Realistic Neutrinogenesis with Radiative Vertex Correction
aa r X i v : . [ h e p - ph ] N ov Realistic Neutrinogenesis with Radiative Vertex Correction
Pei-Hong Gu , ∗ Hong-Jian He , † and Utpal Sarkar ‡ The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy Center for High Energy Physics, Tsinghua University, Beijing 100084, China Physical Research Laboratory, Ahmedabad 380009, India
We propose a new model for naturally realizing light Dirac neutrinos and explaining the baryonasymmetry of the universe through neutrinogenesis. To achieve these, we present a minimalconstruction which extends the standard model with a real singlet scalar, a heavy singlet Diracfermion and a heavy doublet scalar besides three right-handed neutrinos, respecting lepton numberconservation and a Z symmetry. The neutrinos acquire small Dirac masses due to the suppressionof weak scale over a heavy mass scale. As a key feature of our construction, once the heavy Diracfermion and doublet scalar go out of equilibrium, their decays induce the CP asymmetry from theinterference of tree-level processes with the radiative vertex corrections (rather than the self-energycorrections). Although there is no lepton number violation, an equal and opposite amount of CPasymmetry is generated in the left-handed and the right-handed neutrinos. The left-handed leptonasymmetry would then be converted to the baryon asymmetry in the presence of the sphalerons,while the right-handed lepton asymmetry remains unaffected. PACS numbers: 98.80.-k, 11.30.Er, 14.60.Pq arXiv: 0709.1019 [hep-ph]
Strong evidences from neutrino oscillation experi-ments [1] so far have pointed to tiny but nonzero massesfor active neutrinos. The smallness of the neutrinomasses can be elegantly understood via seesaw mech-anism [2] in various extensions of the standard model(SM). The origin of the observed baryon asymmetry [1]in the universe poses a real challenge to the SM, butwithin the seesaw scenario, it can be naturally explainedthrough leptogenesis [3, 4, 5, 6, 7, 8].In the conventional leptogenesis scenario, the leptonnumber violation is essential as it is always associatedwith the mass-generation of Majorana neutrinos. How-ever, the Majorana or Dirac nature of the neutrinos isunknown a priori and is awaiting for the upcoming ex-perimental determination. It is important to note [9, 10]that even with lepton number conservation, it is possibleto generate the observed baryon asymmetry in the uni-verse. Since the sphaleron processes [11] have no directeffect on the right-handed fields, a nonzero lepton asym-metry stored in the left-handed fields, which is equal butopposite to that stored in the right-handed fields, canbe partially converted to the baryon asymmetry as longas the interactions between the left-handed lepton num-ber and the right-handed lepton number are too weakto realize an equilibrium before the electroweak phasetransition, the sphalerons convert the lepton asymmetryin the left-handed fields, leaving the asymmetry in theright-handed fields unaffected [10, 12, 13, 14, 15].For all the SM species, the Yukawa interactions aresufficiently strong to rapidly cancel the stored left- and ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] right-handed lepton asymmetry. However, the effectiveYukawa interactions of the ultralight Dirac neutrinos areexceedingly weak [16, 17] and thus will not reach equi-librium until the temperatures fall well below the weakscale. In some realistic models [12, 14, 15], the effec-tive Yukawa couplings of the Dirac neutrinos are natu-rally suppressed by the ratio of the weak scale over theheavy mass scale. Simultaneously, the heavy particlescan decay with the CP asymmetry to generate the ex-pected left-handed lepton asymmetry after they are outof equilibrium. This new type of leptogenesis mechanismis called neutrinogenesis [10].In this paper, we propose a new model to generatethe small Dirac neutrino masses and explain the originof cosmological baryon asymmetry, by extending the SMwith a real scalar, a heavy Dirac fermion singlet and aheavy doublet scalar besides three right-handed neutri-nos. In comparison with all previous realistic neutrino-genesis models [12, 14, 15], the Dirac neutrino massesin our new model are also suppressed by the ratio ofthe weak scale over the heavy mass scale, but the cru-cial difference is that in the decays of the heavy particles,the radiative vertex