Quasi-Dirac neutrinos in a model with local B−L symmetry
aa r X i v : . [ h e p - ph ] F e b Quasi-Dirac neutrinos in a model with local B − L symmetry A. C. B. Machado ∗ and V. Pleitez † Instituto de F´ısica Te´orica–Universidade Estadual PaulistaR. Dr. Bento Teobaldo Ferraz 271, Barra FundaS˜ao Paulo - SP, 01140-070, Brazil (Dated: 21/11/2012)
Abstract
In a model with B − L gauge symmetry, right-handed neutrinos may have exotic local B − L charge assignment: two of them with B − L = − B − L = 5. Then, itis natural to accommodate the right-handed neutrinos with the same B − L charge in a doubletof the discrete S symmetry, and the third one in a singlet. If the Yukawa interactions involvingright-handed neutrinos are invariant under S , the quasi-Dirac neutrino scheme arise naturally inthis model. However, we will show how in this scheme it is possible to give a value for θ inagreement with the Daya Bay results. For example the S symmetry has to be broken in theYukawa interactions involving right-handed charged lepton. PACS numbers: 14.60.St, 14.60.St, 11.30.Fs ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Usually it is said that neutrino mass eigenstes may be of the Dirac or Majorana type.Notwithstanding, the possibility that the definition of the particle and anti-particle is am-biguous was pointed out many years ago from two different motivations. The first one, ariseswhen Jauch [1] studying the quantization of spinor fields showed that there are anticom-mutation relations which cannot follow from the Schwinger’s action principle [2], and thatit implies the existence of fermion fields with an intermediate nature between those of theDirac and of the Majorana fields. Jauch showed that a real parameter ρ appears in theanticommutation relation between the spinor field at different space-time points and that itmay has values in the close interval [0 , ρ = 0, while ρ = 1corresponds to the Majorana field. The cases with 0 < ρ < aν + bν c ,with | a | + | b | = 1. Moreover, Pauli pointed out the theoretical possibility that processeslike the neutrinoless double beta decay may have cross sections with values between zeroand the theoretical maximum value. The latter point was rediscovery in the context of agauge model in Ref. [6].In modern gauge theories, since there are several neutrino flavors, these sort of fields canbe realized in several ways. They occur when two Majorana neutrinos are mass degeneratedand have opposite parity so they are equivalent to one Dirac neutrino [7]. However, thedegenerescence would be remove by the weak interactions, when quantum corrections aretaken into account the would be a Dirac fermion will split into two Majorana neutrinoswith different masses. But, if the mass splitting is small this Dirac neutrino will became apseudo-Dirac when one of the mass generated neutrinos is active and the other is sterile [8, 9]or pseudo-Dirac if both of them are active neutrinos [6, 10, 11].More recently, a quasi-Dirac neutrino scheme was proposed in Ref. [12]. This model usesthe S discrete symmetry to generate, at the tree level, the so called tribimaximal mixingmatrix in the lepton sector. This was also implemented in a model whith exotic right-handed2eutrinos i.e., they carry non-convencional U (1) B − L charges [13]. The aim of the presentpaper is to illustrate the main characteristics of the scheme and its limitations to produce a θ = 0 in agreement with the recent Daya Bay results [14]. Solar neutrino data may imposetoo strong constraints that it is not possible to generated through quantum corrections theobserved θ .The outline of this paper is as follows: In Sec. II we review the model in which the schemeis implemented. In Sec. III we consider the mechanism for generating charged lepton massesand in Sec. IV we show how the observed θ could be generated. The last section is dedicatedto our conclusions and some remarks. II. MODELS WITH NON-IDENTICAL RIGHT-HANDED NEUTRINOS
