3-3-1-1 model for dark matter
33-3-1-1 model for dark matter
P. V. Dong ∗ and T. D. Tham † Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam
H. T. Hung ‡ Department of Physics, Hanoi University of Education 2, Phuc Yen, Vinh Phuc, Vietnam (Dated: November 5, 2018)We show that the SU (3) C ⊗ SU (3) L ⊗ U (1) X (3-3-1) model of strong and electroweakinteractions can naturally accommodate an extra U (1) N symmetry behaving as a gaugesymmetry. Resulting theory based on SU (3) C ⊗ SU (3) L ⊗ U (1) X ⊗ U (1) N (3-3-1-1) gaugesymmetry realizes B − L = − (2 / √ T + N as a charge of SU (3) L ⊗ U (1) N . Consequently,a residual symmetry, W -parity, resulting from broken B − L in similarity to R -parity insupersymmetry is always conserved and may be unbroken. There is a specific fermion con-tent recently studied in which all new particles that have wrong lepton-numbers are oddunder W -parity, while the standard model particles are even. Therefore, the lightest wrong-lepton particle (LWP) responsible for dark matter is naturally stabilized. We explicitly showthat the non-Hermitian neutral gauge boson ( X ) as LWP cannot be a dark matter. How-ever, the LWP as a new neutral fermion ( N R ) can be dark matter if its mass is in range1 . ≤ m N R ≤ . Z (cid:48) ) mass satisfies2 . ≤ m Z (cid:48) ≤ . H (cid:48) (cid:39) η ) which hastraditionally been studied is only stabilized by W -parity. All the unwanted interactions andvacuums as often encountered in the 3-3-1 model are naturally suppressed. And, the standingissues on tree-level flavor changing neutral currents and CPT violation also disappear. PACS numbers: 12.60.-i, 14.60.St, 95.35.+d
I. INTRODUCTION
One of obvious experimental evidences that we must go beyond the standard model of funda-mental particles and interactions is neutrino oscillations [1], which mean that the neutrinos havehierarchical, small masses and mixing. Among the extensions known, the seesaw mechanism [2] ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ h e p - ph ] M a y is perhaps the most natural one for explaining the above problem with the introduction of heavyright-handed neutrinos ( ν R ) or some kind of new neutral fermions ( N R ). However, while theseassumed particles have been not observed it is useful to ask what is their natural origin. They mayarise as fundamental constituents in left-right models [3] or SO (10) unification [4]. The presenceof these particles might also lead to interesting consequences such as the baryon asymmetry vialeptogenesis [5]. In this work, we will show that they can also exist in a gauge model implying toa class of new particles, odd under a parity symmetry responsible for dark matter.The approach is based on SU (3) C ⊗ SU (3) L ⊗ U (1) X gauge symmetry (thus named 3-3-1) inwhich the last two groups are extended from the electroweak symmetry of the standard model,while the QCD symmetry is retained. The right-handed neutrinos or new neutral fermions mayconstitute in fundamental lepton triplets/antitriplets of SU (3) L to complete the representations, ν L e L ν cR or ν L e L N cR , so-called the 3-3-1 model with right-handed neutrinos [6] or the 3-3-1 model with neutralfermions [7], respectively (see also [8] for a variant). In addition, this approach has intriguingfeatures. The number of fermion families must be an integral multiple of fundamental color num-ber, which is three, in order to cancel SU (3) L anomaly [9]. There are nine flavors of quarks due tothe enlarged electroweak gauge symmetry, so the family number must also be smaller than or equalto five to ensure QCD asymptotic freedom. All these result in an exact family number of three,coinciding with the observation [1]. Since the third family of quarks transforms under SU (3) L differently from the first two, this can explain why top quark is so heavy [10]. The extension canalso provide some insights of electric charge quantization observed in the nature [11].The extended sectors from the standard model in the 3-3-1 models such as scalar, fermionand gauge might by themselves provide dark matter candidates. It is strongly approved by thegauge interactions, minimal Yukawa Lagrangian and scalar potential that normally couple thenew concerned particles (similar to the so-called wrong-lepton particles as defined in the followingsection) in pairs, in similarity to the case of superparticles in supersymmetry. This is automaticallyresulted from the specific structure of 3-3-1 gauge symmetry by itself [6, 8], which is unlike theconclusion in [12, 13]. The first attempts in identifying the dark matter candidates of 3-3-1 modelshave been previously expressed in [14, 15], however a strict treatment on their stability issue andrelic abundance has been not given. The stabilization of dark matter in the 3-3-1 models due toextra symmetries has been firstly discussed in [12, 13]. To this aim, the lepton number symmetryhas been imposed in [12] so that the lightest bilepton particle could be stabilized. It is noteworthythat all the unwanted interactions of Yukawa Lagrangian and scalar potential (other than theminimal interactions mentioned) explicitly violate the lepton number [16] which are naturallysuppressed due to this symmetry (except the coupling of two lepton and one scalar triplets thatviolates only flavor lepton numbers, but leading to an unrealistic neutrino mass spectrum; however,in our model discussed below this coupling explicitly disappears due to the total lepton numberviolation). The Z as introduced therein can be in fact not needed. An alternative problemencountered is that the lepton number should be also violated due to five-dimensional effectiveinteractions responsible for the neutrino masses as cited therein.In [13], the bilepton character of new particles has been arranged to be lost, and the leptonnumber symmetry was no longer to prevent those unwanted interactions from turning on. So, a Z symmetry has been included by hand with appropriate Z representation assignments in orderto eliminate those unwanted interactions, and this symmetry has been regarded as the one forstabilizing the dark matter [13]. However, since the Z that acts on the model multiplets must bespontaneously broken by the Higgs vacuum, there is no reason why the scalar dark matter whichcarries no lepton number cannot develop a VEV and decay then. Also in [13], a continue symmetrycalled U (1) G that acts on component particles, not commuting with the gauge symmetry like thelepton charge before, has been introduced to be equivalently used instead of the Z in interpretingthe dark matter, getting some reason from the gauge interactions. Let us remind the reader thatthese interactions of gauge bosons with fermions, scalars or gauge self-interactions are automaticconsequences of and already restricted by the gauge symmetry by itself as mentioned. Theyalways present, not excluded or added by other interactions in any cases that the U (1) G or evenlepton charge is imposed or not. For the purpose in suppressing those unwanted interactions andvacuums, obviously there are many other symmetries behaving like U (1) G or lepton charge foundout as respective solutions of the gauge interactions’ conservation. However, all these continuesymmetries can face problems below apart from their nature of presence.The continue symmetries above must be supposed as exact symmetries responsible for the darkmatter stability. Therefore, they can be naturally regarded as respective residual symmetries ofhigher symmetries that span the 3-3-1 group (since they do not commute with the gauge symmetryas mentioned) acting at the Lagrangian level, under which the unwanted interactions are explicitlysuppressed. In other words, the minimal Lagrangian of the theory actually contain larger sym-metries spanning the gauge symmetry that shall be spontaneously broken down to those residualsymmetries, respectively. As a specific property of the 3-3-1 models, the lepton charge (or evenany kind of U (1) G if one independently includes, neglected of the lepton charge symmetry) shouldwork as the residual gauge symmetry of some higher symmetry mentioned (as shown in the nextsection) and must be spontaneously broken so that the resulting gauge boson gets a large enoughmass to make a consistency of the theory. On the other hand, this lepton charge symmetry or evena general continue symmetry mentioned is also known to be actually violated due to its anomalies.Therefore, such symmetries (lepton number or U (1) G ) would be no longer to protect the darkmatter