Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Maximal atmospheric neutrino mixing from texture zeros and quasi-degenerate neutrino masses

W. Grimus *, P.O. Ludl

University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Vienna, Austria

ARTICLE INFO

Article history: Received 28 April 2011 Accepted 15 May 2011 Available online 18 May 2011 Editor: A. Ringwald

Keywords: Lepton mixing

Quasi-degenerate neutrino mass spectrum Texture zeros

ABSTRACT

It is well known that, in the basis where the charged-lepton mass matrix is diagonal, there are seven cases of two texture zeros in Majorana neutrino mass matrices that are compatible with all experimental data. We show that two of these cases, namely B3 and B4 in the classification of Frampton, Glashow and Marfatia, are special in the sense that they automatically lead to near-maximal atmospheric neutrino mixing in the limit of a quasi-degenerate neutrino mass spectrum. This property holds true irrespective of the values of the solar and reactor mixing angles because, for these two cases, in the limit of a quasi-degenerate spectrum, the second and third row of the lepton mixing matrix are, up to signs, approximately complex-conjugate to each other. Moreover, in the same limit the aforementioned cases also develop a maximal CP-violating CKM-type phase, provided the reactor mixing angle is not too small.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

It is by now well-established that at least two of the neutrino masses mj (j = 1, 2, 3) are non-zero. The same applies to the angles in the lepton mixing matrix V. Its parameterization is usually chosen in analogy to the CKM matrix as [1]

C13C12

C13S12

sue—is\

s23^13

— c23 s12 — s23 s13c12e c23c12 — s23s13 s12e \ s23s12 — c23s13c12eiS — s23c12 — c23s13s12e's c23^3 /

with cij = cos 0ij

and sj = sin Qj, the Qj being angles of the first

quadrant. While the angles Q12 and Q23 are approximately 34° and 45°, respectively, the angle Q13 is compatible with zero [2,3]. All data on lepton mixing are compatible with the tri-bimaximal matrix

Eq. (2) has lead to the speculation that there is a non-abelian family symmetry behind the scenes,1 enforcing s23 = 1/2 in particular. This speculation is in accord with the finding of [6] that the only extremal angle which can be obtained by an abelian symmetry is Q13 = 0°, i.e., Q23 = 45° cannot be enforced by an abelian symmetry. A favorite non-abelian group in this context is A4 [7]. For recent developments and other favorite groups see the reviews in [8] and references therein, for attempts on systematic studies see [9-11] (the latter paper refers to abelian symmetries).

However, there is an alternative to non-abelian groups. It is not necessary that, for instance, Q23 = 45° is exactly realized at some energy scale. It suffices that such a relation is fulfilled with reasonable accuracy. This could happen without need for a non-abelian symmetry. In order to pin down what we mean specifically we consider the structure of the mixing matrix V. It has two contributions, the unitary matrices Ui and Uv, stemming from the diagonalization of the charged-lepton mass matrix Mi and of the neutrino mass matrix Mv, respectively. Then the matrix

i 2/V6 1/v3 0 s V hps = — 1/v6 1/v3 1/v2 V 1/v6 —1/v3 1/V2,

which has been put forward by Harrison, Perkins and Scott (HPS) [4] already in 2002.

U = U j U v = eil

occurs in the charged-current interaction and the lepton mixing matrix V defined above is obtained by removing the diagonal unitary matrices

= diag( e

= diag( e

* Corresponding author.

E-mail addresses: walter.grimus@univie.ac.at (W. Grimus), patrick.ludl@univie.ac.at (P.O. Ludl).

1 However, recently, it has been argued that tri-bimaximal mixing might nevertheless have an accidental origin [5].

p 2 eia3

0370-2693/$ - see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2011.05.029

