Thus is a central extension of , consisting of elements of the form , with and . Moreover, is a finite subgroup of . Since the only group that can naturally lead to the tribimaximal mixing is a group containing , the same is true for , so the uniqueness of is established even if .
Iv conclusion
We have shown that the horizontal group for leptons is uniquely , or any group containing it. To reach this conclusion, we have used the criterion that a horizontal group should be obtained from the neutrino mixing matrix and vice versa without parameter tuning. To implement this criterion, a purely grouptheoretical link between neutrino mixing matrices and horizontal symmetry groups is established. When this link is applied to the tribimaximal neutrino mixing matrix, emerges as a possible horizontal group. Other finite groups not containing are all ruled out by studying the finite subgroups of and .
Quark mixing and fermion masses are obtained from invariant dynamical models. They are model dependent and their discussion will be postponed to a future publication.
I am grateful to James Bjorken, Ernest Ma, and John McKay for discussions.
References
 (1) Y. Koide, H. Nishiura, K. Matsuda, T. Kikuchi, and T. Fukuyama, Phys. Rev. D66 (2002) 093006; E. Ma, arXiv:hepph/0312192; W. Grimus, A.S.Joshipura, S. Kaneko, L. Lavoura, H. Sawanaka, and M. Tanimoto, Nucl. Phys. B713 (2005) 151; B. Hu, F. Wu, Y.L. Wu, Phys. Rev. D75 (2007) 113003.
 (2) N. Haba, C. Hattori, M. Matsuda, and T. Matsuoka, arXiv:hepph/9511312; A. Ghosal and D. Majumdar, Phys. Rev. D66 (2002) 053004; T. Asaka and Y. Takanishi, arXiv:hepph/0409147; S.L. Chen, M. Frigerio, and E. Ma, Phys. Lett. B612 (2005) 29; W. Krolikowski, Acta Phys. Polon. B36 (2005) 865; arXiv:hepph/0501008; Y. Kajiyama, M. Raidal, and A. Strumia, Phys. Rev. D76 (2007) 117301; C. Luhn, S. Nasri, and P. Ramond, Phys. Letts. B652 (2007) 27.
 (3) C.D. Carone, R.F. Lebed, Phys. Rev. D60 (1999) 096002; R. Dermisek and S. Raby, Phys. Rev. D62 (2000) 015007; Phys. Lett. B622 (2005) 327; H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi, and M. Tanimoto, arXiv:0803.0796; W. Grimus, A.S. Joshipura, S. Kaneko, L. Lavoura, and M. Tanimoto, JHEP 0407 (2004) 078; E. Ma, Fizika B14 (2005) 35; C. Hagedorn and W. Rodejohann, JHEP 0507 (2005) 034; S.L. Chen and Ernest Ma, Phys. Lett. B620 (2005) 151; J. Kubo, Phys. Lett. B622 (2005) 303; M. Honda, R. Takahashi, and M. Tanimoto, JHEP 0601 (2006) 042; C. Hagedorn, M. Lindner and F. Plentinger, Phys. Rev. D74 (2006) 025007; R. Dermisek, M. Harada, and S. Raby, Phys. Rev. D74 (2006) 035011; P. Ko, T. Kobayashi, J.h. Park, and S. Raby, Phys. Rev. D76 (2007) 035005; D76 (2007) 059901(E); A. Blum, R. N. Mohapatra, and W. Rodejohann, arXiv:0706.3801; A. Blum, C. Hagedorn and M. Lindner, Phys. Rev. D77 (2008) 076004; A. Blum, C. Hagedorn and A. Hohenegger, High Energy Phys. 03 (2008) 070; H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi, and M. Tanimoto,Phys. Lett. B662 (2008) 178; Phys. Rev. D77 (2008) 115005; H. Okada, arXiv:0804.0926.
 (4) C.S. Lam, arXiv:0708.3665, Phys. Lett. B656 (2007) 193.
