The neutrino mass matrix and (selected) variants of A4

P

—journal of January 2009

physics pp. 183–193

The neutrino mass matrix and (selected) variants of A

MARTIN HIRSCH

AHEP Group, Instituto de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia, Edificio de Institutos de Paterna, Apartado 22085, E-46071 Val`encia, Spain E-mail: mahirsch@ific.uv.es

Abstract. Recent neutrino oscillation experiments have measured leptonic mixing an- gles with considerable precision. Many theoretical attempts to understand the peculiar mixing structure, observed in these measurements, are based on non-Abelian flavour sym- metries. This talk concentrates exclusively on models based on the non-Abelian symmetry A4. A4 is particularly well suited to describe three family mixing, and allows to explain the near tri-bimaximal mixing observed. Special emphasis is put here on the discussion of the neutrinoless double beta decay observable hmνi. Different models based on A4

with very similar predictions for neutrino angles can yield vastly different expectations forhmνi. Neutrinoless double beta decay can thus serve, in principle, as a discriminator between different neutrino mass models.

Keywords. Neutrinos; flavour symmetries.

PACS Nos 14.60.Pq; 11.30.Fs

1. Introduction

Neutrinos have mass and non-trivial mixing angles, as neutrino oscillation experi- ments have demonstrated [1–5]. If neutrinos are Majorana particles, their mass at low energy is described by a unique dimension-5 operator [6]

mν= f

Λ(HL)(HL). (1)

Using only renormalizable interactions, there are only three tree-level realizations of this operator [7]. The first one is the exchange of a heavy fermionic singlet.

This is the celebrated see-saw mechanism [8–10], usually called see-saw Type-I.

The second possibility is the exchange of a scalar triplet [11,12]. This is commonly known as see-saw Type-II. And lastly, one could also add one (or more) fermionic triplets to the field content of the SM [13]. This is called see-saw Type-III in [7].

The dimension-5 operator of eq. (1) could also be generated at loop level. As the classical examples for loop-generated neutrino masses we only mention the Zee model [14] (1-loop) and the Babu–Zee model [15] (2-loop), although many more

models exist in the literature. A list of generic 1-loop realizations of eq. (1) can also be found in [7].

While all the different realizations of eq. (1) can give us some understanding as to why neutrino masses are so much smaller than all other fermion masses, the see-saw mechanism as such has no predictive power whatsoever for neutrino angles: f in eq. (1) is generally an arbitrary (3,3) matrix of coefficients. It might be possible to estimate the overall magnitude off model by model, but the flavour structure cannot be fixed with gauge symmetries alone. On the other hand, we know from the latest neutrino data [16] that

U =



|[0.75,0.86]| |[0.51,0.63]| |[0,0.22]|

|[0.29,0.61]| |[0.33,0.70]| |[0.57,0.82]|

|[0.14,0.52]| |[0.45,0.77]| |[0.56,0.81]|



 (2)

at 3σ CL. As already suggested in [17], this matrix is very nicely fitted by the following ansatz:

U^HPS =



 q

2 3 √1

3 0

−^√1₆ ^√1₃ ^√1₂

√1

6 −^√1₃ ^√1₂



. (3)

The authors of [17] called this the tri-bimaximal mixing, since (a) the atmospheric mixing angle is maximal in theνµ−ντ sector (i.e. tan²θAtm= 1 and tan²θR= 0) and (b) the second mass eigenstate contains all three observed flavours in an equal amount; this leads to tan²θ¯= ¹₂.

Many attempts to reproduce eq. (3) theoretically based on non-Abelian flavour symmetries have been published in the past few years. Groups as small asS3(only three irreducible representations) [18–20] or as complicated asB₃×Z₂³(with 384 (!) elements) [21] have been used. A lengthy, but still only partial list of references can be found in the recent short review by Ma [22]. In the following, I will concentrate on models based onA4 exclusively [22a].

