Scalar Dark Matter in A_4 based texture one-zero neutrino mass model within Inverse Seesaw Mechanism
aa r X i v : . [ h e p - ph ] F e b Scalar Dark Matter in an Inverse SeesawModel with A Discrete Flavor Symmetry
Rishu Verma ∗ , Monal Kashav † , Surender Verma ‡ and B. C. Chauhan § Department of Physics and Astronomical Science,Central University of Himachal Pradesh, Dharamshala 176215,INDIA.
Abstract
In this paper we propose a model based on A discrete flavor symmetry.We implemented the inverse seesaw mechanism in order to get the right handedneutrino mass at TeV scale and explore neutrino and dark matter sectors. Non-abelian discrete A symmetry spontaneously breaks into Z subgroup and henceprovide stable dark matter candidate. To constrain the Yukawa Lagrangian ofour model, we imposed Z ′ and Z cyclic symmetries in addition to the A flavorsymmetry. In our work we used the recently updated data on cosmological pa-rameters from PLANCK 2018. For the dark matter candidate mass around 45GeV-55 GeV, we obtained the mediator particle mass(right-handed neutrinos)ranging from 138 GeV to 155 GeV. We found Yukawa coupling in the range0.995-1 in order to get observed relic abundance of dark matter. Standard Model of particle physics is a low energy effective theory which has beenastonishingly successful in explaining the dynamics of fundamental particles and theirinteractions. The discovery of Higgs Boson in 2012 at CERN LHC has strengthen ourbelief in this incredible theory. Despite its tremendous success there still remain unan-swered questions such as origin of neutrino mass, dark matter and matter-antimatterasymmetry, to name a few. Neutrino oscillation experiments have been very instru-mental in our quest to understand Standard Model(SM) predictions for the leptonicsector. They have shown at high level of statistical significance that neutrinos have ∗ [email protected] † [email protected] ‡ s [email protected] § [email protected] GeV.Therefore, the seesaw mechanisms explain the tiny non-zero neutrino masses, butthey introduce new physics scale which is beyond the reach of current and futureaccelerator experiments.On the other hand, there are several astonishing astrophysical observations suchas (i) galaxy cluster investigation [7], (ii) rotation curves of galaxy [8], (iii) recentobservations of bullet cluster [9] and (iv) the latest cosmological data from Planckcollaboration [10], which have proven the existence of non-luminous and non-baryonicanatomy of matter known as “Dark Matter”(DM). Apart from the astrophysical en-vironments, it is very difficult to probe the existence of DM in terrestrial laboratory.Alternatively, one can accredit a weak interaction property to the DM through whichit get thermalized in the early Universe which, also, can be examined at terrestriallaboratory. According to the Planck data the current DM abundance is [10]Ω DM h = 0 . ± . A non-abelian discrete symmetry within the framework of inverse seesaw(ISS) [25–27]wherein small neutrino masses emanate from new physics at TeV scale which is withinthe reach of accelerator experiments. The stability of DM is assured by Z symme-try. Type-II seesaw has been implemented to have one-zero in the effective neu-trino mass matrix. Within ISS mechanism, neutrino masses are generated assumingthree right-handed neutrinos N iR and three additional SM singlet neutral fermions S iL ( i = 1 , , N , and S , ) are assumed, to have Yukawacouplings