The possibility that all neutrino flavor eigenstates are part Dirac and part Majorana,as a consequence of a S symmetry, has been put forward recently [12]: two mass eigen-states are Majorana fields and one is a Dirac field at the tree level. This scheme calledbimodal/schizophrenic is just an interesting example of a quasi-Dirac neutrino based on the S symmetry. It is this symmetry that allow to distinguish among lepton generations (dou-blets and right-handed neutrinos). However, if all right-handed neutrinos have the samequantum number their separation in S irreducible representations is arbitrary. This wouldnot be the case if the model has two right-handed neutrinos having a different quantumnumber from that of the third one. It happens in the model of Ref. [13] because of thegauged B − L symmetry. In fact, the quasi-Dirac scheme was already implemented in amodel [15] with local B − L symmetry which has a different scalar content with respect tothat of Ref. [13].Here we will be concerned on how the non-zero θ may be obtained in the scheme whichis also consistent with the Daya Bay results [14]. First, let us briefly review the main feature3f the model which has the following gauge symmetries: SU (3) C ⊗ SU (2) L ⊗ U (1) Y ′ ⊗ U (1) B − L ↓ h φ i SU (3) C ⊗ SU (2) L ⊗ U (1) Y ↓ h Φ i SU (3) C ⊗ U (1) em , (1)where Y ′ is chosen in order to obtain the hypercharge Y of the standard model, given by Y = Y ′ + ( B − L ). Here h φ i and h Φ i denote one (or several) singlets and doublets of SU (2) L , respectively. Thus, in this case, the charge operator is given by Qe = I + 12 [ Y ′ + ( B − L )] . (2)The anomaly cancelation is also implemented if, for quarks, charged leptons and activeleft-handed neutrinos the local B − L charges are as the usual ones, but for right-handedneutrinos this charge is, instead of the usual assigment, B − L = − B − L = 5 for the third one. For this reason we call them exotic right-handed neutrinos.Thus, these neutrinos are, because of the local B − L and Y ′ [= − ( B − L )], charges, naturallysplit in the two irreducible representation of S : = ⊕ . After the breaking of theelectroweak gauge symmetry as is shown in Eq. (2), the usual global U (1) B and U (1) L appearas accidental symmetries as in the standard model. Also, because of the non-standard localcharges, the lepton sector has its own scalar sector and no large hierarchy in the Yukawacouplings is necessary. In this paper, as we said before, we will consider more details of themodel of Ref. [15] and suggest two possibilities of how a non-zero θ can be obtained.The right-handed neutrinos with B − L = − S , say ( n eR , n τR , may beconsider heavy, and the singlet say n µR , is the light right-handed neutrino. It is also assumed,as in the model of Refs. [12, 15, 16] (here we called it case ( a )), D = ( L , L ) = (1 / √ L e − L µ − L τ ) , (1 / √ L µ − L τ )) transforms as a doublet and L S = L = (1 / √ L e + L µ + L τ ) asa singlet. The scalar sector differs slightly from that of [13]: two scalar doublets with weakhypercharge Y = − , = ( ϕ , ϕ − , ) T are singlet of S . The mixing anglesin the ( n eR , n τR ) sector have been absorbed in h and h . There are also scalar singlets of SU (2). For the quantum numbers of these fields see Ref. [15]. We denote h ϕ ( ϕ ) i = v ( v ).4herefore, after integrating the heavy degrees of freedom, the effective Yukawa interac-tions that give neutrino