Neutrino masses through the seesaw mechanism in 3-3-1 models
J. C. Montero,*C. A. de S. Pires,†and V. Pleitez‡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 April 2002兲
Some years ago it was shown by Ma that in the context of the electroweak standard model there are, at the tree level, only three ways to generate small neutrino masses by the seesaw mechanism via one effective dimension-five operator. Here we extend this approach to 3-3-1 chiral models showing that in this case there are several dimension-five operators and we also consider their tree level realization.
DOI: 10.1103/PhysRevD.65.095001 PACS number共s兲: 12.60.⫺i, 14.60.Pq
I. INTRODUCTION
Although recent data on neutrino oscillation experiments 关1–3兴 strongly suggest that neutrinos have a nonzero small mass the issue of explaining its tiny value is still an open question. Several years ago it was noted by Weinberg 关4兴, and independently by Wilczek and Zee关5兴, that the neutrinos may acquire naturally small Majorana masses through dimension-five effective operators such as
liaLc ljbL⌽k (m)⌽
l (n)共 f
abmn⑀ik⑀jl⫹ fabmn
⬘
⑀i j⑀jl兲, 共1兲 the couplings f and f⬘
being of the order of⌳⫺1, where⌳ is a large effective mass related to new physics. When the neu-tral component of the scalar doublet⌽ develops its vacuum expectation value 共VEV兲具
典
it will produce the following mass matrix for the neutrinos:M⫽f
具
典
2⌳ , 共2兲
and since
具
典
is of the order of 100 GeV, small Majorana neutrino masses are generated if ⌳⬎1013 GeV. The inter-esting point in Eq. 共2兲 is that the lightness of the neutrino masses are generated via new physics at an energy scale ⌳ ⰇGF⫺1/2. This is the usual seesaw mechanism关6兴. The real-izations of that operator inside the standard model共SM兲 were already investigated by Ma in Ref.关7兴. All the ways such an operator can be realized at the tree and one loop level were sketched there.Here we rederive step by step such realizations and ex-tend the analysis to 3-3-1 chiral models关8–10兴 in which, like in the SM, neutrinos are massless unless we add right-handed neutrinos or break the lepton number. In this vain, after SuperKamiokande results, several suitable modifica-tions of the model were already proposed in order to generate the neutrino masses关11,12兴.
In this work we will build, in the framework of the 3-3-1 models, an effective dimension-five operator that leads to
seesaw neutrino masses and then investigate how it can be realized at the tree and one loop levels. Although it was noted in the early references 关4,5兴 that it is necessary to renormalize these effective operators, we will not address this issue here 关13兴.
The outline of this work is the following. In Sec. II we review the several ways to generate such sort of operators in the context of the standard model. In Sec. III we consider the generation of those operators in the 3-3-1 model. Our con-clusion will appear in Sec. IV.
II. SEESAW TREE LEVEL REALIZATION IN THE STANDARD MODEL
In the context of the standard model with only the lepton doublets ⌿aL⫽(aL,laL)T,a⫽e,,, and a scalar doublet ⌽⫽(⫹,0)T, there are only three possibilities for imple-menting naturally small neutrino masses with dimension-five effective operators 关7兴. On the other hand, if we add new leptons new possibilities arise. They are the following.
