Lepton masses in a supersymmetric 3-3-1 model

Lepton masses in a supersymmetric 3-3-1 model

J. C. Montero,*V. Pleitez,†and M. C. Rodriguez‡

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, SP, Brazil

共Received 19 December 2001; published 15 May 2002兲

We consider the mass generation for both charginos and neutralinos in a 3-3-1 supersymmetric model. We show that R-parity breaking interactions leave the electron and one of the neutrinos massless at the tree level. However, the same interactions induce masses for these particles at the 1-loop level. Unlike the similar situation in the minimal supersymmetric standard model, the masses of the neutralinos are related to the masses of the charginos.

DOI: 10.1103/PhysRevD.65.095008 PACS number共s兲: 12.60.Jv, 14.60.Pq

I. INTRODUCTION

The generation of neutrino masses is an important issue in any realistic extension of the standard model. In general, the values of these masses 共of the order of, or less than, 1 eV兲 that are needed to explain all neutrino oscillation data关1–3兴 are not enough to put strong constraints on model building. This means that several models can induce neutrino masses and mixing compatible with experimental data. So, instead of proposing models built just to explain neutrino properties, it is more useful to consider what are the neutrino masses that are predicted in any particular model which has a moti-vation other than the explanation of neutrino physics. For instance, the 3-3-1 model was proposed as a possible sym-metry on the lightest lepton sector (␯e,e⫺,e⫹)L 关4兴. Once

that symmetry is assumed it has to be implemented in the rest of the leptons and also in the quark sector. Like in the standard model, if we do not introduce right-handed neutri-nos and/or violation of the total lepton number, the neutrineutri-nos remain massless at any order in perturbation theory. In this vein some effort has been done to produce neutrino masses in the context of that 3-3-1 model and some of its extensions

关5兴.

In this work we consider the generation of neutrino masses in a supersymmetric 3-3-1 model with broken R par-ity. We show that, as an effect of the mixing among all lep-tons of the same charge, at the tree level only one charged lepton and one neutrino remain massless but they gain mass through radiative corrections. In order to compare this model we do the same calculations in the context of the minimal supersymmetric standard model with R broken parity also. In both cases we are not assuming that sneutrinos gain nonva-nishing vacuum expectation values 共VEVs兲; i.e., the only nonzero VEVs are those of the scalars of the nonsupersym-metric models.

The outline of this work is as follows. In Sec. II we re-view the origin of the lepton masses in the minimal super-symmetric standard model context under the same assump-tions that we will use in the case of the 3-3-1 supersymmetric

model. In Sec. III we consider the supersymmetric version of a 3-3-1 model which has only three triplets of Higgs scalars. We explicitly show that leptons gain mass only as a conse-quence of their mixing with gauginos and Higgsinos. Our conclusions are found in the last section.

II. NEUTRINO MASSES IN THE MSSM

Let us consider in this section the lepton masses in the minimal supersymmetric standard model 共MSSM兲 关6兴. In this model the interactions are written in terms of the left-handed 共right-handed兲 Lˆ⬃(2,⫺1) 关lˆc⬃(1,2)兴 leptons, left-handed 共right-handed兲 quarks Qˆ⬃(2,1/3) 关uˆc_⬃(1, ⫺4/3),dˆc_{⬃(1,2/3)兴; and the Higgs doublets Hˆ}

1⬃(2,

⫺1),Hˆ2⬃(2,1). With those multiplets the superpotential that conserves R-parity is given by W2RC⫹W3RC⫹W¯2RC

⫹W¯

3RC, where

W2RC⫽␮⑀Hˆ1Hˆ2,

W_3RC⫽⑀Lˆ_af_abl Hˆ₁lˆ_bc⫹⑀Qˆ_ifu_{i j}Hˆ₂uˆ_jc⫹⑀Qˆ_if_{i j}dHˆ₁dˆc_j, 共1兲

while the R-parity violating terms are given by W2RV

⫹W3RV⫹W¯2RV⫹W¯3RV, where W2RV⫽␮0a⑀LˆaHˆ2, W3RV⫽⑀Lˆa␭abcLˆblˆc c_⫹⑀Lˆ a␭ai j

⬘

Qˆidˆj c_⫹uˆ i c_␭ i jk

⬙

dˆc_jdˆ_kc, 共2兲

and we have suppressed SU(2) indices;⑀is the antisymmet-ric SU(2) tensor. Above, and below in the following, the subindices a,b,c run over the lepton generations e,␮,␶but a superscript c indicates charge conjugation; i, j,k⫽1,2,3 de-note quark generations. We also have to add the soft terms that break the supersymmetry:

*Email address: montero@ift.unesp.br

†_{Email address: vicente@ift.unesp.br} ‡_{Email address: mcr@ift.unesp.br}

Lsoft⫽⫺ 1 2

冉

p

兺

_⫽1 3 m_␭␭_Ap␭_Ap⫹m

⬘

␭B␭B⫹H.c.

冊

⫺ML 2 L ˜†_L˜ ⫺Ml 2 lc ˜†_l˜c_⫺M Q 2 Q˜†Q˜⫺M_u2˜uc†˜uc⫺M_d2˜dc†˜dc ⫺M1 2 H˜1 † H˜1⫺M2 2 H ˜ 2 †

H˜2⫺关ALH1˜ l˜L c⫹AUH2Q˜ u˜c

⫹ADH1Q˜ d˜c⫹M12 2_H

1H2⫹BH2L⫹C1L˜ L˜ l˜c

⫹C2L˜ Q˜ d˜c⫹C3˜uc˜dc˜dc⫹H.c.兴, 共3兲 where p is an SU(2) index and␭_A,␭_Bare the

supersymmet-ric partners of the respective gauge vector bosons but we have omitted generation indices and the gluino-mass terms.