corrections (instead of the self-energycorrections) interfere with the tree-level diagrams to gen-erate the required CP asymmetry and naturally realizeneutrinogenesis.We summarize the field content in Table I, in which ψ L , φ , ν R , D L,R , η and χ denote the left-handed leptondoublets, the SM Higgs doublet, the right-handed neutri-nos, the heavy singlet Dirac fermion, the heavy doubletscalar and the real scalar, respectively. Here ψ L , ν R , D L and D R carry lepton number 1 while φ , η and χ have zerolepton number. For simplicity, we have omitted the fam-ily indices as well as other SM fields, which carry evenparity under the discrete symmetry Z . It should benoted that the conventional dimension-4 Yukawa inter- ν R D L D R ψ L h χ i h φ i h φ i h χ i ην R ψ L FIG. 1: The neutrino mass-generation. The left diagram is the type-I Dirac seesaw while the right one is the type-II Diracseesaw. Fields SU (2) L U (1) Y Z ψ L − / φ − / ν R − D L,R η − / − χ − TABLE I: The field content of our model, where ψ L is theleft-handed lepton doublet, φ is the SM Higgs doublet, ν R is the right-handed neutrino, D L,R is the heavy singlet Diracfermion, η is the heavy doublet scalar, and χ is the real scalar.For simplicity the family indices and the other SM fields (car-rying even Z parity) are omitted from the Table. actions among the left-handed lepton doublets, the SMHiggs doublet and the right-handed neutrinos are for-bidden under the Z symmetry. Our model also exactlyconserves the lepton number, so we can write down therelevant Lagrangian as below, − L ⊃ (cid:8) f i ψ Li φD R + g i χD L ν Ri + y ij ψ Li ην Ri − µχη † φ + M D D L D R + h.c (cid:9) + M η η † η , (1)where f i , g i and y ij are the Yukawa couplings, while thecubic scalar coupling µ has mass-dimension equal one.The parameters M D and M η in (1) are the masses of theheavy singlet fermion D and the heavy Higgs doublet η ,respectively. Note that in the Higgs potential the scalardoublet η has a positive mass-term as shown in the aboveEq. (1), while the Higgs doublet φ and singlet χ both havenegative mass-terms .The lepton number conservation ensures that there isno Majorana mass term for all fermions. As we will The general Higgs potential V ( φ, η, χ ) was given in the Appendixof our first paper in Ref. [15]. discuss below, the vacuum expectation value ( vev ) of η comes out to be much less than the vev of the otherfields. Thus the first two terms generate mixings of thelight Dirac neutrinos with the heavy Dirac fermion, whilethe third term gives the light Dirac neutrino mass term.The complete mass matrix can now be written in thebasis { ν L , D L , ν R , D R } as M = a b c da † c † b † d † , (2)where a ≡ y h η i , b ≡ f h φ i , c ≡ g h χ i and d ≡ M D . Aswill be shown below, d ≫ a, b, c . So, the diagonalizationof the mass matrix (2) generates the light Dirac neutrinomasses of order a − bc/d and a heavy Dirac fermion massof order d .As shown in Fig. 1, at low energy we can integrate outthe heavy singlet fermion as well as the heavy doubletscalar. Then we obtain the following effective dimension-5 operators, O = f i g j M D ψ Li φν Rj χ − µy ij M η ψ Li φν Rj χ + h.c. . (3)Therefore, once the SM Higgs doublet φ and the realscalar χ both acquire their vev s, the neutrinos naturallyacquire small Dirac masses, L m = − ( m ν ) ij ν Li ν Rj + h.c. , (4)where m ν ≡ m Iν + m IIν , (5)with [16] (cid:0) m Iν (cid:1) ij = − f i g j h φ ih χ i M D = − ( bc ) ij d , (6)and [15] (cid:0) m IIν (cid:1) ij = y ij µ h φ ih χ i M η = a ij . (7)To quantify the second equality in (7), we note that dif-ferent from the SM Higgs doublet, the heavy scalar dou-blet η has a positive mass-term in the Higgs potential,so it will develop a tiny nonzero vev until φ and χ bothacquire their vev s [15], h η i ≃ µ h φ ih χ i M η . (8)With this we can derive the neutrino mass formula m IIν = y h η i ≡ a from the Lagrangian (1), which confirmsthe Eq. (7) above. In the reasonable parameter space of M D ∼ M η ∼ µ ≫ h χ i , h φ i and ( f, g, y ) = O (1) , wecan naturally realize a ≪ b, c ≪ d . Furthermore, usingthe second relations in (6) and (7) we can re-express thesummed neutrino mass matrix as m ν ≡ m Iν + m IIν = − bc/d + a . (9)This is consistent with the direct diagonalization of theoriginal Dirac mass matrix (2), which we have mentionedbelow (2).It is clear that this mechanism of the neutrino massgeneration has two essential