masses are − L eff ν = h ¯ L Φ n µR + h m n e [( L c ) R Φ ∗ ][ L L Φ ∗ ]+ h m n τ [( L c ) R Φ ∗ ][ L L Φ ∗ ] + H.c.. (3)From the Yukawa interactions in Eq. (3), the neutrino mass matrix in the basis m m D m m D ( a ) , (4)which has the eigenvalues m , m D , m , m D . At the tree level, there are four massive Majo-rana neutrinos, two of them, an active and the sterile neutrino, are mass degenerated andcorrespond to a (quais)Dirac neutrino.On the other hand, the neutrino mass matrix ( χ ′ ) M νM χ ′ written in the active neutrinobasis χ ′ = N ′ L + ( N ′ L ) c where N ′ L = ( ν eL , ν µL , ν τ ) TL , is of the form M νM = m − − −
13 16 + m m − m m −
13 16 − m m + m m , (5)The eigenvalues of the mass matrix (5) are ( m , , m ) and will be denoted by m M = m , m M = 0, and m M = m . The massive Majorana neutrinos are ν and ν , while ν D has noMajorana mass at tree level, and we have definded m = h v /m n e and m = h v /m n τ .The matrix in Eq. (5) is a consequence of the S symmetry [17]. This matrix is diagonalizedby the tribimaximal matrix [18]: V ′ = U T B Ω ′ = q
23 1 √ − √ √ − √ − √ √ √ Ω ′ , (6)where Ω ′ = diag( e iρ , , N L = ( ν L , ν DL , ν L ) T are related to the flavor basis as N ′ L = U T B N L . Notice that as a result we have only oneMajorana phase because one of the eigenvalues of the matrix (5) is zero.We will understand better this situation if we consider the 4 × χM χ , where χ i = N iL + ( N iL ) c , and N iL = ( ν L ν L ν L n cµL ) T . The eigenvalues of the matrix in Eq. (4)5re m , m D , m , m D where m , m are as given above, and m D = h v / √
2. The respectiveeigenvectors are ν M = χ = (1 , , , ν M = (1 / √ , χ , , χ ), ν M = χ = (0 , , ,
0) and ν M = ( − i/ √ , χ , , − χ ), i.e., we have two Majorana neutrinos, ν M , ν M and a Diracneutrino formed by two mass degenerate Majorana neutrinos ν D = ν M + iν M . The matrixin Eq. (4) has a conserved lepton number induced by the transformation acting only onthe fields χ → e − iβ χ and χ → e − iβ χ , implying the Dirac character of ν D , which is notconserved by the weak interactions, hence we have a quasi-Dirac neutrino formed by anactive and a sterile neutrino (according to the notation discussed in Sec. I).Next, let us also consider, from the Yukawa interaction in Eq. (3), a 4 × χ ′ M ν χ ′ , but now in the basis χ ′ i = N ′ iL + ( N ′ iL ) c where N ′ iL = ( ν e ν µ ν τ n cµ ) TL and M ν given by: M ν = m − − m D √ m −
13 16 + m m − m m m D √ m −
13 16 − m m + m m m D √ m m D √ m m D √ m m D √ m , (7)where m and m are defined as in the matrix in (5) and m D as above. The eigenvalues ofthe matrix (7), denoted by m M e iρ , m M , m M , m M , are m M = m , m M = m M ≡ m D , m M = m , (8)and we see that the two Majorana masses m M and m M are the same as in the case of thematrix in Eq. (5). The four Majorana massive neutrinos have been split as follows: two ofthem have different masses and are purely Majorana fermions; the other two, which are massdegenerated, fuse to form a Dirac massive neutrino. All of this is at tree level. After thebreaking of the gauge B − L symmetry nothing protects neutrinos to gain small Majoranamasses by quantum corrections, in particular, the left- and right- components of the wouldbe Dirac neutrino, ν D . However, it may be rather small and this neutrino will continue tobe, for all practical purposes, a Dirac neutrino (see below).The mass matrix in (7) is diagonalized by the matrix V | ( a ) = q / √ − / √ − / √ / √ − / √ − / √ − / √ / √ / √ − / √
60 1 / √ / √ Ω , (9)6nd Ω = diag( e iρ , , , i ), which can be rewritten as the following matrix product [16] V | ( a ) = q
23 1 √ − √ √ − √ − √ √ √
