Obtaining the Neutrino Mixing Matrix with the Tetrahedral Group

We discuss various"minimalist'' schemes to derive the neutrino mixing matrix using the tetrahedral group $A_{4}.$


I. Neutrino mixing matrix
The neutrino mixing matrix V relates the neutrino current eigenstates (denoted by ν α , α = e, µ, τ, and coupled by the W bosons to the corresponding charged leptons) to the neutrino mass eigenstates (denoted by ν i , i = 1, 2, 3 and endowed with definite masses m i ) according to Thanks to heroic experimental efforts, the neutrino mixing angles have now been determined [1] to be given by sin 2 θ 12 ∼ 0.31, sin 2 θ 23 ∼ 0.50, and sin 2 θ 31 ∼ 0.01, with the mixing angles defined by the standard parametrization (with c 23 ≡ cos θ 23 , s 23 ≡ sin θ 23 , and so forth) (This parametrization may differ slightly from others in that we take det V 23 = det V 12 = −1.) The error bars are such that θ 31 is consistent with 0, in which case the CP violating phase e iφ does not enter. We could suppose either that the entries in V represent a bunch of meaningless numbers possibly varying from village to village in the multiverse landscape as advocated by some theorists of great sophistication or that they point to some deeper structure or symmetry as some theorists with a more traditional faith in the power of theoretical physics might dare to hope for. It is natural to imagine that there is a family symmetry [2] linking the three lepton families.
Starting with the standard model we assign (all fermionic fields are left handed) the lepton doublets ψ a = ν a l a , the lepton singlets l C a (a = 1, 2, 3), and the required Higgs fields to various representations of a family group [3] G F . Indeed, if we guess that s 12 = 1/ √ 3, s 23 = 1/ √ 2, and s 31 = 0, we obtain the attractive mixing matrix first proposed by Harrison, Perkins and Scott [4]. Later, X. G. He and I independently arrived at the same Ansatz [5] Also, this mixing matrix (but curiously, with the first and second column interchanged) was first suggested by Wolfenstein more than 20 years ago [6] based on some considerations involving the permutation group S 3 . It has subsequently been studied extensively by Harrison, Perkins and Scott [7] , and by Xing [8]. Attempts to derive this mixing matrix have been discussed by Low and Volkas [9] [10]. A parametrization of the experimental data in terms of deviation from V is given in [11]. Following Wolfenstein and defining ν x ≡ (ν µ + ν τ )/ √ 2 and ν y ≡ (ν µ − ν τ )/ √ 2, we see that (5) says that the mass eigenstates are given by The basis {ν 1 , ν 2 } is rotated from {ν e , ν x } through arcsin(1/ √ 3) ∼ 35 o . In this paper we will take the neutrinos to be Majorana [12] as seems likely, so that we have in the Lagrangian the mass term L = −ν α M αβ Cν β + h.c. where C denotes the charge conjugation matrix. Thus, the neutrino mass matrix M is symmetric. Also, for the sake of simplicity we will assume CP conservation so that M is real. With this simplification, the orthogonal transformation V T M V produces a diagonal matrix with diagonal elements m 1 , m 2 and m 3 . We are free to multiply V on the right by some diagonal matrix whose diagonal entries are equal to ±1. This merely multiplies each of the columns in V by an arbitrary sign. Various possible phases have been discussed in detail in the literature [13,14].
At present, we have no understanding of the neutrino masses just as we have no understanding of the charged lepton and quark masses. The well-known solar and atmospheric neutrino experiments have determined respectively that ∆m 2 ⊙ = m 2 2 − m 2 1 ∼ 8 × 10 −5 eV 2 and ∆m 2 atm = m 2 3 − m 2 2 ∼ ±2.4 × 10 −3 eV 2 . The sign of ∆m 2 atm is currently unknown, while ∆m 2 ⊙ has to be positive in order for the Mikheyev-Smirnov-Wolfenstein resonance to occur inside the sun. We could have either the so-called normal hierarchy in which |m 3 | > |m 2 | ∼ |m 1 | or the inverted hierarchy |m 3 | < |m 2 | ∼ |m 1 |.