stability from decays. For the stability issue of dark matter, similarly to the R -parity insupersymmetry it is more natural to search for an exact and unbroken residual discrete symmetryof some anomaly-free continue symmetry spanning either the lepton number and other necessarysymmetries such as baryon charge or some kind like U (1) G . Let us remark that among the continuesymmetries analyzed, the one concerning lepton charge is perhaps the most motivative and naturalbecause of the followings: (i) all the unwanted interactions in ordinary 3-3-1 models which shouldbe prevented in fact violate the lepton numbers [16]; (ii) while the discrete symmetry is conservedresponsible for dark matter stability, the lepton or baryon numbers could be broken in several waysnecessarily to account for the observed neutrino masses and baryon-number asymmetry. By theway, we will see that this is similar to enlarge the SU (5) theory to SO (10) in which the B − L charge is naturally gauged. In this work the lepton charge will be taken into account which isdifferent from the U (1) G symmetry ad hoc input.By investigating of nontrivial lepton number behavior and a resulting W -parity (similar to the R -parity in supersymmetry) in a specific 3-3-1 model [7], we show that the theory can contain naturaldark matter candidates. For details, we consider the 3-3-1 model with neutral fermions ( N R ) whichis different from the model of [13]. These neutral fermions possess no lepton number as alreadystudied before in a TeV seesaw extension of the standard model [17] and in the 3-3-1 model withflavor symmetries [7]. We investigate lepton number symmetry, its dynamics and other symmetrieswhich result in a new 3-3-1-1 gauge model. We show that there is an unbroken residual symmetryof such (anomaly-free) 3-3-1-1 theory behaving like the R -parity in supersymmetry under whichalmost the new particles as given are odd. It is interesting that the model can contain several kindsof dark matter such as singlet scalar, fermion and gauge boson as often presented in other extensionsof the standard model and similarity to the conclusion in [13]. However, these candidates may beheavy as some TeV which is unlike light candidates in the standard model familiar extensions.Before [13] and our work, the previous considerations of the 3-3-1 model dark matters recognizedonly scalar singlet [15] and lightest supersymmetric particles of respective supersymmetric 3-3-1versions. The reason behind this may be that the mentioned symmetries such as 3-3-1-1 and W -parity under which the dark matters are dynamically stabilized have been not explored yet. Thedark matter phenomenologies in our models will be different from the other extensions. The modelcan work better under the experimental constraints than the ordinary 3-3-1 model with right-handed neutrinos due to the W -parity mentioned. The above procedure will fail when applying forthe other 3-3-1 models such as the model of [13], the 3-3-1 model with right-handed neutrinos [6]and the minimal 3-3-1 model [8]. In fact, all the particles including the new ones in those modelswould transform trivially under the W -parity. Therefore, this parity may be only realized in theclass of 3-3-1 models with flavor symmetries [7].The rest of this article is organized as follows. In Sec. II we give a review of the 3-3-1 modelwith neutral fermions by stressing on baryon and lepton numbers as well as proposing of wrong-lepton particles. We next construct the 3-3-1-1 gauge symmetry, W -parity, and showing possibledark matter candidates and its direct consequences. We also identify physical scalar particles andgiving a discussion of all the masses. In Sec. III we provide detailed calculations of relic densitiesof possible dark matters and showing constraints. Finally, we summarize our results and makingoutlooks in the last section – Sec. IV. II. PROPOSAL OF 3-3-1-1 MODELA. The 3-3-1 model with neutral fermions and wrong-lepton particles
The gauge symmetry of the 3-3-1 model under consideration is given by SU (3) C ⊗ SU (3) L ⊗ U (1) X , where the first factor is the usual QCD symmetry while the last two SU (3) L ⊗ U (1) X areextended from the electroweak symmetry of the standard model. The electric charge operator is theonly unbroken residual charge of the gauge symmetry and being defined by Q = T − (1 / √ T + X ,where T i ( i = 1 , , , ...,
8) are the SU (3) L charges while X is that of U (1) X . The weak hyperchargeof the standard model is therefore identified as Y = − (1 / √ T + X .The fermion content which is anomaly free is assigned by ψ aL = ν aL e aL ( N aR ) c ∼ (1 , , − / , e aR ∼ (1 , , − , (1) Q αL = d αL − u αL D αL ∼ (3 , ∗ , , Q L = u L d L U L ∼ (3 , , / , (2) u aR ∼ (3 , , / , d aR ∼ (3 , , − / , (3) U R ∼ (3 , , / , D αR ∼ (3 , , − / , (4)where a = 1 , , α = 1 , SU (3) C , SU (3) L , U (1) X ) gauge symmetries, respectively. The N aR and U, D α are the new neutral fermions (which are singlets under the standard model symmetry as the right-handed neutrinos often considered) and exotic quarks, respectively. The electric charges of exoticquarks Q ( U ) = 2 / Q ( D α ) = − / N aR will be taken to be zero: L ( N aR ) = 0. This is due to the fact that the conventionalseesaw mechanism with right-handed neutrinos including that of the 3-3-1 model can require avery high seesaw scale of M R ∼ − GeV [2, 18]. As already shown in [7, 17], if such N R supposed one can have a natural TeV seesaw scale matching the 3-3-1 breaking scale. And, thelepton mixing matrix under flavor symmetries can be naturally explained in those models [7, 19].The presence of N R also implies to a kind of new particles that is odd under a parity symmetry,well-motivated responsible for dark matter candidates as shown below.To break the gauge symmetry and generating the masses, this kind of the 3-3-1 model actuallyrequires three scalar triplets [6] ρ = ρ +1 ρ ρ +3 ∼ (1 , , / , (5) η = η η − η ∼ (1 , , − / , (6) χ = χ χ − χ ∼ (1 , , − / . (7)Here, the gauge symmetry is broken via two stages: the first stage SU (3) L ⊗ U (1) X is broken downto that of the standard model generating the masses of new particles such as exotic quarks U , D α aswell as the new gauge bosons: one neutral Z (cid:48) coupled to the generator that is orthogonal to the weakhypercharge and two charged X / ∗ , Y ∓ corresponding to T ± iT and T ± iT raising/loweringoperators. In the second stage the standard model electroweak symmetry is broken down to U (1) Q responsible for the masses of ordinary particles such as W ± , Z , e a , u a , and d a .The lepton number ( L ) of lepton triplet components is given by (+1 , +1 ,
0) which does notcommute with the SU (3) L gauge symmetry unlike the standard model case:[ L, T ± iT ] = ± ( T ± iT ) (cid:54) = 0 , [ L, T ± iT ] = ± ( T ± iT ) (cid:54) = 0 , (8)which mean that X and Y bosons carry lepton numbers with one unit. This also happens forthe 3-3-1 model with right-handed neutrinos and the minimal 3-3-1 model. It is a characteristicproperty of this kind of the model. Hence, if the lepton number is a symmetry of the theory, itcan be regarded as a residual charge of conserved symmetries, G ≡ SU (3) L ⊗ U (1) L , (9)where the second factor is a new symmetry supposed because the lepton number and the gaugesymmetry generators do not form a closed algebra. [This is because in order for L to be somegenerator of SU (3) L , the L charges of a complete multiplet must add up to zero which is notcorrect. Also, the L and X charges must be distinct because for the SU (3) L fermion singlets,the lepton number and electrical charge generally do not coincide]. The lepton number that is acombination of SU (3) L ⊗ U (1) L diagonal generators (due to the conservation of lepton number asstrictly required by the experiments, in similarity to the case of electric charge operator) can beeasily obtained by acting it on a lepton triplet to be given by L = 2 √ T + L , (10)where the T is the charge of SU (3) L while the L is that of U (1) L [16]. On the grounds of knownlepton numbers, the L charges of all the model multiplets can be easily obtained as given in Table I(notice that the lepton numbers of χ , ρ and η must be zero since these fields are normally requiredto develop VEVs responsible for the gauge symmetry breaking and mass generation). Moreover, it Multiplet ψ aL e aR Q L Q αL u aR d aR U R D αR ρ η χ L −
13 13 − − −
13 23
