Higgs and gauge boson phenomenology of the 3-3-1 model with CKS mechanism
H. N. Long, N. V. Hop, L. T. Hue, N. H. Thao, A. E. Cárcamo Hernández
SSome phenomenological aspects of the 3-3-1 model with CKSmechanism
H. N. Long,
1, 2, ∗ N. V. Hop, † L.T. Hue,
4, 5, ‡ N. H. Thao, § and A. E. C´arcamo Hern´andez ¶ Theoretical Particle Physics and Cosmology Research Group,Advanced Institute of Materials Science,Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam Department of Physics, Can Tho University,3/2 Street, Can Tho 900000, Vietnam Institute for Research and Development,Duy Tan University, Da Nang City 550000, Vietnam Institute of Physics, Vietnam Academy of Science and Technology,10 Dao Tan, Ba Dinh, Hanoi 100000, Vietnam Department of Physics, Hanoi Pedagogical University 2,Phuc Yen, Vinh Phuc 280000, Vietnam Universidad T´ecnica Federico Santa Mar´ıa and Centro Cient´ıfico-Tecnol´ogico de Valpara´ıso,Casilla 110-V, Valpara´ıso, Chile (Dated: June 11, 2019)We perform a comprehensive analysis of several phenomenological aspects of therenormalizable extension of the inert 3-3-1 model with sequentially loop-generatedSM fermion mass hierarchy. Special attention is paid to the study of the constraintsarising from the experimental data on the ρ parameter, as well as those ones resultingfrom the charged lepton flavor violating process µ → eγ and dark matter. Wealso study the single Z (cid:48) production via Drell-Yan mechanism at the LHC. We havefound that Z (cid:48) gauge bosons heavier than about 4 TeV comply with the experimentalconstraints on the oblique ρ parameter as well as with the collider constraints. Inaddition, we have found that the constraint on the charged lepton flavor violatingdecay µ → eγ sets the sterile neutrino masses to be lighter than about 1 .
12 TeV. a r X i v : . [ h e p - ph ] J un In addition the model allows charged lepton flavor violating processes within reachof the forthcoming experiments. The scalar potential and the gauge sector of themodel are analyzed and discussed in detail. Our model successfully accommodatesthe observed Dark matter relic density.
PACS numbers: 12.60.Cn,12.60.Fr
Keywords : Extensions of electroweak gauge sector, Extensions of electroweak Higgssector
I. INTRODUCTION
Despite its great successes, the Standard Model (SM) does not explain the observed massand mixing hierarchies in the fermion sector, which remain without a compelling explanation.It is known that in the SM, the masses of the matter fields are generated from the Yukawainteractions. In addition, the CKM quark mixing matrix is also constructed from the sameYukawa couplings. To solve these puzzles, some mechanisms have been proposed. To thebest of our knowledge, the first attempt to explain the huge differences in the SM fermionmasses is the Froggatt - Nielsen (FN) mechanism [1]. According to the FM mechanism, themass differences between generations of fermions arise from suppression factors dependingon the FN charges of the particles. It has been noticed that in order to implement theaforementioned mechanism, the effective Yukawa interactions have to be introduced, thusmaking this theory non-renormalizable. From this point of view, the recent mechanismproposed by C´arcamo, Kovalenko and Schmidt [2] (called by CKS mechanism) based onsequential loop suppression mechanism, is more natural since its suppression factor arisesfrom the loop factor l ≈ (1 / π ) .One of the main purposes of the models based on the gauge group SU (3) C × SU (3) L × U (1) X (for short, 3-3-1 model) [3–10] is concerned with the search of an explanation for ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] the number of generations of fermions. Combined with the QCD asymptotic freedom, the3-3-1 models provide an explanation for the number of fermion generations. These modelshave nonuniversal U (1) X gauge assignments for the left handed quarks fields, thus implyingthat the cancellation of chiral anomalies is fulfilled when the number of SU (3) L fermionictriplets is equal to the number of SU (3) L fermionic antitriplets, which happens when thenumber of fermion families is a multiple of three. Some other advantages of the 3-3-1 modelsare: i) they solve the electric charge quantization [11, 12], ii) they contain several sources ofCP violation [13, 14], and iii) they have a natural Peccei-Quinn symmetry, which solves thestrong-CP problem [15–18].In the framework of the 3-3-1 models, most of the research is focused on radiative seesawmechanisms, and but some involving nonrenormalizable interactions introduced to explainthe SM fermion mass and mixing pattern (see references in Ref.[19]).The FN mechanism was implemented in the 3-3-1 models in Ref.[20]. It is interestingto note that the FN mechanism does not produce a new scale since the scale of the flavourbreaking is the same as the symmetry breaking scale of the model.The CKS mechanism has been implemented for the first time in the 3-3-1 model with-out exotic electric charges ( β = − / √
3) in Ref. [19]. The implementation of the CKSmechanism in the 3-3-1 model leads to viable renormalizable 3-3-1 model that provides adynamical explanation for the observed SM fermion mass spectrum and mixing parametersconsistent with the SM low energy fermion flavor data [19]. It is worth mentioning thatthe extension of the inert 331 model of Ref. [19] contains a residual discrete Z ( L g )2 leptonnumber symmetry arising from the spontaneous breaking of the global U (1) L g symmetry.Under this residual symmetry, the leptons are charged and the other particles are neutral[19].However, in the mentioned work, the authors have just focused on the data concerningfermions (both quarks and leptons including neutrino mass and mixing), but some questionsare open for the future study.The purpose of this work is to study several phenomenological aspects of the renormal-izable extension of the inert 3-3-1 model with sequentially loop-generated SM fermion masshierarchy. In particular, the constraints arising from the experimental data on the ρ param-eter, as well as those ones resulting from the charged lepton flavor violating process µ → eγ and dark matter. Furthermore our work discusses the Z (cid:48) production at proton-proton col-lider via quark-antiquark annihilation. To determine the oblique ρ parameter constraintson the SU (3) L × U (1) X symmetry breaking scale v χ , which will be used to constrain theheavy Z (cid:48) gauge boson mass, we proceed to study in detail the gauge and Higgs sectors ofthe model. In addition we determine the constraints imposed by the charged lepton flavorviolating process µ → eγ and dark matter on the model parameter space. In what regardsthe scalar potential of the model, due to the implemented symmetries, the Higgs sector israther simple and can be completely solved. All Goldstone bosons and the SM like Higgsboson are defined.The further content of this paper is as follows. In Sect. II, we briefly present particlecontent and SSB of the model. Sect. III is devoted to gauge boson mass and mixing. Takinginto account of data on the ρ parameter, and if only contributions of the gauge bosons arementioned, we will show that the mass of the heavy neutral boson Z (cid:48) will be constrainednearly to the excluded regions derived from other experimental data such as LHC searches,and K, D and B meson mixing. The Higgs sector is considered in Sect. IV. The Higgs sectorconsists of two parts: the first part contains lepton number conserving terms and the secondone is lepton number violating. We study in details the first part and show that the Higgssector has all necessary ingredients. The ρ parameter will be investigated including Higgscontributions. In Sec. V, lepton flavor violating decays of the charged leptons are discussed,where sterile neutral lepton masses are constrained. Sect. VI is devoted to the productionof the heavy Z (cid:48) and the heavy neutral scalar H . In Sect. VII, we deal with the DM relicdensity. We make conclusions in Sect. VIII. The scalar potential of the model is given inAppendix A. II. REVIEW OF THE MODEL
