PHYSICAL REVIEW D, VOLUME 60, 115003 Spontaneous breaking of a global symmetry in a 3-3-1 model 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, S.P., Brazil ~Received 16 March 1999; published 29 October 1999! 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 invisibleZ0 width and that heavy neutrinos can decay sufficiently rapid by Majoron emission, having a lifetime shorter than the age of the universe. @S0556-2821~99!02217-1# PACS number~s!: 14.80.Mz, 12.60.Fr s p ot to k sly al te e de le ht e ly he nd r ts lu s g h c ou e e ea n ou- ata e alar alar al i.e., on l I. INTRODUCTION In chiral electroweak model neutrinos can be massles any order in perturbation theory if both conditions are su plied: no right-handed neutrinos are introduced and the t lepton number is conserved. If we do not assume lep number conservation we have two possibilities: we brea by hand, i.e., explicit breaking or, we break it spontaneou The later possibility implies the existence of a pseudosc Goldstone boson called a Majoron which was first sugges in Ref. @1# where a non-Hermitian scalar singlet~singlet Ma- joron model! was introduced, and in Ref.@2# where a non- Hermitian scalar triplet was introduced~triplet Majoron model!. Since the data of the CERNe1e2 collider LEP @3# the triplet majoron model was ruled out. The original mod only considered one triplet and one doublet. In that mo the majoron is a linear combination of doublet and trip components but it is predominantly triplet. Hence, the lig est scalar (R0) has a mass which is proportional to th vacuum expectation value~VEV! of the triplet and for this reason it is very small. Once we have the decayZ0 →R0M0, whereM0 denotes the Majoron, and since the on decay of the light scalar isR0→M0M0, there is an extra contribution to theZ0-invisible width. Its contribution is ex- actly twice the contribution of a simple neutrino. Since t Higgs scalars have only weak interactions they escape u tected. Hence, any experiment that counts the numbe neutrino species by measuring theZ0-invisible width auto- matically counts five neutrinos@4,5#. There are also possibilities involving only Higgs double and charged singlet scalars but they also need to inc Dirac neutrino singlets. A minimal model of this sort wa proposed in Ref.@6# where an extra doublet scalar-carryin lepton number was added~doublet Majoron model!. The new doublet does not couple to leptons. The LEP data imply t at least a second doublet of the new type has to be introdu @7#. In this sort of model, since there are at least three d blets, the lightest scalarR0 can be assumed naturally to b heavier than theZ0 avoiding the decayZ0→R0M0. In this Majoron doublet model the lepton number violation tak place at the same scale of the electroweak symmetry br ing. It is also possible to consider a Majoron model with o 0556-2821/99/60~11!/115003~6!