The non-Abelian groupA4is the symmetry of the tetrahedron, the group of even permutations of four objects [23]. It has 12 elements and 4 irreducible representa- tions, usually denoted by 1, 1⁰, 1⁰⁰ and 3. It is the smallest discrete group with a triplet representation and thus naturally suited to describe models with three gen- erations of fermions. The most important multiplication rule for the construction ofA4-based models is

3×3 = 1(= 11 + 22 + 33) + 1⁰(= 11 +ω22 +ω²33) +1⁰⁰(= 11 +ω²22 +ω33)

+3(= 23,31,12) + 3(= 32,13,12), (4)

where

ω= exp[2πi/3], i.e. ω³= 1 (5)

i.e. products of triplets can lead not only to singlets but also to new triplet repre- sentations. Note the special feature that 3×3×3 = 1 is possible inA4.

4

There are many different variations of howA4 (or any other flavour symmetry) can be applied to fix the neutrino oscillation data. One can construct models based on see-saw Type-I or see-saw Type-II or a combination thereof. And, of course, neutrino oscillation experiments measure the mismatch between the matrices diag- onalizing the neutrino and the charged lepton mass matrix [23a]. Thus, the flavour structure observed could be due to some non-trivial transformation properties of the neutral or of the charged leptons or, again, a combination of both. It is therefore not surprising that many different constructions can lead to very similar results for the neutrino angles.

Common to nearly all neutrino mass models is the assumption that neutrinos are Majorana particles. So one expects that the rate for neutrinoless double beta decay should be non-zero. However, with the current limits on the absolute neutrino mass scale and the allowed ranges of the neutrino angles the effective neutrino mass hmνi=P

jU_ej²mj ≡Mee, observed in neutrinoless double beta decay, could be as large ashmνi ∼ O(0.5) eV or even arbitrarily small [24]. When it comes to models, on the other hand, the allowed range forhmνiis usually much smaller and different models can have vastly different predictions forhmνi. In the following I will discuss some examples to demonstrate how double beta decay can serve, in principle, as a powerful model discriminator [24a].

2. SelectedA4 models

2.1The early days of A4

The earliest attempts usingA4[26,27] to explain the large (or maximal) atmospheric angle focussed on degenerate neutrinos. Here I just briefly discuss the variant proposed in [27]. In this model theA4 model is broken at a very high scale, such that the low-energy particle content is the same as in the MSSM. The MSSM lepton and Higgs fields are then assigned the following transformation properties underA4: Li = 3, eR = 1,µR = 1⁰, τR= 1⁰⁰ and Φ1,2 = 1. In addition the model contains many new fields at the high scale, to construct a Dirac see-saw for the charged lepton masses. I will not go into the details here. Suffice it to say that the model contains three right-handed neutrino superfields assigned N_i^c = 3. Thus, there is only one MN in the model and N_i^c are degenerate. After see-saw the low-energy neutrino mass matrix takes – at tree-level – the form [27]

Mν = f_N²v²₂ MN



1 0 0 0 0 1 0 1 0



=m0



1 0 0 0 0 1 0 1 0



, (6)

thus also left-handed neutrinos are degenerate, due to the fact that fN also is a constant common to all neutrino generations. The above structure leads to tan²θAtm = 1 and tanθR = 0, but it is not sufficient to explain neutrino data, since there is no mass splitting and no solar angle.

There are several possibilities of how one can amend this texture, such that neutrino data are correctly explained. In [27] this was done by adding 1-loop

(eV) m0

0.0 0.2 0.4 0.6 0.8 1.0

(eV) m0

0.0 0.2 0.4 0.6 0.8 1.0

)2 (eV310×atm2 m∆

0 1 2 3 4 5

(eV) m0

0.0 0.2 0.4 0.6 0.8 1.0

)2 (eV310×atm2 m∆

0 1 2 3 4 5

Figure 1. Allowed value of the parameterm₀ vs. ∆m²_Atm. Note that up to 1-loop correctionshmνi 'm0.

corrections due to supersymmetric thresholds. A detailed study of the phenomeno- logical consequences of this set-up was carried out in [28].

Supersymmetric loops modify the tree-level result of eq. (6) and the solar angle can be correctly fitted [28]. However, the size of the solar angle and the atmospheric mass squared splitting can only be correctly explained if (a) there is sizeable lepton flavour violation in the scalar lepton sector and (b) if m0 is larger than some critical value m^c₀ ≥ 0.3 eV. This leads to two testable predictions of the model.

