Neutrino masses, mixing, and leptogenesis in an S3 model

0.61 MB·

by  Arturo Alvarez Cruz

Neutrino masses, mixing, and leptogenesis in an
S3 model
´
Arturo Alvarez Cruz and Myriam Mondrago´n
Instituto de F´ısica, Universidad Nacional Auto´noma de M´exico
Apdo. Postal 20-364, M´exico 01000 CD MX, M´exico
7
1
January 30, 2017
0
2

n
Abstract
a
Inthisworkweusepreviousresultsonthemassesandmixingofneu-
J
trinos of an S3 model with three right-handed Majorana neutrinos and
7
three Higgs doublets, to reduce one parameter in the case when two of
2
theright-handedneutrinosaremassdegenerate. Wederiveanewparam-

eterization for the V mixing matrix, with a new set of parameters,
] PMNS
h in the more general case where the right-handed neutrino masses are dif-
p ferent. With these results, we calculate leptogenesis and the associated
- baryogenesisinthemodelinthetwodifferentscenarios. Weshowthatit
p
e ispossibletohaveenoughleptogenesistoexplainthebaryonicasymmetry
h with right-handed neutrino masses above 106 GeV.
[

1 1 Introduction
v
9
The Standard Model (SM) is extremely successful, nevertheless the discovery
2
9 of neutrino masses and mixing in neutrino oscillation experiments in 1998 [1],
7 presentedevidencethatisnecessarytogobeyondit. Evenbeforethisdiscovery,
0 the amount of free parameters and the hierarchy problem, among others, have
.
1 prompted attempts to find a more fundamental theory, of which the SM is the
0 low-energy limit [2–4]. Some of the goals of these new models are to under-
7 stand the large differences in the Yukawa couplings of the different fermions,
1
the hierarchy between the fundamental particles, and the amount of CP vio-
:
v lation and the structure of the CKM matrix [5]. A popular way to approach
Xi these problems is to build models with Non-Abelian flavor symmetries, often
supplemented with extra Higgs doublets. Common symmetries in flavor theo-
r
a ries are, among many others, A4, Q6 or S3 [6–11]. The reason is that these
models achieve in a natural way the Nearest Neighbour Interaction textures in
thefermionmassmatrices[12,13]. TheS3extensionoftheSMwiththreeHiggs
doublets(S3-3H)[10,11,14]isamodelinwhichasymmetryonthepermutation
of three objects is imposed, which in additon to the SM particles has another
two Higgs doublets, as well as three right-handed Majorana neutrinos, which
are related to the left ones through the seesaw mechanism (type I).
There has been a lot of work done on various S3 models (see for instance
[15–22]), some of this work reproduces the CKM and PMNS matrices in agree-
ment with the current experimental data [10,11,23–26], and there have been
1
also studies of leptogenesis in a soft breaking S3 model [27]. Nevertheless, most
of this work has been done in the case where two right-handed neutrinos are
degenerate. In this way, it is an interesting question to extend the model and
see the possible new results with a generalization, taking into account both de-
generate and non-degenerate right-handed neutrino masses. Following the idea
ofpreviouswork[28], weextendtheanalysisonthegeneralizationoftheS3-3H
model.
Another question that the SM fails to explain is the observed baryon asym-
metry. It is well know that there are more baryons than antibaryons in the
Universe. Nucleosynthesis is a solid and consistent model of the creation of the
nuclei in the early Universe, which predicts a baryonic density of,
η −η
η = b b =η =(2.6−6.2)×10−10. (1)
η
γ
Measurements of the Cosmic Background Radiation [29–31] show a density of
η =(6.1±0.3)×10−10, (2)
in full agreement with the baryon density of the Nucleosynthesis [30,32].
The idea to explain the baryon asymmetry through a dynamically process was
proposed by Sakharov in 1967 [33]. The present cosmological observations
favour the idea that the matter-antimatter asymmetry of the Universe may
be explained in terms of a dynamical generation mechanism, called baryogen-
esis. Also, it has been realized that a successful model of baryogenesis cannot
occur within the Standard Model (SM).