TABLE I: The L -charge of model multiplets. is able to point out that all the ordinary interactions of the theory (i.e. the minimal interactionsas mentioned) conserve L [16]. For a convenience in reading, we give also the lepton numbersof model particles L in Table II. We see that the standard model particles have lepton numbersas usual. However, almost the new particles such as N R , U , D , X , Y , ρ , η , and χ , possess Particle ν aL e a N aR u a d a U D α ρ +1 ρ ρ +3 η η − η χ χ − χ γ W Z Z (cid:48) X Y − L − − − unnormal lepton numbers in comparison to those of the standard model nature. For example,should L ( U, D ) = 0 like ordinary quarks instead of ±
1. This kind of particles is going to be namedas “wrong-lepton particles” or sometimes “ W -particles” for short.In this work, we suppose that the G symmetry and thus U (1) L responsible for the lepton numberis an exact symmetry. However, since the scalar triplets as given are all charged under G and willget VEVs, the G symmetry must be broken spontaneously (in accompany with the gauge symmetrybreaking). It is also easily shown that the scalar potential can be stabilized by the following solutionof the potential minimization conditions: (cid:104) χ (cid:105) = (cid:104) η (cid:105) = 0 , (11) (cid:104) ρ (cid:105) (cid:54) = 0 , (cid:104) η (cid:105) (cid:54) = 0 , (cid:104) χ (cid:105) (cid:54) = 0 . (12)In fact, this solution of the potential minimization has been formally used in the literature as astandard vacuum structure [6]. Also, with the VEVs as given in (11) and (12), i.e. (cid:104) ρ (cid:105) = 1 √ , v, T , (cid:104) η (cid:105) = 1 √ u, , T , (cid:104) χ (cid:105) = 1 √ , , ω ) T , (13)all the particles in this model (except for ν L and N R ) will get consistent masses at the tree levelin similarity to those of the ordinary 3-3-1 model with right-handed neutrinos [6]. Let us notethat ω is responsible for the first stage of electroweak symmetry breaking SU (3) L ⊗ U (1) X −→ SU (2) L ⊗ U (1) Y providing the masses for new particles, whereas u, v act on the second stage SU (2) L ⊗ U (1) Y −→ U (1) Q giving the masses for ordinary particles. To keep a consistency withthe standard model, we should suppose u , v (cid:28) ω . (14)For the G symmetry, although both the L and T (and all other generators) are broken, thecombination of lepton number L in this case is conserved by the VEVs which can be verifieddirectly from Table II. The brokendown of the G symmetry into the lepton number G = SU (3) L ⊗ U (1) L −→ U (1) L (15)implies the existence of eight Goldstone bosons contained in the scalar sector ρ, η, χ . However,these Goldstone bosons are just those associated with the gauge symmetry breaking SU (3) L ⊗ U (1) X −→ U (1) Q that will be subsequently gauged away (they are unphysical because they arealready the Goldstone bosons of the gauge symmetries as stated).Moreover, by a similar ingredient the baryon number ( B ) may be found to be not commutedwith the gauge symmetry as well as being resulted from some broken exact symmetries, G (cid:48) ≡ SU (3) L ⊗ U (1) B −→ U (1) B , (16)because the baryon numbers of U, D α are unknown and in principle could be arbitrary (in thiscase the unwanted interactions also violate the baryon number and being suppressed due to the B charge conservation). The χ and η will carry the baryon number in this case which can onlybe conserved by the above potential minimization condition. For a simplicity, in this work let ustake B ( U ) = B ( D α ) = 1 / B ( u a ) = B ( d a ) and vanishing for other fields as actually used in theliterature for this kind of the model [16] so that B = B (17)of the theory commuting with the gauge symmetry and being always conserved at the renormal-izable level as in the standard model (i.e., the baryon number of our general theory is always anexact and unbroken symmetry since the unwanted interactions conserve B while the χ and η are neutral under this charge). The B charges of model multiplets are given by Table III. [Let us Multiplet ψ aL e aR Q L Q αL u aR d aR U R D αR ρ η χ B
13 13 13 13 13 13 B -charge of model multiplets. remark on alternative cases: (i) If there was (cid:104) χ (cid:105) (cid:54) = 0 or (cid:104) η (cid:105) (cid:54) = 0, the L would be broken too,along with T and L . (ii) The conservation of L in this model is not an automatic consequence ofthe theory like the standard model. This is because if the U (1) L symmetry was not imposed therewould be the unwanted interactions explicitly violating L as actually seen in the Yukawa sectorand/or scalar sector [16, 18]. (iii) If the baryon numbers of U, D α were alternatively chosen, the(i) and (ii) would also apply for the baryon number.]The above ingredients of lepton and baryon numbers have been presented only for the 3-3-1model with neutral fermions. In general, it can be also applied for the minimal 3-3-1 model [8]0and 3-3-1 model with right-handed neutrinos [6]. Here the crucial discrimination is that in thesemodels wrong-lepton particles differ from the ordinary ones by two units in lepton charge whichhave been also called as bilepton particles, whereas in the present model it differs only one unitdue to a possible nature of the neutral fermions N R . However, they are completely distinguishedwhen replying to the dark matter problems as shown below. B. 3-3-1-1 gauge symmetry and W -parity Let us recall that the lepton numbers L and L which satisfy (10) were primarily introducedin other 3-3-1 models [16], however their dynamical nature has been completely not realized andexamined yet. In the second article of [7], we have given the first notes on this lepton dynamics.And, in the current work it is to be analyzed in more details. It is the fact that since the T is agauged charge of the SU (3) L symmetry the L , thus the L , and vise versa must be subsequentlygauged. This is because by a contrast assumption that both the L and L are global generators,the T ∼ L − L is also global which is incorrect. In this case, the anomalies coupled to L , thusto L , are obviously unable to suppress, which spoils the model’s consistency. To regard the leptonnumber as such a local symmetry for this kind of the model (or leptonic dynamics is considered)we must deal with the leptonic anomaly cancellation issue. All this also applies for the baryonnumber if it satisfies (iii); however it does not by our choice.A solution to canceling the leptonic anomalies is that we can consider the model with gaugedsymmetry N ≡ B − L (where B is the baryon number as given above) as well as introducing threenew right-handed neutrinos transforming as singlets under the 3-3-1 symmetry, ν aR ∼ (1 , , . (18)Here, these particles which have lepton and baryon numbers as usual, L ( ν aR ) = L ( ν aR ) = 1and B ( ν aR ) = B ( ν aR ) = 0, are necessarily included in order to cancel the gravitational anomaly[Gravity] U (1) N (since this anomaly of ν L and N R is not cancelled out). It is explicitly checkedthat the resulting theory is free from all the anomalies as presented in Appendix A. Hence, the newtheory as an important investigation of this article is obtained by the following gauge symmetry: SU (3) C ⊗ SU (3) L ⊗ U (1) X ⊗ U (1) N , (19)so called the 3-3-1-1 model. And, the multiplets of the 3-3-1-1 model as well as their N -chargescan be easily counted to be given in Table IV. Here, the complex scalar 3-3-1 singlet, φ ∼ (1 , , , (20)1with B ( φ ) = B ( φ ) = 0 , L ( φ ) = L ( φ ) = − η, ρ, χ for breaking the3-3-1-1 symmetry and generating the masses in a correct way. Let us remind the reader that the Multiplet ψ aL e aR ν aR Q L Q αL u aR d aR U R D αR ρ η χ φN = B − L − − −
13 13 43 −