/$15.00 60 1150 at - al n it . ar d l l t - e- of de at ed - s k- e complex singlet, a complex triplet, and the usual SU(2) d blet @8#. In this case the Majoron can evade the LEP d since it can be mainly a singlet. II. THE MODEL Here we will consider a model with SU(3)C^ SU(3)L ^ U(1)N symmetry with both exotic quarks and only th known charged leptons@9#. In this model, in order to give mass to all fermions it is necessary to introduce three sc triplets x5S x2 x22 x0 D ;~3,21!, r5S r1 r0 r11 D ;~3,1!, h5S h0 h1 2 h2 1 D ;~3,0! ~1a! and a sextet S5S s1 0 h2 2 A2 h1 1 A2 h2 2 A2 H1 22 s2 0 A2 h1 1 A2 s2 0 A2 H2 11 D ;~6,0!. ~1b! Although we can assign a lepton number to several sc fields we prefer to use the global quantum numberF5L 1B @10#. It is clear that the model needs only one glob quantum number and not four as in the standard model, family lepton numberLi ,(i 5e,m,t with L5( iL i) and the baryonic numberB. The quantum numberF coincides with L andB for the known particles but implies the assignati of a single quantum number to the other particles. The most general gauge andF-conserving scalar potentia is ©1999 The American Physical Society03-1 J. C. MONTERO, C. A. de S. PIRES, AND V. PLEITEZ PHYSICAL REVIEW D60 115003 V5m1 2h†h1m2 2r†r1m3 2x†x1m4 2Tr~S†S!1l1~h†h!21l2~r†r!21l3~x†x!21h†h@l4r†r1l5x†x#1l6~r†r!~x†x! 1l7~h†r!~r†h!1l8~h†x!~x†h!1l9~r†x!~x†r!1l10~TrS†S!21l11Tr@~S†S!2#1Tr~S†S!@l12~h†h!1l13~x†x! 1l14~r†r!#1@l15e i jkx i~Sx†! jhk1l16e i jkr i~Sr†! jhk1l17e i jke lmnSil Sjmhnhk1H.c.#1l18x †SS†x1l19h †SS†h 1l20r †SS†r1F f 1 2 e i jkh ir jxk1 f 2 2 xTS†r1H.c.G . ~2! he tip . - pl he e th - b re s sis Terms like the quartic (x†h)(r†h), xSh†r, hSh†h,xrSS, and the trilinearhS†h,SSSdo not conserve theF quantum number~or the leptonL) and they will not be considered here. The minimum of the potential must be studied after t shifting of the neutral components of the three scalar mul lets. Hence, we redefine the neutral components in Eqs~1! as follows: h0→ 1 A2 ~vh1R1 01 i I 1 0!, r0→ 1 A2 ~vr1R2 01 i I 2 0!, x0→ 1 A2 ~vx1R3 01 i I 3 0!, ~3! and s1 0→ 1 A2 ~vs1 1R4 01 i I 4 0!, s2 0→ 1 A2 ~vs2 1R5 01 i I 5 0!, ~4! where va ~with a5h,r,x,s1 ,s2) are considered real pa rameters for the sake of simplicity. TheF number attribution is the following: F~U22!5F~V2!52F~J1!5F~J2,3!5F~r22!5F~x22! 5F~x22!5F~s1 0!5F~h2 2!5F~H1 22!5F~H2 22! 52, ~5! where J1 (J2,3) are exotic quarks of charge 5/3~24/3! present in the model and we have included them by com tion. For leptons and the known quarksF coincides with the total lepton and baryon numbers, respectively. All the ot fields haveF50. As we said before, all terms in Eq.~2! conserve theF quantum number. However, if we assum that ^s1 0&Þ0 we have the spontaneous breakdown ofF and the corresponding pseudoscalar, the MajoronM0, as we will show below@11#. In a model with several complex scalar fields, as is case of the 3-3-1 model@9#, if the lepton number is sponta neously broken one of the neutral scalars is a Goldstone son associated with the global symmetry breaking. With spect to the SU(2)L ^ U(1)Y gauge symmetry, this model ha naturally three doublets (r1,r0)T,(h0,h2)T, and (h1,s2 0)T, one triplet 11500 - e- r e o- - S s1 0 h2 2 A2 h2 2 A2 H1 22D , ~6! and two singletsx2 22 andx0. The mass matrix in the real sector in the ba (R1 0 ,R2 0 ,R3 0 ,R4 0 ,R5 0)T, is given by the symmetric matrix m115l1vh 22 l16 4A2 vr 2vs2 vh 1 1 4A2vh ~l15vxvs2 2 f 1vr!vx 1 th 2vh , m225l2vr 22 1 8A2vr ~2 f 1vh1 f 2vs2 !vx1 tr 2vr , m335l3vx 22 1 8A2vx ~2 f 1vh1 f 2vs1 !vr1 tx 2vx , m445~l101l11!vs1 2 1 ts1 2vs1 , m555S l101 l11 2 D vs2 2 2 1 4A2vs2 l16vr 2vh2 1 8vs2 ~ f 2vr 1A2l15vh!vx1 ts2 2vs2 , m125 1 2 S l4vh1 