Baryon asymmetry from leptogenesis with four zero neutrino Yukawa textures
aa r X i v : . [ h e p - ph ] J a n Baryon asymmetry from leptogenesis withfour zero neutrino Yukawa textures
Biswajit Adhikary a , b ∗ , Ambar Ghosal a † and Probir Roy a ‡ a) Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, Indiab)Department of Physics, Gurudas College, Narkeldanga, Kolkata-700054, IndiaAugust 8, 2018 Abstract
The generation of the right amount of baryon asymmetry η of the Universe fromsupersymmetric leptogenesis is studied within the type-I seesaw framework with threeheavy singlet Majorana neutrinos N i ( i = 1 , ,
3) and their superpartners. We assumethe occurrence of four zeroes in the neutrino Yukawa coupling matrix Y ν , taken to be µτ symmetric, in the weak basis where N i (with real masses M i >
0) and the chargedleptons l α ( α = e, µ, τ ) are mass diagonal. The quadrant of the single nontrivial phase,allowed in the corresponding light neutrino mass matrix m ν , gets fixed and additionalconstraints ensue from the requirement of matching η with its observed value. Specialattention is paid to flavor effects in the washout of the lepton asymmetry. We alsocomment on the role of small departures from high scale µτ symmetry due to RGevolution. Baryogenesis through leptogenesis [1, 2, 3] is a simple and attractive mechanism to explainthe mysterious excess of matter over antimatter in the Universe. A lepton asymmetry is first ∗ biswajit.adhikary @saha.ac.in † [email protected] ‡ [email protected] enerated at a relatively high scale ( > GeV). This then gets converted into a nonzero η , the difference between the baryonic and antibaryonic number densities normalized to thephoton number density ( n B − n ¯ B ) n − γ , at electroweak temperatures [4] due to B + L violat-ing but B − L conserving sphaleron interactions of the Standard Model. Since the origin ofthe lepton asymmetry is from out of equilibrium decays of heavy unstable singlet Majorananeutrinos [5], the type-I seesaw framework [6, 7, 8, 9, 10], proposed for the generation of lightneutrino masses, is ideal for this purpose. We study baryogenesis via supersymmetric lepto-genesis [11] with a type-I seesaw driven by three heavy ( > GeV) right-chiral Majorananeutrinos N i ( i = 1 , ,
3) with Yukawa couplings to the known left chiral neutrinos throughthe relevant Higgs doublet. There have been some recent investigations [12, 13, 14, 15] study-ing the interrelation between leptogenesis, heavy right-chiral neutrinos and neutrino flavormixing. However, our angle is a little bit different in that we link supersymmetric leptogen-esis to zeroes in the neutrino Yukawa coupling matrix. In fact, we take a µτ symmetric [16]neutrino Yukawa coupling matrix Y ν with four zeroes [17] in the weak basis specified in theabstract.There are several reasons for our choice. First, a seesaw with three heavy right chiralneutrinos is the simplest type-I scheme yielding a square Yukawa coupling matrix Y ν on whichsymmetries can be imposed in a straightforward way. Second, µτ symmetry [18] - [46] in theneutrino sector provides a very natural way of understanding the observed maximal mixingof atmospheric neutrinos. Though it also predicts a vanishing value for the neutrino mixingangle θ , the latter is known from reactor experiments to be rather small. A tiny nonzerovalue of θ could arise at the 1-loop level via the charged lepton sector, where µτ symmetryis obviously broken, though RG effects if the said symmetry is imposed at a high scale [16].Third, four has been shown [17] to be the maximum number of zeroes phenomenologicallyallowed in Y ν within the type-I seesaw framework in the weak basis described earlier. Finally,four zero neutrino Yukawa textures provide [47] a very constrained and predictive theoreticalscheme - particularly if µτ symmetry is imposed [16].The beautiful thing about such four zero textures in Y ν is that the high scale CP violation,required for leptogenesis, gets completely specified here [17] in terms of CP violation thatis observable in the laboratory with neutrino and antineutrino beams. In our µτ symmetricscheme [16], which admits two categories A and B , the latter is given in terms of just onephase (for each category) which is already quite constrained by the extant neutrino oscillationdata. Indeed, the quadrant in which this phase lies - which was earlier unspecified by thesame data - gets fixed by the requirement of generating the right size and sign of the baryon2symmetry. Moreover, the magnitude of this phase is further constrained.In computing the net lepton asymmetry generated at a high scale, one needs to consider notonly the decays of heavy right-chiral neutrinos N i into Higgs and left-chiral lepton doubletsas well as their superpartner versions but also the washout caused by inverse decay processesin the thermal bath. The role of flavor [48, 49, 50, 51] can be crucial in the latter. In theMinimal Supersymmetric Standard Model (MSSM [52]), this has been studied [53] throughflavor dependent Boltzmann equations. The solutions to those equations demonstrate thatflavor effects show up differently in three distinct regimes depending on the mass of thelightest of the three heavy neutrinos and an MSSM parameter tan β which is the ratio v u /v d of the up-type and down-type Higgs VEVs. In each regime there are three N i masshierarchical cases : (a) normal, (b) inverted and (c) quasidegenerate. All these, consideredin both categories A and B , make up eighteen different possibilities for each of which thelepton asymmetry is calculated here. That then is converted into the baryon asymmetry bystandard sphaleronic conversion and compared with observation. These lead to the phaseconstraints mentioned above as well as a stronger restriction on the parameter tan β in somecases.If µτ symmetry is posited at a high scale characterized by the masses of the heavy Majorananeutrinos, renormalization group evolution down to a laboratory energy λ breaks it radia-tively. Consequently, a small nonzero θ λ , crucially dependent on the magnitude of tan β ,gets induced. The said new