with the scalar fields H, φ, φ R and φ S which after spontaneous symmetrybreaking provide diagonal Majorana mass matrix( µ ). The scalar triplets ∆ and∆ are incorporated so that M ν contain one vanishing element, after type-II see-saw implementation. Along with DM abundance, we have also obtained predictionof the model for effective Majorana mass ( | M ee | ) appearing in neutrinoless doublebeta(0 νββ ) decay.The paper is structured as follows. In section 2, we have discussed the inverseseesaw mechanism based on A ⊗ Z ′ ⊗ Z symmetry group and resulting neutrinomass matrices. Section 3 is devoted to the relic density of the DM. In section 4,the prediction of the model for neutrinoless double beta decay is discussed. Finally,conclusions are summarized in section 5. In order to explain the smallness of neutrino mass, different versions of the seesawmechanism play an important role. In our work we have used ISS mechanism, whichis a viable scenario to get the mass of right handed neutrino near the TeV scale. Thisscale is much below the scale which we get from the canonical seesaw. As a require-ment of ISS, the fermion sector is extended by three right handed neutrinos N T =(N ,N ,N ) and three extra singlet fermions S T =(S ,S ,S ). Within ISS mechanismthe mass Lagrangian is written as L = − ¯ v L m D N − ¯ S L mN −
12 ¯ S L µS CL + h.c. (1)where ‘ m D ’, ‘ m ’ and ‘ µ ’ are the 3 × m ’ representslepton number conserving interaction between neutral fermions and right handed neu-trinos and ‘ µ ’ gives the Majorana mass terms for neutral fermions. The corresponding9 × M ν = m D m TD m m T µ . (2)We can obtain standard model neutrinos at sub-eV scale from ‘ m D ’ at electroweakscale, ‘ µ ’ at keV scale and m at TeV scale as explained in [27, 28]. If we consider the3rder µ << m D << m , then after the block diagonalization of above matrix, the 3 × m ν = m D ( m T ) − µm − m TD . (3)It is clear from Eq.3, that there is a double suppression by mass term associated with‘ m ’, which results in the scale that is much below to the one obtained by canonicalseesaw. The essence of inverse seesaw mechanism lies in the fact that we can bringdown the mass of right handed neutrinos to TeV scale by assuming that ‘ µ ’ shouldbe at keV scale [29–31].The symmetry group A has played an important role in understanding particlephysics [32–36]. A is a non-abelian discrete symmetry group of even permutationsof four objects. Order of this group is 12. All the 12 elements are generated from twoelements, S and T which satisfies: S = T = (ST) . It is a symmetry group of regulartetrahedron. It has four conjugacy classes, therefore, four irreducible representations:1,1 ′ ,1 ′′ and 3. The multiplication rules of irreducible representations in T basis are[36, 37]: ′ ⊗ ′ = ′′ , ′′ ⊗ ′′ = ′ , ′ ⊗ ′′ = , ⊗ = ⊕ ′ ⊕ ′′ ⊕ s ⊕ a where,( ⊗ ) = a b + a b + a b , ( ⊗ ) ′ = a b + ωa b + ω a b , ( ⊗ ) ′′ = a b + ω a b + ωa b , ( ⊗ ) s = ( a b + b a , a b + a b , a b + a b ) , ( ⊗ ) a = ( a b − b a , a b − a b , a b − a b ) , the ‘ a ′ i s’ and ‘ b ′ i s’ (i = 1,2,3) here are the basis vectors of the two triplets and ω = e πi .In the model we have taken five right handed neutrinos, three of which N T =(N ,N ,N ) are transforming as triplet under A and rest of the two i.e., N andN are transforming as singlets 1, 1 ′ , respectively. The standard model Higgs ’H’ istransforming as singlet under A and three additional Higgs doublets, i.e η =( η , η , η )are transforming as triplet. In addition to these two Higgs fields, we have extendedthe Higgs sector with three more Higgs fields i.e. φ , φ R and φ S , boosted by Z ′ and Z additional symmetries. The fermion field content and respective charge assignmentsare shown in Table 1, and scalar field content and respective charge assignments areshown in Table 2.Symmetry L e L µ L τ l ce l cµ l cτ N T N N S T S S SU (2) L A ′ ′′ ′′ ′ ′ ′ Z ′ Z ω ω ω ω ω ω ω ω ω ω ω ω Table 1: Fermion field content and respective charge assignments used in the model.4ymmetry H η Φ Φ R Φ S ∆ ∆ SU (2) L A ′ Z ′ Z ω ω ω Table 2: Scalar field content and respective charge assignments used in the model.The leading Yukawa Lagrangian is L I = y e L e l ce H + y µ L µ l cµ H + y τ L τ l cτ H + y ν L e [ N T η ] + y ν L µ [ N T η ] ′′ + y ν L τ [ N T η ] ′ + y ν L e N H + y ν L τ N H + y R [ N T S T ] φ R + y R [ N S ] φ R + y φ [ N T φ ] S + y φ [ N T φ ] ′′ S + y φ [ N T φ ] S T + y s S T S T φ S + y s S S φ S + h.c. (4)We have chosen the following vacuum expectation values h η i ∼ v η (1 , , h φ i ∼ v φ (1 , , h H i = v h , h φ S i = v S , h φ R i = v R .The symmetry A ⊗ Z ′ ⊗ Z is broken down to Z subgroup by the vev h η i ∼ v η (1 , ,
0) [38], since (1,0,0) remains invariant under the A generator S = Diag(1,-1,-1). Therefore, the residual Z symmetry is N → -N , N → -N , S → -S , S → -S , η → − η , η → − η , φ → − φ , φ → − φ .This kind of residual symmetry is responsible for lightest dark matter candidatestability. Since inverse seesaw formula in Eq.3 assumes hierarchy of mass scale µ < 00 0 0 0 0 , m = x l v x h h x l z v , (6)where A = y ν v η , B = y ν v η , C = y ν v η , F = y ν v h , H = y ν v h , y = y S v S , n = y S v S ,x = y R v R , z = y R v R , l = y φ v φ , v = y φ v φ , h = y φ v φ . Within ISS mechanism, the abovematrices lead to the light neutrino mass matrix as follow m ν I = X ′′ , (7)where X = F nz , ∆ = − F Hlnvz and ∆ ′′ = H ( l n + yz ) v z .When we choose a flavor basis in which we are obtaining a diagonal charged leptonmass matrix, only those neutrino mass matrix are allowed where we have at mosttwo zeros. These neutrino mass matrices are consistent with neutrino oscillation re-sults [39]. Since we are getting three zeros in our neutrino mass matrix, we haveintroduced type-II seesaw to reduce the number of zeros in the neutrino mass matrix.The type-II seesaw contribution to the Lagrangian is given as L II = f ( L e L e + L µ L τ + L τ L µ )∆ + f ( L e L τ + L τ L e + L µ L µ )∆ + h.c. (8)Therefore, the ISS + type-II seesaw Lagrangian for our model is given as L = y e L e l ce H + y µ L µ l cµ H + y τ L τ l cτ H + y ν L e N T η + y ν L µ N T η + y ν L τ N T η + y ν L e N H + y ν L τ N H + y R N S φ R + y R N T S T φ R + y φ N T φS + y φ N T φS + y φ N T φS T + y s S T S T φ S + y s S S φ S + f ( L e L e + L µ L τ + L τ L µ )∆ + f ( L e L τ + L τ L e + L µ L µ )∆ + h.c. (9)The SU(2) triplet ∆ and ∆ are transforming as singlets 1 and 1 ′ , respectively.The vacuum alignments h ∆ i = v ∆ , h ∆ i = v ∆ gives m ν II = X ′ ′ ′ X ′ ∆ ′ X ′ , (10)where, X ′ = f v ∆ , ∆ ′ = f v ∆ . Now the final neutrino mass matrix is obtained as, M ν = m ν I + m ν II , is of textureone zero and is given as 6 ν = X + X ′ ′ ′ X ′ ∆ + ∆ ′ X ′ ∆ ′′ . (11)In literature, there are several techniques which are used to reduce the parametersof neutrino mass matrix and texture zeros is one of them [39–58]. We have obtainedone texture