l16 A2 vs2D vr1 f 1 4A2 vx , m135 1 2 S l5vh2 l15 A2 vs2D vx1 f 1 4A2 vr , m145~l121l19! vhvs1 2 , m155 1 2 ~l1222l17!vhvs2 2 l15 4A2 vx 21 l16 4A2 vr 2 , 3-2 n - e eV w t- s E he - hich her t s a ou- vier s the nts, ) plet ve SPONTANEOUS BREAKING OF A GLOBAL SYMMETRY . . . PHYSICAL REVIEW D60 115003 m235 l6 2 vrvx1 f 1 4A2 vh1 f 2 8 vs2 , m245 l14 2 vrvs1 , m255~2l141l20! vrvs2 4 1 l16 2A2 vhvr1 f 2 8 vx , m345 l13 2 vxvs1 , m355 l13 2 vrvs2 2 1 4A2 ~l15vh2l18vs2 !vx1 f 2 8 vr , m455l10vs1 vs2 . ~7! The tadpole equationsta wherea5h,r,x,s1 ,s2 are given in the Appendix. The conditions for an extreme of the pote tial are ta50. Assuming that the matrixmi j above is diago- nalized by an orthogonal matrixO, the relation among sym metry (Ri 0) and mass (H j 0) eigenstates isRi 05O i j H j 0 ; i , j 51,2,3,4,5. The massesmH j can vary, depending on a fin tuning of the parameters, from a few GeV’s up to 2 or 3 T @typical values of the energy scale at which the break do of the SU(3)L symmetry does occur#. Also the arbitrary or- thogonal matrixO is not necessarily almost diagonal. Deno ing the lightest Higgs boson asH1 0 two extreme possibilities are compatible with the LEP data:R45(O 21)41H1 01•••, with (O 21)41!1, if mH1 ,MZ ; or R4'H1 01•••, that is (O 21)41'1, if MH1 .MZ . Intermediate values for the mas mH1 and the mixing angles have been ruled out by the L data~see below!. The symmetric mass matrix of the imaginary part in t basis (I 1 0 ,I 2 0 ,I 3 0 ,I 4 0 ,I 5 0)T reads M1152 l16 4A2 vr 2vs2 vh 1l17vs2 2 1 1 4A2 ~l15vxvh2 f 1vr! vx vh 1 th 2vh , M2252 1 8 SA2 f 1vh1 f 2vs2 vx vr D1 tr 2vr , M3352 1 8 ~A2 f 1vh1 f 2vs2 ! vr vx 1 tx 2vx , M445 ts1 2vs1 , 11500 - n P M555 1 4A2 ~l15vx 22l16vr 2! vh vs2 1l17vh 2 2 f 2 8 vrvx vs2 1 ts2 2vs2 , M1252 f 1 4A2 vx , M1352 f 1 4A2 vr , M1450, M155 l15 4A2 vh2 f 2 8 vs2 , M2352 1 8 ~A2 f 1vh1 f 2vs2 !, M2450, M255 f 2 8 vx , M3450, M355 f 2 8 vr , M4550. ~8! In Eqs.~7! and~8! when allvaÞ0,a5h,r,x,s1 ,s2 then we can useta50. In Eqs.~8! there are three Goldstone bo son. Notice, however, that sinceM4i50, the componentI 4 0 has a zero mass, i.e., it is an extra Goldstone boson w decouples in the sense that it does not mix with the ot CP-odd scalars. Hence,I 4 0 is the Majoron field. Hereafter, i will be denotedM0. The submatrix 434 has still two other Goldstone bosons which are related to the masses ofZ0 and Z80. Hence, although the Majoron in the present model i 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 c plings with fermions~charged leptons and quarks!. More- over, as we said before, the real component can be hea than theZ0. It is easy to understand this. Ifvs 1 050, the tadpole equation in Eq.~A4! must be replaced in the mas matrices in Eqs.~7! and~8!. In this case thes1 0 fields consists of two mass-degenerate fieldsR4 0 and I 4 0 with mass mR4 2 5mI 4 2 5l10vs2 2 1 1 2 ~l121l19!vh 21 l13 4 vx 21 l18 2 vr 2 . ~9! The mass in Eq.~9! can be large because it depends onvx 2 . Whenvs1 Þ0 is used, the degeneration in mass ofR4 0 andI 4 0 is broken, the imaginary part becomes the Majoron and real part has a mixing with the other real neutral compone which include several fields transforming under SU(2L ^ U(1)Y as doublets (h0,r0,s2 0), and one singlet (x0). This also happens in the one-singlet–one-doublet–one-tri model when the triplet does not gain a VEV@12#. Notice that unlike vs1 , if vs2 50 the condition in Eq.