restrictions on tan β coming from η in some cases therefore causestrong constraints on the nonzero value of θ λ which we enumerate.One possible problem with high scale supersymmetric thermal leptogenesis is that of theoverabundance of gravitinos caused by the high reheating temperature. For a decayinggravitino, this can lead to a conflict with Big Bang Nucleosynthesis constraints, while for astable gravitino (dark matter) this poses the danger of overclosing the Universe. The problemcan be evaded by appropriate mass and lifetime restrictions on the concerned sparticles, cf.sec. 16.4 of ref [52]. Such is the case, for instance, with gauge mediated supersymmetrybreaking with a gravitino as light as O (KeV) in mass. In gravity mediated supersymmetrybreaking there are sparticle mass regions where the problem can be avoided – especiallywithin an inflationary scenario. An illustration is a model [54], with a gluino and a neutralinothat are close in mass, which satisfies the BBN constraints. Purely cosmological solutionswithin the supersymmetric inflationary scenario have also been proposed, e.g. [55]. We feelthat, while the gravitino issue is one of concern, it can be resolved and therefore need not3e addressed here any further.The plan of the rest of the paper is as follows. In section 2 we recount the propertiesof the allowed µτ symmetric four zero Y ν textures. Section 3 contains an outline of thebasic steps in our calculation of η . In section 4, η is computed in our scheme for the threedifferent heavy neutrino mass hierarchical cases in the regimes of unflavoured, fully flavoredand τ -flavored leptogenesis for both categories A and B . Section 5 consists of our results onconstraints emerging from η on the allowed µτ symmetric four zero Y ν textures. In section6 we discuss the departures - due to RG evolution down to laboratory energies - from µτ symmetry imposed at a high scale ∼ min ( M , M , M ) ≡ M lowest . Section 7 summarizesour conclusions. Appendices A, B and C list the detailed expressions for η in each of theeighteen different possibilities. µτ symmetric four zero textures of Y ν The complex symmetric light neutrino Majorana mass matrix m ν is given in our basis by m ν = − v u Y ν diag . ( M − , M − , M − ) Y Tν = U diag . ( m , m , m ) U T . (2.1)We work within the confines of the MSSM [52] so that v u = v sin β and the W-mass equals gv , g being the SU (2) L semiweak gauge coupling strength. The unitary PMNS mixingmatrix U is parametrized as U = c − s s c c − s e − iδ D s e iδ D c c s − s c
00 0 1 e iα M e iβ M
00 0 1 , (2.2)where c ij = cos θ ij , s ij = sin θ ij and δ D , α M , β M are the Dirac phase and two Majoranaphases respectively.The statement of µτ symmetry is that all couplings and masses in the pure neutrino part ofthe Lagrangian are invariant under the interchange of the flavor indices 2 and 3. Thus( Y ν ) = ( Y ν ) , (2 . a )( Y ν ) = ( Y ν ) , (2 . b )( Y ν ) = ( Y ν ) , (2 . c )4 Y ν ) = ( Y ν ) (2 . d )and M = M . (2 . µτ symmetry in m ν :( m ν ) = ( m ν ) = ( m ν ) = ( m ν ) , (2 . a )( m ν ) = ( m ν ) . (2 . b )Eqs. (2.5) immediately imply that θ = π/ θ = 0. With this µτ symmetry, it wasshown in Ref. [16] that only four textures with four zeroes in Y ν are allowed. These fall intotwo categories A and B - each category containing a pair of textures yielding an identicalform of m ν . These allowed textures may be written in the form of the Dirac mass matrix m D = Y ν v u / √ a , a , b , b . Category A : m (1) DA = a a a b b , m (2) DA = a a a b
00 0 b , (2 . a ) Category B : m (1) DB = a b b b b , m (2) DB = a b b b b , (2 . b )The corresponding expressions for m ν , obtained via eq.(2.1), are much simplified by a changeof variables. We introduce overall mass scales m A,B , real parameters k , k , l , l and phases¯ α and ¯ β defined by Category A : m A = − b /M , k = (cid:12)(cid:12)(cid:12)(cid:12) a b (cid:12)(cid:12)(cid:12)(cid:12)s M M , k = (cid:12)(cid:12)(cid:12)(cid:12) a b (cid:12)(cid:12)(cid:12)(cid:12) , ¯ α = arg a a . (2 . a ) Category B : m B = − b /M , l = (cid:12)(cid:12)(cid:12)(cid:12) a b (cid:12)(cid:12)(cid:12)(cid:12) s M M , l = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s M M , ¯ β = arg b b . (2 . b )Then the light neutrino mass matrix for each category can be written as [53] m νA = m A k e i ¯ α + 2 k k k k k , m νB = m B l l l e i ¯ β l l e i ¯ β l l e i ¯ β l e i ¯ β + 1 l e i ¯ β l l e i ¯ β l e i ¯ β l e i ¯ β + 1 . (2 . h = m † D m D (2 . Category A as well as for the two textures of Category B . Indeed, it can be given separately for the two categories as h A = | m A | M k x / k k e − i ¯ α x / k k e − i ¯ α x / k k e i ¯ α √ x (1 + k ) √ xk x / k k e i ¯ α √ xk √ x (1 + k ) , (2 . a ) h B = | m B | M l + 2 l x / l e − i ¯ β x / l e − i ¯ β x / l e i ¯ β √ x x / l e i ¯ β √ x , (2 . b )where x = M M . (2 . k , k , cos ¯ α and l , l , cos ¯ β from neutrino oscillation datawere worked out in ref. [16]. The relevant measured quantities are the ratio of the solar toatmospheric neutrino mass squared differences R = ∆ m / ∆ m and the tangent of twicethe solar mixing angle tan 2 θ . One can write R = 2( X + X ) / [ X − ( X + X ) / ] − , (2 . a )tan 2 θ = X X . (2 . b )The quantities X , , are given for the two categories as follows : Category A : X A = 2 √ k [(1 + 2 k ) + k + 2 k (1 + 2 k ) cos 2 ¯ α ] / , (2 . a ) X A = 1 − k − k − k k cos 2 ¯ α, (2 . b ) X A = 1 − k − k − k k cos 2 ¯ α − k . (2 . c ) Category B : X B = 2 √ l l [( l + 2 l ) + 1 + 2( l + 2 l ) cos 2 ¯ β ] / , (2 . d ) X B = 1 + 4 l cos 2 ¯ β + 4 l − l , (2 . e ) X B = 1 − ( l + 2 l ) − l cos 2 ¯ β. (2 . f )6e also choose to define X A,B = ( X A,B + X A,B ) / . (2 . σ level, tan 2 θ is presently known to be [56] between 1.83 and 4.90. For this range,only the inverted mass ordering for the light neutrinos, i.e. ∆ m <
0, is allowed for