With the interactions in Eq.共1兲 it is possible to give mass to all charged fermions in the model共see below兲 but neutri-nos remain massless. Hence, we must introduce R-parity vio-lating terms like those in Eq.共2兲. Some of the coupling con-stants in that expression should be set to zero in order to avoid a too fast proton decay. Defining the basis ⌿_{M SS M}0

⫽(␯e,␯␮,␯␶,⫺i␭A 3_,⫺i␭

B,H˜1 0_,H˜

2

0₎T_{, the mass term is} ⫺(1/2)关⌿M SS M 0T YM SS M 0 _⌿0_{⫹H.c.兴, where Y} M SS M 0 is the mass matrix Y_{M SS M}0 ⫽

冉

0 0 0 0 0 0 ⫺␮0e 0 0 0 0 0 0 ⫺␮_0␮ 0 0 0 0 0 0 ⫺␮_0␶ 0 0 0 m_␭ 0 M_Zs_␤c_W ⫺M_Zc_␤c_W 0 0 0 0 m

⬘

M_Zs_␤s_W ⫺M_Zc_␤s_W 0 0 0 MZs␤cW MZs␤sW 0 ␮ ⫺␮0e ⫺␮0␮ ⫺␮0␶ ⫺MZc␤cW ⫺MZc␤sW ␮ 0

冊

, 共4兲

where s_␤⫽sin␤, sW⫽sin␪W, etc. are defined as tan␤ ⫽v2/v1and␪Wis the weak mixing angle. The matrix in Eq. 共4兲 is generated only by the two usual vacuum expectation

values of the two scalars and by the R-parity breaking terms ␮0a. The mass matrix is similar to that in Refs.关7–9兴 but we have included the three neutrinos and we are neither assum-ing that sneutrinos gain nonzero vacuum expectation values nor introduce sterile neutrinos like in Ref. 关10兴. The mass matrix in Eq. 共4兲 has two zero eigenvalues: it has determi-nant equal to zero and its secular equation which gives the eigenvalues, x, has the form x2 times a polynomial of five degrees; thus there are two neutrinos ␯1,2, which are mass-less at the tree level. Using tan␤⫽1 and M_Z

⫽91.187 GeV, s_W2⫽0.223, ␮_0e⫽␮_0␮⫽0, ␮_0␶

⫽10⫺4 _{GeV 共this value is consistent with that of Ref. 关8兴兲,}

␮⫽100 GeV, m⫽250 GeV, m

⬘

⫽⫺200 GeV, we obtain

besides the two massless neutrinos a massive one with m_␯

3

⫽⫺3⫻10⫺3 _{eV, and four heavy neutralinos with masses}

267.40,⫺199.99,⫺117.40, and 100.0 GeV. These zero ei-genvalues are a product of the matrix structure in Eq.共4兲 and there is no symmetry to protect the neutrinos from gaining mass by radiative corrections. On the other hand, if ␮0a

⫽0, a⫽e,␮,␶, all neutrinos remain massless at the tree level. In this case it is the R-parity and total lepton number conservation that protect neutrinos of gain masses. The neu-tralino masses above are consistent with those of Ref. 关8兴: two states are massless and the other ones have masses of the order O( MZ). More realistic neutrino masses require

radia-tive corrections 关9,11–13兴. Here we will only consider the neutrino masses generated by radiative corrections arisen

from the interactions given in Eqs. 共1兲 and 共2兲 and only two VEVs. We have in this case the interactions

⫺␭abc 3 共¯␯aLlbRl˜c⫹¯␯aR c l_bLc l˜_c*_兲, 共5兲 ⫺␭ai j

⬘

3 共¯␯aLdiR˜dj⫹¯␯aR c d_iLc˜d*_j兲⫹H.c.,

and the 1-loop diagrams like those in Ref. 关7兴 arise. Notice however that if we introduce a discrete symmetry 共called Z₃

⬘

later on兲, Lˆe,␮→⫺Lˆe,␮, and all other fields are even under

this transformation, we have that ␮0e⫽␮0␮⫽0;

␭ebc⫽0,共b,c⫽␮,␶兲;

共6兲 ␭␮bc⫽0, 共b,c⫽e,␶兲;

␭ei j

⬘

⫽␭␮i j

⬘

⫽0, 共i, j⫽1,2,3兲;

and the␯_e,␯_␮ neutrinos will remain massless at all order in perturbation theory. It is also possible to choose the symme-try such as Le,␶→⫺Le,␶while all other fields remain

invari-ant. In this case we have that ␯e and ␯␶ remain massless.

However, if no discrete symmetry is imposed neutrinos gain mass through a 1-loop effect like in Ref. 关7兴.

Next, let us consider the charged sector. With the interac-tions in Eq. 共1兲 it is possible to give mass to all charged fermions. Denoting ␾M SS M⫹ ⫽共e c_,␮c_,␶c_,⫺i␭ W ⫹_,H˜ 2 ⫹_兲T_, ␾⫺M SS M⫽共e,␮,␶,⫺i␭W⫺,H˜1⫺兲 T_{, 共7兲}

where all the fermionic fields are still Weyl spinors, we can

define ⌿_{M SS M}⫾ ⫽(␾_{M SS M}⫹ ,␾_{M SS M}⫺ )T, and the mass term

⫺(1/2)关⌿⫾TM SS MYM SS M⫾ ⌿M SS M⫾ ⫹H.c.兴 where Y⫾is the mass

matrix given by Y_{M SS M}⫾ ⫽

冉

0 XM SS M T X_{M SS M} 0

冊

, 共8兲 with XM SS M⫽

冉

⫺ fee l v1 ⫺ fe␮ l v1 ⫺ fe␶ l v1 0 0 ⫺ fe␮ l v1 ⫺ f␮␮ l v1 ⫺ f␮␶ l v1 0 0 ⫺ fe␶ l v1 ⫺ f_␮␶ l v1 ⫺ f_␶␶ l v1 0 0 0 0 0 m_␭

冑

2 MWc␤ ␮0e ␮0␮ ␮0␶

冑

2 MWs␤ ␮

冊

. 共9兲 With fee l _{⫽2.7⫻10⫺4} , f_␮␮l ⫽3.9⫻10⫺3, f_␶␶l ⫽1.6⫻10⫺2, f_el_␮⫽ f_el_␶⫽ f_␮␶l ⫽10⫺7 we obtain from Eq. 共9兲 the masses 0.0005,0.105,1.777 共in GeV兲 for the usual leptons, and 4.3 and 81 TeV for the charginos. We see by comparing Eq.共4兲 with Eq. 共9兲 that there is no relation between the charged lepton masses and the neutralino masses. Notice also that all charged leptons gain masses at the tree level. We will not consider this model共or some of its extensions兲 further since it has been well studied in literature关7–9,11–13兴.