 (5) S. Pakvasa and H. Sugawara, Phys. Lett. B73 (1978) 61; Y.P. Yao, arXiv:hepph/9507207; L.J. Hall, H. Murayama, Phys. Rev. Lett. 75 (1995) 3985; K. Kang, J.E. Kim, and P. Ko, Z. Phys. C72 (1996) 671; C.D. Carone, Nucl. Phys. Proc. Suppl. 52A (1997) 177; S.L. Adler, Phys. Rev. D59 (1998) 015012; Erratumibid. D59 (1999) 099902; M. Tanimoto, Phys. Rev. D59 (1999) 017304; Acta Phys. Polon. B30 (1999) 3105; Phys. Lett. B483 (2000) 417; Y. Koide, Phys. Rev. D60 (1999) 077301; E. Ma, Phys. Rev. D61 (2000) 033012; R.N. Mohapatra, A. P rezLorenzana, and C.A. de S. Pires, Phys. Lett. B474 (2000) 355; J.I. SilvaMarcos, JHEP 0307 (2003) 012; P.F. Harrison and W. G. Scott, Phys. Lett. B557 (2003) 76; T. Kobayashi, J. Kubo, and H. Terao, Phys. Lett. B568 (2003) 83; K. Hamaguchi, M. Kakizaki, and M. Yamaguchi, Phys. Rev. D68 (2003) 056007; J. Kubo, A. Mondragon, M. Mondragon, and E. RodriguezJauregui, Prog. Theor. Phys. 109 (2003) 795; Erratumibid. 114 (2005) 287; T. Kobayashi, J. Kubo, and H. Terao, Phys. Lett. B568 (2003) 83; J. Kubo, Phys. Lett. B578 (2004) 156; Erratumibid. B619 (2005) 387; Phys. Rev. D70 (2004) 036007; S.L. Chen, M. Frigerio, and E. Ma, Phys. Rev. D70 (2004) 073008; Erratumibid. D70 (2004) 079905; W.l. Guo, Phys. Rev. D70 (2004) 053009; T. Araki, J. Kubo, and E.A. Paschos, Eur. Phys. J. C45 (2006) 465; W. Grimus and L. Lavoura, JHEP 0508 (2005) 013; JHEP 0601 (2006) 018; J. Phys. G34 (2007) 1757; T. Teshima, Phys. Rev. D73 (2006) 045019; Y. Koide, Phys. Rev. D73 (2006) 057901; Eur. Phys. J. C50 (2007) 8009; J.E. Kim and J.C. Park, JHEP 0605 (2006) 017; N. Haba, A. Watanabe, and K. Yoshioka, Phys. Rev. Lett. 97 (2006) 041601; N. Haba and K. Yoshioka, Nucl. Phys. B739 (2006) 254; H. Morisi, arXiv:hepph/0604106; M. Picariello, arXiv:hepph/0611189; R.N. Mohapatra, S. Nasri, and H.B. Yu, Phys. Lett. B639 (2006) 318; R.N. Mohapatra and H.B. Yu, Phys. Lett. B644 (2007) 346; O. Felix, A. Mondragon, M. Mondragon, and E. Peinado, Rev. Mex. Fis. S52 (2006) 67; A. Mondragon, M. Mondragon, and E. Peinado, Phys. Rev. D76 (2007) 076003; J. Phys. A41 (2008) 304035; K.S. Babu, S.M. Barr, and I. Gogoladze, Phys. Lett. B661 (2008) 124; C.Y. Chen and L. Wolfenstein, Phys. Rev, D77 (2008) 093009; M. Mitra and S. Choubey, arXiv:0806.3254.