The first is that lepton flavour violating τ and µ decays should be close to the present experimental limit and the second is thathmνishould be larger than about

>∼0.3 eV (see figure 1).

2.2A4 with triplets

Here I discuss a simple phenomenological model [29] based on a realization of the A4 family symmetry [26,27] in which no right-handed neutrinos are introduced.

Instead, the small neutrino masses arise from the small induced vacuum expectation values (VEVs) generated for the neutral components of triplet Higgs bosons, i.e.

a Type-II see-saw mechanism. The lepton and Higgs particle content and their transformation properties are summarized in table 1.

With these transformation properties, the charged lepton mass matrix is already diagonal in the flavour basis, with

me=h1v1+h2v2+h3v3, mµ=h1v1+ωh2v2+ω²h3v3, mτ =h1v1+ω²h2v2+ωh3v3,

wherehi are charged lepton Yukawa couplings,vi =hφ⁰_iiandωis a complex cubic root of unity satisfying 1 +ω+ω²= 0. The neutrino mass matrix then takes the form

4

Table 1. Lepton and scalar boson quantum numbers under SU(2) andA4

(for discussion, see text).

Li lRi φ1 φ2 φ3 η1 η2 η3 ξ

SU(2) 2 1 2 2 2 3 3 3 3

A4 3 3 1 1⁰ 1⁰⁰ 1 1⁰ 1⁰⁰ 3

Mν =



a+b+c f e f a+ωb+ω²c d e d a+ω²b+ωc



, (7)

where the only non-diagonal entries are those of theA4tripletξ. If one assumes that the conditionsb=c, andd=e=f hold, tan²θAtm = 1 and tan²θR= 0 exactly.

Whereas the former is anad-hocassumption, the latter can be maintained naturally because of a residualZ3symmetry. Allowing forb6=cwill lead to departures from exact maximality in the atmospheric sector. More important phenomenologically, however, is thatd=e=f allows to relate the double beta decay observable to the solar angle. This is shown in figure 2.

Figure 2 shows the lower bound onhmνias a function of tan(2θ¯) (left) and as a function of the Majorana phaseφ₁ (right). Exact TBM mixing, i.e. tan(2θ_¯) = 2√

2 is ruled out by non-observation of neutrinoless double beta decay already.

Similarlyφ1= 0 is not allowed, i.e. current data require already some cancellations in the different terms contributing to hmνi in the model considered here. More interestingly, however, there is an absolute lower bound onhmνi which is around hmνi/p

∆m²_Atm>∼0.2, i.e.hmνi>∼10 meV or so. A tighter determination of tan²θ¯

or a larger central value for ∆m²_Atmwill lead to a larger numerical bound. Allowing for departures from the assumptions b =c, and d=e =f will change the exact number for the lower bound on hmνi. However, departures from these equalities

2 2.5 3 3.5 4

t_2S 0.1

0.2 0.5 1 2 5 10

m ee /

√ 

Inverse Hierarchy

0 0.2 0.4 0.6 0.8 1

cos(φ₁)

0.1 0.2 0.5 1 2 5 10

m ee /

√ 

Normal Hierarchy Inverse Hierarchy

Figure 2. Minimal allowed value for the double beta decay observablehmνi as a function of tan(2θ¯) (left) and as a function of the Majorana phaseφ1

(right).

Table 2. Lepton multiplet structure of a model with a two-zero texture.

There are two possible choices for theA4 assignmet of theSU(2) triplet (see text).

L1 L2 L3 lRi νRi Φi ∆

SU(2) 2 2 2 1 1 2 3

U(1) −1 −1 −1 −2 0 1 2

A4 1 1⁰ 1⁰⁰ 3 3 3 1⁰ or 1⁰⁰

lead to departures of the calculated angles from the desired structure and for values of the neutrino angles within the currently allowed ranges,hmνican never be equal to zero exactly and in general a lower bound of orderhmνi>∼ fewO(meV) persists.

2.3A4 model with a two-zero texture

A very simple model based onA4 that leads to a two-zero texture for the neutrino mass matrix has been discussed in [30]. This texture can nicely fit current neutrino data [31]. The model is a variation which assignsLi to the triplet representation, different from all theA4models proposed previously. Particle content and multiplet assignments are summarized in table 2. Note that the model has been constructed trying to minimize the number of non-SM fields: Only three νR and one SU(2) triplet are needed.

