Simple 3-3-1 model and implication for dark matter
SSimple 3-3-1 model and implication for dark matter
P. V. Dong ∗ Institute of Physics, Vietnam Academy of Science and Technology,10 Dao Tan, Ba Dinh, Hanoi, Vietnam
N. T. K. Ngan † Department of Physics, Cantho University,3/2 Street, Ninh Kieu, Cantho, Vietnam
D. V. Soa ‡ Department of Physics, Hanoi National University of Education,136 Xuan Thuy, Cau Giay, Hanoi, Vietnam (Dated: October 11, 2018)We propose a new and realistic 3-3-1 model with the minimal lepton and scalarcontents, named the simple 3-3-1 model. The scalar sector contains two new heavyHiggs bosons, one neutral H and another singly-charged H ± , besides the standardmodel Higgs boson. There is a mixing between the Z boson and the new neutralgauge boson ( Z (cid:48) ). The ρ parameter constrains the 3-3-1 breaking scale ( w ) to be w >
460 GeV. The quarks get consistent masses via five-dimensional effective in-teractions while the leptons via interactions up to six dimensions. Particularly, theneutrino small masses are generated as a consequence of the approximate lepton-number symmetry of the model. The proton is stabilized due to the lepton-parityconservation ( − L . The hadronic FCNCs are calculated that give a bound w > . PACS numbers: 12.60.-i, 95.35.+d a r X i v : . [ h e p - ph ] O c t I. INTRODUCTION
The standard model has been extremely successful in describing observed phenomena,especially for the outstanding prediction of recently-discovered Higgs boson [1]. However,it must be extended to address unsolved questions such as small masses and mixing ofneutrinos, matter-antimatter asymmetry of the universe, dark matter and dark energy [2].Therefore, we would like to argue that the SU (3) C ⊗ SU (3) L ⊗ U (1) X (3-3-1) gauge theorywhere the color group is as usual while the electroweak group is enlarged [3–6] may be aninteresting choice for the physics beyond the standard model, specially for the dark matter.As a fact, the fermion generations in the standard model are identical, which transformthe same, under the gauge symmetry and each generation is anomaly free. The numberof fermion generations can in principle be arbitrary. All these might be a consequenceof special weak-isospin group SU (2) L that its anomaly vanishes for every chiral fermionrepresentation [7]. By the new weak-isospin symmetry, the SU (3) L anomaly is nontrivialthat is only cancelled if the number of generations is an integer multiple of three [8]. Dueto the contribution of exotic quarks along with ordinary quarks, QCD asymptotic freedomrequires the number of generations lesser than or equal to five. So, the fermion generationnumber is three coinciding with the observations [2].Moreover, the fermion generations in the new model are non-universal that the thirdgeneration of quarks transforms under SU (3) L differently from the two others. This mightprovide a natural solution for the uncharacteristic heaviness of top quark [9]. The quanti-zation of electric charge is a consequence of fermion content under this new symmetry [10].The model can by itself contain a Peccei-Quinn symmetry for solving the strong CP prob-lem [11]. The B − L number behaves as a gauge charge (and R -parity results) since it doesnot commute and nonclosed algebraically with the 3-3-1 symmetry, which provides insightsin the known 3-3-1 model [12, 13]. The neutrino masses, possible leptogenesis [14] and darkmatter [12, 15–18] have been extensively studied.As a result of the new SU (3) L ⊗ U (1) X gauge symmetry, the minimal interactions ofthe theory (including gauge interactions, minimal Yukawa Lagrangian and minimal scalar ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] potential) put the relevant particles (known as wrong-lepton particles [12] or similar ones inother versions) in pairs, similarly to the case of superparticles in supersymmetry. Hence, the3-3-1 model has been thought to give some candidates for dark matter [15–17]. However,the problem is how to suppress or evade the unwanted interactions (almost other than theminimal interactions) and the unwanted vacuums (come from neutral scalar candidates) thatleads to the fast decay of dark matter (for detailed reviews, see [12, 18]).It is easily realized that the new particles in the minimal 3-3-1 model [3] cannot bedark matter because they are either electrically charged or rapidly decayed, even for justminimal Lagrangian. The 3-3-1 model with right-handed neutrinos encounters the sameissue [18]. Even the lepton number symmetry was first regarded as a dark matter stabilitymechanism [16], but it is quite wrong since the generation of neutrino masses violates thelepton number. To overcome the difficulty, Refs. [17] introduced another lepton sector (themodel was changed and called as the 3-3-1 model with left-handed neutrinos). In anotherapproach [12], a mechanism for dark matter stability based on W -parity, similarly to R -parity in supersymmetry, was given. However, this stability mechanism works only withthe particle content of the 3-3-1 model with neutral fermions [19]. Hence, the issue of darkmatter identification and its stability in the typical 3-3-1 models remain unsolved.If the B − L charge is conserved, the typical 3-3-1 models are not self-consistent (sincethe B − L and 3-3-1 symmetries are algebraically nonclosed as mentioned [12, 13]). Thisalso applies for other continuous symmetries imposed such as U (1) G in [17] that do notcommute with the 3-3-1 symmetry. One way to keep the typical 3-3-1 models self-containedis that they have to possess explicitly-violating interactions of lepton number. [Notice thatthe lepton number is thus an approximate symmetry while the baryon number is alwaysconserved and commuted with the 3-3-1 symmetry]. And, a theory for the dark matter inthe typical 3-3-1 models must take this point into account.As a solution to the dark matter issue in the typical 3-3-1 models, we have proposed in theprevious work [18] that if one scalar triplet of the 3-3-1 model with right-handed neutrinos isinert ( Z odd) while all other fields are even, the remaining two scalar triplets (well-knownas the normal scalar sector) will result an economical 3-3-1 model self-consistently [5]. Thismodel provides appropriate masses for neutrinos besides the dark matter as resided in theinert triplet. In this work, we sift such outcome for the minimal 3-3-1 model.The minimal 3-3-1 model has traditionally been studied to be worked with three scalartriplets ρ = ( ρ +1 , ρ , ρ ++3 ), η = ( η , η − , η +3 ), χ = ( χ − , χ −− , χ ) and/nor one scalar sextet S = ( S , S − , S +13 , S −− , S , S ++33 ). The question is which scalars are inert, while the rest ora part of it—the normal scalar sector is appropriate for symmetry breaking, mass generation,and yielding a realistic model on both sides: mathematical and phenomenological. In thiswork, let us restrict our study for the cases with a minimal normal scalar sector so that theinert sector is enriched responsibly for dark matter. Looking in the literature, the reduced3-3-1 model [6] seems to be a candidate. However, this model is encountered with a problemof large FCNCs which is experimentally unacceptable. As an alternative approach, we willindicate that the minimal 3-3-1 model can behave as a so-called “simple 3-3-1 model” thatis based on only the two scalar triplets η and χ (which is different from the reduced 3-3-1model given in [6] due to the scalar and fermion contents). The model will be proved to berealistic rather than the previous version [6].With the proposal of the simple 3-3-1 model, the rest of scalars ( ρ , S ), even the replica-tions of η, χ as well as possible variants of all them including new forms, can be the inertsector ( Z odd) responsible for dark matter. However, the most basic cases that result forthe desirable inert sector can be summarized as1. The triplet ρ is inert ( S is suppressed). However, this candidate ( ρ ) cannot be a darkmatter due to the direct dark matter detection constraints.2. The sextet S is inert ( ρ is suppressed). This sextet does not provide any realistic darkmatter candidate similarly to the previous case. However, a variant of it with U (1) X charge, X = 1, yields a triplet dark matter.3. Introduce an inert scalar triplet as the replication of η ( ρ and S are suppressed). Wehave a doublet dark matter.4. Introduce an inert scalar triplet as the replication of χ ( ρ and S are suppressed). Thiscase yields a singlet dark matter.Note that a combination of the cases above or the whole can be interplayed in a single theorybased on the simple 3-3-1 model, but they will not be considered in the current work.The rest of this work is organized as follows. In Section II we propose the simple 3-3-1model. The identification of physical scalars, Goldstone and gauge bosons is given. Thefermion masses, proton stability and FCNCs are also investigated. In Section III, the darkmatter theories that are based on the simple 3-3-1 model are respectively presented. Thedark matter candidates of the model with ρ inert triplet and of S inert sextet are analyzed torule them out. We will also show that the models with inert triplets as replications of η and χ , respectively, and the model with X = 1 inert scalar sextet can provide realistic candidatesfor dark matter. To be completed, in Section IV we will give a particular evaluation of theimportant dark matter observables and compare them to the experimental data. Finally,we summarize our results and conclude this work in Section V. II. SIMPLE 3-3-1 MODEL
We will re-examine the reduced 3-3-1 model [6] and the minimal 3-3-1 model [3] thatleads to a new and realistic 3-3-1 model with minimal lepton and scalar contents—the so-called simple 3-3-1 model. To make sure this point, the simple 3-3-1 model will be explicitlypointed out to be consistent with the data. By the new approach, the dark matter modelsare emerged to be studied in the next section.