from U. Without loss of generality we will use the convention e'°3 = 1 in the following. Suppose that we have a model in which Ui and Uv are functions of the charged-lepton and neutrino mass ratios, respectively, and that these mass ratios also parameterize the deviations of Ui and Uv from a diagonal form. In Ui these ratios are me/m^, me/mT and m^/mr. Since the mass hierarchy in the charged-lepton sector is rather strong, Ui is approximately a diagonal matrix of phase factors, with the possible exception of the occurrence of m^/mr; if this ratio appears in a square root in analogy to the famous formula sin 6c — Vmd/ms for the Cabibbo angle [12], with quark masses md and ms, then ^Jm^/mx — 0.24 is even larger than sin 6c. The simplest way to avoid such a deviation of Ui from a diagonal matrix is to have a model which, through its symmetries, enforces a diagonal Mi. Switching to Uv, we point out that up to now the type of neutrino mass spectrum is completely unknown [1]. A particularly exciting possibility would be a quasi-degenerate mass spectrum in which case the neutrino mass ratios could be close to one such that effectively Uv is independent of the masses and could look like a matrix of pure numbers, potentially disturbed by phase factors. Thus, with Ui sufficiently close to a diagonal matrix and a quasi-degenerate neutrino mass spectrum it might be possible to simulate a mixing matrix V consisting of "pure numbers", leading for instance to an atmospheric neutrino mixing angle 623 which is in practice indistinguishable from 45°.

The advantage is that such a scenario could be achieved with texture zeros and that texture zeros in mass matrices may always be explained by abelian symmetries [13], at the expense of an extended scalar sector in renormalizable models.2 Let us summarize the assumptions of this Letter:

• Ui is sufficiently close to a diagonal matrix such that in good approximation it does not contribute to V.

• The neutrino mass spectrum is quasi-degenerate.

• The symmetry groups we have in mind are abelian, i.e., we deal with texture zeros.

In the following we will show that these assumptions can indeed lead to a realization of maximal atmospheric neutrino mixing, in the framework which consists of Majorana neutrino mass matrices with two texture zeros and a diagonal mass matrix Mi; two of the viable cases of neutrino mass matrices classified in [15] exhibit precisely the desired features.

In Section 2 we review the viable textures presented in [15] and point out models in which they can be realized. Then, in Section 3, we discuss the phenomenology of the cases B3 and B4 of [15] in the light of the philosophy specified above. The remaining cases are discussed in Section 4. We summarize our findings in Section 5.

2. The framework

Assuming the neutrinos to be Majorana particles, the neutrino mass term is given by

1 ¿C- MvVL + H.c., (5)

v mass — ^ vl C M v

with a symmetric mass matrix Mv. In the basis where the charged-lepton mass matrix is diagonal, there are seven possibilities for an Mv with two texture zeros which are compatible with all available neutrino data, as was shown in [15]. These seven viable cases are listed in Table 1. The phenomenology of those seven

Table 1

The viable cases in the framework of two zeros in the Majorana neutrino mass matrix Mv and a diagonal charged-lepton mass matrix Mi [15].

Case Texture zeros

A1 (Mv)ee = (Mv)e^ = 0

A2 (Mv)ee = (Mv)et = 0

B1 (Mv)w = (Mv)eT = 0

B2 (Mv)tt = (Mv)ep = 0

B3 (Mv)^ = (Mv)e^ = 0

B4 (Mv)tt = (Mv)eT = 0

C (MvW = (Mv)tt = 0

mass matrices has been discussed in [15-17]. Moreover, case C has also been investigated in [18].

There are several ways to construct models where the cases of Table 1 together with a diagonal charged-lepton mass matrix are realized by symmetries. Five of the seven mass matrices, but not B1 and B2, have various embeddings in the seesaw mechanism [19], by placing zeros in the Majorana mass matrix Mr of the right-handed neutrino singlets vR and in the Dirac mass matrix MD connecting the vR with the vL [20]. With the methods described in [13], one can then construct models where the zeros in the various mass matrices, including the six off-diagonal zeros in Mi, are enforced by abelian symmetries.

Four of the seven textures of Table 1, namely Ai, A2, B3, B4, have a realization in the seesaw mechanism with a diagonal MD [20,21]: by suitably placing two texture zeros in MR or, equiv-alently, in M-1, these four textures can be obtained.3 Actually, now we are dealing with 14 texture zeros, namely six in Mi and MD each and two in MR. In order to construct models for these four cases, one Higgs doublet is sufficient, but one needs two scalar gauge singlets in order to implement the desired form of MR [21].

All of the seven cases of Table 1 can be realized as models in scalar-triplet extensions of the Standard Model [18], i.e., in the type II seesaw mechanism [22] without any right-handed neutrino singlets.