To implement the CKS mechanism, only the heaviest particles such as the exotic fermionsand the top quark get masses at tree level. The next - medium ones: bottom, charm quarks,tau and muon get masses at one-loop level. Finally, the lightest particles: up, down, strangequarks and the electron acquire masses at two-loop level. To forbid the usual Yukawainteractions, the discrete symmetries should be implemented. Hence, the full symmetry ofthe model under consideration is SU (3) C × SU (3) L × U (1) X × Z × Z × U (1) L g , (1)where L g is the generalized lepton number defined in Refs. [19, 21]. It is interesting to notethat, in this model, the light active neutrinos get their masses from a combination of linearand inverse seesaw mechanisms at two-loop level.As in the ordinary 3-3-1 model without exotic electric charges, the quark sector containsthe following SU (3) C × SU (3) L × U (1) X representations [19] Q nL = ( D n , − U n , J n ) TL ∼ (3 , ∗ , , Q L = ( U , D , T ) TL ∼ (cid:18) , , (cid:19) , n = 1 , ,D iR ∼ (cid:18) , , − (cid:19) , U iR ∼ (cid:18) , , (cid:19) , i = 1 , , ,J nR ∼ (cid:18) , , − (cid:19) , T R ∼ (cid:18) , , (cid:19) , (cid:101) T L,R ∼ (cid:18) , , (cid:19) , B L,R ∼ (cid:18) , , − (cid:19) , (2)where ∼ denotes the quantum numbers for the three above subgroups, respectively. Notethat the SU (3) L singlet exotic up type quarks (cid:101) T L,R , down type quarks B L,R in the last line ofEq. (2) are added to the quark spectrum of the ordinary 3-3-1 model in order to implementthe CKS mechanism.In the leptonic sector, besides the usual SU (3) L lepton triplets, the model contains extrathree charged leptons E j ( L,R ) ( j = 1 , ,
3) and four neutral leptons, i.e, N jR and Ψ R ( j =1 , , SU (3) C × SU (3) L × U (1) X assignments: L iL = ( ν i , e i , ν ci ) TL ∼ (cid:18) , , − (cid:19) , e iR ∼ (1 , , − , i = 1 , , , (3) E L ∼ (1 , , − , E L ∼ (1 , , − , E L ∼ (1 , , − ,E R ∼ (1 , , − , E R ∼ (1 , , − , E R ∼ (1 , , − ,N R ∼ (1 , , , N R ∼ (1 , , , N R ∼ (1 , , , Ψ R ∼ (1 , , . (4)where ν iL , ν c ≡ ν cR and e iL ( e L , µ L , τ L ) are the neutral and charged lepton families, respec-tively.The Higgs sector contains three scalar triplets: χ , η and ρ and seven singlets ϕ , ϕ , ξ , φ +1 , φ +2 , φ +3 and φ +4 . Hence, the scalar spectrum of the model is composed of the followingfields χ = (cid:104) χ (cid:105) + χ (cid:48) ∼ (cid:18) , , − (cid:19) , (5) (cid:104) χ (cid:105) = (cid:18) , , v χ √ (cid:19) T , χ (cid:48) = (cid:18) χ , χ − , √ R χ − iI χ ) (cid:19) T , TABLE I: Scalar assignments under Z × Z χ η ρ ϕ ϕ φ +1 φ +2 φ +3 φ +4 ξ Z − − i i − − Z − − − − L of fields T L,R J L,R J L,R ν ciL e iL,R E iL,R N iR Ψ R χ χ +2 η ρ +3 φ +2 φ +3 φ +4 ξ i = 1 , , L − − − − − − − − − ρ = (cid:18) ρ +1 , √ R ρ − iI ρ ) , ρ +3 (cid:19) T ∼ (cid:18) , , (cid:19) ,η = (cid:104) η (cid:105) + η (cid:48) ∼ (cid:18) , , − (cid:19) , (cid:104) η (cid:105) = (cid:18) v η √ , , (cid:19) T , η (cid:48) = (cid:18) √ R η − iI η ) , η − , η (cid:19) T ,ϕ ∼ (1 , , , ϕ ∼ (1 , , ,φ +1 ∼ (1 , , , φ +2 ∼ (1 , , , φ +3 ∼ (1 , , , φ +4 ∼ (1 , , ,ξ = (cid:104) ξ (cid:105) + ξ (cid:48) , (cid:104) ξ (cid:105) = v ξ √ , ξ (cid:48) = 1 √ R ξ − iI ξ ) ∼ (1 , , . (6)The Z × Z assignments of scalar the fields are shown in Table I.The fields with nonzero lepton number are presented in Table II. Note that the threegauge singlet neutral leptons N iR as well as the elements in the third component of thelepton triplets, namely ν ciL have lepton number equal − χ and ξ scalar fields. At this step, all new extra fermions, non-SM gauge bosonsas well as the electrically neutral gauge singlet lepton Ψ R gain masses. In addition, theentries of the neutral lepton mass matrices with negative lepton number ( −
1) also getvalues proportional to v ξ . At this step, the initial group breaks down to the direct productof the SM gauge group and the Z × Z ( L g )2 discrete group. The second step is triggered by v η providing masses for the top quark as well as for the W and Z gauge bosons and leaving the SU (3) C × U (1) Q × Z × Z ( L g )2 symmetry preserved. Here Z ( L g )2 is residual symmetry whereonly leptons are charged, thus forbidding interactions having an odd number of leptons.This is crucial to guarantee the proton stability [19]. Thus SU (3) C × SU (3) L × U (1) X × Z × Z × U (1) L g v χ ,v ξ −−−→ SU (3) C × SU (2) L × U (1) Y × Z × Z ( L g )2 v η −→ SU (3) C × U (1) Q × Z × Z ( L g )2 . (7)A consequence of the chain in (7) is v η = v = 246GeV (cid:28) v χ ∼ v ξ ∼ O (10) TeV . (8)The corresponding Majoron associated to the spontaneous breaking of the U (1) L g globalsymmetry is a gauge-singlet scalar and, therefore, unobservable.An explanation for the relation v χ ∼ v ξ ∼ O (10) TeV is provided in the following. Thepresent lower limits on the Z (cid:48) gauge boson mass in 3-3-1 models arising from LHC searches,reach around 2 . SU (3) C × SU (3) L × U (1) X gauge symmetry breaking scale v χ . Furthermore, electroweakdata from the decays B s,d → µ + µ − and B d → K ∗ ( K ) µ + µ − set lower bounds on the Z (cid:48) gaugeboson mass ranging from 1 TeV up to 3 TeV [23–27]. Furthermore, as shown in Ref. [28],the experimental data on K , D and B meson mixings set a lower bound of about 4 TeV forthe Z (cid:48) gauge boson mass in 3-3-1 models, which translates in a lower limit of about 10 TeVfor the SU (3) L × U (1) X gauge symmetry breaking scale v χ .Finally, to close this section we provide a justification of the role of the different particlesof our model:1. The presence of the SU (3) L scalar singlet φ +3 , is needed to generate two loop level downand strange quark masses, as shown in Ref. [19]. Besides that, in order to implementa two loop level radiative seesaw mechanism for the generation of the up, down andstrange quark masses as well as the electron mass, the Z charged SU (3) L scalarsinglets ϕ , ϕ , φ +1 , φ +2 (which do not acquire a vacuum expectation value) are alsorequired in the scalar sector. The Z charged SU (3) L scalar singlet ϕ is also neededto generate one loop level masses for the charm and bottom quarks as well as for thetau and muon leptons. The Z charged SU (3) L scalar singlets ϕ and φ +3 as well as the SU (3) L scalar singlet φ +4 , neutral under Z are also crucial for the implementation oftwo loop level linear and inverse seesaw mechanisms that give rise to the light activeneutrino masses. The SU (3) L scalar singlet ξ is introduced to spontaneously breakthe U (1) L g generalized lepton number symmetry and thus giving rise to a tree-levelmass for the right handed Majorana neutrino Ψ R . It is crucial for generating twoloop-level masses for the down and strange quarks.2. The SU (3) L singlet exotic down type quark, i.e. B , is crucial for the implementation ofthe one loop level radiative seesaw mechanism that generate the bottom quark mass.The SU (3) L singlet exotic up type quarks, i.e., (cid:101) T and (cid:101) T , are needed to generate aone loop level charm quark mass as well as two loop level down and strange quarkmasses. The three SU (3) L singlet exotic charged leptons, i.e., E j ( j = 1 , , N jR ( j = 1 , , R , are crucial for the implementation ofthe two loop level linear and inverse seesaw mechanisms that give rise to the lightactive neutrino masses. III. GAUGE BOSONSA. Gauge boson masses and mixing
After SSB, the gauge bosons get masses arising from the kinetic terms for the η and χSU (3) L scalar triplets, as follows: L gaugemass = ( D µ (cid:104) χ (cid:105) ) † D µ (cid:104) χ (cid:105) + ( D µ (cid:104) η (cid:105) ) † D µ (cid:104) η (cid:105) , (9)with the covariant derivative for triplet defined as D µ = ∂ µ − igA µa λ a − ig X X λ B µ , (10)where g and g X are the gauge coupling constants of the SU (3) L and U (1) X groups, respec-tively. Here, λ = (cid:112) / , ,
1) is defined such that Tr( λ λ ) = 2, similarly as theusual Gell-Mann matrix λ a , a = 1 , , , · · · ,
