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Seesaw Realization of Bi-Large Mixing and Leptogenesis
Nuclear Physics B (Proc. Suppl.) 137 (2004) 236–246 www.elsevierphysics.com
Seesaw Realization of Bi-Large Mixing and Leptogenesis M. Tanimotoa∗ a
Department of Physics, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan The seesaw realization of the bi-large mixing is presented based on the texture zeros of the neutrino mass matrix. By using the obtained Dirac neutrino mass matrix, the leptogenesis is discussed. In the framework of the texture zeros, SO(10)-like model is investigated by focusing on the neutrino masses and mixings.
1. Introduction Neutrino experiments by Super-Kamiokande [1,2] and SNO[3] have brought us an outstanding fact on the neutrino oscillation. Recent results from KamLAND have almost confirmed the large neutrino mixing solution that is responsible for the solar neutrino problem nearly uniquely [4]. We have now common information concerning the neutrino mass difference squared (∆m2atm , ∆m2sun ) and neutrino flavor mixings (sin2 2θatm and tan2 θsun ) [5]. It is remarked that the neutrino mixing is the bi-large and the ratio ∆m2sun /∆m2atm is ∼ λ2 with λ 0.2. A constraint has also been placed on the third mixing angle from the reactor experiment of CHOOZ [6]. The textures with zeros of the neutrino mass matrix have been discussed [7–10] to explain these neutrino masses and mixings [11]. It was found that the two-zero textures are consistent with the experimental data in the basis of the diagonal charged lepton mass matrix [12]. Consequently, the neutrino mass matrix does not display the hierarchical structure as seen in the quark mass matrix [13–19]. Since these textures are given for the light effective neutrino mass matrix, one needs to find the see-saw realization [20] of these textures from the standpoint of the model building. Without fine tunings between parameters of the Dirac neutrino mass matrix mD and the right-handed Majorana neutrino one MR , we obtained several textures of ∗ This work is collaborated with M. Honda, S. Kaneko, M. Bando and M. Obara.
0920-5632/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2004.10.073
mD for the fixed MR [21]. Among them, there are textures of mD and MR which have hierarchical masses without large mixings. These present the seesaw enhancement of mixings, because there is no large mixings in mD and MR , but the bi-large mixing is realized via the seesaw mechanism. The seesaw enhancement are important in the standpoint of the quark-lepton unification, in which quark masses are hierarchical and quark mixings are very small. The general discussions of the seesaw enhancement were given in the case of two flavors [22,23]. Specific cases were discussed in the case of three flavors [24,25] because it is very difficult to get general conditions for the seesaw enhancement of the bi-large mixing. However, the two-zero texture of the neutrino mass matrix Mν are helpful to study the seesaw enhancement of the bi-large mixing. In this study, we present sets of mD and MR to give the seesaw enhancement in the two-zero textures of Mν and discuss the related phenomena, the leptogenesis [26]. Texture zeros of the neutrino mass matrix are presented in the section 2, and the seesaw enhancement of the bi-large mixing are discussed in the section 3. Based on the seesaw enhancement, the thermal leptogenesis is discussed in the section 4. The section 5 is devoted to the SO(10)-like model and the section 6 is devoted to the summary.
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2. Texture Zeros There are fifteen two-zero textures for the neutrino mass matrix Mν , which have five independent parameters. Among these textures, seven acceptable textures with two independent zeros were found for the neutrino mass matrix [12]. Seven two-zero textures of neutrino mass matrix in the basis wherein the charged lepton mass matrix is diagonal. A1
B1
B3
C
0 0 × × × 0 × 0 × × × ×
0 × × × 0 × 0 0 × × 0 ×
× ×, × 0 ×, × × ×, × × × 0
0 A2 × 0 × B2 0 × × B4 × 0
× × × 0 × × × × ×
0 × × × × 0 0 × 0
×: non−zero entry
c12 c13 −c12 s23 s13 − s12 c23 eiδ −c12 c23 s13 + s12 s23 eiδ
s12 c13 −s12 s23 s13 + c12 c23 eiδ −s12 c23 s13 − c12 s23 eiδ
c12 s23 iδ e + s13 s12 c23
which lead to 1 A1 : |Ue3 | tan 2θ12 cot θ23 Rν cos 2θ12 2 1 A2 : |Ue3 | tan 2θ12 tan θ23 Rν cos 2θ12 2 m1 ≥ tan2 θsun min 0.3 . m2
(4)
(5) (6)
where m0 denotes a constant mass and λ 0.2. 3. Seesaw Enhancement of Bi-large Mixing (1)
where and U is parametrized as follows:
=
s13 − 2 c13
The relative magnitude of each entry of the neutrino mass matrix is roughly given for the textures A1 and A2 as follows: 0 0 λ 0 λ 0 Mν m0 0 1 1 , m0 λ 1 1 (7) λ 1 1 0 1 1
Especially, the textures A1 and A2 of ref.[12], which correspond to the hierarchical neutrino mass spectrum, are strongly favored by the recent phenomenological analyses [13,14,19]. Therefore, the two textures are taken in order to discuss the seesaw enhancement in the next section. The neutrino mass matrix is written in terms of the neutrino mixings and neutrino masses as λ1 0 0 (2) Mν = U 0 λ2 0 U T 0 0 λ3 λ1 = m1 e2iρ , λ2 = m2 e2iσ , λ3 = m3
λ2 λ3
(3) s13 e−iδ
s23 c13 c23 c13
For the A1 type, (Mν )ee = (Mν )eµ = 0, we have λ1 s13 s12 s23 iδ = + 2 e − s13 λ3 c13 c12 c23
The neutrino matrix is given in terms of the Dirac neutrino mass matrix mD and the righthanded Majorana neutrino mass matrix MR by the the seesaw mechanism as [20] Mν = mD MR−1 mT D .
(8)
Zeros in mD and MR provide zeros in the neutrino mass matrix Mν of eq.(7) as far as we exclude the possibility that zeros are originated from accidental cancellations among matrix elements. In other words, we take a standpoint that the twozero texture should come from zeros of the Dirac neutrino mass matrix and the right-handed Majorana mass matrix. Possible textures of mD and MR were given in ref. [21]. Among them, we select the set of mD and MR , which reproduce the seesaw enhancement of the bi-large mixing. Let us fix the right-handed Majorana neutrino mass matrix without large mixings. We take simple right-handed Majorana neutrino mass matrix with only three independent parameters. Then, there are 6 C3 = 20 textures. Among them, six textures are excluded because they have a zero eigenvalue, which corresponds to a massless righthanded Majorana neutrino. Other two textures are also excluded because the two-zero textures
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A1 and A2 cannot be reproduced without accidental cancellations. One of the two textures is the diagonal matrix, and another one is the matrix with three zeros in the diagonal elements. We show twelve real mass matrices with three independent parameters with mass eigenvalues |M1 | = λm M3 and |M2 | = λn M3 , where M3 is the mass of the third generation, and m and n are integers with m > n > 1: m −1 0 λ 2 M3 0 λn 0 ai : M R m λ2 0 0 a1 m+n n 0 −λ 2 λ 2 n M3 −λ 2 1 0 m+n 0 0 λ 2 a2 m 2 0 0 λ M3 0 λn 0 m 2 λ 0 −1 a3 m+n 0 0 λ 2 n M3 0 1 −λ 2 (9) m+n n λ 2 −λ 2 0 a4
bi : MR
−λn m+n M3 λ 2 0 0 m+n M3 λ 2 n −λ 2 0 m+n M3 λ 2 0 0 m+n M3 λ 2 0
ci : M R
λm M3 0 0 1 n M3 −λ 2 0 m λ M3 0 0
λ
m+n 2
0 0 λ
m+n 2
0 0 m+n
λ 2 −λn 0 λ
m+n 2
0 n −λ 2 0 −1 n λ2
0 0 1 b1
n −λ 2 0 1 b2 0 0 1 b3 0 n −λ 2 (10) 1 b4
0 n λ2 0 c1 n
−λ 2 0 m+n λ 2 0 0 n λ2
0
λ
m+n 2
0
0 n λ2 −1 c3
c2
1 M3 0 n −λ 2
0 0
λ
m+n 2
n −λ 2 m+n λ 2 (11) 0 c4
where there are no large mixings in twelve matrices since the mass eigenvalues are supposed to be hierarchical. The minus signs in the matrix elements are taken to reproduce signs in the texture A1 and A2 of eq.(7). There are several Dirac neutrino mass matrices to give the textures A1 and A2 in eq.(7) [21]. We show Dirac neutrino mass matrices (mD )ai , (mD )bi , (mD )ci (i = 1 ∼ 4) with maximal number of zeros, which have no large mixings, to give the texture A2 . For the texture A1 , we easily obtain the Dirac neutrino mass matrices by exchanging the second and third rows. For each matrix of (MR )ai , (MR )bi , (MR )ci those are given as follows: λ 0 0 m ai : m D mD0 0 0 λ 2 n 1 λ2 0 a1 n +1 0 0 λ2 m mD0 0 0 λ2 0 1 0 a2 0 0 λ m mD0 λ 2 0 0 n 0 λ 2 1 a3 n 0 0 λ 2 +1 m mD0 λ 2 0 0 (12) 0 1 0 a4