3. Cases B3 and B4

In this section we discuss the cases B3 and B4 which correspond to the Majorana mass matrices

B3: Mv

B4: Mv

The symbol x denotes non-zero matrix elements. The equations which follow from these cases have the form

Y,Va j V a j [ j = Y, Va jVp j [ j = 0 with [ j = mje2iai (7) j=1 j=1

and a = /), where B3 is given by (a,fi) = ([, e) and B4 by (a,fi) = (t, e).

Eq. (7) can be considered from a linear-algebra perspective. Defining line vectors

Za = (V a j ), Zp = (V p j )

2 We emphasize that our approach is different from that of [11] where the Froggatt-Nielsen mechanism [14] is used and, therefore, order-of-magnitude relations among the elements of mass matrices are assumed.

3 There are three more viable cases of texture zeros in Mv 1 which do not correspond to texture zeros in Mv [21].

0.15 0.2

Fig. 1. s23 as a function of mj. In descending order the full curves refer to case B3 (inverted spectrum), case B4 (normal spectrum), case B3 (normal spectrum), and case B4 (inverted spectrum). The dashed line indicates the value 0.5, i.e., maximal atmospheric mixing. In this plot, for sj2, s23, Am^j and Am^ the best-fit values of [3] have been used.

of V, Eq. (7) tells us that, because of the unitarity of V, the line vector

(V V , V ¿3^3)

is orthogonal to both, za and Zß. Therefore, this vector must be proportional to the line vector zy with y = a,ß, and we obtain

Y = (V Y j ) = j Uj) with N2 = £\Vak\2ml

We use Eq. (10) for the discussion of the physical consequences of cases B3 and B4. We begin with case B3 where y = t . Defining e = si3e,&, f 12 = S12/C12 and t23 = s23/c23, from Eqs. (1) and (10) we find the following relations:

V1 V U V

m3 V U V *3

U2 VU V j 2

m3 VU V*3

tl2t23 - g* tl2 + t23g

t23 + tl2g* 1 — tl2t23g

Alternatively, one can use the procedure of [16] to arrive at these expressions.

The analysis of Eq. (11) proceeds in the following way. We define

pj = (2)2 (j=12),

take the absolute values of the two expressions in Eq. (11) and eliminate

Z = 2t12t23S13 cos S

_ ^ 12^23 + S213 — P1 (t212 + t23S13)/t:23

" 1 + P1/t§3 '

Then we end up with a cubic equation for :

t63 + 4^3 + ^23 (^22P1 + S^]

- t23 [S23P1P2 + C23 (S12P1 + <^2)] - P1P2 = 0'

Inspection of this equation shows that it has a unique positive root. Thus, given the neutrino masses m1, m2, m3 and the mixing angles 012 and 013, Eq. (14) determines 623. Using Eqs. (11) and (13), we adopt the following philosophy:

input: mi23, 612,613 ^ predictions: 623,S,2o\2' (15)

Since the Majorana phases 2a12 are not directly accessible to experimental scrutiny, we will later consider instead the observable mpp, the effective mass in neutrinoless double-beta decay.

An approximate solution of Eq. (14) for a quasi-degenerate neutrino mass spectrum is given by

~ 1---

1 1 Am2

or s23 --

where corrections of order (Am^/m^)2 and Am^/mf have been neglected.4 The latter term is small because we know from experiment that Am21/|Am31| ~ 1/30 [2,3]. We observe that the leading correction to t23 is independent of S?2.

Case B4 is treated analogously. We obtain

V1 V h V t 3 t12 + t23g* 1

m3 V U V T1 t12t23 — g t23 ,

U2 V U2 V t 3 1 — t12t23g j 1

m3 V U V T 2 t23 + t12g t23

Comparison with Eq. (11) shows that in the present case the cubic equation for t|3 is obtained from Eq. (14) by the replacement P1 ^ 1/P1, p2 ^ 1/p2. It is then easy to show that the atmospheric mixing angles in the cases B3 and B4 are related by

23 IB4 = v23 IB3

v — 1

= 1 — s23 1B3 •

Accordingly, the curves for B4 in Fig. 1 are obtained from those of B3 by reflection at the dashed line.

