Residual Z 2 × Z 2 symmetries and lepton mixing

We consider two novel scenarios of residual symmetries of the lepton mass matrices. Firstly we assume a Z 2 × Z 2 symmetry G (cid:2) for the charged-lepton mass matrix and a Z 2 symmetry G ν for the light neutrino mass matrix. With this setting, the moduli of the elements of one column of the lepton mixing matrix are ﬁxed up to a reordering. One may interchange the roles of G (cid:2) and G ν in this scenario, thereby constraining a row, instead of a column, of the mixing matrix. Secondly we assume a residual symmetry group G (cid:2) ∼= Z m ( m > 2) which is generated by a matrix with a doubly-degenerate eigenvalue. Then, with G ν ∼= Z 2 × Z 2 the moduli of the elements of a row of the lepton mixing matrix get ﬁxed. Using the library of small groups we have performed a search for groups which may embed G (cid:2) and G ν in each of these two scenarios. We have found only two phenomenologically viable possibilities, one of them constraining a column and the other one a row of the mixing matrix.


Introduction
A group-theoretical philosophy for explaining the phenomenological values of the lepton mixing parameters has emerged during the last few years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. In that philosophy, those values follow from the distinct Abelian symmetry groups-G ℓ and G ν -under which the lepton mass matrices-M ℓ and M ν , respectively-are invariant. 1 Those matrices are defined by the mass terms where ℓ L,R are the left-and right-handed charged-lepton fields, ν L are the light neutrino fields, and C is the charge-conjugation matrix in Dirac space. (We assume the neutrinos to be Majorana particles.) Let H ℓ ≡ M ℓ M † ℓ ; if the mass matrices are diagonalized as U † ℓ H ℓ U ℓ = D ℓ ≡ diag m 2 e , m 2 µ , m 2 τ and U T ν M ν U ν = D ν ≡ diag (m 1 , m 2 , m 3 ), then the lepton mixing matrix is given by U PMNS = U † ℓ U ν . (m 1,2,3 denote the three neutrino masses.) Let the symmetry group G ℓ be generated by a matrix L such that L −1 H ℓ L = H ℓ . 2 If we choose a basis in which L is diagonal and if we assume that the diagonal matrix elements of L are all distinct, then this invariance forces H ℓ to be diagonal. Thus, in that basis U ℓ = ½ 3 (up to a permutation of the charged leptons) and U PMNS = U ν . (½ 3 denotes the 3 × 3 unit matrix.) In the same basis, let a generator N of G ν be a unitary 3 × 3 matrix of order two and with two different eigenvalues, i.e. N 2 = ½ 3 but N = ±½ 3 . Such a matrix can always be written as where γ = ±1 and u = (u 1 , u 2 , u 3 ) T is a normalized column vector, viz.
ν D ν and the neutrino masses are non-degenerate, u must be one of the columns of U ν = U PMNS . It thence follows that |u 1,2,3 | are, up to a reordering of the charged leptons, the moduli of the matrix elements of a column (one may still choose which column) of U PMNS .
The above-mentioned philosophy assumes that there is a finite discrete group G which has both G ℓ and G ν as subgroups. 3 It tries to find a suitable G such that the ensuing |u 1,2,3 | agree with the phenomenological values of the moduli of the matrix elements of one of the columns of U PMNS . This has been done in ref. [2] under the assumption that G is a subgroup of SU(3) of order smaller than 512. In ref. [11] a more complete search has been undertaken, wherein G was assumed to be a subgroup of U(3) of order less than 1536. Both refs. [2] and [11] assume G to possess a faithful three-dimensional irreducible representation. In ref. [11] it was moreover assumed that G fully determines U PMNS , because its subgroup G ν is generated by two commuting matrices N and N ′ , both of the form in eq. (2) but with two mutually orthogonal vectors u and u ′ , respectively. (Thus, ) A variant of this philosophy has been employed in refs. [7,16], where the neutrino mass terms have been assumed to be of the Dirac type and, correspondingly, the matrix N has been assumed to generate a group G ν ∼ = Z n with n > 2.