共I兲 ⌿aLand⌽ form a fermion singlet, so that the effective interaction is L1F e f f⫽fab ⌳ 关⑀共⌿c兲aR⌽兴关⑀共⌿c兲bR⌽兴⫹H.c. ⫽fab ⌳ „共c兲aR0⫺共lc兲aR⫹…共bL0⫺lbL⫹兲⫹H.c., 共3兲
⑀ denotes the antisymmetric SU(2) tensor and we have sup-pressed SU(2) indices. The tree level realization of this ef-fective operator is achieved by the introduction of one or more neutral right-handed singlets, say NbR, which produces the usual seesaw mechanism 关6兴, and is linked to the stan-dard left-handed neutrinos through the interaction
fab⌿¯aL⌽˜ NbR⫹H.c., 共4兲 with⌽˜⫽i2⌽*; this interaction and the bare mass term for NR, (NaR)cMRabNbR, leads to the following Dirac-Majorana matrix
*Email address: montero@ift.unesp.br †Email address: cpires@ift.unesp.br ‡Email address: vicente@ift.unesp.br
冉
0 MD T MD MR冊
, 共5兲
in the basis„L,(NR)c…T. The matrix above, after diagonal-ization, gives the following expression for the neutrino masses:
M⫽MDTMR⫺1MD. 共6兲 From Eq. 共6兲 we recover Eq. 共2兲 by choosing MD ⫽ f
具
典
and taking MR⫽⌳, obtaining in this way the canoni-cal seesaw mechanism, or the now canoni-called type I关6兴, which is the usual realization of the effective operator in Eq.共1兲 found in the literature. Notice that the seesaw here is a relation among fermion masses and NR could be the right-handed neutrinos R. Its realization is depicted in Fig. 1 with the replacements NR→R and具
⌽典
→具
典
.共II兲 ⌿aL and⌿bL form a scalar triplet with the effective interaction L3S e f f⫽fab ⌳ 共⌿c兲aR⌿bL共⑀⌽•⑀⌽兲⫹H.c. ⫽fab ⌳ 关共c兲aRbL00⫺20⫹共¯aR c l bL ⫹ l¯aR c bL兲⫹ l¯aR c lbL⫹⫹兴⫹H.c. 共7兲 The tree level realization in this case is obtained by introduc-ing a complex (Y⫽2) scalar triplet (⫹⫹,⫹,0) or, in ma-trix notation关14,15兴,
ជ•⌶ជ⫽
冉
⫹
冑
2⫹⫹冑
20 ⫺⫹冊
. 共8兲 In this case we have the interaction between the triplet⌶ and the usual leptons,fab⌿¯aL
c 共⑀ជ•⌶兲⌿
bL⫹H.c. 共9兲
From the expression above ⌶ must carry two units of lepton number. After the triplet develops a VEV the neutri-nos gain masses which are given by the following expres-sion:
Mab⫽ fab
具
0典
. 共10兲Differently from the type I seesaw mechanism here the suppression must come from
具
0典
(⬅具
典
). For this we have to study the scalar potential of the model. The complete sca-lar potential composed by the standard Higgs doublet⌽ and the triplet⌶ isV共,兲⫽2⌽†⌽⫹⌶2⌶†⌶⫹共⌽†⌽兲2
⫹⌶共⌶†⌶兲2⌶⌶†⌶⌽†⌽⫹M⌶⌽T⌶†⌽,
共11兲 where the last term breaks explicitly the lepton number. The stationary conditions for this potential give the following constraint equations:
具
典
共2⫹具
典
2⫹具
典
2兲⫽0,具
典
共2⫹
具
典
2⫹具
典
2兲⫹M具
典
2⫽0. 共12兲Assuming that兩兩⬃MⰇ
具
典
, with2⬍0, the second con-straint equation provides the following relation between the vacua of the model:具
典
⬃具
典
2M . 共13兲
Substituting Eq.共13兲 in Eq. 共10兲 we get the seesaw mass relation for neutrinos,
Mab⫽fab
具
典
2M . 共14兲
If the triplet belongs to some ground unified theory 共GUT兲, we can recognize Mas the high scale⌳. This is the
so-called type II seesaw mechanism关16兴. Notice that in this type of seesaw a very heavy scalar could develop a very tiny vacuum关7,17兴, differently of the type I seesaw where a very heavy right-handed neutrino induces tiny mass to the light neutrinos. The tree level realization of this mechanism is shown in Fig. 2.
共III兲 ⌿aLand⌽ form a fermion triplet with Y⫽0 and the effective Lagrangian is then written as
FIG. 1. Tree level realization of the effective dimension-five operator through a heavy neutrino.
FIG. 2. Tree level realization of the effective dimension-five operator through a heavy scalar.
