Predictive Scotogenic Model with Flavor Dependent Symmetry
PPredictive Scotogenic Model with Flavor Dependent Symmetry
Zhi-Long Han ∗ and Weijian Wang † School of Physics and Technology, University of Jinan, Jinan, Shandong 250022, China Department of Physics, North China Electric Power University, Baoding 071003, China (Dated: January 24, 2019)In this paper, we propose a viable approach to realise two texture-zeros in the scotogenic modelwith flavor dependent U (1) B − L α − L β gauge symmetry. These models are extended by two right-handed singlets N Ri and two inert scalar doublets η i , which are odd under the dark Z symmetry.Among all the six constructed textures, texture A and A are the only two allowed by currentexperimental limits. Then choosing texture A derived from U (1) B − L e − L τ , we perform a detailanalysis on the corresponding phenomenology such as predictions of neutrino mixing parameters,lepton flavor violation, dark matter and collider signatures. One distinct nature of such model isthat the structure of Yukawa coupling ¯ L ˜ ηN R is fixed by neutrino oscillation data, and can be furthertested by measuring the branching ratios of charged scalars η ± , . ∗ sps [email protected] † [email protected] a r X i v : . [ h e p - ph ] J a n I. INTRODUCTION
It is well known that the Standard Model (SM) needs extensions to accommodate two missing spices:the tiny but no-zero neutrino masses and the cosmological dark matter (DM) candidates. One way ofincorporating above two issues in a unified framework is the scotogenic model [1–3], where neutrinos areradiatively generated and the DM fields serves as intermediate messengers propagating inside the loopdiagram. With all new particles around TeV scale, the scotogenic model leads to testable phenomenologies[4–25]. Therefore, viable models are extensively studies in recent years [26].On the other hand, the understanding of the leptonic flavor structure is still one of the major openquestions in particle physics. The consensus is that the leptonic mass texture is tightly restricted under thepresent experimental data. An attractive approach is to consider two texture-zeros in neutrino mass matrix( M ν ) so that the number of parameters in the Lagrangian is reduced[27]. The phenomenological analysisof two texture-zeros models have been studied in Ref.[28, 29]. Among fifteen logically patterns, seven ofthem are compatible to the low-energy experimental data.On the theoretical side, the simplest way of realizing texture-zeros is to impose the discrete Z N flavorsymmetry[30]. However, it might be more appealing to adopt gauge symmetries instead of discrete ones,because the latter may be treated as the residual of U (1) gauge symmetry. It is noted that one can notset any restriction on lepton mass matrix by means of fields with flavor universal charges. Thus the flavordependent U (1) gauge symmetry is the reasonable choice. Along this thought of idea, specific models areconsidered in the context of seesaw mechanisms. In Ref.[31], the two texture-zeros are realized based on theanomaly-free U (1) X gauge symmetry with X ≡ B − (cid:80) x α L α ( α = e, µ, τ ) being the linear combinationof baryon number B and the lepton numbers L α per family. In Ref.[32], more solutions are found in thetype-I and/or III seesaw framework.It is then natural to ask if predictive texture-zeros in M ν can be realized in the scotogenic scenario andseveral attempts have been made in this direction. For example, one texture-zero is recently consideredin Ref. [33]. Texture B - B have been discussed in a model-independent way in Ref. [34]. Texture C isobtained by introducing U (1) L µ − L τ gauge symmetry [35–38]. Texture B is realised with U (1) L e + L µ − L τ gauge symmetry in Ref. [39]. If the quark flavor is also flavor dependent, e.g., U (1) xB − xL e − L µ + L τ , thenone can further interpret the R K anomaly with texture A [40]. Other