A. Proposal of the model
The gauge symmetry of the considering model is given by SU (3) C ⊗ SU (3) L ⊗ U (1) X ,where the first factor is ordinary color group while the rest is the extension of the electroweaksymmetry as mentioned. The fermion content which is anomaly free is defined as [3] ψ aL ≡ ν aL e aL ( e aR ) c ∼ (1 , , ,Q αL ≡ d αL − u αL J αL ∼ (3 , ∗ , − / , Q L ≡ u L d L J L ∼ (3 , , / , (1) u aR ∼ (3 , , / , d aR ∼ (3 , , − / ,J αR ∼ (3 , , − / , J R ∼ (3 , , / , where a = 1 , , α = 1 , Q = T − √ T + X , where T i ( i = 1 , , ...,
8) are SU (3) L charges, while X is that of U (1) X (below, the SU (3) C charges will be denoted by t i ). The new quarks possess exotic electriccharges as Q ( J α ) = − / Q ( J ) = 5 / SU (3) L differentlyfrom the first two generations, the FCNCs due to the new neutral gauge boson ( Z (cid:48) ) exchangeis more constrained that yields a low bound of some TeV for the 3-3-1 breaking scale or the Z (cid:48) mass [9]. Such a new physics scale is possibly still in the well-defined region of the theory,limited below the Landau pole of around 5 TeV [20]. By contrast, if the first or second quarkgeneration was arranged differently from the two others like the reduced 3-3-1 model [6],the resulting theory would be ruled out by the large FCNCs, provided that the new physicsenters below the Landau pole. Furthermore, the theory would be invalid (or inconsistent) ifone tried to push the new physics scale far above the Landau pole in order to prevent theFCNCs [9, 21]. All these will also be studied in the last of this section.The model can work with only two scalar triplets [6]. Upon the proposed fermion content,let us impose, however, the following two scalar triplets η = η η − η +3 ∼ (1 , , , χ = χ − χ −− χ ∼ (1 , , − , (2)with VEVs (cid:104) η (cid:105) = 1 √ u , (cid:104) χ (cid:105) = 1 √ w . (3)This yields a dominant tree-level mass for top quark, while some lighter quarks that haveno tree-level mass will get consistent masses via either effective interactions (shown below)or radiative corrections [5]. Otherwise, if the two scalar triplets like [6] which are χ andanother triplet ρ ∼ (1 , ,
1) are retained for this model (in this case the η is suppressed), itwill result a vanishing tree-level mass for the top quark that is unnatural to be induced bysuch subleading quantum effect or effective theory.The original study in [6] gave a comment on the scalar triplets of this model, however thefermion content was never changed that would always face the large FCNC problems. In arecent research [22], the fermion content was changed, but the scalar sector of the reduced3-3-1 model was retained, which would be encountered with a vanishing top quark mass atthe tree-level. Hence, those issues have naturally been solved by this proposal. In otherwords, all the ingredients as stated above recognize an unique 3-3-1 model distinguishedfrom the previous versions such as the reduced and minimal 3-3-1 models [3, 6] due to thedifference in the fermion and/or scalar contents. This is a new observation of this work,which is going to be called as the “simple 3-3-1 model”. B. Scalar sector
The scalar potential of the model is given by V simple = µ η † η + µ χ † χ + λ ( η † η ) + λ ( χ † χ ) + λ ( η † η )( χ † χ ) + λ ( η † χ )( χ † η ) , (4)where µ , have mass-dimensions while λ , , , are dimensionless. The VEVs u, w are givenfrom the potential minimization as u = 2(2 λ µ − λ µ ) λ − λ λ , w = 2(2 λ µ − λ µ ) λ − λ λ . (5)To make sure that1. The scalar potential is bounded from below (vacuum stability),2. The VEVs u, ω are nonzero (for symmetry breaking and mass generation),3. The physical scalar masses are positive,the parameters satisfy µ , < , λ , , > , − (cid:112) λ λ < λ < Min (cid:8) λ ( µ /µ ) , λ ( µ /µ ) (cid:9) . (6)In addition, the VEV w breaks the 3-3-1 symmetry down to the standard model symmetryand provides the masses for new particles, while the VEV u breaks the standard model sym-metry as usual and gives the masses for ordinary particles. Therefore, to keep a consistencywith the standard model, we impose w (cid:29) u .Expanding η, χ around the VEVs, we get η T = ( u √ S + iA √ η − η +3 ) and χ T =(0 0 w √ ) + ( χ − χ −− S + iA √ ). Hence, the physical scalar fields with respective masses areidentified as follows h ≡ c ξ S − s ξ S , m h = λ u + λ w − (cid:113) ( λ u − λ w ) + λ u w (cid:39) λ λ − λ λ u ,H ≡ s ξ S + c ξ S , m H = λ u + λ w + (cid:113) ( λ u − λ w ) + λ u w (cid:39) λ w , (7) H ± ≡ c θ η ± + s θ χ ± , m H ± = λ u + w ) (cid:39) λ w . Here, we have denoted c x = cos( x ) , s x = sin( x ) , t x = tan( x ), and so forth, for any x angle.The ξ is S - S mixing angle, while the θ is that of χ - η . They are obtained as t θ = uw , t ξ = λ uwλ w − λ u (cid:39) λ uλ w . (8)The h field is the standard model like Higgs boson, while H and H ± are new neutraland singly-charged Higgs bosons, respectively, which is unlike [6]. There are eight masslessscalar fields G Z ≡ A , G Z (cid:48) ≡ A , G ± W ≡ η ± , G ±± Y ≡ χ ±± and G ± X ≡ c θ χ ± − s θ η ± thatcorrespond to the Goldstone bosons of eight massive gauge bosons Z , Z (cid:48) , W ± , Y ±± and X ± (see below). In the effective limit, u (cid:28) w , we have η (cid:39) u + h + iG Z √ G − W H + , χ (cid:39) G − X G −− Yw + H + iG Z (cid:48) √ . (9) C. Gauge sector
The covariance derivative is given by D µ = ∂ µ + ig s t i G iµ + igT i A iµ + ig X XB µ , where g s , g and g X are the gauge coupling constants, while G iµ , A iµ and B µ are the gauge bosons, asassociated with the 3-3-1 groups, respectively. On the other hand, in the next section we willintroduce extra scalars that are odd under a Z symmetry (so-called the “inert” scalars).However, the inert scalars do not give the masses for the gauge bosons because they have noVEV due to the Z symmetry. Therefore, the gauge bosons of the model get masses fromthis part of the Lagrangian, (cid:80) Φ= η,χ ( D µ (cid:104) Φ (cid:105) ) † ( D µ (cid:104) Φ (cid:105) ), which results as follows.The gluons G i are massless and physical fields by themselves. The physical charged gaugebosons with masses are respectively given by W ± ≡ A ∓ iA √ , m W = g u , (10) X ∓ ≡ A ∓ iA √ , m X = g w + u ) , (11) Y ∓∓ ≡ A ∓ iA √ , m Y = g w . (12)The W is like the standard model W boson that yields u (cid:39)
