Elsevier

Nuclear Physics B

Volume 733, Issues 1–2, 16 January 2006, Pages 31-47
Nuclear Physics B

SU(3) family symmetry and neutrino bi-tri-maximal mixing

https://doi.org/10.1016/j.nuclphysb.2005.10.039Get rights and content

Abstract

The observed large mixing angles in the lepton sector may be the first signal for the presence of a non-Abelian family symmetry. However, to obtain the significant differences between the mixing of the neutrino and charged fermion sectors, the vacuum expectation values involved in the breaking of such a symmetry in the two sectors must be misaligned. We investigate how this can be achieved in models with an SU(3)f family symmetry consistent with an underlying GUT. We show that such misalignment can be achieved naturally via the see-saw mechanism. We construct a specific example in which the vacuum (mis)alignment is guaranteed by additional symmetries. This model generates a fermion mass structure consistent with all quark and lepton masses and mixing angles. Neutrino mixing is close to bi-tri-maximal mixing.

Introduction

The need to explain the observed pattern of quark and lepton masses and mixing angles remains a central issue in our attempt to construct a theory beyond the Standard Model. Perhaps the most conservative possible explanation is that the symmetry of the Standard Model is extended to include a Grand Unified and/or family symmetry which order the Yukawa couplings responsible for the mass matrix structure.

If one restricts the discussion to the quark sector it is possible to build quite elegant examples involving a spontaneously broken family symmetry which generates the observed hierarchical structure of masses and mixing angles. However, attempts to extend this to the leptons has proved very difficult, mainly because the large mixing angles needed to explain neutrino oscillation are quite different from the small mixing angles observed in the quark sector. Indeed, the present experimental situation is consistent with the Harrison, Perkins and Scott “bi-tri-maximal mixing” scheme [1] in which the atmospheric neutrino mixing angle is maximal (sin2(θ@)=1/2) and the solar neutrino mixing is tri-maximal (sin2(θ)=1/3)). If the mixing indeed proves to be close to bi-tri-maximal mixing it will strongly suggest that the family symmetry is non-Abelian, because equality between mixing angles involving different families requires a symmetry relating the magnitude of the Yukawa couplings of these families, something an Abelian symmetry cannot do.

In this paper we discuss how bi-tri-maximal mixing can emerge in a theory with an underlying SU(3)f family symmetry or with a non-Abelian discrete subgroup of SU(3)f. This is of particular interest as SU(3)f is the largest family symmetry that commutes with SO(10) and so fits nicely with promising Grand Unified extensions of the Standard Model. However, by itself, SU(3)f does not explain why the mixing angles are small in the quark sector while they are large in the lepton sector. If it is to be consistent with an underlying spontaneously broken family symmetry there must be a mismatch between the symmetry breaking pattern in the quark and charged lepton sectors and the symmetry breaking pattern in the neutrino sector. In the quark sector and charged lepton sectors the first stage of family symmetry breaking, SU(3)fSU(2)f, generates the third generation masses while the remaining masses are only generated by the second stage of breaking of the residual SU(2)f. However, in the neutrino mass sector the dominating breaking must be rotated by π/4 relative to this, so that an equal combination of ντ and νμ receives mass at the first stage of mass generation. The subsequent breaking generating the light masses must also be misaligned by approximately the tri-maximal angle in order to describe solar neutrino oscillation.

There has already been significant progress in constructing models with a non-Abelian family symmetry capable of generating bi-tri-maximal mixing. These models are based on a spontaneously broken A4 discrete symmetry. Of course, as just discussed, these models also have to arrange the mismatch of symmetry breaking in the charged and neutral sectors. In [2], Ma demonstrated that, assuming such a pattern of symmetry breaking, there is left an unbroken discrete subgroup of A4 which guarantees the bi-tri-maximal mixing structure. Altarelli and Feruglio [3] have recently analysed the scalar potential for a particular A4 model and shown that, at least in a five-dimensional version of the model, it is possible to achieve naturally the desired pattern of symmetry breaking.