bi :mD
n
λ 2 +1 0 m 0 λ2 mD0 n 0 λ2 n +1 0 λ2 m mD0 0 λ2 0 0 n 0 λ 2 +1 m mD0 λ 2 0 n 0 λ2 n 0 λ 2 +1 m mD0 λ 2 0 0 0
0 0 1 b 1 0 0 1 b2 0 0 1 b 3 0 0 1 b4
(13)
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0 λ 0 n 0 λ2 mD0 0 m λ 2 1 0 c1 n 0 0 λ 2 +1 m 0 λ2 mD0 0 1 0 0 c 2 0 0 λ n mD0 0 λ 2 0 m 0 1 c λ2 3 n +1 2 0 0 λ m mD0 0 λ 2 0 1 0 0 c4
ci :mD
0 m mD0 λ 2 0 0 m mD0 λ 2 0
ci :mD (14)
(15) bi :mD
n
λ 2 +1 λx mD0 n λ2 n +1 λ2 mD0 λx λy
0 m λ2 0 0 m λ2 0
0 λy 1 b3 0 λz 1 b4
x > n2 y>0 x > n2 y > n2 z>0 (16)
4
n
λ 2 +1 λx λy
where mD0 denotes the magnitude of the Dirac neutrino mass and complex coefficients of order one are omitted. Although these matrices have no large mixing among three families, the neutrino mass matrix Mν has the bi-large mixing through the seesaw mechanism. These are so called seesaw enhancement of the bi-large mixing. These Dirac matrices are ones with maximal number of zeros. Without changing the mixings and the mass eigenvalues in the leading order, some zeros can be replaced with small non-zero entries as follows: λ 0 0 m x>0 mD0 λx λy λ 2 ai :mD n y > n2 1 λ2 0 a1 n +1 λ2 0 0 x > n2 m x z mD0 λ λ λ 2 y > n2 y λ 1 0 z>0 a 2 0 0 λ m x > n2 mD0 λ 2 λx λy n y>0 0 λ2 1 a3 n 0 0 λ 2 +1 x > n2 m z x λ y > n2 mD0 λ 2 λ y 0 1 λ z>0 a
n
λ 2 +1 λx n λ2
0 λy 1 b 1 0 λz 1 b2
x > n2 y>0 x > n2 y > n2 z>0
0 λ 0 n x> m 2 mD0 λx λy λ 2 m y > 0 λ2 1 0 c1 n +1 0 x>0 0 λ2 m λy λ 2 y > n2 mD0 λx 1 λz 0 z > n2 c2 0 0 λ n x> m 2 mD0 λx λ 2 λy m y>0 2 λ 0 1 c3 n x>0 0 0 λ 2 +1 m λy y > n2 mD0 λx λ 2 1 0 λz z > n2 c4 (17)
where x, y and z are positive integers. These Dirac neutrino mass matrices are asymmetric ones. However, only the b3 and b4 textures in eq.(16) are adapted to the symmetric texture in the SO(10)-like GUT if y, m and n are relevantly chosen [16,17]. For example, in the b3 case, taking y = n/2 and m = n + 2, the symmetric mass matrix is given, especially, putting n = 8, we have the hierarchical mass matrix such like the up-quark mass matrix. 4. Thermal Leptogenesis Let us examine our textures in the leptogenesis [27–29], which is based on the Fukugita-Yanagida mechanism [26]. The CP violating phases in the Dirac neutrino mass matrix are key ingredients for the leptogenesis while the right-handed Majorana neutrino mass matrix are taken to be real in eqs.(9), (10) and (11). Although the nonzero entries in the Dirac neutrino mass matrix are complex, three phases are removed by the re-definition of the left-handed neutrino fields. There is no freedom of re-definition for the righthanded ones in the basis with the real MR . We
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should move to the diagonal basis of the righthanded Majorana neutrino mass matrix in order to calculate the magnitude of the leptogenesis. Then, the Dirac neutrino mass matrices m ¯ D in the new basis is given as follows:
where v2 = v sin β with v = 174GeV. The lepton asymmetry YL is related to the CP asymmetry through the relation
m ¯ D = PL mD OR
where s denotes the entropy density, g∗ is the effective number of relativistic degrees of freedom contributing to the entropy and κ is the so-called dilution factor which accounts for the washout processes (inverse decay and lepton number violating scattering). In the MSSM with righthanded neutrinos, one gets g∗ = 232.5. The produced lepton asymmetry YL is converted into a net baryon asymmetry YB through the (B + L)-violating sphaleron processes. One finds the relation [31]
(18)
where PL is a diagonal phase matrix and OR is the orthogonal matrix which diagonalizes MR as T OR MR OR in eqs.(9), (10) and (11). Since the phase matrix PL can remove one phase in each row of mD , three phases disappear in m ¯ D. As a typical example, we show the case of the b3 texture in eq.(10). By taking three eigenvalues of MR as follows 2 : M1 = λm M3 ,
M2 = −λn M3 .
(19)
YL =
We obtain the orthogonal matrix OR as cos θ sin θ 0 OR = − sin θ cos θ 0 , tan2 θ = λm−n (20) 0 0 1 Then the Dirac mass matrices mD of b3 in eq.(13) can be parameterized in the new basis as follows: n 0 λ 2 +1 0 m 0 0 OR (21) m ¯ D = mD0 λ 2 n 0 λ 2 eiρ 1 where only one phase ρ remains. The magniby the relation m2D0 tude of mD0 is determined 2 m0 M3 , where m0 ∆matm /2. We examine the lepton number asymmetry in the minimal SUSY model with the right-handed neutrinos. In the limit M1 M2 , M3 , the lepton number asymmetry 1 (CP asymmetry) for the lightest heavy Majorana neutrino (N1 ) decays into l∓ φ± [30] is given by
2 The
Γ1 − Γ1 1 = Γ1 + Γ1 Im[{(m ¯ †D m ¯ D )12 }2 ] M1 3 − 8πv22 M2 (m ¯ †D m ¯ D )11 Im[{(m ¯ †D m ¯ D )13 }2 ] M1 + † M3 (m ¯ Dm ¯ D )11
(22)
minus sign of M2 is necessary to reproduce MR in eq.(10). This minus sign is transfered to mD by the righthanded diagonal phase matrix diag(1, i, 1).
1 nL − nL¯ =κ s g∗
ξ YL ξ−1 8 Nf + 4 NH ξ= 22 Nf + 13 NH
(23)
YB = ξ YB−L =
(24)
where Nf and NH are the number of fermion families and Higgs doublets, respectively. Taking into account Nf = 3 and NH = 2 in the MSSM, we get 8 YL . (25) 15 On the other hand, the low energy CP violation, which is a measurable quantity in the long baseline neutrino oscillations [32], is given by the Jarlskog determinant JCP [33], which is calculated by
YB = −
det[M M† , Mν Mν† ] = −2iJCP (m2τ − m2µ )(m2µ − m2e )(m2e − m2τ ) ×(m23 − m22 )(m22 − m21 )(m21 − m23 )
(26)
where M is the diagonal charged lepton mass matrix, and m1 , m2 , m3 are neutrino masses. Since the CP violating phase is only ρ, we can find a link between the leptogenesis (1 ) and the low energy CP violation (JCP ) in our textures of the Dirac neutrinos. By using the Dirac neutrino mass matrix in eq.(21), we get 1
−
3m2D0 m λ sin 2ρ 8πv22
M. Tanimoto / Nuclear Physics B (Proc. Suppl.) 137 (2004) 236–246
JCP
−8.8 × 10−17 M1 sin 2ρ 1 2 ∆m2atm sin 2ρ λ 64 ∆m2sol
(27)
where M1 is given in the GeV unit and tan β ≥ 10 is taken. It is remarked that 1 only depends on M1 and the phase ρ, and the relative sign of 1 and JCP is opposite. Taking the experimental data ∆m2sol /∆m2atm λ2 and sin 2ρ 1, we predict JCP 0.01, which is rather large and then is favored for the future experimental measurement. The five cases of the Dirac neutrino mass matrix (a1 , a3 , b1 , c1 , c3 ) in eqs.(12), (13) and (14) lead to same results in eq.(27). In other six cases of the Dirac neutrino mass matrix (a2 , a4 , b2 , b4 , c2 , c4 ), the CP violating phases are removed because of only three non-zero entries. Then, we get 1 = 0, but the same result in eq.(27) for JCP . If we use the modified Dirac neutrino mass matrices in eqs.(15), (16) and (17), new CP violating phases appear. However, the contribution to 1 is a next-leading one as far as x 1, y 1, z 1. In order to calculate the baryon asymmetry, we need the dilution factor involves the integration of the full set of Boltzmann equations [34]. A simple approximated solution which has been frequently used is given by [35] −0.6 −3 10 eV m ˜1 ln −3 κ = 0.3 m ˜1 10 eV (10−2 eV ≤ m ˜ 1 ≤ 103 eV)
(28)
where (m ¯ †D m ¯ D )11 . (29) M1 By using this approximate dilution factor and eqs.(23) and (24), we can estimate YB in our textures as follows:
m ˜1 =
YB −2.3 × 10−3 1 κ .
(30)
It is noticed that YB and JCP are same sign since 1 has minus sign. The WMAP has given the new result [36] +0.4 ηB = 6.5−0.3 × 10−10 (1 σ)
which leads to 1 YB ηB . 7
(31)
(32)
241
¯ D )11 = m2D0 λm , In our textures, we have (m ¯ †D m 1 2 which gives m ˜ 1 = 2 ∆matm 0.022. Then we get the dilution factor κ 7 × 10−3 . Putting the observed value into eq.(30), we get M1 sin 2ρ 6 × 1010 GeV .