 (6) S. Pakvasa and H. Sugawara, Phys. Lett. B82 (1979) 105; Y. Yamanaka, H. Sugawara and S. Pakvasa, Phys. Rev. D25 (1961) 1895; E. Ma, hepph/0508231, Phys. Lett. B632 (2006) 352; C. Hagedorn, M. Lindner, and R.N. Mohapatra, JHEP 0606 (2006) 042; Y. Cai and H.B. Yu, Phys. Rev. D74 (2006) 115005; B. Lampe, arXiv:hepph/0610270; F. Caravaglios and S. Morisi, Int. J Mod. Phys. A22 (2007) 2469; H. Zhang, Phys. Lett. B655 (2007) 132; Y. Koide, J. High Energy Phys. 08 (2007) 086; arXiv:0707.0899; S. Nandi and Z. Tavartkiladze, Phys. Lett. B661 (2008) 109; M.K. Parida, Phys. Rev. D78 (2008) 053004.
 (7) E. Ma and G. Rajasekaran, Phys. Rev. D64 (2001) 113012, Mod. Phys. Lett. A16 (2001) 2207; E. Ma, Mod. Phys. Lett. A17 (2002) 289, 627; J. Phys. G29 (2003) 313; Mod. Phys. Lett. A17 (2002) 2361; Phys. Rev. D70 (2004) 031901; New J. Phys. 6 (2004) 104; Mod. Phys. Lett. A20 (2005) 2767; Mod. Phys. Lett. A20 (2005) 2601; Phys. Lett. B632 (2006) 352; Phys. Rev. D73 (2006) 057304; Mod. Phys .Lett. A21 (2006) 2931; Mod. Phys. Lett. A22 (2007) 101; arXiv:hepph/0701016; arXiv:0808.1729; K.S. Babu, Ts. Enkhbat, and I. Gogoladze, Phys. Lett. B555 (2003) 238; K.S. Babu, E. Ma, and J.W.F. Valle, Phys. Lett. B552 (2003) 207; K.S.Babu, T. Kobayashi, and J. Kubo, Phys. Rev. D67 (2003) 075018; M. Hirsch, J.C. Romao, S. Skadhauge, J.W.F. Valle, and A. Villanova del Moral, Phys. Rev. D69 (2004) 093006; G. Altarelli and F. Feruglio, Nucl. Phys. B720 (2005) 64; Nucl. Phys. B741 (2006) 215; S.L. Chen, M. Frigerio, and E. Ma, Nucl. Phys. B724 (2005) 423; M. Hirsch, E. Ma, J.W.F. Valle, and A. Villanova del Moral, Phys. Rev. D72 (2005) 091301; Erratumibid. D72 (2005) 119904; X.G. He, Y.Y. Keum, and R.R. Volkas, JHEP 0604 (2006) 039; B. Adhikary, B. Brahmachari, A. Ghosal, E. Ma, and M.K. Parida, Phys. Lett. B638 (2006) 345; E. Ma, H. Sawanaka, and M. Tanimoto, Phys. Lett. B641 (2006) 301; X.G. He and A. Zee, Phys. Lett. B645 (2007) 427; B. Adhikary and A. Ghosal, Phys. Rev. D75 (2007) 073020; arXiv:0803.3582; L. Lavoura and H. Kuhbock, Mod. Phys. Lett. A22 (2007) 181; S.F. King, M. Malinsky, Phys. Lett. B645 (2007) 351; X.G. He, arXiv:hepph/0612080; Y. Koide, arXiv:hepph/0701018; S. Morisi, M. Picariello, and E. TorrenteLujan, Phys. Rev. D75 (2007) 075015; M. Hirsch, A.S. Joshipura, S. Kaneko, and J.W.F. Valle, arXiv:hepph/0703046; F. Yin, Physics Review D75 (2007) 073010; F. Bazzocchi, S. Kaneko, and S. Morisi, JHEP 03(2008) 063; W. Chao, S. Luo, Z.Z. Xing, and S. Zhou, arXiv:0709.1069; W. Grimus and H. Kuhbock, Phys. Rev. D77 (2008) 055008; F. Bazzocchi, S. Morisi, and M. Picariello, arXiv:0710.2928; L. Lavoura and H. Kuhbock, Eur. Phys. J. C55 (2008) 303; M. Honda and M. Tanimoto, arXiv:0801.0181; B. Brahmachari, S. Choubey, and M. Mitra, arXiv:0801.3554; G. Altarelli, F. Feruglio, and C. Hagedorn, arXiv:0802.0090; T. Fukuyama, arXiv:0804.2107; Y. Lin, arXiv:0804.2867; C. Csaki, C. Delaunay, C. Grojean, and Y. Grossman, arXiv:0806.0356; P.H. Frampton and S. Matsuzaki, arXiv:0806.4592; F. Feruglio, C. Hagedorn, Y. Lin, and L. Merlo, arXiv:0807.3160; H. Ishimori, T. Kobayashi, Y. Omura, and M. Tanimoto, arXiv:0807.4625.