4 Note that Am2, = m2 — m\ > 0 whereas Am^ = m2 — m\ can have either sign:

the inverted order.

Ami. > 0 indicates the normal order of the neutrino mass spectrum and Ami. < 0

B (normal)

0.15 0.2

Fig. 2. cos S as a function of m1. For further details see the legend of Fig. 1.

In Figs. 1 and 2 we have plotted s^3 and cos S versus m1, respectively, for cases B3 and B4. For definiteness, for the solar and reactor mixing angles and the mass-squared differences we

have used the best-fit values listed in [3]: s22 = 0.316

7.64 x 10—5 eV2, which are the same values for both normal and inverted spectrum, and s^3 = 0.017, Am^ = 2.45 x 10—3 eV2 for the normal and s^3 = 0.020, Am^1 = -2.34 x 10—3 eV2 for the inverted spectrum. The two figures illustrate nicely that in all four instances (cases B3 and B4 and both spectra) in the limit m1 we find s23 ^ 1/2 and cos S ^ 0.

Some remarks are at order. First of all, by a numerical comparison it turns out that the approximate formula (16) works quite well. The deviation from the exact value of s^3 is less than 3% at m1 = 0.08 eV and the approximation rapidly improves at larger m1 . Secondly, from Eqs. (11) and (17) we read off that cases B3 and B4 do not allow s13 = 0 because this would lead to [1 = [2. However, this observation does not give a strong restriction on s13, as we find numerically. Thirdly, the lower bound on s13 is correlated with a lower bound on m1 . The reason is that, in our treatment of cases B3 and B4, cosS is computed via Eq. (13) after the determination of s23 by Eq. (14); then the condition | cosS| < 1 leads to the lower bound on m1 . For the inverted spectrum we obtain numerically that the lower bound m1 > 0.05 eV is rather stable for s23 > 0.0001. The normal spectrum allows smaller values of m1, for instance, m1 > 0.03 eV at s13 — 0.0001 and m1 > 0.01 eV at s\3 — 0.01. Therefore, cases B3 and B4 do not automatically entail a quasi-degenerate spectrum which would require something like m1 > 0.1 eV. In accord with the philosophy of this Letter we really have to postulate such a spectrum and only for quasi-degeneracy we obtain an atmospheric mixing angle sufficiently close to 45°.

Computing an approximation for cos S is a bit laborious. It turns out that, due to the smallness of s^3, it is necessary to expand cos S to second order in both

and A2=

in order to obtain a reasonable accuracy. The result is5

Note that the coefficient of A1A2 is zero.

cos S ~ ^

si3tl2 4

1 - pr L12

A1— r A1

where the minus and plus signs correspond to B3 and B4, respectively. At m1 = 0.16 eV the approximation (20) deviates from the exact value by less than 1% (5%) assuming a normal (inverted) spectrum. Actually, the sign difference in Eq. (20) between cases B3 and B4 holds to all orders; with Eqs. (13) and (18) it is easy to show that

cos S|b4 = — cos S|b3 ,

in perfect agreement with the numerical computation.

The general formula for the effective mass in neutrinoless double-beta decay (for reviews see for instance [23]) is given by

mßß = \ (Mv)ee \ = m3

2 ß1 2 ß2

c12 + s12 m3 m3

) + (.

€ *) 2

With ¡x1 and ¡i2 from Eqs. (11) and (17) we specify it to cases B3 and B4, respectively. Inspection of the same equations reveals a simple procedure to switch from B3 to B4:

m3 ßj

(j = 1, 2).

In [1 and we need to insert the numerical values obtained for t23 and S. Eq. (13) determines cosS, therefore, sinS is only determined up to a sign. However, since sin S sin S corresponds to e ^ e*, this sign has no effect on mpp because this observable is computed by an absolute value.

