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

###### Abstract

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 , a subgroup of 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.

## 1 Introduction

The observed neutrino oscillation parameters are consistent with a tri-bi-maximal structure [1]:

(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}^{1}1See [4] for review papers with extensive references on neutrino
models, 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]:

(2) |

(3) |

where the expansion parameters are given by

(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 structure where the
family group is 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}^{2}2Such non-Abelian discrete symmetries often occur in compactified string
models. we use is the semi-direct product group , which is a subgroup ^{3}^{3}3 (where the generators of the distinct don’t
commute) is the group [8] of 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 so much of the structure of the model based on is
maintained. However the appearance of additional terms allowed by but not by 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 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 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 but not by . 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.

## 2 Field content and symmetries

The symmetry of the model is The additional symmetry group 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 is
chosen as a non-Abelian discrete group of in a manner that
preserves the structure of the fermion Yukawa couplings of the associated model of [6]. This means that should be a
non-Abelian subgroup of of sufficient size that it approximates in the sense that most of the leading terms responsible for the
fermion mass structure in the are still the leading terms
allowed by (which being a subgroup, allows further terms which we
want to be subleading). The smallest group we have found that achieves this
is the semi-direct product group . 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 -terms which were very important in
determining the vacuum structure in the model [6].
Due to this, we are able to reduce the total field content of this model,
which in turn only requires an additional to
control the allowed terms in the superpotential ^{4}^{4}4 is an
symmetry and for SUSY purposes plays the same role as parity. (c.f. in [6]).

In choosing the representation content of the theory we are guided by the
structure of the model of [6] which generated a
viable form of all quark and lepton masses and mixing. Since is a discrete subgroup of all invariants of are invariants of Using this we
can readily arrange that the superpotential terms responsible for fermion
masses in the model are present in the model. To implement this we find it convenient to label the
representation of the fields of our model by their transformation properties
under the approximate family group. The Standard Model (SM)
fermions transform as triplets under this group.
The transformation properties of such triplets under the 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 subgroup of and this
is done in Table 2 We also show in Table 2 the
transformation properties under the additional symmetry group . The transformation properties of the SM Higgs, , responsible for electroweak breaking ^{5}^{5}5Two Higgs are required due to SUSY, represented as , they have the same
charges under and . 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 , which transforms as a of an underlying GUT. It has a vacuum expectation value (vev) which breaks but leaves the SM gauge group unbroken. In this model we include 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

At this stage there are no terms generating fermion masses and to complete the model it is necessary to break the family symmetry through the introduction of “flavons” that acquire vevs. To reproduce the results of the phenomenologically viable model [6] we choose a similar but somewhat simplified flavon structure with the antitriplet fields , and and triplet fields , as shown in Table 2, and one triplet field for alignment purposes . The transformation properties of these fields under are shown in Table 1. With this choice, as discussed in the next Section, the Yukawa structure of the model [6] is obtained. One may readily check that the additional terms allowed by the 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 and the absence of the terms associated with a continuous gauge symmetry.

Field | ||
---|---|---|

Field | |||||||
---|---|---|---|---|---|---|---|

## 3 Symmetry breaking

Following [6] the desired pattern of vevs is given by

(5) |

(6) |

(7) |

(8) |

(9) |

where the structure of 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 terms give a scalar potential whose minimum has the desired vacuum alignment. Here however we show that an even simpler mechanism involving terms only achieves the desired alignment.

To understand how this vacuum alignment works note that, unlike the case for the continuous 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 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 symmetry.

To make this more explicit, consider a general triplet field . It will have a SUSY breaking soft mass term in the Lagrangian of
the form which is invariant
under the approximate symmetry. Radiative corrections involving
superpotential couplings to massive states may drive the mass squared
negative at some scale triggering a vev for the field with set
radiatively ^{6}^{6}6The radiative corrections to the soft mass term depend on the details of the
underlying theory at the string or unification scale.. At this stage the
vev of can always be rotated to the direction using the
approximate 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 arising as a
component of the term . In this we have suppressed the coupling
constants and the messenger mass scale (or scales), associated with
these higher dimension operators (which can even be the Planck mass ). The component of the field drives supersymmetry breaking and is the graviton mass ( This term gives rise to two independent quartic invariants
under namely and The former is symmetric and does not remove the vacuum degeneracy. The second
term is not symmetric and does lead to an unique vacuum state.
For the case that the coefficient of is positive the minimum corresponds to
the vev ^{7}^{7}7In general, the phases are different for each entry of this vev. For
simplicity we omit them, as they don’t affect the results. (c.f. eq.(7)). For the case the
coefficient is negative, the vev has the form
(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 -term drive , and negative close to the messenger scale, . The symmetries of the model ensure that the leading terms fixing their vacuum structure are of the form , plus similar terms involving . 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 is positive will acquire a vev in the direction as in eq.(7). If the quartic term involving is negative will acquire a vev in the direction as in eq.(8) where the non zero entry just defines the direction. Finally if the quartic term involving 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 term resolving this ambiguity. If the term dominates and has positive coefficient it will force the vevs of these fields to be orthogonal and so has a vev in the direction, c.f. eq.(5), where again the non zero entry just defines the direction. In a similar manner it is straightforward to see how the fields and align along the direction if the quartic terms and 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 is not very relevant, but with the above terms similar to 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 field with a soft mass squared becoming negative at a scale . For we want the dominant term aligning its vev to be , with positive coefficient. It will then acquire a vev orthogonal to that of . The choice of the particular orthogonal direction will be determined by terms like or . If the latter dominates with a positive coefficient, it will drive orthogonal to - the form given in eq.(6).