Leptogenesisisamechanismwhichgeneratesbaryonasymmetrybycreating
aleptonicasymmetrythroughB+Lviolatingelectroweaksphalerontransitions
[34].
Several things are needed for the occurrence of leptogenesis:
• Heavy right handed neutrinos.
• Majorana type neutrinos.
• Decay of the right handed neutrinos to the left ones.
According to the original proposal of Fukugita and Yanagida [35], this mecha-
nism also satisfies all the Sakharov’s conditions [33] in order to produce a net
baryon asymmetry (for reviews see for instance [36–38]).
In this paper we explore the possibility of leptogenesis in the S3-3H model,
with degenerate and non-degenerate right-handed neutrino masses, and calcu-
late the associated baryogenesis. We first study the case where two of the
right-handed neutrino masses are degenerate, and then the more general case
where all the right-handed neutrino masses are different. We scan the param-
eter space to find the leptogenesis and associated baryogenesis dependence on
the free parameters of the model. We find that there is a region of parameter
space where enough baryogenesis is produced through leptogenesis to explain
the baryon asymmetry of the Universe.
The outline of the paper is organized as follows: In section 2, the S3 model is
2
introducedaswellassomeofitsmostimportantresults. Insection3itisshown
howtoproduceleptogenesisintheS3-3Hmodel,andtheresultantbaryogenesis
is also computed. At the end, in section 4, we conclude summarizing our main
results.
2 S3-3H model
IntheStandardModelanalogousfermionsindifferentgenerationshaveidentical
couplings to all gauge bosons of the strong, weak, and electromagnetic inter-
actions [16]. The group S3 consists of the six possible permutations of three
objects (f ,f ,f ), and is the smallest discrete non-abelian group. It has one
1 2 3
2-dimension irreducible representation (irrep) and two of 1-dimension
F =(f ,f ,f ), F =−f −f +2f , (3)
s 1 2 3 d1 1 2 3
f =f −f . (4)
d2 2 3
We can associate the particles in the model to doublets or to singlets with
the following rules. The direct product of two doublets pT = (p ,p ) and
D D1 D2
qT =(q ,q ) may be decomposed into the direct sum of two singlets r and
D D1 D2 s
r , and one doublet rT where
s(cid:48) D
r =p q +p q r =p q −p q (5)
s D1 D1 D2 D2 s(cid:48) D1 D2 D2 D1
rT =(r ,r )=(p q +p q ,p q +p q ). (6)
D D1 D2 D1 D2 D2 D1 D1 D1 D2 D2
Since the Standard Model has only one Higgs SU(2) doublet, which can only
L
be an S singlet, it gives mass to the particles in the S singlet representation.
3 3
TogivemasstotherestoftheparticlesweextendtheHiggssectorofthetheory,
by adding two more Higgs doublets. The quark and Higgs fields are
QT =(u ,d ),u ,d , (7)
L L r r
Lt =(ν ,e ),e ,ν and H. (8)
L L R R
All of the fields have three species, and we assume that each one forms a re-
ducible representation 1S ⊕2. The first two generations will be assigned to
the doublet S3 irrep, and the third generation to the singlet. This applies to
quarks, leptons, Higgs fields, and right-handed neutrinos. The doublets carry
capital indices I and J, which run from 1 to 2, and the singlets are denoted by
Q ,u ,d ,L ,e ,ν andH . Thesubscript3denotesthesingletrepresen-
3 3R 3R 3 3R 3R S
tation and not the third generation. The most general renormalizable Yukawa
interactions of this model are given by [10]
L =L +L +L +L (9)
Y YD YU YE Yν
where
L =−YdQ H d −YdQ H d
YD l I S IR 3 3 S 3r
−Yd[Q κ H d −Q η H d ] (10)
2 I IJ l JR I IJ 2 JR
−YdQ H d −YdQ H D +h.c.