features: (i) it generatesDirac masses for neutrinos, and (ii) it retains the essenceof the conventional seesaw [2] by making the neutrinomasses tiny via the small ratio of the weak scale over theheavy mass scale. It is thus called Dirac Seesaw [15]. Inparticular, compared to the classification of the conven-tional type-I and type-II seesaw, we may refer to Eqs. (6)and (7) as the type-I and type-II Dirac seesaw, respec-tively.From Eq. (9) we see that both type-I and type-II see-saws can contribute to the 3 × m ν for thelight neutrinos. There are three possibilities in general:(i) m Iν ≫ m IIν , or (ii) m Iν ∼ m IIν , or (iii) m Iν ≪ m IIν .We note that for case-(iii), the type-II contribution alonecan accommodate the neutrino oscillation data even iftype-I is fully negligible; while for case-(i) and -(ii), thetype-II contribution should still play a nontrivial rolefor ν -mass generation because m Iν is rank-1 and addi-tional contribution from m IIν is necessary. The rank-1nature of m Iν = bc/d is due to that there is only onesinglet heavy fermion in our current minimal construc-tion, which means that m Iν has two vanishing mass-eigenvalues. Hence, to accommodate the neutrino os-cillation data [1] in the case-(i) and -(ii) of our minimalconstruction always requires nonzero contribution m IIν from the type-II Dirac seesaw . Let us explicitly ana-lyze how this can be realized for the case-(i) and -(ii). As m Iν is rank-1, we can consider a basis for m Iν where one ofthe two massless states is manifest, i.e., b = c = 0 . For Note that the type-I Dirac seesaw alone can accommodate theoscillation data once we extend the current minimal constructionto include a second heavy fermion D ′ which makes m Iν rank 2;this is similar to the minimal (Majorana) neutrino seesaw studiedbefore [18]. the remaining components of b and c , we choose a genericparameter set , b ≈ b ≈ − c ≈ − c , which naturallyrealizes the maximal mixing angle θ = 45 ◦ for explain-ing the atmospheric neutrino mixing. Including the type-II Dirac-seesaw matrix m IIν = a will then account forthe other mixing angles ( θ , θ ) and the two other neu-trino masses. Thus, we can naturally realize the lightneutrino mass-spectrum via both normal hierarchy (NH)and inverted hierarchy (IH) schemes. To be concrete, theNH-scheme is realized in our case-(i) where the type-IIDirac-seesaw matrix m IIν = a ≡ δ ≪ m ∼ m Iν , with m the neutrino mass scale (fixed by the atmosphericneutrino mass-squared-difference ∆ a with m ≡ p ∆ a )and its relations to the nonzero ( b j , c j ) are defined via b j = p m d/ O ( δ ) and c j = − p m d/ O ( δ ) for j = 2 ,
3. Thus we have m ν = − bcd + a = m
12 12
12 12 + O ( δ ) . (10)It is clear that Eq. (10) predicts the neutrino masses,( m , m , m ) = m (0 , ,
1) + O ( δ ) , consistent with theNH mass-spectrum. Next, the IH-scheme can be real-ized in our case-(ii) where the type-II Dirac-seesaw ma-trix m IIν = a ≡ m diag(1 , ,
0) + δ ∼ m Iν with δ ≪ m ,while the structure of the type-I Dirac seesaw matrix m Iν remains the same, m ν = − bcd + a = m
12 12
12 12 + O ( δ ) . (11)From this equation we deduce the neutrino masses,( m , m , m ) = m (1 , ,
0) + O ( δ ) , consistent with theIH mass-spectrum. So far we have discussed all threepossibilities, the case-(i), -(ii) and -(iii), regarding therelative contributions of the type-I versus type-II see-saw to the neutrino mass matrix m ν for accommodatingthe oscillation data. The question of which one amongthese three possibilities is realized in nature should beanswered by a more fundamental theory which can pre-cisely predict the Yuakawa couplings and masses for D and η as well as the vev of χ . Finally, we also notethat as the neutrinos being Dirac particles, our Dirac-seesaw construction will be consistent with the possi-ble non-observation of the neutrinoless double beta de-cays (0 νββ ) which will be tested in the upcoming 0 νββ -experiments [19].The real scalar χ is expected to acquire its vev nearthe weak scale, so we will set h χ i around O (TeV) .Under this setup, it is straightforward to see that m Iν will Here we consider the difference between any two of the four com-ponents | b j | and | c j | ( j = 2 ,
3) to be much smaller themselves. Here we comment on the cosmological domain wall problem as- be efficiently suppressed by the ratio of the weak scaleover the heavy mass. For instance, we find that m Iν = O (0 .