23 13 13 − N -charges. B − L gauge charge, which can be directly derived from (10) and (17) as follows B − L = − √ T + N, (21)is a residual symmetry of SU (3) L ⊗ U (1) N that does not commute with the 3-3-1 gauge symmetryin similarity to the lepton charge L . The extension from the 3-3-1 gauge symmetry to the new3-3-1-1 gauge symmetry which must also apply for the ordinary 3-3-1 models that respect thelepton number symmetry by this view is very intriguing and quite similar as enlarging the SU (5)theory to SO (10) in which the B − L charge is naturally gauged.While this possibility of a phenomenological 3-3-1-1 model is interesting and worth exploringto be published elsewhere [20], in this work we will focus on only its consequence of a discreteresidual symmetry responsible for the dark matter stabilization as shown below. Therefore, theleptonic and baryonic dynamics as associated with the new gauge charge N will be neglected. Thelepton number will be understood as a consequence of the charge conservation associated with G = SU (3) L ⊗ U (1) L symmetry in which the first factor is a global version of the gauge symmetry.[Namely, in calculating lepton numbers all SU (3) L global quantum numbers for model multipletsare the same gauged ones. And, both T and L in case responsible for the lepton number will betaken as global charges and not gauged, which should be not confused to the similar ones of the SU (3) L ⊗ U (1) X gauge symmetry.] Similarly, the baryon number B will be regarded as an ordinaryglobal charge. Since the general theory is always conserved by the baryon number, talking about L or L is equivalent to the N charge which should be understood in the following discussions.In other words, the 3-3-1 model with neutral fermions and L -charge (plus the new right-handedneutrinos and scalar singlet) is also understood as the 3-3-1-1 model in which the gauge interactionsor dynamics as associated with N -charge (thus B and L ) is omitted in this work.Although the U (1) L and U (1) B symmetries have been imposed and L, B being conserved bythe VEVs of η, ρ, χ as given, it is evident that
B, L should be broken in some way in order toaccount for the matter-antimatter asymmetry of the universe and even neutrino masses included2later. On the other hand, as stated the nature of L in this model is a gauged charge since it is aresult from the T . The theory with L gauged simplest takes N = B − L into account since thisnew charge is necessarily independent of the anomalies, and the complete brokendown of the N charge must be achieved so that its gauge boson Z N gets a large enough mass to escape from thecurrent detectors. All these can be achieved by the scalar singlet φ when it develops a VEV, (cid:104) φ (cid:105) = 1 √ . (22)Therefore, we will assume that the matter parity (quite similar to the MSSM case), a residualdiscrete symmetry of broken B − L = − (2 / √ T + N [or SU (3) L ⊗ U (1) N ], thus R -parity whenincluded spin is an exact and unbroken symmetry of the 3-3-1-1 theory: P = ( − B − L )+2 s = ( − − √ T +3 N +2 s , (23)which still conserves all the vacuum structures above. (For a detailed proof, see Appendix B.) The R -parity of model particles is given in Table V. We see that all the particles having unusual (un- +1 (ordinary or bilepton particle) ν L e u d γ W Z ρ , η , χ φ ν R Z (cid:48) Z N − N R U D ρ η χ , X Y
TABLE V: The R -parity of 3-3-1-1 model particles that separates wrong-lepton particles from ordinaryparticles. Here the family indices for fermions have been suppressed and should be understood. normal) characteristic lepton-number differing from the ordinary one by one unit, e.g. L ( N R ) = 0, L ( U ) = − L ( X ) = 1, L ( ρ ) = −
1, as already named wrong-lepton particles, are odd. Otherwisethe ordinary particles such as the standard model particles or new particles that remain theirwould-be-ordinary properties of the lepton number (or differing from the ordinary ones by an evenlepton number as φ due to the parity symmetry) are even. It is remarkable that the R -parity ( P )which originates in the 3-3-1-1 gauge symmetry is a natural symmetry of wrong-lepton particlesin this model. The lightest wrong-lepton particle (LWP) within the odd ones is stable and ableto contribute to dark matter since this parity is exact, not broken by the VEVs. Simultaneously,as mentioned we can have several violations of L or B (one example is that L = ± R -parity odd particles, wrong-lepton particles, evenin nonsupersymmetric theories like ours. This is due to a possible property of neutral fermions L ( N R ) = 0 as also implemented by a class of 3-3-1 models with flavor symmetries [7]. By contrast,3every particle in the 3-3-1 model with right-handed neutrinos L ( ν R ) = 1 and the minimal 3-3-1model is even under the parity.Also, it can be explicitly pointed out that in the interactions of theory all the odd particles areonly coupled in pairs, so linked to ordinary particles of the standard model due to the 3-3-1 gaugesymmetry, the U (1) L symmetry and the vacuum respecting R -parity (i.e. the 3-3-1-1 symmetrywith conserved R -parity). Let us show now for examples and consequences (the scalar potentialalso possessing these properties to be expressed latter):1. Yukawa sector L Y = h eab ¯ ψ aL ρe bR + h νab ¯ ψ aL ην bR + h (cid:48) νab ¯ ν caR ν bR φ + h U ¯ Q L χU R + h Dαβ ¯ Q αL χ ∗ D βR + h ua ¯ Q L ηu aR + h da ¯ Q L ρd aR + h dαa ¯ Q αL η ∗ d aR + h uαa ¯ Q αL ρ ∗ u aR + H.c. (24)We see the odd scalars ρ , η and χ , do not interact with ordinary fermions which onlycouple to eN , νN , uU , dU , dD , uD with the type of an even-odd particle pair due tothe 3-3-1 symmetry. There are not L -charge violating similar interactions (which lead toviolations of R -parity) such as ¯ ψ aL χν bR , ¯ ψ caL ψ bL ρ , ¯ Q L χu aR , ¯ Q αL χ ∗ d aR , ¯ Q L ηU R and soforth. In addition, since R -parity is conserved the VEVs of η and χ vanish. Due to theseconditions, the ordinary quarks and exotic quarks do not mix which means that the flavorchanging neutral currents at the tree level disappear. The 3-3-1 model with right-handedneutrinos as often considered [6] is strictly improved by this parity. It is also noted that N R do not mix with ν L and ν R due to the parity symmetry.2. Gauge boson sector: The odd gauge bosons X, Y do not couple to the standard model gaugeboson pairs also, except for a similar type as mentioned such as
W X , W Y or other types as W - W -odd-odd, etc due to the 3-3-1 symmetry. This can be verified directly from [21]. Dueto the R -parity symmetry, the neutral gauge boson X cannot mix with Z and Z (cid:48) bosons.The CPT violation at the tree level as stated in [22] is suppressed. Again, constraints onthe 3-3-1 model with right-handed neutrinos [6] are improved by the parity.We notice that in the MSSM, the spins or angular-momenta of component particles withina supermultiplet do not commute with supersymmetry (comparing to our case in an alternativescenario, the lepton number is not commuted with the gauge symmetry). However, the resid-ual R -parity of spin, lepton and baryon numbers (which must also be not commuted with the4supersymmetry) is conserved and unbroken, even though the conservation of its global symme-try (that spans such spin, lepton and baryon numbers known as R -symmetry) is actually brokenalong with the supersymmetry breaking. It is also emphasized that the lepton and baryon numberconservation of the MSSM superpotential is not an automatic consequence of the theory at therenormalizable level unlike the standard model, which must be an assumption in similarity to ourcase with the U (1) L symmetry associated with the lepton number. Our R -parity obviously has adifferent origin of the 3-3-1-1 gauge symmetry as mentioned. This is due to the nature of the leptoncharge nontrivially resided in SU (3) L ⊗ U (1) L , baryon charge U (1) B and spin uniform for bothkinds of respective particles (ordinary and W -particles), instead of those in the MSSM. Particu-larly, the G symmetry or U (1) L of lepton number is broken due to the gauge symmetry breaking.The L -symmetry conservation for this model can be also protected by the R -parity instead. Thebreaking of N = B − L gauge symmetry can happen just above TeV scale or at a very high scale.Therefore, to mark such a different origin we are going to call the R -parity in this model to be W -parity where W means the item “wrong-lepton” as already pointed out.Depending on the parameter space of the model, the LWP may belong to the nature of a vectorparticle ( X ), scalar ( χ or η ), or fermion ( N R ), which must be considered here to be electrically-neutral if they are expected to contribute the dark matter. Before considering those cases detailedin the next section, let us take some comments on the scalar particle identifications and the massesof particles with stressing on those of the wrong-lepton particles. C. Scalar potential and masses