III. A SUPERSYMMETRIC 3-3-1 MODEL

In the nonsupersymmetric 3-3-1 model 关4兴 the fermionic representation content is as follows: left-handed leptons L

⫽(␯a,la,la c

)L⬃(1,3,0), a⫽e,␮,␶; left-handed quarks Q1L

⫽(u1,d1,J)⬃(3,3,2/3), Q␣L⫽(d␣,u␣, j␤)⬃(3,3*,⫺1/3), ␣⫽2,3, ␤⫽1,2; and in the right-handed components we

have u_ic,d_ic,i⫽1,2,3, that transform as in the SM, and the exotic quarks Jc⬃(3*,1,⫺5/3), j_␤⬃(3*,1,4/3). The mini-mal scalar representation content is formed by three scalar triplets: ␩⬃(1,3,0)⫽(␩0,␩1⫺,␩2⫹)T; ␳⬃(1,3,⫹1)

⫽(␳⫹_,␳0_,␳⫹⫹₎T _{and ␹⬃(1,3,⫺1)⫽(␹}⫺_,␹⫺⫺_,␹0₎T_{, and}

one scalar sextet S⬃(1,6,0). We can avoid the introduction of the sextet by adding a charged lepton transforming as a singlet关14,15兴. Notwithstanding, here we will omit both the sextet and the exotic lepton. A seesaw-type mechanism will be implemented by the mixing with supersymmetric part-ners, Higgsinos or gauginos. The complete set of fields in the 3-3-1 supersymmetric model has been given in Refs.关16,17兴. We will denote, as in the previous section, the respective superfields as Lˆ and so on. We recall that in the nonsuper-symmetric 3-3-1 model with only the three triplets the charged lepton masses are not yet the physical ones: 0,m,

⫺m.

We will show how in the present model supersymmetry the R-violating interactions give the correct masses to e,␮

and␶, even without a sextet or the charged lepton singlet. We have the Higgsinos␩˜ ,˜ ,␳ ˜ and their respective primed fields␹ which have the same charge assignment of the triplets ␩,␳ and␹, for details see Ref.关16兴.

Because of the fact that in the supersymmetric model we have the gauginos and Higgsinos共for details on the Lagrang-ian of the model see 关16兴兲, when the R parity is broken we have in analogy with the MSSM, but with important differ-ences, a mixture between the usual leptons and the gauginos and Higgsinos.

One part of the superpotential is given by W2⫹W¯2where

W2⫽␮0aLˆa␩ˆ

⬘

⫹␮␩␩ˆ␩ˆ

⬘

⫹␮␳␳ˆ␳ˆ

⬘

⫹␮␹␹ˆ␹ˆ

⬘

, 共10兲

a⫽e,␮,␶; and W3⫹W¯3 where

W3⫽␭1abc⑀LˆaLˆbLˆc⫹␭2ab⑀LˆaLˆb␩ˆ⫹␭3aLˆa␳ˆ␹ˆ⫹ f1⑀␩ˆ␳ˆ␹ˆ

⫹ f1

⬘

⑀␩ˆ

⬘

␳ˆ

⬘

␹ˆ

⬘

⫹␭␣ai

⬘

Qˆ␣Lˆadˆi c_⫹␭

i jk

⬙

uˆ_icdˆ_jcdˆ_kc⫹␭

⵮

_{i j␤}uˆ_icuˆc_jjˆ_␤c

⫹␭i

⵮⬘

␤dˆi c Jˆcjˆ_␤c⫹␬1iQˆ1␩ˆ

⬘

uˆi c_⫹␬ 2iQˆ1␳ˆ

⬘

dˆi c_⫹␬ 3Qˆ1␹ˆ

⬘

Jˆc ⫹␬4␣iQˆ␣␩ˆ dˆi c_⫹␬ 5␣iQˆ␣␳ˆ uˆi c_⫹␬ 6␣␤Qˆ␣␹ˆ jˆ␤ c_{, 共11兲}

with⑀the completely antisymmetric tensor of SU(3) but we have omitted the respective indices; the generation indices are as follows: a,b,c⫽e,␮,␶ and i, j,k⫽1,2,3.

The gaugino masses come from the soft terms shown in Appendix A, Eq. 共A4兲. The ␮0, ␭1,␭3,␭

⬘

,␭

⬙

,␭

⵮

and ␭⵮⬘ terms break the R parity defined in this model as R⫽ (⫺1)3F⫹2S whereF⫽B⫹L, B(L) is the baryon 共total lep-ton兲 number; S is the spin. The ␭2 term of the superpotential W3 implies interactions like 关see Eq. 共43兲 below兴¯␯aL␩˜2R⫺l˜b* ⫺¯␯_aL␩˜

1R ⫹_l˜

L_␩⫽

冕

d4␪␩_{¯ˆ e}2gVˆ␩_{ˆ , 共12兲} where Vˆ is the superfield related to the Va gauge boson of SU(3)L. This interaction mixes Higgsinos with gauginos as

shown in Ref.关17兴.

The parameters ␮_␩ and ␮_␳ are the equivalent of the ␮ parameter in the MSSM 关6兴. The terms proportional to ␭2 and ␮_␹ have no equivalent in the MSSM. The ␭

⬘

and ␭

⬙

coupling constants are constrained by the proton decay such that 关18兴

␭11j

⬙

␭a2 j

⬘

⬍10⫺24, 共13兲

assuming the superpartner masses in the range of 1 TeV. A. Charged lepton masses

In this model there are interactions like

⫺␭3a 3 关␻共la˜␳ ⫹_{⫹ l¯} a˜␳⫹兲⫹u共la c ␹ ˜⫺_{⫹ l¯} a c ␹ ˜⫺_兲兴 ⫺1 2␮0a关l␩˜1

⬘

⫹_{⫹ l¯␩}˜ 1

⬘

⫹_⫹lc_␩˜ 2

⬘

⫺_{⫹ l¯}c_␩˜ 2

⬘

⫺_{兴, 共14兲}

which imply a general mixture in both neutral and charged sectors. Let us first consider the charged lepton masses. De-noting ␾⫹_⫽共ec_,␮c_,␶c_,⫺i␭ W ⫹_,⫺i␭ V ⫹_,␩˜ 1

⬘

⫹_,␩˜ 2 ⫹_,␳˜⫹_,˜_␹

_⬘

⫹_兲T_, ␾⫺_{⫽共e,␮,␶,⫺i␭} W ⫺_,⫺i␭ V ⫺_,␩˜ 1 ⫺_,␩˜ 2

⬘

⫺_,˜_␳

_⬘

⫺_,␹˜⫺_兲T_, 共15兲

where all the fermionic fields are still Weyl spinors, we can also, as before, define ⌿⫾⫽(␾⫹␾⫺)T, and the mass term

⫺(1/2)关⌿⫾T_Y⫾_⌿⫾_{⫹H.c.兴 where Y}⫾ _{is given by} Y⫾⫽

冉

0 XT X 0

冊

, 共16兲 with X⫽

¨

0 ⫺␭2e␮ 3 v ⫺ ␭2e␶ 3 v 0 0 ⫺ ␮0e 2 0 ⫺ ␭3e 3 w 0 ␭2e␮ 3 v 0 ⫺ ␭2␮␶ 3 v 0 0 ⫺ ␮0␮ 2 0 ⫺ ␭3␮ 3 w 0 ␭2e␶ 3 v ␭2␮␶ 3 v 0 0 0 ⫺ ␮0␶ 2 0 ⫺ ␭3␶ 3 w 0 0 0 0 m_␭ 0 ⫺gv