It might seem that these A4 based models provide examples of the class of discrete subgroups of the SU(3)f family symmetry discussed above.1 However, in these models the A4 is not a subgroup of the SU(3)f family symmetry which commutes with SO(10) because the left-handed leptons are assigned to different A4 representations to those assignments for the charge conjugate of the right-handed leptons. While this is perfectly possible it does mean that there is no straightforward way to embed these models in a Grand Unified structure and so some of the attractive features of such a structure are lost. In particular in the model of [2] the quarks remain massless at the stage the bi-tri-maximal mixing has been generated for the leptons. The quark masses and mixing angles appear only after further symmetry breaking and so are completely unrelated to the lepton mass structure. A possible exception to this approach is [4], which does assign all leptons to the same representation of A4 and can still generate maximal atmospheric angle.

Given this, we consider it interesting to ask whether bi-tri-maximal mixing can emerge in a theory with an underlying SO(10)SU(3)f symmetry,2 capable of preserving the phenomenologically successful GUT relations between quark and lepton masses. The model is heavily constrained because the SU(3)f multiplet assignment of all the quarks and leptons must all be the same. Nevertheless we show that it is possible to build a model capable of describing all quark and lepton masses and mixing angles in which bi-tri-maximal mixing emerges naturally. In this model there is a close relation between quark and lepton masses and the Georgi–Jarlskog relation between charged lepton and quark masses [6] is readily obtained. The symmetric structure of the mass matrices by the multiplet assignments also allows it to reproduce the phenomenologically successful Gatto–Sartori–Tonin (GST) relation [7] for the (1,2) sector mixing.

While completing this paper we received a paper by King [8] who shows how to achieve bi-tri-maximal structure using SO(3) as the family symmetry. The model is similar in general structure to the one presented here both implementing the general strategy explored in [9], [10]. The main difference, apart from the different choice of family symmetry group, is that the model of [8] assigns left-handed states to different SO(3) representations from those of the charge conjugate of the right-handed states, generating non-symmetric mass matrices. As a result the GST relation is lost and the model does not straightforwardly extend to an underlying SO(10) unification.

As discussed above one of the main difficulties in realising bi-tri-maximal mixing in a SU(3)f family symmetry model is the need to explain why the dominant breaking leading to the generation of third generation masses in the quark sector is not the dominant effect in the neutrino sector too. At first sight it appears quite unnatural. However, if neutrino masses are generated by the see-saw mechanism [12] in fact it can readily arise even if all quark and lepton Dirac mass matrices, including those of the neutrinos, have similar forms up to Georgi–Jarlskog type factors. To see this consider the general form of the see-saw mechanismMν=MDνMM−1MDνT, where Mν is the mass matrix for the light neutrino states, MDν is the Dirac mass matrix coupling ν to νc and MM is the Majorana mass matrix coupling νc to νc. We consider the case where the Majorana mass matrix also has an hierarchical structure of the formMM(M1M2M3),M1M2M3. For a sufficiently strong hierarchy this gives rise to sequential domination [13] in which the heaviest of the three light eigenstates gets its mass from the exchange of the lightest right-handed singlet neutrino with mass M1. In this case the contribution to the light neutrino mass matrix of the field responsible for the dominant (3,3) terms in the Dirac mass matrices is suppressed by the relative factor M1/M3 and may readily be subdominant in the neutrino sector. The message from this is that any underlying quark–lepton symmetry is necessarily broken in the neutrino sector due to the Majorana masses of the right-handed neutrino states and, through the see-saw mechanism, this feeds into the neutrino masses and the lepton mixing angles. This example illustrates how this effect can hide an underlying quark–lepton symmetry in the Dirac mass sector.

