Phenomenology of Friedberg Lee Texture in Left-Right Symmetric Model

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1 Commun. Theor. Phys. Beijing China pp c Chinese Physical Society Vol. 50 No. August Phenomenology of Friedberg Lee Texture in Left-Right Symmetric Model LUO Min-Jie 1 and LIU Qiu-Yu 1 1 Department of Modern Physics University of Science and Technology of China Hefei 300 China The Abdus Salams International Center for Theoretical Physics Trieste Italy Received September 1 007; Revised November Abstract We consider that the Higgs triplet Yukawa coupling takes the Friedberg Lee texture and the Higgs doublet Yukawa coupling simply identifies with the diagonal Yukawa coupling of charged lepton in the context of leftright symmetric model. In this scenario the phenomenology including effective neutrino masses mixings and thermal flavor-dependent leptogenesis and lepton flavor violation decays are studied. We investigate the combined constrain of the parameters in this scenario and test its consistency with present data. PACS numbers: 14.0.Lm 14.0.Pq Ry Key words: neutrino mass neutrino mixing flavor-dependent leptogenesis lepton flavor violation Friedberg Lee texture 1 Introduction Current solar [1] atmospheric [] reactor [3] and accelerator [4] neutrino oscillation experiments give us strong evidences that neutrinos have non-zero light masses and two large θ 1 34 θ 3 45 and one small θ 13 < 10 mixing angles. The simplest way to provide for light neutrino masses is the seesaw mechanism [5] and it can be realized naturally in the left-right symmetric model [] based on the gauge group SU L SU R U1 B L since this model contains both left- and right-handed neutrinos one Higgs bi-doublet Φ and two Higgs triplet LR. [] Under a discrete left-right symmetry l L l R c L R and Φ Φ T the invariant Lagrange of the Yukawa interactions between fermions and Higgs sectors is L Y = hl L Φl R + hl L ΦlR + 1 f[l Liτ L l L c + l R c iτ R l R ] + h.c. 1 where Φ = τ Φ τ l LR c T Cl LR with C being the charge-conjugation matrix and a b label different generations. At first stage the symmetry spontaneously broken into SU L U1 Y by a non-zero vacuum expectation value VEV of R leading to a heavy Majorana mass for right-handed neutrinos. The second stage the Φ develops VEV the VEV of LR and Φ are [] L = R = v L 0 v R 0 κ 0 Φ = 0 κ the mass matrices of charged lepton and Dirac neutrino are M e = hκ + hκ M D = hκ + hκ. 3 We get the following form for the mass matrix of neutrinos in the basis of l L c l R fvl M D. 4 fv R M D The effective mass matrix for three light left-handed Majorana neutrinos via Type-II seesaw can be obtained by diagonalizing the above matrix m ν fv L 1 v R M D f 1 M T D. 5 However the flavor pattern of the Higgs doublet Yukawa coupling h h and Higgs triplet Yukawa coupling f remain undetermined constraining on these couplings from phenomenological studies are an important stage in understanding the mass mixing and CP violations. From the oscillation experiments the information of mixing angles only implies the possible textures of effective neutrino mass and inspires many texture buildings in which the Friedberg Lee FL texture [7] b + c b c F = di + b a + b a c a a + c where I denotes the 3 3 identity matrix is not only theoretical [7] but also phenomenological appealing since i it can be diagonalized by a so-called tri-bimaximal [8] mixing matrix which is consistent with current neutrino oscillation data especially ii the inverse of F which appears in seesaw formula has the structure that is parallel to itself. [9] It is therefore possible that F to be the texture of Higgs triplet Yukawa coupling f in Type-II seesaw scenario under such an assumption we will naturally obtain the effective neutrino mass matrix that is nearly of the texture of F under appropriate M D selection. It is clear that M D has similar texture to M e The project supported by National Natural Science Foundation of China under Grant No mjluo@mail.ustc.edu.cn