⬘

0 gu 0 0 0 0 0 m_␭ 0 gv 0 ⫺gw

⬘

0 0 0 gv 0 _⫺␮␩ 2 0 f1w 3 0 ⫺␮0e 2 ⫺ ␮0␮ 2 ⫺ ␮0␶ 2 0 ⫺gv

⬘

0 ⫺ ␮␩ 2 0 ⫺ f₁

⬘

u

⬘

3 0 0 0 ⫺gu

⬘

0 f1

⬘

w

⬘

3 0 ⫺ ␮␳ 2 0 ⫺␭3e 3 u ⫺ ␭3␮ 3 u ⫺ ␭3␶ 3 u 0 gw 0 ⫺ f1u 3 0 ⫺ ␮␹ 2

©

, 共17兲

where we have defined

v⫽ v␩

冑

2, u⫽ v_␳

冑

2, w⫽ v_␹

冑

2, v

⬘

⫽ v␩

⬘

冑

2, u

⬘

⫽ v_␳

⬘

冑

2, w

⬘

⫽ v_␹

⬘

冑

2. 共18兲

The chargino mass matrix Y⫾ is diagonalized using two

unitary matrices, D and E, defined by ␹

˜ i ⫹_⫽D

i j⌿⫹j , ˜␹⫺i ⫽Ei j⌿j⫺, i, j⫽1, . . . ,9 共19兲

(D and E sometimes are denoted, in nonsupersymmetric theories, by U_Rl and U_Ll, respectively兲. Then we can write the diagonal mass matrix as

To determine E and D, we note that

M_{SC M}2 ⫽DXT•XD⫺1⫽E*X•XT_共E*兲⫺1_{, 共21兲}

and define the following Dirac spinors:

⌿共␹˜ i ⫹_兲⫽共␹˜ i ⫹_␹˜¯ i ⫺_兲T_{, ⌿}c_共˜␹ i ⫺_兲⫽共˜_␹ i ⫺_␹˜¯ i ⫹_兲T_{, 共22兲}

where˜␹_i⫹ is the particle and˜␹_i⫺ is the antiparticle关6,17兴. We have obtained the following masses共in GeV兲 for the charged sector:

3186.05,3001.12,584.85,282.30,204.55,149.41, 共23兲 and the masses for the usual leptons 共in GeV兲: me⫽0, m␮ ⫽0.1052 and m␶⫽1.777. These values have been obtained

by using the following values for the dimensionless param-eters:

␭2e␮⫽0.001, ␭2e␶⫽0.001, ␭2␮␶⫽0.393,

␭3e⫽0.0001, ␭3␮⫽1.0, ␭3␶⫽1.0, 共24a兲

f₁⫽0.254, f₁

⬘

⫽1.0, 共24b兲

and for the mass dimension parameters 共in GeV兲 we have used

␮0e⫽␮0␮⫽0.0, ␮0␶⫽10⫺6, 共24c兲 ␮_␩⫽300, ␮␳⫽500, ␮␹⫽700,m␭⫽3000. 共24d兲

We also use the constraint V_␩2⫹V_␳2⫽(246 GeV)2 _coming from MW, where we have defined V_␩2⫽v_␩2⫹v_␩

⬘

2 and V_␳2 ⫽v_␳2_⫹v

␳

⬘

2

. Assuming that v_␩⫽20 GeV, v_␩

⬘

⫽v_␳

⬘

⫽1 GeV,

and 2v_␹⫽v_␹

⬘

⫽2 TeV, the value of v_␳ is fixed by the con-straint above.

Notice that the electron is massless at the tree level. This is again a result of the structure of the mass matrix in Eq.共9兲 and there is not a symmetry that protects the electron to get a mass by loop corrections. Hence, it can gain mass through radiative corrections like that shown in Fig. 1. The interac-tions of the leptons with the sleptons written in terms of Dirac fermions共although we are using the same notation兲 are given by 共and the respective Hermitian conjugate兲

Ll_⫽␭2ab 3 关␩˜2R ⫺ _共l bL˜␯la⫺laL˜␯lb兲⫹␩˜R 0_关共l aLl˜b*⫺lbLl˜a*兲 ⫹共lbL c l˜a⫺laL c l˜b兲兴⫹␩˜1R⫹共laL c _˜␯ bL⫺lbL c _˜␯ aL兲兴 ⫹␭3a 3 关⫺˜␳R ⫺⫺_l aL␹⫺⫹˜¯␳R⫺共laL␹ 0_⫺l aL c _␹⫺⫺兲 ⫺˜␹ R ⫹_共l aL␳⫹⫹⫺lal c ␳0_兲⫺␹˜ R ⫹⫹_l aL c ␳⫹_⫹˜_␳ R 0 l_aLc ␹⫺ ⫹˜␹ R 0 laL␳⫹兴⫹ ␭␣ai

⬘

2 关共u¯␣RlaL⫹ j¯␣RlaL c _兲d˜ i * ⫹d¯iR c_共l aL˜u␣⫹laL c _˜j ␣兲兴. 共25兲

The␭

⬘

interactions generate the low vertices in Fig. 1. On the other hand, the interactions between the squarks, sleptons and scalars, see Appendix A, are given by the scalar poten-tial. The soft part contributes only through the trilinear inter-actions

V_{so f t}⫽⑀_1ab共 l˜_al˜*_b⫺ l˜_a*l˜_b兲␩0_{, 共26兲} while the D terms have only quartic interactions

VD⫽ g2 4

兺

i 共Xi 0 Xi 0_⫹X a

⬘

0 Xi

⬘

0_兲

兺

a

冉

l˜al˜a*⫹ 1 2˜␯a˜␯a*

冊

, 共27兲

where X_i0⫽␹0,␩0;X_i

⬘

0⫽␩

⬘

0,␹

⬘

0; and a⫽e,␮,␶. From the F terms we have contributions to both the trilinear interactions

V3F⫽

冉

␮0c␭1abc 2 ⫹ ␮␹␭2ab 6

冊

共 l˜al˜b*⫺ l˜a*l˜b兲␩

⬘

0⫹H.c., 共28兲

and the quartic ones

V_4F⫽

冉

␭3d␭1dab 3 ⫹ f1␭2ab 9

冊

共 l˜al˜b*⫺ l˜a*l˜b兲␳ 0_␹0 ⫹4␭2ad␭2db 9 共 l˜a*l˜b⫹ l˜al˜b*兲␩ 0_␩0 ⫹␭3a␭3b 9 共 l˜al˜b*⫹˜␯a˜␯b*兲

兺

i X_i0X_i0, 共29兲

where X_i0⫽␹0,␳0 which will contribute to the upper quartic vertex in Fig. 1. Due to the interactions given in Eqs.共25兲–

共29兲, we can generate the appropriate mass to the electron.