246 GeV. The new gaugebosons X and Y have large masses in w scale, satisfying the relation m X = m Y + m W whichcontrasts to [6] and that in the economical 3-3-1 model [5].The photon field A µ as coupled to the electric charge operator is easily obtained, A µ = s W A µ + c W (cid:18) −√ t W A µ + (cid:113) − t W B µ (cid:19) , (13)where s W = e/g = t/ √ t , with t = g X /g , is the sine of Weinberg angle [23]. Thestandard model Z µ boson and the new neutral gauge boson Z (cid:48) µ can be given orthogonally to A µ as follows [23] Z µ = c W A µ − s W (cid:18) −√ t W A µ + (cid:113) − t W B µ (cid:19) , (14) Z (cid:48) µ = (cid:113) − t W A µ + √ t W B µ . (15)The A µ is a physical field ( m A = 0) and decoupled, whereas there is a mixing between Z and Z (cid:48) given by the squared-mass matrix of the form, m Z m ZZ (cid:48) m ZZ (cid:48) m Z (cid:48) , (16)where m Z = g c W u , m ZZ (cid:48) = g (cid:112) − s W √ c W u , m Z (cid:48) = g [(1 − s W ) u + 4 c W w ]12 c W (1 − s W ) . (17)Therefore, we have two physical neutral gauge bosons (beside the photon), Z = c ϕ Z − s ϕ Z (cid:48) , Z = s ϕ Z + c ϕ Z (cid:48) , (18)with the mixing angle t ϕ = √ − s W ) / u c W w − (1 + 2 s W )(1 − s W ) u (cid:39) √ − s W ) / c W u w . (19)0and their masses m Z = 12 [ m Z + m Z (cid:48) − (cid:113) ( m Z − m Z (cid:48) ) + 4 m ZZ (cid:48) ] (cid:39) g c W u , (20) m Z = 12 [ m Z + m Z (cid:48) + (cid:113) ( m Z − m Z (cid:48) ) + 4 m ZZ (cid:48) ] (cid:39) g c W − s W ) w . (21)Because of ϕ (cid:28)
1, we have Z (cid:39) Z and Z (cid:39) Z (cid:48) . The Z is the standard model like Z boson, while Z is a new neutral gauge boson with the mass in w scale. The mixing between Z and Z (cid:48) was not regarded in [6].The contribution to the experimental ρ -parameter can be calculated as∆ ρ ≡ m W c W m Z − (cid:39) m ZZ (cid:48) m Z m Z (cid:48) (cid:39) (cid:18) − s W c W (cid:19) u w . (22)Taking s W = 0 .
231 and ∆ ρ < . w >
460 GeV. Since the other constraintsyield w in some TeV, we conclude that the ρ -parameter is very close to one and in goodagreement with the experimental data [2]. D. Fermion masses and proton stability
Again, the inert scalars as mentioned do not give the masses for fermions since they haveno VEV and no renormalizable Yukawa interactions due to the Z symmetry. Hence, theinteractions that lead to the fermion masses are given only by the two scalar triplets above, L Y = h J ¯ Q L χJ R + h Jαβ ¯ Q αL χ ∗ J βR + h u a ¯ Q L ηu aR + h uαa Λ ¯ Q αL ηχu aR + h dαa ¯ Q αL η ∗ d aR + h d a Λ ¯ Q L η ∗ χ ∗ d aR + h eab ¯ ψ caL ψ bL η + h (cid:48) eab Λ ( ¯ ψ caL ηχ )( ψ bL χ ∗ )+ s νab Λ ( ¯ ψ caL η ∗ )( ψ bL η ∗ ) + H.c., (23)where the Λ is a new scale (with the mass dimension) under which the effective interactionstake place. It is easily checked that h eab is antisymmetric while s νab is symmetric in the flavorindices. The coupling s ν explicitly violates the lepton number by two unit (as also neededfor a realistic 3-3-1 model), while the other couplings h ’s conserve this charge. Notice thatthe effective interactions for quark and neutrino masses start from five dimensions while forthe charged leptons, it is from six dimensions.1Let us remark on the properties of effective interactions.1. No evidence for a GUT and strength of effective interactions : Since the perturbativeproperty of the U (1) X interaction is broken as well as the Landau pole appears at alow scale of some TeV, the model has no origin from a more-fundamental theory suchas GUTs at a higher energy scale. This contradicts to the case of the standard modeland the 3-3-1 model with right-handed neutrinos. Therefore, we do not have such aGUT to compare and to say about the size of the effective interactions.2. Smallness of neutrino masses : The coupling s ν violates lepton number, so it shouldbe very small in comparison to the conserved ones h ’s for charged leptons and quarks, s ν (cid:28) h ’s (since, by contrast, the conservation of lepton number implies s ν = 0 but h ’s (cid:54) = 0). Therefore, the five-dimensional interaction is reasonably to provide the smallmasses for neutrinos in spite of Λ ∼ w in TeV order, which is unlike the canonicalseesaw scale motivated by GUTs [2] due to the above remark. [Notice that Ref. [6]discussed the cases with respect to five- or seven-dimensional interactions, despite thefact that all the effective interactions of this kind give comparable contributions withΛ ∼ w ]. We conclude that the neutrino masses are generated to be naturally-small asa result of the mentioned approximate symmetry of lepton number, characterized by (cid:15) ≡ s ν /h (cid:28) h ’s.3. Lepton parity and proton stability : The lepton number of lepton triplet ( ψ ) compo-nents, for example, is L = diag(1 , , −
1) which does not commute with the gaugesymmetry. In fact, it is an approximate symmetry. Let us introduce a conservedsymmetry as a remnant subgroup of the lepton number, P = ( − L , (24)so-called lepton parity. The lepton parity for the lepton triplet components is P = diag( − , − , −
1) = − P = diag(1 , ,
1) = 1 for scalar triplets, quarktriplets/antitriplets, P = 1 for right-handed quark singlets, in spite of L ( J ) = ± ψ c L Q L ¯ u c R d R that lead to the proton decay, which is unlike the one in [6].2The mass Lagrangian of quarks and charged leptons takes the form − ¯ f aL m fab f bR + H.c. ,where f = J, u, d, e . We have m J = − h J w/ √ J , while m Jαβ = − h Jαβ w/ √ J , . They all have large masses in w scale. The mass matrices of u and d are respectively obtained as m u a = − h u a u √ , m uαa = − h uαa uw , m dαa = − h dαa u √ , m d a = h d a uw . (25)Because of Λ ∼ w , the ordinary quarks u and d all get masses proportional to the weakscale u = 246 GeV. For top quark, we have m t = − h u ×