2 45 LUO Min-Jie and LIU Qiu-Yu Vol. 50 according to Eq. 3. In this paper we will assume that M D = M e = diagm e m µ m τ which are diagonal and real i.e h = h = diagy e y µ y τ and κ 0 κ v 174 GeV. Such an assumption is for simplicity as a consequence the origin of large mixing of neutrino comes mainly from the large mixing of Higgs triplet Yukawa coupling f in this scenario. The textures in the Yukawa sector of the theory are now given phenomenological outcomes can be studied in order to test the consistency of this scenario with present experiment data. In this paper beside the neutrino masses and mixing leptogenesis is also a key factor to understand the flavor parameters especially those in the right-handed neutrino pattern. The flavor independent leptogenesis in this model has been studied in Ref. [9] in which the bound on the lightest right-handed neutrino is too high about GeV to avoid the gravitino problem the flavor-dependent leptogenesis is important in these models which is of the Type II seesaw kind as has been pointed out in Ref. [10]. In this model we will show that the flavor-dependent effects substantially change the results and therefore cannot be ignored the more correct analysis of leptogenesis for this model is needed so we must treat each flavor separately. In addition the predictions of lepton flavor violation LFV decay of this model are briefly discussed. This paper is organized as follows. The prediction of neutrino masses and mixings are discussed in Sec. In Sec 3 the flavor-dependent leptogenesis is studied. LFV decay processes are discussed in Sec. 4. We draw the conclusions in Sec. 5. Neutrino Mass and Mixing Let us start with the studies of predictions on effective neutrino mass matrix in Eq. 5 by the assumptions: i f = F ii M D = diagm e m µ m τ = v diagy e y µ y τ. since v L v R v and y e y µ y τ in which m ν v L F 1 b + c + d b c M D F 1 MD T v L b a + b + d a A 0 0 r 7 v R c a a + c + d 0 r 1 A = yτ dc + d + ab + c + d + bc + d d d Oy + bd + cd + 3bc + a3b + 3c + d τ bc + ab + c + d y µ yµ r = O. 8 dc + d + ab + c + d + bc + d y τ y τ The symmetric matrix F the first term of Eq. 7 can be diagonalized by a matrix that the form of the leading order is the tri-bimaximal matrix U tb = in which sin θ 13 is exactly zero. The second term of the order Oyτ 10 4 slightly departs from Friedberg Lee FL texture and makes the mixing matrix different from tri-bimaximal. Let us begin with considering 3 b + c + d 1 A1 + r A1 r 3 A 3b c 3 U tb m νutb A1 r = v L 3 d 1 3 A1 + r A. 10 A a + 1 b + c + d + 1 A1 r A 3b c 3 We can see clearly from the 13 or 31 element that in the case of b c b-c makes the dominant contribution to the off-diagonal elements with respect to A Oyτ elements with A hence can be safely neglected the effective neutrino mass matrix is just the FL texture the prediction of sin θ 13 from FL texture has been studied in Ref. [11] we will discuss a generic diagonalization of it. Equation 10 now becomes 3 U tb m νutb b + c + d 0 3b c 3 v L 0 d 0 3b c 3 A mixing matrix is parameterized as cos θ 0 sin θ e iδ R = sin θ e iδ 0 cos θ 0 a + 1 b + c + d. 11 e iρ e iσ

3 No. Phenomenology of Friedberg Lee Texture in Left-Right Symmetric Model 453 in which we assume θ is real δ represents the possible Dirac phase and ρ σ for Majorana phases. This parametrization makes no complex phase relate to d. Then we get D 1 0 R U tb m νutbr = 0 D D 3 where = v L 4 e iρ+σ 3c b cosθ 4a + b + c + d e iδ 3b + 3c + d e iδ sin θ D 1 = 1 v L e iρ 3b + 3c + d cos θ + 4a + b + c + d e iδ sin θ + 3b c e iδ sin θ D = v L d D 3 = 1 v L e iσ 4a + b + c + d cos θ + 3b + 3c + d e iδ sin θ 3b c e iδ sin θ. 14 D 13 are all real and positive and = 0 so the parameters a