zero neutrino mass matrix and its phenomenological implications havealso been studied in the literature [54–57] and it has been showed that all the six caseswhich are possible for one texture zero are viable with experimental data. Moreover,we can have larger parametric space for consistency with the data available by usingone texture zero instead of two texture zero. In the early universe, the particles were in thermal equilibrium i.e. the processes inwhich the lighter particles combine to form heavy particles and vice-versa happenedat same rate. At some point of time, the conditions required for thermal equilibriumwere contravened because the density of some particle species became too low. Theseparticles are stated as “freeze-out” and they have a constant density which is knownas relic density, because the abundance of particle remains same. In the process anyparticle χ was in thermal equilibrium, then its relic abundance can be obtained byusing Boltzmann equation [59–62] dn χ dt + 3 H n χ = − < σv > ( n χ − ( n eqbχ ) ) , (12)where ‘ H ’ is the Hubble constant and ‘ n χ ’ is the number density of the DM par-ticle ‘ χ ’. Here, ‘ n eqbχ ’ is the number density of particle ‘ χ ’ when it was in thermalequilibrium. However, < σv > is the thermally averaged annihilation cross-section ofDM particle. For a DM particle with electroweak scale mass, the solution of aboveequation is given as [63] Ω χ h = 3 × − cm s − < σv > , (13)where, Ω χ h gives the relic density of DM particle.From the Lagrangian in Eq.9, it is clear that the interaction of dark matter particlewith right handed neutrinos is there and this is shown in Fig.(1). The cross-sectionformula for this kind of process is given as [64] < σv > = v y m χ π ( m χ + m ψ ) , (14)7here, ‘ y ’ is Yukawa coupling of the interaction between DM and fermions, ‘ m ψ ’and ‘ m χ ’ represent the mass of Majorana fermion and relic particle mass respectively.Here, ‘ v ’ is the relative velocity of two relic particles and is taken to be 0.3c at freezeout temperature. In case of m DM < M W , which indicates the low mass scale of relicparticle, η , η self annihilates via SM Higgs into the SM particles as shown in Fig.(2).The self annihilation cross-section is thus given as below [61, 65] σ xx = ( | Y f | | λ x | πs )( ( s − m f ) / ( s − m x ) / (( s − m h ) + m h Γ h ) ) , (15)where, ‘ Y f ’ is Yukawa coupling of fermions and we have used its recent value thatis 0.308 [66]. Here, ‘Γ h ’ is the SM Higgs decay width and its value used is 4.15 MeV.The ‘ m h ’ is Higgs mass, that is 126 GeV, and ‘ x ’ in the above equation represents η , η and coupling of ‘ x ’ with SM Higgs is represented as ‘ λ x ’. Here, s is thermallyaveraged center of mass squared energy and is given as [65] s = 4 m χ + m χ v . (16)Figure 1: Scattering of DM particle η , .Figure 2: Self annihilation of DM particle η , [67].The neutral component of scalar triplet is our DM candidate as considered in [68,69].In this work we fixed our parameters in Eq.13 to obtain the recently updated con-straints on relic abundance, reported by PLANCK 2018 data. To obtain the correctrelic density of DM, we need to constrain the parameters like, Yukawa coupling, relicmass and mediator mass(right-handed neutrinos in our case). As stated above, we8hose the relic mass much less than the mass of W-Boson. We have done our analysisfor different values of relic mass and obtained different mediator masses and Yukawacouplings, which are shown in Fig.