L3F e f f⫽fab ⌳ 关共⌿c兲aR•⌽兴共⑀⌿bL•⑀⌽兲⫹H.c. ⫽fab ⌳ 关¯aR c lbL⫹0⫺共¯aR c 0⫹ l¯ aR c ⫹兲共l bL⫹ ⫹bL0兲⫹ l¯aR c bL⫹0兴⫹H.c.. 共15兲 In this case, the tree level realization is based on the in-troduction of the Y⫽0 triplet of leptons 关18兴:
ជ•⌬ជL⫽
冉
N
冑
2 P⫹冑
2 P⫺ ⫺N冊
L. 共16兲
This triplet is linked to the standard model content by the interaction
fa关⌽˜Tជ•⌬ជL共⌿aL兲c⫹⌽T共ជ•⌬L兲c⌿aL兴⫹H.c. 共17兲 where ⌽˜⫽⑀⌽*. The interaction in Eq. 共17兲, together with the mass term M⌬(⌬L)c⌬L, yields the mass matrix, in the basis (aL,NL)T,
冉
0 MLML M⌬
冊
. 共18兲
Notice that in this case MaL⫽ fa
具
典
is not an arbitrary mass matrix, hence, after diagonalization, two neutrinos are massless and two have nonzero masses at the tree level. One of the massive neutrinos has a mass proportional to M⌬ and the other one ⬃ fa具
典
2/ M⌬ 关18兴. Although this is also a seesaw relation we see that the neutrino mass spectrum is different from that of the general seesaw mechanism, since two of them are massless at the tree level. Notice also that the interactions in Eq.共17兲 violate explicitly the lepton num-ber. This is the third possibility of realization at the tree level of the effective dimension-five operator that leads to a see-saw relation for the neutrino masses. Even though it leads to the same seesaw relation, as the last two most explored mechanisms explicited in共I兲 and 共II兲, this mechanism has not been widely appreciated in literature. It is depicted in Fig. 1 with NR replaced by (NL)c.An interesting version of the model in which we obtain a general seesaw mass matrix is the one where we identify NL as being (c)Land add three triplets⌬aL, a⫽e,,; since in this case ML⫽ fab
具
典
, we can have a neutrino mass matrix as in Eq. 共6兲 in the canonical seesaw mechanism.Unlike the other two types of seesaw mechanisms, in the present one we must verify if this fermion triplet⌬ does not lead to any implication in the charged lepton masses. The general mass matrix in the charged lepton sector is rather complicated since the interactions in Eq. 共17兲 together with the standard Yukawa interaction⌿¯L⌽lRlead to the follow-ing charged lepton matrix in the basis (lL, PL):
共 l¯L¯PL兲
冉
MD ML 0 M⌬冊冉
lR PR冊
, 共19兲where MDab⫽ab
具
典
. With MaL⫽ fa具
典
共or ML⫽ fab具
典
if we add three triplets兲 we see that the mass matrix in Eq. 共19兲 can be only easily diagonalized if we assume that MD is diagonal. For more details see Ref. 关18兴.共IV兲 However, if we extend the particle content of the standard model, another possibility is that⌿aRc and⌽ form a non-Hermitian fermionic triplet with Y⫽2. In this case, al-though it is not possible to build an effective operator of the form given in Eq. 共3兲, we can introduce a fermionic non-Hermitian (Y⫽2) triplet ⍀⫽(⫹⫹,⫹,0) or ជ•⍀ជ⫽
冉
⫹冑
2⫹⫹冑
20 ⫺⫹冊
L , 共20兲so that we have the interactions
fa关⌽Tជ•⍀ជL共⌿aL兲c⫹⌽˜T共ជ•⍀L兲c⌿aL兴⫹H.c., 共21兲
which together with a mass term M⍀(⍀c)R⍀L give a mass matrix of the form in Eq. 共5兲 with MR→M⍀. Next, we introduce the right-handed doublets as aR⫽(N,E⫺)aR ⬃(2,⫺1), and so it is possible to have
L4F e f f⫽fab
⌳ 共⌿aL•⌽兲共bR•⌽˜兲⫹H.c. 共22兲 Notice that, even if the first three ways of realization of the effective dimension-five operator at the tree level are very different, in the end the neutrino masses receive the same expression in the form of the seesaw relation presented in Eq. 共2兲. In the fourth way it is also possible to generate such a mass matrix but only at the tree level if we do not restrict ourselves to the usual leptonic representation content. The only context in which they can be distinguished is by their consequences at high energy once the type I seesaw favors SO(10) and type II and III favor SU(5) in 15 and 24 representation, respectively. In fact, the first one has been studied more in the literature and it was in this context that the seesaw mechanism was proposed 关6兴. The scalar triplet has also been largely considered in literature 关14兴. Both al-ternatives can be implemented more naturally in the context of the standard model or in some of its extensions since, for instance, the introduction of neutral singlets leaves the model more symmetric relatively to quarks and leptons, or because the scalar representation content is not constrained by the gauge invariance. This is not the case with the lepton triplet. However, recently it has been shown that in a 3-3-1 super-symmetric model 关19兴 there is a fermionic non-Hermitian triplet under the 3-2-1 symmetry that is part of a sextet of Higgssinos transforming as (6,0) under SU(3)L丢U(1)N.