Category A with the allowed interval for R being − . × − eV to − . × − eV . In contrast,the same range of tan 2 θ allows only the normal light neutrino mass ordering ∆ m > Category B with R restricted to be between 2 . × − eV and 3 . × − eV . A thinsliver is allowed [16] in the k − k plane for Category A , while a substantial region with twobranches is allowed [16] in the l − l plane for Category B . Finally, cos ¯ α is restricted to theinterval bounded by 0 and 0 . β is restricted to the interval bounded by 0 and0 . α , ¯ β could be either in the first or in the fourth quadrant. The interesting newpoint in the present work is that the baryogenesis constraint leads to restrictions on sin 2 ¯ α and sin 2 ¯ β to the extent of removing the quadrant ambiguity in ¯ α and ¯ β . Armed with µτ symmetry as well as eqs. (2.8) and (2.10), we can tackle leptogenesis at a scale ∼ M lowest . There are three possible mass hierarchical cases for N i . Case (a) corresponds to anormal hierarchy of the heavy Majorana neutrinos (NHN), i.e. M lowest = M << M = M .In case (b) one has an inverted hierarchy for N i (IHN) with M lowest = M = M << M .Case (c) refers to the quasidegenerate (QDN) situation with M ∼ M ∼ M ∼ M lowest .Working within the MSSM [52] and completely neglecting possible scattering processes [53]which violate lepton number, we can take the asymmtries generated by N i decaying into adoublet of leptons L α and a Higgs doublet H u as ǫ αi = Γ( N i → L Cα H u ) − Γ( N i → L α H Cu )Γ( N i → L Cα H u ) + Γ( N i → L α H Cu ) ≃ πv u h ii X j = i " I αij f ( x ij ) + J αij − x ij , (3.1) I αij = Im [( m † D ) iα ( m D ) αj h ij ] , (3.2) J αij = Im [( m † D ) iα ( m D ) αj h ji ] , (3.3)where x ij = M j /M i (3.4)7nd f ( x ij ) = √ x ij " − x ij − ln 1 + x ij x ij . (3.5)We note here that the J αij term does not contribute to ǫ αi in our scheme since it vanishes [16]on account of µτ symmetry. Further, contributions to ǫ αi from N i decaying into sleptons andhiggsinos and from sneutrinos ˜ N i decaying into sleptons and Higgs as well as into leptons andhiggsinos have been included by appropriately choosing the x ij -dependence in the RHS of eq.(3.5). Observe also that I α j (and hence ǫ α ) gets an overall minus sign from Im( e − i ¯ α , e − i ¯ β ),whereas I α j , I α j (and hence ǫ α , ) get an overall plus sign from Im( e i ¯ α , e i ¯ β ). Except for beingpositive in the region 0 . ≤ x ij <
1, the function f ( x ij ) of eq.(3.5) is negative for all othervalues of its argument. These signs are crucial in determining the sign of η and hence thoseof ¯ α , ¯ β .The decay asymmetries ǫ αi get converted into a lepton asymmetry Y α = ( n αl − ¯ n αl ) s − , s beingthe entropy density and n αl (¯ n αl ) being the leptonic (antileptonic) number density (includingsuperpartners) for flavor α via the washout relation [53] Y α = X i ǫ αi K αi g − ⋆i . (3.6)In eq. (3.6), g ⋆i is the effective number of spin degrees of freedom of particles and antiparticlesat a temperature equal to M i . Furthermore, when all the flavors are active, the quantity K αi is given by the approximate relation [12, 51], neglecting contributions from off-diagonalelements of A , ( K αi ) − ≃ . | A αα | K αi + | A αα | K αi . ! . . (3.7)In eq. (3.7), K αi is the flavor washout factor given by K αi = Γ (cid:16) N → L α H Cu (cid:17) H ( M i ) = | m Dαi | M i M P l . π √ g ⋆i v u , (3.8) M P l being the Planck mass. This follows since the Hubble expansion parameter H ( M i ) at atemperature M i is given by 1 . √ g ⋆i M i M − P l . Moreover, to the lowest order, Γ( N i → L α H Cu )equals | m Dαi | M i (4 πv u ) − . An additional quantity, appearing in eq. (3.7), is A αα , a diagonalelement of the matrix A αβ defined by Y α L = X β A αβ Y β ∆ . (3.9)8ere Y α L = s − ( n α L − ¯ n α L ), n αL being the number density of left-handed lepton and sleptondoublets of flavor α and Y α ∆ = Y B − Y α , Y B being the baryonic number density (normalizedto the entropy density s ) including all superpartners. The precise forms for A αβ in differentregimes of leptogenesis will be specified later.One can now utilize the relation between Y B = ( n B − n ¯ B ) s − and Y l = P α Y α , namely [57] Y B = − n F + 4 n H n F + 13 n H Y l , (3.10)where n F ( n H ) is the number of matter fermion (Higgs) SU (2) L doublets present in thetheory at electroweak temperatures. For MSSM, n F = 3 and n H = 2 so that eq. (3.10)becomes Y B = − Y l . (3.11)The baryon asymmetry η = ( n B − n ¯ B ) n − γ can now be calculated, utilizing the result [58]that sn − γ ≃ .
04 at the present time, to be η = sn γ Y B ≃ . Y B ≃ − . Y l . (3.12)Leptogenesis occurs at a temperature of the order of M lowest and the effective values of A αα and K αi depend on which flavors are active in the washout process. This is controlled [53]by the quantity M lowest (1 + tan β ) − . There are three different regimes which we discussseparately. (1) M lowest ( + tan β ) − > GeV.
In this case there is no flavor discrimination and unflavored leptogenesis takes place. Thus A αβ = − δ αβ and all flavors α can just be summed in eqs. (3.1). Thus ǫ i = P ǫ αi , P α J αij = 0, I ij ≡ P α I αij = Im ( h ij ) and Y = P i ǫ i g − ⋆i K i with K − i = 8 . K − i + ( K i / . . and K i = P α K αi = h ii M P l (6 . π √ g ⋆i M i v u ) − . For the normal hierarchical heavy neutrino(NHN) case (a), M may be ignored and the index i can be restricted to just 1, taking g ⋆ = 232 .
5. For the corresponding inverted hierarchical (IHN) case (b) M can be ignoredand i made to run over 2 and 3 with g ⋆ = 236.25, all quantities involving the index 2being identical to the corresponding ones involving 3. Coming to the quasidegenerate (QDN)heavy neutrino case (c), g ⋆ = 240 and the contributions from i = 1 must be separately addedto identical contributions from i = 2 , (2) M lowest ( + tan β ) − < GeV. A -matrix needs to be taken as [53] A MSSM = − /
110 6 /
55 6 / / − /
30 1 / /
40 1 / − / (3.13)and eqs. (3.6) – (3.8) used for each flavor α . Once again, we consider the different cases(a), (b) and (c) of heavy neutrino mass ordering. Ignoring M for case (a) and with g ⋆ = 232 .
5, we have η ≃ − . × − P α ǫ α K α . Similarly, ignoring M for case (b) andwith g ⋆ = 236 .
25, one gets η ≃ − . × − P α ( ǫ α K α + ǫ α K α ). For case (c), g ⋆ = 240and η ≃ − . × − P α ( ǫ α K α + ǫ α K α + ǫ α K α ). (3) 10 GeV < M lowest ( + tan β ) − < GeV.