~A5! forces f 2 50. All the other VEV’s have to be nonzero in order to ha a consistent breaking of the SU(3) symmetry. 3-3 ith e s al - e r he as e ng dl ns ip ve de an w th o s si y al like e e ass e se s ino - nds en, - n, on e f e- y and cal J. C. MONTERO, C. A. de S. PIRES, AND V. PLEITEZ PHYSICAL REVIEW D60 115003 III. PHENOMENOLOGICAL CONSEQUENCES AND CONCLUSIONS In the present model the interaction of the Majoron w the Z0 ~which is of the formZ0M0H j 0), is given by O4 j ~A2GF!1/2 cW MW~pM02pj ! m, ~10! wherepM0 and pj are the momenta of the Majoron and th physical real scalarsH j 0 , respectively. We see that if it i allowed, the contribution of the decay modeZ0→H1 0M0, whereH1 0 is the lightest Higgs scalar, is 2uO41u2 times that of the neutrino antineutrino. Hence, as we said before, the v of the mixing matrix elementO4 j is constrained appropri ately: 2O 4 j 2 ;1024, to make the model consistent with th LEP data, i.e., nowGZ→H1 0M0 ~whereH1 0 is assumed the lightest scalar! could be reduced to an acceptable level. Mo interesting, however, is the fact that in this model theZ0 →H1 0M0 might be kinematically forbidden sinceH1 can be heavier than theZ0 as was discussed above. It is well known that if neutrinos are massive particles t thermal history of the universe strongly constrains the m of the stable neutrinos, i.e.,mn,100 eV for light neutrinos or a few GeV for heavy ones@13#. One of the ways in which the cosmological constraints on neutrino masses can b tered is when the lepton number is broken globally, givi rise to the Majoron field: heavy neutrinos can decay rapi by Majoron emission, thereby giving negligible contributio to the mass density of the universe@14#. Let us denote nh (n l) heavy ~light! neutrinos and look forn→n81M0 decays in the present model. Those decays, as in the tr Majoron model, are completely forbidden at the tree le too ~there is neithern→n81g nor n→3n8 decays at the lowest order!. Here we will denote, as usual,W1 the vector boson which coincides with the respective boson of the standard mo i.e., it couples to the usual charged current in the lepton the quark sectors and also satisfiesMW 2 /MZ 25cW 2 ; and V1 denotes the vector boson which couples charged leptons antineutrinos or the known quarks with the exotic ones. If lepton number is not spontaneously brokenW1 and V1 do not couple to one another. However, a mixing between b W1 andV1 arises when the lepton number is spontaneou broken. Let us consider this more in detail. In the ba (W1 V1)T the mass square matrix is given by g2 4 S A1vr 2/2 A2vs1 vs2 A2vs1 vs2 A1vx 2/2 D , ~11! whereA5(vh 21vs2 2 12vs1 2 )/2. We see from Eq.~11! that if vs1 50 there is no mixing betweenW1 and V1. The mass eigenstates are given byWi 15U i j Bj 1 , where i , j 51,2 and B1 15W1, B2 15V1, and the orthogonal matrix is given b (N is a normalization factor! 