III. THE EFFECTIVE DIMENSION-FIVE OPERATOR AND ITS REALIZATION IN 3-3-1 CHIRAL MODELS
In this section we build the effective dimension-five op-erator in the context of 3-3-1 chiral models and its realiza-tions at the tree and the 1-loop level. We employ the same approach used for the realizations of Eq.共2兲 in the SM out-lined in the last section. As in the SM, in the minimal 3-3-1 model 关8兴, or in some of its extensions 关12兴, neutrinos are strictly massless due to the conservation of the total lepton number. In these models all leptons come only in triplets. In one version of this model, to generate the mass of all the particle content of the model, three triplets , , and a sextet S of scalars are required.
We have at least two triplets, the leptonic one ⌿aL ⫽(aL,laL,laL
c
)Tand the scalar one⫽(0,1⫺,2⫹)T, both transforming like (3,0) under the electroweak gauge symme-try SU(3)L丢U(1)N. Hence, we can form bilinears like ⌿¯⫽3¯丢3⫽1A丣8 or⌿⫽3丢3⫽3¯A丣6S. In this case we will have the following possibilities for the effective interac-tions:
共A兲 ⌿ and form a fermion singlet, and the effective operator is written as L1F e f f⫽fab ⌳ 关†共⌿c兲aR兴关†⌿bL兴⫹H.c. ⫽fab ⌳ 共¯aR c 0*⫹ l¯ aR c 1 ⫹⫹ l¯ aR2⫺兲共bL0* ⫹lbL1⫹⫹lbL c 2 ⫺兲⫹H.c. 共23兲
At the tree level we can realize this situation by introduc-ing again one or more neutral fermion sintroduc-inglets, NbR’s, and it is also a realization of the usual seesaw mechanism. This is in fact equivalent to case 共I兲 in the previous section. Like there the intermediate heavy particle here should be a right-handed neutrino,R. The realization of this operator withR is similar to that in Fig. 1 with
具
典
replaced by具
典
. The heavy right-handed neutrinos are linked with L by a term like this one:fab⌿¯aL bR⫹H.c., 共24兲
which together with the mass term MR¯RcR leads to the mass matrix defined in Eq. 共5兲 in the basis„L,(R)c… with MDab⫽ fab
具
典
.共B兲 Next, we can form with ⌿aLandan octet of leptons if we define the traceless matrix,
M¯ aij ⫽共⌿c兲aiR† j⫺ 1 3␦i
j„共⌿c兲
aR•†…, 共25兲
where i, j denote SU(3) indices. The effective interaction is in this case given by
L8F e f f⫽fab ⌳Tr共M¯aMb兲⫹H.c. ⫽fab ⌳
冋
1 9共2¯qR c 0*⫺ l¯ aR c 1 ⫹⫺ l¯ aR2⫺兲共2bL0⫺lbL c 1 ⫺⫺l bL2⫹兲⫹¯aR c b:共1⫹1⫺⫹2⫹2⫺兲⫹ l¯aR c lbL ⫻共0*0⫹ 2 ⫹ 2 ⫺兲⫹1 9共2 l¯aR c 1 ⫹⫺¯ aR c 0*⫺ l¯ aR2⫺兲共2lbL c 1 ⫺⫺ bL0⫺lbL2⫹兲 ⫹ l¯aR c l bL c 2 ⫺ 1 ⫹⫹ l¯ aRlbL共0*0⫹1⫹1⫺兲⫹ 1 9共2 l¯aR2 ⫺⫺ l¯ aR c 1 ⫹⫺¯ aR c 0*兲共2l bL2⫹⫺lbL c 1 ⫺⫺ bL0兲册