The effective mass mpp applied to the cases B3 and B4 has the property that, if we do not care that 623 and cos S are actually determined by Eqs. (14) and (13) and simply plug in 623 = 45° and cos S = 0, we obtain the equality m^ = m3. Since for a quasi-degenerate neutrino mass spectrum m1 — m3 holds, this demonstrates that we should expect

mßß ~ m1

in the limit of quasi-degeneracy. Numerically it turns out that the deviation of mpp/m1 from one is very small—even at m1 = 0.05 eV,

for the inverted spectrum, the ratio m^^/mi deviates from one by only -3.2%, at m1 = 0.1 eV the deviation is —1.5 per mill. For the normal spectrum this ratio is even closer to one. This renders a plot mpp vs. m1 superfluous. The smallness of m^^/m1 — 1 is partially explained by the smallness of s^3 which brings the phases of both ¡x1 and ¡i2 close to n [16]. One can check that choosing a large (and thus unphysical) s13 there is indeed a substantial deviation of mpp/m1 from one at the lower end of our range of m1.

4. The remaining cases

Here we will show that the remaining five cases of two texture zeros in Mv are either such that the assumption of a quasi-degenerate spectrum is incompatible with the data or that they do not conform to the philosophy put forward in this Letter.

Cases A1 and A2 are incompatible with quasi-degenerate neutrino masses, as was noticed in [15]. This can be seen in the following way. From (Mv)ee = 0, assuming a quasi-degenerate spectrum and using Eq. (1) we readily find

S13 £ c23 & — s22) = ^13 cos(2012), (25)

in contradiction to our experimental knowledge on 813 and 612.

Next we consider cases B1 and B2. Taking into account that one knows from experiment that s^3 is small, in first order in s13 for B1 one obtains [15]

m --[tfa + S13(e—ist23 + ei5/t23)/t12],

m2 - — [t§3 - S13(e-ist23 + eis/t23)/t12]. (26)

Now we ask the question if the assumption of a quasi-degenerate mass spectrum compellingly leads to t23 — 1. The answer is "no" because we could choose s13/(t23t12) — 1 in order to achieve quasi-degeneracy; even with the experimentally allowed values for s13 and t12 we would obtain a rather small t23 — s13/t12, far from maximal atmospheric neutrino mixing. Case B2 can be discussed analogously.

Case C is a bit more involved—for details see [18]. In the case of the inverted spectrum, maximal atmospheric neutrino mixing is not compelling. For the normal ordering of the spectrum, using the experimental knowledge on the mass-squared differences and the mixing angles 612 and 613 it follows that t23 is extremely close to one and that the spectrum is quasi-degenerate. However, if we do not use the experimental information on s123, we could assume c?3 being small instead which would then admit t23 being smaller than one. This is in contrast to cases B3 and B4 where, for quasi-degeneracy, t23 is always close to one, independently of the values s212 and s213 assume.

5. Conclusions

In this Letter we have considered the possibility that neutrinos differ from charged fermions not only in their Majorana nature but also in a quasi-degenerate mass spectrum, in stark contrast to the hierarchical mass spectra of the charged fermions. The appealing aspect of this assumption is that it is already under scrutiny by present experiments, and more experiments will join in the near future [1]. Such experiments search for neutrinoless doublebeta decay, whose decay amplitude is proportional to the effective mass mpp, and for a deviation in the shape of the endpoint spectrum of the /)-decay of 3H which is, in essence, sensitive to the average of the squares of the neutrino masses if the spectrum is

quasi-degenerate. Moreover, that neutrino mass effects in cosmology have not yet been observed puts already a stringent although model-dependent bound on the sum of the masses.

However, the aspect on which we elaborated in this Letter was the possibility to obtain near maximal atmospheric neutrino mixing from a quasi-degenerate neutrino mass spectrum. The idea is quite simple: if we have a model with symmetries enforcing a diagonal charged-lepton mass matrix and the atmospheric neutrino mixing angle being a function of the neutrino mass ratios, then in the limit of quasi-degeneracy this mixing angle will become independent of the masses. We have found two instances in the framework of two texture zeros in the Majorana neutrino mass matrix where in this limit atmospheric neutrino mixing becomes maximal, namely the cases B3 and B4 of [15]. We have shown that these two textures have the following properties if the neutrino mass spectrum is quasi-degenerate:

1. Using the mass-squared differences as input, the value of s^3 tends to 1/2 irrespective of the values of s\2 and s\3; therefore, maximal atmospheric neutrino mixing has to be considered a true prediction of the textures B3 and B4 in conjunction with quasi-degeneracy.