Finally consider the field with a soft mass squared becoming negative at a scale The leading terms determining its vacuum alignment are and . If the latter dominates with a negative coefficient, will be aligned in the same direction as and have the form given in eq.(7). Note that the term involving is accidental in the sense that it is dependant on the assignments of the field.

In summary, we have shown that higher order 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:

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

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 difference in
the and entries ^{8}^{8}8We take a symmetric form for the mass matrices as would be expected if there
is an underlying GUT [6] 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 element at giving unwanted corrections to the phenomenologically successful Gatto-Sartori-Tonin relation [11] which results if the entry is less than this order [6]. Fortunately, this texture zero is preserved at that order, as the vevs of and 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 .

### 4.2 Majorana terms

The leading terms that contribute to the right-handed neutrino Majorana masses are:

(17) |

(18) |

(19) |

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 controls the hierarchy between (given essentially by eq.(19)) and (from eq.(18)). It is set by
radiative breaking to lie close to the scale of , such that
after seesaw we can fit the ratio of the neutrino squared mass differences ^{9}^{9}9This is different from the model [6] which
predicted the ratio to be associated with the
expansion parameter that was set by the quark sector, and
was consistent with the experimentally measured value .. The hierarchy between the lightest Majorana
mass , and the heaviest, is

(20) |

where is the mass of the messenger responsable for the down quark mass (for details on the messenger sector, we again refer the reader to [6]).

For a viable pattern of neutrino mixing we need to ensure that the hierarchy in eq.(20) is sufficiently strong to suppress the contribution from exchange which would otherwise give an unacceptably large component in the atmospheric (and/or solar) neutrino eigenstates. This requirement on the Majorana hierarchy puts a lower bound on the mass of corresponding right-handed neutrino messenger, as is clear from eq.(20). The light neutrino eigenstates also have an hierarchical mass structure so the heaviest of the light effective neutrinos has a mass given approximately by . Using this, together with eq.(20), we find

(21) |

where is the mass of the messenger responsable for the Dirac neutrino mass.

The final structure of neutrino mixing is very similar to the one in [6], and generates the same predictions for the neutrino mixing angles. The leptonic mixing angles are obtained after taking into account the (small) effect of the charged leptons, yielding nearly tri-bi-maximal mixing [12]:

(22) |

(23) |

(24) |

This leads to the prediction for the reactor angle of , where is the Cabibbo angle, i.e. the prediction is a factor of 3 smaller than the Cabibbo angle due to the Georgi-Jarlskog factor, and also a factor of smaller due to commutation through the maximal atmospheric angle. Also can be related to and the CP violating phase , via the so called neutrino sum rule first derived by one of us in [3]:

(25) |

The above predictions were first shown to follow from the charged lepton
corrections to tri-bi-maximal mixing in the model proposed by one of
us in [3] and later shown to be applicable to a class of models in
[12], including the present model discussed here and in [6] ^{10}^{10}10Note that the prediction for in [6] has been
corrected here..

## 5 Summary and conclusions

We have constructed a complete theory of fermion masses and mixings based on the spontaneous breaking of the discrete non-Abelian symmetry group The model is constructed in a manner consistent with an underlying Grand Unified symmetry with all the members of a family of fermions having the same symmetry properties under the family symmetry group. Many of the properties of the model rely on the approximate symmetry that the discrete group possesses and the model is very close to the continuous family symmetry model of reference [6]. The main difference is a significant simplification in the vacuum alignment mechanism in which the near tri-bi-maximal mixing in the lepton sector directly follows from the 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.

## Acknowledgments

We are grateful to Michal Malinsky for pointing out an error in the vacuum alignment discussion in the original version of this paper.

The work of I. de M. Varzielas was supported by FCT under the grant SFRH/BD/12218/2003.

This work was partially supported by the EC 6th Framework Programme MRTN-CT-2004-503369.

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