Permutation Symmetry, Tri-Bimaximal Neutrino Mixing and the S3 Group Characters

We postulate that the neutrino mass matrix in the lepton flavour basis is an S3 group matrix in the natural representation of S3. This immediately requires one neutrino to be trimaximally mixed, as suggested by the solar neutrino data. We go on to postulate that the charged-lepton mass matrix in the neutrino mass-basis is an S3 class matrix in the natural representation of the S3 class-algebra, leading to exact tri-bimaximal mixing, which is compatible with data overall. The tri-bimaximal mixing matrix is seen to be closely related to the S3 character table, and is properly the S2 C S3 table of induction coefficients, where the S2 corresponds to symmetry under mu-tau interchange in the lepton flavour basis.

Tri-bimaximal mixing is sometimes incorrectly linked (eg. Ref. [12]) with the original rank-one democratic mass matrix (defined by all mass-matrix elements equal [13]). In fact, as we will see, the mass matrices associated with tri-bimaximal mixing are very far from the democratic form (Sections 4-5 below). It is true that the democratic mass matrix has always one trimaximal eigenvector (1/ √ 3, 1/ √ 3, 1/ √ 3) [14] [15] [16], but the problem is that it is always the heaviest mass-eigenstate (or in fact, more generally, the non-degenerate mass-eigenstate, see below) which ends up trimaximally mixed, ie. not what is needed phenomenologically (cf. Eq. 1). In particular, the democratic neutrino mass matrix can in no way be taken as a zeroth-order approximation for a mixing scenario where it is the solar neutrino (normally the intermediate mass neutrino ν 2 ) which is trimaximally mixed, as in the case of tri-bimaximal mixing Eq. 1.
S3 symmetry remains interesting. It has been remarked [17] that the S3 invariance of the democratic matrix is unbroken under the addition of any multiple of the identity matrix to the purely democratic form (if the democratic component has negative sign then an inverted hierarchy results, still with full S3 invariance). Indeed, taking any polynomial function of a matrix preserves all the symmetries, and in general gives any eigenvalues associated with the original eigenvectors in any order (the Vandermonde matrix [18] formed from the original eigenvalues provides the transformation between the required polynomial coefficients and the desired eigenvalues). The problem with the democratic mass matrix here is that (up to a factor) it is 'idempotent', ie. its square is not an independent matrix and so we have in effect only two polynomial coefficients available (with two degenerate eigenvalues, not only are the two corresponding eigenvectors undefined, but the Vandermonde matrix has no inverse).
In this paper, we 'solve' the problem indicated above, by suggesting that the democratic mass matrix be dropped, in favour of an S3 'group matrix' [19] (more precisely, see below, by an element of the S3 group algebra, in the sense of representation theory) in the natural representation of the S3 group. Later in the paper we extend our argument to give a new and succinct prescription leading to tri-bimaximal mixing itself.

Development of Our Approach with a Familiar Example
Although the original trimaximal mixing scheme [20] [21] seems now essentially ruledout by experiment, it cannot be denied that trimaximal mixing occupied a special place in the space of all possible mixings. In the briefest terms, one had only to require that the neutrino mass matrix in the lepton flavour basis (or the charged-lepton mass matrix in the neutrino mass basis) was a C3 group matrix in the natural representation of C3 (see below), and the lepton mixing matrix was completely determined to be the trimaximal mixing matrix, identically the C3 group character table (see Appendix A) up to an overall normalisation factor 1/ √ 3 [22]. (C3 here is the cyclic group on three objects, while S3 above is the corresponding symmetric group).
Explicitly, a C3 group matrix [19] is just an element of the C3 group algebra, ie. an arbitrary linear combination of the three C3 group elements, with arbitrary (complex) coefficients. (In the context of group representation theory, multiplication of group elements by scalars and addition of group elements are understood in the obvious way, within a given matrix representation). In the natural representation of C3 (using cycle notation and taking I to denote the identity) the C3 group elements may be written: From the physics point of view, if we restrict consideration to left-handed fields only, we may (as usual) take our mass matrices to be hermitian (M → M 2 := MM † ), whereby there is nothing to be gained by considering C3 group matrices which are more general than the hermitian combination: where a is real, b is complex, and b * is the complex conjugate of b (ie. making a complex and replacing b * by an arbitrary complex parameter c, simply yields the same form Eq. 4, on taking the hermitian square, M → MM † ).
We see immediately that Eq. 4 is just the familiar 3 × 3 circulant mass matrix [20]: invariant under cyclic (C3) permutations of the generation indicies. Diagonalising Eq. 5 is equivalent to reducing the C3 group algebra to independent idempotents [19] (the projection operators of Ref. [23]) and leads directly to trimaximal mixing [21]: where ω = exp(i2π/3) andω = exp(−i2π/3) are the complex cube roots of unity. We may say that the trimaximal mixing matrix Eq. 6 is the unitary matrix which reduces the natural representation of C3 to its irreducible form. The eigenvectors of the matrix Eq. 5 (appearing as the columns (or the complex-conjugated rows) of the matrix Eq. 6) are simply the character vectors corresponding to the three inequivalent (1-dimensional) irreducible representations of C3 (in which P (αβγ) (or P (γβα)) acts like 1, ω,ω respectively, see Appendix A).