If the scalar singlet φ which is responsible for completely breaking U (1) N as mentioned lives inthe same scale of 3-3-1 breaking (Λ ∼ ω ), it will couple to ordinary scalars η, ρ, χ via the scalarpotential. And, the phenomenologies associated with broken B − L symmetry via Z N happensimultaneously with the new physics of the 3-3-1 model just in TeV scale. This possibility is veryinteresting to be studied [20]. Otherwise, the φ should be very massive that can be integrated outfrom the low-energy effective potential of η, ρ, χ . Also, the Z N will be decoupled from the gaugeboson spectrum. This is the case considered in this work. It is to be noted that the behavior of W -parity by both cases is unchanged. For the φ , we can expand φ = 1 √ R + iI ) . (25)5The imaginary part ( I ) of φ is just the Goldstone boson of Z N , while the real part ( R ) is a newphysical Higgs boson carrying a lepton number of two units and being W -parity even. The massof Z N is proportional to the VEV Λ of φ scalar.At the low-energy, the scalar potential of η, ρ, χ after integrating out the φ that must conservethe 3-3-1 symmetry and W -parity is given by V = µ ρ † ρ + µ χ † χ + µ η † η + λ ( ρ † ρ ) + λ ( χ † χ ) + λ ( η † η ) + λ ( ρ † ρ )( χ † χ ) + λ ( ρ † ρ )( η † η ) + λ ( χ † χ )( η † η )+ λ ( ρ † χ )( χ † ρ ) + λ ( ρ † η )( η † ρ ) + λ ( χ † η )( η † χ )+( f (cid:15) mnp η m ρ n χ p + H.c. ) , (26)where µ , , and f have mass dimension whereas λ , , ,..., are dimensionless. The unwanted termssuch as η † χ , ( η † χ ) , ( ρ † ρ )( η † χ ) and so on which violate L (or L ) are prevented due to the paritysymmetry. Let us note that the f coupling conserves all the natural symmetries of the theory asimposed and there is no reason why it is not presented. In the literature, it was ordinary excluded[23] (see also the first one of [15]). Therefore, it is needed to clarify that its presence makes all theextra Higgs bosons massive, reasonably leading to a phenomenologically consistent scalar spectrumas shown below. For partial solutions of the potential minimization and scalar spectrum, let uscall for the reader’s attention to [12, 13].To identify the scalar particles, let us expand the neutral fields (and the conservation of W -paritymust be maintained as determined above): ρ = ρ +11 √ ( v + S + iA ) ρ +3 , η = √ ( u + S + iA ) η − √ ( S (cid:48) + iA (cid:48) ) , χ = √ ( S (cid:48) + iA (cid:48) ) χ − √ ( ω + S + iA ) , (27)where S (cid:48) , and A (cid:48) , are W -parity odd while S , , and A , , are even. Two kinds of these particlesdo not mix. Similarly, for the charged scalars, ρ and η do not mix with ρ and χ . All these canbe seen by the results given below. The potential minimization conditions are obtained by vµ + v λ + 12 vω λ + 12 vu λ + 1 √ f uω = 0 , (28) uµ + u λ + 12 uv λ + 12 uω λ + 1 √ f vω = 0 , (29) ωµ + ω λ + 12 ωv λ + 12 ωu λ + 1 √ f uv = 0 . (30)The pseudoscalars A , A and A mix because f (cid:54) = 0. One combination of these fields is a6physical pseudoscalar ( A ) with mass, A = u − A + v − A + ω − A √ u − + v − + ω − , m A = − f √ u v + u ω + v ω uvω . (31)We see that f < u, v, ω >
0. The two other fields are massless, orthogonal to A , and identifiedas the Goldstone bosons of Z and Z (cid:48) as given by G Z (cid:48) = − ω − ( u − A + v − A ) + ( u − + v − ) A (cid:112) ( u − + v − + ω − )( u − + v − ) , G Z = − uA + vA √ u + v . (32)The A mass is proportional to ω if f is here (and below) supposed in ω scale ( f ∼ − ω ). At theleading order, ω (cid:29) u, v , we have G Z (cid:48) (cid:39) A and A (cid:39) ( vA + uA ) / √ u + v .The scalars S , S and S mix via the mass Lagrangian:( S S S ) λ u − fvω √ u λ uv + fω √ λ uω + fv √ λ uv + fω √ λ v − fuω √ v λ vω + fu √ λ uω + fv √ λ vω + fu √ λ ω − fuv √ ω S S S . (33)This mass matrix always gives a physical light state to be identified as the standard model Higgsboson ( H ). Since f ∼ − ω , the two other physical states ( H , ) are heavy living in ω scale. At theleading order ( − f, ω (cid:29) u, v ), those physical fields with respective masses can be obtained by H (cid:39) − vS + uS √ u + v , m H (cid:39) − f ω √ (cid:18) uv + vu (cid:19) , H (cid:39) S , m H (cid:39) λ ω , (34) H (cid:39) uS + vS √ u + v , m H (cid:39) λ ( λ u v + λ u + λ v ) − ( √ uv ( f /ω ) + λ u + λ v ) λ ( u + v ) . (35)One combination of S (cid:48) and S (cid:48) is a physical field S (cid:48) = ( ωS (cid:48) + uS (cid:48) ) / √ u + ω with mass m S (cid:48) = (cid:16) λ − fv √ uω (cid:17) ( u + ω ). The orthogonal state G (cid:48) S = ( − uS (cid:48) + ωS (cid:48) ) / √ u + ω is a massless Gold-stone field. Similarly, one combination of A (cid:48) and A (cid:48) is a physical field A (cid:48) = ( ωA (cid:48) − uA (cid:48) ) / √ u + ω with mass m A (cid:48) = (cid:16) λ − fv √ uω (cid:17) ( u + ω ). The orthogonal state G (cid:48) A = ( uA (cid:48) + ωA (cid:48) ) / √ u + ω is a massless Goldstone field. It is easy to realize that the G (cid:48) S and G (cid:48) A are Goldstone bosons ofRe X and Im X gauge fields, respectively. Therefore, their combination can be identified as theGoldstone boson of the X gauge boson: G X = 1 √ G (cid:48) S + iG (cid:48) A ) = ωχ − uη ∗ √ u + ω . (36)Simultaneously, we also have a physical neutral complex field as a combination of S (cid:48) and A (cid:48) (andobviously orthogonal to G X ) with the mass as given: H (cid:48) = 1 √ S (cid:48) + iA (cid:48) ) = uχ ∗ + ωη √ u + ω , m H (cid:48) = (cid:18) λ − f v √ uω (cid:19) ( u + ω ) . (37)7The H (cid:48) is only physical neutral scalar field which is odd under W -parity responsible for dark matteras shown below. It is to be noted that the H (cid:48) mass is always in ω scale. At the leading order( ω (cid:29) u, v ) we have H (cid:48) (cid:39) η (which is a scalar singlet of the standard model) and G X (cid:39) χ .There are two physical charged scalars, one W -parity odd ( H ) and another even ( H ): H − = vχ − + ωρ − √ v + ω , H − = vη − + uρ − √ u + v , (38)with respective masses m H = (cid:18) λ − f u √ vω (cid:19) ( v + ω ) , m H = (cid:18) λ − f ω √ uv (cid:19) ( u + v ) . (39)The orthogonal states to these scalars are Goldstone bosons of the Y and W bosons, respectively: G − Y = ωχ − − vρ − √ v + ω , G − W = uη − − vρ − √ u + v . (40)The masses of H and H are in ω scale. At the leading order, we have H (cid:39) ρ and G Y (cid:39) χ .We conclude that only the standard model like Higgs boson H is light in u, v scale. All theother physical scalars such as A , H , , , and H (cid:48) are heavy in ω scale, while R is in Λ scale. Thenumber of the Goldstone bosons match those of the massive gauge bosons (in the case the U (1) N gauge symmetry is turned on, the extra scalar φ as required will completely break this charge aswell as providing the Goldstone boson I for it). On the other hand, if the f coupling is suppressedas in the literature, the field A ∼ vA + uA becomes a physical massless Goldstone field living inthe doublets of the standard model which is very unrealistic. In addition, the identification in [14]in another 3-3-1 version of Im χ (similar to A in this model) as a dark matter is incorrect sinceit is already the Goldstone boson of Z (cid:48) gauge boson ( G Z (cid:48) ).For the gauge boson sector after integrating out Z N , the masses of the remaining gauge bosonsare given as usual: c W m Z (cid:39) m W = g u + v ) , m Y = g v + ω ) , m X = g u + ω ) , (41)and Z (cid:48) obtaining a mass in ω scale as X and Y (to be specified in the next section), where g is SU (3) L gauge coupling constant. We therefore identify u + v = (246 GeV) . It is noted that W and Y do not mix. Similarly Z , Z (cid:48) and Z N do not mix with X . All these are due to the W -paritysymmetry that forces the VEVs of χ and η vanishing. Moreover, the mass spectrum of neutralgauge boson sector will be changed if Z N gets a mass in the 3-3-1 scale (Λ ∼ ω ). By our conventionas given, this should be skipped in the present work.8For the fermion sector, we first note that the Dirac masses appeared below will be written inthe Lagrangian of form − ¯ f L m f f R + h.c. The masses of exotic quarks are given by m U = − √ h U ω, [ m D ] αβ = − √ h Dαβ ω, (42)which are all in ω scale. The ordinary quarks and charged leptons get consistent masses at thetree-level as in the 3-3-1 model with right-handed neutrinos:[ m u ] αa = 1 √ h uαa v, [ m u ] a = − √ h ua u, (43)for up quarks, [ m d ] αa = − √ h dαa u, [ m d ] a = − √ h da v, (44)for down quarks, and [ m e ] ab = − √ h eab v, (45)for charged leptons. Here, we see that the up quarks do not mix with U , and the down quarks alsodo not mix with D α as already mentioned.The ν L and ν R as coupled to η will have Dirac masses:[ m Dν ] ab = − √ h νab u. (46)However, the right-handed neutrinos ( ν R ) by themselves coupled via φ will get large Majoranamasses (in Λ scale) with the form, − ¯ ν cR m Mν ν R + h.c., where[ m Mν ] ab = −√ h (cid:48) νab Λ . (47)Consequently, the (observed) active neutrinos ( ∼ ν L ) get naturally small masses via a type I seesawmechanism as given by m eff ν = − m Dν ( m Mν ) − ( m Dν ) T ∼ ( h ν ) h (cid:48) ν u Λ . (48)If the Λ is proportional to ω acting on TeV scale, the Dirac mass parameters ( m Dν ) should getvalues in the electron mass range in order for m eff ν in sub eV. In any case, the masses of physicalsterile neutrinos ( ∼ ν R ) are in Λ scale responsible for the U (1) N breaking.Unlike the previous model [6], the N R have vanishing masses at the renormalizable level because ρ does not couple to ψ L ψ L (in addition, χ is also not coupled to ψ L ν R ) due to the L -charge or U (1) N symmetry. [In the 3-3-1 model with right-handed neutrinos [6], the status is not better.9Although the ψ L ψ L ρ coupling is allowed, the tree-level neutrinos have only three Dirac masses, noMajorana type, in which one mass is zero and two others degenerate that are unrealistic under thedata [1]]. Fortunately, the masses of N R can be generated by the scalar content by itself via aneffective operator invariant under the 3-3-1-1 symmetry and W -parity: λ ab M ¯ ψ caL ψ bL ( χχ ) ∗ + h.c. (49)There are no other operators of types ψψηη and ψψηχ as often considered due to the 3-3-1-1symmetry. Consequently, only the neutral fermions get masses via this kind of interactions:[ m N R ] ab = − λ ab ω M . (50)The mass scale of N R is unknown, however it can be taken in TeV order (i.e. M is in or notso high compared to ω ) due to the following facts: (i) In the 3-3-1 model with neutral fermions, m N R were proved to be naturally in ω scale (but the W -parity should be violated) [7], (ii) We canintroduce a new scalar sextet coupled to ψ L ψ L conserving the 3-3-1-1 symmetry and W -parity.The 33 component of sextet provides the masses for N R . However, it is also responsible for the3-3-1 symmetry breaking which should be in the same ω scale. On the other hand, this sextet ifincluded can be also reserved for totally breaking the N -charge and obtaining W -parity due to theVEV of the 11 component carrying a lepton number of two units [7, 18]. The ν L neutrinos alsoget small masses via a type II seesaw by this case. However, let us neglect the scalar sextet by thiswork because the scalar singlet φ as included is just enough for all purposes. Finally, it is notedthat due to W -parity, ν L,R do not mix with N R . Also, the mass sources of N R and ν R might comefrom different kinds of the 3-3-1-1 breaking. III. DARK MATTER ABUNDANCE AND DIRECT DETECTION
Let us note that all the W -particles including the dark matter candidates X , N R and H (cid:48) areheavy particles with the masses proportional to ω . Among the W -particles, supposing X as aLWP ( m X < m N R , m U , m D α , m H , m H (cid:48) ), it will be stabilized responsible for dark matter. Noticethat X cannot decay into Y and wise versa. This is the first case discussed below. For the secondcase, N R will be assumed as a LWP ( m N R < m X , m Y , m U , m D α , m H , m H (cid:48) ) for dark matter. Thework in [13] has presented numerical calculations for relic densities using MicrOMEGAs package.In the following, we will give an analytic evaluation. For the constraints from the direct and indirectdetection experiments, let us call for the reader’s attention to [13]. Below, we provide only one ofthese kinds by analytic calculation so that our conclusions are viable.0The scalar dark matter candidate H (cid:48) , which behaves as a LWP ( m H (cid:48) < m X , m Y , m U , m D α , m H , m N R ), has been traditionally studied in the 3-3-1 models [15]. In the following consideration,this candidate will be neglected. For detailed calculations and experimental constraints, let usrecall the reader’s attention to Refs. [12, 13, 24]. However, let us make some remarks on thisparticle: (i) The previous studies [12, 13, 15] that identify the massive scalar H (cid:48) (cid:39) η as a darkmatter are unnatural since the symmetries protecting it from decay are either neglected or includedin term of lepton charge, G -charge, or even Z which must be broken due to the problems as shownabove. In this case, it will develop a VEV allowing decay channels into the standard model particlessuch as H (cid:48) −→ HH since this field should be naturally heavy (the finetuning in mass was neededin [15] which is very ackword). In our model, by the investigation of W -parity, the stability issueof H (cid:48) has been solved, similar to the standard model extension with a Z odd scalar singlet. (ii) H (cid:48) (cid:39) η is a singlet under the standard model symmetry, and it annihilates into the standardmodel particles via the scalar portal, exotic quarks, and new gauge bosons. A. Relic density of X gauge boson The annihilation of X into the standard model particles is dominated by the following channels XX ∗ −→ W + W − , ZZ, HH, νν c , l + l − , qq c , (51)where ν = ν e , ν µ , ν τ , l = e, µ, τ , and q = u, d, c, s, t, b . Let us consider the channel XX ∗ → W + W − among them. This process is contributed by the diagrams as in Fig. 1. The Feynman rules can X ∗ X W + W − X ∗ X W + W − Y − X ∗ X W + W − Z FIG. 1: Dominant contributions to annihilation of X into W + W − . be found in [21]. To the leading order, the thermal average of the cross-section times relative1velocity [25] is given as follows (cid:104) σv rel (cid:105) v → (cid:39) α m X s W m W . (52)Because m X (cid:29) m W , this result is too large so that the X can meet the criteria of a dark matter.In fact, the relic density of the candidate [25] is bounded byΩ X h (cid:39) . (cid:104) σ tot v rel (cid:105) < . (cid:104) σv rel (cid:105) (cid:39) . × (cid:18) m W m X (cid:19) < . , (53)if we take a previous limit on the mass of X : m X >
440 GeV [26]. The upper bound of the relicdensity is too small to compare to the WMAP data Ω DM h (cid:39) .
11 [1]. The X cannot be a darkmatter. This conclusion is coincided with a mere note in [13]. B. Relic density of neutral fermion N R Among the three neutral fermions, N aR , assuming N R is the lightest particle. In addition,supposing ν and l is the left-handed neutrino and charged lepton that directly couple to N R viathe new gauge bosons X and Y , respectively. There are two other neutrinos and two other chargedleptons to be denoted by ν α and l α , respectively. The annihilation of N R into the standard modelparticles is dominated by the following channels: N N c −→ νν c , l − l + , ν α ν cα , l − α l + α , qq c , ZH, (54)which are given in terms of Feynman diagrams as in Fig. 2. Let us remind the reader that there are NN c X, Y ν c , l + ν, l − NN c Z " ν c , l + , ν cα , l + α , q c , Hν, l − , ν α , l − α , q, Z FIG. 2: Dominant contributions to annihilation of N R . not channels into HH , W − W + and ZZ bosons because ν L and N R do not mix due to W -parity. On2the other hand, there may include also some contributions coming from mediated scalars insteadof the new gauge bosons, but they are all small and thus neglected.The Feynman rules for the above processes can be found in [6, 27] (see also [28]). An evaluationof the thermal average cross-section times relative velocity is given by (cid:104) σv rel (cid:105) (cid:39) g m N R π (cid:18) x F (cid:19) (cid:34) m X + 1 m Y − c W (3 − s W ) m Z (cid:48) (cid:32) m X + 1 m Y (cid:33) + 60 − s W + 196 s W − s W ) m Z (cid:48) (cid:35) + g m N R π (cid:118)(cid:117)(cid:117)(cid:116) − m t m N R (cid:40)(cid:34) x F (cid:32) m t m N R m N R + 2 m t m N R − m t (cid:33)(cid:35) × − s W + 20 s W − s W ) m Z (cid:48) + (cid:34) x F (cid:32) m t m N R − m t (cid:33)(cid:35) s W (3 − s W )3(3 − s W ) m Z (cid:48) m t m N R (cid:41) + g m N R π (cid:118)(cid:117)(cid:117)(cid:116) − m H + m Z m N R c W c W (3 − s W ) m Z (cid:48) (cid:34) m Z − m H m N R + 1 x F (cid:32) m H + 11 m Z m N R (cid:33)(cid:35) , (55)where we have used the facts that m ν , m l , m q ( q (cid:54) = t ) (cid:28) m t , m Z , m H < m N R < m X , m Y , m Z (cid:48) . In addition, the above cross section has been expanded in the non-relativistic limit of N R asusual up to the squared velocity, in which (cid:104) v (cid:105) = 6 /x F and x F ≡ m N R /T F ∼
20 at freeze-outtemperature [25].To have a numerical value for the WMAP data, let us use the condition ω (cid:29) u , v whichfollows the tree-level relation (the mass of Z (cid:48) can be found in [29]), m X (cid:39) m Y (cid:39) − t W m Z (cid:48) . (56)Also let the neutral fermion mass be enough large m N R (cid:29) m t,W,H so that the ratios m t,H,W /m N R can be neglected and the new physics is safe. We have (cid:104) σv rel (cid:105) (cid:39) α (150 GeV) (2557 . m N R m Z (cid:48) , (57)where we have used s W = 0 .