174 GeV, provided that h u a isflavor-diagonal. Therefore, m t = 173 GeV if h u ≈
1. On the other hand, the lighter quarks( u, d, c, s, b ) can be explained by h uαβ < h dab < w < Λ which is more naturalthan the standard model. If the first or second generation of quarks was different under SU (3) L , the mass of top quark would be m t = − h u w Λ ×
123 GeV, which is unnatural toachieve an experimental value of 173 GeV due to the fact that h u < w Λ < m eab = √ u (cid:18) h eab + h (cid:48) eba w (cid:19) . (26)Since Λ ∼ w , the charged leptons have masses in the weak scale. Although h e is antisymmet-ric, h (cid:48) e is a generic matrix in generation indices. Therefore, the charged lepton mass matrixtakes the most general form that can provide consistent masses for the charged leptons insimilarity to the case of the standard model.Finally, the mass Lagrangian of neutrinos is given by − ¯ ν caL m νab ν bL + H.c. , where m νab = − s νab u Λ . (27)To proceed further, let us give a comment of the neutrino masses of the model in [6] thatlook like − κ (cid:48) v ρ (cid:0) v χ Λ (cid:1) . This result that was given from a seven-dimensional interaction issimilar in scale to ours as a fact that v χ is close to Λ. Rising in the dimension of effectiveinteractions may not be a reason of smallness of the neutrino masses. Here, we have arguedthat the effective interaction responsible for the neutrino masses violates the lepton numberas a character for the approximate symmetry of this charge (so that the 3-3-1 model isself-consistent). Whereas, all other mass operators do not have this property. On the otherhand, our effective theory does not have a motivation from GUTs and for such case the3effective interaction strengths such as s ν are unknown. Hence, they just appear due to non-perturbative effects to reflect the observed phenomena. Indeed, using Λ = 5 TeV, u = 246GeV and m νab ∼ eV, we have s νab = (cid:15)h ∼ − . Let us choose the Yukawa coupling ofelectron h = h e ∼ − . We get the lepton number violating parameter (cid:15) ∼ − . (28)The strength of the violating interaction for approximate lepton number is reasonably smallin comparison to the ordinary interactions, and this may be the source why the neutrinomasses are observed to be small. E. FCNCs
Let us give an evaluation of tree-level FCNCs that dominantly come from the gaugeinteractions. With the aid of t = g X /g and X = Q − T + √ T , the interaction of neutralgauge bosons is obtained by L NC = − g (cid:88) Ψ ¯Ψ γ µ [ T A µ + T A µ + t ( Q − T + √ T ) B µ ]Ψ , (29)where Ψ runs over every fermion multiplet of the model. There is no FCNC coupled to Q and T since the flavors ν aL , e aL , e aR , u aL , u aR , d aL , d aR , J αL and J αR are respectivelyidentical under these generators. Hence, the FCNCs happen only with T that are given by L T = − g (cid:88) Ψ ¯Ψ γ µ T ( A µ + t √ B µ )Ψ = − g (cid:112) − t W (cid:88) Ψ L ¯Ψ L γ µ T Ψ L Z (cid:48) µ , (30)where we have used the identities A + t √ B = (1 / (cid:112) − t W ) Z (cid:48) and T (Ψ R ) = 0. In thiscase, there is no FCNC associated with the leptons and exotic quarks because the flavors ν aL , e aL , e aR and J αL correspondingly transform the same under T . Therefore, the FCNCsare only concerned to ordinary quarks ( u aL , d aL ) as a fact that under T the third quarkgeneration is different from the first two. The relevant part is L T ⊃ − g (cid:112) − t W [¯ u aL γ µ T ( u aL ) u aL + ¯ d aL γ µ T ( d aL ) d aL ] Z (cid:48) µ = − g (cid:112) − t W (¯ u L γ µ T u u L + ¯ d L γ µ T d d L ) Z (cid:48) µ = − g (cid:112) − t W [¯ u (cid:48) L γ µ ( V † uL T u V uL ) u (cid:48) L + ¯ d (cid:48) L γ µ ( V † dL T d V dL ) d (cid:48) L ] Z (cid:48) µ , (31)4where T u = T d = √ diag( − , − , u = ( u u u ) T , d = ( d d d ) T , u (cid:48) = ( u c t ) T and d (cid:48) = ( d s b ) T . The V uL and V dL take part in diagonalizing the mass matrices of ordinaryquarks, u L = V uL u (cid:48) L , u R = V uR u (cid:48) R , d L = V dL d (cid:48) L and d R = V dR d (cid:48) R , so that V † uL m u V uR =diag( m u , m c , m t ) and V † dL m d V dR = diag( m d , m s , m b ). The CKM matrix is V CKM = V † uL V dL .Hence, the tree-level FCNCs are described by the Lagrangian, L FCNC = − g (cid:112) − t W ( V ∗ qL ) i √ V qL ) j ¯ q (cid:48) iL γ µ q (cid:48) jL Z (cid:48) µ ( i (cid:54) = j ) , (32)where we have denoted q as u either d .With the above result, substituting Z (cid:48) = − s ϕ Z + c ϕ Z , the effective Lagrangian forhadronic FCNCs can be derived via the Z , exchanges as L effFCNC = g [( V ∗ qL ) i ( V qL ) j ] − t W ) (cid:18) s ϕ m Z + c ϕ m Z (cid:19) (¯ q (cid:48) iL γ µ q (cid:48) jL ) . (33)The contribution of Z is negligible since s ϕ /m Z c ϕ /m Z (cid:39) (1 − s W ) c W u w (cid:39) . × u w (cid:28) , (34)provided that s W = 0 .
231 and u (cid:28) w . Therefore, only Z governs the FCNCs and we have L effFCNC (cid:39) [( V ∗ qL ) i ( V qL ) j ] w (¯ q (cid:48) iL γ µ q (cid:48) jL ) . (35)Interestingly enough, this interaction is independent of the Landau pole 1 / (1 − s W ) (thisis also an evidence pointing out that when the theory is encountered with the Landau pole,the effective interactions take place). It describes mixing systems such as K − ¯ K , D − ¯ D , B − ¯ B and B s − ¯ B s , caused by pairs ( q (cid:48) i , q (cid:48) j ) = ( d, s ) , ( u, c ) , ( d, b ) , ( s, b ), respectively. Thestrongest constraint comes from the K − ¯ K system, given by [2][( V ∗ dL ) ( V dL ) ] w < TeV) . (36)Assume that u a is flavor-diagonal. The CKM matrix is just V dL (i.e., V CKM = V dL ). There-fore, | ( V ∗ dL ) ( V dL ) | (cid:39) . × − [2] and we have w > . . (37)This limit is still in the perturbative region of the model [20] and is in good agreement withthe recent bounds [24].5By contrast, if the first or second generation of quarks is arranged differently from thetwo others under SU (3) L , we have | ( V ∗ dL ) ( V dL ) | (cid:39) | ( V ∗ dL ) ( V dL ) | (cid:39) .