4 3 I IR 5 I I 3R
3
L =−YuQ (iσ H∗u )−YuQ (iσ H∗u )
YU 1 I 2 S IR 3 3 2 S 3R
−Yu[Q κ (iσ H∗u )−Q η (iσ H∗u )] (11)
2 I IJ 2 1 JR I IJ 2 2 JR
−YuQ (iσ H∗u )−YuQ (iσ H∗u R)+h.c.
4 3 2 I IR 5 I 2 I 3
L =−YuQ (iσ H∗u )−YuQ (iσ H∗u )
YU 1 I 2 S IR 3 3 2 S 3R
−Yu[Q κ (iσ H∗u )−Q η (iσ H∗u )] (12)
2 I IJ 2 1 JR I IJ 2 2 JR
−YuQ (iσ H∗u )−YuQ (iσ H∗u R)+h.c.
4 3 2 I IR 5 I 2 I 3
L =−YeL H e )−YeL H e )
YE 1 I S IR 3 3 S 3R
−Ye[L κ H e −L η H e )] (13)
2 I IJ 1 JR I IJ 2 JR
−YeL H e −YeL H D +h.c.
4 3 I IR 5 I I 3R
L =−YνL (iσ H∗ν )−YνL (iσ H∗ν )
Yν 1 I 2 S IR 3 3 2 S 3R
−Yν[L κ (iσ H∗ν )−L η (iσ H∗ν )] (14)
2 I IJ 2 1 JR I IJ 2 2 JR
−YνL (iσ H∗ν )−YνL (iσ H∗ν R)+h.c.,
4 3 2 I IR 5 I 2 I 3
with
(cid:18) (cid:19) (cid:18) (cid:19)
0 1 1 0
κ= η = . (15)
1 0 0 −1
Furthermore, we addto the Lagrangian theMajorana mass termsfor the right-
handed neutrinos
L =−M νT Cν −M νT Cν −M νT Cν . (16)
M 1 1R 1R 2 2R 2R 3 3R 3R
Due to the presence of three Higgs fields, the Higgs potential V (H ,H ) is
H S D
more complicated than that of the Standard Model [39]. In addition to the S3
symmetry,undercertainconditionstheHiggspotentialexhibitsapermutational
symmetryZ2:H1↔H2,whichisnotasubgroupoftheflavorgroupS3[24,25].
ThemodelhasaswellanAbeliandiscretesymmetrythatwewilluseasselection
rules for the Yukawa couplings in the leptonic sector. In this paper, we will
assume that the vacuum respects the accidental Z2 symmetry of the Higgs
potential and that
<H >=<H >. (17)
1 2
With these assumptions, the Yukawa interactions, eqs. (10)-(14) yield mass
matrices, for all fermions in the theory, of the general form
 
µ +µ µ µ
1 2 2 5
M = µ2 µ1−µ2 µ5. (18)
µ µ µ
4 4 3
TheMajoranamassforthelefthandedneutrinosν isgeneratedbythesee-saw
L
mechanism. The corresponding mass matrix is given by
Mν =MνDM(cid:102)−1(MνD)T, (19)
4
whereM(cid:102)=diag(M1,M1,M3). Inprinciple,allentriesinthemassmatricescan
be complex since there is no restriction coming from the S3 flavor symmetry.
The mass matrices are diagonalized by bi-unitary transformations as
U† M U =diag(m ,m ,m ) (20)
d(u,e)L d(u,e) d(u,e)R d(u,e) s(u,e) b(u,e)
UTM U =diag(m ,m ,m ). (21)
ν ν ν ν1 ν2 ν3
Theentriesinthismatrixarecomplexnumbers,sothephysicalmassesaretheir
absolute values. The mixing matrices are, by definition,
V =U† U , V =U† U K. (22)
CKM uL dL PMNS eL ν
Where K is defined as the matrix that take out the phases of the diagonal mass
matrix,
diag(m ,m ,m )=K†diag(|m |,|m |,|m |)K†. (23)
ν1 ν2 ν3 ν1 ν2 ν3
A further reduction of the number of parameters in the leptonic sector may be
achieved by means of an Abelian Z symmetry. A possible set of charge assign-
2
ments of Z , compatible with the experimental data on masses and mixings in
2
the leptonic sector is given in Table I. The Z assignments forbid the following
2
− +
H ,ν H ,L ,L ,e ,e .ν
S 3R I 3 I eR IR IR
Table 1: Z2 assignment in the leptonic sector.