00 0 0 1 √ − √ √ √ Ω . (10)The relation of the mass eigenstates χ i = N iL + ( N iL ) c , with N L = ( ν ν ν ν ) L with theflavor eigenstates χ ′ is given by χ = V | ( a ) χ ′ , where V | ( a ) is given in Eq. (9) or (10). Therespective Majorana mass eigenstates fields are ν M = ν L + ( ν L ) c and ν M = ν L + ( ν L ) c which have masses ˜ m = m e iρ and m , respectively, and the other two Majorana fields ν M = ν L +( ν L ) c , and ν M = ν L +( ν L ) c form a Dirac field, ν D = ν M + iν M , with a Dirac massterm, m D . We define √ ν DL = ν L − iν L ≡ ( n µ ) cL and √ ν cD ) R = ( ν ) cR + i ( ν ) cR ≡ n µR .Note that the mass eigenstates ν and ν , when written on the flavor basis, do not have thecontribution of the fourth neutrino n µR . This is a prediction of the model at tree level, sinceit is the rotation matrix in (10) which determines this outcome.With the masses in (8) we obtain∆ m = m D − m = h v − h v m n e , | ∆ m | = | m − m D | = (cid:12)(cid:12)(cid:12)(cid:12) h v m n τ − h v (cid:12)(cid:12)(cid:12)(cid:12) ≈ | ∆ m | = | m − m | . (11)Experimentally, ∆ m = (7 . ± . × − eV , ∆ m = (2 . ± . × − eV [19]. Justfor an illustration, these values can be fitted by choosing, for the normal hierarchy (here andbelow all parameters with dimension of mass are in eV), h = 0 . , h = 0 . , h = 0 . v = 1 . , v = 10 , and m n e = m n τ = 10 , the neutrino masses are m D = 0 . m =0 . m = 0 . h = 0 . , h = 0 . , h = 0 . v = 1 . , v = 10 , and m n e = m n τ = 10 ,the neutrino masses are m D = 0 . m = 0 . m = 0 . n µR .In fact, it is possible to choose different representations from that we have called case (a),which leads to the effective interactions in Eq. (3), in such a way that the neutrino mass7erm written as ( ¯ N M N ) becomes m m m D m D ( b ) , or m D m m m D ( c ) . (12)The model of [12, 15] corresponds to the case (a), the cases (b) and (c) arise if we defineb): in the right-handed neutrinos sector n τR is now the singlet and ( n eR , n µR ) the doublet,and L S = L and D = ( L , L ). c): n eR is the singlet and ( n µR , n τR ) the doublet, and in L S = L and D = ( L , L ). A similar analysis to that doing for the matrix in Eq. (4), followsfor matrices in Eq. (12).Instead of (10) we have for the cases (b) and (c), respectively: V | ( b ) = q √ − √ − √ √ − √ √ √
00 0 0 1 √ − √ √ √ Ω , (13)with Ω as in Eq. (10), and V | ( c ) = √ q √ − √ − √ √ √ √
00 0 0 1 √ − √ √ √ Ω ′′ , (14)where Ω ′′ = (1 , e iρ , , i ).As we said before, the tribimaximal matrix diagonalized the neutrino mass matrix atleading order. Corrections to that matrix may arise from quantum loop corrections [20] (orevolution with the renormalization group equations [21]) and/or the mixing in the chargelepton sector. In fact, global neutrino data analysis had already suggested that θ = 0 [22],and a first evidence that this is the case has been obtained from the observation, at the 2.5 σ level, of the appearance ν µ → ν e [23, 24]. More recently, the Daya Bay results are moreconclusive, at the 5.2 σ level they found sin θ = 0 . . stat )0 . syst ) [14], whichimplies 0 . ≤ sin θ ≤ .
17 [25]. Finite quantum corrections to the mass matrix in Eq. (5)may be consider but they are strongly suppressed by solar neutrino data because of the8ctive to sterile neutrino oscillation present in this model [26, 27]. It is for this reason thatthe would be Dirac neutrino is for practical proposes a Dirac fermion. This implies a V P MNS matrix of the tribimaxiaml type. The only way to obtain a realistic form of this matrix isto have a nondiagonal mass matrix in the charged lepton sector and using V P MNS = U † l U νL .Here, U νL means the 3 × V P MNS matrix.Before doing it, let us consider the mass matrix of the charged lepton sector.