In this paper we report on two group searches that we have undertaken and which might hold promise of relevant results. In the first search-in section 2.1-we have assumed that G ℓ ∼ = Z 2 × Z 2 (instead of the usual choice G ℓ ∼ = Z m with m > 2) and G ν ∼ = Z 2 . In the second search-in section 2.2-we have assumed that G ν ∼ = Z 2 × Z 2 and that G ℓ ∼ = Z m but with a doubly-degenerate eigenvalue, in such a way that a row (instead of a column) of U PMNS gets fixed. In section 3 the results of our searches are confronted with the phenomenological values. Section 4 contains the conclusions of this work.
We consider in this section a scenario in which the lepton flavour symmetry group G is broken to two residual symmetry subgroups G ℓ ∼ = Z 2 × Z 2 and G ν ∼ = Z 2 . The symmetry group G ℓ holds in the charged-lepton sector while G ν holds in the neutrino sector. We require the embedding group G to be finite and to have a faithful three-dimensional irreducible representation D (G). We assume that −½ 3 ∈ D (G ℓ ) and also −½ 3 ∈ D (G ν ). Furthermore, there must be a mismatch between the residual symmetries G ℓ and G ν , i.e. we require that G ν ⊂ G ℓ .
To summarize, we have searched for groups G which fulfil the following conditions: 1. G is finite.
2. G has a faithful three-dimensional irreducible representation D (G).
3. G has two subgroups, Since we are interested in groups G which have a Z 2 × Z 2 subgroup, ord (G) must be divisible by four. Since we require G to have a three-dimensional irreducible representation, ord (G) must be divisible by three. Thus, we only need to consider groups of order divisible by 12.
Since G is finite, there is a basis in which D (G) consists of unitary matrices. Since G ℓ ∼ = Z 2 × Z 2 is Abelian, a basis can be chosen in which D (G ℓ ) is formed by diagonal matrices. Thus, D (G ℓ ) comprehends ½ 3 and where both α and β may be either +1 or −1. The residual symmetry G ℓ means that Therefore, in the basis where L 1 and L 2 are as in eqs. (3), H ℓ must be diagonal.
In the same basis, the generator N of D (G ν ) is a unitary 3 × 3 matrix of order two, i.e. a matrix of the form in eq. (2), where u = (u 1 , u 2 , u 3 ) T is a normalized column vector. Then, up to a reordering, the |u k | (k = 1, 2, 3) are the moduli of the matrix elements of one column of U PMNS . Given the matrices L 1 , L 2 , and N in an arbitrary basis, one may compute the |u k | 2 without the need to diagonalize L 1 and L 2 ; indeed, Equation (4) is easily verified in the basis where eqs. (2) and (3) hold; since it is written in terms of traces, it holds in any other basis-even in one where D (G) is not formed by unitary matrices. One may thus compute the moduli of the matrix elements of one column of U PMNS just from the knowledge of L 1 , L 2 , and N in an arbitrary basis. 4 The computer algebra system GAP [23] has access to SmallGroups [24], a library of all the groups (up to isomorphisms) of order smaller than 2 000-excluding the 49 487 365 422 groups of order 1 024. Since there are 408 641 062 groups of order 1536 = 12 × 128, we have restricted our search to the 1 336 749 groups of order 12n for n ≤ 127. We have furthermore excluded groups G which are direct products of the form because such groups do not provide any restrictions beyond those already following from the smaller group G ′ . Going through these groups, by constructing their character tables, we have sieved out the groups which have a faithful three-dimensional irreducible representation. We have used the GAP package SONATA [25] to find all the subgroups of the groups under investigation. For those groups which have a Z 2 × Z 2 subgroup and a Z 2 subgroup with trivial intersection, we have explicitly constructed all the non-equivalent faithful three-dimensional irreducible representations D and we have computed all the candidates for pairs (D (G ℓ ) , D (G ν )). When neither D (G ℓ ) nor D (G ν ) contained −½ 3 , we have computed the corresponding |u k | 2 through eq. (4). The results can be found in table 1.