(33)
This result means that M1 is should be larger than 6 × 1010 GeV in order to explain the baryon number in the universe. This value is consistent with previous works [27–29]. It is important to present the discussion from the standpoint of the GUT, which is given after eq.(16). Taking n = 8 and m = 6 in the b3 case of eq.(16) as in the previous discussion, one obtains M3 ∼ 1015 GeV and M1 ∼ 108 GeV taking account of ∆m2atm 2 × 10−3 eV2 . This result does not satisfy the condition of eq.(33). However, the simple SO(10) fermion mass relation may be consistent with the leptogenesis in the case of the more complicated texture of MR , which leads to the two-zero texture A2 , as seen in the work of [37]. We add the discussion of another important problem. In the framework of supersymmetric thermal leptogenesis, there is cosmological gravitino problems. The gravitino with a few TeV mass does not favor M1 ≥ 1010 GeV [38], because M1 should be lower than the maximum reheating temperature of the universe after inflation. In order to keep the thermal leptogenesis in the SUSY model, we may consider the gravitino with O(100)TeV mass, which is derived from the anomaly mediated SUSY breaking mechanism [39]. 5. SO(10)-like model Let us discuss the texture zeros in the SO(10) framework. We take the A2 type of neutrino mass matrix at GUT scale as a minimal model including a phase φ: 0 β 0 ¯ ¯ h Mν = mν β¯ α (34) 0 ¯ h 1 β O(λ) 0 β 0 = mν PνT β eiφ α h Pν , α O(1) h O(1) 0 h 1
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¯ h, ¯ being made positive real numbers, with α, ¯ β, α, β, h by factored out the phases by the diagonal phase matrix Pν . Let us see how the parameters appearing in the neutrino mass matrix at GUT scale are generally constrained from the present experimental neutrino data. For a moment forget about how to derive the parameters of Mν and just see how the parameter regions of h and φ are constrained from the experimental data of sin2 2θatm , tan2 θsun and the ratio of ∆m2sun to ∆m2atm in terms of four parameters α, β, h and φ. To make numerical calculation more strictly, we must take account of the contributions from the charged lepton side. We take the symmetric charged lepton mass matrix in terms of the real matrix (M l )RL and further diagonalized to Mldiag. by Ol [7]: (Ml )RL = PlT (M l )RL Pl , OlT M l Ol = Mldiag. → OlT (PlT )−1 Ml Pl−1 Ol ≡ Mldiag. .
(35)
We use the following symmetric matrix having 2zeros for M l , √ me mµ 0 0 √ √ m me mτ (36) (M l )RL me mµ √ µ 0 me mτ mτ
mass matrices at mZ and MR [40]: 1 0 0 1−e 1 ˜ ν (MR ) × ˜ ν (MZ ) = 0 0M M 1−µ 0 0 1 1 0 0 1−e 1 0 0 (39) 1−µ
0
0
1
˜ ν is the neutrino mass matrix on the bawhere M sis where charged lepton matrix is diagonalized (see eq. (37)). The renormalization factors e and µ depend on the ratio of VEV’s, tan βv . By using the form of eq. (39) we search the region of the parameter set (α, β, h, φ, σ, ρ) which are allowed by experimental data within 3σ. It is found that the phase factor φ should not become large, (|φ| ≤ 70◦ ). This may be important since we have never had the information of the phases appearing in Mν , which is connected to the leptogenesis. Let us explore an example of the allowed region of the parameters in (α, β) plane for the typical value h = 1.3. The allowed region which is consistent with the experimental data is shown in Fig.1, where β is allowed to be in both negative and positive.
where me , mµ , mτ are charged lepton masses at MGUT scale. Here, we ignore the RGE effect from MGUT to MR scale considering that it almost does not change the values of masses for quarks and leptons. On the basis where the charged lepton mass matrix is diagonalized, the neutrino mass matrix at MR scale is obtained as ˜ ν (MR ) = OlT (P −1 )T PνT M ν (MR )Pν P −1 Ol (37) M l l where
0 β 0 M ν (MR ) = β eiφ α h mν 0 h 1 1 0 0 0 . Q ≡ Pν Pl−1 = 0 e−iρ 0 0 e−iσ
(38)
In order to compare our calculations with experimental results, we need the neutrino mass matrix at MZ scale, which is obtained from the following one-loop RGE’s relation between the neutrino
Figure 1. The allowed region on the α − β plane in the case of h = 1.3
So far we have investigated the region of the parameters appearing in the neutrino mass matrix and shown that the parameter region is restricted
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within narrow range by the present experimental data. Here we make a comment whether or not a certain GUT model is consistent with the bi-large mixing with present neutrino mass differences. As an example, let us take a concrete model [17] with the simplest form of right-handed neutrino mass matrix with the phase-factored out diagonal matrix, PR , 0 0 M1 0 0 PR MR = PRT M1 0 0 M2 0 r 0 ≡ mR PRT r 0 0 PR . (40) 0 0 1 This, with the form of 4-zero texture form of MνD , yields also texture-zeros of A2 type with the phase factored out by (MνD )RL = PνTD (M νD )RL PνD , 0 a 0 M νD = a b c mνD → 0 c 1 a2 0 0 r m2 2 ab a νD (41) 2 Mν = a 2 + c c( + 1) r r r mR 1 0 c( ar + 1) where a and c are real numbers and b is complex one. We recognize that, in order to get large mixing angle θ23 , the 23 element must be of the same order as the 33 element, namely c( ar + 1) ∼ 1. Since c 1, ca/r must be of order 1. Thus approximate form of Mν is 2 0 β 0 m2 β ∼ ar Mν ∼ β eiφ α h νD , (42) α ∼ 2ab r mR 0 h 1 h ∼ ca r which clearly shows that none of a, b, c is zero, namely 6-zero texture are already excluded by the experimental neutrino data 3 . Now, one example of the symmetric 4-zero texture with the PatiSalam symmetry [17] provides us with the Dirac neutrino mass matrix at the MGUT scale under a simple assumption of the following Higgs config3 Here, we note that the 6-zero textures for the quark sector have been already ruled out by Ramond, Roberts and Ross [41].
urations:
M νD
0 126 0 10 → MU = 126 10 0 10 126 √ mu mc 0 0 −3 mt √ m m mu c eiφ m −3 mut c mt mt mu 0 −3 mt (43)
in mt unit. By comparing eq. (42) and eq. (43) the parameters α, β are expressed in terms of up-quark masses at the GUT scale. Thus, we can predict α, β from the up-quark masses at the GUT scale, mu = 0.36 ∼ 1.28MeV, mc = 209 ∼ 300MeV, mt = 88 ∼ 118GeV, which are obtained taking account of RGE’s effect to the quark masses at the EW scale [42]. Since the light quark masses are ambiguous because of the non-perturbative QCD effect, the allowed mass region of mu may be enlarged. In the case of mu = 0.36 ∼ 2.56 MeV, we obtain the overlapped region around α 1.24 and β −0.2 with h = 1.3 as seen figure 2. The allowed region on the α − β plane in the case of h = 1.3, which is predicted from a neutrino mass matrix with two zeros. The allowed region of the parameters are very narrow as follows: α = 1.23 ∼ 1.24, β = −0.199 ∼ −0.197 π 7 11 π ∼ , ρ= π∼ π (44) φ=− 18 18 9 9 where h = 1.3 is taken. On the other hand, our results are almost independent of the phase parameter σ. Hereafter we take σ = 0 in our calculations. In these parameters, we can predict Ue3 by including the contribution of the charged lepton sector. Here we stress that Ue3 is crucial to discriminate various models, therefore, we must be careful to estimate it by taking account of the effect of charged lepton mixings as well as CP violating phases. Our formula has already included these contributions. By taking the overlapped region of α and β in Fig.2, we present the prediction of |Ue3 |, JCP and < mee > as follows: |Ue3 | = 0.010 − 0.048 , | < mee > | 2.7meV
|JCP | ≤ 9.6 × 10−3 (45)
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6. Summary
Figure 2. The predicted region (gray region) of the α − β plane, in which h = 1.3 and mu = 0.36 ∼ 2.56 MeV are taken. The black region is the experimentally allowed region predicted from a neutrino mass matrix
where < mee > is the effective neutrino mass in the neutrinoless double beta decay. We hope |Ue3 | can be checked by the neutrino experiments in near future. Since the overlapped region of α and β is restricted in the narrow region, we can predict a set of typical values of neutrino masses and mixings at h = 1.3 as follows; sin2 2θµτ ∼ 0.98, tan2 θµe ∼ 0.28 mν3 ∼ 62 meV, mν2 ∼ 7.5 meV mν1 ∼ 1.4 meV
(46)
with mR = 3.0 × 1015 GeV and r × mR = 1.0 × 109 GeV, which correspond to the Majorana mass for the third generation and those of the second and first generations, respectively. On the other hand, mu 2.56 MeV should be allowed at the GUT scale. Now that our neutrino mass matrix is determined almost uniquely from the up-quark masses at GUT scale, we can make the prediction of leptogenesis once we fix the CP violating phases. Interesting enough is that our form of MR of eq.(40) yields naturally two degenerate Majorana masses with mass r × mR ∼ 109 GeV. In such case the leptogenesis is enhanced by the so-called ”crossing effect” [43].
We have discussed texture zeros with the seesaw enhancement. These textures are important in the standpoint of the quark-lepton unification, in which quark masses are hierarchical and quark mixings are very small. It is very difficult to get general conditions for the seesaw enhancement of the bi-large mixing, however, the two-zero textures of the left-handed neutrino mass matrix Mν are helpful to study the seesaw enhancement of the bi-large mixing. Once the basis of the righthanded Majorana neutrino mass matrix is fixed, one can find some sets of mD and MR , which have hierarchical masses without large mixings, to give the two-zero textures A1 and A2 without fine tuning among parameters of these matrices. These sets present the seesaw enhancement of the bi-large mixing, because there is no large mixings in mD and MR , but bi-large mixing is realized via the seesaw mechanism. We present twelve sets of mD and MR for the seesaw enhancement. Six sets lead to the lepton asymmetry, which depends on only M1 and the phase ρ. Putting the observed value of baryon number in the universe, M1 6 × 1010 GeV is obtained. It is remarked that JCP is the same sign as the YB , and its magnitude is predicted to be 0.01. Other six ones provide the real Dirac neutrino mass matrices, which give no CP asymmetry. Study of modified right-handed Majorana neutrino mass matrices is important for realistic model buildings based on the quark-lepton unification. We have also discussed SO(10)-like model with texture zeros. We have seen that the 4-zero texture with Pati-Salam symmetry restricts the above prameter region to a very narrow region. The precision mesurements, especially, for the solar neutrino mixing angle and the mass squared differences will check if such a texture of geometric form with Pati-Salam symmetry is realized in Nature in the near future.