viable two texture-zeros are system-atically realised in Ref. [41] by considering the U (1) B − L α − L β gauge symmetry with three right-handedsinglets. In this paper, we provide another viable approach. Under same flavor dependent U (1) B − L α − L β gauge symmetry, we introduce only two right-handed singlets but two inert scalars, leading to differenttexture-zeros. In aspect of predicted phenomenology, the texture B considered in Ref. [41] is marginally ν L N ν L η η φ φ ν L N ν L η η φ φ S FIG. 1. Radiative neutrino mass at one-loop. Left pattern is for classic scotogenic model, while right pattern is theadditional contribution in our models. allowed by current Planck result for (cid:80) m i < . eV [42], we thus consider texture A with latest neutrinooscillation data [43] as the benchmark model. In this case, the gauge symmetry is U (1) B − L e − L τ in ourapproach.The rest of this paper is organised as follows. Start with classic scotogenic model in Sec. II, we firstdiscuss the realization of texture-zeros in scotogenic model with U (1) B − L α − L β gauge symmetry in a gen-eral approach. Then the texture A derived from U (1) B − L e − L τ is explained in detail. The correspondingphenomenological predictions, such as neutrino mixing parameters, lepton flavor violation rate, dark mat-ter and highlights of collider signatures are presented in Sec. III. Finally, conclusions are summarised inSec. IV. II. THE MODEL SETUPA. Classic scotogenic model
In the classic scotogenic model proposed by Ma [1], three right-handed fermion singlets N Ri ( i = 1 ∼ and an inert scalar doublet field η = ( η + , η ) are added to the SM. In addition, a discrete Z symmetry isimposed for the new fields in order to forbid the tree-level neutrino Yukawa interaction and stabilize theDM candidate. The relevant interactions for neutrino masses generation are given by L ⊃ h αi L α ˜ ηN Ri + 12 M N N cR N R + 12 λ (Φ † η ) + h.c. . (1)The mass matrix M N can be diagonalized by an unitary matrix V satisfying V T M N V = ˆ M N ≡ diag ( M N , M N , M N ) . (2) Group Lepton Fields Scalar Fields L α (cid:96) αR L β (cid:96) βR L γ (cid:96) γR N R N R Φ η η S S SU (2) L U (1) Y − − − − − −
12 12 12 Z + + + + + + − − + − − + + U (1) B − L α − L β − − − − − − − TABLE I. Particle content and corresponding charge assignments.
Due to the Z symmetry, the neutrino masses are generated at one-loop level, as show in left pattern ofFig. 1. The neutrino mass matrix can be computed exactly, i.e. ( M ν ) αβ = 132 π (cid:88) k h αi V ik h βj V jk M Nk (cid:104) m R m R − M Nk log (cid:0) m R M Nk (cid:1) − m I m I − M Nk log (cid:0) m I M Nk (cid:1)(cid:105) (3)where m R and m I are the masses of √ (cid:60) η and √ (cid:61) η . If we assume m ≡ ( m R + m I ) / (cid:29) M Nk , M ν are then given by ( M ν ) αβ (cid:39) − π λv m (cid:88) k h αi V ik h βj V jk M Nk = − π λv m ( hM N h T ) αβ (4)The neutrino mass matrix M ν is diagonalized as U T PMNS M ν U PMNS = ˆ m ν ≡ diag ( m , m , m ) , (5)where U PMNS is the neutrino mixing matrix denoted as U PMNS = c c s c s − c s s − s c e − iδ − s s s + c c e − iδ s c − c c s + s s e − iδ − s c s − c s e − iδ c c × diag ( e iρ , e iσ , (6)Here, we define c ij = cos θ ij and s ij = sin θ ij ( ij = 12 , , ) for short, δ is the Dirac phase and ρ, σ arethe two Majorana phases as in Ref. [28]. B. Two texture-zeros in scotogenic model
In this section, we demonstrate a class of scotogenic models with G SM × U (1) B − L α − L β × Z gaugesymmetry where two texture-zero structures in M ν are successfully realized. The particle content andcorresponding charge assignments are listed in Tab. I. In the fermion sector, we introduce two right-handed SU (2) L singlets N R and N R and assume they carry the same no-zero B − L α − L β charges as two of SMleptons respectively. Noticeably, if one further introduce one additional N R with zero B − L α − L β charge,the approach considered in Ref. [41] are then reproduced. In terms of gauged U (1) B − L α − L β symmetry,the anomaly free conditions should be considered first and we find all anomalies are zero because [ SU (3) c ] U (1) X : 3 × (cid:16) − − (cid:17) = 0 (7) U (1) Y [ U (1) X ] : 3 (cid:104) (cid:16) (cid:17) − (cid:16) (cid:17) − (cid:16) (cid:17)(cid:105)(cid:16) (cid:17) + (cid:104) (cid:16) − (cid:17) − (cid:16) − (cid:17)(cid:105)(cid:104) ( − + ( − (cid:105) = 0[ SU (2) L ] U (1) X : 12 (cid:104) × (cid:16) (cid:17) + ( −
