Non-Abelian discrete symmetries and neutrino masses: two examples

Two recent examples of non-Abelian discrete symmetries (S3 and A4) in understanding neutrino masses and mixing are discussed.

(φ + , φ 0 ). The quark mixing matrix V CKM is then obtained from the mismatch in the diagonalization of the up and down quark mass matrices. Remarkably, V CKM turns out to be almost identically the unit matrix, i.e. quark mixing angles are all small (for three families, V CKM has three angles and one phase). On the other hand, the analogous U MNS(P) , which is obtained from the mismatch in the diagonalization of the charged-lepton mass matrix and that of the neutrino mass matrix, is far from being the unit matrix. Whereas one angle is indeed small, the other two are definitely large. Indeed, to a good first approximation, In the convention this means that θ 23 π/4, θ 12 which are consistent with the present experimental constraints [1]: sin 2 2θ 23 > 0.91 (90% CL), 0.30 < tan 2 θ 12 < 0.52 (90% CL), sin 2 θ 13 < 0.067 (3σ).
There are a number of approaches in trying to understand the origin of quark and lepton mass matrices. Most of these attempts relate mixing angles with mass ratios. Historically, this was motivated by the phenomenologically successful ansatz θ C √ m d /m s for the Cabibbo angle. One often assumes that there is a symmetry behind this relationship. In addition, a better question to ask is perhaps whether or not there exists a family symmetry, which tells us that V CKM = 1 and U MNS(P) = 1. Obviously, if each family has its own Abelian (continuous or discrete) symmetry, then there is no mixing among families. That works well for V CKM but not for U MNS(P) . If neutrino masses are purely Dirac, then the analogous structure of the quark and lepton sectors would definitely rule out the existence of such a family symmetry. However, if neutrino masses are Majorana, then it is indeed possible to have V CKM = 1 and U MNS(P) as given by equation (1) as the result of a symmetry, as shown below. To fit the experimental data, small (radiative) corrections are needed from physics beyond the Standard Model.

Representations of S 3
The group of permutations of three objects is S 3 . It is isomorphic to the group of three-dimensional rotations of an equilateral triangle to itself, i.e. the dihedral group D 3 . It has six elements and three irreducible representations: 1, 1 and 2. As such, it is ideal for describing two families.
Since 1 × 1 = 1, it is clear that a field transforming as 1 should have a Z 2 parity of −1, i.e. φ → −φ. On the other hand, a doublet (φ 1 , φ 2 ) under S 3 has a choice of representations as long as it satisfies Different representations are simply related by a unitary tranformation. The most convenient representation of S 3 is a complex representation [2,3] such that the products of the doublets φ 1,2 and ψ 1,2 are given by is a doublet. Note also that S 3 has the special property that the symmetric product of three doublets, i.e. φ 1 ψ 1 χ 1 + φ 2 ψ 2 χ 2 is a singlet.
Specifically, the six group elements are the identity: e, the cyclic and anti-cyclic permutations of three objects: g c and g a , and the three interchanges of two objects leaving the third fixed: in 2, respectively, for [e, g c , g a , g 1 , g 2 , g 3 ], where ω = e 2πi/3 .

Quarks and leptons under S 3
Consider a world of only two (i.e. the second and third) families of quarks and leptons. Choosing the convention that all fermions are left-handed, a natural assignment is [4] To allow u, d and l to have Dirac mass terms, two scalar electroweak doublets (φ 0 i , φ − i )(i = 2, 3) transforming as an S 3 doublet are required. The invariant leptonic Yukawa couplings are then resulting in the 2 × 2 mass matrix linking l 2,3 to l c 2,3 below: where v i = φ 0 i . On the other hand, the Majorana neutrino mass matrix depends on the product of L i and L j , i.e. equation (5). Thus, a choice of scalar representations is available. Suppose two which leads to where u i = ξ 0 i . Comparing equations (13) and (15), the lepton mixing matrix is easily obtained (13) is diagonalized by on the left and the unit matrix on the right, which of course means maximal mixing. The chargedlepton mass eigenvalues are then √ 2f 2 v and √ 2f 3 v, whereas the Majorana neutrino mass eigenvalues are hu 2 and hu 3 . They all can be different and yet maximal mixing is ensured. This depends of course on the condition v 2 = v 3 , which can be maintained in the Higgs potential by the interchange symmetry φ 2 ↔ φ 3 . The trilinear scalar couplings to a good approximation even if v 2 = v 3 . In that case, the mixing angle is given by tan −1 (v 3 /v 2 ), which can differ from π/4. In the quark sector, the down-quark mass matrix has the same form as equation (13), i.e.
but the up-quark mass matrix is given by . This means that there is now a mismatch between the two diagonalized mass matrices and the mixing angle is given by [4] In this way, the smallness of the quark mixing between the second and third families is related to the deviation from maximal mixing in the µ-τ sector. To include the first family of quarks and leptons, S 3 singlets must be used. Since either 1 or 1 must be chosen, there has to be mixing of the first family into the 2-3 sector, but its exact form or magnitude cannot be fixed by S 3 alone; see [4] for a specific successful application.