The Neutrino Mass Matrix as an S3 Group Matrix
Forced by experiment to renounce C3 invariance for leptons, we turn with renewed interest to the symmetric group S3. If we are to apply the foregoing argument to S3, we would expect to have to include, in addition, the odd S3 permutations: Constructing a general (hermitian) S3 group matrix amounts to adding an arbitrary (real) linear combination M 2 (odd) of the odd group elements Eq. 7 to the previous linear combination M 2 (even) (:= M 2 from Eq. 4) of the even (C3) operators, Eq. 3: where (as above) there is nothing to be gained by considering non-hermitian S3 group matrices, eg. with x, y, z complex, which always yield the form Eq. 8 on taking the hermitian square (M → MM † ). Notice that, within the natural representation, the six S3 group operators (Eq. 3 together with Eq. 7) are not fully independent: I + P (αβγ) + P (γβα) = P (αβ) + P (βγ) + P (γα) (9) so that the effective number of (real) parameters in Eq. 8 is actually only five. 1 The structure of Eq. 8 may perhaps be best appreciated by noting that the contribution of the odd operators (from Eq. 7) is 'retrocirculant' [24]: The significant property of a retrocirculant here, is that it has non-degenerate eigenvalues in general, and (clearly) always one trimaximal eigenvector (1 it is evident that these properties are not in general invalidated by the inclusion of the circulant (even) contribution already discussed. With S3 being a nonabelian group, Eq. 8 is not invariant under S3 permutations of the generation indices. We observe that it does, however, satisfy an S3 invariant constraint: for all (S3) permutations of the generation indices (α, β, γ = e, µ, τ ). If Eq. 8 is taken to be the neutrino mass matrix in the lepton-flavour basis (as was already anticipated by the subscript on M 2 ν in Eq. 8 and by the introduction of explicit flavour indices in Eq. 11), then, for a suitable choice of the coefficients, we have: ('suitable' only in the sense that the mass-eigenstates should turn out to be ordered appropriately, eg. m 2 1 < m 2 2 < m 2 3 for a conventional neutrino mass hierarchy). Eq. 8 is seen to correspond to the (two-parameter) mixing scheme proposed phenomenologically in Ref. [2] (to interpolate between tri-φmaximal and tri-χmaximal mixing): In Eq. 15 we have used the abreviations: c χ = cos χ, s χ = sin χ, c φ = cos φ, s φ = sin φ, where: The CP -violation parameter J [25] is given by : Clearly, imposing Im b = 0 in Eq. 8 (ie. χ = 0) would imply no CP violation, whereas imposing y = z instead (ie. φ = 0) implies 'mu-tau reflection symmetry' [26]. For both Im b = 0 and y = z (ie. χ = 0 and φ = 0) the mixing matrix Eq. 15 evidently reduces to the tri-bimaximal form [1] [2], as does the mass matrix (Eq. 8), accordingly [4]. The six constants appearing in Eq. 8 may be expressed in terms of the three neutrino masses and the two mixing-matrix parameters as follows: where clearly (by virtue of Eq. 9) any arbitrary constant may be added to Eqs.19-20 provided that the same constant is subtracted from Eqs. 21-23. Note that in this approach, in the case of S3 (cf. the case of C3, Section 2) the resulting mixing matrix (Eq. 15) is not directly the S3 character table. It is simply the generic unitary matrix which reduces the natural representation of S3 to irreducible form. The natural representation of S3 comprises the trivial 1-dimensional representation and a faithful 2-dimensional representation, which is determined only up to similarity transformations (hence the undetermined parameters appearing in Eq. 15). From its present derivation (and to distinguish it clearly from mixing ansatze based on the 'democratic' mass matrix) we will refer to the mixing Eq. 15 as 'S3 group mixing'.