⫹H.c. 共26兲 Hence we need a heavy fermion octet transforming as(1,8,0) under the 3-3-1 factors,
HL⫽
冉
N1 P1⫹ P2⫹ P3⫺ N2 X1 0 P4⫺ X20 N冊
L , 共27兲where N⫽⫺N1⫺N2. We have the interaction
fa关共⌿aL兲cHL⫹†共HL兲c⌿aL兴⫹H.c., 共28兲 omitting SU(3) indices. There is also the bare mass term MH(HL)cHL. These interactions produce a mass matrix for the neutral sector like that in Eq. 共18兲, but now with MaL ⫽ fa
具
典
and M⌬→MH. Two neutrinos remain massless and two gain masses. The tree level realization of this mechanism can be seen in Fig. 1 with NR replaced by (N1,2L)c and具
典
replaced by具
典
. However, if we introduce three of suchoctets and identify, say N1aLwith (c)aL, a⫽e,,, we can generate a general seesaw mass matrix like in Eq. 共5兲 since now MLab⫽ fab
具
典
. Notice that although this case is the analog of case 共III兲 in the previous section, i.e., an Hermit-ian, Y⫽0, fermion triplet, it is also a realization of case 共IV兲 where there is a non-Hermitian, Y⫽2, fermion triplet under SU(2)L丢U(1)Y.As in the case of the leptonic triplet in the standard model, the mass matrix of the charged lepton sector is complicated. Let us assume, for simplicity, that the mass matrix of the standard charged leptons, generated at low energy in the ba-sis of the 3-3-1 symmetries, is diagonal and let us call it Ml, and for the mass matrix of the charged leptons, P1 and P2, matrix MH. With these assumptions and the terms in Eq. 共28兲, we have the following 5⫻5 matrix in the basis (lL, P1L, P2L):
冉
Ml M 0 MH冊
. 共29兲
As in case共III兲 in the previous section, the diagonalization of this matrix is complicated unless we assume that Mlis diag-onal; in this case the usual leptons decouple completely from the heavy ones, being the mass matrix of the former Mland that for the heaviest MH.
共C兲 Finally, we have the possibility that ⌿aR c
and ⌿bL form a symmetric sextet. In this case we have
L6s e f f⫽fab
⌳ „共⌿c兲aR⌿bL††…⫹H.c., 共30兲 and the tree level realization is the introduction of the usual sextet (6,0) 关8兴. Notice that the triplet⑀i jk⌿¯ j
c⌿
kis not
real-ized since the respective triplet ⑀ilmlm⫽0. However, the symmetric sector can contribute for the masses of the charged leptons and also generate the neutrino masses by loop effects.