b and c must be complex to absorb the complex phases which lead to the solutions of θ δ ρ and σ where tan θ = α + β γ + Reα β γ Im[a b c + b c] β α tanδ = Re[a + db c] + b c tan ρ = Im[α γ tan e iδ ] Re[α γ tan e iδ ] tan σ = Im[β + γ tan eiδ ] Re[β + γ tan e iδ ] 15 α = 3 b + c + d β = a b + c + d γ = c b. m ν is then diagonalized the diagonal elements as masses m 1 = 1 v L 3b + 3c + d 3b c tanθ e iδ m = v L d m 3 = 1 v L 4a + b + c + d + 3b c tanθ e iδ. 1 From the mixing matrix V = U tb R three mixing angles can be written as sin θ 1 = 1 + cosθ sin θ 3 = + cosθ 3 sin θ cos δ + cosθ sin θ 13 = sin θ Here the prediction of sinθ 13 is possible to be a testable deviation from zero even up to the order of O0. within the allowed range of θ 3 that deviated from maximal. The numerical outcomes of the effective neutrino masses depending on the values of free parameters a b c d and v L in FL texture can be generated by scanning their possible ranges i a b c are complex we randomly generate their real and imaginary part from 1 to 1 ii d is real and positive from 0 to 1 iii v L from 10 ev to 10 ev keeping b c and other known experimentally constraints 30 θ θ ev < m 1 = m m 1 < ev and ev < m 3 = m 3 m < ev in normal hierarchy or ev < m 3 = m m 3 < ev in inverted hierarchy. The output points of masses are shown in Fig. 1. They are consistent with current bound of neutrino mass. Setting the physical range of δ within [0 π one shows the allowed parameter space of the sin θ 13 θ 3 in Fig. which demonstrates the dependence of the lower bound of sinθ 13 with the deviation of θ 3 from maximal. In addition the prediction of the allowed ranges of the Jarlskog parameter J CP over θ 1 are shown on the right-hand side of Fig.. In the case that b c or the terms involving A Oy τ make the dominant contribution to the off-diagonal elements of m ν thus the texture becomes nearly the form that is invariant under the µ-τ symmetry and it is broken slightly by the non-vanishing off diagonal elements in the 3 block of the second matrix in Eq. 7. Therefore the deviation of sinθ 13 from zero is tiny. We now make a numerical diagonalization three extra Euler angles rather than one in the above case are needed. R θ 1 θ θ 3 U tb m νu tb R θ 1 θ θ 3 = diagm 1 m m 3. θ 1 θ θ 3 as free parameters are generated mildly depart from zero and keep b = c. It is shown that sinθ close to the order of Ar other

4 454 LUO Min-Jie and LIU Qiu-Yu Vol. 50 mixing angles are almost the values given in tri-bimaximal matrix. The prediction of masses are almost the same as the case of b c hence they are not sensitive to the difference between b and c. Fig. 1 Illustrative plot for allowed parameter regions for the effective neutrino mass in normal hierarchy left and inverted hierarchy right. Fig. The allowed ranges of sin θ 13 θ 3 left. The allowed parameter space of J CP θ 1 right in both normal and inverted hierarchy cases. As we have discussed above this scenario gives a possible texture which is consistent with current oscillation experimental data and would be consistent with future experiments for θ 13 since it allows a large room for θ 13 from nearly zero to the upper bound of present constraints. However the dependence of three angles in Eq. 17 are predicted and it would be a possible way to test this scenario in future oscillation experiments. Furthermore because of the smallness of A Oy τ 10 4 the second term that with Yukawa couplings M D in the Type-II seesaw formula plays a subdominant role to the effective neutrino masses and mixing angles phenomenological consequences that depend on the specific texture of Yukawa couplings need to be investigated further. 