(3). This type of studies have already been donein [64, 70]. In order to get correct relic abundance, we did our analysis for DM par-ticle mass around 50 GeV, as suggested by many experimets like XENON1T [71],PandaX-11 [72], LUX [73], SuperCDMS [74] etc. For DM particle mass 45 GeV, 50GeV and 55 GeV, we obtained the mass of right handed neutrinos ranging from 138GeV to 155 GeV. Yukawa coupling is obtained in the range 0.995-1. The results areshown in Table 3. m Ψ = 145 GeV m Ψ = 147 GeV m Ψ = 146 GeV W Χ (a) m Ψ = 140 GeV m Ψ = 139 GeV m Ψ = 138 GeV W Χ (b) m Ψ = 157 GeV m Ψ = 156 GeV m Ψ = 155 GeV W Χ (c) Figure 3: Relic density of DM vs Yukawa coupling plots for (a)DM mass(m χ ) = 50GeV, (b) DM mass(m χ ) = 45 GeV and (c) DM mass(m χ ) = 55 GeV.9r.No. Relic Mass(m χ ) Mediator Mass(m ψ ) Yukawa Coupling Range1 45 GeV 138 GeV 0.995-12 50 GeV 146 GeV 0.998-13 55 GeV 155 GeV 0.996-1Table 3: Constraints on relic(DM) mass, RH neutrino mass and respective YukawaCoupling ranges. ν ββ ) Decay In order to shed the light on the nature of neutrino, the observation of lepton numberviolating neutrinoless double beta decay process is of utmost importance. If futureexperiments confirm the occurrence of 0 νββ process, the nature of neutrino will beMajorana. Texture zero model obtained in Eq.11 is T texture having non-zero con-tribution to neutrinoless double beta decay as identified in literature [20]. We haveopted the same method of phenomenological study for the prediction of the texturezero model on observation of neutrinoless double beta decay. The advantage of com-paring the obtained flavor model with the usual low-energy phenomenology of onezero texture is to distinguish the magnitude of inverse and type-II seesaw contribu-tion to | M ee | . In the charged lepton diagonal mass basis, the Majorana neutrino massmatrix can be written as M ν = U M d U T , (17)where, ‘ M d ’ is diagonal mass matrix containing mass eigenvalues of neutrinos diag ( m , m , m ). ‘ U ’ is neutrino mixing matrix defined as U = V.P where ‘ P ’ isdiagonal phase matrix diag (1 , e iα , e i ( β + δ ) ). In PDG representation, V is given by c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c , (18)where, ‘ δ ’ is Dirac CP violating phase and ‘ α ’, ‘ β ’ are Majorana CP violatingphases. The effective mass parameter in neutrinoless double beta decay is defined as | M ee | = (cid:12)(cid:12) m c c + m s c e iα + m s e iβ (cid:12)(cid:12) . (19)In Eq. 11, the effective mass parameter is non-zero i.e. | M ee | = | ( M ν ) | 6 = 0.Also, due to texture one-zero structure of effective neutrino mass matrix we have( M ν ) = 0, which in explicit form can be written as c ( e i (2 β + δ ) m s s − c m ( c s + c e iδ s s )+ e iα m s ( c c − e iδ s s s )) = 0 . (20)10he above complex constraining equation gives two real constraints which can besolved for ratios m /m ≡ R and m /m ≡ R as R = c ( c s s sin 2 β + c s sin(2 β + δ )) s ( s s s sin 2( α − β ) − c c sin(2 α − β − δ )) , (21)and R = c ( c s ( − c + s s ) sin 2 α + c s s ( − c sin(2 α − δ ) + s sin(2 α + δ ))) s s ( s s s sin 2( α − β ) − c c sin(2 α − β − δ )) . (22)Using these mass ratios, we have two different values of lowest eigenvalue m , equat-ing them gives us R ν ≡ ∆ m | ∆ m | = R − | R + R − | . (23)The parameter space is constrained using the 3 σ range of ‘ R ν ’. Neutrino oscillationparameters ( θ , θ , θ , ∆ m , ∆ m ) are varied in Gaussian distribution