2. If s23 is not exceedingly small, then CP violation in lepton mixing becomes maximal too, i.e., cos 8 tends to zero.

3. Exact vanishing of s\3 is forbidden because this would entail Am21 = 0, however, values as small as s\3 = 0.0001 are nevertheless possible.

The results for the cases B3 and B4 can be understood in the following way. With the usual phase convention (1) for the mixing matrix, in Eq. (10) we have e^ = 1 and, therefore, V¡¡ ~ — V *j for j = 1, 2 and V¡3 ~ V*3 for a quasi-degenerate spectrum. The signs we obtained here are convention-dependent and have no physical significance. That the exact relation V¡ j = V * j for j = 1, 2, 3 is a viable and predictive restriction of V was already pointed out in [24], later on in [25] a model was constructed where this relation is enforced by a generalized CP transformation and softly broken lepton numbers, and it was also shown that such a mixing matrix leads to 623 = 45° and s13 cos 8 = 0 at the tree level. While in [25] the symmetry structure is non-abelian and a type of ¡-t interchange symmetry (see [26] for some early references, and [27] for a recent paper and references therein), the textures B3 and B4 can be enforced by abelian symmetries and, provided the neutrino mass spectrum is quasi-degenerate, we have the approximate relations 623 ~ 45° and cos 8 ~ 0. Therefore, we have shown that maximal atmospheric neutrino mixing could have an origin completely different from ¡-t interchange symmetry, it could simply be a consequence of texture zeros and quasi-degeneracy of the neutrino mass spectrum.

References

[1] K. Nakamura, et al., Particle Data Group, J. Phys. G 37 (2010) 075021.

[2] M. Maltoni, T. Schwetz, M.A. Tórtola, J.W.F. Valle, New J. Phys. 6 (2004) 122, hep-ph/0405172;

G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, Prog. Part. Nucl. Phys. 57 (2006) 742, hep-ph/0506083;

G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo, A.M. Rotunno, Phys. Rev. Lett. 101 (2008) 141801, arXiv:0806.1649;

T. Schwetz, M. Tórtola, J.W.F. Valle, New J. Phys. 10 (2008) 113011, arXiv: 0808.2016;

M.C. Gonzalez-Garcia, M. Maltoni, J. Salvado, J. High Energy Phys. 1004 (2010) 056, arXiv:1001.4524.

[3] T. Schwetz, M. Tórtola, J.W.F. Valle, Global neutrino data and recent reactor fluxes: status of three-flavor oscillation parameters, arXiv:1103.0734.

[4] P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530 (2002) 167, hep-ph/ 0202074.

[5] M. Abbas, A.Yu. Smirnov, Phys. Rev. D 82 (2010) 013008, arXiv:1004.0099.

[6] C.I. Low, Phys. Rev. D 71 (2005 ) 073007, hep-ph/0501251.

[7] E. Ma, G. Rajasekaran, Phys. Rev. D 64 (2001) 113012, hep-ph/0106291; K.S. Babu, E. Ma, J.W.F. Valle, Phys. Lett. B 552 (2003) 207, hep-ph/0206292; G. Altarelli, F. Feruglio, Nucl. Phys. B 741 (2006) 215, hep-ph/0512103;

I. de Medeiros Varzielas, S.F. King, G.G. Ross, Phys. Lett. B 644 (2007) 153, hep-ph/0512313;

X.-G. He, Y. Keum, R.R. Volkas, J. High Energy Phys. 0604 (2006) 039, hep-ph/ 0601001.

[8] G. Altarelli, F. Feruglio, Discrete flavor symmetries and models of neutrino mixing, arXiv:1002.0211;

A.Yu. Smirnov, Discrete symmetries and models of flavor mixing, arXiv: 1103.3461.

[9] H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu, M. Tanimoto, Prog. Theor. Phys. Suppl. 183 (2010) 1, arXiv:1003.3552.

[10] K.M. Parattu, A. Wingerter, Tribimaximal mixing from small groups, arXiv: 1012.2842.

[11] F. Plentinger, G. Seidl, W. Winter, J. High Energy Phys. 0804 (2008) 077, arXiv: 0802.1718.