This work is supported by the Grant-in Aid for Scientific Research No.12047225 and 12047220.
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REFERENCES 1. Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1998); ibid. 82, 2644 (1999); ibid. 82, 5194 (1999); K. Nishikawa, Invited talk at XXI Lepton Photon Symposium, August 10-16,2003, Batavia, USA. 2. Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 86, 5651; 5656 (2001). 3. SNO Collaboration: Q. R. Ahmad et al., Phys. Rev. Lett. 87, 071301 (2001), ibid. 89, 011301 (2002), ibid. 89, 011302 (2002), nuclex/309004. 4. KamLAND Collaboration, K. Eguchi et al., Phys. Rev. Lett. 90, 021802 (2003). 5. G. L. Fogli, E. Lisi, M. Marrone, D. Montanino, A. Palazzo and A.M. Rotunno, Phys. Rev. D67, 073002 (2003); J. N. Bahcall, M. C. Gonzalez-Garcia and C. Pe˜ na-Garay, JHEP 0302, 009 (2003); M. Maltoni, T. Schwetz and J.W.F. Valle, Phys. Rev. D67, 093003 (2003); P.C. Holanda and A. Yu. Smirnov, JCAP 0302, 001 (2003); V. Barger and D. Marfatia, Phys. Lett. 555B, 144 (2003); M. Maltoni, T. Schwetz, M. T´ ortola and J.W.F. Valle, hep-ph/0309130. 6. CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B466, 415 (1999). 7. H. Nishiura, K. Matsuda and T. Fukuyama, Phys. Rev. D60 (1999) 013006. 8. E. K. Akhmedov, G. C. Branco, M. N. Rebelo, Phys. Rev. Lett. 84 (2000) 3535. 9. S.K. Kang and C.S. Kim, Phys. Rev. D63 (2001) 113010. 10. K.T. Mahanthappa and Mu-C. Chen, Phys. Rev. D62 (2000) 113007. 11. Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870. 12. P.H. Frampton, S.L. Glashow and D. Marfatia, Phys. Lett. B536, 79 (2002). 13. Z. Xing, Phys. Lett. B530 (2002) 159. 14. W. Guo and Z. Xing, Phys. Rev. D67 (2003) 053002. 15. R. Barbieri, T. Hambye and A. Romanino, JHEP 0303 (2003) 017.
245
16. K.T. Mahanthappa and Mu-C. Chen, Phys. Rev. D65 (2002) 053010; D68 (2003) 017301. 17. M. Bando and M. Obara, Prog. Theor. Phys. 109 (2003) 995; hep-ph/0212242; M. Bando, S. Kaneko, M. Obara and M. Tanimoto, Phys. Lett. B580 (2004) 229. 18. A. Ibara and G. G. Ross, Phys. Lett. B575 (2003) 279. 19. M. Honda, S. Kaneko and M. Tanimoto, JHEP 0309 (2003) 028. 20. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, Proceedings of the Workshop, Stony Brook, New York, 1979, edited by P. van Nieuwenhuizen and D. Freedmann, North-Holland, Amsterdam, 1979, p.315; T. Yanagida, in Proceedings of the Workshop on the Unified Theories and Baryon Number in the Universe, Tsukuba, Japan, 1979, edited by O. Sawada and A. Sugamoto, KEK Report No. 79-18, Tsukuba, 1979, p.95. 21. A. Kageyama, S. Kaneko, N. Shimoyama and M. Tanimoto, Phys. Lett. B538, 96 (2002). 22. A. Yu. Smirnov, Phys. Rev. D48 (1993) 3264. 23. M. Tanimoto, Phys. Lett. 345B (1995) 477. 24. M. Bando, T. Kugo and K. Yoshioka, Phys. Rev. Lett. 80 (1998) 3004; M. Je˙zabek and Y. Sumino, Phys. Lett. B440 (1998) 327; ibid. B457 (1999) 139. 25. G. Altarelli, F. Ferruglio and I. Masina, Phys. Lett. B472 (2000) 382; E.K. Akhmedov, G.C. Branco and M.N. Rebelo, Phys. Lett. B478 (2000)215; A. Datta, F.-S. Ling, and P. Ramond, Nucl. Phys. B671 (2003) 383; S. Lavignac, I. Masina and C. A. Savoy, Nucl. Phys. B633 (2002) 139; W. Rodejohann, hep-ph/0311142. 26. M. Fukugita and T. Yanagida. Phys. Lett. B175 (1986) 45. 27. M. Flanz, E. A. Paschos and U. Sarkar, Phys. Lett. B345 (1995) 248; W. Buchm¨ uller and M. Pl¨ umacher, Phys. Lett. B389 (1996) 73; ibid. B431 (1998) 354; Int. J. Mod. Phys. A15 (2000) 5047; A. Pilaftsis, Phys. Rev. D56 (1997) 5431. 28. A. S. Joshipura, E. A. Paschos and W. Rodejohann, JHEP 0108 (2001) 029;
246
29.
30. 31.
32.
33. 34. 35.
M. Tanimoto / Nuclear Physics B (Proc. Suppl.) 137 (2004) 236–246
W. Buchm¨ uller and D. Wyler, Phys. Lett. B 521 (2001) 291; J. Ellis and M. Raidal, Nucl. Phys. B643 (2002) 229; S. Davidson and A. Ibarra, Nucl. Phys. B648 (2003) 345; G. Branco, T. Morozumi, B. Nobre and M.N. Rebelo, Nucl. Phys. B 617 (2001) 475; G. Branco, R. Gonz´alez Felipe, F. R. Joaquim and M. N. Rebelo, Nucl. Phys. B 640 (2002) 202; P. Frampton, S. Glashow and T. Yanagida, Phys. Lett. B548 (2002) 119; T. Endoh, S. Kaneko, S. K. Kang, T. Morozumi and M. Tanimoto, Phys. Rev. Lett. 89 (2002) 231601; S. Kaneko, M. Katsumata and M. Tanimoto, JHEP 0307 (2003) 025. D. Falcone, Phys. Rev. D66 (2002) 053001; Z. Xing, Phys. Lett. B545 (2002) 352; W. Rodejohann, Phys. Lett. B542 (2002) 100; M. Fujii, K. Hamaguti and T. Yanagida, Phys. Rev. D65 (2002) 115012. L. Covi, E. Roulet and F. Vissani, Phys. Lett. B 384 (1996) 169. J. A. Harvey and M. S. Turner, Phys. Rev. D42 (1990) 3344; S. Y. Khlebnikov and S. E. Shaposhnikov, Nucl. Phys. B308 (1998) 169. M. Tanimoto, Phys. Rev. D55 (1997) 322; Prog. Theor. Phys. 97 (1997) 901; Phys. Lett. B435 (1998)373; Phys. Lett. B462 (1999) 115; J. Arafune and J. Sato, Phys. Rev. D55 (1997)1653; J. Arafune, M. Koike and J. Sato, Phys. Rev. D56 (1997) 3093; H. Minakata and H. Nunokawa, Phys. Lett. B413 (1997) 369; Phys. Rev. D57 (1998) 4403. For recent proposal of the search for CP violation in neutrino oscillation, see Y. Itow et.al., hep-ex/0106019. C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039. M. Luty, Phys. Rev. D45 (1992) 455; M. Pl¨ umacher, Z. Phys. C74 (1997) 549. E. W. Kolb, M. S. Turner, The early universe, Redwood City, USA: Addison-Wesley (1990),
36.
37. 38. 39.
40.
41. 42. 43.
(Frontiers in physics, 69); M. Flanz and E. A. Paschos, Phys. Rev. D58 (1998) 113009; A. Pilaftsis, Int. J. Mod. Phys. A14 (1999) 1811; H. B. Nielsen and Y. Takanishi, Phys. Lett. B 507 (2001) 241. C. L. Bennett et al., Astrophys.J. Suppl. 148 (2003) 1; D. N. Spergel et al., Astrophys.J. Suppl. 148 (2003) 175; H. V. Peiris et al., Astrophys.J. Suppl. 148 (2003) 213. W. Buchm¨ uller and M. Pl¨ umacher, Phys. Lett. B521 (2001) 291. M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D63 (2001) 103502; astro-ph/0402490. L. Randall and R. Sundrum, Nucl. Phys. B557 (1999) 79; G.F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, JHEP 9812 (1998) 027. N. Haba, Y. Matsui, N. Okamura and M. Sugiura, Eur. Phys. J. C10 (1999) 677; Prog. Theor. Phys. 103 (2000) 145; N. Haba and N. Okamura, Eur. Phys. J. C14 (2000) 347; N. Haba, N. Okamura and M. Sugiura, Prog. Theor. Phys. 103 (2000) 367. P. Ramond, R.G Roberts and G.G Ross, Nucl. Phys. B406, 19 (1993). H. Fritzsch and Z-z. Xing, Prog Part Nucl. Phys. 45, 1 (2000). E. A. Akhmedov, M. Frigerio and A. Yu. Smirnov, JHEP 0309 (2003) 021.
