Triple Seesaw Mechanism ∗ D

Physics Letters B 687 (2010) 400–404

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Triple seesaw mechanism ∗ D. Cogollo a, H. Diniz b,C.A.deS.Piresb, a Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, 58109-970 Campina Grande, Paraíba, Brazil b Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, PB, Brazil article info abstract

Article history: On ﬁtting the type II seesaw mechanism into the type I seesaw mechanism, we obtain a formula to the Received 11 February 2010 neutrino masses which get suppressed by high-scale M3 in its denominator. As a result, light neutrinos Accepted 15 March 2010 are naturally obtained with new physics at TeV scale. As interesting consequence, the mechanism may Editor: M. Cveticˇ be directly probed at the LHC by directly producing the TeV states intrinsic of the mechanism. We show that the 3-3-1 model with right-handed neutrinos realizes naturally such seesaw mechanism. Keywords: © Seesaw mechanisms 2010 Elsevier B.V. All rights reserved. Neutrino masses Standard model 3-3-1 model

1. Introduction ple get interested in such scenario because it may be probed at LHC [8]. However, for we have a seesaw mechanism with M at the One of the major puzzles in particle physics is the understand- TeV scale, we need vew 1 MeV. This is unnatural, unless we pro- ing of the small but nonzero neutrino masses established by the vide a supplementary mechanism of suppression for vew. phenomenon of neutrino oscillation detected in solar [1] and at- An interesting possibility for having a seesaw mechanism work- mospheric [2] as long as in laboratory experiments with reactors ing at TeV scale is fitting the type II seesaw mechanism into the [3] and accelerator [4] neutrinos. type I seesaw mechanism. For this we have to develop type II see- On the theoretical side, seesaw mechanisms is considered the saw mechanism for other scalar multiplets than Higgs doublet. This most elegant way of generating small masses for the neutrinos. In idea was originally proposed by Ma at Ref. [9] and after general- conventional seesaw mechanisms [5–7], neutrino masses get sup- ized by Grimus et al. at Ref. [10]. pressed by a high-scale M in its denominator according to the In this work we review this idea and show that it leads to a 3 v2 neutrino mass formula that get suppressed by high-scale M in formula: m = ew where v is a energy scale parameter at the νl M ew its denominator. Due to this suppression factor, we call this mech- electroweak range. anism “triple seesaw mechanism”. We also show that the 3-3-1 Regions of values for M of interest in particle physics is M model with right-handed neutrinos realizes naturally the triple close to unification scale (1012–14 GeV) and M around TeV scale seesaw mechanism. (1–10 TeV). In the first scenario, neutrino physics is associated to 2 This works is organized as following. In Section 2 we make use vew theories of grand unification and the quotient M lies already at of a toy model to present the idea. In Section 3 we develop the the eV range, explaining in this way the smallness of the neutrino mechanism in an extension of the standard model. In Section 4 we masses. However, the unpleasant point of this scenario is that, due implement the mechanism inside the 3-3-1 model, and finally in to the high value for M, such scenario cannot be tested experi- Section 5 we present a summary of our results. mentally. In the second scenario, i.e., M at the TeV range, for vew at 2. A toy model electroweak scale, we do not have, in fact, a seesaw mechanism 2 2 vew because vew is of order of M which leaves the quotient M close to the unity requiring, in this way, huge fine tuning of the Yukawa For presenting the idea we consider a simple toy model without couplings in order to generate neutrino mass at eV scale. Peo- referenceatfirsttoanygaugemodel.Thetoymodelinvolveone left-handed neutrino, νL , one right-handed neutrino, νR and two scalar fields φ1 and φ2. These scalars do not carry lepton number. * Corresponding author. We assume that the Lagrangian respects the set of discrete sym- E-mail address: cpires@fisica.ufpb.br (C.A. de S. Pires). metry (νL , νR ,φ2) →−(νL , νR ,φ2). Besides, we allow that only νR