22 [2] for both thecases with the K − ¯ K system. Moreover, the new physics scale w is bounded by the Landaupole, w < K − ¯ K systembecomes 1 . × / (10 TeV) that is much greater than the above experimental boundby five order of magnitude. In other words, the experimental bound implies w > . × TeV, provided that the effective interaction (35) works, which contradicts with the fact thatthe model in this region is invalid due to the limit of the Landau pole. Consequently, suchcases should be ruled out due to the large FCNCs that are experimentally unacceptable.The third quark generation should be different from the first two.
III. IMPLICATION FOR DARK MATTER
Let us note that the typical 3-3-1 models [3, 4] are generally supplied with three scalartriplets and/nor one scalar sextet. However, only the two scalar triplets among them (likethe ones given above for the minimal 3-3-1 model or those in [5] for the 3-3-1 model withright-handed neutrinos) are sufficiently for symmetry breaking and mass generation. Hence,we would like to argue that the remaining scalar multiplets or similar ones (which have beendiscarded in the simple versions—the simple 3-3-1 model and the economical 3-3-1 model[5]) can behave as inert multiplets responsible for dark matter. The first work on this searchwas dedicated to the 3-3-1 model with right-handed neutrinos [18].For the case of minimal 3-3-1 model under consideration, the theoretical aspect and darkmatter phenomenology will completely be distinguished from [18] as well as the standardmodel extensions with a singlet, a doublet or a triplet scalar dark matter. For example, inthe model of singlet dark matter, the dark matter interacts with the standard model mattervia only the scalar portal. But, in this model, the singlet dark matter and the standardmodel matter can be coupled via the new gauge portal additionally. Also, the doubletand triplet dark matters can be communicated to the standard model matter by additionalcontributions of new scalars and new gauge bosons.6
A. Simple 3-3-1 model with inert ρ triplet We can introduce into the theory constructed above an extra scalar triplet as ρ = ρ +1 ρ ρ ++3 ∼ (1 , , . (38)This scalar triplet is a part of the minimal 3-3-1 model [3]. However, for the model underconsideration we suppose that it transforms as an odd field under a Z symmetry, ρ → − ρ ,whereas all other fields of the model are even. Therefore, the ρ and its components (includingthe ones proposed below) are all called as inert fields/particles.The normal scalar sector ( η, χ ) which consists of the VEVs, the conditions for parametersand the physical scalars with their masses as obtained above remains unchanged [18]. Forthe inert sector, ρ has vanishing VEVs due to the Z conservation. Moreover, the real andimaginary parts of electrically-neutral complex field ρ = √ ( H ρ + iA ρ ) by themselves arephysical fields. Any one of them can be stabilized if it is the lightest inert particle (LIP)among the inert particles resided in ρ due to the Z symmetry.Unfortunately, we can show that H ρ and A ρ cannot be a dark matter. Indeed, H ρ and H ρ are not separated (degenerate) in mass which leads to a scattering cross-section of H ρ and A ρ off nuclei due to the t-channel exchange by Z boson. Such a large contribution hasalready been ruled out by the direct dark matter detection experiments [25].This kind of model is not favored since it does not provide any dark matter. And, thisis unlike the inert scalar triplet of the 3-3-1 model with right-handed neutrinos [18], eventhought they play equivalently important roles for the typical 3-3-1 models [3, 4]. B. Simple 3-3-1 model with η replication An extra scalar triplet that is a replication of η is defined as η (cid:48) = η (cid:48) η (cid:48)− η (cid:48) +3 ∼ (1 , , . (39)Here, the η (cid:48) and η have the same gauge quantum numbers. However, they differ under a Z symmetry. The η (cid:48) is assigned as an odd field under the Z , η (cid:48) → − η (cid:48) , whereas the η and all7other fields of the simple 3-3-1 model are even.The scalar potential that is invariant under the gauge symmetry and Z is given by V = V simple + µ η (cid:48) η (cid:48)† η (cid:48) + x ( η (cid:48)† η (cid:48) ) + x ( η † η )( η (cid:48)† η (cid:48) ) + x ( χ † χ )( η (cid:48)† η (cid:48) )+ x ( η † η (cid:48) )( η (cid:48)† η ) + x ( χ † η (cid:48) )( η (cid:48)† χ ) + 12 [ x ( η (cid:48)† η ) + H.c. ] (40)Here, µ η (cid:48) has the dimension of mass while x i ( i = 1 , , , ...,
6) are dimensionless. Allthe parameters of the scalar potential are real, except that x can be complex. But, the x ’s phase can be eliminated by redefining the relative phases of η (cid:48) and η . Therefore,this potential conserves the CP symmetry. Moreover, the VEV of η (cid:48) vanishes due to theconservation of Z symmetry. Hence, the CP symmetry is also conserved by the vacuum.All the x , u and w can be considered to be real.Similarly to the previous case, the normal scalar sector ( η, χ ) as identified above thatincludes the minimization conditions, the constraints on u, w , µ ’s, λ ’s and the physicalscalars with respective masses retains unchanged [18]. To make sure that the scalar potentialis bounded from below as well as the Z symmetry is conserved by the vacuum, i.e. (cid:104) η (cid:48) (cid:105) = 0,the remaining parameters of the potential satisfy [18] µ η (cid:48) > , x , > , x + x ± x > . (41)Let us define M η (cid:48) ≡ µ η (cid:48) + x u + x w and η (cid:48) ≡ √ ( H (cid:48) + iA (cid:48) ). It is easily shown that thegauge states H (cid:48) , A (cid:48) , η (cid:48)± and η (cid:48)± by themselves are physical inert particles with the massesrespectively given by m H (cid:48) = M η (cid:48) + 12 ( x + x ) u , m A (cid:48) = M η (cid:48) + 12 ( x − x ) u ,m η (cid:48) = M η (cid:48) , m η (cid:48) = M η (cid:48) + 12 x w . (42)The LIP responsible for dark matter is H (cid:48) if x < Min { , − x , ( w/u ) x − x } , or alterna-tively A (cid:48) if x > Max { , x , x − ( w/u ) x } . Let us consider the case H (cid:48) as the dark mattercandidate (or a LIP). The H (cid:48) transforms as a doublet dark matter under the standard modelsymmetry which is similar to the case of the inert doublet model [26]. However, the H (cid:48) hasa natural mass in the w scale of TeV range. Therefore, this model predicts the large massregion of a doublet dark matter [27]. Its relic density, direct and indirect detections can becalculated to fit the data [28].8 C. Simple 3-3-1 model with χ replication The χ replication has the form χ (cid:48) = χ (cid:48)− χ (cid:48)−− χ (cid:48) ∼ (1 , , − . (43)Let us introduce a Z symmetry so that χ (cid:48) → − χ (cid:48) while all other fields of the simple 3-3-1model are even under this parity. The scalar potential that is invariant under the gaugesymmetry and the Z is given by V = V simple + µ χ (cid:48) χ (cid:48)† χ (cid:48) + y ( χ (cid:48)† χ (cid:48) ) + y ( η † η )( χ (cid:48)† χ (cid:48) ) + y ( χ † χ )( χ (cid:48)† χ (cid:48) )+ y ( η † χ (cid:48) )( χ (cid:48)† η ) + y ( χ † χ (cid:48) )( χ (cid:48)† χ ) + 12 [ y ( χ (cid:48)† χ ) + H.c. ] (44)Similarly to the previous model, we can take y , u and w as real parameters and the CP symmetry is always conserved and unbroken by the vacuum. The normal scalar sectoras obtained retains unchanged. The scalar potential is bounded from below and the Z isconserved by the vacuum if we impose µ χ (cid:48) > , y , > , y + y ± y > . (45)By putting M χ (cid:48) ≡ µ χ (cid:48) + y u + y w and χ (cid:48) ≡ √ ( H (cid:48) + iA (cid:48) ), we have the H (cid:48) , A (cid:48) , χ (cid:48)± and χ (cid:48)±± as physical inert scalar fields by themselves with corresponding masses, m H (cid:48) = M χ (cid:48) + 12 ( y + y ) w , m A (cid:48) = M χ (cid:48) + 12 ( y − y ) w ,m χ (cid:48) = M χ (cid:48) , m χ (cid:48) = M χ (cid:48) + 12 y u , (46)which are all in the w scale of TeV order.Depending on the parameter regime, H (cid:48) or A (cid:48) may be the LIP responsible for dark matter.Let us consider H (cid:48) as the LIP, i.e. y < Min { , − y , ( u/w ) y − y } . The H (cid:48) is a singletdark matter under the standard model symmetry, similarly to the phantom of Silveira-Zeemodel [29, 30]. However, its phenomenology is unique due to the interactions with the newgauge bosons and new Higgs bosons beside the standard model Higgs portal, which lookslike the one in the 3-3-1 model with right-handed neutrinos [18]. It has a natural mass inTeV range, and its relic density as well as the detection cross-sections can be calculated tocompare with the data [28] (see also [18] for the similar ones).9 D. Simple 3-3-1 model with inert scalar sextet
Since the inert scalar multiplets under consideration do not couple to fermions, their U (1) X charges are not fixed. However, these charges must be chosen so that at least onemultiplet component is electrically-neutral for dark matter. Under this view, there are justthree distinct inert scalar triplets ρ , η (cid:48) and χ (cid:48) as already studied. However, there are onlyfive inert scalar sextets since one of them contains up to two electrically-neutral components.In this work, we consider only the two sextets that are correspondingly embedded by thefamiliar scalar triplets with respective hyper-charges Y = (+ / − )1 and Y = 0 under thestandard model symmetry: (6 , X ) = (3 , Y ) ⊕ (2 , Y ) ⊕ (1 , Y ), where Y = −√ T + X can beidentified from the electric charge operator of the model.