Yukawa couplings
Ye =Ye =Yν =Yν. (24)
1 3 1 5
Therefore, the corresponding entries in the mass matrices vanish.
2.1 Mass matrix for the charged leptons
Undertheseassumptions,themassmatrixofthechargedleptonstakestheform
 
µ˜ µ˜ µ˜
2 2 5
Me =mτµ˜2 −µ˜2 µ˜5. (25)
µ˜ µ˜ 0
4 4
The unitary matrix U that enters in the definition of the mixing matrix,
eL
V , is calculated from
PMNS
U† M M†U =diag(m2,m2,m2), (26)
eL e e eL e µ τ
where m ,m and m are the masses of the charged leptons, and
3 µ τ
 2|µ˜2|2 |µ˜5|2 2|µ˜2||µ˜4|e−iδe
MeMeτ =m2τ |µ˜5|2 2|µ˜2|2 0 . (27)
2|µ˜2||µ˜4|eiδe 0 2|µ˜4|2
Noticethatthismatrixonlyhasonephasefactor. Theparameters|µ˜|,|µ˜|and
2 4
|µ˜| may readily be expressed in terms of the charged lepton masses. From the
5
invariants of M M†, we get the set of equations [17]
e e
Tr(M M†)=m2+m2 +m2 =m2[4|µ˜|2+2(|µ˜|2+|µ˜|2)] (28)
e e e µ τ τ 2 4 5
5
ξ(M M†)=m2(m2+m2)+m2m2 (29)
e e τ e µ e µ
=4m4[|µ˜|2+|µ˜|2(|µ˜|2+|µ˜|2)+|µ˜|2|µ˜|2] (30)
τ 2 2 4 5 4 5
(31)
det(M M†)=m2m2m2 =4m6|µ˜|2|µ˜|2|µ˜|2, (32)
e e e µ τ τ 2 4 5
where ξ(M M†) = 1[Tr(∗M M†))2−Tr((M M†)2)]. Solving these equations
e e 2 e e e e
for |µ˜|,|µ˜| and |µ˜|, we obtain
2 4 5
1m2+m2 m2m2
|µ˜|2 = e µ − e µ +β. (33)
2 2 m2 m2(m2+m2)
τ τ e µ
In this expression, β is the smallest solution of the equation
1 x 1 z z2
β3− (1−2y+6 )β2− (y−y2−4 +7z−12 )β− (34)
2 y 4 y y2
1 1z2 3z2 z3
yz− + − =0 (35)
8 2y2 4 y y3
(36)
where y =(m2+m2)/m and z =µ2µ2/µ4.
e µ τ µ e τ
An estimation of β at good order of magnitude is obtained from [40]
m2m2
β (cid:39)− µ e . (37)
2m2(m2 −(m2 +m2))
τ τ τ e
The parameters |µ˜|2 and |µ˜|2 are in terms of |µ˜|2,
4 5 2
1 m2 +m2 m2m2
|µ˜ |2 = (1− µ e +4 e µ )−β) (38)
4,5 4 m2 m2(m2+m2
τ τ e µ
(cid:115)
1 m2 +m2 m2m2 m2m2 1
± ( (1− µ e +4 e µ )−β)2− µ e ) (39)
4 m2 m2(m2+m2 m4 |µ˜|2
τ τ e µ τ 2
(40)
Once M M† has been reparametrized in terms of the charged lepton masses, it
e e
isstraightforwardtocomputeU alsoasafunctionoftheleptonmasses. Here
eL
we will write the result to order (m m /m2)2 and x4 =(m /m )4