III. THE CHARGED LEPTON SECTOR
Let us suppose that the S symmetry also constrains the Yukawa interactions in thecharged leptons. In this case, to obtain the charged lepton mass matrix we add three extrascalar doublets with Y = +1, denoted Φ e , Φ µ , Φ τ , and using the complex representation for S determined by the matrix [28] U ω = 1 √ ω ω ω ω , (15)which change from the irreducible basis and to the reducible one without changing theproduct rules. In this basis L = ( L e , L µ , L τ ), R = ( e R , µ R , τ R ) and S = (Φ e , Φ µ , Φ τ ).The charged lepton sector the Yukawa interactions can be written using the four S singletsformed by the direct products of three triplets ( x i , y i , z i ), i = 1 , , x y z + x y z + x y z , x y z + x y z , x y z + x y z , and x y z + x y z : − L l = G [ ¯ L e Φ e e R + ¯ L µ Φ µ µ R + ¯ L τ Φ τ τ R ] + H ¯ L e ( µ R Φ τ + τ R Φ µ )+ F ( ¯ L µ Φ τ + ¯ L τ Φ µ ) e R + I ( ¯ L µ τ R + ¯ L τ µ R )Φ e + H.c. (16)From (16) we obtain the most general mass matrix for charged leptons (here h ϕ l i = v l / √ M l = 1 √ Gv e Hv τ Hv µ F v τ Gv µ Iv e F v µ Iv e Gv τ . (17)Using the following values for the parameters (dimensional parameters in MeV): v e = 9 . v µ = 2038 . v τ = 34924 . G = 0 .
05 and H = 1 . × − , F = 0 . × − , I = 1 . × − ,9e obtain m e = 0 . m µ = 102 .
155 and m τ = 1746 .
24. The mass matrices are a predictionof the model, hence they are valid at the energies at which all symmetries of the model arerealized, i.e. , at the electroweak scale. For this reason, we use the running lepton massesat µ = M Z which values were taken from Ref. [29]. The U lL unitary matrix diagonalize theHermitian M l M † l matrix, where M l is the mass matrix in (17). We obtain for the values ofthe parameter above: U lL = . − . − . − . − . . . . . . (18)We have consider several set of values for the parameters in the Yukawa interactions (17)and all of them give similar values for the entries of the matrix U lL as in (18). IV. THE PMNS MIXING MATRIX
Notwithstanding, the full PMNS matrix also includes rotations in the charged leptonsector, i.e., it is defined as V P MNS = U l † L U T B . Here, U T B denotes any of the 3 × V P MNS , for the three cases in Eqs. (10),(13) and (14), obtaining, | V P MNS | ( a ) ≈ . . . . . . . . . , (19) | V P MNS | ( b ) ≈ . . . . . . . . . , (20)and | V P MNS | ( c ) ≈ . . . . . . . . . , (21)10espectively. From data we have [25]: | V P MNS | exp = . − .
845 0 . − .
61 0 . − . − .
58 0 . − .
65 0 . − . . − .
43 0 . − .
74 0 . − . . (22)Comparing the three V P MNS in Eqs. (19)-(21) with the matrix in Eq. (22), we see that thereis no agreement with the measured V P MNS .There are a few ways to turn around this trouble. On the one hand, since the neutrinoand charged lepton masses are at the Z -peak [29], it means that the running of these valuesto the low energies have to be done still [21] in the context of the present model. On theother hand, radiactive corrections will induce a non-zero θ . Notwithstanding, it impliesthat the corrections to the neutrino mass matrix are not to small, but since in this casethere are active to sterile neutrino oscillations the mass corrections are strongly constrainedby solar data [27] in such a way that no realistic value for θ arises.Another possible way to overcome this difficulty is by considering the S symmetry tobe break in the charged lepton sector. Thus, a S non-invariant Yukawa interactions givenmass to the charged leptons is − L l = ¯ L iL Y lij l jR v SM √ H.c. (23)where Y l is an arbitrary 3 × H denotes the Higgs boson of the standard modelwhich gives mass to the quarks too. If we solve, simultaneously the following equations V P MNS = U l † L U T B , v SM U lL Y l ( Y l ) † ( U lL ) † = diag( m e , m µ , m τ ) , (24)we obtain that: in case ( a ) has solutions with complex Yukawa parameters, case ( b ) has nosolution and, case ( c ) has solution with real Yukawa parameters. Here we show only thelatter case: with U lL | ( c ) = . . . . − . − . . . . . (25)which implies, from (24), that Y l | ( c ) = .