231 and x F = 20. Because α / (150 GeV) (cid:39) N R (Ω N h (cid:39) . / (cid:104) σv rel (cid:105) (cid:39) .
11) implies m N R (cid:39) m Z (cid:48) . . (58)Since m N R < m Z (cid:48) we derive m Z (cid:48) ≤ . Z (cid:48) mass is needed in order to makethe dark matter candidate N R stable. The several lower limits on Z (cid:48) mass have been given in theliterature as some TeV [1, 30], so let us take the strong one recently studied in the second article3of [30], m Z (cid:48) ≥ . N R is limited by m N R ≥ . N R is a dark matter if it has a mass in the range:1 . ≤ m N R ≤ . . (59)The mass of N R is completely fixed via m Z (cid:48) or the VEV ω as a single-valued function due to therelic density as given which is unlike that in [13] numerically calculated with the MicrOMEGAspackage. Our result above is in agreement with the large range among others as dedicated in [13]. C. Direct detection of dark matter N R The direct detection experiments measure the recoil energy deposited by the scattering of darkmatter with the nuclei in a large detector. At the fundamental level, the scattering is due to theinteractions of dark matter with quarks as confined in the nucleons. In this model, the leadingcontribution to the N R -quark scattering amplitude comes from the t-chanel exchange of Z (cid:48) boson(there may be another contribution of Z boson, however it is very suppressed due to the contrainedsmall mixing of Z − Z (cid:48) ). Therefore, the effective Lagrangian is given by L eff N R − quark = ¯ N R γ µ N R [¯ qγ µ ( α q P L + β q P R ) q ] , (60)where the relevant couplings are evaluated by [6, 27, 28] α u,d,c,s = − g m Z (cid:48) , β u,c = 2 g s W c W − m Z (cid:48) , β d,s = − g s W c W − m Z (cid:48) . (61)In the non-relativistic limit, there are only two operators in the effective Lagrangian surviving (theother operators vanish) as given by [31] L eff N R − quark = λ q,o ¯ N γ µ N ¯ qγ µ q + λ q,e ¯ N γ µ γ N ¯ qγ µ γ q, (62)where λ q,o ≡ ( β q + α q ) / λ q,e ≡ ( β q − α q ) / N R -nucleon amplitudes can be directly converted from the amplitudes above via the nucleonform factors as obtained by [31] L eff N R − nucleon = λ ψ,o ¯ N γ µ N ¯ ψγ µ ψ + λ ψ,e ¯ N γ µ γ N ¯ ψγ µ γ ψ, (63)where ψ is nucleon, ψ ≡ ( p, n ), and λ ψ,e = (cid:80) q = u,d,s ∆ ψq λ q,e with the ∆ ψq values as provided in [31],while λ ψ,o is given by λ ψ,o = (cid:88) q = u,d f ψV q λ q,o , f pV u = 2 , f pV d = 1 , f nV u = 1 , f nV d = 2 . (64)4The λ ψ,o and λ ψ,e are spin-independent (SI) and spin-dependent (SD) interactions, respectively.For the large neuclei, the N R -nucleus scattering cross-section is strongly enhanced due to theSI interaction, while there is no strong enhancement from the SD interaction [31]. Therefore, thedominant contribution to the cross-section comes from the SI interaction as given by σ SI N R − nucleus = 4 µ A π ( λ p Z + λ n ( A − Z )) , (65)where Z is the nucleus charge, A the total number of nucleons, and µ A = m N R m A m N R + m A , λ p = λ p,o s W − g − s W ) m Z (cid:48) , λ n = λ n,o − g m Z (cid:48) . (66)The experimental output for the N R -nucleon cross-section is the above result per a nucleon, σ SI N R − nucleon = 4 µ π (cid:18) λ p ZA + λ n A − ZA (cid:19) , µ nucleon = m N R m p,n m N R + m p,n (cid:39) m nucleon . (67)The strongest limit on the N R -nucleon cross-section presently comes from the XENON100experiment. Taking the Xe nucleon with Z = 54 , A = 131, and m nucleon (cid:39) g = 4 πα/s W with α = 1 /
128 and s W = 0 . σ SI N R − nucleon (cid:39) . × − (cid:18) m Z (cid:48) (cid:19) cm . (68)With the Z (cid:48) mass limit as given above, m Z (cid:48) ≥ . σ SI N R − nucleon ≤ . × − cm . (69)This limit is in good agreement to the constraint from the XENON100 experiment [32] since ourdark matter is heavy with the mass in TeV range. IV. CONCLUSION AND OUTLOOK
We have given a detailed analysis of lepton and baryon numbers in the 3-3-1 model with neutralfermions. If they are (residual) symmetries of the theory, which are strictly respected by the gaugeinteractions, minimal Yukawa Lagrangian and scalar potential, they behave as local symmetries.We have also given a classification of the wrong-lepton particles that have anomalous lepton (evenbaryon) numbers, whereas the ordinary particles including the standard model ones do not havethis property. Moreover, all the unwanted interactions, which lead to the tree-level flavor changingneutral currents, inconsistent neutrino masses, instability of the lightest wrong-lepton particle, andfurther the tree-level CPT violation, are naturally suppressed due to one of those symmetries. It5is generally applied also for the 3-3-1 model with right-handed neutrinos and the minimal 3-3-1model if these theories conserve the lepton and baryon number symmetries.The lepton ( L ) and baryon ( B ) numbers can be as several appearances of and unified in a singlecharge ( N = B − L ) of natural gauge symmetry SU (3) C ⊗ SU (3) L ⊗ U (1) X ⊗ U (1) N independentof all the anomalies such as leptonic and baryonic, recognizing B − L = − (2 / √ T + N as a chargeof SU (3) L ⊗ U (1) N (see also [33] for another 3-3-1-1 symmetry derived from grand unifications).All the unwanted interactions is not invariant under this gauge symmetry which is prevented,while the minimal Lagrangian of the 3-3-1 theory respects it. A direct consequence of the 3-3-1-1 model is that it contains by itself a residual discrete symmetry after breaking, W -parity: P W = ( − − √ T +3 N +2 s , as a nature symmetry of the wrong-lepton particles. The unwantedvacuums, which also lead to the problems as mentioned above, are naturally discarded due to thisparity. The lightest wrong-lepton particle is truly stabilized due to the 3-3-1-1 gauge symmetrywith unbroken W -parity responsible for the dark matter. With the aid of W -parity, it is completelyunderstood why the wrong-lepton particles are only coupled in pairs in the minimal Lagrangianwith the standard vacuum structure, due to the specific 3-3-1 gauge symmetry (which has beenexplicitly shown in the text too). This W -parity has a natural origin of the 3-3-1-1 gauge symmetryby itself which is discriminated from those in the MSSM and others with gauged B − L [3, 4].We have also provided a general analysis of the scalar sector, identifying physical scalars andGoldstone bosons. The trilinear coupling of scalars should present in order to make all extraHiggs bosons massive, keeping the model obviously consistent with the low energy theory. Thepseudoscalar part of χ should be the Goldstone boson of Z (cid:48) gauge boson which is unlike theconclusion in [34]. Therefore, it should be not a dark matter [14]. Finally, this sector will becharged if the U (1) N gauge symmetry is turned on.We have explicitly shown that the non-Hermitian neutral gauge boson ( X ) cannot be a darkmatter. However, the neutral fermion ( N R ) can contribute the dark matter if its mass is givenin the range 1 . ≤ m N R ≤ . Z (cid:48) gauge boson satisfying2 . ≤ m Z (cid:48) ≤ . Z N (it should be assumed to be so massive or weakly interacting). If its mass andcoupling are comparable to those of X, Y, Z (cid:48) , our results may be changed. Also, phenomenologiesin the 3-3-1-1 model such as the baryon number asymmetry, neutrino masses, and new physicsassociated with the Z N gauge boson will be very interesting. All these and the above one aredevoted to further studies to be published elsewhere [20].The W -parity transforms trivially in the 3-3-1 model with right-handed neutrinos, i.e. all the6particles in the model is even under this parity. While the model might