8. By matching gauge the coupling constants atthe SU (3) L × U (1) X symmetry breaking scale, the following relation is obtained [9] t ≡ g X g = 3 √ θ W ( M Z (cid:48) ) (cid:112) − θ W ( M Z (cid:48) ) . (11)Let us provide the definition of the Weinberg angle θ W . As in the SM, one puts g (cid:48) = g tan θ W , where g (cid:48) is gauge coupling of the U (1) Y subgroup satisfying the relation [9] g (cid:48) = √ gg X (cid:112) g − g X . (12)Thus tan θ W = √ g X (cid:112) g − g X . (13)Denoting W ± µ = 1 √ A µ ∓ iA µ ) , Y ± µ = 1 √ A µ ± iA µ ) , X µ = 1 √ A µ − iA µ ) , (14)and substituting (10) and (14) into (9) one gets the following squared masses for thecharged/non-Hermitian gauge bosons m W = g v η , M X = g (cid:0) v χ + v η (cid:1) , M Y = g v χ , (15)where v η = v = 246 GeV, as expected.From Eq.(15) we find the following gauge boson mass squared splitting M X − M Y = m W . (16)For neutral gauge bosons, the squared mass mixing matrix has the form L ngaugemass = 12 V T M ngauge V , (17)where V T = ( A µ , A µ , B µ ) and M ngauge = g v η v η √ − t √ v η (4 v χ + v η ) t √ (2 v χ − v η ) t ( v χ + v η ) . (18)The down-left entries in (18) are not written, due to the fact that the above matrix issymmetric.The matrix in (18) has vanishing determinant, thus giving rise to a massless gauge boson,which corresponds to the photon. The diagonalization of the squared mass matrix for neutralgauge bosons of Eq. (18) is divided in two steps. In the first step, the massive fields areidentified as A µ = s W A µ + c W (cid:32) − t W √ A µ + (cid:114) − t W B µ (cid:33) , Z µ = c W A µ − s W (cid:32) − t W √ A µ + (cid:114) − t W B µ (cid:33) , (19) Z (cid:48) µ = (cid:114) − t W A µ + t W √ B µ , where we have denoted s W = sin θ W , c W = cos θ W , t W = tan θ W . The coupling of thephoton A µ gives e = gs W . After the first step, the squared mass matrix is a block diagonalone in the new basis ( A µ , Z µ , Z (cid:48) µ ), where the entry in the top is zero (due to the masslessnessof the photon), while the 2 × Z µ , Z (cid:48) µ ) in the bottom has the form M × = M Z M ZZ (cid:48) M ZZ (cid:48) M Z (cid:48) . (20)The matrix elements in (20) are given by M Z = g v η c W = m W c W , (21) M ZZ (cid:48) = g c W (cid:112) − s W v η (1 − s W ) ,M Z (cid:48) = g c W − s W ) (cid:20) v χ + v η (1 − s W ) c W (cid:21) . Note that our formula of M Z (cid:48) is consistent with that given in [23].In the last step of diagonalization, the Z − Z (cid:48) mixing angle φ and mass eigenstates Z , are determined as tan 2 φ = 2 M ZZ (cid:48) M Z (cid:48) − M Z , (22) Z µ = Z µ cos φ − Z (cid:48) µ sin φ ,Z µ = Z µ sin φ + Z (cid:48) µ cos φ . (23)Our definition of φ is consistent with that in Ref. [29] needed to study the ρ parameter.The masses of physical neutral gauge bosons are determined as M Z = 12 (cid:110) M Z (cid:48) + M Z − (cid:2) ( M Z (cid:48) − M Z ) + 4( M ZZ (cid:48) ) (cid:3) (cid:111) ,M Z = 12 (cid:110) M Z (cid:48) + M Z + (cid:2) ( M Z (cid:48) − M Z ) + 4( M ZZ (cid:48) ) (cid:3) (cid:111) . (24)In the limit v χ (cid:29) v η , one approximates M Z (cid:39) M Z − ( M ZZ (cid:48) ) M Z (cid:48) + M Z × O (cid:18) v η v χ (cid:19) , (25)1 M Z (cid:39) M Z (cid:48) + ( M ZZ (cid:48) ) M Z (cid:48) + M Z × O (cid:18) v η v χ (cid:19) (cid:39) M Z (cid:48) . (26)tan φ (cid:39) (1 − s W ) (cid:112) − s W c W (cid:18) v η v χ (cid:19) . (27) B. Limit on Z (cid:48) mass from the ρ parameter The presence of the non SM particles modifies the oblique corrections of the SM, thevalues of which have been extracted from high precision experiments. Consequently, thevalidity of our model depends on the condition that the non SM particles do not contradictthose experimental results. Let us note that one of the most important observables in theSM is the ρ parameter defined as ρ = m W c W M Z . (28)For the model under consideration, one-loop contributions of the new heavy gauge bosonsto the oblique correction lead to the following form of the ρ parameter [29] ρ − (cid:39) tan φ (cid:18) M Z (cid:48) m Z − (cid:19) + 3 √ G F π (cid:20) M + M + 2 M M M − M ln M M (cid:21) − α ( m Z )4 π s W (cid:20) t W ln M M + ε ( M + , M )2 + O ( ε ( M + , M )) (cid:21) , (29)where M = M X , M + = M Y + and ε ( M, m ) ≡ M − m m .Combining with Eq. (16), one gets ρ − (cid:39) tan φ (cid:18) M Z (cid:48) m Z − (cid:19) + 3 √ G F π (cid:20) M Y + + m W − M Y + ( M Y + + m W ) m W ln ( M Y + + m W ) M Y + (cid:21) − α ( m Z )4 π s W (cid:20) t W ln ( M Y + + m W ) M Y + + m W M Y + + m W ) (cid:21) , (30)where α ( m Z ) ≈ [30].Taking into account s W = 0 . ρ = 1 . ± . , (31)we have plotted ∆ ρ as a function of v χ in Fig. 1 (the left-panel). From figure 1 (theleft-panel), it follows 3 .
57 TeV ≤ v χ ≤ .
09 TeV . (32)2 ρ ( v (cid:1) ) ρ max ρ min v (cid:0) ( TeV ) ρ v χ ( m Z22 ) v χ max v χ min m Z22 ( TeV ) v χ ( T e V ) FIG. 1: Left-panel: ρ parameter as a function of v χ , upper and a lower horizontal lines are anupper a lower limits in (31) . Right-panel: Relation between v χ and M Z , upper horizontal linesare an upper and a lower limits of v χ , respectively. Substituting (32) into (26) and evaluating in figure 1(the right-panel) we get a bound onthe Z (cid:48) mass as follows 1 .
42 TeV ≤ M Z ≤ .
42 TeV . (33)Then, the bilepton gauge boson mass is constrained to be in the range:465 GeV ≤ M Y ≤
960 GeV , (34)where m W = 80 .
379 GeV [30]. The above limit is stronger than the one obtained from thewrong muon decay M Y ≥
230 GeV [31].It is worth mentioning that the second term in (30) is much smaller the first one. Conse-quently, the limit derived from the tree level contribution is very close to the one obtainedwhen we consider the radiative corrections arising from heavy vector exchange.From LHC searches, it follows that the lower bound on the Z (cid:48) boson mass in 3-3-1 modelsranges from 2 . B s,d → µ + µ − and B d → K ∗ ( K ) µ + µ − [23–27], the lower limit on the Z (cid:48) boson mass ranges from 1 TeV to 3 TeV. We will showthat when scalar contributions to the ρ parameter are included, there will exist allowed m Z values larger than the range given in (33), so that they satisfy the recent lower boundsconcerned from LHC searches.For conventional notation, hereafter we will call Z and Z by Z and Z (cid:48) , respectively.Now we turn into the main subject - the Higgs sector.3 IV. ANALYSIS OF THE LEPTON NUMBER CONSERVING PART OF THESCALAR POTENTIAL
Below we present lepton number conserving part V LNC of the scalar potential of the modelshown in Appendix A. Expanding the Higgs potential around theVEVs, one gets the scalarpotential minimization conditions at tree level as follows w = 0 , (35) − µ χ = v χ λ + 12 v η λ + 12 λ χξ v ξ , − µ η = v η λ + 12 v χ λ + 12 λ ηξ v ξ , (36) − µ ξ = 12 λ χξ v χ + 12 λ ηξ v η + λ ξ v ξ . From the analysis of the scalar potential, taking into account the constraint conditionsof Eq.(35), we find that the charged scalar sector is composed of two massless fields, i.e., η +2 and χ +2 which are the Goldstone bosons eaten by the longitudinal components of the W + and Y + gauge bosons, respectively. The other massive electrically charged fields are φ +1 , φ +2 and φ +4 whose masses are respectively given by: m φ +1 = µ φ +1 + 12 (cid:104) v χ λ χφ + v η λ ηφ + v ξ λ φξ (cid:105) ,m φ +2 = µ φ +2 + 12 (cid:104) v χ λ χφ + v η λ ηφ + v ξ λ φξ (cid:105) , (37) m φ +4 = µ φ +4 + 12 (cid:104) v χ λ χφ + v η λ ηφ + v ξ λ φξ (cid:105) . In addition, the basis ( ρ +1 , ρ +3 , φ +3 ) corresponds to the following squared mass matrix M charged = A + v η ( λ + λ ) 0 v η v ξ λ A + (cid:0) v χ λ + v η λ (cid:1) √ v χ w v η v ξ λ √ v χ w µ φ +3 + B , (38)where we have used the following notations A ≡ µ ρ + 12 (cid:2) v χ λ + λ ρξ v ξ (cid:3) , B i ≡ (cid:16) v χ λ χφi + v η λ ηφi + v ξ λ φξi (cid:17) , i = 1 , , , . (39)From (38), it follows that in the limit v η (cid:28) v ξ , ρ +1 is a physical field with mass m ρ +1 = A + 12 v η ( λ + λ ) , (40)4while the two massive bilepton scalars ρ +3 and φ +3 mix with each other.Now we turn into CP-odd Higgs sector. There are three massless fields: I χ , I η and I ξ .The field I ϕ is a physical state itself with squared mass m I ϕ = µ ϕ + B (cid:48) , (41)where B (cid:48) n ≡ (cid:0) v χ λ χϕn + v η λ ηϕn + v ξ λ ϕξn (cid:1) , n = 1 , . (42)The squared mass matrix of the four remaining CP-odd Higgs fields separate into two blockdiagonal submatrices corresponding to the two original base ( I χ , I η ) and ( I ϕ , I ρ ) , namely m CP odd = λ v η − v χ v η − v χ v η v χ , m CP odd = µ ϕ + B (cid:48) − C v χ v η ( λ − λ ) v χ v η ( λ − λ ) A + λ v η , (43)where C ≡ v χ λ + v η λ + v ξ λ (44)The first matrix in (43) provides two mass eigenstates, where one of them are massless, G = cos θ a I χ + sin θ a I η , m G = 0 ,A = − sin θ a I χ + cos θ a I η , m A = λ v χ θ a , (45)where tan θ a = v η v χ . (46)The physical states relating to the second matrix in (43) are A A = cos θ ρ sin θ ρ − sin θ ρ cos θ ρ I ϕ I ρ , (47)where the mixing angle is given bytan 2 θ ρ = v χ v η ( λ − λ ) (cid:16) µ ϕ − C + B (cid:48) − A − λ v η (cid:17) . (48)Their squared masses are m A , = 12 (cid:26) A + D ∓ (cid:113) ( A − D ) + v η (cid:2) A − D ) λ + v η λ + v χ ( λ − λ ) (cid:3)(cid:27) , (49)5where D = µ ϕ + B (cid:48) − C + v η λ .Next, the CP-even scalar sector is our task. We find that R ϕ is physical with mass m R ϕ = m I ϕ = µ ϕ + 12 (cid:16) v χ λ χϕ + v η λ ηϕ + v ξ λ ϕξ (cid:17) . (50)As mentioned in Ref. [19], the lightest scalar ϕ is a possible DM candidate with lightmass smaller than 1 TeV. Therefore, Eq. (50) suggests a reasonable assumption µ ϕ = − (cid:16) v χ λ χϕ + v ξ λ ϕξ (cid:17) . (51)In this case, the model contains the complex scalar DM ϕ with mass m R ϕ = m I ϕ = λ ηϕ v η .There are other seven CP-even Higgs components which the squared mass matrix sepa-rates into two 2 × × × m CP even = λ v η v χ v η v χ v η v χ , m CP even = A + λ v η − v χ v η ( λ + λ ) − v χ v η ( λ + λ ) µ ϕ + C + B (cid:48) , (52)corresponding to the two original base ( R χ , R η ) and ( R ρ , R ϕ ), respectively. The physicalstates of the first matrix in (52) are determined as follows, R G = cos θ a R χ + sin θ a R η , m R G = 0 ,H = − sin θ a R χ + cos θ a R η , m H = m A = λ v χ θ a . (53)The physical states of the second matrix in (52) are H H = cos θ r sin θ r − sin θ r cos θ r R ρ R ϕ , (54)where the mixing angle is tan 2 θ r = v χ v η ( λ + λ ) (cid:16) µ ϕ + C + B (cid:48) − A − λ v η (cid:17) (55)and their squared masses are m H , = 12 (cid:26) A + D ∓ (cid:113) ( A − D ) + v η (cid:2) A − D ) λ + v η λ + v χ ( λ + λ ) (cid:3)(cid:27) , (56)where D = µ ϕ + B (cid:48) + C + v η λ . R χ , R η , R ξ ) is m CP even = v χ λ v χ v η λ λ χξ v χ v ξ v χ v η λ v η λ λ ηξ v η v ξ λ χξ v χ v ξ λ ηξ v η v ξ λ ξ v ξ , (57)which contains a SM-like Higgs boson found by LHC. The mass eigenstates will be discussedusing simplified conditions.Let us summarize the Higgs content:1. In the charged scalar sector: there are two Goldstone bosons η − and χ − eaten by thegauge bosons W − and Y − . Three massive charged Higgs bosons are φ +1 , φ +2 and φ +4 .The remaining fields ρ +1 , φ +3 and ρ +3 are mixing.2. In the CP-odd scalar sector: there is one massless Majoron scalar I ξ which is denotedby G M . Fortunately, it is a gauge singlet, therefore, is phenomenologically harmless .Two massless scalars I η and I χ are Goldstone bosons for the gauge bosons Z and Z (cid:48) , respectively. There exists another massless state denoted by G , its role will bediscussed below. Here we just mention that in the limit v η (cid:28) v χ , this field is I χ . Themassive CP-odd field are I ϕ , A and other two I ϕ , I ρ are mixing.3. In the CP-even scalar sector: There is one massless field: R G , and in the limit v η (cid:28) v χ ,it tends to R χ . Combination of G and R G is Goldstone boson for neutral bileptongauge boson X , namely G X = √ ( R G − iG ). The massive fields are: R ϕ , H , H and three massive R χ , R η , R ξ and the SM-like Higgs boson h . Note that there existsdegeneracy in Eqs. (50) and (53) when the contribution arising from Z × Z softbreaking scalar interactions is not considered. Thus, the complex scalar ϕ has massgiven by Eq. (50), which is consistent with the prediction in Ref. [19]. To be a DMcandidate, the condition (51) can be used to eliminate the terms with large VEVs suchas v χ and v ξ . As a result, the mass of the DM candidate is m ϕ = 12 v η λ ηϕ . (58)According [30], the WIMP candidate has mass around 10 GeV, implying that λ ηϕ ≈ .
04. To get the second DM candidate, namely, ϕ , we have to carefully chooseconditions.7Eqs. (45) and (53) result in a new complex w defined as follows ω = 1 √ H − iA ) , m ω = λ v χ θ a . (59)Let us rewrite the Higgs content in terms of the mass eigenstates mentioned above: χ (cid:39) G X G Y − √ ( v χ + R χ − iG Z (cid:48) ) , ρ = ρ +11 √ ( R ρ − iI ρ ) ρ +3 , η (cid:39) √ ( v η + h − iG Z ) G W − ω ,ϕ = 1 √ R ϕ − iI ϕ ) ∼ (1 , , , i, , ∼ DM candidate ,ξ = 1 √ v ξ + R ξ − iG M ) ∼ (1 , , . (60) A. Simplified solutions
We have shown that the mass eigenstates of scalars have been determined explicitly,except those relating to the two 3 × ρ parameter. To reducethe arbitrary of the unknown Higgs couplings in the potential (A1), the following relationare assumed firstly λ = λ , λ = λ , λ = λ , w = w . (61)In the next steps, we just pay attention to find the masses and mass eigenstates of the twomatrices (38) and (57). The other will be summarized if necessary.
1. The CP-odd Higgs bosons
Under the assumption (61), the CP-odd scalar sector consists of four massless fields { I χ , I η , G M , G } and four massive fields { A , A , A I ϕ } , as summarized in Table III.8 TABLE III: Squared mass of CP-odd scalars under condition in (61) and v χ (cid:29) v η . Fields I χ = G ∈ G X I χ = G Z (cid:48) I η = G Z I η = A I ρ = A I ϕ = A I ϕ = DM I ξ = G M Squared mass 0 0 0 m A m A m A m I ϕ
2. The CP-even and SM-like Higgs bosons
Now we turn to the sector where the SM Higgs boson exists, i.e., - the matrix in thebasis ( R χ , R η , R ξ ) is given by Eq. (57). Let us assume a simplified scenario worth to beconsidered is characterized by the following relations: λ = λ = λ = λ ξ = λ χξ = λ ηξ = λ, v ξ = v χ . (62)In this scenario, the squared matrix (57) takes the simple form: m CP even = λ x x xx x v χ , x = v η v χ = tan θ a . (63)Because v χ (cid:29) v η , the matrix (63) can be perturbatively diagonalized as follows: R TCP even m CP even R CP even (cid:39) λv η λv χ
00 0 3 λv χ , R CP even (cid:39) − x √ x x − (cid:113) (cid:113) x (cid:113) (cid:113) , (64)Thus, we find that the physical scalars included in the matrix m CP even are: hH H (cid:39) − x x x − (cid:113) (cid:113) √ x (cid:113) (cid:113) R η R χ R ξ , (65)where h is the SM-like Higgs boson with mass 126 GeV identified with that found by LHC,whereas H and H are physical heavy scalars acquiring masses at the breaking scale of the SU (3) L × U (1) X × Z × Z × U (1) L g symmetry. Thus, we find that h has couplings veryclose to SM expectation with small deviations of the order of v η v χ ∼ O (10 − ). In addition,the squared masses of the physical scalars h and H , are given in (64).Now, the content of the CP-even scalar sector is summarized in Table IV.Taking into account mass of the SM Higgs boson equal 126 GeV, from Table IV we obtain λ ≈ . H , once v χ is fixed.9 TABLE IV: Squared masses of CP-even scalars under condition in (62) and v χ (cid:29) v η . Fields R χ ∈ G X R χ (cid:39) H R η = h R η = H R ρ = H R ϕ = H R ϕ = DM R ξ (cid:39) H Squared mass 0 λv χ λv η m H = m A m H m H m R ϕ = m I ϕ λv χ
3. The charged Higgs bosons
The charged scalar sector contains two massless fields: G W + and G Y + which are Goldstonebosons eaten by the longitudinal components of the W + and Y + gauge bosons, respectively.The other massive fields are φ +1 , φ +2 and φ +4 with respective masses given in (39).In the basis ( ρ +1 , ρ +3 , φ +3 ), the squared mass matrix is given in (38). Let us make effortto simplify this matrix. Note that µ χ , µ η , and µ ξ can be derived using relations (35) and(62). In addition, it is reasonable to assume µ ρ = − v χ λ + λ ρξ ) ≈ µ η , µ φ +3 = − v χ λ χφ + λ φξ ) , (66)we obtain the simple form of the squared mass matrix of the charged Higgs bosons, M chargeds = v η ( λ + λ ) 0 λ v η v χ (cid:0) v χ λ + λ v η (cid:1) √ v χ w λ v η v χ √ v χ w v η λ ηφ (67)The matrix (67) predicts that there may exist two light charged Higgs bosons H +1 , withmasses at the electroweak scale and the mass of H +3 which is mainly composed of ρ +3 isaround 3.5 TeV. In addition, the Higgs boson H +1 almost does not carry lepton number,whereas the others two do.Generally, the Higgs potential always contains mass terms which mix VEVs. However,these terms must be small enough to avoid high order divergences (for examples, see Refs.