11500 ue e s al- y let l l, d ith e th ly s U5 1 N S r 2s2A4b21~r 2s!2 2b 1 r 2s1A4b21~r 2s!2 2b 1 D ' 1 As21r 2 S 2s b b sD , ~12! wherer 5vr 2/2, s5vx 2/2 andb5A2vs1 vs2 . This mixing may have interesting cosmologic consequences since there are interactions k(g2/A2) l̄ LgmncWm 2 . Notice that its strength depends on th small parameterk}vs1 vs2 /vx 2 and it can be neglected in th usual processes. In fact, the model has three different m scales sincevs1 !v i!vx with v i5vh ,vr ,vs2 @15#. It means that W1 1'W1, W2 1'V1 with MW 2 /MZ 2cW compatible with the experimental data ifvs1 <3.89 GeV@15#. However, the mixing betweenW1 and V1 is interesting once there ar new contributions to the Majoron emission. In fact, becau of the W1W2M0 vertex we have the neutrino transition (nh)L→(n l)LM0; because of the vertexV1V2M0 we have antineutrino transitions(nh)R→(n l)R . Both contributions could be negligible since they are proportional tovs1 . More interesting is that possible to have neutrino–anti-neutr transitions like the decay (nh)L→(n l)RM0 mediated by the mixing betweenW1 and V1 as shown in Fig. 1. This dia gram is ultraviolet finite in the Feynman gauge and depe quadratically on a low-energy scale that we have chos conservatively, as being themt mass. The latter process im plies a neutrino width which is, in a suitable approximatio dominated by thet lepton contribution and is given by G5 1 8p5 GF 4 uK t3u2uK t1u2mnh mt 6vs2 2 S MW MV D 4 , ~13! where we have neglected a logarithmic dependence mnh (nh can bent or nm with n l5nm ,ne in the first case and n l5ne in the second one!. With reasonable values for th masses in Eq.~13!, that ismnh '1 MeV for the case of thet neutrino,vs2 '100 GeV, andMV'400 GeV, we can get a width of the order of 10221 MeV ~up to the suppression o the mixing matrixK). The age of the universe has a corr spondent width of 10239 MeV, thus it means that the deca can have a lifetime less than the age of the universe could be of cosmological interest. From the cosmologi point of view there are also the processesnh1nh→M0 →n l1n l and n l1n l→M01M0, which occur at the tree FIG. 1. One-loop contribution to the processnL→ n̄R1M0. 3-4 ro ro is its ri- l - , al th itu tr ce ro o fo lo M u- e la t h ys e - en also ı SPONTANEOUS BREAKING OF A GLOBAL SYMMETRY . . . PHYSICAL REVIEW D60 115003 level approximation. The cosmological effect of these p cesses are the same as in Ref.@4#. If the parameters in this model are such that the Majo is irrelevant from the cosmological point of view, there still the possibility that the Majoron may be detected by influence in neutrinoless doubleb decay with Majoron emis- sion nn→e2e2M0 @denoted by (bb)0nM]. However, it needs the Majoron-neutrino couplings in the rangemn /vs1 ;102521023 in order to have the Majoron emission expe mentally observable@16#. Notice that in the present mode the accompanying 01 scalar, which is by definition the light est scalarH1 0, may not be emitted in (bb)0nM if it is a heavy scalar or it is very suppressed by the mixing factor. In our model both the usual neutrinoless (bb)0n decay and also the decay (bb)0nM have new contributions. IfF is conserved in the scalar potential orvs1 50 the mixing among singly charged scalars occurs withh1 2 and r2 and betweenh2 2 andx2. However, if we allowF-breaking terms in the scalar potential orvs1 Þ0 there is a general mixing among the scalar fields of the same charge. For instance trilinear term f 2xTS†r in the potential in Eq.