The dominant contributions, assuming the mass hierarchy mfermionⰆmscalar where fermion means a fermion different from j1,2 and scalar means˜ , l˜,H (H denotes the heaviest␯ Higgs scalar兲 and using the values of the masses and the parameters given in Eqs.共23兲, 共24a兲 and 共24d兲 we obtain that the dominant contribution to the electron mass is, up to loga-rithmic corrections,

FIG. 1. Diagram generating the electron mass. There is also a contribution with v_␹→v_␹⬘. The left- and right-side vertices are proportional to␭_␣ei⬘ /3 and␭_␣⬘_{⬘e j}/3, respectively.

me⬀␭_␣ei

⬘

␭_␣

⬘

_⬘e jVj 2_V b 2_共v ␹ 2_⫹v ␹_⬘ 2 兲mj␣ 9m_b˜ 2, 共30兲

and with all the indices fixed, Vj denotes mixing matrix

el-ements in the two dimension j1,2space; Vb means the same

but in the d-like squark sector. We obtain me ⫽0.0005 GeV if v␹ and v␹

⬘

have the values already given

above, and with␭_␣ei

⬘

␭_␣

⬘

_{⬘e j}⬇10⫺6, which imposes mj_␤ m_b˜ 2 ⬇ 9⫻10⫺4 V2_jV_b2 GeV ⫺1_{, 共31兲}

or mb˜⫽33.33

冑

mj₂VjVb GeV. Using mj⬃250(320) GeV 关19兴, we have mb˜⬃526(596)VjVb GeV. With VjVb ⬃0.14(0.12) we obtain squarks masses of the order of mb˜ ⬃75 GeV 关20兴.

B. Neutral lepton masses

Like in the case of the charged sector, the neutral lepton masses are given by the mixing among neutrinos, induced by

the ␮0a term in Eq. 共10兲, and the neutral Higgsinos and gauginos 关17兴, and also by ␭2 and␭3 in Eq. 共11兲. The first two terms in Eq.共10兲 give the interactions between neutrinos and Higgsinos: 1 2␮0a关␯a␩˜

⬘

0_⫹¯␯ a␩˜

⬘

0兴⫺ ␮␩ 2 关␩˜

⬘

0_␩˜0_⫹␩˜

_⬘

0 _␩˜0_{兴, 共32兲} and from Eq.共11兲 we have also the interactions

␭3a 3 关w共␯a˜␳

0_⫹¯␯

a˜␳0兲⫹u共␯a␹˜0⫹¯␯a␹˜0兲兴. 共33兲

These interactions imply a mass term for the neutrinos and neutralinos. The mass term in the basis

⌿0_⫽共␯ e␯␮␯␶⫺i␭A 3_⫺i␭ A 8_⫺i␭ B␩˜0␩˜

⬘

0˜␳0˜␳

⬘

0␹˜0˜␹

⬘

0兲T, 共34兲 is given by ⫺(1/2)关(⌿0)TY0⌿0⫹H.c.兴 where 共35兲

All parameters in Eq. 共35兲, but m

⬘

, are defined in Eqs.

共18兲, 共24a兲 and 共24d兲; g and g

⬘

denote the gauge coupling constant of SU(3)L and U(1)N, respectively.

The neutralino mass matrix is diagonalized by a 12⫻12 rotation unitary matrix N, satisfying

MN M D⫽N*Y0N⫺1, 共36兲

and the mass eigenstates are ␹ ˜ i 0_⫽N i j⌿j 0 , j⫽1, . . . ,12. 共37兲 We can define the following Majorana spinor to represent the mass eigenstates:

⌿共␹˜ i 0_{兲⫽共˜␹} i 0_␹˜¯ i 0_兲T_{. 共38兲}

As above the subindices a,b,c run over the lepton genera-tions e,␮,␶.

With the mass matrix in Eq. 共35兲, at the tree level we obtain the eigenvalues 共in GeV兲,

⫺4162.22,3260.48,3001.11,585.19,⫺585.19,453.22,

⫺344.14,283.14,⫺272.0, 共39兲

and for the three neutrinos we obtain共in eV兲

m₁⫽0,m₂⬇⫺0.01,m₃⬇1.44. 共40兲 We have obtained the values in Eqs.共39兲 and 共40兲 by choos-ing, besides the parameters in Eqs. 共24a兲 and 共24d兲, m

⬘

⫽

⫺3780.4159 GeV. Notice that the coupling constant g

⬘

and the parameter m

⬘

appear only in the mass matrix of the neu-tralinos; all the other parameters in Eq. 共35兲 have already been fixed by the charged sector, see Eqs. 共17兲, 共24a兲 and

共24d兲. The neutrino masses in Eq. 共40兲 are of the order of

magnitude for the Liquid Scintillation Neutrino Detector

共LSND兲 and solar neutrino data. On the other hand, if we

choose m

⬘

⫽⫺3780.4159 GeV and ␮0␶⫽2⫻10⫺8 GeV, we obtain共in eV兲

m1⫽0.0, m2⬇⫺5.47⫻10⫺5, m3⬇1.32⫻10⫺2, 共41兲 which are of the order of magnitude required by the solar and atmospheric neutrino data. Notice also the sensibility of the neutrino masses in ␮0␶ and m

⬘

, and that if ␮0a⫽0, and a

⫽e,␮,␶, all neutrinos remain massless. The masses of the charged sector are insensible to the values of ␮0a for all a

⫽e,␮,␶ which are suitable for generating the appropriate different neutrino mass spectra 共see Appendix B兲.

We have obtained numerically the unitary matrices E,D and N which diagonalize the mass matrices in Eqs.共17兲 and

共35兲 but we will not write them explicitly. The charged

cur-rent is written in the mass-eigenstate basis as l¯L␥␮VM NS␯LW␮⫺ with the Maki-Nakagawa-Sakata matrix 关21兴 defined as VM NS⫽EL

T_{N, where E and N are the 3⫻3}

submatrices of E and N, respectively. Hence, we have

V_{M NS}⬇

冉

1.000 ⫺0.004 ⫺0.001 0.001 ⫺0.001 0.003

⫺0.004 ⫺0.979 ⫺0.199

冊

. 共42兲

Notice that this leptonic mixing matrix is not orthogonal as it must be since we are omitting the mixture with the heavy charginos and neutralinos and one neutrino remains massless at the tree level. We can always rotate the neutral fields in such a way that the electron neutrino is the one which remains massless; or we can also assume ␮_0e⫽␭_3e

⫽0, so that the electron neutrino decouples from the other

neutrinos and neutralinos. In this case, diagonal and nondi-agonal mass terms in Eq. 共35兲 will be induced by loop cor-rections like that in Fig. 2. Thus, a 3⫻3 nonorthogonal mix-ing matrix will appear in Eq. 共42兲. Here we will only consider the order of magnitude of a mass generated by this process.