 (8) A. Aranda, C.D. Carone, R.F. Lebed, Int. J. Mod. Phys. A16S1C (2001) 896; F. Feruglio, C. Hagedorn, Y. Lin, and L. Merlo, arXiv:hepph/0702194; P.H. Frampton and T.W. Kephart, arXiv:0706.1186; M.C. Chen and K.T. Mahanthappa, arXiv:0710.2118; P.H. Frampton and S. Matsuzaki, arXiv:0710.5928; G.J. Ding, arXiv:0803.2278.
 (9) G.C. Branco, J.M. Gerard, and W.Grimus, Phys. Lett. B136 (1984) 383; I. de Medeiros Varzielas, S. F. King, and G. G. Ross, arXiv:hepph/0607045; E. Ma, Mod. Phys. Lett. A21 (2006) 1917; arXiv:0709.0507; R. Howl and S.F. King, Phys. Rev. B77 (2008) 144527.
 (10) S. F. King and G. G. Ross, Phys. Lett. B574 (2003) 239; T. Appelquist, Y. Bai, and M. Piai, Phys. Rev. D74 (2006) 076001; I. de Medeiros Varzielas, S.F. King, and G.G. Ross, Phys. Lett. B644 (2007) 153; Y. Kiode, rXiv:0707.0899; Riazuddin, arXiv:0707.0912; T.L. Wu, arXiv:0807.3847; W.Y.P. Hwang, arXiv:0808.2091.
 (11) C.S. Lam, arxiv:0804.2622, to appear in the Physical Review Letters.
 (12) P.F. Harrison, D.H. Perkins, and W.G. Scott, Phys. Lett. B458, (1999) 79, hepph/9904297; Phys. Lett. B530, (2002) 167, hepph/0202074.
 (13) H.S.M. Coxeter and W.O.J. Moser, ‘Generators and Relations for Discrete Groups’, (SpringerVerlag, 1980).
 (14) http://en.wikipedia.org/wiki/Presentation_ of_ a_ group.
 (15) G.A. Miller, H.F. Blichfeldt, and L.E. Dickson, ‘Theory and applications of finite groups’, John Wiley and Sons, 1916.
 (16) H.F. Blichfeldt, ‘Finite collineation groups’, University of Chicago Press, 1917.
 (17) J. McKay, Proc. Symp. Pure Math. 37 (1980) 183.
 (18) W.M. Fairbairn, T. Fulton, and W.H. Klink, J. Math. Phys. 5, 1038 (1964).
 (19) A. Bovier, M. Lüling, and D. Wyler, J. Math. Phys. 22, 1543 (1981).
 (20) C. Luhn, S. Nasri, and P. Ramond, arXiv:hepth/0701188, J. Math. Phys. 48, 073501 (2007).
 (21) J.A. Escobar and C. Luhn, arXiv:0809.0639.
 (22) C. Luhn, S. Nasri, and P. Ramond, arXiv:0709.1447.
 (23) E. Ma, hepph/0606039, Pramana 67 (2006) 803.