[33, 34]) and provide baryon asymmetry of Universe by the strong electroweak phase tran-sition (EWPT).Ignoring mixing term containing λ in (67) does not affect other physical aspects, sincethe above mentioned term just increases or decreases small amount of the charged Higgsbosons. Therefore, without lose of generality, neglecting the term with λ satisfies otheraims such as EWPT.Hence, in the matrix of (67), the coefficient λ is reasonably assumed to be zero. Therefore0 TABLE V: Squared mass of charged scalars under condition in (66) and v χ (cid:29) v η . Fields η +2 = G W + χ +2 = G Y + H +1 H +3 H +2 φ +1 φ +2 φ +4 Squared mass 0 0 m H +1 m H +3 m H +2 m φ +1 m φ +2 m φ +4 we get immediately one physical field ρ +1 with mass given by m ρ +1 = 12 v η ( λ + λ ) . (68)The other fields mix by submatrix given at the bottom of (67). The limit ρ +1 = H +1 when λ = 0 is very interesting for discussion of the Higgs contribution to the ρ parameter.The content of the charged scalar sector is summarized in Table V. It is worth mentioningthat the masses of three charged scalars φ + i , i = 1 , , V full = V LNC + V LNV is quite similarto the previous one. There are some differences:1. The masses of the fields receive some new contributions.2. The complex scalar ϕ has the same mass in both cases.3. Majoron does not exist and its mass only arises from lepton number violating scalarinteractions.4. The mixing of the CP-even scalar fields is more complicated. B. Scalar contributions to the ρ parameter The new Higgs bosons may give contribution to the ρ parameter at one-loop level, asshown in many models beyond the SM, such as the simplified models [35], the Two HiggsDoublet Models [36, 37], and the supersymmetric version of the SM [38]. In the 3-3-1 CKSmodel, we will consider the effect of the Higgs contributions to the ρ parameter at one-looplevel. These contributions will be determined in the limit of the suppressed Z − Z (cid:48) mixingand the decoupling of the SM-like Higgs boson with other CP-even neutral Higgs bosons. Asa consequence, the one-loop contribution of the SM-like Higgs boson to the ρ parameter isthe same as in the SM. Excepting for the components of the scalar triplet ρ , the other heavy1CP-even neutral Higgs bosons do not couple with the SM gauge bosons W and Z and thusthey do not provide contributions to the ρ parameter. Contributions of the remaining Higgsbosons can be calculated using the results given in Ref. [38]. In particular, contributions ofany Higgs bosons in our case to ∆ ρ are determined as follows∆ ρ = Π W W (0) M W − Π ZZ (0) M Z , (69)where Π W W (0) and Π ZZ (0) are the coefficients of − ig µν in the vacuum-polarization ampli-tudes of charged and neutral W bosons and Z gauge bosons, respectively. Our case relateswith only the contribution of ”non-Higgs scalars” φ , with masses m , and coupling icφ ∗ ↔ ∂ µ φ V µ ≡ ic [ φ ∗ ∂ µ φ − ( ∂ µ φ ∗ ) φ ] V µ , ( V = W, Z ) , (70)The corresponding contribution isΠ(scalar) = | c | π f s ( m , m ) , (71)where f s ( m , m ) = f s ( m , m ) = m m m − m ln (cid:20) m m (cid:21) + 12 (cid:0) m + m (cid:1) = m f s ( x ) = m (cid:18) x ln( x )1 − x + 1 + x (cid:19) , x ≡ m m . (72)The function in Eq. (72) satisfies f s ( m , m ) = lim m → m f s ( m , m ) = 0 and f s ( m , m ) > m (cid:54) = m . As a consequence, the charged Higgs bosons φ ± , , having vanishing Higgs-gauge couplings with other Higgs bosons give vanishing contributions to the ρ parameter.Nonvanishing contributions now may arise from the charged Higgs bosons H ± , , correspond-ing to the basis ( ρ ± , ρ ± , φ ± ) and the CP-odd neutral Higgs relating with I ρ . The relevantLagrangian is L V HH = ( D µ ρ ) † ( D µ ρ ) + ( D µ φ +3 ) ∗ ( D µ φ +3 ) → ig Z µ (cid:20) − s W c W (cid:16) ρ − ↔ ∂ µ ρ +1 (cid:17) + − s W c W (cid:16) ρ − ↔ ∂ µ ρ +3 (cid:17) + − ic W (cid:16) I ρ ↔ ∂ µ R ρ (cid:17)(cid:21) + − igs W c W Z µ (cid:16) φ − ↔ ∂ µ φ +3 (cid:17) + (cid:20) − ig W + µ ρ − (cid:16) ↔ ∂ µ R ρ − i ↔ ∂ µ I ρ (cid:17) + H . c . (cid:21) . (73)In the scalar-gauge interactions of Eq (73), only the last term contributes to Π W W (0), thusgiving rise to a non-negative contribution to the ρ parameter, which may make the allowed2 M Z (cid:48) mass to move outside the excluded region recently reported by LHC searches [39]. Onthe contrary, all of the remaining terms contributing to Π ZZ (0), give non-positive contribu-tions to the ρ parameter. For illustration, we will consider a simple case where only positivecontributions to the ρ parameter are kept, namely ρ ± , ≡ H ± , , φ ± ≡ H ± , I ρ ≡ A and R ρ are mass eigenstates. Then, all contributions to Π ZZ (0) arising from the charged Higgsbosons are proportional to f s ( m s , m s ) = 0 with s = ρ ± , , φ ± . In addition, the simplifiedcondition (61) with λ (cid:28) m I ρ = m R ρ , leading to a vanishing neutral Higgsboson contribution to Π ZZ (0): f s ( m I ρ , m R ρ ) = 0. The only non-zero contribution has theform ∆ ρ H = g π m W f s ( m H +1 , m R ρ ) = √ G F π f s ( m H +1 , m R ρ ) , (74)where ∆ m ≡ m R ρ − m H +1 = − λ v η ∼ O ( v η ) . (75)Allowed regions of the parameter space for some specific values of ∆ m are shown in Fig. 2. Itcan be seen that the large values of v χ are still allowed, thus implying that no upper boundsare required. The allowed values of v χ and M Z (cid:48) strongly depend on the lower bound of m H ± and m R ρ , which may have previously reported from LHC searches. Unfortunately, theHiggs triplets ρ containing the neutral components ρ with zero VEV, hence all of the threecomponents of ρ do not couple to the two SM gauge bosons Z and W . This Higgs tripletalso does not contribute to the SM-like Higgs boson. As a result, all of the Higgs bosons H ± , R ρ , and I ρ are not affected by the following decays searched by LHC: H ± → W ± Z, W ± h and R ρ , I ρ → W + W − , ZZ, Zh . These Higgs bosons do not couple with the SM quarks [19],can not be produced at LHC from the recent chanel searching [40]. Only the allowed treelevel decays to two SM fermions are leptonic decays: H ± → ν , τ , ν , µ and R ρ , I ρ → ¯ e i e i ( i = 1 , , H ± → W ± γ , R ρ , I ρ → Zγ [41], and R ρ , I ρ → γγ [42, 43]. The heavy neutral Higgs boson masses are predicted to be at the TeV scale,which is outside the LHC excluded regions. Combined with the relation (75), the mass of3 v χ [ TeV ] m H + [ T e V ] Δρ × , m Z ' [ TeV ] , Δ m = FIG. 2: Contour plots of the ρ parameter (dotted-dashed curves) and M Z (cid:48) (black currves) asfunctions of v χ and m H +1 . The green regions are excluded by the recent experimental constrain ofthe ρ . the charged Higgs boson H +1 should also be at the TeV scale. From the figure 2, we can seethat M Z (cid:48) ≥ m is large enough, for example ∆ m ≥ (0 .
246 TeV) .4 V. CHARGED LEPTON FLAVOR VIOLATING DECAY CONSTRAINTS.
In this section we will determine the constraints that the charged lepton flavor violatingdecays µ → eγ , τ → µγ and τ → eγ imposed on the parameter space of our model. Asmentioned in Ref. [19], the sterile neutrino spectrum of the model is composed of two almostdegenerate neutrinos with masses at the Fermi scale and four nearly degenerate neutrinoswith TeV scale masses. These sterile neutrinos together with the heavy W (cid:48) gauge bosoninduce the l i → l j γ decay at one loop level, whose branching ratio is given by: [44–46]: Br ( l i → l j γ ) = α W s W m l i π m W (cid:48) Γ i (cid:12)(cid:12)(cid:12)(cid:12) G (cid:18) m N m W (cid:48) (cid:19) + 4 G (cid:18) m N m W (cid:48) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , ,G ( x ) = − x + 5 x − x − x ) − x − x ) ln x. (76)In our numerical analysis we have fixed m N = 100 GeV and we have varied the W (cid:48) gaugeboson mass in the range 4 TeV (cid:46) m W (cid:48) (cid:46) Z (cid:48) gauge bosonmasses larger than 4 TeV to fullfill the bound arising from the experimental data on K , D and B meson mixings [28]. Figure 3 shows the allowed parameter space in the m W (cid:48) − m N plane consistent with the constraints arising from charged lepton flavor violating decays.As seen from Figure 3, the obtained values for the branching ratio of µ → eγ decay arebelow its experimental upper limit of 4 . × − since these values are located in the range3 × − (cid:46) Br ( µ → eγ ) (cid:46) × − , for sterile neutrino masses m N lower than about 1 . τ → µγ and τ → eγ decays can reach values of the order of 10 − , which is below their correspondingupper experimental bounds of 4 . × − and 3 . × − , respectively. Consequently, ourmodel is compatible with the charged lepton flavor violating decay constraints provided thatthe sterile neutrinos are lighter than about 1 .