In this paper we implement this structure to build a model with near bi-tri-maximal mixing. We consider only the case of a supersymmetric extension of the Standard Model because only in these models is the hierarchy problem associated with a high-scale GUT under control. Rather than work with a complete SO(10)SU(3)f theory (which, in a string theory, may only be relevant above the string scale) we consider here the case where the gauge symmetry is GPSSU(3)f, where GPS is the Pati–Salam group GPSSU(4)PSSU(2)LSU(2)R. The SU(3)f representation assignments are chosen in a way consistent with this being a subgroup of SO(10)SU(3)f. The construction of the model closely follows that of [9] and [10], and proceeds by identifying a simple U(1)U(1) symmetry capable of restricting the allowed Yukawa couplings to give viable mass matrices for both the quarks and leptons. We pay particular attention to an analysis of the scalar potential which is responsible for the vacuum alignment generating bi-tri-maximal mixing.

The Majorana mass matrices are generated by the lepton number violating sector, and we find it to be helpful that the dominant contribution to the Majorana mass matrix for the neutrinos is aligned along the 3rd direction, as is the case for the fermion Dirac matrix. The major difference is the ordering of the hierarchical structure in the two sectors. As we will show, in this case it is possible to achieve bi-tri-maximal mixing very closely, with deviations coming from the charged lepton sector. This type of situation is described in some detail in [11].

The organisation of the paper is as follows. In Section 2 we introduce the symmetry assignment of the states of the Standard Model together with the fields needed to implement a viable pattern of symmetry breaking. We discuss how this symmetry breaking leads to an effective low energy theory of fermion masses through the Froggatt–Nielsen mechanism [9], [14]. In Section 3 we list the dominant Yukawa couplings in the effective theory responsible for fermion masses and compute the Dirac and Majorana mass matrices. The details of the symmetry breaking alignment are presented in Appendix A where we discuss the details of the minimisation of the effective potential, particularly the minimisation and effect of the D-terms and the effect of the soft SUSY breaking masses. In Section 4 we discuss the phenomenological implications of this model, and estimate the magnitude of the corrections to bi-tri-maximal mixing. Finally, in Section 5 we present the conclusions.

Section snippets

Symmetries

As discussed above we will start with the gauge group GPSSU(3)f. We wish to assign our states to representations in a manner consistent with an underlying SO(10)SU(3)f symmetry so we will discuss the representation content as if this is the gauge group even though we will use only the SU(4)PSSU(2)LSU(2)RSU(3)f subgroup in constructing models. The Standard Model (SM) fermions ψi, ψjc (i,j=1,2,3 are family indices) are assigned to a (16,3) representation of SO(10)SU(3)f. The Higgs doublets

The effective superpotential

Having specified the multiplet content and the symmetry properties it is now straightforward to write down all terms in the superpotential allowed by the symmetries of the theory. In this section we concentrate on those terms responsible for generating the fermion mass matrix. Since we are working with an effective field theory in which the heavy modes associated with the various stages of symmetry breaking have been integrated out we must include terms of arbitrary dimension. In practice, it

Phenomenological implications

It is now straightforward to determine the masses and mixing angles in the theory. By construction the form of the up and down quark masses is in agreement with the phenomenological fit of Eq. (15) and Eq. (16). If we further have g=g@, giving a symmetric mass structure, then the (1,1) texture zero gives the successful GST relation [7] relating the light quark masses and CP violating angle to the mixing angle in the (1,2) sector. A symmetric form for the mass matrix is to be expected from the

Summary and conclusions

In this paper we have shown how near bi-tri-maximal mixing in the lepton sector arises from a spontaneously broken SU(3)f family symmetry through vacuum alignment. The model constructed has a phenomenologically acceptable pattern of quark and lepton masses. It generates the successful GST relation between the mixing angles and masses of the first two generations and the Georgi–Jarlskog relations between down quarks and charged leptons. The neutrino sector generates precise bi-tri-maximal mixing

Acknowledgements

The work of I. de Medeiros 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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