1. Inert scalar sextet X = 0 Let us introduce the scalar sextet as often studied in the minimal 3-3-1 model [3] intothe simple 3-3-1 model, S = S S − √ S +13 √ S − √ S −− S √ S +13 √ S √ S ++33 ∼ (1 , , . (47)However, this sextet is odd under a Z symmetry ( S → − S ), while all other fields are even.Notice also that this sextet contains the scalar triplet with Y = − V = V simple + µ S Tr S † S + z (Tr S † S ) + z Tr( S † S ) +( z η † η + z χ † χ )Tr S † S + z η † SS † η + z χ † SS † χ + 12 ( z ηηSS + H.c. ) , (48)where the last terms can explicitly be written as ηηSS = (cid:15) mnp (cid:15) qrs η m η q S nr S ps . To ensurethat the potential is bounded from below as well as the Z symmetry is conserved by thevacuum, i.e. (cid:104) S (cid:105) = 0, we impose µ S > , z > , z > , z + z > ,z + z > , z + 2 z > , z ± z > . (49)0Note that z and the VEVs of η , χ can be chosen to be real due to the CP conservation.Similarly to the above cases, the normal scalar sector as given remains unchanged. Letus put M S ≡ µ S + z u + z w , S ≡ √ ( H S + iA S ) and S ≡ √ ( H (cid:48) S + iA (cid:48) S ). The inertscalar sector yields the physical fields, H S , A S , H (cid:48) S , A (cid:48) S , S ± , S ± ,H ±± = c ζ S ±± − s ζ S ±± , H ±± = s ζ S ±± + c ζ S ±± , (50)where ζ is the S - S mixing angle defined by t ζ = z z u w . The masses of the inert particlesare respectively given by m H S = m A S = M S + 12 z u ,m H (cid:48) S = M S + 14 z w − z u , m A (cid:48) S = M S + 14 z w + 12 z u ,m S = M S + 14 z u , m S = M S + 14 z u + 14 z w ,m H , = M S + 14 z w ∓ (cid:113) z w + 4 z u . (51)All these masses are in the w scale of TeV range.Depending on the parameter space, H S , A S , H (cid:48) S and A (cid:48) S may be dark matter candidates.However, H S and A S belong to the triplet under the standard model symmetry and theyare degenerate in mass. Consequently, they have a t-channel exchange scattering off nucleidue to the contribution of Z boson, which has already been ruled out by the direct darkmatter detection experiments [25], similarly to those in the first dark matter model above.By contrast, H (cid:48) S and A (cid:48) S transform as doublets under the standard model symmetry andare separated in the masses. Unfortunately, they cannot be the LIP because both are muchheavier than the H field: m H (cid:48) S ( A (cid:48) S ) − m H = (cid:112) z w + 4 z u − (+) z u (cid:39) | z | w > H (cid:48) S and A (cid:48) S will rapidly decay that cannot be dark matter [28]. To conclude, the scalarsextet S does not provide realistic dark matter candidates, which is similar to the case ofthe inert triplet model with corresponding scalar triplet as embedded in our sextet [31].To resolve the mass degeneracy of the real and imaginary parts of neutral scalar field inthe sextet (for the current model and even for the inert triplet model) as well as avoid thelarge direct dark matter detection cross-section, let us consider the following model.1
2. Inert scalar sextet X = 1 Let us introduce another sextet with X = 1, σ = σ +11 σ √ σ ++13 √ σ √ σ − σ +23 √ σ ++13 √ σ +23 √ σ +++33 ∼ (1 , , . (52)This sextet is also odd under a Z symmetry, whereas all the other fields are even. It is clearthat the scalar triplet with Y = 0 under the standard model symmetry has been embeddedin the sextet. This scalar triplet has the gauge quantum numbers similarly to the standardmodel gauge triplet, and recently regarded for dark matter [31] (see also [32]).The scalar potential is given by V = V simple + µ σ Tr σ † σ + t (Tr σ † σ ) + t Tr( σ † σ ) +( t η † η + t χ † χ )Tr σ † σ + t η † σσ † η + t χ † σσ † χ + 12 ( t χχσσ + H.c. ) , (53)where all the couplings are real. The results of the normal scalar sector as obtained retain.The potential is bounded from below as well as the Z symmetry is conserved by the vacuumif the new parameters satisfy µ σ > , t + t > , t + t > , t ± t > . (54)Denoting M σ ≡ µ σ + t u + t w and σ ≡ √ ( H σ + iA σ ), we have the physical fields, H σ , A σ , σ ± , σ ±± , σ ±±± ,H ± ≡ c δ σ ± − s δ σ ± , H ± ≡ s δ σ ± + c δ σ ± , (55)where δ is the mixing angle of σ - σ , defined by t δ = − t t w u . The corresponding massesfor the fields are given by m H σ = M σ + 14 t u − t w , m A σ = M σ + 14 t u + 12 t w ,m σ = M σ + 14 t w , m σ = M σ + 14 t u + 14 t w , m σ = M σ + 12 t w ,m H , = M σ + 14 t u ∓ (cid:113) t u + 4 t w (cid:39) M σ + 14 t u ∓ t w ∓ t t u w , (56)2which all have a natural size in the w scale.It is noteworthy that the real and imaginary parts of the neutral scalar field of thestandard symmetry triplet, H σ and A σ , are separated in the masses as a result of the σ - χ interaction via the t coupling. However, the masses of H σ and H as well as those of A σ and H are strongly degenerate, respectively, due to the ( u/w ) (cid:28) | m H ( H ) − m H σ ( A σ ) | (cid:39) (cid:18) t | t | (cid:19) (cid:18) wm H ( H ) + m H σ ( A σ ) (cid:19) (cid:18) . w (cid:19)
10 MeV (cid:46)
10 MeV , (57)which is achieved due to m H ( H ) + m H σ ( A σ ) ∼ w , t ∼ t ∼ u (cid:39)
246 GeV and w > . m H ( H ) − m H σ ( A σ ) (cid:39)