1) + ( − (cid:105) = 0[ U (1) Y ] U (1) X : 3 (cid:104) (cid:16) (cid:17) − (cid:16) (cid:17) − (cid:16) (cid:17) (cid:105)(cid:16) (cid:17) + (cid:104) (cid:16) − (cid:17) − (cid:16) − (cid:17) (cid:105) ( − −
2) = 0 U (1) X : 2( − − − + 2( − − − = 0[ Gravity ] U (1) X : 2( − − −
1) + 2( − − −
2) = 0
Let us now discuss the scotogenic realizations of two texture-zeros in M ν . With two N R components, h and M N are × and × matrices respectively. From Eq.(4), it is clear that the texture-zeros of M ν can beattributed to the texture-zeros in h and M N matrices. In the original scotogenic model with an inert scalardoublet η and two N R fields, the charge assignments for U (1) B − L α − L β gauge symmetry give rise to onlytwo Yukawa terms for h αi L α ˜ ηN Ri ( α = e, µ, τ, i = 1 , . In this case, at least two texture-zeros are placedin the same line of h matrix, being therefore excluded experimentally. In order to accommodate the realisticneutrino mixing data, the scotogenic model are extended where, in scalar sector, two inert doublet η and η are introduced (see Tab. I). In addition, two scalar singlet S and S are added so that U (1) B − L α − L β symmetry is spontaneously breaking after S , get the vacuum expectation value (VEV) (cid:104) S , (cid:105) = v , / √ .Note that N Ri and η i are odd under the discrete Z symmetry. Since we have two inert scalars, the relevantscalar interactions for the loop-induced neutrino masses is given by L S ⊃ λ Λ (Φ η ) S + λ (cid:48) (Φ η ) + h.c. , (8)where Λ is a new high energy scale and the first term is a dimension-five operator guaranteed by the ac-cidental U (1) B − L α − L β symmetry. One can achieve the effective operator by simply adding a new scalarsinglet ρ ∼ (1 , , , − ) so that in scalar sector L S ⊃ µ (Φ † η ) ρ † + µ (cid:48) ρ S is allowed. Then the effectiveinteraction λ (Φ η ) S / Λ is obtained by integrate the ρ field out of L S sector. In the following analysis, weadopt the expression of effective operator in Eq.(8) and do not consider its specific realization in detail.The neutrinos acquire their tiny masses radiatively though the one-loop diagram depicted in Fig. 1. Texture of M ν Group Texture of M ν Group Status A : × × ×× × × U (1) B − L e − L τ A : × × × × × × U (1) B − L e − L µ Allowed B : × × ×× × × U (1) B − L µ − L τ B : × × × × × × U (1) B − L τ − L µ Marginally Allowed D : × × ×× × × U (1) B − L µ − L e D : × × ×× × × U (1) B − L τ − L e ExcludedTABLE II. Two texture-zeros and corresponding U (1) B − L α − L β symmetry. Here, × denotes a nonzero matrix ele-ment. Therefore, the neutrino mass matrix is formulated by two different contribution, namely, ( M ν ) ∝ hM N h T + f M N f T , (9)where h and f are the Yukawa coupling texture for L α ˜ η N Ri and L α ˜ η N Ri with further assumption Λ = (cid:104) S (cid:105) and λ = λ (cid:48) . As a case study, we consider the U (1) B − L e − L τ gauge symmetry under which the flavordependent Yukawa interaction is given by −L Y = h µ ¯ L µ ˜ η N R + h τ ¯ L τ ˜ η N R + f τ ¯ L τ ˜ η N R + f e ¯ L e ˜ η N R (10) + y N cR N R S + y ( N cR N R + N cR N R ) S + h.c. , where from the charge assignment, the texture of fermion Yukawa coupling are h = h µ h τ , f = f e f τ , y = y y y . (11)Provided all the element in M N to be equal, then from the texture structure in Eq.(11) and using Eq.