µ e τ e τ
 (cid:113) 
√1 √m˜µ √1 √m˜µ √1 1√+x2−m˜2µ
 2 1+x2 2 1+x2 2 1+x2 
 (cid:113) 
Me (cid:39)mτ √1 √m˜µ −√1 √m˜µ √1 1√+x2−m˜2µ. (41)
 2 1+x2 2 1+x2 2 1+x2 
(cid:113)m˜e(1+x2) eiδe (cid:113)m˜e(1+x2) eiδe 0 
1+x2−m˜2 1+x2−m˜2
µ µ
The unitary matrix U that diagonalizes M M† and enters in the definition of
eL e e
the neutrino mixing matrix V , equation (22), is
VPMNS
  
1 0 0 O −O O
11 12 13
UeL (cid:39)0 1 0 −O21 O22 O23, (42)
0 0 eiδe −O31 −O32 O33
6
where
 
O −O O
11 12 13
UeL (cid:39)−O21 O22 O23= (43)
−O −O O
31 32 33
 √12x(cid:113)1+m(˜12µ++25mx˜2µ2+−xm˜24µ+−m˜m4µ˜6µ++2mm˜˜21e2e+)12x4 −√12(cid:113)1−m(1˜2µ−+2xm˜22µ++6mm˜˜4µ4µ−−24mm˜˜26µe+)5m˜1e2 √12 
−√12x(cid:113)1+m˜2µ(+1+5x42x−2−m˜m4µ˜−4µ−m˜26µm+˜2me˜)1e2+12x4 √12(cid:113)1−m˜2µ+(1x−2+2m6˜m2µ˜+4µ−m˜44µm)˜6µ+5m˜1e2 √12 =
 −(cid:113)1+2x2−m˜2µ−m˜2e(1+m˜2µ+x2−2m˜2e) −x(cid:113)1+2x2−m˜2µ−m˜2e(1+x2−m˜2µ−2m˜2e) m˜ m˜ √1+x2 
(cid:113) (cid:113) e µ(cid:113)
1+m˜2+5x2−m˜4−m˜6+m˜12+12x4 1−m˜2+x2+6m˜4−4m˜6+5m˜12 1+x2−m˜2
µ µ µ e µ µ µ e µ
(44)
and where m˜ =m /m ,m˜ =m /m and x=m /m .
µ µ τ e e τ e µ
2.2 The mass matrix of the neutrinos
With the Z selection rule (Table 1), the mass matrix of the Dirac neutrinos
2
takes the form
µν µν 0
2 2
MνD =µν2 −µν2 0 (45)
µν µν µν
4 4 3
Then, themassmatrixfortheleft-handedMajorananeutrinosisobtainedfrom
the see-saw mechanism,
 ( 1 + 1 )µ2 ( 1 − 1 )µ2 ( 1 + 1 )µ µ 
Mν =MνDM˜−1(MνD)T = (MM111 − MM122)µ222 (MM111 + MM122)µ222 (MM111 − MM122)µ22µ44,
(M11 + M12)µ2µ4 (M11 − M12)µ2µ4 Mµ242 + Mµ233
(46)
where M are the right handed neutrino masses appearing in eq. (16).
i
The non-Hermitian, complex, symmetric neutrino mass matrix M may be
ν
brought to a diagonal form by a bi-unitary transformation, as
UTM U =diag(m eiφ1,m eiφ2,m eiφ3) (47)
ν ν ν ν1 ν2 ν3
Where U is the matrix that diagonalizes the matrix M .
ν ν
2.2.1 Neutrino matrix with degenerate masses.
In the case where M =M the mass matrix is reduced to [14]
1 2
 ( 1 + 1 )µ2 0 ( 1 + 1 )µ µ 
Mν =MνDM˜−1(MνD)T = M1 0M1 2 (M11 + M11)µ22 M1 M01 2 4.