1) eV for M D = O (10 − ) GeV and ( f, g, y ) = O (0 . −
1) , where h φ i ≃
174 GeV. It also is reasonableto set the trilinear scalar coupling | µ | to be around thescale of the η mass M η . In consequence, the neutrinomass m IIν in (7) will be highly suppressed, similar to m Iν . For example, we derive m IIν = O (0 .
1) eV for M η = O (10 − ) GeV and ( y, µ/M η ) = O (0 . −
1) . So, wecan naturally realize the Dirac neutrino masses around O (0 .
1) eV .We now demonstrate how to generate the observedbaryon asymmetry in our model by invoking the neu-trinogenesis [10] mechanism. Since the sphaleron pro-cesses [11] have no direct effect on the right-handed neu-trinos, and the effective Yukawa interactions of the Diracneutrinos are too weak to reach the equilibrium until tem-peratures fall well below the weak scale, the lepton asym-metry stored in the left-handed leptons, which is equalbut opposite to that stored in the right-handed neutri-nos, can be partially converted to the baryon asymmetryby sphalerons. In particular, the final baryon asymmetryshould be B = 2879 ( B − L SM ) = − L SM , (12)for the SM with three generation fermions and one Higgsdoublet.In the pure type-I Dirac seesaw scenario [14], we cangenerate the CP asymmetry through the interferences be-tween the tree-level decay and the self-energy loops ifthere exist at least two heavy fermion singlets. Similarly,the pure type-II Dirac seesaw model [15] also needs twoheavy scalar doublets to obtain the self-energy loops inthe decays. In the following, we shall focus on the min-imal construction with only one heavy singlet fermionand one heavy doublet scalar to realize the radiative ver-tex corrections for the CP asymmetry, although furtherextensions are allowed in our current scenario.In this framework, depending on the values of themasses and couplings, the leptogenesis can be realized ei-ther from the decay of the heavy singlet fermion or from sociated with spontaneous breaking of a discrete Z symmetry.This problem arises during the phase transition (when the bro-ken discrete symmetry gets restored at the transition tempera-ture) because of the production of topological defects – domainwalls which carry too much energy and trouble the standard big-bang cosmology [20]. This can be avoided by inflation as long asphase transition temperature is above the inflation scale [21, 22].Another resolution [23] to the domain wall problem is realizedby the possibility of symmetry non-restoration at high tempera-ture [24]. It is also very possible that a discrete symmetry like Z is not a basic symmetry but appears as a remnant of a continu-ous symmetry such as U (1) which is free from the domain wallproblem [25]. Finally, the Z symmetry in our model can alsobe replaced by a global U (1) D as in the pure type-I Dirac see-saw model [16], and the phenomenology of the Goldstone bosonassociated with this U (1) D breaking was discussed in [14]. the decay of the heavy doublet scalar. From the decayof the heavy singlet fermion to the left-handed leptonsand the SM Higgs doublet, as shown in Fig. 2, the