(9)we have the M ν as M ν ∝ f e f τ h µ h µ h τ f e f τ h µ h τ f τ , (12)which is texture A allowed by experimental data [28, 29]. Other possible realizations with U (1) B − L α − L β can then be easily obtained in a similar approach. In Tab. II, we summarize all the six textures realisedby U (1) B − L α − L β in our approach. According to Ref. [29], texture A and A predict (cid:80) m i ∼ . eV,hence are allowed by Planck limit (cid:80) m i < . eV [42]. Texture B and B predict (cid:80) m i (cid:38) . eV,thus are marginally allowed if certain mechanism is introduced to modify cosmology data. Texture D and D are already excluded by neutrino oscillation data. Following phenomenological predictions are basedon texture A with U (1) B − L e − L τ .In the mass eigenstate of heavy Majorana fermion N i , the corresponding Yukawa couplings with leptonsare easily obtained by h (cid:48) = hV = h µ V h µ V h τ V h τ V , f (cid:48) = f V = f e V f e V f τ V f τ V . (13)For the Z -even scalars, the CP-even scalars in weak-basis ( √ (cid:60) Φ , √ (cid:60) S , √ (cid:60) S ) mix into mass-basis( h, H , H ) with mass spectrum M h ∼ M H < M H . Without loss of generality, we further assumemixing angle between ( h, H ) being α and vanishing mixing angles between H and h/H for simplicity.The would-be Goldstone boson Φ + , √ (cid:61) Φ , √ (cid:61) S are absorbed by gauge boson W + , Z, Z (cid:48) respectively,leaving √ (cid:61) S a massless Mojoron J . In principle, if we introduce U (1) D gauge symmetry to produce thediscrete Z symmetry, this Mojoron J could be absorbed by the dark gauge boson Z D [44]. For Z -oddscalars, there is no mixing between η and η . Since texture of M ν in Eq. (4) is derived by m (cid:29) M Nk ,only fermion DM is allowed in this paper. III. PHENOMENOLOGYA. Neutrino Mixing
The flavor dependent U (1) B − L e − L τ symmetry leads to texture A [28, 29]. Since ( M ν ) ee = 0 , thepredicted effective Majorana neutrino mass (cid:104) m (cid:105) ee is exactly zero for the neutrinoless double-beta decay.Therefore, only normal hierarchy is allowed [45, 46]. Following the procedure in Ref. [28], we now updatethe predictions of neutrino oscillation data with latest global analysis results [43].In Fig. 2, we show the scanning results of texture A . It is worth to note that the best fit value ofneutrino oscillation parameters by global analysis [43] is only marginally consistent with predictions oftexture A , which is clearly seen in Fig. 2 (a). From Fig. 2 (b), we obtain that m ∼ . eV, m ∼ . eV, and m ≈ √ ∆ m ∼ . eV. The resulting sum of neutrino mass is then (cid:80) m i ∼ . eV, thus it δ ( π ) θ ( ° ) ★ ( a ) δ ( π ) m i ( e V ) m m m ( b ) - - δ ( π ) ρ ( π ) ( c ) - - δ ( π ) σ ( π ) ( d ) FIG. 2. Allowed samples of A texture with neutrino oscillation data varied in σ range of Ref. [43]. In pattern (a),the red star (cid:70) stands for the best fit point from global analysis. satisfies the bound from cosmology, i.e., (cid:80) m i < . eV [42]. The Dirac phase should fall in the range δ ∈ [0 . π, . π ] , meanwhile Fig. 2 (c) and (d) indicate that ρ ≈ δ and σ ≈ δ − π .Instead of the marginally best fit value, we take δ = π and θ = 46 ◦ with other oscillation parametersbeing the best fit value in Ref [43] as the benchmark point for illustration, which leads to the followingneutrino mass structure M ν = . . . . . . eV (14)By comparing the analytic M ν in Eq. (12) and numerical M ν in Eq. (14), one can easily reproduce the
200 400 600 800 100010 - - - - - - - M η ( GeV ) BR τ → μ γ h μ = h μ =
200 400 600 800 100010 - - - - - - - M η ( GeV ) BR τ → e γ h μ = h μ = FIG. 3. Predictions for τ → µγ (left) and τ → eγ (right) with corresponding current bound [49] and future sensitivity[50]. In these figures, we have fixed M N = 200 GeV. observed neutrino oscillation data by requiring h τ h µ : f τ h µ : f e h µ = ( M ν ) µτ ( M ν ) µµ : (cid:115) ( M ν ) ττ ( M ν ) µµ : ( M ν ) eτ (cid:112) ( M ν ) µµ ( M ν ) ττ (15) (cid:39) .
745 : 0 .