Seesaw Realization of Bi-Large Mixing and Leptogenesis M. Tanimotoa∗ a
Department of Physics, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan The seesaw realization of the bi-large mixing is presented based on the texture zeros of the neutrino mass matrix. By using the obtained Dirac neutrino mass matrix, the leptogenesis is discussed. In the framework of the texture zeros, SO(10)-like model is investigated by focusing on the neutrino masses and mixings.
1. Introduction Neutrino experiments by Super-Kamiokande [1,2] and SNO[3] have brought us an outstanding fact on the neutrino oscillation. Recent results from KamLAND have almost confirmed the large neutrino mixing solution that is responsible for the solar neutrino problem nearly uniquely [4]. We have now common information concerning the neutrino mass difference squared (∆m2atm , ∆m2sun ) and neutrino flavor mixings (sin2 2θatm and tan2 θsun ) [5]. It is remarked that the neutrino mixing is the bi-large and the ratio ∆m2sun /∆m2atm is ∼ λ2 with λ 0.2. A constraint has also been placed on the third mixing angle from the reactor experiment of CHOOZ [6]. The textures with zeros of the neutrino mass matrix have been discussed [7–10] to explain these neutrino masses and mixings [11]. It was found that the two-zero textures are consistent with the experimental data in the basis of the diagonal charged lepton mass matrix [12]. Consequently, the neutrino mass matrix does not display the hierarchical structure as seen in the quark mass matrix [13–19]. Since these textures are given for the light effective neutrino mass matrix, one needs to find the see-saw realization [20] of these textures from the standpoint of the model building. Without fine tunings between parameters of the Dirac neutrino mass matrix mD and the right-handed Majorana neutrino one MR , we obtained several textures of ∗ This work is collaborated with M. Honda, S. Kaneko, M. Bando and M. Obara.
0920-5632/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2004.10.073
mD for the fixed MR [21]. Among them, there are textures of mD and MR which have hierarchical masses without large mixings. These present the seesaw enhancement of mixings, because there is no large mixings in mD and MR , but the bi-large mixing is realized via the seesaw mechanism. The seesaw enhancement are important in the standpoint of the quark-lepton unification, in which quark masses are hierarchical and quark mixings are very small. The general discussions of the seesaw enhancement were given in the case of two flavors [22,23]. Specific cases were discussed in the case of three flavors [24,25] because it is very difficult to get general conditions for the seesaw enhancement of the bi-large mixing. However, the two-zero texture of the neutrino mass matrix Mν are helpful to study the seesaw enhancement of the bi-large mixing. In this study, we present sets of mD and MR to give the seesaw enhancement in the two-zero textures of Mν and discuss the related phenomena, the leptogenesis [26]. Texture zeros of the neutrino mass matrix are presented in the section 2, and the seesaw enhancement of the bi-large mixing are discussed in the section 3. Based on the seesaw enhancement, the thermal leptogenesis is discussed in the section 4. The section 5 is devoted to the SO(10)-like model and the section 6 is devoted to the summary.
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2. Texture Zeros There are fifteen two-zero textures for the neutrino mass matrix Mν , which have five independent parameters. Among these textures, seven acceptable textures with two independent zeros were found for the neutrino mass matrix [12]. Seven two-zero textures of neutrino mass matrix in the basis wherein the charged lepton mass matrix is diagonal. A1
B1
B3
C
0 0 × × × 0 × 0 × × × ×
0 × × × 0 × 0 0 × × 0 ×
× ×, × 0 ×, × × ×, × × × 0
0 A2 × 0 × B2 0 × × B4 × 0
× × × 0 × × × × ×
0 × × × × 0 0 × 0
×: non−zero entry
c12 c13 −c12 s23 s13 − s12 c23 eiδ −c12 c23 s13 + s12 s23 eiδ
s12 c13 −s12 s23 s13 + c12 c23 eiδ −s12 c23 s13 − c12 s23 eiδ
c12 s23 iδ e + s13 s12 c23
which lead to 1 A1 : |Ue3 | tan 2θ12 cot θ23 Rν cos 2θ12 2 1 A2 : |Ue3 | tan 2θ12 tan θ23 Rν cos 2θ12 2 m1 ≥ tan2 θsun min 0.3 . m2
(4)
(5) (6)
where m0 denotes a constant mass and λ 0.2. 3. Seesaw Enhancement of Bi-large Mixing (1)
where and U is parametrized as follows:
=
s13 − 2 c13
The relative magnitude of each entry of the neutrino mass matrix is roughly given for the textures A1 and A2 as follows: 0 0 λ 0 λ 0 Mν m0 0 1 1 , m0 λ 1 1 (7) λ 1 1 0 1 1
Especially, the textures A1 and A2 of ref.[12], which correspond to the hierarchical neutrino mass spectrum, are strongly favored by the recent phenomenological analyses [13,14,19]. Therefore, the two textures are taken in order to discuss the seesaw enhancement in the next section. The neutrino mass matrix is written in terms of the neutrino mixings and neutrino masses as λ1 0 0 (2) Mν = U 0 λ2 0 U T 0 0 λ3 λ1 = m1 e2iρ , λ2 = m2 e2iσ , λ3 = m3
λ2 λ3
(3) s13 e−iδ
s23 c13 c23 c13
For the A1 type, (Mν )ee = (Mν )eµ = 0, we have λ1 s13 s12 s23 iδ = + 2 e − s13 λ3 c13 c12 c23
The neutrino matrix is given in terms of the Dirac neutrino mass matrix mD and the righthanded Majorana neutrino mass matrix MR by the the seesaw mechanism as [20] Mν = mD MR−1 mT D .
(8)
Zeros in mD and MR provide zeros in the neutrino mass matrix Mν of eq.(7) as far as we exclude the possibility that zeros are originated from accidental cancellations among matrix elements. In other words, we take a standpoint that the twozero texture should come from zeros of the Dirac neutrino mass matrix and the right-handed Majorana mass matrix. Possible textures of mD and MR were given in ref. [21]. Among them, we select the set of mD and MR , which reproduce the seesaw enhancement of the bi-large mixing. Let us fix the right-handed Majorana neutrino mass matrix without large mixings. We take simple right-handed Majorana neutrino mass matrix with only three independent parameters. Then, there are 6 C3 = 20 textures. Among them, six textures are excluded because they have a zero eigenvalue, which corresponds to a massless righthanded Majorana neutrino. Other two textures are also excluded because the two-zero textures
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A1 and A2 cannot be reproduced without accidental cancellations. One of the two textures is the diagonal matrix, and another one is the matrix with three zeros in the diagonal elements. We show twelve real mass matrices with three independent parameters with mass eigenvalues |M1 | = λm M3 and |M2 | = λn M3 , where M3 is the mass of the third generation, and m and n are integers with m > n > 1: m −1 0 λ 2 M3 0 λn 0 ai : M R m λ2 0 0 a1 m+n n 0 −λ 2 λ 2 n M3 −λ 2 1 0 m+n 0 0 λ 2 a2 m 2 0 0 λ M3 0 λn 0 m 2 λ 0 −1 a3 m+n 0 0 λ 2 n M3 0 1 −λ 2 (9) m+n n λ 2 −λ 2 0 a4
bi : MR
−λn m+n M3 λ 2 0 0 m+n M3 λ 2 n −λ 2 0 m+n M3 λ 2 0 0 m+n M3 λ 2 0
ci : M R
λm M3 0 0 1 n M3 −λ 2 0 m λ M3 0 0
λ
m+n 2
0 0 λ
m+n 2
0 0 m+n
λ 2 −λn 0 λ
m+n 2
0 n −λ 2 0 −1 n λ2
0 0 1 b1
n −λ 2 0 1 b2 0 0 1 b3 0 n −λ 2 (10) 1 b4
0 n λ2 0 c1 n
−λ 2 0 m+n λ 2 0 0 n λ2
0
λ
m+n 2
0
0 n λ2 −1 c3
c2
1 M3 0 n −λ 2
0 0
λ
m+n 2
n −λ 2 m+n λ 2 (11) 0 c4
where there are no large mixings in twelve matrices since the mass eigenvalues are supposed to be hierarchical. The minus signs in the matrix elements are taken to reproduce signs in the texture A1 and A2 of eq.(7). There are several Dirac neutrino mass matrices to give the textures A1 and A2 in eq.(7) [21]. We show Dirac neutrino mass matrices (mD )ai , (mD )bi , (mD )ci (i = 1 ∼ 4) with maximal number of zeros, which have no large mixings, to give the texture A2 . For the texture A1 , we easily obtain the Dirac neutrino mass matrices by exchanging the second and third rows. For each matrix of (MR )ai , (MR )bi , (MR )ci those are given as follows: λ 0 0 m ai : m D mD0 0 0 λ 2 n 1 λ2 0 a1 n +1 0 0 λ2 m mD0 0 0 λ2 0 1 0 a2 0 0 λ m mD0 λ 2 0 0 n 0 λ 2 1 a3 n 0 0 λ 2 +1 m mD0 λ 2 0 0 (12) 0 1 0 a4