166 MeV . (58)Combining the tree-level (57) and loop (58) results, the charged scalars ( H , H ) are actuallyheavier than the neutral ones ( H σ , A σ ), respectively. [Note that the abnormal interactionssuch as ( η † T i η )Tr( σ † T i σ ) and ( χ † T i χ )Tr( σ † T i σ ) can also contribute to the mass differencesof H σ ( A σ ) and H ( H ), respectively. But, these splitting effects are as small as the onesgiven by the minimal scalar potential, which can be neglected.] Therefore, either the H σ orthe A σ can be regarded as the LIP responsible for dark matter. Without lost of generality,in the following let us consider H σ as the dark matter candidate, i.e. t > Max (cid:26) , − t , (cid:2) t ( u/w ) − t (cid:3) , (cid:2) t ( u/w ) − t (cid:3)(cid:27) . (59)The notable consequences are that the contribution of Z boson to the direct dark matterdetection cross-section is suppressed because of the H σ and A σ mass splitting as well as thevanishing H σ A σ Z interaction due to T = Y = 0 for such scalar fields. The mass splitting of H σ and A σ is also necessary to prevent the Z (cid:48) contribution to such processes because the Z (cid:48) boson actually couples to H σ and A σ , by contrast, due to T (cid:54) = 0 for the scalar fields. Indeed,if the contradiction happened ( t = 0), it would give rise to dangerous contributions naivelyproportional to σ SI Z (cid:48) ∼ (cid:0) uw (cid:1) σ SI Z ∼ − cm that is one up to two orders of magnitude largerthan the best experimental bound σ SIexp ∼ − cm − × − cm [33]. Here, we have3used u = 246 GeV, w = 3 . − σ SI Z ∼ − cm as the cross-section for the caseof the scalar triplet with Y = − Z exchange [32]. IV. AN EVALUATION OF DARK MATTER OBSERVABLES
Along the above discussions, we have found the three dark matter candidates: a singletscalar ( H (cid:48) ), a doublet scalar ( H (cid:48) ) and a triplet scalar ( H σ ) under the standard modelsymmetry. And, they are absolutely stabilized due to the Z symmetries as well as the factthat they are the LIPs. In fact, they could be viable dark matters because there always existcorresponding parameter regimes so that their relic densities, direct and indirect detectioncross-sections are experimentally satisfied. Indeed, considering the parameter regimes thatthe candidates are lightest among the new particles of the corresponding models [12, 18], thedark matter observables are dominantly governed and set by the standard model particles,which have been well-established to be in agreement with the data [27, 30, 31]. To beconcrete, in the following we present for the case of the sextet dark matter.Upon the aforementioned regime, the relic density for H σ includes only the processes thatthe candidate as well as the H (co)annihilate into the standard model particles. They aregoverned by the Higgs and gauge portals with the corresponding interactions given by V ⊃
14 ( H σ + 2 H +1 H − ) (cid:26)(cid:18) t + t (cid:19) h + (cid:20) t + t − λ λ ( t − t ) (cid:21) uh (cid:27) , (60)Tr[( D µ σ ) † ( D µ σ )] ⊃ g H σ W + µ W − µ + g H σ ( H +1 W − µ + H − W + µ ) A µ + g | H +1 W − µ − H − W + µ | + g H +1 H − A µ A µ + igH +1 ↔ ∂ µ H − A µ + [ igH σ ↔ ∂ µ H − W + µ + H.c. ] , (61)where we have denoted F ↔ ∂ µ F ≡ F ( ∂ µ F ) − ( ∂ µ F ) F for any F , fields, and A µ = s W A µ + c W Z µ . The modification to the coupling of one h with two inert particles is dueto the h - H mixing, which is at u/w order. However, we have neglected the mixing effectof Z with Z (cid:48) as well as the contribution of the new particles such as H and Z (cid:48) because of u (cid:28) w and the above assumption for the dark matter candidate.There are various channels that might contribute to the relic density such as H σ H σ → hh, tt c , W + W − , ZZ as well as the co-annihilations H σ H ± → ZW ± , AW ± , t ± / b ± / and H ± H ∓ → hh, tt c , W + W − , ZZ, ZA, AA . They are given by the diagrams in Fig. 1 andFig. 2 with respect to the Higgs and gauge portals, respectively. The annihilation cross-4 H σ ( H +1 ) H σ ( H − ) hhhH σ ( H +1 ) H σ ( H − ) hh hH σ ( H +1 ) H σ ( H − ) tt c hH σ ( H +1 ) H σ ( H − ) W − , ZW + , ZH σ hH σ hH σ H +1 hH − hH FIG. 1: Contributions to H σ and/or H ± annihilation via the Higgs portal when they are lighterthan the new particles of the simple 3-3-1 model. There are additionally two u -channels that canbe derived from the corresponding t -channels above. section times relative velocity is defined as (cid:80) ij σ ( H i H j → SM particles) v ij , where i, j = σ, v ij is the relative velocity of the two incoming particles H i and H j . Using the limit m H σ (cid:39) m H ∼ w (cid:29) u ∼ m SM (the relevant masses for the standard model particles) as wellas the freeze-out temperature T F (cid:39) m Hσ (cid:28) m H σ as usual [34], we obtain the leading orderterm for the effective, thermally-averaged annihilation cross-section times velocity, (cid:104) σv (cid:105) (cid:39) α (150 GeV) (cid:34)(cid:18) . m H σ (cid:19) + (cid:18) λ × .
782 TeV m H σ (cid:19) (cid:35) , (62)where the first term in the brackets comes from the gauge portal while the second one isdue to the Higgs portal, λ ≡ t + t /
2, in agreement with [32]. For the above result, we have5 H σ H ± W ± A WH σ H ± W ± A WH σ H ± t ± / b ± / H H σ H ± W ± A H σ ( H +1 ) H σ ( H − ) W − W + H +1 ( H − ) H +1 ( H − ) W + ( W − ) W + ( W − ) H +1 H − A A A H +1 H − W − W + A H +1 H − tt c H H +1 H − A A H σ H +1 H − W + W − H H σ H σ W + W − FIG. 2: Contributions to H σ and/or H ± annihilation via the gauge portal when they are lighterthan the new particles of the simple 3-3-1 model. There remain the u -channel contributions for H +1 H − → A A and H σ H σ → W + W − , respectively, which can be extracted from the correspond-ing t-channel diagrams above. used s W = 0 . α = 1 / α / (150 GeV) (cid:39) h (cid:39) . (cid:104) σv (cid:105) (cid:39) .
11 (where the h is thereduced Hubble constant) [2, 34] that implies m H σ (cid:39) √ .
29 + 0 . λ TeV . (63)If the dark matter–scalar coupling is small λ = t + t / (cid:28)
1, the gauge portal governs theannihilation processes of the dark matter. Simultaneously, the dark matter gets the rightabundance if it has a mass m H σ (cid:39) . λ (cid:38)
1, the Higgs portal gives equivalent contributions, even dominatesover the gauge one. In this case, the dark matter mass depends on the λ parameter asgiven above in order to recover the right abundance. Due to the limit by the Landau pole,say m H σ < λ < .