Representations of A 4
The group of even permutations of four objects is A 4 . It is isomorphic to the group of threedimensional rotations of a regular tetrahedron, one of five perfect geometric solids known to the ancient Greeks and identified by Plato with the element 'fire'. It is thus a discrete subgroup of SO (3). It is also isomorphic to (12), which is a discrete subgroup of SU(3) [5]. It has 12 elements and four irreducible reprsentations: 1, 1 , 1 and 3, with the multiplication rule in analogy to equation (5) for S 3 . As such, it is ideal for describing three families. Specifically, the products of the triplets φ 1,2,3 and ψ 1,2,3 are given by [6] where ω = e 2πi/3 . Note that A 4 also has the special property that the symmetric product of three triplets, i.e.
is a singlet. Specifically, the 12 group elements are divided into four equivalence classes: C 1 contains only the identity, C 2 has 4 elements of order 3, C 3 also has 4 elements of order 3 and C 4 has 3 elements of order 2. The representation matrices in 3 are given by

Quarks and leptons under A 4
In analogy to equations (10) and (11) for S 3 , a natural assignment for three families of quarks and leptons is [6]-[10] To allow u, d and l to have Dirac mass terms, three scalar electroweak doublets (φ 0 i , φ − i ) (i = 1, 2, 3) transforming as an A 4 triplet are required. The invariant leptonic Yukawa couplings are then where 1 × 1 = 1 has been used. The resulting 3 × 3 mass matrix linking l 1,2,3 to l c 1,2,3 is which is diagonalized simply by on the left and the unit matrix on the right for v 1 The charged-lepton mass eigenvalues are then √ 3f 1 v, √ 3f 2 v and √ 3f 3 v, which are of course free to be chosen as m e , m µ and m τ . Since this matrix also diagonalizes the up and down-quark mass matrices, the resulting quark mixing matrix is just the unit matrix, i.e. V CKM = 1.

Three degenerate neutrino masses
Consider now the 3 × 3 Majorana neutrino mass matrix. Since the product of L i and L j is given by equation (21), a choice of scalar representations is available, as for S 3 discussed earlier. The simplest choice is to have one scalar triplet (ξ ++ 1 , ξ + 1 , ξ 0 1 ) transforming as 1 under A 4 . In that case, resulting in three degenerate neutrino masses, i.e.
where m 0 = 2h 1 ξ 0 1 . In the (e, µ, τ) basis, it becomes where the radiative correction matrix is assumed to be of the most general form, i.e.

R
Then using the redefinitions: it becomes Without loss of generality, δ may be chosen real by absorbing its phase into ν µ and ν τ and δ 0 set equal to zero by redefining m 0 and the other δs. As a result, whereδ = δ + (Im δ ) 2 /2δ < 0. Thus, this model explains θ 23 π/4 and predicts three nearly degenerate neutrino masses with neutrinoless double beta decay given by |m 0 |. Since δ is a radiative correction, it cannot be too large. Given that m 2 atm is known to be of order 10 −3 eV 2 , m 0 cannot be much smaller than about 0.3 eV. Remarkably, this is also the upper limit on neutrino mass from the large-scale structure of the Universe [11] and possibly the value of |m 0 | as measured in neutrinoless double beta decay [12].
In the Standard Model, there are no flavour-changing leptonic interactions; thus δ = δ = 0 and equation (47) does not lead to neutrino oscillations at all. However, if there is some new physics which allows all the δs to be nonzero, then equation (47) can be realistic. A recent detailed example [9] is available in the context of supersymmetry with arbitrary soft supersymmetry breaking terms.

Arbitrary neutrino masses
In addition to ξ 1 transforming as 1 under A 4 , consider ξ 2 , ξ 3 and ξ 4,5,6 transforming as 1 , 1 and 3 as well [10]. In that case, M ν in the original basis is given by where a comes from ξ 0 1 , b from ξ 0 2 , c from ξ 0 3 and d from ξ 0 4 , assuming that ξ 0 5 = ξ 0 6 = 0. In the basis where the charged-lepton mass matrix is diagonal, the neutrino mass matrix becomes This matrix has one obvious eigenstate, i.e.
then in the basis defined by this transformation, i.e.
where  In the limit m 4 = 0, equation (56) is diagonal and U becomes the neutrino mixing matrix of equation (1) with the prediction tan 2 θ 12 = 1/2, as well as sin 2 2θ 23 = 1 and θ 13 = 0. This is of course a well-known ansatz [13], but has only just been derived from the symmetry of a complete theory, without arbitrary assumptions regarding its charged-lepton sector, in [10]. Note that m 1,2,3 in the above are all arbitrary. In other words, the mixing angles are determined without regard to the masses, just as in the quark sector. There is however an important difference. Whereas all quark mixing angles are zero, the lepton mixing angles are not. Additional small corrections from physics beyond the Standard Model, such as supersymmetry [9,14], are of course necessary to modify these predictions to coincide with present data.

Conclusion
The non-Abelian discrete symmetry S 3 is ideal for explaining maximal mixing in the µ-τ sector with a normal hierarchy of neutrino masses. In a specific application [4], it also explains why U e3 is small but non-zero. The non-Abelian discrete symmetry A 4 is a natural candidate for describing three families of quarks and leptons. Whereas Dirac fermion masses come from the decomposition Majorana neutrino masses come from the decomposition 3 × 3 = 1 + 1 + 1 + 3.
The mismatch between the quark mass matrices is then naturally given by V CKM = 1, whereas that between the charged-lepton and neutrino mass matrices is definitely not the unit matrix, but rather U MNS(P) of equation (1) in a certain symmetry limit, thus predicting a relationship among θ 23 , θ 12 and θ 13 . Specifically, if θ 13 = 0, then sin 2 2θ 23 = 1 and tan 2 θ 12 = 0.5, independent of the values of the three neutrino masses. (Note that all six quarks and all three charged leptons have names, but the three neutrinos do not, as yet.) To obtain small nonzero quark mixing angles as well as deviations from the lepton-mixing angles constrained by this model, new physics beyond the Standard Model is expected, such as supersymmetry at the TeV scale.