[12] R. Gatto, G. Sartori, M. Tonin, Phys. Lett. B 28 (1968) 128; N. Cabibbo, L. Maiani, Phys. Lett. B 28 (1968) 131.

[13] W. Grimus, A.S. Joshipura, L. Lavoura, M. Tanimoto, Eur. Phys. J. 36 (2004) 227, hep-ph/0405016.

[14] C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B 147 (1979) 277.

[15] P.H. Frampton, S.L Glashow, D. Marfatia, Phys. Lett. B 536 (2002) 79, hep-ph/ 0201008.

[16] Z.-Z. Xing, Phys. Lett. B 530 (2002) 159, hep-ph/0201151.

[17] Z.-Z. Xing, Phys. Lett. B 539 (2002) 85, hep-ph/0205032.

[18] W. Grimus, L. Lavoura, J. Phys. G 31 (2005) 693, hep-ph/0412283.

[19] P. Minkowski, Phys. Lett. B 67 (1977) 421;

T. Yanagida, in: O. Sawata, A. Sugamoto (Eds.), Proceedings of the Workshop on Unified Theory and Baryon Number in the Universe, KEK report 79-18, Tsukuba, Japan, 1979;

S.L. Glashow, in: J.-L Basdevant, et al. (Eds.), Quarks and Leptons, Proceedings of the Advanced Study Institute, Cargese, Corsica, 1979, Plenum, New York, 1981;

M. Gell-Mann, P. Ramond, R. Slansky, Complex spinors and unified theories, in: D.Z. Freedman, F. van Nieuwenhuizen (Eds.), Supergravity, North-Holland, Amsterdam, 1979;

R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.

[20] A. Kageyama, S. Kaneko, N. Shimoyama, M. Tanimoto, Phys. Lett. B 538 (2002) 96, hep-ph/0204291.

[21] L. Lavoura, Phys. Lett. B 609 (2005) 317, hep-ph/0411232.

[22] J. Schechter, J.W.F. Valle, Phys. Rev. D 22 (1980) 2227; J. Schechter, J.W.F. Valle, Phys. Rev. D 25 (1982) 774;

G. Lazarides, Q. Shafi, C. Wetterich, Nucl. Phys. B 181 (1981) 287;

R.N. Mohapatra, G. Senjanovic, Phys. Rev. D 23 (1981) 165;

R.N. Mohapatra, P. Pal, Massive Neutrinos in Physics and Astrophysics, World

Scientific, Singapore, 1991, p. 127;

E. Ma, Utpal Sarkar, Phys. Rev. Lett. 80 (1998) 5716, hep-ph/9802445.

[23] S.R. Elliott, P. Vogl, Ann. Rev. Nucl. Part. Sci. 52 (2002) 115, hep-ph/0202264;

F. Simkovic, Prog. Part. Nucl. Phys. 64 (2010) 219.

[24] P.F. Harrison, W.G. Scott, Phys. Lett. B 547 (2002) 219, hep-ph/0210197.

[25] W. Grimus, L. Lavoura, Phys. Lett. B 579 (2004) 113, hep-ph/0305309.

[26] T. Fukuyama, H. Nishiura, Mass matrix of Majorana neutrinos, hep-ph/9702253; R.N. Mohapatra, S. Nussinov, Phys. Rev. D 60 (1999) 013002, hep-ph/9809415; E. Ma, M. Raidal, Phys. Rev. Lett. 87 (2001) 011802, hep-ph/0102255;

E. Ma, M. Raidal, Phys. Rev. Lett. 87 (2001) 159901, Erratum;

C.S. Lam, Phys. Lett. B 507 (2001) 214, hep-ph/0104116;

K.R.S. Balaji, W. Grimus, T. Schwetz, Phys. Lett. B 508 (2001) 301, hep-ph/

0104035;

E. Ma, Phys. Rev. D 66 (2002) 117301, hep-ph/0207352;

W. Grimus, L. Lavoura, J. High Energy Phys. 0107 (2001) 045, hep-ph/

0105212.

[27] H.-J. He, F.-R. Yin, Common origin of ß-T and CP breaking in neutrino seesaw, baryon asymmetry, and hidden flavor symmetry, arXiv:1104.2654.