Neutrino tri-bi-maximal mixing from a non-Abelian discrete family symmetry

The observed neutrino mixing, having a near maximal atmospheric neutrino mixing angle and a large solar mixing angle, is close to tri-bi-maximal. We argue that this structure suggests a family symmetric origin in which the magnitude of the mixing angles are related to the existence of a discrete non-Abelian family symmetry. We construct a model in which the family symmetry is the non-Abelian discrete group $\Delta(27)$, a subgroup of SU(3) in which the tri-bi-maximal mixing directly follows from the vacuum structure enforced by the discrete symmetry. In addition to the lepton mixing angles, the model accounts for the observed quark and lepton masses and the CKM matrix. The structure is also consistent with an underlying stage of Grand Unification.


Introduction
The observed neutrino oscillation parameters are consistent with a tri-bi-maximal structure [1]: It has been observed that this simple form might be a hint of an underlying family symmetry, and several models have been constructed that account for this structure of leptonic mixing (e.g. [2]). It is possible to extend the underlying family symmetry to provide a complete description of the complete fermionic structure (e.g. [3]) 1 , in which, in contrast to the neutrinos, the quarks have a strongly hierarchical structure with small mixing with Yukawa coupling matrices of the form [5]: where the expansion parameters are given by ǫ u ≈ 0.05, ǫ d ≈ 0.15. (4) A desirable feature of a complete model of quark and lepton masses and mixing angles is that it should be consistent with an underlying Grand Unified structure, either at the field theory level or at the level of the superstring. The family symmetry models which have been built to achieve this are based on an underlying G f ⊗ SO(10) structure where the family group G f is SU (3) f [6,7]. This is very constraining because it requires that all the (left handed) members of a single family should have the same family charge. In this paper we will construct a model based on a non-Abelian discrete family symmetry which preserves the possibility of simple unification by requiring that the discrete symmetry properties of all the members of one family are the same. The discrete non-Abelian group 2 we use is ∆(27), the semi-direct product group Z 3 ⋉ Z ′ 3 , which is a subgroup 3 of SU (3) f . Indeed the dominant terms of the Lagrangian leading to the Yukawa coupling matrices of the form of eq.(2) and eq.(3) are symmetric under SU (3) f so much of the structure of the model based on SU (3) f is maintained. However the appearance of additional terms allowed by Z 3 ⋉ Z ′ 3 but not by SU (3) f determines the vacuum structure and generates the tri-bi-maximal mixing structure. The choice of the multiplet structure ensures that the model is consistent with a stage of Grand or superstring unification and the resulting model is much simpler than that based on the continuous SU (3) f symmetry.
In Section 2 we discuss the choice of the non-Abelian discrete group and the multiplet content of the model. Emphasis is put on obtaining a simplified field content and a reduced auxiliary symmetry compared with the SU (3) f model in [6]. In Section 3 we consider the superpotential terms allowed by the symmetries of the model. Using this we show how the desired vacuum structure arises simply through the appearance of the additional invariants allowed by Z 3 ⋉ Z ′ 3 but not by SU (3) f . Section 4 discusses both the Dirac and Majorana mass matrix structure of the model and the resulting pattern of quark, charged lepton and neutrino masses and mixing angles. Finally in Section 5 we present a summary and our conclusions.

Field content and symmetries
The symmetry of the model is The additional symmetry group G is needed to restrict the form of the allowed coupling of the theory and is chosen to be as simple as possible. As discussed above, the family group G f is chosen as a non-Abelian discrete group of SU (3) f in a manner that preserves the structure of the fermion Yukawa couplings of the associated SU (3) f model of [6]. This means that G f should be a non-Abelian subgroup of SU (3) f of sufficient size that it approximates SU (3) f in the sense that most of the leading terms responsible for the fermion mass structure in the SU (3) f are still the leading terms allowed by G f (which being a subgroup, allows further terms which we want to be subleading). The smallest group we have found that achieves this is ∆(27), the semi-direct product group Z 3 ⋉ Z ′ 3 . The main change that results from using this smaller symmetry group is the appearance of additional invariants which drive the desired vacuum structure and, because we are no longer dealing with a continuous symmetry, the absence of the associated D-terms which were very important in determining the vacuum structure in the SU (3) f model [6]. Due to this, we are able to reduce the total field content of this model, which in turn only requires an additional G = U (1) ⊗ Z 2 ⊗ R to control the allowed terms in the superpotential 4 In choosing the representation content of the theory we are guided by the structure of the SU (3) f model of [6] which generated a viable form of all quark and lepton masses and mixing. Since Using this we can readily arrange that the superpotential terms responsible for fermion masses in the SU (3) f model are present in the Z 3 ⋉ Z ′ 3 model. To implement this we find it convenient to label the representation of the fields of our model by their transformation 2 Such non-Abelian discrete symmetries often occur in compactified string models.
R is an R− symmetry and for SUSY purposes plays the same role as R−parity. Table 1: Transformation properties of SU (3) f anti-triplet fieldsφ i and triplet fields φ i under the non-Abelian discrete group; α is the cube root of unity, α 3 = 1.