12 TeV.
VI. SEARCH FOR Z (cid:48) AT LHC
In this section, we present two typical effects of the LHC, namely, production of single aparticle in proton-proton collisions.5
FIG. 3: Allowed parameter space in the m W (cid:48) − m N plane consistent with the LFV constraints. A. Phenomenology of Z (cid:48) gauge boson In what follows we proceed to compute the total cross section for the production of aheavy Z (cid:48) gauge boson at the LHC via Drell-Yan mechanism. In our computation for thetotal cross section we consider the dominant contribution due to the parton distributionfunctions of the light up, down and strange quarks, so that the total cross section for theproduction of a Z (cid:48) via quark antiquark annihilation in proton proton collisions with centerof mass energy √ S takes the form: σ ( DrellY an ) pp → Z (cid:48) ( S ) = g π c W S (cid:104)(cid:0) g (cid:48) uL (cid:1) + (cid:0) g (cid:48) uR (cid:1) (cid:105) (cid:90) − ln (cid:114) m Z (cid:48) S ln (cid:114) m Z (cid:48) S f p/u (cid:115) m Z (cid:48) S e y , µ f p/u (cid:115) m Z (cid:48) S e − y , µ dy + (cid:104)(cid:0) g (cid:48) dL (cid:1) + (cid:0) g (cid:48) dR (cid:1) (cid:105) (cid:90) − ln (cid:114) m Z (cid:48) S ln (cid:114) m Z (cid:48) S f p/d (cid:115) m Z (cid:48) S e y , µ f p/d (cid:115) m Z (cid:48) S e − y , µ dy + (cid:104)(cid:0) g (cid:48) dL (cid:1) + (cid:0) g (cid:48) dR (cid:1) (cid:105) (cid:90) − ln (cid:114) m Z (cid:48) S ln (cid:114) m Z (cid:48) S f p/s (cid:115) m Z (cid:48) S e y , µ f p/s (cid:115) m Z (cid:48) S e − y , µ dy ✹(cid:0)✁✂ ✄☎✆✝ ✞✟✠✡ ☛☞✌✍ ✎✏✑✒ ✺✓✔✕✵✷✖✗✘✻✙✽✚✶✛✜ ▼❩ ✬❬●✢✣❪✤✭♣✥✦✧★✮✩❢✪✫ FIG. 4: Total cross section for the Z (cid:48) production via Drell-Yan mechanism at the LHC for √ S = 13TeV and as a function of the Z (cid:48) mass.Figure 4 displays the Z (cid:48) total production cross section at the LHC via Drell-Yan mechanismat the LHC for √ S = 13 TeV and as a function of the Z (cid:48) mass, which is taken to range from 4TeV up to 5 TeV. We consider neutral heavy Z (cid:48) gauge boson masses larger than 4 TeV to fullfillthe bound arising from the experimental data on K , D and B meson mixings [28]. For such asa region of Z (cid:48) masses we find that the total production cross section ranges from 85 fb up to 10fb. The heavy neutral Z (cid:48) gauge boson after being produced it will decay into pair of SM particles,with dominant decay mode into quark-antiquark pairs as shown in detail in Refs. [24, 47]. Thetwo body decays of the Z (cid:48) gauge boson in 3-3-1 models have been studied in details in Refs. [47].In particular, in Ref. [47] it has been shown the Z (cid:48) decays into a lepton pair in 3-3-1 modelshave branching ratios of the order of 10 − , which implies that the total LHC cross section for the pp → Z (cid:48) → l + l − resonant production at √ S = 13 TeV will be of the order of 1 fb for a 4 TeV Z (cid:48) gauge boson, which is below its corresponding lower experimental limit arising from LHC searches[39]. On the other hand, at the proposed energy upgrade of the LHC at 28 TeV center of massenergy, the total cross section for the Drell-Yan production of a heavy Z (cid:48) neutral gauge boson gets significantly enhanced reaching values ranging from 2 . . pp → Z (cid:48) → l + l − resonant production at √ S = 28TeV will be of the order of 10 − pb for a 4 TeV Z (cid:48) gauge boson, which corresponds to the order ofmagnitude of its corresponding lower experimental limit arising from LHC searches [39]. ✹(cid:0)✁✂ ✄☎✆✝ ✞✟✠✡ ☛☞✌✍ ✎✏✑✒ ✺✓✔✕✵✖✗✘✙✚✶✛✜✢✣✤✷✥✦✧★✩✸✪✫ ▼❩ ✬❬●✭✮❪✯✰♣✱✲✳✴✻✼✽✾✿ FIG. 5: Total cross section for the Z (cid:48) production via Drell-Yan mechanism at the proposed energyupgrade of the LHC with √ S = 28 TeV as a function of the Z (cid:48) mass. B. Phenomenology of H Heavy Higgs boson
In what follows we proceed to compute the LHC production cross section of the singly heavyscalar H . Let us note that the singly heavy scalar H is mainly produced via gluon fusionmechanism mediated by a triangular loop of the heavy exotic quarks T , J and J . Thus, the totalcross section for the production of the heavy scalar H through gluon fusion mechanism in protonproton collisions with center of mass energy √ S takes the form: σ pp → gg → H ( S ) = α S m H | ( R CP even ) | πv χ S (cid:34) I (cid:32) m H m T (cid:33) + I (cid:32) m H m J (cid:33) + I (cid:32) m H m J (cid:33)(cid:35) × (cid:90) − ln (cid:114) m H S ln (cid:114) m H S f p/g (cid:115) m H S e y , µ f p/g (cid:115) m H S e − y , µ dy where f p/g (cid:0) x , µ (cid:1) and f p/g (cid:0) x , µ (cid:1) are the distributions of gluons in the proton which carry mo-mentum fractions x and x of the proton, respectively. Furthermore µ = m H is the factorizationscale and I ( z ) is given by: I ( z ) = (cid:90) dx (cid:90) − x dy − xy − zxy (77)Figure 6 displays the H total production cross section at the LHC via gluon fusion mechanism ✶(cid:0) ✵✁✂ ✄☎ ✆✝✞ ✟✠ ✡☛☞ ✌✍ ✎✏✑ ✒✓ ✔✕✖ ✷✗ ✘✙✚✛✜✢✣✤✥✦✧★✩✪✫✬✭✮✯✰✱✲✳✴✸✹✺✻✼✽✾✿❀❁❂❃❄❅❆❇❈❉❊❋●❍■❏❑▲▼◆ ✈❖❬P◗❘❪❙❚♣❯❱❲ ❳❨❩❢❭❫ FIG. 6: Total cross section for the H production via gluon fusion mechanism at the LHC for √ S = 13 TeV and as a function of the SU (3) L × U (1) X symmetry breaking scale v χ for thesimplified scenario described in Eq. (62) .for √ S = 13 TeV, as a function of the SU (3) L × U (1) X symmetry breaking scale v χ , which is takento range from 10 TeV up to 20 TeV. The aforementioned range of values for the SU (3) L × U (1) X symmetry breaking scale v χ corresponds to a heavy scalar mass m H varying between 4 . . H in the range 8 TeV (cid:46) M H (cid:46) . W (cid:48) W (cid:48) and Z (cid:48) Z (cid:48) heavy gauge boson pairs. On the other hand, for a heavy scalar field H with mass in the range 4 . (cid:46) M H (cid:46) t ¯ t pair. Furthermore, in the region of H masses considered in our analysis, the H decayinto exotic quark pairs will be kinematically forbidden for exotic quark Yukawa couplings of orderunity. Note that we have chosen values for v χ larger than 10 TeV, which corresponds to a Z (cid:48) gaugeboson heavier than 4 TeV, which is required to guarantee the consistency of 331 models with theexperimental data on K , D and B meson mixings [28]. Here, for the sake of simplicity we haverestricted to the simplified scenario described by Eq. (62) and we have chosen the exotic quarkYukawa couplings equal to unity, i.e, y ( T ) = y ( J ) = y ( J ) = 1. In addition, the top quark mass hasbeen taken to be equal to m t = 173 GeV. We find that the total cross section for the production ofthe H scalar at the LHC takes a value close to about 10 − fb for the lower bound of 10 TeV of the SU (3) L × U (1) X symmetry breaking scale v χ arising from the experimental data on K , D and B meson mixings [28] and decreases when v χ takes larger values. We see that the total cross sectionat the LHC for the H production via gluon fusion mechanism is small to give rise to a signalfor the allowed values of the SU (3) L × U (1) X symmetry breaking scale v χ . A similar situationhappens at the proposed energy upgrade of the LHC with √ S = 28 TeV, where this total crosssection takes a value of 1 . × − fb for v χ = 10 TeV as shown in figure 7. Because of the verysmall H production cross section, we do not perform a detailed study of its decay modes. It isworth mentioning that the smoking gun signatures of the model under consideration will be the Z (cid:48) production and the charged lepton flavor violating decay µ → eγ , whose observation will becrucial to assess to viability of this model. VII. DARK MATTER RELIC DENSITY