develop Majorana mass term. After all this, the Lagrangian we can and at least two scalar doublets H1, H2, plus a scalar singlet S. build with these fields is formed by the terms These scalars do not carry lepton number. To avoid undesirable terms, we assume that the whole Lagrangian respects the discrete L =− ¯ − ¯ C + yνL νR φ1 MνR νR V (φ1,φ2). (1) symmetry (H1, S, νR ) →−(H1, S, νR ). With this, the Lagrangian must involve the terms When φ1 develop vacuum expectation value (VEV) different from zero, v1, the first term in the Lagrangian above generates Dirac ¯ ˜ C L ⊃−yLH1νR − Mν¯ νR + V (H1, H2, S) (8) mass term for the neutrinos. The first two terms of the Lagrangian R = − T above provide the following mass matrix for the neutrinos in the where L (νL , eL ) is the standard lepton doublet. C When H develop nonzero VEV, v , the neutrinos gain Dirac basis (νL , νR ), 1 1 mass term that together with the Majorana mass term for the 0 yv = 1 right-handed neutrino form the following mass matrix in the basis Mν . (2) C yv1 M (νL , νR ), Assuming M v1, we obtain after diagonalize this mass matrix 0 yv1 Mν = . (9) y2 v2 yv1 M m ≈ 1 m ≈ M (3) νl , νh . M Considering M v1, we obtain after diagonalize this mass matrix, This is nothing more than the type I seesaw mechanism. 2 2 y v1 Now let us move to the scalar sector of the model. The main mν ≈ , mν ≈ M. (10) l M h goal here is to arrange things such that v1 get suppressed by a 1 Again, this is the type I seesaw mechanism. factor M . For this we apply the type II seesaw mechanism to φ1. The potential we can construct with all these scalar fields and Now let us move to the scalar sector of the model and apply respect the discrete symmetry mentioned above is composed by the type II seesaw mechanism to the doublet H1. The potential the terms composed by this scalar content and that respect the set of discrete symmetry we discussed above is, = 2 2 + 2 2 + 4 + 4 + 2 2 V (φi) μ1φ1 μ2φ2 λ1φ1 λ2φ2 λ3φ1 φ2 V (H1, H2, S) M1 − 2 + ∗ φ1φ2 H.c. (4) = 2 † + 2 † + 2 2 μ1 H H1 μ2 H H2 μS S S 1 2 † 2 † 2 ∗ 2 The condition for V (φi) develops minimum involves two constraint + + + λ1 H1 H1 λ2 H2 H2 λ3 S S equations. However, what matter for us here is the one related to + † † + † ∗ + † ∗ the VEV of φ1, λ4 H1 H1 H2 H2 λ5 H1 H1 S S λ6 H2 H2 S S M λ − 1 † + 2 2 3 2 2 H1 H2 S H.c. (11) v1 μ + λ1 v + v − M1 v = 0, (5) 2 1 1 2 2 2 The conditions for V (H1, H2, S) develops minimum involve where v1 and v2 are the VEVs of φ1 and φ2, respectively. three constraint equations. What matter for us here is the one re- Thesuppositionswetakeherearethatφ is heavier than φ , 1 2 lated to the VEV of H1, i.e., μ1 μ2 and that μ1 ≈ M1. With this we find that the equation above provides, 2 2 λ4 2 λ5 2 v1 μ + λ1 v 2 + v + v − M1 v2 v S = 0, (12) 1 1 2 2 2 S v2 ≈ 2 v1 . (6) where v2 and v S are the VEVs of H2 and S, respectively. M1 Let us suppose that H1 is heavier than H2 and S, i.e., μ1 ≈ Considering that M1 M and substituting Eq. (6) in (3),we μ2, μS , and that the scale of energy, M1, related to the trilinear obtain, term in the potential above is of the same order of magnitude of ≈ 2 4 the mass of H1, i.e., μ1 M1. With these two suppositions, the y v2 mν ≈ . (7) equation above provides, l M3 v2 vs This is the neutrino mass formula that arises in the triple seesaw v1 ≈ . (13) M mechanism. Perceive that there is a real possibility for we have 1 neutrino