The Charged-Lepton Mass Matrix as an S3 Class Matrix
We have seen in Section 2 that a very succinct way to introduce trimaximal mixing is to demand that one or other of the mass-matrices is a C3 group matrix in the natural representation of C3. The mixing matrix is then essentially the C3 character table, with all states trimaximally mixed [22]. We went on to generalise the argument to S3, finding that it is enough to take one of the mass matrices to be an S3 group matrix in the natural representation of the S3 group, to obtain a mixing matrix where one (and in particular any one) of the eigenvectors is trimaximally mixed, thereby 'solving' the problem of the democratic mass matrix discussed in Section 1.
However, a more predictive (and perhaps more interesting) way to generalise the trimaximal argument is to recognise that, with C3 being an abelian group, there is no distinction between the group elements and the group conjugacy classes in that case. Each C3 group element being individually a class, an arbitrary element of the C3 group algebra is also an arbitrary element of the C3 class algebra. It is therefore not obvious that the better generalisation to S3 should not simply postulate that one or other of the mass matrices should live in the natural representation of the S3 class algebra, rather than the S3 group algebra, which is certainly a signifcantly different idea.
Following this line of thought, we define (normalised) S3 class operators c i : where the precise physical meaning of ξ, η, ζ remains unclear. Evidently, the S3 class multiplication table (by definition commutative) then takes the form: The structure constants in the table themselves provide a matrix representation for the c i (which is the natural representation of the S3 class algebra in terms of the c i ): as is readily verified by direct multiplication of the matrices. We now postulate that the charged-lepton mass matrix in the neutrino mass-basis is a suitable linear combination of the S3 class operators in the above representation: ie. explicitly: From the usual argument (see eg. Section 2) the coefficients p, q, r may be taken to be real. The eigenvalues of the matrix Eq. 29 are then the charged-lepton masses: The coefficients (being 'suitable' only in that 0 < r/ √ 3 < q/ √ 2 < p to order the mass-eigenstates in accord with experiment) are expressible in terms of the masses by: The unitary matrix diagonalising Eq. 29 (independent of the values of p, q and r) is directly the tri-bimaximal mixing matrix: (the neutrino mass-eigenstates having been already implicitly ordered in accord with experiment, by the labelling of the class operators, in Eqs. [24][25][26]. The charged-lepton mass eigenstates (ie. the eigenvectors of Eq. 29) appear as the rows of Eq. 36.
Clearly, the tri-bimaximal mixing matrix Eq. 36 is very closely related to the S3 table of characters (cf. as displayed in Appendix A below). In fact, it differs only by the class-dependent normalisation factors introduced into Eqs. 24-26. Diagonalising an S3 class matrix (such as Eq. 29) is entirely equivalent [27] to determining the S3 group characters, ie. to finding all the irreducible represenations of S3 by reducing the S3 class algebra to independent idempotents. Explicitly: i is the i-th component of the l-th character vector, g i is the order of the class (g i = 1, 2, 3 for i = 1, 2, 3 for S3) and g is the order of the group (g = 6 for S3). Individual character components are simply related [27] to the eigenvalues of the corresponding class operators, whereby the charged-lepton masses may also be expressed (equivalently to Eqs. 30-32) in terms of the S3 group characters and the constants (p 1 , p 2 , p 3 ) := (p, q, r), as follows: where d l is the dimension of the irreducible representation corresponding to the lepton flavour l. The irreducible representations for l = τ and l = µ are the two mutually conjugate 1-dimensional representations (the trivial and alternating representations respectively), while the electron (l = e) is to be associated with the 2-dimensional faithful representation having a self-conjugate tableau. In the extreme hierarchical limit, r/ √ 3 → q/ √ 2 → p in Eqs. 30-32, only the trivial representation has mass. It is perhaps worth re-iterating at this point that 'data on neutrino oscillations point strongly . . . to tri-bimaximal mixing' [1]. We note that from its present derivation, and in view of the need to distinguish it from 'S3 group mixing' (Section 3), tri-bimaximal mixing might reasonably be termed 'S3 class mixing'.

