Spontaneous breaking of a global symmetry in a 3-3-1 model
J. C. Montero, C. A. de S. Pires, and V. PleitezInstituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, S.P., Brazil 共Received 16 March 1999; published 29 October 1999兲
In a 3-3-1 model in which the lepton masses arise from a scalar sextet it is possible to break spontaneously a global symmetry which implies in a pseudoscalar Majoron-like Goldstone boson. This Majoron does not mix with any other scalar fields and for this reason it does not couple, at the tree level, to either the charged leptons or to the quarks. Moreover, its interaction with neutrinos is diagonal. We also argue that there is a set of parameters in which the model can be consistent with the invisible Z0 width and that heavy neutrinos can decay sufficiently rapid by Majoron emission, having a lifetime shorter than the age of the universe.
关S0556-2821共99兲02217-1兴
PACS number共s兲: 14.80.Mz, 12.60.Fr
I. INTRODUCTION
In chiral electroweak model neutrinos can be massless at any order in perturbation theory if both conditions are sup-plied: no right-handed neutrinos are introduced and the total lepton number is conserved. If we do not assume lepton number conservation we have two possibilities: we break it by hand, i.e., explicit breaking or, we break it spontaneously. The later possibility implies the existence of a pseudoscalar Goldstone boson called a Majoron which was first suggested in Ref.关1兴 where a non-Hermitian scalar singlet 共singlet Ma-joron model兲 was introduced, and in Ref. 关2兴 where a non-Hermitian scalar triplet was introduced 共triplet Majoron model兲. Since the data of the CERN e⫹e⫺ collider LEP关3兴 the triplet majoron model was ruled out. The original model only considered one triplet and one doublet. In that model the majoron is a linear combination of doublet and triplet components but it is predominantly triplet. Hence, the light-est scalar (R0) has a mass which is proportional to the vacuum expectation value 共VEV兲 of the triplet and for this reason it is very small. Once we have the decay Z0 →R0M0, where M0denotes the Majoron, and since the only
decay of the light scalar is R0→M0M0, there is an extra contribution to the Z0-invisible width. Its contribution is ex-actly twice the contribution of a simple neutrino. Since the Higgs scalars have only weak interactions they escape unde-tected. Hence, any experiment that counts the number of neutrino species by measuring the Z0-invisible width auto-matically counts five neutrinos关4,5兴.
There are also possibilities involving only Higgs doublets and charged singlet scalars but they also need to include Dirac neutrino singlets. A minimal model of this sort was proposed in Ref. 关6兴 where an extra doublet scalar-carrying lepton number was added共doublet Majoron model兲. The new doublet does not couple to leptons. The LEP data imply that at least a second doublet of the new type has to be introduced 关7兴. In this sort of model, since there are at least three dou-blets, the lightest scalar R0 can be assumed naturally to be heavier than the Z0 avoiding the decay Z0→R0M0. In this Majoron doublet model the lepton number violation takes place at the same scale of the electroweak symmetry break-ing. It is also possible to consider a Majoron model with one
complex singlet, a complex triplet, and the usual SU(2) dou-blet 关8兴. In this case the Majoron can evade the LEP data since it can be mainly a singlet.
II. THE MODEL
Here we will consider a model with SU(3)C丢SU(3)L
丢U(1)N symmetry with both exotic quarks and only the known charged leptons 关9兴. In this model, in order to give mass to all fermions it is necessary to introduce three scalar triplets ⫽
冉
⫺ ⫺⫺ 0冊
⬃共3,ⴚ1兲, ⫽冉
⫹ 0 ⫹⫹冊
⬃共3,1兲, ⫽冉
0 1⫺ 2⫹冊
⬃共3,0兲 共1a兲 and a sextet S⫽冉
1 0 h2⫺冑
2 h1⫹冑
2 h2⫺冑
2 H1⫺⫺ 2 0冑
2 h1⫹冑
2 2 0冑
2 H2⫹⫹冊
⬃共6,0兲. 共1b兲Although we can assign a lepton number to several scalar fields we prefer to use the global quantum number F⫽L ⫹B 关10兴. It is clear that the model needs only one global quantum number and not four as in the standard model, i.e., family lepton number Li,(i⫽e,, with L⫽兺iLi) and the baryonic number B. The quantum numberF coincides with L and B for the known particles but implies the assignation of a single quantum number to the other particles.