masses at eV range with new physics at TeV scale. This is This is nothing more that the type II seesaw mechanism applied so because neutrino masses get suppressed by high-scale M3 in its to v1. ≈ denominator. For example, for M = (1–10) TeV, we have a factor Now, considering M1 M and combining Eq. (13) with (10), of suppression in the range 109–12 which is sufficient to generate we obtain, neutrino masses at around eV. y2 v2 v2 In the next two sections we implement the mechanism in the m ≈ 2 s . (14) νl 3 framework of realistic neutrino models. M Note that, as expected, the mass of the light neutrino get sup- 3. Implementing the mechanism into the standard model pressed by high-scale M3 in its denominator. Moreover, perceive that the formula above involve two VEVs in its numerator. This The way we implement the mechanism here is different from gives us more liberty to arrange the VEVs in conjunction with M the one done in the original work [9].Weextendthestandard in order to obtain neutrino mass at eV range with M at the TeV model (SM) with some new fields. For sake of simplicity, we con- scale. sider only one generation of fermions. The mechanism requires we For example, in this particular extension of the SM, v1 devel- add to the standard field content one right-handed neutrino νR ops exclusively Dirac mass terms for the neutrinos and v2 get 402 D. Cogollo et al. / Physics Letters B 687 (2010) 400–404

→ + + 0 in charge of generating masses for all standard charged fermions S Δ(1,3,YΔ) Φ(1,2,YΦ ) σ(1,1,Y ), (19) 2 σ 0 of the model. This requires v2 ≈ 10 GeV. On the other hand, where Y are the hypercharges of the respective multiplets, with v S is not constrained, which means it can take any value in the − 1 Y =−2, Y =−1 and Y 0 = 0, and electroweak range. Being conservative and taking v S = 10 GeV, Δ Φ σ v ≈ 102 GeV and M = 10 TeV, we obtain 2 Δ0 Δ− Φ0 0 = √1 = √1 √σ Δ − −− ,Φ − , . (20) ≈ 2 2 Δ Δ 2 Φ 2 mνl 0.1y eV. (15) C ∗ We see that the mechanism provides a real possibility for ex- The Yukawa interaction f L S f L can be dismembered into the fol- plaining solar and atmospherics neutrino oscillation with Yukawa lowing ones, couplings of order O(1). ¯C ∗ → ¯ C ∗ + ¯ + ¯ C From a phenomenological viewpoint, this scenario is interest- f L S f L L Δ L LΦνR νR σ0νR , (21) ing because it involves sizable couplings of the TeV new states × × ¯ ˜ when the 3-3-1 symmetry breaks to the SU(3)c SU(2)L U (1)Y with the standard leptons through the term yLH1νR , allowing the symmetry. In the above expression L represents the usual SM dou- mechanism be directly probed at the LHC by directly producing blet of left-handed leptons. It is easy to see that when Δ0, Φ0 and the TeV states intrinsic of the mechanism. σ 0 develop nonzero VEV, then Dirac and Majorana mass terms are generated for the neutrinos. In this work we assume that Δ0 does 4. Implementing the mechanism into the 3-3-1 model with not develop VEV. As we see, the model disposes of all ingredients right-handed neutrinos required by the triple seesaw mechanism. 0 0 When Φ and σ develop nonzero VEV, let us suppose vΦ In this section we show that the triple seesaw mechanism takes and v , the Yukawa interactions h f¯C Sf generate the following σ ab aL bL place naturally in the gauge models based in the SU(3)C ⊗SU(3)L ⊗ C C mass terms for the neutrinos in the basis (νeL , νμL , ντL , νe , νμ , U (1) (3-3-1) symmetry called 3-3-1 model with right-handed R R N νC )T = (ν , νC )T , τR L R neutrinos (331νR) [11]. In it the leptons come in triplet and sin- glets as follows, 1 0 M ν ¯ C ¯ D L ⎛ ⎞ νL , νR C , (22) νa 2 M D M R νR ⎝ ⎠ faL = ea ∼ (1, 3, −1/3), eaR ∼ (1, 1, −1), (16) where c νa L M D = hvΦ and M R = hvσ , (23) with a = 1, 2, 3 representing the three known generations. We are with h being a symmetric matrix formed