II. Family symmetry and the tetrahedral group
For some years, E. Ma [15] has advocated choosing the discrete group A 4 , namely the symmetry group of the tetrahedron, as G F . With various collaborators he has written a number of interesting papers [16] [17] [18] [19] using A 4 to study the lepton sector.
For the convenience of the reader and to set the notation, we give a concise review of the relevant group theory. Evidently, A 4 is a subgroup of SO(3) (which was often used in the early literature on family symmetry but which has proved to be too restrictive.) Since the tetrahedron lives in 3-dimensional space, A 4 has a natural 3-dimensional representation denoted by 3 suggestive of the 3 families observed in nature. The tetrahedron has 4 vertices and thus A 4 is also formed by the even permutations of 4 objects so that A 4 has 4!/2 = 12 elements which could be represented as elements of SO(3). Besides the identity I = The multiplication of representations is easy to work out by using the following trick. Start with the familiar multiplication within SO(3) : 3 × 3 = 1 + 3 + 5. Given two vectors x and y of SO(3), the 3 is of course given by the cross product x× y while the 5 is composed of the symmetric combinations x 2 y 3 + x 3 y 2 , x 3 y 1 + x 1 y 3 , x 1 y 2 + x 2 y 1 , together with the 2 diagonal traceless combinations 2x 1 y 1 − x 2 y 2 − x 3 y 3 and x 2 y 2 − x 3 y 3 . Upon restriction of SO(3) to A 4 the 5 evidently decompose into 5 → 3 + 1 ′ + 1 ′′ with the 3 given by the 3 symmetric combinations just displayed. The 1 ′ and 1 ′′ could be taken respectively as linear combinations of the 2 traceless combinations just given: and with ω ≡ e i2π/3 the cube root of unity so that It is perhaps worth emphasizing the obvious, that while 1 ′ and 1 ′′ furnish 1-dimensional representations of A 4 they are not invariant under A 4 . For example, under the cyclic permutation c, u ′ → ωu ′ and u ′′ → ω 2 u ′′ . Evidently 1 ′ × 1 ′′ = 1, 1 ′ × 1 ′ = 1 ′′ ,and 1 ′′ × 1 ′′ = 1 ′ , and also (1 ′ ) * = 1 ′′ .
Thus, under A 4 we have 3 × 3 = 1 + 1 ′ + 1 ′′ + 3 + 3. It is perhaps also worth remarking that the two 3's on the right hand side may be taken as (x 2 y 3 , x 3 y 1 , x 1 y 2 ) and (x 3 y 2 , x 1 y 3 , x 2 y 1 ). The existence of 3 inequivalent 1-dimensional representations also suggests the relevance of A 4 to the family problem. I can't resist mentioning here the possibly physically irrelevant fact that [20] alone among all the alternating groups A n 's the group A 4 is not simple.

III. A minimalist framework
Given these attractive features of A 4 , there has been, perhaps not surprisingly, a number of recent attempts [15] [21] [22] to derive V using A 4 . In our opinion, they all appear to involve a rather elaborate framework, for example supersymmetry, higher dimensional spacetime, and so on. Within this recent literature Ma [27] has produced a particularly interesting and relatively economical scheme in which the neutrino mixing matrix depends on a parameter such that when that parameter takes on "reasonable" values the matrix V as given in (5) is recovered approximately.
The guiding philosophy of this paper is that we would like to have as minimal a theoretical framework as possible.
Within a minimalist framework, charged lepton masses are generated by the dimension 4 operator Here ϕ denotes generically the standard Higgs doublet, of which we may have more than one. According to a general low energy effective field theory analysis [23] [24] [25] neutrino masses are generated by the dimension 5 operator in the Lagrangian. Here ξ and ξ ′ denote various Higgs doublets that may or may not be the same as the ϕ's. We will suppress the charge conjugation matrix C and the Pauli matrix τ 2 in what follows. It is important to emphasize that the analysis leading up to (11) is completely general and depends only on SU (2) × U (1), and not on which dynamical model you believe in, be it the seesaw mechanism or some other mechanism (such as the model in [26]). We suppose that the family symmetry remains unbroken down to the scale of SU (2) × U (1) breaking, so that the operators O 4 and O 5 have to be singlets under G F . As is completely standard, when ϕ, ξ, and ξ ′ acquire vacuum expectation values, SU (2) × U (1) and G F are broken and the neutrinos acquire masses given by the mass matrix  (1) is given by V = U † l U ν . One difficulty in constructing a theory for V is that it arises from the "mismatch" between two rotations U l and U ν .
As in turns out, in our model building efforts, we often have to forbid the ϕ's that appear in O 4 from appearing in O 5 . This could easily be implemented by imposing a discrete symmetry under which ϕ → e iκ ϕ, l C → e iκ l C (where e iκ = −1 is some appropriate phase factor), with all other fields unaffected. We will leave this implicit in what follows.
Within the minimalist framework outlined here we offer some possible schemes. None of these could be said to be terribly compelling but at least we keep within the usual rules of the model building literature. The various schemes, depending on what representations of A 4 we choose for the various fields ψ, l C , and ϕ, could be listed systematically.