(M11 + M11)µ2µ4 0 Mµ241 + Mµ233
(48)
With this texture is easy to calculate the U matrix that diagonalizes M†M ,
ν ν ν
|A|2+|B|2 0 A∗B+B∗D
Mν†Mν = 0 |A2| 0  (49)
AB∗+BD∗ 0 |B|2+|D|2
7
with A = µ2/M ,B = 2µ µ /M and D = 2µ µ /M +µ2/M , this matrix is
2 1 2 4 1 1 2 1 3 2
diagonalized by
  
1 0 0 cosη sinη 0
Uν =0 1 0  0 0 1. (50)
0 0 eiδν −sinη cosη 0
If we require that the defining equation (47) be satisfied as an identity, we get
the following set of equations:
2(µν)2/M1=m , (51)
2 ν3
2(µν)2/M1=m cos2η+m sin2η, (52)
2 ν1 ν2
2(µν)(µν)/M1=sinηcosη(m −m )eiδnu, (53)
2 4 ν2 ν1
2(µν)2/M1=m , (54)
2 ν3
2(µν)(µν)/M1=sinηcosη(m −m )eiδnu, (55)
2 4 ν2 ν1
2(µν)2/M1+2(µν)2/M3=(m sin2η+m cos2η)e−2iδnu. (56)
4 3 ν1 ν2
(57)
Solving these equations for sinη and cosη, we find
m −m m −m
sin2η = ν3 ν1 cos2η = ν2 ν3. (58)
m −m m −m
ν2 ν1 ν2 ν1
TheunitarityofU constrainssinηtoberealandthus|sinη|≤1,thiscondition
ν
fixes the phases φ and φ as
1 2
|m |sinφ =|m |sinφ =|m |sinφ . (59)
ν1 1 ν2 2 ν3 3
The real phase δ appearing in eq. (50) is not constrained by the unitarity of
ν
U . Therefore the U matrix is,
ν ν
 (cid:113) (cid:113) 
1 0 0  mmνν22−−mmνν31 mmνν32−−mmνν11 0
Uν =0 1 0  0 0 1. (60)
0 0 eiδnu −(cid:113)mν3−mν1 (cid:113)mν2−mν3 0
mν2−mν1 mν2−mν1
Now, the mass matrix of the Majorana neutrinos, M , may be written in terms
ν
of the neutrino masses; from (48) and (55,56,57), we get
 mν3 0 (cid:112)(mν3−mν1)(mν2−mν3)e−iδν
Mν = 0 mν3 0 
(cid:112)
(mν3−mν1)(mν2−mν3)e−iδν 0 (mν1+mν2−mν3)e−2δν
(61)
Theonlyfreeparametersinthesematrices, otherthantheneutrinomasses, are
the phase φ , implicit in m ,m and m , and the Dirac phase δ .
ν ν1 ν2 ν3 ν
Therefore, the theoretical mixing matrix V , is given by
PMNS
 O cosη+O sinηeiδ O sinη−O cosηeiδ −O 
11 31 11 31 21
VPthMNS =−O12cosη+O32sinηeiδ −O12sinη−O32cosηeiδ O22 ×K.
O cosη−O sinηeiδ O sinη+O cosηeiδ O
13 33 13 33 23
(62)
8
Toobtaintheexpressionsforthemixinganglesweneedtomatchthetheoretical
and PDG expressions for the V matrix
PMNS
|Vth |=|VPDG | (63)
PMNS PMNS
meaning |Vth| = |VPDG|. The standard parametrization of the Particle Data
ij ij
Group is
 c12c13 s12c13 s13e−iδCP
VPMNS =−s12c23−c12s23s13eiδCP c12c23−s12s23s13eiδCP s23c13 .
s12s23−c12c23s13eiδCP −c12s23−s12c23s13eiδCP c23c13
(64)
We can straightforwardly read the equation for the mixing angles with
1 1+4x2−m˜4
|sin |=|O |(cid:39) √ x µ , (65)
θ13 21 2 (cid:113)1+m˜2 +5x2−m˜4
µ µ
|O | 1 1−2m˜2 +m˜4
|sin |= 22 (cid:39) √ µ µ , (66)
θ23 (cid:112)1−O221 2(cid:113)1−4m˜2 +x2+6m˜4
µ µ
and
O sin −O cos
tan = 11 η 31 η (67)
θ12 O sin +O cos
31 η 11 η
(cid:113) (cid:113)
(cid:39)−(cid:114)mν2−mν3×( 1+2x2−m˜2µ(1+m˜2µ+x2)− √12x(1+2m˜2µ+4x2) mmνν32−−mmνν13).