by the Yukawa couplings indicating the transformation under 3-3-1 after the similarity sign, h . As it is very well known, from the diagonalization of the mass “∼”. For the quark representation we refer the reader to the origi- ab matrix in Eq. (22),forthecasev v , we obtain, nal work of the model [11]. σ Φ The scalar content capable of generating masses for all charged −1 mν −M D M M D , mh M R , (24) fermions of the model is composed by three scalar triplets as fol- l R lows [11] where mνl is a mass matrix for the left-handed neutrinos and mh ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ is the mass matrix for the right-handed neutrinos. χ 0 ρ+ η0 On substituting the matrices M D and M R giveninEq.(23) into ⎝ − ⎠ ⎝ 0 ⎠ ⎝ − ⎠ χ = χ , ρ = ρ , η = η , (17) Eq. (24), we obtain, 0 + 0 χ ρ η v2 m =−h Φ and m = hv . (25) − 1 νl h σ with η and χ transforming as (1, 3, 3 ) and ρ transforming as vσ − 2 (1, 3, 3 ). Although the scalar content above involves ﬁve neutral This is the type I seesaw mechanism that arises in the 331νR. scalars, it is just necessary that η0, ρ0 and χ 0 develop vacuum Let us move to the scalar sector. In order to avoid a prolifer- expectation value (VEV) for that all the particles of the model, with ation of undesirable terms, we impose the potential respects the exception of the neutrinos, develop their correct mass terms. set of discrete symmetries (χ, ρ) →−(χ, ρ).Afterthis,themost In the 331νR neutrino mass terms must, necessarily, involve complete part of the potential that obeys this set of discrete sym- ¯ c the product f L f L . Considering its transformation by the SU(3)L metry and conserves lepton number is composed by the following ¯ c = ∗ ⊗ ∗ = ⊕ ∗ terms, symmetry, f L f L 3 3 3 6 , we see that, in order to gen- ¯ c erate mass terms for the neutrinos, we must couple f L f L to an 2 2 2 2 2 2 4 4 4 V = μ χ + μ η + μ ρ + λ1χ + λ2η + λ3ρ anti-triplet or to a sextet of scalars. Previous works showed that χ η ρ the ﬁrst case leads to degenerate Dirac mass terms for the neutri- † † † † † † + λ4 χ χ η η + λ5 χ χ ρ ρ + λ6 η η ρ ρ nos [11]. This case is not of interest for us here. In regard to the + λ χ †η η†χ + λ χ †ρ ρ†χ + λ η†ρ ρ†η scalar sextet [12], recent works have employed it to implement the 7 8 9 type I and type II seesaw mechanisms into the model [13]. f + √ ijk + + 2 † + † 2 A remarkable fact about this sextet is that after the SU(3)C ⊗ ηiρ jχk H.c. μS Tr S S λ10 Tr S S ⊗ ⊗ ⊗ 2 SU(3)L U (1)N symmetry breaking to the SU(3)C SU(2)L U (1)Y † 2 † † † † one, the sextet, + λ11 Tr S S + λ12η η + λ13ρ ρ + λ14χ χ Tr S S ⎛ − ⎞ Δ0 Δ Φ0 + λ ijk lmn S S + H.c. + λ † S S† − 15 ρnρk li mj 16 χ χ = √1 ⎝ − −− − ⎠ ∼ 2 S Δ Δ Φ 1, 6, , (18) + λ η† S S†η + λ ρ† S S†ρ , (26) 2 3 17 18 Φ0 Φ− σ 0 while the other part that violates explicitly the lepton number is splits into a triplet plus a doublet and a singlet of scalars, composed by the terms, D. Cogollo et al. / Physics Letters B 687 (2010) 400–404 403

V = λ η†χ η†χ of a mixing among the ordinary quarks with the exotic ones. In 19 view of this, it is natural to expected a low value for vη .Infact, λ20 ∗ λ21 ∗ + √ ijk + √ ijk ηm Smiχ jρk χm Smiη jρk the upper bound vη < 40 GeV was obtained in Ref. [14].Thus,on 2 2 2 −1 being conservative, and taking vη ≈ 10 GeV, vη ≈ 10 GeV and T † T † M = 10 TeV, we obtain − M1η S η − M2χ S χ + H.c. , (27)