while CP phases ( δ , α , β ) are varied randomly in uniform distribution.The neutrino masses can be obtained using mass squared differences for normalhierarchy(NH) and inverted hierarchy(IH) as m = q m + ∆ m , m = q m + ∆ m for NH , and m = q m − ∆ m , m = q m − ∆ m + ∆ m for IH , whereas. lightest neutrino mass ( m (NH), m (IH)) is obtained from mass ratios.11arameter Best fit ± σ range 3 σ rangeNormal neutrino mass ordering ( m < m < m )sin θ . +0 . − . . − . θ . +0 . − . . − . θ . +0 . − . . − . m (cid:2) − eV (cid:3) . +0 . − . . − . m (cid:2) − eV (cid:3) +2 . +0 . − . +2 . − +2 . m < m < m )sin θ . +0 . − . . − . θ . +0 . − . . − . θ . +0 . − . . − . m (cid:2) − eV (cid:3) . +0 . − . . − . m (cid:2) − eV (cid:3) − . +0 . − . − . − − . −4 −3 −2 −1 | M ee | ( e V ) Model Parameter (eV) −4 −3 −2 −1 | M ee | ( e V ) Model Parameter (eV) Figure 4: Variation of effective Majorana neutrino mass | M ee | with Contribution ofinverse seesaw and type-II seesaw model parameters X and X ′ respectively, for bothnormal and inverted hierarchy.In Eq.11, effective Majorana neutrino mass matrix contains addition of distinctseesaw terms contributing to | M ee | , one corresponding to inverse seesaw (X) andother to type-II seesaw (X ′ ). In addition, the type-II seesaw term can be calcu-lated independently from ( M ν ) so that inverse seesaw contribution is calculated as | ( M ν ) − ( M ν ) | (see Eqn. 11 ). In Fig.(4), we have plotted effective mass parame-ter | M ee | with inverse seesaw contribution (X) and type-II seesaw contribution (X ′ ).The sensitivity reach of 0 νββ decay experiments like SuperNEMO [76], KamLAND-12en [77], NEXT [78, 79], nEXO [80] is, also, shown in Fig.(4)The structure of Majorana mass matrix obtained in Eq.11 has ( M ν ) element as asum of inverse seesaw contribution and type-II seesaw contribution. Once the modelis compared with general one zero texture, it ameliorate the different seesaw contribu-tion as shown in Fig.(4). In Fig.(4(a)), it is clear that for NH, higher density of pointsfor type-II seesaw indicate contribution of O (0 . eV ), whereas inverse seesaw contri-bution is of O (0 . eV ). On the other hand, for IH, different seesaw contributionsare of same order as can be seen in Fig.(4(b)). Hence, type-II seesaw contribution to | M ee | is large as compared to inverse seesaw contribution for Normal Hierarchy(NH)for the texture one zero model within the framework of inverse seesaw and type-IIseesaw.For normal hierarchical neutrino mass spectrum, the | M ee | goes below upto the O (10 − eV ) and we did not obtain a clear lower bound as shown in Fig.(4(a)). Oncontrary, for inverted hierarchy, there is a clear cut lower bound for the | M ee | whichcan be probed in future 0 νββ decay experiments. It can be seen from Fig.(4(b)),that the 0 νββ decay experiments like SuperNEMO, KamLAND-Zen can probe theinverted hierarchical spectrum. The particle nature of DM is still a puzzle to be solved by the researchers worldwide.We have used non-abelian discrete flavour symmetry framework to explain the sta-bility of DM. In this work we used A discrete flavor based model because we knowthat A symmetry spontaneously breaks to Z parity, which stabilizes the scalar DM.Our model is an extension of SM by discrete flavor symmetry A ⊗ Z ′ ⊗ Z . Wehave used Z ′ , Z symmetry to avoid extra terms in our Lagrangian. ISS mechanismis used to generate light neutrino mass matrix. Type-II seesaw is also introduced