ThelRi as well as the Higgs doublets responsible for lepton masses transform as A4triplets, while the (undisplayed) quarks and theSU(2) Higgs doublet that gives their masses are all singlets underA4. This leads to the following terms responsible for the lepton masses:

−L=h1L¯1(lRΦ)1+h2L¯2(lRΦ)⁰₁+h3L¯3(lRΦ)⁰⁰₁ +h1DL¯1(νRΦ)1+h2DL¯2(νRΦ)⁰₁+h3DL¯3(νRΦ)⁰⁰₁ +M

2 ν_Ri^TCν_Ri+ H.c., (8)

where the quantities in parenthesis denote products of twoA4-tripletslR(orνR) and Φ forming the representations 1,1⁰,1⁰⁰respectively. Note that eq. (8) includes the most general terms allowed by the symmetry and field content in table 2. Hence, in contrast to many otherA₄models, here one does not need to impose any additional symmetry to forbid unwanted terms.

If ∀hΦ⁰_ii = v the Dirac matrix of the neutrinos is diagonalized by mD = v diag(h1D, h2D, h3D)Uω, with

Uω= 1

√3



1 1 1 1 ω ω² 1 ω² ω



, ω≡e^2πi/3. (9)

As a result, the effective neutrino mass matrix after the see-saw is already in the flavour basis and is given by

4

0.35 0.4 0.45 0.5 0.55 0.6 0.65

sin² Θ₂₃ 0

0.05 0.1 0.15 0.2 0.25 0.3

ÈMeeÈHeVL

Dmatm2<0 Dmatm2<0

Dmatm2>0 Dmatm2>0

B1

B2

Figure 3. Minimal allowed value for the double beta decay observablehmνi as a function of sin²θAtm for the two allowed textures described in the text.

M^I_νf =mDM_R⁻¹m^T_D= v² M



h²_1D 0 0 0 0 h2Dh3D

0 h2Dh3D 0



. (10)

This has the same zero textures as obtained in [27] except that only two (instead of three) neutrinos are degenerate. Again this texture by itself is not yet fully consistent with neutrino data and one needs to modify it. The authors of [30]

choose to introduce a triplet field ∆ transforming either as a 1⁰⁰ or as a 1⁰ under A4, as in table 2.

Then, the total neutrino mass matrix is the sum of the see-saw Type-I contribu- tion of eq. (10), plus a see-saw Type-II part, whose texture is again determined by A4. The two different assignments of table 2 can lead to either

hmνi=



a x 0 x 0 b 0 b y



 or hmνi=



a 0 x 0 y b x b 0



, (11)

where a, band x, y refer to the Type-I and Type-II contributions, respectively. It is possible to modify the assignment of various Li fields among different singlet representations of A₄. This either results in one of the above two textures, or in a texture which is not viable phenomenologically. Thus the realization of the A4 flavour symmetry proposed leads to just two viable two-zero textures. The authors of [33] defined the first of these asB1and the second asB2. Both are very predictive, as I will briefly discuss.

Figure 3 shows the minimal allowed value for the double beta decay observable hmνias a function of sin²θAtm for the two allowed textures described above. Cur- rentlyhmνi>∼0.03 eV and a tighter determination of sin²θAtmwill lead to a larger bound. Note that an exactly maximal atmospheric angle is not allowed in this model. In addition, it is worth mentioning that the model predicts that the Dirac CP-violating phaseδmust be close to maximal, if sin²θR≥10⁻³ [30].

Table 3. Lepton multiplet structure (for discussion, see text).

Fields Li l^c_i ν_i^c h Hi ϕ Φ ξ

SU(2)L 2 1 1 2 2 2 2 1

A4 3 3 3 1 3 1 3 1

Z2 + + − + + − − +

2.4Minimizinghmνiwithin A4

Even though I did not show this variant in my talk, the current discussion would be incomplete without mentioning the recent model(s) presented in [34]. This ansatz deliberately tries to minimize the lower bound onhmνi.