mν =−0.1h eV. (32) where M1 and M2 are energy parameters associated to the explicit l violation of the lepton number. This requires also little fine tuning of the Yukawa couplings h’s We already assumed that only two of the three neutral scalars to generate neutrino masses small enough in order to explain so- that compose the sextet will develop nonzero VEV. In regard to lar and atmospherics neutrino oscillation experiments. Thus the the triplets, they involve five neutral scalars. It was discussed mechanism can be verified at the LHC through the direct detec- above that, in order to generate the correct masses for all charged tion of the TeV states of the 331νR involved in the mechanism. fermions it is just necessary that only three of them develop We say that the triple seesaw mechanism is a natural outcome nonzero VEV, namely 0 0 0 η , ρ and χ . However, as we will see be- of the 331νR for two basic reasons. First, the high-scale M coin- low, the seesaw mechanism we develop here requires that either 0 0 cides with the 331 symmetry breaking scale, and second, the VEV η ,orχ or both develop nonzero VEV. For sake of simplicity, 0 vη that appears in the numerator of the neutrino mass formula is we pick out η . Thus, the set of neutral scalars that will develop phenomenologically small. nonzero VEV is,

η 0, η0, ρ0, χ 0,Φ0, σ 0 5. Summary → √1 + (vη ,η,ρ,χ ,Φ,σ 0 Rη ,η,ρ,χ ,Φ0,σ 0 In this work we developed a new kind of seesaw mecha- 2 nism that is really capable of explaining the smallness of neutrino + iIη ,η,ρ,χ ,Φ0,σ 0 ). (28) masses with new physics just at TeV scale. It arises from the fitting of the type II seesaw mechanism into the type I seesaw Considering the shift of the neutral scalars above, the conditions mechanism. We first presented the idea through a toy model built for the minimum of the composite potential V + V involve six specially to accommodate the mechanism. As realistic scenario, we equations. However, as the idea is to apply the type II seesaw implemented the mechanism in an extension of the SM involving mechanism to Φ0, it is just necessary to analyze the constraint right-handed neutrinos and some new scalar fields. Next we de- equation related to the vΦ , veloped the mechanism in the framework of the 331νR. The main 2 λ10 2 2 λ11 2 2 λ12 2 2 result of such fitting in both models is that neutrino masses for- vΦ μ + v + v + v + 2v + v + v 3 S 2 Φ σ 2 σ Φ 2 η η mula get suppressed by the factor M in its denominator. Thus it is = λ λ λ easy to see that for M 1–10 TeV, neutrino masses get suppressed + 13 2 + 14 2 − 2 + 16 2 by the factor 109–12. This is sufficient to guarantee that neutrino vρ vχ λ15 vρ vχ 2 2 4 will gain masses at eV scale. Due to this suppression factor, we call λ v v v + 17 2 + 2 + η η σ this mechanism “triple seesaw mechanism”. However, we would vη vη 4 vΦ like to emphasize that the mechanism is a natural outcome of the 331νR. From phenomenological viewpoint, the interesting aspect λ20 + λ21 − M1 vη vη − (vη vρ vχ ) = 0. (29) of such mechanism is that its signature may be probed at LHC 4 through direct detection of their intrinsic TeV states. All the massive components of the sextet S and the triplet χ gain mass when the 3-3-1 symmetry breaks to the electroweak Acknowledgements symmetry. The order of magnitude of such masses lie around the 3-3-1 symmetry breaking scale. In view of this we assume This work was supported by Conselho Nacional de Pesquisa μS ≈ μχ ≡ M. Besides, we consider that the energy scale asso- e Desenvolvimento Científico – CNPq and Coordenação de Aper- ciated to the explicit violation of the lepton number, M , M ,also 1 2 feiçoamento de Pessoal de Nível Superior – capes (PNPD/PROCAD). lies around the 3-3-1 symmetry breaking scale, i.e., M1 ≈ M2 ≡ M. After such considerations, the equation above provides the relation, References ∼ vη vη vΦ = . (30) M [1] SNO Collaboration, Q.R. Ahmad, et al., Phys. Rev. Lett. 89 (2002) 011301; Now we came to the main point of this section. On substituting SNO Collaboration, Q.R. Ahmad, et al., Phys. Rev. Lett. 89 (2002) 011302. [2] Super-Kamiokande Collaboration, J. Hosaka, et al., Phys. Rev. D 74 (2006) Eq. (30) in Eq. (25), we obtain the following formula to the neu- 032002. trino masses, [3] KamLAND Collaboration, K. Eguchi, et al., Phys. Rev. Lett. 90 (2003) 021802. [4] K2K Collaboration, M.H. Ahn, et al., Phys. Rev. Lett. 90 (2003) 041801. 