Field
properties under the approximate SU (3) f family group. The Standard Model (SM) fermions ψ i , ψ c j transform as triplets under this group. The transformation properties of such triplets under the Z 3 ⋉ Z ′ 3 discrete group are shown in Table 1. Although the gauge group is just that of the Standard Model it is also instructive, in considering how the model can be embedded in a unified structure, to display the properties of the states under the SU (4) P S ⊗ SU (2) L ⊗ SU (2) R subgroup of SO(10) and this is done in Table 2. We also show in Table 2 the transformation properties under the additional symmetry group G = U (1) ⊗ Z 2 ⊗ R. The transformation properties of the SM Higgs, H, responsible for electroweak breaking 5 are also shown in Table 2.
In a complete unified theory, quark and lepton masses will be related. A particular question that arises in such unification is what generates the difference between the down quark and charged lepton masses. In [6] this was done through a variant of the Georgi-Jarlskog mechanism [9] via the introduction of another Higgs field H 45 , which transforms as a 45 of an underlying SO (10) GUT. It has a vacuum expectation value (vev) which breaks SO(10) but leaves the SM gauge group unbroken. In this model we include H 45 to demonstrate that the model readily Grand Unifies but in practice we only use its vev. This does not necessarily imply that there is an underlying stage of Grand Unification below the string scale but, if not, the underlying theory should provide an alternative explanation for the existence of the pattern of low energy couplings implied by terms involving H 45 .
At this stage there are no terms generating fermion masses and to complete the model it is necessary to break the family symmetry Z 3 ⋉ Z ′ 3 through the introduction of "flavons" that acquire vevs. To reproduce the results of the phenomenologically viable SU (3) f model [6] we choose a similar but somewhat simplified flavon structure with the SU (3) f antitriplet fields θ i ,φ  Table 2, and one triplet field for alignment purposes φ A i . The transformation properties of these fields under Z 3 ⋉ Z ′ 3 are shown in Table 1. With this choice, as discussed in the next Section, the Yukawa structure of the SU (3) f model [6] is obtained. One may readily check that the additional terms allowed by the Z 3 ⋉ Z ′ 3 symmetry are subleading in this sector so the phenomenologically acceptable pattern of fermion masses and mixings obtained in [6] is reproduced here if the flavon vacuum structure is as given in [6]. The main difference between the models is the appearance in the potential determining the vacuum structure of additional invariants allowed by Z 3 ⋉ Z ′ 3 and the absence of the D−terms associated with a continuous gauge symmetry.