CPasymmetry is given by ε I ≡ Γ ( D R → ψ L φ ∗ ) − Γ ( D cR → ψ cL φ )Γ D = 14 π Im (cid:2) Tr (cid:0) f † yg † (cid:1) µ (cid:3) M η (cid:2) Tr ( f † f ) + Tr ( g † g ) (cid:3) M D × ln (cid:18) M D M η (cid:19) , (13)where Γ D = 116 π (cid:20) Tr (cid:0) f † f (cid:1) + 12 Tr (cid:0) g † g (cid:1)(cid:21) M D (14)is the total decay width of D or D c . Here we havetaken M D to be real after proper phase rotation. Fur-thermore, from the decay of the heavy doublet scalar tothe left-handed leptons and the right-handed neutrinos,a CP asymmetry can also be produced. It is given by theinterference of the tree-level process with the one-loopvertex diagram as shown in Fig. 3, ε II ≡ Γ ( η → ψ L ν cR ) − Γ ( η ∗ → ψ cL ν R )Γ η = 14 π Im (cid:2) Tr (cid:0) f † yg † (cid:1) µ (cid:3) M D (cid:2) Tr ( y † y ) M η + | µ | (cid:3) ln M η M D ! (15)where Γ η = 116 π " Tr (cid:0) y † y (cid:1) + | µ | M η M η (16)is the total decay width of η or η ∗ .In the case where the masses of the heavy singletfermion and heavy doublet scalar locate around the samescale, and also their couplings are of the same order ofmagnitude, the two types of asymmetry of Eqs. (13) and(15) can be both important for the neutrinogenesis. Forillustration below, we will analyze two typical scenarioswhere one process dominates over the other.Scheme-1 is defined for M D ≪ M η and f ∼ g ∼ y ,under which the final left- or right-handed lepton asym-metry mainly comes from the pair decays of ( D, D c ). Wecan simplify the CP asymmetry (13) as ε I ≃ π M D M η Im (cid:2) Tr (cid:0) m I † ν m IIν (cid:1)(cid:3) h φ i h χ i Γ D = " π ) g ∗ K D M η M D × M Pl M D Im (cid:2) Tr (cid:0) m I † ν m IIν (cid:1)(cid:3) h φ i h χ i (17) D R ψ L φ ∗ D R D L ψ L φ ∗ ηχν R FIG. 2: The heavy singlet Dirac fermion decays to the left-handed leptons and the SM Higgs boson at one-loop order. η ψ L ν cR η ψ L ν cR D L D R χφ FIG. 3: The heavy doublet scalar decays to the leptons at one-loop order. with K D ≡ Γ D H (cid:12)(cid:12)(cid:12)(cid:12) T = M D (18)as a measurement of the deviation from equilibrium for D . Here H is the Hubble constant, H ( T ) = (cid:18) π g ∗ (cid:19) T M Pl , (19)with g ∗ = O (100) and M Pl ≃ . × GeV. Note that there is a correlation between K D and m Iν , m I ≡ Tr (cid:0) m I † ν m Iν (cid:1) = Tr (cid:0) f † f gg † (cid:1) h φ i h χ i M D = X i | f i | | g i | h φ i h χ i M D < X i | f i | X j (cid:12)(cid:12) g j (cid:12)(cid:12) h φ i h χ i M D = 2 (16 π ) B L B R Γ D h φ i h χ i M D = 2 (4 π ) g ∗ B L B R K D h φ i h χ i M , (20)and hence K D > "
452 (4 π ) g ∗ B L B R M Pl m I h φ ih χ i . (21)Here B L and B R are the branching ratios of the heavyfermion singlet decaying into the left-handed lepton dou-blets and the right-handed neutrinos, respectively. Theysatisfy the following relationship, B L + B R ≡ , ⇒ B L B R . (22)For instance, we may choose the sample inputs, M η = 10 M D = 1 . × GeV ≫ M D , h φ i =174 GeV, h χ i = 400 GeV and ( y, f, g, µ/M η ) ≃ (0 . , . , . , .
01) = O (0 . m I = O ( m Iν ) = O (10 m IIν ) ≃ .
06 eV. In consequence, we can estimate B L B R ≃ . × . ≃ .
011 , and thus K D ≃
88. Thisleads to ε I ≃ − . × − for the maximal CP phase.We then use the approximate relation [21, 26] to deducethe final baryon asymmetry, Y B ≡ n B s ≃ − × . (cid:0) ε I /g ∗ (cid:1) K D (ln K D ) . ≃ − . (23)This is consistent with the current observations [1]. Fur-thermore, the relationship m Iν = O (0 . ≫ m IIν ,shows the dominance of type-I Dirac seesaw. As wementioned earlier, this can be realized via the NH mass-spectrum of the light neutrinos.We also note that even if the Dirac fermion singlet isat a fairly low mass scale, such as TeV, it is still fea-sible to efficiently enhance the CP asymmetry as longas the ratio M η /M D is large enough. In other words,we can realize the low-scale neutrinogenesis without in-voking the conventional resonant effect to enhance theCP asymmetry (which requires at least two heavy Diracfermion singlets).Scheme-2 is defined for the other possibility with M η ≪ M D and f ∼ g ∼ y . Hence the final left- orright-handed lepton asymmetry is dominated by the pairdecays of ( η, η ∗ ). We derive the following CP asymmetryfrom (15), ε II ≃ π M η Im (cid:2) Tr (cid:0) m I † ν m IIν (cid:1)(cid:3) h φ i h χ i Γ η = " π ) g ∗ K η × M Pl M η Im (cid:2) Tr (cid:0) m I † ν m IIν (cid:1)(cid:3) h φ i h χ i , (24)where K η is given by K η ≡ Γ η H (cid:12)(cid:12)(cid:12)(cid:12) T = M η , (25)with the Hubble constant H ( T ) expressed in Eq. (19).Here the parameter K η measures the deviation from theequilibrium for η . We deduce the correlation between K η and m IIν , m II ≡ Tr (cid:0) m II † ν m IIν (cid:1) = Tr (cid:0) y † y (cid:1) | µ | h φ i h χ i M η = (16 π ) B f B s Γ η h φ i h χ i M η = (4 π ) g ∗ B f B s K η h φ i h χ i M , (26)and also K η = " π ) g ∗ B f B s M Pl m II h φ ih χ i , (27)where B f and B s are the branching ratios of the heavyscalar doublet decaying into the light fermions and thescalars, respectively. Similar to Eq. (22), they satisfy B f + B s ≡ , ⇒ B f B s . (28)For instance, given the sample inputs, M η = 26 µ =0 . M D = 2 × GeV ≪ M D , h φ i = 174 GeV, h χ i = 400 GeV and ( f, g, y ) = (0 . , . , .
34) = O (0 . m II = O ( m IIν ) = O (10 m Iν ) ≃ .
05 eV. Subsequently,we derive, B f B s ≃ . × . ≃ . K η ≃ ε II ≃ − . × − , for the maximal CPphase. Using the approximate analytical formula [21, 26]for the baryon asymmetry, we arrive at Y B ≃ − × . (cid:0) ε II /g ∗ (cid:1) K η (cid:0) ln K η (cid:1) . ≃ − , (29)consistent with the present observation [1]. Further-more, we note that in the Scheme-2, the active neu-trino masses are dominated by the type-II Dirac seesaw, m IIν = O (0 . ≫ m Iν , where both the NH and IHneutrino-mass-spectra can be realized.In this paper, we have presented a new possibility torealize the neutrinogenesis in the Dirac seesaw scenario.In our minimal construction, we introduce a real scalar χ , a heavy singlet Dirac fermion D and a heavy dou-blet scalar η besides three right-handed singlet neutrinosto the SM. Therefore, different from previous realisticneutrinogenesis models, the radiative vertex corrections rather than the self-energy corrections interfere with thetree-level diagrams to generate the CP asymmetry in thedecays of the heavy particles. Finally, we note that thereal singlet scalar χ at the weak scale can couple to theSM Higgs doublet φ via the Z -conserving quartic in-teraction χχφ † φ . In consequence, the lightest neutralHiggs boson h is a mixture between φ and χ , lead-ing to non-SM-like anomalous couplings of h with theweak gauge bosons ( W ± , Z ) and the SM-fermions. Thiscan significantly modify the light Higgs boson ( h ) phe-nomenology at the Tevatron Run-2, the CERN LHC and the future International Linear Colliders (ILC) [27]. Asystematical study for the collider phenomenology of φ and χ is beyond the present scope and will be given else-where. [1] Particle Data Group, W.M. Yao et al. , Journal of PhysicsG , 1 (2006); R.N. Mohapatra, A.Y. Smirnov, Ann.Rev. Nucl. Part. Sci. , 569 (2006); A. Strumia and F.Vissani, hep-ph/0606054; and references therein.[2] P. Minkowski, Phys. Lett. B , 421 (1977); T. Yanagida,in Proc. of the Workshop on Unified Theory and theBaryon Number of the Universe , ed. O. Sawada and A.Sugamoto (KEK, Tsukuba, 1979), p. 95; M. Gell-Mann,P. Ramond, and R. Slansky, in
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