2 2 vη vη [5] For type I seesaw mechanism, see Refs.: M. Gell-Mann, P. Ramond, R. Slansky, m =−h . (31) in: P. van Nieuwenhuizen, D.Z. Freedman (Eds.), Supergravity, North-Holland, νl 3 M Amsterdam, 1979; As showed, neutrino masses get suppressed by high-scale M3 in T. Yanagida, in: O. Sawada, A. Sugamoto (Eds.), Proceedings of the Workshop on the Unified Theory and the Baryon number in the Universe, KEK Report No. its denominator. This allows we have neutrinos with masses in the 79-18, Tsukuba, Japan, 1979; eV range with M around few TeVs. R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912; Let us discuss a little the values of the parameters involved in J. Schechter, J.W.F. Valle, Phys. Rev. D 22 (1980) 2227. the triple seesaw mechanism neutrino mass formula above. It is [6] For type II seesaw mechanism, see Refs.: M. Magg, C. Wetterich, Phys. Lett. B 94 (1980) 61; expected in the model that v2 + v2 ≈ (246 GeV)2. On the other ρ η J. Schechter, J.W.F. Valle, Phys. Rev. D 25 (1981) 774; hand, the main consequence of a nonzero VEV for η 0 is the rising R.N. Mohapatra, G. Senjanovic, Phys. Rev. D 23 (1981) 165; 404 D. Cogollo et al. / Physics Letters B 687 (2010) 400–404

E. Ma, U. Sarkar, Phys. Rev. Lett. 80 (1998) 5716. [10] W. Grimus, L. Lavoura, B. Radovcic, Phys. Lett. B 674 (2009) 117. [7] For type III seesaw mechanism, see Refs.: R. Foot, H. Lew, X.G. He, G.C. Joshi, [11] R. Foot, H.N. Long, T.A. Tran, Phys. Rev. D 50 (1994) R34; Z. Phys. C 44 (1989) 441; J.C. Montero, F. Pisano, V. Pleitez, Phys. Rev. D 47 (1993) 2918. E. Ma, Phys. Rev. Lett. 81 (1998) 1171. [12] This scalar sextet was ﬁrst introduced in the second paper in Ref. [11]. Recently [8] Seesaw mechanisms at TeV scale and their effects have being investigated in it was reintroduced in Refs.: N.A. Ky, N.T.H. Van, Phys. Rev. D 72 (2005) 115017; various contexts, see Refs.: P.S. Bhupal Dev, R.N. Mohapatra, Phys. Rev. D 81 A. Palcu, Mod. Phys. Lett. A 21 (2006) 2591; (2010) 013001; P.V. Dong, H.N. Long, Phys. Rev. D 77 (2008) 057302; W. Grimus, L. Lavoura, arXiv:0912.4361; D. Cogollo, H. Diniz, C.A. de S. Pires, P.S. Rodrigues da Silva, Eur. Phys. J. C 58 Zhang He, Shun Zhou, arXiv:0912.2661; (2008) 455. Both works used the sextet to implement the type I seesaw mech- I. Gogoladze, N. Okada, Q. Shaﬁ, Phys. Lett. B 672 (2009) 235; anism into the model. F. Bonnet, D. Hernandez, T. Ota, W. Winter, JHEP 0910 (2009) 076; [13] The type II seesaw mechanism in a version of the 3-3-1 models called the Zhi-zhong Xing, Shun Zhou, Phys. Lett. B 679 (2009) 249; minimal 3-3-1 model was implemented in Refs.: J.C. Montero, C.A. de S. Pires, K. Huitu, S. Khalil, H. Okada, S.K. Rai, Phys. Rev. Lett. 101 (2008) 181802; V. Pleitez, Phys. Lett. B 502 (2001) 167; J. Kersten, A.Y. Smirnov, Phys. Rev. D 76 (2007) 073005; M.B. Tully, G.C. Joshi, Phys. Rev. D 64 (2001) 011301. Concerning these works, D. Atwood, S. Bar-Shalom, A. Soni, Phys. Rev. D 76 (2007) 033004; we would like to emphasize two points. First, that a sextet of scalar is required F. del Aguila, J.A. Aguilar-Saavedra, R. Pittau, JHEP 0710 (2007) 047; also to implement the mechanism and that the mechanism is capable of gen- F.M.L. de Almeida Jr., Y. Do, A. Coutinho, J.A.M. Simoes, A.J. Ramalho, S. Wulck, erating masses to the left-handed neutrinos only; M.A.B. do Vale, Phys. Rev. D 75 (2007) 075002; In regard to the type II mechanism in the 331νR, see: D. Cogollo, H. Diniz, C.A. F. del Aguila, J.A. Aguilar-Saavedra, R. Pittau, J. Phys. Conf. Ser. 53 (2006) 506. de S. Pires, Phys. Lett. B 677 (2009) 338. [9] Ernest Ma, Phys. Rev. Lett. 86 (2001) 2502. [14] C.A. de S. Pires, P.S. Rodrigues da Silva, Eur. Phys. J. C 36 (2004) 397.