As noted earlier [35], when the charged lepton mass matrixMl obeysM^lM^l†= UωM_diag^l2 U_ω^† where Uω is the ‘magic’ unitary matrix of eq. (9) and the neutrino mass matrix has the form

Mν ∼



A 0 0 0 B C 0 C B



, (12)

the resulting lepton mixing matrix has exactly the tri-bimaximal structure given in eq. (3). Consider the see-saw Type-I and the following ansatz for the mass matrices:

M^l∼



α β γ γ α β β γ α



=UωM_diag^l U_ω^†; (13)

mD∼



a 0 0 0 a b 0 b a



; MR∼



1 0 0 0 1 0 0 0 1



. (14)

Note that, the assumed symmetry of the Dirac mass term holds inSO(10) models where it comes from a 16×16×10 Yukawa coupling. There are several constructions which can lead to this new texture [34] discusses two A4-based models. As an example, the lepton and scalar content of the first of these are shown in table 3.

TheA4×Z2 invariant Lagrangian characterizing this model is renormalizable, and given by

L=λ0(Ll^c)h+λ(Ll^cH)

+λ⁰₀(Lν^c)ϕ+λ⁰(Lν^cΦ) +λR(ν^cν^c)ξ,

where the first term involves anA4-invariant couplingλ0 while the second involves a tensorλijk, and similarly for the next two terms. Note that there are additional doublet Higgs scalar bosonsHi,Φi, ϕtransforming non-trivially under the flavour

4

0.027 0.031 0.034 0.037 0.04 10-6

10-5 10-4 10-3 10-2

0.027 0.031 0.034 0.037 0.04 10-6

10-5 10-4 10-3 10-2

hmνi≡|mee|[eV]

α

-Π

€€€€€

-Π 4

€€€€€ - 2

3Π

€€€€€€€€€ 4

- Π 0 Π

€€€€€ 4

Π

€€€€€ 2

3Π

€€€€€€€€€ 4

Π 10^-4

10^-3 10^-2 10^-1 10⁰

-Π

€€€€€

-Π 4

€€€€€ - 2

3Π

€€€€€€€€€ 4

- Π 0 Π

€€€€€ 4

Π

€€€€€ 2

3Π

€€€€€€€€€ 4

Π 10^-4

10^-3 10^-2 10^-1 10⁰

hmνi≡|mee|[eV]

φ12

Figure 4. Left: Lower bound on the 0νββ amplitude parameter mee as a function ofα≡∆m²¯/∆m²Atmfor different values of the Majorana phase. The 1, 2 and 3σranges forαare indicated. Right: meeas function of the Majorana CP phaseφ12 forαwithin the 1σ(yellow) and 2σ(blue) ranges.

symmetry. We assume that these develop non-zero vacuum expectation values (vevs), with the structure

hHii ∼(1,1,1); hΦii ∼(0,0,1).

Similar vev alignment condition has been used in [36]. This leads directly to the lepton mass matrices shown in eqs (13) and (14).

The scheme is extremely predictice, as it involves only the two modulii and the relative phase between aand b. Figure 4 shows the calculated hmνias a function of α ≡ ∆m²_¯/∆m²_Atm (left) and φ12 (right). Only if α = 3/80 and φ12 = π/2 simultaneouslyhmνican be arbitrarily small. Note that this value ofαis currently disfavoured at 2σCL.

3. Conclusions

Many different neutrino mass models based on non-Abelian symmetries can success- fully explain the observed neutrino angles. Although the author has concentrated only on a few A4-based examples, many more models exist in the literature. To distinguish between the different theoretical attempts, additional observables are desperately wanted.

Unfortunately, many of the different models have few-to-none predictions outside the neutrino sector. This leaves essentially neutrinoless double beta decay as a possible model discriminator. In this talk, the predicted lower bounds onhmνiare discussed for the models of [27–30,34]. Apart from [34] all of these models will hopefully be tested by neutrinoless double beta decay within the next round of experiments.

Acknowledgments

The author is grateful to his collaborators: A Joshipura, S Kaneko, E Ma, S Morisi, J C Rom˜ao, S Skadhauge, J W F Valle and A Villanova as without their contribu- tions this paper would not have been written.

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4

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