Symmetry breaking
Following [6] the desired pattern of vevs is given by where the SU (2) R structure of φ 3 has been displayed. The alignment of these vevs can proceed in various ways. By including additional driving fields in the manner discussed in [10] one can arrange their F −terms give a scalar potential whose minimum has the desired vacuum alignment. Here however we show that an even simpler mechanism involving D−terms only achieves the desired alignment.
To understand how this vacuum alignment works note that, unlike the case for the continuous SU (3) f symmetric theory, it is not possible in general to rotate the vacuum expectation value of a triplet field to a single direction, for example the 3 direction. Due to the underlying discrete symmetry the vev will be quantised in one of a finite set of possible minima. However this may only be apparent if higher order terms in the potential are included for the lower order terms may have the enhanced SU (3) f symmetry.
To make this more explicit, consider a general SU (3) f triplet field φ i . It will have a SUSY breaking soft mass term in the Lagrangian of the form m 2 φ φ i † φ i which is invariant under the approximate SU (3) f symmetry. Radiative corrections involving superpotential couplings to massive states may drive the mass squared negative at some scale Λ triggering a vev for the field φ, < φ i † φ i >= v 2 , with v 2 ≤ Λ 2 set radiatively 6 . At this stage the vev of φ can always be rotated to the 3 direction using the approximate SU (3) f symmetry. However this does not remain true when higher order terms allowed by the discrete family symmetry are included. For the model considered here the leading higher order term is of the form m 2 3/2 (φ † φφ † φ) arising as a component of the D− term In this we have suppressed the coupling constants and the messenger mass scale (or scales), M, associated with these higher dimension operators (which can even be the Planck mass M P ). The F component of the field χ drives supersymmetry breaking and m 3/2 is the graviton mass (m 2 3/2 = F † χ F χ /M 2 P ). This term gives rise to two independent quartic invariants under symmetric and does lead to an unique vacuum state. For the case that the coefficient of positive the minimum corresponds to the vev 7 < φ i > T = v(1, 1, 1)/ √ 3 (c.f. eq. (7)). For the case the coefficient is negative, the vev has the form < φ i > T = v(0, 0, 1) (c.f. eq.(9)). Thus we see that, in contrast to the continuous symmetry case, the discrete non-Abelian symmetry leads to a finite number of candidate vacuum states. Which one is chosen depends on the sign of the higher dimension term which in turn depends on the details of the underlying theory. In this paper we do not attempt to construct the full theory and so cannot determine this sign. What we will demonstrate, however, is that one of the finite number of candidate vacua does have the correct properties to generate a viable theory of fermion masses and mixings.
The vacuum alignment needed for this model can now readily be obtained. Suppose that a combination of radiative corrections and the U (1) D-term drive m 2 φ 123 , m 2 φ 1 and m 2 φ 3 negative close to the messenger scale, The symmetries of the model ensure that the leading terms fixing their vacuum structure are of the form m 2 , plus similar terms involvingφ 3 . Provided the unmixed terms of the form of the first two terms dominate the vevs will be determined by the signs of these terms. If the quartic term involving φ 123 is positive φ 123 will acquire a vev in the (1, 1, 1) direction as in eq.(7). If the quartic term involving φ 1 is negative φ 1 will acquire a vev in the (1, 0, 0) direction as in eq. (8) where the non zero entry just defines the 1 direction. Finally if the quartic term involvingφ 3 is also negative it will acquire a vev with a single non-zero entry but the position of this entry will depend on the leading D−term resolving this ambiguity. If the term m 2 3j ) dominates and has positive coefficient it will force the vevs of these fields to be orthogonal and soφ 3 has a vev in the (0, 0, 1) direction, c.f. eq.(5), where again the non zero entry just defines the 3 direction. In a similar manner it is straightforward to see how the fields φ 3 and θ align along the (0, 0, 1) direction if the quartic terms m 2 ) and m 2 3/2 (φ i 3 θ i θ †jφ † 3j ) dominate and have negative coefficients. The scale of their vevs is determined by the scale at which their soft mass squared become negative (the direction of φ 3 is not very relevant, but with the above terms similar to θ it can take the form in eq.(9) and we take it to be so for simplicity).
The relative alignment of the remaining terms follows in a similar manner. Consider the fieldφ 23 with a soft mass squared becoming negative at a scale b < v. Forφ 23 we want the dominant term aligning its vev to be m 2 , with positive coefficient. It will then acquire a vev orthogonal to that of φ 123 . The choice of the particular orthogonal direction will be determined by terms like m 2 ) . If the latter dominates with a positive coefficient, it will drive φ 23 orthogonal to φ 1the form given in eq. (6).
Finally consider the fieldφ 123 with a soft mass squared becoming negative at a scale c ≪ v. The leading terms determining its vacuum alignment are m 2 ) . If the latter dominates with a negative coefficient,φ 123 will be aligned in the same direction as φ 123 and have the form given in eq. (7). Note that the term involvingφ 23 is accidental in the sense that it is dependant on the U (1) assignments of the field.
In summary, we have shown that higher order D−terms constrained by the discrete family symmetry lead to a discrete number of possible vacuum states. Which one is the vacuum state depends on the coefficients of these higher order terms which are determined by the underlying unified GUT or string theory. Our analysis has shown that the vacuum structure needed for a viable theory of fermion masses can readily emerge from this discrete set of states. 4 The mass matrix structure 4

.1 Yukawa terms
We turn now to the structure of the quark and lepton mass matrices. The leading Yukawa terms allowed by the symmetries are: Although of a slightly different from from that in [6] these terms realize the same mass structure and we refer the reader to [6] for the details. It gives a phenomenologically consistent description of all the quark masses and mixing angles and the charged lepton masses, generating their hierarchical structure through an expansion in the family symmetry breaking parameters. The main differences in the way this is achieved lies in eqs. (14, 15, 16). The terms in eqs. (14,15) account for the observed O ǫ 3 d difference in the 12, 21 and 13, 31 entries 8 of the down-type quark mass matrix (c.f. eq.(3)) [5].
The term in eq.(16) is undesirable, but allowed by the symmetries nonetheless. Naively, one expects it would contribute to the 11 element at O ǫ 4 d giving unwanted corrections to the phenomenologically successful Gatto-Sartori-Tonin relation [11] which results if the 11 entry is less than this order [6]. Fortunately, this texture zero is preserved at that order, as the vevs of φ 3 andφ 3 are slightly smaller than the relevant messenger mass scales, and in the eq.(16) there are four such fields, suppressing the term sufficiently. As such, the desired small magnitude of this term can be maintained while keeping the dimensionless coefficients in front of all the allowed Yukawa terms as O(1).

Majorana terms
The leading terms that contribute to the right-handed neutrino Majorana masses are: Note that these terms are different from those in [6] and lead to a different form for the ratios of the Majorana masses. The vev of φ 3 controls the hierarchy between M 1 (given essentially by eq.(19)) and M 2 (from eq.(18)).
non-Abelian discrete group. In addition to the prediction of near tri-bi-maximal mixing the model preserves the Gatto-Sartori-Tonin [11] relation between the light quark masses and the Cabibbo mixing angle, and can accommodate the GUT relations between the down quark and lepton masses. It also provides a explanation for the hierarchy of quark masses and mixing angles in terms of an expansion in powers of a family symmetry breaking parameter.