3 Flavor Dependent Leptogenesis We now consider how leptogenesis works out in which the flavor effect is needed in this scenario. If the righthanded neutrino in the seesaw mechanism exists it will decay to lepton and Higgs boson. The CP asymmetry ɛ = [ΓN l + H c ΓN i l c + H]/[ΓN l + H c + ΓN l c + H] which depends on the structure of Yukawa couplings ˆM D gives us a possible explanation to the baryon number asymmetry of universe BAU via leptogenesis mechanism. [1] For a successful leptogenesis this model should reproduce the observed BAU it is therefore a clue of testing the Yukawa couplings. But the constraints on the Yukawa couplings in seesaw formula is not straightforward yet since there are more parameters than experimental constraints and furthermore the analysis relies on specific models and many additional assumptions. We will assume that i the masses of Higgs triplets are much larger than

5 No. Phenomenology of Friedberg Lee Texture in Left-Right Symmetric Model 455 the normal hierarchical masses of right handed neutrino the CP violation asymmetry is therefore the lightest righthanded neutrino decay dominate. ii The primordial right-handed neutrino is produced thermally by the reheating of universe namely the thermal leptogenesis. iii The matrix F is complex as has been discussed in the previous section the CP phases come from the mixing matrix V = cos θ 0 sin θ e iδ sin θ e iδ 0 cos θ where V FV = diag. iv In this paper we take into account the flavor-dependent effect [13] of leptogenesis. e iρ e iσ Then the CP asymmetry for each lepton flavor denoted as ɛ α = ɛ I α + ɛ II α under the basis that M R is diagonal can be written as [10] ɛ I α = 1 8πv j 1 Im[ ˆM D 1α ˆM ˆM D D 1j ˆM D αj ] [ xj xj + 1 ] ˆM ˆM xj 1 + x j ln D D 11 1 x j x j ɛ II α = 3 M R1 Im[ ˆM D 1α ˆM D g1m L αg ] 1π v g ˆM ˆM 19 D D 11 in which x j = MRj /M R1 ˆMD = M D V is the Dirac neutrino mass defined in the basis where M R is diagonal 3 e iρ m cos θm e e 3 3 eiδ σ sin θm e ˆM D = e iρ cos θ e iρ e iδ sin θ e iδ sin θ m µ m µ 3 e iσ cos θ e iδ sin θ m µ cos θ m m τ τ 3 e iσ cos θ e iδ sin θ m τ 0 ɛ I α is conventional CP asymmetry factor [14] and ɛ II α represents the new contribution from one-loop diagrams where Higges triplets are exchanged in the loop. [15] The baryon asymmetry gives rise to the leptonic asymmetry through the nonperturbative Sphaleron conversion [1] Y B = n B s 1 Y α 1 37 where Y B = 8.7± is baryon asymmetry from observation [17] Y α is the B/3 L α asymmetry in the lepton flavor α which is the solution of flavor-dependent Boltzmann equations. [14] For simplicity we neglect the off-diagonal contribution to the efficient factor i.e. η αα denotes η α which is a good approximation in most cases. It is the efficiency factor that represents the washout effects in the out of equilibrium decay of the lightest right-handed neutrino. Defining that m α = [ ˆM D 1α ˆM D α1 ]/M R1 denotes the effective seesaw neutrino mass m ev the equilibrium neutrino mass κ α = m α /m as wash out parameters exhibiting the out of equilibrium condition on the decay of the right-handed neutrino compared with the Hubble expansion rate at the temperature M R1. In the scenario the m α are m e = cos θm e 3M R1 1 m µ = e iδ 3e iδ cos θ + 3 sin θ 3 cos θ + 3 e iδ sin θm µ 18M R1 1 m τ = e iδ 3e iδ cos θ 3 sin θ 3 cos θ 3 e iδ sin θm τ 18M R1 α Considering that the range of M R1 will be scanned from 10 to 10 1 GeV in the following numerical calculating taking the typical value such as M R GeV we can estimate that κ e 1 and it resides in weakly washed out region κ µ Om µ/m R1 m O1 and κ τ Om τ/m R1 m O10 so they will be possibly in mildly and strongly washed out region respectively. Because we have assumed that the mass of Higgs triplet is much larger than the ones of right-handed neutrino so the efficiencies for flavor-dependent thermal leptogenesis in the case of Type I and Type II are mainly determined by the properties of the right-handed neutrino in this limit the efficiency factors for Type II framework can be computed similarly to the Type I seesaw scenario. The numerical fitting of the Y B from the solution of flavor-dependent Boltzmann equations with vanishing initial N 1 abundance is given as follows: [14] Y B 1 37g [ ɛ e η m e + ɛ µ η 537 m µ 344 ] + ɛ τ η 537 m τ 3