The most general gauge andF-conserving scalar potential is
V⫽1 2†⫹ 2 2†⫹ 3 2†⫹ 4 2 Tr共S†S兲⫹1共†兲2⫹2共†兲2⫹3共†兲2⫹†关4†⫹5†兴⫹6共†兲共†兲 ⫹7共†兲共†兲⫹8共†兲共†兲⫹9共†兲共†兲⫹10共TrS†S兲2⫹11Tr关共S†S兲2兴⫹Tr共S†S兲关12共†兲⫹13共†兲 ⫹14共 †兲兴⫹关 15⑀ i jk i共S †兲 jk⫹16⑀ i jk i共S †兲 jk⫹17⑀ i jk⑀lmnS ilSjmnk⫹H.c.兴⫹18 †SS†⫹ 19 †SS† ⫹20†SS†⫹
冋
f1 2 ⑀ i jk ijk⫹ f2 2 TS†⫹H.c.册
. 共2兲Terms like the quartic (†)(†), S†,
S†,SS, and the trilinear S†,SSS do not conserve theF quantum number 共or the lepton L) and they will not be considered here.
The minimum of the potential must be studied after the shifting of the neutral components of the three scalar multip-lets. Hence, we redefine the neutral components in Eqs. 共1兲 as follows: 0→ 1
冑
2共v⫹R1 0⫹iI 1 0兲, 0→ 1冑
2共v⫹R2 0⫹iI 2 0兲, 0→ 1冑
2共v⫹R3 0⫹iI 3 0兲, 共3兲 and 1 0→ 1冑
2共v1⫹R4 0⫹iI 4 0兲, 2 0→ 1冑
2共v2⫹R5 0⫹iI 5 0兲, 共4兲 where va 共with a⫽,,,1,2) are considered realpa-rameters for the sake of simplicity. TheF number attribution is the following: F共U⫺⫺兲⫽F共V⫺兲⫽⫺F共J 1兲⫽F共J2,3兲⫽F共⫺⫺兲⫽F共⫺⫺兲 ⫽F共⫺⫺兲⫽F共 1 0兲⫽F共h 2 ⫺兲⫽F共H 1 ⫺⫺兲⫽F共H 2 ⫺⫺兲 ⫽2, 共5兲
where J1 (J2,3) are exotic quarks of charge 5/3 共⫺4/3兲
present in the model and we have included them by comple-tion. For leptons and the known quarksF coincides with the total lepton and baryon numbers, respectively. All the other fields have F⫽0. As we said before, all terms in Eq. 共2兲 conserve the F quantum number. However, if we assume that
具
10典
⫽0 we have the spontaneous breakdown of F and the corresponding pseudoscalar, the Majoron M0, as we will show below关11兴.In a model with several complex scalar fields, as is the case of the 3-3-1 model关9兴, if the lepton number is sponta-neously broken one of the neutral scalars is a Goldstone bo-son associated with the global symmetry breaking. With re-spect to the SU(2)L丢U(1)Ygauge symmetry, this model has naturally three doublets (⫹,0)T,(0,⫺)T, and (h⫹,20)T, one triplet
冉
1 0 h2⫺冑
2 h2⫺冑
2 H1⫺⫺冊
, 共6兲and two singlets2⫺⫺ and0.