In table 1 (and in the second column of table 3) one observes that, whenever G ℓ and G ν together generate a group D n with even n, this leads to |u 1 | 2 , |u 2 | 2 , |u 3 | 2 = 0, sin 2 2π m , cos 2 2π m with m = 2n and, possibly, smaller (integer) values of m. 5 The group D n may be defined as consisting of the matrices where α n ≡ 2π/n and p = 0, 1, 2, . . . , n − 1. For even n, this group has a Z 2 × Z 2 subgroup formed by ½ 2 , Y (n/2), X(n/2), and X(0). The group D n is a subgroup of SO(3) through 4 The computation of mixing-matrix elements from invariant traces was pioneered in ref. [22]. 5 The group D 14 is of particular interest, especially for quark mixing, because it nicely fits Cabibbo mixing [26], as can be seen in the second line before the last of table 1.
its reducible triplet representation In this representation of D n , its Z 2 × Z 2 subgroup is formed by where the matrices L 1,2,3 are as in eqs.

Second search:
One may interchange the roles of Klein's four group Z 2 × Z 2 and of the cyclic group Z 2 in section 2.1. When one does that, the neutrino mass matrix M ν is invariant under If we choose the basis where N 1 and N 2 are diagonal, then in that basis M ν will be diagonal too. Let H ℓ possess a residual Z 2 symmetry, i.e. L −1 H ℓ L = H ℓ with L = γ ½ 3 − 2uu † as in equation (2).
Consequently, LH ℓ = H ℓ L and therefore L(H ℓ u) = H ℓ Lu = −γ (H ℓ u). Then, since the eigenvalue −γ of L is non-degenerate, H ℓ u ∝ u. Now, the eigenvalues of H ℓ , viz. the squares of the charged-lepton masses, are non-degenerate. Therefore, u must be a column of the unitary matrix U ℓ diagonalizing H ℓ . Since we are in the basis where M ν is diagonal, U PMNS = U † ℓ up to a permutation of the rows of U PMNS . We have thus found that in this case the residual symmetries constrain a row, rather than a column, of the mixing matrix U PMNS . The possible restrictions on the moduli of the matrix elements of the row are of course precisely the same as those obtained in section 2.1, see table 1.
An important feature of the scenario just described is that the matrix L generating the residual symmetry group of H ℓ has two degenerate eigenvalues and the third eigenvalue is different. The matrix L is, however, restricted by the condition L 2 = ½ 3 , since it generates a group Z 2 . We now lift this restriction and suppose instead that L generates a group Z n with n > 2, i.e. L n = ½ 3 . We thus assume that in the neutrino sector there is a residual symmetry G ν ∼ = Z 2 × Z 2 , represented by D (G ν ) which, in an appropriate basis, is formed by ½ 3 together with In this basis, M ν is diagonal and therefore U PMNS = U † ℓ up to a permutation of rows. In the charged-lepton sector the residual symmetry is Z n , generated by a matrix L with a degenerate eigenvalue σ and another eigenvalue ρ = σ (of course σ n = ρ n = 1). Let, in the basis where eqs. (10) hold, v = (v 1 , v 2 , v 3 ) T denote the normalized eigenvector of L corresponding to the eigenvalue ρ. One may then write The |v k | are, up to a reordering, the moduli of the matrix elements of one row of U PMNS . They may be computed in a basis-independent way through Thus, we have searched for groups G which fulfil the following conditions: 1. G is finite.
2. G has a faithful three-dimensional irreducible representation D (G).
3. G has two subgroups, G ℓ ∼ = Z n (n > 2) and 5. D (G ℓ ) is generated by a matrix L which has a twice degenerate eigenvalue σ and another eigenvalue ρ which differs from σ.
6. The group G ℓ , G ν generated by D (G ℓ ) and D (G ν ) is non-Abelian.
Once again, we have excluded groups of the form G ∼ = Z m × G ′ with m ≥ 2. For each group of order smaller than 6 800 fulfilling the above requirements, we have computed the corresponding |v k | 2 by means of eq. (12). The results can be found in table 2.