The massless neutrino can get a mass from the loop cor-rection like that in Fig. 2 as a consequence of the Majorana mass term of the neutral lepton in the triplet. This is equiva-lent to the mechanism of Ref.关22兴 but now with a triplet of leptons instead of a neutral singlet. For instance, the ␭2 in-teractions will contribute in the left and right vertices in Fig. 2: L␯_⫽␭2ab 3 关␩˜2R ⫺_共␯ aLl˜b⫺␯bLl˜a兲⫹␩˜1R⫹共␯bLl˜a*⫺␯aLl˜b*兲兴 ⫹␭₃3a关␹˜ R ⫹⫹_␯ aL␳⫹⫹⫹˜␳R⫺⫺␯aL␹⫺⫺⫺␹˜R 0_␯ aL␳0 ⫺˜␳ R 0_␯ aL␹ 0_兴⫹␭␣ai

⬘

3 关d¯␣R␯aL˜di c_⫹d¯ iR c _␯ aL˜d␣兴⫹H.c., 共43兲

these interactions generate the lower vertices of Fig. 2. The upper vertex are given in Eqs.共26兲–共29兲. With these interac-tions we can generate the following small mass to the elec-tron neutrino. In fact, assuming a hierarchy of the masses as in the last subsection, we obtain the dominant contribution to the ␯e mass, up to logarithmic corrections:

FIG. 2. Diagram generating the mass for the lightest neutrino. There is another dominant contribution withv_␹→v_␹⬘. Each vertex on the left and right side are proportional to ␭2e␶/3 and ␭2e␶/3,

m_␯ e⬀␭2ea␭2ebEeaEebV␶˜ 2 共v␹2⫹v␹_⬘ 2 兲ma 9m_␶˜2, 共44兲 where all the indices are fixed, E_ea is the mixing matrix element defined in Eqs. 共20兲 and 共21兲, and V_␶˜ denotes the mixing matrix element in the slepton sector. The charged lepton which gives the main contribution is the ␶ lepton: it has a large mass and the mixing angle is not too small, in fact Ee␶⫽0.004. Since, ␭2e␶␭2e␶⬃10⫺6 we obtain an electron neutrino mass of the order of 10⫺3 eV if m_␶˜⬇4

⫻103_V

␶˜ GeV. If V␶˜⫽0.02 we have m␶˜⬃81 GeV 关20兴. IV. CONCLUSIONS

In the nonsupersymmetric 3-3-1 model关4兴 with only three scalar triplets ␩,␳␹ it is not possible to generate the ob-served charged lepton masses. Then, it is necessary to intro-duce a scalar sextet in order to get the appropriate masses. When we supersymmetrize the model and allow R-parity breaking interactions we can give to all known charged lep-tons and neutrinos the appropriate masses even without the introduction of a scalar sextet. Of course, in order to cancel anomalies we have to introduce another set of three triplets ␩

⬘

,␳

⬘

␹

⬘

. In this case, although the correct values for the lepton masses can still be obtained, if the new VEVs u

⬘

,v

⬘

and w

⬘

are zero, it was shown in Ref. 关16兴 that in order to give mass to all the quarks in the model all these VEVs have to be different from zero. Hence, we have considered that the six neutral scalar components got a nonzero VEV.

As can be seen from Eq.共24兲, the charged lepton masses arise from a sort of seesaw mechanism since there are small mass parameters, as in Eq.共24c兲, related with R-parity break-ing interactions, and large ones as in Eq.共24d兲, related with the mass scale of the supersymmetry breaking, this can be better appreciated in Eqs.共B1兲.

The same happens in the neutrino sector, see Eq.共B2兲. In a supersymmetric version of the model in which we add the sextet (6,0), there is a fermionic non-Hermitian triplet under SU(2)丢U(1)Y that is part of a sextet under SU(3)

丢U(1)N. This can also implement a seesaw mechanism for

neutrino masses as was pointed out in Ref.关23兴. The case of a Hermitian fermion triplet was considered in the context of the standard model in Refs.关24兴.

It is interesting to note that in the context of MSSM a Z2 symmetry关7,25兴,

M→⫺M, V→V, X→X, 共45兲 where M ,V,X are matter, vector, and scalar superfields, re-spectively, forbids the R-parity breaking terms in Eq.共2兲. In the present model it happens the same way: the R-parity breaking terms in Eqs.共10兲 and 共11兲 are forbidden. Notwith-standing the Z3 symmetry关25兴,

Lˆ, lˆc→Lˆ,lˆc; Hˆ₁→Hˆ₁, Hˆ₂→Hˆ₂; Qˆ→␻Qˆ , uˆc→␻⫺1uˆc,dˆc→␻⫺1dˆc,

共46兲

where␻⫽e2i␲/3, forbids the B violating terms but allows the L violating ones. This also happens in the present model. However, if we introduce an extra discrete Z₃

⬘

symmetry, such that Lˆe→⫺Lˆe, with all other fields being even under

this transformation, we have that␮_0e⫽␭_2ea⫽␭_3e⫽␭_␣ei

⬘

⫽0, at all orders in perturbation theory. This does not modify the mass matrix in the charged sector in Eqs. 共16兲 and 共17兲, but forbids the electron neutrino to get a mass, at all orders in perturbation theory.

The present model will induce processes contributing to ␮→e␥, ␶→e(␮)␥, (g⫺2)_␮, and other exotic decays. However, these processes can be suppressed mainly by the scalar masses since these scalars do not enter explicitly in the mass matrix at the tree level. Some contributions to those processes are suppressed by the coupling constants them-selves, like ␭2’s in Eq. 共24a兲; other ones which involve

␭3␮,␭3␶which are of the order unity can be suppressed by combining the mixing angles and masses of scalars or charginos sectors. A more detailed study of this issue will be done elsewhere关24兴.

In summary, we have analyzed the charged lepton and neutrino masses in an R-parity breaking supersymmetric 3-3-1 model. Unlike the MSSM model the electron and its neutrino remain massless at the tree level but gain masses at the one loop level. The resulting leptonic mixing matrix V_{M NS} is nonorthogonal.

ACKNOWLEDGMENTS

This work was supported by Fundac¸a˜o de Amparoa` Pes-quisa do Estado de Sa˜o Paulo共FAPESP兲, Conselho Nacional

de Cieˆncia e Tecnologia共CNPq兲 and by Programa de Apoio a Nu´cleos de Exceleˆncia 共PRONEX兲.