The mass matrix in the real sector in the basis (R10,R20,R30,R40,R50)T, is given by the symmetric matrix
m11⫽1v2⫺ 16 4
冑
2 v2v 2 v ⫹ 1 4冑
2v共15vv2⫺ f1v兲v ⫹2t v, m22⫽2v 2⫺ 1 8冑
2v共2 f1v⫹ f2v2兲v⫹ t 2v, m33⫽3v 2⫺ 1 8冑
2v共2 f1v⫹ f2v1兲v⫹ t 2v, m44⫽共10⫹11兲v 1 2 ⫹ t1 2v 1 , m55⫽冉
10⫹ 11 2冊
v2 2 ⫺ 1 4冑
2v 2 16v2v⫺ 1 8v 2 共 f2v ⫹冑
215v兲v⫹ t 2 2v 2 , m12⫽ 1 2冉
4v⫹ 16冑
2v2冊
v⫹ f1 4冑
2v, m13⫽ 1 2冉
5v⫺ 15冑
2v2冊
v⫹ f1 4冑
2v, m14⫽共12⫹19兲 vv 1 2 , m15⫽ 1 2共12⫺217兲vv2⫺ 15 4冑
2v 2⫹ 16 4冑
2v 2,m23⫽ 6 2 vv⫹ f1 4
冑
2v⫹ f2 8 v2, m24⫽ 14 2 vv1, m25⫽共214⫹20兲 vv 2 4 ⫹ 16 2冑
2vv⫹ f2 8 v, m34⫽13 2 vv1, m35⫽ 13 2 vv2⫺ 1 4冑
2共15v⫺18v2兲v⫹ f2 8 v, m45⫽10v1v2. 共7兲The tadpole equations ta where a⫽,,,1,2 are given
in the Appendix. The conditions for an extreme of the poten-tial are ta⫽0. Assuming that the matrix mi j above is diago-nalized by an orthogonal matrixO, the relation among sym-metry (Ri0) and mass (Hj0) eigenstates is Ri0⫽Oi jHj
0; i, j
⫽1,2,3,4,5. The masses mHj can vary, depending on a fine tuning of the parameters, from a few GeV’s up to 2 or 3 TeV 关typical values of the energy scale at which the break down of the SU(3)L symmetry does occur兴. Also the arbitrary or-thogonal matrixO is not necessarily almost diagonal. Denot-ing the lightest Higgs boson as H10 two extreme possibilities are compatible with the LEP data: R4⫽(O⫺1)41H1
0⫹•••,
with (O⫺1)41Ⰶ1, if mH1⬍MZ; or R4⬇H1
0⫹•••, that is
(O⫺1)41⬇1, if MH1⬎MZ. Intermediate values for the mass mH1 and the mixing angles have been ruled out by the LEP data共see below兲.
The symmetric mass matrix of the imaginary part in the basis (I10,I20,I30,I40,I50)Treads M11⫽⫺ 16 4
冑
2 v2v 2 v ⫹17v2 2 ⫹ 1 4冑
2共15vv⫺ f1v兲 v v⫹ t 2v, M22⫽⫺ 1 8冉
冑
2 f1v⫹ f2v2 v v冊
⫹ t 2v, M33⫽⫺ 1 8共冑
2 f1v⫹ f2v2兲 v v⫹ t 2v, M44⫽ t 1 2v 1 , M55⫽ 1 4冑
2共15v 2⫺ 16v 2兲v v 2 ⫹17v 2 ⫺f2 8 vv v 2 ⫹ t2 2v 2 , M12⫽⫺ f1 4冑
2v, M13⫽⫺ f1 4冑
2v, M14⫽0, M15⫽ 15 4冑
2v⫺ f2 8 v2, M23⫽⫺1 8共冑
2 f1v⫹ f2v2兲, M24⫽0, M25⫽ f2 8 v, M34⫽0, M35⫽ f2 8 v, M45⫽0. 共8兲In Eqs.共7兲 and 共8兲 when all va⫽0,a⫽,,,1,2 then
we can use ta⫽0. In Eqs. 共8兲 there are three Goldstone bo-son. Notice, however, that since M4i⫽0, the component I40 has a zero mass, i.e., it is an extra Goldstone boson which decouples in the sense that it does not mix with the other C P-odd scalars. Hence, I4
0
is the Majoron field. Hereafter, it will be denoted M0. The submatrix 4⫻4 has still two other Goldstone bosons which are related to the masses of Z0 and Z⬘0. Hence, although the Majoron in the present model is a triplet under the subgroup SU(2), it does not mix with the other imaginary fields.