In this section we provide a discussion of the implications of our model for DM, assuming thatthe DM candidate is a scalar. Let us recall that our goal in this section is to provide an estimateof the DM relic density in our model, under some simplifying assumptions motivated by the largenumber of scalar fields of the model. We do not intend to provide a sophisticated analysis ofthe DM constraints of the model under consideration, which is beyond the scope of the presentpaper. We just intend to show that our model can accommodate the observed value of the DMrelic density, by having a scalar DM candidate with a mass in the TeV range and a quartic scalar ✶(cid:0) ✵✁✂ ✄☎ ✆✝✞ ✟✠ ✡☛☞ ✌✍ ✎✏✑ ✒✓ ✔✕✖ ✷✗ ✘✙✚✛✜✢✣✤✥✦✧★✩✪✫✬✭✮✯✰✱✲✳ ✈✴❬●✸✹❪✺✻♣✼✽❍ ✾✿❀❢❁❂ FIG. 7: Total cross section for the H production via gluon fusion mechanism at the proposedenergy upgrade of the LHC with √ S = 28 TeV as a function of the SU (3) L × U (1) X symmetrybreaking scale v χ for the simplified scenario described in Eq. (62).coupling of the order unity, within the perturbative regime. We start by surveying the possiblescalar DM candidates in the model. Considering that the Z symmetry is preserved and takinginto account the scalar assignments under this symmetry, given by Eq. (I), we can assign thisrole to either any of the SU (3) L scalar singlets, i.e., Reϕ n and Imϕ n ( n = 1 , ϕ I = Imϕ is the lightest among the Reϕ n and Imϕ n ( n = 1 ,
2) scalar fields andalso lighter than the exotic charged fermions, as well as lighter than Ψ R , thus implying that itstree-level decays are kinematically forbidden. Consequently, in this mass range the Imϕ scalarfield is stable. The relic density is given by (c.f. Ref. [30, 49])Ω h = 0 . pb (cid:104) σv (cid:105) , (cid:104) σv (cid:105) = An eq = T π ∞ (cid:90) m ϕ (cid:88) p = W,Z,t,b,h g p s √ s − m ϕ v rel σ ( ϕϕ → pp ) K (cid:16) √ sT (cid:17) ds T π (cid:88) p = W,Z,t,b,h g p m ϕ K (cid:0) m ϕ T (cid:1) , (78)where (cid:104) σv (cid:105) is the thermally averaged annihilation cross-section, A is the total annihilation rate perunit volume at temperature T and n eq is the equilibrium value of the particle density. Furthermore, K and K are modified Bessel functions of the second kind and order 1 and 2, respectively [49]and m ϕ = m Im ϕ . Let us note that we assume that our scalar DM candidate is a stable weaklyinteracting particle (WIMP) with annihilation cross sections mediated by electroweak interactionsmainly through the Higgs field. In addition we assume that the decoupling of the non-relativisticWIMP of our model is supposed to happen at a very low temperature. Because of this reason, forthe computation of the relic density, we take T = m ϕ /
20 as in Ref. [49], corresponding to a typicalfreeze-out temperature. We assume that our DM candidate ϕ annihilates mainly into W W , ZZ , tt , bb and hh , with annihilation cross sections given by the following relations [50]: v rel σ ( ϕ I ϕ I → W W ) = λ h ϕ π s (cid:16) m W s − m W s (cid:17)(cid:0) s − m h (cid:1) + m h Γ h (cid:114) − m W s ,v rel σ ( ϕ I ϕ I → ZZ ) = λ h ϕ π s (cid:16) m Z s − m Z s (cid:17)(cid:0) s − m h (cid:1) + m h Γ h (cid:114) − m Z s ,v rel σ ( ϕ I ϕ I → qq ) = N c λ h ϕ m q π (cid:115)(cid:18) − m f s (cid:19) (cid:0) s − m h (cid:1) + m h Γ h ,v rel σ ( ϕ I ϕ I → hh ) = λ h ϕ πs (cid:32) m h s − m h − λ h ϕ v s − m h (cid:33) (cid:115) − m h s , (79)where √ s is the centre-of-mass energy, N c = 3 is the color factor, m h = 125 . h = 4 . h mass and its total decay width, respectively. Note that we haveworked on the decoupling limit where the couplings of the 126 GeV Higgs boson to SM particlesand its self-couplings correspond to the SM expectation.The vacuum stability and tree level unitarity constraints of the scalar potential are [51–53]: λ h > , λ ϕ > , λ h ϕ < λ h λ ϕ . (80)
200 300 400 500 6000.00.10.20.30.40.5 m φ [ GeV ] Ω h FIG. 8: Relic density Ω h , as a function of the mass m ϕ of the ϕ scalar field, for several valuesof the quartic scalar coupling λ h ϕ . The curves from top to bottom correspond to λ h ϕ =0 . , . , . , . ,
1, respectively. The horizontal line shows the observed value Ω h = 0 . m ϕ of the scalar dark matter candidate consistent with theexperimental measurement of the dark matter relic density. λ ϕ < π, λ h ϕ < π. (81)The dark matter relic density as a function of the mass m ϕ of the scalar field ϕ I is shown in Fig. 8,for several values of the quartic scalar coupling λ h ϕ , set to be equal to 0 .
7, 0 . . h = 0 . λ h ϕ and the mass m ϕ of the scalar DM candidate ϕ I , as indicated inFig. 9.We find that we can reproduce the experimental value Ω h = 0 . ± . m ϕ of the scalar field ϕ I is in the range 300 GeV (cid:46) m ϕ (cid:46)
570 GeV,for a quartic scalar coupling λ h ϕ in the window 0 . (cid:46) λ h ϕ (cid:46)
1, which is consistent with thevacuum stability and unitarity constraints shown in Eqs. (80) and (81). Note that our range ofvalues chosen for the quartic scalar coupling λ h ϕ also allow the extrapolation of our model athigh energy scales as well as the preservation of perturbativity at one loop level. ✸(cid:0)✁ ✂✄☎ ✹✆✝ ✞✟✠ ✺✡☛ ☞✌✍✵✎✏✑✒✓✔✕✖✗✘✙✚✛✜ ♠✢❬●✣✤❪✥ ❤✷ ✦✧ FIG. 9: Correlation between the quartic scalar coupling and the mass m ϕ of the scalar DM candi-date ϕ , consistent with the experimental value Ω h = 0 . VIII. CONCLUSIONS
We have studied some phenomenological aspects of the extended inert 331 model, which in-corporates the mechanism of sequential loop-generation of the SM fermion masses, explaining theobserved strong hierarchies between them as well as the corresponding mixing parameters. A par-ticular emphasis has been made on analyzing the constraints arising from the experimental dataon the ρ parameter, as well as those ones resulting from the charged lepton flavor violating process µ → eγ and dark matter. Furthermore, we have studied the production of the heavy Z (cid:48) gaugeboson in proton-proton collisions via the Drell-Yan mechanism. We found that the correspondingtotal cross section at the LHC ranges from 85 fb up to 10 fb when the Z (cid:48) gauge boson mass isvaried within 4 − Z (cid:48) production cross section gets significantly enhanced atthe proposed energy upgrade of the LHC with √ S = 28 TeV reaching the typical values of 2 . − . pp → Z (cid:48) → l + l − resonant production cross section reachvalues of about 1 fb and 10 − pb at M Z (cid:48) = 4 TeV, for √ S = 14 TeV and √ S = 28 TeV, re-spectively. These obtained values for the pp → Z (cid:48) → l + l − resonant production cross sections arebelow and of the same order of magnitude of its corresponding lower experimental limit arisingfrom LHC searches, for √ S = 13 TeV and √ S = 28 TeV, respectively. Besides that, we have found that Z (cid:48) gauge bosons heavier than about 4 TeV comply with the experimental constraints on theoblique ρ parameter as well as with the collider constraints. In addition, we have found that theconstraint on the charged lepton flavor violating decay µ → eγ set the sterile neutrino masses tobe lighter than about 1 .