mν3−mν1 (cid:113)1+2x2−m˜2(1+m˜2 +x2)+ √1 x(1+2m˜2 +4x2)(cid:113)mν2−mν3
µ µ 2 µ mν3−mν1
(68)
Wecanexpresstanθ intermsofthedifferencesofthesquareofthemassesas
12
(∆m2 +∆m2 +|m |2cos2φ )1/2−|m ||cosφ |
tan2θ = 12 13 ν3 nu ν3 nu (69)
12 (∆m2 +|m |2cos2φ )1/2+|m ||cosφ |
13 ν3 nu ν3 nu
where ∆m2 =m2 −m2 .
ij νi νj
We can use the experimental values of the masses of the charged leptons and
the differences of the square of the masses to fit the mixing angles,
(sin2θ )th =1.1×10−5, (sin2θ )xp =2.19+0.12×10−2, (70)
13 13 −0.12
and
(sin2θ )th =.499, (sin2θ )xp =.5+0.05. (71)
23 23 −0.05
Fromexpression(69),wemayreadilyderiveexpressionsfortheneutrinomasses
in terms of tanθ ,φ and the differences of the squared masses,
12 ν
(cid:112)
∆m2 1−tan4θ +r2
|m |= 13 12 , (72)
3 2tanθ cosφ (1+tan2θ )(1+tan2θ +r2)
12 ν 12 12
(cid:113)
|m |= |m |2+∆m2 , (73)
1 ν3 13
(cid:113)
|m |= |m |2+∆m2 (1+r2) , (74)
2 ν3 13
9
here r2 = ∆m2 /∆m2 ≈ 3×10−2. This implies an inverted neutrino mass
12 13
spectrum |m | < |m | < |m |. As r2 << 1, the sum of the neutrino masses
ν3 ν1 ν2
is
(cid:88)3 ∆m2 (cid:113)
|m |≈ 13 (1+2 1+2tan2θ (2cos2φ −1)+tan4θ −tan2θ ).
νi 2cosφ tanθ 12 ν 12 12
ν 12
i=1
(75)
The most restrictive cosmological upper bound [41] for this sum is
(cid:88)
|m |≤0.23eV . (76)
ν
This upper bound and the experimentally determined values of tanθ and
12
∆m2 , give a lower bound for
i,j
cosφ ≥0.55 (77)
ν
or 0≤φ ≤57◦. We can use again equation (69) to set the best value of φ, we
ν
find that with φ=50◦ we get,
tanθ =0.665288 (78)
12
Hence, setting φ =50◦ in our formula, we find
ν
m =0.052 eV, m =0.053 eV, m =0.019 eV. (79)
ν1 ν2 ν3
The computed sum of the neutrino masses is
3
(cid:88)
( |m |)th =0.168508 eV, (80)
νi
i=1
below the cosmological upper bound given in eq. (76), as expected. The above
value of φ is in agreement with the requirements for leptogenesis, as we will
show in section 3. One of the successes of the S3-3H model has been to predict
an angle θ different from zero, as well a very accurate angles θ and θ .
13 12 23
Nevertheless new experimental results have shown that the angle θ is greater
13
than the model predicts with degenerate right-handed neutrino masses. This is
the major reason to extend the model further, to the non-degenerate case [28],
and where the angles fit the experimental value.
2.2.2 The mass matrix of the neutrinos without degeneration
In a more extensive analysis than [28], we continue to study the case where the
RHN masses are non-degenerate. The effective neutrino mass matrix m is,
ν
 ( 1 + 1 )µ2 ( 1 − 1 )µ2 ( 1 + 1 )µ µ 
Mν =MνDM˜−1(MνD)T = (MM111 − MM122)µ222 (MM111 + MM122)µ222 (MM111 − MM122)µ22µ44.
(M11 + M12)µ2µ4 (M11 − M12)µ2µ4 Mµ242 + Mµ233
(81)
We are going to assume that the phases of the µ and µ terms are aligned,
3 4
therefore we can write Mν in polar form Mν = PM(cid:103)νP, with M(cid:103)ν, real and
10