Hence, as in the singlet Majoron model ours has no cou-plings with fermions 共charged leptons and quarks兲. More-over, as we said before, the real component can be heavier than the Z0. It is easy to understand this. If v
1
0⫽0, the
tadpole equation in Eq. 共A4兲 must be replaced in the mass matrices in Eqs.共7兲 and 共8兲. In this case the10fields consists of two mass-degenerate fields R4
0 and I4 0 with mass mR 4 2 ⫽m I4 2⫽ 10v2 2 ⫹1 2共12⫹19兲v 2⫹13 4 v 2⫹18 2 v 2. 共9兲 The mass in Eq. 共9兲 can be large because it depends on v2. Whenv
1⫽0 is used, the degeneration in mass of R4
0
and I40 is broken, the imaginary part becomes the Majoron and the real part has a mixing with the other real neutral components, which include several fields transforming under SU(2)L
丢U(1)Y as doublets (0,0,2
0), and one singlet (0). This
also happens in the one-singlet–one-doublet–one-triplet model when the triplet does not gain a VEV关12兴. Notice that unlike v
1, if v2⫽0 the condition in Eq. 共A5兲 forces f2
⫽0. All the other VEV’s have to be nonzero in order to have a consistent breaking of the SU(3) symmetry.
III. PHENOMENOLOGICAL CONSEQUENCES AND CONCLUSIONS
In the present model the interaction of the Majoron with the Z0 共which is of the form Z0M0Hj0), is given by
O4 j
共
冑
2GF兲1/2 cWMW共pM0⫺pj兲, 共10兲
where pM0 and pj are the momenta of the Majoron and the
physical real scalars H0j, respectively. We see that if it is allowed, the contribution of the decay mode Z0→H1
0
M0, where H10is the lightest Higgs scalar, is 2兩O41兩2times that of
the neutrino antineutrino. Hence, as we said before, the value of the mixing matrix element O4 j is constrained appropri-ately: 2O4 j2⬃10⫺4, to make the model consistent with the LEP data, i.e., now ⌫Z→H10M0 共where H10 is assumed the lightest scalar兲 could be reduced to an acceptable level. More interesting, however, is the fact that in this model the Z0 →H1
0
M0 might be kinematically forbidden since H1 can be
heavier than the Z0 as was discussed above.
It is well known that if neutrinos are massive particles the thermal history of the universe strongly constrains the mass of the stable neutrinos, i.e., m⬍100 eV for light neutrinos or a few GeV for heavy ones关13兴. One of the ways in which the cosmological constraints on neutrino masses can be al-tered is when the lepton number is broken globally, giving rise to the Majoron field: heavy neutrinos can decay rapidly by Majoron emission, thereby giving negligible contributions to the mass density of the universe 关14兴. Let us denote
h (l) heavy 共light兲 neutrinos and look for →
⬘
⫹M0
decays in the present model. Those decays, as in the triplet Majoron model, are completely forbidden at the tree level too 共there is neither →
⬘
⫹␥ nor →3⬘
decays at the lowest order兲.Here we will denote, as usual, W⫹the vector boson which coincides with the respective boson of the standard model, i.e., it couples to the usual charged current in the lepton and the quark sectors and also satisfies MW2/ MZ2⫽cW2 ; and V⫹ denotes the vector boson which couples charged leptons with antineutrinos or the known quarks with the exotic ones. If the lepton number is not spontaneously broken W⫹ and V⫹ do not couple to one another. However, a mixing between both W⫹and V⫹arises when the lepton number is spontaneously broken. Let us consider this more in detail. In the basis (W⫹ V⫹)Tthe mass square matrix is given by
g2 4
冉
A⫹v2/2冑
2v 1v2冑
2v 1v2 A⫹v 2/2冊
, 共11兲 where A⫽(v2⫹v 2 2 ⫹2v 1 2)/2. We see from Eq.共11兲 that if v
1⫽0 there is no mixing between W
⫹ and V⫹. The mass eigenstates are given by Wi⫹⫽Ui jB⫹j , where i, j⫽1,2 and B1⫹⫽W⫹, B2⫹⫽V⫹, and the orthogonal matrix is given by (N is a normalization factor兲 U⫽N1
冉
r⫺s⫺冑
4b2⫹共r⫺s兲2 2b 1 r⫺s⫹冑
4b2⫹共r⫺s兲2 2b 1冊
⬇冑
1 s2⫹r2冉
⫺s b b s冊
, 共12兲 where r⫽v2/2, s⫽v2/2 and b⫽冑
2v 1v2.This mixing may have interesting cosmological consequences since there are interactions like
(g2/
冑
2) l¯L␥cW⫺. Notice that its strength depends on the small parameter⬀v1v2/v
2
and it can be neglected in the usual processes. In fact, the model has three different mass scales sincev 1ⰆviⰆvwithvi⫽v,v,v2关15兴. It means that W1⫹⬇W⫹, W2⫹⬇V⫹ with MW 2 / MZ 2 cW compatible with the experimental data ifv
1⭐3.89 GeV 关15兴. However, the