bi :mD
n
λ 2 +1 0 m 0 λ2 mD0 n 0 λ2 n +1 0 λ2 m mD0 0 λ2 0 0 n 0 λ 2 +1 m mD0 λ 2 0 n 0 λ2 n 0 λ 2 +1 m mD0 λ 2 0 0 0
0 0 1 b 1 0 0 1 b2 0 0 1 b 3 0 0 1 b4
(13)
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0 λ 0 n 0 λ2 mD0 0 m λ 2 1 0 c1 n 0 0 λ 2 +1 m 0 λ2 mD0 0 1 0 0 c 2 0 0 λ n mD0 0 λ 2 0 m 0 1 c λ2 3 n +1 2 0 0 λ m mD0 0 λ 2 0 1 0 0 c4
ci :mD
0 m mD0 λ 2 0 0 m mD0 λ 2 0
ci :mD (14)
(15) bi :mD
n
λ 2 +1 λx mD0 n λ2 n +1 λ2 mD0 λx λy
0 m λ2 0 0 m λ2 0
0 λy 1 b3 0 λz 1 b4
x > n2 y>0 x > n2 y > n2 z>0 (16)
4
n
λ 2 +1 λx λy
where mD0 denotes the magnitude of the Dirac neutrino mass and complex coefficients of order one are omitted. Although these matrices have no large mixing among three families, the neutrino mass matrix Mν has the bi-large mixing through the seesaw mechanism. These are so called seesaw enhancement of the bi-large mixing. These Dirac matrices are ones with maximal number of zeros. Without changing the mixings and the mass eigenvalues in the leading order, some zeros can be replaced with small non-zero entries as follows: λ 0 0 m x>0 mD0 λx λy λ 2 ai :mD n y > n2 1 λ2 0 a1 n +1 λ2 0 0 x > n2 m x z mD0 λ λ λ 2 y > n2 y λ 1 0 z>0 a 2 0 0 λ m x > n2 mD0 λ 2 λx λy n y>0 0 λ2 1 a3 n 0 0 λ 2 +1 x > n2 m z x λ y > n2 mD0 λ 2 λ y 0 1 λ z>0 a
n
λ 2 +1 λx n λ2
0 λy 1 b 1 0 λz 1 b2
x > n2 y>0 x > n2 y > n2 z>0
0 λ 0 n x> m 2 mD0 λx λy λ 2 m y > 0 λ2 1 0 c1 n +1 0 x>0 0 λ2 m λy λ 2 y > n2 mD0 λx 1 λz 0 z > n2 c2 0 0 λ n x> m 2 mD0 λx λ 2 λy m y>0 2 λ 0 1 c3 n x>0 0 0 λ 2 +1 m λy y > n2 mD0 λx λ 2 1 0 λz z > n2 c4 (17)
where x, y and z are positive integers. These Dirac neutrino mass matrices are asymmetric ones. However, only the b3 and b4 textures in eq.(16) are adapted to the symmetric texture in the SO(10)-like GUT if y, m and n are relevantly chosen [16,17]. For example, in the b3 case, taking y = n/2 and m = n + 2, the symmetric mass matrix is given, especially, putting n = 8, we have the hierarchical mass matrix such like the up-quark mass matrix. 4. Thermal Leptogenesis Let us examine our textures in the leptogenesis [27–29], which is based on the Fukugita-Yanagida mechanism [26]. The CP violating phases in the Dirac neutrino mass matrix are key ingredients for the leptogenesis while the right-handed Majorana neutrino mass matrix are taken to be real in eqs.(9), (10) and (11). Although the nonzero entries in the Dirac neutrino mass matrix are complex, three phases are removed by the re-definition of the left-handed neutrino fields. There is no freedom of re-definition for the righthanded ones in the basis with the real MR . We
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should move to the diagonal basis of the righthanded Majorana neutrino mass matrix in order to calculate the magnitude of the leptogenesis. Then, the Dirac neutrino mass matrices m ¯ D in the new basis is given as follows:
where v2 = v sin β with v = 174GeV. The lepton asymmetry YL is related to the CP asymmetry through the relation
m ¯ D = PL mD OR
where s denotes the entropy density, g∗ is the effective number of relativistic degrees of freedom contributing to the entropy and κ is the so-called dilution factor which accounts for the washout processes (inverse decay and lepton number violating scattering). In the MSSM with righthanded neutrinos, one gets g∗ = 232.5. The produced lepton asymmetry YL is converted into a net baryon asymmetry YB through the (B + L)-violating sphaleron processes. One finds the relation [31]
(18)
where PL is a diagonal phase matrix and OR is the orthogonal matrix which diagonalizes MR as T OR MR OR in eqs.(9), (10) and (11). Since the phase matrix PL can remove one phase in each row of mD , three phases disappear in m ¯ D. As a typical example, we show the case of the b3 texture in eq.(10). By taking three eigenvalues of MR as follows 2 : M1 = λm M3 ,
M2 = −λn M3 .
(19)
YL =
We obtain the orthogonal matrix OR as cos θ sin θ 0 OR = − sin θ cos θ 0 , tan2 θ = λm−n (20) 0 0 1 Then the Dirac mass matrices mD of b3 in eq.(13) can be parameterized in the new basis as follows: n 0 λ 2 +1 0 m 0 0 OR (21) m ¯ D = mD0 λ 2 n 0 λ 2 eiρ 1 where only one phase ρ remains. The magniby the relation m2D0 tude of mD0 is determined 2 m0 M3 , where m0 ∆matm /2. We examine the lepton number asymmetry in the minimal SUSY model with the right-handed neutrinos. In the limit M1 M2 , M3 , the lepton number asymmetry 1 (CP asymmetry) for the lightest heavy Majorana neutrino (N1 ) decays into l∓ φ± [30] is given by
2 The
Γ1 − Γ1 1 = Γ1 + Γ1 Im[{(m ¯ †D m ¯ D )12 }2 ] M1 3 − 8πv22 M2 (m ¯ †D m ¯ D )11 Im[{(m ¯ †D m ¯ D )13 }2 ] M1 + † M3 (m ¯ Dm ¯ D )11
(22)
minus sign of M2 is necessary to reproduce MR in eq.(10). This minus sign is transfered to mD by the righthanded diagonal phase matrix diag(1, i, 1).
1 nL − nL¯ =κ s g∗
ξ YL ξ−1 8 Nf + 4 NH ξ= 22 Nf + 13 NH
(23)
YB = ξ YB−L =
(24)
where Nf and NH are the number of fermion families and Higgs doublets, respectively. Taking into account Nf = 3 and NH = 2 in the MSSM, we get 8 YL . (25) 15 On the other hand, the low energy CP violation, which is a measurable quantity in the long baseline neutrino oscillations [32], is given by the Jarlskog determinant JCP [33], which is calculated by
YB = −
det[M M† , Mν Mν† ] = −2iJCP (m2τ − m2µ )(m2µ − m2e )(m2e − m2τ ) ×(m23 − m22 )(m22 − m21 )(m21 − m23 )
(26)
where M is the diagonal charged lepton mass matrix, and m1 , m2 , m3 are neutrino masses. Since the CP violating phase is only ρ, we can find a link between the leptogenesis (1 ) and the low energy CP violation (JCP ) in our textures of the Dirac neutrinos. By using the Dirac neutrino mass matrix in eq.(21), we get 1
−
3m2D0 m λ sin 2ρ 8πv22
M. Tanimoto / Nuclear Physics B (Proc. Suppl.) 137 (2004) 236–246
JCP
−8.8 × 10−17 M1 sin 2ρ 1 2 ∆m2atm sin 2ρ λ 64 ∆m2sol
(27)
where M1 is given in the GeV unit and tan β ≥ 10 is taken. It is remarked that 1 only depends on M1 and the phase ρ, and the relative sign of 1 and JCP is opposite. Taking the experimental data ∆m2sol /∆m2atm λ2 and sin 2ρ 1, we predict JCP 0.01, which is rather large and then is favored for the future experimental measurement. The five cases of the Dirac neutrino mass matrix (a1 , a3 , b1 , c1 , c3 ) in eqs.(12), (13) and (14) lead to same results in eq.(27). In other six cases of the Dirac neutrino mass matrix (a2 , a4 , b2 , b4 , c2 , c4 ), the CP violating phases are removed because of only three non-zero entries. Then, we get 1 = 0, but the same result in eq.(27) for JCP . If we use the modified Dirac neutrino mass matrices in eqs.(15), (16) and (17), new CP violating phases appear. However, the contribution to 1 is a next-leading one as far as x 1, y 1, z 1. In order to calculate the baryon asymmetry, we need the dilution factor involves the integration of the full set of Boltzmann equations [34]. A simple approximated solution which has been frequently used is given by [35] −0.6 −3 10 eV m ˜1 ln −3 κ = 0.3 m ˜1 10 eV (10−2 eV ≤ m ˜ 1 ≤ 103 eV)
(28)
where (m ¯ †D m ¯ D )11 . (29) M1 By using this approximate dilution factor and eqs.(23) and (24), we can estimate YB in our textures as follows:
m ˜1 =
YB −2.3 × 10−3 1 κ .
(30)
It is noticed that YB and JCP are same sign since 1 has minus sign. The WMAP has given the new result [36] +0.4 ηB = 6.5−0.3 × 10−10 (1 σ)
which leads to 1 YB ηB . 7
(31)
(32)
241
¯ D )11 = m2D0 λm , In our textures, we have (m ¯ †D m 1 2 which gives m ˜ 1 = 2 ∆matm 0.022. Then we get the dilution factor κ 7 × 10−3 . Putting the observed value into eq.(30), we get M1 sin 2ρ 6 × 1010 GeV .