In this case the tree level realization requires a scalar sex-tet, S: S⫽
冉
s⬘
s1 ⫺冑
2 s2⫹冑
2 s1⫺冑
2 s3⫺⫺ s冑
2 s2⫹冑
2 s冑
2 s4⫹⫹冊
⬃共1,6,0兲. 共31兲The sextet S is linked to the leptons by the interaction
fab⌿¯ a c
S⌿b⫹H.c. 共32兲
When the neutral component s
⬘
of the sextet S develops a VEV, the neutrino mass matrix gets the following form:Mab⫽ fab
具
s⬘典
. 共33兲 The next step is to show how具
s⬘典
gets a tiny value. For this we need to develop the scalar potential of the model taking into account the explicit terms that violate the lepton number. Let us first write the scalar potential that conserves the lepton number:V共,,,S兲⫽2†⫹2†⫹2†⫹S2Tr共S†S兲⫹1共†兲2⫹2共†兲2⫹3共†兲2⫹共†兲„4共†兲⫹5共†兲… ⫹6共†兲共†兲⫹7共†兲共†兲⫹8共†兲共†兲⫹9共†兲共†兲⫹10Tr共S†S兲2⫹ 11„Tr共S†S兲…2 ⫹„12共†兲⫹13共†兲…Tr共S†S兲⫹14共†兲Tr共S†S兲⫹„15⑀i jk共†S兲 ijk⫹16⑀i jk共†S兲ijk ⫹17⑀i jk⑀lmn nkSliSm j⫹H.c.…⫹18 †SS†⫹19†SS†⫹20†SS†. 共34兲
Here, for the sake of simplicity, we impose in the scalar potential the symmetry →⫺ in order to avoid other tri-linear terms besides the one that will generate the seesaw mechanism. There are also four terms permitted by the 3-3-1 gauge symmetry which violate explicitly the lepton number, but for what concerns us here we will just consider only one of them:
MSTS†. 共35兲 Adding this term to the potential in Eq.共34兲, we find the following stationary condition on the VEV of the scalar field s
⬘
:具
s⬘典
冋
S2⫹10具
s典
2⫹冉
12 2 ⫹ 19 2冊
具
典
2⫹13 2具
典
2⫹14 2具
典
2 ⫹冉
10⫹11 2冊
具
s⬘典
2册
⫹M S具
典
2⫽0. 共36兲 We will consider here the case when the charged leptons gain masses, not through a sextet but via the introduction of a heavy charged lepton 关20兴. In this case the sextet can be very heavy. For instance, we can assume S2⬍0 and that 兩S兩⫽MSⰇ具
典
,具
典
,具
典
, so that we have from Eq.共36兲具
s⬘典
⯝具
典
2MS , 共37兲
which gives the following expression to the neutrino mass matrix:
Mab⫽ fab
具
典
2 MS. 共38兲
This is a seesaw mass relation and its realization is equal to that in Fig. 2 with
具
典
replaced by具
s⬘典
and具
典
replaced by具
典
. It is equivalent to the type II seesaw mechanism dis-cussed in case共II兲 in the preceding section and in Ref. 关17兴 where it was assumed that the sextet also gives mass to the charged leptons but the dominant energy scale is具
典
.A sextet like that in Eq.共31兲 disposes of a second neutral component, s, which may contribute to the charged lepton masses when developing a VEV. However, it will not give a large contribution to the charged lepton masses as we can see in the following. From its stationary condition we get the constraint equation:
具
s典
冋
S2⫹10具
s⬘典
2⫹冉
12 2 ⫺17冊
具
典
2⫹冉
13 2 ⫹ 20 4冊
具
典
2 ⫹冉
14 2 ⫹ 18 4冊
具
典
2⫹冉
11 2 ⫹10冊
具
s典
2册
⫹ 15 2冑
2具
典具
典
2 ⫺ 16 2冑
2具
典具
典
2⫽0, 共39兲which gives the following expression to its vacuum:
具
s典
⯝具
典具
典
2S
2 . 共40兲
It implies
具
s典
Ⰶ具
s⬘典
which makes its contribution to the charged lepton masses insignificant in relation to other sources in low energy. Notwithstanding, in the present case the charged leptons obtained mass through the mixing with a heavy charged lepton 关9兴.IV. CONCLUSIONS
We have analyzed the realization at the tree level of an effective dimension-five operator that generates seesaw masses to the neutrinos and have added a new tree level implementation of the seesaw mechanism in the context of the standard electroweak model. From our analysis we can say that all the ways of realizing such an operator in the standard model can be easily implemented in the 3-3-1 model as well. In the particular case 共IV兲 of Sec. II is real-ized in the context of the 3-3-1 model when a fermion octet is added as described in case共B兲 in Sec. III.
Finally, we would like to mention that at the one loop level the effective operator of the sort given in Eq.共1兲 is also easily implemented. Two simple cases are shown in Figs. 3 and 4, but they imply an extension of the 3-3-1 model by adding a pair of exotic lepton singlets. Only the second model is favored by phenomenology. Also we should stress that, except in case 共B兲 the neutrinos and charged lepton masses share a common set of parameters which leave such scenarios interesting.