933 : 0 . . Hence, we can take h µ as free parameters and determine the other three Yukawa coupling by using aboveratios. The overall neutrino mass scale is then determined by λv M N h µ / (32 π m ) ≈ . eV. B. Lepton Flavor Violation
The new Yukawa interactions of the form ¯ L ˜ ηN R will contribute to lepton flavor violation (LFV) pro-cesses [47, 48]. In this work, we take the radiative decay (cid:96) i → (cid:96) j γ for illustration. With flavor dependent U (1) B − L e − L τ symmetry, it is clear from Eq. (10) that η ± ( η ± ) will only induce τ → µγ ( τ → eγ ) atone-loop level. It is worth to note that the most stringent µ → eγ decay is missing at one-loop level. Hence,if the ongoing experiments observe τ → µγ and τ → eγ but no µ → eγ , this model will be favored. Thecorresponding branching ratios are calculated asBR ( τ → µγ ) = 3 α πG F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 ( h µ V i )( h τ V i ) ∗ M η F (cid:18) M Ni M η (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) BR ( τ → µν τ ¯ ν µ ) , (16)BR ( τ → eγ ) = 3 α πG F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 ( f e V i )( f τ V i ) ∗ M η F (cid:18) M Ni M η (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) BR ( τ → eν τ ¯ ν e ) ,
200 400 600 800 100010 - - - - - M η ( GeV ) | Δ a μ | h μ = h μ = FIG. 4. Predictions for | ∆ a µ | . In this figures, we have fix M N = 200 GeV. where the loop function F ( x ) is F ( x ) = 1 − x + 3 x + 2 x − x ln x − x ) . (17)In the limit for degenerate M N , we haveBR ( τ → (cid:96)γ ) ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 V i V ∗ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | ( V V † ) | = 0 , (18)where in the last step, we have considered the fact that V is an unitary matrix. Therefore, large cancellationsbetween the contribution of two N i are also possible even in the case of non-degenerate M N . In Fig. 3, weshow the predictions for τ → µγ and τ → eγ . Although constraint on BR( τ → eγ ) is slightly morestringent than BR( τ → µγ ), the predicted BR( τ → eγ ) is much smaller than BR( τ → µγ ). It is clear thatthe current bound is quite loose, e.g., M η (cid:38) GeV with h µ = 1 can be allowed.Although the Yukawa interaction ¯ L µ ˜ η N i can not induce µ → eγ at one-loop, it does contribute to muonanomalous magnetic moment [51] ∆ a µ = − (cid:88) i =1 | h µ V i | M µ π M η F (cid:18) M Ni M η (cid:19) . (19)Comparing with BR( τ → (cid:96)γ ), there is no cancellations between the contribution of two N i . However, thetotal contribution to ∆ a µ is negative, while the observed discrepancy ∆ a µ = a EXP µ − a SM µ = (261 ± × − is positive [52]. Thus, the Yukawa interaction ¯ L µ ˜ η N R can not explain the ( g − µ anomaly, andsome other new physics is required [51]. On the other hand, since a too large negative contribution to ∆ a µ is not favored, we consider theoretical | ∆ a µ | < − in the following. The results are shown in Fig. 4. Wefind that the bound from | ∆ a µ | < − is actually slightly more stringent than BR ( τ → µγ ) .1
200 400 600 800 10000.0010.0100.100110 M η ( GeV ) Ω h h μ = h μ = M N1 =
200 GeVPlanck
FIG. 5. Predicted relic density as a function of m η , where we have fix M N = 200 GeV. The green line correspondsto the observed relic density Ω h = 0 . ± . [42]. C. Dark Matter
In this work, we consider N is the DM candidate. In the original scotogenic model [1], the viable anni-hilation channel is N N → (cid:96) + (cid:96) − , ¯ νν via the Yukawa interaction ¯ L (cid:96) ˜ ηN [53]. However, such annihilationchannel is tightly constrained by non-observation of LFV [54]. Thanks to relative loose constraints from τ decays, the scanning results of Ref. [54] suggested that N should have a large coupling to L τ . Thus, thedominant annihilation channel is τ + τ − and ¯ ν τ ν τ with M N (cid:46) TeV.Quite different from the original scotogenic model [1], the LFV process is either vanishing or suppressedin this flavor dependent model. Therefore, O (1) Yukawa coupling can be easily realised without tuning.In the following quantitative investigation, we consider a special scenario, i.e., M η = M η = M η forsimplicity. For vanishing lepton masses, the Yukawa-portal annihilation cross section is [53, 55] σv rel = a + bv rel = 0 + (cid:88) α,β (cid:12)(cid:12) h (cid:48) α h (cid:48)∗ β + f (cid:48) α f (cid:48)∗ β (cid:12)(cid:12) r (1 − r + 2 r )24 πM N v rel , (20)where v rel is the relative speed, h (cid:48) and f (cid:48) are defined in Eq. (13), r = M N / ( M η + M N ) . The thermallyaveraged cross section is calculated as (cid:104) σv rel (cid:105) = a + 6 b/x f , where the freeze-out parameter x f = M N /T f is obtained by numerically solving x f = ln (cid:18) . M Pl M N (cid:104) σv rel (cid:105)√ g ∗ x f (cid:19) . (21)2
200 400 600 800 100010 - - - - - - M N1 ( GeV ) σ S I ( pb ) X E N O N T L Z P r o j e c t e d sin α = α = FIG. 6. Spin-independent cross section as a function of M N . The black solid and dashed line correspond to currentXENON1T [68] and future LZ [69] limits, respectively. In this figure, we have set M H = 500 GeV and v S = 10 TeV.