right predict potentialdark matter candidates, let us ask what is the mechanism other than that useless W -parity forstabilizing the dark matter. Back to the past [12], the first notes were assigned for the conservationof lepton number and Z . However, this Z is really broken by the VEVs of scalars while thelepton number should be also violated by five dimensional effective interactions or broken by thevacuum responsible for the neutrino masses. On the other hand, the lepton number respected asa symmetry of the theory is also broken as a gauge symmetry and anomalous. In [13], the leptoncontent was changed and the U (1) G included perhaps to avoid some of those problems. However,this U (1) G takes the same status as the lepton number that acts as a gauge symmetry and alsobroken. Fortunately, by the new lepton content this charge is anomaly free like N = B − L . So,we propose one solution to the stability of the dark matter in their model is by imposing G -parity( P G = ( − G ) which is odd for the G -particles and even for ordinary particles. In this case, thegauge symmetry should be SU (3) C ⊗ SU (3) L ⊗ U (1) X ⊗ U (1) G where G = √ T + G . Now let us turnto the 3-3-1 model with right-handed neutrinos. The U (1) G is also useless as the lepton numberor W -parity since it also yields anomalies. To go over all these difficulties, in the following, wesuppose a new mechanism based on the idea of a potential “inert” scalar triplet naturally realizedby a Z symmetry with the base of the economical 3-3-1 model [28]. The W -parity is explicitlyviolated in this model, and the lepton number is no longer to be regarded as a gauge symmetrysince it is only an approximate symmetry, explicitly violated by the Yukawa interactions.We know that the 3-3-1 model with right-handed neutrinos can work with three scalar triplets( ρ, η, χ ) in which two of them ( η, χ ) have the same gauge symmetry quantum numbers [6]. If weexclude one of these two triplets (assumed η ) it results a new, consistent, predictive model, namedthe economical 3-3-1 model, working with only the two scalar triplets ( ρ, χ ) as recently investigatedin a series of articles [28, 35–38] (note that in those works ρ called as φ instead). Alternative tothat proposal, we can retain η , but introducing a Z symmetry so that the η is odd, while the χ, ρ and all other fields are even. The resulting model will be very rich in phenomenology otherthan the economical 3-3-1 model due to the contribution of η . In fact, the vacuum can also beconserved by the Z as a partial solution of the potential minimization, (cid:104) η (cid:105) = 0, (cid:104) ρ (cid:105) = √ (0 , v, (cid:104) χ (cid:105) = √ ( u, , ω ). We have thus a new economical 3-3-1 model with an inert scalar triplet η that is odd under the exact and unbroken Z symmetry. The scalar triplets χ and ρ can break thegauge symmetry and generating the masses for the particles in a correct way like the economical3-3-1 model [28]. The inert scalar triplet η can provide some dark matter candidates, however theymay belong to a scalar singlet or a scalar doublet under the standard model symmetry. In the7sense the model proposed is quite similar to the two Higgs doublet model in which one doublet isinert, well-known as the inert doublet model [39]. However, this theory provides only a doubletdark matter. The proposal is to be published elsewhere [40]. Acknowledgments
This research is funded by Vietnam National Foundation for Science and Technology Develop-ment (NAFOSTED) under grant number 103.03-2011.35. PVD would like to thank Dr. Do ThiHuong at Institute of Physics, Vietnam Academy of Science and Technology for useful discussions.
Appendix A: Checking the U (1) N anomalies The nontrivial anomalies as associated with U (1) N that are potentially troublesome can belisted as follows: [ SU (3) C ] U (1) N , [ SU (3) L ] U (1) N , [ U (1) X ] U (1) N , U (1) X [ U (1) N ] , [ U (1) N ] ,and [Gravity] U (1) N . The other anomalies associated with the usual 3-3-1 symmetry obviouslyvanish [9] which will not be considered in this appendix.With the fermion content and N -charges of the 3-3-1-1 model as given in Table IV, the mentionedanomalies can be calculated. The first one is proportional to[ SU (3) C ] U (1) N ∼ (cid:88) all quarks ( N q L − N q R )= 3 N Q + 2 × N Q α − N u a − N d a − N U − N D α = 3(2 /
3) + 6(0) − / − / − (4 / − − /
3) = 0 , (A1)which vanishes. The second anomaly also vanishes,[ SU (3) L ] U (1) N ∼ (cid:88) all (anti)triplets N F L = 3 N ψ a + 3 N Q + 2 × N Q α = 3( − /
3) + 3(2 /
3) + 6(0) = 0 . (A2)Here the number of fundamental colors (the 3s in the second and last terms) must be taken intoaccount. In the following the appearance of color numbers should be understood. Notice also thatthe relation Tr[( − T ∗ i )( − T ∗ j ) N ] = Tr[ T i T j N ] has been used.The third anomaly is given by[ U (1) X ] U (1) N = (cid:88) all fermions ( X f L N f L − X f R N f R )8= 3 × X ψ a N ψ a + 3 × X Q N Q + 2 × × X Q α N Q α − × X u a N u a − × X d a N d a − X U N U − × X D α N D α − X e a N e a − X ν a N ν a = 3 × − / ( − /
3) + 3 × / (2 /
3) + 2 × × (0) − × / (1 / − × − / (1 / − / (4 / − × − / ( − / − − ( − − ( − = 0 . (A3)The fourth anomaly is similarly calculated, U (1) X [ U (1) N ] = (cid:88) all fermions ( X f L N f L − X f R N f R )= 3 × X ψ a N ψ a + 3 × X Q N Q + 2 × × X Q α N Q α − × X u a N u a − × X d a N d a − X U N U − × X D α N D α − X e a N e a − X ν a N ν a = 3 × − / − / + 3 × / / + 2 × × − × / / − × − / / − / / − × − / − / − − − − − = 0 . (A4)The U (1) N self-anomaly is[ U (1) N ] = (cid:88) all fermions ( N f L − N f R )= 3 × N ψ a + 3 × N Q + 2 × × N Q α − × N u a − × N d a − N U − × N D α − N e a − N ν a = 3 × − / + 3 × / + 2 × × − × / − × / − / − × − / − − − − = 0 . (A5)The last anomaly is given by[Gravity] U (1) N ∼ (cid:88) all fermions ( N f L − N f R )= 3 × N ψ a + 3 × N Q + 2 × × N Q α − × N u a − × N d a − N U − × N D α − N e a − N ν a = 3 × − /
3) + 3 × /
3) + 2 × × − × / − × / − / − × − / − − − −
1) = 0 . (A6)9These anomalies are only canceled when the right-handed neutrinos are included, in similarity tothe standard model extensions with gauged B − L . Indeed, since B − L = − (2 / √ T + N andthe T obviously independent of anomalies, the cancellation of N anomalies is equivalent to that of B − L . It is noted that the 3-3-1 model with right-handed neutrinos is always free from the U (1) N anomalies due to its fermion content by itself, while the minimal 3-3-1 model like our case is not.Also, if U (1) N is imposed in the model of [13], it is also not free from the gravitational anomaly. Appendix B: Derivation of W -parity The SU (3) L ⊗ U (1) N symmetry is broken down to U (1) B − L by the VEVs of η , ρ and χ becausethe charge B − L = − (2 / √ T + N anihinates these vacuums:( B − L ) (cid:104) η (cid:105) = 0 , ( B − L ) (cid:104) ρ (cid:105) = 0 , ( B − L ) (cid:104) χ (cid:105) = 0 . (B1)This is the first stage of symmetry breaking. In the second stage the B − L will be broken. And,this is achieved by the VEV of φ since ( B − L ) (cid:104) φ (cid:105) (cid:54) = 0 . (B2)It is to be noted that the φ VEV also breaks U (1) N by the first stage. Therefore, the φ vacuumbreaks the N charge totally.Now, let us find an unbroken residual symmetry as a discrete subgroup of U (1) B − L [exactly of SU (3) L ⊗ U (1) N ]. It must satisfy the following condition: e iα ( B − L ) (cid:104) φ (cid:105) = (cid:104) φ (cid:105) , (B3)where α is a parameter of the U (1) B − L Lie group. Because B ( φ ) = 0, L ( φ ) = −
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