68 for the right abundance), the H σ can onlycontribute as a part of the total dark matter relic density, provided that the coupling λ islarge, λ > .
68. In other words, it is only a dark matter component coexisted with otherpotential candidates, which may be a singlet H (cid:48) and/or a doublet H (cid:48) as determined before.The direct searches for the candidate H σ measure the recoil energy deposited by the H σ when it scatters off the nuclei of a large detector. This proceeds through the interactionof H σ with the partons confined in nucleons. Because the H σ is very non-relativistic, theprocess can be obtained by an effective Lagrangian as [35] L eff = 2 λ q m H σ H σ H σ ¯ qq, (64)where the scalar candidate has only spin-independent and even interactions (the interactionswith gluons are loops induced that should be small). The above effective interaction isachieved by the t -channel diagram as mediated by the Higgs boson as Fig. 3. It follows λ q = λ (cid:48) m q m H σ m h , λ (cid:48) ≡ t + t − λ λ ( t − t ) , (65)where the scalar coupling λ (cid:48) that governs the scattering cross-section differs from the λ thatoperates the annihilation cross-section. This separation is due to the term ∼ t − t raisedas a result of the h - H mixing. Hence, the relic density and the direct detection cross-sectionare obviously not correlated, which is a new observation of this work.The H σ -nucleon scattering amplitude is obtained by summing over the quark level in-teractions multiplied by the corresponding nucleon form factors. Thus, the H σ -nucleon7 H σ H σ q qh FIG. 3: Dominant contributions to H σ -quark scattering. cross-section takes the form, σ H σ − N = 4 m r π λ N , N = p, n, (66)where m r ≡ m H σ m N m H σ + m N (cid:39) m N , λ N m N = (cid:88) u,d,s f NT q λ q m q + 227 f NT G (cid:88) c,b,t λ q m q (cid:39) . λ (cid:48) m H σ m h , (67)where f NT G = 1 − (cid:80) u,d,s f NT q as well as the f NT q values were given in [36]. With m N = 1 GeVand m h = 125 GeV [2], we have σ H σ − N (cid:39) (cid:18) . λ (cid:48) TeV m H σ (cid:19) × − cm , (68)which coincides with the current experimental bound σ H σ − N (cid:39) − cm , provided that m H σ (cid:39) . λ (cid:48) TeV in the TeV range [2, 33]. Simultaneously, the H σ can get the rightabundance by this case if we impose λ (cid:48) (cid:39) m H σ / (2 .
494 TeV) (cid:39) √ .
85 + 0 . λ (cid:39) . ÷ | λ | < .
68 as mentioned. Of course, the direct detectioncross-section can also be assigned to a smaller value if the coupling λ (cid:48) is appropriately chosenfor each fixed dark matter mass. V. CONCLUSION
Our aim was to look for a realistic 3-3-1 model with the minimal lepton and scalarcontents in order to solve the dark matter problem of the minimal 3-3-1 model [3] under theguidance of the work in [18]. However, there was not such a theory in the literature despite8the fact that the reduced 3-3-1 model was introduced in [6]. And, for us it has been whatto be investigated in this work.First of all, we have shown that even for a minimal 3-3-1 model with reduced scalarsector the third generation of quarks should transform under SU (3) L differently from thefirst two. This is due to the low limit of some TeV for the Landau pole. In addition, it iswell-known that the mass corrections for some vanishing tree-level quark masses which comefrom quantum effects or effective interactions are subleading. Therefore, the reduced scalarsector must be η and χ (no other case) so that the top quark appropriately gets a tree-leveldominant mass. The simple 3-3-1 model that has been given by such minimal fermion andscalar contents is unique and entirely different from the previous one [6].We have also shown that there are eight Goldstone bosons correspondingly eaten by eightmassive gauge bosons. There remain four physical Higgs bosons h , H and H ± . Here the h is like the standard model Higgs boson with mass in the weak scale while H and H ± are the new heavy Higgs bosons with masses in w scale. Also, there is a small mixingbetween the standard model Higgs boson and the new one, S - S . Our model consists ofonly singly-changed Higgs bosons, not doubly-changed ones as in [6].There are two new heavy charged gauge bosons with the masses in w scale satisfying therelation m X ± = m Y ±± + m W ± , which is unlike [6]. There is a mixing between the standardmodel Z boson and the new neutral gauge boson Z (cid:48) , which was neglected in [6]. The newphysical neutral gauge boson Z has a mass in w scale. From the W mass, we have u (cid:39) ρ parameter, we get w >
460 GeV.Because of the minimal scalar sector, some fermions have vanishing masses at the tree-level. However, they can get corrections coming from the effective interactions. The quarksget consistent masses via the five-dimensional effective interactions, while the charged leptonsgain masses via four- and six-dimensional interactions. The neutrino masses are generated tobe naturally small as a consequence of approximate lepton number symmetry of the model.Notice that the model is only consistent by this way of the lepton charge.Although the lepton charge is an approximate symmetry, we can always find in the theorya conserved residual charge—the lepton parity ( − L . The conservation of lepton parity isjust mechanism for the proton stability. Notice that the model always conserves the globalbaryon charge U (1) B . This may also be regarded as a mechanism for the proton stability.We have calculated the hadronic FCNCs due to the exchange of Z (cid:48) . It is interesting9that the FCNCs are independent of the Landau pole. We have indicated that the strongestconstraint coming from K − ¯ K system can be evaded provided that w > . ρ and S as often studied in the minimal 3-3-1 model, η (cid:48) and χ (cid:48) as the replicationsof the normal ones, the variants of S such as σ as well as the new forms, can be consideredas the inert sectors providing dark matter candidates. We have shown that the simple 3-3-1model with the inert scalar triplet ρ does not contain any realistic dark matter. However,the simple 3-3-1 model with the η or χ replication can yield a doublet dark matter H (cid:48) ora singlet dark matter H (cid:48) , respectively. The simple 3-3-1 model with the inert scalar sextet X = 0 does not provide any realistic dark matter. However, the model with the inert scalarsextet X = 1 can give a triplet dark matter H σ . The dark matter candidates as obtainedcan communicate with the standard model matter via the new Higgs and new gauge bosonsbesides the normal portals as in the ordinary inert triplet and inert doublet models as wellas the Silveira–Zee model.We have pointed out that the parameter spaces of the corresponding dark matter modelscan always contain appropriate parameter regimes so that the dark matter candidates asfound are viable under the data. To be concrete, we have made an evaluation of the impor-tant dark matter observables for the sextet model that possesses the triplet scalar candidate( H σ ). This H σ gets a right abundance if it has a mass as m H σ (cid:39) √ .
29 + 0 . λ TeV (cid:39) . ÷ | λ | < .
68, where the annihilation cross-sections are operated by both theHiggs and gauge portals. The direct detection cross-section, which is governed by anotherscalar coupling λ (cid:48) , is in good agreement with the experiments for the dark matter mass inTeV range. Taking the experimental bound as σ H σ − N (cid:39) − cm , the dark matter massis constrained to be m H σ (cid:39) . λ (cid:48) TeV. The direct detection bound and right abundanceare simultaneously satisfied if λ (cid:48) (cid:39) √ .
85 + 0 . λ (cid:39) . ÷ | λ | < . Acknowledgments
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