(33)
This result means that M1 is should be larger than 6 × 1010 GeV in order to explain the baryon number in the universe. This value is consistent with previous works [27–29]. It is important to present the discussion from the standpoint of the GUT, which is given after eq.(16). Taking n = 8 and m = 6 in the b3 case of eq.(16) as in the previous discussion, one obtains M3 ∼ 1015 GeV and M1 ∼ 108 GeV taking account of ∆m2atm 2 × 10−3 eV2 . This result does not satisfy the condition of eq.(33). However, the simple SO(10) fermion mass relation may be consistent with the leptogenesis in the case of the more complicated texture of MR , which leads to the two-zero texture A2 , as seen in the work of [37]. We add the discussion of another important problem. In the framework of supersymmetric thermal leptogenesis, there is cosmological gravitino problems. The gravitino with a few TeV mass does not favor M1 ≥ 1010 GeV [38], because M1 should be lower than the maximum reheating temperature of the universe after inflation. In order to keep the thermal leptogenesis in the SUSY model, we may consider the gravitino with O(100)TeV mass, which is derived from the anomaly mediated SUSY breaking mechanism [39]. 5. SO(10)-like model Let us discuss the texture zeros in the SO(10) framework. We take the A2 type of neutrino mass matrix at GUT scale as a minimal model including a phase φ: 0 β 0 ¯ ¯ h Mν = mν β¯ α (34) 0 ¯ h 1 β O(λ) 0 β 0 = mν PνT β eiφ α h Pν , α O(1) h O(1) 0 h 1
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¯ h, ¯ being made positive real numbers, with α, ¯ β, α, β, h by factored out the phases by the diagonal phase matrix Pν . Let us see how the parameters appearing in the neutrino mass matrix at GUT scale are generally constrained from the present experimental neutrino data. For a moment forget about how to derive the parameters of Mν and just see how the parameter regions of h and φ are constrained from the experimental data of sin2 2θatm , tan2 θsun and the ratio of ∆m2sun to ∆m2atm in terms of four parameters α, β, h and φ. To make numerical calculation more strictly, we must take account of the contributions from the charged lepton side. We take the symmetric charged lepton mass matrix in terms of the real matrix (M l )RL and further diagonalized to Mldiag. by Ol [7]: (Ml )RL = PlT (M l )RL Pl , OlT M l Ol = Mldiag. → OlT (PlT )−1 Ml Pl−1 Ol ≡ Mldiag. .
(35)
We use the following symmetric matrix having 2zeros for M l , √ me mµ 0 0 √ √ m me mτ (36) (M l )RL me mµ √ µ 0 me mτ mτ
mass matrices at mZ and MR [40]: 1 0 0 1−e 1 ˜ ν (MR ) × ˜ ν (MZ ) = 0 0M M 1−µ 0 0 1 1 0 0 1−e 1 0 0 (39) 1−µ
0
0
1
˜ ν is the neutrino mass matrix on the bawhere M sis where charged lepton matrix is diagonalized (see eq. (37)). The renormalization factors e and µ depend on the ratio of VEV’s, tan βv . By using the form of eq. (39) we search the region of the parameter set (α, β, h, φ, σ, ρ) which are allowed by experimental data within 3σ. It is found that the phase factor φ should not become large, (|φ| ≤ 70◦ ). This may be important since we have never had the information of the phases appearing in Mν , which is connected to the leptogenesis. Let us explore an example of the allowed region of the parameters in (α, β) plane for the typical value h = 1.3. The allowed region which is consistent with the experimental data is shown in Fig.1, where β is allowed to be in both negative and positive.
where me , mµ , mτ are charged lepton masses at MGUT scale. Here, we ignore the RGE effect from MGUT to MR scale considering that it almost does not change the values of masses for quarks and leptons. On the basis where the charged lepton mass matrix is diagonalized, the neutrino mass matrix at MR scale is obtained as ˜ ν (MR ) = OlT (P −1 )T PνT M ν (MR )Pν P −1 Ol (37) M l l where
0 β 0 M ν (MR ) = β eiφ α h mν 0 h 1 1 0 0 0 . Q ≡ Pν Pl−1 = 0 e−iρ 0 0 e−iσ
(38)
In order to compare our calculations with experimental results, we need the neutrino mass matrix at MZ scale, which is obtained from the following one-loop RGE’s relation between the neutrino
Figure 1. The allowed region on the α − β plane in the case of h = 1.3
So far we have investigated the region of the parameters appearing in the neutrino mass matrix and shown that the parameter region is restricted
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within narrow range by the present experimental data. Here we make a comment whether or not a certain GUT model is consistent with the bi-large mixing with present neutrino mass differences. As an example, let us take a concrete model [17] with the simplest form of right-handed neutrino mass matrix with the phase-factored out diagonal matrix, PR , 0 0 M1 0 0 PR MR = PRT M1 0 0 M2 0 r 0 ≡ mR PRT r 0 0 PR . (40) 0 0 1 This, with the form of 4-zero texture form of MνD , yields also texture-zeros of A2 type with the phase factored out by (MνD )RL = PνTD (M νD )RL PνD , 0 a 0 M νD = a b c mνD → 0 c 1 a2 0 0 r m2 2 ab a νD (41) 2 Mν = a 2 + c c( + 1) r r r mR 1 0 c( ar + 1) where a and c are real numbers and b is complex one. We recognize that, in order to get large mixing angle θ23 , the 23 element must be of the same order as the 33 element, namely c( ar + 1) ∼ 1. Since c 1, ca/r must be of order 1. Thus approximate form of Mν is 2 0 β 0 m2 β ∼ ar Mν ∼ β eiφ α h νD , (42) α ∼ 2ab r mR 0 h 1 h ∼ ca r which clearly shows that none of a, b, c is zero, namely 6-zero texture are already excluded by the experimental neutrino data 3 . Now, one example of the symmetric 4-zero texture with the PatiSalam symmetry [17] provides us with the Dirac neutrino mass matrix at the MGUT scale under a simple assumption of the following Higgs config3 Here, we note that the 6-zero textures for the quark sector have been already ruled out by Ramond, Roberts and Ross [41].
urations:
M νD
0 126 0 10 → MU = 126 10 0 10 126 √ mu mc 0 0 −3 mt √ m m mu c eiφ m −3 mut c mt mt mu 0 −3 mt (43)
in mt unit. By comparing eq. (42) and eq. (43) the parameters α, β are expressed in terms of up-quark masses at the GUT scale. Thus, we can predict α, β from the up-quark masses at the GUT scale, mu = 0.36 ∼ 1.28MeV, mc = 209 ∼ 300MeV, mt = 88 ∼ 118GeV, which are obtained taking account of RGE’s effect to the quark masses at the EW scale [42]. Since the light quark masses are ambiguous because of the non-perturbative QCD effect, the allowed mass region of mu may be enlarged. In the case of mu = 0.36 ∼ 2.56 MeV, we obtain the overlapped region around α 1.24 and β −0.2 with h = 1.3 as seen figure 2. The allowed region on the α − β plane in the case of h = 1.3, which is predicted from a neutrino mass matrix with two zeros. The allowed region of the parameters are very narrow as follows: α = 1.23 ∼ 1.24, β = −0.199 ∼ −0.197 π 7 11 π ∼ , ρ= π∼ π (44) φ=− 18 18 9 9 where h = 1.3 is taken. On the other hand, our results are almost independent of the phase parameter σ. Hereafter we take σ = 0 in our calculations. In these parameters, we can predict Ue3 by including the contribution of the charged lepton sector. Here we stress that Ue3 is crucial to discriminate various models, therefore, we must be careful to estimate it by taking account of the effect of charged lepton mixings as well as CP violating phases. Our formula has already included these contributions. By taking the overlapped region of α and β in Fig.2, we present the prediction of |Ue3 |, JCP and < mee > as follows: |Ue3 | = 0.010 − 0.048 , | < mee > | 2.7meV
|JCP | ≤ 9.6 × 10−3 (45)
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6. Summary
Figure 2. The predicted region (gray region) of the α − β plane, in which h = 1.3 and mu = 0.36 ∼ 2.56 MeV are taken. The black region is the experimentally allowed region predicted from a neutrino mass matrix
where < mee > is the effective neutrino mass in the neutrinoless double beta decay. We hope |Ue3 | can be checked by the neutrino experiments in near future. Since the overlapped region of α and β is restricted in the narrow region, we can predict a set of typical values of neutrino masses and mixings at h = 1.3 as follows; sin2 2θµτ ∼ 0.98, tan2 θµe ∼ 0.28 mν3 ∼ 62 meV, mν2 ∼ 7.5 meV mν1 ∼ 1.4 meV
(46)
with mR = 3.0 × 1015 GeV and r × mR = 1.0 × 109 GeV, which correspond to the Majorana mass for the third generation and those of the second and first generations, respectively. On the other hand, mu 2.56 MeV should be allowed at the GUT scale. Now that our neutrino mass matrix is determined almost uniquely from the up-quark masses at GUT scale, we can make the prediction of leptogenesis once we fix the CP violating phases. Interesting enough is that our form of MR of eq.(40) yields naturally two degenerate Majorana masses with mass r × mR ∼ 109 GeV. In such case the leptogenesis is enhanced by the so-called ”crossing effect” [43].
We have discussed texture zeros with the seesaw enhancement. These textures are important in the standpoint of the quark-lepton unification, in which quark masses are hierarchical and quark mixings are very small. It is very difficult to get general conditions for the seesaw enhancement of the bi-large mixing, however, the two-zero textures of the left-handed neutrino mass matrix Mν are helpful to study the seesaw enhancement of the bi-large mixing. Once the basis of the righthanded Majorana neutrino mass matrix is fixed, one can find some sets of mD and MR , which have hierarchical masses without large mixings, to give the two-zero textures A1 and A2 without fine tuning among parameters of these matrices. These sets present the seesaw enhancement of the bi-large mixing, because there is no large mixings in mD and MR , but bi-large mixing is realized via the seesaw mechanism. We present twelve sets of mD and MR for the seesaw enhancement. Six sets lead to the lepton asymmetry, which depends on only M1 and the phase ρ. Putting the observed value of baryon number in the universe, M1 6 × 1010 GeV is obtained. It is remarked that JCP is the same sign as the YB , and its magnitude is predicted to be 0.01. Other six ones provide the real Dirac neutrino mass matrices, which give no CP asymmetry. Study of modified right-handed Majorana neutrino mass matrices is important for realistic model buildings based on the quark-lepton unification. We have also discussed SO(10)-like model with texture zeros. We have seen that the 4-zero texture with Pati-Salam symmetry restricts the above prameter region to a very narrow region. The precision mesurements, especially, for the solar neutrino mixing angle and the mass squared differences will check if such a texture of geometric form with Pati-Salam symmetry is realized in Nature in the near future.