mixing between W⫹ and V⫹ is interesting once there are new contributions to the Majoron emission. In fact, because of the W⫹W⫺M0 vertex we have the neutrino transitions (h)L→(l)LM0; because of the vertex V⫹V⫺M0 we have antineutrino transitions (h)R→(l)R. Both contributions could be negligible since they are proportional tov
1. More
interesting is that possible to have neutrino–anti-neutrino transitions like the decay (h)L→(l)RM0 mediated by the mixing between W⫹ and V⫹ as shown in Fig. 1. This dia-gram is ultraviolet finite in the Feynman gauge and depends quadratically on a low-energy scale that we have chosen, conservatively, as being the m mass. The latter process im-plies a neutrino width which is, in a suitable approximation, dominated by the lepton contribution and is given by
⌫⫽ 1 85GF 4兩K 3兩2兩K1兩2mhm 6 v 2 2
冉
MW MV冊
4 , 共13兲 where we have neglected a logarithmic dependence on mh (hcan beorwithl⫽,ein the first case and
l⫽e in the second one兲. With reasonable values for the masses in Eq.共13兲, that is m
h⬇1 MeV for the case of the
neutrino,v
2⬇100 GeV, and MV⬇400 GeV, we can get a
width of the order of 10⫺21 MeV共up to the suppression of the mixing matrixK). The age of the universe has a corre-spondent width of 10⫺39MeV, thus it means that the decay can have a lifetime less than the age of the universe and could be of cosmological interest. From the cosmological point of view there are also the processes h⫹h→M0 →l⫹l and l⫹l→M0⫹M0, which occur at the tree
level approximation. The cosmological effect of these pro-cesses are the same as in Ref.关4兴.
If the parameters in this model are such that the Majoron is irrelevant from the cosmological point of view, there is still the possibility that the Majoron may be detected by its influence in neutrinoless doubledecay with Majoron emis-sion nn→e⫺e⫺M0 关denoted by ()0M]. However, it
needs the Majoron-neutrino couplings in the range m/v
1
⬃10⫺5⫺10⫺3in order to have the Majoron emission experi-mentally observable 关16兴. Notice that in the present model the accompanying 0⫹scalar, which is by definition the light-est scalar H10, may not be emitted in ()0Mif it is a heavy scalar or it is very suppressed by the mixing factor.
In our model both the usual neutrinoless ()0 decay
and also the decay ()0M have new contributions. IfF is
conserved in the scalar potential or v
1⫽0 the mixing
among singly charged scalars occurs with 1⫺ and⫺ and between2⫺and⫺. However, if we allowF-breaking terms in the scalar potential or v
1⫽0 there is a general mixing
among the scalar fields of the same charge. For instance, the trilinear term f2TS† in the potential in Eq.共2兲 implies the
trilinear interaction f2h2⫺h2⫺⫹⫹, and since there is a general mixing among all scalars of the same charge it means that there are processes where the vector bosons are substituted by scalars since the vertex h2⫹e⫺ does exist and h2⫹ mixes
with all the other singly charged scalars. There are also tri-linear contributions that arise because of the vertices W⫺V⫺H1⫹⫹ and h⫺2h2⫺H1⫹⫹ as in Refs. 关17,18兴. There is also the vertex h2⫺h2⫹M0 contributing to the ()
0M. It
seems that the analysis of both ()0 and ()0M decays is more complicated than those considered in Refs.关15,19兴.
There are also phenomenological constraints on Majoron models coming from a search in the laboratory of flavor changing currents like →e⫹M0 关20兴 or in astrophysics
through processes like ␥⫹e→e⫹M0 which contributes to the energy-loss mechanism of stars 关4兴. However, in the present model the Majoron couples only to neutrinos, for quarks and electrons the couplings arise only at the one-loop level. Hence, all these processes do not constrain the Ma-joron couplings at all共at the lowest order兲. The interaction of the Majoron with neutrinos is diagonal in flavor. The cou-pling between the Majoron and the real scalar field H0j, of the form M0M0Hj0, is
iO4 j共10⫹11兲
v
1
2 , 共14兲
which is a small coupling. Note that since the Majoron de-couples from the other imaginary parts of the neutral scalars there are no trilinear couplings like M0A0Hj0, where A0 de-notes a massive pseudoscalar, hence the model does not have the phenomenological consequences in accelerator physics as the seesaw Majoron model does 关12兴.
Finally, we remark that here we have assumed that there is no spontaneous C P violation. Hence, all vacuum expecta-tion values are real. If we allow complex VEV it has been shown that C P is violated spontaneously 关21兴. If this is the
case, we have a mixing among all the scalars fields and also the majoron mixes with all the other C P-even and C P-odd scalars.
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. C.P. would like to thank Coordenadoria de Aperfeic¸oamento de Pessoal de Nı´ vel Superior共CAPES兲 for financial support.
APPENDIX: CONSTRAINT EQUATIONS
t⫽12v⫹1v 3⫹4 2 v 2 v⫹5 2 v 2 v⫹12 2 共v1 2 ⫹v 2 2 兲v ⫺17v 2 2 v⫹19 2 v1 2 v⫺ 15 2
冑
2v 2 v 2⫹ 16 2冑
2v 2 v 2 ⫹ f1 2冑
2vv, 共A1兲 t⫽22v⫹2v 3⫹4 2 v 2 v⫹6 2 v 2 v⫹14 2 共v1 2 ⫹v 2 2 兲v ⫹冑
16 2v2vv⫹ 20 4 v2 2 v⫹ f1 2冑
2vv⫹ f2 4 v2v, 共A2兲 t⫽32v⫹3v 2⫹5 2 v 2 v⫹6 2 v 2 v⫹13 4 共v1 2 ⫹v 2 2 兲v ⫺15冑
2v2vv⫹ 18 4 v2 2 v⫹ f1 2冑
2vv⫹ f2 4 v2v, 共A3兲 t 1⫽4 2 v 1⫹10共v1 2 ⫹v 2 2 兲v 1⫹11v1 3 ⫹12 2 v 2 v 1 ⫹413v2v 1⫹ 14 2 vv1⫹ 19 2 v 2 v 1, 共A4兲 t 2⫽4 2 v 2⫹10共v2 2 ⫹v 1 2 兲v 2⫹ 11 2 v2 3 ⫹12 2 v 2 v 2 ⫹13 2 v 2 v 2⫹ 14 2 v 2 v 2⫺17v 2 v 2 18 4 v 2 v 2 ⫹20 4 v 2 v 2⫺ 15 2冑
2v 2 v⫹ 16 2冑
2v 2 v⫹f2 4 vv. 共A5兲关1兴 Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, Phys. Lett.
98B, 265共1981兲.
关2兴 G. B. Gelmini and M. Roncadelli, Phys. Lett. 99B, 411 共1981兲. 关3兴 Particle Data Group, C. Caso et al., Eur. Phys. J. C 3, 1
共1998兲.
关4兴 H. Georgi, S. L. Glashow, and S. Nussinov, Nucl. Phys. B193,
297共1981兲.
关5兴 M. C. Gonzalez-Garcia and Y. Nir, Phys. Lett. B 232, 383 共1989兲.
关6兴 S. Bertolini and A. Santamaria, Nucl. Phys. B310, 714 共1988兲. 关7兴 H. Kikuchi and E. Ma, Phys. Lett. B 335, 444 共1994兲. 关8兴 J. Schechter and J. W. F. Valle, Phys. Rev. D 25, 774 共1982兲. 关9兴 F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 共1992兲; R. Foot,
O. F. Herna´ndez, F. Pisano, and V. Pleitez, ibid. 47, 4158
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关10兴 V. Pleitez and M. D. Tonasse, Phys. Rev. D 48, 5274 共1993兲. 关11兴 This case was considered briefly by M. B. Tully and G. C.
Joshi, hep-ph/9810282, but no detail of the Majoron was shown.
关12兴 M. A. Diaz, M. A. Garcia-Jareno, D. A. Restrepo, and J. F. W.
Valle, Nucl. Phys. B527, 44共1998兲.
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