IV. Model A
We first try the assignment ψ ∼ 3, l C ∼ 1, 1 ′ , and 1 ′′ , and ϕ ∼ 3. The Lagrangian then contains the terms It is natural for the 3 < ϕ α >= v α 's to be equal since A 4 requires that the coefficients of ϕ † α ϕ α and of (ϕ † α ϕ α ) 2 in the potential be independent of α = 1, 2, 3. (See the Appendix for a more detailed analysis.) If so, then upon spontaneous gauge symmetry breaking we obtain with m e = h 1 v and so on. It is useful to define the "magic" matrix [28] A = Then The crucial observation at this point is that the sum of the first and third columns in A gives then U † l U ν = V Φ with V the desired mixing matrix in (5) and the diagonal phase matrix Φ with the diagonal elements −1, 1, and −i. Thus, if we could obtain U ν we would achieve our goal of deriving V.
We recognize that U ν is just a rotation through 45 o in the (1 − 3) plane. Recalling that U ν is determined by requiring U T ν M ν U ν = D ν be diagonal we see that if we could obtain an M ν of the form (note that the 2 × 2 matrix in the (1 − 3) sector has equal diagonal elements) then we are done. Our discussion here overlaps with that given recently by Babu and He [22]; however, their discussion is given in the context of a much more elaborate scheme involving supersymmetry. Referring to (11) we see that by imposing a discrete symmetry K 2 under which ψ 2 → −ψ 2 , ϕ 2 → −ϕ 2 , with all other fields unaffected, or equivalently a discrete symmetry K 13 under which ψ 1 → −ψ 1 , ψ 3 → −ψ 3 , ϕ 1 → −ϕ 1 , ϕ 3 → −ϕ 3 , with all other fields unaffected, we can obtain the texture zeroes in (17), but unfortunately this does not imply that ( M ν ) 11 = ( M ν ) 33 . Furthermore, K 13 is just the element r 2 of A 4 and so it does not commute with A 4 . Note that upon the ϕ's acquiring equal vacuum expectation values, A 4 is broken down to a Z 3 generated by {I, c, a} and unfortunately r 2 does not belong in Z 3 . Perhaps there is a more attractive scheme in which a reflection symmetry like K 2 could emerge effectively.
(This is essentially the same as the analysis of an A 4 invariant Higgs potential given in the Appendix.) Upon ξ 2 acquiring a vacuum expectation value, we obtain respectively ψ 2 ψ 2 , ψ 2 ψ 2 , 0, ψ 1 ψ 1 and ψ 3 ψ 3 . Unfortunately, the effective coupling constants in front of the operator in (21) and (22) are in general not equal to each other and thus we obtain an M ν of the form rather than the M ν in (17). To set ε to 0 we would have impose a discrete interchange symmetry P 13 which interchanges the indices 1 and 3 but unfortunately, just as before for K 13 , P 13 does not commute with A 4 . At this point, we could only suppose that ε is small compared to β, in which case U ν is perturbed from the desired U ν in (16) to The resulting deviation from the V in (5) may be interesting phenomenologically. In particular, V e3 ≃ − ε √ 6β is no longer identically 0. In [11] it was advocated that experimental data be parametrized as a deviation from V in (5) as discussed in section III there.
In this scheme, the neutrino masses come out to be α − β 2 + ε 2 ,γ, and α + β 2 + ε 2 and thus both the normal hierarchy and the inverse hierarchy could be accomodated by suitable tuning, but there is no true understanding of neutrino masses as remarked earlier.