98 1 . − . .
886 18 .
51 33 . − . − . − . . (26)11nd we obtain | V P MNS | ( c ) = .
79 0 .
56 0 . .
53 0 .
65 0 . .
33 0 .
58 0 . , (27)which in agreement with (22). V. CONCLUSIONS
The quasi-Dirac neutrino scheme of Refs. [12, 15] is interesting in its own. Although thereis a light sterile neutrino, its spectrum is not of the form ”3+1”, since the fourth massiveneutrino is quasi degenerated with one of the active neutrinos. Hence, the effect of the extramass square difference might appear only in neutrinos coming from long distance sourceslike supernovas and not as a solution to some possible neutrino anomalies [30].The mass matrix in equation (17) is the best mass matrix that can be obtained withoutviolating the S symmetry and contribute to the value of θ = 0. However, without violatingthe S symmetry in the charged lepton Yukawa interactions we cannot explain the Daya Baydata. since the right-handed charged lepton do not have exotic values for the gauge B − L symmetry, the S symmetry is not naturally incorporated in that interactions. Hence, thevalidity of S symmetry only in the neutrino sector can be a prediction of the model thatmade the B − L symmetry a gauge symmetry and exotic right-handed neutrinos. On theother hand, if the charged lepton interactions violate the symmetry S , we can get a generalmass matrix, as that in Eq. (23) and, as we have shown, it is possible to fit a realistic PMNSmixing matrix in agreement with the Daya Bay’s results.Some final remarks are in order. 1) If a Majorana mass for the sterile neutrino, say n µR , isallowed the 44 entry in the mass matrix (7) is non-zero. Notwithstanding, the Majorana massfor the n µR arises only from non-renormalizable interactions since the operator ( n µR ) c n µR isnot allowed by the B − L attribution of the model and the Z symmetries, see Ref. [15] for therespective quantum numbers. Hence, the operator with the lower dimension generating thismass term is one of dimension seven: λ M (Φ SM ǫ Φ φ φ ∗ x / Λ )( n µR ) c n µR . In this case M = λ M v SM v u v x / Λ . Just for illustration, if λ M ∼ O (1), u < ∼ Λ, M ≈ v SM v v x / Λ is rathersmall using v SM ∼
100 GeV, v = 10 − GeV and Λ = 1 TeV we have M ≈ − ( v x / GeV).Moreover, v x is not necessarily large since it is not the responsible for breaking the B − L M value is rather small and satisfies the constraint fromsolar data [27].2) The present model involves Higgs scalar doublets which couple mainly to leptons andhave small VEVs. Moreover, the model has also scalar singlets and some of them may haveVEVs lower than the TeV scale. Tiny VEVs have been proposed before [31] and after [32]the models in Ref. [13, 15]. Then, the question concerning on the stability of the tree levelVEVs arises. This has been done recently [33] in the context of the particular 2DHM ofRef. [32]. This question is rather model dependent and a similar analysis in the context ofthe present model will be considered elsewhere.3) The model in Ref. [13] was proposed just as a new solution to the anomaly cancellationwhen B − L is a local symmetry, and it has interesting features by its own. For instance,beside implementing the bimodal scheme without fine tuning in the neutrino Yukawa inter-actions, it is a model which also implement naturally the features of the so called leptophilictwo Higgs doublet model (L2DHM) [34] and the neutrino specific 2HDM one [32, 35]. Inthe latter models those Higgs doublets were introduced ad hoc . In general the supersym-metric versions of this sort of models has interesting features in accelerator physics [36]and in cosmology [37]. Recently, a model with quasi-Dirac and Majorana neutrinos in thecontext of supersymmetric standard model with the extra symmetries S ⊗ ( Z ) has beenproposed [38]. However, we would like to stress that all these features, for instance dou-blets given masses just for neutrinos, arise naturally when we consider the anomaly free B − L gauge symmetry and they are not assumed ad hoc . This avoids extreme fine tuningin the Yukawa coupling of the Dirac neutrino as h = 0 . . h ∼ − as in [12, 37].4) Finally, we stress that the existence of such a very light sterile neutrino is a predictionof the model, and it is not motivated by possible anomalies observed in neutrino experi-ments [30]. Since it is (almost) mass degenerate with one of the active neutrinos its effectmay be only observed in extragalactic neutrinos [16]. Acknowledgments
One of the author (ACBM) was supported by CAPES and (VP) was partially supportedby CNPq and FAPESP. 13 ppendix A: charged leptons
The charged lepton masses in Ref. [12] is generated by a dimension five operator and itis diagonal at tree level. Thus the V P MNS is just the matrix that diagonalized the neutrinomass matrix and this is just the tribimaximal one. To obtain those authors introduce threegauge singlet scalars σ e , σ µ and σ τ these fields and the right-handed charged letons transformlike the left-handed doublets and for avoiding a general mixing it is necessary to impos Z n symmetries: Z n,e ⊗ Z n,µ ⊗ Z n,τ in such a way that the right-handed leptons transfom as ω pe,µ,τ and the singlets scalars as ω − pe,µ,τ . This case has been rouled out by recent neutrino data.Case 1. The right-handed components trasfoorm as e S = e R , e D = ( µ R , τ R ) and theYukawa coupling is given by − L l = h e ¯ L e S H + h µτ [ ¯ L D e D ] H + H.c. (A1)where H is the usual SM Higgs doublet. This case is not favored because the µ and τ aremass degenerated and the matrix which diagonalize the mass matrix is the tribimaximal,hence V P MNS = U T B = .Case 2. Now e S = 1 √ e R + µ R + τ R ) , e D = (cid:20) √ e R − µ R − τ R ) , √ µ R − τ R ) (cid:21) , (A2)with the Yukawa interactions as in Eq. (A1). In this case the matrix which diagonalize thecharged leptons mass matrix is again the the tribimaximal and m µ = (2 / m τ .Case 3. The right-handed components of the charged leptons transform as in Case 1 butwe introduce three Higgs doublets, Φ e , Φ µ , Φ τ , transforming under S as a singlet H S = Φ e and a doublet H D = (Φ µ , Φ τ ). The Yukawa interction is − L l = h e ¯ L e S H S + h [ ¯ L e D ] H D ] + h [[ ¯ L D H S ] e D ] + h [[ ¯ L D H D ] e D ] + h [[ ¯ L D H S ] e D ] + h [ ¯ L D H D ] e S + H.c. (A3)In this case we have that Det M l = 0, hence the electron remains massless at tree level.Case 4. Introduce Higgs scalars as in Case 3 and all right-handed charged lepton transformas singlet under S . The Yukawa interactions is − L l = ( h i ¯ L S H S + h ′ i [ ¯ L D H D ] ) l iR + H.c. (A4)14n this case we obtain as in the Case 1, the trimaximal matrix in the charged lepton sectorand two leptons remains massless at tree level.Case 5. Now the e R , µ R , τ R transform as in Eq. (A2) and the three Higgs doublets asin Case 3 and the Yukawa interactions are also given in Eq. (A3). This case is difficult toanalyse analitically but numerical calculations indicate altought we can fit the three chargedlepton masses the mixing matrix is given by U lL = − .
382 0 .
621 0 . .
360 0 .
582 0 . .
851 0 .
525 0 . (A5)and the PMNS matrix | V P MNS | = .
51 0 .
06 0 . .
06 0 .
99 0 . .
86 0 .
03 0 . (A6)does not fit the experimental values, see Eq. (22).All these cases allow a diagonal mass matrix in the charged lepton sector if extra Z n symmetries are added as in Ref. [12]. [1] J. M. Jauch, 1954, Helv. Phys. Acta V27 , P89[2] J. Schwinger 1951 Phys. Rev. ibid ibid B186
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