toget the desired neutrino mass matrix. By choosing the appropriate values of Yukawacoupling, right handed neutrino mass and DM mass, we produced the observed relicabundance of DM. To obtain the required relic abundance, we should have relic massaround 50 GeV, the right handed neutrino mass around 146 GeV and the Yukawacoupling range 0.998-1. The comparison of obtained texture one-zero model withknown phenomenology of texture one-zero results that type-II seesaw contribution to | M ee | is more as compared to inverse seesaw contribution. We have obtained lowerbound of | M ee | for inverted hierarchy, which can be probed in future 0 νββ decayexperiments like like SuperNEMO, KamLAND-Zen. Acknowledgments R. Verma acknowledges the financial support provided by the Central University ofHimachal Pradesh. B. C. Chauhan is thankful to the Inter University Centre for As-tronomy and Astrophysics (IUCAA) for providing necessary facilities during the com-13letion of this work. M. K. acknowledges the financial support provided by Depart-ment of Science and Technology, Government of India vide Grant No. DST/INSPIREFellowship/2018/IF180327. The authors, also, acknowledge Department of Physicsand Astronomical Science for providing necessary facility to carry out this work. References [1] R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23, 165 (1981).[2] G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181, 287 (1981).[3] C. Wetterich, Nucl. Phys. B 187, 343 (1981).[4] J. Schechter and J.W.F. Valle, Phys. Rev. D 25, 774 (1982).[5] B. Brahmachari and R. N. Mohapatra, Phys. Rev. D 58, 015001 (1998).[6] R. Foot, H. Lew, X. G. He and G. C. Joshi, Z. Phys. C 44, 441 (1989).[7] F. Zwicky, Helv. Phys. Acta, 6, 110 (1933) , Gen. Relativ. Gravit., 41, 207 (2009).[8] V. C. Rubin and W. K. Ford Jr., Astrophys. J. 159, 379 (1970).[9] D. Clowe, M. Bradac, A.H. Gonzalez, M. Markevitch, S.W. Randall, C. Jonesand D. Zaritsky, Astrophys. J. 648, L109 (2006).[10] N. Aghanim et al. [Planck Collaboration], Astron. Astrophys. A6, 641 (2020).[11] P. H. Frampton, S. L. Glashow and D. Marfatia, Phys. Lett. B 536, 79 (2002).[12] Z. -Z. Xing, Phys. Lett. B 530, 159 (2002).[13] W. Guo and Z. -Z. Xing, Phys. Rev. D 67, 053002 (2003).[14] S. Dev and S. Kumar, Mod. Phys. Lett. A 22, 1401 (2007).[15] S. Dev, S. Kumar, S. Verma and S. Gupta, Phys. Rev. D 76, 013002 (2007).[16] S. Dev, S. Verma, S. Gupta and R. R. Gautam, Phys. Rev. D 81, 053010 (2010).[17] S. Verma, Adv. High Energy Phys., 2015, 385968 (2015).[18] S. Dev, S. Kumar and S. Verma, Mod. Phys. Lett. A 24, 2251 (2009).[19] S. Verma, M. Kashav and S. Bhardwaj, Nucl. Phys. B 946, 114704 (2019).[20] S. Verma and M. Kashav, Mod. Phys. Lett. A 35, 20, 2050165 (2020).[21] S. Kaneko, M. Katsumata and M. Tanimoto, JHEP 0307, 025 (2003).1422] T. Hugle, M. Platscher and K. Schmitz, Phys. Rev. D 98, 2, 023020 (2018).[23] L. M. G. de la Vega, R. Ferro-Hernandez and E. Peinado, Phys. Rev. D 99,055044 (2019).[24] T. Kitabayashi, Phys. Rev. D 98, 083011 (2018).[25] R. N. Mohapatra, Phys. Rev. Lett, 56, 561 (1986).[26] M. C. Gonzalez-Garcia and J. W. F. Valle, Phys. Lett. B, 216, 360 (1989).[27] F. Deppisch and J. W. Walle, Phys. Rev. D 72, 036001 (2005).[28] P. B. Dev and R. Mohapatra, Phys. Rev. D 81(1), 013001 (2010).[29] A. G. Dias, C. A. de S. Pires and P . S. Rodrigues da Silva, Phys. Rev. D 84,053011 (2011).[30] F. Bazzocchi, Phys. Rev. D 83, 093009 (2011).[31] E. Ma, Phys. Rev. D 80, 013013 (2009).[32] G. Altarelli and F. Feruglio, Nucl. Phys. B 741, 215 (2006).[33] G. Altarelli, F. Feruglio, Rev. Mod. Phys. 82, 2701 (2010).[34] E. Ma, Phys. Rev. D 73, 057304 (2004).[35] B. Brahmachari, Sandhya Choubey and Manimala Mitra, Phys. Rev. D 77,119901 (2008).[36] I. de M. Varzielas and O. Fischer, JHEP 01, 160 (2016).[37] H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto,Prog. Theor. Phys. Suppl., 183, 1 (2010).[38] M. Hirsch, S. Morisi, E. Peinado and J. W. F. Valle, Phys. Rev. D 82, 116003(2010).[39] P. H. Frampton, S. L. Glashow, and D. Marfatia, Phys. Lett. B 536, 79 (2002).[40] M. Singh, G. Ahuja, and M. Gupta, PTEP, 2016, 8, 123 (2016).[41] Z. Z. Xing, Phys. Lett. B 530, 159–166 (2002).[42] R. Bipin, R. Desai, D. P. Roy, R. Alexander, and R. Vaucher, Mod. Phys. Lett.A 18, 1355 (2003).[43] A. Merle and W. Rodejohann, Phys. Rev. D 73(7), 073012 (2006).[44] S. Dev, S. Kumar, S. Verma, and S. Gupta, Nucl. Phys. B 784, 103 (2007).1545] M. Randhawa, G. Ahuja, and M. Gupta, Phys. Lett. B 643, 175 (2006).[46] G. Ahuja, S. Kumar, M. Randhawa, M. Gupta and S. Dev, Phys. Rev. D 76,013006 (2007).[47] S. Kumar, Phys. Rev. D 84, 7, 077301 (2011).[48] P. O. Ludl, S. Morisi, and E. Peinado, Nucl. Phys. B 857, 411 (2012).[49] H. Itoyama and N. Maru, Int. J. of Modern Physics A 27, 1250159 (2012).[50] D. Meloni and G. Blankenburg, Nucl. Phys., B 867, 749 (2013).[51] W. Grimus and J. Ludl, Journal of Physics G: Nuclear and Particle Physics, 40,125003 (2013).[52] P. O. Ludl and W. Grimus, JHEP, 2014 (2014).[53] H. Fritzsch, Z. Z. Xing, and S. Zhou, Journal of High Energy Physics, 2011, 083(2011).[54] E. I. Lashin and N. Chamoun, Phys. Rev. D 85, 11, 113011 (2012).[55] K. N. Deepthi, S. Gollu, and R. Mohanta, The European Physical Journal C72,2, 1 (2012).[56] J. Liao, D. Marfatia, and K. Whisnant, Phys. Rev. D 87, 7, 073013 (2013).[57] J. Liao, D. Marfatia, and K. Whisnant, Phys. Rev. D 88, 3, 033011 (2013).[58] H. Fritzsch and Z. Z. Xing, Progress in Particle and Nuclear Physics, 45, 1 (2000).[59] K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991).[60] E. W. Kolb and M.S. Turner, Front. Phys., 69, 1 (1990).[61] J. Edsjo and P. Gondolo, Phys. Rev. D 56, 1879 (1997).[62] G. Gelmini and P. Gondolo, Nucl. Phys. B 360, 145 (1991).[63] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rep., 267, 195 (1996).[64] Y. Bai and J. Berger, JHEP 11, 171 (2013).[65] N. F. Bell, Y. Cai and A. D. Medina, Phys. Rev. D 89, 115001 (2014).[66] M. Hoferichter, P. Klos, J. Menendez, and A. Schwenk, Phys. Rev. Lett., 18,181803, 119 (2017).[67] H. K. Dreiner, H. E. Haber and S. P. Martin, Phys. Rep. 494, 1 (2010).1668] M. S. Boucenna, S. Morisi, E. Peinado, J. W. F. Valle and Y. Shimizu, Phys.Rev. D 86, 073008 (2012).[69] M. S. Boucenna, M. Hirsch, S. Morisi, E. Peinado, M. Taoso and J.W.F. Valle,JHEP 05, 037 (2011).[70] Y. Bai and J. Berger, JHEP 08, 153 (2014).[71] E. Aprile et al. [XENON], Phys. Rev. Lett. 121, 11, 111302 (2018).[72] X. Cui et al. [PandaX-II], Phys. Rev. Lett. 119, 18, 181302 (2017).[73] D. S. Akerib et al. [LUX], Phys. Rev. Lett. 118, 2, 021303 (2017).[74] R. Agnese et al. [SuperCDMS], Phys. Rev. Lett. 120, 6, 061802 (2018).[75] I. Esteban, et al. , arXiv: 2007.14792[hep-ex].[76] A. S. Barabash, J. Phys. Conf. Ser., 375, 042012 (2012).[77] A. Gando et al. , Phys. Rev. Lett., 117, 082503 (2016).[78] F. Granena et al. , arXiv:0907.4054[hep-ex].[79] J. J. Gomez-Cadenas et al.et al.