6 45 LUO Min-Jie and LIU Qiu-Yu Vol. 50 for the region of M R GeV in which µ and τ Yukawa couplings are in equilibrium. Y B 1 [ 417 ɛ η 37g 589 m + ɛ τ η 390 ] 589 m τ 4 where ɛ = ɛ e + ɛ µ m = m e + m µ is for 10 9 M R GeV in which only τ is in equilibrium and should be treated separately in the Boltzmann equations while e and µ are indistinguishable. The effective number of the degrees of freedom g = in the Standard Model. When κ τ 1 and µ flavor suffers κ µ 1 based on the estimation of κ the efficiency factor is fitted by the formula [14] η m α [ m α ev ev ] m α In the case of a thermal initial abundance one has the approximation [18] η α 1 e κ αz B κ α / κ α z B κ α z B κ α + 4κ 0.13 α e.5/κ α Y α = ɛ α η α Y eq N 1 T M N ɛ α η α Y B α ɛ α η α where the Y eq N 1 is the thermal population of the lightest right-handed neutrino N 1. Fig. 3 Prediction for Y B as a function of M R1 in the case of vanishing initial N 1 abundance points and the case of thermal initial abundance circles. The dashed line represents the observed value of Y B. Now we perform the numerical prediction of Y B in such two cases and only concern the region 10 M R GeV and see whether the flavor leptogenesis can produce successful BAU in this region. When the region is above 10 1 GeV all flavors are indistinguishable the one flavor approximation becomes valid. The only free parameters are a b c d v L and v R we generate the former five ones in their ranges like the previous section the v R is scanning from 10 GeV to 10 1 GeV. Three CP phases δ ρ and σ are determined by Eq. 15 three unknown right-handed neutrino masses can be deduced from diagm R1 M R M R3 = v R V FV and possess a hierarchical mass spectrum. The numerical result of the baryon asymmetry Y B over M R1 is demonstrated in Fig. 3 in the case of b c and we find that the Y B is insensitive to the small difference between b and c. This scenario can generate successful BAU within the scale of M R1 from 10 9 GeV to 10 1 GeV in both vanishing and thermal initial N 1 abundance. Unfortunately there is no constraints on the CP phases δ ρ σ and particularly if the flavor is taken into account no longer a bound on neutrino mass scale from the requirement of successful BAU. We have got that the mass matrix of the right-handed neutrino takes FL texture the prediction of the other two heavy masses are approximately fixed from the constraint of the lightest one the numerical prediction of the masses of right-handed neutrino are plotted in Fig. 4. The main difference between the flavor-independent approximation and the correct flavor-dependent treatment is the fact that in the latter case the baryon asymmetry is a function of the sum over each individual asymmetry weighted by the corresponding efficiency factor: α ɛ αη α but rather the total asymmetry times the efficiency factor that obtains from the Boltzman equation neglecting flavor effects: α ɛ α η α κ α. It will lead to an explicitly change in this model. First of all with the help of the estimation of κ α κ τ O10 and locates in the strongly washed out region ηκ τ O10 4 while κ µ O1 and is mildly washed out in both vanishing and thermal initial abundance ηκ µ O10 we have ηκ µ /ηκ τ O10. We find the ɛ II α in this model is also important for the CP asymmetry the estimation of total contribution of ɛ α = ɛ I α+ɛ II ɛ α < M R GeV we obtain α can be deduced from [13] m α α m 7 α m ev ɛ e M R GeV ɛ µ M R GeV ɛ τ M R GeV m ev m ev m ev. 8 ɛ µ /ɛ τ O10 so Oɛ µ η µ Oɛ τ η τ. But because m τ m µ m e η α κ α ηκ τ. Therefore the