ACKNOWLEDGMENTS
This work was supported by Fundac¸a˜o de Amparo a` 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.
关1兴 Homestake Collaboration, B.T. Cleveland et al., Astrophys. J.
496, 505共1998兲; Kamiokande Collaboration, Y. Fukuda et al.,
Phys. Rev. Lett. 77, 1683共1996兲; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B 477, 127共1999兲; SAGE Collabo-ration, J.N. Abdurashitov et al., Phys. Rev. Lett. 77, 4708 共1996兲; Phys. Rev. C 60, 055801 共1999兲.
关2兴 SuperKamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81, 1562共1998兲; 82, 2644 共1999兲.
关3兴 LSND Collaboration, C. Athanassopoulos et al., Phys. Rev. Lett. 77, 3082共1996兲; 81, 1774 共1998兲.
关4兴 S. Weinberg, Phys. Rev. Lett. 43, 1566 共1979兲.
关5兴 F. Wilczek and A. Zee, Phys. Rev. Lett. 43, 1571 共1979兲.
FIG. 3. One loop realization of the effective dimension-five op-erator.
FIG. 4. Another one loop realization of the effective dimension-five operator.
关6兴 M. Gell-Mann, P. Ramond, and R. Slansky, in Supergravity, edited by P. van Nieuwenhuizen and D. Z. Freedman 共North-Holland, Amsterdam, 1979兲; T. Yanagida, in Proceedings of
the Workshop on the Unified Theory and the Baryon Number in the Universe, edited by O. Sawada and A. Sugamoto共KEK
Report No. 79-18, Tsukuba, Japan,1979兲; R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912共1980兲.
关7兴 E. Ma, Phys. Rev. Lett. 81, 1171 共1998兲.
关8兴 F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 共1992兲; P.H. Frampton, Phys. Rev. Lett. 69, 2889 共1992兲; R. Foot, O.F. Hernandez, F. Pisano, and V. Pleitez, Phys. Rev. D 47, 4158 共1993兲.
关9兴 J.C. Montero, C.A.de S. Pires, and V. Pleitez, hep-ph/0103096. 关10兴 J.C. Montero, C.A.de S. Pires, and V. Pleitez, Phys. Rev. D 64,
096001共2001兲.
关11兴 P.H. Frampton, P.I. Krastev, and J.T. Liu, Mod. Phys. Lett. A 9, 761共1994兲.
关12兴 U. Okamoto and M. Yasue, Phys. Lett. B 466, 267 共1999兲; T. Kitabayashi and M. Yasue, Phys. Rev. D 63, 095002 共2001兲; Phys. Lett. B 508, 85 共2001兲; Nucl. Phys. B609, 61 共2001兲;
Phys. Rev. D 63, 095006 共2001兲; J.C. Montero, C.A.de S. Pires, and V. Pleitez, hep-ph/0112203.
关13兴 See, however, E.J. Chun and S. Pokorski, Phys. Rev. D 62, 053001共2000兲; S. Antusch, M. Dress, J. Kersten, M. Lindner, and M. Ratz, Phys. Lett. B 519, 238 共2001兲, and references therein.
关14兴 T.P. Cheng and L.-F. Li, Phys. Rev. D 22, 2860 共1980兲. 关15兴 G.B. Gelmini and M. Roncadelli, Phys. Lett. 99B, 411 共1981兲. 关16兴 R.N. Mohapatra and G. Senjanovic, Phys. Rev. D 23, 165 共1981兲; C. Wetterich, Nucl. Phys. B187, 343 共1981兲; E. Ma and U. Sarkar, Phys. Rev. Lett. 80, 5716共1998兲.
关17兴 J.C. Montero, C.A.de S. Pires, and V. Pleitez, Phys. Lett. B
502, 167共2001兲; M.B. Tully and G.C. Joshi, Phys. Rev. D 64,
011301共2001兲.
关18兴 R. Foot, H. Lew, X. -G. He, and G. C. Joshi, Z. Phys. C 44, 441共1989兲.
关19兴 J.C. Montero, V. Pleitez, and M.C. Rodriguez, Phys. Rev. D 共to be published兲, hep-ph/0112248.