APPENDIX A: THE SCALAR POTENTIAL

The interactions between the scalars of the theory are given by the scalar potential that is written as

V331⫽VD⫹VF⫹Vsoft, 共A1兲 where the VDterm is given by

VD⫽⫺LD⫽ 1 2共D a_Da_⫹DD兲 ⫽g

⬘

2 2

冉

2 3Q˜1 † Q˜1⫺ 1 3Q˜␣ † Q˜_␣⫺2 3˜ui †c u ˜ 1 c ⫹1₃d˜_i†c˜d₁c⫹␳†␳⫺␹†␹⫺␳

⬘

†␳

⬘

⫹␹

⬘

†␹

⬘

冊

2 ⫹g 2 8

兺

i, j 共L˜i †_␭ i j a_L˜ j⫹Q˜1i †_␭ i j a_Q˜ 1 j ⫹␩i †_␭ i j a_␩ j⫹␳i †_␭ i j a ␳j⫹␹i †_␭ i j a ␹j ⫺Q˜ ␣i † _␭ i j *a_Q˜ ␣ j⫺␩i

⬘

†_␭ i j *a_␩ j

⬘

⫺␳i

⬘

†_␭ i j *a_␳ j

⬘

⫺␹i

⬘

†_␭ i j *a_␹ j

⬘

共A2兲

V_F⫽⫺L_F⫽

兺

m F_m*F_m ⫽

兺

i, j,k

冋

冏

␮0 2 ␩i

⬘

⫹␭1⑀i jk˜Lj˜Lk⫹ 2␭2 3 ⑀i jk␩jL˜k⫹ ␭3 3 ⑀i jk␹j␳k

冏

2 ⫹

冏

␮␩ 2 ␩i

⬘

⫹ ␭2 3 ⑀i jkL˜jL˜k⫹ f1 3 ⑀i jk␳j␹k⫹ ␬4␣i j 3 Q˜␣d˜j c

冏

2 ⫹

冏

␮₂␳␳i

⬘

⫹ f1 3 ⑀i jk␹j␩k⫹ ␬5␣i j 3 ˜Q␣˜uj c_⫹␭3 3 ⑀i jkL˜j␹k

冏

2 ⫹

冏

␮₂␹␹i

⬘

⫹ f1 3 ⑀i jk␳j␩k⫹ ␬6␣i␤ 3 ˜Q␣˜j␤ c_⫹␭3 3 ⑀i jk˜Lj␳k

冏

2 ⫹

冏

␮␩ 2 ␩i⫹ f₁

⬘

3 ⑀i jk␳

⬘

j␹k

⬘

⫹ ␬1i j 3 ˜Q1˜uj c

冏

2 ⫹

冏

␮␳ 2 ␳i⫹ f₁

⬘

3 ⑀i jk␹

⬘

j␩k

⬘

⫹ ␬2i j 3 Q˜1˜dj c

冏

2 ⫹

冏

␮␹ 2 ␹i⫹ f₁

⬘

3 ⑀i jk␳j

⬘

␩k

⬘

⫹ ␬3i 3 ˜Q1˜J c

冏

2 ⫹

冏

␬1i j 3 ␩i

⬘

u˜j c_⫹␬2i j 3 ␳i

⬘

˜dj c_⫹␬3i 3 ␹i

⬘

˜J c

冏

2 ⫹

冏

␬4␣i j 3 ␩i˜dj c_⫹␬5␣i j 3 ␳i˜uj c_⫹␬6␣i␤ 3 ␹i˜j␤ c_⫹␭␣i j

⬘

3 ˜Li˜dj c

冏

2 ⫹

冏

␬1i j 3 Q˜1␩i

⬘

⫹ ␬5␣i j 3 Q˜␣␳i

⬘

⫹ ␭i jk

⬙

3 ˜di c d ˜ k c_⫹␭i j

⵮

␤ 3 ˜ui c j ˜ ␤ c

冏

2 ⫹

冏

␬2i j 3 ˜Q1␳i

⬘

⫹ ␬4␣i j 3 Q˜␣␩i⫹ ␭_{␣i j}

⬘

3 ˜Q␣L˜i⫹ 2␭_{i jk}

⬙

3 ˜di c_˜u k c_⫹␭

⵮

j␤⬘ 3 ˜J c_˜j ␤ c

冏

2

册

. 共A3兲

Finally, the soft term is 共the following soft terms do not include the exotic quarks兲 Vso f t⫽⫺Lso f t ⫽1₂

冉

m_␭

兺

w⫽1 8 ␭A w_␭ A w_⫹m

_⬘

_␭ B␭B⫹H.c.

冊

⫹mL 2 L ˜†_L˜⫹m Q₁ 2 Q˜₁†Q˜1⫹

兺

␣⫽2 3 m_Q ␣ 2 Q˜_␣†Q˜_␣⫹

兺

i⫽1 3 m_u i 2 u ˜ i c† u ˜ i c ⫹

兺

i⫽1 3 m_d i 2 d ˜ i c† d ˜ i c_⫹m ␩ 2_␩†_␩⫹m ␳ 2 ␳†_␳⫹m ␹ 2 ␹†_␹⫹m ␩_⬘ 2 ␩

⬘

†_␩

_⬘

_⫹m ␳_⬘ 2 ␳

⬘

†_␳

_⬘

_⫹m ␹_⬘ 2 ␹

⬘

†_␹

_⬘

⫹

冋

M2

兺

i⫽1 3 L ˜ i␩i †_⫹␧ 0

兺

i⫽1 3

兺

j⫽1 3

兺

k⫽1 3 ⑀i jkL˜i˜Lj˜Lk⫹␧1

兺

i⫽1 3

兺

j⫽1 3 ⑀i jkL˜iL˜j␩k⫹␧3

兺

i⫽1 3

兺

j⫽1 3

兺

k⫽1 3 ⑀i jkL˜i␹j␳k ⫹k1⑀i jk␳i␹j␩k⫹k1

⬘

⑀i jk␳i

⬘

␹j

⬘

␩k

⬘

⫹

兺

i⫽1 3 Q ˜ 1共␨1i␩

⬘

˜ui c_⫹␨ 2i␳

⬘

˜di c_兲⫹

兺

␣⫽2 3 Q˜_␣

冉

兺

i⫽1 3 ␻1␣i␩˜di c_⫹␻ 2␣i␳˜ui c

冊

⫹

兺

i⫽1 3

兺

j⫽1 3

兺

k⫽1 3 ␵1i jkd˜i c d ˜ j c u ˜ k c_⫹H.c.