7 No. Phenomenology of Friedberg Lee Texture in Left-Right Symmetric Model 457 flavor-component µ will give another important contribution to the Y B from ɛ µ η µ in flavor-dependent scenario which is absent in the flavor independent analysis. The extra factor making the bound of the lightest right-handed neutrino depends on the wash out parameters in flavors. Because the ɛ e is too small compared with ɛ µτ although ηκ e is very different in the region of κ e 1 when it is in the vanishing and thermal initial abundance the contribution of ɛ e η e is small we obtain a result that does not very sensitive to the initial abundance shown in Fig. 3. Finally the flavor-dependent analysis enhances the magnitude for Y B several times. Compared with the result in Ref. [9] a more relaxed lower bound on M R1 than M R GeV in the flavorindependent analysis is obtained the most favored scale is about GeV which is low enough to avoid the gravitino problem and improves the consistency of this model and the bound on the BAU increases with increasing scale of M R1 as plotted in Fig. 3 in contrast to the flavor independent analysis where it decreases. Of course the analysis is special in many aspects. It is also possible that the mass spectrum of right-handed neutrinos is nearly degenerate in this scenario. In this special case the approximation of the dominance of the lightest right-handed neutrino decay in leptogenesis should be reconsidered in the framework of flavor-dependent resonance leptogenesis [19] in which the self-energy diagram dominates. The CP asymmetry is resonantly enhanced when M Ri M Rj M Ri and will lead to a more lower bound for M R. 4 Lepton Flavor Violation Decay The existence of neutrino oscillation implies the individual lepton charges e µ τ are not conserved namely LFV processes of the charge leptons such as µ e + γ. This kind of observation is highly suppressed in SM by the small ratios of neutrino mass to the W boson mass. But one of the striking phenomenological implications of seesaw model in SUSY is the prediction of sizable LFV. For simplicity we will work in the framework of minimal supergravity msugra extension of the SM. The low energy LFV processes l j l i +γ come from the REG effects of the slepton mixing. In the basis where M e and M R are diagonal the branching ratio of l j l i +γ approximately reads [0] BRl j l i + γ α3 tan β G F m8 s 3m 0 + A Cij 0 8π v sin 9 β where G F is the Fermi constant α is the fine structure constant m 0 is the universal scalar soft mass A 0 the trilinear term at Λ GUT m s is supersymmetric leptonic masses and approximately m 8 s 0.5m 0M1/ m M 1/ [1] and C ij = k ˆM D ik ˆ M D kj ln Λ GUT M Rk. 30 Fig. 4 Prediction for normal hierarchical right-handed neutrino masses. The circles and points show the allowed ranges of M R3 and M R respectively. a For vanishing initial N 1 abundance; b For thermal initial abundance. Note that the branching ratios of LFV are very sensitive to the off-diagonal elements of C ij and hence the structure of Yukawa coupling ˆM D = M D V at the scale of M R. To get an estimation for the branching ratio we adopt typical Snowmass Points and Slops SPS in Ref. [] for the parameters m 0 A 0 tanβ adding Λ GUT 10 1 GeV BRl j l i + γ C ij. 31 Taking the typical values of M Ri from Fig. 4 we estimate C µe < 0. m e m µ C τe < 0. m e m τ and C τµ <.3 m µ m τ.7 10 and obtain BRµ e + γ < BRτ e + γ < BRτ µ + γ < which are far below the current experimental bounds on the LFV decay [3] and highly suppressed BR exp µ e + γ < BR exp τ e + γ <

8 458 LUO Min-Jie and LIU Qiu-Yu Vol. 50 BR exp τ µ + γ < We note that the simple assumption for the diagonal structure of M D is of course too safe to violate the flavor charge so that a more complicated non-diagonal M D which can also suppress the branching ratio of LFV decay and produce the BAU successfully might be allowed. 5 Summary and Conclusion In this paper we have studied the phenomenology including masses mixings of neutrino the thermal flavordependent leptogenesis and lepton flavor violation under the scenario of the left-right symmetric model in which the Higgs triplet and the Higgs doublet Yukawa couplings take the form of Friedberg Lee texture and the form identified with the diagonal charged lepton M e respectively. The choice of the Higgs doublet Yukawa coupling of neutrino is for simplicity the dominant contribution to the large mixing of neutrino is therefore due to the Higgs triplet Yukawa couplings with FL texture. The phenomenological predictions and parameter constraints have been investigated. By now this scenario is consistent with current available data and still have potential to fit future experiments. It is a possible scenario to realize the nearly tri-bimaximal mixing pattern for neutrino in the left-right symmetric model. The flavor-dependent leptogenesis can generate successful baryon number asymmetry of the universe. The flavor-dependent analysis for this model is important which magnifies the magnitude for BAU several times. The most favored scale is M R GeV which is lower than the flavor-independent case. The branching ratio of LFV decay is highly suppressed with respect to the current bounds. References [1] SNO Collaboration Q.R. Ahmad et al. Phys. Rev. Lett [] For a review see: C.K. Jung et al. Ann. Rev. Nucl. Part. Sci [3] KamLAND Collaboration K. Eguchi et al. Phys. Rev. Lett [4] KK Collaboration M.H. Ahn et al. Phys. Rev. Lett [5] P. Minkowski Phys. Lett. B ; T. Yanagida in Proceedings of the Workshop on Unified Theory and the Baryon Number of the Universe eds. O. Sawada and A. Sugamoto KEK Tsukuba 1979 p. 95; M. Gell-Mann P. Ramond and R. Slansky in Supergravity eds. F. van Nieuwenhuizen and D. Freedman North Holland Amsterdam 1979 p. 315; S.L. Glashow in Quarks and Leptons ed. M. Lévy et al. Plenum New York 1980 p. 707; R.N. Mohapatra and G. Senjanovic Phys. Rev. Lett [] J.C. Pati and A. Salam Phys. Rev. D ; R.N. Mohapatra and J.C. Pati Phys. Rev. D ; Phys. Rev. D ; G. Senjanovic and R.N. Mohapatra Phys. Rev. D [7] T.D. Lee arxiv:hep-ph/005017; Chinese Phys ; R. Friedberg and T.D. Lee High Energy Phys. Nucl. Phys arxiv:hep-ph/ [8] P.F. Harrison D.H. Perkins and W.G. Scott Phys. Lett. B ; Z.Z. Xing Phys. Lett. B ; P.F. Harrison and W.G. Scott Phys. Lett. B ; X.G. He and A. Zee Phys. Lett. B [9] Wei Chao Shu Luo and Zhi-Zhong Xing arxiv:hepph/ [10] S. Autusch arxiv:hep-ph/ [11] Z.Z. Xing H. Zhang and S. Zhou Phys. Lett. B ; Shu Luo and Zhi-Zhong Xing Phys. Lett. B [1] M. Fukugita and T. Yanagida Phys. Lett. B [13] A. Abada S. Davidson F.X. Josse-Michaux M. Losada and A. Riotto J. Cosm. Astropart. Phys arxiv:hep-ph/ [14] A. Abada S. Davidson A.Ibarra F.X.Josse-Michaus M.Losada and A. Riotto arxiv:hep-ph/ [15] T. Hambye and G. Senjanovic Phys. Lett. B ; S. Antusch and S.F. King Phys. Lett. B ; P.H. Gu and X.J. Bi Phys. Rev. D [1] E. Nardi Y. Nir E. Roulet and J. Racker JHEP ; arxiv:hep-ph/001084; G. Engelhard Y. Grossman E. Nardi and Y. Nir arxiv:hep-ph/01187; E. Nardi arxiv:hep-ph/070033; Y. Nir arxiv:hepph/ [17] D.N. Spergel et al. [WMAP Collaboration] Astrophys. J. Suppl [18] Steve Blanchet and Pasquale Di Bari arxiv:hepph/007330; F.X. Josse-Michaux and A. Abada arxiv:hep-ph/ [19] S. Blanchet and P. Di Bari arxiv:hep-ph/003107; hep-ph/007330; Z.Z. Xing and S. Zhou arxiv:hepph/ [0] J.A. Casas and A.Ibarra Nucl. Phys. B ; J. Hisano T. Moroi K. Tobe M. Yamaguchi and T. Yanagida Phys. Lett. B ; J. Hisano T. Moroi K. Tobe and M. Yamaguchi Phys. Rev. D [1] S.T. Petcov S. Profumo Y. Takanishi and C.E. Yaguna Nucl. Phys. B [] B.C. Allanach et al. Eur. Phys. J. C [3] MEGA Collaboration M.L. Brooks et al. Phys. Rev. Lett ; BABAR Collaboration B. Aubert et al. Phys. Rev. Lett ; BABAR Collaboration B. Aubert et al. Phys. Rev. Lett