~2! implies the trilinear interactionf 2h2 2h2 2x11, and since there is a gener mixing among all scalars of the same charge it means there are processes where the vector bosons are subst by scalars since the vertexh2 1e2n does exist andh2 1 mixes with all the other singly charged scalars. There are also linear contributions that arise because of the verti W2V2H1 11 and h2 2h2 2H1 11 as in Refs.@17,18#. There is also the vertexh2 2h2 1M0 contributing to the (bb)0nM . It seems that the analysis of both (bb)0n and (bb)0nM decays is more complicated than those considered in Refs.@15,19#. There are also phenomenological constraints on Majo models coming from a search in the laboratory of flav changing currents likem→e1M0 @20# or in astrophysics through processes likeg1e→e1M0 which contributes to the energy-loss mechanism of stars@4#. However, in the present model the Majoron couples only to neutrinos, quarks and electrons the couplings arise only at the one- level. Hence, all these processes do not constrain the joron couplings at all~at the lowest order!. The interaction of the Majoron with neutrinos is diagonal in flavor. The co pling between the Majoron and the real scalar fieldH j 0 , of the formM0M0H j 0 , is iO4 j~l101l11! vs1 2 , ~14! which is a small coupling. Note that since the Majoron d couples from the other imaginary parts of the neutral sca there are no trilinear couplings likeM0A0H j 0 , whereA0 de- notes a massive pseudoscalar, hence the model does no the phenomenological consequences in accelerator ph as the seesaw Majoron model does@12#. Finally, we remark that here we have assumed that th is no spontaneousCP violation. Hence, all vacuum expecta tion values are real. If we allow complex VEV it has be shown thatCP is violated spontaneously@21#. If this is the 11500 - n the at ted i- s n r r op a- - rs ave ics re case, we have a mixing among all the scalars fields and the majoron mixes with all the otherCP-even andCP-odd scalars. ACKNOWLEDGMENTS This work was supported by Fundac¸ão de Amparo a` Pes- quisa do Estado de Sa˜o Paulo~FAPESP!, Conselho Nacional de Ciência e Tecnologia~CNPq! and by Programa de Apoio a Núcleos de Exceleˆncia ~PRONEX!. C.P. would like to thank Coordenadoria de Aperfeic¸oamento de Pessoal de N´ vel Superior~CAPES! for financial support. APPENDIX: CONSTRAINT EQUATIONS th5m1 2vh1l1vh 31 l4 2 vr 2vh1 l5 2 vx 2vh1 l12 2 ~vs1 2 1vs2 2 !vh 2l17vs2 2 vh1 l19 2 vs1 2 vh2 l15 2A2 vx 2vs2 1 l16 2A2 vr 2vs2 1 f 1 2A2 vrvx , ~A1! tr5m2 2vr1l2vr 31 l4 2 vh 2vr1 l6 2 vx 2vr1 l14 2 ~vs1 2 1vs2 2 !vr 1 l16 A2 vs2 vhvr1 l20 4 vs2 2 vr1 f 1 2A2 vhvx1 f 2 4 vs2 vx , ~A2! tx5m3 2vx1l3vx 21 l5 2 vh 2vx1 l6 2 vr 2vx1 l13 4 ~vs1 2 1vs2 2 !vx 2 l15 A2 vs2 vhvx1 l18 4 vs2 2 vx1 f 1 2A2 vhvr1 f 2 4 vs2 vr , ~A3! ts1 5m4 2vs1 1l10~vs1 2 1vs2 2 !vs1 1l11vs1 3 1 l12 2 vh 2vs1 1 l13 4 vx 2vs1 1 l14 2 vrvs1 1 l19 2 vh 2vs1 , ~A4! ts2 5m4 2vs2 1l10~vs2 2 1vs1 2 !vs2 1 l11 2 vs2 3 1 l12 2 vh 2vs2 1 l13 2 vx 2vs2 1 l14 2 vr 2vs2 2l17vh 2vs2 l18 4 vx 2vs2 1 l20 4 vr 2vs2 2 l15 2A2 vx 2vh1 l16 2A2 vr 2vh1 f 2 4 vrvx . ~A5! 3-5 et C as . ev. . D s. J. C. MONTERO, C. A. de S. PIRES, AND V. PLEITEZ PHYSICAL REVIEW D60 115003 @1# Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, Phys. L 98B, 265 ~1981!. @2# G. B. Gelmini and M. Roncadelli, Phys. 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