This work is supported by the Grant-in Aid for Scientific Research No.12047225 and 12047220.
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REFERENCES 1. Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1998); ibid. 82, 2644 (1999); ibid. 82, 5194 (1999); K. Nishikawa, Invited talk at XXI Lepton Photon Symposium, August 10-16,2003, Batavia, USA. 2. Super-Kamiokande Collaboration, S. Fukuda et al., Phys. Rev. Lett. 86, 5651; 5656 (2001). 3. SNO Collaboration: Q. R. Ahmad et al., Phys. Rev. Lett. 87, 071301 (2001), ibid. 89, 011301 (2002), ibid. 89, 011302 (2002), nuclex/309004. 4. KamLAND Collaboration, K. Eguchi et al., Phys. Rev. Lett. 90, 021802 (2003). 5. G. L. Fogli, E. Lisi, M. Marrone, D. Montanino, A. Palazzo and A.M. Rotunno, Phys. Rev. D67, 073002 (2003); J. N. Bahcall, M. C. Gonzalez-Garcia and C. Pe˜ na-Garay, JHEP 0302, 009 (2003); M. Maltoni, T. Schwetz and J.W.F. Valle, Phys. Rev. D67, 093003 (2003); P.C. Holanda and A. Yu. Smirnov, JCAP 0302, 001 (2003); V. Barger and D. Marfatia, Phys. Lett. 555B, 144 (2003); M. Maltoni, T. Schwetz, M. T´ ortola and J.W.F. Valle, hep-ph/0309130. 6. CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B466, 415 (1999). 7. H. Nishiura, K. Matsuda and T. Fukuyama, Phys. Rev. D60 (1999) 013006. 8. E. K. Akhmedov, G. C. Branco, M. N. Rebelo, Phys. Rev. Lett. 84 (2000) 3535. 9. S.K. Kang and C.S. Kim, Phys. Rev. D63 (2001) 113010. 10. K.T. Mahanthappa and Mu-C. Chen, Phys. Rev. D62 (2000) 113007. 11. Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870. 12. P.H. Frampton, S.L. Glashow and D. Marfatia, Phys. Lett. B536, 79 (2002). 13. Z. Xing, Phys. Lett. B530 (2002) 159. 14. W. Guo and Z. Xing, Phys. Rev. D67 (2003) 053002. 15. R. Barbieri, T. Hambye and A. Romanino, JHEP 0303 (2003) 017.
245
16. K.T. Mahanthappa and Mu-C. Chen, Phys. Rev. D65 (2002) 053010; D68 (2003) 017301. 17. M. Bando and M. Obara, Prog. Theor. Phys. 109 (2003) 995; hep-ph/0212242; M. Bando, S. Kaneko, M. Obara and M. Tanimoto, Phys. Lett. B580 (2004) 229. 18. A. Ibara and G. G. Ross, Phys. Lett. B575 (2003) 279. 19. M. Honda, S. Kaneko and M. Tanimoto, JHEP 0309 (2003) 028. 20. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, Proceedings of the Workshop, Stony Brook, New York, 1979, edited by P. van Nieuwenhuizen and D. Freedmann, North-Holland, Amsterdam, 1979, p.315; T. Yanagida, in Proceedings of the Workshop on the Unified Theories and Baryon Number in the Universe, Tsukuba, Japan, 1979, edited by O. Sawada and A. Sugamoto, KEK Report No. 79-18, Tsukuba, 1979, p.95. 21. A. Kageyama, S. Kaneko, N. Shimoyama and M. Tanimoto, Phys. Lett. B538, 96 (2002). 22. A. Yu. Smirnov, Phys. Rev. D48 (1993) 3264. 23. M. Tanimoto, Phys. Lett. 345B (1995) 477. 24. M. Bando, T. Kugo and K. Yoshioka, Phys. Rev. Lett. 80 (1998) 3004; M. Je˙zabek and Y. Sumino, Phys. Lett. B440 (1998) 327; ibid. B457 (1999) 139. 25. G. Altarelli, F. Ferruglio and I. Masina, Phys. Lett. B472 (2000) 382; E.K. Akhmedov, G.C. Branco and M.N. Rebelo, Phys. Lett. B478 (2000)215; A. Datta, F.-S. Ling, and P. Ramond, Nucl. Phys. B671 (2003) 383; S. Lavignac, I. Masina and C. A. Savoy, Nucl. Phys. B633 (2002) 139; W. Rodejohann, hep-ph/0311142. 26. M. Fukugita and T. Yanagida. Phys. Lett. B175 (1986) 45. 27. M. Flanz, E. A. Paschos and U. Sarkar, Phys. Lett. B345 (1995) 248; W. Buchm¨ uller and M. Pl¨ umacher, Phys. Lett. B389 (1996) 73; ibid. B431 (1998) 354; Int. J. Mod. Phys. A15 (2000) 5047; A. Pilaftsis, Phys. Rev. D56 (1997) 5431. 28. A. S. Joshipura, E. A. Paschos and W. Rodejohann, JHEP 0108 (2001) 029;
246
29.
30. 31.
32.
33. 34. 35.
M. Tanimoto / Nuclear Physics B (Proc. Suppl.) 137 (2004) 236–246
W. Buchm¨ uller and D. Wyler, Phys. Lett. B 521 (2001) 291; J. Ellis and M. Raidal, Nucl. Phys. B643 (2002) 229; S. Davidson and A. Ibarra, Nucl. Phys. B648 (2003) 345; G. Branco, T. Morozumi, B. Nobre and M.N. Rebelo, Nucl. Phys. B 617 (2001) 475; G. Branco, R. Gonz´alez Felipe, F. R. Joaquim and M. N. Rebelo, Nucl. Phys. B 640 (2002) 202; P. Frampton, S. Glashow and T. Yanagida, Phys. Lett. B548 (2002) 119; T. Endoh, S. Kaneko, S. K. Kang, T. Morozumi and M. Tanimoto, Phys. Rev. Lett. 89 (2002) 231601; S. Kaneko, M. Katsumata and M. Tanimoto, JHEP 0307 (2003) 025. D. Falcone, Phys. Rev. D66 (2002) 053001; Z. Xing, Phys. Lett. B545 (2002) 352; W. Rodejohann, Phys. Lett. B542 (2002) 100; M. Fujii, K. Hamaguti and T. Yanagida, Phys. Rev. D65 (2002) 115012. L. Covi, E. Roulet and F. Vissani, Phys. Lett. B 384 (1996) 169. J. A. Harvey and M. S. Turner, Phys. Rev. D42 (1990) 3344; S. Y. Khlebnikov and S. E. Shaposhnikov, Nucl. Phys. B308 (1998) 169. M. Tanimoto, Phys. Rev. D55 (1997) 322; Prog. Theor. Phys. 97 (1997) 901; Phys. Lett. B435 (1998)373; Phys. Lett. B462 (1999) 115; J. Arafune and J. Sato, Phys. Rev. D55 (1997)1653; J. Arafune, M. Koike and J. Sato, Phys. Rev. D56 (1997) 3093; H. Minakata and H. Nunokawa, Phys. Lett. B413 (1997) 369; Phys. Rev. D57 (1998) 4403. For recent proposal of the search for CP violation in neutrino oscillation, see Y. Itow et.al., hep-ex/0106019. C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039. M. Luty, Phys. Rev. D45 (1992) 455; M. Pl¨ umacher, Z. Phys. C74 (1997) 549. E. W. Kolb, M. S. Turner, The early universe, Redwood City, USA: Addison-Wesley (1990),
36.
37. 38. 39.
40.
41. 42. 43.
(Frontiers in physics, 69); M. Flanz and E. A. Paschos, Phys. Rev. D58 (1998) 113009; A. Pilaftsis, Int. J. Mod. Phys. A14 (1999) 1811; H. B. Nielsen and Y. Takanishi, Phys. Lett. B 507 (2001) 241. C. L. Bennett et al., Astrophys.J. Suppl. 148 (2003) 1; D. N. Spergel et al., Astrophys.J. Suppl. 148 (2003) 175; H. V. Peiris et al., Astrophys.J. Suppl. 148 (2003) 213. W. Buchm¨ uller and M. Pl¨ umacher, Phys. Lett. B521 (2001) 291. M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D63 (2001) 103502; astro-ph/0402490. L. Randall and R. Sundrum, Nucl. Phys. B557 (1999) 79; G.F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, JHEP 9812 (1998) 027. N. Haba, Y. Matsui, N. Okamura and M. Sugiura, Eur. Phys. J. C10 (1999) 677; Prog. Theor. Phys. 103 (2000) 145; N. Haba and N. Okamura, Eur. Phys. J. C14 (2000) 347; N. Haba, N. Okamura and M. Sugiura, Prog. Theor. Phys. 103 (2000) 367. P. Ramond, R.G Roberts and G.G Ross, Nucl. Phys. B406, 19 (1993). H. Fritzsch and Z-z. Xing, Prog Part Nucl. Phys. 45, 1 (2000). E. A. Akhmedov, M. Frigerio and A. Yu. Smirnov, JHEP 0309 (2003) 021.