V. Model B
Following Ma [27], we take ψ ∼ 3, l C ∼ 3, and ϕ ∼ 1, 1 ′ , and 1 ′′ . In other words, we have 3 Higgs doublets ϕ each transforming as a singlet under A 4 . The Lagrangian then contains the terms Upon the ϕ's acquiring vacuum expectation values v we obtain a diagonal charged lepton mass matrix, with the charged lepton masses given by the absolute values of All that matters here for our purposes is that we have enough freedom to match the observed masses m e , m µ , and m τ . The salient point here is that U l = I, so that we only have to worry about getting the desired U ν .
As is obvious and as was discussed in [11] and in [5], in a basis in which the charged lepton mass matrix is already diagonal, the neutrino mass matrix M ν is of course determined in terms of the three neutrino masses and the neutrino mixing matrix V. Call the three column vectors in the mixing matrix v i . Then M ν is given by In particular, if we believe in the V in (5) we have With A 4 it is natural to obtain the matrix M D ≡   1 1 1 1 1 1 1 1 1   and the identity matrix. In particular, if we introduce a Higgs doublets ξ transforming as 3 under A 4 and arrange the Higgs potential such that the 3 vacuum expectation values < ξ 1 >=< ξ 2 >=< ξ 3 > are equal. We then see from the list of operators of the form (ξϕ)(ξϕ) given in (18)(19)(20)(21) at the end of the last section that we obtain for M ν an arbitrary linear combination of M D and the identity matrix, which is not what we want.
In [5], in discussing the neutrino mass matrix, we proposed a basis of 3 matrices other than those that appear in (27). First, the 3 column vectors in V are the eigenvectors of the matrix with eigenvalues m 1 = m 2 = 2a, and m 3 = −4a. (The parameter a merely sets the overall scale.) Thus, with M 0 as the mass matrix ∆m 2 21 = 0 and this pattern reproduces the data |∆m 2 21 |/|∆m 2 32 | ≪ 1 to first approximation. Because of the degeneracy in the eigenvalue spectrum, V is not uniquely determined. To determine V, and at the same time to split the degeneracy between m 1 and m 2 , we perturb M 0 to M = M 0 + εaM D . The matrix M D is evidently a projection matrix that projects the first and third columns in V to zero. Thus, the eigenvalues are given by m 1 = 2a, m 2 = 2a(1 + 3ε/2), and m 3 = −4a, where to the lowest order ε = ∆m 2 21 /∆m 2 32 and a 2 = ∆m 2 32 /12. Finally, to break the relation |m 3 | = 2|m 1 | ≃ 2|m 2 | we can always add to M a term proportional to the identity matrix. But it seems difficult to get the matrix in (28) using A 4 alone.

VII. Other possibilities and conclusion
Given that A 4 has only 4 distinct representations we could of course systematically go through all possibilities. Thus, next we could take ϕ ∼ 3, l C ∼ 3, and ψ ∼ 1, 1 ′ , and 1 ′′ . The charged lepton mass term would have a form analogous to that given in (12). But clearly, if we now assume the < ϕ α >'s to be equal, we once again get the matrix A but now acting on l C instead of on ψ. Note that if we assign ψ 2 ∼ 1 and ψ 1 , ψ 3 to 1 ′ and 1 ′′ respectively and introduce a Higgs doublets χ transforming as 1, we get via the operator O 5 a neutrino mass matrix M ν of the form in (17) but with α = 0.
Another possibility is to assign ψ, ϕ, and l C all to the 3 in which case the charged lepton mass matrix is generated by two terms, . If we assume the < ϕ α >'s to be equal, then the three charged lepton masses are given in terms of only two parameters.
In conclusion, we have discusssed various schemes to obtain a particularly attractive neutrino mixing matrix that closely approximates the data. Instead of detailed models, we use a low energy effective field theory approach, allowing only Higgs doublets to survive down to the electroweak scale. We have also explicitly made the restrictive assumption that A 4 survives down to the SU (2) × U (1) breaking scale. Of course, if Higgs triplets could also be used, as for example in [16], or if A 4 is broken at higher scale (for example by the coupling of the ϕ's to the singlet scalar field h in the model in [26]), then many more possibilities open up and one could go beyond the discussion given here. We have been intentionally restrictive here. Ultimately, of course, any disucssion of neutrino mixing shuld be given in a grand unified framework (for recent attempts, see for example [29] [30] in which neutrino masses, as well as quark masses and mixing, are also "explained." We do not attempt this more ambitious program in this paper. It is then straightforward though tedious to calculate the value of V and the eigenvalues Ω of the second derivative matrix ∂ 2 V ∂vα∂v β evaluated at the three mimina of interest: E : {v 1 = v 2 = v 3 = v}, U : {v 1 = v, v 2 = v 3 = 0}, and P : {v 1 = v 2 = v, v 3 = 0}. We find E : v 2 = µ 2 2(λ+λ) , V | E = − 3µ 4 4(λ+λ) , Ω = [4µ 2 , 2µ 2 (2λ−λ) We note that by choosingλ < 0 and sufficiently close to −λ or by not doing this we could set V | E much lower than V | U or vice versa. On the other hand, for P to be a minimum, we needλ − 2λ > 0, which would make V | U lower than V | P . It appears that in this simple one Higgs doublet case, P is never the true minimum. Of course, in all the models we discussed, we have to introduce more than one Higgs doublets and so presumably almost anything is possible by coupling the various doublets together.