In this regime the τ - flavor decouples first while the electron and muon flavors act indistin-guishably. The latter, therefore, can be summed. Now effectively A becomes a 2 × A given by [53] ˜ A = − /
761 152 / / − / ! (3.14)and acting in a space spanned by e + µ and τ . Indeed, we can define K e + µi and ˜ K τi by( K e + µi ) − = 8 . | ˜ A | ( K ei + K µi ) + | ˜ A | ( K ei + K µi )0 . ! . , (3 . a )( ˜ K τi ) − = 8 . | ˜ A | K τi + | ˜ A | ( K τi )0 . ! . . (3 . b )Now, for case (a) with g ⋆ = 232 . η ≃ − . × − [( ǫ e + ǫ µ ) K e + µ + ǫ τ ˜ K τ ]. Case (b) has g ⋆ = 236.25 and η ≃ − . × − P k =2 , [( ǫ ek + ǫ µk ) K e + µk + ǫ τk ˜ K τk ]. Finally, case (c), with g ⋆ = 240, has η ≃ − . × − P i [( ǫ ei + ǫ µi ) K e + µi + ǫ τi ˜ K τi ]. (1) Regime of unflavored leptogenesis As explained in Sec. 3, there is no flavor discrimination if M lowest (1 + tan β ) − > GeV.The lepton asymmetry parameters ǫ i can now be given after summing over α . Additional10implifications can be made by taking v u = v sin β with v ≃
246 GeV and substituting | m | = (∆ m /X ) / . (4.1)The relevant expressions for the two categories then are the following Category A : ǫ A ≃ − . × − M GeV k √ xf ( x ) sin 2 ¯ αX / A sin β , (4 . a ) ǫ A = ǫ A ≃ . × − M GeV k k
22 1 √ x f ( x ) sin 2 ¯ α (1 + k ) X / A sin β = 1 . × − M GeV k k f ( x ) sin 2 ¯ α (1 + k ) X A / sin β . (4 . b ) Category B : ǫ B ≃ − . × − M GeV l √ xf ( x ) sin 2 ¯ β ( l + 2 l ) X / B sin β , (4 . a ) ǫ B = ǫ B ≃ . × − M GeV l f ( x ) sin 2 ¯ βX / B sin β = 1 . × − M GeV l √ xf ( x ) sin 2 ¯ βX / B sin β . (4 . b )Note that x was defined in eq.(2.11). We are now in a position to discuss the three N i mass hierarchical cases. For case (a), with the much heavier M = M ignored and only M contributing, we can give the following expressions for the flavor-summed washout factors. Category A : K A ≃ . k √ g ⋆ X / A sin β , (4 . a ) K A = K A ≃ .
36 (1 + k ) √ g ⋆ X / A sin β . (4 . b ) Category B : K B ≃ .
36 ( l + 2 l ) √ g ⋆ X / B sin β , (4 . c ) K B = K B ≃ . √ g ⋆ X / B sin β . (4 . d )11onsequently, η NHNA ≃ − . × − ( ǫ A K A ) g ⋆ =232 . , (4 . a ) η NHNB ≃ − . × − ( ǫ B K B ) g ⋆ =232 . , (4 . b )with the dependence on the category ( A or B ) coming both through ǫ and K occuring in K . For case (b), one can ignore M and hence ǫ and K . Thus we have η IHNA ≃ − .
06 ( ǫ A K A ) g ⋆ =236 . , (4 . a ) η IHNB ≃ − .
06 ( ǫ B K B ) g ⋆ =236 . , (4 . b )where once again the category dependence comes in through ǫ and K occuring in K .Finally, for case (c) with all three M ′ s contributing, η QDNA,B ≃ − . × − ( ǫ A,B K A,B + 2 ǫ A,B K A,B ) g ⋆ =240 . (4 . K , in terms of K , , have already been given in Sec. 3. Detailedexpressions for the right hand sides of eqs. (4.5), (4.6) and (4.7) are given in appendix A. (2) Regime of fully flavored leptogenesis If M lowest (1 + tan β ) − < GeV, all leptonic flavors become active causing fully flavoredleptogenesis, cf. Sec. 3. We now need to resort to eqs. (3.1) – (3.8) to compute the lepton(flavor) asymmetry Y α . However, J αij vanishes explicitly for all the four cases of four zerotextures of m D being considered by us. Thus we need be concerned only with the I αij term ineq. (3.1). Even some of the latter vanish on account of the zeroes in our textures. However,let us first draw some general conclusions about the two categories of textures before takingup the three N i hierarchical cases separately. Category A :It is clear from eq. (2.6a) that the presence of two zeroes in rows 2 and 3 in both textures m (1) DA and m (2) DA implies the vanishing of ( m D ) † iµ ( m D ) µj and ( m D ) † iτ ( m D ) τj for i = j . As aresult, I µij = I τij = 0 which imply that ǫ µiA = ǫ τiA = 0 . (4 . K µiA and K τiA do not contribute to η . The expressions for the pertinent nonvanishingquantities are given by ǫ e A ≃ − . × − M GeV k √ xf ( x ) sin 2 ¯ αX / A sin β , (4 . a )12 e A = ǫ e A ≃ . × − M GeV k k f ( x ) sin 2 ¯ α (1 + k ) X / A sin β , (4 . b ) K e A ≃ . k √ g ⋆ X / A sin β , (4 . c ) K e A = K e A ≃ . k √ g ⋆ X / A sin β . (4 . d ) Category B :In this case, each allowed texture of m D in eq.(2.6b) has two zeroes in the first row inconsequence of which ( m D ) † ie ( m D ) ej vanishes for i = j . Therefor, I eij = 0 because of which ǫ eiB = 0 . (4 . h B ) and ( h B ) being zero, I µ,τ and I µ,τ vanish here forboth textures m (1) DB and m (2) DB . An additional point is that, for the texture m (1) DB , I µ = 0 = I τ but I τ = 0 = I µ while, for m (2) DB , I µ = 0 = I τ but I µ = 0 = I τ . Consequently, ǫ µ B isthe same for both allowed textures and so is ǫ τ B . Moreover, for m (1) DB , ǫ µ B and ǫ τ B vanishbut ǫ τ B and ǫ µ B do not while, for m (2) DB , ǫ τ B and ǫ µ B vanish but ǫ µ B and ǫ τ B do not. In fact,explicitly one has ǫ (1) µ B = ǫ (1) τ B = ǫ (2) τ B = ǫ (2) µ B = 0 , (4 . a ) ǫ (1) µ B = ǫ (1) τ B = ǫ (2) µ B = ǫ (2) τ B ≃ − . × − M GeV l √ xf ( x ) sin 2 ¯ β ( l + 2 l ) X / B sin β , (4 . b ) ǫ (1) τ B = ǫ (1) µ B = ǫ (2) µ B = ǫ (2) τ B ≃ . × − M GeV l
22 1 √ x f ( x ) sin 2 ¯ βX / B sin β . (4 . c )In these equations and henceforth the superscripts (1),(2) refer to m (1) D , m (2) D respectively.Coming to the washout factors, one sees a similar pattern. For m (1) DB , K µ B and K τ B vanishwhile for m (2) DB , K τ B and K µ B are zero. Explicitly, K (1) µ B = K (1) τ B = K (2) µ B = K (2) τ B ≃ . l √ g ⋆ X / B sin β . (4 . a ) K (1) e B = K (2) e B = K (1) µ B = K (1) τ B = K (2) τ B = K (2) µ B = K (1) e B = K (2) e B = 0 , (4 . b ) K (1) τ B = K (1) µ B = K (2) µ B = K (2) τ B ≃ . √ g ⋆ X / B sin β . (4 . c )13et us finally draw attention to an important consequence of eqs. (4.11) and (4.12). Since K αi is just a known function of A αα as well as K αi and since A µµ equals A ττ , the combination ǫ µ B K µ B + ǫ µ B K µ B + ǫ τ B K τ B + ǫ τ B K τ B (4 . m (1) DB and m (2) DB and is a characteristic of just Category B .Now, for the normal N i -hierarchical case (a), with M , neglected, we have the followingexpression for the baryon asymmetry. Category A : η NHNA ≃ − . × − ǫ e A ( K e A ) g ⋆ =232 . . (4 . a ) Category B : η NHNB ≃ − . × − [( ǫ µ B K µ B + ǫ τ B K τ B ) − . × − ( ǫ µ B K µ B )] g ⋆ =232 . , (4 . b )where µτ symmetry has been used in the last step. For the inverted N i - hierarchical case(b), with M neglected, the results are given below. Category A : η IHNA ≃ − . × − ( ǫ e A K e A + ǫ e A K e A ) g ⋆ =236 . ≃ − . × − ǫ e A ( K e A ) g ⋆ =236 . . (4 . a ) Category B : η IHNB ≃ − . × − ( ǫ µ B K µ B + ǫ τ B K τ B + ǫ µ B K µ B + ǫ τ B K τ B ) g ⋆ =236 . ≃ − . × − ( ǫ µ B K µ B + ǫ µ B K µ B ) g ⋆ =236 . . (4 . b )In eq. (4.15b), the first (second) term in the RHS bracket vanishes for m (1) DB ( m (2) DB ); the non-vanishing terms have identical expressions for both textures. Lastly, for the quasidegeneratecase (c), the expressions for the baryon asymmetry are as follows. Category A : η QDNA ≃ − . × − ( ǫ e A K e A + 2 ǫ e A K e A ) g ⋆ =240 . (4 . a ) Category B : 14 QDNB ≃ − . × − ( ǫ µ B K µ B + ǫ τ B K τ B + ǫ µ B K µ B + ǫ τ B K τ B + ǫ µ B K µ B + ǫ τ B K τ B ) g ⋆ =240 ≃ − . × − ( ǫ µ B K µ B + ǫ µ B K µ B + ǫ µ B K µ B ) g ⋆ =240 . (4 . b )The second (third) term within the RHS bracket vanishes for m (1) DB ( m (2) DB ), while the remain-ing terms are identical for both textures of Category B . Detailed expressions for the righthand sides of eqs. (4.14), (4.15) and (4.16) appear in appendix B. (3) Regime of τ -flavored leptogenesis We have discussed in Sec. 3 that, with 10 GeV < M lowest (1 + tan β ) − < GeV, thereis flavor active leptogenesis in the τ -sector but the electron and muon flavors can be summed.Thus, use can be made here of the flavor dependent results of Regime (2), but there is aproviso : both the generation and washout of Y L take place in a flavor subspace spanned by e + µ and τ , cf. eqs. (3.13) and (3.14). Using the notation of eq. (3.15), we can then writethe consequent baryon asymmetry as η ≃ − . X i g − ⋆i [( ǫ ei + ǫ µi ) K e + µi + ǫ τi ˜ K τi ] . (4 . N i -hierarchical cases (a), (b) and (c) for each of the four texturesusing the subscripts A, B for the category and subscripts (1) , (2) for the textures. Category A , m (1) DA . Now ǫ (1) µiA = 0 = ǫ (1) τiA , cf. eq. (4.8). But, in addition, we have0 = K (1) µ A = K (1) τ A = K (1) µ A = K (1) τ A . (4 . ǫ (1) e A , ǫ (1) e A = ǫ (1) e A , K (1) e A and K (1) e A = K (1) e A are as given by eqs. (4.9a)– (4.9d). Additionally, K (1) e A = 86 . √ g ⋆ k X / A sin β , (4 . a ) K (1) e A = K (1) e A = 86 . √ g ⋆ k X / A sin β , (4 . b ) K (1) µ A = K (1) τ A = 86 . √ g ⋆ X / A sin β . (4 . c )15ow for the NHN case (a), we have η (1) NHNA ≃ − . × − ǫ e A (cid:20)(cid:16) K e + µ A (cid:17) K µ A =0 (cid:21) g ⋆ =232 . (4 . K e + µ A calculated as per eq. (3.15a) but setting K µ A = 0. For the IHN case (b), we canwrite η (1) IHNA ≃ − . × − (cid:20) ǫ e A (cid:16) K e + µ A (cid:17) K µ A =0 + ǫ e A K e + µ A (cid:21) g ⋆ =236 . . (4 . K e + µ A is calculated by putting K µ A = 0.For the QDN case (c), the expression is η (1) QDNA ≃ − . × − (cid:20) ǫ e A (cid:16) K e + µ A (cid:17) K µ A =0 + ǫ e A (cid:16) K e + µ A (cid:17) K µ A =0 + ǫ e A K e + µ A (cid:21) g ⋆ =240 . (4 . K e + µiA . Category A , m (2) DA .Again, ǫ (2) µiA = 0 = ǫ (2) τiA , but the vanishing washout factors now are0 = K (2) µ A = K (2) τ A = K (2) µ A = K (2) τ A . (4 . ǫ (2) e A , ǫ (2) e A = ǫ (2) e A , K (2) e A , K (2) e A and K (2) e A are thesame as for m (1) DA . In addition, K (2) µ A = K (2) τ A = 86 . √ g ⋆ X / A sin β . (4 . η (2) NHNA ≃ − . × − ǫ e A (cid:20)(cid:16) K e + µ A (cid:17) K µ A =0 (cid:21) g ⋆ =232 . , (4 . η (2) IHNA ≃ . × − (cid:20) ǫ e A K e + µ A + ǫ e A (cid:16) K e + µ A (cid:17) K µ A =0 (cid:21) g ⋆ =236 . , (4 . K e + µ A is calculated fully but K e + µ A by setting K µ A = 0. This expression turns out to bethe same as for m (1) DA . 16inally, the QDN case (c) has the baryon asymmetry as η (2) QDNA ≃ − . × − (cid:20) ǫ e A (cid:16) K e + µ A (cid:17) K µ A =0 + ǫ e A K e + µ A + ǫ e A (cid:16) K e + µ A (cid:17) K µ A =0 (cid:21) g ⋆ =240 (4 . K e + µiA , as shown. Again, this turns out to beequal to that for m (1) DA . Detailed expressions for the right hand side of eqs. (4.20) and (4.25),which are identical, appear in appendix C. We make the same statement for eqs. (4.21) and(4.26) as well as for eqs. (4.22) and (4.27). Category B , m (1) DB .Here, ǫ (1) eiB = 0 = ǫ (1) µ B = ǫ (1) τ B and K (1) e B cf. eqs. (4.10) and (4.11a), while the vanishingwashout factors are K (1) µ B , K (1) τ B , K (1) e B . The pertinent nonzero quantities, as given in eqs.(4.11b), (4.11c) and (4.12a), (4.12c), are ǫ (1) µ B = ǫ (1) τ B , ǫ (1) τ B = ǫ (1) µ B and K (1) µ B = K (1) τ B , K (1) τ B = K (1) µ B . Additionally, K (1) e B = 86 . √ g ⋆ l X / B sin β , (4 . a ) K (1) µ B = K (1) τ B = 86 . √ g ⋆ l X / B sin β . (4 . b ) K (1) τ B = K (1) µ B = 86 . √ g ⋆ X / B sin β . (4 . c )Therefore, for the NHN case (a), η (1) NHNB ≃ − . × − h ǫ µ B K e + µ B + ǫ τ B ˜ K τ B i g ⋆ =232 . . (4 . η (1) IHNB ≃ − . × − (cid:20) ǫ µ B (cid:16) K e + µ B (cid:17) K µ B =0 + ǫ τ B ˜ K τ B (cid:21) g ⋆ =236 . , (4 . K µ B = 0. For the finalQDN case (c), the expression is η (1) QDNB ≃ − . × − (cid:20) ǫ µ B K e + µ B + ǫ µ B (cid:16) K e + µ B (cid:17) K e B =0 + ǫ τ B ˜ K τ B + ǫ τ B ˜ K τ B (cid:21) g ⋆ =240 , (4 . K e + µ B is calculated with K e B set to vanish. Category B , m (2) DB . 17ere we have ǫ (2) eiB = 0 = ǫ (2) τ B = ǫ (2) µ B from eqs. (4.10) and (4.11a), while the washout factors K (2) τ B , K (2) µ B , K (2) e B , K (2) e B vanish. The remaining nonzero quantities of relevance, as appearin eqs. (4.11b), ( 4.11,c) and (4.12a) (4.12c), are ǫ (2) µ B = ǫ (2) τ B , ǫ (2) µ B = ǫ (2) τ B and K (2) µ B = K (2) τ B , K (2) τ B = K (2) µ B . In addition, K (2) e B has the same expression as K (1) e B i.e. K (2) e B = 86 . √ g ⋆ l X / B sin β . (4 . η (2) NHNB ≃ − . × − h ǫ µ B K e + µ B + ǫ τ B ˜ K τ B i g ⋆ =232 . , (4 . m (1) DB . For the IHN case (b), the baryon asymmetry reads η (2) IHNB ≃ − . × − (cid:20) ǫ τ B (cid:16) ˜ K τ B (cid:17) + ǫ µ B (cid:16) K e + µ B (cid:17) K e B =0 (cid:21) g⋆ =236 . (4 . m (1) DB . Finally, for the QDN case (c), thebaryon asymmetry is η (2) QDNB ≃ − . × − (cid:20) ǫ µ B K e + µ B + ǫ µ B (cid:16) K e + µ B (cid:17) K e B =0 + ǫ τ B ˜ K τ B + ǫ τ B ˜ K τ B (cid:21) g⋆ =240 (4 . m (1) DB . Thus the baryon asymmetry in each of thethree N i -hierarchical cases has the same expression for both m (1) D and m (2) D in Category A and the same statement holds for Category B . Detailed expressions of η in Category B forthe NHN, IHN and QDN cases are given in appendix C. We had earlier deduced [16] from neutrino oscillation data with 3 σ errors the constraints0 ≤ cos ¯ α ≤ . ≤ cos ¯ β ≤ . α and ¯ β of Categories A and B respectively. Thus each phase could have been in either the first or the fourth quadrant with89 o ≤ | ¯ α | ≤ o and 87 o ≤ | ¯ β | ≤ o . The new requirement of matching the generated baryonasymmetry η A ( η B ) for Category A ( B ) with its observed value in the 3 σ range 5 . × − to 7 . × − [57]–[62] puts restrictions on sin 2 ¯ α (sin 2 ¯ β ) which fix both the magnitude andthe sign of ¯ α ( ¯ β ). To be specific in our numerical analysis, we choose x = M /M for thedifferent hierarchical cases as follows : (a) for NHN, x ≥
10, (b) for IHN, x ≤ .
1, (c) forQDN, 0 . ≤ x ≤
10. So far, we did not dwell on the mass ordering (normal or inverted) of18he right handed heavy neutrinos N i in the QDN case. For the normal ordering (NON) case,we take 1 . ≤ x ≤
10, while for an inverted ordering (ION), our choice is 0 . ≤ x ≤ .
9. Asmentioned earlier, the function f ( x ) is positive for 0 . ≤ x < . x = 1 which corresponds to the complete degeneracy of the N i , i.e. M = M = M since f ( x ) diverges at this point. The inclusion of finite widthcorrections to propagators of right handed neutrinos in the one loop decay diagrams avoidsthis problem. Now, both the previously divergent part of the modified f ( x ) and the leptonasymmetry vanish there. We also avoid the near x = 1 region, 0 . < x < .
1, to excludethe so called resonant leptogenesis [63] since that is not part of our scenario. Tables 1 – 3enumerate the emergent constraints on ¯ α , ¯ β in consequence of matching η A , η B for each Category A Parameters NHN IHN QDNNON ION¯ α ¯ α < α > α < α > . o − . o . o − . o . o − . o . o − . o x − . − . . − . . − . β −
60 2 − −
60 2 − . × . × × . × M lowest GeV — — — —4 . × . × . × . × Category B Parameters NHN IHN QDNNON ION¯ β ¯ β < β > β < β > . o − . o . o − . o . o − . o . o − . o x − . − . . − . β − −
12 2 −
60 2 − . × . × . × . × M lowest GeV — — — —8 . × . × . × . × Table 1: Allowed ¯ α , ¯ β and other parameters for unflavored leptogenesisof the eighteen different possibilities described earlier with corresponding restrictions on the19 ategory A Parameters NHN IHN QDNNON ION¯ α ¯ α < α > α < α > . o − . o . o − . o . o − . o . o − . o x − . − . . − . . − . β −
60 22 −
60 2 −
60 2 − . ×
23 10 M lowest GeV — — — —3 . × . × . × . × Category B Parameters NHN IHN QDNNON ION¯ β ¯ β < β > β < β > . o − . o . o − . o . o − . o . o − . o x − . − . . −
10 0 . − . β −
60 24 −
60 6 −
60 7 − . × . × . × . × M lowest GeV — — — —3 . × . × . × . × Table 2: Allowed ¯ α , ¯ β and other parameters for fully flavored leptogenesisparameters x , tan β and M lowest as shown. We would like to make the following commentson the information contained in tables 1 – 3.1. Signs of phase angles : We have a positive baryon asymmetry in our universe. Fromthe formulae for all NHN cases in the Apendices, we can say that sign of f ( x ) sin 2( ¯ α, ¯ β )has to be positive in order to generate such a positive asymmetry. But f ( x ) is negativein the NHN region of x ≥
10. So, ¯ α, ¯ β have to be negative for all NHN cases. Onthe contrary, for all IHN cases, there is an overall negative sign in the formulae for η since Im( h ) = Im( h ) here is opposite in sign to Im( h ) = Im( h ) that come infor the NHN case. So, for a positive η , a negative sign of f ( x ) sin 2( ¯ α, ¯ β ) is neededin all IHN cases. Again, f ( x ) is negative in the NHN region of x ≤ .