The Neutrino Mass Matrix as an S3 ⊃ S2 Class Operator.
Alerted to the relevance of class operators, we may now return to reconsider the neutrino mass matrix in the flavour basis. According to Section 3, the charged-lepton flavour basis (α, β, γ = e, µ, τ ) carries the natural representation of the S3 group (it was also noted that tri-bimaximal mixing requires a particular S3 group matrix with Im b = 0 and y = z). Clearly any representation of a group also provides a representation for the classes, and seeking consistency with the results of Section 4, we now postulate that the neutrino mass matrix in the flavour basis is a class operator for the canonical subgroup chain S1 ⊂ S2 ⊂ S3 in the natural representation of the S3 group (class operators for successive subgroups clearly commute). The individual class operators may be written: (class normalisation factors are not needed here since they may be absorbed into the coefficients s, t, u below, with no change of basis involved). The S2 class operator C(2) has been chosen to be the µ − τ interchange operator [26]. Of course, C(3) is familiar as the 'democratic' mass matrix.
The most general (hermitian) S1 ⊂ S2 ⊂ S3 class operator may be written: Explicitly: where s, t, u are real. The eigenvalues of the matrix Eq. 41 are the neutrino masses: The coefficients 0 ≤ 3u ≤ −2t ≤ 2s (for m 2 1 ≤ m 2 2 ≤ m 2 3 ) are given in terms of the neutrino masses by: now with no arbitrary constant involved (cf. Eqs. 19-23). The extreme hierarchical limit for the neutrino masses is approached as u → 0 and t → −s, when only the ν 3 has mass. It may be remarked that it is the 'democratic' component C(3) which has the (numerically) smallest coefficient (u) in Eq. 41, vanishing in the hierarchical limit.
In retrospect, the original circulant mass matrix [20] leading to trimaximal mixing might have been seen as a class operator for the group chain S3 ⊃ C3. Of course we now know that, for the neutrino mass matrix in the flavour basis, an S3 ⊃ S2 class operator Eq. 41, is preferred experimentally (the '⊃ S1' in fact carries no additional symmetry information and is dropped here in accord with usual practice).

Discussion
We have been to a large degree logically led, from the original trimaximal hypothesis, first to 'S3 group mixing' Eq. 15, and then on to 'S3 class mixing' or 'S3 ⊃ S2 mixing', ie. to tri-bimaximal mixing. The two levels of generalisation are not inconsistent: the latter is clearly more restrictive, in that exact tri-bimaximal mixing requires the charged-lepton mass matrix to be an S3 class matrix in the neutrino mass-basis, and also requires the neutrino mass matrix in the lepton flavour basis to be a particular S3 group matrix (with Im b = 0 and y = z), ie. an S3 ⊃ S2 class operator. For a discussion of the forms of both mass-matrices in an intermediate basis see Ref. [2].
Thus, while 'S3 group mixing' is regarded as an interesting mixing ansatz in its own right [2], our main results relate to tri-bimaximal mixing, and the link to the S3 group characters [19] via Eq. 37 (Section 4) and to the S3 induction coefficients [27] (Section 5). In the first case the neutrino mass eigenstates are associated with the normalised S3 class operators Eqs. 24-26 (ν i ∼ c i ), while the charged-lepton masseigenstates are in correspondence with the S3 irreducible representations. Then, in the flavour basis, the charged leptons e, µ, τ are in correspondence with the C3 classes c 0 , c − , c + respectively (viewed as the coset representatives with respect to the µ − τ exchange subgroup) while the neutrino mass-eigenstates are in correspondence with the irreducible basis vectors of the corresponding induced representation of S3. Clearly classes (and hence linear combinations of classes) are always permutation invariants.
Finally, we remark that the notion of the yukawa couplings here being related to the structure constants of a permutation class algebra, is not so different in character from the established notion of the couplings between gauge bosons being the structure constants of a lie algebra. Of course as always, experiment will be the ultimate judge, with the detailed experimental predictions of exact tri-bimaximal mixing (eg. P (e → e) → 5/9 ≃ 0.56 in KAMLAND [9], zero CP violation, no high-energy matter resonance etc.) being already documented in the literature [1].

Acknowledgement
We thank P. Slodowy for helpful explanations on group characters. This work was supported by the UK Particle Physics and Astronomy Research Council (PPARC).

Appendix A: Group Character Tables for the C3 and S3 Groups
For ease of reference, the character tables for the cyclic group C3 on three symbols, and for the corresponding symmetric group S3, are given below.
For C3 there are three irreducible representations, all 1-dimensional, where the generator of (say 'clockwise') cyclic permutations acts like 1, ω orω, which are referred to here as the trivial, ω andω-representations respectively. The three classes (c 0 , c + , c − ) comprise the identity, clockwise and anti-clockwise cyclic permutations, respectively.
For S3 there are likewise three irreducible representations, two of which are 1-dimensional. In the trivial representation all group elements act like (+1), while in the alternating representation, elements corresponding to odd permutations act instead like (-1). There is a faithful 2-dimensional representation which may be written [28]: