Neutrinoless double beta decay with and without Majoron-like boson emission in a 3-3-1 model
J. C. Montero*
Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil C. A. de S. Pires†
Department of Physics, University of Maryland, College Park, Maryland 20742-4111
V. Pleitez‡
Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil
共Received 28 March 2000; revised manuscript received 17 April 2001; published 17 September 2001兲
We consider the contributions to the neutrinoless double beta decays in a SU(3)L丢U(1)N electroweak
model. We show that for a range of parameters in the model there are diagrams involving vector-vector-scalar and trilinear scalar couplings which can be potentially as contributing as the light massive Majorana neutrino exchange one. We use these contributions to obtain constraints upon some mass scales of the model, such as the masses of the new charged vector and scalar bosons. We also consider briefly the decay in which, in addition to the two electrons, a Majoron-like boson is emitted.
DOI: 10.1103/PhysRevD.64.096001 PACS number共s兲: 23.40.⫺s, 12.60.Cn, 14.60.St I. INTRODUCTION
The issue of neutrino mass continues to be a golden plate in elementary particle physics. Although data coming from solar 关1兴, atmospheric 关2兴, and the Liquid Scintillation Neu-trino Detecter 共LSND兲 关3兴 neutrino experiments strongly suggest that neutrinos must be massive particles, direct mea-surements did not obtain any positive result 关4兴. It is a very well known fact that if neutrinos are massive Majorana par-ticles the neutrinoless double beta ()0decay should exist
关5,6兴. If the neutrino mass is the main effect that triggers this
decay, the decay lifetime is proportional to 共for the case of light neutrinos兲
具
M典
⫽兺
i
Uei2m
i, 共1兲
where Uei, i⫽1,2,3 denote the elements of a mixing matrix that relates symmetry␣,␣⫽e,,and mass eigenstatesi through the relation␣⫽兺iU␣ii, and mi are the neutrino masses. Experimentally a half-life limit T1/20⬎1.8⫻1025 yr implies关7兴
具
M典
⬍0.2 eV. 共2兲The important point is that the ()0 decay probes the physics beyond the standard model. In particular the obser-vation of this decay would be evidence for a massive Majo-rana neutrino although it could say nothing about the value of the mass. This is because although right-handed currents and/or scalar bosons may affect the decay rate, it has been shown that whatever the mechanism of this decay is, a non-vanishing neutrino mass is required for the decay to take
place 关8兴. However, this does not mean that the neutrino mass is necessarily the main factor triggering this decay. In some models the ()0 decay can proceed with arbitrary small neutrino mass via scalar boson exchange 关9兴. The mechanism involving a trilinear interaction of the scalar bosons was proposed in Ref. 关10兴 in the context of a model with SU(2)丢U(1) symmetry with doublets and a triplet of
scalar bosons. However, since in these types of models there is no large mass scale, it was shown in Ref. 关11兴 that the contribution of the trilinear interactions is, in fact, negligible. In general, in models with that symmetry, a fine tuning is needed if we want the trilinear terms to give important con-tributions to the ()0 decay 关8,12兴. Here we will show that in a model with gauge symmetry SU(3)c丢SU(3)L 丢U(1)N 共3-3-1 model for short兲 关13兴, which has a rich Higgs bosons sector as in the multi-Higgs extensions of the standard model, there are new contributions to the ()0 decay. However, unlike the latest sort of models, a fine tun-ing of the parameters of the 3-3-1 model is not necessary since some trilinear couplings, which have mass dimension, could imply an enhancement of the respective amplitudes
共see Sec. III兲. We will use the following strategy: First, we
consider the several new contributions to the ()0 decay introduced by the 3-3-1 model. Next, once this decay has not experimentally been seen, we will consider the usual stan-dard model amplitude共which would arise with massive Ma-jorana neutrinos兲 as the reference amplitude and make the assumption that all the new amplitudes are at most as con-tributing as this one. Hence, we can obtain constraints on some typical mass scale 3-3-1 parameters. The new contri-butions to the ()0 decay are of the short-range type关14兴. Since the respective matrix elements are different from those of the long-range contributions 共the exchange of a light-Majorana neutrino兲 our results should be considered only as an indication of the possible large contributions to this decay in the context of the 3-3-1 model. The outline of the paper is the following. In Sec. II we introduce the interactions that are relevant to the present study. The model with
具
10
典
⫽0, *Email address: [email protected]†Email address: [email protected] ‡Email address: [email protected]
which in some cases has a Majoron-like Goldstone boson, is also discussed. In Sec. III we consider the more important contributions to the ()0 decay and the constraints upon some masses of the model. In Sec. IV we show that if we add a neutral scalar singlet to the minimal model a Majoron-like Goldstone boson is consistent with the Z0 invisible width and we also discuss briefly the Majoron emission process ()0M comparing the relative strength of two amplitudes. Our conclusions are in Sec. V.
II. THE MODEL
Here we will consider the 3-3-1 model with the leptons belonging to triplets (ll lc)T, l⫽e,, and in which a sex-tet of scalar bosons
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3兲is necessary to give to the charged leptons a mass if
具
20典
⬅v2⫽0 关13兴. Most of the phenomenological studies of the
model has been done by considering
具
10典
⬅v1⫽0. The case
when
具
10典
⫽0 was considered in Ref. 关15兴, where the other scalar multiplets are explicitly given. The main difference in the latter case with respect to the former one is that there is a mixing between the vector bosons W⫹ and V⫹:共W⫹ V⫹兲
冉
MW2 ␦ ␦ MV2冊冉
W⫺ V⫺冊
, 共4兲 where␦⫽(g2/2)(2v 1v2), MW 2 and M V2 are the mass eigen-values when␦⫽0; if␦⫽0 共i.e., when v
1⫽0) the mass of
the physical fields are now 2 M1,2⫽共MW 2⫹M V 2兲⫾关共M W 2⫺M V 2兲2⫹4␦2兴1/2 共5兲 and we have defined MW2⫽(g2/2)(v2⫹v2⫹v
2 2 ⫹v 1 2 ), MV2 ⫽(g2/2)(v 2⫹v 2⫹v 2 2 ⫹v 1
2 ), and g is the SU(3)
Lcoupling constant which is numerically equal to the coupling of
SU(2)Li.e., g2⫽8MW
2G
F/
冑
2. We have denoted byv,v, andvthe vacuum expectation values of the neutral compo-nents of the triplets. Notice that M1→MW and M2→MVwhen␦→0. The vector bosons W⫹and V⫹are related to the
new mass eigenstates W1⫹and W2⫹ as
冉
W⫹ V⫹冊
⫽冉
c s ⫺s c冊
冉
W1⫹ W2⫹冊
共6兲with tan 2⫽⫺2␦/( MW2⫺MV2). We can obtain an upper bound on ␦ by assuming that the main contribution to the
MW2 mass is given by v
2⬇246 GeV and that v1 has its
maximum value 3.89 GeV allowed by the value of the parameter 关16兴. In fact if v
2 were the main contribution to
the MW mass we would have ␦/ MW
2⬇(2v
1/v2)⬍0.032.
The constraint on the mixing angle is
0⭐s2⫽1 2
冉
1⫺ MV2⫺MW2 关4␦2⫹共M V 2⫺M W 2兲2兴1/2冊
⬍ 1 2. 共7兲 Some illustrative values for sare obtained by using typical values for the parameters. For instance, for v1
⫽3.89 GeV, v2⫽10 GeV, MW⫽80.41 GeV, and MV
⫽100 (300) GeV, we get s2⫽1.9⫻10⫺5
(3.4⫻10⫺8); or if
MV⫽100 GeV and if v2 has its maximal value v2
⫽246 GeV we have s2⫽1.1⫻10⫺2. We see that only for values of MV⬇MW the s2 is almost 0.5 but this light vector boson may be not phenomenologically safe. However, ifv
1
is of the same order of magnitude of the neutrino mass smaller values for the mixing angle are obtained. Hence, it may not be relevant for collider physics and low-energy pro-cesses like the ()0 decay at all and in practice W1⫹
⬇W⫹, W 2
⫹⬇V⫹, but this could not be the case in astro-physical processes 关15兴. Next, we consider several tions that are present in this model. The scalar-quark interac-tions are ⫺LY u⫺d⫽
冑
2 兩v兩D ¯ LVCKM † MuUR⫺⫹冑
2 兩v兩U¯LVCKMMdDR1⫹ ⫹D¯ L共VL d兲T⌬ V L uMuU R冋
冑
2 兩v兩1⫺⫺冑
2 兩v兩 ⫺册
⫹U¯L共V L u兲T⌬ V L dMdD R冋
冑
2 兩v兩 ⫹⫺冑
2 兩v兩1 ⫹册
⫹H.c., 共8兲 with ⌬⬅冉
0 0 0 0 0 0 0 0 1冊
, 共9兲 and VL u,dare unitary mixing matrices, Mu,d are the diagonal mass matrices of the u-like and d-like quark sectors, and
VCKM denotes the usual mixing matrix of Cabibbo-Kobayashi-Maskawa. The Yukawa interactions in the lepton sector are ⫺LY l⫽ 1
冑
2 ¯LK1lRH 1 ⫹⫹ 1冑
2l¯LK2R c H2⫺⫹1 2l¯LK3共lL兲 cH 1 ⫺⫺ ⫹1 2共l c兲 LK4lRH2⫹⫹⫹2¯LK1⬘
lR1⫹⫺2 l¯LK2⬘
R c 2 ⫺ ⫹H.c., 共10兲where K1⫽EL†GER, K2⫽EL†GEL*; K3⫽EL†GEL, K4 ⫽ER T GER; K1
⬘
⫽EL†G⬘
ER, K2⬘
⫽EL † G⬘
EL*; G and G⬘
are symmetric and antisymmetric共they can be complex兲 matri-ces, respectively. ER,EL,EL are the right- and left-handed mixing unitary matrices in the lepton sector relating symme-try eigenstates 共primed fields兲 with mass eigenstates共unprimed fields兲 关17兴:
lR
⬘
⫽ERlR, lL⬘
⫽ELlL, L⬘
⫽ELL. 共11兲 Some of the couplings in Eq. 共10兲 do not depend on the charged lepton masses and since all matrices in Eq. 共10兲 are not unitary, the model breaks the lepton universality but it can be shown that, for the massless neutrino case no strong constraints arise from exotic muon and tau decays 关18兴. In the scalar sector we also have mixing angles. In the singly charged sector we have i⫺⫽兺lOi jH⫺j , where i⫺⫽1⫺,2⫺,⫺,⫺,h1⫺,h2⫺ and Hj⫺, j⫽1, . . . ,6 denotes the respective mass eigenstate field; similarly in the doubly charged sector we have ⌽i⫺⫺⫽兺lOi j⌿j⫺⫺, with ⌿i⫺⫺
⫽⫺⫺,⫺⫺,H 1 ⫺⫺,H 2 ⫺⫺ and ⌿ j ⫺⫺, j⫽1, . . . ,4 the respec-tive mass eigenstate fields. However, in the following we will use H⫺ and H⫺⫺ as typical mass eigenstates of the respective charged fields and omit the scalar mixing param-eters. We recall that the model conserves theF⫽L⫹B quan-tum number, L is the total lepton number, and B is the baryon number. The assignments are
F共U⫺⫺兲⫽F共V⫺兲⫽F共⫺⫺兲⫽F共⫺⫺兲⫽F共 2 ⫺兲 ⫽F共⫺兲⫽F共 1 0兲⫽F共h 2 ⫺兲⫽F共H 1 ⫺⫺兲 ⫽F共H2⫺⫺兲⫽2, 共12兲
and all other scalar fields with F⫽0. The charged currents coupled to the vector bosons are given by
LCC⫽⫺ g
冑
2共U ¯ L␥uVCKMDLW⫹⫺¯L␥VWlLW⫹ ⫹ l¯L c␥V UVW † LV⫹⫺ l¯L c␥V UlLU⫹⫹兲⫹H.c., 共13兲with the mixing matrices defined as VCKM⫽(VL u
)†VLd in the quark sector, and VW⫽EL †EL, VU⫽ERTEL in the leptonic sectors. We have the trilinear interactions involving one-vector and two scalar bosons which are of the form共up to a
ig/
冑
2 factor兲 LV2S⫽⫹⫺⫺W ⫹⫹⫺⫹⫹W ⫺⫹h 2 ⫹H 1 ⫺⫺W ⫹ ⫹H 1 ⫹⫹h 2 ⫺W ⫺⫹⫺⫹⫹V ⫺⫹⫺⫺⫹V ⫹ ⫹h1⫹H2⫺⫺V⫹⫹h⫺1H2⫹⫹V⫺⫹共2⫹1⫹ ⫹1⫹2⫹⫹h2⫹h1⫹⫹h1⫹h2⫹兲U⫺⫺⫹H.c. 共14兲There are also trilinear interactions involving two-vector and one scalar bosons 共the ⫺ scalar couples to ordinary and exotic quarks and for this reason it is not of concern here兲. They are given by 共up to a g2/2 factor兲
L2VS⫽v1 2 H1 ⫹⫹W⫺•W⫺⫹v1 2 H2 ⫹⫹V⫺•V⫺⫹
冉
v ⫹⫹ ⫹v⫹⫹⫹ v 2 2 共H1 ⫹⫹⫹H 2 ⫹⫹兲冊
W⫺•V⫺. 共15兲 Notice that there is a coupling which is proportional to vand hence it will be the dominant one. Next, we write down the trilinear interactions among three vector bosons
L3V⫽i g
冑
2共W ⫹V⫹U⫺⫺⫹V ⫹V⫹U⫺⫺ ⫹W⫹V⫹U ⫺⫺兲, 共16兲where W⫽W⫺W and so on. Finally, we have tri-linear interactions among scalar bosons only
L3S⫽f1 2 ⑀ i jk ijk⫹ f2 2 TS†⫹H.c. 共17兲 The couplings f1,2have dimension of mass but both of them are arbitrary parameters 共see the next section兲. Other terms, such as the trilinears f3TS† and f
4⑀SSS and the quartic interactions ⑀(S*), †† and ⑀SS, violate the
conservation of F. However, as we will show in Sec. IV, when discussing the Majoron emission, the model must be modified by adding a scalar singlet in order to be consistent with the LEP data.
III. THE NEUTRINOLESS DOUBLE BETA DECAY Some of the more relevant diagrams of the ()0 decay in the present model are shown in Figs. 1– 6. Our goal is to analyze the order of magnitude of each diagram and to ob-tain constraints on some mass scales of the model. We will consider the diagram in Fig. 1 as the reference one, i.e., it is the diagram that already exists in the standard model frame-work with massive Majorana neutrinos and which is param-etrized by two effective four-fermion interactions. The other contributions will be considered as being at most equally as important as the standard one.
The strength of the diagram in Fig. 1 is given by A共1兲⬀ g 4
具
M 典
MW4具
p2典
c 4⫽32GF 2具
M 典
具
p2典
c 4, 共18兲where
具
M典
is the effective mass defined in Eq.共1兲 and具
p2典
is the average of the four-momentum transfer squared, which is of the order of (100 MeV)2. Below we will use a small␦ so that M1→MW and M2→MV. In Eq. 共18兲 and hereafter we will omit for simplicity the mixing parameters. Only in the vertices we will take care about the mixing between W and V defined in Eq.共6兲 but in the propagator we will use the masses of W and V.
Next, let us consider the diagram in Fig. 2 which has the strength given by A共2兲⬀32GF2
冉
MW MV冊
2 c 3 s冑
具
p2典
, 共19兲 and we have the ratioA共2兲 A共1兲⫽
冉
MW MV冊
2冑
具
p2典
具
M典
tan, 共20兲and if A(2)/A(1)⬍1 we have that
MV⬎2.2⫻104MW
冑
tan⫽1.79⫻106冑
tan GeV. 共21兲 We recall that a lower limit of 440 GeV is obtained for MV from the muon decays but when only the bilepton contribu-tions to those decays are considered 关19兴. However, in the minimal 3-3-1 model the scalar-boson contributions cannot be negligible since some of the charged scalar bosons can belighter than the vector bilepton boson V⫺. Hence, a lighter vector boson V may still be possible but this subject deserves a more detailed study of the muon decay considering both vector and scalar contributions. A contribution similar to that in Fig. 1 but with two V⫺bosons instead of two W⫺bosons may be not negligible but it does not constrain the mass MV as much as those in Eq.共21兲 since the condition that its ratio to the A(1) amplitude be less than one gives the condition
MV⬎MW
冑
tan. All the Lagrangian interactions in Eqs.共8兲,共10兲, 共13兲, 共14兲, 共15兲, and 共16兲 are written in terms of
sym-metry eigenstates. We have assumed Yukawa couplings of the order of unity. As we are not considering the mixing among the scalar fields our constraints are valid only for the main component of the symmetry eigenstate scalar fields. It means that H⫺ and H⫺⫺ denote the dominant mass eigen-states of the singly and doubly charged scalar fields, respec-tively. The amplitude of the diagram in Fig. 3 is
A共3兲⬀
具
M典
具
p2典
MH4⫺. 共22兲The scalar contribution in Fig. 3 can be as important as the standard one in Fig. 1. We have
A共3兲
A共1兲⫽
1 32GF2MH4⫺c4
, 共23兲
and assuming that A(3)/A(1)⬍1 and c⫽1 we get
MH⫺⬎124 GeV. 共24兲
From Eq.共15兲 we see that the contribution v⫹⫹W⫺V⫺is the dominant one in diagrams like that in Fig. 4. As we said before we will omit the mixing angles, i.e., assuming ⫺⫺
⬇H⫺⫺. Hence we have FIG. 2. The same as in Fig. 1 but with one bilepton vector boson
V⫺replaced by a vector boson W⫺.
FIG. 3. Charged scalars contribution to the ()0decay.
FIG. 4. New contribution to the ()0 decay in the 3-3-1
model.
A共4兲⬀ v MW2MV2MH2⫺⫺c
2s
2. 共25兲
Next, we note that
A共4兲 A共1兲⬀ v
具
M典
具
p2典
32GF2MW2MV2MH2⫺⫺ tan2 ⬇5.33⫻1015tan2共1 GeV兲 4 MV2MH2⫺⫺, 共26兲where we used
具
M典
⫽0.2 eV 关4兴, v⫽3 TeV, and具
p2典
⫽(100 MeV)2. If A(4)/A(1)⬍1 it impliesMV⬎7.3⫻107tan
共1 GeV兲2
MH⫺⫺
. 共27兲
A similar analysis arises by considering Fig. 5; however, Fig. 5 is less enhanced than the contribution of Fig. 4 because instead ofvis the momentum of one of the vector bosons,
p⬃
冑
具
p2典
, that appears in it.More interesting are the contributions involving trilinear scalar interactions given in Eq. 共17兲 like that of the diagram in Fig. 6. We have in this case
A共6兲⬀ f
MH4⫺MH2⫺⫺, 共28兲
where MH⫺ represents a typical mass of the singly charged scalar bosons, say 124 GeV, MH⫺⫺is the mass of the doubly charged scalar boson, and f is the trilinear coupling f1 or f2 in Eq. 共17兲 with dimension of mass. The ratio of these am-plitudes is A共6兲 A共1兲⬀ f
具
p2典
32GF2MH4⫺MH⫺⫺ 2具
M典
c4 ⬇冉
f /GeV MH2⫺⫺/GeV2冊
4.8⫻107 c4 . 共29兲If A(6)/A(1)⬍1, and assuming c⫽1 and MH⫺
⫽124 GeV, we obtain the constraint
f
MH2⫺⫺⬍2.1⫻10
⫺8 GeV⫺1. 共30兲
For an arbitrary U(1)N charge for the scalar multiplets the symmetry of the potential is SU(3)L丢关U(1)兴2. If the triplet and the sextet S have both N⫽0, as is the case for the present model, the trilinear couplings f1,2 break the extra
U(1) symmetry. We have verified that if both f1,2⫽0 there is indeed a pseudo-Goldstone boson关20兴. It means that f1,2are arbitrary parameters and in principle they can be small共say, 1 GeV兲, or large 共say, 1 TeV兲 mass scales. We see that if f
⫽1 (10⫺3) TeV then M
H⫺⫺is greater or of the order of 300
共10兲 TeV. For this value for the mass of the doubly charged
scalar field and small the constraint given in Eq. 共21兲 is stronger than that of Eq. 共27兲. For instance, if tan⫽10⫺8 we have from Eq. 共21兲 that MV⭓179 GeV.
There is also a diagram in which the doubly charged sca-lar field in Fig. 6 is substituted by a vector boson U⫺⫺. Although the interactions in Eq. 共14兲 are proportional to g they are also derivative and proportional to the momentum
p⬃
冑
具
p2典
; hence it is suppressed with respect to the diagram in Fig. 6.IV. MAJORON EMISSION
If the F quantum number is spontaneously broken as in the present model, it means that a Majoron-like boson does exist. Since the scalar field that is responsible for the break-down of that continuous symmetry is10, and it belongs to a triplet of the subgroup SU(2)丢U(1), this Majoron-like
Goldstone has similar couplings to that of the triplet Majoron model of Ref.关21兴. It is well known that this sort of Majoron model has been ruled out by the CERN e⫹e⫺ collider LEP data 关22兴. Apparently, since the Higgs sector of the present model is rather complicated having a neutral scalar singlet
关under SU(2)丢U(1)兴, 0, it seems that the Majoron-like Goldstone in this case will be able to avoid the LEP con-straints as claimed in Refs.关15,23兴. However, we will show that this is not indeed the case. The mass matrices of the scalar and pseudoscalar in this model have been given in Ref. 关15兴. Here we will only give the results of the mass eigenvalues and the respective mixing matrix in the C P-even scalar sector. The argument in Ref. 关15兴 was the following. Let us begin with the relation R40⫽兺jO4 j
o
H0j, where H0j, j
⫽1, . . . ,5, denotes the mass scalar eigenstates and R4 the real component of the scalar field10according to the general shifting of the neutral scalar fields in the scalar potential of the form Xi0→(1/
冑
2)(vXi⫹Ri⫹iIi), i⫽1,2,3,4,5 where Xi0
⫽,,,1,2, respectively. In this case if H1 0
denotes the lightest scalar boson ( MH1⬍MZ, we are assuming a mass spectrum where MHi⬍MHj if i⬍ j), the contribution to the decay mode Z0→H10M0 is ⌫H 1 0M0 Z ⫽2兩O41 o兩2⌫ ¯ Z . Hence, if 兩O41 o兩⬍10⫺2
the model would be consistent with the LEP data; i.e., now ⌫H
1 0M0
Z
would be reduced to an acceptable level. First of all recall that as shown in Ref. 关15兴 the Majoron-like boson decouples from the other pseudoscalar FIG. 6. Pure scalar Higgs bosons contribution the ()0decay
fields, i.e., Im1 0⬅I
4⫽M0. For instance, using the same val-ues of the vacuum expectation valval-ues 共VEVs兲 and f1,2⫽
⫺1 TeV and with the dimensionless constant of the scalar
potential given in Ref. 关15兴 with k⫽0.1 for k
⫽1,2,3,4,5,6,7,8,9,18,20, m⫽0.01 for m⫽13,14,16,17, n
⫽0.001 for n⫽10,11,12,19, and 15⫽0.05, we obtain the following masses in the scalar sector (0.056,102,1342,3626,4325) GeV and the mixing matrix共up to three decimal places兲:
Oo⫽
冉
0.0 0.081 ⫺0.010 0.996 0.021 0.0 0.995 ⫺0.029 ⫺0.082 0.039 0.0 ⫺0.030 ⫺0.999 ⫺0.008 0.004 1.0 0.0 0.0 0.0 0.0 0.0 0.040 ⫺0.005 0.017 ⫺0.999冊
. 共31兲This pattern of mixing matrix remains the same for several values of the parameters provided that v
1 is a small VEV
restricted to the condition that it has to be smaller than 3.89 GeV关16兴. From Eq. 共31兲 it can be seen that the scalar partner of the Majoron is always mainly the lightest scalar, i.e.,
兩O41
o兩⬃1 and it would be always produced at LEP. We see that the Majoron in the minimal 3-3-1 model has been also ruled out by the LEP data. One possibility to recover consis-tency with the LEP data is to break explicitly theF symme-try by adding trilinear terms like f3TS, f4TS† in the scalar potential, see Eq.共17兲. In this case there is no Majoron at all and although v
1 still has a small value, due to f3 all
scalars are heavy enough to not be produced at the LEP energies关24,25兴. Of course, in this case there is no contribu-tion to Majoron emission in the neutrinoless double beta de-cay. However, our results in Sec. III are still valid since they depend only on the small value of v
1. Another possibility
that we will consider here is to modify the model by intro-ducing a scalar singlet⌺0, which carriesF⫽2 共or L⫽2), in the same way as considered in Ref.关26,27兴 in the context of a SU(2)丢U(1) model. In this case we have to add the
fol-lowing terms to the scalar potential in Ref.关15兴:
V共Xi,⌺兲⫽5 2⌺2⫹ 21⌺4⫹
兺
i 关Xi Tr共Xi†Xi兲⌺2 ⫺TS†⌺⫹H.c.兴, 共32兲where Xi denotes any triplet,, or the sextet S, and we will denote Xi as 22,23,24,25, respectively, and ⬎0. The neutral Higgs sector contains six C P-even scalars and three massive C P-odd pseudoscalar beside the massless C P-odd Majoron. The neutral scalar singlet also gains a VEV, i.e.,
⌺⫽(v⌺⫹R6⫹iI6)/
冑
2, and the mass term is given by M2/2, where M2 in the pseudoscalar sector in the basisI1,I2,I3,I4,I5,I6 is given by 共the constraint equations ap-pear in the Appendix兲
M11⫽⫺ 16 2
冑
2 v2v 2 v ⫹217v2 2 ⫹ 1 2冑
2共15vv2⫺ f1v兲 v v ⫹2v 1v⌺⫹ t v, M22⫽⫺1 4共冑
2 f1v⫹ f2v2兲 v v⫹ t v, M33⫽⫺1 4共冑
2 f1v⫹ f2v2兲 v v⫹ t v, M44⫽ v2v⌺ 2v 1 ⫹ t1 v 1 , M55⫽ 1 2冑
2共15v 2⫺ 16v 2兲v v 2 ⫹217v 2⫺f2 4 vv v 2 ⫹ t2 v 2 , M66⫽ v2v 1 2v⌺ ⫹ t⌺ v⌺, M12⫽⫺ f1 2冑
2v, M13⫽⫺ f1 2冑
2v, M14⫽⫺vv⌺, M15⫽ 1 2冑
2共15v 2⫺ 16v 2 兲⫹217vv2, M16⫽vv1, M23⫽⫺1 4共冑
2 f1v⫹ f2v2兲, M24⫽0, M25⫽ f2 4 v, M26⫽0, M34⫽0, M35⫽ f2 4 v, M36⫽0, M45⫽0, M46⫽⫺ v2 2 , M56⫽0. 共33兲The mass matrix above has two true Goldstone bosons G1,20 and the Majoron-like one, M0, and three massive C P-odd pseudoscalar bosons. The massless ones are given by
G10⫽共0,v/v,⫺1,0,0,0兲/共1⫹v2/v2兲1/2, G20⫽
冉
v v 2 ,⫺vv 2 V1 ,⫺v 2 v V1 , 2v 1v⌺ 2 V2 ,⫺1, ⫺2v1 2 v⌺ V2冊
冒
N, M0⫽共0,0,0,v 1/v⌺,0,1兲/共1⫹v1 2 / v⌺2兲1/2, 共34兲 where V1⫽v2(v 2⫹v 2 ), V2⫽v2(v 1 2 ⫹v ⌺ 2 ); N is the nor-malization factor that we will omit here. We have verifiedthat M0 in Eq. 共34兲 is in fact the Majoron: by adding an explicitF-violating term, like f3TS, it gets a mass while the other two G1,2remain massless. The massive pseudosca-lars for the parameters used before have the following masses共in GeV兲: 174, 3625, and 4325. On the other hand, if
v
1⫽0, which forces ⫽0, the Majoron is purely singlet
and the real and imaginary parts of 10 are mass degenerate, i.e., form a complex field, with mass
m 1⫽4 2⫹ 10v2 2 ⫹共12⫹19兲 v2 2 ⫹13 v2 2 ⫹14 v2 2 ⫹25 v⌺2 2 . 共35兲
We see that in this case the Majoron has no doublet compo-nents at all and it is mainly singlet. Hence it is possible to keep consistent with LEP data. Although there are astro-physical constraints 共the Majoron emission implies a differ-ent rate for the stellar cooling兲 that have to be taken into account关28兴, in the basis we have chosen they are less severe since we have avoided the doublet component of the Ma-joron. Anyway, since these constraints have been already considered in Ref.关27兴 and they imply that v
1⬍0.33 GeV if
v⌺⫽1 TeV, we will use these values for v
1,v⌺. Once we
have shown in what situation there is a safe Majoron-like boson in the present model we can consider the emission of this Goldstone boson in the neutrinoless double beta decay. In fact, as in the triplet Majoron model, in the present model it is possible to have the neutrinoless double beta decay with Majoron emission: 2n→2p⫹2e⫺⫹M0 关29兴, denoted here by ()0M. We will denote the strength of the amplitude of the diagram i of the ()0M decay by B(i). This decay proceeds via the diagram in Fig. 7 and it has a strength proportional to B共7兲⬀ m共v 1/v⌺兲 MX4⫺
具
p2典
v 1 , 共36兲where X⫺can be a scalar or a vector boson, i.e., the diagram in Fig. 7 can be formed with any one of Figs. 1, 2, or 3 with a Majoron attached to the neutrinos. The couplings between neutrinos and the Majoron are diagonal and given by
m/v
1. Notice that in Eq. 共36兲 the truly neutrino mass
ap-pears instead of the effective mass
具
M典
defined in Eq.共1兲.However, we still can assume that neutrinos have small masses and numerically m⬇
具
M典
. We will assume also that the contribution to the ()0M decay in Fig. 7 with X⫺⫽W⫺ is the reference contribution. This diagram depends only on the neutrino masses and mixing angles, and we will compare it with other contributions like the one in Fig. 8. The couplings of the Majoron to the vector bosons are pro-portional tov
1 and so they are negligible. We will consider
only the diagram with the Majoron coupled to the scalar H⫺ since it is proportional to the trilinear f2 shown in Eq.共17兲. We have
B共8兲⬀ f f2
MH6⫺MH2⫺⫺, 共37兲
with f that can be f1 or f2. Let us consider the ratio
B共7兲 A共1兲⬀
mQ
32GF2MX4
具
M典
c4v⌺, 共38兲where we have introduced the factor Q which denotes the available energy. It implies that the diagram in Fig. 7 is a potentially important contribution when X is the W vector boson since for Q⬃3 MeV 关30兴 the suppression of B(7) will depend mainly on the value ofv⌺. If B(7)/A(1)⬍1 we obtain that v⌺⬎1.65⫻10⫺2 GeV which is automatically satisfied.
On the other hand, comparing the amplitudes of the dia-grams in Figs. 7 and 8 we have
B共8兲 B共7兲⬀ f f2具p2
典
MX4v⌺ MH6⫺MH⫺⫺ 2 m, 共39兲and for MX⫽MV⫽400 GeV and v⌺⫽1 TeV, using typical values as f⫽ f1⫽ f2⫽⫺1 TeV, MH⫺⫽124 GeV, and
MH⫺⫺⫽500 GeV, and the other parameters in Eq. 共39兲, we have that B(8)/B(7)⬇1.3⫻109 or B(8)/B(7)⬇1.5⫻103 if
f1⫽ f2⫽ f ⫽⫺10⫺3 TeV. The relative importance of the processes in Figs. 7 and 8 will depend on the values of the trilinear parameters and on the value ofv⌺.
V. CONCLUSIONS
We see that in the 3-3-1 model, as in other models with complicated Higgs sector 关8,12兴, besides the well known mechanism of exchanging massive Majorana neutrinos be-FIG. 7. Contribution to the Majoron emission ()0M decay.
X⫺can be a scalar or vector boson.
FIG. 8. Trilinear scalar coupling contributing to the ()0M
tween two standard model V⫺A vertices, there are new con-tributions involving the exchange of scalar bosons. However, unlike a similar mechanism in the context of extensions of the standard model there is no need for fine tuning in order to have trilinear scalar couplings giving large contributions to the several neutrinoless double beta decay modes. Notice that effective interactions from diagrams such as those in Fig. 3 are still parametrized in the form of two general four-fermion effective interactions共they are pointlike at the Fermi scale兲 exchanging a light neutrino in between 关31兴. However, contributions involving trilinear interactions like those in Figs. 4, 5, and 6 necessarily need a six-fermion effective interaction parametrization. Another important point to be stressed here is that in the present model the double Majoron emission 2d→2p⫹2e⫺⫹2M0 may be as contributing as the decay with only one Majoron boson. This decay is ex-pected to be important in supersymmetric models关32,33兴. In the present model it can occur because in diagrams like that in Fig. 8 a second Majoron can be attached to the scalar lines. Since this coupling is proportional to the trilinear f2it is still possible that the suppression coming from the mass square in the denominator does not sufficiently suppresses this process共there is also an important contribution coming from the vertex v⫺1⫹M
0). There are also contributions similar to the one in Fig. 8 but now the scalar-Majoron ver-tex above is substituted by the verver-tex W⫺V⫹M0, which is proportional to g2v
2, and for this reason it is not necessarily
suppressed. It is interesting to note that this sort of contribu-tion to the ()0MM decay, coming from adding another trilinear coupling in the diagram in Fig. 8, which is not de-rivatively suppressed, was not considered in Ref. 关33兴. Re-cent experimental data on Majoron emission decays have been constrained only the effective Majoron-neutrino cou-pling constant 关30兴. Other processes like the double K cap-ture 关29兴 can also be important in the present model. Some comments are now in order.共i兲 We have not considered pos-sible cancellations among several contributions to each dia-gram. It means that our constraints are valid, as we said in Sec. III, for the main component of each scalar field of the singly and doubly charged scalar sectors. 共ii兲 Our results were obtained assuming that all new contributions, say to the ()0 decay, are at most as important as the contribution due to a light massive Majorana neutrino exchange which is proportional to
具
M典
. However, we can wonder what would be the value of the effective mass具
M典
if we use the oscil-lation data and direct measurements on neutrino masses. Re-cent analyses show that by assuming a normal mass hierar-chy the effective mass parameter can take any value from zero to the present upper limit 关34兴. In fact, if the data on oscillation is put together with that from ()0 decay and tritium beta decay, it was shown that if the minimum of具
M典
with respect to the mixing angles is greater than the present bound of 0.2 eV, then neutrinos are quasi-Dirac par-ticles关35兴. As discussed previously, the black box of Ref. 关8兴 may induce a negligible Majorana mass to the neutrinos and in the context of the present model we must interpret this situation as an indication of the fact that the main contribu-tion to the ()0 decay is not the diagram in Fig. 1. In thiscase the neutrinos would be almost Dirac particles and the constraints on the several mass scales of the model should be obtained by directly comparing these contributions with the lower bound on the half life T1/20⬎1.8⫻1025 yr 关7兴. 共iii兲 In the basis we have chosen, see Eq.共34兲, the Majoron couples at the tree level only to neutrinos, and hence the constraints on the--M vertex, coming from muon (→eM ), pion,
and kaon ⫹(K⫹)→l¯ M decays, are the same as in Ref.
关33兴. The existence of the vertex e⫺V⫹, which is propor-tional to sin and for this reason is not relevant for labora-tory processes, may have, as we said before, important astro-physical consequences关15兴. 共iv兲 Phenomenology of nonzero initial electric charge processes, like e⫺e⫺ and hadronic ones, will furnish constraints on the trilinear vertices appear-ing in Figs. 4, 5, and 6 but this will be considered elsewhere. The 3-3-1 model has a rich scalar sector indeed. This implies that it may be, in principle, difficult to separate in a given process the contributions of all fields belonging to a charged sector. However, it has been shown that in lepton-lepton col-liders the left-right asymmetries are not sensible to the scalar contributions. It means that those asymmetries are the appro-priate observable for the doubly charged vector bilepton dis-covery 关36兴. We see that the opposite occurs in the ()0 decay: it is possible that the main contribution comes from the doubly charged scalar boson, through the diagram in Fig. 6, while the respective vector boson contribution seems to be negligible. Finally we would like to compare our Majoron model with that of Schechter and Valle 关26兴. Firstly, we no-tice that although our model has two singlet, three doublets, and a triplet of scalars under the subgroup SU(2)丢U(1),
the respective scalar potential is not reduced to the scalar potential invariant under the standard SU(2)丢U(1)
symme-try, involving the same multiplets. For instance, in our model there are cubic invariants that are not present in the former. Secondly, we have not introduced right-handed neutrinos and for this reason we have only light neutrinos. It means that the singlet⌺ does not couple to the leptons and that the coupling of neutrinos to Majoron and Z0 are diagonal. Thus, the de-caysH→L⫹M0 andH→L⫹L
⬘
⫹L⬘
are not induced at the tree level, whereH, although light, is heavier thanL. The decay H→Lc⫹M0 is produced at the one-loop level due to the mixing between W and V. The vertex is propor-tional to (g2/
冑
2)(v2v1/v⌺); then even with v2 of the
order of 10 GeV andv⌺of the order of 1 TeV the lifetime is of the order of the age of the universe.关The decay H→L
⫹M0 also occurs but the vertex involved is proportional to (g2/
冑
2)(v1
2 /
v⌺).兴 Notice also that in the basis given in Eq.
共34兲 the Majoron does not couple to the charged leptons so
there is not the process␥⫹e→M0⫹e at the tree level which imposes severe astrophysical constraints inv
1 as has been
noted in Ref. 关27兴.
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.
APPENDIX: CONSTRAINT EQUATIONS OF THE SCALAR POTENTIAL
Here we show the constraint equations that must be satis-fied by the scalar potential:
t⫽1 2 v⫹1v 3⫹4 2 v 2 v⫹5 2 v 2 v⫹12 2 共v1 2 ⫹v 2 2 兲v ⫺ 15 2
冑
2v 2 v 2⫹ 16 2冑
2v 2 v 2⫺17v2 2 v⫹19 2 v1 2 v ⫺vv 1v⌺⫹ f1 2冑
2vv, t⫽22v⫹2v3⫹ 4 2 v 2 v⫹6 2 v 2 v⫹14 2 共v1 2 ⫹v 2 2 兲v ⫹16冑
2v2vv⫹ 20 4 v2 2 v⫹23 vv⌺2 2 ⫹ f1 2冑
2vv ⫹f42v 2v, t⫽32v⫹3v 3⫹5 2 v 2 v⫹6 2 v 2 v⫹13 2 共v1 2 ⫹v 2 2 兲v ⫺15冑
2v2vv⫹ 18 4 v2 2 v⫹24 vv⌺2 2 ⫹ f1 2冑
2vv ⫹f2 4 v2v, t 1⫽4 2 v 1⫹10共v1 2 ⫹v 2 2 兲v 1⫹11v1 3 ⫹12 2 v 2 v 1 ⫹213v2v 1⫹ 14 2 v 2 v 1⫹ 19 2 v 2 v 1⫹ 25 2 v1v⌺ 2 ⫺2v2v⌺, 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⫺ 15 2冑
2v 2 v⫹ 16 2冑
2v 2 v ⫺17v 2 v 2⫹ 18 4 v 2 v 2⫹ 20 4 v 2 v 2⫹ 25 2 v2v⌺ 2 ⫹f2 4 vv, t⌺⫽52v⌺⫹21v⌺ 3⫹22 2 v 2 v⌺⫹23 2 v 2 v⌺⫹24 2 v 2 v⌺ ⫹25 2 共v1 2 ⫹v 2 2 兲v ⌺⫺ 2v 2 v.关1兴 Homestake Collaboration, K. Lande, in Neutrinos ’98,
Pro-ceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Japan, 1998, edited by Y. Suzuki and T. Tosuka关Nucl. Phys. B 共Proc. Suppl.兲 共in press兲兴; Ka-miokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 77, 1683共1996兲; GALLEX Collaboration, T. Kirsten, in Neutrinos ’98; SAGE Collaboration, V. Gavrin, in Neutrinos ’98; Su-perKamiokande Collaboration, Y. Suzuki, in Neutrinos ’98.
关2兴 SuperKamiokande Collaboration, Y. Fukuda et al., Phys. Lett.
B 433, 9共1998兲; Phys. Rev. Lett. 81, 1562 共1998兲; Phys. Lett. B 436, 33共1988兲.
关3兴 C. Athanassopoulos et al., Phys. Rev. Lett. 77, 3082 共1996兲;
81, 1774共1998兲.
关4兴 Particle Data Group, D.E. Groom et al., Eur. Phys. J. C 15, 1 共2000兲.
关5兴 H. Primakov and S.P. Rosen, Rep. Prog. Phys. 22, 111 共1969兲. 关6兴 For a recent review see H.V. Klapdor-Kleingrothaus, Int. J.
Mod. Phys. A 13, 3953共1998兲; P. Fisher, B. Kayser, and K.S. McFarland, Annu. Rev. Nucl. Part. Sci. 49, 481 共1999兲; for early references see M. Doi, T. Kotani, and E. Takasugi, Suppl. Prog. Theor. Phys. 83, 1共1985兲.
关7兴 L. Baudis et al., Phys. Rev. Lett. 83, 41 共1999兲.
关8兴 J. Schechter and J.W.F. Valle, Phys. Rev. D 25, 2951 共1982兲. 关9兴 V. Pleitez and M.D. Tonasse, Phys. Rev. D 48, 5274 共1993兲. 关10兴 R.N. Mohapatra and J.D. Vergados, Phys. Rev. Lett. 47, 1713
共1981兲.
关11兴 W.C. Haxton, S.P. Rosen, and G.J. Stephenso, Jr., Phys. Rev. D
26, 1805共1982兲; L. Wolfenstein, ibid. 26, 2507 共1982兲.
关12兴 C.O. Escobar and V. Pleitez, Phys. Rev. D 28, 1166 共1983兲. 关13兴 F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 共1992兲; P.
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14兴 M. Hirsch, H.V. Kkapdov-Kleingrothaus, and S.G. Kovalenko,
Phys. Rev. Lett. 75, 17共1995兲; Phys. Rev. D 53, 1329 共1996兲; H.V. Klapdor-Kleingrothaus and H. Pa¨s, ibid. 62, 117301
共2000兲.
关15兴 J.C. Montero, C.A. de S. Pires, and V. Pleitez, Phys. Rev. D 60,
115003共1999兲.
关16兴 J.C. Montero, C.A. de S. Pires, and V. Pleitez, Phys. Rev. D 60,
098701共1999兲.
关17兴 J.T. Liu and D. Ng, Phys. Rev. D 50, 548 共1994兲. 关18兴 M.M. Guzzo et al., hep-ph/9908308.
关19兴 M.B. Tully and G.C. Joshi, Int. J. Mod. Phys. A 13, 5593 共1998兲.
关20兴 S. Weinberg, Phys. Rev. Lett. 29, 1698 共1972兲.
关21兴 G.B. Gelmini and M. Roncadelli, Phys. Lett. 99B, 411 共1981兲. 关22兴 M.C. Gonzalez-Garcia and Y. Nir, Phys. Lett. B 232, 383
共1989兲.
关23兴 F. Pisano and S. Shelly Sharma, Phys. Rev. D 57, 5670 共1998兲. 关24兴 J.C. Montero, C.A. de Sousa Pires, and V. Pleitez, Phys. Lett.
B 502, 167共2001兲.
关25兴 E. Ma, Phys. Rev. Lett. 81, 1171 共1998兲.
关26兴 J. Schechter and J.W.F. Valle, Phys. Rev. D 25, 774 共1982兲. 关27兴 M.A. Diaz, M.A. Garcia-Jareno, D.A. Restrepo, and J.F.W.
Valle, Nucl. Phys. B527, 44共1998兲.
关28兴 D.A. Dicus, E.W. Kolb, V.L. Teplitz, and R. Wagoner, Phys.
Rev. D 18, 1829共1978兲; R.N. Mohapatra and P.B. Pal, Massive Neutrinos in Physics and Astrophysics共World Scientific, Sin-gapore, 1991兲, and references therein.
关29兴 H. Georgi, S.L. Glashow, and S. Nussinov, Nucl. Phys. B193,
297共1981兲.
关30兴 J. Bockholt et al., Phys. Rev. D 54, 3641 共1996兲.
关31兴 H. Pa¨s, M. Hirsch, H.V. Klapdor-Kleingrothaus, and S.G.
Kov-alenko, Phys. Lett. B 453, 194共1999兲.
关32兴 R.N. Mohapatra and E. Takasugi, Phys. Lett. B 211, 192 共1988兲.
关33兴 P. Bamert, C. Burguess, and R.N. Mohapatra, Nucl. Phys.
B449, 25共1995兲.
关34兴 W. Rodejohann, Phys. Rev. D 62, 013011 共2000兲; H.V.
Klapdor-Kliengrothaus, H. Pa¨s, and A.Yu. Smirnov, ibid. 63, 073005共2001兲.
关35兴 M. Czakon, J. Gluza, and M. Zralek, hep-ph/0003161. 关36兴 J.C. Montero, V. Pleitez, and M.C. Rodriguez, Phys. Rev. D
58, 094026共1998兲; 58, 097505 共1998兲; Int. J. Mod. Phys. A 16, 1147共2001兲.