1. For this20 ategory A Parameters NHN IHN QDNNON ION¯ α ¯ α < α > α < α > . o − . o . o − . o . o − . o . o − . o x − . − . . − . . − . β −
60 2 −
60 2 −
60 2 − . ×
50 100 100 M lowest GeV — — — —4 . × . × . × . × Category B Parameters NHN IHN QDNNON ION¯ β ¯ β < β > β < β > . o − . o . o − . o . o − . o . o − . o x − . − . . − . . − . β −
60 2 −
60 2 −
60 2 − . × . × . × . × M lowest GeV — — — —2 . × . × . × . × Table 3: Allowed ¯ α , ¯ β and other parameters for τ -flavored leptogenesisreason, ¯ α, ¯ β are positive in all IHN cases. For QDN cases we need to discuss thepossibilities of normal and inverted ordering of M i separately. Here there are twoterms with f ( x ) and − f (1 /x ) along with an overall factor sin 2( ¯ α, ¯ β ). For the NONregion 1 . ≤ x ≤ . f ( x ) is negative while − f (1 /x ) is negative for 1 . ≤ x ≤ . . ≤ x ≤ .
5, ¯ α, ¯ β are required to be negative. For the remainingpart of the NON region 2 . ≤ x ≤ f ( x ) is negative and − f (1 /x ) positive but the f ( x ) term dominates over the − f (1 /x ) term. So, negative signs also are needed for¯ α, ¯ β , in the region 2 . ≤ x ≤
10. Thus all QDN cases with NON require negativesign of ¯ α, ¯ β . Again, for QDN with ION, both f ( x ) and − f (1 /x ) are positive in theregion 0 . ≤ x ≤ .
9. In the rest of the ION region 0 . ≤ x ≤ . f ( x ) is negativeand − f (1 /x ) is positive. But, now the latter term dominates over the former one. So,21ositive ¯ α, ¯ β are needed in all QDN cases with ION. In fact, we see (tables 1 – 3) thatfor all normal (both hierarchical and quasidegenrate) mass ordering cases of M i , thephases are negative whereas, for all inverted (both hierarchical and quasidegenrate)mass ordering cases, they are positive. One may also note that in all cases and regimesthe size of the allowed range of tan β is correlated with that of the phase ¯ α / ¯ β .2. Magnitudes of phase angles and other parameters : Neither ¯ α nor ¯ β can be strictly90 o since η then vanishes. Therefore, a nonzero η is incompatible in Category A withtribimaximal mixing which requires [16] ¯ α = π/
2. The numerical value of η is mostsensitive to the values of sin 2( ¯ α, ¯ β ), M lowest ( M for normal mass ordering, M forinverted mass ordering) and to some extent to the function f (and hence x ) for ac-ceptable ranges of k , k ( Category
A) and l , l ( Category B ). The latter are ofcourse restricted [16] by the neutrino oscillation data. For unflavored leptogenesis with M lowest > (1 + tan β )10 GeV, the minimum value of M lowest is 5 × GeV, whilewe cut the maximum value at 5 × GeV to avoid the GUT scale whereabouts allproduced asymmetry gets washed out by inflation. Such a large value of M lowest forcesa small value of sin 2( ¯ α, ¯ β ) in order to have the baryon asymmetry in the right range.In Category A , the range of | ¯ α | is restricted to 89 o ≤ | ¯ α | ≤ o so that sin 2 ¯ α is smallthere. In the IHN case of Category A , other associated factors including f (1 /x ) causefurther restrictions on ¯ α , cf. Table 1. In Category B the range 87 o ≤ | ¯ β | ≤ o iscurtailed to | ¯ β | > . o due to the large value of M lowest in flavor independent lepto-genesis except the QDN (NON) case where other factors are responsible for necessarysuppression.3. The quadrants of ¯ α , ¯ β do not change between unflavored, fully flavored and τ -flavoredleptogenesis, nor is there any dependence of them on the value of tan β . They onlydepend on whether N i have a normal ( M < M ) or inverted ( M > M ) massordering. For the former, ¯ α and ¯ β are always in the fourth quadrant ( <
0) since ǫ always has a minus sign in front, while the latter always forces them to be in the firstquadrant ( >
0) since ǫ = ǫ always has a plus sign in front.4. The constraints on sin 2 ¯ α , sin 2 ¯ β - extracted from η A,B - restrict the allowed intervalsfor | ¯ α | , | ¯ β | more stringently than do constraints on cos ¯ α , cos ¯ β obtained [16] fromneutrino oscillation phenomenology. 22 Effect of radiative µτ symmetry breaking While explaining a maximal value for θ , exact µτ symmetry predicts a vanishing θ .The latter will make the CP violating Dirac phase δ D unobservable in neutrino oscillationexperiments, many of which are being planned to study CP violation in the neutrino sector.Thus it may be desirable to have a nonzero θ , however small.Suppose µτ symmetry is exact at a high energy Λ ∼ GeV characterizing the heavyMajorana neutrino mass scale. Running down to a laboratory scale λ ∼ GeV, via one-loop renormalization group evolution, one picks up small factorizable departures from µτ symmetry, induced by charged lepton mass terms, in the elements of the light neutrino massmatrix m ν . These cause small departures from 45 o in θ λ and tiny nonzero values for θ λ .Neglecting m µ,e in comparison with m τ , one obtains [16] that m λν ≃ − ∆ τ m Λ ν − ∆ τ , (6 . m Λ ν is µτ symmetric and the deviation ∆ τ is given in MSSM by∆ τ ≃ m τ π v (1 + tan β )ln Λ λ ≃ × − (1 + tan β ) . (6 . τ , the phenomenological consequences of eq.(6.1), derived from extant neutrino oscillation data, were worked out in ref.[15]. The allowedregions in the k − k ( l − l ) plane for Category A ( B ) get slightly extended. Moreover,one finds that θ λ ≤ o as well as 0 ≤ θ λ ≤ . o for Category A and 45 o ≤ θ λ as well as0 ≤ θ λ ≤ . o for Category B . The upper bounds on θ λ in both categories correspond totan β = 60.RG evolution from Λ to λ has no direct effect on the baryon asymmetry η . The leptonasymmetry Y l , produced at the heavy Majorana neutrino mass scale, remains frozen till thetemperature comes down to the weak scale where it is converted to η . The requirementof the latter being in the observed range leads to correlated constraints on x , M lowest andtan β , vide tables 1 – 3. While the constraints on x and M lowest have some effects on themagnitude of Λ, they are numerically quite weak. Such is, however, not the case with thetan β constraints, owing to eq. (6.2). In particular, the bounds on θ λ can be significantlyaffected by restrictions on tan β . 23et us discuss the consequent effects on the said bounds in the three regimes.(1) Flavor independent leptogenesis. Here tan β can go from 2 to 60, as taken in Ref.[15],for the NHN and QDN cases of Category A and the QDN (NON) case of Category B , cf.Table 1. Therefore the range of θ λ remains unchanged for those cases. But the strongerrestrictions on tan β given in Table 1 for the IHN case of Category A and the NHN, IHN andQDN (ION) cases of Category B force the corresponding θ λ and θ λ to be practically equalto 0 o and 45 o respectively for those two situations.(2) Fully flavored leptogenesis. We can deduce from the information given in table 2 thatthe ranges of θ λ are affected here for either category in each case. The results are given intable 4. Category A Category B NHN IHN QDN NHN IHN QDNtan β −