The relic density is then calculated as [56] Ω h = 1 . × GeV − M Pl x f √ g ∗ a + 3 b/x f , (22)where M Pl = 1 . × GeV is the Planck mass, g ∗ is the number of relativistic degrees of freedom. Thenumerical results are depicted in Fig 5. Provided the mass of the DM candidate is M N = 200 GeV, thenthe observed relic density is interpreted by h µ = 1 , M η = 366 GeV or h µ = 1 . , M η = 640 GeV. Thatis to say, h µ ∼ O (1) is required to obtain correct relic density, and the larger h µ is, the larger the masssplitting M η − M N is.In addition to the Yukawa-portal interaction, N can also annihilate via the Higgs-portal and Z (cid:48) -portalinteractions [57–63]. In these two scenarios, M N (cid:39) M h,H / or (cid:39) M Z (cid:48) / are usually required to realizecorrect relic density [64]. If the additional scalar singlet scalar H is lighter than N , then the annihilationchannel N N → HH with H → b ¯ b is able to explain the Fermi-LAT gamma-ray excess at the Galacticcenter [65, 66].The spin-independent DM-nucleon scattering cross section is dominantly mediated by scalar interac-tions, which is given by σ SI = 4 π (cid:18) M p M N M p + M N (cid:19) f p , (23)3
200 400 600 800 10000.00.51.01.52.02.53.03.5 M η ( GeV ) h μ τ→μγ | Δ a μ | LHC Ω h = M N1 =
200 GeV
200 400 600 800 10002004006008001000 M η ( GeV ) M N ( G e V ) M N1 > M η | Δ a μ | LHC h μ = h μ = h μ = . h μ = . FIG. 7. Combined results for the Yukawa-portal DM. Left pattern: in the h µ - M η plane; right pattern: in the M N - M η plane. The green lines satisfy the condition for correct relic density. And the blue regions are excluded by LHCdirect search, which will be discussed in Sec. III D. where M p is the proton mass and the hadronic matrix element f p reads f p M p = (cid:88) q = u,d,s f pT q α q M q + 227 − (cid:88) q = u,d,s f pT q (cid:88) q = c,b,t α q M q . (24)and the effective vertex α q M q = − y N √ v sin 2 α (cid:32) M h − M H (cid:33) , (25)Here, y N = y V is the effective Yukawa coupling of N with S . For proton, the parameters f pT q areevaluated as f pT u = 0 . ± . , f pT d = 0 . ± . and f pT s = 0 . ± . [67]. Fig. 6 showsthe numerical results for direct detection. It is obvious that the predicted σ SI with sin α = 0 . lies belowcurrent XENON1T limit, but the range of M N (cid:38) GeV is within future LZ’s reach. However, if no directdetection signal is observed by LZ, then sin α (cid:46) . should be satisfied.In Fig. 7, we show the combined results from LFV, | ∆ a µ | , relic density and LHC search. In left patternof Fig. 7, it indicates that for M N = 200 GeV, the only exclusion region is from LHC search. Hence, either M η (cid:46) GeV with h µ (cid:46) . or M η (cid:38) GeV with h µ (cid:38) . is required. In right pattern of Fig. 7,two benchmark value h µ = 1 . , . are chosen to illustrate. For h µ = 1 . , we have GeV (cid:46) M N (cid:46) M η ∼ GeV. Therefore, the only viable region is M N ∼ M η (cid:46) GeV for h µ (cid:46) . Meanwhile for h µ = 1 . , M N (cid:38) GeV with M η (cid:38) GeV is able to escape LHC limit.4
200 400 600 800 10000.010.050.100.501 M H ( GeV ) BR ( H ) sin α = JJWWhhZZt t
200 400 600 800 10000.010.050.100.501 M H ( GeV ) BR ( H ) sin α = JJWWhhZZt tN N FIG. 8. Branching ratios of scalar singlet H for sin α = 0 . (left) and sin α = 0 . (right). In this figures, we havealso fix M N = 200 GeV and v S = 10 TeV. Note in left pattern, BR ( H → N N ) is less than 0.01, thus is notshown in the plot. D. Collider Signature
In this part, we highlight some interesting collider signatures. Begin with the newly discovered 125 GeVHiggs boson h [72, 73]. The existence of massless Mojoron J will induce the invisible decay of SM Higgsvia h → J J [74]. The corresponding decay width is evaluated as Γ( h → J J ) (cid:39) M h sin α πv S . (26)Then, the branching ratio of invisible decay is BR( h → J J ) = Γ( h → J J ) / (Γ( h → J J ) + Γ SM cos α ) ,where Γ SM = 4 . MeV [75]. Currently, the combined direct and indirect observational limit on invisibleHiggs decay is BR( h → J J ) < . [76]. Typically for sin α = 0 . , v S = 10 TeV, we have BR( h → J J ) = 4 . × − , which is far below current limit. Meanwhile, if M H < M h , then h → H H with H → J J will also contribute to invisible Higgs decay [77].In this paper, we consider the high mass scenario M H > M h . In addition to the usual H → SMfinal states as real singlet model [78], the heavy scalar singlet can also decay into Majoron pair H → J J and DM pair H → N N . Fig. 8 shows the dominant decay branching ratios of H . The invisibleBR ( H → J J ) is less than 0.02 when sin α = 0 . , therefore H appears as a SM heavy Higgs withBR ( H → hh ) ≈ BR ( h → ZZ ) ≈ BR ( H → W W ) ≈ . While for sin α = 0 . , the invisibleBR ( H → J J ) increases to about . , reaching the same order of visible V V, hh decay. And the otherinvisible decay H → N N maximally reaches about . at M H ∼ GeV. The dominant production5 q ¯ q e + e − µ + µ − τ + τ − νν N N H H U (1) B − L e − L τ gauge boson Z (cid:48) , where we have show the lepton flavor individ-ually. channel of H is via gluon fusion at LHC, which can be estimated as σ ( gg → H ) ≈ sin α × σ ( gg → h ) , (27)where σ ( gg → h ) is the SM Higgs production cross section but calculated with M h = M H . At present, sin α ∼ . [79] leads to the promising signatures as H → W W → eνµν [80], ZZ → (cid:96) [81] and hh → b γ [82, 83], etc. In the future, if no DM direct detection signal is observed, then the signature ofheavy scalar H will be much suppressed by tiny value of sin α .Next, we discuss the gauge boson Z (cid:48) associated with U (1) B − L e − L τ . Its decay branching ratios areflavor-dependent, which makes it quite easy to distinguish from the flavor-universal ones, such as Z (cid:48) from U (1) B − L [84]. Considering the heavy Z (cid:48) limit, its partial decay width into fermion and scalar pairs aregiven by Γ( Z (cid:48) → f ¯ f ) = M Z (cid:48) π g (cid:48) N fC ( Q fL + Q fR ) , (28) Γ( Z (cid:48) → SS ∗ ) = M Z (cid:48) π g (cid:48) Q S , (29)where N fC is the number of colours of the fermion f , i.e., N l,νC = 1 , N qC = 3 , and Q X is the U (1) B − L e − L τ charge of particle X . In Tab. III, we present the branching ratio of Z (cid:48) . The dominant channel is Z (cid:48) → e + e − with branching ratio of . , and no Z (cid:48) → µ + µ − . The B − L e − L τ nature of Z (cid:48) predicts definite relationbetween quark and lepton final states, e.g.,BR ( Z (cid:48) → b ¯ b ) : BR ( Z (cid:48) → e + e − ) : BR ( Z (cid:48) → µ + µ − ) : BR ( Z (cid:48) → τ + τ − ) = 13 : 4 : 0 : 1 , (30)which is also an intrinsic property to distinguish Z (cid:48) of U (1) B − L e − L τ from other flavored gauge bosons[85].In the framework of U (1) B − L e − L τ , one important constraint on Z (cid:48) comes from the precise measure-ment of four-fermion interactions at LEP [86], which requires M Z (cid:48) g (cid:48) (cid:38) . (31)Since Z (cid:48) couples to both quarks and leptons, the most promising signature at LHC is the dilepton signature pp → Z (cid:48) → e + e − . Searches for such dilepton signature have been performed by ATLAS [70] and CMS6 - - - - M Z ' ( GeV ) R σ g' = = @ M Z ' ( GeV ) g ' LEP LHC
FIG. 9. Left pattern: predicted cross section ratios in U (1) B − L e − L τ and corresponding limit from LHC. Rightpattern: allowed parameter space in the g (cid:48) - M Z (cid:48) plane. collaboration [87]. Because of no Z (cid:48) → µ + µ − channel, we can only take the results from CMS, whichprovides a limit on the ratio R σ = σ ( pp → Z (cid:48) + X → e + e − + X ) σ ( pp → Z + X → e + e − + X ) . (32)The theoretical cross section of the dilepton signature are calculated by using MadGraph5 aMC@NLO [88].Left pattern of Fig. 9 shows that the dilepton signature has excluded M Z (cid:48) (cid:46) . . TeV for g (cid:48) = 0 . . .Then comparing the theoretical ratio with experimental limit, one can acquire the exclusion limit in the g (cid:48) − M Z (cid:48) plane as shown in right pattern of Fig. 9. Obviously, LHC limit is more stringent than LEP when M Z (cid:48) (cid:46) TeV.The inert charge scalars η ± , are also observable at LHC. They can decay into charged leptons andright-hand singlets via the Yukawa interactions as Γ( η ± → (cid:96) ± N i ) = M η ± π (cid:12)(cid:12) h (cid:48) (cid:96)i (cid:12)(cid:12) − M N i M η ± , (33) Γ( η ± → (cid:96) ± N i ) = M η ± π (cid:12)(cid:12) f (cid:48) (cid:96)i (cid:12)(cid:12) − M N i M η ± . (34)From Eq. (10), we aware that η ± decays into µ, τ final states, while η ± decays into e, τ final states. Theelectron-phobic nature of η ± and muon-phobic nature of η ± make them quite easy to distinguish. Mean-while, their decay branching ratios are related by neutrino oscillation data through the Yukawa coupling7 Final state e ± N µ ± N τ ± N e ± N µ ± N τ ± N η ± η ± η ± , . h (cid:48) , f (cid:48) . Considering the benchmark point in Eq. (15), the predicted branching ratios are shown in Tab. IVin the heavy scalar limit. The dominant decay channel of η ± is µ ± N , and τ ± N for η ± . So η ± is ex-pected easier to be discovered. Produced via the Drell-Yan process pp → η +1 η − , η +2 η − , the decay channel η ± , → (cid:96) ± N then leads to signature (cid:96) + (cid:96) − + (cid:0)(cid:0) E T . Exclusion region by direct LHC search for such signature[71] has been shown in right pattern of Fig. 7. To satisfy the direct LHC search bounds, one needs either M N (cid:46) M η < GeV or M η > GeV.