12 TeV. We have found that the obtained values of the branching ratiofor the µ → eγ decay are located in the range 3 × − (cid:46) Br ( µ → eγ ) (cid:46) × − , whereasthe obtained branching ratios for the τ → µγ and τ → eγ decays can reach values of the order of10 − . Consequently, our model predicts charged lepton flavor violating decays within the reachof future experimental sensitivity. We found that the total cross section for the production of the H scalar at the LHC with √ S = 13 TeV takes a value close to about 10 − fb for the lower boundof 10 TeV of the SU (3) L × U (1) X symmetry breaking scale v χ required by the consistency of the ρ parameter with the experimental data. This value is increased to 1 . × − fb at the proposedenergy upgrade of the LHC with √ S = 28 TeV, thus implying that the total cross section at theLHC for the H production via gluon fusion mechanism is very small to give rise to a signal for theallowed values of the SU (3) L × U (1) X symmetry breaking scale v χ even at the proposed energyupgrade of the LHC. We also analyzed in detail the scalar potential and the gauge sector of themodel.The general Higgs sector is separated into two parts. The first part consists of lepton numberconserving terms and the second one contains lepton number violating couplings. The first part ofpotential was considered in details and the SM Higgs boson was derived and as expected, mainlyarises from η . We have showed that the whole scalar potential, excepting its CP-even sector, hasa quite similar situation since the resulting physical scalar mass spectrum is similar in both cases.The scalar spectrum contains enough number of Goldstone bosons for massive gauge bosons. Inthe CP-odd scalar sector, there are four massive bosons and one of them is a DM candidate. TheCP-even scalar mass spectrum consists of seven massive fields including the SM Higgs boson and aDM candidate. The singly electrically charged Higgs boson sector contains six massive fields. Twoof them have masses at the electroweak scale and the remaining one has a mass around 3 . φ + i , i = 1 , , Z symmetry ourmodel has the stable scalar dark matter candidates Reϕ n and Imϕ n ( n = 1 ,
2) and the fermionicdark matter candidate Ψ R . In this work we assume that ϕ I = Imϕ is the lightest among the Reϕ n and Imϕ n ( n = 1 ,
2) scalar fields and also lighter than the exotic charged fermions and than Ψ R , which implies that it is stable and thus it is the dark matter candidate considered in thiswork. To reproduce the Dark matter relic density, the mass of the scalar dark matter candidatehas to be in the range 300 GeV (cid:46) m ϕ (cid:46)
570 GeV, for a quartic scalar coupling λ h ϕ in thewindow 0 . (cid:46) λ h ϕ (cid:46)
1. In addition, it has been shown in Ref. [19] that requiring that the DMcandidate ϕ lifetime be greater than the universe lifetime τ u ≈ . m ϕ ∼ > × GeV. Thus we conclude that under theabove specified conditions the model contains viable fermionic Ψ R and scalar ϕ DM candidates.A sophisticated analysis of the DM constraints of our model is beyond the scope of the presentpaper and is left for future studies.
Acknowledgments
This research has been financially supported by Fondecyt (Chile), Grants No. 1170803, CON-ICYT PIA/Basal FB0821, the Vietnam National Foundation for Science and Technology Devel-opment (NAFOSTED) under grant number 103.01-2017.356. H. N. L. is very grateful to theBogoliubov Laboratory for Theoretical Physics, JINR, Dubna, Russia for the warm hospitalityduring his visit.
Appendix A: The scalar potential
The renormalizable potential contain three parts: the first one invariant under group G in (1)is given by V LNC = µ χ χ † χ + µ ρ ρ † ρ + µ η η † η + (cid:88) i =1 µ φ + i φ + i φ − i + (cid:88) i =1 µ ϕ i ϕ i ϕ ∗ i + µ ξ ξ ∗ ξ + χ † χ ( λ χ † χ + λ ρ † ρ + λ η † η ) + ρ † ρ ( λ ρ † ρ + λ η † η ) + λ ( η † η ) + λ ( χ † ρ )( ρ † χ ) + λ ( χ † η )( η † χ ) + λ ( ρ † η )( η † ρ )+ χ † χ (cid:32) (cid:88) i =1 λ χφi φ + i φ − i + (cid:88) i =1 λ χϕi ϕ i ϕ ∗ i + λ χξ ξ ∗ ξ (cid:33) + ρ † ρ (cid:32) (cid:88) i =1 λ ρφi φ + i φ − i + (cid:88) i =1 λ ρϕi ϕ i ϕ ∗ i + λ ρξ ξ ∗ ξ (cid:33) + η † η (cid:32) (cid:88) i =1 λ ηφi φ + i φ − i + (cid:88) i =1 λ ηϕi ϕ i ϕ ∗ i + λ ηξ ξ ∗ ξ (cid:33) + (cid:88) i =1 φ + i φ − i (cid:88) j =1 λ φφij φ + j φ − j + (cid:88) j =1 λ φϕij ϕ j ϕ ∗ j + λ φξi ξ ∗ ξ + (cid:88) i =1 ϕ i ϕ ∗ i (cid:88) j =1 λ ϕϕij ϕ j ϕ ∗ j + λ ϕξi ξ ∗ ξ + λ ξ ( ξ ∗ ξ ) + (cid:110) λ (cid:0) φ +2 (cid:1) (cid:0) φ − (cid:1) + λ (cid:0) φ +2 (cid:1) (cid:0) φ − (cid:1) + λ (cid:0) φ +3 (cid:1) (cid:0) φ − (cid:1) + w (cid:0) ϕ (cid:1) ϕ + w χ † ρφ − + w η † χξ + w (cid:0) ϕ (cid:1) ϕ ∗ + w φ +3 φ − ϕ + w φ +3 φ − ϕ ∗ + χρη ( λ ϕ + λ ϕ ∗ ) + χ † ρφ − (cid:0) λ ϕ + λ ϕ ∗ (cid:1) + λ η † ρφ − ξ + λ φ +1 φ − ϕ ξ + (cid:0) λ φ − φ +4 + λ φ +3 φ − (cid:1) (cid:0) ϕ (cid:1) + λ (cid:0) ϕ (cid:1) ϕ ∗ + (cid:32) λ χ † χ + λ ρ † ρ + λ η † η + (cid:88) i =1 λ i φ + i φ − i + (cid:88) i =1 λ i ϕ i ϕ ∗ i + λ ξ ∗ ξ (cid:1) ( ϕ ) + h.c. (cid:9) (A1)The second part is a lepton number violating one (the subgroup U (1) L g is violated) V LNV = µ χη (cid:16) χ † η + η † χ (cid:17) + (cid:104) λ ( χ † χ ) + λ ( ρ † ρ ) + λ ( η † η ) (cid:105) ( χ † η + η † χ )+ λ (cid:104) ( χ † η ) + ( η † χ ) (cid:105) + λ (cid:104) ( η † ρ )( ρ † χ ) + ( χ † ρ )( ρ † η ) (cid:105) + (cid:40) ξ (cid:32) w χ † χ + w ρ † ρ + w η † η + (cid:88) i =1 w i φ + i φ − i + (cid:88) i =1 w i ϕ i ϕ ∗ i + w ξ ∗ ξ (cid:33) + ξ (cid:104) w χ † η + w (cid:0) ϕ (cid:1) + w (cid:0) ϕ ∗ (cid:1) + w (cid:0) ξ (cid:1) (cid:105) + w η † ρφ − + w φ − φ +1 ϕ + (cid:0) ξ (cid:1) (cid:34) λ χ † χ + λ ρ † ρ + λ η † η + (cid:88) i =1 λ i φ + i φ − i + (cid:88) i =1 λ i ϕ i ϕ ∗ i + λ ξ ∗ ξ + λ (cid:0) ϕ (cid:1) + λ (cid:0) ϕ ∗ (cid:1) (cid:105) + χ † η (cid:34) (cid:88) i =1 λ i φ + i φ − i + (cid:88) i =1 λ i ϕ i ϕ ∗ i + λ ξ ∗ ξ + λ (cid:0) ϕ (cid:1) + λ (cid:0) ϕ ∗ (cid:1) + λ (cid:0) ξ (cid:1) + λ (cid:0) ξ ∗ (cid:1) (cid:105) + η † ρ (cid:0) λ φ − ϕ + λ φ − ϕ ∗ + λ φ − ξ ∗ (cid:1) + ρ † χφ +3 (cid:0) λ ξ + λ ξ ∗ (cid:1) + λ (cid:0) φ +1 (cid:1) φ − φ − + φ +1 φ − (cid:0) λ ϕ ϕ ∗ + λ ϕ ∗ ϕ ∗ + λ ϕ ξ ∗ (cid:1) + φ +3 φ − (cid:0) λ ϕ ξ + λ ϕ ξ ∗ + λ ϕ ∗ ξ + λ ϕ ∗ ξ ∗ (cid:1) + (cid:0) ϕ (cid:1) (cid:0) λ ϕ ξ + λ ϕ ξ ∗ + λ ϕ ∗ ξ + λ ϕ ∗ ξ ∗ (cid:1) + h.c. (cid:111) (A2)The last part which breaks softly Z × Z , is given by L scalarsgsoft = µ ϕ ϕ + µ ϕ ϕ ∗ + µ (cid:0) ϕ (cid:1) + µ φ +2 φ − + µ φ +2 φ − + µ φ +3 φ − + h.c. (A3) The total potential is composed of three above mentioned parts V = V LNC + V LNV + L scalarssoft . (A4)The scalar interactions needed for quark and charged lepton mass generation, read as follows L Higgsqcl = λ χρηϕ + λ η † ρφ − ξ + λ φ +1 φ − ϕ ξ + w (cid:0) ϕ (cid:1) ϕ + w χ † ρφ − + h.c . (A5)For the neutrino mass generation, beside the first term in (A5), the additional part is given as L Higgsneutrino = λ ( χ † χ ) + λ ( χ † χ )( η † η ) + (cid:104) λ ( ρ † ρ )( χ † η + η † χ ) + µ φ − φ +3 + h.c (cid:105) . 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