60 2 −
60 16 −
60 24 −
60 6 −
60 (NON)7 −
60 (ION) θ λ . ◦ − . ◦ . ◦ − . ◦ ◦ − . ◦ . ◦ − . ◦ . ◦ − . ◦ ◦ − . ◦ Table 4: Effect on θ λ of the more restricted range of tan β in fully flavored leptogenesis.(3) τ -flavored leptogenesis. There is no additional restriction on tan β here as comparedwith unflavored leptogenesis, vide table 3. Hence the ranges of θ λ stand unchanged in eithercategory for the NHN, IHN and QDN cases.Now that there is a nonzero θ λ , one has CP violation in the neutrino sector which canbe measured from the difference in oscillation probabilities P ( ν µ → ν e ) − P ( ¯ ν µ → ¯ ν e ) [64].For the CKM CP phase δ λ , we find the 3 σ range of its value to be 1 . o ≤ δ λ ≤ o ( Category A ) and 1 . o ≤ δ λ ≤ o ( Category B ) for both flavored and unflavored leptogenesis in allregimes. The sign of δ λ is opposite to the sign of ¯ α / ¯ β for Category A / B and hence it doeschange from one regime for M lowest (1 + tan β ) − to another for a given mass ordering of N i .24 Conclusion
In this paper we have studied the generation of the observed amount of baryon asymmetry η in our scheme of µτ symmetric four zero neutrino Yukawa textures within the type-I seesaw.For each of the two categories A and B of our scheme, we have identified three regimesdepending on the value of M lowest (1 + tan β ) − and have studied the normal-hierarchical(NHN), inverted-hierarchical (IHN) and quasidegenerate (QDN) cases for the masses of theheavy Majorana neutrinos N i . The requirement of matching the right value of η forces thephases ¯ α ( Category A ) and ¯ β ( Category B ) to be in the fourth quadrant for the NHN andQDN cases and in the first quadrant for the IHN case in each regime. Restrictions on smallbut nonzero θ , arising from radiative µτ symmetry breaking, have also been worked out. We thank K. S. Babu for suggesting this investigation. P. R. has been supported by a DAERaja Ramanna fellowship.
Note added
A new paper on supersymmetric leptogenesis appeared [65] after this work was completed.The authors of ref. [65] have highlighted certain additional contributions to Y ∆ . Thesearise from soft supersymmetry breaking effects involving gauginos and higgsinos as well asanomalous global symmetries causing a different pattern of sphaleron induced lepton flavormixing. While some of the numerical coeffcients – given in the various expressions for η inour analysis – are likely to change if these effects are included, their overall signs will not.Consequently, there will be no alteration in our conclusions on the quadrants of the phases¯ α and ¯ β which remain robust. 25 Baryon Asymmetry in flavor independent leptogen-esis
Category
A η
NHNA ≃ . × − M GeV k sin 2 ¯ αX / A sin β M M f (cid:16) M /M (cid:17) × .
46 sin βX / A k + . k X / A sin β ! . − . (A.1) η IHNA ≃ − . × − M GeV k k sin 2 ¯ α (1 + k ) X / A sin β M M f (cid:16) M /M (cid:17) × . k ) − X / A sin β +
28 (1 + k ) X / A sin β ! . − . (A.2) η QDNA ≃ . × − k sin 2 ¯ αX / A sin β ( M GeV M M f (cid:16) M /M (cid:17) × . k ) − X / A sin β + . k ) X / A sin β ! . − − M GeV k (1 + k ) M M f (cid:16) M /M (cid:17) × . k ) − X / A sin β + . k ) X / A sin β ! . − ) . (A . Category Bη NHNB ≃ . × − M GeV l sin 2 ¯ β ( l + 2 l ) X / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin β ( l + 2 l ) + . l + 2 l ) X / B sin β ! . − . (A . IHNB ≃ − . × − M GeV l sin 2 ¯ βX / B sin β f (cid:16) M /M (cid:17) × . X / B sin β + . X / B sin β ! . − . (A . η QDNB ≃ . × − ( M GeV l sin 2 ¯ β ( l + 2 l ) X / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin β ( l + 2 l ) + . l + 2 l ) X / B sin β ! . − − M GeV l sin 2 ¯ βX / B sin β f (cid:16) M /M (cid:17) × . X / B sin β + . X / B sin β ! . − ) . (A . B Baryon Asymmetry in fully flavored leptogenesis
Category
A η
NHNA ≃ . × − M GeV k sin 2 ¯ αX / A sin β M M f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − . (B . η IHNA ≃ − . × − M GeV k k sin 2 ¯ α (1 + k ) X / A sin β f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − . (B . QDNA ≃ . × − ( M GeV k sin 2 ¯ αX / A sin β M M f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − − M GeV k k sin 2 ¯ α (1 + k ) X / A sin β f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − ) . (B . Category Bη NHNB ≃ . × − M GeV l sin 2 ¯ β ( l + 2 l ) X / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin βl + . l X / B sin β ! . − . (B . η IHNB ≃ − . × − M GeV l sin 2 ¯ βX / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin β + . X / B sin β ! . − . (B . η QDNB ≃ . × − ( M GeV l sin 2 ¯ β ( l + 2 l ) X / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin βl + . l X / B sin β ! . − − M GeV l sin 2 ¯ βX / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin β + . X / B sin β ! . − ) . (B . Baryon asymmetry in τ -flavored leptogenesis Category
A η
NHNA ≃ . × − M GeV k sin 2 ¯ αX / A sin β M M f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − . (C . η IHNA ≃ − . × − M GeV k k sin 2 ¯ α (1 + k ) X / A sin β M M f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − + . X / A sin β (1 + k ) + . k ) X / A sin β ! . − . (C . η QDNA ≃ . × − k sin 2 ¯ αX / A sin β ( M GeV M M f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − − M GeV k (1 + k ) M M f (cid:16) M /M (cid:17) × . X / A sin βk + . k X / A sin β ! . − + .
08 sin βX / A (1 + k ) + . k ) X / A sin β ! . − ) . (C . Category Bη NHNB ≃ . × − × M GeV l sin 2 ¯ β ( l + 2 l ) X / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin β ( l + l ) + . l + l ) X / B sin β ! . − + . X / B sin βl + . l X / B sin β ! . − . (C . η IHNB ≃ − . × − M GeV l sin 2 ¯ βX / B sin β M M f (cid:16) M /M (cid:17) × . X / B sin βl + . l X / B sin β ! . − + . X / B sin β + . X / B sin β ! . − . (C . QDNB ≃ . × − l sin 2 ¯ βX / B sin β M GeV 1( l + 2 l ) M M f (cid:16) M /M (cid:17) × . X / B sin β ( l + l ) + . l + l ) X / B sin β ! . − + . X / B sin βl + . l X / B sin β ! . − − M GeV M M f (cid:16) M /M (cid:17) × . X / B sin βl + . l X / B sin β ! . − + . X / B sin β + . X / B sin β ! . − ! . (C . References [1] M. Fukugita and T. Yanagida,
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