册

. 共A4兲

APPENDIX B: NUMERICAL ANALYSIS OF MASS MATRICES

Here we show explicitly the numerical values of each entry of the mass matrices in Eqs.共17兲 and 共35兲 using the parameters given in Eq. 共24兲 and m

⬘

⫽⫺3780.4159 GeV. For the charged sector we have

X⫽

冢

0.0 ⫺0.005 ⫺0.005 0.0 0.0 0.0 0.0 ⫺0.024 0.0 0.005 0.0 ⫺1.851 0.0 0.0 0.0 0.0 ⫺235.702 0.0 0.005 1.851 0.0 0.0 0.0 0.0 0.0 ⫺235.702 0.0 0.0 0.0 0.0 3000.0 0.0 ⫺0.462 0.0 80.071 0.0 0.0 0.0 0.0 0.0 3000.0 0.0 9.237 0.0 ⫺923.707 0.0 0.0 0.0 9.237 0.0 ⫺150.0 0.0 59.868 0.0 0.0 0.0 0.0 0.0 ⫺0.462 0.0 ⫺150.0 0.0 ⫺0.236 0.0 0.0 0.0 ⫺0.462 0.0 471.405 0.0 ⫺250.0 0.0 ⫺0.004 ⫺40.864 ⫺40.864 0.0 461.854 0.0 ⫺10.379 0.0 ⫺350.000

冣

共B1兲

and for the neutral sector Y0_⫽

¨

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.024 0.0 0.004 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 235.702 0.0 40.864 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 235.702 0.0 40.864 0.0 0.0 0.0 0.0 3000.0 0.0 0.0 6.532 ⫺0.327 ⫺56.619 0.327 0.0 0.0 0.0 0.0 0.0 0.0 3000.0 0.0 3.771 ⫺0.189 32.689 ⫺0.189 ⫺377.102 754.204 0.0 0.0 0.0 0.0 0.0 ⫺3780.416 0.0 0.0 99.391 ⫺0.573 ⫺573.290 1146.580 0.0 0.0 0.0 6.532 3.771 0.0 0.0 ⫺150.0 ⫺59.868 0.0 10.379 0.0 0.0 0.0 0.0 ⫺0.327 ⫺0.189 0.0 ⫺150.0 0.0 0.0 ⫺471.405 0.0 0.236 0.024 235.702 235.702 ⫺56.619 32.689 99.391 ⫺59.868 0.0 0.0 ⫺250.0 ⫺1.197 0.0 0.0 0.0 0.0 0.327 ⫺0.189 ⫺0.573 0.0 ⫺471.405 ⫺250.0 0.0 0.0 ⫺0.236 0.004 40.864 40.864 0.0 ⫺377.102 ⫺573.290 10.379 0.0 ⫺1.197 0.0 0.0 ⫺350.0 0.0 0.0 0.0 0.0 754.204 1146.580 0.0 0.236 0.0 ⫺0.236 ⫺350.0 0.0

©

. 共B2兲

Here we show that the relevant parameters for the leptons masses are␭2,3. We note that there are four types of param-eters in the mass matrices in Eqs. 共24兲. First we have the dimensionless Yukawa couplings in the usual leptons ␭2,3 and in the supersymmetric partners f1, f1

⬘

. We also have the mass dimension parameters ␮0a and␮␩,␳,␹ and m␭and m

⬘

which are soft terms in Eq.共A4兲. Of all these parameters we expect that the relevant ones in the charged lepton and neu-trinos are ␭2,3. To show this we consider several choices of the parameters as follows 共below all masses are in GeV兲.

Case 1:

␭2e␮⫽0.0, ␭2e␶⫽0.0, ␭2␮␶⫽0.0,

␭3e⫽0.0, ␭3␮⫽0.0, ␭3␶⫽0.0,

␮0e⫽␮0␮⫽0.0;␮0␶⫽0.0 共in GeV兲. 共B3兲 Charged sector masses:

3186.03,3001.10,557.17,196.55,149.30,16.85,0,0,0. Neutral sector masses:

⫺4162.22,3260.47,3001.10,557.79,⫺557.17,450.14, ⫺330.68,17.18,⫺17.01,0,0,0. Case 2: ␭2e␮⫽0.0, ␭2e␶⫽0.0, ␭2␮␶⫽0.0, ␭3e⫽0.0, ␭3␮⫽0.0, ␭3␶⫽0.0, ␮0e⫽␮0␮⫽0.0;␮0␶⫽2⫻10⫺8 共in GeV兲. 共B4兲 Charged sector masses:

3186.03,3001.10,557.17,196.55,149.30,16.85,1.92

⫻10⫺12_,0,0.

Neutral sector masses:

⫺4162.22,3260.47,3001.10,557.79,⫺557.17,450.14, ⫺330.68,17.18,⫺17.01,2.80⫻10⫺21_,0,0. Case 3: ␭2e␮⫽0.0, ␭2e␶⫽0.0, ␭2␮␶⫽0.0, ␭3e⫽0.0001, ␭3␮⫽1.0, ␭3␶⫽1.0, ␮0e⫽␮0␮⫽0.0;␮0␶⫽0.0 共in GeV兲. 共B5兲 Charged sector masses:

3186.05,3001.11,584.85,282.30,149.41,204.55,2.10

⫻10⫺10_,0,0.

Neutral sector masses:

⫺4162.22,3260.47,3001.10,585.18,⫺585.18,453.22, ⫺344.14,283.14,⫺271.99,1.23⫻10⫺11_,0,0. Case 4: ␭2e␮⫽0.001, ␭2e␶⫽0.001, ␭2␮␶⫽0.393, ␭3e⫽0.0, ␭3␮⫽0.0, ␭3␶⫽0.0, ␮0e⫽␮0␮⫽0.0;␮0␶⫽0.0 共in GeV兲. 共B6兲 Charged sector masses:

3186.03,3001.10,557.17,196.55,149.30,16.85,1.85,1.85,0. Neutral sector masses:

⫺4162.22,3260.47,3001.10,557.79,⫺557.17,450.14, ⫺330.68,17.18,⫺17.01,0,0,0.

Case 5:

␭2e␮⫽0.001, ␭2e␶⫽0.001, ␭2␮␶⫽0.393,

␭3e⫽0.0001, ␭3␮⫽1.0, ␭3␶⫽1.0,

␮0e⫽␮0␮⫽0.0;␮0␶⫽0.0 共in GeV兲. 共B7兲 Charged sector masses:

3186.03,3001.11,584.85,282.30,204.55,149.41,1.78,0.105,0.

Neutral sector masses:

⫺4162.22,3260.47,3001.10,585.19,⫺585.19,453.22, ⫺344.14,283.14,⫺271.99,1.23⫻10⫺11_,0,0.

Notice that the values of the masses in the charged sector are not significantly affected by the values of␮0a.

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