IV. CONCLUSION
The scotogenic model is an elegant pathway to explain the origin of neutrino mass and dark matter.Meanwhile, texture-zeros in neutrino mass matrix provide a promising way to under stand the leptonicflavor structure. Therefore, it is appealing to connect the scotogenic model with texture-zeros. In this paper,we propose a viable approach to realise two texture-zeros in the scotogenic model with flavor dependent U (1) B − L α − L β gauge symmetry. These models are extended by two right-handed singlets N Ri and twoinert scalar doublets η i , which are odd under the dark Z symmetry. Six kinds of texture-zeros are realisedin our approach, i.e., texture A , A , B , B , D and D . Among all the six texture-zeros, we find thattexture A and A are allowed by current experimental limits, while texture B and B are marginallyallowed. Besides, texture D and D are already excluded by neutrino oscillation data.Realization of texture-zeros in the scotogenic model makes the model quite predictive. And we havetaken texture A derived from U (1) B − L e − L τ for illustration. Some distinct features are summarized in thefollowing:• The texture A predicts vanishing neutrinoless double beta decay rate. And only normal neutrinomass hierarchy is allowed. It predicts m ∼ . eV, m ∼ . eV, and m ≈ √ ∆ m ∼ . eV, then (cid:80) m i ∼ . eV. There are also strong correlation between the Dirac and Majorana phases,i.e., ρ ≈ δ and σ ≈ δ − π .8• The ratios of corresponding Yukawa couplings are also predicted by neutrino oscillation data, e.g., h τ h µ : f τ h µ : f e h µ (cid:39) .
745 : 0 .
933 : 0 . . • Due to specific Yukawa structure, the LFV process µ → eγ is missing at one-loop level. Meanwhile,large cancellations are possible for τ → µγ and τ → eγ with degenerate right-handed singlets. Morestringent constraint comes from muon anomalous magnetic moment ∆ a µ . Although O (1) Yukawacouplings are easily to avoid such limit.• Satisfying all constraints, correct relic density of dark matter N is achieved for M N (cid:46) M η < GeV with h µ (cid:46) or M η > GeV with h µ > .As for direct detection, we have shown thatthe predicted spin-independent DM-nucleon cross section σ SI with sin α = 0 . satisfies the currentXENON1T limit, but is within future reach of LZ.• The massless Mojoron J contributes to invisible decay of SM Higgs. The additional scalr singlet H can be probe in the channel gg → H → W + W − , ZZ at LHC. Decays of charged scalars η ± , leadto pp → η +1 , η − , → (cid:96) + (cid:96) − + (cid:0)(cid:0) E T signature. Note that the corresponding branching ratios are alsocorrelated with neutrino oscillation parameters.• The neutral gauge boson Z (cid:48) is promising via the di-electron signature pp → Z (cid:48) → e + e − . Its B − L e − L τ nature can be confirmed byBR ( Z (cid:48) → b ¯ b ) : BR ( Z (cid:48) → e + e − ) : BR ( Z (cid:48) → µ + µ − ) : BR ( Z (cid:48) → τ + τ − ) = 13 : 4 : 0 : 1 , In a nutshell, the scotogenic model with flavor dependent U (1) B − L α − L β symmetry predicts distinctand observable phenomenology, which is useful to distinguish from other models. V. ACKNOWLEDGEMENTS
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