3 The case (1/4, 1/4, 1/2) One sees in tables 1 and 2 that most predicted columns or rows of U PMNS contain a zero matrix element. Such a situation is phenomenologically excluded 7 and therefore most of the data in those tables seem irrelevant for our purposes. The remaining cases are more encouraging. The possibility |u 1 | 2 , |u 2 | 2 , |u 3 | 2 = , in the last line of table 1, was recently discovered and constitutes a viable prediction for the first column of U PMNS [27]. On the other hand, the possibility |u 1 | 2 , |u 2 | 2 , |u 3 | 2 = (1/4, 1/4, 1/2) gives a rather poor fit to the second column of U PMNS . 6 We have stopped this search at a rather low group order because the construction of the irreducible representations becomes, for large groups, extremely expensive in terms of computer time. 7 One might consider the possibility where our predictions only hold as a first approximation and are corrected by other effects-for instance, suppressed terms in the Lagrangian and/or the renormalizationgroup evolution of the parameters of U PMNS . We shall not entertain such possibilities here.
The difference between eqs. (13b) and (13c) yields a prediction for cos δ: Since c 12 > s 12 , cos δ is predicted to be negative. 8 Moreover, |cos δ| is quite large; the bound cos 2 δ ≤ 1 gives This implies that θ 12 and θ 13 cannot be both within their 1σ intervals of ref. [28] and can only marginally be both within their 2σ intervals, see fig. 1. Anyway, the angle δ should be close to either 0 or π, i.e. CP violation in lepton mixing is predicted to be small.

Conclusions
In this work, using the software GAP and the SmallGroups Library, we have looked for finite groups G which have a faithful three-dimensional irreducible representation D (G) and have two subgroups, Z n and Z 2 × Z 2 , with a trivial intersection. Moreover, D (Z n ) should have a twice degenerate eigenvalue and neither D (Z n ) (for n = 2) nor D (Z 2 × Z 2 ) should contain the matrix −½ 3 . When n = 2 we have taken the search up to group order 1536 but for n > 2 we only reached group order 800.  Figure 1: The area in the sin 2 θ 12 -sin 2 θ 13 plane allowed by our prediction in eq. (17). The dotted lines represent the 2σ phenomenological bounds of ref. [28] on those parameters; the shaded area extends to their 3σ bounds.
Applying the results of our search to the prediction of lepton mixing, we have noticed that almost all the groups that we have found lead to a zero mixing matrix element, which is phenomenologically disallowed. There are only two exceptions. In one of them, the groups [60, 5] ∼ = A 5 and [1080, 260] ⊃ [60, 5] may lead to the first column of the lepton mixing matrix having elements with moduli squared (0.6545, 0.25, 0.0955); this is viable and had already been found in a previous paper [27]. In the other exception, many groups-see tables 1 and 2-may lead to either the second or the third row of the lepton mixing matrix having elements with moduli 1/2, 1/2, 1/ √ 2 ; the consequences of this prediction are a very close to maximal atmospheric mixing angle and |cos δ| straddling 1.  Table 1: In the first column, the groups resulting from the search described in section 2.1; the symbol [g, j] denotes the j-th group of order g in the SmallGroups Library. In the second column, the corresponding values for the |u k | 2 (k = 1, 2, 3). In the third column, the symbol G ℓ , G ν denotes the group generated by D (G ℓ ) and D (G ν ), i.e. the smallest finite group having G ℓ and G ν as subgroups. A characterization of the occurring groups can be found in table 3. G |v 1 | 2 , |v 2 | 2 , |v 3 | 2 G ℓ G ℓ , G ν Table 2: In the first column, the groups resulting from the search described in section 2.2 and of order smaller than 800. In the second column, the corresponding values of the |v k | 2 (k = 1, 2, 3). The group G ℓ is shown in the third column and the smallest finite group having G ℓ and G ν as subgroups is listed in the fourth column. A characterization of the occurring groups can be found in table 3.